Math Overflow-ites! Hopefully, this question fits here better than places like Space and Physics. And sorry if I ramble a little - I'm rather new to this all.
My understanding of Lambert's theorem is that the time of flight of a transfer orbit depends on only on the semi-major axis, the sum of the radii and the linear distance between both points. I've seen equations roughly like these: $$ T = F(r_{1} + r_{2}, a, c). $$ What I have read is that since the time does not depend on the eccentricity, so I'm pretty sure that you can change the eccentricity while keeping these three parameters the same and the value of $T$ would remain the same.
So if I 'squash' the elliptic orbit with the same parameters into a perfect circle, it makes sense that the problem is still solvable, except that instead of a phase difference of $\theta$ there's a phase change of $\Phi$, and if I can find it I would have an easier time calculating $T$, because its a circular orbit.
And if it is a circular orbit I can redefine the distances to the start and destination as $r'$, such that:
$$2r'= r_{1} + r_{2} $$ and $$a = \frac{r_{1} + r_{2}}{2}$$
And if you equate the different values of $c^2$, you should be able to calculate $\Phi$ from $r_1$, $r_2$, $\theta$ and $a$. $$ 2a^{2}\big(1 - \cos(\Phi)\big) = r_{1}^2 + r_{2}^2 - 2r_{1}r_{1}\cos(\theta). $$ And if I can find what $\Phi$ is from numerical methods I can find what $T$ is from my value for the semi-major axis and the gravitational constant.
However, for some reason, this does not work, and I'm not very sure why. Is there anything that I am doing wrong?