Constellation of Two Orbiters Around Mars

Home , Argument of periapsis, Molniya orbit

Trans. JSASS Aerospace Tech. Japan Vol. 8, No. ists27, pp. Pd_23-Pd_28, 2010 Original Paper

Constellation of Two Orbiters around Mars

By Naoko OGAWA 1), Mutsuko Y. MORIMOTO1), Yasuhiro KAWAKATSU1,2) and Jun’ichiro KAWAGUCHI1,2)

1)JAXA Space Exploration Center, Japan Aerospace Exploration Agency, Sagamihara, Japan 2)Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Japan

(Received July 21st, 2009)

In this paper, we consider constellation of two orbiters around the Mars. We assume two orbiters, whose orbits are required to be orthogonal. After Mars orbit insertion, it will be needed to adjust their orbital planes appropriately for required conﬁguration under perturbation. We discuss how to transfer S/C to the seperated orbits after simultaneous insertion, and how to maintain these orbits during the mission phase. We adopt the frozen orbit for one orbiter to ﬁx the orbit axis, and utilize J2 perturbation on the argument of periapsis or the right ascension of ascending node for keeping the orthogonality between two orbital planes. Maneuver plan is also reported.

Key Words: Constellation, Mars, Perturbation

Nomenclature normal vector and MOB’s eccentricity vector should be parallel. Such constellation can be actually required in some scientiﬁc a :semimajor axis missions including both global and in-situ observation of the e :eccentricity planet1). i :inclination Ω :right ascension of the ascending node

ω :argument of periapsis MOA ω� :projection of ω onto the equatorial plane n :mean motion e :eccentricity vector of an orbital plane n :normal vector of an orbital plane θ :orthogonality index (OI) rp :periapsis ra :apoapsis Rm :Mars radius Subscripts A :Mars Orbiter A (MOA) B :Mars Orbiter B (MOB)

MOB 1. Introduction Fig. 1. Constellation of two orbiters around the Mars. Recent progress in planetary science and space exploration technology has resulted in increasing demand for larger scale of After Mars orbit insertion, it will be needed to adjust missions, such as combined exploration by multiple spacecraft their orbital planes and eccentricity vectors appropriately for (S/C). In this paper, as one example of preliminary study on assumed constellation under perturbation. This paper deals with missions by multiple S/C, we consider orthogonal constellation a preliminary study on how to insert S/C into these orbits and of two orbiters around the Mars. The purpose of this paper how to maintain them. is to investigate a rational solution to establish and maintain There have not been so many previous works on constellation orthogonal constellation. around the Mars or other bodies2–6), and most of them have We assume a mission where two orbiters will be launched focused on the low-orbit constellation for the global navigation and injected into the Mars orbit simultaneously. One orbiter, satellite system or communication satellites. There have or Mars Orbiter A (MOA), has a low altitude polar orbit, seemed no missions in which multiple orbiters are inserted while the other, Mars Orbiter B (MOB), has a highly elliptical simultaneously into a celestial body and then transfer to each orbit. As illustrated in Fig. 1, it is assumed that MOB orbit, except for Kaguya (SELENE)7) or Bepi-Colombo8). looks down upon MOA’s orbital plane at MOB’s apoapsis for simultaneous observation of atmosphere; i.e., the two orbits should be preferably orthogonal. More exactly, MOA’s

