Can't solve differential equation for atmosphere density

[steenreem]

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:

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

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

 

[matematikawan]

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

 

[AUMathTutor]

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.