Estimate the mass of the Coma cluster

[leonne]

Homework Statement

Use the results we derived in class based on the virial theorem to estimate the mass of
the Coma cluster of galaxies, which has a radial velocity dispersion of o = 977 km s1
and a radius R = 3 Mpc. You can assume an isothermal sphere model for the mass
distribution and that the measured radial velocity dispersion is based on isotropic
orbits of identical galaxies.

Homework Equations

M=3B/n (Ro²/G

The Attempt at a Solution

I found this formula in my notes said it was the virial mass estimate so would i just plug in what i have into this? Seems too easy that's why I am asking if i have to do something else.
B and n would equal to 1 because its a isothermal sphere

 

[zhermes]

The answer should be very simple... this is not how I would have done it however.
What are B and n?

I would just use the virial theorem... what does the virial theorem state?

 

[leonne]

Hey in the class notes they did use virial theorem and than sub potential and kinetic to get that formula i said idk what B and N are didnt fully read the notes but said when its a isothermal sphere that b=n=1

 

[zhermes]

Okay, well in general the virial theorem let's you relate the time average kinetic energy to the potential energy. For the case of gravity it says that:
$$<T> = -\frac{1}{2} <U>$$
So if you plug in equations for kinetic and potential energies... what do you get?

 

[leonne]

well you get 3Bmo-nGM²/R=0
Was told that kinetic energy is 3/2BMo²
"We have made two assumptions: the motion is isotropic, and the stars are identical. Dropping
those assumptions... where B is a dimensionless factor of order unity (e.g., b = 1 if we are considering isotropic orbits of identical particles).
then potentail is -n(GM² /R)
where n is a dimensionless factor of order unity that depends on the density profile (e.g.,
n = 1 for an isothermal sphere, and n = 3/5 for a constant density sphere)

So from 3Bmo-nGM²/R=0 just solve for M and you get 1/m=nG/3o²RB
than just flip it to gte m=3BRo²/nG

 

[leonne]

got the mass as 1.997e15 solar mass which seems right it also asks about .."if you used a constant density model" If the mass will increase or what looking in the note found that ... n = 3/5 for a constant density sphere so i am guess just replace the 1 with 3/5 in the formula i got before right? or is there something more?

 

[zhermes]

This isn't quite the virial theorem I'm used to... but it sounds like you're doing it exactly right---based on the prescription you've been given. The answer shouldn't be any more complicated.

 

[leonne]

kk thxs for the help