JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 33, No. 4, July–August 2010

Engineering Notes

Five Special Types of Orbits sensing satellites. Lagrangian planetary equations, which give the mean precession rate of the line of node, provide the theoretical Around Mars foundation of sun-synchronous orbits. Boain described basic mechanical characteristics of Earth’s sun-synchronous orbits and ∗ † ‡ provided several design algorithms to satisfy practical mission Xiaodong Liu, Hexi Baoyin, and Xingrui Ma requirements [4]. Several Martian satellites have been placed into Tsinghua University, sun-synchronous orbits, such as MGS [1] and MRO [2]. 100084 Beijing, People’s Republic of China Orbits at the critical inclination can keep eccentricity and argument of perigee invariable on average. For example, the Molniya DOI: 10.2514/1.48706 and Tundra orbits applied such conditions to stop the rotation of argument of perigee. Orlov introduced the concept of the Earth Nomenclature critical inclination [5], and Brouwer eliminated short-period terms based on canonical transformations [6]. Later, Coffey et al. analyzed a = semimajor axis, km the averaged Hamilton system reduced by the Delaunay normal- e = orbital eccentricity ization and provided geometrical interpretation of the critical i = orbital inclination, deg. inclination for Earth’s artificial satellites [7]. Some research has J2 = second-order zonal harmonic regarded orbits at the critical inclination as types of frozen orbits J3 = third-order zonal harmonic [8,9]. J4 = fourth-order zonal harmonic Frozen orbits have always been an important focus of orbit design, J22 = second-degree and -order tesseral harmonic because their average eccentricity and argument of perigee remain M = mean anomaly, deg. constant. Cutting et al. first proposed the idea of frozen orbits in their 1 n = mean angular velocity, s research regarding orbit analysis of the Earth satellite SEASAT-A ’ ns = Mars s mean motion around the sun [10]. Coffey et al. discovered three families of frozen orbits in the p = semiparameter, km averaged zonal problem up to J9 for satellites close to an Earth-like Re = reference radius of Mars, km planet [8]. Aorpimai and Palmer analyzed the conditions of Earth r = position of the spacecraft, km frozen orbits using epicycle elements, and they extended analytic T = orbital period, days analysis to arbitrary zonal terms [9]. Folta and Quinn used numerical = right ascension of the ascending node, deg. simulations combined with a traditional differential correction = latitude of body-fixed coordinate system process to obtain frozen orbits around the moon in the full gravity 22 = longitude of the major axis of the elliptical equator 3 2 model [11]. = Martian gravitational constant, km =s Repeating ground track orbits are defined as orbits with periodi- ’ = longitude of body-fixed coordinate system cally repeating ground tracks, meaning that the trajectory ground ! = argument of perigee, deg. track will repeat after a whole number of revolutions within a whole !m = rotational angular speed of Mars number of days. A repeating ground track is the key factor for the temporal resolution of Earth satellites, and this concept is one of the major design considerations of sun-synchronous orbits [4]. Earth’s I. Introduction satellites are often placed into repeating ground track orbits in order ARS’S similarity to Earth has aroused considerable curiosity to achieve better coverage properties [12,13]. M for a long time. Recently, interest in searching for evidence Stationary orbits around Mars are usually referred to as areo- that water and extraterrestrial life previously existed on Mars has stationary orbits, even though no satellites have been placed in these made the planet a major target of planetary exploration. orbits so far. However, stationary orbits around Earth have been Since the 1990s, several Mars-orbiting space missions have been

