Science a GoGo's Home Page Forums General Discussion General Science Discussion Forum Can two planets combine without melting all crust?

Hello, preearth especially, I hope you would like to participate in my new thread.

It seems from your idea (www.preearth.net) that when the two planets collide, the Preearth has to keep most of its crust in the solid state. If it were molten then as it expanded, presumably it wouldn't break into pieces with well defined edges that still have matching shapes today.

I see you and ImagingGeek have both calculated (preearth website and post #34510) the minimum energy released by the two planets combining, based only on potential energy as

Potential energy lost = 10^31 J

You've also calculated a temperature increase of 3500 degC.

And that's where it seems to stop.

This still doesn't answer the important question of whether the crust would melt or not. How about some of these possiblities?

1) The heat is applied instantaneously everywhere leading to temperature of 3500degC and complete melting of the entire earth (1200degC melting point for crust).

2) The heat is applied under the surface and is conducted through the crust then radiated into space slowly enough that the crust never reaches melting point.

3) The collision occurs slowly so the heat is radiated into space before the crust ever reaches melting point.

4) Any other possibilities?

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This is the post from my forum:

http://www.preearth.net/phpBB3/search.php?search_id=newposts

that Kellog refers to above. The 2,468 degree rise turns out to be much too high as various factors have not been taken into account.

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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,...

Can you detail those factors? I don't want to rake over what's already been done. At this stage it doesn't clearly say yay or nay. It really needs more work.


I'm thinking of putting all that heat in the middle where it might not damage the crust, then seeing how far it would have cooled down by today.