# Gravity and Planetary Differentiation

During solar system formation, many bodies achieved hydrostatic equilibrium, a spherical shape where their self gravitational force was balanced by internal pressure. Many also achieved differentiation, where a body is seperated into layers of different density (core, mantle and crust). Differentiation was mainly the result of the melting heat from isotopes like Al 26 and Fe 60, with half lives of ~10^6 years, coupled with the body's gravity. My question is, can differentiation occur without the presence of radioactive isotopes? Can a body's mass alone create sufficient gravitational force to induce melting, say by the attraction of meteoroids, and develop differentiation? Of course stars are differentiated, but I'm interested in their orbiting bodies.

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Yes, a body's mass is sufficient to cause melting. For example, suppose the moon, instead of orbiting the earth, fell onto the earth.

The moon weighs about 7.35E22 kg, and is a distance of about 3.8E8 m from the earth which weighs about 6E24 kg and has a radius of around 6.4E6 m. So the difference in potential energy $U = -Gm_1m_2/R$ for the earth-moon system at 3.8E8m and 6.4E6m is. $$-G(7.35\times 10^{22})(6\times 10^{24})(1/3.8\times 10^8-1/6.4\times 10^6) = 4.5\times 10^{30}\; \textrm{Joules}$$ where $G= 6.7\times 10^{-11}$ is the gravitational constant. The combined mass is still about 6E24 kg so the collision creates heat of $$4.5\times 10^{30}/6.0\times 10^{24} = 750,000\;\textrm{Joules/kg}$$

To heat a kilogram of iron up by 1 degree kelvin requires about 460 Joules. Granite needs about 790 while basalt is 840. Other materials making up most of the earth have similar specific heats. So the collision will heat up the material by something around 1000 degrees kelvin.

Now that was just for the moon whose mass is only a tiny fraction of the earth's. If a larger mass collided with the earth the temperature would be proportionally larger. If the body were about 4x the mass of the moon, the temperature increase would be around 4000 K and that would certainly thoroughly melt the planet.

So yes, it is possible for gravity alone to melt planets.

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Carl: please explain why the R term is (1/3.8×10^8 − 1/6.4×10^6 ) instead of (3.8x10^8 - 6.4x10^6), which would result in a two magnitude reduction in potential energy? I must be missing something. – Michael Luciuk Apr 3 '11 at 14:29
I think the real issue is how to get the heat to distribute throughout the body of the planet. If the planet accrues mass via a series of small impacts, most of the impact energy will end up near the surface, and can be radiated away. Conduction is much too slow a process to melt the interior, so unless it forms so rapidly that material can't melt before it is buried by new material it seems unlikely. Another method might be an impact so violent it disrupts the body, which reforms so rapidly it can't cool off. – Omega Centauri Apr 3 '11 at 15:23
Omega,Centauri: I agree. Carl's scenario is one feasible option. If early in solar system evolution. a massive number of meteoroids pounded an emerging planetoid, suficient heat may be retained throughout its center to create differentiation as it accretes mass. – Michael Luciuk Apr 3 '11 at 15:51
@Michael; I used the formula for gravitational potential energy. The calculation you're suggesting approaches infinity as the distance to the moon is increased. That's not physical because gravitational potential energy is finite. – Carl Brannen Apr 3 '11 at 22:21