About a minute after liftoff every launch commentary makes the same call: "max-Q". It is the moment the air is pushing on the vehicle hardest, and it is not at liftoff, where the air is thickest, nor at burnout, where the vehicle is fastest. It is somewhere in between, at around ten kilometres, and a well-flown rocket often goes through it with its engines deliberately turned down.

This lesson is about why. What the air does to a rocket depends on one quantity, the dynamic pressure, and that quantity rises and then falls on every ascent for reasons that come straight out of how the atmosphere is built. Where it peaks, and how high the peak is, sets the loads the airframe must be designed for, and the flight program has a say in both.

Speed against thinning air

Hold a hand out of the window of a moving car. The push grows much faster than the speed: twice as fast is four times the force. Now imagine the car driving up a mountain so steep that the air thins as fast as the speed rises. The push would grow at first, because the speed is doing most of the changing, and then fade, because the air runs out.

A rocket is that car. It starts from rest in the thickest air it will meet, and accelerates into air that thins by half every five kilometres or so. For the first few tens of seconds speed wins. Later, climbing a few hundred metres every second, the vehicle is leaving the air behind faster than its speed can make up for it. The peak between those two regimes is max-Q.

Dynamic pressure

Air approaching a body at speed $v$ carries kinetic energy $\frac{1}{2} ρ v^{2}$ per unit volume, where $ρ$ is its density. Where the flow is brought to rest — at the tip of the nose, the stagnation point — that energy reappears as pressure. For steady flow slow enough that the air's density does not change, Bernoulli's equation along a streamline says $p + \frac{1}{2} ρ v^{2}$ is constant, so the stagnation pressure exceeds the surrounding pressure by exactly

q = \frac{1}{2} ρ v^{2}

That is the dynamic pressure. Above about Mach 0.3 compressibility adds to the stagnation pressure, and above Mach 1 a shock stands in front of the nose, but the convention survives both: every aerodynamic force is written as $q$ times a reference area $S$ times a dimensionless coefficient that carries the shape and the Mach number. Drag is $q S C_{D}$ , normal force is $q S C_{N}$ . So $q$ is the single number that says how hard the air can push, and its unit is the pascal. At 35 kPa, a third of an atmosphere, the air pushes on the 21.2 m² cross-section of a 5.2 m fairing with a characteristic force $q S$ of about 750 kN.

The exponential atmosphere

The density falls with height because each layer of air is squeezed by the weight of all the air above it. A thin horizontal slab of air of thickness $d h$ is pressed down by its own weight, $ρ g d h$ per unit area, and held up by the difference in pressure across it, so

\frac{d p}{d h} = - ρ g

where $p$ is the pressure and $g$ the acceleration due to gravity. Air is close to an ideal gas, $p = ρR T$ , with $R = 287.05$ J/(kg·K) for dry air and $T$ the absolute temperature. Eliminating $ρ$ :

\frac{d p}{p} = - \frac{g}{R T} d h

If the temperature were the same at every height, this would integrate to

p = p_{0} e^{- h / H}, ρ = ρ_{0} e^{- h / H}, H = \frac{R T}{g}

with $p_{0}$ and $ρ_{0}$ the sea-level values. $H$ is the scale height: the climb over which pressure and density fall by a factor of $e$ . It is 8.4 km at the sea-level standard temperature of 288 K and 6.3 km at the 217 K of the lower stratosphere.

The real atmosphere is not isothermal, and the density falls a little differently from the pressure when the temperature changes with height: in general $d ρ / ρ = - (g / R T + (d T / d h) / T) d h$ . In the troposphere, where the temperature drops by 6.5 K per kilometre, the second term partly offsets the first and the density's own scale height is about 10.4 km at sea level, shrinking to about 8 km just below the tropopause; above the tropopause at 11 km the temperature stops falling and it is 6.3 km. Taken together, the 1976 US Standard Atmosphere gives:

sea level: 1.225 kg/m³
5 km: 0.736 kg/m³
10 km: 0.414 kg/m³, a third of sea level
20 km: 0.0889 kg/m³, 7 %
30 km: 0.0184 kg/m³, 1.5 %
40 km: 0.0040 kg/m³
50 km: 0.0010 kg/m³

Over the first 20 km that is an average scale height of 7.6 km — density halving every 5.3 km. By 50 km there is less than a thousandth of the air left.

Why it rises and then falls

Write the dynamic pressure with an exponential atmosphere and take the time derivative along the trajectory. With $ρ = ρ_{0} e^{- h / H}$ , the derivative of the density is $\overset{ρ}{˙} = - ρ \dot{h} / H$ , where $\dot{h}$ is the climb rate, so

\frac{q ˙}{q} = \frac{2 v ˙}{v} - \frac{h ˙}{H}

The first term is how fast $v^{2}$ is growing, as a fraction of itself; the second is how fast the density is falling. At liftoff the speed is near zero, so any acceleration is an enormous fractional growth and $q$ rises steeply. As the vehicle gathers speed that fraction shrinks, while the climb rate, and with it the fractional fall in density, grows. Dynamic pressure peaks when the two are equal:

\frac{2 v ˙}{v} = \frac{h ˙}{H}

After that the vehicle is climbing out of the air faster than its speed can make up for, and $q$ falls away, until by first-stage burnout, some 60 km up, it is a few hundred pascals and the air has stopped mattering. Nothing in this depends on the details of the vehicle: any rocket that accelerates while it climbs out of an exponential atmosphere has one peak, and it comes when the climb rate is large enough to beat the acceleration.

