ANALYSIS of PERFORMANCE of SOLAR SAIL SPACECRAFT a Thesis Presented to the Faculty of the Department of Mechanical Engineering C

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ANALYSIS OF PERFORMANCE OF SOLAR SAIL SPACECRAFT

A Thesis

Presented to the faculty of the Department of Mechanical Engineering

California State University, Sacramento

Submitted in partial satisfaction of the requirements for the degree of

MASTER OF SCIENCE

Mechanical Engineering

David Shira

FALL 2019

David Shira

ANALYSIS OF PERFORMANCE OF SOLAR SAIL SPACECRAFT

A Thesis

David Shira

Approved by:

______, Committee Chair Ilhan Tuzcu, Ph.D.

______, Second Reader Akihiko Kumagai, Ph.D.

______Date

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Student: David Shira

I certify that this student has met the requirements for format contained in the University format manual, and that this thesis is suitable for shelving in the Library and credit is to be awarded for the thesis.

______, Graduate Coordinator ______Troy D. Topping, Ph.D. Date

Department of Mechanical Engineering

Abstract

ANALYSIS OF PERFORMANCE OF SOLAR SAIL SPACECRAFT

David Shira

Solar sails offer potential for future space flight without the need for added fuel. Science has nearly reached the limits of the chemical effectiveness of rocket fuel and long term and deep space missions will require an alternate propellant. Solar sails solve this need without adding mass, thereby allowing more scientific payload. Applications have been limited mostly to orbits of Earth and analysis has been done as a rigid body. For long term missions, particularly including a rendezvous with other bodies, a high degree of accuracy is necessary, and as such, flexible analysis will be needed. A sail was analyzed in a solar orbit at 1 AU sufficiently far from Earth to avoid gravitational pull. Finite element modeling was used for a square sail, a design primarily used in current applications, with the payload concentrated at the center of the sail. A very small amount of deformation was seen symmetric about the center, resembling a wide bowl. This decreased slightly the effective area of the sail and thus lowered the acceleration. The tension in the sail material was varied while all other variables were kept constant. For the initial value of tension, a 26% difference in acceleration for the flexible analysis was observed. The values observed indicate that a flexible analysis of the sail will be required

to accurately project a long-term trajectory of a deep-space solar sail. Potential applications include reaching the Kuiper Belt in as little as six years and reaching the

Oort cloud within our lifetime.

______, Committee Chair Ilhan Tuzcu, Ph.D.

______Date

ACKNOWLEDGEMENTS

To my wonderful wife, Becca, for being so supportive through this journey. You encouraged me to apply and were an amazing support throughout all of this. Along the way, we added a beautiful daughter, Abigail, and lost my appendix. There is another one at the end of this paper.

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TABLE OF CONTENTS Page Acknowledgments ...... vii List of Figures ...... ix Chapter 1 INTRODUCTION ...... 1 1.1 History of Space Flight ...... 1 1.2 Fundamentals and Limitations of Rockets ...... 2 1.3 Alternative Space Flight Methods ...... 4 2 BACKGROUND ...... 7 2.1 Solar Sail Concept ...... 7 2.2 Sail Design ...... 9 2.3 Potential Applications ...... 13 2.4 Relativistic Effects ...... 19 2.5 Past and Future Missions ...... 21 3 EXPERIMENT ...... 25 3.1 Equations of Motion ...... 25 3.2 Analysis of Body ...... 31 3.3 Body Specifications ...... 34 4 RESULTS ...... 36 5 CONCLUSION ...... 40 6 FURTHER STUDY ...... 41 Appendix ...... 43 References ...... 59

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LIST OF FIGURES

Figures

1-Layout of various solar sail designs ...... 11

2-Earth-Jupiter-Sun trajectory proposal ...... 14

3-Lagrange Point of Earth ...... 16

4-Comparison of perihelion and lifetime Δv ...... 20

5-Apogee and Perigee over time of LightSail 2 ...... 23

6-Cubesat section of LightSail 2 with stowed sails ...... 24

7-FEA model of Solar Sail ...... 33

8-Natural Modes of Vibration of a sail ...... 37

9-Tension of sail vs. Flexible-Rigid Error ...... 38

10-Tension vs acceleration of a flexible sail ...... 39

CHAPTER 1

INTRODUCTION

1.1 History of Space Flight

Space exploration has always been a desire of mankind. Well before Nicolaus

Copernicus discovered the heliocentric solar system and that the universe did not rotate around the Earth, legends of ancient civilizations desiring to explore the stars have existed. The tale of Icarus flying too close to the sun has origins of over 4,000 years ago

[1] and demonstrates the yearning to explore objects far from Earth.

But the origins of the rocket, the main source of propulsion in today’s space crafts, date

back to at least 50 BC when Hero of Alexandria constructed the aeolipile, a suspended

sphere that was allowed to spin by torque generated from steam. Some ancient scientists

saw this as a model for how wind works but it is not clearly known if there was a

practical use for this or it was more of a gimmick. Either way, it is generally accepted

that these ancient civilizations did not understand the fundamentals of rocket mechanics.

Regardless, this early device was an ancestor to the rocket [1], [2].

China was the first to utilize the rocket for warfare in 900 AD, but it wasn’t until the late

19th century that they were proposed for space travel. Konstatine Tsiolkovsky, a math

teacher in rural Russia, proposed not only using rockets to escape earth, but also

theorized the idea of the solar sail as early as the 1920s [1].

Leaps forward in earth-bound aviation from balloons to controlled flight in the late 19th and early 20th centuries assisted in seeing this dream realized. Rockets developed by the

Germans toward the end of World War II, as well as the Cold War between the United

States and Russia pushed forward many advances that enabled mankind to explore the depths of space.

1.2 Fundamentals and Limitations of Rockets

Rockets employ the fundamentals of conservation of momentum, = , to not only escape Earth, but to gain velocity and make trajectory adjustments𝑝𝑝 while𝑚𝑚𝑚𝑚 in space. Best described as a controlled bomb exploding in one direction [1], ignited chemical thrust is pushed out of a nozzle on the bottom of the rocket, pushing the spacecraft in the opposite direction until orbit, escape velocity, or trajectory correction is accomplished.

There are several limits to this approach of space travel. Safety is a large concern. When

NASA first proposed to launch men into space on top of a rocket, current rocket technology was such that half of all launches failed in some way [3]. Advances in our understanding of rockets have improved since then, but accidents do still occur during rocket utilization, most notably, the Challenger disaster where a failed O-Ring seal caused hot liquid fuel to penetrate an external tank and ignite, destroying the shuttle and killing all passengers 71 seconds after launch [4].

