{'fraction_non_alphanumeric': 0.046485305023223136, 'fraction_numerical': 0.037761992115842476, 'mean_word_length': 3.9988293824992684, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
INVESTIGATING SLIM DISK SOLUTIONS FOR HLX-1 IN ESO 243-49

The hyperluminous X-ray source HLX-1 in the galaxy ESO 243-49, currently the best intermediate-mass black hole (BH) candidate, displays spectral transitions similar to those observed in Galactic BH binaries, but with a luminosity 100–1000 times higher. We investigated the X-ray properties of this unique source by fitting multi-epoch data collected by Swift, XMM-Newton, and Chandra with a disk model computing spectra for a wide range of sub- and super-Eddington accretion rates assuming a non-spinning BH and a face-on disk (i = 0°). Under these assumptions we find that the BH in HLX-1 is in the intermediate-mass range (∼2 × 104 M☉) and the accretion flow is in the sub-Eddington regime. The disk radiation efficiency is η = 0.11 ± 0.03. We also show that the source does follow the LX ∝ T4 relation for our mass estimate. At the outburst peaks, the source radiates near the Eddington limit. The accretion rate then stays constant around 4 × 10−4 M☉ yr−1 for several days and then decreases exponentially. Such “plateaus” in the accretion rate could be evidence that enhanced mass-transfer rate is the driving outburst mechanism in HLX-1. We also report on the new outburst observed in 2011 August by the Swift X-Ray Telescope. The time of this new outburst further strengthens the ∼1 year recurrence timescale.


Introduction
Ultra Luminous X-ray sources (ULXs) are defined as off-nucleus extragalactic sources showing X-ray luminosity exceeding 3×10 39 erg s −1 assuming isotropic emission (see Roberts et al. 2007). Even if their nature is still in dispute, it is likely that their huge luminosity is produced by accretion of matter onto a black hole (BH). Three explanations for their nature have been considered.
(1) ULXs may be X-ray stellar-mass BH binaries (BHBs) similar to those observed in our Galaxy, but in a more extreme version of the very high state (e.g. Remillard & McClintock 2006), the ultraluminous or wind-dominated state (e.g. Gladstone et al. 2009). In such states, the source would be able to radiate above the Eddington limit. It is still to be understood why this ultraluminous state is so rarely observed in X-ray binaries. (2) King et al. (2001) proposed that the emission of ULXs is highly anisotropic. In this case, the requirement to have super-Eddington emission is alleviated. The nature of this anisotropy could be due to either geometrically thick disks funnelling the X-ray photons produced in the inner parts of the accretion disks (King 2009) or due to relativistic beaming of a jet or strong outflow. However, the discovery of several optical and radio nebulae around ULXs (e.g. Pakull & Grisé 2008) and QPOs in M82 X-1 (Strohmayer & Mushotzky 2003) argue against strong beaming. (3) The final and the most exciting explanation is that some ULXs are accreting intermediate mass BHs (IMBHs) with masses ranging from ∼ 100 M ⊙ to ∼ 10 5 M ⊙ (e.g. Colbert & Mushotzky 1999). The existence of such IMBHs will naturally alleviate the need for super-Eddington emission. It is clear now that the IMBH interpretation is not valid to explain the ULX population as a whole (see e.g. Roberts 2007). However, the most luminous ULXs, the so-called Hyper Luminous X-ray sources (HLX - Gao et al. 2003) with X-ray luminosities above 10 41 erg s −1 , are good candidates. Finding convincing evidence for the existence of IMBHs is important for the growth of supermassive BHs via mergers or accretion episodes (e.g. Micic et al. 2007), dark matter studies (e.g. Fornasa & Bertone 2008, but see also Bringmann et al. 2009), cosmology (e.g. Trenti & Stiavelli 2007) and gravitational wave detection (e.g. Matsubayashi et al. 2004, Amaro-Seoane & Santamara 2010. Farrell et al. (2009) reported the serendipitous discovery of a ULX candidate 2XMM J011028.1-460421, referred to hereafter as HLX-1, located in the outskirts of the edge-on spiral galaxy ESO 243-49 at a redshift of 0.0224 (Wiersema et al. 2010). From its maximum luminosity reaching ∼ 1.3 × 10 42 erg s −1 at peak and assuming that the source luminosity reached up to ten times the Eddington limit (L Edd ), Farrell et al. (2009) derived a BH mass of more than 500 M ⊙ . HLX-1 is so far the best candidate to harbour an IMBH. Apart from its extreme luminosity which has been observed many times over the past 3 years by different Xray satellites (Swift, XMM-Newton & Chandra), HLX-1 is unique amongst other ULXs because it is the only one for which clear spectral hysteresis similar to those observed in Galactic BHBs (GB-HBs) are seen. Indeed, Servillat et al. (2011) using XMM-Newton, Chandra and Swift data confirmed the results presented in Godet et al. (2009) that HLX-1 underwent transitions from the high/soft state to the low/hard state. The Swift-X-ray Telescope (XRT) lightcurve covers so far four outbursts from 2008 to 2011 (hereafter P0, P1, P2 and P3 -see Fig. 1). While we had only one observation in October 2008 during the P0 outburst, the P1 and P2 outbursts display well-sampled Fast-Rise-Exponential-Decay (FRED)-like temporal profiles. Thanks to our dedicated Swift-XRT ToO, we caught the rise of the P3 outburst from 15 th August 2011 (Godet et al. 2011). A precursor could be seen prior to the P3 outburst peak in Fig. 1. Two reflare events are also visible during the P1 and P3 outbursts. The outbursts appear to be separated by a recurrence timescale of nearly a year. Recently, Lasota et al. (2011) interpreted the X-ray light-curve as the result of enhanced mass-transfer rate onto a pre-existing accretion disk around an IMBH, when an asymptotic giant branch star orbiting along an eccentric orbit with a period of ∼ 1 year is tidally stripped near periastron.
In order to further investigate the nature of this unique source, it is essential to put some constraints on the accretion flow and the BH mass. Lower and upper limits from radio, optical and Xray observations have been derived (see Wiersema et al. 2011and Webb et al. 2011. Davis et al. (2011) using their relativistic accretion disk model BHSPEC to fit X-ray spectra when the source was in various spectral states put some constraints on the BH mass. They showed that the inclination (i) has a strong influence on both the mass and the spin (a * ) of the BH derived by the model. Due to degeneracies in their best-fit parameters, they were only able to give a range of possible BH masses from 3000 M ⊙ (where the limit corresponds to i = 0 • , LX L Edd = 0.7 and a * = −1) to 3 × 10 5 M ⊙ (where the limit corresponds to i = 90 • and a * = 0.99). However, the inclination is likely to be less than 60 − 70 • due to the lack of observed eclipse dips assuming HLX-1 is a binary system. These results were obtained neglecting the effects of radial advection that are important at luminosities above the Eddington limit L Edd with L Edd = 1.3 × 10 38 M M⊙ erg s −1 . Radial advection is a mechanism enabled to stabilize disks in the super-Eddington accretion regime (see e.g. Abramowicz et al. 1988). This regime is reached when the accretion rate (Ṁ ) is larger thanṀ Edd = η −1 L Edd c 2 with c the speed of light and η the radiation efficiency. In this regime, the vertical disk structure could be geometrically thick (H/R > 0.5 with H and R the scale height and the radius of the disk, respectively). However, for a range of moderate super-EddingtonṀvalues there are still disk solutions for which the vertical disk structure can be considered as relatively thin. They are often referred to as slim disk solutions (Abramowicz et al. 1988). For large accretion rates (ṁ ≥ 10 withṁ =Ṁ c 2 L Edd ), electron scattering (opacity and comptonization) have several effects on the emergent spectra: i) they can be highly distorted and no longer look like a multicolor black-body (BB) spectrum (e.g. Kawaguchi 2003, hereafter K03); ii) the L ∝ T 4 relation with T the colour temperature at the inner disk radius for standard α-disk (Shakura & Sunayev 1973) is no longer valid (see Fukue 2000, K03). In addition, slim disks may extend beyond the innermost stable circular orbit (ISCO) due to non-negligible pressure support resulting in higher disk temperatures (e.g. Abramowicz et al. 1988Abramowicz et al. , 2010Watarai et al. 2000).
In this paper, we make use of the K03 disk model in order to: i) investigate the accretion flow and BH properties in HLX-1 through spectral fitting of a multi-epoch and multi-instrument dataset using a model that computes a wide range of sub-and super-Eddington accretion disk solutions; ii) investigate whether radial advection and electron scattering have important effects on the emergent spectra; iii) compare our results with those performed on less luminous ULXs using the same model that always favoured critical or super-Eddington accretion onto a stellar mass BH (e.g. Yoshida et al. 2010, Foschini et al. 2006, Vierdayanti et al. 2006, Okajima et al. 2006.
The paper is organised as follows. In Section 2, we describe the data used to perform the spectral analysis, as well as the data reduction. Section 3 presents some possible observational evidence for effects of radial advection and/or electron scattering as well as the detection of possible X-ray lines.
In Section 4, we present the fitting results obtained with the K03 disk model. In Section 5, we discuss the implications of the fitting results on the BH mass, the accretion rate and the disk structure.
The main conclusions are given in Section 6.

