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ASTR 5110, Majewski [FALL 2017]. Lecture Notes

ASTR 5110 (Majewski) Lecture Notes

Observing Through Earth's Atmosphere:

Turbulence and Seeing

"If the Theory of making Telescopes could at length be fully brought into Practice, yet there would be certain Bounds beyond which telescopes could not perform. For the Air through which we look upon the Stars, is in perpetual Tremor; as may be seen by the tremulous Motion of the Shadows cast from high Towers, and by the twinkling of the fix'd Stars... The only Remedy is a most serene and quiet Air, such as may perhaps be found on the tops of the highest Mountains above the grosser Clouds." Isaac Newton

Some References:

  • Coulman, C.E. 1985, Annual Reviews of Astronomy & Astrophysics, 23, 19.

  • Section 2.1 and pp. 89-97 of Kitchin, Astrophysical Techniques, Fourth Edition.

  • Sections 2.1, 2.6, 4.4 of Lena, Observational Astrophysics.

  • Section 3.6, Chapter 16 of Schroeder .

  • Section 1.3 of Bely, The Design and Construction of Large Optical Telescopes.

  • Section 5.8.1 of Hecht, Optics.

  • Chapter 5 by Quirrenbach in Lawson (ed.), Principles of Long Baseline Stellar Interferometry.

Atmospheric turbulence is produced by a cascading transfer from larger to smaller eddies, all which contribute to poor seeing (van Gogh's Starry Night).

A. Introduction to the Problem of "Seeing"

All ground-based astronomical observations are hobbled by the fact that the light must pass through the Earth's atmosphere.

  • Since the atmosphere is layered by (or consists of varying gradients in) temperature and pressure, it has refractive power.

    We have discussed the effects of the atmosphere's global refractive power elsewhere.

  • Worse than the presence of its net global refractive power is the fact that atmospheric layering is not smooth.

    • Wind and convection and other currents create turbulence, which mixes layers with differing indices of refraction in non-uniform and constantly changing ways.

    • The net result has a serious effect (e.g., tilting, bending and corrugating) on transiting, initially plane-parallel wavefronts.

      From Schroeder, Astronomical Optics.

    • This is the source of the "twinkling" phenomenon we are familiar with.

The observed negative impacts of the turbulent atmosphere on astronomical images are encompassed globally under the expression "seeing".

Understanding the physics of seeing allows us to:

  • Improve site selection of telescopes for better image quality/stability.

  • Improve the design of observatories to reduce the local effects of seeing.

  • Improve the conditions at existing observatories by reducing the local effects of seeing.

  • Design active/dynamical means for overcoming the image degradation from atmospheric effects.

  • Understand how to optimize our observing for image quality.

B. Physics of Turbulence and Seeing

Index of Refraction of the Air

We have previously adopted the index of refraction of air as n=1.

But in fact the index of refraction of air has a small variability, depending on its physical state and composition.

The variable part of the refraction index is given by Cauchy's formula (extended by Lorenz to account for humidity):

where λ is the wavelength of light, p is the atmospheric pressure (mbars), T is the absolute temperature (K) and v is the water vapor pressure (mbars).

  • The dominant terms in this equation translate to :

    n - 1 = 77.6 X 10-6 p / T

  • To give some sense of the small variability in this index, for 500 nm light, n = 1.0003 at sea level and n = 1.0001 at 10 km altitude.

In terms of seeing, what we care about are changes in this quantity affecting the transiting wavefront, or, more insidiously, differentially affecting the wavefront on small scales (turbulence).

  • Fluctuations in water vapor have no significant effect on the refractive index at optical wavelengths, except in extreme situations, such as in fog or just above sea surface.

    Water vapor can affect the radiative transfer properties of the air, and therefore alter the convective properties of air columns.

    The effect of water vapor (latter term in equation above) is generally small for modern astronomical observatories, which are typically built in very dry sites, and which already typically have other weather-related observing problems when the humidity is high (e.g., condensation on mirror surfaces, etc.).

