Astronomical Seeing

Part 1: The Nature of Turbulence

Seeing is the astronomer's term for the relative optical quality of the Earth's atmosphere. Optical quality is defined as the steadiness and absence of distortion in a telescopic image across an interval of observation. A motionless and optically perfect image indicates excellent seeing; a rapidly changing and grossly distorted image indicates poor seeing.

The cause of degraded or poor seeing is thermal turbulence in the atmosphere. Seeing has nothing to do with whether the night air is cloudy or clear, warm or cool, or even whether it is windy or calm. The critical issue is only whether temperature differences in the atmosphere are in motion.

The effect of mixing air of different temperatures can be seen in the appearance of objects behind convection currents, such as the air rising from an asphalt road on a hot day. Warm air rising through cooler air produces a characteristic wavering or undulation in the appearance of objects behind the thermal currents, similar to the appearance of objects below the surface of rippling water. In the atmosphere, as in air over a hot asphalt road, thermal turbulence is the cause of poor seeing.

Before delving that topic, however, it is important to note that atmospheric aerosols (water vapor, dust, volcanic ash, coal and oil combustion byproducts) can significantly degrade astronomical images. Aerosols create a diffuse directional glow visible from the Moon, bright planet or bright star when the object is completely outside the field of view in binoculars or a telescope. Significant diffusion can be present even when the sky appears dark and faint stars are easily visible.

Thermal turbulence causes image perturbations on the order of 10^–5 to 10^–4 radians (2 to 20 arcseconds); the radius of the forward scatter caused by diffusion can be extremely large, on the order of a radian (~60°, image, right of water vapor diffusion).

To assess diffusion, cover the disk of the Sun or Moon with your thumb at arm's length. The amount of stray light visible around the obstruction is an indication of the amount of diffusion. Viewing a bright star through a telescope will also show a nimbus of glare if diffusion is present. (Be sure that the objective and eyepiece are not fogged or dewed.)

When the object is centered in the field of view, aerosols can significantly contribute to image blurring and contrast reduction, even when thermal turbulence is negligible. And aerosols can be the primary source of image degradation even when significant thermal turbulence is present.

A Topic of Recent Interest

Scientists have been aware of optical turbulence since English naturalist Robert Hooke in 1665 attributed the twinkling of stars to "small, moving regions of the atmosphere having different refracting powers which act like lenses." Astronomer William Herschel was aware of optical turbulence and explicitly adopted measures to cope with it, and observational analyses of the problem appear in the late 19th century. But the scientific study of astronomical seeing really takes off in the 1950's, when photoelectric photometry, photomultipliers, oscilliscopes and sensitive photographic films made detailed measurement of turbulence possible. Academic papers do not cite research on the topic much before 1950 and review articles, summarizing previous studies of astronomical seeing, do not appear until around 1960.

Despite the fact that Harvard Observatory astronomers around 1900 had identified atmospheric turbulence as a "factor of prime importance" in planetary astronomy, amateur astronomical sources printed before 1950 either treat the problem not at all or only in passing — as the difficulty of viewing under a "tremulous atmosphere" (Webb's Celestial Objects, 1917). Among the first science based descriptions of optical turbulence available to amateur astronomers were two articles on "Seeing" in Sky & Telescope (January/February, 1950; see Further Reading).

This relatively recent focus on atmospheric turbulence can be traced in the evolving treatment of the topic found in Norton's Star Atlas and Telescopic Handbook, then and now. Here is the sole mention of the issue under the heading "Atmospheric Conditions" in the first edition (1910, p.17):

When the stars twinkle much it is an indication that the air is unsteady and not altogether satisfactory for observation.

