Coming up with the critical density formula

[makosheva7]

Homework Statement

"Imagine a galaxy with mass m at a distance r away from the center of a sphere, within
which a total mass M reside. As viewed by an observer in the center,
the galaxy appears to be receding according to the Hubble's law, v = H0r. To heuristically derive the critical density m0, we associate a kinetic energy to the galaxy's Hubble
ow, symbolically,1/2 mv^2, and balance this against its gravitational energy. If the density in the sphere is equal to the critical density, the total binding energy (kinetic plus gravitational) is zero. Show that this yields the following expression for the critical density."

Homework Equations

What am I doing wrong in my process to find the critical density formula based on the given information?

The Attempt at a Solution

My logic is:
1/2(M-m)v^2 = GMm/r
So using v = H0r and the volume of a sphere, I plug density(volume of sphere) into M-m, and H0r into v.
after doing this, my solution looks like this.
density = GMm/2∏r^6H0^2
I can't figure out the next step.

 

[Dick]

Homework Statement

"Imagine a galaxy with mass m at a distance r away from the center of a sphere, within
which a total mass M reside. As viewed by an observer in the center,
the galaxy appears to be receding according to the Hubble's law, v = H0r. To heuristically derive the critical density m0, we associate a kinetic energy to the galaxy's Hubble
ow, symbolically,1/2 mv^2, and balance this against its gravitational energy. If the density in the sphere is equal to the critical density, the total binding energy (kinetic plus gravitational) is zero. Show that this yields the following expression for the critical density."

Homework Equations

What am I doing wrong in my process to find the critical density formula based on the given information?

The Attempt at a Solution

My logic is:
1/2(M-m)v^2 = GMm/r
So using v = H0r and the volume of a sphere, I plug density(volume of sphere) into M-m, and H0r into v.
after doing this, my solution looks like this.
density = GMm/2∏r^6H0^2
I can't figure out the next step.

For one thing the kinetic energy of the galaxy is mv^2/2. I don't know why you are using (M-m). For another I would plug density(volume of sphere) into M. After that you really need to straighten up your algebra. Show you you got what you got.