  • If one takes derivatives of the above equation with respect to temperature and pressure, one finds that variations in temperature are far more important than variations in pressure, and, assuming adiabatic conditions and a perfect gas

    dN / dT ∝ (p / T 2)

  • Thus, the primary source of variations in the index of refraction are attributable to thermal variations.

  • In terms of seeing, what we care about is small scale variations in T, or thermal turbulence.

Sources of Thermal Turbulence

Atmospheric turbulence is created on a variety of scales from several origins:

  • Convection: Air heated by conduction with the warm surface of the Earth becomes buoyant and rises into cooler air, while the cooler air descends.

  • Wind shear: High winds, particularly the very fast ones associated with the jet stream, generate wind shear and eddies at various scales, and create a turbulent interface between other layers that are in laminar (i.e., non-turbulent) flow.

  • Disturbances: Large landform variations can create turbulence, particularly on the lee side of mountains where the air flow becomes very non-laminar.

    From Quirrenbach's chapter in Lawson (ed.), Principles of Long Baseline Stellar Interferometry.

Turbulence and the Eddy Cascade

The properties of fluid flows are characterized by the Reynold's number, a dimensionless quantity relating inertial to viscous forces

Re = V L / ν = (Inertial forces) / (Viscous forces)

where V is the fluid velocity, L is a characteristic length scale, and ν is the kinematic viscosity (m2/s) of the fluid.

  • Determines whether the flow will be:

    • Re low --- dominated by viscosity and characterized as smooth / laminar / constant; or

    • Re high --- dominated by inertial forces and therefore turbulent and full of vortices and eddies.

  • For air, viscosity is very small: ν = 1.5 X 10-5 m2/s.

  • Thus, for typical wind speeds and length scales of meters to kilometers, Re > 106 and the air is moving turbulently.

One can think of turbulence as being made up of many eddies of different sizes.

The nature of thermal turbulence is created by a process of Eddy transfer:

  1. Kinetic energy is deposited into turbulence starting with the large scale air flow processes of convection or wind shear.

    • The characteristic scale over which the energy is deposited, L0 is called the external scale or outer scale.

      L0 is generally larger than the aperture of a telescope, but there is considerable debate over its typical value.

      Somewhere between 1 to 100 meters.

  2. The large turbulent eddies created by the above processes create wind shears on a smaller scale.

  3. These still smaller eddies, in turn, spawn still smaller eddies, and so on, in a cascade to smaller and smaller scales.

    This intermediate range of cascading turbulent scales is called the inertial range, and is where:

    • inertial forces dominate and energy is neither created or destroyed but simply transferred from larger to smaller scales.

    • all of the thermal fluctuations relevant to seeing occur.

  4. The cascade continues until the shears are so large relative to the eddy scale (Re ~ 1) that the small viscosity of air takes over and the kinetic energy is "destroyed" (converted into heat).

    • Happens on scale, l0 , of a few millimeters.

    • Called the internal or inner scale.

    • The cascade stops.

    • The temperature fluctuations are smoothed out.

A Model for Turbulence: The Kolmogorov Spectrum

It is perhaps somewhat surprising that for the inertial range there is a universal description for the turbulence spectrum (the strength of the turbulences as a function of eddy size, usually expressed in terms of wave number κ).

  • In 1941, Kolmogorov found that in the above process of an eddy cascade, the energy spectrum had a characteristic shape:

    The inertial range follows a spectrum of κ-5/3 (the Kolmogorov spectrum or Kolmogorov Law).

    From Lena, Observational Astrophysics.

  • When turbulence occurs in an atmospheric layer with a temperature gradient (differing from the adiabatic one) it mixes air of different temperatures at the same altitude and produces temperature fluctuations.

    Hence, the above spectrum also describes the expected variation of temperature in turbulent air.

  • Aside: The three-dimensional version of the spectrum follows κ-11/3.

The Structure Function

The theory of atmospheric seeing was developed from the 1960s to 1980s and turns out to work remarkably well as a physical description of the process.

To understand turbulence in the context of seeing, we need to translate the above thermal turbulence spectrum into a spatial context -- i.e., how the atmosphere is going to transform wavefronts.

  • Generally and conveniently described in terms of the structure function -- a statistical measure of the fluctuation over a spatial span of r.

