How soon does the Earth's surface re-solidify after the red-giant Sun is replaced with a different star?

Question

(Note: This is a follow-up question to my previous one: Moved into further orbits to protect them, how much damage do Earth and Moon take when the Sun expands?)

Thanks to clever stellar engineering by a group of aliens (see below.), the Sun has been induced to end its red giant stage early by turning into a blue-white (B-type) subdwarf. These have lifespans of less than 200 million years, plus another 20-40 million as a bluer O-type subdwarf, before cooling toward the white dwarf stage.

One of my questions on Astronomy.SE notes that sdB stars originate from main-sequence stars with mass in the range $0.5M_{\odot} \leq M \leq 2M_{\odot}$. Certainly our Sun is in that range, though I'm trying to find out if tighter bounds are known to astronomers!

I'm assuming the subdwarf is a fairly typical star of that type. Mass between $0.29$ and $0.53M_{\odot}$, surface temperature between 27,000 and 36,000 K (I don't know why stars at the higher end of that range weren't filed as O-type instead of B, but they exist.), luminosity $22.9-34 L_{\odot}$, and the star is rotating (though as I type this, I don't have a range of values for just how fast to consult.)

My question:

How soon after the red giant Sun lost its hydrogen envelope and turned into this star would it take for the damaged Earth from my previous question (which had been moved to a 1.15 AU orbit) to cool down enough that there is once again a solid crust, with continents people can walk on? (Probably wearing protective clothing.)

Notes:

1. The stellar engineering involved stealing either:

   • a gas giant similar to Saturn - but much larger, somewhere between 1 and 5 times the size of Jupiter, or
   • a brown dwarf.

and parking it close enough to the main-sequence Sun to form a binary. The planet's core survived engulfment, but its presence inside the star caused the Sun to lose its hydrogen envelope prematurely, turning it abruptly from a red-giant into a B-subdwarf.

This is what was meant to have happened with Kepler-70 (aka KIC 05807616). The exoplanets were the remains of the core or cores of the Hot Jupiter gas giants involved. Though more recent research has suggested that they may not in fact exist.

2. These are not the alien explorers from my previous question, they're another group. But the explorers have realised "This star shouldn't have reached the white dwarf stage so soon!" and are getting ready to travel back in time and find out what happened.

3. I do have some information on how long the Moon took to solidify when it was intially formed, thanks to a 2011 paper. According to this, 80% of the Moon's magma ocean solidified in about 1000 years, however the plagioclase crust that had formed atop it acted as a "conductive lid". Counterintuitively, this slowed the remainder of the cooling process down significantly. Tidal heating from the Earth slowed down the remainder of the process even more significantly, melting portions of the crust and causing new eruptions. The total time was approximately 220 million years, but would have been only about 10 million without the tidal effects.

   The 220 million may still be an underestimate - a later 2015 paper suggests that it may be roughly 300 million.

   In another question on this site, I discuss the geology of the exposed layer of the smaller moon. You can see it at:

   (The Earth and Moon resolidify under a bluer star, their outer layers evaporated and burned away. What do they look like now?)

   In brief, the plagioclase is now burned away, and there isn't enough aluminium left in the iron-rich remains of the moon to form another plagioclase crust. From the 2011 paper, we discover that there isn't another way for the Moon to form a conductive lid, so the solidification process should now be faster than before. How much faster isn't clear, but the aforementioned papers plus a 2010 paper and a paywalled 2008 paper suggest that even with the tidal effects it should be a few tens of millions of years at most.

Sources:

Dorman, B., Rood, R., & O'Connell, R. (1993). Ultraviolet Radiation from Evolved Stellar Populations--I. Models. arXiv preprint astro-ph/9311022. For a version that includes the diagrams but doesn't allow you to select text, see here.

Elkins-Tanton, L. T. (2008). Linked magma ocean solidification and atmospheric growth for Earth and Mars. Earth and Planetary Science Letters, 271(1-4), 181-191. I'm afraid this one's paywalled.

Heber, U. (2009). Hot subdwarf stars. Annual review of Astronomy and Astrophysics, 47, 211-251. There are also slides.

