Prograde and Retrograde Black Holes: Whose Jet is More Powerful?

Introduction

Various astrophysical systems—active galactic nuclei (AGN), black hole binaries (BHBs), and gamma-ray bursts (GRBs)—produce relativistic outflows. Mediated by large scale magnetic fields, outflows can be powered by the central BHs via Blandford & Znajek 1977 mechanism or by the inner regions of accretion discs via Blandford & Payne 1982 mechanism.

Large-scale magnetic fields extract BH spin energy at a rate,

For fixed $\Phi_{\rm BH}$, $P_{\rm BZ}$ is the same for prograde ($a>0$) and retrograde ($a<0$) BHs. However, a massive accretion disc introduces a preferred spin/rotation direction. Does this lead to large differences in $\Phi_{\rm BH}$ and jet powers of prograde and retrograde BHs? Understanding this is crucial, as the link between jet power and BH spin has been confirmed in BHBs (Narayan & McClintock 2012, but see Fender et al. 2010) and jet power is increasingly often used for inferring BH spin (Daly 2011; Martínez-Sansigre & Rawlings 2011; Buliga et al. 2011; Lei & Zhang 2011; Bambi 2012).

To quantify jet strength in an accretion system, we define BZ efficiency as jet power (eq. 1) in units of mass accretion rate, $\dot{M}$,

where $\phi_{\rm BH}=\Phi_{\rm BH}/(\langle\dot{M}\rangle r_{g}^{2}c)^{1/2}$ is BH dimensionless magnetic flux, and $\langle...\rangle$ is a time average. Similarly, we define magnetic flux enclosed by a toroidal ring, $(r,\theta)$, via $\Phi(r,\theta)=\iint_{\theta^{\prime}<\theta,\varphi^{\prime}}B^{r}dA_{\theta^{\prime}\varphi^{\prime}}$, and its dimensionless version, $\phi(r,\theta)=\Phi(r,\theta)/(\langle\dot{M}\rangle r_{g}^{2}c)^{1/2}$, where the integral is over a polar cap, $\theta^{\prime}<\theta$, of a sphere of radius $r$.

Is it at all possible to obtain a well-defined value of jet efficiency, free from the uncertainties in large-scale magnetic flux content of the initial simulation setup? Tchekhovskoy et al. 2011 showed that a promising approach is to start with a large vertical magnetic flux in the disc, more than the accreting gas can push into the BH. The excess flux remains outside, impedes the accretion, and leads to a magnetically-arrested disc (MAD, Bisnovatyi-Kogan & Ruzmaikin 1974; Narayan et al. 2003; Igumenshchev et al. 2003; Igumenshchev 2008; 44). Estimates show that many astrophysical systems contain enough large-scale magnetic flux to naturally form MADs (Narayan et al. 2003; McKinney et al. 2012). The inner disc properties of MADs are shown to be independent of the initial value of large-scale magnetic flux. So, we can reliably determine $\eta$ for prograde and retrograde BHs and thereby test the flux-trapping “gap” model. In §2 we describe our numerical method and present our results and in §3 we conclude.

Numerical Method and Results

We carried out time-dependent 3D general relativistic non-radiative MHD simulations using a high-order version of Godunov-type shock-capturing code HARM (Gammie et al. 2003; McKinney & Gammie 2004; Tchekhovskoy et al. 2007; Tchekhovskoy et al. 2009; McKinney & Blandford 2009) in modified spherical polar coordinates. We use logarithmically spaced radial grid, $dr/r={\rm constant}$, for $r\lesssim r_{\rm br}$ (see Table 1 for $r_{\rm br}$ values). For $r\gtrsim r_{\rm br}$, the grid becomes progressively sparser, $dr/r=4(\log r)^{3/4}$, with a smooth transition at $r_{\rm br}$. We choose grid inner radius, $R_{\rm in}$, such that there are at least $9$ grid cells between the inner radial boundary and the BH horizon and place the outer radial boundary at $R_{\rm out}=10^{5}r_{g}$, which is larger than the light travel distance in a duration of the simulation (see Table 1 for $R_{\rm in}$ values). This ensures that both radial boundaries are causally disconnected. We use standard boundary conditions (see 44): outflow in $r$-, reflecting in $\theta$-, and periodic in $\varphi$-directions.

