Spring-training games in the Cactus League are a unique joy, especially for baseball fans (like me) who hail from colder climes. Unlike the Grapefruit League, which features stadiums separated by hundreds of miles of humid Florida air, the Cactus League consists of a compact cluster of stadiums bathed in sunshine and desert-dry air. Spectators and players alike can enjoy the spring conditions (and for some, including myself and Carson Cistulli, Barrio Queen guacamole and sangria) in the Valley of the Sun for weeks before teams return to their home stadiums across the country in late March.

Figure 0: Your author enjoying the 82-degree sunshine (and probably a juicy IPA, not pictured) at Hohokam Stadium, March 2017

Some teams will return to relatively warm and dry climates (Arizona Diamondbacks, who have to trudge the 20 freeway miles to Chase Park), but others will return to retractable domes (Seattle Mariners) or cold conditions where snowed-out games are certainly not out of the question (Cleveland). Given that the point of spring training is to get players ready for 81 games at their home ballpark, are two months of baseball in dry, sunny paradise the best way to prepare players for opening day at home? Short of building exact climate-controlled replicas of Kauffman Stadium and Wrigley Field in the Phoenix Metro, how could teams better prepare their players for the start of the season at their own home ballpark? Enter an unlikely hero, the great “Rocky Mountain equalizer”: the humidor.

Figure 1: Climatology of Phoenix, AZ (Feb-Mar) and the home locations (ICAO Airport codes) of the 15 Cactus League teams (Apr-May)

Just by eyeballing the graphs in Figure 1, without wading into the different lines and the specific airports (some lines switch to larger airports with RH), no stadium’s meteorological conditions are close to those in the Phoenix area. With the exception of the Rangers, no team plays in a stadium with an average May high temperature greater than the average March high temperature in Arizona. And only the “high desert” of Colorado comes close in RH to the dry air in Arizona March. Clearly, the opening day meteorological conditions will be significantly different from those Cactus League players see during spring training (Figure 2).

Figure 2: Changes in climate between April (major airport nearest home stadium) and March (PHX), with larger markers indicating larger temperature differences (dotted markers indicate increased T) and blue markers indicating more humid conditions (orange being drier)

This drastic change in temperature and humidity (Figure 2) is likely to have a major impact on how the ball plays once teams leave Arizona. Like many baseball physics researchers before me, I will once again heavily rely on the work previously done by Dr. Alan Nathan to inform my physical exploration herein. As shown in Nathan, et al. (2011), the two crucial meteorological factors of temperature (T) and relative humidity (RH) have a strong impact on both aerodynamic factors (such as drag) AND contact factors (such as coefficient of restitution, COR) that determine how far a batted ball travels. Rather than run afoul of the copyright of the American Journal of Physics by reproducing the figures here, I highly encourage you to check out Figures 2-4 in Nathan, et al. (2011) to see these relationships.

Equation Block 1: Calculating the effect of COR changes on “effective” exit velocity of a batted ball

The eternally relevant Baseball Trajectory Calculator developed by Alan Nathan has the ability to adjust aerodynamic factors associated with stadium altitude, barometric pressure, temperature, and relative humidity. Combined with the equations from Block 1 above, the changes in COR as a result of meteorological changes can be simply approximated in the Nathan Calculator as a manual change in the rebound (exit) velocity of the ball off the bat.

Great, simply smash aerodynamic and COR changes together and we’re in business, right? Well, almost…it seems every baseball physics article could have all the baseball-specific details stripped out and what would remain is a meditation on linearity and covariance. This example is no different. While we might expect meteorologically-induced aerodynamic and contact factors to vary independently, in real on-the-field situations, balls will be affected by not only their current conditions but also their recent history of past conditions. Absent experimental data on the time scale of such internal ball changes, we can still get a general sense of what could happen when multiple changes overlap. Let’s dive into some colorful 3-D contour plots of results using the default batted ball parameters of the Trajectory Calculator (100 mph pitch, 100 mph exit velocity, 30 degree launch angle) and see what happens!

