Delta V Rendezvous⚠ unverified

Understanding the Parameters

• Two problems in one: matching and meeting — rendezvous is harder than a transfer because it is not enough to reach the target's orbit; you must arrive at the same point at the same time. The first term ($|v_2 - v_1|$) handles the orbit-matching (like a Hohmann arrival); the phasing term handles the "meet-in-time" part.

• The phase angle $\phi$ — the angular lead or lag between chaser and target around the orbit. If the chaser is directly behind the target it must "catch up"; if ahead, it must let the target catch up. Closing this gap is done by loitering in a slightly different orbit (a lower/faster or higher/slower phasing orbit) for one or more revolutions — trading time for the angle.

• Why the phasing term is only a rough proxy — the real cost of closing a phase angle depends on how many orbits you allow: taking more revolutions makes the phasing orbit closer to the target orbit, so the required $\Delta v$ shrinks toward zero. The registry's $\phi\,v_1/360$ ignores this time-vs-fuel trade entirely, assigning a fixed cost per degree — hence "heuristic."

• The counter-intuitive chase — to catch a target ahead of you, you must slow down, dropping to a lower, faster orbit that laps the target, then climb back up. Speeding up sends you higher and slower, falling further behind. This backwards-seeming logic is the crux of orbital rendezvous, and no simple speed-plus- phasing formula captures it faithfully.

• What a real estimate needs — the Clohessy–Wiltshire equations linearise the relative motion of two nearby orbits and give closed-form $\Delta v$ for a two-impulse rendezvous as a function of transfer time. That, not this formula, is the tool used for actual mission planning; this page's value is in explaining the problem and flagging the placeholder.

Derivation (Approaching a Proof)

There is no rigorous derivation of the registry formula — it is an assembled heuristic, and the honest account is to show the real structure and where the formula sits relative to it.

The speed-matching term is genuine. To settle into the target's orbit the chaser must null the relative speed $|v_2 - v_1|$, exactly as the arrival burn of a transfer nulls the apogee-to-circular speed gap (see Hohmann Transfer Dv2). This term is a legitimate lower bound on the matching cost.

The phasing term is the placeholder. Real phasing works like this: to close a phase angle $\phi$ in $N$ revolutions, the chaser enters a phasing orbit whose period differs from the target's by $\Delta T = \phi/(360N)$ of an orbit. Entering and leaving that phasing orbit costs a $\Delta v$ that decreases as $N$ grows:

$$\Delta v_{\text{phasing}} \sim \frac{2\pi\,\phi}{360\,N}\cdot(\text{orbital speed scale}),$$

so with many orbits the phasing cost tends to zero (at the expense of time). The registry instead writes a fixed

$$\Delta v_{\text{phasing}} = \frac{\phi}{360}\,v_1,$$

which corresponds loosely to the "one-revolution, no-time-budget" worst case and carries no $N$ dependence. It has the right scaling (proportional to $\phi$ and to orbital speed) but not the right physics.

The rigorous route replaces both terms with the Clohessy–Wiltshire (Hill's) equations, a linearised solution of the two-body relative-motion problem about a circular reference orbit. Their state-transition solution gives the two impulses that move the chaser from its current relative position and velocity to coincidence with the target in a chosen time — the actual rendezvous $\Delta v$. The registry formula should be read as a back-of-envelope surrogate for that machinery.

Dimensional check. $|v_2 - v_1|$ is m/s; $\phi\,v_1/360$ is (dimensionless degrees ratio)$\times$m/s = m/s, so $\Delta v$ is m/s ✓. (Dimensionally sound even though the phasing physics is only approximate.)

History and Development

• Clohessy and Wiltshire (1960). W. H. Clohessy and R. S. Wiltshire published their linearised equations of relative orbital motion in 1960, giving mission designers a tractable tool for the terminal phase of rendezvous. They built on George William Hill's 19th-century lunar-theory equations, so the relations are also called Hill's equations.

• Gemini and the first rendezvous (1965–66). Orbital rendezvous was one of the hardest problems of the Apollo programme's build-up. Gemini 6A and 7 achieved the first crewed rendezvous in December 1965, and Gemini 8 the first docking (1966) — proving the counter-intuitive orbital-chase mechanics could be flown by astronauts. Buzz Aldrin's MIT doctoral thesis on rendezvous earned him the nickname "Dr. Rendezvous."

• Why it mattered for the Moon. Lunar-orbit rendezvous — the mission mode chosen for Apollo — made the whole programme feasible by avoiding a direct ascent, but it staked the crew's survival on a successful rendezvous in lunar orbit. Every crewed docking since, from Apollo–Soyuz to the ISS and commercial crew vehicles, rests on the relative-motion theory this simple formula only gestures at.

Related Concepts: Hohmann Transfer Dv1, Hohmann Transfer Dv2, Orbital Velocity, Inclination Change Delta V, Orbital Period, Circular Orbit Velocity

Notes: Heuristic, not rigorous — real rendezvous $\Delta v$ comes from the Clohessy–Wiltshire (Hill's) equations and depends on transfer time / number of phasing orbits, which this formula ignores. Speed-matching term $|v_2-v_1|$ is genuine; the $\phi v_1/360$ phasing term is a fixed-cost placeholder (right scaling, wrong physics). To catch a target ahead, slow down (drop to a lower, faster orbit). $\phi$ in degrees (mislabelled dimensionless).