# Wide-Field Astrophotography Calculations

## Field of view

The formula for determining the angular field of view of a lens (and this works for any film format) is:

V = 2 * arctan[S/(2*F)]

where V is the field of view, S is the film size (a 35mm frame is 36mm by 24mm), and F is the focal length of the lens. This formula breaks down for fish-eye lenses, because with respect to the sky they all have a 180 degree field of view.

## Plate scale

Using the same definitions, plate scale is given by

P = V/S.

With this, you can determine how large an object will appear on the film by taking the size of the object divided by the plate scale. This also doesn't work for fish-eye lenses because the plate scale isn't constant across the image. In the table at the bottom of the page, I express the plate scale in arcmin per mm because the numbers are more convenient.

## Effective aperture

The formula for the effective aperture of a lens at a given f-ratio is simply

d = F/f,

where d is the aperture, F is the focal length of the lens, and f is the f-ratio. The effective collecting area, A, (directly proportional to the number of photons of light that will hit the film) is given by pie are square ;-)

## Relative stellar limiting magnitude

Since stars are point sources, given a particular film, the effective aperture of the lens determines the limiting magnitude for recording stars (or planets or asteroids) and not the f-ratio. Given a film and exposure time combination that reveals a 8.0 magnitude star with a 50mm lens at f/2.8 (a couple minutes on fast film), the 'Mag' column gives the limiting stellar magnitude for an identical exposure for each lens/aperture combination. Basically, if the lens area is doubled (one f-stop), a star twice as bright is visible. Magnitudes are on a logarithmic scale, so that a change in one magnitude corresponds to a factor of 2.512 change in brightness. The formula is still pretty simple, though:

Mag = 8.0 + 2.512 * log10(A2/A1)

where A1 is the reference collecting area (252 mm^2 in this case), and A2 is the collecting area of the lens in question. This formula can be used for any lens combinations and reference magnitudes. However, given an exposure time to reach 8.0 for a certain lens, determining how long it will take to reach 8.0 on another lens will not generally give a good result. This is due to reciprocity failure, the tendency for films to become less sensitive the longer they are exposed, which for long exposures is not very well determined for films. Also, if you have light pollution in your area, sky fog can foul things up.

nd now the most important caveat: These times do not take in consideration lens aberrations at large apertures. A wide-open lens will suffer from at least a small amount of aberration and this ruins the point source argument, because aberrations will cause the stellar image to appear more extended. It's best to experimentally determine this for your lenses and then compare to the values given in the table. On top of this, longer focal length lenses are harder to track accurately, and this enlargens the stellar images. Thus, it's best to apply these comparisons between lens/aperture combinations to lenses of similar focal lengths. The actual limiting magnitude you will reach in an exposure depends upon the quality of the optics, the quality of the tracking, the quality of the sky, and the characteristics of the film. Experiment.

For extended objects, it's the f-ratio that counts. Basically, the brightness of an extended object on film is proportional to the f-ratio squared. Thus, while a 135mm lens at f/2.8 will show stars two magnitudes fainter than a 50mm lens at f/2.8, extended objects will appear the same brightness. However, the image will be almost 3 times as large, with a similar increase in the amount of detail visible. For tips on specific extended objects see the suggested targets page.

## Tracking accuracy in time

The next-to-last column gives an estimate of the time accuracy in the tracking. This is most useful for people using hand-powered barndoors. These trackers require that you turn a screw a set amount every few seconds. The fewer times you do this during an exposure the less chance you have of bumping the tracker enough to affect the image. However, you must rotate the screw often enough to prevent the star images from trailing significantly. A typical medium speed film has a maximum possible resolution of 100 lines/mm, and that's what I'm starting with. However, the inherent vibrations that will occur during any type of tracking, and the fact that you are not shooting at the optimum f-ratios for the best images prevents that sort of resolution from being acheived. My criteria is a factor of 3 worse than the maximum resolution. The formula is then:

T = P / 8.33

where T is the maximum time between adjustments, P is the plate scale in arcmin per mm, and 8.33 is the 0.25 arcmin per second rotation rate of the earth times the 33.33 lines/mm resolution. Note that if you are shooting on a particularly fine-grain film you may need to be more accurate. Conversely, if you are shooting a very fast film with grain the size of a small farm animal you have more leeway. However, if the limiting factor really is vibrations, that will dominate any film grain effects.

If you've been paying attention, you may have noticed that something's missing from the tracking accuracy calculations; the declination of the object. The closer you are to the celestial poles, the less accurate tracking needs to be. The times in the table should be divided by the cosine of the declination. Thus, for a 50mm lens an exposure at the celestial equator needs better than 7.9 second time accuracy, at 45 degrees the needed accuracy is 11.2 seconds. I recommend turning the screw on a time interval equal to about 1/2-3/4 of the maximum allowable. Personally, when shooting a 50mm lens I turn the screw every 4 seconds, and for a 135mm lens I turn the screw every 2 seconds.

## Maximum image size for sharp stars

Based upon the plate scale and the 33.33 lines/mm resolution, you can determine the maximum amount of deviation from absolutely perfect tracking that will still yield sharp images. This is given by

M = P / 33.33

and is listed in the final column of the table. Note that this is the total of all possible effects including polar alignment, vibrations, bad tracking, etc.

## The table

The following table contains values for 35mm film and typical lens settings (the plate scale is given in arcminutes per mm). I don't give numbers for fish-eye lenses because the plate scale isn't constant across the image, and limiting magnitude numbers are pretty meaningless. Except for the longer telephotos lenses, I stop at f/4 since most good lenses shorter than 135mm are f/2.8 or faster and thus you will usually be shooting at f/4 or slower. If you notice a missing lens/aperture combination faster than f/4, let me know and I'll put them in the table.

