# Astrophysical black holes

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## Preliminary

### Problem 1: Newtonian collapse time

Calculate in the frame of Newtonian mechanics the time of collapse of a uniformly distributed spherical mass with density $\rho_0$.

### Problem 2: enter white dwarfs: why not green giants?

Why are stars of a certain type called "white dwarfs"?

### Problem 3: nature of WDs

What is the physical reason for the stopping of thermonuclear reactions in the stars of white dwarf type?

### Problem 4: averages

Estimate the radius and mass of a white dwarf.

### Problem 5: typical example

What is the average density of a white dwarf of one solar mass, luminosity one thousandth of solar luminosity and surface temperature twice that of the Sun?

### Problem 6: explosion!

Explain the mechanism of explosion of massive enough white dwarfs, with masses close to the Chandrasekhar limit.

### Problem 7: no hydrogen in type I SNe.

Thermonuclear explosions of white dwarfs with masses close to the Chandrasekhar limit lead to the phenomenon of supernova explosions of type I. Those have lines of helium and other relatively heavy elements in the spectrum, but no hydrogen lines. Why is that?

### Problem 8: SNe of type II

A supernova explosion of type II is related to the gravitational collapse of a neutron star. There are powerful hydrogen lines in their spectrum. Why?

### Problem 9: neutron star enters

Estimate the radius and mass of a neutron star.

### Problem 10: magnetic fields in NS

Why do neutron stars have to possess strong magnetic fields?

### Problem 11: redshift

Find the maximum redshift of a spectral line emitted from the surface of a neutron star.

### Problem 12: gravitational radius of the Universe

What is the gravitational radius of the Universe? Compare it with the size of the observable Universe.

### Problem 13: Salpeter time

What is the time (the Salpeter time) needed for a black hole, radiating at its Eddington limit, to radiate away all of its mass?

### Problem 14: AGN typical size

The time scales of radiation variability of active galactic nuclei (AGNs) are from several days to several years. Estimate the linear sizes of AGNs.

### Problem 15: rotation of SMBHs

What mechanisms can be responsible for the supermassive black hole (SMBH) in the center of a galaxy to acquire angular momentum?

### Problem 16: the mass of Milky Way's SMBH

The Galactic Center is so "close" to us, that one can discern individual stars there and examine in detail their movement. Thus, observations carried out in 1992-2002 allowed one to reconstruct the orbit of motion of one of the stars (S2) around the hypothetical SMBH at the galactic center of the Milky Way. The parameters of the orbit are: period $15.2$ years, maximum distance from the black hole $120$ a.u., eccentricity $0.87$. Using this data, estimate the mass of the black hole.

### Problem 17: BHs are not necessarily "very dense"

Using the results of the previous problem, determine the density of the SMBH at the Galactic Center.

### Problem 18: BH in equilibrium with gas.

Show that for a black hole of mass $M$ the temperature of the surrounding hot gas in thermal equilibrium is proportional to $T\sim {{M}^{-1/4}}.$

### Problem 19: why X-rays

Show that luminosity of a compact object (neutron star or black hole) of several solar masses is mostly realized in the X-rays.

### Problem 20: masses of AGNs

In order to remain bound while subject to the rebound from gigantic radiative power, AGNs should have masses $M>{{10}^{6}}{{M}_{\odot }}$. Make estimates.

### Problem 21: another estimate

AGNs remain active for more than tens of millions of years. They must have tremendous masses to maintain the luminosity $L\sim {{10}^{47}}\mathit{erg/sec}$ during such periods. Make estimates for the mass of an AGN.

### Problem 22: energy release at BH merger.

What maximum energy can be released at the merger of two black holes with masses ${{M}_{1}}={{M}_{2}}=\frac{M}{2}$?

### Problem 23: BH do not multiply by fission

Show that it is impossible to divide a black hole into two black holes.

J. Wheeler noticed that in the frame of classical theory of gravity the existence of black holes itself contradicts the law of entropy's increase. Why is that?

### Problem 25: entropy of the interior?

What is the reason we cannot attribute the observed entropy's decrease (see the previous problem) to the interior of the black hole?

### Problem 26: Smarr formula

Find the surface area of a stationary black hole as a function of its parameters: mass, angular momentum and charge.

## Quantum effects

### Problem 27: densities of realistic BHs.

Estimate the maximum density of an astrophysical black hole, taking into account that black holes with masses $M<{{10}^{15}}g$ would not have lived to our time due to the quantum mechanism of evaporation.

Particles and antiparticles of given mass $m$ (neutrinos, electrons and so on) can be emitted only if the mass $M$ of the black hole is less than some critical mass ${M}_{cr}$. Estimate the critical mass of a black hole $M_{cr}(m)$.