A standard reference for the field of view of a Galilean telescope is a short paper published in 1920, in the Transactions of the Optical Society, by H.A. Hughes and P.F. Everitt. (H. A. Hughes and P. F. Everitt, “On the Field of View of a Galilean Telescope,” Trans. Opt. Soc. 22, 15-19 (1920)). In their notation the focal length of the objective is f1 and the focal length of the eyepiece is –f2. That is, f2 is a positive number. The magnification m is also a positive number equal to f1/f2. The diameter of the objective is 2P and the diameter the iris of the eye is 2p. The height of the exit pupil is the size of the objective divided by the magnification, or 2P/m. Their results can be understood in terms of the next diagram.
The vertical line on the right represents the exit pupil of the telescope, with a total height of 2P/m, where 2P is the diameter of the objective. The vertical line on the left is the pupil of the eye, with a total diameter of 2p. The distance between the two lines is labeled l. This must be slightly greater than the distance of the exit pupil from the eyepiece. Later I shall approximate it by this value. Then
As light enters the objective traveling at an angle θ to the axis, rays leave all points on the exit pupil traveling at an angle mθ to the axis. If the ray leaving the top of the exit pupil passes through the bottom point of the eye pupil, it will be the only ray getting into the eye. This corresponds to an angle given by
If θ is reduced until the ray from the top of the exit pupil hits the top of the eye pupil, then the eye will be fully illuminated. The value of θ is given by
These are the two extreme cases. Hughes and Everitt also define a median ray corresponding to the ray from the top of the exit pupil passing through the center of the eye. The value is given by
These are the results given by Hughes and Everitt. If you completely neglect p compared with P/m, the three formulas become equivalent. If you also make a small angle approximation and further replace l with the distance from the eyepiece to the exit pupil, the half-field of view becomes
The field of view is directly proportional to the diameter of the objective. This is something that Galileo clearly believed, at least at the start of his investigations.
Hughes and Everitt made it clear that they were interested in field glasses of the Galilean design that were popular before modern prismatic binoculars were introduced. In the examples that they looked at, the exit pupil was larger than the eye pupil. I have a small opera glass made by Vivitar, described as 4 x 30. The 30 is the diameter of the objective in millimeters. The magnification is 4. The diameter of the exit pupil is then 7.5 mm. In daylight a typical human eye has an opening of about 3 mm so that it is small compared with the exit pupil. The formulas given above are then accurate. This was also very probably the case for the first telescope that Galileo made, with a magnification of 3. However, he rapidly developed telescopes with magnifications of 20 and perhaps 30. The objective lenses were of the order of 25 mm in diameter and the exit pupil was around 1 mm. This situation is qualitatively shown in the next figure.
The light leaving the telescope now emerges as a narrow pencil and it is obvious that a dark-adapted eye, with a diameter of about 6 mm, can never be fully illuminated. The condition for any ray at all to enter the eye is unchanged and given in eq. (2). The ray from the top of the exit pupil passes through the bottom of the eye pupil, just as in the previous case. The condition that all the rays leaving the telescope enter the eye corresponds to the ray from the bottom of the exit pupil entering the bottom of the eye. The condition on θ is
As a median value, the condition for the ray from the center of the exit pupil to enter the eye is
In the limit that the size of the exit pupil is negligible, which is rather good in this case, the three formulas become equal. With the small angle approximation and the approximate value for l they give a half-field of view of
In this limit the field of view is proportional to the diameter of the eye opening. However, there are rays of light leaving the telescope that don't enter the iris of the eye. You can see these by moving your eye around, up or down, right or left. This gives you access to a much wider field of view without any movement of the telescope. It is often stated that Galileo was not able to see the whole face of the Moon without resetting his telescope. This is probably not true. For a fixed position of the eye, the field of view might typically be only 14 or 15 minutes of arc, whereas the angular size of the Moon is about 30 minutes. However, for the replica telescope that I constructed, the field of view that can be seen by "cheating" the eye around is about 1.6 degrees. The telescopes that Galileo used for his discoveries have not survived. I can calculate the total accessible field of view for the two telescopes that are attributed to him. One has magnification of 20 but the eyepiece is generally believed to be a replacement. For that telescope I calculate a total field of just over one degree. For the other telescope, which was on display at the Franklin in Philadelphia in 2009, the magnification is 14 so that it does not seem to have been one that he used for his discoveries. It has quite a small eyepiece. I calculate the total field accesssible without moving the telescope to be just over 30 minutes.
The total field of view that can be accessed is determined by the diameter of the eyepiece. The diagram is very similar to the second case that I worked through, except that the vertical line on the left is now the width of the eyepiece.
I have used p' for the half-height of the eyepiece, and its distance from the exit pupil is given by equation (1). There are, of course, formulas corresponding to the three cases I used earlier, but if I neglect the size of the exit pupil and make a small angle approximation they reduce to
Actually, this is exactly the formula for the field of view of a Keplerian telescope except that the separation of the lenses in that case is the sum of the focal lengths. I have used the notation set up by Hughes and Everitt, where P, p, and p' are half-widths and θ is a half-field of view. When putting in numbers, I find it easiest to use the full width of the lenses and then θ comes out as the full field of view.