Implementation of a radial disk ionization profile in the RELXILL_NK model
Askar B. Abdikamalov, Dimitry Ayzenberg, Cosimo Bambi, Honghui Liu, Yuexin Zhang
IImplementation of a radial disk ionization profile in the relxill nk model
Askar B. Abdikamalov,
1, 2
Dimitry Ayzenberg, Cosimo Bambi, ∗ Honghui Liu, and Yuexin Zhang Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 200438 Shanghai, China Ulugh Beg Astronomical Institute, Tashkent 100052, Uzbekistan Theoretical Astrophysics, Eberhard-Karls Universit¨at T¨ubingen, D-72076 T¨ubingen, Germany Kapteyn Astronomical Institute, University of Groningen, 9747 AD Groningen, The Netherlands
Very steep reflection emissivity profiles in the inner part of accretion disks are commonly found inthe analysis of X-ray observations of black hole binaries and AGNs, but there is some debate abouttheir exact origin. While steep reflection emissivity profiles can be naturally produced by compactcoronae close to black holes, the measured radial emissivity parameter can be further increased bythe radial disk ionization profile when the theoretical model assumes a disk with constant ionization.In this paper, we implement the possibility of a radial disk ionization profile in the reflection model relxill nk and we analyze a
NuSTAR observation of the black hole binary EXO 1846–031, whichwas previously found to have a very high inner emissivity index. We find that the model with aradial disk ionization profile improves the fit, but the impact on the estimate of the black hole spinparameter and on the constraint of the deformation parameter is modest.
I. INTRODUCTION
Relativistic reflection features are common in the X-ray spectra of accreting black holes [1–5]. They arethought to be generated by illumination of the accretiondisk by a hot corona, which is some energetic plasmanear the black hole [6, 7]; for a pedagogical review, see,e.g., Ref. [8]. More specifically, thermal photons fromthe accretion disk can inverse Compton scatter off freeelectrons in the corona. A fraction of the Comptonizedphotons can then illuminate the disk, generating the re-flection spectrum.In the rest-frame of the gas in the disk, the reflectionspectrum is characterized by a soft excess below 1 keV,narrow fluorescent emission lines in the 1-10 keV band(including the iron K α line at 6-7 keV, depending onthe ionization of the iron ions), and a Compton humppeaked at 20-30 keV [9–12]. The spectrum of the wholedisk observed far from the source is the result of relativis-tic effects occurring in the strong gravity region aroundthe black hole (Doppler boosting, gravitational redshift,and light bending) [1, 13]. The narrow fluorescent emis-sion lines in the gas rest-frame are thus broadened andskewed for a distant observer. Since the reflection spec-trum is mainly produced from the very inner part of theaccretion disk, the analysis of relativistic reflection fea-tures in the spectra of accreting black holes can be usedto study the accretion process around these compact ob-jects, measure black hole spins, and possibly test funda-mental physics [14–19].In the past decade, there has been significant progressin the analysis of relativistic reflection spectra, thankseither to the development of more sophisticated theoret-ical model to calculate synthetic reflection spectra andthe advent of new observational facilities more suitablefor the study of these features. However, even the latest ∗ Corresponding author: [email protected] and most advanced theoretical models rely on a numberof simplifications [20], so caution is necessary in the at-tempts to get precision measurements of the properties ofaccreting black holes from the analysis of their reflectionfeatures.A number of studies have reported very steep reflectionemissivity profiles in the inner part of accretion disks, ei-ther for stellar-mass black holes in X-ray binaries and su-permassive black holes in AGNs; see, e.g., [21–25]. Steepemissivity profiles in the inner part of an accretion diskcan be naturally produced by a compact source very closeto the black hole [26, 27]. In such a case, the strong lightbending near the black hole can drive the radiation emit-ted from the corona to illuminate better the inner partof the accretion disk closer to the event horizon.The ionization of the disk plays an important role inthe shape of the reflection spectrum. The ionization pa-rameter, normally indicated with the letter ξ and mea-sured in units of erg cm s − , is defined as ξ = 4 πF X ( r ) n e ( r ) , (1)where F X is the X-ray flux from the corona illuminatingthe disk and n e is the disk electron density. In general, ξ , F X , and n e are all functions of the radial coordinate r . The radial profile of F X is determined by the coronalgeometry. For n e , the radial profile depends on the diskproperties.Theoretical models for the analysis of relativistic re-flection features in the spectra of accreting black holesnormally assume a constant ionization parameter overall of the disk. From an observational point of view, formost data the fit simply does not require any ionizationgradient. When the inner edge of the accretion disk isvery close to the black hole and the corona is compactand low, the Componized photons are highly focused ona small portion of the very inner part of the accretiondisk, which we can expect to be approximated well by aone-ionization region. For instance, the study in Ref. [28]finds a very high inner emissivity index when the data are a r X i v : . [ a s t r o - ph . H E ] J a n fit with a model with a broken power-law emissivity pro-file and a constant disk ionization, but when they allowfor