# Pressure & Temperature in Saturns Atmosphere

1. Apr 4, 2009

### CaptainEvil

1. The problem statement, all variables and given/known data

At the top of the NH3 clouds in Saturn’s atmosphere, the temperature To = 110 K and the pressure Po is about 0.5 bar = 0.5 atm. Below this level at a radius of ro=60268 km, the atmosphere is convective and the temperature changes with radius at a constant rate of dT/dr = -7 x 10-4 K m-1 (which you may find convenient to call by a name, say L, in the question below). The gravity at Saturn’s cloud tops is 1.06 times that on the surface of the Earth.

a. Write down and explicit expression for the value of the temperature as a function of
depth T(r) and the expression for the equation of hydrostatic equilibrium including
this explicit variation of T(r). Eliminate the density from the equation of hydrostatic
equilibrium using the ideal gas law and assuming that Saturn’s atmosphere is pure H2.
State any assumptions you use.

b. Integrate your equation of hydrostatic equilibrium to find an explicit expression for
the pressure P(r) as a function of depth below the NH3 cloud tops.

c. Use the ideal gas law and your expressions for P(r) and T(r) to find an expression for
the density ρ(r) as a function of depth.

d. The separation of the two nuclei in H2 is about 7.4 x 10-10m , so two molecules will essentially touch when their separation is of order 10-10 m. When the density is high enough for the average separation between molecules to be of this order, H2 will sure be essentially liquid rather than gaseous. Imagine for counting purposes that the molecules in a liquid are arranged in a simple lattice of rows, columns and layers with all the spacing between adjacent molecules being 10-10 m. What is the density of H2 in this state, in kg m-3?

e. Use your estimate from parts c) and d) to estimate the depth below the NH3 cloud tops at which Saturn’s interior changes from gas to a molecular liquid. What is the
temperature there?

2. Relevant equations

3. The attempt at a solution

To = 110k
Po = 0.5 atm
ro = 60268 km
gs = 1.06g = 10.388 m/s2
L = -7x10-4 K/m

a) T(r) = To + Lr

hydrostatic quilibrium = dP/dr = -ρg

density ρ can be eliminated via I.G.L => ρ = $$\mu$$P/RT(r) where $$\mu$$ is the molecular weight, R is the gas constant and P pressure.

dP/dR = -$$\mu$$Pg/RT(r)

I can then separate these and integrate to solve for Pressure, but I am having a lot of trouble.

Could someone help me with the integration (if I'm even on the right track) and part d (where I don't even know where to start)

Thanks.

P.S. my teacher gave me a hint for the integration that I should get something like dP/dr = P*f(r) and so I will have to integrate 1/P dP = f(r) dr

Thanks