Design and Analysis of Optimal Operational Orbits around Venus

for the EnVision Mission Proposal

Marta Rocha Rodrigues de Oliveiramarta.rocha.oliveira@tecnico.ulisboa.pt

Instituto Superior Tecnico, Lisboa, Portugal

November 2015

Abstract

In space exploration, planetary orbiter missions are essential to gain insight into planets as a whole,and to help uncover unanswered scientific questions. In particular, the planets closest to the Earthhave been a privileged target of the world’s leading space agencies. EnVision is a mission proposalwith the objective of studying one of these planets. Designed for Venus and competing for ESA’s nextlaunch opportunity, the proposal already went through the selective technical review for the M4 launchopportunity, and will now be submitted for the M5 call, incorporating feedback from ESA. The maingoal is to study geological and atmospheric processes, namely surface processes, interior dynamics andatmosphere, to determine the reasons behind Venus and Earth’s radically different evolution despitethe planets’ similarities. In this thesis, we propose to study and improve the design of the operationalorbit around Venus for the EnVision mission proposal. The constraints and scientific requirements thataffect the orbit will be examined in order to develop a computational model adapted to the missionobjectives. Finally, the orbit optimization is applied for the parameters with more influence in themission planning.Keywords: operational orbit, orbit design, scientific requirements, targets coverage, orbit optimization,genetic algorithms.

1. Introduction1.1. Thesis Objective

The main objective of this thesis is to study andimprove the design of a science orbit around Venusfor EnVision, Europe’s new medium class missionproposal for the European Space Agency’s CosmicVision 2015-2025 M5 call. The different constraintsand requirements that affect the orbit design willbe analyzed in order to select and refine selectedmission features, in particular the observation ofselected targets and mission duration.

1.2. Studying Venus from Orbit with EnVision

The study of a planet from orbit is essential to gainknowledge about the planet as a whole. A typ-ical orbiter mission can provide data on a broadspectrum of elements such as a planet’s atmosphere,weather, surface, gravity field, magnetic fields, ele-mental composition, and internal structure.

To achieve these goals different spacecraft sub-systems are needed, and an operational orbit mustbe selected. The latter is the orbit from which themission’s scientific observations will be made. Afteran analysis of the space environment (third bodies,solar radiation pressure, micrometeorites, space de-bris, the planet’s atmospheric drag, gravity), the

operational orbit can be selected to satisfy instru-ments and scientific requirements (data resolution,coverage, revisit time, link budgets, eclipse dura-tion, etc.). Often these requirements are contra-dictory and a prioritization balance must be made[1, 2].

Terrestrial planets closest to Earth are populartargets for obter missions [3]. Venus is often consid-ered Earth’s twin in terms of its size, distance fromthe Sun and bulk composition; though a great num-ber of fundamental questions, such as the planet’sgeology and its correlation to the atmosphere arestill unanswered [4]. There have been numerous in-ternational orbiters, atmospheric probes, and lan-ders that have explored Venus. Venus Express wasthe first ESA mission to Venus. The mission aimedat a global investigation of the planet’s atmosphere,plasma environment, and some surface properties[5].

EnVision is a mission proposal that follows VenusExpress to pursue its findings and research. Itsmain goal is to outline the state of geological ac-tivity on Venus and its relation to the atmosphere[4]. It will also provide gravity and geoid data, as

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well as new spin rate measurements, and new in-sight into the planet’s interior.

The EnVision mission is to be launched on aSoyuz-Fregat on December 27 2024. Following aer-obreaking, the orbiter was initially planned to reacha low circular science orbit of an altitude of 258 km.The payload includes [4]:

• A phased array synthetic aperture radar (Ven-SAR);

• A subsurface radar sounder (SRS);

• An infrared mapper and spectrometer (VEM)

• And a 3.2 m X-band steerable antenna.

Figure 1: Preliminary operational configuration ofEnVision’s orbiter.[4]

To conclude, EnVision will provide global imag-ing, topographic and subsurface data with a betterresolution than previous missions, and may uncoverthe reasons for the radically different evolution ofVenus and Earth.

2. Operational Orbit Design FundamentalsIn this section, the fundamental concepts behindthe orbit design will be introduced to support theorbit analysis and computation.

