Radiation from an accretion disk with jets

Relativistic outflows are very nicely described by a model of the small-scale symbiosis between the accretion disk and jets [11]. Donea and Biermann [9] have used such a model for an accretion disk with jets starting at the inner region of a disk, and were able to reproduce the UV bump in quasars. By fitting observed spectra they derived upper limits for the radius of the base of the jet, $R_{\rm jet}$, and they found that $R_{\rm jet}$ cannot be too far from $R_{\rm ms}$. The jet base, or ``footring'', is actually the thin layer between $R_{\rm ms}$ and radius $R_{\rm jet}$, the jet being approximated by a hollow cylinder of inner radius $R_{\rm ms}$ and outer radius $R_{\rm jet}$, such that $R_{\rm ms} \le R_{\rm jet} \ll R_{\rm out}$, where $R_{\rm out}$ is the outer radius of the disk.

The ADJ model assumes that the gravitational potential energy available between $R_{\rm ms}$ and $R_{\rm jet}$ is the energy reservoir of the jets. The total power of the jets is strongly dependent on the accretion mass rate in the disk and on the size of its footring (we not include here the interaction between the black hole and jets which could also result in it putting a non-negligible fraction of its energy into the jets [18]). The total power of the jets is

$$Q_{\rm jet} = L_{\rm disk} - L_{\rm disk}^{\rm jet}$$ (2)

where $L_{\rm disk}^{\rm jet}$ is the luminosity of the disk with jets and $L_{\rm disk}$ would be the luminosity of the disk if the conditions necessary to drive outflows were not met, i.e., the case of a standard relativistic disk.

A coupled jet-disk system must obey the laws of conservation of mass and angular momentum. We assume that the jet is fed with mass by the accretion disk, and that the flow of mass into the jet per unit of time $\dot{M}_{\rm jet}=q_{ m}\dot{M}$ is a fraction $q_{ m} \le 1$ of the accretion mass rate into the disk $\dot{M}=-2 \pi R \Sigma u^R$, where $\Sigma$ is the surface density of mass in the disk and $u^R$ is the radial velocity of gas at given radius $R$. The equation for conservation of mass requires $\dot {M}=\dot {M}_{\rm disk}+\dot {M}_{\rm jet}$. Derivation of the relevant equations can be found in Donea and Biermann [9], who followed the standard method of calculating the emission spectrum from an accretion disk [12]. If there is no angular momentum and mass loss into the jet the equations used for mass and angular momentum transport in the disk become the standard equations [12], with $R_{\rm jet} \to R_{\rm ms}$, $Q_{\rm jet} \to 0$ and $L_{\rm disk}^{\rm jet} \to L_{\rm disk}$. In this paper, we shall adopt $q_m=0.1$ as used by Falcke et al. [19] in interpreting the radio-UV correlation in AGN.

The local physics at the inner radius of the disk (radius $R_{\rm jet}$) is directly related to the extraction of angular momentum from the in-falling gas, and so modifies the structure of the relativistic disk [20]. From this one calculates the dissipation energy at radius $R$, $D^*(R)$, which must be done numerically in the case of a Kerr black hole, and a detailed discussion of this would serve no useful purpose here. Instead, for the purpose of illustration, and for the sake of simplicity we give here only the relation for the simpler case of a Schwarzschild black hole:

$$D^*(R) = \frac{3 G M \dot M}{8 \pi R^3} \bigg[ (1-q_m) - (1-q_m) {\bigg( \frac {R_{\rm jet}}{R} \bigg) }^{1/2} \bigg]$$ (3)

where $q_m = \dot{M}_{\rm jet}/\dot{M}$ and $M$ is the black hole mass; nevertheless, in the present paper we use $D^*(R)$ for the disk of a Kerr black hole. A reasonable outer radius of the footring of the jet would be $R_{\rm jet} \le 10 R_{\rm g}\approx 10^{-4} M_8$ pc where $M_8=M/10^8 M_{\rm\odot}$.

The disk/jet symbiosis is reflected mainly in a modified photon spectrum from the inner region of the accretion disk where the jet is anchored. The important result is that the spectrum from a Kerr accretion disk is cut off at high frequencies, from extreme UV to soft X-rays. In Fig. 1 we plot disk luminosities for the ADJ model, versus the thickness of the footring of the jet for Kerr black holes with masses $M=10^8 M_{\odot}$ and $M=10^9 M_{\odot}$, and for different mass accretion rates given by $\dot m=\dot M/\dot M_{\rm edd}$, where $M_{\rm edd}$ is the Eddington accretion rate. The case $R_{\rm jet}=R_{\rm ms}$ corresponds to an accretion disk without jets (``standard'' accretion disk) and $Q_{\rm jet} =0$. As can be seen, for thicker jet bases more energy is available to power jets (dotted curves show $Q_{\rm jet}$ increasing with $R_{\rm jet}-R_{\rm ms}$) and less energy is radiated by the disk (solid curves show $L_{\rm disk}$ versus thickness). We note that a small variation of the geometry at the coupling between jet and disk at radii $R_{\rm jet}$ less than $\sim 3 R_g$ would induce large variations in the power of the jet, possibly causing flare activity in blazars. This is because, in a Kerr metric, at small radii close to the black hole there is a large amount of gravitational potential energy available for dissipation into the jet.

