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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 densityThe mass of the sphere, inside that shell is (4/3)πr3ρ GPE = -G (Mass inside shell)(Mass of shell)/r=-4Gπr3ρ4πr2ρdr/3r
At the moment I think it is supposed to be contributing about half of the net heat loss of the earth
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.