Expense is also a factor. NASA launches dispose of rockets after one launch. SpaceX is

3 attempting to master the reusable rocket to reduce costs by a controlled landing of the rocket after separating from the payload, but the technique is not yet perfected. Even with this expense removed, the fuel to achieve liftoff is quite large.

Thrust from rockets are inherently limited by the mass of the fuel contained in the rockets. Once the rocket burns all its fuel, no more thrust can be created. The only way to create more thrust is to develop larger rockets that contain more fuel. This is cost prohibitive, as the added mass increases the necessary thrust to move the rocket. The needed mass ratio of payload to thrust increases exponentially and will eventually converge to an infinite amount of thrust needed to lift the rocket [1].

Of course, by the conservation of momentum, if one increased the velocity which the thrust exits the rocket, that would increase the momentum that can be used to move the rocket in the opposite direction, either lifting it from earth or increasing its space trajectory velocity. However, it appears that rocket nozzle design and chemical reactions are near their performance limit [1].

If long range missions are desired, most notably exploration of the Kuiper Belt, the Oort

Cloud, or Alpha Centauri, over four lightyears from earth, space agencies must look at alternatives that produce enough total thrust to accelerate space craft to much higher velocities.

Some alternatives have been experimented with to reach escape velocity, including the nuclear rockets. Environmental pollution and fallout are a serious consideration, as the added expense to contain the radiation makes this an impractical approach. Lenses can be added to create a solar-thermal rocket, using the power of the sun to heat the exhaust molecules, but this creates an acceleration of 1% of Earth’s acceleration, making it impractical to get off the ground [1].

1.3 Alternative Space Flight Methods

While chemical rocket technology still appears to be the best method with our current science to leave earth, there are promising new technologies that can be utilized while in spaceflight that do not use rockets. The idea is to place a payload on top of a rocket.

Once the craft has left earth, it would detach from the rocket and employ a secondary form a thrust, one that may not work under Earth’s influence of gravity but a low thrust option viable while in space.

Once a spacecraft has left earth, whether in orbit or on another trajectory, atmospheric drag declines significantly, even disappearing after certain distances. Gravitational acceleration experienced near the surface becomes less of a factor as well. In fact, depending on trajectory goals, gravitational influences of other celestial bodies can be used advantageously, most notably slingshots around the sun or Jupiter. Because of the declining influence of these factors on a spacecraft, low thrust alternatives become a viable option.

Some proposals involve creating a thermo-nuclear rocket, essentially detonating a nuclear

device on the back side of a rocket, propelling the payload the opposite direction. A main

issue in this approach is the added mass to protect passengers on board. This added mass

slows down the gained acceleration significantly [1].

Ion drives are another proposal. The method employs ionizing fuel by an

electromagnetic field to expel the fuel at very high speeds. This is a comparatively safe

but very low thrust maneuver, about 0.01% of Earth’s acceleration so this would need to

be employed for long periods to achieve a high velocity [1]. The novel, The Martian,

explored this concept. On a mission to Mars, the Hermes, the spacecraft carrying the

passengers, employed ion thrusters to slowly accelerate the craft to the Martian orbit.

High velocity was achieved with minimal thrust. The velocity was high enough that the

deceleration period would take one month to slow the craft enough to re-enter Earth’s orbit [5].

While the novel theorized that ion drives could be used for long manned missions, ion drives are not merely the work of science fiction. Not just a subject of science-fiction, the European Space Agency employed an ion drive on the SMART-1 in 2003 to launch a lunar probe. This probe reached the moon in 14 months but only used 59 kilograms of propellant [1].

This method, like the others, are not without its problems. This method requires a fuel

that ionizes easily. But if a fuel ionizes quickly and easily, obtaining the fuel in a non-

ionized state creates its difficulties. Storing the fuel for long missions is also

problematic. Inherently, the fuel ionizes easily and is sometimes unstable unless pricy

measures are employed to isolate it. Many of the ideal candidates, particularly cesium

and mercury, are toxic and environmental regulations make this difficult. Xenon appears to be the ideal candidate. It is an inert noble gas, very stable and ionizes easily.

Unfortunately, the gas is very rare on Earth, limiting the application of a long-range

spacecraft [1].

CHAPTER 2

BACKGROUND

2.1 Solar Sail Concept

All the propulsion methods mentioned in the previous section rely on a mass to propel the

craft. A major drawback in this is that one can only design a propulsion system with

finite mass and propellant will eventually run out. Light itself shows promise to

accelerate spacecraft to speeds high enough to reach interstellar space, perhaps in our

lifetime. It has been long theorized that light not only exhibits behavior like a wave, but

also as a particle. Like a moving particle, it contains energy, in the case of light,

proportional to its wavelength. It was theorized by Planck that light was made up of

individual quanta that carried energy, and further explained by Einstein theory of

relativity [1]. This energy and momentum of light was characterized by Planck’s Theory in the equations below.

= (1) ℎ𝑐𝑐 𝐸𝐸 𝜆𝜆 = (2) ℎ 𝑃𝑃 As such, light could transfer momentum with anything𝜆𝜆 it came in contact. Put a different

way, light particles, known as photons, can interact with objects in a Newtonian manner.

This effect is commonly referred to as Solar Radiation Pressure (SRP) and has the

potential to dwarf the effects of solar wind or power generated by solar panels.

This effect is more difficult to see on Earth. The momentum transfer is rather small and

8 even a little air resistance inside Earth’s atmosphere is enough to dim its effects significantly. However, in space, the effect, small as it may be, is much more observable and grow cumulatively over time. An example of this is the VIKING program. NASA launched VIKING 1 and VIKING 2 in the 1970s. Had they ignored the effects of SRP, the probes would have missed Martian orbit by about 9,300 miles [6]. Of course, with the law of conservation of momentum, the lighter the spacecraft, the greater the transfer of momentum and therefore, the greater the effect. Manned missions over shorter distances, the moon for example, would experience significantly less effects of this phenomenon.

The intensity of the light is also a factor in how much momentum is exchanged. The sun is a great momentum exchanger in space, at least in the inner solar system. As the sun radiates light in all directions simultaneously, the amount of energy over a given surface of the sun is a finite amount. As the sunlight travels, the surface area of the light it produces increases but the energy emitted in the original surface is constant. As such, the amount of energy concentration over a given surface decreases as the distance from the original light source, the sun in this case. Because of this, the momentum exchange is greatly reduced in the outer solar system.

