A spectrally stratified hot accretion flow in the hard state of MAXI J1820+070
MMNRAS , 1–9 (2021) Preprint 24 February 2021 Compiled using MNRAS L A TEX style file v3.0
A spectrally stratified hot accretion flow in the hard state of MAXIJ1820 + Marta A. Dziełak, (cid:63) Barbara De Marco , and Andrzej A. Zdziarski Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, PL-00-716 Warszawa, Poland Departament de Física, EEBE, Universitat Politècnica de Catalunya, Av. Eduard Maristany 16, E-08019 Barcelona, Spain
24 February 2021
ABSTRACT
We study the structure of the accretion flow in the hard state of the black-hole X-ray binaryMAXI J1820 +
070 with
NICER data. We use the power spectra to reconstruct the energyspectra of the variability components peaking at four di ff erent time scales. We find that thespectrum changes as a function of time scales. The two variability components peaking atlonger time scales have similar shape, while the two peaking at the shorter time scales di ff ersignificantly. In particular, the one corresponding to the shortest time scales has the hardestspectrum. Both the variability spectra and the time-averaged spectrum are well-modelled bya disc blackbody and thermal Comptonization, but the presence of (at least) two Comptoniza-tion zones with di ff erent temperatures and optical depths is required. The disc blackbodycomponent is highly variable, but only in the variability components peaking at the longesttime scales ( > ∼ Key words: accretion, accretion discs–X-rays: binaries–X-rays: individual (MAXIJ1820 + The majority of known black-hole (BH) X-ray binaries (XRBs) aretransient (Coriat et al. 2012). They spend most of their time in aquiescent state, characterized by low / undetectable levels of the X-ray luminosity, L X / L Edd (cid:46) − (where L Edd is the Eddington lu-minosity). After some years of quiescence, they go through X-raybrightening episodes, or outbursts, lasting from a few to tens ofmonths. During an outburst the X-ray spectral and timing proper-ties change dramatically (e.g. Belloni et al. 2005; Homan & Belloni2005; Dunn et al. 2010; Muñoz-Darias et al. 2011; Heil et al. 2015),and the source goes through a sequence of di ff erent accretion states.At the beginning and at the end of an outburst the source is ob-served in a hard state, spectrally dominated by a hard X-ray ther-mal Comptonization component (e.g. a review by Done et al. 2007).During the hard state the source becomes increasingly brighter atalmost constant spectral hardness, typically over periods of weeksto a few months. Apart from the dominant hard X-ray Comptoniza-tion component, broad band X-ray data show spectral complexity,requiring additional components. Many hard state spectra show asoft component, which can be ascribed, at least partly, to the hard-est part of the disc thermal emission (due to intrinsic dissipationand / or hard X-ray irradiation, e.g. Zdziarski & De Marco 2020), (cid:63) E-mail: [email protected] with a typical inner temperature of kT in ∼ . R g , where R g = GM / c is thegravitational radius and M is the BH mass) from low to high hard-state luminosities (e.g. McClintock et al. 2001; Esin et al. 2001;Done & Diaz Trigo 2010; Kolehmainen et al. 2014; Plant et al.2015; Basak & Zdziarski 2016; Dziełak et al. 2019). The latter re-sults are in line with theoretical models which predict evaporationof the inner disc (Meyer et al. 2000; Petrucci et al. 2008; Begelman& Armitage 2014; Kylafis & Belloni 2015; Cao 2016).One way to understand the origin of these discrepancies is toconsider more complexity in the underlying Comptonization con-tinuum. The continuum is usually assumed to result from scatteringof disc photons in an homogeneous region near the BH, which inmost cases results in a flat or convex spectral shape. On the otherhand, inhomogeneity of the Comptonization region is able to pro-duce additional curvature, resulting in a concave underlying contin-uum, as often required by the data (Di Salvo et al. 2001; Frontera © a r X i v : . [ a s t r o - ph . H E ] F e b M. A. Dziełak et al. et al. 2001; Ibragimov et al. 2005; Makishima et al. 2008; Nowaket al. 2011; Yamada et al. 2013; Basak et al. 2017; Zdziarski et al.2021b). Such conditions can account for both a part of the Fe Kline red wing and a part of the reflection hump at (cid:38)
10 keV, whichallow the need for extreme relativistic reflection parameters, as wellas a super-solar Fe abundance (often found in the fits, Fürst et al.2015; García et al. 2015; Parker et al. 2015, 2016; Walton et al.2016, 2017; Tomsick et al. 2018) to be relaxed. The presence ofa radially stratified and spectrally inhomogeneous Comptonizationregion is independently supported by X-ray variability studies. Inparticular, the common detection of frequency-dependent hard X-ray lags (hard X-ray variations lagging soft X-ray variations) inenergy bands dominated by the primary hard X-ray continuum canbe explained by mass accretion rate fluctuations propagating in-ward (Lyubarskii 1997; Arévalo & Uttley 2006). However, in or-der to detect a net hard lag, the zone these perturbations prop-agate through must have a radially-dependent emissivity profile,with the inner regions emitting a harder spectrum (Kotov et al.2001). Spectral-timing techniques, which allow resolving the spec-tral components contributing to variability on di ff erent timescales(frequency-resolved spectroscopy) have been applied to BH XRBs,showing