A plane change paid at the worst speed there is

Three rungs of this anchor have been about turning an orbit that already exists, and each found the same answer in a different form: do it where the vehicle is slow. Turn at apogee rather than in the low orbit. Combine the rotation with a burn that is happening anyway. Fly further out to make a slower place.

The most expensive plane change anybody performs is not in orbit at all, and it is not optional.

It is not optional because it is not a manoeuvre. It is a consequence of where the vehicle started, and by the time anybody can act on it the vehicle is already moving at seven and a half kilometres a second.

Where the relation comes from #

An orbit’s plane contains the centre of the Earth, and a launch site is on the surface at latitude φ. The vehicle’s velocity at insertion is horizontal and points along an azimuth A, measured east of north.

Spherical trigonometry on the right triangle formed by the equator, the meridian through the site and the orbital track gives

$$\cos i = \sin A \cos \varphi$$

and that is the whole of it. The inclination is minimised when sin A is largest, which is A = 90° — due east — and there cos i = cos φ, so i = φ. There is no azimuth that does better, because sin A cannot exceed one.

The consequence is a hard floor. A vehicle launched from latitude φ enters an orbit with i ≥ |φ|, whatever it does on the way up, and the bound is geometric rather than energetic: the orbital plane must contain both the centre of the Earth and the launch site, and a plane through a point at latitude φ has inclination at least φ. No amount of propellant changes it while the vehicle is still attached to the launch site.

What removing it costs #

An inclination acquired at launch has to be removed in orbit, and orbit is where the vehicle is fastest.

At 400 km the circular speed is 7.669 km/s, so a plane change of Δi costs 2 × 7.669 × sin(Δi/2). Removing 5.2° costs 0.696 km/s. Removing 28.5° costs 3.775. Removing 45.6° costs 5.94, which is more than the 3.18 km/s that would leave Earth entirely from the same orbit.

The ordering in that caption is the practical rule for the whole subject. A launch site’s latitude is not a cost paid at launch; it is a cost deferred to apogee, where it is a fifth as expensive. What a high-latitude site actually costs a geostationary mission is the difference between two apogee burns, and for Baikonur against Kourou that is 1.254 − 0.145 = 1.11 km/s — which on a five-tonne satellite is about a tonne of propellant, or four years of station-keeping, or a smaller satellite.

The dog-leg, and why it is not a solution #

A launch does not have to fly a constant azimuth. It can turn during ascent — a dog-leg — and the question is whether that helps.

It does not, and the reason is the thing this ladder has been about. A turn during ascent is a plane change performed at whatever speed the vehicle has reached, and by the time the trajectory is being shaped that speed is a substantial fraction of orbital. Turning at 3 km/s costs 2 × 3 × sin(Δi/2), which for ten degrees is 523 m/s — cheaper than the same turn at 7.7 km/s, and far dearer than at apogee.

Worse, a dog-leg is flown in the atmosphere or just above it, so it is a steering manoeuvre at nonzero angle of attack with the structural and drag penalties those carry. A launcher that dog-legs pays the plane change and pays a steering loss on top of it.

Dog-legs are flown anyway, and the reason is never optimisation. They are flown when the direct azimuth would drop spent stages on somewhere they may not fall — which is why Baikonur’s flown minimum inclination is 51.6° rather than its latitude’s 45.6°, and why launches from Vandenberg fly a corridor over open ocean rather than the azimuth the mission would prefer.

There is one further thing a dog-leg cannot do, and it is the reason the constraint is worth calling geometric rather than economic. A dog-leg changes the azimuth, and the relation cos i = sin A cos φ bounds the inclination below by φ for every azimuth. Turning during ascent moves along that curve; it does not leave it. The floor is not a floor on what is affordable. It is a floor on what exists.

What the latitude gives back #

One term runs the other way, and it is the reason equatorial sites are valuable twice over rather than once.

The Earth’s surface rotates east at 465 m/s at the equator and 465 cos φ elsewhere. A vehicle launched due east starts with that velocity already, and it is a genuine reduction in the Δv the launcher must supply — about a third of what an ascent loses to gravity and drag. At Kourou the bonus is 463 m/s; at Kennedy 409; at Baikonur 325; at a polar launch aimed north, zero.

So latitude enters twice and both times in the same direction. A high-latitude site starts with less free velocity and delivers an inclination that costs more to remove. Between Kourou and Baikonur the two effects come to 138 m/s at launch plus 1.11 km/s at apogee — about 1.25 km/s of difference for a geostationary mission, from geography alone.

