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Ex2.7 Black holes, white dwarfs and neutron stars-Shapiro, Teukolsky

  1. Nov 18, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that from (*) that for a nonrelativistic Maxwell-Boltzmann gas,
    [itex]n=g\bigg(\frac{nkT}{2\pi\hbar^2}\bigg)^{\frac{3}{2}}e^{\frac{\mu-mc^2}{kT}}[/itex]
    [itex]P=nkT[/itex]
    [itex]e=nmc^2+\frac{3}{2}nkT[/itex] [itex][/itex]

    2. Relevant equations
    (*): [itex]f(E)=e^{\frac{\mu-E}{kT}}[/itex]
    [itex]E=\sqrt{p^2c^2+m^2c^4}[/itex]
    [itex]n=\frac{g}{h^3}\int f(E)d^3p[/itex]
    (#) [itex]h=2\pi\hbar[/itex]
    3. The attempt at a solution
    So I suppose that nonrelativistic mean [itex]E=mc^2[/itex], than [itex]n=\frac{g}{h^3}\int e^{\frac{\mu-mc^2}{kT}}4\pi p^2dp=ge^{\frac{\mu-mc^2}{kT}}\frac{4\pi}{3h^3}p_f^3[/itex]. [itex]E_k=mv^2/2=p^2/2m=3kT/2[/itex]=>[itex]p=\sqrt{3mkT}[/itex]. Employ this in previous [itex]n=ge^{\frac{\mu-mc^2}{kT}}\frac{4\pi}{3h^3}(3mkT)^{3/2}[/itex] due (#) [itex]n=ge^{\frac{\mu-mc^2}{kT}}\bigg(\frac{3^{1/3}mkT}{2^{2/3\pi^{4/3}}\hbar^3}\bigg)^{3/2}[/itex]. Do somebody see some mistake?
    Thank you very much.
     
  2. jcsd
  3. Nov 23, 2014 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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