Angular Location of the \(n^{th}\) Einstein Ring at Large n

We perform a matched asymptotic expansion to find an analytic formula for the trajectory of a light ray in a Schwarzschild metric, in a power series expansion in the deviation of the impact parameter from its critical value. We present results valid to second sub leading order in this expansion. We use these results to find an analytic expansion for the angular location of the \(n^{th}\) $n^{\mathit{th}}$ Einstein Ring (at large n) resulting from a star that lies directly behind a black hole but not necessarily far from it. The small parameter for this expansion is \(e^{-\pi (2n+1)}\) $e^{-\pi (2n+1)}$ : our formulae are accurate to third order in this parameter.