Equatorial stability analysis of dust particle orbits within a charged rotating disc of dust

TL;DR

The paper addresses the equatorial stability of circular dust particle orbits within a charged, rigidly rotating disc of dust in the Einstein-Maxwell framework. It employs a post-Newtonian expansion up to tenth order in the relativity parameter $g\in[0,1]$ and a constant specific charge $ε\in[0,1]$, with the spacetime expressed in Weyl-Lewis-Papapetrou coordinates. An effective potential $\mathcal{U}$ for equatorial motion of charged test particles is constructed, and stability is assessed via $\mathcal{U}_{,\rho\rho}>0$ (equivalently $\mathcal{S}<0$). The main result is that all interior equatorial circular orbits are stable for $ε<1$, while for $ε=1$ dust is in a marginally stable state; the rim at $\rho=\rho_0$ is unstable but not populated due to vanishing surface density. This work demonstrates that the charged rotating disc of dust can realize all stable interior orbits, and it points to future investigations of vertical perturbations and collective disc dynamics, which will likely require numerical methods.

Abstract

Stability of circular orbits of the dust particles within a charged rotating disc of dust with respect to perturbations in the equatorial plane is analyzed. Within Einstein-Maxwell theory, the charged rotating disc of dust is an axisymmetric, stationary solution given in terms of a post-Newtonian expansion. The disc solution is characterized by a rigid rotation around the axis of symmetry and a constant specific charge $ε$. It is found that for $ε<1$ all realized dust particle orbits are stable and for $ε=1$ all dust particles are in marginally stable states.

Figures (6)

• Figure 1: $\mathcal{S}$ with $g=0.3$ (top) and $g=0.5$ (bottom), shown for $\epsilon\in\{0, 1/4, 1/2, 3/4, 1\}$.
• Figure 2: $\mathcal{S}$ with $g=0.7$ (top) and $g=0.9$ (bottom), shown for $\epsilon\in\{0, 1/4, 1/2, 3/4, 1\}$.
• Figure 3: $\mathcal{U}$ (top) and $\rho_{0}^2\,\mathcal{U}_{,\rho\rho}$ (bottom), with $\tilde{L}=\tilde{L}(0)$ and $\tilde{E}=\tilde{E}(0)$, shown for $g=0.7$ and $\epsilon\in\{0, 1/4, 1/2, 3/4, 1\}$.
• Figure 4: $\mathcal{U}$ (top) and $\rho_{0}^2\,\mathcal{U}_{,\rho\rho}$ (bottom), with $\tilde{L}=\tilde{L}(\frac{\rho_{0}}{2})$ and $\tilde{E}=\tilde{E}(\frac{\rho_{0}}{2})$, shown for $g=0.7$ and $\epsilon\in\{0, 1/4, 1/2, 3/4, 1\}$.
• Figure 5: $\mathcal{U}$ (top) and $\rho_{0}^2\,\mathcal{U}_{,\rho\rho}$ (bottom), with $\tilde{L}=\tilde{L}(\rho_{0})$ and $\tilde{E}=\tilde{E}(\rho_{0})$, shown for $g=0.7$ and $\epsilon\in\{0, 1/4, 1/2, 3/4, 1\}$.