Did the force of the moon’s collision knock the Earth onto its side?

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*10²³ kg*v_Theia = 5.972*10²⁴ kg * 29800 m/s + 7.348*10²² kg * 1000 m/s, so Theia must have been traveling at a speed of approximately 3*10⁵ 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).