Downloaded by Tsinghua University on December 10, 2012 | http://arc.aiaa.org DOI: 10.2514/1.48706 widely used, especially for communications satellites and navigation launched, such as Mars Global Surveyor (MGS) [1] and Mars satellites. Clarke proposed the notion of stationary orbits for Earth, Reconnaissance Orbiter (MRO) [2]; additional Mars exploration and he considered it applicable for communications satellites [14]. missions are currently being planned, such as Mars Atmosphere and Musen and Bailie proved that four equilibrium solutions for Volatile Evolution [3] and Mars Trace Gas Mission. For these geostationary orbits exist: two are stable, and two are unstable [15]. missions, orbit design is a very important issue, and research in this Elipe and López-Moratalla analyzed the stability of the stationary area is vitally important for the success of these missions. points for any celestial body [16]. Typically, satellites orbiting around a celestial body are placed into In the present study, analytical formulations and numerical one of five special types of orbits: sun-synchronous orbits, orbits at simulations are used to analyze these five types of orbits around the critical inclination, frozen orbits, repeating ground track orbits, Mars. The analytical formulations are based on mean element theory, and areostationary orbits. Sun-synchronous orbits are orbits with a for which Kozai [17], Brouwer [18], Lyddane [19], and Iszak [20] precession rate of the orbital plane equal to the planetary revolution contributed research. For the numerical verifications, the PSODE around the sun, so these orbits are especially suitable for remote algorithm, which combines particle swarm optimization (PSO) with differential evolution (DE) [21], is used. The only perturbation Received 29 December 2009; revision received 22 March 2010; accepted considered is the gravitational asphericity effect, because this is the for publication 23 March 2010. Copyright © 2010 by the American Institute most basic factor that encourages or influences these orbits. Other of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper perturbations, including atmospheric forces, solar gravitational may be made for personal or internal use, on condition that the copier pay the force, and solar radiation pressure, are not considered. $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood fi Drive, Danvers, MA 01923; include the code 0731-5090/10 and $10.00 in This study investigates the characteristics of ve special types of correspondence with the CCC. Martian orbits and provides analytical and numerical methods to ∗Ph.D. Candidate, Department of Aerospace Engineering. achieve them. These orbits are also compared with their Earth †Associate Professor, Department of Aerospace Engineering. counterparts. It is expected that the survey results could be useful for ‡Professor, Department of Aerospace Engineering. the orbit design of future Mars exploration missions. 1294 J. GUIDANCE, VOL. 33, NO. 4: ENGINEERING NOTES 1295

9 2 4 5 5 3 p 35 1 3 II. Mars Gravity Model nJ2 Re 2 1 2 J4 2 ; A1 4 e e 2 e The Mars gravity model used in this study is the Goddard Mars 4p 3 24 2 6J2 2 4 Model 2B, which contains spherical harmonic coefficients up to 3 2 3 2 1 3 1 p nJ2Re 1 J2Re 2 1 2 80 80 [22]. Some of the zonal harmonics and tesserals used B1 2 2 e e 3 5 2p 2p 6 8 2 are given as J2 1:95545 10 , J3 3:14498 10 , J4 1 53774 105 6 30692 105 74 7447 5 1 3 : , J22 : , and 22 : .For J4 2 ; 2 e Earth, these zonal harmonics and tesserals are given as 2J2 2 4 3 6 J2 1:08263 10 , J3 2:53266 10 , J4 1:61962 5 6 C1 n (6) 10 , J22 1:81534 10 , and 22 14:9288 [23]. s It is clear that the term of oblateness J2 of Mars is dominant among cos cos the harmonic coefficients, but it is not as dominant as Earth’s J2. The It can be seen that f1 i is the cubic function of i, so Eq. (5) other first few harmonic coefficients are also strong for Mars: about probably has one, two, or three distinct real roots in the interval 1–2 orders of magnitude lower than J2; for Earth, the other first few ; . It is necessary to judge whether all roots of Eq. (5) are harmonic coefficients are about 3–4 orders of magnitude lower than meaningful for different semimajor axes. Based on the theory of the J2. Because of these characteristics, the motion of spacecraft orbiting cubic equation, Eq. (5) only has three real roots if the discriminant fi around Mars and Earth will be quite different. satis es the condition 2 3 C1 B1 0 III. Mean Element Theory 2 3 (7) 4A1 27A1 The motion of a spacecraft under gravitational perturbation can be described using mean element theory, which is a modified Additionally, in order to avoid impact with the Martian surface, perturbation method. It divides the perturbation variables into three eccentricity should yield different terms (secular terms, long-period terms, and short-period 1 terms), and it approximates satellite motion by separating periodic e< Re=a (8) terms from secular terms [17,18]. In this study, analytical formu- lations contain periodic perturbations of the first order and secular Figure 1 shows the values of the discriminant for a 2 3497 103 397 km 0 1 perturbations up to the second order. ; ; and e 2 ; Re=a. Note that the values of the discriminant are always greater than zero. Therefore, it is IV. Sun-Synchronous Orbits concluded that Eq. (5) has one meaningful root at the range of 3497 103 397 km 0 1 fi a 2 ; ; and e 2 ; Re=a; in other words, the Mars has a richly aspherical gravity eld, with swelling at the conditions for three roots will result in an impact with the surface of equator. The equatorial bulge is responsible for the fact that the line of Mars. Thus, it is evident that there usually only exists one sun- node, which is the intersection of the orbital plane of the spacecraft synchronous orbit at a certain semimajor axis with normal eccen- and the Mars equatorial plane, remains in constant rotation. The tricity. Figure 2 shows a wide range of sun-synchronous circular changing trend can be expressed in the form of the nodal precession _ orbits with combinations of altitudes and inclinations based on rate . Eq. (4). It is evident that, given the same altitude, the required According to mean element theory, the mean nodal precession rate inclination of Martian sun-synchronous orbits is always smaller than is composed of two parts: the secular perturbations of the first order Earth’s. and those of the second order. The nodal precession rate arising from For Martian sun-synchronous circular orbits, when a 3897 km, the secular perturbations of the first order [1] is the corresponding i is 93.242, based on Eq. (4). Figure 3a shows the 2 evolution of for a sun-synchronous orbit with this inclination, and 3nJ2R _ e cos Fig. 3b shows that the difference between the variations of the right 1 2 2 1 2 2 i (1) a e ascension of the ascending node and Mars’s rotation angle around the sun is about 1.8 over 1 yr. The nodal precession rate arising from the secular perturbations of By applying PSODE algorithm with inclination as the optimized the second order [18] is variable, the difference between the variations of the right ascension 9 2 3 1 p of the ascending node and Mars’s rotation angle around the sun are _ nJ2 2 2 2 e 1 e 4p4 2 6 reduced further. The optimized i is obtained as 93.216 when

