Taking a deep dive into cylindrical pools
The cylindrical pool we are talking about is not quite like the cheap plastic one commonly found in backyards. The crucial differences are:
- it is bigger: starting from 28 m in diameter and 30 m long
- the water is not located at the cylinder base but instead naturally settles on the “side” of the cylinder, the curved surface
- it is a bit more expensive, with a price tag at least in the $500 million range. Why? Because it is in space, and has artificial gravity
As a picture is worth 10³ words¹, we have illustrated the cylindrical pool concept in the article featured image. The scene is a rendering of a pool 14 m in radius and 30 m long, with 2 m deep water. The camera is located at one of the bases of the cylinder. The model was done in Blender and rendered with Blender’s integrated cycles engine. This engine is well known for its unremarkable inability to render caustics, the shiny “light filaments” that form at pools bottom when it is sunny. As a result the sunlit area look unrealistic and washed out. However, with a small effort of imagination, one can easily notice:
- the tiled pool bottom on the cylinder side all around with the water surface “above it”
- the entering sunlight which appears to come from the “bottom right” and the beautiful starry sky (proof that we really are in space!)
- the highly transparent window in front of the starry sky, without which we would be suffocating long before reaching the end of this article
Artificial Gravity and rotational gravity
The pool is using rotational gravity, also called pseudo gravity or centrifrugal gravity, because it does not even need science fiction to work. The other kind of artificial gravity is the “star trek” type, using generators that magically create a gravitational field based on a mass that does not exist. This is required if you want to cancel out the inertial force generated by a rapidly accelerating faster than light spaceship and not instantly turn into human mash.
Rotational gravity is in contrast easy to achieve. You have already experienced it if you ever played on a merry-go-round. A $5 salad spinner uses it. It is not currently used by astronauts on-board the international space station (as of 2019) because anything in space with humans is extremely expensive, but the research interest is here. The rotation creates an inertial force in the rotating frame of reference, with a strength equal to g_pool = r ω_pool², in the direction away from the rotation axis, where r is the distance from the axis, and ω_pool (“omega pool”) is the cylinder rotation rate. Figure 1 shows the inertial force field across a section of the cylinder.
Applying this inertial force field to the cylindrical pool explains why the water settles on the sides: the downward direction is outward from the rotation axis, at every point. Anywhere on the cylinder wall, the force is also perpendicular to the wall, which becomes the artificial ground, hence the name pseudo gravity. A notable feature of this gravity field is how it increases proportionally with the distance from the axis.
We will assume the following parameters for the pool:
- ω_pool = 4.42 rpm (rotations per minute)
- r_pool = 12 m (radius down to water surface)
- v_rim = 5.5 m/s (~20 km/h, tangential speed at the water surface, or rim speed)
- g_pool = 0.26 g (26% of Earth gravity, pseudo gravity at the pool surface)
We are striking a balance between having enough gravity to provide a detectable sense of up and down², a low enough rotation rate to avoid canal sickness³ and keeping the radius to reasonable values for obvious cost reasons.
The potato dive
With our basic intuition about pseudo/rotational gravity let’s imagine diving from the rotation axis, down into the water. Ideally the rotation axis would be materialized by a small physical axis, from which you can push yourself away. Diving from a 5 m high diving board can already be a bit frightening, at least for the first few times. Our pool has a radius of 12 meters. Now just take a moment to imagine yourself in that position, with a wall a water 12 m “down” all around you. Given the pseudo gravity field, every direction is down, this would be very obvious while looking around at swimmers. But fear not, as the gravity will only increase linearly with the distance from the axis and up to 0.26 g, so your fall will be much more gentle than from 12 m high on Earth… or at least that is the theory. So you gently push yourself off from the axis at 1 m/s (3.6 km/h), and off you go.