And here the entire discussion under the heading "Twinkling of Stars" given in the fourteenth edition (1959, p.38):

Though purely atmospheric in its origin, this phenomenon is of interest to astronomers, as it is affected by the nature of the light emitted by each star, e.g., by its spectrum. White stars (Types B and A) twinkle most; yellow stars (Types F to K) slightly less, and red stars (Type M) least of all. Twinkling is least at the zenith, and in settled and calm weather; greatest toward the horizon, and in unsettled and stormy weather; there is also a seasonal waxing and waning from mid summer to mid winter and vice versa. Planets do not usually twinkle except when near the horizon — supposed to be due to the fact that they have discs of an appreciable size.

Finally, here is just the opening paragraph from the extended discussion of "Seeing" in Norton's nineteenth edition (1998, p. 29):

Seeing is a term used to indicate the steadiness of the air, as judged by the appearance of the telescope image. The two are connected by the fact that air currents are caused by masses of air at different temperatures, and the refractive index of air changes with temperature: therefore the currents cause the image to flicker.

Fortunately a significant body of research has accumulated since 1970, primarily motivated by the need to improve the yield from optical surveillance and mapping satellites and to analyze turbulence at candidate sites for the modern generation of 10 meter and larger terrestrial telescopes. This page summarizes some of the key principles.

The Structure of Turbulence

All substances that transmit light also refract or bend the direction of the light by an amount proportional to the refractive index of the medium. The refractive index of air changes with its density, which near the surface of the earth depends primarily on temperature and to a lesser extent on humidity: warmer air, and more humid air, is less dense and therefore refracts light less than cooler, drier air.

Two bodies of air of different temperatures and/or humidities create a refractive boundary that bends light in the same way as the boundary between air and water or air and glass. If this boundary is distorted into turbulence, it has a similar (though weaker) optical effect as the surface of water disturbed into ripples by the wind, or glass with a randomly irregular surface (image, left).

Astronomers describe this turbulence statistically, using an optical analysis developed by V.I. Tatarski from the mathematical description of turbulence cascades by Andrei Kolmogorov. This Kolmogorov-Tatarski model begins with the fact that turbulence in flowing media such as air or water depends on three factors: (1) the velocity of the flowing medium; (2) the boundary width or spatial dimensions of the flow; and (3) the kinematic viscosity of the medium.

Because the medium is in motion, it creates friction against the boundaries around it. At low velocity and/or high viscosity, this friction only impedes the outer layer of the flow: the inner flow simply slides over this laggard outer layer, creating layered or laminar flow. This distributes friction farther into the flow, layer by layer, the way playing cards slide over one another when the deck is spread out on a table.

The velocity limit at which laminar flow can no longer dissipate friction is evaluated as a Reynolds number, calculated as the average viscosity and physical dimension of the flow as a proportion of the flow velocity. Air has a very low kinematic viscosity of around 0.15 cm² per second, so even when the physical scale of the flow is as large as several hundred meters, turbulence appears at velocities of only a few kilometers per hour. Layers of moving air are therefore almost always turbulent.

Turbulence develops as increasing thermal energy — heat from the sun or heat rising from the earth — breaks laminar flows into very large cells that roll over themselves as whorls or eddies. Because these whorls are relatively inefficient at dissipating energy, the increased flow velocity breaks them into smaller and more efficient whorls, and so on until the flow viscosity impedes smaller divisions. At that point, the flow energy can only be dissipated as heat from viscous friction. This turbulence cascade creates a turbulence frequency spectrum from the largest, highest energy vortices to the smallest, low energy eddies or whorls, which randomly emerge and mix within the flow. Louis Fry Richardson wittily summarized the turbulence cascade in a couplet:

Big whorls have little whorls that feed on their velocity,
And little whorls have lesser whorls and so on to viscosity.

