Difference between revisions of "Dark Matter Halo"

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(Problem 4)
(Problem 5)
 
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
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     <p style="text-align: left;">\[\rho \left( r \right) = \frac{v_\infty ^2}{4\pi G}\frac{1}{r_c^2  + r^2} ;\quad v(r) = v_\infty  \sqrt{ 1 - \frac{r_c }{r} \arctg {\frac{r}{r_c}}}  \]
+
     <p style="text-align: left;">\[\rho \left( r \right) = \frac{v_\infty ^2}{4\pi G}\frac{1}{r_c^2  + r^2} ;\quad v(r) = v_\infty  \sqrt{ 1 - \frac{r_c }{r} \arctan{\frac{r}{r_c}}}  \]
  
 
For $v_\infty  = 220\mbox{km/s},\;r_c  = 2.6\mbox{kpc},\,r_0 = 8\mbox{kpc}$
 
For $v_\infty  = 220\mbox{km/s},\;r_c  = 2.6\mbox{kpc},\,r_0 = 8\mbox{kpc}$
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M(r)=-\frac{rkT}{Gm_p}\frac{d\ln\rho}{d\ln r}.
 
M(r)=-\frac{rkT}{Gm_p}\frac{d\ln\rho}{d\ln r}.
 
$$
 
$$
Using results of the problem \ref{DM16}
+
Using results of the [[#DM16|problem]]
 
$$
 
$$
\rho(r)  = \frac{{v_\infty ^2 }}{{4\pi G\left( {r_c^2  + r }
+
\rho(r)  = \frac{v_\infty ^2}{4\pi G\left( {r_c^2  + r }\right)},
\right)}},
+
 
$$
 
$$
 
one obtains
 
one obtains

Latest revision as of 10:13, 4 October 2012




Problem 1

Estimate the local density of the dark halo in the vicinity of the Earth, assuming that its density decreases as $\rho_g=C/r^2$.


Problem 2

Build the model of the spherically symmetric dark halo density corresponding to the observed galactic rotation curves.


Problem 3

In frames of the halo model considered in the previous problem determine the local dark matter density $\rho_0$ basing on the given rotation velocities of satellite galaxies at the outer border of the halo $v_\infty\equiv v(r\rightarrow\infty)$ and in some point $r_0$.


Problem 4

For the halo model considered in problem about halo model obtain the dependencies $\rho(r)$ and $v(r)$ in terms of $\rho_0$ and $v_\infty$. Plot the dependencies $\rho(r)$ and $v(r)$.


Problem 5

Many clusters are sources of X-ray radiation. It is emitted by the hot intergalactic gas filling the cluster volume. Assuming that the hot gas ($kT\approx10keV$) is in equilibrium in the cluster with linear size $R=2.5Mpc$ and core radius $r_c=0.25Mpc$, estimate the mass of the cluster.