Mathematical description of r₀ and t₀

Simulated negative image showing what a single (point-like) star would look like through a ground-based telescope with a diameter of 2r₀. The blurred look of the image is because of diffraction, which causes the appearance of the star to be an Airy pattern with a central disk surrounded by hints of faint rings. The atmosphere would make the image move around very rapidly, so that in a long-exposure photograph it would appear more blurred.

Simulated negative image showing what a single (point-like) star would look like through a ground-based telescope with a diameter of 7r₀, on the same angular scale as the 2r₀ image above. The atmosphere makes the image break up into several blobs (speckles). The speckles move around very rapidly, so that in a long-exposure photograph the star would appear as a single blurred blob.

Simulated negative image showing what a single (point-like) star would look like through a ground-based telescope with a diameter of 20r₀. The atmosphere causes further atomization of the image into many blobs (speckles). As above, the speckles move around very rapidly, so that in a long-exposure photograph the star would appear as a single blurred blob.

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 ${\displaystyle r_{0}}$ (defined below). ${\displaystyle r_{0}}$ is a commonly used measurement of the astronomical seeing at observatories. At visible wavelengths, ${\displaystyle r_{0}}$ 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 center 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 ${\displaystyle \psi }$. For monochromatic plane waves arriving from a distant point source with wave-vector ${\displaystyle \mathbf {k} }$:

$${\displaystyle \psi _{0}\left(\mathbf {r} ,t\right)=A_{u}e^{i\left(\phi _{u}+2\pi \nu t+\mathbf {k} \cdot \mathbf {r} \right)}}$$

where ${\displaystyle \psi _{0}}$ is the complex field at position ${\displaystyle \mathbf {r} }$ and time ${\displaystyle t}$, with real and imaginary parts corresponding to the electric and magnetic field components, ${\displaystyle \phi _{u}}$ represents a phase offset, ${\displaystyle \nu }$ is the frequency of the light determined by ${\textstyle \nu ={\frac {1}{\pi }}c\left|\mathbf {k} \right|}$, and ${\displaystyle A_{u}}$ is the amplitude of the light.

The photon flux in this case is proportional to the square of the amplitude ${\displaystyle A_{u}}$, and the optical phase corresponds to the complex argument of ${\displaystyle \psi _{0}}$. 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 ${\displaystyle \psi _{p}}$ may be related at any given instant to the original planar wavefront ${\displaystyle \psi _{0}\left(\mathbf {r} \right)}$ in the following way:

$${\displaystyle \psi _{p}\left(\mathbf {r} \right)=\left(\chi _{a}\left(\mathbf {r} \right)e^{i\phi _{a}\left(\mathbf {r} \right)}\right)\psi _{0}\left(\mathbf {r} \right)}$$

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

The Kolmogorov model of turbulence

A description of the nature of the wavefront perturbations introduced by the atmosphere is provided by the Kolmogorov model developed by Tatarski,^[2] based partly on the studies of turbulence by the Russian mathematician Andrey Kolmogorov.^[3][4] This model is supported by a variety of experimental measurements^[5] and is widely used in simulations of astronomical imaging. The model assumes that the wavefront perturbations are brought about by variations in the refractive index of the atmosphere. These refractive index variations lead directly to phase fluctuations described by ${\displaystyle \phi _{a}\left(\mathbf {r} \right)}$, 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 infrared wavelengths the instantaneous imaging performance is dominated by the phase fluctuations ${\displaystyle \phi _{a}\left(\mathbf {r} \right)}$. The amplitude fluctuations described by ${\displaystyle \chi _{a}\left(\mathbf {r} \right)}$ have negligible effect on the structure of the images seen in the focus of a large telescope.

