11. Mass Driver Appendices

Purdue University Arch Pleumpanya Mass Driver Appendix | 1

Suhas Anand1, Alexander J. Chapa2, Kevin Huang3, Nicholas Martinez-

Cruces4, Arch Pleumpanya5, Dylan Pranger6, Peter Salek7, William

Sanders8, Erick Smith9, and Natasha Yarlagadda10

Purdue University, West Lafayette, Indiana, 47906, United States

1 Power and Thermal 2 Controls 3 Human Factors, Associate Editor for Mass Driver 4 Mission Design 5 Propulsion 6 CAD 7 Power and Thermal 8 CAD 9 CAD 10 Propulsion Purdue University Arch Pleumpanya Mass Driver Appendix | 2

Appendix. Mass Driver

A. Sources:

[1] Bilby, Curt R., and Nathan Nottke. "A Superconducting Quenchgun for Delivering Lunar

Derived Oxygen to Lunar Orbit - NASA-CR-185161." 1990.

[2] Burton, R. R., Crisman, R. P., Alexander, W. C., Grisset, J. D., Davis, J. G., and Brady, J. A.,

“Physical Fitness Program to Enhance Aircrew G Tolerance,” https://apps.dtic.mil/, Mar.

1988, pg 13, Available: https://apps.dtic.mil/dtic/tr/fulltext/u2/a204689.pdf.

[3] C. Frueh, Space Traffic Management, AAE590 course script, Purdue University, 2019 Hall,

N. (Ed.). (2015, May 5).

[4] Centripetal Acceleration. (n.d.). Retrieved March 3, 2020, from http://hyperphysics.phy-

astr.gsu.edu/hbase/cf.html.

[5] Davey, K., “Designing with null flux coils,” IEEE Transactions on Magnetics, vol. 33, Sep.

1997, pp. 4327–4334.

[6] Elert, G. (n.d.). Equations of Motion. Retrieved January 2020, from

https://physics.info/motion-equations/.

[7] Green, Michael A. “Cooling Large HTS Magnet Coils Using a Gas Free-Convection Cooling

Loop Connected to Coolers.” IOP Conference Series: Materials Science and

Engineering, vol. 502, 2019, p. 012100., doi:10.1088/1757-899x/502/1/012100.

[8] Guo, Li, and Zhou, “Study of a Null-Flux Coil Electrodynamic Suspension Structure for

Evacuated Tube Transportation,” Symmetry, vol. 11, Mar. 2019, p. 1239. Purdue University Arch Pleumpanya Mass Driver Appendix | 3

[9] He, J., and Rote, D. M., “Computer Model Simulation of Null-Flux Magnetic Suspension and

Guidance,” Center for Transportation Research, Energy Systems Division, Argonne

National Laboratory, Jun. 1992.

[10] He, J., Rote, D., and Coffey, H., “Survey of foreign maglev systems,” US Army Corps of

Engineers, Jan. 1992.

[11] Jokic, M. D., & Longuski, J. M. (2004). Design of Tether Sling for Human Transportation

System Between Earth and Mars. Journal of Spacecraft and Rockets, 41(6), 1010–1015.

doi: 10.2514/1.2413.

[12] Katorgin, B., Chvanov, V., Chelkis, F., Ford, R., and Tanner, L., “Atlas with RD-180

now,” 37th Joint Propulsion Conference and Exhibit, Salt Lake City, UT, 2001.

[13] Lunar Reconnaissance Orbiter. (2019, July 11). Retrieved January 2020, from

https://solarsystem.nasa.gov/missions/lro/in-depth/.

[14] “Maglev: Magnetic Levitating Trains,” Electrical and Computer Engineering Design

Handbook Available: https://sites.tufts.edu/eeseniordesignhandbook/2015/maglev-

magnetic-levitating-trains/.

[15] Mars Atmosphere Model - Metric Units. Retrieved January 2020, from

https://www.grc.nasa.gov/WWW/K-12/airplane/atmosmrm.html.

[16] Mars Odyssey. (2019, July 23). Retrieved January 2020, from

https://solarsystem.nasa.gov/missions/mars-odyssey/in-depth/.

[17] Masugata, K., “Hyper velocity acceleration by a pulsed coilgun using traveling magnetic

field,” IEEE Transactions on Magnetics, Vol. 33, No. 6, pp. 4434-4438. Purdue University Arch Pleumpanya Mass Driver Appendix | 4

[18] NASA: Lunar Reconnaissance Orbiter. (2013). LUNAR RECONNAISSANCE ORBITER:

Detailed Topography of the Moon. Retrieved from

https://lunar.gsfc.nasa.gov/images/lithos/LRO_litho8-lunar_topography.pdf.

[19] Nasar, S., and Boldea, I, “High-Speed Linear Induction Motors,” Linear motion electric

machines, Wiley New York, 1976, pp. 53-133.

[20] “New England Wire Technologies,” New England Wire Technologies Available:

https://www.newenglandwire.com/application/the-repulsive-attractive-world-of-maglev/.

[21] Ohsaki, H., “Review and update on MAGLEV,” European Cryogenics Day 2017

[22] Power. (n.d.). Retrieved January 2020, from

https://www.physicsclassroom.com/class/energy/Lesson-1/Power.

[23] R., G., P., A., and H., W., “Artificial gravity as a countermeasure for mitigating

physiological deconditioning during long-duration space missions,” Frontiers In Systems

Neuroscience Available:

https://www.frontiersin.org/articles/10.3389/fnsys.2015.00092/full.

[24] Redd, N. T. (2017, December 9). Olympus Mons: Giant Mountain of Mars. Retrieved

January 2020, from https://www.space.com/20133-olympus-mons-giant-mountain-of-

mars.html.

[25] “Shape Effects on Drag.” NASA, NASA, www.grc.nasa.gov/www/k-

12/airplane/shaped.html. Purdue University Arch Pleumpanya Mass Driver Appendix | 5

[26] Slemon, G. R., “Linear induction motors,” Encyclopædia Britannica Available:

https://www.britannica.com/technology/electric-motor/Linear-induction-motors.

[27] Tarantola, A., “Why the Human Body Can't Handle Heavy

Acceleration,” Gizmodo Available: https://gizmodo.com/why-the-human-body-cant-

handle-heavy-acceleration-1640491171.

[28] The Topography of Mars. (2009, January 12). Retrieved January 2020, from

https://www.asc-csa.gc.ca/eng/astronomy/mars/topography.asp.

[29] Vasantha Kumar, K., and Norfleet, W. T., “Issues on Human Acceleration Tolerance After

Long-Duration Space Flights,” NASA Technical Memorandum, Oct. 1992, pg 11-24,

Available: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930020462.pdf.

[30] Wyrick, B., and Brown, J. R., “Acceleration in Aviation: G-Force,” faa.gov Available:

https://www.faa.gov/pilots/safety/pilotsafetybrochures/media/acceleration.pdf. Purdue University Arch Pleumpanya Mass Driver Appendix | 6

B. Coilgun Acceleration Profile Analysis

Initially when we were studying coilgun designs for our system, an analysis was done on the acceleration profile delivered by an induction coilgun. An induction coilgun exploits repulsive magnetic force to drive a loop diamagnetic material mounted to a non-magnetic projectile. This is in contrast to the reluctance coilgun which exploits attractive magnetic force to drive a ferromagnetic projectile.

The analysis followed the equations outlined by Matsugata [17]. The mutual inductance between the projectile coil and the driving coil (each section within the coilgun) is the main parameter to be calculated. The driving force, 퐹 (and thus the acceleration, if the projectile mass is known), can be calculated using Eq. (A.1).

1 푑푀 2 퐹 = 퐼푝 퐿푝 (A1) 푀 푑푥푐

Figure 5 from Ref. [1] was reproduced to verify the MATLAB script used for this analysis and given here as Fig. A1. Our plot matches the original satisfactorily hence validating the script for further analysis.

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Fig. A1: Force on Projectile by Coilgun. This plot is a reproduced result from Ref [1] of force profile in a single coilgun section.

The architecture of our coilgun needed to be immense in scale relative to the original analysis. The dimensions were scaled up to match the taxi vehicle’s size. The launch stack was assumed to be 269 Mg. We chose each coilgun section to be 11 m in diameter and 50 long containing 50 coils. The coils are circular and wound such that no free space is permitted between them, therefore the coil wire is 1 m in diameter. The diamagnetic wire loop that needs to be mounted to the projectile is 9 m in diameter and 1 m long. The driving coil is energized at 785 kA which is 20% the maximum current rating for copper wires (500 A/cm2). The acceleration profile resulting from this design is given in Fig. A2.

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Fig. A2: Acceleration for Single-Stage Coilgun. This plot shows the acceleration profile where coilgun is constantly energized.

It can be observed that acceleration only occurs once the projectile has passed through the center of the coil section. From the entrance to the center (corresponding to -0.5 on the horizontal axis in Fig. A2), only deceleration occurs. Downrange of the exit (corresponding to 0.5 on the horizontal axis in Fig. A2), the projectile is still being accelerated at a decaying rate despite not being physically enclosed by the coil section anymore. It can be concluded that constantly energizing the coils will not produce net acceleration in the direction we desire. Once we synchronize the capacitor discharge such that the coil becomes energized right at the instant the projectile passes the center of the coil section, net acceleration can occur reaching almost 4 g at peak, as seen in Fig. A3.

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Fig. A3: Acceleration for Single-Stage Coilgun. This plot shows the acceleration profile where coilgun is energized only when the projectile has passed the center of the section.

Thus far, we have examined a single-stage coilgun. To qualitatively assess the acceleration profile of a multi-stage coilgun, we append the acceleration profiles and sum up overlapping portions. Figure A4 is an approximation of the acceleration profile of a multi-stage coilgun with five sections arranged in series. Thrust ripples can be observed clearly as the acceleration peaks near the exit of each section and drops almost instantaneously as it enters the successive section.

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Fig. A4: Acceleration for Multi-Stage Coilgun. This plot shows the acceleration profile of a multi-stage coilgun with five sections arranged in series displaying pronounced thrust ripples.

C. Kinematics of Launch

A script was written to analyze the kinematics of a constant acceleration motion like the one our mass driver needs to execute. The script plots a value of acceleration and the associated duration of launch for varying track lengths. The analysis was done on both Luna and Mars. If a mass driver were to be constructed on Luna, the track length and launch duration associated with the limit of 2 g constant acceleration are 159 km and 2 minutes 7 seconds. The required forces on

Luna and Mars are equal since the gravitational acceleration is almost perpendicular to the direction of launch.

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Fig. A5: Launch Acceleration and Duration on Luna. This is a plot of launch acceleration and duration as a function of track length on Luna.

D. Track Length, Duration, and Required Force Calculation for Launch

The acceleration is constrained to be constant at 2 g. The launch stack goes from rest to a launch velocity (푣푙푎푢푛푐ℎ) of 5 km/s. The track length can be calculated from the kinematic equation

Eq. (A2).

푣2 ∆푥 = 푙푎푢푛푐ℎ (A2) 푐 2푎

The track length is calculated to be 637.3 km. The launch duration can be calculated from kinematic equation Eq. (A3).

푣 ∆푡 = 푙푎푢푛푐ℎ (A3) 푎

The launch duration is be calculated to be 4 minutes and 14 seconds. The required force to achieve 2 g acceleration can be calculated by Newton’s Second Law (퐹 = 푚푎) to be 5.88 MN.

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Gravitational force of Mars was neglected in the direction of launch. At a launch slope of 3° and local acceleration of approximately 3.71 m/s2, gravitational acceleration opposing the launch acceleration is a mere 0.194 m/s2. To put into perspective, the opposing gravitational acceleration is less than one percent of launch acceleration. Therefore, neglecting gravitational force in the launch direction is a valid assumption.

E. Track Length, Duration, and Required Force Calculation for Cradle Arrest

The driving force is doubled to be 11.8 MN. Since the mass of the cradle is 80 Mg, the deceleration is 147.5 m/s2. This translates to 15.0 g. From Eq. (A1), the track length is calculated to be 84.7 km. From Eq. (A2), the arrest duration is calculated to be 34 seconds.

F. Calculation of Redundant Track Sections

We assumed a 5% failure rate for each 10 km track section. For electronics, 5% is rather high but for extremely high-load electrical components in our case, this is a conservative estimate.

The probability associated with track sections can be modeled as a binomial distribution with failure rate 푝 among 푛 + 푟 sections of track. There are 푛 marginal sections and 푟 redundant sections. The probability that all 푟 or more redundant sections will fail during a given launch can be expressed by Eq. (A4).

푛 + 푟 푃(푋 ≥ 푟) = ∑푛+푟 ( ) 푝푘(1 − 푝)푛+푟−푘 (A4) 푘=푟 푘

For the launch phase, the failure rate 푝 = 0.05 among 푛 = 64 marginal number of sections.

To achieve a 1% probability of failure, we are required to have 푟 = 9 redundant sections. The resulting probability of failure associated with the launch phase is 1.243%.

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75 75 푃(푋 ≥ 9) = ∑ ( ) 0.95𝑖0.0575−𝑖 = 1.243% 푖 𝑖=9

For the arrest phase, the failure rate 푝 = 0.05 among 푛 = 9 marginal number of sections.

To achieve a 1% probability of failure, we are required to have 푟 = 3 redundant sections. The resulting probability of failure associated with the launch phase is 1.957%.

12 12 푃(푋 ≥ 3) = ∑ ( ) 0.95𝑖0.0512−𝑖 = 1.957% 푖 𝑖=3

G. Propellant Saved Calculation

The use of mass driver does not require propellant for acceleration as opposed to conventional chemical rockets. The propellant saved can be calculated using the rocket equation,

Eq. (A5).

푚0 ∆푣 = Isp푔0 ln (A5) 푚푓

2 Standard gravity (푔0) is 9.80665 푚/푠 . The specific impulse (퐼푠푝) and propellant mass fraction was assumed to be 338 seconds and 0.93, according to Ref. [19]. The specific impulse was referenced from RD-180 engine employed on Atlas V which is comparable in launch capacity to our mass driver.

푚 λ = 푝 (A6) 푚0

The definition of propellant mass fraction is expressed in Eq. (A6). We assume that the propellant is completely expended such that

mf = 푚0 − 푚푝 (A7)

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Combining Eq. (A5), Eq. (A6), and Eq. (A7) we arrive at Eq. (A8)

훥푣 푒푥푝( )−1 푔0퐼푠푝 푚푝 = 휆푚0 [ 훥푣 ] (A8) 1−(1−휆) 푒푥푝( ) 푔0퐼푠푝

To achieve a velocity gain (훥푣) of 5 km/s, propellant mass required for a conventional chemical rocket was calculated by Eq. (A8) to be 906 Mg. In other words, the mass driver saved

906 Mg of propellant per launch of taxi vehicle. It is worth noting that the 906 Mg of propellant was assumed to be produced on Mars. If the propellant needed to be transported from Earth, additional propellant would be required. The additional propellant for transportation would be an order of magnitude more mass than its payload of 906 Mg propellant for escaping Earth’s escape velocity of 11.2 km/s, performing orbital correction maneuvers, and accommodating for inert mass used as re-entry heat shielding and propulsive braking systems. The disproportionate demand for propellant required in an all-conventional rocket system is infeasible and thus justifies the utilization of alternative launch technologies like our mass driver system.

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H. Initial “Divot” Design for Launch Separation Event

Fig. A6: Launch Stack Launch Process. This diagram depicts the launch stack accelerating, the separation of launch stack, and the deceleration of the cradle.

Consult Fig. A6 above for the initial design of launch sequence. Initially, a truly propellant- less launch system was conceived. A small section along Olympus Mons was to be dug out, forming a divot on the exact spot the separation event was calculated to occur. There is a fundamental flaw associated with this design. There exists a centrifugal force associated with any motion with curvature including in this design where the track dips down. The null-flux coils are designed to take a loading force up to a certain value. The maximum centrifugal force is also thus equal to this same limit. The centripetal force equation, Eq. (A9), dictates the value of force 퐹푐 perpendicular to the motion of the launch stack moving in a curved path of radius 푟푙푎푢푛푐ℎ.

2 푚푠푡푎푐푘푣푙푎푢푛푐ℎ 퐹푐 = (A9) 푟푙푎푢푛푐ℎ

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The null-flux coils are designed to levitate the launch stack weight (푚푠푡푎푐푘푔푀푎푟푠) as expressed in Eq. (A10).

퐹푛푓푐 = 푚푠푡푎푐푘푔푀푎푟푠 (A10)

Combining Eq. (A9) and Eq. (A10), keeping in mind that 퐹푛푓푐 = 퐹푐, Eq. (A11) is derived.

2 푣푙푎푢푛푐ℎ 푟푙푎푢푛푐ℎ = (A11) 푔푀푎푟푠

Martian gravitational acceleration is approximately 3.71 m/s2. From Eq. (A11), the required curvature radius required to be at least 6,738 km. The curvature is too large to be feasible even when optimized by allowing a larger loading on the null-flux coils or decreasing dip angle.

This design also fixes the separation point to a specific location along Olympus Mons’ slope. This is troublesome since the separation point cannot be precisely predicted because the sections of track will fail in random. Calculated from Eq. (A4), there is a 97.6% probability that at least one of the track sections will fail. Even if one section fails, the separation point already becomes uncertain.

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I. Levitation Technology

The following are the calculations behind determining levitation force and specifications for the figure-eight null flux coils and ReBCO High Temperature Superconducting Magnets. We first solve for the total mass of the system of interest – the launch stack – using Eq. (A12) below.

푀푠푡푎푐푘 = 푀푡푎푥𝑖 + 푀푐푟푎푑푙푒 (A12)

The mass of the taxi vehicle is approximated to be 200 Mg and the mass of the cradle

(including magnets and explosive bolts) is taken as about 50% of the taxi weight, at 100 Mg.

Though this is a slight overestimate compared to the exact taxi weight of 189.167 Mg, the extra levitation force accounts for any electrical resistance resulting from the coils, which is not taken into consideration in any further detail during analysis. The total mass of the system is 300 Mg.

Knowing the total mass of the system, we can easily solve for the required levitation force

퐹푙푒푣𝑖푡푎푡𝑖표푛, using Eq. (A13).

퐹푙푒푣𝑖푡푎푡𝑖표푛 = 푀푠푦푠푡푒푚 × 푔푀푎푟푠 (A13)

With Eq. (A13), and knowing that the standard gravitational constant of Mars, 푔푀푎푟푠, is

푚 3.711 , we calculated a required levitation force of 1.11 MN. From research, we find that a certain 푠2 maglev system on Earth which weighs 17 Mg and uses similar magnets, uses 12 magnets per vehicle in their system. With Eq. (A13) again and the standard gravitational constant on Earth as

푚 푔 , is 9.81 , we calculate their 퐹 as 1.67 MN. With Eq. (A14), and knowing their system 0 푠2 푙푒푣𝑖푡푎푡𝑖표푛 uses 12 SCMs, we can find the force per magnet, which turns out to be 14 kN:

퐹푙푒푣𝑖푡푎푡𝑖표푛 퐹푆퐶푀 = (A14) 푛푚푎푔푛푒푡푠

Purdue University Peter J. Salek, Suhas Anand Mass Driver Appendix | 18

Since we are using similar magnets, we now know that each ReBCO magnet can reasonably produce 14 kN of lift force (퐹푆퐶푀). With rearranging Eq. (A14) we can solve for the number of ReBCO magnets we need for our system, which turns out to be 79.5. We round this up to 80 magnets required for the system. The magnet dimensions are 1.7 m by 0.5 m and are placed in rows of 8, with 40 magnets per side of the launch stack. The magnet specifications are tabulated in Table A1 below.

Table A1: ReBCO High Temperature Superconducting Magnet specifications

Dimensions (m) Lift per SCM (kN) No. of SCM (#) Spacing (m) 1.7 x 0.5 14 80 0.4

With more research, into the null-flux coils, we use experimental plots in determining certain parameters for our coils. We decide on a coil width of 0.31 m and a height of 0.55 m. From extrapolating experimental data, we find that increasing the distance between the upper loop and lower loop increases the lift force per SCM, but it levels out after 12 cm. Therefore, we choose this as our displacement distance between the upper and lower loops, as well as an air gap of 5 cm, after which the lift force per SCM also begins to decrease. The specifications and information for the null flux coil parameters are shown in Table A2.

Table A2: Figure-Eight Null Flux Coil specifications Distance between Dimensions (m) Air Gap (m) upper & lower coil Spacing (m) (m) 0.31 x 0.55 0.05 0.12 0.15

The following are detailed calculations for the number of coils used on each side of the

Purdue University Peter J. Salek, Suhas Anand Mass Driver Appendix | 19 track, for the three different track locations. We assume that the same launch stack is used in all three locations, so the orientation and number of magnets at each location stays the same. We continue to use 80 magnets (40 on each side of the launch stack) in five rows of eight, as these dimensions with a 0.4 m spacing between the magnets fit the cradle dimensions of length, width, and height as 25 m, 30 m, and 8 m respectively.

For the Olympus Mons launch track, we know that the track length, 푙푙푎푢푛푐ℎ,푀푎푟푠, is 880 km total, with 750 km for acceleration and 130 km for deceleration. The width of each null flux coil,

푤푐표𝑖푙, is 0.31 m with a spacing distance, 푤푠푝푎푐𝑖푛푔,푐표𝑖푙, of 0.15 m between each coil. The total room taken up by each coil is the sum of those values, which is 0.46 m. Using Eq. (A15) we can calculate the number of coils needed per side of the Olympus Mons launch track.

푙푙푎푢푛푐ℎ,푀푎푟푠 푛푐표𝑖푙푠 = (A15) 푤푐표𝑖푙+푤푠푝푎푐𝑖푛푔,푐표𝑖푙

This turns out to be 1.91 million coils per side of the Olympus Mons launch track.

We can do a similar analysis to find the number of coils necessary for the Luna launch track whose length, 푙푙푎푢푛푐ℎ,퐿푢푛푎, is 159 km. Luna requires a much shorter track than Mars because

푘푚 of its lower escape velocity of 2.38 . Again, we use the same width of the null flux coil and 푠 spacing distance, as well as Eq. (A15) to get that we need 346,000 coils on each side of the track.

The Mars landing track requires a bit of a different calculation since it is circular. We know that the average radius of the track is 697 m, 푟푎푣푔,푙푎푛푑, and the track is about as wide as the cradle, which is 30 m - 푤푐푟푎푑푙푒. To calculate the radii of the inner and outer portions of the track, we use the coupled Eq. (A16) and Eq. (A17):

푤 푟 = 푟 + 푐푟푎푑푙푒 (A16) 표푢푡푒푟,푙푎푛푑 푎푣푔,푙푎푛푑 2

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푤 푟 = 푟 − 푐푟푎푑푙푒 (A17) 𝑖푛푛푒푟,푙푎푛푑 푎푣푔,푙푎푛푑 2

We find that the outer track has a radius of 712 km, while the inner track has a radius of

682 km. These radii are then fed into Eq. (A18) and Eq. (A19) to calculate the circumference of each loop – the track length.

푙표푢푡푒푟,푙푎푛푑 = 2 × 휋 × 푟표푢푡푒푟,푙푎푛푑 (A18)

푙𝑖푛푛푒푟,푙푎푛푑 = 2 × 휋 × 푟𝑖푛푛푒푟,푙푎푛푑 (A19)

Calculating the outer track length to be 4473.62 km and the inner track length to be 4285.13 km, we can use Eq. (A15) and substitute the inner and outer track lengths as the numerators while keeping the same coil width and spacing. There are 9.73 million coils required for the outer track and 9.32 million coils required for the inner track.

The number of coils for each track are tabulated below. We continue to keep a spacing of

0.12 m between the coils and hold the assumption that the curvature of the Mars landing track is negligible and does not affect the orientation or function of the coils.

Table A3: Launch and landing track coil specifications

Track Length (km) No. of coils (#)

Olympus Mons Launch 880 1.91 million

Luna Launch 159 346,000

4473.62 (outer) 9.73 million Mars Landing 4285.13 (inner) 9.32 million

A simple code was created to do all the above calculations for ease of updating levitation specifications should any of the components such as track length or launch stack weight change.

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J. Drag Calculations on the Mass Driver

One of the most important aspects of the designing the mass driver was understanding the

Martian atmosphere. This was done using the equations shown below.

푇 = −31 − 0.000998 ∗ ℎ ℎ < 7,000: { (A20) 푃 = 0.699 ∗ 푒−0.00009∗ℎ

푇 = −23.4 − 0.00222 ∗ ℎ ℎ ≥ 7,000: { (A21) 푃 = 0.699 ∗ 푒−0.00009∗ℎ

Eq. (A20) and Eq. (A21) define the Martian atmosphere and if we assume the atmosphere to be an ideal gas, then we can use these equations to solve for density. Now that we have density as a function of altitude we can start delving deeper into the kinematics of the mass driver. Since we know that the mass driver has a constant acceleration of 2 g’s, and that it starts at rest, the velocity and the distance can be determined as a function of time. The final velocity is 5.1521 km/s and so the acceleration lasts for 4 minutes and 14 seconds. The important component to determine is the velocity as a function of time as it will be used in the drag equations. Once we have the velocity profile and the distance traveled as a function of time we can use the assumption that

Olympus Mons as a slope of 3 degrees in order to solve for the altitude of the mass driver as it accelerates. This means that we can use Eq. (A20) and Eq. (A21) in order to find the density of the atmosphere at every time step that we calculated. The Taxi Vehicle has a drag coefficient of 0.295 due to its shape. We designed the Taxi Vehicle to have a similar shape to that of a bullet, so a bullet’s drag coefficient applies in this case [25]. The final drag profile can be seen below.

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Fig. A7: Drag Profile. This plot shows the aerodynamic drag profile of the Taxi during the mass driver Acceleration.

K. Analysis of Power on Luna Figure A8 shows a plot relating power consumed by the mass driver on the Moon and time.

Fig. A8: Mass Driver total power vs time on the Moon.

Purdue University Nicolas Martinez Cruces Mass Driver Appendix | 23

Since the Moon does not have any air drag, the relationship between total power and time remains linear throughout the vehicles entire flight. The power requirements for the mass driver on the Moon can be seen in Table A4.

Table A4: Power Requirements.

Location Maximum Power (GW) Total Energy (GJ)

The Moon 19.5 1,240 This table show the power and energy requirements for the mass driver on the Moon.

Table A5 shows the area of each solar panel placed on the Moon, and Table A6 shows the number of battery cells placed on the Moon.

Table A5: Solar Panel Area.

Location Solar Area (Km2)

The Moon 0.0035 This table lists the area of each solar panel on the Moon.

Table A6: Battery Cell Count.

Location Number of Cells (Million)

The Moon 0.92 This table lists the number of individual battery cells on the Moon, by millions.

Purdue University Megan Brown Comm. Sat Appendix | 24

12. Communication Satellites Appendices

Megan M. Brown11, Chuhao Deng12, Yashowardhan Gupta13, Hanson-

Lee N. Harjono14, Jordan I. Mayer15, Colin A. Miller16, Nicholas C. W.

Oettinga, Rachel L. Roth17, Peter J. Salekc, Eric P. Smitha, and Brady D.

Walter18

Purdue University, West Lafayette, Indiana, 47906, United States

11 Communications. 12 Propulsion. 13 Power & Thermal. 14 CAD. 15 Mission Design, Associate Editor for Communication Satellites. 16 Mission Design. 17 Structures. 18 Controls.

Purdue University Megan Brown Comm. Sat Appendix | 25

I. Nomenclature

α = absorptivity

G = antenna gain

AREO = areosynchronous (Mars-synchronous) orbit

AU = astronomical unit

M’ = atomic mass

B = bandwidth

C = bit rate

TB = brightness temperature q = charge of an electron dB = decibel

ΔV = ΔV

D = diameter of aperture

DR = diameter of receiver antenna

DT = diameter of transmitter rtr = distance between transmitter and receiver e = eccentricity

ε = emissivity

EHz = exaHertz ue = exhaust velocity va = exhaust velocity just after acceleration mf = final mass

LS = free-space path loss

Purdue University Megan Brown Comm. Sat Appendix | 26

FSPL = free-space path loss

GEO = geosynchronous orbit

Gb = gigabit g = gravitational acceleration on Earth surface

G = gravitational constant

θ = half-power bandwidth

Q̇ Albedo = heat flux due to albedo

Q̇ IR = heat flux due to infrared radiation

Q̇ Diss = heat flux due to power dissipated by satellite

Q̇ Solar = heat flux due to solar irradiance

HD = high-definition

As = illuminated surface area of satellite

I = inclination m0 = initial mass

Ki = integral gain

L4 = 4th (L4) Lagrange point

L5 = 5th (L5) Lagrange point

LTT = laser transceiver telescope xa = length of acceleration m = mass

M = mass of planet mppu = mass of power supply mpr = mass of propellant

Purdue University Megan Brown Comm. Sat Appendix | 27

mu = mass of useful payloads

Ix = moment of inertia about x axis

Iy = moment of inertia about y axis

Iz = moment of inertia about z axis

N = noise power

N0 = noise power density

P = power

PSSM = power supply specific mass

Kp = proportional gain

RF = radio frequency r = radius

Rsun = radius of Sun

RCS = reaction control system(s) a, SMA = semi-major axis s = side length of cube

SNR = signal-to-noise ratio

HSun = solar irradiance (power density) at surface of Sun

H0 = solar irradiance at satellite

Isp = specific impulse

μ = standard gravitational parameter

σ = Stefan-Boltzmann constant

T = temperature

FT = thrust

Purdue University Megan Brown Comm. Sat Appendix | 28

η = thruster efficiency

τ = torque

θ* = true anomaly

ε0 = vacuum permitivity

λ = wavelength

II. Table of Contents

I. Nomenclature ...... 25

II. Table of Contents ...... 28

Appendix A: Ground Station Analysis ...... 29

Appendix B: Data on Deep Space Network ...... 33

Appendix E: Equations for Eclipse Calculations ...... 59

Appendix F: Future Work for Mission Profile ...... 60

Appendix G: Additional Figures for Mission Profile ...... 62

Appendix I: Suggestions and Possible Improvements for Propulsion Systems ...... 74

Appendix J: Calculation of External Torques ...... 76

Appendix K: Further Explanation of Simulink Modeling ...... 77

Appendix L: Power Calculations ...... 79

Appendix M: Thermal Calculations ...... 80

Appendix N: Additional Tables on Satellite Structure ...... 81

Purdue University Megan Brown Comm. Sat Appendix | 29

Appendix O: Equations for Moment of Inertia Calculations ...... 87

References ...... 89

Appendix A: Ground Station Analysis

Purdue University Megan Brown Comm. Sat Appendix | 30

Figure A1 This handwritten work shows power calculations from ground stations at

Morehead State and University of Alaska, Fairbanks directly to the cycler orbiting Luna.

Purdue University Megan Brown Comm. Sat Appendix | 31

Figure A2 This is the first of two pages of handwritten work showing power calculations

from Earth ground stations to GEO.

Purdue University Megan Brown Comm. Sat Appendix | 32

Figure A3 This is the second of two pages of handwritten work showing power calculations

from Earth ground stations to GEO.

Purdue University Megan Brown Comm. Sat Appendix | 33

Appendix B: Data on Deep Space Network

In the case that the chosen ground station antennas fail to communicate with the satellites in

GEO, the Deep Space Network (DSN) houses five satellites that are able to communicate directly with Mars [A1, A2].

Table 1. DSN antennas that can contribute to the overall communication network

Antenna Location Range, 106 km Diameter, m

Antenna 55 Madrid, Spain 262.67 34

Antenna 25 Goldstone, USA 262.67 34

Antenna 26 Goldstone, USA 253.15 34

Antenna 43 Canberra, Australia 925.42 70

Antenna 35 Canberra, Australia 303.88 34

We choose these satellites based on the fact that the longest distance from Earth to Mars, when both are at apihelion, is 401 million km, and the average distance from Earth to Mars is 225 million km.

In the case that the chosen ground station antennas fail to communicate with the satellites in GEO, the DSN houses four satellites that are also able to communicate directly with the satellites in GEO [A3, A4].

Purdue University Megan Brown Comm. Sat Appendix | 34

Table 2. DSN antennas that are able to communicate with relay satellites in GEO

Antenna Location Range Diameter, m

Antenna 1 White Sands, New LEO/GEO 18.2

Mexico

Antenna 2 White Sands, New LEO/GEO 18.2

Mexico

Antenna 3 White Sands, New LEO/GEO 18.2

Mexico

Antenna 1 Guam LEO/GEO 11

Antenna 2 Guam LEO/GEO 4.5

Purdue University E. C. Smith Comm. Sat Appendix | 35

Appendix C: Calculations for Communications

From the required bit rate, we use the Shannon-Hartley theorem, which gives the bit rate as a function of bandwidth and Signal to Noise Ratio (SNR), to find the required SNR for the selected bandwidth. The Shannon-Hartley theorem is,

퐶 = 퐵 푙표푔2 (1 + 푆푁푅) (1)

Where C is the required bit rate in bit/s, B is the bandwidth in Hz, and SNR is the ratio of the signal power to the noise power at the receiver [A5].

To compute the SNR of a link, the received signal power and the received noise power must be calculated. The two sources of noise focused on for communications satellites are thermal noise, as well as solar noise. The solar irradiance at 1550 nm at 1 AU from the sun is about 0.3 W/(m2 nm) [A6]. At further distances however, the power decreases according to the inverse square law, so the irradiance of the sun at Mars (approximately 1.53 AU) is about 0.128 W/(m2 nm) . The units of the irradiance tell us that a larger area telescope or larger bandwidth filter will increase the solar noise power.

At Ka bands the sun has a brightness temperature (TB) of about 6000 degrees Kelvin (K) [A7].

Using the Rayleigh-Jeans approximation, we can find the noise power density of the sun at Ka band is about -160.8 dBm/Hz.

푁0 = 푘푇퐵 (2)

Purdue University E. C. Smith Comm. Sat Appendix | 36

Since the noise power is proportional to the noise power density and the bandwidth, larger bandwidths would also increase the noise. However, since large bandwidths allow more data to be transmitted at a given SNR as can be seen in the Shannon-Hartley theorem, this effect balances out.

Another source of noise is the temperature of the satellite components themselves. If the temperature of the components is 290 K, then the noise power density of the thermal emissions is about -174 dBm/Hz. This value is commonly used for thermal noise in communications link budgets [A5]. Cooling the spacecraft is another way to reduce the noise power from thermal emissions of the satellite components, but since there are many components that require this temperature to operate, this value is good to use for calculating the link budget.

To find the received signal power, a link budget is used to incorporate any gains or losses to the transmitted power in the system.

The main source of loss in the link budget is free-space path loss (퐿푠). The equation for the free-space loss is,

휆 2 퐿 = ( ) (3) 푠 4휋푟

Where 휆, is the wavelength in meters and r is the distance between the transmitter and receiver

[1]. For interplanetary communications, the distances between the transmitter and receiver are very large, which means the free-space loss is also very large. However, this equation reveals that smaller wavelengths reduce the free space loss, which makes optical communications very attractive for communicating over long distances. Free-space path loss is discussed further in the following subsection.

Purdue University E. C. Smith Comm. Sat Appendix | 37

The main way to overcome free-space path loss is through antenna gain. Antenna gain, G, tells us how much power the antenna will transmit or receive. In general, the gain is related to the diameter of the aperture, D, by the relation,

휋2퐷2 퐺 = (4) 휆2

So larger antennas provide better links. Shorter wavelengths are again beneficial for high-bitrate communications because they improve the gain of an antenna which is the same size. It should be noted that in general large antennas for optical communications are more difficult to manufacture than for longer wavelengths, but as technology progresses, the improvement of using optical frequencies outweighs this difficulty.

A loss that is related to the size of the antenna is the pointing loss. A 3 dB loss relates to a reduction in the transmitted power by half. The half-power beamwidth of an antenna or telescope tells us how much error we can have when pointing the antennas. As the name suggests, the half- power beamwidth, which we will just call the beamwidth from here, is the angle where the power radiated or received by the antenna is about half of that radiated or received along the bore sight.

For a parabolic reflector the equation for the beam width is

1.22 휆 휃 = (5) 퐷

Purdue University E. C. Smith Comm. Sat Appendix | 38

휃 is the beamwidth in radians, 휆 is the wavelength in meters, and D is the diameter of the aperture in meters [A8]. We can see from the equation for beamwidth, that the pointing tolerance of our antenna will be tighter for smaller wavelengths and larger antennas.

Purdue University E. C. Smith Comm. Sat Appendix | 39

Appendix D: Link Budgets for Communication Satellites

Table 3. Parameters used for Earth ground station to GEO satellite communication link

Parameter Value Units

Bandwidth 30 GHz

Transmit Power 100 W

Receive Antenna Diameter 2 m

Transmit Antenna Diameter 2 m

Transmit Distance 73030.19025 km

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Noise Density -160.8176547 dBm/Hz

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 0 dB

Polarization Loss 0 dB

Electrical Efficiency 0.2

Purdue University E. C. Smith Comm. Sat Appendix | 40

Table 4. Link budget for Earth ground station to GEO satellite link, showing the required

margin is met

Link Parameter Value Units

Transmitter Power 16.99 dBW

Transmitter Gain 69.95 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -214.66 dB

Pointing Loss -3.00 dB

Atmospheric Loss -4.00 dB

Polarization Loss 0.00 dB

Receiver Gain 43.93 dB

Receive Antenna Efficiency -3.01 dB

Received Power -102.20 dBW

Link Margin 4.41 dB

Required Electrical Power 250.00 W

Purdue University E. C. Smith Comm. Sat Appendix | 41

Table 5. Parameters used for GEO satellite to Earth ground station link

Parameter Value Units

Bandwidth 30 GHz

Transmit Power 50 W

Receive Antenna Diameter 10 m

Transmit Antenna Diameter 0.5 m

Transmit Distance 43000 km

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Noise Density -161 dBm/Hz

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 4 dB

Polarization Loss 0 dB

Electrical Efficiency 0.2

Purdue University E. C. Smith Comm. Sat Appendix | 42

Table 6. Link budget for GEO satellite to Earth ground station, showing the required

margin is met

Link Parameter Value Units

Transmitter Power 16.99 dBW

Transmitter Gain 43.93 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -214.66 dB

Pointing Loss -3.00 dB

Atmospheric Loss -4.00 dB

Polarization Loss 0.00 dB

Receiver Gain 69.95 dB

Receive Antenna Efficiency -3.01 dB

Received Power -102.20 dBW

Link Margin 4.41 dB

Required Electrical Power 250 W

Purdue University E. C. Smith Comm. Sat Appendix | 43

Table 7. Parameters used for GEO satellite to GEO satellite link

Parameter Value Units

Bandwidth 30 GHz

Transmit Power 100 W

Receive Antenna Diameter 2 m

Transmit Antenna Diameter 2 m

Transmit Distance 73030 km

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Noise Density -161 dBm/Hz

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 0 dB

Polarization Loss 0 dB

Electrical Efficiency 0.2

Purdue University E. C. Smith Comm. Sat Appendix | 44

Table 8. Link budget for GEO satellite to GEO satellite, showing the required margin is

met

Link Parameter Value Units

Transmitter Power 20.00 dBW

Transmitter Gain 55.97 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -219.26 dB

Pointing Loss -3.00 dB

Atmospheric Loss 0.00 dB

Polarization Loss 0.00 dB

Receiver Gain 55.97 dB

Receive Antenna Efficiency -3.01 dB

Received Power -102.20 dBW

Link Margin 4.89 dB

Required Electrical Power 500 W

Purdue University E. C. Smith Comm. Sat Appendix | 45

Table 9. Parameters used for GEO satellite and Earth Lagrange satellite link

Parameter Value Units

Wavelength 1550 nm

Transmit Power 20 W

Receive Antenna Diameter 3 m

Transmit Antenna Diameter 3 m

Transmit Distance 1 AU

Bandwidth (nm) 0.1 nm

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Irradiance (W m^-2 0.3 W/m^2 nm nm^-1)

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 0 dB

Polarization Loss 0 dB

Purdue University E. C. Smith Comm. Sat Appendix | 46

Table 10. Link budget for GEO satellite and Earth Lagrange satellite, showing the

required margin is met

Link Parameter Value Units

Transmitter Power 13.01 dBW

Transmitter Gain 135.68 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -361.68 dB

Pointing Loss -3.00 dB

Atmospheric Loss 0.00 dB

Polarization Loss 0.00 dB

Receiver Gain 135.68 dB

Receive Antenna Efficiency -3.01 dB

Received Power -93.06 dBW

Link Margin 5.76 dB

Required Electrical Power 100 W

Purdue University E. C. Smith Comm. Sat Appendix | 47

Table 11. Parameters used for the Earth Lagrange satellite to Mars Lagrange satellite link

Parameter Value Units

Wavelength 1550 nm

Transmit Power 25 W

Receive Antenna Diameter 4 m

Transmit Antenna Diameter 4 m

Transmit Distance 2.68 AU

Bandwidth (nm) 0.1 nm

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Irradiance 0.128 W/m^2 nm

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 0 dB

Polarization Loss 0 dB

Purdue University E. C. Smith Comm. Sat Appendix | 48

Table 12. Link budget for Earth Lagrange satellite to Mars Lagrange satellite link,

showing the required margin is met

Link Parameter Value Units

Transmitter Power 13.98 dBW

Transmitter Gain 138.18 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -370.24 dB

Pointing Loss -3.00 dB

Atmospheric Loss 0.00 dB

Polarization Loss 0.00 dB

Receiver Gain 138.18 dB

Receive Antenna Efficiency -3.01 dB

Received Power -94.25 dBW

Link Margin 4.36 dB

Required Electrical Power 125 W

Purdue University E. C. Smith Comm. Sat Appendix | 49

Table 13. Parameters used for the Mars Lagrange satellite to Earth Lagrange satellite link

Parameter Value Units

Wavelength 1550 nm

Transmit Power 60 W

Receive Antenna Diameter 4 m

Transmit Antenna Diameter 4 m

Transmit Distance 2.68 AU

Bandwidth (nm) 0.1 nm

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Irradiance 0.3 W/m^2 nm

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 0 dB

Polarization Loss 0 dB

Purdue University E. C. Smith Comm. Sat Appendix | 50

Table 14. Link budget for Mars Lagrange satellite to Earth Lagrange satellite, showing the

required gain margin is met

Link Parameter Value Units

Transmitter Power 17.78 dBW

Transmitter Gain 138.18 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -370.24 dB

Pointing Loss -3.00 dB

Atmospheric Loss 0.00 dB

Polarization Loss 0.00 dB

Receiver Gain 138.18 dB

Receive Antenna Efficiency -3.01 dB

Received Power -90.57 dBW

Link Margin 4.48 dB

Required Electrical Power 300 W

Purdue University E. C. Smith Comm. Sat Appendix | 51

Table 15. Parameters used for the Mars Lagrange satellite and AREO satellite link

Parameter Value Units

Wavelength 1550 nm

Transmit Power 20 W

Receive Antenna Diameter 2.5 m

Transmit Antenna Diameter 2.5 m

Transmit Distance 1.53 AU

Bandwidth (nm) 0.1 nm

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Irradiance 0.128 W/m^2 nm

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 0 dB

Polarization Loss 0 dB

Purdue University E. C. Smith Comm. Sat Appendix | 52

Table 16. Link budget for Mars Lagrange satellite and AREO satellite, showing the

required margin is met

Link Parameter Value Units

Transmitter Power 13.01 dBW

Transmitter Gain 134.10 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -365.37 dB

Pointing Loss -3.00 dB

Atmospheric Loss 0.00 dB

Polarization Loss 0.00 dB

Receiver Gain 134.10 dB

Receive Antenna Efficiency -3.01 dB

Received Power -98.28 dBW

Link Margin 4.12 dB

Required Electrical Power 100 W

Purdue University E. C. Smith Comm. Sat Appendix | 53

Table 17. Parameters used for Mars ground station to AREO link

Parameter Value Units

Bandwidth 30 GHz

Transmit Power 6.5 W

Receive Antenna Diameter 0.5 m

Transmit Antenna Diameter 10 m

Transmit Distance 17000 km

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Noise Density -160.8176547 dBm/Hz

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 3.4 dB

Polarization Loss 0 dB

Electrical Efficiency 0.2

Purdue University E. C. Smith Comm. Sat Appendix | 54

Table 18. Link budget for Mars ground station to AREO, showing the required margin is

met

Link Parameter Value Units

Transmitter Power 8.13 dBW

Transmitter Gain 69.95 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -206.60 dB

Pointing Loss -3.00 dB

Atmospheric Loss -3.40 dB

Polarization Loss 0.00 dB

Receiver Gain 43.93 dB

Receive Antenna Efficiency -3.01 dB

Received Power -102.20 dBW

Link Margin 4.21 dB

Required Electrical Power 32.5 W

Purdue University E. C. Smith Comm. Sat Appendix | 55

Table 19. Parameters used for AREO satellite to Mars ground station link

Parameter Value Units

Bandwidth 30 GHz

Transmit Power 6.5 W

Receive Antenna Diameter 10 m

Transmit Antenna Diameter 0.5 m

Transmit Distance 17000 km

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Noise Density -160.8176547 dBm/Hz

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 3.4 dB

Polarization Loss 0 dB

Electrical Efficiency 0.2

Purdue University E. C. Smith Comm. Sat Appendix | 56

Table 20. Link budget for AREO satellite to Mars ground station, showing the required

margin is met

Link Parameter Value Units

Transmitter Power 8.13 dBW

Transmitter Gain 43.93 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -206.60 dB

Pointing Loss -3.00 dB

Atmospheric Loss -3.40 dB

Polarization Loss 0.00 dB

Receiver Gain 69.95 dB

Receive Antenna Efficiency -3.01 dB

Received Power -102.20 dBW

Link Margin 4.21 dB

Required Electrical Power 32.5 W

Purdue University E. C. Smith Comm. Sat Appendix | 57

Table 21. Parameters used for AREO satellite to AREO satellite link

Parameter Value Units

Bandwidth 30 GHz

Transmit Power 20 W

Receive Antenna Diameter 2 m

Transmit Antenna Diameter 2 m

Transmit Distance 35382.3339 km

Require Data Rate 1.00E+09 bit/s

System Temp 290 K

Solar Noise Density -160.8176547 dBm/Hz

Transmit Antenna Efficiency 0.5

Receive Antenna Efficiency 0.5

Illumination Efficiency 0.8

Pointing Loss 3 dB

Atmospheric Loss 0 dB

Polarization Loss 0 dB

Electrical Efficiency 0.2

Purdue University E. C. Smith Comm. Sat Appendix | 58

Table 22. Link budget for AREO satellite to AREO satellite, showing the required margin

is met

Link Parameter Value Units

Transmitter Power 13.01 dBW

Transmitter Gain 55.97 dB

Illumination Efficiency -0.97 dB

Transmit Antenna Efficiency -3.01 dB

Free Space Loss -212.97 dB

Pointing Loss -3.00 dB

Atmospheric Loss 0.00 dB

Polarization Loss 0.00 dB

Receiver Gain 55.97 dB

Receive Antenna Efficiency -3.01 dB

Received Power -102.20 dBW

Link Margin 4.19 dB

Required Electrical Power 100 W

Purdue University C. A. Miller Comm. Sat Appendix | 59

Appendix E: Equations for Eclipse Calculations

To find the orbital angular velocity, we use

3 푎3 푇 = 2휋 ∗ √ (6) µ

2휋 휔 = (7) 푇

The angle in eclipse is given by

푟푝푙푎푛푒푡 휑 = arcsin ( ) (8) 푟푠푡푎푡𝑖표푛푎푟푦

Thus, the time in eclipse is

2휑 푡 = (9) 푒푐푙𝑖푝푠푒 휔

Purdue University C. A. Miller Comm. Sat Appendix | 60

Appendix F: Future Work for Mission Profile

There were some limitations to the analysis we could do for this mission. One of the primary limitations was time and forced us to focus only on higher-level analysis. One of the aspects where much more work could go into was the station-keeping and longitude analysis. Running more monte carlo simulations might have revealed more longitudes which were stable for both Earth and Mars and might further reduce our need for ∆V over the lifespan of the mission. This would make our satellites last longer or might allow us to make them smaller and lighter or allow any number of benefits from reduced and weight and size of a particular subsystem. In addition to this, some of the more precise simulation show a change to the inclination of the AREO satellites that would need to be corrected to maintain their orbit. More time could be spent building a controller to mitigate not only the longitude changes but the inclination changes as well. It may also be possible that these inclination changes may not affect the ability of the AREO satellites to observe the Martian ground station or the Martian Lagrange point relays. To verify this a further simulation would be needed analyzing the line-of-sight and outage periods of those satellites to the other nodes in the network. The lines of changing longitude around Mars appear much thicker than those around Earth. In fact, there are high frequency oscillations with a period of around a Martian day that change the longitude ±5º each half period. Without an extraordinary amount of fuel and with the current controller it might be impossible to cancel out those oscillations. More time could go into designing a controller that only accounts for the average longitude of the AREO satellite such that only the low frequency, high amplitude oscillations around Mars are correct for though there are likely non-linearities that would make this difficult. There may also be more fuel-efficient ways to implement the controller that we implement to change the longitude or to use more esoteric methods for station-keeping like a solar sail. There are many directions in which further analysis

Purdue University C. A. Miller Comm. Sat Appendix | 61 could go, but this is all primarily centers around the topic of station-keeping for both Earth and

Mars.