Observation log & Data reduction
We consider X-ray data coming from 3 observatories: XMM-Newton, Chandra (ACIS) and Swift. The data used in this study cover the different spectral states observed during the outbursts of HLX-1 (see Table 3). We used the same XMM-Newton and Chandra spectra as those presented in Servillat et al. (2011). The nomenclature used here is the same as that in Servillat et al. (2011) and Farrell et al. (2009) for the XMM-Newton data (see Table 3). For the XMM-Newton data, we only used the EPIC-pn data because it offers the best statistics for a given observation. Please refer to Servillat et al. (2011) for the details of the data reduction on the XMM-Newton and Chandra data. In order to investigate for the presence of X-ray lines during the outbursts, we also used the Reflection Grating Spectrometer (RGS) data from the XMM2 observation (see Table 3). The RGS data have been reduced using the standard XMM-Newton Science Analysis Software (SAS) task RGSPROC and the most recent calibration files released in February 2011. For the spectral analysis, we only considered the first order of the RGS data.
All the Swift-XRT Photon Counting data were processed using the HEASOFT v6.11 and the lattest calibration files (CALDB version 3.8). This new CALDB includes a new gain file enabled to correct the data for charge traps that accumulate on the CCD due to radiation damage (see Pagani et al. 2011). These traps can induce some energy offsets. The data were processed using the tool XRTPIPELINE v0.12.6. We inspected all segments of data to search for epochs when the background was enhanced by the presence of hot pixels induced by a high CCD temperature (T > −55 • ). For these time intervals, we checked whether there were some hot pixels in the extraction regions used to extract the spectra. Thus, we excluded the data from segments 00031287125 (0.8 ks) and 00031287129 (0.2 ks), because the data were severely contaminated by hot pixels due to a high (T > −51 • C) CCD temperature. We also excluded the first orbit (∼ 0.6 ks) of segment 00031287023 for the same reason. These latter bad data induce an artificially high count rate in the X-ray light-curve at MJD = 55209 (during the P1 outburst -see Fig. 1). However, the re-flare event seen around that time is not due to hot pixel contamination. We used the grade 0-12 events, giving slightly higher effective area at higher energies than the grade 0 events, and a 20 pixel (47.2 arcsec) radius circle to extract the source and background spectra using XSELECT v2.4b. The background extraction region was chosen in order to be close to the source extraction region and not to contain any XMM-Newton sources. The ancillary response files were generated using XRTMKARF v0.5.9 and exposure maps generated by XRTEXPOMAP v0.2.7. The response file SWXPC0TO12S6−20010101V012.RMF is used to fit the spectra. To extract the Swift-XRT spectra, we divided the data over different intensity ranges for the outbursts P0, P1 and P2. The work on the first three outbursts being completed before the most recent outburst P3, we decided to compare the latest outburst with the previous three by dividing the data over time instead. Table 4 summarises the nomenclature used for the Swift-XRT data.

