The black widow pulsar J1641+8049 in the optical, radio and X-rays

PSR J1641+8049 is a 2 ms black widow pulsar with the 2.2 h orbital period detected in the radio and $\gamma$-rays. We performed new phase-resolved multi-band photometry of PSR J1641+8049 using the OSIRIS instrument at the Gran Telescopio Canarias. The obtained data were analysed together with the new radio-timing observations from the Canadian Hydrogen Intensity Mapping Experiment (CHIME), the X-ray data from the Spectrum-RG/eROSITA all-sky survey, and all available optical photometric observations. An updated timing solution based on CHIME data is presented, which accounts for secular and periodic modulations in pulse dispersion. The system parameters obtained through the light curve analysis, including the distance to the source 4.6-4.8 kpc and the orbital inclination 56-59 deg, are found to be consistent with previous studies. However, the optical flux of the source at the maximum brightness phase faded by a factor of $\sim$2 as compared to previous observations. Nevertheless, the face of the J1641+8049 companion remains one of the most heated (8000-9500 K) by a pulsar among the known black widow pulsars. We also report a new estimation on the pulsar proper motion of $\approx$2 mas yr$^{-1}$, which yields a spin down luminosity of $\approx$4.87$\times 10^{34}$ ergs s$^{-1}$ and a corresponding heating efficiency of the companion by the pulsar of 0.3-0.7. The pulsar was not detected in X-rays implying its X-ray-luminosity was<3 $\times$ 10$^{31}$ erg s$^{-1}$ at the date of observations.


INTRODUCTION
Among about 3400 pulsars discovered to date, more than 550 belong to the class of millisecond pulsars (MSPs; Manchester et al. ★ E-mail: aida@astro.unam.mx(2005)) 1 .These objects have short spin periods ( < 30 ms) and low spin-down rates (  ∼ 10 −20 -10 −18 s s −1 ).The most generally accepted scenario implies that MSPs are old neutron stars (NSs) which were spun-up (or 'recycled') through angular momentum transfer by accretion from their main-sequence companions during a lowmass/intermediate-mass X-ray binary stage (Bisnovatyi-Kogan & Komberg 1974;Alpar et al. 1982).The binary MSP population comprises several classes depending on the type of the companion.In the so-called 'spider' systems with tight orbits (  ≲ 1 d), low-mass companions are heated and ablated by the pulsar wind of relativistic particles and radiation (e.g.Manchester 2017).Evaporated material often causes eclipses of the pulsar radio emission.Black widows (BWs), which represent a subclass of such binaries, have very low-mass (  ≲ 0.05M ⊙ ) almost ablated degenerate companions.The origin and formation of such systems is not well understood, but it is actively discussed (Chen et al. 2013;Benvenuto et al. 2014Benvenuto et al. , 2015;;Ablimit 2019;Ginzburg & Quataert 2021;Guo et al. 2022).
About 70 BWs have been discovered so far thanks to radio and -ray observations.Half of them reside in the Galactic disk (Swihart et al. 2022), while others are associated with globular clusters2 .According to the recent census (Swihart et al. 2022), only about 20 BWs have been detected in the optical.However, optical studies allow one to determine fundamental parameters of BWs such as spectral type and temperature of a companion, irradiation efficiency, distance, as well as masses of its components when they cannot be derived from radio timing observations alone.
The binary MSP PSR J1641+8049 (hereafter J1641) was discovered in the Green Bank North Celestial Cap (GBNCC) pulsar survey (Stovall et al. 2014;Lynch et al. 2018).This is an eclipsing radio pulsar which is also presented in the list of Fermi-   2023) who performed its phase-resolved multiband photometry with the HiPERCAM instrument at the 10.4-m Gran Telescopio Canarias (GTC) in 2019.They analysed the multiband light curves and obtained the fundamental parameters of the system, including the inclination, the Roche-lobe filling factor, the companion mass and the temperature gradient over the companion surface, as well as the distance to the system.
In this paper, we report the results of our independent phaseresolved multi-band optical observations of J1641 obtained with the OSIRIS instrument at the GTC in 2020.We analyse the OSIRIS, HiPERCAM, LDT, and the 1 m telescope data of J1641 all together.In addition, we present the updated parameters of the system derived from ongoing observations with the Canadian Hydrogen Intensity Mapping Experiment (CHIME) telescope, and report an upper limit on the pulsar X-ray flux based on observations with eROSITA (Predehl et al. 2021) aboard the Spectrum-RG (SRG) orbital observatory (Sunyaev et al. 2021).The paper is organised as follows: observations and data reduction are described in Sec. 2, the radio timing analysis is presented in Sec. 3, while the modelling of the light curves is described in Sec. 4. Discussion and conclusions are given in Sec. 5.

