What about the Roche Limit? Would the planet breakup?

The hypothesis being that Earth formed from the collision of two medium sized planets.

What about the Roche Limit? Would the planet breakup?

Postby preearth » Tue Jun 22, 2010 5:58 am

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Models of Heaven and PreEarth that we can work with.

Since, Heaven and PreEarth were the original binary pair, we will make the natural assumption that both Heaven and PreEarth have had similar histories since their formation.

This means that they would have had a similar, core iron to silicate rock, ratio.

To have had the same core iron to silicate rock ratio, PreEarth must have had fifty eight percent of Earth's core and Heaven must have had the remaining forty two percent. For future reference,

M_E = Earth's Total Mass = 5.97369 x 10^24 kg,
C_E = Earth's Core Mass = 1.90675 x 10^24 kg.

We will assume that before the collision both planets had a uniform temperature of 300 degrees K.

The cold density profile for PreEarth.

In this model of PreEarth we are assuming,

1) a radius of 5200 km,
2) an initial temperature of 300 degrees K and
3) that PreEarth's core is 58% percent of Earth's core.

The radius of 5200 km is estimated from the area of continental crust (see the section "Determining Their Size" of http://preearth.net).

Using the Earth as our model, we guess the following density profile for PreEarth, namely;

Image

With this density profile we have the following numbers:

M_P = Total Mass = 3.483 x 10^24 kg
R_P = Radius = 5200 km
C_P = Core Mass = 1.106 x 10^24 kg
B_P = Binding Energy = 10.379 x 10^25 MJ
P_P = Pressure at center of PreEarth = 268 GP
G_P = Surface Gravity = 8.5966 m/s^2

The binding energy, pressure at center and surface gravity are derived from the density profile.

The cold density profile for Heaven.

In this model of Heaven we are assuming,

1) a radius of less than 5050 km (see http://preearth.net),
2) an initial temperature of 300 degrees K and
3) that Heaven's core is 42% percent of Earth's core.

Image

With this density profile we have the following numbers:

M_H = Total Mass = 2.48456 x 10^24 kg
R_H = Radius = 4680 km
C_H = Core Mass = 0.80058 x 10^24 kg
B_H = Binding Energy = 5.84173 x 10^25 MJ
P_H = Pressure at center of Heaven = 200 GP
G_H = Surface Gravity = 7.5712 m/s^2

The radius of Heaven is fixed by the assumptions 1), 2), 3) and the density profile we choose.

The binding energy, pressure at center and surface gravity are derived from the density profile.

With these quite reasonable models of PreEarth and Heaven, we can make various calculations and predict certain aspects of the collision.

What about the Roche Limit (as regards the collision).

The Roche limit, sometimes referred to as the Roche radius, is the distance within which a celestial body, held together only by its own gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction.

Roche Limit = R_H*(2*M_P/M_H)^(1/3)
= 4680*(2*3.48280 x 10^24/2.48456 x 10^24)^(1/3)
= 4680*1.41006 = 6600 km

Since 6600 is less than the distance between the centers = 5200 + 4680 = 9880 of the planets, at impact, the Roche Limit is not reached until after the collision and is thus irrelevant.
preearth
 
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