# If Earth's Moon Were Ganymede-Like, Part I: Rotation

I know that in the past I posted questions on if specific bodies larger than the moon--Mars in one and Titan in another--but this series deals with questions regarding Earth's effects by a natural satellite like Ganymede in two ways--a diameter of 3,274 miles and orbiting its parent from a distance of about 665,000 miles. The major fundamental difference is that this moon has a five-mile deep mantle separating its all-iron core from its rocky crust.

In Episode 1, we look at how the moon creates Earth's rotation. Before the Theia impact event, Earth spins at a daily rate of six hours. After Theia, the moon started life orbiting Earth from a distance of 15,000 miles and leaving orbit at a speed of 1.6 inches--four centimeters--per year. As a result, a day has been getting longer by 2.3 milliseconds since the eighth century before the Common Era. Whereas sea scorpions and trilobites swarmed the oceans under a 21-hour day, Stegosaurus may have been munching on cycads under a 23-hour day. Now that this 2,159-mile-wide body of rock currently orbits Earth at a distance of 238,900 miles, we walk and run under a day lasting 24 hours.

Can an alternate Earth still have a 24-hour day with the moon described in the first paragraph above? And how will its larger size affect the rate of disintegration?

This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

## 1 Answer

Short answer: "All else being equal", it looks to me like making Bizarro-Moon twice as big and three times as far away as the actual Moon can't happen -- not even if you spin the Earth down completely (ie. to the point where the Earth is tidally locked to the Moon, the way the Moon is already locked to the Earth).

So all else can't be equal. But there's a lot of things you could change, and you can get just about any answer you want. So 24 hour days is just fine.

Long answer: Bizarro-Moon is about twice as big and three times as far awaya, so we're talking three or four times the orbit angular momentum (which is proportional to mass and to the square-root of distancec).

The actual Earth lost about three quarters of its spin angular momentum (which is inversely proportional to spin periodd): days went from 6 hoursb to 24 hours. And the Moon also lost its own spin completely and is tidally locked -- but the Moon is a lot smaller than the Earth (angular momentum proportional to massd), so the Earth's spin angular momentum is a lot bigger. Bizarro-Moon is somewhat bigger but still a lot smaller than the Earth.

So: most of the Moon's orbit angular momentum came from the Earth's spin, most of that spin is gone now, and the result only got the actual-sized Moon to its actual position. You need three or four times more of that to get Bizarro-Moon where you want it -- so it won't get there, even if the Earth becomes completely tidally locked.

But maybe Bizarro-Earth is somewhat bigger than the actual Earth, and maybe the collision that produced Bizarro-Moon was imparted faster initial spin to Bizarro-Earth and/or a greater initial orbital angular momentum to Bizarro-Moon.

There's also the problem that Bizarro-Moon might be orbiting so far from Earth that its orbit is unstable due to Jupiter perturbing it over timee. But again, maybe Bizarro-Earth is somewhat bigger, so objects can have larger stable orbits.

You also talked about rate of disintegrationf, which I take to mean how much slower the Earth spins each year.

"All else being equal", if Bizarro-Moon is twice as massive, the rate of slowdown will be four times as fast (quadratic relation). If it's three times as far away, the rate of spin slowdown will be hundreds times slower (negative sixth power)g.

There's also an effect from the internal composition of Bizarro-Earth. Maybe Bizarro-Earth has a different composition. Note that the internal composition of Bizarro-Moon won't matter to the Earth's spinh, but it would have mattered to how fast Bizarro-Moon lost its own spin (eg. how big are Ganymede's oceans reallyi?)

The dominant effect for the current rate is the fact that it is three times further away now, so transfer of momentum is going to be much much slower today. But, in the past, when Bizarro-Moon was closer to Bizarro-Earth, the other things matter.

Notes:

a Using OP's figures... After writing all of this, I later realized this wasn't clear. Just looking at OP's figures for diameter, you might conclude Bizarro-Moon about three times more massive than the actual Moon, not two. (about 50% bigger diameter than the actual Moon, and mass is proportional to diameter cubed). But I was also assuming that Bizarro-Moon has a similar density to Ganymede, which (not stated by OP) is about 2/3 the density of the Moon.

b Again using OP's figures, for the impact event -- I am not endorsing the giant impact hypothesis.

c Using the basic definition of angular momentum and Kepler's 3rd law: $$L = mr^2\omega$$ and $$\omega^2r^3 = GM$$.

d For spin angular momentum, $$L = \alpha mr^2\omega$$, which is the same as orbit angular momentum, other than the fudge factor $$\alpha$$, which will be somewhat less than 0.4 (exactly 2/5 for a sphere of uniform density), but not much less, for a rocky moon or planet (it can be quite small for a gas giant or star). See moment of inertia factor.

e Basically, outside 1.5 million km, you are orbiting the Sun not the Earth. OP's figure for Bizarro-Moon is still inside this sphere. However it is not 'comfortably' so (about 2/3 the way) -- this sphere is a theoretical limit based on three bodies (Moon orbiting Earth orbiting Sun), while in reality other planets (especially Jupiter) perturb things in a complicated way. It is believed that something 1/2 the way or more to the theoretical limit will end up being unstable over long periods of time. See Hill Sphere.

f Although the Moon's orbit is spiraling outwards, it is not really "disintegrating" in the sense that it will eventually cease to orbit the Earth. As previously explained, it has already gotten most of the Earth's spin angular momentum, so in the future when the Earth becomes tidally locked to the Moon, it won't be all that much further way than it is already, ie. still be comfortably inside the Hill sphere of the Earth (see previous note).

g For tidal torque see here, esp. formulas 6.92 and 6.87.

h See previous note and link for tidal torque. The formulas cited contain fudge factors: these are the "tidal phase angle" $$\delta$$ and the "effective rigidity" $$\tilde{\mu}$$, and these relate to the structure of the spinning object that is slowing down, not the structure of the object producing the tides that cause the slow down.

i Oceans matter! These have an effect on the "tidal phase angle". See previous two notes. Also see here for Ganymede specifically.

• Welcome to Worldbuilding! As this is a 'hard-science' question, do you have any references for the conclusions you've got here? – Mithrandir24601 Nov 1 '18 at 22:26
• @Mithrandir24601 -- thank you, glad to be here! For a lot of it I was just taking OP's figures and assumptions uncritically. I am not endorsing his or anyone else's version of the giant impact hypothesis. For the rest I have added some inline notes to be a bit more clear as to how I got to that conclusion. Hope this helps! – TimeTravellyParadoxySciFiSmeg Nov 2 '18 at 20:32