Instantaneous recycling approximation (IRA)

The instantaneous recycling approximation treats short-lived stars as returning their ejecta immediately after formation while long-lived stars retain their mass indefinitely. It turns delayed chemical enrichment into local differential equations.

Under the instantaneous recycling approximation, forming mass $dM$ locks $ds=(1-\beta )dM$ into stars. If a fraction $f$ of returned material remains in the interstellar medium, the retained original metal mass is $\beta fZdM$, and retained newly made metal mass is $fyds$, where $y$ is the stellar yield per locked mass. Subtracting the metal mass $ZdM$ initially consumed gives the displayed equation. Pristine galactic gas inflow adds gas but no metals; enriched inflow adds its metal mass separately. The retained fraction applies to both old and newly made metals in this idealization.

Let $S=s/s_{\infty }$, let the interstellar medium have mass $g=s_{\infty }S(1-S)$, and retain fraction $f=S$ of returned material. For $0\le \beta <1$ put $q=(2-\beta )/(1-\beta )$. The integrating factor for the metal mass is $S^{1/(1-\beta )}/(1-S)$. A finite initial gas-phase metallicity removes the homogeneous solution and gives the displayed expression, equivalently $Z=y{\sum }_{n\ge 1}{S}^n/(q+n)$ for $0\le S<1$. The increasing initial gas mass requires pristine galactic gas inflow: gas accounting gives $d{M}_{\mathrm{in}}/ds=q(1-S)$. A fixed initial mass function makes the birth metallicity distribution function proportional to $ds/dZ$ when enrichment is monotone.

The stellar yield is the newly synthesized metal mass returned to gas per unit mass locked into long-lived stars, under a specified recycling convention and initial mass function.

Let a fraction $R$ of gross stellar birth mass return immediately, and let the new metal source be $bR\Psi$. The yield per permanently locked mass is then $y=bR/(1-R)$. Metal conservation in an initially pristine closed box gives $Z=y\ln (M_0/M_g)$. A coefficient defined by the metal source must not be confused with a yield normalized per locked stellar mass; the source convention must be stated, especially when taking $R\to 0$ or $R\to 1$ limits.