The complex texture of single turbulent boundaries is exquisitely visible in the light scattering contours of cumulus clouds — which form as convection currents of warm, moist air surge into drier, cooler air above — and in computer simulations of turbulent media. The images (below) illustrate phenomena described above: turbulence produced by convection currents, turbulence resulting from boundary friction within a single moving layer, and the complex result of these factors in atmospheric turbulence.

computer simulation of thermal turbulence produced by combustion at (left to right) low to high temperatures

computer simulation of boundary turbulence within a single flowing layer, viewed from the side (left) and from above (right)

comupter simulation of geostrophic (atmospheric) turbulence in air layers of two different temperatures

There are two boundaries to the turbulence frequency spectrum. The largest dimension or outer scale of the turbulence (L_o) usually represents the thickness of the entire flowing medium, which in the atmosphere can be a layer of air 100 or more meters thick. The smallest eddies define the inner scale of turbulence (l_o), which has been estimated to be as small as a few millimeters. Between these limits the turbulence forms a distribution of whorls where the number of small whorls increases exponentially.

What is the effect of this turbulence on the light from a star? The diagram (right), adapted from the Lucky Imaging Web Site, shows that these tumbling eddies disrupt and refract the light in a complex but self similar or fractal pattern: from the largest to the smallest scale, the light fluctuates by random amounts across random intervals of time.

Across time (temporal frequency), the largest variations in the turbulence (L₀) can extend across intervals of 20 seconds or more, while the troughs between those peaks are churned by successively smaller and more rapid fluctuations down to the minimum time interval (l₀) shown by a single vertical line, which represents fluctuations that occur several hundred times a second. Within the image, the size or amplitude of the optical distortions varies from largest to smallest in the same way.

In addition, atmospheric turbulence often shows intermittency or gusts of higher turbulence separated by intervals of less turbulence.

Temperature differences as small as 0.1 to 1 K can produce noticeable optical effects, but only in air masses warmer than about 10°F (–12°C). In addition to providing the energy that creates the turbulence, wind shear and convection currents also move the turbulence across the landscape and telescope line of sight, sometimes at high speed.

Atmospheric turbulence is the random combination of two separate types of variation: amplitude, or the amount of change in refracting effect (produced by the width of the eddies and the difference in temperature between them), and frequency, or the time interval between amplitude changes of the same size (produced by the movement of eddies across the optical path). Both the mathematical models and visual inspection of star images show that the refracting effect (red brackets) and temporal spacing (blue brackets) of atmospheric turbulence fluctuate randomly across scales exceeding 100,000 to 1.

The Location of Turbulence

The Komolgorov-Tatarski model represents turbulence at a single boundary between thermally different layers of air. But turbulence actually arises in several different locations, across several different atmospheric layers.

The simplest meteorological model of atmospheric turbulence was proposed by Hufnagel (1974) and revised by Valley (1979), and (with allowances for local geography and climate) this model has been generally supported in subsequent research — with the caution that measurements at specific sites around the year can depart from it widely, and turbulence will be concentrated in atmospheric layers at almost any altitude up to the tropopause.

The Hufnagel Valley model locates most atmospheric turbulence in two regimes (chart, left): turbulence within the surface boundary layer, which occurs in dense, relatively low velocity (up to 50 kilometers per hour) convection and layered air currents moving within a kilometer or two of the earth's surface, and high altitude turbulence around the tropopause, which occurs in relatively rarefied, high velocity (up to 500 kilometers per hour) air currents at the temperature inversion between the troposphere and stratosphere.

François Roddier (1981), adopting the discussion in Jean Texereau (1961; 1984), elaborated this model into four categories: "turbulence associated with the telescope and the dome, turbulence in the surface boundary layer or due to ground convection, turbulence in the planetary boundary layer or associated with orographic [mountain] disturbances, and turbulence in the tropopause or above" (1981, p. 288). Schematically I will refer to these as instrument, surface, geographic and high atmosphere turbulence.

1. Instrument turbulence occurs inside the telescope and any structure that shelters it. It is most often produced by convection layers rising at the surface of a reflecting mirror generated by heat inside the cooling glass (mirror seeing), by air currents crawling along the sides of a closed telescope tube (telescope structure seeing), by convection currents from the observer's body wafting across the optical path (especially in cold weather), by heat rising through the restrictive opening of an observatory dome (structure seeing), and by heat rising from pavement, masonry or metal immediately under and around the telescope (site seeing). Several studies suggest that mirror seeing degrades the image in a 25 cm telescope by about 0.1 arcsecond for every degree Centigrade that mirror temperature exceeds ambient temperature; the effect is less in larger apertures.