Downloaded by Tsinghua University on December 10, 2012 | http://arc.aiaa.org DOI: 10.2514/1.48706 a 3897 km. Figure 3c shows the evolution of over one year with 5 5 3 p sin2i e2 1 e2 this inclination, and Fig. 3d shows that the difference between the 3 24 2 variations of the right ascension of the ascending node and Mars’s 35 6 9 3 9 rotation angle around the sun is about 0.29 over one year. J4 2 sin2 2 cos 2 e i e i (2) 18J2 7 7 2 4

Therefore, the mean nodal precession rate is _ _ _ 1 2 (3)

For sun-synchronous orbits, the rate of node precession is equal to the Mars mean motion around the sun; that is,

_ ns (4) This is a useful equation for achieving a wide range of sun- synchronous orbits. It can be seen that, if values of a and e are set, the value of i can be determined. Equation (4) can be further rearranged as

3 f1cos iA1cos i B1 cos i C1 0 (5) Fig. 1 The discriminant of Eq. (5) for different values of semimajor axis where and eccentricity. 1296 J. GUIDANCE, VOL. 33, NO. 4: ENGINEERING NOTES

106 5 sin2 2 0 2 i (11) 104 Thus, the critical inclination, in the traditional sense, is 102 i 63:435 or 116:565 For Earth (12) 100 The method previously mentioned is useful for Earth, but it is not 98 accurate enough for Mars because of special properties of the Mars gravity field, where the J2 term does not exceed J3 by even 100 times. Inclination (deg) For Mars 96 By contrast, in Earth’s gravity field, J2 exceeds J3 by more than 1000 times. The changes of e and ! over one year are shown by the phase 94 space diagram (where e is radial and ! is counterclockwise) in Fig. 4 for i 63:435. It can be seen that ! keeps rotating during the year, 92 0 500 1000 1500 2000 the magnitude of which is approximately 30. Altitude (km) To find a better initial condition, the secular perturbations of the Fig. 2 Sun-synchronous altitude vs inclination, circular orbits. second order need to be considered in addition to those of the first order. The average variation rates of e and ! arising from the secular perturbations of the second order are [18] Comparing Figs. 3b and 3d, it is evident that the optimization under fi fi the full gravity eld signi cantly improves the synchronous e_ 2 0 (13) characteristics. The difference in i between the analytical and numerical solutions is 0.026, corresponding to the position 9 2 4 7 p precision of the spacecraft at about 1.8 km. This position precision nJ2Re 2 2 !_ 2 4 e 2 1 e can be achieved at present. p4 12 103 3 11 p sin2 2 1 2 V. Orbits at the Critical Inclination i 12 8 e 2 e The Martian equatorial bulge can lead to rotation of argument of 215 15 15 p sin4 2 1 2 perigee and variation of eccentricity, which will negatively affect i 48 32 e 4 e missions to Mars. These disadvantages can be avoided by selecting orbits at the critical inclination. 35J4 12 27 93 27 e2 sin2i e2 The traditional method for analyzing orbits at the critical 18J2 7 14 14 4 2 inclination only considers the J2 term (i.e., the secular perturbations 21 81 of the first order). Using this method, the average variation rates of sin4i e2 (14) eccentricity and argument of perigee are 4 16

