**Question:** Is air quality on airplanes better or worse than in other environments?

**Short answer:** Yes.

**Long answer:** In our previous post, we reported on the dynamics of airplane farts. Specifically, when an airplane takes off, the cabin pressure drops precipitously, and this triggers a sharp increase in farts. Interestingly, the fart smell dissipates after about an hour, after which the quality of the air is actually better than what is found in most people’s houses. In fact, by some measures the air quality on a plane is similar to that in a hospital operating room, which is to say that it is quite good. Nevertheless, there is a clearly detectable fart stink that fluctuates through time. What causes these fluctuations?

To a first approximation, the dynamics of fart smell is governed by the interplay of two factors, fart emissions and the airplane’s ventilation system:

We can capture these dynamics with a first-order ordinary differential equation:

Here *f* is the amount of fart particles in the plane, *a* is the rate at which they are removed by the ventilation system, and *I* is the rate at which they are released by the passengers and crew. If we assume that the farts begin immediately after takeoff and continue at a constant rate for a duration of *h* minutes, we have the following equation for the concentration of farts over time:

This is simply the solution to a first-order differential equation that responds to a box function input, with the constant *M* added to capture the overall smelliness of the farts.

We reasoned that, to the degree that this equation fits the observed data, the best-fitting parameters should inform us about the dynamics of fart smell on airplanes.

Here are the data from an airplane trip taken recently by one of the members of the College:

Takeoff occurred about 70 minutes into the data record, and as in the previous examples, this was accompanied by an abrupt decrease in cabin pressure (red line). The decreased pressure was in turn followed by an increase in fart smell (blue line), which spiked to about 600% of its pre-takeoff level. As discussed elsewhere, the increased farting is likely a result of Boyle’s Law.

Through numerical simulations, we have found that we can reproduce these fluctuations in fart smell with the equation above by setting *h* = 41 and *a* = 0.02, which yields an acceptable fit to the VOC data:

In this plot, the data are shown in red, and the model output in blue. The accuracy of the model suggests that our simple differential equation captures the key dynamics of airplane farts. In particular, our results indicate that the increase in fart output begins within a few minutes of takeoff and lasts for just over 40 minutes. The associated smell is removed with a time constant of 1/*a* = 50 minutes. As a result, fart smell persists for about 3 hours, after which it is largely eliminated from the cabin, leaving behind exceptionally fart-free air.