Design the approach trajectory of a chaser spacecraft to a target satellite for rendezvous and docking. Sweep target altitude, initial relative position and transfer time, and the closed-form solution of the CW equations returns the two-impulse Δv budget in real time, so you can pick an approach path that fits the fuel margin.
Parameters
Target orbit altitude
km
Altitude of the target on its circular orbit (above Earth's surface).
Along the target velocity vector. Positive = ahead.
Initial range Z (cross-track)
km
Out-of-plane direction. Solved as an independent oscillator.
Transfer time
min
Time of flight between the first and second impulses.
Maneuver type
Approach direction (label only).
Results
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Target orbit period (min)
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Mean motion n (rad/s)
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Initial range (km)
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First impulse Δv₁ (m/s)
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Second impulse Δv₂ (m/s)
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Total Δv (m/s)
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LVLH frame — chaser relative trajectory
x = radial (outward), y = along-track (velocity direction). Yellow dot = initial position, green dot = target (origin), red line = approach trajectory, blue arrow = Δv direction.
Relative range vs time
Δv vs transfer time (optimization curve)
Theory & Key Formulas
$$\ddot{x} - 3n^2 x - 2n\dot{y} = a_x,\quad \ddot{y} + 2n\dot{x} = a_y,\quad \ddot{z} + n^2 z = a_z$$
Clohessy-Wiltshire equations. n = target mean motion (rad/s); x = radial, y = along-track, z = cross-track. The (x,y) in-plane channel is coupled; the (z) out-of-plane channel is an independent oscillator.
$$n = \sqrt{\frac{\mu}{a^3}},\quad a = R_\oplus + h,\quad T = \frac{2\pi}{n}$$
μ = 398600.4418 km³/s² (Earth's gravitational parameter), R_⊕ = 6378.137 km, h = orbit altitude. These give the orbital period T and mean motion n.
Closed-form two-impulse rendezvous from the state-transition matrix. The first impulse Δv₁ puts the chaser on a trajectory from r₀ to the target r_f = 0; the residual velocity at arrival is killed by Δv₂.
"Rendezvous" — that's when a spacecraft docks with the ISS, right? If you want to catch up, can't you just throttle up the engine?
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That's the weird part of orbital mechanics. On the ground we'd say "to catch the car ahead, hit the gas." In space, if you accelerate prograde (forward), you add energy, your orbit puffs up, and you actually slow down and fall behind. Decelerate, and your orbit drops and you speed up. Pure intuition will lose you in seconds. The Clohessy-Wiltshire (CW) equations were built in 1960 exactly to take that intuition out and let you compute approach trajectories analytically.
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So ground intuition is useless. The trick is to set a frame fixed on the target and reason about everything relative to it?
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Exactly — that's the LVLH frame, short for Local-Vertical/Local-Horizontal. It rotates with the target. x = geocentric-to-target (radial, positive outward), y = velocity vector (along-track, positive forward), z = orbit-normal (cross-track). Linearise the relative motion in that frame and you get x'' - 3n²x - 2ny' = ax and friends, where n is the target's mean motion — for LEO around 0.001 rad/s, an orbit period near 90 min.
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What's the "two-impulse rendezvous"? Moving the sliders, I see Δv₁ and Δv₂ pop out separately…
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It's the textbook strategy. Fire a first impulse so the chaser drifts onto a trajectory that reaches the target position at t_f. Coast unpowered until you arrive, then fire a second impulse to zero out the residual relative velocity. Because the CW equations are linear, the state-transition matrix Φ(t) gives the clean closed form Δv₁ = -Φ_rv⁻¹·Φ_rr·r₀. Clohessy and Wiltshire wrote this by hand in 1960 — no computers — and it directly fed the Gemini rendezvous designs.
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Total Δv changes a lot with transfer time. The chart shows several valleys — how do I pick one?
