Astronomical seeing
Astronomical seeing refers to the amount of apparent blurring and twinkling of astronomical objects like stars due to turbulent mixing in the atmosphere of Earth, causing variations of the optical refractive index. The seeing conditions on a given night at a given location describe how much Earth's atmosphere perturbs the images of stars as seen through a telescope.
The most common seeing measurement is the full width at half maximum (FWHM) of the optical intensity across the seeing disc (the point spread function for imaging through the atmosphere). The FWHM of the point spread function (loosely called seeing disc diameter or "seeing") is the best possible angular resolution that can be achieved by an optical telescope in a longexposure image, and corresponds to the FWHM of the fuzzy blob seen when observing a pointlike source (such as a star) through the atmosphere. The size of the seeing disc is determined by the seeing conditions at the time of the observation. The best conditions give a seeing disk diameter of ~0.4 arcseconds and are found at highaltitude observatories on small islands such as Mauna Kea or La Palma.
Seeing is one of the biggest problems for Earthbased astronomy. While large telescopes have theoretically milliarcsecond resolution, the real image is limited to the average seeing disc during the observation. This can easily mean a factor of 100 between the potential and practical resolution. Starting in the 1990s, new adaptive optics have been introduced that can help correct for these effects, dramatically improving the resolution of groundbased telescopes.
Effects
Typical shortexposure negative image of a binary star (Zeta Boötis in this case) as seen through atmospheric seeing. Each star should appear as a single Airy pattern, but the atmosphere causes the images of the two stars to break up into two patterns of speckles (one pattern above left, the other below right). The speckles are a little difficult to make out in this image due to the coarse pixel size on the camera used (see the simulated images below for a clearer example). The speckles move around rapidly, so that each star appears as a single fuzzy blob in long exposure images (called a seeing disc). The telescope used had a diameter of about 7r_{0} (see definition of r_{0} below, and example simulated image through a 7r_{0} telescope). 
Astronomical seeing has several effects:
 It causes the images of point sources (such as stars), which in the absence of atmospheric turbulence would be steady Airy patterns produced by diffraction, to break up into speckle patterns, which change very rapidly with time (the resulting speckled images can be processed using speckle imaging)
 Long exposure images of these changing speckle patterns result in a blurred image of the point source, called a seeing disc
 The brightness of stars appears to fluctuate in a process known as scintillation or twinkling
 Atmospheric seeing causes the fringes in an astronomical interferometer to move rapidly
 The distribution of atmospheric seeing through the atmosphere (the C_{N}^{2} profile described below) causes the image quality in adaptive optics systems to degrade the further you look from the location of reference star
The effects of atmospheric seeing were indirectly responsible for the belief that there were canals on Mars. In viewing a bright object such as Mars, occasionally a still patch of air will come in front of the planet, resulting in a brief moment of clarity. Before the use of chargecoupled devices, there was no way of recording the image of the planet in the brief moment other than having the observer remember the image and draw it later. This had the effect of having the image of the planet be dependent on the observer's memory and preconceptions which led the belief that Mars had linear features.
The effects of atmospheric seeing are qualitatively similar throughout the visible and near infrared wavebands. At large telescopes the long exposure image resolution is generally slightly higher at longer wavelengths, and the timescale (t_{0}  see below) for the changes in the dancing speckle patterns is substantially lower.
Measures
There are three common descriptions of the astronomical seeing conditions at an observatory:
 The full width at half maximum (FWHM) of the seeing disc
 r_{0} (the size of a typical "lump" of uniform air within the turbulent atmosphere[1]) and t_{0} (the timescale over which the changes in the turbulence become significant)
 The C_{N}^{2} profile
These are described in the subsections below:
The full width at half maximum (FWHM) of the seeing disc
Without an atmosphere, a small star would have an apparent size, an "Airy disk", in a telescope image determined by diffraction and would be inversely proportional to the diameter of the telescope. However, when light enters the Earth's atmosphere, the different temperature layers and different wind speeds distort the light waves, leading to distortions in the image of a star. The effects of the atmosphere can be modeled as rotating cells of air moving turbulently. At most observatories, the turbulence is only significant on scales larger than r_{0} (see below—the seeing parameter r_{0} is 10–20 cm at visible wavelengths under the best conditions) and this limits the resolution of telescopes to be about the same as given by a spacebased 10–20 cm telescope.
The distortion changes at a high rate, typically more frequently than 100 times a second. In a typical astronomical image of a star with an exposure time of seconds or even minutes, the different distortions average out as a filled disc called the "seeing disc". The diameter of the seeing disk, most often defined as the full width at half maximum (FWHM), is a measure of the astronomical seeing conditions.
It follows from this definition that seeing is always a variable quantity, different from place to place, from night to night, and even variable on a scale of minutes. Astronomers often talk about "good" nights with a low average seeing disc diameter, and "bad" nights where the seeing diameter was so high that all observations were worthless.
