What does seeing cause the image to look like?

The ``quality'' of an image can be described in many different ways. The overall shape of the distribution of light from a point source is specified by the point spread function. Diffraction gives a basic limit to the quality of the PSF, but any aberrations or image motion add to structure/broadening of the PSF.

Another way of describing the quality of an image is to specify it's modulation transfer function (MTF). The MTF and PSF are a Fourier transform pair. Turbulence theory gives:

$$MTF(\nu) = \exp [-3.44 (\lambda\nu/r_o)^{5/3}]$$

where $\nu$ is the spatial frequency.

Note that a gaussian goes as $\nu^2$, so this is close to a gaussian. The shape of seeing-limited images is roughly Gaussian in core but has more extended wings Empirically, seeing may be fit with a gaussian or, better, a Moffat function :

$$I = p_1 (1 + (x-p_2)^2/p_4^2 + (y-p_3)^2/p_5^2) ^{-p_6}$$

Another way of characterizing the PSF is by giving the encircled energy as a function of radius, or at some specified radius.

A final way of characterizing the image quality, more commonly used in adaptive optics applications, is the Strehl ratio. The Strehl ratio is the ratio between the peak amplitude of the PSF and the peak amplitude expected in the presence of diffraction only.

Measuring the Seeing

Typically seeing is characterized by the FWHM. Usually the implication is that the form is Gaussian with sigma = FWHM /2.354 . Recall that FWHM doesn't fully specify a PSF, and one should always consider how applicable the quantity is.

Options: measure FWHM directly; fit a Gaussian or other function (we'll talk later about fitting).

However, beware of effects of sampling of the PSF: you're really getting the PSF integrated over pixels, not the PSF!