# Constraining the evolution of the unstable accretion disk in SMC X-1 with
NICER

McKinley C. Brumback Cahill Center for Astronomy and Astrophysics, California
Institute of Technology, 1216 E California Blvd, Pasadena, CA 91125, USA
Department of Astronomy, University of Michigan, 1085 S. University Ave. Ann
Arbor, MI 48109 USA  Georgios Vasilopoulos Department of Physics, National and
Kapodistrian University of Athens, University Campus Zografos, GR 15783,
Athens, Greece  Joel B. Coley Department of Physics and Astronomy, Howard
University, Washington, DC 20059, USA  CRESST/Mail Code 661, Astroparticle
Physics Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
Kristen Dage  Department of Physics, McGill University, 3600 University
Street, Montréal, QC H3A 2T8, Canada  Jon M. Miller  Department of Astronomy,
University of Michigan, 1085 S. University Ave. Ann Arbor, MI 48109 USA

###### Abstract

Neutron star high mass X-ray binaries with superorbital modulations in
luminosity host warped inner accretion disks that occult the neutron star
during precession. In SMC X-1, the instability in the warped disk geometry
causes superorbital period “excursions:” times of instability when the
superorbital period decreases from its typical value of 55 days to $\sim$40
days. Disk instability makes SMC X-1 an ideal system in which to investigate
the effects of variable disk geometry on the inner accretion flow. Using the
high resolution spectral and timing capabilities of the Neutron Star Interior
Composition Explorer (NICER) we examined the high state of four different
superorbital cycles of SMC X-1 to search forchanges in spectral shape and
connections to the unstable disk geometry. We performed pulse phase-averaged
and phase-resolved spectroscopy to closely compare the changes in spectral
shape and any cycle-to-cycle variations. While some parameters including the
photon index and absorbing column density show slight variations with
superorbital phase, these changes are most evident during the intermediate
state of the supeorbital cycle. Few spectral changes are observed within the
high state of the superorbital cycle, possibly indicating the disk instability
does not significantly change SMC X-1’s accretion process.

††software: HEAsoft (v6.29c; HEASARC 2014), NuSTARDAS, Stingray (Huppenkothen
et al. 2019), HENDRICS (Bachetti 2018), MaLTPyNT (Bachetti 2015)

##  1 Introduction

Among high-mass X-ray binaries (HMXBs), super-giant X-ray binaries (SgXRBs)
where the donor star overfills the Roche lobe offer the ideal laboratory for
studying the interaction of the pulsar radiation with the accretion disk
because the mass transfer via Roche-lobe overflow is relatively stable. The
study of these systems has offered insight into the morphology of their
accretion disk and its temporal stability. In particular, it is now well
established that SgXRBs host warped disks (e.g. Ogilvie & Dubus 2001; Hickox &
Vrtilek 2005; Brumback et al. 2020), which cause superorbital modulation on
timescales of 40-100 days as the warped disk edge partially obscures the
compact object during its precession. Neutron star SgXRBs with superorbital
modulation present a unique opportunity to probe the structure of the inner
accretion disk because an observer simultaneously views the neutron star beam
(dominated by hard X-ray radiation) and the soft, reprocessed emission from
the inner accretion disk. Previous works by Hickox & Vrtilek (2005), Brumback
et al. (2020), and Brumback et al. (2021) have demonstrated the success of
using soft ($<1$ keV) pulse profiles to perform reverberation mapping in these
sources, allowing them to model the geometry and kinematics of the warped
accretion disks.

Perhaps the best systems for studying the “soft-excess” are SgXRBs in the
Magellanic Clouds. These systems have known distances and suffer from little
foreground absorption. In this respect SMC X-1 is potentially the best
candidate for further study of warped accretion disks. SMC X-1 is a SgXB
consisting of a 1.21(12) M⊙ NS with a $\sim 0.7$ s spin period orbiting an 18
M⊙ B0 supergiant companion every 3.9 days (Leong et al. 1971; Schreier et al.
1972; Webster et al. 1972; Liller 1973; Falanga et al. 2015). A radiation
driven warp in the inner accretion disk, combined with the high orbital
inclination ($i\sim 70$∘; Reynolds et al. 1993), gives rise to a superorbital
modulation in luminosity because the warped edge of the disk will partially
obscure the pulsar during its precession (Pringle 1996). The disk precession
causes three distinct superorbital states: the high state, which is
characterised by maximum source flux, when the neutron star is unobscured by
the disk, the low state, or time of minimum flux when the neutron star is
occulted by the warped disk, and the intermediate state, which is the period
of increasing or decreasing flux that marks the transition between high and
low state. The period of this superorbital cycle varies between 40–60 days
(e.g. Wojdowski et al. 1998; Vrtilek et al. 2005), which Ogilvie & Dubus
(2001) suggest arises from an instability in the accretion disk warp that
causes the disk shape to vary between competing configurations. The
superorbital cycle typically remains around 55 days in length, but during
these periods of instability (known as “excursions”), the cycle length can
decrease down to 40 days (e.g. Hu et al. 2019; Dage et al. 2019).

The study of warped accretion disks in SgXRBs has potentially wider
implications for a variety of systems. Super-orbital modulation of unknown
origin is found in ultra-luminous X-ray systems (e.g. Vasilopoulos et al.
2021; Gúrpide et al. 2021; Brightman et al. 2022), which are a poorly
understood class of X-ray binaries that host the brightest X-ray pulsars
($L_{X}>10^{39}$ erg/s or $>$10L Eddington; e.g. King et al. 2023). Warped
accretion disks have also been observed in active galactic nuclei such as the
Circinus galaxy and NGC 4945, making the geometry of these warped disk
structures significant across mass and luminosity scales (e.g. Greenhill et
al. 2003; Martin 2008).

