Quasars versus Microquasars: Scaling and Particle Acceleration

The first quasars recognized were 3C 48 and 3C 273 (Schmidt 1963). Salpeter (1964) explained their enormous luminosities as the result of accretion onto supermassive black holes. Quasars are typically accompanied by twin opposed radio lobes, often with narrow jets along the paths of energetic electrons accelerated near the black hole that power the synchrotron emission of the extended radio lobes. Particle acceleration is ubiquitous in low density astronomical plasmas, but no predictive theory exists (Katz 1991).

Micro-quasars are scaled-down analogues of quasars with black holes of stellar mass. The type specimens are 1E1740.7−2942 (Mirabel et al. 1992) and GRS 1915+105 (Mirabel & Rodriguez 1994; Türler et al. 2004; Chaty 2023). Despite their qualitative similarities (variability, nonthermal emission and double synchrotron radio lobes), quasars and microquasars differ quantitatively: quasars convert a much larger fraction of their bolometric luminosity to particle acceleration. I summarize the data for a few exemplars and offer a qualitative explanation derived from accretion disk scaling laws. Laor & Behar (2008) found a related but distinct correlation between the radio and X-ray luminosities of a number of classes of astronomical objects.

A very large number of quasars exist because most massive galaxies appear to have supermassive black holes at their centers, and accretion onto these black holes makes a quasar. In contrast, recent summaries (Chaty 2023) list only about 19 microquasars. The available data are incomplete for most objects in both categories. Table 1 shows some of the most famous and best-studied quasars and those microquasars for which bolometric and radio luminosities (the result of acceleration of relativistic electrons) are known.

The Table shows the ratio of radio luminosity, an estimate of the power that accelerates energetic electrons averaged over their (long) lifetime in the radio lobes, to bolometric luminosity, taken as the X-ray luminosity of microquasars and the visible luminosity of quasars, for exemplars of each type (considering Sco X-1 as a microquasar; Fomalont et al. 1983). Both categories are heterogeneous and show a wide range of values, and the sources are individually very variable, but these data show that quasars convert orders of magnitude greater fractions of their bolometric luminosity to accelerating relativistic electrons than do microquasars.

In order to accelerate energetic particles, necessary both for incoherent emission (as in active galactic nucleus and radio sources) and for coherent emission (as in pulsars and fast radio bursts, if these particles drive a plasma instability), it is necessary that they gain energy from an electric field faster than they lose it to interaction with an ambient medium by "Coulomb drag" (Atzeri et al. 2009). For a relativistic electron the ratio of these quantities defines an acceleration parameter

$$\begin{eqnarray}&&A\approx \displaystyle \frac{{{Em}}_{e}{c}^{2}}{4\pi {e}^{3}{n}_{e}\mathrm{ln}{\rm{\Lambda }}},\end{eqnarray} \tag{ 1 }$$

where E is the electric field, n_e the medium's electron density and the Coulomb logarithm Λ = 2m_e c²/I, where I is the ionization potential in a neutral medium or m_e c²/ℏ ω_p in a plasma. In most astronomical environments $\mathrm{ln}{\rm{\Lambda }}\approx 20$. A > 1 is necessary for particle acceleration.

In the midplane of an accretion disk, a simple scaling model may be used to estimate the scaling of the magnetic induction B. The accretion rate

$$\begin{eqnarray}&&\dot{M}=2\pi h\rho {{rv}}_{r},\end{eqnarray} \tag{ 2 }$$

where h is a disk thickness, ρ the mass density and v_r a radial velocity. The shear stress

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{{B}^{2}}{8\pi }\end{eqnarray} \tag{ 3 }$$

and the torque it exerts

$$\begin{eqnarray}&&{ \mathcal T }=\displaystyle \frac{{B}^{2}}{8\pi }2\pi {r}^{2}h.\end{eqnarray} \tag{ 4 }$$

The torque must remove the specific angular momentum ${\ell }=\sqrt{{GMr}}$ of the accreting matter:

$$\begin{eqnarray}&&{ \mathcal T }=\dot{M}{\ell }=\sqrt{{GMr}}\dot{M},\end{eqnarray} \tag{ 5 }$$

where M is the black hole mass. The accretion rate $\dot{M}$ is related to the bolometric luminosity $L=\epsilon \dot{M}{c}^{2}$ by the radiative efficiency , a dimensionless quantity plausibly independent of M. Combining the two expressions for the torque,

$$\begin{eqnarray}&&\displaystyle \frac{{B}^{2}}{8\pi }=\displaystyle \frac{\dot{M}}{2\pi }\sqrt{\displaystyle \frac{{GM}}{{r}^{3}}}\displaystyle \frac{1}{h}.\end{eqnarray} \tag{ 6 }$$

A characteristic value of the electric field in the near-vacuum above the accretion disk is

$$\begin{eqnarray}&&E=\displaystyle \frac{v}{c}B=\sqrt{\displaystyle \frac{{GM}}{{{rc}}^{2}}}B=\sqrt{\displaystyle \frac{{GM}}{{{rc}}^{2}}\displaystyle \frac{4\dot{M}\sqrt{{GMr}}}{{r}^{2}h}}.\end{eqnarray} \tag{ 7 }$$

