Few things are more generally disappointing than one’s first glimpse of the Heavens through a telescope. The novice is fed up with maps of Mars as a great disc full of intricate markings, and he generally sees a little wriggling ball of light with no more visible detail than an egg. It is almost impossible to believe that, at a fair opposition, Mars under the power of even the smallest astronomical telescope really looks as big as the full moon. Again, one looks at a double star to see not two brilliant little discs resplendent in color, but an indeterminate flicker void of shape and hue.

The fact is, that most of the time over most of the world seeing conditions are bad, so that the telescope does not have a fair chance, and on the whole the bigger the telescope the worse the chance. One famous English astronomer, possessed of a fine refractor that would be reckoned large even now-a-days, averred that he had seen but one first class night in fifteen years past.

The case is really much less bad than this implies, for even in rather unfavorable climates many a night, at some o’clock or other, will furnish an hour or two of pretty good seeing, while now and then, without any apparent connection with the previous state of the weather, a night will turn up when the pictures in the popular astronomies come true, the stars shrink to steady points set in clean cut rings, and no available power seems too high.

One can get a good idea of the true inwardness of bad seeing by trying to read a newspaper through an opera glass across a hot stove. If the actual movements in the atmosphere could be made visible they would present a strange scene of turbulence—rushing currents taking devious courses up and around obstacles, slowly moving whirlpools, upward slants such as gulls hug on the quarter of a liner, great downward rushes dreaded by the aviator, and over it all incessant ripples in every direction.

And movements of air are usually associated with changes of temperature, as over the stove, varying the refraction and contorting the rays that come from a distant star until the image is quite ruined.

The condition for excellence of definition is that the atmosphere through which we see shall be homogeneous, whatever its temperature, humidity, or general trend of movement. Irregular refraction is the thing to be feared, particularly if the variations are sudden and frequent. Hence the common troubles near the ground and about buildings, especially where there are roofs and chimneys to radiate heat—even in and about an observatory dome.

Professor W. H. Pickering, who has had a varied experience in climatic idiosyncrasies, gives the Northern Atlantic seaboard the bad preËminence of having the worst observing conditions of any region within his knowledge. The author cheerfully concurs, yet now and then, quite often after midnight, the air steadies and, if the other conditions are good, definition becomes fairly respectable, sometimes even excellent.

Temperature and humidity as such, seem to make little difference, and a steady breeze unless it shakes the instrument is relatively harmless. Hence we find the most admirable definition in situations as widely different as the Harvard station at Mandeville, Jamaica; Flagstaff, Arizona 7000 feet up and snow bound in winter; Italy, and Egypt. The first named is warm and with very heavy rainfall and dew, the second dry with rather large seasonal variation of temperature, and the others temperate and hot respectively.

Perhaps the most striking evidence of the importance of uniformity was noted by Evershed at an Indian station where good conditions immediately followed the flooding of the rice fields with its tendency to stabilize the temperature. Mountain stations may be good as at Flagstaff, Mt. Hamilton, or Mt. Wilson, or very bad as Pike’s Peak proved to be, probably owing to local conditions.

In fact much of the trouble comes from nearby sources, atmospheric waves and ripples rather than large movements, ripples indeed often small compared with the aperture of the telescope and sometimes in or not far outside of the tube itself.

Aside from these difficulties, there are still others which have to do with the transparency of the atmosphere with respect to its suspended matter. This does not affect the definition as such, but it cuts down the light to a degree that may interfere seriously with the observation of faint stars and nebulÆ. The smoke near a city aggravates the situation, but in particular it depends on general weather conditions which may be persistent or merely temporary.

Often seeing conditions may be admirable save for this lack of transparency in the atmosphere, so that study of the moon, of planetary markings and even of double stars, not too faint, may go on quite unimpeded. The actual loss of light may reach however a magnitude or more, while the sky is quite cloudless and without a trace of fog or noticeable haziness by day.

There have been a good many nights the past year (1921) when Alcor (80 UrsÆ Majoris) the tiny neighbor of Mizar, very nearly of the 4th magnitude, has been barely or not at all visible while the seeing otherwise was respectably good. Ordinarily stars of 6^m should be visible in a really clear night, and in a brilliant winter sky in the temperate zones, or in the clear air of the tropics, a good many eyes will do better than this, reaching 6^m.5 or even 7^m, occasionally a bit more.

