# Is there an upward limit to the size of spacecraft with "Centrifugal" gravity?

I'm in the middle of creating a RPG one-shot for some friends, and I'm thinking of doing an Aliens meets Space Hulk style "Big-Gribbly Aliens VS Humans" game.

Is there an upper limit to the size of a craft with centrifugal style gravity? I've only ever seen them in fiction presented with single floor pods or rings around a central hub. You can obviously get more floorspace by creating a cylinder around the central spire, but could you go out and create more floors?

If so, would this alter the relative gravity per floor, and by how much?

(tagged with "Science-Based" as I'm not sure how "hard" the centrifugal gravity concept is)

• Are you talking about centrifugal antigravity or centrifugal gravity?
– o.m.
Commented Jul 5, 2016 at 15:28
• Good point. I'll edit the question (D'oh!) Commented Jul 5, 2016 at 15:31
• Centrifugal gravity is very hard, There is a size limit that depends on material strength. It can't have a radius more than the free break length of the material but for nano tubes this is 1000's of km. Commented Jul 5, 2016 at 15:50
• One issue that is difficult enough to quantify that it is usually ignored in speculative discussions but would be a major issue in any real design. Needed velocity goes up with radius and the energy released on failure goes up with the square of velocity. Damage caused and features needed for safety go up with energy. Cost of safety would scale NlogN or even N² I think? I expect that in any real design that needs to think about the safety margins for the people that would be the actual limiting factor. Just a side note and possibly totally wrong, but for realism failure modes are big. Commented Jul 6, 2016 at 4:11
• One issue, for ships is that they have to maneuver, not just sit there and spin (anyone remember the Sit-n-Spin?). That means that they have to be over built to take the stresses of accelerating and turning. Commented Mar 6, 2018 at 17:55

That is certainly possible, and enough science fiction settings cover them.

• The centrifugal force depends on the radius and the rotation speed. It is proportional to the radius, so a ring with a large radius allows several decks with very similar "gravity."
• A large diameter allows a low rotation speed, which means less distraction through the coriolis force.

Follow-Up:

Imagine a cylinder with a diameter of 40 m (i.e. a radius of 20 m) that rotates three times a minute. You stand on the inside of the cylinder. You would move at 6.3 m/s and feel 0.2 g.

• If you walk spinward around the cylinder, you will feel more than 0.2 g.
• If you walk antispinward around the cylinder, you will feel less than 0.2 G.
• if you walk parallel to the axis, you will feel 0.2 g.

"Up" will always be towards the spin axis, but the little changes can make you dizzy.

So the direction of movement on the same deck changes the felt gravity.

• If the deck height is 2 m and you stand one deck higher, you will feel only 0.18 g.
• OK, so it's possible - my math is horrendous (particularly when it comes to this!) would it be possible for you to explain this a bit more laymans terns than a = w^2 * r? I understand that the W is the angular velocity (I.E. speed the ring rotates at), A is the gravity produced and R is the radius of the ring. I also understand that this would mean that you'd have to walk at right angles to the spacecraft (i.e on the "sides" of the ring, rather than the "bottom") so the center would be "Up". Commented Jul 5, 2016 at 16:07
• sorry- my edits won't save. Am I right in my assumption that the gravity would change per floor? as I understand it, as the radius increases, the angular velocity decreases for the same value of "a" - therefore, the gravity would increase as you go down floors? Commented Jul 5, 2016 at 16:27
• If we use the velocity v of a point on the ring rather than the angular velocity w, the formula is $a = v^2 /r$, you can solve that for v to give $v = \sqrt{a * r}$. So the statement "A large diameter allows a low rotation speed" is possibly misleading--it does allow a lower angular speed (angle swept out by a given point per second), but the regular speed in meters/second becomes higher the larger the diameter, assuming we keep the artificial gravity constant. Commented Jul 6, 2016 at 1:37
• How was this selected as the answer when it does not address the upper limit aspect of the question and merely explains the principles of centrifugal gravity? And the example given implies a noticeable impact on apparent gravity if additional decks are built, but that is only true of very small structures.
– rek
Commented Mar 6, 2018 at 20:41
• @rek, I suspect timing. It was an early answer.
– o.m.
Commented Mar 7, 2018 at 6:06

O'Neill, G. K: The Colonization of Space, Physics Today, vol. 27, no. 9, Sept. 1974, pp. 32-40.

Up to 16km radius is doable with usual materials. Conservative with old steel (1920) 3.2km

$\tiny \text{Source, O'Neill, G. K: The Colonization of Space, Physics Today, vol. 27, no. 9, Sept. 1974, pp. 32-40.}$

One foreseeable development is the use of near-frictionless (for example, magnetic) bearings between a rotating cylinder and its supporting structure, which need not be spun. For eight tons per square meter of surface density and a tensile strength of 300,000 psi, R would be 16 km, the total area would 50,000 km2, and the population would be between five million (low density) and 700 million (the ecological limit, the maximum population that can be supported).

This is with common materials, with 1g, and normal pressure.
The internal pressure of air is a big part of forces, and if it will be 1/3 of normal (used in some spacecraft's), and 1/3g (mars like) - it will be bigger, 3 times at least.

The main problem here is this surrounding external supporting structure(how much material we may have) and interaction of internal structure with that external one, bigger is radius bigger is the difference in speeds between internal and external structure(there are some tricks though).

