Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 349 / 2

The galaxy mass--metallicity relation is the observed tendency for more massive galaxies to have larger gas-phase metallicity and stellar metallicity. Gas metallicity is commonly inferred from nebular emission-line ratios in star-forming H II regions, often quoted as $12+{\log }_{10}(\mathrm{O}/\mathrm{H})$ and measured within a finite spectroscopic aperture. Stellar metallicity comes from stellar absorption features or population-synthesis fits and is luminosity weighted unless the analysis explicitly reconstructs a mass-weighted distribution. Galaxy stellar mass is inferred from photometry or a spectral-energy-distribution fit and depends on the adopted initial mass function. Radial metallicity gradients, dust, line calibration, and aperture selection must consequently be matched before samples are compared.

The usual physical explanation is that a shallow potential well lets a low-mass galaxy lose a larger fraction of newly synthesized metals in galactic outflows. The closed-box model of galactic chemical evolution is therefore replaced by a leaky-box model of galactic chemical evolution withwhere $\lambda$ is the mass-loading factor. Under the instantaneous recycling approximation, let $y_z$ be the stellar yield, absorb the returned mass fraction into the definitions, and suppose the escaping gas has the current gas metallicity $Z_g$. Then mass conservation and metal conservation areSubstitution of the first equation into the second cancels the terms that merely transfer pre-existing metals and leavesBecause the mass of metals locked into stars obeys $d{M}_{z,\ast }=Z_{g}d{M}_{\ast }$, integration gives $M_{z,\ast }=M_{\ast }{Z}_{\ast }=\int Z_{g}d{M}_{\ast }$. The total newly made metal mass is partitioned between present gas, stars, and the outflow:Writing the gas-to-stellar mass ratio as $r_g=M_g/M_{\ast }$ therefore producesThus simultaneous gas and stellar metallicities, together with the gas fraction and an assumed nucleosynthetic yield, estimate the integrated mass loading. The corresponding effective yield is $y_{\mathrm{eff}}=y_z/(1+\lambda )$, and the leaky box has

The G-dwarf problem is that the local Milky Way disk contains far fewer low-metallicity long-lived G dwarfs than the constant-yield closed-box metallicity distribution predicts. A yield that rises with metallicity may initially sound promising because enrichment would accelerate after the first generations. In fact it worsens the problem. In a closed box, $d{M}_{\ast }=-d{M}_g$ andFor $y_z=kZ$ and a nonzero initial metallicity $Z_0$ at gas mass $M_{g,0}$,Hence the cumulative mass of stars born below metallicity $Z$ isIf the system begins at $Z_0=0$, the assumed yield also vanishes and enrichment never starts. For $Z_0>0$, the metallicity distribution function haswhich puts still more stellar mass near the low-metallicity floor. Metal-poor gas inflow, pre-enrichment, and selective outflow are therefore more plausible ingredients in resolving the G-dwarf problem.