Where did the energy released due to gravitational binding energy of the Earth go?

The gravitational binding energy of the Earth is $2×10^{32} J$, so the same amount of energy must have been released during the Earth's history.

According to this and this, the current internal energy of the Earth is ~ $1.5×10^{31} J$, and according to this source, the amount of heat loss due to outward radiation by the Earth during the entire lifetime of the planet is about $0.45×10^{31} J$.

So, by adding those two numbers we get the Earth internal energy + energy radiated = ~ $2×10^{31} J$, which is an order of magnitude less than what we should expect. We are also ignoring the fact that 50-90% of the current internal energy of the Earth is due to radioactive decay. So where did the rest of the energy go ?

• Not that this helps, but Wikipedia claims the binding energy is actually more like $2.5\times10^{32}\ \mathrm{J}$. (A centrally concentrated distribution will always be more bound than a uniform sphere.) I can confirm that I get the same answer numerically integrating the model given in the referenced paper. – user10851 Dec 13 '14 at 4:02
• Cross posted earthscience.stackexchange.com/questions/3032/… – user60063 Dec 26 '14 at 0:12

Terrific question. You had it right in your first sentence: “the same amount of energy must have been released during the Earth's history,” but then it gets a little mixed up when you look at various energies, some of which aren’t related to the question at hand (for example, the current internal energy contributes positive mass-energy to the Earth, rather than counting toward the negative binding energy).

Bear in mind that the gravitational binding energy represents the total potential energy required to separate all of the matter of the Earth out to infinity against the attraction of gravity. Playing that movie backward, we see that the gravitational binding energy has been radiating into space since before the Earth began to coalesce from the primordial dust cloud of the nascent solar system (and ultimately, all the way back to the end of the inflationary period). So a full accounting of that energy loss would be a long and difficult calculation. But we don’t need to do that calculation, which would require looking at even the previous generations of stars before the Sun, because we can derive it from the gravitational field strength we measure, and our best approximations of the Earth’s matter content and distribution.

As matter accrues into a gravitationally bound region, and eventually forms a solid object, it acquires kinetic energy in exact proportion to the potential energy lost to its new position deeper inside the gravity well. If none of the matter collided, and all the particles simply orbited the center of gravity, the kinetic energy of the masses would exactly balance their loss of potential energy to the gravity well, and the total energy of the system would remain unchanged by the presence of the gravitational field (this is just another way of describing a conservative field). But matter does collide, heat, and radiate some energy away as it loses kinetic energy. That radiated energy is the gravitational binding energy that we measure and calculate – it accounts for the mass that’s missing, aka the “mass defect.”

The internal energies of spin and internal heat all count toward the positive mass of the Earth, by the mass-energy equivalence relation.

So the answer is that the sources you’ve cited simply aren’t going back far enough in time to account for all of the energy radiated into space by the Earth’s matter as it coalesced from “infinity.” A thorough derivation of the gravitational binding energy can be seen here.

• "the current internal energy contributes positive mass-energy to the Earth" -- sure, but since the rest-mass energy of Earth is over $5\times10^{41}\ \mathrm{J}$, I'd say this is pretty negligible. – user10851 Dec 13 '14 at 3:15
• True, but for conceptual clarity it's important to keep the "debits" vs. the "credits" straight in the energy accounting, and they were getting mixed up when the question was posed: "by adding those two numbers we get the Earth internal energy + energy radiated = ~ $2×10^{31} J$" – Thomas M Dec 13 '14 at 3:27
• The answer in short is "Earth and Sun were formed from a galactical cloud, collapsed into a solar system disk". Those parts which spinned closer to the center formed the Sun, the outer layers formed the planets. The Earth was formed from the layer that was already at the right distance from the Sun and it does not need to exradiate its energy to bind with it. So, the true question remains unanswered is where the energy goes during the cloud collapse in space. Atoms come closer to each other. Where does the energy go? – Little Alien Sep 8 '16 at 9:29
• I guess it must be exradiated as el-m/heat waves. Otherwise, if cloud molecules, gravitationally accelerated towards each other, collide elastically, i.e. if they do not convert their kinetic energy into the light in collision, they will recede again into original large distances between them and cloud will fail to form. So, binding energy was exradiated as heat light during solar system formation from the interstellar gas. – Little Alien Sep 8 '16 at 9:31

The rest of the energy went into space. Without that energy loss the planet would not even have condensed and the gas/dust cloud would have stayed a cloud. Having said that, the details of these condensation processes in planetary clouds seem to be non-trivial and, from what I have read, are not fully understood, as of yet.