Centrifugal Force in A.C. Clarke's "Rama"

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Centrifugal Force in A.C. Clarke's "Rama"

Postby RichterCa » Wed Mar 21, 2012 1:03 am UTC

I'm in the process of re-reading Arthur C. Clarke's "Rendezvous With Rama", and I just got nerd sniped. Unfortunately, I'm not that up on my physics, especially centrifugal force, so I have no idea if Clarke screwed up, or I did. (I know, it's probably me, but I can't get my head around this.)

For anyone who hasn't read the book, Rama is an alien-made artifact that enters the Solar system. Picture an empty tin can, 8 km in radius, 50 km long. The entire thing is spinning fast enough to provide approximately 0.6G of centrifugal force to anyone standing on the inner surface. The characters in the book enter Rama through an air lock at the hub, where they are weightless; they then descend a series of ladders and stairs, getting heavier as they go, until they reach full weight at the surface.

The part that sniped me was this:

One character comes forward with a contraption designed to fly in 0 to 1/3 gravity. He sets off from the hub, weightless, with the intent to ride along the axis the entire length of Rama. He is warned not to drop down towards the surface, as and lowering at all will increase the weight (or centrifugal force) he feels. But I'm not sure if this is true.

Centrifugal force is not like gravity, which reaches out to pull you down. If there is nothing pushing him along the rotation, he shouldn't feel any centrifugal force at all, right? Of course, the air itself is going to be moving along with Rama's rotation, but would the air near the hub be moving fast enough to exert a serious sideways force on him, causing him to take part in Rama's rotation? Theoretically, if there was no air, he could lower himself almost all the way to the surface (which from his vantage point would be moving by very fast), and still remain weightless the entire time.

So, am I thinking about this entirely wrong? Or is Clarke?
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby gmalivuk » Wed Mar 21, 2012 1:49 am UTC

You seem to be understanding the situation correctly. I'd handwave it with the moving air explanation, but it's possible Clarke missed it all together.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Copper Bezel » Wed Mar 21, 2012 2:17 am UTC

Would the air itself be spinning? Or the platform from which the plane launched? If the atmosphere is stationary, it seems like you'd have odd things going on at "ground level." If everything is rotating uniformly, then leaving the center to move toward "ground level" would seem hazardous for more reasons than the one mentioned. (For that matter ... being at the center would be hazardous for more reasons than the one mentioned.)

Edit: I'd thought the idea for these cylindrical habitats was, the atmosphere itself is squashed to the sides by centrifugal force, rather than evenly pressurized, so the air thins as you go up.

Edit: Which makes no bloody sense, because there's nothing heavier to be - I have no idea what I meant here.

Edit: Because spinning is not gravity.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby ++$_ » Wed Mar 21, 2012 2:19 am UTC

So, how about that air? What would the flow pattern be like?

The air at the surface would be moving at 219 meters per second relative to the air at the axis.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Copper Bezel » Wed Mar 21, 2012 8:17 am UTC

We've seen that recently, though - just replace "baffles" with "buildings."

If you run partitions down the cylinder, does that help? Or does that just make a larger number of small, spinning cylinders? If you make the compartments small enough, on both axes that aren't along the axis of rotation, there wouldn't be any serious relative differences in motion pressure anywhere within the compartment, and the air would just accelerate with the compartment, right? (Or would there be, like, a constant pressure gradient along the axis of the acceleration?)
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby idobox » Wed Mar 21, 2012 11:30 am UTC

In the book, Clarke considers air is not significatly moving relative to the walls (and things start to get bad when convection currents mess up with that).
It means the angular velocity of air is the same everywhere, and since the tangential velocity of the guy relative to air is almost zero, he must be spinning at the same velocity. Air friction plays the same role as solid friction for people on the "ground", pushing him away from the center of rotation.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Tass » Wed Mar 21, 2012 11:43 am UTC

++$_ wrote:So, how about that air? What would the flow pattern be like?

The air at the surface would be moving at 219 meters per second relative to the air at the axis.


Without an external supply of energy the air would quickly reach an equilibrium where it would be spinning uniformly with the cylinder. If you have uneven heating on the inside you get updrafts, winds and in general weather, with the huge Coriolis force in this cylinder weather would be extremely complicated, so lets forget that. If it is at uniform temperature then the air is spinning with the cylinder, and you feel no wind when standing on the inside.

