The paper to read is (Tonry and Davis, 1979) and of course the fxcor help page: http://stsdas.stsci.edu/cgi-bin/gethelp.cgi?fxcor

The help page is not useful for the error estimation. It is just saying that it follows the paper. It would be useful to remember the following.

The R statistic

It is using the porperties of the cross-correlation funtion:

\[\epsilion = \frac{k}{1+r}\]

\[r = \frac{h}{\sqrt{2}\sigma_a}\]

Where h is the peak height of the normalized cross-correlation function c(n), \(\sigma_a\) is the rms of its antisymmetric component and l a proportionailty. k empirically was found to be \(k = 0.5 w\). In Iraf I think it is 0.7. Where \(w\) is the FWHM of the cross-correlation peak. \(\epsilion\) is given in pixels. This is then multuply by the Velocity Dispersion given in Km/sec/pixel. I think this factor can be calibrated as mentioned here https://arxiv.org/pdf/0912.4755.pdf

Finally, the FWHM values are related to the velocity dispersion by the creation of a calibration curve. This is detailed in Appendix A. Briefly, this is constructed by correlating the original template spectrum with itself, but convolved with a Gaussian of set width. By doing this several times, a calibration curve can be made, and a calibration function interpolated with it.

TO see the antisymmetric part press a in iraf. The FWHM is given as a sigma^2 in the correalation fit or in the velocity information. This can be seen pressing v in iraf.

This is also mentioned here:

The galaxy spectra were Fourier cross-correlated in fxcor against each standard star in turn. The observed FWHM of the cross-correlation peak was transformed into a value for \(\sigma \)by direct calibration with broadened template spectra, using the preocedure outlined by Baggley (1996).

Some pacgakes use 3./8. like the SAO packae and I have seen to use 0.75 like I think iraf do. FOr example this paper Gaskell and Peterson 1987 he accuracy of cross-correlation estimates of quasar emission-line region sizes

How to calculate the Velocity per pixel dispersion?

from (http://iraf.net/forum/viewtopic.php?showtopic=102972) i Joe, The FXCOR task uses a relativistic redshift equation so the velocity computation is still valid at high z values, namely vel = C * (10 ** (wpc * shift) - 1) = 299792.5 * (10 ** (8.3926e-5 * 1) - 1) = 299792.5 * (1.00019324 - 1) = 299792.5 * 0.00019324 ~= 57.93 km/sThis can also be calculated as dV = wpc * C * ln(10) but I don’t have the derivation handy. If you like I can dig up some mail discussing how this should all be done or I’d be happy to discuss it further, unfortunately I don’t remember many of the details at the moment. Does this answer your question?Regards, Mike Fitzpatrick

The wpc is done from the rebinning with by default is a 5th order polynomial

According to http://aas.aanda.org/articles/aas/full/1999/21/ds8990/node4.html Astron. Astrophys. Suppl. Ser. 140, 327-344 K.R. Müller et al 1999

All spectra were rebinned, using onedspec.dispcor,