X-ray and optical observations of the millisecond pulsar binary PSRJ1431-4715
D. de Martino A. Phosrisom V. S. Dhillon D. F. Torres F. Coti Zelati R. P. Breton T. R. Marsh A. Miraval Zanon N. Rea A. Papitto








  X-ray and optical observations of PSR J1431-4715
	D. de Martino et al.



INAF-Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, I-80131 Naples, Italy

         domitilla.demartino@inaf.it
           Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK
        Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, UK
	Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain
        Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, E-08193 Barcelona, Spain
        Institut d'Estudis  Espacials de Catalunya (IEEC), Carrer de Gran Capitá 2-4, E-08034 Barcelona, Spain
        Institució Catalana de Recerca i Estudis Avançats (ICREA), E-08010 Barcelona, Spain
        INAF-Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807 Merate (LC), Italy
        Department of Physics, University of Warwick, Coventry CV4 7AL, UK
        ASI - Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Rome, Italy
        INAF-Osservatorio Astronomico di Roma, Via Frascati 33, I-00040 Monte Porzio Catone (RM), Italy
	






We present the first X-ray observation of the energetic millisecond pulsar binary
PSR J1431-4715, performed with and complemented with fast optical 
multi-band photometry acquired with the instrument at ESO-NTT. 
It is found as a faint X-ray source without a significant orbital modulation. This
contrasts with the majority of systems that instead display substantial X-ray orbital
variability.
The X-ray spectrum is dominated by non-thermal emission and, due to the 
lack of orbital modulation, does not favour an origin in an intrabinary shock between the pulsar and
companion star wind. While thermal emission from the neutron star polar 
cap cannot be excluded in the soft X-rays, the dominance of synchrotron 
emission favours an origin in the  pulsar magnetosphere that we describe 
at both X-ray and gamma-ray energies with a synchro-curvature model. 
 
The optical multi-colour light curve folded at the 10.8 h orbital period is 
double-humped, dominated by ellipsoidal effects, but also affected by 
irradiation. The light
curves are fit with several models encompassing direct heating and a cold 
spot, or heat redistribution after irradiation either through convection or 
convection plus diffusion. 
Despite the inability to constrain the best irradiation models, the fits 
provide consistent system parameters, giving an orbital inclination of 
59±6^∘ and a distance of 3.1±0.3 kpc.
The companion is found to be an F-type star, underfilling its Roche lobe 
(f_ RL=73±4%), 
with a mass of 0.20±0.04  M_⊙, confirming the redback status,
 although hotter than the majority of redbacks. 
The stellar dayside and 
nightside temperatures of 7500 K and 7400 K, respectively, indicate a 
weak irradiation effect on the companion, likely due to its high 
intrinsic luminosity. Although  the pulsar mass cannot be precisely derived, 
a heavy  (1.8-2.2 M_⊙) neutron star is favoured.





X-ray and optical observations of the millisecond pulsar binary PSR J1431-4715
    D. de Martino1
	A. Phosrisom2
	V. S. Dhillon3,4
	D. F. Torres5,6,7
	F. Coti Zelati5,6,8
	R.P. Breton2
	T.R. Marsh9Deceased
	A. Miraval Zanon10,11
	N. Rea5,6
	A. Papitto11
	
    Received 3 June 2024 / Accepted 2 September 2024
================================================================================================================================================================================





§ INTRODUCTION



Millisecond pulsar (MSP) binaries are compact systems consisting of a
fast spinning neutron star (NS) and a low-mass companion star. Those in a tight
orbit (P_orb 1 d)  are dubbed "spiders" 
<cit.> and, depending on the companion mass (M_c),  
 are subdivided into "black widows" (M_c  0.1 M_⊙)
and "redbacks" (M_c ∼ 0.1-0.4  M_⊙). These old NSs in binaries
are believed to be spun-up to very short spin periods during a previous
Gyr-long phase of mass accretion from an evolved companion. 
According to the recycling
scenario <cit.> during the accretion phase, MSP binaries were
Low-Mass X-ray Binaries (LMXBs) and turned into radio and gamma-ray pulsars when mass
accretion ceased. The first observational evidence of transitions from a LMXB 
to a rotation-powered pulsar state was found in the MSP PSR J1023+0038 <cit.>.
However, the subsequent detections of state transitions from and/or to a disc state
in IGR J1825-2452, in the M28 globular  cluster <cit.>, in 
the Galactic field X-ray source XSS J12270-4859 
<cit.> and again in PSR J1023+0038
<cit.> imply that
 transitions between accretion-powered and rotation-powered states can
occur on timescales much shorter than secular evolution, likely due to
changes in the mass transfer rate from the companion star and possibly 
controlled by the interplay between the pulsar spin-down power and the 
companion star wind or by its magnetic activity 
<cit.>.  

In spiders, the interaction of the relativistic pulsar wind with the 
companion star wind produces an intrabinary shock (IBS), where particles 
are accelerated, evaporating the late-type star <cit.>. 
Depending on the energy of the particles, the IBS cools via inverse Compton or 
synchrotron radiation. Also, 


depending on the pulsar spin down power and companion wind momentum, the IBS
assumes different orientations, wrapping around the companion star in black widows 
or the pulsar in redbacks <cit.>. 
X-ray orbital modulations are indeed observed 
in a number of black widows and redbacks  with different orbital phasing indicating the 
different IBS geometry <cit.>. Furthermore redbacks are found to be  more luminous
on average in the X-rays than black widows and isolated MSPs, indicating that a larger fraction of
pulsar wind is intercepted at the shock compared to black widows 
<cit.>.

The late-type companions are generally found to be strongly 
irradiated in black widows,
whilst in redbacks the effect is not always detected
<cit.>. The irradiation pattern in both black widows
and  redbacks is not always found to be consistent with direct irradiation 
by the pulsar, sometimes requiring illumination from the IBS as well <cit.>, or 
even additional heating due to magnetic activity of the companion star <cit.>.


The number of spiders and candidates of both types has largely increased 
recently, thanks to several and deep radio pulsar searches 
 <cit.> and to the gamma-ray Fermi-LAT 
survey <cit.>. The latter, in particular,  has allowed 
efficient detection of MSP binaries, which are unaffected at high energies by
eclipse effects from the intrabinary material <cit.>.
Redbacks are of particular interest because they include the recently discovered
subclass of transitional millisecond pulsar binaries. Among  the 
newly identified redbacks a number of them still lack multi-band
observations to derive the physical parameters of the IBS and the 
companion star.  

