# Can't solve differential equation for atmosphere density

## Main Question or Discussion Point

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:

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

which can be simplified to:
$$G*PlanetMass/r^2 * \rho * (dr^2+2rdr) * PI = - C*(\rho*dr + d\rho * (r+dr))*d\theta$$
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?

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Probably if you use latex then your problem will be much clearer to everyone.

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.