2. Surface turbulence extends from the ground up to a few hundred meters in the landscape around the telescope, which when viewing at a zenith angle above 60° is within half a kilometer of the observing site. Surface turbulence often represents up to half of all the observed optical distortion; it is largely due to convection currents rising from heat stored in the sunlit earth during the day. Particular concentrations of convection currents can arise from nearby residences, paved roads, surfaces of masonry or concrete, commercial buildings, and from turbulence between low lying layers that form temperature inversion boundaries. At many locations, surface turbulence follows a diurnal cycle from a minimum just after sunrise, steeply rising to a peak during early afternoon, declining to a secondary minimum shortly after sunset, increasing during the early evening to a secondary peak at around midnight, before returning to a minimum in the hour or two before morning.

3. Geographic turbulence extends from a few hundred meters to a few kilometers above the ground; for viewing at a zenith angle of 60° or higher this implies a geographic radius from the observing site of up to 7 km. Geographic turbulence typically forms as several overlying layers of air 100 to 200 meters thick that can extend horizontally for several kilometers; above 4 km it is generally independent of the landscape and becomes less significant up to a minimum at around 6 to 9 kilometers. It is caused not only by air currents forced upward by mountainous terrain but by the disposition of other large landscape features — large bodies of water, expanses of bare ground or sand, conurban development, large areas of snow — as these shape the thermal and moisture content of the weather bearing atmosphere.

4. High atmosphere turbulence is primarily associated with the jet stream, which is normally confined to latitudes above 30° north or south of the equator at altitudes of around 10 to 15 km. (At higher latitudes the jet stream altitude is much less, and near the poles it disappears near the surface.) Stratospheric layers above about 20 km are rarefied and thermally homogenous, and have a negligible effect on seeing. The jet stream contributes to turbulence both directly through its high velocity movement against lower atmospheric layers, and indirectly through the quantity of cold or moist air it brings from northern latitudes and ocean surfaces, the impact of its motion on the formation of high and low pressure areas, and the weather turbulence produced by the energetic mixture of moisture, temperature and barometric pressure. The jet stream is high enough so that, even if it is not directly overhead, it can cause significant differences in the amount of turbulence seen in opposite directions of the sky — at distances of up to 25 km from the observing site when viewing at a 60° zenith angle.

The relative scale and location of these four sources of turbulence is nicely summarized in predictive models of optical turbulence (using the turbulence structure index C_n², explained in the next section) developed by Trinquet & Vernin (2009; below).

Although this is the graphical representation of a forecasting model, not of actual measurements, it reproduces the pattern of turbulence as it has been measured at various sites around the world and as summarized in the previous graph: large, undulating turbulence (red) near the ground, and small, vibrating turbulence (cyan) in the high atmosphere. It also illustrates the remarkable variability in seeing over time, both across days and within a single evening. This reveals the chaotic quality of optical turbulence across larger physical and temporal scales. (Compare with the simulation of boundary turbulence, above.)

The Optics of Turbulence

To the naked eye, optical turbulence produces the twinkling of stars. In telescopes, turbulence produces a range of effects on the image of stars and planets that has been variously described as "wavering" and "wobbling" in small telescopes and "boiling" or "churning" in large telescopes. This is a clue that the optical effects of turbulence vary with the aperture of the observing instrument.

The simplest model of optical turbulence represents it as the boundary between two layers of air at different temperatures (as diagrammed by Dorrit Hoffleit, right). The refracting effect of a single small area of the boundary is equivalent to the refracting effect of an air/glass optical surface. The entire layer disrupts the light from a star into moving light and dark bands, similar to the caustics or network of light bands and shadow cells visible at the bottom of a rippling swimming pool on a sunny day. These shadows can be seen in a telescope superimposed on the image of a bright star brought far out of focus; the focused images of extended surfaces such as the Moon appear as if under moving water. (See for example this brief animation of turbulence imaged in a large telescope.)