Spectral analysis
All the spectra were grouped at a minimum of 20 counts per bin to provide sufficient statistics to use the χ 2 minimisation technique. The XMM3, Swift-XRT and Chandra spectra were fitted in the 0.3-10 keV energy range within XSPEC v12.7.0 (Arnaud 1996), while the XMM1 and XMM2 spectra were fitted in the 0.2-10 keV energy range. The Chandra data being moderately piled-up were fitted using the PILEUP model (Davis 2001) within XSPEC with a frame time fixed at 0.8 s (see Servillat et al. 2011 for more details). For each model, we determined the appropriate value of the grade morphing parameter. For the data used in this paper, the statistical errors are dominant over the instrumental systematics. The RGS1 and RGS2 spectra were binned at twice the resolution of the instrument (∆λ = 0.2Å). Since with this choice of binning there are fewer than 20 counts per resolution bin, we used the C-statistic (Cash 1979) available within XSPEC for the spectral fit.
The total column density was fixed at the bestconstrained value of N H = 4 × 10 20 cm −2 from the XMM2 observation (see Farrell et al. 2009). The Galactic absorption column in the direction of the source is equal to 1.8 × 10 20 cm −2 (e.g. Kalberla et al. 2005). For all fits, the absorption is modeled using the WABS photo-electric absorption model. The source redshift being 0.0224 (Wiersema et al. 2010), we adopted a source distance of d L = 95 Mpc using the cosmological parameters from the WMAP5 results (H 0 = 71 km s −1 Mpc −1 , Ω M = 0.27 & Ω Λ = 0.73). All the errors quoted below are given at a 90% confidence level for one parameter of interest (i.e. ∆χ 2 = 2.706).

Hardness-Intensity diagram
We fitted the Swift-XRT spectrum using a multi-color disk DISKBB and/or POWERLAW model. This phenomelogical model is often used to fit the spectra of GBHBs. In that context, the powerlaw component is often interpreted as emission produced by a corona of hot electrons located in the inner regions of the accretion disk, while the DISKBB component is interpreted as the emission from the accretion disk (see Remillard & McClintok 2006). The photon index of the powerlaw was tied together for the S CR0,1,2,3,4,5 spectra, because we found Γ-values consistent within the errors when fitting the spectra individually. The powerlaw normalisation and the parameters of the DISKBB model were left free to vary independently between spectra. The S t1,2 spectra are well fitted by an absorbed DISKBB model. We did not obtain a good fit (χ 2 /dof = 14.5/8) using an absorbed DISKBB model for the S t3 spectrum. There are residuals left at high energies. When adding a powerlaw component with Γ free, the Γ parameter was not constrained. So, we decided to freeze this parameter at the same value as found for the S CR0,1,2,3,4,5 spectra (Γ = 2.2). The fit is improved (∆χ 2 = 5 for 1 dof i.e. an improvement at a > 2 σ significance level). For each spectrum, we then computed the hardness ratio (HR) of the 0.3-1.5 keV observed flux over the 1.5-10 keV flux. This is different from the Swift hardness ratio presented in Godet et al. (2009) andServillat et al. (2011) that was based on count rate. The HR values, the best-fit spectral parameters and the 0.2-10 keV unabsorbed luminosity (L = 4π d 2 L F with F , the 0.2-10 keV unabsorbed flux) for each spectrum are summarised in Table 5. Fig. 2 shows the Swift-XRT hardness ratio versus the 0.2-10 keV unabsorbed luminosity. The points from the Chandra (kT = 0.21 ± 0.01 keV, N = 37 +11 −8 ), XMM2 (kT = 0.17 ± 0.01 keV, N = 40 +16 −10 , Γ = 2.1 ± 0.4, N Γ = 1.9 +0.8 −0.6 × 10 −5 ph cm −2 s −1 keV −1 ) and XMM3 1 (Γ = 2.2 +0.4 −0.6 and N Γ = 2.9 ± 0.7 × 10 −6 ph cm −2 s −1 keV −1 ) observations are also reported on the plot. Fig. 2 shows that we have a good agreement between the different instruments. The HLX-1 track in the Hardness-Intensity Diagram (HID) is clearly reminiscent of that observed in GBHBs. Between S 1p+2p and S af ter 2p , there is a softening of the source at nearly constant luminosity (∼ 1.2 × 10 42 erg s −1 ). Such a track in the HID is observed in GBHBs (see e.g. Meyer-Hofmeister et al. 2009). However, the situation appears to be different because the disk temperature near the peak of the outbursts P1, P2 and P3 varies at nearly constant luminosity (from 0.24 ± 0.02 keV to 0.18 ± 0.02 keV -see Table 5). Note that if we consider the 3σ errors on the temperature (i.e. ∆χ 2 = 9 for one parameter of interest), then we found that all data have consistent kT -values given the limited statistical quality of the X-ray data used. However, when merging the S 1p & S 2p spectra (S 1p+2p ) and the S t1 & S t2 spectra (S t1+t2 ), the kT -values (kT = 0.240 +0.023 −0.020 for S 1p+2p and kT = 0.20 +0.019 −0.017 for S t1+t2 ) are no longer consistent within the 3σ errors.

Investigating the L − T relation
Fig. 3 displays the disk temperature versus the unabsorbed 0.2-10 keV luminosity. We fitted our L−kT points using a L ∝ T a relation where a was left as a free parameter. We obtained a good fit with log(L) = (2.4±0.7) log T +43.7±0.5 (1 σ errors). Feng & Kaaret (2007) showed that the luminosity L varies as L ∝ T −3.1±0.5 in the ULX NGC 1313 X-2. Using a larger sample of ULXs Kajava & Poutanen (2008) found a L ∝ T −4 relation. King (2009) suggested that the L ∝ T −4 relation is expected in the super-Eddington regime provided that the geometrical beaming factor varies as b ∝ṁ −2 . From Fig. 3, the observed trend in the L − kT space does not follow such a correlation. Mizuno et al. (2001) found that some ULXs do 1 We also added a MEKAL component with kT = 0.44 +0.16 −0.12 keV to take a possible contamination from the galaxy into account (see Servillat et al. 2011). Only the powerlaw component was used to compute the hardness ratio. present a L ∝ T 2 relation. They interpreted such a correlation as a possible signature for moderate super-Eddington accretion (see also Fukue 2000;K03). However, we cannot exclude a T 4 relationship at the 3σ level (see also Servillat et al. 2011). In Fig. 3, we also show the best-fit obtained using the L ∝ T 4 relation.

