Measuring a gravitomagentic effect with the triple pulsar PSR J0337+1715

To the first post--Newtonian order, the orbital angular momentum of the fast--revolving inner binary of the triple system PSR J0337+1715, made of a millisecond pulsar and a white dwarf, induces an annular gravitomagnetic field which displaces the line of apsides of the slower orbit of the other, distant white dwarf by $-1.2$ milliarcseconds per year. The current accuracy in determining the periastron of the outer orbit is $63.9$ milliarcseconds after 1.38 years of data collection. By hypothesizing a constant rate of measurement of the pulsar's times of arrivals over the next 10 years, assumed equal to the present one, it can be argued that the periastron will be finally known to a $\simeq 0.15$ milliarcseconds level, while its cumulative gravitomagnetic retrograde shift will be as large as $-12$ milliarcseconds. The competing post--Newtonian gravitolectric periastron advance due to the inner binary's masses, nominally amounting to $74.3$ milliarcseconds per year, can be presently modelled to an accuracy level as good as $\simeq 0.04$ milliarcseconds per year. The mismodelling in the much larger Newtonian periastron rate due to the quadrupolar term of the multipolar expansion of the gravitational potential of a massive ring, whose nominal size for PSR J0337+1715 is $0.17$ degrees per year, might be reduced down to the $\simeq 0.5$ milliarcseconds per year level over the next 10 years. Thus, a first measurement of such a novel form of gravitomagnetism, although challenging, may be somehow feasible in a not too distant future.

1. INTRODUCTION PSR J0337+1715 (Ransom et al. 2014) is a peculiar hierarchical triple system made of three astrophysical compact objects: an inner, tight binary S composed by a 2.7 ms millisecond pulsar A and a white dwarf B revolving one around each other in a relative circular orbit with orbital period P b 1.6 d, and another white dwarf C moving about S along a wider circular path with orbital period P 327 d and coplanar with that of S itself, both inclined by1 I b = I = 39.2 • to the plane of the sky, assumed as reference {x, y} plane.Their masses are M A = 1.44M , M B = 0.2M and M C = 0.4M , respectively, where M is the mass of the Sun.
So far, PSR J0337+1715 is the sole relatively tight member so far discovered of the class of hierarchical triple systems hosting stellar corpses, apart from B162-26 whose pulsar has a white dwarf as inner companion, and a roughly Jupiter-mass at 35 au as outer orbiter (Thorsett et al. 1999).According to Ransom et al. (2014), 1% of the millisecond pulsars population resides in stellar triples, and 100 such systems exist in the Galaxy.In general, it is expected that, in triple systems hosting a millisecond pulsar, dubbed as triple pulsars in the following, the orbit of the distant companion is more eccentric due to dynamical interactions between the stars during stellar evolution (Portegies Zwart et al. 2011).The coplanarity and circularity of the orbits of PSR J0337+1715, along with the fact that P/P b 200, imply that the current configuration is stable on long timescales, which favored its discovery.It is likely that the angular momentum vectors of the inner and outer orbits were torqued into near alignment in a phase, lasted about 1 Gyr, during which the outermost member evolved and transferred mass onto the inner binary, perhaps within a common envelope (Larwood and Papaloizou 1997).
On the one hand, PSR J0337+1715 proved unsuitable, at least until now, to perform the usual tests of the General Theory of Relativity (GTR) done with some tight binary pulsars (Kramer et al. 2006;Wex and Kramer 2020;Kramer et al. 2021) like, lorenzo.iorio@libero.itL. Iorio e.g., the Hulse-Taylor pulsar PSR B1913+16 (Hulse and Taylor 1975) and the double pulsar PSR J0737-3039A/B (Burgay et al. 2003;Lyne et al. 2004) because of its orbital configuration.Indeed, the argument of periastron ω is badly defined because of the almost vanishing eccentricity e of the orbit of the outer white dwarf.Furthermore, the mass of the inner one is far too small for the gravitational redshift of the pulsar signal to be measurable.Finally, the Shapiro delay is negligible; the radio waves traveling along the line of sight toward the Earth pass very distant from its companion because the orbital plane is far from being seen edgeon.On the other hand, PSR J0337+1715 turned out a valuable laboratory to put the tightest constraints so far on the (absence of the) Nordtvedt effect (Nordtvedt 1968b,a).The latter is an orbital consequence of a possible violation of the strong equivalence principle which would take place should bodies with different amounts of gravitational self-energy, just like a neutron star and a white dwarf, fell with different accelerations in an external gravitational field.As a consequence, if the falling objects orbit one around each other while moving altogether about a third body, their barycentric orbits should experience a differential elongation toward the source of the external field.Actually, no Nordtvedt effect was found in PSR J0337+1715 to a relative accuracy of two parts per million at 95% confidence level (Archibald et al. 2018;Voisin et al. 2020).
Does PSR J0337+1715 offer the possibility of testing some other predictions of GTR in a not too far future?The answer may become positive within a reasonable timeframe, as soon as a sufficiently large number of times of arrivals (TOAs) have been collected.
The paper is organized as follows.Section 2 is devoted to the orbital precessions of a test particle induced by the gravitomagnetic field of a distant, inner rotating massive ring.In Section 3, they are calculated for PSR J0337+1715 (Section 3.1), and an evaluation of the experimental accuracy obtainable in a not too remote future is given (Section 3.2).The competing effects due to the Schwarzschild-like gravitoelectric field of S (Section 3.3), and to the expansion of the Newtonian gravitational field of a massive ring to the quadrupolar order (Section 3.4) are worked out as well.Section 4 summarizes our findings and offers conclusions.