2.1. Orbit RepresentationTo describe the motion of a spacecraft orbiting amain body in a Keplerian orbit, we assume thatgravity is the only force, since the mass of the space-craft is negligible when compared to the mass of theorbited planet. Also the planet is spherically sym-metric with uniform density, so it can be treatedas a point mass, and other perturbations such asgravitational interactions with other bodies or at-mospheric drag are neglected [6].

The six Classical Orbital Elements (COEs) thatdescribe the ideal Kepler orbit [7] are the following:

• Longitude of the ascending node Ω;

Figure 2: Classical Orbital Elements.

• Argument of perigee ω;

• Inclination i;

• True anomaly ν0;

• Semi-major axis a;

• Eccentricity e.

In this thesis, we will use near circular and cir-cular orbits. In the case of the circular orbit, thereis no periapsis, and consequently no argument ofperiapsis, or true anomaly. In order to correct theabsence of the periapsis as a reference, we use theargument of latitude u, which can be related to theargument of peripasis and true latitude through thefollowing expression: u = ω+ν. Essentially, the ar-gument of latitude u is measured from the ascendingnode to the spacecraft’s position in the direction ofthe spacecraft’s motion.

2.2. Orbit PropagationThe simplest solution to propagate an orbit is givenby Kepler’s equation [6]:

M = E − e sin(E) (1)

This equation doesn’t consider perturbations(gravitational pull of third bodies, thruster forces,solar pressure, planet’s oblateness, etc.). These per-turbations correspond to any changes to these clas-sical orbital elements due to other forces beyond thesimplified gravity model we have considered. How-ever for the purpose of this thesis the perturbationsare small enough or corrected in a way that makesit possible to propagate the elements directly.

2.3. Design of an Operational Orbit around VenusTo design an operational orbit around Venus, it isessential to look into a few details of the planet.

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Mass 4.9.1024 kgRadius 6051.8 km

Surface Temperature 462 CRevolution Period 224 daysRotation Period 243 days

Number of Moons noneAtmosphere carbon dioxide, nitrogen

Table 1: Venus facts summary.

Venus has no natural satellite, which immedi-ately simplifies the gravitational effects. The ro-tation of the planet is retrograde (clockwise andcontrary to the rotation direction of the Sun andthe Earth). Also, the planet’s path around the Suntakes around 224 days but takes 243 days to com-plete a full rotation around its axis, which resultsin a Venus year being shorter than a Venus day.Indeed the planet’s rotation is extremely slow andhas the slowest angular velocity in the Solar System(2.99× 10−7 [rad/sec]) [8].

In Venus the gravitational perturbations are verysmall. Indeed the gravitational perturbations termJ2 (4.458 × 10−6) is the largest term its value isstill only about 0.4 % of Earth’s value [3]. Thisreduced perturbations effect is related to the factthat Venus’ flattening coefficient is very close to 0[9]. Essentially, Venus is the most spherical planetin the Solar System.

3. Orbit Design and Optimization Approachfor EnVision

In this section, the procedure adopted to implementthe orbit optimization will be described, from theprovisional orbital parameters computation to thealgorithm selection and application.

3.1. Mission Constraints

We will focus on one of the main objectives: ob-serving the specified sites of interest, which give usthe ground truth. The Venera and Vega Russianlanders are the most significant targets to observeon the surface of Venus. In Table 2, the targetsare listed with the corresponding priority level (1being the highest level). Because of their metallicnature, they appear as bright spots (more than 6dB brighter than the surrounding surface) [4]. Be-yond the sites of interest, we will also focus on theNorth Pole interferometry measurements that willbe necessary for many science goals (such as spinaxis and rate).

The scientific requirements demand a well-controlled near circular orbit (with a maximum ec-centricity of 0.001) [4]. Also, the altitude should beas low as possible since the resolution of the grav-ity field declines rapidly with altitude. In terms ofthe atmospheric drag effect, previous missions de-

Target Latitude Longitude Priority

Vega 1 7.2 N 177.8 E 2Vega 2 7.14 S 177.67 E 2

Venera 5 3 S 18 E 3Venera 6 5 S 23 E 3Venera 7 5 S 351 E 3Venera 8 10.70 S 335.25 E 2Venera 9 31.01 N 291.64 E 1Venera 10 15.42 N 291.51 E 1Venera 11 14 S 299 E 3Venera 12 7 S 294 E 3Venera 13 7.5 S 303 E 1Venera 14 13.25 S 310 E 1

Table 2: Target sites.

tected sensible atmosphere below 200 km altitude:an altitude above 230 km is desirable.

