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ASTR 3130, Majewski [SPRING 2015]. Lecture Notes

ASTR 3130 (Majewski) Lecture Notes


REFERENCE: Chapter 6, Birney, Gonzalez & Oesper.

  • Another important function of telescopes is to provide increased angular resolution, which means reducing the angular size of he smallest measurable detail.

  • The resolution of a telescope is related to the diffraction pattern created by the objective.

  • Recall the diffraction from a slit of width a :

    • By Huygen's Principle, we can consider the slit as a series of new individual sources, S1, S2, ... SN, each propagating new wavefronts. For now consider them to be equally spaced (though this is not necessary).

    • Now imagine a source S1, that has waves that are 180° out of phase with a source Sc at the center of the slit at angle θ; this happens when the path difference is λ/2. Thus, we find that there is a minimum in the interference of S1 and Sc in the direction of θmin, where:

    • By analogy, the hypothesized S2 source will destructively interfere with the source Sc+1 at the same angle θ, S3 will then destructively interfere with Sc+2, etc.; then the entire wavefront will be diffracted to a minimum at .

    • Note that the above thought experiment concerns the first minimum from center in the interference pattern, and other minima occur that correspond to path length differences of 3λ/2, 5λ/2, ..., where we have similar relations that hold regarding the locations of minima:

  • A similar, but slightly more detailed, analysis allows us to calculate the relative amplitude of the total wavefront when viewed at any other angle, and we find:

    Because we observe the intensity, which is given by the square of the amplitude of the field, then:

    If we put up a screen to intercept diffracted wavefronts, we see the interference pattern, I, as a set of fringes:

    The minima occur at, so that if the aperture a is larger, the is smaller.

    Thus, the larger the slit or aperture, the narrower the central fringe and the closer together in angle are the dark and bright fringes.

    DEMO: Laser / expanding slit width

  • Now, in reality we use circular (i.e., 2-dimensional) apertures in astronomy (i.e. objective of telescope) rather than 1-dimensional slits.

  • A similar, but more complicated 2-D analysis yields a radially symmetric analogue of the 1-D fringe patterns, given by:

    where J1(2m) is a Bessel function of the first kind of order unity (Bessel functions commonly show up in 2-D circular analyses, e.g. waves on a drumhead).

    Amplitude of waves (radially) in first order from Bessel function of first kind.

    Intensity (amplitude2) of diffracted waves shown above reveals fringes in circular pattern

  • The fringe pattern from a circular aperture is called an "Airy pattern".

  • The central bright spot (fringe) is the "Airy disk".

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

  • The minima for the Airy pattern actually occur at:

  • Compare these to the fringe minima for the 1-d slit:

  • We normally call the "size" of the Airy disk as that diameter across the Airy disk from minimum to minimum, or:

    Remember that for small angles (in radians):

  • In Lab 2, one experiment you will do is to see the change in the Airy disk as you change the effective aperture of a telescope.

  • NOTE: it is sometimes hard to see a minimum of light, and it is much easier to judge the location of the first bright Airy ring. The first bright Airy ring has radius:

  • In Lab 2, you will also measure the sizes of bright Airy rings as well as the Airy disk.


  • If we observe two point sources separated in the sky by an angle , you get a superposition of their Airy disks:

    If the two point sources are very well separated, there is not much overlap of their Airy patterns, and the sources are easily "resolved", as in the left pink figure below.

  • Now imagine the angle between the sources to be small enough that the peak of one Airy disk is in the first Airy minimum of the other (and vice versa), as seen in the middle image below:


    In this case, the individual Airy disks are well blended and in principle can just barely be resolved individually -- this is Rayleigh's criterion for the best theoretical resolution of a telescope:

    = 1.22 λ / d

    Point sources separated smaller than this angle are in principle unresolvable.

    For yellow light (5500 Angstroms) and a telescope diameter, d, Rayleigh's criterion translates to:

    α = 140 / d(mm) = 5.5 / d(inches)

  • Rayleigh's criterion gives the theoretical limiting resolution a telescope - and obviously, this resolution is better for larger d.
    • Clark 6" telescope = 0.92"
    • Space telescope = 0.056"
    • 10-m Keck telescope = 0.014"
  • In Lab 2, you will attempt the Rayleigh test on closely spaced double stars.
  • Note that skilled observers can resolve stars that are in fact a bit closer than the theoretical resolving power as defined by the Rayleigh criterion.

