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?
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
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)
T = (G*M*μ*h)/(R*r^2)
Image Credit: NASA/JPL/USGS