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Christian M. Fromm

Publications and source records attributed to Christian M. Fromm.

At least 37 records · Page 2Linked to original sources

Relativistic reflection within an extended hot plasma geometry

The reflection of X-rays at the inner accretion disk around black holes imprints relativistically broadened features in the observed spectrum. Aside from the black hole properties and the ionization and density of the accretion disk, these features also depend on the location and geometry of the primary source of X-rays, often referred to as the corona. We present a fast general relativistic model for spectral fitting of a radially extended, ring-like corona above the accretion disk. A common approach used to explain observed X-ray reflection spectra is the lamp post geometry, which assumes a point-like source on the rotational axis of the black hole. While it is typically able to explain the observations, this geometric model does not allow for any constraint to be placed on the radial size of the corona. We therefore extended the publicly available relativistic reflection model relxill by implementing a radially extended, ring-like primary source. With the new RELXILL model allowing us to vary the position of the primary source in two dimensions, we present simulated line profiles and spectra and discuss the implications of carrying out a data fitting, in comparison to the lamp post model. We applied this extended RELXILL model to XMM-Newton and NuSTAR data of the radio-quiet Seyfert-2 active galactic nucleus (AGN) ESO 033-G002. The new model describes the data well and we are able to constrain the distance of the source to the black hole to be less than three gravitational radii, while the angular position of the source is poorly constrained. We show that a compact, radially extended corona close to the innermost stable circular orbit is able to explain the observed relativistic reflection as well as the lamp post corona does. This model has been made freely available to the community.

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Probing the disk-jet coupling in M87

Context. Recent GMVA observations of M 87 at event horizon scales revealed a ring-like structure which is 50% larger at 86 GHz than the ring observed by the Event Horizon Telescope at 230 GHz. Aims. In this paper, we study a possible origin of the increased ring size at 86 GHz. We specifically aim to study the role the nonthermal electron population plays in the observed event horizon scales. Methods. We carry out 3D general relativistic magnetohydrodynamic simulations followed by radiative transfer calculations. We incorporate into the latter synchrotron emission from both thermal and nonthermal electrons. To better compare our results to observations, we generate synthetic interferometric data adjusted to the properties of the observing arrays. We fit geometrical models to this data in Fourier space through Bayesian analysis to monitor the variable ring size and width over the simulated time span of years. Results. We find that the 86 GHz ring is always larger than the 230 GHz ring, which can be explained by the increased synchrotron self-absorption at 86 GHz and the mixed emission from both the accretion disk and the jet footpoints, as well as flux arcs ejected from a magnetized disk. We find agreement with the observations, particularly within the error range of the observational value of M/D for M 87. Conclusions. We show that state-of-the art 3D GRMHD simulations combined with thermal and nonthermal emitting particles can explain the observed frequency-dependent ring size in M 87. Importantly we found that MAD events triggered in the accretion disk can significantly increase the lower frequency ring sizes.

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Probing jet base emission of M87* with the 2021 Event Horizon Telescope observations

We investigate the presence and spatial characteristics of the jet base emission in M87* at 230 GHz, enabled by the enhanced uv coverage in the 2021 Event Horizon Telescope (EHT) observations. The addition of the 12-m Kitt Peak Telescope and NOEMA provides two key intermediate-length baselines to SMT and the IRAM 30-m, giving sensitivity to emission structures at scales of $\sim250~μ$as and $\sim2500~μ$as (0.02 pc and 0.2 pc). Without these baselines, earlier EHT observations lacked the capability to constrain emission on large scales, where a "missing flux" of order $\sim1$ Jy is expected. To probe these scales, we analyzed closure phases, robust against station-based gain errors, and modeled the jet base emission using a simple Gaussian offset from the compact ring emission at separations $>100~μ$as. Our analysis reveals a Gaussian feature centered at ($Δ$RA $\approx320~μ$as, $Δ$Dec $\approx60~μ$as), a projected separation of $\approx5500$ AU, with a flux density of only $\sim60$ mJy, implying that most of the missing flux in previous studies must arise from larger scales. Brighter emission at these scales is ruled out, and the data do not favor more complex models. This component aligns with the inferred direction of the large-scale jet and is consistent with emission from the jet base. While our findings indicate detectable jet base emission at 230 GHz, coverage from only two intermediate baselines limits reconstruction of its morphology. We therefore treat the recovered Gaussian as an upper limit on the jet base flux density. Future EHT observations with expanded intermediate-baseline coverage will be essential to constrain the structure and nature of this component.

