Modeling Polarized Emission from Black Hole Jets: Application to M87 Core Jet

1. Introduction

AGN jets are believed to be powered by the accretion of material from their host galaxy onto a supermassive black hole [1]. The long standing question is: What are the roles of the accretion disk, magnetic fields and the black hole spin in the formation of these relativistic flows?

We investigate the physical conditions in the jet launching zone in global three-dimensional, general relativistic magnetohydrodynamical (3-D GRMHD) simulations. The numerical models follow the dynamics of fully ionized plasma and magnetic fields down to the black hole event horizon. Similar simulations have been performed by several groups [2,3,4,5,6,7,8,9]. A recent breakthrough in GRMHD models of black hole accretion and jets comes from modeling the radiation from plasma near the black hole [10,11,12]. Until recently, the least understood part of these models was a proper treatment of radiating electrons. Finally, however, this issue has been addressed [13,14,15,16,17,18,19]. With new and more detailed observations, we are beginning to directly compare the numerical simulations of accretion flows to real astronomical systems such as Galactic center Sgr A* and the core of M87 galaxy [20].

2. GRMHD Models of Jets

Here, we describe a fiducial 3-D GRMHD model of accretion disk with a jet. Example simulations are carried out with the HARM-3D code [4,6] by [21]. The code solves ideal-MHD equations in a fixed Kerr metric. These models do not include radiative losses; hence, they are applicable to sources that are rather underluminous ($L/L_{\mathrm{Edd}}<{10}^{-7}$). Models with radiative looses taken into account selfconsistently shall be explored in the future [22,23].

At $t=0$, the plasma is confined to a geometrically-thick donut-shaped torus. The plasma density distribution, internal energy and velocity are computed using an analytical torus model presented in [24]. The inner edge of the torus is $r_{\mathrm{in}}=12GM/c^2$, and the position of the plasma pressure maximum is $r_{\max }=24GM/c^2$. The size of the computational domain extends to $r_{\mathrm{out}}=240GM/c^2$. We follow standard procedures and seed the initial plasma with a sub-thermal magnetic field ($\beta =P_{\mathrm{gas}}/P_{\mathrm{mag}}=10-100$), with its geometry aligned with the iso-density surfaces of the torus (the so-called single loop scenario). The free parameter is the black hole spin chosen to be $a_{*}=cJ/G{M}^2=0.9375$.

Jets appear almost naturally in our simulations. The rotational energy of the black hole and accretion disk is extracted through a combination of the well-known Blandford-Znajek and Blandford-Payne mechanisms [25,26]. Typically, the models described here are called SANE in the current jargon, which stand for Standard And Normal Evolution, having low-power jets. Other solutions with stronger magnetic fields and more powerful jets are equally possible [27,28].

3. Electron Treatment and Modeling Polarized Emission from GRMHD Jets

The GRMHD simulations provide only the fluid pressure, which is dominated by the protons. In a perfect fluid, the pressure in a grid zone gives a proton temperature. For underluminous accretion flows (in Sgr A* and M87), protons and electrons are not necessarily well coupled. We have to assume an electron temperature as they are not self-consistently computed in the considered GRMHD simulations, but they are essential in calculating the synchrotron emission.

Inspired by early models for radio cores in quasars [1] and the Solar corona plasma models [19,29,30], we have developed a simple parametric description for electron thermodynamics in the simulation of accretion flows. In combination with radiative transfer model, this finally makes our numerical simulations of accretion flows resemble observations of underluminous accreting black holes [13,14,16,18,31].

