Problem Set 4. Due T 31 Oct

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Please show your working on this sheet, and write your answers on this sheet. Attach extra sheets if you need them.

**1. Quasar**

**(a) Schwarzschild radius**

The black holes which power quasars have masses typically about
10^{8} M_{sun}.
The Schwarzschild radius
*R* of a black hole is proportional
to its mass *M*,
with the Schwarzschild radius of a 1 M_{sun} object being 3 km.
What is the Schwarzschild radius of a 10^{8} M_{sun} black hole?
Express your answer in km, and then in Astronomical Units
(1 AU = 1.5 ×10^{8} km).

The Schwarzschild radius of a 10^{8} M_{sun} black hole is
______________________ km = ______________________ AU.

**(b) How long to fall in?**

If you fell (woops) through the Schwarzschild radius of a
10^{8} M_{sun} black hole,
how many minutes would it take you to fall to the central singularity?
[Hint: From your own point of view,
you would fall at the speed of light, *c* = 3 ×10^{8} m/s.]

I would hit the singularity in ______________________ minutes.

**(c) Tidal force**

The tidal force between your head and your toes
at the Schwarzschild radius of a 1 M_{sun} black hole is
about 10^{9} *g* (a giga-gee).
What is the tidal force at the Schwarzschild radius of a
10^{8} M_{sun} black hole?
Would you get ripped apart while entering the Schwarzschild radius
of a 10^{8} M_{sun} black hole?
Would you get ripped apart somewhere inside the black hole
(explain)?
[Hint:
The tidal force
at distance *R* from the center of an object of mass *M*
is proportional to *M*/*R*^{3}.
Remember that the Schwarzschild radius *R* is proportional to *M*.]

The tidal force between head and toes at the Schwarzschild radius of a
10^{8} M_{sun} is ______________________ *g*.

I would/would not be ripped apart at the Schwarzschild radius of a
10^{8} M_{sun} black hole.

I would/would not be ripped apart somewhere inside the black hole because

**2. Hawking Radiation**

**(a) Hawking Temperature**

By combining quantum mechanics and general relativity,
Steven Hawking showed that black holes are not quite black,
but emit a Planck spectrum.
This radiation is called `Hawking radiation',
and its temperature and luminosity are called the `Hawking temperature'
and `Hawking luminosity'.
The peak wavelength
l_{peak} of the Planck spectrum
is equal to the Schwarzschild radius *R* of the black hole.
Using Wien's Law,
l_{peak} = 0.0029 m K / *T*,
determine the Hawking temperature of a 1 M_{sun} black hole.

The Hawking temperature of a 1 M_{sun} black hole is
______________________ K.

**(b) Hawking Luminosity**

Would the Hawking luminosity of a 1 M_{sun}
black hole be more or less than the luminosity of the Sun? Why?
[Hint:
The Stefan-Boltzmann law *L* µ *R*^{2} *T*^{4} applies in both cases.
Don't bother to figure out the numbers, but state what the argument is.]

**(c) Schwarzschild Radius**

What would be the Schwarzschild radius of a black hole whose temperature is that of the Sun? [Hint: In Problem Set 2, you found that the Sun's spectrum peaks in the yellow, at 500 nm.]

The Schwarzschild radius of a yellow black hole would be ______________________ nm.

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On 17 Oct 2000, 04:20.