Monte Carlo simulations of the Nickel K α fluorescent emission line in a toroidal geometry
aa r X i v : . [ a s t r o - ph . H E ] N ov Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 23 October 2018 (MN L A TEX style file v2.2)
Monte Carlo simulations of the Nickel K α fluorescentemission line in a toroidal geometry Tahir Yaqoob and Kendrah D. Murphy Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218. Department of Physics, Skidmore College, 815 North Broadway, Saratoga Springs, NY 12866.
Accepted 2010 November 9. Received 2010 November 9; in original form 2010 October 5
ABSTRACT
We present new results from Monte Carlo calculations of the flux and equivalent width(EW) of the Ni K α fluorescent emission line in the toroidal X-ray reprocessor model ofMurphy & Yaqoob (2009, MNRAS, 397, 1549). In the Compton-thin regime, the EWof the Ni K α line is a factor of ∼
22 less than that of the Fe K α line but this factorcan be as low as ∼ α line in the optically-thin limit. Whenthe reprocessor is Compton-thick and the incident continuum is a power-law with aphoton index of 1.9, the Ni K α EW has a maximum value of ∼ ∼
250 eV fornon-intercepting and intercepting lines-of-sight respectively. Larger EWs are obtainedfor flatter continua. We have also studied the Compton shoulder of the Ni K α lineand find that the ratio of scattered to unscattered flux in the line has a maximumvalue of 0.26, less than the corresponding maximum for the Fe K α line. However, wefind that the shape of the Compton shoulder profile for a given column density andinclination angle of the torus is similar to the corresponding profile for the Fe K α line. Our results will be useful for interpreting X-ray spectra of active galactic nuclei(AGNs) and X-ray binary systems in which the system parameters are favorable forthe Ni K α line to be detected. Keywords : galaxies: active - line:formation - radiation mechanism: general - scattering - X-rays: general
The Ni K α fluorescent emission line has the potential to offer complementary diagnostics to the Fe K α fluorescent line inactive galactic nuclei (AGNs) and some X-ray binary systems in which the Fe K α line is detected (e.g., see Torrej´on et al.2010, and references therein). In AGNs, the narrow Fe K α emission line is a ubiquitous feature of the X-ray spectrum of bothtype 1 and type 2 sources (e.g. see Shu, Yaqoob, & Wang 2010, and references therein). However, since the abundance of Niis more than an order of magnitude less than that of Fe, the Ni K α line is expected to be weak. Nevertheless, it has beendetected in a few AGNs. Three of the best examples are the Circinus galaxy (Molendi, Bianchi, & Matt 2003), NGC 6552(Reynolds et al. 1994), and NGC 1068 (Matt et al. 2004; Pounds & Vaughan 2006). In all of these sources the equivalent widthof the Fe K α line is large (hundreds of eV, to over 1 keV) because the X-ray spectrum is dominated by, or has a relativelylarge contribution from, reflection in Compton-thick matter. Naturally, such sources are the most likely to yield detections ofthe Ni K α line because its EW will also be correspondingly larger, in tandem with that of the Fe K α line EW. The Ni K α linehas also been detected in a source that is not reflection-dominated, but still moderately absorbed (Centaurus A), albeit witha lower statistical significance of detection (Markowitz et al. 2007). Improvements in the sensitivity of X-ray detectors in the ∼ −
10 keV band aboard forthcoming X-ray astronomy missions such as
NuStar and
Astro-H will likely reveal detections ofthe Ni K α emission line in a larger number of accreting X-ray sources and will therefore open up the opportunity to use theline as a diagnostic tool in conjunction with the Fe K α line. c (cid:13) Tahir Yaqoob & Kendrah D. Murphy