As the momentum of light is proportional to Planck’s constant (order of 10-34) divided by wavelength (visible light being of the order 10-7), it can quantitatively be shown that the momentum transfer of a perfectly reflective sail is of the order 10-27, rather very small.

A few factors must be considered to design an effective solar sail in addition to the material properties that all space vehicles must possess to survive in the extreme environments of space

2.2 Sail Design

Sail area is very important. The larger the scale, the more photons will interact with the sail and as such, more momentum will be transferred. This increases costs not only for material, but construction and stowing of the sail. As will be discussed later, deployed sail area must be many times greater than the stowed area.

Of course, mass must be considered. Manned missions with enough room and life support to sustain humans will require sails with widths measured in kilometers, while sails for small space probes can be measured in meters. Mass of the payload must be minimized to achieve the greatest effect. As such, the sail must also be very thin. We are in the early stages of development of sails that have been launched into space and with current material science we are limited to sails being measured in microns [7]. Future sails may be measured in nanometers which, for a sail of great size, has the potential to greatly reduce the mass of the sail [1].

For calculation purposes, the mass of the spacecraft is measured as a combination of the mass of the sail and the payload per the sail area as shown below

t + m = (3) A ρsail sail payload σ sail The payload needs to be minimized as much as possible, while the structure of the sail must be strong, but light. Carbon fibers are promising in this area as their lightweight structure is very sturdy and can withstand the harsh environment of space [1].

The reflectivity of the material must also be considered. While absorbed light will generate thrust, it is analogous to an inelastic collision in classical mechanics. Causing the light to reflect light back would generate the effect of an elastic collision causing the struck object to gain more momentum. It is of course difficult to reflect 100% of the light, and some of the absorbed light will be absorbed or emitted on the other side of the sail, causing the sail to lose momentum. This results partially in the Poynting-Robertson

Effect which will be discussed later. As such, making the sail as reflective as possible is important.

The sail must also be as flat and wrinkle-free as possible. Wrinkles and surface roughness will cause photons to be reflected in unpredictable directions and will greatly affect the performance over long periods.

The shape of the sail has many possibilities and the supporting structure vary for the designs, and each have their advantages. A summary of these shapes is shown in Figure

1 below.

Figure 1-Layout of various solar sail designs [1]

The heliogyro has an advantageous shape for deployment. By simply rotating the craft, centrifugal acceleration will cause deployment. However, as one can easily see, there is significantly less surface area. To generate equivalent thrust, the sails would need to be extremely long.

The spinning disc sails uses a similar approach while maintaining a larger surface area and mounts the payload in the center. The hoop support is also similar except its structure is on the outside. The payload would need to be mounted on the outer rim of

12 the sail and as such, great care would be needed to distribute the mass of the payload evenly so as not to create a dynamic imbalance.

Parachute sails suspend the payload on the sun facing side. High strength cables are needed to suspend the payload and must be very lightweight. The parabolic sail, a variation of the parachute sail contains two sails and operates so that one sail is always generating thrust. The main disadvantage of these is the complexity of the structure. As suggested by Vulpetti, et al [1], these may be more utilized if space agencies were able to construct in orbit rather than launching them from Earth and attempting to unfurl after launch.

Not shown above is the lattice structure. This design would be similar to a square sail but composed of strong, lightweight wires which are interwoven like a tapestry. The gaps between the wires would be less than the wavelength of light, ensuring the sail receives maximum momentum from photons. While this design is extremely light, manufacturing is expensive and time consuming and is not currently a practical design.

To date, the shape utilized with the most success is the square sail. This shape extends booms diagonally from the normal surface of the payload and unfurls the four sails laterally.

Attitude control is very important in the operation of the sail. Ground control or a pre-

loaded program based on intended trajectory, or perhaps some sort of artificial

intelligence, must be able to rotate the craft about all three axes so that the sail can

receive thrust in the intended directions. Early sails are achieving this using gyroscopes

and magnetic torque rods based on light data received from solar sensors. Another

possibility is to have a motorized weight that will move about the sail to achieve proper

rotation [8].

Attitude control is analogous to adjusting sails on a sailboat. While it is not possible to gain thrust sailing directly into the wind or the direction of sunlight, one can manipulate the sails of a boat and a solar sail to receive thrust at a different angle to manipulate thrust and direction. Orienting the sail so that sunlight will collide with the edge of the thin layer minimizes the loss of momentum by a sail when facing or traveling toward the sun

[1].

2.3 Potential Applications

As noted earlier, long term and interstellar missions are limited by the mass of propellant

in a traditional rocket or other methods using mass propellant. Drag in space is negligible

for distances sufficiently far from large bodies so a spacecraft will not slow down, but its

acceleration will stop when propellant runs out. Adding more propellant becomes a very

pricey endeavor. As previously mentioned, some of the added propellant must be used to

launch the craft and the momentum exchange is less impactful with more fuel in the

rocket. Conversely, wherever there is light, there is available thrust for a solar sail with

no added mass. This makes solar sail use for interstellar missions plausible.

Utilizing a solar slingshot seems to be the most efficient method. Rockets could send it to the sun using traditional methods. Once the craft reaches its perihelion, its closest distance to the sun, it can slingshot around the sun, maximizing orbital velocity, then deploy its sails to receive thrust in the most intense region of space, near the sun.

Increasing the eccentricity of the flyby is the best way to make a close approach to the sun while minimizing the energy lost in escaping its gravity well. To increase the eccentricity, Ancona [9] suggests a Jupiter slingshot initially before going to the sun as shown in Figure 2

Figure 2-Earth-Jupiter-Sun trajectory proposal [9]

Another possibility is harnessing desorption. On a mission to include a solar slingshot,

one could coat the surface of the sail with a thin layer of special material. This material

would have a lower desorption temperature than the sail itself. Once the sail reaches a

close enough distance to the sun and therefore a high enough temperature, the thin layer

would desorb. This process is assumed to be less than one second and in the opposite

direction of the motion of the sail. This procedure would increase the initial velocity at

the perihelion of the trajectory as the mass of the desorbed material exchanges

momentum with the sail. This would increase the velocity to a value that would

eventually be reached by sailing alone but much quicker.

Embracing these techniques, it becomes possible for a spacecraft to reach the Kuiper Belt

in as little as 6 years, about half the time required for a rocket to accomplish this [9].

