Solving the PDE for a Sphere in Vacuum: Temperature over Time

[MarkL]

Is there a PDE describing temperature as a function of time for a sphere in a vacuum (a moon in space)?

Since it can only radiate, the boundary would be a function of temperature to the fourth power (nonlinear), right?

Is there a book or thread out there I can reference?

Maybe this can be handled easily numerically.

Thank you

Mark

 

[sylas]

Is there a PDE describing temperature as a function of time for a sphere in a vacuum (a moon in space)?

Since it can only radiate, the boundary would be a function of temperature to the fourth power (nonlinear), right?

Is there a book or thread out there I can reference?

Maybe this can be handled easily numerically.

Thank you

Mark

You will need some assumptions about the energy coming in, and about the heat capacity of the surface. The latter tells you how quickly temperature drops as energy is radiated. It can be more complicated if the planet is water covered, or otherwise has some heat capacity below the surface.

Cheers -- sylas

 

[Integral]

Apply the http://en.wikipedia.org/wiki/Heat_equation" . Due to the symmetry of a sphere it can be done as a one dimensional problem with the radius of the sphere as the variable.