The results of model calculations of the flux and EW of a Ni K α fluorescent line that originates in neutral matter, asexpected in AGNs, have been reported in the literature for disk and spherical geometries (e.g., Reynolds et al. 1994; Matt,Fabian, & Reynolds 1997). In the present paper we study the theoretical properties of the Ni K α line produced by the toroidalX-ray reprocessor model of Murphy & Yaqoob (2009; hereafter, MY09). The paper is organized as follows. In § α line fluxand EW in § § § α emission line. We summarize our conclusions in § Here we give a brief overview of our model Monte Carlo simulations and the key assumptions that they are based upon (furtherdetails can be found in MY09). We assume that the reprocessing material is uniform and neutral (cold). X-ray spectroscopy ofAGNs shows overwhelming evidence for the narrow Fe K α line peaking at ∼ . that line is essentially neutral (e.g. Sulentic et al. 1998; Weaver, Gelbord, & Yaqoob 2001; Page et al. 2004;Yaqoob & Padmanabhan 2004; Jim´enez-Bail´on et al. 2005; Zhou & Wang 2005; Jiang, Wang, & Wang 2006; Levenson et al.2006; Shu et al. 2010). Although emission lines from ionized species of Fe are observed in some AGN (e.g. Yaqoob et al. 2003;Bianchi et al. 2005, 2008), the present paper is concerned specifically with modeling the Ni K α fluorescent emission line thatoriginates in the same material as the Fe K α line component that is centered around 6.4 keV. We note that this Fe K α lineat ∼ . θ , and the equatorial column density, N H (see Fig. 1 in MY09). If a is the radiusof the circular cross-section of the torus, and c + a is the equatorial (i.e. maximum) radius of the torus then ( a/c ) is a coveringfactor such that ( a/c ) = [∆Ω / (4 π )]. Here, ∆Ω is the solid angle subtended by the torus at the X-ray source, which is assumedto be located at the center of the system, emitting isotropically. The mean column density, integrated over all incident anglesof rays through the torus, is then ¯ N H = ( π/ N H . The inclination angle between the observer’s line of sight and the symmetryaxis of the torus is denoted by θ obs , where θ obs = 0 ◦ corresponds to a face-on observing angle and θ obs = 90 ◦ corresponds toan edge-on observing angle. In our calculations we distribute the emergent photons in 10 angle bins between 0 ◦ and 90 ◦ thathave equal widths in cos θ obs , and refer to the face-on bin as θ for which we have calculated a comprehensive set of models is 60 ◦ , for N H in the range 10 cm − to10 cm − , valid for input spectra with energies in the range 0.5–500 keV (see MY09 for details). Our model employs afull relativistic treatment of Compton scattering, using the full differential and total Klein-Nishina Compton-scattering cross-sections. For θ = 60 ◦ , the solid angle subtended by the torus at the X-ray source, ∆Ω, is 2 π , so that [∆Ω / (4 π )] = ( a/c ) = 0 . τ T = KN H σ T ∼ . N where N is the column density in units of10 cm − . Here, we have employed the mean number of electrons per H atom, (1+ µ ), where µ is the mean molecular weight.With the abundances of Anders & Grevesse, K = 1 . A Ni , is 1 . × − relativeto H. A more recent determination by Scott et al. (2009) yields a value of 1 . × − but the statistical and systematicuncertainties do not exclude the Anders & Grevesse (1989) value. We use the latter for consistency with our previous resultson the Fe K α emission line (MY09; Yaqoob et al. 2010; Yaqoob & Murphy 2010).The Ni K α fluorescent emission line consists of two components, Kα and Kα , at energies 7 . . Kα : Kα = 2 : 1 (Bearden 1967). These line energies are appropriate for neutralmatter. In the Monte Carlo simulations we used a single line for Ni K α , at a rest-frame monoenergetic energy, E , of 7.472 keV(obtained from weighting the Kα and Kα values with the branching ratio). We used a fluorescence yield, ω K , for Ni of0.414 (see Bambynek et al. 1972) and a Ni K β to Ni K α ratio of 0.135 (consistent with results in Bambynek et al. 1972).Compared to MY09, the results in the present paper have a substantially higher statistical accuracy because they arebased on Monte Carlo simulations with higher numbers of injected rays at each energy, and the calculations employ themethod of weights (as opposed to following individual photons). Throughout the present paper we present results for power-law incident continua (in the range 0.5–500 keV), characterized by a photon index, Γ, by integrating the basic monoenergeticMonte Carlo results (Greens functions– see MY09). α LINE FLUX
In this section we discuss the flux of Ni K α emission-line photons that escape the torus without any interaction with it(the zeroth-order, or unscattered line photons). In § c (cid:13) , 000–000 onte Carlo simulations of the Nickel K α fluorescent emission line in a toroidal geometry − − − N i K α li n e f l ux ( pho t on s c m − s − ) N H (10 cm −2 ) edge−onface−on bin Figure 1.
The Ni K α line flux versus N H for Γ = 1 .