  • In one dimension, the structure function is defined as:

    where T(x) and T(x+r) represent the temperatures for two points separated by r.

    The brackets denote a mean squared difference in temperature.

  • For a Kolmogorov spectrum:

    where CT2 is the temperature structure constant (or coefficient).

    The size of CT2 gives the intensity of the spatial thermal turbulence.

  • Ultimately, of course, we want the spatial variation in the index of refraction, and this can be described by an index of refraction structure constant, now derived by using the relation for n = n(P,T) above:

Connection to Fried Parameter

As we shall show below, there is a characteristic transverse linear scale over which we can consider the atmospheric variations to flatten out, and in which plane parallel wavefronts are transmitted.

This scale is known as the Fried Parameter, r0 , and is central to a description of the effects of turbulence on images of stellar sources (it fully describes the PSF).

  • As one might expect, the quality of an image depends on how much turbulence there is along a line of sight.

  • The Fried parameter can actually be well predicted by integrating Cn2 along a line of sight though the atmosphere:

    where z is the altitude, and γ is the zenith distance (the angle between the line of sight and the zenith).

    An example of a Cn2 profile. Integrating this profile through the airmass allows one to derive the Fried parameter. One sees the largest Cn2 are near the ground, although certain layers of the atmosphere, such as the altitudes of the jet stream (about 104 meters) also contribute large turbulence. From Bely, The Design and Construction of Large Telescopes.

  • A larger Fried parameter is better, hence the inverse dependence on Cn2.

  • Note the direct dependence on cos(γ), which shows that the Fried parameter gets smaller as cos(γ) gets smaller, which happens as one looks through more atmosphere with increasing zenith angle.

  • The Fried parameter, ro is inversely proportional to the size of the PSF, θ, transmitted by the atmosphere.

    Thus, remarkably, the FWHM of the observed PSF (in arcsecs) can be predicted if one knows how Cn2 varies along the line of sight. From the previous equation, and assuming now that the seeing is given by the diffraction limit of one Fried cell (see below) so that θ ~ λ/ro , one gets:

    Again, note the dependence on zenith distance (in the sense that the PSF is smaller for smaller zenith distance) and wavelength (in the sense that the PSF is smaller for longer wavelengths).

    Or, for λ = 500 nm and typical mountain top observatory conditions:

Time Dependence of the Turbulence

At a given point in the atmosphere, the temperature is a random function of time.

We can characterize the time frequency character of seeing phenomena considering the above physical description.

  • A simple model (the Taylor Hypothesis) is to consider the turbulence along a line of sight as "frozen" with given spatial power spectrum and configuration, and assuming that a uniform wind translates the column of air laterally with a velocity V.

    (In effect this is the basis of the Fried description of seeing we adopt below.)

  • The physical basis of the Taylor hypothesis is simply that the timescales involved in the development of turbulence are much longer than the time for a turbulent element displaced by wind to cross the telescope aperture.

    Rather than think of "frozen turbulence", probably more correct to think of a "frozen phase screen" over short timescales.

  • The temporal cut-off frequency (the quickest timescale for observed changes in image deformation) can then be given by order of magnitude as:

    fc = V / l0

    For l0 = 10 mm and V = 10 m/s, we get fc = 103 Hz.

    Cross-correlations of the wavefront from a star reaching a telescope pupil at intervals of 40 ms. The displacement of the "phase screen" is obvious. From Lena, Observational Astrophysics.

  • In reality, it is found that the turbulence at any particular site generally consists of a superposition of this "frozen" turbulence (generally from higher layers) and a local turbulence which has a significant vertical component, generally from convection near the telescope from telescope, dome, other local features, etc.

    Still, adopting the Taylor hypothesis allows us to gain some important insights using the Fried description, as we did above and will again below.

C. The Turbulent Layers of the Atmosphere

Overall Vertical Structure of Atmosphere

To first order, the Earth's atmosphere is in radiative equilibrium, with a balance of the solar flux and the energy re-radiated into space.

But there are important cyclic variations (diurnal, annual) superposed.