e_ 1 0 (9) The condition of Eq. (13) is also naturally met, so only the average variation rate of ! needs to be considered and is set equal to zero: 3 2 5 _ nJ2R2 2 ! !_ 1 !_ 2 0 (15) !_ 1 sin i 2 (10) 2a21 e22 2 Substituting Eqs. (10) and (14) into Eq. (15), Eq. (15) can be The condition that the drift rate of the mean eccentricity equals zero is rearranged as naturally met, and rotation of argument of perigee can be stopped if sin2 sin4 sin2 0 the inclination satisfies the equation f2 iA2 i B2 i C2 (16)

250 2 Ω 200 Ω +n t 1.5

Downloaded by Tsinghua University on December 10, 2012 | http://arc.aiaa.org DOI: 10.2514/1.48706 0 s 150 1 t (deg) s (deg)

100 -n 0.5 Ω ∆Ω 50 0

0 -0.5 0 100 200 300 400 0 100 200 300 400 Time (day) Time (day) Ω ∆Ω a) Evolution of for i =º 93.242 b) Difference between and ns t for i =º93.242

250 0.3 Ω 200 Ω +n t 0 s 0.2 150 t (deg)

s 0.1 (deg)

100 -n Ω

∆Ω 0 50

0 -0.1 0 100 200 300 400 0 100 200 300 400 Time (day) Time (day) Ω ∆Ω c) Evolution of for i =º 93.216 d) Difference between and ns t for i =º93.216   Fig. 3 Evolution of and the difference between and nst over one year for a ˆ 3897 km and i ˆ 93:242 , 93.216 . J. GUIDANCE, VOL. 33, NO. 4: ENGINEERING NOTES 1297

90 0.15 120 60 0.1 150 30 0.05

180 0

210 330

240 300 270 f Fig. 4 e-! evolution over one year for i ˆ 63:435. Fig. 5 2 1† as a function of the semimajor axis and eccentricity.

where 90 0.15 90 0.15 120 60 9nJ2R4 215 15 15 p 120 60 2 e 2 1 2 0.1 0.1 A2 4 4 48 32 e 4 e p 150 30 150 30 0.05 35 21 81 0.05 J4 2 ; 2 e 18J 4 16 180 0 2 180 0 3 2 3 2 103 3 11 p nJ2Re J2Re 2 2 B2 5 e 1 e 4p2 p2 12 8 2 210 330 210 330 35J4 31 9 2 e ; 240 300 240 300 6J2 14 4 2 270 270 3 2 3 2 7 p nJ2Re J2Re 2 2 a) i = 63.310º b) i = 63.311º C2 4 4 e 2 1 e 4p2 p2 12 Fig. 6 e-! evolution over one year. 5J4 9 4 e2 (17) 6J2 2 2 without any maintenance. The difference in inclination between 2 2 analytical and optimized solutions is only 0.001, corresponding to It can be seen that f2sin i is the quadratic function of sin i,so Eq. (16) may have two or four real roots in the interval ; .Itis the position precision of the spacecraft at approximately 70 m. To fi necessary to clarify whether all roots of Eq. (16) are meaningful for gain the bene t shown, it would be required to navigate to this level different semimajor axes. Setting x sin2i, Eq. (16) can be rewritten of precision, the feasibility of which depends on the navigation as performance of the spacecraft. 2 0 f2xA2x B2x C2 (18) VI. Frozen Orbits So, the condition that Eq. (16) has four real roots in the interval To keep e and ! constant on average, frozen orbits are also ; is equivalent to the condition that Eq. (18) has two distinct advantageous, because a and i can be set independently. For frozen e ! real roots in the interval [0, 1]. It can be easily proved that A2 > 0, orbits, the variations of and can be stopped by choosing a small B2 < 0, and C2 > 0 for Mars. Thus, Eq. (18) only has two distinct real eccentricity and argument of perigee to be around either 90 or 270 . fi roots in the interval [0, 1] if Because of the small eccentricity, long-period terms of the rst fi

