11institutetext: International Space Science Institute (ISSI), Hallerstrasse
6, 3012 Bern, Switzerland
11email<EMAIL_ADDRESS>22institutetext: Institut für Astronomie
und Astrophysik, Kepler Center for Astro and Particle Physics, Universität
Tübingen, Sand 1, 72076 Tübingen, Germany 33institutetext: School of Physics
and Astronomy, University of Southampton, Southampton, Hampshire SO17 1BJ, UK
44institutetext: Department of Astronomy, University of Geneva, Chemin
d’Ecogia 16, 1290, Versoix, Switzerland 55institutetext: Institut de Recherche
en Astrophysique et Planétologie, UPS-OMP, CNRS, CNES, 9 avenue du Colonel
Roche, BP 44346, F-31028 Toulouse Cedex 4, France 66institutetext: DTU Space,
Technical University of Denmark, Elektrovej 327-328, DK-2800 Lyngby, Denmark
77institutetext: European Space Agency (ESA), European Space Astronomy Centre
(ESAC), Camino Bajo del Castillo s/n, 28692 Villanueva de la Cañada, Madrid,
Spain 88institutetext: MIT Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
99institutetext: Embry Riddle Aeronautical University, Department of Physics
Prescott Campus, 3700 Willow Creek Road, Prescott, AZ 86301 1010institutetext:
Center for Astrophysics and Space Sciences, University of California San
Diego, La Jolla, California 92093 1111institutetext: Dipartimento di Fisica,
Università degli Studi di Cagliari, SP Monserrato-Sestu km 0.7, 09042
Monserrato, Italy 1212institutetext: Remeis-Observatory and Erlangen Centre
for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
Sternwartstr. 7, 96049 Bamberg, Germany

# The unaltered pulsar: GRO J1750-27, a supercritical X-ray neutron star that
does not blink an eye

C. Malacaria 11 L. Ducci 22 M. Falanga 11 D. Altamirano 33 E. Bozzo 44 S.
Guillot 55 G. K. Jaisawal 66 P. Kretschmar 77 M. Ng 88 P. Pradhan 99 R.
Rothschild 1010 A. Sanna 1111 P. Thalhammer 1212 J. Wilms 1212

When accreting X-ray pulsars (XRPs) undergo bright X-ray outbursts, their
luminosity-dependent spectral and timing features can be analyzed in detail.
The XRP GRO J1750-27 recently underwent one such episode, during which it was
observed with NuSTAR and monitored with NICER. Such a data set is rarely
available, as it samples the outburst over more than 1 month at a luminosity
that is always exceeding ${\sim}5\times 10^{37}\,$erg/s. This value is larger
than the typical critical luminosity value, where a radiative shock is formed
above the surface of the neutron star. Our data analysis of the joint spectra
returns a highly ($N_{H}\sim(5-8)\times 10^{22}\,$cm-2) absorbed spectrum
showing a K$\alpha$ iron line, a soft blackbody component likely originating
from the inner edge of the accretion disk, and confirms the discovery of one
of the deepest cyclotron lines ever observed, at a centroid energy of
${\sim}44\,$keV corresponding to a magnetic field strength of $4.7\times
10^{12}\,$G. This value is independently supported by the best-fit physical
model for spectral formation in accreting XRPs which, in agreement with recent
findings, favors a distance of 14 kpc and also reflects a bulk-Comptonization-
dominated accretion flow. Contrary to theoretical expectations and
observational evidence from other similar sources, the pulse profiles as
observed by NICER remain remarkably steady through the outburst rise, peak and
decay. The NICER spectrum, including the iron K$\alpha$ line best-fit
parameters, also remain almost unchanged at all probed outburst stages,
similar to the pulsed fraction behavior. We argue that all these phenomena are
linked and interpret them as resulting from a saturation effect of the
emission from the accretion column, which occurs in the high-luminosity
regime.

###### Key Words.:

X-ray: binaries –stars: neutron –pulsars: individual: GRO J1750-27 –accretion,
accretion disks–magnetic fields

##  1 Introduction

Accreting X-ray pulsars (XRPs) are binary systems consisting of a neutron star
(NS) and a donor companion star. In these systems, the NS can accrete matter
supplied by the companion either via stellar wind or Roche-lobe overflow, and
therefore emit in the X-ray domain. Most of these systems pertain to the
subclass of Be/X-ray binaries (BeXRBs), in which the donor star is of B
spectral type, expelling its wind under the form of a circumstellar decretion
disk, characterized by H$\alpha$ Balmer emission lines. For recent reviews of
such systems see, e.g., Malacaria et al. (2020) and Mushtukov & Tsygankov
2022.

GRO J1750-27 is an XRP discovered with the BATSE observatory (Wilson et al.,
1995), which also detected pulsations at about 4.4 s and an orbital period of
about 30 days (Scott et al., 1997). More recently, the spin period was
measured at ${\sim}4.45\,$s (Shaw et al., 2009; Devaraj & Paul, 2022).
Although no optical counterpart has yet been unambiguously identified, the
X-ray timing behavior (Scott et al., 1997) and the infrared spectral
properties (Lutovinov et al., 2019) hint at a BeXRB system at a distance of
between 14 and 22 kpc. The closest Gaia counterpart in the Early Data Release
3 (Fabricius et al., 2021, EDR3) is found at $2.8\,$arcsec from the Chandra
source position determined by Lutovinov et al. (2019) and outside of its 90%
confidence level (c.l.) ellipse region, thus the two sources are likely not
associated. In addition, parallax nor photogeometric distance has been
measured for the Gaia counterpart (Bailer-Jones et al., 2021). More recently,
Sharma et al. (2022) used accretion torque modeling to further constrain the
distance to 13.6-16.4 kpc.

During the BATSE era, the only outburst observed from GRO J1750-27 was the
same that led to its discovery, and only a few additional outbursts have been
observed by the currently operating all-sky monitors (Krimm et al., 2008;
Finger & Wilson-Hodge, 2014; Boissay et al., 2015). The most recent outburst
was observed in 2021 (Malacaria et al., 2021a). In 2008, the source exhibited
an outburst that was monitored with INTEGRAL and Swift/XRT, and peaked at
about 250 mCrab (Shaw et al., 2009). The INTEGRAL ISGRI and JEM-X1 joint
spectra ($3-50\,$keV) at the peak of the outburst were fit with a cutoff
power-law ($\Gamma=-0.15,E_{cut}=6.0\,$keV) or a Comptonization model (CompTT,
Titarchuk 1994, with plasma temperature kT${}_{e}=4.6\,$keV, and plasma
optical depth $\tau=6.4$).

During the 2021 outburst, we initiated a monitoring campaign with NICER and
NuSTAR. Preliminary results from those observations allowed to identify an
electron cyclotron resonant scattering feature (CRSF) for the first time in
this source at ${\sim 43}\,$keV (Malacaria et al., 2022b; Devaraj & Paul,
2022; Sharma et al., 2022). Here we further investigate the implications of
the spectral analysis carried out with phenomenological and physical models,
and take advantage of NICER data to explore the timing behavior and pulse-
profile evolution of the source throughout the outburst.

Figure 1: Swift/BAT (15-50 keV) daily average light curve of GRO J1750-27
during the outburst in 2021 (black dots with gray error bars). Start times of
each pointed observation are marked by vertical colored lines as detailed in
the legend. NICER ObsIDs showing evidence of a secondary peak in the pulse
profile (see text) are marked with a longer line.

