Ty France’s Roof Banger: Home Run or Not?

Michael McLoone-Imagn Images Every now and then, something takes place in a major league baseball game that causes my inbox to light up. That’s what happened on October 4, when then-Padres first baseman Ty France hit a towering fly ball down the left field line during Game 1 of the NLDS at American Family Field. The ballpark has a retractable roof, which was closed at the time. As a result, the ball hit a roof cable in fair territory, slowing its progress to the point where it was caught on the warning track by Brewers left fielder Jackson Chourio. Since the ball struck the cable in fair territory, it was in play according to the American Family Field ground rules, so the caught ball was ruled an out. Nobody disputes those facts. But here’s the question everyone seemed to be asking me: Had the ball not hit the cable, would it have cleared the left field fence in fair territory for a home run? Note that the fence is 342 feet from home plate and eight feet high. Note also that the ball was hit down the line. So the question of whether it would have been a home run reduces to two separate questions regarding the location of the ball once it reached 342 feet from home plate: Was the height of the ball greater than eight feet? Was the ball to the field side of the foul line? Normally, given the exit velocity and launch angle, determining whether France’s ball would have been a home run would be easy. I would simply use my Trajectory Calculator with default values for the unknown parameters to calculate the full trajectory. Those default parameters are based on analysis of Statcast batted ball data from past major league seasons. However, this particular batted ball was somewhat unusual in that, while the exit velocity was a respectable 105.3 mph, the launch angle, 48.7 degrees, was considerably steeper than is typical for a major league home run. Had the ball been hit at close to the optimum launch angle of about 30 degrees, it would have carried over 400 feet, making the answer to our first question a resounding “yes.” But that is not what happened. In fact, the default parameters for my Trajectory Calculator would have the ball falling well short of the fence due to the large expected spin rate on the ball. It is expected that a ball hit at such a steep launch angle would have a high backspin rate. Moreover, a ball hit down the line would have a high sidespin rate. Given that the drag coefficient Cd on the ball increases with spin, the larger-than-typical spin would lead to a larger-than-typical Cd, which would reduce the total distance. As such, the initial answer to my questions, absent additional information, was that the ball would not have been a home run. It would have fallen short of the fence and, due to the large expected sidespin, it would easily have hooked into foul territory. Of course, the key phrase in the previous paragraph is “absent additional information.” Fortunately, I was able to gather non-public data from a source with the league; it is included here with MLB’s permission. With that information in hand, let’s revisit my initial analysis. Analysis For this analysis, I used a modified version of the HitTracker method, which was developed by my good friend Greg Rybarczyk to estimate home run distances in the pre-Statcast era. Typically a home run lands somewhere in the stands, whereas the distance people quote is how far the ball would have traveled had it been free to arrive at ground level. Greg’s technique was to use his maps of major league ballparks and video to find a reasonably precise location of where the ball hit the stands, as well as a measure of the flight time from contact with the bat to contact with the stands. Using that information along with his model for drag and spin rate, the launch parameters were adjusted so that the trajectory passed through the landing location at the measured time. The resulting full trajectory could then be extrapolated to ground level to find the home run distance. My modification to that technique during the Statcast era was to use Greg’s landing location and flight time along with the measured launch parameters of exit velocity, launch angle, and launch direction to find the full trajectory. The technique uses a non-linear least-squares fitting method (the so-called Levenberg-Marquardt algorithm, a technique I have been using since my grad student days) to adjust the backspin rate, sidespin rate, and Cd value, so that the trajectory passes through the (X,Y,Z) coordinates of the landing point at the observed flight time T. This results in a nearly model-independent determination of the ball’s trajectory and therefore the home run distance. To apply that technique here requires the additional information provided by MLB that I mentioned earlier: the (X,Y,Z) coordinates of the impact location and the flight time T from bat contact to impact. Those and other parameters are given in Table I. The (X,Y,Z) coordinates are based on observation and careful lidar-based maps of the stadium. The angle of the impact location relative to the y axis, ϕC=arctan(Y/X), is equal to -43.8 degrees, indicating that the impact point was in fair territory (note that in