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EFFECTS OF GRAVITY PERTURBATIONS ON OPTIMAL LOW-THRUST TRANSFER SOLUTIONS FOR LOW

By ALEXA HORN

AN HONORS THESIS PRESENTED TO THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE MAGNA CUM LAUDE UNIVERSITY OF FLORIDA 2019 ⃝c 2019 Alexa Horn This thesis is dedicated to my friends and family who have supported me throughout my academic journey. I would not be submitting this project without them. ACKNOWLEDGMENTS This thesis would not have been possible without the mentorship and guidance of my thesis chair, Dr. Anil Rao. Thank you as well to Dr. William Hager and Dr. Richard Lind for serving on my thesis committee. Finally, thank you to the University of Florida and the friends and family who have supported me throughout my education.

4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...... 4 LIST OF FIGURES ...... 6 ABSTRACT ...... 7

CHAPTER 1 INTRODUCTION ...... 8 2 Background ...... 9 2.1 Modified Equinoctial Elements ...... 9 2.2 Optimization Problem Definition ...... 9 2.3 Initial and Other Parameters ...... 12 2.4 Survey of Earth-Centered Perturbations ...... 12 2.4.1 Zonal Harmonics ...... 13 2.4.2 Analysis ...... 13 2.4.3 Selection ...... 14 3 Methodology ...... 17 3.1 GPOPS-II ...... 17 3.2 Comparing Solutions With and Without J2 ...... 17 3.3 Variations with Inclination ...... 18 3.4 Variations with Eccentricity ...... 18 3.5 Final Orbit Selection ...... 18 4 Results ...... 19 4.1 Near-Equatorial Orbits ...... 19 4.1.1 Eccentricity of 0.1 ...... 19 4.1.2 Eccentricity of 0.2 ...... 20 4.1.3 Eccentricity of 0.3 ...... 20 4.2 Polar Orbits ...... 25 4.2.1 Eccentricity of 0.1 ...... 25 4.2.2 Eccentricity of 0.2 ...... 25 4.2.3 Eccentricity of 0.3 ...... 26 4.3 Tundra Orbit ...... 31 5 Conclusions ...... 33 BIOGRAPHICAL SKETCH ...... 36

5 LIST OF FIGURES Figure page 2-1 Longitude of the ascending node versus time in hours for a final orbit with an eccentricity of 0...... 15 2-2 versus time in hours for a final orbit with an eccentricity of 0...... 15 2-3 Longitude of the ascending node versus time in hours for a final orbit with an eccentricity of 0.737...... 16 2-4 Argument of periapsis versus time in hours for a final orbit with an eccentricity of 0.737...... 16

4-1 Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for Near-Equatorial orbit with eccentricity 0.1...... 22

4-2 Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for Near-Equatorial orbit with eccentricity 0.2...... 23

4-3 Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for Near-Equatorial orbit with eccentricity 0.3...... 24

4-4 Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for a with eccentricity 0.1...... 28

4-5 Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for a Polar orbit with eccentricity 0.2...... 29

4-6 Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for a Polar orbit with eccentricity 0.3...... 30

4-7 Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for a Tundra orbit...... 32

6 Abstract of Honors Thesis Presented to the University of Florida in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science Magna Cum Laude EFFECTS OF GRAVITY PERTURBATIONS ON OPTIMAL LOW-THRUST TRANSFER SOLUTIONS FOR LOW EARTH ORBITS By Alexa Horn May 2019 Chair: Dr. Anil V. Rao Major: Aerospace Engineering This thesis explores the effects of gravity perturbations on the optimal solutions for transfers in . The focus is on very low-thrust maneuvers, which are not time dependent, but are capable of minimizing fuel consumption. Using optimal control software, GPOPS-II, transfer orbits are calculated from a space shuttle to various final orbits. The final orbits, Near-Equatorial, Polar, and Tundra, are tested over a range of inclinations. Solutions are found and compared that include and exclude the zonal harmonic, J2. Calculating the numerical differences in the transfer orbit’s orbital parameters for both cases leads to the conclusion that J2 can, at certain final orbit inclinations, have a significant effect on the optimal transfer solution. It is, in some cases, imperative

to consider J2 in spacecraft trajectory optimization in order to realistically model earth-centric maneuvers. It is important to note that this thesis only seeks to demonstrate

the influence of J2. Additional, in-depth study is required to ascertain why J2 has the effects it does on these select orbits.

