Horses aren't spherical. Neither are footballs.
2015-11-06
If you haven’t heard of DeflateGate by now, congratulations! I’m not sure how you avoided all mention of it; if you’re that person, feel free to go Google a summary.
If you ask a physicist to predict the winner of a horse race, the first words out of their mouth might be, “Well, if you assume a spherical horse…” Probably not very helpful, but most of the physics that the rest of us remember are in the spherical horse category; the rules work really well on paper, but are, at best, imprecise models when describing or predicting events in the real world.
Take the Ideal Gas Law (please!). PV = nRT; in a closed system filled with an ideal gas (far from a liquid), the product of the pressure exerted on the boundary and the volume enclosed by the boundary moves linearly with the product of the number of molecules of the gas and the temperature of the gas. R is just a constant, and it changes based on which units you want to measure your values in. A lot of ink has been spent on asserting what the Ideal Gas Law predicts should have happened to those footballs that night; but most of it assumes a spherical horse.
And when you assume a spherical horse, you get a little surprised when your horse has hooves and a tail. A simple running of the Ideal Gas Law with some assumptions about that night (Start temp=71F, end temp=48F, start air pressure is 12.5psi) projects a low end pressure of 11.32. And looking at the halftime measurements of the balls, even using the favorable Logo gauge, we see three balls are under that number - one as low as 10.90psi. That seems a little surprising, but only if you assumed a perfectly spherical ball. Some other factors might be at play. Here are a few.
Of course, of less significant value is the air that leaks out every time the ball is measured - in this case, three times. From Exponent page 36, we see that every needle insertion removes 0.01 psi of air. Seemingly minor, but this 0.03 drops us to 11.22. That’ll round down to 11.2
And we see this in the chart on page 42 of the Exponent Report:
The wet balls do have a lower air pressure. Referencing back to the Ideal Gas Law, the means either the balls got larger (leather stretches, and may reduce force on the bladder), some gas escaped (unlikely, but if so, this whole scandal ought to be moot), or the temperature in the balls is lower - as evaporative cooling suggests. Unfortunately, Exponent doesn’t tell us how much cooler the balls effectively are, but we can eyeball the chart to guess. At 69 degrees (the two hour mark), the pressure appears to be about 14.48. After being on the field, we see the wet ball as having a pressure about 11.26, and the dry ball having a pressure around 11.36. That suggests that the air inside the wet ball was equivalent to about 45.46 degrees, while the dry ball was 47.39 degrees. Actual air temperature was 48 degrees; so the wetness of the ball causes around 2.5 degrees of temperature loss (or the equivalent growth of the ball).
Plugging in our estimated effective temperature in, we now expect a gauge pressure of around 11.09.
At 3:45, the time of the initial measurement, the air pressure is 29.9 inches (of mercury!), which corresponds to 14.686 psi. By halftime (8:30), it’s 29.65 inches, or 14.56 psi. This should make our balls appear to get more “inflated” at halftime, and they do - we’re back up to 11.22.
And what was the temperature in the officials’ locker room pregame? Exponent measured it on February 7th, when the high was 28 degrees; with a crowd of people and a warmer day, was it a bit warmer? And since the balls were measured in the shower room, had anyone showered in there (warming the start temperature still further)? Had the shower room been at 74 degrees, and the halftime HVAC-assisted air pressure been at 14.725 psi, then we’d expect to see a lowest gauge reading at 10.90. But we’re now in the realm of “we didn’t measure it at the time, so we can’t now what happened.”
Were any of these balls fatigued and lose pressure? Again, not something that we’re likely to know. But there is enough uncertainty around the shape of these horses that we shouldn’t just expect them to be perfect spheres.
If you weren’t already convinced, then you’re probably not convinced further.
If you ask a physicist to predict the winner of a horse race, the first words out of their mouth might be, “Well, if you assume a spherical horse…” Probably not very helpful, but most of the physics that the rest of us remember are in the spherical horse category; the rules work really well on paper, but are, at best, imprecise models when describing or predicting events in the real world.
Take the Ideal Gas Law (please!). PV = nRT; in a closed system filled with an ideal gas (far from a liquid), the product of the pressure exerted on the boundary and the volume enclosed by the boundary moves linearly with the product of the number of molecules of the gas and the temperature of the gas. R is just a constant, and it changes based on which units you want to measure your values in. A lot of ink has been spent on asserting what the Ideal Gas Law predicts should have happened to those footballs that night; but most of it assumes a spherical horse.