The mean large-scale structure of the atmosphere shows variations in pressure and temperature:

  • The pressure drops off exponentially as:

    P(z) = P0 e-z/H

    where the scaleheight, H ~ 8 km near the ground.

    From Lena, Observational Astrophysics.

  • The temperature structure is more complicated:

    • In the troposphere, dT / dz < 0 in general (it gets colder as you go away from the ground).

    • Above this the tropopause, where dT / dz ~ 0.

    • Then enter stratosphere, where dT / dz > 0 in general (it gets warmer as you go higher).

  • Note also:

    • Height of tropopause depends on latitude, almost touching "ground level" over Antarctica (altitude of South Pole already 3 km, and the scaleheight of the atmosphere more compressed there).

    • At all latitudes, and especially near ground level, there can be significant deviations from average temperature distribution.

    • Of importance for our understanding of turbulence are inversion layers, where dT / dz changes sign.

    • Also of importance (particularly near the ground) is the strength of a temperature gradient compared to the adiabatic gradient:

      ( dT / dz )ad = - (Tm / H) (Cp - Cv ) / Cp

      • H is the scaleheight

      • Tm is a mean temperature over that distance

      • Cp , Cv are specific heats of the air.

      If a temperature gradient larger than the above is created there is convective instability and vertical currents are created from buoyant forces.

Atmospheric Layers in the Context of Seeing

When discussing seeing, there are four basic atmospheric layers of relevance, with gradual transitions of properties between them:

  • The surface or ground layer is the layer within a few to several tens of meters (depending on the site) above the ground.

    • Here turbulence is generated by wind shear due to frictional and topographic effects at the ground surface.

    • Strongly influenced by site geometry and "roughness" of ground from boulders, crags, trees.

    • Best sites have a surface layer as small as 5 meters (e.g., Paranal) or 6-10 meters (Mauna Kea).

    • Pays to have telescope, enclosure, primary mirror above this layer.

  • The planetary boundary layer extends to of order 1 km above sea level.

    • Here there are still significant frictional effects from the Earth's surface, but also significant vertical motion due to diurnal heating/cooling cycles of the ground and air in contact with it.

    • Convective effects are in play up to the inversion layer, which tops the planetary boundary layer, at about 1 km.

    • Modern observatories are generally situated above this layer (most desirable, to get above planetary layer turbulence).

      From Bely, The Design and Construction of Large Telescopes.

      From Lena, Observational Astrophysics.

  • The atmospheric boundary layer is the next major air mass up, and is separated from the planetary boundary layer by the thermal inversion layer.

    • No longer dominated by convection, but may still be effected by the ground, both thermally and mechanically, e.g., by large mountains or ranges that poke through the planetary boundary layer.

    • Abrupt changes in topography can create gravity waves or buoyancy waves, that can create turbulent flow.

      From Bely, The Design and Construction of Large Telescopes.

      Gravity waves in air, like water waves on the ocean, are waves in layers of a fluid medium with the property of a decreasing density with altitudes and where gravity provides a restoring force (opposite buoyancy) when a parcel of the fluid (air) is displaced, resulting in an oscillation of the parcel about an equilibrium state.

      An interesting video showing passage of gravity waves is shown here.

      Gravity waves transfer momentum from the troposphere to higher layers of the atmosphere, with the amplitudes of the waves being larger in the more rarefied layers.

    • To avoid these long wavelength waves, best to be near the sea (over which there is laminar flow) on side of prevailing winds (e.g., foothills of coastal ranges).

    • Best sites are also above the atmospheric boundary layer, but at many existing sites, this is the dominant source of seeing.

      • E.g., at ESO site at La Silla, Chile, 80% of seeing is from the 500 m of the atmospheric boundary layer still above the telescopes.

    • The atmospheric boundary layer above water generally exhibits smaller thermal fluctuations than over land because the temperature contrast between water and air is generally smaller.

      • Thus, observatory sites on islands or shorelines tend to be better.

      • Because lower layers of an airstream quickly adapt to land conditions, close proximity to sea is best.

    The figure below demonstrates how orographic irregularities can create turbulence on a variety of scales, affecting the surface layer (e.g., L1 ), planetary boundary (e.g., L2 ), and atmospheric boundary layers (e.g., L3 ).

