Two Major Constraints on the Inner Radii of Accretion Disks

A major constraint on the source geometry in the hard state follows from considering the flux irradiating the cold medium in the system, ${F}_{\mathrm{irr}}$ (energy-integrated and per unit area). That flux is partly Compton-reflected, and partly absorbed and then re-emitted. If that re-emission would perfectly thermalize, it would appear as a blackbody at the effective temperature corresponding to the absorbed flux. However, the reprocessing does not lead to the full thermodynamic equilibrium, and the re-emitted spectrum contains lines and edges on top of a quasi-thermal continuum. Still, we have a strict lower limit on the shape of the re-emitted spectrum: its average energy has to be above the average energy of the blackbody at the effective temperature, ${T}_{\mathrm{eff}}$, given by the Stefan–Boltzmann law,

$$\begin{eqnarray}&&\sigma {T}_{\mathrm{eff}}^{4}=(1-a){F}_{\mathrm{irr}}+{F}_{\mathrm{intr}},\end{eqnarray} \tag{ 1 }$$

where σ is the Stefan–Boltzmann constant, a is the albedo for backscattering, and ${F}_{\mathrm{intr}}$ is the intrinsic flux generated by an internal dissipation.

Figure 1 compares some published re-emitted spectra from irradiation with blackbody spectra at their corresponding ${T}_{\mathrm{eff}}$. They demonstrate that a blackbody can indeed roughly approximate the observed curvature of the reprocessed continuum, especially in cases with high irradiating fluxes. We show incident and re-emitted spectra for irradiation of strongly ionized slabs ($\xi \sim {10}^{3}$ erg cm s⁻¹) by an isotropic source above it, i.e., the reflection fraction (defined as the ratio of the irradiating flux to that emitted outside in a local frame) is ${ \mathcal R }=1$. Since ${F}_{\mathrm{intr}}=0$ was assumed, the energy conservation requires then that both spectra contain the same energy-integrated flux. We use the re-emitted spectra from Figure 7 of García et al. (2016; hereafter G16)¹ calculated with their reprocessing code xillverD and from Figure 6 of Jiang et al. (2019b; hereafter J19a), their spectrum LF7 calculated with the reprocessing code reflionx (Ross et al. 1999; Ross & Fabian 2007) renormalized to ${ \mathcal R }=1$. We compare the re-emitted spectra for the bolometric irradiating fluxes of ${F}_{\mathrm{irr}}\approx 4\times {10}^{16}$, $4\times {10}^{20}$ and $4\times {10}^{23}$ erg cm⁻² s⁻¹ with those of a re-emitted blackbody component at ${T}_{\mathrm{eff}}$. The values of ${F}_{\mathrm{irr}}$ follow from the given values of the ionization parameter, ξ, defined as

$$\begin{eqnarray}&&\xi \equiv 4\pi {F}_{\mathrm{irr}}^{{\prime} }/n,\end{eqnarray} \tag{ 2 }$$

where ${F}_{\mathrm{irr}}^{{\prime} }$ is the irradiating flux measured at the source in either the 0.01–100 keV or 0.1–1000 keV photon energy ranges, and n is either the H or electron density in the reflection codes reflionx, and xillver, relxill, and associated codes (García et al. 2013; G16; Dauser et al. 2016), respectively. We note that the re-emitted spectra shown in Figure 1, as well as those in Ross & Fabian (2007) and Vincent et al. (2016), are for ${{kT}}_{\mathrm{eff}}\lesssim 1\,\mathrm{keV}$. There remains an uncertainty about the form of the re-emitted spectra for hotter reprocessing media; they may become somewhat broader due to stronger effect of Compton scattering in upper layers of the reflectors. However, as implied by our analysis below, values of ${{kT}}_{\mathrm{eff}}$ higher than ∼2 keV are not expected in the hard state of BH binaries.

