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# DeflateGate January 2015

Robert C. Hilborn
American Association of Physics Teachers
January 28, 2015

Crowd-Sourcing Deflate-Gate

Unless you have been out of the country for the past few weeks, you certainly know that a major controversy has arisen concerning the purported under-inflation of footballs used in the January 18, 2015 American Football Conference championship game between the New England Patriots and the Indianapolis Colts.  Near the end of the first half, the game’s officials suspected that the Patriots’ footballs were under-inflated (that is, the football’s air pressure was less than the 12.5 pounds per square inch (psi) specified by the National Football League’s rules.  See the Appendix of this document for the full set of Ball Rules.). Pressure measurements of those footballs at half time confirmed that 11 of the 12 footballs provided by the Patriots were “significantly underinflated,” with various news reports claiming that the pressure was 1.4 to 2 psi under the minimum pressure.

Although the details of the situation remain unclear, the question naturally arises:  Could the “under-inflation” have been due to simply a temperature difference between when the balls were inflated by the Patriots and verified by the officials before the game and when the pressure measurements were made at half time?  The alternative is, of course, that the Patriots deflated the balls, presumably to enhance handling, in the time between the officials’ verification and the beginning of the game even though the balls are supposedly under the officials’ control in the time between verification and the kickoff.

A Challenge for Physics Teachers and Physics Students

Many physics teachers and physics professors have provided online analyses of the Deflate-Gate situation using reasoning based on the ideal gas law.  In the following section, I give a simple summary of that argument.  But the main purpose of this note is to suggest “crowd sourcing” what has been largely missing from the online debate: some measurements to see what happens with real footballs.  This situation provides students with an opportunity to provide scientific data and analysis to help resolve an interesting sports controversy and, along the way, to re-enforce their understanding of the behavior of gases.  I encourage physics teachers and physics professors to have their students decide on what kinds of measurements are needed to settle the temperature issue. I suggest working with athletic departments to set up and carry out the measurements.  I encourage teachers to send a two-page description of the results to me at rhilborn@aapt.org with the subject line Deflate-Gate Data.  I will summarize and then publicize the results.

Ideal Gas Law Analysis

In this section, I summarize a simple analysis that suggests that the under-inflation might be largely explained by the temperature difference between when the balls’ pressure was first verified by the Referee and when the balls were re-measured at half time.  I assume that the air inside the football is reasonably described by the ideal gas law under the conditions used for the football preparation and during the game.  Let me denote the initial pressure and temperature at which the footballs were prepared as Pi and Ti, respectively.  Similarly, I denote the “final” pressure and temperature (during the game) as Pf and Tf.  As almost every chemistry and physics student knows, the ideal gas law tells us that

Pf  = Pi (Tf / Ti),                                (1)

under the assumption that the volume of the football and the number of molecules inside the football bladder remain the same, conditions which are at least approximately correct for the game situation. Knowing the initial pressure, the initial temperature, and the final temperature, it is easy to calculate the final pressure.  I note that if the final temperature (at half time) is lower than the initial temperature (when the balls’ pressure was first verified), the final pressure will be lower than the initial pressure.  At the lower temperature, the air molecules are moving less rapidly (on average) and hence exert a lower pressure on the football bladder in which the air is trapped.

Before using numbers in Eq. (1), I need to point out two crucial issues.  First, the pressure used in Eq. (1)  is the total air pressure inside the football.  Most pressure gauges measure the pressure relative to the surrounding atmospheric pressure.  This relative pressure is often called the “gauge pressure.”  To get the total pressure from the gauge pressure, I need to add the surrounding atmospheric pressure to the gauge pressure.  The atmospheric pressure is typically about 14.7 psi.  Details can be found in the local weather records.  A common mistake, even among scientists, is to use gauge pressure rather than total pressure in Eq. (1).

The second important point is that the temperatures in Eq.  (1) must be absolute temperature values (the so-called Kelvin scale).  Fortunately, there are many web sites that make it easy to convert temperatures in degrees Fahrenheit (which is what is available from local weather reports) to Kelvin scale temperatures.  Any units of pressure may be used as long as they are consistent from one side of the equation to the other.

