Cheat sheet · Engineering & Science · Spin gravity
Spin a ring and the floor pushes back on your feet. That push is weight. The radius and the spin rate set how much of it you get, and how well people tolerate the spin. Enter a radius and either a spin rate or a target gravity to get rim speed, head-to-foot gradient, walking Coriolis and the published comfort band, then compare five historical designs with Concordia.
§ 1 · Quick reference
| Quantity | Formula | Concordia design value |
|---|---|---|
| Gravity at radius r | g = ω²r = v²/r | 0.0626² × 2,500 = 9.80 m/s² = 1.00 g |
| Spin rate | rpm = 60ω / 2π | 0.0626 rad/s = 0.598 rpm |
| Rim speed | v = ωr | 156.6 m/s (564 km/h) |
| Period | T = 2π / ω | 100.3 s per turn |
| Radius for a target g | r = g / ω² | 2,500 m at 0.0626 rad/s |
| Gravity at height h above the floor | g(h) = g₀ (1 − h/R) | 0.82 g at the 450 m sky |
| Head-to-foot gradient | Δg / g = height / R | 1.8 / 2,500 = 0.072 % |
| Coriolis on a walker | a = 2ωv, as a share of local weight 2v/(ωr) | 1.4 m/s: 0.0179 g (1.8 % of weight) |
| Weight of a rider moving along the rim | (ωR ± v)² / R, + spinward, − antispinward | 0.66 g at 100 km/h antispinward; 1.33 g at 90 km/h spinward (metro, 2,455 m) |
| Radius for 1 g at a given rpm | 1 rpm 894 m · 2 rpm 224 m · 3 rpm 99 m · 4 rpm 56 m · 6 rpm 25 m · 10 rpm 9 m | 0.598 rpm needs 2,500 m |
The rule-of-thumb radii use g₀ = 9.80665 m/s². Globus and Hall's table (section 6) uses a slightly different g and gives 895.47 m at 1 rpm. The Coriolis ratio is twice the relative speed over the habitat's tangential speed (Hall's SpinCalc states the same ratio).
§ 2 · Spin calculator and spin chart
Pick a preset or type your own. The orange marker on the chart follows your inputs; each dot is a design from section 7.
§ 3 · How spin makes weight
Nothing pulls you toward the rim. Seen from outside, a free object keeps moving in a straight line while the rotating floor swings round and meets it.
| Radius | Spin rate | Period | Rim speed | Head-to-foot gradient | Walking Coriolis |
|---|---|---|---|---|---|
| 10 m | 9.46 rpm | 6.34 s | 9.9 m/s | 18 % | 0.283 g (28 % of weight) |
| 56 m | 4.00 rpm | 15.0 s | 23.4 m/s | 3.2 % | 0.120 g (12 %) |
| 224 m | 2.00 rpm | 30.0 s | 46.9 m/s | 0.80 % | 0.060 g (6.0 %) |
| 2,500 m Concordia design value | 0.598 rpm | 100.3 s | 156.6 m/s | 0.072 % | 0.0179 g (1.8 %) |
Concordia's own page on the same physics: Artificial gravity, with a dropped-coin demonstration.
§ 4 · Gravity fades with height
Gravity is ω²r, so it falls in a straight line from the floor to zero at the axis. Climb a tower and you move to a smaller circle that turns at the same rate, so you go slower and weigh less.
| Height above ground | Radius | Gravity g₀(1 − h/R) | Air pressure vs ground |
|---|---|---|---|
| Ground | 2,500 m | 1.00 g (0.999) | 100 % |
| 250 m | 2,250 m | 0.90 g | 97 % |
| 450 m, the artificial sky | 2,050 m | 0.82 g (0.819) | 95 % |
| 1,000 m | 1,500 m | 0.60 g | 91 % |
| 1,250 m | 1,250 m | 0.50 g | 90 % |
| 2,000 m | 500 m | 0.20 g | 87 % |
| Axis | 0 m | 0 g | 86 % (87.6 kPa) |
Air pressure: the atmosphere turns with the rotor, so pressure rises with radius. The column is an isothermal model, p = p₀ exp(−ω²(R² − r²) / (2 × 287 J/(kg K) × 293 K)), from 101,325 Pa at the ground. It is a derived estimate, not a Concordia specification.
§ 5 · Coriolis, gradients and the inner ear
Concordia's train-weight page shows the same rule on its metro at 100 km/h: 0.66 g one way, 1.33 g the other.
