Caption: A diagram of the Keplerian telescope, a classic refractor telescope.

Features:

1. A refractor telescope consists of two converging lenses (here represented by vertical lines) aligned along and centered on an optical axis. The one to the left is the objective (AKA primary) lens and the one to the right is the eyepiece.

2. The setup here has the telescope focused at optical infinity: i.e., focused for point sources of light so far away that light rays from them are effectively parallel. This setup is normal for astronomical observations.

3. Our analysis is in the Gaussian optics limit of geometrical optics which itself is one limit of physical optics.

   Gaussian optics assumes the paraxial approximation. This assumption means that light rays make sufficiently small angles to the optical axis that trigonometric functions can always be treated in the small angle approximation.

   Geometrical optics itself is the limit of physical optics in which diffraction is neglected and light is treating just as propagating light rays.

4. The primary forms a real image at its focal plane which is at the focal length f_p along the optical axis.

   Speaking of telescopes in general, the primary is the light-gathering optical device of a telescope AND is usually the first optical device the in the formation of the observed image.

   The main parameter (controlling variable) of the primary (and also of the the telescope as a whole) is its diameter. The light-gathering power varies as collecting area of the primary, and thus as the the square of the diameter.

   The primary is a mirror for a reflector telescope and a lens for a refractor telescope.

5. The real image is the vertical line downward from the optical axis of length h. Somewhere off to the left at optical infinity is the REAL OBJECT which is a vertical line that stands upwards from the optical axis.

   For clarity, we only show a representative set of light rays that form the tip of the real image. The tip is a point real image.

   The tip of the REAL OBJECT emits a continuum family of parallel light rays that impact the primary which is the aperture of the telescope.

   Light rays that miss the aperture, of course, don't get redirected to help form the tip of the real image.

6. The eyepiece captures diverging light rays from the real image and converts them into parallel light rays.

   The observer to the right sees these parallel light rays from the real image, and thus his/her human eye sees the real image at optical infinity.

7. The telescope magnification of the Keplerian telescope follows from simple geometry and trigonometry with the small angle approximation for the tangent function.

   The derivation is given in the diagram itself.

   The formula for telescope magnification is M = -f_p/f_e .

8. Since magnification is a ratio of like quantities, it can be regarded as a dimensionless number: i.e., a number that has no units.

   But there is a sort of conventional unit: the magnification unit X. The unit X which means "multiplies the angular object by X" or the like. So magnification 10 X means magnification by a factor of 10.

9. A few comments about the telescope magnification M:
   1. There are several different kinds of optical magnification.

   2. Telescope magnification is an angular optical magnification: the angular size of an object is magnified by a factor M to become the final image size that the observer sees.

   3. We have derived the telescope magnification formula for a Keplerian telescope, but the formula is general, in fact, as long as the meaning of f_p is generalized to effective primary focal length.

   4. Clearly from the diagram, M increases linearly with primary focal length f_p since the real image increases linearly with f_p.

      The primary focal length is usually a fixed parameter of telescope, and so usually cannot be changed for a given telescope.

   5. The telescope magnification increases as 1 over (i.e., is inverse linearly with) the eyepiece focal length because the angular size of the real image seen in the eyepiece increases this way.

   6. Eyepieces can easily be changed on most telescopes, and so changing eyepieces is the main way of changing telescope magnification for a given telescope.

      Nota bene: the smaller the eyepiece focal length (conventionally given in millimeters), the greater the telescope magnification.

      Smaller focal length means the eyepiece has greater converging power.

   7. The diagram makes clear some of the limits on telescope magnification.

      If eyepiece focal length f_e becomes too small, θ_e (the angular size of the real image) get so large that the tip of the real image will effectively disappear out of the field of view (FOV). You probably can still see something looking at a high angle from the optical axis, but this is not a good observation posture.

      In fact, the effective FOV decreases as f_e decreases (i.e., as telescope magnification increases).

      Also in fact, if θ_e becomes so large that Gaussian optics fails, then our analysis fails.

   8. The FOV is also limited by the fact that if the angular size of the real image gets too large, the light rays that form it will hit the telescope tube (not shown).

   9. The minus sign in the telescope magnification formula ( M = -f_p/f_e ) indicates that the final image seen by the observer is point inverted (i.e., rotated by 180°) from what is seen on the sky.

      The diagram just shows the real image gets flipped. However, the diagram is just a cross-sectional view of a physical system with rotational symmetry about the optical axis. Therefore, the diagrammatic flip implies point inversion.

      The minus sign can be suppressed if the point inversion is taken as understood.

10. And that concludes our story of the Keplerian telescope.