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Primate Studies — Where BTF's Members Investigate the Grand Old Game Saturday, June 27, 2009Home Runs and Ballparks
Baseball is unusual in that there is not a standard playing field. Outfield distances vary from 200 feet (Little League) to infinity (many high school fields). Major League outfield distances vary from 302 feet to 435 feet and outfield fence heights vary from 4 feet to 37 feet. Given these variations, it is of interest to estimate the effect of Major League ballpark configurations on the ease of hitting home runs.
Such an estimate is obtained by calculating ^{1}the minimum energy required to hit a midJuly home run in the different ballparks under a set of reasonably typical conditions for each ballpark. More specifically, the minimum energy for each ballpark is determined for home runs hit down the foul lines, the power alleys, and to dead center. These five minimum energies are then averaged to provide a measure for each ballpark. A horizontal tail wind is arbitrarily directed toward center field, as illustrated in Figure 1.
Distances to the outfield fences and the fence heights at the 30 ballparks are listed in Table 1 ^{2}. On the average, the American League distances are significantly shorter to the left field, rightcenter field, and right field. However, only the average left field fence is significantly higher, primarily due to the Red Sox green monster.
In determining the minimum energy required to hit a home run, forces on the baseball must be computed as the ball travels to the outfield fence. These forces, indicated in Figure 2, are (1) the horizontal wind pushing the baseball toward the fence, (2) gravity which pulls the baseball toward the ground, (3) air drag which is the braking action of the atmosphere, and (4) the Magnus force which results from the spin. The wind and gravity forces are constant during the flight of the baseball. Furthermore, the gravity force is the same for all ballparks.
The air drag force is proportional to the air density, the velocity squared, and the drag coefficient. The drag coefficient is a constant 0.50 below 40 mph, decreasing to 0.29 at 120 mph. The high altitude Rockies have the lowest value air density at 0.79 atmospheres. The air density of the other ballparks varies from the sea level value of 1.00 atmospheres (several low elevation teams) to 0.90 atmospheres (Diamondbacks). The air drag force is always in direct opposition to the velocity.
Batted balls usually have appreciable backspin, which is fixed here at 17.60 times the initial velocity, e.g., a 100 mph baseball would have an initial spin rate of 1760 rpm. Due to wind resistance, this rate is slowed throughout the flight. As an estimate of this decrease, the spin is linearly ramped to zero in five seconds, although the ball may continue in flight.
The spin causes a pressure differential on opposite sides of the baseball that deflects the baseball in the direction perpendicular to both the velocity and the spin axis. This Magnus force, which acts to curve thrown balls, depends on the spin rate, the velocity, and the drag coefficient. For a pure backspin with the spin axis parallel to the ground, the vertical component of this force increases the height of the trajectory throughout its flight by increasing the vertical velocity, as shown in Figure 3. However, until the ball reaches its maximum height, the horizontal component of the Magnus force decreases the horizontal velocity. After the ball reaches its maximum height the horizontal component of the Magnus force increases the horizontal velocity for the remainder of the flight, albeit with a reduced effect since the velocity and spin rate are less on the downward flight than on the upward flight. The net effect is to decrease the minimum energy by about 2%.
At the beginning of the baseball’s flight, the gravity and air drag force are roughly the same, with the Magnus force being about 75% less. The bat provides the main force, but it only lasts for around 0.005 seconds. It propels the baseball at the initial angle relative to the ground. The calculations do not include the details of the force provided by the bat or the compression of the baseball but effectively includes their effects by assuming an initial velocity of the baseball at a height of 39 inches above home plate. The calculations vary the initial velocity and initial angle until the energy is minimized.
A typical optimized trajectory is shown in Figure 4 for a home run hit along the right field foul line of the Pirates ballpark. The initial velocity is 95.4 mph at a 40.0degree angle. At 320 feet the ball just clears the 21foot fence (21 feet to honor the great Pirates right fielder, Roberto Clemente, who wore the number 21). This figure also shows how the velocity decreases over the first 220 feet, due primarily to air drag, and then increases at longer distances, due to gravity, until it is 65.2 mph at the outfield fence.
