Rules of thumb for planetary scopes, part 1.
In some circles, in fact, these marvels of glass technology and design have earned a reputation as the best of all instruments for the Moon and planets. But they are expensive, leaving many observers "stuck" with supposedly less desirable telescope types. Even the classical f/15 doublet refractor is often recommended for planetary work, despite its residual color aberration.
For some time "aperture fever" has gripped the amateur world. Now we also see occasional cases of "refractor fever," along with its unspoken corollary, "reflector despair." Amateurs have every right to wonder why there is so much confusion about image quality. Large sums of money often hinge on the decision of what telescope to buy. However, in all fairness, only rather late in the history of the telescope -- since the 1940s -- has optical science found a means to quantify an instrument's performance on extended objects.
In this two-part article, I will attempt to make these research results more widely known in the amateur community. A few simple formulas based on these results can go a long way toward quantifying a telescope's performance. In particular, I'll explore the differences between refracting and reflecting telescopes for observing the planets.
CONTRAST: KEY TO PERFORMANCE
If asked what characterizes a good telescopic image, a well-informed amateur astronomer might answer, "High contrast." No truer words can be spoken when it comes to glimpsing subtle surface markings and shadings. The crucial factor is a telescope's ability to preserve the contrast already available in the object being viewed. Contrast performance is generally better for coarse detail and worse for fine. It also depends on telescope aperture, configuration, and optical quality.
Compounding the problem, however, is the human eye. Contrasts must exceed a particular minimum level in order to be perceptible. We call this the visual contrast threshold. Detail in the telescopic image exists whether we observe it or not, for it is a real aspect of the pattern of light sitting at the telescope's focal plane. But if the contrast it makes with the surrounding planetary disk is below the visual threshold (less than about 0.1), the detail or feature will be invisible. The contrast threshold depends, among many other factors, on the angular size of the feature being observed.
Any discussion of a telescope's performance must, then, consider three independent factors: the intrinsic contrast of an extended object like a planet, the contrast performance of the telescope, and the contrast threshold of the human eye. We'll begin by taking up a major issue in the reflector-versus-refractor debate, the effect of placing a small obstruction in the telescope's light path.
Most common reflecting telescope designs call for a secondary mirror in the incoming light bundle. As is widely known, this partial blockage of the circular aperture alters the diffraction patterns of stars and, in the case of the Moon and planets, "reduces contrast." What is less well known is the exact amount by which the contrast is reduced.
The top diagram on the next page plots the contrast performance of four high-quality telescopes, three with central obstructions and one without. As expected, the larger the secondary mirror the worse the effect. (Curiously, however, the performance of the obscured telescopes is very slightly better for fine detail close to the resolution limit.)
Can these results be summarized in a simple formula? At first glance the plots appear too complicated for that. But if we take into account the fact that most planetary detail is low contrast, and include the threshold of the human visual system, we can ignore the more complicated behavior of the curves near the resolution limit.
The diagram below shows how a centrally obstructed telescope handles an extended object of low intrinsic contrast -- for example, Mars. That part of the curve above the visual threshold almost exactly matches the contrast curve of a somewhat smaller, unobstructed instrument! In other words, the performance of a centrally obstructed telescope on low-contrast detail is the same as that of an unobstructed telescope of somewhat smaller diameter.
Thus we can think in terms of an "effective diameter," smaller than the true diameter, that applies to any instrument with a secondary mirror. As is evident from the plot, this simple rule applies:
|D.sub.effective~ = |D.sub.primary~ - |D.sub.secondary~,
where D stands for diameter. For example, a 10-inch telescope with a 3-inch secondary mirror will perform the same on planetary detail as a 7-inch unobstructed telescope of equal quality. (The rule is slightly too pessimistic for very small obstructions.)
This startling result puts the matter of a central obstruction into a perspective that the observer can deeply and intuitively grasp. Conventional wisdom, that a secondary mirror reduces contrast, is recast in terms of effective diameter. Doing so allows users of reflectors of otherwise high quality to shake off the sense that somehow their scopes are inadequate for planetary observing.
We see that a central obstruction is not a fatal flaw; high-quality planetary performance is not the sole domain of the refractor. A topnotch 6-inch reflector with a 1-inch diagonal should be able to outperform any 4-inch apochromatic refractor. If you don't believe this, read the comments by Terence Dickinson and Douglas George, two very experienced observers (S&T: March 1992, page 253).
In fact, a 6-inch Newtonian reflector with a 1-inch diagonal mirror is really a 5-inch telescope, as far as its potential performance on the planets is concerned. The diagonal is less harmful to light grasp, where the unobstructed area is what counts. Here the effective aperture is about 5.9 inches (with the best available coatings). The table on page 93 shows a similar calculation for several popular commercial instruments.
The diffraction effects of spider vanes have been vilified in amateur astronomy circles for generations. Most of the discussion relates to the "spikes" they add to star images. The complete absence of a spider in refractors and tilted-component telescopes is generally regarded as a distinct advantage. If bright stars acquire narrow spikes that are relatively easy to see, there must be something harmful that is done to planetary images as well. True, but does this justify the belief that the vanes must be avoided, or curved, as some observers claim?