9. Curves are shown for each of the 10 θ obs bins and are color coded and labeledby the angle bin number (see Table 1 in MY09). The angle bins correspond to equal solid angle intervals in the range 0 ◦ to 90 ◦ . Anglebins 1–5 correspond to lines-of-sight that do not intercept the torus and angle bins 6–10 correspond to lines-of-sight that intercept thetorus. The normalization of the line flux corresponds to a power-law incident continuum that has a monochromatic flux at 1 keV of1 photons cm − s − keV − . The dashed line corresponds to the optically-thin limit for the relation between the Ni K α line flux and N H (equation 1). Compton shoulder). In practice it may not actually be possible to observationally distinguish the zeroth-order component ofan emission line from its Compton shoulder. The finite energy resolution of the instrument and/or the velocity broadening(of all the emission-line components) may confuse the two blended components of a line (see discussion in Yaqoob & Murphy2010 for the Fe K α line). The Monte Carlo results for the flux of the zeroth-order component of the Ni K α emission line areshown in Fig. 1 as a function of the equatorial column density, N H , for each of the 10 angle bins in θ obs . The line flux, I Ni K α ,has been normalized to an incident continuum that has a monochromatic flux of 1 photon cm − s − keV − at 1 keV.As N H is increased, the Ni K α line flux first increases but then turns over, reaching a maximum for N H somewhere inthe range ∼ − × cm − , depending on the inclination angle. This is because the escape of Ni K α line photons fromthe medium after they are created is significantly impeded by the absorption and scattering opacity that is relevant at theline energy. For the edge-on angle bin the maximum Ni K α line flux is attained well before the medium becomes Compton-thick , at only ∼ × cm − . For the face-on angle bin the maximum line flux is attained at a higher column density( ∼ × cm − ), but still before the medium becomes Compton-thick. The position of the turnover can be understood asapproximately corresponding to a situation when the average optical depth to absorption plus scattering for the zeroth-orderNi K α line photons is of order unity. The behavior of the Ni K α zeroth-order line flux as a function of N H and θ obs is in factvery similar to that of the flux of the Fe K α line, for which a detailed discussion can be found in Yaqoob et al. (2010). α line flux in the optically-thin limit In the optically-thin limit, for which absorption and scattering optical depths in the ∼ − ≪
1, we can obtainan approximate analytic expression for the Ni K α line flux. Following Yaqoob et al. (2001), we get I Ni K α ∼ . × − (cid:16) ∆Ω4 π (cid:17) (cid:16) ω K . (cid:17) (cid:16) ω Kα ω K (cid:17) × (cid:16) A Ni . × − (cid:17) (cid:16) σ . × − cm (cid:17) × (cid:16) .
61Γ + α − (cid:17) (8 .
348 keV) (1 . − Γ) ¯ N photons cm − s − (1)The quantity [∆Ω / (4 π )] is the fractional solid angle that the line-emitting matter subtends at the X-ray source. Note thatequation 1 utilizes the mean (angle-averaged) column density , not the equatorial column density. Thus, ¯ N = ( π/ N H / (10 cm − )].The K-shell fluorescence yield is given by ω K , and ω Kα is the yield for the Ni K α line only. Using our adopted value of 0.135for the Ni K β /Ni K α line ratio, ω Kα /ω K = 0 . A Ni is the Ni abundance relative to Hydrogen (1 . × − ,Anders & Grevesse 1989). The quantity σ is the Ni K shell absorption cross-section at the Ni K photoelectric absorptionedge threshold energy, E K , and α is the power-law index of the cross-section as a function of energy. For the Verner et al. c (cid:13) , 000–000 Tahir Yaqoob & Kendrah D. Murphy − − . . N i K α li n e e qu i v a l e n t w i d t h ( k e V ) N H (10 cm −2 )
109 876 543 2 1 edge−onface−on bin
Figure 2.
The Ni K α line equivalent width (EW) versus N H for Γ = 1 .
9. Curves are shown for each of the 10 θ obs bins and are colorcoded with the same scheme as in Fig. 1, and labeled by the angle bin number (see Table 1 in MY09). The angle bins correspond toequal solid angle intervals in the range 0 ◦ to 90 ◦ . Angle bins 1–5 correspond to lines-of-sight that do not intercept the torus and anglebins 6–10 correspond to lines-of-sight that intercept the torus. The dashed line corresponds to the optically-thin limit for the relationbetween the Ni K α line EW and N H (equation 3). (1996) data that we have adopted, E K = 8 .
