Max q

For other uses, see Max q (disambiguation).

The max q condition is the point when an aerospace vehicle's atmospheric flight reaches maximum dynamic pressure. This is a significant factor in the design of such vehicles because the aerodynamic structural load on them is proportional to dynamic pressure. This may impose limits on the vehicle's flight envelope.

Dynamic pressure

Dynamic pressure, q, is defined mathematically as

$q={\tfrac {1}{2}}\,\rho \,v^{2},$

where ρ is the local air density, and v is the vehicle's velocity; the dynamic pressure can be thought of as the kinetic energy density of the air with respect to the vehicle. This quantity appears notably in the drag equation.

For a car traveling at 90 km/h (25 m/s) at sea level (where the air density is about 1.225 kg/m^3 ), the dynamic pressure on the front of the car is 383 Pa, about 0.38% of the static pressure (101,325 Pa at sea level).

For an airliner cruising at 828 km/h (230 m/s) at an altitude of 10 km (where the air density is about 0.4135 kg/m^3), the dynamic pressure on the front of the plane is 10937 Pa, about 41% of the static pressure (26,500 Pa at 10 km).

Max q in a rocket launch

For a launch of a rocket from the ground into space, dynamic pressure is:

• zero at lift-off, when the air density ρ is high but the vehicle's speed v = 0
• zero outside the atmosphere, where the speed v is high, but the air density ρ = 0
• always non-negative, given the quantities involved

During the launch, the rocket speed increases but the air density decreases as the rocket rises. Therefore, (by Rolle's theorem) there is a point where the dynamic pressure is maximum.

In other words, before reaching max q, the dynamic pressure change due to increasing velocity is greater than that due to decreasing air density so that the dynamic pressure (opposing kinetic energy) acting on the craft continues to increase. After passing max q, the opposite is true. The dynamic pressure acting against the craft decreases as the air density decreases, ultimately reaching 0 when the air density becomes zero.

This value is significant since it is one of the constraints that determines the structural load that the body rocket must bear. For many rockets, if launched at full throttle, the aerodynamic forces would be higher than what they can withstand. For this reason, they are often throttled down before approaching max q and back up afterwards, so as to reduce the speed and hence the maximum dynamic pressure encountered along the flight.

Rocket launch examples

During a normal Space Shuttle launch, for example, max q value of 0.32 atmospheres occurred at an altitude of approximately 11 km (36,000 ft). The three Space Shuttle Main Engines were throttled back to about 60–70% of their rated thrust (depending on payload) as the dynamic pressure approached max q; combined with the propellant grain design of the solid rocket boosters, which reduced the thrust at max q by one third after 50 seconds of burn, the total stresses on the vehicle were kept to a safe level.

During a typical Apollo mission, the max q (also just over 0.3 atmospheres) occurred between 13 and 14 kilometres (43,000–46,000 ft) of altitude; approximately the same values occur for the SpaceX Falcon 9.

The point of max q is a key milestone during a rocket launch, as it is the point at which the airframe undergoes maximum mechanical stress.

See also

• Prandtl–Glauert singularity
• Ideal gas law
• Gravity turn
• Gravity drag