Integrate LHS by parts 2000111mrrmr dtF rdtττττττ⋅−=⋅∫∫±±The first term is zero – since the quantity has the same value at 0 and τ. Thus 2TF= −⋅rwhere denote time average of the quantity within brackets. but dVkrFrrVdrrV−⋅==== −⋅∇hence 2TV= −but 22VVETVV=+= −+=hence 2V=Ebut 2kEconstanta= −=and 012kEEdtEaττ=== −∫so 2kEa= −Thus: kVa= −as before and therefore 122kTVa= −=6.21The energy of the initial orbit is 2122kkmvEra−== −(1) 221kvmra=−Since (1ara)ε=+at apogee, the speed , at apogee is 1v()()()2112111kkvmaamaεεε−=−=++To place satellite in circular orbit, we need to boost its speed to
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Celestial mechanics, Semi-major axis, Hyperbolic trajectory, denote time average