- #1

Vrbic

- 407

- 18

## Homework Statement

It is my idea so I hope there is no problem in assignment.

How long takes small black hole to eat an ordinary star, if the black hole sit in the center of star?

## Homework Equations

We probably should suppose that star is ideal fluid (incompressible).

(1) ##\frac{dm}{dt}=A\rho v##, where ##\frac{dm}{dt}## is mass falling onto black hole per unit time, ##A## is area of the hole a ##\rho## is density of the star and ##v## is speed of falling matter on the horizon.

(2) ##A=4\pi R^2##

(3) ##R_g=\frac{2Gm}{c^2}##, where ##m## is mass of black hole, ##G## is gravitation constant and ##c## is speed of light.

##F=G\frac{m_1m_2}{r^2}=m_2a => a=G\frac{m_1}{r^2}##

(4) ##v=G\frac{m}{r^2}t##

## The Attempt at a Solution

So if I put (1) - (4) together I got

##\frac{dm}{m}=2\pi G\rho t dt##

and from that integrating from ##m_0## mass of black hole to ##m_0+M_0## where ##M_0## is mass of star, I have got ##t=\sqrt{\frac{1}{k}\ln{\frac{m_0+M_0}{m_0}}}##.

What do you mean about it?

a) If I would use ##\rho(r)## and better formulae for ##v##? Is equation (1) general?

b) May I use such speed of falling into black hole as a reasonable approximation?

c) Generally, is it alright?