Figure 1: Luminosities of disks with jets vs. the thickness of the jet's footring $(R_{\rm jet} - R_{\rm ms})$ (solid curves). The dotted lines show $Q_{\rm jet}$ vs. the thickness of the footring. ( $M_8\equiv M/10^8 M_{\odot }$, $q_{m}=0.1$).

Because of the jet/disk coupling, the ADJ model predicts a lower disk luminosity and a softer photon spectrum, and some of the implications of this for interactions of accelerated electrons or protons are discussed in a separate paper [21]. As mentioned earlier, a flaring state could arise from enhanced AGN central activity whereby the jets get more energy from the disk. In this case, the disk changes from an ADJ with weak jets into an ADJ with a spectrum cut off at high frequencies but with correspondingly stronger jets. Since, in this case, the average energy of photons from an ADJ with strong jets (flaring) becomes lower than the average energy of photons from an ADJ with weak jets this will change the shape of the resulting $\gamma$-ray spectrum in models [2,22] where disk photons are inverse-Compton scattered by relativistic electrons in the jet.

Luminosities of disks in blazars are probably $L_{\rm disk} < 10^{46}$ erg/s, while quasars typically have $L_{\rm disk}=10^{46}$-$10^{48}$ erg/s. The ADJ model gives a simplified approach to the symbiosis between the disk and jets, and could describe low-luminosity accretion disks for quasars and some blazars. However, for some blazars the ADJ model could still give UV fluxes much higher than those observed, and in this case an ejection dominated accretion flow (EDAF) model, which is an ADJ model with a wind [23], may be better. An EDAF model has been applied to the central activity of Sgr A* where it was shown that a wind plus jets extracts energy from the disk more efficiency. Since the EDAF model does not leave much energy to be dissipated in the accretion disk, the radiation field from the disk would become even less important for GeV and TeV $\gamma$-ray absorption. Alternatively, the jets could extract more energy from the accreting gas if the disk with a jets turns into an advection dominated accretion flow (ADAF) [24].

Figure 2: Optical depths from $z=0.01$ pc to infinity for $\gamma$-rays produced in the jet interacting with UV disk photons for the case of an ADJ model with $q_m=0.1$ and $R_{\rm jet}=10R_g$: (a) for various black hole masses and accretion rates indicated by the labels attached to the curves; (b) optical depth normalized to disk luminosity in units of $10^{46}$ erg s$^{-1}$ showing the probable range of $\tau _{\gamma \gamma }$ for blazars ``Bl'' and quasars ``Q''.

Fig. 2(a) shows the optical depths from $z=0.01$ pc to infinity for absorption of GeV-TeV photons produced in the jet and interacting with photons from an ADJ with $q_m=0.1$. We see that at 1 TeV photons are only absorbed for high black hole masses and mass accretion rates close to the Eddington limit. This could apply to quasars where the disk radiation field has a high density around the base of the jet. For low accretion mass rates with $\dot M \le 0.1 M_{\rm edd}$, and lower black hole masses, $M \le 10^8M_\odot$, the photon-photon optical depth is $\tau_{\gamma \gamma} <1$ at 1 TeV. We note that the disk luminosity corresponding to the case $\dot{m}=1$ for $M_8=1$ lies roughly at the boundary between blazars and quasars. In Fig. 2(b) we plot the optical depth divided by $L_{46} \equiv L_{\rm UV}/10^{46}$ erg s$^{-1}$, and give the likely range for blazars and quasars using $\dot{m}=1$ for $M_8=1$ as the boundary. In blazars we may have ADJ models with accretion rates as low as $\dot{m}=0.01$-0.001 - at lower accretion rates an ADAF or EDAF model including jets may be more realistic. We use $M_8=1$ and $\dot{m}=0.01$ as the low luminosity boundary of blazars with an ADJ in Fig. 2(b). Similarly, we use a supermassive black hole ($M_8=10$) and accretion close to the Eddington limit ($\dot{m}=1$) for the upper luminosity boundary of quasars in Fig. 2(b). Of course, these boundaries are by no means rigid and there will be some overlap between the two populations, particularly when we plot $\tau_{\gamma\gamma}/L_{46}$.

The ADAF model seems to explain better very low luminosity blazars, but it cannot energize the BLR or heat efficiently dust in the torus; the lack of observed BLR and torus activity for some blazars has been sometimes interpreted as suggesting that blazars do not have tori or BLR. We consider that if blazars are to be included in the unification schemes they should also have BLR and dusty tori, but possibly with lower emission than in quasars, and we shall discuss this possibility later.