Parameter distributions of Keplerian orbits

TL;DR

This paper derives probability distributions for Keplerian orbital elements assuming uniform orientation, providing tools for analyzing planetary observations and demonstrating key statistical independencies.

Contribution

It introduces new analytical expressions for orbital element distributions based on arbitrary eccentricity and semi-major axis distributions, enhancing modeling of planetary systems.

Findings

Derived expressions for orbital element distributions.

Validated the independence of phase angle distribution.

Provided methods for calculating probability density functions in planetary observations.

Abstract

Starting with just the assumption of uniformly distributed orbital orientations, we derive expressions for the distributions of the Keplerian orbital elements as functions of arbitrary distributions of eccentricity and semi-major axis. We present methods for finding the probability density functions of the true anomaly, eccentric anomaly, orbital radius, and other parameters used in describing direct planetary observations. We also demonstrate the independence of the distribution of phase angle, which is highly significant in the study of direct searches, and present examples validating the derived expressions.