On the Zeeman Effect in Magnetically-Arrested Disks

TL;DR

This paper proposes that Zeeman splitting of Fe Kα lines in X-ray spectra can serve as an observational signature of magnetically arrested disks (MAD), potentially confirming their presence in accreting systems.

Contribution

It introduces a novel method to detect MAD through Zeeman splitting of Fe lines, linking magnetic field measurements to accretion physics.

Findings

Zeeman splitting could be observable with future X-ray satellites.

MAD induces specific spectral signatures in Fe Kα lines.

Turbulence broadens but does not obscure Zeeman splitting signals.

Abstract

Magnetically arrested disk (MAD) has been argued as the key accretion phase to realize the formation of relativistic jets. However, due to the lack of magnetic field measurements of accreting systems, MAD has not been observationally confirmed yet. Here we propose that a strong magnetic field accompanied by MAD would induce the Zeeman splitting of relativistically broadened Fe K$\alpha$ fluorescence lines in X-ray binaries and active galactic nuclei, where we consider a two-phase medium in the inner accretion disk, magnetically dominated hot corona and cold reflector. Such a geometrical configuration is suggested from X-ray observations and recently confirmed by numerical simulations. Although turbulence in accretion flows would broaden the split lines, future X-ray high-energy resolution satellites, XRISM and Athena, would be capable of seeing the Zeeman effect on the Fe lines in X-ray binaries in the case with the MAD configuration. The signature of the Zeeman split lines would provide observational evidence for MAD.

Equations

$${\Phi }_{\mathrm{BH}}\equiv {\varphi }_{\mathrm{BH}}({\dot{M}}_{\mathrm{BH}}{R}_g^{2}c{)}^{1/2},$$

$$B(R)$$

$$=$$

$$\simeq$$

$$\Delta E_{\mathrm{split}}\approx\frac{e\hbar}{2m_{e}c}\left(M_{L}+2M_{S}\right)B\simeq 11.6~{}\mathrm{eV}~{}\biggl{(}\frac{B}{10^{9}~{}\mathrm{G}}\biggr{)}$$

$$\Delta E_{\mathrm{split}}\simeq 27~{}\mathrm{eV}~{}\biggl{(}\frac{\phi}{30}\biggr{)}\biggl{(}\frac{m}{10}\biggr{)}^{-1/2}\biggl{(}\frac{\dot{m}}{0.3}\biggr{)}^{1/2}\biggl{(}\frac{r}{1}\biggr{)}^{-p}.$$

$$\Delta E_{\mathrm{shift}}=\frac{e^2{a}_0^2}{8Z^2{m}_e{c}^2}{n}^4(1+M_L^2)B^2$$

$$\displaystyle\simeq 9.6\times 10^{-3}~{}\mathrm{eV}~{}\biggl{(}\frac{\phi}{30}\biggr{)}^{2}\biggl{(}\frac{m}{10}\biggr{)}^{-1}\biggl{(}\frac{\dot{m}}{0.3}\biggr{)}\biggl{(}\frac{r}{1}\biggr{)}^{-2p},$$

Full text

1 Introduction

It has been more than a century since astrophysical jets were first observed (Curtis, 1918). Astrophysical jets are collimated relativistic magnetized plasma outflows launched from compact accreting objects. They are found in stellar-mass black holes (BH; Fender et al. (2004)) and supermassive BHs (Blandford et al., 2019). Powers of some relativistic jets can exceed the Eddington limit of BHs, requiring highly efficient energy conversion processes from accretion to outflows. However, the formation mechanism of powerful relativistic jets has yet to be answered.

Theoretically, the Blandford-Znajek (BZ) mechanism (Blandford & Znajek, 1977) is believed as the plausible explanation for the jet launch. In the BZ mechanism, the jet power is extracted by the rotation of BHs with the support of the magnetic fields threading the central BH. General relativistic magnetohydrodynamic (GRMHD) simulations confirm this process as a plausible and efficient jet power extraction mechanism (see, e.g., Komissarov et al. (2007); Tchekhovskoy et al. (2010, 2011); McKinney et al. (2012); Takahashi et al. (2016); Avara et al. (2016); Liska et al. (2022)).

