RELATIVISTIC LINES AND REFLECTION FROM THE INNER ACCRETION DISKS AROUND NEUTRON STARS

The data reduction was performed using HEASOFT v6.6.2, and the latest calibration files (as of 2009 June). Suzaku consists of both a set of soft X-ray CCD detectors (XIS) as well as a separate hard X-ray detector (HXD). Details of the data reduction for the XIS and HXD are given below. For all sources we adopted the following standard data reduction method, except where explicitly stated. Since our original analysis of Ser X-1, GX 349+2, and 4U 1820 − 30 (Cackett et al. 2008), there has been significant advances in the analysis tools for Suzaku, for instance, one can now generate response files for any source region (allowing for pile-up correction).

3.1.1. XIS Data Reduction

3.1.2. HXD/PIN Data Reduction

We processed the observation data files (ODFs) for each observation using the XMM-Newton Science Analysis Software (v8). For the analysis here we only use timing mode data from the PN camera as it is the appropriate observing mode for the sources of interest. Calibrated event lists were created from the ODFs using the PN processing tool epproc. Exceptions to this procedure are the observations of SAX J1808.4 − 3658 and GX 340+0. We obtained the calibrated event lists for these observations directly from the XMM-Newton Science Operations Center.

Before extracting spectra, we checked for periods of high background by extracting a light curve from an off-source strip with energy >10 keV. We note where we found high background levels below. Using only times of low background, we extracted spectra with the following criteria: quality flag = 0, pattern ⩽4 (singles and doubles) and in the energy range 0.3–12 keV. The source extraction regions used were 24 pixels wide (in RAWX), covered all RAWY values and were centered on the brightest RAWX column. In a few cases, where the source was particularly bright, we had to exclude the brightest two RAWX columns (details noted below). The corresponding rmf and arf were created using the rmfgen and arfgen tools following the Scientific Analysis System (SAS) procedures.

Observation-specific details follow.

Ser X-1. During the first observation, there is only a slightly high background at the beginning, which we choose to include. There is no background flaring in the other two observations.

4U 1636 − 53. There were three X-ray bursts during the first observation, one during the second observation, and one during the third observation, all of which were excluded. Only during the second observation there was some background flaring. Removing this reduced the exposure time by 3.5 ks.

4U 1705 − 44. One type-I X-ray burst occurred during this observation, and was excluded. We find no background flaring. Note that this is not the data set presented in Di Salvo et al. (2009c), which is still proprietary at the time of writing. Other archival XMM-Newton observations of this source were observed in imaging mode, and so are not analyzed here.

GX 340+0. Some low-level background flaring that reduces the exposure time by 7 ks. The source is particularly bright during this observation (see light curve in D'Aì et al. 2009); we therefore remove the two brightest columns from the extract region. Given that D'Aì et al. (2009) demonstrate that the Fe K line properties do not vary significantly with source state, we chose not to split the observation based on source state, but instead prefer to analyze the entire time-averaged spectrum.

GX 349+2. The source count rate is high during this observation (Iaria et al. 2009). Following Iaria et al. (2009), we therefore exclude the two brightest columns from the extraction region. There is no significant background flaring. Another archival observation of this source is also available, however, it was observed in imaging mode, and thus is not analyzed here.

SAX J1808.4 − 3658. We use the spectra from Cackett et al. (2009a) and do not reprocess the data here.

Our first step is to fit the spectra with phenomenological models (throughout the paper we use XSPEC v12; Arnaud 1996). There has been debate for many years as to which is the most appropriate continuum model to fit to neutron star LMXBs (e.g., White et al. 1988; Mitsuda et al. 1989), as it is often the case that several different models fit the data equally well (e.g., Barret 2001). A recent investigation by Lin et al. (2007) studied two atoll sources over a wide range in luminosity. They investigated the luminosity dependence of measured temperatures for different continuum models, looking to see for which model choice the measured temperatures followed L ∝ T⁴. Based on this, they suggested a prescription for spectral fitting, using an absorbed disk blackbody (xspec model: diskbb; Arnaud 1996) plus single temperature blackbody plus a (broken) power law for the soft and intermediate states, and a blackbody plus broken power law for the hardest states. We have also found this model to fit the spectra of neutron star LMXBs well (Cackett et al. 2008, 2009a, 2009b). Here, we adopt this same model (though we use a power law, rather than a broken power law), testing that the addition of each component statistically improves the fit.

We interpret the hot blackbody component as the boundary layer between the accretion disk and neutron star surface (e.g., Inogamov & Sunyaev 1999; Popham & Sunyaev 2001). One can also successfully model this component with a Comptonized model, such as comptt. When doing this, however, one gets high optical depths—the Comptonized component is close to a Wien spectrum. Rather than fitting the more complicated Comptonization model, we chose the simpler blackbody model.

The millisecond X-ray pulsar, HETE J1900.1 − 2455, has a much harder spectrum than the other sources we study here. In fact, we find that its broadband continuum spectrum can be well fit by a single power law only.

We found that several of the sources have an apparent emission feature at around 1 keV, as has previously been observed in a number of neutron star LMXBs, with a number of different missions (e.g., Vrtilek et al. 1988; Kuulkers et al. 1997; Schulz 1999). It has suggested that this feature is a blend of a number of emission lines from Fe and O (Vrtilek et al. 1988). Here, if the feature is present, we fit with a single Gaussian emission line.

The presence of a broad Fe K emission line in eight of these sources has already been established elsewhere (see references earlier). Of the other two sources (GX 17+2 and HETE J1900.1 − 2455) a Gaussian Fe K emission has previously been reported in GX 17+2 (Di Salvo et al. 2000b; Farinelli et al. 2005; Cackett et al. 2009b) and the fact that a broad red wing has not been previously observed is likely due to sensitivity (Cackett et al. 2009b). Here we show that the profile is asymmetric. In HETE J1900.1 − 2455, we detect a broad Fe K emission line for the first time, though the line can be fit equally well by a broad Gaussian or a relativistic line due to the low signal-to-noise ratio. Therefore, all 10 objects studied here are seen to have broad Fe K lines.

To fit these emission lines, we use the relativistic line model for a Schwarzschild metric (diskline; Fabian et al. 1989). The Schwarzschild metric should be a good approximation for neutron stars—for neutron stars with reasonable equations of state and spin frequencies less than 600 Hz, the dimensionless angular momentum parameter is expected to be less than 0.3; thus any deviations from the Schwarzschild metric are minor (see Miller et al. 1998 for further discussion of this). We note that the diskline profile does not account for light bending, and so does not fully account for all relativistic effects. However, it still remains the most appropriate available model here.

We include this model in addition to the continuum model discussed above. The emission-line parameters are the line energy, disk emissivity index, inner disk radius, outer disk radius, and inclination. We constrain the line energy to be within 6.4–6.97 keV, the allowed range for Fe Kα emission. The disk emissivity index and inner disk radius are left as free parameters in the fits. In most cases, the inclination is left as a free parameter (we discuss the exceptions to this later). Finally, we fix the outer disk radius at 1000 GM/c². Typical disk emissivities mean that the Fe K emission is quite centrally concentrated, and hence the outer disk radius is usually not constrained well and has little effect on the measured inner disk radius.

Several of the sources considered here have multiple observations by the same, or different telescopes. The inclination of the source should not change over time; thus, we should expect to recover the same inclination from the Fe K line modeling for each observation. In order to investigate the extent to which this is true, we chose to let the inclination parameter vary from observation to observation.

For the Suzaku spectra, we fit the XIS spectra over the energy range 1.0–10 keV, ignoring 1.5–2.5 keV where large residuals are always seen (regardless of the count rate), presumably of an instrumental nature. The PIN spectra are fit from 12 keV upward, with the upper energy depending on where the background dominates over the source spectrum. The combined front-illuminated spectrum, XIS 1 spectrum, and HXD/PIN spectra are all fit simultaneously, with a floating constant used to account for any absolute flux calibration offset between them. When fitting the XMM-Newton PN spectra, we fit over the 0.8–10 keV range. Source specific details including exceptions to this standard procedure follow.