Figure 3: Effects of meteorological T and RH on fly ball distance, including COR effects equal to ambient conditions (as if balls were kept in the same conditions)

We aren’t too far afield from the basic variables one can change in the Nathan Calculator, so the results from Figure 3 aren’t terribly surprising. Baseballs travel further through warm and dry air. In addition, dry/warm baseballs are bouncier than cold/wet baseballs. It’s unlikely that equipment managers are keeping baseballs outside, so they probably aren’t going to actually experience changes in COR associated with extreme conditions due to the time necessary for water vapor to diffuse into the guts of the baseballs and soften them. But absent a sense of how equipment managers store baseballs, let’s explore the possible impact that a spring training humidor could have.

Figure 4: Effects of humidor-like T and RH on fly-ball distance, with aerodynamic effects equal to PHX March average but COR changing with humidor conditions

Figure 4 shows what would happen if we changed the internal ball T and RH but continued to play in the average Phoenix-area meteorological conditions in March. The weakness of the temperature effect compared to the strength of the humidity effect can be predicted with the slope of each experiment in Nathan, et al. (2011). It’s unlikely, though, that T and RH both have, when combined, a linear effect on COR. For example, it’s unclear whether this linear model captures the hot/wet and cold/dry combinations correctly. This indicates the need to inspect the covarying relationship between T and RH on COR (and therefore, fly-ball distance) more deeply than the simple linear combination I used in this model.

Table 1: Monthly climate, elevation, default fly ball distance using the Nathan Calculator and monthly climate, and scale factors for conversion of March fly ball distance (at PHX) to April fly ball distance (at home).

With the data from Figures 3-4, we can figure out an appropriate scaling factor (Table 1) to translate the dimensions of each team’s spring training stadium and compare them to the dimensions of their home stadium (Figure 5).

Figure 5: Surprise Stadium (KC) and Scottsdale Stadium (SF) scaled to April climatology in KC and SF (no humidor)

After comparing the “effective dimensions” of the Cactus League stadiums to the home stadiums of each team, one can’t help but wonder if the teams had a hand in the way the stadiums in Arizona were constructed. Some teams, such as the Royals, share a stadium with another team (Texas Rangers); therefore, this clearly can’t explain all of the similarities between stadium shapes.

Figure 5 shows that in Arizona during the month of March, the spring training stadiums play much “smaller” compared to other stadiums than their physical dimensions might indicate. By slightly lowering the COR of the ball by using a humidor, teams could cause their spring training stadiums to play with effective dimensions approximately equal to those of their home stadiums. If the Royals were to store their spring training baseballs in a humidor at approximately 70% RH, the differences between the distance up the lines (longer at Surprise than Kauffman) and the distance to straightaway center (shorter at Surprise than Kauffman) would yield around the same “effective surface area” of the scaled outfield.

This analysis, much like my earlier piece on fly-ball precession, neglects many physical variables that would impact the actual games being played. In this example, I have neglected the effects of wind and day-to-day changes in barometric pressure. Prevailing winds due to stadium orientation and location would make this experiment much more realistic. For variations in pressure due to synoptic weather systems (cold fronts, warm fronts, etc.), however, “averages” over an entire month inform us less in terms of the baseline environments of each stadium than monthly averages of temperature and relative humidity. The model also assumes that the balls are essentially stored in temperatures and humidities equal to the ambient conditions in the home stadiums; equipment managers likely store them in some indoor location, but it’s unclear whether they are treated to the exquisite RH control seen with the humidor at Coors Field. Such confounding factors will be explored in future follow-ups to this piece.

In addition to physical assumptions made here, it’s quite possible that baseball operations departments in teams have goals in spring training other than closely approximating the hitting conditions in their home stadiums. But if they want to see who will have power that plays well in their home stadium, the humble humidor could play a key role in moderating the enhanced fly-ball distance that comes naturally with the warm, dry spring air of paradise (Cactus League baseball, that is).

