<p>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.</p>

CC BY-SA 3.0
2015-08-21T02:15:10.433Z
2015-08-21T02:15:10.433Z
Stan Liou
<p>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 $H_0=70 km s^{-1} Mpc^{-1}$. </p>

<p>Taking the Virgo cluster to be spherically symmetric with a radial density profile given by </p>

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

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

<hr>

<p>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?</p>

CC BY-SA 3.0
2015-08-21T11:32:46.333Z
I'm voting to close this question as off-topic because it looks like your homework.
ProfRob
CC BY-SA 3.0
2015-08-21T15:41:03.230Z
@RobJeffries Hmm, I don't think this site has a coherent homework policy, but it should probably get one at some point.
Stan Liou
CC BY-SA 3.0
2015-08-21T19:01:41.313Z
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
mnmakrets
CC BY-SA 3.0
2015-08-21T19:53:35.773Z
@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.
ProfRob
CC BY-SA 3.0
2015-08-21T20:06:16.427Z
@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](http://meta.astronomy.stackexchange.com/questions/87/homework-policy), and this has moved on a bit since then.
Stan Liou
<galaxy><galaxy-cluster><density>
Radial Density Profile Equation
mnmakrets
# Radial Density Profile Equation

(Asked by: username_0 on 2015-08-21)

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.

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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?

**Tags:** galaxy, galaxy-cluster, density

> **Comments:**
> **username_1**: I'm voting to close this question as off-topic because it looks like your homework.  
> **username_2**: @RobJeffries Hmm, I don't think this site has a coherent homework policy, but it should probably get one at some point.  
> **username_0**: 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  
> **username_1**: @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.  
> **username_2**: @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](http://meta.astronomy.stackexchange.com/questions/87/homework-policy), and this has moved on a bit since then.  

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## 1 answers

### Answer by username_2 on 2015-08-21. Score: 0 | Accepted answer

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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