A worked example: Aster 5

Take a first stage sized like Vivapse's Aster 5, the simulator's Falcon 9-class preset: 563 t at liftoff, nine Merlin 1D engines giving 7.6 MN at sea level (thrust-to-weight 1.38) and 8.39 MN in vacuum, a 5.2 m fairing. Fly it through the standard atmosphere at full throttle, on the flight-path profile the reference program uses, and read off $q = \frac{1}{2} ρ v^{2}$ along the way:

T+40 s, 3.8 km, 218 m/s. The air is at 0.838 kg/m³ and $q = \frac{1}{2} \times 0.838 \times 21 8^{2} = 19.9$ kPa, and still climbing fast.
T+64 s, 11.0 km, 446 m/s, Mach 1.51. The density has fallen to 0.366 kg/m³, a third of the sea-level value, while $v^{2}$ has grown fourfold: $q = 36.4$ kPa. This is the peak. The speed is growing by 11.8 m/s every second, so $v^{2}$ is rising at 5.3 % per second; the vehicle is climbing at 389 m/s, and the density it flies in is falling at about the same 5.3 % per second.
T+90 s, 23.8 km, 846 m/s. The density is down to 0.049 kg/m³ and $q$ has halved, to 17.4 kPa, although the vehicle is nearly twice as fast.

The simulator flying the real Aster 5 at full throttle from Cape Canaveral, in calm air, puts its peak at 35.4 kPa at T+68.5 s, at 12.6 km and Mach 1.67 — the same peak, a little later and higher, because its full six-degree-of-freedom flight differs in detail from this point-mass sketch.

At the peak, the drag is $q S C_{D}$ with a drag coefficient of about 0.45 at Mach 1.5: roughly 350 kN, four per cent of the thrust. For a vehicle this size drag is a small tax — integrated over the whole first-stage burn it costs only about 40 m/s. The dynamic pressure matters for a different reason.

What the airframe feels

Max-Q is not the moment of highest acceleration — that comes near burnout, when the stage is light — and it is not the moment of highest heating, which goes as $ρ v^{3}$ and peaks higher up. It is the moment when every aerodynamic load is at its largest for a given attitude, and several of them arrive together.

Bending. A rocket is never pointed exactly into the relative wind. The angle between its axis and the oncoming air, the angle of attack $α$ , produces a sideways force along the body. Slender-body theory gives the nose alone a normal force of about $2 q A α$ , with $A$ its base area and $α$ in radians, and the body adds more. That force, acting well ahead of the centre of mass, bends the long, thin, pressurised tank structure, and the bending moment scales with the product $q α$ . Launch vehicle engineers therefore quote the load as $q α$ , in kilopascal-degrees, and design against a limit on it. Real launchers are built for roughly 100 to 200 kPa·°. At 35 kPa, 200 kPa·° is only 5.7° of angle of attack.

Wind. The angle of attack is set by the air's motion as well as the vehicle's. A crosswind $w$ at speed $v$ tilts the relative wind by roughly $arctan (w / v)$ without the vehicle turning at all. Near max-Q the vehicle is at the altitude of the jet stream: a 50 m/s crosswind at 12 km, met at 486 m/s, is 5.9° of angle of attack, and at 35.4 kPa that is 208 kPa·° — the whole budget of a real vehicle from the weather alone. That is why launch-weather rules look at winds aloft as well as at the ground, and why guidance flies load relief, turning the vehicle into the wind to keep $α$ small instead of holding a planned attitude.

The transonic region. Max-Q usually falls near Mach 1. Nobody arranges it that way: it is where a vehicle of ordinary thrust-to-weight happens to be when $2 \overset{v}{˙} / v$ and $\dot{h} / H$ cross. Around Mach 0.8 to 1.2 shock waves form and move over the body, the drag coefficient peaks — the simulator's nose-first table rises from 0.30 subsonic to 0.62 at Mach 1.15 — and the flow buffets. Peak $q$ and transonic buffet arriving together is what makes this stretch of flight the hardest on the structure.

Control. The aerodynamic moment that turns a rocket away from the wind grows with $q$ too, and a rocket is aerodynamically unstable. At max-Q the engine gimbal is working hardest to hold the vehicle straight. That is the subject of the next lesson.

Why vehicles throttle down

Every one of those loads is proportional to $q$ , and the structure has to be built for the largest $q$ it will see. A lower peak means lighter tanks and interstages, or more margin for wind on the day. The flight program can lower the peak without touching the airframe: throttle the engines down as the vehicle approaches max-Q, so that it passes through the densest part of its climb more slowly, and throttle back up once the air has thinned.

