---
abstract: 'We report detection of the likely companion of the binary millisecond pulsar 0740  with the Gran Telescopio Canarias in the $r''$ and $i''$ bands.  The position of the detected starlike source coincides with the pulsar coordinates within the $1\sigma$ uncertainty of $\approx$ 0.2 arcsec.  Its magnitudes are $r''=26.51\pm0.17$ and $i''=25.49\pm0.15$. Comparing the data with the white dwarf cooling tracks suggests that it can be an ultracool helium-atmosphere white dwarf with the temperature $\lesssim3500$ K and cooling age $\gtrsim5$ Gyr.  The age is consistent with the pulsar characteristic age corrected for kinematic effects.  This is the reddest source among known white dwarf companions of millisecond pulsars.  Detection of the source in other bands would be useful to clarify its properties and nature.' 
author:
- |
    D. M. Beronya$^{1}$[^1], A. V. Karpova$^{1}$, A. Yu.  Kirichenko$^{2,1}$, S. V. Zharikov$^{2}$, D. A. Zyuzin$^{1}$, Yu.  A. Shibanov$^{1,3}$ and A. Cabrera-Lavers$^{4,5}$\
    $^{1}$Ioffe Institute, Politekhnicheskaya 26, St. Petersburg, 194021, Russia\
    $^{2}$Instituto de Astronomía, Universidad Nacional Autónoma de México, Apdo.  Postal 877, Baja California, México, 22800\
    $^{3}$Peter the Great St. Petersburg Polytechnic University, Politekhnicheskaya 29, St. Petersburg, 195251, Russia\
    $^{4}$Instituto de Astrofísica de Canarias, La Laguna 38200, Tenerife, Spain\
    $^{5}$Departamento de Astrofísica, Universidad de La Laguna, La Laguna 38206, Tenerife, Spain
bibliography:
- 'msp4.bib'
date: 'Accepted XXX.  Received YYY; in original form ZZZ'
title: 'The ultracool helium-atmosphere white dwarf companion of PSR 0740?' 
---

\[firstpage\]

binaries: general – pulsars: individual: PSR 0740

Introduction
============

Millisecond pulsars (MSPs) are neutron stars (NSs) with short rotation periods $P$, typically below 30 ms.  They have small spin-down rates ($\dot{P}\lesssim10^{-19}$ s s$^{-1}$) and constitute a special subclass on the $P-\dot{P}$ diagram, distinct from the population of ‘normal’ pulsars [e.g., @manchester2017].  The first MSP, PSR B1937+21, was discovered in 1982 [@backer1982].  It is believed that these objects are formed in binary systems: the initially more massive of the two binary components star turns into a NS which is then spun-up (or ‘recycled’) due to accretion of matter from the secondary star [@Bisnovatyi-Kogan1974; @alpar1982; @tauris2011].  This is supported by the fact that a high ($>50$ per cent) fraction of MSPs has companions and by discoveries of several MSPs switching between the rotation-powered and accretion-powered pulsar states [@Archibald2009; @Papitto2013; @Bassa2014; @Roy2015].  Moreover, the NS binary 1FGL J1417.7–4402 was recently found to be in the late stages of the canonical MSP recycling process [see @swihart2018 and references therein]. 

Spin periods of MSPs are extraordinarily stable which may be related to their weak external magnetic fields [@manchester2017].  The period stability makes MSPs important in different areas of astrophysics.  For instance, studies of the binary pulsar PSR B1913+16, the double-pulsar system PSR J0737$-$3039A/B and especially the triple system PSR J0337+1715 allow one to carry out unprecedented tests of general relativity [@kramer2018; @archibald2018; @weisberg2016 and references therein].  A set of MSPs (or ‘Pulsar Timing Array’) was proposed for detection of low-frequency gravitational waves and spacecraft navigation [@manchester2017b; @becker2018].  MSP timing is also used to obtain physical characteristics of stellar clusters [@freire2017; @prager2017].  Studies of MSPs are essential for exploration of binary and stellar evolution and also reveal properties of the interstellar medium (ISM).  Binary systems containing a MSP allow for most accurate determination of NSs masses which plays a key role in constraints on equation-of-state (EoS) of the superdense matter in NSs interiors [@lattimer2012; @antoniadis2013]. 