In this case he would indeed feel the "gravity" becoming stronger the lower he flied. If he is capable of going fast in relation to the air then you need to factor in the Coriolis force. Flying with the rotation would make him feel heavier, while flying against it makes him lighter. If his engine is capable of pushing him through the air at 219m/s then he could even fly right above the ground without even requiring wings, since from an inertial viewpoint he is not moving, just standing still while countering the air drag trying to push him. In the rotating frame this same conclusion is explained by the Coriolis effect of his high speed exactly countering the centrifugal force.

If the top speed of the plane is low compared to the rotational speed (seems likely if it is only capable of supporting itself against 0.3g), then indeed we can ignore Coriolis effects and treat the centrifugal force like a static gravity field which decreases with height.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Zamfir » Wed Mar 21, 2012 12:06 pm UTC

Copper Bezel wrote:We've seen that recently, though - just replace "baffles" with "buildings."

Not really comparable, I am afraid. In a mixing vessel, the rotor makes the flow rotate with respect to the walls and baffles. In a space cylinder the flow rotates together with the walls and buildings.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Copper Bezel » Wed Mar 21, 2012 4:05 pm UTC

Why, though? I mean, why would it become uniform? A constant angular momentum for the object is still is constant acceleration for every individual particle. What's accelerating the air particles so uniformly? I just see a (round) glass of ice water. Rotating the glass around the vertical axis doesn't turn the ice, obviously.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby chenille » Wed Mar 21, 2012 4:27 pm UTC

If you try it, you'll find that the liquid actually does end up rotating thanks to friction with the glass. Spin it fast enough and you can see the surface becomes curved; the shape is what provides the balance between the centrifugal acceleration, which varies with the radius, and gravity. You can even make parabolic mirrors this way. The same thing should happen with a cylinder of gas, only the changing acceleration will be end up being provided by a pressure gradient instead; essentially, once air is moving with the ground then it will pile up against it, and that gradient provides the centripetal force you need.

If you're not convinced the gas will rotate with the container, consider the earth, which is more or less the same thing with air on the outside. The poles stand still, and as you move closer to the equator the ground gradually rotates faster, up to nearly 1700 km/h; surface winds are nothing near that, so the air must always end up moving very close to the same speed.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby gmalivuk » Wed Mar 21, 2012 4:55 pm UTC

Copper Bezel wrote:Why, though? I mean, why would it become uniform? A constant angular momentum for the object is still is constant acceleration for every individual particle. What's accelerating the air particles so uniformly? I just see a (round) glass of ice water. Rotating the glass around the vertical axis doesn't turn the ice, obviously.
The constant acceleration only means the density won't be uniform.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby idobox » Wed Mar 21, 2012 5:13 pm UTC

Tass wrote:Without an external supply of energy the air would quickly reach an equilibrium where it would be spinning uniformly with the cylinder. If you have uneven heating on the inside you get updrafts, winds and in general weather, with the huge Coriolis force in this cylinder weather would be extremely complicated, so lets forget that. If it is at uniform temperature then the air is spinning with the cylinder, and you feel no wind when standing on the inside.

That happens later in the book, when Rama comes closer to the sun and starts to heat up from inside out.

Tass wrote: If his engine is capable of pushing him through the air at 219m/s then he could even fly right above the ground without even requiring wings

The flying machine in question is human propelled. Some kind of futuristic flying bike for low gravity environements.
And he his flying roughly along the main axis, so Coriolis force is not an issue. He just has to make sure he doesn't go too low, because he would get tired faster, and might not be able to climb up again.

There is a reason he can fly exactly along the axis, but I don't remember. I think it's something to do with the design of the machine requiring some gravity.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby gmalivuk » Wed Mar 21, 2012 5:34 pm UTC

idobox wrote:There is a reason he can fly exactly along the axis, but I don't remember. I think it's something to do with the design of the machine requiring some gravity.
Do you mean can't fly along the axis? Wasn't that where the "sun" was?
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Tass » Wed Mar 21, 2012 6:35 pm UTC

idobox wrote:And he his flying roughly along the main axis, so Coriolis force is not an issue.


