THE COOL ACCRETION DISK IN ESO 243-49 HLX-1: FURTHER EVIDENCE OF AN INTERMEDIATE-MASS BLACK HOLE

In this work, we focus exclusively on the BHSPEC model. We are motivated by the observational evidence of a thermally dominant soft X-ray component in the spectra of HLX-1 considered here and the similarity of these spectra to those observed in Galactic BH X-ray binaries. Hence, we assume from the outset that the emission is from a radiatively efficient, thin accretion disk, and derive constraints on the parameters of interest, most importantly the BH mass, M. We refer readers interested in comparisons with other models to O. Godet et al. (2011, in preparation), which fits a wider array of models and attempts to differentiate between various interpretations.

In addition to log M, the BHSPEC model has four fit parameters: a normalization, the BH spin a_*, the cosine of the inclination i, and the log of the Eddington ratio ℓ = L/L_Edd, where L_Edd = 1.3 × 10³⁸(M/M_☉) erg s⁻¹ is the Eddington luminosity. The normalization can be fixed using the known distance to the source of D ∼ 95 Mpc, leaving four parameters to fit. We have computed models for log ℓ = −1.5 to 0, log M/M_☉ = 3.25 to 5.5, cos i = 0 to 1, and a_* = −1 to 0.99. We assume a Shakura & Sunyaev (1973) stress parameter α = 0.01, but discuss the implications of this choice in Section 4.

Although the spectra under consideration are thermally dominated, there is a tail of emission at high energies which is not accounted for by BHSPEC. We model this emission with SIMPL (Steiner et al. 2009), which adds two free parameters: the power-law index Γ and the fraction of scattered photons f_sc relative to the BHSPEC model. Photoelectric absorption by neutral gas along the line of sight is accounted for by the PHABS model in XSPEC. We leave the neutral hydrogen column N_H free, adding a single additional parameter.

We begin with a collection of prospective disk-dominated observations of HLX-1. This includes two observations with XMM-Newton (2004 November 23: XMM1; 2008 November 28: XMM2), two observations with Swift (2008 November 24: S1; 2010 August 30: S2), and one observation with Chandra (2010 September 6).

The XMM-Newton observations were performed with the three EPIC cameras in imaging mode with the thin filter. The two MOS cameras were in full-frame mode for both observations. The pn was operated in full-frame mode for the first observation (ObsID: 0204540201) and small window mode for the second (ObsID: 0560180901). The data were reduced using the XMM-Newton Science Analysis System (SAS) v8.0 and event files were processed using the epproc and emproc tools. The event lists were filtered for event patterns in order to maximize the signal-to-noise ratio against non X-ray events, with only calibrated patterns (i.e., single to double events for the pn and single to quadruple events for the MOS) selected. Events within a circular region of radius 60'' around the position of HLX-1 were extracted from the MOS data in both observations and from the pn data in the first observation. Background events were extracted for the same data from an annulus around the source position with an inner radius of 60'' and an outer radius of 8485. As the pn was in the small window mode during the second observation, a smaller circular extraction region with a radius of 40'' was used for extracting source events. Background events were in turn extracted from a circular region with radius 40'' from a region the same distance from the center of the chip as HLX-1, which appeared to have a similar background level in the image. Source and background spectra were extracted in this way for each camera, with response and ancillary response files generated in turn using the SAS tools rmfgen and arfgen.

Although the MOS and pn spectra are consistent down to ∼0.5 keV, below this boundary the pn spectrum deviates significantly, showing a sharp, absorption-like feature that is absent in the MOS spectra. Since this indicates a problem with the low energy response of the pn camera in this observation, we follow Farrell et al. (2009) and ignore channels with energies below 0.5 keV when fitting the pn spectrum. For the MOS spectra we fit channels with energies greater than 0.2 keV.