a non-constant ionization profile the data do not re-quire any ionization gradient. On the other hand, if theradial profile of F X is steep, even the profile of the ion-ization parameter ξ should be steep for any reasonabledisk density profile n e . This point was investigated inRef. [29] and then in Ref. [30]. The conclusion of bothstudies is that fitting the data with a theoretical modelemploying a constant ionization parameter may lead tooverestimate the steepness of the inner emissivity profile,which, in turn, can lead to inaccurate black hole spinmeasurements; see also Ref. [31].The latest versions of the relativistic reflection models relxill [27, 32], kyn [33], and reltrans [34] offer the op-tion to have a non-trivial ionization profile in the disk. Inthe present paper, we implement a non-trivial ionizationprofile in our reflection model relxill nk [35–37], whichis an extension of the relxill package to non-Kerr space-times [38]. We then use this new version of relxill nk to analyze a 2019 NuSTAR observation of the black holebinary EXO 1846–031, as previous analyses of these datahad assumed a constant ionization parameter and founda very steep inner emissivity index. For this source, wefind that the model with a non-trivial ionization gradientcan provide a better fit, but the impact on the estimatesof the black hole spin or of the deformation parameterare weak and, in the end, negligible.Our manuscript is organized as follows. In Section II,we describe the implementation of a radial ionization pro-file in relxill nk . In Section III, we present the spectralanalysis of the 2019
NuSTAR observation of EXO 1846–031 with constant and non-constant disk ionization pro-files. We discuss our results in Section IV.
II. RADIAL DISK IONIZATION PROFILES
The relxill nk package has so far assumed a con-stant ionization parameter across the entire disk. Sinceit also employs a constant electron density, from Eq. (1)it would follow that F X is constant too. The questionis whether such an approximation can be used withoutintroducing undesirable large systematic uncertainties inthe estimate of the model parameters, and in particularon the black hole spin and the deformation parameterof the relxill nk model. This issue can be particularlyrelevant when the fit finds a very high inner emissivityprofile, which is also the case in which we can get moreprecise measurements of the black hole spin and the de-formation parameter. relxillion nk is the new flavor of the relxill nk package that offers the possibility of a non-constant radialionization profile. We employ an empirical power-lawform for the radial profile of the ionization parameter ξ ( r ) = ξ (cid:18) R in r (cid:19) α ξ , (2) where ξ is the value of the ionization parameter at the in-ner edge of the accretion disk, R in , and α ξ is the extra pa-rameter (ionization index) in relxillion nk to describethe ionization gradient. The standard relxill nk modelwith constant disk ionization is recovered for α ξ = 0,while for any positive value of α ξ the value of the ion-ization parameter decreases as the radial coordinate r increases.The accretion disk is divided into 50 annuli and thevalue of the ionization parameter at the center of everyannulus derived from Eq. (2). For every annulus, thereflection component is extracted from the xillver tableaccording to the ionization state of that region. Thereflection spectrum from each annulus is then convolvedto obtain the reflection spectrum detected by the distantobserver and the spectra of all annuli are summed uptogether to obtain the total spectrum.Figs. 1-4 illustrate the impact of the ionization gradi-ent of the accretion disk on the reflection spectrum of anaccreting black hole. In all plots, we assume that the in-ner edge of the accretion disk is at the innermost stablecircular orbit (ISCO), so R in = R ISCO in Eq. (2). Theconstant ionization case ( ξ = ξ , blue curves) is comparedwith the spectra calculated assuming the ionization in-dex α ξ = 1 (orange curves) and α ξ = 2 (black curves)for various values of the black hole spin parameter a ∗ ,the Johannsen deformation parameter α , the viewingangle ι , and the ionization parameter at the ISCO ξ . III. THE CASE OF EXO 1846–031
EXO 1846–031 is a black hole candidate discoveredin 1985 by the
EXOSAT mission [40]. After spendingmore the 30 years in quiescence, it went through a newoutburst in 2019 [41].
NuSTAR [42] observed EXO 1846–031 on 3 August 2019 (ObsID 90501334002) with a netexposure time of 22.2 ks. By analyzing its disk reflectionfeatures in the intermediate state of this outburst, it wasfound the system consists of a fast-rotating black holeand an accretion disk with high inclination angle [43].We reduce the same
NuSTAR data as in [43] using the nupipeline and nuproducts routines in NuSTARDAS.The calibration database is CALDB 20200813. We ex-tract the source spectra from circular regions with radiiof 180 (cid:48)(cid:48) on the FPMA and FPMB detectors. The back-ground regions are of the same size but at the farthestdiagonal place from the source regions to avoid the influ-ence of the source photons. We group the spectra using grppha in order that each energy bin contains at least 30counts. Since the new CALDB corrects the calibrationin the 3-7 keV band, in the fitting we exclude the table nuMLIv1.mod used in [43]. In this paper, we use the version of relxill nk employing theJohannsen metric with non-vanishing deformation parameter α [39] as background metric. FIG. 1. Synthetic relativistic reflection spectra in the Johannsen spacetime for the spin parameter a ∗ = 0, ionization parameterat the ISCO log ξ = 3 .