that those spectra generally harden with decreasing timescale. This also provides evidence for the di ff erences in the ob-served spectral shape to be related to their di ff erent distances fromthe BH, see, e.g. Revnivtsev et al. (1999); Axelsson & Done (2018).Following these findings, Mahmoud & Done (2018) and Mahmoudet al. (2019) created theoretical models reproducing those observa-tions by multi-zone models coupled with propagation mass accre-tion rate fluctuations.We study the BH XRB system MAXI J1820 + Neutron star Interior Composition ExploreR ( NICER ; Gen-dreau et al. 2016) observations taken during the rise of its 2018outburst (see De Marco et al. 2021, for a systematic analysis of allthe first part of the outburst). We perform frequency-resolved spec-troscopy method in the formulation outlined by Axelsson & Done(2018). We fit frequency-resolved energy spectra using Comp-tonization models, with the goal to constrain the structure of thehot flow, and verify the possible presence of multiple Comptoniza-tion zones.MAXI J1820 +
070 was first detected in the optical wave-lengths by the All-Sky Automated Search for SuperNovae on 2018-03-06 (ASASSN-18ey, Tucker et al. 2018). It was detected in X-rays five days later by the Monitor of All-sky X-ray Image (MAXI;Matsuoka et al. 2009) on board of
International Space Station (Kawamuro et al. 2018). The two detections were connected to thesame source by Denisenko (2018). It was then identified as a BHcandidate (Kawamuro et al. 2018; Shidatsu et al. 2019). Torres et al.(2019) confirmed the presence of a BH by dynamical studies of thesystem in the optical band, and Torres et al. (2020) measured in de-tail the parameters of the system. The donor is a low-mass K starand the orbital period is 0 . ± . M ≈ (5 . ± . (cid:12) / sin i (Torres et al.2020), anticorrelated with the binary inclination, i . Those authorsestimated the latter as 66 ◦ < i < ◦ . On the other hand, Atri et al.(2020) estimated the inclination of the jet as i ≈ ± ◦ . Basedon their radio parallax, Atri et al. (2020) determined the distance toMAXI J1820 +
070 as d ≈ . ± . We note that the two inclinations can be di ff erent, and, in fact, are sig-nificantly di ff erent in some BH XRBs, e.g. GRO 1655–40, (Hjellming &Rupen 1995; Beer & Podsiadlowski 2002). Data Release 2 parallax, yielding d ≈ . + . − . kpc (Bailer-Jones et al.2018; Gandhi et al. 2019; Atri et al. 2020). MAXI J1820 +
070 is arelatively bright source, with the peak 1–100 keV flux estimated byShidatsu et al. (2019) as ≈ . × − erg cm − s − , which corre-sponds to the isotropic luminosity of ≈ . d / × erg s − .The value of Galactic hydrogen column density toward the sourcehas been estimated in the range of (0.5–2) × cm − (Uttley et al.2018; Kajava et al. 2019; Fabian et al. 2020; Xu et al. 2020). For our analysis, we choose two relatively long observations ofMAXI J1820 +
070 carried out by
NICER , see Table 1. Hereafter, werefer to the studied observations using only the last three digits oftheir identification number (ID), i.e. 103 and 104. They correspondto the initial phases of the 2018 outburst, when the source was inthe rising hard state (see Figure 1 in De Marco et al. 2021). We es-timate their energy fluxes to be ≈ ≈ . × − erg cm − s − ,respectively, in the 0.3–10 keV energy band.The data were reduced using the NICERDAS tools in hea - soft v.6.25, starting from the unfiltered, calibrated, all Measure-ment / Power Unit (MPU) merged files, ufa . We applied the stan-dard screening criteria (e.g. Stevens et al. 2018) through the
NIMAKETIME and
NICERCLEAN tools. We check for periods of highparticle background, i.e., with rate > − , by inspecting lightcurves extracted in the energy range of 13–15 keV, in which thecontribution from the source is negligible because of the drop inthe e ff ective area (Ludlam et al. 2018).Of the 56 focal plane modules (FPMs) of the NICER
X-rayTiming Instrument (XTI), four (FPMs 11, 20, 22, and 60) are notoperational. Additionally, we filter out the FPMs 14 and 34, sincethey are found to occasionally display increased detector noise .Thus, the number of FPMs used for our analysis is 50. In orderto correct for the corresponding reduction of e ff ective area, the re-ported source fluxes were rescaled by the number of used FPMs.The shortest good time intervals, with the length <
10 s, wereremoved from the analysis, resulting in the net, on-source, expo-sure times reported in Table 1. Filtered event lists were barycentre-corrected and used to extract the light curves and spectra using
XSELECT v.2.4e. We used publicly distributed ancillary responseand redistribution matrix files as of 2020-02-12 and 2018-04-04,respectively. Fits were performed using the X-Ray Spectral FittingPackage (X spec v.12.10.1; Arnaud 1996). Hereafter uncertaintiesare reported at the 90 per cent confidence level for a single param-eter. We first compute and fit the power spectral density (PSD) for eachof the observations in two broad energy bands, 0.3–2 keV and 2–10 keV, in order to identify components contributing to the ob- We have used a model including a disc blackbody plus a powerlaw, both absorbed by the cold gas in the interstellar medium;
TBabs(diskbb+powerlaw) in X spec . https://heasarc.gsfc.nasa.gov/docs/nicer/data_analysis/nicer_analysis_tips.html https://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/data/nicer/xti/index.html MNRAS , 1–9 (2021) tratified hot accretion flow Table 1.