The home run results which gave the lowest average hit energy show that the optimum hit angle varies from 36.6 degrees (Diamondbacks) to 41.7 degrees (Red Sox). The optimum initial ball velocity, to just clear the outfield fence, varies from 98 mph (Red Sox) to 110 mph (Brewers). The shortest flight time is for the Angels (4.23 seconds) and the longest flight time is for the Nationals (4.56 seconds).
The calculations produce the energy expended from the time the ball is hit until it clears the fence. However, most of the energy expended by the batter is lost in friction. This effect is roughly estimated by multiplying the calculated energy by 4.7. The final result is then an estimate of the minimum energy that a batter has to expend to hit a home run in one of the directions in Figure 1. The final energy values are in units of ftlbs. To put the values in more understandable units, the energies are expressed in terms of how many pounds of weight a person would have to lift 30 inches (a typical bench press length) to expend the determined energy in ftlbs.
The minimum energy results, averaged over the five directions, for all thirty ballparks are given in Table 2. Included are the citydependent parameters. The city elevations are from the pmiusa web site. City temperatures are mean July values given at the National Climatic Data Center web site. Elevations and temperatures are needed to calculate the air density ^{3}. Each wind speed is the ground level July values given in the Wind Energy Resource Atlas. Wind speeds in parentheses are for the city, but the calculations assume the wind speed for these ballparks is 0 mph since they are enclosed.
The energy results show a 19% difference between the lowest and highest values (only a 10% difference if enclosed ball parks are excluded). The Red Sox (short outfields) and Rockies (low air density) have the easiest ballparks. The hardest ballparks in which to hit home runs are enclosed ballparks where there is no wind increased by 2.2% due to the greater density of cool air. However, wind has a significant effect; when the Giants 13 mph wind is reduced to 0 mph, the required initial velocity is increased by 7.7% and the energy by 16.0%. This effect explains why it is easier to hit home runs with even a small tail wind in an openair ballpark than in an enclosed ballpark (even ballparks with retractable roofs have such high supporting walls that there will be little wind on the field).
The easiest home run (162 lbs) is along the Red Sox right field foul line (shortest distance) and the hardest home run (329 lbs) is to the Astros center field (longest distance). The longest measured home run (565 ft) was hit by Mickey Mantle in 1953 at the Griffith Stadium in Washington D.C. With the recorded steady wind of 20 mph, this home run would require a lift weight of 408 lbs – 19% more than needed for an Astros center field home run and the initial baseball velocity would have been 145 mph! These high values cast some doubt on the authenticity of this record.
The results indicate that most of the ballparks have about the same degree of difficulty. They also indicate that Barry Bonds hit primarily in an easy ballpark, Mark McGwire in an average ball park, and Sammy Sosa in a slightly harder ballpark (although the Cubs ballpark is famous for its variable winds).
This analysis gives estimates of relative difficulties in hitting home runs during a typical July day. It is based entirely on the ballpark dimensions and locations, plus nominal temperatures and wind velocities. The analysis could be improved if actual wind values and directions were available for each ballpark. However, relevance to actual home run production will always be limited by managers who routinely alter their lineups and strategies to accommodate ballpark conditions.
Additional subjects for discussion are provided by the 2007 Major League hit distribution given in the 2008 ESPN Baseball Encyclopedia and reproduced in Table 3. In spite of the designated hitter in the American League, the percentage of home runs is essentially the same in both leagues – although the percentage of singles is slightly higher. However, the most striking feature is that Major League outfield fences make it five times easier to hit a home run than a triple. The big mystery is therefore why do triples count less than home runs? Furthermore, why do rare insidethepark home runs only count as much as the much easier outsidethepark home runs?
^{1}The baseball trajectory equations are given by R. K. Adair in his book: The Physics of Baseball.
Brian R. Taylor and Lyle H. Taylor
Posted: June 27, 2009 at 12:44 AM  26 comment(s)
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1. Zoppity Zoop Posted: June 27, 2009 at 01:02 AM (#3234703)Those who attend more games at AT&T;Park can correct me, but I don't believe the wind ever blows straight out to center. Most commonly there's a cross breeze. In the absence of any wind, AT&T;would be the most difficult HR park in baseball by these numbers. I don't know what effect the cross breeze would have.