1, deformation parameter α = −
2, 0, and 2, and viewing angle ι = 20 ◦ and 70 ◦ . The spectra withconstant ionization parameter are in blue ( α ξ = 0 and thus ξ = ξ ), those with ionization index α ξ = 1 are in orange, and thespectra with α ξ = 2 are in black.FIG. 2. As in Fig. 1 for the spin parameter a ∗ = 0 .
998 and the Johannsen deformation parameter α = − .
3, 0, and 0.3.
FIG. 3. As in Fig. 2 for the ionization parameter at the ISCO log ξ = 0 . ξ = 4. We perform the spectral fitting in XSPEC12.10.1s [44]. We first fit the spectra with an ab-sorbed cut-off power-law, with a floating constantmatching the slight discrepancy between FPMA andFPMB. The residuals below 4 keV indicate a diskthermal component, while the residuals around 6-7 keVand 10-30 keV indicate a reflection component. We thususe diskbb [45] and relxill nk [35, 36] to model theseresiduals, respectively. The final model set in XSPEC is const × tbabs × (diskbb+relxill nk) .For the reflection component, we use the flavor relxillion nk described in the previous section. Weassume that the inner radius of the disk is at the ISCOand the outer radius is fixed at 400 gravitational radii( r g ). Since EXO 1846–031 is a Galactic black hole, theredshift is set to 0. Other parameters are freely fit in diskbb and relxillion nk . We fit the data with fourmodels :Model 1: α ξ = 0 and α = 0.Model 2: α ξ = 0 and α free.Model 3: α ξ free and α = 0.Model 4: α ξ free and α free.Table I shows the best-fit values of the four fits. Best-fit models and ratio plots are in Fig. 5. Our results arediscussed in the next section. IV. DISCUSSION AND CONCLUSIONS
The analysis of relativistic reflection features in blackhole binaries and AGNs shows that steep and very steepradial emissivity profiles are common in the very innerpart of the accretion disk of black holes. While steepemissivity profiles can be naturally produced by compactcoronae very close to the black hole, there is some debatewhether current measurements of inner emissivity indicesare overestimated. In particular, from Eq. (1) we shouldexpect that even the radial profile of the ionization pa-rameter ξ is steep for any reasonable radial profile of thedisk electron density n e , while most analyses assume aconstant ionization parameter over all of the disk andseveral past attempts to fit the data with a non-trivialionization profile found that an ionization gradient is notrequired by the data.In the present work, we have presented an extensionof our reflection model relxill nk to include the pos-sibility of a non-trivial radial disk ionization profile. Inthe new flavor, called relxillion nk , the disk ionizationprofile is described by a power-law, so we have now the We remind the reader that the case α = 0 corresponds to theKerr solution, while deviations from the Kerr metric are presentfor a non-vanishing value of α . ionization parameter at the inner edge of the disk, ξ ,and the ionization index α ξ ( > NuSTAR observation of the black hole binaryEXO 1846–031. First, we have fit the data assuming aconstant ionization for all of the disk, α ξ = 0, and abroken power-law emissivity profile, either imposing theKerr metric ( α = 0) and relaxing such an assumption( α free). In both cases, we find a very steep inneremissivity index, with q in stuck to the maximum valueallowed in our fit, and an almost vanishing outer emis-sivity index q out . We note that such an emissivity patterncould be generated by a number of coronal geometries,see Refs. [23, 46, 47]. The second model with α freedoes not improve the fit (∆ χ = 0 .
19 with one less dof),so the data are consistent with the hypothesis that theblack hole in EXO 1846–031 is a Kerr black hole as pre-dicted by General Relativity.The
NuSTAR data are then fit with a free ionizationindex α ξ and still assuming a broken power-law emissiv-ity profile. As before, first we impose the Kerr metric( α = 0) and then we repeat the analysis with α free.The difference between the two fits is marginal (∆ χ =0 .
