Heat Flow from planets and moons
• near the surface of the earth, the average geothermal gradient in the continents is about 30 K per km [Garland, Introduction to Geophysics, 1979]
• at the base of the continents (35 km depth), this would extrapolate to a temperature of over 1300 K, well above the melting point of rock
Tearth(D) = Tsurface + 30*D
Tearth(D=35 km) = 288 K + 30*35 K
Tearth(35 km) = 1338 K
• since a great deal of heat production occurs in the crust, the temperature gradient decreases with depth, so that the temperature inside the Earth is estimated to reach about 2000 K at a depth of 1000 km.
• all moons and planets will behave similarly: under pressure, materials at depth are warmer.  In addition, any heat sources (radioactivity, tidal heating, gravitational compression) at depth increase internal temperatures.  Thus, all objects will be warmer inside than outside.

Tio(D) = 124 K + dT/dD * D

• what is dT/dD?  what are heat sources?
• if dT/dD = 5 K per km, at what depth D does rock melt?
Tio(D) = 124 K + 5 * D
D = [Tio(D) -124 K] / 5 (k per km)
if the melting point of rock is 1200 K, then we want to solve for the situation at which
Tio(D) = 1200 K
then
D = [1200 K -124 K]  / 5 (K per km) = 215 km

Teuropa(D) = 124 K + dT/dD * D

• what is dT/dD?  what are heat sources?
• if dT/dD = 1 K per km, at what depth D does ice melt?
Teuropa(D) = 124 K + 1 * D
D = [Teuropa(D) -124 K] / 1 (K per km)

if  the melting point of water is 300 K, then we wan to solve for the situation at which

Teuropa(D) = 300 K

then

D = [300 K -124 K]  / 1 (K per km) = 176 km

so as long as a thermal gradient of 1 K per km is possible, and provided water exists within Europa, we would get liquid water at a depth of only 176 km.

Heat Flow Models:
• assume uniform distribution of material in planet/moon
• assume radiogenic component at Time = 0
• assume equation of state (pressure, volume as function of temperature)
• assume heat flow properties of materials
• gravity/pressure establishes an initial temperature gradient
• heat generated as function of time
• heat radiated from surface as function of time, temperature