arXiv subjects

Dengke Zhou

Publications and source records attributed to Dengke Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Spectro-Polarimetric Properties of CHIME FRB Sources

Fast radio bursts (FRBs) are enigmatic millisecond-duration radio transients whose polarization properties offer crucial insights into their origins and environments. In particular, low-frequency depolarization---quantified by the parameter $\sigma_{\mathrm{RM}}$---probes the complex magneto-ionic medium surrounding the progenitor, and has been observed across a population of repeating FRBs. We present a systematic spectro-polarimetric analysis of repeating and non-repeating FRBs using observations from the Canadian Hydrogen Intensity Mapping Experiment (CHIME). For 28 repeating FRBs, we measure $\sigma_{\mathrm{RM}}$, expanding the known sample from 14 to 36 sources (an increase by a factor of 2.6). The kernel density estimate (KDE) of the repeating population peaks at $1.3\ \mathrm{rad\,m^{-2}}$, with approximately 70\% of the sources showing $\sigma_{\mathrm{RM}} \gtrsim 1\ \mathrm{rad\,m^{-2}}$, implying that most reside in complex magneto-ionic environments. For 70 non-repeating FRBs, we investigate four spectro-polarimetric models; no source exhibits significant depolarization with $\sigma_{\mathrm{RM}} \gtrsim 5\ \mathrm{rad\,m^{-2}}$. Roughly half of the non-repeaters are consistent with a constant linear polarization fraction across frequency. We caution, however, that these results may be affected by the limited frequency coverage of CHIME. Future ultra-wideband polarimetry, spanning widely separated frequencies, will overcome current observational biases, enable precise $\sigma_{\mathrm{RM}}$ measurements, and substantially deepen our understanding of FRB environments.

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arXiv2026

No Strong Evidence for Plasma Lensing in FRB 20240114A

FRB~20240114A is an extremely active repeating fast radio burst for which plasma lensing has been proposed to explain its burst-rate variations, spectral evolution, and apparently ``carbon-copy'' burst pairs. Using FAST data and publicly available Parkes observations, we test this interpretation with a one-dimensional Gaussian plasma-lens model. Although the burst-rate enhancements can be fitted separately, the corresponding magnification peaks and demagnification troughs are offset by far more than predicted and show no consistent periodicity. Moreover, with more than 10,000 bursts detected, a few apparently ``carbon-copy'' pairs can readily occur by chance. The burst bandwidth is not systematically narrower during the proposed lensing interval, nor are the burst energies significantly enhanced during the predicted magnification interval. These results provide no compelling evidence that a single Gaussian plasma lens explains the observed variability, which is more likely dominated by intrinsic source activity.

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arXiv2026

FRTSearch: Unified Detection and Parameter Inference of Fast Radio Transients using Instance Segmentation

The exponential growth of data from modern radio telescopes presents a significant challenge to traditional single-pulse search algorithms, which are computationally intensive and prone to high false-positive rates due to Radio Frequency Interference (RFI). In this work, we introduce FRTSearch, an end-to-end framework unifying the detection and physical characterization of Fast Radio Transients (FRTs). Leveraging the morphological universality of dispersive trajectories in time-frequency dynamic spectra, we reframe FRT detection as a pattern recognition problem governed by the cold plasma dispersion relation. To facilitate this, we constructed CRAFTS-FRT, a pixel-level annotated dataset derived from the Commensal Radio Astronomy FAST Survey (CRAFTS), comprising 2{,}392 instances across diverse source classes. This dataset enables the training of a Mask R-CNN model for precise trajectory segmentation. Coupled with our physics-driven IMPIC algorithm, the framework maps the geometric coordinates of segmented trajectories to directly infer the Dispersion Measure (DM) and Time of Arrival (ToA). Benchmarking on the FAST-FREX dataset shows that FRTSearch achieves a 98.0\% recall, competitive with exhaustive search methods, while reducing false positives by over 99.9\% compared to PRESTO and delivering a processing speedup of up to $13.9\times$. Furthermore, the framework demonstrates robust cross-facility generalization, detecting all 19 tested FRBs from the ASKAP survey without retraining. By shifting the paradigm from ``search-then-identify'' to ``detect-and-infer,'' FRTSearch provides a scalable, high-precision solution for real-time discovery in the era of petabyte-scale radio astronomy.