The Swift XRT data were processed using the tool XRTPIPELINE v0.12.3.4, as discussed in Godet et al. (2009). The Chandra ACIS-S data reduction is discussed in more detail in M. Servillat et al. (2011, in preparation). Their modeling indicates that the data likely suffer from mild pile-up, so we include a pile-up model based on Davis (2001) in our spectral fits, following the guidelines in the Chandra ABC Guide to Pileup.⁷ The grade migration parameter (denoted by α in the guide, and not to be confused with the accretion disk stress parameter) was left as a free parameter in our initial fits. These fits generically favored the maximum allowed value of 1. Our best-fit BHSPEC parameters were mildly sensitive to variations in this α, but α ∼ 1 yields results which are consistent with the S1 and S2 data sets. Hence, we fixed α = 1 for subsequent analysis. The frame time was set to 0.8 s to match the observational setup, and all other parameters were left at their XSPEC defaults.

All spectra were rebinned to require a minimum of 20 counts per bin.

Using XSPEC⁸ (v12.6.0q), we fit each observation independently to determine its suitability for modeling with BHSPEC. Specifically, we evaluate the level of disk dominance for typical best-fit parameters. For both of the XMM-Newton observations, we fit all EPIC data sets (MOS1, MOS2, and pn) simultaneously. An additional hard X-ray component is necessary to obtain a good fit for these data sets. Fits with SIMPL typically find a best-fit scattering fraction f_sc ≳ 0.5 for the XMM1 data and f_sc ≲ 0.15 for XMM2. These results are consistent with those of Farrell et al. (2009) who find an acceptable fit to the XMM1 data with an absorbed power law, but require an additional soft component (modeled with DISKBB; Mitsuda et al. 1984) for the XMM2 data. Our fits suggest that the XMM1 spectrum is more typical of a steep power-law state, while the XMM2 spectrum is consistent with a thermally dominated state. A large f_sc in the fit with SIMPL is problematic for our accretion disk model, which does not account for the effects of irradiation by neighboring corona. Hence, we only report our results from the XMM2 analysis below. We note that the XMM1 data generally favor a best-fit M that is higher than XMM2 for the same i. This yields a cooler disk model, consistent with the fact that more of the high energy flux is accounted for with the SIMPL component in these data. However, the joint confidence contours on a_* and M are much larger for the XMM1 data, so the best-fit values are still consistent with the XMM2 values at 90% confidence.

The Swift and Chandra observations caught the source at higher luminosities than the XMM2 observation, but the overall signal-to-noise is still lower because of the lower effective area of the Swift XRT and the short duration (10 ks) of the Chandra ACIS-S observation. Due to the lower signal-to-noise and stronger disk dominance, a suitable fit is provided by the (absorbed) BHSPEC model alone. The addition of the SIMPL model only provides a marginal improvement to the fit and the best-fit SIMPL parameters are poorly constrained. We also fit the combined S1 and S2 observations. In this case we tie all BHSPEC parameters together, except for ℓ, which is allowed to vary independently for S1 and S2. Even for the combined data set, we find a good fit with BHSPEC alone, and poor constraints on the SIMPL model parameters. Therefore, we do not include the SIMPL model in subsequent analysis of the combined S1 and S2 data sets or in our analysis of the Chandra data.

We first consider the XMM2 observation. The soft thermal component is largely characterized by only two parameters: the energy at which the spectrum peaks and the overall normalization. In practice, this leads to degeneracies in the best-fit parameters of BHSPEC (see, e.g., Davis et al. 2006), unless all but two of the parameters can be independently constrained. Since we can only constrain D independently, we will need to consider joint variation of the remaining parameters.

For completeness, our best-fit parameters for various choices of i or a_* are summarized in Table 1. We find acceptable fits for all i. The best-fit model and data for i = 0 are plotted in Figure 1. We report the 90% confidence interval for a single parameter, but due to the significant degeneracies in the model, the joint confidence contours better illustrate the actual parameter uncertainties. Hence, they are the focus of this work.

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Since M and a_* are of primary interest to us, we first examine their joint confidence contours, and consider the joint confidence of M and i in Section 4. For illustration, it is useful to consider several sets of contours for different choices of (fixed) i, leaving ℓ as a free parameter. These best-fit joint confidence contours are shown in Figure 2. We consider five choices for cos i, evenly spaced from 0 to 1. Each set of contours has three curves corresponding to 68%, 90%, and 99% confidence. These are determined by the change in χ² relative to the best-fit values listed in Table 1.