I know, it was just to explain to people how the rotating frame still works even if you are actually standing still in an inertial frame. If he can fly fast then he could make himself lighter by flying against the rotation.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby idobox » Thu Mar 22, 2012 9:40 am UTC

gmalivuk wrote:
idobox wrote:There is a reason he can fly exactly along the axis, but I don't remember. I think it's something to do with the design of the machine requiring some gravity.
Do you mean can't fly along the axis? Wasn't that where the "sun" was?

He can't fly, your right, but there is nothing at the axis.
There are three row of lights on the shell that work as the sun.

You should read the book, it's one of my favourites, and it's quite short.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Technical Ben » Thu Mar 22, 2012 10:23 am UTC

Perhaps he gets motion sickness in the centrer (lacks reference frame). :D
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby gmalivuk » Thu Mar 22, 2012 2:10 pm UTC

idobox wrote:You should read the book, it's one of my favourites, and it's quite short.
I read all (4?) of the Rama books. But it was like 15 years ago, so my memory of the details isn't that spectacular.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Tass » Thu Mar 22, 2012 5:12 pm UTC

idobox wrote:
gmalivuk wrote:
idobox wrote:There is a reason he can fly exactly along the axis, but I don't remember. I think it's something to do with the design of the machine requiring some gravity.
Do you mean can't fly along the axis? Wasn't that where the "sun" was?

He can't fly, your right, but there is nothing at the axis.
There are three row of lights on the shell that work as the sun.

You should read the book, it's one of my favourites, and it's quite short.


Ah, so it is pretty much exactly an O'Niell cylinder. Then there likely is going to be wind because of the uneven heating over the alternating "ground" and "sky" parts. To my knowledge nobody has ever simulated the air flow in those conditions.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby idobox » Thu Mar 22, 2012 5:27 pm UTC

Tass wrote:
idobox wrote:
gmalivuk wrote:
idobox wrote:There is a reason he can fly exactly along the axis, but I don't remember. I think it's something to do with the design of the machine requiring some gravity.
Do you mean can't fly along the axis? Wasn't that where the "sun" was?

He can't fly, your right, but there is nothing at the axis.
There are three row of lights on the shell that work as the sun.

You should read the book, it's one of my favourites, and it's quite short.


Ah, so it is pretty much exactly an O'Niell cylinder. Then there likely is going to be wind because of the uneven heating over the alternating "ground" and "sky" parts. To my knowledge nobody has ever simulated the air flow in those conditions.


I don't how accurate that is, but in the book there is a kind of hurricane for some time, then the wind slows down, and things stabilize.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Copper Bezel » Thu Mar 22, 2012 9:19 pm UTC

chenille & gmalivuk, thanks for the clarification!

(Hey, does anyone know how steep the pressure gradient would be?)
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby chenille » Fri Mar 23, 2012 2:36 am UTC

Someone should probably check my math, but I think you can use the Navier-Stokes equation with fictitious forces, so for an ideal gas with k = ρ/P (density/pressure):

∇P = ρv2/r
∂P/∂r = kP(4π2r)/T2
dP/P = (4π2k/T2) r dr
log P = (2π2k/T2) r2 + constant
P = P0 e^((2π2k/T2) r2)

Here r is the distance out and T is the period of rotation. To make things a little easier to see, if the centrifugal acceleration at the ground is g, the total pressure change will be: Pr/P0 = e^(kgr/2)

For comfortable air k is about 1.2 × 10-5 s2/m2. So for instance if you were simulating Earth's gravity, a 10 km radius should mean the pressure at the ground is a little less than twice that at the axis.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby jmorgan3 » Fri Mar 23, 2012 7:08 am UTC

chenille wrote:Someone should probably check my math

Checked. Your math is correct.
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Re: Centrifugal Force in A.C. Clarke's "Rama"

Postby Tass » Fri Mar 23, 2012 12:33 pm UTC

chenille wrote:For comfortable air k is about 1.2 × 10-5 s2/m2. So for instance if you were simulating Earth's gravity, a 10 km radius should mean the pressure at the ground is a little less than twice that at the axis.


Which is also easily seen from the fact that pressure on earth drops to about half every five kilometers. In a cylinder "gravity" decreases linearly with radial distance from center, so the average "gravity" over a full radius is equal to half the surface "gravity".
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