    • If you see the image corresponding to Rayleigh's criterion above, you notice that there is in fact a small dip between the two peaks at the 20% level.

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

    • To the experienced observer, the peaks corresponding to the two point sources in this situation are distinguishable and together will resemble a "dumbbell", as shown in the righthand image below.

      From .

    • This fine discriminatory power has been empirically gauged, and gives rise to Dawes' criterion for the absolute limit of perceptible resolution, given roughly by:

      α = 115 / d(mm) = 4.5 / d(inches)

      • Clark 6" telescope = 0.76"
      • Space telescope = 0.046"
      • 10-m Keck telescope = 0.012"


  • Aside: We can use interferometry to increase d (and therefore improve resolution) in a "cheap" way, by coherently combining the light from multiple collecting apertures, which together act as one aperture of a larger diameter (in a resolution sense).
  • Can dramatically increase resolution with interferometry much more easily than can dramatically increase light gathering power (latter requires more surface area whereas resolution only requires large spacing of individual parts of the receiving aperture).
  • EXAMPLE: Large Binocular Telescope (UVa's largest telescope).

    Twin 8.4-m mirrors working together:

    • Light gathering power equivalent to single 11.8-m.

    • Resolving power equivalent of a single 22.8-m.


REFERENCE: Birney, Gonzalez & Oesper, pp. 136-143. Roy & Clarke, section 19.7.3. Kaler, section 13.23.

The theoretical limiting resolution of a (ground-based) telescope is not always (generally is not) achieved. The primary reason is the blurring effects of seeing.

  • Because of variations in temperature and density, the Earth's atmosphere has variations in index of refraction.

  • Because of turbulence, the atmosphere develops eddies and pockets that corrugate the boundaries of layers with different indices of refraction and this, in turn, has the effect of corrugating the incoming wavefronts, as if the atmosphere was made up of a series of moving lenses (or cells) of different power.

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

  • The typical cell size, called the Fried parameter, or r0 , is about 10 cm in size (note, this is a diameter, not a radius) and is at a typical distance of about 7 km above sea level (though turbulence at all levels contributes).

    From Bely, The Design and Construction of Large Telescopes.

    • When the atmosphere is very turbulent, the Fried parameter is smaller, so that the wavefronts are much more corrugated.

      When things are calm, r0 is larger, and the wavefronts are smoother and less corrugated.

    • Note, as shown in the above figure, anything within the angle subtended by one cell of size the Fried parameter is brought to the same focus.

      This angle is called the isoplanatic angle or isoplanatic patch, and is on the order of a few arcseconds for r0 = 10 cm and distance about 7 km.

      The size of the patch is actually larger for longer wavelengths and smaller for shorter wavelengths.

      Better observing sites have larger Fried cells/isoplanatic patches than poorer observing sites.

  • Because the atmosphere of Earth is constantly moving, the distortions are making random corrugations and distortions of point source wavefronts on the timescale of 10s of Hz.
  • There are basically three results of this churning motion:


    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 (like a telescope or a human eye).

    • Scintillation only is obvious when the aperture/pupil diameter is of order r 0 or less.

      E.g., the human eye pupil is generally always less than r 0.

      A telescope of order 10 cm = 4 inches will still have scintillation effects.

      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 (Agitation)

    Another obvious effect for apertures smaller than r 0 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 passes through the line of sight.

      Thus, a sharp Airy disk -- with central FWHM ~ λ / D -- can be formed from each r 0 cell, and if the telescope aperture is smaller than r 0, 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.

    • Thus, to instantaneously achieve the ultimate, diffraction-limited performance of a telescope, one only needs to ensure that the telescope aperture be smaller than r 0 (by about a factor of 1.6). (Of course, the diffraction limit of a smaller telescope is a broader Airy pattern -- but that Airy pattern will be the imaging limit in this case.)

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

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

      ... or a tip-tilt mirror system to correct out the wavefront tilts.

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

      Tip-tilt systems are now being sold for small amateur telescopes to remove image wander.

      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 > r 0), 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 (but of course, smaller telescopes have larger Airy disks).

    • 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, and each is an individual image of the star from a different isoplanatic patch.

      Seen together, the speckles give a shimmering blur.

      In this case, the ensemble of speckles will have an envelope given by FWHM ~ λ / r 0.