Optical data
The phase-resolved photometric observations 4 of the J1641 field were carried out during two observing runs in the Sloan  ′ ,  ′ and  ′ bands with the Optical System for Imaging and low Resolution Integrated Spectroscopy (OSIRIS) instrument at the GTC.In order to reduce the CCD readout time and increase the efficiency of the phase-resolved observations, we used windowing.The target was exposed on CCD1, and the windowed FoV was 3.05×3.05arcmin 2 .To avoid effects from CCD defects, 5 arcsec dithering between the individual exposures was used in both observing runs.The observations roughly covered two orbital periods in total, i.e. one orbital period per each observing run.The first period was observed in the  ′ band only, whereas the second one was covered one month later using the alternating  ′ and  ′ bands.The log of observations is given in Table 1.The  ′ -band image of the pulsar field is presented in Fig. 1, where the inserts demonstrate the variability of the pulsar companion.
Using the Image Reduction and Analysis Facility (iraf) package, we performed a standard data reduction, including bias subtraction and flat-fielding.The cosmic rays were removed from all images with the L.A.Cosmic algorithm (van Dokkum 2001).Astrometric referencing was performed using a single 180-s  ′ -band image and a set of stars from the Gaia DR3 Catalogue (Gaia Collaboration et al. 2023).Given the reduced image size in the windowing mode, only five field stars detected by Gaia appeared to be suitable for the astrometric purposes.Using these stars, we computed the astrometric solution with the formal rms uncertainties ΔRA < ∼ 0.13 arcsec and ΔDec < ∼ 0.14 arcsec.Photometric calibration was performed using the Sloan photometric standards SA111-1925 for the  ′ band and PG1528+062B for the  ′ and  ′ bands (Smith et al. 2002) observed during the same nights as the target.Using their instrumental magnitudes and the site extinction coefficients   ′ = 0.15(2),   ′ = 0.07(1), and   ′ = 0.04(1) (Cabrera-Lavers et al. 2014), we calculated the zero points   ′ = 28.53(2),  ′ = 28.94(1), and   ′ = 28.42(1).To verify these zero points, we checked the sky transparency variability by comparing the instrumental magnitudes of a field star during the observations of the target.The variation did not exceed errors of the flux measurements for the used star.Nevertheless, since the standards and the target were observed at different sky positions, we also compared the magnitudes of several field stars with those from the Sloan Digital Sky Survey (SDSS) Release 14 Catalogue (Abolfathi et al. 2018) and the Pan-STARRS Catalogue (Flewelling et al. 2020).To convert the Pan-STARRS magnitudes to the SDSS photometric system, we used the equation (6) from Tonry et al. (2012).The stars used in this analysis are shown in Fig. 1 and listed in Table 2. Their catalogue  ′ -band magnitudes were found to be consistent within uncertainties with those calibrated using the photometric standard.However, in case of the  ′ and  ′ bands, we found a slight discrepancy between the respective magnitudes.In addition, we calculated the colour term corrections and found that they are negligible in the  ′ and  ′ bands, and only slightly affect the  ′ -band measurements.Accounting for all of the mentioned corrections, the resulting zero 4 Proposal GTC11-20AMEX, PI A. Kirichenko points are   ′ = 28.51(2),  ′ = 29.03(3),and   ′ = 28.34(3).The point source 3 upper limits in individual  ′  ′  ′ exposures are  ′ = 26.4, ′ = 25.8, and  ′ = 24.8.We note that all magnitudes presented in this paper are in the AB system.