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arXiv2026

The Real and Pseudo Dispersion Measures of FRB~20220912A

Fast radio bursts (FRBs) are millisecond-duration radio transients. As they propagate through the interstellar medium, they interact with free electrons, resulting in dispersion. The corresponding dispersion measure (DM) is referred to as the real DM (DM$_{\rm real}$). In practice, however, the dispersion measure derived from modeling (DM$_{\rm model}$) is often contaminated by intrinsic burst morphology, giving rise to a pseudo DM component (DM$_{\rm pseudo} = {\rm DM}_{\rm model} - {\rm DM}_{\rm real}$). In this work, we focus on the highly active repeating FRB~20220912A and utilize its microshots -- extremely short-duration (typically tens of microseconds), broadband emissions -- to investigate its DM$_{\rm real}$ and DM$_{\rm pseudo}$. We adopt two assumptions: first, that FRB~20220912A resides in a non-magneto-ionic environment and that its DM$_{\rm real}$ variation is smaller than $10^{-2}$\,pc\,cm$^{-3}$ over a few years; and second, that microshots have a negligible intrinsic morphological time delay. By identifying two new microshots and combining them with previously reported ones, we find that all four microshots exhibit remarkably consistent DM values over a one-month timescale, with an average of $219.380 \pm 0.004\,\mathrm{pc\,cm^{-3}}$. We define this value as the DM$_{\rm real}$ of FRB~20220912A. We further show that bright, narrow bursts with a width of less than 2\,ms also yield DM estimates consistent with the microshot-based DM$_{\rm real}$. A survey of five repeating FRBs reveals that DM$_{\rm pseudo}$ is a common phenomenon, with variations typically spanning a range of approximately $10\,\mathrm{pc\,cm^{-3}}$ at 1.2\,GHz. These findings highlight the importance of accounting for morphological contributions in DM interpretation and demonstrate that microshots and narrow bursts are powerful tools for probing DM$_{\rm real}$.

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arXiv2026

FAST Polarization Catalog of FRB 20240114A

Polarization measurements of fast radio bursts (FRBs) probe the magnetized plasma surrounding their central engines. FRB~20240114A is an exceptionally active repeating source, with 17,356 bursts detected between 2024 January 28 and 2025 May 30 by FAST, enabling time-resolved polarimetric studies. In this work, we present a polarimetric catalog of 6,131 bright bursts (with a signal-to-noise ratio S/N $\geq$ 20, 35.3% of the total sample), including arrival time (MJD$_{\text{topo}}$), dispersion measure (DM), burst width (W$_{\text{eff}}$), bandwidth, Faraday rotation measure (RM), linear and circular polarization degrees (DOL, DOC), and intrinsic polarization angle (PA$_0$). We detect a clear temporal evolution of RM: after an initial stable phase, it decreases linearly by $\sim$200 $\rm rad\ m^{-2}$ over 200 days, forming a bimodal distribution, whereas DM remains stable at 528.9 $\rm pc\ cm^{-3}$. The linear polarization fraction is generally high, with the 3$\sigma$ lower bound around 76%, while circular polarization is low, with 1,157 of 17,356 bursts (6.67%) having DOC $\geq$10%. We perform a power-law fit between $|\textrm{V}|$/I and $|\textrm{RM}|$, which yields an index of $-2.98 \pm 0.80$. It is found that the combined 2D distribution of L/I versus V/I remains stable, implying that the emission mechanism is largely invariant. Our PA$_0$ measurements show a broad, non-uniform distribution, implying a complex emission geometry. These results suggest that FRB~20240114A resides in a dynamically evolving magneto-ionic environment. This catalog provides a foundation for studies of repeating FRB progenitors and their environments.

