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I am having trouble trying to derive the air pressure at a given height. So far, I have considered a 1m^2 patch of area, and the pressure is the weight of all of the air above this patch.

So [itex]P= \int_R^{\infty}g(x)\rho(x)dx[/itex]

So [itex]P= GM\int_R^{\infty}\frac{1}{x^2}\rho(x)dx[/itex]

But then I don't know what to do because the density will depend on the pressure at a given point? So I feel like I am going around in circles...

Any help will be much appreciated! :) I have a feeling that I am missing something really obvious.

EDIT: here I was finding the pressure at ground level, hence the limits of integration, although I would find a general expression by changing the lower limit from R to a given height R+h.

So [itex]P= \int_R^{\infty}g(x)\rho(x)dx[/itex]

So [itex]P= GM\int_R^{\infty}\frac{1}{x^2}\rho(x)dx[/itex]

But then I don't know what to do because the density will depend on the pressure at a given point? So I feel like I am going around in circles...

Any help will be much appreciated! :) I have a feeling that I am missing something really obvious.

EDIT: here I was finding the pressure at ground level, hence the limits of integration, although I would find a general expression by changing the lower limit from R to a given height R+h.

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