Obviously, this exercise can't take into account factors such as hitter's background.
Technically not true, right? You've got Coors. And then there's whatever stadiums Prince Fielder and Bartolo Colon are playing in that night.
What benefit is there to having the wind in the model that outweighs this?
Earlier this year the Rockies' announcers pointed out that Coors Field was right in the middle of the pack as far as allowing homers this year. So I don't know how seriously we should take this whole exercise.
This assumes that a 100 mph batted ball (as a HR possibility) would 1760 rpm worth of backspin.
Suppose a pitcher (with one of the livest arms in the majors) threw an overhand "rising" fastball. What rpm would you expect on the backspin of that?
(1) some parks might have these tailwinds, but many do not.
(2) certainly, fly balls are not distributed evenly around the park.
I'd like to see home run PF split by batter handedness and pitcher handedness (over years of data).
I wouldn't really expect them to, even if the data was perfect. Remember, basic HR park factors generally track the overall rate of home runs, not the overall rate that fly balls become home runs, so factors that would cause the player to have a harder or easier time making contact in the first place would have an effect on the HR factor.
Using the NL numbers listed above, and average run values for events...
Rate x Run Value = Runs
1B 15.6 x .46 = .072
2B: 4.9 x .75 = .037
3B: 0.5 x 1.03 = .0052
HR: 2.7 x 1.40 = .038
Homeruns and doubles are just about even, while triples are far too uncommon, and singles far too common. Singles are about twice as common as they'd need to be, while triples are about 1/7 as common as needed to even everything up. Pushing the fences back would increase triples, but increase homeruns. This is complicated by homeruns cutting into singles, doubles and triples, and vice versa. Each influences the other, which would make it hard to balance.
I think a stadium that evened up all these factors would actually move fences IN in general while making fences higher, which would lower singles and increase homers, while hopefully increasing doubles at a rate high enough to keep pace with homeruns. At the same time these stadiums would have to have areas where the fences went almost straight back towards very deep, 430ish, allies to allow for triples. This is starting to remind me of Fenway, but maybe even more extreme. I'll have to look up Fenways park factors now to see the effects.
Well, not only that, but you're neglecting the fact that run values would change, too. When you start messing with the occurrence rates of building block events, you can't just start with 2008 NL run values and stick with it. You'd need to adjust the values because you're wildly adjusting the environment in which the event occurs.
Off the top of my head I would imagine that the value of a single would go down if you adjusted triples upward in occurrence. A single will more than likely score a runner from second, but not always, while it is for all intents 100% at scoring a guy from third. This would seem to argue for the value going up, but with that many more guys on third from hitting triples, I *think* you'd get that eaten up by a drastic increase in SFs.
Once my brain starts trying to figure out the effect the new occurrence rates would have on baserunning strategies and the subsequent effect on run values, or how composition of the timelines (triple,double,single will score less than triple,single,double and both will score less than single, double, triple) affects the whole matrix, it starts to hurt and I black out.
That number is from some other part of the city, which is quite hilly (maybe the airport?), but certainly not alongside the rivers where PNC Park is located.
The elevation for Turner Field is pretty close (1057 ft. rather than 1026).
Ya I was thinking that as the occurrence rates change the value of each event changes. I didn't get into it simply because, as you said, my head starts to hurt and I black out. haha. A decrease in singles would seemingly decrease the run values of all other events, because we are moving from an OBP weighted scoring to more SLG based. Proportionally I would think run scoring would drop a bit and the marginal values of extra bases would increase. Of course who knows what the run scoring environment would eventually be since fluctuating event values keep us from knowing the break even points. It's probably something that a nifty simulation system could try and figure out.
1. The authors purport to calculate the "energy expended" by the ball in flight. What exactly is meant by that? Is it the initial energy minus the final energy? And what is the relevance of this quantity? I simply do not know how to interpret the "Weight" numbers in Table 2.