The primary instrument carried by EnVision isVenSAR, which has five operating modes. Themodes used in this thesis are interferometry andhigh resolution strip-mode. The geometries forthese modes depend primarily on the swath widthand incidence angles. The area covered by the an-tenna is the footprint, and the swath width refers tothe strip of Venus’ surface from which the missiondata is being covered (Figure 4). VenSAR has afixed axis of maximum radiated power, also knownas boresight, of 32 (off-nadir angle).

Parameter Interferometry Strip-mode

Swath Width 43 km 40 kmIncidence Angle (near) 38.2 36.3

Incidence Angle (far) 44.1 42.2

Table 3: VenSAR operating modes parameters andcoverage.

3.2. Orbit Dynamics with Provisional ParametersTo test and simulate the orbit dynamics using Mat-lab we first considered the provisional EnVison or-bit parameters.

Ω e i u a (altitude)

0 0 88 195 6309.8 km (258 km)

Table 4: Provisional Orbit Parameters.

In the visualization of the spacecraft’s groundtrack we included the target sites to have a moreintuitive tool regarding the observation of these tar-gets. The North Pole reference point is just repre-sentative, it corresponds to all longitudes. The sitesof interest plotted were the landers Vega 1,2 andVenera 5, 6, 7, 8, 9, 10, 11, 12, 13, 14.

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Figure 3: Ground track plot for 5 days (∼ 80 orbits)at 100 s step for provisional parameters.

The Venus rotation speed is very low as wasstated before. Modeling the mission time can berather slow, since the time step shouldn’t be greaterthan 100 seconds. The discontinuous appearance inthe trajectory lines for the 5 days (∼ 80 orbits) isrelated to the fact that we chose a time step of 100s.

3.3. Observations ComputationIn order to compute the observations of the targetsites with VenSAR, it is necessary to examine validgeometry approximations.

The SAR antenna operates as a continuous swath(strip-mode) to detect the targets, and the targetsobservations computation was scripted with Matlaband was approached as follows:

• We considered an approximation of a flat sur-face model that comes with a total error ofaround 4km when compared with the sphericalmodel swath results from the proposal (Figure5);

• The footprint was assumed to be equal to theswath width, i.e. if the target is inside theswath width, we can assume it can be observed;

Figure 4: VenSAR and ground tracks.

• In order to take into account the deviations andto make sure that the target site is not justintersected at the swath edge we considered awidth of 30 km, equivalent to 0.3 (33 km forthe North Pole) for the initial scenario with theprovisional parameters;

• Since for testing the observations, the VenSARwith nadir direction is interesting to considerfor its simplicity, both nadir and off-nadir ge-ometries were calculated (only the off-nadir ge-ometry represents the operational SAR);

• The off-nadir geometry corresponding to thehigh resolution mode that will be observing thetargets was scripted with incidence angles of36.3 and 42.2 (44.1 and 42.2 for the NorthPole interferometric measurements);

• Finally, the swath width is given by y−x, andfrom the altitude and the tangent relations thatgive us x and y, we can estimate the swath andthe distance between the “VenSAR sub-point”(VSSP) longitude and the spacecraft sub-point

(SSP) longitude dV SSP = x+swath

2.

The “VenSAR sub-point” (VSSP) represented infigure 5 corresponds to the spacecraft sub-point(SSP) corrected for the VenSAR observations track.In brief, since the strip SAR image dimension is lim-ited across track, if the VSSP is between [Target’s

longitude -swath

2; Target’s longitude +

swath

2],

the target is considered observed.

3.4. Optimization Problem FormulationWe want to observe the targets with high priority assoon as possible in a minimum time, while satisfyingthe orbit constraints (near polar, near circular, lowaltitude). This way we ensure that the main targetsare observed at the beginning of each mission cycle(1 cycle corresponds to a Venus period) and thatthe operations planning for VenSAR will be easier.