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Quasi-Periodic Polarized Emissions from Kink Structure in Magnetized Relativistic Jets

Recent polarimetric observations of blazars indicate the development of current-driven (CD) kink instability after passing the recollimation shocks in the relativistic jets and association with quasi-periodic oscillations (QPOs). To investigate multi-wavelength polarized features of CD kink instability in jets, we develop {\tt RaptorP}, a new special relativistic module of the polarized General Relativistic Radiative Transfer (GRRT) code {\tt RAPTOR}. Based on 3D SRMHD simulations of over-pressured magnetized jets, we find that jet images vary at different frequencies. At low frequencies, the emission comes from the turbulent ambient medium surrounding the jet that obscures the inner jet structure. Electronic Vector Position Angle (EVPA) patterns are perpendicular to the jet propagation direction, indicating a dominance of the poloidal magnetic field. At high frequencies, bright knots and twisted kink structures appear, and EVPA patterns are consistent with a toroidal magnetic field. We also find that QPOs in light curves of intensity and linear polarization (degree and angle). The peak frequency in Power Spectral Densities (PSDs) is well-matched with the rotation period of the kink structure in relativistic jets. It shows an anti-correlation between total intensity and the degree of polarization at a lower inclination angle. Our findings, based on realistic polarized radiation calculations, will explain the observational signatures seen in blazars.

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Magnetically Driven Retrograde Precession in Misaligned Black Hole Accretion Flows

Observations of accreting black hole (BH) systems, such as microquasars and supermassive black holes, often reveal a precessing jet with changing directions, indicating a misaligned accretion flow relative to the BH spin. The precession is commonly attributed to the Lense-Thirring (LT) effect, which arises from the BH's rotation twisting the surrounding spacetime and accretion flow. In the strongly magnetized regime, which is preferred accretion flow conditions for M~87$^*$ and likely other jet-producing systems, the large-scale magnetic field can significantly influence the flow dynamics. Here, we perform large-scale three-dimensional general relativistic magnetohydrodynamic simulations of tilted accretion onto a rotating BH, and find a never-seen-before new retrograde precession. This precession arises from a magnetic torque on the disk generated by the poloidal magnetic field aligned with the BH's rotation, opposing the LT torque. This finding highlights the unique property of highly magnetized accretion flows around BHs and provides a new interpretation of jet precession observed in many systems.

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Millimeter VLBI constraints on the central magnetic field and symmetric jet production in the twin-jet galaxy NGC 1052

This paper investigates the symmetry and magnetic field properties of the jets in the radio galaxy NGC 1052, with particular attention to the impact of the ionized torus that surrounds the central region on the emitted radiation. Our study is based on three new 43 GHz Very-Long-Baseline Interferometry (VLBI) observations and one 86 GHz observation conducted between April 2021 and April 2022. We derive key jet parameters, such as speed, width, and flux density for both jets at the two frequencies and compare them with those obtained from previous VLBI campaigns. Additionally, we present the first (43-86) GHz spectral index image of NGC 1052, which is crucial to assess the role of the torus at high frequencies. Finally, we leverage the derived observational parameters to constrain the magnetic field strength and configuration in the launched jets. We observe variability in the jet morphology at 43 GHz across the three epochs, which can be associated with the propagation of jet knots launched from the nuclear region. The stacked 43 GHz image reveals that the western and receding jet is approximately three times fainter than its eastern (approaching) counterpart in the sub-mas region. This asymmetry, together with the (43-86) GHz spectral index map, suggests that free-free absorption may affect the 43 GHz emission. On the contrary, the jets appear highly symmetric at 86 GHz. From the stacked images at 43 GHz and 86 GHz, we extract the jet width, which is consistent with previous VLBI studies and supports the presence of a parabolic jet profile on very compact scales. Overall, our results suggest that the jets are intrinsically launched symmetrically, and that the observed time-dependent asymmetries may result from free-free absorption by the torus and the downstream propagation of jet components, a scenario supported by previous theoretical studies.