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Even for fixed i, there is a clear correlation of a_* and M. At 99% confidence, the entire range of a_* (−1 < a_* < 0.99) is allowed, and the corresponding M varies by a factor of 4–5 over this range. The only exception is for nearly edge-on systems (cos i ∼ 0), for which low spins are disallowed. The correlation of best-fit M with i is very strong as well, consistent with previous modeling of other ULX sources (Hui & Krolik 2008). This is illustrated more clearly in Figure 3, which shows the joint confidence contours of M and cos i for fixed a_* = −1, 0, 0.7, and 0.95. The corresponding best-fit values are listed in Table 1. At fixed a_*, there is up to a factor of 10 change in M as inclination varies from 0° to 90°. The combined overall uncertainty in the mass is nearly a factor of 100.

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Despite these degeneracies, we can still place a firm lower limit of M ≳ 3000 M_☉ on the mass of the BH in HLX-1. The limiting mass is obtained for the case of a maximally spinning BH with a counter-rotating disk viewed nearly face-on. An independent lower limit on the mass may be obtained under the assumption that the source luminosity does not exceed the Eddington limit. This gives the same answer, M ≳ 3000 M_☉, which is purely coincidental. At the high mass end, the limit we obtain from our fits is M ≲ 3 × 10⁵ M_☉, where the limiting value is obtained for nearly maximal spins in the case where disk rotation is spin-aligned and the system is viewed nearly edge-on. However, edge-on systems may be ruled out by the lack of X-ray eclipses if one assumes the accreting matter is provided by a binary companion. If one limits i ≲ 75°, one obtains M ≲ 10⁵ M_☉. The above limits on the mass place HLX-1 in the IMBH regime, though the very highest masses in our allowed range approach the lower end of the mass distribution inferred in active galactic nuclei (e.g., Greene & Ho 2007).

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We plot the variation of the other free parameters in Figure 4 for the specific example of cos i = 0.5. The variation of ℓ is the most interesting, as there are fairly clear correlations with M and a_*. Following the best-fit contour, ℓ decreases as M and a_* increase. This correlation arises because ℓ has a role in determining both the location of the spectral peak and its overall normalization, a point we elaborate on in Section 4.

The bottom two panels show the variation of the SIMPL parameters. In contrast to ℓ, these parameters tend to vary across (rather than along) the confidence contours, indicating that they are highly correlated with χ². A similar behavior is present in the absorption column. For the best-fit values we find N_H ≃ (1–4) × 10²⁰ cm⁻², Γ ≃ 2.2–2.7, and a scattered fraction ≲ 12%. The allowed N_H range is roughly equivalent to or slightly greater than the Galactic value (1.7 × 10²⁰ cm⁻²; Kalberla et al. 2005). The ranges for the two SIMPL parameters are generally consistent with fits to the thermal state of Galactic BH binaries and confirm that the bolometric luminosity of the best-fit models is dominated by the BHSPEC component. For typical parameters within the confidence contours, the accretion disk accounts for 80%–95% of the unabsorbed model luminosity from 0.3 to 10 keV. Comparison with Table 1 of Socrates & Davis (2006) shows that this is a much higher disk fraction than typically inferred in other bright ULXs, in which values 10%–50% are more typical.

In Figure 5, we plot the best-fit joint confidence contours of a_* and M for the combined Swift (S1 and S2) data sets. Figure 6 is the corresponding plot for the Chandra data set. As discussed in Section 2.2, we do not include a SIMPL component in fitting these data sets and we tie all the BHSPEC parameters together, except ℓ, which is allowed to vary independently for each observation. The corresponding best-fit parameters are depicted with crosses and summarized in Tables 2 and 3, where ℓ₁ and ℓ₂ correspond to the S1 and S2 observations, respectively. The reduced χ² for the Chandra data indicate a slightly poorer fit than with the XMM-Newton or Swift data. This is primarily due to narrow residuals near 0.6, 1.1, and 1.3 keV in fits to the ACIS-S data. Since these residuals are not highly significant and their presence does not significantly impact the best-fit BHSPEC parameters, we do not attempt to model them with additional components. Further discussion can be found in M. Servillat et al. (2011, in preparation).

