216 J. Johnson et al. In order to capture the irregularities of the gravity model in three dimensions, a spherical harmonic model of the gravitational potential is used, as shown in Eq. 22.1. U D r 1 XnD0 n XmD0 RCB r n Pnmsin cos.m / CnmC RCB r n Pnmsin cos.m / Snm (22.1) where U is gravitational potential. The coordinates r, , and are the radial distance, latitude, and longitude of the spacecraft in a coordinate system fixed to the object’s center of mass. RCB is the mean radius for the body and is the object’s gravitational parameter. The functions Pnm are the normalized Legendre polynomials, and Cnm and Snm are the gravity coefficients of degree n and order m[7]. Itokawa has been studied extensively and a degree and order four model was built based on the data captured by Hayabusa [8]. Studying the natural environment about Itokawa, with no control measures used, it was found that a stable prograde orbit can be achieved using a high-inclination orbit with an orbit radius between 1.0 and 1.5 km [9]. Any closer to the surface of Itokawa and the orbit becomes disrupted by the spherical harmonic gravity model. Outside of this range the orbit becomes disrupted by other effects, such as solar radiation pressure (SRP). For Itokawa’s gravity model, the C20, C22, C42, andC44 coefficients are the most significant [9]. These values correspond with the effects of the asteroid’s oblateness and its ellipcity (Fig. 22.6). A body’s oblateness and its effect on orbits has been characterized and studied extensively. For example, this perturbation is observed in Earth orbits. Due to the Earth’s angular velocity as it spins about its polar axis, there is a bulge around equator. The effect this bulge has on orbits (known as the J2 effect) causes a precession in the orbital plane, not unlike a spinning top that is about to fall; the orbital plane wobbles as it spins. An inclined orbit about an oblate body has the same wobble as the orbital plane twists around the body. Specifically, the orbit’s right ascension of the ascending node (RAAN) ( ) rotates westward for prograde orbits around the Earth, and the argument of periapsis (¨) rotates in the direction of the spacecraft’s motion. Semi-major axis (a), eccentricity (e), and inclination (i) suffer no long-term perturbations from oblateness (Fig. 22.7) [10]. A body’s ellipcity has more dramatic effects on the orbit and can cause the spacecraft to transition from a safe orbit into an impacting or ejecting orbit within a few periods. The ellipcity of the body causes changes in the orbit semi-major axis, eccentricity, and inclination while effecting both the orbit’s energy " D =.2a/ and angular momentumhD[ a(1-e)]1/2 [11]. In previous studies, it has been observed that prograde orbits experience much larger changes in energy and angular momentum for each orbit, where retrograde orbits experience little, if any, changes per orbit [12]. This paper studies the difference in stability between prograde and retrograde orbits around a small asteroid with various rotation rates by looking at the excitation frequencies seen in the dynamic system. Fig. 22.6 Effects of degree (n) and order (m) on the spherical harmonic gravity model [8]
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