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DeflateGate and the Temperature of the Playing Field

Overview

During the Patriots - Colts game on January 18, 2015, the temperature of the turf field is expected to have been colder than the ambient temperature.  It would be expected to have an impact on the temperature/pressure of the balls used on the playing field, as modern artificial turf fields are designed to maintain a reservoir of cold that is transmitted to the surface using the artificial grass.  This keeps the playing surface cool on hot days.  Using reasonable assumptions, the distribution in the pressures and the lower absolute pressure of the Patriots footballs at halftime is consistent with this expectation; however, experimentation is required to confirm this expectation. 

Inclusion of this effect removes the requirement of a person removing air from the Patriots footballs.  It can be entirely explained by the weather.

This effect was not discussed in the Wells Report. 

Description

Before the game 13 Patriots footballs were measured by a referee to have a pressure of 12.5psi at a temperature of 70oF.

They were put into a ball bag, and carried out to the ball field with an air temperature of 50oF which dropped to 48oF by halftime.

Using the Ideal Gas Law, and knowing that a dry football in open air exponentially will go from a warmer temperature to a colder temperature with a time constant of 15 minutes  (see Wells report), we expect the footballs exponentially approached 11.4psi, dropping  63% in the first 15 minutes, and were 98% of the way there in 1 hour.  [Note: this assumes there is no ball bag effect – i.e. balls in a ball bag might take longer to begin changing in temperature.   The bag and the air inside the bag may insulate the balls from the outside air].

The ball game started approximately 2 hours after the balls were brought to the field.

After the start of the game, with the teams on the field, a ball for the team on offense was placed on the field at the line of scrimmage in between plays.  These balls were regularly exchanged as needed by ball boys in between some plays. 

The air was damp, misting, and rain was expected in a couple of hours.  The temperature had risen from 30oF to 50oF in the last 6 hours.  It was 6pm.  The temperature for the last week had been in the teens and 20’s.

Artificial Turf

So what temperature was the field?  The field is artificial turf.  An ad for it shows a side on view of small rubber pellets mixed with sand, with most of the sand sitting under the pellets, with grass like plastic feathers coming up from under.

Undoubtedly, under the turf field was frozen earth, probably matching the average temperature of the last week, about 25oF (see Appendix B below).  The grass blades would be damp from the mist.  They are not frozen – the composite substance provides some level of thermal insulation.  In the literature the artificial turf is described as being of high heat capacity, with a heavy total system weight providing that heat capacity.  Physically, that heat capacity is probably provided by the sand.  We might expect the sand is still in a sub-freezing state, since it has been cloudy.  The rubber particles provide the springy earth like feeling and are probably insulating.  The blades of artificial grass pass up through the sand and rubber pellets.  They are designed to bring some of the cooling power of the sand to the surface, keeping it cool on high temperature days.

So what is the temperature of the field?  It is probably a gradation of temperatures, starting near the temperature of the surrounding air near the top of the grass, but quickly falling to match the temperatures of the substrate underneath.

Water collected on the grass, and would have been cooled by the presence of the cooled substrate below.

Painting the Football with Cold Dampness

Would the damp grass have painted cold water on the ball as the ball hit the ground or the center spun it into position?  Did contact with the ground bring the temperature down faster than the exponential law for the open air?  It seems easy to answer yes to these questions.  We just don’t know what the temperature was, because we do not have a temperature analysis of the field.

Is it possible the field has warmed up in the surprisingly warm( 50oF) afternoon of an otherwise frigid January in Boston?  I could not find a detailed analysis of modern artificial turf thermal characteristics in the literature that might give that answer.   There are descriptions of early turf fields being unbearably hot in the sun.  The modern fields talk about their high heat capacity which keeps them cool.  That is about as much as I could find.  But given the timing of the atmospheric temperature changes that day, it may be enough to assume (given the literature) that the turf is colder than the air temperature, and its high heat capacity is what keeps it cool, and the artificial grass which transmits that coolness to the surface.

The Final Patriots Drive of the First Half

The first half was played over a time period of about 2 hours.  The first half wound up with a sustained drive by the home team, with multiple balls used as they got replaced by the ball boys.  Every ball that was used became wet with the cold dampness of the field.  They sat on the field at the line of scrimmage in between each play.  The center leaned on them and spun them in the grass.