The log of the
NICER observations used for the analysis. Expo-sures correspond to the e ff ective on-source time after data cleaning.Obs. ID Start time Exposure[yyyy-mm-dd hh:mm:ss] [s]1200120 served variability on di ff erent timescales. We extract light curveswith a time bin of 0.4 ms in these energy bands. The light curveswere split into segments of 200 s length. We calculate the PSD ofeach segment and average them, in order to obtain more accurateestimates of the PSD of each single observations. The chosen timebin and the segment length allow us to cover the range of frequen-cies of 0.005–1250 Hz. This range well samples both the broad-band noise intrinsic to the source and the Poisson noise level. Thelatter was fit at frequencies of ν >
300 Hz and subtracted from thePSDs at all frequencies. We adopt a logarithmic rebinning in orderto improve the statistics at high frequencies.We normalize the PSDs using the fractional squared root-mean-square units (( rms / mean rate) Hz − , Belloni & Hasinger1990; Miyamoto et al. 1992). The PSDs are shown in Fig. 1. Theircomplex structure hints at the presence of several variability com-ponents. As commonly done in the literature (Belloni et al. 1997;Nowak 2000), we fit each PSD with a sum of Lorentzian compo-nents, for which we use X spec . A Lorentzian is described by P ( ν ) = K σ π ν − ν ) + ( σ/ , (1)where K is the normalization, σ is the full width at half maximum,and ν is the centroid frequency. However, the maximum power (i.e.the peak of ν P ( ν )) is observed at the frequency ν max > ν (Belloniet al. 1997) ν max = (cid:114) ν + (cid:18) σ (cid:19) . (2)We jointly fit the resulting four PSDs, we tie ν and σ pa-rameters, but allow for di ff erent normalization. We start with fit-ting a single Lorentzian, and add a new one if significant residu-als remained. We find that the hard band is well described by fourLorentzian components, but an addition of the highest-frequency,fourth, Lorentzian in the soft band PSD yields a normalization con-sistent with zero. We also find that letting ν and σ vary indepen-dently for each PSD does not lead to significant fit improvements.Therefore, our final joint model consists of four Lorentzians (here-after L1, L2, L3, and L4), with the parameters given in Table 2. ThePoisson noise subtracted PSDs with the best-fit models are shownin Fig. 1.We check that the PSDs of the two observations have com-patible shape and consistent fractional normalization in each of thetwo energy bands ( ≈
32 and ≈
27 per cent in the soft and hard band,respectively). This means that there is no significant deviation fromstationarity between the two observations. Therefore, we decide tocombine the two observations in order to obtain a higher signal-to-noise for the remainder of our analysis.
We then create energy spectra of the variability componentscomponents (hereafter rms spectra) contributing to the di ff erenttimescales using the Lorentzian fits parametrized in Section 3.1. Table 2.
The results of the joint fits of the PSDs of the observations 103 and104 in the 0.3–2 keV and 2–10 keV energy bands with the model consist-ing of four Lorentzians, see equation (1). We also give the resulting peakfrequency of ν P ( ν ), equation (2), and the resulting reduced χ .L1 L2 L3 L4 ν [Hz] 0 + . + . − . + . − . + . − . σ [Hz] 0.045 + . − . + . − . + . − . + . − . ν max [Hz] 0.023 + . − . + . − . + . − . + . − . χ ν − − . . F r equen cy x P o w e r (r m s / m ean ) Frequency (Hz)
L1 L2L3 L4 − − . . Frequency (Hz)
L1 L2L3 L4 F r equen cy x P o w e r (r m s / m ean ) Figure 1.
The Poisson noise subtracted PSDs of the two analyzed obser-vations in the two energy bands, 0.3–2 keV (black symbols) and 2–10 keV(red symbols). The total best-fit models (solid lines) and the Lorentziancomponents of the model (dotted curves) are also shown.
