R R Hence ,= A=2(a2-a1) (41) A separate relationship between a, and a, comes from (39): sIn where vi, v2 denote the initial and final orbital velocities Combining(41)and(42) al 1-C0s-Ai sn-△ In a, sin -Al VcOS I Ai The most important quantity is the optimized Av From(35), a。-假 dR/R 2入1 2dα (cot a, -cot a,) 4 rsna八(cos TaSnα 16.522, Space Propulsion Lecture 3 Prof. Manuel martinez-Sanchez Page 7 of 916.522, Space Propulsion Lecture 3 Prof. Manuel Martinez-Sanchez Page 7 of 9 2 2 i R = sin , 2 ⎛ ⎞ π µ α ⎜ ⎟ λ ⎝ ⎠ so dR 2d 2d sin ( ) = =. R sin tan α α α α Hence, 2 1 i 2 = d α α ∆ α π ∫ i ( ) 2 ∆ αα = 2 1 π − (41) A separate relationship between α1 and 2 α comes from (39): 2 1 1 2 sin R v = = sin R v 2 1 α α (42) where v1, v2 denote the initial and final orbital velocities. Combining (41) and (42), 1 2 sin + i v = sin v 1 1 ⎛ ⎞ π ⎜ ⎟ α ∆ ⎝ ⎠ 2 α ; 1 2 v cos i + cot sin i = v 1 ⎛⎞ ⎛⎞ π π ∆ α∆ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ 2 2 1 2 v - cos i v cot = sin i 1 π ∆ 2 ⇒ α π ∆ 2 2 1 1 2 2 sin i or sin = v v 1 - 2 cos i v v 1 ⎛ ⎞ ⎜ ⎟ π ∆ 2 α ⎛ ⎞ π ⎜ ⎟ ∆ 2 ⎝ ⎠ ⎝ ⎠ + (43) The most important quantity is the optimized ∆V . From (35), 2 2 1 1 R R R R 3 1 dR 1 dR R V= = 2 cos 2 R cos R µ µ ∆ α α ∫ ∫ ( ) 2 2 1 1 R i i R 2 = = = cot - cot 1 1 2d d 2 22 i 2 sin cos tan sin α 1 2 α ⎛ ⎞ λ λλ ⎛ ⎞⎛ ⎞ α α ⎜ ⎟⎜ ⎟⎜ ⎟ α α ⎝ ⎠ πα α α π π ⎝ ⎠⎝ ⎠ α ∫ ∫ (44)