The BZ mechanism has two key parameters: the spin parameter of the BH and the large-scale magnetic field threading in the BH. Numerical simulations suggest that strong magnetic fields are required to realize observed powerful jets (e.g., McKinney et al. (2012)). Such large magnetic flux accumulation is expected to appear in the magnetically arrested disk (MAD) scenario (Narayan et al. (2003); Igumenshchev et al. (2003), see also Bisnovatyi-Kogan & Ruzmaikin (1974, 1976)), where magnetic field dominates the dynamics of the inner disk. Although the event horizon telescope (EHT) has resolved inner accretion disks of M 87 and Sgr A* with unprecedented spatial resolutions, it is still not conclusive whether their accretion processes are dominated by MAD or other processes (Event Horizon Telescope Collaboration et al., 2021, 2022; Blandford & Globus, 2022). Therefore, observational evidence of MAD in accretion systems is still lacking.

Here, the presence of a magnetic field induces spectral line splitting by the Zeeman effect due to the interaction of the magnetic dipole moment of an electron with the magnetic field. The Zeeman effect has been applied to measure magnetic fields of various astrophysical systems such as sunspots (Hale, 1908), active stars (Donati et al., 1997), molecular clouds (Nakamura et al., 2019), and an outer maser disk of an AGN (Modjaz et al., 2005). Detectability of the Zeeman effect in the X-raying accreting neutron stars has also been argued in the literature (Sarazin & Bahcall, 1977; Loeb, 2003).

BH-powered X-ray binaries (XRBs) and active galactic nuclei (AGNs) ubiquitously have the Fe K$\alpha$ fluorescence line in their X-ray spectra. Broaden Fe K$\alpha$ lines imply the location of the cold reflecting medium near the BHs and have been used for the BH spin measurements (Reynolds, 2021). Here, hot plasma, namely coronae, should also exist in the vicinity of BHs to reproduce X-ray continuum spectra of accreting BHs. Geometrical configurations of hot coronae and cold reflectors have been debated in literature (see e.g., Done et al. (2007); Meyer-Hofmeister & Meyer (2011)). Recently, by performing two temperature GR-radiation-MHD simulations, Liska et al. (2022) demonstrated that a geometrically thin accretion disk transitions into a two-phase medium of cold gas clumps and a hot, magnetically dominated corona, when the thin disk is threaded by large-scale poloidal magnetic fields. This numerical result can naturally explain the coexistence of broaden Fe K$\alpha$ line and hard X-ray continuum emission in XRBs and AGNs.

In this Letter, we consider the Zeeman effect on the Fe K$\alpha$ lines of XRBs and AGNs in the MAD state assuming the two-pahse medium in the inner accretion disk. Future X-ray satellite missions such as X-Ray Imaging and Spectroscopy Mission (XRISM; Tashiro et al. (2020)) and Advanced Telescope for High ENergy Astrophysics (Athena; Nandra et al. (2013)) will carry an X-ray microcalorimeter with an energy resolution down to several eV at 6 keV. We also discuss whether future X-ray missions can probe the MAD via the Zeeman effect.

2 Zeeman Effect on MAD

Magnetic flux $\Phi_{\mathrm{BH}}$ threading a BH is described as

[TABLE]

where $\phi_{\mathrm{BH}}$ is the dimensionless magnetic flux, $\dot{M}_{\mathrm{BH}}$ is the accretion rate onto the BH, and $R_{g}=GM_{\mathrm{BH}}/c^{2}$ is the gravitational radius. $M_{\mathrm{BH}}$ is the BH mass, $G$ is the gravitational constant, and $c$ is the speed of the light. $\Phi_{\mathrm{BH}}$ in accretion flows are always nonzero since the magnetic flux is transported inward via accretion. $\phi_{\rm BH}$ is typically 20–50 depending on accretion rates based on GRMHD simulations (e.g., McKinney et al. (2012); Avara et al. (2016); Liska et al. (2022)). We set $\phi_{\rm BH}=30$ as a fiducial value, which is based on the recent GR-radiation-MHD simulation accounting for both hot corona and cold medium at $\sim 35$% of the Eddington luminosity (Liska et al., 2022).

