Why a rocket turns instead of steering

Orbit is not high; it is fast. At 200 km the Earth's gravity still has 94 per cent of its strength at the surface. What keeps a satellite up there is that it moves sideways at 7.8 km/s, falling around the Earth as fast as it falls towards it. Reaching orbit means building up an enormous horizontal speed. Yet every rocket starts by going straight up, and the ones that reach orbit take a long, curving path between the two.

Why not go sideways from the start? Why not go straight up and turn at the top? And why does a rocket not point wherever it likes? The answers are three kinds of loss, and one shape of trajectory that keeps all three small at once.

Three ways to waste Δv

The rocket equation gives the Δv a vehicle could produce. On the way to orbit, part of it never becomes speed.

Gravity loss. A rocket hovering on its engines burns propellant at full rate and goes nowhere: all the Δv it spends goes into holding itself up. A rocket climbing straight up pays the same toll, 9.8 m/s of Δv for every second it spends pointing at the sky, on top of whatever speed it gains. Gravity loss is the part of the budget spent holding the rocket up rather than speeding it along. It is the largest of the three.

Drag loss. The air resists the rocket, more the faster it goes and the denser the air. Every newton of drag has to be matched by a newton of thrust that does nothing else.

Steering loss. Thrust that does not point along the direction of motion does not all go into speed. Part of it turns the path instead. Turning is sometimes exactly what you want, but it is not free.

The three pull against each other. Getting gravity off your back quickly means going sideways early. Getting out of the thick air quickly means going up early. Forcing the path from one to the other means steering. The pitch programme — the plan for how the nose tilts from vertical towards horizontal — is how a rocket trades them.

Along the path and across it

To see the trade, split the forces on the rocket into the part along its path and the part across it. Two angles are needed. The flight-path angle $\gamma$ is the angle of the velocity above the local horizontal: 90° straight up, 0° level. The angle of attack $\alpha$ is the angle between the rocket's nose — and so its thrust — and its velocity.

Along the path, thrust contributes its component $F\cos \alpha$, drag $D$ pulls straight back, and gravity pulls back with its component $mg\sin \gamma$: all of it when the rocket is climbing vertically, none of it when it is level. Divide by the mass $m$ and you have the rate at which the speed changes, in metres per second gained each second:

Across the path, the thrust's sideways component $F\sin \alpha$ bends the path towards the nose, and gravity's other component $mg\cos \gamma$ bends it down. One more term appears because the Earth is round. A rocket moving sideways finds the ground curving away beneath it, so its local horizontal keeps tilting, and that alone lifts the flight-path angle at a rate $v\cos \gamma /r$, where $r$ is the distance from the Earth's centre. Together, and ignoring any lift from the air:

The first equation holds all three losses. Add it up second by second over the whole burn — which is what the integral sign $\int \dots \mathrm{d}t$ means — and the speed actually gained is

where the only step was to write $\cos \alpha$ as $1-(1-\cos \alpha )$. The first term is everything the engines produced. Since $F=\dot{m}c$, it is the same sum as in course 01, and for a constant $c$ it comes to the rocket equation's $c\ln (m_0/m_f)$. The other three are the losses, in order: steering, drag and gravity.

The second equation says how the path bends. Put the nose exactly on the velocity, $\alpha =0$, and the thrust has no sideways part at all. The path still bends, because gravity pulls it over, at a rate

That is a gravity turn. The rocket does not steer; gravity turns it.

Three things follow from that rate. It is fastest when the rocket is slow, so the path bends quickly just after lift-off and hardly at all later. It is zero when the path is vertical, since $\cos 90°=0$, so a rocket pointing exactly up stays exactly up, and something has to tip it over to start the turn. And it falls to zero when $v^2/r=g$, the speed at which the ground curves away as fast as gravity pulls the path down. At 200 km that speed is 7.78 km/s. It is orbital speed.

With the angle of attack at zero, only gravity turns the path. Raise the speed and watch the turn rate shrink towards nothing at orbital speed. Then tilt the nose off the velocity. At 10° the thrust's sideways part turns the path hard, yet the steering loss is only $1-\cos 10°=1.5$ per cent of the thrust, because a cosine hardly falls for small angles. Steering in vacuum is cheap. The expensive part of steering is somewhere else.