Each time a ball gets replaced, it gets brought back to the bag, where it starts to warm back up to the air temperature of 48oF.  The time constant for warming of a dry ball is 15 minutes like before, but experimental measurements done later (see an analysis of Fig 21 of the Wells report below) show the time constant of a wet ball is more like 20-25 minutes -- probably due to the layer of water on its surface needing to heat up as well.  And those 20-25 minutes only bring it to 63% of the way to the ambient temperature in the ball bag.

At the end of the half, all the balls were put in the bag, and are brought back inside, into the 70oF locker room, the ball bags unzipped, and the pressure was measured in all the balls within 3-8 minutes.

The Pressures in the Balls

So how do we model the pressure in the balls?

Shouldn’t we expect to see a distribution of ball pressures?  Shouldn’t we find that the ball most recently used in the game will have the lowest pressure, a set of balls which have had more time to warm up, but are still slightly damp having a range of pressures, and winding up with a set of balls that are still dry because they were not used at the pressure suggested by the Ideal Gas Law as applied to the ambient air temperature?

Here is the distribution of Patriots football pressures reported in the Wells report represented graphically:

Figure 1. Pressure in the Patriot footballs, Gauge 1.   This figure was created using the Pressure gauge the Referee recalled as the one he used.

Figure 2. Pressure in the Partriot footballs, Gauge 2. Footballs to the left are the ones we expect were recently used in the field of play.  The three to the right may have been dry and not used.

This top figure was created using the ball measurements using the Pressure gauge Referee Walt Anderson recalled was the one he used.  Since there was some ambiguity in which gauge was used, the measurements are shown for both gauges.  The Wells report noted an average of .35-.4psi difference in the measurements of the balls.   The uncertainty in the actual measurement seems to be between .1psi and .2 psi.

Based on this distribution, we might expect that the 3 balls at the high end of the range were dry, and not used in the game.  The balls used in the game were slightly damp, and would be at the lower end of the range.  Additionally, the Wells report displayed an empirical difference of .1psi to .2psi difference between balls that were damp, and those that remained dry as they settled to the lower pressure (see Appendix A.)

What was the Field Temperature?

Predicting the absolute pressures require that we know the effective temperature that the grass will reduce the football down to (we don’t have this data, so here is a guess).  Remember that the ball sits on the grass at the line of scrimmage in between plays which sometimes can be several minutes.  This direct contact will allow for quickly reaching equilibrium faster than might be expected if it is just air.  If we assume a temperature of between 25oF and 30oF for the substrate, is it reasonable to assign a value of 38oF to the football that rests on the grass for some time?  This is 10oF lower than the ambient temperature. 

Such a temperature would bring the expected pressure to 10.75psi (see Appendix A).  This is below all the pressures measured with one gauge, and all but two of the balls with the other gauge.  From there, the pressure is entirely dependent upon the amount of warm-up time before the half-time measurement.  The amount of time, 5-20 minutes, that we might give for that warm-up entirely explains the distribution of pressures we see in the Patriots balls.

Four of the Colts balls were measured as well.  However, we don’t have measurements for all their balls, and we don’t know anything about whether those 4 balls were used in game conditions or not.  The 4 balls all measured at the high end of the range, to the right, and lie within measurement uncertainty at the same value.  If the above theory is correct, we might find that those balls had not been used in the game, or used very early in the game.  In any case, without having all the balls measured, we find that given the timing of the measurements, they are consistent with having been dry balls or relatively unused balls, and directly follow the conclusions of the Wells report.

Conclusion

There is strong reason to believe that the turf field was at a lower temperature than the surrounding air.  If so, we would expect to see a distribution of ball pressures in the Patriots ball bag at halftime due to the Patriots sustained drive leading up to halftime.  In fact, a distribution of pressures was measured at half-time. With reasonable assumptions, the actual ball pressures are consistent with the Patriot ball pressures measured in the locker room at the beginning of the game.  However, an investigation modelling this scenario must be done before making a final determination.

 

In addition, with limited information available, and the fact that the Colts balls were not on the field in a way similar to the Patriots drive leading up to halftime, the Colts ball pressures measured in the locker room are consistent with the pressures measured at halftime by the four Colts balls, as has already been determined in the Wells Report.

Appendix A

Analysis of Figure 21 of the Wells Report.

This graph is the result of experimentally modelling the changes in temperature experienced by balls, done in a controlled environment by the Wells Report scientists.  This is their game day scenario.  (This curve does not take into account the effect of a cold artificial turf field.)