Another possible item of interest would be the end-of-life operations and replacement of each of the satellites. For this work the only interest is where the spacecraft are placed and ensuring they meet their mission requirements. There is no analysis for the trajectory to place each satellite in its orbit, nor is there significant end-of-life analysis beyond moving each spacecraft with its last portion of fuel to a graveyard orbit. There would also some check on if there are other spacecraft at the selected longitude around Earth or Mars, though this is difficult to do now as this mission is planned for the far future and the spacecraft in GEO and AREO will change.

Purdue University J. I. Mayer Comm. Sat Appendix | 62

Appendix G: Additional Figures for Mission Profile

The following pages provide a collection of figures and plots that further demonstrate the design and analysis of the mission profiles of the interplanetary relay satellites. Figure 1 provides another representation of the placement of the interplanetary satellites, this one including lines-of-sight between Earth, Mars, and each satellite. Figs. 49-54 show simulation results for communication distances over time.

Figure 1 With four interplanetary relay satellites located at the Sun-Earth and Sun-Mars

L4 and L5 points, our system provides multiple possible paths of communication between

Earth and Mars. Here, Earth’s orbit is depicted in cyan, Mars’s orbit is depicted in red, and lines-of-sight are depicted in white, green, and pale blue. The lines-of-sight from Earth

to the Sun-Earth Lagrange points are depicted in white on the left side of the figure. The lines-of-sight from Mars to the Sun-Mars Lagrange points are depicted in pale blue on the

Purdue University J. I. Mayer Comm. Sat Appendix | 63 right side of the figure. The lines-of-sight between the Sun-Earth and Sun-Mars Lagrange

points are depicted in green and green-blue, forming an hourglass shape in the center of

the figure. Credit for the figure goes to Nick Oettig of Communications.

Figure 2 This plot depicts the distances from Earth to Mars and from Earth to the Sun-

Mars L4 and L5 points over the 15-year simulated timespan.

Purdue University J. I. Mayer Comm. Sat Appendix | 64

Figure 3 This plot depicts the distances from Mars to Earth and from Mars to the Sun-

Earth L4 and Sun-Earth L5 points over the 15-year simulated timespan.

Purdue University J. I. Mayer Comm. Sat Appendix | 65

Figure 4 This plot depicts the distances from each of the Sun-Earth L4 and L5 points (EL4

and EL5) to each of the Sun-Mars L4 and L5 points (ML4 and ML5) over the 15-year

simulated timespan.

Purdue University J. I. Mayer Comm. Sat Appendix | 66

Figure 5 This plot depicts the 2 closest distances between the Sun-Earth L4 and L5 points and the Sun-Mars L4 and L5 points over the 15-year simulated timespan. In other words,

this plot shows the 2 closest distances between the ‘Earth Lagrange relays’ and the ‘Mars

Lagrange relays’ over time.

Purdue University J. I. Mayer Comm. Sat Appendix | 67

Figure 6 This plot depicts the distances from a cycler vehicle to all other points of communication (Earth, Mars, and each of the 4 interplanetary relay satellites) over the 15-

year simulated timespan.

Purdue University J. I. Mayer Comm. Sat Appendix | 68

Figure 7 This plot depicts the 2 closest distances between a cycler vehicle and any point of

communication (Earth, Mars, or an interplanetary relay satellite) over the 15-year

simulated timespan. Note that, because other simulations verify that Earth and Mars can always communicate, and each interplanetary relay satellite can always communicate with

at least one of Earth and Mars, a cycler vehicle need only be capable of communication with Earth, Mars, or one interplanetary relay satellite in order to be able to communicate with both Earth and Mars. Although this plot shows only the results for one cycler vehicle,

our analysis is based on simulation results for all cycler vehicles.

Purdue University C. Deng Comm. Sat Appendix | 69

Appendix H: Equations Used in Propulsion Analysis

All equations are from Physics of Electric Propulsion written by Robert G. Jahn [A2].

∆푉 ∆푉 2 2 퐼푠푝푔 푚푢 − 푃푆푆푀푔 퐼푠푝(1 − 푒 ) 퐼푠푝푔 = 푒 + ∆푉 (10) 푚0 2휂∆푡푒퐼푠푝푔

퐹푇푔퐼푠푝 푃 = (11) 2휂

Below are the steps for deriving equation (10). From the definition of initial mass and final mass, we know

푚0 = 푚푝푟 + 푚푝푝푢 + 푚푢 (12)

푚푓 = 푚푝푝푢 + 푚푢 (13)

퐹푇∆푡 푚푝푟 = 푚̇ ∆푡 = (14) 퐼푠푝푔

푚푝푝푢 = 훼푃 (15)

From equation (12), we can express mass of useful payloads as

푚푢 = 푚0 − 푚푝푟 − 푚푝푝푢 (16)

Purdue University C. Deng Comm. Sat Appendix | 70

Combining equations (14), (15), and (16), we can get

푚푢 푚0 푚푝푟 푚푝푝푢 퐹푇∆푡 훼푃 = − − = 1 − − (17) 푚0 푚0 푚0 푚0 퐼푠푝푔푚0 푚0

The rocket equation is

푚0 ∆푉 = 푢푒ln ( ) (18) 푚푓

From equation (18), we can express initial mass as

∆푡 푢 (19) 푚0 = 푚푓푒 푒

Substituting equations (13) and (16) into equation (19), we get

∆푡 푢 (20) 푚0 = (푚0 − 푚̇ ∆푡)푒 푒

Rearranging equation (20), we get

∆푡 ∆푡 푢 푢 푚0 (1 − 푒 푒) = −푚̇ ∆푡푒 푒 (21)

Substituting equation (14) into equation (21), we get

Purdue University C. Deng Comm. Sat Appendix | 71

∆푡 ∆푡 퐹 ∆푡푒푢푒 푢 푇 (22) 푚0 (1 − 푒 푒 ) = − 퐼푠푝푔

Rearranging equation (22), we get

∆푡 푢 퐹푇∆푡푒 푒 푚0 = − ∆푡 (23) 푢 퐼푠푝푔 (1 − 푒 푒)

Substituting equation (23) into equation (17), we get

∆푡 ∆푡 푢 푢 퐼푠푝푔 (1 − 푒 푒) 퐼푠푝푔 (1 − 푒 푒 ) 푚푢 퐹푇∆푡 (24) = 1 + ∆푡 + 훼푃 ∆푡 푚0 퐼푠푝푔 푇∆푡푒푢푒 푇∆푡푒푢푒

Rearranging equation (24), we get equation (25), which is identical to equation (10).

∆푉 ∆푉 2 2 퐼푠푝푔 푚푢 − 훼푔 퐼푠푝(1 − 푒 ) 퐼푠푝푔 = 푒 + ∆푉 (25) 푚0 2휂∆푡푒퐼푠푝푔

Below is the derivation for equation (11).

By definition,

Purdue University C. Deng Comm. Sat Appendix | 72

푚̇ 푢2 푃 = 푒 (26) 2

퐹푇 = 푚̇ 푢푒 (27)

Using equations (27) and (26), we get

퐹 푢 푃 = 푇 푒 (28) 2

Since the power use is not 100%, there is an efficiency factor:

푃 = 휂푃 (29)

From equations (28) and (29), we get

퐹 푢 퐹푇푔퐼푠푝 푃 = 푇 푒 = (30) 2휂 2휂

Below is the calculation of the ΔV for orbit transfer, using the Hohmann orbit transfer equation.

휇 2푟1 ∆푉 = √ (1 − √ ) (31) 푟2 푟1 + 푟2

Below is the equation for fuel analysis.

Purdue University C. Deng Comm. Sat Appendix | 73

푢푒 = 푔퐼푠푝 (32)

For 1-D electrostatic acceleration,

푃 4휀 2푞 푉5/2 0 1/2 0 (33) = ( ) 2 퐴 9 푀′ 푥푎

2푞푉 푣 = √ 0 (34) 푎 푀′

Rearranging equation (33), we get

2 9푥푎 푃 푀′ 2 푉0 = ( √ )5 (35) 4휀0 퐴 2푞

Substituting equation (35) into equation (34), we get

2푞 9푥2 푃 푀′ 2 푎 (36) 푣푎 = √ ( √ )5 푀 4휀0 퐴 2푞

This is the equation of atomic mass.

Purdue University C. Deng Comm. Sat Appendix | 74

Appendix I: Suggestions and Possible Improvements for Propulsion Systems

It should be noted that our design of the electric propulsion system is still very top-level. It is assumed to be an ion thruster. However, it could be a magnetoplasmadynamic accelerator, a hall thruster or a pulsed plasma accelerator. The electric thruster equation used in Appendix G is a generally equation for all types of electric thrusters. It is inaccurate and not specific.

During our analysis for choosing the propellant, we used 1-D cross-field acceleration, also known as electrostatic acceleration, and it is almost the most simplified version of plasma acceleration, which makes it unrealistic. There is no mention of what kinds of material to be used for the ion thruster. Typically, the temperature of the ion thruster is high enough to melt copper, so a proper material choice is necessary. No analysis to the temperature could also cause problems to the whole power and thermal system of the satellite.

Furthermore, although the end-of-life plan is canceled, we still think that it is worthy to be considered as part of the design because 6-year operating time is short compared to how long our project is going to last, so replacement of the satellite will be frequently needed. However, the model used in our solution to the end-of-life plan is not accurate enough. Different from the chemical propulsion system, the electric propulsion system provides low level of thrust, which means the satellite cannot decrease to a certain speed quickly. As a result of that, the actual orbit transfer figure is more similar to the one shown below.

Purdue University C. Deng Comm. Sat Appendix | 75

Figure 8 This figure shows a more realistic orbit transfer for the end-of-life plan.

In the future, we can consider the orbit transfer shown above and make our design more realistic.

Purdue University B. D. Walter Comm. Sat Appendix | 76

Appendix J: Calculation of External Torques

To calculate gravity gradient torque, we use the expression given below, in equation (37). These values also include torques from bodies other than the primary body the satellites orbit, although these are significantly smaller than the primary sources of torque. In this equation, μ represents the gravitational constant of the body, r represents the distance from the center of the body to the satellite, and I is the moment of inertia of the satellite about the rotational axis. θ is the angle between the pointing of the satellite and the vector normal to the surface of the body, such that when the satellite is pointed directly at the body it is orbiting, θ is 0 degrees.

3 푎3 휏 = 2휋 ∗ √ (37) µ

Purdue University B. D. Walter Comm. Sat Appendix | 77

Appendix K: Further Explanation of Simulink Modeling

Figure 9 This figure shows the control scheme of the satellite, modeled in Simulink. The satellite uses the pointing angle as a feedback variable to ensure correct satellite orientation

(repeated from Controls Architecture section of this chapter).

The torque limit in the Simulink model (the saturation block) is -.17 and .17, representing the maximum torque that a reaction wheel could provide in either direction. The random block uses a variance equal to the maximum expected torque/3. This ensures that a majority of the torques applied from the random block can be reasonably expected over the course of the mission.

Moments of inertia used in the gain values are from the CAD models of the satellites.

We found the gains in the controller by increasing the derivative gain until an acceptable response was achieved. Increasing the proportional gain proved to be not useful due to the limit placed on the torque thanks to the saturation block. It was crucial to limit the overshoot of the response, since the external torques on their own were generally not enough to push the satellite out of its acceptable range. To do this, we used the very high derivative gain.

Purdue University B. D. Walter Comm. Sat Appendix | 78

Table 23. Controller gains

Gain Value

Kp 99

Ki 0

Kd 12,040

Future considerations for the project would be developing more precise gains- such a high derivative gain would potentially cause problems with the hardware. A controller for each individual satellite and axis should be developed as well. For our case, we simply designed a controller for the most important axis (yaw) for the satellite subject to the greatest external torque.

We aimed to demonstrate that our control hardware is able to handle the mission, rather than designing a comprehensive control scheme that would be operational for an actual satellite with this mission.

Purdue University P. J. Salek Comm. Sat Appendix | 79

Appendix L: Power Calculations

We used solar irradiance values to calculate the different solar panel sizing based on location and power requirements of the location. To determine the solar radiation at different distances from the sun, we assumed a circular orbit for each location and that the sun emits uniform radiation.

We used the equation shown below to calculate the solar irradiance at each location.

푅푠푢푛2 푆표푙푎푟 퐼푟푟푎푑푖푎푛푐푒 = ∗ 퐻푠푢푛 (38) 퐷푖푠푡푎푛푐푒2

To determine the square footage required for each solar panel, we used the equation below.

푃표푤푒푟 푅푒푞푢푖푟푒푑 푆표푙푎푟 퐴푟푒푎 = (39) 푆표푙푎푟 퐼푟푟푎푑푖푎푛푐푒 ∗ 푆표푙푎푟 푃푎푛푒푙 퐸푓푓푖푐푖푒푛푐푦

Purdue University Y. Gupta Comm. Sat Appendix | 80

Appendix M: Thermal Calculations

We calculate solar irradiance by using a distance ratio [10].

(40)

We find Hsun by using Stefan-Boltzmann’s equation:

(41)

To form a thermal equilibrium for the satellites, we perform a simple, steady state thermal analysis and equate the satellite system’s radiation to find the temperature.

(42)

and H0 is the Solar Irradiance at that location in space.

Using a rearrangement of equation (43) we can get an expression to describe the ratio α/ε:

(43)

Purdue University R. L. Roth Comm. Sat Appendix | 81

Appendix N: Additional Tables on Satellite Structure

Table 24. Structural mass and volume calculations

Structural Component Mass, Mg Volume, m3

Wall structure 0.604 0.215 GEO/AREO Trusses 0.610 0.217

Sun-Earth/Sun-Mars Wall structure 0.717 0.255

Lagrange Points Trusses 0.691 0.246

Total GEO Satellite Structure 1.214 0.432

Total Sun-Earth Lagrange Points Structure 1.408 0.501

Total Sun-Mars Lagrange Points Structure 1.408 0.501

Total AREO Satellite Structure 1.214 0.432

Purdue University R. L. Roth Comm. Sat Appendix | 82

Table 25. GEO satellite mass and volume calculations

Component Mass, Mg Volume, m3

Structure 1.214 0.432

RF GEO to Earth 0.008 External

RF GEO to GEO 0.030 External Communications × 2

Equipment Optical Link GEO

to Sun-Earth 0.085 External

Lagrange Points

Control 0.092 1.181

Propellant 8.679

Propulsion Thrusters 0.118 External

RCS External

Solar Panels External 0.028 Power and Thermal Battery 0.006

Thermal 0.108 1.000

Total GEO Satellite Structure 1.713 11.298

Purdue University R. L. Roth Comm. Sat Appendix | 83

Table 26. Sun-Earth Lagrange point satellite mass and volume calculations

Component Mass, Mg Volume, m3

Structure 1.408 0.501

Optical Link GEO

to Sun-Earth 0.085 External

Lagrange Points

Communications Optical Link Sun-

Equipment Earth Lagrange

Points to Sun- 0.201 External

Mars Lagrange

Points

Control 0.011 1.091

Propellant 4.340

Propulsion Thrusters 0.095 External

RCS External

Solar Panels External 0.037 Power and Thermal Battery 0.003

Thermal 0.097 1.000

Total Sun-Earth Lagrange Points Satellite 1.934 6.935 Structure

Purdue University R. L. Roth Comm. Sat Appendix | 84

Table 27. Sun-Mars Lagrange point satellite mass and volume calculations

Component Mass (Mg) Volume (m³)

Structure 1.408 0.501

Optical Link Sun-

Earth Lagrange

Points to Sun- 0.050 External

Communications Mars Lagrange

Equipment Points

Optical Link Sun-

Mars Lagrange 0.201 External

Points to AREO

Control 0.011 1.091

Propellant 4.340

Propulsion Thrusters 0.095 External

RCS External

Solar Panels External 0.037 Power and Thermal Battery 0.003

Thermal 0.097 1.000

Total Sun-Mars Lagrange Points Satellite 1.899 6.935 Structure

Purdue University R. L. Roth Comm. Sat Appendix | 85

Table 28. AREO satellite mass and volume calculations

Component Mass (Mg) Volume (m³)

Structure 1.214 0.432

RF AREO to 0.008 External Mars

RF AREO to 0.030 External Communications AREO × 2

Equipment Optical Link

AREO to Sun- 0.050 External Mars Lagrange

Points

Control 0.092 1.181

Propellant 19.09

Propulsion Thrusters 0.095 External

RCS External

Solar Panels External 0.033 Power and Thermal Battery 0.002

Thermal 0.076 1.000

Total AREO Satellite Structure 1.628 21.705

Purdue University R. L. Roth Comm. Sat Appendix | 86

Table 29. Al 7075-T3751 material properties [A8]

Property Value

Density 2810 kg/m3

Tensile strength, 505 MPa ultimate

Tensile strength, yield 435 MPa

Modulus of Elasticity 72 GPa

Elongation at Break 13%

1 Fracture Toughness 20 – 32 MPa-m2

Melting Point 477 – 635 ℃

Thermal Conductivity 155 W/m-K

Purdue University H. N. Harjono Comm. Sat Appendix | 87

Appendix O: Equations for Moment of Inertia Calculations

For a cube:

1 퐼 = 퐼 = 퐼 = 푚푠2 (44) 푥 푦 푧 6

For a rectangular prism:

1 퐼 = 푚(푦2 + 푧2) (45) 푥 12

1 퐼 = 푚(푥2 + 푧2) (46) 푦 12

1 퐼 = 푚(푥2 + 푦2) (47) 푧 12

For a rectangular plate:

푚푥푦3 퐼 = (48) 푥 12

Purdue University H. N. Harjono Comm. Sat Appendix | 88

푚푥3푦 퐼 = (49) 푦 12

For a circular disk:

푚휋푟4 퐼 = (50) 푟 4

Purdue University Comm. Sat Appendix | 89

References

Note that references 31-35 do not appear specifically in the report, but are sources of values and methods that we used to write some of the code for our analysis.

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[2] Hamid Hemmati, Deep Space Optical Communications, Jet Propulsion Laboratory,

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[3] Wallace, Megan. “LLCD: 2013-2014.” NASA, NASA, 27 Mar. 2018,

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[16] Yang, L., Li, Q., Kong, L., Gu, S., and Zhang, L., “Quasi-All-Passive Thermal Control

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American Institute of Aeronautics and Astronautics, 2003.

[18] Price, M. K., Mass and Power Modeling of Communication Satellites, Cleveland, OH:

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[19] National Aeronautics and Space Administration. (2014). Structural Design Requirements

and Factors of Safety for Spaceflight Hardware (JSC 65828). Retrieved from

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[21] Federal Aviation Administration. (2003). Metallic Materials Properties Development and

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1, ????, pp. 1–22. https://doi.org/10.1023/A:1024048429145.

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[29] Jahn, R. G. (2006). Physics of electric propulsion. Mineola: Dover.

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[32] Silva, Juan J., and Pilar Romero. “Optimal Longitudes Determination for the Station

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Purdue University Cycler Appendix | 93

13. Cycler Vehicles Appendix

Riley Franklin19, Jacob Leet20, Beverly Yeo, David Fox21, Aaron

Engstrom22, Griffin Pfaff23, Jacob Nunez-Kearney24, Adam Brewer25,

Adam Wooten26, Alexey Zenin3, Eli Sitchin4, Jennifer Bergeson27,

Megan Brown8, Sarah Culp3, Kaitlyn Hauber3, Suhas Anand6, Walter

Manuel3

Purdue University, West Lafayette, Indiana, 47906, United States

19 Controls, Associate Editor for Cycler Vehicle 20 Controls 21 Human Factors 22 CAD 23 Propulsion 24 Power and Thermal 25 Structures 26 Communications 27 Mission Design

Purdue University Cycler Appendix | 94

Nomenclature

푑휔 ( ) = Maximum rate of change of angular velocity of the tether 푑푡 푚푎푥

∅푝 = Angle of Pyramid Configuration

µ = standard gravitational parameter (constant for each celestial body)

a = centripetal acceleration (artificial gravity)

A = area

abody = Semi-major axis of orbit of a planetary body

abody = Semi-major axis of orbit of any body

acycler = Semi-major axis of the hyperbolic orbit of the cycler with respect to the

planetary body

AEclipsing body = Projected surface area of the eclipsing body

ASun = Projected surface area of the Sun

atransfer = Semi-major axis of the hyperbolic orbit of the spacecraft with respect to

the planetary body

BN = Boron Nitride

BNNT = Boron Nitride Nanotubes

d = Distance from Sun (m)

d = diameter

Purdue University Cycler Appendix | 95

e = Eccentricity of the hyperbolic orbit of the cycler with respect to the

planetary body

etransfer = Eccentricity of the hyperbolic orbit of the spacecraft with respect to the

planetary body

F = force

F vtransfer = Velocity of the spacecraft the moment it separates from the cycler

F = Emissive flux from the Sun at a given radial distance from the Sun

FEarth = Emissive flux from the Sun at Earth’s semi-major axis of orbit

Freduced = Reduced Solar flux due to blackout

FSun = Surface flux of the Sun

g = Earth’s gravitational acceleration (constant)

GCR = Galactic Cosmic Radiation

h = height

Hcycler = Hyperbolic anomaly of the cycler

Htransfer = Hyperbolic anomaly of the spacecraft

I = Moment of Inertia

IPhobos = Moment of inertia of Phobos

Iy = moment of inertia about y-axis (minor axis)

Iz = moment of inertia about z-axis (major axis)

L = Luminosity of the Sun (3.86*1026 W)

L = length

L = Angular Momentum

l = length

Purdue University Cycler Appendix | 96

LEO = Low Earth Orbit

LMO = Low Mars Orbit

m = mass

mtaxi = Mass of the taxi

mtether = Mass of the tether

n = Mean motion

NCRP = National Council on Radiation Protection and Measurements

P = Power

p = pressure

p = Burn power on a 0-1 scale where 1 is full power

Ploss = Percentage of power loss due to blackout of inner planet

PTD = Plastic Track Detector

r = radius

R = distance from spacecraft to celestial body (planet or sun)

R = radius from center of rotation

r: Distance between the spacecraft and the center of the planetary body

rbody = Radius of a planetary body

rcirc = Radius of circular low body orbit

rp, transfer = Periapsis of the hyperbolic orbit of the spacecraft with respect to the

planetary body

rp = Periapsis of the hyperbolic orbit of the cycler with respect to the planet

RPM = rotations per minute

rSun = Radius of the Sun

Purdue University Cycler Appendix | 97

rtether = Radius of the tether

S = Solar Constant

SPE = Solar Particle Event

Sv = Sievert

T = Temperature (K)

t = thickness

t = Time

T = Torque

TEPC = Tissue Equivalent Proportional Counter

TLD = Thermoluminescence Detector

Tmax, tether = Maximum torque on Phobos exerted by the tether

TOF = Time of Flight

V = velocity

vinf, cycler = Velocity of the cycler with respect to a planetary body at infinity

Δt = Amount of time the spacecraft will take to reach periapsis

Δt = Spin-up or spin-down time of the tether

Δtactual = Actual burn time for a single burn

Δtbody = Amount of time a spacecraft spends in the shadow angle behind a body

Δteq = Equivalent full power burn time for a single burn

Δttot = Total equivalent full power burn time between refueling

ΔV = Change in Velocity

ΔωPhobos, max = Maximum change in angular velocity of Phobos

θ = Shadow angle of local planet

Purdue University Cycler Appendix | 98

μ = Gravitational parameter of the planetary body

μSun = Gravitational parameter of the Sun

ρ = density

σy = yield strength

ω = angular velocity

ωescape = Angular velocity necessary for the surface of Phobos to reach escape

velocity

ωPhobos = Angular velocity of Phobos currently

푄 = Heat Energy (J)

휀 = Emissivity

휂 = Radiator Efficiency

휎 = Stefan Boltzmann Constant (5.67 * 10-8 W/m2K4)

휔 = angular velocity

Appendix

Structural Analysis a. Superstructure

For a closed, cylindrical pressure vessel, there are two stresses that develop: one along the longitudinal direction (longitudinal stress) and one around the circumference of the cylinder

(hoop stress). These two stresses are given by equation 1 and equation 2, respectively.

푝푑2 (1) 휎 = 푙 (푑 + 2푡)2 − 푑2

푝푑 (2) 휎 = ℎ 2푡

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In both of these equations, d is the mean diameter, or the mean of the outer and inner diameter.

Since t is small compared to d, then the longitudinal stress can be approximated as given by equation 3.

푝푑 (3) 휎 ≈ 푙 4푡

This shows that, for a thin-walled pressure vessel, the hoop stress will always be higher than the longitudinal stress. In fact, the hoop stress is twice as high as the transverse stress. For this reason, only the hoop stress will be analyzed. Furthermore, since only the inner cylinder of the superstructure is pressurized (due to micrometeorite shielding), only this cylinder will be analyzed.

For this inner cylinder, the pressure is 101.3 kPa (atmospheric pressure at sea level), the thickness is 3.5 mm, and the mean diameter is 9.39 m. Plugging these values into equation 2 gives a maximum stress of 67.9 MPa, which in turn gives a minimum factor of safety of 3.7. b. Tether

For the static analysis of the tethers connecting the habitation modules to the superstructure, we assume that the only non-negligible forces are the centripetal force from the habitation module as it rotates about the superstructure, and the centripetal force from the tether itself.

These forces induce a tension in the tether that causes stress, which we assume to be concentrated at the root of the tether. Centripetal force is given by equation 4.

푀푣2 (4) 퐹 = 푐 푟

In this equation, M is the mass of the habitation module and r is the length of the tether.

Since v = ωr, then equation X.1 can be rewritten as equation 5.

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2 퐹푐 = 푀휔 푟 (5)

As a conservative model, the habitation module is modelled as a point mass. Since there are four tethers per habitation module, then the stress due to the centripetal force of the habitation module in each tether can be written as equation 6.

푀휔2푟 (6) 휎 = ℎ 4퐴 where A is the cross-sectional area of the tether.

For the stress due to the centripetal force of the tether, we examine an infinitesimal segment of the tether’s length. The mass of this segment is given by equation 7.

푑푚 = 휌퐴푑푥 (7) where x is directed outward along the length of the tether. Combining equations 7 and 5 and integrating along the length of the tether, we find the total centripetal force from the tether, given in equation 8.

푟 2 (8) 퐹푡 = ∫ 휌퐴휔 푥푑푥 0

Solving this integral and dividing by the cross-sectional area gives the stress due to the centripetal force of the tether, shown by equation 9.

1 (9) 휎 = 휌휔2푟2 푡 2

In this equation, ρ is the density of the tether material.

Assuming that the tension from the two centripetal forces, and therefore the stresses, can be added linearly, the total stress is given by equation 10.

푀휔2푅 1 (10) 휎 = + 휌휔2푟2 푡표푡푎푙 4퐴 2

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For this analysis, the tether cross sectional area is 1 m2, the rotational rate is 3 RPM (0.314 rad/s), the tether length is 400 m, and the habitation module total mass is 4036.5 Mg (1009.2 Mg per tether). Putting these values into the code cycler_tether_stress.m, we find that the maximum stress is 52.2 MPa. Since Zylon has an ultimate tensile strength of 5.8 GPa, this gives a factor of safety of 111.2.

Cycler Superstructure Configuration

Because particles in outer space have a wide variety of compositions, sizes, speeds, and trajectories, it is difficult, if not impossible, to derive analytical equations for micrometeorite shielding. This is further complicated by the fact that different configurations will result in different types of damage (even when using the same material for protection) and that different levels of damage may be acceptable in different applications. However, since the cycler is a pressurized vessel with humans on board, it is prudent to design shielding that can withstand a wide range of projectiles, moving at a wide range of speeds, with no damage to the innermost layer.

The most commonly used method of shielding from micrometeorites is a Whipple shield; that is, two layers of material (commonly aluminum) separated by an air or vacuum barrier. A

1993 NASA paper [1] describes equations for calculating the thickness of such a shield based on experimental data. For a Whipple shield with no damage to the interior layer, the thicknesses of the outer and inner walls, respectively, are given by the equations 11 and 12 below. Note that in these equations, the subscript b refers to the outer wall, w refers to the inner wall, and p refers to the projectile.

휌푝 (11) 푡푏 = 0.25 푑 휌푏

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1 1 0.5 푉 휎푦 (12) 푡 = 0.16 푑0.5 (휌 휌 )6 푚3 푛 ( ) 푤 푝 푏 푝 푆0.5 70

In order to protect against a wide range of micrometeorites, a set of very conservative assumptions were made. It was assumed that a projectile would have a diameter of 1 cm, a mass of 1.5 g, a uniform density of 3.5 g/cm3 (3500 kg/m3), and would be travelling at a speed of 7 km/s, normal to the spacecraft. These values were the largest tested in the paper by Christiansen, and therefore should provide adequate shielding in a worst-case scenario.

Both the inner and outer barriers, being constructed from aluminum 7075-T6, have a density of 2810 kg/m3 and a yield strength of 503 MPa. S, the distance between the two barriers, was chosen to be 30 cm, the largest distance tested in the paper by Christiansen.

Plugging all of these values into the code whipple_shield.m shows that the safest Whipple shield has an outer thickness of 3.1 mm, an inner thickness of 3.5 mm, and a 30 cm gap between them. This yields a total thickness of 30.66 cm.

To calculate the empty mass of the superstructure, one simply needs to multiply the volume of the superstructure hull by the density of its material. The mass of a capped hollow cylinder is given by equation 13.

2 2 2 푚 = 휌[휋ℎ(푟표 − 푟𝑖 ) + 2휋푟표 푡] (13)

For the outer cylinder, the height is 50 m, the outer radius is 5 m, and the inner radius is

4.9969 m (the outer diameter minus the outer thickness). The inner cylinder has a height of 4.4 m

(the total height minus the spacing between barriers, doubled to account for both ends), an outer radius of 4.6969 m, and an inner radius of 4.6934. These numbers can be used to calculate the mass of both inner and outer cylinders, which gives a total hull mass of 17.7 Mg.

References

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[1] Christiansen, E. L. (1993). Design and performance equations for advanced meteoroid and debris shields. International Journal of Impact Engineering, 14(1-4), 145–156. doi:

10.1016/0734-743x(93)90016-z

a. Cycler Connective Structure Configuration

Preliminary estimates for Cycler mass and volume required an accurate measure of the radius of the vehicle. The larger the radius, the slower the habitation module has to rotate to generate artificial gravity and the less likely passengers are to realize they are rotating. Limiting factors to increasing the radius are connective structural mass, complications from traveling large distances to the superstructure, and material limits. With collaboration between the structures, human factors, and CAD team, we choose a radius of 400 meters as a balance between structural load and travel time from the habitation modules to the superstructure.

푅 = 푎 / 휔2 = 400 푚 (14)

Although mission requirements specify only Martian gravity is required on the cycler (푎 =

0.376푔), the Cycler is built to withstands loads of Earth’s gravity and operates as such.

푎 = 푅휔2 = 푔 = 9.81 푚/푠2 (15)

휔 = √푎/푅 = 0.1556 푟푎푑/푠푒푐 = 1.496 푅푃푀 (16)

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Figure XX. In this figure the window of artificial gravity is plotted with Earth’s gravity in blue and Martian gravity in red. Rotational speed cannot exceed 3 RPM. Vehicle radius cannot be under 38 meters. See the code Cycler_Rotation.m for more information.

Mission Profile

The purpose of structuring this mission in the way we have is to provide a sustainable infrastructure for repeatable Martian missions, and to reduce the amount of propellant required to complete these missions. Because of this purpose, we decide to investigate exactly how much propellant ΔV the reusable infrastructure would save us.

Determining the ΔV savings from this mission is not as simple as taking the tether and mass driver ΔV’s and replacing them with propellant. If this mission is completed using entirely propellant, the whole mission profile would look different. Because of this difference, the first task we address is determining the propellant based mission profile. A few main assumptions and differences exist between the tether-based profile and the propellant based profile. First of all, as addressed in the section on the electrodynamic tether, the specific LEO that the taxi starts from is severely limited by the electrodynamic tether’s range of motion. If this mission were completed

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Purdue University Cycler Appendix | 1 0 5 using entirely propellant, this LEO restriction would disappear. In order to give the propellant- based mission as much credit as possible, we assume that in a propellant-based mission the taxi could start from any inclination LEO. We also choose to neglect the third-body impacts of moons and the Sun on the calculation of the propellant-based mission profile, because those effects are outside the scope of this project. We compare our design to a propellant-based mission using the same orbits. Based on these assumptions we create a new propellant-based mission profile that skips the rendezvous with Luna and Phobos because they are unnecessary.

The new profile consists of a taxi trip from LEO to the passing cycler, followed by the requisite

4- to 8-month trip on the cycler. Once the cycler nears Mars the taxi again departs the cycler and heads straight for Mars, where it executes a propellant-based Martian entry. All the mission phase intersections (i.e. departure from LEO, etc.) are done using the taxi propulsion system.

We calculate the details of the propellant ΔV using the patched conics method from orbital mechanics. We assume the cycler will pass by Earth and Mars on a hyperbola with respect to each planet, and that a taxi in a low body orbit will execute an impulse burn to place itself on a hyperbola with the same velocity at infinity as the cycler. This assumption is relatively conservative and allows us to come up with conservative estimates for propellant savings. The equations below dictates the ΔV values for a spacecraft departing low-body-orbit to the cycler.

2 푣 휇 휇 (17) 훥푉 = 푠푞푟푡 (2 ( 𝑖푛푓,푐푦푐푙푒푟 + )) − 푠푞푟푡 ( ) 2 푟푝 푟푐𝑖푟푐

Finding the ΔV for a spacecraft departing a body to the cycler is fairly straightforward.

However, finding the ΔV value for a spacecraft departing the cycler to a body is less simple. Our biggest problem for determining the ΔV for a taxi leaving the cycler and directly arriving at a body is determining the distance from the body at which the taxi and cycler would separate. This issue is particularly tricky because the closer to the body that the cycler is before it separates

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Purdue University Cycler Appendix | 1 0 6 from the taxi, the more ΔV will be required. However, the farther from the body that the cycler is at separation, the more time spent in the taxi will be required. It is important that we reach a happy medium between these two tradeoffs. Based on orbital and human factors requirements of the standard tether-based mission, the maximum time of flight allowed for a taxi trip is set to be

4 days. This maximum TOF is used as the upper limit for the propellant-based mission as well in order to keep the missions as comparable as possible. In order to resolve this tradeoff issue, we implement an algorithm where a separation point estimate is taken to be the point on the cycler trajectory when the cycler is 3 days away from its closest approach to the body. This point is used because the taxi needs to have a time of flight less than 4 days, and the time of flight for the intercept trajectory of the taxi and its low body orbit is longer than the time of flight for the cycler. This separation point is determined by simultaneously solving the following three equations for a, e, and H.

2 푣 휇 (18) 𝑖푛푓,푐푦푐푙푒푟 = 2 2∗푎푏푠(푎푐푦푐푙푒푟)

휇 (19) 푠푞푟푡 ( 3) ∗ 훥푡 = 푒푠푖푛ℎ(퐻푐푦푐푙푒푟) − 퐻푐푦푐푙푒푟 (푎푏푠(푎푐푦푐푙푒푟))

푟푝 = 푎푏푠(푎푐푦푐푙푒푟)(푒 − 1) (20)

Once this separation point estimate is known, we can iterate on the exact point in order to find a separation point that yield a 3.75 -day taxi flight time after it departs the cycler. We do this using the following equations.

2 푣 휇 (21) 푣 = 푠푞푟푡 (2 ∗ ( 𝑖푛푓, 푐푦푐푙푒푟 + )) 푐푦푐푙푒푟 2 푟

푣푡푟푎푛푠푓푒푟 = 1 + 푣푐푦푐푙푒푟 (22)

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휇 푎푏푠(푎푡푟푎푛푠푓푒푟) = 2휇 (23) 푣2 − 푡푟푎푛푠푓푒푟 푟

푟 ( +1) (24) 푎푏푠(푎 ) 퐻 = ( 푡푟푎푛푓푒푟 ) 푡푟푎푛푓푒푟 푒_푡푟푎푛푠푓푒푟

푟푝, 푡푟푎푛푠푓푒푟 (25) 푒푡푟푎푛푠푓푒푟 = + 1 푎푏푠(푎푡푟푎푛푠푓푒푟)

푒푡푟푎푛푠푓푒푟푠𝑖푛ℎ푠𝑖푛ℎ (퐻푡푟푎푛푠푓푒푟) −퐻푡푟푎푛푠푓푒푟 (26) 훥푡 = 휇 푠푞푟푡( ) 푎푏푠(푎)3

From the equations above the iteration to find r of separation is executed as follows: Eq. (21) is plugged into Eq. (22) which is plugged into Eq. (23) which is plugged into Eq.s (24) and (25).

Eq.s (24) and (25) are plugged into Eq. (26), and the resulting value from Eq.(26) is checked with the desired amount of time. If it is not acceptable, r is varied and the process is repeated.

Using this separation point is important because it allows the taxi to leave the cycler when it is far enough from the local body that ΔV can be minimized, but at the same time it allows a quarter of a day for adjustments and deorbit before the acceptable taxi flight time is exceeded.

Now that we have addressed the methods and logic behind the mission profile of the trade study propellant-based mission, it is clear that our propellant and tether missions look a bit different. The propellant mission involves three legs, whether the initial body is Mars or Earth.

The taxi begins in low body orbit around the initial body. It then executes a burn which places it on an intercept hyperbola with the cycler. It then stays with the cycler until it nears the final body, where it executes another burn to take it off of the cycler trajectory and place it on an intercept trajectory for low body orbit of the final body. The tables below present the ΔV results for the intersections of each of these mission legs. Table 4 provides the average ΔV values for the intersections of each of the three legs (body to cycler, trip along with cycler, cycler to body)

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Purdue University Cycler Appendix | 1 0 8 and Table 5 provides the maximum ΔV values for the intersections of each of the three legs. The second column of each table gives the ΔV values for the alternative propellant based mission.

The third column gives the propellant ΔV values that our infrastructure still requires for each segment, and the fourth column gives the ΔV savings when these values are accounted for.

Table 4: Average ΔV from Propellant for Equivalent Mission

Taxi Segment Average ΔV (km/s)

LEO to Cycler (outbound) 4.30

Cycler to LMO (outbound) 3.18

LMO to Cycler (inbound) 2.80

Cycler to LEO (inbound) 4.88

Table 5: Maximum ΔV for Equivalent Propellant Mission

Taxi Segment Maximum ΔV (km/s)

LEO to Cycler (outbound) 4.65

Cycler to LMO (outbound) 3.83

LMO to Cycler (inbound) 3.55

Cycler to LEO (inbound) 5.18

a. Calculation of Solar Power Loss Due to Inner Planets

The inner planets of Mercury and Venus both have an impact on the power received by the cycler vehicle. Because these planets have orbits that are closer to the Sun than Earth and Mars, from time to time the planets will pass between the Sun and the cycler vehicle. Because of this passage, it is important to determine the effect that these planets’ “blackouts” will have on the

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Purdue University Cycler Appendix | 1 0 9 cycler vehicle. In the main body of the report we stated that the power loss due to these planets was small enough to be negligible. Here we present the equations for the calculations behind this statement. The percentage of solar power at the cycler which is lost due to the inner planets is calculated using a few different assumptions. First of all, we assume that the Sun emits energy uniformly and that the planets are spherical. We also assume that the Sun and inner planets are sufficiently far away that the inner planets can be modeled as a cylinder superimposed on the

Sun’s uniform power emission sphere. The first step is to determine the solar flux at any point.

Essentially this is the power per unit area due to the radiative emission from the Sun at any radial distance from the Sun. The emissive flux at the surface of the Sun is assumed to be 63

[MW/m2]2, and knowing this we are able to calculate the emissive flux at any distance using the equation

2 푟푠푢푛 (27) 퐹 = 퐹푠푢푛 ∗ 2 푎푏표푑푦

This equation gives us the total solar flux at any given location. In order to find the reduced flux (reduced due to inner planet power blockage) we need to find the total flux at the point we are observing. We choose to calculate our reduced flux at the radial distance of the

Earth’s semi-major axis of orbit. This choice is made because the radial distance of the Earth from the Sun is the closest that the cycler will ever be to the Sun when it is crewed and thus using maximum power. The closer to the Sun, the larger the effect that the inner planetary power loss will have. Thus, by taking the Earth’s semi-major axis as our radial distance of investigation we make a conservative estimate and explore the inner planets’ largest possible effect. That being said, we need to calculate the total Solar flux at the Earth’s semi-major axis (FEarth). This value is calculated using Eq. (27), and is then used to calculate the reduced flux that the cycler

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Purdue University Cycler Appendix | 1 1 0 will experience at that radial distance due to the presence of an inner planet. The reduced flux is calculated using the equation

퐴푒푐푙𝑖푝푠𝑖푛푔 푏표푑푦 (28) 퐹푟푒푑푢푐푒푑 = 퐹퐸푎푟푡ℎ ∗ (1 − ) 퐴푠푢푛

We further use the reduced flux value and total flux value previously calculated to determine the percentage of power lost due to the inner planet. The equation for this power loss percentage is

퐹푟푒푑푢푐푒푑 (29) 푃푙표푠푠 = (1 − ) ∗ 100 퐹퐸푎푟푡ℎ

By plugging Eq.s (27) and (28) into Eq. (29) we are able to obtain the worst -case percentage of power lost at the cycler due to the blackout effect of the inner planets. As was shown in the main body of the report, this percentage turned out to be low enough to be negligible, and thus the inner planets were ignored when performing blackout time calculations for the cycler vehicle. c. Calculation of Blackout Times

The blackout times for each Mars and Earth employ a few different assumptions. We assume that the two -body approximation is valid, and thus the patched conics approach can be taken.

We also assume that the planets are spherical and have an average distance from the Sun which is equal to their semi-major axes of orbit. Finally, we assume that for the time in which the cycler is behind a planet, its angular velocity with respect to the Sun is constant and equivalent to the

Solar orbit’s mean motion. (For those not familiar with orbital mechanics, mean motion is the average angular rate at which a spacecraft orbits a central body.) Based on all of these assumptions, the blackout time is calculated using the following procedure. Because we are assuming that the angular velocity of the spacecraft in the vicinity of each planet is equal to the mean motion of that spacecraft’s solar orbit, the results of this procedure will actually be an overestimate, since entering the near vicinity of the planet would in reality accelerate the

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Purdue University Cycler Appendix | 1 1 1 spacecraft and cause the blackout time to be shorter. Thus, the results obtained here are a safe estimate and tend to the conservative side. First, the angle of shadow (θ) is calculated using

푟푏표푑푦 (30) 휃 = 2 ∗ ( ) 푎푏표푑푦 and the mean motion is calculated using

.5 휇푠푢푛 (31) 푛 = ( 3 ) 푎푏표푑푦

The units on mean motion are rad/s, and the units on angle of shadow are radians. Thus, the following equation can be developed to calculate the amount of time Δtbody that the cycler spends in the angle of shadow.

