A General Quadrature Solution for Relativistic, Non-relativistic, and Weakly-Relativistic Rocket Equations

TL;DR

This paper presents a unified quadrature approach to solve rocket equations across non-relativistic, relativistic, and weakly-relativistic regimes, providing explicit correction terms and broad applicability.

Contribution

It introduces a general quadrature framework that encompasses classical and relativistic rocket equations, with series expansions for relativistic corrections and special cases analysis.

Findings

Low order correction products accurately approximate relativistic equations up to 0.5c.

The method unifies non-relativistic and relativistic rocket equations within a single framework.

Explicit series expansions reveal how relativistic effects modify mass ratio calculations.

Abstract

We show the traditional rocket problem, where the ejecta velocity is assumed constant, can be reduced to an integral quadrature of which the completely non-relativistic equation of Tsiolkovsky, as well as the fully relativistic equation derived by Ackeret, are limiting cases. By expanding this quadrature in series, it is shown explicitly how relativistic corrections to the mass ratio equation as the rocket transitions from the Newtonian to the relativistic regime can be represented as products of exponential functions of the rocket velocity, ejecta velocity, and the speed of light. We find that even low order correction products approximate the traditional relativistic equation to a high accuracy in flight regimes up to $0.5c$ while retaining a clear distinction between the non-relativistic base-case and relativistic corrections. We furthermore use the results developed to consider the case where the rocket is not moving relativistically but the ejecta stream is, and where the ejecta stream is massless.