The brown curves show the Patriots balls starting at 12.5psig. The blue curves show the Colts balls starting at 13.0psig.  The pressure drop at 120 minutes (balls brought to the field) appears to be an exponential curve, and doing a rough fit, has a time constant of 15 minutes (i.e. it drops 63% in about 15 minutes).  The pressure rise at 240 minutes (halftime) does not as closely match an exponential curve, but with another rough fit, using the outer parts of the curve, the wet ball curves have a time constant of 20 minutes.  This extra time is significant because of the timing of the last Patriots drive, and subsequent measuring of the ball pressures at halftime.

Extrapolating to the Presence of a Cold Field

So where would a ball recently used on the field be on this chart?  Suppose the pressure was reduced by .65psi by the field (effectively a 10oF temperature drop for .5psi, and then an additional .15psi because of the wet ball effect as shown in this chart).   It should be starting out 5 to 10 minutes before halftime at the 10.75psig point (well off the bottom of the chart). 

 If we had a sequence of such balls, starting at the 220^th minute, and rising towards  the 11.25-11.4 psi pressure, we would find a range of pressures when the ball bag is finally opened, and the pressures start to be measured in the Officials Locker room.

Appendix B

January Boston area temperatures (from The Weather Underground -- wunderground.com).

Figure 3. Weather chart showing the temperatures in the Boston area during the month of January, 2015.  From Weather Underground (wunderground.com)

Figure 4. Weather chart, expanding the section from Figure 3, to highlight game day on January 18th.  Note the dramatic rise in temperature (the red line) shortly before noon on that day.  The temperature leading up to that time averaged near 25oF.  The ground underneath the Artificial Turf, and the sand within the turf are expected to have been near that average temperature.

 Appendix C

An alternative explanation to the data was reached by the Wells report.  

The Wells Report reached the conclusion that a person must have released some air from the Patriots footballs sometime between the initial measurement and the halftime measurement. 

The motive for this was to allow the Quarterback better handling of the ball. 

 

By examining the distribution of pressures we find that this person must have let the air out of some balls and not others, and then relatively random amounts were removed, leaving the Quarterback with a wide distribution of pressures in the balls.

In their report, the uncertainty in the pressure measurements put the average pressure in the Patriots balls just outside the predicted values, close enough, so the verdict from the Wells report scientists was only “probably” this happened.

Inclusion of another source of cold temperatures (i.e. the Artificial Turf) should cause this model to be re-examined, and may change that "probably" rating.

More Football Physics -- The Wells Report -- What They Missed!

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:

1. 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?
2. 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 (67^o F - 71^o 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?

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.

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 37^oF.  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 40^oF.

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 36^oF.



Is 36^oF 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 50^oF 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.

Football Physics - Local Weatherperson "Person of Interst" in Patriots DeflateGate

Climate conditions can entirely explain the mysterious 2.0psi pressure drop in the Patriots footballs.

• -1.52 psi  Temperature -- A drop from 81^o F to 51^o F. 
• -0.16 psi  Rain  -- Wet leather stretching the ball diameter 1/32" 
• -0.47 psi  Humidity -- Using humid air from the locker room to inflate the football

The weather did it.

And without trying place blame on any particular individual, the person of interest in this case - the person the NFL most needs to talk to - is a local weatherperson.

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. At the beginning of the game each side selected 12 footballs to be used by their team for the game.  By NFL rule, each ball must be inflated to a pressure between 12.5psi and 13.5psi. The refs checked the balls before the game, and confirmed the balls were within the correct pressure range.  At halftime, the refs determined that the Patriots had 11 footballs which were under-inflated by 2 psi, and re-inflated them.

At the end of the game, a major controversy erupted, with people accusing the Patriots (who won the game) of cheating.  The implication was that a lower pressure football gave an advantage to people gripping the football.  The Patriots presumably deflated the balls prior to their use in the game, and after the refs measurement.

The easier grip may be true for us amateurs.  This may or may not be true at the professional football level where speed, accuracy, and feel of the football may be paramount to the quarterback.  A computer simulation done by Popular Science of the ball with the pressure differences yielded no effective change in grip or flight speed would have occurred.

Nonetheless, if the Patriots did deflate the balls after the refs initially checked them, which the press and most stand-up comedians assume they did, they deserve to be called cheaters.