In the chase to break the story of the “smoking gun” behind the recent surge in MLB home runs, many a gallon of digital ink hath been spilt exploring possible modifications to the MLB balls, home-run-optimized swing paths, and even climate change. In my field of Earth Science (atmospheric chemistry, to be more exact), it’s rare that a trend in observations can be easily attributed to a single causal factor. Air quality in a city is driven by emissions of pollutants, wind conditions, humidity, solar radiation, and more; this typically leads to a jumble of coupled differential equations, each with a different capacity to impact overall air quality. To my untrained eye, agnostic to the contents of the confidential research commissioned by MLB and others, this problem is no different: a complex mixture of factors, some compounding each other and some canceling others, is likely fueling the recent home-run spike.

This article will examine the potential for a change in the MLB ball minimally explored thus far: reduction of precession due to decreased internal mass anisotropy. What a mouth full! “Precession” and “anisotropy” don’t have the same ring as “juiced ball” or “seam height” (though they may be on par with “coefficient of restitution”). But these words can be replaced with a more familiar (though funny-sounding) word: wobble. This wobble can occur for many reasons, but the most probable explanation in baseball is that the internal baseball guts are slightly shifted from the center of the ball. This could be due to manufacturing imperfection, or in the course of a game, contact-induced deformation of the ball.

Precession, in general, occurs when the rotational axis of an object changes its own orientation, whether due to an external torque (such as gravity) or due to changes in the moment of inertia of the rotating object (torque-free). Consider a spinning top: the top spins about its own axis (symmetrically spinning about the “stem” of the top) while the rotational axis itself (as visualized by the movement of the stem) can trace out a coherent pattern. If imparted with the same initial “amount” of spin in different ways, the total angular momentum (from both rotation and precession) of the top will be the same whether it’s spinning straight-up or precessing (wobbling) in an elliptical path.

Figure 0: Perhaps the most hotly debated spinning top in the world

As with other potential explanations relating to a physical change in the ball, a change in mass distribution could have occurred unintentionally due to routine improvements in manufacturing processes. By getting the center of mass (approximately, the cork core of the baseball) closer to the exact geometric center of the ball, backspin originally “lost” to precession (in the form of wobble-inducing sidespin) could remain as backspin while conserving total angular momentum; increased backspin has been shown to increase the “carry” of a fly ball, therefore increasing the distance (potentially extending warning-track shots over the fence). A deeper discussion of angular momentum can be found in any mechanics textbook or online resource (such as MIT OCW handouts), but the key takeaway when considering a particular batted fly ball is that productive backspin gets converted to non-productive precession (roughly approximated as sidespin in one axis) when mass is not isotropically (uniformly from the center in every direction) distributed. This imparts a torque-free precession on the spinning ball, causing the rotational axis to trace out a coherent shape.

Precession in baseball has not been deeply studied; in fact, when explicitly mentioned in seminal baseball physics resources, it is noted as a potential factor that will be ignored to simplify the set of physical equations. Together, dear reader, we shall peek behind the anisotropic veil and explore how precession might impact fly-ball distance, and by extension, home-run rates.

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For those of us with some experience throwing a football, even just in the park, we can picture the ideal “backyard Super Bowl” pass: a tight spiral that neatly falls into the outstretched hands of the intended receiver. The difficulty of executing such a perfect throw is evident in the number of nicknames for imperfect throws that wobble (precess) on their way up the field short of their intended target (see “throwing ducks” re: Peyton Manning). In football, the wobbly precession of a ball in flight is typically blamed on the passer or credited to a defender for deflecting it (or in some cases, allegedly, a camera fly wire). It’s not as easy to imagine such behavior in baseball: even in slow-motion video shots of fly balls, the net spin of the ball is dominated by backspin. In addition, the nearly-spherical shape of a spinning baseball has significantly different aerodynamics than the tapered ellipsoid used in football. However, even a small amount of precession has the potential to shave yards off the distance of a football pass; therefore, impacts of precession are certainly worth exploring in the game of baseball.

