Gravitational potential energy of a multi-component galactic disk

TL;DR

This paper extends a method to calculate the gravitational potential energy of multi-component galactic disks, revealing that each component's energy remains unchanged from the single-component case, with implications for galactic stability.

Contribution

It derives an explicit expression for the gravitational potential energy of multi-component galactic disks, extending Camm's method and analyzing the energy distribution in coupled star-gas systems.

Findings

Energy of each component equals that in the single-component case.

Coupled systems are more tightly bound near the mid-plane.

Harder to disturb the system externally due to increased binding.

Abstract

We calculate ab initio the gravitational potential energy per unit area for a gravitationally coupled multi-component galactic disk of stars and gas, which is given as the integration over vertical density distribution, vertical gravitational force, and vertical distance. This is based on the method proposed by Camm for a single-component disk, which we extend here for a multi-component disk by deriving the expression of the energy explicitly at any galactocentric radius R. For a self-consistent distribution, the density and force are obtained by jointly solving the equation of vertical hydrostatic equilibrium and the Poisson equation. Substituting the numerical values for the density distribution and force obtained for the coupled system, in the derived expression of the energy, we find that the energy of each component remains unchanged compared to the energy for the corresponding single-component case. We explain this surprising result by simplifying the above expression for the energy of a component analytically, which turns out to be equal to the surface density times the squared vertical velocity dispersion of the component. However, the energy required to raise a unit test mass to a certain height z from the mid-plane is higher in the coupled case. The system is therefore more tightly bound closer to the mid-plane, and hence it is harder to disturb it due to an external tidal encounter.