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RADIO ASTRONOMY

Radio emission from a pulsar’smagnetic pole revealed bygeneral relativityGregory Desvignes1,2*, Michael Kramer1,3, Kejia Lee4, Joeri van Leeuwen5,6,Ingrid Stairs7, Axel Jessner1, Ismaël Cognard8,9, Laura Kasian7,Andrew Lyne3, Ben W. Stappers3

Binary pulsars are affected by general relativity (GR), causing the spin axis of each pulsarto precess.We present polarimetric radio observations of the pulsar PSR J1906+0746 thatdemonstrate the validity of the geometrical model of pulsar polarization. We reconstructthe (sky-projected) polarization emission map over the pulsar’s magnetic pole and predictthe disappearance of the detectable emission by 2028. Two tests of GR are performedusing this system, including the spin precession for strongly self-gravitating bodies. Weconstrain the relativistic treatment of the pulsar polarization model and measure thepulsar beaming fraction, with implications for the population of neutron stars and theexpected rate of neutron star mergers.

Pulsars are fast-spinning neutron starsmea-suring ~1.2 to 2.2 solar masses (M⊙) withstrong magnetic fields that emit a beamof radio waves along their magnetic axesabove each of their oppositemagnetic poles.

According to Einstein’s theory of general relativi-ty (GR), space-time is curved by massive bodies.Predicted effects of this theory include relativisticspin precession in binary pulsars (1). This preces-sion arises from any misalignment, by an angled, of the spin vector of each pulsar with respectto the total angular momentum vector of thebinary, most likely caused by an asymmetricsupernova (SN) explosion imparting a kick ontothe neutron star(s) (2, 3). This precession causesthe viewing geometry to vary, which can be testedobservationally.

As a pulsar rotates, its radio beams sweep thesky. If one of the beams crosses our line of sight(LOS), its emission is perceived as being pulsed.When averaged over several hundreds of pulsarrotations, the pulses typically form a stable pulseprofile. Evidence for a variable pulse profile attrib-uted to changes in the viewing geometry causedby spinprecessionhavebeenobserved andmodeledfor the binary pulsar PSR B1913+16 (4, 5). Polar-ization information can provide an additional,independent tool to study relativistic spin preces-sion (6, 7). We expect that the position angle (PA)

sweep of a pulsar’s linearly polarized emissionresulting from geometrical effects can be de-scribed by the rotating vector model (RVM) (8).This simple model, which assumes a magneticdipole centered on the pulsar, relates the PA tothe projection of the magnetic field line direc-tion as the pulsar beam rotates and crosses ourLOS. The resulting gradient of the PA sweep as afunction of the pulsar rotational phase (9) dependsonly on the magnetic inclination angle a and theimpact parameter b (i.e., the angle of closest ap-proach between the observer direction and themagnetic axis). For LOSs crossing opposite sidesof the same magnetic pole, the RVM predictsopposite slopes of the PA swing, which becomessteeper for smaller b (10). RVMhas been extendedto include rotational and relativistic effects be-tween the pulsar and observer frame (11, 12), inprinciple allowing emission heights for the ob-served radio emission to be estimated. Althoughthe RVM matches observations of young pulsarsthat present a smooth PA swing (13), deviationsare also observed, so for large emission heights,the pure dipole approximation may not be validfor a rotating plasma-loaded magnetosphere (14).Evidence for the central assumption of all thesemodels, i.e., the geometrical meaning of the PAsweep, has so far been missing.PSR J1906+0746 (right ascension 19h06m48.86s,

declination +07°46′25.9′′, J2000 equinox) is ayoung pulsar with spin period Ps ~ 144 ms in a4-hour orbit around another neutron star. Whenit was discovered in 2004 (15), PSR J1906+0746showed two polarized emission components sep-arated by nearly half a period (or ~180° of pulselongitude). The main pulse (MP) and interpulse(IP) indicated anearly orthogonal geometrywhereemission from both magnetic poles is visible fromEarth. Comparison with archival data from theParkes Multibeam Pulsar Survey (PMPS) (16)revealed that only the stronger MP had been