Why not steer in the air

A rocket is a long thin tube, built to be squeezed end to end by its own thrust. Tilt it into the airflow and the air pushes it sideways along its whole length, and the tube bends. The sideways force grows with the angle of attack and with the dynamic pressure

where $\rho$ is the density of the air: roughly the pressure a flat plate facing the oncoming air would feel. Dynamic pressure is zero on the pad and rises as the rocket speeds up. But the air thins as it climbs — its density halves in the first 7 km, and again in every 4 to 6 km above that — and before long the thinning wins, so $q$ peaks and falls away. The peak is max-Q, usually around a minute after lift-off, 7 to 13 km up.

The bending load grows with the product of the two, $q\alpha$, and every airframe has a limit on it. Real launchers are built for something like 100 to 200 kPa·°. Vivapse is more forgiving, at 250 kPa·°, and above that the vehicle breaks up. At a max-Q of 30 kPa, 250 kPa·° allows about 8° of angle of attack; a real vehicle would allow nearer 3 to 7.

So in the thick air a rocket has almost no freedom to point anywhere but along its velocity. Whatever turning it needs has to be done before the air matters — in the first seconds, while it is slow — or by gravity itself, with the nose held on the velocity. That is the gravity turn's real virtue. It is not the most efficient path through vacuum. It is the path that keeps the angle of attack at zero through max-Q, so the airframe is only ever squeezed along its length.

The pitch programme

A launch therefore runs like this.

1. Vertical rise. The rocket climbs straight up for a few seconds, to clear the tower.
2. Pitch-over. At a few tens of metres per second, the nose is tipped a few degrees towards the direction of flight. The air is dense but the rocket is slow, so $q$ is tiny and the brief angle of attack costs almost nothing.
3. Gravity turn. Once the velocity has swung round to meet the nose, the nose is held on it. Gravity bends the path over, through max-Q and out of the thick air.
4. Steering above the air. Once the dynamic pressure has fallen away, usually by the time the first stage is spent, the upper stage is free to point wherever its guidance says, and steers to reach the orbit exactly; closed-loop ascent guidance is about how.

The shape of the whole ascent is set almost entirely by the pitch-over. Because gravity turns the path fastest at low speed, a small change in the kick becomes a large change in the trajectory by the time the rocket is fast. Too small a kick and the rocket climbs nearly vertically, pays gravity loss all the way, and arrives high but slow and steep. Too large, and it flattens early, spends too long fast in thick air, and hands the upper stage a path too low to finish the job — or turns back into the ground.

Start from the 3° kick. The first stage burns out 93 km up at 2.94 km/s, climbing at 36°, having lost 1.16 km/s of its 4.11 to gravity and only 20 m/s to drag. Cut the kick to 1° and it burns out 27 km higher but 210 m/s slower, still climbing at 69°; the extra gravity loss is almost exactly the missing speed. Raise the kick to 6° and gravity loss falls to 0.78 km/s, but the stage burns out at 42 km, nearly level, with a coast apogee of only 49 km: the upper stage would start inside the atmosphere and have to climb on its own engine. At 8° the path turns down into the air before burnout.

So the best kick is not the one with the smallest losses in this figure. It is the one that gives the next stage the best start, which is why a real pitch programme is tuned against the whole ascent, not the first stage alone.

Now switch the programme to on a clock. This one ignores the velocity and tilts the nose at a steady rate, like the hand of a clock. At 0.45° per second it reaches much the same burnout as the 3° gravity turn — 91 km, 2.96 km/s, 32° — but its bending load rises to 195 kPa·° near max-Q, against 4 for the gravity turn. At 0.6° per second it passes 250 kPa·° 51 seconds after lift-off, and the vehicle breaks up. Paths that look alike in the upper plot can load the airframe very differently. The lower plot is the one that decides whether it survives.

A worked example: where Aster's Δv goes

The figure is a sketch. The simulator is not, so here is the real accounting: Aster flown from Cape Canaveral into a 200 km orbit by the reference program on a calm day, with the three losses added up step by step against the rocket's speed relative to the ground.

Four things stand out.

Gravity is almost the whole cost. The rocket spent 1.5 km/s — 17 per cent of everything its engines produced — holding itself up on the way.