348 keV, σ = 2 . × − cm , and α = 2 .
71 (obtained from fitting the K-shellcross-section up to 30 keV with a power-law model).The optically-thin limit for the Ni K α line flux from equation 1 is shown in Fig. 1 (dashed line). It can be seen thatthe Monte Carlo curves converge to this optically-thin limit, but only for column densities < × cm − . Note that theoptically-thin limit for the Ni K α line flux is independent of the details of the geometry .We can obtain a simple result in the optically-thin limit from equation 1 for the ratio of the Fe K α to Ni K α line flux.Neglecting the small difference in the energy dependence of the K-shell cross-section in Ni and Fe ( ∼ E − . and ∼ E − . respectively), we have I Fe K α I Ni K α ∼ . (cid:16) .
124 keV8 .
348 keV (cid:17) (1 . − Γ) [ A Fe /A Ni ][ A Fe /A Ni ] AG89 . (2)In equation 2, 7.124 keV is the neutral Fe K shell threshold edge energy in Verner et al. (1996), [ A Fe /A Ni ] AG89 is theAnders & Grevesse (1989) Fe to Ni abundance ratio (26.3), and [ A Fe /A Ni ] is the actual Fe to Ni abundance ratio in the source.What is interesting about equation 2 is that not only is it independent of geometry, it is independent of the covering factor . α LINE EQUIVALENT WIDTH
In Fig. 2, we show the EWs of the unscattered (zeroth-order) component of the Ni K α line as a function of the column densityof the torus, N H , calculated for Γ = 1 .
9. The lower set of curves show the results for the non-intercepting angle bins, and theupper set of curves show the results for the intercepting angle bins, as indicated by the color-coded angle bin numbers. It canbe seen in Fig. 2 that inclination-angle effects become important for N H greater than ∼ × cm − . For inclination anglesthat do not intercept the torus, the EW peaks between ∼ × cm − and 10 cm − , then decreases by more than 50% ofthis peak value at N H = 10 cm − . For these non-intercepting lines-of-sight, the peak value for the EW of the Ni K α line,for Γ = 1 .
9, is ∼ α line EW reaches its maximum value between N H ∼ − × cm − , becoming as high as ∼
250 eV for the edge-on angle bin. Overall, the behavior of the EW as afunction of N H and θ obs is analogous to that of the Fe K α line, for which a detailed discussion can be found in MY09. α line EW in the optically-thin limit Overlaid on the curves in Fig. 2 is the theoretical optically-thin limit (dashed line), given by dividing equation 1 by E − Γ (recallthat the line flux in equation 1 is normalized to a power-law continuum normalization at 1 keV of 1 photon cm − s − keV − ).Thus, the EW of the Ni K α line in the optically-thin limit is c (cid:13) , 000–000 onte Carlo simulations of the Nickel K α fluorescent emission line in a toroidal geometry E W ( Γ = . ) N i K α E W ( Γ = . ) / non−intercepting angle bins 12 3 45 E W ( Γ = . ) N i K α E W ( Γ = . ) / E W ( Γ = . ) N i K α E W ( Γ = . ) / E W ( Γ = . ) N i K α E W ( Γ = . ) / E W ( Γ = . ) N i K α E W ( Γ = . ) / E W ( Γ = . ) N i K α E W ( Γ = . ) / N H (10 cm −2 ) intercepting angle bins 1098 76 E W ( Γ = . ) N i K α E W ( Γ = . ) / N H (10 cm −2 )0.1 1 101.522.5 E W ( Γ = . ) N i K α E W ( Γ = . ) / N H (10 cm −2 )0.1 1 101.522.5 E W ( Γ = . ) N i K α E W ( Γ = . ) / N H (10 cm −2 )0.1 1 101.522.5 E W ( Γ = . ) N i K α E W ( Γ = . ) / N H (10 cm −2 ) Figure 3.
Ratios of the Ni K α line equivalent width (EW) for Γ = 1 . .