Evidence for possible line features
When fitting the S 1p+2p spectrum, we found some strong residuals around 0.6 keV (see the top left panel in Fig. 4). There may be some lower significance residuals at higher energy as well. We separately investigated the S 1p and S 2p spectra, and both show some residuals around 0.6 keV in particular for S 2p . Given that the residuals are close to the instrumental oxygen edge, we investigated if the residual may have an instrumental origin. We extracted a background spectrum using different parts of the field of view and encompassing the time when the source was at the peak. The fit of the background spectrum does not reveal any residuals around the O edge. We checked the images and found no evidence of contamination by hot pixels. When the observations were performed the source location on the CCD was not close to the bad columns. The application of the charge trap correction did not change the residuals. So, the residuals are unlikely to be instrumental. We added a Gaussian line redshifted at the galaxy distance (z = 0.0224) with an intrinsic width equal to zero, because this parameter is otherwise not constrained. The fit was improved by ∆χ 2 = 12.7 for 2 d.o.f. The line centroid is equal to 0.618 +0.045 −0.042 keV and the equivalent width is EW = 93 +75 −51 eV (see the top right panel in Fig. 4). The disk parameters do not change with kT = 0.25±0.01 keV and N = 16 +6 −4 (see Table 5 for a comparison). In the rest of the paper, we did not include the Gaussian line when fitting the S 1p+2p spectrum, because it did not change the spectral parameters of the K03 disk model considered in Section 4 significantly.
In order to further investigate the soft X-ray emission at ∼ 0.6 keV, we inspected the RGS data available from the XMM2 observation. The RGS 1 & 2 spectra being background-dominated below 0.5 keV and above 1.3 keV, we only considered this energy range in our analysis. In the 0.5-1.3 keV energy band a total of 1230 and 1170 counts were collected in the RGS1 and RGS2, respectively (with 520 and 605 net counts). We first used an absorbed DISKBB plus a power-law component to fit the data fixing the photon index of the powerlaw component and the absorbing column density to the values derived from the XMM2 data ). We found a temperature (kT = 0.19 ± 0.02 keV) consistent with that derived from the EPIC data (see Table 5; see also Farrell et al. 2009. Although the fit is statistically acceptable (C-Statistic = 160.4 for 113 bins) a first inspection of the RGS1 & 2 residuals (see the bottom panel in Fig. 4) reveals that features are present at ∼ 0.6 keV, ∼ 0.9 keV and ∼ 1 keV.
We then added to the baseline continuum three unresolved Gaussian emission lines redshifted at the galaxy distance. The improvement in the fit due to the addition of the ∼ 0.6 keV emission line is ∆C/dof = 26.2/2 (corresponding to a detection significance > 99.9% confidence). The emission line is detected at an energy centroid of E = 0.642±0.002 keV, a flux of F = (1.91±0.6)×10 −5 photons cm −2 s −1 and an equivalent width of EW = 20±7 eV. The energy centroid is consistent with that derived from the Swift-XRT data, and the line equivalent width seems to decrease with the luminosity of the continuum. An additional line is detected at E = 0.880 +0.008 −0.002 keV (∆C/dof = 12.2/2), with a flux of F = (0.93 ± 0.6) × 10 −5 photons cm −2 s −1 (EW = 27 ± 12 eV). Finally, the addition of the third Gaussian line at E = 0.98 ± 0.01keV (EW = 31 ± 22 eV) gave a much lower statistical significance (∆C/dof = 7.0/2). Even if additional data are needed to further investigate the origin of these lines, we speculate that the most likely identification for the ∼ 0.6 keV and ∼ 0.9 keV lines is OVIII Lyα and FeXVIII-FeXIX. The third line, if any, might be associated with NeX Lyα.

A simple approach
In order to investigate whether the L X ∝ T ∼2.4 relation we found in Section 3.2 could be the result of a slim disk surrounding the BH, we first used the DISKPBB model instead of the DISKBB model. The DISKPBB model is a multi-temperature blackbody disk model where local disk temperature is given by T (r) ∝ r −p with p, a free parameter (e.g. Mineshige et al. 1994, Hirano et al. 1995, Watarai et al. 2000. When p = 0.75, the model is equivalent to a DISKBB model. For p < 0.75, radial advection starts to become important (e.g. Fukue 2000). Since the spectral shape of the model could change significantly with the p-value, we fitted the XMM1 and XMM2 spectra as well as those from Swift and Chandra. The XMM3 spectrum was not considered because any evidence for a disk component is only marginal (see Servillat et al. 2011). The results are summarised in Table 6. In most cases, the p-values are not well constrained. So, we cannot tell from this model whether or not radial advection plays a role.

Description of the model
Kawaguchi (2003) computed disk spectra in a self-consistent way, taking into account the effects of electron scattering (opacity and disk Comptonization) and the effects of the relativistic correction (i.e. gravitational redshift and transverse Doppler shift) on the disk effective temperature in the inner part of the accretion disk. Doppler boosting being not considered implies that disk solutions are mostly seen face-on (i = 0 • ). The model table of the disk spectra 2 , from sub-to super-Eddington accretion rates with BH masses of 1 − 10 3 M ⊙ , was successfully used for analysis of several ULXs (e.g. Foschini et al. 2006, Vierdayanti et al. 2006, Okajima et al. 2006, Yoshida et al. 2010. We used here a new table including additional computation for higher BH masses, extending up to 10 5 M ⊙ . The key parameters of the model are the BH mass (M ), the accretion rate (Ṁ ) and the viscosity parameter (α). For sub-Eddington accretion rates (i.e. in the case of the standard disk), the emergent disk spectra are αinsensitive. Near and at super-Eddington rates, the disk spectra however become α-sensitive, because electron scattering, which is density sensitive (and density is α-sensitive), begins to affect the emergent spectra quite strongly. The normalisation is fixed using the source distance (d = 95 Mpc). To perform the fits, we considered the model option 7 that takes into account the effects of advection, electron scattering on opacity, Comptonization and relativistic effects (see the appendix A.1 for more details about the influence of the different model options on the spectral parameters from HLX-1 data). It is important to keep in mind that the K03 code automatically computes at each radius how much advection is present for a given accretion rate. All the models are computed assuming a non-rotating BH.

