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An image scale formula

Posted by Gary Hooper   06/05/2011 02:47PM

Most of the mathematical formulas dealing with optics are above my level of comprehension. Twenty years ago, when I was using 35 mm format astrophotography, I used a simple formula to determine the field of view captured with various telescopes and lenses. The formula gives the relationship between focal length and image scale. There are many "plug in the numbers" type programs around that calculate field of view for eyepieces or imaging, but I am putting this out there just in case someone is interested. I originally came across this formula in a publication called Newtonian Notes by Kenneth Novak, a telescope accessories dealer from the 1970's.

The formula is: A = 3438 L/F
A=Angular field of view in minutes of arc
L=Linear size as measured at the focal plane
F= Focal length of optical system
Note: F and L must be in the same units of measurement. For my use at the time, I simplified the formula by dividing the right side of the equation by 60 so the outcome is in degrees rather than minutes of arc. For the outcome to be in degrees, the new version is: A= 57.3 L/F.

I used various focal lengths but always the same 35 mm format, so I factored the film dimensions into the formula by multiplying 57.3 times the film dimensions. This makes two formulas, each with a singe math step. For the 36 mm dimension of the film, the formula is: A = 2063/F, and for the 24 mm dimension of the film the formula is A=1375/F. Note: Because the L dimensions have been factored in millimeters, the focal length must always be in millimeters. As can be seen without doing any math, a focal length of 2063 mm will record a field of view exactly one degree wide across the wide dimension of the film, and by doing a little math about 2/3 of a degree across the narrow dimension of the film. It is an easy task to make similar formulas for any image chip or eyepiece field stop dimension.

A few other things came to my attention while working with these simple formulas. I came to realize that these formulas are not really necessary. The same thing can be calculated in a more straightforward and logical way. Since I cannot draw a diagram for you, you will have to make this a thought experiment. First imagine a parabolic mirror, and the image of the sky formed at the focal point. As the mirror scans around 360 degrees, it will have formed a corresponding image of the sky encircling the mirror. In fact if it scanned the whole sky, it would form an image sphere surrounding the mirror. This sphere is a miniature version of the sky that we examine with our eyepieces with the field of view limited by the size of the eyepiece field stop. The dimensions of this image circle/sphere are easy to calculate by simple geometry. The focal length is the radius of this image circle/sphere, the diameter is twice the focal length, and the circumference is equal to the diameter times Pi. Image scale can then be calculated by dividing the circumference by 360 degrees to get units of measure per degree, or the reciprocal of this number is the number of degrees per unit of measure.