##  2 Data reduction

A log of all used observations is shown in Table 1, while a Swift/BAT light
curve of the outburst is shown in Fig. 1. The outburst started around
September 15 2021 (MJD ${\sim}59470$). Swift/BAT data present several
observational gaps during the decaying phase, but according to continuous
monitoring by
Fermi/GBM111https://gammaray.nsstc.nasa.gov/gbm/science/pulsars/lightcurves/groj1750.html.,
the source finalized its outburst decay only in January 2022 (MJD
${\sim}59590$).

All data were reduced using instrument-specific pipelines provided by HEASOFT
v6.29c. Spectral data were analyzed using XSPEC v12.12.0 (Arnaud, 1996).

Table 1: Log of the source observations used in this work. | ObsID | MJD | Exposure
---|---|---|---
|  | (Start) | [ks]
NuSTAR | 90701331002 | 59484.2 | 29.9
NICER | 4202350101 | 59481.8 | 1.7
| 4202350104 | 59491.8 | 1.2
| 4202350105 | 59491.9 | 0.6
| 4202350106 | 59493.2 | 2.3
| 4202350107 | 59495.2 | 1.7
| 4202350108 | 59497.5 | 1.1
| 4202350109 | 59500.5 | 0.1
| 4202350110 | 59504.9 | 0.1
| 4202350111 | 59505.1 | 0.9
| 4202350112 | 59506.9 | 0.9
| 4202350113 | 59507.4 | 0.9
| 4202350114 | 59509.1 | 1.6
| 4202350115 | 59511.3 | 1.6
| 4202350116 | 59513.2 | 2.1
| 4202350117 | 59515.2 | 1.4
| 4202350118 | 59519.2 | 1.8
| 4202350119 | 59520.0 | 1.3
| 4202350120 | 59521.3 | 0.9
| 4202350121 | 59522.1 | 1.9

###  2.1 NuSTAR

NuSTAR (Harrison et al., 2013) was launched in 2012 and is currently the only
X-ray mission with a telescope able to focus hard X-rays up to 79 keV. NuSTAR
consists of two identical co-aligned telescopes that focus X-ray photons onto
two independent Focal Plane Modules, FPMA and FPMB. At the focus of each
telescope module are four ($2\times 2$) solid-state cadmium zinc telluride
(CdZnTe) imaging detectors. These provide wide-band (3–79 keV) energy coverage
with a FWHM of $18\arcsec$ and a spectral resolution of 400 eV at 10 keV, for
a field of view of 10 arcmin at 10 keV.

NuSTAR observed GRO J1750-27 on September 27, 2021 (ObsID 90701331002, MJD
59484.2) following a Director’s Discretionary Time (DDT) request. The filtered
total exposure time was about 30 ks. NuSTAR data were reduced with NUSTARDAS
v2.1.1 and using the CALDB 20211020 (Madsen et al., 2021). Cleaned events were
obtained following the standard NuSTAR guidelines. Source spectra were
extracted through the NUPRODUCTS routine. The source extraction region was a
$65\arcsec$ radius circular region centered on the Chandra source position.
Due to the position angle (PA) observational constraints and the vicinity of
other bright sources, the NuSTAR field of view for both modules is
contaminated by stray light and ghost rays. A careful background extraction
was therefore applied, in concert with the instrument team. Given that NuSTAR
internal background varies from detector to detector, the background should be
extracted from the same detector as the source when possible (Madsen et al.,
2015). We therefore extracted the background from a source-free and
contamination-free region of comparable radius to that of the source
extraction region on the same detector (Det $0$) and compared it to the
background spectrum of other detectors. This allowed us to verify that the
background spectrum was free from artificial, detector-dependent features.
Moreover, a spectrum of the contaminating flux was extracted and analyzed.
Compared to the source spectrum, the contaminating spectrum was found to be
about 300 times fainter, free from prominent features, and dominated by the
background counts above 20 keV. We therefore concluded that our NuSTAR source
spectrum can be safely considered free from contamination. The energy band of
NuSTAR spectral data was limited to the $4-60\,$keV energy range, above which
the background counts dominate the spectrum, and to prevent calibration
uncertainties in the $3-4\,$keV energy band.

Table 2: Best-fit results of GRO J1750-27 joint analysis for the joint NuSTAR
and NICER (ObsID 4202350104) spectral data with different best-fit solution
for the Model I (Deep and Smooth)
tbabs*(bbodyrad+highecut*pow+gaussKα[+gaussbroad])*gabs, Model II
tbabs*(bbodyrad+cutoffpl+gaussKα)*gabs, and Model III
tbabs*(bbodyrad+bwcyc+gaussKα)*gabs (see text). All reported errors are at
$1\sigma\,$c.l. and based on the MCMC chain values.

