Schwarzschild Law

updated:  2024-04-03

The Schwarzschild effect  or Schwarzschild law is a phenomenon that occurs during exposure in analog or chemical photography.

The reciprocity law of Bunsen and Roscoe (1862) states that mathematically equal products of exposure time and intensity result in the same blackening: 

I: Intensity (e.g. of the starlight )

t: Exposure time 

If, for example, the aperture is closed by one value (or the Intensity halved) but the exposure time is doubled, the film material should have the same level of blackness.

In 1899, the astronomer and physicist Karl Schwarzschild discovered that the sensitivity of a photographic layer decreased exponentially with exposures of more than one second (the time depends on the film material used), i.e. the expected blackness did not match the achieved blackness. 

The images were underexposed - with the same light dose but longer exposure times. To compensate for this effect, the images must be exposed for longer. For this purpose, the manufacturers offer data sheets from which the corresponding exposure times can be taken.

The reciprocity law of Bunsen and Roscoe was brought by Schwarzschild to the form 

which agrees well with experimental experience, for the range of low intensity and long time exposures he investigated,

Color films are particularly affected by the Schwarzschild effect, as the different emulsion layers for the individual primary colors show this effect to different degrees, which can result in color casts. These already occur from around 1/30 s. For exposure times above 1 s, tungsten color films are recommended for which no exposure time extensions are required up to 5 s due to the black shield effect.

This effect does not occur in digital photography, as the CCD or CMOS image sensors used do not lose sensitivity. However, other effects, such as dark noise, are noticeable in long exposures with low light.

Schwarzschild law

Karl Schwarzschild developed a formula that is used to determine the intensity of the blackening of a photographic emulsion.[1] This formula contains an exponent, which he named the Schwarzschild law:


i: Blackening of the photographic plate 

f: Blackening curve (see gradation and saturation) 

I: Intensity (e.g. of the starlight )

t: Exposure time 

p: Schwarzschild exponent, for real films always p<1 (usually just under 0.9) 

So that a faint galaxy can be imaged twice as large in astrophotography (with the same telescope aperture and the same blackening of the photographic plate), the exposure time must not just be quadrupled, but approximately quintupled. 

The formula does not match the mechanism of the Schwarzschild effect, is therefore only valid for small variations and only has historical significance.

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