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2. Orbit Design 2.3. Condition for keeping orthogonality In the actual orbit, several external perturbation forces act on 2.1. Assumption for orbits the S/C to change orbital elements. and the orthogonality may We set the following assumption for orbits of two S/C: be lost within several months. Thus we must choose orbital elements carefully so that the orthogonality is kept during the 1. MOA and MOB should be injected simultaneously into the mission phase. Mars orbit and then separated. The most dominant perturbation is the J2 term in the Mars’ 2. MOA: low-altitude near-polar orbit. gravitational potential. According to Goddard Mars Model 9) 2B , The perturbation by J2 is 100-1,000 times larger than 3. MOB: highly elliptical orbit. J3 or solar perturbation at 4Rm. Thus, for simplicity, here we consider only the J perturbation, ignoring other perturbations 4. MOA’s normal vector and MOB’s eccentricity vector 2 (See Appendix). Because the J term affects only on the right should be parallel. 2 ascension of ascending node Ω, the argument of periapsis ω, 2.2. Orthogonality index and the mean anomaly M, we focus on perturbation only on Here we introduce an index for quantitative evaluation of the these elements, especially on Ω and ω: constellation of two orbiters. As mentioned above, the normal dΩ 3 nJ cosi R 2 vector of MOA’s orbital plane, nA, and the eccentricity vector = 2 m , (4) dt −2 (1 e2)2 a of MOB’s orbital plane, eB, should be parallel in the ideal − � � constellation. Thus the angle θ between these two vectors can be appropriate for the index. Figure 2 shows the definition of θ. 2 2 dω 3 nJ2(1 5cos i) Rm = − . (5) dt −4 (1 e2)2 a − � � The two vectors nA and eB are composed of Ω and ω, and affected by J2 perturbation. If two vectors nA and eB rotate toward the same direction at the same rate, the orthogonality will be maintained. Here we estimate the rotation rates of nA and eB, as illustrated in Fig. 3. Assuming that the MOA is in a polar orbit, i.e. iA is around 90 degrees, the rotation rate of nA can be approximated by Ω˙ A. As for the MOB, the rotation rate of eB is affected not only by Ω˙ B but also by ω˙B. Let us introduce a variable ωB� as shown in Fig. 3, the projection of ωB onto the equatorial plane. ωB� and its time derivative are as follows: Fig. 2. Definition of the orthogonality index. ωB� = arctan(tanωB cosiB), (6) dω cosi dω Using orbital elements, we can write down the two vectors B� = B B , (7) dt 2 2 2 dt nA and eB as: cos ωB + sin ωB cos iB ˙ sinΩA siniA and the rotation rate of e˙B will be approximated by ΩB + ω˙B� . nA = cosΩA siniA , (1) When rotation of the nA and eB is synchronized, the ⎛ − ⎞ cosiA following relation should be satisfied: ⎝ ⎠ cosωB cosΩB sinωB sinΩB cosiB dΩ dΩ dω − A = B + B� (8) eB = cosωB sinΩB + sinωB cosΩB cosiB , (2) ⎛ ⎞ dt dt dt sinωB siniB dΩ cosi dω = B + B B . (9) ⎝ ⎠ dt cos2 ω + sin2 ω cos2 i dt where Ω is the right ascension of the ascending node, ω is the B B B argument of periapsis, and i is the inclination of each orbit. In Eq. (9), the left side, Ω˙ A, and the first term of the right The orthogonality index (OI) θ is defined as: side, Ω˙ B, are time-invariant. The second term of the right side, ω˙B� , is time-variant, however, because its coefficient includes nA eB θ = arccos · . (3) ωB, which changes according to the J2 perturbation. It indicates nA eB �| || |� that Eq. (9) is valid only if the second term of the right side is θ around 0 degree or 180 degrees indicates that the two orbital constant in the following three cases. planes are properly placed. iB = 0 or 180 degrees When iB is zero or 180 degrees, the coefficient becomes 1, ± and the equation becomes simple:

dΩ dΩ dω A = B B . (10) dt dt ± dt

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Orbital Plane

Orbital Plane MOB

equatorial plane MOA

equatorial plane

Fig. 3. Deﬁnition of ωB� and synchronization of two orbits.

If the apoapsis and periapsis of two orbits are given, we can As the second term in the left side of Eq. (12) equals to zero in derive iA appropriate for synchronization by substituting Eq. (4) the Molniya orbit, we derive and Eq. (5) into Eq. (10). A drawback of this case is huge dΩ 3 nJ cosi R 2 delta-V to create large difference of about 90 degrees between B 2 B m 0 5240 degrees/day. (13) = 2 2 = . inclinations of the two orbits. dt −2 (1 e ) aB − B � � iB = 63.4 or 116.5 degrees From the equation above, possible combinations of (aB,eB) are When iB equals to 63.4 or 116.5 degrees, Eq. (5) becomes obtained. Actually we permitted 5 degrees deviation in θ zero. It means that ωB is ﬁxed. These orbits are called Molniya ± within the 0.5 Martian year. orbit or frozen orbit. We have only to choose iA so as to realize Figure 4 shows the possible region for rpB and raB to realize dΩA dΩB SMO. SMO with 5 degrees deviation in θ can be achieved = . (11) ± dt dt only within the extremely small region between two lines close to each other. It implies that the r and r to realize SMO are i = 90 or 270 degrees pB aB B highly restricted. iB of 90 or 270 degrees also makes the coefﬁcient zero. In this case, however, actually ωB rotates in the north-south direction 10 and the orthogonality is not maintained. We will not deal with 9 this case in this paper. 8 2.4. Sun-synchronous molniya orbit In this section, we investigate whether a Sun-synchronous 7 Molniya orbit (SMO) is possible or not, for an optional study. 6 5