The relation of air waves and such like irregularities to telescopic vision was rather thoroughly investigated by Douglass more than twenty years ago (Pop. Ast. 6, 193) with very interesting results. In substance, from careful observation with telescopes from 4 inches up to 24 inches aperture, he found that the real trouble came from what one may call ripples, disturbances from say 4 inches wave length down to ¾ inch or less. Long waves are rare and relatively unimportant since their general effect is to cause shifting of the image as a whole rather than the destruction of detail which accompanies the shorter waves.

This rippling of the air is probably associated with the contact displacements in air currents such as on a big scale become visible in cloud forms. Clearly ripples, marked as they are by difference of refraction, located in front of a telescope objective, produce different focal lengths for different parts of the objective and render a clean and stable image quite out of the question.

In rough terms Douglass found that waves of greater length than half the aperture did not materially deteriorate the image, although they did shift it as a whole, while waves of length less than one third the aperture did serious mischief to the definition, the greater as the ripples were shorter, and the image itself more minute in dimension or detail.

Hence there are times when decreasing the aperture of an objective by a stop improves the seeing considerably by increasing the relative length of the air waves. Such is in fact found to be the case in practical observing, especially when the seeing with a large aperture is decidedly poor. In other words one may often gain more by increased steadiness than he loses by lessened “resolving power,” the result depending somewhat on the class of observation which chances to be under way.

And this brings us, willy-nilly, to the somewhat abstruse matter of resolving power, depending fundamentally upon the theory of diffraction of light, and practically upon a good many other things that modify the character of the diffraction pattern, or the actual visibility of its elements.

When light shines through a hole or a slit the light waves are bent at the margins and the several sets, eventually overlapping, interfere with each other so as to produce a pattern of bright and dark elements depending on the size and shape of the aperture, and distributed about a central bright image of that aperture. One gets the effect well in looking through an open umbrella at a distant street light. The outer images of the pattern are fainter and fainter as they get away from the central image.

Without burdening the reader for the moment with details to be considered presently, the effect in telescopic vision is that a star of real angular diameter quite negligible, perhaps 0.001 of arc, is represented by an image under perfect conditions like Fig. 154, of quite perceptible diameter, surrounded by a system of rings, faint but clear-cut, diminishing in intensity outwards. When the seeing is bad no rings are visible and the central disc is a mere bright blur several times larger than it ought to be.

The varying appearance of the star image is a very good index of the quality of the seeing, so that, having a clear indication of this appearance, two astronomers in different parts of the world can gain a definite idea of each other’s relative seeing conditions. To this end a standard scale of seeing, due largely to the efforts of Prof. W. H. Pickering, has come into rather common use. (H. A. 61 29). It is as follows, based on observations with a 5 inch telescope.

STANDARD SCALE OF SEEING

1. Image usually about twice the diameter of the third ring.

2. Image occasionally twice the diameter of the third ring.

3. Image of about the same diameter as the third ring, and brighter at the centre.

4. Disc often visible, arcs (of rings) sometimes seen on brighter stars.

5. Disc always visible, arcs frequently seen on brighter stars.

6. Disc always visible, short arcs constantly seen.

7. Disc sometimes sharply defined, (a) long arcs. (b) Rings complete.

8. Disc always sharply defined, (a) long arcs. (b) Rings complete all in motion.

9. Rings, (a) Inner ring stationary, (b) Outer rings momentarily stationary.

10. Rings all stationary, (a) Detail between the rings sometimes moving. (b) No detail between the rings.

The first three scale numbers indicate very bad seeing; the next two, poor; the next two, good; and the last three, excellent. One can get some idea of the extreme badness of scale divisions 1, 2, 3, in realizing that the third bright diffraction ring is nearly 4 times the diameter of the proper star-disc.

It must be noted that for a given condition of atmosphere the seeing with a large instrument ranks lower on the scale than with a small one, since as already explained the usual air ripples are of dimensions that might affect a 5 inch aperture imperceptibly and a 15 inch aperture very seriously.

Douglass (loc. cit.) made a careful comparison of seeing conditions for apertures up to 24 inches and found a systematic difference of 2 or 3 scale numbers between 4 or 6 inches aperture, and 18 or 24 inches. With the smallest aperture the image showed merely bodily motion due to air waves that produced serious injury to the image in the large apertures, as might be expected.

There is likewise a great difference in the average quality of seeing as between stars near the zenith and those toward the horizon, due again to the greater opportunity for atmospheric disturbances in the latter case. Pickering’s experiments (loc. cit.) show a difference of nearly 3 scale divisions between say 20° and 70° elevation. This difference, which is important, is well shown in Fig. 182, taken from his report.

