The angular magnification, M, of a telescope compares the angle subtended at the eye by the image with the angle subtended at the unaided eye by the object:M = angle subtended by image at eye / angle subtended by object at unaided eye = β / α Astronomical objects subtend tiny angles, so a telescope makes them look larger by increasing the angle, not by making a bigger image on a screen.
In normal adjustment (final image at infinity), the geometry of the two lenses gives:M = fo / fe where fo is the focal length of the objective and fe is the focal length of the eyepiece. For large magnification you need a long focal length objective and a short focal length eyepiece. The angles are small, so they are measured in radians and tan θ ≈ θ.Worked example: a telescope has fo = 1.5 m and fe = 0.025 m.
• M = 1.5 / 0.025 = 60
• The Moon subtends 9.0 × 10-3 rad at the unaided eye, so its image subtends 60 × 9.0 × 10-3 = 0.54 rad.Rearranging: fe = fo / M. For example, a telescope with M = 40 and fo = 2.0 m needs an eyepiece of focal length 2.0 / 40 = 0.050 m. Swapping eyepieces is how one telescope gives several magnifications.
In normal adjustment (final image at infinity), the geometry of the two lenses gives:• M = 1.5 / 0.025 = 60
• The Moon subtends 9.0 × 10-3 rad at the unaided eye, so its image subtends 60 × 9.0 × 10-3 = 0.54 rad.Rearranging: fe = fo / M. For example, a telescope with M = 40 and fo = 2.0 m needs an eyepiece of focal length 2.0 / 40 = 0.050 m. Swapping eyepieces is how one telescope gives several magnifications.