Spin-gravity formulas and reference numbers

The physics used throughout the thread, with worked numbers that contributors checked against each other. More formulas are in formulas and calculators.

Key results

Centripetal acceleration

  • a = ω²·r, with ω in rad/s. In practical units: a [m/s²] = 0.01097 · r [m] · n² [rpm] dror, 2014-03-20.
  • At a given g, r ∝ 1/n²: doubling the rpm allows a quarter of the radius Hop_David, 2018-07-04.
  • Checked reference points:
Case Numbers
1 g at 50 m 4.22 rpm, rim speed 22.1 m/s RanulfC, 2014-03-21
1 g below 2 rpm radius >224 m RanulfC, 2014-03-21
1 g at 2 rpm / 1 rpm r ≈ 224 m / 894 m Coastal Ron, 2015-10-26
Mars g at 1.84 rpm 100 m radius RanulfC, 2014-03-21
Moon g at 1 rpm ≈149 m radius RonM, 2015-03-31
Mars g at 2 rpm / 0.67 g at 2 rpm 85 m (17.8 m/s) / 150 m (31.4 m/s); the larger station has about 3× the disc area Coastal Ron, 2026-03-13
1 g, 400 m diameter rim speed ≈63 m/s LarryCanuck, 2023-03-02
1 g, 100 km diameter 0.1337 rpm, rim ≈700 m/s whitelancer64, 2023-06-15
2001's Space Station V 150 m radius, 1 rev/61 s → ~1/6 g (as filmed) Hop_David, 2018-07-04

Gradient and floors

  • g scales linearly with radius, so each floor of a multi-deck module has its own g.
  • Paul451's Starship floors at r = 7, 11, 15 and 19 m give 0.20/0.31/0.42/0.53 g at 5 rpm Paul451, 2022-12-31.
  • A 24 m radius cylinder has 4 m ceilings at 1 g, so the next floor up is at 0.83 g.
  • Floor sag in a BA330 module: 5.6 cm at 223 m radius, 1.4 cm at 890 m Roy_H, 2014-02-19.
  • Flat vs curved floors. A flat floor segment feels tilted away from its centre. Physically curved floors are "gravitationally flat" but differ with radius, which prevents standard racks Paul451, 2023-06-13.
  • With 8 straight segments the gravity vector is up to 22° off normal at the ends; with 36 segments, ±5° omegahacker, 2025-01-10.

Moving in the rotating frame

  • Coriolis acceleration = 2ω × v. It is felt only when moving, not at rest Paul451, 2016-01-13.
  • Tangential motion changes apparent g: a_apparent = (v_rim ± v)² / r. Jogging spinward at 22% of rim speed changes g by about ±50% mikelepage, 2022-10-23. A 103 m ring at 0.5 g feels about 1 g when running prograde at 6 m/s mikelepage, 2022-10-15.
  • Falling objects keep their tangential velocity and appear to drift antispinward. Near the hub, collisions tend to eject loose objects outward Twark_Main, 2023-03-06.

Energy, momentum and spin-up

Structural size limits (self-weight)

  • A rotating hoop of material supports only its own weight up to a radius set by specific strength.
  • mikelepage: plain steel at 1 g tops out at ~2.5 km radius with no margin mikelepage, 2023-06-15.
  • Forum lore: titanium to ~14 km radius at 0.25 rpm; ordinary steel ~3.4 km diameter; carbon nanotubes ~1000 km whitelancer64, 2018-10-31.
  • Spinning Ceres (as in The Expanse) is impossible, at ~2 GPa stress Lampyridae, 2016-01-18.
  • Equivalent tangential "pressure" from internal mass (Twark_Main): P_t = ρ_internal · a · R. At 100 kg/m³ and R = 175 m (2.25 rpm, 1 g) this is already ~1.7 atm, before any shielding Twark_Main, 2023-09-29.

Open questions

  • None of these formulas answer what rpm humans tolerate (see comfort limits). They only convert between rpm, radius and g.

