# Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

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1 Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This pape compaes the Cicula Resticted Thee-Body Poblem model to the Thid Body Petubation model fo the puposes of analyzing spacecaft tajectoies in the Mas Phobos system. The pupose of this is to help design obits fo missions to Phobos. Compute models have been developed fo each method and compaing thei outputs to each othe and known behavios of the Mas-Phobos system. These models ae then used to investigate mission citical scenaios about Phobos. These scenaios include possible obits about Phobos and fomation flying with Phobos. The geneal behavio of spacecaft attempting to obit o fly in fomation with Phobos was defined. While obiting Phobos was found to be geneally impossible, flying in fomation was found to be possible and stable ove long peiods. It was also demonstated that the Thid-Body Petubation model was bette than the Cicula Resticted Thee-Body Poblem fo geneal investigation of spacecaft behavio. Nomenclatue = mass atio in CRBP Model Mas = Gavitational Paamete of Mas (km /s 2 ) Phobos = Gavitational Paamete of Phobos (km /s 2 ) Mas Sat = Position Vecto fom Mas to the hypothetical spacecaft (km) Mas Phobos = Position Vecto fom Mas to the hypothetical spacecaft (km) Phobos Sat a v = Position Vecto fom Mas to the hypothetical spacecaft (km) = semi-majo axis (km) = scala velocity of cicula obit (km/s) I. INTRODUCTION Space exploation has poven to be a geat diving foce of innovation, industy and scientific gowth. Fo this eason, humanity looks to the stas fo the answes to the challenges of today as well as those in the futue. Though getting to those stas has poven beyond the each of moden spacecaft, it is not only possible but a well-documented fact that unmanned missions can go to the planets and moons of Eath s sola system to the benefit of all. Howeve, seveal of the bodies closest to the Eath have gone unvisited and lagely unstudied. Among the most notable of these bodies is the pimay Matian moon, Phobos. Though thee was an attempt to visit and study this moon on the pat of the Russian space pogam via the Phobos-Gunt mission in 20 [6], this mission failed befoe leaving Eath obit due to a systems failue. Undegaduate Student, Depatment of Aeospace Engineeing, 229 Hammond Building, AIAA Student Membe

2 With the failue of the Phobos-Gunt mission, Phobos still emains a mysteious figue in the sola system. It poses an inteesting study in geology and undestanding its oigins may help in efining cuent models of the development of the sola system. Cuent theoies tend to look at Phobos as a captued asteoid [], but nothing shot of sampling its egolith will help in confiming o debunking this. Phobos may also be a location whee fuel and othe esouces could be mined o othewise geneated. Buzz Aldin has epeatedly and adamantly pushed fo the development of Phobos into a fowad opeating base in the Matian system fo this vey eason. He contends that Phobos would be the ideal point to station astonauts to begin the pocess of human settlement of the ed planet using teleobotics [2]. This makes Phobos an inteesting taget fom an engineeing and mission planning standpoint. To investigate these points, howeve, it will be necessay to send a mission to Phobos. At pesent, thee is inteest by NASA and the ESA to visit Mas closest companion fo diffeent pogams. NASA is looking to exploe Phobos unde the outline of the Unified Space Vision, and may wish to implement the plans poposed by Buzz Aldin to make Phobos a fowad base of opeations [2]. The ESA is consideing a visit to Phobos unde the banne of thei ecently unveiled Phootpint pogam, which would function as a sample-etun mission [5]. While sending missions to the Matian system has been done on many occasions, sending a satellite to obit Mas o deploying a lande to the Matian suface is vey diffeent fom visiting a Matian moon. To do this, it is necessay to have a clealy defined launch window and undestand when the vehicle will be aiving in the aea of the moon. It is also necessay to know how to position the vehicle as to keep it in the aea of the moon. This second point is the basis fo this eseach. Using the Cicula Resticted Thee-Body Poblem model and the Thid-Body Petubation model, this pape detemines the geneal chaacteistics of a spacecaft in the vicinity of Phobos. This pape also identifies which of these gavitational mechanics models is bette suited fo geneal mission planning nea Phobos. This pape investigates a link between the investigated obits and the solutions posed by Wiesel as stable obits about Phobos. The solutions in question popose stable, etogade obits about Phobos that could be used to investigate Phobos [7]. The eason fo eviewing these solutions is to see if they stand up to less pecise methods, and theeby gauge moe geneal utility. If the simulations used in this pape disagee with Wiesel s wok, it may point to an instability in the obits. II. CIRCULAR RESTRICTED THREE-BODY PROBLEM MODEL The Cicula Resticted Thee-Body Poblem (CRBP) model begins by making an assumption that thee ae only thee objects in the system that can exet a gavitational foce, and one of the thee is massless []. The thee objects in this case ae Mas, Phobos and the spacecaft, which is the object consideed to be massless. This is a valid assumption because the mass of the spacecaft is so small when compaed with that of Mas o Phobos that it has essentially no effect on eithe body. Figue futhe shows some additional elationships in the Cicula Resticted Thee-Body Poblem model. The distance between the centes of the two bodies with mass is nomalized to have a value of one distance unit. The mass of the smalle object, hee Phobos, is divided by the oiginal value of the distance between the two lage masses and called µ. The lage mass, hee Mas, is given the value ( - µ). The cente of mass fo this system, denoted by the cicle with an x though it, is the oigin fo this model and is a distance µ fom Mas and a distance ( - µ) fom Phobos. This whole system is otating and has a peiod of 2π, with the i axis going fom the cente of mass of the system to the cente of the pimay mass and the j axis being pependicula to the i axis. Fo the emainde of this pape, plots using the i-j axies denote a otating system and plots using the x-y axies denote an inetial system. Fo this model, the x axis and y axis would also have thei oigins at the cente of mass. 2