Animation 1 is your dive numerically simulated in Python, and takes into account the air drag. To keep the computation of the air drag simple you are assumed to have the shape of a sphere of relative density 1 (by definition the density of water) which is within a few percents, equal to that of a potato, or any living human that is not about to die within seconds. You are also assumed to have a mass of 60 kg (rounding the world average human mass of 62 kg). This yields a “diver-sphere” of 0.48 m in diameter⁴. In reality with a non spherical human the air drag would vary both in amplitude and direction with your orientation to the relative wind during the fall: belly first, “superman-like” or random and rapidly changing, depending on your diving skills.
The view from within the cylinder shows that as our pseudo gravity field theory thoroughly failed to predict, you will not fall in a straight line while accelerating ever more rapidly under the increasing pseudo gravity. Instead you go into a crazy spiral. However the spiral is very easy to understand if we look from the static/non rotating frame of reference: you are simply moving in a straight line at constant speed while the pool is rotating. The air drag is weak and has almost no impact on your trajectory, except near the water, where your trajectory bends slightly to the right. While going down, you encounter an increasing crosswind (of speed v=rω) as the air is also rotating with the pool as a block.
Your final velocity at contact with the water and relative to the water is 5.6 m/s (20 km/h), whereas for comparison the terminal speed of a 12 m fall on Earth would be 10.8 m/s. Additionally, in the cylindrical pool you enter the water at a very low angle of attack of 10°, defined as the angle between the water surface and your velocity vector in the pool frame of reference. In other words you arrive almost skimming the water.
Your first contact with water occurs under “no effective pseudo gravity”: you are not rotating with the pool, just slowly drifting towards the water. You may slowly bounce several times before really falling into the water, only after acquiring some tangential speed and as a result experiencing some pseudo gravity. The entry in the water will be different than down here on Earth, it will take longer, be unpredictable and much more fun. Given the angle of attack and entry speed the diving skill required for not hurting yourself on arrival is not greater than that of a potato.
Rescuing the pseudo gravity force field
It turns out we were not ready at all to properly understand rotational gravity. Our pseudo gravity force field “theory” seems to have taken on water rather quickly, is there anything we can do to rescue it?
Dear reader, if you have a life-threatening allergy to equations, you are under no obligation to embark on this perilous mission. At any time you can skip to the next part (Intuitive Coriolis), which is a summary of the rescue mission findings presented in a more intuitive way.
We have interpreted the pseudo gravity field to mean that in the rotating cylinder, one would be subject to it, plus the familiar accelerations caused by your self-motion (e.g. accelerations caused by walking, jumping or swimming). This is the correct way to understand the force field, however in the rotating cylinder case, the pseudo gravity is not the only inertial force, we have forgotten the Coriolis forces. To derive these additional forces we have to start from the expression for the acceleration of an object in cylindrical coordinates (r,θ):
The dot over letter notation indicate derivation with time. One and two dots indicate respectively the first and second times derivatives, for example: ṙ
is equal to dr/dt and is pronounced “r dot”. Unfortunately Unicode does not support all weird physicists notations and medium does not support latex or inline images, so brace yourself for some funny spelled out inline variable notations.
{r dot dot} and {r θ dot dot} are the “natural” acceleration terms, directly resulting from the definition of acceleration (a=dv/dt), but projected unto the radial and tangential directions instead of x and y. The two remaining terms {r θ dot squared} and {2 r dot θ dot} originate from the curved geometry of cylindrical coordinates and are oriented respectively in the radial and tangential directions. To understand each one of these 4 terms in isolation, 4 simple cases are presented, in a setup with a rod rotating around a central point O, and a mobile M that can move along the rod.
To obtain the expression of the Coriolis inertial forces, another step is needed. The acceleration of the mobile/object must be expressed as a function of rₒ and θₒ, the radial and angular position in the pool (rotating) frame of reference. Thus, let θₒ be the angular coordinate of the object in the pool frame and θ_pool the angular position of the pool in the static frame. The two are related by:
θ = θₒ + θ_pool.