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Figure 5: Geometry approximation for VenSAR instrip-mode.

3.5. Optimization Method SelectionTo solve our optimization problem, we need globaloptimization, i.e. a method that seeks to find theglobal best solution of an objective function thattranslates our goals while satisfying the orbit con-straints. We refer to the optimization as globalsince our objectives will likely lead to an objectivefunction with multiple local optima.

In the presence of multiple local minima, if weuse traditional local scope search methods we willoften find locally optimal solutions [10]. To obtaina globally optimal solution, there are exact meth-ods, in particular deterministic methods, which al-ways produce the same output for a given input anddon’t involve randomness, and stochastic methods,that use randomness. But there are also heuristicmethods, which look for solutions among all possi-ble ones, but do not guarantee that the best solutionis found [10].

Recently, genetic algorithms have proven to bea very successful way of getting solutions in orbitdesign optimization problems similar to the one ad-dressed in this thesis. We have summarized someof the advantages of selecting the genetic algorithmto optimize the orbit design:

• The proven success of the method for problemsthat are not convex, having many local minima[11, 12];

• The possibility of combining code for integer,real values and options [11];

• The fact that the algorithm’s randomness canaccelerate the progress of the optimization, andcan make the method less sensitive to modelingerrors [13];

• Finally, the possibility that the randomnesscan lead to a global optimum by escaping alocal minimum.

3.6. Genetic Algorithm ImplementationThe unknown variables considered for the optimiza-tion are the following:

• Longitude of the ascending node Ω;

• Argument of perigee ω;

• Inclination i;

• True anomaly ν0;

• Semi-major axis a;

• Eccentricity e;

• And finally, the time it takes to observe themission’s targets tf.

To fulfill the main objective identified, we need toselect the fitness function that will be minimized bythe genetic algorithm for the input boundaries wedefine. We want to observe the targets with highpriority (Venera 8, 9, 10, 13, 14, Vega 1, 2) as soonas possible in a minimum time, while satisfying theorbit constraints. Additionally, we want to observethe landers Venera 5, 6, 7, 11, 12. To evaluate ourmain objective we used the following fitness func-tion Fi:

Fi = −αNi(Ω, ω, i, ν0, tf, a, e)

N+ β

tfiTV enus

(2)

Where Ni and tfi are the fitness terms for theith design point, Ni is the total number of observedsites in tfi seconds, α and β are the fitness weightparameters (values between 0 and 1), which trans-late the relative importance of the fitness terms(maximizing number of covered sites versus mini-mizing the time needed), N is the total number oftarget sites considered, and finally TV enus is Venus’period.

We chose the fitness function for its simplicityand consequent computational efficiency. As wasstated before, modeling mission times for Venus isvery slow, because the time step has to provideenough time to pass above the area of interest. Dueto extremely different scales of number of intersec-tions and mission time the genetic algorithm willtend to kill populations with high mission times ifthe fitness function weights don’t compensate, thatis why α and β are necessary. This effect couldresult in an optimal solution with minimum timewindow but losing focus of the number of intersec-tions, which is why it is necessary to use differentcombinations of the fitness function.

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The total number of observed sites N is obtainedfrom the observations computation, i.e. if the targetis inside the swath width, we can assume it wasobserved. Considering only the sites with priority,we have N =

∑8n=1 Tn with Tn = 1 if the target

n is observed, and Tn = 0 if the target n is notobserved. Ni is a dependent variable that can becomputed given the candidate orbit and the set ofsites to select from. The reason why we divide Ni

by N and tfi by TV enus is so that we get betterscaled fitness function values.

To implement the genetic algorithm function inMatlab it is fundamental to go over a few optionsthat were made. We defined the population sizeto 100 individuals in each generation, since it wasa good compromise between convergence efficiencyand computational cost. The algorithm runs un-til the average relative change in the fitness func-tion value above the minimum stall generations (de-fault value of 50) is less than the function tol-erance TolFun (with set value of 1 × 10−4). If|f(xi)–f(xi+1)| < TolFun, the iterations stop [14].

4. Optimization Results4.1. Algorithm Tests and ValidationWe tested the script in order to evaluate the fitnessfunction’s convergence with the conditions in Table5.

