Kepler fits Compton

1 Kepler fits Compton Adolf Cusmariu [email protected] A photon bouncing at angle q off a stationary electron undergoes unexpectedly a wavelength increase [1], the impact also recoiling the electron at angle f ; see graphic below (photon in red, electron in blue). After years of experiments and theoretical efforts to explain this increase, Compton derived the now-famous expression for the positive change Dl as Dl = l(q) - l 0 = l c(1 - cos q ) where q = Photon bounce angle l c = Compton wavelength of the Electron = 0.002426 nm using some powerful modern physics; truly ‘standing on the shoulders of giants’, as Newton might say. Now, given his angular data set Deg = [0 45 90 135] (or in radians) Rad = [0 0.7854 1.5708 2.3562] and the corresponding wavelength measurements (in nanometers) Lam = [.0709 .0715 .0731 .0749] what might Kepler have done to fit a model here, as he finally fit an ellipse through orbital measurements on the planet Mars; that is, to find a functional relationship between wavelength and angle: Lam = f(Rad) ? A simple plot of wavelength vs. angle isn’t particularly suggestive (see below) q f l(q) l0 2 Although wavelengths corresponding to non-zero angular measurements appear to line up, there is no obvious functional dependence overall. After various false starts, Kepler might perhaps have stumbled on a way to linearize, by changing angles trigonometrically as cos(Rad) Then a plot of wavelength vs. cos(angle) is promisingly linear indeed! Even better, 1 - cos(rad) would make things positive and allow an easy estimate of slope (see plot below): The intercept of this (almost) line is clearly Lam(1) = 0.0709. A model would now be available to Kepler as Lam - Lam(1) = Slope*(1 - cos(rad)) And a Slope estimate would be easy: Slope = range(Lam(2:4))/range{1-cos(Rad(2:4))} ≈ .002404 nm very close indeed to l c = 0.002426 nm. Of course, the physical meaning of this Slope would escape Kepler (we think). [1] A. H. Compton, ‘A Quantum Theory of the Scattering of X-Rays by Light Elements’, Phys. Rev, 21, 1923.