An Arc-Length Approximation For Elliptical Orbits

TL;DR

This paper introduces an analytical method to approximate the arc-length of celestial bodies in elliptical orbits using a parameterization based on the Lagrange inversion theorem, enabling precise calculations of orbital distances over time.

Contribution

It presents a novel formalism for computing elliptical orbit arc-lengths through a parameterization and numerical routine, enhancing orbital analysis accuracy.

Findings

Provides a new analytical relation for orbit arc-lengths.

Enables generation of consistent celestial ephemerides.

Offers a method to estimate orbit perimeter from intrinsic properties.

Abstract

In this paper, we overlay a continuum of analytical relations which essentially serve to compute the arc-length described by a celestial body in an elliptic orbit within a stipulated time interval. The formalism is based upon a two-dimensional heliocentric coordinate frame, where both the coordinates are parameterized as two infinitely differentiable functions in time by using the Lagrange inversion theorem. The parameterization is firstly endorsed to generate a dynamically consistent ephemerides of any celestial object in an elliptic orbit, and thereafter manifested into a numerical integration routine to approximate the arc-lengths delineated within an arbitrary interval of time. As elucidated, the presented formalism can also be orchestrated to quantify the perimeters of elliptic orbits of celestial bodies solely based upon their orbital period and other intrinsic characteristics.