Disturbance Torques, PMAC & B-Dot Detumbling

Shrinking a spacecraft from 1U (L = 10 cm) to 1P (L = 5 cm) inverts the LEO disturbance hierarchy. Gravity-gradient torque T_g = (3μ / 2r³)·|ΔI|·sin 2θ scales with inertia difference as L⁵, dropping 32-fold to ≈ 0.027 nN·m at 525 km for ΔI = 1.5 × 10⁻⁵ kg·m² (ineffective on a symmetric 1P cube, though exploited on 6P STECCO), while aerodynamic torque T_a = ½·ρ·v²·C_D·A·l_cp (≈ 0.23 nN·m bare hull at 525 km moderate solar flux, larger with a 171 mm UHF monopole) and solar radiation pressure T_srp = (S₀/c)·(1 + q)·A·l_sp (≈ 0.1 nN·m) both scale as L³.

Residual magnetic dipole torque T_m = m_res · B scales weakly with size (L⁰ to L²) because battery cans, current loops, and fasteners do not shrink cubically. Above ~450 km, residual magnetism exceeds environmental torques by two orders of magnitude: at 525 km (equatorial B ≈ 23 µT, polar B ≈ 47 µT), m_res = 1.5 mA·m² produces 65–75 nN·m (~47 nN·m per mA·m² polar) and tilts PMAC alignment by Δθ = arctan(m_res,⊥ / m_PMAC).

Keeping m_res ≤ 0.5–1.0 mA·m² (and meeting the CDS 0.5 gauss static envelope limit) requires strict layout rules: routing solar return traces beneath cell strings cancels the 0.26 mA·m² loop of 150 mA around a 42 × 42 mm panel, aluminium-laminate LiPo pouches replace nickel-plated steel 18650 cans and nickel tabs, and Ti-6Al-4V or passivated A4-80 (316) screws plus brass RF connectors replace cold-worked 304/A2 hardware.

Passive Magnetic Attitude Control (PMAC) aligns one axis along the geomagnetic vector B_⊕ (dip angle tan I = 2·tan λ_m, rotating twice per polar orbit) with zero power, sensors, or software, oscillating at libration period T_lib = 2π·√(I / (m·B)). Enforcing a 10× torque margin (k_m = 10) at equatorial B_min ≈ 23 µT (525 km) via m ≥ k_m·T_dist / (B_min·sin θ_max) needs only 0.015–0.030 A·m² against drag or 0.02–0.10 A·m² including residual dipoles.

Magnet shape and alloy stability set flight dipole: squat AlNiCo-5 blocks self-demagnetize. Grizu-263A flew four 8 mm AlNiCo-5 cubes (N = 1/3), operating near 0.12 T (~15% remanence) for m ≈ 0.31 A·m² and a 10–23 s libration period, whereas stacking them into an 8 × 8 × 32 mm column (N = 0.108) triples output to 0.91 A·m²; AlNiCo 5 and SmCo also avoid NdFeB's ~12%/100 K thermal drift.

Oversizing the magnet inside a 50 mm cube causes severe near-field pinning because magnet-to-rod spacing d shrinks to 20–35 mm (vs. 60–150 mm in 1U–3U). Along an orthogonal rod at axial offset δ from the magnet mid-plane, peak stray field H_∥,max = 0.859·m·δ / [4π·(d² + δ²)²] ∝ m·δ / d⁴ jumps 25× from d = 45 mm to 20 mm. At d = 20 mm and δ = 1 mm, 0.31 A·m² drives ≈ 130 A/m (saturating dampers; on Delfi-C3, pinned rods stretched despin to ~3 months) and 3.8–7.6 mT bias across ±0.4–4.9 mT MEMS magnetometers, requiring right-sized magnets and mid-plane placement (δ ≈ 0).

Soft permeable rods convert spin energy into lattice heat as tumbling sweeps B_⊕ along their axis, cycling alloys in lib/physics/magnetics.ts: 80% Ni–Fe HyMu-80 (H_c = 0.96 A/m, J_s = 0.74 T; twelve 0.2 × 0.2 × 33.5 mm rods flew on Grizu-263A), amorphous FeBSi ribbon (H_c = 3.0 A/m, J_s = 1.56 T), and grain-oriented FeSi (H_c = 7.5 A/m, J_s = 2.0 T).

With relative permeability μ_r ~ 10⁴–10⁵ and cylindrical demagnetizing factor N_d ≈ [1 + (4/√π)·(L/D)]⁻¹ (N_d·μ_r ≫ 1), internal field H_i = H_a − N_d·M is demagnetization-limited: peak polarization caps at J_p = min[μ₀·(H_a − H_c) / N_d, J_s] with apparent saturation H_sat,app = N_d·J_s / μ₀. Thin 0.04 g FeBSi strips dissipate 4× more energy per cycle than 0.31 g rods; with L⁵ inertia scaling (I_t = 1.1 × 10⁻⁴ kg·m²), the /lab/pmac 1P reference (45 mm³ rod volume, H_c = 1.2 A/m, B_s = 0.75 T) despins 15 °/s in 0.93 hours with 583 nN·m torque at 10°.

Below coercivity, loops collapse and leave an undamped dead-band θ_db ≈ arcsin(H_c / H_⊥) (±3.7° at H_c = 1.2 A/m, H_⊥ = 18.5 A/m). Monograph SQ-R-002 notes that rigid epoxy potting across 60 K induces 50–105 MPa stress that widens H_c, favouring PTFE sleeves and hybrid sets pairing unbonded FeBSi strips for fast capture with final-shape hydrogen-annealed HyMu-80.

Spacecraft omitting permanent magnets detumble actively via the B-dot law, commanding three-axis PCB magnetorquers m = clip(−k·Ḃ_B, −m_max, +m_max) directly from body magnetometer differences without attitude estimation. Since Ḃ_B ≈ −ω × B_B above orbital rate, kinetic power dT/dt = ω · (m × B) = −k·‖B‖²·‖ω_⊥‖² ≤ 0 strictly drains transverse spin while inclined orbits rotate B_⊕ across all three axes.

Although 42 × 42 mm inner-layer PCB coils yield 0.005–0.015 A·m² (vs. 0.05–0.2 A·m² in 1U), 1P inertia is ~22× smaller (1.0 × 10⁻⁴ vs. 2.2 × 10⁻³ kg·m²). In RK4 runs from lib/physics/bdot.ts (500 km, 97.4° SSO, B₀ = 31.2 µT) from 9.6 °/s ([6, −5, 5.5] °/s), a 0.005 A·m² 1P detumbles in ~1 orbit—matching POQUITO from 20 °/s—while /lab/bdot (0.012 A·m² from 25 °/s) crosses 1 °/s in 47.2 min, settling near 2·n ≈ 0.13–0.19 °/s (n ≈ 0.063 °/s).

Flight firmware inserts coil-off magnetometer guard windows, samples at 5–10 Hz to avoid 1 Hz aliasing at 180 °/s tip-off, applies 1 °/s / 3 °/s hysteresis, and checks polarity in a Helmholtz cage. Once damped, 2P–3P buses hand over to sun sensors, MEMS gyros, TRIAD/QUEST or MEKF estimators, and miniature reaction wheels (7 g, 25 mW; 3P Unicorn-2 achieved ~5° pointing and 24 m GSD imaging).