The FWHM of the seeing disc (or just "seeing") is usually measured in arcseconds, abbreviated with the symbol (″). A 1.0″ seeing is a good one for average astronomical sites. The seeing of an urban environment is usually much worse. Good seeing nights tend to be clear, cold nights without wind gusts. Warm air rises (convection), degrading the seeing, as do wind and clouds. At the best highaltitude mountaintop observatories, the wind brings in stable air which has not previously been in contact with the ground, sometimes providing seeing as good as 0.4".
r_{0} and t_{0}
The astronomical seeing conditions at an observatory can be conveniently described by the parameters r_{0} and t_{0}.
For telescopes with diameters smaller than r_{0}, the resolution of longexposure images is determined primarily by diffraction and the size of the Airy pattern and thus is inversely proportional to the telescope diameter.
For telescopes with diameters larger than r_{0}, the image resolution is determined primarily by the atmosphere and is independent of telescope diameter, remaining constant at the value given by a telescope of diameter equal to r_{0}. r_{0} also corresponds to the lengthscale over which the turbulence becomes significant (10–20 cm at visible wavelengths at good observatories), and t_{0} corresponds to the timescale over which the changes in the turbulence become significant. r_{0} determines the spacing of the actuators needed in an adaptive optics system, and t_{0} determines the correction speed required to compensate for the effects of the atmosphere.
The parameters r_{0} and t_{0} vary with the wavelength used for the astronomical imaging, allowing slightly higher resolution imaging at longer wavelengths using large telescopes.
The seeing parameter r_{0} is often known as the Fried parameter (pronounced "freed"), named after David L. Fried. The atmospheric time constant t_{0} is often referred to as the Greenwood time constant, after Darryl Greenwood.
Mathematical description of r_{0} and t_{0}
Mathematical models can give an accurate model of the effects of astronomical seeing on images taken through groundbased telescopes. Three simulated shortexposure images are shown at the right through three different telescope diameters (as negative images to highlight the fainter features more clearly—a common astronomical convention). The telescope diameters are quoted in terms of the Fried parameter (defined below). is a commonly used measurement of the astronomical seeing at observatories. At visible wavelengths, varies from 20 cm at the best locations to 5 cm at typical sealevel sites.
In reality, the pattern of blobs (speckles) in the images changes very rapidly, so that longexposure photographs would just show a single large blurred blob in the center for each telescope diameter. The diameter (FWHM) of the large blurred blob in longexposure images is called the seeing disc diameter, and is independent of the telescope diameter used (as long as adaptive optics correction is not applied).
It is first useful to give a brief overview of the basic theory of optical propagation through the atmosphere. In the standard classical theory, light is treated as an oscillation in a field . For monochromatic plane waves arriving from a distant point source with wavevector : where is the complex field at position and time , with real and imaginary parts corresponding to the electric and magnetic field components, represents a phase offset, is the frequency of the light determined by , and is the amplitude of the light.
The photon flux in this case is proportional to the square of the amplitude , and the optical phase corresponds to the complex argument of . As wavefronts pass through the Earth's atmosphere they may be perturbed by refractive index variations in the atmosphere. The diagram at the topright of this page shows schematically a turbulent layer in the Earth's atmosphere perturbing planar wavefronts before they enter a telescope. The perturbed wavefront may be related at any given instant to the original planar wavefront in the following way:
where represents the fractional change in wavefront amplitude and is the change in wavefront phase introduced by the atmosphere. It is important to emphasise that and describe the effect of the Earth's atmosphere, and the timescales for any changes in these functions will be set by the speed of refractive index fluctuations in the atmosphere.
The Kolmogorov model of turbulence
A description of the nature of the wavefront perturbations introduced by the atmosphere is provided by the Kolmogorov model developed by Tatarski,[2] based partly on the studies of turbulence by the Russian mathematician Andreï Kolmogorov.[3][4] This model is supported by a variety of experimental measurements[5] and is widely used in simulations of astronomical imaging. The model assumes that the wavefront perturbations are brought about by variations in the refractive index of the atmosphere. These refractive index variations lead directly to phase fluctuations described by , but any amplitude fluctuations are only brought about as a secondorder effect while the perturbed wavefronts propagate from the perturbing atmospheric layer to the telescope. For all reasonable models of the Earth's atmosphere at optical and infrared wavelengths the instantaneous imaging performance is dominated by the phase fluctuations . The amplitude fluctuations described by have negligible effect on the structure of the images seen in the focus of a large telescope.
For simplicity, the phase fluctuations in Tatarski's model are often assumed to have a Gaussian random distribution with the following secondorder structure function:
where is the atmospherically induced variance between the phase at two parts of the wavefront separated by a distance in the aperture plane, and represents the ensemble average.
