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## Homework Statement

Consider an asteroid with an iron core [itex] \rho_m = 8000 kg\; m^{-3} [/itex] covered by a thin silicate mantle [itex] \rho_m = 3500 kg\; m^{-3} [/itex] with a thickness of 20% the raidus R of the asteroid.

Assume that the internal temp is [itex] T_i = 600K[/itex] and is constant throughout the core due to high thermal conductivity of iron. Take the thermal energy in the core to be [itex] 3k T_i[/itex] per atom and assume that the thermal conductivity is [itex] k_c = 2 W\; m^{-1} K^{-1} [/itex]. Ignore the heat capacity of of the mantle. If the surface has a temp of [itex] T_s = 200K [/itex], find the value of R for which the cooling rate is about 1 K per million years

## Homework Equations

[tex] \frac{dQ}{dt} = -k_c A \frac{dT}{dx} [/tex]

A is the surface area

## The Attempt at a Solution

There's a few things that are confusing me right off the bat. What is the k is the thermal energy of the core [itex] 3k T_i[/itex]? There is no reference to it in any of the literature, so I find it rather ambiguous which doesn't help my situation.

I started by saying that [itex] \frac{dT}{dx} = \frac{ T_i - T_s}{0.8 R} [/itex] in an approximating sense, using that the iron core is 80% of the asteroid. Then I assumed a spherical asteroid and subbed this into the above equation to get

[tex]\frac{dQ}{dt} = 5 \pi R k_c (T_i - T_s) [/tex]

Now I'm kinda stuck. I want to say something about the heat production like

[tex] \frac{dQ}{dt} = ML [/tex] where M is the mass and L is the energy production, however I'm not sure how to find energy production (something to do with the [itex] 3k T_i [/itex]?). And then where does the 1K per million years come in?