The three lower curves were from Cambridge observations, the others obtained at various Jamaica stations. They clearly show the systematic regional differences, as well as the rapid falling off in definition below altitude 40°, which points the importance of making provision for comfortable observing above this altitude.

The relation of the diffraction pattern as disclosed in the moments of best seeing to its theoretical form is a very interesting one. The diffraction through a theoretically perfect objective was worked out many years ago by Sir George Airy who calculated the exact distribution of the light in the central disc and the surrounding rings.

This is shown from the centre outwards in Fig. 183, in which the ordinates of the curve represent relative intensities while the abscissÆ represent to an arbitrary scale the distances from the axis. It will be at once noticed that the star image, brilliant at its centre, sinks, first rapidly and then more slowly, to a minimum and then very gradually rises to the maximum of the first bright ring, then as slowly sinks again to increase for the second ring and so on.

For unity brightness in the centre of the star disc the maximum brightness of the first ring is 0.017, of the second 0.004 and the third 0.0016. The rings are equidistant and the star disc has a radius substantially equal to the distance between rings. One’s vision does not follow down to zero the intensities of the rings or of the margin of the disc, so that the latter has an apparent diameter materially less than the diameter to the first diffraction minimum, and the rings themselves look sharper and thinner than the figure would show, even were the horizontal scale much diminished. The eye does not descend in the presence of bright areas to its final threshold of perception.

One gains a somewhat vivid idea of the situation by passing to three dimensions as in Fig. 184, the “diffraction solid” for a star, a conception due to M. AndrÉ (Mem. de l’Acad. de Lyon 30, 49). Here the solid represents in volume the whole light received and the height taken at any point, the intensity at that point.

A cross section at any point shows the apparent diameter of the disc, its distance to the apex the remaining intensity, and the volume above the section the remaining total light. Substantially 85% of the total light belongs to the central cone, for the theoretical distribution.

Granting that the eye can distinguish from the background of the sky, in presence of a bright point, only light above a certain intensity, one readily sees why the discs of faint stars look small, and why shade glasses are sometimes useful in wiping out the marginal intensities of the solid. There are physiological factors that alter profoundly the appearance of the actual star image, despite the fact that the theoretical diffraction image for the aperture is independent of the star’s magnitude.

Practically the general reduction of illumination in the fainter stars cuts down the apparent diameters of their discs, and reduces the number of rings visible against the background of the sky.

The scale of the diffraction system determines the resolving power of the telescope. This scale is given in Airy’s original paper (Cambr. Phil. Trans. 1834 p. 283), from which the angle a to any maximum or minimum in the ring system is defined by

sin a = n?/R

in which ? is numerically the wave length of any light considered and R is the radius of the objective.

We therefore see that the ring system varies in dimension inversely with the aperture of the objective and directly with the wave length considered. Hence the bigger the objective the smaller the disc and its surrounding ring system; and the greater the wave length, i.e. the redder the light, the bigger the diffraction system. Evidently there should be color in the rings but it very seldom shows on account of the faintness of the illumination.

Now the factor n is for the first dark ring 0.61, and for the first bright ring 0.81, as computed from Airy’s general theory, and therefore if we reckon that two stars will be seen as separate when the central disc of one falls on the first dark ring of the other the angular distance will be

Sin a = 0.61 ?/R'

and, taking ? at the brightest part of the spectrum i.e., about 560 , in the yellow green, with a taken for sin a, we can compute this assumed separating power for any aperture. Thus 560 being very nearly 1/45,500 inch, and assuming a 5 inch telescope, the instrument should on this basis show as double two stars whose centres are separated by 1.1 of arc.

In actual fact one can do somewhat better than this, showing that the visible diameter of the central disc is in effect less than the diameter indicated by the diffraction pattern, owing to the reasons already stated. Evidently the brightness of the star is a factor in the situation since if very bright the disc gains apparent size, and when very faint there is sufficient difficulty in seeing one star, let alone a pair.

The most thorough investigation of this matter of resolving power was made by the Rev. W. R. Dawes many years ago (Mem. R.A.S. 35, 158). His study included years of observation with telescopes of different sizes, and his final result was to establish what has since been known as “Dawes’ Limit.”

To sum up Dawes’ results he established the fact that on the average a one inch aperture would enable one to separate two 6th magnitude stars the centers of which were separated by 4.56. Or, to generalize from this basis, the separating power of any telescope is for very nearly equal stars, moderately bright, 4.56/A where A is the aperture of the telescope in inches.