Using the high resolution spectral and timing capabilities of the Neutron Star
Interior Composition Explorer (NICER; Gendreau et al. 2016), we examined the
high state of different superorbital cycles of SMC X-1 to search for short
term changes in spectral shape that might be caused by fluctuations in the
unstable accretion disk geometry. Our data set, described in Section 2,
consisted of 18 NICER observations covering four superorbital cycles. We
performed pulse phase-averaged and pulse phase-resolved, and spectral analysis
analysis to closely compare the spectral characteristics and any cycle-to-
cycle variations (see Section 2 for our analysis description and Section 3 for
our results and discussion).

Table 1: Description of NICER Observations ObsID | Start Time (MJD) | Exposure (ks) | Orbital Phase |  $F_{\text{0.6--10keV}}$ (erg cm-2 s${}^{-1})$ | Pulse Frequency (Hz)
---|---|---|---|---|---
3638010101 | 59172.26 | 5.58 | 0.32 | (8.09$\pm$0.02)$\times 10^{-10}$ | 1.432756(5)
3638010102 | 59173.10 | 5.15 | 0.54 | (8.84$\pm$0.02)$\times 10^{-10}$ | 1.432652(5)
3638010103 | 59174.02 | 3.54 | 0.89 | (8.75$\pm$0.03)$\times 10^{-10}$ | 1.43298(3)
3638010104 | 59175.57 | 0.12 | 0.17 | (8.2$\pm$0.1)$\times 10^{-10}$ | 1.4324(3)
3638010105 | 59176.27 | 3.42 | 0.29 | (8.81$\pm$0.02)$\times 10^{-10}$ | 1.432756(1)
3638010201 | 59216.25 | 8.17 | 0.62 | (4.08$\pm$0.01)$\times 10^{-10}$ | 1.432809(1)
3638010202 | 59217.73 | 3.84 | 0.0 | N/AaaThe orbital phases, fluxes, and light curves of this observation indicate that it occurred during either binary eclipse or a pre-eclipse dip. For this reason, this observation was not included in our analysis. | N/AaaThe orbital phases, fluxes, and light curves of this observation indicate that it occurred during either binary eclipse or a pre-eclipse dip. For this reason, this observation was not included in our analysis.
3638010203 | 59218.52 | 5.70 | 0.21 | (6.64$\pm$0.02)$\times 10^{-10}$ | 1.433064(3)
3638010204 | 59219.03 | 5.43 | 0.33 | (7.40$\pm$0.02)$\times 10^{-10}$ | 1.432821(5)
3638010301 | 59368.64 | 0.55 | 0.78 | (8.87$\pm$0.07)$\times 10^{-10}$ | 1.43355(4)
3638010302 | 59369.02 | 0.48 | 0.88 | N/AaaThe orbital phases, fluxes, and light curves of this observation indicate that it occurred during either binary eclipse or a pre-eclipse dip. For this reason, this observation was not included in our analysis. | N/AaaThe orbital phases, fluxes, and light curves of this observation indicate that it occurred during either binary eclipse or a pre-eclipse dip. For this reason, this observation was not included in our analysis.
3638010303 | 59370.25 | 6.00 | 0.19 | (9.15$\pm$0.03)$\times 10^{-10}$ | 1.433382(7)
3638010304 | 59371.02 | 8.43 | 0.39 | (9.54$\pm$0.01)$\times 10^{-10}$ | 1.432996(5)
3638010305 | 59372.00 | 2.07 | 0.64 | (9.23$\pm$0.03)$\times 10^{-10}$ | 1.433183(2)
3638010306 | 59373.80 | 3.18 | 0.11 | (8.80$\pm$0.04)$\times 10^{-10}$ | 1.433564(9)
3638010307 | 59373.99 | 0.77 | 0.15 | (9.32$\pm$0.07)$\times 10^{-10}$ | 1.43344(7)
3638010401 | 59409.48 | 8.28 | 0.27 | (1.007$\pm$0.003)$\times 10^{-9}$ | 1.433350(4)
3638010402 | 59409.99 | 10.99 | 0.40 | (1.026$\pm$0.002)$\times 10^{-9}$ | 1.433151(1)

##  2 Observations and Data Analysis

###  2.1 NICER Observations

Our data set consisted of 18 NICER observations taken between November 2020
and July 2021. The observations are grouped into four epochs (hereafter
referred to as Epochs 1–4), with each epoch corresponding to a different
superorbital cycle in either the bright or intermediate. Each epoch contains a
varying number of NICER visits; Epoch 1 contains 5 visits, Epoch 2 contains 4
visits, Epoch 3 contains 7 visits, and Epoch 4 contains 2 visits. The
observation information including obsID, exposure, and start time is contained
in Table 1. For the sake of brevity, we refer to each observation by the last
three digits of its obsID. In this format, the first digit corresponds to the
epoch number and the second two digits refer to the visit number (e.g.
Observation 302 is the second visit in the third epoch).