Rocket missions would need to continually expel fuel to counter the gravitational effects

of certain missions. Solar sails make these missions possible by adjusting the sails to

increase thrust in a given direction. Doing so in any type of orbit near the sun is

obviously very easy due to the proximity of the star. This does become a little tricky for

exploration of the outer solar system, but not impossible.

Non-Keplerian orbits of large bodies become much easier as well. Spacecraft launched from Earth will stay very near the equatorial plane of the solar system and using rocket thrust to move an orbit up or down is limited to the propellant available. Sails could be used to guide a craft to an orbital plane much closer to the poles [10].

Additionally, stationary positions can be held above the pole of the sun or other celestial

body by continually adjusting sails. This would enable a spacecraft to be stationary at a

pole and continually monitor the same section of the sun. Current knowledge of the sun

at its poles is rather limited to orbits near its equatorial region and being able to

manipulate the sails to explore polar regions would dramatically increase our knowledge of the sun [1], [10].

Figure 3-Lagrange Point of Earth [11]

Solar sails have the potential to act as if they have a closer L1 Lagrange point between

Earth and the sun. Lagrange points are points in Earth’s orbit or in line with Earth and the sun where an object will be in a gravitational balance of Earth and the sun and will

orbit at Earth’s rate. There are five such points in an orbit shown in Figure 3 [11].

The L1 point of Earth and sun is 0.01 AU from Earth. If a sail were placed in orbit

between Earth and the sun, its L1 point could effectively be significantly closer to the sun due to the continual thrust the sail could generate. As before, this is not possible with a rocket mission where thrust is finite [1].

The benefits of this are many, most notably in detecting solar flares and storms where charged particles and plasma erupt from the sun and travel to Earth at speeds of 1,000 km per second. These storms interfere with communication satellites and electrical systems.

Particularly, this is dangerous for flying aircraft being unable to communicate with ground control. With an L1 point of 0.99 AU, warning times of solar storms are limited to 1 hour. This offers very little time to ground aircraft and take other necessary precautions. Theorized by Vulpetti et al, a sail with a radius of 230 meters could be placed at 0.7 AU and be in continual orbit with Earth. Solar storm warning times could be increased up to 31 hours, allowing ample time to take precautions [1].

One could take this a step further and place solar sails at L3, L4, and L5, pointing the

sails to have points closer to the sun than traditional satellites and could continually

monitor the sun for flares and storms and further increase knowledge of these events.

Mercury, while close to Earth, currently has been mostly unexplored. Due to its close

proximity to the sun, any satellite close enough to orbit Mercury will be drawn strongly

to the sun. As such, a rocket mission requires great cost to explore the planet for

extended periods. However, a solar sail would enable a long-term observation without being too drawn toward the sun. The MESSENGER mission to Mercury actually utilized

SRP on the solar panels to make minute attitude adjustments that were not feasible with

rocket thrust [12].

While Mars has been explored, no samples have been returned. One proposal is to sail to

Mars, collect a sample, and return. Thrust would only be required to escape Earth’s orbit,

land controlled on the Martian surface, and launch from Mars, then sail back to Earth.

This mission would require significantly less thrust and cost than a traditional mission

while still being completed in a reasonable amount of time [12].

Comets and asteroid rendezvous become more practical as well. Accelerating a rocket to

catch an object, slowing down to match the velocity, and controlling landing all require

significant thrust, and every kilogram of thrust is less room for scientific payload. Solar

sails could be used to meet these objectives at significantly less expense [1].

Moreover, a solar sail has the ability to explore multiple bodies. One could adjust the

sails of the craft to create an orbital transfer. Minimal rocket thrust would be required to

land the payload and relaunch from the surface, but orbital transfers would be simple and

would require no rocket thrust [1].

2.4 Relativistic Effects

An interesting phenomenon occurs with small, light objects, such as a solar sail known as

the Poynting-Robertson Effect. This only affects objects in a bound orbit with the sun

and takes many orbits to see the long-term impact of the effect. This originates from

absorbed radiation at an angle with respect to the radial direction. The result is a force

component in the direction opposite of motion, slowing down the orbit and causing the

object to slowly spiral inward until a solar impact. It is assumed that objects smaller than

1 micrometer will be blown away by SRP and that larger objects collide with a different

object before this occurs [13].

But for solar sails, the effect is noticeable. This was studied by Kezerashvili and

Vazquez-Portiz [13]. For an orbit of 0.5 AU, a sail reflective efficiency of 0.85, and a

sail density of 0.00131 kg/m2, the effect can be negated by adjusting the pitch angle with

respect to the radial direction 5.6 x 10-6 radians. This level of precision is very difficult to achieve under the best of circumstances, and as suggested by Kezerashvili and

Vazquez-Portiz, it is more realistic to make periodic adjustments to the sail to correct the orbit [13].

For escape trajectories, the effect is much more noticeable. Solving equations derived in

Section 3 results in a comparison of one-day cruising velocity (ΔV) for varying

perihelion distances and initial velocities is shown in the graph below for sails with

2 σ=0.003 kg/m and v0 values calculated by Kezerashvili [14].

Figure 4-Comparison of perihelion and lifetime Δv, graphed using equations in Section 3.1 With the assumption of accelerating mostly in the first day of sailing [14], this small difference has a very large cumulative effect. Over a 30-year journey, a sail with the parameters mentioned above with a perihelion of 0.1 AU, this translates to a difference in solar distance of about 2.6 million kilometers. This is a result of only 2.8 m/sec difference in cruising velocity. Of course, if a rendezvous with a Kuiper belt object rotating about the sun, such as Pluto, is desired, overlooking this effect will certainly result in missing the object.

Another phenomenon to be aware of is the Lense-Thirring effect. This is a relativistic effect of rotating large bodies, such as Earth or the sun. Due to angular momentum, large

rotating bodies will tend to have an equatorial bulge, a region near the center that bulges,

making the shape oblate. The sun and Earth contain such regions. An object rotating in a

non-Keplerian orbit will tend to orbit these sections due to the increased mass and

therefore higher gravitational pull. However, as noted by Kezerashvili [13], this is a

small effect, only 1.45 x 10-7 radians per year and like the Poynting Robertson Effect, it

may be easier to make periodic adjustments to correct any desired trajectories.

2.5 Past and Future Missions

Like many other components to space exploration, initial tests resulted in various failures.

Antennas snagged delicate sails, separation commands failed, and sails were unfurled incompletely due to communication issues [1].