For radial coordinates, things are very simple: rₒ = r_pool = r, and all their time derivatives are also equal. We can now express the radial and tangential accelerations as a function of θₒ and θ_pool, by replacing θ and its derivatives into equation (1) and expanding, which yields:
We recognize here the same terms as for the acceleration in cylindrical coordinates, but with θₒ substituted into the place of θ. These two terms are labeled aᵣᵖᵒᵒˡ and a_θᵖᵒᵒˡ. They would be the only nonzero terms if the pool was not rotating (ω_pool = 0), they correspond to the acceleration caused by self motion, in cylindrical coordinates, in the pool frame. In addition to this “expected” acceleration, there are 3 other terms: the pseudo gravity, the vertical Coriolis and the horizontal Coriolis. The corresponding inertial forces are the opposite of the acceleration terms. An easy way to convince yourself is the simple case of a linear acceleration. If you are in a elevator at rest, gravity provides a downward force of 1 g. If the Earth suddenly disappears and you want to replicate the same force in the same direction, the elevator need to accelerate upward (at 1 g), not downward.
Dear reader, if you find yourself confused by this section, you might want to read it again but in a different frame of reference, one where you scroll more slowly. And maybe also carefully read the Wikipedia entry on inertial forces.
Intuitive Coriolis
The first of the two additional inertial forces is the horizontal Coriolis:
This effect is triggered by radial motion, i.e. vertical motion with respect to the pseudo gravity. It cause an apparent force to the side and can be understood intuitively as a “tangential speed lag”. When moving up for example, you access a part of the cylinder where the tangential speed “at rest inside the rotating cylinder” (V_θ = rω_pool) is lower, as illustrated by figure 3. In this example ω_pool is constant and r is decreasing. Your excess tangential speed will push you toward the prograde direction (the direction where the cylinder is rotating forward). And vice versa, the push is retrograde for a downward motion.
The second of the two additional inertial forces is the vertical Coriolis:
This is caused by tangential motion i.e horizontal motion with respect to the pseudo gravity, but only in the direction of the rotation. When you move in the prograde direction you are adding your own rotation speed to the cylinder’s which increases the effective centrifugal force or effective pseudo gravity. And vice versa, the pseudo gravity is decreased with a retrograde motion.
Figure 4 summarizes the direction of the Coriolis force, for each direction of motion. Vertical (or radial) motion is in green, horizontal motion along the rotation axis is in blue, and horizontal motion perpendicular to rotation axis (or tangential) is in red. Black arrows indicate the direction of the Coriolis forces.
The magnitude of the horizontal Coriolis is proportional to both vᵣ and ω_pool. The faster you move, the fastest your speed becomes desynchronized with the cylinder’s and the stronger the apparent force. The magnitude of the vertical Coriolis is proportional to both your tangential speed in the pool frame (noted v_θₒ) and ω_pool. At small tangential speeds, i.e with dθₒ/dt = ωₒ << ω_pool, the effect is an increase or decrease in apparent gravity. In the prograde direction, this increase would be in accordance with the natural intuition. A downward push is expected due to moving along a curved path. However the push will be much stronger than the naive expectation. Because you are already rotating, the gravity increase is compounded by that prior motion. In the retrograde direction, the result is counter intuitive, as you feel lighter instead of heavier. In the extreme case of an object moving at exactly the rotation speed in the retrograde direction: Fᵣ = -2rω_pool². Not only does the object appear to cheat the pseudo gravity, but even to experience it in reverse because it is orbiting the central axis. Of course this is all an illusion, the whole pool is in fact rotating while the object is static.
For such cases it is not really practical to think in terms of pseudo gravity plus Coriolis forces. Let’s come back to the original centripetal acceleration term in cylindrical coordinates: a = -rω². Here ω is our old θ-dot from equation 1, it is the rotational speed of the object in the static frame of reference. We are now thinking like we could see things from outside the pool. It becomes obvious from there that an object with ω = 0 would not fall down. Also the square on ω implies that if you can cancel even just a modest fraction of the rotational speed with self motion, you remove large chuck of the pseudo gravity. For example at half the rim speed towards retrograde, 3/4 of the pseudo gravity is gone.
