Stefan's Fit of the Dulong and Petit Data for the Velocity of Cooling in a Vacuum

  In March 1879 Stephan pointed out that the velocity of cooling data obtained by Dulong and Petit could be calculated with an radiation function involving the 4th power of the absolute temperatures. How might he have discovered this? We start as he did with the velocities of cooling for an ambient temperature of 0 °C. In a manner similar to that previously used we can assume the difference of the two rates is Δv=A+Be^λT where T is the absolute temperature of thermometer in the vacuum given in Kelvins. If we assume a value for λ we can use linear least squares to find the remaining coefficients and search for a value of λ which will minimize the error.

Next we subtract A from Δv to get an estimate of the rate for emission of heat radiation. The data appears to be fairly linear in a log-log plot so we look for another set of coefficients for a second fit. A little math allows us to convert the constant term into a factor.

This fit is fairly good but noticing the value of B we are tempted to replace it with an integer, n, to see what we get for a third fit. Assuming n=4 the only unknown is the coefficient a and a search can be used to obtain the best fit. The search works better than averaging ratios of Δv/(T⁴-T₀⁴) to find a.

One gets calculated values and errors very close to those of Stefan.

Supplemental (Apr 1): The averaging of the ratios procedure shows that using some other integer n to compute the coefficient a verifies that n=4 gives the least error.

Edit (Apr 1): Caught an error in the first fit so had to redo it. The fit assumed the formula shown above rather than the original formula shown, Δv=Ae^λT-B. This affected the second fit also since the coefficients were confused and B was added to Δv while A should have been subtracted. Using a log-log fit can can bias the errors somewhat so it is best to work with the original data. The second fit above suggests a 3rd power law and one finds mention of it in some publications. The mistake didn't affect the last fit or the comparison of power laws.

Supplemental (Apr 1): Curvature in a semi-log plot might have suggested the quadratic emission power law mentioned by Dulong and Petit. The slight curvature in the plot of the data in the second suggests the presence of a systematic error in the log of the velocity and even greater fit errors in the velocities themselves.