Given certain simplifying assumptions, a "linear tangent" steering law is optimal for orbital insertion of a launch vehicle:
$$\tan \psi = a\cdot t + b$$
i.e. the tangent of the flight path angle changes linearly from the point at which insertion guidance starts until circular orbit is achieved. My question is then how to compute the correct values for $a$ and $b$.
Assuming that an open-loop pitch guidance program is used for the initial ascent, linear-tangent guidance will take over from a given velocity vector and position, and our goal is to reach a given altitude with the correct horizontal velocity for circular orbit:
Is there a more direct way to determine the $a$ and $b$ coefficients than by a trial-and-error search algorithm? My application is for a generalized launch-to-LEO simulation tool.