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Long-Term Perturbations Due to a Disturbing Body in Elliptic Inclined Orbit

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Long-Term Perturbations Due to a Disturbing Body in Elliptic Inclined Orbit Long-term perturbations due to a disturbing body in elliptic inclined orbit Xiaodong Liu1, Hexi Baoyin2, and Xingrui Ma3 School of Aerospace, Tsinghua University, 100084 Beijing, China Email: [email protected]; [email protected]; [email protected] Abstract In the current study, a double-averaged analytical model including the action of the perturbing body’s inclination is developed to study third-body perturbations. The disturbing function is expanded in the form of Legendre polynomials truncated up to the second-order term, and then is averaged over the periods of the spacecraft and the perturbing body. The efficiency of the double-averaged algorithm is verified with the full elliptic restricted three-body model. Comparisons with the previous study for a lunar satellite perturbed by Earth are presented to measure the effect of the perturbing body’s inclination, and illustrate that the lunar obliquity with the value 6.68º is important for the mean motion of a lunar satellite. The application to the Mars-Sun system is shown to prove the validity of the double-averaged model. It can be seen that the algorithm is effective to predict the long-term 1 PhD candidate, School of Aerospace, Tsinghua University 2 Associate Professor, School of Aerospace, Tsinghua University 3 Professor, School of Aerospace, Tsinghua University behavior of a high-altitude Martian spacecraft perturbed by Sun. The double-averaged model presented in this paper is also applicable to other celestial systems. Keywords: Third-body perturbations; Restricted problem of three bodies; Obliquity; Averaged model; Earth-Moon system; Mars-Sun system 1. Introduction The problem of the third-body perturbation has been studied for many years. Plenty of papers contributed research, the details of which were well reviewed in (Broucke 2003; Prado 2003). In the past century, the method of averaging technique to deal with the third-body perturbation already aroused great attentions (Musen et al. 1961; Cook 1962; Lorell 1965; Harrington 1969; Williams and Benson 1971; Lidov and Ziglin 1974; Ash 1976; Collins and Cefola 1979; Hough 1981; Šidlichovský 1983; Kwok 1985; Lane 1989; Kinoshita and Nakai 1991; Kwok 1991; Ferrer and Osacar 1994; Delhaise and Morbidelli 1997). Recently, the subject of averaged third-body perturbation is still a popular topic. A single averaged model over the short period of the satellite was used to study the effect of lunisolar perturbations on high-altitude Earth satellites (Solórzano and Prado 2007). The perturbing potential was doubly averaged first over one spacecraft orbit and second over one orbit of the third body (Scheeres 2001). The lunisolar effect that is double-averaged based on Lie transform was analyzed to deal with the resonance on a satellite’s motion of the oblate Earth (Radwan 2002). With the disturbing function expanded in Legendre polynomials at the second-order term, the double-averaged method was applied to analyze the long-term effect of a third body, and two important first integrals (energy and angular momentum) were used to discuss and classify properties of the perturbed orbits (Broucke 2003). An analytical double-averaged model with the expansion of Legendre polynomials up to the fourth-order term was developed, and a numerical study in the full elliptic restricted three-body model was made (Prado 2003). In the above papers, it was assumed that the perturbing body is in a circular orbit around the main body. Further, the analytical expansion to study the third-body perturbation was extended to the case where the perturbing body is in an elliptical orbit (Domingos et al. 2008). The averaged models found wide applications in celestial mechanics, including analyzing long-term perturbations on lunar satellites (Folta and Quinn 2006; Carvalho et al. 2009a; Winter et al. 2009; Carvalho et al. 2010b), design of high altitude lunar constellations (Ely 2005; Ely and Lieb 2006), missions to Europa (Paskowitz and Scheeres 2005; Paskowitz and Scheeres 2006a; Paskowitz and Scheeres 2006b; Carvalho et al. 2010a), and missions to Ganymede and several other planetary moons (Russell and Brinckerhoff 2009). In most above-mentioned papers, the xy-plane was defined as the orbital plane of the perturbing body instead of the equatorial plane of the main body, in which way the mean motion could be expressed in a simple form when only the third-body perturbation is concerned. However, when the gravitational non-sphericity effect is present or other perturbations are considered, the averaged models for third-body perturbations used in the above-mentioned papers may need redundant spherical trigonometric manipulations (Kinoshita and Nakai 1991; Yokoyama 1999) in order to be added into the spacecraft’s equations of motion. To solve the