Ser X-1. Suzaku/PIN spectrum fit from 12 to 25 keV. In the XMM-Newton spectra, there are significant residuals around 1 keV that are fit well by a Gaussian emission line.

4U 1636 − 53. The Fe line profile observed in this source (Pandel et al. 2008) is unlike what is seen in any of the other sources. It has a blue wing that extends to high energies, suggestive of a high inclination. Our spectral fitting does give a best fit with a high inclination (we find that all cases peg at 90°) and Pandel et al. (2008) also reported high inclinations (greater than 64° in all cases). However, no eclipses have been detected in this source and optical measurements are not especially constraining (36°–74°; Casares et al. 2006).

4U 1705 − 44. Suzaku/PIN spectrum fit from 12 to 30 keV. The first Suzaku observation has quite low sensitivity through the Fe K band (see Reis et al. 2009b). We therefore fix several of the line parameters.

When fitting the XMM-Newton data of 4U 1705 − 44, we fit in the range 1.2–10 keV, as a relatively high N_H and low source flux lead to low statistics below 1.2 keV. We found that both the diskbb + bbody and the bbody + power-law model fit the continuum spectra well, with the diskbb + bbody model having a marginally better χ² value. The spectrum is of fairly low quality through the Fe K region, and thus a simple Gaussian models the Fe K line well. However, to allow comparison with other observations we still fit the line with the diskline model. When including the diskline model both choices of the continuum model give consistent Fe K parameters, with almost identical χ² values: 1762.4 versus 1762.5 (dof = 1752 in both cases) for diskbb + bbody and bbody + power-law continuum choices, respectively.

An unphysically high inclination is determined from the XMM-Newton observation. This source is not known to be eclipsing or dipping. There is a different inclination determined in the Suzaku observations (Reis et al. 2009b) than from Di Salvo et al. (2009c). We revisit this issue later when discussing systematic uncertainties in Fe K modeling.

4U 1820 − 30. Suzaku/PIN spectrum fit from 12 to 25 keV. With the improved calibration, the combined front-illuminated spectrum has a significantly higher signal-to-noise ratio than the spectra presented by Cackett et al. (2008). With this improved spectrum, the detected emission line is found to be weaker than before, but still a significant detection: with the emission line the best fit has χ² = 1269 for 1108 degrees of freedom (dof), whereas with no emission line the best fit has χ² = 1362 for 1113 dof.

GX 17+2. Suzaku/PIN spectrum fit from 12 to 40 keV.

GX 340+0. This source is hampered by a very large column density (N_H ∼ 10²³ cm⁻²). This means that even the neutral Fe K edge at ∼7 keV is significant. Unfortunately, a large neutral Fe K edge can be a big problem when fitting the Fe K emission line. If there is any small change in ISM Fe abundance toward the source, it will change the depth of the edge, and consequently the appearance of the emission line. Additionally, the edge is likely to be more complicated than the simple step-like function used in absorption models. Detailed structure has been observed for Fe L, and several Ne and O edges with sensitive grating spectra (e.g., Juett et al. 2004, 2006). Thus, with a large N_H, structure in the edge might affect results.

When fitting these data, D'Aì et al. (2009) chose to allow the ISM Fe abundance to vary, effectively changing the size of the Fe K edge. This produces a good fit, and the resulting Fe K emission line profile looks good. We, similarly, achieve good fits by varying the Fe abundance. Nevertheless, there is not a unique model that fits the data well (see the line profiles in Figure 1). Figure 1(a) shows the resulting line profile when the continuum is fit from 2.5 to 5 keV and 8 to 10 keV, ignoring the Fe K region (and assuming standard abundances). Evidently, the resulting residuals do not look like a typical Fe K emission line profile, with a clear dip at around 7 keV. As mentioned above, D'Aì et al. (2009) successfully correct for this by assuming that the ISM Fe abundance toward this source is lower (see Figure 1(c)). With a lower Fe abundance, the absorption edge is weaker; thus the line profile no longer shows a dip.

Zoom In Zoom Out Reset image size

Here, we alternatively suggest that this can also be interpreted as an Fe xxvi absorption line at 6.97 keV superposed on a broad Fe K emission line (see Figure 1(b) and Table 2). There are precedents for such a scenario. For instance, Boirin et al. (2005) find Fe xxv and Fe xxvi absorption lines superposed on a broad Fe K emission line in the dipping LMXB 4U 1323 − 62, and recently similar absorption lines superposed on a broad Fe K emission line were observed in the black hole binary GRO J1655 − 40 (Díaz Trigo et al. 2007).

As is apparent in Figure 1, the two models lead to differing emission-line profiles. While in the model with varying Fe abundance the line profile looks asymmetric, it is not as obviously asymmetric when including an absorption line. Fitting the emission line profiles with the diskline model we also achieve different measurements of R_in: 9.3 ± 0.2GM/c² (consistent with the analysis of D'Aì et al. 2009) compared to 22 ± 4 GM/c² for the alternative model with an absorption line included. Due to the ambiguous nature of the line profile in this source we do not consider this spectrum further here, though parameters from the above spectral fits can be found in Table 2.

GX 349+2. We fit the two Suzaku spectra separately, even though the observation times are close together, in order to investigate any variability. The Suzaku/PIN spectrum was fit from 12 to 25 keV. In the XMM-Newton spectrum, there are large residuals at ∼ 1 keV, which are well fit by a Gaussian emission line (Iaria et al. 2009).

Cyg X-2. XIS spectra fitted from 0.8 to 10 keV, ignoring 1.5–2.5 keV. The Suzaku/PIN spectrum was fit from 12 to 25 keV. We find large residuals at ∼ 1 keV (in all XIS spectra), that are well fit by a single Gaussian emission line. This spectral feature has previously been observed in this source (e.g., Vrtilek et al. 1988; Piraino et al. 2002; Shaposhnikov et al. 2009), and may be a complex of blended emission lines (Vrtilek et al. 1988; Schulz et al. 2009).

SAX J1808.4 − 3658. We do not re-fit the Suzaku and XMM-Newton spectra of this source here, and refer the reader to Cackett et al. (2009a) regarding specifics.

HETE J1900.1 − 2455. The Suzaku/PIN spectrum was fit at 12–30 keV. This spectrum is quite noisy, though an Fe K emission line is apparent. A broad (σ = 0.8 keV) Gaussian fits the line well (χ² = 1148, dof = 1228), and a relativistic disk line is not statistically required. We do, however, fit the diskline to allow comparison with other data, and it gives a similarly good fit (χ² = 1146, dof = 1226). Also note that the continuum model for this source only requires a power-law component and no disk blackbody or blackbody is required.

4.1.1. Investigating the Effects of Pile-up

Of all the sources observed, Cyg X-2 had the highest count rate observed by Suzaku, leading to severe pile-up. The continuum shape changes due to pile-up, becoming artificially harder (e.g., Davis 2001). In order to investigate the robustness of the Fe K line profile in Cyg X-2, we extracted spectra from multiple different regions. In all cases, we used a box of 250 ×  400 pixels, but had a circular exclusion region of varying radius centered on the source. The radius of this circle ranged from 0 pixels to 90 pixels. We fitted the spectra from 0.8 to 10 keV, ignoring the 1.5–2.5 keV energy range, as usual. The spectra were allowed to have completely different continuum models, fit using an absorbed disk blackbody, plus a single temperature blackbody. Additionally, we had to include a narrow Gaussian at 1 keV. Here, we do not fit the PIN data; thus no power-law component is required. We do not fit the PIN data as pile-up significantly affects the shape of the XIS spectrum; hence the broadband spectrum will not fit well unless the XIS spectra are corrected for pile-up.