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The Physical Origin and Time Lag of Multi-Frequency Flares from SgrA*

Sagittarius~A$^*$, the supermassive black hole at the center of our galaxy, exhibits flares across various wavelengths, yet their origins remain elusive. We performed 3D two-temperature General Relativistic Magnetohydrodynamic (GRMHD) simulations of magnetized accretion flows initialized from multi-loop magnetic field configuration onto a rotating black hole and conducted General Relativistic Radiative Transfer (GRRT) calculations considering contributions from both thermal and non-thermal synchrotron emission processes. Our results indicate that the polarity inversion events from the multi-loop magnetic field configurations can generate $138\,\rm THz$ flares consistent with observations with the help of non-thermal emission. By tracing the intensity evolution of light rays in GRRT calculations, we identify the precise location of the flaring region and confirm that it originates from a large-scale polarity inversion event. We observe time delays between different frequencies, with lower-frequency radio flares lagging behind higher frequencies due to plasma self-absorption in the disk. The time delay between near-infrared and 43 GHz flares can reach up to $\sim 50$ min, during which the flaring region gradually shifts outward, becoming visible at lower frequencies. Our study confirms that large-scale polarity inversion in a Standard And Normal Evolution (SANE) accretion flow with a multi-loop initial magnetic configuration can be a potential mechanism driving flares from Sgr~A$^*$.

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Origin of the ring ellipticity in the black hole images of M87*

We investigate the origin of the elliptical ring structure observed in the images of the supermassive black hole M87*, aiming to disentangle contributions from gravitational, astrophysical, and imaging effects. Leveraging the enhanced capabilities of the Event Horizon Telescope (EHT) 2018 array, including improved $(u,v)$-coverage from the Greenland Telescope, we measure the ring's ellipticity using five independent imaging methods, obtaining a consistent average value of $τ= 0.08_{-0.02}^{+0.03}$ with a position angle $ξ= 50.1_{-7.6}^{+6.2}$ degrees. To interpret this measurement, we compare against General Relativistic Magnetohydrodynamic (GRMHD) simulations spanning a wide range of physical parameters including thermal or non-thermal electron distribution function, spins, and ion-to-electron temperature ratios in both low and high-density regions. We find no statistically significant correlation between spin and ellipticity in GRMHD images. Instead, we identify a correlation between ellipticity and the fraction of non-ring emission, particularly in non-thermal models and models with higher jet emission. These results indicate that the ellipticity measured from the \m87 emission structure is consistent with that expected from simulations of turbulent accretion flows around black holes, where it is dominated by astrophysical effects rather than gravitational ones. Future high-resolution imaging, including space very long baseline interferometry and long-term monitoring, will be essential to isolate gravitational signatures from astrophysical effects.

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The putative center in NGC 1052

Many active galaxies harbor powerful relativistic jets, however, the detailed mechanisms of their formation and acceleration remain poorly understood. To investigate the area of jet acceleration and collimation with the highest available angular resolution, we study the innermost region of the bipolar jet in the nearby low-ionization nuclear emission-line region (LINER) galaxy NGC 1052. We combined observations of NGC 1052 taken with VLBA, GMVA, and EHT over one week in the spring of 2017. For the first time, NGC 1052 was detected with the EHT, providing a size of the central region in-between both jet bases of 250 RS (Schwarzschild radii) perpendicular to the jet axes. This size estimate supports previous studies of the jets expansion profile which suggest two breaks of the profile at around 300 RS and 10000 RS distances to the core. Furthermore, we estimated the magnetic field to be 1.25 Gauss at a distance of 22 μas from the central engine by fitting a synchrotron-self absorption spectrum to the innermost emission feature, which shows a spectral turn-over at about 130 GHz. Assuming a purely poloidal magnetic field, this implies an upper limit on the magnetic field strength at the event horizon of 26000 Gauss, which is consistent with previous measurements. The complex, low-brightness, double-sided jet structure in NGC 1052 makes it a challenge to detect the source at millimeter (mm) wavelengths. However, our first EHT observations have demonstrated that detection is possible up to at least 230 GHz. This study offers a glimpse through the dense surrounding torus and into the innermost central region, where the jets are formed. This has enabled us to finally resolve this region and provide improved constraints on its expansion and magnetic field strength.

