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Alan Nathan is Professor Emeritus of Physics at the University of Illinois at Urbana-Champaign. His principal area of research is the physics of baseball. He maintains a web site devoted to this topic at go.illinois.edu/physicsofbaseball. His younger colleagues at Complete Game Consulting have bestowed upon him the exalted title of Chief Scientist.
Some of you may already know me from the blogosphere, primarily at Tom Tango’s Inside The Book site and as a serious researcher in the physics of the game we all love. I am not primarily a baseball analyst. However, I am of the firm belief that a physicist has potentially useful things to contribute to baseball analysis, and I am appreciative of the opportunity to demonstrate so in this article.
Coors Field in mile-high Denver has been long viewed as a batter’s paradise and a pitcher’s nightmare. Because the air density in Denver is approximately 80 percent of that at sea level, fly balls hit there carry farther and pitches thrown there have less movement, both of which contribute to an increase in a variety of offensive statistics, particularly home runs. For the first seven seasons at Coors, there were 3.20 home runs hit per gamecompared to 1.93 per Rockies away game. However, beginning in 2002 the Colorado Rockies began to store their baseballs in a humidor at a constant 50 percent relative humidity and 700F, as opposed to the more typical 30 percent humidity in Denver. During the period from 2002-2010 the Coors ratio decreased to 2.39, a reduction of 25 percent, while the away game ratio stayed constant at 1.86. Is it plausible that the reduction in home runs can be attributed to the humidor? The primary goal of this article is to answer that question.
Over the past few years, many articles have been written addressing this issue, so it is a fair question to ask why this topic might still be of interest. Here are three reasons. First, there are new data that I will discuss on the effect of the humidor on the properties of the baseball. Second, there are data available today on the trajectories of fly balls that allow a more definitive study than has been heretofore possible. And third, there have been media reports that the Arizona Diamondbacks are considering storing their baseballs in a humidor, so it is important that those involved in making that decision have the best information available to them on the possible consequences. So as a secondary goal, I will address the issue of a humidor at Chase Field.
Before going through the analysis, let me first give an outline of my approach. I will first assume that in the absence of a humidor, the baseballs at Coors are stored at the ambient relative humidity of 30 percent, as opposed to the 50 percent value when stored in the humidor. The approach involves two distinct issues that need to be investigated:
- How does elevated humidity change the properties of the baseball, resulting in a reduced batted-ball speed? To answer this question requires both careful laboratory measurements and a physics analysis of the ball-bat collision.
- Given the reduced batted-ball speed, by how much is the distance on a long fly ball reduced and how does that translate into fewer home runs? My approach is to utilize actual fly-ball data to answer these questions.
First I look at the reduction in batted-ball speed (BBS), which gets us into a brief discussion of the physics of the ball-bat collision. Last year I wrote an article about this topic for Baseball Analysts, with an emphasis on how properties of the bat affect the BBS. The interested reader should go back to this article, since the same essential physics applies when considering the effect of the ball on the BBS. When baseballs are stored at an elevated humidity, two effects occur that decrease the BBS: the weight increases and the coefficient of restitution (COR) decreases. The increase in weight is due to the absorption of water by the ball. To see how that affects the BBS, realize that what really matters in the ball-bat collision is the ratio of ball weight to bat weight. The smaller the ratio, the harder the ball will be hit. Just as for a given bat speed a heavier bat will hit the ball harder than a lighter bat, so too will a given bat hit a lighter ball harder than a heavier ball.
The COR is the “bounciness” of the ball. It would be 1 for a perfectly elastic superball and 0 for a totally dead ball. For a baseball, if falls right in the middle of the two extremes at approximately 0.5. When the ball absorbs water, it gets “mushier,” resulting in a reduced COR and a lower BBS. The effect of increased weight and decreased COR on the BBS are pretty easy to understand qualitatively. Physics allows us to determine these effects quantitatively, at least within the limits that I will discuss later. A third effect occurs that does not affect the BBS but does affect the flight of the ball. Namely, at elevated humidity the size of the ball increases, and this will increase the air drag. However, the resulting effect on the trajectory has been shown to be small and I will not consider it further.
So, how do we determine the dependence of weight and COR on humidity? Well, we do what physicists have been doing for centuries: careful laboratory experiments. Two such experiments have been done in recent years, one of which was published and the other of which is about to be published in the American Journal of Physics. The experiments found that when the relative humidity is increased from 30 percent to 50 percent, the weight of the ball increases by 1.6 percent and the COR decreases by 3.7 percent. Together, these results can be used to calculate that for a hard-hit ball typical of home runs in Major League Baseball, the BBS is reduced by 2.8±0.5 mph (or about 2.8 percent), where the error bar is due to the uncertainty in the COR measurement and its affect on the BBS. More on this a little later. Most of the reduction comes from the reduced COR (2.2 mph), with the remainder coming from the increase in weight (0.6 mph).