We found that even though each spectrum had different continuum shapes (due to pile-up effects), the Fe K line profile was remarkably robust. This is clearly demonstrated in Table 3 and Figure 2. In Table 3, we give the continuum and Fe K line parameters from fitting the spectra with different size exclusion regions. Figure 2 shows the resulting line profiles for each spectrum. The Fe K line is both quantitatively and qualitatively consistent, regardless of the pile-up correction. Thus, while with pile-up the continuum may vary significantly from the "true" (not piled-up) continuum shape (in the most extreme cases it is obvious that the continuum model parameters are not realistic), it appears that the Fe K line shape can be robustly recovered as long as the continuum can be fit well. To further demonstrate this, we also show the Fe K profile from the Suzaku observation of Ser X-1 (Figure 2). We again see that the line profile remains consistent as the radius of the circular exclusion region is varied from 0 pixels to 90 pixels.

Zoom In Zoom Out Reset image size

Table 3. Investigating Pile-up in Cyg X-2: Spectral Parameters for Different Extraction Regions

Parameter	Excluded Circle Radius
	0 pixels	30 pixels	60 pixels	90 pixels
NH(1022 cm−2)	0.147 ± 0.003	0.136 ± 0.004	0.128 ± 0.004	0.130 ± 0.005
Disk kT (keV)	1.595 ± 0.006	1.555 ± 0.002	1.52 ± 0.02	1.45 ± 0.04
Disk normalization	124 ± 2	151 ± 1	165 ± 3	190 ± 20
Blackbody kT (keV)	9 ± 1	4.1 ± 0.1	2.9 ± 0.2	2.2 ± 0.2
Bbody normalization (10−2)	41 ± 2	7.0 ± 0.1	4.2 ± 0.1	4.4 ± 0.6
Line Energy (keV)	6.97−0.02	6.97−0.02	6.97−0.02	6.97−0.02
Emissivity, q	5.8 ± 0.6	5.7 ± 0.5	6.3 ± 0.8	6.5+1.0−1.5
Rin (GM/c2)	8.0 ± 0.2	7.8 ± 0.2	7.9 ± 0.3	8.0 ± 0.5
i (°)	24.0 ± 0.5	24.3 ± 0.6	23.9 ± 0.6	23 ± 1
Line normalization (10−3)	7.2 ± 0.4	8.4 ± 0.4	8.7 ± 0.6	8.4 ± 1.0
EW (eV)	69	76	79	74

Notes. Spectral parameters from fitting the XIS 3 data. While the continuum is variable due to pile-up effects, the Fe K line parameters are remain consistent. The normalization of the disk blackbody is in R²_innercos i/(D10)² where R_inner is the inner disk radius in km, D10 is the distance to the source in units of 10 kpc, and i is the disk inclination. The normalization of the blackbody component is L39/(D10)², where L39 is the luminosity in units of 10³⁹ erg s⁻¹. The normalization of the diskline is the line flux in photons cm⁻² s⁻¹.

Download table as:  ASCIITypeset image

4.1.2. Results from Phenomenological Fitting

The spectral parameters from this phenomenological modeling are given in Tables 4 and 5 (the tables have been split into continuum and line parameters). Table 6 gives the 0.5–25 keV unabsorbed fluxes and luminosities. A summary of all the Fe K lines observed in neutron star LMXBs can be seen in Figure 3.

Zoom In Zoom Out Reset image size

Table 4. Phenomenological Model Parameters: Continuum

Source	Mission	Obs. ID	NH	Disk Blackbody		Blackbody		Power-law		Gaussian
			1022 (cm−2)	kTin (keV)	Norm.	kT (keV)	Norm. (10−2)	Γ	Norm.	Line Energy (keV)	σ (keV)	Norm. (10−2)
Serpens X-1	Suzaku	401048010	0.67 ± 0.04	1.26 ± 0.01	97 ± 5	2.27 ± 0.02	5.23 ± 0.06	3.4 ± 0.3	0.9 ± 0.2	...	...	...
	XMM-Newton	0084020401	0.35 ± 0.01	0.95 ± 0.01	205 ± 1	1.76 ± 0.01	4.46 ± 0.01	...	...	1.10 ± 0.01	0.11 ± 0.01	1.3 ± 0.01
	XMM-Newton	0084020501	0.35 ± 0.01	0.99 ± 0.01	175 ± 3	1.81 ± 0.01	3.13 ± 0.02	...	...	1.10 ± 0.01	0.14 ± 0.01	1.4 ± 0.02
	XMM-Newton	0084020601	0.35 ± 0.01	0.95 ± 0.01	203 ± 2	1.76 ± 0.01	3.80 ± 0.01	...	...	1.10 ± 0.01	0.13 ± 0.01	1.2 ± 0.01
4U 1636 − 53	XMM-Newton	0303250201	0.36 ± 0.01	0.21 ± 0.01	(1.0 ± 0.2) × 104	1.97 ± 0.03	0.15 ± 0.01	1.9 ± 0.1	0.22 ± 0.01	...	...	...
	XMM-Newton	0500350301	0.51 ± 0.01	0.89 ± 0.01	101 ± 4	2.00 ± 0.01	1.19 ± 0.01	3.8 ± 0.1	0.54 ± 0.01	...	...	...
	XMM-Newton	0500350401	0.50 ± 0.01	0.96 ± 0.01	103 ± 4	2.03 ± 0.01	1.57 ± 0.01	3.8 ± 0.1	0.58 ± 0.01	...	...	...
4U 1705 − 44	Suzaku	401046010	2.0	0.09 ± 0.01	(2.3+7.2−1.6) ×  107	1.05 ± 0.05	0.063 ± 0.005	1.65 ± 0.01	0.124 ± 0.002	...	...	...
	Suzaku	401046020	1.95 ± 0.05	1.16 ± 0.02	56 ± 2	2.12 ± 0.03	2.40 ± 0.02	2.85 ± 0.08	0.64 ± 0.06	...	...	...
	Suzaku	401046030	2.02 ± 0.05	0.82 ± 0.02	82 ± 5	1.95 ± 0.03	0.95 ± 0.01	3.00 ± 0.06	0.49 ± 0.03	...	...	...
	XMM-Newton	0402300201	1.53 ± 0.01	1.44 ± 0.03	2.5 ± 0.2	3.2 ± 0.3	0.25 ± 0.01	...	...	...	...	...
4U 1820 − 30	Suzaku	401047010	0.21 ± 0.01	1.18 ± 0.01	103 ± 5	2.44 ± 0.01	7.7 ± 0.1	2.3 ± 0.1	0.37 ± 0.03	...	...	...
GX 17+2	Suzaku	402050010	2.2 ± 0.1	1.75 ± 0.02	86 ± 5	2.66 ± 0.02	6.1 ± 0.2	2.6 ± 0.6	0.4 ± 0.4	...	...	...
	Suzaku	402050020	2.14 ± 0.05	1.74 ± 0.02	92 ± 4	2.55 ± 0.02	9.2 ± 0.3	2.4 ± 0.3	0.2 ± 0.2	...	...	...
GX 349+2	Suzaku	400003010	0.77 ± 0.01	1.53 ± 0.02	112 ± 4	2.24 ± 0.02	12.5 ± 0.2	1.3 ± 0.7	0.01+0.05−0.01	...	...	...
	Suzaku	400003020	0.85 ± 0.02	1.48 ± 0.02	105 ± 4	2.30 ± 0.01	9.6 ± 0.1	2.3 ± 0.1	0.31 ± 0.01	...	...	...
	XMM-Newton	0506110101	0.67 ± 0.01	1.17 ± 0.01	258 ± 6	1.85 ± 0.01	16.7 ± 0.3	...	...	1.08 ± 0.01	0.09 ± 0.01	3.9 ± 0.4
Cyg X-2	Suzaku	403063010	0.20 ± 0.02	1.50 ± 0.01	163 ± 4	2.21 ± 0.02	3.6 ± 0.1	2.8 ± 0.1	60 ± 20	1.01 ± 0.01	0.11 ± 0.01	9.0 ± 0.6
SAX J1808.4 − 3658	Suzaku	903003010	0.046 ± 0.04	0.48 ± 0.01	5.9 ± 0.3	1.00 ± 0.02	0.127 ± 0.001	1.93 ± 0.01	0.20 ± 0.01	...	...	...
	XMM-Newton	0560180601	0.23 ± 0.01	0.23 ± 0.01	20400+13200−5000	0.40 ± 0.01	0.125 ± 0.004	2.08 ± 0.01	0.33 ± 0.01	...	...	...
HETE J1900.1 − 2455	Suzaku	402016010	0.032 ± 0.04	...	...	...	...	2.14 ± 0.01	0.063 ± 0.001	...	...	...

