Question bank — Rendezvous and proximity operations — Clohessy-Wiltshire equations

False. A +y burn raises the orbit; a higher orbit is slower, so via the along-track Coriolis coupling ($\ddot{y}+2n\dot{x}=0$) you eventually fall behind.

False. $\ddot{z}+n^{2}z=0$ is pure simple harmonic motion; it stays a bounded oscillation forever with no secular term, since $z$ is fully decoupled from $x,y$.

False. Only if ${\dot{y}}_0=-2n{x}_0$. Otherwise the $-6n{x}_{0}t$ and $-3{\dot{y}}_{0}t$ terms make $y$ walk off linearly in time.

False. The derivation needs $n$ constant; on an ellipse $n$ (angular rate) varies with position, so you must use the Tschauner–Hempel Equations instead.

False. It is physics: the target is in a balanced circular orbit, so gravity exactly supplies its centripetal need ($\mu /R_0^2=n^2{R}_0$). That balance is why the origin can hover.

False. All coefficients depend on $n=\sqrt{\mu /R_0^3}$, which shrinks as $R_0$ grows, so the whole relative-motion timescale slows.

True. All initial conditions are zero, so every term in the CW solution is zero for all $t$ — the chaser is co-orbiting the target perfectly.

False. The along-track ($y$) amplitude is twice the radial ($x$) amplitude, so the long axis lies along-track.

The error: $x$ and $y$ are coupled by Coriolis ($\pm 2n\dot{\cdot }$). They are not independent oscillators; solving requires integrating the $y$-equation and substituting, which is what produces the secular drift.

The error: the drift is entirely a linear effect — it lives in the exact solution of the linear CW system. It comes from a differential orbital period, not from any nonlinearity.

The error: on V-bar ($x_0=0$, zero velocity) there is no secular drift and no forcing to first order, so the chaser loiters passively — no continuous thrust needed (to first order).

The error: it is just Kepler's third law rearranged — gravity = centripetal, $\mu /R_0^2=n^2{R}_0$. Nothing new is assumed.

The error: only the oscillatory parts return; the secular parts accumulate. Net along-track shift per orbit is $-12\pi x_0$ (with ${\dot{y}}_0=0$), not zero, unless $x_0=0$ and ${\dot{y}}_0$ is balanced too.

The error: CW is a first-order linearization valid only for $r\ll R_0$ and a circular target orbit. It is an excellent approximation at close range, never exact.

In an inertial frame the target flies around Earth and the chaser tracks a huge curved path; co-rotating pins the target at the origin so only the small relative motion — what the pilot cares about — survives.

A radial offset changes your orbital radius, hence your period; a slightly different period compounds every orbit into an ever-growing along-track lead or lag ($-6n{x}_{0}t$).

It's the exact along-track velocity that matches the orbital period difference implied by the radial offset $x_0$, so the two spacecraft share a period and the secular $t$-terms cancel.

The rotation axis is $\hat{z}$, so Coriolis and centrifugal act only in the orbital plane; motion perpendicular to the plane feels only the linearized restoring gravity, giving clean SHM.

It is the leading gravity term at the target's radius; it is exactly cancelled by the centrifugal $+n^2{R}_0$ because the target is in force balance — leaving only the first-order tidal term $3n^{2}x$.

Picture the six-number state (position + velocity) as a point in a 6D space. Because the CW forces are linear in that state, evolving forward in time simply stretches and rotates that space in a fixed way — no term ever depends on where you started nonlinearly. So "the state at time $t$" is one fixed matrix multiplying "the state at time 0," and that matrix (the state transition matrix) is the same for every chaser.

All coupling and restoring terms vanish; motion becomes force-free straight lines, $x=x_0+{\dot{x}}_{0}t$, etc. — the limiting free-drift case, as expected physically.

Pure SHM in $z$ only: the chaser bobs above and below the orbital plane at frequency $n$, staying put in-plane forever.

A closed 2:1 ellipse (the football) that repeats every orbit — the knife-edge between forward drift and backward drift.

The linearization breaks: the neglected $1/r^2$ nonlinear terms become significant, so CW loses accuracy and one must integrate full two-body relative dynamics.

No. With $x_0=0$ the bounded condition needs ${\dot{y}}_0=0$; any nonzero ${\dot{y}}_0$ leaves the $-3{\dot{y}}_{0}t$ secular term, so it drifts.

The neglected time-varying $n$ and radial acceleration cause errors that grow with eccentricity and time; the Tschauner–Hempel form restores correctness.