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Great catch! When τ = n·t approaches π, 2π, 3π…, the state-transition matrix becomes singular (det ≈ 0) and the first impulse blows up. So it's not just "faster = more Δv, slower = less Δv" — valleys and peaks alternate. In practice designers pick a stable valley around half to one full orbit period. For an ISS approach that's typically 90-180 min split into phase-up → V-bar approach → final hold. Dragon does exactly the same.
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V-bar, R-bar, Natural Drift — what's the difference, and when should I pick each?
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It's the approach direction and intent. V-bar Hop comes in along-track — stable attitude, easy to control. R-bar Hop climbs from below, and it's fail-safe: if the engine quits, gravity passively separates the chaser, so the ISS sometimes requires it. Natural Drift uses the free CW solution — the so-called "football" orbit — to close phase with essentially zero thrust. Cheapest fuel-wise, slowest in time. Large cargo vehicles like HTV combine Natural Drift and V-bar to save propellant.
Frequently Asked Questions
The Clohessy-Wiltshire (CW) equations, published in 1960, are a linearized system of three coupled ODEs that describe the motion of a chaser spacecraft relative to a target on a circular orbit. Expressed in the LVLH (Local-Vertical/Local-Horizontal) frame attached to the target (x = radial, y = along-track, z = cross-track), they read x'' - 3n²x - 2ny' = ax, y'' + 2nx' = ay, and z'' + n²z = az. For short ranges (a few hundred km) and nearly circular orbits, they have a closed-form analytical solution and are the workhorse of fast Δv-budget design. Soyuz-ISS, SpaceX Dragon, JAXA HTV and every other docking spacecraft use CW for the final approach phase.
Write the state-transition matrix as Φ(t) = [Φ_rr Φ_rv; Φ_vr Φ_vv]. The first impulse that brings the chaser from initial relative position r0 to target r_f=0 at time t_f is Δv1 = -Φ_rv^(-1)·Φ_rr·r0. The residual relative velocity at arrival is cancelled by a second impulse Δv2, and the fuel budget is total Δv = |Δv1| + |Δv2|. This tool builds the matrix entries as functions of τ = n·t_f, solves the coupled 2×2 in-plane system, and treats the out-of-plane (z) channel independently to return Δv values.
A V-bar Hop approaches along the y-axis (along-track, target velocity direction), while an R-bar Hop comes in along the x-axis (radial, geocentric-to-satellite line). V-bar is the textbook final-approach geometry with stable attitude and easy relative-velocity control. R-bar is sometimes mandated for large targets such as the ISS because, on engine failure, gravity passively pushes the chaser away (fail-safe trajectory) and plume impingement on the target structure is reduced. Natural Drift uses the homogeneous CW solution to close phase difference with essentially zero thrust — fuel-optimal but slow.
The CW equations assume (1) a nearly circular target orbit (eccentricity e<0.01), (2) relative range much smaller than the orbit radius (typically up to a few hundred km), (3) short propagation times (a few orbits), and (4) negligible J2, drag and other perturbations. For eccentric orbits use the Tschauner-Hempel equations; for long range or long times use higher-order Hill-CW or full numerical propagation; for low LEO with significant drag, add LEO-correction terms. This tool targets standard LEO rendezvous design and covers altitudes between 200 and 2000 km.
Real-World Applications
ISS resupply missions: SpaceX Dragon, Northrop Grumman Cygnus, JAXA HTV / HTV-X, Roscosmos Progress and ESA ATV all run a CW-based final approach phase. From ground, the Δv budget is set up from orbit altitude and phase offset, the chaser hops to hold points (a few km out) via V-bar or R-bar, and then steps down to 200 m / 30 m / 10 m while attitude and relative velocity are checked. The Δv numbers this tool prints are what engineers compute first on a mission-planning whiteboard.
Crew Dragon and Soyuz crewed docking: Crew Dragon (NASA) and Soyuz (Roscosmos) execute the CW solution autonomously (KURS / DragonEye + LIDAR) while leaving the crew a "Manual Mode" override inside 200 m. The CW-predicted trajectory is continuously cross-checked against rendezvous-and-docking sensor measurements, and any deviation outside tolerance triggers an automatic Hold. Designs therefore add 20-30 % margin on top of the analytical Δv budget to cover navigation noise and abort.
