Formulas and calculators
A quick reference to the equations and tools used in the thread. The derivations and discussion are on the linked topic pages.
Spin gravity
| Quantity | Formula | Notes |
|---|---|---|
| Centripetal acceleration | a = ω² r = (2π·rpm/60)² · r | 1 g at 1 rpm ≈ 895 m radius; at 4 rpm ≈ 56 m Habitant, 2026-09-09 |
| Radius for a given g and rpm | r = a / ω² ≈ 895 · (a/g) / rpm² m | Radius ∝ period² Paul451, 2015-02-13 |
| Rim speed | v = ω r | 50 m at 1 g → 4.22 rpm, 22.1 m/s RanulfC, 2014-03-21 |
| Gravity gradient (head–foot) | Δa/a = h / r | The "1% rule": r ≥ 100 h for <1% mikelepage, 2015-09-01; A_M_Swallow, 2015-09-02 |
| Coriolis acceleration | a_c = 2 ω × v | Walking spinward at 22% of rim speed changes perceived g by ±50% mikelepage, 2022-10-23 |
Paul451's diameter table Paul451, 2015-03-31:
| g-level | 1 rpm | 2 rpm | 3 rpm | 5 rpm | 7 rpm | 10 rpm |
|---|---|---|---|---|---|---|
| Earth (1 g) | 1800 m | 450 m | 200 m | 70 m | 35 m | 20 m |
| Mars (0.38 g) | 680 m | 170 m | 75 m | 30 m | 15 m | 7 m |
| Moon (0.16 g) | 300 m | 75 m | 30 m | 12 m | 6 m | — |
Moments of inertia and stability
- Thin torus: I_z = m/4 · (4R² + 3r²). Rectangular toroid/annulus: I = m/2 · (r_in² + r_out²) Twark_Main, 2023-01-30.
- Solid rectangular toroid, density form: I = ½ π ρ w (R_out⁴ − R_in⁴); shell = solid − hollow Paul451, 2023-02-02. Square and round tori agree within ~10% Paul451, 2023-02-03.
- Use the parallel-axis theorem for barbells, spokes and tanks Twark_Main, 2023-02-04.
- Passive stability: spin about the maximum-inertia axis, with a margin; I_spin / I_other ≥ ~1.2 is commonly cited, and Globus's cylinder rule is length < 1.3 × radius lamontagne, 2026-09-18. Details on stability.
- Three-arm balance: the short arm length is 2 sin((180° − angle between long arms)/2) Twark_Main, 2024-09-02.
Structures
- Hoop stress: cylinder σ = P r / t; sphere σ = P r / (2t) lamontagne, 2022-10-13; lamontagne, 2026-09-15.
- Rotating ring: σ = ω² ρ (r₁² + r₁ r₂ + r₂²) / 3 lamontagne, 2022-10-13.
- Equivalent pressure from rotating contents: P_t = ρ_internal · a · R Twark_Main, 2023-09-29.
- Shield mass factor: (r + t)² / r² − 1 Twark_Main, 2023-09-30.
Angular momentum and orientation
- L = I ω. Sun-pointing requires rotating L through 360° per year, so the propellant impulse is ≈ 2π · L per year: torque τ = 2π L / (1 yr) Twark_Main, 2025-04-29; Twark_Main, 2026-09-19.
- Earlier "≈ 4 × spin-up" estimates mikelepage, 2022-11-28 were superseded by the 2π·L result.
- Spin-up propellant: Δm ≈ L / (r_thruster · I_sp · g₀).
Calculators and spreadsheets shared in the thread
| Tool | Author | Use | Source |
|---|---|---|---|
| SpinCalc (artificial-gravity.com/sw/SpinCalc) | T. Hall | Comfort chart, rpm/radius | RanulfC, 2014-03-21 |
| calctool centrifugal calculator | — | Quick a/ω/r | Aussie_Space_Nut, 2014-03-21 |
| "Torque to spin up Rotating spacecraft.xlsx" (+ updated2) | mikelepage | Spin-up torque and propellant | mikelepage, 2023-01-27; mikelepage, 2023-02-01 |
| Station.xlsx | lamontagne | Ring structure, EML-1 logistics | lamontagne, 2022-10-16 |
| "Barbell vs cylinder.xlsx" | lamontagne | Barbell vs torus hull mass | lamontagne, 2023-09-21 |
| "Torque calculation.xlsx" | lamontagne | Sun-pointing torque | lamontagne, 2023-09-21 |
| "Rad vs heat pump" / "Rad vs stirling.xlsx" | lamontagne, Twark_Main | Thermal trades | lamontagne, 2023-10-03; Twark_Main, 2023-11-13 |
| "Modular Space Station.xlsx", "Vessel design (4).xlsx" | lamontagne | Station catalogue | lamontagne, 2019-03-28 |
| Arcadia.xlsx | lamontagne | 100 m station spin-up | lamontagne, 2026-09-18 |
| SpinStationTorqueCalcs1b.xlsx | Peter_GST | Propulsive spin cycling | Peter_GST, 2025-04-11 |
| "Numbers Forward" toy model | Twark_Main | Torus self-support | Twark_Main, 2022-11-13 |
| Coriolis force simulator (Godot) | — | Visualization | Habitant, 2025-10-20 |
| OLTARIS (NASA) | — | Radiation transport | LMT, 2018-07-08 |
Open questions
- Is there an open, validated simulation tool for rotating-station dynamics? The thread repeatedly asked for one.
