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.

 

[thomate1]

I would approach this problem by first considering the basic principles of thermodynamics and atmospheric physics. I would also take into account the known properties of Pluto, such as its mass and radius, and the composition of its atmosphere.

To estimate the temperature of Pluto's atmosphere, I would use the ideal gas law, which states that the pressure (P), volume (V), and temperature (T) of a gas are related by the equation PV = nRT, where n is the number of moles of gas and R is the gas constant.

In this case, we know that the atmospheric density at a height of 50 km is one-third the density at the surface. This can be expressed as P2 = (1/3)P1, where P1 is the density at the surface and P2 is the density at 50 km. We can also assume that the number of moles of gas remains constant, since there is no indication of any chemical reactions or changes in the amount of gas present.

Using the ideal gas law, we can rearrange the equation to solve for temperature: T2 = P2V2/nR. We know the volume (V2) at 50 km is equal to the surface area of Pluto (4πr^2) multiplied by the height (50 km). We also know the mass (m) of Pluto and the composition (N2) of its atmosphere, which allows us to calculate the number of moles (n) of gas present.

Using these values and the gas constant, we can estimate the temperature of Pluto's atmosphere at 50 km above the surface. However, this is only an approximation and does not take into account factors such as atmospheric composition, variations in temperature at different altitudes, and other complexities. Additional data and observations would be needed to obtain a more accurate estimation of the temperature of Pluto's atmosphere.