The observations in Epochs 1 and 2 occurred during the intermediate state of
SMC X-1’s superorbital cycle, as the source was increasing in brightness.
Epochs 3 and 4 occured during the high state, when the source was at its
brightest (see Figure 1). We determined the length of the superorbital cycle
containing each of our four epochs using the zero-crossing method (e.g. Smale
& Boyd 2012) with the Swift/BAT light curve binned to SMC X-1’s orbital
period. Taking the average of the cycle length found by the up-crossing and
down-crossing times, we determined that the supeorbital cycles hosting Epochs
1–4 were approximately 46$\pm$3 days, 30$\pm$3 days, 44$\pm$3 days, and
46$\pm$3 days, respectively. From these cycle lengths and the excursion
measurements of Hu et al. (2023) it is clear that SMC X-1 was within a period
of superorbital excursion during our observations (see also Hu et al. 2019;
Dage et al. 2019).

The observations were reduced using the nicerl2 pipeline within HEASoft
v6.29c. To reduce contamination from background effects, we selected an
underonly count range of 0–250 and an overonly count range of 0–1.5. Spectra
and light curves were extracted using Xselect for each observation. Because
SMC X-1 has a variety of known time dependent behaviors including binary
eclipse and occasional pre-eclipse dips, we examined each light curve to
determine the quality of the observation. Sixteen of the eighteen observations
showed NICER count rates of approximately 250 counts s-1 with little
variability, consistent with viewing the source out of eclipse (e.g., Brumback
et al. 2020).

Because SMC X-1 is an eclipsing binary with known pre-eclipse dips, we
examined the light curve of each observation to exclude times of low counts.
Observations 202 and 302 showed low NICER count rates (less than 100 counts
s-1) and high variability consistent with eclipse and pre-eclipse dips.
Calculating the orbital phase of these observations (using the Falanga et al.
(2015) ephemeris) indicated that Observation 202 occured at orbital phase 0
and Observation 302 occurred at orbital phase 0.88 (see Table 1). Falanga et
al. (2015) defines the eclipse in SMC X-1 to be between orbital phases
0.939(9) and 0.066(5), where the values in parenthesis represent the error on
the last digit and the time of mid-eclipse is orbital phase 0. By these
calculations, Observation 202 occured during binary eclipse. As seen in the
light curves in Brumback et al. (2022), SMC X-1 sometimes exhibits pre-eclipse
dips around orbital phase $\sim$0.9, therefore it is likely that the low count
rates in Observation 302 are caused by a pre-eclipse dip. Because these
observations fall during eclipse or a pre-eclipse dip, we excluded these two
observations from the remainder of our analysis.

Further examination of the NICER light curves revealed that several
observations (Observations 103, 201, 203, 303) had light curves that were
predominantly bright but showed short intervals at the beginning or end of the
observation in which the source decreased in flux. Since these intervals
likely arise from the observation partially occurring during an eclipse or
pre-eclipse dip, we filtered these observations by time in Xselect to remove
the intervals of low count rates from our analysis.

Figure 1: The Swift/BAT 15–50 keV light curve for SMC X-1 (black) showing the
superorbital modulation around the time of our NICER observations (blue dashed
lines). The observations occurred during the bright high state of four
supeorbital cycles, referred to in this work as Epoch 1-4. Epoch 1 contains
five NICER visits, Epoch 2 contains four NICER visits, Epoch 2 contains seven
NICER visits, and Epoch 4 contains two NICER visits for a total of 18
observations.

For our final data set of sixteen observations, response files were generated
with the nicerarf and nicerrmf tools. We generated background spectra using
the nibackgen3C50 model (Remillard et al. 2022) and binned spectra using the
FTOOL ftgrouppha with the optimum binning regime described in Kaastra &
Bleeker (2016).

Figure 2: Phase-averaged spectra for Observations 307 (left) and Observations
402 (right) with the best fit spectral model
tbabs*(cutoffpl+blackbody+apec+gaus) (top residual panel) The bottom two
residual panels show alternative model components. The middle residual panels
show the use of pow instead of cutoffpl which fails to accurately describe the
power law tail in all spectra. The bottom residual panels show the best fit
model without apec, the strength of which varies strongly between
observations. This effect is illustrated here by the significant low energy
residuals in Observation 402 (demonstrating a strong requirement for apec)
compared to the minimal change in residuals in Observation 307. However, even
in the case of Observation 307 the data prefer the presence of apec with a
$\Delta\chi^{2}$of 8.6.

###  2.2 Phase-averaged spectroscopy

We used Xspec to find the best fit phase-averaged spectral model (Arnaud
1996). We found that the soft spectral continuum was well described by a
thermal blackbody, but prominent residuals below 2 keV remained in most
spectra. Broad emission line features below 2 keV were also seen in previous
XMM-Newton EPIC-pn observations of SMC X-1 (Brumback et al. 2020). Chandra
ACIS observations of SMC X-1 reveal these lines to be blends of Ne IX, Ne X,
and O VIII (Vrtilek et al. 2001, 2005). With our NICER spectra, we found that
adding the collisionally-ionized diffuse gas model apec to our model provided
a good fit to the soft continuum (see Fig. 2). The apec model component varied
in strength between different observations (as illustrated by Fig. 2), and
generally was more strongly required in the brighter spectra. Two of the
faintest spectra, Observations 104 and 306, had poorly constrained apec
parameters that did not significantly improve the $\Delta\chi^{2}$of the fit,
and therefore we removed apec from the model in these two cases.

Additionally, we applied the photoelectric absorption model tbabs to model
X-ray absorption between NICER and SMC X-1. We used abundances from Wilms et
al. (2000) and cross-sections from Verner et al. (1996). We attempted to add a
second absorption component with abundances set to 20% Solar values, as has
been used in the past to represent absorption within the SMC itself, but found
that this component was not required by the model (normalization set to zero).