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1Max-Planck-Institut für Radioastronomie, Auf dem Hügel,69 D-53121 Bonn, Germany. 2Laboratoire d’ÉtudesSpatiales et d’Instrumentation en Astrophysique,Observatoire de Paris, Université Paris-Sciences-et-Lettres, Centre National de la Recherche Scientifique,Sorbonne Université, Université de Paris, 5 place JulesJanssen, 92195 Meudon, France. 3Jodrell Bank Centre forAstrophysics, School of Physics and Astronomy, The Universityof Manchester, Manchester M13 9PL, UK. 4Kavli Institutefor Astronomy and Astrophysics, Peking University, Beijing100871, People’s Republic of China. 5ASTRON, TheNetherlands Institute for Radio Astronomy, Postbus 2,7990 AA Dwingeloo, Netherlands. 6Astronomical InstituteAnton Pannekoek, University of Amsterdam, Science Park904, 1098 XH Amsterdam, Netherlands. 7Department ofPhysics and Astronomy, University of British Columbia,Vancouver, BC V6T 1Z1, Canada. 8Laboratoire de Physiqueet Chimie de l’Environnement et de l’Espace, CentreNational de la Recherche Scientifique–Universitéd’Orléans, F-45071 Orléans, France. 9Station deradioastronomie de Nançay, Observatoire de Paris, CentreNational de la Recherche Scientifique, Institut national dessciences de l’Univers, F-18330 Nançay, France.*Corresponding author. Email: gdesvignes@mpifr-bonn.mpg.de

Fig. 1. Geometry ofPSR J1906+0746.Thepulsar rotates with aspin period Ps = 144 msaround its spin vector Sshown with the verticalred arrow. S precesseswith a period of Pp =360°/Wp ~160 yraround the total angu-lar momentum vector(misaligned by theangle d from S) thatcan be approximatedby the orbital momen-tum vector X, perpen-dicular to the Y–Zorbital plane. Theorbital inclination angleis i. As the pulsar spins, its magnetic pole corresponding to the MPemission, BMP, and inclined with anangle aMP sweeps the sky along the dashed blue trajectory.The MP beam, with the extent pictured bythe dotted blue circle, crosses our LOS represented by theKvector if jbMPj < 22°.This allows us to observea cut, shown with the red curve, through the MP beam. After half a rotation of the pulsar, our LOS canalso potentially cut through the IP beam if jbIPj < 22°. For clarity, only the MP beam is shown.

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visible in 1998 (15), suggesting effects of rela-tivistic spin precession. Timing analysis of thispulsar (17) has measured three relativistic cor-rections to the orbit, the post-Keplerian (PK)parameters. Two of the PK parameters, the peri-astron advance w

�

and the time dilation g, allowthe determination of the pulsar and companionmasses,mp = 1.29 ± 0.01 M⊙ andmc = 1.32 ± 0.01

M⊙ , respectively, assuming that GR is correct.The third PK parameter, the orbital decay dueto gravitational wave emission P

�

b, is consistentwith these measurements, providing a test ofGR with 5% precision (17). GR also provides anestimate of the orbital inclination angle i= 43.7° ±0.4° or 136.3° ± 0.4° (due to a 180° degeneracy)and the spin precession rateWp = 2.234° ± 0.014°

per year. Independent measurements of thesetwo terms are potentially observable in precess-ing systems, allowing these GR-predicted valuesto be tested. Observations of the double pulsar(18) and PSR B1534+12 (7, 19) match the GR pre-dictions of spin precession with precisions of 13and 20%, respectively.We monitored PSR J1906+0746 from 2012 to

2018 with the 305-m William E. Gordon Areciboradio telescope using the Puerto Rico UltimatePulsar Processing Instrument (PUPPI) tuned at acentral frequency of 1.38GHz.We supplement thoseobservations with archival data from the Nançayand Arecibo telescopes recorded between 2005and 2009 (17, 20). In total, our dataset comprises47 epochs spanning from July 2005 to June 2018.A previously observed trend of changing sepa-

ration betweenMP and IP profiles with time (17)has continued. Our data show that the slope ofthe PA under the MP gradually flattens withtime, whereas the MP becomes simultaneouslyweaker and subsequently undetectable towardthe end of 2016. This indicates that our LOS hasmoved out of the MP emission beam.Theprogressive steepening of thePA curveunder

the IP eventually leads to a flip of the PA aroundMay 2014 (although it appears flat because of the180° ambiguity in the PA), followed by a flattening.This behavior is in agreement with the RVM for apole crossing our LOS. The observed change in thePA curve provides evidence that the PA swing hasa geometrical origin linked to the magnetic field.The presence of MP and IP emission, covering

a wide range of pulse longitudes, allows a precisedetermination of the viewing geometry as a

Desvignes et al., Science 365, 1013–1017 (2019) 6 September 2019 2 of 5

Fig. 2. Mass-massdiagram.The blacklines delimit the 1� scontours from themeasurements of theorbital period decay P

�

b,the periastronadvance w

� , and thetime dilation g (17).The red dotted anddashed lines delimitour two additionalconstraints, the mea-surements of spin-precession rate Wp andinclination angle i,respectively. The grayparameter space isexcluded by the massfunction becausesin i ≤ 1. All parametersare consistent withmp ¼ 1:29 T 0:01 M⊙

and mc¼ 1:32 T

0:01 M⊙.