THE GRAVITOMAGNETIC ORBITAL PRECESSIONS DUE TO THE ORBITAL ANGULAR MOMENTUM OF THE INNER BINARY
The effect in question, to the first post-Newtonian (1pN) order, would not only be the usual gravitoelectric periastron advance due solely to the masses of the system, but, in principle, also a particular form of gravitomagnetic orbital shift.
At first, this might sound strange, given that the generalizations of the well-known Lense-Thirring (LT) orbital precessions (Lense and Thirring 1918) to a binary system (Soffel 1989) induced by the spin angular momenta J A and J B of both bodies through are notoriously orders of magnitude smaller than the 1pN gravitoelectric periastron rate In Equations ( 1)-( 5 are three mutually orthogonal unit vectors directed along the line of nodes in the reference {x, y} plane, a direction in the orbital plane perpendicular to the latter, and the orbital angular momentum, respectively, in such a way that lb × mb = ĥb .Suffice it to say that, for, e.g., the double pulsar PSR J0737-3039A/B (Burgay et al. 2003;Lyne et al. 2004), the gravitoelectric periastron precession of Equation ( 5) is as large as (Kramer et al. 2006) while the gravitomagnetic one, mainly due to the angular momentum of PSR J0737-3039A, is expected to be as small as (Hu et al. 2020 and recalling that the spin period of the A component of the double pulsar is 22.7 ms (Kramer et al. 2006).
As pointed out in Iorio (2022), a hierarchical triple system offers, in principle, the possibility of measuring the orbital precessions due to the gravitomagnetic field generated by the mass current generated by the fast orbital motion of the inner binary, seen as a rotating matter ring.In other words, one has to replace3 J b in Equations ( 1)-( 3) with the orbital angular momentum L b of the inner binary.Its magnitude is where is the reduced mass, and is the total mass of the inner binary.Usually, L b can be orders of magnitude larger than the spin angular momentum of any of the inner binary's components; in the case of PSR J0337+1715, it is while the pulsar's spin angular momentum J should be about three orders of magnitude smaller.From Equations ( 1)-( 3) it straightforwardly follows that, for a coplanar triple system, i.e., if Lb • ĥ = 1, Lb • l = Lb • m = 0, the inclination and the node stay constant, while the periastron precesses at a rate L. Iorio