Downloaded by Tsinghua University on December 10, 2012 | http://arc.aiaa.org DOI: 10.2514/1.48706 order must be considered. The main parts of the rst-order long- f21A2 B2 C2 > 0 (19) period terms of e and ! [18] are

Additionally, in order to avoid an impact on the Martian surface, 1 J3Re sin 1 2 sin el i e ! (20) eccentricity should satisfy the inequality (8). Figure 5 shows the 2J2p values of f21 for a 23497; 103; 397 km and e 20; 1 R =a. e Note that the values of f21 in Fig. 5 are always less than zero, which 2 2 1 J3R sin i e cos i cos ! ! e means that Eq. (18) has only one root in the interval [0, 1]. It can be l 2 sin (21) concluded that there exist two critical inclinations at the given J2p i e wide range of a and e, and that the conditions for the four critical Thus, the average variation rates of e and ! are given by inclinations will result in a Martian impact. For Earth, the same critical inclinations of 63.435 and 116.565 3 3 sin 5 _ 1 1 nJ3Re i 2 e e_1 e_ e_2 e_ sin i 2 cos ! exist for different semimajor axes and eccentricities, while for Mars, l l 4a31 e22 2 the critical inclinations are a function of the semimajor axes and eccentricities. For example, if a 3897 km and e 0:1, the critical (22) inclinations are 63.310 and 116.690. Figure 6a presents the e-! evolution over one year for i 63:310, and it shows that excursions _ 1 ! !_ 1 !_ !_ 2 in e and ! are reduced. l 3 2 5 sin2 2cos2 sin By employing the PSODE algorithm, with i as the optimized nJ2Re 2 sin2 1 J3Re i e i ! fi 2 i variable in the full gravity eld, an optimized critical inclination is 2p 2 2J2p sin i e 63 311 found (i.e., i : ). Figure 6b presents the e-! evolution over 3 2 J2Re one year for the optimized critical inclination. It can be seen that 2 D (23) excursions in e and ! maintain their local location over one year 2p 1298 J. GUIDANCE, VOL. 33, NO. 4: ENGINEERING NOTES

where 90 0.015 90 0.015 120 60 120 60 7 p 0.01 0.01 4 2 2 1 2 D 12 e e 150 30 150 30 0.005 0.005 103 3 11 p sin2 2 1 2 i 12 8 e 2 e 180 0 180 0 215 15 15 p sin4 2 1 2 i 48 32 e 4 e 210 330 210 330 35J4 12 27 93 27 240 300 240 300 e2 sin2i e2 18J2 7 14 14 4 270 270 2 21 81 a) ω =270º, e = 0.0063414 b) ω = 270.54477º, e = 0.0075389 sin4i e2 (24) 4 16 Fig. 8 e-! evolution over one year for different e and !.

