A Possible Phase-dependent Absorption Feature in the Transient X-Ray Pulsar SAX J2103.5+4545

Using the NuSTARDAS pipeline, we extracted a phase-averaged spectrum from the source region of each observation for both Focal Plane Modules A and B (FPMA and FPMB), which we rebinned using the FTOOLS command grppha. We rebinned the spectra above 10 keV, using variable bin sizes to keep the errors in the residuals above 10 keV approximately constant. Several continuum models including a power law with a high-energy cutoff (White et al. 1983), FDCut (Fermi–Dirac CUT-off power law, Tanaka 1986), and negative and positive power law with an EXponential cutoff (NPEX, e.g., Mihara et al. 1998) were applied to the phase-averaged spectra using Xspec (version 12.8.2, Arnaud 1996). In Observation I, the best fit (${\chi }^{2}$ of 857.10 for 915 degrees of freedom) was obtained for the NPEX model, which is defined in Xspec as

$$\begin{eqnarray*}&&f(E)={n}_{1}({E}^{-{\alpha }_{1}}+{n}_{2}{E}^{-{\alpha }_{2}}){e}^{-E/{kT}}.\end{eqnarray*}$$

However, it should be noted that Xspec requires that $-10\lt {\alpha }_{2}\lt 0$, and the second power law has a positive exponent. The power law with a high-energy cutoff failed to fit the hard energy tail of the data, and resulted in a higher ${\chi }^{2}$ of 881.17 for 918 degrees of freedom. Additionally, this model has been known to produce artificial absorption features around the cutoff energy that must be smoothed by the addition of Gaussians (e.g., Coburn et al. 2002) or polynomials (e.g., Burderi et al. 2000), thus adding additional free parameters to the model. The FDCut model also failed to fit the hard tail and resulted in a ${\chi }^{2}$ of 1217.53 for 918 degrees of freedom. Figure 2 shows the phase-averaged spectrum for each observation fitted with the best-fit NPEX model.

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In addition to the NPEX model, the data required a soft thermal component, two Gaussian emission lines, and a Gaussian absorption line that is discussed in Section 4. Adding the blackbody component to the model reduced the ${\chi }^{2}$ from 910.23 to 857.10 for Observation I. In Observation II, the blackbody component is not as strongly significant, but we included it in the Observation II model so that we can more directly compare the two observations.

In Observation I, narrow lines corresponding to the Fe Kα line at 6.4 keV and a highly ionized iron line at 6.9 keV are present. In Observation II, the highly ionized iron line is no longer detected; instead, we find the best fit with a broad and a narrow line at 6.4 keV. We found that the narrow line is required by the data by running an f-test simulation using the tcl script simftest where the continuum model without the narrow Gaussian component was the null hypothesis. We created 500 simulated spectra and, using the test statistic ${\rm{\Delta }}{\chi }^{2}$, found that the observed ${\rm{\Delta }}{\chi }^{2}$ was greater than the simulated values by an order of magnitude in all cases. We conclude that the narrow Fe line feature is strongly required by the data.

For both observations, the model was applied and fitted jointly to the NuSTAR FPMA and FPMB spectra. The parameters in the model for the FPMB spectrum were tied to the parameters in the FPMA spectrum and related by a cross-calibration constant. Table 1 contains the results of the spectral fits shown in Figure 2. In all models we used elemental abundances and cross sections given by Wilms et al. (2000) and Verner et al. (1996), respectively.