Østensen, R. H. (2010). Observational asteroseismology of hot subdwarf stars. Astronomische Nachrichten, 331(9‐10), 1026-1033.

Meyer, J., Elkins-Tanton, L., & Wisdom, J. (2010). Coupled thermal–orbital evolution of the early Moon. Icarus, 208(1), 1-10.

Elkins-Tanton, L. T., Burgess, S., & Yin, Q. Z. (2011). The lunar magma ocean: Reconciling the solidification process with lunar petrology and geochronology. Earth and Planetary Science Letters, 304(3-4), 326-336.

Charpinet, S., Fontaine, G., Brassard, P., Green, E. M., Van Grootel, V., Randall, S. K., ... & Telting, J. H. (2011). A compact system of small planets around a former red-giant star. Nature, 480(7378), 496-499. This is the paper which announced the discovery of the Kepler-70 exoplanets, before research in later years provided a strong counterargument and suggested that they didn't in fact exist. It also reveals that Kepler-70 has been a B-subdwarf for 18.4 million years so far.

Bear, E., & Soker, N. (2012). A tidally destructed massive planet as the progenitor of the two light planets around the SDB star KIC 05807616. The Astrophysical Journal Letters, 749(1), L14. This is the one that suggested the Kepler-70 exoplanets might not be the remains of two separate Hot Jupiter gas giants, but one. The theory being that the core of that planet did not completely survive engulfment, and was split into two.

Suckale, J., Elkins‐Tanton, L. T., & Sethian, J. A. (2012). Crystals stirred up: 2. Numerical insights into the formation of the earliest crust on the Moon. Journal of Geophysical Research: Planets, 117(E8).

Schindler, J. T., Green, E. M., & Arnett, W. D. (2015). Exploring stellar evolution models of sdB stars using MESA. The Astrophysical Journal, 806(2), 178. This one's particularly relevant to the question of Subdwarf B lifespans.

Planetary candidates around the pulsating sdB star KIC 5807616 considered doubtful. J. Krzesinski A&A, 581 (2015) A7 DOI: https://doi.org/10.1051/0004-6361/201526346. This is the one which provided evidence that the things which had seemed to indicate exoplanets in 2011... probably didn't. As someone who loves the idea of planets orbiting blue stars, you have no idea how disappointed I was to read this!

Kamata, S., Sugita, S., Abe, Y., Ishihara, Y., Harada, Y., Morota, T., ... & Matsumoto, K. (2015). The relative timing of Lunar Magma Ocean solidification and the Late Heavy Bombardment inferred from highly degraded impact basin structures. Icarus, 250, 492-503.

Heber, U. (2016). Hot Subluminous Stars. arXiv preprint.

Sleep, N. H. (2016). Asteroid bombardment and the core of Theia as possible sources for the Earth's late veneer component. Geochemistry, Geophysics, Geosystems, 17(7), 2623-2642.

Deca, J., Vos, J., Németh, P., Maxted, P. F. L., Copperwheat, C. M., Marsh, T. R., & Østensen, R. (2018). Evolutionary constraints on the long-period subdwarf B binary PG 1018–047. Monthly Notices of the Royal Astronomical Society, 474(1), 433-442.

Analysis of putative exoplanetary signatures found in light curves of two sdBV stars observed by Kepler. A. Blokesz, J. Krzesinski and L. Kedziora-Chudczer A&A, 627 (2019) A86 DOI: https://doi.org/10.1051/0004-6361/201835003

Answer 1

Okay... it's not a precise answer, but here goes. I'll start by discussing my research into the Moon's solidification, since it's relevant, especially as the Moon took longer to solidify than the Earth originally. (I edited all this into the question as supporting information.) Then I'll go from there to the case of the Earth.

According to a 2011 paper, 80% of the Moon's magma ocean solidified in about 1000 years. However, after this point, the plagioclase crust that had formed atop it acted as a "conductive lid". Counterintuitively, this slowed the remainder of the cooling process down significantly. Tidal heating from the Earth also slowed down the remainder of the process significantly, melting portions of the crust and causing new eruptions.