For frozen orbits, the average variation rates of e and ! are set ! 270:54477. Figure 8b presents the evolutions of e and !. It can equal to zero: be seen that the numerical solution is close to the analytical solution, and the variations of e and ! are not minimized significantly after _ 0 e (25) applying the PSODE algorithm. Therefore, analytical formulations can provide good initial conditions for Martian frozen orbits. !_ 0 (26) VII. Repeating Ground Track Orbits Solving Eqs. (25) and (26), where it is approximated that e2 0, the required e and ! are obtained as Repeating ground track orbits have the valuable property that their ground track repeats periodically. These orbits can provide a good 2 sin sin J3Re= J2a i ! appearance for orbital covering, which is significant for remote e 1 3 2 2 5sin2 4 (27) f J2ReE=a i g sensing satellites. For a satellite, the interval of the adjacent ground track in the ! 90 or 270 (28) equator is _ where TN!M (30) 169 395 6 sin2 sin4 where TN is the nodal period of the motion of the spacecraft, which E 12 i 48 i can be expressed as 35J4 12 93 21 2 sin2 sin4 T (31) 18 2 7 14 i 4 i (29) N _ _ J2 M ! Based on mean element theory, the average rates of , M, and ! can From the above analysis, it can be seen that a and i can be set be expressed as independently for frozen orbits, which will provide great _ _ _ convenience for different missions. Figure 7 shows values of frozen M n M1 M2 (32) e for combinations of a and i. Here, the twin peaks correspond to zeros of the denominator of Eq. (27). _ _ _ To make e meaningful, ! must be set to about 90 for Earth, ! !1 !2 (33) because J2 and J3 have the opposite sign; otherwise, e is negative. For Mars, ! must be set to around 270 , because J2 and J3 have the same _ _ _ 1 2 (34) sign. If given a 3897 km and i 60, the corresponding e is Downloaded by Tsinghua University on December 10, 2012 | http://arc.aiaa.org DOI: 10.2514/1.48706 0.0063414 and ! is 270 , based on Eqs. (27) and (28). This result where [18] achieves a good property as an approximate frozen orbit in the full fi 3 2 3 p gravity eld, as shown in Fig. 8a. _ nJ2Re 2 2 M 1 1 sin i 1 e (35) Using the PSODE algorithm with e and ! as optimized variables, 2p2 2 the optimized e and ! are obtained, where e 0:0075389 and 9 2 4 p 1 3 2p _ nJ2 Re 2 2 2 M2 1 e 1 sin i 1 e 4p4 2 2 5 10 19 26 2 sin2 2 2 3 e i 3 3 e 233 103 sin4 2 i 48 12 e e4 35 35 315 sin2i sin4i 1 e2 12 4 32 35 9 45 45 J4 2 sin2 sin4 2 e i i (36) 18J2 14 14 16

To realize global coverage, engineers usually adopt repeating Fig. 7 Frozen eccentricity as a function of the semimajor axis and ground track orbits for which the periods are multisol. The condition inclination. for repeating ground track orbits can be written as J. GUIDANCE, VOL. 33, NO. 4: ENGINEERING NOTES 1299