And when you assume a spherical horse, you get a little surprised when your horse has hooves and a tail. A simple running of the Ideal Gas Law with some assumptions about that night (Start temp=71F, end temp=48F, start air pressure is 12.5psi) projects a low end pressure of 11.32. And looking at the halftime measurements of the balls, even using the favorable Logo gauge, we see three balls are under that number - one as low as 10.90psi. That seems a little surprising, but only if you assumed a perfectly spherical ball. Some other factors might be at play. Here are a few.
Gauge calibration
From the Exponent Report in the Wells Report, page 28, we see that the Logo gauge doesn’t match true relative air pressure. A measurement of 12.5 on the Logo gauge corresponds to 12.17 on a true gauge. The measurement we’d expect to see at halftime (using the initial calculations) becomes 11.276 (corresponding to 11.01 of true pressure).Measurement errors
There are two ways that the gauge might impact the measurement. Rounding error comes into play - the gauges only measure in twentieths of psi. Note that 11.276 above ought to round up to 11.3; but what if a ball didn’t start at 12.5, but instead was at 12.475? That drops our halftime prediction down to 11.25.Of course, of less significant value is the air that leaks out every time the ball is measured - in this case, three times. From Exponent page 36, we see that every needle insertion removes 0.01 psi of air. Seemingly minor, but this 0.03 drops us to 11.22. That’ll round down to 11.2
Temperature, and a swamp cooler
Ever sweat, and then feel cooler as a breeze blows past? Or blow a fan across a bowl of water, and watch the air temperature drop? What’s happening is /evaporative cooling/ - as a liquid evaporates, it needs a substantial amount of energy to change states from liquid to gas. It draws that energy from its surroundings - cooling things down - to make its phase change. So even if the ambient temperature was 48 degrees Fahrenheit, a wet ball might exhibit as cooler.And we see this in the chart on page 42 of the Exponent Report:
The wet balls do have a lower air pressure. Referencing back to the Ideal Gas Law, the means either the balls got larger (leather stretches, and may reduce force on the bladder), some gas escaped (unlikely, but if so, this whole scandal ought to be moot), or the temperature in the balls is lower - as evaporative cooling suggests. Unfortunately, Exponent doesn’t tell us how much cooler the balls effectively are, but we can eyeball the chart to guess. At 69 degrees (the two hour mark), the pressure appears to be about 14.48. After being on the field, we see the wet ball as having a pressure about 11.26, and the dry ball having a pressure around 11.36. That suggests that the air inside the wet ball was equivalent to about 45.46 degrees, while the dry ball was 47.39 degrees. Actual air temperature was 48 degrees; so the wetness of the ball causes around 2.5 degrees of temperature loss (or the equivalent growth of the ball).
Plugging in our estimated effective temperature in, we now expect a gauge pressure of around 11.09.
Actual air pressure
Each of these calculations has assumed that the ambient air pressure was 14.7 psi, which is the approximation for standard atmospheric pressure. Fortunately for our weather systems, air pressure changes; unfortunately for an argument in favor of the Patriots, it’s moving in an unhelpful direction that day:At 3:45, the time of the initial measurement, the air pressure is 29.9 inches (of mercury!), which corresponds to 14.686 psi. By halftime (8:30), it’s 29.65 inches, or 14.56 psi. This should make our balls appear to get more “inflated” at halftime, and they do - we’re back up to 11.22.
Unknowns
And here’s where it gets hard to measure: the balls aren’t outside, but are inside Gillette Stadium. If pregame, the outside temperature was in the 70s, it’s possible that the building HVAC was barely running. As it drops to 48 degrees, if the HVAC is engaged at halftime, that’ll raise the ambient air pressure a bit, and lowering our anticipated halftime measurement.And what was the temperature in the officials’ locker room pregame? Exponent measured it on February 7th, when the high was 28 degrees; with a crowd of people and a warmer day, was it a bit warmer? And since the balls were measured in the shower room, had anyone showered in there (warming the start temperature still further)? Had the shower room been at 74 degrees, and the halftime HVAC-assisted air pressure been at 14.725 psi, then we’d expect to see a lowest gauge reading at 10.90. But we’re now in the realm of “we didn’t measure it at the time, so we can’t now what happened.”
Were any of these balls fatigued and lose pressure? Again, not something that we’re likely to know. But there is enough uncertainty around the shape of these horses that we shouldn’t just expect them to be perfect spheres.
If you weren’t already convinced, then you’re probably not convinced further.