problem, some papers (Carvalho et al. 2008; Carvalho et al. 2009a; Carvalho et al. 2009b; Winter et al. 2009; Carvalho 2010b; Carvalho 2011; Lara 2010; Lara 2011; Lara et al. 2009) neglected the inclination of the main body’s equatorial plane with respect to its orbital plane (also known as obliquity or axial tilt); Other papers used the so-called Earth Orbit Frame to express the effect of the perturbing body in Earth-Moon system conveniently (Folta and Quinn 2006; Ely 2005; Ely and Lieb 2006). However, in solar system, there exist some celestial bodies with non-ignorable obliquities. Table 1 lists the values of obliquities for different celestial bodies: Moon, Mars, Earth, Saturn, Uranus, Neptune, Pluto, Iapetus, Phoebe, and Nereid. The values of obliquities for these celestial bodies could not be considered as zero. The previous literature pointed out that the obliquity is probably the key to provide some important variations in eccentricities, and determines the extension of the chaotic zone (Yokoyama 1999). Table 1 Obliquities for different celestial bodies Celestial Bodies Moon Mars Earth Saturn Uranus Neptune Pluto Obliquity (deg) 6.68 25.19 23.44 26.73 97.77 28.32 122.53 Table 1—Continued Celestial Bodies Iapetus Phoebe Nereid Obliquity (deg) 15.215 26.723 30.011 This study extends the investigations of the double-averaged analytical model done by (Broucke 2003; Prado 2003; Domingos et al. 2008), and includes the action of the perturbing body’s inclination, where the xy-plane is defined as the main body’s equatorial plane. This paper is organized as follows. In Section 2, the dynamical model of the system is established. A double-averaged algorithm with respect to the period of the spacecraft and the period of the perturbing body is developed. In Section 3, this algorithm is applied to the Earth-Moon system and the Mars-Sun system. The full elliptic restricted three-body problem is also considered to verify the efficiency of the double-averaged algorithm. In Subsection 3.1, comparisons with the previous study for a lunar spacecraft are given in order to show the effect of the perturbing body’s inclination. In Subsection 3.2, the application of the double-averaged algorithm to the Mars-Sun system is demonstrated and proved effective. The algorithm presented in this paper is also applicable to other celestial systems. Using this double-averaged model, the action of the third-body perturbation can be conveniently added to spacecraft’s equations of motion when multiple perturbations are present. 2. Dynamical model The system considered in this paper consists of three bodies: a main body with mass m0, a massless spacecraft m, and a perturbing body (also known as a third body) with mass m′. All three bodies are assumed to be point masses. The reference frame Oxyz is established with the origin O located at the center of the main body, the xy-plane coinciding with the equatorial plane of the main body, the x-axis along the intersection line between the equatorial plane of the main body and the orbital plane of the third body, and the z-axis along the north pole of the main body. It is assumed that the perturbing body is in an elliptic inclined three-dimensional Keplerian orbit around the main body with semimajor axis a′, eccentricity e′, inclination i′, argument of pericentre ω′, right ascension of the ascending node Ω′, and mean motion n′ (given 23 by the expression na Gm 0 m , where G is the gravitational constant). It is obvious that Ω′=0 due to the definition of the reference frame. The spacecraft is in a three-dimensional Keplerian orbit with semimajor axis a, eccentricity e, inclination i, argument of pericentre ω, right ascension of the ascending node Ω, and mean motion 23 n (given by the expression na Gm0 ) around the main body, and is perturbed by the third body. The illustration of the system is presented in Fig. 1. Note that although the central body is assumed to be the point mass in the simplified model, when the model is applied to actual celestial systems, the equatorial plane of the central body and the orbital plane of the third body is different, and the inclination of the third body with respect to the equatorial plane of the central body takes effect. z maei ,,, , , M r maei ,,, , , M r m0 y Equatorial plane x Fig. 1 Illustration of the system. Based on the theory of celestial mechanics (Murray and Dermott 1999), the disturbing function R due to the action of the perturbing body can be obtained as 1 r RGm r , (1) 3 rr r where r and r′ are the orbital radius vectors of the spacecraft and the perturbing body, respectively. Further, the disturbing function R can be expanded in the form of Legendre polynomials n Gm 0 m r RP n cos , (2) rrn2 where φ is the angle between the radius vectors r and r′, Pn are the Legendre polynomials, and mmm/ 0 . Note that the P0 cos term is omitted because it is independent on orbital elements of the spacecraft, and the P1 cos term is eliminated after simple algebraic operations.
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