Radiatively Inefficient Hot Flows

A radiatively inefficient hot accretion flow keeps much of its dissipated energy in the gas instead of emitting it locally. Low density weakens Coulomb coupling and radiation, the ions can remain near the virial temperature, and the flow becomes geometrically thick and transonic. The central result is a timescale ordering: if inflow outruns electron–ion equilibration and cooling, a one-temperature thin-disk model cannot close the energy budget.

Required background. Fluid and Kinetic Theory: continuum kinematics and balance laws supplies advective energy transport and kinetic relaxation.

Helpful background. Black-Hole Geometry supplies the relativistic inner boundary; Relativistic Matter and Compact Objects supplies plasma and radiative processes.

A two-temperature energy budget

Let a fraction $\delta$ of turbulent or collisionless heating act directly on electrons. A schematic local closure is

$$(1-\delta)q^+-q_{ie}=q_{{\rm adv},i},$$ $$\delta q^+ + q_{ie}-q_{\rm rad}=q_{{\rm adv},e},$$

where $q^+$ is total dissipative heating per volume, $q_{ie}$ transfers energy from ions to electrons, and $q_{\rm rad}$ includes synchrotron, inverse Compton, and bremsstrahlung losses. Adding the equations gives $q^+-q_{\rm rad}=q_{{\rm adv},i}+q_{{\rm adv},e}$: Coulomb exchange redistributes energy between species but creates none. These equations are the simplest two-temperature moment model; a nonthermal tail or species-dependent collisionless heating requires additional closure.

Three clocks organize the regime:

$$t_{\rm in}\sim\frac{R}{|v_R|}, \qquad t_{ie}\sim\frac{u_i}{q_{ie}}, \qquad t_{\rm cool}\sim\frac{u_e}{q_{\rm rad}}.$$

Here $u_i$ and $u_e$ are internal-energy densities. For ions hotter than electrons, $t_{ie}$ estimates ion energy loss to Coulomb transfer; it is not an exact temperature-equilibration time near equal temperatures. When $t_{\rm in}\lesssim t_{ie}$, ions can retain their heat during inflow. The electron cooling time is a separate diagnostic: electrons may cool efficiently even while weak ion–electron exchange leaves most ion heat advected. Global radiative inefficiency follows from the combined energy equations, not from an electron clock alone. Vertical support gives $H\sim c_s/\Omega$, so near-virial ions can support $H/R\sim1$ Yuan and Narayan 2014, § 2.2, equations (2)–(3), and § 2.5.

The usual “ADAF” name describes an advection-dominated limiting solution, not every faint source. Other hot solutions differ in entropy gradient, convection, magnetization, electron heating, and outflow. The required checks are conservation of rest mass, energy, and angular momentum; a regular sonic transition; and a cooling calculation using the same density, field, and electron distribution as the dynamics.

Radius-dependent inflow

Height-integrated wind models often parameterize the mass rate remaining in the accreting component as

$$\dot M(R)=\dot M_{\rm out} \left(\frac{R}{R_{\rm out}}\right)^s, \qquad 0\le s<1.$$

With $H\propto R$ and $|v_R|\propto R^{-1/2}$, continuity gives

$$n(R)\propto \frac{\dot M(R)}{RH|v_R|} \propto R^{-3/2+s}.$$

Thus a Bondi-scale or outer supply rate cannot simply be inserted at the horizon. For $s\gt0$, progressively less mass remains in this inflowing component, and processes proportional to $n^2$, such as thermal bremsstrahlung, change strongly.

This declining rate is not the conserved net flux through a complete sphere. Defining positive inward and outward components from the signs of the radial velocity gives

$$\dot M_{\rm net} =\dot M_{\rm in}-\dot M_{\rm out}.$$

In a steady state without mass sources between spheres, $\dot M_{\rm net}$ is independent of radius and equals the horizon rest-mass rate. Both separately measured components can nevertheless grow outward, and turbulent circulation can contribute to both without becoming an escaping wind Yuan and Narayan 2014, § 3.4, equations (32)–(35). The density scaling above uses a disk-inflow model, not a claim that full-sphere mass conservation fails.

Yuan and Narayan 2014, §§ 3.2 and 3.4–3.6 reviews magnetic stresses, winds, and collisionless effects. A trustworthy numerical comparison reaches inflow equilibrium across the quoted radial range, resolves the magnetorotational instability, varies floors and boundaries, and closes shell-integrated energy. A shallow density profile before those tests is not by itself evidence for a physical wind.

Sgr A* as a constrained example

The public 2017 Sgr A* campaign provides a sharply defined data release rather than a hand-assembled historical spectral energy distribution. Use the Event Horizon Telescope 1.3 mm observations from 2017 April 6–7 together with the contemporaneous 22, 43, and 86 GHz, near-infrared, and X-ray products. Treat the brighter April 11 X-ray flare and enhanced millimeter variability as a different state, not as extra points in a stationary quiescent SED. Event Horizon Telescope Collaboration 2022, §§ 2–5 specifies the observing campaign, calibration, multiwavelength selection, and public products.