Here are the key known facts:

• The temperature on the field was 51^o F at the start of the game.  According to weather charts there was a slight change in temperature towards the end of the game, of maybe 2 degrees.  The barometric pressure was 30.16 inches.
• The balls were measured to have 10.5 psi of pressure at half-time.
• The balls were measured to have 12.5 psi of pressure 2 hours before game time.
• The balls were moved from inside to outside well before game time, and were outside for most of those 2 hours.
• It was rainy and wet for the entire game.
• The ball-boys were rubbing the balls in the Patriots locker room to make for better handling prior to their being inspected by the refs.
• The height above sea level of the game field is 257 feet.

What we do not know:

1. What was the starting temperature of the balls when measured to be 12.5 psi? 
2. Did the rubbing increase the ball temperatures by friction from the ball-boys?
3. How much does football leather stretch as it becomes wet?
4. Does leather shrink when it becomes cold?
5. What was the uncertainty in the pressure measurements?
6. What was the relative humidity of the air that was pumped into the footballs?

Our Suspicious Questions:

1. Did a nefarious ball-boy remove some pressure from the balls as they were transported outside?
2. Did Tom Brady gain an advantage by having deflated footballs?
3. Or, did the Patriots gain an advantage by having only their balls re-inflated at halftime?  And why didn’t the refs give the same opportunity to the Colts?
4. There were reports that the Colts balls were measured and "within the legal range", although we don't have precise values as we have for the Patriots.  Did the refs really measure the Colts balls?  Given a temperature drop from 76oF to 51oF is it possible for them to still be in the "legal range" without some nefarious ball-boy adding pressure to the balls?  Or perhaps the Colts were keeping their balls warm on the sideline?

And the big question:

• Can the Patriots deflation be explained innocently just using science???

Science and the Results of Calculations

The Ideal Gas Law, familiar to most high school physics students states that there is a relationship between Pressure, Volume, and Temperature for a fixed amount of gas given by the following equation:

PV = nRT 

(For details see the Appendix below on the calculations)

The Scientific Results

Applying this equation to the known conditions above, we find the following results:

1. Assuming no stretching of the football due to the rain, dry air in the ball,  and precise measurements, the starting temperature of the football when the ref measured the pressure should have been 91^o F +/- 3^o F.
    
2. TEMPERATURE: Starting at 76^o F, and 12.5psi, and otherwise dry conditions with no stretching of the football, the football pressure outside at 51^o F will be 11.23psi.
   
   Delta P (Temperature change 76 -> 51^o F) = -1.27psi
    
3.  WET LEATHER DUE TO RAIN: Assuming the football stretches when wet, it would have to stretch by .5% (about 1/32" change in 7" diameter of the football)  to give a .16psi change in the pressure of the football.  Larger stretching of the ball could entirely account for the additional drop in pressure required here.  However, it may be unlikely that the ball stretched that amount.
   
   Delta P (0.5% Stretch of leather when wet at 51^o F) = -.16psi
    
4. HUMID AIR IN THE FOOTBALL: Assuming the ball was pumped up with air at 100% humidity at 76^o F, then condensation within the ball as the temperature dropped would have resulted in an additional .47 psi loss in ball pressure at 51^o F.  So humidity of the air pumped into the ball could be a significant factor.
   
   Delta P ( 100% humidity air pumped into ball at 76^o F ) =  -.47psi
    
5. UNCERTAINTY: Assuming a .1psi uncertainty in the measurement of the football pressures, (i.e. 10.5 psi could have really been 10.6 psi, and 12.5 psi could have really been 12.4 psi), then averaging the measurement over the 11 ball measurements results in an uncertainty of temperature of +/- 2^o F.  In addition, the temperature measurement of 51oF was done at the start of the game, but the pressure measurement was at halftime. Assuming that gives us a 2^o F uncertainty, the temperature uncertainty including that due to the pressure measurements is about +/- 3^o F.
   
   Uncertainty P = +/- .1psi
   Uncertainty T = +/-  3^o F

Scientific Comments and Speculations

• The Results shown above are dependent on a 76^o F starting temperature.  If indeed the starting temperature was 88^o F, or some temperature in between, then the resulting change in pressure due to temperature may be partially or entirely accounted for. For example 81oF would increase the Temperature deflation amount from -1.27psi to -1.52psi.
   