It is not free. A throttled engine burns for longer, and every extra second of burn is a second spent holding the vehicle up against gravity: the gravity loss from the gravity turn grows. The drag loss shrinks, but for a large vehicle it was small to begin with. The trade is a few tens of metres per second of performance for a structure that sees a fifth less load.

Where the bucket sits matters as much as how deep it is. Throttle down too late and the peak has already passed. Throttle up too early and there is a second, later peak when the full thrust returns. Throttle down too early, and the vehicle spends the bucket low and slow, and meets denser air at higher speed afterwards — a worse max-Q, not a better one. And too deep a bucket early in flight stops the climb: at T+40 s, 40 % throttle gives Aster 5 about two-thirds of its weight in thrust. The Space Shuttle is the best-known example of a vehicle flown this way: its main engines throttled back through the region of maximum dynamic pressure and returned to full power about a minute into flight.

Figure · dynamic pressure on the way up

SHOW

THROTTLE DOWN ATT+50 s

FOR22 s

TO78 %

MAX-Q: 29.3 kPa
AT: T+50 s · 6.3 km
MACH THERE: 0.96
VERSUS FULL THROTTLE: −20 %
GRAVITY LOSS: 1136 (+48) m/s
DRAG LOSS: 36 (−2) m/s
CUTOFF: T+143 s

A first stage sized like Aster 5, flown through the 1976 standard atmosphere on the reference program's pitch profile. The dashed curve is the same flight at full throttle. The throttle cannot go below 40 % (a Merlin's floor) and slews at 1 per second. Gravity and drag losses are integrated to cutoff, which comes when 380 t of propellant are gone — later, if the bucket held the engines back.

The default setting is close to what Vivapse's reference program does: 78 % throttle from T+50 s for 22 s. The peak drops from 36.4 kPa to 29.3 kPa, now at T+50 s and Mach 0.96, at a cost of 48 m/s of extra gravity loss, against 2 m/s saved in drag, and a cutoff 5 s later. Move the bucket to start at T+30 s for 30 s at 60 % and the peak arrives later and higher, at 45 kPa. Switch the view to density and speed to see the two curves whose product $q$ is: $ρ$ falling away from the start, $v^{2}$ barely moving for half a minute and then climbing steeply, and their product peaking where one overtakes the other.

In Vivapse

The atmosphere is src/sim/atmosphere.ts: the 1976 US Standard Atmosphere from the ground to 1,000 km — the same layer formulas as the figure above — modified by the day's weather. With live weather the measured temperature profile and surface pressure replace the standard ones in the lower atmosphere, and the air co-rotates with the Earth and carries the wind. The physics model describes it, and Weather and launch conditions explains why wind matters most around max-Q.

A program reads fc.dynamicPressure (Pa), which is $\frac{1}{2} ρ v^{2}$ with $v$ the speed relative to the moving air, along with fc.airDensity, fc.mach and fc.aoa. The bending load is fc.qAlpha, in Pa·°, and the limit it must stay under is fc.limits.qAlpha: 250 kPa·°, set in src/sim/aero.ts. Above it the vehicle breaks up. That limit is deliberately more generous than a real launcher's 100–200 kPa·°, a choice recorded in fidelity and its limits. The simulation notes the peak itself: src/sim/sim.ts records the highest $q$ and emits a maxq event once it has fallen 10 % below it, so the console line arrives a little after the peak it describes.

The reference program, src/programs/full-mission.js, flies the ideas of this lesson. Its throttle bucket is not a time window but a limit on $q$ itself:

let thr = fc.dynamicPressure > 30e3 ? 0.78 : 1;

so the engines drop to 78 % whenever $q$ exceeds 30 kPa, and the throttle's own slew smooths the switching into a plateau. Above 150 m/s of airspeed it also steers within a cone round the relative wind whose half-angle is 80 kPa·° divided by the dynamic pressure — about 2.7° on the 30 kPa plateau — which keeps the commanded $q α$ near a third of the break-up limit.

Try it

Choose the Aster 5 preset and launch from Cape Canaveral with the full-mission program, on custom weather with the wind, gusts and turbulence set to zero. When the vehicle is through the thick air the console reports Max-Q: 30.1 kPa at 7.0 km. Press T to open the details and look at the dynamic pressure chart: instead of a sharp peak it has a flat top from about T+53 to T+72 s, the bucket holding $q$ at its 30 kPa limit.

Now find the bucket line in the program and change it to

let thr = 1;

and fly again. The chart becomes a single sharp peak and the console reports about 35.4 kPa at 12.6 km, at T+68 s — 18 % more load on the structure, fifteen seconds later and more than five kilometres higher. Log fc.qAlpha through the peak on both flights. Then switch to live weather, fly the full-throttle version again on a day with a strong jet stream, and see how much of the 250 kPa·° is left.

What carries forward

Max-Q is where the air's grip on the vehicle is strongest, and the vehicle is built to be pushed there nose-first. It is also where an unstable airframe is hardest to hold straight. Attitude control in six degrees of freedom is about the torques that do the holding and the loop that decides how much of them to use.