Wednesday, January 7, 2009

The rotating spaceship

The trouble with space, among other things, is that there's no gravity.* Weightlessness is known to cause a variety of health effects, the most obvious being muscle atrophy. So how can we create an artificial gravity? There aren't really any other forces which behave quite like gravity, which always has a strength proportional to your mass. But wait, there's the centrifugal force! If you're on a large rotating spaceship, the centrifugal force is what flings you away from the center of rotation.

*Technically, there is still gravity if you're in orbit, but you can't feel it because an orbit is basically a free-fall trajectory.

At this point, I am obligated to defend my use of "centrifugal force". Some overzealous science teachers have insisted that there is no such thing as the "centrifugal force"; there is only the "centrifugal effects" and "centripetal acceleration". Take it from me, that is baloney. The centrifugal force is what we call a "fictitious force", a force that depends on our reference frame. But "fictitious" is a bit of a misnomer. Fictitious forces have effects that are equally real as any other force. It's also worth noting that under Einstein's General Relativity, gravity is also essentially a fictitious force. Gravity is still real, is it not? We're all physicists here, not solipsists!

This is Space Station V, as seen in 2001: A Space Odyssey. It is a fictional example of a spaceship which harnesses the centrifugal force to create artificial gravity. No real examples exist, not yet. On the Space Station V, there are two independently rotating wheels. Unless any rocket fuel is expended, angular momentum (which is a measure of rotation) is a conserved quantity. Therefore, to preserve fuel, the spaceship would have an overall angular momentum of zero. Therefore, the space station is built so that the two wheels rotate in opposite directions, canceling each other's angular momentum.

[ETA: They also could use fuel to make the entire space station rotate in the same direction. I'm not sure which is the case for Space Station V.]

To consider the physics of the spaceship, let us consider just one of the wheels, and consider a rotating reference frame. It is important to remember that a rotating reference frame is not an inertial reference frame. That means that Newton's Laws do not apply! An object at rest does not necessarily remain at rest. Not every force is necessarily accompanied by an equal and opposite force. How can we even make sense of the physics of our rotating reference frame? It turns out that we can make sense of it rather easily. We simply need to assume that a new force, the centrifugal force, is acting on all objects. The centrifugal force pushes all objects outwards, away from the center of rotation.

This situation should make sense in both the rotating reference frame and the stationary reference frame.
  • Rotating reference frame: The centrifugal force pushes astronauts outwards, but the floor of the spaceship keeps him from being flung away.
  • Stationary reference frame: The astronaut is circling around inside the rotating spaceship. He would continue flying in a straight line away from the spaceship, except the floor is stopping him.
Let's consider another thought experiment. Let's say the astronaut is not inside the spaceship, but just outside of it, stationary. Let's look at what's happening in the two frames. Remember, both frames should make sense.

Image not to scale
  • Stationary reference frame: The spaceship is rotating, but the astronaut is not rotating with it. No forces are acting on him, so he remains stationary just outside the spaceship.
  • Rotating reference frame: The astronaut is circling the spaceship. The centrifugal force is pushing him outwards, and yet, he remains in the circular path. How?
The answer is that there are not one but two fictitious forces associated with a rotating reference frame. We have the centrifugal force, which pushes the astronaut outwards, and the Coriolis force, which pushes the astronaut inwards. You can all breathe a sigh of relief--we haven't broken physics after all.

The Coriolis force works on all objects which appear to be moving in the rotating reference frame. That's why we didn't have to include it when we were talking about the astronaut inside the spaceship--he was stationary with respect to the rotating spaceship. The force does not always pull you inwards, but it is always perpendicular to the object's motion. Let's say you are inside the spaceship, facing the direction of rotation. If you run forward, you will feel heavier. If you run backwards, you will feel lighter. And this is a little harder to imagine: if you jump up, the Coriolis force pushes you slightly forward, and when you fall back down the Coriolis force pushes you slightly backwards.

If that made any sense, you can congratulate yourself for successfully performing a thought experiment, in very much the same spirit as Albert Einstein. If it didn't make any sense, it's probably my fault, and you can ask me in the comments. :)


Anonymous said...

Are you sure the wheels rotate in opposite directions? I'm pretty sure they don't have to, to preserve angular momentum. Once spun up to speed, no additional fuel would be required to keep it spinning (friction of the upper atmosphere notwithstanding).

miller said...

There are many rotating spaceship designs which have counter-rotating parts specifically to preserve angular momentum. However, they could instead just use fuel to rotate the entire ship. I admit that I'm not actually sure whether Space Station V is the former or the latter.

Anonymous said...

Again, not sure why counter-rotation is necessary to preserve angular momentum. In the absence of friction, once you spin it up it will continue to spin indefinitely.

miller said...

Rocket fuel is by nature heavy, and it costs money to send it over to the space station, even if you only have to do it once. Depending on the details, it may be cheaper to simply equip it with solar panels (which would also have other uses). Solar panels, unlike rocket fuel, cannot change the overall angular momentum of the spaceship, so you'd have to have rotating and counter-rotating components.

Anonymous said...

Ah, I see what you mean -- either two counter-rotating wheels, or a single wheel with an internal flywheel to spin in the other direction.

Anonymous said...

I think: "If you run forward, you will feel heavier. If you run backwards, you will feel lighter." is not correct.
But "And this is a little harder to imagine: if you jump up, the Coriolis force pushes you slightly forward, and when you fall back down the Coriolis force pushes you slightly backwards." is correct.

miller said...

As far as I can tell, I spoke correctly. When I said you are "facing the direction of rotation", I meant that from a stationary reference frame, it appears that the rotation is pushing you forward. Was that a source of confusion?

