Difference between revisions of "Astrophysical black holes"

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(Problem 16.)
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nothing here yet </p>
 
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=== Problem 21. ===
 
=== Problem 21. ===
 
AGNs remain active for more than tens of millions of years. They must have tremendous masses to maintain the luminosity
 
AGNs remain active for more than tens of millions of years. They must have tremendous masses to maintain the luminosity
\[L\sim {{10}^{47}}\text{erg/sec}\]
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\[L\sim {{10}^{47}}\mathit{erg/sec}\]
 
during such periods. Make estimates for the mass of an AGN.
 
during such periods. Make estimates for the mass of an AGN.
 
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no hints here either </p>
 
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todo </p>
 
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Using the explicit form of the Kerr-Newman metric, in the same way as was computed for the Kerr case (problem \ref{BlackHole61}), one eventually obtains
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Using the explicit form of the Kerr-Newman metric, in the same way as was computed for the Kerr case (problem [[Kerr_black_hole#BlackHole61]]), one eventually obtains
 
\[A = 4\pi \left( {2{M^2} - {Q^2} + 2M\sqrt {{M^2} - {Q^2} - {J^2}/{M^2}} } \right).\] </p>
 
\[A = 4\pi \left( {2{M^2} - {Q^2} + 2M\sqrt {{M^2} - {Q^2} - {J^2}/{M^2}} } \right).\] </p>
 
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==Quantum effects==
 
==Quantum effects==
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<div id="BlackHoleQ1"></div>
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=== Problem 27. ===
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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.
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Maximum density corresponds to the minimum possible mass:
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\[\rho _{\max } = \frac{M}{\tfrac{4}{3}\pi r_g^3}
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\approx 2 \times {10^{16}}\left( {\frac{M_\odot}{M_{\min}}\right)^2}\frac{g}{{cm}^3}
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\sim {10^{52}}\frac{\mathit{g}}{\mathit{cm^3}}.\] </p>
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  </div>
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</div>
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<div id="BlackHoleQ2"></div>
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=== Problem 28. ===
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Determine the lifetime of a black hole with respect to thermal radiation.
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Hawking has shown that a black hole radiates as a black body with surface area $A$ and temperature $T_H$ given by
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\[{T_H} = \frac{{\hbar {c^3}}}{{8\pi GM{k_B}}},
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\quad A = 4\pi r_g^2.\]
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Consequently
 +
\[\frac{dM}{dt}
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= \frac{\sigma T_H^4 \times 4\pi r_g^2}{c^2}
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= \frac{1}{512}\frac{\hbar c^4}{G^2}
 +
\frac{1}{M^2}.\]
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For simplicity we have omitted the factor depending on the number of states of particles that are radiated.  Then for the lifetime of black hole we obtain
 +
\[{T_H} \approx 512\pi
 +
\frac{G^2}{\hbar c^4}{M^3}.\] </p>
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  </div>
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</div>
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<div id="BlackHoleQ3"></div>
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=== Problem 29. ===
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Determine the temperature of a black hole (Hawking temperature) of one solar mass, and the temperature of the supermassive black hole at the center of our Galaxy.
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Using the general formula from the previous problem,
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\[{T_{ \odot H}}
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\approx 6 \times {10^{ - 7}}K,\quad {T_{HG}}
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\approx 2 \times {10^{ - 13}}K.\] </p>
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  </div>
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</div>
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<div id="BlackHoleQ4"></div>
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=== Problem 30. ===
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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 $M_{cr}(m)$.
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A particle can be emitted only if its reduced Compton wavelength is greater than the Schwarzschild radius of the black hole. Hence
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\[\frac{\hbar }{{mc}} > \frac{{2MG}}{{{c^2}}}
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\quad\Rightarrow\quad
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M_{cr} = \frac{M_{Pl}^2}{2m}.\]
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For instance, thermal emission of electrons and positrons is possible only when $M < {10^{16}}\;kg$ (which is $16$ orders of magnitude smaller than the mass of the Sun). </p>
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Revision as of 12:43, 23 July 2012

Preliminary

Problem 1.

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

Problem 2.

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

Problem 3.

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

Problem 4.

Estimate the radius and mass of a white dwarf.

Problem 5.

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.

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

Problem 7.

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.

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.

Estimate the radius and mass of a neutron star.

Problem 10.

Why do neutron stars have to possess strong magnetic fields?

Problem 11.

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

Problem 12.

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

Problem 13.

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.

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.

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

Problem 16.

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.

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

Problem 18.

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.

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.

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.

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.

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

Problem 23.

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

Problem 24.

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.

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.

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

Quantum effects

Problem 27.

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.

Problem 28.

Determine the lifetime of a black hole with respect to thermal radiation.

Problem 29.

Determine the temperature of a black hole (Hawking temperature) of one solar mass, and the temperature of the supermassive black hole at the center of our Galaxy.

Problem 30.

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 $M_{cr}(m)$.