For the Gaussian random approximation, the structure function of Tatarski (1961) can be described in terms of a single parameter :
indicates the strength of the phase fluctuations as it corresponds to the diameter of a circular telescope aperture at which atmospheric phase perturbations begin to seriously limit the image resolution. Typical values for I band (900 nm wavelength) observations at good sites are 20–40 cm. also corresponds to the aperture diameter for which the variance of the wavefront phase averaged over the aperture comes approximately to unity:[6]
This equation represents a commonly used definition for , a parameter frequently used to describe the atmospheric conditions at astronomical observatories.
can be determined from a measured C_{N}^{2} profile (described below) as follows:
where the turbulence strength varies as a function of height above the telescope, and is the angular distance of the astronomical source from the zenith (from directly overhead).
If turbulent evolution is assumed to occur on slow timescales, then the timescale t_{0} is simply proportional to r_{0} divided by the mean wind speed.
The refractive index fluctuations caused by Gaussian random turbulence can be simulated using the following algorithm:[7]
where is the optical phase error introduced by atmospheric turbulence, R (k) is a twodimensional square array of independent random complex numbers which have a Gaussian distribution about zero and white noise spectrum, K (k) is the (real) Fourier amplitude expected from the Kolmogorov (or Von Karman) spectrum, Re[] represents taking the real part, and FT[] represents a discrete Fourier transform of the resulting twodimensional square array (typically an FFT).
Turbulent intermittency
The assumption that the phase fluctuations in Tatarski's model have a Gaussian random distribution is usually unrealistic. In reality, turbulence exhibits intermittency.[8]
These fluctuations in the turbulence strength can be straightforwardly simulated as follows:[9]
where I (k) is a twodimensional array which represents the spectrum of intermittency, with the same dimensions as R (k), and where represents convolution. The intermittency is described in terms of fluctuations in the turbulence strength . It can be seen that the equation for the Gaussian random case above is just the special case from this equation with:
where is the Dirac delta function.
The profile
A more thorough description of the astronomical seeing at an observatory is given by producing a profile of the turbulence strength as a function of altitude, called a profile. profiles are generally performed when deciding on the type of adaptive optics system which will be needed at a particular telescope, or in deciding whether or not a particular location would be a good site for setting up a new astronomical observatory. Typically, several methods are used simultaneously for measuring the profile and then compared. Some of the most common methods include:
 SCIDAR (imaging the shadow patterns in the scintillation of starlight)
 LOLAS (a smallaperture variant of SCIDAR designed for lowaltitude profiling)
 SLODAR
 MASS
 MooSci (11channel lunar scintillometer for ground level profiling)[10]
 RADAR mapping of turbulence
 Balloonborne thermometers to measure how quickly the air temperature is fluctuating with time due to turbulence
 V2 Precision Data Collection Hub (PDCH) with differential temperature sensors use to measure atmospheric turbulence
There are also mathematical functions describing the profile. Some are empirical fits from measured data and others attempt to incorporate elements of theory. One common model for continental land masses is known as HufnagelValley after two workers in this subject.
Overcoming atmospheric seeing
The first answer to this problem was speckle imaging, which allowed bright objects with simple morphology to be observed with diffractionlimited angular resolution. Later came NASA's Hubble Space Telescope, working outside the atmosphere and thus not having any seeing problems and allowing observations of faint targets for the first time (although with poorer resolution than speckle observations of bright sources from groundbased telescopes because of Hubble's smaller telescope diameter). The highest resolution visible and infrared images currently come from imaging optical interferometers such as the Navy Prototype Optical Interferometer or Cambridge Optical Aperture Synthesis Telescope, but those can only be used on very bright stars.
Starting in the 1990s, many telescopes have developed adaptive optics systems that partially solve the seeing problem. The best systems so far built, such as SPHERE on the ESO VLT and GPI on the Gemini telescope, achieve a Strehl ratio of 90% at a wavelength of 2.2 micrometers, but only within a very small region of the sky at a time.
A wider field of view can be obtained by using multiple deformable mirrors conjugated to several atmospheric heights and measuring the vertical structure of the turbulence, in a technique known as Multiconjugate Adaptive Optics.
Another cheaper technique, lucky imaging, has had good results on smaller telescopes. This idea dates back to prewar nakedeye observations of moments of good seeing, which were followed by observations of the planets on cine film after World War II. The technique relies on the fact that every so often the effects of the atmosphere will be negligible, and hence by recording large numbers of images in realtime, a 'lucky' excellent image can be picked out. This happens more often when the number of r0size patches over the telescope pupil is not too large, and the technique consequently breaks down for very large telescopes. It can nonetheless outperform adaptive optics in some cases and is accessible to amateurs. It does require very much longer observation times than adaptive optics for imaging faint targets, and is limited in its maximum resolution.