5, versus N H . Curves areshown for each of the 10 θ obs bins and are color coded with the same scheme as in Fig. 1 and Fig. 2, and labeled by the angle binnumber (see Table 1 in MY09). The angle bins correspond to equal solid angle intervals in the range 0 ◦ to 90 ◦ . The upper panel showsangle bins 1–5, corresponding to lines-of-sight that do not intercept the torus, and the lower panel shows angle bins 6–10 correspondingto lines-of-sight that intercept the torus. Note that the vertical axis scale is different for the two panels. The dashed line shows theoptically-thin limiting value of the EW ratio, obtained from equation 3. EW Ni K α ∼ . (cid:16) ∆Ω4 π (cid:17) (cid:16) ω K . (cid:17) (cid:16) ω Kα ω K (cid:17) × (cid:16) A Fe . × − (cid:17) (cid:16) σ . × − cm (cid:17) × (cid:16) .
61Γ + α − (cid:17) (cid:16) E E K (cid:17) (Γ − . ¯ N eV . (3)As in equation 1, the column density in equation 3 is the mean, angle-averaged column density ( ¯ N = ( π/ N H / (10 cm − )]).In equation 3 E is the Ni K α line centroid energy. The ratio ( E /E K ) is (7 . / . . α line, (6 . / . . α line and the EW of the Ni K α line in the optically-thin limit, that is independent of the shape of the intrinsic continuum . In analogy to equation 2, we get EW Fe K α EW Ni K α ∼ . A Ni /A Fe ][ A Ni /A Fe ] AG89 . (4)As was the case for the line flux ratio, the EW ratio in the optically-thin limit is independent of geometry and thecovering factor. Note that in the Compton-thick regime, for lines-of-sight that intercept the torus, the Ni K α line EW is largerrelative to the Fe K α line EW than simple linear scaling of the optically-thin case. In other words, the EW ratio in equation 4becomes smaller as N H increases, for non-intercepting inclination angles. We find that the ratio has its smallest value ( ∼ N H ∼ − × cm − and an edge-on inclination angle.We note another important aspect of the Ni K α line EW versus N H curves in Fig. 2. That is, fortuitously, the relation forthe optically-thin limit (dashed line) happens to give excellent agreement for the edge-on inclination angle bin all the way upto N H = 6 × cm − . This gives a very convenient way to analytically estimate the EW of the Ni K α line for an edge-onorientation and a column density less than 6 × cm − .We find that smaller values of Γ yield larger values of the Ni K α EW; this is expected as there are relatively more photonsin the continuum above the Ni K edge for flatter spectra. Fig. 3 shows the ratio of the Ni K α line EW for Γ = 1 . .
5, versus N H , for each of the 10 inclination-angle bins (see Table 1 in MY09). In the optically-thinregime, this ratio can simply be obtained by evaluating equation 3 for each value of Γ and taking the ratio. We get a value of1.465, and this is shown in Fig. 3 (dashed line), from which it can be seen that there is excellent agreement with the MonteCarlo results. For the non-intercepting angle bins this ratio does not increase above ∼ .
72 even in the Compton-thick regime.However, for the edge-on angle bin, the ratio has its maximum value (with respect to all the angle bins and N H values) of ∼ .
5, for N H ∼ × cm − .The EW versus N H curves have an explicit dependence on the assumed opening angle of the torus. This is becausedifferent opening angles correspond to different solid angles subtended by the torus at the source and to different projection- c (cid:13) , 000–000 Tahir Yaqoob & Kendrah D. Murphy . . N i K α ( s ca tt e r e d f l ux / ze r o t h − o r d e r f l ux ) N H (10 cm −2 ) Figure 4.
The ratios of the total number of scattered Ni K α line photons to the number of zeroth-order Ni K α line photons, versus N H .Ratios are shown for an input power-law continuum with Γ = 1 .
9, for the face-on inclination angle bin (lower curve) and the edge-oninclination angle bin (upper curve). angle effects. In the optically-thin regime, this dependence is linear. In the Compton-thick regime, there is a more complicateddependence that must be determined by additional Monte Carlo simulations, which will be the subject of future investigation. α LINE COMPTON SHOULDER
In addition to the zeroth-order (unscattered) core of the Ni K α emission line, the shape and relative magnitude of the scatteredcomponent of the Ni K α emission line (i.e. the Compton shoulder) are also sensitive to the properties of the reprocessor (e.g.,see Sunyaev & Churazov 1996; Matt 2002; Watanabe et al. 2003; Yaqoob & Murphy 2010). Fig. 4 shows plots of the ratioof the total number of scattered Ni K α line photons to zeroth-order Ni K α line photons (hereafter, CS ratio) versus N H .The ratios are shown for an input power-law continuum with Γ = 1 .