Radio data
J1641 is being observed by the CHIME telescope (CHIME; CHIME Collaboration et al. 2022).For our present work, we processed and analysed high-cadence timing data recorded with the pulsartiming backend built for CHIME (CHIME/Pulsar Collaboration et al. 2021).The CHIME/Pulsar backend generates folded profiles evaluated over 10-s integrations and 1,024 frequency channels that span the 400-800 MHz range.These data are coherently dedispersed in real time and prior to folding, using the value of DM listed in Table 3.
The CHIME/Pulsar data set on J1641 currently spans ∼3 years of observations between early 2020 and 2023, and is being collected in support of ongoing GBNCC analyses (McEwen & et al. prep).All data were processed -through statistical cleaning of radiofrequency interference and downsampling to 32 channels -using the psrchive (van Straten et al. 2012) andclfd (Morello et al. 2019) analysis suites.Once initially cleaned, a subset of data was co-added to form a high-significance pulse profile that turned into a 'standard template' after de-noising; this standard template was then used to compute times of arrival (TOAs) for the entire CHIME/Pulsar data set, yielding 32 TOAs per epoch that are each evaluated across 32 downsampled frequency channels.Any TOAs with S/N values less than 8.0, as determined with the pat utility in psrchive, were excluded from analysis based on sub-optimal detection statistics.For the remaining data set, a small, additional amount of TOAs (≪1 per cent) were excised during the timing analysis due to corruption of the pulse profile from sub-threshold interference that was not detected during the data-preparation process.

X-ray data
The J1641 field was observed in the course of SRG/eROSITA allsky survey in four visits spanning between April 9, 2020 and Oct. 13, 2021 with the total exposure time of 2.8 ks.No source was statistically significantly detected at the pulsar position.The derived upper limit on the unabsorbed flux is ≈ 1.3 × 10 −14 erg s −1 cm −2 in the 0.5-10 keV range (90 per cent confidence), assuming a power law (PL) model with the photon index Γ = 2.5 (the average value for BWs, Swihart et al. 2022) and the absorbing column density  H = 7 × 10 20 cm −2 .The latter was derived using the reddening  ( − ) = 0.08 mag obtained for J1641 from the extinction map of Green et al. (2019) and the empirical relation from Foight et al. (2016).Note that the reddening is equal to that obtained from the optical light curve modelling (see below).