Satellite servicing and debris removal: CW is also central to refueling failed satellites and Active Debris Removal of large space junk. Even with non-cooperative targets (no markers or reflectors), ground radar gives orbit elements precise enough for the CW solver to plan an approach. Northrop Grumman MEV (Mission Extension Vehicle) has already docked to Intelsat satellites commercially, using a Yamanaka-Ankersen extension of CW for eccentric orbits.
Smallsat formation flying: NASA MMS (4-spacecraft formation), CanX-4/5, PRISMA and similar missions that keep multiple spacecraft in loose relative geometry rely on CW for operations design. The homogeneous CW solution is periodic — a closed elliptic "football" orbit — and Natural Drift mode lets the formation stay in shape with essentially no propellant. The Natural Drift option in this tool illustrates that behaviour.
Common Misconceptions and Pitfalls
The biggest trap is forgetting the "target orbit is circular" assumption. CW is derived for eccentricity e ≈ 0 and is accurate for ISS-class or sun-synchronous orbits with e < 0.001, but it cannot be applied to elliptic targets such as upper-stage tanks in mid-transfer or periodic comets. Errors become visible above e > 0.01 and Δv predictions can be off by 10-50 % above e > 0.1. For eccentric rendezvous use the Tschauner-Hempel equations, the Yamanaka-Ankersen closed form, or a full numerical propagator. This tool is for nearly circular LEO.
Next, assuming "longer transfer time always means smaller Δv". The Δv vs transfer-time chart below shows that whenever τ = n·t passes through π, 2π, 3π…, the state-transition matrix determinant collapses and the first impulse blows up. This is not a quirk of the analytical solution — it is a physical singularity at half, full and 1.5 orbit periods where the geometric relationship between initial and target position degenerates. Rendezvous designers from Gemini and Apollo onward maintained tables of these "blow-up times" to avoid. In real practice you pick the minimum-Δv valley out of several candidates.
Finally, do not equate Δv budget with propellant mass directly. The Δv this tool prints is an ideal impulsive sum, whereas in flight you must add (1) gravity losses from finite burn duration, (2) attitude-control RCS consumption, (3) reserves for navigation error (typically 20-30 %) and (4) Δv set aside for an Approach Abort. Even then, identical Δv on a hypergolic thruster (Isp around 280 s, as on Dragon / ATV) and on an electric thruster correspond to very different propellant masses — translate via Tsiolkovsky's equation Δm/m = 1 - exp(-Δv/(Isp·g)) when sizing the tank.
How to Use
Set target altitude (e.g., 400 km LEO) and initial chaser position offset in X, Y, Z coordinates (typically ±5 km relative to target satellite)
Run simulation to compute mean motion n and orbital period for the target's circular orbit
Simulator calculates phasing maneuver Δv₁ and final rendezvous impulse Δv₂ using Clohessy-Wiltshire linearized equations of motion
Review trajectory preview showing relative position evolution and total Δv budget required for approach completion
Worked Example
Target ISS altitude = 408 km, orbital period = 92.9 min, mean motion n = 0.001104 rad/s. Chaser positioned 2.5 km ahead (X), 0.8 km below (Z), initial separation = 2.65 km. First impulse Δv₁ = 1.8 m/s establishes phasing; second impulse Δv₂ = 2.1 m/s performs final approach at contact. Total Δv budget = 3.9 m/s for safe docking corridor entry within 50 m/s relative velocity.
Practical Notes
Use negative X-offset (trail position) to plan overtaking rendezvous; positive X accelerates approach closure rate proportional to 3n²·separation
Z-offset (out-of-plane) requires separate inclination correction impulses; minimize Z deviation to reduce total Δv—typical budget assumes Z < 1 km
Clohessy-Wiltshire equations assume circular orbits and small relative distances (<10 km); validate with high-fidelity propagation for autonomous spacecraft rendezvous operations