In order to calculate A it is necessary to estimate the electron density n_e . The component of gravity normal to the disk midplane is

$$\begin{eqnarray}&&g=\displaystyle \frac{{GM}}{{r}^{2}}\displaystyle \frac{h}{r},\end{eqnarray} \tag{ 8 }$$

so the disk pressure is approximated

$$\begin{eqnarray}&&P=\displaystyle \frac{{GM}}{{r}^{3}}\rho {h}^{2}.\end{eqnarray} \tag{ 9 }$$

Then the fractional disk thickness

$$\begin{eqnarray}&&\displaystyle \frac{h}{r}=\sqrt{\displaystyle \frac{P}{\rho }\displaystyle \frac{r}{{GM}}}.\end{eqnarray} \tag{ 10 }$$

The shear stress

$$\begin{eqnarray}&&\displaystyle \frac{{B}^{2}}{8\pi }=\alpha P=\alpha \displaystyle \frac{{h}^{2}}{{r}^{3}}{GM}\rho =\displaystyle \frac{\dot{M}{\ell }}{2\pi {r}^{2}h},\end{eqnarray} \tag{ 11 }$$

where α is the ratio of shear stress to pressure.

Combining Equations (2) and (11), the radial velocity

$$\begin{eqnarray}&&{v}_{r}=\alpha \displaystyle \frac{{h}^{2}}{{r}^{2}}\sqrt{\displaystyle \frac{{GM}}{r}}.\end{eqnarray} \tag{ 12 }$$

Using the right hand equality of Equation (11),

$$\begin{eqnarray}&&\rho =\displaystyle \frac{\dot{M}{\ell }r}{2\pi {h}^{3}{GM}\alpha }\end{eqnarray} \tag{ 13 }$$

and the electron density

$$\begin{eqnarray}&&{n}_{e}=\displaystyle \frac{\rho }{\mu {m}_{p}}=\displaystyle \frac{\dot{M}{\ell }r}{2\pi {h}^{3}{GM}\alpha \mu {m}_{p}}\propto {M}^{-1},\end{eqnarray} \tag{ 14 }$$

where m_p is the proton mass and μ the molecular weight per electron. Then

$$\begin{eqnarray}&&A=\displaystyle \frac{{m}_{e}{c}^{2}\alpha {m}_{p}\mu }{{e}^{3}\mathrm{ln}{\rm{\Lambda }}}\sqrt{\displaystyle \frac{\epsilon \kappa }{4\pi }}{\left(\displaystyle \frac{{GM}}{{{rc}}^{2}}\right)}^{1/4}{\left(\displaystyle \frac{h}{r}\right)}^{5/2}\sqrt{\displaystyle \frac{{L}_{{\rm{E}}}}{L}}\sqrt{{GM}}.\end{eqnarray} \tag{ 15 }$$

The opacity κ appears because the luminosity L is scaled to the Eddington luminosity L_E ≡ 4π cGM/κ.

The preceding relations are not quantitative but may give useful information about how their properties scale. The first factors in Equation (15) are either fundamental constants or dimensionless parameters that may be independent of M. The final factor gives the scaling of A:

$$\begin{eqnarray}&&A\propto {M}^{1/2}.\end{eqnarray} \tag{ 16 }$$

More massive black holes provide a more favorable environment for the acceleration of energetic electrons.

The central mystery of quasars is why so much of their power appears as particle acceleration, energizing their double radio lobes, rather than being radiated thermally by the accretion disk. If we did not know that that accretion onto supermassive black holes makes powerful nonthermal radio sources, we might expect them to be thermal emitters. This cannot explain acceleration of relativistic particles or the existence of their double radio lobes. The low density corona or funnel of the disk around a supermassive black hole, too dilute for magnetohydrodynamics to be applicable, offers a favorable environment for the acceleration of energetic particles. Equation (16) explains why supermassive black hole accretion is a more favorable environment for particle acceleration than accretion onto stellar mass black holes, as indicated by the data in the Table. The electron density n_e must be orders of magnitude less than the mid-plane estimate Equation (14) used in deriving Equation (15) in order that A ≫ 1. Such small n_e may be found in a gravitationally stratified disk corona or centrifugally stratified disk funnel.

In denser plasma electric fields produce current flow that screens out time-varying magnetic fields and reduces the electric field from the value in Equation (7), but in very low density plasma above the accretion disk or in its throat the current density is limited to nce and may be insufficient to exclude the magnetic field, justifying Equation (7) as a scaling relation.

At lower plasma densities "anomalous" resistivity, resulting from two-stream electron–positron or ion-acoustic instability, may be large, preventing the cancellation of the induced electric field by current flow. These instabilities accelerate energetic particles. In the coronæ or axial throats of accretion disks, particularly those around supermassive black holes in which the electron density is comparatively small, the "anomalous" resistivity may be high or the current saturated because of the low density. Then Equation (7) may be, at least approximately, valid and Equation (15) useful for scaling.

This theoretical study generated no new data.