We will use the solution in equation (3) to derive the response of the continuum stellar system to point particles. The effect of a point particle on the system is then the combined effect of the potential due to point particle and its response. This has been called a dressed point particle by Rostoker & Rosenbluth (1960) who first derived the properties of a plasma of dressed particles.
Following Weinberg (1989), I will expand the
perturbation in a biorthogonal series whose basis is constructed from
eigenfunctions of the Laplacian. The potential or density, then,
trivially follow from the expansion coefficients and the
potential-density pairs, , solve the Poisson equation
explicitly. I will deviate from traditional notation and define the
true space density corresponding to to be . The biorthogonal expansion, then, is in potential and
times the density. This leads to the convenient
The upper limit of the integral in equation (4) may be infinite, depending on the pairs. I will set G=1 throughout.
Using this, the spherical harmonic expansion coefficients for a point
mass of mass at is
The gravitational potential of the point mass, the perturbing Hamiltonian, is then