Related pages
Sources
Forum posts
- Aussie_Space_Nut, 2014-03-21 (reply #238, 1 like): calctool. Post. Attachments: Centrifugal Force Calculator.jpg.
- RanulfC, 2014-03-21 (reply #239): SpinCalc link. Post.
- RanulfC, 2014-03-21 (reply #242, 1 like): RanulfC numbers. Post.
- Paul451, 2015-02-13 (reply #272): Radius ∝ period². Post.
- Paul451, 2015-03-31 (reply #558, 1 like): Diameter table (key). Post.
- mikelepage, 2015-09-01 (reply #577): 1% rule. Post.
- A_M_Swallow, 2015-09-02 (reply #587): 1% was a guess. Post.
- LMT, 2018-07-08 (reply #898, 1 like): OLTARIS. Post.
- lamontagne, 2019-03-28 (reply #1414, 2 likes): Catalogue spreadsheets. Post. Attachments: Modular Space Station.xlsx, Vessel design (4).xlsx, Dumbell station x-ray.jpg.
- lamontagne, 2022-10-13 (reply #2046): Hoop stress. Post.
- lamontagne, 2022-10-13 (reply #2049): Ring stress. Post.
- lamontagne, 2022-10-16 (reply #2112): Station.xlsx. Post. Attachments: Station.xlsx.
- mikelepage, 2022-10-23 (reply #2180): Spinward running. Post. Attachments: gravComfort03-target.jpg.
- Twark_Main, 2022-11-13 (reply #2377): Toy model. Post.
- mikelepage, 2022-11-28 (reply #2477, 2 likes): mikelepage key (2 likes). Post.
- mikelepage, 2023-01-27 (reply #3126, 3 likes): Spin-up spreadsheet (3 likes). Post. Attachments: Calculator screenshot.png, Torque to spin up Rotating spacecraft.xlsx.
- Twark_Main, 2023-01-30 (reply #3192): Torus MoI formulas. Post.
- mikelepage, 2023-02-01 (reply #3213): Updated. Post. Attachments: Torque to spin up Rotating spacecraft-updated2.xlsx.
- Paul451, 2023-02-02 (reply #3215, 2 likes): Paul451 MoI (2 likes). Post. Attachments: poloidally diamond toroidal hexagon.png.
- Paul451, 2023-02-03 (reply #3220): Square vs round. Post. Attachments: Square vs round torus rot-inertia.PNG, horn torus.gif.
- Twark_Main, 2023-02-04 (reply #3237): Parallel axis theorem. Post.
- lamontagne, 2023-09-21 (reply #4007): Barbell vs cylinder. Post.
- lamontagne, 2023-09-21 (reply #4015, 2 likes): Torque calc (2 likes). Post. Attachments: Torque calculation.xlsx.
- Twark_Main, 2023-09-29 (reply #4083): P_t = ρaR. Post.
- Twark_Main, 2023-09-30 (reply #4085): Shield mass factor. Post.
- lamontagne, 2023-10-03 (reply #4118): Rad vs heat pump. Post. Attachments: Rad vs heat pump.xlsx.
- Twark_Main, 2023-11-13 (reply #4347, 1 like): Rad vs Stirling (1 like). Post. Attachments: Rad vs stirling.xlsx.
- Twark_Main, 2024-09-02 (reply #4779): Balance formula. Post.
- Peter_GST, 2025-04-11 (reply #4850): GST calcs. Post.
- Twark_Main, 2025-04-29 (reply #4917, 2 likes): 2π·L per year (2 likes). Post.
- Habitant, 2025-10-20 (reply #5022, 1 like): Coriolis simulator. Post.
- Habitant, 2026-09-09 (reply #5091, 1 like): SpinCalc values. Post. Attachments: screenshot3.jpg.
- lamontagne, 2026-09-15 (reply #5107, 3 likes): Design checks (3 likes). Post. Attachments: Image51_002.jpg.
- lamontagne, 2026-09-18 (reply #5130, 2 likes): Arcadia (2 likes). Post. Attachments: Arcadia.xlsx.
- lamontagne, 2026-09-18 (reply #5131): 1.3 × radius. Post.
- Twark_Main, 2026-09-19 (reply #5153): Correct factor. Post.