Ω bounds [0 - 360 ]ω bounds [0 - 360 ]i bounds [-90 - 90 ]ν0 bounds [0 - 360 ]tf bounds [1000000s - 10000000s]a bounds [6300km - 6400km]e bounds [0 - 0.01]

α 1β 1

Targets Venera, Vega landers, North PoleGeometry Nadir

Table 5: Fitness function test conditions.

Ω 272.779

ω 342.080

i -49.714

ν0 126.598

tf 5670319.015sa 6390.834kme 0.001

Targets 12/13

Table 6: Fittest solution Fi = −0.653 for fitnessfunction test conditions.

We plotted the minimum, maximum, and mean

fitness function values in each generation for theselected fitness function Fi, and observed the con-vergence (|f(xi)–f(xi + 1)| < TolFun) after a fewmore than 50 generations. As expected the Best,Worst values bars decrease progressively until thefittest solution is reached.

We also wanted to see how Fi varied with thechange in design variables, and for that purpose sep-arate metric studies were performed to investigatethe variation of the fitness function with the orbitalelements i, Ω, ω and the time variable tf (Figures7 and 8). It is very clear from these studies thatthe objective function is a multi-minima function.It is also interesting to highlight that in Figure 7,we can observe an expected symmetry in the fitnessfunction dependency of inclination, and no targetsare covered for an equatorial orbit.

Figure 6: Minimum, maximum, and mean fitnessfunction values versus generations for fitness func-tion test conditions.

Furthermore, in Figure 7 we observe that for thenear polar inclinations the fitness function valueincreases. This is related to the fact that not asmany targets are covered for the these inclinationsfor short durations, which is expected since in thiscase the optimization is taking into account equalweights both fitness terms. Most targets are lo-cated near the equator, so for shorter mission timesthe algorithm should lead to a fittest solution witha lower inclination. The ground track of an orbitwith lower inclination covers a larger area aroundthe equator than near polar orbits.

Naturally, with these conditions, we only get 12out of the 13 targets covered, since the North Pole is

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not observed. The fittest solution obtained has aninclination of -49.714 (Table 6). In figure 7 we canobserve a peak around that inclination value, cor-responding as expected to a global minimum peak.The Ω value associated to that peak corresponds tothe fittest solution value. In figure 8, we can alsoidentify the global minimum peak associated to thefittest solution’s tf and ω values.

Figure 7: Fi versus inclination i and longitude ofascending node Ω for the fitness function test con-ditions.

Figure 8: Fi versus time window tf and argumentof perigee ω for fitness function test conditions.

After observing the fitness function, it is also nec-essary to further test the algorithm implementation.For that purpose, we used a simple test with a largerand fixed footprint of 5 to target just the Veneralanders with the nadir geometry for its intuitive na-ture (Table 7). These set conditions provide us theopportunity to quickly test the algorithm for shortdurations, since it should very quickly find a solu-tion that covers these targets with a low inclinationorbit (the same weights were attributed to the fit-ness terms). These conditions are set just to test thevalidity of the algorithm implemented, since it doesnot satisfy many of the mission constraints (polarcoverage, e inferior to 0.001).

We plotted the best and mean function values in

function of the generation to observe the conver-gence (Figure 9). In this case, as expected, due tothe large footprint considered, the lower the inclina-tion the faster the Venera landers near the equatorare covered, which explains the parametric plot inFigure 10 and the fittest solution obtained (Table8).

Ω bounds [0 - 360 ]ω bounds [0 - 360 ]i bounds [-90 - 90 ]ν0 bounds [0 - 360 ]tf bounds [10000s - 100000s]a bounds [6300km - 6400km]e bounds [0 - 0.01]

α 1β 1

Targets Venera landers, North PoleGeometry Nadir altered with fixed 5 footprint

Table 7: Test conditions for short durations.

Ω 246.470

ω 233.501

i 9.281

ν0 151.388

tf 10595.253sa 6390.834kme 0.001

Targets 8/11

Table 8: Fittest solution Fi = −0.727 for short du-rations test conditions.

Figure 9: Best and mean fitness function values ver-sus generation for short durations test conditions.

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Figure 10: Fi versus orbital parameters i and Ω forshort durations test conditions.