§ 6 · Comfort limits
Four sets of numbers, oldest first in each table. They disagree, and the disagreement is the point: the data come from short-radius rooms and small centrifuges.
| Criterion | Value | Where |
|---|---|---|
| Gravity | 0.95 ± 0.05 g (text: between 0.9 g and 1 g, "an arbitrary 10 percent variation") | Ch. 3 text; Table 3-3. The web transcription of Table 3-3 prints "± 0.5", an evident typo. |
| Rotation limit | 1 rpm, upper limit for the principal living quarters | Ch. 3; Table 3-3 lists ≤ 1 rpm |
| Adaptation | People can eventually adapt to rates below 3 rpm after prolonged exposure; frequent changes of rate may inhibit adaptation | Ch. 3 |
| Coriolis example | 0.55 m jump from the mid-deck lands about 5.3 cm to one side | Ch. 3, figure 3-1 |
| Gravity gradient | No limit stated | Ch. 3 does not discuss it |
| Relaxed case | 0.85 ± 0.15 g at 1.9 rpm: torus major radius 209 m, shield 1.0 Mt | Ch. 4, "Lower Simulated Gravity and Higher Rotation Rates", Table 4-2(a) |
| Band | Up to | Residents | Visitors | Radius for 1 g |
|---|---|---|---|---|
| 1 | 2 rpm | No problem | Little adaptation | 224 m |
| 2 | 4 rpm | No problem | Some training and/or a few hours to perhaps a day or so | 56 m |
| 3 | 6 rpm | Unlikely to be a problem | May need extensive training and/or multiple days; particularly susceptible people may have great difficulty | 25 m |
| 4 | 30 rpm | Adaptation achieved with specific training | The radius is so small it is hard to imagine anyone living there permanently or raising children | 1.0 m |
The authors call these recommendations quite conservative, because the underlying data come from rotation radii of a few metres and negative effects decline as radius grows. Caveats they list: most studies had 10 or fewer subjects, only adults, a few weeks or less, and were mostly done on Earth. The radius column is derived with g₀ = 9.80665; the paper's own radius-for-1 g table follows.
| rpm | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| radius (m) | 895.47 | 223.87 | 99.50 | 55.97 | 35.82 | 24.87 | 18.27 | 13.99 | 11.06 | 8.95 |
The paper notes that beyond about 6 rpm each extra rpm saves very little radius (6 m between 6 and 7 rpm).
| Chart | Min radius (m) | Max spin (rpm) | Min rim speed (m/s) | Min gravity (g) | Max gravity (g) |
|---|---|---|---|---|---|
| Hill and Schnitzer 1962 | not stated | 4 | 6 | 0.035 | 1.0 |
| Gilruth 1969 | 12 | 6 (optimum 2) | not stated | 0.3 | 0.9 |
| Gordon and Gervais 1969 | 12 | 6 | 7 | 0.2 | 1.0 |
| Stone 1973 | 4 | 6 | 10 | 0.2 | 1.0 |
| Cramer 1985 | not stated | 3 | 7 | 0.1 | 1.0 |
The rim-speed column is a minimum: a faster rim keeps the Coriolis-to-gravity ratio (2 × relative speed / rim speed) small. Nobody in this table goes above 6 rpm; SP-413's 1 rpm is stricter than all five.
Swipe the table sideways to see every column.
