Closed Form Expressions for the Potentials and Accelerations of Generalized Ring Models

TL;DR

This work derives closed-form expressions for gravitational potentials $V$ and accelerations $\vec{a}$ due to ring- and arc-like mass distributions, including uniform and eccentric configurations, as well as mass distributions averaged over Keplerian orbits. The authors express these fields in terms of elliptic integrals (including incomplete and complete first, second, and third kinds) and employ a Weierstrass substitution and Carlson forms to manage complex arguments and branch structure, with careful patching to obtain continuous definite integrals. Key contributions include analytic results for uniform circular arcs, eccentric orbit-averaged rings/arcs, and eccentric time-averaged rings, along with a robust treatment of elliptic-jump corrections ($\Delta_F$, $\Delta_E$, etc.) and continuous antiderivatives $F_c$ and $\Pi_c$. The methodology enables fast, accurate modeling in celestial mechanics for diverse ring-like mass distributions, and the authors provide reproducible code at the associated GitHub repository. This work advances analytic celestial mechanics by delivering compact, checkable closed-form expressions that surpass many traditional series approaches, especially at high eccentricities or with time-averaged distributions.

Abstract

We present several closed-form expressions of useful mass distributions. These include the potentials and accelerations of circular rings and arcs, the potentials of uniform density rings and arcs at arbitrary eccentricities, and the potentials and accelerations of rings and arcs when the mass is time-averaged over a Kepler orbit. We show that these expressions can be expressed, often simply, in terms of elliptic functions of complex arguments. We show that in a few limiting cases, the expressions are entirely real. We expect that these expressions will allow for more rapid modeling in many areas of celestial mechanics.

Figures (6)

• Figure 1: The top left panel shows the result of a numerical integration $\phi_{num}$, the top center panel shows the analytic form $\phi_{ana}$ given by Eq \ref{['eq:arcpot']} and the top right panel shows the log of their difference. The agreement is good to machine precision far from the ring and good to $10^{-12}$ near it, due to growing integration error caused by the steepening slope of the potential. Finally, the acceleration components given by Eq \ref{['eq:arc_acc']} are shown in the bottom row. All quantities were computed assuming $G = M = a = 1$,$\delta\theta = 1.0$, $\Theta_0 = 0.0$ and $z_0=0.1$
• Figure 2: The top left panel shows the result of a numerical integration $\phi_{num}$, the top center panel shows the analytic form $\phi_{ana}$ given by Eq \ref{['eq:ringpot']} and the top right panel shows the log of their difference. The agreement is good to machine precision far from the ring and good to $10^{-13}$ near it, due to growing integration error caused by the steepening slope of the potential. Finally, the acceleration components given by Eq \ref{['eq:ring_acc']} are shown in the lower rows. All quantities were computed assuming $G = M = a = 1$,$\delta\theta = 1.0$, $\Theta_0 = 0.0$ and $z_0=0.1$
• Figure 3: We show an example of the discontinuous behavior of the $\boldsymbol{F}$, $\boldsymbol{\Pi_1}$ and $\boldsymbol{\Pi_2}$ functions described in Eq \ref{['eq:etref']}. In each case, we show the function itself in red, the function times $S(t)$ in blue, and the continued, patched function in green. All quantities were computed assuming $G = M = a = 1$, $r_0=0.5$, $e=0.5$, $\omega = 2.0$, $\Theta_0 = 0.0$ and $z_0=0.1$
• Figure 4: The left panel shows the result of a numerical integration $\phi_{num}$, the center panel shows the analytic form $\phi_{ana}$ given by Eqs \ref{['eq:lim1_minus']},\ref{['eq:lim1_plus']},\ref{['eq:Fc']},\ref{['eq:indef']} and the right panel shows the log of their difference. All quantities were computed assuming $G = M = a = 1$,$\delta\theta = 1.9$, $\Theta_0 = 0.0$, $e=0.5$, $\omega=2.0$ and $z_0=0.1$
• Figure 5: The top left panel shows the result of a numerical integration $\phi_{num}$, the top center panel shows the analytic form $\phi_{ana}$ and the top right panel shows the log of their difference. The lower rows shows the accelerations in $r$, $\theta$,$z$ directions. All quantities were computed assuming $G = M = a = 1$,$\delta\theta = \pi$, $\Theta_0 = 0.0$, $e=0.5$, $\omega=2.0$ and $z_0=0.1$