Many years of experience have emphasized the usefulness of this approximate rule, but that it is only approximate must be candidly admitted. It is a limit decidedly under that just assigned on the basis of the theory of diffraction for the central bright wave-lengths of the spectrum. Attempts have been made to square the two figures by assuming in the diffraction theory a wave length of 1/55,000 inch, but this figure corresponds to a point well up into the blue, of so low luminosity that it is of no importance whatever in the visual use of a telescope.

The fact is that the visibility of two neighboring bright points as distinct, depends on a complex of physical and physiological factors, the exact relations of which have never been unravelled. To start with we have the principles of diffraction as just explained, which define the relation of the stellar disc to the center of the first dark ring, but we know that under no circumstances can one see the disc out to this limit, since vision fails to take cognizance of the faint rim of the image. The apparent diameter of the diffraction solid therefore corresponds to a section taken some distance above the base, the exact point depending on the sensitiveness of the particular observer’s eye, the actual brilliancy of the center of the disc, and the corresponding factors for the neighboring star.

Under favorable circumstances one would not go far amiss in taking the visible diameter of the disc at about half that reckoned to the center of the first dark ring. This figure in fact corresponds to what has been shown to be within the grasp of a good observer under favorable conditions, as we shall presently see.

On the other hand, if the stars are decidedly bright there is increase of apparent diameter of the disc due to the phenomenon known as irradiation, the spreading of light about its true image on the retina which corresponds quite closely to the halation produced by a bright spot on a photographic plate.

If, on the contrary, the stars are very faint the total amount of light available is not sufficient to make contrast over and above the background sufficient to disclose the two points as separate, while if the pair is very unequal the brighter one will produce sufficient glare to quite over-power the light from the smaller one so that the eye misses it entirely.

A striking case of this is found in the companion to Sirius, an extremely difficult object for ordinary telescopes although the distance to the companion is about 10.6 and its magnitude is 8.4, making a superlatively easy double for the very smallest telescope save for the overpowering effect of the light of the large star. Another notoriously difficult object for small telescopes is d Cygni, a beautiful double of which the smaller component falls unpleasantly near the first diffraction maximum of the primary in which it is apt to be lost.

“Dawes’ Limit” is therefore subject to many qualifying factors. Lewis, in the papers already referred to (Obs. 37, 378) did an admirable piece of investigation in going through the double star work of about two score trained observers working with telescopes all the way from 4 inches to 36 inches aperture.

From this accumulation of data several striking facts stand out. First there is great difference between individual observers working with telescopes of similar aperture as respects their agreement with “Dawes’ Limit,” showing the effect of variation in the physiological factors as well as instrumental ones.

Second, there is also a very large difference between the facility of observing equal bright pairs and equal faint pairs, or unequal pairs of any kind, again emphasizing the physiological as well as the physical factors.

Finally, there is most unmistakable difference between small and large apertures in their capacity to work up to or past the standard of “Dawes’Limit.” The smaller telescopes are clearly the more efficient as would be anticipated from the facts just pointed out regarding the different effect of the ordinary and inescapable atmospheric waves on small and large instruments.

The big telescopes are unquestionably as good optically speaking as the small ones but under the ordinary working conditions, even as good as those a double star observer seeks, the smaller aperture by reason of less disturbance from atmospheric factors does relatively much the better work, however good the big instrument may be under exceptional conditions.

This is admirably shown by the discussion of the beautiful work of the late Mr. Burnham, than whom probably no better observer of doubles has been known to astronomy. His records of discovery with telescopes of 6, 9.4, 12, 18½ and 36 inches show the relative ease of working to the theoretical limit with instruments not seriously upset by ordinary atmospheric waves.

With the 6 inch aperture Burnham reached in the average 0.53 of Dawes’ limit, quite near the rough figure just suggested, and he also fell well inside Dawes’ limit with the 9.4 inch instrument. With none of the others did he reach it and in fact fell short of it by 15 to 60%. All observations being by the same notably skilled observer and representing discoveries of doubles, so that no aid could have been gained by familiarity, the issue becomes exceedingly plain that size with all its advantages in resolving power brings serious countervailing limitations due to atmosphere.

But a large aperture has besides its possible separating power one advantage that can not be discounted in “light grasp,” the power of discerning faint objects. This is the thing in which a small telescope necessarily fails. The “light grasp” of the telescope obviously depends chiefly on the area of the objective, and visually only in very minor degree on the absorption of the thicker glass in the case of a large lens.

