The Big Bang Theory of Dolbear’s Law

Season 03, Episode 02: “The Jiminy Conjecture”

In the “The Jiminy Conjecture” episode, Raj (Kunal Nayyar), Howard (Simon Helberg) and Sheldon (Jim Parsons) are having dinner when they hear the sound of a cricket chirping. Sheldon claims to know the species from the cricket’s chirping speed and the room’s ambient temperature. Not to be outdone and still angry from losing a bet in the comic book store, Howard wagers his rare Fantastic Four #48 (“The Coming of Galactus”) against Sheldon’s The Flash #123 “Flash of Two Worlds”.

Dolbear’s Law

Sheldon is able to determine the room’s temperature by using Dolbear’s Law, an equation that states the relationship between air temperature and the rate at which a cricket chirps. This relationship was formulated by an American physicist, Amos Dolbear in 1897 in his article “The Cricket as a Thermometer”. While Dolbear didn’t specify the species of cricket in his article, it is generally believed to the snowy tree cricket.

Dolbear's Law

Amos Emerson Dolbear (November 10, 1837 – February 23, 1910) was an American physicist and inventor. He is noted for his pioneering work in wireless telegraphy — several years before Guglielmo Marconi. He is also known for finding a relatoinship between a cricket’s chirp rate and ambient temperature.

If we look closely we can see Dolbear’s equations written on the whiteboard in the background. They all describe the relationship between temperature in Fahrenheit to the number of chirps different species of cricket make in one minute.

\mathrm{Field~Cricket:} \; T_{f} &= 50 + \frac{(N-40)}{4} \\
\mathrm{Snowy~Tree~Cricket:} \; T_{f} &= 50 + \left[\frac{(N – 92)}{4.7}\right] \\
\mathrm{Katydid:} \; T_{f} &= 60 + \left[\frac{(N – 19)}{3}\right]

It is a popular myth that crickets chirp by rubbing their legs together but that couldn’t be further from the truth. Only the males chirp and they produce sounds through the process of stridulation i.e by rubbing certain body parts together. In the case of crickets, there is a stridulatory organ, a large vein running along the bottom of each wing, that is covered with “teeth” or serrations like a comb. The chirping sound is created when the cricket runs the top of its wing along the bottom serrated edge of the other. As the cricket does this, it holds its wings up and open so they act as acoustical sails.

Insects are cold-blooded and take the temperature of their surroundings. As a result, a cricket’s chirp rate depends on its metabolism. This can be determined by the Arrhenius equation which describes the activation energy needed to induce a chemical reaction. The less energy there is, the slower a chemical reaction and hence metabolism. As temperatures rise, more energy is available for muscle contractions. This accounts for the relation of observed by Dolbear.

Measuring Dolbear’s Law

We know by the end of the show that Howard was right and Sheldon was wrong. We can calculate the number of chirps with temperature for the various cricket species seen on the whiteboard.

Table showing Temperature and Number of Cricket Chirps per Minute
Temperature(°F) Temperature(°C) Field Cricket Snowy Tree Cricket Katydid
60 16 80 139 19
65 18 100 163 34
70 21 120 186 49
75 24 140 210 64
80 27 160 233 79
85 29 180 257 94
90 32 200 280 109

There are noticeable differences in chirp rates between the snowy tree cricket and the field cricket–the snowy tree cricket has a faster chirp rate. But how could Sheldon been wrong? According to the roommate agreement, the apartment is supposed to be 71°F(22°C) which translates to a chirp rate of 191 for the snowy tree cricket. Assuming Sheldon counted the chirps correctly, that would translate to a much warmer temperature of 88°F(31°C) for the field cricket. Could someone have changed the temperature in violation of the roommate agreement? Surely this would have been noticed without looking at the thermostat. Your guess is as good as mine.