| Model I | Model II | Model III
---|---|---|---
| Deep | Smooth | Cutoffpl | BWcyc
CFPMA (fixed) | 1 | 1 | 1 | 1
CFPMB | $1.018^{+0.002}_{-0.002}$ | $1.018^{+0.002}_{-0.002}$ | $1.018^{+0.008}_{-0.002}$ | $1.018^{+0.002}_{-0.003}$
CNICER | $1.414^{+0.010}_{-0.007}$ | $1.408^{+0.010}_{-0.007}$ | $1.408^{+0.008}_{-0.008}$ | $1.407^{+0.006}_{-0.008}$
N${}_{\textrm{H}}$ [$10^{22}\,$cm-2] | $4.64^{+0.08}_{-0.05}$ | $8.24^{+0.33}_{-0.09}$ | $4.74^{+0.07}_{-0.06}$ | $6.2^{+0.1}_{-0.2}$
kT${}_{\rm BB}\,$[keV] | $1.27^{+0.02}_{-0.03}$ | $0.137^{+0.004}_{-0.006}$ | $1.29^{+0.02}_{-0.02}$ | $0.134^{+0.003}_{-0.004}$
normBB | $9.7^{+0.5}_{-0.5}$ | $(6.4^{+5.5}_{-1.2})\,{\times 10^{6}}$ | $9.8^{+0.4}_{-0.4}$ | $(4.1^{+0.7}_{-0.5})\,{\times 10^{5}}$
E${}_{\rm K\alpha}\,$[keV] | $6.47^{+0.02}_{-0.02}$ | $6.45^{+0.02}_{-0.02}$ | $6.47^{+0.02}_{-0.02}$ | $6.46^{+0.01}_{-0.01}$
$\sigma_{\rm K\alpha}\,$[keV] | $0.21^{+0.02}_{-0.02}$ | $0.18^{+0.03}_{-0.02}$ | $0.21^{+0.02}_{-0.02}$ | $0.18^{+0.01}_{-0.02}$
normKα [$10^{-4}$ ph/cm2/s] | $6.3^{+0.5}_{-0.5}$ | $5.3^{+0.6}_{-0.4}$ | $6.1^{+0.4}_{-0.4}$ | $5.2^{+0.1}_{-0.1}$
$\Gamma$ | $-1.2^{+0.1}_{-0.1}$ | $1.43^{+0.09}_{-0.01}$ | $-1.2^{+0.1}_{-0.1}$ | –
norm${}_{\Gamma}^{*}$ | $(1.2^{+2.9}_{-1.9})\,{\times 10^{-3}}$ | $(8.1^{+0.1}_{-0.2})\,{\times 10^{-2}}$ | $(1.8^{+0.4}_{-0.3})\,{\times 10^{-3}}$ | –
HighEcut [keV] | – | – | $6.1^{+0.3}_{-0.3}$ | –
cutoffE [keV] | $3.2^{+0.5}_{-0.4}$ | $19.1^{+0.8}_{-1.9}$ | – | –
foldE | $6.6^{+0.3}_{-0.1}$ | $13.6^{+2.5}_{-0.5}$ | – | –
E${}_{\rm Gauss}\,$[keV] | – | $5.4^{+0.6}_{-0.9}$ | – | –
$\sigma_{\rm Gauss}\,$[keV] | – | $10.49^{+1.36}_{-0.05}$ | – | –
normGauss [ph/cm2/s] | – | $0.106^{+0.018}_{-0.004}$ | – | –
$\xi$ | – | – | – | $1.35^{+0.03}_{-0.04}$
$\delta$ | – | – | – | $1.9^{+0.1}_{-0.1}$
B [$10^{12}\,$G] | – | – | – | $4.62^{\star}$
$\dot{M}$ [$10^{17}\,$g/s] | – | – | – | $6.74^{+0.11}_{-0.08}$
Te [keV] | – | – | – | $4.19^{+0.04}_{-0.05}$
r0 [m] | – | – | – | $87.6^{+5.2}_{-8.0}$
d [kpc] | – | – | – | 14.0 (fixed)
normbwcyc | – | – | – | $(7.7^{+0.6}_{-0.9})\,{\times 10^{-2}}$
E${}_{\rm cyc}\,$[keV] | $44.0^{+0.8}_{-0.3}$ | $43.3^{+1.0}_{-0.5}$ | $44.0^{+0.6}_{-0.6}$ | $43.6^{+0.3}_{-0.2}$
$\sigma_{\rm cyc}\,$[keV] | $10.0^{+0.6}_{-0.2}$ | $6.3^{+1.4}_{-0.5}$ | $9.6^{+0.6}_{-0.5}$ | $10.5^{+0.9}_{-0.2}$
Strength${}_{\rm cyc}^{\ddagger}$ ($\tau_{cyc}^{\ddagger}$ [keV]) | $50.6^{+1.3}_{-0.7}\,(2.02^{+0.09}_{-0.14})$ | $19.8^{+6.5}_{-3.1}\,(1.6^{+0.4}_{-0.2})$ | $46.2^{+4.9}_{-5.6}\,(1.9^{+0.3}_{-0.3})$ | $46.7^{+4.2}_{-2.7}\,(1.8^{+0.2}_{-0.2})$
Flux† (1-60 keV) | $2.336^{+0.005}_{-0.004}$ | $2.52^{+0.01}_{-0.01}$ | $2.346^{+0.005}_{-0.005}$ | $2.427^{+0.007}_{-0.006}$
$\chi^{2}$/d.o.f. | $3156/3395$ | $3308/3392$ | $3364/3396$ | $3435/3393$
Goodness-of-fit | $44\%$ | $45\%$ | $40\%$ | $45\%$
Null-hypothesis P | $70\%$ | $85\%$ | $68\%$ | $28\%$

222 ∗In units of photons/keV/cm2/s at 1 keV. ⋆Linked to the Ecyc parameter
(see text). ‡Line depth (optical depth) as defined in XSPEC. †Unabsorbed flux
calculated for the entire model (in units of $10^{-9}\,$erg cm${}^{-2}\,$s-1),
obtained using the cflux model from XSPEC as resulting from FPMA.

###  2.2 NICER

NICER (Gendreau & Arzoumanian, 2017) is an X-ray telescope deployed on the
International Space Station (ISS) in 2017 June. NICER X-ray Timing Instrument
(XTI) has 56 aligned focal plane modules FPMs (52 currently operational), each
made up of an X-ray concentrator optic associated with a silicon drift
detector. The peak collecting area of all FPMs combined is $1900\,$cm2 at 1.5
keV, with a field of view of 30 arcmin2. NICER is capable of fast-timing
observations in the $0.2-12.0$ keV band, with timing accuracy of time-tagged
photons to better than 100 ns (Prigozhin et al., 2016; LaMarr et al., 2016;
Gendreau & Arzoumanian, 2017; Okajima et al., 2016). The spectral resolution
is about 140 eV at 6 keV.

Upon Target of Opportunity (ToO) request, NICER monitored GRO J1750-27 also in
coordination with NuSTAR. However, simultaneous NICER observations were
prevented by ISS visibility constraints. NICER started observing GRO J1750-27
on September 24, 2021 (ObsID 4202350101, MJD 59481.8), and continued to
monitor the source with variable cadence until November 4, 2021 (ObsID
4202350121). The total exposure time was about 24 ks after data cleaning.
Hereinafter, NICER ObsIDs are identified by their last two digits, that is
42023501XX. We also verified that NICER data were not contaminated by other
sources in the relatively large field of view. In fact, the field of view
includes the supergiant fast X-ray transient IGR J17503-2636 (Ferrigno et al.,
2019). However, the same sky region was also monitored by INTEGRAL, and IGR
J17503-2636 was not X-ray active during our NICER observations.

NICER data were processed with HEASoft version 6.29c and the NICER Data
Analysis Software (nicerdas) version 8 (2021-08-31_V008c) with Calibration
Database (CALDB) version xti20210720, adopting standard calibration and
screening criteria from the nicerl2 tool. The adopted version of nicerdas also
produces proper response files (RMF and ARF) for each ObsID. The background
spectrum was obtained using the nibackgen3C50 v7
tool333https://heasarc.gsfc.nasa.gov/docs/nicer/tools/nicer_bkg_est_tools.html
(Remillard et al., 2022). Hot detectors and additional off detectors in a few
ObsIDs were also identified and manually excluded during the data-reduction
pipeline. ObsIDs 02 and 03 were not considered here due to their null exposure
time. As advised by the instrument team, a systematic error of 1% has been
applied for all NICER spectra. The energy band of NICER spectra was limited to
$0.7-10\,$keV, below which background counts dominate the spectrum, and to
prevent calibration uncertainties in the $10-12\,$keV energy band.

Table 3: Best-fit results of GRO J1750-27 NICER-only spectral analysis using a
cutoff power-law model cutoffpl with an iron K$\alpha$ line. All reported
errors are at $1\sigma\,$c.l., obtained using the err tool from XSPEC.

