2.3. Telescope magnification

Telescope magnification is given by a ratio of the image size produced on the retina when looking through a telescope, versus retinal image size with the naked eye. As FIG. 6 shows, image size on the retina in both cases is proportional to the apparent angle of view, giving telescope magnification as MT= ε/α, ε and α being the apparent and true (semi) angle of view, respectively. For sufficiently small ε, the angles relate nearly as their tangents. Replacing the two angles - with their tangents (tanε=h'/ƒE and tanα=h'/ƒO) gives telescope magnification as:

with ƒO, ƒE being the objective and eyepiece focal length, respectively. For simplicity, both telescope and eyepiece focal length will be considered numerically positive. Also, since most telescope objectives form reversed object image, which is not changed by the eyepiece, their magnification is, by definition, numerically negative; for simplicity, it will be given as numerically positive here, since it is not used for other (broader) calculations.

Since this relation assumes small angle of view, it is strictly accurate only for small angular objects, not larger than about 10 degrees in the eyepiece. With the tangent increasing faster than the angle, large angular objects in the eyepiece will have lower actual magnification than indicated by this formula. For instance, a 1 arc minute object magnified to 1° apparent size will have actual magnification of 60, exactly as the formula indicates. But a 30 arc minute object magnified to 30° apparent size will also have actual magnification of 60, while the formula indicates 61.4.

Telescope magnification can be split into two components: (1) magnification of the objective and
(2) magnification of the eyepiece. Magnification of the image formed by the objective is either relative to the object imaged (absolute, or optical magnification), or relative to its apparent size in the naked eye (apparent magnification). The former is expressed with a simple formula:

with ƒ being the telescope focal length, and O the object distance (FIG. 6). Obviously, it is very small for astronomical objects, due to their enormous distances. Apparent magnification of the objective is given by the ratio of the viewing angle of its object-image from the least distance of distinct vision (250mm average) to the viewing angle of the object observed directly. Since these angles are sufficiently small, they can be replaced with their tangents, giving the apparent objective magnification as

for the focal length ƒ in mm.

In terms of image-to-objective separation I (also assumed numerically positive) the apparent telescope magnification is Ma=(I-ƒ)/ƒ, which determines the object distance in terms of I and ƒ as O=Iƒ/(I-ƒ). Due to enormous distances of astronomical objects - thus with I only negligibly greater than ƒ - their absolute magnification in a telescope approaches zero.

The eyepiece acts as a magnifying glass, effectively allowing the eye to observe the object-mage formed by the objective from the distance of eyepiece's focal length (ƒE in Fig. 6). It makes the image apparently larger by a factor ME= ε/β, with ε and b being the apparent viewing angle in the eyepiece and in the naked eye (the latter also called "true" angle, or field of view), respectively (Fig. 5-6). Again, for small to moderate viewing angle ε, we can replace angles with their tangents (tanβ=h'/V~h'/250 for h' in mm), giving the eyepiece magnification factor as

for the eyepiece f.l. ƒE in mm. Thus, apparent telescope magnification is a product of the two magnifications - the initial apparent magnification by the objective, and the final one by the eyepiece - resulting in Mt=MaMe~ƒ/ƒE. Of course, the eyepiece magnifies by enlarging the apparent angle of rays converging toward the eye, but it produces collimated beams and, thus, no actual image. It is the eye that focuses these collimated beams into point images.

As noted, for large viewing angles in the eyepiece, use of the tangent results in higher than actual magnification figure. For instance, a 5mm high object-image on the optical axis is viewed through a 50mm f.l. eyepiece at an angle α given by tanα=5/50, giving α=5.7 degrees apparent viewing angle. And that same object-image observed through a 10mm f.l. eyepiece has the apparent viewing angle α'=26.6 degrees. While the tangent (i.e. eyepiece f.l.) based magnification factor is 5, actual magnification factor for the 10mm vs. 50mm eyepiece is 26.6/5.7=4.7.

While there is no single optimum magnification for all types of astronomical objects and individuals, there is a range of so called useful magnification. On the low side of this range, the limit is set by the size of eye pupil. It is not to be smaller than the "exit pupil" of a telescope - an image of the entrance pupil (objective) formed by the eyepiece. Exit pupil appears as a bright circle of light floating in front the eyepiece eye lens (the eyepiece lens facing the eye). In order to capture all the light entering telescope, the eye lens has to be placed at the location of exit pupil and, of course, in order to avoid light loss, eye pupil has to be at least as wide as the exit pupil. Since exit pupil diameter is given by P=D/Mt=ƒ'/F, the lowest magnification that still preserves light-gathering power is Mt=D/ƒ'. For an average eye pupil maximum of 6mm, it comes to D/6 for D in mm.

Limit to magnification increase is set primarily by image imperfections, but also by dimming, loss of field, vibrations and eye physiology. As we have seen, even perfect optics will not produce perfect images, due to the effect of diffraction. Point-object image is smeared into a pattern of finite size. Magnified enough (to about 4 to 5 arc minutes) it becomes visible to the eye, and there is no benefit in resolution from further magnification increase. Taking the standard resolution limit for a pair of near equally bright point sources of 4.5/D (D in inches) arc seconds (1/13D in arc minutes), needed magnification to 5 arc minutes apparent center separation for a pair of relatively bright point-object images at the resolution limit is given as ~67D, or 67x per inch of aperture (~2.7D for D in mm).

The often quoted 50x per inch of aperture limit to useful magnification is based on empirically determined average resolution limit of 4 arc minutes for the naked eye and two near-equal intensity moderately bright stars.

However, just as the "standard" resolution limit is strictly valid only for a very particular object type, as explained above, so is the high magnification limit derived from it. Some objects - generally those with lower inherent contrast - will dictate lower maximum useful magnification, while some other objects will require higher. Practical limit to useful magnification on the high side will be lower, the larger aperture the more so - due to increase in wavefront aberrations in general, and especially so called "seeing error": the deterioration of image quality caused by atmospheric turbulence. Not to forget, optimum magnification varies rather significantly individually, due to differences in eyesight quality and observing experience.

◄ 2.2. Telescope resolution ▐ 3. TELESCOPE ABERRATIONS ►

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