An approximate inverse of Kepler's Equation at the parabolic limit e → 1 using a hyperbolic function structure.

The paper outlines an approximate inverse of M = E - sinE, which is Kepler's equation where e = 1. It does not describe a parabolic orbit, which is the physical system that an eccentricity being equal to 1 describes (which would use Barker's equation). Instead, this is Kepler's elliptical equation, but described at its highest limit, being the point where the equation ceases to describe a physical system. Mapping the elliptical Kepler equation at this high limit essentially means that, using a transformed version of the inverse function outlined in the paper, it may be easier to model an orbit with an eccentricity of 0.999 than it is to model an orbit of 0.9. This is the opposite of how most models function, where lower eccentricities are typically easier to model. The function in question possesses a hyperbolic function structure as well as a machine learned transformation model, denoted by the symbol \chi(M), which can take on one of multiple forms depending on the priority (High accuracy and complexity, low complexity at the cost of accuracy, accuracy focused at certain regions such as periapsis etc.).