Moved into further orbits to protect them, how much damage do Earth and Moon take when the Sun expands?

Question

Short version A far future catastrophe burns/melts away large chunks of the Earth and Moon. For reasons related to my story's fantasy elements, the remains of the Moon have about one fifth their original diameter, and I'd like to know what the corresponding reduction in size for the Earth would be.

Long version:

In the far future, humanity (or whichever species now occupies the Solar System) have been desperately working on technology capable of moving the planet to a further orbit. The Sun is expanding into a subgiant, and they want Earth far enough out to survive its expansion into a red giant.

It would be simpler to relocate to other planets - and they are in fact preparing to evacuate - but Earth and the Moon are of tremendous symbolic importance and they would like to save their planet of birth from destruction. The Sun's increase in brightness over time has already rendered the Earth incapable of supporting life, so it's too late for that, but the future people still want it to continue existing as a monument to their origins.

A 2008 research paper considers the question of just how far from the sun Earth would need to be to survive:

planet Earth will not be able to escape engulfment, despite the positive effect of solar mass-loss. In order to survive the solar tip-RGB phase, any hypothetical planet would require a present-day minimum orbital radius of about 1.15 AU. The latter result may help to estimate the chances of finding planets around White Dwarfs.

A paper from 2012, which does not cite the first, states that the present-day radius for a terrestrial planet orbiting a star of one solar mass would need to exceed 1.5 AU. It also states that by the time the Sun has become a white dwarf, the remains of the planet would be about 2.5 AU distant.

And that's just to avoid engulfment! The habitable zone will have been moving further out, and 1.15AU will not be enough to keep the Earth suitable for supporting life. By the time the sun is starting to expand, the HZ has moved further out to start at 1.29 AU (source: 2008 paper.) By the time it reaches the point of maximum expansion, the HZ would start at 49.4 AU! (source: 2008 paper again)

Cut to an even longer time in the future. The Sun has now shrunk significantly. Maybe it's on the horizontal branch and will eventually expand again in the AGB. Or maybe that's already happened, the Sun has ejected its planetary nebula and is now a white dwarf.

Whatever the case, alien explorers onboard their ship, the U.S.S. Rocinante, have just arrived in the Solar System. It's clear the humans didn't manage to move Earth far enough. Maybe to very nearly 1.5 AU, or to only very slightly more. (replace 1.5 with 1.15 if you prefer the 2008 paper.)

The Earth and Moon have lost a lot of their mass. They were very close to the red giant, and so some of their surface may well have heated up enough to evaporate. Even if this didn't happen, hot and ionized gases given off by the Sun may have burned away material from the surface.

For the purposes of the story, I want this to reduce the Moon's diameter to about 1/5 of what it was. This would imply that its volume is only 1/125 of the present-day Moon's.

(I'm assuming the Earth and Moon are still close by each other, please let me know if I'm wrong!)

My main question is:

How much loss of volume would the catastrophe that did this have caused to the Earth?

My initial guess would be that the Earth, like the Moon, would lose about 2779.84 km of its average diameter, and calculations indicate that it would then have 0.478 x its original volume. I'm afraid I'm not as scientifically knowledgeable as some of my characters, so I don't really have confidence in this guess and have turned to the good people of the Planet Stackexchange for help!

I do also wonder about a few other properties of the smaller Earth - would it be able to generate a magnetic field? How far does the Sun need to recede before it can start to regain some sort of atmosphere (albeit non-oxygen-rich)? But the question about the Earth's loss of volume is my primary topic.

Sources:

• K.-P. Schröder, Robert Connon Smith, Distant future of the Sun and Earth revisited, Monthly Notices of the Royal Astronomical Society, Volume 386, Issue 1, May 2008, Pages 155–163, https://doi.org/10.1111/j.1365-2966.2008.13022.x

• Mustill AJ, Villaver E. Foretellings of Ragnarök: world-engulfing asymptotic giants and the inheritance of white dwarfs. The Astrophysical Journal. 2012 Dec 4;761(2):121.

Answer 1

My initial guess would be that the Earth, like the Moon, would lose about 2779.84 km of its average diameter, and calculations indicate that it would then have 0.478 x its original volume.

My initial guess is more like "this is really hard".

In order to remove mass from a body, you need to deliver enough energy to a portion of it to allow it to achieve escape velocity. Escape velocity depends on mass and distance:

$$v_e = \sqrt{\frac{2GM}{r}}$$

where $G$ is the gravitational constant, $M$ is the mass of the body and $r$ is the distance from the body's barycenter of the thing that's trying to escape. Earth is heavier and denser than the moon, which is why its escape velocity at the surface is about 4.7x higher. Kinetic energy being proportional to velocity squared, it'll require ~22 times more energy to kick a particle out of Earth's gravitational grasp than the equivalent particle on the Moon.

At a lazy first guess, then, I'd go for a loss of closer to 130km. It might even be lower than that. The moon has clearly suffered a catastrophic mass loss, and as such its escape velocity has dropped dramatically during the process. The Earth would appear to be losing much less mass over the same time period, so the reduction in escape velocity will be lower.

I'm not going to work out how much lower, as the whole thing seems a bit too hard for my meagre maths ability. I suspect it would be enough to blow off the lithosphere (the crust, continental and otherwise, plus all the oceans) but not the asthenosphere (the squidgy upper mantle) because the latter is several hundred kilometres thick on average.

There are a few other things tipping the balance in favour of the erosion being much lower, including the Earth's density (1.65x higher) implying more heavy elements that are harder to boil away and Earth's magnetosphere which will help reduce the effects of the increased solar wind. I won't even attempt to quantify these, however.

would it be able to generate a magnetic field?

If my estimates are in the right ballpark, then maybe! All of the useful bits of the geodynamo would still be very much intact, and the infusion of heat that roasted off the crust might even have delivered a bit more energy to it.

Thing is though, the dynamo is slowing down and at some point it will presumably stop. I can't find a handy reference suggesting when that might be, or whether it is possible to kickstart it again if it did stop before being reheated. There's a chance that it will have simply stopped and been unable to restart, given the timescales involved in your scenario.

That warrants a bit more research, anyway.

How far does the Sun need to recede before it can start to regain some sort of atmosphere (albeit non-oxygen-rich)?

There's a not-unreasonable chance that it formed a rather Venusian atmosphere as the roasting effect of the Sun was reduced. Given that it may well have held on to much of its mass, that thick atmosphere may well have stuck around.