equivalent S/N = exposure-time scaled by f-ratio, regardless of aperture.
relative S/N = exp_time / f-ratio^2.
Example: a 10-minute exposure with a 10” f/10 scope is equivalent to a 5-minute exposure with a 10” f/7 scope.
This is false!
Varying the f-ratio of a constant aperture has little or no affect on real S/N, except in certain limited circumstances. The relationship of exposure-time and f-ratio only holds true for equivalent focal lengths, which means the aperture must be varied to produce a given f-ratio.
The “CCD f-ratio myth” originates in people’s experience with film-based photography, where the exposure-time/f-ratio relation is practically considered a “law of nature”. But that “law” is actually a consequence of some peculiar properties of film emulsions that largely do not affect CCD imaging (though there are some potentially analogous properties).
Film emulsion response is sensitive to spatial-flux (photons/area) and, to a lesser extent, temporal-flux (the “reciprocity” effect). Film response diminishes with both low and high spatial-flux so it is highly desirable to optimize photons per area. This is done by varying exp-time &/or f-ratio, which gives rise to the “exposure-time / f-ratio law”.
CCD response is basically insensitive to spatial-flux and time-flux. A photon’s probability of detection is determined solely by QE regardless of spatial or temporal flux (or pixel size or number of pixels). That’s why CCD is said to be a “linear detector”. The “exposure-time / f-ratio law” is inappropriate for CCD, although there are two CCD characteristics (saturation and readout-noise) that can roughly mimic this relationship under certain circumstances.
Varying CCD exposure-time or f-ratio varies the number of photons per pixel. If the number of photons approaches the holding capacity of the pixel (“full well”) then saturation occurs and the detector is no longer linear. Thus a fast f-ratio &/or long-exp-time may result in an “over-exposure” that is roughly analogous to film.
CCD low-flux situations are complicated by camera noise (primarily readout noise and to a much lesser extent, dark noise). If the noise from the object and sky-glow is significantly larger than the camera noise then that camera noise has little effect on the image (due to the quadratic nature of noise). But if the object and sky are both low-level (dim or short exp) then camera noise may become significant and degrade the image. Because camera noise is pixel based, this potential degradation is sensitive to the number of pixels used to capture an object or sky-area, and thus it is sensitive to f-ratio. This effect can roughly mimic film’s “under-exposure” in some circumstances (e.g. very short exposures), though the CCD S/N function is actually very different from film’s response function.
One way to look at it might be to say that film has a narrow “sweet spot” that requires a certain spatial flux, but CCD accommodates a much larger “zone”.
True S/N (Object S/N)
All of the above is very interesting but it actually says almost nothing about the relationship of exposure-time, aperture, f-ratio, etc. to “true S/N”!
“True S/N” (or “object S/N”) refers to the actual information content of the image. Object S/N measures information about the target-object and determines important qualities of the image, such as limiting magnitude and feature contrast and visibility. Object S/N is primarily determined by object brightness, aperture, and camera QE. F-ratio itself has virtually no effect on object S/N, except for some potential secondary camera noise effects (discussed below).
Information about an astronomical object (star, galaxy, nebula, features of galaxy or nebula, etc.) is contained in the light that falls onto Earth. That light consists of a certain number of photons per second per square meter of earth’s surface.
The quality of information from an object depends on how many photons are captured and measured by the instrument. The number of object photons available to the camera is solely determined by:
1) Object flux (photons/second/square-meter)
2) Aperture size (square-meters) and efficiency
3) Exposure time
Focal length (and thus f-ratio) has absolutely no effect on the number of photons collected and delivered.
Note that the collected information also contains noise from 2 sources:
1) Poisson noise of the object (square-root of the number of photons)
2) Poisson noise of the sky-glow that occupies the area of the target
A perfect/ideal scope and camera would detect all of those photons and contribute no noise, thus yielding the full S/N delivered by the scope:
Image S = ObjectFlux * Time * Aperture
Image N = sqrt(S + SkyGlow)
But the situation is different for real scopes and cameras, with QE (Quantum efficiency) and pixel noise (primarily readout).
Image S = ObjectFlux *QE * Time * Aperture * ScopeEfficiency
Image N = sqrt(S + SkyGlow + (PixNoise^2 * numPix))
I should also mention here another limitation of real cameras – detector size. Obviously the detector should be larger than the object’s projected size (determined by the object’s angular extent and the scope’s focal length).
F-ratio has no effect on S/N from the perfect/ideal camera. But f-ratio can affect the S/N from a real camera by varying the number of pixels used to capture the object (numPix). This effect is often miniscule, especially for images of bright objects or long exposure deep space images (where SkyGlow dominates noise). But for very short exposures or narrow-band images of dim objects, the pixel noise can be very significant. (This is why binning is so beneficial for very short exposures, an effect that is often erroneously attributed to increased “pixel S/N” of combined pixels.)
Thus there is an actual relationship between S/N and f-ratio, but it is not the simple characterization of the “f-ratio myth”.
Below is an illustration of the principle. Each image is a 10-minute exposure using the same aperture (Tak CN-212) at vastly different f-ratios. If the “f-ratio myth” was true then the f/3.9 exposure should be 10 times “better”!
© Stan Moore 2005