Fig. 3. Beam mapsof the radio emis-sion. (A to E) showthe PA measurements(in black) and the fit bythe precessional RVM(the red curve)corresponding to dif-ferent LOSs throughthe IP. (F) and(G) show the two-dimensional contoursof the reconstructedStokes I emissionmaps (on a logarith-mic color scale), asprojected on the sky,for the MP and the IP,respectively. Thereference frame ofthese maps is fixed onthe vertical spin vectorS at longitude zero and180° pointing towardnegative latitude.Thered crosses at (0,0)and (180,0) indicatethe pulsar's magneticpole on the MP and IP maps, respectively, and the dashed circles show increments of 2° in the beam map.The dotted lines represent the LOSs at a given year,indicated on the left side, and, for the IP, refer to the PA measurements (A to E).The hatched areas correspond to the parts of the maps that were not observed.

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function of time. We performed a simultaneousmodeling of the polarimetric profiles using theRVM including the effects of relativistic spin pre-cession (21), hereafter referred to as the preces-sional RVM. A total of 53 parameters are includedin this model (see table S1); the main parametersare the angle between the rotation axis and themagnetic axis of the MP, aMP; the misalignmentangle, d; the inclination angle, i; and the pre-cession rate,Wp. The phase of the inflection pointof the RVM under the MP, f0MP;k

, for each epochkwas also included in the analysis as a free para-meter. The geometry of the system is illustratedin Fig. 1. We used the Bayesian sampling toolPolyChord (22) to explore the parameter space ofthemodel (20). The 1s uncertainty intervals werederived from the one-dimensional marginalizedposterior distributions (shown in fig. S3), givingthe 68% confidence level on each parameter.We measure aMP = 99°.41 ± 0°.17 (and there-

fore aIP = 180° − aMP = 80°.62 ± 0°.17), which

combined with a separation between the MPand IP close to 180° confirms that PSR J1906+0746 is an orthogonal rotator and that theMPand IP emissions originate from opposite mag-netic poles. The large d =104° ± 9° is consistentwith the pulsar having formed in an asymmetricSN explosion, producing a tilt of the spin axis ofthe younger pulsar in the binary pair (17) relativeto the pre-SN orbit (3). As the spin vector con-tribution to the total angular momentum vectoris negligible, we also interpret d as the angle be-tween the orbital angular momentum vector andthe pulsar spin vector.With d ~ 104°, this suggeststhat the pulsar spin axis lies close to the orbitalplane of this binary system.This analysis independently determines Wp =

2.17° ± 0.11° per year and i = 45° ± 3° in additionto the three previously measured PK parame-ters (17). This allows us to perform two additionalself-consistent tests of GR in this system (seeFig. 2). Both values agree with the GR predic-

tions within 1s. The constraint onWp tests GR ata 5% uncertainty level, which is tighter than theprecession ratemeasurement in the double pulsarsystem (18). The parametrization of our modelallows us to determine i without ambiguity, incontrast to information obtained solely frompulsar timing where only sin i can be derived.From our fitted model, we can describe the

evolution of the impact parameters for both theMP and IP, bMP(t) and bIP(t) (see fig. S4) andthe latitudinal extent of the pulsar beam (i.e., thelargest value of jbj for which emission is ob-served). Our model predicts that in 1998, bMP ~5° and bIP > 22°, coinciding with the strongdetection of the MP and the nondetection ofthe IP in the PMPS data (15). The appearance ofthe IP between 1998 and 2004 corresponds tobIP(2001) ~ +20°. By 2018, our LOS to the IP hadtraversed to bIP(2018) ~ −6°, so we infer a latitu-dinal extent for the IP of ~20°. For theMP, we areonly able tomap the southern part of the emission

Desvignes et al., Science 365, 1013–1017 (2019) 6 September 2019 3 of 5

A B

C D

Fig. 4. Polarization beam maps. Same as Fig. 3, F and G, but showing fractional linear polarization for the MP (A) and IP (B) and fractional circularpolarization for the MP (C) and IP (D).