Our parametric model for electrons assumes a thermal relativistic (Maxwell-Jüttner) distribution function. The electron temperatures, $T_{\mathrm{e}}$, are computed assuming that the proton-to-electron coupling depends on plasma magnetization [14,16,18]:

$$\frac{T_{\mathrm{p}}}{T_{\mathrm{e}}}=R_{\mathrm{low}}\frac{1}{1+{\beta }^2}+R_{\mathrm{high}}\frac{{\beta }^2}{1+{\beta }^2},$$

(1)

where $\beta =P_{\mathrm{gas}}/P_{\mathrm{mag}}$ is ratio of the gas pressure to magnetic field pressure $P_{\mathrm{mag}}=B^2/2$ (where B and hence $P_{\mathrm{mag}}$ and $P_{\mathrm{gas}}$ are in the HARM code units). $R_{\mathrm{low}}$ and $R_{\mathrm{high}}$ are two free parameters. In a strongly magnetized plasma, $\beta \ll 1$ and $T_{\mathrm{p}}/T_{\mathrm{e}}\to R_{\mathrm{low}}$. In a weakly magnetized plasma, $\beta \gg 1$ and so $T_{\mathrm{p}}/T_{\mathrm{e}}\to R_{\mathrm{high}}$. We set $R_{\mathrm{low}}=1$ and $R_{\mathrm{high}}=100$ so that the electrons are always cooler in the disk but hotter towards the jet, making the jet more visible.

A ray-tracing approach is used to construct mock observations of GRMHD simulations. The radiative transfer equations through changing plasma conditions for a single photon frequency $\nu$ are integrated along null-geodesic paths. This approach is valid only if the plasma index of refraction is one. The ray-tracing radiative transfer methods can be also used only if the photon frequency is much larger than the plasma cyclotron frequency, and is larger than the plasma frequency, ${\nu }_p$ (i.e., $\nu \gg {\nu }_c=2.8\times {10}^{6}B$, $\nu \gg {\nu }_p=8980n_e^{1/2}$ Hz). In our GRMHD models, typically, $n_e={10}^2$ – ${10}^7{\mathrm{cm}}^{-3}$ and $B=0.1$ – 100 Gauss, so our plasma ${\nu }_c=2.8\times {10}^{-4}$ – $0.28$ GHz and ${\nu }_p=8.9\times {10}^{-5}$ – $0.028$ GHz. Here we model emission at $\nu =43-230$ GHz ($\lambda$=7 – 1mm), so the approach is valid.

Several existing numerical codes can generate polarized images of radiative plasma near black holes (e.g. [32,33]). In this proceeding we present results based on our new radiative transfer code ipole [34]. The unique feature of our scheme is that it is fully covariant, and so it is suitable for parallel transport of polarized light rays in arbitrary spacetimes and coordinates. We also use the analytical solution to evolve Stokes vector that guarantees stability of the radiative transfer solutions regardless of physical conditions in the plasma. Our model is therefore applicable to plasma with high optical and Faraday depth, which are likely common in jets embedded in cooler accretion flows. We have demonstrated that our integration scheme is stable and accurate; it reproduces known analytic radiative transfer solutions in our method paper [34].

4. Simulated Emission from M87 Core Jet

Figure 1 displays the apperance of the simulations scaled to M87 central supermassive black hole with its mass $M_{\mathrm{BH}}=6.2\times {10}^9{\mathrm{M}}_{\odot }$. We show the model in polarized light at three different wavelengths ($\lambda$= 7.0, 3.5 and 1.3 mm) at which the source is often observed [35,36,37]. The 1.3 mm synthetic images have been published in [16,18]. Here, we have extended our previous results to the other wavelengths.

Synchrotron emission is intrinsically highly polarized. For ultra-relativistic thermal electrons, the linear polarization fraction measured locally can reach nearly 100% [38]. In our images, the brightest regions are typically linearly depolarized and polarization ticks are scrambled. Both effects are due to strong Faraday effects. We found that the Faraday optical depth in our models is ${\tau }_F\gg 1$. This suggests that even though the Advection Dominated Aaccretion Flows (ADAF)-type accretion flows are optically thin, they can still be Faraday thick (for details see [18]). Interestingly, in our models we also observe non-zero circular polarization. The theoretical synchrotron maps should be convolved with instrumental effects before direct comparison to the Very Long Baseline Interferometric images on M87 core jet [37,39].