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We have approximately determined the values of the albedo based on the spectra (blue solid curves) as the fraction of the re-emitted flux in the range where the re-emitted spectrum is dominated by backscattering (rather than reprocessing). This happens above E ≈ 0.2, 1.0 and 10 keV in Figures 1(a)–(c), yielding a ≈ 0.73, 0.66 and 0.34, respectively. While this is a crude method, the resulting blackbodies provide fair approximations to the overall shape of the quasi-thermal low-E feature, especially in the high-flux cases b and c. On the other hand, a part of the emission at higher energies is from reprocessing as well, in particular O viii K and Fe K complexes in Figures 1(a) and (b), and thus our albedo estimates are conservative. Furthermore, the treatment of G16 (based on García & Kallman 2010 and used in xillver and xillverD) does not correctly treat the Klein–Nishina effects (see footnote 1), which effects strongly reduce the albedo for hard spectra and the high-energy cutoffs at ≳100 keV, characteristic of the hard state. Zdziarski et al. (2003) considered a spectrum from Comptonization with the photon index of Γ = 1.5 and the electron temperature of ${{kT}}_{{\rm{e}}}=120\,\mathrm{keV}$ reflecting from a fully ionized medium, and found $a\approx 0.48$, much less than a = 1 of the non-relativistic case, and not much larger than $a\approx 0.34$ corresponding to reflection of the same spectrum from a neutral medium. With the Klein–Nishina effects taken exactly into account, the values of the albedo for the cases of Figures 1(a) and (b) would be reduced by ∼1/2 (because the present re-emitted spectra miss ∼1/2 of the incident flux; see footnote 1). The results of Zdziarski et al. (2003) also imply that even fully ionized disks in the hard state would show quasi-thermal features when irradiated.

Figures 1(b) and (c) imply that the reprocessed spectral features in those high-flux cases have the average energy relatively close to the average energy of the blackbody at ${T}_{\mathrm{eff}}$, i.e., their ratio (the so-called color correction, κ; Shimura & Takahara 1995; Davis et al. 2005) is only slightly larger than unity. In particular, $\kappa \approx 1.3$ in the case c. Also, the seen broadening of that quasi-thermal feature as compared to the blackbody is similar to those of accretion disk atmosphere spectra from intrinsic dissipation (Hubeny et al. 2001).

In order to use the above constraint, we need to estimate the irradiating flux. We use the observed bolometric flux, ${F}_{\mathrm{obs},0}$, the distance to the source, D, and its characteristic size scale, R. Assuming isotropy of the irradiating (primary) source, its luminosity, L₀, equals $4\pi {D}^{2}{F}_{\mathrm{obs},0}$. We initially consider a simple planar geometry with the primary source on each side of the disk emitting ${L}_{0}/2$ away from the disk and ${L}_{0}/2$ toward it (thus, the total luminosity is $L=2{L}_{0}$). The emission toward the disk is either Compton backscattered or absorbed and re-emitted, and each side of the disk emits in steady state ${L}_{0}/2$. The flux irradiating the disk can be written as ${F}_{\mathrm{irr}}={L}_{0}/(\zeta {R}^{2})$, where ζ is a geometry-dependent factor. For example, for two isotropic point sources (lampposts) irradiating the disk at a distance R at an angle θ between the ray and the disk normal, $\zeta =2\pi /\cos \theta$. For an isotropic corona above and below a ring between the inner radius R and $R+{\rm{\Delta }}R$, $\zeta \approx 4\pi {\rm{\Delta }}R/R$. Then we consider a radially stratified isotropic and optically thin corona above and below the disk. We point out that unless the coronal scale height is large compared to the radius, we do not need to take it into account in the estimates since the corona emits a half of its power toward the disk and a half away, regardless of the scale height. The irradiating flux at the inner radius, ${R}_{\mathrm{in}}$, can be expressed as

$$\begin{eqnarray}&&{F}_{\mathrm{irr}}({R}_{\mathrm{in}})=\displaystyle \frac{4\pi {D}^{2}{F}_{\mathrm{obs},0}}{\zeta {R}_{\mathrm{in}}^{2}}=\displaystyle \frac{4\pi {{\ell }}_{0}{m}_{{\rm{p}}}{c}^{5}}{\zeta {r}_{\mathrm{in}}^{2}{\sigma }_{{\rm{T}}}{GM}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\approx \,4.6\times {10}^{24}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}\displaystyle \frac{{{\ell }}_{0}/0.2}{{(\zeta /2\pi )(M/10{M}_{\odot })({r}_{\mathrm{in}}/2)}^{2}},\end{eqnarray} \tag{ 4 }$$