Results
To avoid calibration uncertainties between instruments affecting the fits (see Tsujimoto et al. 2011), we decided to fit the spectra from XMM-Newton, Chandra and Swift separately.
We first fitted the Swift-XRT spectra S 1p+2p , S af ter 2p , S CR0 and S CR3 by tying together the BH mass between them. We proceeded in the same way with the viscosity parameter α, while the accretion rate was left as an independent parameter between the different spectra. For the S CR0 and S CR3 spectra, we added a powerlaw component of which the photon index was tied together between the two spectra. This is because we found consistent values when fitting individually the two spectra. When we fixed the α-value to the default K03 model (α = 0.01) we did not obtain a good fit because some spectra display strong residuals. In order to reproduce the spectra, it was necessary to leave the viscosity parameter α free. We then obtained a good fit (χ 2 /d.o.f. = 108/104). The best-fit parameters are given in Table 7. The derived BH mass is M = 1.8 +0.2 −0.1 × 10 4 M ⊙ . Fitting the data from the outburst P3 using the K03 disk model, we found M = 1.9 +1.9 −0.2 × 10 4 M ⊙ . These estimates are consistent and well inside the IMBH mass range. We note that the powerlaw component was not needed to obtain a good fit for the S t3 spectrum when leaving the viscosity parameter free to vary. Leaving the N H -value free to investigate the sensitivity of the BH mass with respect to the N H value did not change the spectral parameters much and they are still consistent within the errors with the values derived with N H = 4 × 10 20 cm −2 . Indeed, we found N H = 6.5 +3.8 −2.5 × 10 20 cm −2 and M = 2.8 +1.7 −0.9 × 10 4 M ⊙ from the outbursts P0-P2, while we found N H = 4.1 +2.8 −2.7 × 10 20 cm −2 and M = 1.9 +3.2 −0.6 × 10 4 M ⊙ from the outburst P3. Second, we fitted the Chandra spectrum. For the K03 disk model, we estimated for the PILEUP model a grade morphing parameter of 0.33. Note that this parameter does not have a strong impact on the spectral parameters derived here. The derived BH mass is M = 1.9 ± 0.2 × 10 4 M ⊙ . We then left the N H -value free and we found N H = 3.7 +3.9 −2.5 × 10 20 cm −2 and a BH mass estimate (M = 1.8 +1.0 −0.4 × 10 4 M ⊙ ) consistent with those derived from Swift.
Even if Farrell et al. (2009) showed that the XMM1 spectrum is well fitted by a steep powerlaw (Γ ∼ 3.4) suggesting that the source was in the steep-powerlaw state as seen in some GB-HBs, we decided to fit the XMM1 spectrum as well as the XMM2 one with the K03 disk model. This is because the emergent spectrum from the K03 disk model can have a shape strongly different from that of a simple MCD spectrum (see Appendix A.1). We tied the BH mass between the two spectra, but we left α andṀ to vary independently. First, we keep N H fixed at 4 × 10 20 cm −2 . For XMM2, it was necessary to add a powerlaw component to obtain a good fit, while the addition of a powerlaw for XMM1 did not improve the fit. The derived BH mass is then M = 1.4 ± 0.1 × 10 4 M ⊙ ; which is not consistent with the Swift-XRT and Chandra estimates within the 90% errors. If we consider the 3σ errors, then the estimates from the different instruments are all consistent within the errors. Then, in order to investigate the sensitivity of the BH mass with respect to the N H value, we left free the N H parameter. We obtained a value of N H = 5.5 +1.1 −1.0 × 10 20 cm −2 and a BH mass of M = 1.9 +1.3 −0.3 × 10 4 M ⊙ . The photon index of the powerlaw component in both cases is consistent within the 90% errors between the XMM-Newton and Swift-XRT spectra. The bestfit results are given in Table 7.
For XMM1, the accretion rate (ṁ ∼ 4.4) and the luminosity that is very different from that derived using a powerlaw (L ∼ 1.3×10 42 erg cm −2 , a luminosity which was observed at the peak of the three outbursts) are commensurate with the values found for the XMM2 and S CR0 spectra (see Table 7). This however implies a large value of α = 0.13 ± 0.07, even if all α-values are consistent within the 3σ errors. If we force the viscosity parameter to the lowest possible value of the K03 model (α = 0.01) keeping the mass fixed at M = 1.8 ×10 4 M ⊙ and N H free, we did not obtain a good fit (χ 2 /dof = 47.3/24). The addition of a powerlaw improves the fit (χ 2 /dof = 23.3/22), but in this case the powerlaw component dominates over the disk component. In this case, we found similar Γ and N H values to those given in Farrell et al. (2009).