      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.

      Speckles of the star Vega photographed in the groundbreaking experiments by Labeyrie on the 200-inch telescope in the 1970s. Each speckle is a diffraction-limited image of the star.

    Speckle images of double stars from WIYN telescope, from Matthew Hoffman's webpage, Note how the speckle patterns look identical in each double star, showing that the images of the double star are passing through the same Fried cells.

    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.

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

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

  • Because of "SEEING" we rarely achieve diffraction limit of (large) telescopes but resolution instead limited by atmosphere.
    • Best Earth seeing has a seeing disk size ~ 0.5"
      • Typically good sites with laminar airflow.
      • Telescopes typically placed in foothills of coastal ranges where prevailing winds are from sea to land.
      • For example: California, Baja, Chile, Hawaii, Canary Islands.
    • Typical seeing in C-ville ~ 1-3"
  • If the telescope diameter is of order or slightly larger than r 0, will see several speckles, but occasionally can chance upon a moment when the seeing turns very good (r 0 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.

    Roy and Clarke (Section 19.7) calculate the frequency with which such good moments would occur depending on telescope aperture (happens with lower frequency as telescope diameter increases).

    Individual video frames of the Russian Cosmos 1076 satellite (top) taken by Ron Dantowitz using a 12-inch Meade Schmidt-Cassegrain and a video-camera, as presented in the August 1998 issue of Sky & Telescope. The best video frames were coadded to produce the rightmost image, which shows the satellite structure.

    (left) A single frame of the Space Shuttle Atlantis in orbit (left), but a composite image (right) was assembled from selected, high resolution pieces of frames (center). The images were actually taken in daytime. (right) The Shuttle docked at the Russian Mir station in 1996.
  • In general, increasing the telescope aperture will collect more light (i.e. more speckles), but will not improve the resolution (PSF).

  • A strategy for taking advantage of the speckles to recover the theoretical diffraction limit of a large aperture telescope is called speckle interferometry.

  • 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.

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

    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.


    Modern astronomy is developing ways to overcome seeing effects (apart from sending spacecraft above the atmosphere).

    Active Optics

    • A technique that removes image wander (from seeing or from movement of telescope, e.g., from wind buffeting, and correcting on 1-10 Hz timescales.

    • Done primarily with a single, flat, wobbling mirror put into the beam that exactly compensates for motion.

    • Requires picking off part of the focal plane to send to an image analyzer that calculates the instantaneous position of one bright star that is monitored.

      From Bely, The Design and Construction of Large Optical Telescopes.

    • Sometimes have some additional broad compensation for wavefront direction/shape using actuators on back of primary mirror (as shown above).

    • Some amateur CCD cameras are now being sold with tip-tilt capability!

    Adaptive Optics
    • Much more complicated/technically challenging.

    • Uses deformable/shapeable optics ("rubber mirrors") to counteract corrugations in wavefronts and working on few to 1000 Hz timescales.

      From Hecht, Optics.

    • After early work by astronomers (first theoretical paper in 1953), astronomers and Department of Defense were working independently in the 1970s to develop adaptive optics.

      The technology was so advanced and expensive that DOD got there first.

    • Results of this work was discouraging at first:

      • Everyone discovered how expensive this was going to be.

      • Also, it was found that one could only apply the corrections to small isoplanatic angles around bright reference stars on angular scales of 10s of arcseconds.

        Thus, only small fields of view could be corrected, and these fields had to be around bright stars.

    • But more recently a resurgence due to several developments:

      • Idea of synthetic, laser reference stars proposed by military in 1982, prototypes in 1984, declassified in 1991.

        Allows one to "make" a "star" anywhere!

        Until the early 1990s, the Starfire Optical Range at Kirtland Air Force Base in Albuquerque, New Mexico, was one of the U.S. Air Force's most closely guarded secrets. Here, laser beams probe the atmosphere above the system's 1.5-m telescope, allowing minute computer-controlled changes to be made to the mirror surface thousands of times each second. (b) The improvement in seeing produced by such systems can be dramatic, as can be seen in these images acquired at another military observatory atop Mount Haleakala in Maui, Hawaii, employing similar technology. The uncorrected image (left) of the double star Castor is a blur spread over several arc seconds, with little hint of its binary nature. With adaptive compensation applied (right), the resolution is improved to a mere 0.10 and the two components are clearly resolved. (R. Ressmeyer; MIT Lincoln Laboratory). Image and caption from

        Laser guide stars were independently proposed by Labeyrie & Foy in 1985.