where $\zeta =2\pi$ for ${F}_{\mathrm{irr}}(R)\propto {R}^{-3}$, ${{\ell }}_{0}\equiv {L}_{0}/{L}_{\mathrm{Edd}}$, ${L}_{\mathrm{Edd}}$ is the Eddington luminosity (for pure H), ${r}_{\mathrm{in}}\equiv {R}_{\mathrm{in}}/{R}_{{\rm{g}}}$, ${m}_{{\rm{p}}}$ is the proton mass, and ${\sigma }_{{\rm{T}}}$ is the Thomson cross section. Profiles $\propto {R}^{-3}$ approximately correspond to both dissipation of the gravitational energy in a disk and disk irradiation by a central source. This implies that the flux irradiating the inner radius of a disk extending close to the ISCO of a fast-rotating $10{M}_{\odot }$ BH in the bright hard state ($20 \% {L}_{\mathrm{Edd}}$ or more) is $\sim 5\times {10}^{24}$ erg cm² s⁻¹, i.e., $\sim {10}^{8}$ higher than values of ${F}_{\mathrm{irr}}$ used for the luminous hard state if the standard models relxill and the public version of reflionx are employed; see Figure 1(a). This flux is also factors of $\sim {10}^{4}$ and ∼10 higher than that shown in Figures 1(b) and (c), respectively (though for some geometries ${F}_{\mathrm{irr}}$ needs to be multiplied by the reflection fraction ${ \mathcal R }$; see below). This implies the presence of a quasi-thermal feature with ${{kT}}_{\mathrm{eff}}\approx 1\,\mathrm{keV}$, as shown in Figures 1(b) and (c). Indeed, at ${F}_{\mathrm{intr}}=0$ and ζ = 2π, we have the inner color temperature of ${T}_{\mathrm{in}}\equiv \kappa {T}_{\mathrm{eff}}({R}_{\mathrm{in}})$ given by

$$\begin{eqnarray}&&{{kT}}_{\mathrm{in}}\approx 3.1\displaystyle \frac{\kappa {(1-a)}^{1/4}{{\ell }}_{0}^{1/4}}{{r}_{\mathrm{in}}^{1/2}{(M/10{M}_{\odot })}^{1/4}}\,\mathrm{keV}.\end{eqnarray} \tag{ 5 }$$

We stress that ${T}_{\mathrm{eff}}$ depends mostly on the irradiating flux, with only a secondary dependence on the density through the albedo, a. For ${F}_{\mathrm{irr}}(R)\propto {R}^{-3}$, ${T}_{\mathrm{eff}}\propto {R}^{-3/4}$, which smooths the quasi-thermal feature, and makes it resemble a disk blackbody spectrum (which has the same temperature profile). Given that the observed hard-state inner disk blackbody temperatures are in the ≈0.2–0.5 keV range (e.g., Basak & Zdziarski 2016; Wang-Ji et al. 2018), we need at least ${r}_{\mathrm{in}}\gtrsim 10$ at $a\sim 0.5$ and ${{\ell }}_{0}=0.1$.

These estimates are modified by GR, e.g., radiation of a lamppost is focused toward both the disk and the horizon. The focusing toward the disk enhances the irradiating flux near the inner edge (Martocchia & Matt 1996) with respect to that of Equations (3)–(4), making our constraint stronger. If the lamppost is at a large height, $H\gg {R}_{{\rm{g}}}$, in which case the characteristic distance from the primary source to the reflector is ∼H rather than $\sim {R}_{\mathrm{in}}$, the relativistic broadening of the fluorescent features would be only modest even if the disk extends to the ISCO, similarly to the case of a truncated disk (Fabian et al. 2014). The corona can also be outflowing (Beloborodov 1999). In that case, the irradiating flux of Equation (3) will be multiplied by the reflection fraction, ${ \mathcal R }\lt 1$, resulting from the Doppler de-boosting, but the rest of the estimates will remain unchanged. The dissipation profile, $\propto {R}^{-q}$, can have the index, q, different from 3. If q ≲ 2, the disk can extend to the ISCO with a weak relativistic broadening, similarly to the case of a lamppost at $H\gg {R}_{{\rm{g}}}$. However, rather large values of q are fitted in literature, e.g., $q\gt 7.2$ in García et al. (2018). Furthermore, there is strong evidence from timing that the hard-state accretion disks do have intrinsic dissipation (e.g., Wilkinson & Uttley 2009; Uttley et al. 2011). Both $q\gt 3$ and ${F}_{\mathrm{intr}}\gt 0$ would further increase ${T}_{\mathrm{eff}}$.

