Constraint: Spin-Up Energy¶
Overview¶
A rotating habitat must be spun from rest to its operating angular velocity \(\omega\). The rotational kinetic energy \(E = \frac{1}{2}I\omega^2\) must be supplied by external power systems (solar arrays, nuclear reactors, or ion thrusters). This constraint checks that the spin-up can be completed within an acceptable time given available power.
Physics¶
Rotational Kinetic Energy¶
where \(I_z\) is the moment of inertia about the spin axis (cylinder longitudinal axis).
Moment of Inertia Components¶
The total rotating mass has four components, each with a different radial distribution:
1. Hull barrel — thin cylindrical shell at radius \(r\):
2. Endcaps — two flat disks of thickness \(t\):
(solid disk about its axis)
3. Radiation shielding — distributed on the outer shell surface:
where \(A_{\text{total}} = 2\pi r L + 2\pi r^2\) (barrel + endcaps) and \(\sigma_{\text{shield}}\) is the areal density (kg/m²). Since shielding sits at or near radius \(r\):
4. Atmosphere — fills the interior as a uniform-density gas:
where \(\rho_{\text{air}} = P / (R_{\text{specific}} \cdot T)\) with \(R_{\text{specific}} = 287\) J/(kg·K) for air and \(T \approx 293\) K (20°C):
(solid cylinder of gas about its axis)
Total Moment of Inertia¶
Simplification for 1g¶
At constant gravity \(g\), \(\omega = \sqrt{g/r}\), so:
For the dominant barrel terms (\(I \approx m r^2\)):
Energy scales linearly with radius at constant gravity — a key insight for comparing habitat sizes.
Spin-Up Time¶
Assuming constant applied power \(P_{\text{avail}}\):
This is an idealized lower bound — real spin-up involves variable torque, structural settling, and atmospheric drag losses. A practical estimate would add 20–50% overhead, but we use the ideal value for the constraint.
Reference Numbers¶
Reference design (\(r = 982\) m, \(L = 1{,}276\) m, steel \(t = 0.2\) m)¶
| Component | Mass (Mt) | \(I_z\) (kg·m²) |
|---|---|---|
| Hull barrel | 12.4 | \(1.20 \times 10^{16}\) |
| Endcaps | 9.6 | \(4.63 \times 10^{15}\) |
| Shielding (barrel) | 35.4 | \(3.42 \times 10^{16}\) |
| Shielding (caps) | 27.2 | \(1.31 \times 10^{16}\) |
| Atmosphere | 3.6 | \(1.74 \times 10^{15}\) |
| Total | 88.2 | \(5.59 \times 10^{16}\) |
At 1g: \(\omega = 0.0999\) rad/s, \(E = 2.79 \times 10^{14}\) J = 279 TJ
| Available Power | Spin-Up Time |
|---|---|
| 1 GW | 3.2 days |
| 10 GW | 7.7 hours |
| 100 GW | 46 minutes |
O'Neill cylinder (\(r = 3{,}200\) m, \(L = 32{,}000\) m)¶
Total mass ~6,000 Mt → \(E \approx 9.4 \times 10^{16}\) J = 94 PJ
| Available Power | Spin-Up Time |
|---|---|
| 1 GW | 3.0 years |
| 10 GW | 109 days |
| 100 GW | 11 days |
Constraint Definition¶
Pass condition: Spin-up time ≤ maximum allowed spin-up time.
Default Assumptions¶
| Parameter | Default | Rationale |
|---|---|---|
| \(P_{\text{avail}}\) | 10 GW | ~37 km² solar array at L5 (20% efficiency) |
| \(t_{\max}\) | 1.0 year | Engineering patience for colony-scale construction |
The constraint is soft — spin-up time is always finite and adjustable by adding more power. But it provides useful engineering feedback: a design requiring 10 years of spin-up with available power is impractical even if structurally sound.
Output Details¶
The constraint reports:
- total_rotating_mass_kg: sum of all mass components
- moment_of_inertia_kg_m2: total \(I_z\)
- kinetic_energy_j: rotational KE
- spinup_time_s: time at available power
- spinup_time_days: same, in days
- power_w: available power used
References¶
- O'Neill, Gerard K. "The Colonization of Space." Physics Today, vol. 27, no. 9, 1974, pp. 32–40.
- NASA. Space Settlements: A Design Study (SP-413). 1977, ch. 5.
- Johnson, Richard D., and Charles Holbrow, eds. Space Settlements: A Design Study. NASA SP-413, 1977.