Expression for orbital eccentricity

[Robaj]

Homework Statement

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).

Homework Equations

(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.

 

[Filip Larsen]

I don't think it can be made significantly simpler if it has to be expressed as a function of r, v, and γ. At least one of my textbooks has a similar unruly expression for e (although it uses a instead of v).

 

[Robaj]

Ok, I'll leave it as it is. Thanks a lot.