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Pressure Gradient in Hydrostatic Equilibrium

  1. Oct 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider an isothermal atmosphere (T = const.) over a sufficiently small range of radii, so that you can assume that the gravitation acceleration g is constant. Use the equation for the gas pressure gradient in hydrostatic equilibrium to show that the gas pressure decreases exponentially with height.


    2. Relevant equations

    [itex]\frac{dP}{dr} = -g\rho[/itex]


    3. The attempt at a solution

    so I solve the differential equation for P and I get

    [itex]P = -\rho gr = \frac{-GM\rho}{r}[/itex]

    I think I'm doing something really dumb here...
     
  2. jcsd
  3. Oct 15, 2012 #2
    Hi Shishkabob! :biggrin:

    You need to re-check solving your differential equation. Is the density independent of pressure? Can you simply use it as a constant??

    (Hint : Think about manipulating the ideal gas equation to form a relation)
     
  4. Oct 15, 2012 #3
    oh okay, thanks for the tip. I found a relationship between the number density of the gas and the pressure, and so I had a differential equation that was dn/dr. Which I solved for the number density as a function of r and then just resubstituted stuff in for pressure.
     
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