For simplicity, the phase fluctuations in Tatarski's model are often assumed to have a Gaussian random distribution with the following second-order structure function:

$${\displaystyle D_{\phi _{a}}\left(\mathbf {\rho } \right)=\left\langle \left|\phi _{a}\left(\mathbf {r} \right)-\phi _{a}\left(\mathbf {r} +\mathbf {\rho } \right)\right|^{2}\right\rangle _{\mathbf {r} }}$$

where ${\displaystyle D_{\phi _{a}}\left({\mathbf {\rho } }\right)}$ is the atmospherically induced variance between the phase at two parts of the wavefront separated by a distance ${\displaystyle {\boldsymbol {\rho }}}$ in the aperture plane, and ${\displaystyle \langle \cdot \rangle }$ represents the ensemble average.

For the Gaussian random approximation, the structure function of Tatarski (1961) can be described in terms of a single parameter ${\displaystyle r_{0}}$:

$${\displaystyle D_{\phi _{a}}\left({\mathbf {\rho } }\right)=6.88\left({\frac {\left|\mathbf {\rho } \right|}{r_{0}}}\right)^{5/3}}$$

${\displaystyle r_{0}}$ 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 ${\displaystyle r_{0}}$ values for I band (900 nm wavelength) observations at good sites are 20–40 cm. ${\displaystyle r_{0}}$ also corresponds to the aperture diameter for which the variance ${\displaystyle \sigma ^{2}}$ of the wavefront phase averaged over the aperture comes approximately to unity:^[6]

$${\displaystyle \sigma ^{2}=1.0299\left({\frac {d}{r_{0}}}\right)^{5/3}}$$

This equation represents a commonly used definition for ${\displaystyle r_{0}}$, a parameter frequently used to describe the atmospheric conditions at astronomical observatories.

${\displaystyle r_{0}}$ can be determined from a measured C_N² profile (described below) as follows:

$${\displaystyle r_{0}=\left(16.7\lambda ^{-2}(\cos \gamma )^{-1}\int _{0}^{\infty }C_{N}^{2}(h)dh\right)^{-3/5}}$$

where the turbulence strength ${\displaystyle C_{N}^{2}(h)}$ varies as a function of height ${\displaystyle h}$ above the telescope, and ${\displaystyle \gamma }$ is the angular distance of the astronomical source from the zenith (from directly overhead).

If turbulent evolution is assumed to occur on slow timescales, then the timescale t₀ is simply proportional to r₀ divided by the mean wind speed.

The refractive index fluctuations caused by Gaussian random turbulence can be simulated using the following algorithm:^[7]

$${\displaystyle \phi _{a}(\mathbf {r} )={\mbox{Re}}[{\mbox{FT}}[R(\mathbf {k} )K(\mathbf {k} )]]}$$

where ${\displaystyle \phi _{a}(\mathbf {r} )}$ is the optical phase error introduced by atmospheric turbulence, R (k) is a two-dimensional square array of independent random complex numbers which have a Gaussian distribution about zero and white noise spectrum, K (k) is the (real) Fourier amplitude expected from the Kolmogorov (or Von Karman) spectrum, Re[] represents taking the real part, and FT[] represents a discrete Fourier transform of the resulting two-dimensional square array (typically an FFT).

Astronomical observatories are generally situated on mountaintops, as the air at ground level is usually more convective. A light wind bringing stable air from high above the clouds and ocean generally provides the best seeing conditions (telescope shown: NOT).

Turbulent intermittency

The assumption that the phase fluctuations in Tatarski's model have a Gaussian random distribution is usually unrealistic. In reality, turbulence exhibits intermittency.^[8]

These fluctuations in the turbulence strength can be straightforwardly simulated as follows:^[9]

$${\displaystyle \phi _{a}(\mathbf {r} )=\operatorname {Re} [{\mbox{FT}}[(R(\mathbf {k} )\otimes I(\mathbf {k} ))K(\mathbf {k} )]]}$$

where I(k) is a two-dimensional array which represents the spectrum of intermittency, with the same dimensions as R(k), and where ${\displaystyle \otimes }$ represents convolution. The intermittency is described in terms of fluctuations in the turbulence strength ${\displaystyle C_{n}^{2}}$. It can be seen that the equation for the Gaussian random case above is just the special case from this equation with:

$${\displaystyle I(k)=\delta (|k|)}$$

where ${\displaystyle \delta ()}$ is the Dirac delta function.