• The Results shown above can easily account for the 2.0psi difference in ball pressure observed at the game.  However, what we don't know is how humid the air was that was pumped into the balls.  And in fact the balls come from the factory preset to about 13.0 psi, so it might be that they just let some air out of the ball, and never put any additional in.  But it does seem possible that the Patriots locker room is hot and humid.
   
• Could the ball have stretched because it got wet?  I do not know what the hydrothermal properties of the football leather are.  When surfing the net, you find people who regularly stretch leather between 2% and 5% by getting it wet.  Some even claim 10%.  It confirms my feeling about leather jackets.  An interview with the football manufacturer revealed that they initially stretch the leather using steam.  However the stretch depends on the type of leather, the grain of the leather, the tanning method, etc.  Could the football leather have been stretched 1%? (Note: The expansion/shrinkage of leather as a function of temperature is approximately known.  Calculations using this produced negligible change).
    
• Can a ball boy heat a ball using friction by rubbing it?  Sure!  The temperature of their hands can also help that.  But we don’t know if they could have added the 13^o F required to get within the uncertainty range assuming they start at 76^o F. That seems like alot of rubbing.  It might be plausible, but needs to be verified.
   
• What about the uncertainty?  My wife runs a town soccer league, and has a power pump with a meter for “precisely” inflating balls to a certain pressure.  The uncertainty in that measurement (a dial guage) is at least 2% .  Anyone who has inflated a tire probably has noted the lack of precision in the gauges which measure tire pressure.  In our estimation, however, we note that the NFL is not cheap with their equipment.  There are digital gauges with hand pumps explicitly used to pump balls and measure pressure accurately to within .1psi.  So that is what we assume their pressure gauge was. ( As an aside about soccer, the legal pressure range is wide – somewhere between 8.7psi and 15.6psi, and the refs vary it according to the skill level of the players, and climatic conditions.  Younger players generally have the lower pressure, and the more skilled professional player will be at the high end of the range.  Perhaps we should all take note of this wide range allowed for a soccer ball when we think about football.)

Some Conclusions

So where does this leave us? 

Our scientific analysis, unless given some additional information, cannot rule out the purely innocent scenario for the Patriots, and in fact it is completely plausible.  A combination of temperature change, wet balls, and humidity of the air in the ball can account for the pressure drop.

It also cannot rule out that someone released some pressure in the balls prior to the game. 


It depends a lot on the starting temperature.  For most of these calculations, we assumed a temperature of 76^o F.  That could easily have been more than 81^o F in a hot locker room or if the balls were placed near a radiator.  It also depends on the humidity of the air used to pump up the balls, and the amount of stretch that occurred on the leather as they got wet.  Was the air in the locker room humid?

Plausibility

Does it seem plausible that Tom Brady instructed someone to secretly release .5 psi of air between the time the ball left the refs in the locker room and the game field?
  
Or is it more pausible that a combination of ball-boys rubbing the balls heating them up, and the stretching of the leather because of the rain and humidity of the air, and just uncertainty in ball pressure measurement?  Or were the balls originally sitting near a heat source just before the original measurement?

My own subjective sense sees that the former is far less likely scenario than the other possibilities – however I am a Patriots fan.  Many other non-Patriot fans will want to believe the former, which is a plot led by Tom Brady working with some ball-boys to release pressure in the footballs.


I suspect that Brady's answer will ultimately be that he instructed his ball boys to work the balls over, which they did in a warm locker room, and then instructed them to also make sure the balls were at the proper 12.5psi just before the refs measured them, and in fact the refs would have adjusted the pressure to make it 12.5psi if they were not.   Then they just brought the balls out to the field and assumed they will be fine for the rest of the game.

So my answer is that we cannot prove or demonstrate an exact answer, because we do not know all the starting conditions.  Some additional investigation may determine more about some of the speculations given above.  

Recommendations for the future

In the future, if the NFL wants to regulate the pressure in the balls, they need to think about temperature.

My recommendation would be that they not change much, but that they require a pre-game filling of the balls at a temperature of 72^o F and a pressure between 12.5psi and 13.5psi using dry air.

What the teams want to do from that point is up to them.  On cold days they may want to keep the balls in a heated box.  On hot days they may want to keep the balls cool.  Or maybe they won't care.

The refs could still have the right at any point in the game to inspect a ball, measuring temperature and pressure and make sure we're not having an equipment malfunction.  The ball would have to be okay according to a chart that correctly uses the Ideal Gas Law, and not just between two fixed pressure points, but uses temperature as well as pressure measurement.