Note that deﬁnition of “Sun-synchronous” for MOB in this Rm] paper is that the orbit axis synchronizes to the Sun and different rp [ 4 from usual deﬁnition, while “Sun-synchronous” for MOA has 3 the usual meaning, i.e., the orbital plane synchronizes to the 2 Sun. 1 Here we will derive the possible conditions. First, the Invalid 0 Molniya orbit requires iB = 116.5 degrees for MOB. We cannot 0 1 2 3 4 5 6 7 8 9 10 set iB = 63.4 degrees, because the sun-synchronous orbit can be ra [ Rm] realized only in the retrograde orbit. Next, the rotation rate of Fig. 4. The possible region for rpB and raB to realize SMO. SMO with the orbit axis should equal to 0.5240 degrees/day, which is the 5 degrees deviation in θ can be achieved only within the extremely 1 ± mean motion of the Mars : small region between two lines. dΩ dω B + B� = 0.5240 degrees/day. (12) dt dt 2.5. Numerical simulations We performed some numerical experiments to verify the 1The sidereal mean motion of Mars, μ, is 1886.52 arcseconds per mean solar day 10), which equals to 0.5240 degrees per day. orbits using numerical analysis software (MATLAB 2007b, The

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Table 1. Orbital parameters for MOA and MOB. MOA MOB 180 SMO non-SMO SMO non-SMO 160

Apo ra 3,000 km +Rm 3 Rm 4 Rm 140

Peri rp 150 km +Rm 1.5 Rm 120

Incl i 96.08 degrees 93.78 degrees 116.5 degrees 100

MathWorks Inc.). The parameters of orbits for MOA and MOB 60 are shown in Table 1. For simplicity, ωA = ωB = ΩA = 0 orthogonality index [deg] 40 and ΩB = 90 degrees initially so that two orbits are initially orthogonal. We assumed two phases, SMO phase and non-SMO 20 0 phase. The inclination of MOA was set so as to synchronize 0 50 100 150 200 250 300 MOB’s rotation using Eq. (9). time [days] Fig. 5-10 show the simulation results in each phase. In both Fig. 7. The proﬁle of the OI in the Molniya orbit phase. phases, OI is ﬁxed and orthogonality is properly maintained. In the SMO phase, it is shown that the Sun-synchronous orbit is achieved.

1 2

0 1

-1

Y [Rm] 0

-2 Y [Rm]

-1 -3

-2 -1 0 1 2 3 X [Rm]

Fig. 5. Monthly orbits of MOA and MOB in the Molniya orbit phase -1 0 1 2 drawn in the Mars equatorial coordinate system. X [Rm]

Fig. 8. Monthly orbits of MOA and MOB in the SMO phase drawn in the Mars equatorial coordinate system.

Sun

Fig. 6. Monthly orbits of MOA and MOB around the Sun in the Molniya orbit phase. The orbit sizes are magniﬁed about 60 times to make relation to the Sun visible. Sun

In actual missions, Ω and ω are given according to the Fig. 9. Monthly orbits of MOA and MOB around the Sun in the SMO insertion orbit. Dependency of the OI on Ω and ω is to be phase. The orbit sizes are magniﬁed about 60 times to make relation to investigated. the Sun visible. Fig. 11 shows the difference of MOA and MOB inclinations during synchronization with respect to the apoapsis of MOA

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180

160

140

120

100

orthogonality index [deg] Fig. 12. A possible sequence to establish the orthogonal constellation. 40 Left: initial orbit, middle: intermediate orbit, right: ﬁnal orbit. 20