In a binary system masses of both the NS and its companion can be inferred with a high precision through the measurement of the general relativistic Shapiro delay from pulsar radio timing observations [@shapiro1964; @jacoby2005].  However, the effect is strong for edge-on systems; otherwise, additional optical observations are necessary.  The latter are also crucial for determination of an MSP companion nature [e.g. @vankerkwijk2005].  Low-mass white dwarfs (WDs) are the most frequent MSP companions [@tauris2012].  In such a case, comparison between the optical data and WD evolutionary sequences can provide the type, temperature, cooling age and mass of the WD.  Knowing the companion mass, the mass function (from pulsar timing observations) and the mass ratio (from the optical phase-resolved spectroscopy), one can estimate the pulsar mass and the system inclination.  Unfortunately, due to the weakness of the optical counterparts, identifications of MSP companions are not numerous.  Nevertheless, increasing the number of such identifications is important for understanding the nature and origin of these systems.  The situation has been improving thanks to the world’s largest telescopes and deep sky surveys [e.g., @bassa2016; @dai2017; @karpova2018; @kirichenko2018]. 

  ------------------------------------------------------------------------- ---------------------
  Right ascension $\alpha$ (J2000)                                             07404579492(2)
  Declination $\delta$(J2000)                                                  +6620335593(2)
  Galactic longitude $l$ (deg)                                                      149.7
  Galactic latitude $b$ (deg)                                                       29.6
  Epoch (MJD)                                                                       57017
  Proper motion $\mu_\alpha =\dot{\alpha}{\rm cos}\delta$ (mas yr$^{-1}$)        $-$10.3(2)
  Proper motion $\mu_\delta$ (mas yr$^{-1}$)                                     $-$31.0(2)
  Spin period $P$ (ms)                                                       2.88573641085019(2)
  Period derivative $\dot{P}$ ($10^{-20}$ s s$^{-1}$)                             1.2184(4)
  Characteristic age $\tau$ (Gyr)                                                   3.75
  Spin-down luminosity $\dot{E}$ (erg s$^{-1}$)                              $2.0\times10^{34}$
  Orbital period $P_b$ (days)                                                  4.766944619(1)
  Projected semi-major axis (lt-s)                                              3.9775602(2)
  Eccentricity (10$^{-6}$)                                                         4.9(1)
  Mass function $f_{\rm M}$ ()$^b$                                                  0.003
  Minimum companion mass $M_{c, \rm min}$ ()$^b$                                     0.2
  Dispersion measure (DM, pc cm$^{-3}$)                                             15.0
  Distance $D_{\rm YMW}$ (kpc)                                                      0.93
  Distance $D_{\rm NE2001}$ (kpc)                                                   0.68
  Distance $D_{p}$ (kpc)                                                     0.4$^{+0.2}_{-0.1}$
  ------------------------------------------------------------------------- ---------------------

  : Parameters of the  system (from @arzoumanian2018).  Current values are available in the NANOGrav Data Repository$^a$. The dispersion measure distances $D_{\rm YMW}$ and $D_{\rm NE2001}$ were estimated using the YMW16 [@ymw] and NE2001 [@ne2001] models for the distribution of free electrons in the Galaxy, respectively.  The distance $D_p$ corresponds to the timing parallax of 2.3(6) mas.  Numbers in parentheses are 1$\sigma$ uncertainties related to the last significant digits quoted. 

\[tab:param\]

$^a$https://data.nanograv.org/

$^b$Parameters are provided by @lynch2018.  $M_{c,\rm min}$ is calculated assuming the inclination angle $i=90$ deg and the pulsar mass $M_p=1.4$ . 

The subject of this paper, the binary MSP 0740 (hereafter ), was discovered in the radio in the frame of the Green Bank North Celestial Cap Pulsar Survey [@stovall2014; @lynch2018].  The pulsar was also identified in gamma-rays with the   telescope [@laffon2015; @guillemot2016].  Table \[tab:param\] summarises the parameters of the pulsar system.  Basing on the computed minimum companion mass of $\approx0.2$ , @lynch2018 suggested that the pulsar companion is a He WD.  Using the magnitude lower limits from the PanSTARRS 3$\pi$ Steradian Survey bands, they constrained the WD temperature $<4200$ K and age $>3.2$ Gyr. 

To find the optical counterpart of the pulsar companion, we performed deep optical observations with the Gran Telescopio Canarias (GTC).  Here we report a likely identification of the pulsar companion.  The details of the observations and data reduction are described in Sect.  \[sec:data\], the analysis and results are presented in Sect.  \[sec:results\] and discussed in Sect.  \[sec:discussion\]. 