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The Swift and Chandra data seem to place much tighter overall constraints on a_* and M, with a minimum a_* at each i. The overall minima of M ∼ 6000 M_☉ for Swift and M ∼ 4000 M_☉ for Chandra both correspond to i = 0°. Fits with retrograde accretion flows (a_* < 0) are inconsistent with the Swift data and models with a_* ≲ −0.5 are ruled out by the Chandra spectrum. For values of a_* allowed by both data sets, the position of the confidence contours for a given i is consistent with each other. They are also broadly consistent with the XMM2 results, although offset to slightly higher M. However, due to the lower signal-to-noise of the Swift and Chandra data sets, the range of allowed M for a given i is larger than in Figure 2.

The differences between the XMM-Newton fits and the Swift or Chandra results at low (negative) a_* are driven primarily by the larger luminosities of the disk component in the Swift and Chandra observations. With the exception of nearly edge-on systems (cos i = 0), we find ℓ < 1 within the 99% confidence contours in Figure 2. In the Swift and Chandra data, ℓ ⩾ 1 is required for all i, at sufficiently low a_*. As we discuss further in Section 4, reductions in a_* and M must be offset by increases in ℓ, but our models are capped at ℓ = 1, which sets a lower limit on the allowed a_* and M for a given choice of i. Hence, in contrast to the XMM2 observation, these tighter constraints depend strongly on the assumption that luminosity does not exceed the Eddington limit: ℓ ⩽ 1.

We also fit the combined Swift, Chandra, and XMM-Newton data sets assuming no correction was needed to the relative effective area of the various instruments. In principle, a constant offset could be included based on cross-calibration analysis (e.g., Tsujimoto et al. 2011), but even without such a correction, we obtain good fits to the combined data set. The range of allowed a_* is similar to that found with the Swift data alone, but the widths of the contours are comparable to those from the XMM2 fits plotted in Figure 2. At 99% confidence, the minimum allowed M was about 6000 M_☉, again for i = 0° and ℓ = 1.

To first approximation, thermal state disk spectra are characterized by only two parameters: a peak energy and an overall normalization. Indeed the popular DISKBB model (Mitsuda et al. 1984), with only two parameters, provides an acceptable fit to the soft thermal component of the XMM2 data (Farrell et al. 2009). Although we obtain all our constraints by fitting a relativistic accretion disk model directly to the spectrum, our fit results are easily understood in terms of these two constraints. These observables can be mapped onto a characteristic color temperature T_obs (≈T_in from DISKBB), and since the distance D is known, a characteristic luminosity L_obs.

The characteristic temperature is related to the peak effective temperature T_eff of the accretion disk. For illustration, we assume this is equivalent to the effective temperature at the inner edge of the disk. Ignoring relativistic terms and constants of order unity, the flux near the inner edge of an accretion disk can be approximated by

$$\begin{equation} \sigma T_{\rm eff}^4 \simeq \frac{{\it G M}\! \dot{M}}{R_{\rm in}^3,}, \end{equation} \tag{ 1 }$$

where G is Newton's constant and $\dot{M}$ is the accretion rate. The inner radius of the disk R_in corresponds to the innermost stable circular orbit (ISCO) in our model.

Due to deviations from blackbody emission and relativistic effects on photon propagation, T_obs ≠ T_eff. We parameterize the deviations from blackbody via a spectral hardening factor (or color correction) f_col and the relativistic energy shifts with δ so that T_obs = f_colδT_eff. In practice, δ is primarily a function of i and a_* with δ ≳ 1 typical for most a_* and i in our models (i.e., Doppler blueshifts are generally more important than Doppler and general relativistic redshifts). In contrast, f_col is dependent on M, a_*, and ℓ and largely independent of i. Using this parameterization and Equation (1) we obtain

$$\begin{equation} T_{\rm obs} \simeq T_0 f_{\rm col} \delta \left(\frac{\ell }{r_{\rm in}^2 m}\right)^{1/4}, \end{equation} \tag{ 2 }$$

where, m = M/M_☉, r_in = R_in/R_g, T₀ = (c⁵/GM_☉κ_esσ)^1/4, R_g = GM/c² is the gravitational radius, and κ_es is the electron scattering opacity. Here, $\ell =\eta \dot{M}c^2/L_{\rm Edd}$, and for simplicity we have approximated η ∼ 1/r_in.

