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Constellation of Two Orbiters Around Mars

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Constellation of Two Orbiters Around Mars 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 Copyright© 2010 by the Japan Society for Aeronautical and Space Sciences1 and ISTS. All rights reserved. Pd_23 Trans. JSASS Aerospace Tech. Japan Vol. 8, No. ists27 (2010) 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 2 Pd_24 N. OGAWA et al.: Constellation of Two Orbiters around Mars 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.
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