The high energy continuum was described as a non-thermal continuum with a
Gaussian emission line at 6.4 keV corresponding to Fe K$\alpha$. We trialed
several different power law models to fit the non-thermal continuum above 2
keV including a power law (pow), a cutoff power law (cutoffpl), a power law
with a high energy cutoff pow*highecut, and a negative and positive
exponential (NPEX, Mihara et al. 1998). The best fit was provided by cutoffpl,
although the value of the high energy cutoff was poorly constrained due to
NICER’s upper energy limit of 12 keV and this affected the accuracy of our
photon index values. Despite the poor constraints on the cutoff energy, the
cutoffpl model was strongly preferred by the data over the pure power law
model pow, which failed to correctly model the high energy emission (see Fig.
2).

In order to better constrain the photon index, we attempted to fix the cutoff
energy to the value given by archival NuSTAR data. To determine this, we used
the two high state NuSTAR Observations 30202004002 and 30202004008 and fit
them with a high energy X-ray continuum model of
constant*(tbabs*(cutoffpl+gaus)). By taking the average of the cutoff energies
from the two observations, we found a value of 11.06 keV for our NICER data
set. However, fixing the cutoff energy to this value resulted in
overestimation of the NICER data at high energies. To best fit the data while
constraining the high energy cutoff, we therefore allowed the cutoff energy to
vary around the NuSTAR value of 11.06 keV within the range of only 8–15 keV,
and we recognize that our data are not sensitive to this parameter.

####  2.2.1 Degeneracy between blackbody temperature and column density

With a best fit continuum model of tbabs*(cutoffpl+blackbody+apec+gaus), we
discovered degeneracy between the absorbing column density and the blackbody
temperature when both parameters were left free. We first used the HEASARC NH
Calculator to estimate the column density towards SMC X-1 from the HI4PI
survey, which suggested a density of $4.57\times 10^{21}$ cm-2 (HI4PI
Collaboration et al. 2016). This value was approximately four times larger
than the average column density required by our model with both the
$N_{\text{H}}$ and blackbody temperature free ($\sim 1\times 10^{21}$ cm-2).
Fixing the $N_{\text{H}}$ to the value from the HI4PI survey resulted in
significantly larger $\chi^{2}$ and poorer fits.

Previous and ongoing NICER studies of SMC X-1 suggest that the blackbody
temperature in SMC X-1 is a more constant spectral parameter than
$N_{\text{H}}$ (Dage et al. 2022). In keeping with these findings, we limited
the degeneracy between $N_{\text{H}}$ and blackbody temperature by only
allowing both parameters to be free in the brightest observation from each
Epoch (Observations 105, 204, 304, and 402). In all other observations, we
fixed the blackbody temperature to the value found in the brightest
observation from that Epoch. This method still allowed us to check for cycle-
to-cycle variations in blackbody temperature while limiting the degeneracy
with $N_{\text{H}}$.

After these tests, best fit phase averaged spectral model was
tbabs*(cutoffpl+blackbody+apec+ gaus), where the blackbody temperature was
frozen to the value determined by the brightest observation in each Epoch. We
limited the cutoff energy between 8–15 keV. The variations in key spectral
parameters are plotted in Figure 3 and given in Table 2. Fluxes for each
observation, calculated from the flux command in Xspec, are shown in Table 1.

###  2.3 Pulse profiles

In preparation for timing analysis and making pulse profiles, the data were
corrected to the Solar System barycenter using the FTOOL barycorr. The data
were further corrected for the orbital motion of SMC X-1 using the ephemeris
from Falanga et al. (2015), which gives the mid-eclipse time($T_{0}$) as
52846.688810(24) MJD, the orbital period at this epoch ($P_{\text{orb}}$) as
3.891923160(66) days, and the orbital decay
($\dot{P}_{\text{orb}}/P_{\text{orb}})$ as -3.541(2)$\times
10^{-6}\text{yr}^{-1}$.

