DOC // NL-TN-2026-011VECTOR -Z1P / 1U · MATERIALS · ORBITAL

STATUS: PUBLISHED // REV 0.3

Disturbance torques at PocketQube scale

How L⁵, L³ and L⁰ scaling laws invert the environmental torque hierarchy between 1U CubeSats and 1P PocketQubes

Author: nadir lab.
Published: 17 October 2026
Reading time: 1 min
Level: Engineering
Tags: Disturbance torques
Scaling laws
ADCS
Magnetic cleanliness
PocketQube

Abstract

We evaluate the four environmental disturbance torques—gravity gradient, aerodynamic drag, solar radiation pressure and residual magnetic dipole—for a 50 mm 1P PocketQube across 350 to 700 km altitude and show why residual current loops and battery ferromagnetic cases dominate picosatellite attitude dynamics.

Contents

Scaling laws of the four environmental torques

When a spacecraft shrinks by a linear factor $L$ at constant bulk density ( $m \propto L^{3}$ ), the four environmental torques scale at different powers of $L$ [1]:

Gravity-gradient torque ( $\propto L^{5}$ ): $T_{g} = \frac{3 μ}{2 r ^{3}} ∣Δ I ∣ sin 2 θ$ . Because principal moments of inertia scale as $m L^{2} \propto L^{5}$ , halving the edge from $100 mm$ ( $1 U$ ) to $50 mm$ ( $1 P$ ) reduces $T_{g}$ by a factor of $2^{5} = 32$ [2].
Aerodynamic torque ( $\propto L^{3}$ ): $T_{a} = \frac{1}{2} ρ v^{2} C_{D} A l_{cp}$ . Cross-section $A \propto L^{2}$ and centre-of-pressure offset $l_{cp} \propto L$ yield $L^{3}$ scaling—unless a deployed UHF tape antenna ( $171 mm$ at $437 MHz$ ) introduces an unscaled lever arm[3].
Solar radiation pressure torque ( $\propto L^{3}$ ): $T_{srp} = \frac{S}{c} (1 + q) A l_{sp}$ , which sits near $0.1 nN \cdot m$ for a $1 P$ cube.
Residual magnetic torque ( $\propto L^{0}$ to $L^{2}$ ): $T_{m} = m_{res} B$ . Current loops on solar panels, nickel-plated battery cans and steel fasteners produce $m_{res} \sim 0.5 - 2.0 mA \cdot m^{2}$ almost regardless of hull size[4].

Worst-case 1P PocketQube disturbance torques vs. altitude (residual dipole 1.5×10⁻³ A·m², ΔI = 1.5×10⁻⁵ kg·m², CP–CM offset 4 mm)

Altitude (km)	Gravity gradient (N·m)	Aerodynamic (N·m)	Solar radiation (N·m)	Magnetic residual (N·m)	Dominant source
320	2.98 × 10⁻¹¹	1.32 × 10⁻⁸	5.67 × 10⁻¹¹	7.93 × 10⁻⁸	Magnetic residual
400	2.88 × 10⁻¹¹	2.45 × 10⁻⁹	5.67 × 10⁻¹¹	7.65 × 10⁻⁸	Magnetic residual
500	2.76 × 10⁻¹¹	3.64 × 10⁻¹⁰	5.67 × 10⁻¹¹	7.32 × 10⁻⁸	Magnetic residual
525	2.73 × 10⁻¹¹	2.34 × 10⁻¹⁰	5.67 × 10⁻¹¹	7.24 × 10⁻⁸	Magnetic residual
600	2.64 × 10⁻¹¹	6.72 × 10⁻¹¹	5.67 × 10⁻¹¹	7.01 × 10⁻⁸	Magnetic residual
700	2.53 × 10⁻¹¹	1.55 × 10⁻¹¹	5.67 × 10⁻¹¹	6.72 × 10⁻⁸	Magnetic residual

Design consequences for 1P attitude control

Above $450 km$ , a parasitic dipole of $1.5 mA \cdot m^{2}$ generates $65 - 75 nN \cdot m$ in the polar geomagnetic field—more than three orders of magnitude larger than gravity-gradient torque ( $0.027 nN \cdot m$ at $525 km$ for $Δ I = 1.5 \times 1 0^{- 5} kg \cdot m^{2}$ ) and around two orders of magnitude above aerodynamic torque ( $0.23 nN \cdot m$ at $525 km$ under moderate solar activity). Routing solar-string return paths directly beneath cell interconnects (back-wiring) and replacing nickel-plated steel fasteners with titanium or A4-80 stainless steel are mandatory before sizing or magnets[3].