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The
Limits of Telescopic Performance By
Lenny Abbey, FRAS There
can be little doubt that the telescope is the basic tool of astronomy.
Practically all our knowledge of the physical nature of the heavenly
bodies can be attributed to this instrument, by itself or as the basic
component of a system of instruments. The theoretical limits of
telescopic performance have been discussed by many authors; however
little of this information seems to have filtered down to the amateur.
Even though what an observer sees is largely dependent on his ability
and experience as an observer, there are certain limits imposed upon his
observations by the size and quality of his telescope. To the observer,
a knowledge of exactly what his telescope will and will not do is of
great importance. Only
the telescope used in conjunction with the human eye will be considered
here, since most amateurs are concerned primarily with visual
observation. The
performance of the eye as an optical instrument is well understood. Most
amateur astronomers are familiar with the gross features of the eye; how
incident light is refracted through the cornea, aqueous humor, lens, and
vitreous humor in turn, to form a real inverted image on the
light-sensitive retina. The structure of the retina is of the utmost
importance to the subject of visual telescopic performance, since the
formation on its surface of a brighter or enlarged image of the object
under view is the primary purpose of the telescope. The
light sensitive elements of the retina are the rods and cones. The
information from the image on the retina is transmitted to the brain in
the form of impulses from the rods and cones; thus what the brain
perceives is a mosaic of signals from individual receptors. The rods are
most sensitive to faint illumination, and are concentrated near
the edges of the retina. This is why an observer can see faint objects
by means of "averted" vision which cannot be seen by direct
vision. The cones, on the other hand, are sensitive to brighter
illumination and to colors. An object can be sharply seen only when it
is focused on the center of the retina, where cones are found
exclusively. Both rods and cones are bathed in rhodopsin, or visual
purple, which is produced only when comparatively little light falls on
the retina. It is believed that the presence of rhodopsin is responsible
for the sensitivity of the rods to dim light sources. The
most common defects of the eye are myopia (short sightedness),
hypermetropia (far sightedness), and astigmatism. Since both myopia and
hypermetropia are defects of focus, the observer can compensate for them
by changing the focus of the telescope, and thus need not wear glasses
while observing. An observer with a moderate amount of astigmatism can
also observe without glasses. When such an observer uses high powers, in
which case the astigmatism might be expected to be objectionable
– it is not. This
is because only a small part of the eye’s lens is being used by the
narrow exit pupil of the high-power eyepiece. If a strong astigmatic
condition is present however, glasses are necessary for a clear image. Other
defects which are characteristic of all eyes are spherical and chromatic
aberration. Fortunately, these defects are not serious enough to impose
meaningful limitations on the quality of the images formed by the eye. The
light-gathering power of a telescope is the instrument's ability to make
faint sources of illumination visible. For a point source, incident
light from the object under view which falls on the objective lens is
concentrated in the image. A larger lens has a greater area and
consequently more light from the source falls on it. This increased
amount of light makes the image brighter than it would be in smaller
telescopes. Thus, the brightness of the image of a point source is
proportional to the area of the objective (or to the square of its
diameter). By making faint point sources appear to be brighter, the
telescope brings them above the visibility threshold of the eye. The
limiting magnitude – the magnitude of the faintest star that can be
seen with a given instrument – is
the most convenient and meaningful measure of its light-gathering
power. The most widely used formula for finding the limiting magnitude
is:
where
M is the limiting magnitude, d the diameter of the pupil of the
observer's dark adapted eye, D the aperture of the instrument, and 6.5
the assumed limiting magnitude of the unaided eye. This formula takes
advantage of the fact that the light-gathering power of the
eye-telescope system is greater than that of the eye alone by a factor
which is determined by the ratio of the apertures of the two systems.
Since it is known that the pupil of the average (young!) eye will open
to about 7.5mm (0.3 inches) when fully dark-adapted, the above formula
can be simplified to:
where
D is in inches. The factor 9.1 in this formula can be thought of as the
limiting magnitude of a one-inch telescope. As
might be expected, the above formulae, which are based on pure
theoretical considerations, do not hold true for all observers. This is
probably because the pupillary openings and retinal sensitivities vary
from individual to individual, resulting in differences of as much as
1.5 magnitudes in the limiting magnitude of a given instrument when used
by various observers. An interested observer can determine his personal
limiting magnitude with any instrument by determining the faintest star
that is visible to him (under optimum conditions) with a one-inch
telescope. By substituting this number in place of 9.1 in (2), he can
personalize the equation. Such
conditions as bad seeing and atmospheric and instrumental absorption
have been neglected in the above considerations. Bad seeing blurs and
enlarges the stellar images so that all of the light col1ected by the
objective is no longer concentrated into a tiny point. Faint stars
appear to be fainter, and the faintest stars that were formerly visible
escape detection. Atmospheric absorption is determined by the amount of
air through which the light has passed and is therefore inversely
proportional (by a complex relationship) to the altitude of the object
under view above the horizon. This is easily calculated with the aid of
tables in standard reference works. Telescopic absorption depends on the
type and condition of the instrument. Small telescopes, both reflectors
and refractors, transmit about 80% of the incident light. For sizes over
five inches, however, the reflector has an edge on the refractor. As
aperture increases, light loss due to absorption in the lens increases
as the lens becomes thicker, while the percentage of light reflected by
mirrors remains the same. This effect is generally negligible in
instruments under about twenty inches in aperture. Extended
sources behave quite differently when viewed with the telescope. Such
objects as nebulae and planets actually are fainter, per unit area, in
the telescope. They seem to be brighter because they are enlarged to an
appreciable size, while the brightness of the sky background is actually
decreased by the telescope. This is demonstrated by the surprising fact
that the Veil nebula, one of the more difficult gaseous nebulae, is just
as easy to see with a six-inch as with a sixteen-inch telescope! Resolving
power is by far the most misunderstood telescopic function. There is
much confusion in the literature available to the amateur as to just
what the resolving power of a telescope is, and how it limits the amount
of detail that can be seen on planets and the separation of double stars
that can be resolved. The subject has often been approached from an
empirical standpoint (i.e. observers reporting very delicate details
they have seen), but before a precise solution of the problem can be
formulated (if, indeed, it can be), the underlying optical principles
involved must be understood. The
resolving power of a telescope is its ability to form distinguishable
images of two objects of small angular separation, and is proportional
to the diameter of the telescope’s objective. The resolving power of a
telescope in relation to double stars will be considered first, as the
other cases are merely extensions and reapplications of the principles
pertaining to the separation of doubles. The stars are so far away that they may be considered to be true geometrical points, having no detectable size, but only position and brightness. If the image of a star in a telescope were a true point, there would be no limit to the theoretical resolving power of a given instrument. To resolve any pair, it would only be necessary to apply sufficient magnification to make the two images clearly separate to the eye. Unfortunately this is not the case. Due to the wave nature of light, rays striking different parts of the objective interfere with each other as they are brought to a focus and a circular image of finite size surrounded by a number of concentric faint rings (alternately light and dark) is formed. This is called a diffraction pattern.