According to the conventional scale of star magnitudes as now in universal use, stars are classified in magnitudes which differ from each other by a light ratio of 2.512. a number the logarithm of which is 0.4, a relation suggested by Pogson some forty years ago. A second magnitude star therefore gives only about 40% of the light of a first magnitude star, while a third magnitude star gives again a little less than 40% of the light of a second magnitude star and so on.

But doubling the aperture of a telescope increases the available area of the objective four times and so on, the “light grasp” being in proportion to the square of the aperture. Thus a 10 inch objective will take in and deliver nearly 100 times as much light as would a 1 inch aperture. If one follows Pogson’s scale down the line he will find that this corresponds exactly to 5 stellar magnitudes, so that if a 1 inch aperture discloses, as it readily does, a 9th magnitude star, a 10 inch aperture should disclose a 14th magnitude star.

Such is substantially in fact the case, and one can therefore readily tabulate the minimum visible for an aperture just as he can tabulate the approximate resolving power by reference to Dawes’ limit. Fig. 186 shows in graphic form both these relations for ready reference, the variation of resolving power with aperture, and that of “light grasp,” reckoned in stellar magnitudes.

It is hardly necessary to state that considerable individual and observational differences will be found in each of these cases, in the latter amounting to not less than 0.5 to 1.0 magnitude either way. The scale is based on the 9th magnitude star just being visible with 1 inch aperture, whereas in fact under varying conditions and with various observers the range may be from the 8th to 10th magnitude. All these things, however convenient, must be taken merely at their true value as good working approximations.

Even the diffraction theory can be taken only as an approximation since no optical surface is absolutely perfect and in the ordinary refracting telescope there is a necessary residual chromatic aberration beside whatever may remain of spherical errors.

It is a fact therefore, as has been shown by Conrady (M.N. 79 575) following up a distinguished investigation by Lord Rayleigh (Sci. Papers 1 415), that a certain small amount of aberration can be tolerated without material effect on the definition, which is very fortunate considering that the secondary spectrum represents aberrations of about ½,000 of the focal length, as we have already seen.

The chief effect of this, as of very slight spherical aberration, is merely to reduce the maximum intensity of the central disc of the diffraction pattern and to produce a faint haze about it which slightly illuminates the diffraction minima. The visible diameter of the disc and the relative distribution of intensity in it is not however materially changed so that the main effect is a little loss and scattering of light.

With larger aberrations these effects are more serious but where the change in length of optical path between the ray proceeding through the center of the objective and that from the margin does not exceed ¼? the injury to the definition is substantially negligible and virtually disappears when the image is focussed for the best definition, the loss of maximum intensity in the star disc amounting to less than 20%.

Even twice this error is not a very serious matter and can be for the most part compensated by a minute change of focus as is very beautifully shown in a paper by Buxton(M. N. 81, 547), which should be consulted for detail of the variations to be effected.

Conrady finds a given change dp in the difference in lengths of the optical paths, related to the equivalent linear change of focus, df, as follows:—

df = 8dp(f/A)²

A being the aperture and f the focal length, which indicates for telescopes of ordinary focal ratio a tolerance of the order of ±0.01 inch before getting outside the limit ? for variation of path.

For instruments of greater relative aperture the precision of focus and in general the requirements for lessened aberration are far more severe, proportional in fact to the square of this aperture. Hence the severe demands on a reflector for exact figure. An instrument working at F/5 or F/6 is extremely sensitive to focus and demands great precision of figure to fall within permissible values, say ¼? to ½?, for dp.

Further, with a given value of dp and the relation established by the chromatic aberration, i.e., about f/2000, a relation is also determined between f and A, required to bring the aberration within limits. The equation thus found is

f = 2.8A²

This practically amounts to the common F/15 ratio for an aperture of approximately 5 inches. For smaller apertures a greater ratio can be well used, for larger, a relatively longer focus is indicated, the penalty being light spread into a halo over the diffraction image and reducing faint contrasts somewhat seriously.

This is one of the factors aside from atmosphere, interfering with the full advantage of large apertures in refractors. While as already noted small amounts of spherical aberration may be to a certain extent focussed out, the sign of df must change with the sign of the residual aberration, and a quick and certain test of the presence of spherical aberration is a variation in the appearance of the image inside and outside focus.

