# What minimal radius is needed for rotation to simulate gravitation without averse effects on humans?

A quite common idea to provide "gravitation" in space stations is to make them rotate, so the centrifugal force gives an effective gravitation. A possible design is a ring-shaped space station.

Now of course it is easy to calculate how fast a space station has to rotate, as function of the radius, in order to provide a given g-value. Of course, the smaller the radius, the larger the needed angular velocity to provide the gravitation:

$$\omega = \sqrt{\frac{g}{r}}$$

However, a small, fast rotating a space station has two disadvantages:

• Due to the dependence of the centrifugal force of the radius, there's a difference in the gravitational strength between head and feet. For a human of height $h$, when the floor is on radius $r$ (assuming $r>h$, of course) you get

$$\Delta g = \frac{g h}{r}$$

This difference might give problems; but I doubt they will be the main problem. Also, it falls off quite quickly with $r$, so with any halfway reasonable size of the space station, I guess it should be no issue.

• Due to the rotation you get a Coriolis force. This should mess with the human sense of balance. Moreover, since it is proportional to $\omega$ (more exactly, for running perpendicular to the rotation axis, it's $2\omega v$), it only falls off with the square root of the radius, so I guess that is the determining factor to decide which radius is needed.

• Also an effect is that when moving, the direction of "gravitation" will change as you walk around the ring. This I guess will tend to cause you to stumble as soon as you walk, let alone run, if the radius is too low. I have no idea how well humans can adapt to this (or if that question has even been studied).

So my question is:

What would be the minimal radius for a space station, if there should be no problematic effects for humans?

Assume aiming for earth-like gravity ($g=10\,\rm m/s^2$), normal size humans (height up to not much more than 2 meters) and people may run (from 100 meter sprint times, one may assume a maximum speed of about 10 m/s). Also, we can assume there's only one floor (no "upstairs" with different radius).

Note that this is not really a question about physics (how to calculate the physical quantities is clear to me) but more a question about human physiology (how weak do we have to make the effects to not cause problems), thus the tag.

• I don't know how accurate his points are, but William Gibson in Neuromancer speaks of how the most precise athletes (which for the scope of the story means assassins) must account for coriolis force, especially when using missile weapons. From memory, Freeside is a few hundred metres in radius (and cylindrical, rotating about its long axis). Assuming he's correct, your answer strongly depends whether "adverse" to you means "unable to put an arrow through my enemy" (you need a very large station or to train for it) or "my heart suddenly stops due to the stress" (smaller is fine) :-) – Steve Jessop Dec 9 '14 at 1:50
• Just a quick note: Coriolis effects won't come into play with a ring-shaped space station. Coriolis acceleration is perpendicular to the rate of change of radius multiplied by the angular velocity. If your characters are walking on the interior surface of a rotating ring, their radius is fixed. – Doresoom Oct 8 '15 at 17:47
• @Doresoom: Wrong: Coriolis force comes into play whenever you move perpendicular to the rotation axis. That's true both for radial movement and for movement along the ring. – celtschk Oct 8 '15 at 18:50
• @celtschk I suppose if you want to look at tangential velocity with respect to the ring's surface as a separate component from overall angular velocity, then your Coriolis acceleration will be additive to your normal acceleration. I was considering Coriolis from the perspective of it acting in a lateral direction, which would unbalance space station occupants, rather than adding to their perceived weight. – Doresoom Oct 8 '15 at 21:20
• @celtschk You yourself mentioned "BALANCE" in your original question. The only way to achieve that lateral unbalancing force is with a rate of change of radius. So my statement above is not wrong, just approaching the problem from a different perspective - one that you alluded to, whether your realize it or not. – Doresoom Oct 8 '15 at 21:21

There is a very nice source written by Theodore W. Hall, where you can calculate parameters of a rotating habitat. Even more importantly, they give comfort areas for different parameters based on many scientific sources. The parameter ranges are:

Because centripetal acceleration – the nominal artificial gravity – is directly proportional to radius, inhabitants will experience a head-to-foot “gravity gradient”. To minimize the gradient, maximize the radius.

Above 12 m is comfortable, 4 m - 12 m might work

Angular Velocity

The cross-coupling of normal head rotations with the habitat rotation can lead to dizziness and motion sickness. To minimize this cross-coupling, minimize the habitat’s angular velocity.

