\frac{d Z}{d M _{g}} = - \frac{Y}{M _{g}} .

The logarithm appears on integrating from

M_{0}

and

Z = 0

Z = - Y \int_{M_{0}}^{M_{g}} \frac{d m}{m} = Y ln \frac{M _{0}}{M _{g}} .

μ = M_{g} / M_{0}

, this is

Z = Y ln (1/ μ) .

It gives

Z = 0

μ = 1

and increasing gas-phase metallicity as gas is consumed, independently of the time dependence of the star formation rate. For this closed initially star-free system,

M_{0} = M_{g} + M_{s}

, so this initial-reservoir fraction equals the present gas-to-total baryonic fraction. That equality fails in an open system.

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