Table 1.  Phase-averaged Spectral Parameters^a

Parameter	Observation I	Observation II
${N}_{{\rm{H}}}$ ($\times \,{10}^{22}$ cm−2)	4.4 ± 1.4	2.9 ± 0.8
${{kT}}_{\mathrm{BB}}$ (keV)	0.35 ± 0.05	0.2 (fixed)
${n}_{\mathrm{BB}}$ (keV)	0.008 ± 0.007	0.62 ± 0.04
${\alpha }_{1}$	0.61 ± 0.06	0.64 ± 0.1
${\alpha }_{2}$	−2.1 ± 0.2	−2.2 ± 0.3
${n}_{{\alpha }_{2}}$	(8.2 ± 3.3) $\times \,{10}^{-5}$	(7.9 ± 7.0) $\times \,{10}^{-5}$
${{kT}}_{\mathrm{fold}}$ (keV)	10.7 ± 0.8	9.1 ± 1.2
log10(${F}_{3-40\mathrm{keV}}$)	−8.650 ± 0.004	−9.191 ± 0.004
E${}_{{\mathrm{Fe}}_{1}}$ (keV, fixed)	6.4	6.4
${\sigma }_{{\mathrm{Fe}}_{1}}$ (keV, fixed)	0.1	0.05
${n}_{{\mathrm{Fe}}_{1}}$	(5.8 ± 0.5) $\times \,{10}^{-4}$	(6.9 ± 2.5) $\times \,{10}^{-5}$
${E}_{{\mathrm{Fe}}_{2}}$ (keV, fixed)	6.9	6.4
${\sigma }_{{\mathrm{Fe}}_{2}}$ (keV)	0.1 (fixed)	0.65 ± 0.15
${n}_{{\mathrm{Fe}}_{2}}$	(1.7 ± 0.5) $\times \,{10}^{-4}$	(3.2 ± 0.7) $\times \,{10}^{-4}$
${c}_{\mathrm{FPMA}}$ (fixed)	1	1
${c}_{\mathrm{FPMB}}$	1.021 ± 0.003	1.021 ± 0.004
${\chi }^{2}$	857.10	881.85
Degrees of Freedom	915	885

Note.

^aFor the continuum model constant * tbnew * (cflux * (npex * gabs) + bbody + Gauss + Gauss). All errors are 90% confidence intervals. The absorption feature is discussed in Section 4.

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Using the X-ray and optical distance measurements for SAX J2103.5+4545, we calculated the 3–40 keV luminosity of Observation I to be between $(0.5\mbox{--}1.1)\times {10}^{37}$ erg s⁻¹ and the 3–40 keV luminosity of Observation II to be between $(1.5\mbox{--}3.2)\times {10}^{36}$ erg s⁻¹, where the range in luminosities results from the different distance measurements to the source.

The pulse period of J2103 was found using the FTOOL efsearch (Leahy et al. 1983), which folds the light curve over a range of test periods and produces a chi-squared plot where the maximum is the best pulse period. The uncertainty in the pulse period was found using the pulse profile to simulate 500 light curves, and finding the range of simulated pulse periods. We found 349.23 ± 0.01 s as the best pulse period for Observation I and 349.20 ± 0.01 s as the best pulse period for Observation II. Pulse profiles were created using the FTOOL efold, which fold the light curve over the desired period to create a pulse profile. The pulse profiles are shown in Figure 3, where, for clarity, the phases have been chosen so that the peaks align.

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The pulse profile for Observation I shows a three-peaked profile where the peaks increase in count rate as a function of phase. In Observation II, the first peak is no longer visible, yielding a pulse profile with only two peaks. To check the energy dependence associated with these pulse profile changes, the cleaned event files for Observations I and II were filtered into five energy bands, and the pulse profiles were re-extracted in the same way as before. The energy-filtered pulse profiles, shown in Figure 4, were normalized by subtracting the average count rate in each band and dividing by the respective standard deviation. In both observations, the strongest pulse peak is dominated by hard X-rays. In Observation I, the first peak is predominantly soft X-rays, and disappears entirely in the harder energy bands. The first peak in Observation II, which seemed to have disappeared in the full energy band pulse profile (Figure 3), is shown to still exist at harder energies, although it has also shifted in relative phase.

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The phase-averaged source event files were filtered into 10 equal phase bins with xselect. The phase-resolved spectra were extracted from these phase bins, and grouped with grppha to have 100 counts per spectral bin. We fitted each spectrum in the same way as the phase-averaged spectra using the same NPEX continuum model in the of 3–40 keV range.

In initial fits to Observation I and Observation II, the ${N}_{{\rm{H}}}$ appeared to vary with pulse phase. These fluctuations were most likely caused by degeneracy in the model between the parameters for ${N}_{{\rm{H}}}$ and the values of the blackbody normalization. In order to limit this degeneracy, we fixed the ${N}_{{\rm{H}}}$ to its phase-averaged value in all phase bins.

Within each model the power-law index ${\alpha }_{1}$, folding energy, flux, and normalizations of the blackbody component and Gaussian lines were variable. We fixed the power-law index ${\alpha }_{2}$ to its phase-averaged value to reduce degeneracy between the two power-law indices and the folding energy. Although it would have further reduced degeneracy in the model to fix the folding energy as well, we found that doing so resulted in a poor fit to the data. An error analysis of each variable parameter was also conducted in Xspec, yielding values with 90% confidence intervals. To check for pulse-phase dependence, these parameters were plotted against pulse phase (see Figure 5).