We define rates of accretion of rest mass, $\dot{M}$, and rest mass energy, $F_{M}\equiv\dot{M}c^{2}$, by $\dot{M}=-\iint_{\theta,\varphi}\rho u^{r}dA_{\theta\varphi}\equiv F_{\rm M}/c^{2}>0$, where $\rho$ is fluid-frame mass density, $u^{r}$ is $r$-component of contravariant $4$-velocity, and the integration is over all $\theta,\varphi$ on the BH horizon, $r=r_{\rm H}$. Figure 1(c),(h) shows that after the start of the simulation $\dot{M}$ settles to a steady state at $t\gtrsim 5000r_{g}/c$. BH magnetic flux, $\phi_{\rm BH}$, increases until $t\sim 6000r_{g}/c$, beyond which $\phi_{\rm BH}$ saturates (Figure 1d,i). Flux accumulates outside the BH, impedes the accretion, and leads to a magnetically-arrested accretion disc (MAD, Narayan et al. 2003). Some of the BH flux occasionally escapes from the BH via magnetic interchange (Stehle & Spruit 2001; 44), leading to oscillations of $\phi_{\rm BH}$ in time (Fig. 1d,i). Figure 1(b),(g) shows the $t$- and $\varphi$-average of the flow over the MAD period ($t_{\rm avg}$ in Table 1). In order to get to the BH, gas “diffuses” through the vertical magnetic flux that reached size $\sim 25r_{g}$ in a duration of our simulations. At large $r\gtrsim 30r_{g}$, all our discs have very similar values of angular thickness, $\langle h/r\rangle_{30}\approx 0.3$ (see Table 1). At smaller $r$, funnel magnetic fields compress the disc vertically and lead to smaller $h/r$.

We define total energy accretion rate (as measured at infinity), $F_{E}=\iint_{\theta,\varphi}{T^{r}}\!_{t}\,dA_{\theta\varphi}$, where ${T^{\mu}}\!_{\nu}$ is stress-energy tensor, the integral is over all $\theta,\varphi$ on BH horizon, and $F_{E}$ is positive if energy flows into the BH. We define energy outflow efficiency $\eta$ as the energy return rate to infinity divided by the time-average mass accretion rate:

Time-dependence of $\eta$ for our fiducial models, A$-0.9$f and A$0.9$f, is shown in Figure 1(e),(j). The outflow efficiency saturates at $t\gtrsim 6000r_{g}/c$ and clearly correlates with $\phi_{\rm BH}$. This is not surprising since we will see below that most of the outflowing energy is carried by BZ-driven jets, hence $\eta\approx\eta_{\rm BZ}\propto\phi_{\rm BH}^{2}$ (eq. 2). A retrograde BH traps $\simeq 30$ units of $\phi_{\rm BH}$ (panel d), while a prograde BH traps $\simeq 50$ units (panel i). Correspondingly, the retrograde BH produces outflows $(50/30)^{2}\approx 3$ times less efficiently (see eq. 2), with $\eta=34\pm 3$% (panel e), than the prograde BH, which has $\eta=102\pm 10$% (panel j and Table 1). It is interesting to compare these results to those for thicker discs, $\langle h/r\rangle_{30}\approx 0.6$, and a somewhat larger magnitude of BH spin, $\left|a\right|=0.9375$. The retrograde model has $\eta=88\pm 75$% and the prograde model has $\eta=354\pm 43$% (see Table 1 and McKinney et al. 2012). Thicker discs reach higher $\eta$, and prograde BHs have higher $\eta$ than retrograde BHs.

