Chapter 5 - Part 1 Orbit Perturbations D.Mortari - AERO-43
Orbital Elements Orbit normal i North Orbit plane Equatorial plane ϕ P O ω Ω i Vernal equinox Ascending node D. Mortari - AERO-43
Introduction Orbits are not Keplerian: 1) the hypothesis of spherical gravitational field is never satisfied, ) other forces are always present. Kepler solution can be considered as a basic solution, to be perturbed. dr μ General equation is: = r + a 3 p dt r a indicates the perturbation (not the disturb!) p Analytical Methods or General Perturbations (constant variations, coordinate variations, geometric) Numerical Methods or Special Perturbations (Cowell, Enke, Herrick) D. Mortari - AERO-43 3
Primary Perturbations Non spherical Earth gravitational Equatorial bulge (J ) is the dominant term Lunar and solar gravitational Causes N-S drift of geosynchronous orbits Atmospheric drag Dominant perturbation below 400 Km Solar radiation Leaking thrusters Out gassing D. Mortari - AERO-43 4
Semi-major axis variation Let ( a, a, a ) be the perturbing accelerations acting on the satellite in the radial, r s n tangential and perpendicular to the orbit plane directions. When other forces act on the satellite the orbital elements aeiωω,,,, are no longer constants, they vary out-of-plane directions. μ a perturbing acceleration, a = e + a where a = a e + a e + a e r = = = = + + ϕ p r p p r r s s n n 1 v v v a r v a r μr a r μ r μ a r μ r μ r μ Therefore we have ( ) with time. We want the equations for their variations. The method of variation of parameters is used to obtain the equations of motion. As an example, consider the semi-major axis. Let e, e, e be the unit-vectors in the radial, tangential, and ( ) r s n a a = rar + rϕa μ ( ) ( ra r a ) D. Mortari - AERO-43 5 s r s
Semi-major axis variation r a The equation was: a = ( ra r + rϕ as) μ From angular momentum we can compute rϕ ( 1+ ecosϕ) ( 1+ ecosϕ) r ϕ h μ rϕ= = = μ p = + e ϕ r r p p sin sin si ( 1 cos ) From the expression of the radius we can compute r dr pe ϕ r e ϕ re = = ϕ= ϕ= dt p n ϕ μ μ ( 1+ ecosϕ ) = esinϕ p p p a μ substituting a = esin ϕ ar + ( 1+ ecos ϕ) a μ p a a = esin ϕ ar + ( 1+ ecos ϕ) a h s s D. Mortari - AERO-43 6
Lagrange s Planetary Equations da esinϕ a 1 e a = a [ sin (1 cos ) ] r + as = e ϕ ar + + e ϕ as dt n 1 e nr h de 1 e sinϕ 1 e a (1 e ) = ar + r a s dt na nea r dm 0 1 r 1 e (1 e ) r dn = cos ϕ ar 1 sin a s t dt na a e nae + ϕ a(1 e ) dt di r cos( ω+ϕ) dω rsin( ω+ϕ) = a and dt na 1 e n = a 1 n dt na e sini dω 1 e + ecosϕ rcotisin( ω+ϕ) = cos ϕ a + sin ϕa a dt nae 1 ecos + ϕ h r s n D. Mortari - AERO-43 7
Observations From these equations we can corroborate some of our previous conclusions on orbit changes. From the da / dt equation we see that with a tangential impulse the maximum change in a occurs when an impulse is applied at perigee, where (1+ecosϕ) is max. From the di / dt equation we see that the maximum change in the inclination occurs with an out of plane impulse at the equator, where cos(ω+ϕ) is max. From the dω / dt equation the maximum change in the right ascension occurs with an out-of-plane impulse at the antinode, where sin(ω+ϕ) is max. For eccentricity corrections, tangential impulses at perigee or apogee are twice as efficient as the radial impulses at ϕ=π/ or ϕ=3π/ (see next slide). Corrections of semi-major axis and eccentricity are done together. D. Mortari - AERO-43 8
Eccentricity term de 1 e sinϕ 1 e a (1 e ) = ar + r a dt na nea r substituting r = a(1 e ) we obtain: 1 + ecos ϕ de 1 e sinϕ 1 e 1 e = ar + 1 ecos a dt na nea + ϕ 1+ ecosϕ de 1 e sinϕ 1 e 1 1 e = ar + cos a dt na na ϕ+ e e(1+ ecos ϕ) de 1 e e + cos ϕ = (sin ϕ ) ar + cos ϕ+ as dt na 1+ ecosϕ de 1 (for e 0) we have = ( arsin ϕ+ ascos ϕ) dt na D. Mortari - AERO-43 9 s s s
Non Spherical Earth Gravitational Forces a = U, where U = gravitational potential n n n μ RE R U = 1 Jn Pn cosφ + Pnm cosφ Cnm cosλ+ Snm sinλ r n= r n= m= 1 r φ = latitude, λ = longitude R E = Earth radius E ( ) ( )( ) P n, P nm are Legendre polynomials of the 1 st and nd kind J = 1.086 10 and J, J,, C, S O(10 ) 3 6 3 4 nm nm Thus, J is the dominant perturbation except for resonant or repeating ground track or compatible orbits. These orbits are characterized with an orbit period that is a rational number times the Earth rotation period, i.e., m T = TE nt = mt n E D. Mortari - AERO-43 10