3 Figue : CRBP Model The Cicula Resticted Thee-Body Poblem model is by its default natue set in a otating fame of efeence. It is govened by the following equations: x 2 y x x x y z 2 x y z z 2 y y 2 2 () (2) () ( x ) y z (4) ( x ) y z (5) In these equations, x, y and z ae the displacements of the spacecaft elative to the cente of mass in the i-j-k coodinate fame, as seen in Figue. Note that the k o z diection is pependicula to both the i axis and j axis shown in Figue. Futhemoe, this eseach only consideed the case of the spacecaft being in the same plane as Mas and Phobos. The values and 2 ae the displacements fom Mas and Phobos to the spacecaft, espectively. III. THIRD-BODY PERTURBATION MODEL The Thid-Body Petubation model does not equie any nomalizing convesions to set up. In this model, the system is by default in an inetial fame of efeence. Its oigin and cente is located at the coe of the pimay body [4], which is Mas in this instance as seen in Figue 2. In Figue 2, X and Y axies define the inetia fame while i and j axies define the otational fame. This model is moe modula and can accommodate the influences of moe bodies that may be elevant. Howeve, fo this eseach poject the only thid-body consideed was Phobos. The spacecaft was again teated as a massless object.

4 Figue 2: Coodinate Fames fo the Thid-Body Petubation Model The elationship between Mas, Phobos and the spacecaft in this model is defined by the following equation: Mas Mas Sat Sat Phobos Phobos Mas Sat Sat Phobos Mas Phobos Mas Phobos (6) This equation is the base fom of the thid-body petubation model and is the fom that was used to ceate the simulation. It can, howeve, be expanded to include tems epesenting the influence Deimos, the Sun o any othe sufficiently impotant body. IV. SIMULATIONS Two simulations wee built, one fo using the Cicula Resticted Thee-Body Poblem model and the othe using the Thid-Body Petubation model. These simulations wee built and un in MATLAB as it is vey easy to display the geneated data in a numbe of helpful ways. Given that thee ae no analytical solutions fo eithe model, it was necessay to numeically integate the govening equations using MATLAB s ode45 function. Fo the puposes of both models, Mas and Phobos wee said to be point masses and the spacecaft was said to be massless. It was futhe assumed in both models that the centes of Mas and Phobos wee always 975 km fom each othe at all times, whee 975 km is thei aveage distance. As peviously stated, this distance was nomalized to having a value of distance unit fo use in the Cicula Resticted Thee-Body Poblem model. Thee wee thee geneal scenaios tested that gave the theoetical satellite the equied velocities to place it into a cicula obit about Phobos, flying in fomation, in the same obital path, as Phobos and flying in a adially offset fomation with Phobos. With the fomation flying scenaios, thee was also the point of the spacecaft being ahead o behind Phobos obit o of it being adially in o out of Phobos obit. To detemine the velocities equied in all instances, the vis-viva integal, equation (7), was used. Hee, is the adius of the obit, v is the speed, a is the semimajo axis and µmas is the gavitational paamete of Mas. v Mas 2 Mas (7) a With fully defined simulation conditions and the govening equations set up, it was possible to stat testing. Both simulations wee used to look at a spacecaft placed at 0 km, 0 km, 00 km and 000 km, and seveal othe intemediate altitudes, above the suface of Phobos fo each scenaio vaiation. Fo the Thid-Body Petubation model it was necessay to use 0.0 km instead of 0 km simply because it is impossible to divide by zeo. The time it took to geneate the data on each individual un was also measued. 4