The first real successful application in space was in 2004 by ISAS (Institute of

Astronautical Science), now unified with other agencies as JAXA (Japan Aerospace

Exploration Agency). Its sole purpose was to demonstrate capabilities of unfurling sails

in space. This was followed up by the IKAROS (Interplanetary Kite-craft Accelerated by

Radiation Of the Sun), launched by JAXA in 2010. Launched together on a probe to

Venus, the IKAROS separated and demonstrated the small but noticeable effects of SRP

on the spacecraft [15]. The IKAROS is presumably named after the Greek legend,

Icarus, who constructed wings of goose feathers and candlewax and met his demise by

flying too close to the sun.

NASA attempted to execute a mission to demonstrate the first solar sail deployment

while in orbit with the NanoSail-D. This craft was lost when a SpaceX rocket failed in

2008, but NASA had a backup craft, NanoSail-D2. NanoSail-D2, which was launched in

2010 successfully deployed its sails. NASA did not possess any control systems for the sail and it orbited uncontrolled for 240 days before falling out of orbit [16].

In 2015, the Planetary Society entered the solar sail race. This crowdsourced project first attempted a solar sail with the Cosmos 1. However, to save on costs, they utilized a surplus Russian rocket, which failed to reach orbit. LightSail 1 was the next attempt.

This craft reached orbit but due a communication error, unfurling of the sails was problematic. While the mission had its challenges, overall the test was declared a success

[16].

LightSail 2, launched June 25, 2019 [17], sought to demonstrate that a satellite in Earth’s orbit could increase its orbit gradually, in this case, 500 meters per day. On July 23,

2019, sails were successfully unfurled by small robotic motors over the course of several minutes [18], [19]. The craft with sails stowed was the size of a loaf of bread, approximately 4” x 4” x 20”. After deployment, the craft was the size of a boxing ring, about 18.4’ square. LightSail 2 successfully demonstrated orbital raising by light alone as shown in the Figure 5 below that shows the craft’s apogee and perigee.

Figure 5-Apogee (Blue) and Perigee (Orange) over time of LightSail 2 [25]

Note: Published drawings incorrectly indicate dimensions in inches, implying a sail

length of 462 feet [20]. Personal correspondence with the Planetary Society stated this was incorrect and dimensions were actually in millimeters [21].

The mass of LightSail 2 is about 5 kg and it has a sail area of 32 m2 [21]. This equates to

an areal density of 156 g/m2. This is much larger than the experimental numbers

typically used, which are around 1-3 g/m2. This is an acceptable number to demonstrate

the fundamentals of sailing and orbit raising, but for long term missions, this number will

need to drop significantly.

Figure 6 below shows the configuration of LightSail 2. Three magnetic torque rods and a

momentum wheel are used to control attitude. Attitude corrections are determined by

four magnometers and four sun sensors. These sensors consist of the four panels and an

additional panel on the rear of the craft. The nine inputs are averaged, and the attitude of

the craft is then adjusted to maximize thrust. This craft orbited Earth for nearly a month

before unfurling sails to ensure that attitude control could be monitored and corrected by

ground control [22].

Figure 6-Cubesat section of LightSail 2 with stowed sails [22]

Many analyses of solar sails are performed as a rigid model, even the LightSail 2. The complex six degree-of-freedom Simulink model of LightSail 2 included many aspects of the craft, including sensors, actuators, attitude control system, and the guidance system, but did not include any flexibility of the sail [22].

For experimental missions, such as focusing on sail deployment and orbit raising, this may be a valid estimation for modeling, but if manned missions and long-term missions are to be considered, the flexibility of the sail must be analyzed.

CHAPTER 3

EXPERIMENT

3.1 Equations of Motion

A sail receives acceleration in three components: light reflected, light absorbed, and light reemitted. Light can be reemitted on both sides of the sail and is a comparatively small portion of the light. If one assumes that light is emitted equally on the sun facing side as well as the anti-sun facing side, the effects of the reemitted light will have a net zero effect on the sail and can be ignored.

As noted earlier, the momentum and energy of a single photon is given in the equations

below

= (1) ℎ 𝑃𝑃𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝜆𝜆 = (2) ℎ𝑐𝑐 𝐸𝐸𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 Solar luminosity, Ls, is the amount of power, or the𝜆𝜆 quantity of photons given off by the

sun for a given time. While the sun is not a perfect sphere, it is a close approximation

and we will analyze it as such. The surface area of a sphere is given by the equation

below.

= 4 (4) 2 𝑠𝑠𝑠𝑠ℎ𝑒𝑒𝑒𝑒𝑒𝑒 The quantity of photons emitted over𝑆𝑆 𝐴𝐴a given area𝜋𝜋 is𝑟𝑟 given by irradiance and is simply the

solar luminosity divided by the surface area of a theoretical sphere with a radius value

measured as the distance from the center of the sun. Therefore, the quantity of photons

26 over an area at a particular distance from the sun for a set amount of time is given by

= (5) 4 𝐿𝐿𝑠𝑠 𝐼𝐼𝑠𝑠 2 The value of k is near one and accounts for variation𝜋𝜋𝑟𝑟 𝑘𝑘 of the Earth-Sun distance. For our analysis, we will assume a value of 1. The value of r refers to the distance from the sun.

The acceleration of photons over a given area is thus given by the equation below.

= (6) 𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑆𝑆 𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑃𝑃 𝐼𝐼 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑛𝑛 𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 This equation reduces to 𝐸𝐸 𝜎𝜎

= ℎ 𝐿𝐿𝑠𝑠 = (7) 2 4 𝑃𝑃𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝐼𝐼𝑆𝑆 𝜆𝜆 ∗ 4𝜋𝜋𝐴𝐴 𝐿𝐿𝑠𝑠 ℎ𝑐𝑐 𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑙𝑙 2 𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸 𝜎𝜎 𝜆𝜆 ∗ 𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜋𝜋𝑟𝑟 𝑐𝑐𝑐𝑐 Before a collision, the photon is traveling at the speed of light. As light bounces off a perfectly reflective sail, the photon is now traveling at the same speed of light in the opposite direction. With the conservation of the momentum, this causes the acceleration of the sail to be doubled, resulting in the below equation.

= (8) 2 𝐿𝐿𝑠𝑠 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 2 The efficiency, or reflectiveness, of the sail, η, has 𝜋𝜋a𝑟𝑟 value𝑐𝑐𝑐𝑐 between 0.5 and 1. This is because a sail that does not reflect at all, corresponding to a value of 0.5, will still gain momentum by absorbed light, analogous to an inelastic collision, but will be half the value of a sail that reflects all light. Thus, the efficiency factors are given by the equations below.