| Experiment | Setup | Finding | Source |
|---|---|---|---|
| Pensacola Slow Rotation Room, 1969 | Ten men aged 18 to 30; room 15 ft (4.57 m) diameter, 7 ft high; speed raised in 1-rpm steps to 10 rpm | Six of the ten reached 10 rpm, four could not; the authors recommend that rotation may safely commence at 2 rpm | Reason and Graybiel 1969, NTRS 19700005540 |
| Graybiel 1977, same room | Slow rotation room at 1.0, 3.0, 5.4 and 10 rpm | 1.0 rpm: even highly susceptible subjects symptom-free or nearly so. 3.0 rpm: symptoms, no significant handicap. 5.4 rpm: only low-susceptibility subjects performed well. 10 rpm: pilots with no air-sickness history had not fully adapted in 12 days | as quoted by Hall's SpinCalc |
| Lackner and DiZio 2003 | Repeated identical movements in rotation | Sensory-motor adaptation to 10 rpm can come relatively easily and quickly when subjects repeat the same movement | as quoted by Hall's SpinCalc |
| Gemini XI tether, 14 Sept 1966 | Capsule tethered to an Agena by a 100 ft line, slowly rotated with thrusters | About 0.00015 g of artificial gravity. The rotation rate is not stated in the NASA page | NASA Gemini XI mission page |
| NASA bed-rest centrifuge pilot, 2007 | 15 men, 21 days of 6° head-down bed rest; 8 got 1 h/day on a 3 m radius centrifuge (2.5 g at the feet, 1.0 g at the heart) | Beneficial effects on bone, muscle and cardiovascular function, no harmful effects on vestibular, immune or cognitive function; 153 of 160 sessions completed. A pilot, not conclusive | Paloski and Young, NTRS 20070016631 |
| JAXA mouse centrifuge on the ISS, 2017 | 12 male mice for 35 days, microgravity group vs an artificial 1 g group | Microgravity mice lost femur bone density and soleus and gastrocnemius weight; the 1 g mice matched the ground control | Shiba and others 2017, Scientific Reports |
§ 7 · Rotating habitat designs compared
Oldest first. d means derived by this page from the sourced radius and spin rate (g₀ = 9.80665); "not stated" means the source gives no value. Click a design's chip in section 2 to load it into the calculator.
Swipe the table sideways to see every column.
| Design | Year, source | Shape and size | Radius (m) | Spin (rpm) | Rim g | Rim speed (m/s) | Population | Shielding | Band |
|---|---|---|---|---|---|---|---|---|---|
| O'Neill cylinder, Model 4 | 1974, Physics Today, Table 1 | Cylinder pair, each 32 km long (the text says about 16 miles), 4 miles across | 3,200 | 0.526d | 0.99d | 176d | 0.2 to 20 million for the pair (the top figure is the ecological limit with conventional agriculture) | Atmosphere, soil and steel structure; 7.5 t/m² averaged surface density of the structure | 1 |
| Stanford torus | 1975, NASA SP-413, ch. 4 and 5 | Torus: major radius 830 m, tube 130 m across (65 m minor radius), six spokes 15 m across. Design gravity 0.95 ± 0.05 g | 830 (895 outerd) | 1 | 0.93 to 1.00d | 87 to 94d | 10,000 | Passive slag shield separate from the habitat, rotated far below 1 rpm or not at all; 9.9 Mt at 4.5 t/m² | 1 |
| Island One (Bernal sphere) | 1976, L5 News, Sept 1976 | Sphere, equator "nearly a mile in circumference"; Earth-normal gravity at the equator, falling to zero at the poles | about 256d | about 1.9 | 1.03d | 51d | about 10,000 | Non-rotating shell of industrial slag; mass not stated | 1 |
| Island Three | O'Neill, as summarized by NSS | Counter-rotating pair, each about 20 miles long and 4 miles across | about 3,220d | not stated | not stated | not stated | several million | not stated | n/a |
| Kalpana One | 2007 (revised), Globus and others | Cylinder: radius 250 m, length 325 m, spin capped at 2 rpm | 250 | at most 2 (1.89d for 1 g) | 1.00d at 1.89 rpm | 49.5d | 3,000 | Up to 10 t/m² of lunar or asteroid regolith inside the hull, about 7 Mt | 1 |
| Concordia design value | Design study set in 2045 | Four habitation rotors, each radius 2,500 m and length 2,500 m, plus a counter-rotating industrial rotor | 2,500 | 0.598 | 1.0 | 156.6 | about 1 to 1.2 million (all four rotors) | Stationary shield outside the rotating pressure vessels; thickness not fixed (worked example: 1 m at 1.6 t/m³, about 750 Mt) | 1 |
| Concordia industrial rotor design value | same | Radius 2,300 m, length about 3,500 m; industry and agriculture, not housing | 2,300 | about 0.42 | about 0.45 | about 101 | not residential | Open framework with pressurized modules | 1 |
| Model | Length (km) | Radius (m) | Period (s) | Spin (rpm) | Population (double unit) | Earliest date |
|---|---|---|---|---|---|---|
| 1 | 1 | 100 | 21 | 2.86d | 10,000 | 1988 |
| 2 | 3.2 | 320 | 36 | 1.67d | 100,000 to 200,000 | 1996 |
| 3 | 10 | 1,000 | 63 | 0.95d | 0.2 to 2 million | 2002 |
| 4 | 32 | 3,200 | 114 | 0.53d | 0.2 to 20 million | 2008 |
Why the stages scale together: each stage is sized for about 1 g, so the period grows as √radius. Model 1's 21 s is as printed; exactly 1 g at 100 m needs 20.1 s. The dates are the paper's "earliest estimated" dates, now long past.
