It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts. – Sherlock Holmes (Sir Arthur Conan Doyle)
Overview
On a cold rainy afternoon of January 18, 2015, the New England Patriots played the Indianapolis Colts in a football game to determine who gets to go to the superbowl. The Colts lost, but during the game complained that the Patriots were using under inflated footballs. This has become known as the infamous DeflateGate.Recently (4 months after the fact) a report from the NFL, called the Wells Report, came out and declared that the Patriots "probably" had cheated. Subsequent to that the NFL has decided to fine the Patriots some money, ban the Patriots QB Tom Brady from the first 4 games of next season, and eliminate some draft picks for next season.
Physics was critical to the Wells report conclusion. Without the data, scientific tests, theories, and scientific models that were used in the Wells report, the condemnation from the Wells report of the Patriots would be baseless.
However, the report fails to question why any quarterback would want randomly inflated (or deflated) footballs.
Instead, the cold, wet, nearly frozen playing surface at Gillette Stadium, the harshness of the New England weather leading up to the game, and the play on the field in the late minutes of the first half can account for the low pressures and the scatter in the pressures in the halftime football pressure measurements of the Patriots.
In this paper, I will present an argument which can account for the pressure measurements of both teams in a straightforward manner, without having an Officials Locker Room Attendant deflate balls. I welcome people to test this. If it doesn't happen now, then we can wait until next season, when we will again sometime have climate conditions similar to that January 18th game.
The Questions
When reading the report, the following questions came to mind:
- There is no answer in the report that satisfactorily explains the scatter in the pressure readings of the Patriots. The theory of the nefarious person deflating footballs might do so, but this would require random and inconsistent pressure release on this persons part, which seems unlikely. Why would a quarterback want a set of randomly inflated balls?
- The showpiece of the report deals with a model of transient Temperatures and Pressures that occur before the game, and lead up to the pressure measurements at halftime. This is what seems to have brought the scientific experts here to their final conclusion. However, a critical fact was ignored. During the time leading up to the halftime, and within a time window that would make a difference (the last 15 minutes), a number of balls were in direct contact with a grassy wet cold playing field which would have been significantly colder than the ambient temperature.
In this paper I will stay away from the text messages and their interpretation. Although entertaining, they are of little consequence if the scientific evidence and pressure measurements cannot support the allegations.
Hooray for Data
The Wells report finally released to the public the data that we have wanted for so long. It throws out many of the wild and crazy theories we have all heard, including some of my own in a previous paper. I apologize to the referees for the rating I gave them in that paper. They appear to have done a good job in these circumstances, except for sticking with the balls when Mr. McNally walked off with them down the corridor.
Reading of the Report
When reading the report, the first time we really encounter some data to sink our teeth into is about 8 pages into the report when the half-time data is presented showing the measured pressures in the balls.
I had to gasp. First I was expecting to find that most of the Patriots balls measured 2 psi under the 12.5psi initial pressure for the Patriots. This is what had been spread to the press and so fervently fed to us.
In fact the data was scattered heavily, and averaged only 1.2psi under the original 12.5psi pressure measurement. My second gasp was when I realized the Colts balls averaged only about .46psi under their initial 3.0 psi starting point. (Note: I averaged over both gauge measurements to get these numbers).
Understand this – that based on the temperature values revealed in this report, the expected pressure drop by the Ideal Gas Law was 1.1psi. Based on the average measurement of the Patriots balls, and the uncertainty in the scatter, the Patriots were a little on the low side, but still right on target! My reaction was – good for the Patriots, but how in the world did the Colts avoid obeying the Ideal Gas Law with their balls!! Their .45psi drop was well under the amount expected by the Ideal Gas Law. This had suddenly become a nail-biter of a report for the Colts. Were they warming their balls on the sidelines?? In fact, later in the report the scientists were able to come up with a good explanation for that, their transient Temperature and Pressure analysis which works well for the Colts, but became the reason the report no longer supported the Patriots.
The scatter in the Patriot football measurements was also disturbing, but the fact that two different pressure gauges showed significantly different readings initially made me think that there was some significant measurement uncertainty in this process that would be explained later in the report.
However, one of the important things the Wells report did was demonstrate the relative accuracy of the gauges. And thereby has made that scatter a real phenomenon that needs to be explained. Unfortunately, this fact was used to support a theory of a nefarious Locker Room Attendant who was not very consistent in the amount of air he let out of the balls. This doesn't seem very likely, and even more unlikely that it would be a benefit to the Patriots quarterback.