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In both observations, the line energies and widths for the two Gaussian components were fixed to their phase-averaged values because they could not be constrained in the phase-resolved analysis. In Observation I the iron line strengths are consistent with being constant with pulse phase. In Observation II, the iron lines are required by the data, as their absence results in reduced ${\chi }^{2}$ values of 1.2 or higher. However, due to the reduced signal to noise in these spectra and degeneracy in the continuum model, it is challenging to quantify the fluctuation of iron line strength with spin phase in a meaningful way.

The phase-resolved spectroscopy with 10 phase bins ultimately does not yield conclusive results about the behavior of spectral parameters with pulse phase. The lack of coherent pulsations in Figure 5 could be caused by degeneracies between the power-law indices and the folding energy.

Since phase-resolved spectroscopy using fine phase bins did not yield conclusive results, we used the shape of the pulse profile to motivate a different binning selection. Both observations were divided into three phase bins, each of which covered a pulse peak within the pulse profile (see the black lines in Figure 4). These phase bins were labeled Peak 1, Peak 2, and Peak 3, respectively, where Peak 3 is the strongest pulse in the pulse profile. We extracted three spectra for each observation according to this phase selection, and use the same binning and energy range restrictions as in the phase-averaged analysis.

For both observations, we fit the spectra from the three pulse-peak phase bins simultaneously and refer to parameters that are tied together, so as to have the same value in all phase bins, as global parameters (as in Ballhausen et al. 2017, see method description in Kühnel et al. 2015, 2016). We also fixed the Fe line energies and widths to their phase-averaged values in all phase bins to reduce the number of free parameters. We fitted each of the three pulse-peak spectra with the continuum model used on the phase-averaged spectra (constant * tbnew (bbody + cflux * npex + gaus + gaus)). We defined the column density ${N}_{{\rm{H}}}$, the folding energy, and the FPMB normalization to be global parameters since we expect these to vary little with pulse phase. Additionally, the blackbody temperature is a global parameter in Observation I; however, this parameter is poorly constrained in Observation II and had to be fixed to its phase-averaged value in all phase bins.

We found the following global parameters for Observations I and II, respectively: the ${N}_{{\rm{H}}}$ values are $(4.6\pm 1.4)\times {10}^{22}\ {\mathrm{cm}}^{-2}$ and $(3.0\pm 0.7)\times {10}^{22}\ {\mathrm{cm}}^{-2}$, the folding energies are 10.6 ± 0.6 keV and 8.0 ± 0.5 keV, and the FPMB normalizations are 1.021 ± 0.003 and 1.021 ± 0.004. The blackbody temperature for Observation I is 0.37 ± 0.04 keV. The continuum parameters, found in Table 2, generally do not vary strongly with pulse phase.

Table 2.  Pulse-peak-resolved Spectral Parameters^a,b

Observation	Parameter	Peak 1	Peak 2	Peak 3
Observation I	${n}_{\mathrm{BB}}$	0.007 ± 0.004	0.004 ± 0.003	0.010 ± 0.005
	${\alpha }_{1}$	0.63 ± 0.06	0.70 ± 0.06	0.55 ± 0.06
	${\alpha }_{2}$	−2.1 ± 0.2	−2.0 ± 0.2	−2.2 ± 0.2
	${n}_{{\alpha }_{2}}$	(6.1 $\pm 4.9)\times {10}^{-5}$	$({9.0}_{-4.2}^{+7.7})\times {10}^{-5}$	(6.9 $\pm 3.4)\times {10}^{-5}$
	log10(${F}_{3-40\mathrm{keV}}$)	−8.755 ± 0.005	−8.689 ± 0.005	−8.562 ± 0.004
	${n}_{{\mathrm{Fe}}_{1}}$	(6.1 $\pm 0.8)\times {10}^{-4}$	(5.5 $\pm 1.0)\times {10}^{-4}$	(5.8 $\pm 0.8)\times {10}^{-4}$
	${n}_{{\mathrm{Fe}}_{2}}$	(1.7 $\pm 0.7)\times {10}^{-4}$	(2.0 $\pm 1.0)\times {10}^{-4}$	$(1.7\pm 0.8)\times {10}^{-4}$
Observation II	${n}_{\mathrm{BB}}$	0.12 ± 0.03	0.03 ± 0.03	0.10 ± 0.05
	${\alpha }_{1}$	0.45 ± 0.07	0.65 ± 0.07	0.54 ± 0.07
	${\alpha }_{2}$	−2.5 ± 0.2	−2.7 ± 0.2	−2.5 ± 0.2
	${n}_{{\alpha }_{2}}$	$({1.1}_{-0.3}^{+0.6})\times {10}^{-4}$	(${2.1}_{-0.8}^{+1.3})\times {10}^{-6}$	$({5.7}_{-1.9}^{+3.0})\times {10}^{-5}$
	${\mathrm{log}}_{10}({F}_{3\mbox{--}40\mathrm{keV}})$	−9.315 ± 0.004	−9.279 ± 0.004	−9.056 ± 0.003
	${n}_{{\mathrm{Fe}}_{\mathrm{broad}}}$	$(1.6\pm 0.8)\times {10}^{-4}$	$(4.3\pm 1.0)\times {10}^{-4}$	$(3.2\pm 1.2)\times {10}^{-4}$
	${n}_{{\mathrm{Fe}}_{\mathrm{narrow}}}$	$(7.5\pm 3.4)\times {10}^{-5}$	$(6.8\pm 4.1)\times {10}^{-5}$	$(6.1\pm 5.2)\times {10}^{-5}$

