Closed-box model of galactic chemical evolution

The closed-box model evolves a fixed total baryonic mass with no inflow or outflow. Under instantaneous recycling and constant yield it predicts $Z=y\log (1/\mu )$ for gas fraction $\mu$.

A finite-metal-fraction variant of the closed-box model of galactic chemical evolution uses effective stellar yield $Y(1-Z_g)$, rather than constant yield. With pristine initial gas and $Y>0$, metal accounting gives $d{Z}_g/d\log (1/\mu )=Y(1-Z_g)$ and hence $Z_g=1-{\mu }^Y$. The mass-weighted mean stellar metallicity is $[Y-(Y+1)\mu +{\mu }^{Y+1}]/[(Y+1)(1-\mu )]$. At final gas fraction ${\mu }_f$, the metallicity distribution function is $(1-z{)}^{1/Y-1}/[Y(1-{\mu }_f)]$ on $0<z<1-{\mu }_f^Y$. For trace abundances, $Z_g\simeq -Y\log \mu$ and the usual closed-box metallicity distribution is recovered to leading order.

For a pristine closed-box model of galactic chemical evolution, $M_{g}dZ=Yd{M}_s$ and $d{M}_s=-d{M}_g$ imply $M_g=M_0{e}^{-Z/Y}$ independently of the star formation rate. The resulting metallicity distribution function is exponential on $0\le Z\le Z_f$. Its normalized density is $e^{-Z/Y}/[Y(1-e^{-Z_f/Y})]$. A fixed initial mass function and survival selection are needed to turn locked stellar mass into a stellar number distribution. Logarithmic abundance intervals have the additional factor $(\ln 10)Z$.

With gross star formation rate $\psi =k{M}_g$, constant returned mass fraction $F$, and the instantaneous recycling approximation, the net locked mass rate is $(1-F)\psi$. Mass conservation gives ${\dot{M}}_g=-(1-F)k{M}_g$. For constant stellar yield $Y$ per unit locked mass and initially pristine gas, metal conservation gives $M_g\dot{Z}=Y(1-F)\psi$, so gas-phase metallicity grows as $Z=Y(1-F)kt$. Gross star formation and permanent locking have different normalizations.

The leaky-box model adds a galactic outflow to the closed box. If the outflow has the ambient gas metallicity and constant mass-loading factor $\lambda$, its effective yield is reduced to $y/(1+\lambda )$ when returned fractions are absorbed into the definitions.

For initially pristine gas, zero bulk mass return, a constant stellar yield $y$ and a well-mixed galactic outflow of mass-loading factor $\nu$, gas conservation gives $d{M}_g=-(1+\nu )d{M}_s$ and metal accounting gives $M_{g}dZ=yd{M}_s$. Consequently $M_g=M_{g,0}{e}^{-Z/y_e}$ with $y_e=y/(1+\nu )$. Since $d{M}_s/dZ=M_g/y$, complete depletion gives a normalized mass-weighted metallicity distribution function $p(Z)=e^{-Z/y_e}/y_e$ and mean stellar metallicity $y_e$. A time-independent initial mass function makes the distribution of surviving tracers proportional to this distribution; a sample selected by light need not have the same weights.

The effective yield is the yield that a simple closed-box relation would need to reproduce an observed gas fraction and metallicity. Outflow and pristine inflow generally lower it below the nucleosynthetic yield.

The G-dwarf problem is the deficit of low-metallicity long-lived G dwarfs near the Sun relative to the prediction of a constant-yield closed-box model. Prolonged accretion of metal-poor gas is a standard ingredient in its resolution.

A metallicity distribution function gives the number or mass of stars per metallicity interval. Its interpretation depends on sample selection, stellar lifetimes, spatial migration, and whether metallicity means total heavy-element abundance or a particular abundance ratio.

Changing from linear stellar metallicity to its logarithm multiplies the metallicity distribution function by its Jacobian determinant. A flat distribution in linear metallicity rises exponentially in logarithmic metallicity. The commonly used iron abundance coordinate is equivalent to total-metallicity scaling only if relative element abundances are fixed.