## Aperture, Focal-Length, and F-Ratio Facts

### by Robert J. Vanderbei at Princeton University

Regarding prime-focus imaging, we have the following:
• For extended sources (i.e., everything except stars), the average number of photons arriving at the image plane per second per unit area (in the focal plane) is proportional to the brightness of the object times the square of D/f. Here, D is aperture and f is focal length so that f/D is f-ratio. The photon arrival rate, of course, is inversely related to minimum exposure time required to achieve a "nice" image.
• For point sources that are under sampled (i.e., stars that fall entirely on one pixel---something one normally tries to avoid), the average number of photons arriving at the image plane per second accumulating in that one pixel is proportional to the brightness of the object times the square of D. But, if the f-ratio is larger than some threshold, even stars become extended sources. That threshold depends on the pixel size---see the next bullet.
• The null between the Airy disk and the first diffraction ring is at 1.2 l f / D distance from the center of the Airy disk. Here, l is wavelength of visible light. l f / D has units of length. For green light (l = 0.5 microns) and f/D = 10 (typical SCT), this works out to 6 microns, which is a little bit smaller than the pixel size of current generation CCD cameras.
• The field of view as an angular measure is given by L/f, where L is either the length or width of your camera's CCD chip. If L and f are in common length units, such as millimeters, the angle is in radians. To convert radians to degrees, multiply by 360 and divide by 2 p. Example: for a chip that is 10mm on a side and a telescope with a 2000mm focal length, the field of view is (10/2000)*360/(2 p) = 0.28 degrees = 17 arcminutes.
• Limiting depth is determined solely by background sky glow. The three most effective ways to combat light pollution are to use a light pollution filter or special purpose narrow-band filters (such as Ha and O-III) or relocate to a dark-sky location.
So, ignoring ancillary factors like brightness of the source or size of the CCD chip, we can summarize as follows.
• F-ratio determines speed at which a nice non-grainy image can be formed.
• Focal length determines field of view (for a given chip size).
• Aperture determines Rayleigh resolution, although in practice resolution is usually limited by seeing.

Here's how to determine how long it will take to get a "nice" picture:

• After centering the object and focusing, take a short-exposure test image. Use, say a one-minute exposure.
• Stretch the image so that the black point and white point are about where they would be in a properly stretched final version.
• Determine the "range", which we define as the difference between the white set point and the black set point.
• Ideally, we'd like to integrate long enough that the range is about 60,000 (so that the S/N is roughly 250, which is about equal to the number of different brightness levels the eye is sensitive to). So, if the one-minute exposure gives a range of 1000, then you will need 60 minutes. In other words, take 60,000 and divide by the range achieved in a one-minute exposure to get the number of minutes required.
• If the exposure required is too long, consider using a focal reducer. A 0.5x focal reducer will get you to 60000 in one-fourth the time. But it will dramatically change the field of view which might not be what you want.
• Of course, a one-hour exposure should be broken into a sequence of subexposures. If the subexposures are too short, then read noise could limit the depth of the final image. If the subexposures are too long, then the risk of airplane and satellite trails and tracking errors become an issue.
Here's an anecdote:
My first telescope was an 3.5" f/15 Questar.
Thought it would be fun to do astrophotography with it.
Bright things, especially the planets and globular clusers, came out well but faint things took forever.
Why, you ask? Is it the small aperture of only 3.5" or is it the slow f-ratio of f/15?
Well, it's not easy to increase the aperture (without buying a whole new system) but it is easy to change the f-ratio with a focal reducer.
So, I bought a Meade 0.33x focal reducer. This is a positive lens.
It concentrates the light so that an area that was originally 9 (=3x3) times larger now actually falls on the CCD chip.
The result is a three times wider field of view and nine times as many photons per second landing on the chip.
So, it is the f-ratio that determines how long one must integrate.

Finally, for visual observation the basic facts are different as the eyepiece and human eye complicate the story. Simply put, as long as the magnification is sufficiently high that the exit pupil does not overfill your eye's pupil, then aperture determines image brightness (at a given magnification). In this regime, f-ratio is largely unimportant. See http://skyandtelescope.com/howto/visualobserving/article_1671_1.asp for a lengthier discussion of the relevant principles for visual observers.