9, for the face-on and edge-on inclination angle bins(see Table 1 in MY09). It can seen that the CS ratio peaks at N H ∼ − × cm − , reaching a maximum of ∼ . ∼ .
26 (edge-on). These are ∼
75% and ∼
70% of the corresponding ratios for the Fe K α line (see MY09and Yaqoob & Murphy 2010). For the face-on inclination angle, the CS ratio for the Ni K α line remains at ∼ .
22 once themaximum is reached (even if the column density is increased further) since the Compton shoulder photons escape from withina Compton-depth or so from the illuminated surfaces of the torus for lines-of-sight that are not obscured. For the edge-oninclination angle the CS ratio for the Ni K α line declines as a function of column density after reaching its maximum value,due to a higher probability of absorption at higher column densities.We found that for all values of θ obs for the torus, there was no detectable difference in the CS ratio as a function of Γ up tothe N H value that gives the maximum CS ratio (for a given value of θ obs ). After that, the CS ratios diverge for different valuesof Γ, with flatter incident continua giving larger CS ratios. For the face-on inclination angle the CS ratio at N H = 10 cm − varies between ∼ .
21 to ∼ .
22 as Γ varies from 2.5 to 1.5. For the edge-on case, the CS ratio at N H = 10 cm − variesbetween ∼ .
22 to ∼ .
23 as Γ varies from 2.5 to 1.5. Flatter spectra have relatively more continuum photons at higher energiesso that the Ni K α line photons are produced deeper in the medium, increasing the average Compton depth for zeroth-orderline photons to scatter before escaping. These variations in the CS ratio with Γ are likely to be too small to be detectable inpractice.The shape of the Compton shoulder of a fluorescent emission-line escaping from the torus also has a dependence on thecolumn density and inclination angle of the torus. We found that the shapes of the Compton shoulder profiles for the Ni K α line are practically indistinguishable from the shapes of the Fe K α line Compton shoulder profiles (see MY09 and Yaqoob& Murphy 2010). Fig. 5 illustrates the shapes of the Ni K α line Compton shoulder (solid lines) for a power-law incidentcontinuum with Γ = 1 .
9, for two column densities (10 and 10 cm − ) and two inclination angles of the torus (face-onand edge-on). Corresponding Compton shoulder profiles are also shown for the Fe K α line (dotted lines) for comparison.The Compton shoulder shapes shown in Fig. 5 have no velocity broadening applied to them. The Compton shoulders areshown in wavelength space in units of the dimensionless Compton wavelength shift with respect to the zeroth-order rest-frameenergy of the emission line. In other words, if E is the energy of a line photon, and E is the zeroth-order line energy,∆ λ = (511 keV /E ) − (511 keV /E ). In order to facilitate a direct comparison of the Compton shoulder profile shapes fordifferent column densities and inclination angles, all of the profiles in Fig. 5 have been normalized to a total flux of unity. It c (cid:13) , 000–000 onte Carlo simulations of the Nickel K α fluorescent emission line in a toroidal geometry . F l ux ( a r b it r a r y un it s ) ∆λ N H = 10 cm −2 face−on . F l ux ( a r b it r a r y un it s ) ∆λ N H = 10 cm −2 edge−on . F l ux ( a r b it r a r y un it s ) ∆λ N H = 10 cm −2 face−on . F l ux ( a r b it r a r y un it s ) ∆λ N H = 10 cm −2 edge−on Figure 5.
The Ni K α emission-line Compton shoulders (solid curves) for a power-law incident continuum with Γ = 1 .