The calibration model must include station gains and closure quantities at 1.3 mm, the interstellar-scattering kernel, absolute flux scales, extinction in the infrared, X-ray response and background, aperture differences, and each instrument’s clock. The selection records observing windows, baseline and frequency flags, quiescent versus flaring intervals, and the allowed non-simultaneity. Combining measurements with different apertures or epochs without that bookkeeping creates a fictitious one-zone spectrum.

A minimal physical likelihood predicts each native observable from a two-temperature synchrotron–Compton–bremsstrahlung flow. Its parameters include the horizon accretion rate, $s$, electron-heating prescription, magnetic pressure ratio, thermal and nonthermal electron content, inclination, and black-hole mass and distance. Millimeter visibilities constrain both flux and angular size; infrared limits constrain the nonthermal tail; X-rays constrain bremsstrahlung or Compton emission and variability. Fit visibility data and instrument counts, not merely flux points read from a figure.

The statistical uncertainty is the finite-data likelihood and its posterior covariance. The dominant model and systematic uncertainty includes electron heating, magnetic topology, a nonthermal tail, $\dot M(R)$, foreground extinction and scattering, gain calibration, aperture contamination, and non-simultaneity. The 2017 data show intrinsic horizon-scale variability on minutes-to-hours timescales and an approximately $50\,\mu{\rm as}$ blurred-ring scale Event Horizon Telescope Collaboration 2022, § 5. Those results restrict emitting size and variability; they do not uniquely choose an electron-heating or outflow prescription.

For reproduction, download the released calibrated visibility tables and multiwavelength products, record checksums and software versions, rebuild the April 6–7 SED and size likelihood, and refit at least: (i) thermal electrons with weak outflow, (ii) stronger outflow with altered electron heating, and (iii) a thermal-plus-nonthermal distribution. Leave each band out in turn and test whether the held-out flux and size are predicted. A parameter that shifts across these alternatives is model dependent even if every individual posterior is narrow.

The observational comparison includes results through 2026-08-18. A later EHT release, scattering calibration, or contemporaneous multi-band campaign triggers a new comparison.

Interpretation and boundaries

Hot-flow models explain how very low radiative efficiency can coexist with appreciable supplied gas. They do not imply that all retained energy crosses the horizon: some can emerge in winds or jets. Nor does a successful Sgr A* fit identify a universal prescription for low-luminosity active nuclei; mass scale, external supply, jet contribution, and electron physics differ.

Detailed spectra and instrument likelihoods are beyond this page’s physical boundary. Radiation, Spectra, and Variability develops component-by-component radiative inference, and Magnetospheres, Jets, and Outflows develops the jet and wind channels. Here the observational data test the density, temperature, size, and timescale consequences of a hot inflow.

Common pitfalls

Equating low luminosity with low supply. The inference requires both a mass-flow model and a radiative efficiency; winds can separate the outer and horizon rates by orders of magnitude.

Assuming one temperature. Weak Coulomb coupling permits hot ions and cooler radiating electrons, so the spectrum depends on the electron-heating closure.

Calling turbulent circulation a wind. Net unbound mass and energy fluxes must remain outward after time and angular averaging and must be insensitive to the diagnostic radius.

Diagnose a hot-flow ordering

For Sgr A* take $M=4.0\times10^6M_\odot$, $R=20r_g$, and $|v_R|=0.03c$. Suppose local microphysics gives $t_{ie}=20\,{\rm h}$ and $t_{\rm cool}=10\,{\rm h}$. Compute $t_{\rm in}$ and classify the ordering. If $s=0.5$, by what factor does $\dot M$ fall between $R_{\rm out}=10^5r_g$ and $R=10r_g$, and what density slope follows?

Solution

The gravitational time is

$$t_g=\frac{GM}{c^3}\simeq 4.9255\,\mu{\rm s}\left(\frac{M}{M_\odot}\right) \simeq19.7\,{\rm s}.$$

Therefore

$$t_{\rm in} =\frac{20r_g}{0.03c} =\frac{20}{0.03}t_g \simeq1.31\times10^4\,{\rm s} \simeq3.65\,{\rm h}.$$

Since $t_{\rm in}<t_{\rm cool}<t_{ie}$, neither complete electron cooling nor ion–electron equilibration occurs before inflow; advection and a two-temperature treatment are required. The mass-flux ratio is

$$\frac{\dot M(10r_g)}{\dot M(10^5r_g)} =\left(10^{-4}\right)^{1/2}=10^{-2}.$$

Only one percent of the outer accreting-component rate remains at $10r_g$ in this parameterization. The density scales as $n\propto R^{-3/2+s}=R^{-1}$. These are regime diagnostics, not a proof that the real flow has exactly $s=0.5$.

References

• Event Horizon Telescope Collaboration. “First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multi-wavelength Observations, Data Processing, and Calibration.” Astrophysical Journal Letters 930 (2022): L13. DOI.
• Yuan, Feng, and Ramesh Narayan. “Hot Accretion Flows around Black Holes.” Annual Review of Astronomy and Astrophysics 52 (2014): 529–588. DOI.