We repeated our fiducial models in a reduced azimuthal wedge, $\Delta\varphi=\pi$. We refer to them as A$-0.9$ and A$0.9$ and find $\eta=43\pm 8$% and $\eta=96\pm 17$%, respectively, in agreement with our fiducial models. We carried out a number of resolution studies (see Table 1) and find they agree with our fiducial models within $\eta$ measurement uncertainty (see Table 1): the mean efficiency of our retrograde models is $\eta=38\pm 4$% and of prograde models is $\eta=106\pm 7$%. Hence, $\eta$ values are converged in our fiducial models, which at $r=10r_{g}$ have $Q_{z}\simeq 95$ and $Q_{\varphi}\simeq 20$ cells per the fastest growing $z$- and $\varphi$- MRI wavelengths, respectively, and an equatorial cell aspect ratio, $\delta r\!:\!r\delta\theta\!:\!r\delta\varphi\approx 2\!:\!1\!:\!7$. Our highest $\varphi$-resolution model, A$0.9h^{2}_{\varphi}$, has $Q_{z}\approx Q_{\varphi}\approx 100$ and $\delta r\!:\!r\delta\theta\!:\!r\delta\varphi\approx 2\!:\!1\!:\!2$ and is very well resolved according to MRI resolution criteria (Hawley et al. 2011; Sorathia et al. 2011, e.g.,).

Why do outflow efficiencies of prograde and retrograde BHs differ? Can this be due to differences in initial conditions amplified by accretion flow turbulence? To test this, we ran the prograde model A$0.9$ until $t=14207r_{g}/c$ and instantaneously flipped the direction of BH spin. We refer to this model as A$-0.9$flip, which gives $\eta=40\pm 8$%, consistent with our previous retrograde results. But what about large-scale magnetic flux? Since its value is conserved, the differences in the flux cannot be erased by turbulence; can they affect the outcome? To test this, we carried out a series of simulations different only by the magnitude of the magnetic field, i.e., with $\beta_{\rm min}=\{25,50,100,200\}$. We refer to these simulations as models A$0.9$N25, A$0.9$N50, A$0.9$f (our fiducial prograde model), A$0.9$N200, respectively. The initial fluxes in these models differ by a factor of $\approx 3$. Does this lead to a similar difference in the magnetic flux that reaches the BH? Figure 2 shows that all models have equatorial magnetic flux profiles, $\langle\phi(r,\theta=\pi/2)\rangle$, that agree to $\lesssim 10\%$, and $\eta$ that agree to $\lesssim 20$% (see Table 1). This demonstrates that $\phi_{\rm BH}$ and $\eta$ are independent of the initial magnetic flux content of the flow. We also verified that our results are insensitive to the initial position of the torus (see model A$0.9$R20 in Table 1).

So far we considered the total energy output of the BH. What fraction of this energy goes into jets and into winds? The $t$- and $\varphi$-averages of our fiducial models, A$-0.9$f and A$0.9$f, shown in Figure 3, have similar hourglass shapes. The equatorial part of the accretion flow directly reaches the BH, while the inflow at higher latitudes turns around and forms weakly magnetized disc winds. These winds confine the highly magnetized relativistic polar jets that directly connect to the BH. Figure 3(a) shows that in the retrograde model, A$-0.9$f, the efficiency of the wind is $\eta_{\rm wind}(-0.9)\approx 7$%. The remainder comprises jet efficiency, $\eta_{\rm jet}(-0.9)\approx 34-7=27\%$. Similarly, Figure 3(b) shows that wind efficiency for the prograde model, A$0.9$f, is $\eta_{\rm wind}(0.9)\approx 16$%, so jet efficiency is $\eta_{\rm jet}(0.9)\approx 102-16=86$%. In both cases, the jet, which is predominantly powered by the BZ mechanism, carries most ($\simeq 80$%) of the total power output, and the power of the wind, launched by a BP-like mechanism, is subdominant. Note that all energy flow streamlines start on the BH and none start in the body of the disc. This is a manifestation of energy conservation: energy can either come from the BH via the release of gravitational binding energy or Penrose/BZ effect, or it can be advected inward from large distances by accreting material (if the accretion flow is unbound at large radii).