This table uses the PRE markup as opposed to a real HTML table. Maybe I'll convert it at some point.

```F(mm) f-ratio  d(mm)   V(deg) P('/mm) A(mm^2) Mag  T(sec) M(')
17    f/2.8    6.1   93 x 70   155    29.2   5.6  18.6   4.7
17    f/3.5    4.9   93 x 70   155    18.9   5.2  18.6   4.7
17    f/4      4.3   93 x 70   155    14.5   4.9  18.6   4.7

20    f/2.8    7.1   84 x 62   140    39.6   6.0  16.8   4.2
20    f/3.5    5.7   84 x 62   140    25.5   5.5  16.8   4.2
20    f/4      5.0   84 x 62   140    19.6   5.2  16.8   4.2

24    f/2     12.0   74 x 53   123     113   7.1  14.8   3.7
24    f/2.8    8.6   74 x 53   123    58.1   6.4  14.8   3.7
24    f/3.5    6.9   74 x 53   123    37.4   5.9  14.8   3.7
24    f/4      6.0   74 x 53   123    28.3   5.6  14.8   3.7

28    f/2     14.0   65 x 46   109     154   7.5  13.1   3.3
28    f/2.8   10.0   65 x 46   109    78.5   6.7  13.1   3.3
28    f/3.5    8.0   65 x 46   109    50.3   6.2  13.1   3.3
28    f/4      7.0   65 x 46   109    38.5   6.0  13.1   3.3

35    f/1.8   19.4   54 x 38   90.7    296   8.2  10.9   2.7
35    f/2     17.5   54 x 38   90.7    241   8.0  10.9   2.7
35    f/2.8   12.5   54 x 38   90.7    123   7.2  10.9   2.7
35    f/3.5   10.0   54 x 38   90.7   78.5   6.7  10.9   2.7
35    f/4      8.8   54 x 38   90.7   60.8   6.4  10.9   2.7

45    f/2     22.5   44 x 30   72.7    398   8.5   8.7   2.2
45    f/2.8   16.1   44 x 30   72.7    204   7.8   8.7   2.2
45    f/3.5   12.9   44 x 30   72.7    131   7.3   8.7   2.2
45    f/4     11.3   44 x 30   72.7    100   7.0   8.7   2.2

50    f/1.2   41.7   40 x 27   66.0   1366   9.8   7.9   2.0
50    f/1.4   35.7   40 x 27   66.0   1001   9.5   7.9   2.0
50    f/1.7   29.4   40 x 27   66.0    679   9.1   7.9   2.0
50    f/2     25.0   40 x 27   66.0    491   8.7   7.9   2.0
50    f/2.8   17.9   40 x 27   66.0    252   8.0   7.9   2.0
50    f/3.5   14.3   40 x 27   66.0    161   7.5   7.9   2.0
50    f/4     12.5   40 x 27   66.0    123   7.2   7.9   2.0

58    f/1.2   48.3   34 x 23   57.5   1832  10.2   6.9   1.7
58    f/1.4   41.4   34 x 23   57.5   1346   9.8   6.9   1.7
58    f/1.7   34.1   34 x 23   57.5    913   9.4   6.9   1.7
58    f/2     29.0   34 x 23   57.5    661   9.1   6.9   1.7
58    f/2.8   20.7   34 x 23   57.5    337   8.3   6.9   1.7
58    f/3.5   16.6   34 x 23   57.5    216   7.8   6.9   1.7
58    f/4     14.5   34 x 23   57.5    165   7.5   6.9   1.7

85    f/1.7   50.0   24 x 16   39.9   1963  10.2   4.8   1.2
85    f/2     42.5   24 x 16   39.9   1419   9.9   4.8   1.2
85    f/2.8   30.4   24 x 16   39.9    726   9.2   4.8   1.2
85    f/3.5   24.3   24 x 16   39.9    464   8.7   4.8   1.2
85    f/4     21.3   24 x 16   39.9    356   8.4   4.8   1.2

105    f/2     52.5   19 x 13   32.4   2165  10.3   3.9   0.97
105    f/2.8   37.5   19 x 13   32.4   1104   9.6   3.9   0.97
105    f/3.5   30.0   19 x 13   32.4    707   9.1   3.9   0.97
105    f/4     26.3   19 x 13   32.4    543   8.8   3.9   0.97

135    f/2     67.5   15 x 10   25.3   3578  10.9   3.0   0.76
135    f/2.8   48.2   15 x 10   25.3   1825  10.2   3.0   0.76
135    f/3.5   38.6   15 x 10   25.3   1170   9.7   3.0   0.76
135    f/4     33.8   15 x 10   25.3    897   9.4   3.0   0.76
135    f/5.6   24.1   15 x 10   25.3    456   8.6   3.0   0.76

200    f/2.8   71.4   10 x 7    17.1   4004  11.0   2.1   0.51
200    f/3.5   57.1   10 x 7    17.1   2561  10.5   2.1   0.51
200    f/4     50.0   10 x 7    17.1   1963  10.2   2.1   0.51
200    f/5.6   35.7   10 x 7    17.1   1001   9.5   2.1   0.51

300    f/2.8  107.1  6.9 x 4.6  11.4   9010  11.9   1.4   0.34
300    f/3.5   85.7  6.9 x 4.6  11.4   5768  11.4   1.4   0.34
300    f/4     75.0  6.9 x 4.6  11.4   4418  11.1   1.4   0.34
300    f/5.6   53.6  6.9 x 4.6  11.4   2256  10.4   1.4   0.34
```