Rocket Physics, the Hard Way: Max q and Bernoulli’s Principle

Header image: Computational fluid dynamics simulation of a NASA X-43A hypersonic aircraft (image source: NASA.)

Recall the last time you watched a rocket launch. About a minute into ascent, you may have heard Launch Control call out “Max q!”. But what does dynamic pressure mean?

Welcome back to Rocket Physics, the Hard Way. Although this installment will be relatively light compared to some of our more in-depth examinations, we will nevertheless be covering an important concept in aerodynamics that has sparked curiosity in the mind of every space fan.

One of the most popular space exploration films (and books), The Martian, features a dramatic opening sequence: A violent dust storm strikes the landing site of a crewed mission to Mars, threatening to tip over their ascent vehicle. This is an unrealistic circumstance, and we will discover why shortly. The answer is connected to ‘max q’ and a concept known in fluid mechanics as dynamic pressure.

There are two kinds of pressure: static pressure and dynamic pressure. To explain the difference, first imagine that you are standing in a field on a pleasant spring day, and the air is completely still.

The Earth’s atmosphere presses down on you with a pressure of one atmosphere, or 101,325 Pascals (14.7 pounds per square inch, for American readers.) That is static pressure, which is what you see in your daily weather report. At Martian zero elevation datum, the atmosphere’s static pressure is a much lower 0.006 atmospheres, or 610.5 Pascals (0.09 pounds per square inch.)

Imagine now that a strong wind begins to blow in your face, causing you to feel an extra pressure that threatens to push you over. This increased pressure is known as dynamic pressure (also known as stagnation pressure.) When a fluid flow encounters an obstruction (like your body), it is forced to slow down. In doing so, it exchanges its kinetic energy for pressure energy. Hence, its pressure rises as its velocity falls. This is a consequence of the conservation of energy, which is described for fluids (under certain conditions¹) by Bernoulli’s Principle.

For example, standing in a 100 km/h gale (about 60 mph) on Earth would subject you to a dynamic pressure of 473 Pascals. This is a force equivalent to 48 kilograms per square metre, enough to uproot trees and damage buildings.

However, this same object in a 100 km/h wind on Mars would only experience a dynamic pressure of 7.7 Pascals. This is equivalent to about 0.8 kilograms per square metre – a light breeze.

As stated earlier, in the dramatic beginning of The Martian, a dust storm hits. It is so powerful that it threatens to tip the Mars Ascent Vehicle, forcing an emergency launch, stranding astronaut Mark Watney on Mars and kick-starting the plot.

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However, this would be unlikely in real life. The Martian atmosphere has such a low density that even a gust of 400 km/h would have the same dynamic pressure as a 50 km/h wind on Earth – merely a stiff breeze, rather than a gale. This means that even the most powerful windstorm Mars can muster would apply the dynamic pressure of a modest Earth gust.

(Besides, if the dynamic pressure of the windstorm is powerful enough to tip the vehicle, is it really safe to launch?)

This is because dynamic pressure depends on two things: the speed of the fluid and its density. Slow-moving water can knock you over easily, because it is eight hundred times denser than air. On the other hand, the International Space Station orbits in the extremely thin upper atmosphere at 28,000 kilometres per hour, yet experiences almost no air resistance. These effects are captured in the dynamic pressure equation:

This extra, unbalanced pressure pushing against a flow obstruction is the main reason blunt objects experience air resistance – objects like people, trees, and space capsules. For these shapes, friction only contributes a small part to air resistance. The faster a rocket flies through an atmosphere, the greater the dynamic pressure on its nose becomes.

What does this have to do with ‘max q’?

(As an aside, this compression of the air in the vehicle’s path heats it up, which is why high-speed aircraft experience aerodynamic heating. Contrary to popular belief, air compression is the chief reason that spacecraft experience high temperatures during reentry. Friction only contributes a small amount of heat.)

Let’s take an example from the Space Shuttle’s launch profile.

At the moment of lift off, the shuttle is in the densest part of the Earth’s atmosphere. However, it is moving slowly, and so experiences low dynamic pressure. As it climbs, the density of the air falls, but its speed continues to increase. When the shuttle reaches an altitude of about 11 km (36,000 ft) and a speed of 1,600 km/h, dynamic pressure reaches its peak of 3.3 tonnes per square metre. It is at this moment, a very specific intersection of airspeed and atmospheric density, that we find our answer. This is ‘max q’.

Max q is one of the most dangerous times in a rocket’s ascent, because that is when aerodynamic stresses on its structure are the greatest. Around this time, the Space Shuttle’s autopilot would throttle down the main engines to about 60-70% power. This keeps its speed from increasing too much, reducing the maximum dynamic pressure.

Once the vehicle reaches a higher altitude (where the atmosphere is thinner), it becomes safe to fly faster. Those who have seen shuttle launches may then recall Launch Control saying, “Go for throttle-up.” This meant that after safely passing through max q, the shuttle’s engines would be throttled back up to full power.

The booster rockets are jettisoned at an altitude of about 45 km (146,000 ft) and a speed of 4,600 km/h. By this point, the air is so thin that the dynamic pressure is only about one percent of the Earth’s atmospheric pressure.

The concept of dynamic pressure is also important for living on Mars. Martian dust storms can have wind speeds of up to 100 km/h and gusts of up to 400 km/h, but with such a thin atmosphere, the greatest threats they pose are dust infiltration into moving parts and reduced Solar power.

This is also important for wind turbine design. Because the atmosphere carries so little momentum, wide blades are required to generate significant amounts of power from the Martian atmosphere. For details, read Generating Energy on Mars: ISRU Part 3.

The phenomenon of dynamic pressure is an important one, and a consequence of how momentum and energy are transported by a fluid flow. So the next time you walk outside and feel the wind in your face, think of the Space Shuttle drilling through the atmosphere on its way to space. And a few years from now, when the first Starship lifts off from the red planet and punches through the Martian sky, keep your ears open for mission control calling out, “Max q!”

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Footnotes and further reading

¹ Bernoulli’s principle only applies for isentropic flows, or flows where ‘irreversible’ effects are negligible. This means that it ignores the effects of friction and of energy exchange with the environment. While a thorough explanation of fluid entropy is too complex to fit in a footnote, one can say that these ‘irreversible’ effects affect the way in which the fluid’s energy is dissipated into the environment. For modelling the energy balance of a single streamline in a brisk airflow, Bernoulli’s principle can be helpful.

For a more in-depth yet easy-to-understand introduction to fluid dynamics, the Khan Academy fluids unit is strongly recommended.