UPDATED RADIO TIMING OF PSR J1641+8049
We refined the timing model of J1641, based on the solution developed by Lynch et al. (2018), using the tempo pulsar-timing package to obtain updated estimates of the spin, astrometric, and orbital parameters based on the CHIME/Pulsar data set alone.We also incorporated 355 additional degrees of freedom to fit for DM in contiguous time bins of 1.5-day extent that span the timespan of the data set.Due to the turbulent environments often observed in BW  3. systems, we also explored the fitting of parameters that quantify variations in the orbital elements.
A summary of the best-fitting CHIME/Pulsar timing residuals for J1641 and DM timeseries is shown in Fig. 2, and timing model parameters are reported in Table 3.One important result from our updated modelling is that the CHIME/Pulsar data yielded statistically different estimates of the proper motion than those obtained by Lynch et al. (2018), with the new magnitude of proper motion  = 2.02(10) mas yr −1 being lower by a factor of ∼20.We attribute this difference to the high-cadence nature of the CHIME/Pulsar TOAs, which allow for better estimates of short-term variations that are typically observed in other BW systems.This change in proper motion mainly impacts the derived estimate of 'intrinsic' spin-down of the pulsar, i.e., the time rate of change in spin frequency corrected for biases induced by proper motion and acceleration in the Galactic potential (e.g., Nice & Taylor 1995).The corrected spin-down we derive for J1641 is discussed in Sec. 5 as it depends on results obtained from the optical analysis presented below.
Our timing model of orbital motion uses the ELL1 formalism to describe low-eccentricity orbits (Lange et al. 2001).The bestfit ELL1 model indicates that deviations from purely-Keplerian motion are detectable in the CHIME/Pulsar data set.The inclusion of one time-derivative in orbital frequency   = 2/  as a degree of freedom improves the fit of the timing model by an amount Δ 2 ≈ 17, 000.We found that fitting for two time-derivatives in   , along with the five Keplerian elements, yielded an optimal fit to the timing data.The interpretation of these parameters in terms of macroscopic quantities (e.g., mass and/or geometry) is nontrivial due to the complex environment in BW systems that produce stochastic orbital variations (e.g., Shaifullah et al. 2016).Future analysis of orbital evolution in the J1641 system will be performed once several years of additional CHIME/Pulsar data are obtained.
The reliable detection of J1641 indicates minimal eclipsing of radio signal at superior conjunction.However, the DM variations in Fig. 2 show secular and quasi-periodic trends over time.The estimate of  for J1641 nonetheless remains comparable whether we use a many-bin DM model or a polynomial expansion in DM, which differ in functional form and by hundreds of fit parameters.In order to better assess these variations, we generated a separate timing solution based on data acquired in the MJD 59000-59400 range5 and setting maximum DM-bin extents to be 0.5 days, i.e., to ensure a single DM is estimated for each observing epoch.Despite the shortened data set, we fit for the same parameters reported in Table 3 in order to separately measure variations in celestial position, DM and orbital motion.
The results of this data-subset modelling is shown in Fig. 3, which shows this residual subset as function of time and orbital phase.In the ELL1 binary model, the orbital phase Φ =   ( −  asc )/(2), where  asc is the epoch of passage through the longitude of ascending node specified in Table 3; superior conjunction corresponds to Φ = 0.25 in Figure 3.The fitting of per-epoch DMs allows for better resolution of periodic variations, though requires at least 270 DM-bin fit parameters for the MJD 59000-59400 portion of the data alone.Nonetheless, the per-epoch DMs exhibit clear periodic variations that occur on a timescale equal to the orbital period for J1641; the ∼monthly variation in the DM timeseries is a manifestation of aliasing due to CHIME/Pulsar observations occuring once every sidereal day.While an excess in DM coincides with superior conjunction, the DM appears to modulate over the whole orbit and thus indicates a structured circumbinary medium.Further analysis of the DM variations over the entirety of the J1641 data set, as well as other BW systems discovered and monitored by GBNCC, will be presented in future work.