7 CHAPTER 1 INTRODUCTION Low-thrust trajectories have, historically, been appealing to missions that depend on minimizing fuel consumption. As long as the duration of the trajectory isn’t an issue, low-thrust propulsion methods such as electric thrusters and solar sails produce more delta-V with the same amount of fuel than chemical engines. One caveat of low-thrust maneuvers, however, is that forces due to aerodynamic drag and perturbations can have an outsized impact on the very low thrust [1]. The focus of this project is an optimization problem of an earth-centered, low-thrust orbit transfer with a goal of minimizing fuel consumption. The thrust model that is used simulates the low thrust that is produced by electric propulsion. The objective is to demonstrate just how significant the impact of perturbations is on optimal low-thrust trajectories and to explore how that impact can vary with orbital parameters like inclination. The project finds that gravity perturbations can have a noticeable, and sometimes egregious, effect on low-thrust orbit transfers, depending on final orbit inclination.

8 CHAPTER 2 BACKGROUND The low-thrust optimization problem utilized by John Betts in Ref. [2] serves as the basis for the optimization of various low-thrust transfers in this thesis. In addition to the problem definition, the parameters for the initial parking orbit are also borrowed. Just as in Betts’ problems, Modified Equinoctial Elements (MEE) are used here because of their greater simplicity in calculating transfer orbit solutions than Classical (COE). 2.1 Modified Equinoctial Elements

The use of MEE (p, f , g, h, k, L) eliminates the singularities that arise in COE (a, e, i, Ω, ω, ν) when e=0 and i=0, 90◦ [1]. While it is easier to calculate solutions with MEE, it is easier to visualize and input orbits in their COE form. The following equations are used to transfer between the two sets of elements.

p = a(1 − e2) a = p/(1 − f 2 − g2) √ f = ecos(ω + Ω) e = f 2 + g2 √ 2 h2+k2 g = esin(ω + Ω) i = arctan( 1−h2−k2 ) k h = tan(i/2)cos(Ω) Ω = arctan( h ) gh−fk k = tan(i/2)sin(Ω) ω = arctan( fh+gk ) L = Ω + ω + ν ν = L − Ω − ω

2.2 Optimization Problem Definition

The dynamics are defined by the state and the control variables given below. [yT , w] = [p, f , g, h, k, L, w]

T u = [ur , uθ, uh] The equations of motion used for a vehicle with varying thrust are: ˙y=A(y)∆+b

˙w = −T [1 + 0.01τ]/Isp

9 0 =∥u∥−1

τL ≤ τ ≤ 0

where w is weight, T is the maximum thrust, τ is the throttle factor and Isp is the engine’s specific impulse. The equinoctial dynamics are defined by:

 √  2p p  0 q 0  √ √ µ √     p sinL p 1 [(q + 1)cosL + f ] − p g [hsinL − kcosL]  µ µ q µ q  √ √ √   p p 1 − p g −   µ cosL µ q [(q + 1)sinL + g] µ q [hsinL kcosL] A =  √   2   0 0 p s cosL   µ 2q   √   p s2sinL   0 0   √ µ 2q  p 1 − 0 0 µ q [hsinL kcosL] √ T q 2 b = [0 0 0 0 0 µp( p ) ] where q = 1 + fcosL + gsinL

p r = q α2 = h2 − k 2 √ χ = h2 + k 2 s2 = 1 + χ2. The following expressions relate the equinoctial coordinates to Cartesian coordinates.

