## FANDOM

157 Pages

Let us consider a tidal disruption event of a star with mass $M_s$ and a radius $R_s$ by a black hole of mass $M_h$. The tidal radius is defined as $R_t \approx R_s \left(M_h/M_s \right)^{1/3}$ and the dimensionless impact parameter is defined as $\beta = R_t/R_p$ where $R_p$ is the periapse (i.e. the closest the star gets to the black hole). For very massive black holes the minimum periapse is given by the gravitational radius

$R_{p,\min} \approx \frac{G M_h}{c^2}$

and the maximum dimensionless impact parameter is given by

$\beta_{\max} = \frac{R_t}{R_{p,\min}} = \frac{c^2 R_s}{G M_s} \left( \frac{M_h}{M_s}\right)^{-2/3}$

In the limit of a very small black hole, the minimal impact parameter is limited by the radius of the disrupted star

$R'_{p,\min} \approx R_s$

and the maximum dimensionless impact parameter in this case is given by

$\beta'_{\max} = \frac{R_s}{R_t} = \left(\frac{M_h}{M_s} \right)^{1/3}$

It is therefore clear that the maximum $\beta$ is attained when

$\frac{M_h}{M_s} \approx \frac{c^2 R_s}{G M_s}$

The right hand side of the previous equation can be interpreted as the ratio between the actual radius of the star and the gravitational radius of the same mass. The corresponding maximal $\beta$ is

$\beta_{\max} \approx \left( \frac{c^2 R_s}{G M_s} \right)^{1/3}$