0 0 50 100 150 200 250 300 time [days] So as to reduce the amount of the total delta-V, we propose to utilize several strategies. The inclination maneuver is reduced Fig. 10. The proﬁle of the OI in the SMO phase. by a higher apoapsis in the initial orbit. Rotation of the orbital plane will be carried out by utilizing J2 perturbation. The and MOB. Difference increases if the apoapsis of MOB descent maneuver can be supported by aerobraking. becomes greater, or that of MOA becomes smaller. In actual 3.1.1. Inclination maneuver at higher apoapsis missions, we have to design orbits so as not to have large Changing of the inclination requires a huge delta-V. Thus we difference between two inclinations. should adopt the frozen orbit for MOB, where the difference of inclination between MOA and MOB is less than the case of 40 MOA Apo 3,000km iB = 0 or 180 degrees. Moreover, the inclination maneuver will 35 MOA Apo 5,000km be performed at the apoapsis. It will be easier if the S/C has a MOA Apo 7,000km higher apoapsis. Therefore, this time we assume initial insertion 30 of the S/C into a high orbit with a 30Rm apoapsis. 25 3.1.2. Rotation by J2 20 Rotation of ΩA will be performed after change of iA. When both of two orbiters are in the 30Rm 1.5Rm orbit, and iA = 15 × Inclination Diff [deg] 96.08 degrees and iB = 116.5 degrees respectively, then Ω˙ A is 10 0.0024 degrees/day, and Ω˙ B is 0.5053 degrees/day. It means

5 that it will take 179 days to form orthogonal constellation by J2. 3.2. A possible maneuver sequence 0 2 3 4 5 6 7 8 9 10 A possible scenario with the amount of delta-V is as follows. MOB Apoapsis [ ] x Rm In this scenario, the SMO phase is initially realized, and then Fig. 11. Difference of MOA and MOB inclinations during the non-SMO phase is started after the phase transition. The synchronization with respect to the apoapsis of MOA and MOB. parameters are same as those shown in Table 1.

1. Simultaneous Mars orbit insertion. 30Rm 1.1Rm, incl.: 116.5 degrees. × 2. Changing i from 116.5 degrees to 96.08 degrees. ΔV = 61 m/s. 3. How to Establish Orthogonal Constellation A 3. MOA’s “walk-in” into the atmosphere to prepare for aerobraking. Descend from 1.1Rm to (135 km +Rm). ΔV = 4.6 m/s. We propose a possible scenario to establish the orthogonal 4. MOA’s aerobraking (30Rm (3,000 km +Rm)) and rotation of → constellation, and show an example of sequence. ΩA by J2 perturbation for 179 days. 3.1. A possible scenario 5. MOA’s “walk-out” from 135-km to 150-km periapsis in altitude. ΔV = 3.0 m/s. We assume that the two orbiters will be inserted simultane- 6. MOB’s “walk-in” from 1.1Rm to (135 km +Rm). ΔV = 4.6 m/s. ously into the Mars orbit. In order to realize orthogonal constel- 7. Aerobraking of MOB (30Rm 3Rm). → lation, several orbit control maneuvers are necessary. A possible 8. MOB’s “walk-out” from (135 km +Rm)to1.5Rm periapsis. ΔV scenario will be as follows as shown in Fig. 12: = 202 m/s. 9. SMO completed. Primary observation phase. 1. Mars Orbit Insertion (MOI): two orbiters are inserted into 10. MOB’s ascending the apoapsis from 3Rm to 4Rm. ΔV = 148 m/s. the same initial orbit. 11. Changing iA from 96.08 degrees to 93.78 degrees. ΔV = 87 m/s. 2. Inclination Maneuver: the inclination of one orbiter is 12. Molniya orbit completed. Secondary observation phase. changed The total ΔV is 513 m/s except ΔV for MOI. This can be reduced 3. Orbital Plane Rotation: the orbital plane of one orbiter by further optimization. is rotated, or ΩA is rotated, so as the two orbits become orthogonal. 4. Descent: MOA descents to the ﬁnal orbit.