  r 2  2 (cosL + α cosL + 2hksinL)  s    r =  r (sinL − α2sinL + 2hkcosL)   s2  2r − s2 (hsinL kcosL)

10  √  − 1 µ (sinL + 2sinL − 2hkcosL + g − 2fhk + 2g)  s2 p α α   √   1 µ 2 2  v = − 2 (−cosL + α cosL − 2hksinL − f − 2ghk + α f )  s p √  2 µ s2 p (hcosL + ksinL + fh + gk)

The disturbing acceleration is defined by

∆ = ∆g + ∆T

where ∆g is due to oblate earth effects and ∆T is from thrust. The coordinate axes of

the disturbing acceleration are given as[ ] r (r × v) × r r × v Q = [i i i ] = r r θ h || r || || r × v |||| r || || r × v || in a rotating radial frame. A local horizontal reference frame for oblate gravity models is

δg = δgnin − δgr ir where e − (eT i )i i = n n r r . n || T || en(en ir )ir

Ignoring tesseral harmonics and focusing solely on the first four zonal harmonics, gravitational perturbations can be calculated with

µ cos ϕ ∑4 Re δg = − ( )k P′ J n r 2 r k k k=2 µ ∑4 Re δg = − (k + 1)( )k P J r r 2 r k k k=2

where ϕ is the geocentric , Re is the radius of the earth, Pk is the k-th order

Legendre polynomial and Jk are the zonal harmonic coefficients. Gravity perturbations in the rotating radial frame are therefore found with

T ∆g = Qr δg.

Finally, the thrust acceleration is described thus,

g T [1 + 0.1τ] ∆ = 0 u T w

11 where g0 is the mass to weight conversion factor and u is the direction of the thrust acceleration. 2.3 Initial Orbit and Other Parameters

The goal of Betts’ problem is to find the optimal transfer between two earth-centered orbits, a parking orbit and a , while minimizing the spacecraft’s fuel consumption (or maximizing its final weight). The initial orbit is similar to a ”standard space shuttle park orbit” [2] with an eccentricity of 0 and an inclination of 28.5◦. The table below contains the initial orbit parameters (in MEE) as well as other key constants used both in Betts’ problem and in this thesis. The only exception is the value τ, which is -50 in Betts’ problem and -25 in this analysis.

p = 21837080.052835 ft f = 0 g = 0 h = −0.25396764647494

k = 0 L = π

−3 Isp = 450 sec T = 4.446618 × 10 lb

16 3 2 2 µ = 1.407645794 × 10 ft /s g0 = 32.174 ft/s

−6 Re = 20925662.73 ft J2 = 1082.639 × 10

−6 −6 J3 = −2.565 × 10 J4 = −1.608 × 10 w(0) = 1 1b τ = −25

One other important value to note is the parameter p for the final orbit. Betts’ final orbit is a Molniya orbit with p = 40007346.015232 ft. While a Molniya orbit is not included in this project, this p is used for the final orbits that are tested. 2.4 Survey of Earth-Centered Perturbations

There are a number of abnormalities that cause disruptions in the trajectories of spacecraft, but this project looks specifically at perturbations caused by distortions in the earth’s gravitational field. The earth is not a perfect sphere. The distance between the north and south poles is shorter than the distance between opposite points on the

12 - in other words, the earth has an equatorial bulge which causes similar distortions in the gravitational field [3]. 2.4.1 Zonal Harmonics

The perturbations considered in this project are the zonal harmonics for earth, J2,

J3, and J4, and they are surveyed to determine which of them has the largest measurable impact on orbit transfers. It is important to note that, for analysis purposes, there is a focus on transfers of a relatively small time-scale.

J2 primarily affects the longitude of the ascending node and the argument of periapsis of an orbit. For an eastward orbit, for example, the longitude of the ascending node precesses westward due to the increased gravitational attraction at the equator [3]. The node regresses for inclinations between 0 and 90 degrees. Additionally, all orbits are said to have a critical inclination of 63.4◦ where the rate of change of the argument of periapsis is zero [3].

