On the nodal distance between two Keplerian trajectories with a common focus

TL;DR

This paper investigates the range of possible nodal distances between two non-coplanar Keplerian orbits sharing a focus, providing bounds based on orbital elements to aid in celestial observation and orbit determination.

Contribution

It extends previous work on orbit distance estimates from circular to elliptic trajectories, deriving bounds for nodal distances based on orbital parameters.

Findings

Derived optimal bounds for nodal distance as functions of orbital elements.

Extended previous circular orbit estimates to elliptic cases.

Provides insights into observability of celestial bodies along specific trajectories.

Abstract

We study the possible values of the nodal distance $\delta_{\rm nod}$ between two non-coplanar Keplerian trajectories ${\cal A}, {\cal A}'$ with a common focus. In particular, given ${\cal A}'$ and assuming it is bounded, we compute optimal lower and upper bounds for $\delta_{\rm nod}$ as functions of a selected pair of orbital elements of ${\cal A}$, when the other elements vary. This work arises in the attempt to extend to the elliptic case the optimal estimates for the orbit distance given in (Gronchi and Valsecchi 2013) in case of a circular trajectory ${\cal A}'$. These estimates are relevant to understand the observability of celestial bodies moving (approximately) along ${\cal A}$ when the observer trajectory is (close to) ${\cal A}'$.