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The Virgo Galaxy Cluster has a mass of $10^{14} M_{\odot}$ and its centre is $16Mpc$ from Earth. The large elliptical galaxy $M87$ lies at the centre of the Virgo cluster. $M87$ has a supermassive black hole at it's centre with an estimated mass of $6 \times 10^9 M_{\odot}$. Take Hubble's constant to be $H_0=70 km s^{-1} Mpc^{-1}$.

Taking the Virgo cluster to be spherically symmetric with a radial density profile given by

$\rho(r)=\rho_0 (\frac{r}{1Mpc})^{-2}$,

Determine the value of the constant $\rho_0$ is S.I units assuming the radius of the Virgo cluster is 1Mpc.


I am confused with how to approach this question, I know that density $\rho=\frac{M}{\frac{4}{3} \pi r^3}$, when I substitute it into the given radial density profile, the $r$ variable doesn't cancel, should I substitute the radius of the cluster into $r$? Is it really that straight forward?

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    $\begingroup$ I'm voting to close this question as off-topic because it looks like your homework. $\endgroup$ – ProfRob Aug 21 '15 at 11:32
  • $\begingroup$ @RobJeffries Hmm, I don't think this site has a coherent homework policy, but it should probably get one at some point. $\endgroup$ – Stan Liou Aug 21 '15 at 15:41
  • $\begingroup$ This is actually a question from one of the previous years exam papers for one of my university modules, I am not assessed on it in any way. I was just looking for help on it to further my knowledge on the subject. @RobJeffries $\endgroup$ – mnmakrets Aug 21 '15 at 19:01
  • $\begingroup$ @StanLiou As someone that frequently sets such basic questions for year 1 ug students, it's my policy. What bit of astronomy/astrophysics is this asking about? It could be any density distribution for any spherical object. $\endgroup$ – ProfRob Aug 21 '15 at 19:53
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    $\begingroup$ @RobJeffries I'm not dissenting about moving to close, but I am noting that this site does need to make it clear how homework questions are to be received. I have no problem with your policy being adopted by this site, since the last time such things were considered was two years ago, and this has moved on a bit since then. $\endgroup$ – Stan Liou Aug 21 '15 at 20:06
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If the density is radially symmetric and given by $\rho = \rho(r)$, then a shell of radius $r$ and infinitesimal thickness $\mathrm{d}r$ has volume $\mathrm{d}V = 4\pi r^2\, \mathrm{d}r$, and therefore a mass of $\mathrm{d}M = \rho\,\mathrm{d}V$. The total mass enclosed by some radius $R$ is of course the sum of the masses of all shells up to that radius. I think this should enable you to finish the problem yourself.

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