Synchrotron radiation from hot gas near a black hole results in a polarized image. The image polarization is determined by effects including the orientation of the magnetic field in the emitting region, relativistic motion of the gas, strong gravitational lensing by the black hole, and parallel transport in the curved spacetime. Now, Narayan and colleagues explored these effects using a simple model of an axisymmetric, equatorial accretion disk around a Schwarzschild black hole. Their study recently appeared in the Astrophysical Journal .
The Event Horizon Telescope (EHT) Collaboration has recently published the first images of a black hole. These images achieve a diffraction limited angular resolution that corresponds to approximately 5GM/c², where M is the mass of the black hole. They reveal a bright ring of emission with a twisting polarization pattern and a prominent rotationally symmetric mode.
The polarization structure in the EHT images depends on details of the emitting plasma, principally the magnetic field geometry. However, it is also affected by the strongly curved spacetime near the black hole. Over the past few decades, simulated polarimetric images of black holes have been studied as a means to understand astrophysical properties of their surrounding accretion flows and to infer the disk inclination and black hole spin through the effects of parallel transport. While they are becoming increasingly realistic, these simulations are generally difficult to use for broad parameter surveys because of their computational cost, and they often provide little insight into how to decouple astrophysical and relativistic effects.
Now, Narayan and colleagues develop a simple toy model to understand polarimetric images of black holes. This model consists of a ring of magnetized fluid orbiting a Schwarzschild black hole. Their model allows arbitrary emission radius, magnetic field geometry, equatorial fluid velocity, and observer inclination.
“We test this model using currently favored general relativistic magnetohydrodynamic simulations of M87*, using ring parameters given by the simulations.”
They considered an accretion disk around a Schwarzschild black hole of mass M. They used standard geometrized units: G = c = 1. The fluid radiates from the equatorial plane within a narrow range of radii centered on a dimensionless radius R, measured in units of M (or GM/c², including the physical constants). With respect to a distant observer, the ring is tilted from a face-on orientation by an angle θo. They assumed that the tilt is towards the North, so that the line-of-nodes between the ring orbital plane and the observer’s sky plane is in the East-West direction. They take the sky angular coordinate x to be oriented towards the West (i.e., to the right), and the coordinate y towards the North (i.e., towards the top). The fluid has radial and tangential components of velocity in the plane of the ring, but no vertical velocity. In the comoving frame of the fluid, the magnetic field has radial, azimuthal and vertical components.
Later, they computed the following primary observables: (1) the shape of the ring as viewed by the distant observer, (2) the variation of the polarized intensity around the observed ring, and (3) the orientation and pattern of the polarization vectors around the ring. And, then compared EHT polarimetric image of M87 with their ring model.
1) Of Primary Observables
For first case they found that, the shape of ring appears to be flat because of tilt. Means, it appears elliptical in shape, with a minor axis radius equal to cos θo ≈ 1 − (1/2) sin²θo times the original ring radius. The sin θo terms describe the effect of tilt on lensing. Geodesics reaching the observer from the upper half of the ring (0 < φ < π) travel a longer distance near the black hole and suffer more deflection (this is the case shown schematically in Fig. 2), while geodesics from the lower half (π < φ < 2π) experience less deflection. This causes an upward shift of the observed ring, i.e., a net positive bias in y. The shift is of the order of sin θo in units of GM/c².
While for second case, they found that, observed polarized intensity depends on the Doppler factor δ as well as the path length lp and the angle ζ between the photon wave-vector ~k(F) in the fluid frame and the local magnetic field B~. For small tilt angles, the variation in the path length is small and not very important.
Finally, for third case, they found that, for a radial velocity (χ = π), the phase of β2 is π, i.e., the polarization vectors should be tangentially oriented. This is indeed seen in the brightest part of the ring in the Bottom Left panel in Fig. 3. Similarly, for a tangential velocity (χ = −π/2), the phase of β2 = 0 and the polarization ticks should be radial, as seen in the Top Right panel of Fig. 3. Finally, if there is no velocity but we consider strong lensing (small R), then equation (40) shows that β2 has phase = π and the polarization should be tangential, as in the Bottom Right panel.