In this paper we  focus on the poorly studied MSP binary
PSR J1431-4715 (henceforth J1431). It was discovered in the  
High Time Resolution Universe Survey (HTRU) by <cit.> as one of the
fastest (2.01 ms) and most energetic pulsars with a spin-down power 
Ė = 6.8× 10^34 erg s^-1. Its pulsed radio 
emission was found to be  affected by strong  eclipses at the binary
10.8h orbital period. The radio orbital solution classified  
J1431 as a MSP binary with a non-degenerate donor, with a minimum mass of
M_c = 0.12 M_⊙. J1431 has also recently been detected at high 
energies by Fermi-LAT and catalogued as 4FGL J1431.4-4711 in the 
4FGL-DR3 12-yr and 4FGL-DR4 14-yr LAT catalogs 
<cit.>, respectively,
with L_γ =1.26-8.11× 10^33 erg s^-1

for a distance range of 1.5-3.8kpc, encompassing distance values 
derived from dispersion measure (DM) <cit.> and 
Gaia <cit.>[The Gaia DR3 geometric distance 
ranges beteen 1.61-2.58 kpc]. 









A deep search for pulsations has also identified J1431 as a gamma-ray pulsar at the ∼ 7σ level 
<cit.> and thus it is included  in the recent 3^rd Fermi-LAT  catalog 
of gamma-ray pulsars  <cit.>, joining  the sample of 14 confirmed redbacks 
with radio and gamma-ray detections.  Medium time resolution optical photometry and 
spectroscopy was carried out by <cit.> that revealed  
the orbital motion of the companion star and found it to be affected by mild 
irradiation. The binary parameters were not constrained due to the degeneracy
between inclination and filling factor. 

We present here a multi-band analysis of J1431 consisting of an X-ray 
observation of J1431 performed with , that has allowed the detection 
for the first time of the X-ray emission, as well
as optical multicolour fast photometry acquired at ESO La Silla 
with the instrument mounted at the NTT telescope. The
observations are presented in Sect. <ref>, the X-ray timing and
spectral analysis are reported in Sect. <ref> and the optical
photometric study in Sect. <ref>. We discuss the results
in Sect. <ref> and compare them with other redback properties.




§ OBSERVATIONS AND DATA REDUCTION






 §.§ The observation



J1431 was observed by on 2021 Jan. 21 (OBSID: 0860430101) with the
EPIC-pn camera <cit.> in the Large Window imaging mode and with the
thin optical blocking filter for an  exposure time of 92.3 ks and with the EPIC-MOS1 
and MOS2 cameras <cit.> in Partial Window imaging mode with the thin filter 
for an  exposure time of 93.3 and 94.6 ks, respectively. The presence
of background flaring activity at the beginning and end of the pointing  
reduced the effective exposures to 78.9 ks and 84.4 ks for the EPIC-pn and MOS 
cameras, respectively[The filtering of the event files was performed creating good time intervals (GTIs)
by setting a cut in the background rate time series without source contributions].
The Optical Monitor (OM) <cit.> was set in 
fast window mode using the V (5100-5800Å) and UVW1 
(2450-3200Å) filters. Nine OM science windows of ∼4400 s each were
acquired sequentially in the V and UVW1 filters, totalling an exposure 
of 39.6ks in each of them, respectively. 
The log of the observations is reported in Tab. <ref>


The data were processed and analysed using the Science Analysis Software
(SAS v.20.0) with the latest calibration files. The photon arrival
times from EPIC cameras were corrected to the Solar System barycentre 
using the JPL DE405 ephemeris and the nominal position of J1431 reported 
by <cit.>[The radio position is fully consistent with the
Gaia DR3 position <cit.>]. For the EPIC cameras, we
extracted events using a 16" radius circular region centred on the source
and using a background region of the same size located on the same CCD chip. 
To improve the S/N the data were filtered by selecting pattern pixel events up to 
double with zero-quality flag for the EPIC-pn data and up to quadruple for 
the EPIC-MOS data. Background subtracted light curves were produced using 
the epiclccorr task in the whole 0.2-12 keV range and in two
bands covering the 0.2-4 keV and 4-12 keV ranges. Given the faintness of the
source, the light curves were binned into  1200 s and 2400 s-long intervals, 
respectively. 
The EPIC spectra were extracted in the 0.3-10 keV range, filtering the events using good
time intervals obtained by selecting low background epochs with the task gtigen. 
Response matrix and ancillary files were generated using the tasks rmfgen and 
arfgen, respectively. The spectra were rebinned with a minimum
of 30 counts in each bin using the specgroup routine before fitting. 

The OM V and UVW1 fast window data were processed and background light
curves were generated with the task omfchain with a bin time of 400 s and
1100 s, respectively. Given the small size of the fast window mode, the centering
of the target was inspected for each window. Due to a drift of the telescope, 
 the target was found to be partially outside the window of the last V band 
observation 
and the first UVW1 band window and  therefore these were disregarded. 
The light curves
were then corrected to the Solar System barycentre in the same manner as the 
EPIC data. The OM V and UVW1 count rates were also converted into AB 
magnitudes using the count rate magnitude conversion available at the SOC[https:/www.cosmos.esa.int/web/xmm-newton/sas-watchout-uvflux]. 








 §.§ The observations



J1431 was observed with the ESO 3.5m New Technology Telescope (NTT) 
at La Silla (Chile) equipped with the photometer <cit.> that allows simultaneous 
high speed three-colour photometry for three consecutive nights from 2019 April
12 to April 14 in relatively good seeing (1-1.5") conditions.
 
During the first two nights the Super SDSS 
u_ s,g_ s,r_ s filters
were used, while on the third night the photometry was acquired with the 
u_ s,g_ s,i_ s filters. The Super SDSS filters are described in <cit.>. 
Single exposure times were 60.6 s for the u_ s 
filter and 20.2 s for the g_ s,r_ s, i_ s filters. The instrument was set in 
full-frame and no-clear mode giving a dead time between each frame of 0.024 s
with a GPS time stamp of each frame to an absolute accuracy of 1 ms. Due
to a rotator flip at transit, the science exposures were stopped for a few minutes. 
The log of  the observations is reported in Tab. <ref>. 


The data were reduced using the improved reduction pipeline developed for 
HIPERCAM <cit.>. The images were debiased using master 
bias and dark frames and flat-fielded using a master twilight flat field.
Aperture photometry with variable sized object apertures scaled to the 
seeing in each frame was applied to extract counts. The sky was 
determined from a clipped mean in an annulus around the object aperture 
and subtracted. 
Several comparison stars in the same field of view were also extracted to 
perform differential photometry. Among the comparison stars two
of them have similar colour to J1431[The comparison stars colours 
were checked against Gaia DR3], which then have been used to calibrate 
the target using their magnitudes from the SkyMappper Southern 
Survey DR2 catalogue <cit.>. These have also been checked
against variability and we chose the more stable reference star 
SMSS J143149.5-471538.1  with AB magnitudes u=17.61±0.03, 
g=15.89±0.01, r=15.50±0.01 and i=15.261±0.008.
The light curves in the different bands were inspected against degradation
due to twilight at the end of the observing runs and the affected points
were removed from the light curves. Barycentric correction was
also applied to the time series in the different filters. 




