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Astrophysics: pressure in a supernova as in explodes

  1. Mar 17, 2012 #1
    1. The problem statement, all variables and given/known data
    In its early phase, the supernova is an explosion within the star’s own envelope; about half the energy is thermal and half is kinetic. Estimate the ratio of radiation pressure to gas pressure in the star at the moment the shock reaches the surface. (Not in the pre-explosion star!)
    Known values: radius = 50R⊙, M = 16M⊙, energy of the explosion = [itex]10^{51}[/itex] ergs, about M = 14M⊙ are ejected; the rest collapse.

    2. Relevant equations
    The equations for pressures that I know are:

    [itex]P_{rad}[/itex] = a/3 * [itex]T^{4}[/itex]
    [itex]P_{gas}[/itex] = ρ/μ * kT

    Where μ is mean molecular mass.

    I also have [itex]E_{thermal}[/itex] = 3/2 M/μ kT (where M/μ is to estimate the number of particles).

    3. The attempt at a solution


    I have substituted mass/volume for ρ and divided the two equations by each other getting:
    [itex]P_{rad}[/itex]/[itex]P_{gas}[/itex] = 4πaμ[itex]T^{3}[/itex][itex]R^{3}[/itex] / (9Mk)

    Evaluating it all I get the ratio to be 3.23 * [itex]10^{-11}[/itex] μ[itex]T^{3}[/itex] (with SI units used throughout, not cgs).

    If I use E = [itex]10^{51}[/itex] ergs = [itex]10^{44}[/itex] J given in the setup of the question and equate half of that to [itex]E_{thermal}[/itex] to try to get the value of T, I come up with some terribly small estimates, like [itex]10^{-11}[/itex]K. This is obviously wrong, but I don't quite get what should I change to get the correct result.

    Any pointers appreciated.
     
  2. jcsd
  3. Mar 19, 2012 #2
    I think you are just calculating it wrong. Lets do a ballpark approximation.

    The number of particles N ~ 10^31*10^27 = 10^58
    Thermal energy E ~ 2*10^43 J ~ 10^66 K
    -> Temperature ~ 10^66K/10^58 =10^8 K.
     
  4. Mar 19, 2012 #3
    This is called sleep is good for you! The next morning I redid my calculation and got something in that range - turns out I was accidentally multiplying by Boltzmann constant instead of dividing. :)
     
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