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Orbital Mechanics: 8. Introduction To Orbit Perturbations
Orbital Mechanics: 8. Introduction To Orbit Perturbations
Orbital Mechanics: 8. Introduction To Orbit Perturbations
Orbital Mechanics: 8. Introduction To Orbit Perturbations
Orbital Mechanics: 8. Introduction To Orbit Perturbations
Orbital Mechanics: 8. Introduction To Orbit Perturbations
Orbital Mechanics: 8. Introduction To Orbit Perturbations
Orbital Mechanics: 8. Introduction To Orbit Perturbations
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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.
  • 4. 188 ORBITAL MECHANICSattractions of the rings will create a torque about the line of nodes tending toturn the satellite ring into the ecliptic. The gyroscopic effect of the torque on thespinning satellite ring will induce a gyro precession of the orbit about the poleof the ecliptic, specifically a regression of the nodes along the ecliptic. Similarly,the moon will cause a regression of the orbit about an axis normal to the moonsorbit plane, which has a 5-deg inclination with respect to the ecliptic plane with anode rate of one rotation in 18.6 yr. The equations of nodal regression and rate ofargument of perigee are (from Ref. 3) 3 n 2 (1 ÷ (3/2)e 2) = cos i (3 cos 2 i3 - 1) (8.6) 8 n ~/1 - e 2 (8.7) dg= 4 n ~ e 2where n3 and i3 are the mean motion and inclination with respect to Earth equato-rial plane, respectively. For low-altitude orbits, the coefficient (n2/n) is very smallcompared to J2 effects, and sun-moon effects can be neglected. For orbits withperiods equal to 12 h or longer, the lunisolar effects are significant and should beincluded.8.4 Radiation Pressure Effects The effect of solar radiation on particles moving through interplanetary spacehas been investigated for many years. The first radiation pressure effect studiedwas related to very small meteorites or dust particles. It was first developed byPoynting in 1920 and was refined in keeping with the principles of relativity byRobertson in 1937. The net effect of this so-called Poynting-Robertson forceis the influence of the impinging photon momentum or radiation pressure on aspace vehicle. At one A.U., the solar radiation pressure constant P0 is 4.7 × 10 -5dyne/cm 2. This value may fluctuate slightly (< 1%) as a result of variations in thesolar activity index. The typical radiation pressure effect on satellite orbits is the long-term sinusoidal(yearly for geosynchronous orbits) variations in eccentricity. The magnitude ofthe variation is proportional to the effective area, surface reflectivity, and inverseof the satellite mass. For a typical communication satellite at geosynchronousaltitude, the eccentricity may vary from 0.001 to 0.004 in six months as a result ofsolar radiation pressure effects. For low-altitude orbits, the period of the long-termvariation in eccentricity is governed by the combined rates of the longitude of themean sun, nodal regress, and argument of perigee. Resonance conditions mayoccur when one of the combined rates is vanishingly small. In summary, radiation pressure induces periodic variations in all orbital ele-ments, even exceeding the effects of atmospheric drag at altitudes above 900 kin.The induced changes in perigee height can have significant effects on the satelliteslifetime. For example, the 30-m ECHO balloon satellite in its 1852-kin-altitudecircular orbit displayed the greatest perturbation due to radiation pressure, approx-imately 3.5-km/day initial decrease in perigee height.
  • 5. INTRODUCTION TO ORBIT PERTURBATIONS 1898.5 Atmospheric Drag When the orbit perigee height is below 1000 kin, the atmospheric drag effectbecomes increasingly important. Drag, unlike other perturbation forces, is a non-conservative force and will continuously take energy away from the orbit. Thus,the orbit semimajor axis and the period are gradually decreasing because of theeffect of drag. The orbit velocity is increasing, however, because Keplers law(Ix = n2a 3 = const) must be satisfied. This relationship leads to the "drag para-dox": the effect of atmospheric friction is to speed up the motion of the satelliteas it spirals inward. Since the drag is greatest at perigee, where the velocity and atmospheric densityare greatest, the energy drain is also greatest at this point. Under this dominantnegative impulse at perigee, the orbit will become more circular in each revolution.Figure 8.3 shows the time history of the perigee and apogee altitude of a decayingsatellite. The elliptic orbit first becomes circular as the apogee altitude decreasesto the same value of perigee and then rapidly spirals into the dense atmosphere. For near-circular orbits, the orbit semimajor axis decay rate can be computedby the following simple equation: da - na2pB (8.8) dtwhere n = mean motion p = atmosphere density at that altitude B = ballistic coefficient = C a A / m PERIGEE APOGEE DEOBY HISTORY [LIFETIMEI I i i - , i i 600, . . . . . . . . . . . . . i . . . . . . . . . . . . ............. i .............. i ........... i . . . . . . . . i . . . . . . . . i ........... i ........ i . . . . . . S ~ . . .~POGEE TUDE .... kLT i i i :, I i i i i i i 2 450. ............................ I .............. i ...................................... ~ ...................... : ........... i ......... . I i i i i i J i i ! 8 300. l& o 18 o TIME IN DAYS FROH EPOCH Fig. 8.3 Perigee and apogee decay due to atmospheric drag effects.
