Translated notes on visual observing from “Praxishandbuch Deep Sky”, by Vereinigung der Sternfreunde e.V., published by Kosmos publishers.
With the eye we see that which enters in the pupil, which is an aperture in the iris. Its size is known as the “Entrance Pupil”. This concept, is generalizable to all sorts of “openings”, so it’s also used for telescopes (where one would refer to it as “Aperture”). The entrance pupil depends not only on the dark adaptation, but also on the physiology of the observer. It can typically amount to a maximum of 6–8mm, younger people generally having it better. Thus, if after complete dark adaptation we consider observation with the naked eye, we can assume that the maximum exit pupil is about 7mm (this shall henceforth be the value we will assume to calculate).
What happens then, if we’re observing through a telescope? Let’s for a moment ignore the eye and consider the telescope. A parallel pencil of light rays emerges from the eyepiece. This beam furnishes a (demagnified) image of the telescope’s aperture. This can be readily seen when one observes at a distance of 30cm from the eyepiece (best in daylight). There will be a bright, round spot visible, which does not fill the eyepiece lens: this is the image of the telescope’s aperture. If we move around, so does this image. The object imaged in the telescope becomes visible only when one approaches the eyepiece very closely with the eye.
The aforementioned round spot is the “Exit Pupil”. This is the diameter of the exiting parallel light beam and thereby also the amount of “light information”. The exit pupil only depends on the parameters of the telescope and the eyepiece used. Before we concern ourselves with the calculations, let us examine how the exit pupil (telescope) and entrance pupil (eye) fit together. The exiting parallel light beam falls perpendicularly on the entrance pupil and enters it. If the diameter of the light beam (i.e. the exit pupil) is smaller than the entrance pupil, then all the light enters into the eye. However, if it is larger, then the entrance pupil cuts the “overflowing” part out mercilessly – it acts like a simple diaphragm! This means that when exit pupil > entrance pupil, light is lost. When entrance pupil < exit pupil, all the light comes through, but we don’t get any extra light! Then, is exit pupil = entrance pupil perfect? Yes, particularly if we first ensure that the entrace pupil is maximum (i.e. dark adaptation) and we’re using an eyepiece with the corresponding exit pupil. We cannot receive any more light on the retina – that’s the whole secret!
Now comes an important assertion for observation of extended deep-sky objects (under which comes pretty much everything except for quasars, stars, details in extended objects and stellar planetary nebulae). It can be reduced to a single formula for the visual intensity of a luminous surface (SB):
SB = (Exit Pupil / Entrance Pupil)2
What does this mean? Suppose we are first observing with the naked eye. What is the exit pupil of the sky? It doesn’t matter how big it is, at any rate, the parallel light from the depths of the cosmos is trimmed by our entrance pupil! We can therefore set Exit Pupil = Entrance Pupil in the formula, which simply gives us SB = 1. SB can effectively never exceed 1. Now let us bring our telescope into the picture. As we have seen, Exit Pupil > Entrance Pupil is pointless. So if it isn’t the case that Exit Pupil = Entrance Pupil (as could be achieved by appropriate eyepiece selection, as we shall see later), then the Exit Pupil can only be smaller than the Entrance Pupil, which means that SB < 1. That however brings us to the following (surprising, but fully correct) insight: One can never see a luminous surface in a telescope brighter than it would appear to the naked eye – no matter whether we are using a department store telescope or a 5m mirror.
The sole “knob” that one can control (and must) is dark adaptation, which brings us to the maximal entrance pupil of about 7mm. Thereafter one picks a suitable eyepiece: one which provides an exit pupil of 7mm (one can thus talk about “maximal exit pupil”). Unfortunately the exit pupil value is not printed on the eyepiece! If we were to randomly grab an eyepiece from our case, perhaps we would find one which provides an exit pupil that’s only half as big as our entrance pupil. We’ve then hit some bad luck: we obtain SB = (½)2 = (¼). The object would shine four times dimmer in this eyepiece than it actually is.