Now let me use some numbers to see if temperature changes are sufficient to explain the reported under-inflation.  Let’s suppose that the Patriots initially inflated the footballs to a gauge pressure of 12.5 psi, the minimum allowed by the NFL rules (the maximum is 13.5 psi) and that pressure was verified by the Referee (as specified in the NFL Rules: See http://www.nfl.com/rulebook or the Appendix in this document).  I assume those measurements were made in the officials’ locker room whose temperature was probably about 70 oF.  (When the NFL issues its official report on the incident, we may have a clearer idea of what that temperature was.)  The weather report for Foxboro, MA, where the Patriots’ stadium is located, and the game-day commentary indicated that the air temperature on the field at game time was about 50 oF, rather warm for a January day in New England.  Those numbers give us 27.2 psi for Pi, 294 K for Ti, and 283 K for Tf resulting in Pf = 26.2 psi for the total pressure and a corresponding gauge pressure of 11.5 psi, about 1 psi below the NFL lower limit.

Water Vapor Effects

Another possible contribution to the lower pressure inside the football when the temperature is reduced comes from the change in water vapor pressure with temperature.  January 18 was a very rainy day in Massachusetts so the air had high humidity.  Presumably the footballs were inflated with that humid air.  (In motor racing sports, tires are often inflated with “dry” nitrogen, or argon, to reduce the effects due to changes of water vapor pressure with temperature.)  Data tables of water vapor pressure as a function of temperature indicate that water vapor changes its pressure by about 0.2 psi as the temperature drops from 70 oF to 50 oF.  Combining the air temperature change effect with that of water vapor gives us a total of 1.2 psi, not too far from the figure of 1.4 psi given in one of the early NFL statements.

A conservative conclusion is that temperature changes alone can account for much of the under-inflation, but until more accurate information is available about the preparation of and measurement of the footballs’ air pressure, we should not assert that conclusion with a great deal of certainty.  Note that the NFL rules do not specify the temperature at which the footballs are prepared and verified prior to the game.  The ideal gas law analysis shows that even modest temperature changes can cause the football pressure to fall outside the rules limits even if the balls were prepared according to specifications before the game.

Other Issues

It is also important to know more about the devices the NFL officials use to measure the football air pressure.  The Ball Rules state that the home team is to supply a pump (presumably with some sort of pressure gauge), but we should be skeptical; we have all experienced the different automobile tire air pressure readings you get depending on the device used:  from simple pop-up pressure gauges to fancy digital devices.  How accurate (and precise) are the devices used by the NFL?

Another issue that can be easily addressed by some measurements:  Does the volume of the football change significantly when the gauge pressure changes by 1-2 psi?

A final research question:  How long does it take a football to come to equilibrium if it finds itself in an environment with a different temperature?  For example, how long does it take a room temperature football to come to equilibrium if it is placed in a refrigerator whose internal temperature is, say, 40 oF?  This issue is important in estimating the temperature of the footballs when the half time measurements were made.

Appendix

From the NFL Rule Book available at http://www.nfl.com/rulebook

Rule 2 The Ball

Section 1

BALL DIMENSIONS

The Ball must be a “Wilson,” hand selected, bearing the signature of the Commissioner of the League, Roger Goodell. The ball shall be made up of an inflated (12 1/2 to 13 1/2 pounds [sic]) urethane bladder enclosed in a pebble grained, leather case (natural tan color) without corrugations of any kind. It shall have the form of a prolate spheroid and the size and weight shall be: long axis, 11 to 11 1/4 inches; long circumference, 28 to 28 1/2 inches; short circumference, 21 to 21 1/4 inches; weight, 14 to 15 ounces.

The Referee shall be the sole judge as to whether all balls offered for play comply with these specifications. A pump is to be furnished by the home club, and the balls shall remain under the supervision of the Referee until they are delivered to the ball attendant just prior to the start of the game.

Section 2

BALL SUPPLY

Each team will make 12 primary balls available for testing by the Referee two hours and 15 minutes prior to the starting time of the game to meet League requirements. The home team will also make 12 backup balls available for testing in all stadiums. In addition, the visitors, at their discretion, may bring 12 backup balls to be tested by the Referee for games held in outdoor stadiums. For all games, eight new footballs, sealed in a special box and shipped by the manufacturer to the Referee, will be opened in the officials’ locker room two hours and 15 minutes prior to the starting time of the game. These balls are to be specially marked by the Referee and used exclusively for the kicking game.

In the event a home team ball does not conform to specifications, or its supply is exhausted, the Referee shall secure a proper ball from the visitors and, failing that, use the best available ball. Any such circumstances must be reported to the Commissioner.

In case of rain or a wet, muddy, or slippery field, a playable ball shall be used at the request of the offensive team’s center. The Game Clock shall not stop for such action (unless undue delay occurs).

Note: It is the responsibility of the home team to furnish playable balls at all times by attendants from either side of the playing field.