Technicalities: Watney, Smeaton and Rho

As you probably know, the movie The Martian concerns an astronaut who gets stranded on Mars and contrives to survive there until rescuers can come to get him. It has a kind of Potemkin village verisimilitude: a facade of scientific-sounding talk concealing ramshackle structures of crossed fingers and nonsense. Screenwriters are not under oath, but unfortunately a lot of kids probably get their ideas about physics from pseudoscientific movies like this one.

The setup for the plot is a huge and violent dust storm that blows the hero, Mark Watney, off his feet, impales him with a piece of flying metal, and obliges the other crew members to blast off, leaving Watney for dead, to prevent their tall rocket from being knocked over by the mighty wind.

Now, there’s a problem right away. The strongest wind recorded by Mars rovers is reported online — I admit to the unreliability of the source — to be 60 mph. Not exactly a hurricane, but wait, there’s more. The Martian atmosphere is only 1/100 as dense as ours. Because wind pressure is a function of the square of velocity, a 60 mph gale on Mars feels like a 6 mph puff on Earth. Some sources suggest Martian winds may reach 200 mph, but they would still feel like no more than a stiff Earth breeze.

The pressures really available from the Martian atmosphere need not concern moviemakers, for whom the important thing is that something look powerful, not that it be possible. And real Martian dust storms do ­indeed look powerful, sometimes enveloping most of the planet in a shroud of dust so fine — more like smoke than dust — that it takes weeks to settle back to the surface.

Even though we know that it is a vast, vacant lot full of orange rocks, we seem to be hellbent on going to Mars. One of the schemes for exploring more of the Martian surface before the first deluded colonists get there is to use drones. They would have to be very light and have a large wingspan to fly in the wispy Martian atmosphere, which is the equivalent of Earth’s somewhere above 100,000 feet, but it helps that things on Mars weigh only three-eighths of what they do on Earth. On Mars, a drone with a wing area of 10 square feet and an indicated airspeed of 20 knots could weigh 10 Earth pounds or more — easy work for the structure’s team. More of a challenge would be achieving an indicated speed of 20 knots in the first place; the true airspeed would have to be around 200 knots. The problem is that at a given indicated airspeed, the power required is proportional to the true airspeed; that’s why it takes more fuel to indicate 150 knots at 10,000 feet than at 5,000. Air particles are hitting the airplane with the same force, but in a given period of time more of them hit it.

Another awkward thing about Mars is that internal-combustion engines won’t work there. There is almost no oxygen in the atmosphere, which contains so much carbon dioxide that it’s toxic to both plants and power plants.

An airplane named Helios, built for NASA by the ingenious California firm AeroVironment, actually flew at almost 97,000 feet on Earth — the highest altitude at which anything has sustained itself here on wings for any length of time. Powered by 14 electric motors and thousands of solar cells, Helios was a flying wing that spanned 250 feet with a wing loading of less than 1 pound per square foot. Its indicated airspeed in cruise was around 17 knots; its true airspeed was 10 times that.

“Indicated airspeed” is an airman’s term for dynamic pressure: how hard moving gas hits you. It is of fundamental importance to aerodynamics. It probably first concerned the builders of windmills, who would have relied on practical experience until the middle of the 18th century, when a British engineer named John ­Smeaton attempted to ascertain a baseline value. Working in the English system, he determined the pressure exerted upon a flat plate of 1 square foot by a wind of 1 mile per hour to be 1/200 of a pound. This was off by a factor of 2, but still, it was a start. In the absence of any alternative, that value, 0.005, was generally ­accepted and became known as ­“Smeaton’s coefficient.”

One source of error for Smeaton, and later for the Wrights (who did their own investigation and came much closer to the right answer), was that the pressure of wind on a surface of 1 square foot is greater than the true dynamic pressure of the wind because additional air is displaced by spillage around the ­edges. If the true dynamic pressure of a wind is, say, 10 pounds per square foot, a flat plate feels almost 13 pounds of pressure.

What Smeaton measured was really the density of air — though he did not call it that — which we now abbreviate by the Greek letter rho, which sounds like “R” and looks like ρ. Rho is now known to be 0.002379 slugs per cubic foot at sea level on a standard day.

What? Slugs? When did we start using garden pests as units of measure?

Slugs are more beloved of physicists than of greengrocers, though even physicists shun them in favor of metric units. (Go Lincoln Chafee!) A slug is 32.2 pounds mass, so the weight of sea-level air is 32.2 ­multiplied by 0.002379, or about 0.08 pounds per cubic foot or 1 pound per 12.5 cubic feet. The air inside your refrigerator weighs a pound or so.

A convenient way to remember the dynamic pressure of moving air in the range of speeds relevant to airplanes is to recall that at 100 mph at sea level the dynamic pressure is roughly 25 pounds per square foot. At 50 mph it’s a quarter of that, around 6 pounds per square foot.

The magical thing about wings is that they are able to produce a lifting force that is greater than the dynamic pressure of air. For example, a simple unflapped wing moving at 50 mph can support not just 6 but 8 or 9 pounds per square foot. The ratio between the lift force that is actually produced and the dynamic pressure available is called the “lift coefficient.” The maximum lift coefficient, just before the wing stalls, is typically 1.5 or so for simple plain wings and can go above 3 for wings with good flaps.

By the same token, the ratio of the drag that is actually produced by an object and the drag that is expected based on its frontal area is called the object’s “drag coefficient.” The drag of a streamlined spindle, for instance, might be 1/20 of that of an idealized — meaning free of those edge effects — flat plate of equal frontal area; its drag coefficient is said to be 0.05. For wings, because their drag is principally due to surface friction rather than displacement and wake turbulence, the drag coefficient is referred to the wing area rather than the frontal area. For example, suppose a wing of 160 square feet has a drag coefficient of 0.008 at 100 mph. Its drag — not including the “induced” component due to producing lift — would be about 0.008 multiplied by 25 multiplied by 160, or 32 pounds.

What these numbers mean is that manned flight is pretty convenient on this Goldilocks planet of ours. It’s not so on Mars, where thin air offers little support and fainter sunlight makes even charging solar batteries a slower process than on Earth.

If you want to fly, stay here.