The dolphin leap
By taking advantage of the small rim velocity and the square relation between the pseudo gravity and the tangential speed, you can leap around like a dolphin.
Our pool rim speed is 5.52 m/s. By using a monofin underwater, it is possible to reach a speed of 3.61 m/s, and by using bifins at the surface a speed of 3.30 m/s (50 m world records). However using a monofin properly with a good dolphin kick technique is difficult. In contrast bifins only use the more basic flutter kick. Let’s assume that a fit individual can reach 2.2 m/s (7.9 km/h) when exiting water with bifins. This speed only need to be maintained for a few seconds before the jump, and only requires 30% of the power of the 3.3 m/s record, as power needed is proportional to the cube of speed.
The dolphin leap maneuver is as follows: you dive underwater and then accelerate in the retrograde direction. As you accelerate, your apparent gravity decreases, while you are still underwater. You thus experience a differential Archimedes force that pushes you upward, as the water around you is still under nominal pseudo gravity. However, as you are climbing up, the Coriolis force pushes you in the prograde direction. Some trial and error may be necessary before you find how to max out your exit speed and how to exit at the desired angle. If you exit the water at 2.2 m/s with a 45° angle, your tangential speed of 1.5 m/s will cancel 49% of the pseudo gravity, already weak at 0.26 g, allowing for a nice leap. Below is an animation of the simulated leap with these parameters. The air time and length of the leap are respectively 2.1 s and 2.6 m.
Differential Archimedes sounds like fun but a warning is in order. It will make you float like a champ if you swim in the retrograde direction, but if you try to swim the other way you will sink miserably. Drowning in a cylindrical pool will be dead easy if you start to panic and try to regain a foothold by swimming the wrong way. If you find yourself in such a situation, don’t panic and instead calmly remember the expression of acceleration in cylindrical coordinates!
The space Jesus
What if you could somehow leap with a tangential speed equal to the rim speed? You would then stay above the water forever… until the head wind of 5.5 m/s slow you down after a few seconds, ending your grace period. Leaping at that speed seems difficult though. But what if you can accelerate progressively using fins: the faster you go, the lighter you become, the more you can pull yourself out of the water, the less water drag, the faster you go, and so on until you reach rim speed or close to, at which point you can try to stand up and run over the water! Well, not quite… as you are are now close to no apparent gravity, running would be complicated, but instead you can try to tumble while scooping water behind you to compensate for the air drag. That would look nowhere near as graceful as Jesus walking on water but that is the best you can do without breaking physics. Alternatively, you can try the transition from the potato dive to the space Jesus for an easier start.
It remains to be determined empirically at which parameters it is humanly possible to perform the space Jesus for any meaningful length of time. The current pool parameters might need to be tweaked to a lower pseudo gravity, slower rim speed or even an air pressure lower than 1 bar.
The thrill seeker parameters
As evidenced by numerous youtube compilations of the “World Barefoot center”, face planting in water at high speed at near 0 angle of attack seems relatively benign. That opens up the possibility of the super potato dive. Barefooting is a version of water skiing but without skis, and requires higher speeds of 14 to 20 m/s. So let’s abandon any semblance of reasonableness toward costs, and generously increase the pool radius to get a rim speed around those values:
- ω_pool = 4.42 rpm (rotations per minute)
- r_pool = 43.2 m (radius down to water surface)
- v_rim = 20 m/s
- g_pool = 0.93 g
In this larger (lake sized) pool, the terminal angle of attack is 9° and terminal speed is 16.6 m/s. The air drag obviously affects the trajectory, but the tangential speed (in the static frame of reference) at the end is still small compared to the rim speed, and thus the angle of attack is still small. Amazingly, even at this larger radius diving is not that dangerous.
Bigger still?
