Accretion Rate

Equation (2.26) indicates that the gas shell with a large $\bar{\rho}$ reaches the center earlier than that with a small $\bar{\rho}$. Imediately, this means a spherical cloud with a uniform density $\rho_0$ contracts uniformly and all the mass reaches the center at $t=t_{\rm ff}=(3\pi/32 G\rho_0)^{1/2}$. In this case, the mass accretion rate to a central source becomes infinity at an epoch $t=t_{\rm ff}$. In contrast, consider a cloud whose density gradually decreases outwardly. In this case, the outer mass shell has smaller $\bar{\rho}$ than the inner mass shell. Therefore even when the inner mass shell collapses and reaches the center, the outer mass shell are contracting and does not reach the center. This gives a smaller mass accretion rate than a uniform cloud. If the gas pressure is neglected, the accretion rate is determined by the initial spatial distribution of the density. We will compare the accretion rate derived here with results of hydrodynamical calculation in $\S$4.7