Less than 2 rotations/minute is comfortable, 2 rpm - 6 rpm should work

Tangential Velocity

When people or objects move within a rotating habitat, they’re subjected to Coriolis accelerations that distort the apparent gravity. For relative motion in the plane of rotation, the ratio of Coriolis to centripetal acceleration is twice the ratio of the relative velocity to the habitat’s tangential velocity. To minimize this ratio, maximize the habitat’s tangential velocity.

More than 10 m/s is comfortable, 6 m/s - 10 m/s should work

Centripetal Acceleration

The centripetal acceleration must have some minimum value to offer any practical advantage over weightlessness. One common criterion is to provide adequate floor traction. The minimum required to preserve health remains unknown. For reasons of cost as well as comfort, the maximum should generally not exceed 1 g.

0.3 g - 1 g is comfortable, 0.1 g - 0.3 g might work

Based solely on the calculator, if you want centripetal acceleration 1 g, radius 220 m would be very comfortable and radius below 25 m would be already very uncomfortable. If you want acceleration 0.3 g, radius 70 m would be very comfortable and radius below 13 m would be very uncomfortable.

Edit 1: Recently, I found very interesting related link - TidalWave's commentary about experiments with centrifuges, to his answer at Space Stackexchange.

• A very informative answer, and a great link. Thank you. – celtschk Nov 22 '14 at 14:59

It's highly uncertain.

The limiting factor appears to be Coriolis effects, which means the key parameter is rotation rate rather than radius. Studies have given tolerable values ranging from 0.1 RPM to 23 RPM, and corresponding radiuses ranging from as high as 90km to as low as 4m.

• Got a link / reference? – user3082 Nov 23 '14 at 5:53
• @user3082, are the studies linked to in my post insufficient? – Mark Nov 23 '14 at 6:12

I think the short answer is that the radius should be as large as is pragmatically possible, because there will always be unforseen reasons for making the centrifuge bigger. Also, there is no real reason to limit the cylinder to one level only. For a trip to another astronomical body, at least one level could be devoted to the new gravity, or something in-between if it's going to be lower gravity.

The Earth Gravity simulator would only need be used for exercise periods, which I assume would increase in frequency and duration during a return trip to Earth out of a sense of impatience if nothing else. Routine operation of a space ship could be done either in a nonrotating portion of the ship or in the central core, the latter of which would be best served by the least possible speed of rotation.

As far as the ill effects of varying moment(um)s, habits will undoubtedly develope to deal with this problem. Again, the slower the speed, the easier it will be to do so.

The added mass of a larger cylinder should not be a great problem in a zero gravity-zero atmosphere other than the additional fuel required for acceleration, and since the larger diameter cylinder need not be of significant weight if it's used only for exercise, this problem could be minimized.

• Welcome to Worldbuilding.SE! Please have a look at the Tour and the Help Center! Please improve the readability of your answer by formatting the text into paragraphs. Otherwise, this is a good first answer! – DarthDonut Mar 19 at 11:39
• Done - it takes two returns to make a paragraph. I'd forgotten that since the last forum I posted in. Thanks! – Leech Phillip Mar 19 at 11:48
• Of course another effect of larger size/mass is larger building cost. And I think the weight of the outer hull at a given size will be determined mostly by the fact that it should protect against radiation and micrometeorites. But good point on the unforeseen reasons. – celtschk Mar 20 at 17:18

Coriolis forces and other effects aside, you will want your space station to have a level of simulated gravity similar to that of planets that the inhabitants might be from or be destined to visit. At least in some part of the station.

There would be no point having a station orbiting Mars that could not simulate at least 0.38g, the approximate surface gravity on Mars and Mercury, for example. It could be scaled to simulate higher gravity at the rim with the design including another major level at a radius that offers 0.38g.

In the Earth-Moon region, it would make more sense to keep it large enough to have comfortable full Earth gravity at the rim because of the need for people to acclimatize to full gravity as they return to Earth. Lunar gravity could easily be simulated on the same station at a smaller radius, or in a separately rotating section.

For the solar system as a whole, with two major planets having 0.38g on the surface, that seems like a reasonable minimum value to aim at. The only way I see that being too low is if there turns out to be a serious medical effect of it being that low. That may not matter if reproduction or long-term residence is not intended.