ObsID | NH | $\Gamma$ | HighEcut | norm${}^{*}_{\Gamma}$ | Flux† | $\chi^{2}/\rm d.o.f.$ | E${}_{\rm K\alpha}\,$ | $\sigma_{\rm K\alpha}\,$ | norm${}_{\rm K\alpha}^{\star}$ |
---|---|---|---|---|---|---|---|---|---|---
|  $[10^{22}\,$cm${}^{-2}]$ |  | [keV] |  |  |  | [keV] | [keV] |  |
4202350101 | $4.8(2)$ | $0.06(12)$ | $10.3_{-1.9}^{+3.0}$ | $1.6(2)$ | $0.593(9)$ | $1078/944$ | – | – | – |
4202350104 | $5.2(1)$ | $0.01(12)$ | $7.7_{-1.1}^{+1.4}$ | $2.8(2)$ | $1.11(2)$ | $1035/942$ | $6.53(8)$ | $0.16(6)$ | $0.05(2)$ |
4202350105 | $5.4(2)$ | $0.1(2)$ | $8.9_{-1.8}^{+3.2}$ | $3.4(4)$ | $1.14(2)$ | $1097/935$ | $6.59(9)$ | $0.21(8)$ | $0.1(3)$ |
4202350106 | $5.3(1)$ | $0.03(8)$ | $8.5_{-0.9}^{+1.1}$ | $3.4(2)$ | $1.24(1)$ | $1138/942$ | $6.56(4)$ | $0.20(4)$ | $0.11(2)$ |
4202350107 | $5.3(1)$ | $0.03(8)$ | $8.0_{-0.9}^{+1.2}$ | $3.8(2)$ | $1.31(1)$ | $1066/942$ | $6.53(4)$ | $0.14(4)$ | $0.08(2)$ |
4202350108 | $5.3(1)$ | $-0.05(10)$ | $7.4_{-0.9}^{+1.2}$ | $3.7(2)$ | $1.39(1)$ | $1070/942$ | $6.61(8)$ | $0.20(7)$ | $0.09(2)$ |
4202350109 | $6.1(4)$ | $0.6(2)$ | $33(unc.)$ | $6.3(6)$ | $1.55(7)$ | $1006/887$ | $6.33(3)$ | $0.1(unc.)$ | $0.04(3)$ |
4202350110 | $5.7(5)$ | $0.3(3)$ | $14_{-5}^{+15}$ | $4.7(9)$ | $1.49(7)$ | $1136/877$ | $6.47(7)$ | $0.4(1)$ | $0.3(1)$ |
4202350111 | $5.5(2)$ | $-0.01(12)$ | $7.2_{-0.9}^{+1.3}$ | $4.7(4)$ | $1.58(2)$ | $1036/919$ | $6.56(9)$ | $0.19(8)$ | $0.09(3)$ |
4202350112 | $5.2(2)$ | $-0.17(12)$ | $6.1_{-0.7}^{+0.9}$ | $3.8(3)$ | $1.45(2)$ | $992/942$ | $6.6(1)$ | $0.3(1)$ | $0.098(4)$ |
4202350113 | $5.3(2)$ | $-0.06(12)$ | $7.2_{-0.9}^{+1.3}$ | $3.8(3)$ | $1.40(2)$ | $1133/941$ | $6.4(1)$ | $0.01(1)$ | $0.04(1)$ |
4202350114 | $5.4(1)$ | $0.08(9)$ | $8.7_{-1.1}^{+1.4}$ | $4.3(3)$ | $1.46(2)$ | $1093/942$ | $6.50(4)$ | $0.17(4)$ | $0.10(2)$ |
4202350115 | $5.0(1)$ | $-0.17(9)$ | $6.2_{-0.6}^{+0.7}$ | $3.9(3)$ | $1.48(1)$ | $1045/942$ | $6.52(6)$ | $0.23(6)$ | $0.12(3)$ |
4202350116 | $5.3(1)$ | $-0.22(9)$ | $5.4_{-0.4}^{+0.5}$ | $3.3(3)$ | $1.23(1)$ | $1076/942$ | $6.62(6)$ | $0.23(6)$ | $0.09(2)$ |
4202350117 | $5.4(1)$ | $0.08(9)$ | $8.8_{-1.2}^{+1.7}$ | $4.2(3)$ | $1.44(1)$ | $987/942$ | $6.51(5)$ | $0.15(5)$ | $0.09(2)$ |
4202350118 | $5.5(1)$ | $0.13(9)$ | $9.4_{-1.3}^{+1.8}$ | $3.7(3)$ | $1.22(2)$ | $1027/942$ | $6.45(6)$ | $0.16(8)$ | $0.08(2)$ |
4202350119 | $5.4(1)$ | $0.06(11)$ | $8.3_{-1.2}^{+1.6}$ | $3.5(3)$ | $1.18(2)$ | $926/932$ | $6.56(9)$ | $0.17(9)$ | $0.05(2)$ |
4202350120 | $5.7(2)$ | $0.17(14)$ | $9.1_{-1.7}^{+2.7}$ | $3.8(4)$ | $1.13(2)$ | $1055/892$ | $6.52(6)$ | $0.02(9)$ | $0.03(2)$ |
4202350121 | $5.7(1)$ | $0.40(9)$ | $21_{-6}^{+12}$ | $3.9(3)$ | $1.18(1)$ | $945/932$ | $6.64(8)$ | $0.29(9)$ | $0.10(2)$ |

444 ∗In units of $10^{-2}\,$photons/keV/cm2/s at 1 keV. †Unabsorbed flux
calculated for the entire model in the $0.5-10\,$keV energy band and reported
in units of $10^{-9}\,$erg cm${}^{-2}\,$s-1. ⋆In units of
$10^{-2}\,$photons/cm2/s. unc. Unconstrained value.

##  3 Data analysis and results

###  3.1 Spectral analysis

####  3.1.1 Spectral setup

Both background and source spectra were rebinned to have at least one count
per bin (similarly to, e.g., Snios et al. 2020) in order to use the
C-statistic (Cash, 1979) with Poissonian background (W-stat) as a fit
statistic. Such rebinning addresses possible biases when using
W-stat555https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/XSappendixStatistics.html.
Given the non-Poissonian nature of NICER background (Remillard et al., 2022),
the pgstat statistic was used for NICER data. On the other hand, the
$\chi^{2}$ was employed as a test statistic. For all tested models, the
photoelectric absorption component was set according to Wilms et al. (2000,
tbabs in XSPEC) to account for photoelectric absorption by neutral
interstellar matter (or column density NH), and we assumed model-relative
(wilm) solar abundances. The Galactic NH in the direction of the source is
about $1.1\times 10^{22}\,$cm-2 (HI4PI Collaboration et al., 2016).

We performed spectral analysis both for the joint NICER and NuSTAR spectra
(using data from the neighbor NICER ObsID 04, see Sect. 3.1.2), and for the
single NICER observations (see Sect. 3.1.3). In the following, errors are
reported at $1\sigma\,$c.l. in tables and figures. Errors for the joint NICER
and NuSTAR spectral analysis are calculated through MCMC simulations using the
Goodman-Weare algorithm of length $2\times 10^{5}$ with 20 walkers and
$10^{4}$ burn-in steps. For the joint fit, the goodness-of-fit parameter is
also reported as resulting from the goodness tool from XSPEC (with fit and
nosim options), indicating the percentage of simulated spectra with a test
statistic better than the one obtained from real data. In those cases, for
each relevant model, we also report the null-hypothesis probability (NHP) as
resulting from the XSPEC fitting procedure, which represents the probability
of the data being drawn from the best-fit model. For the single NICER ObsIDs
spectral analysis, errors are calculated through the err tool in XSPEC without
any chains loaded, as this method is less computational expensive (compared to
MCMC simulations) and, at the same time, reliable for steady spectra such as
those analyzed here (which do not show any local minima in their best-fit
solution). Our spectral analysis results are reported in Sect. 3.1.2 and Table
2 for the joint NICER and NuSTAR analysis and in Sect. 3.1.3 and Table 3 for
NICER-only analysis.