Table 2: Phase-averaged spectral parameters

ObsID | $N_{\text{H}}$ | Photon | Cutoff EnergyaaThe cutoff energy is limited between 8–15 keV to provide constraints for this parameter. When the cutoff energy fits to the lower limit (8 keV) then it has only a positive error value, whereas when it fits to the upper limit (15 keV) it has only a negative error value. We emphasize that this parameter is poorly constrained in the NICER energy bandpass. | cutoffpl | kT${}_{\text{blackbody}}$ | blackbody | kT${}_{\text{apec}}$ | apec | Gaussian | d.o.f. | $\chi^{2}$ | Reduced
---|---|---|---|---|---|---|---|---|---|---|---|---
| ($10^{22}\text{cm}^{-2}$) | Index | (keV) | norm. | (keV) | norm. | (keV) | norm. | norm. |  |  | $\chi^{2}$
101 | 0.171(5) | 0.53(3) | 10(1) | 0.0427(5) | 0.21(fixed | 0.00081(4) | 1.01(3) | 0.0028(4) | 0.0012(2) | 136 | 221.82 | 1.63
102 | 0.169(5) | 0.57(3) | 11(1) | 0.0469(5) | 0.21(fixed) | 0.00089(5) | 1.08(7) | 0.0021(5) | 0.0009(2) | 136 | 184.09 | 1.35
103 | 0.177(6) | 0.44(4) | 8.2(8) | 0.0458(6) | 0.21(fixed) | 0.00101(4) | 1.2(1) | 0.003(1) | 0.0006(3) | 131 | 146.24 | 1.12
104 | 0.19(3) | 0.62(6) | 15(-7) | 0.042(3) | 0.21(fixed) | 0.0010(2) |  $\cdots$ |  $\cdots$ | 0.002(1) | 106 | 80.93 | 0.76
105 | 0.16(2) | 0.51(6) | 11(2) | 0.044(1) | 0.218(9) | 0.00097(8) | 1.11(8) | 0.0028(7) | 0.0004(3) | 128 | 148.88 | 1.16
201 | 0.122(5) | 0.37(5) | 8.4(9) | 0.0189(3) | 0.25(fixed) | 0.00035(2) | 0.89(7) | 0.0006(2) | 0.0004(1) | 129 | 158.42 | 1.23
203 | 0.117(5) | 0.36(1) | 8.0(+2) | 0.0312(1) | 0.25(fixed) | 0.00055(2) | 0.95(3) | 0.0022(3) | 0.0008(2) | 137 | 184.95 | 1.35
204 | 0.12(2) | 0.39(7) | 9(1) | 0.034(1) | 0.25(1) | 0.00062(4) | 1.00(4) | 0.0028(3) | 0.0007(2) | 128 | 209.22 | 1.63
301 | 0.13(1) | 0.5(1) | 15(-3) | 0.045(2) | 0.222(fixed) | 0.0011(1) | 1.0(1) | 0.003(1) | 0.0014(7) | 115 | 123.57 | 1.07
303 | 0.141(4) | 0.49(3) | 10(1) | 0.0444(5) | 0.222(fixed) | 0.00122(4) | 1.06(8) | 0.0021(5) | 0.001(2) | 139 | 265.33 | 1.91
304 | 0.14(1) | 0.51(4) | 11(1) | 0.046(1) | 0.222(5) | 0.00122(5) | 1.07(6) | 0.0031(4) | 0.0010(2) | 141 | 271.92 | 1.93
305 | 0.129(7) | 0.55(5) | 15(-2) | 0.0459(9) | 0.222(fixed) | 0.00106(7) | 1.10(7) | 0.0037(9) | 0.0008(4) | 130 | 154.64 | 1.19
306 | 0.120(5) | 0.43(4) | 10(1) | 0.0413(6) | 0.222(fixed) | 0.00096(4) |  $\cdots$ |  $\cdots$ | 0.0005(3) | 133 | 223.10 | 1.68
307 | 0.15(1) | 0.53(7) | 15(-3) | 0.046(1) | 0.222(fixed) | 0.0011(1) | 0.8(2) | 0.0016(9) | 0.0007(6) | 120 | 118.78 | 0.99
401 | 0.134(3) | 0.54(3) | 13(1) | 0.0488(5) | 0.226(fixed) | 0.00118(4) | 1.06(3) | 0.0038(4) | 0.0009(2) | 141 | 228.20 | 1.62
402 | 0.14(1) | 0.52(4) | 12(1) | 0.050(1) | 0.226(4) | 0.00124(5) | 1.07(4) | 0.0036(3) | 0.0009(2) | 142 | 316.03 | 2.23

Note. — Phase averaged spectral parameters for the base model of
tbabs*(cutoffpl+blackbody+apec+gaus). In Observations 104 and 306, the apec)
component was poorly constrained, with normalizations of effectively 0, so in
these two observations we removed the apec) model component. Astericks (*)
denote the brightest observation in each Epoch.

Figure 3: The key phase-averaged spectral parameters for our data set as a
function of time. Panel A (grey points): The Swift/BAT 15–50 keV light curve
for SMC X-1 (grey). Panel B (purple triangles): the absorbing column density
($N_{\text{H}}$). Panel C (navy circles): the blackbody temperature. Only one
point for each epoch, corresponding to the brightest observation, shows the
error bars. In the other observations, the temperature is fixed to the value
from the brightest observation. Panel D (blue squares): the photon index.
Panel E (light blue triangles): the plasma temperature from apec. Epochs 1 and
2, which occur during the intermediate state of the superorbital cycle, show
the most significant differences in spectral parameters between cycles. Epochs
3 and 4 have values more consistent with each other.

For each observation, we used the HENDRICS (Bachetti 2018) and Stingray
(Huppenkothen et al. 2019) software packages to perform the timing analysis.
First, we confirmed the presence of pulsations in each of our observations by
creating power density spectra in HENDRICS, which showed the presence of
strong pulsations at approximately 1.4 Hz. We used the HENDRICS folding search
to determine the frequency of the pulsations and the frequency first
derivative. In all observations, the spin frequency first derivative was
consistent with zero. We calculated the uncertainty of the spin frequency as
the change in observed pulse phase over the course of each observation
($\delta\nu=\delta\text{phase}/\delta\text{time}$). To calculate the change in
pulse phase we used Xselect to extract pulse profiles from short time
intervals at the beginning and end of each observation and measured the
difference in phase between the profiles. We present the pulse frequency in
Table 1 where the number in parentheses is the uncertainty on the last digit.

The entire light curve was used to make the final pulse profiles for each
observation, except in the case of light curves that had been previously
filtered to remove low count rate intervals as described in section 2.1. Final
pulse profiles in the entire NICER bandpass were made using the Stingray epoch
folding tool, the pulse frequency described above, and the start time of the
observation as the starting epoch. The pulse profiles are shown in Figure (4).

Figure 4: Pulse profiles for our data set in the full NICER bandpass. Pulse
profiles have been shifted to align the primary pulse peak and all plotted
with the same y-axis scale. All profiles show the double peaked structure
characteristic of SMC X-1, but the relative strength of the pulse peaks varies
between observations.

