The velocity dispersion of giant molecular clouds. II. Mathematical and numerical refinements

It has been proposed by Fukunaga and Jog & Ostriker that gravitational scattering of clouds off one another in encounters caused by differential rotation can account for the velocity dispersion of the largest molecular clouds in our Galaxy. Angular momentum transfer in cloud-cloud scatterings increases the eccentricity, or epicyclic amplitude, of the clouds. This input of random energy is ultimately balanced by dissipative cloud collisions, leading to equilibrium. Here we recalculate the energy input to the clouds using the proper linearized equations of motion, including the Coriolis force and allowing for changes in the guiding center. Perturbation theory gives a result in the limit of distant encounters and small initial epicyclic amplitudes. Direct integration of the equations of motion allows us to study the strong encounter regime. Our perturbation theory result differs by a factor of order unity from that of Jog & Ostriker. The result of our numerical integrations for the two-dimensional (planar) velocity dispersion, adopting the same model for energy loss as Jog & Ostriker, is σ ≃ 0.94(GM_cl κ)^1/3 = 5.1 km s^-1 for our fiducial cloud of mass 5 × 10⁵ M_⊙ at R₀/2, slightly smaller than the value obtained by Jog & Ostriker. In an Appendix we calculate the accretion rate for a molecular cloud in the galactic disk.