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arXiv2026

SwinYNet: A Transformer-based Multi-Task Model for Accurate and Efficient FRB Search

In this study, we present a transformer-based multi-task model for Fast Radio Burst (FRB) detection, signal segmentation, and parameter estimation directly from time-frequency data, without requiring computationally expensive de-dispersion preprocessing. To overcome the scarcity of labeled observational data, we develop an FRB simulator and a rule-based automatic annotation pipeline, enabling training exclusively on simulated data. Evaluations on the FAST-FREX dataset show that our model achieves an F1 score of 97.8%, recall of 95.7%, and precision of 100%, outperforming both conventional tools (e.g., PRESTO, Heimdall) and recent AI-based baselines (e.g., RaSPDAM, DRAFTS) in both accuracy and inference speed. The model supports pixel-level signal segmentation and yields reliable estimates for dispersion measure (DM) and time of arrival (ToA). Large-scale blind searches on CRAFTS data further demonstrate robustness, with an average false positive rate of 0.28% and minimal human verification required. This search has already led to the identification of two pulsar candidates, both confirmed as known pulsars. Processing benchmarks indicate that the model enables real-time searches on a single consumer-grade GPU, making petabyte-scale blind searches feasible. The code is publicly available on GitHub, and the model can be easily integrated with existing tools to automate and streamline radio data analysis beyond FRB or pulsar searches.

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arXiv2025

Revealing the Temporally Stable Bimodal Energy Distribution of FRB 20121102A with a Tripled Burst Set from AI Detections

Active repeating Fast Radio Bursts (FRBs), with their large number of bursts, burst energy distribution, and their potential energy evolution, offer critical insights into the FRBs emission mechanisms. Traditional pipelines search for bursts through conducting dedispersion trials and looking for signals above certain fluence thresholds, both of which could result in missing weak and narrow-band bursts. In order to improve the completeness of the burst set, we develop an End-to-end DedispersE-agnostic Nonparametric AI model (EDEN), which directly detect bursts from dynamic spectrum and is the first detection pipeline that operates without attempting dedispersion. We apply EDEN to archival FAST L-band observations during the extreme active phase of the repeating source FRB 20121102A, resulting in the largest burst set for any FRB to date, which contains 5,927 individual bursts, tripling the original burst set. The much enhanced completeness enables a refined analysis of the temporal behavior of energy distribution, revealing that the bimodal energy distribution remains stable over time. It is rather an intrinsic feature of the emission mechanisms than a consequence of co-evolving with burst rate.

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arXiv2025

The magnetar model's energy crisis for a prolific repeating fast radio burst source

Fast radio bursts (FRBs) are widely considered to originate from magnetars that power the explosion through releasing magnetic energy. Active repeating FRBs have been seen to produce hundreds of bursts per hour and can stay active for months, thus may provide stringent constraints on the energy budget of FRBs' central engine. Within a time span of 214 days, we detected 11,553 bursts from the hyper-active FRB 20240114A that reached a peak burst rate of 729 hr$^{-1}$. This is the largest burst sample from any single FRB source, exceeding the cumulative total of all published bursts from all known FRBs to date. Assuming typical values of radio efficiency and beaming factor, the estimated total isotropic burst energy of this source exceeds 86% of the dipolar magnetic energy of a typical magnetar. The total released energy from this source exceeds that of other known repeaters by about one and a half orders of magnitude, yielding the most stringent lower limit of $4.7\times10^{32}$ G cm$^3$ for the magnetar's magnetic moment. The source remained active at the end of this observation campaign. Our findings thus require either the FRB's central magnetar engine's possessing exceptionally high emission efficiency or a more powerful compact object than a typical magnetar.

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arXiv2025

Investigating FRB 20240114A with FAST: Morphological Classification and Drifting Rate Measurements in a Burst-Cluster Framework

This study investigates the morphological classification and drifting rate measurement of the repeating fast radio burst (FRB) source FRB 20240114A using the Five-hundred-meter Aperture Spherical Telescope (FAST). Detected on January 14, 2024, FRB 20240114A exhibited an exceptionally high burst rate, revealing unique properties. Through observational campaigns over several months, we selected a dataset comprising 3,203 bursts (2,109 burst-clusters) during a continuous monitoring session (15,780 seconds) on March 12, 2024. Improving upon previous work, we clarify the definitions of sub-bursts, bursts and burst-clusters. Using an average dispersion measures (DM) of 529.2 pc cm$^{-3}$, we classified the burst-clusters into Downward Drifting, Upward Drifting, No Drifting, No Evidence for Drifting, Not-Clear, and Complex burst-clusters. Among the 978 burst-clusters that exhibit drifting behavior, 233 (23.82%) show upward drifting. Additionally, if 142 upward drifting single-component burst-clusters are excluded, upward drifting double- and multi-component burst-clusters still account for 10.89% of the 836 burst-clusters exhibiting drifting behavior, equating to 91 burst-clusters. Furthermore, if only upward drifting burst-clusters with consecutive time intervals (or upward drifting bursts) are considered, only 9 bursts remain. Drifting rate comparisons with other physical quantities reveal that the drifting rate increases with peak frequency for single-component burst-clusters with drifting behavior. Moreover, in single-component burst-clusters, those with upward drifting exhibit smaller effective widths, bandwidths, and fluxes than their downward drifting counterparts. A Kolmogorov-Smirnov test further indicates that upward drifting burst-clusters possess longer consecutive time intervals than downward drifting ones, suggesting distinct underlying physical mechanisms.