Notes. Continuum spectral fitting parameters from phenomenological modeling. All uncertainties are 1σ. Where no uncertainty is given, that parameter was fixed. The χ² values for the fits are those in Table 5. Normalization of the disk blackbody is in R²_innercos i/(D10)² where R_inner is the inner disk radius in km, D10 is the distance to the source in units of 10 kpc, and i is the disk inclination. Normalization of the blackbody component is L39/(D10)², where L39 is the luminosity in units of 10³⁹ erg s⁻¹. Normalization of the power-law component gives the photons keV⁻¹ cm⁻² s⁻¹ at 1 keV. Finally, the normalization of the Gaussian is the total number of photons cm⁻² s⁻¹ in the line.

Download table as:  ASCIITypeset image

Table 5. Phenomenological Model Parameters: Fe K Line

Source	Mission	Obs. ID	Diskline						χ2/dof
			Line Energy (keV)	Emissivity Index	Rin (GM/c2)	Inclination (°)	Norm. (10−3)	EW (eV)
Serpens X-1	Suzaku	401048010	6.97−0.02	4.8 ± 0.3	8.0 ± 0.3	24 ± 1	5.6 ± 0.4	98	1442/1105
	XMM-Newton	0084020401	6.66 ± 0.02	1.8 ± 0.2	25 ± 8	32 ± 3	1.8 ± 0.1	43	2125/1831
	XMM-Newton	0084020501	6.97−0.03	3.7 ± 4	14 ± 1	13 ± 2	1.6 ± 0.1	50	1961/1831
	XMM-Newton	0084020601	6.66 ± 0.02	2.0 ± 0.2	26 ± 8	27 ± 2	1.4 ± 0.1	38	2151/1831
4U 1636 − 53	XMM-Newton	0303250201	6.4+0.02	2.7 ± 0.1	6.1+0.4	>80	1.2 ± 0.9	184	1939/1832
	XMM-Newton	0500350301	6.4+0.04	2.8 ± 0.1	7.7 ± 0.6	>77	1.2 ± 0.1	106	2112/1832
	XMM-Newton	0500350401	6.4+0.02	2.8 ± 0.1	6.0+0.2	>84	2.5 ± 0.1	164	2406/1832
4U 1705 − 44	Suzaku	401046010	6.62 ± 0.03	3.0	10.0	24	<0.44	<76	953/1122
	Suzaku	401046020	6.94 ± 0.06	4.8 ± 0.6	7.3 ± 0.4	24 ± 4	3.3 ± 0.3	120	1266/1120
	Suzaku	401046030	6.9 ± 0.1	3.7 ± 0.3	6+0.2	15 ± 3	1.3 ± 0.1	94	1152/1120
	XMM-Newton	0402300201	6.43 ± 0.03	2.1 ± 0.1	6+1	>74	0.34 ± 0.05	151	1760/1752
4U 1820 − 30	Suzaku	401047010	6.97 − 0.03	3.3 ± 0.1	6.0+0.1	<8	2.6 ± 0.3	36	1269/1108
GX 17+2	Suzaku	402050010	6.8 ± 0.1	2.6 ± 0.2	7 ± 3	15 ± 15	6.0 ± 0.5	46	636/606
	Suzaku	402050020	6.75 ± 0.07	4.6 ± 0.7	8.0 ± 0.4	27 ± 2	14 ± 1	81	628/606
GX 349+2	Suzaku	400003010	6.97−0.01	3.9 ± 0.2	7.5 ± 0.4	26 ± 1	16 ± 1	105	1268/1106
	Suzaku	400003020	6.64 ± 0.03	2.3 ± 0.1	10 ± 2	42 ± 2	9.2 ± 0.6	79	1262/1106
	XMM-Newton	0506110101	6.76 ± 0.01	1.9 ± 0.1	6.0+0.5	55 ± 2	12.1 ± 0.3	79	2531/1831
Cyg X-2	Suzaku	403063010	6.97−0.02	6.5 ± 1.3	8.1 ± 0.9	25 ± 3	7.0 ± 0.6	65	1305/1129
SAX J1808.4 − 3658	Suzaku	903003010	6.40+0.04	2.9 ± 0.2	12+7−1	50+7−2	0.70 ± 0.06	134	2987/2550
	XMM-Newton	0560180601	6.4+0.06	3.0 ± 0.2	13 ± 4	59 ± 4	0.68 ± 0.04	118	2124/1950
HETE J1900.1 − 2455	Suzaku	402016010	6.97−0.2	5 ± 1	6+1	20 ± 4	0.16 ± 0.02	122	1146/1226

Notes. Fe K line parameters from phenomenological modeling. All uncertainties are 1σ. Where no uncertainty is given, that parameter was fixed. In the diskline model, the emissivity of the disk follows a power-law radial dependence, R^−q, where q is the emissivity index we report here. The normalization of the diskline is the line flux in photons cm⁻² s⁻¹.

Download table as:  ASCIITypeset image

Table 6. Source Fluxes and Luminosities in the 0.5–25 keV Range

Source	Mission	Obs. ID	Phenomenological		Reflection
			Unabs. Flux	Luminosity	Unabs. Flux	Luminosity
			(10−8 erg s−1 cm−2)	(1037 erg s−1)	(10−8 erg s−1 cm−2)	(1037 erg s−1)
Serpens X-1	Suzaku	401048010	1.19 ± 0.01	10.0 ± 5.0	1.32 ± 0.08	11.1 ± 5.6
	XMM-Newton	0084020401	0.70 ± 0.01	5.9 ± 3.0	0.71 ± 0.01	6.0 ± 3.0
	XMM-Newton	0084020501	0.59 ± 0.01	5.0 ± 2.5	0.60 ± 0.01	5.1 ± 2.5
	XMM-Newton	0084020601	0.65 ± 0.01	5.5 ± 2.7	0.66 ± 0.01	5.6 ± 2.8
4U 1636 − 53	XMM-Newton	0303250201	0.17 ± 0.01	0.73 ± 0.05	0.18 ± 0.02	0.78 ± 0.09
	XMM-Newton	0500350301	0.39 ± 0.01	1.7 ± 0.1	0.37 ± 0.01	1.6 ± 0.1
	XMM-Newton	0500350401	0.48 ± 0.01	2.1 ± 0.1	0.46 ± 0.03	2.0 ± 0.1
4U 1705 − 44	Suzaku	401046010	0.17 ± 0.01	0.68 ± 0.06	0.57 ± 0.05	2.3 ± 0.3
	Suzaku	401046020	0.61 ± 0.01	2.5 ± 0.2	0.49 ± 0.01	2.0 ± 0.1
	Suzaku	401046030	0.30 ± 0.01	1.2 ± 0.1	0.31 ± 0.01	1.2 ± 0.1
	XMM-Newton	0402300201	0.043 ± 0.001	0.17 ± 0.01	0.042 ± 0.001	0.17 ± 0.01
4U 1820 − 30	Suzaku	401047010	1.21 ± 0.01	8.4 ± 0.9	1.21 ± 0.13	8.4 ± 1.3
GX 17+2	Suzaku	402050010	2.37 ± 0.01	27.2 ± 2.2	2.27 ± 0.01	26.1 ± 2.1
	Suzaku	402050020	2.60 ± 0.01	29.9 ± 2.4	2.62 ± 0.01	30.1 ± 2.5
GX 349+2	Suzaku	400003010	2.35 ± 0.03	7.0 ± 3.5	2.36 ± 0.13	7.1 ± 3.6
	Suzaku	400003020	1.97 ± 0.01	5.9 ± 2.9	1.92 ± 0.01	5.7 ± 2.9
	XMM-Newton	0506110101	2.38 ± 0.01	7.1 ± 3.6	2.40 ± 0.09	7.2 ± 3.6
Cyg X-2	Suzaku	403063010	2.22 ± 0.01	32.1 ± 11.7	2.30 ± 0.01	33.3 ± 12.1
SAX J1808.4 − 3658	Suzaku	903003010	0.20 ± 0.01	0.29 ± 0.02	0.27 ± 0.02	0.40 ± 0.04
	XMM-Newton	0560180601	0.25 ± 0.01	0.37 ± 0.03	0.26 ± 0.01	0.38 ± 0.03
HETE J1900.1 − 2455	Suzaku	402016010	0.034 ± 0.001	0.053 ± 0.015	0.032 ± 0.005	0.050 ± 0.016

Note. Distances used are given in Table 1.