The Scatter in the Patriot Data
With some careful investigation, the scientists have good estimates for the starting temperature (67o F - 71o F) . They have also done some investigations into how footballs behave under conditions of changing temperature, the stress of heavy pressure from bodies falling on them, some of the effects of moisture, etc. This investigation appeared to be thorough and repeatable. They also checked the accuracy and repeatability of the measurements using the actual gauges used by the referees.
According to the Referee, thirteen balls were set to be 12.5psi for the Patriots, and twelve balls were set to 13.0 psi for the Colts.
Based on repeatability of measurements from the scientists, and this report from the Referee, we would expect the ball pressures of most of the balls to be within .2psi of each other throughout the time of the game if the balls were allowed the time to equilibrate with the ambient air. This narrow range was not the case. In fact, just picking one of the gauges, it measured a half-time pressure of 10.9 psi on one ball, and 12.3 psi on another. This is not even close, and neither of them is very close to the Ideal Gas Law prediction of 11.4psi. Were these statistical fluctuations of the measurement? The results of the scientists on the repeatability and accuracy of the gauges would suggest not! We are dealing with something real here.
From a scientific perspective this is exciting, because this indicates an as yet undetermined phenomena is at play here. But until we know what it is, to reach the conclusions this report reaches seems dangerous.
The Temperature of the Playing Field
The ambient temperature was between 48oF and 51oF. But what about the temperature of the playing surface itself? After all the balls came in contact with that surface, and in fact that surface contained a lot of water on that particular day which the balls would pick up. That would cool the balls down. January is a cold month in the Boston area. In fact in the week leading to, and even up to the very afternoon of the game itself, the temperature had barely reached 40oF, and most of the time had been well below freezing. (See Climate Data) The playing surface is a heat sink which does not increase or decrease temperature rapidly and must have been cold, reflecting that lead up to the game.The curves measured by the Wells report show that balls heat up or cool down exponentially due to air temperature changes (as expected) with a t0 factor of 15 minutes (i.e. exp(-t / t0) ). This means that it takes 15 minutes for a ball to reach 62% equilibrium, 30 minutes to reach 87% equilibrium, ...
This means balls in substantial contact with wet cold grass within 15-20 minutes of the halftime measurements would be expected to measure lower pressures. (I expect that direct ground/wet grass contact with cold water would cool the ball faster than ambient air -- example: a beer thrown into a bucket of ice water cools faster than a beer put in a cold refrigerator)
The Patriots had a long sustained drive before halftime. They probably had a number of balls that spent considerable amount of time in contact with the wet grass which may have been colder than the ambient air. But at what temperature. Based on the climate data, and assuming the ground does not heat up quickly, it is hard to imagine that the ground temperature was above 38 degrees. Wasn't it misty down by the field? Is it likely to surmise that at least a few of the Patriots balls became significantly colder than the ambient temperature within a 15-20 minute window of being measured by the ref's at halftime?
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| Pressure distrubution (psi). Each ball is represented by a blue rectangle. The balls with the lower pressure are expected to be the balls that were used in the game. The 3 balls at the higher pressure might be those that sat in the bag, and were not used. Only one gauge was used in this display. |
In fact if you look at the distribution of the ball measurements there are a small cluster of balls at a high psi, probably those that were dry and not used in the game. The group at the lower pressure end, the colder damp balls were probably the ones used in the game! Those down at the very low end could be the last one or two balls used in that drive just before half time. And the others may have been used in that drive as well.
All the Patriots balls were measured, but unfortunately only 4 of the Colts balls were measured. Were the bags in contact with the ground as well? Would it be expected that the Colt's 4 balls were taken from the top of the bag, or perhaps balls that had not sat in close proximity to the ground? Certainly the Colts balls did not spend as much time on the playing field as the Patriots did shortly before halftime. The report uses the Colt's 4 balls as the "control" for the experiment. Which balls were these? Using these as a control may have been a questionable choice.
So the scatter and the low temperature may have been simply caused by the balls being in play in the game!
The Transient Temperature/Pressure Model
As I mentioned earlier, there is a problem accounting for the Colts data being well above the predicted Ideal Gas Law value. The Wells report shows how this may be accounted for by how the data was collected at half-time. The balls were brought in their ball bags back into a warm room, and the measurements took 10-13 minutes to occur. During that time the balls were warming up, and may have warmed up sufficiently to be in the proper range. We need a transient model to account for this.