PSR J1641+8049 OPTICAL LIGHT CURVES AND THE SYSTEM PARAMETERS
The resulting  ′ ,  ′ ,  ′ -band light curves folded with the orbital period are presented in Fig. 4, left.In the right panel of Fig. 4 we show the HiPERCAM data obtained in the  ′  -,  ′  -,  ′  -,  ′  -, and  ′ bands about one year before our observations (Mata Sánchez et al. 2023).The shapes of the light curves are found to be consistent, nevertheless the object appears to be slightly brighter and bluer at the maximum of the light curves in the HiPERCAM data.To demonstrate this, in Fig. 5 we present the source broad-band optical spectra in the maximum of the light curves, corresponding to the OSIRIS (blue) and the HiPERCAM (green) data.The slopes 6 of the spectra  ∼   are  OSIRIS = −1.74(21) and  HiPERCAM = −2.01(18),respectively.As it can be seen, there is a brightnessdecreasing tendency together with a relative reddening of the object spectrum on the time scale of these observations.We also note that the flux measurements close to the photometric maximum obtained with the McDonald Observatory 1 m telescope were brighter than those reported for the GTC/HiPERCAM observations, whereas the LDT measurements were mostly close to them (Lynch et al. (2018) and D. Kaplan, private communication).For comparison, in Fig. 6 we show the r ′ -band light curves with all available data points.
To estimate the system parameters, we fitted the light curves using the emission model of a binary system described in Zharikov et al. (2013Zharikov et al. ( , 2019)).The model consists of an NS as the primary which heats a low-mass companion as the secondary.The spectrum of each surface element of the companion is approximated by a blackbody with an effective temperature which is distributed nonuniformly over the star surface accounting for its heating by the pulsar.The contribution of the pulsar into the observed optical flux is negligible for any expected distance and NS brightness values.Following Zharikov et al. (2019), the effective irradiation factor of the secondary is related to the heating efficiency  and the spin-down luminosity of the pulsar as and it defines the effective radiate flux  in transferred from the pulsar to the secondary: where  norm is the angle between the incoming flux and the normal to the surface, Ω =  2 NS / 2 is the solid angle from which the pulsar is visible from the surface element Δ of the companion,  NS = 13 km is the NS radius and  is the orbit separation.The corresponding 'day-side' temperature of the companion star surface element is where  is the Stefan-Boltzmann constant.The phase-resolved light curves were calculated by integrating the flux from all visible elements of the secondary in the corresponding band.The gradient descent method was used to find the minimum of  2 defined as where   is the number of observations in a given filter,   ,   , and 6 Without the  ′  band for the HiPERCAM data.
are the observed and the calculated magnitudes, and the error of the observed magnitude, respectively.The free fitted parameters were the distance , the reddening  ( − ), the binary system inclination , the Roche lobe filling factor   defined as a ratio of distances from the centre of mass of the secondary to the star surface and to the Lagrange point  1 , the 'night-side' temperature  n of the secondary, the effective irradiation factor  irr [ergs s −1 cm −2 sr −1 ], the pulsar mass  NS , and the component mass ratio.
The best-fitting model light curves for the OSIRIS data are shown in the left panel of Fig. 4 by solid lines.The model parameters are given in the second column of Table 4.The uncertainty of each fitted parameter was calculated following the method proposed by Lampton et al. (1976).The geometry of the system and the distribution of the effective temperature at the companion surface for the OSIRIS data are shown in Fig 7 .For comparison, we also used the model for the HiPERCAM  ′  -,  ′  -,  ′  -band light curves.We limited our analysis of the HiPERCAM data to the three close optical bands because a simple blackbody spectrum approximation for radiation from a star surface element used in the model cannot describe the companion spectrum in a wide spectral range from the  ′  -up to the  ′  -band.On the other hand, this approach was applied to simplify the comparison of the fit results achieved using the same model setup in both cases.Since the source was slightly brighter and bluer in the HiPERCAM data compared to the OSIRIS data, we first fixed all model parameters excluding the secondary heating at the values given in Table 4 for the OSIRIS data, and then fitted the HiPERCAM data.The fit provided a hotter dayside part of the secondary with a maximum temperature of about 9200 K.After that, we thawed all parameters as in the case of the OSIRIS data.The best-fitting parameters of the last fit are given in the third column of Table 4.In general, they are close to or inside the 1 error range of the OSIRIS data fit results, except for the effective irradiation factor  irr , which, in turn, gives a higher day-side temperature of the secondary.Another difference is related to the mass of the pulsar.The fitting of the two data sets results in a significantly lower mass in the HiPERCAM data compared to that in the OSIRIS data.Nevertheless, their 1 uncertainties overlap.We note that the light curves shapes and fluxes are mainly defined by the size of the companion Roche lobe, its filling factor, temperature distribution and system inclination.The Roche lobe size is changed by 15 per cent at the variation of the pulsar mass within its reasonable limits.To get the observed light curves, these changes can be compensated by variations of the temperature and filling factor.Thus, these parameters are correlated and additional information is needed to better constrain them.
For instance, we can utilise the mass function from the new radio timing measurements (see Table 3).It decreases the numbers of free parameters, because it links the pulsar mass, the inclination and the mass of the companion.Taking into account the mass function we repeated the fitting of the data from both instruments.The results are presented in the last two columns of Table 4.They are very close to those obtained by Mata Sánchez et al. (2023) who used the icarus code (Breton et al. 2012) to model the HiPERCAM data (see table 3 in the respective paper).However, we note that the formal  2 values are higher as compared to the cases when the masses of components and inclination are free parameters.All results are summarised in Fig. 8 and discussed below.