      • Better detectors (CCDs) were invented, increasing their sensitivity, which means shorter exposures could be taken and so the system can work faster.

      • Scientists realized that adaptive optics work better at near0infrared wavelengths because r 0 is proportional to λ 6/5.

        • This means that in the near-infrared, the number of required adaptive elements decreases.

          Also means that the required temporal control frequencies decrease --- easier to do the compensations.

        • Array detectors that work in the infrared were developed.

        • The telescope diffraction limit is approachable at NIR wavelengths (Strehl ratios approaching 1).

          The Strehl ratio is the observed intensity of the center of the point-spread-function (PSF) to that expected from the theoretical limit of the Airy function.

          • This can be as poor as a few percent in the case of an uncompensated star image.

          • Adaptive optics can sometimes bring results as good as Streahl ratio > 0.5.

      • It was found that better sites with better seeing and the reduction of local seeing (around the telescope and dome) decrease the complexities of the process even more.

      Now many observatories are implementing adaptive optics systems, many with laser guide star components.

    An animation showing a set of images of a star observed through turbulent atmosphere without any correction, with only tip-tilt correction with a fast steering mirror, and closed loop adaptive optics with a deformable mirror. From the Max Planck Institute for Astrophysics webpage: AO/INTRO/AOWFSintro.html.

    An image made by summing together the frames from the first third of the above animation (no seeing compensation) and showing the "long exposure" image, which is defined by the size of the speckle envelope. By Guillermo Damke.



    The practical resolution of an optical system is also affected by the limitations of the smallest "bits" of information allowed to be "seen" / resolved by the detector.
    • The ability of a detector to record fine detail is a function of the size of the individually acting photoreceptors (e.g. grains of film, pixels of camera, neurons of eye).
    • CCD pixel (in this class) = 0.024 mm

      Photographic grain = 0.003 - 0.010 mm
    To translate these smallest linear-sized bits into angles requires us to understand how angles in sky are translated to physical sizes in the focal plane.

    • This is called the plate scale of the system.
    • Typical units are (" / mm).
    • Once you know the size of your pixels (in mm), can translate this to (" / pixel).


    Note that the linear size of an image on the focal plane is given by:

    y = Fo sin() ~ Fo()

    where is in radians if you use the skinny triangle formula.

    The scale or plate scale of an image is the angular size on the sky projected to each millimeter. Thus, based on the figure above:

    scale = / y = / ( Fo ) = 1 / Fo

    So, the bigger the focal length, the bigger the image at the focal plane

    -- but note, when the light spreads over larger area --> fainter.

    • Telescopes for faint imaging --> small Fo , big scale, brighter image.
    • Telescopes for accurate positions --> larger Fo , small plate scale, easier to measure positions accurately.
    Note that to put into standard arcseconds/mm units, have to convert plate scale from radians / mm by noting that there are 206265 arcsec / radian.

    • Thus, the plate scale in arcsec/mm is given by 206265 / F(mm).

    • Example: 6-inch, f/12 Clark refractor has Fo = 1830 mm.

      plate scale = (1 radian)/(1830 mm) = 5.46 X 10-4 radian / mm

      plate scale = [(1 radian)/(1830 mm)][(206265 arcsec)/radian] = 112.7 arcsec / mm

    Aside: UVa traditionally interested in astrometry, the measurement of accurate positions of stars on the sky (for parallaxes and proper motions of stars).

    • 26-inch Clark refractor, f/14.9;

      focal length 9.93 m;

      plate scale 20.75 arcsec / mm.

    • 1-m Fan Mountain Observatory, f/13.5;

      focal length 13.8 m;

      plate scale = 15.0 arcsec / mm.


    In modern astronomy we typically use detectors to record the images in the focal plane.

    These detectors have individual photoreceptor elements with sizes typically as:

    • optical CCD detectors pixel sizes: 9-30 µ = 0.009-0.030 mm
    • photographic emulsion grain ~ 3-10 µ = 0.003-0.010 mm

    Thus, we can determine the pixel scale (arcsec/pixel) of these devices as the amount of angle subtended by each photoreceptor in the focal plane, given as:

    pixel scale = (pixel size)(plate scale)

    Often we need to consider the balance between the pixel scale and the seeing scale, which we want to match well.