The total time was somewhere roughly in the time range of 220 million years to 300 million years. If the tidal effects were not present but the conductive lid was, it would only have been about 10 million years. I don't have any figures for a situation where the tidal effects are present but not the conductive lid.

In another question on this site, I discuss the geology of the exposed layer of the smaller moon. You can see it at this link:

The Earth and Moon resolidify under a bluer star, their outer layers evaporated and burned away. What do they look like now?

In brief, the plagioclase is now burned away, and there isn't enough aluminium left in the iron-rich remains of the moon to form another plagioclase crust. From the 2011 paper, we discover that there isn't another way for the Moon to form a conductive lid, so the solidification process should now be faster than before. How much faster isn't clear, but the aforementioned papers plus a 2010 paper and a paywalled 2008 paper ("Linked magma ocean solidification and atmospheric growth for Earth and Mars.") suggest that even with the tidal effects it should be a few tens of millions of years at most.

I don't feel able to use a stronger word than "suggest", though.

According to a 2012 paper, these conductive lids aren't expected to form on most planets. In addition, a 2005 paper shows in Table 3 that the Earth's mantle doesn't have much aluminium in it to form a plagioclase lid anyway.

(I should warn you, Table 3 of that paper can be a bit hard to understand - I've posted on a Wikipedia talk page because I thought it contradicted something in that article, when in fact it didn't.)

Moving from the Moon to the Earth, we now come back to our 2008 paper, "Linked magma ocean solidification and atmospheric growth for Earth and Mars." This one is behind a paywall, and if anyone has a non-paywalled link to it, please edit it into this answer! It's a very highly cited and interesting paper that I think will be of interest to several worldbuilders.

Table 3 of that paper (Ooh, we've got a lot of Table 3s here!), in a case where there's no initial $H_{2}O$, gives various scenarios relating to the depth of the magma ocean and the amount of $CO_2$ in the atmosphere. Importantly, some of these cover cases where there's no water vapour in the atmosphere or magma ocean. The Earth takes longer to solidify in these cases, carbon dioxide being a more powerful greenhouse gas than water vapour. But anyway, the longest the Earth takes to reach 98% solidification in these is 5.3 Myr. (Mars takes 2.8 Myr under similar conditions.)

It's said that it should be at least five million, and at most some value on the order of tens of millions of years, to reach "clement" conditions after that's happened. The no-water-vapour case wasn't one of a few cases modelled in more depth than the others, but the paper does seem to be referring to all possible cases when it claims this, especially in the abstract at the start.

With the lid, the 2010 paper by Elkins-Tanton et al explicitly states that the Moon took longer than the Earth to resolidify, and that the lid was the reason. That language suggests that without the lid, the tidal effects wouldn't have slown down the Moon's cooling by enough to keep it molten much longer than the Earth.

As seen in my comments on Zeiss Ikon's answer, the smaller blue Sun is probably supplying less than a third as much heat to the Earth and Moon as before. A typical B subdwarf with 1/5 the Sun's radius has 1/25 the surface area. They are hotter - the Sun has $5772K$ surface temperature, the hottest sdB star I know about has $\leq 36,000K$, and that's a 6.237-fold temperature difference, but once you divide that by 25 you get about 0.25. Now, I have seen an unsupported claim that O-type subdwafs can go up to 100,000K, but even then the reduced surface area means that it's still supplying less heat to Earth than the original Sun did. (Only about 0.693 times as much as before). So that should also help the Earth cool and solidify faster than it did the first time.

As should the fact that there's no Theia impact or Late Heavy Bombardment this time.

Finally, we take another look at the 2012 paper I mentioned earlier. In section 5, the author makes the assumption that it took Earth roughly 50 million years to cool down to "clement" (inc. solid) conditions after the Theia impact. I don't know if there's a scientific consensus on whether Earth actually had solidified prior to the Theia impact energy melting the crust again, though. Figure 7 gives a total cooling time of 55 million years, relying on a very steep temperature dropoff in the final 5 million. I think I may have to increase this for the no-water-vapour atmosphere, though, especially if a carbon dioxide atmosphere isn't convective enough.