###  2.4 Pulse phase-resolved spectroscopy

We employed pulse phase-resolved spectroscopy in order to examine how spectral
parameters vary with spin phase. Each pulse profile was separated into four
phase bins corresponding to the primary and secondary peaks of the pulse
profile and the two troughs between them. We filtered by these spin phase bins
in Xselect to create four pulse phase-resolved spectra.

As expected, the pulse phase-resolved spectra are of weaker signal to noise
than the phase-averaged spectra. Even for the spectra corresponding to the
primary and secondary pulse peaks, where the signal is highest, we did not
detect emission lines present in the phase-averaged spectra. Therefore, we
removed the apec and gaussian model components corresponding and fit all pulse
phase-resolved spectra with the simplified tbabs*(cutoffpl+bbody) model. To
assist our spectral modeling, we also froze the blackbody temperature and the
cutoff energy to their phase-averaged values for each pulse phase-resolved
spectrum. The resulting spectral model allowed us to test how the
$N_{\text{H}}$, photon index, power law normalization, and blackbody
normalization varied with pulse phase.

For the sake of brevity, we plot pulse phase-resolved spectral parameters from
Observations 307 and 402, which illustrate the types of variations seen in our
data, in Figure 5.

Figure 5: Pulse phase-resolved spectral parameters for Observations 307 and
402 as an example of the behavior seen in the full data set. In all panels,
the solid black line is the full bandpass NICER pulse profile. Both the pulse
profile and the parameter values have been normalized against their mean
values to enable plotting on a uniform scale. First panel (blue squares): The
photon index shows strong variation with pulse phase, increasing in strength
during the pulse trough and decreasing during the pulse peak. Second panel
(purple triangles): The $N_{\text{H}}$ does not show changes with pulse phase
in some observations, like 307, while in others it does show changes as in
Observation 402. Third panel (navy circles) the normalization of the blackbody
component follows the shape of the pulse profile. Fourth panel (red squares):
The normalization of the power law component follows the shape of the pulse
profile.

##  3 Results and Discussion

###  3.1 Spectral variations with superorbital cycle

Our spectral analysis suggests that SMC X-1 does exhibit some changes in
spectral shape between superorbital cycles, but that these changes may depend
more on supeorbital phase than cycle length. The greatest difference can be
seen between the $N_{\text{H}}$, blackbody temperature, and photon indices of
Epochs 1 and 2, which both take place during SMC X-1’s intermediate state as
it is increasing in brightness (Figure 3). Epochs 3 and 4, which occur during
the brightest parts of their superorbital cycles when the source flux is not
changing as rapidly, show spectral parameters that are generally consistent
with each other.

The observations in Epoch 1 and Epoch 3 show the greatest overall distribution
in phase-averaged spectral parameters (Figure 3). This is potentially caused
by the varying exposure times of the observations in these epochs; both epochs
contain long exposures (greater than 2 ks) and short exposures (less than 1
ks), resulting in a wide range in spectral quality.

Our 16 phase-averaged spectral fits produced an average photon index of 0.49
and an average $N_{\text{H}}$ of 1.4 $\times 10^{21}\text{cm}^{-2}$. These
values are both smaller than the average photon index and $N_{\text{H}}$ found
by Dage et al. (2022) (0.69 and 2.48 $\times 10^{21}\text{cm}^{-2}$ ,
respectively), whose NICER observations of SMC X-1 occurred later during the
2020-2022 superorbital excursion. The difference in these values could arise
from differences in the spectral modeling; Dage et al. (2022) restricted the
cutoff energy in their fits to vary between 12 and 20 keV and we used a range
of 8 to 15 keV. The cutoff energy is poorly constrained in both our data set
(see large fluctuations in Table 2) and in Dage et al. (2022) (see their
Section 3.1). Additionally, Dage et al. (2022) used Gaussian emission lines to
model the soft emission line blends where we used the collisionally ionized
plasma model apec.

To test whether the difference between our average $N_{\text{H}}$and photon
index values compared to Dage et al. (2022) was a result of differing spectral
model, we applied their same spectral model to our 16 spectra. When using the
Dage et al. (2022) model on our data set, the mean $N_{\text{H}}$was
1.85$\times 10^{21}\text{cm}^{-2}$ and the mean photon index was 0.59. These
values are closer to the values of Dage et al. (2022) but not in full
agreement, which implies that our differing spectral model was only partly
responsible for the discrepancy.

One possible explanation of this discrepancy is spectral selection criteria.
Our data set comprises of both superorbital high and intermediate state
spectra that vary in exposure time and overall data quality. Dage et al.
(2022) modeled only high quality spectra from the superorbital high state.
Indeed, when we examine our $N_{\text{H}}$and photon index values from our
brightest observations fit with the Dage et al. (2022) spectral model, we find
individual photon indices in better agreement with their values. An
alternative explanation for the difference in averaged photon index and
$N_{\text{H}}$ values between our work and Dage et al. (2022) is that SMC X-1
has undergone an intrinsic spectral change as it progresses through its
excursion and period of disk instability. A direct analysis of all the NICER
data of SMC X-1 during excursion would determine whether or not this is the
case, but we defer such an analysis for future work.

Despite the difference in average value between our continuum parameters and
those of Dage et al. (2022), our analysis does confirm their finding that the
spectral shape of the bright state observations are general consistent. Dage
et al. (2022) modeled 11 spectra from the bright state of SMC X-1’s
superorbital cycle between April 2021 and January 2022 and found very little
fluctuation in their spectral parameters over this time period (see Table 2 of
Dage et al. 2022). We see similar consistency within our Epoch 3 and Epoch 4
bright state data. The overall consistency of SMC X-1’s bright state spectrum
with time appears to imply that the inner accretion flow may be largely
unaffected by changes to the accretion disk geometry during excursion.