휃푏표푑푦 (32) 훥푡푏표푑푦 = 푛푚푎푟푠∗60

By plugging Eq. (30) and Eq. (31) into Eq. (32), we are able to determine the blackout time of the cycler each time it passes behind either Earth or Mars. The only numbers which we need are the radii of Earth and Mars, their semi-major axes of orbit around the Sun, and the gravitational parameter of the Sun. d. Additional Trajectory Information

Below we present the trajectory plots for cyclers 2, 3, and 4. These plots are for the identical type of cycler trajectory as was presented in the trajectory information of the cycler, however, they have different encounter dates and locations because they are for different cyclers. Fig. 4 presents the trajectory of cycler 2, Fig. 5 presents the trajectory of cycler 3, and Fig. 6 presents the trajectory of cycler 4.

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Fig. #4: In this figure the trajectory of cycler 2 is plotted (blue line) and the locations of the planetary encounters are added (red dots). The first planetary encounter is denoted ‘E-1’, and the last planetary encounter is denoted ‘M-26’.

Fig. #5: In this figure the trajectory of cycler 3 is plotted (blue line) and the locations of the planetary encounters are added (red dots). The first planetary encounter is denoted ‘E-1’, and the last planetary encounter is denoted ‘M-26’.

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Fig. #6: In this figure the trajectory of cycler 4 is plotted (blue line) and the locations of the planetary encounters are added (red dots). The first planetary encounter is denoted ‘E-1’, and the last planetary encounter is denoted ‘M-26’. e. Phobos Perturbation Investigation

Because of this issue, we investigate the impact that our tether will have on Phobos in order to determine if we will either despin Phobos, or spin it up to the point where matter begins to fly from its surface. In order to perform these calculations, we make a few assumptions. Primarily, we assume that the mass of the Phobos tether is concentrated at a distance r/3 from the center of the tether, and we assume that Phobos and the tether are axis-aligned, meaning that their angular momentum vectors are coincident. We also assume that the maximum rate of change of angular velocity of the tether is equivalent to ten times the average rate of change of the tether

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Purdue University Cycler Appendix | 1 1 4 throughout a spin-up cycle. Based on these assumptions, the following equation is used to determine the maximum torque exerted on Phobos by the tether.

2 2 푟푡푒푡ℎ푒푟 푑휔 (33) 푇푚푎푥, 푡푒푡ℎ푒푟 = (푚푡푎푥𝑖푟푡푒푡ℎ푒푟 + 푚푡푒푡ℎ푒푟 ( ) ) ( ) 3 푑푡 푚푎푥

From this equation we are able to obtain the maximum torque exerted on Phobos, and we can use this maximum torque to generate an overestimate of the change in angular velocity of

Phobos from one spin-up or spin-down cycle of the tether. The equation for the change in angular velocity of Phobos is

푇푚푎푥, 푡푒푡ℎ푒푟 (34) 훥휔푝ℎ표푏표푠, 푚푎푥 = 훥푡 퐼푝ℎ표푏표푠

This equation is a significant overestimate, which allows us to be certain that the effect it yields is larger than the actual effect we will observe. Based on this equation, Table 66666 below presents the effect that our tether will have on the rotational motion of Phobos.

Table 6: Phobos Tether Effect on Phobos’ Rotational Motion

ωPhobos (rad/s) ωescape (rad/s) |ΔωPhobos| (rad/s) % change in ωPhobos (%)

-4 -3 -6 1.82*10 2.23*10 7.20*10 3.95

Table 6 shows the change in angular velocity of Phobos during a single spin-up or spin- down cycle. For reference, it also presents the angular velocity of Phobos currently, as well as the angular velocity necessary for the surface of Phobos to reach escape velocity. As we show in the table, the change in angular velocity of Phobos during any one spin-up or spin-down cycle is a fairly small percentage of the total angular velocity. Depending on the direction of the spin cycle, the change in angular velocity could be positive or negative. If a proper configuration of cycle directions is used, the net effect on Phobos’ angular motion should be able to be

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Purdue University Cycler Appendix | 1 1 5 minimized. Thus, we conclude that the effect of the Phobos tether sling on the motion of Phobos is acceptably small.

Burn Time Calculations

A fairly simple algorithm was used to determine the equivalent maximum burn times for the cycler trajectory. First, the amount of full power burn time for each burn was calculated using the following equation.

훥푡푒푞 = 훥푡푎푐푡푢푎푙 ∗ 푝 (35)

This value was then integrated from one refueling Earth flyby to the next refueling Earth flyby in order to find the equivalent burn time between refueling. A refueling flyby was considered to be any Earth flyby on which a crew either embarked or disembarked from the cycler. The following equation was used to determine the total equivalent burn time between refueling flybys.

푒푛푐표푢푛푡푒푟 2 (36) 훥푡푡표푡 = ∫푒푛푐표푢푛푡푒푟 1 훥푡푒푞푢𝑖푣푎푙푒푛푡

References:

[1] Potter, R., Longuski, J., and Saikia, S., “Survey of Low-Thrust, Earth-Mars Cyclers”

AAS 19-799, Portland, ME: 2019.

[2] “Energy from the Sun”, American Chemical Society, retrieved on February 11, 2020. https://www.acs.org/content/acs/en/climatescience/energybalance/energyfromsun.html

Bioregenerative Life Support System (BLSS) a. Determining the requirements for the BLSS system and Lettuce Study

Assuming 30 days per lettuce cycle:

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푘푔 푘푔 푂 = 푛 ∗ 푂 ∗ 푡 = 70 푝푒표푝푙푒 ∗ 0.84 ∗ 30 푑푎푦푠 = 1764 2 2 푝푒푟푠표푛 ∗ 푑푎푦 푚표푛푡ℎ

푊ℎ푒푟푒 푛 − 푁푢푚푏푒푟 표푓 푝푒표푝푙푒 [푝푒표푝푙푒]

−1 −1 푂2 − 푂푥푦푔푒푛 푟푒푞푢푖푟푒푚푒푛푡푠 [푘푔 푑푎푦 푝푒푟푠표푛 ]

푡 − 푛푢푚푏푒푟 표푓 푑푎푦푠 [푡]

푘푔 푘푔 퐶푂 = 푛 ∗ 퐶푂 ∗ 푡 = 70 푝푒표푝푙푒 ∗ 1 ∗ 30 푑푎푦푠 = 2100 2 2 푝푒푟푠표푛 ∗ 푑푎푦 푚표푛푡ℎ

푊ℎ푒푟푒 푛 − 푁푢푚푏푒푟 표푓 푝푒표푝푙푒 [푝푒표푝푙푒]

−1 −1 퐶푂2 − 퐶푎푟푏표푛 퐷푖표푥푖푑푒 푟푒푞푢푖푟푒푚푒푛푡푠 [푘푔 푑푎푦 푝푒푟푠표푛 ]

푡 − 푛푢푚푏푒푟 표푓 푑푎푦푠 [푡]

Output of 1 pot of Lettuce 푔 퐿푒푡푡푢푐푒 = 72 표푓 푏푖표푚푎푠푠 표푢푡 푝표푡

푘푔 퐻 푂 = 1.07 2 𝑖푛 푚2

푘푔 퐻 푂 = 0.067 2 푝푙푎푛푡푠 푚2

푘푔 퐻 푂 = 0.7924 2 푟푒푐표푣푒푟푒푑 푚2

Scaling

푚2 퐴푟푒푎 = 푃 ∗ 퐴 = 45 푝표푡푠 ∗ 0.0(2) = 1 푚2 푝표푡

푊ℎ푒푟푒 푃 − 푛푢푚푏푒푟 표푓 푝표푡푠 [푝표푡푠]

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퐴 − 퐴푟푒푎 표푓 1 푝표푡 [푚2]

푘푔 퐻 푂 = 퐻 푂∗ ∗ 푃 = 48.2 (45 푝표푡푠) 2 𝑖푛 2 𝑖푛 푚2

푔 푘푔 퐿푒푡푡푢푐푒 = 72 ∗ 45 푝표푡푠 = 3.240 표푓 푏푖표푚푎푠푠 (45 푝표푡푠) 표푢푡 푝표푡 푚2

푘푔 퐻 푂 = 퐻 푂∗ ∗ 푃 = 3.015 (45 푝표푡푠) 2 푝푙푎푛푡푠 2 푝푙푎푛푡푠 푚2

푘푔 퐻 푂 = 퐻 푂∗ ∗ 푃 = 35.658 (45 푝표푡푠) 2 푟푒푐표푣푒푟푒푑 2 푟푒푐표푣푒푟푒푑 푚2

Oxygen Production per unit area scaled

푘푔 푂 = 0.4717 2 표푢푡 푚표푛푡ℎ ∗ 푚2

푘푔 퐶푂 = 0.3066 2 𝑖푛 푚표푛푡ℎ ∗ 푚2

Cycler requirements of plants for one month

푘푔 푂 1764 푂 = 2 = 푚표푛푡ℎ = 3 739.7 푚2 2 푓𝑖푛푎푙 푂 푘푔 2 표푢푡 0.4717 푚표푛푡ℎ ∗ 푚2

푘푔 퐶푂 2100 퐶푂 = 2 = 푚표푛푡ℎ = 6 849.3 푚2 2 푓𝑖푛푎푙 퐶푂 푘푔 2 𝑖푛 0.3066 푚표푛푡ℎ ∗ 푚2

Thus, the required plants area in case of using lettuce for sustainable atmosphere is chosen from the final value of the Carbon Dioxide, since it requires greater area.

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Estimation of the food and water from the BLSS area

푘푔 푘푔 푂 = 푂 ∗ 퐶푂 = 0.4717 ∗ 6 849.3 푚2 = 3 230.8 2 푓𝑖푛푎푙 2 표푢푡 2 푓𝑖푛푎푙 푚표푛푡ℎ ∗ 푚2 푚표푛푡ℎ

푘푔 푀푔 퐿푒푡푡푢푐푒 = 3.240 ∗ 6849.3 푚2 = 22 191.732 푘푔 = 22.2 표푢푡 푚2 푚표푛푡ℎ

푘푔 푀푔 퐻 푂 = 35.658 ∗ 6849.3 푚2 = 244.2 2 푟푒푐표푣푒푟푒푑 푚2 푚표푛푡ℎ

푘푔 푀푔 퐻 푂 = 48.2 ∗ 6849.3 푚2 = 330 2 𝑖푛 푚2 푚표푛푡ℎ

Power Estimation:

PHYTOFY system is used, thus:

푆푦푠푡푒푚′푠 푢푛푖푡 푎푟푒푎 = 0.21 푚2

푆푦푠푡푒푚′푠 푀푎푥푖푚푢푚 푃표푤푒푟 = 150 푊

푀푎푠푠 = 8.9 푘푔

2 퐶푂2 푓𝑖푛푎푙 6849.3 푚 푈푛푖푡푠 푛푒푒푑푒푑 = = = 6850 푆푦푠푡푒푚′푠 푢푛푖푡 푎푟푒푎 1 푚2

푀푎푠푠 표푓 푃퐻푌푇푂퐹푌 = 6850 ∗ 8.9 푘푔 = 61 푀푔

푃표푤푒푟 = 6850 ∗ 150 푊 = 1.03 푀푊

Mass Estimation:

퐵퐿푆푆′푠 푚푎푠푠 = 푃퐻푌푇푂퐹푌 + 푊푎푡푒푟 + 푆푡푟푢푐푡푢푟푒푠 = 61 푀푔 + 330 푀푔 + 100 푀푔

= 491 푀푔

Area Estimation:

퐵퐿푆푆′푠 퐴푟푒푎 = 6849.3 푚2

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The design with the use of the Lettuce was not chosen, since the new design was outperforming this in every aspect, especially the flexibility of the system and the optimization parameters to be balanced with food, water, power, and atmosphere sustainability. f. Chlorella Pyrenoidosa Cultivationtion Study

Please refer to the MatLab solution in the MatLab files (BLSS_Study.m).

Radiation Shielding a. Galactic Cosmic Radiation (GCR) Description

Galactic Cosmic Radiation is comprised of low flux ionized particles, including many heavy nuclei, that come from outside the solar system at relatively steady rate [1]. GCR is strongest at the solar minimum of the solar cycle because the sun’s magnetic field deflects GCR away from the solar system [2]. The maximum possible GCR exposure for 1 year is 0.6 Sv, based on measurements taken at the solar minimum [3]. GCR can be even more dangerous due to the presence of secondary radiation. This means that when the cosmic rays hit a barrier, such as the hull of a spacecraft, rather than being fully reflected, the heaviness of the particles in the rays will give the collision greater force, which will break atoms and release an even larger amount of particles. g. Solar Particle Event (SPE) Description

Solar Particle Events are made of particles emitted from the sun, most of which are protons

[1]. Although the sun releases radiation at a steady rate, SPE radiation is known to be unpredictable, especially when compared to GCR, because solar flares or coronal mass ejections can release a large number of particles with little warning. Because of its direct relationship to the sun, SPE is strongest at the solar maximum of the solar cycle. While SPE radiation consists of a large number of particles, those particles have lower energies, which allows SPE to be

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Purdue University Cycler Appendix | 1 2 0 shielded relatively easily compared to GCR. To be clear, mitigating SPE must still be taken into account in spacecraft design, but the consensus is that known methods exist to effectively shield almost all incoming SPE, whereas no method or material has yet demonstrated the ability to fully guard against GCR in deep space. h. Effects of Radiation Exposure

Long-term effects of chronic or acute radiation exposure can include cataracts, increased risk of cancer, sterility, nervous system damage, and cardiovascular disease [4]. Acute doses of radiation can be very harmful, and in some cases, fatal to humans. Table 1 summarizes some of the deleterious biological effects that different levels of exposure can cause [5].

Table 1: Threshold Doses For 50% Of Population Following Acute Exposures

Effect Effective Dose (Sv)

Blood Count Changes 0.2-0.5

Vomiting/Nausea 1.0

Death (minimal medical treatment provided) 3.2-3.6

Death (medical treatment provided) 4.8-5.4

Death (Autologous stem cell transplant performed) 11.0

Permanent Sterility - Males 3.5

Permanent Sterility - Females 2.5

Cataracts 2.0-5.0 a. Adapted from Cucinotta, F. A. (n.d.). Overview of integrated space radiation protection program. Retrieved from NASA Johnson Space Center website: https://www.nasa.gov/centers/johnson/pdf/514218main_FC- 01_Overview_Integrated_SRP_Program.pdf

One of the most significant harmful effects of radiation is the increased risk of cancer. Table

2 provides additional information on exactly how much 0.2 Sv of radiation can increase the risk

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Purdue University Cycler Appendix | 1 2 1 of dying from cancer. Similarly, Table 3 provides quantifiable data on how much 0.2 Sv of radiation can increase the risk of contracting cancer.

Table 2: Percent Probability of Excess Cancer Mortality for Chronic Radiation Exposure of 0.2 Sv In One Year Age 35 45 55

Sex Male Female Male Female Male Female

Solid Cancers 0.38 % 0.68 % 0.26 % 0.48 % 0.20 % 0.32 %

Leukemia 0.13 % 0.05 % 0.08 % 0.06 % 0.06 % 0.04 %

Total 0.51 % 0.73 % 0.34 % 0.54 % 0.26 % 0.36 % i. https://www.nasa.gov/centers/johnson/pdf/514218main_FC- 01_Overview_Integrated_SRP_Program.pdf

Table 3: Percent Probability of Excess Cancer Morbidity for Chronic Radiation Exposure of 0.2 Sv In One Year Age 35 45 55

Sex Male Female Male Female Male Female

Solid 0.78 % 1.42 % 0.42 % 1.20 % 0.32 % 0.76 %

Cancers

Leukemia 0.09 % 0.06 % 0.10 % 0.12 % 0.08 % 0.06 %

Total 0.87 % 1.48 % 0.52 % 1.32 % 0.40 % 0.82 %

a. Adapted from Cucinotta, F. A. (n.d.). Overview of integrated space radiation protection program. Retrieved from NASA Johnson Space Center website: https://www.nasa.gov/centers/johnson/pdf/514218main_FC- 01_Overview_Integrated_SRP_Program.pdfAdapted from Cucinotta, F. A. (n.d.). Overview of integrated space radiation protection program. Retrieved from NASA Johnson Space Center website: j. History of Space Travel and Radiation

Table 4 shows the observed radiation dose by NASA on different human space missions, as well as an estimate by NASA for a manned Mars mission. The estimated radiation dose for a

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Mars mission appears to be in accordance with yearly radiation exposure in deep space. Based on proposed NASA mission plans, it is reasonable to assume that this hypothetical three-year Mars mission is projecting approximately one year of transit to and from Mars each way, and a one year stay on the Martian surface [6]. It is not advisable to assume that radiation exposure is negligible on Mars because the exposure levels are still 2.5 times that the levels that astronauts are exposed to while on the International Space Station due to Mars’ thinner atmosphere (relative to Earth) and lack of magnetosphere [7]. This would mean that while spending a year on Mars, astronauts would be exposed to 0.8 Sv of radiation. This does not appear to be accounted for by the NASA estimates unless they are expecting to have highly effective shielding in place.

However, since the Project Elevator mission profile includes negligible time on the Martian surface, the amount of radiation exposure on the surface is not relevant to the scope of this specific project. Therefore, if we take the assumption that than the total radiation exposure would be based on spending two years in space, this would lead to a total of 1.2 Sv at most, based on a maximum possible annual GCR exposure of is 0.6 Sv in deep space, which matches the projection made by NASA shown in Table 4. The data shown in Table 4 gives a sense of how the scale of the radiation problem is much greater for a Mars mission compared to what human space travelers have faced on past missions. The duration of a Mars mission will exceed any prior mission, and crew members will not be afforded the protection of the Earth’s atmosphere and magnetosphere. This necessitates in-depth studies of the radiation problem and development of effective countermeasures against it prior to undertaking a manned Mars mission.

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Table 4: Observed Radiation Dose on Different Human Space Missions Mission Type Radiation Dose (Sv)

Space Shuttle Mission 41-C 0.00559 (8-day mission orbiting the Earth at 460 km)

Apollo 14 0.0114 (9-day mission to the Moon)

Skylab 4 0.178 (87-day mission orbiting the Earth at 473 km)

ISS Mission 0.160 (up to 6 months orbiting Earth at 353 km)

Estimated Mars mission 1.20 (3 years)

a. Adapted from NASA. (2008). The Radiation Challenge. https://www.nasa.gov/pdf/284273main_Radiation_HS_Mod1.pdf

Table 5 presents more data related to past radiation exposure of NASA astronauts [5]. Of note is the high variability in radiation doses, as the maximum dose observed during a program can be far greater than the average.

Table 5: Radiation Doses for Missions from Different NASA Programs

Program Average Inclination Total Dose per Dose-rate*

Altitude* (km) (degrees) Mission* (cSv) (cSv/day)

Gemini 454 30 0.053 (0.47) 0.087

(1370) (0.47)

Apollo^ N/A N/A 1.22 (3.3) 0.13

(0.39)

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Skylab 381 (435) 50 7.2 (17.0) 0.12

(0.21)

STS (Altitude > 570 28.5 2.65 (7.8) 0.32

450 km) (0.77)

STS (Altitude < 337 28.5 0.21 (0.71) 0.023

450 km) (0.04)

STS/Mir 341 (355) 51.6 9.9 (14.0) 0.072

(0.10)

ISS 51.6 10-20 0.06-

0.12 a. * – Maximum value in parenthesis, all other values are averages. b. ^ – Altitude and inclination data is irrelevant for the Apollo missions because the majority of them left the Earth’s orbit. c. Adapted from Cucinotta, F. A. (n.d.). Overview of integrated space radiation protection program. Retrieved from NASA Johnson Space Center website: https://www.nasa.gov/centers/johnson/pdf/514218main_FC- 01_Overview_Integrated_SRP_Program.pdf

Table 6 provides another view of the history of the space program, this time by breaking down radiation exposure by source. It is interesting to note that space travel was not the only contributor to their radiation dosage. The diagnostic X-Rays in particular had a large impact on the radiation dosage, until the Carter administration issued a directive to scale down the usage of diagnostic X-Rays in the 1980s.

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Table 6: Radiation Exposure for Astronauts by Source Over Different Time Periods (cSv) Source 1957- 1970- 1980- 1990- Total

1969 1979 1989 1999

Space (Total) 20 111 42 273 446

Space (Average) 0.46 4.0 0.26 0.734 0.738

Diagnostic X- 141 179 52 15 387

Rays (Total)

Diagnostic X- 0.095 0.082 0.027 0.007 0.05

Rays (Average)

Air Flight 20 32 50 75 180

Background 15 25 35 55 130

Total 202 359 187 420 1,168

NASA’s ability to track radiation both actively and passively has increased and improved over the years, as shown by Table 7. Having complete and accurate dosimetry data is essential to both keeping crew members safe in the present and the future.

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Table 7: History of Radiation Dosimetry in NASA Programs Area Dosimetry Crew Dosimetry

Program Active Passive Active Passive

Mercury None None None TLD*

Gemini None None None TLD

Apollo None Plastic track None TLD,

(Cellulose PTD* (cellulose

Nirate) nitrate), BioD*

Skylab None TLD, Plastic None TLD,

PTD* (cellulose

nitrate)

STS None TLD None TLD,

PTD (CR-39)

STS-Mir TEPC* Particle TLD/Plastic None TLD

Telescopes Track (CR-39) PTD (CR-39)

Bio.D.

ISS TEPC Particle TLD/Plastic Silicon Chip TLD

Telescopes Track (CR-39) Detector Bio.D.

a. * – TLD = Thermoluminescence detector, PTD = Plastic Track Detector, Bio.D = Biodosimetry TEPC = Tissue Equivalent Proportional Counter b. Adapted from Cucinotta, F. A. (n.d.). Overview of integrated space radiation protection program. Retrieved from NASA Johnson Space Center website: https://www.nasa.gov/centers/johnson/pdf/514218main_FC- 01_Overview_Integrated_SRP_Program.pdf

Figure 1 presents a very troubling reality. The one group of astronauts who have left the confines of the Earth’s orbit have shown a much higher risk of dying from cardiovascular disease

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Purdue University Cycler Appendix | 1 2 7 compared to their peers on other missions [8]. As stated in Appendix C: Effects of Radiation

Exposure, cardiovascular disease can be caused by radiation exposure. This means installing proper radiation protection will be even more important if we are to send humans back to the moon and onto Mars, as Project Elevator entails.

Fig. 1: Magnetic Radiation Shield Concept. The proportional mortality rate due to cardiovascular disease in the United States among individuals age 55–64 years, non-flight astronauts, astronauts that flew only low Earth orbit missions, all flight astronauts, and Apollo astronauts that flew missions to the Moon. The high rate of the Apollo lunar astronauts is a

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Purdue University Cycler Appendix | 1 2 8 statistically significant difference from the US population 55–64 years of age at the time of death, and the non-flight astronaut group, within a 95% confidence interval, and from the LEO astronaut group within a 99% confidence interval. Adapted from Delp, M. D., Charvat, J. M., Limoli, C. L., Globus, R. K., & Ghosh, P. (2016). Apollo lunar astronauts show higher cardiovascular disease mortality: Possible deep space radiation effects on the vascular endothelium. Scientific Reports, 6(1). doi:10.1038/srep29901 k. NASA Exposure Limits

NASA determines radiation exposure limits for its astronauts based on both yearly and career exposure totals [9]. These career limits are summarized in Table 8.

Table 8: Career Exposure Limits for NASA Astronauts by Age and Gender (in Sv) Age (years) 25 35 45 55

Male 1.50 2.50 3.25 4.00

Female 1.00 1.75 2.50 3.00 a. Adapted from NASA. (2008). The Radiation Challenge. https://www.nasa.gov/pdf/284273main_Radiation_HS_Mod1.pdf

NASA bases its radiation standards on recommended dose limits from the National Council on Radiation Protection and Measurements (NCRP), a non-profit, scientific organization [5]. The

NCRP received a congressional charter in 1964, but it is a nongovernmental, private corporation.

Its mission is to collect, produce, and distribute information and guidance related to radiation protection and measurements [10]. The NCRP determined these career limits based the dosage that would give a maximum of 3% excess risk of cancer mortality for the astronauts. For astronauts in low-Earth orbit, the monthly exposure limit is 0.25 Sv and the yearly limit is 0.50

Sv. NASA has proposed changes to make its administrative guidelines stricter than the NCRP guidelines. Those changes include having an administrative limit of 0.2 Sv per year and only a

1% excess risk of cancer mortality [5].

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To give those numbers some context, a person on Earth typical receives an average yearly dose of 0.0036 Sv from normal background, ambient radiation. Also, the typical doses from the most powerful medical scans are on the order of 10-2 Sv [11]. Some other comparisons for radiation exposure on Earth are in Table 9 below.

Table 9: Annual Terrestrial Exposures Exposure Type Dose (Sv) Dose-Rate

Chest X-Ray 0.0001 High

Atomic Bomb Survivor 0-2 High

Airline Pilots 0.001-0.04 Low

Nuclear Power Workers 0.001-0.04 Low

Cancer Therapy 10-80 High a. Adapted from Cucinotta, F. A. (n.d.). Overview of integrated space radiation protection program. Retrieved from NASA Johnson Space Center website: https://www.nasa.gov/centers/johnson/pdf/514218main_FC- 01_Overview_Integrated_SRP_Program.pdf

The exposure limits set by the NCRP for astronauts are higher than those which they have set for those who work on the ground in close proximity to radioactive material, such as a nuclear reactor. These workers are allowed doses of 0.05 Sv per year by NCRP standards. They are also allowed career limits of [Worker’s Age x 0.01 Sv]. The rationale given by NASA and the NCRP to explain the discrepancy in short-term (1 year or less) dose limits is that the exposure patterns are different for ground workers and astronauts. Astronauts may take 1-5 missions over the course of a 15 year career, while ground workers are exposed daily over the course of 30 years.

It should also be noted that the career limits increase, not decrease with age. This is because

NASA assumes that exposure to high levels of radiation early in astronauts’ career could put them at greater risk once they reach old age. Basically, the principle is that the more years a

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Purdue University Cycler Appendix | 1 3 0 person has left to live, the more time that person has to develop radiation-induced health problems. Female astronauts have lower career limits than their male counterparts due to their differences in biology, primarily the potential complications to reproduction and the ability to give birth that radiation can cause.

Table 10 provides additional information and context for what NASA considers to be healthy limits for its Astronauts and the general public. When we refer to human radiation exposure limits, what we are really referring to is the amount of radiation that can penetrate 5 cm into a person’s body, which is the first column of data in Table 10. Notice that the amount of radiation exposure decreases as the depth increases While more radiation might impact a person at an exterior level, such as the skin and eyes, it is the radiation that impacts blood forming organs which has the potential to cause serious health problems discussed in Appendix C: Effects of

Radiation Exposure. This begs the question – what makes Astronauts different from the general public? At the very least, what makes astronauts different from workers operating in high- radiation environments on Earth? How much hazard is acceptable to subject them to? Answering those questions is beyond the scope of this project, yet they are important questions to consider for all parties involved, including the general public who supports government-funded space programs, the astronauts themselves who are the ones who will actually suffer the adverse effects, and the leadership at NASA and the private space industry. We should not take an actual mission to Mars without a full understanding of the risks.

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Table 10: Depth of Radiation Penetration and Exposure Limits for Astronauts and the General Public (in Sv)

Exposure Blood Forming Eyes Skin

Interval Organs (5 cm depth) (0.3 cm depth) (0.01 cm depth)

Astronauts 30 Days 0.25 1.0 1.5

Annual 0.50 2.0 3.0

Career 1-4 4.0 6.0

General Public Annual 0.001 0.015 0.05 a. Adapted from NASA. (2008). The Radiation Challenge. https://www.nasa.gov/pdf/284273main_Radiation_HS_Mod1.pdf

l. Parker’s Alternative Radiation Shielding Designs

We cannot say with certitude that we have the ability to protect crew members from space radiation, specifically radiation categorized as Galactic Cosmic Radiation. While NASA has established yearly and career radiation exposure limits for its astronauts, there are some opinions in the scientific community that believe the threat is of greater magnitude. Perhaps the most notable of those is Dr. Eugene Parker, an expert on solar wind and radiation, and the namesake of the Solar Parker Probe launched in 2018. He has written about the effects of radiation on deep-space travelers and what steps could potentially be used to prevent it [12]. He discussed three solutions: magnetic shielding, electrostatic shielding, and material shielding. The following is an overview of those three and how they compare to our cycler design.

The first option is a magnetic shield. This operates on the concept of generating a magnetic field strong enough to deflect incoming cosmic rays. Those cosmic rays can have energies of up to 2 billion electron volts, which is why Parker claims that the field would need to have a strength of 20 teslas. To give a sense of scale, the field would be more than 600,000 times the

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Purdue University Cycler Appendix | 1 3 2 strength of Earth’s magnetic field at the equator, which is about 0.0000305 teslas [13]. This design raises questions of practicality as well as concerns related to human factors. Anecdotal reports give evidence of 0.5 tesla fields causing flashes of light in the eyes and an acid taste in the mouth. This would suggest that a field 40 times greater in strength, such as the proposed 20 tesla magnetic shield concept, could potentially pose a serious threat to human biology, possibly as dangerous as that introduced by the GCR radiation, if the crew members were directly exposed to it. One proposed method to prevent the magnetic field from having deleterious effects on the health of crew members is installing an opposing electromagnet. This electromagnet can generate a concentric opposing field and be positioned so that the net effect in the habitation module is neutral, or close to neutral. The primary field would still be present and undiminished on the exterior of the cycler in order to block incoming GCR. However, adding a second magnetic field would add an unprecedented degree of complexity to an already complex system.

One other point to note is that a magnetic field is not completely uniform, and the wider it is spread out, the wider the “gaps” there will be in between magnetic field lines.

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Fig. 2: Magnetic Radiation Shield Concept. This is a 2-dimensional sketch of what a cycler with magnetic radiation shielding would look like, with the gray body representing the cycler as actually designed. The cycler would be rotating in the plane of the page. In order to generate the magnetic field, a toroidal cylinder is needed, which is represented by the two gold rings. The outer ring generates the magnetic field to shield the cycler from GCR, while the inner ring, if included, generates the secondary magnetic field to balance the effects in the habitation module.

The second option is to create an electrostatic shield. As Parker noted, some researchers have put forth the idea to give the spacecraft an electric charge. As stated in the discussion of the magnetic shield above, the cosmic ray protons carry energies up to 2 billion electron volts. Therefore, an electrostatic shield design would require the walls of the spacecraft to be charged to the same level; 2 billion volts. An electrostatic shield provides multiple advantages over the magnetic shield concept, namely the electrostatic shield will not have any gaps in coverage, and the harmful side effects on human biology that the magnetic shield causes will be nonexistent.

However, the electrostatic shield concept does suffer from two main deficiencies. The first is that the presence and effects of ions and electrons in space appears to have been neglected or unaccounted for. The electrostatic shield would actually pull these electrons in towards the cycler. The electrons would then not only impact the hull of the cycler, but they would do so with an energy equivalent to that of the cosmic rays, and then produce gamma rays when reflected off. Thus, solving the problem of repelling the cosmic rays with an electric field would create an artificial field with an intensity equal to or greater than the natural cosmic flux, as well as harmful gamma rays. Furthermore, the power estimated to charge the cycler to 2 billion volts would be 2,000 MW. For comparison, our final cycler design requires a maximum of 3.2 MW of power total.

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Fig. 3: Electrostatic Radiation Shielding Concept. This is a 2-dimensional sketch of what a cycler with electrostatic radiation shielding would look like, with the gray body representing the cycler as actually designed. The cycler would be rotating in the plane of the page. The blue arrows represent the electric field emanating from the charged spacecraft, and it is important to note that the field would cover the entire exterior surface area of the cycler (or at least its habitation modules) so that there are no gaps in coverage.

The last option mentioned by Parker is a material shield. Rather than blocking the radiation with other forms of energy and currents, this method proposes to simply obstruct it by putting mass in the way. This method is the most reliable because it has been proven to work on smaller scale laboratory testing, as well as applications to other fields, such as nuclear energy. However, the simplicity of this concept is diminished when considering the scale necessary to execute it for our cycler vehicle. Parker claims that a material shield needs to have an areal density of 500 grams/cm2 in order to be effective in protecting crew members. This is equivalent to the air mass above an altitude of 5,500 meters, so it is not exactly the same as being on the surface of the

Earth when considering the protection we receive from Earth’s magnetic field and atmosphere.

Parker believes that shields of areal density lower than 500 g/cm2 would be counterproductive, because the shielding material would deflect the initial wave of GCR radiation, but not the secondary particles and rays that develop as a result of that reflection. Parker proposed using water as the shielding material due to its high hydrogen content, which would require the shield

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Purdue University Cycler Appendix | 1 3 5 to be 5 meters thick. For our cycler vehicle, this would result in an additional mass of 84,000

Mg. For comparison, the total cycler mass is 9400 Mg and the total wall thickness of the habitation modules is 0.65 meters (0.15 meters of Aluminum and 0.5 meters of polyethylene). To give a comparison to a tangible object on Earth, the Washington Monument has a mass between

80,000 and 100,000 Mg [14]. Parker mentioned the possibility of using polyethylene as an alternate material, but that would also have similar issues of mass and volume, based on the standards he determined as the requisite amounts necessary to adequately protect the crew.

Figure 4: Material Radiation Shielding Concept. This is a 2-dimensional sketch of what a cycler with material radiation shielding would look like, with the gray body representing the cycler as actually designed. The thick blue outline represents the water walls that would cover the entire exterior surface area of the cycler (or at least its habitation modules).

Table 11: Summary of Alternative Radiation Shielding Concepts Concept Description Advantages Disadvantages

Magnetic Magnetic field -Low mass -Strong magnetic field may be Shielding of 20 teslas dangerous

-Adding inner ring would be complex

Electrostatic Electric field of -No gaps in coverage -Dangerous influx of negatively Shielding 2 billion volts. charged particles -No hazardous magnetic field -Needs to supply 2000 MW of Power

Material 5-meter thick -High chance of success -High Mass (84,000 Mg) Shielding water walls

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Purdue University Cycler Appendix | 1 3 6 m. Materials Used in Radiation Shielding

Many different materials have been researched in an effort to find an answer to the radiation problem, or more specifically, the GCR problem [15] [16]. Figures 5, 6, and 7 present data from different sources of the effectiveness of some of these materials, including Aluminum, Boron

Nitride (BN), Heat Melt Compactor (HMC) tiles, polyethylene, and water. In general, the most effective shielding options are those with the most mass and the most hydrogen content, which is why artificial compounds like Polyethylene and hydrogenated Boron Nitride Nanotubes (BNNT) that contain hydrogen perform well. It should be noted that liquid hydrogen is included as an ideal case, but only as a reference; it is impractical at this time to construct the cycler hull out of pure liquid hydrogen. Liquid water also performs well despite having similar impracticalities.

Aluminum has been used on most spacecraft hulls traditionally due to its structural properties, but it does worse than the hydrogen-based materials in terms of radiation protection. Ultimately, a layered aluminum and polyethylene design was chosen in order to balance structural integrity with effectiveness.

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Figure 5: Female effective dose for 1977 solar minimum galactic cosmic ray environment. This compares the effective dose received based on Areal density for three types of material: Aluminum, Heat Melt Compactor (HMC) tiles, and polyethylene. It shows that the polyethylene performs best out of the three, and the HMC material is very close to the polyethylene. Adapted from Adapted from Bahadori, A., Semones, E., Ewert, M., Broyan, J., & Walker, S. (2017). Measuring space radiation shielding effectiveness. EPJ Web of Conferences, 153, 04001. doi:10.1051/epjconf/201715304001

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Figure 6. Calculated exposure to galactic cosmic radiation (GCR) with shielding by a wall of the same thickness (30 cm) made of various materials. The Boron Nitride Performs the best out of all the materials. Adapted from [15] Thibeault, S. A. (2012). Radiation Shielding Materials Containing Hydrogen, Boron, and Nitrogen: Systematic Computational and Experimental Study - Phase I Final Report. Retrieved from NASA website: https://www.nasa.gov/sites/default/files/atoms/files/niac_2011_phasei_thibeault_radiationshieldi ngmaterials_tagged.pdf

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Fig. 7: GCR Dose Equivalent for Various Shielding Materials. Again the BN compounds perform best. Unfortunately, its early stage of development and low structural integrity will not allow us to use it on the cycler exterior. Adapted from Thibeault, S. A. (2012). Radiation Shielding Materials Containing Hydrogen, Boron, and Nitrogen: Systematic Computational and Experimental Study - Phase I Final Report. Retrieved from NASA website: https://www.nasa.gov/sites/default/files/atoms/files/niac_2011_phasei_thibeault_radiationshieldi ngmaterials_tagged.pdf

n. Calculation of Material Shield Thickness and Effective Dose

Aluminum has a density of 2700 kg/m3. In order to provide the optimal amount of shielding based on our capabilities, the aluminum needs to have an areal density of 40 g/cm2, which is equivalent to 400 kg/m2. This target areal density was selected by observing the trend on Fig. 5.

After 40 g/cm2, adding more thickness has a very minor effect on increasing the effects of shielding, and the mass to effective dose ratio becomes disproportionate. The requisite thickness to accommodate both the density and the selected areal density is 0.1481 m. Based on the data

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Purdue University Cycler Appendix | 1 4 0 presented in Fig. 5, an areal density of 40 g/cm2 will result in an effective dose of .2920 Sv/year, which is below the NASA yearly limits for LEO. Adding an interior layer of polyethylene will reduce the effective dose by 6%, so the final projected effective dose is 0.2745 Sv/year, which is almost half of the NASA yearly exposure limit [17][9]. Again, it must be stressed that we cannot fully guarantee the safety of this material radiation shielding at this time.

Sources

[1] NASA. (2019, August 5). How NASA Will Protect Astronauts from Space Radiation at the

Moon. Retrieved from https://www.nasa.gov/feature/goddard/2019/how-nasa-protects-

astronauts-from-space-radiation-at-moon-mars-solar-cosmic-rays

[2] NASA SSERVI. (n.d.). Next solar maximum may be safest time for manned missions to

Mars. Retrieved from https://sservi.nasa.gov/articles/next-solar-maximum-may-be-safest-

time-for-manned-missions-to-mars/

[3] Rais-Rohani, M. (2004). On Structural Design of a Mobile Lunar Habitat with Multi-Layered

Environmental Shielding. Retrieved from NASA Faculty Fellowship Program website:

https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20050215340.pdf

[4] NASA. (2015, September 30). How to Protect Astronauts from Space Radiation on Mars.

Retrieved from https://www.nasa.gov/feature/goddard/real-martians-how-to-protect-

astronauts-from-space-radiation-on-mars

[5] Cucinotta, F. A. (n.d.). Overview of integrated space radiation protection program. Retrieved

from NASA Johnson Space Center website:

https://www.nasa.gov/centers/johnson/pdf/514218main_FC-

01_Overview_Integrated_SRP_Program.pdf

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[6] NASA Space Science Data Coordinated Archive. (2015, November 16). A crewed mission to

Mars. Retrieved from https://nssdc.gsfc.nasa.gov/planetary/mars/marsprof.html

[7] NASA Jet Propulsion Laboratory. (2004). MEPAG-Mars radiation environment. Retrieved

from https://mepag.jpl.nasa.gov/topten.cfm?topten=10

[8] Delp, M. D., Charvat, J. M., Limoli, C. L., Globus, R. K., & Ghosh, P. (2016). Apollo lunar

astronauts show higher cardiovascular disease mortality: Possible deep space radiation

effects on the vascular endothelium. Scientific Reports, 6(1). doi:10.1038/srep29901

[9] NASA. (2008). The Radiation Challenge.

https://www.nasa.gov/pdf/284273main_Radiation_HS_Mod1.pdf

[10] National Council on Radiation Protection and Measurements. (2015, May 25). NCRP

Mission. Retrieved from https://ncrponline.org/about/mission/

[11] US Food and Drug Administration. (2017, December 5). What are the radiation risks from

CT?. Retrieved from https://www.fda.gov/radiation-emitting-products/medical-x-ray-

imaging/what-are-radiation-risks-ct

[12] Parker, E. N. (2006). Shielding Space Travelers. Scientific American, 294(3), 40-47.

doi:10.1038/scientificamerican0306-40

[13] Tufts University. (2010). NASA's Cosmos: The invisible buffer zone with space -

atmospheres, magnetospheres and the solar wind. Retrieved from

https://ase.tufts.edu/cosmos/view_chapter.asp?id=3&page=6

[14] National Park Service. (2015, April 10). Washington monument: frequently asked

questions. Retrieved from https://www.nps.gov/wamo/faqs.htm

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[15] Bahadori, A., Semones, E., Ewert, M., Broyan, J., & Walker, S. (2017). Measuring space

radiation shielding effectiveness. EPJ Web of Conferences, 153, 04001.

doi:10.1051/epjconf/201715304001

[16] Thibeault, S. A. (2012). Radiation Shielding Materials Containing Hydrogen, Boron, and

Nitrogen: Systematic Computational and Experimental Study - Phase I Final Report.

Retrieved from NASA website:

https://www.nasa.gov/sites/default/files/atoms/files/niac_2011_phasei_thibeault_radiationshi

eldingmaterials_tagged.pdf

[17] Li, X., Warden, D., & Bayazitoglu, Y. (2018). Analysis to Evaluate Multilayer Shielding of

Galactic Cosmic Rays. Journal of Thermophysics and Heat Transfer, 32(2).

Retrieved from DOI: 10.2514/1.T5292

Cycler Control Design a. Cycler Dynamics and Stability

1. Gravity gradient torque calculation

The torque on the spacecraft resulting from the gravity gradient of a celestial body (only planets and the sun is considered in calculation for the cycler) is given by Equation (32):

3휇 (32) 푇 = ∣ 퐼 − 퐼 ∣푠푖푛 푠푖푛 (2휃) 푔푔 2푅3 푦 푧

The numerical values of the above parameters are summed up in Table 6 below.

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Table 3066666: Parameters used for gravity gradient torque calculation Parameter Sun Mars Earth

2 Iy (kg m ) 1.319e+12 1.319e+12 1.319e+12

2 Iz (kg m ) 1.322e+12 1.322e+12 1.322e+12

Ravg (m) 1.845e+8 *a *a

Rmin (m) 1.282e+8 3.697e+6 7.378e+6

µ (m3 s−2) 1.327124e+20 4.282837e+13 3.986004e+14 a. Since the cycler shuttles between Earth and Mars over the course of the mission, a calculation of average gravity gradient torque for Earth and Mars would not be meaningful; therefore, such calculation was excluded and only the maximum near-Earth and near-Mars gravity gradient torques were calculated. This was done using the minimum distance from the cycler to each planet (Rmin).

Equation (32) was derived from Wertz (1978) [3][3] on the basis that the moment of inertia about the second minor axis (x-axis), Ix is of an order of magnitude that is negligible. This

9 assumption holds because the order of magnitude of Ix is 10 , which is far smaller than the order

12 of magnitude of Iy and Iz which are 10 (i.e. 1000 times larger).

We also restrict θ, the maximum angular deviation from the z-axis (axis of rotation), to be

0.1° in order to ensure pointing accuracy for the communications devices and antennae.

Finally, in order to calculate the gravity gradient torque at each stage of the mission, we sum up the gravity gradient torque due to each relevant celestial body at that stage, according to the following table.

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Table 3177777: Relevant gravity gradient torques at each stage of mission Location Gravity gradient torques added

Near-Earth Sun (average), Earth

Near-Mars Sun (average), Mars

Interplanetary (average torque) Sun (average)

Interplanetary (max. torque) Sun (closest distance from cycler to sun)

Fuel slosh torque model

The equation of motion given by Mason and Starin (2005) [2]is reproduced here as Equation

(33). The relevant parameters in Equation (33) are the slosh mass ms as well as the slosh displacement angle θ.

2 (푚푠퐿푠 + 퐼푠)휃̈ + 퐶휃̇ − 푎퐿푠 푠푖푛 푠푖푛 (휃) (33)

For convenience, the pendulum model and relevant parameters are reproduced in the illustration shown in Figure 1 below.

Figure 122222: Pendulum model for fuel slosh with torsional damping [2]. The slosh angle θs is defined as the angle between the fuel slosh mass blob and the vertical axis. Meanwhile, the masses ms and m0 respectively denote the sloshing mass (akin to the pendulum bob) and the rest of the non-sloshing fuel mass. Also, Ls is the “length” of the pendulum and Is is the moment of inertial of the fuel slosh mass.

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The model proposed in Equation (33) treats the fuel slosh motion equivalent to a swinging pendulum in a circular motion, with the mass of the pendulum being the slosh mass. This is since most of the fuel mass is stationary in the fuel tank.

As it is not known what is the proportion of fuel mass that sloshes as well as the precise slosh dynamics (including displacement angle and rate of slosh), we assume that the same percentage of fuel mass contributes to slosh in both the Solar Dynamics Observatory (SDO) as well as the cycler. We also assume the same slosh dynamics are present in the cycler as for the SDO.