Observations and data reduction {#sec:data}
===============================

The observations of the  field were performed on December 26, 2017[^2] in the Sloan $r'$ and $i'$ bands using the GTC/OSIRIS[^3] instrument.  The conditions were photometric and seeing varied from 0.8 to 1.3 arcsec.  The instrument field of view (FOV) was 7.8 arcmin $\times$ 7.8 arcmin with the pixel scale of $0.254$ arcsec.  The pulsar was exposed on the CCD2 chip of the two OSIRIS CCD chips.  The dithered science frames of $120$ s were taken, resulting in the total exposure times of $3.7$ and $2.4$ ks for the $r'$ and $i'$ filters, respectively.  The short 15 s exposures were obtained in each band to avoid saturation of bright stars that were further used for precise astrometry.  The observing log is presented in Table \[tab:gtc\]. 

   Filter   Exposure time, s   Airmass   Seeing, arcsec
  -------- ------------------ --------- ----------------
    $r'$      $15\times1$       1.32          0.9
             $120\times31$      1.29          0.9
    $i'$      $15\times1$       1.26          0.8
             $120\times20$      1.26          0.8

  : Log of the GTC observations.  The mean airmass and seeing values are presented.[]{data-label="tab:gtc"}

The data reduction and analysis were carried out within the Image Reduction and Analysis Facility ([iraf]{}) package.  The raw data included 10 bias frames and 5 sky flats for each filter, which were used to create the master bias and master flats frames.  The bias subtracted and the flat-fielded science frames were obtained with the [ccdproc]{} routine, aligned to the frame with the best seeing, and then stacked for each filter using the [imcombine]{} task.  Accounting for known OSIRIS geometrical distortions and background gradients in the images towards the FOV boundaries[^4], we ignored the areas outside 2.5 arcmin from the target to avoid the influence of these effects on the calibration and further results. 

To perform the astrometric transformation, we used the [iraf ccmap]{} task and 9 bright non-saturated reference stars detected with a high signal-to-noise ratio in the $r'$-band short-exposure frame.  The best coordinates of the stars are contained in the Gaia DR2 catalogue [@gaia2016; @gaia2018b] and have the catalogue conservative accuracy of $\approx0.7$ mas [@lindegren2018].  The resulting $1\sigma$ [*rms*]{} uncertainties of the astrometric fit are 0.08 arcsec in RA and Dec. The Gaia catalogue uncertainties are substantially smaller and can be neglected.  The resulting WCS solution was applied to the combined images in the $r'$ and $i'$ bands.  Selecting other reference stars did not change the fit result. 

For the photometric calibration, we used the Landolt standard star G163-50 [@landolt1992] observed during the same night as the target.  The derived photometric zeropoints for the CCD2 chip are $z_{r'}=28.88\pm0.01$ and $z_{i'}=28.52\pm0.02$. The values were obtained by comparing the standard star magnitudes from @smith2002 with its instrumental magnitudes, corrected for the finite aperture and atmospheric extinction.  We used the atmospheric extinction coefficients[^5] $k_{r'}=0.07\pm0.01$ and $k_{i'}=0.04\pm0.01$. The 3$\sigma$ detection limits on the resulting images are $r'<27.5$ and $i'<26.5$.

Results {#sec:results}
=======

Figure \[fig:gtc\] shows the sections of the $r'$ and $i'$ images, containing the  position.  The radio coordinates of the pulsar corrected for the proper motion (Table \[tab:param\]) to the epoch of our observations (58114 MJD) are RA=0740457898(1) and Dec=+6620334661(6).  Accounting for the astrometric referencing uncertainty, the radius of the  1$\sigma$ position error circle on the optical images is $\approx 0.08$ arcsec.  In the main panels of Figure \[fig:gtc\] we show the 3$\sigma$ circle for convenience.  A faint starlike source is detected in both bands, which overlaps with the circle.  Its coordinates RA=07404576(1) and Dec=+6620334(2) were derived as a mean of the source positions in both bands.  The errors account for the position uncertainties of the object in the frames, the difference in the positions in the two bands, and the astrometric referencing uncertainty.  The object coordinates coincide with the pulsar position at the $1\sigma$ significance level.  This is demonstrated in the inset of the right panel of Figure \[fig:gtc\] where the $1\sigma$ position circles for the pulsar and the object overlap. 