Next I investigate how the reduced BBS affects fly-ball distance and home-run production. My approach to quantifying this effect is to rely as much as possible on actual home-run data from the 2009 and 2010 seasons, which come from two sources. First we have the precise landing location and hang time, as well as the location and height of the nearest fence, from Greg Rybarczyk, proprietor of HitTracker. Second we have the BBS, vertical launch angle, and spray angle from Sportvision’s HITf/x data. Of the 359 home runs hit at Coors in 2009-10, there are 336 for which I am able to match up the HitTracker and HITf/x data. These data and the much larger sample of 8801 home runs from all parks potentially provide a wealth of information about fly balls in MLB. As an example, I had earlier used such data to compare the “carry” of a fly ball in different parks.
Armed with the initial velocity, an aerodynamics model can be fine-tuned to reproduce the landing point and flight time, with the result being that the entire trajectory can be reconstructed to a high level of accuracy. In particular, we can find the height of the ball as it crosses the nearest fence and the total distance that the fly ball would have carried when extrapolated to ground level. Then we can use that same aerodynamics model with a reduced BBS and increased ball weight (the latter since the effect of air drag is reduced on a heavier ball) to recalculate the trajectory and see if the ball makes it over the nearest fence. In that manner, we can find the reduction in home runs as a result of the humidor.
The mean home-run distance at Coors is 414.8 ft. After applying our 2.8 percent reduction in BBS and 1.6 percent increase in weight, the mean distance is reduced to 401.6 ft, a change of 13.2 ft, or 3.2 percent. With those changes, the number of home runs in Coors is reduced from 336 to 235, a (30±6) percent reduction. Most of it (27 percent) comes from the reduced COR, with the remainder (3 percent) coming from the increased weight. Another way to express this result is that for each 1 ft reduction in the fly-ball distance, the home-run probability is reduced by 2.3 percent.
Let me digress a bit to talk about the two major sources of uncertainty in the result. First, the part of the calculation that associates a reduced BBS with a reduced COR requires knowledge of additional parameters of the ball-bat collision that we do not know with certainty, such as the swing speed, the impact location, etc. In my calculation, I used values that I would expect to apply to the kinds of well-hit balls that lead to home runs. The ±0.5 mph in the BBS reduction arises in part from my estimation of the uncertainty in the collision dynamics and in part from the actual COR measurements. Second, the 336 home runs that were studied are subject to the usual random fluctuations associated with a finite sample size. Together, these two sources are combined to arrive at the uncertainty of ±6 in the home-run reduction percentage.
In summary, the calculation gives a reduction of (30±6) percent, whereas the actual reduction is 25 percent. The closeness of the calculation to the actual result suggests that it is very plausible that the reduction in home runs at Coors Field can be attributed to the humidor. That is the primary conclusion of this investigation.
Let me turn now to Chase Field, where the typical relative humidity is even lower than in Denver, on the order of 20 percent. Therefore, we need to investigate the effect of changing from 20 percent to 50 percent, an increase that is 1.5 times the increase at Coors. Here I will present only a rough analysis, which ought to provide a ballpark estimate (so to speak) of the reduction we might expect at Chase. The experimental data show that the dependence of both weight and COR on relative humidity is approximately linear. The analysis of the home-run data shows that the dependence of fly-ball distance on BBS is also linear. Finally, the 2.3 percent-per-foot reduction for Coors is essentially the same as for Chase. Putting these things together, I would expect to find a reduction at Chase approximately 1.5 times that at Coors, or a whopping (45±9) percent! That’s not a small number, folks.
The technique used in the present analysis allows us to investigate only factors that decrease home run production, since we have accurate fly-ball data only for home runs and not for near misses. Nevertheless, one can speculate that for small changes in BBS the reduction might also apply in the reverse direction, allowing a prediction of increases in home-run production. For example, given that the typical humidity in most MLB parks is greater than 50 percent, storing the baseballs at 50 percent in every park would result in an increase in home-run production by an amount that we could predict using the techniques employed here. As another example, the consequence of moving the fences uniformly closer by 5 ft would be an increase in home-run production by 13 percent. Other examples abound, but I’ll leave those for another day.
In conclusion, I hope I have been able to convince you that applying physics techniques to baseball analysis can lead to useful insights about the game.
I’d like to take this opportunity to thank Greg Rybarczyk and Ryan Zander for providing the HitTracker and HITf/x data, respectively. I also thank both Greg and Mike Fast for some helpful suggestions on this article. Finally, I thank my colleague Prof. Lloyd Smith for his careful measurements of the effects of humidity on the COR and weight.