THE CASE OF PSR J0337+1715
3.1.The effect of the 1pN gravitomagnetic annular field In the particular case of PSR J0337+1715, Equation ( 17) predicts for it a gravitomagnetic precession of the order of ωLT = −1.2mas yr −1 , where mas stands for milliarcseconds, while the current level of uncertainty in measuring its periastron over 1.38 yr during which TOAs were collected, can be calculated from Table 1 of Ransom et al. (2014) to be σ 0 ω 63.9 mas.(20)

Prospects for future accuracy improvements
By provisionally assuming that about same number of TOAs as given by Equation ( 19) will be collected in 1.38 yr over, say, the next 10 years, the resulting accuracy would be improved down to4 σ ω 0.15 mas (21) with respect to the figure quoted in Equation (3.2), while the total gravitomagnetic shift would amount to ∆ω LT −12 mas.( 22)

The 1pN gravitoelectric periastron precession
The 1pN gravitoelectric apsidal rate of the orbit of the outer companion around the inner binary, whose nominal value is could be safely subtracted from the measured total periastron precession since its uncertainty, calculated by propagating in Equation ( 5) the errors of P, µ, e from Table 1 of Ransom et al. (2014), is currently as little as 3.4.The Newtonian periastron precession due to the quadrupolar term of the gravitational potential of a matter ring By expanding the Newtonian gravitational potential of a circular massive ring, calculated in points far from the latter and lying in its plane (Ciftja et al. 2009), to the quadrupolar order, an extra-potential falling as 1/r 3 arises; in the case of the inner binary of PSR J0337+1715, it can be cast into the form (Demetrian 2006) where the ring's mass and radius are assumed to be equal to the total mass M b and to the semimajor axis a b of S, respectively.It can be straightforwardly calculated that the periastron of the distant companion is shifted by Equation (25) according to (Adkins and McDonnell 2007;Iorio 2012 where is the total standard gravitational parameter of PSR J0337+1715.The nominal precession of Equation ( 26) amounts to ωqp = 636775 mas yr −1 = 0.17 deg yr −1 .
It turns out that the main sources of uncertainty in Equation ( 26) are the masses of the pulsar A and of the outer white dwarf C. By assuming that their errors will reduce by a factor of 1/ √ N in the next 10 yr, where N is calculated as in Section 3.2, the mismodeling in Equation (28) should finally be σ ωqp 0.5 mas yr −1 . (29)

SUMMARY AND CONCLUSIONS
By assuming a rate of collection of TOAs over the next 10 years equal to the present one, it can be guessed that the accuracy in determining the periastron precession of PSR J0337+1715 may reach the 0.15 mas level, allowing, in principle, for a measurement of its cumulative gravitomagnetic shift of −12 mas due to the orbital angular momentum of the inner binary.The competing gravitoelectric advance should be subtracted from the total periastron variation without problems since its uncertainty is as of now as little as 0.04 mas yr −1 .On the other hand, relying upon the aforementioned assumptions, it can be argued that the accuracy in modelling the nominally much larger Newtonian periastron precession due to the quadrupolar term of the multipolar expansion of the gravitational potential of a massive ring should reach the 0.5 mas yr −1 level.
The situation may be more favorable for a hypothetical triple system, yet to be discovered, whose outer companion's orbit had a smaller size and was more eccentric, a scenario that should not be deemed as unrealistic.Furthermore, if Lb and ĥ were misaligned, also the inclination and the node precessions would come into play, as per Equations ( 1)-( 2).
), G is the Newtonian constant of gravitation, µ b := G (M A + M B ) is the standard gravitational parameter of the binary, c is the speed of light in vacuum, a b , e b , ω b and Ω b are the semimajor axis, the eccentricity, the argument of pericentre and the longitude of the ascending node of the binary's relative orbit, respectively, while n b := µ b /a 3 b is the Keplerian mean motion related to the orbital period by P b = 2π/n b .Furthermore, lb := {cos Ω b , sin Ω b , 0} , (6) mb := {− cos I b sin Ω b , cos I b cos Ω b , sin I b } , (7) ĥb := {sin I b sin Ω b , − sin I b cos Ω b , cos I b }