See also
 Atmosphere and Telescope Simulator, a simulator of atmospheric turbulence
 Clear Sky Chart, web charts that include weather forecasts for astronomical seeing
 Mirage
 Planetary boundary layer
 Transient lunar phenomenon
References
Much of the above text is taken (with permission) from Lucky Exposures: Diffraction limited astronomical imaging through the atmosphere, by Robert Nigel Tubbs
 Chromey, Frederick R. (2010). To measure the sky : an introduction to observational astronomy (1. publ. ed.). Cambridge: Cambridge University Press. p. 140. ISBN 9780521763868.
 Tatarskiĭ, V. I. (1961). R.A. Silverman (ed.). Wave Propagation in a Turbulent Medium. University of Michigan: McGrawHill Books. p. 285. Bibcode:1961wptm.book.....T.
 Kolmogorov, A. N. (1941). "Dissipation of energy in the locally isotropic turbulence". Comptes Rendus de l'Académie des Sciences de l'URSS. 32: 16–18. Bibcode:1941DoSSR..32...16K. JSTOR 51981.
 Kolmogorov, A. N. (1941). "The local structure of turbulence in incompressible viscous fluid for very large Reynold's numbers". Comptes Rendus de l'Académie des Sciences de l'URSS. 30: 301–305. Bibcode:1941DoSSR..30..301K. JSTOR 51980.

BUSCHER, D. F.; ARMSTRONG, J. T.; HUMMEL, C. A.; QUIRRENBACH, A.; MOZURKEWICH, D.; JOHNSTON, K. J.; DENISON, C. S.; COLAVITA, M. M.; SHAO, M. (February 1995). "Interferometric seeing measurements on Mt. Wilson: power spectra and outer scales". Applied Optics. 34 (6): 1081–1096. Bibcode:1995ApOpt..34.1081B. doi:10.1364/AO.34.001081. PMID 21037637.
 NIGHTINGALE, N. S.; BUSCHER, D. F. (July 1991). "Interferometric seeing measurements at the La Palma Observatory". Monthly Notices of the Royal Astronomical Society. 251: 155–166. Bibcode:1991MNRAS.251..155N. doi:10.1093/mnras/251.1.155.
 O'BYRNE, J. W. (Sep 1988). "Seeing measurements using a shearing interferometer". Publications of the Astronomical Society of the Pacific. 100: 1169–1177. Bibcode:1988PASP..100.1169O. doi:10.1086/132285.
 COLAVITA, M. M.; SHAO, M.; STAELIN, D. H. (October 1987). "Atmospheric phase measurements with the Mark III stellar interferometer". Applied Optics. 26 (19): 4106–4112. Bibcode:1987ApOpt..26.4106C. doi:10.1364/AO.26.004106. PMID 20490196.

 FRIED, D. L. (1965). "Statistics of a Geometric Representation of Wavefront Distortion". Journal of the Optical Society of America. 55 (11): 1427–1435. Bibcode:1965JOSA...55.1427F. doi:10.1364/JOSA.55.001427.*NOLL, R. J. (March 1976). "Zernike polynomials and atmospheric turbulence". Journal of the Optical Society of America. 66 (3): 207–211. Bibcode:1976JOSA...66..207N. doi:10.1364/JOSA.66.000207.
 The effect of temporal fluctuations in r0 on highresolution observations , Robert N. Tubbs Proc SPIE 6272 pp 93T, 2006

 BATCHELOR, G. K., & TOWNSEND, A. A. 1949 (May).
 Baldwin, J. E.; Warner, P. J.; Mackay, C. D., The point spread function in Lucky Imaging and variations in seeing on short timescales, Astronomy and Astrophysics V. 480 pp 589B.
 The effect of temporal fluctuations in r0 on highresolution observations, Robert N. Tubbs Proc SPIE 6272 pp 93T, 2006
 Villanueva, Steven Jr.; Depoy, D. L.; Marshall, J.; Berdja, A.; Rheault, J. P.; Prieto, G.; Allen, R.; Carona, D. (July 2010). "MooSci: a lunar scintillometer" (PDF). Groundbased and Airborne Instrumentation for Astronomy III. Edited by McLean, Ian S.; Ramsay, Suzanne K.; Takami, Hideki. Proceedings of the SPIE. Groundbased and Airborne Instrumentation for Astronomy III. 7735: 773547. doi:10.1117/12.857413.
 "A Mix of Colours and Wonder". Retrieved 15 June 2015.
External links
 Free 72hour seeing prediction for every location on earth (Click on 'Charts&Tools' and then 'Astronomy Seeing')
 The Royal Astronomical Society of Canada Calgary Centre  Atmospheric "Seeing". Includes animated illustrations of effects of seeing.
 Seeing forecasts for North America
 Seeing forecasts for Mauna Kea, Hawaii