    From Lena, Observational Astrophysics.

  • The free atmosphere is the bulk of the atmosphere shown in the density/pressure plot at the beginning of Section C above.

    • Essentially unaffected by frictional influence of the ground (by definition).

    • This layer mainly driven by very large-scale air flows from, e.g.:

      • at lower altitudes, tropical trade winds and "the westerlies" at mid-latitudes.

      • in high layers, the jet stream.

      These winds create mechanical turbulence in layers having temperature gradients, creating variations in CN2.

    • The jet stream is strongest at mid-latitudes.

      From Bely, The Design and Construction of Large Telescopes.

    • Were the jet stream the only consideration, then one would expect better seeing (from the free atmosphere) near the equator and poles (but see future web page on site selection for other considerations).

      The contribution to seeing from free atmosphere is about 0.4 arcsec on average.

Diurnal Effects in the Low Altitude Layers

There are significant daily changes in the thermal profile of the lower atmospheric layers over land, which drive the changing convective properties within these layers.

The nature of this profile is important to the design of observatories, both night time and solar.

A typical example of the profile variation is as follows:

From Coulman, 1985, ARAA, 23, 19.

  • Solar heating of the Earth's surface during daytime leads to conductive heating of air in contact with it.

  • By mid-afternoon, the heated air creates convective currents that extend the planetary boundary layer to a typical maximum of 1.3-1.5 km, as shown by the purple, 3 PM curve above.

    • In the tropics or in disturbed weather, cumulus cloud growth leads to a planetary boundary layer as high as 3 km or more.

    • Moist air is especially sensitive to vertical displacement, which can induce condensation, clouds, precipitation.

  • After sunset, the ground rapidly cools by radiative transfer. Convective mixing becomes greatly reduced.

    • Radiative transfer is even faster in moist air.

  • A nocturnal inversion layer then develops near the surface.

  • Night time conditions are more convectively stable (red shaded lines for 9 PM = "21" and 3 AM = "03" in the plot), given the constantly increasing or neutral gradient with altitude.

  • With sunrise, ground heating begins again, and a convective planetary boundary layer develops again.


  • The above represents an average profile, whereas large thermal fluctuations from this average can occur on short timescales.

  • The thermal turbulence is much larger in convective updrafts than in downdrafts, and so:

    • on average, the fluctuation amplitudes usually decrease with altitude.

    • but because there are typically bulk updraft and downdraft cells, significant and quick changes can occur in the seeing with the lateral shifting of the cells.

  • The Cn2 profile is generally given by the temperature variance through the convective layers, and so, in principle, can be estimated. A schematic view of thermal variance, incorporating mainly convective effects, gives rise to something like this:

    From Coulman, 1985, ARAA, 23, 19.

  • As seen in the above figure for daytime turbulence, it pays to locate a solar telescope as high as possible above surrounding terrain, to get above the lower turbulence.

In fact, the actual Cn2 profile is more complicated than this.

  • Observations show the existence of thin layers having high wind shear or other properties creating greatly enhanced Cn2.

    From Coulman, 1985, ARAA, 23, 19.

    For example:

    • the jet stream near 104 m.

    • the planetary boundary capping layer (daytime inversion layer) near 103 m.

  • The existence of these discrete, high Cn2 layers suggests that seeing compensation (e.g., with adaptive optics) might be best accomplished by optically complementing these discrete layers.

    • This is the basis of multi-conjugate adaptive optics (MCAO) (to be discussed later).

  • The diurnal variation in the Cn2 profile near the ground is shown by the thick solid lines for midday (line 3), late afternoon (line 2) and a good seeing night (line 1).

  • A simplified version of Coulman's figure is shown in Bely:

    From Bely, The Design and Construction of Large Telescopes.

  • Here is a version of this plot (with a linear altitude scale) -- showing primarily free atmosphere effects above the VLT site at Paranal (which is above most of the other layers):

    From Quirrenbach's chapter in Lawson (ed.), Principles of Long Baseline Stellar Interferometry.

