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Five Special Types of Orbits Around Mars 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.
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