On the link between nuclear star cluster and globular cluster system mass, nucleation fraction and environment
We present a simple model for the host mass dependence of the galaxy nucleation fraction ($f_{nuc}$), the galaxy's nuclear star cluster (NSC) mass and the mass in its surviving globular clusters ($M_{GC,obs}$). Considering the mass and orbital evolution of a GC in a galaxy potential, we define a critical mass limit ($M_{GC,lim}$) above which a GC can simultaneously in-spiral to the galaxy centre due to dynamical friction and survive tidal dissolution, to build up the NSC. The analytic expression for this threshold mass allows us to model the nucleation fraction for populations of galaxies. We find that the slope and curvature of the initial galaxy size-mass relation is the most important factor (with the shape of the GC mass function a secondary effect) setting the fraction of galaxies that are nucleated at a given mass. The well defined skew-normal $f_{nuc} - M_{gal}$ observations in galaxy cluster populations are naturally reproduced in these models, provided there is an inflection in the {initial} size-mass relation at $M_{gal} \sim 10^{9.5} {\rm M_{\odot}}$. Our analytic model also predicts limits to the $M_{gal} - M_{GC,tot}$ and $M_{gal} - M_{NSC}$ relations which bound the scatter of the observational data. Moreoever, we illustrate how these scaling relations and $f_{nuc}$ vary if the star cluster formation efficiency, GC mass function, galaxy environment or galaxy size-mass relation are altered. Two key predictions of our model are: 1) galaxies with NSC masses greater than their GC system masses are more compact at fixed stellar mass, and 2) the fraction of nucleated galaxies at fixed galaxy mass is higher in denser environments. That a single model framework can reproduce both the NSC and GC scaling relations provides strong evidence that GC in-spiral is an important mechanism for NSC formation.