The Sideways Tide | Hacker News Reader

The Sideways Tide

solipsys.co.uk

40 pointsheadalgorithm24 comments

There's a (relatively) famous problem in astrodynamics called "Apples in a Spacecraft". I wrote a blog post about it a little while back [1], but the gist of the problem is this:

Suppose you're in a rectangular spacecraft orbiting around the Earth and you release a bushel of apples so that they are uniformly distributed and stationary relative to your spacecraft. If you don't disturb the apples, what will their long-term distribution look like?

There was a bit of a debate in the literature as to the answer to this problem and the physicist Alfvén (better known for Alfvén waves [2]) ended up getting the wrong answer. The correct answer (due to Hénon, who is better known for the Hénon map [3]). Hénon showed that half the apples would end up at the top back corner, and the other half would end up at the bottom front corner of the spacecraft.

The reason for this is that the apples at the top of the spacecraft are in a slightly larger orbit and therefore have a slower velocity than the center of mass of the spacecraft. This causes them to drift to the back of the spacecraft. Similarly, the apples that start on the bottom of the spacecraft have a higher velocity and start to drift towards the front of the spacecraft.

[1]: https://joe-antognini.github.io/astronomy/apples-in-a-spacec...

[2]: https://en.wikipedia.org/wiki/Alfv%C3%A9n_wave

[3]: https://en.wikipedia.org/wiki/H%C3%A9non_map

Wouldn't the answer be a circulation of apples, depending on how many apples we're talking? Half the apples would pile up in a top corner, which'd push some of the apples into the bottom corner instead, making them traverse to the other side of the spaceship.

This is of course ignoring that the spaceship itself might be bumped off-course by the apples or start spinning because of them.

"What happens if we throw the ball forward or backward? Will we again have a fictitious force?"

Yes, you will. Only that force will be in the plane of the orbit. Throw the ball forward and you raise its orbit - so it will start going forward, end up going "up" and then cycle back down exactly one orbital period later.

Throw it backwards and you will lower its orbit. It will go back, down, the up again. Unless you throw it too hard, in which case it will re-enter and burn up.

Yes, but in either case you will change its orbital period slightly.

> If you get the direction exactly right (and the tolerances are inhumanly small) then the ball will go away from you, yes, in a straight line (more-or-less), yes, but it will slow down. After about 23.2 minutes it will come to a complete stop, and start to come back.

Ok, this makes me wonder, in the movie "Gravity", when people accidentially got separated from the space station, they were not forever lost?

> when people accidentally got separated from the space station, they were not forever lost?

Only if going perfectly sideways, i.e. keeping the same distance from the center of the gravity well ("and the tolerances are inhumanly small").

Go down, you'll spiral to the bottom; go up, you'll float away, maybe to the Sun or Jupiter or just the Ooort cloud and beyond.

(I don't think you'll find another stable orbit without some sort of maneuvering jets or very well tossed bits of spare mass you may have on you. Even so your distance from the station is probably unrecoverable.)

If orbital mechanics were this unstable, we wouldn't ever manage to have any sattellites up in orbit around the Earth (or have the Earth orbit the Sun, and the Sun orbit the centre of the galaxy..). The important part of your orbit is the sideways velocity (For example, the International Space Station orbits at around 27,600 km/h). Let's say you're launched from the ISS (and that the ISS's orbit is perfectly circular), straight down towards the Earth at about 30km/s. Initially, you'll get closer to the Earth, but at the same time your orbital velocity will increase as you're going down, which eventually ends up sending you back up to the same orbit as the ISS (although I'm not sure if you'll end up ahead of the ISS, behind it or right back at it). You'll have an upwards speed of 30km/s, and will keep going up at a decreasing speed until you hit some maximum point, and start coming back down to repeat the cycle. Your orbital path will end up tracing out an ellipse, and at no point will you spiral in or away.

Ultimately, to actually come back down to Earth you have to burn off a lot of sideways speed (especially if you don't want to be going too fast when you enter the atmosphere). And, to leave Earth's gravity, it takes quite a lot of acceleration too.

While your parent comment does have a ridiculous idea of how orbits fundamentally work, orbital mechanics in near earth do end up being that unstable, hence the orbital stationkeeping fuel budget all satellites in the region have. It's just the toll of a faint atmosphere, solar radiation, mountainous terrain. You'd have to go out to around the geostationary area for a "clean" orbit, and even that one's still going to be tugged around by the moon.

On the plus side all those weird interactions with real life allow you to do fun things like change orbit with electronic tethers or orbit around nothing by exploiting Lagrange points

Orbits in such heights (400km) are quite stable no matter the direction of small perturbation from the circular orbit. The body will be following an ellipse in all cases. Only after a long time, the rare atmosphere friction will bring you down.

Sadly, the tolerances are as inhumanly small as they stated. The intuition is that the fractional change in orbital period will equal the fractional change in orbital speed (times 3, but we're only interested in orders of magnitude).

Thus, in low earth orbit, at a speed of 8000 m/s, a relative speed of 8 mm/s is a fractional change of 1 in 1 million. The orbital period is 90 minutes, a millionth of which is half a second. That doesn't seem like much, but objects in orbit move really, really fast. Arriving at the same spot half a second later means you'll miss the other object by 4 kilometers! Without active guidance, a rendezvous in orbit is near impossible.

Actually, you dropped a decimal point somewhere there: 5400 seconds divided by 1 million is .0054 second, times 8000 meters.sec is 43.2 meters, not quite as much of a stretch as 4 km.

Perhaps I'm misunderstanding, but does this mean that the sci-fi cliche of going for a spacewalk and accidentally being knocked off your ship and drifting away forever is wrong? Actually, given a short wait you'll just stop and drift right back to the ship again?

(Yes, in orbit only, I understand that freefall is different from zero gravity scenarios)

You'll pass through the ship's orbit again and again.

If you pushed in a direction exactly perpendicular to the orbit, you'll reach the ship again. If you pushed somewhat into or against the orbit direction, you will get in front or behind the ship.

You'd have to have enough oxygen for one [edit: half] orbital period, mind you. The unfortunate astronaut would likely still die.

Ah, yes, thank you. This is the part I didn't take in. You'll return to pass through the orbit, but the ship will most likely not be at the same point.

Yes, sort of, while in orbit. If your velocity relative to the spacecraft has a prograde/retrograde component, you'll desync the orbits - which means that after a single full orbit you'll not return to the spacecraft, but might find yourself many kilometers from it instead. Unless you or your ship's crew have some fuel available for maneuvers, you'll still die.

If you wish to read this article in novel form, Incandescence by Greg Egan.

I would recommend Seveneves by Neal Stephenson on this topic.

That book has the major problem that it's three-quarters of a good book stapled to half of a mediocre one, and the mediocre one is the one that the author was apparently actually interested in.

That's fair to say, but I think it's true of most of Stephenson's books - and he still manages to cover so much interesting ground that they turn out to be good reads despite the long-winded plot holes.

Or maybe better, Anathem. Because there's more exposition.

I would absolutely recommend playing a game called Kerbal Space Program if you have any questions on how orbital mechanics works.

I wonder if it can be simulated in Universe Sandbox (2).

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