####  3.1.2 Joint NICER and NuSTAR spectral analysis

Malacaria et al. (2022b) reported the GRO J1750-27 spectral fitting results
for the joint NICER and NuSTAR data analysis. The best-fit continuum spectrum
presented by those authors consists of an absorbed power law with high-energy
cutoff (highecut*powerlaw) modified by an ad hoc broad Gaussian emission
component (gauss) that serves two purposes. First, it smooths the steep high-
energy cutoff, thus providing a good fit of data around the cutoff energy.
Second, the spectrum also shows an absorption feature at about 43 keV, modeled
with a Gaussian absorption line and interpreted as an electron CRSF. However,
the cyclotron line depth is found to be unusually large (i.e., optical depth
$\tau>2$) if the ad hoc broad component is not added to the model spectrum.
The addition of the broad Gaussian emission component therefore mitigates the
otherwise exceptionally deep cyclotron line.

The best-fit spectrum also includes an iron K$\alpha$ line at 6.4 keV.
Finally, positive residuals remaining around 1 keV were modeled with a
blackbody component, which returned a lower $\chi^{2}$ compared to a partial
covering component. The entire model (see Fig. 2 and “Model I Smooth” in Table
2) fits the data well, but one disadvantage is that it needs the broad
Gaussian emission component, which mimics a high-energy bump but has no
physical interpretation. For completeness, Table 2 also reports the results of
the best-fit model without the broad Gaussian emission component, which
results in a deeper CRSF (“Model I Deep”).

Figure 2: NICER (green) and NuSTAR (black and red) combined spectrum of GRO
J1750-27 . Top panel: Data (crosses) and best-fit model (continuous lines).
Middle: Ratio residuals (data divided by model) using Model I Smooth (see
Table 2) but without the inclusion of a cyclotron line. Bottom: Residuals from
the best-fit model including a Gaussian absorption line near 43 keV to account
for the cyclotron resonant scattering feature. Data were rebinned for plotting
purposes. Figure 3: Same as Fig. 2, but for the bwcyc model (see Table 2).
Figure 4: Evolution of the investigated fundamental quantities for GRO
J1750-27 as observed by NICER in each ObsID (x-axis) of the outburst monitored
in 2021. Top panel (a): Flux in the $1-10\,$keV band. Middle panel (b):
Hardness ratio [${4-10}\,$keV/${1-4}\,$keV]. Bottom panel (c): Pulsed
fraction.

Building upon those results, we also searched for alternative and possibly
more physically consistent spectral models (i.e., not requiring the ad hoc
broad Gaussian emission component) that also return a more commonly observed
cyclotron line depth. Our analysis takes advantage of the broader energy
coverage including NICER data with respect to Sharma et al. (2022) and Devaraj
& Paul (2022), whose analysis is limited to the NuSTAR energy band. This is
significant given the peculiar steepness of the spectral continuum and the
complex cyclotron line profile, both of which can be better constrained thanks
to the lower-energy continuum data. The following models were therefore
tested: an absorbed Comptonization model of soft photons in a hot plasma
(CompTT in XSPEC, Titarchuk 1994), cutoffpl, highecut*powerlaw, a power-law
modified by a Fermi-Dirac cutoff powerlaw*FDcut (Tanaka, 1986), and the
Negative and Positive power laws with a common EXponential cutoff NPEX (Mihara
et al., 1998). Of these, the best-fit model showing the highest NHP is the
cutoff power-law model (see Table 2). The spectral model highecut*powerlaw was
also employed in other works to fit the continuum of GRO J1750-27 as observed
by NuSTAR, although modified by the inclusion of different components, such as
a nonharmonic cyclotron line (Sharma et al., 2022) or a smoothing absorption
feature centered at the cutoff energy (Devaraj & Paul, 2022). However, we also
separately tested the nonharmonic cyclotron line and the smoothing absorption
feature around the cutoff energy but, as also reported by the respective
authors, we noticed that their inclusion only slightly reduces the fundamental
line depth. Therefore, we opted to focus on a different model that does not
include additional absorption components, and to report the results of the
cutoff power-law model in Table 2.

Moreover, two physical models for spectral formation based on bulk and thermal
Comptonization in accreting XRPs were also tested, namely the bwcyc model
(Becker & Wolff, 2007; Ferrigno et al., 2007) and the compmag model (Farinelli
et al., 2012). Of these, the bwcyc model returned a $\chi^{2}$ value much
lower than the compmag model (the difference in test statistic between the two
models was $\Delta\chi^{2}\sim 400$), and the former was therefore employed in
our analysis (see Fig. 3 and Table 2). Following the official usage
guidelines, we kept the following model parameters frozen during the fit: NS
radius RNS and mass MNS, distance to the source $D$, and magnetic field
strength $B$. For those parameters we chose the following values: R${}_{\rm
NS}=12\,$km, M${}_{\rm NS}{=}1.4\,\rm M_{\odot}$, $\rm D{=}14\,$kpc, ${\rm
B{=}E_{cyc}^{best-fit}}((1+z)/11.6){\times}10^{12}\,$G (where $\rm E_{\rm
cyc}^{\rm best-fit}$ is the best-fit centroid energy of the cyclotron line at
about 44 keV, and $z\approx 0.24$ is the gravitational redshift). Among the
other model parameters are the so-called similarity parameters $\xi$ and
$\delta$. These describe, respectively, the importance of the photons escape
through the accretion column and the relative importance of bulk and thermal
Comptonization:

$\quad\quad\quad\quad\xi=\frac{\pi
r_{0}m_{p}c}{\dot{M}\sqrt{\sigma_{\parallel}\sigma_{\perp}}},\quad\quad\quad\delta=4\frac{y_{bulk}}{y_{therm}}\propto\frac{\sigma_{\parallel}}{\bar{\sigma}}.$
(1)

The other model parameters are: the accretion column radius $r_{0}$, the mass
accretion rate $\dot{M}$, and the electron temperature of the Comptonizing
electrons $T_{e}$, while $m_{p}$ is the proton mass, $y_{bulk}$ and
$y_{therm}$ are the Compton $y-$parameters as defined in Becker & Wolff
(2007), and $\bar{\sigma}$ and $\sigma_{\parallel}$ are the electron
scattering cross-sections angle-averaged and parallel to the magnetic field,
respectively. The above-mentioned frozen set of values represents the best fit
when compared to other configurations of the same model where the distance was
frozen at 16 or 18 kpc.

####  3.1.3 NICER ObsIDs spectral analysis

Figure 5: Heat map of NICER pulse profiles from GRO J1750-27 as monitored
during the 2021 outburst. The black vertical dashed line at spin phase
${\sim}0.95$ marks the pulse profile peak for ObsID 01. The bottom horizontal
bar shows the color-coded normalized intensity. The color-coded ObsIDs
reported on the left y-axis show the bolometric luminosity (as derived by
Swift/BAT count rates, see text) at each ObsID as illustrated by the color bar
to the right. The vertical yellow dashed line at spin phase ${\sim}0.6$
confines the ObsIDs where the secondary peak appears more prominently. Pulse
profiles are plotted twice for clarity and bins have been smoothed with a
Gaussian filtering for visibility purpose.

NICER spectral data from each ObsID have also been modeled to investigate
spectral variations throughout the observed outburst stages. Due to the
restricted energy band, only a relatively simple model was tested, namely an
absorbed cutoff power-law model with the addition of a Gaussian emission line
for the Fe K$\alpha$ around 6.4 keV. Results are reported in Table 3. The
model can fit the data at all ObsIDs, returning best-fit spectral continuum
parameters that are roughly consistent throughout all observations.

