Question: What fraction of fart smell reaches the human nose?
Short answer: Less than 1%.
Long answer: A few weeks ago, we discussed the smelliest fart ever recorded, which arrived with a concentration of 891 ppm of hydrogen sulfide. For reference, the olfactory threshold for humans to detect H2S is 0.1 ppm, and levels above 500 ppm are considered lethal.
Now, our measurement equipment does not actually allow us to quantify concentrations greater than 60 ppm, so the figure of 891 was reached by extrapolation from available measurements, constrained by a diffusion-based model of fart spread. In this regard, our characterization of fart smelliness is similar to the ways in which scientists describe the origins of the universe. In both cases, we have a model that accounts for the data we observe sometime after the event, and we work backwards in time, until we reach a space-time singularity. From this kind of exercise, we can understand the formation of the universe, as it evolved in time relative to the Big Bang:
For the equations governing fart diffusion, this singularity occurs as time approaches zero in the following equation:
Near this time, the smelliness M approaches infinity, in much the same way that the density and temperature of the universe approach infinity near the primordial singularity known colloquially as the Big Bang. Just after the singularity, in the period known as the Planck epoch in cosmology, the models become unreliable, but we can analyze the subsequent time points to infer what actually happened.
This allows us to answer the question posed at the beginning of the post. Recall that we measure fart smell using a collection tube with sensors spaced at different distances from the source of the fart. The collection tube, rotated for comparison with the graphic above, is depicted here:
In previous posts, we showed that measurements obtained at the sampling points (black circles) register an average smelliness of around 3 or 4 ppm. As such, they are readily detectable by the human nose. Mathematically, we can work backwards from this data to understand what happens in the milliseconds just after the release of the fart. Typically, we see something like this:
Each horizontal line in the plot corresponds to time evolution of the measurement at one of our sensors; the rest of the data points are inferred from the parameters recovered by the model. In this visualization, the singularity is the transition from the pre-fart period (black) to the sudden appearance of fart smell (orange).
Clearly, the smelliest part of the fart actually never reaches our collection tube and in fact remains suspended within one inch of the source. At greater distances from the source, we see a slow diffusion, as the fart smell gradually reaches the more distant reaches of the collection tube. The full process is actually rendered for some farts as the animation function in our fart browser.
If we consider only the peak smelliness values obtained near the space-time singularity, we arrive at the following distribution for a sample of 304 farts:
This reveals that the average smelliness near the singularity is actually 167.8 ppm, a level which the CDC classifies as “immediately dangerous to life or health.” Because we have not attempted to measure fart smell so close to the source, these values remain theoretical for the moment. Nevertheless, given that the typical fart smell experienced at a distance of one foot from the source is 0.84 ppm, we can conclude that the fart smell that we perceive is a faint echo of the original output.
From this perspective, each fart is its own universe, a complex panoply of chemicals and gasses that emerges gradually from a single hot, dense core whose true nature may never be fully understood.