Temperature of Pluto's Atmosphere

[viviolet]

Homework Statement

In 1988 telescopes viewed Pluto as it crossed in front of a distant star. As the star emerged from behind the planet, light from the star was slightly dimmed as it went through Pluto’s atmosphere. The observations indicated that the atmospheric density at a height of 50 km above the surface of Pluto is about one-third the density at the surface. The mass of Pluto is known to be about 1.5x10^22 kg, and its radius is about 1200 km. Spectroscopic data indicate that the atmosphere is mostly nitrogen (N2). Estimate the temperature of Pluto’s atmosphere. State what approximations and/or simplifying assumptions you made.

Homework Equations

C= deltaEatom/detlaT
Average energy of a diatomic molecule = translational energy + vibrational energy + rotational energy + gravitational energy
= 1.2mvx^2 + 1/2 mvy^2 + 1/2mvz^2 + (1+ m1/m2)(p1^2/2m1) + 1/2kS^2 + L^2rotx/2I + L^2roty/2I + Mgycm/
...?

The Attempt at a Solution

I don't think I even understand how to approach this - there is no change in energy given. None of the equations I have seem to explain how to figure out something's temperature based only on its size... The only think I can think of is trying to equate gravitational energy and temperature somehow.

 

[mgb_phys]

For the atmosphere to stay attached the molecular speed has to be less than escape velocity. That gives you an upper limit on the temperature.