Orbit costs 7.8 and a launch buys 9.4

Every rung of this anchor has quoted a number without deriving it. Low Earth orbit costs 9.4 km/s, the essays say — the first rung marks it on the propellant-fraction curves, the third divides it between stages, the fourth prices what each kilometre a second of it costs in mass.

Orbital speed at 200 km is 7.78 km/s. The difference is a fifth of the budget and rather more than a fifth of the propellant, since the exponential is unforgiving at that end, and it is worth knowing what it is made of.

It is also worth saying why the gap cannot simply be designed away. Every term in it is a consequence of launching from the surface of a planet with an atmosphere into a gravitational field, and none of them exists for a vehicle already in orbit. A stage that begins in low orbit and burns to escape pays no gravity loss worth measuring, because the burn is short and the vehicle is already moving horizontally; the same stage on the pad would pay a thousand metres a second. The 9.4 is not a property of orbit. It is a property of getting there from here.

Three integrals, and what each one is #

The rocket equation delivers an ideal velocity change: what the vehicle’s speed would become if nothing acted on it but its own engine. Along a real ascent three things do.

The gravity loss is ∫g sin γ dt, where γ is the flight-path angle above the horizon. It is the component of the vehicle’s own weight along its velocity, and it is not a loss in the sense of energy going somewhere useless — the potential energy is really gained and really needed. What it is is Δv spent to reach altitude rather than to reach speed, and it is large: over a thousand metres a second on any real launch.

The drag loss is ∫(D/m) dt, the deceleration from pushing air aside, integrated over the ascent. It is genuinely dissipative and it is small compared with gravity — a hundred metres a second or so for a real vehicle.

The steering loss is ∫(T/m)(1 − cos α) dt, where α is the angle between the thrust and the velocity. It is what a vehicle pays for pointing its engine anywhere except along its own motion, and in the trajectory integrated here it is identically zero, because a gravity turn flies at zero angle of attack after one initial kick and lets gravity do the turning. A real ascent has some — tens of metres a second, from the pitch programme not being an exact gravity turn.

There is a fourth term that is not a loss in the same sense. A sea-level nozzle is over-expanded near the pad, so the engine delivers less thrust than its vacuum rating; that shows up as a lower exhaust speed early in the burn rather than as a deceleration, and it is why the previous rung had to distinguish two specific impulses for the same engine.

The trade, and why it does not decide anything #

The two losses that can be traded move in opposite directions with the same knob.

A vehicle with a high thrust-to-weight leaves the pad quickly. It spends less time with a large flight-path angle, so ∫g sin γ dt is smaller. But it reaches high speed lower in the atmosphere, where the density is higher, so ∫(D/m) dt is larger. The integration puts numbers on both: at a thrust-to-weight of 1.15 the split is 1,263 and 119 m/s; at 2.2 it is 381 and 1,874.

The sum has a minimum, and at the vehicle drawn it is around 1.5, at about 1,200 m/s of total loss. What is more useful than where the minimum is, is that no real launcher is where it is because of this curve. Lift-off thrust-to-weight ratios cluster between about 1.2 and 1.5, and they are there because a vehicle needs enough margin over one to leave the pad decisively and because thrust is expensive — engines are heavy, and an engine sized for 2.2 is carrying its own mass for the rest of the flight.

Why the gravity loss is so large #

Over a thousand metres a second is a fifth of a launcher’s budget spent on being lifted, and the reason it is that large is a comparison of two times.

The ascent takes of order five hundred seconds. Gravity removes 9.8 m/s of vertical velocity every second of it, so if the vehicle spent the whole ascent pointing straight up the loss would be nearly 5 km/s. It does not: the flight-path angle falls from 90° to zero, and sin γ falls with it, so the integral is a few tenths of that. But the scale is set by g times the burn time, and the burn time is set by how much Δv is being delivered at what acceleration.

That gives the loss a shape worth stating as a rule: the gravity loss is roughly g times the time spent climbing, and reducing it means either accelerating harder or turning sooner. Both have limits — acceleration costs engine mass, and turning sooner means flying faster in thicker air.

It also explains an asymmetry between ascent and everything else in this collection. A transfer between two orbits has no gravity loss at all in the impulsive limit, because the burns are instantaneous and the vehicle is never fighting anything. Gravity loss is a property of a long burn in a strong field, and launch is the only manoeuvre in ordinary spaceflight that is both.