_ Denoting 22 by , the radial, tangential, and normal motion RT ! R N 2 (37) N M equations of the spacecraft are, respectively, 8 2 2 2 or 2 _ 2 1 3J2Re 2 9J22Re 2 >r rcos ’ r’_ 2 4 3sin ’ 1 4 cos ’cos2 <> r 2r r 6 2 RT ND (38) d 2 2 _ J22Re 2 N N r cos ’ 4 cos ’sin2 >dt r : 2 3 2 6 2 _ d 2 _ 1 2 sin2 _ J2Re sin cos J22Re cos sin cos2 2 d r ’2r ’ 4 ’ ’ 4 ’ ’ where N and R are positive integers, and DN =!M .In t r r this condition, the ground track will repeat after the satellite runs R (41) circles in N sols. In engineering practice, the ground track repetition parameter From Eq. (41), it can be seen that the zonal term produces normal and Q is often used to describe repeating ground track orbits, which is radial perturbations. Normal perturbations are zero when the orbital defined as plane is in accordance with the equatorial plane; radial perturbations increase the gravity of Mars, and thus enlarge the radius of stationary R orbits. The tesseral term mainly produces tangential and radial Q (39) N perturbations. Tangential perturbations can lead to longitudinal drift; radial perturbations caused by the tesseral term are much smaller than Figure 9 shows the relation between altitude and inclination for those caused by the zonal term, which can always be neglected. repeating ground track circular orbits with different Q values. For Four special solutions of Eq. (41) can be obtained as practical planetary exploration, repeating ground track orbits are often sun-synchronous, in which case Eq. (4) must be satisfied. Thus, r r01; 01 90 ; 270 ; ’ 0 (42) if the values of R and N are fixed, a and i can be determined. For example, for a circular five-sol repeating ground track orbit with 41 r r02; 02 0 ; 180 ; ’ 0 (43) revolutions in one period, the corresponding H and i can be 1627 395 km 97 809 calculated: H : and i : . It is found that where r01 and r02 are roots of the first expression of Eq. (41), which fi the ground track was offset by 3.328 after ve sols in the full can be rewritten as fi gravity eld. 3 2 By using the PSODE algorithm with H and i as optimized 2 Re J2 1634 019 km r0! 3J22 (44) variables, the optimized H and i can be obtained: H : m r2 r4 2 and i 97:845, and the offset of the ground track is reduced to 0 0 0.227 after five sols. Between analytical and numerical solutions, where the signs and before J22 correspond to r01 and r02, the difference in inclination is 0.036 , corresponding to the position respectively. precision of the spacecraft at about 2.5 km. This position precision The two roots r01 and r02 denote that the spacecraft is located can be achieved at present. above the minor and major axes of the elliptical equator, respectively. Substituting the Martian constants into Eq. (44), the equilibrium 20 428 095533 km VIII. Areostationary Orbits solutions are obtained, where r01 ; : and r02 20; 428:309266 km. The ground track of stationary orbits remains stationary, which The longitude of r01 is 15:255 or 164.745 , and the longitude of arouses great interest to operators of communications and weather r02 is 74.745 or 254.745 (for Mars, the prime meridian is defined as satellites. Extensive research has been conducted on geostationary the longitude of the crater Airy-0). It can be easily proved that orbits. However, the Mars gravity field is different from Earth’s, Eqs. (42) and (43) correspond to central points and saddle points, which will produce distinctions in their stationary orbits. respectively, of which the former is linearly stable. For geostationary For stationary orbits, tesseral harmonic terms (except for zonal orbits, the longitude of r01 is 75.071 or 255.071 , and the longitude of harmonics) must be taken into account. Taking the second-order and r02 is 14:929 or 165.071 . Let -degree harmonics, the gravity potential function of Mars can be simplified as r r0 r; 0 ; ’ ’0 ’; 0 0 22 2 3 2 (45) 1 J2Re 3sin2 1 J22Re cos2 cos 2 U 2 2 ’ 2 ’ 22 r r r where 0 and ’0 are the initial longitude and latitude, and 0 is the Downloaded by Tsinghua University on December 10, 2012 | http://arc.aiaa.org DOI: 10.2514/1.48706 (40) initial angular distance from the spacecraft to the equatorial major axis. Substituting Eq. (45) into Eq. (41), the disturbance equations are derived after linearization: 200 2 6 2 36 2 2 J2Re J22Re 2 _ r !m 3 5 5 r r0!m r0 r0 r0 18 2 J22Re 150 sin 20 Q=14 Q=13.5 4 Q=13 Q=12.5 Q=12 r0 3 2 9 2 2 J2Re J22Re cos 2 ; 2 r0!m 4 4 0 100 r0 2r0 r0 12 2 24 2 J22Re cos 2 2 _ J22Re sin 2 r0 4 0 !m r 5 r 0 Inclination (deg) Inclination r0 r0 50 6 2 J22Re sin 2 ; 4 0 r0 3 2 6 2 2 J2Re J22Re cos 2 0 0 r0 ’ r0!m ’ 4 ’ 4 ’ 0 (46) 0 100 200 300 400 500 600 r0 r0 Altitude (km) Fig. 9 Altitude vs inclination for repeating ground track circular orbits It is evident from Eq. (46) that the motion in the orbital plane and with different Q. the motion of the orbital plane are decoupled. Introducing new 1300 J. GUIDANCE, VOL. 33, NO. 4: ENGINEERING NOTES

20 40

0 20

-20 0

Longitude (deg) -40 -20 Longitudinal drift (deg) -60 -40 0 200 400 600 800 0 200 400 600 800 Time (day) Time (day) a) The point with the longitude −15.255º b) The point with the longitude −15.255º

x 10-3 -16 5

-17

-18 0

-19 Longitude (deg) Longitudinal drift (deg) -20 -5 0 200 400 600 800 0 200 400 600 800 Time (day) Time (day) c) The point with the longitude −18.138º d) The point with the longitude −18.138º Fig. 10 Longitudinal evolution and drift over two years for the point located above the equatorial minor axis ( ˆ15:255) and the optimized longitude ( ˆ18:138).

