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

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Unformatted text preview: Application: main problem Consider the simplest geopotential perturba- tion problem: J 2 only. In this case the poten- tial is: U ( r, ) = r- J 2 R 2 2 r 3 (3sin 2 - 1) | {z } Disturbing function= R . The satellite latitude is not a convenient quantity. Using spherical trigonometry, write sin = sin i sin( + ) = sin i sin u u = + is the argument of latitude and is the angle along the orbit from the equator. So the disturbing function is R =- J 2 R 2 2 r 3 (3sin 2 i sin 2 u- 1) . L. Healy ENAE404 Spring 2007 Lecture 28 (May 8) 1 Perturbations: secular and periodic If we expand in a Fourier series in the mean anomaly M , R = R s + X 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 Z 2 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- cos2 u ) . L. Healy ENAE404 Spring 2007 Lecture 28 (May 8) 2 Fourier series Compute constant and periodic terms in u R =- J 2 R 2 2 r 3 3 2 sin 2 i [1- cos2 u ]- 1 separately. Integrate over M ; the constant term is proportional to 1 2 Z 2 1 r 3 dM = 1 a 3 (1- e 2 ) 3 / 2 , and the periodic term in u is proportional to 1 2 Z 2 1 r 3 cos2 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 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 R 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 dt = 1 na 2 q 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 R 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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