Galactic gas regulator

A gas-budget model in which inflow competes with star formation, gas return and outflows. A zero instantaneous gas derivative defines a balance nullcline; it need not define a fixed point of the evolving stellar and black-hole masses.

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.

In a system where gas and stellar masses both evolve, the locus ${\dot{M}}_g=0$ gives the instantaneous gas-balance nullcline. Motion along the other state variables generally changes the equilibrium reservoir, so the nullcline is not itself an exact constant-gas trajectory.

For an Eddington-luminous hole with $M_{\mathrm{BH}}=\beta M_s$ and wind mass flux $\lambda L$, define $A=\lambda \beta (4\pi G{m}_{p}c/{\sigma }_T)$. The gas budget ${\dot{M}}_g=\Phi -ϵM_g-A{M}_s$ has positive instantaneous balance only for $M_s<\Phi /A$. A positive constant-gas trajectory is incompatible with ongoing ${\dot{M}}_s=ϵM_g$ unless the secular black-hole coupling is neglected.