# Finding the pressure at a given height?

21joanna12
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 $P= \int_R^{\infty}g(x)\rho(x)dx$

So $P= GM\int_R^{\infty}\frac{1}{x^2}\rho(x)dx$

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.

Last edited:

Homework Helper
Gold Member
density will depend on the pressure at a given point
Can you think of any relation between density and pressure? Perhaps that would allow a change of variables?

21joanna12
Staff Emeritus
Homework Helper
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 $P= \int_R^{\infty}g(x)\rho(x)dx$

So $P= GM\int_R^{\infty}\frac{1}{x^2}\rho(x)dx$

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.
Like a lot of natural things, the variation of atmospheric pressure with altitude is quite complicated, due to a variety of factors:

http://en.wikipedia.org/wiki/Atmosphere_of_Earth

The mass density of the atmosphere varies with altitude in a roughly linear fashion up to about 70 km above the earth's surface.

The problem you are trying to solve, the variation of atmospheric pressure with altitude, leads to what is known as the barometric formula:

http://en.wikipedia.org/wiki/Barometric_formula

and a derivation of this relationship is included in the article.

21joanna12
21joanna12
Like a lot of natural things, the variation of atmospheric pressure with altitude is quite complicated, due to a variety of factors:

http://en.wikipedia.org/wiki/Atmosphere_of_Earth

The mass density of the atmosphere varies with altitude in a roughly linear fashion up to about 70 km above the earth's surface.

The problem you are trying to solve, the variation of atmospheric pressure with altitude, leads to what is known as the barometric formula:

http://en.wikipedia.org/wiki/Barometric_formula

and a derivation of this relationship is included in the article.
Thank you for the link! Although there is something that is bugging me: in the Barometric formula, it is assumed that the temperature is constant, but it actually varies with altitude? I suppose it doesn't really vary in a predictable way because of the different layers of the atmosphere, so nothing can really be done about this unless you considered each temperature section separately and summed them all up to find the total pressure?

Staff Emeritus
Homework Helper
Thank you for the link! Although there is something that is bugging me: in the Barometric formula, it is assumed that the temperature is constant, but it actually varies with altitude? I suppose it doesn't really vary in a predictable way because of the different layers of the atmosphere, so nothing can really be done about this unless you considered each temperature section separately and summed them all up to find the total pressure?

There is a range of altitudes for which the Barometric formula is applicable. The Barometric formula is sometimes called the Isothermal atmosphere because it assumes constant temperature in the various layers of the atmosphere to which it is applicable.

In the Wiki article, there is a table which gives the values of various constants for the six different layers of atmosphere below 71 km altitude. In the article on the Atmosphere of Earth, there is a plot which shows that the variation on temperature with altitude is quite nonlinear. The temperatures used in the table agree pretty closely with the graph for the reference altitudes for each layer.

I think the best you can say is that the Barometric formula is an attempt to develop a mathematical relationship between atmospheric pressure and altitude using a few key data points.

21joanna12
Mentor
The differential equation that applies is:
$$\frac{dp}{dz}=-\frac{p(z)M}{RT(z)}g(z)$$
The solution to this equation is:
$$p(z)=p(0)e^{-\frac{M}{R}\int_0^z{\frac{g(z')}{T(z')}}dz'}$$
where z' is a dummy variable of integration. This equation takes into account the dependence of g and the dependence of T on altitude z. So, to apply it, you need to know how g and T vary with altitude. The dependence of T on z comes from observational data.

Chet

21joanna12