How close to each other can Earth-mass planets plausibly form from the protoplanetary disk?

We occasionally get questions and answers that discuss how close to each other planets can be and still meet some criteria. For example, this answer to the question ''Habitable'' planet close to a star.

For the purposes of this question, I am not interested in answers discussing rogue planets which have been captured by the star. I am only interested in planets that form from a star's protoplanetary disk and remain gravitationally bound to that star.

I am also not interested in co-orbiting planets, such as anything at another planet's L4 or L5 point, or dual-planet pairs (not entirely unlike our Pluto and Charon). The planets I am talking about should each be in distinct orbits around the star, which differ in more than just phase.

Quite simply, given what we know about planetary formation and material density in a star's protoplanetary disk, what is the smallest difference in orbital radius where distinct, rocky planets similar in size to Earth can plausibly form?

I'm looking primarily for the formation of planets of similar total mass (let's say to within ±30% of Earth's mass), but bonus points if they can also plausibly have a similar composition (which, as also pointed out in a comment, I suspect is the easier part).

I realize that the answer will to some extent depend on the orbital radius of the planet that is forming, since there's more distance to cover around a circle and thus at least the potential for more material to go around the farther from the focal point you are. Therefore, let's limit this to final planets' orbits with radii of 0.5 to 1.5 AU from the central star. For simplicity, it is acceptable to assume that all orbits in the resulting system are perfectly circular, though if you want to do proper elliptical orbits, then go ahead.

The planets should ideally be able to hold together on their own over an astronomical timeframe. If you want to go down that rabbit hole, I suppose some such formed planets being ejected (or having their orbits changed to the point of, say, falling into the star) by perturbations from other planets could be acceptable, but it'd be a nice touch if the orbit is stable over the long term.

• @aCVn - would you accept very small Red Dwarf stars like TRAPPIST-1? (average distance between earth-like planets here is 0.004AU - 0.015AU) Jul 18, 2019 at 14:32
• @Raisus If a red dwarf has a different protoplanetary disk matter density compared to other stars, sure. (Otherwise, the stellar classification of the star shouldn't matter much...) I don't see why that would necessarily be the case, though, so in that case you might want to provide a citation for that assumption.
– user
Jul 18, 2019 at 19:42
• The problem is what is meant by "form". Does the OP mean when the mass of the planet is gravitationally bound together, or must enough energy have dissipated to allow a rocky scum to form on top of the molten interior? How thick must the scum be before it is a rocky planet? How similar to earth must the planets be? Oceans? Weather? Plate tectonics? Magnetosphere? Atmosphere? Life? At what point are we to judge that a rocky planet, similar to Earth, has been formed?
– cmm
Jul 22, 2019 at 19:28
• Another potential issue with this question is the lack of more concrete definition of "similar" total mass. You provided an acceptable range for orbital distance, but a defined range for mass would also help, as the mass of the planets will affect the Hill Sphere's which is significant to orbital distances. Jul 24, 2019 at 21:42
• @Dalila I've added a number to quantify "similar mass".
– user
Jul 25, 2019 at 8:07

Planets will likely form in 2:1 resonances

From Zhu, et al., 2018, simulations showed that multiple planet formation from a circumstellar dust cloud results very strongly in 2:1 planetary resonances, at least upon initial formations.

In most cases, the planets are trapped into 2:1 resonances between each pair of adjacent planets...Whenever the resonant angle librates between the interval [-π,π] the planets are trapped in a 2:1 resonance. The smaller the amplitude of libration, the deeper resonance locking.

All exceptions to this resonance ended up causing one or the other proto-planet to be dispersed by gravitational interactions before it was fully formed.

There are two cases, however, where planets experience a close encounter and are gravitationally scattered.

Over their million years simulation, formations of three or more planets cause migration out of the original resonances, as we would expect. On the other hand, this model specifically deals with the formation of giant planets, so the possibility of smaller, more Earth-like planets exists.

Notes

I'll add more potential answers as (if) I find them. My review so far makes it seem unlikely that there is much settled science to be found in this area.

• Not quite true... Again; look at the Resonances of TRAPPIST-1 (TL/DR Sic: The relative orbital periods (proceeding outward) approximate whole integer ratios of 24/24, 24/15, 24/9, 24/6, 24/4, 24/3, and 24/2, respectively, or nearest-neighbor period ratios of about 8/5, 5/3, 3/2, 3/2, 4/3, and 3/2 (1.603, 1.672, 1.506, 1.509, 1.342, and 1.519).) Jul 29, 2019 at 12:44
• @Raisus That reflects post-formation planetary migration. This article discusses the initial formation from a uniform circumstellar dust cloud. Of course, there is no saying that a dust cloud would be uniform. All in all, this isn't a well developed topic. Jul 29, 2019 at 12:52
• Of course, if it were; someone might win a Nobel Jul 30, 2019 at 12:39

I am by no means an expert in planetary formation, but I actually think this boils down something called the Hill Sphere. For circular orbits, the radius of the Hill Sphere is given by $$r_H\approx a \sqrt[3]{\frac{m}{3M}}$$. For Earth and the Sun, this equates to about 1.5 million km or around 0.01 AU.

