What Size Must Asteroids Be to Exceed Melting Temperatures of Ice and Rocks?

[Mattbro]

Consider main-belt chondritic asteroids. How large should be an asteroid so that the
maximal internal temperature exceeds the melting temperature of ice? Of rocks? For your
calculations, use the present-day chondritic heat production w = 4×10-12 W kg-1, typical
thermal conductivity of slightly fractured rock k = 2 W K-1m-1, and density r = 2700 kg
m-3 representative of chondritic materials. Make (and explain) a reasonable guess about
the surface temperature Ts and melting temperatures Tm of ice and rocks

 

[mgb_phys]

Welcome to PF - you have to show some attempt to work out the homework before we can help.

 

[Mattbro]

well i know heat flux = -k(dt/dr) this is change in temperature with depth and thermal diffusivity x= k/pC (here p is density and C is heat capacity). I just don't know how to apply heat production. I feel like I'm missing an equation and i have no idea what it is.

 

[mgb_phys]

Heat production is given per mass, so with the density it's easy to work out heat production/volume. If you assume a spherical asteroid you can get a heat prouction / radius relation

 

[Mattbro]

the units for that would be W/m^3. this is almost heat flux (W/m^2). where would i go from there?

 

[Mattbro]

asteroid size

How large should be the body so that the maximal internal temperature reaches given
temperature Tm (melting temperature)?

I'm thinking i can use F= K*[(Ta-Tb)/(Ra-Rb)]. where F is heat flux, K is thermal conductivity, T is change in temperature and R is change in depth.
I'm thinking the radius i need should be R=(K*(Tm))/F, but I'm not sure if it is that easy.

I need some reassurance.

Thank you

 

[berkeman]

(two threads on the same question merged into one)

(thread moved to Advanced Physics Homework Help)