2 2 variables, !mt, J22Re=r0 and r1 r=r0, the first solution can be obtained, which is near to the point located above the two expressions of Eq. (46) can be simplified as equatorial minor axis with the longitude 18:138. Figure 10c presents two years of simulation, and it shows that this point is the 00 3 2 0 18 2 sin 2 9 2 cos 2 ; r1 r1 0 0 exact equilibrium solution in the full gravity field. Figure 10d shows 00 2 0 2 that, after optimization, the amplitude of the longitude can be 12 cos 20 2r 24 r1 sin 20 1 reduced significantly, to about 0.00435. Thus, the point with the 2 6 sin 20 (47) longitude 18:138 can be used as the initial condition for areo- stationary orbits, where stationkeeping is not necessary. where 0 represents the derivative with respect to . The characteristic Eq. (47) can be written as IX. Conclusions 2 2 2 2 2 s 12 sin 20s 1s 12 sin 20s 36 cos 200 There exist five special types of orbits around Mars: sun- (48) synchronous orbits, orbits at the critical inclination, frozen orbits, repeating ground track orbits, and areostationary orbits. The By approximating that 4 0, the characteristic roots of Eq. (48) can constraint conditions for achieving these special orbits have been be calculated as found based on analytical formulations and numerical simulations in 80 80 fi 2 an Mars gravity eld. It has been shown that the behaviors of s1 2 6 sin 20 i ; p (49) these orbits are different from those of their Earth counterparts. 6 2 sin 2 6 cos 2 s3;4 0 0 Martian sun-synchronous orbits may have one, two, or three distinct

Downloaded by Tsinghua University on December 10, 2012 | http://arc.aiaa.org DOI: 10.2514/1.48706 inclinations when given semimajor axis and eccentricity. However, When the spacecraft is located above the major axis of the equator, only one meaningful inclination exists; the other inclinations might cos 20 1, where the characteristic equation contains a positive be expected to result in an impact with the Martian surface. The real root, so two balance positions above the major axis of the equator required inclination of Martian sun-synchronous orbits is always are unstable. When the spacecraft is located above the minor axis of smaller than that of Earth’s with the same altitude. For Mars, critical the equator, cos 20 1, where the roots of the characteristic inclinations are functions of the semimajor axis and eccentricity. For equation are all pure imaginary numbers, so two balance positions a given semimajor axis and eccentricity, there may exist either two or above the minor axis of the equator are linearly stable. From the four critical inclinations. However, only the two critical inclinations preceding analysis, the deviation of the spacecraft from the minor are meaningful; the four critical inclinations might result in an impact axis is periodic. Based on the theory of ordinary differential with the Martian surface. Martian frozen orbits always have small equations, the characteristic roots s1;2 determine the short period, eccentricity, and argument of perigee must be set to approximately which is one day; the characteristic roots s3;4 determine the long 270 , while for Earth, argument of perigee must be set to about 90 . period, which is 126.204 days. For geostationary orbits, the short Repeating ground track orbits around Mars can also be realized and, period is also one day, but the long period is 819.230 days, which is in engineering practice, they are often sun-synchronous. For much longer than that of areostationary orbits. satellites orbiting around Mars, there exist four stationary solutions: Taking the initial condition r r01, 01 90 , and ’ 0 [i.e., the two stable and two unstable. Stable solutions are located at the minor point above the minor axis of the equator (longitude: 15:255)], the axis of the elliptical equator, making areostationary orbits possible. longitudinal evolution over two years can be seen in Fig. 10a. The long oscillation period of areostationary orbits is about 126 days, Figure 10b shows that the point located above the equatorial minor which is much shorter than that of geostationary orbits. axis is not the exact equilibrium solution in the full gravity field, and The analysis also shows that the restrictive definition of these the drift amplitude is approximately 35. orbits cannot be precisely satisfied in the 80 80 Mars gravity field, Using the PSODE algorithm with the initial longitude of the but it is possible to select these special orbits for Mars that can reduce spacecraft as the optimized variable, the more accurate equilibrium or eliminate the need for stationkeeping. These orbits are of special J. GUIDANCE, VOL. 33, NO. 4: ENGINEERING NOTES 1301

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