Proportional-infall model of galactic chemical evolution (${\dot{M}}_{\mathrm{in}}=\nu \psi$)

With well-mixed gas-phase metallicity $Z$, zero mass return, star formation rate $\psi$, pristine galactic gas inflow $\nu \psi$, and constant stellar yield $y$, mass conservation and metal accounting give $\dot{M}=-(1-\nu )\psi$ and $M\dot{Z}=(y-\nu Z)\psi$. Thus, for $0<\nu <1$,The initial gas-phase metallicity $Z_0=0$ gives the usual pristine-infall expression. Neglecting returned bulk mass while retaining a nonzero metal yield is a trace-metal idealization.

For $\nu =1/2$ and initially pristine gas, $M=M_0(1-Z/(2y))$ and the formed stellar mass is $S=2(M_0-M)=M_{0}Z/y$. If a fixed initial mass function supplies $\kappa$ surviving tracers per unit formed mass, then $N(<Z)=\kappa M_{0}Z/y$ and $dN/dZ=\kappa M_0/y$ until the current gas abundance. Complete exhaustion gives a uniform normalized metallicity distribution function on $0<Z<2y$. This is uniform in linear metallicity, not in its logarithm.