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A multi-frequency study of sub-parsec jets with the Event Horizon Telescope

The 2017 observing campaign of the Event Horizon Telescope (EHT) delivered the first very long baseline interferometry (VLBI) images at the observing frequency of 230 GHz, leading to a number of unique studies on black holes and relativistic jets from active galactic nuclei (AGN). In total, eighteen sources were observed: the main science targets, Sgr A* and M87 along with various calibrators. We investigated the morphology of the sixteen AGN in the EHT 2017 data set, focusing on the properties of the VLBI cores: size, flux density, and brightness temperature. We studied their dependence on the observing frequency in order to compare it with the Blandford-Königl (BK) jet model. We modeled the source structure of seven AGN in the EHT 2017 data set using linearly polarized circular Gaussian components and collected results for the other nine AGN from dedicated EHT publications, complemented by lower frequency data in the 2-86 GHz range. Then, we studied the dependences of the VLBI core flux density, size, and brightness temperature on the frequency measured in the AGN host frame. We compared the observations with the BK jet model and estimated the magnetic field strength dependence on the distance from the central black hole. Our results indicate a deviation from the standard BK model, particularly in the decrease of the brightness temperature with the observing frequency. Either bulk acceleration of the jet material, energy transfer from the magnetic field to the particles, or both are required to explain the observations.

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Impact of radiative cooling on the magnetised geometrically thin accretion disk around Kerr black hole

It is believed that the spectral state transitions of the outbursts in X-ray binaries (XRBs) are triggered by the rise of the mass accretion rate due to underlying disc instabilities. Recent observations found that characteristics of disc winds are probably connected with the different spectral states, but the theoretical underpinnings of it are highly ambiguous. To understand the correlation between disc winds and the dynamics of the accretion flow, we have performed General Relativistic Magneto-hydrodynamic (GRMHD) simulations of an axisymmetric thin accretion disc with different accretion rates and magnetic field strengths. Our simulations have shown that the dynamics and the temperature properties depend on both accretion rates and magnetic field strengths. We later found that these properties greatly influence spectral properties. We calculated the average coronal temperature for different simulation models, which is correlated with high-energy Compton emission. Our simulation models reveal that the average coronal temperature is anti-correlated with the accretion rates, which are correlated with the magnetic field strengths. We also found that the structured component of the disc winds (Blandford-Payne disc wind) predominates as the accretion rates and magnetic field strengths increase. In contrast, the turbulent component of the disc winds ($B_{\rm tor}$ disc wind) predominates as the accretion rates and magnetic field strengths decrease. Our results suggest that the disc winds during an outburst in XRBs can only be understood if the magnetic field contribution varies over time (e.g., MAXI J1820+070).

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Demographics of black holes at $<$100 R$_{\rm g}$ scales: accretion flows, jets, and shadows

Using the Event Horizon Telescope (EHT), the gravitationally lensed rings around the supermassive black holes (SMBHs) in Messier 87 (M87) and Sagittarius A* (Sgr A*) have now been successfully imaged at a resolution under 10 gravitational radii (R$_{\rm g}$ $ = \rm{GM/c^2}$). To expand studies beyond M87 and Sgr A*, we have constructed the Event Horizon and Environs (ETHER) sample, a comprehensive database encompassing approximately 3.15 million SMBH mass estimates, $\sim$ 20,000 Very-Long Baseline Interferometry (VLBI) radio flux densities, and $\sim$ 36,000 hard X-ray flux densities. This database is designed to identify and optimize target selection for the EHT and its upgrades on the ground and in space. We have identified a Gold Sample (GS) of nearby low-luminosity Active Galactic Nuclei (AGNs) within it that are ideal for studying jet bases and potentially imaging black hole shadows. We observed 27 of these AGNs using the EHT from 2022 to 2024, providing an opportunity to resolve and image accretion flows and jets at resolutions of $\leq$ 100 R$_{\rm g}$. Only a few SMBHs have sufficiently high enough flux density to be imaged at scales of $\leq$ 50 R$_{\rm g}$ with the present EHT. Among these are M87, Sgr A*, NGC4594 (Sombrero/M104), NGC4261, and NGC4374 (Messier 84/M84). Of these, NGC4261, Sombrero, and M84 have been observed and/or are scheduled for deep imaging with EHT+ALMA from 2023 to 2025. Sombrero, NGC4261, M84, NGC4278, and NGC5232 are clearly detected in our EHT+ALMA observations in 2022, indicating that the 230 GHz flux density from the accretion flows is significantly high. Ongoing imaging of the ETHER GS will enable measurements of black hole mass and spin, help constrain General Relativity, and enrich our understanding of jet launching and accretion inflows across a broad multi-parameter space, including black hole mass, spin, accretion rate, and orientation.

