This week, I went to the Boston Museum of Science to watch
the film Moons: Worlds of Mystery. This movie got me thinking, and I started
reading up a little on the formation of Earth's moon. A few weeks ago, a new
article published in the New York Times described the original collision of
the Earth with the moon. The article cited a recent
paper published in Nature, which modified the giant-impact theory.
According to the giant-impact theory (also known as “the big whack”), a
catastrophic collision between the Earth and a protoplanet created chunks of
debris, which eventually consolidated to form the moon (See picture below).

However, this model no longer makes sense, given what we now
know about the moon’s Earth-like composition of isotopes. Another key
discrepancy is the five-degree tilt of the moon’s orbit compared with the
orbits of pretty much everything else in the solar system. The Nature article
proposes a new theory: maybe the moon’s orbit is still tilted, because Earth’s
orbit used to be tilted, prior to the “big whack.” The model relied on complex numerical
simulations to support this hypothesis. While I was unable to understand much
of the math behind this model, I did use my rudimentary physics knowledge to
estimate the speed with which Theia, the protoplanet that would eventually form
the moon, must have been traveling when it collided with Earth.

Assuming that friction and drag don’t exist in space, the only
force acting on the moon should be the force of gravity. The collision was both
elastic and inelastic, since some of the material from Theia remained as part
of the Earth, while the rest formed the debris that would eventually become the
moon. (See animation below)

Assuming conservation of momentum and that the proto-Earth
was at rest prior to the collision, m

_{Theia }v_{Theia }= m_{Earth}v_{Earth }+ m_{moon}v_{debris}, where the mass of Earth is m_{Theia}+ m_{proto-Earth}- m_{moon}. Of course, this makes many assumptions, for instance assuming that all debris has the same speed (and assuming that the Earth was, in fact, at rest). Making even more assumptions that the speeds haven’t changed in the 4.31 billion years post-collision (therefore ignoring the gravitational effects of the sun, the moon, and Earth), ignoring all rotational motion, and substituting in the masses of Theia (assumed to be 0.1 times the mass of Earth), the Earth, and the moon, the equation becomes: 5.972*10^{23}kg*v_{Theia }= 5.972*10^{24}kg * 29800 m/s + 7.348*10^{22}kg * 1000 m/s, so Theia must have been traveling at a speed of approximately 3*10^{5}m/s prior to the collision.
While this is probably not a very
accurate estimate of Theia’s actual speed, it does give a new appreciation for
the complexity involved in computational astronomy. Interestingly, research suggests that the post-collision angular momentum is the same as the Earth and
moon’s combined angular momentum now (I would have used this approach, but I
couldn’t find the initial rotational velocity of proto-Earth).

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