by Chris Lord
Astronomers refer to "seeing" either rather loosely or if you are involved in correcting its effects somewhat more rigorously.
Seeing as amateurs using moderate apertures understand it may be broken down into two basic causes, the "syrup" analogy, in which density and hence refractive index, is changed by temperature fluctuations in the air-mass, and sonic waves that produce higher frequency density fluctuations. Both combine, largely due to the relatively sluggish movement of large tropospheric air-masses, to produce image motion, or "oscillation", and from gas motions at high altitude, fluctuation of brightness caused by phase retardation, known as "scintillation".
Oscillation causes a change in the position of an image in the focal plane. The defect does not necessarily effect visual resolution, although it is obviously a nuisance from the photographer's viewpoint.
Oscillation may be considered as being caused by the passage of various lens shaped or prism shaped modules of atmosphere, across the objective. When the air module is larger than the objective, little image movement will arise, and a good image may be seen whenever the objective is covered by a single homogeneous module. (so called "slow seeing")
Irregularities in the density of gases at high altitude (above the Tropopause) produce a complex pattern of lights and darks, known as shadow bands. These irregularities diffract the light and cause rarification and reinforcement of wavefronts at various points along the ground. (so called "fast seeing")
Visual observers especially know that the theoretical diffraction pattern of a star can only be seen with telescopes of small aperture in good "seeing." With instruments of moderate aperture (~1m plus) such theoretical resolution is rarely, if ever achieved. Using a 1m aperture example, the theoretical resolution limit would be less than 0".15arc. In practice, however, the starlight is usually spread over a "seeing disk" from 2"arc to 5"arc diameter. The "seeing disk" is simply the circle of confusion for the rays reaching focus (physical optics disregarded for the moment), and its diameter is a measure of the lack of parallelism of the rays when they arrive at the objective from the star. The "seeing disk" can be assumed to be the summation of the diffraction patterns formed by each element of the air column before the objective. These elementary diffraction patterns are in rapid oscillatory movement both along and perpendicularly to the optical axis of the telescope.
The crucial point is this: if the aperture of a large telescope is stopped down to the aperture which would yield the theoretical limit of resolution equal to the actual resolution of the system, the usual diffraction rings will become clearly visible and sharply defined. The amplitude of the brightness scintillation will be noted to have increased roughly inversely to the ratio of the unstopped and stopped diameters. This leads to the conclusion that the "seeing disk" is a phenomenon largely independent of brightness scintillation.
The angular radius of the diffraction disk is 0.25*lambda (microns) / D (m), so if the radius of the seeing disk is 1"arc and the wavelength, lambda = 0.5 microns, the seeing disk will equal the AIry disk in a 120-mm telescope.
The effect of the oscillatory part of seeing is a function of aperture (not focal ratio). In good seeing, with an aperture less than 100-mm, the Airy disk moves randomly about a mean position in the focal plane with excursions typically 1"arc - 2"arc. (Variations in the wavefront curvature may be analysed using a technique devised by Roddier - see the Roddier Yahoo group for details).
Professional astronomers use a statistical parameter to characterize the seeing known as the "Fried parameter" or "Fried length". Simply, the Fried length r0 is the diameter of the bundle of rays from an source at infinity which travel through the various turbulent atmospheric layers and arrive, still parallel and in phase, at the entrance pupil of the telescope.
A telescope of aperture r0 would primarily suffer from image motion (as the tilt of the ray bundle changes), but not much from image blur. For diffraction limited performance of a quasi-perfect telescope, r0 must be slightly larger than the telescope aperture, typically about 1.6 times. At a wavelength of 0.5 microns (500nm), in the best sites, r0 varies from 100-mm to 300-mm, with seeing oscillations varying from 1"arc to 0".35arc.