Sources

Forum posts

  • Roy_H, 2014-02-19 (reply #77): Floor sag BA330 at 223 m and 890 m. Post. Attachments: Bigelow.png.
  • dror, 2014-03-20 (reply #222, 1 like): a = 0.01097 r n². Post.
  • RanulfC, 2014-03-21 (reply #242, 1 like): SpinCalc reference points. Post.
  • RonM, 2015-03-31 (reply #557): Moon g at 1 rpm ≈149 m. Post.
  • Coastal Ron, 2015-10-26 (reply #735): 1 g @2 rpm 224 m, @1 rpm 894 m. Post.
  • Paul451, 2016-01-13 (reply #812): No lean at constant rotation. Post. Attachments: Spin ship - acceleration.jpg.
  • Lampyridae, 2016-01-18 (reply #831, 1 like): Spinning Ceres ~2 GPa. Post.
  • Hop_David, 2018-07-04 (reply #860, 3 likes): Spin g ∝ ω²; 2001 station ~1/6 g. Post.
  • lamontagne, 2018-08-06 (reply #1188): Wheel KE 2500 MJ; small station 70× less. Post.
  • whitelancer64, 2018-10-31 (reply #1243): Material size limits lore. Post.
  • mikelepage, 2019-04-07 (reply #1501, 2 likes): Spin-up/down 10–40 m/s per Starship. Post.
  • mikelepage, 2022-10-15 (reply #2104, 3 likes): Run prograde 6 m/s to feel ~1 g. Post.
  • mikelepage, 2022-10-23 (reply #2180): Prograde running changes g ±50%. Post. Attachments: gravComfort03-target.jpg.
  • Paul451, 2022-10-25 (reply #2206): Starship spin-up ~15 m/s tangential. Post.
  • Paul451, 2022-12-31 (reply #2736): Starship floor g table. Post.
  • lamontagne, 2023-01-15 (reply #2904): Ring vs disc MoI; outer ring dominates. Post. Attachments: Station (3).xlsx.
  • mikelepage, 2023-01-27 (reply #3126, 3 likes): Spin-up torque calculator (3 likes). Post. Attachments: Calculator screenshot.png, Torque to spin up Rotating spacecraft.xlsx.
  • lamontagne, 2023-01-27 (reply #3130): 57 t, 450 N, 100 h. Post.
  • JohnFornaro, 2023-01-27 (reply #3139): Ring spin-up ~21 kt propellant. Post. Attachments: Ring-Station-Spin-up-Torque.xls.
  • mikelepage, 2023-01-28 (reply #3149): Spin-up/down < one tanker; sun-facing costs more. Post.
  • JohnFornaro, 2023-01-28 (reply #3154): Eight Raptors spin up in 11 h. Post.
  • Twark_Main, 2023-01-30 (reply #3192): Torus MoI formula; spreadsheet bug. Post.
  • Paul451, 2023-02-02 (reply #3215, 2 likes): Rectangular toroid MoI formulas (2 likes). Post. Attachments: poloidally diamond toroidal hexagon.png.
  • Paul451, 2023-02-03 (reply #3220): Square vs round torus MoI within ~10%. Post. Attachments: Square vs round torus rot-inertia.PNG, horn torus.gif.
  • LarryCanuck, 2023-03-02 (reply #3455): 400 m dia 1 g → 63 m/s. Post. Attachments: Hall AG Spin Speed.png, Hall AG Spin Speed 2.png.
  • Twark_Main, 2023-03-06 (reply #3493): Loose objects ejected outward; boundary layer wind. Post.
  • Paul451, 2023-06-13 (reply #3858, 1 like): Flat vs curved floor trade (1 like). Post.
  • whitelancer64, 2023-06-15 (reply #3873): 100 km station numbers. Post.
  • mikelepage, 2023-06-15 (reply #3874, 1 like): Steel hoop limit ~2.5 km radius. Post.
  • Twark_Main, 2023-09-29 (reply #4083): Tangential pseudo-pressure ρaR. Post.
  • omegahacker, 2025-01-10 (reply #4811): Straight segments: 22° off-normal (8), ±5° (36). Post.
  • Twark_Main, 2025-04-29 (reply #4917, 2 likes): Use angular momentum, not energy. Post.
  • Coastal Ron, 2026-03-13 (reply #5042, 3 likes): 0.38 g vs 0.67 g at 2 rpm. Post.
  • lamontagne, 2026-09-18 (reply #5130, 2 likes): 20,000 t spin-up 70 t methalox / 11 t ion (2 likes). Post. Attachments: Arcadia.xlsx.

External references

  • SpinCalc (Ted Hall), artificial-gravity.com/sw/SpinCalc/.
  • Wikipedia "List of moments of inertia".