Finally, the geometry can be different from either pure disk corona or lamppost. In particular, if the disk (with a corona) is truncated at a ${R}_{\mathrm{in}}\gt {R}_{\mathrm{ISCO}}$ (the ISCO radius), it is very likely that the flow below ${R}_{\mathrm{in}}$ is dissipative and hot, e.g., Yuan & Narayan (2014). Then the irradiating flux will be lower than that estimated above, and the observed reflection fraction will follow from the fraction of the emission still taking place in the corona above the disk, the fraction of the hot-flow emission irradiating the disk, and the degree of scattering of the reflection in the corona.

Both the reflection and quasi-thermal features would then be partly Compton upscattered by the corona. The latter would result in the total spectrum consisting of the quasi-thermal feature (resembling a disk blackbody) and its Comptonization, as well as reflection components, the same as in the currently standard model fitted to hard-state data. The only difference would be the ${{kT}}_{\mathrm{eff}}$ of that feature, ∼1 keV if the disk extends close to the ISCO in the luminous hard state, higher than the observed range of ∼0.2–0.5 keV.

We illustrate the above constraint in the case of a luminous occurrence of the hard state in the BH X-ray binary GX 339–4. We study its brightest hard-state observation by XMM-Newton, on 2010 March 28. We use the EPIC-pn data in the timing mode, fit the energy range of 0.7–10 keV, and exclude the 1.75–2.35 keV range (to discard the range affected by calibration uncertainties), as in Basak & Zdziarski (2016). The XMM-Newton observation was accompanied by a contemporaneous observation by the Rossi X-Ray Timing Explorer (RXTE), for which only the Proportional Counter Array (PCA) data are available. We fit jointly both data sets in order to better constrain the underlying slope of the spectrum and to estimate the total flux. However, as shown in Basak & Zdziarski (2016), there are significant differences in the residuals in the joint fit of these data. This appears to be due to the PCA spectral calibration being significantly different from that of the EPIC-pn at $E\lesssim 10\,\mathrm{keV}$, while the two are in good agreement at higher energies (see Figure 3 in Kolehmainen et al. 2014). Therefore, in our joint fit, we use the PCA data only at $E\gt 9\,\mathrm{keV}$. On the other hand, these data become very noisy at $E\gt 35\,\mathrm{keV}$, which we discard.

We fit the joint data by the model tbabs(diskbb+ reflkerr). The first component accounts for the interstellar medium (ISM) absorption (Wilms et al. 2000). The relativistic reflection model reflkerr (Niedźwiecki et al. 2019) assumes the thermal Comptonization model of Poutanen & Svensson (1996) as the incident spectrum, and we use the option geom=0, which corresponds to the primary source being isotropic in the local frame. We assume the dimensionless BH spin of ${a}_{* }=0.998$ (at which ${R}_{\mathrm{ISCO}}\approx 1.237{R}_{{\rm{g}}}$), and the emissivity $\propto {R}^{-3}$ (used here for relativistic broadening only). That model uses the static reflection model xillver (García et al. 2013), which assumes $n={10}^{15}$ cm⁻³. At the fitted ionization parameter of $\xi \sim {10}^{3}$ erg cm s⁻¹ (Table 1), the irradiating flux is very low (Equation (2)), at which the bulk of the reprocessed emission is at ≲0.2 keV; see Figure 1(a). The much higher ${F}_{\mathrm{irr}}$ of our fitted model implies the presence of a reprocessed soft component at higher energies. Reflection models suitable for high density ($\gt {10}^{19}$ cm⁻³) in the disk would be needed in order to self-consistently account for that. However, such models are currently not publicly available (J19a). Therefore, we use the diskbb disk blackbody model (Mitsuda et al. 1984) to approximate the reprocessed soft component, with the temperature of blackbody seed photons for Comptonization equal to that at the disk inner radius. The relative similarity of the disk blackbody emission to that from reprocessing in an accretion disk is pointed out in Section 2.2. The models are used within the X-ray fitting package xspec (Arnaud 1996).