the Statcast coordinate system, ϕC=-45 degrees is the third base line, so that angles less negative than -45 degrees are in fair territory). The time T is based on video observation. Table I: Input Parameters Exit Velocity (mph) Launch Angle (deg) Direction (deg) X (ft) Y (ft) Z (ft) ϕC (deg) T (sec) 105.3 48.7 -41.4 -132.4 138 161.5 -43.8 2.9 Input parameters for the analysis using the standard Statcast coordinate system. Note that a direction of -45 degrees is the left field foul line, so the direction in the table is 3.6 degrees to the field side of the line. The parameters (X,Y,Z) and ϕC are the location and direction, respectively, where the ball struck the roof cable and T is the flight time. The results of the least-squares method are given in Table II. Table II: Results Backspin (rpm) Sidespin (rpm) Cd ϕ (deg) H (ft) R (ft) 1364 338 0.404 -45.09 19 352 Results from the least-squares minimization process, where ϕ and H are the direction and height above ground level of the ball at the fence distance of 342 feet. The range R is the full distance extrapolated to ground level. Whereas the ball easily clears the fence, since H is greater than eight feet, it passes the fence just barely in foul territory. Indeed, it passes about six inches to the left of the foul pole. So, given the input parameters, the fly ball would not have been a home run. Interestingly, the fitted spin values are significantly smaller than those expected for batted balls at such extreme launch and direction angles. As a result, the drag coefficient is much smaller than that used in my Trajectory Calculator, and the ball carries farther. It is not the carry of the ball that leads to my conclusion, but rather the direction. Note also from Table I that, since the impact location ϕC is less than three degrees from the initial direction, there is very little hook on the ball, suggesting very small sidespin, as confirmed in Table II. So I find that, taken at face value, the input parameters from Table I imply no home run — but just barely. Surely these parameters are not known with infinite precision, so it is useful to investigate how reasonable changes in these parameters might affect the ultimate conclusion. I investigated this using a Monte Carlo simulation, where the four parameters (X,Y,Z,T) were each selected from a normal distribution centered on the values in Table I and with standard deviations of one foot for (X,Y,Z) and 0.1 second for T. Here I’ll note that the values chosen for these standard deviations are just guesses. The goal of the Monte Carlo is to show the sensitivity of the primary conclusion to the input parameters. For each of the 10,000 runs of the simulation, the new parameters were used as input into the minimization routine and new values of the parameters in Table II were calculated. FIG. 1: Distribution of 10,000 Monte Carlo input values for the flight time T and the cable direction ϕC. The events in red result in a home run; those in black are fair but fail to clear the fence; those in green clear the fence but are foul; and those in blue are both foul and fail to clear the fence. The horizontal and vertical lines correspond to the values in Table I. and the intersection of those lines is in the green zone, indicating no home run. Partial results of the simulation are shown in Figure 1, which is a color-coded scatterplot of T and ϕC for the 10,000 events. Those events that result in a home run are denoted in red, while those that exceed the fence height but end up foul are denoted in green. Other outcomes are denoted in blue and black, as explained in the caption. The horizontal and vertical lines correspond to the T and ϕC values in Table I, respectively. Several important features follow from the simulation: The most important parameter for determining whether or not there is a home run is the direction of the impact location ϕC, which is -43.8 degrees from Table I, or only 1.2 degrees from the foul line. The crosshair is very close to the red-green boundary between fair and foul. A change in ϕC by 0.05 degrees in the field direction (to the right on the plot) would result in a home run. That would correspond to moving the impact point by about two inches! For the given value of ϕC, no amount of change in T would result in a home run. If the flight time were as long as about 3.0 seconds, then the ball would not have cleared the fence regardless of ϕC. The sharp boundary between red and green for T ≤ 2.9 seconds suggests that other parameters play a much smaller role than ϕC in determining whether or not there is a home run. Over the range of variation in the simulation, a home run is achieved about 30% of the time. Conclusion Using the estimates of the cable location and flight time in Table I, I conclude that France’s fly ball would not have been a home run. However, it is a close call, proving once again that baseball is a game of inches. The crucial determining factor is not whether the ball would have cleared the fence, but whether it was fair or foul, which in turn depends primarily on the precise direction of the impacted cable. I’d like to thank my sources at MLB for providing me with the necessary data and for allowing me to share it publicly.