    As pointed out in the caption, this particular data set does not show the typical strong turbulence associated with the jet stream turbulence.

D. The Fried Model and Isoplanaticity

In 1665, Robert Hooke, in his Micrographia, first suggested the existence of "small, moving regions of the atmosphere having different refracting powers which act like lenses" to explain scintillation.

In 1966 David L. Fried showed that the atmosphere can indeed be modeled in this fairly simple way.

  • One can assume that at any moment the atmosphere behaves like a compressed, horizontal array of small, contiguous, wedge-shaped refracting cells.

  • These act on the plane parallel incoming waves from an astronomical source by locally tilting the wavefront randomly over the size scales of the cells.

    Each cell imposes its own tilt to the plane waves, creating local isoplanatic patches within which the wavefront is fairly smooth (~ λ / 17) and has little curvature.

    Thus, each isoplanatic patch transmits a quality (though perhaps shifted) image of the source.

    From Kitchin, Astrophysical Techniques.

  • The critical scale over which images begin to lose quality (i.e., the size of the isoplanatic patch) is called the Fried parameter or Fried length and is usually designated as ro .

    This is not meant to be a radius, but is more like a diameter of the patch.

    It describes the scale over which the wavefront maintains its parallel (though possibly tilted) nature and the diameter over which bundles of rays still arrive parallel and in phase.

  • The Fried parameter is a statistical description that characterizes the seeing.

  • The more turbulence there is, the smaller is ro and the poorer is the seeing.

  • A typical value for the Fried parameter in the optical is about 10 cm.

  • However, the Fried parameter varies with wavelength as λ6/5.

    This means that the seeing will always be better (i.e., PSF will always be sharper) at longer wavelengths.

  • If we carry the model further, and assume that the turbulence is occurring in a single layer, can view the changing aspects of seeing in terms of the speed that wind moves the cells across the line of sight.

    The coherence time -- the time for the transit of one cell in the Fried description -- is then given by

    τo ~ ro / v

    From Bely, The Design and Construction of Large Telescopes.

    The coherence time is typically 10's of Hz. (Note that this is longer than the "quickest" timescale derived using the inner scale of the turbulence.)

  • Another important characteristic of seeing is the angular size of the isoplanatic patch.

    This isoplanatic angle is the angle on the sky over which the effects of turbulence are uniform/correlated, given by:

    θo ~ 0.6 ro / h

    where h is the altitude of the primary turbulence layer over the telescope.

E. Observed Seeing Effects: Scintillation, Image Wander, Image Blur

There are various manifestations of the observed effects of turbulence (seeing).

Scintillation changes the apparent brightness of a source, whereas image wander and image blur degrade the long term image of a source.

Which of these effects comes into play depends on the telescope aperture size relative to the Fried parameter.


Scintillation ("twinkling") is the result of a varying amount of energy being received by a pupil over time.

  • Variations in the "shape" of the turbulent layer results in moments where it mimics a net concave lens that defocuses the light and other moments where it is like a net convex lens that focuses the light.

    This curvature of wavefront results in moments of more of less light being received by a fixed pupil.

  • Scintillation only is obvious when the aperture/pupil diameter is of order ro or less.

    E.g., the human eye pupil is generally always less than ro .

    Larger apertures average out the affects and the "twinkling" is diminished.

  • Since scintillation is ultimately an interference phenomenon, it is highly chromatic.

    You may have noticed this by observing stars close to the horizon changing color and strongly twinkling on timescales of seconds.

From Lena, Observational Astrophysics.

Image Wander

Another obvious effect for apertures smaller than ro is image wander or agitation.

  • This is the motion of the instantaneous image in the focal plane due to changes in the average tilt of the wavefront with time.

  • But because the wavefront is locally plane parallel, the image can actually be diffraction-limited for each isoplanatic patch that passes through the line of sight.

    Thus, a sharp Airy disk -- with central FWHM ~ λ / D -- can be formed from each ro cell, and if the telescope aperture is smaller than ro, it will be common to see a sharp Airy disk in the focal plane...

    ... but the center of that diffraction-limited Airy disk will wander around the image plane as different cells imposing different wavefront tilts pass over the aperture.

