◄ 2.2. Telescope resolution   ▐    3. TELESCOPE ABERRATIONS ►
 

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. 5B). 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. 5A-B). 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 dates back to the 1940s, when Allyn Thompson used a mixture of theoretical diffraction resolution limit for point-like sources, anecdotal accounts of the limit to naked-eye stellar resolution (ε Lyrae), and the results of group's own experiments with naked-eye resolution limits for pairs of 0.0003-inch illuminated pinholes, to come up with four arc minutes as the approximate average resolving limit for point-like objects (Making your own telescope, p173-174). Combining it with the Dawes' limit formula for the minimum resolvable stellar separation in arc seconds α=4.56/D for the aperture D in inches, for the corresponding magnification M needed to enlarge this limit to four arc seconds (i.e. 240 arc seconds), produced M=240/α=52.6D.

In line with this, higher magnifications than ~50x per inch of aperture would not produce additional benefit.

However, Thompson's consideration missed to recognize that telescopic and naked-eye point-source resolution are distinctly different. The former is limited by diffraction, and the latter by aberrations. At 4mm pupil diameter, which was the estimated pupil size level in their experiments with pinhole resolution (the naked-eye pupil size while resolving ε Lyrae is at least as large), eye not corrected with ophthalmic lenses averages about 1 wave RMS of (mainly) combined defocus and astigmatism. The resulting diffraction blur is roughly a dozen times larger than the Airy disc and, more importantly, its angular size is over 10 arc minutes, nearly four times larger than the maximum angular size still perceived by the eye as point-object.

Unlike the naked-eye observer, telescope user has the benefit of eye defocus error corrected by refocusing the eyepiece. This reduces the RMS error by a factor of four, or so, with the diffraction blur shrinking nearly as much, to about 3 arc minutes. Hence average telescope user needs only about 1/4 of the magnification calculated by Thompson - about 13x per inch of aperture - for the theoretical resolution of two point objects at the Dawes' limit. The corresponding eyepiece exit pupil - and the effective eye pupil - is about 2mm in diameter, at which the average eye becomes nearly diffraction-limited. In reality, due to inherent and induced telescope aberrations, particularly seeing error, the needed magnification is somewhat higher, but not likely more than 20x per inch.

Obviously, taking the resolution limit for this pupil size as a criterion for telescopic resolution was inappropriate, resulting in the considerably lower limit to resolution assumed for the telescope user and, consequently, significantly higher resolving magnification estimate.

A simple consideration based on the retinal physiology (FIG. 11A left) indicates that foveal resolution of two patches of light requires them to be separated by at least a single non-illuminated cone, as the illustration below shows. With the smallest cones being less than half arc minute in diameter (about 2 microns), the limit of resolution of two diffraction discs that don't exceed about 1/2 arc minute in diameter is approximated by about twice the cone diameter, or 0.8 arc minute. Since the discs' centers are separated by twice their diameter, the corresponding angular separation is about twice the disc's angular diameter. Taking that this diameter is FWHM of the PSF, or λ/D, on average, (somewhat smaller for faint stars, and somewhat larger for bright stars), this separation corresponds to double the stellar diffraction limit of resolution, λ/D.

In other words, twice the diffraction resolution limit for point-sources is attainable with the diffraction FWHM magnified to about 0.4 arc minute. In a 4-inch aperture, with the diffraction FWHM of 1.14 arc seconds, the corresponding magnification is about 21, or 5.25x per inch (for simplicity, will round these off to 20 and 5, respectively). An 8-inch aperture needs twice as high nominal magnification for the diffraction FWHM to reach 0.4 arc minute, but it results in twice better resolution. In general, the minimum magnification for achieving twice (200%) the limit to stellar resolution, scales with the aperture size, and can be approximated by M~5x per inch of aperture, for the aperture D in inches.

Taking into account eye aberrations, enlarging the PSF FWHM, needed magnification for this level of resolution is only slightly higher, 6-7x per inch of aperture.

Further increase in magnification at this angular separation enlarges discs' apparent diameters, which makes them easier to separate visually. Also, higher magnification effectively diminishes relative size of the minimum separation between two FWHM needed for resolution, allowing better limiting resolution. As long as the separation between their boundaries remains nearly unchanged, at about the smallest cone diameter, the smallest angular separation between two resolved diffraction FWHM, in units the diffraction limit, decreases as 1+(1/R), where R=M/5 is the FWHM diameter in units of the smallest cone diameter. So, at 10x per inch R=10/5=2, and the minimum resolvable angular separation is 1.5, or 150% of the diffraction limit.