In developed systems the Hill Sphere is "the region in which an astronomical body dominates the attraction of sattelites." The upshot of this is that any body that forms within this radius would either become a satellite or be ejected. Whether it's reasonable that a second planet could form within this distance is determined by the properties of the protoplanetary disk, but if the conditions are right to form a pair of planets so close together, I'm fairly confident this is your absolute lower bound.

The planet Theia formed near or at the L4 or L5 Lagrange point of the Earth.

I see that the OP rules out planets that are very close to each other.

I am also not interested in co-orbiting planets, such as anything at another planet's L4 or L5 point, or dual-planet pairs (not entirely unlike our Pluto and Charon).

I am not sure why these are excluded. Planetary formation is planetary formation, so I am going to answer with the Theia example anyway. If two planets can form that close to each other it would seem to me to set the inner limit of how close to one another planets can possibly form. The linked wikipedia article states Charon is a big moon that was knocked off of Pluto by an ancient impact much as our own moon; not really relevant to the question of planet formation.

Theia was an ancient planet the size of Mars which collided with and merged with Earth, in the process forming the moon. I here assert that Theia formed from the protoplanetary disk at the L4 or L5 point of the ancient Earth.

Evidence.

Prior to merging with the Earth, Theia is thought to have orbited at L4 or L5.

https://en.wikipedia.org/wiki/Theia_(planet)

Theia is thought to have orbited in the L4 or L5 configuration presented by the Earth–Sun system, where it would tend to remain. In that case, it would have grown, potentially to a size comparable to Mars. Gravitational perturbations by Venus could have eventually put it onto a collision course with the Earth.[7]

Theia hit Earth gently. It was not swooping in from distant reaches of the solar system.

https://en.wikipedia.org/wiki/Giant-impact_hypothesis

In astronomical terms, the impact would have been of moderate velocity. Theia is thought to have struck the Earth at an oblique angle when the Earth was nearly fully formed. Computer simulations of this "late-impact" scenario suggest an impact angle of about 45° and an initial impactor velocity below 4 km/s.[16]

That makes sense if it was only at L4 before impact and eventually had its orbit perturbed by Jupiter or Venus.

Earths Titanium Twin

By contrast, Zhang et al.1 find that the Earth and the Moon are identical in their titanium isotopic compositions within errors of 0.0004% — almost the limit of detectability. This is not the first time the giant impact hypothesis has been challenged by isotopes. During the past decade, similarities between lunar and terrestrial rocks have been identified for oxygen6, silicon7, chromium8 and tungsten9 isotopes. The latter three can be brought into accordance with the latest giant impact simulations5, if one assumes that Theia had a composition similar to Mars — possibly the only surviving planetary embryo from which the larger terrestrial planets accreted10. However, the oxygen isotopic compositions of terrestrial and lunar rocks are so similar that, if Theia had a Mars-like composition, it cannot have contributed more than a few per cent of material to the Moon-forming disk6. Zhang et al. demonstrate that titanium isotopes are similarly constraining.

The Earth and the Moon have the same isotopic composition. Where is the contribution from Theia? It is there. The Earth, the Moon and Theia all had the same composition.

On the origin and composition of Theia

If the FeO content of the Earth and the Moon is indeed inherited from the proto-Earth and Theia, then by implication Theia must have had an Earth-like isotopic composition (similar to enstatite chondrites, aubrites or other Earth-like materials like NWA 5400). This is possible if both the Earth and Theia, but not Mars, were part of an early inner disk uniform reservoir (IDUR; Dauphas et al., 2014), or if the inner disk region has been isotopically homogenized in the time between the isolation of Mars from the disk and the Giant Impact that formed the Moon.

From OP: How close to each other can Earth-mass planets plausibly form from the protoplanetary disk?

Theia formed right next to Earth from materials in the same region of the disk. The near-identical isotopic composition of the Earth and Moon precludes the possibility of Theia being some weird asteroid, or even Marslike in composition. Theia is formed of Earthlike materials. It formed closer to Earth than Mars is. I cannot see how two planets can form any closer than that.

Nothing is going to form closer than the Lagrange points because it would fall into the planet. The example of Theia and Earth forming next to each other sets the minimum distance from one another that planets can form from the protoplanetary disk.