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beam,�5° < bMP ≲�22°, after which the emis-sion is no longer detected. The appearance ofthe IP and the disappearance of the MP at thesame jbj suggests that the MP and IP share thesame latitudinal extent of ~22°. We thereforepredict that the IP will disappear from our LOSaround 2028 and will reappear between 2070and 2090 (fig. S4). The MP should reappear be-tween 2085 and 2105.With our determination of the geometry of the

system, we can use the observed pulse profilesrecorded at different epochs, including the re-processed PMPS data (20), to reconstruct (20)the sky-projected beammaps of the radio emis-sion from both magnetic poles (Fig. 3) and itspolarization properties (Fig. 4). Previous attemptsat mapping the pulsar emission have been pub-lished for other pulsars [e.g., (23)]. Such plotsmust not be viewed as strict maps of active fieldlines anchored on the polar cap but rather asprojections of the emission on the sky.The map shows emission from both sides of

the IP magnetic pole, ruling out models predict-ing that radio emission is restricted to one sideof the pole [e.g., (24)]. The IP emission patternis not symmetric in the latitudinal direction orwith the MP. The IP weakens when our LOScrosses the magnetic pole. This finding matchestheoretical predictions of the current density inthe polar cap for an orthogonal rotator, albeit atlow radio emissionheight (25, 26). Radio emissionis produced at a height above the pulsar whereany higher-order magnetic field componentsshould have diminished. In a standard scenariowhere emission is produced in the entire openfield-line region, wewould expect the radio beamto be circular. Yet the observed beam is elongatedand smaller than expected in the longitudinaldirection (Fig. 3).The passage of the LOS over the IP magnetic

pole coincides with a drop in fractional linearpolarization. At the same time, the circular pola-rization (Stokes V) appears to change sign whencrossing the magnetic pole (Fig. 4 and fig. S5)(20). The fractional linear polarization of the MPdecreases as our LOS moves away from the cen-ter of the MP beam, whereas some unpolarizedpulse components are shown to appear and dis-appear on time scales of months (20).If we assume that the emission region is sym-

metric around the magnetic meridian and thatthe radiation is beamed in the forward direc-tion of the relativistic charge flow along the mag-netic field lines, then the rotation of the pulsarmagnetosphere in the observer frame causes thepulse profile to precede the magnetic meridianand the PA to lag behind it.Models predict that the total phase shift DfS

between the midpoint of the pulse profile andthe inflection point of the PA swing is given byDfS ≈ 4hem=RLC , where hem is the emissionheight and the light cylinder radius RLC = cPs/2p,with c being the speed of light (11, 12). This rela-tionship is expected to be valid for small emissionheights, hem ≲ 0:1RLC . But these models predictdifferent contributions to the aforementionedshifts in the pulse profile and the PA curve. There

is also a shift in absolute value of the PA, but thiscan be neglected here as a ~ 90° (27).Measuring the phase shifts DfSMP;k

for the MPat a given epoch k [and its impact parameterb(k)] gives an estimate of the MP emissionheights hemMP;k , assuming all shifts are attribut-able to rotational effects. The emission heightsfor the IP hemIP;k can be derived in the same way,also using the RVM inflection points for the IPgiven by f0IP;k ¼ f0MP;k

þ 180°. These results showthat the emission heights range from close to thesurface of the pulsar, for low impact parameter b(when our LOS is atop the magnetic pole), to~320 km for large b, matching the theoreticalprediction (28) (Fig. 5), although this theoreticalprediction only depends on a and b. Deviationsare observed for LOSs close to the beam edge ofthe MP, and large b for the IP, where we mayobserve a different active patch of the IP beam assuggested by the polarization properties (e.g., thesign of Stokes V under the IP has flipped be-tween 2009 and 2012) (see Fig. 4 and fig. S5).The latitudinal extent of a pulsar beam deter-

mines the pulsar beaming fraction (i.e., the por-tion of sky illuminated by the pulsar beam). Thisparameter affects the predicted number of Galac-tic double neutron stars (DNSs) and, hence, theexpected gravitational wave detection rate forneutron star mergers (29). Usually, the latitudi-nal extent of the beam cannot be independentlydetermined and so the expected beam radius rfor a conal emission beam is used. This beamshape is typically expected, although its internal

structure and/or filling factor is still a matter ofdebate (10). On the basis of beam cuts given bypulsar profiles, various studies [e.g., (30)] havesuggestedr ∼ 6:5°Ps

