Re: escape velocity question

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Mon, 15 Mar 1999, William J. Larson wrote:

> A great question Jonathan.  May I put it on my physics final exam (on
> Thursday)?
>
> > I was doing some research on 40 Eridani B (a white dwarf).  I read that
> > it has 37,000 times the surface gravity of the Earth.  Since Earth's
> > escape velocity is 11.2 kilometers per second, the white dwarf's escape
> > velocity should be 414,400 kilometers per second, exceeding the speed of
> > light.
> >
> > Shouldn't this make the star invisible?  Perhaps the data is wrong.  Or
> > perhaps the light is just deflected. ...

To which Jack wrote:

>        Try calculating the escape kinetic energy (relativistically) so
> you don't get an absurd result.

No relativity required here.  Escape velocity is given classically by

  v_esc = (2 g R)^(1/2)

For an escape velocity equal to the speed of light with g = 37,000 g_earth
we would find R = 120 million km (the size of a red giant) and
M = 42 million solar masses.  A white dwarf has (I think) a radius closer
to that of Earth so the escape velocity would be "only"
37000^(1/2) x 11.2 km/s = 2.2 x 10^6 m/s, less than 1/100 the speed of
light.

John
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