  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.
