# What is planetary surface temperature given constant sub-surface temperature?

If a planet of radius R1 has a constant sub-surface temperature T0 at R0 < R1, what is the long-term equilibrium surface temperature T1? Say we assume constant thermal diffusivity of the planet material, surface emissivity 1.0, no atmosphere, and no incoming radiation. I figure the temperature profile is harmonic, i.e. T=a+b/R, for constants a and b, and that we can use (R0,T0) to eliminate one of these constants. Is there enough information to obtain the other constant?

-

You want to equate the conductive energy flux at the surface which is a constant times T1-T0 -given your geometry and conductivity you can determine the coupling factor, with the radiative heat flux sigma*T**4. You could also add in the CMB, which is simply the same sigma*T**4 using the CMB temperature. Then you simply have a nonlinear algebraic equation in T0 to solve. It should be solvable by iteration without much effort.

-
So I need to find a coupling factor k, depending on R0, R1 and the conductivity, then solve k*(T1-T0)=sigma*T1**4 for T1? That's great! Do you have any pointers about figuring out k? A reference? –  SteveBrooklineMA Jan 16 '11 at 23:37
Fourier's equation gives $Q= -k Area dT/dR = -k 4 pi R^2 dT/dR$, So integrating gives $Q \int_{R0}^{R1} 1/R^2 dR = -4 pi k (T1-T0)$ or $Q = -4 pi k (T1-T0) R0 R1 / (R1 - R0).$ Setting equal to radiated flux from the surface gives: $-4 pi k (T1-T0) R0 R1 / (R1 - R0) = sigma T1^4 4 pi R1^2.$ So we need to solve that for for $T1.$ Does that seem about right? –  SteveBrooklineMA Jan 17 '11 at 2:12

Sorry for the confusion. R0 is some given particular radius between 0 and R1, e.g. R0=5000, R1=6000 km. R is the general radial variable, as in T=a+b/R. T0 is the temperature at position R0, assumed constant throughout time.

-
How can the temperature at R=R0 stay constant if at the same time you assume the body at R=R1 to radiate into a zero temperature environment? Where is the energy coming from? –  Johannes Jan 16 '11 at 21:06
Let's say an internal energy source. –  user1385 Jan 16 '11 at 22:00
Ok, in that case the temperature at surface is zero, so: a = -RO T0/(R1-R0) and b = R0 R1 T0/(R1-R0). –  Johannes Jan 16 '11 at 23:27