Reciprocity (photography) - Schwarzschild Law

Schwarzschild Law

In response to astronomical observations of low intensity reciprocity failure, Karl Schwarzschild wrote (circa 1900):

"In determinations of stellar brightness by the photographic method I have recently been able to confirm once more the existence of such deviations, and to follow them up in a quantitative way, and to express them in the following rule, which should replace the law of reciprocity: Sources of light of different intensity I cause the same degree of blackening under different exposures t if the products are equal."

Unfortunately, Schwarzschild's empirically determined 0.86 coefficient turned out to be of limited usefulness. A modern formulation of Schwarzschild's law is given as

where E is a measure of the "effect of the exposure" that leads to changes in the opacity of the photosensitive material (in the same degree that an equal value of exposure H = It does in the reciprocity region), I is illuminance, t is exposure duration and p is the Schwarzschild coefficient.

However, a constant value for p remains elusive, and has not replaced the need for more realistic models or empirical sensitometric data in critical applications. When reciprocity holds, Schwarzschild's law uses p = 1.0.

Since the Schwarzschild's law formula gives unreasonable values for times in the region where reciprocity holds, a modified formula has been found that fits better across a wider range of exposure times. The modification is in terms of a factor the multiplies the ISO film speed:

Relative film speed

where the t + 1 term implies a breakpoint near 1 second separating the region where reciprocity holds from the region where it fails.