ψ = k M_{g}

, with

k > 0

F

returns promptly, while a fraction

1 - F

stays in long-lived stars and remnants. Assume

0 \leq F < 1

, instantaneous mixing, and no galactic gas inflow or galactic outflow. If

M_{s}

denotes the permanently locked stellar mass, mass conservation gives

\dot{M}_{s} = (1 - F) ψ, \dot{M}_{g} = - (1 - F) ψ = - (1 - F) k M_{g} .

λ = (1 - F) k

. Integration, with

M_{g} (0) = M_{0}

M_{g} (t) = M_{0} e^{- λ t}, ψ (t) = k M_{0} e^{- λ t} .

The locked mass is

M_{s} = M_{0} (1 - e^{- λ t})

, so

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

at every time. The decay time is

[(1 - F) k]^{- 1}

, longer than the gross gas-processing time

k^{- 1}

because returned gas can form subsequent generations. The formal

F = 1

limit retains all gas and locks no mass; a finite stellar yield defined per locked mass then needs separate treatment.

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