    From Roy and Clarke's Astronomy: Principles and Practice.

  • Thus, to instantaneously achieve the ultimate, diffraction-limited performance of a telescope, one only needs to ensure that the telescope aperture be smaller than ro (by about a factor of 1.6). Of course, this means you are forcing yourself to lower light-gathering power and resolution!

    This image wander is a characteristically observed phenomenon when looking through small telescopes.

    To obtain the diffraction limit over a long timescale, one needs a guiding system that can remove the shifting image position...

    ... or a tip-tilt mirror system to correct out the wavefront tilts (the basis of active optics seeing compensation -- see future lecture).

    In this (relatively simple) way the primary atmospheric degradation for small apertures can be removed.

    From Bely, The Design and Construction of Large Telescopes.

  • Image wander diminishes as telescope aperture grows, because larger apertures average over more isoplanatic patches.

Image Blurring

Things are more complicated for a large telescope (D > ro), since many isoplanatic patches will be in the beam of the telescope, and image blurring or image smearing dominates.

  • Because many parts of the corrugated wavefront are contributing, larger telescope apertures suffer from a larger image spread from the accumulated contributions of differently pointed, wavefronts.

  • At any given time, if looking at the image of a single star in a large telescope, each isoplanatic patch creates its own diffraction-limited Airy disk (FWHM ~ λ / D). These individual Airy spots are called speckles.

    Seen together, the speckles give a shimmering blur.

    In this case, the ensemble of speckles will have an envelope given by FWHM ~ λ / ro .

    From Lena, Observational Astrophysics. Note the error in the figure in that the number of areas of coherence is of course

    N ~ (area of pupil) / (area of Fried cell), or the inverse of what is shown.

    If integrating over long times compared to the coherence timescale, the resultant image, which is the superposition of all of the speckle patterns during the integration, will have this size (FWHM ~ λ / ro) for the long exposure seeing (PSF) -- a significant degradation of image quality.

  • If the telescope diameter is of order or slightly larger than ro , you will see several speckles, but occasionally can chance upon a moment when the seeing turns very good (ro is temporarily larger).

    From Hecht, Optics.

    In principle, one could fast shutter the telescope, leaving only these moments of good seeing get to the detector and shuttering out the bad moments to get crisp images.

    But, as mentioned in the caption above, the larger the aperture, the likelihood of a clear moment decreases exponentially with aperture diameter.

  • For a given small telescope of a set aperture size, what stellar images will look like depends entirely on the seeing cell size compared to the telescope aperture.

    For example, the figure below shows drawings of a star observed with the same small telescope under different seeing conditions. As may be seen, under good conditions with ro larger than the aperture we see an Airy pattern, as in the rightmost drawing, but as ro shrinks to about the size of the telescope aperture, one sees something more like the middle panel. Under terrible conditions, when ro is much smaller than the telescope aperture, one can see speckles even in a small telescope (with each speckle coming from a different Fried cell parcel of air).
    From the Celestron CPC Series Instruction Manual.

  • In general, increasing the telescope aperture will collect more light (i.e. more speckles), but will not improve the resolution (PSF) beyond the atmospheric, long exposure seeing limit.

  • A strategy for taking advantage of the speckles to recover the theoretical diffraction limit of a large aperture telescope is called speckle interferometry (see next lecture webpage).

  • Note that because the Fried parameter itself goes as λ6/5, the seeing PSF goes as λ-1/5 -- i.e., the seeing is always better in the red.

Short and Long Exposure Seeing

Imagine that you could capture and pick apart the speckled image (and we will!) from one instant in time (an exposure time a fraction of the coherence time).

  • Each speckle presents a diffraction-limited image of the target source.

  • Thus the MTF of each speckle contains high frequency information to the diffraction limit ~ λ / D (solid line in the image below).

    The solid line below then represents the short exposure seeing for a speckle taken through a telescope of diameter D.

From Lena, Observational Astrophysics.

Imagine then that you collect an image from the telescope for much longer than the coherence timescale.

  • This is like averaging over many of the short exposure speckle patterns.