DISCUSSION AND CONCLUSIONS
As it was mentioned before, the J1641 orbital period of 2.18 h is one of the shortest among the known BWs (Swihart et al. 2022), and its highly modulated optical light curves are typical for such tight binary systems with MSPs.The shapes of the light curves do not demonstrate significant changes on a one-year time scale between the HiPERCAM and OSIRIS observations.However, the fluxes at the photometric maximum exhibit a significant decrease between the observations by up to a factor of two.This can indicate variability of the pulsar wind on a short (days) and long (years) time scales.We note that variable heating (increases and decreases of the companion irradiation by the pulsar) is observed for some other MSP binary systems, such as, e.g., the BW candidate 4FGL J0935.3+0901(Halpern 2022) or the redback PSRs J1048+2339 (Yap et al. 2019) and J2129-0429 (Bellm et al. 2016).Follow-up observations are needed to confirm and study this effect for J1641.The previously reported proper motion  = 39(3) mas yr −1 for J1641 led to a negative value of the pulsar intrinsic spin-down luminosity, implying a spin-up scenario (Lynch et al. 2018;Mata Sánchez et al. 2023), which could indicate the presence of ac- cretion.However, the latter would complicate detection of radio pulsations from the pulsar and affect the optical light curves of the system.Neither the HiPERCAM nor the OSIRIS observations support this scenario.In addition, using the distance obtained through the HiPERCAM light-curve modelling, Mata Sánchez et al. (2023) concluded that the maximum proper motion for this system, which would allow to avoid the spin-up scenario, is  ≤ 19 mas yr −1 .Indeed, the updated proper motion  = 2.02(10) mas yr −1 derived from the new radio observations (see Sec. 2.2) is significantly lower than the one provided by Lynch et al. (2018).It yields a new estimation on the intrinsic period derivative,   and the intrinsic spin-down luminosity   .Considering the distance D = 4.6(2) kpc obtained from our optical light curve modelling, we estimated the Shklovskii correction (Shklovskii 1970),  Shk ≈ 0.08 × 10 −21 s s −1 .Taking into account the corrections due to the differential Galactic rotation and the pulsar acceleration, the corresponding intrinsic period derivative is   = 10.2 × 10 −21 s s −1 , and the spin-down luminosity  is   = 4.87 × 10 34 erg s −1 , rejecting the spin-up and accretion scenario.We note that in the case of this particular BW system, the Shklovskii correction is subdominant leading to the fact that the intrinsic spin-down luminosity is ≈ 4 per cent higher than the observed one.In addition, taking into account the newly determined proper motion, we can now estimate the J1641 transverse velocity.Considering the distance 4.6 kpc derived from the light-curve modelling and following Verbunt et al. (2017) to estimate the contribution of the Galactic rotation and solar peculiar velocity, we obtain   ≈ 36 km s −1 in the local standard of rest of the pulsar.This velocity is typical for pulsar binary systems (Hobbs et al. 2005).
According to Mata Sánchez et al. (2023), J1641 has one of the heaviest companions ( c =0.055 +0.016 −0.014 M ⊙ ) among the known BWs.More massive secondaries were found for, e.g., PSR J1555−2908 (0.060 +0.005 −0.003 M ⊙ ; Kennedy et al. 2022) and PSR J1810+1744 (0.065(1) M ⊙ ; Romani et al. 2021).However, our fit for all free parameters resulted in a lower companion mass of 0.035(13) M ⊙ and 0.025(4) M ⊙ for the OSIRIS and HiPER-CAM data, respectively.These values are close to the average mass m = 0.026(15) of pulsar companions in BW systems (Swihart et al. 2022, see table 5 therein).However, the value for the HiPER-CAM fit is inconsistent with the mass function derived from the radio timing (see Fig. 8), while the value for the OSIRIS fit agrees with it.Taking the mass function into account we got the mass of the companion of 0.046(10) M ⊙ for both data sets.This is in agreement with the values reported by Mata Sánchez et al. (2023).Thus, the companion indeed can be rather heavy in comparison with other BW systems.
Another distinct feature of J1641 is the high day-side temperature of the companion, 8200-9500 K, making it one of the five most heated sources among the known BWs (see Fig. 9).The other four sources are PSR J1311−3430 with  d ≳ 12000 K (Romani et al. 2012(Romani et al. , 2015)), the BW candidate ZTF J1406+1222 with  d ≈ 10500 K (Burdge et al. 2022), PSR J1810+1744 with  d ≈ 9400 K (Romani et al. 2021) and PSR J1555−2908 with  d ≈ 9400 K (Kennedy et al. 2022).The J1641 day-side temperature is ≳ 2 times larger than the respective temperatures of, e.g., PSR J0251+2606 (≈3400 K) or PSR J0636+5129 (≈4600 K) (Mata Sánchez et al. 2023) near the low end of the source temperature distribution.It seems natural that the number of companions with a lower day-side temperature is larger than that with a hotter one, as the latter have to be evaporated faster.However, it remains unclear how the companion heating is related to the 'spin-down flux' defined as   −4/3  (Zharikov et al. 2019, see figure 5 and table 5 therein).It is possible that the absence of a clear dependence on the spin-down luminosity and/or system separation indicates the importance of the pulsar wind instability and/or asymmetry in the companion heating.
Using the intrinsic spin-down luminosity   and the irradiation factor  irr , we estimated the irradiation efficiency range  ≈ 0.3-0.7.These values seem to overshoot those typically observed for BWs (e.g., Draghis et al. 2019).An irradiation luminosity even larger than  ( > ∼ 1), calculated assuming a canonical momentum of inertia value of 10 45 g cm 2 , was derived, e.g., for BW PSR J1810+1744 by Romani et al. (2021).It is also an additional argument that the true one can be lower due to the beaming factor (see Draghis et al. 2019;Romani et al. 2021).
The derived companion mass of J1641 (Table 4) is close to typical masses of brown dwarfs (0.01-0.07 M ⊙ ) implying the possible origin of the companion.However, a field brown dwarf with an age of > ∼ 1 Gyr, similar to the J1641 characteristic age (Table 3), would be twice more compact ( < ∼ 0.1 R ⊙ ) and colder ( < ∼ 1500 K, Marley et al. 2021) as compared to the derived night-side temperature of J1641.This disfavours the brown dwarf nature of the companion.Nevertheless, such a discrepancy appears to be not a unique property of J1641.As was noted before, most BW companions, especially the strongly heated, are bloated up by a factor of two in comparison with the Galactic field brown dwarfs (Kandel & Romani 2023, see table 6 therein).The strong irradiation by the pulsar can affect

