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.

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.

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.

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.

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?

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.

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.

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.

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 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?

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.

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.

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).

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.

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.

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.

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.

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).

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.

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 http://webusers.npl.illinois.edu/~a-nathan/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 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.

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 http://webusers.npl.illinois.edu/~a-nathan/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.

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.

#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?

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.

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.

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:
http://webusers.npl.illinois.edu/~a-nathan/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.