The iron K$\alpha$ line in GRO J1750-27 as observed by NICER has an energy of
about $6.3-6.6\,$keV. This is consistent with emission of fluorescent lines
from neutral or weakly ionized iron. The only observation that did not require
an iron line component (i.e., normalization consistent with zero) is the ObsID
01, at the lowest flux observed by NICER. The nondetection of the iron line in
ObsID 01 was also tested through Monte Carlo simulations. To this aim, the
XSPEC simftest routine was employed, which allows the user to simulate a given
number of spectra based on the actual data and test the resulting
$\Delta\chi^{2}$ between each instance of the fit with and without the model
component to be tested (the Gaussian line in our case). A total of $10^{4}$
simulations were carried out. The obtained $\Delta\chi^{2}$ from fitting the
actual data with and without the iron line is about 6, but about $10\%$ of the
simulations show a higher $\Delta\chi^{2}$, thus suggesting that the observed
improvement is likely due to statistical fluctuations (significance
$<2\sigma$).

To further investigate the spectral behavior, we explored the spectral
evolution in a model-independent way, that is by means of the hardness ratio
(HR). We define the HR as the count-rate ratio in two different energy bands,
i.e., HR$\,=4-10$ keV / $1-4$ keV. Fig. 4(a and b) shows the evolution of the
HR as observed during NICER observations. Except for the first ObsID, there is
no significant variation of the HR along the outburst.

Figure 6: Selection of representative NICER pulse profiles. The labels on the
bottom left corner of each panel represent the correspondent ObsID. The red
dashed vertical line marks the secondary peak that is more relevant in ObsID
11 at $\phi=0.6$ (see also Fig. 5). The blue dashed line shows the pulse
profile from ObsID 01 rescaled for reference. Pulse profiles are plotted twice
for clarity and normalized by the average source intensity in a given ObsID.

###  3.2 Timing analysis

To analyze the NICER pulse profiles at different outburst stages, barycentered
events were extracted in the $1-10\,$keV energy band using JPL-DE405 Solar
System ephemeris. We then searched for pulsations for each ObsID using the
epoch folding method (Leahy et al., 1983) as employed by the efsearch tool in
HEASOFT. For reference, a spin period of 4.45150(1) was obtained for ObsID 01
(the pulse period uncertainty was estimated through simulations following the
method outlined in Lutovinov et al. 2012; Boldin et al. 2013). Events from
each ObsID were then folded with their correspondent pulse period to create
pulse profiles. To phase-align the pulse profiles, a cross-correlation was
performed between two consecutive ObsID profiles. The phase-aligned pulse
profiles are shown in the heat map in Fig. 5.

By comparing the flux obtained by the joint NICER and NuSTAR spectral analysis
($2.346(5)\times 10^{-9}\,$erg cm${}^{-2}\,$s-1, see Table 2, Model II) with
the contemporary Swift/BAT count rate (0.015(2) cnt cm${}^{-2}\,$s-1) at MJD
58484, we obtain a conversion factor $C_{f}=1.5(2)\times 10^{-7}\,$erg cm-2 to
estimate the $1-60\,$keV flux from the Swift/BAT count rate, which we use as a
proxy for the bolometric flux. Such an estimate represents a valid
approximation if the spectral emission does not change its shape, a condition
that is well satisfied here (see Sect. 3.1.3). By applying this conversion
factor, and assuming a distance value of 14 kpc (see Sect. 3.1.1), a
corresponding luminosity value was associated to each Swift/BAT data point and
therefore to each correspondent NICER observation. This information was
included in the NICER pulse-profile heat map in Fig. 5. A selection of
representative NICER pulse profiles is also shown in Fig. 6.

Finally, we also investigated the evolution of the pulsed fraction (PF) during
the outburst. For this, the PF was defined as ($I_{\rm max}-I_{\rm
min}$)/($I_{\rm max}+I_{\rm min}$), where $I_{\rm max}$ and $I_{\rm min}$ are
the maximum and minimum pulse-profile count rates, respectively. The observed
PF evolution is shown in Fig. 4c.

##  4 Discussion

###  4.1 The broadband spectrum

The joint NuSTAR and NICER spectral analysis of GRO J1750-27 allowed us to
confirm the discovery of a cyclotron line at about 44 keV. For cyclotron
lines, the centroid energy of the fundamental line is linked to the magnetic
field as E${}_{cyc}\approx\,B_{12}\,11.6\,E_{cyc}/(1+z)$ (where B12 is the
magnetic field strength in units of $10^{12}\,$G). This corresponds to
B$\simeq 4.7\times 10^{12}\,$G. This value is in agreement with recent
findings from (Malacaria et al., 2022b; Devaraj & Paul, 2022; Sharma et al.,
2022). The cyclotron line also shows an unusually large optical depth
($\tau\simeq 2$, see Table 2). For comparison, some of the deepest fundamental
cyclotron lines were observed in A0535+26 (Grove et al., 1995) and V 0332+53
(Mowlavi et al., 2006), with optical line depths $\tau=1.8$ and $\tau=2.11$,
respectively666The optical depth values are derived here following the
definition of the gabs function in XSPEC.. V 0332+53 also shows some of the
deepest harmonic lines (Pottschmidt et al., 2005), with $\tau=3.3$ for the
second harmonic at $\sim 74\,$keV. More commonly, the average cyclotron line
depth is $\tau=0.5-1$ (Staubert et al., 2019; Malacaria et al., 2021b, 2022a),
although one must also account for the anti-correlation of the line depth with
luminosity (Mowlavi et al., 2006; Tsygankov et al., 2010). However, it is also
worth noting that even super-Eddington XRPs do not show unusually large values
of the cyclotron line depth (Jaisawal & Naik, 2016; Kong et al., 2022).
Moreover, we note that the line depth in GRO J1750-27 remains large with most
spectral models tested here and elsewhere (Sharma et al., 2022; Devaraj &
Paul, 2022). All these aspects support the interpretation that the large line
depth has a physical origin and is not merely an artefact of the imperfect
spectral modeling.

Several spectral models have been tested for the broadband emission from GRO
J1750-27. For all tested spectral models, the best-fit column absorption value
is several times larger than the Galactic absorption in the direction of the
source, in agreement with Lutovinov et al. (2019). We found that a cutoff
power-law model can fit the data with a negative power-law photon index (see
Model I Deep and Model II in Table 2). Although a positive power-law photon
index is typically observed in XRPs, a negative photon index has been observed
in a few sources (Iyer et al., 2015; Aftab et al., 2019) where the spectrum is
generally harder than the typical XRP spectrum. Here, when the analysis is
limited to the narrower NICER energy band (see Sect. 3.1.3 and Table 3), the
obtained photon index is in general much softer than that obtained when the
broadband spectrum is considered (i.e., Model I Deep). However, the negative
photon index in Model I Deep is softened when the emission excess around the
cutoff energy is modeled with a broad Gaussian emission component (see Model I
Smooth in Table 2). This way, the obtained photon index value is in agreement
with the more commonly observed positive values. Such a hard bump is
reminiscent of the 10-keV feature observed in many accreting XRPs (Coburn et
al., 2002). Even so, the hard bump feature in GRO J1750-27 is broader than the
10-keV width observed elsewhere (see, e.g., Klochkov et al. 2007; Ferrigno et
al. 2009). Just as there is no commonly accepted interpretation of the 10-keV
feature (that is, its physical origin in unknown), the hard bump observed in
GRO J1750 is likely due to the inadequacy of the adopted phenomenological
model, but it nonetheless highlights the fact that the observed spectral shape
for GRO J1750-27 differs from those commonly observed in accreting XRPs.