###  3.2 Pulse profiles and pulse phase-resolved spectroscopy

In Figure 4 we show the NICER pulse profiles for the 16 observations that we
analyzed. Although all the pulse profiles have double peaked shape
characteristic of SMC X-1, both the strength of the pulsations and the
relative strengths of the pulse peaks vary with time. Similar to our spectral
results, changes in pulse profile shape are small within a single epoch and
larger between epochs, although overall changes in pulse strength can be seen
as exposure times vary, especially in Epoch 1.

Our spectral analysis indicated very little change between the spectral shape
of Epochs 3 and 4, and our timing analysis provides further evidence of the
similarity between high state properties. Our pulse profiles from Epochs 3 and
4 (Fig. 4, right panel) show the least variation both in shape and strength.

In our pulse phase-resolved spectral analysis, we filtered each observation by
pulse phase into four bins corresponding to the two peaks and two troughs of
the pulse profile. We used a simplified spectral model of
tbabs*(bbody+cutoffpl) with the blackbody temperature and cutoff energy fixed
to their phase-averaged values to reduce degeneracy in the pulse phase-
resolved spectra.

In all observations, the photon index showed a clear anti-correlation with
pulse profile (Figure 5, top panels); the photon index increased during the
troughs of the pulse profile and decreased during the pulse peaks. We
interpret this as viewing the accretion column during the pulse peaks, which
results in a harder X-ray spectrum.

The pulse phase-resolved results for the $N_{\text{H}}$ are more challenging
to interpret. For some observations (as illustrated by Observation 307 in
Figure 5), the $N_{\text{H}}$ is consistent with being constant across pulse
phase. In other observations (as illustrated by Observation 402), significant
changes in $N_{\text{H}}$ are seen with pulse phase, but these changes do not
show a clear trend with pulse phase. The fact that we needed to freeze the
blackbody temperature in order to obtain meaningful information from the
$N_{\text{H}}$ means that some of these variations could be due to changes in
blackbody temperature, such as seeing the neutron star’s hot spot rotate in
and out of the line of sight. Alternatively, the changes in $N_{\text{H}}$
could be more sensitive to binary phase and the position of the stellar wind.
Ultimately, the variations in $N_{\text{H}}$ with spin phase are not
sufficiently clear to draw physical conclusions.

Finally, the blackbody and power law normalization values follow the shape of
the overall pulse profile as it varies in flux, which is consistent with our
expectations for brighter spectra from the pulse peaks.

##  4 Conclusions

In this work we analyzed 16 NICER observations of the SgXB SMC X-1. These
observations spanned four different superorbital cycles, with Epochs 1 and 2
occurring during the intermediate state of the superorbital cycle and Epochs 3
and 4 occurring during the bright state. We performed both phase-averaged and
pulse phase-resolved spectral analysis and also examined the pulse profiles
for each of our observations. We summarize our primary findings here.

The pulse phase-averaged spectral parameters from the superorbital high state
(Epochs 3 and 4) show little difference from one another. This indicates that
the soft X-ray spectrum, which tracks the inner accretion flow, is likely
insensitive to changes in superorbital cycle length and accretion disk
geometry. This result is supported by our timing analysis, which indicates
that the pulse profiles from Epochs 3 and 4 show little variation in shape and
strength.

The pulse phase-averaged spectral parameters from the intermediate state of
the superorbital cycle (Epochs 1 and 2) show more variation between cycles and
significant differences in pulse profile shape. This indicates that the soft
X-ray spectrum of the superorbital intermediate state is less consistent,
possibly due to fact that, during this part of the cycle, we witness the
neutron star emerging from behind the accretion disk so our view of the
neutron star is rapidly changing. Our pulse phase-resolved spectroscopy also
showed a strong anti-correlation between pulse peak and photon index, however
variations in $N_{\text{H}}$ are less clear.

We would like to thank the anonymous referee for comments and suggestions that
strengthened the analysis and manuscript. MCB acknowledges support from NASA
grant 80NSSC21K0263. GV acknowledges support by H.F.R.I. through the project
ASTRAPE (Project ID 7802). KCD acknowledges fellowship funding from Fonds de
Recherche du Québec $-$ Nature et Technologies, Bourses de recherche
postdoctorale B3X no. 319864. Joel B. Coley acknowledges support by NASA award
number 80GSFC17M0002.