  • 6. 190 O R B I T A L M E C H A N I C S 2,g ,, • 2o :::::::::::::::::::::::::::::::::: ..... : ................ : ............... r ............... .......................................................................... [ ................ ~ :1 i--~-.-...--~ ~ " ~ " , i .............. ............... T ............... ~ .............. : ................ i ~................ : ............... ...i ................ i ................ i ............... i ............... ............... ~.......................... ~ ............... , .................................................... i.......z.....~ i t i""1 i~:~ ............... ............... . , "f"""~ ~ ............................... i ........................................... i .......................... t ........... i.-....-~ ...................................... 1 .... ~ ................ i........................................ . . . . . . . ~ /. ............................................................. ~.~ ...................... .................................... ~ ~ i -l. ~ ~ ~ ~ • ,ae ............................................. :""l ............................ "~ ......................................................... tl ................ : ............. ~~ ............................... ,,, ........... i "...................4 . ........................ i : ............... :.....-.~........: ................ : ............... ~ ............ I.i ................ ~ .............. ~- ............... : .............. - i / i i i i i /i i ~ i ~e ........ :" .................................. .J 1...........! ............... .~ .......... ~...! ................ ! ............... i ......... J ..................................... "4 ............. i ............... z,ll ...... ~.................................. : I i I • . ......................................... ~-~ ........................................ I ............................................. ~.......... i ............... iz8 ............................................................................ ~:< ................................. I ................................................ ~-K:: .............. ............... i ............... i ................ i ---~--i ............... .............. ............................................... ............... . . . . . . . . . . . . ,, :~-:i ......... " .............. i ............................... " ............. ~................ i ............... i ............... i " ............... ~i .............. "~,~ ................... r ~ ............... -i ................ ~"~,- .......... " ............... ............... " : ................ : ............... i ................ i""~: ....... - .............. ! ............... " ............... i ................ , ix .~-::::~, . it ........................................... . . i J ................ i.........~,~ ~o ..... ~................ : ............... , .............. ~................ ~ ............... r ............... ~ ................ i ............... .." ............... i ................ ~ ............... ..- ............... ! ............ ,o ..... ! ................ ~............... - ............... i ................ i ............... f ............... i ................ i ............... - ............... ! ................ i ............... f ............... ~.............. ~o ..... i ................ i ............... f ............... i ................ ~ ............... f ............... ! ................ i ............... " ............... i ................ i ............... .~ ............... ! .............. ,, ..... ~................ ~ - ~ ..... ! ................ i .............. " ....................................................................................... - .............................................. ":: ............... i .............. ...................... ............... ; ................................ ...................................... .................... °r i i ] . i . i . i . i . i . i . i . i . i . i . - i~02 198" t~eS t~e° lP~e t~;2 1904 199~ ~R 28e° 2lie2 ~ee, 2ee( 2eeeFig. 