2. The discussion about the forces on a baseball (Fig. 2) is wrong. The "wind" is not an additional force. Rather, the wind affects the speed of the baseball with respect to the air The statement that the air drag is opposite to the velocity is correct. However the velocity in that case is the velocity of the ball with respect to the air, not with respect to the ground. It is that relative velocity that affects both the drag and the Magnus forces. It would appear that the authors did not take into account the affect of wind on the Magnus force. Finally, the estimate of Adair in his book of the Magnus force is almost surely wrong and underestimates the effect of spin on the flight of a baseball by a lot. For a dicussion, see http://webusers.npl.illinois.edu/~anathan/pob/AJPFeb08.pdf.
3. The results regarding the optimum launch angle cannot possible be right. There is no way that the optimum launch angle is greater than 40 degrees. An inspection of home run data from hittrackeronline.com shows that very few home runs are hit with a launch angle that steep.
4. The assumption about the magnitude of the backspin is probably not right, as it is wellknown that the backspin is a function of the launch angle. Generally, the larger the launch angle, the larger the backspin.
5. The commment about the ballbat contact time being 0.005 sec is not even close. It is more like 0.001 sec.
6. If the authors want to contact me privately at my email address, I would be happy to carry on a dialogue with them about baseball aerodynamics.
7. The spindown time constant of 5 seconds that Adair has in his book is also probably not correct. The time constant is much longer, probably more like 25 seconds, an number based on actual (albeit a bit crude) data as well as scaling from careful measurements on golf balls. See http://webusers.npl.illinois.edu/~anathan/pob/spindown.pdf
8. I think I finally figured out what the "energy expended" is. If I am not mistaken, it corresponds to the minimum velocity needed for the ball to clear the fence. The idea for the calculation is a good one. The work could be improved with a better aerodynamics model.
 There aren't any 11 foot fences in AT&T;Park in LCF (nor anywhere in the park)
 The LF fence at Dolphin Stadium (Land Shark now) is considerably less than 33 feet.
 The Mets fences listed are incorrect whether they are intended to represent Shea Stadium or Citi Field.
 There aren't any 4 foot high fences in San Diego
 The Angels RCF fence should be 18, and the RF fence less than that (it varies a bit, which is why using only 5 heights is a bad idea)
 CF at Fenway Park is not 9 feet, except for a tiny section where the fence goes from the back left corner of the home bullpen to the front left corner of the bullpen. Silly to use that number for all of CF there.
 The Metrodome LF fence is not 13 feet high. Perhaps when the plexiglass was up, but that was removed quite some time ago...
 The fence heights for new Yankee Stadium are all 8 feet, not the variety of numbers listed. There isn't a 14 foot fence anywhere in the new or old Yankee Stadium, although you might have been able to find one back before the renovation in the early 70's.
Seriously, I'll echo pobguy above and suggest that the authors have made some fundamental, and avoidable, mistakes here.
Such an estimate is obtained by calculating1 the minimum energy required to hit a midJuly home run in the different ballparks under a set of reasonably typical conditions for each ballpark. More specifically, the minimum energy for each ballpark is determined for home runs hit down the foul lines, the power alleys, and to dead center.
Am I confusing two different things here?
http://webusers.npl.illinois.edu/~anathan/pob/v0_by_park.gif.
For each park, the bar graphs shows the mean SOB for all home runs hit there during the first 6 weeks of the 2009 season. A total of 819 home runs make up the data base, so there are roughly 27 per park. Not great statistics but it is the best we can do for now. We will do much better with a full season of data. The error bar shows the standard error on the mean. Coors and Fenway have the lowest mean SOB (about 98.8 mph) while Turner and Chase have the highest (102.8 mph). That's a difference of 4 mph between lowest and highest, which is a 4% spread in SOB, corresponding to an 8% spread in "initial energy" of the ball.
FYI: hitf/x is the latest from Sportvision, the company that does the technology for pitchf/x. The same camera images that are analyzed to determine the pitched ball trajectory can also be analyzed to determine the initial part of the batted ball trajectory. In particular, the initial SOB and the launch and spray angles are determined.
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