With stronger materials, not know at that time 1975, structure may be bigger, linear proportion here 5 times stronger material, 5 times bigger radius. With the strongest material as we know at the moment(>100GPa), it may be 100 times bigger - so 320km radius (or bigger with 1/3g and 1/3 pressure)

But that's still not the limit, but probably good enough for the size of a craft with centrifugal style gravity.

A rough estimation of limit is actually external structure and it's ability to hold 1bar pressure. And roughly there is a linear proportion between thickness of walls and radius of cylinder, more complex internal structure of cylinder also may help (remove pressure of air stress to walls - tethers inside with central axis - so making cylinder itself more robust helps)

The cylinder is more limited with the material strength we have and the amount of that material, and our wishes to move so much material.

## P.S.

Here is some NASA stuff, about choosing sizes, masses of possible habitats, crew. It's from Space Settlements: A Design Study broader and more practical overview of that topic.

• See my answer below for a structure built to the theoretical limits of molecular nanotechnology.
– rek
Commented Mar 6, 2018 at 18:47
• @rek yes, quite an okay answer. However, I would not call it theoretical limits of molecular nanotechnology. It is just an option for a combination of technology and material. idk, maybe it may be interesting to you, you can take look at SmartMatter, and there are options to increase the size of such space habitat by using the material. I have material for Q3 about the smart matter which would make the option more understandable, but I'm not sure when(if) I will add this information. Commented Mar 7, 2018 at 16:49
• O'neill's numbers seem high, 152ksi working stress in particular. JFE Engineering claim 780 MPa for the steel they used in the Akashi Kaikyo Bridge (1997). That 113 ksi on a modern bridge. Commented Jul 20, 2019 at 10:35
• @Andrew yes interesting link, maybe, in any case, it some sort of alloy stuff so it will depends on the situation, but composite materials in general and it may not necessarily be steel. Also, it sort of cables has lesser tensile strength than a ticker block as an example. For an estimation, it is an okayish number, so as dependance of radius and tensile strength is linear so it can be easily adjusted on any number one thinks is realistic. Commented Jul 21, 2019 at 5:24

Tom McKendree came the conclusion that an O'Neill-style cylinder constructed with carbon nanotubes could have a radius of approximately 461 km.

Quoting the above (footnotes omitted):

The maximum radius of such an O'Neill style colony is limited by the hoop stress of the spinning structure, and the tensile strength to density ratio of the material. The formula is

$$R < \frac{HoopStress}{gG}$$

Where R is the radius, g is the acceleration of pseudo-gravity at the rim, and G is the density. [Molecular nanotechnology (MNT)] offers a 5 x 10$$^{10}$$ Pa tensile strength. Using the design rule of 50% safety factors for O'Neill style colonies, a 3.3 x 10$$^{10}$$ Pa design tensile strength is reasonable. The associated material density is 3.51 x 10$$^3$$ kg/m$$^3$$. One goal of the architecture is for g to equal 9.8 m/s$$^2$$. This all gives a possible space station radius of 9.6 x 10$$^5$$ m, or nearly 1000 km. For comparison, the corresponding feasible radius for titanium is 14 km, and even at its ultimate tensile strength with no safety factor, the titanium limit would be 23 km.

At the 9.6 x 10$$^5$$ m radius, the entire available strength (at the safety factor) of the MNT-based material is being used to prevent the rotating structure from bursting, and there is no strength left over to hold the space station's contents, including an atmosphere. To do so, a lower radius must be set.

In the section immediately following, McKendree arrives at a radius of 461 km when atmosphere and fixtures are accounted for:

One can directly solve for the structure radius where the shell is 5000 kg/m$$^2$$. Using MNT materials, the structure will be 461 km in radius. For comparison, the equivalent number for titanium is 6.6 km, or for a titanium shell is at its ultimate tensile strength with no safety factor, 11 km.

A number of sci-fi space colonies use the rotating cylinder concept, with an implied diameter of a few kilometres. For a 3km radius cylinder, Earth-like gravity would require a rotation rate of about 1 spin per 110 seconds. Rim speed for the cylinder is a bit over 170 m/s.

Apparent gravity variation is directly proportional to the radius, so a few tens of metres up or down out of 3 km isn't going to make a huge difference.

The Ringworld (also echoed in the Halo game concept) uses a giant ring millions of km across. Rim speeds are ridiculous, and material stresses in the ring would be infeasible in real-world materials.

SpinCalc is useful.

• Welcome to Worldbuilding Joffan!
– fi12
Commented Jul 5, 2016 at 18:30
• @fi12 thanks, some interesting discussions here. Commented Jul 5, 2016 at 18:58

Yes. The material strength needed goes up with the radius. Eventually you can't hold it together.

You can create a modified version, though--the strength is all in an outer non-rotating shell, the station rotates within this on a maglev system. Even this will eventually reach a limit as you can't support the floors (just like there's a limit on how tall a building you can build.)

• I don't get why, if one needs to exert 1g, the strength expected by the material would increase.
– L.Dutch
Commented May 31, 2019 at 3:29
• @L.Dutch Take a metal ring, the material weighs 1kg/m. A 1m radius ring thus weighs 6.28kg, it's spinning at 1g, thus 6.28kg of force is trying to pull it apart. 2m ring, 12.56kg of force. 1km ring, 6.28 tons of force. Eventually the force pulling the ring apart snaps it. You can't beef the ring up because that increases the force as much as it makes it stronger. Commented May 31, 2019 at 16:12