§ DATA ANALYSIS AND RESULTS





 §.§ The X-ray emission



The observation finds J1431 as a faint source (Tab. <ref>) and
thus we restrict to the more sensitive EPIC-pn data for the timing analysis, 
after inspection of no improvement in the S/N  when summing the rates from 
the three  EPIC cameras.   
In Fig. <ref> the EPIC-pn light curve in the total 0.2-12 keV is displayed 
together with those in   
the soft 0.2-4 keV and hard 4-12 keV ranges. Although the 
length of the observation covers twice the 10.8 h binary period, there is no
clear orbital modulation in the total band. The soft 0.2-4 keV  bands may hint
at a possible variability with a marginal increase in the flux at about 28 ks
from the start of the observation, not observed in the hard (4-12 keV) range. 
The hardness ratio HR, defined as the ratio of  count rates between the 4-12 keV and 0.2-4 keV bands,  
may indicate a softening at that time, but also at earlier and later times as depicted in Fig. <ref>, 
although at these epochs the hard band count rates are consistent with zero given the 
uncertainties on the background subtracted rate.








The EPIC-pn energy resolved light curves were also folded at the 10.8 h binary orbital 
period using  the <cit.> ephemeris (see left panels of Fig. <ref>), where phase zero corresponds
to the time of the NS ascending node[The radio solution is highly 
accurate (dϕ/ϕ = 1.15× 10^-10) allowing the 
folding of the data]. The soft band  hints to a  variability with
a semi-amplitude of 14±12%  (χ^2_ν/d.o.f. = 0.94/9), while a fit with a constant 
gives χ^2_ν/d.o.f. =0.99/11. The signficance of an orbital variability estimated with an F-test is only 
at 1.6σ confidence level, thus not statistically 
significant. The light curve in the harder (4-12 keV) range appears to be double-humped
with unequal maxima at ϕ_orb∼ 0.25-0.4 and ∼0.8-1.0. These phases
are close but slightly later than the superior and inferior conjunction of the pulsar, respectively. 
Fitting the hard band orbital light curve with the fundamental and harmonic frequencies, the amplitude is 34± 14% 
(χ^2_ν/d.o.f. = 0.60/9), while a fit with a constant gives χ^2_ν/d.o.f. =1.20/11. The significance
of the modulation estimated with an F-test is at 4σ confidence level. 
The hardness ratios indicate a hardening at these two maxima (bottom left panel in Fig. <ref>). 




On the other hand, the light curve in the 
total range does not reveal significant orbital modulation 
(right upper panel in Fig. <ref>) being consistent with a constant (χ^2_ν/d.o.f. = 0.50/11)
likely due to the higher count rate in the soft than in the hard band. 
While a detailed comparison with the optical emission is performed in
 Sect. <ref>, the simultaneous UV and optical photometry  
(Fig. <ref>, right bottom panels) 
shows a weak double-humped variability with amplitudes of 8±6% in the 
UVW1 band and 8±3% in 
the V band with unequal maxima at the ascending and descending nodes of the NS,  although an F-test gives 
significance at 3σ and 2.5σ levels.
The different behaviour will be discussed in Sect. <ref>.

The X-ray  0.3-10 keV spectrum averaged over the whole observation is 
featureless with no sign of a cut-off. 
The EPIC-pn and MOS1,2 spectra were analysed using xspec 
package <cit.> and fitted together. We first  adopt a simple model consisting of an 
absorbed power law, const*tbabs*powerlaw, where tbabs accounts
for the ISM absorption with abundances from <cit.>, and const
accounts for intercalibration among the three instruments. This model  gives a reasonable fit 
(χ^2_ν/d.o.f. = 1.01/26), with a power law index 
α =1.63_-0.18^+0.29 (uncertainties are 90% confidence level
on one interesting parameter) and an unconstrained neutral hydrogen 
column density N_H ≤ 8×10^20 cm^-2.



This upper limit is roughly consistent with    
the total ISM hydrogen column density in the direction of the source
(N_H,ISM = 1.1× 10^21 cm^-2 <cit.>)  and the column
density 
from the dispersion measure DM=59.35pc cm^-3 of <cit.>, 
N_H,DM = 1.8(6)× 10^21 cm^-2,
using the empirical relation of <cit.>, as well as with the column 
density 









of ∼ 1.1× 10^21 cm^-2,
derived from the recent 3D-N_H tool of <cit.> that combines dust maps and distance to estimate
the optical reddening and X-ray absorption adopting the same distance range.   
Fixing the column density to 
N_H,ISM = 8× 10^20 cm^-2 the fit provides similar results 
(χ^2_ν/d.o.f. = 1.08/27) for the power law index 
α= 1.88±0.19 (see Fig. <ref>)[The results do not change adopting a column 
density of 1.0× 10^21 cm^-2]. 
The unabsorbed flux in the 0.3-10 keV
range is 2.6±0.3× 10^-14 erg cm^-2 s^-1.




At the distances of 1.53-3.8 kpc the corresponding luminosity is 
L_X = 0.7-5.0× 10^31 erg s^-1. 

Although the simple power law model is formally acceptable, we also included a 
blackbody component, using bbodyrad  to inspect
whether the presence of the thermal emission of the NS is required. We do not 
attempt to use a NS atmosphere model (e.g. nsa) given the low statistics
of the spectra.  
Keeping fixed the hydrogen column density as before we find a power law index
α=1.31_-0.38^+0.36, a blackbody temperature 
kT_BB= 0.15±0.04 keV and normalisation N_BB= 1.12_-0.72^+2.63
(χ^2_ν/d.o.f. = 0.83/25) (see Fig. <ref>). 
The latter implies an emitting radius 
R_BB∼100-740 m for a distance in the range of 1.53-3.8 kpc.




The 0.3-10 keV unabsorbed 
fluxes of the two components result in: F_BB=6.4_-0.5^+0.3× 
10^-15 erg cm^-2 s^-1 and F_pow=2.6_-0.2^+0.1× 
10^-14 erg cm^-2 s^-1. The total luminosity in the 0.3-10 keV range
then results  L_X ∼ 0.9-5.55× 10^31 erg s^-1 for the
above distance range.