    • In general, we wish to have the pixels mapped to a size whereby they are about two times less than the expected image resolution delivered by the optics + atmosphere.
    • This rule of "two pixels per resolution element" derives from the Nyquist sampling theorem, which says that optimally sampling all of the information contained in an image requires about two pixels per resolution element.
      • Sampling the resolution finer than this does not yield you more information and can be considered "wasteful".
      • Sampling more coarsely means you are not sensitive to all of the find detail in the picture and you are losing information.
      • See here for a demonstration of how the sampling limit comes about.
    • Example 1: Imagine we place the ST-8 CCD camera (with 9 µ pixels) sometimes used in this course on the Clark 6-inch telescope:

      pixel scale = (0.009 mm/pixel)(112.7 arcsec/mm) = 1.01 arcsec/pixel
      • In this case, if the seeing is 2 arcseconds, the pixels in the CCD camera are a good match to the resolution and we can sample all of the information delivered to the focal plane.
      • However, should the seeing drop to 1 arcsecond, the pixels in the camera would be too big and we would lose information (not Nyquist sample); this is called undersampling and the image we would collect would be pixel-limited.
      • If the seeing ballooned up to 5 arcseconds, the 1 arcsecond pixels would be overkill, since we would be oversampling the delivered resolution.

        In this case, the resolution is seeing-limited .
    • Example 2: Imagine putting the same ST-8 CCD camera (with 9 µ pixels) on the 26-inch Clark refractor, which has a 20.75 arcsec/mm pixel scale:

      pixel scale = (0.009 mm/pixel)(20.75 arcsec/mm) = 0.19 arcsec/pixel

      • As you can see, under almost any typical seeing conditions at McCormick Observatory, the pixels will oversample the seeing.
      • Unfortunately, it is not possible to change the physical pixel scale of our detector. Instead -- if possible -- we generally design the optics of our instruments with the expected typical seeing and the expected detector pixel sizes in mind.
      • Such calculations do not always stand the test of time as prevailing seeing conditions and detector technology changes (as was the case for the McCormick telescopes, designed for human eye detectors!).
    • Example 3: Space Telescope, with 58-m focal length has plate scale of about 4 arcsec/mm.
      • No atmospheric seeing in space, so can achieve theoretical resolution limit, or

        1.22 (5500 Angstroms)(206265 arcsec/radian)/(2.4-m) = 0.05 arcsec.
      • Hubble Space Telescope cameras are actually undersampled.

        For example, the 15 micron pixels in the old WF/PC2 camera gave either 0.05 arcsec/pixel (1 chip) or 0.10 arcsec/pixel (3 chips) -- so not Nyquist sampled.
      • In this case the decision to not sample to the limit was dictated by desire to have a reasonable FIELD OF VIEW.

        • The old detectors with 800 x 800 pixel sizes gave only an 80 arcsecond FOV at 0.10 arcsec/pixel.

    SUMMARY: What affects resolution?

    In reality stars have true angular sizes as seen from the Earth << 1 arcsec.

    We typically do not achieve anything near this kind of resolution due to several limiting factors (in order of usual importance):

    1. Atmosphere of Earth

      • Typical seeing is ~ 0.5-5 arcsec.
      • On Earth, we are usually in the seeing-limited resolution domain.
    2. Pixel size

      • Normally design instruments to select pixel sizes optimized to expected typical seeing (Nyquist sample).
      • For imaging at most sites, pixel scales of about 0.3-0.6 arcsec/pixel typical.
      • Typical professional CCD pixels about 21 µ = 0.021 mm.
      • Oversampling generally better than undersampling to avoid being pixel-limited and losing detail (but at a cost of image area for a given detector).
    3. Theoretical limit of telescope objective

      • Physical limit due to diffraction by aperture.
      • Goal of Earth-bound astronomers is to reach the theoretical resolution limit of their telescopes.

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    Twinkle image is copied from Nick Strobel's Astronomy Notes. Go to his site at for the updated and corrected version. The lens and seeing image is from Marta Lewandowska's web pages for Nassau Station, Case Western Reserve University. All material copyright © 2002,2006,2008,2012,2015 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 313 and Astronomy 3130 at the University of Virginia.