Putting the results of §§2.1-2.3 together, we can derive the Laplace transform of and using equations (5)--(8) to get the distribution function of the dressed particle.
First, for a particular spherical harmonic, the action-angle transform
where is a some general well-behaved function of time. The quantity is shorthand for W with . Because the motion of a star on a regular orbit is quasi-periodic, we will consider terms with pure sinusoidal dependence: . Substituting this into equation (9) and Laplace transforming gives the desired result:
where the Laplace transform of is