Roxana Dior writes:

From this news article, I found out about this paper, “Global warming, home runs, and the future of America’s pastime,” by Christopher Callahan, Nathaniel Dominy, Jeremy DeSilva, and Justin Mankin, which suggests home runs in baseball have become more numerous in recent years due to climate change, and will be scored more frequently in the future as temperatures rise.

Apart from the obvious question—when will Moneyball obsessed general managers look at optimizing stadium air density when their team is at bat?—is this a statistically sound approach? I am no baseball aficionado, but air density changes due to temperature seems like it would have a miniscule affect on home runs scored, as I assume that the limiting factor in scoring one is the “cleanness of contact” with the bat, and that most batters hit the ball with sufficient power to clear the boundary when they do. There are probably a hundred other confounding variables to consider, such as PED usage etc. but the authors seem confident in their approach.

They end with:

More broadly, our findings are emblematic of the widespread influence anthropogenic global warming has already had on all aspects of life. Warming will continue to burden the poorest and most vulnerable among us, altering the risks of wildfires, heat waves, droughts, and tropical cyclones (IPCC, 2022). Our results point to the reality that even the elite billion-dollar sports industry is vulnerable to unexpected impacts.

I think I agree with the sentiment, but this feels like a bit of a reach, no?

From the abstract to the paper, which recently appeared in the Bulletin of the American Meteorological Society:

Home runs in baseball—fair balls hit out of the field of play—have risen since 1980, driving strategic shifts in gameplay. Myriad factors likely account for these trends, with some speculating that global warming has contributed via a reduction in ballpark air density. Here we use observations from 100,000 Major League Baseball games and 220,000 individual batted balls to show that higher temperatures substantially increase home runs. We isolate human-caused warming with climate models, finding that >500 home runs since 2010 are attributable to historical warming. . . .

My first thought on all this is . . . I’m not sure! As Dior writes, a change of 1 degree won’t do much—it should lower the air density by a factor 1/300, which isn’t much. The article claims that a 1 degree C rise in temperature is associated with a 2% rise in the number of home runs. On the other hand, it doesn’t take much to turn a long fly ball into a homer, so maybe a 1/300 decrease in air density is enough to do it.

OK, let’s think about this one. The ball travels a lot farther in Denver, where the air is thinner. A quick Google tells us that the air pressure in Denver is 15% lower than at sea level.

So, if it’s just air pressure, the effect of 1 degree heating would be about 1/50 of the effect of going from sea level to Denver. And what would that be? A quick Google turns up this page by physicist Alan Nathan from 2007, which informs us that:

There is a net force on the ball that is exactly opposite to its direction of motion. This force is call the drag force, although it is also commonly referred to as “air resistance”. The drag plays an extremely important role in the flight of a fly ball. For example, a fly ball that carries 400 ft would carry about 700 ft if there were no drag. The drag plays a less significant — but still important — role in the flight of a pitched baseball. Roughly speaking, a baseball loses about 10% of its speed during the flight between pitcher and catcher, so that a baseball that leaves the pitcher’s hand at 95 mph will cross the plate at about 86 mph. If the baseball is also spinning, it experiences the Magnus force, which is responsible for the curve or “break” of the baseball. . . .

Both the drag and Magnus forces . . . are proportional to the density of the air. . . . the air density in Denver (5280 ft) is about 82% of that at sea level. . . . the drag and Magnus forces in Coors will be about 82% of their values at Fenway.

What about the effect of altitude? Here’s Nathan again:

The reduced drag and Magnus forces at Coors will have opposite effects fly balls on a typical home run trajectory. The principal effect is the reduced drag, which results in longer fly balls. A secondary effect is the reduced Magnus force. Remember that the upward Magnus force on a ball hit with backspin keeps it in the air longer so that it travels farther. Reducing the Magnus force therefore reduces the distance. However, when all is said and done, the reduced drag wins out over the reduced Magnus force, so that fly balls typically travel about 5% farther at Coors than at Fenway, all other things equal. . . . Therefore a 380 ft drive at Fenway will travel nearly 400 ft at Coors. . . .

Also, Nathan says that when the ball is hotter and the air is dryer, the ball is bouncier and comes faster off the bat.

The next question is how will this affect the home run total. Ignoring the bouncy-ball thing, we’d want to know how many fly balls are close enough to being a home run that an extra 20 feet would take them over the fence.

I’m guessing the answer to this question is . . . a lot! As a baseball fan, I’ve seen lots of deep fly balls.

And, indeed, at this linked post, Nathan reports the result an analysis of fly balls and concludes:

For each 1 ft reduction in the fly-ball distance, the home-run probability is reduced by 2.3 percent.

So making the air thinner so that the ball goes 20 feet farther should increase the home run rate by about 46%. Or, to go back to the global-warming thing, 1/50th of this effect should increase the home run rate by about 1%. This is not quite the 2% that was claimed in the recent paper that got all this publicity, but (a) 2% isn’t far from 1%, indeed given that 1% is the result from a simple physics-based analysis, 2% is not an unreasonable or ridiculous empirical claim; (b) the 1% just came from the reduced air pressure, not accounting for a faster speed off the bat; (c) the 1% was a quick calculation, not directly set up to answer the question at hand.

And . . . going to Nathan’s site, I see he has an updated article on the effect of temperature on home run production, responding to the new paper by Callahan et al. He writes that in 2017 he estimated that a 1 degree C increase in temperature “results in 1.8% more home runs.” Nathan’s 2017 paper did this sort of thing:

I don’t like the double y-axis, but my real point here is just that he was using actual trajectory data to get a sense of how many balls were in the window of being possibly affected by a small rise in distance traveled.

Callahan et al. don’t actually refer to Nathan’s 2017 paper or the corresponding 1.8% estimate, which is too bad because that would’ve made their paper much stronger! Callahan et al. run some regressions, which is fine, but I find the analysis based on physics and ball trajectories much more convincing. And I find the combination of analyses even more convincing. Unfortunately, Callahan et al. didn’t do as much Googling as they should’ve, so they didn’t have access to that earlier analysis! In his new article, Nathan does further analysis and continues to estimate that a 1 degree C increase in temperature results in 1.8% more home runs.

So, perhaps surprisingly, our correspondent’s intuition was wrong: a small change in air density really can have noticeable effect here. In another way, though, she’s kinda right, in that affects of warming are only a small part of what is happening in baseball.

Relevance to global warming

The home runs example is kinda goofy, but, believe it or not, I do think this example is relevant to more general concerns about global warming. Not because I care about the sanctity of baseball—if you got too many home runs, just un-juice the ball, or reduce the length of the game to 8 innings, or make them swing 50-ounce bats, or whatever—but because it illustrates how a small average change can make a big change on the margin. In this case, it’s all those balls that are close to the fence but don’t quite make it over. The ball going 5% farther corresponds to a lot more than a 5% increase of home runs.

Elasticities are typically between 0 and 1, so it’s interesting to see this example where the elasticity is much greater than 1. In the baseball example, I guess that one reason there are so many fly balls that are within 20 feet of being home runs, is that batters are trying so hard to hit it over the fence, and often they come close when they don’t succeed. The analogy to environmental problems is that much of agriculture and planning is on the edge in some way—using all the resources currently available, building right up to the coast, etc.—so that even small changes in the climate can have big effects.

I’m not saying the baseball analysis proves any of this, just that it’s a good example of the general point, an example we can all understand by thinking about those batted balls (a point that is somewhat lost in the statistical analysis in the above-linked paper).

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