Download table as:  ASCIITypeset image

The distribution of measured inner disk radii is shown in Figure 4. We only use one measurement per source, choosing the one with the smallest fractional uncertainty. The vast majority of the measured inner disk radii fall into a relatively narrow range of 6–15 GM/c². Only two observations (of Ser X-1) fall outside this range, and even then the inner disk radii from those observations are only a little more than 1σ away. Note that there are a number of fits in which R_in pegs at the lower limit of the model (6 GM/c², the innermost stable circular orbit (ISCO) in the Schwarzschild metric). This corresponds to about 12 km, assuming a 1.4 M_☉ neutron star. Given that this lower limit is very similar to predicted neutron star radii, smaller inner disk radii are not expected. The lines in the observations with R_in = 6 GM/c² are fit well by the model, and do not show large residuals in the red wing of the line (which would be expected if R_in were smaller than measured).

Zoom In Zoom Out Reset image size

One possibility for the large number of sources that reach the lower limit of the model is that there is some contribution to the broadening of the line from Compton scattering. Reis et al. (2009b) found that for 4U 1705 − 44 the measured inner disk radius increased when such effects were taken into account using reflection modeling.

We note that the differences in equivalent width (EW) between our original Suzaku analysis of Ser X-1, GX 349+2, and 4U 1820 − 30 and this analysis can be attributed to updated responses. During the original analysis the standard response matrices had to be used, which did not fully account for the window mode, and therefore gave incorrect fluxes.

As noted earlier, the Fe Kα emission line is only the most prominent feature expected from disk reflection where hard ionizing flux irradiates the accretion disk leading to line emission. In order to more self-consistently model the spectra, we fit reflection models which include line emission from important astrophysical metals (the strongest line being Fe Kα), as well as effects such as Compton broadening.

As is clear from the phenomenological modeling of the spectra (and as has been noted before, e.g., Cackett et al. 2008; Iaria et al. 2009), in the atoll and Z sources, the blackbody component (potentially associated with the boundary layer) is often dominating between 8 and 20 keV. Thus, it is the emission that provides the vast majority of ionizing flux that illuminates the accretion disk forming the Fe K line (and reflection spectrum). In order to model the reflection component, one therefore needs to employ a reflection model where a blackbody component provides the illuminating flux (Ballantyne & Strohmayer 2004; Ballantyne 2004). This differs from the standard reflection models used to model black hole sources (e.g., Ballantyne et al. 2001; Ross & Fabian 2007), which assume illumination by a power-law spectrum. The resulting reflection spectra are quite different (see Figure 5 and Ballantyne 2004, for details), with the blackbody reflection model dropping off quickly above 10 keV compared to the power-law reflection.

Zoom In Zoom Out Reset image size

Here, we use models created to study the Fe K emission line observed in 4U 1820 − 30 during a superburst, where a constant-density slab is illuminated by a blackbody⁹ (Ballantyne & Strohmayer 2004; Ballantyne 2004). We use the model assuming solar abundances for all sources except 4U 1820 − 30, where we use the model for a He system (Ballantyne & Strohmayer 2004). We note here that the density used in these models is n_H = 10¹⁵ cm⁻³. This is significantly lower than the density expected at the inner disk in neutron star systems. Ballantyne (2004) tested increasing the density used in the model, and found that it did not significantly change the Fe K emission, though changes at lower energies (<1 keV) were more important. While a higher density model would be more appropriate here, there is not yet one available for use.

We use a model where the irradiating blackbody component is included in the reflection model, in order to calculate the reflection fraction, R, which is given in Table 8 with respect to that for a point source above a slab. The overall model is given by a combination of the reflected spectrum and this blackbody. In the model, the ionization parameter is defined as

$$\begin{equation} \xi = \frac{4\pi F_{\rm x}}{n_{\rm H}}, \end{equation} \tag{ 1 }$$

where F_x is the irradiating flux (erg cm⁻² s⁻¹) in the 0.001–100 keV range and n_H is the hydrogen number density. The ionization parameter, ξ, therefore has units erg cm s⁻¹. The normalization of the reflection model is simply a scale factor that sets the overall strength of the combined blackbody and reflection spectrum, whereas the reflection fraction sets the relative strength of the two. The observed, unabsorbed flux (erg cm⁻² s⁻¹) of the blackbody plus reflection spectrum from the source in the 0.001–100 keV range is therefore given by

$$\begin{equation} F_{\rm obs, unabs} = {\rm Norm}\,\times \, F_{\rm x}(1+R), \end{equation} \tag{ 2 }$$

where Norm is the normalization of the model. Rearranging gives

$$\begin{equation} {\rm Norm} = \frac{4\pi F_{\rm obs, unabs}}{\xi n_{\rm H}(1+R)} \end{equation} \tag{ 3 }$$

and the density used in the models, n_H = 10¹⁵ cm⁻³, is substituted.

To make these spectral fits more easily reproducible by other models we have calculated the equivalent blackbody normalization (using the definition of the XSPEC model bbody), from the best-fitting value of the reflection model normalization and ξ. The strength of the reflection component relative to this is given by the reflection fraction. So that the results are reproducible using the same model as used here, we also quote the normalization value directly from the model (as defined above).

The strength of the Fe K line is dependent on both the ionization parameter and the temperature of the irradiating blackbody. As the blackbody temperature increases, the number of photons above the Fe K edge increases, leading to a greater EW. The EW peaks both at low ionization parameters, log ξ = 1.0, where Fe Kα is at 6.4 keV, and then at log ξ ∼ 2.7 where the line is at 6.7 keV. At the peak of the higher ionization parameter, the EW is between ∼100 and 300 eV for a blackbody temperature in the range 1.5 keV–3.0 keV. This is shown clearly in Figure 3 of Ballantyne (2004). These EWs are for a reflection fraction of 1. Lower reflection fractions lead to smaller EWs.

We relativistically blur this reflection model by convolving with the rdblur model in XSPEC. rdblur is simply a convolution with the diskline model, allowing the entire reflection component to be blurred by the relativistic effects present at the inner accretion disk. This blurred reflection component is then included in the model instead of the diskline model used in the phenomenological fits. This modeling acts to self-consistently model the entire reflection spectrum (multiple emission lines and continuum) rather than just treating the Fe K line alone.

We are able to achieve good fits with this reflection modeling, generally finding similar inner disk radii as with phenomenological modeling (see spectral fit parameters in Tables 7 and 8). An example of our reflection fits is shown in Figure 6. It is important to note that because the blackbody component drops off over 8–20 keV, so too does the reflection component. The typical strong Compton hump seen in the black hole sources (both in active galactic nucleus and X-ray binaries) is therefore not apparent because of this.

Zoom In Zoom Out Reset image size

The two sources where we do not fit this blackbody reflection model are the accreting millisecond pulsars (SAX J1808.4 − 3658 and HETE J1900.1 − 2455). Both sources show particularly hard spectra, where the high-energy continuum is dominated by a power law. We therefore fit these spectra with the constant density-ionized disk reflection model (Ballantyne et al. 2001), where a power law irradiates a slab. As with the blackbody reflection model, the model normalization is dependent on ξ, and is given by the same definition as above. We therefore have calculated the equivalent power-law normalization at 1 keV so that these results are reproducible using other reflection models. Again, we also report the best-fitting normalization directly from the model so that these results can be easily reproduced using the same model. This model fits the data well in both cases. HETE J1900.1 − 2455 requires only the power-law reflection component, and no other continuum components are needed. SAX J1808.4 − 3658, on the other hand, requires both a disk blackbody and a blackbody in addition to the power-law reflection component.