Table 1.  Results of the Spectral Fit with tbabs(diskbb+reflkerr)

NH	y	kTe	rin	ZFe	i	${ \mathcal R }$	${\mathrm{log}}_{10}\xi$	N	kTin	Ndisk	APCA	χ2/ν
$({10}^{21}{\mathrm{cm}}^{-2})$		(keV)			(deg)		(erg cm s−1)		(keV)	104
${7.4}_{-0.1}^{+0.2}$	${1.25}_{-0.01}^{+0.01}$	${42}_{-3}^{+2}$	${15.3}_{-2.7}^{+3.4}$	${4.9}_{-0.2}^{+0.1}$	${34}_{-2}^{+1}$	${0.28}_{-0.02}^{+0.01}$	${3.40}_{-0.02}^{+0.02}$	0.74	${0.207}_{-0.004}^{+0.003}$	${7.2}_{-0.4}^{+0.3}$	${1.31}_{-0.01}^{+0.01}$	172/186

Note. $y\equiv 4\tau {{kT}}_{{\rm{e}}}/{m}_{{\rm{e}}}{c}^{2}$, ${Z}_{\mathrm{Fe}}\leqslant 5$ is imposed, N is the flux density of reflkerr at 1 keV, and ${A}_{\mathrm{PCA}}$ is the relative PCA/EPIC-pn flux normalization. The uncertainties are for 90% confidence, ${\rm{\Delta }}{\chi }^{2}\approx 2.71$.

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Our fit results are given in Table 1, and the unfolded spectrum with the absorbed model and the unabsorbed model components are shown in Figures 2(a) and (b), respectively. We find a truncated disk, with ${r}_{\mathrm{in}}\approx 15$, and the reflection fraction of ${ \mathcal R }\approx 0.28$. The fitted high reflector Fe abundance, ${Z}_{\mathrm{Fe}}$, is probably an artifact of assuming a low disk density (Tomsick et al. 2018). The fitted electron temperature of ${{kT}}_{{\rm{e}}}\approx 42\,\mathrm{keV}$ is similar to the values found in high-L hard states of GX 339–4 by Wardziński et al. (2002), whose fits included the Compton Gamma Ray Observatory/Oriented Scintillation Spectrometer Experiment (CGRO/OSSE) data with significant source detection up to ≈500 keV, thus well constraining the actual value of ${{kT}}_{{\rm{e}}}$.

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We assume $M=8{M}_{\odot }$, D = 10 kpc (see Heida et al. 2017 and model D2 of Zdziarski et al. 2019). The 0.1–10³ keV absorption-corrected flux is $\approx 2.4\times {10}^{-8}$ erg cm⁻² s⁻¹, which corresponds to the isotropic luminosity of $L\approx 2.9\times {10}^{38}{(D/10\mathrm{kpc})}^{2}$ erg s⁻¹, and 24% of ${L}_{\mathrm{Edd}}$ at those M and D. We estimate the flux from the XMM-Newton data rather than from the PCA (which would yield the fluxes ≈1.3 times higher). The flux in the Comptonization component is ${F}_{\mathrm{obs},0}\approx 1.6\times {10}^{-8}$ erg cm⁻² s⁻¹. We estimate the albedo in the same way as for Figure 1, obtaining a ≈ 0.63. This implies that the remaining flux has to be emitted as reprocessed emission at energies higher than those corresponding to kT_eff, see Equations (1) and (5).