The
dark rings are areas where the interference is destructive, while light
rings are areas where it is constructive. The radius of the inner and
most conspicuous dark ring is given by:
where
λ is the wavelength of the light, and D the diameter of the
objective. R is in radians. Substituting the value of λ for which
the eye is most sensitive (5500 Angstroms), and converting to seconds of
arc, (3) becomes:
where
D is in inches and S is in seconds of arc. The central (or Airy) disk is
somewhat smaller than the innermost dark ring, and gradually fades into
it. It is impossible for a telescope to form an image of a bright object
smaller than the Airy disk. If the Airy disk of a faint star appears to
be smaller than (4) predicts, it is because the outer edge, where the
light is tapering off, becomes imperceptible, even though it is still
present. About 85% of the energy from the source collected by the
objective is concentrated in the Airy disk. The first bright ring is
about 1.7% as bright as the Airy disk, and outer bright rings become
successively fainter. For faint stars, the rings may be so dim that they
are invisible, and then only the Airy disk is seen. A
double star appears in the telescope as a set of diffraction patterns.
As successively closer doubles are viewed, it will be seen that the two
patterns approach each other, overlap, and finally merge. The two stars
are "resolved" as long as the observer can be sure that two
diffraction patterns are present. The standard laboratory definition of
resolution is that when the center of the Airy disk of one pattern falls
on the first dark ring of the other, the pair can be considered to be
resolved. The limit of resolution would then be given by (4). But this
is not quite good enough for astronomical purposes. When the conditions
for laboratory resolution are satisfied, the two Airy disks overlap and
there is about a 20% intensity dip between their centers. This is
because the light intensity is greatest at the center of the Airy disk,
and diminishes towards its edge. The disks can be brought closer
together and still have an intensity dip between them. Well trained
observers can detect a dip of only 5%, and when two stars are so close
together that this is the case, their separation is given by:
Somewhat
closer pairs can be detected by noting the elongated diffraction
pattern, but they are not really “split." Equation (5) is the
famous Dawes’ limit, which was introduced as the result of a long
series of observations by W. R. Dawes, one of the nineteenth century's
most skilled observers. Strictly speaking, Dawes’ limit is valid only
for two yellow, sixth-magnitude stars, viewed in a six-inch telescope.
However, the equation is accurate except for pairs differing greatly in
brightness. Dawes’ limit applies to observations of double stars only. The
surface of an extended source, such as the Moon and planets, can be
considered to be a mosaic of an infinite number of point sources. Each
of these points forms an image, or diffraction pattern, in the focal
plane of the telescope, the size of which is approximately given by (3).
Dark areas on such surfaces are areas where there are no point sources.
Gross dark features are well defined, but when the angular size of a
feature approaches that of the individual diffraction patterns, it
becomes increasingly difficult to obtain a clear view of it because it
is swamped by overlapping neighboring patterns. Small
bright objects appear larger in telescopes than they really are because
the diffraction patterns from points near the edges project beyond it.
Dark areas surrounded by bright areas appear smaller for the same
reason. However, no bright object, however small, can appear to be
smaller than the Airy disk for the telescope in which it is viewed. On
the contrary, it is not impossible for a dark marking to appear smaller
than the Airy disk. Consider a long thin marking on the surface of a
planet. As long as the thickness of the marking is considerably greater
than the resolving power of the instrument it can be seen clearly. Now
let the marking become narrower until the outer rings of the diffraction
patterns from the bright points on either side Considerations similar to those above indicate that a dark round object can be seen against a bright background if its diameter is greater than one-third Dawes’ limit.
The
angular sizes of the four large satellites of Jupiter are very near the
angular size of the diffraction patterns for a six-inch telescope. In a
telescope of that size the satellites appear to be twice as large as
they really are. Any detail present would be so much smaller than the
diffraction pattern, that it would be clearly impossible to detect it.
The details would be completely covered by the Airy disks from the
adjacent bright spots. It
would be virtually impossible to give exact limits of resolution for the
above types of observations. Whether
or not an object smaller than the limit of resolution of the telescope
can be seen depends to a large extent on the observer, but if the limits
set by theory are greatly exceeded, such observations must, at best, be
suspect.
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