We also notice that the blackbody component best-fit parameters return two
different solutions according to the employed continuum model. One solution
(see Model I Deep and Model II in Table 2), similar to the solution obtained
by Sharma et al. 2022, consists of a blackbody component with $kT\sim
1.3\,$keV and an emitting radius of about 4 km (assuming a distance of 14
kpc). Sharma et al. 2022 ascribe this component to the NS surface. However, as
also pointed out by Hickox et al. 2004 for example, it is worth noting that
the blackbody emission from the NS surface is typically colder (i.e., $\sim
0.1\,$keV) and only extends to the accreting polar cap (i.e., $\sim
10^{2}\,$m, see also below). Even when hot (e.g., $1.3\,$keV) blackbody
components have been ascribed to the NS surface, their emitting radius was
found to be consistent with the polar cap radius (see, e.g., La Palombara et
al. 2012). The other solution (see Model I Smooth and Model III in Table 2)
returns a blackbody component with $kT\sim 0.1\,$keV and an emitting radius of
the order of $10^{3}\,$km. This is consistent with reprocessing by optically
thick gas at the inner edge of the accretion disk, assumed to be at the
magnetospheric radius $R_{m}\sim 10^{8}\,$cm (see Hickox et al. 2004).

The physical model bwcyc also fits the data adequately. The best-fit
configuration for the bwcyc spectral fit was obtained with the distance
parameter frozen to the value of 14 kpc, thus favouring recent independent
findings from Sharma et al. (2022). The best-fit parameters are in agreement
with those found for other similar sources (Thalhammer et al., 2021; Wolff et
al., 2016). Also, the relatively large value of the $\delta$ parameter
confirms that the accretion process is dominated by bulk motion
Comptonization. Moreover, the best-fit accretion column radius, $r_{0}\sim
90\,$m, is much smaller than the blackbody-emitting area ascribable to the NS
surface (see above and Model I Deep and Model II in Table 2). This result
therefore disfavors the interpretation of the blackbody-emission component as
originating from the polar cap. Similarly to Thalhammer et al. (2021), we can
use the similarity parameters $\xi$ and $\delta$ to derive the cross-sections
$\bar{\sigma}\sim 3.5\times 10^{-5}\sigma_{T}$ and $\sigma_{\parallel}\sim
5\times 10^{-6}\sigma_{T}$ (while the cross-section perpendicular to the
magnetic field, $\sigma_{\perp}$, is fixed to the Thomson cross-section within
the model). The results are consistent with the expectation that
$\sigma_{\parallel}<\bar{\sigma}<\sigma_{\perp}$ (Canuto et al., 1971).
Additionally, the model was able to fit the data with the magnetic field
strength parameter linked to the value derived by the cyclotron line best-fit
centroid energy, which supports the cyclotron line finding and the magnetic
field value derived by it.

###  4.2 A steady spectrum throughout the outburst

Figure 7: Pulsed fraction (y-axis) as a function of luminosity (x-axis in
units of $10^{37}\,$erg/s and for a distance of 14 kpc) for GRO J1750-27 as
observed by NICER along the outburst in 2021.

The spectral analysis of single NICER ObsIDs reveals a spectrum that is almost
unchanging throughout the probed outburst stages (see Table 3), both in its
continuum (such as the photon index $\Gamma$), and in its discrete features
(that is the Fe K$\alpha$ line best-fit parameters).

In XRPs, the iron line is nearly ubiquitously observed (see, e.g., Nagase
1991). It is believed to be due to fluorescence emission resulting from
reflection of the primary X-ray emission usually off the accretion disk (but
also off the stellar wind or the surface of the donor star, Basko et al. 1974;
Inoue 1985). The iron line parameters observed in this work do not show any
clear correlation with flux (see Table 3). Several other XRPs, on the
contrary, show a clear dependence of the iron line flux with the observed flux
from the X-ray source (see, e.g., Reig & Nespoli 2013; Jaisawal et al. 2019).

The spectral steadiness is also confirmed by the model-independent
investigation of the HR (see Fig. 4b). This is contrary to other sources
where, even for small luminosity variability (i.e., $\Delta L_{X}{\sim}10\%$),
a hardening or softening of the spectrum is observed according to their
accretion regime (see, e.g., Klochkov et al. 2011; Reig & Nespoli 2013; Fürst
et al. 2014; Malacaria et al. 2015). Furthermore, distinct accretion regimes
are separated by a certain critical luminosity (Basko & Sunyaev, 1976; Becker
et al., 2012; Mushtukov et al., 2015b), where the luminosity becomes strong
enough that a radiation shock forms and stops the accretion flow above the NS
surface. As a reference, the critical luminosity obtained employing the model
by Becker et al. (2012, see their Eq. 55) is:

$\begin{split}L^{a}_{\rm crit}=&1.28\times
10^{37}\,erg/s\left(\frac{\Lambda}{0.1}\right)^{-7/5}\left(\frac{M_{\rm
NS}}{1.4M_{\odot}}\right)^{29/30}\\\ &\left(\frac{R_{\rm
NS}}{10\,km}\right)^{1/10}\left(\frac{E_{\rm cyc}}{10\,\rm
keV}\right)^{16/15},\end{split}$ (2)

where $\Lambda$ is the accretion flow geometry constant, and MNS and RNS are
the mass and radius of the NS, respectively. Adopting $\Lambda=0.1$ (for disk
accretion), R${}_{\rm NS}=12\,$km, M${}_{\rm NS}{=}1.4\,\rm M_{\odot}$ and
$E_{cyc}=44\,$keV in Eq. 2 results in a critical luminosity of $L^{a}_{\rm
crit}=6.3\times 10^{37}\,$erg/s. On the other hand, for a value of the
cyclotron line energy of E${}_{\rm cyc}=44\,$keV, Mushtukov et al. (2015b)
predicts a critical luminosity value $L_{\rm crit}^{b}\sim 2\times
10^{37}\,$erg/s (see, e.g., their Fig. 7 for the case of pure X-mode
polarization).

However, Postnov et al. (2015) observed a flattening of the hardness ratio
above the critical luminosity in several sources (i.e., 4U 0115+63, V 0332+53,
EXO 2030+375, A 0535+26 and MXB 0656-072). Kühnel et al. (2017) also observed
indications of a photon index break at high luminosity in GRO J1008-57 (see
their Fig. 6), implying a possible constant spectrum above the critical
luminosity. The spectral results presented here in Table 3 and Fig. 4 are
therefore consistent with the saturation effect of the spectral hardness
described in Postnov et al. (2015). Their interpretation is based on the
fraction of radiation reflected off the NS atmosphere with respect to the
total emission. Since Compton scattering is the most important process
affecting the reflected radiation (see, e.g. Poutanen et al. 2013), softer
photons are absorbed while harder photons are scattered. Therefore, the
reflected radiation from the NS atmosphere is harder than its incident
emission, and the spectrum becomes harder as the luminosity increases.
However, such a trend breaks down in the supercritical accretion regime, where
the accretion column grows so high that the fraction of reflected radiation
from the NS surface to the total emission becomes increasingly small, thus
producing a flattening of the spectral HR. As the GRO J1750-27 observations
analyzed here were all carried out at a high-luminosity regime, and the
observed HR remains remarkably flat, we might interpret the flat HR as the
plateau observed by Postnov et al. (2015) and as due to the saturation effect
at work in the supercritical regime. This might imply that the accretion
column has reached the maximum height imposed by the accretion physics
limitations described in Poutanen et al. (2013) and Mushtukov et al. (2015a).
Following their predictions, we infer a column height of $\sim 2-5\,$km for
the highest luminosity observed in this work from GRO J1750-27. Additional
evidence supporting the saturation effect scenario is provided by the timing
analysis and related discussion (Sect. 3.2 and 4.3, respectively).

