Proto-Earth was ~230 Earth Masses

[Widdekind]

Terrestrial worlds are 28% iron by mass*. The Solar nebula was 0.11-0.13% iron by mass**. Assuming that the early Earth kept all of its iron:

M_sub-nebula x 0.0012 = M_earth x 0.28​

Thus:

M_sub-nebula = M_earth x 230​

There are ~2.0 Earth-masses of rocky bodies in the Inner Solar System. Thus, those ~2.0 Earth masses of rock condensed out of ~460 Earth-masses of sub-nebular gases. This is comparable to the combined masses of Jupiter & Saturn. So, perhaps Jupiter & Saturn accumulated their material from the Inner Solar System. This also suggests that Jupiter's rocky core is only about 1.5 Earth-masses, that Saturn's rocky core is only about 0.5 Earth-masses, while those of Uranus & Neptune are more like Mars in size.

* S.N. Raymond, et al. Exotic Earths: Forming Habitable Worlds with Giant
Planet Migration, pg. 13. (See: http://arxiv.org/PS_cache/astro-ph/pdf/0609/0609253v1.pdf )
** http://www.kayelaby.npl.co.uk/chemistry/3_1/3_1_3.html ; http://astro.uni-tuebingen.de/~rauch/TMAP/UserGuide/UserGuidese12.html​

X = 0.735
Y = 0.248
Z = 0.017​

S.F. Green, et al. An Introduction to the Sun and Stars, pg. 46.

 

[Vanadium 50]

If you pick some other element, you get a different number. I don't think this tells you anything except the Earth has a lot of iron in it.

 

[Widdekind]

True, but picking other elements (eg. Carbon & Oxygen) is inappropriate, b/c Earth lost many of its volatiles. Conversely, I assumed that Earth retained all of its Iron.

For example, the mass-ratio of Carbon-to-Iron is ~2, while the mass-ratio of Oxygen-to-Iron is ~5. Meanwhile, ~0.3 of the Earth is Iron. Thus, all of the C, O, & Fe which were originally present amounted to ~2 M_earth (!). So, the Earth lost at least ~1 M_earth of C & O to space.

Even if the Earth only retained 80-90% of its Iron, the above figure is still accurate to w/in 10-20%. So, the proto-Earth's sub-nebula must have originally contained ~200 M_earth.

However, if you can show that the numbers work out (significantly) differently for, say, Nickel (which must have been lost / retained at substantially the same rate as Iron), that would undermine my assertion. How much mass of Nickel does the Earth contain?

 

[Vanadium 50]

Fine. Pick, say tungsten and rhenium, two elements right next to each other so volatility is not an issue. The Earth is tungsten-rich and rhenium-poor, so your protocloud gets two different values.

 

[Widdekind]

The Earth is, or merely the Earth's Crust ?

Rhenium is denser than Tungsten, by about ~10% (21.02 : 19.25). Denser materials preferentially sink down towards the core, leaving a "deficit" up in the crust.

What about Nickel? Iron & Nickel always seem to go "hand-in-hand", as far as I've read. What is Earth's mass fraction of Nickel?

 

[Widdekind]

Earth is 32.1% Iron & 1.8% Nickel*, a ratio of 17.8. The Solar ratio is ~16.5**, in substantial agreement. Thus, to w/in ~10%, the Earth's sub-nebula once contained ~230 M_earth.

* http://en.wikipedia.org/wiki/Earth#Chemical_composition
** http://astro.uni-tuebingen.de/~rauch/TMAP/UserGuide/UserGuidese12.html​

 

[D H]

You are making a very big assumption here: That the proto-Earth had the exact same composition as the solar nebula. There is no justification for this assumption, and in fact modelers of the development of solar systems assume just the opposite. See, for example, http://arxiv.org/abs/astro-ph/0602217.

 

[Widdekind]

Thank you very much for the reference!

However, that article's abstract discusses the Snow Line, and Gas Giant accretion in the "Ice Zone" beyond ~2 AU. How does that affect the formation of the Earth, inside of the Snow Line, in the "Water Zone" ? If necessary, would you please quote the actual article (?), as I cannot access it.

According to Carroll & Ostlie, Jupiter & Saturn show essentially Solar Abundances. All I'm saying is, Earth would too, if it had kept its massive envelope of Volatiles.

Certainly, Earth accumulated from ~230 M_earth of proto-Solar-nebular material. And, the other terrestrial planets amount to about another M_earth of rock. Thus, the rocky worlds of the Inner Solar System condensed out of ~460 M_earth of proto-Solar-nebular material.

What happened to those ~460 M_earth ? Did the Sun accrete all that gas, ripping it away from the planets ?

 

[Widdekind]

200 M_earth would have been a rather big proto-planetary sub-nebula. With all that gas & dust swirling around, if one proto-planetary core condensed (Earth), why not two such cores (Earth & Moon)?

Moreover, even if the Moon strayed in from another orbit, all that gas & dust could have acted as a "Third Body", to absorb all the energy/angular momentum necessary to trap the Moon in the Earth's gravity-well.

So, assuming that the Earth-Moon system formed sufficiently early on, when all that gas / dust was still present, then the Earth could have trapped the Moon, w/ all the excess energy / momentum "slack" being siphoned off by the gaseous sub-nebula.

More generally, if there were other (sufficiently large?) gravitating bodies in the area, they too could have helped the Earth capture the Moon. If the Earth-Moon system formed early enough, when the region was still "busy" w/ "traffic" of all kinds of impactors, perhaps they could have been flung off for the Moon to fall into orbit.