From Prussing and Conway (Q1.11): derive an expression for the eccentricity e in terms of the initial speed v, radius r, and flight path angle x (they use gamma).
(1) h^2 = mu*a*(1-e^2) [a is semimajor axis, mu is gravitational parameter]
(2) h = r*v*cos(x)
(3) v^2 = mu*((2/r)-(1/a))
The Attempt at a Solution
Rearrange (1): a = h^2/(mu*(1-e^2))
Sub in (2): a = (r^2*v^2*cos^2(x)) / (mu*(1-e^2))
Rearrange for e: e^2 = 1-[(r^2*v^2*cos^2(x)) / (mu*a)]
Rearrange (3) in terms of a and sub in: e^2 = 1-[(r^2*v^2*cos^2(x)) / mu]*[(2/r)-(v^2/mu)]
This seems far too unwieldy.. have I misunderstood the basic geometry or relationships behind the problem? I've been unable to find an expression for e = f(r,v,x) anywhere .
Thanks for your help.