CRITICALITY of FOCUSING:
Criticality of focusing is related to the Strehl ratio. The Strehl Intensity Ratio for an incoherent ray bundle, and a circular unvignetted pupil, are given a series of polynomial (Zernike) coefficients. When the Strehl ratio = 0.8, the zero order (defocus) coefficient is 0.25 lambda, and likewise the first & second order (tilt). The third order (spherical aberration) = 0.24 lambda, and the sum of the two gives ±0.95 lambda. (This condition is equivalent to the Rayleigh limit)
As the Strehl ratio of the system approaches 1, the criticality of focus as measured by coefficients order zero through three, decreases.
DEPTH of FOCUS:
Depth of focus is a concept usually associated with photography, rather than visual observing, although there is an equivalence since the eye acts as a re-imaging system.
The depth of focus is the tolerance of the axial position of a detector relative to the best optical focus. (And I have just mentioned that the best optical focus is related to the system, Strehl Intensity Ratio)
In the case of Rayleigh limited optics, the light beam near the focus has a tunnel shape due to diffraction effects (e.g. as shown in Suiter's "Star Testing Astronomical Telescopes.") Using Couder's figures for the Rayleigh 1/4 wave rule, the wavefront error at the distance Delta z from the geometrical focus is given, to the first order, by Delta z / 8 * f/#^2, and the depth of focus is defined as ±2* lambda*f/#^2.
The formula only applies to Rayleigh limited optics and the purpose of my article was to show how depth of focus decreased as the wavefront error of the optical system decreased.
Seeing comprises two components, oscillation (image shift) which is a result of a tilt to the wavefront, and whose integrated effects produce the seeing disk itself. and scintillation which produces brightness fluctuations in the seeing disk.
Seeing (in particular oscillation) is a function of aperture alone. A tilt (or defocus) of ±n wave before the OG remains ±n wave at the focal plane regardless of the OG's focal length (in the absence of OG wavefront errors).
Wavefront distortions occur when a ray bundle is variously refracted so parts of it are no longer parallel. The likelihood of this occurring is dependent on the seeing cell sizes and telescope aperture. Apertures smaller than 100-mm are by and large immune to the effects of seeing and nearly always show stars as Airy discs. For a telescope to show a clean Airy disc the seeing cell size (Fried length) needs to be about 1.6 times bigger than the telescope aperture.
Criticality of focus is related to the Strehl Intensity ratio of the telescope. It approaches zero as the Strehl ratio approaches 1.
Depth of focus as applicable to any diffraction limited telescope, is a function of focal ratio, not aperture, and is calculated from an acceptable "circle of confusion", rather than the size of the Airy disk. The actual size of the circle of confusion is determined by the integrated effects of scintillation, although in photography is it usually a fixed size related to the granularity of the film or the pixel size.
I hope my exposition helps clarify the various aspects of seeing and depth of focus.
STAR COLOURS at focus and either side of focus:
The other aspects of this discussion seem to me to have conflated three quite separate phenomena,
atmospheric dispersion, scintillation and chromatic aberration.
Dispersion, also known as differential refraction, is the result of the atmosphere acting as a base down prism. Red light is refracted through a slightly smaller angle than blue light. It may be calculated for mean temperatures and pressures for any particular zenith distance. It may be corrected using a Risley wedge.
Atmospheric refraction is given to the first order of approximation by R = k tan z, where z is the zenith distance. (the formula applies between z = 0¼ to 70¼ at MSL/STP in the Sodium D line). The value I adopt is k=60".2arc, but it varies - see http://www.asterism.org/tutorials/tut28-1.htm
Dispersion at MSL/STP between the C & F lines for various zenith distances is as follows:
Z RD RF - RC
25¼ 0' 27" 0".3
45¼ 0' 57" 0".6
65¼ 2' 02" 1".2
72¼ 2' 54" 1".8
76¼ 3' 45" 2".4
79¼ 4' 46" 3".0
A white object has its colours separated out into a short spectrum. Its apparent zenith distance decreases, and because red light is refracted less than blue light, the red image will have a bigger zenith distance than the blue image. In other words, relative to the horizon, the blue image will be above the red image. Hence if you observe Venus in a dark sky when its zenith distance is about 70¼, using an inverting telescope, it will be topped by a red and bottomed by a blue fringe. Similarly bright Lunar craters near the terminator will be fringed top & bottom with red and blue. (The actual "top" to the telescope field will be altered by use of a diagonal).