8.4 Observed and predicted solar flux index (F10.7) and index of magneticactivity (Ap).and m = total mass of spacecraft, kg A = effective cross-section area, m 2 Cd = drag coefficient = 2.0 An accurate prediction of the satellite motion under the influence of drag re-quires a good density model of the upper atmosphere. Over the past three decades,various density models have been developed, with varying degrees of complexityand fidelity. The commonly used models are the Jacchia 64 and 71 models. Thesetwo models take into account the diurnal variation, the 27-day fluctuation (sunsrotation period), the annual variation, and the 11 -year solar cycle. Figure 8.4 showsthe predicted 11 -year solar cycle variations in F107 (solar flux index) and Ap (indexof magnetic activity) for the period between 1990 and 2008. Based on the Jacchia1971 model, the estimated orbit lifetimes at various initial orbit altitudes and F]0.7values are plotted in Fig. 8.5. Figure 8.6 shows the estimated stationkeeping AVper year for maintaining the orbit altitude in an active atmosphere. Both Figs. 8.5and 8.6 are for orbits with small eccentricity.8.6 Tidal Friction Effects and Mutual Gravitational Attraction The magnitude of tidal friction effects on the artificial satellites is very small,and it usually is not included in the perturbation equations. However, tidal fric-tion plays an important role in the evolution of the massive satellites of Jupiter,Saturn, and other outer planets. The coupling effects between tidal friction andmutual gravitational pulls among the massive satellites of the outer planet areresponsible for the existence of some of the resonance phenomena discovered
  • 7. INTRODUCTION TO ORBIT PERTURBATIONS 191 LIFETIMES FOR CIRCULAR ORBITS (Normalized to CdA/W = 0.2044m2/kg) 1000 I I I I Quiet atmosphere, F10.7=75t 03 rr < I F10,7=100 / ~ UJ >- 100 z ~F107=150 L ~- ~ .11~ .ffOJ --L " I ">~ ~ - -~ ! . ~ ///r-UJLL 10,_J£3UJa._J<n- I " " J J " "" i " t A ctive F107=250Oz --/ I i I I Lifetime = Normalized lifetime x (0.2044/CDA/w) 0,1 i ~ i i J i 600 650 700 750 800 850 900 950 1000 ALTITUDEIN KM LIFETIMES FOR CIRCULAR ORBITS (Normalized to W/CdA = 1 Ib/ft**2) 10000 I i ! r i ~uiet atmosphere, F10.7=75~__< 1000 I F10"7=100 ""4tmz I I "~J / ,./" //UJ 100F-LULI. FlO 7 = 1 5 0 L i--~-ff --" -._1tmLU 10N ""//" "/"/~ "- Ii F10.7 =200<n- /,///, ~.~ i I iO .~/~,/~./,// ! Active, F10.7=250z ~ ~ Lifetime= Normalizedlifetimex (0.2044/CDMw) 0.1 i 200 225 250 360 350 460 4/50 560 550 600 ALTITUDE IN KM Fig. 8.5 Estimated orbit lifetime for average and active atmosphere.
  • 8. 192 ORBITAL MECHANICS 0.5 0.45cJ 0.4 ,," , .." , ,E-0.35 ]E 0.3 jf~ ...... 700 km ]~ 0.25• / 600 kmo 0.2 / / ~ - - - 500 km0•-~ 0.15 7 / / •"" / 0.1rn 0.05 .... ......... 1--- 0 0 10 20 30 40 50 Drag makeup delta V in m/s/year for an average atmosphere (MSIS90)Fig. 8.6 Drag makeup AV for an average MSIS90 atmosphere (F10.7 = 150,a, = 15) at various orbit altitudes and CaA/m.through the first telescope built by Galileo. These effects will be discussed in detailin Chapter 10. References 1Roy, A. E., Orbital Motion, 3rd ed., Adam Hilger, Bristol, UK, 1988. 2Michielsen, H. J., and Webb, E. D., "Stationkeeping of Stationary Satellites MadeSimple," Proceedings of the First Western Space Conference, 1970. 3Chao, C. C., "An Analytical Integration of the Averaged Equations of Variation Due toSun-Moon Perturbations and Its Application," The Aerospace Corp., Tech. Rept. SD-TR-80-12, Oct. 1979. Problem8.1. A low altitude Earth satellite moves in near circular orbit with the fol-lowing elements at time to: a -- 6800km, e = 0.002, i --- 50dog., co = 95 dog.,f2 = 120 dog., M = 20 dog. Determine the secular rates of the last three angular elements (eg., &, ~2,/~I) ofthe above set due to J2 effects. Selected Solution8.1. & = 4.2446 deg./day ~2 = - 5 . 1 1 9 5 deg./day lVl = 5573.6783 + 0.9537 = 5574.6320 deg./day = 15.4851 Rev/day

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