= 2 1 (9)

𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑦𝑦 = 1𝜂𝜂 − (10)

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 Taking the value of 1, it can be𝑒𝑒 𝑒𝑒𝑒𝑒shown𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 that𝑦𝑦 all light will −be𝜂𝜂 reflected, and no light will be

absorbed while taking a value of 0.5, it can be shown that all light will be absorbed, and no light will be reflected. The accelerations become

= (2 1) (11) 2 𝐿𝐿𝑠𝑠 𝑎𝑎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 2 𝜂𝜂 − 𝜋𝜋𝑟𝑟 𝑐𝑐𝑐𝑐 = (1 ) (12) 2 𝐿𝐿𝑠𝑠 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 2 − 𝜂𝜂 For calculations with current technology, 0.75𝜋𝜋 𝑟𝑟is 𝑐𝑐a𝑐𝑐 reasonable estimate [14].

If a sail is tilted at an angle α, the effective sail area seeing the light is reduced by a factor

of cos α. And as the sunlight vector for reflected light will now be hitting the sail the

same angle α, the force vector associated with the collision in the radial direction will be

reduced by another factor of cos α. Absorbed light will see this effect in the radial

direction as well as the transverse direction, but by a factor of sin α.

(2 1) = cos (13) 2 𝐿𝐿𝑠𝑠 𝜂𝜂 − 2 𝑎𝑎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 2 𝛼𝛼 𝑟𝑟̂ (1 ) 𝜋𝜋𝑟𝑟 𝑐𝑐𝑐𝑐 = cos + (14) 2 𝐿𝐿𝑠𝑠 − 𝜂𝜂 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 2 𝛼𝛼 �𝑐𝑐𝑐𝑐𝑐𝑐𝛼𝛼 𝑟𝑟̂ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜙𝜙�� The angle α is defined by 𝜋𝜋𝑟𝑟 𝑐𝑐𝑐𝑐

= sin (15) 𝜙𝜙 −1 𝑣𝑣 𝛼𝛼 As previously noted, reflected light will cause the𝑐𝑐 interaction of an elastic collision while

28 absorbed light will act as an inelastic collision. Both of these holds true in the radial direction, but in the transverse direction, the absorbed light acts as a drag force against the motion of the sail. This is the source of the force against the direction of motion responsible for the Poynting-Robertson Effect mentioned earlier. Acceleration due to reflected light is entirely away from the sun and thus does no contribute to the Poynting-

Robertson Effect.

In addition to the light-based acceleration, the acceleration due to the gravity of the central body must be considered. This is defined in the equations below.

= (16) 𝐺𝐺𝐺𝐺 𝑎𝑎𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2 𝑟𝑟̂ In these equations, G refers to the gravitational cons𝑟𝑟 tant and M refers to the mass of the central body.

For bound solar orbits, the values of rs and rc are equivalent. For other orbits, rs will refer to the solar distance and rc will refer to the central body. The acceleration equations become

cos = (17) 2 𝑟𝑟 𝜂𝜂𝜂𝜂 𝛼𝛼 𝐺𝐺𝐺𝐺 𝑎𝑎 2 − 2 (1 𝑟𝑟𝑠𝑠 ) sin 𝑟𝑟𝑐𝑐cos = (18) 𝜙𝜙 − 𝜂𝜂 𝜅𝜅 𝛼𝛼 𝛼𝛼 𝑎𝑎 − 2 𝑟𝑟𝑠𝑠 = (19) 2 𝐿𝐿𝑠𝑠 𝜅𝜅 𝑠𝑠 Since vφ<

reduces to α.

= (20) 𝑟𝑟 𝜂𝜂𝜂𝜂 𝐺𝐺𝐺𝐺 2 2 𝑎𝑎 𝑠𝑠 − 𝑐𝑐 (𝑟𝑟1 𝑟𝑟) = (21) 𝜙𝜙 𝜙𝜙 − 𝜂𝜂 𝜅𝜅𝑣𝑣 2 𝑎𝑎 − 𝑠𝑠 The general form for equations of motion in polar𝑐𝑐𝑐𝑐 coordinates are

= (22) 𝑟𝑟 2 𝑎𝑎 = 𝑟𝑟̈ −+𝑟𝑟2𝜙𝜙̇ (23) 𝜙𝜙 Inserting our equations of motion and𝑎𝑎 rearranging𝑟𝑟𝜙𝜙̈ 𝑟𝑟termṡ𝜙𝜙̇ , this becomes

+ = 0 (24) 𝐺𝐺𝐺𝐺 𝜂𝜂𝜂𝜂 2 2 2 𝑟𝑟̈ 𝑐𝑐 − 𝑠𝑠 − 𝑟𝑟𝜙𝜙̇ 𝑟𝑟 𝑟𝑟 (1 ) = (25) 2 𝜙𝜙 𝑑𝑑�𝑟𝑟 𝜙𝜙̇� − 𝜂𝜂 𝜅𝜅𝑣𝑣 − 𝑠𝑠 Using = , the second equation𝑑𝑑𝑑𝑑 can be reduced𝑐𝑐𝑟𝑟 to the resulting system of 𝜙𝜙 differential𝑣𝑣 equations.𝑟𝑟𝜙𝜙̇

+ = 0 (26) 𝐺𝐺𝐺𝐺 𝜂𝜂𝜂𝜂 2 2 2 𝑟𝑟̈ 𝑐𝑐 − 𝑠𝑠 − 𝑟𝑟 𝜙𝜙̇ 𝑟𝑟 (𝑟𝑟1 ) = ( ) ( ) (27) 𝑣𝑣0𝑟𝑟0 𝑐𝑐 𝜅𝜅 − 𝜂𝜂 �𝜙𝜙𝑠𝑠 − 𝜙𝜙0 𝑠𝑠 � 2 2 𝜙𝜙̇ 𝑐𝑐 − 𝑠𝑠 Here, v0, r0, and φ0 refer to the initial𝑟𝑟 velocity, heliocentric𝑐𝑐𝑟𝑟 distance, and orbital position relative to the sun. For simple calculations, φ0 can be zero. φ refers to the angle of orbit

with respect to the initial position The angle φ is calculated by Kezerashvili [10] [23] in

the equations below as a Newtonian value and a relativistic approximation. In these

equations, M represents the mass of the central body and represents the effective mass

𝑀𝑀�

the sail observes.