5 V. RESULTS AND DISCUSSION A. Analysis of Cicula Obit Scenaios The simulation based on the Cicula Resticted Thee-Body Poblem model showed clealy that to put a spacecaft into obit about Phobos is faily impactical. Consideing that the sphee of influence of Phobos has a adius of about 7 km and the aveage adius of Phobos itself is about. km, it is to be expected that objects would not obit Phobos. The inability to obit Phobos can be seen in Figue which shows the movement of the spacecaft ove the couse of two of Phobos obits of Mas. Using this simulation, it is also clea that objects placed diectly onto the suface of Phobos without secuing themselves also dift off into space. Figue : CRBP Model Obit Scenaio Stating at an Altitude of 0 km This obsevation also holds tue fo the simulation based on the Thid-Body Petubation model. As can be seen in Figue 4, the spacecaft is moving away fom Phobos. While diffeent fom the motion seen in Figue, both paths have a simila tendency oveall which is that the satellite difts away fom Phobos. In both cases this obit ends up as a Phobos independent, low eccenticity obit of Mas. 5

6 Figue 4: THIRD-BODY PERTURBATION Model Obit Scenaio Stating at an Altitude of 0 km fo Two Obits B. Analysis of In-Line and Radial Fomation Flying Scenaios Putting the spacecaft into fomation with Phobos, adially o within the same obital path, poves to be moe effective in staying nea Phobos. This holds tue fo both the Resticted Cicula Thee-Body Poblem model and the Thid-Body Petubation model. Figue 5 is a good example fo the behavio of a spacecaft in fomation in the same obital path with Phobos, at an altitude of 5 km, fom the Resticted Cicula Thee-Body Poblem simulation. This figue also includes the spacecaft s motion with espect to Mas. As should be clea all of the discussed, and yet to be discussed, situations esult in an obit about Mas. Fo this case, the obit is mostly cicula. Figue 5: CRBP Model Fomation Scenaio Stating at an Altitude of 5 km 6

7 Due to the fact that the spacecaft is only 5 km fom the suface, Phobos gavity altes the obit of the spacecaft ove time. Moving the spacecaft to a highe altitude takes it away fom Phobos gavity, allowing it to continue on its path with less vaiation. It is woth noting in the adial fomation flying case, shown in Figue 6 stating at an altitude of 000 km in the Mas centic inetial view, the basic behavios and effects ae the same except fo the fact that the obit of the spacecaft about Mas is elliptical. Figue 6: CRBP Model Radial Fomation Scenaio Stating at an Altitude of 5km C. Identifying Useful Obits Discounting attempting diect obits via the Cicula Obits method leaves the two foms of fomation obits peviously discussed as the pefeed couse of focus. Figue 7 shows meshes that look at the displacement of a spacecaft fom Phobos acoss time, specifically the time it takes Phobos to complete two obits about Mas, and stating altitudes, km to 00 km. This ange was selected fo analysis as it is whee the moe inteesting behavios and potentially useful obits emege. An inteesting, ecuing featue of the meshes in Figue 7 is the valleys, maked by dak blue, that occu with the obits stating within the fist 40 km of Phobos. While these egions suggest obits that do not stay too fa fom Phobos, they also have a citical flaw: they all intesect the suface of Phobos ealy in thei obit. The obits that ae guilty of having this citical flaw all have the dakest blue somewhee along thei line in the mesh. Due to the colo scale used in these meshes, it can be had to spot those touble egions at fist glance, especially given that tighte obits ae within the dak blue egions. Obits that develop in height in the meshes, and theefoe become lighte blue though ed ae also to be avoided, as the show obits whee the altitude of the spacecaft ove Phobos becomes between 600 km and 900 km. These ae not paticulaly useful altitudes to occupy if the objective is to investigate the moon. 7

8 Figue 7: Thid-Body Petubation Model Fomation Displacements fo Time and Stat Altitudes Looking close at the behavio of the spacecaft in the ange fom a stating altitude of 0 km to an altitude of 40 km fo the adial fomation case yields positive esults. Stating at an altitude of km, a spacecaft taveling 2.27 km/s pependicula to the line between the centes of Mas and Phobos will ente into an obit, which fom Phobos appeas to be a stable obit. Figue 8 shows the behavio of a spacecaft that beginning its obit nea Phobos with the mentioned conditions. This when scaled fo inceasing altitude, this type of behavio emains tue fo all othe pogade obits, though at futhe and futhe distances fom Phobos. These obits seem to emain stable fo long peiods of time and have been shown to be stable fo at least 20 days. Figue 8: Stable Obit Relative to Motion of Phobos ove Twenty Days 8