J3 can have a measured effect on orbits with smaller eccentricities. These orbits will experience long-term variation in eccentricity, which will also induce a long-term variation in inclination. However, for a study with a limited temporal scope, these changes will

not be as noticeable as those caused by J2. J4 has an even smaller value than J3, so it is

unlikely that its effect on transfer orbits would be as significant as J2. 2.4.2 Analysis

Using the initial orbit parameters outlined in the previous section and two different final orbits, the longitude of the ascending node and the argument of periapsis are tracked over the duration of the solution transfer orbits. The first final orbit has an eccentricity of 0 and an inclination of 63.4◦. The rest of the orbital parameters are kept constant from the initial orbit. The results are shown in Figures 2-1 and 2-2. The dark blue line marks the solution computed with

no perturbations. The light blue line represents the solution including J2 perturbations. Two other lines exist but are difficult to separate from the light blue line as they represent

13 J2 and J3 perturbation effects together and J2, J3, and J4 effects together, respectively.

Since J2 is so dominant over the other zonal harmonics, its line is the only one visible. The second final orbit is a Molniya orbit. It has an eccentricity of 0.737 and an inclination of 63.4◦. The rest of the parameters are, again, kept the same as the initial orbit. The color coding is the same for these results as stated before. J2 is also extremely dominant here among the other perturbations. See Figures 2-3 and 2-4 for the results. 2.4.3 Perturbation Selection

It is evident from Figures 2-1 through 2-4 that the addition of J2 perturbations to these select orbit transfers can drastically alter the longitude of ascending node and the argument of periapsis over time. It is also important to note that the transfer to the

Molniya orbit is less effected than the transfer to an equatorial orbit. Since the J3 and J4 perturbations have no discernible effect on the orbit transfers apart from the effects of J2, they are disregarded for the remainder of the study. J2 is the only perturbation that will be considered from now on.

14 Figure 2-1: Longitude of the ascending node versus time in hours for a final orbit with an eccentricity of 0.

Figure 2-2: Argument of periapsis versus time in hours for a final orbit with an eccentricity of 0.

15 Figure 2-3: Longitude of the ascending node versus time in hours for a final orbit with an eccentricity of 0.737.

Figure 2-4: Argument of periapsis versus time in hours for a final orbit with an eccentricity of 0.737.

16 CHAPTER 3 METHODOLOGY Now that an orbital perturbation with a significant impact on transfer orbits has been selected, the optimization problem is established. Using Betts’ initial orbit, optimal transfer solutions are calculated to various final orbits including and excluding the effects of the gravity perturbation J2. The goals are to determine just how significant the effects of J2 are on various transfer orbits and to study how these perturbation effects change with the inclination of the final orbit. 3.1 GPOPS-II

To accomplish this, nonlinear optimal solutions have to be found for a number of transfer orbits. The optimal control software, GPOPS-II, is utilized in Matlab to do just that. GPOPS-II approximates a continuous-time optimal control problem as a sparse nonlinear programming problem using variable-order Gaussian quadrature methods [4]. By defining the initial and final orbits and the appropriate boundary conditions and initial guesses, GPOPS-II can produce the transfer orbit that consumes the least amount of fuel.

3.2 Comparing Solutions With and Without J2 Using GPOPS-II, two different solutions can be found for each orbit transfer: one with J2 effects and one with no perturbation effects. These two solutions can then be compared side-by-side to illustrate the extent of J2’s influence. The optimal solutions are given in the form of MEE’s over the period of time required for the transfer. The MEE’s are translated to COE’s before they are compared for easier comprehension.

