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After looking at it, I remember why I previously abandoned this. Taking the e->1 limit can be done relatively easily for many parts of the solution. For example, the polar solution reverts to the trivial Schwarzschild solution. The however, the limit of the t solution appears non-trivial. My first attempt yielded a divergent answer. Probably the best route is to take the limit of the EOMs and then solve the resulting elliptic integrals.
It looks like parabolic orbits (e=1) are not working:
returns a bunch of 1/0 errors.
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