Therefore, the fuel slosh torque value for the SDO, given as 0.08 Nm, was simply scaled directly according to the fuel mass ratio:

푚푓푢푒푙,푐푦푐푙푒푟 126600 (34) 푇푠푙표푠ℎ,푐푦푐푙푒푟 = ∙ 푇푠푙표푠ℎ,푆퐷푂 = ∙ 0.08 = 8.4050 푁푚 푚푓푢푒푙,푆퐷푂 1205

Again, it is pertinent to note here that this is the peak value of fuel slosh torque experienced by the cycler vehicle. It will likely experience fuel slosh torque far less than this value throughout the course of the mission – fuel is burnt over the mission, and the pendulum model also predicts damping of the fuel slosh torque.

o. Cycler dynamics & stability

From Wertz (1978) [3] Chobotov (1996) [4] and Chobotov (2008) [5] the Euler equations for a generic spacecraft are reproduced as Equation (35):

[퐿 푀 푁 ] = [퐼푥휔̇ 푥 + 휔푦휔푧(퐼푧 − 퐼푦) 퐼푦휔̇ 푦 + 휔푥휔푧(퐼푥 − 퐼푧) 퐼푧휔̇ 푧 + 휔푥휔푦(퐼푦 − 퐼푥) ] (35)

∆퐿 퐼푧 − 퐼푦 ∆푀 퐼푥 − 퐼푧 ∆푁 (36) ∆휔̇ (푡) = [ − 푛∆휔푦(푡) − 푛∆휔푥(푡) ] 퐼푥 퐼푥 퐼푦 퐼푦 퐼푧

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The response of the cycler vehicle to any disturbance torque ∆푇 = [∆퐿 ∆푀 ∆푁 ]−1 can hence be derived from the linearized equation given in Equation (36). For the sake of brevity, such derivation will not be included in this report; however, the reader is encouraged to refer to Wertz

(1978) [3][3] or Chobotov (2008) [5][5] for further detail.

In the case of spin-stabilization, the angular spin rate 휔̇ 푧 is constant, meaning that we want

∆휔̇ 푧 = 0 (i.e. zero angular acceleration). However, examining Equation (A36), this is clearly impossible with the presence of a perturbing torque acting about the major axis i.e. ∆푁 ≠ 0.

Therefore, we conclude that the rate of rotation of the cycler vehicle must increase in the presence of a perturbance torque.

Over the course of the mission, perturbance torques accumulate by continuously increasing the rate of rotation of the cycler. In the case of spin stabilization, this is clearly a problem over the course of a long mission, as a higher rotation rate results in increased forces acting on the contents of the cycler vehicle (including the crew and passengers).

However, for a three-axis-stabilized cycler vehicle, the perturbance torques can be dissipated with reaction control wheels and control moment gyroscopes. These devices accumulate the perturbance torques in the form of angular momentum, and only need to be desaturated occasionally using thrusters or other forms of momentum dissipation.

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Orbit Trajectory Control

Table 2: Maximum ΔV Required to Maintain Desired Course for Cycler #1 V V V V V Leg # Leg # Leg # Leg # Leg # (m/s) (m/s) (m/s) (m/s) (m/s)

1 32.1327 6 47.4512 11 32.3354 16 25.3863 21 65.3726

2 25.4906 7 21.545 12 62.0681 17 29.9995 22 20.7046

3 50.0393 8 31.6475 13 21.0424 18 46.2636 23 31.1263

4 20.0361 9 68.9178 14 31.9393 19 29.684 24 66.9257

5 29.9931 10 39.5336 15 48.0343 20 30.9847 25 16.7436

Table 3: Maximum ΔV Required to Maintain Desired Course for Cycler #2 V V V V V Leg # Leg # Leg # Leg # Leg # (m/s) (m/s) (m/s) (m/s) (m/s)

1 43.402 6 41.9761 11 67.5124 16 39.0353 21 55.9023

2 49.5736 7 63.3963 12 13.8174 17 24.2888 22 51.3013

3 33.0449 8 39.4413 13 28.5357 18 34.8718 23 14.7492

4 34.818 9 38.4944 14 21.1652 19 45.4748 24 21.4155

5 63.9657 10 65.3613 15 56.9076 20 28.6254 25 31.8788

Table 4: Maximum ΔV Required to Maintain Desired Course for Cycler #3

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V V V V V Leg # Leg # Leg # Leg # Leg # (m/s) (m/s) (m/s) (m/s) (m/s)

1 28.4488 6 29.4207 11 43.6815 16 21.8258 21 44.3428

2 64.0707 7 29.3335 12 67.4035 17 23.0305 22 22.6407

3 40.4651 8 37.1932 13 29.291 18 57.0084 23 26.5064

4 31.1752 9 12.7597 14 26.6596 19 58.0366 24 69.508

5 49.0415 10 36.5353 15 41.3945 20 61.0084 25 19.8226

Table 5: ???? V V V V V Leg # Leg # Leg # Leg # Leg # (m/s) (m/s) (m/s) (m/s) (m/s)

1 26.0276 6 68.428 11 27.3853 16 25.2967 21 61.0638

2 40.2633 7 57.7127 12 33.845 17 38.0555 22 65.9149

3 54.1665 8 27.0331 13 43.5521 18 50.2001 23 33.299

4 34.9617 9 50.0773 14 47.7216 19 33.013 24 22.4694

5 45.1075 10 68.4344 15 33.3867 20 29.386 25 31.0312

References

Chamberlin, A. B. (n.d.). HORIZONS Web-Interface. Retrieved February 19, 2020, from https://ssd.jpl.nasa.gov/horizons.cgi#top

Longuski, J. M., Todd, R. E., & Konig, W. W. (1992). Survey of Nongravitational Forces and Space Environmental Torques: Applied to the Galileo. Journal of Guidance, Control, and Dynamics, 15(3).

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Attitude Control System a. System Overview

The cycler will undergo perturbations that will disturb both its attitude and its trajectory. In this section we seek strategies that will ensure that the attitude of the cycler will stay controlled and safe for inhabitants on the cycler. The attitude of the cycler is one of the most important mission factors. If the cycler is not pointed in the correct direction, the thrusters will not be able to make trajectory corrections, the solar panels will not get enough power, and the outside perturbation forces could worsen. Also, if the rotation rate of the cycler is not correct, the inhabitants of the cycler could lose artificial gravity, or be killed from rapid rotation rates. For all these reasons we must control the attitude of the cycler. p. Requirements

The attitude control system of the cycler must be capable of maintaining the correct attitude of the cycler throughout the course of the mission. It must be able to do this in the face of the expected perturbations that we have modeled, and in the face of unexpected torques. The attitude control system must also be able to adapted for a mission extension.

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Figure #.#: Control Moment Gyroscope

The attitude of the cycler will be controlled by four Control Moment Gyroscopes (CMGs). A

CMG is a system composed of a flywheel and two interlocking gimbals. This system has proven to be effective on several spacecraft in the past including the ISS and is used particularly to counteract consistent and predictable torques that span a long mission.

A CMG accumulates angular momentum by changing the direction of the flywheel in the center. The flywheel is a disk made of stainless steel that accounts for most of the mass of the system. Therefore, when we rotate this wide heavy disk very quickly, it has a large amount of angular momentum. By changing this angular momentum using the interlocked gimbals, we can exert a torque on the body that the CMG is attached to. This phenomenon can also be described

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Purdue University Cycler Appendix | 1 5 1 as the CMG absorbing angular momentum from the body attached it is attached to, therefore ensuring that the angular momentum of the system is conserved.

After accumulating a certain amount of angular momentum, the gimbals of a CMG will become parallel, therefore saturating the CMG. This means that the CMG is no longer able to accumulate angular momentum in that direction and is rendered useless. In order to prevent this gimbal lock from happening there will be a small reaction control system on the cycler habitation modules that will be able to unload the CMGs over the course of the mission. The CMGs are unloaded by rotating the gimbals back towards the initial position before angular momentum was accumulated. This involves applying a torque on the body the CMG is attached to that is opposite in direction to the perturbing torque that resulted in the stored angular momentum.

Figure #.#: Example of Gimbal Lock

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Above the light and dark blue gimbals are stuck parallel, resulting in a configuration that would act as a single gimbal system. q. Assumptions

In order to simplify the problem, we assume that most of the important torques are in the z direction, so we can treat the problem as one-dimensional. We assume a hardware safety factor of 1.2 that accounts for all the detailed components that would fasten the gimbals and flywheel together, and the CMGs to the cycler. This safety factor is used for both volume and mass calculations. Also, we assume that the gimbals of the CMGs do not contribute significantly to the mass or volume of the CMG, as the flywheel will be significantly larger.

The cycler is huge, and because of the long elevators with a large habitation unit at the end, has an enormous moment of inertia. This means that it is extraordinarily difficult to incite a change in angular velocity of the cycler. For this reason, we assume that the cycler is already spun up when we begin our analysis. r. CMG Design Methodology

The design for the Attitude Control System of the Cycler is primarily concerned with the constant torques that the cycler will experience throughout the mission time frame. The CMGs could be sized so that they would be constantly unloaded, and therefore could be very small and still handle the relatively small expected torque. There is also the concern of a large-scale angular impulse that could disrupt the function of the cycler. In the event of a large shift in angular velocity of the cycler, the CMGs designed for constant unloading would be saturated in a matter of moments. For this reason, the CMGs are designed so that they will be able to hold all the expected angular momentum that the cycler will experience throughout the length of the

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Purdue University Cycler Appendix | 1 5 3 mission. Therefore, if there is a large angular impulse event, the CMGs will be able to absorb much more angular momentum before being saturated. s. CMG Design Breakdown

To simplify things, we use angular momentum as the base unit to measure the capacity of the

CMGs. Using the average torque that we will be experienced through the course of the mission and the length of the mission we can find the required angular momentum storage capacity of the

CMGs.

퐿푅푒푞 = 푡 ∗ 푇푧 (37)

In order to find the size of the CMGs from the above requirement, we have to decide the relationship between the orientation of the CMGs and their angular momentum storage in each direction (momentum envelope). After looking into several configuration options for the CMGs it was decided that we would use a pyramidal configuration because it gives a momentum envelope that is best suited for storing angular momentum in any direction. From research we have found the following equation that for a pyramidal configuration of the CMGs gives us the momentum envelope.

퐿푆 = [2 ∗ 퐿푓 ∗ (1 +푐표푠 푐표푠 (∅푝 )) 2 ∗ 퐿푓 ∗ (1 +푐표푠 푐표푠 (∅푝 )) 2 ∗ 퐿푓 ∗ (38)

푠푖푛 푠푖푛 (∅푝) ]

The above equation describes the orientation of the CMGs as they sit as the vertices on the square face of a square pyramid, and the angle used describes the pointing of the CMGs from the square face to each triangle face of the pyramid. The figure below shows this.

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Figure #.#: Pyramid orientation

Equation 38 gives us the momentum envelope in each direction. It is possible to design this pyramid angle so that the amount of storage in each direction is the same and at a maximum so that the CMGs will be ready for any perturbation in any direction. Solving for the pyramid angle that gives the most angular momentum storage in every direction or spherical momentum envelope, we get that the angle should be 53.1 degrees.

Now that we have found the relationship between angle of orientation of the CMGs and how that relates to the momentum storage capabilities of the CMG system, we size the flywheels of the CMGs. From Equation 38 we see that the angular momentum of the flywheel affects the angular momentum storage capacity of the CMG. First, we set the angular velocity of the

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Purdue University Cycler Appendix | 1 5 5 flywheel to a constant 6000 rad/s. This number being similar to proven systems on the ISS and other spacecraft. Then using the definition of angular momentum below:

퐿푓 = 퐼푓 ∗ 휔푓 (39)

We calculate the moment of inertia of the flywheel to get the angular momentum. To size the CMG correctly we plug equation 39 into equation 38, and the Ls in all three directions must be equal to the Lreq in equation #.1. For this derivation, the moment of inertia is the variable that must be investigated. Using the moment of inertia calculation for a disk and assuming the radius of the flywheel is much larger than its thickness,

1 1 1 (40) 퐼 = [ ∗ 푚 푟2 ∗ 푚 (푟2 + ℎ2 ) ∗ 푚 (푟2 + ℎ2 ) ] 푑𝑖푠푘 2 푓 푓 12 푓 푓 푓 12 푓 푓 푓 we then increment the radius of the flywheel until we reach the required moment of inertia that in turn gives the required angular momentum storage. From there it is simple to get the volume and mass of the flywheel. The volume of the flywheel is calculated using the volume equation of a disk. Then we multiply this number by the density of stainless steel and the hardware safety factor of 1.2 to find the mass of each CMG. However, the flywheel should be able to rotate in any direction inside the CMG, so in order to estimate the actual volume of each CMG we use a cube with side length equal to the diameter of the flywheel times the hardware safety factor.

Finally, the power draw of the CMG system is found by averaging the control inputs from the simulation of the controller that will be explained shortly.

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Table #.#: Mass, Volume, and Power of Each CMG CMG Characteristic Value

Mass (kg) 81.43

Power (KW) 16.19

Volume of Each CMG(Mg) 46.66

t. Reaction Control System Design Methodology

The CMG system is sized so that it would be able to control the cycler for the entire length of the mission without needing to be unloaded. However, in the event of a large perturbing torque towards the end of the mission, the CMGs could be very close to saturation in the direction of the perturbation, resulting in gimbal lock and very possibly mission failure. Also, in the event of a power failure, the CMGs could possible not be able to collect enough torque and once again this could result in an uncontrolled cycler attitude and therefore mission failure.

For the above reasons we have included a Reaction Control System (RCS) composed of four ion thrusters identical to the ones that will be used as main propulsion thrusters on the cycler.

We reused these thrusters because the force requirement they have are similar to what is required to unload the CMGs efficiently, and to keep the design of the cycler simple by not having many different types of thrusters. u. RCS Design Breakdown

The RCS system must be able to unload the CMGs of angular momentum caused by expected torques at any point in the mission. Therefore, we have a number for angular

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Purdue University Cycler Appendix | 1 5 7 momentum that must be counteracted, and the amount of fuel that must be available is found by multiplying the expected torque by time.

퐿푟푒푞 = 푇푒푥푝푒푐푡푒푑 ∗ 푡푟푒푓푢푒푙 (41)

The time frame that we will use is the maximum amount of time between refueling opportunities for the cycler. This is found to be 1917 days from the flight plan. Using this value for time and the definition of angular momentum we can plug in the force supplied by the thrusters and the length of the arm of the cycler to find an amount of time spent burning in the case of maximum refuel time.

퐿푟푒푞 = 푇푒푥푝푒푐푡푒푑 ∗ 푡푟푒푓푢푒푙 = 퐹푇ℎ푟푢푠푡푒푟 ∗ 푙푐푦푐푙푒푟 푎푟푚 ∗ 푡푏푢푟푛 (42)

From Equation 42 we find the required amount of fuel using the characteristics of the thrusters, we can find the size and mass of the fuel and tank required to fuel the thrusters.

The RCS sizing above leaves some fuel in reserve during the legs that have a shorter refuel time that will allow us to correct unexpected torques and to reduce the loads of the CMGs.

Potentially allowing time to devote to the maintenance of the CMGs. Also, the required burn time for each leg will be far shorter than the time of each leg, so the thrusters can be burned at times that are the most convenient for the power requirements of the cycler. The thrusters can be activated at points when the main thrusters are offline, ensuring that the system essentially requires no increase in the power demands for the cycler. v. Controller Design

The cycler will undergo constant perturbations that could possibly destabilize the system and saturate the CMGs. In order to ensure that this does not happen, there needs to be a method to when and how the CMG’s absorb angular momentum. We will use a controller in the control of

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Purdue University Cycler Appendix | 1 5 8 the cycler. There are constant and complex perturbations that would be to difficult for a human to control, and the controller will be able to react to any perturbation almost instantly.

In order to develop a controller that can accurately maneuver the cycler, we must first develop a model of the motion of the cycler. Once again, we will make several assumptions so that we will be able to solve this problem. First, we assume that the only direction we need to control is rotation about the z axis. This is because most of the torque that we expect to encounter will be in the z-direction, we will be rotating in the z-direction, and the CMG are able to absorb angular momentum equally in all directions. This means that we would easily be able to translate this controller to each other direction.

The simplified equation of motion is below:

푤3 ∗ 퐼 = 퐿 (43)

We also know that,

퐿̇ = 푇 (44)

The controller for the cycler uses the angular momentum and angular velocity as a basis.

Starting from the current angular velocity, we subtract the guidance angular velocity. This guidance angular velocity is described as the angular velocity required for the people in the habitation modules of the cycler to feel one g of acceleration. After the subtraction, we have an error in angular velocity. We multiply this error by gain values for a PI controller that were tuned by trial and error, and we have the basic idea of our controller.

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Figure #.#: Basic Model of the Motion of the Cycler.

However, the perturbation torques on the cycler have not yet been modeled. To do this we have to convert this error into a quantity that we can combine with torques. To do this we will convert the error in angular velocity into error in angular momentum using Equation #.7. Then, we can integrate the torque that we anticipate changing it to angular momentum using Equation

#.8. We can then model this torque as disturbances and add them to the error that we previously calculated.

One of the requirements that was mentioned early in this report was that the controller for the cycler must be able to react to torques beyond what is expected. In order to accomplish this, we add random torques using a random number generator and observe how the controller reacts to these torques. By tuning the gains, we confirm that the controller can deal with random torques larger than the ones that are expected.

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Figure #.#: Basic Model of the Motion of the Cycler.

Now that the controller is confirmed to work for random torques and for our expected torques we use it to find useful quantities. We need to find a power value for their second to second use. We originally used Equation 44 to find the power that the CMGs would draw.

푃 = 2 ∗ 푇푚푎푥 ∗ 휔푔𝑖푚푏푎푙 (45)

In this equation the two is the hardware efficiency factor and the T is the largest torque that could be produced from the CMG configuration. The issue with the above equation is that the moment to moment power draw will not be nowhere close to as large as the maximum torque that could be exerted by the CMGs. The maximum torque supplied by the CMG is found using

Equation 46.

푇푚푎푥 = 휔푔𝑖푚푏푎푙 푥 휔푓푙푦푤ℎ푒푒푙 ∗ 퐼푓푙푦푤ℎ푒푒푙 (46)

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The gimbal rate is a value that comes from research as being safe for continued use, the other two values are already known and with this calculation we get a maximum torque possible of over a million N*M. After observing the error in angular velocity that happens in the moment to moment operation of the CMGs, we can transfer this angular velocity error to angular momentum and find the required Torque that corresponds with this error at each point. This allows for us to average the torque over the operation time of the CMG and then we can use

Equation 45 to find average power draw.

References

(1) Currie, B. J., “Control of a Spacecraft Using Mixed Momentum Exchange Devices.”

(2) “List of moments of inertia,” Wikipedia Available: https://en.wikipedia.org/wiki/List_of_moments_of_inertia.

(3) “Control Moment Gyro (CMG) Sizing and Cluster Configuration ...” Available: https://www.researchgate.net/publication/311665872_Control_Moment_Gyro_CMG_Sizing_and

_Cluster_Configuration_Selection_for_Agile_Spacecraft.

(4) Nagabhushan, V., “Development of Control Moment Gyroscopes for Attitude Control of

Small Sattelites.”

(5) Gurrisi, C., Seidel, R., Dickerson, S., Didziulis, S., Frantz, P., and Ferguson, K., “Space

Station Control Moment Gyroscope Lessons Learned,” Proceedings of the 40th Aerospace

Mechanisms Symposium, NASA

Cycler Power Systems

References

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[1] Surampudi, S. (2011). Overview of the Space Power Conversion and Energy Storage

Technologies. Jet Propulsion Laboratory, Pasadena.

[2] Beauchamp, P. (2015). Solar Power and Energy Storage for Planetary Missions. Jet

Propulsion Laboratory, Pasadena.

[3] Surampudi, S., et al. (2017). Solar Power Technologies for Future Planetary Science

Missions. Jet Propulsion Laboratory, Pasadena.

[4] Ragheb, M. (2011). RadioIsotopes Power Production. Stanford, CA.

[5] Palac, D. (2016). Nuclear Systems Kilopower Overview. Los Alamos National

Laboratory.

[6] Advanced Stirling Radioisotope Generator (ASRG). (2013). National Aeronautics and

Space Administration.

Cycler Docking Arm References

Ruel, S., Luu, T., & Berube, A. (2012). Space shuttle testing of the TriDAR 3D rendezvous and docking sensor. Journal of Field robotics, 29(4), 535-553.

English, C., Zhu, S., Smith, C., Ruel, S., & Christie, I. (2005, September). Tridar: A hybrid sensor for exploiting the complimentary nature of triangulation and LIDAR technologies.

In Proceedings of the 8th International Symposium on Artificial Intelligence, Robotics and

Automation in Space (Vol. 1).

Yaskevich, A. (2014). Real time math simulation of contact interaction during spacecraft docking and berthing. J. Mech. Eng. Autom., 4, 1-15.

Miele, A., Weeks, M. W., & Ciarcia, M. (2007). Optimal trajectories for spacecraft rendezvous. Journal of optimization theory and applications, 132(3), 353-376.

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Ho, C. C. J., & McClamroch, N. H. (1993). Automatic spacecraft docking using computer vision-based guidance and control techniques. Journal of Guidance, Control, and

Dynamics, 16(2), 281-288.

Philip, N. K., & Ananthasayanam, M. R. (2003). Relative position and attitude estimation and control schemes for the final phase of an autonomous docking mission of spacecraft. Acta

Astronautica, 52(7), 511-522.

Singla, P., Subbarao, K., & Junkins, J. L. (2006). Adaptive output feedback control for spacecraft rendezvous and docking under measurement uncertainty. Journal of guidance, control, and dynamics, 29(4), 892-902.

Cycler Thermal Systems a. Components of a Thermal System

As stated above, the components of a thermal system are the heat exchangers, heaters, pumps, coldplates and radiators. This section goes over what each of these components are and how they can be used with the thermal system. As stated before, the thermal system is made up of loops which are pipes filled with coolant that move heat around the Cycler. This heat is either dissipated through radiators which use radiation as their method of transferring heat or it is transferred via conduction to the cabin in order to keep the crew at a habitable temperature. The two types of heat movement can be categorized as heat rejection loops and heat acquisition loops. The Cycler does not have a lot of time in the eclipse of a planet where the sun is blocked by a planet so the heat acquisition is not as robust as the Space Shuttle or ISS’s heat acquisition loops. We predict that the Cycler will have under 10 minutes of its mission where it will be in the

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Purdue University Cycler Appendix | 1 6 4 ecliptic of a planet. This is much less than the time in the ecliptic for the ISS and the Space

Shuttle, so the thermal system deviations from previous designs are justified. w. Heat Acquisition Loops

Heat Acquisition is made up of Heat Exchangers and Coldplates. Coldplates are metal plates that have channels for coolant to flow through. These plates are exposed to cold environments to cool the coolant quickly. However, this device can be used to heat coolant as well if the surrounding temperature is warmer than the coolant. This is a very commonly used device for acquiring Heat into the system. The Heat exchangers are necessary for transferring heat from one coolant system into another which would be used in conjunction with the coldplate in this case. The heat exchangers that we are using in this case are phase change heat exchangers. The

CAD Model for the heat exchangers can be seen below.

Figure 3. The Heat Exchanger for the Cycler Thermal System.

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Purdue University Cycler Appendix | 1 6 5 x. Heat Rejection Loops

Heat Rejection is necessary to accommodate the amount of heat generated by the spacecraft.

This is done through simple radiators that are positioned on the outside of the spacecraft and use convection, conduction and radiation to dispel the heat generated by the electronics and other systems on board. Heat Rejection Loops on the Cycler are very important as they help move heat generated by the electronic components. The heat rejection loops are simply made up of radiators and heat exchangers as well as a significant amount of tubing. y. Heat Transport

The heat transport system is made up of all the pumps, tubes and fluid that transports the heat around to the various systems that either dispel heat or acquire heat. This system is made up of the pipes, pumps, valves, control instruments and the Freon fluid inside the pumps. This system can be very hard to size due to the immense size of the Cycler but the sizing can be approximated using an overall assessment of the Thermal Control System. This analysis can be found later in the paper about the Thermal System Sizing. z. Passive Thermal Control

Passive Thermal Control helps maintain the temperature of the Cycler close to the habitable temperature for the crew and the Active Thermal System helps maintain the temperature within a smaller range for the crew and the electronics on board. This system includes insulation around the entire spacecraft as well as Louvers which are shutters for the Sun to radiate onto the surface and close when insulation is necessary. Louvers are very important to make sure that heat is not dissipated or absorbed when they are not required. The louvers on the Cycler can be seen below.

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Purdue University Cycler Appendix | 1 6 6

Figure 4. The Louvers for the Cycler Thermal System.

Heaters are necessary when the spacecraft is in low power mode and needs some thermal energy to regulate the temperature of the cabin. Heat pipes provide a low energy solution for moving heat around by using concentric pipes that have fluid on the outside and vapor on the inside. Passive Thermal Control Techniques are very useful when insulating and maintaining a wide range of temperatures for a vehicle.

aa. Sizing for the Thermal Control System

Sizing the Thermal Control System is primarily dependent on the electronic components on board; however, due to the immense size of the Cycler, the solar heating on the Cycler is also a significant factor in determining the thermal load. The thermal load from the sun can be determined using the equation below.

푄 = 휀휎퐴푇4 (47)

The equation above shows the heat transfer equation for radiation. Radiation is the only way that heat can be transferred in a vacuum like space. Because of the conditions of space, this

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Purdue University Cycler Appendix | 1 6 7 equation defines the solar heat transferred to the Cycler. The equation is defined by a constant as well as the surface area and the emissivity. The emissivity is the ratio of the energy emitted by a substance to that of a perfect radiator. The Cycler is covered in Multi-Layer Insulation which is a silver or gold foil that has a very low emissivity. This is ideal for the spacecraft as we only want the coldplates and the radiators to absorb or release heat. This means that the coldplates and radiators are covered in black paints in order to increase emissivity as black plates have an emissivity close to 1. From this equation, we can size the surface area of the radiators using the equation below.

푄 (48) 퐴 = 4 4 [𝜎휀휂(푇푟 −푇푒 )]

In this case, 푇푟 is the maximum temperature for the Radiator and 푇푒 is the effective temperature for the thermal environment. We can also determine the heat added to the Cycler from the sun using the Lumosity of the sun and assuming that the solar constant at a certain position is related to the heat flux.28

퐿 (49) 푆 = 4 ∗ 휋 ∗ 푑2

The thermal control system has to be able to handle the hottest condition that the Cycler will be in so we plug in the smallest distance from the sun during the mission. This distance is

1.24*1011 meters. The Solar Constant is equal to 1986.2 W/m2. If we assume that half of the

Cycler’s surface area is facing the Sun, then the heat energy rate to the Cycler can be determined

1. 28 Juhasz, A.j. “An Analysis and Procedure for Determining Space Environmental Sink Temperatures with Selected Computational Results.” Collection of Technical Papers. 35th Intersociety Energy Conversion Engineering Conference and Exhibit (IECEC) (Cat. No.00CH37022), doi:10.1109/iecec.2000.870928.

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Purdue University Cycler Appendix | 1 6 8 to be 39.4 MW of energy. Using equation 49, we can solve for the surface area of the radiators to be 1279.9 m2. This is a crucial step in sizing the thermal control system. Using the figure shown below, the total sizing can be determined.

Figure #. The sizing factors for the components for the Thermal Control System.

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Purdue University Cycler Appendix | 1 6 9

These factors are put into a Matlab Code in order to size all the components and the overall total size of the Cycler Thermal System.

Table X: PDR CDR Final Values

Principal moments of

inertia throughout the

project

10 10 9 퐼푥푥 4.643 x 10 4.526 x 10 7.635 x 10

12 12 12 퐼푦푦 5.082 x 10 7.048 x 10 1.319 x 10

12 12 12 퐼푧푧 5.06 x 10 7.047 x 10 1.322 x 10

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Fig. X: We optimized the cycler’s angular velocity over the course of the project.

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14. Tether Slings Located on Phobos, Mars, and Luna Appendix

S. Adeniji29, V. Bartels30, A. Brewer31, A. Chapa32, N. Cruces33, J.

Cuellar34, N. DeAngelo35, S. Lach36,

S. Ness37, V. Richard38, E. Schott39, E. Smith40, S. Yang41

Purdue University, West Lafayette, Indiana, 47906, United States

29 CAD

30 Power and Thermal

31 Structures

32 Controls

33 Mission Design

34 Mission Design

35 Communication

36 Structures

37 Mission Design

38 Mission Design

39 Human Factor

40 Erick Smith

41 Propulsion

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M. Appendix

M.1. Tether Cross-sectional Area Input Effects

The cross-sectional area at tether tip relies on the taxi mass, maximum acceleration, ultimate tensile strength. The maximum velocity and length terms partially cancel out, leaving us with the maximum acceleration. Decreasing the taxi mass is the easiest way to decrease the cross- sectional area since decreasing it does not negatively affect any other parameters. The ultimate tensile strength is fixed for the most part depending on what material is chosen for the tether in the first place.

The cross-sectional area at the base of the tether relies on the taxi mass, maximum acceleration and ultimate tensile strength as well, but it also adds the density of the tether material and the maximum velocity at the tether tip. Like the cross-section at the tip, the cross-section at the base is decreased by decreasing the taxi mass, but it decreases at faster rate when decreasing the maximum velocity at the tether tip. Decreasing the velocity at the tether tip also provides favorable results for the other tether parameters. The material density is relatively fixed along with the ultimate tensile strength.

M.2. Tether Mass Input Effects

The tether mass relies on the taxi mass, maximum velocity at the tether tip, density of the material, and ultimate tensile strength. The density and the ultimate tensile strength are relatively fixed as discussed in pervious sections. The relationship of the tether mass to the taxi mass and maximum velocity at the tether tip is shown in Fig. 62

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Fig. 67: A comparison of the relationship of the tether mass to the maximum tether tip velocity and the taxi mass

As we see, the maximum tether tip velocity affects the tether mass more than the taxi mass does and this is especially pronounced above 5 km/s. The jump from 5 to 6 km/s results in a 10,000

Mg gain. The Luna tether is the only tether operating in this range. The tether mass can be reduced by a few thousand metric tons if the maximum velocity is reduced to around 4 or 4.5 km/s which should be attainable.

M.3. Torque Arm and Hub Sizing Details

The goal in sizing the tether sling and torque arm is to size the torque arm such that the applied stresses do not cause it to fail, while also keeping the mass of the torque arm and hub to a reasonable amount. This analysis is complicated by several factors. First, increasing the cross- sectional area of the torque arm (which would decrease stress) would also increase the mass of the torque arm, thereby increasing the stress on the torque arm due to gravity loading. Second, increasing the cross-sectional area of the torque arm also necessarily increases the height of the central hub, which increases the total mass of the system. Due to these competing factors, an optimization study could be performed to find the minimum mass of the system. However, as this

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Purdue University Tether Sling Appendix | 1 7 4 project is a feasibility study only, we use the first configuration that results in a total mass of less than one million metric tons.

We model the torque arm as a cantilever beam subjected to 3 loads: gravity, centripetal force due to the rotating taxi, and centripetal force due to the rotating tether. There are also several assumptions we make in this analysis. First, the stress from each of these loads can be added linearly, and the resultant stress is concentrated at the root of the torque arm. We also assume that the limiting case is when the taxi is rotating at maximum velocity and neglect the effect of creep during spin-up, as the temperature of the torque arm will not get high enough for creep to be applicable. Finally, we assume that the loading due to centripetal force is not equivalent to the entire centripetal force. Figure 7.63 illustrates this in further detail.

Figure 7.63: Representation of the tether sling system. Because the centripetal force is directed towards the center of the hub, rather than along the length of the tether, only part of the centripetal force loads the torque arm. Not to scale.

To calculate the stress due to gravity loading, we begin with equation 7.38, which gives the stress in a cantilever beam caused by a bending moment.

푀푦 (7.38) 휎푔 = − 퐼푐 In this equation, Ic is the moment of inertia of the cross-sectional area, and y is the distance from the neutral axis to the plane of interest. Since this is being analyzed as a cantilever under bending stress, then the neutral axis is the center of the torque arm, and the plane of interest is the very top

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Purdue University Tether Sling Appendix | 1 7 5 of the arm, meaning that y is equivalent to the outer radius. We assume that the cross section is a hollow circle. Therefore, equation 7.39 gives the moment of inertia for the torque arm.

휋 퐼 = (푟2 − 푟2) (7.39) 푐 4 표 𝑖 To calculate the bending moment, we must analyze an infinitesimal segment of the torque arm. The infinitesimal mass is found by multiplying cross sectional area, density, and infinitesimal length, and is given by equation 7.40.

2 2 푑푚 = 휌휋(푟표 − 푟𝑖 )푑푥 (7.40) In this equation, ρ is the material density of the torque arm, and x is directed along the length of the torque arm.

Next, we multiply the mass by the acceleration from gravity, g, which results in a force.

To get the infinitesimal moment, this force is multiplied by the distance from the base of the torque arm, x. Integrating this moment along the length of the torque arm results in equation 7.41.

1 (7.41) 푀 = 푔휌휋(푟2 − 푟2)퐿2 2 표 𝑖 In this equation, L is the total length of the torque arm.

To get the total bending stress from gravity, we combine equations 7.41, 7.39, and 7.38, and substitute ro for y. This results in equation 7.42.

2 2 2푔휌푡표푟푞푢푒휋푟표(푟표 − 푟𝑖 ) (7.42) 휎푔 = 4 4 (푟표 − 푟𝑖 ) Centripetal force is given by equation 7.43, where r is the radius of the circle of rotation— in this case, the distance from the taxi to the center of the hub.

푚푉2 (7.43) 퐹 = 푐 푟 Observing the geometry seen in Figure 7.63 above, we can find an expression for the component of centripetal force acting along the length of the tether. Substituting this expression and the appropriate variables into equation 7.43 gives equation 7.44, the tension force in the tether due to the centripetal force of the taxi.

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2 푚푡푉 −1 퐿 (7.44) 퐹푡푎푥𝑖 = cos (tan ( )) √퐿2 + 푙2 푙 In this equation, l is the length of the tether and mt is the mass of the taxi. Dividing this expression by the cross-sectional area of the torque arm gives the stress from this force, shown in equation

7.45.

퐹푡푎푥𝑖 (7.45) 휎푡푎푥𝑖 = 2 2 휋(푟표 − 푟𝑖 ) Calculating the stress due to the centripetal force of the tether follows a similar process.

We begin with an expression for the infinitesimal mass of the tether, given by equation 7.46.

2 2 2 푚푡푉 푉 휌 푥 (7.46) 푑푚 = 휌푡푒푡ℎ푒푟 exp [ (1 − 2 )] 푑푥 휎푦푙 2휎푦 푙 Note that in this equation, mt is still the mass of the taxi. In addition, σy is the yield stress of the tether material, Dyneema.

To find the force acting along the length of the tether, we use the expression for centripetal force in terms of angular velocity, given by equation 7.47.

2 퐹푐 = 푚휔 푟 (7.47) Next, we combine equation 7.47 with equation 7.46 and the expression for the force component along the length of the tether. This results in an expression for infinitesimal force, shown in equation 7.48.

푉 2 (7.48) 푑퐹 = 푑푚 ( ) √퐿2 + 푥2 푡푒푡ℎ푒푟 푙 Unfortunately, the exact total force cannot be found, since the integral that results from equation 7.48 does not have an analytic solution. Because of this, we solve the integral using an

Euler method approximation, that is, the expression shown in equation 7.49.

퐹푡푒푡ℎ푒푟,푛+1 = 퐹푡푒푡ℎ푒푟,푛 + 푑퐹푡푒푡ℎ푒푟Δ푥 (7.49) In this way, we sum the “pieces” of force from x = 0 to x = l, in intervals of Δx = 0.01. Finally, by dividing this quantity by the cross-sectional area, we get the stress from the centripetal force of the tether, shown in equation 7.50.

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퐹푡푒푡ℎ푒푟 (7.50) 휎푡푒푡ℎ푒푟 = 2 2 휋(푟표 − 푟𝑖 ) Adding the three stresses together gives the total stress in the torque arm, shown in equation 7.51.

휎푡표푡푎푙 = 휎푔 + 휎푡푎푥𝑖 + 휎푡푒푡ℎ푒푟 (7.51) This expression leaves L as an unknown variable. This can be solved for by comparing

σtotal to the yield stress of the material. Due to the complexity of this expression, this is not done analytically, but by use of the Matlab script torque_arm_stress.m.

Once the length of the torque arm is known, the other relevant quantities can be determined.

The radius of the hub is given by equation 7.52.

푟ℎ푢푏 = 퐿푡표푡푎푙 − 퐿 (7.52) In this equation, Ltotal is the total length of the system. By default, this is 1000 m, but is less in some locations in order to prevent the system from becoming prohibitively massive. Next, the mass of the torque arm is given by equation 7.53.

2 2 푚푡표푟푞푢푒 = 휌휋(푟표 − 푟𝑖 )퐿 (7.53) As stated previously, we assume that the height of the hub is equivalent to the outer diameter of the torque arm. This leads to equation 7.54, the mass of the central hub. Note that we also assume that the hub is solid.

2 푚ℎ푢푏 = 휌휋푟ℎ푢푏푟표 (7.54) To find the total mass of the system, we combine equations 7.53 and 7.54 to get equation 7.55.

푚푡표푡푎푙 = 푚푡표푟푞푢푒 + 푚ℎ푢푏 (7.55) As stated previously, the Matlab script torque_arm_stress.m was used to determine the dimensions and masses of the systems in each location. Table 7.28 lists all of the input values for each location. For completeness, this table also lists the input parameters for the ED tether in low Earth orbit, which was analyzed using the same method.

Table 7.28: Input values at each tether sling location Location Gravity (m/s2) ΔV (km/s) Tether length (km) ED (LEO) 0 3.75 573.56 Luna 1.62 4.0218 825

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Mars 3.711 5.1515 1352.9 Phobos 0.0057 3.6508 678

In addition, the taxi mass used is 137.36 Mg. The “yield stress” for the tether material in this analysis is 3.325 GPa. This is because Dyneema, being a synthetic polymer, does not have a yield stress, so the ultimate tensile stress is used instead. The density of Dyneema is 970 kg/m3. The torque arm material used is aluminum 6061-T6. The density of aluminum 6061-T6 is 2700 kg/m3, and its yield strength is 184 MPa. To account for unforeseen complications, a factor of safety of

1.5 is used, meaning the effective yield strength used in this analysis is 122.67 MPa. Finally, the

Luna tether sling has an inner diameter of 2 m, so that passengers can travel through it on the way to the taxi.

With these inputs, we use a trial-and error approach to determine the size of each system.

A variety of outer diameters (now the independent variable) are input into the script until a system with a mass of less than one million metric tons was found. If needed, the total distance is halved

(e.g. from 1000 m to 500 m, 500 m to 250 m, etc.).

M.4. Piling Depth Estimate Calculation

Piling depths are usually calculated using soil properties at a building location. In this case, we do not have access to the types of soil properties at each building site that we need to make those calculations. We decide that we need to do a relative comparison to an existing piling depth instead. We use the Burj Khalifa as our analog because its empty weight of 500,000 Mg is similar to what the total mass of our tether sling systems and the very weak, sandy soil gives us a worst- case scenario of sorts. First, we input the overall tether sling mass. Second, we calculate the gravitational force acting upon the Burj Khalifa at its empty weight. Then, we calculate the gravitational force acting on the tether sling. We use the Earth’s gravity for the calculation purposes because the tether slings are subjected to a large lateral force from the swinging tethers

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Purdue University Tether Sling Appendix | 1 7 9 despite the celestial bodies not having as large of a gravitational field as Earth. We do this to ensure the pilings are deep enough to prevent the tether sling from tipping over when subjected to the lateral loads. We create a ratio between the tether sling gravitational force and the Burj Khalifa gravitational force that allows us to directly relate the system masses to the piling depths. Finally, we calculate the tether sling piling depth estimate by multiplying the ratio of gravitational forces by the Burj Khalifa piling depth and the safety factor. As mentioned before, this is a very crude estimate since the Burj Khalifa and its location are likely completely different aside from the overall mass.

M.5. Docking Mechanisms Strengths and Weaknesses

Fig.68: Zoomed view of the Figure 15 near the arm length limit.

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The different docking mechanism options are listed below:

• Clamp

o Strengths: Adapted to withstand relatively high radial forces. Needs little to no

mechanism on the taxi apart from structural components.

o Weaknesses: Requires a certain amount of precision. Needs structural

improvements to withstand tangential forces during deceleration.

• Snap – fit

o Strengths: Once docked, the system is tightly locked. It is also tolerant to high

compressive forces and has little to no backlash.

o Weaknesses: Requires a high precision and impact force. Important internal

constraints due to the absence of backlash. Not applicable for departures.

• Hook

o Strengths: Once locked, the mechanism behaves like a “ball-joint” so it is really

tolerant to change in direction of constraints. The 3 degrees of freedom remaining

mitigates the constraints to some extent.

o Weaknesses: Requires a high precision to dock in. The taxi part of the mechanism,

most probably some sort of ring, is not aerodynamic. Also not really adapted for

departures.

• Magnets

o Strengths: Tolerant to the docking orientation. Magnets intensity can be modulated

to dock smoothly. Less friction and fatigue compared to other mechanisms.

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o Weaknesses: The electric energy dependency of the mechanism requires an

additional safety factor. Requires to also add a magnet on the Taxi. This will add a

lot of weight and power consumptions to the taxi, which we want to avoid.

• “Classical space docking” (APAS, CBM, …)

o Strengths: Mature and flight-proven technology.

o Weaknesses: Requires a relatively high impact force. High use of the Taxi’s RCS

system that can be reduced or even avoided with other mechanisms.

• Net

o Strengths: Requires an extremely low accuracy.

o Weaknesses: Hardly predictable behavior and dangerous. Departures from a net is

highly uncertain.

• Berthing mechanism

o Strengths: Reduces the propellant consumption of the Taxi. Six Degrees of freedom

arm ensures a good level of security and capability of docking.

o Weaknesses: Moving the arm at the rendezvous speeds in an atmosphere would

require a really strong and massive arm.

• Grapple (NASA “Boom” [1])

o Strengths: Rewindable grapple reduces collision probability at docking. The

grapple can have 6 degrees of freedom if mounted on a gimbal. May reduce the taxi

mass the same way as the clamp does. Can soften the tension spike when capture

by releasing the cable. Could be used for rendezvous and in a lesser extent for

departures.

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Once all the strengths and weaknesses defined, we made our choice using a decision

matrix. The weightings have been chosen qualitatively.

Table 7.29 Mechanism decision matrix

Mechanism Reusability Reliability Mass Resilience Tolerance Total

Weights 4 Weights 4 Weights 3 Weights 2 Weights 2

Clamp / Hook / Grapple 3 3 3 3 2 43

Snap-fit 1 2 2 2 2 26

Magnet 3 2 1 3 3 35

"Classical space docking" 3 2 2 3 1 34

Net 2 1 3 1 3 29

Berthing Mechanism 3 2 3 2 3 39

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M.6. Taxi Landing Hub

The first concept we came up with was more than just a square; it was a new system within the project. The landing hub is such a solution. The hub contains three landing pads with a rail system that would deliver taxies to the mass driver on Mt. Olympus.

The landing pad itself is the key part in the project’s success. To fix the problem of buoyancy the landing pad will use a strong magnet that will hook onto the taxi and bring the spacecraft down onto the landing pad safely. The best way to describe it would be a tractor beam from sci-fi movies. This will keep the spacecraft from bouncing around after touchdown and prevent serious damage to passengers and the spacecraft.

Once the taxi lands an electromagnetic cradle will arrive and attach to the taxi. The cradles job is to bring the taxi down from a vertical plane to a horizontal plane. This will help passengers disembark like an airplane plus get the taxi in the ready position to load into the mass driver.

The final part of the design of the landing hub is the underground gate. After the taxi lands and attaches to the cradle, the spacecraft is then lowered below ground to a taxi gate. This below- ground garage will protect the spacecraft from being damaged from the environment on the surface. This will also enable passengers to have a safer disembark from the cabin in an air sealed compartment below ground. The bunker’s design is encased with metal to help ensure that the

Martian soil will not interfere with the hub’s job and take out the problem of the soil degradation on our system.

The hub also includes an emergency runway and an air traffic control bunker to house the coding for the system. We also included a power supply south of the hub to ensure power is

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Figure 7.64: Top down view of the Taxi Hub on Mars

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Figure 7.65 Cross sectional cutout of the Landing Hub

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Figure 7.66 The first design of the cradle

Table 7.30 Cradle Design 1 Information Mass 100 tons Material Aluminum 6061 Length 25 meters Magnet Width 15 meters Width 36 meters ______

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M.7. Sample Taxi Transfer Calculations

Calculating the periapsis velocity for the Phobos tether:

푘푚3 휇 = 42828.31 푀 푠2

푟푝 = 푆푒푚푖 − 푀푎푗표푟 퐴푥푖푠 표푓 푃ℎ표푏표푠 = 9376 푘푚

푘푚 푉 = 5.0 = 푀푎푥푖푚푢푚 퐹푙푦푏푦 푉푒푙표푐푖푡푦 표푓 푡ℎ푒 퐶푦푐푙푒푟 푟푒푙푎푡푖푣푒 푡표 푀푎푟푠 ∞ 푠

푉2 25 푘푚2 퐸푛푒푟푔푦 = ∞ = = 12.5 2 2 푠2

휇 42828.31 푘푚 푃푒푟푖푎푝푠푖푠 푉푒푙표푐푖푡푦 = 푉 = √2 ∗ (퐸푛푒푟푔푦 + 푚) = √2 ∗ (12.5 + ) = 5.84 푝 푟푃 9376 푠

Calculating the Circular Velocity of Phobos:

휇푀 42828.31 푘푚 푉푐 = √ = √ = 2.14 푟푝 9376 푠

Now that the circular velocity of Phobos has been calculated as well as the required velocity at periapsis, we can calculate the velocity that will be required by the tether. As the circular velocity of the acts in the same direction of the velocity required by the tether, we can subtract the circular velocity from the periapsis velocity.

푘푚 푉 = 푉 − 푉 = 5.84 − 2.14 = 3.7 푟푒푞푢𝑖푟푒푑 푝 푐 푠

This result is mentioned in the text as the first velocity calculated for the Phobos tether.

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Velocity Requirement Improvement Calculation:

In this new analysis, the length of the tether is added onto the periapsis distance to calculate a new periapsis velocity.