Comparing the contrast performance of a telescope with and without a spider easily resolves this question. For several different vane widths the results are illustrated on page 93. I've included a curve for a spider-free telescope.
The curves in this diagram yield our second performance formula, that relating to diffraction by a four-vane spider:
Max. Contrast Loss = 2 |T.sub.vane~ / |D.sub.primary~,
where |T.sub.vane~ is the vane thickness and |D.sub.primary~ is, again, the diameter of the primary mirror. This rule of thumb gives the fractional loss in contrast for detail TABULAR DATA OMITTED twice the size of the resolution cutoff. (Losses are even less for other detail sizes.) To convert this value to a percentage, multiply by 100. For example, a 6-inch telescope with 0.03-inch vanes will suffer a mere 1 percent loss in contrast!
It is clear that the loss in contrast due to spider vanes, contrary to popular belief, is negligible. The potential for significant loss in performance still exists, however. Twisted or misaligned vanes, not to mention excessively thick ones, can indeed cause a significant loss in contrast. The effect of spider diffraction can also be expressed roughly as follows:
|D.sub.effective~ = |D.sub.actual~ - 2 |T.sub.vane~.
To eliminate spider vanes altogether, the secondary mirror can be supported by an optical window -- or, in the case of Schmidt-Cassegrains and Maksutovs, by the corrector lens. When this is done in a Newtonian reflector, the tradeoff is that the window adds yet another element to the system. Its optical quality can never be perfect, though good windows, even nearly perfect ones, are available for a price.
However, experienced planetary observers widely tout the benefits of such a window. So what is really responsible for the improved images? It can't be the absence of spider vanes because, as we have seen, the change they produce in contrast is imperceptible (assuming they are reasonably thin). A more plausible reason is that closing off the tube, as occurs naturally in a refractor, causes a distinct improvement in another arena of telescope performance: local seeing.
Over the years other anecdotal knowledge, most of it subjective, has accumulated about telescope performance. An exception is the famous Dawes limit of angular resolution, which works well for double stars but does not apply to bright extended objects. Rayleigh's quarter-wave criterion is certainly quantitative as to wavefront error -- but it is also a poor guide for discerning details on a planet's disk.
We'll have more to say about these issues, and offer additional rules of thumb, in the continuation of this article.
WILLIAM P. ZMEK 138 Millville Ave. Naugatuck, CT 06770
ANYONE who fiddles with the knobs on a TV set or enjoys fine photographs already has an intuitive grasp of contrast. In a scene with a light and a dark area, the contrast between them is defined as the difference in their brightnesses divided by the sum. The higher the number, the more visible and "harsher" the feature becomes. A contrast of 1.0 is the maximum possible.
Evaluating a telescope's contrast performance is just a matter of plotting the ratio of image contrast to object contrast for details in a whole range of angular sizes. In the technical literature, contrast is often called modulation, and "contrast performance" is the same as the "modulation transfer function" or MTF. Strictly speaking, a plot of this type applies to a pattern of brightness varying as a sine wave across a planet's disk rather than to an isolated spot or marking.
The graphs in this article show the contrast performance of telescopes with an aperture of 113 millimeters (4.4 inches). It is customary to plot detail size along the horizontal axis in inverse units. For example, 0.2 represents a detail 5 arc seconds across. To make any graph valid for a larger telescope, simply multiply the numbers along the horizontal axis by D/113, where D is the new diameter in millimeters. (The plotted curve remains the same.) On the vertical axis, a contrast performance of 0.5 means that an image detail appears with only half the contrast it has in the object itself.
Three important points can be made about these plots:
* Image contrast is always less than object contrast, even for a perfect telescope. The finer the detail in the object, the weaker that detail appears in the image. Thus, Encke's gap in Saturn's A ring is much harder to see in a backyard telescope than the much wider Cassini's division. These features, which at opposition have angular widths of about 0.74 and 0.05 arc second, respectively, have similar intrinsic contrasts above 0.9. But the image contrast of the narrower one might be close to zero.
* A resolution cutoff exists at the point labeled 1.0 on the horizontal axis, where the curve drops to zero. There will be no image detail visible in a 113-mm telescope that is smaller than 1 arc second, even though a celestial object may be loaded with features of this size or smaller. For example, Saturn's rings contain very fine divisions ("phonograph grooves") that were discovered during the Voyager mission. They can never be seen from Earth with modest-size telescopes, even if optics and seeing are perfect, because they are finer than the resolution cutoff for such instruments.
* Contrast performance can be exactly calculated, even measured in the laboratory. Telescopes suffering from defects of figuring or alignment, for instance, will exhibit performance curves worse (lower) than those shown in the plots. Wavefront distortion from figuring error, misalignment, and seeing, as well as scattered light and obstructions, all produce their own specific curves of contrast performance.
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|Title Annotation:||Telescope Making; includes related article|
|Author:||Zmek, William P.|
|Publication:||Sky & Telescope|
|Date:||Jul 1, 1993|
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