As a simplification most useful to the visual astronomer, the optical effects of this turbulence layer can be contrasted as three types of distortion in the star diffraction artifact (diagram, right):

• Oscillating is a wavering or jumping of the star image around an average location within the image field, which is slow enough that the eye can perceive and follow a single coherent star image. The "undistorted" star image appears mostly intact, as a recognizable Airy disk and first diffraction ring, but it is in continual motion from place to place. Oscillations are caused by moderate energy (medium scale) turbulence where F = H and v is not fast, and is typical of turbulence within a kilometer or two of the ground; it is also characteristic of poor seeing in small aperture (below ~10–20 cm) telescopes. The angular displacement produced by oscillation is usually small, less than a few arcseconds. As a result, star images do not seem to oscillate when L > D — in small aperture telescopes and the naked eye; instead a shadow band momentarily fills the small aperture, which produces a brief dimming or twinkling in brightness by around 10%, known as scintillation. In apertures greater than or approaching the diameter of the turbulence cells (L = D) the tilting occurs entirely within the aperture diameter and the focused image appears to wobble or dance around a central point.

• Speckling is produced by high energy, high frequency turbulence (small angular size and very rapid fluctuation) where F < H and v is so rapid that motion blurs. It is usually located at high altitudes and becomes dominant when D >> L so that the aperture can sample the images from many turbulence cells simultaneously. This breaks the star image into multiple, simultaneous Airy disks superimposed on each other at random small distances from a fixed central location within the image field. The same wave interference that produces the dark rings in the undistorted star diffraction artifact creates dark boundaries between the superimposed Airy disks of the simultaneous star images, creating numerous visibly distinct beads of light, called speckles. Because these images of the star are produced simultaneously, the "dancing" locations of the star are combined as a single image — resulting in a bloated, boiling mass of speckles that remains fixed at a single location. At this scale the angular width of turbulence cells is so small that even closely spaced binary stars of equal magnitude will show different speckle patterns moment to moment, and matched magnitude double stars will merge into an unresolved oblong mass.

Temporal scale contributes to the scintillation appearance: image fluctuations or flickers that are faster than about 50 cycles per second are not visible to the eye, which instead perceives the flickers as a continuous light, but this threshold declines to a rate of a few flickers per second in very faint images. Consequently the visible "boiling" of speckles in poor seeing has a characteristic maximum perceptible rate at different visual magnitudes, and appears most vigorous and incoherent in bright stars viewed at high magnification with a large aperture.

Note that this temporal scale means that naked eye twinkling is diagnostic only of low frequency turbulence (usually, heat from the ground or rapidly changing weather) which may have little effect on the telescopic image; the amount of low frequency turbulence is also not indicative of the amount of high frequency turbulence. If most of the atmospheric turbulence is high frequency, the scintillation can be too rapid to be discerned by the naked eye in bright stars, although the telescopic image will be seriously degraded.

• Flashing is an abrupt expansion of the star image accompanied by a loss of focus and increased illumination in the surrounding aerosol diffusion. It caused by very large fluctuations in turbulence where F > H and the angular air speed v is slow. These conditions imply that the optical path is through a thickening in a refracting air layer or an unusually large turbulence cell, which can be either in instrument or surface turbulence turbulence or (rarely) in high altitude turbulence. If the flashing does not also produce a simultaneous brightening in the aerosol illumination (the diffusion nimbus around the star image), the turbulence is probably in the instrument or observatory structure.

In general, all three types of distortion can combine in different proportions to produce the complex and moment to moment changes in a star image, although it is not unusual for a visual astronomer to experience distortion as a mixture of adjacent forms in the diagram. Thus, on exceptionally fine nights, an undistorted star image will be disturbed by brief dancing or rippling movement, on average nights a dancing Airy disk will combine with speckling of the diffraction rings, and on poor nights there can be frequent flashing in a scintillating star image.