퐿푒푛푔푡ℎ 표푓 푡ℎ푒 푇푒푡ℎ푒푟 = 푙 = 700 푘푚

푟푝 = 푆푒푚푖 − 푀푎푗표푟 퐴푥푖푠 표푓 푃ℎ표푏표푠 + 푇푒푡ℎ푒푟 퐿푒푛푔푡ℎ = 9376 푘푚 + 700 푘푚

푟푝 = 10076 푘푚

휇 42828.31 푘푚 푃푒푟푖푎푝푠푖푠 푉푒푙표푐푖푡푦 = 푉 = √2 ∗ (퐸푛푒푟푔푦 + 푚) = √2 ∗ (12.5 + ) = 5.79 푝 푟푃 10076 푠

휇푀 42828.31 푘푚 푉푐 = √ = √ = 2.14 푟푝 9376 푠

푘푚 푉 = 푉 − 푉 = 5.79 − 2.14 = 3.65 푟푒푞푢𝑖푟푒푑 푝 푐 푠

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M.8. Tether Length and Centripetal Acceleration

When we start looking at the mission overview, we determine the tether length. This is because the period of one revolution of a tether depends on the radius of the circular motion.

Therefore, the tether length effects the timing of the catch and release of taxis. The relationship for tether length is [1]

v2 lt = (7.56) amax

We determine the tether length by assigning a value for amax. This number is the maximum force from acceleration tolerable by the human body. We determine our human limitations to set a maximum acceleration. Anti-g suits may be a way to raise the tolerance for acceleration by 1.5 g’s [3], however there is not a lot of research on the effects of high amounts of acceleration for more than a few minutes. Typically, humans are able to withstand 2 g’s for 24 hours or less [4].

There are some safety precautions that can be used to prevent the heart rate and arterial pressure changes that come with high g-force. These include lying perpendicular to the acceleration, wearing an anti-g suit, and lying in the “eye-ball inward” position [4]. This helps prevent blood from pooling in the feet or brain. Based on these recommendations, we set the maximum centripetal acceleration of the tether slings to 2 g’s. We also set 24 hours as maximum time limit for a taxi to be attached to the tether during one rendezvous event.

The remaining variable v is still undefined. This is the velocity on the tip of the tether (also called end-tip velocity). It must be determined through the different transfers along the taxi’s mission. Since the velocity requirements change based on the mission, we use the maximum velocity requirement to set the tether length to a safe length. Using some of the maximum velocity requirements for each tether, we determine each length for the different locations: Phobos is 700 km, Mars is km, and Luna is km.

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M.9. Pre-spin-up Analysis

In the early stages of design, we determined how the tether system would be affected if the tether had no spin-up or down times and was instead always spinning. The maximum ∆V that needs to be delivered from the Phobos tether is 3.7051 km/s. In a situation where the tether is spun- up before the taxi arrives, it is necessary to determine how much a of jerk is felt on the passengers.

The taxi is launched from Mars to Phobos and at the end of its journey has a velocity of zero relative to Phobos and follows Phobos’ orbit. The end of the spinning tether then connects with the taxi. Below is a diagram of how this interaction may look from a view above the tether looking down:

Fig 7.67: Diagram of Tether Sling with attached spring

The centripetal acceleration (in the direction towards the center of the circular motion) is set at 1 g. The relationship below shows the tether length. In this case, the collision means that the

∆V is going to be less than the initial end tip velocity. To find this required velocity, we use the conservation of momentum, also shown below:

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푚푡푒푡ℎ푒푟 ∗ 푣푡푒푡ℎ푒푟 + 푚푡푎푥𝑖 ∗ 푣푡푎푥𝑖 = 푚푡표푡푎푙 ∗ 푉 (7.57)

In this analysis, the material used is Zylon. When the taxi rendezvous with the tether, it must go from 0 to 3.962 km/s in a very short amount of time. It’s similar to a car going 8,862 miles per hour crashing into a stationary car. One possible way to mitigate the effect of the sudden jump in velocity, is to add a spring on the end of the taxi to absorb some of the collision. The figure shows what this spring may look like.

The goal now is to determine the displacement and spring constant that would be safe for humans. Assuming a spring acceleration of 1 g, the equation below gives us the spring force.

퐹푠푝푟𝑖푛푔 = −푘푥 = 푚푡푎푥𝑖 ∗ 푎 (7.58)

There are two unknowns and only one equation. The next relationship used comes from the conservation of energy, shown below.

1 1 1 1 푚 ∗ 푣2 + 푚 ∗ 푣2 = 푚 ∗ 푉2 + 푘푥2 (7.59) 2 푡푎푥𝑖 푡푎푥𝑖 2 푡푒푡ℎ푒푟 푡푒푡ℎ푒푟 2 푡표푡푎푙 2

The spring potential energy is the last part of the equation. Plugging in all the other values, we have a way to solve for spring constant and displacement. We find the spring would have to displace 1,500,000 km with a spring constant of 6.55E-07 N/m. This is one insane spring.

There are not a lot of ways to mitigate the effects of such a strong jerk. However, the spring idea could possibly work. There are a lot more elements to add to the problem and it becomes very complex. The spring motion may affect the final orbit, and where can we find a spring with the exact requirements we need. Another possibility is to give the taxi an extra boost of velocity, so it is moving closer to the speed of the tether at the time of attachment. In the end, it is not feasible to have the tether pre spun-up for the ∆V required and we ultimately chose the option of having a spin-up time.

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M.10. Propellant Analysis Variations of Inputs

Table 7.31 Variation of Inputs and Corresponding Outputs for Tether Slings on Phobos

∆V Density UTS 푀푝 2 2 푀푅푡푝 푀푡 (Mg) L (km) 퐴푡 (푐푚 ) 퐴0 (푐푚 ) (km/s) (kg/푚3) (GPa) (Mg)

3.7053 970 3.325 193 123 3.4 x 104 700 113.85 843.41

3.6 970 3.325 193 111 3.0 x 104 660.78 113.85 753.89

3.5 970 3.325 193 101 2.6 x 104 624.58 113.85 679.72

3.2 970 3.325 193 76 1.7 x 104 522.09 113.85 506.99

3.7053 1340 62 193 2 6.3 x 104 700 6.11 7.08

3.7053 2460 3.1 193 6673 1.9 x 106 700 122.11 28348.3

3.7053 2330 7 193 177 4.9 x 104 700 54.08 531.31

Table 7.32 Convert Table 7.31 Data into the Percentage Form.

∆V (%) Density (%) UTS (%) 푀푝 (%) 푀푅푡푝 (%) 푀푡 (%) L (%) 퐴푡 (%) 퐴0 (%)

3 0 0 0 10 14 6 0 11

6 0 0 0 18 25 11 0 19

14 0 0 0 38 50 25 0 40

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0 -38 -1765 0 98 -85 0 95 99

0 -154 7 0 -5325 -5309 0 -7 -3261

0 -140 -111 0 -44 -44 0 52 37 a. A positive percentage represents the decreased amount. b. A negative percentage represents the increased amount.

Table 7.33 Variation of Inputs and Corresponding Outputs for Tether Slings on Luna

∆V Density UTS 푀푝 2 2 푀푅푡푝 푀푡 (Mg) L (km) 퐴푡 (푐푚 ) 퐴0 (푐푚 ) (km/s) (kg/푚3) (GPa) (Mg)

5.1521 1550 5.8 193 469 2.2 x 105 1353.37 65.26 2264.93

5 1550 5.8 193 389 1.7 x 105 1274.65 65.26 1842.67

4.5 1550 5.8 193 219 8.3 x 104 1032.46 65.26 976.79

4 1550 5.8 193 131 4.1 x 104 815.77 65.26 553.57

5.1521 1500 5.93 193 377 1.8 x 105 1353.37 63.83 1832.5

5.1521 1340 62 193 3 1.3 x 103 1353.37 6.11 8.13

5.1521 2230 7 193 1256 5.9 x 105 1353.37 54.08 4483.26

Table 7.34 Convert Table 7.33 Data into the Percentage Form.

∆V (%) Density (%) UTS (%) 푀푝 (%) 푀푅푡푝 (%) 푀푡 (%) L (%) 퐴푡 (%) 퐴0 (%)

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3 0 0 0 17 21 6 0 19

13 0 0 0 53 63 24 0 57

22 0 0 0 72 82 40 0 76

0 3 -2 0 20 20 0 2 19

0 14 -969 0 99 99 0 91 100

0 -44 -21 0 -168 -168 0 17 -98 a. A positive percentage represents the decreased amount. b. A negative percentage represents the increased amount.

M.11. Propellant Analysis Suggestions for Improvements

M.11.a. Research Targets

Besides of analyzing the chemical propellant saving, we explore how to make improvements on our current design of the tether sling system to decrease MR푡푝. We decide to research how to lower the MR푡푝 during the flight phases: P to C and L to C, when the chemical propellant is LOX/LH2 for the following reasons.

First, the mass ratios of LOX/LH2 are the highest among the mass ratios of other propellants. After the highest set of MR푡푝 is decreased, the other set of MR푡푝 will also be decreased with the improved tether sling system. Thus, it is reasonable to focus on the chemical propellant which results in the highest MR푡푝. Second, because the highest set of mass ratios happens during

L to C, there is need to research how to decrease the mass ratios. Third, the mass ratios during P

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Purdue University Tether Sling Appendix | 1 9 5 to C and M to P are relatively high. The two sets of mass ratios are very close, so it is fine to choose either set of data as a research target. We decide to choose the flight phase, P to C. Lastly, since there is no need to further decrease the lowest set of mass ratios during C to M, the flight phase, C to M, is not a research target for design improvements.

Overall, it is reasonable to research how to lower the MR푡푝 when the propellant is

LOX/LH2 during L to C and P to C, in order to finally lower the MR푡푝 of all flight phases.

M.11.b. Research Processes

Based on our preliminary designs, for the flight phase: P to C, the ∆V 푃ℎ표푏표푠 푡표 퐶푦푐푙푒푟 is

3.7053 km/s, and the tether material is Dyneema. For the flight phase: L to C, the ∆V 퐿푢푛푎 푡표 퐶푦푐푙푒푟 is 5.1521 km/s, and the tether material is Zylon. We assume the payload or taxi mass is 193 Mg.

Since the payload mass is not a variable during the research, the assumption is effective.

Fig. 68 shows the MR푡푝 during P to C and L to C based on the preliminary designs. The mass ratios shown in Fig. 68 are slightly larger than those shown in Fig. 2 because the data of Fig.

2 is based on the final design which is improved based on the preliminary design. Due to the large mass ratios, the research objects are to reduce MR푡푝 and refine key dimensions for the tether sling.

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57 HTPB 185

64 NTO/Aerozine 50 216

72 LOX/RP-1 248

76 NTO/MMH 264

LOX/LH2 123 Propellant Type Propellant 469

0 100 200 300 400 500 Mass Ratio

Fig. 7.68: Mass Ratio of Tether to Chemical Propellants. The plot shows the mass ratio of tether to five chemical propellants during the flight phases: from Phobos to Cycler and from Luna to

Cycler. In this figure two series of data are plotted, Series1 (Blue) and Series2 (Orange). Series1 represents the flight phase, P to C. Series2 represents the flight phase, L to C.

During our research, the inputs are ∆V of each flight phase, density and ultimate tensile strength of tether material, and a taxi mass. The outputs are MR푡푝, mass of a tether sling, length of a tether sling, and cross section areas at tip and base of a tether. We firstly change the input, ∆V, to observe the percentage changes of the outputs. Then, we change the input, tether material, to observe the percentage changes of the outputs. Finally, by analyzing the outputs of varied inputs, we find the effective methods of decreasing MR푡푝.

M.11.c. Research Results

Table 7.31, Table 7.32, and Table 7.33 show the final research results. The raw data of

Table 7.12 and Table 7.13 are shown in Table 7.31, Table 7.32, Table 7.33, and Table 7.34 in the

Appendix. Table 7.35 presents the percentage change of the outputs when the only varied input is

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∆V. Table 7.36 presents the percentage change of the outputs when the only varied input is tether material.

Table 7.35: Change of Outputs VS. ∆V

∆V Decreased Decreased Decreased Decreased Decreased Decreased

(km/s) ∆V (%) 퐌퐑퐭퐩 (%) 퐌퐭 (%) L (%) 퐀ퟎ (%) 퐀풕 (%)

3.6 3 10 14 6 11 0

3.5 6 18 25 11 20 0

3.2 14 38 50 25 40 0

5 3 17 21 6 19 0

4.5 13 53 63 24 57 0

4 22 72 82 40 76 0 a. The first three rows with numerical values are the data for the flight phase, P to C. b. The last three rows with numerical values are the data for the flight phase, L to C.

According to Table 7.35, the change in ∆V does not affect the cross section area at the tip of the tether because of the zero percentage shown in the last column. We also find that a small decrease in ∆V can result in large decreases in MRtp and key dimensions of the tether for both flight phases: P to C and L to C. For example, a 13 percent of decrement in ∆V for the flight phase,

L to C, leads to a 53 percent of decrement in MRtp, a 63 percent of decrement in Mt, and a 57 percent of decrement in cross section area at the base of the tether. So, one solution is to refine the flight path from one location to another location to decrease the maximum ∆V required for each flight phase to further decrease MRtp and key dimensions of the tether.

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Table 7.36: Change of Outputs VS. Tether Material

Tether Decreased Decreased Decreased Decreased Decreased

Material 퐌퐑퐭퐩 (%) 퐌퐭 (%) 퐀퐭 (%) 퐀ퟎ (%) L (%)

Carbon Nanotube 98 98 95 99 0

Boron -5,325 -5,309 -7 -3,261 0

m-Si -44 -44 52 37 0

Carbon Nanotube 99 99 91 100 0

Carbides 20 20 2 19 0

m-Si -168 -168 17 -98 0 a. The first three rows with numerical values are the data of the tether on Phobos. b. The last three rows with numerical values are the data of the tether on Luna. c. Monocrystalline silicon is abbreviated as m-Si.

Table 7.37: Tether Material VS. 퐌퐑퐭퐩

Tether Material 푀푅푡푝

Dyneema 123

Carbon Nanotube 2

m-Si 177

Boron 6673

Zylon 469

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Carbon Nanotube 3

Carbides 377 m-Si 1256 a. The first four rows with numerical values are the data of the tether on Phobos. b. The last four rows with numerical values are the data of the tether on Luna.

Table 7.36 and Table 7.37 show the research results when tether material is the only variable. For the preliminary design, we choose Dyneema as the tether material for the tether on

Phobos and Zylon as the tether material for the tether on Luna. In Table 7.35, a negative percentage means the increased amount of an output, which is not what we expect. Thus, we will not use boron or monocrystalline silicon (m-Si) as the new tether material for the tether on Phobos. Carbon nanotube can be considered as a new tether material for the tether on Phobos and Luna.

Additionally, m-Si is not a desired tether material for the tether on Luna. Carbides can also be considered as a new tether material for the tether on Luna. Since the outputs are significantly reduced if we use carbon nanotube, the solution is to use carbon nanotube as the new tether material to decrease MRtp and key dimensions of the tether.

For the tether on Phobos and Luna, if we use carbon nanotube as the tether material, all the outputs, except the tether length, are reduced by over 90 percent. However, based on secondary research and the discussion with Steven Lach from the structures team, since there is not a settled and secured method of manufacturing such small ropes of carbon nanotube for our project, currently we can only use Dyneema and Zylon as the tether material.

M.11.d. Suggestions

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One suggestion generated in was to decrease the maximum ∆V required of each flight phase. According to the comparison between the preliminary design and the final design, the maximum ∆V required for each flight phase is indeed decreased slightly. However, due to the limitation of current technology, MRtp can only be slightly reduced. The final MRtp of each flight phase is shown in Fig. 68.

Another suggestion is to use carbon nanotube (CN) as the tether material, although it is currently impractical to use CN, the idea may become feasible after 20 years. If we assume that we are able to use CN which has an extremely high ultimate tensile strength, 62 GPa, the new

MRtp and key dimensions of the tether during four flight phases are shown in Table 7.38. Table

7.38 also shows the MRtp and key dimensions of the tether based on our final design.

Table 7.38: Tether Material VS. 퐌퐑퐭퐩 and Key Dimensions of Tether

Flight Tether 2 2 MRtp Mt (Mg) L (km) At (푐푚 ) A0 (푐푚 ) Phase Material C to M Zylon 92 24,239 670 64.2 372 M to P Dyneema 134 41,307 825 64.2 557 P to C Dyneema 117 31,168 680 111.91 781.96 L to C Zylon 468 217,863 1,353 64.15 2,225 C to M CN 2 593 670 6 6.9 M to P CN 2 746 825 6 7.2 P to C CN 2 602 680 6 6.93 L to C CN 3 1,322 1,353 6 7.99 a. The first four rows are the data based on the final design. b. The last four rows are the data based on the assumption that carbon nanotube is applicable.

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M.12. Phobos Perturbation Investigation (J. Bereson)”.

Phobos Perturbation Investigation:

Tmax, tether: Maximum torque on Phobos exerted by the tether

mtaxi: Mass of the taxi

rtether: Radius of the tether

mtether: Mass of the tether

푑휔 ( ) : Maximum rate of change of angular velocity of the tether 푑푡 푚푎푥

ΔωPhobos, max: Maximum change in angular velocity of Phobos

IPhobos: Moment of inertia of Phobos

Δt: Spin-up or spin-down time of the tether

ωPhobos: Angular velocity of Phobos currently

ωescape: Angular velocity necessary for the surface of Phobos to reach escape velocity

Our tether slings on Mars and Luna do not require significant body perturbation analysis, because Mars and Luna are so large compared to the tethers that they simply act as fixed bodies that the tether is spinning on. However, our tether sling on Phobos is a very different story, because the largest radius of Phobos is in fact much smaller than the length of our Phobos tether. Because of this issue, we investigate the impact that our tether will have on Phobos in order to determine if we will either despin Phobos, or spin it up to the point where matter begins to fly from its surface. In order to perform these calculations we make a few assumptions. Primarily, we assume that the mass of the

Phobos tether is concentrated at a distance r/3 from the center of the tether, and we assume that

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Phobos and the tether are axis-aligned, meaning that their angular momentum vectors are coincident.

We also assume that the maximum rate of change of angular velocity of the tether is equivalent to ten times the average rate of change of the tether throughout a spin-up cycle. Based on these assumptions, the following equation is used to determine the maximum torque exerted on Phobos by the tether.

2 2 푟푡푒푡ℎ푒푟 푑휔 푇푚푎푥, 푡푒푡ℎ푒푟 = (푚푡푎푥푖푟푡푒푡ℎ푒푟 + 푚푡푒푡ℎ푒푟 ( ) ) ( ) (17) 3 푑푡 푚푎푥

From this equation we are able to obtain the maximum torque exerted on Phobos, and we can use this maximum torque to generate an overestimate of the change in angular velocity of Phobos from one spin-up or spin-down cycle of the tether. The equation for the change in angular velocity of Phobos is

푇푚푎푥, 푡푒푡ℎ푒푟 훥휔푝ℎ표푏표푠, 푚푎푥 = 훥푡 (18) 퐼푝ℎ표푏표푠

This equation is a significant overestimate, which allows us to be certain that the effect it yields is larger than the actual effect we will observe. Based on this equation, Table 1 below presents the effect that our tether will have on the rotational motion of Phobos.

ωPhobos (rad/s) ωescape (rad/s) |ΔωPhobos| (rad/s) % change in ωPhobos (%)

-4 -3 -6 1.82*10 2.23*10 7.20*10 3.95

Table 6: Phobos Tether Effect on Phobos’ Rotational Motion Table 6 shows the change in angular velocity of Phobos during a single spin-up or spin-down cycle. For reference, it also presents the angular velocity of Phobos currently, as well as the angular velocity necessary for the surface of Phobos to reach escape velocity. As we show in the table, the change in angular velocity of Phobos during any one spin-up or spin-down cycle is a fairly small percentage of the total angular velocity. Depending on the direction of the spin cycle, the change in

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Purdue University Tether Sling Appendix | 2 0 3 angular velocity could be positive or negative. If a proper configuration of cycle directions is used, the net effect on Phobos’ angular motion should be able to be minimized. Thus, we conclude that the effect of the Phobos tether sling on the motion of Phobos is acceptably small.

Additional Trajectory Information:

Fig. #4: In this figure the trajectory of cycler 2 is plotted (blue line) and the locations of the planetary encounters are added

(red dots). The first planetary encounter is denoted ‘E-1’, and the last planetary encounter is denoted ‘M-26’.

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Fig. #5: In this figure the trajectory of cycler 3 is plotted (blue line) and the locations of the planetary encounters are added

(red dots). The first planetary encounter is denoted ‘E-1’, and the last planetary encounter is denoted ‘M-26’.

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Fig. #6: In this figure the trajectory of cycler 4 is plotted (blue line) and the locations of the planetary encounters are added

(red dots). The first planetary encounter is denoted ‘E-1’, and the last planetary encounter is denoted ‘M-26’.

Burn Time Calculations:

Δteq: Equivalent full power burn time for a single burn

Δtactual: Actual burn time for a single burn p: Burn power on a 0-1 scale where 1 is full power

Δttot: Total equivalent full power burn time between refueling

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훥푡푒푞 = 훥tactual ∗ p (19)

푒푛푐표푢푛푡푒푟 2 훥푡푡표푡 = ∫푒푛푐표푢푛푡푒푟 1 Δ푡푒푞푢𝑖푣푎푙푒푛푡 (20)

References:

[1] Potter, R., Longuski, J., and Saikia, S., “Survey of Low-Thrust, Earth-Mars Cyclers” AAS 19-799,

Portland, ME: 2019.

[2] “Energy from the Sun”, American Chemical Society, retrieved on February 11, 2020. https://www.acs.org/content/acs/en/climatescience/energybalance/energyfromsun.html

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(S. Lach)

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[1] Jokic, M.D., Longuski, J.M., “Design of Tether Sling for Human Transportation Systems

Between Earth and Mars,” Journal of Spacecraft and Rockets, Vol. 41, No. 6, November-

December 2004, pp. 1010-1015

[2] “Zylon®(PBO fiber) Technical Information (2005),” Toyobo Co., LTD., F0739K, 2005

[3] Finckenor, M.M., “Comparison of High-Performance Fiber Materials Properties in Simulated and Actual Space Environments,” NASA/TM-2017-219634, January 2017

[4] Salvetat, J.P., Bonard, J.M., Thomson, N.H., Kulik, A.J., Forró, L., Benoit, W., Zuppiroli, L.,

“Mechanical Properties of Carbon Nanotubes” Applied Physics A Vol. 69, 1999, pp. 255-260

(V. Richard)

[1]: Steven G. Tragesser, Luis G. Baars, "Dynamics and Control of a Tether Sling Stationed on a

Rotating Body", Journal of Guidance and Control, and Dynamics, Vol. 37, No. 1, January-

February 2014

(N. Cruces)

Centripetal Acceleration. (n.d.). Retrieved March 3, 2020, from http://hyperphysics.phy- astr.gsu.edu/hbase/cf.html

The Topography of Mars. (2009, January 12). Retrieved January 2020, from https://www.asc- csa.gc.ca/eng/astronomy/mars/topography.asp

C. Frueh, Space Traffic Management, AAE590 course script, Purdue University, 2019

Hall, N. (Ed.). (2015, May 5). Mars Atmosphere Model - Metric Units. Retrieved January 2020, from https://www.grc.nasa.gov/WWW/K-12/airplane/atmosmrm.html

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NASA: Lunar Reconnaissance Orbiter. (2013). LUNAR RECONNAISSANCE ORBITER:

Detailed Topography of the Moon. Retrieved from https://lunar.gsfc.nasa.gov/images/lithos/LRO_litho8-lunar_topography.pdf

The Topography of Mars. (2009, January 12). Retrieved January 2020, from https://www.asc-

csa.gc.ca/eng/astronomy/mars/topography.asp

Elert, G. (n.d.). Equations of Motion. Retrieved January 2020, from https://physics.info/motion

equations/

Redd, N. T. (2017, December 9). Olympus Mons: Giant Mountain of Mars. Retrieved January

2020, from https://www.space.com/20133-olympus-mons-giant-mountain-of-mars.html

Mars Odyssey. (2019, July 23). Retrieved January 2020, from https://solarsystem.nasa.gov/missions/mars-odyssey/in-depth/

Lunar Reconnaissance Orbiter. (2019, July 11). Retrieved January 2020, from https://solarsystem.nasa.gov/missions/lro/in-depth/

Power. (n.d.). Retrieved January 2020, from https://www.physicsclassroom.com/class/energy/Lesson- 1/Power

Jokic, M. D., & Longuski, J. M. (2004). Design of Tether Sling for Human Transportation

System Between Earth and Mars. Journal of Spacecraft and Rockets, 41(6), 1010–1015. doi: 10.2514/1.2413

(S. Ness)

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[1] Puig-Suari, J., Longuski, J.M., and Tragresser, S.G., “A Tether Sling for Lunar and

Interplanetary Exploration,”

Acta Astronautica, Vol. 36, No. 6, 1995, pp. 291-296.

[2] Jokic, M.D., and Longuski, J.M., “Design of Tether Sling for Human Transportation

Systems Between Earth and

Mars,” Journal of Spacecraft and Rockets, Vol. 41, No. 6, 2004, pp. 1010-1015.

[3] Perez, S.A, “Cardiovascular Effects of Anti-G Suit and Cooling Garment During Space

Shuttle Re-entry and

Landing” 2003

(S. Yang)

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https://monroeengineering.com/info-general-guide-tensile-strength.php

2. Puigsuari, J., Longuski, J., & Tragesser, S. (1995). A tether sling for lunar and

interplanetary exploration. Acta Astronautica, 36(6), 291-295.

3. Rees, D. (2009). Mechanics of optimal structural design minimum weight structures.

Chichester, West Sussex, U.K. ; Hoboken: J. Wiley.

4. Robert, A. (2008). Rocket Propellants. Retrieved February 3, 2020, from

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YA100, January 2010.

(N. DeAngelo)

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[1] Lunar Distance Calculator Available: https://www.lpi.usra.edu/lunar/tools/lunardistancecalc/.

[2] “Mars Fact Sheet,” NASA Available: https://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html.

(V. Bartels)

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[2] Shoykhet et al. (2008, May). Development of Ultra-Efficient Electric Motors . Retrieved February 12, 2020, from https://www.osti.gov/servlets/purl/973932

[3] Mars Atlas: Olympus Mons. (n.d.). Retrieved April 4, 2020, from https://mars.nasa.gov/gallery/atlas/olympus-mons.html

[4] Surampudi, S., et al. (2017). Solar Power Technologies for Future Planetary Science

Missions. Jet Propulsion Laboratory, Pasadena.

[5] Azurspace. (2019). 32% Quadruple Junction GaAs Solar Cell Type: Qj Solar Cell 4G32C -

Advanced. Retrieved from http://www.azurspace.com/images/0005979-01-

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[6] SolAero. (2018). Imm-α Space Solar Cell. Retrieved from https://solaerotech.com/wp- content/uploads/2018/04/IMM-alpha-Preliminary-Datasheet-April-2018-v.1.pdf

[7] Spectrolab. (n.d.). Xte-Lilt (Low Intensity Low Temperature) Space Qualified Triple Junction

Solar Cell. Retrieved from https://www.spectrolab.com/photovoltaics/XTE-LILT Data Sheet

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[8] Carufel, G. de, Li, Z. Q., Crues, E. Z., & Bielski, P. (2016). Lighting Condition Analysis for

Mars’ Moon Phobos. Retrieved February 12, 2020, from https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20150019633.pdf

[9] How HOMER Calculates the PV Cell Temperature. (n.d.). Retrieved March 4, 2020, from https://www.homerenergy.com/products/pro/docs/latest/how_homer_calculates_the_pv_cell_tem perature.html

[10] Hyers, R. W., Tomboulian, B. N., Craven, P. D., & Rogers, J. R. (2012). Lightweight, High-

Temperature Radiator for Space Propulsion. Retrieved from https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20130001608.pdf

[11] Tomboulian, Briana N., "Lightweight, High-Temperature Radiator for In-Space Nuclear-

Electric Power and Propulsion" (2014). Doctoral Dissertations. 247. https://scholarworks.umass.edu/dissertations_2/247

(Co-authored)

[1]: Joseph A. Bonometti, "Boom Rendezvous Alternative Docking Approach", Space 2006

Conference, AIAA 2006-7239, San Jose, CA, September 2006

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15. Electrodynamic Hybrid Tether Sling

Joe Tiberi42, Juliann Mahon43, Brady Walter44, Yashowardhan Gupta45,

Steven Lach46, Valentin Richard47, Adam Wooten48, and Kristen

Fleher49

Purdue University, West Lafayette, Indiana, 47906, United States

42 Propulsion 43 CAD 44 Human Factors 45 Power and Thermal 46 Structures 47 Mission Design 48 Communications 49 Propulsion

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Appendix. Electrodynamic/Tether Sling Hybrid

I. Mission Design Appendix

A. Choosing the orbit

The orbit we use to place the ED tether must be stable in time, allow us to create thrust using the magnetic field and offer regular launch window to Luna. We first considered using an orbit coplanar with the Moon because it would give us the maximum frequency of launch opportunities. The inclination of such orbit varies from 18 to 28° over 19 years long period of time. This ended up being impossible, here are the reasons why. To re-boost the orbit, we need to apply an external force to the Tether that is intensive enough to create a measurable acceleration. We also need that force to be oriented toward the prograde direction, if not it will either decelerate the ED-tether or change the orbital plane. The Lorentz force we use can be computed as follows: 퐹̅ = 퐽̅ × 퐵̅ (1) The cross product means the force is always perpendicular to the magnetic field. If we want prograde thrust, the local magnetic fields lines must be perpendicular to our forward velocity vector. The magnetic field is not constant, and its magnitude/orientation vary around Earth.

퐵 퐵̅ = 0 ∙ (3 푟̂ (푚̂ ∙ 푟̂) − 푚̂) [3] (2) 푟3 With B the local magnetic field and 푟̂ the unit vector going from the earth center to the spacecraft and 푚̂ the magnetic dipole of Earth (unit vector from North pole to South pole). In MATLAB script ‘ED_Thrust.m’ we calculated the angle between the velocity vector of spacecraft travelling around Earth on an orbit inclined of 18°-28° and the local magnetic field. This yielded the following results:

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Figure 1. Velocity and Magnetic field are orthogonal at only two specific position around the orbit (highlighted by red dots). When spacecraft reaches highest / lowest latitude.

We see here there are only two points on the orbit where orthogonality condition between V and B is met. Electromagnetic thrust is by definition very low in magnitude and the ED-tether is massive. Re-boost maneuver will be spread over long period of time, we cannot consider doing impulsive maneuvers… since the only locations where prograde thrust is possible are two discrete points rather than portion of the orbit, low-thrust, electromagnetic maneuvers will not give us satisfaction on an inclined orbit.. They will create an out of plane force that will disrupt the orbital plane. Stability is our other concern: if we use an orbit coplanar with the Moon, we want it to stay this way over a long period of time. The focus here is J2 perturbation. The oblateness of the Earth create a component in the gravity force that is non-radial and out of the orbital plane. This small disturbance tends to make inclined orbits’ plane rotate around the rotation axis of Earth. We called these orbit “walking orbits”. The Moon experiences such motion, called nodal precession. Its orbital plane rotates westward at a pace of a 19 degrees per year [1]. For our orbit to remain coplanar with the Moon, we want it to have the same rate of Nodal Precession[2] :

2 푑훺 3 푅퐸 휇 = − ∙ 2 ∙ 퐽2 ∙ √ ∙ cos(푖) (3) 푑푡 2 (푎(1−푒2)) 푎3

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Using an 18-28° interval for i, a radius of 7368 km (1000 km altitude) and a value of J2 -3 [2] equal to 1.082645 x 10 we get a nodal precession rate of 5.3-5.6° per day. This rate is far greater than the Moon’s natural nodal precession rate. If we place the Tether on such orbit, it will drift in a few hours if it is no corrective maneuver is performed. Plane change maneuvers require thrust in a direction normal to the orbital plane, meaning we would need the magnetic field line to be in the orbital plane, which is not the case (they are roughly perpendicular to it). Additionally, the inclination of the Moon relative to the equator changes over time because of the Nodal Precession of the natural satellite. This inclination ranges from 18 to 28° with a period of 19 years. (Complex motion of the moon depicted in Fig. 4 below) Plane correction and station keeping maneuver will therefore be very complicated using the ED-Thrust. Especially because it’s a low-thrust solution. Low thrust maneuvers stretch over long periods of time and their effects are very hard to predict, especially when thrusting out of plane. The analysis needed to design such maneuvering process is out of the scope of this project, we will therefore not consider using this inclined orbit. We will instead use a Low circular orbit in the equatorial plane (inclination is zero). There will be no Nodal Precession at this inclination and the magnetic field will be almost perfectly perpendicular to the plane.

Nodes

Figure 2: The moon's orbit undergoes inclination oscillations (yellow arrow) and Nodal Precession (red arrow). The orbit of the tether would have to follow the same pattern in order to stay coplanar.

Transfers windows to Luna will not be open at all time but they will appear every time the moon crosses the equator’s plane. In Fig. 2 above, these locations are called “Nodes”. Luna passes there every half period, meaning every 13.5 days, which is very reasonable considering the period of cycler arrivals and departures are longer than a year.

MATLAB script used for this section: - ‘ED_thrust.m’

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B. Orbit Degradation

Every time the Tether launches a taxi, it exchanges momentum with it. Momentum conservation is a fundamental of mechanics and tells us what follows. In the absence of external forces, a systems’ momentum is conserved. Here, our system is {EDtether + Taxi}. We assume that the motion of both elements is rectilinear. Taxi has a mass mtaxi, a velocity Vtaxi1 before docking with the tether and a velocity Vtaxi2 after launch (after the acceleration). Tether has a mass mtether, a velocity Vtether2 before and Vtether2 after launch. Conservation of momentum can be written:

푚푡푎푥𝑖푉푡푎푥𝑖1 + 푚푡푒푡ℎ푒푟푉푡푒푡ℎ푒푟1 = 푚푡푎푥𝑖푉푡푎푥𝑖2 + 푚푡푒푡ℎ푒푟푉푡푒푡ℎ푒푟2 (4)

푚푡푎푥𝑖 푉푡푒푡ℎ푒푟2 − 푉푡푒푡ℎ푒푟1 = (푉푡푎푥𝑖2 − 푉푡푎푥𝑖2) (4a) 푚푡푒푡ℎ푒푟

∗ ∆푉푡푒푡ℎ푒푟 = 푚 × ∆푉푡푎푥𝑖 (4b)

푚 with 푚∗ = 푡푎푥𝑖 (5) 푚푡푒푡ℎ푒푟

The velocity lost by the tether is directly linked to the velocity gained by the taxi and the mass ratio. The heavier the Tether, the less speed it is going to lose. From this DeltaV calculation, we can derive the expression for the altitude the tether will lose. In orbit, losing velocity means losing kinetic energy and translate into an altitude loss. The relationship between velocity and orbital radius is:

2 1 푉2 = 휇 ( − ) V: instantaneous velocity (6) 푟 푎 μ: gravitational we isolate semi major axis a: parameter r: instantaneous radius 휇푟 푎 = a: semi major axis (6a) 2휇−푟푉2 rp: radius of perigee ra: radius of apogee and 푟푝 = 2푎 − 푟푎 (푐) (7)

Right after the launch, the tether still has the same orbital radius as initially, but its velocity decreased from Vtether1 to 푉푡푒푡ℎ푒푟1 − ∆푉푡푒푡ℎ푒푟. We can compute the new perigee radius of the tether’s orbit by plugging “r” and the new velocity value into the equation. We then can plot the perigee radius (or altitude) for different taxi DeltaV and different mass ratio.

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Sea Level

Figure 3. The Perigee of the Tether for mass ratios greater than 1/10 drop below the ground. The system will need to be massive or fly higher than 1000 km to stay in orbit.

Because increasing the orbit radius is not a viable option (see Appenxi I.A) our only way of mitigating the perigee drop is by adding mass. We first decided the momentum bank (which role is to store angular momentum) could serve as a linear momentum bank too. We sized the mass of the momentum bank so that the sum of {tether mass + momentum bank mass} is enough to maintain a reasonable altitude, even with the high DeltaV the taxi leg to the moon requires. To do so we defined arbitrarily a maximum altitude loss ∆ℎ. Using eqn. (7) again we derive the maximum DeltaV acceptable to respect this altitude constraint:

2 2 푉푡푒푡ℎ푒푟2 = √휇 ( − ) (8) 푟푡푒푡ℎ푒푟 푟푡푒푡ℎ푒푟+(푟푡푒푡ℎ푒푟− ∆ℎ)

2 2 Then comes: ∆푉푡푒푡ℎ푒푟 = √휇 ( − ) − 푉푡푒푡ℎ푒푟1 (8a) 푟푡푒푡ℎ푒푟 푟푡푒푡ℎ푒푟+(푟푡푒푡ℎ푒푟− ∆ℎ)

We now introduce the mass of the momentum bank in the equation using (5b) & (6).

2 2 푚푡푎푥𝑖 ∆푉푡푒푡ℎ푒푟 = √휇 ( − ) − 푉푡푒푡ℎ푒푟1 = × ∆푉푡푎푥𝑖 푟푡푒푡ℎ푒푟 푟푡푒푡ℎ푒푟+(푟푡푒푡ℎ푒푟− ∆ℎ) 푚푡푒푡ℎ푒푟

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푚 ∙ ∆푉 Which yields: 푚 = 푡푎푥𝑖 푡푎푥𝑖 – 푚 (9) 푚표푚푒푛푡푢푚 2 2 푡푒푡ℎ푒푟 √휇( − )−푉푡푒푡ℎ푒푟1 푟푡푒푡ℎ푒푟 푟푡푒푡ℎ푒푟+(푟푡푒푡ℎ푒푟− ∆ℎ)

The momentum bank mass required to limit the perigee drop to ∆ℎ = 100 푘푚 for a high DeltaV requirement (6.7 km/s – See Appendix I.C) was calculated to be 30,000 Mg. However, later structural analysis ended up returning an overall system mass in the tens of thousands of tons, mainly due to the torque arm and the hub mass. The taxi being only 190 tons, this gave Mission -3 Design a mass ratio of 1.7 x 10 and substantially reduced the perigee drop. Because of these new system mass, momentum bank mass is no longer tied to altitude loss requirements. The final mission profile shows a difference in altitude between initial and final orbit of less than 50 km (see Appendix I.D).

MATLAB script used for this section: - ‘EDtether_PerigeeDrop.m’: (simple analysis of ED-tether perigee drop for various mass

ratios and deltaVs)

- ‘ED_PerigeeDrop_MomBank’: (analysis with tether’s parameters and momentum bank

mass calculations, not used in final design)

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C. Calculating Taxi DeltaV and Tether Length

The tether communicates additional velocity to the Taxi by swinging it around like a sling and releasing it in the forward direction. The taxi must reach the Moon (far away) therefore, it will have to leave the Tether with a very high speed. The relationship for tip velocity Vtip of the tether is:

푉푡𝑖푝 = 푉푡푒푡ℎ푒푟 + 휔푡푒푡ℎ푒푟 × 퐿푡푒푡ℎ푒푟 (10)

Tip velocity (or taxi velocity at ejection) increases with either the length of tether Ltether or its angular velocity ωtether. If we want our taxi to fly fast, we need the tether to be very long or to spin at a fast rate. Swinging an object around at the tip of a tether or a sling creates a centrifugal acceleration. This centrifugal acceleration is defined as:

2 푎푐 = 휔푡푒푡ℎ푒푟 × 퐿푡푒푡ℎ푒푟 (11)

This acceleration can be harmful to humans if it is too intense, sustained for too long or in a wrong direction. (See HUMAN FACTOR). Since it increases in intensity with the square of angular rate, we will have to increase the length of the tether to keep this centrifugal acceleration within a safe range. By injecting (11) in (10) and isolating L we get:

2 (푉푡𝑖푝− 푉푡푒푡ℎ푒푟)^2 (푉푡푎푥𝑖−푉푡푒푡ℎ푒푟) 퐿푡푒푡ℎ푒푟 = = (12) 푎푐 푎푐

The length of the tether is now defined as a function of taxi velocity (at release), tether orbital velocity, and centrifugal acceleration. We know from Human Factor that an acceleration of 2g is only sustainable for less than an hour. If the acceleration of the taxi takes less than an hour, we can tolerate higher acceleration. For now, we will follow a conservative approach and use 2g’s as a limit (twice the Earth’s pull on the human body). To compute the length Ltether, we still need both velocities Vtaxi and Vtether. The taxi velocity is driven by its journey to the Moon. Using orbital equations of motion, we can compute the minimum velocity required in Low Earth Orbit to travel to the Moon within a desired time interval. The twist is, the moon is at a very high altitude, to get there from LEO one must accelerate up to a velocity very close to Earth’s Escape Velocity (velocity for which you leave Earth orbit and end up into interplanetary space). This means we must use two different mathematical model to compute a trajectory to the Moon. For the lowest end of the velocity range (longer time of flights) we will use the elliptical model, for the highest end of the velocity range (shorter time of flights) we will use the hyperbolic model.

푎3 1−푟 Elliptical model: 푇푂퐹 = 푡 − 푡 = √ (푢 − 푝 푠푖푛(푢)) (13) 0 휇 푎

푎−푟 with 푢 = acos ( ) (13a) r: radius of spacecraft position 푎−푟푝 u: eccentrical anomaly of S/C position 2 1 a: semi major axis of transfer orbit 푉푝 = √휇 ( − ) (14) 푟푝 푎 rp: perigee radius of transfer orbit 219

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푎3 푟 Hyperbolic model: 푇푂퐹 = 푡 − 푡 = √ [( 푝 + 1) sinh(푢) − 푢] (15) 0 휇 푎

푎+푟 with 푢 = acos ( ) (15a) 푟푝+푎

푟푝 ( +2)휇 푎 푉푝 = √ (16) 푟푝

Using a range of value for semi major axis “a” in both models, we can compute an array containing different TOF and their related perigee velocity Vp. These calculations require a value for perigee radius “rp”, which is the radius at which the taxi will start its journey to the Moon. It is also the radius at which it will decouple from the tether. This radius is yet to be defined. To do so we considered two possibilities:

1) Either the tether swings down when it releases the taxi. In this case rp is shorter than the

tether orbital radius. Let’s set it to 400 km in this case (above the atmosphere, around the

ISS altitude). We will call this sequence Low Swing.

2) Or the tether swings up when it releases the taxi. In this case rp is greater than the tether

orbital radius. Let’s set it to 1500 km in this case. We know from back-of-the-envelope

calculations that the tether’s length will be 100’s of kilometers. Therefore, if the tether

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spins below the taxi’s release altitude, this altitude will need to be around 1500 km. We

will call this sequence High Swing.

Figure 4. In the sequence shown on the left, the tether flies below the taxi's release altitude (High Swing). On the right it is the opposite (Low Swing)

With these taxi’s perigee altitude, we know have everything we need to compute the taxi’s velocity at release, or Vp. We will do so for both sequence (Low and High Swing) and compare. D. Tether Length

We saw earlier that tether length is a function of Vtether and Vtaxi. We have Vtaxi, we need Vtether . In orbit, velocity is a function of altitude. We also know that tether altitude is a function of tether length and taxi altitude (see figure 6 above):

푟푡푒푡ℎ푒푟 = 푟푡푎푥𝑖 ± 퐿푡푒푡ℎ푒푟 (17)

휇 휇 Which leads to: 푉푡푒푡ℎ푒푟 = √ = √ (18) 푟푡푒푡ℎ푒푟 푟푡푎푥𝑖±퐿

We will derive the equations for the High Swing sequence with 푟푡푒푡ℎ푒푟 = 푟푡푎푥𝑖 − 퐿푡푒푡ℎ푒푟

2 휇 2 (푉푝−√ ) (푉푡푎푥𝑖−푉푡푒푡ℎ푒푟) 푟푡푎푥𝑖 – 퐿 Back to (12): 퐿푡푒푡ℎ푒푟 = = (12a) 푎푐 푎푐

Straight forward calculations lead to a 4th order polynomial equation with L as the unknown:

2 4 2 3 2 2 2 푎푐 퐿 + 2푎푐 (푉푝 − 푎푐 푟푝 )퐿 + [(푉푝 + 푎푐 푟푝 ) + 2푎푐 (푉푝 푟푝 + 휇)]퐿 +

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2 2 2 2 2 2 [4푉푝 휇 − 2(푉푝 + 푎푐 푟푝)(푉푝 푟푝 + 휇)]퐿 + (푉푝 푟푝 + 휇) − 4푉푝 휇 푟푝 = 0 (19)

We will use an iterative script on MATLAB to solve for L instead of trying to solve this equation analytically. (→ script: ‘EDtether_High_vs_Low_sequence.m’) The script works as follow: first we set taxi altitude, we compute the (TOF / Perigee velocity) range to Luna from this altitude. We then arbitrarily set an initial length for the tether (500 km), compute the tether altitude with equation (17), the tether velocity with (18) and we check the constraint equation (12). If length does not respect the centrifugal acceleration constraint, we adjust the length value and reiterate. This process is repeated until length converges and is applied to both High Swing and Low Swing sequence. The results are shown in figure 7 below:

The High Swing kinematics yields to a significantly shorter tether than the low swing profile does. For a single-day transfer to the moon, a High Swing sequence requires 570 km of tether while a Low Swing tether would stretch up to 950 km. That is why we use High Swing in our design, with a 570 km tether.

