SatBase

Reaction Wheel Sizing

Wheel torque and momentum from the four environmental disturbance torques and a rest-to-rest slew, anywhere from low Earth orbit to the outer planets.

Inputs

Orbit

km

Spacecraft

kg·m²
kg·m²
kg·m²
deg
A·m²

Slew requirement

A rest-to-rest maneuver: accelerate for half the time, brake for the other half.

deg
s

Sizing

orbits
%

Results

Standard model. Solar pressure and aerodynamic drag are not included.

Wheel sizing

Required torque 0.02504mN·m
Required momentum 0.002239N·m·s
Per-wheel torque 0.02504mN·m
Per-wheel momentum 0.002239N·m·s
Torque before margin 0.01669mN·m
Momentum before margin 0.001493N·m·s

Disturbance torques

Gravity gradient 0.00702µN·m
Magnetic 0.9785µN·m
Total disturbance 0.9855µN·m
Dominant source Magnetic

Momentum & slew

Cyclic storage 0.0009818N·m·s
Secular accumulation 0.00003985N·m·s
Storage total 0.001022N·m·s
Slew torque 0.01571mN·m
Slew momentum 0.0004712N·m·s
Peak slew rate 1°/s
Dump dipole needed 0.0001435A·m²

Environment

Orbit radius 6,878.1km
Orbit period 94.6 min
Orbital velocity 7,612.7m/s
Magnetic field (polar) 48.925µT

Resolved inertia

Moment about X 0.025kg·m²
Moment about Y 0.027kg·m²
Moment about Z 0.005kg·m²
Largest moment 0.027kg·m²

Method

Worst-case environmental disturbance torques and a rest-to-rest slew, after SMAD 3rd ed. Table 11-9A. Every term is an upper bound: the Sun normal to the sunlit face, the magnetic field at its polar peak, and the gravity gradient at whatever off-nadir angle you enter.

hsize = (hcyc + hsec + hslew) · (1 + M/100) The torque requirement is the matching sum, Tsize = (Tg + Tsp + Tm + Ta + Tslew) · (1 + M/100). They add rather than compete: the wheel slews while it is already rejecting the disturbance and already holding what it has stored. The Standard model leaves out the solar pressure and aerodynamic torques, and Slew only leaves out every disturbance torque.

Derivation

1
Resolve the orbit Mean motion comes first, and the period and orbital velocity follow from it. n = √(μ / r³)  ·  P = 2π / n  ·  v = n · r
2
Resolve the environment Solar irradiance falls with the square of the distance from the Sun, so an outer-planet spacecraft feels almost no solar pressure. The magnetic field is a centered dipole at its polar value; an equatorial orbit sees half of it, so this is the conservative choice. Φ = S0 / d²  ·  B = 2 · 𝕄 / r³
3
Evaluate the four disturbance torques Gravity gradient peaks at 45° off the local vertical, not at 90°, which is why a larger angle can lower it. Solar pressure and drag are each a force times a lever arm (the distance from the pressure center to the center of mass), so balancing the vehicle is the lever with the most travel in it. Tg = 1.5 · n² · |Imax − Imin| · |sin 2θ|
Tsp = (Φ / c) · As · (1 + q) · cps
Tm = D · B
Ta = ½ · ρ · Cd · Af · v² · cpa
4
Split the total by pointing mode and store it A torque that rotates in the body frame comes back every orbit and only has to be held to its peak; one that holds a body-frame direction accumulates until something dumps it. A nadir-pointing spacecraft sees a steady gravity gradient and a sweeping Sun line; an inertially-pointing one sees the reverse. Magnetic and aerodynamic torques are cyclic either way. hcyc = 0.707 · Tcyclic · P / 4
hsec = Tsecular · P · N
5
Size the slew A bang-bang profile accelerates for half the time and brakes for the other half, so torque rises with the square of how quickly you want the maneuver and peak momentum only with the first power. The largest of the three moments is used, which over-sizes a slew about a lighter axis. Tslew = 4 · φ · Imax / t²  ·  hslew = 2 · φ · Imax / t
6
Add, apply margin, and share out Three orthogonal wheels each own an axis outright. In a four-wheel pyramid two wheels straddle each transverse axis and split it; all four share the pyramid axis, which needs only 1/(4 cos β), so the transverse case is what sizes the wheel. The 54.7356° skew is the cube’s body diagonal: it equalizes each wheel’s projection onto the three body axes, which is a different objective from equalizing available torque (that one is atan 2 = 63.43°). This is the all-four-healthy figure: lose one wheel and each survivor must supply twice as much. three orthogonal: hwheel = hsize
four-wheel pyramid: hwheel = hsize / (2 sin β)