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4. Summary A more precise version of Eq. (5) incorporating J3 is written as11, 12): In this paper, we discussed how to control and maintain 2 2 dω 3 nJ2(1 5cos i) Rm orthogonal orbits of two S/C around the Mars using J2 = − dt −4 (1 e2)2 a perturbation. It was shown that some special cases for � � 3 − inclination allow us to keep orthogonality between two S/C 3 nJ3Rm sinω + 3 2 3 considering J2 perturbation. It was also indicated that such 2 a (1 e ) esini − constellation can be realized with rational maneuvers. This sin2 i 35 1 5cos2 i + e2 1 sin2 icos2 i . is just a preliminary example, and more precise analysis with × 4 − − 4 � � �� consideration of the J3 term and solar perturbations, coverage � � (14) analysis and footprints, thrust optimization and feasibility of aerobraking are to be discussed for future works. Even if the critical inclination is adopted, the second term will be non-zero, but can be neglected in some cases. For example, References the second term in an orbit of 8Rm 150 km is about 0.0014 × − 1) Terada, N., Imamura, T., Seki, K., Yamazaki, A., Satoh, T., deg/day at most. If the mission duration is assumed to be about Matsuoka, A., Okada, T. and Ogawa, N.: Synergetical Atmospheric one or two Martian years, it is small enough compared to the Science by 2 Mars Orbiters, in Proc. 27th International Symposium on perturbation by J2. Space Technology and Science (ISTS 2009), 2009. 2) Ely, T. A., Anderson, R., Bar-Sever, Y. E., Bell, D., Guinn, J., Jah, M., Kallemeyn, P., Levene, E., Romans, L. and Wu., S.-C.: Mars network constellation design drivers and strategies, in Proc. 1999 AAS/AIAA Astrodynamics Specialist Conf., 1999. 3) Nann, I., Izzo, D. and Walker., R.: A reconfigurable Mars constellation for radio occultation measurements and navigation, in Proc. 4th Int. Workshop on Satellite Constellation and Formation Flying, 2005. 4) Talbot-Stern, J.: Design of an integrated Mars communication, navigation and sensing system, in Proc. 38th Aerospace Sciences Meeting & Exhibit, 2000. 5) Pirondini, F. and Fernandez, A. J.: A new approach to the design of navigation constellations around Mars: The MARCO POLO evolutionary system, in Proc. Int. Astronautical Congress 2006, 2006. 6) Roh, K.-M., Luehr, H., Rothacher, M. and Park, S.-Y.: Investigat- ing suitable orbits for the Swarm constellation mission – the frozen orbit, Aerospace Science and Technology, (2008). 7) Katoh, T., Terada, H., Tanaka, K., Ohtani, K., Kamikawa, E., Mat- suoka, M., Matshumoto, S., Takizawa, Y., Ogawa, M., Kawakatsu, Y., Kasuga, K., Ikegami, S. and Yamamoto, M.: Orbital maneuver plan and operation results of “KAGUYA” during lunar transfer orbit and lunar orbit injection, in Proc. 26th Int. Symp. Space Technlogy & Science (ISTS2008), 2008. 8) Yarnoz, D. G., Jehn, R. and Pascale, de P.: Trajectory design for the Bepi-Colombo mission to Mercury, in Proc. 57th International Astronautical Congress 2006 (IAC 2006), 2006. 9) Lemoine, F. G., Smith, D. E., Rowlands, D. D., Zuber, M. T., Neumann, G. A. and Chinn, D. S.: An improved solution of the gravity field of Mars (GMM-2B) from Mars Global Surveyor, J. Geophysical Research, 106 (2001), pp.23359–23376. 10) National Astronomical Observatory of Japan, ed.: Rika Nenpyo (Chronological Scientific Tables 2008), Maruzen, 2008. 11) Kiedron, K. and Cook, R.: Frozen orbits in the J2 + J3 problem, in Proceedings of the AAS/AIAA Astrodynamics Conference 1991, 1992. 12) Chobotov, V. A.: Orbital Mechanics, AIAA Education Series, American Institute of Aeronautics and Astronautics, 2002.

Appendix: Consideration of J3 Term

In the discussion above, only J2 perturbation among many types of perturbations was taken into account. Of course the J2 term of Mars is the most dominant perturbations, but the J3 term or the Sun gravity cannot be negligible in some cases.

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