The <closed-box model of galactic chemical evolution> assumes a fixed total baryonic mass, no gas inflow or outflow, instantaneous and homogeneous mixing, a constant <stellar yield>, a fixed <initial mass function>, and usually the <instantaneous recycling approximation>. It predicts $Z=y\log(1/\mu)$ for gas fraction $\mu$, but real galaxies violate these assumptions.

Evidence for <galactic outflows> includes blueshifted absorption, broad or split emission lines, extraplanar ionized and molecular gas, X-ray bubbles, and metal-enriched circumgalactic material. The low baryon fractions and low effective yields of dwarf galaxies, together with the <galaxy mass--metallicity relation>, provide indirect evidence for preferential gas and metal loss. Sustained star formation for longer than a closed reservoir's depletion time, low-metallicity high-velocity clouds, metallicity dilution during starbursts, the G-dwarf problem, and circumgalactic or intergalactic accretion signatures imply continuing <galactic gas inflow>.

Let $f(t)=Ae^{-t/\tau}$ be the inflow rate, let the outflow rate be $\lambda\psi$, and let $R$ be the promptly returned fraction. The total baryonic, gas, and metal masses obey
$$
\boxed{\dot M_{\rm tot}=f-\lambda\psi},
$$
$$
\boxed{\dot M_g=-(1-R+\lambda)\psi+f},
$$
$$
\boxed{\frac d{dt}(ZM_g)
=y_Z(1-R)\psi-Z(1-R+\lambda)\psi+Z_{\rm prim}f}.
$$
The first metal term is newly synthesized material; the second locks existing metals into long-lived stars and removes them in an ambient-composition wind. Expanding the final derivative and substituting $\dot M_g$ cancels those common terms, leaving
$$
\boxed{\dot Z
=\frac{y_Z(1-R)\psi
+Ae^{-t/\tau}(Z_{\rm prim}-Z)}{M_g}}.
$$

The observed <Kennicutt–Schmidt law> relates star-formation and gas surface densities approximately by $\Sigma_{\rm SFR}\propto\Sigma_g^{1.4}$, with a nearly linear relation to molecular gas over many resolved regimes. For the simple integrated model requested here, take a constant depletion coefficient $S$ so that
$$
\psi(t)=SM_g(t).
$$
Then, with $\alpha=S(1-R+\lambda)$,
$$
\dot M_g+\alpha M_g=Ae^{-t/\tau}.
$$
The <integrating factor> $e^{\alpha t}$ gives, for $\alpha\ne1/\tau$,
$$
\boxed{M_g(t)=M_g(0)e^{-\alpha t}
+\frac{A}{\alpha-1/\tau}
\left(e^{-t/\tau}-e^{-\alpha t}\right)}.
$$
At resonance, $\alpha=1/\tau$, the continuous limit is
$$
\boxed{M_g(t)=[M_g(0)+At]e^{-\alpha t}}.
$$
The first term is depletion of the initial reservoir; the second is gas supplied by the exponentially declining inflow and subsequently consumed or expelled.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 349 / 3 / Solution

Galactic gas inflow supplies a galaxy from its circumgalactic medium, cooling halo, or intergalactic filaments. It can sustain <star formation> and dilute the metallicity of the existing interstellar gas.

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 Incoming links: Galactic gas inflow