Figure 1 .
Figure 1.The 3.05×3.05arcmin 2 J1641 FoV imaged with the GTC/OSIRIS in the  ′ band.The J1641 vicinity is shown by the box in the centre of the image, and the optical companion is indicated by the arrow.The left and right inserts correspond to the enlarged J1641 vicinity imaged near the companion maximum and minimum brightness phases.The stars from the Pan-STARRS catalogue listed in Table2and used for the photometric calibration, are marked by the capital letters.
detected pulsars(Ray 2023).Its flux in the 0.1-100 GeV range is 2.0(3) × 10 −12 erg s −1 cm −2(Abdollahi et al. 2022).J1641 was not observed in X-rays.It has an orbital period of 2.18 h which is one of the shortest among the known BWs.The companion's minimum mass was estimated to be 0.04 M ⊙(Lynch et al. 2018).The dispersion measure (DM) distances to the pulsar are  YMW16 = 3.0 kpc and  NE2001 = 1.7 kpc based on the YMW16 (Yao et al. 2017) and NE2001 (Cordes & Lazio 2002) models for the distribution of free electrons in the Galaxy, respectively.Lynch et al. (2018) found a faint optical counterpart ( = 24.0(3)) to J1641 in the archival data, and performed photometric observations of the pulsar with the 4.3-m Lowell Discovery Telescope 3 (LDT) in the , ,  and  filters and with the McDonald Observatory 1 m telescope in the  ′ and  ′ filters.The counterpart revealed strong brightness variations tied to the orbital period, confirming the optical identification of the pulsar companion.Further optical studies of J1641 were recently reported by Mata Sánchez et al. (

