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I need some help rearranging some orbital mechanics formulas.

All images have been borrowed from http://www.braeunig.us/space/orbmech.htm which has a through treatment of orbital equations, but provides no further insight on my question.

Given the following:

Position equations

Apsis equations

I have $r$, $\phi$ and $R_{a}$ or $R_{p}$ (only one will be relevant in each instance). I need to derive $a$, $e$ and $v$.

I've had no luck at all trying to rearrange these formulas to produce those results. I can't even assert one way or the other if it is possible, however knowing the system they represent it seems like there should be a single solution in each instance.

Background:

The goal here is to calculate a two tangent transfer from an arbitrary elliptical orbit to a circular orbit. The outputs $a$, $e$ and $v$ give the core parameters of the transfer orbit. The supplied $r$ and $\phi$ are the altitude and flightpath angle at the point of the first tangential burn. These are preserved during the burn while $a$ and $e$ are changed. $R_{a}$ / $R_{p}$ is the radius of the destination orbit. Since the destination orbit is circular meeting it at either the apoapsis or periapsis of the transfer orbit will will be another tangential burn. The choice of which to use is dependent on whether we are ascending or descending.

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    I believe that it's well known that you have to use approximation methods here. You might try asking it over on physics SE.2011-04-05
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    That would be annoying, I already have an iterative solver for a more general version of which this is a subset. I was hoping that knocking out half the free variables would let me lose the iterative solver and all the performance issues that go with it.2011-04-05

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