Aerodynamic Forces

It seems strange to be talking about aerodynamic effects on spacecraft, because space is a vacuum. Right? Well, as far as people are concerned, as living and breathing creatures, space is effectively a vacuum; if you stepped out of a space station in orbit and took off your helmet, then the consequences would reinforce this notion. However, for an orbiting spacecraft the effects of air drag are encountered at heights up to around 1000 km. At these altitudes, there is an atmosphere, but it is extremely thin. To describe how thin it is, let's think about the atmospheric density at an altitude of, say, 800 km (500 miles) at mean solar activity (for a discussion on the effects of solar activity on atmospheric density, see Chapter 6). In every cubic meter of volume at this height there is a mass of air of around 0.000 000 000 000 01 kg, which explains why you can't breath it! Compare this with the density of air at sea level, which is around 1.2 kg/m3.

The next question is, How can such a thin atmosphere produce aerodynamic effects on spacecraft that perturb their orbital motion (see also the discussion about drag forces on launchers in Chapter 5)? The key to this is to realize that the aerodynamic force on an object is dependent not only on the air density but also on how fast the object is moving through the air. For example, we know that sufficient aerodynamic force can be exerted on a garden fence to knock it down in a winter storm, provided the wind speed is high enough. This force, known as dynamic pressure, actually depends on the square of the wind speed. If the wind speed doubles, the force on the fence increases by a factor of four (22), if it trebles the force is nine times as big (32), and so on. No wonder storm-force winds make short work of fences!

Taking the argument a little higher than a garden fence, we can think about airplanes moving about in the atmosphere. They encounter high speed winds, but this time the wind is produced by their own motion through the air. The flight of an airplane through the air is resisted by an aerodynamic drag force. This force is measured in a number of different units, but the most common is the Newton, named after Isaac Newton. A Newton of force has a formal definition: it is the force required to accelerate a 1 kg mass by 1 m/sec2. As we explained in Chapter 1, this is an increase in speed of 1 m/sec for every second that the force is applied. Another perhaps easier way to get a feeling for a Newton of force is to adopt an approximate and informal definition, which is that it is about the weight of a (smallish) apple! Returning to our airplane, and thinking about a civil airliner at cruising altitude, the level of dynamic pressure acting on it due to its motion through the air is on the order of 10,000 to 15,000 Newtons for every square meter of area presented to the air flow. A metric tonne weight is about equal to about 10,000 Newtons, so that's quite a lot of aerodynamic drag, which of course needs to be overcome by the thrust from the jet engines to keep the airplane in the air.

Finally, climbing even higher to orbital altitude, the same principle applies to spacecraft. They encounter a level of aerodynamic drag force that is much smaller than that of an airplane, but it is nevertheless tangible because, although the air density at orbit height is small, the speed of the spacecraft through the atmosphere is high. For each square meter of spacecraft area presented to the air flow, the level of aerodynamic force varies from about 1/100th of a Newton at a 200-km altitude to a tiny 0.000 000 05 of a Newton at a 1000-km altitude. These small forces are produced by the molecules of atmosphere impacting on the spacecraft. The forces seem too small to be of any consequence, but the point is that they act all the time, in a retrograde direction, that is, in a direction opposed to the motion of the spacecraft, producing a small but steady decrease in the orbit height day after day. For example, if the spacecraft resides in a 200-km-altitude circular orbit, this steady decay will lead to the spacecraft's reentering the atmosphere within a short period of time—a few days to a few weeks depending on the characteristics of the vehicle.

We can now begin to understand how aerodynamic drag perturbations change a typical satellite's orbit. The main effects are to reduce the size of the orbit and to change the orbit's shape, making it more circular. To see this, we imagine a spacecraft in an eccentric orbit, as shown in Figure 3.8a, with an initial apogee at point 1, and a perigee height low enough to allow the spacecraft to dip into Earth's thin upper atmosphere. During each perigee passage, the aerodynamic drag forces take energy out of the orbit, so that it does not quite reach the same height on each subsequent apogee (points 2,3, and so on). In the figure we see two things happening: the size of the orbit is

Figure 3.8: (a) The drag effects occur around the perigee of the orbit, mainly causing a change in apogee height. As a consequence, both the orbit size and eccentricity decrease. (b) The evolution of a moderate eccentricity orbit due to air drag, starting at the outer ellipse and ending at the inner circular orbit.

Figure 3.8: (a) The drag effects occur around the perigee of the orbit, mainly causing a change in apogee height. As a consequence, both the orbit size and eccentricity decrease. (b) The evolution of a moderate eccentricity orbit due to air drag, starting at the outer ellipse and ending at the inner circular orbit.

decreasing, and it is becoming more circular. It should be noted that the figure is not drawn to scale, and the sort of decrease in apogee height illustrated would not occur in two orbit revolutions, but in fact would require perhaps hundreds of orbits revolutions. Figure 3.8b shows a similar orbit evolution for an orbit of moderate eccentricity. Again, the major effect is a decrease in apogee height, with a smaller lowering of perigee altitude.

For a spacecraft in a circular orbit, the drag force causes the orbit height to decrease on each orbit revolution, producing a kind of spiral trajectory of diminishing altitude. By how much the orbit height changes depends on the characteristics of the spacecraft. For example, if it is small and massive like the cannonball shot from Newton's cannon in Chapter 2, then the drag effects are relatively small. On the other hand, if the spacecraft is large and of low mass, like a balloon satellite, then the drag effect on orbit height is large. A number of such balloon satellites were launched in the 1960s to bounce radio waves off in early space communications experiments, but their orbit lifetime was generally short due to their vulnerability to drag perturbations. In "ballpark" numbers, for a typical spacecraft the height change per orbit at a 800-km altitude may be on the order of a few centimeters or a few tens of centimeters, whereas at 200-km altitude the height can change by as much as a kilometer or more for each orbit revolution. Clearly, spacecraft in low-altitude circular orbits do not stay in orbit for long.

Another curious feature about drag force on a spacecraft in a low circular orbit is that, although the force acts in the direction opposed to its motion, it actually causes the spacecraft's speed to increase. This is something we do not experience often, and it seems quite counterintuitive. However, a little thought resolves the puzzle. Any force acting in a retrograde sense on a spacecraft will take energy out of the orbit and cause the orbit height to decrease. And as we saw in Chapter 2, the lower the orbit, the faster the spacecraft moves. What's happening here is that, given the decrease in height on each orbit revolution, the spacecraft is actually flying "downhill." The situation is entirely analogous to riding a bicycle down a slope. Gravity pulls it forward and tends to increase its speed, but at the same time we can feel the wind on our faces, which produces a drag force that acts against the motion, tending to slow us down. In the case of the satellite, the gravity pulling it forward is larger than the drag force slowing it down, resulting in the net increase in orbital speed as its orbit height decreases. Surprisingly, I have seen statements in professional journals along the lines of ""the aerodynamic drag force decreases the orbit velocity,'' so you can see that sometimes even the experts get it wrong!

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