A problem arises when trying to directly compare the solutions with and without J2. Since time of flight is not bounded, the optimal solution sets are based on different time scales. To make the sets more comparable, the solutions with J2 are interpolated to fit the time scale established by the solutions with no perturbations. Now that they can be compared, the deviations in the two solutions are measured by calculating the maximum numerical difference between the COE’s of the solutions over

17 time. The result is a single value for each COE. For example, eccentricity may differ by as much as 0.002 between the two solution transfer orbits for a particular final orbit. 3.3 Variations with Inclination

The inclination of the final orbit can have a considerable impact on the scale of

J2’s effects due to the uneven distribution of gravitational attraction latitudinally. For example, as stated previously, orbits under the influence of J2 perturbations have a critical inclination of 63.4◦ at which the argument of periapsis is a constant. To measure just how much J2’s effects vary with inclination, the final orbit of the transfer is tested at multiple inclinations. The resulting maximum differences in COE’s between the J2 and no-J2 solutions are compared over the range of inclinations tested. 3.4 Variations with Eccentricity

Since the focus of this project is trajectories in Low Earth Orbit (LEO), the eccentricities of the final orbit are kept within the range 0.1 to 0.3. Eccentricities smaller than 0.1 are computationally challenging and any value greater than 0.3 is unusual for LEO orbits. Some of the selected final orbits are tested over multiple eccentricities to discern if the shape of an orbit influences the impact of gravity perturbations on transfer orbits. 3.5 Final Orbit Selection

To assess a wide range of orbital transfer solutions, the initial equatorial orbit is kept constant while various final orbits are tested over a select range of inclinations and, in some cases, eccentricities. The final orbits are chosen for their prevalence in LEO. The orbits selected for testing are Near-Equatorial and Polar orbits, and the Tundra orbit.

18 CHAPTER 4 RESULTS 4.1 Near-Equatorial Orbits

Orbits that are Near-Equatorial have small inclinations. As such, the range of values tested are i = 1◦ to i = 15◦, with a step size of 1◦. This array of inclinations is analyzed for three different eccentricities: e = 0.1, 0.2, and 0.3. The results for each scenario are below. 4.1.1 Eccentricity of 0.1

The final orbit’s input parameters are included in the table below. They are presented as COE because that is how they are entered into the code and it is simpler to visualize. The results are presented in Figure 4-1.

p = 40007346.015232 ft e = 0.1 a = 40411460.621446 ft i = 1 : 1 : 15◦

Looking at Figure 4-1, there is stark cut-off between the results for inclinations at 6 and 7 degrees. For all five COE, there are measurable differences in the solution transfer

◦ ◦ orbits that include and exclude J2 perturbations for inclinations of 1 through 6 , but there is no measurable difference in COE for the larger inclinations. This indicates that

◦ ◦ for an inclination range of 7 to 15 , J2 has no discernible effect on the optimal transfer between a LEO parking orbit and a Near-Equatorial orbit with an eccentricity of 0.1. Remembering that the inclination of the initial orbit is 28.5◦, it might be possible that the cut-off exists because of the amount of inclination change required by the transfer. Other points of interest are that the measurable max differences in longitude of the ascending node are all equal for the 6 different inclinations in which they appear. It is possible that inclination has an effect on J2’s influence on the transfer orbit only so much that there is a threshold whereupon the J2 solution is different from the solution without it.

19 The argument of periapsis changes the most at 6◦ inclination by approximately 430◦. The maximum change in the semi-major axis of the transfer orbit is also at 6◦ inclination. There, it changes by almost 90,000 ft. The largest change in eccentricity is by 0.003 at an inclination of 1◦. The is altered by, at most, 7.5◦ at the 6◦ inclination. 4.1.2 Eccentricity of 0.2

The input parameters are included in the table below.

p = 40007346.015232 ft e = 0.2 a = 41674318.765867 ft i = 1 : 1 : 15◦

Figure 4-2 displays the results and it is immediately obvious that they are similar to the results from the Near-Equatorial orbit with eccentricity of 0.1. Just as in Figure 4-1, these transfer solutions do not exhibit any differences in solution for inclinations of