Figure 2 .
Figure 2. Best-fit timing residuals (R, top) and DM values determined for the CHIME/Pulsar data set described in Section 2.2.The DM values are plotted as changes relative to the value reported in Table3.

Figure 3 .
Figure 3. Per-epoch measurements of DM for J1641 plotted as a function of time (top) and orbital phase (bottom), estimated by fitting to all CHIME/Pulsar data within the MJD 59000-59400 range.In the top panel, the grey line is a best-fit estimate of the secular variation over time that is presumed to be linear.In the bottom panel, the DM values have been corrected for the secular variation.

Figure 4 .
Figure 4. Top-left: Multi-colour light curves of J1641 obtained with the OSIRIS/GTC ( ′ ,  ′ ,  ′ ) folded with the orbital period.The phase zero corresponds to the orbital phase when the companion day-side is facing an observer.Two orbital cycles are shown for clarity.Solid lines represent the result of the best fit of the data by the model where the companion is heated by the pulsar.Horizontal dashed lines show 3 detection limits of the observations.Photometric bands are marked by different colours.Top-right: The HiPERCAM/GTC multi-band data of J1641 and the best fit of the  ′  ,  ′  ,  ′  light curves by the model.Bottom panels: Fit residuals calculated as the difference between the observed () and the calculated () magnitudes for each data point in terms of the magnitude error .

Figure 5 .
Figure 5.The broad-band optical spectra of the J1641 optical component at the photometric maximum for different epochs of observations.

Figure 6 .
Figure 6.The r ′ -band light curves folded with the orbital period obtained with the GTC, LDT, and McD.The orbital phase  = 0.0 corresponds to the maximum of the GTC/HiPERCAM light curve and it is shifted to -0.76 from the radio convention orbital phase, where  = 0.0 corresponds to the ascending node.LDT and McDonald 1-m telescope data were obtained in March 2017.

Figure 7 .
Figure7.The Roche lobe of the system components and a magnification of the secondary.The colours mark the effective temperature distribution on the secondary surface.The image corresponds to the OSIRIS data modelling, when all parameters are free.

Figure 8 .
Figure 8.The masses of the pulsar and its companion from the fits of the optical light curves.The lines show the relation between masses at different orbit inclinations based on the mass function found from the pulsar radio timing.The black triangles correspond to the fits when the masses and inclinations are free parameters.The red triangle marks the fits which take into account the mass function.The green triangle shows the result from MataSánchez et al. (2023).

Table 1 .
Log of the J1641 observations with GTC/OSIRIS.

Table 2 .
Stars from the Pan-STARRS Catalogue detected in the J1641 field and their magnitudes.

Table 3 .
The J1641 parameters derived from observations with the CHIME/Pulsar backend.