All spectral parameters from this reflection fitting are given in Tables 7 and 8, and source fluxes and luminosities are given in Table 6. We also show the range of measured inner disk radii from the reflection component in Figure 4. We find that a large number of sources have their measured inner disk radii pegged at the model limit of 6 GM/c², which we discuss in more detail later.

The inner disk radii that we measure mostly fall into a small range: 6–15 GM/c². Also consistent with this range is a recent independent measurement of the inner disk radius in 4U 1705 − 44 (R_in = 14 ± 2; Di Salvo et al. 2009c). Where there are multiple observations of a source, in most cases there is not a large change in the inner disk radius, which may suggest that the inner disk radius does not vary with state. While we note this from observations at different epochs, similar results have been found previously when looking at how the line profile changes with source state during one observation (D'Aì et al. 2009; Iaria et al. 2009).

The only source where we see variations in inner disk radii is Ser X-1, and this is easily apparent when one compares the line profile observed by Suzaku with one of the XMM-Newton observations (see Figure 3). Comparing these observations, the spectrum from the Suzaku observation of Ser X-1 is harder than the XMM-Newton observations—although both the blackbody and disk blackbody components are hotter, the relative normalization of the blackbody component relative to the disk blackbody component is higher. The ionizing flux during the Suzaku observation is therefore stronger, which would explain the higher line flux. In all cases, though, the emitting radius inferred from the blackbody component is very similar. The 0.5–10 keV source flux is also highest during the Suzaku observation (10.3 ×  10⁻⁹ erg cm⁻² s⁻¹ compared to 6.3 ×  10⁻⁹, 4.4 ×  10⁻⁹, and 4.9 ×  10⁻⁹ erg cm⁻² s⁻¹ for the XMM-Newton observations). This may tentatively suggest that the disk extends closer to the stellar surface at the highest luminosities, though this is not confirmed when looking at all sources (Figure 7).

Zoom In Zoom Out Reset image size

We have also looked to see if there is any significant change in the inner disk radius with 0.5–25 keV luminosity for all sources (see Figure 7). There is no apparent systematic trend with luminosity, and the inner disk radius is found to be close to 6 GM/c² for the majority of sources/observations over almost three orders of magnitude in luminosity. Similarly, we see no clear difference between atolls, Z sources, and accreting millisecond pulsars, as can be seen in Figure 4.

There is typically consistency between the inner disk radii measured from phenomenological and reflection spectral fits. Nevertheless, in both cases many of the inner disk radius measurements are pegged at the lower limit of the model (the ISCO). In 4U 1705 − 44, Reis et al. (2009b) found that fitting with a model for a Kerr metric (relevant for maximally spinning black holes; Laor 1991) gave inner disk radii much smaller than the expected neutron star radii. They suggested that as the disks in these systems are ionized one must also take broadening due to Comptonization into account by using relativistic ionized reflection models. When they did this, the inner radius measured was significantly higher, and no longer smaller than the expected stellar radii. The reflection models employed here also take Compton broadening into account, yet, we still find that the inner disk radius pegs at the ISCO in a number of sources.

This may not necessarily be a problem. As neutron stars are spinning, their angular momentum will change the radius of the ISCO making it slightly smaller for a corotating disk (see Figure 1(a) of Miller 2007 to see how the ISCO changes with angular momentum). Potentially, one could then measure the dimensionless angular momentum parameter, under the assumption that the inner disk is truncated at the ISCO and not at the boundary layer or stellar surface. To test this, we fit a line profile with a variable angular momentum parameter (Brenneman & Reynolds 2006) to several of the best data sets in our sample. However, in each case we found that this parameter was not well constrained, being consistent with zero. Better, more sensitive data may be able to constrain this further.

We can also compare the inner disk radii from the Fe line fitting with that implied from the disk blackbody fits. The normalization of the disk blackbody used here (diskbb) is defined as N = R²_innercos i/(D10)², where R_inner is the inner disk radius in km, D10 is the source distance in units of 10 kpc, and i is the disk inclination. Therefore, from the fitted normalization and the inclination determined from the Fe line fits, we can calculate the inner disk radius in kilometers. Assuming a reasonable range for the neutron star mass (here we use 1.4–2.0 M_☉) we can convert to units of GM/c² for direct comparison with the Fe line fits. The disk radii determined this way are given in Table 11. Where only an lower/upper limit for the inclination is found from the Fe line fits, we just use this limit for the inclination in the calculation.

Comparing the inner disk radii from the continuum (diskbb) and Fe line fitting, we find that the continuum inner disk radius is generally smaller than the Fe line value, though there are a small number of cases where the values are consistent (when assuming a mass of 1.4 M_☉). It is important to note that we have not used any spectral hardness factor or a correction for the inner boundary condition for the continuum radii. These two factors are expected to lead to a factor of 1.18–1.64 increase between the radius measured by diskbb and the real inner disk radius (e.g., Kubota et al. 2001). This would bring more of the values to be consistent, though a large number would still disagree. Where there are large disagreements, it could be possible that the continuum model chosen is not physical. For disagreements where the continuum radius is smaller than the Fe line radius, one can imagine that disk photons may be scattered out of the line of sight by a corona, diminishing the strength of the corona, and leading to a smaller measured continuum radius. Additionally, the continuum method relies on knowing the absolute flux of the disk component. Therefore, if the absolute flux calibration of the telescope or the continuum model (column density or other parameters) is incorrect, then this can lead to an incorrect inner disk radius.

We found in many cases that there are several local χ² minima within the diskline parameter space. There is usually one solution that is clearly the global minimum, with a significantly better χ² value. However, we found that in one instance (GX 349+2, second Suzaku observation) the difference was as small as Δχ² ∼ 2. This issue is due to a degeneracy where a model with a combination of a low-inclination, high emissivity index and high line energy can look similar to a higher inclination, lower emissivity index and a lower line energy model. Usually the inner disk radius remains consistent between the solutions. This is apparent in Table 5, see particularly the GX 349+2 and Ser X-1 observations. We demonstrate this issue by showing the dependence of the χ² confidence contours on inclination and emissivity index for the second GX 349+2 Suzaku observation in Figure 8. Here, one clearly sees two minima with only a small difference in χ². The χ² minimum is 1262.6 for the low emissivity index, high inclination minimum compared to 1265.0 for the high emissivity index, low inclination local minimum (both have 1106 dof). For the first Suzaku GX 349+2 we also find two minima. In this case, however, the global minimum has a high emissivity index, low inclination opposite to the second observation (χ² = 1268.1 for low inclination compared to 1273.6 for higher inclination).

Zoom In Zoom Out Reset image size

As discussed, comparing all observations of GX 349+2, the inclination and emissivity index from spectral fitting varies significantly (see Table 5). This was first pointed out by Iaria et al. (2009) when comparing their XMM-Newton results with the Cackett et al. (2008) results. As they also noted, while it is possible that the emissivity index, inner disk radius, and the line energy may vary with source state, the source inclination obviously does not. As we have demonstrated, this difference is not because the lines look very different. In fact, if all three observations of GX 349+2 are fit simultaneously with the inclination tied between observations (best fit value i = 44⁺¹¹₋₄ deg), then a good fit is achieved, and the lines look almost identical (see Figure 9). Plainly, a priori knowledge of source inclination is of significant benefit. However, joint fitting of multiple observations of the same source potentially resolves this issue. As can been seen in Figure 8 when the three observations of GX 349+2 are fit jointly, there is no longer a local χ² minimum with a high emissivity index and low inclination.