As a sometimes-teacher (I have taught two laboratory classes at MIT), I strongly believe in the power of simple physical models to qualitatively inform trends in the not-so-simple real world. Therefore, for the first step of exploring the effect of ball precession in the game of baseball, I have turned to the wonderful Trajectory Calculator developed by Dr. Alan Nathan. The Calculator numerically solves the trajectory of a batted ball by computing key physical properties in discrete time steps. While many physical attributes of the ball are calculated in the various colored fields, any of them can be overwritten with custom values.

Figure 1: Fly Ball Distance with Nathan Trajectory Calculator defaults, conversion of backspin to sidespin

In Figure 1, I use the Trajectory Calculator to explore the effect of sidespin conversion on a single fly ball with the same initial contact conditions as the default (100mph exit velocity, 30-degree launch angle, default meteorological conditions), with the total spin set to 240 radians per second. Backspin is not converted to sidespin in a one-to-one fashion: because of the Pythagorean relationship between these factors, total spin is equal to the square root of the sum of the squares of sidespin and backspin. Therefore, to conserve angular momentum, a 10% reduction in backspin (216 rad/s) yields 104.6 rad/s of sidespin, which together lead to a ~1% decrease in fly ball distance from 385.3 ft to 381.3 ft.

With all of the assumptions made here, notably that introduction of precession can be simulated as pure conversion to sidespin to conserve angular momentum, the effect of precession on the flight path is clear but rather modest in this simple approach. However, the Calculator results show that by reducing the “wobble” in a ball’s trajectory, it will carry further. A league-wide reduction in precession would mean that balls would, on average, travel further, leading to an uptick in home runs. If decreased precession would also decrease the effective drag the ball experiences in flight, the effect of increased fly-ball distance could be even further enhanced.

A more realistic exploration of precession will require further modification to the modeling tools at hand. Following Brancazio (1987), which studied the effects of precession on the trajectory of a football, and additional follow-on work, a precession-only physical model can be developed to explore more complex aspects of the problem posed here. Elements of this precession-only model can be fed back into the Nathan Trajectory Calculator, but without a full understanding of some unconstrained physical constants and mechanical aspects of the pitch-contact-trajectory sequence, a tidy figure in the style of Figure 1 will be difficult to produce.

Again, as I mentioned above, I find simple models to be effective tools for teaching concepts. Therefore, let’s consider a “perfect” baseball to be a completely uniform, isotropic sphere, as in Figure 2. This perfect ball is axially symmetric and should not have any precession in its trajectory due to changes in its moment of inertia (I). Now, let’s add a small “spot mass” (that doesn’t add roughness to the surface) on the surface of the ball along the axis of rotation corresponding to pure backspin (the x-axis here). This ball with a spot mass should approximately represent an otherwise-perfect sphere whose center of mass is slightly shifted in the x-direction.

Figure 2: (A) real baseball, (B) perfect sphere, (C) sphere with a point mass at the surface, and (D) sphere with slightly offset center of mass approximately equivalent to (C)

If the model ball has a mass m1 that is isotropically distributed through the entire sphere, and a point mass with mass m2 that is located on the surface along the x-axis, the moment of inertia can be calculated in each direction, summing the contributions from the bulk mass m1 and the point mass m2 (Figure 3).

Figure 3: Moments of inertia for isotropic ball (mass m1) with a point mass (m2) at the surface

Of course, the mass of a real baseball isn’t isotropically distributed, and there is no such thing as a “point mass” in reality; however, by exploring different combinations of m1 and m2 that sum to to mass of an actual MLB baseball (5.125 oz, as used in the Nathan Trajectory Calculator), the ball can be distorted in a controlled manner to explore the effects on precession and fly-ball distance.  Using a set of equations derived from Brancazio (1987) Equation #7, the initial backspin of a ball (omega_x0) can be calculated given an initial total spin (omega), the variable B (the “spin-to-wobble” ratio indicating the number of revolutions about the x-axis per precession-induced “wobble”, a function of the moments of inertia I_x and I_yz), and the angle of precession (built into the variable C, with theta being the angle between the x-axis and the vector of angular momentum when precessing, similar to the angle between a table and the “stem” of a spinning top).

