Cut the big blue marble in half...

May 23, 2012 5:37 AM

What would happen to the earth's rotation, orbit, etc. if the planet were half as large, diameter-wise, as it is now?

My daughter wants to know what would happen to the planetary motion of the earth if it were suddenly to shrink down to half its current size (further questioning reveals she is thinking about it being roughly half as large in terms of diameter). Would its axial rotation speed up? How would its solar orbit be affected? Would it heat up?

Signed,
I'm a SOCIAL scientist, dammit.

posted by drlith to science & nature (11 answers total)

of the earth if it were suddenly to shrink down to half its current size

Does the mass change? Or are you just compressing the same matter into a small space?
posted by spaltavian at 5:43 AM on May 23, 2012

If the mass stays the same:

it's going to spin faster, due to conservation of angular momentum. (Think of a figure skater holding her arms in to spin faster)
The orbit of itself and the moon would be unaffected since gravitating bodies can be treated as point masses at the center of gravity, and that wouldn't change.
It would almost certainly heat up because of the increased pressure and density. I think it would probably become a molten rock again, actually and would become unlivable.

If the mass also halved, then it's speed of rotation would stay the same I believe, and the moon would start drifting away, though I think the orbit around the sun would be roughly the same since the mass of the earth is negligible in comparison to the sun.
posted by empath at 5:45 AM on May 23, 2012 [1 favorite]

If mass is kept constant, halving the diameter would mean that the total mass would be reduced to 1/8, I think.
posted by carter at 6:02 AM on May 23, 2012

I think she envisions the mass being reduced proportionally.
posted by drlith at 6:06 AM on May 23, 2012

IF the diameter is halved, the mass would decrease by (1/2)^3=1/8. The earth's crust would decrease in strength by a factor of (1/2)^2=1/4. This would make the tides much stronger in terms of crust deformation.
posted by notsnot at 6:12 AM on May 23, 2012

How would its solar orbit be affected?

It wouldn't make an appreciable difference with or without a mass change (unless the mass starts approaching the mass of the Sun)

Essentially, planetary orbits are independent of mass. Kepler's third law relates orbit radius and period without ever referring to mass.
posted by vacapinta at 6:14 AM on May 23, 2012

You're describing Mars, within margin of error, and Mars behaves exactly the same in its orbit as the Earth does in its.

If you have one object orbiting another, and the smaller one is massively outweighed by the larger one, then the characteristics of the orbit are determined solely by the mass of the primary. The mass of the smaller one has almost no effect.

That's why they were able to determine that Eris is heavier than Pluto. Pluto has a moon named Charon, and Eris has a moon named Dysnomia. Once their orbits were characterized, it was possible to calculate the masses of Eris and Pluto very accurately. (Which led to a crisis in astronomy about six years ago, and a redefinition of the word "planet".)
posted by Chocolate Pickle at 7:08 AM on May 23, 2012

Earth would then have only about ten times the mass of the moon. Depending on where the moon was in relation to the sun and earth when this event happened, earth's orbit could very well be changed (i.e., not only would the moon drift away from earth, but earth would drift away from the moon, too, a little bit). That would almost certainly change the severity of the seasons (unless we were very very lucky) ... but that wouldn't matter much because by the time anyone noticed most of the atmosphere would have boiled away.

(PS: your daughter just made my day :)
posted by labberdasher at 7:39 AM on May 23, 2012

FYI you don't need to guess--there are some simulators out there, and they are a lot of fun to play around with. Just cut the mass of the Earth in half, by by 7/8, or whatever, and see what happens.

This simulator can easily do sun/earth/moon type simulations--unfortunately orbit only, not rotation.
posted by flug at 9:11 AM on May 23, 2012

With the Earth and Moon, they rotate around the middle of each other like two squirrels chasing each other around a tree. You can take them both together like a spinning dumbbell. At the moment with the Earth so much more massive the center of rotation is IIRC inside the Earth so it's easier to think of the Moon orbiting the Earth. Reduce the Earth's mass and the center of rotation of the Earth-Moon system would move outside the Earth. It is this center that is orbiting the Sun. The Earth would get closer and farther from the Sun on the new monthly cycle as it orbits this new center, but the scale of that is small compared to the distance from the Sun. Seasons are much more a factor of the tilt of the Earth's axis; in the northern hemisphere we're farther from the Sun in summer than in winter, but tilted towards the Sun vs closer to the Sun in winter but tilted away.
posted by zengargoyle at 11:43 AM on May 23, 2012

The Moon would do more than merely drift away from Earth, I think.

Escape velocity from the Earth at the orbit of the Moon is ~1.4 km/sec, and the orbital velocity of the Moon is already ~1.0 km/sec, so if the Earth were suddenly only 1/8th as massive, the Moon would suddenly be moving at more than escape velocity.

However, 30.8 km/sec (1 km/sec + 29.8 km/sec, Earth's orbital velocity) is much less than the escape velocity from the Sun at the orbit of the Earth (42.1 km/sec), so even though Earth and Moon would no longer be bound together gravitationally, they would both be orbiting the Sun, and the orbit of the Moon would at least roughly intersect the orbit of the Earth. I don't see anything which would rule out a collision between the two at some time in the future.

Also, since gravitational attraction is GMm/r², plugging 1/8M and 1/2r into that equation gives GMm/2r, so everything would weigh half of what it does now if Earth's diameter were half what it is now.
posted by jamjam at 5:54 PM on May 23, 2012