4.2. Optimal Solution

When searching for an optimal orbit solution, wetook into account the priority established for thetarget sites, and only considered Vega 1, 2, Venera8, 9, 10, 13, 14 and the North Pole observations. Inorder to ensure that all targets with high prioritywere covered, the fitness function weight coefficientsconsidered were α=0.7 and β=0.3. The other con-ditions are summarized in Table 9.

Ω bounds [0 - 360 ]ω bounds [0 - 360 ]i bounds [87 - 90 ]ν0 bounds [0 - 360 ]tf bounds [1000000s - 10000000s]a bounds [6300km - 6400km]e bounds [0 - 0.001]Geometry Off-Nadir

Table 9: Test Conditions for EnVision.

Ω 285.789

u 351.669

i 88.163

tf 5571678.038 sa 6310.828 kme 0.000

Targets 8/8

Table 10: Fittest solution Fi = −0.700 for an opti-mal orbit solution.

Figure 11: Best and mean fitness function valuesversus generation for an optimal orbit solution.

In a next step, we checked separately with thetargets observation script that with this orbit it ispossible to cover all 13 targets just after 1000 or-bits, so almost 2 times faster than with the provi-sional parameters considered in the EnVision pro-posal. Furthermore, the first observation is imme-diately achieved during the mission’s first orbit.

We calculated a few of the main features of thisoptimal orbit:

• Spacecraft altitude: 259.028 km

• Spacecraft velocity: 7.1747 km/s

• Orbital period: 5526.65 s

Even though the spacecraft’s orbital period re-mained closely the same as in the previous proposal,the interest sites are optimally covered at the startof the first cycle, and it is possible to reduce thetotal number of cycles needed for the mission andstill repeat the coverage of the sites.

To visualize the optimal orbit we obtained theground track plot (Figure 12). After 155 Orbitswe can check on the ground track that 4 sites wereintersected (Venera 9, 10, 12 and North Pole).

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Figure 12: Ground track plot for 10 days (∼ 155orbits) at 100 s step for optimal orbit.

On the overall, the optimality of the method wasverified in the sense that the heuristic returned anoptimal solution. We evaluated the quality of thesolutions through numerous algorithm tests and theconvergence of the method was determined by Tol-Fun. With the optimal solution we observe all tar-gets in an optimal time and as soon possible.

5. Conclusions

In this thesis, we investigated EnVision’s opera-tional orbit and optimized it for the observationof selected targets. The problem formulation wasdeveloped to design an orbit that covers as manyof the target sites as possible, while minimizing thetime window in which these observations are per-formed. A genetic algorithm was implemented toevaluate the combinatorial coverage problem.

On the overall, the developed optimizationmethod was a success in finding a fit solution toEnVision’s challenging case study. We also per-formed separate metrics studies to investigate thedependence of the algorithm’s fitness function andorbital elements and time variable which showedconsistency with the main results.

Even though the metric, constraints and prior-ities considered are specific to the EnVision mis-sion, the main scripts developed in this thesis canbe adapted to other orbiter missions to Venus andeven to other planets with the right adaptations,as the intersection computation and optimizationprocedure might be similar.

The results of this thesis were first introducedin the EnVison session during the European Plan-etary Science Congress in September 2015, and areto be included in the mission proposal to ESA. Fur-ther research on this subject will be developed asthe mission’s science and payload team refine theirdata. It will be possible to evaluate other objectivessuch as telecommunications, spotlight mode cover-age, and propulsion metrics applied to the naturalorbit.

AcknowledgementsI would like to express my sincere gratitude to mysupervisors Professor Paulo Gil from Instituto Su-perior Tecnico and Professor Richard Ghail fromImperial College London. Without their help, coun-sel, and generous transmission of knowledge, thisthesis would not have been possible.

I must also thank the EnVision team for givingme the unique opportunity of working with suchan exciting Venus project, and for letting me con-tribute to an outstanding ESA proposal.

Furthermore, I would like to express my appreci-ation to Dr. Edward Wright and to the NAIF teamfrom JPL for the exceptional support provided.

For the very welcomed inspiration, I must thankmy friends, in particular the ISU community whoencouraged me to pursue my work when I wasreaching a breaking point.

This thesis is dedicated to my parents and mysister, who give meaning and purpose to all my pur-suits.

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