44,904 research outputs found
The radical of a vertex operator algebra associated to a module
The radical of a vertex operator algebra associated to a module is defined
and computed.Comment: Latex 14 pages. This is part of the original paper with a new titl
Atmospheres and Spectra of Strongly Magnetized Neutron Stars II: Effect of Vacuum Polarization
We study the effect of vacuum polarization on the atmosphere structure and
radiation spectra of neutron stars with surface magnetic fields B=10^14-10^15
G, as appropriate for magnetars. Vacuum polarization modifies the dielectric
property of the medium and gives rise to a resonance feature in the opacity;
this feature is narrow and occurs at a photon energy that depends on the plasma
density. Vacuum polarization can also induce resonant conversion of photon
modes via a mechanism analogous to the MSW mechanism for neutrino oscillation.
We construct atmosphere models in radiative equilibrium with an effective
temperature of a few \times 10^6 K by solving the full radiative transfer
equations for both polarization modes in a fully ionized hydrogen plasma. We
discuss the subtleties in treating the vacuum polarization effects in the
atmosphere models and present approximate solutions to the radiative transfer
problem which bracket the true answer. We show from both analytic
considerations and numerical calculations that vacuum polarization produces a
broad depression in the X-ray flux at high energies (a few keV \la E \la a few
tens of keV) as compared to models without vacuum polarization; this arises
from the density dependence of the vacuum resonance feature and the large
density gradient present in the atmosphere. Thus the vacuum polarization effect
softens the high energy tail of the thermal spectrum, although the atmospheric
emission is still harder than the blackbody spectrum because of the non-grey
opacities. We also show that the depression of continuum flux strongly
suppresses the equivalent width of the ion cyclotron line and therefore makes
the line more difficult to observe.Comment: 21 pages, 21 figures; MNRAS; corrected minor typo
Atmospheres and Spectra of Strongly Magnetized Neutron Stars
We construct atmosphere models for strongly magnetized neutron stars with
surface fields G and effective temperatures K. The atmospheres directly determine the characteristics
of thermal emission from isolated neutron stars, including radio pulsars, soft
gamma-ray repeaters, and anomalous X-ray pulsars. In our models, the atmosphere
is composed of pure hydrogen or helium and is assumed to be fully ionized. The
radiative opacities include free-free absorption and scattering by both
electrons and ions computed for the two photon polarization modes in the
magnetized electron-ion plasma. Since the radiation emerges from deep layers in
the atmosphere with \rho\ga 10^2 g/cm, plasma effects can significantly
modify the photon opacities by changing the properties of the polarization
modes. In the case where the magnetic field and the surface normal are
parallel, we solve the full, angle-dependent, coupled radiative transfer
equations for both polarization modes. We also construct atmosphere models for
general field orientations based on the diffusion approximation of the
transport equations and compare the results with models based on full radiative
transport. In general, the emergent thermal radiation exhibits significant
deviation from blackbody, with harder spectra at high energies. The spectra
also show a broad feature (\Delta E/\Ebi\sim 1) around the ion cyclotron
resonance \Ebi=0.63 (Z/A)(B/10^{14}{G}) keV, where and are the atomic
charge and atomic mass of the ion, respectively; this feature is particularly
pronounced when \Ebi\ga 3k\Teff. Detection of the resonance feature would
provide a direct measurement of the surface magnetic fields on magnetars.Comment: 29 pages, 11 figures; corrected factor of 2 in He models: minor
changes to figs 4 and 9 as a result; other very minor change
Framed vertex operator algebras, codes and the moonshine module
For a simple vertex operator algebra whose Virasoro element is a sum of
commutative Virasoro elements of central charge 1/2, two codes are introduced
and studied. It is proved that such vertex operator algebras are rational. For
lattice vertex operator algebras and related ones, decompositions into direct
sums of irreducible modules for the product of the Virasoro algebras of central
charge 1/2 are explicitly described. As an application, the decomposition of
the moonshine vertex operator algebra is obtained for a distinguished system of
48 Virasoro algebras.Comment: Latex, 54 page
- …