Plane Changes and Rendezvous

Simple and combined plane changes, launch-site geometry and phasing for rendezvous

One degree

A satellite is in a 400 km circular orbit at 7.67 km/s. It wants its orbital plane tilted by one degree. Not a different altitude. Not a different shape. The same circular orbit, at the same height, at the same speed, tilted by an angle you could not distinguish from zero in any figure in this course. The cost is 134 m/s. For comparison, raising that satellite's altitude from 400 km to 500 km — a change you could actually see on a diagram — costs about 56 m/s. Tilting by one degree costs nearly two and a half times as much. And a launch from Cape Canaveral at 28.5° N, trying to reach the equatorial geostationary belt, does not need one degree. It needs 28.5 of them. Geometry is not being fair here, and the reason it is not fair is worth understanding, because it shapes the entire commercial launch industry.

Watch first · presented by Shoaib Aziz

The numbers in this film are the labs' own: tilting a 400 km orbit by 1° costs 134 m/s against 56 m/s to raise it 100 km; Δv = 2v sin(Δi/2), so Cape Canaveral's 28.5° costs 3.78 km/s in low orbit but 0.80 at transfer apogee (4.74 times less at every angle), 60° costs the whole orbital speed and 90° costs 10.85 km/s; to GEO, turning first costs 7.63 km/s, turning at GEO 5.37 and folding the turn into the second Hohmann burn 4.22 (24.4 t against 8.0 t at launch for 2 t delivered); the turn at apogee costs 0.15 km/s from Kourou, 0.80 from Cape Canaveral, 1.25 from Baikonur and 1.69 from Plesetsk; and a GEO rendezvous needs a 100.4° lead, while two orbits 20 km apart line up only once every 14.5 days.

The reason is a triangle. A pure plane change does not alter the speed — the satellite is going just as fast afterwards — it alters the direction of the velocity vector. So the required Δv is the third side of an isosceles triangle whose two equal sides are the old and new velocity vectors:

\Delta v = 2v\sin\frac{\Delta i}{2}

The plane-change cost. Note what it is proportional to: not the altitude, not the mass, but your current speed v. Going faster makes turning more expensive.

Figure 6.10

The velocity triangle. Both vectors have the same length v; only the direction differs by Δi. The chord joining their tips is the required Δv, and simple trigonometry gives 2v sin(Δi/2). At Δi = 60° the chord equals v itself — turning by 60° costs as much as your entire orbital speed.

💡The plane-change equation

Δv scales with your current speed. So change planes where you are moving slowest. That is the whole strategy, and it follows straight from the formula. Compare a 28.5° plane change at two points on a geostationary transfer orbit: In LEO, at

v = 7.67

km/s:

\Delta v = 2(7.67)\sin(14.25°) = \mathbf{3.78}

km/s. At GTO apogee, at

v = 1.62

km/s:

\Delta v = 2(1.62)\sin(14.25°) = \mathbf{0.80}

km/s. A saving of three kilometres per second — which, through the rocket equation, is roughly a factor of three in vehicle mass — for doing exactly the same manoeuvre in a different place. This is the mirror image of the Oberth effect, and the pair of them make a single rule: Energy changes are cheap where you are fast. Direction changes are cheap where you are slow. Half of manoeuvre design is arranging to make each kind of change in the right place.

Figure 6.11

Plane-change Δv against inclination change, for LEO speed and for GTO apogee speed. The two curves have the same shape and wildly different scales. Note that above 60° the LEO curve exceeds orbital speed itself — a 90° plane change in LEO costs 10.85 km/s, more than it cost to get to orbit in the first place.

There is one further saving, and it is not small. If you need to change both the speed and the plane — as you do at GEO insertion — do not do them as two separate burns. Combine them into one, and the cost is the vector difference rather than the sum. The law of cosines does the work:

\Delta v = \sqrt{v_1^2 + v_{2}^2 - 2v_1v_2\cos\Delta i}

The combined manoeuvre. Because it is a triangle rather than two sides walked in sequence, the result is always less than doing the two burns separately — often by several hundred m/s.

📐Worked Example: Cape Canaveral to GEO, three ways

Launch to a 400 km parking orbit at

i = 28.5°

. Target: circular GEO at

i = 0°

. Option 1 — plane change first, then Hohmann. Plane change in LEO at 7.673 km/s:

\Delta v = 2(7.673)\sin(14.25°) = 3.777

km/s. Then the full Hohmann: 3.857 km/s. Total: 7.634 km/s. Catastrophic. Option 2 — Hohmann first, then plane change at GEO. Hohmann: 3.857 km/s. Plane change at GEO speed 3.075 km/s:

\Delta v = 2(3.075)\sin(14.25°) = 1.513

km/s. Total: 5.370 km/s. Much better — the plane change happens where the satellite is slower. Option 3 — combine the plane change with the circularisation burn. First burn unchanged: 2.400 km/s. Second burn now does both jobs at once. At apogee

v_{t2} = 1.617

km/s, and we need

v_{c2} = 3.075

km/s rotated by 28.5°:

\Delta v_2 = \sqrt{1.617^2 + 3.075^2 - 2(1.617)(3.075)\cos 28.5°} = \sqrt{2.616 + 9.454 - 8.741} = \sqrt{3.329} = 1.824

km/s. Total: 4.224 km/s. The comparison. Option 3 beats the naive option 1 by 3.41 km/s. Through the rocket equation with

I_{sp} = 311

s, that is the difference between a mass ratio of 12.2 and 4.0 — the naive plan needs three times the launch mass for the same delivered satellite. This is not a subtle optimisation. It is the difference between a viable commercial launch and an impossible one, and it is why every GEO mission flies option 3.

