A planet of radius r and mass M has a homogeneous atmosphere (density independent of height). The atmosphere has a height h and the atmosphere’s gas has a molar mass μ.

What is the surface temperature of the planet?

###### Solution.

The ideal gas law is often written as

P*V = (m/μ)*R*T

where P is the pressure of the gas, V is the volume of the gas, m is the mass of the gas, μ is the molar mass, R is the ideal (universal) gas constant and T is the absolute temperature of the gas.

Mathematically, density is defined as mass divided by volume

ρ = m/V.

Thereby, we can rewrite the ideal gas law as

P = (ρ/μ)*R*T

or

T = P*μ/(ρ*R) (1)

The pressure due to the gas of constant density (homogeneous atmosphere) is represented by the following formula

P = ρ*g*h (2)

where g is the acceleration of gravity of this planet.

We can calculate g by using the following formula

g = G*M/r^2 . (3) (Learn more How do you calculate g?)

Now, we substitute (3) into (2) and get

P = ρ*G*M*h/r^2 . (4)

Finally, we substitute (4) into (1) and get

T = (ρ*G*M*h/r^2)*μ/(ρ*R)

or

T = (G*M*μ*h)/(R*r^2)

Image Credit: NASA/JPL/USGS