1 1 = cos 1 (28) 2 2 −1 𝑣𝑣𝑜𝑜 𝑟𝑟0 ϕN � − � − �� 𝐺𝐺𝑀𝑀� 𝑟𝑟 𝑟𝑟0 ( ) ( ) 𝑅𝑅 29 𝑑𝑑𝑑𝑑 2 𝜙𝜙 𝑅𝑅 ≈ 𝐿𝐿 �𝑟𝑟0 1 1 𝑟𝑟 √ℎ = 2 + (30) 2 𝑓𝑓0 𝑓𝑓 ℎ 𝐺𝐺𝑀𝑀� � − � 𝐿𝐿 � 2 − 2� 𝑟𝑟 𝑟𝑟0 2 𝑟𝑟0 𝑟𝑟 = 1 (31) 𝐺𝐺𝐺𝐺 𝑓𝑓 − 2 𝑐𝑐 𝑟𝑟 . = (32) 2 −0 5 𝑣𝑣0 𝐿𝐿 𝑣𝑣0𝑟𝑟0 �𝑓𝑓0 − 2 � 𝑐𝑐 = (33) 𝜅𝜅 𝑀𝑀� 𝑀𝑀 − The integral for the relativistic approximation is rather𝐺𝐺 cumbersome and numerical

integration can be used to calculate it. The difference in angles between the Newtonian

value and the relativistic approximation results the Poynting-Robertson Effect [14].

For orbits around Earth, the value of becomes extremely small and the term can

0 be ignored. The solution for earth-bound𝜙𝜙 − 𝜙𝜙orbit acceleration can be reduced to the equation below.

1 1 1 = ( ) ( ) + ( ) (34) 2 3 0 0 2 2 𝑟𝑟̈ 𝑐𝑐 𝑣𝑣 𝑟𝑟 𝑐𝑐 − 𝑐𝑐 𝐺𝐺𝐺𝐺 𝑠𝑠 𝜂𝜂𝜂𝜂 For approximations in Earth orbits,𝑟𝑟 rs can be 𝑟𝑟assumed to 𝑟𝑟be 1 Astronomical Unit.

The solution of these differential equation is also rather cumbersome and is not practical

to solve exactly to model a craft. For actual sail missions, an exact solution may be

necessary, but for modeling purposes, this can be approximated by numerical solutions

such as Euler’s Method. This approximation is shown below.

= = (35)

𝑅𝑅 =𝑟𝑟̇ 𝑣𝑣 (36)

=𝑅𝑅̇ +𝑟𝑟̈ ( ) (37)

𝑘𝑘+1 𝑘𝑘 𝑘𝑘 𝑟𝑟 =𝑟𝑟 + Δ𝑡𝑡 𝑅𝑅 (38)

𝑘𝑘+1 𝑘𝑘 The numerical solution this yields 𝑅𝑅would offer𝑅𝑅 aΔ good𝑡𝑡�𝑅𝑅̇ �approximation of a sail orbit.

For a solar orbit, as Kezerashvili notes, the areal density of the spacecraft plays a great

role in effects seen by the Poynting-Robertson Effect [10]. The lighter the craft, the

greater the effect. The areal mass is, of course, dependent on the area of the sail that experiences thrust. While rigid analysis of the spacecraft and sails would have no effect on the area, if one were to analyze it as a flexible body, the areal mass would be affected and as such, the Poynting-Robertson Effect would have a different impact. A greater areal density will be slowed less by the Poynting-Robertson Effect, but will sail slower due to its increased density. For long term missions involving any solar orbits or slingshots, these effects need to be considered

3.2 Analysis of Body

Current solar sails being experimented with are composed of the payload, typically a

Cube Satellite, booms, and sails. Material selection is obviously very important. Strong

materials are desired as they must be able to tolerate the extreme environment associated

with space travel, particularly a solar slingshot. Conversely, they must be lightweight to

fully utilize the thrust available from photon momentum.

The cube has many working components to control attitude, sail deployment, and

communication with ground control. These components will be analyzed as part of the

cube. The Cube is a small fraction of the sail area, less than 5% on LightSail 2. On

longer missions and especially for manned missions, this percentage will be desired to be

much smaller. Thus, the density of the sail will be considered constant throughout and

the payload can be considered a concentrated mass at the sail’s center.

The booms are typically very small, solid rods 0.1 mm thick on the LightSail 2 [20], but

are usually a very sturdy material. Carbon fiber materials are being developed for use as

booms. Inflatable booms are in development as well. Some of these booms are designed

to become more rigid in cold environment and are easier to store compact prior to sail

deployment, making them a desirable candidate for sails [1]. For this assessment, these

booms will be analyzed as hollow cylindrical rods. It is possible to create a rotation in

the craft to maintain the tension in the sail rather than constructing booms, but this

requires more power and very precise calculations to be done properly [1].

Vibration related to attitude control is present when a craft rotates. Rotation is necessary to ensure optimum photon interaction and to stabilize the orbit. Yaw, pitch, and roll of

33 the craft are all present [8], [24]. This rotation is very slow, seldom exceeding 5 degrees per second and usually below 1 degree per second on the LightSail 2 [25].

A finite element analysis of the craft will be analyzed. Triangular elements will be used to analyze the sail. The booms will be analyzed as beam segments. A diagram of the element and nodes is below in Figure 7.

Figure 7-FEA model of Solar Sail

In this diagram example, the sail is divided into 292 triangular elements, fourteen squares high and fourteen squares wide, then divided in half as two equal triangles. The booms would be represented by the longest diagonal and as a line perpendicular to it, intersecting the opposite corners. Of course, more nodes and smaller elements will create a more accurate picture of the sail but will conversely add significant computing time as the number of nodes increases.

A heliocentric orbit 1 AU from the sun will be analyzed. The orbit will follow Earth’s orbit at 1 AU, approximately 90 degrees around the orbit from Earth, so that any gravitational force of Earth on the sail can be negated. Thus, the solar radiation pressure and gravitational acceleration of the sun will be the only forces present on the body.

3.3 Body Specifications

The following values will be used to analyze the sail. Sail efficiency will be assumed to be 100% for simplicity purpose.

