On the relation between specific star formation rate and metallicity

In this paper, we present an exact general analytic expression Z\Z(circle dot)+I linking the gas metallicity Z to the specific star formation rate (sSFR), which validates and extends the approximate relation put forward by Lilly et al. (L13), where y(z) is the yield per stellar generation, I >(sSFR) is the instantaneous ratio between inflow and star formation rate expressed as a function of the sSFR and I is the integral of the past enrichment history, respectively. We then demonstrate that the instantaneous metallicity of a self-regulating system, such that its sSFR decreases with decreasing redshift, can be well approximated by the first term on the right-hand side in the above formula, which provides an upper bound to the metallicity. The metallicity is well approximated also by Z(Z\Z(circle dot)=Z\Z(circle dot)}(L13 ideal regulator case), which provides a lower bound to the actual metallicity. We compare these approximate analytic formulae to numerical results and infer a discrepancy < 0.1 dex in a range of metallicities (Z\Z(circle dot)) in 1.5,0], for y(z) a parts per thousand Z(aS (TM)) = 0.02) and almost three orders of magnitude in the sSFR. We explore the consequences of the L13 model on the mass-weighted metallicity in the stellar component of the galaxies. We find that the stellar average metallicity lags similar to 0.1-0.2 dex behind the gas-phase-metallicity relation, in agreement with the data.