Difference between revisions of "Astrophysical black holes"

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[[Category:Black Holes|5]]
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__TOC__
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==Preliminary==
 
==Preliminary==
 
<div id="BlackHole88"></div>
 
<div id="BlackHole88"></div>
=== Problem 1. ===
+
=== Problem 1: Newtonian collapse time ===
 
Calculate in the frame of Newtonian mechanics the time of collapse of a uniformly distributed spherical mass with density $\rho_0$.
 
Calculate in the frame of Newtonian mechanics the time of collapse of a uniformly distributed spherical mass with density $\rho_0$.
 
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<div class="NavFrame collapsed">
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<div id="BlackHole89"></div>
 
<div id="BlackHole89"></div>
=== Problem 2. ===
+
 
 +
=== Problem 2: enter white dwarfs: why not green giants? ===
 
Why are stars of a certain type called "white dwarfs"?
 
Why are stars of a certain type called "white dwarfs"?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole90"></div>
 
<div id="BlackHole90"></div>
 
+
=== Problem 3: nature of WDs ===
=== Problem 3. ===
+
 
What is the physical reason for the stopping of thermonuclear reactions in the stars of white dwarf type?
 
What is the physical reason for the stopping of thermonuclear reactions in the stars of white dwarf type?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole91"></div>
 
<div id="BlackHole91"></div>
=== Problem 4. ===
+
=== Problem 4: averages ===
 
Estimate the radius and mass of a white dwarf.
 
Estimate the radius and mass of a white dwarf.
 
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<div class="NavFrame collapsed">
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<div id="BlackHole92"></div>
 
<div id="BlackHole92"></div>
  
=== Problem 5. ===
+
=== 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?
 
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?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole93"></div>
 
<div id="BlackHole93"></div>
  
=== Problem 6. ===
+
=== Problem 6: explosion! ===
 
Explain the mechanism of explosion of massive enough white dwarfs, with masses close to the Chandrasekhar limit.
 
Explain the mechanism of explosion of massive enough white dwarfs, with masses close to the Chandrasekhar limit.
 
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<div class="NavFrame collapsed">
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<div id="BlackHole94"></div>
 
<div id="BlackHole94"></div>
=== Problem 7. ===
+
=== 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?
 
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?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole95"></div>
 
<div id="BlackHole95"></div>
  
=== Problem 8. ===
+
=== 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?
 
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?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole96"></div>
 
<div id="BlackHole96"></div>
=== Problem 9. ===
+
=== Problem 9: neutron star enters ===
 
Estimate the radius and mass of a neutron star.
 
Estimate the radius and mass of a neutron star.
 
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<div class="NavFrame collapsed">
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<div id="BlackHole97"></div>
 
<div id="BlackHole97"></div>
  
=== Problem 10. ===
+
=== Problem 10: magnetic fields in NS ===
 
Why do neutron stars have to possess strong magnetic fields?
 
Why do neutron stars have to possess strong magnetic fields?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole98"></div>
 
<div id="BlackHole98"></div>
=== Problem 11. ===
+
=== Problem 11: redshift ===
 
Find the maximum redshift of a spectral line emitted from the surface of a neutron star.
 
Find the maximum redshift of a spectral line emitted from the surface of a neutron star.
 
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<div class="NavFrame collapsed">
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<div id="BlackHole99"></div>
 
<div id="BlackHole99"></div>
=== Problem 12. ===
+
=== Problem 12: gravitational radius of the Universe ===
 
What is the gravitational radius of the Universe? Compare it with the size of the observable Universe.
 
What is the gravitational radius of the Universe? Compare it with the size of the observable Universe.
 
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<div class="NavFrame collapsed">
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<div id="BlackHole100"></div>
 
<div id="BlackHole100"></div>
  
=== Problem 13. ===
+
=== 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?
 
What is the time (the Salpeter time) needed for a black hole, radiating at its Eddington limit, to radiate away all of its mass?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole101"></div>
 
<div id="BlackHole101"></div>
=== Problem 14. ===
+
=== 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.
 
The time scales of radiation variability of active galactic nuclei (AGNs) are from several days to several years. Estimate the linear sizes of AGNs.
 
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<div class="NavFrame collapsed">
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<div id="BlackHole102"></div>
 
<div id="BlackHole102"></div>
=== Problem 15. ===
+
=== 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?
 
What mechanisms can be responsible for the supermassive black hole (SMBH) in the center of a galaxy to acquire angular momentum?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole103"></div>
 
<div id="BlackHole103"></div>
=== Problem 16. ===
+
=== 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.
 
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.
 
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<div class="NavFrame collapsed">
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<div id="BlackHole104"></div>
 
<div id="BlackHole104"></div>
  
=== Problem 17. ===
+
=== 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.
 
Using the results of the previous problem, determine the density of the SMBH at the Galactic Center.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
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<div id="BlackHole105"></div>
 
<div id="BlackHole105"></div>
=== Problem 18. ===
+
=== 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
 
Show that for a black hole of mass $M$ the temperature of the surrounding hot gas in thermal equilibrium is proportional to
 
\[T\sim {{M}^{-1/4}}.\]
 
\[T\sim {{M}^{-1/4}}.\]
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<div id="BlackHole106"></div>
 
<div id="BlackHole106"></div>
=== Problem 19. ===
+
=== 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.
 
Show that luminosity of a compact object (neutron star or black hole) of several solar masses is mostly realized in the X-rays.
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
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<div id="BlackHole107"></div>
 
<div id="BlackHole107"></div>
=== Problem 20. ===
+
=== Problem 20: masses of AGNs ===
 
In order to remain bound while subject to the rebound from gigantic radiative power,
 
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.
 
