First measurement of the total gravitational quadrupole moment of a black widow companion

Abstract

We present the first measurement of the gravitational quadrupole moment of the companion star of a spider pulsar, namely the black widow PSR J2051−0827. To this end we have re-analysed radio timing data using a new model which is able to account for periastron precession caused by tidal and centrifugal deformations of the star as well as by general relativity. The model allows for a time-varying component of the quadrupole moment, thus self-consistently accounting for the ill-understood orbital period variations observed in these systems. Our analysis results in the first detection of orbital precession in a spider system at ω = −68.6 +0.9 −0.5 deg/yr and the most accurate determination of orbital eccentricity for PSR J2051−0827 with e = (4.2 ± 0.1) × 10 −5 . We show that the variable quadrupole component is about 100 times smaller than the average quadrupole momentQ = −2.2 +0.6 −1 × 10 41 kg m 2 . We discuss how accurate modelling of high precision optical light curves of the companion star will allow its apsidal motion constant to be derived from our results.

Multicolour optical light curves of the companion star to the millisecond pulsar PSR J2051−0827

“…Clark et al 2021); allowing for heat redistribution due to convective flows on the stellar surface (e.g. Voisin et al 2020, Stringer et al 2021. Although acceptable fits could be obtained by adding free parameters in this way, the various methods did not point to a consistent set of fit parameters: in particular, the best-fit inclinations and Roche-lobe filling factors varied widely between models.…”

“…With a reliable estimate of the Roche-lobe filling factor, we can now determine the apsidal motion constant, k2, from the orbital precession measurement of PSR J2051−0827 by Voisin et al (2020), constraining it to the range 0.0036 < k2 < 0.0047. 8 The apsidal motion constant describes how centrally condensed an object is, with higher central concentrations corresponding to smaller values of k2.…”

“…Models for the origin of the asymmetry include ducting of the particles in the shock between the pulsar and companion star winds (the intra-binary shock, IBS) onto 8 This value has been calculated using the mass ratio given in Table 3, not by using Fig. 1 of Voisin et al (2020). Note also that the ordinate of the latter figure is incorrect due to an error of a factor of 10 in the assumed mass ratio.…”

“…Note also that the ordinate of the latter figure is incorrect due to an error of a factor of 10 in the assumed mass ratio. the magnetic poles of the companion star (Sanchez & Romani 2017), re-processed radiation from the IBS (Romani & Sanchez 2016), hot and cold spots on the companion star's surface (van Staden & Antoniadis 2016, Clark et al 2021, and redistribution of energy on the companion star's surface by convection (Voisin et al 2020.…”

“…Levinson & Eichler 1991). Knowing the Roche-lobe filling factor also allows the apsidal motion constant of the companion to be determined from the detection of its gravitational quadrupole moment by Voisin et al (2020), enabling the internal structure of the star to be modelled.…”