Black hole inside a star -- How long for it to consume the star?

[Vrbic]

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?

 

[Buzz Bloom]

Hi Vrbic:

I am not an expert, and I am curious about what others might respond. Since no one else has yet responded, I thought I might offer a few comments.

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

I would expect a star to be mostly a hot gaseous atmosphere and therefore compressible. I do not know at what temperature and pressure the atmosphere might have a phase change to an incompressible liquid.

Re (1): (a) If the BH mass M is >> star mass m, then it may be OK to assume that M is a constant. However, If M is not much much greater than M, then M will increases as star mass falls pass the BH event horizon (EH). (b) I assume that by A you mean the surface area of the BH EH. A will also increases as star mass falls across the EH. (c) As these variables increase, the rae of star mass faliing across the EH also increases.

I hope this helps some.

Regards,
Buzz

 

[Vrbic]

Hi Vrbic:

I am not an expert, and I am curious about what others might respond. Since no one else has yet responded, I thought I might offer a few comments.I would expect a star to be mostly a hot gaseous atmosphere and therefore compressible. I do not know at what temperature and pressure the atmosphere might have a phase change to an incompressible liquid.

Re (1): (a) If the BH mass M is >> star mass m, then it may be OK to assume that M is a constant. However, If M is not much much greater than M, then M will increases as star mass falls pass the BH event horizon (EH). (b) I assume that by A you mean the surface area of the BH EH. A will also increases as star mass falls across the EH. (c) As these variables increase, the rae of star mass faliing across the EH also increases.

I hope this helps some.

Regards,
Buzz

Thank you for your comment, I hope someone will help us :-)