That figure is the one genuinely surprising thing on this rung. A higher specific impulse buys a smaller mass ratio and costs a larger gravity loss, because it delivers the same Δv over a longer time at the same thrust-to-weight. It is one of the reasons hydrogen first stages are rare quite apart from the density argument, and it is invisible to anybody working from the rocket equation alone — which sees no time in it at all.

What a gravity turn is, and why it is used #

The trajectory integrated here is not steered. After one pitch kick at 60 m/s, the vehicle holds zero angle of attack and gravity alone rotates the velocity vector from vertical to horizontal.

The name is a slight misnomer and the misnomer is instructive. Nothing about the manoeuvre is passive: the engine is at full thrust throughout, and the vehicle’s speed is climbing the whole way. What is passive is the steering. The vehicle points where it is going and gravity decides where that is, so the shape of the trajectory is an output rather than a command — which is why one number, the pitch kick, determines the whole of it.

The mechanical reason is structural. A launch vehicle is a long thin tube optimised for axial loads, and flying at an angle of attack through the atmosphere puts a bending moment on it that its structure is not sized for. The product of dynamic pressure and angle of attack — “q-alpha” — is a monitored quantity on every ascent, and the trajectory is designed to keep it small through the region of maximum dynamic pressure. Zero angle of attack does that by construction.

The economic reason is the steering loss. Thrust not along the velocity contributes nothing to speed, and the loss goes as one minus the cosine, so it is second order in small angles and grows quickly beyond ten or fifteen degrees. A gravity turn pays none of it.

What a gravity turn cannot do is choose where the vehicle ends up. The whole trajectory is fixed by one number — the initial kick — so the burnout altitude and speed are outputs rather than targets, and the figures here solve for the kick that leaves the vehicle horizontal at burnout. A real ascent is a gravity turn only approximately, with small steering commands to hit an insertion state, and those commands are the tens of metres a second of steering loss.

The habit #

The structure worth extracting is that a quantity presented as a single number is often a sum of terms with completely different physics, and that pricing them separately changes what can be done about it.

“Orbit costs 9.4 km/s” is a budget. Taken as one number it invites one response, which is to build a bigger rocket. Split into 7.8 of orbital speed, 1.2 of gravity, 0.1 of drag and 0.4 of a launch site’s latitude, it invites four different responses, of which two are engineering, one is geography and one is not negotiable at all. The split is what makes the question answerable.

This collection meets the same move repeatedly. Mercury’s perihelion advance is a sum of five terms of which four are Newtonian and one is not, and it is only the residual that means anything. A distance error is a sum of contributions from each rung of a ladder, and knowing which rung dominates says where to spend an observing programme. In each case the decomposition is the result and the total was never the interesting quantity.

What the picture cannot show #

The integration is a single-stage vehicle burning continuously from the pad to burnout, and no launcher does that. The three departures are worth stating because two of them move the answer in the same direction.

There is no staging. A real vehicle sheds most of its dry mass partway up, which raises its acceleration discontinuously and reduces the remaining gravity loss; the model’s single vehicle carries its whole structure to burnout. That makes the gravity loss here an overestimate.

There is no coast. A real ascent burns the first stage, coasts on a ballistic arc while the second stage waits, and then burns again at apogee — which is a much cheaper way of reaching altitude than climbing under power, because the coast costs no Δv at all. The model’s continuous burn is again the more expensive option, so again the loss is an overestimate.

And the burnout is at the wrong place. The trajectories here end horizontal at a few tens of kilometres, on an ellipse whose perigee is well inside the Earth, and reaching a stable orbit from there needs a further burn at apogee that the model does not perform. A real profile lofts higher and inserts higher, and the extra altitude costs some of the gravity loss the model saves elsewhere.

Taken together, the number the figure produces — around 1,200 m/s of loss at a realistic thrust-to-weight — is the right size for a real launch and it is right for slightly wrong reasons. What the figure is for is the shape of the trade and the ordering of the terms, both of which survive all three simplifications: gravity dominates drag by an order of magnitude, the two move oppositely with thrust-to-weight, and the minimum of the sum is shallow.

