Effective absorbing column density in the gamma-ray burst afterglow X-ray spectra

We investigate the scaling relation between the observed amount of absorption in the X-ray spectra of gamma-ray burst afterglows and the absorber redshift. Through dedicated numerical simulations of an ideal instrument, we establish that this dependence has a power-law shape with index 2.4. However, for real instruments, this value depends on their low-energy cut-off, spectral resolution and on the detector spectral response in general. We thus provide appropriate scaling laws for specific instruments. Finally, we discuss the possibility to measure the absorber redshift from X-ray data alone. We find that 10⁵-10⁶ counts in the 0.3-10 keV band are needed to constrain the redshift with 10 per cent accuracy. As a test case, we discuss the XMM-Newton observation of GRB 090618 at z = 0.54. We are able to recover the correct redshift of this burst with the expected accuracy.