�0:5, where the dependenceonPs is consistentwith the expectation for dipolarfield lines.Previous studies that estimated the DNS pop-

ulation number andmerger rate (29) from a sam-ple of pulsars dominated by only two pulsars(PSR J0737−3039A and PSR J1906+0746) haveassumed for PSR J1906+0746 a uniform radioluminosity across a conal beam of r ~ 17°. Thisvalue is smaller than our observed latitudinalextent of 22° that, combined with aMP = 99°.4,gives a large beaming fraction of 0.52 but withsubstantial luminosity variations. The simple conalbeam model with uniform luminosity cannot beused to reliably constrain the DNS merger ratefor PSR J1906+0746.

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Fig. 5. Radio emission heights. Phase shift DfS of the PA from Stokes I (left axis) with the derivedemission height (right axis) for the MP (blue data points) and IP (red data points) as a functionof b. The blue and red dashed curves represent the theoretical emission height as a function of b.

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ACKNOWLEDGMENTS

We thank N. Wex and S. Johnston for useful discussions, A. Ridolfi forsharing the flux calibration observations for PUPPI and the PALFAcollaboration for recording some of the early Arecibo observations. Weacknowledge the use of the Hercules cluster hosted at the Max PlanckComputing and Data Facility in Garching. This research has madeextensive use of NASA’s Astrophysics Data System. Funding: G.D. andM.K. gratefully acknowledge support from European Research Council(ERC) Synergy Grant “BlackHoleCam” Grant Agreement Number610058. K.L. received support from 973 program 2015CB857101,NSFC U15311243, XDB23010200, 11690024, and funding fromMax-Planck Partner Group. J.v.L. acknowledges funding from theNetherlands Organisation for Scientific Research (NWO) under project“CleanMachine” (614.001.301) and from the ERC under the EuropeanUnion’s Seventh Framework Programme (FP/2007-2013)/ERCGrant Agreement 617199. Pulsar research at UBC is supported by anNSERC Discovery Grant and by the Canadian Institute for Advanced

Research. The Arecibo Observatory is operated by SRI Internationalunder a cooperative agreement with the NSF (AST-1100968), and inalliance with Ana G. Méndez-Universidad Metropolitana, and theUniversities Space Research Association. The Nançay RadioObservatory is operated by the Paris Observatory, associated with theFrench Centre National de la Recherche Scientifique (CNRS) andthe Université d’Orléans. Author contributions: G.D. and M.K. led theanalysis and the writing of the manuscript. K.L. and A.J. providedtheoretical input and analysis. J.v.L., I.S., and L.K. performed theArecibo observations. I.C. performed the Nançay observations. Allauthors commented on the manuscript. Competing interests: Theauthors declare no competing interests. Data and materialsavailability: The Parkes data are available from (31), filePM0055_01341.sf . The Nançay and Arecibo datasets are availablefrom Zenodo (32). The MODELRVM software is also available fromZenodo (33). The output parameters of our modeling are listed intables S1 and S2.

SUPPLEMENTARY MATERIALS

science.sciencemag.org/content/365/6457/1013/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S6Tables S1 and S2References (34–46)

23 October 2018; accepted 8 August 201910.1126/science.aav7272

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Radio emission from a pulsar's magnetic pole revealed by general relativity

Kasian, Andrew Lyne and Ben W. StappersGregory Desvignes, Michael Kramer, Kejia Lee, Joeri van Leeuwen, Ingrid Stairs, Axel Jessner, Ismaël Cognard, Laura

DOI: 10.1126/science.aav7272 (6457), 1013-1017.365Science 

, this issue p. 1013Scienceangle, the angular region in which a radio observer can detect a pulsar.out of our line of sight and disappeared. Mapping the radio emission across the magnetic pole determines the beaming companion. In 2005, two pulses per rotation were visible, one from each magnetic pole, but by 2018 one had precessedradio pulses vary. General relativity causes precession of the rotation axis, because of the influence of a binary

monitored a pulsar for more than a decade, observing how itset al.pulses if the beam points toward Earth. Desvignes Pulsars are rotating neutron stars that emit beams of radio waves along their magnetic poles, seen as regular

General relativity reveals pulsar beams

ARTICLE TOOLS http://science.sciencemag.org/content/365/6457/1013

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2019/09/04/365.6457.1013.DC1

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