To this aim, we use the technique proposed by Axelsson et al.(2013). We create a logarithmic energy grid with 47 bins in the 0.3–9.9 keV energy range. We then calculate the PSD in absolute units((counts − ) Hz − , Vaughan et al. 2003) for each of the bins. We thenfit the PSD in each band with the same Lorentzian components asthe best-fit model to the PSD of the broad energy bands (Table 2),keeping the values of ν and σ fixed, and allowing only the nor-malization of each Lorentzian to vary. The best-fit PSD models foreach narrow energy band are then used to extract the rms variabilityamplitude spectrum of each Lorentzian. To this aim we analyticallyintegrate the best-fit Lorentzians within the frequency range 0.001– MNRAS000
To this aim, we use the technique proposed by Axelsson et al.(2013). We create a logarithmic energy grid with 47 bins in the 0.3–9.9 keV energy range. We then calculate the PSD in absolute units((counts − ) Hz − , Vaughan et al. 2003) for each of the bins. We thenfit the PSD in each band with the same Lorentzian components asthe best-fit model to the PSD of the broad energy bands (Table 2),keeping the values of ν and σ fixed, and allowing only the nor-malization of each Lorentzian to vary. The best-fit PSD models foreach narrow energy band are then used to extract the rms variabilityamplitude spectrum of each Lorentzian. To this aim we analyticallyintegrate the best-fit Lorentzians within the frequency range 0.001– MNRAS000 , 1–9 (2021)
M. A. Dziełak et al. . . k e V ( P ho t on s c m − s − k e V − ) Energy (keV)
L1 L2 L3 L4
Figure 2.
The rms spectra of each of the four Lorentzians, unfolded to aconstant model. rms = (cid:115)(cid:18) K π (cid:19) arctan (cid:32) ( ν − ν ) σ/ (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = ν = . . (3)We plot these values as a function of energy to obtain the rms spectra. The resulting rms spectra will then show the spectralcomponents that contribute to the variability associated with eachLorentzian. We report these spectra in Fig. 2. We note that sincethese four components were obtained from the power spectrum,which gives rms , they were assumed to be completely uncorre-lated, and, in order to obtain the total rms spectrum, the single rms spectra have to be summed in quadrature. We note that the rms spectra have the redistribution matrix obtained by rebinningthe matrix for the average spectrum.The errors on the rms spectra were calculated based on the 1 σ deviation of the normalization of the best-fit Lorentzians. To thisaim, for each Lorentzian and in each energy band, we draw tworandom numbers distributed according to a Gaussian distributionwith the mean equal to the best-fit value of the normalization, K ,and the standard deviation equal to its 1 σ lower / upper error of K .This procedure was repeated 100 times for each Lorentzian compo-nent, resulting in two sets of 100 simulations each. We use each setto estimate the upper and lower limits on the rms in every energychannel. In order to have symmetric errors (as required in X spec )we take the average of the obtained upper and lower errors. Theunconstrained points, i.e., those resulting in upper limits on the rms (e.g. as in the case of the normalization of L4 in the softest energybands) were omitted in the fits and are not presented in the plots. In our models, absorption due to the interstellar medium (ISM) ismodeled using tbabs (Wilms et al. 2000), with the elemental abun-dances from the same paper. We model thermal Comptonization us-ing the model
ThComp (Zdziarski et al. 2020). Since this is a convo-lution model, the covered energy range has to be extended beyondthat of the
NICER spectra in order to get a full range of energies for the seed photons; we use the range of 0.01 keV–2000 keV. Since athigh energies the sensitivity of the instrument is limited to E < ∼ kT e =
50 keV, which closely resembles the values found fromsimultaneous fits of
NuSTAR data (Zdziarski et al. 2021a).We first consider single Comptonization models, with thedisc thermal emission (Mitsuda et al. 1984) being the sourceof seed photons. The ISM hydrogen column density is fixedat N H = . × cm − (which is the best-fit value obtainedfrom our fit to the time-averaged spectrum, see Table 4 below). ThComp has the fraction of the seed photons incident on the Comp-tonization region, f , as a free parameter. However, we hereafterset f = spec notation of the modelis TBabs(diskbb +ThComp(diskbb )) . We note that such addi-tive form of the variability model spectrum assumes that the twospectral components are fully correlated. In our first case, ModelA, we fit rms spectra of each Lorentzian (L1, L2, L3, L4) withthe same parameters except for the normalizations. The normal-ization of thComp is given by the normalization of the convolved diskbb model. The other free parameters are the inner disc tem-perature, kT in , and the low-energy spectral index of the Comp-tonization spectrum, Γ (defined by the energy flux, F ( E ) ∝ E − Γ ).The obtained best-fit parameters are given in Table 3 and the ra-tios of the data to the model are shown in Fig. 3 (top panel). Wefind that this