Although turbulence can be described in terms of atmospheric models, for example as superimposed turbulence layers of different frequencies, it is not convenient to apply it to imaging on those terms. Instead, subsequent research has analyzed the cumulative effects of turbulence on the wavefront and as optical distortions in the image. To briefly summarize this highly technical literature: R.E. Hufnagel and N.R. Stanley (1964) derived the changes in the modulation transfer function (MTF) and Strehl ratio or point spread function (PSF) that result from the transmission of a diffraction limited image through turbulent media. David Fried (1965, 1966) applied their work to the problem of "looking down" through the atmosphere with optical (military or mapping) satellites. These developments were summarized and applied to the astronomical problems of "looking up" through the atmosphere by François Roddier (1981).

Vladimir Sacek's web page on Atmospheric Turbulence summarizes the thrust of a strict optical analysis, which attributes oscillation effects to wavefront tilt and scintillation effects to roughness. Roughness in turn can be described in terms of traditional optical aberrations — a random mixture of defocus, spherical aberration, astigmatism and coma.

Rather than approach the technical analysis, it is helpful to visualize the optical effects of atmospheric turbulence in terms of a Newtonian reflector mirror divided up into thousands of tiny hexagonal plane mirror cells. Undisturbed, these tiny mirrors align to produce a perfect paraboloid surface that creates a diffraction limited telescopic image.

However each mirror can oscillate independently from side to side by a miniscule angle that represents the refraction added by the atmosphere. If we looked across the mirror from one side, we would see the surface appear to ripple continuously, with an overall movement across the mirror from one side to the other but with a variety of smaller disturbances and eddies in the flow. These oscillations deflect the light falling on each mirror away from the optical path necessary for perfect focus by the entire aperture.

When a large proportion of these mirrors tilt in the same direction at the same moment across the entire objective diameter, the image at the focus is displaced to one side ("tilt") and oscillation results. When some of the mirrors tilt at the same moment toward or away from the optical axis by an amount proportional to their distance from the optical axis, defocus or flashing results. When mirrors on one side of the aperture tilt to a shorter focus than mirrors on the opposite side, coma results. When the mirrors along one diameter tilt outward at the same time mirrors along the perpendicular diameter tilt inwards, astigmatism results. Extremely complex and chaotic patterns of turbulence can in this way be attributed to distinct aberration categories, and as each category effect is subtracted from the whole the remainder can be explained by other types of aberration, until we are left with the mirror cells in perfect alignment again.

The Statistical Description of Turbulence

Atmospheric turbulence has been successfully analyzed and described in separate models of turbulent flows and optical aberrations. These can be used to derive descriptive statistics for the degree of turbulence, which can be done both for the turbulence itself and its effect on image quality.

Because optical turbulence and optical aberrations are separate complex phenomena produced by many different sources of distortion, it is not feasible to connect them in detail. Instead, the connection is conveniently defined as an average turbulence in relation to an average image aberration.

A star diffraction artifact consists of the Airy disk surrounded by concentric diffraction rings. This Airy disk has an angular (visual) radius fixed by the telescope aperture and a linear (photographic) radius fixed by the telescope relative aperture. In perfect optical system, the Airy disk comprises about 86% of the total light from the star. The cumulative effect of the many small mirrors (in the previous analogy) tilting in divergent directions is to randomly shift light that would form inside a compact Airy disk away from the optically perfect location, spreading the light across a wider area. This lowers the Strehl ratio (the ratio of total light that is focused into the Airy disk) and inflates the point spread function, both technical indicators of degraded optical quality.

Because optical speckling bloats the star image, and oscillation shifts the image around on the image plane during a time exposure, the full width half maximum of a star image (explained in the page on seeing measurement methods) is a concrete and empirical measure of the overall impact of atmospheric turbulence in a given aperture. Seeing that is below a FWHM of 1 arcsecond is considered excellent, and seeing that is above 5 arcsecond is considered poor. (Note that FWHM is based on the cumulative image dispersion measured by a CCD sensor and is not directly a measure of appearance to the visual astronomer.)

