Analyses of dimensionless science

I read with interest the feature article “Dynamic similarity, the dimensionless science” by Diogo Bolster, Robert Hershberger, and Russell Donnelly. The authors give a simple example of dimensional analysis of the deflection of a light beam passing through a star’s gravitational field. But the example stretches the possibilities of dimensional analysis beyond its true limits. In fact, the deflection angle could be any nontrivial function of the dimensionless fraction Gm/c²r. Thus at least one more assumption outside dimensional reasoning must be made—the simplest solution, as the authors imply in the given example.

A nontrivial solution may be demonstrated by performing the same exercise with the Planck law, which gives the radiation spectrum emitted by a blackbody. The relevant quantities are wavelength λ, temperature T, spectral emittance B_λ, speed of light c, Planck’s constant h, and Boltzmann’s constant k. According to the Buckingham Pi theorem, two independent dimensionless quantities can be formed using this list of quantities. An example of a solution is

One dimensionless quantity forms the left-hand side of the equation; the second, independent one occurs in the exponent of the denominator of the Planck law. Such a result could not be found by dimensional reasoning alone.