PS2%202010%20Soln[1]

PS2%202010%20Soln[1] - AAE 532 Orbit Mechanics Problem Set...

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AAE 532 – Orbit Mechanics Problem Set 2 Due: 9/8/10 Problem 1: Defining a system of 6 bodies (modeled here as particles), lines up as indicated, Sun as origin: Sun — Venus — s/c — Moon — Earth — Jupiter And assume the distance between the bodies to be consistent with values of semi-major axis ( a i ) as given in the table, so that for example : 108,208,927 ra == ♀♀ km 149,587,457km 384,400km a =−= ☼e e 149,203,057km = …….and so on Also assume that 6 // 1.5006 10 km, 82kg sc rm = (a) Then the c.m. of the system is determined by: / / Jup cm s c Jup TTT Gm Gm Gm Gm Gm Gm rr rrr Gm Gm Gm Gm Gm Gm =+ + ++ + ☼e where T Gm is the total Gm values: / Ts c J u p Gm Gm Gm Gm Gm Gm Gm = +++ Note “ Gm values for planets are given in the Table of Constants in units of km 3 /s 2 and () 11 3 2 8 3 2 / 6.67300 10 m /(kg-s ) 82kg =5.4719 10 m /s Gm −− × × or 18 3 2 / 5.4719 10 km /s Gm negligibly small Noting that the origin is the Sun as defiend above, ˆ iii r x ☼☼ (since all bodies are in the positive ˆ x direction from Sun) 5 ˆˆ 7.43700 10 km 1.068550 solar radii cm rx x = ( s/c made no noticeable contribution to this!) The vector differential equation for the s/c is: ˆ x
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/ / / 3 / 1 / sc j js c j js c ji mm n mr G r r = =− ±± where ,,,, jJ u p = ☼e (Note that the base point of the vector / s c r is the c.m., so the equation applies as written since the c.m. is inertially fixed due to conservation of linear momentum of the system.) (b) Thus, the acceleration of the s/c is: // / / / / / / 33 3 / / / Jup s c Jup s cs c s c s c J u p s c J u p s c r Gm Gm r Gm r Gm r Gm r r Gm r Gm r Gm r Gm r Gm r ⊕⊕
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PS2%202010%20Soln[1] - AAE 532 Orbit Mechanics Problem Set...

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