A calculation helps to prevent such unwarranted surprises. For that we would firstly need the f-ratio “N” of the telescope. This is a characteristic value of the telescope, only depending on its mirror, or rather, objective (as opposed to the eyepiece). One obtains the f-number by dividing the focal length “f” (in mm) by the diameter of the objective “D” (in mm):
N = f / D
Example: For a telescope with 30cm main mirror diameter and a 1.5m focal length, one obtains N = 1500mm/300mm = 5. Now, the exit pupil (in mm) is obtained if we divide the eyepiece focal length f' (in mm) by the f-ratio:
Exit Pupil = f' / N
Example: An eyepiece of 15mm focal length on the aforementioned telescope (N = 5) provides exit pupil = 15mm/5 = 3mm. If we do not change the telescope, so that N is a fixed number, the exit pupil is determined solely by the choice of the eyepiece (i.e. f') – and f' is stamped on the eyepiece! To note: the larger the eyepiece focal length, so is the exit pupil larger.
We can also alternatively calculate the exit pupil using the magnification M. M is defined as the quotient of the objective’s focal length (f) to the eyepiece focal length (f'):
M = f/f'
Thereby from the above relations, we obtain for the exit pupil
Exit Pupil = D / M.
In our example, M = 1500mm/15mm = 100x, so the Exit Pupil = 300mm/100 = 3mm (as we had earlier). This formula is particularly useful for binoculars, where D and M are already stamped. For a 10x50 binocular, one easily obtains Exit Puipil = 50mm/10 = 5mm. For a telescope, since D is fixed, the Exit Pupil in this case is soleley determined by the magnification. Note: The smaller the the magnification, the larger the exit pupil. Corresponding to the maximal exit pupil, we thus obtain a minimum magnification (with D in mm):
Mmin = D / 7.
We have seen that the optimum detection of extended objects is achieved if the entrance pupil is maximal (~7mm) and we choose an eyepiece such that Exit Pupil = Entrance Pupil. Extended nebulae can never be seen brighter through a telescope than with the naked eye. What then with the many objects, that we can see through a telescope, but not with the naked eye? Theoretically, we should not be able to detect much more in a telescope than Andromeda Galaxy. As we shall see, there are two additional factors that are crucial: contrast and magnification.
Suppose that the eye and telescope are (in the mentioned sense) optimally matched, the success of observation must then solely depend upon what the deep-sky object itself emits (i.e. surface brightness). Unfortunately, this is only half the story, for we must consider the sky background too. Really, we should call it the “sky foreground”, since the source of its surface brightness lies in the atmosphere (straylight due to light pollution, airglow etc.) and in a small fraction in the planetary system (Zodiacal light). The key quantity is the “Contrast” between object and the night sky, technically one speaks of a Signal-to-Noise Ratio. An observation succeeds only when the object stands out against the noise.
How large is this noise? An “absolute” dark night sky has a surface brightness of V' = 13.1mag/arcmin2 = 22mag/arcsec2 in the visual band. In the B-band, it is yet another magnitude darker (B' = 14.1mag/arcmin2). This is far from perfect darkness! If one holds, for example, a stretched out hand against the sky, it appears significantly darker. The eye is in a position to be able to even distinguish surfaces with a fraction of the night sky brightness. For example, if we let the night sky illuminate the surface of our hand, one can distinguish structures, even though only 25% is reflected in this case (the difference from the sky amounts to scarcely 1.5mag). The surface brightness of the night sky corresponds to an intensity, which we shall denote by I0 (“Noise”). As we have seen in Section 1.1, the brightness corresponds (on physiological ground) to the logarithm of the intensity.
What is the signal? A measurement of the intensity of the object results in not the object’s intensity alone, but rather invariably in the sum of the intensities of the object (I) plus the night sky (I0), i.e. Itot = I + I0. This total intensity is our signal. Expressed in surface brightnesses, it means the following: The true surface brightness of the object is obtained when one “subtracts” the night-sky from the observed value. This is to note, if one reads values such as V' = 14.5mag/arcmin2 for faint galaxies – that is 1.6mag/arcmin2 fainter than the night sky! The observed surface brightness is the “sum” of 13.1mag/arcmin2 and 14.5mag/arcmin2, which adds up to 12.8mag/arcmin2 in this case (one must add the intensities first). The object thus is 0.3mag/arcmin2 brighter than the night sky. With the true intensity I = Itot - I0 of the object, the contrast is obtained through the relation: C = I / I0.