The inclusion of the blackbody using  
an F-test is however significant only at a confidence level of 2.5σ.

J1431 displays a similar spectrum to  other MSP binaries in
the rotation-powered state, with a power
law index within the range α=1.0-1.9. 
The X-ray emission is therefore dominated by synchrotron radiation with
a luminosity consistent with that derived in redbacks and higher than
those of black widows <cit.>. 
Although not required from the fits, a blackbody component with 
temperature of ∼0.15 keV and size of a few hundreds of meters is reasonably
consistent with those found in MSPs, where the NS thermal
emission from the heated pole is detected 
<cit.>.  
The ratio of the unabsorbed fluxes between the putative thermal component and 
the power law in the 0.3-10 keV is ∼0.25 
and the bolometric luminosity of the thermal component 
in the range 2.3-15.0× 10^30 erg s^-1 are also consistent

with those found in MSPs <cit.>. 


Whether the non-thermal emission originates in the IBS or from the pulsar magnetosphere 
will be discussed in  Sec. <ref>. 


 


 §.§ The optical light curves



The time series clearly show a low-amplitude periodic variability
in the u_ s,g_ s,r_ s,i_ s bands. Using the longer time coverages in the 
u_ s,g_ s,r_ s  filters, the Lomb-Scargle periodograms <cit.> 
reveal the presence of strong  power at the harmonic of the 10.8 h orbital period 
(see Fig. <ref>). 
A sinusoidal fit composed of the fundamental and harmonic  
to the light curves gives a period 
P_u_s^Ω=0.45024(22) d, P_g_s^Ω=0.449499(48) d and
P_r_s^Ω=0.44950(12) d fully consistent with the more precise orbital
period of 0.4497391377(7) d derived from radio timing by <cit.>. 
The modulation amplitudes in the four bands are  A_u_s^Ω = 0.0193(8) mag, 
A_u_s^2Ω = 0.0734(2) mag; A_g_s^Ω= 0.0215(1) mag, 
A_g_s^2Ω = 0.0682(1) mag, A_r_s^Ω=0.0153(2) mag, 
A_r_s^2Ω = 0.0630(1) mag and 
A_i_s^Ω= 0.0117(3) mag, A_i_s^2Ω = 0.0601(2) mag, 
where for the i_ s band the orbital frequency of <cit.> has been adopted.

The observed light curves, including the OM UVW1 and V band filters 

folded at the orbital period using the accurate ephemeris of 
<cit.> are shown in Fig. <ref>. 
The double-humped shape clearly demonstrates the presence of ellipsoidal
variations due to the tidal distortion of the companion star, which however
are not symmetrical and do not show equal minima.  These minima occur
at ϕ_orb∼ 0.25 and 0.75. Also slight phase shifts are observed
at the two maxima, one anticipating at ϕ_orb=1.0 and the other lagging at ϕ_orb=0.5.
The wavelength dependent amplitudes also hint at colour effects, 
likely due to irradiation as pointed out by <cit.>. 
It is worth of note the different shape of the V band orbital modulation observed in 2021 with
respect to the light curves acquired in 2019, possibly indicating
a change in irradiation efficiency, although the lack of a deeper minimum
at ϕ_orb∼ 0.25 could also be due to the lower  accuracy of 
the OM photometry.  We then limit ourselves to the analysis of
the high time resolution data. 











  §.§.§ Modeling of the optical light curves



The light curves, complemented with the
radial velocity curve obtained from the H_α absorption line 
by <cit.> for the companion star,
were analysed with the icarus[https://github.com/bretonr/Icarus] 
binary modeling code <cit.> 
to constrain system parameters and the temperature profile of the companion. 
This software is used along with ATLAS9 stellar atmosphere grids <cit.>
to construct a photometric grid of synthetic atmosphere models,
which in turn are folded through the filter transmissions to 
obtain the observed flux at a given distance. The MultiNest 
nested sampling algorithm  <cit.> was used to explore the 
parameter space, which provides the Bayesian evidence (Z) of a model. 






The precise pulsar radio-timing solution derived by <cit.> was adopted 
to fix the input parameters: the time of the ascending node of the pulsar 
(T_asc= BMJD 55756.1047771), the orbital period 
(P_orb=0.4497391377 d) and the projected 
semimajor axis of the pulsar orbit (x=a_ psr sin i/c=0.550061 lt-s). 
The pulsar mass function f( M_psr) relates to the system parameters
as f( M_psr)= M_psr sin^3i/(1+1/q)^2 = 
q^3 x^3 4 π^2/G  P_orb^2 where q=M_psr/M_c is the  
pulsar-to-companion mass ratio, i is the binary inclination. Since the mass 
function is related to the companion star radial velocity amplitude K_c 
as f( M_psr)=P_orb K_c^3 /2 π G, the parameters to 
be fit  are i, K_c.


The icarus code constructs a stellar surface whose radius is 
parametrized by the Roche lobe filling factor f_ RL, 
defined as the ratio between the
radius of the companion in the direction of the pulsar and the distance 
between the companion centre of mass and the inner Lagrangian point 
(L_1). f_ RL is then another fit parameter.
The  surface stellar temperature T_base is defined as the 
temperature of the 
pole prior to irradiation and,  accounting for gravity darkening, 
T_base is
multiplied by (g/g_ pole)^β, where g is the acceleration 
gravity and
β is the gravity darkening  exponent.  This parameter is fixed at 
β=0.08 and β=0.25, which are appropriate for convective and 
radiative envelopes, respectively <cit.>. 
The effect of the pulsar heating is represented by the 
irradiating temperature T_irr, defined such that a flux 
σ T_irr^4 at the star centre of mass is received 
at a distance a from the pulsar. With the assumption that the irradiating
flux is immediately thermalised and re-radiated <cit.>, each surface 
element at a distance r from the pulsar, with its normal vector 
forming an angle θ with respect the vector pointing to the pulsar, 
has a temperature such that 
T^4 = [T_base^4 (g/g_ pole)^4β + T_irr^4 cos θ  (a/r)^2]^1/4.
This model assumes direct heating (DH) from the pulsar and applies gravity 
darkening prior to irradiation, here termed pre-irradiation gravity darkening (pre-IGD) <cit.>.
Gravity 
darkening applied after irradiation and heat redistribution on the companion
surface, here termed post-irradiation gravity darkening (post-IGD), 
as done in <cit.> and <cit.>, has also been investigated. 