###  4.3 A steady pulse profile and pulsed fraction

Our timing analysis of GRO J1750-27 as observed by NICER during the outburst
in 2021 reveals a remarkably steady pulse profile (see Fig.s 5, 6) and pulsed
fraction (see Fig. 4c). The pulse-profile heat map shows a dominant single
peak at pulse phase ${\sim}1.0$. Nonetheless, a transient feature appears at
$\phi{\sim}0.6$ only for a few consecutive NICER observations during the
outburst rising stage, in the form of a secondary peak. The feature shows up
not only at ObsIDs 09 and 10, where the secondary peak might be an artefact
due to the short exposure times, but also at ObsID 11, where in fact it
appears more prominently. Moreover, a broadening of the main peak as the
outburst evolves is also apparent in Figs. 5 and 6. According to our derived
conversion factor $C_{f}$ (see Sect. 3.2), the bolometric luminosity
corresponding to those ObsIDs is about $9(1)\times 10^{37}\,$erg s-1, while a
peak luminosity of about $1.1(1)\times 10^{38}\,$erg s-1 is reached during
ObsID 15. However, the same feature is not present during the decline of the
outburst, when a similar value of the luminosity is crossed again. It is worth
noting that a similar secondary peak, anticipating the main peak in phase by
about $\sim 0.3$, also appears in the NuSTAR $9-18\,$keV pulse profile (see
Fig. 5 from Devaraj & Paul 2022). Apart from this transient feature, the pulse
profiles do not show any appreciable luminosity dependence. This is contrary
to several other sources (see, e.g., Klochkov et al. 2008; Malacaria et al.
2015; Epili et al. 2017; Koliopanos & Vasilopoulos 2018; Wilson-Hodge et al.
2018; Wang et al. 2022), where a drastic change of the pulse profile is
observed as a function of luminosity, and is interpreted as a change of the
accretion column beaming pattern (Basko & Sunyaev, 1976; Becker et al., 2012;
Mushtukov et al., 2015b). Here, despite the probed outburst stages ranging
from a bolometric luminosity of about $5\times 10^{37}\,$erg s-1 to a peak
luminosity that is about $60\%$ of the Eddington luminosity for a 1.4 M⊙ NS
(see Fig. 5), the pulse profile remains almost unchanged, suggesting that the
source is most likely always observed by NICER in the same, supercritical
accretion regime. This also can be interpreted as an indication of the
saturation effect at work in the supercritical regime as indicated by Postnov
et al. (2015) and already invoked in Sect. 4.2. In fact, at the probed high-
luminosity values the pulse profile is not affected by dramatic beaming
pattern changes or by emission reflected off the NS surface, which might alter
its overall shape. As a consequence, the pulse profile appears almost
unaltered throughout the observed outburst stages.

Finally, we notice that the PF behavior also seems to follow a similar
saturation effect to that shown by the spectral HR (see Sect. 3.1.1). Indeed,
the PF shows only a minor decrease during the rising part of the outburst, and
then flattens around $25\%$ (see Fig. 4c). This suggests a similar
interpretation where, above a certain flux level, the beaming pattern
contributions do not significantly change as a function of the accretion rate,
nor does the radiation reflected off the NS surface affect the PF at any
appreciable level. Moreover, we report the pulsed-fraction variability as a
function of luminosity in Fig. 7. Several XRPs have been observed to show a
luminosity-dependent PF, which is often interpreted in terms of beaming
pattern and geometrical changes (Lutovinov & Tsygankov, 2009; Yang et al.,
2018; Gupta et al., 2019; Lutovinov et al., 2021). The relatively large error
bars and lack of data for intermediate-luminosity stages in Fig. 7 prevent us
from drawing any firm conclusions. However, except for the lowest luminosity
data point, the data suggest a PF that is insensitive to the observed
bolometric luminosity.

##  5 Summary

We performed spectral and timing analysis of the accreting XRP GRO J1750-27 as
observed by NICER and NuSTAR at the peak of its latest outburst toward the end
of 2021. Our main results can be summarized as follows:

  * •

We tested unexplored spectral models and confirm the discovery of a cyclotron
line from this source at about 44 keV, implying a magnetic field strength of
${\sim}4.7\times 10^{12}\,$G (consistent with recent independent findings).
The cyclotron line depth is possibly amongst the largest ever observed. Our
results also highlight a highly absorbed source ($N_{H}\sim(5-8)\times
10^{22}\,$cm -2), and a soft blackbody component likely originating from
reprocessing at the inner edge of the accretion disk.

  * •

The spectral data can be fitted with a physical model based on bulk and
thermal Comptonization (bwcyc), favoring a distance of 14 kpc and supporting
the magnetic field strength derived by the cyclotron line. This model also
features a bulk-Comptonization-dominated spectrum, a relatively narrow
accretion column, and a scattering cross-section parallel to the magnetic
field that is severely reduced relative to the Thomson cross-section.

  * •

The NICER monitoring shows an almost immutable spectrum throughout all
observations. This is interpreted here as a saturation effect occurring above
the critical luminosity and due to emission from the accretion column
reflected off the surface of the NS (Postnov et al., 2015).

  * •

The 0.5-10 keV pulse profiles as monitored by NICER show a simple one-peaked
shape that remains almost identical throughout the outburst. The hint of a
secondary peak appears at an intermediate luminosity during the outburst rise
(but not during the decay). This is similar to what is observed by NuSTAR at
harder energies (i.e., 9-18 keV).

  * •

The pulsed fraction also appears to be independent of time and luminosity,
supporting the saturation effect interpretation.

To the best of our knowledge, this might be the first time that the saturation
effect is observed in the spectral properties, the pulsed fraction, and the
pulse profile shape at the same time. Given the elusive nature of several
physical effects at work in GRO J1750-27, of which the present analysis
reveals only a mere glimpse, an investigation into the luminosity dependence
of the cyclotron line at lower accretion rates is encouraged, together with an
X-ray high- and low-luminosity spectropolarimetry study, which could help us
to discern the accretion column beaming pattern and the amount of radiation
reflected off the surface of the NS.

###### Acknowledgements.

This research has made use of data and software provided by the High Energy
Astrophysics Science Archive Research Center (HEASARC), which is a service of
the Astrophysics Science Division at NASA/GSFC and the High Energy
Astrophysics Division of the Smithsonian Astrophysical Observatory. We
acknowledge extensive use of the NASA Abstract Database Service (ADS). This
work used data from the NuSTAR mission, a project led by the California
Institute of Technology, managed by the JPL, and funded by NASA, and has
utilised the NUSTARDAS software package, jointly developed by the ASI Science
Data Center, Italy, and the California Institute of Technology, USA. DA
acknowledges support from the Royal Society. This work was supported by NASA
through the NICER mission and the Astrophysics Explorers Program.

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