Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 65 / 1

Let $G$ be the remaining gas mass, $S$ the formed stellar mass, and $G_0$ the initial gas mass; initially $S=0$ and the gas-phase metallicity is zero. Take the stellar yield $y$ and the mass-loading factor $\nu$ to be constant, with homogeneous mixing and no galactic gas inflow. With the specified zero bulk return, mass conservation givesNew metals add $ydS$, while star formation and the galactic outflow remove metals at the existing gas-phase metallicity $Z$. ThusThe leaky-box model of galactic chemical evolution therefore givesFor the usual gas fraction of a galaxy, $\mu =G/(G+S)$, the denominator is the baryonic mass still in the system, which decreases under the galactic outflow. Since $G_0=G+(1+\nu )S$, we have $G_0/G=1+(1+\nu )(1-\mu )/\mu$. In the remaining-mass convention,If instead $\mu$ denotes $G/G_0$ relative to the fixed initial reservoir, the same calculation gives $\boxed{Z=y\ln (1/\mu )/(1+\nu )}$. Stating the denominator matters: these two gas fraction of a galaxy conventions coincide for the closed-box model of galactic chemical evolution, recovered at $\nu =0$, but differ when gas escapes. Neglecting bulk return while retaining a nonzero stellar yield is the intended trace-metal approximation.

A star inherits the gas-phase metallicity at its birth. Set $y_e=y/(1+\nu )$; the preceding calculation gives $G(Z)=G_0{e}^{-Z/y_e}$ and $dS/dZ=G(Z)/y$. If the initial mass function is fixed, let $\kappa$ be the number of the counted stars per unit formed stellar mass; this includes any restriction to surviving long-lived tracers. The metallicity distribution in a leaky box is thereforeThe total eventual formed stellar mass is $G_0/(1+\nu )$, so $N_{\infty }=\kappa G_0/(1+\nu )$ and the equivalent normalized metallicity distribution function isIf $N(Z)$ means a cumulative count rather than a count per stellar metallicity interval, integration yieldsAt a finite observation time these expressions end at the current gas-phase metallicity $Z_f$ and the normalized distribution divides by $1-e^{-Z_f/y_e}$. A distribution per logarithmic stellar metallicity instead multiplies $dN/dZ$ by $(\ln 10)Z$; it is not the same shape as the distribution per linear stellar metallicity.

All gas being used or expelled means $G\to 0$, so $Z\to \infty$ in the formal leaky-box model of galactic chemical evolution and the final stellar mass is $S_{\infty }=G_0/(1+\nu )$. Weight the stellar metallicity by formed stellar mass, using $dS/dZ=(G_0/y)e^{-Z/y_e}$:Here ${\int }_0^{\infty }u{e}^{-u}du=1$, by integration by parts. The final mass-weighted mean isAs a metal-budget check, newly produced metal mass is $y{S}_{\infty }$; stars retain $\int ZdS=y{G}_0/(1+\nu {)}^2$, while the galactic outflow carries $\nu \int ZdS$. Their sum is exactly $y{S}_{\infty }$. An observed luminosity-weighted stellar metallicity need not equal this mass-weighted mean.

Assume the stellar yield is common to the compared stellar populations and compare their mass-weighted stellar metallicities. Inverting the metallicity distribution in a leaky box result givesFor example, with $y=Z_{\odot }$, a solar-metallicity massive elliptical galaxy requires $\nu \simeq 0$, whereas $⟨Z{⟩}_{\ast }<0.01Z_{\odot }$ in a dwarf galaxy requires $\nu >99$. A supersolar stellar metallicity requires a correspondingly supersolar stellar yield: outflow alone cannot raise $⟨Z{⟩}_{\ast }$ above $y$. If $y=2Z_{\odot }$, a massive elliptical galaxy with $⟨Z{⟩}_{\ast }$ between $Z_{\odot }$ and $2Z_{\odot }$ has $0\le \nu \le 1$, while the same low-metallicity dwarf galaxy needs $\nu >199$.

For a yield of order solar, the required mass loading runs from order zero or unity in massive spheroids to order one hundred or more in faint dwarfs. Without a specified stellar yield, the robust comparison isThis explains the sense of the galaxy mass--metallicity relation; galactic gas inflow, incomplete depletion, and changes in the initial mass function can also change observed stellar metallicity.

The escape-speed scaling of wind mass loading makes the proposed trend dynamically plausible: the shallower Newtonian gravitational potential of a dwarf galaxy lets stellar feedback expel more gas per unit stellar mass. If coupled feedback supplies specific energy $e_{\ast }$, escape requires $\nu v_{\mathrm{esc}}^2/2\lesssim e_{\ast }$; fixed coupling suggests $\nu \propto v_{\mathrm{esc}}^{-2}$ for an energy-driven outflow. Fixed specific momentum $p_{\ast }$ instead gives $\nu v_{\mathrm{esc}}\lesssim p_{\ast }$, or $\nu \propto v_{\mathrm{esc}}^{-1}$ for a momentum-driven outflow.

Since escape velocity has $v_{\mathrm{esc}}^2\sim GM/R$, galaxies whose characteristic radius grows more slowly than their gravitating mass have increasing binding energy per unit mass with increasing mass. For $R\propto M^a$ with $a<1$, the energy-driven outflow estimate gives $\nu \propto M^{a-1}$. Equivalently, greater velocity dispersion in massive spheroids implies greater escape energy and more effective retention of metals.

For scale, one core-collapse supernova of $10^{51}\mathrm{erg}$ per $100M_{\odot }$ formed supplies about $5\times 10^{15}\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-2}$ before coupling losses. At ten percent coupling the energetic upper estimate $2e_{\ast }/v_{\mathrm{esc}}^2$ is about $110$ for $v_{\mathrm{esc}}=30\mathrm{km}{\mathrm{s}}^{-1}$, but about $0.4$ for $500\mathrm{km}{\mathrm{s}}^{-1}$. Efficient loss from shallow potentials and retention in deep potentials can span the required range. These are illustrative energetics rather than a unique prediction: cooling, entrainment, fallback and the actual initial mass function determine the escaping mass-loading factor.