## References

  * Arnaud (1996) Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
  * Bachetti (2015) Bachetti, M. 2015, MaLTPyNT: Quick look timing analysis for NuSTAR data, Astrophysics Source Code Library.  http://ascl.net/1502.021
  * Bachetti (2018) —. 2018, HENDRICS: High ENergy Data Reduction Interface from the Command Shell.  http://ascl.net/1805.019
  * Brightman et al. (2022) Brightman, M., Bachetti, M., Earnshaw, H., et al. 2022, ApJ, 925, 18, doi: 10.3847/1538-4357/ac3829
  * Brumback et al. (2020) Brumback, M. C., Hickox, R. C., Fürst, F. S., et al. 2020, ApJ, 888, 125, doi: 10.3847/1538-4357/ab5b04
  * Brumback et al. (2021) —. 2021, ApJ, 909, 186, doi: 10.3847/1538-4357/abe122
  * Brumback et al. (2022) Brumback, M. C., Grefenstette, B. W., Buisson, D. J. K., et al. 2022, ApJ, 926, 187, doi: 10.3847/1538-4357/ac4d24
  * Dage et al. (2019) Dage, K. C., Clarkson, W. I., Charles, P. A., Laycock, S. G. T., & Shih, I. C. 2019, MNRAS, 482, 337, doi: 10.1093/mnras/sty2572
  * Dage et al. (2022) Dage, K. C., Brumback, M., Neilsen, J., et al. 2022, MNRAS, 514, 5457, doi: 10.1093/mnras/stac1674
  * Falanga et al. (2015) Falanga, M., Bozzo, E., Lutovinov, A., et al. 2015, A&A, 577, A130, doi: 10.1051/0004-6361/201425191
  * Gendreau et al. (2016) Gendreau, K. C., Arzoumanian, Z., Adkins, P. W., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99051H, doi: 10.1117/12.2231304
  * Greenhill et al. (2003) Greenhill, L. J., Booth, R. S., Ellingsen, S. P., et al. 2003, ApJ, 590, 162, doi: 10.1086/374862
  * Gúrpide et al. (2021) Gúrpide, A., Godet, O., Vasilopoulos, G., Webb, N. A., & Olive, J. F. 2021, A&A, 654, A10, doi: 10.1051/0004-6361/202140781
  * HI4PI Collaboration et al. (2016) HI4PI Collaboration, Ben Bekhti, N., Flöer, L., et al. 2016, A&A, 594, A116, doi: 10.1051/0004-6361/201629178
  * Hickox & Vrtilek (2005) Hickox, R. C., & Vrtilek, S. D. 2005, ApJ, 633, 1064, doi: 10.1086/491596
  * Hu et al. (2019) Hu, C.-P., Mihara, T., Sugizaki, M., Ueda, Y., & Enoto, T. 2019, ApJ, 885, 123, doi: 10.3847/1538-4357/ab48e4
  * Hu et al. (2023) Hu, C.-P., Dage, K. C., Clarkson, W. I., et al. 2023, MNRAS, 520, 3436, doi: 10.1093/mnras/stad384
  * Huppenkothen et al. (2019) Huppenkothen, D., Bachetti, M., Stevens, A. L., et al. 2019, ApJ, 881, 39, doi: 10.3847/1538-4357/ab258d
  * Kaastra & Bleeker (2016) Kaastra, J. S., & Bleeker, J. A. M. 2016, A&A, 587, A151, doi: 10.1051/0004-6361/201527395
  * King et al. (2023) King, A., Lasota, J.-P., & Middleton, M. 2023, New A Rev., 96, 101672, doi: 10.1016/j.newar.2022.101672
  * Leong et al. (1971) Leong, C., Kellogg, E., Gursky, H., Tananbaum, H., & Giacconi, R. 1971, ApJ, 170, L67, doi: 10.1086/180842
  * Liller (1973) Liller, W. 1973, ApJ, 184, L37, doi: 10.1086/181283
  * Martin (2008) Martin, R. G. 2008, MNRAS, 387, 830, doi: 10.1111/j.1365-2966.2008.13275.x
  * Mihara et al. (1998) Mihara, T., Makishima, K., & Nagase, F. 1998, Advances in Space Research, 22, 987, doi: 10.1016/S0273-1177(98)00128-8
  * Ogilvie & Dubus (2001) Ogilvie, G. I., & Dubus, G. 2001, MNRAS, 320, 485, doi: 10.1046/j.1365-8711.2001.04011.x
  * Pringle (1996) Pringle, J. E. 1996, MNRAS, 281, 357, doi: 10.1093/mnras/281.1.357
  * Remillard et al. (2022) Remillard, R. A., Loewenstein, M., Steiner, J. F., et al. 2022, AJ, 163, 130, doi: 10.3847/1538-3881/ac4ae6
  * Reynolds et al. (1993) Reynolds, A. P., Hilditch, R. W., Bell, S. A., & Hill, G. 1993, MNRAS, 261, 337, doi: 10.1093/mnras/261.2.337
  * Schreier et al. (1972) Schreier, E., Levinson, R., Gursky, H., et al. 1972, ApJ, 172, L79, doi: 10.1086/180896
  * Smale & Boyd (2012) Smale, A. P., & Boyd, P. T. 2012, ApJ, 756, 146, doi: 10.1088/0004-637X/756/2/146
  * Vasilopoulos et al. (2021) Vasilopoulos, G., Koliopanos, F., Haberl, F., et al. 2021, ApJ, 909, 50, doi: 10.3847/1538-4357/abda49
  * Verner et al. (1996) Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, ApJ, 465, 487, doi: 10.1086/177435
  * Vrtilek et al. (2001) Vrtilek, S. D., Raymond, J. C., Boroson, B., et al. 2001, ApJ, 563, L139, doi: 10.1086/338733
  * Vrtilek et al. (2005) Vrtilek, S. D., Raymond, J. C., Boroson, B., & McCray, R. 2005, ApJ, 626, 307, doi: 10.1086/429718
  * Webster et al. (1972) Webster, B. L., Martin, W. L., Feast, M. W., & Andrews, P. J. 1972, Nature Physical Science, 240, 183, doi: 10.1038/physci240183b0
  * Wilms et al. (2000) Wilms, J., Allen, A., & McCray, R. 2000, ApJ, 542, 914, doi: 10.1086/317016
  * Wojdowski et al. (1998) Wojdowski, P., Clark, G. W., Levine, A. M., Woo, J. W., & Zhang, S. N. 1998, ApJ, 502, 253, doi: 10.1086/305893