Reciprocity Failure Caused by Long Exposure Time
Fact or Fiction?

Great astronomer Karl Schwarzschild published his famous formula to describe reciprocity failure in 1900 (ApJ, 11, 89).  In his article, he wrote "sources of light of different intensity I cause the same blackening under different exposure t if the product I * t^0.86 are equal."  Nowadays we don't use the term "blackening" to describe what is on the plate/film.  Instead, we use "density." For different kind of emulsion, Schwarzschild's formula can be generalized to

E = I * t ^p,                                        (1)

where E is effective exposure that causes constant density on the plate/film and p is called Schwarzschild factor or p factor.  The Schwarzschild factor varies from emulsion to emulsion.

Mathematically, equation (1) can be written in another form:

E = I ^(1+q) * t,                                    (2)

where q satisfies p=1/(1+q).  Although equations (1) and (2) are mathematically equivalent, their interpretation might be different.  By comparing equation (1) with the law of reciprocity E = I * t, we know the simplest interpretation of equation (1) is that the film speed is decreased by a factor of R, where

R = t ^(p-1).                                          (3)

Similarly, equation (2) implies

R = I ^q.                                              (4)

Which of the two equations (3) or (4) is correct? Both are correct if we strictly follow Schwarzschild's instruction: "... cause the same blackening ...," i.e., the same density.  Many people compare different effective exposures that cause different densities, but they still use Schwarzschild’s original formula (1) or its direct interpretation (3).  This is the source of a long lasting disaster misunderstanding that has been widely spread among photographers: film loses its speed as exposure time gets longer and longer.  This is simply wrong.

What is correct?  I highly recommend to you the book written by Robert Reeves, "Wide-Field Astrophotography."  In page 219, Robert wrote "The root of reciprocity problems is not the length of the exposure, but the intensity of the light we try to record during it.  Reciprocity failure is caused by the emulsion becoming progressively less efficient in recording the light as the intensity of that light decreases…"  As Robert cautioned, the key is intensity I, not exposure time t.  Only when the effective exposure is fixed (thus the density is also fixed), exposure time and light intensity can be translated into each other by using Schwarzschild's original equation (1).  Only in this case, is it (mathematically) correct to say that film speed loss is caused by long exposure time.  But this is not the case in most astrophotography.  Let me use a textbook example to explain this.

Example 1.

In the book "Astrophotography for the Amateur" written by Michael A. Covington, section 10.7, the author stated that under no reciprocity failure and constant aperture, in order to get one-stop increases in densities, the exposures would be 5, 10, 20, 40... minutes.  This is correct.  However, then the author further stated that under reciprocity failure and constant aperture, in order to get one-stop increases in density, the exposure would have to be 5, 15, 45, 135... minutes.  The exposure time difference between each step becomes a factor of 3 and the author called this reciprocity failure.  This is wrong.  As long as the aperture is fixed, the light intensity on the focal plane is fixed, and therefore the film speed is fixed.  This is explicitly written in Robert’s book, page 220.  Under reciprocity failure, at a fixed aperture, the exposures to get one-stop increase in densities should be still 5, 10, 20, 40... minutes.

Why do many people (including Michael) think that longer exposure time produces greater film speed loss?  The reason is that they didn't follow the condition that Schwarzschild set for his equation.  In the example above, Michael aimed to get series of densities that is differed by one stop.  This violates Schwarzschild's "constant density" criterion and therefore equation (3) and equations in page 180 of Michael's book are not valid any more.

Schwarzschild's formula describes how intensity and exposure time are related, under constant density (and thus constant effective exposure).  When the density or effective exposure is not a constant any more, the actual behavior of film is described by equation (2) and (4), not (1) and (3).  Equation (4) determines film speed loss based on light intensity but not exposure time.  This is what reciprocity failure is.  The film speed loss is caused by low light intensity, but not long exposure time.

Example 2.

This example explains why exposure time is not a real indicator of reciprocity failure. People use to say, we need to start to worry about reciprocity failure when the exposure time is longer than 1 second.  Suppose the light meter of a camera suggests an exposure time of 1/4 second, under a specific aperture and some lighting condition. What happens if we intentionally use an exposure time of 2 second without changing the aperture? Will there be any reciprocity failure simply because the exposure time is longer than 1 second? No. We will get a 3-stop over-exposed film, as over-exposed as we expect from no reciprocity failure. There will be no film speed loss at all, because the light intensity is not changed. 

If so, why do people (including film data sheets published by Kodak, Fujifilm, Agfa, … etc.) talk about reciprocity failure in terms of exposure time? Don't forget the fact that these exposure times are the "correct" exposure times suggested by a light meter. Light meter is a machine that strictly obeys the reciprocity law, or the p=1.0 version of Schwarzschild's formula. A light meter always aims to obtain a constant density. This satisfies the condition that Schwarzschild set for his formula. Therefore for a metered exposure, it is OK to talk about reciprocity failure in terms of either exposure time or light intensity. Do we use a light meter for deep sky photography? In example 2, by over exposing 3 stops, do we still aim for the same density as the light meter does? No.

Example 3

This example explains what is valid and what is not, for Schwarzschild's formula. Suppose a film has p=0.80 and therefore q=0.25.  A combination of exposure time and light intensity, t₀ and I₀, gives the desired effective exposure and density.  After the light intensity is decreased by a factor of 2.0 (i.e., I' = 0.5I₀), what exposure time should we use to reach the same density?  No matter we use equation (1) or (2), we get the conclusion that the exposure time t' must be 2.38 times t₀.  This shows the equivalence of the equations (1) and (2), provided that we are talking about a fixed effective exposure or a fixed density.

Let's go back to example 1.  What can we do if we want to double the effective exposure (i.e., we want a one-stop increase in density)?  Suppose the light intensity is fixed and we only want to change exposure time.  Equation (1) says that t'=2.38t₀ is needed in order to double the effective exposure, while equation (2) says only t'=2.0t₀ is needed.  Which one is correct?  The answer is equation (2).  Now we are not aiming at a constant effective exposure so we cannot use equation (1) any more.  In this case, since reciprocity failure is caused by low light intensity and the light intensity is not changed here, the film speed remains constant.  We only need to double the exposure time to double the effective exposure.

If we fix the exposure time and want to double the effective exposure, what should we do to the light intensity?  We hopefully had learned that equation (2) is the right one to use.  According to equation (2), we need to use I' = 1.74I₀.  Because we increase the light intensity, the film speed loss decreases.  Not surprisingly, we double the effective exposure by increasing the intensity for less than 2 times.

I’m not saying that Schwarzschild's formula and p factor are wrong.  They are perfectly correct if the users pay attention to Schwarzschild's constant density criterion.  However, example 3 does show that equation (1) is hard to generalize.  It only works for constant density.  Between equations (1) and (2), it is historically unfortunate that Schwarzschild chose (1) but not (2).  This causes a misunderstanding that lasts for a century.  It's not Schwarzschild's fault. In 1900, people still did't know the cause of reciprocity failure.  It seemed equally fine to put a power index on either I or t in 1900.  Now we know what causes reciprocity failure.  Equations (2) and (4) are better ways to describe reciprocity failure and film speed loss.