Tobin, a specialist in condensed matter physics with a long-time interest in the physics of baseball, will publish his results in an upcoming issue of the American Journal of Physics. As Tobin's paper notes, between 1998 and 2006, players hit more than 60 home runs in a season six times. The explosion in home runs coincides with the dawn of the "steroid era" in sports, and that surge quickly dropped to normal levels in 2003, when Major League Baseball instituted steroid testing.

Many have wondered why home-run hitting would be particularly vulnerable to performance enhancement, or whether it is even physically and physiologically plausible that steroids could produce effects of the magnitude observed. Tobin’s results point to 'yes' for both questions.

"A change of only a few percent in the average speed of the batted ball, which can reasonably be expected from steroid use, is enough to increase home run production by at least 50%," he says. This disproportionate effect arises because home runs are relatively rare events on the tail of the statistical range distribution of batted balls.

Explains Tobin, "It's a well-known statistical property of such distributions that a relatively small shift in the center point can produce a much larger proportional change in the number of values well above or below the center. Because the distribution's tail is particularly sensitive to small changes in the peak and/or width, home run records can be more strongly affected by steroid use than other athletic accomplishments."

Previous studies of steroid use led Tobin to conclude that steroids could boost muscle mass, the force exerted by those muscles, and the kinetic energy of the bat by about 10%. His calculations show the speed of the bat as it strikes the pitched ball will be about 5% higher than without the use of steroids. The speed of the ball as it leaves the bat will be about 4% higher.

The ultimate impact on home run production comes from an analysis of models for baseball trajectory, accounting for gravity, air resistance and lift-force caused by ball spin. While there was considerable variation among the models, "the salient point," Tobin says, "is that a 4% increase in ball speed, which can reasonably be expected from steroid use, can increase home run production by anywhere from 50 to 100%."

Tobin applied a similar, though less extensive, mechanical analysis to pitching. He calculated that a 10% increase in muscle mass should raise the speed of a thrown ball by about 5%, or four to five miles-per-hour for a pitcher with a 90 mph fastball. That translates to a reduction in earned run average of about 0.5 runs/game.

"That is enough to have a meaningful effect on the success of a pitcher, but it is not nearly as dramatic as the effects on home run production," says Tobin. "The unusual sensitivity of home run production to bat speed results in much more dramatic effects, and focuses attention disproportionately on the hitters." Tobin points out that many other changes, including adjustments in ballpark dimensions and lowering of the pitcher’s mound, could affect major league batting — although he says none of those changes coincide with the sudden burst of home runs in the mid-1990s.

"Physics cannot tell us whether a particular home run was steroid-assisted, or even whether an extraordinary individual performance indicates the use of illicit means," says Tobin. But analysis of the physics, combined with physiology, yields telling results. "These results certainly do not prove that recent performances are tainted, but they suggest that some suspicion is reasonable," he concludes.

In his study, Tobin used two kinds of math models (by Adair and Sawicki) for a rapidly spinning baseball’s drag coefficient as a function of speed. This gave the data shown here depicting how home run rate depends on the fractional change in the average speed of a batted ball for various ball speeds and launch angles from the bat.