◦ ◦ 7 through 15 . In other words, the inclusion of J2 does not change the outcome of the transfer solution for inclinations between 7◦ and 15◦. For the inclinations that are affected, the max differences are not the same as the 0.1 eccentricity case. For the 0.2 eccentricity case, the max difference in eccentricity is relatively constant for each inclination up to 6◦, at a value just below 0.003. The change in the true anomaly is also constant, with a difference of a little more than 12◦. Interestingly, the changes in the remaining COE’s vary with inclination in an almost wave-like pattern up to 6◦. The max change in longitude of the ascending node peaks twice at just under 400◦ (at inclinations of 2◦ and 5◦) and otherwise holds steady at 350◦. The difference in argument of periapsis peaks around 450◦ at an inclination of 3◦. The semi-major axis is altered by as much as 88,000 ft when the inclination is set at 2◦ and fluctuates to a change of 79,000 ft at a 5◦ inclination. 4.1.3 Eccentricity of 0.3

The table below contains the final orbit’s parameters. The results are given by Figure 4-3.

20 p = 40007346.015232 ft e = 0.3 a = 43964116.500255 ft i = 1 : 1 : 15◦

The case of the Near-Equatorial orbit with an eccentricity of 0.3 differs quite a bit from the results of the 0.1 and 0.2 eccentricity cases. The changes in longitude of the ascending node and argument of periapsis are the most similar to the previous scenarios. They exhibit max differences that fluctuate in a wave-like way for inclinations of 1◦ to 6◦, and exhibit no differences for inclinations of 7◦ and greater. Both values peak near 450◦. The true anomaly results are still relatively similar with differences near 12◦, although there are low points of 9◦ at 3◦ inclination and 6◦ at 6◦ inclination. The changes in the semi-major axis and the eccentricity appear to be related. The max differences in the semi-major axis are all at or near zero for every inclination except

◦ that of 3 . In that case, the semi-major axis is different for the J2 and no-J2 cases by about 2.3 million feet. While the eccentricity differences are familiar values of 0.003 for the inclinations 1◦, 2◦, 4◦, 5◦, and 6◦, significantly, it is different by more than 0.05 at the 3◦ inclination. This is a whole order of magnitude greater than the biggest difference in eccentricity for the other Near-Equatorial cases.

21 Figure 4-1: Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for Near-Equatorial orbit with eccentricity 0.1.

22 Figure 4-2: Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for Near-Equatorial orbit with eccentricity 0.2.

23 Figure 4-3: Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for Near-Equatorial orbit with eccentricity 0.3.

24 4.2 Polar Orbits

Polar orbits consist of high inclinations, often close to 90◦. The range of inclinations, therefore, is between 70◦ and 90◦, with a step size of 2◦. The final orbit is tested for the same range of eccentricities as the Near-Equatorial orbits: e = 0.1, 0.2, and 0.3. 4.2.1 Eccentricity of 0.1

The final orbit’s input parameters are below. The results are displayed in Figure 4-4.

p = 40007346.015232 ft e = 0.1 a = 40411460.621446 ft i = 70 : 2 : 90◦

Like with the Near-Equatorial orbits, the transfer solutions to Polar orbits appear to have a cutoff point, this time at an inclination of 80◦. At inclinations larger than 80◦, the transfer solution’s COE do not have a measurable difference.

◦ ◦ For the inclinations that are affected by J2, 70 through 80 , semi-major axis and eccentricity fluctuate in wave-like patterns. The former hovers between 300,000 ft and 325,000 ft, the latter between 0.0025 and 0.0031. Longitude of the ascending node, argument of periapsis and true anomaly, on the other hand, have distinct jumps in the values of their max differences before cutting off after 80◦. Max change in longitude of the ascending node is 450◦ for the first four inclinations and around 360◦ for 78 and 80 degrees. True anomaly follows the same pattern except with values of 8◦ difference for the first four inclinations and values around 6.5◦ for 78◦ and 80◦. Argument of periapsis jumps down three times from 70◦ to 80◦ inclination, with values of around 360◦, 170◦, and 40◦. 4.2.2 Eccentricity of 0.2