Notes.

^aAll errors are 90% confidence intervals. The energy and width of the Fe lines were fixed to their phase-averaged values in all phase bins. The ${N}_{{\rm{H}}}$, ${{kT}}_{\mathrm{BB}}$, folding energy, and FPMB normalization were defined as global parameters. ^bThe reduced ${\chi }^{2}$ values were 1.029 and 1.005 for Observations I and II, respectively.

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In Observation I, Figure 4 indicates that there are only slight variations in continuum shape between pulse peaks. For Observation I, the pulse profiles indicate that Peak 1 and Peak 2 are both softer than Peak 3. However, as seen in Figure 6, the steepness of the power law does not change significantly across the three pulse peaks. The continuum parameters for Observation I (Table 2) are generally consistent with being constant.

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In Observation II, the largest change with pulse phase can be seen in the ${\alpha }_{1}$ parameter, which varies significantly between Peak 1 and Peak 2. This change can be seen in Figure 6, where the Observation II Peak 1 and Observation II Peak 2 spectra have different slopes. The change in slope could also be driven by the relative strength of the second power law in NPEX, which is significantly stronger in Observation II Peak 1 than in the other two phase bins.

NuSTAR's hard X-ray sensitivity also allows us to search the hard X-ray spectrum for cyclotron lines. The CRSF appears as an absorption-like feature at hard energies that occurs due to resonant scattering of continuum photons by electrons on quantized Landau energy levels in the strong magnetic field of the neutron star's accretion column. Because the energy spacing between Landau levels can be given by ${E}_{\mathrm{CRSF}}={\hslash }{eB}/{m}_{e}c=11.57{B}_{12}$ keV (where B₁₂ is the magnetic field strength in units of 10¹² G), the line energy at which the CRSF appears is a direct measurement of the strength of the pulsar's magnetic field close to the neutron star surface. Understanding the magnetic properties of neutron star X-ray binaries like J2103 is necessary for modeling both accretion and emission models (e.g., Becker & Wolff 2007).

Despite the lack of an obvious CRSF in the phase-averaged spectra, we attempted to find an upper limit on a cyclotron line feature at any energy. We added the Xspec Gaussian absorption model component gabs to the continuum model and stepped the line energy from 8 to 60 keV. In each observation, we found an improvement in ${\chi }^{2}$ at specific energies (∼30 keV for Observation I and ∼15 keV for Observation II).

We verified this ${\rm{\Delta }}{\chi }^{2}$ by running steppar on the absorption line energy while the line strength and continuum parameters (NPEX: α1, α2, folding energy, cross normalization; cflux: log of flux) were variable. All other parameters in the model were fixed to their phase-averaged values. Since steppar returns the ${\chi }^{2}$ for the model fit as it steps through ${E}_{\mathrm{CRSF}}$, we compared the maximum and minimum ${\chi }^{2}$ to obtain a second measurement of ${\rm{\Delta }}{\chi }^{2}$.