The statistical description of the average atmospheric turbulence begins with the temperature structure coefficient, which is a variance defined on a three dimensional average squared distance. It is calculated as:

C_T² = [T(x) – T(x+r)]²/r^2/3

where x is any point within the turbulence, and r is the three dimensional distance from x to any second point of the same temperature. Referring to the middle series of turbulence simulations (above), it is evident that higher energy turbulence breaks isothermal cells into smaller units, reducing the average distance between points of the same temperature: as turbulence becomes more extreme, the average C_T² becomes smaller.

This can be applied to estimate the refractive index structure coefficient C_n² of the turbulence, which is the average difference in the refractive index between between two points in the turbulent layer. Mathematically it is simply the temperature structure coefficient scaled by the average air pressure and temperature:

C_n² = C_T²·[7.9x10^-5·P / T²]²

where P is the atmospheric pressure in millibars and T is the temperature (in Kelvins). The refractive index becomes smaller as the size of turbulence cells decreases, but exponentially smaller as pressure decreases and temperature increases.

Next the refractive index coefficient can be used to estimate the average physical width of a turbulence cell having a uniform optical effect, known variously as Fried's seeing parameter, Fried's coherence length or Fried's r₀ (pronounced R naught):

r₀ = [0.184λ^6/5·(cos γ)^3/5∫dh·C_n²(h)]^–3/5 meters

σ² = 1.0299·(d / r₀)^5/3

where h is altitude in meters, γ is the zenith angle, and C_n²(h) is the refractive index structure coefficient at a specific altitude, integrated across all atmospheric layers in the optical path; σ² is the two dimensional variance of the coherence length around the average. The key points are that the optical effects of seeing are less (Fried's r₀ gets larger) at smaller zenith angles and longer wavelengths, but that turbulence effects are additive (cumulative) across the entire atmospheric layer to maximum h, from the mirror surface to the stratosphere.

Finally the isoplanatic angle θ_o of Fried's r₀ is the average angular width of an isoplanatic patch along the optical path of a ground based observer:

θ_o = 0.6*r_o/h

where h is the altitude of the turbulent layer.

Fried's r₀ is roughly the average physical dimension of a single isoplanatic patch (a turbulence cell refracting light in the same direction). Observational results indicate r₀ varies from about 2 to 21 cm for visible wavelengths viewed at the zenith, with 10 cm used as a rule of thumb average value. The following table, adapted from Roddier (1981), equates Fried's r₀ with direct measures of seeing at wavelengths around λ = 500 nm ("blue green").


Fried's parameter r₀ (cm)	Coherence area λ²σ (cm²)	Seeing angle ω (arcsecond)

21.1	152	0.62
16.5	94	0.79
13.9	66	0.94
10.9	41	1.20
9.2	29	1.4
8.0	22	1.6
6.3	14	2.1
5.3	9.6	2.5
4.2	5.9	3.2
3.5	4.2	3.7
2.7	2.6	4.8
2.0	1.4	6.5

The diffraction limit DL in turbulence of a given r₀ at wavelength λ for a focal length ƒ and spatial frequency ν_c is estimated as:

DL_r₀ = 1/ν_c·1/ƒ = 2.1·λ/r₀

and the diameter and number of speckles as:

N_s = (D / r₀)²

D_s = (λ / D_o)·206265 arcseconds

This can be compared to the unperturbed diffraction limit λ/D_o to determine the relative degradation in average resolution. However, if seeing is stated in terms of an image FWHM, a rough rule of thumb is that Fried's r₀ is 22 cm when the FWHM seeing is about 0.5 arcseconds, is 11 cm when the seeing is about 1 arcsecond, and so on.

The chart (right) is based on several years of atmospheric measurements at the ESO observatory at Cerro Paranal, Chile; the basic form of the curve is typical of atmospheric measurements at other locations.

