An analytic relation for the thickness of accretion flows

We take the vertical distribution of the radial and azimuthal velocities into account in spherical coordinates, and find that the analytic relation c _s0/ν _K Θ=[(γ−1)/2γ]^1/2 is valid for both geometrically thin and thick accretion flows, where c _s0 is the sound speed on the equatorial plane, ν _K is the Keplerian velocity, Θ is the half-opening angle of the flow, and γ is the adiabatic index.