To emphasize the importance of exact knowledge of existing aberrations note Fig. 187, which shows the results of Hartmann tests on a typical group of the world’s large objectives. All show traces of residual zones, but differing greatly in magnitude and position as the attached scales show. The most conspicuous aberrations are in the big Potsdam photographic refractor, the least are in the 24 inch Lowell refractor. The former has since been refigured by Schmidt and revised data are not yet available; the latter received its final figure from the Lundins after the last of the Clarks had passed on.

Now a glance at the curves shows that the bad zone of the Potsdam glass was originally near the periphery, (I), hence both involved large area and, from Conrady’s equation, seriously enlarged df due to the large relative aperture at the zone. An aberrant zone near the axis as in the stage (III) of the Potsdam objective or in the Ottawa 15 inch objective is much less harmful for corresponding reasons. Such differences have a direct bearing on the use of stops, since these may do good in case of peripheral aberration and harm when the faults are axial. Unless the aberrations are known no general conclusions can be drawn as to the effect of stops. Even in the Lowell telescope shown as a whole in Fig. 188, the late Dr. Lowell found stops to be useful in keeping down atmospheric troubles and reducing the illumination although they could have had no effect in relation to figure. Fig. 188 shows at the head of the tube a fitting for a big iris diaphragm, controlled from the eye-end, the value of which was well demonstrated by numerous observers.

There are, too, cases in which a small instrument, despite intrinsic lack of resolving power, may actually do better work than a big one. Such are met in instances where extreme con[Pg 268]
[Pg 269]trast of details is sought, as has been well pointed out by Nutting (Ap. J. 40, 33) and the situation disclosed by him finds amplification in the extraordinary work done by Barnard with a cheap lantern lens of 1½ inch diameter and 5½ inches focus (Pop. Ast., 6, 452).

The fact is that every task must seek its own proper instrument. And in any case the interpretation of observed results is a matter that passes far beyond the bounds of geometrical optics, and involves physiological factors that are dominant in all visual problems.

With respect to the visibility of objects the general diffraction theory again comes into play. For a bright line, for example, the diffraction figure is no longer chiefly a cone like Fig. 183, but a similar long wedge-shaped figure, with wave-like shoulders corresponding to the diffraction rings. The visibility of such a line depends not only on the distribution of intensity in the theoretical wedge but on the sensitiveness of the eye and the nature of the background and so forth, just as in the case of a star disc.

If the eye is from its nature or state of adaptation keen enough on detail but not particularly sensitive to slight differences of intensity, the line will very likely be seen as if a section were made of the wedge near its thin edge. In other words the line will appear thin and sharp as the diffraction rings about a star frequently do.

With an eye very sensitive to light and small differences of contrast the appearance of absolutely the same thing may correspond to a section through the wedge near its base, in other words to a broad strip shading off somewhat indistinctly at the edges, influenced again by irradiation and the character of the background.

If there be much detail simultaneously visible the diffraction patterns may be mixed up in a most intricate fashion and one can readily see the confusion which may exist in correlating the work of various observers on things like planetary and lunar detail.

In the planetary case the total image is a complex of illuminated areas of diffraction at the edges, which may be represented as the diffraction solid of Fig. 185, in which the dotted lines show what may correspond fairly to the real diameter of the planet, the edge shading off in a way again complicated by irradiation.

Fancy detail superimposed on a disc of this sort and one has a vivid idea of the difficulty of interpreting observations.

It would be an exceedingly good thing if everyone who uses his telescope had the advantage of at least a brief course in microscopy, whereby he would gain very much in the practical understanding of resolving power, seeing conditions, and the interpretation of the image. The principles regarding these matters are in fact very much the same with the two great instruments of research.

Aperture, linear in the case of the telescope and the so-called numerical in the case of the microscope, bear precisely the same relation to resolution, the minimum resolvable detail being in each case directly proportional to aperture in the senses here employed.

Further, although the turbulence of intervening atmosphere does not interfere with the visibility of microscopic detail, a similar disturbing factor does enter in the form of irregular and misplaced illumination. It is a perfectly easy matter to make beautifully distinct detail quite vanish from a microscopic image merely by mismanagement of the illumination, just as unsteady atmosphere will produce substantially the same effect in the telescopic image.

In the matter of magnification the two cases run quite parallel, and magnification pushed beyond what is justified by the resolving power of the instrument does substantially little or no good. It neither discloses new detail nor does it bring out more sharply detail which can be seen at all with a lower power.