Density of sail and booms-1660 kilograms per cubic meter

Sail

Shape-square

Length-20 meters

Width-20 meters

Thickness-2.5 micrometers

Tension per unit length-50 Newtons per micrometer*

*This value was varied to demonstrate change in flexibility

Booms

Shape-hollow cylinder

Young’s Modulus of boom-1 gigapascal

Thickness of boom-0.1 millimeters

Radius-5 millimeters

Payload Shape-concentrated point Mass-120 kilograms

CHAPTER 4

RESULTS

The initial analysis of the full body with all degrees of freedom resulted in a highly non- linear and very complex output. Simplifications were made to easily understand the effects of flexibility.

First, the rotation of the sail was assumed to be zero in yaw and roll axes. The pitch rotation was assumed to be small as the craft rotated around the sun. As the cosine of this angle was cubed in the equations of motion, small angle approximation was used, and this value was assumed to be 1.

Additionally, the initial analysis allowed for elastic deformation in all three directions in addition to the six degrees of freedom of motion. This was simplified by only looking at elastic deformation in the direction which the photons collided with the sail, or the solar direction.

When a sail first is placed in orbit, it will initially have unsteady deformation. After a certain period of time, the body will eventually reach steady state deflection. This is the area that was analyzed, and the initial stages of orbit were ignored.

Triangular finite element analysis was performed by dividing the sail into fourteen

squares long by fourteen squares high and then splitting each of the squares in half. If

greater accuracy was desired, a higher number of elements could be used. However, the

Mathematica code used for the analysis requires very long computing time for sixteen

elements. As the payload mass was concentrated at the central node, this required an

even number of elements to be used. Thus, a model 14 elements wide by 14 elements

high was selected. Mathematica was able to model the natural modes of vibration for 16

elements wide, shown in Figure 8 below.

Figure 8-Natural Modes of Vibration of a sail, graphed using Mathematica

The size and mass of the payload, sail, and booms, along with the stiffness of the booms

were kept constant. The rigidity of the sail material was increased from the initial value.

With each rigidity increase, the steady-state acceleration of the flexible sail was compared to the acceleration of the rigid sail.

In the initial case, it was observed that there was a 26% difference in predicted

acceleration values with the rigid beam having greater acceleration. Rigid body analysis

resulted in an acceleration of 0.02993 mm/sec2 while flexible body analysis resulted in an acceleration of 0.02373 mm/sec2. This is a result of the sail being deformed to resemble

a wide symmetric bowl. This would appear from above or from the side as a parabola

with a large amplitude. This in turn decreases the sail area that interacts with photons

and lowers acceleration. As tension in the material increased, the difference in

acceleration between the flexible and rigid models decreased. When tension was

increased tenfold, the flexible body analysis resulted in an acceleration of 0.0296

mm/sec2, an error of only 1.13%. The first graph shows the decrease in error between a

flexible and rigid analysis as the tension increases.

Figure 9-Tension of sail vs. Flexible-Rigid Error

The second graph, shown below, demonstrates the acceleration calculated in the flexible analysis as a function of tension in the sail. As tension increases, the acceleration approaches 0.02993 mm/sec2, the value calculated for the rigid analysis.

Figure 10-Tension vs acceleration of a flexible sail

CHAPTER 5

CONCLUSION

The applications of solar sails are many. As science reaches the limits of chemical propulsion, they offer the ability for long term missions with no rocket fuel once they are in space, allowing for more mass to be dedicated to scientific payload. Sails could potentially closely explore the sun, Mercury, and Venus while at the same time, can be

used for deep space missions, much like the Voyager probes of the 1970s.

While simplifications were made in the analysis of the sail, the basic fundamentals of

solar sail motion were maintained and the difference in performances was easily

observed. The performance of the flexible sail when compared to the rigid sail was found

to be significant. The effects of the flexibility are decreased with an increase in tension

in the sail. This has a theoretical upward limit as the sail material is very thin and

delicate and will tear beyond a certain tension.

In past and current missions which are focusing on a conceptual understanding of sails, it

is unnecessary to analyze as a flexible body. However, in the not-too-distant future, when mankind may attempt to travel to the outer reaches of our solar system, or even to rendezvous with another body, a flexible analysis of the spacecraft should be considered.

CHAPTER 6

FURTHER STUDY

As SRP follows the inverse square law, the effectiveness of this thrust drops significantly in the outer solar system. As such, solar slingshots are the primary source of theoretical thrust generation. But one object in the outer solar system may be utilized for sailing.

The surface of Enceladus, a moon of Saturn, is covered in clean, white ice. This ice

reflects nearly all of light that reaches it and is the most reflective body in our solar

system [26]. As a result of not absorbing much light, its surface temperature is

approximately 62 Kelvin.

Also of interest is that higher temperatures have been observed near the pole and further

examination revealed large oceans likely exist under the ice. It also appears that icy jets

from the surface are the source of Saturn’s E-Ring. While extreme surface temperature

and variation is interesting and certainly worthy of observation, there is also speculation

that Enceladus may contain life in its warm oceans [26].

It is plausible that light reflected not only from the sun, but also the surface of Saturn,

may provide enough light for a solar sail satellite to orbit Enceladus for a longer period

than a traditional satellite, offering greater insight to the moon’s composition and

potential life, as well as Saturn’s rings. Additionally, it would be interesting to observe

the impact of the light reflected as it would compare to the sun at that distance. It may be

that Enceladus produces a notable amount of light capable of sailing. This could be used

42 for a long-term orbit of Saturn and its many moons.

Furthermore, it has been proposed that a high-powered laser could assist in creating thrust in the outer solar system. A laser beam would be more concentrated than sunlight as it would not disperse. An ideal beam would not be subject to the inverse square law, but some scattering of an actual beam over several AU would be expected. Placing such a laser in space rather than in the atmosphere would further prevent scattering of the beam.

A lunar mount would cut down on orbital costs for this endeavor. This would slightly limit times the laser could reach the sail but the benefits of providing stronger thrust in the outer solar system would outweigh any downtime [1].

APPENDIX-WOLFRAM MATHEMATICA CODE

REFERENCES

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[2] M. Gruntman, Blazing the Trail: The Early History of Spacecraft and Rocketry. Virginia: American Institute of Aeronautics and Astronautics, 2008.

[3] “Success, Failure, and NASA Culture | APPEL Knowledge Services.” [Online]. Available: https://appel.nasa.gov/2008/09/01/success-failure-and-nasa-culture/. [Accessed: 01-Aug-2019].

[4] W. Rogers, “ReportReport to the President By the Presidential Commission On the Space Shuttle Challenger Accident to the President,” Jun. 1986.

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