The MAD magnetic field at a distance $R\equiv rR_{g}$ from the BH is, assuming $BR^{p}=\mathrm{const.}$,

[TABLE]

where $m\equiv M_{\mathrm{B}H}/M_{\odot}$ and $\dot{m}\equiv\dot{M}_{\mathrm{BH}}/\dot{M}_{\mathrm{Edd}}$ with a 10% radiative efficiency. $\dot{M}_{\mathrm{Edd}}$ is the Eddington accretion rate for the mass of $M_{\mathrm{BH}}$.

The energy separation by the Zeeman effect is given by

[TABLE]

where $M_{L}$ and $M_{S}$ are the quantum numbers of the orbital angular momentum and the spin angular momentum. We consider the split transition lines associated with changes of $\Delta M_{L}=\pm 1$ and $\Delta M_{S}=0$.

In the accreting system, Eq. 5 can be replaced as

[TABLE]

Therefore, we can expect the line splitting of Fe K$\alpha$ lines at the level of several tens of eV for XRBs and $10^{-3}$ eV for AGNs with $\phi_{\mathrm{BH}}=30$, $\dot{m}=0.3$, and $p=2$.

The energy resolution of current X-ray CCDs at the Fe K$\alpha$ line is $\sim 120$ eV (Ezoe et al., 2021), larger than the typically expected split. With this resolution, we may infer $B<10^{10}$ G for any available observations. Next-generation X-ray telescopes such as XRISM and Athena will have X-ray micro-calorimeter instruments. The planned energy resolution is 7.0 eV and 2.5 eV at 6 keV for XRISM and Athena, respectively. Figure 1 shows the expected Fe K$\alpha$ line split for various BH masses and accretion rates with $\phi_{\mathrm{BH}}=30$, $p=2$, and $r=2$. Figure 1 also shows the energy resolutions of XRISM and Athena. Next-generation X-ray telescopes will enable us to see the Zeeman line splitting in XRBs with the presence of MAD.

About 20 XRBs have dynamically confirmed BHs (Corral-Santana et al., 2016). Among them, low-mass black holes such as GX 339-4 and GRO J1655-40, whose BH mass is $5.8\pm 0.5~{}M_{\odot}$ (Hynes et al., 2003) and $5.4\pm 0.3~{}M_{\odot}$ (Beer & Podsiadlowski, 2002), respectively, would be possible candidates to see the Zeeman effect on MAD. Another candidate is a persistent X-ray binary Cyg X-1 having the BH mass of $14.8\pm 1.0~{}M_{\odot}$ (Orosz et al., 2011)111Recent radio astrometric observation suggests its mass of $m=21.2\pm 2.2$ (Miller-Jones et al., 2021). A broad iron K$\alpha$ line has also been reported for Cyg X-1 (Duro et al., 2011, 2016), with the inner edge of the disk of $\sim 1.6R_{g}$ (Fabian et al., 2012). Although the accretion rate depends on the states, it is about 1% even in the low/hard state (Yamada et al., 2013). The expected $\Delta E_{\mathrm{split}}$ is $2.7$ and $8.5$ eV for $\dot{m}=0.03$ and $0.3$, respectively. Therefore, XRISM and Athena would see the Zeeman effect on MAD even for a relatively massive stellar BH object Cyg X-1.

