Curveballs in space go backwards

Colin McNally

In a near-vacuum, the aerodynamic effect which makes a spinning ball's trajectory curve reverses sign and the trajectory curves the opposite direction from where it does in a thick atmosphere. In space, curveballs, "bent" soccer/football kicks, and cricket spin-bowls, all go backwards. At OrchardNumerical, working out which situation a problem is really in, and so which rules apply, is the first thing we do.

The reason for the behaviour in the rarefied near-vacuum of space is simple, or at least simpler than the more down-to-earth fluid case. At high Knudsen number ($\mathrm{Kn} \gg 1$, gas particle mean free path much larger than the object size) the interaction of the ball with the gas molecules reduces to particle collisions with the surface. When particles impact the surface, their collisions are not perfectly elastic. They exchange energy and partially equilibrate their momentum and energy with the ball's surface (this is called accommodation in the jargon of the field). In essence one can think of this as the gas particles slightly sticking to the ball's surface before rebounding, so that they rebound, but they thermally equilibrate a little, and get a bit of a sideways kick from the motion of the ball's surface. The forces on the ball results from the sum of exchange of momentum of gas particles with the surface of the ball over all these collisions. Because of the gas particles getting a sideways nudge from the rotation of the ball, the surface of the ball gets a drag (the reaction to the force exerted on the gas particles). If you were to consider a ball sitting still in such a rarefied gas, this would just slow the spin of the ball down (Epstein 1924). However, if we consider a ball moving through the rarefied gas, then the collisions with gas particles are more common on the leading side. This introduces an asymmetry to the situation, and there is not just a torque slowing the ball down, but a net force on the ball as a reaction to its kicking incoming gas particles off to one side. This is particularly clear if we consider the extreme limit of a ball moving very fast compared to the (thermal) velocity of the gas particles (hypersonic), where the flow reduces to streams of gas particles hitting only the leading side of the ball and being kicked off to one side by the spin as they bounce off, as drawn in Figure 1 (Borg et al. 2003).

From the directness of this explanation, it's surprising that the description of the slowing down of rotation first appears in Epstein (1924), but the inverse Magnus effect does not seem to appear until Wang (1972). Perhaps in the intervening years many people came across it but thought it too trivial to report.

If the same spinning ball were flying through a dense gas like ground-level air, the conventional fluid Magnus effect would act in the opposite direction. The mechanism in a fluid is quite different: The flow of gas past the ball is dragged by friction with the surface, leading to it being asymmetrical, with gas making a fast long trip around one side of the ball, and a slow short trip around the other — that familiar situation with an airfoil. There's a couple ways of intuitively describing the origin of the lift due to the Magnus effect here. One is that difference in the flow speed induces a pressure difference (Bernoulli's principle) with high pressure on the side of the ball with slow moving air, and low pressure on the side with fast moving air, and then this pressure difference acts on the ball to cause the lift force. Another way is to observe that the flow coming off the ball is directed off to one side, and intuit the lift on the ball as the reaction to the force being exerted to redirect this wake.

Of course, even a "dense" part of space is so close to a vacuum that there's not much gas to bend the trajectory. If a typical baseball curveball pitch (speed $36$ m/s or $80$ mph in baseball units, spin 2,500 RPM) was thrown in a molecular cloud (dense part of space, star forming region) where representative conditions are a gas density of $3.3\times 10^{-15}$ kg/m$^3$ ($20$ molecules per cm$^3$) the radius of the curved trajectory due to the Magnus effect force would be about $4\times10^{5}$ AU (an AU is roughly the distance from earth to the sun), which is in the ballpark of the distances to the nearest stars to the sun. This force would be equivalent to the gravitational pull of an earth mass at a distance of $880$ AU, which is about the distance to the Oort cloud, the source of comets.

Wang (1972), as a graduate student, seems to be the person who introduced the characterization of the lift force on a spinning ball in free molecular flow as an inverse Magnus effect, but the underlying process and most of the analysis had been done before. It seems that there are up to three independent rediscoveries, first in the Sputnik and early satellite era studies of the forces on bodies in the upper atmosphere appearing in the Russian literature Beletskii (1963; Gaikin 1965b) (as pointed out by Ivanov and Yanshin (1980))¹, then with Wang's work in 1972 in the English language aeronautical world, and then it managed to pop up in the fluid dynamics literature as a rediscovery in Borg et al. (2003). By the third time it was rediscovered and published, it sparked some back and forth about the priority of the claim (Dubinsky and Elperin 2004; Borg et al. 2004).

A few equations, all approximations. The Magnus effect is a force perpendicular to the velocity of the ball $F_\perp$, which can be expressed as:

with $\rho$ the gas density, $v$ the ball speed, $R$ the ball radius. Now, on earth, most interesting sport balls are at a reasonably high Reynolds number such that the flow is quite turbulent and difficult to summarize cleanly (for a baseball: Briggs 1959). But the canonical understanding of the fluid Magnus effect comes from low Reynolds, so let's allow ourselves to consider that simpler regime. The appropriate low Reynolds number (laminar flow) expression for $C_\mathrm{L}$ is (Rubinow and Keller 1961):

where $\omega$ is the speed of the wall's rotation (in rad/s). In the free molecular flow regime this is replaced by $C_\mathrm{L,FM}$ as (Wang 1972):

where $\sigma$ is what is termed the tangential momentum accommodation coefficient, describing how particles hitting the surface reflect off in a statistical sense varying from perfect specular reflection without picking up any momentum from the surface motion ($\sigma=0$) to diffusely with momentum dominated by the motion of the surface ($\sigma=1$) and we are sweeping some details of the flow under the carpet. Looking at the relation between the two regimes we simply have

Which, remarkably, says that despite the microscopic physics being entirely different, the result is about the same, just with a minus sign.

Another surprising regime change I found: very small rocks can levitate in a hot, dusty gas (optically thick photophoresis). We love problems like these: odd ones, where the rules change somewhere and there's no existing solution to reach for. If you have one, get in touch.