model provides a rather poor description of the data, χ ν ≈ . / Γ , to varyamong di ff erent Lorentzians. This means that we assume the pres-ence of four di ff erent Comptonization zones with di ff erent Thom-son optical depths, τ T , since the spectral index is a function of both kT e and τ T (see eq. 14 in Zdziarski et al. 2020; and Sunyaev &Titarchuk 1980 for a general discussion).In Model C, we also allow for changes among kT in of the seedsfor Comptonization, and allow them to be di ff erent from the di-rectly observed disc blackbody. However, we find the seed photonstemperature of the rms spectrum of L1 to be unconstrained, so wekeep it equal to kT in of the directly observed disc. Model C assumesthe presence of four di ff erent disc blackbodies. While it is unreal-istic, we include it as a phenomenological description in order tohighlight that the data do require the seed photons to have di ff erentcharacteristic energies for di ff erent components. The best-fit pa-rameters of Models B and C are given in Table 3, while the ratiosof the data to the corresponding best-fit models are shown in Fig. 3(middle and bottom panel).Comparing results from the spectral fits of Model A with thoseof B and C, we see significant evidence for changes of the spec-tral properties between the rms spectra of the di ff erent Lorentzians.In particular, the fit improves when letting an increasing numberof parameters free to vary among the di ff erent variability compo-nents (from model A to model C, ∆ χ ≈ − rms spectrum of the highest fre-quency Lorentzian (L4) is best fit by a much higher seed photons The X spec commands energies extend low 0.001 and energiesextend high 2000 were used. This avoids modifying the energy grid ofthe data. MNRAS , 1–9 (2021) tratified hot accretion flow temperature ( kT in2 in Table 3) than the lower frequency compo-nents. This hints at a di ff erent source of seed photons for L4, thussuggesting that the innermost parts of the accretion flow are fueledby photons with higher temperature than those fueling the outerparts.Given the clear indications of changes in spectral propertiesamong di ff erent Lorentzian components, we model now the rms spectra and the time-averaged spectrum simultaneously, with theaim of obtaining more robust and self-consistent constraints on thestructure of the Comptonization region. For consistency with the rms spectra, the time-averaged spectra of observations 103 and 104were averaged (using MATHPHA in ftools ). The resulting spectrumwas rebinned so as to have a minimum of 3 original energy chan-nels and signal-to-noise ≥
50 in each new energy channel. To ac-count for uncertainties in the current calibration, a systematic errorof 1 per cent was added. (Note that this step is not required forthe rms spectra, owing to their lower resolution.) Given the limitedbandpass of
NICER , we did not include complex reflection modelsin the fits, which would result in overfitting the rms spectra. There-fore, we model the fluorescent Fe K line with a simple Gaussiancomponent.We now assume that the hot flow can be described by twoComptonization zones. The model, denoted as D, comprises a di-rectly visible disc (zone I), which photons are upscattered in theouter Comptonization zone II, and those upscattered photons arepartly directly observed and partly are upscattered in the innerComptonization zone III, as illustrated in the drawing in Fig. 4(top panel). The X spec notation is given in the caption to Table4. Again, the underlying assumption is that the three componentsare fully correlated. The unabsorbed best-fit model to the time-averaged spectrum with these components is presented in Fig. 4(bottom panel). In this model, we assume that all the parametersgoverning the shape of the above spectral components are the samefor both the rms and the time-averaged spectra, and allow only theirnormalizations to di ff er. Since the Fe line is unconstrained in the rms spectra, we require their normalizations to be ≤ than that in thetime-averaged spectrum. The best-fit parameters are given in Table4, and the best-fit model and the data-to-model ratios are shown inFig. 5. We find that model D can simultaneously describe the time-averaged and rms spectra well, with χ ν ≈ . / Low frequency Lorentzians (L1 and L2) : In model D, the rms spectra of L1 and L2 show significant contributions fromall of their spectral components (see normalizations in Table 4).Still, most of the photons related to the variability component pik-ing at frequencies (cid:46) ≈
63 per cent of the to-tal (time-average) disc blackbody flux at the best fit. In zone II,the variable components actually exceed the corresponding time-average, which implies an inaccuracy resulting from the assumedspectral model (see Section 4). Still, it indicates that most of thephotons in this zone are variable.
High frequency Lorentzians (L3 and L4) : These variabil-ity components show no direct emission from the disc at the bestfit. This means that the disc gives low contribution to the variabilitycomponents peaking at the time scales (cid:38) ≈ / ≈ / ABC r a t i o r a t i o r a t i o Figure 3.