Perspective scaling — turbulence becomes visually smaller the farther it is from the viewer — accounts for basic dependence of turbulence size on altitude. This is illustrated by the orange curve, which plots the apparent angular width to a ground observer of a constant 1 meter width located at the matching altitudes. Comparing the angular width of the two curves at the same altitude shows that the average dimension of atmospheric turbulence can be quite large near the ground, on the order of several meters or more, but decreases in size from the surface boundary up to an altitude of about 5 km.

Fried's r₀ is often interpreted as the telescope aperture at which the transition occurs from scintillation or oscillation to speckling as the dominant image distortion. Thus large r₀ means relatively good seeing, and for any specific telescope, when D < r₀, the atmosphere does not significantly degrade the image, though it may affect the stability the image (as oscillation). In that situation the telescope is said to be diffraction limited rather than seeing limited. The consensus experience is that telescopes above about 10–20 cm in aperture are, on most nights, continuously seeing limited.

The context for the previous "small mirrors" analogy is that some of the newest generation of large aperture (10+ meters) telescopes are designed as small segmented mirrors attached to servomechanisms that can independently tilt each mirror segment to compensate for local tilt aberrations measured in real time in the side to side variations in the location of a laser beam reflected off a refracting layer high in the atmosphere. These adaptive optics mechanisms cannot respond quickly enough to eliminate the "roughness" components of turbulence, but they can cancel the effects of the relatively broad and sluggish "tilt" oscillations. At the other extreme, speckle interferometry allows analysis of extremely small details through the computational filtering of speckle patterns backwards to isolate the original Airy disk (or equivalent object detail).

If the astronomical object is a bright star or planet, the alternative strategy of lucky imaging can be applied: a very large number of images are captured and only those images are used in which the turbulence has lapsed for a fraction of a second into a glimpse of minimal distortion. This is made possible by modern CCD imaging that allows continuous, rapid video capture of telescope images, which can be evaluated, selected and averaged ("stacked") into a single image by computer software. The probability of obtaining a short exposure diffraction limited image is related to

Prob ~ 5.6*exp[–0.1557*(D/r₀)²]

which implies a probability of 1 in 1 when D = 3.5r₀, about 1 in 400 when D = 7r₀ and becomes exponentially impractical (1 in 1 million) when D = 10r₀. (In these analyses the exposure time is roughly 1/3 the fluctuation period of a single Fried length, which is most feasibly obtained with video capture.) These steeply escalating costs of large apertures are a principal reason for the development of adaptive optics.

Astronomers evaluate seeing by a variety of methods: radar imaging, imaging the variations in a laser reflected off a high atmosphere layer, or observing a star simultaneously through parallel apertures of different sizes (which therefore sample r₀ at different values). Statistical analyses applied to each kind of data can isolate the different distances or spatial and temporal frequencies in the image, and from these estimate the relative vertical location, scale parameters and lateral velocity of the turbulence.

Some of the best observing sites in the world average an r₀ of 10 to 15 cm, which means that all telescopes of aperture greater than about 6" would perform at less than their diffraction limits.

How often do nights of excellent seeing occur? The graph (left) summarizes four years of night r₀ measurements at the William Herschel Telescope site in the Canary Islands. Even this superb viewing location has many nights of relatively poor seeing: the distribution is positively skewed, and the cumulative probability shows only a 50% chance on any night of values of r₀ above 14 cm, and only a 10% chance of values above 25 cm or 10 inches. Thus, at this excellent site, a 10 inch telescope will experience at least some seeing limitation on 9 out of 10 nights.

Since this graph defines a site chosen as having the best seeing among many possible sites in the world, it can be generalized to other locations by shifting the average or median value to the right (to represent worse average seeing). However the basic shape of the graph — when seeing is very good, it much much better than average seeing, but when seeing is bad, it is only slightly worse than average seeing — may be typical of all sites at all times. Certainly the low probability of a large value of r₀ at La Palma is consistent with astronomer E.M. Antoniadi's comment that nights of excellent seeing are only "one in fifty".