86) but the difference with the two fits with constantionization parameter is not negligible (∆ χ ≈ α ξ = 0 and α ξ free, Table I, we see that the ion-ization gradient has quite a modest impact on the mea-surements of the black hole spin parameter a ∗ and on thedeformation parameter α . There is some difference inthe estimate of q in , which is not stuck to the boundaryany longer. The estimate of the hydrogen column den-sity in tbabs is a bit lower when α ξ is free. The best-fitvalues of the iron abundance are lower with α ξ free, butactually the uncertainties on the estimates are larger, sowe cannot really say that the model with a radial diskionization profile can find a lower iron abundance. Theestimate of the high energy cut-off E cut is higher with α ξ free. In general, the uncertainties in the best-fit val-ues are larger in the fits with α ξ free than in those with α ξ = 0.We note that in Ref. [31] the authors analyzed the2012 NuSTAR observation of the black hole binaryGRS 1915+105 and found that the ionization profile hasan impact on the estimate of the black hole spin, in agree-ment with the claim in Ref. [30], where the authors onlyworked on simulations and limited their study to the 1-10 keV band. The conclusion of those authors is in appar-ent disagreement with our results, where we do not findany bias on the black hole spin measurement (and thedeformation parameter). However, the shape of relativis-tic reflection spectra is determined by many parametersand it is possible that in certain regions of the parameterspace the ionization gradient can be ignored and in otherregion it cannot, as well as that in some regions the cor-rect ionization gradient has an impact on the estimate of
Model 1 Model 2 Model 3 Model 4 tbabs N H /10 cm − . +0 . − . . +0 . − . . +0 . − . . +0 . − . diskbb kT in [keV] 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . norm [10 ] 2 . +0 . − . . +0 . − . . +1 . − . . +1 . − . relxillion nk q in . − . − . +(P) − . . +0 . − . q out . +0 . . +0 . . +0 . . +0 . R br [ r g ] 6 . +0 . − . . +0 . − . . +4 . − . +16 − a ∗ . +(P) − . . +(P) − . . +0 . − . . − . α ∗ . +0 . − . ∗ − . +0 . − . ι [deg] 74 . +1 . − . . +0 . − . +5 − . +6 . − . Γ 1 . +0 . − . . +0 . − . . +0 . − . . +0 . − . log ξ . +0 . − . . +0 . − . . +0 . − . . +(P) − . α ξ ∗ ∗ . +1 . − . . +0 . − . A Fe . +0 . − . . +0 . − . . +0 . − . . +2 . − . E cut [keV] 80 +6 − +6 − +17 − +15 − R f . +0 . − . . +0 . − . . +0 . − . . +1 . − . norm [10 − ] 2 . +0 . − . . +0 . − . . +0 . − . . +0 . − . χ / dof 2727 . / . / . / . . . . χ = 2 . ∗ means the value is frozen in the fit. When there is no lower/upper uncertainty, the parameter isstuck at the lower/upper boundary of the range in which it is allowed to vary. (P) indicates that the 90% confidence levelregion reaches the boundary. The ionization parameter ξ is in units of erg cm s − . some model parameters and in other region it does not.Note also that both those papers used a lamppost model,while here we use a broken power-law for the emissivityprofile. Acknowledgments –
This work was supported bythe Innovation Program of the Shanghai MunicipalEducation Commission, Grant No. 2019-01-07-00-07-E00035, the National Natural Science Foundation of China (NSFC), Grant No. 11973019, and Fudan Uni-versity, Grant No. JIH1512604. D.A. is supportedthrough the Teach@T¨ubingen Fellowship. Y.Z. acknowl-edges the support from China Scholarship Council (CSC201906100030). C.B. and H.L. are members of the Inter-national Team 458 at the International Space Science In-stitute (ISSI), Bern, Switzerland, and acknowledge sup-port from ISSI during the meetings in Bern. [1] A. C. Fabian, M. J. Rees, L. Stella and N. E. White,Mon. Not. Roy. Astron. Soc. , 729-736 (1989).[2] Y. Tanaka et al. , Nature , 659 (1995).[3] K. Nandra, I. M. George, R. F. Mushotzky, T. J. Turnerand T. Yaqoob, Astrophys. J. , 602 (1997)[arXiv:astro-ph/9606169 [astro-ph]].[4] D. J. Walton, E. Nardini, A. C. Fabian, L. C. Gallo andR. C. Reis, Mon. Not. Roy. Astron. Soc. , 2901 (2013)[arXiv:1210.4593 [astro-ph.HE]].[5] A. Tripathi, Y. Zhang, A. B. Abdikamalov, D. Ayzen-berg, C. Bambi, J. Jiang, H. Liu and M. Zhou,[arXiv:2012.10669 [astro-ph.HE]].[6] A. C. Fabian, K. Nandra, C. S. Reynolds, W. N. Brandt, C. Otani, Y. Tanaka, H. Inoue and K. Iwasawa, Mon.Not. Roy. Astron. Soc. , L11-L15 (1995) [arXiv:astro-ph/9507061 [astro-ph]].[7] G. Risaliti, F. A. Harrison, K. K. Madsen, D. J. Wal-ton, S. E. Boggs, F. E. Christensen, W. W. Craig,B. W. Grefenstette, C. J. Hailey and E. Nardini, et al.
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