Foundations — Rendezvous and proximity operations — Clohessy-Wiltshire equations

This page assumes nothing. Before you touch the parent topic, we build every symbol, one at a time, each on top of the last. Read top to bottom.

1. Position, and what a vector arrow means

Everything starts with where something is.

Look at the figure: the black dot is Earth's centre (our home point), and the arrow reaches out to a spacecraft. That single arrow, $\vec{r}$, is the raw material of the whole subject.

Why the topic needs it. Orbital mechanics is entirely about how these arrows change over time. If you cannot picture "an arrow from Earth to the ship", nothing later makes sense.

• A component is the shadow the arrow casts on one axis. If $\vec{r}$ has components $(x,y,z)$ that means: go $x$ along the first axis, then $y$ along the second, then $z$ along the third, and you arrive at the tip.
• $r$ without the arrow (plain italic) means the length of $\vec{r}$: $r=\sqrt{x^2+y^2+z^2}$. That square-root formula is just Pythagoras stretched into 3D.

2. Velocity and acceleration — the dot notation

Positions change. We need words for how fast and how fast the fast is changing.

Why the topic needs it. Newton's law is about acceleration ($\ddot{\vec{r}}$), and the CW equations are literally three lines of the form "acceleration = stuff". You cannot even read them without knowing $\dot{}$ and $\ddot{}$.

3. Gravity as a formula: $\mu$ and $1/r^2$

Why do orbits happen at all? One force: gravity pulling the ship toward Earth.

Why the topic needs it. The whole difficulty of orbital mechanics is that $1/r^2$ is nonlinear — it curves. The CW trick is to straighten it out near one radius (that is what "linearize" means, §7). See Two-Body Problem for where this force law comes from.

4. A circular orbit, radius $R_0$, and mean motion $n$

Now specialise: the target spacecraft rides a perfect circle.

Why does a circle need gravity of exactly the right size? Going in a circle requires a constant inward pull (centripetal) of size $n^2{R}_0$. Setting gravity equal to that requirement:

$\underset{\text{gravity pull}}{\underbrace{\frac{\mu }{R_0^2}}}=\underset{\text{needed inward}}{\underbrace{n^2{R}_0}}⟹n=\sqrt{\frac{\mu }{R_0^3}}.$

Why the topic needs it. $n$ is the heartbeat of every CW formula — it sets the frequency of the oscillations and appears in every term. And the identity $\mu /R_0^3=n^2$ is the algebraic hinge that later makes the target "hover".

5. Frames: fixed vs. rotating (the "chase-cam")

Here is the conceptual leap.

• Fixed (inertial) frame: you float in deep space watching both ships whirl around Earth. Exhausting to track.
• Rotating (LVLH / Hill) frame: you sit on the target and spin with it. Now the target is nailed to the origin and you only watch the nearby chaser drift. This is the docking pilot's view.

The axes of this rotating frame:

• $x$ = radial, straight up away from Earth,
• $y$ = along-track, the direction of travel ("forward"),
• $z$ = cross-track, sideways out of the orbit plane.

Why the topic needs it. CW equations are defined in this frame. The price of the convenient view is two invented "fake" forces, which we meet next.

6. The two fictitious forces: Coriolis and centrifugal

When your camera itself spins, straight-line motion looks curved. To keep Newton's law working, you add two correction terms — the standard rotating-frame terms.

Why the topic needs it. These two terms are the entire difference between naive gravity and the CW equations. Coriolis is why "burn toward the target" fails; centrifugal is what cancels the leftover gravity to let the target hover.

7. Linearization — keeping only "first order"

The last tool. Gravity's $1/r^2$ is curvy; CW wants straight-line (linear) equations.

Concretely, for small $x$ we use the first-order Taylor expansion (see Linearization and Taylor Expansion): $(R_0+x{)}^{-3}\approx R_0^{-3}\left(1-\frac{3x}{R_0}\right).$ We keep the constant and the term linear in $x$; anything with $x^2$ or higher is "too small to matter" when $x\ll R_0$.

Why the topic needs it. This single approximation is what converts intractable orbital gravity into three clean linear ODEs. It is also the topic's Achilles heel: it holds only for small separations and near-circular orbits — otherwise you need the Tschauner–Hempel Equations. Once you have linear equations, the whole solution can be packaged as a State Transition Matrix.

8. How it all feeds the topic

Read it as: raw arrows and their rates + gravity + a circular orbit give you $n$; a rotating frame spawns the two fake forces; linearizing gravity + adding those forces = the CW equations.

Equipment checklist

Test yourself — you are ready for the parent note only if you can answer all of these out loud.