The data-to-model ratios of models A, B, C (from top to bottom;see Table 3). The same colour coding as in Fig. 2 is used: L1 – red, L2 –green, L3 – blue and L4 – orange. region (zone II) are used as seed photons for the inner Comptoniza-tion region (zone III).In the model, emission from the inner Comptonization regiondominates the bolometric X-ray flux. The outer Comptonization re-gion is significantly softer than the inner one, i.e., Γ > Γ . Overall,these results show that the structure for the accretion flow in thehard state is quite complex, with stratified Comptonization regions. In our approach, spectral components derived from variability ondi ff erent time scales are thought to originate from regions of theaccretion flow related to these time scales. Generally, we expect thecharacteristic variability time scale to decrease with the decreasingradial distance from the BH. However, connecting variability timescales to radii is far from simple. Detailed spectro-timing modelsfor Cyg X-1 and GX 339–4, in which such radii were identified,were performed by Mahmoud & Done (2018) and Mahmoud et al.(2019). In particular, the former work found evidence for the exis-tence of discrete regions of enhanced turbulence corresponding tohumps in the power spectra.Here, we simply compare the characteristic Keplerian fre-quency, ν K ≡ ( GM ) / / (2 π R / ) to the characteristic frequenciesof our power spectrum, to identify the corresponding characteristicradii. Since this frequency corresponds to the fastest possible vari-ability, it sets an upper limit on the actual emitting radii. This canbe written as RR g (cid:46) c (2 π GM ν ) / ≈ (cid:32) M (cid:12) (cid:33) − / (cid:18) ν (cid:19) − / . (4)For the highest peak frequency in the power spectrum, ≈ MNRAS000
The data-to-model ratios of models A, B, C (from top to bottom;see Table 3). The same colour coding as in Fig. 2 is used: L1 – red, L2 –green, L3 – blue and L4 – orange. region (zone II) are used as seed photons for the inner Comptoniza-tion region (zone III).In the model, emission from the inner Comptonization regiondominates the bolometric X-ray flux. The outer Comptonization re-gion is significantly softer than the inner one, i.e., Γ > Γ . Overall,these results show that the structure for the accretion flow in thehard state is quite complex, with stratified Comptonization regions. In our approach, spectral components derived from variability ondi ff erent time scales are thought to originate from regions of theaccretion flow related to these time scales. Generally, we expect thecharacteristic variability time scale to decrease with the decreasingradial distance from the BH. However, connecting variability timescales to radii is far from simple. Detailed spectro-timing modelsfor Cyg X-1 and GX 339–4, in which such radii were identified,were performed by Mahmoud & Done (2018) and Mahmoud et al.(2019). In particular, the former work found evidence for the exis-tence of discrete regions of enhanced turbulence corresponding tohumps in the power spectra.Here, we simply compare the characteristic Keplerian fre-quency, ν K ≡ ( GM ) / / (2 π R / ) to the characteristic frequenciesof our power spectrum, to identify the corresponding characteristicradii. Since this frequency corresponds to the fastest possible vari-ability, it sets an upper limit on the actual emitting radii. This canbe written as RR g (cid:46) c (2 π GM ν ) / ≈ (cid:32) M (cid:12) (cid:33) − / (cid:18) ν (cid:19) − / . (4)For the highest peak frequency in the power spectrum, ≈ MNRAS000 , 1–9 (2021)
M. A. Dziełak et al.
Table 3.
The joint fit results for models with a single Comptonization zone for each of the Lorentzians (L1, L2, L3, L4). The X spec notation is
TBabs(diskbb +ThComp(diskbb )) . The normalization of thComp is given by the normalization of the convolved diskbb model. In model A, the Comp-tonization plasmas have the same parameters for all of the Lorentzians, and in model B, they can di ff er in the optical depth, resulting in di ff erent spectralindices, Γ . Model C is a phenomenological one, in which the inner disc temperatures, kT in2 , of the seed photons for Comptonization can also di ff er betweenthe Lorentzians. Hereafter, the parameters assumed to be fixed are denoted by (F).A B CComponent Parameter L1 L2 L3 L4 L1 L2 L3 L4 L1 L2 L3 L4 TBabs N H [10 cm − ] 1.4 (F) 1.4 (F) 1.4 (F) diskbb kT in1 [keV] 0.24 + . − . + . − . + . − . N [10 ] 15.8 + . − . + . − . + . − . + . + . − . + . − . + . + . + . − . + . − . + . + . ThComp Γ + . − . + . − . + . − . + . − . + . − . + . − . + . − . + . − . + . − . kT e [keV] 50 (F) 50 (F) 50 (F) diskbb kT in2 [keV] = kT in1 = kT in1 = kT in1 + . − . + . − . + . − . N [10 ] 8.1 + . − . + . − . + . − . + . − . + . − . + . − . + . − . + . − . + . − . + . − . + . − . + . − . χ ν /
160 232.7 /
157 185.9 / Table 4.