Doubling magnification to 20x per inch of aperture (R=4), allows reaching 125% of the stellar diffraction limit, and tripling it to 30x per inch (R=6) allows reaching 117% of the limit. Formally, it would take infinitely high magnification to achieve diffraction resolution limit, but for all practical purposes it is achieved when the minimum resolvable separation drops to 105% of the limit, which gives R=20, corresponding to about 100x per inch magnification, or higher. This equals 400x and 800x magnification for 4-inch and 8-inch aperture, respectively.

At this point, the FWHM is nearly 8 arc minutes in diameter, more than 1/4 the apparent Moon's size. Doubling this magnification to 200x per inch (R=40) results in only 2.5% gain in resolution, which is for all practical purposes negligible. On the other hand, going down to, say, 50x per inch (R=10) lowers the resolution only marginally, by about 5% vs. 100x per inch magnification. At a 5 arc minutes average angular FWHM (65x per inch of aperture magnification), which is the size at which most people can clearly recognize the shape of diffraction disc, a telescope still reaches down to 108% of its stellar resolution limit.

Actual neural processing involved in eye's visual function is, of course, much more complex. But this simplistic concept suggest that gain in stellar resolution does not scale linearly with magnification increase. It is higher when magnification increases in the lower range, gradually decreasing with higher magnifications, and falling to negligible nearing 50x per inch of aperture, and beyond, as the above graph shows.

However, by assuming diffraction-limited resolution, this concept neglects the effect of atmospheric error on the diffraction pattern. Magnifications enabling diffraction resolution are seldom attainable in field conditions; reaching 105% of the stellar diffraction limit would require 100x per inch of aperture magnification, or about 400x and 800x for 4-inch and 8-inch aperture, respectively. As aperture increases, typical seeing causes break-down and expansion of the diffraction FWHM, causing telescopic resolution - and corresponding magnification - to become seeing limited. In the typical 2 arc seconds seeing, apertures from about 8 inch in diameter up, will have their resolution limit and magnification needed to reach it progressively decreasing. Long-exposure expansion is approximated by a (r0/D) ratio, where r0 is the atmospheric coherence length (nearly 3 inches for 550nm wavelength in 2 arc sec seeing, changing inversely to the seeing).

Arbitrary approximate adjustment is made for the visual FWHM, which in the low D/r0 range is closer in size to the short-exposure FWHM, gradually becoming nearly as large as long-exposure FWHM beyond D/r0~5. The resulting seeing-limited optimum magnification, as well as needed magnifications for diffraction-limited stellar resolution are as plotted at left (FIG. 11B). It is assumed that stellar resolution is near diffraction-limited (i.e. aperture-limited) for D/r0<2, gradually deteriorating to the seeing limited (i.e. r0 limited) resolution for D/r0~5 and larger, based on the relation between r0 and seeing FWHM.

Note that the magnification plot (red line) indicates the level at which the stellar resolution limit is approached, not the highest possible magnification. For instance, a 16-inch in 2" seeing has D/r0=5.3, has the resolution limit gravitating toward that of an aperture equaling r0, i.e. 3 inch, which is the effective aperture diameter for stellar resolution (Deff). Thus the nominal magnification required for reaching 110% of the resolution limit is MN~500Deff/(110-100)=150, same as if it is for 3-inch aperture in perfect seeing (as indicated by extending the level of magnification indicated by the red line to the left, where its intersection with magnification needed for 110% resolution in perfect seeing determines the corresponding effective aperture).

This is a very approximate model, but it does indicate, even roughly, the magnitude of seeing-induced reduction in the limiting stellar resolution for the range of apertures. Keep in mind that seeing constantly fluctuates, and so do the effective aperture and corresponding magnification needed for 110% of the stellar resolution limit (in 2" seeing average fluctuations are probably between 1" and 4" seeing).

In addition, 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 bright extended objects with low inherent contrast, like planets - will dictate lower maximum useful magnification, while more contrasty, like Moon or brighter doubles, as well as dim deep-sky objects, will allow - or demand - higher magnification level. The former gravitate toward one half of the 50x per inch criterion, while the later can be twice as high, or even higher.

As mentioned, the practical limit to useful relative magnification on the high end will be generally lower, the larger aperture the more so - due to the 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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