  • We have seen above that the ensemble of speckles will have an envelope given by FWHM ~ λ / ro .

    Thus, the net PSF of the image of a star will have an extent given by this envelope, FWHM ~ λ / ro .

  • As seen in the MTF above, the long exposure seeing profile (dashed line) will now have a cutoff frequency given by ro / λ.

    As the exposure time is lengthened past a fraction of the coherence timescale, higher spatial frequency information begins to be lost (the region of the MTF between the dashed and solid lines).

  • The corresponding comparison for what you see in the focal plane is shown below, with a diffraction-limited image (as one would get from short exposure seeing) compared to an image from long exposure seeing ("seeing disk").

    If integrating over long times compared to the coherence timescale, the resultant image, which is the superposition of all of the speckle patterns during the integration, will have this size for the long exposure seeing (PSF) -- a significant degradation of image quality. From Roy and Clarke's Astronomy: Principles and Practice.

  • The image presented by the telescope with D > ro is that presented by a telescope of diameter ro .

    It is seeing-limited.

    The size of the resulting, degraded image is called the seeing disk.

    In this case, increasing the diameter of the telescope will not increase the resolution (decrease the size of the PSF)!

As we saw in the discussion of atmospheric layers above, much of the seeing is contributed by the surface layer.

  • The improvement in the seeing as one moves away from the Earth surface is shown by the long and short exposure MTFs presented below at 2, 10, 30 and 100 meters above the Earth surface.

    From Coulman, 1985, ARAA, 23, 19.

  • The improvement in the cutoff frequency in the long exposure seeing is obvious.

    There is no increase in the cutoff frequency for the short term seeing (it is the diffraction limit).

    The tendency for the long term MTF/long term seeing to become more like the short term seeing is one rationale for moving telescopes off the ground.

  • Note that the above figure is for daytime conditions when boundary layer convective currents are worse, so the improvement is a bit overstated in the situation for night time conditions.

Net Effects of Scintillation, Image Wander and Image Blur

Below is an enlarged image of the bright star Betelgeuse seen though a large telescope.

Click on the image for a video of the seeing effects.

Note the numerous speckles, the image wander and imagine the net image blur one would see in the long exposure image of this star.

Click on the above image to see "seeing" produced by Earth's atmosphere in a series of quick snapshots. Note that in each snapshot you can see individual speckles images of the star at the resolution limit of the telescope, similar to the cartoon representation above.

A movie borrowed from the website showing the effects of bad seeing on images of the lunar crater Clavius.

Natural Site Seeing

When one discusses "the seeing" one is generally referring to the long exposure FWHM of the PSF.

One can characterize the typical image quality results by compiling statistics on the seeing from night to night.

From Bely, The Design and Construction of Large Telescopes.

Seeing quality from Gemini South (Chile, left) and Gemini North (Mauna Kea, right). From

The seeing statistics can be converted to other useful image quality criteria:
From Bely, The Design and Construction of Large Telescopes.

Many sites rely on special instruments called Differential Image Motion Monitors (DIMMs) to measure their site seeing.

  • DIMMs traditionally measure the differential image motion (which are caused by wavefront slope differences) between stellar images formed between several sub-apertures of a small telescope, and use this to calculate the Fried parameter.

  • Small prisms help to separate the sub-pupil images in the focal plane.

  • Because this is a differential measurement, it is immune to wind shake, earthquakes (!) and errors in tracking.

(Left) A view of the four sub-pupils for a DIMM monitor made by the RoboDIMM company. (Middle) The DIMMs are normally placed above the atmospheric ground layer in a small tower, such as this one by the RoboDIMM company. (Right) A DIMM in place at La Palma (the building on the right, shown on a peak overlooking another telescope).

A sequence of DIMM images of Aldebaran taken in 1.75 arcsec (poor) seeing with a DIMM at the Telescopio Nazionale Galileo on La Palma. The images, and how they are used to reconstruct the seeing, are taken from and described here. The website also shows an image of the dual pupil DIMM used to make the images.

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Unless otherwise attributed, material copyright © 2005,2007,2009,2011,2013,2015,2017 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 511 and Astronomy 5110 at the University of Virginia.