Constant-inflow gas regulator solution

With constant inflow $\Phi$, no mass return or outflow, and $\Psi =ϵM_g$, mass conservation gives $M_g=\Phi /ϵ+(M_0-\Phi /ϵ)e^{-ϵt}$. Positive $ϵ$ makes the gas equilibrium stable, with perturbations decaying on timescale ${ϵ}^{-1}$. The star formation rate tends to the inflow, not necessarily to zero.

For ${\dot{M}}_g+ϵM_g=\Phi (t)$, multiplication by $e^{ϵt}$ and integration gives $M_g=M_0{e}^{-ϵt}+{\int }_0^t{e}^{-ϵ(t-s)}\Phi (s)ds$. Exponential memory filters the past inflow; the constant-inflow equilibrium is one specialization.

For initially no stars, $M_{\star }=\Phi t+(M_0-\Phi /ϵ)(1-e^{-ϵt})$. Adding the gas solution verifies $M_g+M_{\star }=M_0+\Phi t$. Thus equilibrium of the gas reservoir coexists with continuing linear growth of stellar mass.