The microscopist early learns to shun high power oculars, both from their being less comfortable to work with, and from their failing to add to the efficiency of the instrument except in some rare cases with objectives of very high resolving power. Furthermore in the interpretation of detail the lessons to be learned from the two instruments are quite the same, although one belongs to the infinitely little and the other to the infinitely great.

Nothing is more instructive in grasping the relation between resolving power, magnification, and the verity of detail, than the study under the microscope of some well known objects. For example, in Fig. 189 is shown a rough sketch of a common diatom, Navicula Lyra. The tiny siliceous valve appears thus under an objective of slightly insufficient resolving power. The general form of the object is clearly perceived, as well as the central markings, standing boldly out in the form which suggests the specific name. No trace of any finer detail appears and no amount of dexterity in arranging the illumination or increase of magnifying power will show any more than here appears, the drawing being one actually made with the camera lucida, using an objective of numerical aperture just too small to resolve the details of the diatoms on this particular slide.

Figure 189a shows what happens when, with the same magnifying power, an objective of slightly greater aperture is employed. Here the whole surface of the valve is marked with fine striations, beautifully sharp and distinct like the lines of a steel engraving. There is a complete change of aspect wrought by an increase of about 20% in the resolving power. Again nothing further can be made out by an increase of magnification, the only effect being to make the outlines a little hazier and the view therefore somewhat less satisfactory.

Finally in Fig. 189b we have again the same valve under the same magnifying power, but here obtained from an objective of numerical aperture 60% above that used for the main figure. The sharp striÆ now show their true character. They had their origin in lines of very clearly distinguished dots, which are perfectly distinct, and are due to the resolving power at last being sufficient to show the detail which previously merely formed a sharp linear diffraction pattern entirely incapable of being resolved into anything else by the eye, however much it might be magnified.

Here one has, set out in unmistakable terms, the same kind of differences which appear in viewing celestial detail through telescopes of various aperture. What cannot be seen at all with a low aperture may be seen with higher ones under totally different aspects; while in each case the apparent sharpness and clarity of the image is somewhat extraordinary.

Further in Fig. 189b in using the resolving power of the objective of high numerical aperture, the image may be quite wrecked by a little carelessness in focussing, or by mismanagement of light, so that one would hardly know that the valve had markings other than those seen with the objectives of lower aperture, and under these circumstances added magnification would do more harm than good. In precisely the same way mismanagement of the illumination in Fig. 189a would cause the striÆ to vanish and with Navicula Lyra, as with many other diatoms, the resolution into striÆ is a thing which often depends entirely on careful lighting, and the detail flashes into distinctness or vanishes with a suddenness which is altogether surprising. For “lighting” read “atmosphere,” and you have just the sort of conditions that exist in telescope vision.

With respect to magnifying powers what has already been said is sufficient to indicate that on the whole the lowest power which discloses to the eye the detail within the reach of the resolving power of the objective is the most satisfactory.

Every increase above this magnifies all the optical faults of the telescope and the atmospheric difficulties as well, beside decreasing the diameter of the emergent pencil which enters the eye, and thereby causing serious loss of acuity. For the eye like any other optical instrument loses resolving power with decrease of effective aperture, and, besides, a very narrow beam entering it is subject to the interference of entoptic defects, such as floating motes and the like, to a serious extent.

Figure 190 shows from Cobb’s experiments (Am. Jour. of Physiol., 35, 335) the effect of reduction of ocular aperture upon acuity. The curve shows very plainly that for emergent pencils below a millimeter (1/25 inch) in diameter, visual acuity falls off almost in direct proportion to the decreasing aperture. Below this figure there can be only incidental gains, such as may be due to opening up double stars and simultaneously so diminishing the general illumination as to render the margins of the star discs a little less conspicuous.

An emergent pencil of this diameter is not quite sufficient for the average eye to utilize fully the available resolving power and some excess of magnification even though it actually diminishes visual acuity materially, may be of some service.

Increased acuity will of course be gained for the same magnification in using an objective of greater diameter, to say nothing of increased resolving power, at the cost, of course, of relatively greater atmospheric troubles.

To come down to figures as to the resolving power of the eye, often repeated experiments have shown that two points offering strong contrast with the background can be noted as separate by the normal eye when at an angular separation of about 3' of arc. People, as we have seen, differ considerably in acuity so that now and then individuals will considerably better this figure, while others, far less keen sighted, may require a separation of 4' or even 5'.