Using the above definitions the observed luminosity is

$$\begin{equation} L_{\rm obs} \simeq L_0 l m \mu, \end{equation} \tag{ 3 }$$

where L₀ ≡ 4πGM_☉c/κ_es is the Eddington luminosity for a one solar mass BH and μ is a variable that encapsulates all of the angular dependence of the radiation field. In an isotropically emitting, Newtonian disk μ = cos i, accounting for the inclination dependence of the disk projected area. In our models, two other effects are important as well: electron scattering and relativistic beaming which tend to make the disk emission more limb-darkened and limb-brightened, respectively. The latter depends significantly on the spin so μ can be a strong function of a_* as well as i. Lensing by the BH can also be important, particularly for nearly edge-on systems where it places a lower limit on the effective projected area of the disk.

Equations (2) and (3) can be solved for m and ℓ, yielding

$$\begin{equation} M \simeq M_{\odot }\left(\frac{T_{\rm obs}^4 L_0}{T_0^4 L_{\rm obs}} \right)^{1/2} \frac{f_{\rm col}^2 \delta ^2}{\mu ^{1/2} r_{\rm in}}. \end{equation} \tag{ 4 }$$

The quantities in parentheses on the right-hand side of Equation (4) are observational constraints or constants of nature. Uncertainties in D enter through L_obs∝D², so we approximately have M∝D⁻¹. Therefore, the relatively small uncertainty in D contributes only a very modest additional uncertainty in M. The remaining quantities are functions of model parameters: r_in(a_*), μ(a_*, i), δ(a_*, i), and f_col(a_*, ℓ, M).

The dependence of r_in on a_* is the primary driver of the correlation of M with a_* in Figure 2. As a_* increases from −1 to 0.99, r_in decreases from 9 to 1.45, driving corresponding increases in M. This variation is strongest as a_* → 1, leading to the "bend" in the confidence contours at high a_*. Both δ and f_col increase with a_*, and contribute to the correlation as well.

The spectrum in BHSPEC is calculated directly, so there is no explicit f_col, but we can estimate f_col by taking the best-fit BHSPEC models and fitting them with the KERRBB model, for which f_col is a parameter.⁹ The variation of f_col is shown for cos i = 0.5 in Figure 7. Although f_col can vary by more than a factor two over the parameter range of interest, its change within the best-fit contours is significantly less. For cos i ∼ 0.5, δ increases modestly with a_* due to Doppler blueshifting, so the product of f_col and T_eff must decrease to maintain agreement with T_obs. Since T_eff is the primary driver of variations in f_col, and f_col and T_eff are positively correlated, both T_eff and f_col must decrease as a_* increases. The same argument applies outside the best-fit contours, but leads to a much larger variation in f_col.

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The confidence contours in Figure 3 confirm a strong anti-correlation of M with cos i or, equivalently, a correlation of M with i. This correlation is driven primarily through the i dependence of δ and μ. As i increases, the projected area of the disk decreases while limb darkening shifts intensity to lower i. The combined effects lead to a significant decrease in μ as i increases. For higher a_*, these effects are somewhat mitigated by relativistic beaming, which tends to shift intensity to larger i. In addition, Doppler blueshifts due to the Keplerian motion cause δ to increase with i. Both effects are present and drive a positive correlation of M and i, consistent with Figure 3. The anti-correlation of i and μ dominates at low spin, while the correlation of δ with i dominates at higher spin because rotational velocities are a large fraction of the speed of light. The effects contribute comparably for a_* ∼ 0.7.

We note that Hui & Krolik (2008) found a similar correlation in their analysis of several ULX sources. To explain their results, they attribute the correlation to Doppler shifts (i.e., the dependence of δ on i), which is consistent with the fact that their best-fit models favored high a_*.

