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.
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.
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