However the presence of 
asymmetries and shifts in the light curves of J1431, already noticed by 
<cit.>, cannot be accounted for by the simple DH approach. 
These features
can instead be modeled either by adding hot or cold spots or by including 
heat redistribution  within the outer layers of the star by convection 
only (C) and/or diffusion and convection (D+C)  
<cit.>. These models have also been inspected by 
applying both pre-IGD and post-IGD. 

The icarus code also encodes as fit parameters the optical 
interstellar reddening defined as E(B-V) in the direction of J1431 
determined in each band using dust maps and extinction 
vectors from <cit.> as well as the source distance d. 
Hence the fit parameters for all these models are: E(B-V), d, K_c, 
i, f_ RL, T_base, T_irr. For the models accounting 
for heat 
redistribution with convection (C) and diffusion plus convection (D+C),  
additional fit parameters are the convective strength parameter C_ ampl,  
which determines the profile of the convective wind 

and the diffusion parameter κ_ diff, which describes the amplitude of 
a linear diffusion effect <cit.>. In the case of spot model 
a single surface hot or cold spot is added to the DH model with fitted 
temperature difference T_ spot, angular size R_ spot, and position angles 
θ_ spot, ϕ_ spot <cit.>.
 

The main priors for the fit parameters were choosen such that they are 
physically or geometrically  motivated.
Among them, the distance of J1431 whose prior is constructed combining
the ranges derived by  <cit.>, <cit.>, 
<cit.> and Gaia DR3 (see Sec. <ref>), but allowed to vary from 0.1 to 20 kpc. 
Also a Gaussian prior is applied to the interstellar 
extinction E(B-V), centred at 0.166 with standard deviation of 0.16 but allowed
to vary over a larger range from 0 to 1. The base and irradiation temperatures
were constrained to lie between 2100 K and 12000 K, and between 2300 K 
and 15000 K, respectively.
For the binary inclination angle i, a uniform prior in cos i has been 
adopted. As for the Roche lobe filling factor f_ RL, a uniform prior has
been adopted between 0.2 and 1.0. 
The spectroscopic K_c constraints are provided by the observed radial
velocity curves obtained from cross-correlation of the H_α absorption line in the
optical spectrum of J1431 by <cit.>. Here the icarus code 
simulates spectra at specific orbital phases and the radial velocities of the model are 
compared to the observed values <cit.>.    

Among the several model fits, those with post-IGD prescription outperform the pre-IGD in all cases, 
except identical results are found for the the DH+Spot model adopting either 
pre-IGD or post-IGD heat redistribution prescription.
In Tab. <ref> the results of the different 
modelling are reported for the post-IGD prescription only and for the two values of the 
gravity darkening exponent β. Models with β=0.25 have much lower 
Bayesian evidence (log Z) with respect to those adopting β=0.08, and hence are discarded. 
Here we note that the transition temperature from convective to radiative envelopes 
falls in the range ∼ 7000-7900 K  <cit.>, justifying the investigation with both β values. 
The fitted parameters with β=0.08  are found to be all consistent 
within 1σ for those models with  
higher Bayesian evidence, namely that with convection (C), with diffusion and convection (D+C) 
and with direct heating (DH) with a cold spot. 

A slight preference for the DH with a cold spot is found. This model is shown in 
Fig. <ref> where the fitted 
multi-colour light curves (panel A) are displayed together with the surface 
temperature map of the companion (panel B) as well as the observed and 
model radial velocity curves (panel C). Corner plots for the best  three
models (C), (D+C) and DH+spot with post-IGD are reported
in Fig. <ref>, Fig. <ref> and 
Fig. <ref>, respectively. 
The similarities of the results obtained with these three  best fit models provide 
evidence that the derived parameters are reliably constrained despite the
inability of determining whether heat redistribution models or a spotted
surface represent the physical state of the companion star.






The distance of J1431 has not been well constrained 
by several methods <cit.>, allowing 
for a wide range from 1.5 kpc to 3.8 kpc. Here we find consistency for all
the three models, which, adopting the higher and lower bounds of the derived parameters, 
provide  a distance to J1431 of 3.1±0.3 kpc, well within the above
range and that we adopt in the forthcoming section. The interstellar extinction 
E(B-V)=0.17-0.19, giving A_V=0.52-0.59
is also consistent with the upper limit to the 
hydrogen column density found from the X-ray spectral fits (see 
Sec. <ref>) using the <cit.> relation. 
The binary system parameters are discussed in Sec. <ref>.
 




§ DISCUSSION







 §.§ A mildly irradiated companion



The parameters obtained with the icarus code for the three  best fit models
give consistent values within 1σ for the binary inclination,  
the companion Roche lobe filling factor and mass. By averaging among the three models we
obtain  i=59±6^∘, f_ RL=73±4%, M_c=0.20±0.04 M_⊙, which
are broadly consistent with the preliminary results obtained by 
<cit.> but more accurate. These values confirm the redback status of J1431. 

The companion surface is found to have  hemisphere-averaged nightside and 
dayside temperatures  T_night∼7400 K and 
T_day∼7500 K, both higher than previously estimated 
(∼6500-6600 K) by <cit.>.  We here note that this lower temperature 
estimate is based on spectra covering a narrow spectral range between 5500-6730 Å
encompassing only H_α and not a wider wavelength range as our multi-band photometry. 
We also verified whether the extinction prior could affect the temperature determination adopting
a flat prior for the reddening. The posterior distributions of the fit are found similar to those 
adopting a Gaussian prior, indicating that the reddening does not affect the temperature determination.
The spectral type of the  companion would then be an early-F star,
 much hotter than the majority of redbacks, which are instead found in the range of 2800-6300 K <cit.>, with only
PSR J1816+4510 found to be extreme (16000 K) <cit.>.
The companion of J1431 could be an outlier due to an unusual binary evolution,  
or hot redback companions are still to be discovered. This shows that the characterization of the optical companions of newly discovered
redbacks is an important aspect of investigation.
The irradiating temperature 
T_irr∼ 3900 K  implies an equivalent irradiation 
luminosity L_irr= 4 π a^2 σ T_irr^4 
∼ 7.8× 10^33 erg s^-1, where a=2.17× 10^11 cm 
having adopted a_ psr sin i from <cit.> and the derived 
values for the inclination i=59^∘ and
mass ratio q=10.4.  This luminosity when compared
to the spin-down power Ė = 6.8× 10^34 erg s^-1 from
<cit.> or that derived in Sec.<ref> below,  gives
an efficiency η_irr=L_irr/Ė∼ 12-13%, 
not unusual in rebacks, although at the low end of 
irradiation efficiencies <cit.>. To estimate 
the heating effect
on the companion we consider the expected heating impinging on the companion 
in the case of an isotropic pulsar wind L_heat = f_Ω Ė
where f_Ω is the geometric factor ∼ 0.5 (1-cosθ) with 
θ= arctan  ( R_c/a), neglecting albedo. Adopting the derived companion
radius R_c =0.6 R_⊙ and the above value for a, 
f_Ω=0.009 and then 
L_heat = 6.1× 10^32 erg s^-1. The ratio between the 
expected heating and the companion unheated luminosities,
L_c,base = 4π R_c^2 σ T_base^4 = 4×
10^33 erg s^-1, then   results in  
η_heat=L_heat/L_c,base= 0.15. This indicates that
the companion luminosity is about 6.5 times larger than the heating, 
hence implying weak irradiation  as indeed evidenced by the observed 
double-humped light curves, indicating that they are dominated by ellipsoidal effects.
The fact that MSP binaries with intrinsic companion  
luminosities larger or comparable to that expected from pulsar irradiation
show negligible heating effects was already pointed out by 
<cit.> and it is further supported by the results 
by <cit.> on a larger number of spiders with a wide 
range of orbital periods. 
The long orbit of J1431 and hence large
separation together with the high companion luminosity can explain the
lack of strong irradiation in this system.






















