ASTRONOMICAL SEEING

'Astronomical seeing' refers to the blurring and twinkling of astronomical objects such as stars caused by turbulence in the Earth's atmosphere. The ''astronomical seeing'' conditions on a given night at a given location describe how much the Earth's atmosphere perturbs the images of stars as seen through a telescope. The most common seeing measurement is the diameter (technically full width at half maximum or FWHM) of the ''seeing disc'' (the point spread function for imaging through the atmosphere). The point spread function diameter (seeing disc diameter or "''seeing''") is a reference to the best possible angular resolution which can be achieved by an optical telescope in a long photographic exposure, and corresponds to the diameter of the fuzzy blob seen when observing a point-like star through the atmosphere. The size of the seeing disc is determined by the ''astronomical seeing'' conditions at the time of the observation. The best conditions give a seeing disk diameter of ~0.4 arcseconds and are found at high-altitude observatories on small islands such as Mauna Kea or La Palma. A detailed description of the seeing disc can be found in the FWHM of the seeing disc subsection of the following article.
Seeing is one of the biggest problems for Earth-based astronomy: while the big telescopes have theoretically milli-arcsecond resolution, the real image will never be better than the average seeing disc during the observation. This can easily mean a factor of 100 between the potential and practical resolution.

Contents

The effects of astronomical seeing

Measures of astronomical seeing

The full width at half maximum (FWHM) of the seeing disc

r0 and t0

Mathematical description of r0 and t0

The Kolmogorov model of turbulence

The CN2 profile

Overcoming atmospheric seeing

See also

References

External links

The effects of astronomical seeing

The FWHM of the seeing disc (or just ''Seeing'') is usually measured in arcseconds, abbreviated with the symbol ("). A 1.0" seeing is a good one for average astronomical sites. The seeing of an urban environment is usually much worse. Good seeing nights tend to be clear, cold nights without wind gusts. Warm air rises (convection) degrading the seeing as does wind and clouds. At the best high-altitude mountaintop observatories the wind brings in stable air which has not previously been in contact with the ground, sometimes providing seeing as good as 0.4".

r₀ and t₀

The astronomical seeing conditions at an observatory can be well described by the parameters r₀ and t₀. For telescopes with diameters smaller than r₀, the resolution of long-exposure images is inversely proportional to the telescope diameter. For telescopes with diameters larger than r₀, the image resolution is independent of telescope diameter, remaining constant at the value given by a telescope of diameter equal to r₀. r₀ also corresponds to the length-scale over which the turbulence becomes significant (10-20 cm at visible wavelengths at good observatories), and t₀ corresponds to the time-scale over which the changes in the turbulence become significant. r₀ determines the spacing of the actuators needed in an active optics system, and t₀ determines the correction speed required to compensate for the effects of the atmosphere.
r₀ and t₀ vary with the wavelength used for the astronomical imaging, allowing slightly higher resolution imaging at longer wavelengths using large telescopes.
r₀ is often known as the ''Fried parameter'' (pronounced freed), named after David L. Fried.

Mathematical description of r₀ and t₀

Mathematical models can give an accurate model of the effects of astronomical seeing on images taken through ground-based telescopes. Three simulated short-exposure images are shown at the right through three different telescope diameters (as negative images to highlight the fainter features more clearly -- a common astronomical convention). The telescope diameters are quoted in terms of the Fried parameter $r\_\{0\}$ (defined below). $r\_\{0\}$ is a commonly used measurement of the astronomical seeing at observatories. At visible wavelengths, $r\_\{0\}$ varies from 20 cm at the best locations to 5 cm at typical sea-level sites.
In reality the pattern of blobs (''speckles'') in the images changes very rapidly, so that long exposure photographs would just show a single large blurred blob in the centre for each telescope diameter. The diameter (FWHM) of the large blurred blob in long exposure images is called the seeing disc diameter, and is independent of the telescope diameter used (as long as adaptive optics correction is not applied).
It is first useful to give a brief overview of the basic theory of optical
propagation through the atmosphere. In the standard classical theory,
light is treated as an oscillation in a field $psi$. For
monochromatic plane waves arriving from a distant point source with
wave-vector $mathbf\{k\}$:
$$
psi_{0} left(mathbf{r},t
ight)
= A_{u}e^{ileft (phi_{u} + 2pi
u t + mathbf{k}cdotmathbf{r}
ight )}