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Probing jet dynamics and collimation in radio galaxies. Application to NGC 1052

Context. Radio galaxies with visible two-sided jet structures, such as NGC 1052, are sources of particular interest to study the collimation and shock structure of active galactic nuclei jets. High-resolution very-long-baseline interferometry observations of such sources can resolve and study the jet collimation profile and probe different physical mechanisms. Aims. In this paper, we study the physics of double-sided radio sources at parsec scales, and in particular investigate whether propagating shocks can give rise to the observed asymmetry between jet and counterjet. Methods. We carry out special relativistic hydrodynamic simulations and perform radiative transfer calculations of an over-pressured perturbed jet. During the radiative transfer calculations we incorporate both thermal and nonthermal emission while taking the finite speed of light into account. To further compare our results to observations, we create more realistic synthetic data including the properties of the observing array as well as the image reconstruction via multifrequency regularized maximum likelihood methods. We finally introduce a semi-automatized method for tracking jet components and extracting jet kinematics. Results. We show that propagating shocks in an inherently symmetric double-sided jet can lead to partially asymmetric jet collimation profiles due to time delay effects and relativistic beaming. These asymmetries may appear on specific epochs, with one jet evolving near conically and the other one parabolically (width profile evolving with a slope of 1 and 0.5, respectively). However, these spurious asymmetries are not significant when observing the source evolve for an extended amount of time. Conclusions. Purely observational effects are not enough to explain a persisting asymmetry in the jet collimation profile of double-sided jet sources and hint at evidence for asymmetrically launched jets.

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Numerical Investigation of Instabilities in Over-pressured Magnetized Relativistic Jets

Context. Relativistic jets from Active Galactic Nuclei are observed to be collimated on the parsec scale. When the pressure between the jet and the ambient medium is mismatched, recollimation shocks and rarefaction shocks are formed. Previous numerical simulations have shown that instabilities can destroy the recollimation structure of jets. Aims. In this study, we aim to study the instabilities of non-equilibrium over-pressured relativistic jets with helical magnetic fields. Especially, we investigate how the magnetic pitch affects the development of instabilities. Methods. We perform three-dimensional relativistic magnetohydrodynamic simulations for different magnetic pitches, as well as a two-dimension simulation and a relativistic hydrodynamic simulation served as comparison groups Results. In our simulations, Rayleigh-Taylor Instability (RTI) is triggered at the interface between the jet and ambient medium in the recollimation structure of the jet. We found that when the magnetic pitch decreases the growth of RTI becomes weak but interestingly, another instability, the CD kink instability is excited. The excitement of CD kink instability after passing the recollimation shocks can match the explanation of the quasi-periodic oscillations observed in BL Lac qualitatively.

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The impact of resistivity on the variability of black hole accretion flows

Context. The accretion of magnetized plasma onto black holes is a complex and dynamic process, where the magnetic field plays a crucial role. The amount of magnetic flux accumulated near the event horizon significantly impacts the accretion flow behavior. Resistivity, a measure of how easily magnetic fields can dissipate, is thought to be a key factor influencing this process. This work explores the influence of resistivity on accretion flow variability. We investigate simulations reaching the magnetically arrested disk (MAD) limit and those with an initial multi-loop magnetic field configuration. Methods. We employ 3D resistive general relativistic magnetohydrodynamic (GRMHD) simulations to model the accretion process under various regimes, where resistivity has a global uniform value. Results. Our findings reveal distinct flow behaviors depending on resistivity. High resistivity simulations never achieve the MAD state, indicating a disturbed magnetic flux accumulation process. Conversely, low resistivity simulations converge towards the ideal MHD limit. The key results are: i) For the standard MAD model, resistivity plays a minimal role in flow variability, suggesting that flux eruption events dominate the dynamics. ii) High resistivity simulations exhibit strong magnetic field diffusion into the disk, rearranging efficient magnetic flux accumulation from the accretion flow. iii) In multi-loop simulations, resistivity significantly reduces flow variability, which was not expected. However, magnetic flux accumulation becomes more variable due to frequent reconnection events at very low resistivity values. Conclusions. This study shows that resistivity affects how much the flow is distorted due to magnetic field dissipation. Our findings provide new insights into the interplay between magnetic field accumulation, resistivity, variability and the dynamics of black hole accretion.

