On the relation between specific star formation rate and metallicity

Abstract

In this paper, we present an exact general analytic expression Z(sSFR) = yZ/δ(sSFR) + I (sSFR) 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 yz is the yield per stellar generation, δ(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 ZidL13 = Z(sSFR) = yZ/1+η+sSFR/v (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 (log(Z/Z) ∈ [-1.5, 0], for yz = Z⊙ = 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 ~0.1-0.2 dex behind the gas-phase-metallicity relation, in agreement with the data. © 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.