Black hole accretion disks with outflows II. Time dependent Green's function solutions in Newtonian gravity

TL;DR

This work derives Green's-function solutions for time-dependent Newtonian thin-disk evolution in the presence of outflows, providing exact analytical expressions for zero-torque, torquing, and inflow–outflow coupling scenarios. The Fourier-domain analysis reveals that outflows suppress inner-disk variability and long-timescale fluctuations while leaving high-frequency diffusion-dominated behavior largely intact, and they flatten the disk temperature profile, accelerating bolometric luminosity decay. The authors identify two analytically tractable wind classes—a radius-dependent wind with $t_w$ tied to the viscous time and a constant wind efficiency $\varepsilon_w$—and show that mass conservation drives the late-time relation $|\dot M_{acc}(r,t)| \approx |\dot M_{out}(<r,t)|$. These results offer a powerful, scalable framework for interpreting observations of accreting black holes and lay groundwork for extending to Kerr geometry; they may also inform protoplanetary disk studies in related outflow regimes.

Abstract

We present Green's function solutions of the Newtonian time-dependent thin disk equations in the presence of outflows, showing that simple and exact analytical expressions exist in various natural limits of the problem. These Green's functions are mathematically very similar to the classical Lynden-Bell & Pringle solutions in the absence of outflows, but differ strongly in their precise physical details and observational implications. Solutions are presented for phenomenological radius-dependent outflows which both do and do not torque the local accretion flow, and for outflows which are launched proportional to the local accretion rate. Generically, outflows lead to a more rapid decay of the bolometric luminosity of the disk, flatten the radial dependence of the disk temperature, and suppress variability in the accretion rate at small radii and low frequencies (on long timescales). Observational implications of these four results are discussed in detail.

Figures (5)

• Figure 1: Left: the evolving surface density (arbitrarily but consistently normalised) of Green's function solutions to the constant $\varepsilon_w$ coupled inflow-outflow disk evolution equation, at three different dimensionless times (denoted by line style) and three different outflow parameters $\varepsilon_w$ (denoted by color). Stronger outflows strongly suppress the surface density of the inner disk regions. The evolution of the outer disk regions is almost unaffected by the outflows. Right: the evolving disk temperature of the same disk solutions. Outflows flatten the disk temperature profile markedly, and strongly suppress the peak value of the inner disk temperature.
• Figure 2: Left: the evolving bolometric luminosity (arbitrarily but consistently normalised) of Green's function solutions of the coupled inflow-outflow disk equation for different values of the outflow parameter $\varepsilon_w$ (see legend), plotted as a function of dimensionless time. Right: the (arbitrarily but consistently) normalised spectral energy distributions of the Green's function solutions plotted in Figure \ref{['fig:dens+temp']}, plotted against dimensionless observing frequency. Outflow parameters $\varepsilon_w$ are denoted by color, while the dimensionless times are denoted by line style. Outflows strongly suppress high energy emission, resulting from the flattened disk temperature profiles. Outflows can also modify the mid frequency spectral slope of the disk, but do not strongly impact emission in the Rayleigh Jeans tail of the disk.
• Figure 3: The accretion rate (arbitrarily but consistently normalised) resulting from Green's function solutions of the coupled inflow-outflow disk equations for different values of the wind efficiency $\varepsilon_w$. Left: the spatial dependence of the accretion rate for different times (denoted by line styles) and efficiencies (denoted by color). Note the strong suppression of the inner disk accretion rate for large wind efficiencies. Right: the evolution (in dimensionless time) of the accretion rate at $r/r_0=1/10$, showing both the suppression as well as the more rapid time evolution for larger wind efficiencies.
• Figure 4: Left: the mass outflow rate from all radius interior to the initial radius of the flow (arbitrarily but consistently normalised) for different wind efficiencies (denoted by color), as a function of dimensionless time. Mass outflow peaks at early times, when the accretion rate is highest. Right: the integrated mass budgets for inflow (matter which passes through $r = r_0/10$, solid curves) and outflow (matter launched from $r < r_0$, dashed curves) as a function of dimensionless time. The curves are normalised to sum to the accreted mass in the absence of winds. We see that even moderate wind efficiencies can prevent a large fraction of matter from reaching small radii.
• Figure 5: The Fourier transform of the mass accretion rate for different wind efficiencies (denoted by color). Left: the (absolute value of) the variability in the mass accretion rate at $r/r_0=1/3$, showing that outflows generically suppress variability in the accretion rate. Right: the radial dependence of the accretion rate variability for different wind efficiencies, at low frequencies compared to the local viscous frequency $f = 0.01/t_v$. Showing that the larger the separation between two disk radii the stronger the suppression in the accretion rate variability (on long timescales) in the presence of outflows.