Astrophysical black holes - Revision history

Igor at 08:26, 1 May 2013

2013-05-01T08:26:27Z

2012-10-13T17:26:08Z

‎Problem 1: Newtonian collapse time.

2012-10-13T17:25:41Z

2012-07-24T20:54:25Z

2012-07-23T21:55:52Z

2012-07-23T13:00:09Z

2012-07-23T12:44:49Z

‎Quantum effects

2012-07-23T12:43:00Z

2012-07-23T12:33:57Z

‎Problem 16.

2012-07-23T12:32:59Z

‎Problem 12.

@@ Line 4: / Line 4: @@
 ==Preliminary==
 <div id="BlackHole88"></div>
-=== Problem 1: Newtonian collapse time. ===
+=== Problem 1: Newtonian collapse time ===
 Calculate in the frame of Newtonian mechanics the time of collapse of a uniformly distributed spherical mass with density $\rho_0$.
 <div class="NavFrame collapsed">
@@ Line 42: / Line 42: @@
 <div id="BlackHole89"></div>
 === Problem 2: enter white dwarfs: why not green giants? ===
 Why are stars of a certain type called "white dwarfs"?

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 Maximum density corresponds to the minimum possible mass:
 \[\rho _{\max } = \frac{M}{\tfrac{4}{3}\pi r_g^3}
-  \approx 2 \times {10^{16}}\left( {\frac{M_\odot}{M_{\min}}\right)^2}\frac{g}{{cm}^3}
+  \approx 2 \times {10}^{16} \left(\frac{M_\odot}{M_{\min}}\right)^2 \frac{g}{{cm}^3}
 	\sim {10^{52}}\frac{\mathit{g}}{\mathit{cm^3}}.\]	</p>
    </div>
@@ Line 492: / Line 492: @@
      <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}}}
+\[\frac{\hbar }{mc} > \frac{2MG}{c^2}
 	\quad\Rightarrow\quad
 	M_{cr} = \frac{M_{Pl}^2}{2m}.\]

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    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">
-problem solution	</p>
+nothing here yet	</p>
    </div>
 </div>
@@ Line 363: / Line 363: @@
 === Problem 21. ===
 AGNs remain active for more than tens of millions of years. They must have tremendous masses to maintain the luminosity
-\[L\sim {{10}^{47}}\text{erg/sec}\]
+\[L\sim {{10}^{47}}\mathit{erg/sec}\]
 during such periods. Make estimates for the mass of an AGN.
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@@ Line 369: / Line 369: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">
-problem solution	</p>
+no hints here either	</p>
    </div>
 </div>
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    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">
-problem solution	</p>
+todo	</p>
    </div>
 </div>
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    <div style="width:100%;" class="NavContent">
      <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>
    </div>
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 ==Quantum effects==

@@ Line 286: / Line 286: @@
      <p style="text-align: left;">
 According to the third Kepler law,
-\[M = \frac{{{a^3}}}{{{T^2}}}\frac{{4{\pi ^2}}}{G}.\]
+\[M = \frac{a^3}{T^2}\frac{4\pi^2}{G}.\]
 Here $a$ is the major semiaxis of the ellipse, related to the minimal distance to the black hole though
-\[a = \frac{{1 + e}}{{1 - {e^2}}}{r_{\min }}.\]
+\[a = \frac{1 + e}{1 - e^2}{r_{\min }}.\]
 Then
 \[{M_{SMBH}} \approx 3 \times {10^6}{M_ \odot }.\]	</p>
@@ Line 295: / Line 295: @@
 <div id="BlackHole104"></div>
 === Problem 17. ===
 Using the results of the previous problem, determine the density of the SMBH at the Galactic Center.

@@ Line 221: / Line 221: @@
 &{r_{gVU}} = 2G{M_{VU}},\qquad
 {R_{VU}} = {H^{ - 1}};\\
-&{M_{VU}} = \frac{{4\pi }}{3}R_{VU}^3\rho ;\\
+&{M_{VU}} = \frac{4\pi }{3}R_{VU}^3\rho ;\\
-&{H^2} = \frac{{8\pi G}}{3}\rho
+&{H^2} = \frac{8\pi G}{3}\rho
 	\quad\Rightarrow\quad
-	\rho  = \frac{{3{H^2}}}{{8\pi G}};\\
+	\rho  = \frac{3{H^2}}{8\pi G};\\
 &{M_{VU}}
-	= \frac{{4\pi }}{3}R_{VU}^3\frac{{3{H^2}}}{{8\pi G}}
+	= \frac{4\pi }{3}R_{VU}^3\frac{3{H^2}}{8\pi G}
-	= \frac{{{R_{VU}}}}{{2G}};\\
+	= \frac{R_{VU}}{2G};\\
 &{R_{VU}} = 2G{M_{VU}} = {r_{gVU}}.
 \end{align*}	</p>
@@ Line 234: / Line 234: @@
 <div id="BlackHole100"></div>
 === 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?