Drag is tiny for a rocket this size. Nineteen metres per second. The drag on a rocket grows with its cross-section, which scales with the square of its size, while its mass scales with the cube, so the bigger the rocket, the less the air slows it. The simulator's little Sparrow, 1.3 m across and 23 t at lift-off, loses 54 m/s to drag during its first-stage burn; the 9 m, 5,247 t Colossus II loses 11 over its whole ascent. The reason to respect the atmosphere is not the Δv it takes. It is the load it can put on the airframe.

Steering is almost free. Twenty-five metres per second, and the first stage, flying a gravity turn, spent only 1 of them. The rest went on the upper stage above the air, pointing its nose a little below its velocity to bend a steep path flat.

The Earth helped. Cape Canaveral moves east at 409 m/s with the Earth's rotation, and a rocket launched east keeps that speed. By insertion the rotation was contributing 421 m/s, a little more at 200 km up, where the same turning carries a point further round. That is why launch sites sit as close to the equator as their countries allow, and why rockets launch east.

A note on comparing numbers. Published loss figures are usually measured against the rocket's velocity in space rather than relative to the ground. Measured that way, the same flight shows 951 m/s of gravity loss and 591 m/s of steering loss, because in the first minute the rocket climbs almost straight up while the Earth's rotation carries it east, and thrust at right angles to that eastward motion counts as steering. The total barely changes — 1,557 m/s of losses measured that way, 1,569 measured this one — but the split between the columns does. When you compare loss figures, check which velocity they were measured against.

In Vivapse

Nothing in the simulator flies a gravity turn for you. The pitch programme is part of your flight program, and the reference programs show two ways to write it.

The starter template does it in one line:

if (fc.surfaceSpeed > 100) fc.steer(Math.min(85, fc.prograde), 0);

At 100 m/s it tips the nose 5° from vertical. Once the velocity has swung round to meet it, it holds the nose on fc.prograde, the direction of travel relative to the ground. Gravity does the rest.

The reference full-mission.js is more careful. It ramps the kick in between 55 and 115 m/s, to 8° off vertical, then keeps the nose within 3° of the velocity. On a real launch site it instead steers the turn onto a table of flight-path angles against speed, leading or lagging the velocity by up to 5°, so that the wind of the day cannot send the whole ascent off course. Through the dense air it keeps the nose inside a cone around fc.airPrograde, the direction of travel relative to the air rather than the ground, no wider than $80\text{ kPa⋅°}/q$ and never more than 10°. And while the dynamic pressure is above 30 kPa it throttles back to 78 per cent.

The simulator enforces the load limit. fc.qAlpha is the bending load now and fc.limits.qAlpha the limit, 250 kPa·°; above it the vehicle is lost, and its loss card says aerodynamic break-up. fc.dynamicPressure and fc.aoa are the other readings a pitch programme needs, and the steering and guidance page explains the guidance plane that fc.steer() measures its pitch in.

Try it

Load the Starter template and fly it on Aster. On the default mission it reaches the edge of space on its first stage, with max-Q about 37 kPa and a peak q·α of about 140 kPa·°.

Then replace its gravity-turn line with a pitch programme on a clock:

fc.steer(Math.max(20, 90 - 1.0 * fc.t), 0);

This tilts the nose one degree per second from lift-off, whatever the velocity does. On the default mission the rocket flattens into thick air, drives the dynamic pressure to about 100 kPa — nearly three times the starter's max-Q — and breaks up about 75 seconds after lift-off. The loss card shows q·α at the limit with an angle of attack of only 2.4°: at that dynamic pressure even a small angle is too much.

Finally, put the gravity turn back and change fc.prograde to fc.airPrograde. The starter's peak q·α falls from about 140 to about 53 kPa·°. Most of the bending load in the original came from the wind: by following the ground, the nose was flying at an angle to the moving air. The atmosphere is not only something to climb out of, but something to steer relative to.

What carries forward

A gravity turn ends with the rocket above the air, flying nearly level, fast enough that gravity bends its path no more steeply than the Earth curves away. Shut the engines down there and the path closes on itself. That closed path is an orbit, and the next course is about its shape: the curves every unpowered trajectory follows, and the six numbers that name one.