For photometry of this faint object, we used the circular source aperture with $\approx$ 0.76 arcsec (3 pixels) radius, which is consistent with the seeing value and provides the maximum signal-to-noise ratio on the source.  To minimise possible contributions from wings of nearby sources, backgrounds were taken from a narrow annulus centred on the source with the inner and outer radii of $\approx$ 0.8 and 1.5 arcsec.  As a result, we obtained the source magnitudes $r'=26.51\pm0.17$ and $i'=25.49\pm0.15$ corrected for the finite aperture and the atmospheric extinction.  Alternatively, for both bands we constructed the point spread functions (PSFs) using about a dozen of unsaturated isolated field stars and the [daophot]{} package, and performed the PSF-fitting photometry of field sources applying the [allstar]{} routine with the same source aperture but a wider background annulus with the inner and outer radii of $\approx$ 2.5 and 5 arcsec[^6].  This yielded the same object magnitudes with only marginally smaller errors.  Our object is perfectly PSF-subtracted without any residuals that could indicate a possible presence of fainter sources hidden in its profile.  Finally, we performed the aperture photometry on the images where all neighbors, excluding the object of interest, were PSF-subtracted.  Independently of the annulus width this led again to the results consistent with the above aperture and PSF magnitude measurements.  We thus adopt the above values as the most conservative ones. 

We estimated the probability of detecting a random object at the pulsar position using the Poisson distribution $P=1-exp(-{\sigma}S)$, where $\sigma$ is the surface number density of objects within the GTC field of view and $S$ is the area of the pulsar error ellipse.  As a result, we found the probability of $\approx2\times10^{-3}$ for an accidental coincidence of the pulsar and an unrelated object with a brightness of $>$ 20 mag.  This indicates the source is a likely optical counterpart to the  binary companion. 

To obtain intrinsic colours and magnitudes of the presumed companion we used the empirical correlation between the distance $D$ and the interstellar reddening $E(B-V)$ based on the Pan-STARRS 1 and 2MASS photometric data [@dustmap2018].  The expected $E(B-V)$ in the pulsar direction is $\approx0.04\pm0.02$ for the most conservative distance range of 0.3 – 0.9 kpc following from Table \[tab:param\].  This value was transformed to the extinction correction values $A_{r'}=0.09\pm0.05$ and $A_{i'}=0.07\pm0.03$ using the conversion coefficients presented by @schlafly2011.  The resulting intrinsic magnitudes and colour index are $r'_0=26.42\pm0.18$, $i'_0=25.42\pm0.15$, and $(r'-i')_0=1.0\pm0.2$, where the errors include uncertainties of the reddening and magnitude measurements.  For the above distance range, the absolute magnitude $M_{r'}$ varies from 16.4 to 19.2.  The specific values for the timing parallax distance and for the maximum model depended DM distance of $930$ pc are $M_{r'}=18.4^{+0.8}_{-1.1}$ and $16.6^{+0.2}_{-0.2}$, respectively. 

As it has been suggested, the  companion is a WD [@lynch2018].  Therefore, we compared the derived colour and magnitudes of the likely companion with the cooling sequences of hydrogen (DA) and helium (DB) white dwarfs.  The corresponding colour-magnitude and colour-colour diagrams are presented in Figures \[fig:col-mag\] and \[fig:col-col\], respectively.  Cooling tracks for WDs with hydrogen and helium atmospheres [known as Bergeron models: @holberg2006; @kowalski2006; @tremblay2011; @bergeron2011] are shown by different line types[^7].  The data points of companions to some other MSPs are presented: J0348+0432 [@antoniadis2013], J0614$-$3329 [@bassa2016], J0621+2514 [@karpova2018], J0751+1807 [@bassa2006], J1012+5307 [@nicastro1995], J1231$-$1411 [@bassa2016] and J2317+1439 [@dai2017].  We also include two ultracool isolated WDs: SDSS J110217.48+411315.4 [hereafter WD J1102; @hall2008; @kilic2012] and WD 0346+246 [@oppenheimer2001].  For WD 0346+246 and the PSR J0751+1807 companion, the $UBVRI$ magnitudes were converted to the Sloan Digital Sky Survey (SDSS) system using transformations by Lupton[^8].  The magnitudes of other objects were adopted either from the cited articles or the SDSS catalogue [@abolfathi2018].  The corresponding values of interstellar reddening $E(B-V)$ of the sources were obtained using the models by @drimmel2003 and @dustmap2018 together with the distances from DM and parallax measurements or model-predicted WD distances.  For PSR J2317+1439, the parallax information was updated by @arzoumanian2018, and for this reason the reddening and absolute magnitude from @dai2017 were recalculated.  For PSR J0621+2514, we combined the results for two DM distance estimates [@karpova2018]. 