where $psi\_\{0\}$ is the complex field at position $mathbf\{r\}$ and
time $t$, with real and imaginary parts corresponding to the electric
and magnetic field components, $phi\_\{u\}$ represents a phase offset,
$$
u is the frequency of the light determined by $$
u=cleft |
mathbf{k}
ight | / left ( 2 pi
ight ), and $A\_\{u\}$ is the
amplitude of the light.
The photon flux in this case is proportional to the square of the
amplitude $A\_\{u\}$, and the optical phase corresponds to the complex
argument of $psi\_\{0\}$. As wavefronts pass through the Earth's
atmosphere they may be perturbed by refractive index variations in the
atmosphere. The diagram at the top-right of this page shows schematically a turbulent layer in the
Earth's atmosphere perturbing planar wavefronts before they enter a
telescope. The perturbed wavefront $psi\_\{p\}$ may be related at any
given instant to the original planar wavefront $psi\_\{0\}$
left(mathbf{r}
ight) in the following way:
$$
psi_{p} left(mathbf{r}
ight) = left ( chi_{a} left(mathbf{r}
ight)
e^{iphi_{a} left(mathbf{r}
ight)}
ight ) psi_{0} left(mathbf{r}
ight)

where $chi\_\{a\}\; left(mathbf\{r\}$
ight) represents the fractional
change in wavefront amplitude and $phi\_\{a\}\; left(mathbf\{r\}$
ight)
is the change in wavefront phase introduced by the atmosphere. It is
important to emphasise that $chi\_\{a\}\; left(mathbf\{r\}$
ight) and
$phi\_\{a\}\; left(mathbf\{r\}$
ight) describe the effect of the Earth's
atmosphere, and the timescales for any changes in these functions will
be set by the speed of refractive index fluctuations in the atmosphere.

The Kolmogorov model of turbulence

A description of the nature of the wavefront perturbations introduced
by the atmosphere is provided by the ''Kolmogorov model'' developed
by Tatarski (1961), based partly on the studies of turbulence by the
Russian mathematician Andreï Kolmogorov
(see references below by Kolmogorov). This model is supported by a variety of
experimental measurements
(see e.g. references below by Buscher et al 1995, Nightingale and Buscher 1991, O’Byrne 1988, Colavita et al 1987) and is widely used in
simulations of astronomical imaging. The model assumes that the
wavefront perturbations are brought about by variations in the
refractive index of the atmosphere. These refractive index variations
lead directly to phase fluctuations described by $phi\_\{a\}$
left(mathbf{r}
ight), but any amplitude fluctuations are only
brought about as a second-order effect while the perturbed wavefronts
propagate from the perturbing atmospheric layer to the telescope. For
all reasonable models of the Earth's atmosphere at optical and
infra-red wavelengths the instantaneous imaging performance is
dominated by the phase fluctuations $phi\_\{a\}$
left(mathbf{r}
ight). The amplitude fluctuations described by
$chi\_\{a\}\; left(mathbf\{r\}$
ight) have negligible effect on the
structure of the images seen in the focus of a large telescope.
The phase fluctuations in Tatarski's model are usually assumed to have
a Gaussian random distribution with the following second order
structure function:
$$
D_{phi_{a}}left(mathbf{
ho}
ight) = left langle left | phi_{a} left (
mathbf{r}
ight ) - phi_{a} left ( mathbf{r} + mathbf{
ho}
ight )
ight | ^{2}
ight
angle _{mathbf{r}}

where $D\_\{phi\_\{a\}\}\; left\; (\{mathbf\{$
ho}}
ight ) is the
atmospherically induced variance between the phase at two parts of the
wavefront separated by a distance $mathbf\{$
ho} in the aperture
plane, and represents the ensemble average.
The structure function of Tatarski (1961) can be described in terms
of a single parameter $r\_\{0\}$:
:$$
D_{phi_{a}} left ({mathbf{
ho}}
ight )
= 6.88 left ( rac{left | mathbf{
ho}
ight |}{r_{0}}
ight ) ^{5/3}

$r\_\{0\}$ indicates the ''strength'' of the phase fluctuations as it
corresponds to the diameter of a circular telescope aperture at which
atmospheric phase perturbations begin to seriously limit the image
resolution. Typical $r\_\{0\}$ values for I band (900 nm wavelength)
observations at good sites are 20---40 cm. Fried (1965) and
Noll (1976) noted that $r\_\{0\}$ also corresponds to the aperture
diameter for which the variance $sigma\; ^\{2\}$ of the wavefront phase
averaged over the aperture comes approximately to unity:
$$
sigma ^{2}=1.0299 left ( rac{d}{r_{0}}
ight )^{5/3}

This equation represents a commonly used definition for $r\_\{0\}$, a parameter frequently used to describe the atmospheric conditions at astronomical observatories.
$r\_\{0\}$ can be determined from a measured C_N² profile (described below) as follows:
:$r\_\{0\}=left\; (\; 16.7lambda^\{-2\}(\; cos\; gamma\; )^\{-1\}int\_\{0\}^\{infty\}dh\; C\_\{N\}^\{2\}(h)$
ight )^{-3/5}
where the turbulence strength $C\_\{N\}^\{2\}(h)$ varies as a function of height $h$ above the telescope, and $gamma$ is the angular distance of the astronomical source from the zenith (from directly overhead).
The timescale t₀ is simply proportional to r₀ divided by the mean wind speed.