The joint fit results for model D with disc blackbody and two Comptonization zones for each of the rms components and the time-averaged spectrum,S av . The X spec notation is TBabs(diskbb +ThComp (diskbb )+ThComp (ThComp (diskbb ))+Gaussian) . The single diskbb emission is split betweenthree parts with the same kT in and di ff erent normalizations, N , N and N . The Roman numbers correspond to the zones shown in Fig. 4. The componentfluxes, F , are unabsorbed and bolometric (measured in the energy range 0.001-2000keV). DComponent Parameter L1 L2 L3 L4 S av TBabs N H [10 cm − ] 1.4 + . − . I diskbb kT in [keV] 0.18 + . − . N [10 ] 52.7 + . − . + . − . + . + . + . − . F [10 − erg cm − s − ] 1.21 + . − . + . − . + . + . + . − . II ThComp Γ + . − . kT e1 [keV] 0.34 + . − . diskbb N [10 ] 8.9 + . − . + . − . + . − . + . + . − . F [10 − erg cm − s − ] 0.47 + . − . + . − . + . − . + . + . − . III
ThComp Γ + . − . kT e2 [keV] 50 (F) diskbb N [10 ] 13.5 + . − . + . − . + . − . + . − . + . − . F [10 − erg cm − s − ] 14.61 + . − . + . − . + . − . + . − . + . − . Gaussian E [keV] 6.5 + . − . σ [keV] 0.39 + . − . N [10 − cm − s − ] 0 + . + . + . + . + . − . χ ν /
150 213.571 / / in Table 2), we have R / R g (cid:46) spec . Using the total normalization in the time-averaged spectrumin model D, ( N + N + N ), we can estimate the disc inner radius, R in . We find, R in ≈ . + . − . (cid:18) κ . (cid:19) d (cid:18) cos i cos 65 ◦ (cid:19) − / cm ≈ (5) ≈ + − (cid:32) M (cid:12) (cid:33) − (cid:18) κ . (cid:19) d (cid:18) cos i cos 65 ◦ (cid:19) − / R g , (6)where κ ≈ . MNRAS , 1–9 (2021) tratified hot accretion flow . . k e V ( P ho t on s c m − s − k e V − ) Energy (keV)
I II III
Figure 4. (Top panel): a drawing of the geometry assumed in model D.(Bottom panel) the unabsorbed best-fit to the time-averaged spectrum (solidcurve) and its components (dotted curves) marked by the zones shown in thetop panel and Table 4. and III can fit downstream of R in , showing the self-consistency ofthis aspect of model D.The question our research directly addressed was the structureof the accretion flow. Detailed modelling of Comptonization in theaccretion flow appears not unique. Clearly, this flow does not con-sist of a single uniform plasma cloud, as shown by the failure ofour model A to describe the rms spectra. Our model B (Table 3)shows that a significant improvement is achieved when allowingthe slopes of the rms spectra to di ff er between the components, inagreement with what is seen in Fig. 2. This can be due to the pa-rameters of the Comptonizing cloud, namely the optical depth andelectron temperature, evolving along the accretion flow, and form-ing distinct zones.Our model C points to the average energy of the seed pho-tons for Comptonization to vary between the rms spectra of eachLorentzian component. The inner temperature for L1 is weaklyconstrained, but consistent with being the same as that of the di-rectly observed disc blackbody; those for L2 and L3 are somewhatlarger, and the seed photons for L4 have significantly higher tem-perature, kT in ≈ . ff than that in the fitted model, see Fig. 3. Disc black-body emission is significant in the rms spectra of the two lowerfrequency Lorentzians (L1 and L2), meaning that the disc stronglyvaries on long time scales ( ν max (cid:46) rms spectra of L3 and L4, meaning that the disc doesnot contribute significantly to the high frequency variability com- . . k e V ( P ho t on s c m − s − k e V − ) . . k e V ( P ho t on s c m − s − k e V − ) . . k e V ( P ho t on s c m − s − k e V − ) . . k e V ( P ho t on s c m − s − k e V − ) . . Energy (keV) D r a t i o Figure 5.