That is why the Sea Launch platform sailed to the equator before firing, and why the location of a launch site is one of the few decisions in this subject that cannot be revisited.

There is a mirror-image site worth naming for contrast. A launch aimed west — a retrograde orbit — pays the eastward rotation twice: it forgoes the 465 cos φ bonus and has to cancel it, which is 2 × 465 cos φ of extra Δv, or 930 m/s at the equator. Retrograde orbits are therefore launched from mid-latitude sites where the penalty is smaller, and Vandenberg’s 34.7° is a compromise between that penalty and the range safety that requires a southward or westward corridor.

Which is the second time in this rung that the site has been chosen against a constraint that has nothing to do with orbital mechanics. Baikonur’s inclination is set by where the stages fall; Vandenberg’s exists because the Pacific is to the west. The geometry says what is possible and the geography says what is allowed, and the second is usually the binding one.

The inclination that is chosen rather than inherited #

Not every inclination is a cost. One of them is a design parameter with a job, and it is above ninety degrees.

The Earth’s equatorial bulge exerts a torque on any orbit not in the equatorial plane, and the result is a steady regression of the ascending node — the orbit’s plane rotates about the polar axis at a rate proportional to cos i. For a prograde orbit the node regresses westward; for a retrograde one, i > 90°, it advances eastward.

At about 98° of inclination for a low orbit, the advance is 0.9856° per day, which is exactly 360° per year. An orbit at that inclination therefore keeps a fixed angle to the Sun as the Earth goes round it, so the satellite crosses the equator at the same local solar time on every pass, forever, without spending anything. That is a sun-synchronous orbit, and it is what every Earth-observation and weather satellite uses.

The inclination is being paid for — 98° is further from any launch site’s latitude than 45° is, and a launch to it forgoes the whole eastward rotation bonus and pays a retrograde penalty besides, which is several hundred metres a second of the launcher’s budget. What it buys is a lighting condition that would otherwise require continuous station-keeping against a perturbation, and the perturbation is doing the work instead.

That is the one place in this anchor where an inclination is worth more than it costs, and it is worth noticing what makes it so: the vehicle is not fighting the oblateness, it is using it. Everything else on this ladder is about paying to undo something.

The window, which is the other half of the plane #

An inclination is one number and an orbital plane is two. The second is the right ascension of the ascending node — where the plane crosses the equator, measured against the stars — and a launch site’s relationship to it is a clock rather than a geometry.

The site rotates with the Earth. Once a day it passes through any given orbital plane, twice if the plane’s inclination exceeds the site’s latitude, and a launch can only insert into that plane while the site is in it. That is what an instantaneous launch window is: a moment, or a few minutes either side of it, when the site lies in the target plane and the vehicle can fly into it without a plane change.

Everything a rendezvous mission does is governed by that. A flight to the International Space Station launches into a plane the station is already in, which means launching at the minute the site passes through it — and the station’s 51.6° inclination is Baikonur’s flown minimum, chosen so that Russian launches face no plane change at all.

Missing the window costs a plane change at low-orbit speed, and the geometry is unforgiving: the plane’s node regresses about 5° a day under the oblateness while the Earth turns 360°, so waiting a day brings the site back into a plane that has moved. A launch delayed by an hour is out of plane by 15° of the Earth’s rotation, which is a rotation of several degrees in the orbital plane and costs hundreds of metres a second to fix. This is why launches to a target are scrubbed rather than delayed.

Two numbers that both come from the latitude #

It is worth collecting the whole effect of a site’s latitude in one place, because it enters the mission budget three times and the three are usually discussed separately.

At launch. The eastward rotation bonus is 465 cos φ, so a site at 45.6° starts 140 m/s behind one at 5.2°.

In orbit, as an inclination. The vehicle arrives at i ≥ φ, and removing it at geostationary apogee costs 2 × 1.618 × sin(φ/2) km/s: 145 m/s from Kourou’s 5.2°, 1,254 from Baikonur’s 45.6°. That is the large term.

And in the coverage. A satellite’s ground track reaches latitudes up to its inclination and no further, so a launch from a high-latitude site cannot place a payload in a low-inclination orbit without paying the second term, and cannot place one in a high-inclination orbit from a low-latitude site without paying a plane change in the other direction. The constraint is two-sided even though only one side is usually discussed.

