## Gainer reciprocity compared to Schwarzschild

Well, I couldn't let this go without comparing the results of Gainer's equations based on Bond's testing with the old standard Schwarzschild formula and tests done by Covington and Reeves.

For those unfamiliar with the Schwarzschild exponent, I'll just briefly explain it (using "spreadsheet mathematical notation"). Let p be the Schwarzschild exponent and t equal to exposure time in seconds. For a given effective film speed to be corrected in the range where the reciprocity law fails, the following equation applies:

Effective speed = Standard speed * t^(p-1)

Covington (page 180, Astrophotography for the Amateur, 2nd ed. ISBN 0-521-62740-0) says the following is more accurate, especially nearer the range where long exposure reciprocity is just beginning to fail:

Effective speed = Standard speed * (t+1)^(p-1)

This equation can be reworked to give revised exposure times as opposed to film speed:

Corrected exposure time = (Metered exposure time+1)^(1/p)-1

I've picked one film tested by both Covington and Bond, Ilford Delta 100, which Gainer gives a reciprocity factor of 0.046 using his equation. Covington's 1996 tests on Delta 100 assign an approximate Schwarzschild exponent of 0.85. Covington's tests are approximations to about 1/3 stop. I haven't read the Bond article, so I don't know his method, but from his other work, I expect accuracy and meticulous testing to be the norm, probably to a higher degree than Covington for several reasons. Astrophotographers are plagued by a host of other factors that rapidly overtake attempts to be very highly accurate when calculating exposure times, so it's pointless in that application to be overly careful with such testing. They mostly do this for relative comparsion of emulsions in deciding which film to use, not to nail down exposure times to a fraction of a stop with a particular film. Covington tests with this in mind and warns against using his results for elaborate calculations.

Given all that, I forged ahead and graphed the results from Covington's 1996 emulsion test and Bond's 2003 tests on Ilford Delta 100, using the appropriate Covington revision of the Schwarzschild formula and the Gainer formula describing Bond's data. The attached graph is the result. Covington's results run about 1/3 stop more correction early on, but Gainer's formula crosses over at a bit over 400 seconds metered exposure time. In practical terms the difference in using either equation in this instance is about 1/3 stop or less. Covington reports that reciprocity failure variations from batch-to-batch emulsion changes can be greater than this.

Covington (and Reeves in Wide-Field Astrophotography, ISBN 0-943396-64-6) list Schwarzschild exponents for a number of films in their books, but all on pre-2000 emulsions. Reeves includes 45 film tests, 12 of them B&W.

As Pat Gainer says, the testing is time consuming, but you could use the methods of Covington (also adopted by Reeves) to do some rough testing using a few frames on each roll during "normal" photography to get a data point for determining a Gainer film factor, or a Schwarzschild exponent for a given film. As always, doing this with sheet film will require more work, and very consistent processing methods.

Lee