 §.§ A heavy neutron star



 Noteworthy is the derived  pulsar mass M_psr ranging from 1.8 to 2.2
M_⊙, considering all three models. Although we are unable to 
discriminate among the various models employed, which limits the precision to which this 
parameter is constrained, a massive NS in J1431 is favoured.
In the recent years, NS masses in spiders have been subject of several investigations 
<cit.>. Comparing the NS mass derived for J1431 with the latest compilations of 
NS masses in <cit.>, J1431 joins the 
group of massive NSs, which includes the redbacks PSR J2215+5135, PSR J1622-0315 and PSR J1816+4510 and the 
black widows PSR J1653-0158, PSR J1810+1744, PSR J0952-0607 and 
PSR JB1957+20.  Noteworthy is the case of PSR J1622-0315, a mild irradiated redback potentially hosting a massive
NS <cit.>. These systems are particularly attractive because they 
allow lower limits be set on the maximum NS mass and hence have 
important implications for the dense matter equation of state (EoS).

The spin-down power of the pulsar, defined as 
Ė=4 π^2 I Ṗ/P^3, where P and Ṗ are the spin period
 and its derivative  and
I = 2/5  M_NS R_NS^2  is the momentum of inertia of the NS,
was estimated by <cit.> as Ė = 6.8× 10^34 erg s^-1, 
adopting the canonical NS moment of inertia I=10^45 g cm^2
and correcting the spin period derivative for the Shklovskii effect.
The proper motion of J1431 is now better constrained by 
Gaia DR3 <cit.> μ=18.7(2)  mas yr^-1, which 
allows a more accurate determination of the Shklovskii effect. 
We then correct the observed spin period derivative Ṗ for 
this effect:
Ṗ_Sh/P = v_t^2/(c d), where v_t is the transverse velocity, 
and d the distance. Adopting d =3.1±0.3 kpc, we derive
Ṗ_corr = Ṗ_obs - Ṗ_Sh = 8.7(4) 
× 10^-21 s s^-1, where P_obs=1.411×10^-20 s s^-1 is the observed
spin period derivative by <cit.>.















Given the evidence of a higher NS mass than 1.4M_⊙ (see Tab. <ref>) , 
we adopt conservatively the lower limit of M_psr=1.8 M_⊙ and, lacking
knowledge of EoS,  
a NS radius of 10 km <cit.>, giving
 Ė =  5.8-6.4× 10^34 erg s^-1, slightly lower 
than the previous determination. It is not too far from the value
of 5.2±0.3 × 10^34 erg s^-1  derived
by <cit.> adopting the canonical moment of inertia 
I=10^45 g cm^2 and the lower Gaia distance of ∼1.6-2.6 kpc. 

Further observations, such as optical spectroscopy, would be desirable
to better investigate the irradiation model and hence constrain the 
NS mass. 




 §.§ An X-ray dim redback 



 Our first deep X-ray observation of J1431 has detected a faint X-ray source  without signficant variability
at its orbital period. The X-ray emission in the soft (0.2-4 keV) is unmodulated and in the hard (4-12 keV) 
range only a hint at 4σ level of a double peaked variability is found. This contrasts with the majority
of spiders displaying substantial X-ray orbital variations, which are often found double-peaked 
due to Doppler boosting of the IBS synchrotron emission. In redbacks the non-thermal emission 
is Doppler boosted at inferior conjunction of the NS and de-boosted at superior conjunction. 
The separation of the two peaks and their intensity strongly depends on the bulk Lorentz factor, 
viewing angle and shock radii <cit.>. In black widows the maximum of X-rays occurs 
at superior conjunction of the NS, while in the majority of rebacks it is the opposite 
<cit.>. This has been interpreted as different geometry of the IBS, which wraps the pulsar/companion in 
redbacks/black widows, respectively  depending to the momentum ratio between the companion and the pulsar 
winds <cit.>.

In J1431 the companion is found to be luminous, suggesting a higher wind momentum and it is conceivable that
the IBS wraps the pulsar as the majority of redbacks. In this case an orbital modulation is expected and 
affected by Doppler boosting at inferior conjunction of the NS. 
The lack of an orbital modulation in J1431 could arise if the system is viewed at low inclination angles. However
the optical analysis reveals an intermediate binary inclination of ∼60^∘.
A low inclination is also not favoured by the presence of long radio eclipses at superior conjunction of the pulsar, 
lasting ∼ 0.3 in orbital phase <cit.>. 
However, the possibility that an IB shock dominates the X-ray emission in J1431, not producing an X-ray orbital modulation
cannot be discarded, but determining the properties of the IB shock is not an easy task given the dependence on the binary inclination,
the shock location, opening angle and bulk velocity (β=v/c) of accelerated particles.
In the thin-shell approximation <cit.>, generally adopted for spiders, for a moderate inclination angle,
shadowing of the companion is not expected to produce signficant obscuration of the shock emission if the shock is located
close to the pulsar and if most of the emission originates close to the shock nose and the opening angle is small.  
Near the shock nose, the bulk velocity is low and Doppler-boosting is not efficient <cit.>. 
Given the lack of higher energy coverage above 10 keV that would allow the  inspection of a 
spectral break to estimate the shock location as well as the lack of coverage of the radio eclipses at low frequencies 
<cit.> to derive the maximum shock opening angle, it is not possible to verify this hypothesis.