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arXiv2025

A comprehensive search for Long and Short Periodic Features from an Extremely Active Cycle of FRB 20240114A

Possible periodic features in fast radio bursts (FRBs) may provide insights into their astrophysical origins. Using extensive observations from the Five-hundred-meter Aperture Spherical radio Telescope (FAST), we conduct a multi-timescale periodicity search for the exceptionally active repeater FRB~20240114A. Our analysis is based on different datasets for different timescales: for short-timescale periodicity in Time of Arrivals (TOAs), we use 57 observations from January to August 2024; for long-timescale periodicity, we employ an extended TOA dataset comprising 111 observations spanning from January 2024 to October 2025; and for burst time series analysis, we utilize individual burst data from the 57 FAST observations. We identify three candidate short-timescale periodic signals (0.673~s, 0.635~s, and 0.536~s) with significances of $3.2\sigma$--$6\sigma$, each detected in two independent observations. On longer timescales, we detect a significant $143.40\pm7.19$-day periodicity with $5.2\sigma$ significance, establishing FRB~20240114A as a periodic repeater. In burst time series, we find quasi-periodic oscillations in the few hundred Hz range ($3.4\sigma$ and $3.7\sigma$) and periodic burst trains with periods of several to tens of milliseconds ($3\sigma$--$3.9\sigma$), though these periodic features appear transient and short-lived. The detection of periodic signals at these different time scales indicates that FRB 20240114A exhibits intriguing periodic self-similar characteristics. Despite the comprehensive dataset, no definitive periodicity linked to the source's rotation is confirmed, placing stringent constraints on the intrinsic source properties and the modulation mechanisms. All data are available via the Science Data Bank.

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arXiv2025

Multi-year Polarimetric Monitoring of Four CHIME-Discovered Repeating Fast Radio Bursts with FAST

In this study, we report multi-year polarization measurements of four repeating FRBs initially discovered by CHIME: FRBs~20190117A, 20190208A, 20190303A, and 20190417A. We observed the four repeating FRBs with FAST, detecting a total of 66 bursts. Two bursts from FRB~20190417A exhibit a circular polarization signal-to-noise ratio greater than 7, with the highest circular polarization fraction recorded at 35.7%. While the bursts from FRBs 20190208A and 20190303A are highly linearly polarized, those from FRBs~20190117A and 20190417A show depolarization due to multi-path propagation, with σ_{\mathrm{RM}} = 2.78 \pm 0.05 rad m$^{-2}$ and 5.19 \pm 0.09 rad m$^{-2}$, respectively. The linear polarization distributions among five repeating FRB--FRBs~20190208A, 20190303A, 20201124A, 20220912A, and 20240114A--are nearly identical but show distinct differences from those of non-repeating FRBs. FRBs~20190117A, 20190303A, and 20190417A exhibit substantial rotation measure (RM) variations between bursts, joining other repeating FRBs in this behavior. Combining these findings with published results, 64% of repeating FRBs show RM variations greater than 50 rad m$^{-2}$, and 21\% exhibit RM reversals. A significant proportion of repeating FRBs reside in a dynamic magneto-ionic environment. The structure function of RM variations shows a power-law index of $γ\sim (0-0.8)$, corresponding to a shallow power spectrum $α= -(γ+ 2) \sim -(2.0-2.8)$ of turbulence, if the RM variations are attributed to turbulence. This suggests that the variations are dominated by small-scale RM density fluctuations. We perform K-S tests comparing the RMs of repeating and non-repeating FRBs, which reveal a marginal dichotomy in the distribution of their RMs.We caution that the observed dichotomy may be due to the small sample size and selection biases.