Rating the “Scientists”

Now I would like to be subjective.  Acting like a professor, I would like to grade some of the science that has gone on in this whole affair.
Starting at the top:


A+: A group of people at Carnegie Mellon who not only did the correct calculations with Ideal Gas Law, but then did the actual experiment including wetting 12 brand new NFL footballs, to get the stretch effect on the leather.  The result was convincing.  The pressure on a ball starting at 75^o F and ending at a temperature of 50^o F dropped an average 1.8psi.  By the Ideal Gas Law this should have been just 1.25psi.  The stretching of a wet ball or the humidity (not reported in this report) contributed another .55 psi.  Congratulations to this group!  Excellent job!

A-:  Bill Belichick for hypothesizing that climatic conditions including temperature and wetness can affect the pressure in a football.  He also gets an A+ for effort in presenting the findings of others about this, but it was a B presentation largely because this was such unfamiliar territory for him, and I don’t think any Bill Belichick press conference can be rated above a B.  He did mention the correct number that a 20^o F change in temperature results in a 1 psi change in the football.  So the resulting grade is an A-.

B: Tom Brady’s “I don’t know” press conference.  He may have been trying to speak the truth here, but it was not overly convincing.

C: Bill Nye the science guy.  One of my favorite guys.  But he got his physics wrong.  He gets a D- because he came up with a 6% change in psi from 80^o F to 51^o F when the correct answer is about 1.5psi / 12.5psi which is about  12%.  He made the same mistake as many news reporters and high school physics students do in the application of the Ideal Gas Law.  But he did do a good thing in trying to change the focus to global warming instead of deflategate.  That was an A+ effort on global warming, but a little off topic.  So we average it out, and give him a C.

D-: Popular Science article by Chad Orzel on the science of football pressure in which the ideal gas was mentioned, but he never presented the results of any calculations.   He then over-inflated a couple of footballs and put them in the freezer, got a pressure difference of 2 psi as they froze.  He then concluded from this experiment that because the game time conditions did not match his experiment that it must have been some devious doings that deflated the balls, and that Bill Belichick has psychological problems.  He did publish some later columns on this, explained that he did not totally understand the results of his experiment, and expressed confusion over his pressure gauge.  But at least his charts had straight lines that indicated that the Ideal Gas Law applied. Then he gave a lot of correct statements about the Ideal Gas Law and the type of computations needed, but stopped short of doing the correct computations for the Patriots case.

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Appendicitis A. The Calculations

The Ideal Gas Law, familiar to most high school physics students states that there is a relationship between Pressure, Volume and Temperature for a fixed amount of gas given by the following:

PV = nRT
 

The trick here is that the Pressure (P) is the absolute pressure which includes the weight of the atmosphere (1 atm, or about 14.696 psi at sea level, and about 14.55 psi at Foxborough stadium which is 257 feet above sea level, and 14.62psi if we adjust for the barometric pressure of 30.18 on January 18  -- I use 14.7psi in the calculations below.). This means that a measurement of 12.5 psi on the football is actually about 27.12 psi.  Likewise Temperature (T) is the absolute temperature measured from Absolute Zero, which is about -459^o F.  So a value for T at 51^o F would be 510^o.

So in all our calculations we have make the proper adjustment for the “absolute” values.

Other than that, the calculation is easy. 

We can take our initial condition T1, P1, and V1 to be the halftime measurement.  The value of n which is related to the number of gas molecules, and R which is a constant, will be constant (unless the football leaks), then we can set up two equations if we want to figure out what happens at any condition P2, T2, and V2.

P1 * V1 = n * R * T1

P2 * V2 = n * R * T2

Dividing one equation by the other, we find that:

   (P1/P2)  *( V1/V2) = (T1/T2)

The parentheses are added for emphasis to show that the ratios are what we are concerned with.  This saves us a lot of work with units.

Assuming there is no volume change (i.e. no stretching of the football), then V1/V2 is just the number 1, and:

    P1/P2 = T1/T2

Given that we know P1 and T1, we can solve for either P2 or T2 if we know one or the other numbers.  In particular if we know that P2 is the pressure measured inside the locker room, then we can solve for T2.  The correct answer here for the game conditions is 91^o F.  If you can get that answer, then you will know that you have correctly converted from absolute temperatures and absolute pressures to our usual temperature and pressure units.