Figure 5. Tether for a 'High Swing' sequence is significantly shorter than for the 'Low Swing'. For a single day ToF, tether should be 570 km

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E. Final Mission Profile

We went through all individual analysis (taxi velocity, tether length, orbit degradation, orbit choice) in the previous Appendices (A to C). Now we can use design parameters from other team members to get an overview of the whole mission. Structural analysis (S.Lachs, Structure) gave us mass for all the ED tether components (Tether, Hub and Torque Arm). Work from Taxi Team gave us a final mass for the taxi. Derivations by (J.Tiberi) gave us the position of the system Center of mass {EDtether + Taxi} along the tether. Propulsion analysis by (K.Fleher) proved we can re-boost the orbit between two taxi launches. The shift of the center of mass reduces the rotational radius of the tether, meaning we need to spin in faster to compensate. The centrifugal acceleration increases and now exceeds the 2g’s limit. This is no threat because since the taxi is ejected in less than one tether rotation, the crew won’t experience the acceleration for more than minutes. We could increase the length beyond 570 km, but this would bring more issues than it would solve Consequently, we will increase angular rate (and acceleration) instead of length. The last piece of code (‘EDtether_MissionProfile_Final.m’) compiles in its first section (I) all previous result to compute key variables that parametrize the taxis and tether Motion. The second section (II) computes a history of the tether and taxi positions in orbit around launch time. It then generates a GIF file with an animated rendering of the RendezVous and Launch of a taxi. Equations used in this script are the same as the ones shown or derived in previous Appendices (A, B, and C). To be sure the ED-Tether integrates smoothly with its counterpart on the lunar North pole, we propagate a transfer trajectory on GMAT using the orbital parameters of the taxi trajectory returned by the MATLAB script. Time of departure (Epoch) and ejection angle (Argument of Perigee) are adjusted to give a nice transfer arc with Lunar fly by above the Northern Hemisphere.

Taxi 1

Taxi 2

Figure 6. Taxi 1 and 2 are launched 13 days and encounter the Moon with a polar fly by without any corrective maneuver. We can fly from the ED-to the Lunar Tether Sling propellant-less.

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From this GMAT propagation we can extract the velocity history of the Taxi when it flies by the Moon. Maximum velocity relative to Luna is reached at Lunar Perigee and is 4.65km/s. This value is within the scope of what the Lunar tether sling can accept for an incoming payload (according to Lunar Sling Tether team). The table that follow lists all parameters of interests for the Mission Profile.

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Table 1 Final Mission Profile Parameters

Parameter Value Units Tether total radius 570 Km Tether rotational radius 544 Km Tether initial altitude 956 Km Taxi 1 release altitude 1500 Km Taxi 1 release velocity 10.73 Km/s Taxi 1 rendezvous altitude 412 Km Taxi 1 rendezvous velocity 4 Km/s Tether 1st altitude drop 45 Km Taxi 2 release altitude 1456 Km Taxi 2 release velocity 10.75 Km/s Taxi 2 rendezvous altitude 367 Km Taxi 2 rendezvous velocity 4.1 Km/s Tether 2nd altitude drop 45 Km Tether initial orbit (before 1st launch) 956 x 956 Km Tether intermediate orbit (before 2nd launch) 956 x 911 Km Tether Final orbit (before Re-Boost) 911 x 912 Km Taxi Lunar flyby maximum velocity 4.65 Km/s

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Bibliography :

[1] Moon Nodal Precession

US Department of Commerce, « Tidal Datums And Applications”. NOAA Special Publications NOS CO-OPS1, Silver Spring, Maryland June 2000 (page 9) https://tidesandcurrents.noaa.gov/publications/tidal_datums_and_their_applications.pdf [2]Orbital mechanics equations and constants E. Bourgeois, D.A. Handschuch, Orbital Mechanics Lecture Notes 2019-2020, ESTACA Engineering School [3] Magnetic Field Model Longuski, James Michael and Muterthies, Michael James. Modeling and Analysis of the Electrodynamic Tether Sling. AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, Minnesota, 2012.

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II. Propulsion Elements - Tether

A. Equation Derivations

Figure 7. Spin-up velocity profile equation.

Figure 8. Spin-up Power requirement equation.

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B. Plots

Figure 8. Spin-up procedure velocity profile.

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Figure 9. Spin-up power requirements

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Figure 10. Spin-up time to achieve 3.35 km/s dependent on torque arm length

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Figure 11. Spin-up power to achieve 3.35 km/s dependent on torque arm length.

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III: Propulsion Elements - Electrodynamic Boost

The following equations are used to find the force from the electrodynamic wires:

퐹⃗ = 퐼퐿 × 퐵 (20)

푅 3 퐵 = 퐵 ∗ ( 퐸) (21) 0 푟

−5 퐵0 = 3.12 × 10 푇 (22)

The following values are found in the analysis:

Table 2 Hybrid Tether Values

Variable Value Time of Boost 5.1 days Current in Wire 50 Amps Collective Length of Wire 5.704 km Power Required 514.6 MW

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IV: Control Appendix

Matlab Code edTetherCalcs.m edtorque.m

Simulink Models

EDTetherSim.mdl

Model:

Figure 12. Simulink model of the tether rotation. The Simulink model simulates the rotation of

the tether subject to gravity gradient torque. It can be used to find motor torque and motor

power consumption.

The Simulink model above simulates the movement of the tether under the variable torque model. It follows the equation of motion given in equation 23, below. The gain value is

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Purdue University Brady Walter ED Tether Appendix | 234 simply 1/I, where I is the moment of inertia about the center of mass. The model demonstrates how the angle is used to determine gravity gradient torque, as well as the torque from the motor.

It also outputs motor torque exerted and the power, which is a function of torque and angular velocity. No controller is used in this mock-up since we did not conduct a quantitative analysis on perturbing torques, and as such it would not have any effect on the response of the tether rotation.

The tether angular equation of motion including gravity gradient is given below, in equation 23.

3 휇 퐼훼 − 퐼sin(2휃) = 푇 (23) 2 푟3 푚표푡표푟

Angular acceleration is calculated using the desired spin up time and angular rate.

Knowing this, motor torque becomes a very clear expression. In the spin maintenance phase, I is adjusted to include the taxi at the end of the tether, and 훼 is set to 0. The Simulink model is most helpful in determining torque for spin up, as this becomes a differential equation. The basic equation used to calculate power is given in equation 24.

푃 = 휔 ∗ 푇푚표푡표푟 (24)

System energy was calculated by integrating the power over the duration of the system spin. Sources: Longuski, James Michael and Muterthies, Michael James. Modeling and Analysis of the Electrodynamic Tether Sling. AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, Minnesota, 2012.

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V: Power System Appendix A. Comparisons

High Power system

Generation Transmission Storage

Eclipse Time Power Loss Heating Effects Considerations

Current and Size and Mass Size and Mass Voltage Constraints Constraints Limitations

Efficiency Heating Effects Constraints

Figure 13. Power system constraints. This diagram shows how restraints on the power system

affect each other.

Table 3 Comparison of transmission lines

Power Voltage Current (A) Examples (MW) (kV) 20704.2 24 14 AWG Most 14614.7 34 12 AWG Common in 9555.8 52 10 AWG Households 1806.9 275 3/0' Heavy Duty 1325.1 375 4/0' Wires 100 4969.0 496.9 500 993.8 HV Transmission (High Voltage) 800 621.1 1,000 496.9 China 1,100 451.7 (Maximum) with 12 GW UHVDC Power 1,400 354.9

Table 4 Solar Array Comparison

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Solar Power Solar Power Solar Array Transmission Array Mass Required Array Produced Type Efficiency Size (Mg) (MW) Efficiency (MW) (km2)

IMM-α Space 32% 1686.4 1.233 604.1 Solar Cell

MonoCrystalline 496.9 20% 92.08% 2698.2 1.973 966.5 Silicon

PolyCrystalline 16% 3372.7 2.466 1208.2 Silicon

B. Code

1. S_Irradiance_ED.m

This code calculates the solar irradiance at an orbit of 955 km from the Earth’s Surface. It

uses the Stefan Boltzmann blackbody radiation equation to solve for the Solar Intensity.

(25)

This is then used to calculate at a distance from the Sun by a ratio condition.

(26)

By using this and orbit parameters of Earth and Sun, we calculate a mean irradiance.

2. Power_ED.m

This code calculates:

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i. Power Required

ii. Power Supplied by Arrays

iii. Power Supplied by Batteries

iv. Sizing of Arrays

v. Sizing of Batteries

vi. Total power supplied and required

Majority of the equations used are simple Volume, Energy, Power, Mass calculations.

C. References

1. MATLAB CODE, total power

2. Ost, I., “Space-Based Solar vs. Conventional Solar – How are They Different?,” Solar

Learning Center, July 2018.

https://www.solar.com/learn/space-based-solar-vs-conventional-solar-how-are-they-

different/

3. “IMM- Space Solar Cell”, SolAero Technologies Corp., April 2018

4. Kasangala, F.M., and Atkinson-Hope, G., “Losses and costs associated with HVDC and

UHVDC transmission lines”, EE Publishers, published online 10 Sept. 2018.

https://www.ee.co.za/article/losses-costs-associated-hvdc-uhvdc-transmission-lines.html

5. Daware, K., “HVDC vs HVAC transmission,” ElectricalEasy, retrieved on 3 April 2020.

https://www.electricaleasy.com/2016/02/hvdc-vs-hvac.html

6. “Lithium-Ion Battery”, University of Washington, retrieved on 3 April 2020.

https://www.cei.washington.edu/education/science-of-solar/battery-technology/

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7. Mikhaylik, Y. , Kovalev, I., Scordilis-Kelley, C., Liao, L., Laramie, M., Schoop, U., and

Kelley, T., “650 Wh/kg, 1400 Wh/L Rechargable Batteries for New Era of Electrified

Mobility”, NASA Aerospace Battery Workshop, 2018

8. “How Hubble got its wings”, European Space Agency, 13 Dec. 2010.

http://www.esa.int/Enabling_Support/Space_Engineering_Technology/How_Hubble_got

_its_wings

9. “AWG Wire Gauges Current Ratings,” Engineering ToolBox, 2003

https://www.engineeringtoolbox.com/wire-gauges-d_419.

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VI: Thermal System Appendix A. Code:

Thermal_ed.m

This code uses Liquid Nitrogen and calculates the mass required for the entire system on board the hub.

B. References:

1. “Technical Brochure: Dyneema® in marine and industrial applications.” , Dyneema

DSM LLC, retrieved on 3 April 2020

http://www.pelicanrope.com/pdfs/DyneemaSK75_Tech_Sheet.pdf

2. “Atmosphere Properties”, Rocket & Space Technology, retrieved on April 3 2020.

http://www.braeunig.us/space/atmos.htm

3. Gilmore, D.G., “ Spacecraft Thermal Control Handbook Volume I”, The Aerospace

Press, California, 2002

4. “Melting point of Metals and Alloys”, All Metals & Forge Group, retrieved on April 3

2020

https://www.steelforge.com/literature/metal-melting-ranges/

5. Johnson, D., “Heliostat Design Concepts”, 14 July 2017

http://www.solarmirror.com/fom/fom-serve/cache/43.html

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VII: Structural Analysis Appendix

A. Hybrid Tether Material Selection Trade Study

Proceeding under the spin up design, we conduct a trade study on possible tether materials. We follow the same procedure as the tether sling for this trade study. The material properties accounting for environmental effects are shown in Table X1.

Table 5 Material properties accounting for environmental effects

Material Dyneema Zylon Kevlar Hexcel IM7 Ultimate Tensile 3.325 2.03 2.044 4.338 Strength, GPa Density, 970 1560 1450 1550 kg/m3

The exact environmental effects are covered in the tether sling section. In summary,

Zylon had the largest strength reduction followed by Kevlar, Hexcel IM7 and then Dyneema. We calculate the tether length, mass, and cross-sectional area at the base and the tip with the updated material properties. These characteristics are displayed in Table X2.

Table 6 Comparative tether characteristics using material properties accounting for environmental effects.

Material Zylon Dyneema Kevlar Hexcel IM7 Length, km 573 573 573 573 Mass, kg 520000 20600 360000 34000 Cross-Sectional 14000 580 9800 638 Area at the Base, cm2 Cross-Sectional 184 110 182 86 Area at the Tip, cm2

Based on the results shown in this table, it is clear Dyneema fiber holds up the best in

Ultraviolet (UV) and visible light. Dyneema fiber is also abrasion resistant which helps reduce

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Purdue University Steven Lach, Adam Brewer ED Tether Appendix | 241 the wear on the fibers under loading. While the atomic oxygen degradation is concerning and needs some level of protection in this environment, we determine that atomic oxygen protection is beyond the scope of this feasibility study. It is important to note that Zylon fiber performs better than Dyneema in mass and cross-sectional area if the environmental effects were neglected due to its much higher UTS. If the environmental effects are eliminated, Zylon makes the most sense, however, it has a much lower abrasion resistance than Dyeema.[1]

We rule out a conducting tether that spins up using the Earth’s magnetic field because the mass required to achieve the velocity we need is too high. We calculate the estimated mass of a tether made from aluminum, for conducting purposes, to be on the order of 1030 Mg.

We also consider carbon nanotubes as a material since they are much stronger than the materials we are currently comparing. The ultimate tensile strength has been measured in very small-scale tests to be over 20 GPa. At this strength, the Luna tether sling would be just 6800

Mg. However, we rule carbon nanotubes out because this strength is theoretical when we scale up to the tether size. For large fibers, the ultimate tensile strength is approximately 1 GPa because it is difficult to manufacture carbon nanotubes at a large scale without having large defects.[2] We can reconsider carbon nanotubes for tether application once the large scale manufacturing of carbon nanotubes advances to a level that consistently delivers high ultimate tensile strengths.

B. Electrodynamic Tether Characteristics

1. Length

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Just like the tether slings, we can find the required length of each tether with just the maximum tip velocity and the maximum acceleration at the tip of the tether. The length is represented as

2 푙 = 푣푚푎푥/푎푚푎푥 (27)

We observe that the length is solely on velocity and acceleration and not the payload mass.

Although we apply a safety factor of 10 to the tether calculations, the length is unaffected because it is not reliant on the ultimate tensile strength. We calculate the length of the electrodynamic tether to be 574 km long. While, this is a shorter length than any of the tether slings, the electrodynamic tether will be orbiting at 980 km so the tip of the tether will reach as low as 406 km above the Earth’s surface. The calculation of vmax is further broken down in

Appendix C.

2. Cross-Sectional Area

The electrodynamic tether is also tapered to match the cross-sectional area with the tension along the length of the tether like the tether slings. A tapered tether minimizes the mass of the tether by ensuring that the cross-sectional area always remains at an optimal level.[3] The cross-sectional area at the tether tip is represented as

푣2 퐴 = 푚 푚푎푥 (28) 푡푡 푇푎푥𝑖 𝜎푙

Whereas the cross-sectional for any given point is represented as

2 2 𝜌푣푚푎푥 푥 ( )(1− 2 ) 퐴푥 = 퐴푡푡푒 2𝜎 푙 (29)

We see that the Ax equation adds the extra exponential term to account for the force from the mass of the tether itself. More analysis on how input values affect cross-sectional area is included in Appendix 2X. The cross-sectional area at the base of the electrodynamic tether is 576 cm2 and the cross-sectional area at the tip of the tether is 112 cm2.

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C. Cross-sectional Area Input Effects

The cross-sectional area at tether tip relies on the taxi mass, maximum acceleration, ultimate tensile strength. The maximum velocity and length terms partially cancel out, leaving us with the maximum acceleration. Decreasing the taxi mass is the easiest way to decrease the cross-sectional area since decreasing it does not negatively affect any other parameters. The ultimate tensile strength is fixed for the most part depending on what material is chosen for the tether in the first place.

The cross-sectional area at the base of the tether relies on the taxi mass, maximum acceleration and ultimate tensile strength as well, but it also adds the density of the tether material and the maximum velocity at the tether tip. Like the cross-section at the tip, the cross- section at the base is decreased by decreasing the taxi mass, but it decreases at faster rate when decreasing the maximum velocity at the tether tip. Decreasing the velocity at the tether tip also provides favorable results for the other tether parameters. The material density is relatively fixed along with the ultimate tensile strength.

3. Mass

The mass of the tether is calculated with the tether material properties as well as the maximum velocity at the tether tip in the same manner as the tether slings. The tether mass is represented as

𝜌푣2 𝜌 ( 푚푎푥) 𝜌 푚 = 푚 푣 √ 휋푒 2𝜎 erf (푣 √ ) (30) 푇푒푡ℎ푒푟 푇푎푥𝑖 푚푎푥 2𝜎 푚푎푥 2𝜎

We note that the safety factor must be applied to the tether mass after calculating the mass by multiplying the mass calculated by the safety factor. By including the safety factor after calculating the mass, we are effectively calculating the mass for one tether and adding more tethers to the mass to achieve the safety factor. We apply the safety factor after mass calculation

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Purdue University Steven Lach, Adam Brewer ED Tether Appendix | 244 because the error function (erf) will calculate incorrectly if the safety factor is applied directly to the ultimate tensile strength. We discuss further input impacts in Appendix D.

We calculate the tether mass to be 20,600 Mg. The mass calculations follow the same pattern as the previous length and cross-sectional area calculations because the maximum tether tip velocity is the most impactful input. The masses included in the table above are for one tether only. Each location will have three separate tethers attached to the hub. We multiply each of the masses by three to get the total mass of the tethers at each location.

D. Tether Mass Input Effects

The tether mass relies on the taxi mass, maximum velocity at the tether tip, density of the material, and ultimate tensile strength. The density and the ultimate tensile strength are relatively fixed as discussed in Appendix A. The relationship of the tether mass to the taxi mass and maximum velocity at the tether tip is shown in Fig. 14.

Fig 14: A comparison of the relationship of the tether mass to the maximum tether tip velocity and the taxi mass. As taxi mass and tether tip velocity increase, tether mass also increases. As we see, the maximum tether tip velocity affects the tether mass more than the taxi mass does and this is especially pronounced above 5 km/s. Since the electrodynamic tether is

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Purdue University Steven Lach, Adam Brewer ED Tether Appendix | 245 operating at just over 3.5 km/s, the mass of the tether is comparatively small compared to the

Luna and Mars tethers.

E. Tether Sling Torque Arm Length Tradeoff Study

The torque arm is the component that allows the tether to spin. If the tether were attached to the center of rotation, it would just wind itself around the rod that it is attached to. The torque arm is designed to transfer the torque generated at the center of rotation to the tether itself, causing the tether to spin around. There is a tradeoff between the spin up time and the power required to spin the tether up.

The relationships between the torque arm length and the spin up time as well as the torque arm length and the power required for the electrodynamic tether are shown in Fig. X2.

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Figure 15. Torque arm length tradeoff between spin up time and power required for the electrodynamic tether. As the torque arm length increases, spin-up time decreases, but power increases. All the torque arm lengths in the plot are viable from the spin up time and power required standpoint. We do not have to worry as much about the gravitational forces that would limit the size of the torque arm and hub in low Earth orbit (LEO). We choose to test values close to 1000 m because that would reduce the spin up time. We must consider that large quantities of power is more difficult to produce in LEO than on a celestial body because of the limited surface area that solar panels can be placed on.

References [1] “Zylon®(PBO fiber) Technical Information (2005),” Toyobo Co., LTD., F0739K, 2005 [2] Salvetat, J.P., Bonard, J.M., Thomson, N.H., Kulik, A.J., Forró, L., Benoit, W., Zuppiroli, L., “Mechanical Properties of Carbon Nanotubes” Applied Physics A Vol. 69, 1999, pp. 255-260 [3] Jokic, M.D., Longuski, J.M., “Design of Tether Sling for Human Transportation Systems Between Earth and Mars,” Journal of Spacecraft and Rockets, Vol. 41, No. 6, November- December 2004, pp. 1010-1015

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VIII. Docking Mechanism Appendix

A. Mechanism Selection

After defining the guidelines, we came up with a list of mechanism ideas based on space applications but also other that proven their efficiency on Earth. We will quickly go through each of them and discuss their strengths and weaknesses relative to our application, qualitatively.

❖ Clamp

Strengths: Adapted to withstand relatively high radial forces. Needs little to no mechanism on the taxi apart from structural components.

Weaknesses: Requires a certain amount of precision. Needs structural improvements to withstand tangential forces during deceleration.

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❖ Snap - fit

Strengths: Once docked, the system is tightly locked. It is also tolerant to high compressive forces and has little to no backlash.

Weaknesses: Requires a high precision and impact force. Important internal constraints due to the absence of backlash. Not applicable for departures.

❖ Hook

Strengths: Once locked, the mechanism behaves like a “ball-joint” so it is really tolerant to change in direction of constraints. The 3 degrees of freedom remaining mitigates the constraints to some extent.

Weaknesses: Requires a high precision to dock in. The taxi part of the mechanism, most probably some sort of ring, is not aerodynamic. Also not really adapted for departures.

❖ Magnets

Strengths: Tolerant to the docking orientation. Magnets intensity can be modulated to dock smoothly. Less friction and fatigue compared to other mechanisms.

Weaknesses: The electric energy dependency of the mechanism requires an additional safety factor. Requires to also add a magnet on the Taxi. This will add a lot of weight and power consumptions to the taxi, which we want to avoid.

❖ “Classical space docking” (APAS, CBM, …)

Strengths: Mature and flight-proven technology.

Weaknesses: Requires a relatively high impact force. High use of the Taxi’s RCS system that can be reduced or even avoided with other mechanisms.

❖ Net

Strengths: Requires an extremely low accuracy.

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Weaknesses: Hardly predictable behavior and dangerous. Departures from a net is highly

uncertain.

❖ Berthing mechanism

Strengths: Reduces the propellant consumption of the Taxi. 6 Degrees of freedom arm ensures a

good level of security and capability of docking.

Weaknesses: Moving the arm at the rendezvous speeds in an atmosphere would require a really

strong and massive arm.

❖ Grapple (NASA “Boom” [1])

Strengths: Rewindable grapple reduces collision probability at docking. The grapple can have 6

degrees of freedom if mounted on a gimbal. May reduce the taxi mass the same way as the clamp

does. Can soften the tension spike when capture by releasing the cable. Could be used for

rendezvous and in a lesser extent for departures.

Once all the strengths and weaknesses defined, we made our choice using a decision

matrix. The weightings have been chosen qualitatively.

Table 7. Mechanism Decision Matrix

Mechanism Reusability Reliability Mass Resilience Tolerance Total

Weights 4 Weights 4 Weights 3 Weights 2 Weights 2

Clamp / Hook / Grapple 3 3 3 3 2 43

Snap-fit 1 2 2 2 2 26

Magnet 3 2 1 3 3 35

"Classical space docking" 3 2 2 3 1 34

Net 2 1 3 1 3 29

Berthing Mechanism 3 2 3 2 3 39

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IX. Communications Appendix

[1] Butterfield, A., & Szymanski, J. (2018). Shannon–Hartley theorem. In A Dictionary of

Electronics and Electrical Engineering. : Oxford University Press. Retrieved 29 Jan. 2020, from https://www.oxfordreference.com/view/10.1093/acref/9780198725725.001.0001/acref-

9780198725725-e-4260.

[2] Thuillier, Hersé, Labs, Foujols, Peetermans, Gillotay, . . . Mandel. (2003). The Solar Spectral

Irradiance from 200 to 2400 nm as Measured by the SOLSPEC Spectrometer from the Atlas and

Eureca Missions. Solar Physics, 214(1), 1-22.

[3] Hemmati, H. (2006). Deep space optical communications (Deep-space communications and navigation series). Hoboken, N.J.: Wiley-Interscience.

[4] Satellite dish data sheet for a 1.2 m diameter antenna (used to estimate mass of antenna & systems) https://newerasystems.net/wp- content/uploads/2017/01/AvLModel12mModel1050PIBSpecSheet2015-07-31.pdf

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X. System Overview Appendix

The propulsion of this system follows the same concepts as the Electrodynamic Hybrid

Tether Sling’s electrodynamic portion. The Cargo System is a separate system that utilizes the same propulsion as the boost used in the Hybrid Tether to send it to the moon. The equations used in the propulsion forces of this system are as follows:

퐹⃗ = 퐼퐿 × 퐵 (31)

푅 3 퐵 = 퐵 ∗ ( 퐸) (32) 0 푟

−5 퐵0 = 3.12 × 10 푇 (33)

The cargo system could be scaled up, with a larger mass or a smaller time. Both of these changes would increase the power required from the system, which could be made up in the addition of batteries to the solar array. The following values are found in the analysis:

Table 8 Hybrid Tether Values

Variable Value Payload Mass 40,000 Mg ED Wire Mass 12,515 Mg Collective Length of Wire 5 km Current through Wire 2,000 A Power Required 137.7 kW Time of Flight 274.2 days

Sources

Cosmo, M.L. and Lorenzini, E.C., Tethers in Space Handbook, 3rd Ed,

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Smithsonian Astrophysical Observatory, Cambridge MA, December 1997 p 77.

Cosmo, M.L. and Lorenzini, E.C., Tethers in Space Handbook, 3rd Ed,

Smithsonian Astrophysical Observatory, Cambridge MA, December 1997 p 146.

Cron, Alfred, Applications of Tethers in Space, Compuler General Research

Coorporation, McLean, Virginia, June 1983

Walt, M., Introduction to Geomagnetically Trapped Radiation, Cambridge Atmospheric and

Space Science Series.

“Wire Resistance & Properties,” Paige Wire Available: http://www.paigewire.com/wire_resistance-prop.aspx?AspxAutoDetectCookieSupport=1.,mn0

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16. Taxi Vehicle Appendices

Emily Schottxx, William Sandersyy, Michael Porterzz, Sarah Culpaaa,

Caroline Krenbbb, Yash Mishraccc, Sidharth Prasadddd, Dean Lontoceee,

Joshua Schmeidlerfff, Anna N. Liuggg, Alex Moorehhh, Pierre Veziniii,

Natasha Yarlagaddajjj

Purdue University, West Lafayette, Indiana, 47906, United States

xx Human Factors, Associate Editor for Taxi Vehicle yy CAD zz Mission Design aaa Human Factors bbb Propulsion ccc Controls ddd Controls eee Power Design fff Thermal Design ggg Structures hhh Communications iii Mission Design jjj Propulsion

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Table of Contents

I. Appendix ...... 255

1. System Design and Configuration ...... 255

A. Mission Profile ...... 256

a. Lambert Arc and general orbital equations used for lambert iterator ...... 256

1. Full Lambert Analysis Data tables for Taxi Mission Profiles ...... 259

2. Asteroid Feasibility Study...... 268

B. Propulsion Elements ...... 278

a. Taxi Propulsion Design Theory and Calculations ...... 278

2. Propellant-feed system ...... 296

C. Controlling Mars Reentry ...... 305

a. Equations and Figures ...... 305

D. Structural Analysis ...... 310

II. References ...... 314

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Appendices

1. System Design and Configuration

Figure51: Old taxi design before shape alterations.

Figure52: Old taxi shape dimensions.

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Mission Profile a. Lambert Arc and general orbital equations used for lambert iterator

Space Triangle:

r̅ ∙ r̅ (5) TA = transfer angle = cos−1 ( 1 2 ) |r̅1||r̅2|

2 2 (6) c = √|r̅1| + |r̅2| − 2|r̅1||r̅2| cos(TA)

s = 0.5 (|r̅1| + |r̅2| + c) (7) s a = (8) min 2

αmin = π (9)

(10) −1 s − c βmin = 2 sin √ 2 ∗ amin

Determining Elliptical or Hyperbolic Transfer:

Type 1: TA < 180° (11)

Type 2: TA > 180° (12)

(13) 1 2 3 3 If Type 1: TOF = √ [s2 − (s − c)2] par 3 μ

(14) 1 2 3 3 If Type 2: TOF = √ [s2 + (s − c)2] par 3 μ

Lambert Elliptical Case:

3 (15) √μ (t2 − t1 ) = a2 [ (α − β ) − (sin α − sin β )]

Where α and β depend on the Ellipse Type are:

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s (16) α = 2 sin−1 √ 0 2a

(17) s − c β = 2 sin−1 √ 0 2a

If 1A: α = α0 β = β0 If 1B: α = 2π − α0 β = β0 (18)

If 2A: α = α0 β = −β0 If 2B: α = 2π − α0 β = −β0 (19)

And Ellipse Types are defined as follows:

3 a2 (20) TOFmin = [ (αmin − βmin ) − (sin αmin − sin βmin )] √μ

If Type A: TOF < TOFmin IfType B: TOF > TOFmin (21)

Lambert Hyperbolic Case

3 ′ (22) √μ (t2 − t1 ) = |a|2 [ (sinh α′ − α′) − (sinh β′ − β )]

Where α′ and β′ depend on the Hyperbolic Type are:

If 1H: α′ = α0′ β′ = β0′ (23)

If 2A: α′ = α0′ β′ = −β0′ (24)

Orbit Determination

If Hyperbolic:

4 |a | (s − r )( s − r ) α′ ± β′ (25) P = solved 1 2 sinh2( ) 1,2 c2 2

If Elliptical:

4 a (s − r )( s − r ) α ± β (26) P = solved 1 2 sin2( ) 1,2 c2 2

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If type 1A, 1H, or 2B:

Pchosen = max (P1, P2) (27)

If type 1B, 2H, 2A:

Pchosen = min (P1, P2) (28)

Orbital Equations

(29) P e = √1 − a

(30) a3 Period = 2π √ seconds μ

h̅ = r̅ × v̅ (31)

ṙ = v̅ ∙ r̂ (32)

ω = θ − θ∗ (33)

If ṙ > 0: ascending If ṙ < 0: descending (34)

f and g functions:

r r r r̅ = {1 − [1 − cos( θ∗ − θ∗ )]} r̅ + 0 sin( θ∗ − θ∗ )v̅̅̅ (35) p 0 0 √μ p 0 0

(36) r̅0 ∙ v̅̅0̅ ∗ ∗ 1 μ ∗ ∗ v̅ = { [1 − cos( θ − θ0 )] − √ sin( θ − θ0)} r̅̅0̅ p r0 r0 p

r + {1 − 0 [ 1 − cos(θ∗ − θ∗ )]} v̅̅̅ p 0 0

Orbital Equations:

P = a ( 1 − e2) (37)

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p r = (38) 1 + e cos θ∗

θ∗ e + 1 1 E (39) tan = ( )2 tan 2 1 − e 2

1 p (40) θ∗ = cos−1 ( ( − 1 ) ) e r

M = E − e sin E (41)

θ = ω + θ∗ (42)

μ (43) M = √ (t − t ) a3 p

r = a ( 1 − e cos E) (44)

cΩ cθ − sΩ ci sθ −cΩ sθ − sΩsicθ sΩ si (45) rθh xyz C = [sΩ cθ + cΩ ci sθ −sΩ sθ + cΩ ci cθ −cΩ si] si sθ si cθ ci

***Column Vector format

i.Full Lambert Analysis Data tables for Taxi Mission Profiles

Table 21: Table for TOFS < 1 day for Phobos to Cycler generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 23.8320 2.4866 2.0490 4.5356

1 2 23.4240 24.1420 13.0370 37.1790

1 3 23.4240 1663.3306 1625.6704 3289.0010

1 4 23.4240 27.6153 3.7353 31.3506

1 5 23.4240 26.3891 8.8569 35.2459

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1 6 23.4240 1682.7771 1440.0688 3122.8459

1 7 20.0880 1.2969 15.1925 16.4894

1 8 23.8320 4.0255 0. 7561 4.7816

1 9 23.4240 1187.6414 1144.6903 2332.3316

Table 22: Table for TOFS < 1 day for Cycler to Mars generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 22.5840 5.3107 2.9065 8.2171

1 2 23.8320 17.9962 15.5932 33.5894

1 3 23.8320 1578.9895 1633.1687 3212.1582

1 4 22.1760 27.5785 11.7935 39.3720

1 5 23.8320 13.1320 12.2623 25.3943

1 6 23.8320 1381.1857 1663.4035 3044.5892

1 7 4.6560 4.4111 12.6685 17.0796

1 8 23.8320 4.9889 1.4850 6.4738

1 9 23.8320 1136.9487 1170.7763 2307.7249

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Table 23: Table for TOFS < 1 day for Cycler to Phobos generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 22.5840 5.2107 0.8305 6.0412

1 2 22.9920 18.6188 15.1310 33.7498

1 3 23.8320 1578.8835 1633.0847 3211.9682

1 4 22.1760 27.6902 9.8183 37.5085

1 5 23.8320 13.1474 14.2890 27.4364

1 6 23.8320 1381.1882 1665.3465 3046.5347

1 7 4.6560 4.5056 13.9942 18.4998

1 8 23.8320 5.3818 1.0596 6.4414

1 9 17.5920 1136.9665 1172.8723 2309.8389

Table 24: Table for TOFS < 1 day for Cycler to Moon generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 23.8320 16.0575 15.7950 31.8525

1 2 23.833 12.4567 22.5924 35.0490

1 3 23.833 12.4306 7.9911 20.4217

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1 4 23.833 14.4681 10.8000 25.2681

1 5 23.833 7.3039 18.8259 26.1298

1 6 23.833 229.4365 13.6465 243.0830

1 7 19.667 4.5293 1.7185 6.2478

1 8 19.250 29.5113 5.3843 34.8956

1 9 23.833 7.4053 7.3680 14.7733

Table 25: Table for TOFS < 1 day for Moon to Cycler generated using the lambert iterator

Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 23.833 17.3473 15.3942 32.7415

1 2 23.833 24.0452 19.5020 43.5473

1 3 23.833 11.5115 3.2001 14.7117

1 4 23.417 15.4469 10.8402 26.2872

1 5 23.833 20.6891 5.5563 26.2454

1 6 16.750 4.2981 219.3690 223.6671

1 7 23.833 17.4533 0.6480 18.1013

1 8 23.833 9.7463 15.8272 25.5735

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1 9 20.500 27.9404 8.0851 36.0254

Table 26: Table for TOFS < 3 day for Phobos to Cycler generated using the lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 44.667 0.3688 2.3350 2.7037

1 2 71.333 8.0113 0.5516 8.5628

1 3 71.750 541.1937 507.7367 1048.9305

1 4 65.500 2.1449 27.6542 29.7992

1 5 71.333 8.2217 2.2247 10.4464

1 6 71.750 548.3065 306.8656 855.1721

1 7 65.500 6.1798 9.8624 16.0423

1 8 66.750 2.5051 0.0829 2.5880

1 9 71.750 386.7402 345.4640 732.2043

Table 27: Table for TOFS < 3 day for Cycler to Mars generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 44.667 4.9663 3.4384 8.4047

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1 2 71.750 8.0737 5.1802 13.2539

1 3 71.750 499.9104 553.5536 1053.4640

1 4 71.750 32.2283 5.7394 37.9677

1 5 71.750 5.6121 4.0731 9.6852

1 6 71.750 285.7681 566.2618 852.0299

1 7 71.750 12.3817 9.4569 21.8386

1 8 71.750 4.1376 0.4933 4.6309

1 9 71.750 364.2356 397.7779 762.0135

Table 28: Table for TOFS < 3 day for Cycler to Phobos generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 71.750 1.2209 2.4024 3.6233

1 2 71.750 9.9229 0.5842 10.5071

1 3 71.750 542.1565 507.6418 1049.7983

1 4 71.750 3.6403 27.7082 31.3485

1 5 71.750 10.8319 1.5429 12.3748

1 6 71.750 549.7306 306.2328 855.9634

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1 7 71.750 0.9085 16.5016 17.4102

1 8 71.750 3.9931 0.2271 4.2202

1 9 71.750 387.9838 344.8115 732.7953

Table 29: Table for TOFS < 3 day for Cycler to Moon generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 71.750 6.2390 5.1754 11.4144

1 2 25.083 13.5027 21.4753 34.9780

1 3 71.750 8.7436 2.3852 11.1288

1 4 71.750 7.5609 3.2789 10.8398

1 5 71.750 5.5227 6.3283 11.8511

1 6 29.250 231.7835 11.3651 243.1486

1 7 24.250 4.8217 1.4385 6.2602

1 8 27.583 28.6954 4.7981 33.4935

1 9 71.750 4.5147 2.3874 6.9021

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Table 30: Table for TOFS < 3 day for Moon to Cycler generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 71.750 7.8745 4.3323 12.2068

1 2 71.750 6.3782 35.1621 41.5403

1 3 71.750 7.1257 1.1119 8.2376

1 4 69.250 7.6495 3.1484 10.7979

1 5 71.750 5.8144 5.2361 11.0505

1 6 24.250 2.9771 220.6952 223.6724

1 7 71.750 6.9479 0.3682 7.3161

1 8 55.500 4.3024 20.7592 25.0616

1 9 71.750 7.7783 1.0306 8.8089

Table 31: Table for TOFS < 4 day for Cycler to Mars generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 44.667 4.9663 3.4384 8.4047

1 2 95.917 6.9323 3.8754 10.8077

1 3 95.917 364.4252 417.7772 782.2024

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1 4 94.250 33.1068 4.4391 37.5459

1 5 95.917 4.8256 3.0468 7.8724

1 6 92.583 164.1148 442.4227 606.5375

1 7 95.917 12.4966 7.8323 20.3289

1 8 95.917 4.0354 0.3691 4.4045

1 9 95.917 268.2577 301.7693 570.0270

Table 32: Table for TOFS < 4 day for Cycler to Moon generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 95.083 5.2000 3.8067 9.0067

1 2 25.083 13.5027 21.4753 34.9780

1 3 94.667 8.1759 1.8573 10.0332

1 4 95.083 6.8229 2.2529 9.0758

1 5 90.917 4.9694 5.0032 9.9726

1 6 29.250 231.7835 11.3651 243.1486

1 7 24.250 4.8217 1.4385 6.2602

1 8 27.583 28.6954 4.7981 33.4935

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1 9 95.917 4.4357 1.7011 6.1367

Table 33: Table for TOFS < 6 day for Cycler to Mars generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 5 143.833 4.1518 2.0316 6.1834

1 8 143.833 2.5428 0.9638 3.5066

Table 34: Table for TOFS < 6 day for Cycler to Mars generated using lambert iterator Cycler Leg TOF (hours) 푘푚 푘푚 푘푚 ∆ ( ) ∆푉 ( ) ∆푉 ( ) 푑푒푝 푠 푎푟푟 푠 푡표푡 푠

1 1 143.833 4.2251 2.2383 6.4634

1 7 96.333 5.7881 0.8515 6.6396

1 9 143.833 4.4633 0.9158 5.3791

ii.Asteroid Feasibility Study

Outside the scope of the other designs, our team wants to explore the potential benefits of adding a new body to the existing architecture. The idea is to add an asteroid into the cislunar system that serves as an additional launch location. The location serves as a steppingstone from

Luna to the Cycler or vice-versa. The asteroid leg would allow lower ∆푉푠 needed at each location, reducing the size of the Lunar architecture and alleviating fuel carried by the taxi.

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After establishing the concept of the asteroid leg, our team recognized one major problem: how is the asteroid placed in cislunar space. We recognize that there is currently no solution to moving a large asteroid today, but we begin the asteroid feasibility study by assuming the asteroid exists in the desired orbit in cislunar space. We want to know if the asteroid architecture could add benefits to the system, even if the act of moving the asteroid proves to be too far- fetched.

With the first crucial assumption made, the team identifies three major question: What type of orbit should be used? What will be the natural attitude stability of the asteroid? Where/which asteroid would be chosen?

First, we will investigate the type of orbit to use. First, the team investigates a halo orbit. To gain more information on halo orbits, we consult with Brian McCarthy, a PhD student at Purdue

University who did his master’s thesis on the use of Near Rectilinear halo orbits in cislunar space. From the meeting, halo orbits are classified as stable or unstable. Unstable halo orbits are very hard to model, but an unstable halo orbit is achieved with low ∆푉 requirements. The orbit requires constant station keeping, or the asteroid will unwind from the orbit. Station keeping needs to be nearly non-zero for our application though, because applying station keeping to a large asteroid would be very difficult, if not impossible. If station keeping is not successfully kept, the unwinding of an unstable halo orbit is very unpredictable and could lead to the asteroid crashing into the Earth or Luna. The second type of halo orbit is a stable orbit. A stable halo orbit is very hard to design and enter, but there are added benefits. The stable halo orbit will require substantially less station keeping, but station keeping still exists. Furthermore, the same consequences apply if station keeping is not maintained.

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There was one major benefit of using a halo orbit though. A halo orbit is a three-body orbit, so it resides outside of the Lunar/Earth gravity wells, meaning the orbit will have a much lower

∆푉 to escape the cislunar system. The orbit allows the asteroid architecture to provide benefits to the overall system by serving as a steppingstone from either the Earth or Luna on the way to the cycler or vice-versa. A new orbit still needs to be selected due to the consequences of orbit degradation (potential of crashing into Earth or Luna).

We believe a different three-body orbit should be investigated, as the benefit of sitting outside the gravity wells is still be true. McCarthy suggests we investigate a distant retrograde orbit (DRO), which is still a three-body orbit. DROs are extremely stable and can maintain an orbit without station keeping for over a hundred years. The NASA Asteroid Redirect mission (a

NASA mission planning on placing an asteroid into cislunar space for investigation) even planned on using a DRO before mission cancellation.

As suggested by McCarthy, we started digging on the DRO question. We learned that there is actually a wide variety of DROs. Their orbital amplitude, if I may call them so, are ranging from around 20,000 km to more than 600,000 km. The estimates ΔVs and TOFs to transfer into DROs from LEO varies from the aimed DRO and transfer maneuvers.

The goal of this study is to discuss some of those transfer maneuvers, tradeoffs between ΔVs and TOFs in order to evaluate how much using an asteroid in the Earth/moon system could be beneficial to our mission.

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Figure53: Illustration of three different maneuver to insert in a DRO. L1 and L2 represents the Earth-Moon Lagrange points. “E” and “L” respectively represents the Earth and Luna (not to scale). The red lines represent three different possible paths to get into a DRO. The red stars at the end of the paths represent the last insertion burns of each maneuver. The small bean- shaped orbit inside the DRO orbit is a L1-Lyapunov orbit. The direct following of this report will focus on these different ways of inserting into a DRO and their major differences. .

Intermediate L1 – Lyapunov insertion

This maneuver is composed of three burns. The first burn is the departure maneuver from

LEO, it is a high-energy arc, similar to a Hohmann transfer. The second burn is the insertion in an intermediate orbit around the first Lagrange point L1, called L1 Lyapunov. the time spent around this orbit is actually quite long, ranging from 5.84 days to 9.40 days! The third and last burn is the insertion maneuver, which is quite small compared to the other burns.

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Lunar far-side insertion

This maneuver just consists in a direct transfer from LEO to the lunar DRO, thus it only requires two burns. The departure burn from LEO is also similar to a Hohmann transfer, the second burn is the insertion maneuver. An interesting and counter-intuitive fact is that DRO insertions on the lunar far-side are actually quicker than lunar flyby insertions (which is the third maneuver than will be discussed in this report). However, far-side insertions may require more propellant as the apoapsis of the far-side transfer remarkably further for large DROs.

Close Lunar flyby

This maneuver consists of first getting from LEO to a flyby around the moon. Once flying by the moon, a second burn is made to correct the trajectory towards the DRO. The third burn is, as in other cases, an insertion burn.

Table 35. ΔV and TOF abstract [74] Insertion maneuver ΔV (km/s) TOF (Days)

Min Max Min Max

Intermediate L1-Lyapunov 3.576 4.334 9.999 15.02

Lunar far-side 3.456 3.785 5.19 10.90

Close Lunar flyby 3.365 3.393 Up to 20+ days*

* Only precise data found is the TOF for a Lunar DRO at C (Jacobi Constant) = 2.91 is 21.934 days.

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We have to keep in mind that those are value ranges. Thus, the best insertion maneuver on a given DRO might be the worst on a different one. These values are most valuable in giving an overview of the order of magnitude of the problem and the possible benefits it may bring.

Nevertheless, the last parameter we have to consider is leaving a DRO orbit.

David Conte et al. [75] showed that leaving a 61,500 km amplitude DRO to escape the Earth-

Moon system would require about 1.01 km/s. From this, getting into LMO (Low Mars Orbit) would require an additional ΔV ranging from 0.7 to 0.8 km/s.

As a matter of comparison, going directly from LEO to LLO (from a simple Hohmann transfer) would require about 3.959 km/s, which is the same order of magnitude than going in a

DRO. However, having an extra launch point on an asteroid may give us more flexibility on transfer windows. Hence the asteroid may be used as an intermediate refueling station and launch point to meet with the cycler, coming from the Earth or Luna, or the other way around.

From further research on DRO’s, we learn that DRO’s have a stability region within the

60,000-70,000 km amplitude range over 30-year simulations [76]. Furthermore, a DRO with a

61,500 km 푥̂ amplitude (where x represents a fixed line drawn between Earth and Luna) maintains stability for over 500 years whilst accounting for solar, Earth, asymmetric Lunar gravity fields, Venus, Jupiter, and solar radiation pressure perturbations [76]. To be useful, we remember the orbit needs to have a natural attitude stability as well. If the attitude is not naturally stable, the asteroid will constantly rotate, making it nearly impossible to properly launch and capture the taxi at the asteroid. We cannot rely on thrusters to provide attitude stabilization due to the large expected size of the asteroid. After further investigation, a DRO

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Purdue University M. Porter Taxi V. Appendices | 274 needs to have a period larger than 13.22 days for a natural, repeating attitude stability that can be modeled by the blue rectangles in Figure 54 [77].

Figure54: DRO Natural Attitude Stability. The blue rectangles represent the spacecraft. The black and red arrows represent body-fixed unit vectors. The blue oval represents the DRO orbit in a fixed Earth-Moon frame. The figure represents the natural attitude stability through the course of one revolution about the Earth-Moon fixed frame of a DRO of period = 13.22 days. The attitude rotates 180 degrees about the body for every 90 degrees processes about the fixed Earth-Moon orbit of a DRO. Figure adapted from [77].

The period of the 61,500 km orbit is below 13.22 days. Using GMAT, we design a new

DRO. The new DRO combines initial conditions for stable DRO’s in a paper by Turner [78] and an iterative guess and check within GMAT. The resulting orbit is stable for 100 years with asymmetric lunar gravity field, solar, Saturn, Jupiter, and Earth perturbations. The new 푥̂ amplitude of 69,500 km stays within the general DRO stability region cited by Bezrouk and

Parker [1], and the attitude stability region cited by Guzzetti and Howell [77].