A direct application of our polarized images is to model an observed rotation measure (RM) and provide a model dependent constrain on the mass accretion/outflow rate of the M87 black hole. There are a few observations of RM near supermassive black holes [40,41,42,43,44], including the core of M87 with $\left|\mathrm{RM}\right|<7.5\times {10}^5\mathrm{rad}{\mathrm{m}}^{-2}$ [36]. All these RM are measured at millimeter wavelengths with non-VLBI observations thus the sources are not resolved. The observed RM was then used to constrain the mass accretion rate onto the central object. The usual procedure to interpret the observed RM is to integrate

$$\mathrm{RM}={10}^4\frac{e^3}{2\pi m_{\mathrm{e}}^2{c}^4}\int f_{\mathrm{rel}}{n}_{\mathrm{e}}{B}_{\Vert }dl\left[\mathrm{rad}{\mathrm{m}}^{-2}\right]$$

(2)

in the radial direction with a power-law profiles for $n_e$ and B (usually from equipartition condition) from a semi-analytical ADAF model (e.g., [45]). In Equation (2), all physical constants are in c.g.s. units while ${10}^4$ factor converts RM from c.g.s. to SI units. Notice also that Equation (2) assumes that the polarized source behind is an ’external’ Faraday screen. The accretion rate onto the black hole is then provided by the ADAF model [46]. In Sgr A*, the rotation measure of $-5.6\pm 0.7\times {10}^5\mathrm{rad}{\mathrm{m}}^{-2}$ was used to estimate $\dot{M}=5\times {10}^{-9}-2\times {10}^{-7}{\mathrm{M}}_{\odot }{\mathrm{yr}}^{-1}$ where the lower and upper limits correspond to different slopes of the electron density radial profiles, which depend on the presence and properties of an outflow assumed to be produced by accretion process [40,41,46]. Following the same procedure, the upper limit for RM in the M87 core led to an accretion rate constraint of $\dot{M}<9\times {10}^{-4}{\mathrm{M}}_{\odot }{\mathrm{yr}}^{-1}$ [36,47].

Our radiative transfer model allow us to directly calculate the change of polarization angle $\chi \equiv arg(Q+iU)/2$ as a function of wavelength and compute the model $\mathrm{RM}=({\chi }_{{\lambda }_1}-{\chi }_{{\lambda }_2})/({\lambda }_1^2-{\lambda }_2^2)$ without using approximate formulas, such as Equation (2). Contrary to many previous RM modelings, in the current RM model all relativistic effects will be included; the observer viewing angle effects and the magnetic field geometry are naturally taken into account; the mass accretion rate onto the black hole is self-consistently calculated within the GRMHD simulation. Based on radiative transfer calculations at 220–230 GHz our model predict observed RM=$6\times {10}^5\mathrm{rad}{\mathrm{m}}^{-2}$ which is consisitent with observations. Interestingly, our model accretion rate is $\dot{M}=9\times {10}^{-3}{\mathrm{M}}_{\odot }{\mathrm{yr}}^{-1}$, which is an order of magnitude larger than previous estimates for this source.

5. Discussion

The polarized emission models demonstrate that observed changes in total polarization angle should be interpreted more carefully as they could be model dependent. The previous constraints of accretion rate onto underluminous black holes assumed that the polarized emission is produced near the black hole, and is Faraday rotated in the extended accretion flow. In Sgr A*, the compact emission region [48], observed scaling of $\chi \sim {\lambda }^2$, and $\simeq 10$ per cent RM variability [41] support this scenario and motivate the use of Equation (2). In the M87 jet models the accretion flow is simultaneously the source of synchrotron radiation and the Faraday screen. Consequently, the polarization plane position angle $\chi$ does not have to be constant with ${\lambda }^2$ (as if it were the polarized source behind an ’external’ Faraday screen) and Equation (2) may not be directly applicable. This is because the accretion flow has complex structure where the self-absorption and depolarization of radiation take place. Additionally, especially in M87, the emission near the black hole may not be produced by an accretion flow such as ADAF, but, e.g., by a jet; hence, the existing measurements would be probing the plasma flowing out from the central region (also because of our viewing angle of the jet). In our models, the polarized flux is produced and Faraday rotated in the forward jet. As a consequence the accretion rate onto the black hole is allowed to be higher.