🛰Why Kourou exists, and why [Sea Launch](https://en.wikipedia.org/wiki/Sea_Launch) was tried

Because the plane-change penalty scales with launch-site latitude, a few degrees of geography are worth a great deal of money. Taking the cost as a pure rotation at GTO apogee, where every operator does it: Kourou, French Guiana — 5.2° N. About 0.15 km/s. Europe put its spaceport in South America for exactly this reason, and it is one of the most commercially valuable coordinates on the planet. Cape Canaveral — 28.5° N. About 0.80 km/s. Chosen in 1949 for range safety over the Atlantic, not for latitude. Baikonur — 45.6° N. About 1.25 km/s. Russia has never been able to compete for commercial GEO on fuel grounds, which is a large part of why Proton was priced the way it was. Plesetsk — 62.9° N. About 1.69 km/s, which is more than the entire circularisation burn. Nobody launches GEO from there. Sea Launch — 0° N. A converted oil platform, towed to the equator in the Pacific, launching Zenit rockets from the exact latitude that costs nothing at all. The physics was perfect. The business went bankrupt twice. Folding the rotation into the circularisation burn shrinks all of these — Cape Canaveral's 28.5° adds only 0.37 km/s to the apogee burn rather than 0.80 — but the ordering never changes, and neither does the conclusion. The cheapest plane change is the one you never make, and the only way to arrange that is to launch from the right latitude.

Now the second half of this lesson, and the thing that makes rendezvous different from transfer.

A Hohmann transfer takes you to the right orbit. It says nothing about being at the right place on that orbit at the right time — and the target is not waiting for you.

So rendezvous is a phasing problem. The transfer takes a known time

t_t

, during which the target sweeps through a known angle. For the two to arrive together, the target must be at a specific lead angle ahead of you at the moment you depart:

\phi_{\text{lead}} = \pi - \omega_2 t_{t} = \pi\left(1 - \left(\frac{a_t}{a_2}\right)^{3/2}\right)

The required phase angle at departure. You arrive at the far apsis half a revolution later, so the target must start exactly that far short of it.

Figure 6.12

The phasing geometry. The chaser departs when the target leads by φ. Both coast — the chaser along the transfer ellipse, the target along its circle — and they meet at the far side. Get φ wrong by a few degrees and you arrive to find empty space where the target should have been.

💡Rendezvous: slowing down to catch up

If the phase angle is wrong, you cannot simply speed up to catch the target. Try it and something perverse happens. Burn prograde to chase a target ahead of you and you raise your orbit. A higher orbit has a longer period. You slow down, in the sense that matters, and fall further behind. So: To catch up, burn retrograde. Drop into a lower, faster orbit, gain on the target from below, then raise back up when the phasing is right. To fall back, burn prograde. Rise into a higher, slower orbit and let the target pull ahead. This is the single most counterintuitive fact in orbital mechanics, and it kills people in simulators. The intermediate orbit used to fix the phasing is called a phasing orbit, and the amount of phase you correct per revolution depends on how far you drop — which is a Δv-versus-time trade exactly like every other one in this unit. Gemini 4 discovered this the hard way in 1965: James McDivitt tried to close on the spent Titan upper stage by pointing at it and thrusting. The stage receded. He burned more propellant failing to rendezvous than the entire mission had budgeted, and the attempt was abandoned.

The same phasing logic governs launch windows. A launch to the ISS is not merely a matter of reaching 51.6° inclination — it must reach the correct plane, which means launching at the moment the site rotates through the orbital plane. That happens twice a day, and the window is minutes wide. Miss it and you wait, because fixing the plane afterwards costs the several km/s this lesson opened with.

For interplanetary launch the same constraint applies with the phase angle of a whole planet, and the window recurs at the synodic period — every 26 months for Mars. That is the subject of Unit 8.

↓But there is a problem

Six units in. We can compute the shape of any orbit, the position at any time, the transfer between any two, and the exact price of every manoeuvre in propellant. It is a complete and internally consistent theory. It is also, in a specific and quantifiable sense, wrong — and it has been wrong since Unit 2, where we wrote down four assumptions and dated them for demolition. Earth is not a sphere. It is 21 km fatter across the equator than pole to pole, and that bulge slowly rotates the orbital plane of every satellite ever launched. Space is not empty. There is measurable atmosphere up to a thousand kilometres, and it has brought down space stations. Sunlight is not weightless. Photons carry momentum, and over years they push spacecraft thousands of kilometres off course. And the Moon and Sun pull on everything, all the time. Every orbit in this course has been a Keplerian fiction. Real orbits drift, precess, decay, and eventually fail — unless someone spends propellant, continuously, forever, to hold them in place. Unit 7 is where the model breaks.