We then consider how to account for the obtained low fractional reflection, ${ \mathcal R }\approx 0.28$. One possibility is a reduction of the observed reflection due to scattering in the corona (Steiner et al. 2017). We have tested this by Comptonizing a fraction, ${f}_{\mathrm{sc}}$, of the reflected spectrum using the convolution model ThComp (Zdziarski et al. 2020). The best fit was at a negligible scattering fraction, and f_sc ≲ 0.02 at 90% confidence. On the other hand, the fits to a spectrum from RXTE PCA averaged over a few outbursts of GX 339–4 by Steiner et al. (2017) using their code simplcut allowed f_sc ≲ 0.5 with ${ \mathcal R }$ increasing then to ∼1. Thus, we consider this explanation still possible in principle. In that case, F_irr at the disk surface would be given by Equations (3)–(4).

Another possibility is that the primary source consists of two main parts; one forming a corona above the disk (and emitting ∼50% toward it), and the other forming a hot flow inside the truncation radius, emitting mostly away from the disk, as for a partial overlap between hot and cold flows (see Figure 1(f) in Poutanen et al. 2018). The presence of the hot inner flow is supported by our fit with R_in ≫ R_ISCO. Neglecting complications associated with details of this geometry, we assume that a fraction ${ \mathcal R }$ of the primary luminosity, L₀, is emitted by the corona, whose reflection is then observed, and the remainder is emitted by the hot inner flow, without any associated reflection. We thus multiply F_irr of Equations (3)–(4) by ${ \mathcal R }$, which is a conservative choice, minimizing T_eff. We further assume F_intr = 0, and obtain

$$\begin{eqnarray}&&{{kT}}_{\mathrm{in}}\approx 1.12\,\mathrm{keV}\displaystyle \frac{\kappa }{1.3}{\left(\displaystyle \frac{{r}_{\mathrm{in}}}{2}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{D}{10\mathrm{kpc}}\right)}^{\tfrac{1}{2}}{\left(\displaystyle \frac{M}{8{M}_{\odot }}\right)}^{-\tfrac{1}{2}},\end{eqnarray} \tag{ 6 }$$

where we scaled ${T}_{\mathrm{eff},\mathrm{in}}$ to r_in = 2 similar to those obtained by García et al. (2015) for GX 339–4 and used κ for the case of Figure 1(c). The observed disk blackbody flux in this component would then be

$$\begin{eqnarray}&&{F}_{\mathrm{obs},\mathrm{disk}}\approx 2\cos i(1-a){ \mathcal R }{F}_{\mathrm{obs},0},\end{eqnarray} \tag{ 7 }$$

where i is the viewing angle (Table 1), and the blackbody angular flux distribution is assumed. Then ${F}_{\mathrm{obs},\mathrm{disk}}\approx 3.3\cos i\times {10}^{-9}$ erg cm⁻² s⁻¹, and i ≤ 60° is most likely (Muñoz-Darias et al. 2013), which constraint we impose. We thus added a disk blackbody component from the estimated irradiation in Figure 3 in order to illustrate its appearance expected at a low r_in. Since the observed spectrum shows almost no curvature above 2 keV, adding such a component is in a strong conflict with the data. We have still tried to fit it, by allowing kT_in and the flux of this component to be ≥1.12 keV, ≥3.3 cos i × 10⁻⁹ erg cm⁻² s⁻¹, respectively, tying the temperature of the seed photons for Comptonization to T_in and allowing another disk blackbody at a lower temperature (which is strongly required by the data). The obtained fit was indeed very poor, with ${\chi }^{2}/\nu =5071/185$. Thus, the presence of such a high-T_in spectral component from irradiation appears incompatible with the data. This is also in agreement with previous fits of high-flux hard states of GX 339–4 (García et al. 2015; Basak & Zdziarski 2016; Dziełak et al. 2019), which show no trace of such a component. On the other hand, given that we have no access to a self-consistent spectral model for high illuminating fluxes (and high n), we do not attempt to set a quantitative limit on r_in for this observation. We can only state that r_in ≫ 2 is required for coronal models with the emissivity law q ≳ 2–3 or lampposts at heights less than a few tens of R_g.

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