The  presumed companion is the reddest among sources shown in the diagrams (Figures \[fig:col-mag\],\[fig:col-col\]).  The Bergeron models presented here are constructed for CO-core WDs.  However, it is generally believed that most of the MSP companions are He-core WDs, due to stripping a companion envelope during the mass-transfer binary phase [see e.g. @rivera-sandoval2015; @cadelano2015].  For hydrogen-atmosphere WDs, the comparison of the Bergeron models with those of the He-core ones created by @althaus2013 shows that the presumed  companion cannot certainly belong to DA WDs, considering any core composition.  This is because the upper limit for $(r'-i')_0$ indices of the hydrogen atmosphere sequences is $\la0.5$ for both core types and all DA tracks turn towards blue colours at high ($M_{r'}\gtrsim16$) absolute magnitudes/low temperatures [see also @bassa2016].  At the same time, the colour index of the presumed companion is about one (Figure \[fig:col-mag\]).  On the other hand, for cool WDs with pure helium atmospheres colours continue to redden with decreasing temperatures.  Unfortunately, we have not found respective He-core magnitude sequences for these WDs.  Nevertheless, calculations show that the luminosity of a He-core WD at a given age has to be significantly higher than that of the CO-core WD with the same mass [e.g., @oirschot2014].  This is due to the fact that the global specific heat capacity of the former is larger than that of the latter.  However, at ages $\ga 1$ Gyr the luminosity difference for low mass WDs becomes less than a half of magnitude which could not be resolved within a factor of four larger intrinsic magnitude uncertainty of the presumed companion.  This is mainly caused by the large distance uncertainty (light-blue rectangle in Figure \[fig:col-mag\]).  Moreover, the spectral energy distributions of cool helium-atmosphere WDs become close to those of blackbodies (BBs) [e.g. @kilic2011hoard].  As an example, in the diagrams we present the cooling sequence for a 0.4  WD with the BB spectrum, describing the cooling from 6000 to 2000 K. One can see that the presumed  companion position in the diagrams is in agreement with this track.  Therefore, it most likely belongs to the class of ultracool WDs with helium atmospheres and has a temperature of $\lesssim3500$ K and a cooling age of $\gtrsim5$ Gyr. 

Discussion and conclusions {#sec:discussion}
==========================

We detected the likely companion to the MSP .  The comparison of the photometry results with the WD evolutionary sequences showed that the source is most probably an ultracool WD with a pure helium atmosphere.  It can be the reddest WD among the known MSP companions.  The WD-core composition remains unclear due to the large distance and magnitude uncertainties. 

The presumed WD cooling age is $\gtrsim 5$ Gyr.  If it has a low mass, this age limit can be larger since after the Roche-Lobe detachment a proto-WD goes through the contraction phase until it reaches its cooling track [@istrate2014; @istrate2016].  The duration of this phase increases as the mass of the proto-WD decreases, and may last as long as $\sim$ 2 Gyr.  The  characteristic age of $3.75$ Gyr (Table \[tab:param\]) is smaller than the WD age estimate.  However, the observed pulsar period derivative and consequently its characteristic age can be biased by kinematic effects, i.e. the effects of the pulsar proper motion [Shklovskii effect; @shklovskii1970], the acceleration towards the Galactic plane and the acceleration due to differential Galactic rotation [@nice1995].  Using the  proper motion value from Table \[tab:param\], the Sun’s galactocentric velocity and the distance [240 km s$^{-1}$ and 8.34 kpc, respectively; @reid2014], we calculated these corrections to the pulsar period derivative: $\dot{P}_{\rm S}=3.0\times10^{-21}$, $\dot{P}_{\rm G,\perp}=-1.6\times10^{-22}$, $\dot{P}_{\rm G,p}=3.8\times10^{-23}$ for the minimum and $\dot{P}_{\rm S}=6.9\times10^{-21}$, $\dot{P}_{\rm G,\perp}=-2.2\times10^{-22}$, $\dot{P}_{\rm G,p}=8.9\times10^{-23}$ for the maximum pulsar distance estimates.  The corresponding intrinsic characteristic ages are $\tau_i\sim5$ Gyr and $\sim 8.5$ Gyr, which are compatible with the cooling age[^9].  Thus, the considered binary system indeed can be very old and the presumed WD can be ultracool.  This is not a unique situation.  There are other examples of the objects with similar characteristics.  One of them is the WD companion of PSR J0751+1807 [@bassa2006].  Its colours (see Figure \[fig:col-mag\] and \[fig:col-col\]) indicate that the WD has a pure helium or mixed H/He atmosphere with a temperature $T\sim$ 3500 – 4300 K. Other examples are isolated ultracool white dwarfs WD J1102 [@hall2008; @kilic2012] and WD 0346+246 [@oppenheimer2001].  These WDs have temperatures of about 3650 and 3300 K, respectively, and are best explained by the mixed atmosphere models [@gianninas2015]. 