Zoom In Zoom Out Reset image size

Such a degeneracy between the line parameters is self-evident when studying how each parameter affects the line profile (see Figure 1 of Fabian et al. 1989). The inclination shifts the peak of the blue wing of the line profile. Therefore, a higher line energy can be offset by a lower inclination. Moreover, a higher inclination makes the line broader and less "peaked," and the emissivity index also has a similar effect. Thus, a higher inclination and lower emissivity index can look similar to a lower inclination and higher emissivity index.

We find a similar problem with the inclination determined from fitting the Cyg X-2 spectra, where we measure an inclination of 25°. This inclination is not consistent with the value measured from optical observations, where lower and upper inclination limits are 49°–73°, with a best-fitting value of 63° (Orosz & Kuulkers 1999; Elebert et al. 2009). If we constrain the inclination to be within the optical values, we do not achieve as good a fit (χ² = 1337 compared to χ² = 1305), and find the inclination pegs at the lower bound of 49°. With the inclination constrained, the line parameters we find are E = 6.40^+0.01 keV, q = 2.9 ± 0.2, R_in = 25 ± 4 GM/c², i = 49^+1°. Again, there is a pattern where a higher inclination leads to a lower line energy and emissivity index.

Note that in several cases we find high emissivity indices and/or very low inclinations when fitting the reflection models. Both a high emissivity index and a low inclination try to make the line profile more peaked. We tested whether this can be solved with a higher Fe abundance by using a reflection model with two times solar abundance, but retrieved similar results, and worse fits. Light bending can potentially lead to a high emissivity index, causing the irradiation of the disk to be more centrally concentrated. In the most extreme case of spinning black holes an index of ∼ 5 is expected (Miniutti et al. 2003). These effects should not be as strong in neutron stars, and thus a lower index would be expected.

It is also important to note that in the reflection modeling the inclination is only introduced by the relativistic blurring, and the reflection spectrum itself is angle averaged. However, reflection is sensitive to angle, with the reflection strength weakening with higher inclination. The use of an angle-averaged model may therefore bias the results obtained.

In black hole systems, the ionizing flux is dominated by a power law. The Compton back-scattering part of the reflection spectrum is easily detected against a power-law continuum. Disk reflection is more difficult to detect against a thermal spectrum since the continuum flux drops precipitously with higher energy. In the case of neutron stars, then, disk reflection modeling may only yield improved physical constraints in the limit of extraordinary high energy sensitivity. The fact that many of our reflection fits imply small radii and low inclinations may be partially due to a combination of modest high energy sensitivity, coupled with the limitations of angle-averaged disk reflection models. Deeper observations with Suzaku aimed at clearly detecting the Compton back-scattering hump will help to obtain better physical constraints.

Reflection modeling allows us to infer the ionization state of the inner accretion disk. The reflection spectrum is highly dependent on the ionization parameter, ξ (see Figure 1 of Ballantyne 2004). For low ionization states many emission lines are present which disappear as the ions become more highly ionized. The Fe Kα line, in particular, evolves significantly with the ionization parameter. For neutral reflection, the 6.4 keV line is strong and as the ionization increases this line becomes weaker. The 6.7 keV line then becomes prominent until it disappears as iron becomes fully ionized. This evolution is displayed clearly in Figure 3 of Ballantyne (2004), where the dependence of the EW of the Fe Kα on the ionization parameter is shown. The EW peaks either for neutral reflection when the 6.4 keV line is strong or for log ξ = 2.6–2.8 when the 6.7 keV line is dominant. From our reflection fitting (Tables 7 and 8) we find quite a narrow range of ionization parameters. Most spectra are best fit with an ionization parameter in the range log ξ = 2.6–2.8, where the Fe Kα EW peaks.

We also consider the strength of the reflection component, which directly relates to the solid angle subtended by the reflector, Ω. The reflection fraction, R = Ω/2π, is given in Table 8. A reflection fraction of 1 corresponds to an isotropic source above the slab. Most of the observations here have a reflection fraction in the range R = 0.1–0.3, suggesting that a reasonable fraction of the ionizing flux is intercepted by the accretion disk and re-emitted. If we interpret the blackbody component as the boundary layer, this may also suggest that the boundary layer has quite a similar extent in these sources. As discussed above, Figure 3 of Ballantyne (2004) shows a strong dependence of the Fe Kα line EW and the blackbody temperature. Most of the blackbody temperatures we find have kT > 2 keV, which equates to an EW of >200 eV for R = 1. From the phenomenological fits (see Table 5), we get EWs that are mostly less than 100 eV; thus, the reflection fractions that we find generally seem reasonable.

There are two cases (one spectra of 4U 1705 − 44 and one spectra of 4U 1636 − 53) where the best-fitting reflection fractions are at the upper limit of the model (R = 5). In these spectra, either the blackbody temperature or the blackbody normalization is very low. Both these lead to a low EW, and therefore for the model to achieve the required line strength a large reflection fraction is required. Both these sources are atolls which do transit between hard and soft states (see, e.g., Gladstone et al. 2007), and these two particular observations are in the hard state. As discussed earlier, spectral deconvolution in atoll and Z sources is not unique (e.g., Lin et al. 2007), and the high values of the reflection fraction suggest that our particular choice of the model here is unlikely the best physical model. We have also tried replacing the blackbody reflection model with a power-law reflection model. While this achieved more reasonable reflection fractions it did not provide as good a fit to the data. D'Aì et al. (2010) have recently successfully modeled the hard state in 4U 1705–44 using an alternative model involving a Comptonization model to fit the hard component, and such a model may be more appropriate for these two particular observations, though we do not investigate this here.

It is well known that a hot blackbody (or alternatively a Comptonized component) dominates from around 7–20 keV (e.g., Barret et al. 2000). In all sources but the accreting millisecond pulsars, our spectral fitting agrees with this. This blackbody component may potentially be due to emission from the boundary layer between the inner accretion disk and the neutron star surface (see, e.g., Sunyaev & Shakura 1986; Inogamov & Sunyaev 1999; Popham & Sunyaev 2001; Grebenev & Sunyaev 2002; Gilfanov et al. 2003; Revnivtsev & Gilfanov 2006, for discussions about the boundary layer). The boundary layer is the transition region between the rapidly spinning accretion disk and the more slowly spinning neutron star, and should account for a significant fraction of the accretion luminosity.

As neutron star spectra show that the blackbody component dominates the flux able to ionize iron, it suggests that the boundary layer may be the dominant source of ionizing flux illuminating the inner accretion disk, leading to Fe Kα emission. Our spectral analysis with reflection models shows that this picture is consistent with the data.

It is not a new idea that the boundary layer or neutron star surface can illuminate the disk leading to Fe K emission. For instance, Day & Done (1991) were the first to note that X-ray bursts on the surface of the neutron star can lead to a disk-reflection component. Furthermore, Brandt & Matt (1994) use a geometry where an inner spherical cloud of Comptonized gas acts to illuminate the inner accretion disk as a basis for calculating Fe K profiles expected in Z sources. Moreover, detailed calculations of the inner disk and boundary layer led Popham & Sunyaev (2001) to conclude that the X-ray flux from the hot boundary layer incident on the accretion disk would be large enough to lead to line emission and a reflection spectrum.

Observationally, there is not a large body of previous work fitting broadband neutron star spectra with reflection models where the boundary layer irradiates the accretion disk. However, Done et al. (2002) successfully fit Ginga data of Cyg X-2 with a reflection model that illuminates the disk with a Comptonized continuum, and similarly Gierliński & Done (2002) achieved good fits to the RXTE spectra of atoll 4U 1608 − 52 with the same model. However, the data in both cases had significantly lower spectral resolution to those presented here. Di Salvo et al. (2000a) also successfully fit the broadband BeppoSAX spectrum of MXB 1728 − 34 with a reflection model which illuminates the disk with a Comptonized continuum. Furthermore, Barret et al. (2000) suggest that the boundary layer provides the X-ray flux irradiating the disk from spectral fitting of RXTE data.