The anti-correlation of best-fit ℓ with M and a_* follows directly from Equations (2) and (3). As M increases, ℓ must decrease to keep L_obs approximately constant, but keeping T_obs constant then requires an increase in a_* (i.e., a reduction in r_in). This is why high M and low a_* (and vice versa) yield poor fits for cos i = 0.5 in Figure 4.

With this understanding of how M, i, ℓ, and a_* correlate, it is useful to consider what ultimately sets the minimum allowed M in the two data sets. The L_obs constraint (Equation (3)) allows M to decrease as long as there is a corresponding increase in μ or an increase in ℓ. For fits to the XMM2 data, ℓ is set by the T_obs constraint (Equation (2)). Since ℓ increases as M decreases, r_in must increase. Hence, the maximum ℓ and minimum M are obtained for a_* = −1. This argument leads to a minimum M for any fixed i and since μ is maximum for face-on disks, the "global" minimum of M occurs at i = 0° and corresponds to ℓ slightly less than unity. Hence, the constraint that M ≳ 3000 M_☉ is independent of the argument that ℓ < 1.

For the Swift and Chandra data sets, which consist of observations with higher L_obs, this is not the case. Our models are capped at ℓ = 1 due to inconsistencies in the underlying model assumptions for ℓ ≳ 1. For some minimum M (M ∼ 6000 M_☉: Swift; M ∼ 4000 M_☉: Chandra), Equation (3) cannot be satisfied for ℓ ⩽ 1, even with i = 0°. This corresponds also to a lower limit on a_* since further increases in r_in cannot be offset by decreases (increases) in M (ℓ) to keep T_obs constant. If we ignored the internal inconsistencies in thin disk assumptions and extended our models to ℓ > 1, we expect we would obtain reasonable fits for lower M and a_* as Equations (2) and (3) could still be satisfied. Hence, the more stringent lower limit on M and a_* for the Swift and Chandra fits is not independent of the Eddington limit.

For the XMM2 data, the minimum allowed M is reached for models with ℓ ∼ 0.7, just below the Eddington limit.¹⁰ The assumptions underlying the thin accretion disk model are likely to break down for ℓ ∼ 1 although it is difficult to precisely estimate the ℓ value where our spectral constraints are no longer reasonable. General relativistic magnetohydrodynamic simulations (e.g., Penna et al. 2010; Kulkarni et al. 2011) and slim disk models (Sądowski et al. 2011) suggest that the thin disk spectral models remain reliable to (at least) ℓ ≲ 0.3. Observations of Galactic X-ray binaries suggest that there are no abrupt changes at this ℓ (Steiner et al. 2010), so it is plausible that models remain basically sound even for somewhat higher ℓ.

The strong correlation of χ² with Γ and f_sc in Figure 4 (as well as a similar one for N_H) results from the failure of the BHSPEC model to adequately approximate the thermal emission for M and a_* outside the confidence contours. If BHSPEC is a poor match to the thermal spectrum, the fit compensates by adjusting N_H or the SIMPL parameters. For example, at low M and a_*, the disk is too cold to match the XMM2 data, so SIMPL adjusts by increasing the scattering fraction and making the power-law steeper, to better fit the high energy tail of the thermal emission. To prevent an excess at lower energies, the N_H increases simultaneously. For the lower signal-to-noise Swift and Chandra data, N_H adjusts in a similar manner, even though the SIMPL component is absent. The width of the confidence contours appears to be set largely by the effectiveness of these compensation mechanisms. Hence, the extent of the confidence contours for a given i could presumably be reduced with better statistics at high energies to constrain the SIMPL parameters and independent constraints on N_H.

