Whatever happened to the original point of all this, which I seem to remember (could be wrong!) was something like:
Preearth says the gravitational potential energy of two planets moving from very close to combining may not be converted into enough heat to melt the crust.
Somebody else saying the gravitational potential energy of the two planets moving from very close to combining would have to be converted to enough heat to melt the crust.
How about a simple calculation like this:
1) Find the gravitational potential energy of the two planets just touching (using several methods if need be)
2) Find the gravitational potential energy of the combined planet (using the same method(s) as in 1).
3) Find the difference, and make sure it's the same for all methods. If not, got back and identify the wrong ones.
4) Assume all the energy is converted to heat and so find the amount of heat energy created
5) Estimate an upper bound on the time taken to combine.
6) Apply a radiative cooling model (or a suitable approximation), and calculate the temperature gradient through the surface of the combined planet.
7) See if the surface of the combined planet has a layer of similar thickness to continental crust which never exceeds the melting point of that material.
8) Conclude that the energy calculation either disproves Preearth's hypothesis, or is compatible with it.
Here is something on this from way back.
Here is a cute calculation that comes up in showing the impact between the two planets (i.e., Heaven and PreEarth) would not necessarily melt the entire surface of the larger planet that coalesced from them (i.e., Earth).
It is simple enough for high schools students to understand and it illustrates the power of (even simple) mathematics.
The problem is to estimate the rise in the temperature caused by placing the planet PreEarth next to the planet Heaven and letting gravity transform them into the Earth.(Note, that if two large enough planets are placed side by side and left alone, then gravity will pull them into one planet. So here we are placing PreEarth next to Heaven and gravity is pulling these two into a new planet which just happens to be the Earth.)
We will make the simplifying assumption that both the planets Heaven and PreEarth were spherical with uniform density.
This makes the math simple and the argument easy to follow.
The gravitational binding energy of a planet, U, is the energy released by the assembly of the planet from atoms which were originally an infinite distance away. Or, alternatively, it is the energy needed to disassemble the planet into atoms by moving each an infinite distance away.
The gravitational binding energy of a spherical planet with uniform density, is given by the formula;
U = 0.6GM^2/R, where
G = 6.67428 x 10^-20 km^3/(kg s^2) is the gravitational constant,
M is the mass of the planet in kg,
R is its radius in km.
U is here measured in megajoules, MJ.
Earth Radius R_E = 6371 km.
Earth Mass M_E = 5.97369 x 10^24 kg.
Approximate Earth Binding Energy = 0.6*G*M_P^2/R_P = 22.430 x 10^25.
PreEarth Radius R_P = 5200 km.
PreEarth Mass M_P = 3.48280 x 10^24 kg.
Approximate PreEarth Binding Energy = 0.6*G*M_P^2/R_P = 9.341 x 10^25 MJ.
Heaven Radius R_H = 4680 km.
Heaven Mass M_H = 2.48456 x 10^24 kg.
Approximate Heaven Binding Energy = 0.6*G*M_H^2/R_H = 5.282 x 10^25 MJ.
The energy necessary to separate PreEarth and Heaven to infinity, is:
G*M_P*M_H/(5200+4680) = G*M_P*M_H/9880 = 5.846 x 10^25 MJ.
The idea is to take PreEarth and Heaven at the point of first contact, that is, when they are just 9,880 kilometers apart, dissemble them to infinity, then bring everything back from infinity and assemble Earth.
So, the energy released from the point of contact through the formation of the Earth, is:
Energy Released = (22.430 - 9.341 - 5.282 - 5.846) x 10^25 = 1.961 x 10^25 MJ.
This is (1.961 x 10^25)/(5.97369 x 10^24) = 3.2827 megajoules per kilogram.
Suppose an average specific heat of 1330 J/kg°K.
Then we have that the Earth experiences a 3282700/1330 = 2,468 degree (average) rise in the temperature.
I emphasize that this temperature rise is due solely to the energy released by just placing PreEarth next to Heaven and letting gravity transform them into the Earth.
It is worth mentioning that the temperature rise will not be uniform.
Any pre-impact kinetic energy of Heaven (relative to PreEarth) will cause an additional rise in temperature.
To be continued,...
