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Coming up with the critical density formula

  1. Feb 28, 2013 #1
    1. The problem statement, all variables and given/known data
    "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."

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

    3. 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.
  2. jcsd
  3. Feb 28, 2013 #2


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    Science Advisor
    Homework Helper

    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.
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