Together those come to about 1.25 km/s between the fleet’s two extremes, which is roughly thirteen per cent of a launcher’s whole velocity budget. Nothing else about a launch site — its altitude, its weather, its infrastructure — is worth a tenth of that.

What the picture cannot show #

The relation cos i = sin A cos φ is exact for an instantaneous insertion from a point on a spherical Earth, and three things move real launches away from it.

The vehicle does not insert above the launch site. It flies downrange for several hundred to a couple of thousand kilometres before reaching orbital speed, so the insertion point is at a different latitude from the site — a due-east launch from 28.5° inserts at nearly the same latitude, but a launch on a northerly azimuth inserts several degrees further north, and the achieved inclination differs from the formula’s by a fraction of a degree.

The Earth is not spherical and its rotation contributes a velocity that is not along the azimuth. The insertion velocity is the vehicle’s own plus the site’s eastward motion, and the two add as vectors — so the achieved azimuth of the velocity differs from the flown azimuth by a few degrees, more at high latitude. The formula uses the velocity’s azimuth, and quoting the flown one is a common way to get a small answer wrong.

And an inclination is not a target on its own. A geostationary mission needs an inclination of zero and a specific longitude; an Earth-observation mission needs an inclination and a local solar time; a rendezvous needs the target’s plane exactly, including its node, which means launching within a window of a few minutes each day. The plane is two numbers and this rung has only priced one of them.

The rule the four rungs add up to #

The four rungs of this anchor say one thing four times, and it is worth stating in the form that covers all of them.

A plane change costs the speed at which it is done, and nothing else. Not the angle — the angle enters only through a sine that saturates. Not the altitude, except through the speed. Not the propellant or the engine. The whole subject is a question about where, and every technique on this ladder is a way of arranging to be somewhere slower when the rotation happens.

Turn at apogee: 21 per cent of the speed. Combine it with a burn already happening: the difference between a chord and a sum. Fly further out first: as slow as patience allows. And, at launch, do not turn at all — because there is nowhere slower than the pad and yet a launch cannot use it, since the constraint at the pad is geometric rather than energetic.

That last inversion is the one worth carrying. Everywhere else on this ladder, being slow is an advantage. At the launch site the vehicle is at rest and the plane change is impossible rather than cheap, because the orbit does not exist yet and its plane is fixed by where the vehicle stands.

What a site is chosen for now #

The four sites in the figures were built between 1955 and 1968, and the calculus that placed them is not quite the calculus that would place one today.

Kourou at 5.2° is as close to the equator as a European launch site could be, and its value is exactly what this rung has priced: 463 m/s of free eastward velocity and 5.2° of inclination that costs 145 m/s to remove at apogee. Baikonur was placed for radar coverage of missile tests and a downrange corridor over unpopulated steppe, and its 45.6° has cost every Russian geostationary mission about 1.25 km/s ever since.

What has changed is the mix of missions. Geostationary satellites — the traffic this arithmetic is about — are a shrinking share of what is launched, and the growing share is low-inclination or sun-synchronous constellations in low orbit, for which the site’s latitude matters far less: a constellation at 53° is served by a mid-latitude site rather than penalised by one, and a sun-synchronous mission at 98° pays a retrograde penalty from anywhere.

So the premium on equatorial sites is falling, and the reason has nothing to do with the mechanics on this page. The mechanics are unchanged and always will be; what changed is which orbits people want.

Where this ladder goes next #

Four rungs have now taken a plane change from its formula to its three cheap tricks to the one place it cannot be avoided. What is left on the anchor is where the manoeuvre stops being a manoeuvre.

One direction is the plane change as a term inside a larger solve: a Lambert problem whose two ends differ in plane has an optimal split between the departure and arrival burns, which is the second rung’s interior optimum arriving inside a two-point boundary-value problem rather than as a standalone question.

Another is Edelbaum’s result for continuous low thrust, where the rotation is spread over a spiral rather than applied at points and the closed form looks nothing like the chord formula — a combined plane change and orbit raise costs √(v₁² + v₂² − 2v₁v₂cos(πΔi/2)), with a π/2 in it that no impulsive analysis produces.

And the third is the one that removes the cost entirely: a gravity assist. A flyby rotates a trajectory for nothing, and a lunar one turns a geocentric orbit’s plane by tens of degrees — a trick that has recovered satellites stranded in the wrong plane. Where a suitable third body exists, every number on this page is the price of not using it.