Adopting the derived distance d=3.1±0.3 kpc, 
the X-ray luminosity ranges between 1.4-2.1× 10^31 erg s^-1 in the 
2-10 keV band, or between 2.1-3.0× 10^31 erg s^-1 in the 
0.5-10 keV range. When compared the X-ray luminosity to those observed in spiders, 
J1431 is located between the redbacks and black widows and in 
particular among low X-ray luminosity redbacks <cit.>. 








The efficiency in converting spin-down power to X-ray luminosity in J1431 is 
η_x= L_x/Ė∼ 0.02-0.04% (2-10 keV) or 0.03-0.05% (0.5-10 keV).
Although caution should be taken adopting empirical relations, 
this luminosity ratio locates J1431 in between  the so-called "inefficient" and "efficient"  IBS 
tracks derived by <cit.>.




All this indicates a very weak contribution of the IBS, if any. 

Noteworthy, J1431 not only joins the low X-ray luminosity redbacks but also shares with 
these systems a double-humped optical light curve, dominated by ellipsoidal variability.
Hence, it is very likely that the weak companion heating is not due to 
X-ray irradiation. Indeed our time-resolved X-ray light curve does not
allow us to conclude whether the non-thermal emission originates in an IBS.




The spectral analysis indicates that the emission is dominated by non-thermal radiation, described
with a power-law model with index 1.3-1.9, similar to those observed in spider pulsars 
<cit.>. Although formally not required by the spectral fits,
a thermal component at kT_BB∼ 0.15 keV 
originating at the heated NS polar cap (R_BB∼ 180-660 m for the
distance range 2.8-3.4 kpc derived in Sec.<ref>) 
could be also present. Similar values for this thermal component are found in other spiders 
<cit.>,  contributing ∼20% to the total X-ray flux.
Hence, it is also possible that the dominant X-ray non-thermal emission originates in the NS magnetosphere.
In this case the X-ray emission would be pulsed at the spin period of the NS, but the X-ray data were acquired in 
imaging  mode  with a temporal resolution of 47.7 ms, preventing any search of spin pulses. Worth of note is that
searches for X-ray spin pulsations in the rotation-powered state of the transitional MSP XSS J12270-4859 led negative
detection even using high time resolution data <cit.>.



The gamma-ray emission of J1431 is typical of pulsars with a 
spectral shape modelled with a  power law with a superexponential 
cutoff <cit.>
giving a 100 MeV-100 GeV flux of 4.7× 10^-12 erg cm^-2 s^-1 
and a  gamma-ray luminosity for a distance range of 2.8-3.4 kpc  
L_γ = 4.4-6.5× 10^33 erg s^-1.




The ratio of  gamma-ray luminosity




to spin down power  η_γ = L_γ/Ė∼ 7-11%  
indicates an efficiency of conversion consistent with the median of 10% of 
redbacks <cit.>. 
The gamma-ray emission in pulsar binaries is generally dominated by the pulsar 
magnetospheric radiation
and should not be modulated at the orbital period for moderately low 
binary inclinations. 
However if gamma-rays are also produced in
an IBS an orbital modulation should be observed such as in the bright systems 
XSS J12270-4853, PSR J2339-0533 and PSR J2039-5617 <cit.>, 
in which also the X-rays
are strongly modulated at the orbital period <cit.>. 
In J1431  there is no firm evidence of variability in the X-rays and searches in the gamma-rays
do not provide evidence of either an orbital  modulation or eclipses <cit.>.
The X-ray (0.5-10 keV) to gamma-ray flux ratio  F_x/F_γ∼ 0.5% is lower than 
that found in  most redbacks and similar to PSR J1908+2105 (∼ 0.6%), surpassing only 
 PSR J1622-0315 (∼ 0.3%) and PSR J1816+4510 (∼ 
0.05%) <cit.>. It is then 
plausible that the gamma-ray emission in J1431 originates in the pulsar wind.



We constructed the broad band X-ray and gamma-ray spectral energy distribution (SED) using our X-ray
spectral analysis adopting the composite blackbody and power law model and the Fermi-LAT 
SED as retrieved from the recent 3^rd pulsar catalog <cit.> along with the 
typical pulsar superexponential cutoff
power law model  PLEC4 with  peak energy E_p = 0.44±0.24 GeV, spectral slope Γ=2.63±0.16
spectral curvature  d=0.8±0.3 and superexponential index fixed at the
canonical value b=2/3 due to the source faintness (see details in <cit.>). 
The SED (Fig. <ref>) shows that
the X-ray spectrum, when extrapolated to higher energies, is higher than that predicted by the  
PLEC4 model that, however, cannot be reliably evaluated at lower energies due to the fixed b parameter
at the canonical value. 















We then explored whether a significant part of the gamma-ray and X-ray 
emission could be the direct result of the pulsar magnetospheric emission. 
Curvature and synchrotron emission from accelerated particles in the magnetospheric
gaps or reconnection in the current sheet producing synchrotron radiation at high energies 
are possible mechanisms at the origin of non-thermal pulses. The synchro-curvature emission model 
developed by <cit.> to describe the high energy spectra of pulsars is based on the assumption
that near the light cylinder of a pulsar with determined spin period and period derivative there is
a gap with a significant component of the electric field parallel to the magnetic lines and this field
accelerates particles. The model follows the particle time evolution, 
solving the equation of motion and balancing acceleration and losses. 
It has been successfully applied to the high energy spectra
of gamma-ray and  X-rays detected pulsars <cit.> and has been 
further updated 
with improved treatment of particle injection to generate
synthetic spectra and gamma-ray light curves of pulsars 
<cit.>.
We hence attempt to describe the observed broad-band SED of J1431 with the updated version of the 
synchro-curvature model. This allows us to compare the results with those  obtained for other gamma-ray 
pulsars detected. It also has the advantage that it is a rather austere spectral model
having only  three main physical parameters and a normalisation, which has proven 
useful to represent X-ray and gamma-ray spectra
of pulsars, the appearance of sub-exponential cutoffs at high energies,
or the flattening of the X-ray spectra at soft energies.
The parameters of the model are the electric field parallel to the 
magnetic field lines, E_||, assumed  constant throughout an accelerating region;
the magnetic gradient b,  such that the local magnetic field 
B= B_s (x/R_s)^-b, where B_s

is the surface polar magnetic field estimated from radio timing 
(∼ 2× 10^8 G) and R_s is the NS radius; and the spatial extent 
x_0/R_lc[
R_lc=c /Ω=96 km
is the radius of the light cylinder of the pulsar in J1431] 
of the emitting region for particles injected in a
given injection point x_in.

The parameter x_0/R_lc plays the role of a weighting function representing the
reduction of the number of emitting particles directed
towards the observer at a distance from their injection point.