Finally, it is worth considering the degree to which our correlations could be reproduced by a non-relativistic analysis, but assuming r_in is equal to the ISCO radius. For example, assuming constant f_col ∼ 1.7, δ ∼ 1, and μ ∼ cos i in Equation (4), one can recover some aspects of the inferred correlations of M with a_* and i. The sensitivity of M to a_* would be approximated well for low-to-moderate a_*, but the dependence of δ, f_col, and μ on a_* can lead to modest discrepancies as a_* → 1. In contrast, the correlation of M with i would only be crudely reproduced since the above prescription would suggest M∝(cos i)^−1/2. This underestimates the sensitivity of M to i for low-to-moderate i, but overestimates it as i → 90°, where the projected area goes to zero in a non-relativistic model. Since the low-to-moderate range is probably more relevant for observed systems, the overall uncertainty of M would be underestimated. We also note that although the assumption of a constant f_col ∼ 1.7 turns out to be approximately correct (f_col ≈ 1.8 ± 0.1 in Figure 7), it was not clearly justified a priori and may not be as good of an assumption for other ULX sources. In principle, one could improve this analysis by estimating f_col, μ, and δ from BHSPEC (or some similar model), but at that level of sophistication, it seems more sensible to fit the relativistic model directly.

There are a number of assumptions present in the BHSPEC model that could have some impact on f_col. In particular, magnetic fields (and associated turbulence) may play a role in modifying the disk vertical structure and radiative transfer (e.g., see Davis et al. 2005; Davis & Hubeny 2006; Blaes et al. 2006; Davis et al. 2009). Another assumption of interest is our choice of α = 0.01. For α ≲ 0.01, the models depend very weakly on α for the parameter range relevant to our fit results. For higher values of α, the typical color correction is larger. For low-to-moderate ℓ and a_*, f_col increases by less than 25% as α increases from 0.01 to 0.1. Much larger shifts can occur if both ℓ and a_* are larger (a_* ≳ 0.8 and ℓ ≳ 0.3; Done & Davis 2008), but models in this range overpredict T_obs and are irrelevant to our results. Hence, if the characteristic α associated with real accretion flows is larger (as some models of dwarf novae and some numerical simulations suggest; King et al. 2007), the effect would be to shift our best-fit contours to higher M, but only by a modest amount. For the parameters corresponding to the lower M limit (i = 0°, ℓ = 0.7, and a_* = −1), BHSPEC yields f_col ∼ 2. From Equation (4) we see that reducing f_col = 1 (the absolute minimum) only reduces M by a factor of four, still placing HLX-1 in the IMBH regime.

Alternatively, one could make the disk around a low M BH look cooler by truncating it at larger radius. Equation (4) suggests that decreasing M to a value near 30 M_☉ would require a factor of 100 increase to r_in ∼ 900. Such an interpretation would need to explain why the flow does not radiate inside this radius. Since the energy does not come out in the hard X-rays, it cannot be a transition to an advection dominated accretion flow, which is often invoked to explain the low state of Galactic X-ray binaries (e.g., Esin et al. 2001). Furthermore, since η ∼ 1/r_in, the required $\dot{M}$ would increase by a factor of 100 to $\dot{M}\sim 10^{-2} \, M_{\odot }\; \rm yr^{-1}$. For $\dot{M}\sim 10^{-4} \, M_{\odot }\; \rm yr^{-1}$, the accretion rate and time variability of HLX-1 present a challenge to standard models of mass transfer (Lasota et al. 2011). Hence, it is unlikely that such a high rate is even feasible in a binary mass transfer scenario.

Finally, one could plausibly obey the Eddington limit by assuming a large beaming factor (e.g., King et al. 2001; Freeland et al. 2006; King 2008) so that L_obs ≫ L_iso, which (in this formalism) is equivalent to increasing μ for some narrow range of i. Obeying the Eddington limit with M ∼ 30 M_☉ would require μ ∼ 100, but note that this is insufficient for explaining T_obs due to the μ^1/2 dependence in Equation (4). Fixing ℓ = 1 and decreasing M by a factor of 100 yields a factor of three increase in T_obs. Maintaining agreement with T_obs requires M and ℓ to decrease proportionately, which requires a factor of 1000 increase in μ. Any scenario with such large beaming factors probably requires a relativistic outflow or very different accretion flow geometry so these simple scalings may not strictly apply. Nevertheless, we emphasize that explaining the soft emission in HLX-1 presents a serious challenge to any beaming model, but is naturally explained by the IMBH interpretation.