The extent to atmospheric fringing has nothing to do with the chromatic correction of the objective. It is not effected by the chromatic errors either. The angular separation of the blue and red components of the image will remain constant (for any given zenith distance) regardless whether the image is focussed, or you observe slightly inside or outside focus.
The visibility of dispersion is dependent on the zenith distance and telescope resolution. If your telescope resolves to 1"arc (Rayleigh Limit) then dispersion less than 0".5"arc will be very difficult to see, even in a brilliant object like Venus. If you observe within 25D & 50D colour fringing due to dispersion will be seen in a mid-aperture apo on bright objects at zenith distances exceeding 45¼.
Scintillation produces a rapid variation in the apparent brightness of an object. A bright star like Sirius will have its component colours separated into a short spectrum due to differential refraction. In perfect seeing, and at a high zenith distance (greater than 70¼), the Airy disc would theoretically be smeared into a short blur, red at one end and blue at the other. In practice, since perfect seeing rarely happens, you see a dithering diffraction pattern tinged with spectral colours, and the Airy disc centre also appears to randomly change colour from bluish through yellowish to reddish.
Yet again, these colours have nothing to do with the chromatic correction of the objective.
The visual appearance of a white star in an achromatic and an apochromatic refractor:
No refractor has perfect colour correction. (I think we have adequately dealt with that topic already ref forum article #23.
A standard achromat, by which I mean a crown-flint doublet corrected at the C & F lines, will have the following distribution of secondary colour: at minimum focus (i.e. the focus, visually) a white star will show a yellowish diffraction pattern with little surrounding colour; the image of a very bright white star (Vega) will be surrounded by a halo of out-of-focus secondary colour, bluish-violet or possibly with a tinge of purple (depending on your relative violet and deep red sensitivity); inside focus a red fringe will develop as the rings proliferate; just outside focus a minute red point will develop at the centre of the diffraction pattern; further out this vanishes again, and the whole diffraction pattern will be suffused with blue 'flare'.
If the OG is under-corrected, the image at focus will be surrounded by a red fringe which becomes increasingly conspicuous as the eyepiece is racked in; if it is over-corrected, no red fringe will develop around the intrafocal image.
NB An ED doublet will exhibit the same colour distribution, but to a noticeably lesser extent.
A contact triplet (be it air-spaced, cemented or oiled) corrected between the r & g lines, will have the following distribution of secondary colour (or should that be tertiary?): at minimum focus (i.e. the focus, visually) a white star will show an almost pure white diffraction pattern with little surrounding colour; the image of a bright white star (Vega) will be surrounded by a faint halo of out-of-focus secondary colour, violet, or possibly with a tinge of purple; inside focus a bluish point fringed by yellow-green will develop as the rings expand; just outside focus a yellow-orange point will develop at the centre of the diffraction pattern, further out this vanishes and the whole diffraction pattern will be suffused with a barely visible deep violet 'flare'.
I have never observed through a noticeably under-corrected or over-corrected triplet apochromat, but I would have thought the effects on colour distribution to be similar to an achromat, but less because of the smaller colour focus shift.
The curves of an object glass are calculated to give optimum colour correction with a particular magnification, usually in the neighbourhood of 60D to 70D with instruments of moderate aperture (where D is aperture in inches). Since the natural under-correction of the eyepiece and the eye is exaggerated when the diameter of the pencil with which they are dealing is increased, it follows that the degree of over-correction in the objective necessary to counteract this under-correction at about 60D will be insufficient to do so at lower magnifications. With low magnifications, therefore, an objective will appear to be under-corrected, and (though this is much less noticeable or significant) with very high magnifications, over-corrected.