Anyway... at this point, there's a lot of evidence for "on the order of tens of millions of years" as the answer for full solidification, although I don't think it's proven conclusively for this particular scenario. And for large amounts of partial solidification but a world too hot for humans to inhabit unprotected, it's even less, 5.3 million years at most, maybe sufficiently solid in only 1000 years!

So my answer is "on the order of a few tens of millions of years, probably quite a bit less than 100 million years, but even then I'm still not 100% sure."

Sources:

Workman, R. K., & Hart, S. R. (2005). Major and trace element composition of the depleted MORB mantle (DMM). Earth and Planetary Science Letters, 231(1-2), 53-72.

Elkins-Tanton, L. T. (2008). Linked magma ocean solidification and atmospheric growth for Earth and Mars. Earth and Planetary Science Letters, 271(1-4), 181-191. I'm afraid this one's paywalled.

Meyer, J., Elkins-Tanton, L., & Wisdom, J. (2010). Coupled thermal–orbital evolution of the early Moon. Icarus, 208(1), 1-10.

Elkins-Tanton, L. T., Burgess, S., & Yin, Q. Z. (2011). The lunar magma ocean: Reconciling the solidification process with lunar petrology and geochronology. Earth and Planetary Science Letters, 304(3-4), 326-336.

Elkins-Tanton, L. T. (2012). Magma oceans in the inner solar system. Annual Review of Earth and Planetary Sciences, 40, 113-139.

Suckale, J., Elkins‐Tanton, L. T., & Sethian, J. A. (2012). Crystals stirred up: 2. Numerical insights into the formation of the earliest crust on the Moon. Journal of Geophysical Research: Planets, 117(E8).

Kamata, S., Sugita, S., Abe, Y., Ishihara, Y., Harada, Y., Morota, T., ... & Matsumoto, K. (2015). The relative timing of Lunar Magma Ocean solidification and the Late Heavy Bombardment inferred from highly degraded impact basin structures. Icarus, 250, 492-503.

Sleep, N. H. (2016). Asteroid bombardment and the core of Theia as possible sources for the Earth's late veneer component. Geochemistry, Geophysics, Geosystems, 17(7), 2623-2642.

Answer 2

Given that the inner Solar System has been pretty thoroughly cleaned by the swollen red giant phase of the Sun's senescence, there could be no Late Heavy Bombardment to remelt the intial crust that begins to form almost immediately after the engulfment that cause the melting is halted.

Whatever primordial crust Earth had predated the Moon-forming event. We have no Theia impact to form the Moon (in the process, completely or almost completely remelting the Earth), no (currently in dispute) millions of years of asteroidal bombardment -- in fact, I'd question whether the existing crust would melt completely from a few million years orbiting inside the fairly rarified (albeit pretty hot) outer envelope of the Sun. Dating of the oldest known rocks on both Earth and Moon suggests the Earth had begun to reform a crust less than a hundred million years after the Theia impactn -- and given that nearly all of that "second crust" has since been recycled by erosion and subduction, that figure might well have been much shorter.

On that reasoning, I'd say there ought to be a solid surface to land and walk on within, at most, a few million years of the Sun's induced sub-dwarf collapse (because of the small radiating surface, the radiation of the blue to blue-white sub-dwarf would be trivial compared to the cooling effect of the magma surface radiating to space). Fresh magma flows form a weight bearing crust in days or weeks, though that surface is still hot enough to melt boot soles for many times that span. The far deeper heat beneath the surface of a fully melted crust would extend that figure significantly, but rock isn't a very good heat conductor -- hence why there are still rocks hot enough to boil water hundreds of thousands of years after the most recent eruption at Yellowstone.

Best wear heavily insulated vacuum suit boots; the ground will be nearly as hot as the crust on a fresh magma flow. Also be careful where you land, as there might be thin spots in the crust that aren't up to supporting a starship or heavy lander, and plate drift might locally be real-time visible.