Non Spherical Earth Gravitational Forces The orbital elements are no longer constant, they vary with time. We have three types of variations: 1) Short period - vary with orbital period. ) Long period - varies with period from days to months. 3) Secular - linear with time. (Period is the period of the perigee rotation) D. Mortari - AERO-43 11
Element Variations 1.6 1.4 1. 1 Secular Short period Long period Sum 0.8 0.6 0.4 0. 0-0. 0 100 00 300 400 500 600 Time D. Mortari - AERO-43 1
Definitions Mean or averaged elements: those obtained from averaging the elements over one orbit or from using the averaged disturbing potential. These are the elements usually provided by Space Command. Osculating elements: those elements are obtained from computing the orbital elements from position and velocity using the two body equations. They are instantaneous values. D. Mortari - AERO-43 13
Equatorial Bulge (J ) Effect Disturbing potential: r R r = r + 3 r Lagrange s Planetary Equations in terms of the disturbing potential R are: = = + dt na M dt na e ω na e M da R de b R b R 3 4 0 0 di 1 R cosi R dω 1 R = + = dt nabsin i Ω nabsin i ω dt nabsin i i dω cosi R b R dm 0 R b R = + = 3 4 dt nabsini i na e e dt na a na e e μ 3 J = 1.083 10 μ RE R= J P( cos φ) where 1 r r P ( cos φ ) = 3sin ( ϕ+ω) sin i 1 D. Mortari - AERO-43 14
Equatorial Bulge (J ) Effect The sin(λ), cos(λ) terms vary as the orbit frequency. These are short period variations that occur in all the orbital elements. They do not affect long term behavior. To obtain the long term effects average the RHS of the equation over one orbit. Since the only terms varying rapidly one can average the disturbing potential. T 0 μ R where E 3sin ( ) sin 1 1 1 R = Rdt R J i T = ω+ϕ r r rdϕ= hdt(the angular momentum integral), and T= n π r nr R = Rdϕ= J E i 0 3/ π h 4(1 e ) (3sin ) π n D. Mortari - AERO-43 15
Equatorial Bulge (J ) Effect da de di = 0, = 0, = 0, dt dt dt dω 3 RE dω 3 RE = J ncos i, = J n 5cos i 1 dt p dt 4 p dm 3 RE n= = n0 1 + J ( 3sin i) 1 e dt 4 p ( ) Thus, the primary effects due to J are the nodal precession, argument of perigee precession, and the change in period. Periods: Anomalistic - perigee to perigee. Nodal - equator crossing to equator crossing. D. Mortari - AERO-43 16 π (Anomalistic period) TA = n π (Nodal period) TN = ω+ n
J Effects 0 Altitude = 400 km 15 Perigee drift rate 10 critical inclination 5 0 0 0 40 60 80 100 10 140 160 180-5 Rt. Ascension precession rate -10 D. Mortari - AERO-43 17 Inclination - deg
Dominant Perturbations on LEO R Atmospheric drag - important below 600 Km Equatorial Bulge 3 3 a sin i 1 = n RE J r 3 dω 1 R = dt nabsin i i 3 RE Ω= J cos n i a(1 e ) 3 RE ω= J n i 4 a(1 e ) (5cos 1) μ where n= and J 3 = 1.08 10 a Critical inclination implies ω = 0 = 63.4 5cos i = 1 i= = 116.6 D. Mortari - AERO-43 18 3
Leo Satellite (MIR) i = 51.6 Period 91 min 350 Km D. Mortari - AERO-43 19
Perturbation effects at GEO Longitude dependent gravitational forces cause a longitude drift. Two stable points, 75 E, 5 E East-West station-keeping requirements. Sun and Moon cause an oscillation of the inclination. Rate is about 0.7 deg/year. Amplitude is 14 degrees. Period is 57 years. North-South station-keeping requirements require considerably more fuel than East-West station-keeping D. Mortari - AERO-43 0
Geosynchronous i = 0.3 (TDRS-7) Perigee: 35,774 Apogee: 35,795 Intelsat F7-6 D. Mortari - AERO-43 1
Equatorial Bulge (J ) Effect Anomalistic period is T A = π π, Nodal period is TN n = n + ω 3 RE For circular orbits: n+ω= n0 1 + J 3 4sin i p Sun Synchronous Orbits ( ) Orbits for which the orbit plane orientation with respect to the Earth-Sun line is fixed. Equator crossings occur at the same local time each orbit. Defined by ΔΩ = 360/365.5 deg/day. 3 RE 4 1 a(1 e ) J ncosi cosi 1.7 10 a(1 e ) n RE Ω= = D. Mortari - AERO-43
Sunsynchronous (Landsat 7) i = 98.8 Period 98 min 700 Km D. Mortari - AERO-43 3
Sunsynchronous Orbits Ω 1deg/day Orbit plane maintains the same orientation with respect to the Sun, Satellite crosses the equator at the same local time each day View at the same altitude at the same local time (same sun rays angles or illuminating conditions) D. Mortari - AERO-43 4
Sun Synchronous Orbits 106 105 104 Inclination (deg) 103 10 101 100 99 98 97 e=0 96 00 400 600 800 1000 100 1400 1600 1800 000 Altitude (Km) - km D. Mortari - AERO-43 5