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If all objects in the solar system suddenly went cold to the core, including the Sun, would it gravity play any different role? I guess what i am trying to ask, is there any relation between energy and gravity?

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up vote 4 down vote accepted

This is really just an expansion on Alfred's answer (which is entirely correct :-).

In most cases the only element in the stress-energy tensor that matters is the energy density, and for a body that can be treated as point-like this is the mass + the energy. You add the contribution of the energy by dividing it by $c^2$ according to Einstein's famous equation $E = mc^2$.

Working out what happens when the Sun cools is hard because it's so complicated, so let's take the Earth as a simple example. The mass of the Earth (call this $M_0$) is 5.97219 × 10$^{24}$ kg. If we let the Earth cool to absolute zero we have to take out some energy $E$, and this will reduce the mass of the Earth by $E/c^2$. The gravitational potential at some distance $r$ will fall to:

$$ V(r) = - \frac{G(M_0 - E/c^2)}{r} $$

Actually calculating the value of $E$ is near impossible. According to the Engineering Toolbox the specific heat of basalt is 0.84 kJ/kg.K and the specific heat of iron is 0.46 kJ/kg.K. Lets take an average figure of 0.7 and assume the average temperature of the Earth is 2500K then just from cooling the value of $E$ we get is about 10$^{28}$J, and dividing by $c^2$ gives an equivalent mass of about 10$^{11}$ kg. So the mass of the Earth would decrease by about 1 part in 10$^{13}$, which is immeasurably small.

Even this calculation is flawed because I haven't included the latent heat of fusion of the molten iron core, and you also need to include the energy given off as all the radioactive elements in the Earth decay. Trying to do the same calculation for the Sun is vastly harder because it is continuously converting matter to energy, plus the Sun continuously ejects matter as the solar wind. However hopefully this gives you some idea of how the calculation would be done.

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I guess what i am trying to ask, is there any relation between energy and gravity?

Indeed, the relation between energy and gravity is the core of General Relativity.

From the Wiki article Stress-energy tensor:

The stress–energy tensor (sometimes stress–energy–momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

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Okey thank you, if there was a little less math in that page, I might understand what happens if all bodies in solar system freeze. – user25675 Jun 12 '13 at 2:01
@user25675, math is the language of physics. You might think that you could understand, that you could know what would happen if all bodies in the solar system freeze, without the language of math, but, in fact, you wouldn't; you would only be parroting what someone else said. If you really want to know, rather than parrot, you must "grok" the language. – Alfred Centauri Jun 12 '13 at 2:11
Even if every body in solar system freezes, it won't change their mass. So there should be no effect on their motion. – anuragsn7 Jun 12 '13 at 2:11
@anuragsn7, energy, whatever the form, gravitates. – Alfred Centauri Jun 12 '13 at 2:15

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