The input parameters are in the table below. The results can be found in Figure 4-5.

p = 40007346.015232 ft e = 0.2 a = 41674318.765867 ft i = 70 : 2 : 90◦

25 At first glance, the results for the case of a Polar orbit with an eccentricity of 0.2 doesn’t seem to follow any previously discussed pattern. The change in longitude of ascending node and the change in true anomaly do behave similarly though. They both have significant and constant values for the first three inclinations and maximum values at 76◦ and 80◦. The max change is 8◦ for true anomaly and 450◦ for longitude of the ascending node. Both cut off after 80◦ like in previous cases. Changes in semi-major axis and eccentricity, however, all hover near zero except for maximums at 78◦, where the difference in semi-major axis is 18 million feet and change in eccentricity is 0.08. Both of these values are much higher than the maximums in previous cases, signifying that 78◦ is a key inclination. Change in argument of periapsis has two notable values of 350◦ at 70◦ inclination and 200◦ at 78◦ inclination. The final orbits with inclinations of 72◦, 74◦, 76◦ and 80◦, had small differences in their solutions of around 40◦. 4.2.3 Eccentricity of 0.3

The table following table contains the final orbit’s parameters. The results are displayed in Figure 4-6.

p = 40007346.015232 ft e = 0.3 a = 43964116.500255 ft i = 70 : 2 : 90◦

Once again, the results demonstrate a clear cut-off inclination where, after 80◦, no discernible difference can be found between the J2 and no-perturbation solution. The value of longitude of the ascending node differs most for the solutions with inclinations of 78◦ and 80◦, with a familiar change of 450◦. Similarly, the largest difference in true anomaly occurs for final orbits with inclinations of 78◦ and 80◦, with a difference of 8◦. The change in argument of periapsis is consistently around 360◦ for inclinations before the cut-off. The change in eccentricity, meanwhile, hovers near 0.0025 before the cut-off. The change in the value for semi-major axis is much less predictable than the others. It

26 has the largest maximum difference at 70◦ inclination with a value of 225,000 ft and the smallest non-zero difference at 74◦ inclination with a change of 100,000 ft.

27 Figure 4-4: Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for a Polar orbit with eccentricity 0.1.

28 Figure 4-5: Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for a Polar orbit with eccentricity 0.2.

29 Figure 4-6: Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for a Polar orbit with eccentricity 0.3.

30 4.3 Tundra Orbit

The Tundra orbit is a specific orbit with an inclination close to 63.4◦ and an eccentricity between 0.2 and 0.3. To test it, an eccentricity of 0.25 is used along with a range of inclinations between 53.4◦ and 73.4◦, with a step size of 2◦. The results are given in Figure 4-7.

p = 40007346.015232 ft e = 0.25 a = 42674502.416247 ft i = 53.4 : 2 : 73.4◦

Like in previously discussed cases, the Tundra orbit has a cut-off inclination. Unsurprisingly, the solutions with inclinations greater than 63.4◦ have no measurable difference in COE values (this orbit is useful because its inclination near 63.4◦ minimizes the perturbations on a spacecraft). For inclinations less than or equal to 63.4◦, however, there are significant fluctuations in the differences in COE. The changes in longitude of the ascending node are around 350 and 400 degrees, with the largest difference belonging to the final orbit with a 53.4◦ inclination. The largest change in argument of periapsis exists at 53.4◦ inclination as well, with a value of almost 400◦. Notably, while there is a large difference in the parameter for the 63.4◦ inclination, there is a minimum, nonzero change in argument of periapsis at 61.4◦ with a value around 25◦. As the inclination increases and approaches 63.4◦, the max differences in semi-major axis and eccentricity decrease. Both have the largest changes at the lowest inclination and the smallest (nonzero) changes at 63.4◦. At the cut-off, the semi-major axis is only altered by about 10,000 ft and the eccentricity only by about 0.002. It is worth noting that at the lowest inclination, the changes in solution based on the inclusion of J2 perturbations are extreme for these two parameters with values at 53.4◦ of 3.5 million feet for semi-major axis and 0.065 for eccentricity.