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First Very Long Baseline Interferometry Detections at 870μm

The first very long baseline interferometry (VLBI) detections at 870$μ$m wavelength (345$\,$GHz frequency) are reported, achieving the highest diffraction-limited angular resolution yet obtained from the surface of the Earth, and the highest-frequency example of the VLBI technique to date. These include strong detections for multiple sources observed on inter-continental baselines between telescopes in Chile, Hawaii, and Spain, obtained during observations in October 2018. The longest-baseline detections approach 11$\,$G$λ$ corresponding to an angular resolution, or fringe spacing, of 19$μ$as. The Allan deviation of the visibility phase at 870$μ$m is comparable to that at 1.3$\,$mm on the relevant integration time scales between 2 and 100$\,$s. The detections confirm that the sensitivity and signal chain stability of stations in the Event Horizon Telescope (EHT) array are suitable for VLBI observations at 870$μ$m. Operation at this short wavelength, combined with anticipated enhancements of the EHT, will lead to a unique high angular resolution instrument for black hole studies, capable of resolving the event horizons of supermassive black holes in both space and time.

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Dynamics and Emission Properties of Flux Ropes from Two-Temperature GRMHD Simulations with Multiple Magnetic Loops

Flux ropes erupting from the vicinity of the black hole are thought to be a potential model for the flares observed in Sgr\,A$^*$. In this study, we examine the radiative properties of flux ropes that emerged from the vicinity of the black hole. We have performed three-dimensional two-temperature General Relativistic Magnetohydrodynamic (GRMHD) simulations of magnetized accretion flows with alternating multiple magnetic loops, and General Relativistic Radiation Transfer (GRRT) calculations. In GRMHD simulations, two different sizes of initial magnetic loops are implemented. In the small loop case, magnetic dissipation leads to a weaker excitement of magneto-rotational instability inside the torus which generates a lower accretion rate compared to the large loop case. However, it makes more generation of flux ropes due to frequent reconnection by magnetic loops with different polarities. By calculating the thermal synchrotron emission, we found that the variability of light curves and emitting region are tightly related. At $230\,\rm GHz$ and higher frequency, the emission from the flux ropes is relatively stronger compared with the background, which is responsible for the filamentary structure in the images. At lower frequencies, e.g. $43\,\rm GHz$, emission comes from more extended regions, which have a less filamentary structure in the image. Our study shows self-consistent electron temperature models are essential for the calculation of thermal synchrotron radiation and the morphology of the GRRT images. Flux ropes contribute considerable emission at frequencies $\gtrsim 230\,\rm GHz$.

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Impacts of nonthermal emission on the images of a black hole shadow and extended jets in two-temperature GRMHD simulations

The recent 230 GHz observations from the Event Horizon Telescope collaboration can image the innermost structure of the M87 galaxy showing the shadow of the black hole, a photon ring, and a ring-like structure that agrees with thermal synchrotron emission from the accretion disc. However, at lower frequencies, M87 is characterized by a large-scale jet with clear signatures of nonthermal emission. It is necessary to explore the impacts of nonthermal emission on black hole shadow images and extended jets, especially at lower frequencies. In this study, we aim to compare models with different electron heating prescriptions and investigate how these prescriptions and nonthermal electron distributions may affect black hole shadow images and the broadband spectrum energy distribution (SED) function. We performed general relativistic radiative transfer (GRRT) calculations in various two-temperature general relativistic magnetohydrodynamic (GRMHD) models utilizing different black hole spins and different electron heating prescriptions coupled with different electron distribution functions (eDFs). Through a comparison with GRRT images and SEDs, we found that when considering a variable kappa eDF, the parameterized prescription of the R-beta model with Rh = 1 is similar to the model with electron heating in the morphology of images, and the SEDs at a high-frequency. This is consistent with previous studies using the thermal eDF. However, the nuance between them could be differentiated through the diffuse extended structure seen in GRRT images, especially at a lower frequency, and the behavior of SEDs at low frequency. The emission from the nearside jet region is enhanced in the reconnection heating case and it will increase the contribution from the regions with stronger magnetization is included or if the magnetic energy contribution to kappa eDF mainly in the magnetized regions is considered.

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