The C_N² profile

A more thorough description of the astronomical seeing at an observatory is given by producing a profile of the turbulence strength as a function of altitude, called a C_N² profile. C_N² profiles are generally performed when deciding on the type of adaptive optics system which will be needed at a particular telescope, or in deciding whether or not a particular location would be a good site for setting up a new astronomical observatory. Typically, several methods are used simultaneously for measuring the C_N² profile and then compared. Some of the most common methods include:
# SCIDAR (imaging the ''shadow patterns'' in the scintillation of starlight)
# LOLAS (a small aperture variant of SCIDAR designed for low-altitude profiling)
# SLODAR
# RADAR mapping of turbulence
# Balloon-borne thermometers to measure how quickly the air temperature is fluctuating with time due to turbulence
There are also mathematical functions describing the C_N² profile. Some are empirical fits from measured data and others attempt to incorporate elements of theory. One common model for continental land masses is known as Hufnagel-Valley after two workers in this subject.

Overcoming atmospheric seeing

The first answer to this problem was speckle imaging, which allowed bright objects to be observed with very high resolution. Later came NASA's Hubble Space Telescope, working outside the atmosphere and thus not have any seeing problems and allowing observations of faint targets for the first time (although with poorer resolution than speckle observations of bright sources from ground-based telescopes because of Hubble's smaller telescope diameter). The highest resolution visible and infrared images currently come from imaging optical interferometers such as the Navy Prototype Optical Interferometer or Cambridge Optical Aperture Synthesis Telescope.
Starting in the 1990s, many telescopes have begun to develop adaptive optics systems that partially solve the seeing problem, but none of the systems so far built or designed completely removes the atmosphere effect, and observations are usually limited to a small region of the sky surrounding relatively bright stars.
Another cheaper technique, Lucky Imaging, has had very good results. This idea dates back to pre-war naked-eye observations of moments of good seeing, which were followed by observations of the planets on cine film after World War II. The technique relies on the fact that every so often the effects of the atmosphere will be negligible, and hence by recording large numbers of images in real-time, a 'lucky' excellent image can be picked out. This technique can outperform adaptive optics in many cases and is even accessible to amateurs. It does, however, require a longer observation time than adaptive optics for imaging faint targets.

See also

★ Transient lunar phenomenon

★ Mirage

References

Much of the above text is taken (with permission) from ''Lucky Exposures: Diffraction limited astronomical imaging through the atmosphere'', by Robert Nigel Tubbs

★ Interferometric seeing measurements on Mt. Wilson: power spectra and outer scales, , D. F., BUSCHER, Applied Optics, 1995

★ Atmospheric phase measurements with the Mark III stellar interferometer, , M. M., COLAVITA, Applied Optics, 1987

★ Statistics of a Geometric Representation of Wavefront Distortion, , D. L., FRIED, Optical Society of America Journal, 1965

★ Dissipation of energy in the locally isotropic turbulence, , A. N., KOLMOGOROV, Comptes rendus (Doklady) de l'Académie des Sciences de l'U.R.S.S., 1941

★ The local structure of turbulence in incompressible viscous fluid for very large Reynold's numbers, , A. N., KOLMOGOROV, Comptes rendus (Doklady) de l'Académie des Sciences de l'U.R.S.S., 1941

★ Interferometric seeing measurements at the La Palma Observatory, , N. S., NIGHTINGALE, Monthly Notices of the Royal Astronomical Society, 1991

★ Zernike polynomials and atmospheric turbulence, , R. J., NOLL, Optical Society of America Journal, 1976

★ Seeing measurements using a shearing interferometer, , J. W., O'BYRNE, Publications of the Astronomical Society of the Pacific, 1988

★ Wave Propagation in a Turbulent Medium, , V. I., TATARSKI, McGraw-Hill Books, 1961,

External links

★ Seeing forecasts for North America

★ Seeing forecasts for Mauna Kea, Hawaii