Please show your working on this sheet, and write your answers on this sheet. Attach extra sheets if you need them.
(a) Schwarzschild radius
The black holes which power quasars have masses typically about 108 Msun. The Schwarzschild radius R of a black hole is proportional to its mass M, with the Schwarzschild radius of a 1 Msun object being 3 km. What is the Schwarzschild radius of a 108 Msun black hole? Express your answer in km, and then in Astronomical Units (1 AU = 1.5 ×108 km).
The Schwarzschild radius of a 108 Msun black hole is ______________________ km = ______________________ AU.
(b) How long to fall in?
If you fell (woops) through the Schwarzschild radius of a 108 Msun 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 ×108 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 Msun black hole is about 109 g (a giga-gee). What is the tidal force at the Schwarzschild radius of a 108 Msun black hole? Would you get ripped apart while entering the Schwarzschild radius of a 108 Msun 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/R3. Remember that the Schwarzschild radius R is proportional to M.]
The tidal force between head and toes at the Schwarzschild radius of a 108 Msun is ______________________ g.
I would/would not be ripped apart at the Schwarzschild radius of a 108 Msun 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 lpeak of the Planck spectrum is equal to the Schwarzschild radius R of the black hole. Using Wien's Law, lpeak = 0.0029 m K / T, determine the Hawking temperature of a 1 Msun black hole.
The Hawking temperature of a 1 Msun black hole is ______________________ K.
(b) Hawking Luminosity
Would the Hawking luminosity of a 1 Msun black hole be more or less than the luminosity of the Sun? Why? [Hint: The Stefan-Boltzmann law L µ R2 T4 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.