Pressure Gradient in Hydrostatic Equilibrium

[SHISHKABOB]

Homework Statement

Consider an isothermal atmosphere (T = const.) over a sufficiently small range of radii, so that you can assume that the gravitation acceleration g is constant. Use the equation for the gas pressure gradient in hydrostatic equilibrium to show that the gas pressure decreases exponentially with height.

Homework Equations

$\frac{dP}{dr} = -g\rho$

The Attempt at a Solution

so I solve the differential equation for P and I get

$P = -\rho gr = \frac{-GM\rho}{r}$

I think I'm doing something really dumb here...

 

[Infinitum]

Hi Shishkabob!

You need to re-check solving your differential equation. Is the density independent of pressure? Can you simply use it as a constant??

(Hint : Think about manipulating the ideal gas equation to form a relation)

 

[SHISHKABOB]

oh okay, thanks for the tip. I found a relationship between the number density of the gas and the pressure, and so I had a differential equation that was dn/dr. Which I solved for the number density as a function of r and then just resubstituted stuff in for pressure.