Symbols

Input Description Unit
h Altitude above the body’s mean radius km
Ix,y,z Principal moments of inertia kg·m²
θ Maximum deviation from the local vertical deg
D Residual magnetic dipole of the vehicle A·m²
As Sunlit area m²
q Reflectance: 0 absorbs, 1 reflects —
cps Solar center of pressure to center of mass m
ρ Atmospheric density at altitude: entered, not modeled kg/m³
Cd Free-molecular drag coefficient —
Af Frontal area facing the velocity vector m²
cpa Aerodynamic center of pressure to center of mass m
φ Slew angle, rest to rest deg
t Time allowed for the slew s
N Orbits between momentum dumps orbits
M Sizing margin on torque and momentum alike %
S0 Solar constant at 1 AU: 1361, fixed W/m²
β Pyramid skew from the pyramid axis: 54.7356, fixed deg
Derived Description Unit
r Orbit radius: body radius plus altitude m
n Mean motion rad/s
P Orbit period s
v Circular orbital velocity m/s
Φ Solar irradiance at the body’s distance W/m²
B Magnetic flux density, polar worst case T
Tg Gravity-gradient torque N·m
Tsp Solar radiation pressure torque N·m
Tm Magnetic torque N·m
Ta Aerodynamic torque N·m
hcyc Momentum stored by the cyclic share N·m·s
hsec Momentum accumulated by the secular share N·m·s
Tslew Torque the maneuver demands N·m
hslew Momentum at the halfway point of the slew N·m·s
hsize Momentum one axis must hold N·m·s

Reference values

Figures for sanity-checking inputs and placing a result.

Body and altitude Density kg/m³
Earth, 200 km 2×10⁻¹⁰ – 4×10⁻¹⁰
Earth, 300 km 1×10⁻¹¹ – 5×10⁻¹¹
Earth, 400 km 1×10⁻¹² – 1×10⁻¹¹
Earth, 500 km 3×10⁻¹³ – 3×10⁻¹²
Earth, 600 km 8×10⁻¹⁴ – 1×10⁻¹²
Earth, 800 km 1×10⁻¹⁴ – 2×10⁻¹³
Earth, 1000 km 3×10⁻¹⁵ – 5×10⁻¹⁴
Mars, 300–400 km 1×10⁻¹⁴ – 5×10⁻¹⁴
Venus, 300 km ~2×10⁻¹³
Moon, Mercury, GEO 0
Wheel class Momentum N·m·s Torque mN·m
CubeSat wheel 0.001–0.03 0.1–2
Small satellite wheel 0.1–1 5–30
ESPA-class / minisatellite 1–15 20–100
Large GEO / science bus 25–100 100–300

Every torque here is a worst case, and the four are added as scalars even though they act about different axes. Real sizing is per-axis. Excluded: albedo and planetary infrared radiation pressure, which together add roughly half again to the solar-pressure torque in low orbit; eclipse, so secular solar pressure accumulates over the whole orbit; atmospheric co-rotation; wheel friction and zero-speed crossing; and internal disturbances (array and antenna drives, propellant slosh, thruster misalignment), which on a real bus routinely exceed all four external torques together. The pyramid is sized with all four wheels healthy. Treat these results as first-pass sizing, not an ADCS design.