# What was earth's mass before the collision with theia?

Assume the "Big Splash" or "Giant Impact" (whichever name you prefer) theory is more or less correct.

EDIT: With so many variations of this theory, with so many names and qualifiers on them, such as "direct impact theory" and "glancing blow hypothesis", etc., let's assume that the impact was direct enough that nothing of Theia (other than Earth and the Moon) was left that was large enough to for its gravity to compress it into a hydrostatically equilibrious shape, and scatters too much to coalesce (on its own) in to anything large enough either.

Assume "Theia" was the size of Mars (some variations on the theory use other sizes, but for the sake of this question, let' stick with Mars)

How big was Earth before the impact?

It can't be as simple as "Earth+moon-Theia(mars)=pre-impact size", can it? Because there must have been some debris ejected completely from the earth+moon system, right? or was it such a small amount that it can be considered negligible?

• This kind of depends on whether you believe the glancing blow hypothesis or direct impact hypothesis. I would think the latter would have ejected enough material at escape velocity for it never to return. However, in the former hypothesis it's doubtful anything more than negligible material would have escaped the system. – BillDOe Dec 28 '18 at 21:47
• @BillDOe I've seen some of those terms used in a variety of ways and it seems to me that, in the end, it's really impossible to actually draw a solid line between any of them. Sure, 180 degree difference in direction is certainly "direct", but what about 145 degrees? 125? 110? 95? At what point does it change from 'direct' to 'glancing'? And then there's the ratios of the surface areas that come in contact, with similar variation possibilities. The most common uses, that I've seen, for "big splash" seem to rule out most of the uses I've seen for "glancing", so if I have to pick, that's it – Dalila Dec 28 '18 at 21:58
• @BillDOe: None of those are different hypotheses. It's one and the same theoretical framework, that happens to reproduce the Earth/Moon mantle compositions best when one of the free paramters, the impact parameters, is an oblique angle. – AtmosphericPrisonEscape Dec 29 '18 at 1:08
• I've read one article (can't remember the author) who claimed a more direct, head-on impact, rather than oblique, better explained the earth-moon compositions. Personally, I prefer the oblique angle explanation. I also prefer the hot-earth explanation for the difference between the near side and far side composition of the Moon's crust. – BillDOe Dec 29 '18 at 21:39

$$M_{piE}=\text{Greatest possible mass of pre-impact Earth}\\ M_E=\text{Mass of Earth today}=6.0\times10^{24}kg\\ M_M=\text{Mass of the Moon}=7.3\times10^{22}kg\\ M_T=\text{Mass of Theia}\approxeq6.4\times10^{23}kg\\$$ $$\begin{eqnarray*} M_{piE}&=&M_E+M_M-M_T\\ &=&(6.0\times10^{24})+(7.3\times10^{22})-(6.4\times10^{23})\\ &=&5.4\times10^{24}kg \end{eqnarray*}$$ So it wouldn't have been more than about $$5.4\times10^{24}kg$$.