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Saturday, June 27, 2009

Home 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


  the minimum energy required to hit a mid-July 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


.  On the average, the American

League distances are significantly shorter to the left field, right-center 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 40.0-degree

angle. At 320 feet the ball just clears the 21-foot 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 ft-lbs. 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 ft-lbs.

The minimum energy results, averaged over the five directions, for all thirty ballparks are given in Table 2. Included are the city-dependent 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


.  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
Temperature has a small effect; when the Diamondbacks temperature is reduced by 20 degrees, the required initial velocity is only increased by 1.0% and the energy

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 open-air 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 inside-the-park home runs only count as much as the much easier outside-the-park home runs?


1The baseball trajectory equations are given by R. K. Adair in his book: The Physics of Baseball.


3Humidity differences are not included in the calculations. Humid air is less dense than dry air but humid conditions also increase the baseball weight and
inelasticity. The net effect is that humidity decreases the flight distance slightly.


Brian R. Taylor and Lyle H. Taylor Posted: June 27, 2009 at 12:44 AM | 26 comment(s) Login to Bookmark
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   1. Zoppity Zoop Posted: June 27, 2009 at 01:02 AM (#3234703)
Good read. I like when articles like these have little pictures for people like me who haven't thought about physics in 20 years and think the Magnus Effect is a Scandinavian Death Metal band.
   2. Mefisto Posted: June 27, 2009 at 01:31 AM (#3234739)
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%.

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.
   3. Dan Szymborski Posted: June 27, 2009 at 01:38 AM (#3234746)
It might be interesting to get some additional game-specific data on wind and temperature.

Obviously, this exercise can't take into account factors such as hitter's background.
   4. Jeff K. Posted: June 27, 2009 at 02:33 AM (#3234838)
Furthermore, the gravity force is the same for all ballparks.

Technically not true, right? You've got Coors. And then there's whatever stadiums Prince Fielder and Bartolo Colon are playing in that night.
   5. Jeff K. Posted: June 27, 2009 at 02:41 AM (#3234855)
I don't see why a wind is applied. I don't see an explanation, and it's going to skew the data. They have it blowing to center, which is going to drift a ball that is anywhere off the straight middle towards center field, of course. Think about a ball hit to left field, 25% of the arc between the foul pole and center (or 50% of the way between the gap and the pole, if that makes more sense.) The wind will push it towards the gap. As such, the mere presence of that wind is going to skew the values in favor of outfield fences featuring less curvature. If they had chosen the wind blowing crossways left to right, it favors less curvature in left field but more in right field.

What benefit is there to having the wind in the model that outweighs this?
   6. puck Posted: June 27, 2009 at 03:45 AM (#3234898)
What's the "Elev. ft" column in table 2--it can't be the straight altitude of the field, right? I didn't think there was anywhere in Denver at 5800 ft.
   7. Tom Nawrocki Posted: June 27, 2009 at 04:38 AM (#3234934)
There's that purple ring of seats near the top of Coors Field that's at exactly 5280 feet, so there can't be any part of the stadium at 5800.

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.
   8. puck Posted: June 27, 2009 at 04:45 AM (#3234939)
I remember the announcers saying that, but I can't remember how specific they got. They could have been counting total HR, so the middle of the pack thing wouldn't have been surprising given how few home games they've played.
   9. puck Posted: June 27, 2009 at 04:57 AM (#3234944)
Huh. Are the bb-ref "ballpark" splits for batting and pitching for NL 2009 messed up? Shouldn't batting HR and pitching HR in a park be equal? They're not for most if not all the parks.
   10. OCF Posted: June 27, 2009 at 06:11 AM (#3234977)
I have a side question about the physics.

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?
   11. KJOK Posted: June 27, 2009 at 06:31 AM (#3234986)
Huh. Are the bb-ref "ballpark" splits for batting and pitching for NL 2009 messed up? Shouldn't batting HR and pitching HR in a park be equal? They're not for most if not all the parks.
I believe the calculation takes into consideration that the batters and pitchers do not face their own batters/pitchers, so the factor applied to normalize home runs may be different for a team's batters vs. pitchers.
   12. KJOK Posted: June 27, 2009 at 06:37 AM (#3234987)
And I know this is a minor thing, but 'left-center' and 'left-center power alley' really shouldn't be used interchangeably. Left-center (and right-center) generally means 30 degrees from the foul lines, splitting the OF into 3 equal zones. The "power alleys" generally define the distance at the points between foul lines and dead CF, 22 degrees from the foul lines.
   13. Danlby Posted: June 27, 2009 at 12:55 PM (#3235037)
Good physics, though with anything like this, you have to make assumptions and can't capture all the variables. The results don't jive very well with HR park factors (Wrigley and Arlington routinely have high HR PF, Fenway quite the opposite). I expect for a variety of reasons:

(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).
   14. Dan Szymborski Posted: June 27, 2009 at 12:59 PM (#3235039)
The results don't jive very well with HR park factors (Wrigley and Arlington routinely have high HR PF, Fenway quite the opposite). I expect for a variety of reasons:

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.
   15. Erik, Pinch-Commenter Posted: June 29, 2009 at 01:40 AM (#3236137)
The last portion of this article has me wondering. What would a stadium that was designed to balance the occurrence of each hit type (single, double, triple, homerun) with their corresponding average run value look like. In this theoretical stadium the total number of singles would create the same amount of runs as the total amount of homeruns, and so on. I'm not sure this is actually possible.

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.
   16. Jeff K. Posted: June 29, 2009 at 02:33 AM (#3236185)
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.

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.
   17. Justin Turner Overdrive Posted: June 29, 2009 at 03:37 AM (#3236314)
The elevation of PNC Park in Pittsburgh is 730 ft., not the 1223 ft. cited.

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.
   18. Justin Turner Overdrive Posted: June 29, 2009 at 03:45 AM (#3236327)
Likewise, Kaufmann Stadium in Kansas City is at 750 ft., not 1026.

The elevation for Turner Field is pretty close (1057 ft. rather than 1026).
   19. Erik, Pinch-Commenter Posted: June 29, 2009 at 04:39 AM (#3236347)
Re: Jeff K.

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.
   20. pobguy Posted: June 30, 2009 at 03:22 PM (#3237688)
Some comments from a physicist who has done extensive work in the physics of baseball:

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

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 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 well-known 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 ball-bat 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 e-mail address, I would be happy to carry on a dialogue with them about baseball aerodynamics.
   21. pobguy Posted: June 30, 2009 at 06:30 PM (#3237947)
One more comment that I forgot in the previous post:

7. The spin-down 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

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.
   22. gator92 Posted: July 01, 2009 at 01:58 AM (#3238468)
Maybe I shouldn't pile on, but a lot of the fence heights are wrong as well.

- 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.
   23. Jeff K. Posted: July 03, 2009 at 11:05 PM (#3241655)
#21/22, I'm not really a physics guy, oddly not really exposed all that much but I can do 90% of the math behind it. So I'm likely the most dangerous guy, the one who thinks he knows more than he does, but your question about what energy expended means (and your followup possible answer), I thought was explained in the second paragraph:

Such an estimate is obtained by calculating1 the minimum energy required to hit a mid-July 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?
   24. pobguy Posted: July 04, 2009 at 03:45 AM (#3241943)
Re Jeff K. #23: Well, I thought I had figured out what was referred to as "energy expended" but I was wrong. I thought it was the initial kinetic energy of the ball just after leaving the bat. From the table, I take a typical "weight" as 230 lb, which means the energy is the amount of energy need to lift a weight of 230 lb. to a height of 30" (according to the text). If I equate that energy to the kinetic energy of the ball, I find the speed of the ball is about 230 mph! Clearly that is an absurd result, so that means I don't know what is meant by "energy expended." Without knowing what that is, it is hard to evaluate what was actually calculated in the paper. The authors of the paper don't seem to want to respond to any of my posts, even privately, so I guess it will remain a mystery. And aside from that difficulty, there are serious errors in the aerodynamics calculations, as I already noted in #20,21 and as gator92 confirms in #22.
   25. Gregg Posted: July 04, 2009 at 12:01 PM (#3242013)
Interesting study, but measuring five distances (down the line and to the power alleys and to center) doesn't take into account irregularities in shape and therefore doesn't always give an accurate reading of the degree of difficulty for homers. The extreme example is Fenway, which has the shortest distance down the line in right, and the longest to the right-field power alley. That would statistically work out to average difficulty to right. However, the 302 down the line is statistically misleading, because the wall juts almost straight out. Similarly, Wrigley has the longest distances down the line, which is deceptive because it is the only park where the wall curves IN as it moves toward center. I think the study would be more accurate measuring the area of the outfield, or at least using far more measurements than five.
   26. pobguy Posted: July 04, 2009 at 04:38 PM (#3242083)
There is another way to approach the issue of home run park factors. Namely, use hitf/x data to find the initial speed off bat (SOB) for every home run, then sort by parks. I have done such an analysis. Since I don't know how to post an image to this site, I'll just give a link to the image:
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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