Bernal 1929. J. D. Bernal's The World, the Flesh and the Devil describes a spherical shell "ten miles or so in diameter" for twenty or thirty thousand inhabitants (p. 23, as quoted in arXiv 1806.01322). It is generally described as a non-rotating globe; the spinning "Bernal sphere" is the 1976 Island One, which borrowed the name. This page did not check the book's text for the question of spin.
§ 8 · Field guide: life inside a spinning world
Each card: the surprise, the number, why. Numbers are for Concordia's 2,500 m rotor design value unless a card says otherwise.
Walk 8 km along the rim and you have turned 183°: your feet point where your head was. One lap is 2π × 2,500 m = 15.7 km, a 95 minute walk at 1.4 m/s.
The street is a circle seen from inside, so it never feels like a slope.
On the metro's 2,455 m rail head, a rider at 100 km/h against the spin weighs (ωr − v)²/r = 0.658 g; at 90 km/h with the spin, 1.326 g. Standing still: 0.98 g.
Riding with the spin adds to your speed in inertial space, so the seat must push harder.
Drop it 2 m: 0.64 s later it lands 5.3 cm antispinward of the point below. From 100 m the offset is 19.7 m after 4.7 s. The same 2 m drop lands 18 cm aside at 2 rpm (224 m) and 1.07 m aside in a 10 m centrifuge at 1 g.
The coin keeps the tangential speed of the height it left, ω(R − h), which is slower than the floor's. Exact: t = √(R² − r₀²) / (ωr₀), lag = R (ωt − arccos(r₀/R)), with r₀ = R − h.
g = g₀(1 − h/R): 0.82 g at the 450 m artificial sky, 0.60 g at 1,000 m.
You move to a smaller circle that turns at the same rate, so you go slower.
From the sky level (radius 2,050 m) the far ground is 2,050 + 2,500 = 4,550 m away: a second city with its floor facing you, upside down.
Everything inside a rotor faces the axis, so the opposite side is a ceiling that is also someone's street.
In Concordia's glider model (a 120 kg hang glider, estimated design values, not a real aircraft), least-power flight with the spin takes about 2.0 kW at the ground and about 250 W at 2,000 m, where weight is 0.20 g.
The wing carries a fifth of the load. At 14 m/s it feels 1.19 g with the spin and 0.83 g against it at the ground.
Concordia's radial lifts climb about 2,380 m in 595 s (9.9 minutes), from 0.98 g at the 50 m deck to 0.028 g at the hub terminal, 70 m from the axis.
Gravity is ω²r, so it reaches zero only at r = 0. The terminal still weighs 2.8 % of normal; the transfer to the stationary axis spins down first.
If all four rotors turned the same way, turning the 21 km station 1° a day would take about 4.1 MN at each end of the hull. With counter-rotating pairs as designed it takes about 4.5 kN, roughly 1,000 times less.
A body with net angular momentum resists being turned like a gyroscope; pairs cancel it.
At 20 m/s (45 mph) along the rim, a ball behaves as if its weight were 1.27 g thrown spinward and 0.76 g antispinward: (ωr ± v)²/r. At 4 rpm (56 m radius) the same antispinward throw weighs about 0.02 g, because its speed in inertial space is almost zero.
Rule: spinward curves floorward, antispinward lifts off. Players who adapt to it have a home-field advantage.
A 20 cm pour is pushed aside by 0.17 cm at Concordia, 1.1 cm at 4 rpm (56 m) and 2.7 cm in a 10 m centrifuge at 1 g.
The same fall geometry as the coin. It ignores splashing and air drag, so it is a lower bound on what a liquid does.
The air turns with the rotor, so pressure falls from the ground to the axis: about 95 % of ground pressure at the 450 m sky and about 86 % (87.6 kPa) at the axis, in an isothermal model at 293 K.
Pressure follows p₀ exp(−ω²(R² − r²) / (2 Rgas T)); the same centrifugal field that makes weight sorts the air.
§ 9 · Engineering consequences of spin
§ 10 · Common mistakes
§ 11 · Explore it
Concordia is a design study set in 2045, built so you can stand at 1 g, ride the metro that weighs 0.66 g one way, and climb to the 0.82 g sky, all in your browser. This page only links to it; nothing here loads the simulation.
Sources