Generally the following thumb-rule holds, that objects with V' = 14.5mag/arcmin2 lie on the threshold of visual observation for medium-sized telescopes in an absolutely dark sky. Unforunately, the German night-sky is mostly brighter than 13.1mag/arcmin2, so the contrast drops out lower. In a night sky of 12.1mag/arcmin2 (near a city), objects with V' = 13.5mag/arcmin2 are already in this critical range. In the threshold region, by the way, the apparent (integrated) magnitude plays no major role anymore, and even the brighter objects “sink” into the night sky. M 31 with V = 3.5mag and V' = 13.4mag/arcmin2 is still directly visible to the naked-eye under these city-outskirt conditions. M 33 with V = 5.7mag and V' = 14.1mag/arcmin2 requires however a night-sky which is darker than V' = 12.7mag/arcmin2 (a good country sky). Thomas Pfleger has incorporated these considerations into his program “Eye & Telescope” and has defined the observability of objects through the “contrast reserve”. This is the difference between the contrast of the object and a minimal contrast (“contrast threshold”) that the eye requires at least to detect an extended object.
What role does magnification play then? In the above formula, the contrast is clearly independent of the magnification. If we change the magnification (and thereby the exit pupil), it affects the surface brightness of the object and the background equally, so the contrast remains the same. Then why do we see faint nebulae (of any kind) nevertheless better with higher magnification? The reason is of physiological nature. We shall now compare three different situations with (maximal) exit pupil = 7mm. According to the formula, Exit Pupil = D / M, we may achieve this with a 70mm refractor at 10x magnification, but also with a 50cm Dobsonian and about 70x magnification – and of course also with the eye with D = 7mm and M = 1x.
The surface of the nebula is nominally of the same brightness in both telescopes and to the naked eye. Can we therefore conclude that all three situations are therefore the same? No: with the naked eye we see nothing at all, in the small refractor probably a small fleck, whereas in the Dobsonian, the object appears large and structured. Clearly, the eye honors the size of the nebula, insofar as it does not fill the field of view – and the size is determined by the magnification. With the naked eye, the object is stellar and vanishes, like a faint star, into the noise. The same holds at low magnifications (a phenomenon well-known in photographs). If the object is sufficiently magnified, the eye is much better served. This is clearly attributable to a larger “visual angle”, by which we mean the size of the image subtended on the retina. According to Ricco’s law, the threshold of detection of the eye is constant up to a visual angle of about 10'. For larger visual angles, Piper’s law applies. According to this, the sensitivity (to averted vision) increases proportional to the visual angle. Hence, a 1o size image on the retina renders about 1 magnitude fainter objects visible. This rule holds until about 8o, whereby one gains (theoretically) 3 magnitudes.
By magnification (whence increasing the visual angle), nebulae with small angular extents become observable; however the surface brightness of smaller (farther) galaxies and bigger (nearer) galaxies are mostly not very different. We therefore owe it to our eyes, that we can observe far out galaxies too – it doesn’t end with just the Andromeda Galaxy! Film or CCD are not subject to these effects, and they can integrate light unlike the eye. The eye in contrast is hardly accumulative, but is instead sensitive to movement or changes in intensity.
Another implication is noteworthy: If we wish to observe large and faint nebulae that do not necessitate, or tolerate, a high magnification because their extents, a small telescope fully suffices. A few examples. The elliptical companions of M 31, NGC 147 and NGC 185, have integrated brightness of 9.5mag, whereas V' = 14.7mag/arcmin2. Whereas NGC 185 has a relatively bright nucleus, NGC 147 is quite diffuse and accordingly challenging. A 6-incher (D = 150mm) at maximum exit pupil (M = 20x) and dark sky however achieves the goal here (for an f/6, one would require a 45mm eyepiece). But whether one can also use it to see the Ursa Minor dwarf galaxy (V' = 18.3mag/arcmin2), is suspect. Another historical example: Herschel has described 52 extended, extremely faint “nebulous wisps”. His telescope was, at 18.7", not exactly small, and the large f-ratio of f/12.7 was completely inappropriate for observation of such objects, moreover he possessed no filter. Here, the minimum magnification would be 68x, which would call for a 90mm eyepiece, but that he did not have! So what had Herschel seen at the (surely dark skies of) Slough near Windsor? A few of those objects have been identified, and apparently amongst them is “Barnard’s Loop” in Orion!
Maximal exit pupil is definitely important for large, faint nebulae; but one doesn’t always get lucky. If the sky is brightened by stray light, it might be better to use a higher magnification, thereby choosing a smaller exit pupil, so that the sky background is rendered darker and the contrast of the faint nebula is increased.
We have hitherto only spoken to extended objects, so how does it work with point objects? With the maximal exit pupil, faint nebulae are best seen, however this does not apply to the faintest stars. For the unaided eye, the stellar limiting magnitude does not depend on its entrance pupil. It is actually better to reduce the exit pupil, i.e. to use higher magnifications. What’s responsible for this?