The DRO has a resonance of 1:1:2, which means Luna passes the Earth once every time the asteroid passes the Earth once, but the asteroid will cross between Luna and Earth twice in this

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Purdue University M. Porter Taxi V. Appendices | 275 time. A full summary of chosen orbit for the asteroid architecture can be found in Table 36: A summary of the selected orbit for the asteroid architecture.

Table 36: A summary of the selected orbit for the asteroid architecture.

Solution 1: Orbit Parameters [79]

Type DRO

X-Amplitude 69,500 km

Resonance 1:1:2

Period 13.785 days (Luna)

Stability Over 100 years

Now that we have determined a safe, usable orbit that is advantageous to the overall mission architecture, we now return to the final question, is there an asteroid near Earth that is usable. To guide the search, we constrain the asteroid to be similar in size to Phobos. Since a tether sling architecture on Phobos is already being developed, we assume the asteroid tether sling architecture will resemble the Phobos architecture. Using the JPL database for Near Earth

Objects [80], we search for asteroids like Phobos. The asteroid 433 Eros nearly matches Phobos in size. Eros passes by Earth at similar closest approach distances every 43 years [80]. Table 37:

General mass and closest approach elements of the asteroid chosen summarizes the important

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Purdue University M. Porter Taxi V. Appendices | 276 characteristics of Eros while Table 38: The Keplerian elements about the sun for the asteroid chosen summarizes the Keplerian elements of Eros about the sun.

Table 37: General mass and closest approach elements of the asteroid chosen.

Solution 2: Asteroid Chosen [80]

Name 433 Eros (A898 PA)

Mass 6.687 x 1012 Mg

Diameter 22 - 49 km

Closest Approach (⨁) ~ 0.17362 AU

푘푚 푣∞ (relative to ⨁) 5.92 푠

Table 38: The Keplerian elements about the sun for the asteroid chosen.

Solution 2: Asteroid Chosen Keplerian Elements

Epoch 2019-Apr-27.0

e 0.22273

a 1.4581 au

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i 10.83 deg

훺 304.31 deg

휔 178.82 deg

M 47.23946 deg

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Propulsion Elements a. Taxi Propulsion Design Theory and Calculations

1. Introduction

This appendix is to provide the theory and calculations behind the main propulsion systems of the OMS and RCS configurations for the taxi vehicle. The main requirements in the design of the MPS is that MPS tanks need to fit in the structural allotment of

푘푚 6.436푚 푥 6.436푚 푥 10.5푚, and a minimum ∆푉 of 1.529 needs to be achieved. These 푠푒푐 constraints are put in place to achieve appropriate rendezvous, orbit trajectory transfers, and attitude adjustments of the taxi. It is noted that refueling is determined to be at every tether sling location and on the cycler, therefore all of the RCS propellant for the entire duration of the mission does not need to be carried on the taxi at one time. This saves on taxi mass and size.

Derivation of the Rocket Equation [81]

Using a system of particles approach, we start with the principle of linear impulse and linear momentum, as seen in Equation 46 and by simplifying by cancelling like terms and setting second order terms to zero, this results in Equation 47.

푡2 2 (46) ⃑ 푒 ⃑ 푂퐶 푡2 푒 ⃑ 푂푃𝑖 푡2 ℱ = ∫ 퐹푑푡 = 푚 푉 |푡1 = ∑ 푚𝑖 푉 |푡1 푡1 𝑖=1

⃑0 = (푚 − ∆푚)(푉 − ∆푉)푒̂1 + ∆푚(푉 − 푉푒)푒̂1 − 푚푉푒̂1

= 푚푉 + 푚∆푉 − ∆푚푉 − ∆푚∆푉 + ∆푚푉 − ∆푚푉푒 − 푚푉 = 0

푚∆푉 = ∆푚푉푒

푑푉 푚 = 푚̇ 푉 푑푡 푝푟표푝 푒

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퐹 = 푚̇ 푝푟표푝푉푒 (47)

A more useful form of Equation 47 is shown in the derivation below and in Equation 48, which we use in our code to determine the ∆푉 that our OMS configuration can achieve.

푚̇ 푝푟표푝 푚̇ 푑푣 = 푉 = 푉 푚 푒 푚 푒

푚̇ 푉 푑푚 = −푉 푑푡 = − 푒 푒 푚 푚

푉2 − 푉1 = −푉푒(ln(푚2) − ln(푚1))

푚1 푚1 ∆푉 = 푉푒 ln ( ) = 퐼푠푝푚 ln ( ) 푚2 푚2

푚𝑖푛𝑖푡𝑖푎푙 (48) ∆푉 = 퐼푠푝푤푔0푙푛 ( ) 푚푓𝑖푛푎푙

Determining Constraints (∆푉 Minimum & Structural Allotment)

Before the design of the MPS for the OMS and RCS, we need to determine what constraints are imposed on the taxi vehicle that will be integral in determining the specifics of the MPS design. With the design chosen from the structures team, we have a total rectangular prism shape of 6.436푚 푥 6.436푚 푥 10.5푚 allotted for the entirety of the overall RCS propulsion, to be located in the aft portion of the taxi structure. We currently define ‘RCS’ to be the combined

OMS and RCS configurations that make up the overall RCS.

We determine the ∆푉 minimum from values given by the mission design team using a 2D approximation approach. As for the majority of the legs of the journey, since the taxi is assisted by tether slings, electrodynamic tethers, or mass drivers, we only need to focus on the

∆푉 required from RCS alone. Therefore, by summing the ∆푉 required from RCS for a transfer

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from the Cycler to Low Mars Orbit (LMO) and from Mars to Phobos: ∆푉푐푦푐푙푒푟−푡표−퐿푀푂 =

푘푚 푘푚 0.868 kkk and ∆푉 = 0.661 lll, respectively, we determine an estimate of 푠푒푐 푀푎푟푠−푡표−푃ℎ표푏표푠,푅퐶푆 푥푒푐

푘푚 ∆푉 = 1.529 for the taxi vehicle between each refuel period. It should be noted that the 푚𝑖푛 푠푒푐 value given for ∆푉푀푎푟푠−푡표−푃ℎ표푏표푠,푅퐶푆 is the ∆푉 required by only the RCS, as the majority of the

∆푉 for that transfer will be given by a mass driver. Additionally, the transfer from the Cycler to

LMO is achieved by only RCS.

From this point on, ‘RCS’ alone refers to the individual RCS configuration, not the combined

OMS and RCS configurations, as was defined previously. This ∆푉푚𝑖푛 is delivered by the OMS only, as the RCS will provide the steering of the taxi vehicle while the OMS will provide the majority of the ∆푉 to move in the new desired direction. By optimizing the OMS MPS configuration to achieve all the necessary ∆푉푚𝑖푛, we allow the RCS configuration to not take away from the OMS thrustings, and can therefore maximize RCS adjustments. Therefore, the

RCS MPS configurations are solely for RCS maneuvers, and not for OMS maneuvers, which we define the latter to be the calculated maneuvers of the taxi vehicle.

Designing The OMS MPS For The Taxi Vehicle

Designing the OMS MPS configuration requires a scaled re-design using the OMS MPS configuration used on the Space Shuttle Orbiter, denoted SSOPC. Each SSOPC pod consisted of the P&ID components shown in Figure55, and each pod supplied one OMS engine.

kkk Whitcomb, M., AAE 450 Presentation March 6, 2020, AAE 450 Project, 2020. lll Cruces, N. AAE 450 Presentation March 13, 2020, AAE 450 Project, 2020.

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Figure55: SSOPC OMS P&ID. The P&ID of the OMS configuration used on the Space Shuttle Orbiter is shown, with the fuel as MMH, oxidizer as NTO, pressurant as He, and purging substance as N. The propellant feed system is pressure-fed.

By estimating the RCS system mass and including the mass estimates for other components of the taxi system, upon running the code titled:

Delta_v_calculations_pod_config_REV3.m, we determine the necessary number of Shuttle pods needed for our system. The result of the running of this code is shown in Figure56.

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Figure56: ∆푽 trend. The effect of increasing propellant on deltaV of using SSOPC is shown.

As shown in Figure56, and further proven in Appendix O3a2, the effect of propellant on ∆푉 is not linear, it is instead exponential. Therefore, as you increase the number of pods (and therefore available propellant), the achieved ∆푉 increases in such a way that the change between each ∆푉 decreases as you increase propellant. As this was the starting point for the design of the

OMS MPS for our taxi vehicle, we determine that it is best to overestimate the ∆푉푚𝑖푛, as the

SSOPC will not be the same as our design for the taxi OMS. Therefore, we choose a new goal for our design, using the SSOPC as a basis, by choosing a point from the plot in Fig.56. We choose an estimation of the propellant amount of twenty SSOPC pods, which achieve a

푘푚 ∆푉 = 2.146 . This corresponds to making our design have the amount of fuel of 푆푆푂푃퐶,푔표푎푙 푠푒푐 twenty SSOPC MMH tanks, oxidizer of twenty SSOPC NTO tanks, pressurant of twenty SSOPC

He tanks, and purgant of twenty SSOPC N tanks. To determine the specific values of this

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SSOPC estimate for our design, we apply Table 39, which shows the SSOPC volume data of each tank.

Table 39: The volume data of tanks on the Space Shuttle Orbiter’s OMS (SSOPC), per each tank, is shown. [82] Volume Variable Volume Value (푚3)

표푙 푉푓푢푒푙 푡푎푛푘,푆푆푂푃퐶 2.545

표푙 푉표푥𝑖푑𝑖푧푒푟 푡푎푛푘,푆푆푂푃퐶 2.545

표푙 푉퐻푒 푡푎푛푘,푆푆푂푃퐶 0.482

표푙 −4 푉푁 푡푎푛푘,푆푆푂푃퐶 9.832 ∗ 10

By employing Equation 49, by knowing the density of the substances, we can find the mass in the tanks. To find the density of the substances, due to MMH and NTO being storable liquids,

푘푔 푘푔 their densities can be easily approximated as 휌 = 880 and 휌 = 1440 , respectively. 푀푀퐻 푚3 푁푇푂 푚3

If we assume that Helium and Nitrogen are ideal gases, we can use the Ideal Gas Equation of

State to determine the masses in their tanks, and essentially their densities, using Equation 50.

With the SSOPC operating pressure and temperature of 4,875 psia and 200℉ for He,

푘푔 respectively, we find 휌 = 44.115 . For Nitrogen, we have data on the operating pressure but 퐻푒 푚3 not of the temperature, and therefore if we assume a temperature of 199.13℉ and an operating

푘푔 pressure of 3,000 psia for N, we find 휌 = 190.234 [83]. 푁 푚3

푚 = 푉표푙 ∗ 휌 (49)

푃 ∗ 푉표푙 = 푚 ∗ 푅 ∗ 푇 (50)

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By applying Equations 49 and 50 to the densities previously mentioned, this results in the masses in each of the individual tanks of the SSOPC used for our estimation. We then multiply these mass values by twenty (i.e. the number of SSOPC pods we need to achieve ∆푉푆푆푂푃퐶,푔표푎푙) to find the total masses of propellant, pressurant, and purgant we design our taxi OMS configuration to have, the values of which are shown in Table 40.

Table 40: The total masses of substances required to achieve ∆푽푺푺푶푷푪,품풐풂풍 is shown. Mass Variable Mass Value (푀푔)

푚푓푢푒푙,푡표푡,푆푆푂푃퐶 44.799

푚표푥𝑖푑𝑖푧푒푟,푡표푡,푆푆푂푃퐶 73.31

푚퐻푒,푡표푡,푆푆푂푃퐶 0.425

−3 푚푁,푡표푡,푆푆푂푃퐶 3.74 ∗ 10

We do not want to have twenty SSOPC stacks in our taxi, as that would not be an effective use of our alloted space. To optimize this space, we choose that our taxi design employs four pods, with the pods made up of tandem-stacked tanks. These pods, also referred to as column stacks and are each referred to in the P&ID shown in Figure57, are designed for our taxi OMS, denoted as TOC. To design the TOC, we first take the 푚푡표푡 values given in Table 40, and divide all these values by four, to find the mass in each tank in each column stack, as each column stack will include one tank of each substance. Applying the previously mentioned density values of

MMH, NTO, He, and N, and Equation 49, we find the volume of each tank in each TOC pod,

표푙 denoted 푉"s" tank, TOC, where “s” is the substance in the tank.

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Figure57: Taxi OMS P&ID. The P&ID of the OMS configuration for the taxi vehicle, with the fuel as MMH, oxidizer as NTO, pressurant as He, and purging substance as N is shown. The propellant feed system is pressure-fed. .

The next step is determining the shape of the tanks. Spherical tanks are the optimum shape for tanks because when considering high pressure systems, a uniform pressure distribution is preferred for structural stability as compared to domed cylindrical tanks. Domed cylindrical tanks are made up of a cylinder portion with a hemisphere on the top and bottom. The problem with spherical tanks is that they are not optimal for carrying the maximum amount of substance, due to their shape. Because we want to minimize the inert mass that comes from the empty tanks and maximize the structural allotment given for propulsion, we decide to use domed cylindrical

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Purdue University C. Kren Taxi V. Appendices | 286 tanks for the propellant tanks and spherical tanks for the pressurant and purging tanks. The reason for this is because the propellant tanks are kept at lower operating pressures than the He and N tanks (~313 psia vs. 3,000+ psia). We can safely make this decision as Space Shuttle

Orbiter used domed cylindrical tanks for their propellant tanks as well.

To determine the dimensions of the tanks used for the TOC, we keep the ratio shown in

Equation 51 the same for all of the tanks. This is to ensure that the structural stability has a factor of safety the same as that of the Space Shuttle Orbiter, which has been flown successfully.

퐷푆푆푂푃퐶 퐷푇푂퐶 푟 = = = 0.5094 (푑표푚푒푑 푐푦푙. ) & = 1 (푠푝ℎ) (51) ℎ푡표푡푎푙,푡푎푛푘,푆푆푂푃퐶 ℎ푡표푡푎푙,푡푎푛푘,푇푂퐶

표푙 표푙 Having calculated the previously mentioned values of 푉푓푢푒푙 푡푎푛푘,푇푂퐶, 푉표푥𝑖푑𝑖푧푒푟 푡푎푛푘,푇푂퐶,

표푙 표푙 푉퐻푒 푡푎푛푘,푇푂퐶, and 푉푁 푡푎푛푘,푇푂퐶, we start with determining the dimensions of the pressurant and purging tanks. Using the volume for a sphere, shown in Equation 52, we apply the volumes of the He and N tanks to this spherical shape to find the diameters of the Nitrogen tank to be

퐷푁 푡푎푛푘,푇푂퐶 = 0.211 푚 and the Helium tank to be 퐷퐻푒 푡푎푛푘,푇푂퐶 = 1.66 푚. 4 퐷 3 푉표푙 = 휋 ( ) (52) 푠푝ℎ푒푟푒 3 2

Since the ratio for the spherical tanks is 1, Equation 51 does not affect our design of the spherical tanks. Keeping in mind that we need to stay within the maximum height of 10.5m

(within the structural allotment), we now solve for the dimensions of the domed cylindrical tanks, which include the parameters of height of the cylinder portion of the tanks and the diameter of the tanks. We start with the equation of the volume of a cylinder, shown in Equation

53, and the volume of a domed cylinder, shown in Equation 54.

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퐷 2 (53) 푉표푙 = 휋 ( ) ℎ 푐푦푙𝑖푛푑푒푟 2 푐푦푙𝑖푛푑푒푟

퐷 2 4 퐷 3 푉표푙 = 푉표푙 + 푉표푙 = 휋 ( ) ℎ + 휋 ( ) (54) 푑표푚푒푑 푐푦푙𝑖푛푑푒푟 푐푦푙𝑖푛푑푒푟 푠푝ℎ푒푟푒 2 푐푦푙𝑖푛푑푒푟 3 2

Because Equation 51 relates the diameter of a tank to the total height of the tank, we apply that to Equation 55, to remove the unknown of ℎ푡표푡푎푙,푡푎푛푘, which shows the height of a cylinder.

ℎ푐푦푙𝑖푛푑푒푟 = ℎ푡표푡푎푙,푡푎푛푘 − 퐷푠푝ℎ푒푟𝑖푐푎푙 푝표푟푡𝑖표푛 (55)

Therefore, we find that the equation to find the diameter of a domed cylinder for our TOC fuel and oxidizer tanks is shown in Equation 56.

3 푉표푙 퐷 = 2 ∗ √ 푑표푚푒푑 푐푦푙𝑖푛푑푒푟 (56) 푑표푚푒푑 푐푦푙𝑖푛푑푒푟 3.259 ∗ 휋

Applying the calculated volumes of the propellants to Equations 54 and 56, we find

퐷푓푢푒푙 푡푎푛푘,푇푂퐶 = 2.15 푚 and 퐷표푥𝑖푑𝑖푧푒푟 푡푎푛푘,푇푂퐶 = 2.15 푚. We then employ Equation 55 to find the ℎ푐푦푙𝑖푛푑푒푟,푓푢푒푙 푡푎푛푘,푇푂퐶 = 2.07 푚 and ℎ푐푦푙𝑖푛푑푒푟,표푥𝑖푑𝑖푧푒푟 푡푎푛푘,푇푂퐶 = 2.07 푚.

Upon checking the total height of one TOC (i.e. one column stack for the taxi OMS), we find that through summing the height of one fuel tank, one oxidizer tank, one He tank, and one N tank, as shown in the P&ID in Figure57, ℎ푡표푡,푐표푙푢푚푛 푠푡푎푐푘,푇푂퐶 = 10.32 푚 which is less than the maximum of 10.5m. There will be four equivalent column stacks in the TOC, placed in the planar locations shown in Figure58.

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Figure58: The placement of the OMS column stacks is shown, with each OMS encirclement showing where one stack will be placed from a top view. All values are in m.

Having determined the dimensions of all the individual tanks included in the TOC, we can now reapply Equation 49 to check our work that our masses match those of the twenty pods from

SSOPC. It is important to remember that when doing this calculation, the mass in each tank per substance must be multiplied by four, as we include four column stacks (meaning four tanks of each substance total) for the TOC. The exception to this rule is the fuel and oxidizer tank, which need to have ullage applied. Ullage volume is the space in tanks that is not filled with the liquid, and is usually kept between 3-10% [83]. For our taxi design, we consider an ullage volume of

5% [84]. When calculating the result of ullage, we take the volume of the propellant tanks (as it is only applied for the fuel and oxidizer) and multiply by (1 − 푢푙푙푎푔푒 푝푒푟푐푒푛푡푎푔푒). Therefore to find the mass in each propellant (fuel and oxidizer) tank we apply Equation 57.

표푙 푚푢푠푎푏푙푒,푝푟표푝푒푙푙푎푛푡 푝푒푟 푡푎푛푘,푇푂퐶 = (푉푝푟표푝푒푙푙푎푛푡 푡푎푛푘,푇푂퐶 ∗ 0.95) ∗ 휌푝푟표푝푒푙푙푎푛푡 (57)

The piping specifics are beyond the scope of this project, and therefore this design does not take into consideration components such as lines, valves, actuators, etc. But, to complete the

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TOC, we need to determine the mass of the empty tanks, as these values have some influence on the ∆푉 we can achieve. As the determination of materials and thickness of the tanks is deemed beyond the scope as well, to determine these values we made an estimate by scaling up the

SSOPC tank masses. By taking the mass of each empty fuel tank, oxidizer tank, He tank, and N tank as part of the SSOPC, and multiplying these values by twenty (recall this is the number of

Space Shuttle Orbiter pods we choose to base our masses off of), we then use this the summation of these values for the total mass of all empty tanks in our taxi design for OMS. Table 41 shows the final total mass values of the substances and empty tanks upon completion of the design of the taxi OMS Configuration. We now must design the RCS configuration for the taxi, so we may determine what ∆푉 we can achieve with our design.

Table 41: The total mass of substances and of empty tanks for entire TOC are shown.

Mass Variable Mass Value (Mg)

푚푓푢푒푙,푡표푡푎푙,푇푂퐶 42.56

푚푓푢푒푙 푡푎푛푘푠,푒푚푝푡푦,푡표푡푎푙,푇푂퐶 2.27

푚표푥𝑖푑𝑖푧푒푟,푡표푡푎푙,푇푂퐶 69.64

푚표푥𝑖푑𝑖푧푒푟 푡푎푛푘푠,푒푚푝푡푦,푡표푡푎푙,푇푂퐶 2.27

푚퐻푒,푡표푡푎푙,푇푂퐶 0.425

푚퐻푒 푡푎푛푘푠,푒푚푝푡푦,푡표푡푎푙,푇푂퐶 2.47

−3 푚푁,푡표푡푎푙,푇푂퐶 3.74 ∗ 10

−3 푚푁 푡푎푛푘푠,푒푚푝푡푦,푡표푡푎푙,푇푂퐶 5.03 ∗ 10

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Designing The RCS MPS For The Taxi Vehicle

To prevent repetition, only the differing parts of the design process as compared to the design of the OMS MPS for the taxi will be included in this section. Because the RCS takes advantage of the same propellants as the OMS, all of the density values for propellants stay the same as in

Appendix O3a4. The RCS configuration for the taxi (denoted TRC) similar to the Space Shuttle

Orbiter, does not include Nitrogen as purgant, and only uses Helium. This Helium operates at a pressure of 3,600 psia and an assumed temperature similar to that of SSOPC of 200℉, therefore

푘푔 we change the helium density for the RCS compared to the OMS to 휌 = 32.58 . 퐻푒 푚3

With the OMS tanks placed within the structural allotment on four corners of the squared face of the rectangular prism, we find that the max allowable diameter for any RCS tank can be

0.99 m, to allow for some room for auxiliary components between the OMS and RCS. We determine that like the OMS for taxi, there will be four column stacks, and seek to maximize the amount of propellant we can fit within these stacks. To do this we employ the P&ID shown in

Figure59, which can be compared to the P&ID used for the Space Shuttle Orbiter’s RCS also shown in Figure59. It is important to note that unlike the Space Shuttle Orbiter’s RCS (SSRPC), we are using domed cylindrical tanks for our propellant, not spherical tanks. This decision is safe to make as the operating pressures of MMH and NTO are the same in our RCS configuration as those in our OMS configuration, and thus we can use the same shape of the propellant tanks as those used in the TOC.

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Figure59(a) Fig 59(b) Figure59: SSRPC & Taxi RCS P&IDs. The P&IDs of the RCS configurations used on the Space Shuttle Orbiter (left) and the Taxi (right), with the fuel as MMH, oxidizer as NTO, and pressurant as He are shown. The propellant feed systems are pressure-fed.

We apply a similar process as that shown in Appendix O3a4 to determine the dimensions of the tanks and masses of substance held by the tanks. The major difference is we do not use the

SSRPC volume data to estimate the amount of total mass we need. Instead, we start by using our determined value of 퐷푚푎푥 = 0.99푚 and apply this value to the propellant tanks. We then rearrange Equations 54 through 56 to find the volume of the propellant tanks, and thus can find the mass of the propellant in the tanks as well through Equation 49. The SSRPC data comes into consideration to determine the dimensions of the He tanks, as we define a new ratio, that of the volume of He to the volume of propellant in one tank in the SSRPC to be the same as that ratio

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in the TRC (see Equation 58). Having just determined the volume of the TRC propellant tanks, we can plug that into the Equation 58 to find the volume of the He tank for the TRC. From this point, we use Equation 52 to determine the diameter of the spherical tank. Having the dimensions of the Helium tanks, we can employ Equation 49 to find the mass of Helium in the tanks.

표푙 표푙 푉퐻푒 푡푎푛푘,푆푆푅푃퐶 푉퐻푒 푡푎푛푘,푇푅퐶 푟퐻푒 푑푒푡푒푟푚𝑖푛푎푡𝑖표푛 = 표푙 = 표푙 = 0.1104 (58) 푉푓푢푒푙 푡푎푛푘,푆푆푅푃퐶 푉푓푢푒푙 푡푎푛푘,푇푅퐶

Please note that ullage has been taken into consideration for the propellants for the RCS configuration as well. We find that we can include two P&ID stacks in one column stack for the

RCS to maximize space usage while staying within the structural allotment max height. Each column stack for the RCS includes four helium tanks, two fuel tanks, and two oxidizer tanks (the equivalent of two P&IDs tandem-stacked together), with four column stacks included in the total

RCS design (placement shown in Figure58). This is the maximized design for taking advantage of allowable space post-OMS design. The top portion of the RCS MPS will feed to the front RCS thrusters while the bottom portion of the RCS MPS will feed to the bottom RCS thrusters.

As the RCS of the taxi is not estimated based on the SSOPC pods, we employ a different method to determine the mass of the empty tanks used in the RCS. We determine a scaling factor through Equation 59.

푚푠푢푏푠푡푎푛푐푒 푝푒푟 푡푎푛푘,푇푅퐶 (59) 푆 = ∗ #푅퐶푆 푃&퐼퐷 푠푡푎푐푘푠 ∗ #푅퐶푆 푐표푙푢푚푛 푠푡푎푐푘푠 푚푠푢푏푠푡푎푛푐푒 푝푒푟 푡푎푛푘,푆푆푅푃퐶

= 19.5199

This scaling factor (S) is found by comparing the mass in each RCS tank in the taxi design to the mass in each RCS tank in Space Shuttle Orbiter’s RCS design, and scaling it up to how many tanks we have included in our design.

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Other than the changes mentioned in this portion of the Appendix, the rest of the process for designing the RCS MPS is the same as that mentioned in Appendix O3a4. The volume data from the SSRPC and results of these calculations are shown in Tables 42 and 43, respectively.

Table 42: The volume data of tanks on the Space Shuttle Orbiter’s RCS (SSRPC), per each tank, is shown. [85] Volume Variable Volume Value (푚3)

표푙 푉푓푢푒푙 푡푎푛푘,푆푆푅푃퐶 0.509

표푙 푉표푥𝑖푑𝑖푧푒푟 푡푎푛푘,푆푆푅푃퐶 0.509

표푙 −2 푉퐻푒 푡푎푛푘,푆푆푅푃퐶 5.62 ∗ 10

Table 43: The total mass of substances and total mass of empty tanks for entire TRC are shown. Mass Variable Mass Value (Mg)

푚푓푢푒푙,푡표푡푎푙,푇푅퐶 8.31

푚푓푢푒푙 푡푎푛푘푠,푒푚푝푡푦,푡표푡푎푙,푇푅퐶 0.682

푚표푥𝑖푑𝑖푧푒푟,푡표푡푎푙,푇푅퐶 13.59

푚표푥𝑖푑𝑖푧푒푟 푡푎푛푘푠,푒푚푝푡푦,푡표푡푎푙,푇푅퐶 0.682

−2 푚퐻푒,푡표푡푎푙,푇푅퐶 7.15 ∗ 10

푚퐻푒 푡푎푛푘푠,푒푚푝푡푦,푡표푡푎푙,푇푅퐶 0.425

Calculating ∆푉 For the Taxi Vehicle

By including all of the calculated values using the processes and sample calculations depicted in Appendices O3a4-5, we create a code, titled:

Mass_Delta_v_calculations_propulsion_systems_NPC2OMS_NPC1RCS.m to calculate the final

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∆푉 achieved by the OMS system, through including all of the masses from non-propulsion components and propulsion components (OMS and RCS). When employing Equation 48, we treat 푚푓𝑖푛푎푙 = 푚𝑖푛푡𝑖푎푙 − 푚푓푢푒푙,푡표푡푎푙,푇푂퐶 − 푚표푥𝑖푑𝑖푧푒푟,푡표푡푎푙,푇푂퐶 once again reiterating that fact that only OMS is used to achieve the ∆푉 required between each leg of the mission.

푘푚 With the fully configured system, we achieve a ∆푉 = 2.22 , surpassing our minimum 푡푎푥𝑖 푠푒푐

푘푚 requirement of 1.529 . 푠푒푐

It should be noted that, as mentioned in the Mission Profile portion of this report, when the

∆푉 determination method was investigated with a 3D approximation approachmmm, the Cycler to

푘푚 Mars and Cycler to Moon RCS transfers did not fall within the ∆푉 = 1.529 determined 푚𝑖푛 푠푒푐 from the 2D approximation method. Given the time constraints of this project, it is determined that either modifying the current propulsion system (which meets the ∆푉푚𝑖푛 based on the 2D approximation method) to meet the most recent projections based on the 3D approximation method, or, further investigating other trajectory transfers that may meet the current propulsion system’s limits will be left as open-ended work for future missions

Codes Used In Report

1. Mass_calculations_propulsion_systems_pod_config_REV3.m

a. Referenced in Code 2 and 3

2. Delta_v_calculations_pod_config_REV3.m

a. Referenced in Code 3

3. Mass_Delta_v_calculations_propulsion_systems_NPC2OMS_NPC1RCS.m

mmm Porter, M., AAE 450 Presentation March 26, 2020, AAE 450 Project, 2020.

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4. Test_of_Shuttle.m

Please note that descriptions of these codes are in the beginning lines of each code

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i.Propellant-feed system

Prior to settling on the tether system for catching and safely landing the taxi vehicle, the team first approached landing on Mars in a more traditional sense – using a chemical landing system.

The following explains the analysis and calculations behind a chemical propellant landing and reiterates the benefits of moving away from this system. We investigated this system at the very beginning of Project Escalator and the concept was not pursued further, so there are assumptions which were not addressed and refined in the final system.

Additionally, since this analysis was done at the beginning, it is important to note that certain vehicle parameters have changed from that point to the final specifications and recommendations. The calculations will use the current taxi vehicle weight so that we can easily compare the amount of propellant saved in our current design. However, the taxi vehicle configuration in these calculations is kept as the same rocket-like design as in the beginning of the project. This is because the current taxi configuration cannot be used with a chemical propulsion system, as it was not designed taking one into account.

We started with the assumption that the taxi would use the OMS and RCS systems to enter the Martian atmosphere and align vertically. Since the atmosphere of Mars is thin, we will need a propulsion system – more powerful than OMS and RCS – that can help decelerate the taxi as it descends to the surface. We also assumed that the engines we would consider would operate at a

100% power level and that no heat shielding would be taken into consideration at this point of the analysis.

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Research was done looking into multiple systems that were either a similar weight to the taxi

vehicle at the time (ie. SpaceX Falcon 9), or had been designed for entry, descent, and landing in

the Martian atmosphere (ie. Mars Insight). Table 39 shows a snapshot of the systems and

parameters that were considered to be scaled up for the taxi system.

Table 44: Initial systems research for taxi Mars landing

System Thrust Engine Mission Engine Propellant Weight (Mg) (kN) Weight (Mg) Mars Insight [86] 0.61 AR MR-107K Hydrazine 0.222 0.00091

Mars Curiosity [87] 0.9 AR MR-80B Hydrazine 3.113 0.00851

SpaceX Falcon 9 [88] 30.39 Merlin 1D LOX/Kerosene 854 0.47

Blue Origin New 34 BE-3 LOX/LH2 90 N/A Shepard [89]

Space Shuttle [90] 77 RS-25 LOX/LH2 1860 3.17

We looked at this table and saw very clearly that for a taxi vehicle of our mass, 189.17 Mg,

the larger systems are easier to scale up and are also coupled with much more powerful engines.

Since we were considering landing the vehicle vertically, we would need a large thrust force

which could be provide by an engine system similar to the Merlin 1D or RS-25.

To calculate the thrust force needed, we scaledf the Falcon 9 booster landing, as this is

similar to how we were going to land on Mars. The Falcon 9 booster weighs 30 Mg, which

makes our taxi vehicle about 6.5 times heavier. The Falcon 9 booster also uses 3 Merlin 1D

engines to fire upon landing, each of which produce a thrust of 854 kN. This means the total

thrust to land their 30 Mg booster is 7,686 kN. By using Eq. (60) and scaling this up for our taxi

vehicle weight of around 190 Mg, we require roughly 49.96 MN of thrust to land.

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퐹푡ℎ푟푢푠푡,푟푒푞푢𝑖푟푒푑 = 6.5 × 퐹퐹푎푙푐표푛 9 푏표표푠푡푒푟 (60)

Taking this information, we analyzed two types of engines that could provide this amount of thrust: the Merlin 1D engine and the RS-25 engine. Eq. (61) provides us with the number of engines we would need given the thrust levels, 퐹푒푛푔𝑖푛푒, of each system. The Merlin 1D, as mentioned previously, produces 854 kN of thrust, and the RS-25 engine produces 1860 kN of thrust.

퐹푡ℎ푟푢푠푡,푟푒푞푢𝑖푟푒푑 (61) 푛푒푛푔𝑖푛푒푠 = 퐹푒푛푔𝑖푛푒

From this, we find that we need 59 Merlin 1D engines or 27 RS-25 engines. We also calculated the additional weight that those engines would add to the system overall, by taking the number of engines and multiplying them by their respective weights shown in Table 44 above.

We add an additional 27.73 Mg with the Merlin 1D engines and an additional 85.59 Mg to the system with the RS-25 engines. Though the latter add much more weight to the system, we continued with a propellant analysis to determine stoichiometric ratios of propellant combinations that would be used with either a system similar to Merlin 1D or with a system similar to the RS-25.

Another requirement that was used in evaluating the options was from the Human Factors team, who wanted to reuse the H2O byproduct of the system. Therefore, systems which produced more water as a byproduct were beneficial in multiple ways. We first tested a combination of liquid oxygen (LOX) and kerosene (RP-1) as used by Merlin 1D. Most systems with this propellant combination operate at a mixture ratio of 2.5:1. This means the ratio of oxidizer to fuel is 2.5. The chemical reaction equation for LOX and RP-1 was shown as Eq. (62)

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퐶퐻1.97 + 푎푂2 → 푏퐶푂2 + 푐퐻2푂 (62)

where a = 1.4925, b = 1, and c = 0.985 for a stoichiometric reaction. This mixture ratio of 2.5 that the combustion normally occurs at was found to be fuel rich compared to the stoichiometric reaction. After taking the excess fuel in the products into account, the balanced equation was calculated to be Eq. (63) where a = 0.73, b = 0.72, and c = 0.2687. This is shown below for ease of viewing.

퐶퐻1.97 + 1.0914푂2 → 0.73퐶푂2 + 0.72퐻2푂 + 0.27퐶퐻1.97 (63)

A similar calculation was done with a LOX and liquid hydrogen (LH2), which is normally used in the RS-25 at a mixture ratio of 6:1 [91]. The chemical reaction for this propellant combination is shown below in Eq. (64):

퐻2 + 푎푂2 → 푏퐻2푂 (64)

The stoichiometric reaction was found to have coefficients a = 0.5 and b = 1. Once again, the mixture ratio of 6 was fuel rich compared to the calculated stoichiometric mixture ratio of 8. The balanced equation was calculated with coefficients a = 0.75 and b = 0.25. It is shown below for ease of viewing in Eq. (64).

퐻2 + 0.375푂2 → 0.75퐻2푂 + 0.25퐻2 (65)

From the balanced reactions, we can see that for every mole of fuel in the LOX/RP-1 reaction, we produce 0.72 moles of water. The LOX/LH2 combination produces 0.75 moles of water. Additionally, LOX/LH2 engines have a higher specific impulse, Isp, which means that they produce more thrust for less propellant. We also considered the fact that kerosene is a denser propellant compared to LH2, so a smaller, lighter tank would be needed. At the same time,

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kerosene produces less thrust than LH2 so more propellant would be needed for the specified thrust. Since it produces more water and more thrust per kg of propellant, we chose to use the

LOX/LH2 propellant combination in a system similar to the RS-25 engines.

Since we chose this propellant combination and based thrust levels of the RS-25 engine, we based propellant and tank volumes off scaling the Space Shuttle (SS) systems. While the SS systems did not exactly fit our configuration in terms of the way they stored their fuel and oxidizer, they provided a simple place to start with scaling tank volumes.

Specifications for the SS components were also readily available online from NASA [92].

The SS System Stack consisting of the orbiter, engines, solid rocket boosters (SRBs), external tank, and propellant weighed 2000 Mg, and the SRBs weighed 571 Mg each [93]. Since our system won’t be using solid rocket propellant, we remove the solid rocket booster components from the total system stack mass. This results in a weight of 858 Mg which we use in scaling volumes.

The volume of LOX on the space shuttle can be found using the simple Eq. (66) below.

푀퐿푂푋,푆푆 (66) 푉퐿푂푋,푆푆 = 휌퐿푂푋

Here, the mass of LOX on the space shuttle is 617763.778 kg and the density of LOX is 1141

푘푔 3 , resulting in a volume of 541.42 푚 . A similar equation is used to find the volume of LH2 on 푚3 the space shuttle, but we swap the mass of LOX for the mass of fuel on the space shuttle, and

푘푔 swap the density of LOX with the density of LH2, which is 70.8 . With a fuel mass of 푚3

103256.22 kg, the volume of fuel on the space shuttle is 1458.42 푚3. We use Eq. (67) as a

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Purdue University N. Yarlagadda Taxi V. Appendices | 301 generic equation to scale the volumes of fuel and oxidizer to our system, which weighs 189.17

Mg.

푀 푉푝푟표푝푒푙푙푎푛푡 (67) 푡푎푥𝑖 = 푀푆푆 푉푝푟표푝푒푙푙푎푛푡,푆푆

The mass of the SS that we will use is 858 Mg, and 푉푝푟표푝푒푙푙푎푛푡,푆푆 refers to the volume of either fuel or oxidizer in the space shuttle (solved for above). Above that is the volume of fuel or oxidizer in our system, the variable we are solving for in this equation. With our known values,

3 the volume of LOX in our system is 119.37 푚 , and the volume of LH2 in our system is 321.55

푚3.

Using Eq. (65) from above, we can rearrange it and solve for the mass of fuel and oxidizer in our system as well. The mass of LOX in our system, using a volume of 119.37 푚3 and LOX

푘푔 density of 1141 , is 136.2 Mg. Similarly, still using a generic, rearranged form of Eq. (65), an 푚3

3 푘푔 LH2 volume of 321.55 푚 , and an LH2 density of 70.8 , we get a fuel mass of 22.77 Mg. 푚3

The calculations are continued further to find the volumes of the propellant tanks for fuel and oxidizer, as these will add extra weight to the system and need to be accounted for. We first find the total propellant volume, combining the 119.37 푚3 of oxidizer and the 321.55 푚3 of fuel to get a total propellant volume of 440.92 푚3. We then multiply this by a factor of 1.05 to get the total tank volume. We choose to multiply by 1.05 under the assumption that the tank volume is

5% greater than the propellant volume. This provides room for ullage in the propellant tanks.

Ullage is the extra space in the propellant tanks which is not filled with fuel or oxidizer. In designing propellant tanks, we generally overestimate the volume to account for this space.

Doing this, we get a total tank volume of 462.97 푚3.

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Using Eq. (68) below, we can relate the volumes of oxidizer and fuel in our system to the total volume and set it equal to the tank volume to find the breakdowns of the tank volumes for oxidizer and fuel separately. In the equation, 푉푝푟표푝 refers to the volume of either fuel or oxidizer, and correspondingly, 푉푡푎푛푘 refers to the volume of the LH2 and LOX tank.

푉푝푟표푝 푉 (68) = 푡푎푛푘 푉푝푟표푝,푡표푡푎푙 푉푡푎푛푘,푡표푡푎푙

Substituting in a total propellant volume of 440.92 푚3 and a total tank volume of 462.97 푚3, we start with calculating the LOX tank. With a LOX volume of 119.37 푚3, we calculate a LOX

3 3 tank volume of 125.34 푚 . With an LH2 volume of 321.55 푚 , we get an LH2 tank volume of

337.63 푚3. We then proceeded to calculate the mass of the tank. We chose to estimate the thickness of the tank as 75 mm as that is a common thickness for cryogenic propellant tanks.

Additionally, at this point we assumed the tank material consisted of an Aluminum-Lithium (Al-

푘푔 Li) alloy, with a density of 2685 [94]. 푚3

Assuming cylindrical tanks, we use a code to calculate the height of the tank, taking the taxi vehicle as 4.2 m wide. This constraint of 4.2 m was provided by the Structures team, who had designed the entire seating and storage section off a vehicle width of 4.2 m. Using the code, the height of the tank for a volume of 462.97 푚3 was calculated to be 33.21 m. The tank thickness of

75 mm was added to both the height and the width of the tank, resulting in a new height of 33.36 m and a new width of 4.35 m. The new volume was calculated as 495.79 푚3. Details for this calculation can be seen in the code as well.

When calculating the mass of the tank, we essentially need to find the volume of the tank, which is the area which contains material. To do this, we use Eq. (69) below.

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푉푡푎푛푘,푚푎푡푒푟𝑖푎푙 = 푉푡푎푛푘,표푢푡푒푟 − 푉푡푎푛푘,𝑖푛푛푒푟 (69)

This results in an internal tank volume of 32.82 푚3, which can then be used along with the density of Al-Li, 휌푡푎푛푘 to calculate the mass of the tank, 푀푡푎푛푘. This is done by Eq. (70):

푀푡푎푛푘 = 푉푡푎푛푘,푚푎푡푒푟𝑖푎푙 × 휌푡푎푛푘 (70) which produces a tank weight of 88.12 Mg.

These results are tabulated in Table 40 and can also be reproduced in a code written for doing all the calculations above. The code provides an easy way to update propellant and tank volumes and masses should the taxi vehicle weight change or a different type of engine is chosen.

Table 45: Taxi chemical landing propulsion system specifications Component Propellant Propellant Mass Tank Volume Tank Mass Engine Mass (Mg) (푚3) (Mg) (Mg) Fuel LH2 22.77 321.55 -- --

Oxidizer LOX 136.2 125.34 -- --

Total -- 158.9 446.89 88.12 85.59

Taking the components of propellant, tank, and engine mass from Table 40, we can use Eq.

(71) to get the total mass of the propulsion system for a chemical landing of the taxi vehicle on

Mars.

푀푝푟표푝 푙푎푛푑𝑖푛푔 = 푀푒푛푔𝑖푛푒푠 + 푀퐿푂푋 + 푀퐿퐻2 + 푀푡푎푛푘 (71)

This turns out to be a total of 332.61 Mg. The amount of propellant it takes to land a vehicle as heavy as our taxi in an environment with little atmosphere to help with deceleration is quite large. Currently, we have forgone this approach of a traditional chemical propellant system to

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Purdue University N. Yarlagadda Taxi V. Appendices | 304 land and are using a tether system. The tether system catches and lands the taxi on a maglev track, which slows it down to rest. The total weight of the cradle used to transport the taxi around the track is 100 Mg, which includes the superconducting magnets used to levitate the cradle plus taxi (launch stack) above the track. This is much less than the chemical propulsion system analyzed above.

Additionally, one of the goals of Project Escalator is to minimize the use of propellant. The propellant saved analysis from using the tether can be found in section G on tether slings, but it should be noted that there is minimal propellant required when using the tether slings and maglev systems. Based on this requirement, we choose to move away from the chemical propellant approach and focus on the design of tether slings in catching and landing the taxi vehicle safely on Mars.

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Controlling Mars Reentry a. Equations and Figures

푟̇ = 푉푠푖푛(훾) (72)

푉푐표푠(훾)푐표푠(ψ) (73) θ̇ = 푟푐표푠(ϕ)

푉푐표푠(γ)푐표푠(ψ) (74) 휙̇ = 푟푐표푠(ϕ)

q V̇ = + 푔 푠푖푛(γ) + 푔 푐표푠(γ)푐표푠(ψ) + 푔 푐표푠(γ)푠푖푛(ψ) (75) β 푟 휃 휙

+ Ω2푟푐표푠(ϕ)(푠푖푛(γ)푐표푠(ϕ) − 푐표푠(γ)푠푖푛(ϕ)푠푖푛(ψ))

푞(퐿/퐷) 1 (76) γ̇ = 푐표푠(σ) + (푔 푐표푠(γ) + 푔 푠푖푛(γ)푐표푠(ψ) + 푔 푠푖푛(γ)푠푖푛(ψ)) 푉훽 푉 푟 휃 휙

푉푐표푠(훾) Ω2푟 + + 푐표푠(ϕ)(푐표푠(γ)푐표푠(ϕ) 푟 푉

+ 푠푖푛(γ)푠푖푛(ϕ)푠푖푛(ψ))

푞(퐿/퐷) 푠푖푛(휎) 1 (77) ψ̇ = + (− 푔 푠푖푛(ψ) + 푔 푐표푠(ψ)) 푉훽 푐표푠(σ) 푉푐표푠(훾) 휃 휙

푉 Ω2푟 − 푐표푠(훾)푐표푠(휓)푡푎푛(휙) + 푠푖푛(ϕ)cos(ϕ)cos(ψ) 푟 푉푐표푠(훾)

+ 2Ω(tan(γ)cos(ϕ)sin(ψ) − sin(ϕ))

푠̇ = 푉푐표푠(훾) (78)

1 (79) 퐿 = 휌푉2퐴퐶 2 퐿

1 (80) 퐷 = 휌푉2퐴퐶 2 퐷

β = m/(퐶퐷퐴) (81)

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Purdue University S. Prasad Taxi V. Appendices | 306

1 휕푈 (82) 푞 = 휌푉2푔 = 2 푟 휕푟

1 휕푈 (83) 푔 = 휃 푟푐표푠(휙) 휕휃

1 휕푈 (84) 푔 = 휙 푟 휕휙

2 3 휇푝 푟푝 1 3 푟푝 3 5 (85) 푈 = [1 + 퐽 ( ) ( − 푠푖푛2(휙)) + 퐽 ( ) ( 푠푖푛(휙) − 푠푖푛3(휙)) 푟 2 푟 2 2 3 푟 2 2

4 푟푝 3 15 35 + 퐽 ( ) (− + 푠푖푛2(휙) − 푠푖푛4(휙))] 4 푟 8 4 8

A rough approximation of the equation for U was used, in which the J terms were neglected,

휇 resulting in 푈 = 푝. As a result, the gravitational acceleration terms become easier to handle, 푟

휇 with 푔 = 푝, and 푔 = 푔 = 0. This makes some of the equations of motion now equal to 푟 푟2 휃 휙 the following simplified form:

푞 푉̇ = + 푔 푠푖푛(훾) + + Ω2푟푐표푠(휙)(푠푖푛(훾)푐표푠(휙) − 푐표푠(훾)푠푖푛(휙)푠푖푛(휓)) (86) β 푟

푞(퐿/퐷) 1 푉푐표푠(훾) Ω2푟 (87) 훾̇ = 푐표푠(휎) + 푔 푐표푠(훾) + + + 푐표푠(휙)(푐표푠(훾)푐표푠(휙) 푉훽 푉 푟 푟 푉

+ 푠푖푛(훾)푠푖푛(휙)푠푖푛(휓))

푞(퐿/퐷) 푠푖푛(휎) 푉 (88) ψ̇ = − 푐표푠(훾)푐표푠(휓)푡푎푛(휙) 푉훽 푐표푠(휎) 푟

Ω2푟 + 푠푖푛(휙)cos(ϕ)cos(ψ) + 2Ω(tan(γ)cos(ϕ)sin(ψ) 푉푐표푠(훾)

− sin(ϕ)

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Purdue University S. Prasad Taxi V. Appendices | 307

Figure60: Velocity vs. Altitude BC 300. A higher Ballistic significantly changes dynamics, note that with higher gammas the vehicle has significant velocity while hitting the ground. This is not that much of an issue as the tether sling will be responsible for slowing it down to zero velocity.