Where the loss actually accumulates #

The integral is dominated by a particular stretch of the ascent, and knowing which one explains most of what launch trajectories look like.

Gravity loss is g sin γ per second, so it is largest when the vehicle is pointing steeply up — which is the first minute or two, before the turn has developed. Roughly half of the total is accumulated in the first ninety seconds, when the vehicle is slow, heavy and nearly vertical. Nothing about the rest of the trajectory can recover it.

That is why the pitch-over happens as early as it does, and why the timing of it is the most consequential decision in an ascent profile. Turning too early means flying at a shallow angle through thick air, which raises the drag loss and the structural load; turning too late means paying gravity for the delay at a thousand metres a second per two minutes. Real vehicles pitch over between about 50 and 150 metres a second, within the first half-minute of flight, and then do nothing further until the atmosphere is behind them.

Drag loss accumulates somewhere else entirely: almost all of it in the ten or fifteen seconds either side of maximum dynamic pressure, at around 11 kilometres and 400 metres a second. Dynamic pressure is ½ρv², the density falls exponentially with a scale height of 8.5 km while the speed climbs, and the product has a sharp maximum. Above it the air is thin enough that drag is negligible however fast the vehicle goes, which is why hypersonic speed inside the atmosphere is expensive and hypersonic speed just outside it is free.

The number that is not a loss #

One term in a launch budget goes the other way and is worth naming, because it is the only free velocity available anywhere in this subject.

The Earth’s surface at the equator moves east at 465 m/s. A vehicle launched due east starts with that, and it is a genuine reduction in the Δv required rather than a bookkeeping convenience. At Kennedy’s 28.5° of latitude the bonus is 465 cos(28.5°) = 409 m/s; at Baikonur’s 45.6° it is 325; at a polar launch site aimed north it is zero, and at a retrograde launch it is negative twice over.

Against the 1,200 m/s of losses computed here, a few hundred metres a second is a third of them, gained or lost by geography alone. It is why equatorial launch sites are valuable, why the never-built sea-launch platforms were placed on the equator, and why the inclination a site can reach and the velocity it starts with are two consequences of the same latitude pulling in the same direction.

Two vehicles that measured this #

The loss budget is not only a computation. It is published for flown vehicles, from telemetry, and the numbers are worth setting beside the figures.

A Saturn V reaching a 185-kilometre parking orbit spent about 9,240 m/s of ideal Δv for an orbital speed of 7,793: 1,447 m/s of losses, of which roughly 1,220 was gravity, 40 drag, and the rest steering and nozzle back-pressure. The Space Shuttle, with a much draggier stack and a longer time in the atmosphere, spent about 1,200 of gravity loss and 110 of drag.

Both sit close to the figures here, and the agreement is worth reading carefully. The model overestimates the gravity loss because it has no staging and no coast; it underestimates it because it inserts at 42 kilometres rather than 185. Those errors are of comparable size and opposite sign, which is why the total lands in the right place, and it is not a reason to trust the model further than the shape it draws. A model that agrees with reality by two cancelling errors is a model that will disagree as soon as one of them is removed.

What both vehicles confirm without ambiguity is the ordering. Gravity loss is ten to thirty times drag loss on every launcher ever flown, drag is a rounding error, and the popular account in which a rocket “fights through the atmosphere” has the two terms the wrong way round. The atmosphere costs a launcher about one per cent of its budget. Being in a gravitational field while doing it costs fifteen.

Where the ladder goes next #

Everything above is Newtonian and every number in it is a few kilometres a second. The rocket equation itself is not restricted to those speeds, and it is worth knowing what it says when it is not.

The next rung is the relativistic form. Momentum conservation in special relativity replaces the velocity change with the rapidity — the quantity that adds when velocities are combined — so the mass ratio becomes exponential in artanh(v/c) rather than in v/c. The equation does not break at the speed of light: it diverges there, smoothly, which is worse. Even a photon rocket, whose exhaust leaves at c and which converts its propellant entirely to directed radiation, needs a mass ratio of 4.4 to reach nine tenths of light speed.

Further rungs: nuclear thermal and nuclear pulse propulsion, which raise the exhaust speed by a factor of two and of thirty respectively; aerobraking, which buys Δv from an atmosphere for free and pays in heat; and solar and laser sails, which have no propellant and therefore no rocket equation at all.