So

T2 = T1 * P2/P1 = (459 + 51) * (14.7 + 10.5)/(14.7 + 12.5)  = 550  =>  91^o F

Subtracting off the 459o of absolute temperature yields 91^o F.

STARTING IN A 76 degree LOCKER ROOM

Okay, that answered one question.  Now, can we start in a 76oF locker room, and get a 2psi decrease in pressure if it is 51^o F at halftime.

First, assuming no stretching of a wet football, no problems with humidity and condensation inside the football, and perfect measurements.  Just switching around the calculation we did above we find that the temperature change alone reduces the pressure by 1.25psi.

Delta-T ( 76 -> 51 )    =>    -1.27psi

What about the stretching of football leather as it gets wet?  Well that will cause a change in volume.  Since we are only interested in the change in volume, and the ratio V1/V2, we can express the change in a percentage.

Suppose the leather stretches by 1% when it gets wet.  This stretch only refers to one dimension, not the three dimensions required by volume.  So a 1% increase in size would cause the circumference of the ball to increase by 1%.  Since the volume will be proportional to the cube of the radius (the circumference is directly proportional to the radius), we have to cube this increase.  Thus:

V1/V2 = 1.01^3 ~ 1.03

Remember that V1 refers to the volume at half-time.  If we set T2 (the temperature of the locker room measurement) to be 76^o F, then

This means the volume would increase by 3%
.

(P1/P2)  *( V1/V2) = (T1/T2)

Solving for P1 in this equation

P1 = P2 * (V2/V1) * (T1/T2) = (14.7 +12.5) * (1/1.03) * (459+51)/(459+76) = 25.174     =>  10.47 psi

P1 = P2 * (V2/V1) * (T1/T2) = (14.7 +12.5) * (459+51)/(459+76) = 25.929      =>  11.23 psi

So the

Delta-V ( 1% stretch)    =>  -0.76psi

Humidity

If a ball at 76^o F is filled with air at 100% humidity, we know that some of that water vapor will condense if lower the temperature to 51^o F.   

To solve this we can look a different form of the Ideal Gas Law, which can be written this way.

PV = NkT

where N is the number of gas molecules.  At 76^o F, that N is composed of molecules of air and molecules of water.    As the temperature is decreased some of those molecules of water will condense and become water droplets on the inside of the ball.  They are no longer part of the gas.

What we really care about here is a density of molecules in the ball, and the relative densities of the air density to that of the water.  We can look up these numbers in a chart of saturated vapor pressure versus temperature done at atmospheric pressure.  (Note: this is not strictly correct.  It would be better to get these numbers at atmospheric pressure + 12.5 psi, but in the absence of having that information, this is the best approximation we’ve got – and it should be pretty good).

Accordingly, SVP( 51 ) = 9.53, and SVP(76) = 23.16.  See these tables for info.

The ratio

N1 / N2 =  760/(760-(23.16 – 9.53)) = .982

Using the formula

P1/P2 = (N1/N2)(T1/T2)

We solve for P1, and get

P1 = P2 * (N1/N2)(T1/T2) = (14.7 +12.5) * .982 * (459+51)/(459+76) = 25.46  =>  10.76psi

Delta-N (100% humidity at 76oF)  => 10.76 – 11.23 =  -.47psi

In the event that we don’t start at 100% humidity, then condensation will not occur until we reach a temperature where 100% humidity will occur.  This needs to be taken in account here.  If we start at 50% relative humidity, then we won’t reach 100% humidity until you get to about 57^o F.  The remaining change will result in much less change in psi.  It will not be a linear scale.

The Uncertainty.

If we use a gauge with an uncertainty of .1psi, the regular uncertainty principals apply.  We made two measurements, so the resulting uncertainty is the square root of the sum of the squares, so we get .14psi. 
This means our measurement of 12.5psi – 10.5psi = 2.0psi +/- .14psi.  This results in a temperature uncertainty (going to the full 91oF ) of  +/- 5.6^o F. 
However, we made 11 such measurements (the 11 footballs), so we divide this uncertainty by the sqrt(11), and round up a bit to get an uncertainty of +/- 2oF.  The initial measurement of 51^o F also has an uncertainty, which we might guess is +/- 2^o F.  Again using the sum of the squares argument, we get

Uncertainty (T)  = +/- 3^o F