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Gravitational potential energy of a multi-component galactic disk

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.