31 Figure 4-7: Maximum deviations in COE’s for J2 and no-perturbation solutions as a function of inclination for a Tundra orbit.

32 CHAPTER 5 CONCLUSIONS Studying all of the results together, it is clear that each type of orbit has a distinct ”cut-off” inclination, greater than which the transfer solutions are unchanged with the

◦ addition of J2 perturbations. For the Near-Equatorial final orbits, that cut-off is at 6 inclination. For the Polar and Tundra orbits, that threshold is at inclinations of 80◦ and 63.4◦, respectively. For inclinations that have a marked difference in transfer orbital parameters, the maximum differences tend to vary in linear or wave-like patterns. Notable exceptions include the maximum differences in semi-major axis and eccentricity for the Near-Equatorial orbit with eccentricity 0.3. Other outliers are present at 78◦ inclination for the Polar orbit of eccentricity 0.2. The argument of periapsis, semi-major axis and eccentricity all have unusually high

◦ differences in J2 and no-perturbation solutions at 78 . Perhaps significantly, these values are present right before the cut-off inclination. In general, the eccentricity tends to change by around 0.003 (excluding the Tundra orbit) and the longitude of the ascending node often has a maximum difference between 350 and 450 degrees. The true anomaly consistently displays max differences in the range of 6◦ to 12.5◦ for all of the orbits. The semi-major axis, on the other hand, experiences drastically different effects depending on the final orbit, with differences in solutions ranging from 90,000 feet to 18 million feet at the Polar Orbit’s outlier. The changes in the argument of periapsis also depend heavily on the final orbit of the optimization problem. The objective of this project is not to determine why the solutions to these optimization problems differ the way they do when J2 gravity perturbations are considered.

It is, however, its intention to demonstrate the imperative of acknowledging J2 in transfer orbit calculations and mission design. The optimal solution for a low-thrust transfer between low earth orbits can obviously be altered significantly by the inclusion, depending

33 on the final orbit and its inclination. Therefore, any solutions calculated without the perturbations may be dramatically off course from the truly optimal path that would minimize fuel consumption. Furthermore, this project verifies the usefulness of popular orbits such as Polar and Tundra orbits. The Polar orbit results clearly demonstrate why inclinations closer to 90◦ are advantageous for minimizing perturbation effects. It is also apparent that the Tundra

◦ orbit, which is defined by an inclination of 63.4 , utilizes its inclination to minimize J2 effects. (Interestingly, the minimum perturbation effects for the Tundra orbit were present at the inclinations just higher than 63.4◦, not at 63.4◦.) For transfers to less common orbits, though, J2’s effects are less predictable and should not be neglected.

34 REFERENCES [1] Betts, J. T., “Very Low Thrust Trajectory Optimization Using a Direct SQP Method,” Journal of Computational and Applied Mathematics, Vol. 120, 2000, pp. 27–40. [2] Betts, J. T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, SIAM Press, Philadelphia, 2nd ed., 2009. [3] Chobotov, V. A., editor, , AIAA Education Series, Reston, Virginia, 3rd ed., 2002. [4] Patterson, M. A. and Rao, A. V., “GPOPS-II: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using hp-Adaptive Gaussian Quadrature Collocation Methods and Sparse Nonlinear Programming,” ACM Trans. Math. Softw., Vol. 41, No. 1, Oct. 2014, pp. 1:1–1:37.

35 BIOGRAPHICAL SKETCH The author studied at the University of Florida for the Bachelor of Science degree in Aerospace Engineering. Their undergraduate research sprung out of an interest in astrodynamics and spacecraft mission design. Of particular interest was the unique orbital circumstances that surround different planets and bodies and how that would effect the design of mission.

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