Cooling of a hot body in the vacuum of space

[Thecla]

In the 19th century Lord Kelvin made the first numerical calculation of the age of the Earth not based on the Bible.From his initial guess that the Earth started as a molten rock and that today the temperature of the interior increases at a certain rate as you approach the center, he got an age of about 100 million years.(Part of the reason for his error was that he had no way of knowing that radioactive decay was an additional source of heat)
I want to ask a simpler question; How would you calculate the surface temperature of a 10,000
mile solid sphere of a very hot rock initially at 5000 degrees K in the vacuum of space. Assume it is of uniform density and a perfect black body.Also assume no atmosphere and the sun is 10 billion miles away and contributes nothing. What is its surface temperature after one million years or 10 million years.

 

[russ_watters]

The way you worded that it sounds a lot like homework, so let's treat it as homework. What phenomena are involved? What equations are there related to them?

 

[Drakkith]

I want to ask a simpler question; How would you calculate the surface temperature of a 10,000
mile solid sphere of a very hot rock initially at 5000 degrees K in the vacuum of space. Assume it is of uniform density and a perfect black body.Also assume no atmosphere and the sun is 10 billion miles away and contributes nothing. What is its surface temperature after one million years or 10 million years.

That depends a great deal on how we model the thermodynamics of the interior. If we say our rock has extremely high thermal conductivity, enough so that the internal temperature stays roughly the same throughout, then we have greatly simplified the problem.

However, if we model the body using more realistic properties, the calculations can become much more difficult, as we have to take into account how quickly heat can transfer out from the inside. This can range from relatively simple if we keep our sphere close to homogenous, to very difficult if we want to add layers like a crust or a mantle that would affect how heat flows through the body.

 

[jeffery_winkler]

A homework problem would not have started off by talking about Lord Kelvin, and would not have said that the Sun was 10 billion light years away. If it was worded like a homework problem, it would have said, "Calculate the surface temperature..." etc. I believe that the original poster was genuinely curious how Lord Kelvin reached his conclusion. Lord Kelvin was unaware of the interior structure of the Earth, unaware of radioactivity, and chose to ignore convection, and instead only considered conduction, which is part of why he underestimated the age of the Earth. His estimate met with immediate resistance from geologists who said mountains could not formed in such a short time, and biologists who said that the diversity of life on Earth could not have evolved in such a short time.

 

[mfig]

If we assume that the Earth's interior is homogeneous and isotropic, then the question is asking for the the temperature T=T(r,t) which solves

$\frac{\partial T}{\partial t} = \frac{1}{\alpha r^2} \frac{\partial}{\partial r} (r^2 \frac{\partial T}{\partial r} )$

Subject to the conditions

$T(r,0) = T_0$

$\frac{\partial T(0,t)}{\partial r}=0$

$-k\frac{\partial T(R,t)}{\partial r}=\sigma T(R,t)^4$

The solution is likely found in Carslaw and Jaeger. Note this assumes radiation into space at 0K.

 

[Thecla]

The sun is only 10 billion miles away. It is not a homework problem; I was just curious. I knew that radiation is emitted as the 4th power of temperature and put the sun at some distance away so its contribution can be ignored. I also wanted the hot planet to have the most simple structure.
By the way Lord Kelvin's result upset both biblical creationists who took the bible literally because they they thought the calculation of the age of the Earth was too old and evolutionists who thought the calculation was too young.

 

[jeffery_winkler]

I am sorry. When I first read it, I thought you said "light year" as a way of saying, "Ignore energy input from the Sun".

 

[davenn]

The sun is only 10 billion miles away.

which Sun ?

our Sun is only 93 million miles away

 

[jeffery_winkler]

That's what I meant. I meant to say "would not have said that the Sun was 10 billion miles away". However, I just glanced at his original question, and was not paying attention, so initially I thought he said "light year", and I assumed it was intended as a joking way of saying "assume that the Sun is so far away that it can be ignored".