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arXiv2025

Polarization Characteristics of the Hyperactive FRB 20240114A

Fast radio bursts (FRBs) are transient radio bursts of extragalactic origin characterized by millisecond durations and high luminosities. We report on observations of FRB 20240114A conducted with the Robert C. Byrd Green Bank Telescope (GBT) at frequencies ranging from 720 to 920 MHz. A total of 437 bursts were detected, with a single observation recording 365 bursts over 1.38 hours, corresponding to a burst rate of 264 bursts per hour. The average rotation measures (RMs) were $347.0 \pm 1.0$ rad m$^{-2}$ on February 23, 2024, and $353.7 \pm 0.6$ rad m$^{-2}$ on March 1, 2024. Of the 301 bursts with detected RMs, 81% have a linear polarization fraction greater than 90%, and 14% exhibit circular polarization with a signal-to-noise ratio $> 5$. Our sample also displayed polarization angle swings. We compared the linear polarization fraction of FRB~20240114A with those of the repeating sources FRB~20201124A and FRB~20220912A. Our analysis reveals that all three exhibit similar distributions in both linear and circular polarization fractions. These results indicate that the three sources share the same radiation mechanism. We analyze the fluence and waiting-time distributions of FRB~20240114A, revealing a right-skewed fluence distribution and a bimodal waiting-time structure, suggesting intrinsic emission timescales and potential multiple burst populations. Additionally, We present a novel method to determine the frequency range of bursts based on their spectral characteristics. This algorithm is independent of spectral models and remains unaffected by the removal of interference-affected channels in the data, ensuring robust performance.

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arXiv2025

Fermi-LAT and FAST observation of the gamma-ray binary HESS J0632+057

Using 15 years of data from the Fermi Large Area Telescope (Fermi-LAT), we performed a comprehensive analysis on the gamma-ray binary HESS J0632+057. Its spectrum in 0.1-300 GeV band is well described by a power law model with an index of $2.40\pm0.16$, leading to an energy flux of (5.5$\pm$1.6$)\times$ 10$^{-12}$ erg cm$^{-2}$ s$^{-1}$. The GeV Spectral Energy Distribution (SED) of HESS J0632+057 hints for a spectral turn-over between $\sim$10-100 GeV. Orbital analysis reveals a flux enhancement during the phase range of 0.2-0.4, consistent with the X-ray and TeV light curves, indicating an origin of a common particle population. We carried out six deep radio observations on HESS J0632+057 with the Five-hundred-meter Aperture Spherical Telescope (FAST), evenly distributed across its orbit, reaching a detection sensitivity of 2$μ$Jy. However, no radio pulsation was detected within these observations. The absence of radio pulsation may be attributed to the dense stellar wind environment of HESS J0632+057.

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arXiv2024

An extremely active repeating fast radio burst source in a likely non-magneto-ionic environment

Fast radio bursts (FRBs) are bright radio bursts originating at cosmological distances. Only three repeating FRBs FRB 20121102A, FRB 20190520B and FRB 20201124A among $\sim$ 60 known repeating FRBs have circular polarization. We observed the FRB 20220912A with the Robert C. Byrd Green Bank Telescope (GBT) at L-band on 24 October 2022 and detected 128 bursts in 1.4 hours, corresponding to a burst rate of about 90 hr$^{-1}$, which is the highest yet for FRBs observed by the GBT. The average rotation measure (RM) was $-$0.4$\pm$0.3$\,$rad$\,$m$^{-2}$ with negligible intraday RM change, indicating a likely non-magneto-ionic environment. 61% bursts have linear polarization fraction greater than 90%. Approximately 56% of the bright bursts have circular polarization. A downward drift in frequency and polarization angle swings were found in our sample. The characterization of FRB 20220912A indicates that the circular polarization is unlikely to be caused by the magneto-ionic environment for at least some of the repeating FRB population.