In the Observation I phase-averaged spectrum, the best-fit absorption feature had a line energy of 31 keV and a depth of 0.2. The line width was frozen to 4 keV, a typical width for CRSFs (e.g., Tendulkar et al. 2014), due to this parameter being poorly constrained. This line was detected with a ${\rm{\Delta }}{\chi }^{2}$ of 2.5, which is not strong enough to be considered significant. In the Observation II phase-averaged spectrum, the best-fit absorption feature had a line energy of 15.3 keV, a depth of 0.3, and a width of 3.3 keV, and resulted in a ${\rm{\Delta }}{\chi }^{2}$ of 6.4.

We used the Xspec tcl script simftest to determine the significance of the absorption feature in the Observation II spectrum. We used the continuum NPEX model without gabs as our null hypothesis, and simulated fake spectra from this model. Simftest then fitted the continuum NPEX model with and without gabs and recorded the change in ${\chi }^{2}$ for 10,000 trials. As a test statistic, we used a ratio of the normalized ${\chi }^{2}$ for the model fit without the absorption feature to the normalized ${\chi }^{2}$ for the model fit with the absorption feature, where the normalized ${\chi }^{2}$ is the ${\chi }^{2}$ divided by the associated degrees of freedom (in other words, the random variate of the F-distribution. We found that the observed normalized ${\chi }^{2}$ ratio was greater than 96.2% of the ratios from the sample of 10,000 simulated data sets, corresponding to $\sim 1.8\sigma$ (one-tail test).

The 10 phase-resolved spectra had too few counts in each bin to effectively search for a weak absorption feature. We therefore checked for phase dependence in the feature using the three pulse-peak spectra from each observation. For both observations, as was done with the phase-averaged spectra, we stepped the energy of the absorption feature from 8 to 60 keV and performed a fit at each energy. For both observations, we found that the absorption lines in the phase bins Peak 1 and Peak 3 were not significant detections. However, the absorption features in Peak 2 were significant in both observations, and the line parameters were similar. In Observation 1, the best-fit Peak 2 absorption feature had a central energy of 11.2 keV, a line width of 4 keV, and a depth of 0.5. In Observation II, the best-fit Peak 2 feature had a central energy of 12 keV, a width of 3.3 keV, and a depth of 0.7. In both of these cases, the widths were taken from the phase-averaged spectral line fits. The line parameters for each phase bin and the ${\rm{\Delta }}{\chi }^{2}$ improvement are listed in Table 3. We found that adding an absorption feature resulted in a ${\chi }^{2}$ improvement of 11 in Observation I and 34 in Observation II. In Figure 7, we show the fit to the Observation II Peak 2 spectrum with and without the gabs model component. The absorption feature can be seen as a slight dip in the residuals in the bottom panel.

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Table 3.  Pulse-peak-resolved Spectral Parameters^a

Observation	Parameter	Peak 1	Peak 2	Peak 3
Observation I	${E}_{{\mathtt{gabs}}}$ (keV)	12.4	11.4	29
	${d}_{{\mathtt{gabs}}}$	${0.2}_{-0.2}^{+0.3}$	1.0 ± 0.4	${0.2}_{-0.2}^{+0.3}$
	${w}_{{\mathtt{gabs}}}$ (keV, fixed)	4	4	4
	Maximum ${\rm{\Delta }}{\chi }^{2}$ from adding gabs	0.4	11	0.8
	Significance	⋯	99.2%	⋯
Observation II	${E}_{{\mathtt{gabs}}}$ (keV)	⋯	12.0 ± 0.9	${16.1}_{-1.5}^{+1.3}$
	${d}_{{\mathtt{gabs}}}$	⋯	0.8 ± 0.3	0.4 ± 0.2
	${w}_{{\mathtt{gabs}}}$ (keV, fixed)	⋯	3.3	3.3
	Maximum ${\rm{\Delta }}{\chi }^{2}$ from adding gabs	⋯	34	6
	Significance	⋯	99.8%	⋯

Note.

^aWhere no uncertainty is given, these values were found to be the best fit in Xspec, but needed to be frozen to ensure a stable error analysis.

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We ran simftest and calculated the normalized ${\chi }^{2}$ ratio in the same way as done for the phase-averaged spectrum to test the significance of the absorption features in the Peak 2 phase bin of both observations. We found that, for Observation I our observed normalized ${\chi }^{2}$ ratio was greater than 99.2% of the ratios from the simulated spectra. In Observation II, the observed ratio was greater than 99.8% of the ratios from the simulated spectra.

The $\approx 2.4\sigma$ and $\approx 2.9\sigma$ (one-tail test) presence of this absorption feature within the same phase bin of both observations, and at approximately the same energy, suggests that this feature is not a calibration error or statistical phenomenon.