Anonymous said...

To feel the Coriolis force on earth, you have to move perpendicular to the rotation (north or south). If you move with constant latitude no Coriolis force can be expirienced and hence no different weight. I dont think that the direction (centripetal or -fugal) does matter.

miller said...

To feel the Coriolis force, you have to move perpendicular to the axis of rotation. Otherwise my second thought experiment wouldn't have worked.

The Coriolis force is expressed mathematically as -2m*(v x w) where m is the mass, v is the velocity, x is a cross product, and w is the rotation vector (pointing in the direction of the rotation axis, using the right-hand rule).

Anonymous said...

You are right.
On earth moving north has only an effect because there is a component pointing to the axis of rotation. Coriolis corrects for a difference in angular velocity when you change the radius.
When you don't approach the axis as in your case (still moving perpendicular du the rotation axis) then Coriolis has to correct the angular velocity which changes because you change the radius in a tangential movement. And this yields this bizarre "weight change".

Hugo said...

I do agree the "heavier when you run forward" (in the direction you're already rotating, effectively rotating you faster). And I enjoyed pondering briefly the effect of jumping. Sweet!

Yup, fictitious things can most certainly have real world effects. But... let's not derail this one. ;)

Two links in my collection:

Classic xkcd:

The Bad Astronomer on centrifugal force:

Hope I'm allowed two links.

Back in my undergrad years, when we were doing some applied maths and working with the "force of gravity", I decided we're actually all accelerating outwards at 9.9m/s^2, due to the force of the earth's surface against our feet. (Assuming we're standing.) Being "at rest" in terms of maintaining constant velocity, would require us falling inward according to the bending of space-time, but the earth is in the way, accelerating us outwards. (I mean, we feel the force pushing against us, we can't be "at rest"! :-P) I'm hoping here's a place I can say this without being looked at strangely? Do you find it to be an interesting way of looking at it? It would have interesting effects on e.g. the ballistic trajectory calculations we did in applied maths.

miller said...

I was actually thinking of that xkcd comic when I mentioned the centrifugal force. I even used the same word, "overzealous".

Hugo, that's actually the basic idea behind General Relativity. General Relativity models gravity as a bending of space-time. Because of the curvature of space-time, all free-fall trajectories are actually considered inertial reference frames (ie, "at rest"). When we change from a free-fall reference frame to an Earth's surface reference frame, we must introduce a fictitious force which we call "gravity".

As for the implications of General relativity... Unfortunately undergrads are considered too dumb to learn it, so I'm not exactly sure. It implies that light and the passage of time are affected by gravity. It predicts gravitational radiation, and it explains an anomaly in Mercury's rate of precession. Outside of that, I cannot say.

Anonymous said...

Can you please define the relationship between spacecraft and centrifugal force?

miller said...

If you consider a rotating reference frame (such as the frame in which the rotating spacecraft is stationary), then you must include the centrifugal force in your equations in order for it to be consistent.

You'll have to phrase your question more specifically if you want a better answer.

Anonymous said...

In the unfortunate event that something went "loopy" with regard to the torus space station, what would occurr if the rotation were to suddenly halt? Assumption, astronauts suddenly floating around?

Also, Why has this not been considered yet by NASA or some private company? Seems as though this would beat the balls off of the ISS or any future floating boring box we will put up in space anytime in a near year. Kubrick's 1968 fictional rotating space station will just have to due for now I suspect.

miller said...

If the space station suddenly stopped spinning, all objects inside would suddenly be thrown forward.

I'm told that a space station like this has been considered by NASA. However, they don't have the budget for it. Furthermore, they want to be able to conduct micro-gravity experiments, which you can't do on a rotating wheel space station.

Aleks said...

I found your article very interesting. I especially enjoyed Jeffrey's idea of the station not being required to be constantly spun mechanically. I have no idea if such an outcome is true or not but it seems very likely!(and amazing!)

Erandall Nelson said...

I am curious how angular momentum would be "handled" in a craft similar to the Leonov in 2010: Odyssey Two. Could it use a counter-rotating flywheel of equal momentum(?).

miller said...

I'm not familiar with the details of Leonov, but if it rotates there are two possibilities. Either it used rocket fuel to change its overall angular momentum, or there is a counter-rotating part such that the overall angular momentum is about zero.

ERandall Nelson said...

I do not know if this makes a difference: the spacecraft design is linear (drive engines in rear) with the habitat "barbell" rotating perpendicular to the vehicle's axis.

philip metts said...

there's a little trick with the jumping up and falling down. it has to do with your velocity. when you jump up you will not fall back down but travel in a strait line until you intersect with the floor that is following a curved path. since you are not pulled back down by the floor that is why they do not call centrifugal effect a force.

miller said...

Phillip Metts,

Yes, there is a way of considering the system as if there is no centrifugal force. There is also a way of considering it with the centrifugal force. Your argument just doesn't follow.

ERandall Nelson said...

A quick follow up. I recently discovered one of the art directors on the series Babylon 5 lifted the Leonov design for the EarthForce "destroyers." It included gimballed engines to compensate for the directional effects the rotating section's angular inertia caused. Does this "problem" presume the absence of a counter-rotating flywheel (or section)?

Santiago said...

Take a peek to 2001 Space Odyssey, both rings rotate in the same direction. Once the station acquires the necessary angular speed no more fuel is expended. Remember angular momentum conservation? There is no friction there and nothing arrives or is launched tangentially . Any ship boarding the station will do it at its axis after acquiring the same angular speed.