Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: lyner on 10/06/2008 14:19:19
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Establishment Lackeys, such as I are often accused of accepting the figures that we read in textbooks. It is because we tend believe in the basics - because they work. Nevertheless, it is sometimes nice to verify things for oneself.
I was busy spouting off about the energy which was available as the Earth was formed by the accretion of a large volume of bits of the Solar System into one lump. This energy would then have been dissipated by radiative cooling after melting all the bits together.
So I did the following calculation.
I think the logic of the following is OK but would appreciate any errors being pointed out.
Start off with a large sphere of material of low density (and uniform density to make it simple). Work out the Gravitational Potential for the initial, large, sphere and the GPE of the final sphere (i.e. the present Earth). The difference between the two will be the energy available to heat the Earth up, melt it together etc. and, the heat which has been lost since it was formed.
The GPE of a piece of the sphere at distance r from the centre is provided by the material nearer to the centre (the effect of the material further out all cancels out).
The mass of a shell, radius r and thickness dr will be
4πr2*ρ*dr.
where ρ is the average density
The mass of the sphere, inside that shell is
(4/3)πr3ρ
GPE = -G (Mass inside shell)(Mass of shell)/r
=-4Gπr3ρ4πr2ρdr/3r
where G is the Gravitational Constant
The density is the total mass / the volume of the sphere
so
ρ = 3M/(4πD2π)
where D is the radius of the whole sphere.
This gives the GPE of the shell as:
-3Gr4M2dr/4D6
which gets rid of the ρ.
You can simplify this into
GPE of the shell = -3Gr4M2dr/4D6
To find the total GPE of the sphere you have to integrate over all the shells - this gives you
Total GPE = -3GM2/20D
This looks OK as the units of the final result are in energy!
It also gives the GPE of the Earth with its present radius (say d)
The difference between the two will be
=-3GM2/20D +3GM2/20d
If D is, say, a thousand times d (it must have been at least as much)then the formula becomes
-3GM2/20d
Putting in the numbers, you get the available energy as 6e31J.
That's a lot of energy! Easily enough to do all the melting and fusing and stuff. It's about ten million year's worth of Solar energy.
Cooling. If the Earth had the same albedo as it has today and assuming that the surface temperature was similar to present day (300k; you've got to start somewhere), using Stefan's law tells you the rate the Earth will radiate energy if the surface were matt black. That's about 10e17W.
At that rate, the energy would last 6e31/10e17seconds - only about 10 million years.
It must imply that, for billions of years, the surface temperature must have been dominated by the Sun's radiation. I'm ignoring the fact that it would cool exponentially and that the initial rate would be much higher.
Interesting, though!
There is the other issue of how fast heat can travel through thousands of km of mantle and crust.
[mod - please try to make the subjects of new posts into questions as this makes the forum more intuitive to use. Thanks. CS]
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The GPE of a piece of the sphere at distance r from the centre is provided by the material nearer to the centre (the effect of the material further out all cancels out).
The mass of a shell, radius r and thickness dr will be
4πr2*ρ*dr.
where ρ is the average density
The mass of the sphere, inside that shell is
(4/3)πr3ρ
GPE = -G (Mass inside shell)(Mass of shell)/r
=-4Gπr3ρ4πr2ρdr/3r
I agree with you up to here, which simplifies to
GPE=-16Gπ2r4ρ2dr/3
I think you made a couple of transcription errors in the density equation, which should be (as M/V),
ρ = 3M/(4πD3)
which, plugged into the GPE above, gets you
GPE=-3GM2r4/D6dr
(I don't get the factor of 4 on the bottom as you did, but otherwise it's the same.)
The final GPE without this is just 4 times what you got:
Total GPE = -3GM2/(5D).
Of course, someone should double check mine as well, since it's easy to make a mistake with all this math.
The rest of your logic seems good--that energy had to go somewhere as the earth condensed, and most of it probably went into temperature. Very interesting to see the magnitude of some of these quantities, and how long you'd expect the earth to (roughly) radiate.
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Remember there is also going to be a lot of radioactive heating as well.. At the moment I think it is supposed to be contributing about half of the net heat loss of the earth and if you go back 4.5 billion years there must have been a lot more elements with a half life of a few hundred of millions of years kicking about such as 235U and even about twice as much 238
Although overall solar radiation will have dominated for most of the life of the earth...
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At the moment I think it is supposed to be contributing about half of the net heat loss of the earth
I did some sums on this too.
Stephan's law suggests 2e17W for a black radiator the size of the Earth. But it isn't black.
We intercept about 1e17W from the Sun.
That sort of agrees with your idea but you have to bear in mind that the emissivity at 300K is probably very different from the albedo of the Earth's surface (about 0.4).
Dunno what the infra red emission is like tho. Where do 'they' get the 50% radioactivity contribution from? I guess it must all be in the detailed satellite measurements of radiated energy.
I liked the 10 million years of solar energy corresponding to the GPE of all the bits.
jp
I think you're right about the factor of 4. Thanks for checking - I'm glad it took at least one person's fancy.
Amazingly I had a comment on another thread about it which likened it to Mickey Mouse Science!!? And there was me thinking it was a piece of real textbook stuff. Perhaps I'd better learn some more proper Science.
The two 4's in the density do cancel with the two 4's on the top line.
That would make the above into 40 million years. Not bad value for paying the electricity bill.
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In 1862, the physicist William Thomson (who later became Lord Kelvin) of Glasgow published calculations that fixed the age of the Earth at between 24 million and 400 million years. He assumed that the Earth had been created as a completely molten ball of rock, and determined the amount of time it took for the ball to cool to its present temperature. His calculations did not account for the ongoing heat source in the form of radioactive decay, which was unknown at the time.
http://en.wikipedia.org/wiki/Age_of_the_Earth
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It's a great pity that the Moon doesn't rotate quickly. Its surface temperature is dominated by solar radiation and we could make an easy comparison. But the slow (monthly) timescale must make it more difficult.