It is clear that a single colour index is not enough to confidently assert that the source is the optical counterpart of the pulsar companion.  Although the probability of the accidental coincidence is low, however, it cannot be excluded.  As seen from the colour-magnitude diagram presented in Figure \[fig:col-mag\_field\], the presumed companion is not very distinct from the distribution of other stellar sources detected within the GTC FOV.  It is located near the red locus of the main sequence distribution.  Below we consider possible alternative interpretations of the source nature. 

Firstly, the source colour is also in agreement with that of red/brown dwarfs which are cool low-mass ($M\sim$ 0.08 – 0.6 ) main-sequence stars.  According to the correlation between the colour and the spectral type, based on the SDSS sample of M, L, and T dwarfs [@hawley2002], the source can be a M0–M3V red dwarf or a T-type brown dwarf.  The corresponding distance of a red dwarf is $\gtrsim 8$ kpc, and the object with $b=+29.6$ deg (see Table \[tab:param\]) must be located at $\gtrsim$ 4 kpc above the Galactic disk.  A T-type brown dwarf would be very close to the Sun at a distance $\lesssim50$ pc.  Both values are not consistent with the distance to  and, in this case, the source must be rejected as the MSP companion. 

Another possibility is that the source is a red subdwarf.  Red subdwarfs are the metal-poor analogues of late-type dwarfs [see e.g. @savcheva2014; @zhang2013].  Absolute magnitudes of most of the detected late-type subdwarfs are $9\lesssim M_{r'}\lesssim14$, which infer a distance of $\gtrsim3$ kpc to the source.  This is also incompatible with the distance to  and implies the object is outside the Galactic disk. 

Finally, the extragalactic origin of the source cannot be excluded.  The total Galactic extinction in this direction $E_{B-V}\lesssim0.06$ is relatively low and such colour is expected, e.g., for distant E/S0/Sbc-type galaxies with redshifts $z\leq1$ [see e.g. @georgakakis2006]. 

If the source is not related to the , we can consider the derived magnitudes as lower limits for the pulsar binary companion.  In this case it could be even cooler and older though we cannot determine the type of its atmosphere.  Deep infrared observations are necessary to clarify the properties and nature of the detected source.  For a cool WD with a pure helium atmosphere one can expect the blackbody-like spectral energy distribution.  Future proper motion measurements can confirm the association of the likely counterpart with . 

Acknowledgements {#acknowledgements .unnumbered}
================

We thank the anonymous referee for useful comments and P. Bergeron for helpful discussions.  The work of DMB, AVK and DAZ (Sec.  \[sec:data\] and \[sec:results\]) was supported by the Russian Foundation for Basic Research, project No.  18-32-00781 mol\_a.  AYuK and SVZ acknowledge PAPIIT grant IN-100617 for resources provided towards this research.  The work is based on observations made with the Gran Telescopio Canarias (GTC), installed at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma. [ iraf]{} is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. [ This work has 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.  DAZ thanks Pirinem School of Theoretical Physics for hospitality.]{} 

\[lastpage\]

[^1]: E-mail: daria.beronya@gmail.com

[^2]: Proposal GTCMULTIPLE2A-17BMEX, PI A. Kirichenko

[^3]: [http://www.gtc.iac.es/instruments/osiris/]{}

[^4]: http://www.gtc.iac.es/instruments/osiris/media/OSIRIS-USER-MANUAL\_v3\_1.pdf

[^5]: http://www.iac.es/adjuntos/cups/CUps2014-3.pdf

[^6]: The latter is appropriate as the routine iteratively subtracts brighter neighbors in a source group before the photometry of fainter ones. 

[^7]: http://www.astro.umontreal.ca/$\sim$bergeron/CoolingModels

[^8]: https://www.sdss.org/dr14/algorithms/sdssUBVRITransform/

[^9]: It is well known that the characteristic age is just an estimate of the pulsar true age and can significantly deviate from it [e.g. @psrhandbook].  However, it is usually the only available estimate which is the case of .  The pulsar true age can be even larger than $\tau_i$ if the braking index is $<$ 3.