Equation Block 1: Derivations from Brancazio (1987) used in a simple model of baseball precession

The limitation of this approach is that in order to explore the theta-m2 phase space, we must prescribe a priori an angle theta at which the precession occurs. By instead solving for theta from equation 5 above (Figure 4), we can get a sense of the possible values for theta by prescribing the fraction of omega that is converted to precession (the variable A, a mixture of omega_y and omega_z, also called “effective sidespin”).

Figure 4: Contour plot of theta (degrees) with respect to ranges of m2 and variable A (effective total sidespin)

Figure 4 shows that angles between 0 and 6 degrees are reasonable for the conditions explored using the approach from Brancazio (1987) as translated to baseball. So let’s turn to equation 6, using a range of angles from 0 to 6 degrees, to explore the effects of precession on backspin omega_x (Figure 5).

Figure 5: Contour plots of backspin (omega_x) and effective sidespin (variable A) with respect to m2 (as % of m) and theta (degrees)

Great, the effect of a point mass along the x-axis of the ball can be quantified in this model! The effect is modest, but has the potential to slightly decrease the distance of an identically struck isotropic ball. But there is one major limitation to the model as currently shown: when the angle theta is chosen a priori, there is no capacity of the model to correct to a more physically stable angle. In fact, along the entire x-axis of the plots in Figure 5, where m2 = 0, the ball should be completely isotropic and therefore no precession would occur; a small initial theta would likely be damped out over a small number of time steps. In addition, the contours of constant omega_x in Figure 5a curve in the opposite sense than might be expected: increasing m2 should lead to more pronounced procession. On the other hand, this very simple model does not take into account the possible effects of torque-induced precession caused by gravity (extending the effect of mass anisotropy alone), nor does it account for additional drag impacting a precessing ball. More study is needed to further elucidate the possibility of precession having a considerable impact on fly-ball distance; however, unlike the sometimes-empty calls for “further exploration” of minimally promising leads in academic journal articles, I intend to execute such investigation.

All of these limitations are inherent in the fact that, without outside data to constrain the physics of precession as it applies to baseball, the problem we are trying to solve with this simple model is an ill-posed problem in which there is not a unique solution for a given set of initial conditions. Luckily for us, we live in the Statcast age where position, velocity, and spin of the baseball are all continuously measured (if not fully publicly available). In addition to benefits gained from Statcast data, this problem can also be further constrained by experimental data on MLB balls. Finally, an opportunity to put my skills as an experiment-first, computational-modeling-second scientist, to use! Stay tuned to these pages for follow-up experiments and data analysis in this vein.

The conspiratorial allure of an intentional ball modification directly induced by Commissioner Rob Manfred is visible on online comment sections far and wide; however, many of the most credible explanations for ball changes are benign in Commissioner intent and perhaps attendant with improvements in ball-manufacturing processes. In any case, there are likely multiple facets to the current home-run surge. Ball trajectory effects due to precession have traditionally been ignored to simplify the problem at hand; this initial exploration shows that due to the difficulty of the problem, that was likely a good trade-off given the data available in the past. In the future, however, past work in diverse areas from planetary dynamics to mechanics of other sports can be used alongside new and emerging data streams to help determine the impact of precession on fly-ball distance.

Python code used to generate Figures 4-5 can be found at https://github.com/mcclellm/baseball-fg

Special thanks to Prof. Peko Hosoi (MIT) and Dr. Alan Nathan for providing feedback on early versions of this idea, which was born on a scrap of paper at Saberseminar 2017.