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# Orbital Mechanics: 8. Introduction To Orbit Perturbations

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• 1. 8 Introduction to Orbit PerturbationsThis chapter provides an introduction to, and an overview of, the orbit perturba-tions, the perturbing sources, and the physical phenomena associated with orbitalmotion. Simplified examples and key equations should give readers an easy-to-access reference for understanding and solving typical perturbation problems. Chapter 9 provides an in-depth description of the mathematical foundationsand derivations of the perturbing functions and equations. Chapter 10 gives moredetailed descriptions and discussions on various perturbing sources and theirspecial effects on orbital motion. The chapter also includes actual examples ofmission analysis and design utilizing the unique orbit-perturbation properties.8.1 A General Overview of Orbit Perturbations What are orbit perturbations? By definition, those small deviations from thetwo-body orbit motion are called orbit perturbations. The two-body orbit motioncan be expressed by the conic solutions (ellipse, hyperbola, and parabola) in closedform, which has been explained in detail in Chapters 3 and 4. The equations of two-body motion and its solutions were derived through Newtons law of gravitationand Keplers laws of orbit motion under the assumption of point mass or masswith spherically symmetrical distribution. The equations of motion of a two-body problem can be given in the relativeform as d2r /z -- (8.1) dt 2 r 3rwhere r is the position vector of the satellite measured from the center of theprimary body, and/~ is the gravitational constant of kZ(ml q- m2). Because of the presence of various perturbing forces, Eq. (8.1) can be usedonly as an approximation of the actual motion. The accuracy of the approximationdecreases as the time of propagation increases. Those perturbing forces includeEarth gravity harmonics (deviations from a perfect sphere), the lunisolar gravi-tational attractions, atmospheric drag, solar radiation pressure, and Earth tides.For natural satellites with sizable mass such as the inner satellites of Jupiter andSaturn, the mutual gravitational attraction among the satellites is another sourceof perturbation. The general form of equations of motion, including perturbations,can be expressed as follows: d2r L~ -- ~~r d- ap (8.2)where ap is the sum of all the perturbing accelerations. In the solar system, themagnitude of all the perturbing accelerations is at least one order of magnitudeless than the two-body acceleration. That is why the term "perturbations" is used. 185
• 2. 186 ORBITAL MECHANICSWith the perturbing accelerations included, the solutions of Eq. (2) can no longerbe expressed by the closed-form conic equations. As an astronomer put it, "Inproblems of celestial mechanics, the simple solutions are not good and the goodsolutions are not simple." It will be shown in the next chapter the reasons why thegood solutions, including the perturbations, are not so simple.8.2 Earth Gravity Harmonics Earth gravity harmonics are derived from the gravity potential through thepotential theory, which will be explained in detail in the next chapter. Thoseharmonics are the terms of a mathematical expansion through which the deviationsfrom a sphere can be represented. The commonly encountered gravity harmonicsare J2 and J22, which are the largest terms of the zonal and tesseral harmonics,respectively. The coefficient of the second harmonic J2 is related to Earth equatorialoblateness through Earth rotation, and the estimated difference between the polarradius and equatorial radius is about 22 km. J2 is responsible for the secular ratesof the right ascension of ascending node, the argument of perigee, and a smallcorrection to the mean motion of the orbit. These rates can be computed by thefollowing equations~: (2 3 J2R2 -- - - h cos i (8.3) 2 p2 3 J2R2 [ 7 n ,2- 5 sin2 i ) _ (8.4) h = # 1 + ~--~-~1- ~sin2i ( 1 - e 2 ) -~ (8.5)where (2 = rate of ascending node & = argument of perigee rate h = orbit mean motion with J2 correction J2 = 0.00108263 R = Earth equatorial radius i = orbit inclination # = gravitational constant a0 = semimajor axis at epoch e = eccentricity p = a0(1 - e 2) The second tesseral harmonic ./22 is related to the ellipticity of the Earth equa-torial plane and is responsible for the long-term (860-day) resonance effects ongeosynchronous orbits. The long-term longitude oscillation of the communica-tion satellites, which must be controlled by periodic stationkeeping maneuvers,is caused by the J22 effects. The magnitude and frequency of stationkeepingmaneuvers depend on satellite longitude and control tolerance. Figure 8.1, from
• 3. INTRODUCTIONTO ORBIT PERTURBATIONS 187 2.0 1.5 AII I lHarmonics 1.0 0.5 0 / s1 ~ s2 U 0.5 v 1.0 1.5 2.0 I I I ! I 1 I ! 30 60 90 120 150 180 210 240 270 300 330 East Longitude, dee Fig. 8.1 Annual AV expenditure for triaxiality correction.Ref. 2, shows the annual A V expenditure for longitude stationkeeping of geosyn-chronous satellites. The maximum values occur 45 deg from the four equilibriumpoints (sl, s2, ul, u2); sl and s2 are stable points, and Ul and u2 are unstablepoints. The typical longitude control tolerance is +0.1 deg, and the frequency ofmaneuver is about once every 14 days, depending on the longitude.8.3 Lunisolar Gravitational Attractions To understand the long-term behavior of a satellite orbit under the influenceof the sun, imagine both the satellite and the sun smeared out into ellipticalrings coinciding with their respective orbits (Fig. 8.2). The mutual gravitational Ecliptic Pole h~ ~ " . . . . "~" Gyro Precession i / solar Ring Fig. 8.2 Gyro precession of a satellite orbit.