The Popham & Sunyaev (2001) model for the boundary layer is the one where the hot boundary layer gas is radially and vertically extended. Popham & Sunyaev (2001) showed that the hot boundary layer is much thicker than the inner disk (see their Figure 1). As mass accretion rate changes they find that both the radial and vertical extent of the boundary layer will change, and thus the fraction of X-ray emission from the boundary layer that is intercepted by the disk will vary. An alternative model for the boundary layer is one where matter spreads over the neutron star surface in a latitudinal belt (Inogamov & Sunyaev 1999). In this spreading layer, the belt width is dependent on the mass accretion rate. At low mass accretion rates (3 ×  10⁻³) the width is of the order of the thin-disk thickness; however, the belt disappears at high Eddington fractions (∼0.9) as the entire surface is radiating. It therefore seems reasonable that even the spreading layer may plausibly illuminate the inner accretion disk.

We now further investigate the notion that the boundary layer is the source of ionizing flux. Using ionization parameters typical to what we find from spectral fitting, we determine the maximum height of the boundary layer, for a disk extending close to the neutron star surface. The ionization parameter at the inner accretion disk, ξ, is given by

$$\begin{equation} \xi = \frac{L_{\rm BL}}{nR^2}, \end{equation} \tag{ 4 }$$

where L_BL is the boundary layer luminosity, n is the number density in the disk, and R is the distance from the ionizing source to the disk. For the purposes of this estimation we use the distance between the top of the boundary layer and the inner accretion disk. That distance is given by

$$\begin{equation} R = \sqrt{R_{\rm in}^2 + Z^2}, \end{equation} \tag{ 5 }$$

where R_in is the inner disk radius, and Z is the height of the ionizing source above the disk. The distance will, of course, be higher for other regions of the accretion disk, and for boundary layer emission closer to the equator. However, combining these two equations and re-arranging leads to the height of the ionizing source above the disk being defined as

$$\begin{equation} Z^2 = \frac{L_{\rm BL}}{n\xi } - R_{\rm in}^2 \,. \end{equation} \tag{ 6 }$$

From spectral fitting and reflection modeling we determine L_BL, R_in, and ξ, and from standard disk theory we can put reasonable constraints on n. From Shakura & Sunyaev (1973), we find n > 10²¹ cm⁻³. For log ξ = 3, L_BL = 10³⁷ erg s⁻¹ and R_in = 20 km, we find Z < 24 km. It is therefore completely reasonable that the boundary layer provides the ionizing flux to produce the Fe K emission lines observed. A subsequent detailed analysis of multiple spectra from 4U 1705 − 44 has also concluded that the boundary layer likely illuminates the disk, at least in high flux states (D'Aì et al. 2010).

A further check can be performed by looking at the inferred blackbody emitting radius from the spectral fits. In Table 12, we show the calculated emitting radius and assumed distances from the blackbody normalizations. We use the same distances as given in Table 11. As the boundary layer emission is more correctly described by a Comptonized component as opposed to a simple blackbody we can maybe not interpret these numbers directly. However, they do provide a good consistency check, with radii typically being a significant fraction of the stellar radius, as expected theoretically (Popham & Sunyaev 2001). In a couple of cases the radii are extremely high, in those few cases it seems likely that while the model may be a good fit to the data it is not physically correct.

In many cases, we find that the hot blackbody component contributes substantially to the overall flux. Substantial boundary layer emission is consistent with observed spins in neutron stars in LMXBs, which spin slower than the Keplerian angular speed in the inner accretion disk. If a neutron star had significantly higher angular speed, the boundary layer emission is expected to tend to zero as the angular speed approached the Keplerian angular speed (see, for example, Figure 5 from Bhattacharyya et al. 2000).

We interpret the observed Fe K emission line as originating in the inner accretion disk, with relativistic effects giving rise to the asymmetric line profile (as is generally accepted). There has, however, been a suggestion that these asymmetric Fe K lines can originate from an optically thick high-velocity flow in these systems (Titarchuk et al. 2003; Laming & Titarchuk 2004; Laurent & Titarchuk 2007; Shaposhnikov et al. 2009; Titarchuk et al. 2009). In this model, a diverging wind with a wide opening angle is launched from the disk. The Fe K line is formed in a narrow wind shell that is illuminated by X-rays from the accretion disk. The asymmetric profile arises when Fe K photons scatter off electrons in the partly ionized wind, becoming redshifted in the process. These photons undergo a large number of multiple scatterings to form the red wing of the line. Such a model appears to fit some Fe K line profiles well (Shaposhnikov et al. 2009; Titarchuk et al. 2009).

To reproduce the observed line profiles requires that there be a very large number of electron scatters in the wind, and hence the wind had a high optical depth (Titarchuk et al. 2009 find τ_wind = 1.6 for neutron star LMXB Ser X-1 and τ_wind = 4.9 for black hole X-ray binary GX 339 − 4). The mass outflow rate required for GX 339 − 4 is comparable to the Eddington mass inflow rate. In the low/hard state considered by Titarchuk et al. (2009), this means that the outflow rate is 30–100 times higher than the inferred mass inflow rate. The outflow rate required to produce the line in Ser X-1 is also comparable to the Eddington mass inflow rate. Not only are high outflow rates required, but the outflow must also not be coupled to the inflow rate given that we see similar line profiles in the neutron star systems over more than two orders of magnitude in luminosity.

Wide-angle, optically thick outflows should have observable consequences in X-rays and in other bands. In X-rays, ionizing flux from the central engine should generate absorption lines for any line of sight through the wind. Yet when absorption lines are observed in black holes, they are always slow, with velocities less than approximately 1000 km s⁻¹ (see, e.g., Miller et al. 2006b, 2008a on GRO J1655 − 40 and Ueda et al. 2009 on GRS 1915 + 105). In some cases, absorption lines are seen in the absence of any relativistic line (e.g., H 1743 − 322 and 4U 1630 − 472; Miller et al. 2006c; Kubota et al. 2007). The absorption lines detected in most neutron star systems (the edge-on "dipping" sources) are often consistent with no velocity shift (e.g., Sidoli et al. 2001; Parmar et al. 2002; Boirin et al. 2005; Díaz Trigo et al. 2006). Clearly, observed absorption spectra are in strong disagreement with the wind model.

In optical and IR bands, optically thick diverging outflows should also have consequences. Whether in a black hole, neutron star, or white dwarf system, the outflow should serve to obscure our view of the outer accretion disk and/or companion star, at least during some intervals of the binary period. Along particular lines of sight, the outer disk should be obscured for nearly all of the binary period. Obscuration of this kind has not been reported, though it would surely be easily detected. Doppler tomography reveals no evidence of such outflows as accretion disk, stream/hot spot, and companion star emission are often clearly revealed (e.g., Marsh et al. 1994; Steeghs & Casares 2002).

Finally, it is worth noting that relativistic lines are not detected in white dwarf systems. Titarchuk et al. (2009) fit their model to the XMM-Newton CCD spectrum of GK Per, where they only observe one emission line. This, however, ignores the fact that high-resolution Chandra grating observations clearly separated the emission into three separate Fe components, as is seen in many magnetic cataclysmic variables (CVs; Hellier & Mukai 2004). The spectra of CVs are best described in terms of partially obscured emission from the boundary layer, which can be fitted with cooling-flow models that predict multiple iron charge states (Hellier & Mukai 2004).

To begin to put meaningful constraints on the neutron star equation of state, it is useful to combine both mass and radius measurements. Of the sources in our sample, Cyg X-2 is the only one with a well-constrained mass measurement, M_NS = 1.5 ± 0.3M_☉ (Elebert et al. 2009). Combining this mass measurement with our stellar radius upper limit (8.1 GM/c²) can be seen in Figure 10. This does not rule out any of the possible equations of state shown here, and tighter constraints on mass and radius would be needed.

Zoom In Zoom Out Reset image size

Significantly improved sensitivity and spectral resolution from future X-ray missions, such as Astro-H and the International X-ray Observatory (IXO), will certainly provide tighter constraints from modeling Fe K emission lines, and with the High Timing Resolution Spectrometer on IXO even the potential to study simultaneous spectral and timing evolution on extremely short timescales (Barret et al. 2008).