AGNs should have masses $M>{{10}^{6}}{{M}_{\odot }}$. Make estimates.
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   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
 
     <p style="text-align: left;">
 
     <p style="text-align: left;">
problem solution </p>
+
nothing here yet </p>
 
   </div>
 
   </div>
 
</div>
 
</div>
  
 
<div id="BlackHole108"></div>
 
<div id="BlackHole108"></div>
=== Problem 21. ===
+
=== Problem 21: another estimate ===
 
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
\[L\sim {{10}^{47}}\text{erg/sec}\]
+
\[L\sim {{10}^{47}}\mathit{erg/sec}\]
 
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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   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
 
     <p style="text-align: left;">
 
     <p style="text-align: left;">
problem solution </p>
+
no hints here either </p>
 
   </div>
 
   </div>
 
</div>
 
</div>
  
 
<div id="BlackHole109"></div>
 
<div id="BlackHole109"></div>
=== Problem 22. ===
+
=== Problem 22: energy release at BH merger. ===
 
What maximum energy can be released at the merger of two black holes with masses ${{M}_{1}}={{M}_{2}}=\frac{M}{2}$?
 
What maximum energy can be released at the merger of two black holes with masses ${{M}_{1}}={{M}_{2}}=\frac{M}{2}$?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole110"></div>
 
<div id="BlackHole110"></div>
=== Problem 23. ===
+
=== Problem 23: BH do not multiply by fission ===
 
Show that it is impossible to divide a black hole into two black holes.
 
Show that it is impossible to divide a black hole into two black holes.
 
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   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
 
     <p style="text-align: left;">
 
     <p style="text-align: left;">
problem solution </p>
+
todo </p>
 
   </div>
 
   </div>
 
</div>
 
</div>
  
 
<div id="BlackHole111"></div>
 
<div id="BlackHole111"></div>
=== Problem 24. ===
+
=== Problem 24: entropy paradox ===
 
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?
 
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?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole112"></div>
 
<div id="BlackHole112"></div>
=== Problem 25. ===
+
=== 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?
 
What is the reason we cannot attribute the observed entropy's decrease (see the previous problem) to the interior of the black hole?
 
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<div class="NavFrame collapsed">
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<div id="BlackHole113"></div>
 
<div id="BlackHole113"></div>
=== Problem 26. ===
+
=== Problem 26: Smarr formula ===
 
Find the surface area of a stationary black hole as a function of its parameters: mass, angular momentum and charge.
 
Find the surface area of a stationary black hole as a function of its parameters: mass, angular momentum and charge.
 
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   <div style="width:100%;" class="NavContent">
 
     <p style="text-align: left;">
 
     <p style="text-align: left;">
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
+
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>
 
   </div>
 
   </div>
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==Quantum effects==
 
==Quantum effects==
 +
<div id="BlackHoleQ1"></div>
 +
=== 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.
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
Maximum density corresponds to the minimum possible mass:
 +
\[\rho _{\max } = \frac{M}{\tfrac{4}{3}\pi r_g^3}
 +
\approx 2 \times {10}^{16} \left(\frac{M_\odot}{M_{\min}}\right)^2 \frac{g}{{cm}^3}
 +
\sim {10^{52}}\frac{\mathit{g}}{\mathit{cm^3}}.\] </p>
 +
  </div>
 +
</div>
 +
 +
<div id="BlackHoleQ2"></div>
 +
=== Problem 28: BH lifetime ===
 +
Determine the lifetime of a black hole with respect to thermal radiation.
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
Hawking has shown that a black hole radiates as a black body with surface area $A$ and temperature $T_H$ given by
 +
\[{T_H} = \frac{{\hbar {c^3}}}{{8\pi GM{k_B}}},
 +
\quad A = 4\pi r_g^2.\]
 +
Consequently
 +
\[\frac{dM}{dt}
 +
= \frac{\sigma T_H^4 \times 4\pi r_g^2}{c^2}
 +
= \frac{1}{512}\frac{\hbar c^4}{G^2}
 +
\frac{1}{M^2}.\]
 +
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>
 +
  </div>
 +
</div>
 +
 +
<div id="BlackHoleQ3"></div>
 +
=== Problem 29: typical temperatures ===
 +
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.
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
Using the general formula from the previous problem,
 +
\[{T_{ \odot H}}
 +
\approx 6 \times {10^{ - 7}}K,\quad {T_{HG}}
 +
\approx 2 \times {10^{ - 13}}K.\] </p>
 +
  </div>
 +
</div>
 +
 +
<div id="BlackHoleQ4"></div>
 +
=== Problem 30: particle production limit ===
 +
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 $M_{cr}(m)$.
 +
<div class="NavFrame collapsed">
 +
  <div class="NavHead">solution</div>
 +
  <div style="width:100%;" class="NavContent">
 +
    <p style="text-align: left;">
 +
A particle can be emitted only if its reduced Compton wavelength is greater than the Schwarzschild radius of the black hole. Hence
 +
\[\frac{\hbar }{mc} > \frac{2MG}{c^2}
 +
\quad\Rightarrow\quad
 +
M_{cr} = \frac{M_{Pl}^2}{2m}.\]
 +
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>
 +
  </div>
 +
</div>

Latest revision as of 08:26, 1 May 2013

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.

Problem 24: entropy paradox

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.

Problem 28: BH lifetime

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

Problem 29: typical temperatures

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: particle production limit

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 $M_{cr}(m)$.