Exploring relations between BCG and cluster properties in the SPectroscopic IDentification of eROSITA Sources survey from 0.05

We present a sample of 329 low to intermediate redshift ($0.05 < z < 0.3$) brightest cluster galaxies (BCGs) in X-ray selected clusters from the SPectroscopic IDentification of eRosita Sources (SPIDERS) survey, a spectroscopic survey within Sloan Digital Sky Survey-IV (SDSS-IV). We define our BCGs by simultaneous consideration of legacy X-ray data from ROSAT, maximum likelihood outputs from an optical cluster-finder algorithm and visual inspection. Using SDSS imaging data, we fit S\'ersic profiles to our BCGs in three bands (\textit{g}, \textit{r}, \textit{i}) with \textsc{SIGMA}, a \textsc{GALFIT}-based software wrapper. We examine the reliability of our fits by running our pipeline on ${\sim}10^{4}$ psf-convolved model profiles injected into 8 random cluster fields, we then use the results of this analysis to create a robust subsample of 198 BCGs. We outline three cluster properties of interest: overall cluster X-ray luminosity ($L_{X}$), cluster richness as estimated by \textsc{redMaPPer} ($ \lambda $) and cluster halo mass ($M_{200}$), which is estimated via velocity dispersion. In general, there are significant correlations with BCG stellar mass between all three environmental properties, but no significant trends arise with either S\'ersic index or effective radius. There is no major environmental dependence on the strength of the relation between effective radius and BCG stellar mass. Stellar mass therefore arises as the most important factor governing BCG morphology. Our results indicate that our sample consists of a large number of relaxed, mature clusters containing broadly homogeneous BCGs up to $z \sim 0.3$, suggesting that there is little evidence for much ongoing structural evolution for BCGs in these systems.