Let us consider the bright globular cluster M 13. Here, in the previously considered comparison of instruments, the 70mm refractor is disadvantaged in comparison to the 50cm Dobsonian. With the same exit pupil, the Dobsonian at 70x magnification shows (on the outskirts) individual stars, whereas in contrast the refractor at 10x magnification shows only a bright, round nebulous spot. When it comes to resolution, it is clearly unnecessary that the image of M 13 exhibit the same (observed) surface brightness in both cases. Moreover, the limiting magnitude increases with the magnification. The 70x magnification in the Dobsonian here is only the bare minimum: 500x magnification would clearly be better (!) – take note, we are talking about stars. The exit pupil would then be about 1mm. With a dark-adapted entrance pupil of 7mm and an exit pupil of 1mm, the brightness of the sky background in the eyepiece amounts to only (1/7)2, i.e. it appears almost 50 times darker than with the naked eye. A faint nebula would then also be 50 times fainter, invisible. For a star, this attenuation does not apply, it doesn’t succumb to the “SB-law”, and only now does it begin to appear correctly. It separates out from the “noise” and the contrast increases. Furthermore, the eye can adapt a lot better with the darker background. Magnification and adaptation make it possible even to see stars during the daytime through a telescope. Also interesting is the (night-time) observation of Heber Curtis: He could see stars down to 8mag with the naked eye through a tube with a 5' field of view! This corresponds to the theoretical detection threshold of the eye. However, anyone who claims to see stars with a brightness of 16th magnitude with a 6-incher is bonkers, because it would mean that he would reach a limiting magnitude of 9 with the naked eye, which corresponds to 25 stars of the Pleiades – that is too good to be true.
How high can one magnify? That depends most importantly on the size of the diffraction disk. This plays a role starting at an exit pupil of 0.7mm, which defines an “Optimal magnification” (whereby we have Vopt = 10 Vmin). It is only up to this value (which is 714x for our Dobsonian) that stars appear point-like, insofar as the seeing allows this and the telescope is in optimal optical condition. With further higher magnifications, the star becomes “extended” and (like a nebula) becomes fainter with smaller exit pupils – the “SB-rule” applies once again. The optimal magnification thus also determines the limiting magnitude in a telescope (for our Dobsonian, about 17.5mag). With very good air one can certainly attempt the “Maximum magnification”. Vmax is given roughly by 3 times the objective diameter (in mm). This is deployed, for example, to observe details in bright planetary nebulae. It is known that both William Herschel as well as Lord Rosse used magnifications over 2000x.
Stars are (nominally) the objects with the highest surface brightness. The above discussed (higher magnification, smaller exit pupil) also applies for other objects with high surface brightness, like rather small planetary nebulae, compact galaxies or quasars. Many objects are however “mixed”: there are bright, condensed patches, surrounded by regions with lower surface brightness. Examples for inhomogenous brightness distribution are bright galaxies (with many “knots”), galactic nebulae which are clearly structured, or Seyfert galaxies. These (Seyfert galaxies) possess a bright core, surrounded by a faint halo. These register per calculation a lower (intermediate) surface brightness, therefore suggesting a lower magnification. However, to find it warrants a higher magnification, because only then is the stellar core recognizable. Only then can one use a smaller magnification to hunt for the diffuse envelope. For detail-rich nebulae, the following holds: one sees the structures best with higher magnification (smaller exit pupil) – at the expense of the detection of the faint outer regions, which require a large exit pupil. This shows that for many objects, there is no single appropriate magnification. Only different magnifications bring the maximum observing success – an important insight for description and sketching.
An interesting tool is the program “Optimum Detection Magnification” (ODM) by Mel Bartels, that is based on the findings of Roger Clark (Visual Astronomy of the Deep Sky). With this, it is possible to compute the observability of an extended object (in the sense of “contrast reserve”) easily. The inputs are: the surface brightness of the night sky (e.g. 21MPSAS for country skies, 18MPSAS for city skies), telescope diameter (in inch), object brightness and the major and minor diameters (in arcminute). The contrast difference (contrast reserve) is calculated; positive values imply visibility, negative values imply invisibility. In the case of visibility, the suitable magnification (ODM) is provided. As is elucidated in this section, one realizes by doing so clearly that the quality of the night sky is more important than a large aperture.