Notes_LL%20BP - LL 1 Transfer Orbits: Lambert Arcs Transfer...

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Transfer Orbits: Lambert Arcs Transfer Orbit Design (special class of boundary value problem) 1. Geometrical relationships Conic paths connecting two points that are fixed in space with focus at the attracting center 2. Analytical Relationships 3. Lambert’s Theorem LL 1
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Lambert’s Theorem Know a lot about possible orbits connecting two points But analytical relationships rely on “ a how to get it? Must somehow select a an additional specification about the transfer path What to specify? From a number of options, choose TOF 11 1 () s i n p nt t E e E =− 22 2 s i n p E LL 2
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Given TOF, this relationship contains unknowns: 12 ,, , EEea Must be rewritten in terms of only one unknown Æ a HOW? Define: 11 (1 cos ) ra e E =− 22 ) e E = 1 2 [2 (cos cos )] c o s c o s 2[ 1 c o s c o s ] pM rr a e E E EE ae a e E E += + ⎡⎤ +− ⎛⎞ ⎜⎟ ⎢⎥ ⎝⎠ ⎣⎦ LL 3 measured from center 1 ˆˆ px e y p = + 2 e y p = + 21 cp p =
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21 22 2 2 1 2 2 2 2 2 1 2 2 2 2 1 ˆˆ () ( ) (c o s c o s ) (s i n s i n ) (cos cos ) (1 ) (sin sin ) (cos cos ) ) (sin sin ) cp p xx eyy p ca E a E b E b E aE E a e E E aEE eE E =− +− + + ⎡⎤ + −− ⎣⎦ cos cos 2sin sin A BA B AB + ⎛⎞ −= ⎜⎟ ⎝⎠ cos cos sin p M EE E E sin sin 2cos sin A B + sin sin sin p M E E = 2 2 2 2 2 2 2 2 2 2 2 2 4sin sin (1 )4cos 4s i n s i n c o s c o s i n 1 c o s pM Mp p p E E e E E E E e E E e E =+ LL 4
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Define ok because e < 1 [ ] () 12 22 2 2 21c o sc o s 4s i n 1 c o s OR 2s i n s i n M M M rr a E ca E caE η += =− = So c o s c o s 2 s i n s i
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This note was uploaded on 12/22/2010 for the course A&AE 532 taught by Professor Kathleenhowell during the Spring '10 term at Purdue University-West Lafayette.

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Notes_LL%20BP - LL 1 Transfer Orbits: Lambert Arcs Transfer...

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