Intuition The ONE core ideaA spinning wheel and the spacecraft body share a fixed "amount of turning" — push one, the other spins back, but the total never changes on its own. To actually change that total you must reach outside (Earth's magnetic field or a thruster), and until you do, any steady outside nudge silently fills the wheel up until it can spin no faster.
This page assumes nothing . Before you read the parent note , every symbol it throws at you is built here from a picture. Read top to bottom; each idea uses only the ones above it.
Everything on this topic is about spinning things . So first: how do we measure spin?
θ angle . We measure it not in "hours" or degrees but in radians : one radian is the angle you get when the arc you traced is exactly as long as the radius.
Full circle = 2 π ≈ 6.28
Symbol: θ
ω θ t rate of turning is how much angle per second:
ω = change in t change in θ [ radians per second ] ω ω = 0
Why the topic needs this: the spacecraft body turns at rate ω b body ) and the wheel spins at rate Ω w wheel ). Same idea, two objects. We use a capital Ω
ω Ω ω Ω
Definition The dot notationx ˙ how fast that symbol is changing per second ." It is shorthand for the same "change ÷ time" idea we just used.
Ω ˙ w speed is changing = the wheel is speeding up or slowing down.ω ˙ b
Intuition Why a dot and not a fraction every timeWriting d t d Ω w
The tool being introduced: this dot is the derivative — the mathematical machine that answers the question "at this instant, how fast is my quantity moving?" We choose it (rather than plain subtraction) because the torques and speeds here change continuously, moment to moment, not in jumps.
The derivative (dot) breaks a total into a rate. The integral does the exact opposite: it stitches a rate back into a total.
∫ 0 t f d t ′ add up all the little contributions of f 0 t " The stretched-S symbol ∫ is an old-style "S" for "Sum." The d t ′
Intuition Why we need the integral hereA steady tiny outside push (a torque) adds a rate of momentum . To know how much momentum has piled up after hours or days, we must sum that rate over all that time — that is exactly what ∫ τ ext d t ′
Definition Moment of inertiaI push something in a straight line. Moment of inertia I twist something into (or out of) a spin. A fat, heavy, spread-out object has large I I
I b body (whole spacecraft) — big, because the spacecraft is large.I w wheel — tiny, because the wheel is small and light.
Spin a broomstick about its length (easy, tiny I I I I b ≫ I w 500 0.05 slowly .
Now we combine rate and inertia into the single quantity the entire topic conserves.
Definition Angular momentumH inertia × rate :
H = I ω [ kg⋅m 2 / s = N⋅m⋅s ] H is its H
Intuition Why it adds as plain numbers along one axisWhen the body and the wheel spin about the same line (the spin axis), their stored spins simply add:
H tot = H body I b ω b + H wheel I w Ω w + −
The deep fact — proven in Conservation of Angular Momentum — is that with nothing pushing from outside, H tot That single sentence is the engine of the whole topic.
Mnemonic The see-saw of momentumH tot other way. That is the reaction wheel in one image.
τ torque is a twisting effort — a force applied with leverage that tries to change something's spin. It is to angular momentum what a push is to ordinary motion:
τ = d t d H = H ˙ H H
Intuition Two flavours of torque in this topic
Internal / motor torque τ m moves H External torque τ ext sunlight , gravity gradient , thin-air drag . These do change H tot
Why the topic separates them so carefully: the entire "momentum management" problem is precisely that internal torque can't undo what external torque does — you must fetch a second external torque (a magnetorquer or thruster) to bleed it back out.
The dump law uses τ = m × B × cross product , and we need it because torque here has a direction that depends on two other directions.
Definition Vector (the little arrow)v both size and direction (not just a number). B m
a × b third arrow, perpendicular to both , whose length is
∣ a × b ∣ = ∣ a ∣ ∣ b ∣ sin θ θ a b
sin θ sin θ = 1 perpendicular (θ = 9 0 ∘ zero when they are parallel (θ = 0 no twist along the field direction. That geometric fact is the whole reason a magnetorquer "can't touch the component along B sin θ Magnetorquers and Earth's Magnetic Field .
These aren't new maths, just named situations of Ω w
Definition Three wheel states
Saturation : Ω w Ω m a x H H w = H m a x Bias speed Ω bias Zero-crossing : the moment Ω w 0 bearing friction reverses and the wheel jerks — a jitter source we design around .
Integral means sum over time
Angular momentum H = I omega
Torque = rate of change of H
Reaction wheel body-wheel coupling
Magnetorquer momentum dump
Saturation bias and zero-crossing
Reaction Wheels topic 3.5.48
Cover each answer and test yourself. If any stump you, re-read that section.
What does an angle θ How far something has turned; measured in radians (full circle = 2 π
What is ω The angular rate — how much angle is swept per second.
Why is the wheel's rate written Ω w ω b Only to distinguish the fast wheel from the slow body; same physical quantity.
What does a dot over a symbol, like Ω ˙ w The rate of change per second of that quantity (a derivative).
In words, what does ∫ 0 t τ d t ′ The total momentum piled up by adding the torque's contribution over every sliver of time from 0 t
For a constant torque c ∫ 0 t c d t ′ c ⋅ t c t
What does moment of inertia I How hard it is to change an object's spin (rotational "heaviness").
Why is I b ≫ I w The whole spacecraft body is large and heavy; the wheel is small and light.
Write angular momentum in terms of I H = I ω
Why do body and wheel momenta add as plain numbers? They spin about the same axis, so direction reduces to a + −
What single law says H tot Conservation of angular momentum — no external torque means frozen total.
What is torque τ The rate of change of angular momentum, τ = H ˙
Difference between internal motor torque and external torque? Internal only shuffles H H tot
What does the cross product m × B A vector perpendicular to both, of length m B sin θ
When is a magnetorquer's torque maximum, and when zero? Maximum when
m ⊥ B (
sin 9 0 ∘ = 1 ); zero when parallel (
sin 0 = 0 ).
What is saturation? The wheel has hit its max speed and can absorb no more momentum.
Why keep a wheel at a bias speed instead of zero? To avoid zero-crossing, where friction reverses and jitters the pointing.