Most objects warm up or cool down by exponentially approaching a final temperature from a starting temperature. This is because the rate of temperature change is proportional to the difference in temperature. I can go through the physics of this elsewhere, but it does involve calculus, so we will leave it out of this paper. This is the resulting equation below for the simple ideal case, but it is written in terms of Pressure because in this case of no leaks and constant volume Pressure is directly proportional to Temperature via the Ideal Gas Law
P(t) = P1 – (P1-P0) * exp( -t/t0)
In words, this says that the Pressure as a function of time equals the final pressure P1 minus the difference between the two pressures P1 and initial pressure P0 times an exponential function which is decreasing in time, with a special timing constant t0.
The constant t0 is important. If t0 is 15 minutes, that will mean that the pressures will be roughly 63% towards equilibrium in 15 minutes, after 30 minutes will be 87% towards equilibrium, and after 1 hour will be 98% towards equilibrium.That means balls on the field can be expected to maintain a significant Pressure differences resulting from the field even after 15 minutes.
With any object of mixed composition such as a football, and in an environment which might vary itself this can be a little difficult to exactly predict. What the Wells report people did was to just measure what that was, rather than try to predict it. Their result is shown in the graph below, which is their figure 21.
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| From the Wells Report, Figure 21 of the Exponent Appendix |
At time = 120 minutes, the balls were brought to the field. By this chart, the Patriots balls (in brown) should have begun their exponential approach to the expected 11.4psi. And they do. By eyeballing the chart carefully, the t0 value for a football appears to be 15 minutes.
But what is this? There is a light brown curve with a wet ball that seems to be approaching 11.3psi. Does dampness affect the pressure and bring it below the Ideal Gas Law point?? And for the Colts ball their wet ball this effect is even more pronounced. And when we try to bring the balls back into the original warmth, the wet and dry balls do not come back together even at the 360th minute after 2 hours of waiting. (As an aside, to be discussed at some other time, this kind of change is consistent with football leather stretching when wet, which as seen above will produce different results with different balls.)
Another problem with this curve is that the dry balls while warming up at the 240th minute do not seem to follow a good exponential curve. There is nothing wrong with this because of the composite material, but it should be carefully investigated whether a systemic problem, like a HVAC unit which is warming the air is not overdoing itself, and pushing them up faster than expected. Over the next few pages of the report the period of time between 240 minute and 255 minutes is scrutinized carefully, and if there is a systemic problem with the model's data, that should be accounted for.
Notice also that the wet balls rise more slowly than dry balls. Now the moisture on the outside also has to be brought up to temperature, and that takes longer. Our wet balls from the cold below ambient temperature field may thereby remain at the colder temperature longer.
Can We Predict the Temperature of the Field?
As with any theory its strength is in its predictive power. So, based on the data, can we predict the temperature of the field?
There are many uncertainties, but it is necessary to see if a reasonable temperature can be assumed.
So let's try. The second Patriots ball was measured to be 10.85psi, well below the expected 11.4 psi value of the Ideal Gas Law at an ambient temperature of 48 degrees. We'll assume this ball did reach equilibrium with the field. Based on the Ideal Gas Law, this temperature would be 37oF. But wait, this ball was probably wet, and as we saw on the chart above, about .15psi is due to the ball being wet. So making that adjustment, we find a temperature of 40oF.
There is another gotcha here though. It must have been at least 6 minutes before this temperature was measured. During that 6 minutes it spends 3 minutes outside and 3 minutes inside. During the outside time it is moving towards the lower temperature (11.4psi) minus the .15 due to a wet ball. In the inside time, it is moving towards the Room Temperature pressure in accordance with the Transient graphs generated by the Wells report scientists. The total change here seems to be something around .2-.3psi. If we do the computation again, we get about 36oF.
Is 36oF a reasonable temperature for the field? I expect it probably is. Certainly January in New England is cold, and these modern turf fields are known for their ability to keep cool. (Advertisements for turf fields tout how cool they feel on hot days -- and 50oF in January is hot when the preceding days and nights were in the low 30's and below). There have been studies done of the heat capacity of these fields, what happens with cool ground underneath, etc. This may warrant further research. Note also that the very top surface may measure close to the ambient temperature, but since these footballs get ground into the surface, the deeper down temperature is the temperature that is relevant.
Summary
The Wells scientists clearly wanted to do a complete job. The idea that the field temperature could make a difference seems to have been overlooked.
It should be reasonable enough to ask the scientists of the Wells Report to take into account any data they can figure out regarding the temperature of the field, and rethink their results and their conclusions.
Certainly the simple estimates from above point to a very different conclusion for this game than what the Wells report concluded. Both teams were operating fairly with their equipment and it was simply the climatic conditions, and the playing time that determined the scatter in the ball pressures that were measured.
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