Constraints on the BH mass
To be able to derive a dynamical measurement of the BH mass in HLX-1 will be very challenging given the distance of the source. So, we have to rely on indirect estimates such as spectral fitting of the X-ray data. Zampieri & Roberts (2009) showed that the BH mass estimate derived using spectral fitting can be highly variable depending on the disk model used (see their Table  2). In the case of HLX-1, Davis et al. (2011) used their advanced relativistic disk model to fit spectra with different shapes. They found a BH mass within the IMBH range (from 3000 M ⊙ to 3 × 10 5 M ⊙ ) with extreme and opposite assumptions. In this paper, we used the K03 disk model to fit X-ray spectra of HLX-1 when the source was in various spectral states. We note that this model includes spectra for a wide range of accretion rates from sub-to super-Eddington rates. In all cases for ULXs, the fits favoured super-or near-Eddington accretion onto a stellar mass BH. However, the fits to HLX-1 spectra do favour an IMBH solution with a BH mass of M = 1.8 +1.6 −0.5 × 10 4 M ⊙ . This is an interesting result because we get three independent and nonetheless consistent mass estimates from three different instruments (Swift-XRT, XMM-Newton EPIC-pn and Chandra/ACIS) and over different spectral shapes. This estimate is consistent with the observational lower and upper limits on the BH mass . So, using the K03 model which assumes a nonspinning BH and a face-on accretion disk, we find that an IMBH solution is favoured with an estimated mass of ∼ 1.8 × 10 4 M ⊙ . The BH accretes at the Eddington limit (ṁ ∼ 10) and radiates close to the Eddington luminosity at the outburst peak (L Edd = 2.3 +2.0 −0.6 × 10 42 erg s −1 ). This corresponds to an Eddington fraction of f Edd = 1.1 +0.6 −0.5 considering a bolometric luminosity of 2.5 × 10 42 erg s −1 on average (see Fig. 5). Davis et al. (2011) Using their BHSPEC model, Davis et al. (2011) studied the dependency of the BH mass in HLX-1 with the inclination and the BH spin. We can only compare our results with their work for a * = 0 and i = 0 • . In that configuration, they found a consistent mass estimate for XMM2 and Chandra with log M ∼ 3.8 ± 0.1 (Davis 2011, private communication). However, no solution could be found in this case for the Swift spectra, because the f Edd = L X /L Edd ratio was pegged to the maximum permitted value in their BHSPEC model (i.e. f Edd = 1, Davis 2011, private communication). From Figures 2, 5 and 6 in Davis et al. (2011), the best-fit contours for fixed inclination in the a * versus log M plots overlap well between the simulations for the range of allowed parameters. Thus, they found a consistent mass estimate between the three instruments for i = 0 • & a * = 0.7 with log M ∼ 4.0 for Swift & Chandra and log M ∼ 3.9 for XMM2. The mass values derived by Davis et al. (2011) from the XMM2 and Chandra spectra for a * = 0 & i = 0 • are smaller than the one derived using the K03 disk model. We note that the Eddington ratios (f Edd ) they obtained are large (and close to 1). From our fitting results using the K03 model, we found similar high f Edd values near the outburst peak. We believe that for such high f Edd values the effects of advection on the emergent spectra should be considered.

Comparison with the results from
Evaluating the level of systematic errors of the K03 model by comparison with alternative models by exploring the whole parameter space is beyond the scope of the present paper and deserves a dedicated paper. Nevertheless, we estimated the level of systematic errors of the K03 model by comparison with the KERRBB (Li et al. 2005) and BHSPEC models assuming a non-spinning BH (a * = 0) and a face-on disk (i = 0 • ) in the limit of low mass accretion rate (i.e. a few percent of the Eddington limit). In that limit, the accretion disk is expected to be very close to the relativistic model of Novikov & Thorne (1973). We found that the level of systematic in the BH mass between the K03 and KERRBB models is less than 45% when only considering the relativistic effects in both models (see Appendix A.2). When comparing the results between the K03 and BHSPEC models taking into account the relativistic effects and the effects due to Comptonization and electron scattering, we found that the level of systematic errors in the BH mass is less than 24% (see Appendix A.2). In all cases, the values of the accretion rate derived using the K03, KERRBB and BHSPEC models agree within the errors. So, providing similar level of systematics errors over the whole parameter space, we are confident that our BH mass and accretion rate estimates are reliable. Fig. 5 shows a linear correlation between the 0.01-20 keV bolometric disk luminosity (L) and the accretion rate so that L = ηṁ L Edd with radiation efficiency η = 0.11 ± 0.03. Watarai et al. (2000) found a similar linear correlation forṁ < 20. This implies that advection does not play a significant role, and that the slim disk solutions are not needed. At all times during the outbursts, the disk aspect ratio H/R is less than 0.2, from Fig. 5 in K03. Note that H/R is mostly independent from the BH mass and the viscosity parameter.

The disk structure
From Fig. 1 in K03, the disk appears to be radiation-pressure dominated in its inner regions for all theṁ-values found. Such disks might be viscously and thermally unstable, and they might give rise to outbursts (Taam & Lin 1984;Honma et al. 1991, Lasota & Pelat 1991Xue et al. 2011). However, Hirose et al. (2009) showed via MHD simulations that such disks could be thermally stable (but see Xue et al. 2011). Most observations of Galactic X-ray binaries in outburst radiating above 10% of the Eddington luminosity for a stellar mass BH do not show any evidence for such an instability except maybe for GRS 1915+105 (Belloni et al. 1997;Xue et al. 2011). Fig. 6 shows the evolution of the accretion rate through the different outbursts. At the peak of the three outbursts and for a few weeks (∼ 3 and ∼ 2 − 3 weeks for the outbursts P2 and P3, respectively), the BH accretes at the Eddington limit (ṁ ∼ 10). This "plateau" is seen in the X-ray light-curve of the outbursts P2 and P3 (see the right panel in Fig. 1). Recently, Lasota et al. (2011) proposed that the HLX-1 outbursts may be due to enhanced mass-transfer rate onto a pre-existing accretion disk when an asymptotic giant branch star orbiting along an eccentric orbit with a period of ∼ 1 year passing at periastron is tidally stripped. Such a mechanism is known to produce "plateaus" in light-curves (Bath & Pringle 1981;Esin, Lasota & Hynes 2000). The latter authors considered an unstable disk that became stable after an enhanced mass-transfer rate phase. In the case of HLX-1, the situation is different since the disk is stable. Therefore, quasi-constant enhanced mass-transfer rate during a given time interval would naturally produce a "plateau" in the light-curve. After the "plateau", the accretion rate drops exponentially with a decay time of 182 ± 30 days and 92 ± 15 days for the outbursts P1 and P2, respectively. The difference in the decay time is because there was a small re-flare event at the end of the outburst P1 (see Fig. 1). From the evolution of the accretion rate over time, we calculated the mass accreted during the outbursts P1 and P2 assuming that the accretion rate is the same throughout the disk. If it is not the case, then the values derived below are lower limits. We found an accreted mass of ∼ 1.2 × 10 −4 M ⊙ and ∼ 8.2 × 10 −5 M ⊙ for the outbursts P1 and P2, respectively.