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arXiv2024

A discovery of Two Slow Pulsars with FAST: "Ronin" from the Globular Cluster M15

Globular clusters harbor numerous millisecond pulsars, but long-period pulsars ($P \gtrsim 100$ ms) are rarely found. In this study, we employed a fast folding algorithm to analyze observational data from multiple globular clusters obtained by the Five-hundred-meter Aperture Spherical radio Telescope (FAST), aiming to detect the existence of long-period pulsars. We estimated the impact of the median filtering algorithm in eliminating red noise on the minimum detectable flux density ($S_{\rm min}$) of pulsars. Subsequently, we successfully discovered two isolated long-period pulsars in M15 with periods approximately equal to 1.928451 seconds and 3.960716 seconds, respectively. On the $P-\dot{P}$ diagram, both pulsars are positioned below the spin-up line, suggesting a possible history of partial recycling in X-ray binary systems disrupted by dynamical encounters later on. According to timing results, these two pulsars exhibit remarkably strong magnetic fields. If the magnetic fields were weakened during the accretion process, then a short duration of accretion might explain the strong magnetic fields of these pulsars.

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arXiv2023

Blinkverse: A Database of Fast Radio Bursts

The volume of research on fast radio bursts (FRBs) observation have been seeing a dramatic growth. To facilitate the systematic analysis of the FRB population, we established a database platform, Blinkverse (https://blinkverse.alkaidos.cn), as a central inventory of FRBs from various observatories and with published properties, particularly dynamic spectra from FAST, CHIME, GBT, Arecibo, etc. Blinkverse thus not only forms a superset of FRBCAT, TNS, and CHIME/FRB, but also provides convenient access to thousands of FRB dynamic spectra from FAST, some of which were not available before. Blinkverse is regularly maintained and will be updated by external users in the future. Data entries of FRBs can be retrieved through parameter searches through FRB location, fluence, etc., and their logical combinations. Interactive visualization was built into the platform. We analyzed the energy distribution, period analysis, and classification of FRBs based on data downloaded from Blinkverse. The energy distributions of repeaters and non-repeaters are found to be distinct from one another.

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arXiv2022

An Insight-HXMT view of the mHz quasi-regular modulation phenomenon in the black hole X-ray binary 4U 1630-47

Here we report the spectral-timing results of the black hole X-ray binary 4U 1630-47 during its 2021 outburst using observations from the Hard X-ray Modulation Telescope. Type-C quasi-periodic oscillations (QPOs) in 1.6--4.2 Hz and quasi-regular modulation (QRM) near 60 mHz are detected during the outburst. The mHz QRM has a fractional rms of 10%--16% in the 8--35 keV energy band with a Q factor (frequency/width) of 2--4. Benefiting from the broad energy band of hxmt, we study the energy dependence of the 60 mHz QRM in 1--100 keV for the first time. We find that the fractional rms of the mHz QRM increases with photon energy, while the time lags of the mHz QRM are soft and decrease with photon energy. Fast recurrence of the mHz QRM, in a timescale of less than one hour, has been observed during the outburst. During this period, the corresponding energy spectra moderately change when the source transitions from the QRM state to the non-QRM state. The QRM phenomena also shows a dependence with the accretion rate. We suggest that the QRM could be caused by an unknown accretion instability aroused from the corona.

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arXiv2022

Insight-HXMT Study of the Inner Accretion Disk in the Black Hole Candidate EXO 1846--031

We study the spectral evolution of the black hole candidate EXO 1846$-$031 during its 2019 outburst, in the 1--150 keV band,with the {\it {Hard X-ray Modulation Telescope}}. The continuum spectrum is well modelled with an absorbed disk-blackbody plus cutoff power-law, in the hard, intermediate and soft states. In addition, we detect an $\approx$6.6 keV Fe emission line in the hard intermediate state. Throughout the soft intermediate and soft states, the fitted inner disk radius remains almost constant; we suggest that it has settled at the innermost stable circular orbit (ISCO). However, in the hard and hard intermediate states, the apparent inner radius was unphysically small (smaller than ISCO), even after accounting for the Compton scattering of some of the disk photons by the corona in the fit. We argue that this is the result of a high hardening factor, $f_{\rm col}\approx2.0-2.7$, in the early phases of outburst evolution, well above the canonical value of 1.7 suitable to a steady disk. We suggest that the inner disk radius was close to ISCO already in the low/hard state. Furthermore, we propose that this high value of hardening factor in the relatively hard state is probably caused by the additional illuminating of the coronal irradiation onto the disk. Additionally, we estimate the spin parameter with the continuum-fitting method, over a range of plausible black hole masses and distances. We compare our results with the spin measured with the reflection-fitting method and find that the inconsistency of the two results is partly caused by the different choices of $f_{\rm col}$.

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