## FANDOM

167 Pages

Let us consider a small pebble moving toward a gravitating mass $M$. We assume that the velocity of the pebble at infinity is $v_{\infty}$ and its impact parameter is $b$. The closest approach of the pebble to $M$, $a$ can be related to impact parameter via

$v_{\infty} b = v_a a$

$\frac{1}{2} v_{\infty} = \frac{1}{2} v_a^2 - \frac{G M}{a}$

hence

$b = a \sqrt{1 + \frac{2 G M}{a v_{\infty}^2}}$

If the radius of the gravitating mass is $R$, then pebbles whose closest approach is smaller then the radius will collide with the gravitating mass. The maximum impact parameters for which a pebble will collide with the gravitating mass is

$b_{\max} = R \sqrt{1+\frac{v_e^2}{v_{\infty}^2}}$

where $v_e = 2 \frac{G M}{R}$ is the escape velocity. The cross section for collision is therefore

$\sigma = \pi b_{\max}^2 = \pi R^2 \left(1 + \frac{v_e^2}{v_{\infty}^2} \right)$

We can now write an equation for the mass accretion rate

$\frac{d M}{d t} = \sigma v_{\infty} \rho = \rho v_{\infty} \pi R^2 \left( 1 + \frac{v_e^2}{v_{\infty}^2} \right)$

where $\rho$ is the density of the ambient medium. If matter does not compress, then the mass can be related to the radius through

$M = \frac{4 \pi}{3} \rho R^3$

Although this equation can be solved analytically, we prefer to focus on two limiting cases. In the first case, we assume the escape velocity is much smaller than the typical velocity of the pebbles $v_{\infty} \gg v_e$. In this case the equation can simplified to

$\frac{d}{d t} \left(\frac{4 \pi}{3} R^3 \rho \right) = \pi R^2 \rho v_{\infty}$

and the solution is

$R = \frac{1}{4}v_{\infty} t$

So the radius increases linearly with time.

In the other limit, the escape velocity is much larger then the velocities of the pebbles $v_e \gg v_{\infty}$. In this case the equation can be simplified to

$\frac{d}{d t} \left( \frac{4 \pi}{3} \rho R^3 \right) = \pi R^2 v_{\infty} \frac{v_e^2}{v_{\infty}^2} = \pi R^2 v_{\infty}^{-1} \cdot \frac{2 G}{R} \cdot \frac{4 \pi}{3} \rho R^3$

and the solution is

$R = \frac{3}{2 \pi} \frac{v_{\infty}}{G \rho \left(t_0 - t \right)}$

so the radius (and mass) diverge at a finite time.