2) Of Comparison
They also compared EHT polarimetric image of M87 with their ring model and found that the fractional polarization of their model is significantly higher than that seen in EHT images of M87* (EHTC VII). This may indicate significant sub-beam depolarization, potentially from strong internal Faraday effects (EHTC VIII). If so, observations at higher frequencies, where Faraday effects are suppressed, may show significantly higher image polarizations, while observations at lower frequencies are expected to show a heavily depolarized “core.”
Their polarized ring model provides intuition and insights about how a black hole’s accretion flow and spacetime combine to produce a polarized image. It also provides a pathway to constrain these physical properties through direct comparisons with data and images from the EHT, GRAVITY, and future X-ray polarimetry studies.
“Further studies which examine the capability of the model in matching snapshots of GRMHD simulations with similar magnetic field and flow conditions will elucidate how readily field geometries may be directly inferred from polarized images.”— concluded authors of the study
Reference: Ramesh Narayan, Daniel C. M. Palumbo, Michael D. Johnson, Zachary Gelles, Elizabeth Himwich, Dominic O. Chang, Angelo Ricarte, Jason Dexter, Charles F. Gammie, Andrew A. Chael, The Event Horizon Telescope Collaboration: Kazunori Akiyama, Antxon Alberdi, Walter Alef, Juan Carlos Algaba, Richard Anantua, Keiichi Asada, Rebecca Azulay, Anne-Kathrin Baczko, David Ball, Mislav Balokovic, John Barrett, Bradford A. Benson, Dan Bintley, Lindy Blackburn, Raymond Blundell, Wilfred Boland, Katherine L. Bouman, Geoffrey C. Bower, Hope Boyce, Michael Bremer, Christiaan D. Brinkerink, Roger Brissenden, Silke Britzen, Avery E. Broderick, Dominique Broguiere, Thomas Bronzwaer, Do-Young Byun, John E. Carlstrom, Chi-kwan Chan, Shami Chatterjee, Koushik Chatterjee, Ming-Tang Chen, Yongjun Chen, Paul M. Chesler, Ilje Cho, Pierre Christian, John E. Conway, James M. Cordes, Thomas M. Crawford, Geoffrey B. Crew, Alejandro Cruz-Osorio, Yuzhu Cui, Jordy Davelaar, Mariafelicia DeLaurentis, Roger Deane, Jessica Dempsey, Gregory Desvignes, Sheperd S. Doeleman, Ralph P. Eatough, Heino Falcke, Joseph Farah, Vincent L. Fish, Ed Fomalont, H. Alyson Ford, Raquel Fraga-Encinas, Per Friberg, Christian M. Fromm, Antonio Fuentes, Peter Galison, Roberto Garcıa, Olivier Gentaz, Boris Georgiev, Ciriaco Goddi, Roman Gold, Jose L. Gomez, Arturo I. Gomez-Ruiz, Minfeng Gu, Mark Gurwell, Kazuhiro Hada, Daryl Haggard, Michael H. Hecht, Ronald Hesper, Luis C. Ho, Paul Ho, Mareki Honma, Chih-Wei L. Huang, Lei Huang, David H. Hughes, Shiro Ikeda, Makoto Inoue, Sara Issaoun, David J. James, Buell T. Jannuzi, Michael Janssen, Britton Jeter, Wu Jiang, Alejandra Jimenez-Rosales, Svetlana Jorstad, Taehyun Jung et al., “The Polarized Image of a Synchrotron Emitting Ring of Gas Orbiting a Black Hole”, Astrophysical Journal, ApJ 912(1), pp. 1-29, 2021. DOI: 10.3847/1538-4357/abf117
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