On the star formation rate - brightest cluster relation: estimating the peak star formation rate in post-merger galaxies

We further the recent discussion on the relation between the star formation rate (SFR) of a galaxy and the luminosity of its brightest star cluster (SFR versus M(V)(brightest)). We first show that the observed trend of SFR versus M(V)(brightest) is due to the brightest cluster in a galaxy being preferentially young (<= 15 Myr - for a constant SFR) and hence a good tracer of the current SFR, although we give notable exceptions to this rule. Archival Hubble Space Telescope (HST) imaging of high-SFR galaxies, as well as additional galaxies/clusters from the literature, is used to further confirm the observed trend. Using a series of Monte Carlo simulations, we show that a pure power-law mass function with index alpha = 2 is ruled out by the current data. Instead, we find that a Schechter function (i.e. a power law with an exponential truncation at the high-mass end) provides an excellent fit to the data. Additionally, these simulations show that bound cluster formation (in M(circle dot) yr(-1)) represents only similar to 8 +/- 3 per cent of the total star formation within a galaxy, independent of the SFR. From this, we conclude that there is only a single mode of cluster formation which operates over at least 6 orders of magnitude in the SFR. We provide a simple model of star/cluster formation feedback within dwarf galaxies (and star-forming complexes within spirals) which highlights the strong impact that a massive cluster can have on its surroundings. Using this relation, we can extrapolate backwards in time in order to estimate the peak SFR of major merger galaxies, such as NGC 7252, 1316 and 3610. The derived SFRs for these galaxies are between a few hundred and a few thousand solar masses per year. The inferred far-infrared luminosity of the galaxies, from the extrapolated SFR, places them well within the range of ultraluminous infrared galaxies (ULIRGs) and for NGC 7252 within the hyperluminous infrared galaxy (HLIRG) regime. Thus, we provide evidence that these post-merger galaxies passed through a ULIRG/HLIRG phase and are now evolving passively. Using the current and extrapolated past SFR of NGC 34, we infer that the ULIRG phase of this galaxy has lasted for at least 150 Myr.