The best-fit models and data of the Lorentzian spectral compo-nents (from top to bottom: L1, L2, L3, L4), and the data-to-model ratios(bottom panel) of model D. The colour code is the same as in Fig. 2, andthe time-averaged spectrum and its residuals are shown in grey.MNRAS000
The best-fit models and data of the Lorentzian spectral compo-nents (from top to bottom: L1, L2, L3, L4), and the data-to-model ratios(bottom panel) of model D. The colour code is the same as in Fig. 2, andthe time-averaged spectrum and its residuals are shown in grey.MNRAS000 , 1–9 (2021)
M. A. Dziełak et al. ponents ( ν max > (cid:46) ≈ ≥ rms spectra show more uniformity (Fig. 2).In model D, we explored an alternative scenario in which eachof the rms spectra is fitted by two Comptonization zones. Followingresults from fits of model B and C, we ascribe the disc blackbodyemission to the outer disc zone I (Fig. 4). The electrons in the outerComptonization zone (zone II in Fig. 4) upscatter the disc black-body photons, while those in the inner Comptonization zone (zoneIII in Fig. 4) upscatter the photons created in the outer Comptoniza-tion zone. This follows the hints for such complexity from the re-sults of the fitting of the previous models B and C. Model D cansimultaneously describe the time-averaged and rms spectra quitewell. The fractional variability of the directly observed disc black-body emission is as high as (cid:38)
60 per cent (model D; Table 2). Thisvalue is consistent with estimates for GX 339-4 in the hard state(De Marco et al. 2015). The variability of the disc blackbody pho-tons is almost entirely in the low-frequency components L1 andL2. Still, the components L1 and L2 also show variability in theComptonization zones. The spectrum L3 has significant contribu-tions from both the outer and inner Comptonization zones but nonefrom the disc, while L4 is formed almost exclusively in the innerzone.The behaviour described above strongly supports the pictureof propagating fluctuations, with the dominant variability movingfrom the outer disc through the outer hot flow to the inner one whilethe dominant characteristic variability frequency increases. Our re-sults agree with the detection of hard X-ray lags in this source (Karaet al. 2019; De Marco et al. 2021). Indeed, such lags are commonlyascribed to propagation of mass accretion rate fluctuations in theaccretion flow (Lyubarskii 1997). In order to produce lags, the per-turbations have to propagate through a spectrally inhomogeneousregion, with the hardest spectrum produced at the smallest radii(Kotov et al. 2001), as we find from our fits.Still, this model has a number of caveats. It shows significantresiduals at the low-energy cuto ff of the spectrum L4, similarly asin the previous models. Such residuals suggest that the emissionfrom zone III may be underestimated in the spectrum of L1, L2, andL3. This would ultimately lead to overestimate emission from zoneII, resulting in this component exceeding the corresponding timeaverage contribution. A solution for this would be to include furtherstratification of the hot flow, but our data are statistically not su ffi -cient for such complex models. Moreover, since we assumed thatthere are only two Comptonization zones, the rms spectrum of eachLorentzian is modelled assuming the same slope for each Comp-tonization zone, and only the relative normalizations vary. Whilethis provides a good fit to the data at ≤
10 keV, it fails to describethe hardening observed at (cid:38)
10 keV in this source (Zdziarski et al.2021a). Such hardenings are also observed in other BH XRBs, e.g.Cyg X-1 (Nowak et al. 2011) or XTE J1752–223 (Zdziarski et al.2021b). We note that the outer Comptonization zone has a very lowelectron temperature, kT e ≈ . + . − . keV. This means that this islikely to be a warm corona above the disc rather than a hot flow.The implied Thomson optical depth is τ T ≈
48 (as follows for the used spectral model from eq. 14 of Zdziarski et al. 2020), whichis quite unrealistic. Looking at the spectral components in Fig. 4,we see that this component resembles quite well another blackbodycomponent rather than a genuine Comptonization zone. While thispoints to a deficiency of our model, more model complexity couldnot be reasonably constrained by our data.
We studied the spectral structure of the Comptonization region inthe hard state of MAXI J1820 +
070 using
NICER data. To thisaim we extracted the energy spectra of each Lorentzian describingthe humped shape of the PSD of the source. The distribution overtimescales of these variability components is thought to resemblethe spatial distribution of energy dissipation zones in the hot flow,with the highest frequency Lorentzians corresponding to variabilityproduced in the innermost regions.The main result of our analysis is the evidence of a spec-tral stratification of the hot flow, which can be clearly appreciatedin a model-independent way by simply looking at the significantchanges in the rms spectra of the di ff erent Lorentzians (Fig. 2).Going from the lowest to the highest frequency Lorentzian (L1 andL4 respectively) a net hardening of the hard X-ray spectrum is ob-served.We modeled the rms spectra of each Lorentzian in order tocharacterize the spectral structure of the source. Our simple modelsB and C (Sect. 3.3 and Table 3) show that the slope of the Comp-tonization component significantly changes among Lorentzians.This means that the physical properties of the Comptonization re-gion change as a function of radius. In particular, we found that thehighest frequency Lorentzian L4 has a significantly harder spec-trum than L3 (Table 3). The two lowest frequency Lorentzians L1and L2 do not show significant changes of spectral index and havea softer / harder spectrum than L4 / L3 (Table 3). Also, our model Cprovides evidence for hotter seed photons ( kT in = . + . − . keV) inthe fastest variability component L4 compared to the other compo-nents ( kT in ∼ . rms spectra of eachLorentzian and the time-averaged spectrum of the source. Themodel (D, Table 4) comprises an outer Comptonization region fu-eled by thermal photons from the cool disc, and an inner Comp-tonization region fueled by a fraction of the upscattered photonsfrom the outer Comptonization region. We find that this model candescribe data spectra well. ACKNOWLEDGEMENTS
We have benefited from discussions during Team Meetings ofthe International Space Science Institute in Bern, whose supportwe acknowledge. We also acknowledge support from the PolishNational Science Centre under the grants 2015 / / A / ST9 / / / B / ST9 / MNRAS , 1–9 (2021) tratified hot accretion flow DATA AVAILABILITY
The data underlying this article are available in HEASARC, at https://heasarc.gsfc.nasa.gov/docs/archive.html . REFERENCES
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