Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 349 / 3 / Solution

The Kennicutt–Schmidt law is the empirical relationfor disk-averaged total gas, with a nearly linear molecular-gas relation in many resolved observations. Atomic gas is mapped through the H I 21-cm line, molecular gas mainly through carbon-monoxide line emission and a CO-to-${\mathrm{H}}_2$ conversion factor, and star formation through combinations of ultraviolet continuum, H-alpha recombination emission, and infrared dust emission. Inclination, dust attenuation, the initial mass function, tracer lifetimes, and conversion factors must be treated consistently.

The gas-depletion time $M_g/{\dot{M}}_{\ast }$ is typically of order a gigayear, whereas a giant molecular cloud has a dynamical or free-fall time of order a few megayears. Star formation is therefore inefficient per collapse time, commonly at the percent level, rather than converting an entire cloud in one free fall.

For the first closed-box model of galactic chemical evolution, neglect returned mass or absorb it into the definitions. Thenand henceThus a region reaching $Z_{\odot }=0.014$ after an enrichment time $t_{\odot }$ with $y_Z=0.006$ hasabout $4.3\mathrm{Gyr}$ for a fiducial $t_{\odot }\simeq 10\mathrm{Gyr}$ enrichment age of the Galactic disk.

The second prescription implies ${\dot{M}}_{\ast }=M_g^2/(M_{g0}{\tilde{\tau }}_{\ast })$. Solving the gas-consumption equation givesTherefore

For long-lived stars, $dN$ is proportional to $d{M}_{\ast }=-d{M}_g$. In the exponential model,In the second model, $1+t/{\tilde{\tau }}_{\ast }=e^{Z/y_Z}$; multiplying its star-formation rate by $dt/dZ$ gives exactly the same result:The metallicity distribution is fixed by the closed-box relation $M_g/M_{g0}=e^{-Z/y_Z}$ and is independent of the star-formation history. Merely changing the time law therefore does not cure the G-dwarf problem; gas inflow, outflow, variable yields, or selection effects must alter the closed-box assumptions.