9 D. Investigation of Wiesel s Solutions Wiesel s solutions wee plugged into the simulation that uses the Thid Body Petubation Method to attempt to eceate the obits that he identified. It became immediately clea, howeve, that thee is something pofoundly diffeent between the simulations, as all of Wiesel s solutions yield obits pimaily aound Mas with no egad fo Phobos. Table is a table of Wiesel s stating altitudes and velocities that wee fed into the simulation. All of these yielded the same type of esult of an obit about Mas that was uncoupled fom Phobos. This begs the question as to what caused the dispaity in the esults. Going though Wiesel s deivations, it can be seen that his solutions fom analyzing the Mas-Phobos system used a non-zeo tem fo Phobos eccenticity, which is commonly ecoded as It is easonable to conclude that this is a souce of the diffeence in esults. As discussed in pevious sections, Phobos has a vey stong effect on spacecaft at altitudes up to aound 50 km especially in the fomation flying obits. Within this ange of values, a change of a kilomete o less leads to wildly diffeent behavios anging including neamisses, impacts and stable obits. Based on this, alteing the tajectoy of Phobos even slightly, in the cosmic sense, would damatically alte its inteactions with the spacecaft such that nea-misses could feasibly become stable obits o vice vesa. Table : Wiesel s Stating Paamete s fo Stable Peiodic Obits Altitude (km) Velocity (km/s) VI. CONCLUSION AND FUTURE WORK Upon eview the data geneated duing this study, it becomes clea that placing an object in diect obit about Phobos is geneally unlikely to succeed. Given Phobos weak gavity, especially when compaed with that of Mas gavity, it is clea that any object attempting to obit Phobos diectly is likely to dift away. Even objects on the suface of Phobos will have a had time emaining on the suface, equiing some fom of anchoing to stay on the suface. The moe stable, and theefoe pefeable, option is to fly in fomation with Phobos. Fomation flying has been shown to be even moe stable at highe altitudes, at least fo the in-line case. Flying adially in fomation poves to yield even bette esults as the motion of the spacecaft, acoss a wide ange of stating altitudes, as seen by Phobos look like peiodic obits. The outcomes of this eseach, howeve, go against the well documented esults of Wiesel. This can likely be attibuted to the diffeence in basic assumptions. Namely, Wiesel used a non-zeo value of eccenticity fo Phobos whee this study assumed a cicula obit. While the eccenticity of Phobos is only about 0.05, the system has shown to be faily sensitive such that a change of a faction of a kilomete may esult in a vey diffeent behavio. That said, the solution shown hee may still be a viable and useful type of obit that could be used instead of a diect obit, if any should exist. If this eseach wee to be expanded upon, the focus should be in econstucting the simulation using a non-zeo eccenticity in an attempt to eceate Wiesel s solutions to attempt to veify the sensitivity of the oveall system. Though it lagely comes down to ease of use, the Thid-Body Petubation model was lagely the bette model to wok with. A lage pat as to why that was the case is the fact that eveything is in eal-wold units and no convesions ae equied to make use of the esulting data points, which has the added benefit of peseving accuacy bette. This model also allows the system to have its oigin at Mas cente as opposed to having it at the cente of mass fo the Mas-Phobos system. This in tun allows fo the data to be easily manipulated and displayed in helpful ways with ease. It also seemed that the Thid-Body Petubation model equied less computational time when compaed with the Resticted Cicula Thee-Body model. 9

10 VII. ACKNOWLEDGEMENT I would like to thank the Pennsylvania State Univesity College of Engineeing and the College of Engineeing Reseach Initiative fo making this study possible. Additionally, I would like to thank my eseach adviso, D. David B. Spence, fo helping me to define my study and assisting me to ovecome the challenges my study pesented. VIII. REFERENCES [] Andet, T. P., Rosenblatt, P., Paetzold, M., Haeusle, B., Dehant, V., Tyle, G. L., & Maty, J. C. (200). Pecise mass detemination and the natue of phobos. Geophysical Reseach Lettes, 7(09) doi:http://dx.doi.og/0.029/2009gl04829 [2] Aldin, B. (20). Mission to Mas: My Vision fo Space Exploation ( st ed.). Washington, D.C.: National Geogaphic Society [] Cutis, H. D. (200). Obital Mechanics fo Engineeing Students (2 nd ed.). Bulington, Massachusetts: Elsevie. [4] Vallado, D. A. (200). Fundamentals of Astodynamics and Applications (2 nd ed.). El Segundo, Califonia: Micocosm Pess. [5] Walke, R. (204, Januay 7). Why Phobos Might be the Best Place to go fo a Sample Retun fom Mas Right Now. doi: [6] Wall, M. (202, Januay 5). Failed Russian Mas Pobe Cashes Into Pacific Ocean: Repots. doi:http://www.space.com/4242-ussia-spacecaft-phobos-gunt-cash-eath.html [7] Wiesel, W. (99). Stable obits about the matian moons. Jounal of Guidance, Contol, and Dynamics, 6(), doi:http://seach.poquest.com/docview/ ?accountid=58 0

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