Fitting trends in quasar emission and absorption line redshifts

The spectrum of a quasar consists of a few emission lines whose wavelengths are shifted by similar redshifts and numerous absorption lines whose wavelengths are shifted by different redshifts. Hence each quasar is characterised by an emission line redshift and the absorption lines redshifts are all less than the emission line redshift. The distribution of observed absorption line redshifts ($z_{abs}$) with respect to emission line redshift ($z_{em}$) for a large sample of quasars shows a systematic trend as pointed out by \citet{2016arXiv160901593K}. They noticed that increase in $z_{em}$ is accompanied by a monotonic increase in the lowest detected value of $z_{abs}$ and inferred that the emission and absorption lines were all formed in the quasar. This study focuses on modeling the systematic trend in the observed $z_{em} \rightarrow z_{abs}$ distribution. We considered the redshift data of absorption lines of singly ionized magnesium (denoted by MgII) and triply ionized carbon (denoted by CIV) for a large sample of quasars. We find that the envelope of data points defining the lowest value of the MgII absorption line redshift (which we denote by $z_{MgIImodel}$) for a given $z_{em}$ satisfies $z_{MgIImodel} = (0.418 \pm 0.008) z_{em} - (0.482 \pm 0.02)$ with an $R^2$ value of 0.99. The model can be used to predict the lowest expected MgII absorption line redshift for any $z_{em}$. We find a similar model for the lowest expected redshift of triply ionized carbon lines for any $z_{em}$ which is $z_{CIVmodel} = 0.845 (\pm 0.0002) z_{em} - 0.153 (\pm 0.0006)$.