The evolution of the accretion rate
At the end of the outbursts P1 and P2 i.e. when the source undergoes a transition from the high/soft state to the low/hard state, the count rate drops very quickly over a timescale less than a month (see Fig. 1). As shown by Servillat et al. (2011), any disk component in the spectrum of HLX-1 in the low/hard state is marginal. Given the XMM3 luminosity of ∼ 2 × 10 40 erg s −1 derived from the powerlaw component and assuming a radiation efficiency in the low/hard state of 0.11 as during the outbursts, we could compute an upper limit on the accretion rate in the low/hard state ofṁ < 0.09 i.e.Ṁ < 3.6 × 10 −6 M ⊙ yr −1 . This is nicely consistent with the results obtained by Esin et al. (1997Esin et al. ( , 1998 using the Advection Dominated Accretion Flow model. According to that model, this must be connected with a varying (receding) inner disc radius (e.g. Dubus et al. 2001). Higher statistical quality of the data when the source transits to the low/hard state will be needed to further investigate that possibility.

The L − T relation
In the luminosity-temperature space, we found a correlation L ∝ T ∼2.4 . Fukue (2000) and Kawaguchi (2003) showed that near the Eddington limit (ṁ = 10) and above a departure from the L − T 4 relation is expected, due to the change of the radial profile of the disk temperature and thė m-sensitive disk colour temperature via the effects of electron scattering. However below this limit, the disk is still expected to follow the L ∝ T 4 relation found in the standard disk model. Apart from the peak of the outbursts when the accretion rate is at the Eddington limit, the accretion rate is below this limit. Therefore, we should expect to have a L ∝ T 4 correlation. The behaviour near the peak is probably responsible for the flatter correlation. To check that, we fitted our L−kT points for the XM M 2, S CR1 , S CR2 , S CR3 , S t3 observations (for whichṁ < 10 -see Table 7) using the relation for a standard disk. We found: log(L) = 4 log T + 44.95 ± 0.06 (1 σ error). From Eq. 1 in Lasota et al. (2011) giving L as a function of the colour disk temperature at the inner disk radius and the BH mass, we derived the following relation: log(L) ∼ 4 log T + 45.1 using η ∼ 0.11, M = 1.8 × 10 4 M ⊙ and R ISCO = 3 R S (for a non-rotating BH). This demonstrates that the disk in HLX-1 follows the L ∝ T 4 relation.

Conclusion
We investigated in detail the X-ray spectral properties of the best IMBH candidate, HLX-1, using multi-epoch data collected by three different X-ray instruments (Swift-XRT, XMM-Newton EPIC-pn and Chandra). To do so, we used the K03 disk model in order to constrain the BH mass, the accretion rate and the disk structure in a non biased way. Indeed, this disk model has the advantage of taking into account the effects of radial advection and covering a wide range of accretion rates (from sub-to super-Eddington) assuming a non-rotating BH and a face-on disk inclination (i = 0 • ). We found that our multi-epoch data are consistent with sub-Eddington accretion onto a nearly 2 × 10 4 M ⊙ BH. At the peak, the X-ray luminosity is near the Eddington luminosity. The derived radiation efficiency is η ∼ 11%. The disk solution we found for HLX-1 is different from those derived for other ULXs using the same disk model (e.g. Vierdayanti et al. 2006, Okajima et al. 2006, Yoshida et al. 2010). Indeed, for other ULXs the spectral fitting favours super-or near Eddington accretion onto a stellar mass BH. Here, the disk in HLX-1 likely undergoes sub-and near Eddington accretion onto an IMBH and appears to stay relatively thin. The inner regions of the accretion disk are dominated by radiation-pressure. We also showed that the source globally follows the L ∝ T 4 relation. At the outburst peak and for a further few days, the BH accretes at near the Eddington limit (ṁ ∼ 10 i.e. ∼ 4 × 10 −4 M ⊙ yr −1 ). The occurence of this "plateau" at the outburst peaks could provide some evidence that the outburst mechanism is driven by enhanced mass transfer rate onto a pre-existing accretion disk as proposed by Lasota et al. (2011). After the "plateau", the accretion rate decreases exponentially until the end of the outburst. However, it is not clear from our data what is the geometry of the accretion flow when the source transits to the low/hard state, even if the presence of an optically thin advection dominated flow would avoid a dramatic variation in the accretion rate by more than two orders of magnitude from the peak to the low/hard state. Finally, the recent outburst starting in August 2011 gives more credit to the ∼ 1 year recurrence timescale proposed by Lasota et al. (2011). Our dedicated multi-wavelength (radio, optical and X-ray) observations of this outburst will shed further light on the nature of this unique source as well as on the outburst mechanism.

Acknowledgments
We thank the anonymous referee for his useful comments that helped to improve the paper. We are grateful to Ken Ebisawa and Takashi Okajima for creating a new fits model for XSPEC incorporating the extended black hole mass ranges. We thank Shane Davis for his useful suggestions during the writing of this paper. MS acknowledges supports from NASA/Chandra grants GO0-11063X, DD0-11050X and NSF grant AST-0909073. SAF acknowledges funding from the UK Science and Technology Funding Council. SAF is the recipient of an Australian Research Council Postdoctoral Fellowship, funded by grant DP110102889. VB acknowledge support from the UK STFC research council. JPL acknowledges support from the French Space Agency CNES.