9, for two columndensities and two inclination angle bins, as indicated ( N H = 10 and 10 cm − , each for face-on and edge-on orientations of the torus).The dotted curves show the corresponding Compton shoulder profiles for the Fe K α emission line, for the same column densities andinclination angles. No velocity broadening has been applied. Note that in order to directly compare the Compton shoulder shapes, thetotal flux for each shoulder has been renormalized to the same value . The line flux (in units of normalized flux per unit wavelength shift)is plotted against the dimensionless Compton wavelength shift with respect to the zeroth-order rest-frame energy of the emission line( E ), ∆ λ = (511 keV /E ) − (511 keV /E ). should be remembered that the absolute flux of the Compton shoulder varies significantly with column density, and the fluxratio for two column densities can be estimated using Fig. 4.Yaqoob & Murphy (2010) discussed the dependence of the shape of the Fe K α line Compton shoulder on N H andinclination angle in considerable detail. Differences in the shapes of the Compton shoulder profiles for the Ni K α line andthe Fe K α line for the same model parameters only become apparent for N H ≫ cm − and edge-on inclination angles.However, even at N H = 10 cm − , the differences are less than 15%. Therefore, since the shapes of the Compton shoulderprofiles for the Ni K α line are similar to the Fe K α line Compton shoulder profiles within the statistical uncertainties of theMonte Carlo simulations, we do not discuss the Ni K α line Compton shoulder further. The discussion and interpretation ofthe Fe K α line Compton shoulder in Yaqoob & Murphy (2010) can be applied to the Ni K α line. We have presented some new results for the flux and EW of the Ni K α fluorescent emission line from Monte Carlo simulationsof a toroidal reprocessor illuminated by a power-law X-ray continuum. Our results cover values of the equatorial columndensity, N H , of 10 cm − to 10 cm − , and the calculations were performed for a global covering factor of 0.5 and cosmicelemental abundances. As might be expected, the behavior of the Ni K α line flux and EW as a function of the column densityand inclination angle of the torus is similar to that of the Fe K α line. However, the EW of the Ni K α line is a factor of ∼ α in the Compton-thin regime. In the Compton-thick regime, the EW of the Ni K α line reachesa maximum of ∼ α EW can beas high as ∼
250 eV. The ratio of the Fe K α to Ni K α line EW in the Compton-thick regime, for intercepting lines-of-sight,can be significantly less than the optically-thin limit, as low as ∼
6. The above results pertain to an incident power-law X-raycontinuum with a photon index of 1.9. Flatter continua give larger EWs and steeper continua give smaller EWs. Varying Γin the range Γ = 1 . α EW by up to ∼
70% in the Compton-thick regime.We have given analytic expressions for the Ni K α flux and EW in the optically-thin limit. We have also given simpleanalytic expressions, in the optically-thin limit, for the ratio of the Fe K α to Ni K α line flux, as well as the ratio of the Fe K α to Ni K α line EW. Both of these ratios are independent of the geometry and covering factor of the reprocessor. Moreover, wehave found that the ratio of the Fe K α to Ni K α line EW is independent of Γ, depending only on the Fe to Ni abundanceratio (in the optically-thin limit).We have also investigated the Compton shoulder of the Ni K α line and we found that the ratio of the flux in the Comptonshoulder to that in the zeroth-order component of the line has a maximum value of ∼ .
22 and ∼ .
26 for face-on and edge-oninclination angles respectively. These are less than the corresponding maxima for the Fe K α line. However, we have foundthat the shapes of the Ni K α and Fe K α line Compton shoulder profiles are indistinguishable within the statistical accuracyof the Monte Carlo results, except for edge-on inclination angles and N H ≫ cm − . However, even for N H as high as10 cm − , the differences are less than 15%. c (cid:13) , 000–000 Tahir Yaqoob & Kendrah D. Murphy
Our calculations of the Ni K α line flux, EW, and Compton shoulder are meant to serve as a baseline reference because thedetailed results, especially in the Compton-thick regime, depend on a number of factors that have not been investigated here.The opening angle of the torus (or effective covering factor) may of course be different to the value used in the calculations.Also, as suggested by Lubinski et al. (2010), part of the torus may be shielded from the X-ray continuum by the accretiondisk. However, quantifying this is subject to uncertainties in the geometry of the X-ray source and accretion disk system.Further deviations from the baseline model could occur if the torus does not have a circular cross-section and/or if the torusis clumpy (e.g., Krolik & Begelman 1988; Nenkova, Ivezi´c & Elitzur 2002). Extension of the parameter space for our modelwill be the subject of future work.AcknowledgmentsPartial support (TY) for this work was provided by NASA through Chandra
Award TM0-11009X, issued by the Chandra X-rayObservatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the NASA undercontract NAS8-39073. Partial support (TY) from NASA grants NNX09AD01G and NNX10AE83G is also acknowledged. Theauthors thank Andrzej Zdziarski for helpful comments for improving the paper.
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