The pair of double stars e₁, e₂, LyrÆ, separated by 3' 27 mags. nearly 4 and 5 respectively, can be seen as separate by those of fairly keen vision, while Mizar and Alcor, 11' apart, seem thrown wide to nearly every one. On the other hand the writer has never known anybody who could separate the two components of Asterope of the Pleiades, distant a scant 2½ but of mags. 6.5 and 7.0 only, while Pleione and Atlas, distance about 5¼', mags. 6.5 and 4, are very easy.

Assuming for liberality that the separation constant is in the neighborhood of 5' one can readily estimate the magnification that for any telescope will take full advantage of its resolving power. As we have already seen this resolving power is practically 4.56/A for equal stars moderately bright. An objective of 4.56' inches aperture has a resolving constant of 1 and to develop this should take a magnification of say 300, about 65 to the inch of aperture, requiring a focal length of ocular about 0.20 to 0.25 inch for telescopes of normal relative aperture, and pushing the emergent pencil down to little more than 0.02 inch,—rather further than is physiologically desirable. Except for these extreme stunts of separation, half to two thirds this power is preferable and conditions under which one can advantageously go above this limit are very rare indeed.

A thoroughly good objective or mirror will stand quite 100 magnification to the inch without, as the microscopist would say, “breaking down the image,” but in at least nine cases out of ten the result will be decidedly unsatisfactory.

As the relative aperture of the instrument increases, other things being equal, one is driven to oculars of shorter and shorter focus to obtain the same magnification and soon gets into trouble. Very few oculars below 0.20 inch in focus are made, and such are rarely advisable, although occasionally in use down to 0.15 inch or thereabouts. The usual F/15 aperture is a figure quite probably as much due to the undesirability of extremely short focus oculars as to the easier corrections of the objective.

In the actual practice of experienced observers the indications of theory are well borne out. Data of the habits of many observers of double stars are of record and the accomplished veteran editor of The Observatory, Mr. T. Lewis, took the trouble in one of his admirable papers on “Double Star Astronomy” (Obs. 36, 426) to tabulate from the original sources the practice of a large group of experts. The general result was to show the habitual use with telescopes of moderate size of powers around 50 per inch of aperture, now and then on special occasions raised to the neighborhood of 70 per inch.

But the data showed unequivocally just what has been already indicated, that large apertures, suffering severely as they generally do from turbulence of the air, will not ordinarily stand their due proportion of magnification. With the refractors of 24 inches aperture and upwards the records show that even in this double star work, where, if anywhere, high power counts, the general practice ran in the vicinity of 30 per inch of aperture.

Analyzing the data more completely in this respect Mr. Lewis found that the best practise of the skilled observers studied was approximately represented by the empirical equation

m = 140 vA

Of course the actual figures must vary with the conditions of location and the general quality of the seeing, as well as the work in hand. For other than double star work the tendency will be generally toward lower powers. The details which depend on shade perception rather than visual acuity are usually hurt rather than helped when magnified beyond the point at which they are fairly resolved, quite as in the case of the microscope.

Now and then they may be made more distinct by the judicious use of shade glasses. Quite apart from the matter of the high powers which can advantageously be used on a telescope, one must for certain purposes consider the lowest powers which are fairly applicable. This question really turns on the largest utilizable emergent pencil from the eye piece. It used to be commonly stated that ? inch for the emergent pencil was about a working maximum, leading to a magnification of 8 per inch of aperture of the objective. This in view of our present knowledge of the eye and its properties is too low an estimate of pupillary aperture. It is a fact which has been well known for more than a decade that in faint light, when the eye has become adapted to its situation, the pupil opens up to two or three times this diameter and there is no doubt that a fifth or a fourth of an inch aperture can be well utilized, provided the eye is properly dark-adapted. For scrutinizing faint objects, comet sweeping and the like, one should therefore have one ocular of very wide field and magnifying power of 4 or 5 per inch of aperture, the main point being to secure a field as wide is practicable. One may use for such purposes either a very wide field Huyghenian, or, if cross wires are to be used, a Kellner form. Fifty degrees of field is perfectly practicable with either. As regards the rest of the eyepiece equipment the observer may well suit his own convenience and resources. Usually one ocular of about half the maximum power provided will be found extremely convenient and perhaps oftener used than either the high or low power. Oculars of intermediate power and adapted for various purposes will generally find their way into any telescopic equipment. And as a last word do not expect to improve bad conditions by magnifying. If the seeing is bad with a low power, cap the telescope and await a better opportunity.

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