Polarization evolution in strong magnetic fields

Extremely strong magnetic fields change the vacuum index of refraction. Although this polarization-dependent effect is small for typical neutron stars, it is large enough to decouple the polarization states of photons travelling within the field. The photon states evolve adiabatically and follow the changing magnetic field direction. The combination of a rotating magnetosphere and a frequency-dependent-state decoupling predicts polarization phase lags between different wavebands, if the emission process takes place well within the light cylinder. This QED effect may allow observations to distinguish between different pulsar-emission mechanisms and to reconstruct the structure of the magnetosphere. Key words: magnetic fields ± polarization ± stars: neutron. 1 I N T R O D U C T I O N Understanding the structure of the magnetic fields surrounding neutron stars may provide a key in developing models for the radio and X-ray emission from pulsars, pulsar spin-down, softgamma repeaters and the generation of the magnetic fields themselves. Although the magnetic field is instrumental in models of many phenomena associated with neutron stars, measuring its structure over a range of radii is problematic. Observations of the thermal emission from the surface may constrain the magnetic field geometry near the star, and the slowing of the rotation of the pulsar may yield an estimate of the strength of the field near the speed-of-light cylinder. Connecting these regimes is difficult. The intense magnetic fields associated with neutron stars influence many physical processes ± cooling (Shibanov et al. 1995; Heyl & Hernquist 1997c), atmospheric emission (Pavlov et al. 1994; Rajagopal, Romani & Miller 1997) and the insulation of the core (Hernquist 1985; Schaaf 1990; Heyl & Hernquist 1998a,b). Even stronger fields such as thought to be found near anomalous X-ray pulsars (AXPs) and soft-gamma repeaters (SGRs) alter the propagation of light through the magnetosphere by way of quantum-electrodynamic (QED) processes and may further process the emergent radiation (Adler 1971; Baring 1995; Baring & Harding 1995; Heyl & Hernquist 1997a,b), and distort our view of the neutron-star surface (Shaviv, Heyl & Lithwick 1999). Even for weaker fields, QED renders the vacuum anisotropic. The speed at which light travels through the vacuum depends on its direction, polarization and the local strength of the magnetic field. Although for neutron stars with B , 10 G this effect is too weak to grossly affect images and light curves of these objects, it is strong enough to decouple the propagating modes through the pulsar magnetosphere. To lowest order in the ratio of the photon energy to the electron rest-mass energy, the index of refraction of a photon in a magnetic field is independent of frequency. Near the pair-production threshold, the photon propagation adiabatically merges with a postironium state (Shabad & Usov 1986). However, well above the pair-production threshold for weak fields …mec2 ! E ! mec 4:4 10 G=B† the low-energy results again provide a good approximation (Tsai & Erber 1975). In the context of the field surrounding a neutron star, only photons with sufficiently high wavenumbers (the optical and blueward) will travel through a portion of the rapidly weakening magnetic field without their two polarization modes mixing. At radio frequencies, the plasma surrounding the neutron star produces a similar effect (e.g. Cheng & Ruderman 1979; Barnard 1986). One would expect that radio emission will be initially polarized according to the direction of the local magnetic field. When one observes a pulsar at a particular instant, one sees emission from regions with various magnetic field directions; therefore, one would expect the polarization to cancel out substantially. However, one finds that pulsars exhibit a significant linear polarization. As the polarized radiation travels from its source, its polarization direction changes as the direction of the local magnetic field changes. At a distance from the star that is large when compared with its radius, the local magnetic field in the plane of the sky is parallel across the observed portion of the star. Because the field changes gradually, the polarization modes are decoupled, and the disparate linear polarizations can add coherently. Previous authors have focused on the propagation of polarized radio waves through the magnetosphere. At these frequencies, the vacuum polarization is safely neglected. Furthermore, they have assumed that the coupling of the two polarization modes occurs instantaneously. In this paper, we will treat the problem of vacuum polarization in particular and how the gradual coupling of the polarization modes affects the final polarization of the emergent radiation. At sufficiently high frequency the plasma only q 2000 RAS w E-mail: [email protected] (JSH); [email protected] (NJS) 556 J. S. Heyl and N. J. Shaviv negligibly affects the radiation as it travels through the magnetosphere. The precise frequency at which the vacuum birefringence begins to dominate depends on the charge density of the magnetospheric plamsa. If we assume the Goldreich±Julian value (Goldreich & Julian 1969), we find that for Ephoton . 0:035 eV B 1012 G P