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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?