A. Appendix material
A.1. Impact of the K03 disk model computing options on the shape of the emergent spectra, the BH mass and the accretion rate The K03 model makes use of different options to compute the emergent spectra starting from a local BB and then including different physical effects (advection, electron opacity, relativistic effects and Comptonization). First, we investigate the impact of these different options on the shape of the emergent spectra as well as on the M andṀ parameters. We take the example of the S 1p+2p spectrum as an illustration, but what is described below is also valid for the other spectra. Fig. 7 shows the modification of the S 1p+2p spectrum when selecting different model options: 1) standard disk & local BB; 2) slim disk & local BB; 5) slim disk including relativistic effects on the BB; 6) slim disk including the effects of electron scattering on opacity and relativistic effects; 7) slim disk including the effects of electron scattering on opacity, Comptonization and relativistic effects. Here, the term "slim disk" implies that the K03 code automatically computes (at each radius) how much advection is present for a given accretion rate.
First, we just changed the option parameters without running a new fit for option 1 in order to see the effects of the different model components. When using option 1, we got for the S 1p+2p spectrum a BH mass around 1.8 × 10 3 M ⊙ andṁ ∼ 32 assuming a viscosity parameter of α = 0.01. Such highṁ-values mean super-Eddington accretion. However, in this case, radial advection is expected to start playing a significant role. The large derivedṀ -value also implies that electron opacity and Comptonisation should play an important role (see Fig.2 in Kawaguchi 2003 for instance). This results in more spectral boosting towards high energies. The spectrum in red in Fig. 7 shows the global shape of the data. In order to get such a shape when including all the above effects in a slim disk (option 7), it is clear that the effects of both electron scattering on opacity and Comptonisation have to be reduced. To do so, the accretion rate (in the unit of the Eddington rate) has to decrease; which in turn results in an increase of the BH mass.
To check this, we fitted the S 1p+2p spectrum for each chosen option leaving free the model parameters. The evolution of the model parameters are summarised in Table 8. From option 1 to option 2, there is a relatively small variation of the M andṁ parameters (M ∼ 3.4 × 10 3 M ⊙ andṁ ∼ 38). From option 2 to option 5 (including the relativistic effects), we obtained a significant impact on the accretion rate (ṁ ∼ 200) and a slight decrease of the BH mass (M ∼ 2.3 × 10 3 M ⊙ ). Given the large accretion rate (ṁ ∼ 38) derived in option 2, the accretion disk extends below the last stable orbit (e.g. Watarai et al. 2000). The relativistic effects (gravitational redshift and transverse Doppler redshift) when applying option 5 strongly suppress the emission coming from these regions; which in turn results in the model underestimating the data. To have a good match between the data and the model, the accretion rate has to be increased. From option 5 to option 6, the spectral parameters significantly change with an increase in the BH mass by one order of magnitude and a decrease of the accretion rate by more than one order of magnitude. From option 6 to option 7, the parameters do not change significantly (M ∼ 1.8 × 10 4 M ⊙ andṁ ∼ 10).
From option 5, the accretion flow is in the super-Eddington regime. The effects of Comptonisation and electron scattering (option 6) are likely to be significant given the largeṁ value (see Fig. 2 in K03). This would in turn result in a stronger hardening of the model than what is seen in Fig. 7. However, in order to obtain a good fit given the shape of the S 1p+2p spectrum, the effects of both Comptonisation and electron scattering has to be strongly minimised. This is achieved by significantly decreasing the accretion rate (the accretion flow is then in the sub-Eddington regime) and therefore by significantly increasing the BH mass. If we force the source distance from 95 Mpc (measured distance) to an arbitrary lower value (e.g. d = 3.5 Mpc), such a dramatic change in the BH mass and the accretion rate is not seen because the accretion flow always stays in the sub-Eddington regime (see Table 9).

A.2. Comparison of the K03, KERRBB & BHSPEC results in the limit of the Novikov & Thorne (1973) relativistic disk model
We compare the results from the K03 disk model with those obtained with KERRBB and BHSPEC assuming a non-spinning BH (a * = 0) and a face-on disk (i = 0 • ) in the limit of low mass accretion rate (i.e. less than 10% of the Eddington limit). In that limit, the accretion is expected to follow the predictions of the relativistic disk model from Novikov & Thorne (1973). In the K03 model, this limit corresponds tȯ m ≤ 1.6 L Edd /c 2 . For the comparisons with KERRBB, we limit ourselves by setting the hardening factor to 1 in the model (i.e. we did not take the effects of electron scattering and Comptonisation into account). We also did not consider the effects of limb-darkening and self-irradiation since they are not included in the K03 disk model. This configuration corresponds to option 5 in the K03 disk model (only including the relativistic effects). For the comparisons with BHSPEC, we used the option 7 (including the effects for Comptonisation and electron scattering) in the K03 disk model. We used the table of BHSPEC models bhspec − mass − 0.01.fits 3 considering BH mass up to 300 M ⊙ .
To perform the comparisons, we proceeded in both cases as follows: • we fitted the S 1p+2p spectrum (i.e. when the source luminosity is maximum) using the WABS*KERRBB and WABS*BHSPEC model with a * = 0, i = 0 • and N H = 4 × 10 20 cm −2 for different and arbitrary values of the source distance D so that theṁ values derived were below 10% of the Eddington limit.
• from the best-fits, we simulated a model spectrum with an exposure time of 38 ks using the Swift-XRT response files and the fakeit command in Xspec.
• we fitted the faked spectra with the WABS*K03 model with alpha fixed to 0.01, the same source distance D as used for the KERRBB and BHSPEC fits and the option 5 and 7 when comparing the results with KERRBB and BHSPEC, respectively.
Tables 1 and 2 summarise the results of the comparisons with KERRBB and BHSPEC, respectively. From Table 1, we show that there is a systematic discrepancy (less than 45% in the range considered) in the BH mass between the two models, while the accretion rate values appear to be consistent within the errors. From Table 2, we obtained a good agreement between the BHSPEC and K03 models within the 3 σ errors except for the BH mass estimate for a distance of 2.9 Mpc. Based on that latter case, the "systematic errors" in the BH mass are less than 24%. We also show in Tables 1 and 2 comparisons withṁ values larger than 10% of the Eddington limit. Again, we emphasize the good agreement between the K03-KERRBB models and K03-BHSPEC models. So, we conclude that the K03 disk model provides reliable estimates of accretion disk parameters (mass and accretion rate). (1.08 +0.04 −0.03 ) • The effects of Comptonisation, electron scattering, limb-darkening and self-irradiation were not taking into account.
△ We used the option 5 in the K03 model that only includes the relativistic effects. † Accretion rate value derived from the K03 disk model. * Consistent at the 3 σ level. △ We used the option 7 in the K03 model that includes the relativistic effects as well as the effects of Comptonisation and electron scattering.
• Accretion rate value derived from the K03 disk model. * Consistent at the 3 σ level. † Marginally consistent at 3 σ. ‡ Lower error bar pegged to the lowest permitted value of the model (ṁ = 1).