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Purdue University S. Prasad Taxi V. Appendices | 308

Figure61: Altitude over time for Gamma of 13. Note how the taxi vehicle exits orbit.

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Fig 62: Altitude vs downrange. This plot basically gives a 2D visualization of the fight possibilities for the uncontrolled taxi. Note that there is about a 150 km radius of possibilities at the height of Olympus mons.

휕푅 휕푅 (89) 푅 = 푅 + (퐷 − 퐷 ) − (푟̇ − 푟 ̇ ) 푝 푅푒푓 휕퐷 푅푒푓 휕푟̇ 푟푒푓 퐿/퐷 (90) 휙 = 푐표푠−1 ( 퐶) ∗ 퐾2푅푂퐿퐿 퐶 퐿/퐷 퐿 퐿 퐾3(푅 − 푅푝) (91) ( ) = ( ) + 퐷 퐶 퐷 푅푒푓 퐿 휕푅/휕 (퐷)

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Purdue University A. N. Liu Taxi V. Appendices | 310

Structural Analysis

Figure63: Iteration 1 structural CAD.

Figure64: Phobos Tether – One Point Contact (200mm).

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Figure65: Phobos Tether – One Point Contact (300mm).

Figure66: Phobos Tether – Three Point Contact (200mm).

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Figure67: Phobos Tether – Five Point Contact (200mm).

Figure68: Phobos Tether – Seven Point Contact (200mm).

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Figure69: Mars Tether – Five Point Contact (300mm).

Figure70: Mars Tether – Seven Point Contact (300mm).

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Purdue University A. N. Liu Taxi V. Appendices | 314

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17. Risk Analysis Appendix

Michael Lagrange66

Purdue University, West Lafayette, Indiana, 47906, United States

66 Assistant Project Manager

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I. Points of Failure Investigation

A. Tether Sling Structural Integrity

Tether sling systems on the scale of those discussed in this report have never been constructed nor tested. To provide comparable data, the failure rates on suspension bridge cables under repeated loading were modified for our use. In the 25 year period under which the Study conducted by Dr. Wesley Cook was considered, a survey of 17,300 bridges found that there were

92 failures, 40 of which were total collapses of the bridge [1, pg. 24]. For the purposes of this analysis partial collapse would be considered a necessity to abort the mission, while total collapse would cause a LOC. These failures were divvied up into their causes, with 19.57 % of the failures occurring due to effects similar to those our tether sling would experience, such as overloading, fatigue, and material deterioration [1, pg. 24]. Impacts of vehicles into the supports were ignored in this calculation as failure of our flight controllers is addressed in another failure mode. Additional changes made to the failure rates per the direction of the tether sling structural

SMEs include: The 10x safety factor of the tether slings resulted in the assumed reduction of failure rate by a factor of 2 as suspension bridge cables are expected to have a minimum safety factor of 5. The Martian tether sling, based on its heat loading due to drag, is expected to receive additional fatigue. This increased the failure rate by a factor of 10, this change includes taking the increased environmental resistance of the Dyneema material into account. The Luna and

Phobos tethers do not expect to receive a significant material deterioration due to vacuum operation when compared to the atmospheric weathering of suspension cables. This reduced the expected failure rates of Phobos by 2 dues to the Dyneema material and 1.5 for Luna due to the

Zylon material. The ED Tether will receive atmospheric drag heating at its lowest point, for this

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Purdue University M. Lagrange Risk Appendix | 328 reason the Dyneema material only changes the failure rate by 1.7. The final failure rates for the various tether sling sturures are shown below.

Table 32: Tether Slings Failure per Hour of Operation

Mission Itinerary Accident Rate LOM Accident Rate LOC (Fail/Hour (Fail/Hour Op) Op) Martian Tether 2.376 E-8 1.033 E-8

Lunar Tether 1.584 E-9 6.887 E-10

Phobos Tether 1.188 E-9 5.165 E-10

ED Tether 1.398 E-9 6.076 E-10

B. Taxi, Tether Rendezvous

Rendezvous between a tether sling and a vehicle like our taxi have not been attempted, however rendezvous between similarly small and large fast moving objects have been. Initially information on rendezvous between space systems were considered but a decision was made that the relative maneuverability of these two systems did not provide a complete error comparison.

Information on the accident percentage of rendezvous and aerial refueling of fighter aircraft was used instead. Fighter aircraft were used as opposed to larger aircraft for this comparison as the difference in maneuverability between nimble fighter aircraft and large aerial tankers is more akin to the difference in maneuverability between our taxi and the tether. Additionally, only accidents involving later F series aircraft were considered as developments in flight controllers drastically reduced the accident percentage over the period of the study. All failure types of the F series aircraft were considered. These include errors of the fighter aircraft pilot, errors of the boom operator, errors of the tanker aircraft pilot, mechanical failures in either aircraft, mechanical failures in the boom, and avionics failures. The results of this analysis show that over

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10 year period 30 accidents involving later generation fighter aircraft occurred [2, pg. 27]. This was over the 37,140 sorties flown as part of Operation Enduring Freedom [3]. It was determined that LOM does not necessarily result in loss of crew and that a second attempt could be made.

After the failure of the second attempt the crew was considered lost. No changes were made to the final accident percentage, reduced risk due to additional maneuverability of the taxi was estimated to roughly be canceled out by the increased risk due to the complexity of the taxi tether rendezvous maneuver in comparison to an aerial refueling. The final failure rates for the tether sling rendezvous is shown below:

Table 33: High Speed Rendezvous Accident Rate

Mission Itinerary Accident Rate LOM Accident Rate LOC (Fail/Per Attempt) (Fail/Per Attempt) Taxi Tether Rendezvous 6.462 E-4 4.176 E-7

C. Taxi Maneuvering Near Cycler

The maneuvering of an object similar to our taxi using an OMS system does have existing historical data. This part of the analysis was based on legacy data taken from the Apollo program and additional risk analysis programs conducted by NASA as part of the Exploration

Systems Architecture Study (ESAS). The closest parallel in this data was determined to be the failure rates for docking procedure for the lunar surface module rendezvousing with the command module in lunar orbit after its trip from the surface. This set of failure rates were chosen for the following reasons. The slower moving lunar module would have to rendezvous with the faster moving command module in lunar orbit without disrupting the orbit significantly; this is similar to our taxi system having to rendezvous with the cycler without significantly

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Purdue University M. Lagrange Risk Appendix | 330 altering its path. The lunar operation was chosen over a LEO operation due to the increased risk factors from having low abort capabilities in deep space; this is similar to our taxi system’s rendezvous with the cycler as any error at this point would be critical. Ascent docking was chosen as by this point the Apollo crew would have been fatigued due to their mission, this is similar to our taxi system’s rendezvous with the cycler as by this point the crew would have been in space for quite a bit of time and would be expected to be fatigued. The results of the failure rate for the lunar docking are 0.26% chance of failure and 100% fatality upon failure [4, pg.

580]. Consulting with appropriate SMEs confirmed that this number is about what is expected, and no change were made. The final failure rates for the taxi, cycler docking is shown below.

Table 34: Cycler Taxi Docking Failure Rate

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Per Attempt) (Fail/Per Attempt) Taxi Cycler Rendezvous 2.600 E-3 2.600 E-3

D. Accidental Release / Damage on Release

The release of an object traveling at high speeds, similar to the interaction between the tether sling and the taxi or the cycler and the taxi upon release, does have comparable existing historical data. This part of the analysis was based on legacy data taken from the Apollo program, the Space Shuttle Program, the International Space Station and additional risk analysis programs conducted by NASA as part of the Exploration Systems Architecture Study (ESAS).

The closest parallel in this data was determined to be the mean failure rates for outbound docking procedures with no EVA assistance. This set of failure rates were chosen for the following reasons. Outbound docking procedures for two fast moving objects in orbit are very similar to

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Purdue University M. Lagrange Risk Appendix | 331 the maneuvers that the taxi system would undergo as it departs the cycler. Included in this analysis are failure rates for Apollo command module separation during the Apollo era, Space

Shuttle departure from the International Space Station, and additional risk analysis data from

NASA testing. More extensive data was available to create a mean failure rate for outbound docking procedures, for this reason it was used rather than a specific mission in order to eliminate outliers. The results of the mean failure rate for undocking procedures are 0.435% chance of failure for LOM [4, pg.598]. Consulting with the appropriate SMEs confirmed that the number for LOM was about what was expected, and no changes were made to that value.

Additionally, after consultation the LOC value was determined to be 0.0435% as it was estimated that 10% of failures cause fatalities. For the deposit of the taxi onto the Martian landing track these numbers are increased by a factor of two as these systems are under significant bending stress and may snap up upon release. The final failure rate for taxi undocking from the tether system are shown below.

Table 35: Taxi Release Failure Rate

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Per Attempt) (Fail/Per Attempt) Taxi Release (General) 4.350 E-3 4.350 E-4

Taxi Release 8.700 E-3 8.700 E-4 (Martian Touchdown)

E. Cycler BLSS and ECLSS System Failure rates

As described in section F.f Cycler System Human Factors of this report they cycler life support systems are composed of a primary BLSS system and a partially redundant secondary

ECLSS system. For this reason, the failure rates of the overall life support system required a

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Purdue University M. Lagrange Risk Appendix | 332 hybrid analysis between the two. ECLSS systems similar to the one used on the cycler have been used extensively and have a great deal of existing data. No BLSS system has ever been constructed similar to the one used on the cycler, for this reason the system failure rates will be compared to existing controlled growing environments that do exist on earth. This set of failure rates was selected due to the massively increased crop success rate of closely monitored control grown crops as opposed to open grown crops; our cycler will use a similarly closely monitored control grown system. According to the United States Department of Agriculture National

Agricultural Statistics service, the number of acres of potatoes grown in Idaho from 2015 is

323,000 acres, and the number of acres of potatoes harvested were 322,000 [5]. This gives a failure rate of 3.096 E-3 for a single crop. As we have 5 separate types of crops and require two of them to fail completely for our life support system to exceed its redundancy of 25%. The

LOM failure rate would be the failure of one crop and the redundant system where the LOC failure rate would be the failure rate of two crops and the redundant system. For the purpose of this analysis ECLSS system failure data for the International Space Station shall be used as it is a similar closed loop recycling based system. Recycling systems are useful for long term missions but have issues in reliability due to complexity. From the NASA paper on the design of reliable deep space life support systems the failure rate of such recycling based ECLSS system is

0.001219 chance of a critical failure per day [6, pg.10] . This include Oxygen production, Carbon

Dioxide Removal, Carbon Dioxide Reduction, and Water Processing. Only non-reparable, critical failures of the system were considered. As this is a redundant system however it is not expected to be in operation during the entirety of the cycler voyage, only during crop failure. For the purposes of this analysis however we will assume it is needed for the entirety of the habited parts of the cycler legs. Consultation with SMEs in aeroponic yielded the following results, as

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Purdue University M. Lagrange Risk Appendix | 333 our system is operating in a closely monitored grow environment we reduce the failure rate by a factor of 100. The cycler mission leg lengths considered are 169 days for Earth to Mars, and 153 for Mars to Earth. Consultation with ECLSS SMEs confirmed no changes were required in this case. The final cycler life support failure rates are shown below.

Table 36: Cycler Life Support Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Mission Leg) (Fail/Mission Leg) Earth to Mars flyby 6.378 E-6 1.975 E-10 (169 days) Mars to Earth flyby 5.774 E-6 1.788 E-10 (153 days)

F. Airlock Failure Rates

The airlock systems used by the Taxi, Cycler, and Tether Crawler systems are similar in design to those used by the International Space Station. As such extensive testing and operation data exists for such systems. The failure rates for the systems in this analysis were calculated using the failure rate of the International Space Station airlocks as a baseline. Data for this analysis was taken from the NASA ESAS analysis section on failures aboard the International

Space Station [4]. From this section only those failures due to docking were considered for this part of the analysis. Of those they were divided into those that contributed to loss of mission and those that contributed to loss of crew. The values found for LOM failure rates were 3.4 E-3 and for LOC were 2 E-5. SMEs on human factors from each system concurred with these results. The final values for the failure rates of airlocks per docking instance are shown below.

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Table 37: Airlock Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Per Docking) (Fail/Per Docking) Airlock Docking 3.400 E-3 2.000 E-5 Operation

G. Taxi ECLSS System Failure Rates

An open loop ECLSS system, similar to the one in use on the taxi system is something that has been constructed and has extensive his/torical data to be compared to. For the purposes of this analysis the taxi system’s is modeled as an open loop system that uses elements stored in tanks as its means of providing an environment. Legacy data from the Space Shuttle, the

International Space Station and Apollo were used to calculate this system’s failure rate. This data was chosen as these systems had either a primary or a secondary open loop ECLSS system that used elements stored in tanks to create an environment, similar to our taxi. The work conducted by NASA Ames Research center [6] was able to model the effects of adding extra tankage for this system to add redundancy. Only non-repairable failures of the system that put the mission in jeopardy were considered. The rate of these LOM failures was 1.06 E-6 fail/1000 days for the two times safety factor on the taxi, and LOC were 1.06 E-6 fail/1000 days as well [6, pg.9]. The numbers are the same as unrepairable failures were also considered to have 100% lethality for deep space voyages. Conferring with the ECLSS SMEs yielded the following results, the lethality of failures at this point was reduced to 90% as though air quality is reduced failures after a certain point are survivable.

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Table 38: Taxi ECLSS System Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Mission Leg) (Fail/Mission Leg) LEO to Luna 2.000 E-9 2.000 E-9

Luna to Cycler 3.000 E-9 3.000 E-9

Cycler to Phobos 3.000 E-9 3.000 E-9

Phobos to Mars 1.000 E-9 1.000 E-9

Mars to Phobos 1.000 E-9 1.000 E-9

Phobos to Cycler 3.000 E-9 3.000 E-9

Cycler to Moon 6.000 E-9 6.000 E-9

H. Waste Processing System Failure Rates

Recycling waste processing systems similar to the ones used by our cycler have already been design and constructed. The International Space Station is the closest parallel and its systems have undergone extensive operation, leading to a great deal of test and operational data.

A failure rate was connected from this baseline provided by the international space station. The research conducted into a reliable deep space life support system by NASA Ames research center put the failure rate of the ISS Urine Recycler System at 3.710 E-4 chance of a critical failure per day of operation [6]. This rate was used for both LOM and LOC as this rate was based on unrepairable failures and those that directly compromise the crew. Consulting SMEs on Human

Factors determined that the operational time for the urine processing system would be 24 days for the Earth to Mars trip and 22 days for the Mars to Earth trip. The failure rates are shown below.

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Table 39: Cycler Waste Processing Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Mission Leg) (Fail/Mission Leg) Earth to Mars flyby 8.904 E-3 8.904 E-3 (24 days) Mars to Earth flyby 8.162 E-3 8.162 E-3 (22 days)

I. Failure of Taxi Propulsion Systems

Our Taxi system uses OMS and RCS systems similar in architecture and operation to those used on the Space Shuttle. Extensive test and operational data exist for the firing of these propulsion systems, as such our system borrow from their data to construct its failure rates. The shuttle systems have a LOM failure chance of 0.01 and a LOC failure chance of 0.0008 from the historical data found in the NASA ESAS analysis [4]. These values are taken per critical burn on an outbound leg. Consulting with our Taxi Propulsion SME did not yield any significant changes to these values, so they were kept for the final calculation. The final failure rates are shown below.

Table 40: Taxi Propulsion System Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Per Leg) (Fail/Per Leg) Taxi Propulsion System 1.000 E-2 8.000 E-4 Critical Burn

J. Cycler Push Propulsion System Failures

The failure rates for the push drives used in the cycler were considered separately from failure rates of the ion drives used in the communication satellites as those system are considered as part of the overall communication system failures in Appendix #.j Ion propulsion systems

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Purdue University M. Lagrange Risk Appendix | 337 have been tested and have a reasonable amount of operational reliability data. Ion drives on the scale of our push drives have yet to be constructed and extensively tested. As such comparative operational reliability data from the ion drives of various satellites has been used to form a baseline for the failure rates of the push drive. A paper on electric propulsion reliability for the

Georgia Institute of Technology [7] provides a value of 162 failures of all types in 430 days of total flight experience. Of these error types, this analysis only considers those of Class I and

Class II which are immediate loss of system or unrepairable loss of engine performance respectively. This decision was made as this analysis assumes that any reparable faults can be repaired mid-flight. No Class I faults were reported, Class II faults accounted for 36 % of the total faults. Due to no faults being recorded, Class I faults were assumed to be 10% of Class II faults. The total amount of time the cycler is expected to burn over its two inhabited leg is 0.989 days for Earth to Mars and 1.53 days for Mars to Earth. After consulting with cycler SMEs no additional changes were made to any values. The final failure rates for the cycler propulsion system are shown below.

Table 41: Cycler Propulsion System Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Mission Leg) (Fail/Mission Leg) Earth to Mars flyby 3.221 E-2 3.221 E-3 (0.989 days) Mars to Earth flyby 4.982 E-2 4.982 E-3 (1.53 days)

K. Mass Driver Launch Failures

The details of this analysis are covered in section [Insert Arch’s Section]. This analysis includes failure rates due to magnetic coil failure, improper balancing of the magnets, and battery and electrical system failure. From this analysis the LOM failure is 0.01243 per launch and LOC

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Purdue University M. Lagrange Risk Appendix | 338 failure is 0.01243 per launch. These numbers are the same due to the 100% fatality rate of failures during mass driver launch. Conferring with SMEs on mass driver propulsion yielded no changes to the values. The final failure rates for mass driver launch are shown below.

Table 42: Mass Driver Launch Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Per Launch) (Fail/Per Launch) Mass Driver Launch to 1.243 E-2 1.243 E-2 Phobos

L. Explosive Bolts Failure Rates

As mentioned in the mass driver analysis explosive bolts are used to hold the taxi in place until it is released. The bolts used in our analysis are functionally the same as those used in legacy system. As such there exists extensive test and operational data for these systems. For this analysis’ failure rate bolt failure data was collected from the NASA technical memorandum on pyrotechnic system failures. Both Apollo and Shuttle data were considered but only Shuttle data was used for this analysis as advances in reliability of the explosive bolt technology reduced the number of failures to a point where it would have been inaccurate to include older data.

Additionally, only failures that resulted from the failures of the bolts themselves and not from operator error were considered. It was found that a total of 7 bolt failures over an eleven year period from 1976 caused failure of flight [8]. This was then compared to the amount of missions that occurred over the same period of time. This was then compared to the amount of explosive bolt systems that are present on a shuttle which was found to be 8 groups of explosives. This was then used to calculate the failure rate of a particular system of explosive bolts. Consulting with

SMEs for the mass driver cradle confirmed there were two groups of explosive bolts holding the

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Purdue University M. Lagrange Risk Appendix | 339 taxi to the mass driver cradle. Additionally, a decision was made based on SME information that there would be no LOC statistic as the only common failure modes are fail to detonate which would only result in the Taxi remaining attached for deceleration. The final failure rates are show below for the two bolt system.

Table 43: Mass Driver F Bolt Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Per Launch) (Fail/Per Launch) Taxi Explosive Bolt 1.530 E-2 N/A Separation

M. Cycler Robotic Arm Failure

The robotic arm used for the docking of the taxi with the cycler has never been constructed nor extensively tested as such these systems will be relying on comparative legacy data. The Canada Arm on the Space Shuttle is an existing system of comparable operation with extensive test and operational data. As such, data from this system will be used as a baseline for this analysis. From a Department of the Airforce Thesis on the reliability of Mechanical Arms it was found that the operation mean time between failure of the Canada Arm was 10,000 hours

[9]. Using this statistic to find the failure rate during any given hour of operation found that the failure rate was 1 E-4 per hour of operation for LOM. No LOC statistic could be found as abort systems were in place to save crew in any dangerous failure involving the arm. Consultation with the Cycler Arm SMEs determined that our failure rate should be about half of the Canada Arm’s due to the Cycler Arm only having half of the degrees of freedom. The final failure rates for the

Cycler arm are shown below.

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Table 44: Cycler Arm Failure Rate

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Per Launch) (Fail/Per Launch) Cycler Arm Docking 5.000 E-5 N/A

N. Avionics Failures

It was determined that the mechanical features of the control systems beginning to feed incorrect or corrupted data were a much more likely point of failure than the control systems themselves. This analysis includes failures of all avionic systems, the gyros and reaction wheels that govern the attitude and any additional laser or pressure sensors that might fail, based on the most limiting factors. As the specific design of the avionics system beyond allocation of weight is outside the scope of this analysis, reliability data from a comparable system was needed. The

Hubble Space telescope is a sufficiently large system requiring the same general avionics as this analysis would require as such its failure rate was used as a baseline for this analysis. A report by the National Academies of Science Engineering and Medicine on the assessment of options for the extending the life of the Hubble Telescope stated that the Hubble space telescope sensors had a reliability lifetime of 12 years with 50% confidence in accuracy of the system at 4.5 years.

Using this data, the LOM failure rate was calculated as 2.283 E-4 failure per day and a LOC failure rate of 2.283 E-4 failure per day. LOC was considered to be equivalent to LOM as serious avionics failures severely hinders the craft’s ability to control itself. Consulting with the SME on avionics and controls the following changes were made. Firstly, the avionics were broken into the failure rates of the gyroscopes and reaction wheels as they are notoriously fault prone and the rest of the sensors and avionics. Accordingly, the gyroscopes and reaction wheels were changes to have an expected reliability lifespan of 6 years; the remaining avionics were given an expected

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Purdue University M. Lagrange Risk Appendix | 341 reliability lifespan of 14 years. Additionally, the LOC failure rate of the system was reduced to

10% of its original number for the general avionics and 50% of its original number for the gyroscopes and reaction wheels as this was the expected rate of unrepairable failures. The final failure rates for avionics are listed below.

Table 45: Avionics Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Day Operation) (Fail/Day Operation) General Avionics 1.957 E-4 1.957 E-5

Gyroscopes and Reaction 4.566 E-4 2.283 E-4 Wheels

O. Communication System Failure Rates

Satellites similar to those used in our communication system have been built and extensive test and operational data has been collected for them. As none of our communication systems have been built or tested the operational data from these satellites will be used as a baseline for the failure analysis. The limiting factor for communication systems is the 6 year lifespan of the laser communication systems which defines the overall system lifespan. This is further described in section [Section on COMMSATS]. The overall lifespan was used to calculate the overall failure rate of the system. The double redundancy of our communication system means two satellites would have to fail for a communication system blackout. The final failure rates are shown below.

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Table 46: Communication System Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Per Day) (Fail/Per Day) Communication Systems 5.213 E-8 N/A

P. Taxi/Tether Connector Structural Failure Rates

In selecting an equivalent system to the various Taxi-Tether connectors an appropriately similar system in terms of structural requirements as well as precision needed to be selected. A comparable system in the form of dock loading cranes was selected for their similar functionality and need to deal with heavy loads repeatedly. As dock loading cranes are a well established technology extensive data was available for their failure rates. A paper from the New Mexico

Institute of Mining and Technology on the failure probability of a crane system found a failure rate to be 5 E-6 per operation [11]. For this analysis failures due to human error are not considered as they are outside the scope of this analysis. LOC and LOM are both considered to be 5 E-6 as any failure of the grabbing arm would make it impossible for the Taxi to dock with the cycler. Consultation with the taxi-tether connector SMEs yielded no additional changes. The final failure rates are shown below.

Table 47: Taxi Tether Structural Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Per Attempt) (Fail/Per Attempt) Taxi Connection 5.000 E-6 5.000E-6

Q. Tether Sling Motor Failure Rates

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Nothing on the scale of the motor that would be needed to power the tether sling has ever been constructed as such no test data exists to find a failure rate. A comparative system used to form a baseline in large industrial motors used in a variety of applications. These systems were used for their comparative failure modes that accompany large and powerful motors. Based on data gather from a US Department of Energy Reliability Analysis the failure rate of large industrial motors is 2.770 E-5 / hour in operation [12]. For the purposes of this calculation the motors are only considered in operation during the periods they are spinning up or spinning down as otherwise there would be little operational friction. No LOC failure rate was considered as stopping the motor at any point would still allow the crew to be within the range of rescue.

Based on consultations by our SME this number was reduced by 50% due to the expected increase in performance from HMS motors. Additionally, each tether was given an increase in failure rate of 10% based on its expected torque not in the direction of its rotation. The final failure rates are listed below for each tether system.

Table 48: Tether Motor Failure Rate

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Hour Operation) (Fail/Hour Operation) ED Tether 1.524 E-5 N/A

Lunar Tether 1.524 E-5 N/A

Phobos Tether 1.524 E-5 N/A

Martian Tether 1.524 E-5 N/A

R. Cycler Kilopower Nuclear System Failure Rates

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The Kilopower nuclear system similar to ones used aboard the cycler has been tested in a lab environment. However, the system does not have sufficient data to form a proper failure rate a and some legacy system must be used to form a baseline. In order to form this baseline, the failure rates of various components from an industry nuclear power system were used. This data was found from a reliability analysis by the US Atomic Energy Division and takes into account a variety of failure modes [13, pg.D-3-D-5]. This data includes all failures over the 10^6 hours of operation these systems underwent, as such it is assumed that unrepairable failure is 10% of the total failures, and lethal failures are 10% of those. This gives a total failure rate for LOM of

1.508E-5 per day and 1.508 E-6 per day for the Kilopower systems. Consultations with the

SMEs of the cycler power systems reduced the failure rates by a further factor of 10 as the kilo power system is estimated to be much safer than a legacy nuclear reactor. The final failure rates are shown below.

Table 49: Cycler Kilopower Nuclear System Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/Mission Leg) (Fail/Mission Leg) Earth to Mars flyby 2.548 E-4 2.548 E-5 (169 days) Mars to Earth flyby 2.307 E-4 2.307 E-5 (153 days)

S. Solar Power System Failure Rates

Solar panel arrays similar to those in use on the cycler system are already in use and have extensive test and operational data available. For this analysis, the closest system parallel, the

International Space Station’s solar array, is used to create a baseline. An article in the Progress in

Photovoltaics journal about photovoltaic failure and degradation estimates the mean time between failure for photovoltaic arrays on the international space station to be 24333.333 hrs

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[14], based on the decay rate until 20% of the original power is left. Using this statistic a LOM failure rate of 4.110 E-5 per day was found, no statistic was available for LOC so it was assumed that they comprised 10% of the LOM failures. Conferring with Power and Thermal SMEs confirmed that these numbers were similar to what is expected. The final failure rates for solar panels are shown below.

Table 50: Solar System Failure Rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/ Day) (Fail/Day) Solar Panel Operation 4.110 E-5 4.110 E-6

T. Heat Shield Failure Rates

The heat shield in use by our taxi is based heavily off of the TPS system in use by the

Space Shuttle as such there is extensive test and operational data that can be used to form a baseline failure rate for our system. This analysis uses data collected as part of the NASA ESAS report from the section on aerodynamic stability [4]. From the data on space shuttle direct earth entry to failure rate for LOM is 4 E-4 per attempt and failure rate for LOC is 4 E-4. These numbers are the same due to the guaranteed lethality of TPS failure during re-entry. Consultation with TPS SMEs increased both of these failure rates by a factor of 10 due to the incredible amount of increased drag from the taxi connector bar. The final failure rates for TPS during reentry and launch are shown below.

Table 51: TPS failure rates

Mission Itinerary Failure Rate LOM Failure Rate LOC (Fail/ Day) (Fail/Day) Re-entry/Launch 4.000 E-3 4.000 E-3

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II. References

[1] Cook, W. “Bridge Failure Rates, Consequences and Predictive Trends.” Utah State

University All Graduate Theses and Dissertations, Logan, UT, 2014

[2] Thomas, M. L., “Analysis of the Causes of Inflight Refueling Mishaps with the KC-135,”

Dept. AF AFIT, Wright-Patterson Air Force Base, OH, December 1989.

[3] Lasley, E., “Refueling Through the Century,” Air Mobility Command History Office, June

2017.

[4] “NASA’s Exploration Systems Architecture Study,” NASA 214062, November 2005

[5] “National Agricultural Statistics, Idaho (2015),” USDA Database, retrieved 1 April, 2020. https://quickstats.nass.usda.gov/

[6] Jones, H. W., “Design and Analysis of Flexible, Reliable Deep Space Life Support System,”

NASA Ames Research Center, Moffett Field, CA, May 2016.

[7] Saleh, H. S., Geng, F., Ku, M., Walker II, M. L. R., “Electric Propulsion Reliability:

Statistical Analysis of On-Orbit Anomalies and Comparative Analysis of Electric vs Chemical

Propulsion Failure Rates.

[8] Bement, L. J., “Pyrotechnic System Failures: Causes and Prevention” NASA Langley

Research Center, Hampton, VA, June 1988.

[9] Capt. Schneider, D. L. “Reliability and Maintainability of Modular Robot System”

Department of the Air Force, Wright-Patternson AFB, OH, August 1993.

[10] “Assessment of Options for Extending the Life of the Hubble Space Telescope: Final

Report” National Academy of the Sciences, Washington, DC. 2005.

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[11] Greenfield, M. A., Sargent, T. J. “Probability of Failure of the Trudock Crane System at the

Waste Isolation Pilot Plant (WIPP),” NMIMT Thesis program, NM, May 2000.

[12] Penrose H. W., “Electric Motor Energy and Reliability Analysis Using the US Department of Energy’s Motor Master+,” BJM CORP All-TEST Division, Old Saybrook, CT.

[13] Garrick, B. J., Gelker, W. C, Goldfisher, L., Karcher, R. H., Shimizu, B., Wilson, J. H.,

“Reliability Analysis of Nuclear Power Plant Protective Systems” U. S. Atomic Energy

Commission, Idaho Falls, ID, May 1967.

[14] Jordan, D. C., Silverman, T. J., Wohlgemuth, J. H., Kurtz, S. R., VanSant, K. T.,(2017)

“Photovoltaic Failure and Degradation Modes,” Progress in Photovoltaics: Research and

Applications, 25(4), 318-326.

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III. Points of Failure/Key Events List

A. Introduction

This list contains all of the events that were considered as part of the risk analysis and all of the modes of failure investigated in Appendix 10.A. These Points of failure were chosen as either the most likely or the most dangerous for our design. Each Key Event is labeled with numbers based on the system it is referring to. Each point of failure is labeled with lower case letters based on their Key Event.

B. List Analysis

LEO Luna Cycler Phobos Mars

I. LEO to Luna A. ED Tether Sling 1.Rendezvous in LEO with ED tether a. Taxi is at incorrect altitude to meet tether b. Tether is at an incorrect speed to meet taxi without harm c. Taxi/ Tether connection fails structurally d. Taxi/ Tether misses the catch 2.Reach Required ΔV for lunar leg p a. ΔV reading is incorrect and taxi is launched b. Tether undergoes structural failure during spin up c. Tether Motor Failure 3.Achieve required orbital parameters for launch a. Launch window is missed, must wait for next one b. Parameters are incorrect and launch occurs 4.Taxi is released and begins journey a. Failure to release b. Release is unintentionally delayed c. Release causes damage to Taxi/Tether connection 5.ED tether momentum bank de-orbits expected amount a. ED tether bank de-orbits more than expected b. ED tether bank de-orbits less than expected 6.ED tethers activate and bring the momentum bank to appropriate altitude a. ED tether bank fails to activate

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B. Taxi 1.In transit to Luna and uses supplies appropriately a. ECLSS systems are damaged/fail 2.Makes corrective burns as required a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure 3.Arrives at Luna and makes ready for rendezvous with tether 4.Lunar tether slows to receive taxi 5.Achieves correct speed to rendezvous with lunar tether a. Tether speed is incorrect to meet taxi without harm b. Taxi speed is in incorrect to meet taxi without harm 6.Rendezvous with lunar tether a. Taxi/ Tether connection fails structurally b. Taxi/ Tether misses the catch 7.Crew is disembarked from taxi a. Airlock does not seal correctly, crew cannot disembark b. Depressurization c. Crawler is inoperable cannot be used II. Luna to Cycler A. Lunar Tether Sling 1.Taxi is resupplied from lunar stockpiles 2.Reach Required ΔV for Luna to Cycler leg a. ΔV reading is incorrect and taxi is launched b. Tether undergoes structural failure during spin up c. Motor Failure 3.Crew is embarked onto the taxi a. Airlock does not seal correctly cannot embark b. Depressurization c. Crawler is inoperable cannot be used 4.Achieve required orbital parameters for launch a. Launch window is missed, must wait for next one b. Parameters are incorrect and launch occurs 5.Taxi is released and begins journey a. Failure to release b. Release is unintentionally delayed c. Release causes damage to Taxi/Tether connection B. Taxi 1.In transit to Cycler and uses supplies appropriately a. ECLSS systems are damaged/fail 2.Makes corrective burns as required a. Propulsive system fails to activate b. Propulsive system burns incorrectly

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c. Propulsive system suffers critical failure 3.Reaches cycler and begins for rendezvous with Cycler a. Misses cycler location 4.Reaches appropriate spin rate to dock with cycler 5.Navigates to correct location for docking procedure a. Taxi maneuvering causes damage to cycler Cycler 6.Docking arm is extended to appropriate position a. Docking arm fails 7.Docking arm successfully attaches to taxi in a way as to not cause unnecessary force on crew a. Docking arm movements cause damage to taxi/ arm 8.Docking arm moves taxi to airlock a. Structural failure in docking arm 9.Airlock is sealed correctly on both ends a. Airlock is not sealed correctly, cannot disembark b. Depressurization 10. Crew is unloaded a. Harm occurs to crew during unloading 11. Crew move to quarters and prepare for journey 12. Adjusts to taxi and crew changes to weight and spin stability a. Failure of Gyros/ RCS 13. Journey is continued along cycler orbit 14. Journey to Mars is conducted using stores and supplies appropriately a. Hydroponic systems function appropriately i. Hydroponic system fails, supplies are insufficient b. Waste management systems function appropriately i. Waste management system fails, waste piles up c. Radiation Protection systems function appropriately i. Radiation systems fail during radiation event d. Power systems function appropriately i. Main power systems fail ii. Backup power systems fail (May need to add these) e. Thermal Systems function appropriately i. Environmental control systems fail ii. Equipment thermal management systems fail f. Structural integrity is maintained i. Tether to superstructure undergoes structural failure ii. Habitation modules undergo structural failure g. Communication systems remain online i. Communication systems fails/ is damaged 15. Makes corrective ion pulses as required a. Ion propulsion system fails/ is damaged

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16. Cycler arrives at location for departure a. Cycler in off course III. Cycler to Mars A. Initialization 1.Supplies are loaded onto the taxi a. Harm occurs to crew during loading 2.Crew and passengers enter the taxi and prepare for journey 3.Airlock is successfully sealed on both ends a. Airlock is not sealed correctly, cannot embark b. Depressurization 4.Docking arm separates taxi from Cycler a. Arm experiences structural failure 5.Docking arm releases from taxi a. Arm causes damage to taxi 6.Taxi moves clear of the cycler a. Taxi causes damage to cycler as it maneuvers 7.Taxi begins journey to Phobos B. Taxi 1.In transit to Phobos and uses supplies appropriately a. ECLSS systems are damaged/fail 2.Makes burns as required to enter the correct trajectory a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure 3.Successfully tracks trajectory and arrives at mars a. Taxi is off course 4.Makes burns to enter Phobos orbit a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure C. Phobos Tether Sling 1.Arrives at Phobos and makes ready for rendezvous with tether 2.Phobos tether sling makes ready to receive taxi 3.Achieves correct speed to rendezvous with phobos tether a. Tether speed is incorrect to meet taxi without harm b. Taxi speed is in incorrect to meet taxi without harm 4.Rendezvous with Phobos tether a. Taxi/ Tether connection fails structurally b. Taxi/ Tether misses the catch 5.Phobos tether sling slows taxi to required speed for Martian entry a. ΔV reading is incorrect and taxi is launched b. Tether undergoes structural failure during spin up c. Motor Failure

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6.Achieves optimal angle to deorbit a. Angle is too shallow b. Angle is too steep 7.Begins to deorbit from Martian orbit 8.Successfully travels through Martian atmosphere a. Failure in heat shielding i. Nose Cone probably point of failure ii. Other Heat Shield Failing 9.Tether Sling is correctly spun up for rendezvous a. Tether Sling fails to start, motor failure 10. Levels off and prepares to rendezvous with Martian tether. a. RCS system fails 11. Taxi successfully engages with tip of Martian tether a. Taxi is at an incorrect speed to meet taxi w/out harm b. Tether is at an incorrect speed to meet taxi without harm c. Taxi/ Tether connection fails structurally d. Taxi/ Tether misses the catch e. Tether experiences structural failure due to heat load. D. Martian Tether 1.Begins Spin down of taxi a. Tether experiences structural failure due to heat load b. Taxi heat shielding fails 2.Reaches required spin rate to deposit taxi on track a. Overshoots required velocity, taxi crashes into ground 3.Taxi engages with track cart a. Taxi is damaged on touch down with cart b. Taxi is put down too hard, Cart is pushed into track causing immense friction 4.Taxi disengages from tether sling a. Snap up of tether sling damages taxi b. Snap up of tether sling causes failure in tether sling c. Tether sling fails to disengage 5.Tether sling tip is clear of taxi 6.Taxi is moved to shuttle center 7.Crew is disembarked from taxi a. Airlock is not sealed correctly, cannot disembark b. Depressurization

Mars Phobos Cycler Luna LEO

IV. Mars to Phobos (Mass Driver)

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A. Initialization 1.Taxi is supplied from Martian stockpiles or manufacturing facilities 2.All systems are checked out and all crew are suitably strapped down a. Crews or materials are not suitably strapped down for launch 3.Taxi is correctly loaded into Martian mass driver system a. Mounting of taxi system incorrect, causes incorrect acceleration 4.Mass Driver is turned on and begins to accelerate craft a. Magnetic forces are imbalanced causes craft to accelerate in unexpected direction b. Power loss causes craft to fail to accelerate c. Shock from sudden electricity shorts an important system 5.Travel the length of the Olympus Mons mass driver without issue a. Wheels do not retract properly b. Power loss causes magnetic forces to go down 6.Reach optimal speed for travel to Phobos a. Speed is too low b. Speed is too high 7.Successfully exit the mass driver a. Cart fails to separate from taxi 8.Successfully travel through Martian atmosphere a. Heat shielding fails at some point on mass driver 9.Makes first plane change burn (Recoverable) a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure 10. Makes second burn to enter Phobos orbit (More Dire) a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure V. Mars to Phobos (Tether Sling) A. Initialization 1.Taxi is supplied from Martian stockpiles or manufacturing facilities 2.All systems are checked out and all crew are embarked a. Material in incorrectly secured 3.Tether sling begins accelerating to required length to meet taxi on track a. Tether sling whips at start up causing damage to itself or hub 4.Taxi moves along maglev track to meet tether tip speed a. Magnetic forces are imbalanced causes craft to accelerate in unexpected direction b. Power loss causes craft to fail to accelerate c. Shock from sudden electricity shorts an important system(may remove 5.Taxi rendezvous with tether tip a. Shock of connection causes damage to connector on taxi or tether

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6.Taxi separates from maglev cart a. Taxi fails to separate B. Tether Sling 1.Accelerates to reach required ΔV for journey to Martian orbit a. ΔV reading is incorrect and taxi is launched b. Tether undergoes structural failure during spin up due to high heat load 2.Achieve required orbital parameters for launch a. Launch window is missed, must wait for next one b. Parameters are incorrect and launch occurs 3.Taxi is released and begins journey a. Failure to release b. Release is unintentionally delayed c. Release causes damage to Taxi/Tether connection 4.Successfully travel through Martian atmosphere a. Heat shielding fails at some point on mass driver 5.Makes first plane change burn (Recoverable) a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure 6.Makes second burn to enter Phobos orbit (More Dire) a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure VI. Rendezvous with Phobos Tether 1.Enter Phobos orbit at correct orbital parameters for tether sling a. Orbital parameters are incorrect 2.Tether sling is at correct speed to meet with taxi a. Taxi is at incorrect distance to meet tether b. Tether is at an incorrect speed to meet taxi without harm 3.Taxi successfully rendezvous with end of tether sling a. Taxi/ Tether connection fails structurally b. Taxi/ Tether misses the catch 4.Taxi is successfully rotating about tether sling 5.Taxi is resupplied and continues spinning VII. Phobos to Cycler 1.Reach Required ΔV for Phobos to Cycler leg a. ΔV reading is incorrect and taxi is launched b. Tether undergoes structural failure during spin up c. Motor Failure 2.Achieve required orbital parameters for launch a. Launch window is missed, must wait for next one b. Parameters are incorrect and launch occurs

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3.Taxi is released and begins journey a. Failure to release b. Release is unintentionally delayed c. Release causes damage to Taxi/Tether connection B. Taxi 1.In transit to Cycler and uses supplies appropriately a. ECLSS systems are damaged/fail 2.Makes corrective burns as required a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure 3.Reaches cycler and begins for rendezvous with Cycler a. Misses cycler location 4.Reaches appropriate spin rate to dock with cycler 5.Navigates to correct location for docking procedure a. Taxi maneuvering causes damage to cycler C. Cycler 1.Docking arm is extended to appropriate position a. Docking arm fails 2.Docking arm successfully attaches to taxi in a way as to not cause unnecessary force on crew a. Docking arm movements cause damage to taxi/ arm 3.Docking arm moves taxi to airlock a. Structural failure in docking arm 4.Airlock is sealed correctly on both ends a. Airlock is not sealed correctly, cannot disembark b. Depressurization 5.Crew is unloaded a. Harm occurs to crew during unloading 6.Crew move to quarters and prepare for journey 7.Adjusts to taxi and crew changes to weight and spin stability 8.Journey is continued along cycler orbit 9.Journey to Earth is conducted using stores and supplies appropriately a. Hydroponic systems function appropriately i. Hydroponic system fails, supplies are insufficient b. Waste management systems function appropriately i. Waste management system fails, waste piles up c. Radiation Protection systems function appropriately i. Radiation systems fail during radiation event d. Power systems function appropriately i. Main power systems fail ii. Backup power systems fail (May need to add these) e. Thermal Systems function appropriately

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i. Environmental control systems fail ii. Equipment thermal management systems fail f. Structural integrity is maintained i. Tether to superstructure undergoes structural failure ii. Habitation modules undergo structural failure g. Communication systems remain online i. Communication systems fails/ is damaged 10. Makes corrective ion pulses as required a. Ion propulsion system fails/ is damaged 11. Cycler arrives at location for departure a. Cycler in off course VIII. Cycler to Luna A. Initialization 1.Supplies are loaded onto the taxi a. Harm occurs to crew during loading 2.Crew and passengers enter the taxi and prepare for journey 3.Airlock is successfully sealed on both ends a. Airlock is not sealed correctly, cannot embark b. Depressurization 4.Docking arm separates taxi from Cycler a. Arm experiences structural failure 5.Docking arm releases from taxi a. Arm causes damage to taxi 6.Taxi moves clear of the cycler a. Taxi causes damage to cycler as it maneuvers 7.Taxi begins journey to Luna B. Taxi 1.In transit to Luna and uses supplies appropriately a. ECLSS systems are damaged/fail 2.Makes corrective burns as required a. Propulsive system fails to activate b. Propulsive system burns incorrectly 3.Propulsive system suffers critical failure a. Tether speed is incorrect to meet taxi without harm b. Taxi speed is in incorrect to meet taxi without harm 4.Rendezvous in lunar tether a. Taxi/ Tether connection fails structurally b. Taxi/ Tether misses the catch 5.Crew is disembarked from taxi a. Airlock does not seal correctly, crew cannot disembark b. Depressurization c. Crawler is inoperable cannot be used IX. Cycler to LEO

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A. Initialization 1.Supplies are loaded onto the taxi a. Harm occurs to crew during loading 2.Crew and passengers enter the taxi and prepare for journey 3.Airlock is successfully sealed on both ends a. Airlock is not sealed correctly, cannot embark b. Depressurization 4.Docking arm separates taxi from Cycler a. Arm experiences structural failure 5.Docking arm releases from taxi a. Arm causes damage to taxi 6.Taxi moves clear of the cycler a. Taxi causes damage to cycler as it maneuvers 7.Taxi unspins itself and begins journey to LEO B. Taxi 1.In transit to LEO and uses supplies appropriately a. ECLSS systems are damaged/fail 2.Makes corrective burns as required a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure 3.Arrives at LEO and makes ready for rendezvous with ED tether a. Taxi is off course 4.ED tether reaches speed to meet taxi a. Propulsive system fails to activate b. Propulsive system burns incorrectly c. Propulsive system suffers critical failure 5.Achieves correct speed to rendezvous with ED tether a. Tether is at an incorrect speed to meet taxi without harm b. Motor Fail 6.Rendezvous with ED tether a. Taxi is at incorrect altitude to meet tether b. Taxi/ Tether connection fails structurally c. Taxi/ Tether misses the catch

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