3 Discussions

The Fe K$\alpha$ lines appear broadened and skewed by the Doppler effect and gravitational redshift (Fabian et al., 1989; Laor, 1991). Those relativistic effects would blur the Zeeman split lines. Figure 2 shows a simulated Fe K$\alpha$ line in the MAD. We apply the kerrdisk code (Brenneman & Reynolds, 2006) with emissivity indices of $\alpha_{1}=\alpha_{2}=0$, inclination of $i=30^{\circ}$, dimensionless spin parameter of $a=0.9$, inner and outer radii of $r_{\mathrm{min}}=r_{\mathrm{ms}}$ and $r_{\mathrm{max}}=100~{}r_{\mathrm{ms}}$, and redshift $z=0$. $r_{\mathrm{ms}}$ is the marginally stable orbit radius. We include three lines split by the Zeeman effect assuming $\Delta E_{\mathrm{split}}=24~{}\mathrm{eV}$, corresponding to $B\simeq 2\times 10^{9}~{}\mathrm{G}$, we also set $p=0$. As the figure shows, three split lines would be distinguishable within $\Delta E_{\mathrm{split}}=24$ eV.

There also exists Doppler broadening due to turbulent motions of accreting flows. The turbulence speed is characterized by the sound speed. The broadening by turbulence is expected at the level of $\sim 10$ eV at the temperature of $\sim 10^{7}$ K, which can be comparable to the expected Zeeman split (Eq. 6). Three split lines would be more broaden by turbulence than shown in Figure 2. In addition, the effect of the continuum is not included in Figure 2 and the energy split would depend on radius if $p\neq 0$. Further detailed spectral simulations, including instrument response functions of XRISM and Athena, will be needed.

Broad Fe K$\alpha$ lines are commonly reported in various XRBs and AGNs (Reynolds, 2021). However, line broadening is known to depend on the spectral modeling (see, e.g., Done & Zycki (1999); Makishima et al. (2008)) and disk winds further would disturb the ionization states (see e.g., Tomaru et al. (2019)). If the location of reflecting iron atoms is further away from the BH, the expected line split drops with $r^{-p}$. Determination of the reflecting medium is necessary to probe the MAD through X-ray Zeeman effect measurements. Therefore, the Zeeman effect on the Fe K$\alpha$ line is realized only with the MAD state and the near BH reflector case. In other words, if we do not see the Zeeman effect even with sufficient energy resolutions, it will imply that MAD is absent or the reflector is distant.

With a strong magnetic field like MAD, we would also have the quadratic Zeeman effect producing displacements of lines toward shorter wavelengths (Jenkins & Segrè, 1939; Schiff & Snyder, 1939; Preston, 1970; Loeb, 2003). The energy shift is given as

[TABLE]

where $n$ is the principal quantum number, $a_{0}$ is the Bohr radius, and $Z$ is the nuclear charge. We set $n=1$, $M_{L}=1$ and $Z=26$ in Eq. 8. Thus, several orders of magnitude better energy resolution would be necessary to see the quadratic energy shift.

4 Summary

In this Letter, we consider the Zeeman effect on the MAD state. MAD is expected to be associated with the jet production (Tchekhovskoy et al., 2010, 2011; McKinney et al., 2012). In the black hole accretion systems, broad Fe K$\alpha$ fluorescence lines have been often reported. A strong magnetic field environment by MAD would induce line splitting of the Fe K$\alpha$ line by the Zeeman effect (Eq. 6). Next-generation X-ray telescopes such as XRISM and Athena will be able to see the Zeeman splitting of Fe lines in XRBs, if reflectors exist near BHs. The detection of the Zeeman effect would be clear evidence of the MAD in the BH accretion systems. If the Zeeman effect does not appear even with sufficient energy resolutions, it would imply that MAD is absent or the iron reflector is distant.

{ack}

We would like to thank the anonymous referee for thoughtful and helpful comments. We would also like to thank Roger Blandford, Chris Done, Norita Kawanaka, Katsunori Kusakabe, Shin Mineshige, Hirokazu Odaka,and Shinsuke Takasao for useful discussions and comments. Y.I. is supported by JSPS KAKENHI Grant Number JP18H05458, JP19K14772, and JP22K18277. This work was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan.