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Can't solve differential equation for atmosphere density

  1. May 10, 2009 #1
    Hi I'm a third year physics student currently working on my bachelorstage, and I have a differential equation I want to solve, but can't =).

    Given an planet with an atmosphere so thick relative to the planet radius, that gravitation in the atmosphere cannot be seen as a constant but decreases with the distance from the planet. I'd say the differential equation for the atmosphere density of a perfect gas would be:

    [tex]
    $
    Gravitation(r) * \rho * dA = Pressure(r)*d\theta*r - (Pressure + dPressure)*d\theta*(r+dr)
    $
    [/tex]
    , where dA = ((r+dr)^2-r^2)*PI
    , Pressure = Constant*rho
    , and Gravitation(r) = G*PlanetMass/r^2

    which can be simplified to:
    [tex]
    $
    G*PlanetMass/r^2 * \rho * (dr^2+2rdr) * PI = - C*(\rho*dr + d\rho * (r+dr))*d\theta
    $
    [/tex]
    I guess I can just remove the dr^2 right?

    Normally I'd try to solve this by bringing both variables to one side of the equation, say rho to the left and r to the right side, and then integrate. But I can't do that now because of the two messy terms on the RHS (right hand side).

    Any help on this? Maybe I should just neglect some terms in order to make this more easily solvable?
     
  2. jcsd
  3. May 13, 2009 #2
    Probably if you use latex then your problem will be much clearer to everyone. :wink:
     
  4. May 13, 2009 #3
    So you know the gravitation law and you want the density?

    Well, if the situation is perfectly spherically symmetric, just note that the gravity at radius r from the center is:

    g(r) = GM(r)/r^2

    M(r) is the mass inside a spherical shell centered around the center of the planet. THis can take on various forms, but in the most complicated case, it would look something like this:

    M(r) = M_core + M_atmosphere(r)
    where
    M_atmosphere(r) is the integral from r_0 to r of 4PIr^2 density(r) dr

    Put all of this together, and you should be able to solve: given gravity => find density; given density => find gravity.
     
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