The model solves the equation of motion of the particles to obtain their trajectories, 
and computes the emission of particles along each point. The latter is dependent on
the local kinetic properties, like Lorentz factor, the pitch angle and position, 
and the local properties of the magnetic field <cit.>. 

Fig. <ref> (left panel) shows the results  of the model with 
log E_||=9.31 ± 0.04 V m^-1, log (x_0/R_lc)=-2.34 ± 0.04 and 
b= 1.24±0.23 (χ^2_ν/d.o.f.=2.29/23).
The fit is not perfect, but reasonably close to the data.
The X-ray emission is the worse described by the model, 
and, related to that, the magnetic gradient is low in comparison with other 
fitted pulsars <cit.>. This can come as a result of two related effects.
First, the X-ray spectrum is steeply rising at soft energies, more than usually found for 
other pulsars, and as a result the peak in X-rays is comparable to the peak in gamma-rays,
so that the model is trying to keep up with a large synchrotron emission.
Second, this may also come as an effect of considering a more extended region. 
Farther from the pulsar, the magnetic field still needs to be high –decay more slowly– 
in order for synchrotron emission to proceed. In addition, 
the possible NS thermal contamination would mostly affect at these low energies. 


Given such possibility, the SED has also been fit excluding 
the soft X-ray portion of the spectrum, and hence below 2 keV.  Fig. <ref> (right panel)
shows the results with log E_||=9.56 ± 0.04 V m^-1, 
log (x_0/R_lc)=-2.59 ± 0.04 and
b=0.99±0.23 (χ^2_ν/d.o.f.=0.72/7), which describe much better the
observed SED. The higher value of the parallel electric field and the lower b are 
the consequence of the steep X-ray slope. Since there are not many MSPs with both X-ray and gamma-ray data 
<cit.>, it is not possible to state whether this steep spectrum is a common or unusual
property of MSPs. 

In summary, a magnetospheric origin for the X-ray to gamma-ray emission seems to be plausible.





§ CONCLUSIONS



We have analysed for the first time the X-ray and optical emission of the energetic
MSP pulsar binary J1431 to assess the system parameters and characterize its multi-band
emission. Here we summarise the main results:



  * Our first deep X-ray observation of J1431 has detected a dim X-ray
source  without signficant  variability at its 10.8 h orbital period. 
 This contrasts with the majority
of spiders displaying substantial X-ray orbital variations, which are often found double-peaked
due to Doppler boosting of the IBS synchrotron emission. 






  * The X-ray spectrum is featureless and consistent with non-thermal emission
with a power law photon index Γ= 1.6^+0.3_-0.2 and negligible absorption 
(N_h ≤ 8×10^20 cm^-2). While not statistically
significant, the presence of a thermal component (kT_BB =0.15±0.04 keV) may hint
at the contribution of the heated polar cap (R_BB∼ 180-660 m) in the soft X-rays
together with a harder (Γ=1.3±0.4) power law component dominating at higher energies. 


  * Given that the X-ray timing and spectral analyses do not strongly favour
a  dominant contribution from an IBS, we inspected whether a magnetospheric origin from the MSP can describe
the X-ray and gamma-ray SED and found a reasonable match. 
This suggests that an IBS in this system, if present, has a negligible contribution.


  * 

 The companion star is found to be an early F-type star with a mass of 0.20±0.04 M_⊙
which  underfils its Roche lobe (f_ RL=73±4%), confirming the redback nature of J1431,
 although hotter than the majority of redbacks.
The multi-colour optical light curves  display typical 
ellipsoidal modulation at the long orbital period but are also affected by irradiation. 
Among the several models attempted to fit the multi-colour light curves and radial velocity
curve,  similar parameters are found with a model that includes direct heating from the 
pulsar and a cold spot with respect to the base stellar temperature (Δ T ∼ -500 K),  
as well as with models that 
include heat redistribution after irradiation encompassing convection and also 
both diffusion and convection. The resulting
temperatures of the dayside and nightside faces of the companion are found to be ∼7500 K
and 7400 K, respectively, indicating a mild irradiation temperature T_irr∼3900 K.
The lack of a strong irradiation is likely due to the high luminosity of the star which
is about 6.5 times larger than that expected from heating at the orbital distance 
from the pulsar. 


  * 

 The binary inclination is found to be i= 59±6^∘, explaining
the lack of orbital modulation in the X-rays and gamma-ray regimes. Furthermore, a 
distance of 3.1±0.3 kpc is found, which locates J1431 at larger distance than
previously estimated.


  * 

 Despite the inability to derive the best irradiation model, which limits
the precision to which the NS mass is constrained, a massive (1.8-2.2 M_⊙)
NS in J1431 is favoured. 
 



This work is based on observations obtained with , am ESA science mission
with instruments and contributions directly funded by ESA Member States and
NASA and on observations collected at the 
European Southern Observatory under ESO programme 0103.D-0241.
This work has made use of data from SkyMapper, a facility funded through 
ARC LIEF grant LE130100104 from the Australian Research Council, 
awarded to the University of Sydney, the Australian National University, 
Swinburne University of Technology, the University of Queensland, the 
University of Western Australia, the University of Melbourne, Curtin 
University of Technology, Monash University and the Australian 
Astronomical Observatory. SkyMapper is owned and operated by The 
Australian National University's Research School of Astronomy and 
Astrophysics. 
This work has also  made use of data from the European Space Agency (ESA) mission
Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia
Data Processing and Analysis Consortium (DPAC,
<https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.
The data are publicly available at the European Space Agency (ESA) archive
http://nxsa.esac.esa.int/nxsa-web. The data can be obtained by contacting
the corresponding author or the team (V.S. Dhillon).  

DdM, AP, FCZ, AMZ acknowledge financial support from the Italian Institute for Astrophysics 
(INAF) Research Grant 2022:"FANS". AP is supported by the Italian Ministry
of University and Research (MUR), PRIN 2020 (prot.2020BRP57Z) "Gravitational 
and Electromagnetic-wave Sources in the Universe with current and next generation
detectors (GEMS).  
FCZ is supported by a Ramón y Cajal fellowship (grant agreement RYC2021-03088-I).
VSD and operations are funded by the Science and Technology Facilities Council (grant ST/Z000033/1).
RPB acknowledges support from the European Research Council (ERC) 
under the European Union's Horizon 2020 research and innovation program
(grant agreement N.715051; Spiders).
We also acknowledge the support of the PHAROS COST Action (CA 16214).

We acknowledge useful comments from the anonymous referee.


 


aa
 













§ CORNER PLOTS FOR THE THREE BEST FIT MODELS WITH POST-IGD