Our results differ from semi-analytical flux-trapping “gap” models (Garofalo 2009), which predict for $\left|a\right|=0.9$ that retrograde BHs produce $\gtrsim 10\times$ more powerful jets than prograde BHs. The “gap” models make use of a sharp transition at the ISCO to a plunge flow that sweeps into the BH all flux from an area (the “gap”) between the BH horizon and the ISCO. Since retrograde BHs have the largest ISCO area, in the “gap” model they receive the largest flux and produce the most powerful jets. However, poloidal flux distributions in our prograde and retrograde simulations are strikingly similar and show no “gap” (see Figures 2, 3): clearly, the “gap” model does not apply to thick radiatively-inefficient discs (unlike used by Garofalo 2009; Garofalo et al. 2010). This is not surprising because the sharp transition to the plunge at the ISCO, which the “gap” model relies on, is pronounced only for very thin ($h/r\lesssim 0.05$) radiatively-efficient discs (Penna et al. 2010). In addition, a key piece of physics is missing from flux-trapping “gap” models: “gap” models assume that magnetic field is weak and has no back-reaction on the plunging inflow. However, BH magnetic flux builds up to a natural saturation point at which the field does back-react: the magnetic flux not only modifies the rotation rate of plasma in the plunging region, it also escapes from the BH via magnetic interchange (Stehle & Spruit 2001; 44; McKinney et al. 2012). In our simulations, this non-axisymmetric mechanism fills-in the ISCO region with magnetic flux. No “gap” forms because the inflow of magnetic flux happens just as fast as the outflow of magnetic flux, unlike in the flux-trapping “gap” models. “Gap” models have not yet accounted for these effects, which dominate in MADs and, hence, in accretion systems with the most efficient jets.

Conclusions

We set out to determine whether prograde or retrograde BHs produce the most efficient outflows. Accretion discs with large-scale vertical magnetic fields naturally evolve into a magnetically-arrested disc (MAD) state in which the central BH is saturated with magnetic flux and the outflow efficiency ($\eta$) is maximum. Our simulations show that in this state, $\eta$ only depends on the BH spin, $a$, and the angular thickness of the accretion disc, $h/r$, and is independent of the initial magnetic flux content of the disc (see §2).

We find that for medium-thickness discs, $h/r\approx 0.3$, and an absolute value of spin, $\left|a\right|=0.9$, a retrograde BH produces jets and winds with an efficiency, $\eta=38\pm 4$%, which is a few times smaller than a prograde BH, $\eta=106\pm 7$%. In both cases, most of the energy ($\simeq 80$%) emerges in the form of BZ-powered, relativistic, highly magnetized collimated polar jets (see §2). We expect the above values of $\eta$ to serve as upper bounds on jet efficiency for lower BH spins, $\left|a\right|<0.9$, and thinner discs, $h/r<0.3$. We also find that a two-fold increase in disc thickness (to $h/r\approx 0.6$) leads to about a three-fold increase in $\eta$ (see §2). These results can be used to place limits on the spin of central BHs from the observed values of jet efficiency. However, based on jet energetics alone it will be challenging to tell apart prograde and retrograde BHs due to their similar efficiencies. Future studies should consider radiative effects, which are not included in the current work, and investigate spin-dependence of $\eta$ for thinner discs, with $h/r\lesssim 0.1$.

Acknowledgments

AT was supported by a Princeton Center for Theoretical Science Fellowship. AT is grateful to Perimeter Institute for hospitality. We thank A. Broderick, D. Caprioli, I. Contopoulos, D. Giannios, L. Lehner, M. Lyutikov, R. Narayan, R. Nemmen, D. Proga, J. Steiner, J. Stone, and D. Uzdensky for fruitful discussions. We thank the referee, C. Reynolds, for useful suggestions. We acknowledge NSF support via TeraGrid resources: NICS Kraken and Nautilus, where simulations were carried out and data were analyzed, and NCSA MSS and TACC Ranch, where data were backed up, under grant numbers TG-AST100040 (AT) and TG-AST080025N (JCM).

References

Supporting Information

Additional Supporting Information may be found in the online version of this article: Movie files. Movies of models A$-0.9$f and A$0.9$f (click for movies).