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# lecture28 - Application main problem Consider the simplest...

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Application: main problem Consider the simplest geopotential perturba- tion problem: J 2 only. In this case the poten- tial is: U ( r, φ ) = μ r - J 2 μR 2 2 r 3 (3 sin 2 φ - 1) “Disturbing function”= R . The satellite latitude φ is not a convenient quantity. Using spherical trigonometry, write sin φ = sin i sin( ω + θ ) = sin i sin u u = ω + θ is the argument of latitude and is the angle along the orbit from the equator. So the disturbing function is R = - J 2 μR 2 2 r 3 (3 sin 2 i sin 2 u - 1) . L. Healy – ENAE404 – Spring 2007 – Lecture 28 (May 8) 1

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Perturbations: secular and periodic If we expand in a Fourier series in the mean anomaly M , R = R s + j =1 R cj cos( jM ) + R sj sin( jM ) the secular perturbations arise from the con- stant term R s given by R s = 1 2 π 2 π 0 R dM. So to find contribution of secular perturbation of motion, look at R s . Note physical pertur- bations usually have both kinds of effects. The quantities u and r are dependent on mean anomaly M ; everything else is independent. To get Fourier terms, expand trigonometric ex- pressions in u in the disturbing function, using sin 2 u = 1 2 (1 - cos 2 u ) . L. Healy – ENAE404 – Spring 2007 – Lecture 28 (May 8) 2
Fourier series Compute constant and periodic terms in u R = - J 2 μR 2 2 r 3 3 2 sin 2 i [1 - cos 2 u ] - 1 separately. Integrate over M ; the constant term is proportional to 1 2 π 2 π 0 1 r 3 dM = 1 a 3 (1 - e 2 ) 3 / 2 , and the periodic term in u is proportional to 1 2 π 2 π 0 1 r 3 cos 2 u dM = 0 . So in fact there is no periodic term in u from the J 2 perturbation at this order. L. Healy – ENAE404 – Spring 2007 – Lecture 28 (May 8) 3

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Calculate secular term R s Reinstating the proportionality factor to the constant term above, the secular part of the disturbing function is R s = - J 2 μR 2 2 a 3 (1 - e 2 ) 3 / 2 3 2 sin 2 i - 1 . Note that it depends only on a , e , and i . We will now study the influence the secular term has on the elements by using the LPE. The dominant secular effect comes from the J 2 term, so this is a reasonable simplification. L. Healy – ENAE404 – Spring 2007 – Lecture 28 (May 8) 4
Apply planetary equations Apply the Lagrange planetary equations, d Ω dt = 1 na 2 1 - e 2 sin i R ∂i , and the partial derivative of the secular dis- turbing function R s is R s ∂i = - 3 J 2 μR 2 2 a 3 (1 - e 2 ) 3 / 2 sin i cos i.

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lecture28 - Application main problem Consider the simplest...

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