Admittedly, this new simulation is mostly an excuse to run the engine in conditions where the air finally play an important role. We have been deeply frustrated to create such a wonderful simulator only to find that air drag was predictably negligible with the smaller pools. In animation 3, the new parameters are those of the “Stanford torus”:
- ω = 1 rpm (rotations per minute)
- r = 890 m
- v_rim = 93.45 m/s
- g_pool = 1 g
At about 200 m from the axis, the angle of attack goes through a minimum of 16° and then starts increasing again, because at this point the air drag is significant (air speed of 13 m/s). Because of air drag, the object has already acquired some tangential velocity and now accelerates more quickly downward (in the rotating frame of reference) because of the pseudo gravity, than it accelerates forward because of the Coriolis effect. The terminal speed and angle of attack after the 107 seconds fall are respectively 46 m/s and 35°. Water or not the arrival will be brutal, don’t try this at home. Just in case you were wondering, for the even larger O’Neill cylinder, with a radius of 3560 m and rotation rate of 0.5 rpm (also generating 1 g), the terminal speed and angle of attack after a 262 seconds fall are 59 m/s and 56°.
Cost (12 m radius version)
Surely, constructing such a ludicrous and lavish recreational facility would not only infuriate ecologists for many generations, but also bankrupt humanity. Let’s see how much it would cost just to haul all that water into low Earth orbit.
A cylindrical pool with a water surface radius of 12 m, a length of 30 m and a depth of 2 meters contains 4900 m³ of water. You can visualize the cylindrical pool as an XL Olympic pool wrapped on itself like a burrito. Their volumes are similar: the Olympic pool dimensions are 50 m × 25 m × 2 m (2500 m³ in volume).
Hauling this much water thus requires 49 reusable Big Falcon Rocket launches (100 tons payload). Assuming an optimistic cost per launch of $10 million, the cost comes at $490 million. If you are less optimistic, you can use a BFR launch cost of $100 million: then the cost balloons to $4.9 billion. If you are downright pessimistic and use the upcoming NASA space launch system (also ~100 tons to LEO), with an estimated launch cost of $500 million, then you can forget about your orbital pool.
In conclusion
We have found the Goldilocks parameters for an optimally fun and family-friendly cylindrical pool, although we have little confidence that the results will withstand the test of experimentation with real human subjects.
We have proposed 3 playful activities that can be done in a cylindrical pool, in order of increasing difficulty:
- the potato dive, safely available for anybody courageous enough to launch himself from the axis
- the dolphin leap, for which some swimming skills are needed to make a beautiful leap. Anyone can safely try but the result might only be a “whale leap”.
- the space Jesus, which requires power, balance and an absence of fear of the ridicule
We find that assuming a working BFR and basic on-orbit assembling capabilities, the cylindrical pool could be affordable by a 6 star space resort.
Footnotes & references
[1] In this article we unapologetically use the metric system and scientific notations. If you happen to have the misfortune of being exclusively accustomed to the imperial system: 1 m = 3.2808 ft. Caution: this will only work for British feet, if you are American, God bless you.
[2] Bukley 2006. Physics of Artificial Gravity.
[3] Globus 2015. Space Settlement Population Rotation Tolerance.
The recommandations on rotation rate from the paper are the following:
- Up to 2 rpm should be no problem for residents and require little adaptation by visitors.
- Up to 4 rpm should be no problem for residents but will require some training and/or a few hours to perhaps a day of adaptation by visitors.
- Up to 6 rpm is unlikely to be a problem for residents but may require extensive visitor training and/or adaptation (multiple days). Some particularly susceptible individuals may have a great deal of difficulty.
- Up to 10 rpm adaptation has been achieved with specific training. However, the radius of a settlement at these rotation rates is so small (under ~20 m for seven rpm) it’s hard to imagine anyone wanting to live there permanently, much less raise children.
[4] A sphere of human equivalent density and mass would have a substantially smaller air drag than the corresponding human, so we have cheated and changed the sphere air drag coefficient value from 0.47 (true coefficient for a sphere) to 1.2 so that the terminal velocity of the sphere falling in Earth atmosphere and gravity is 67 m/s, somewhere between the speed of a typical skydiver falling belly-to-earth (55 m/s), or head first (80 m/s). The air drag direction of the equivalent sphere is always aligned with the relative wind, any lift or Magnus effects are neglected.
