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Multipolar electromagnetic ﬁelds around neutron stars: exact vacuum solutions and related properties

The magnetic ﬁeld topology in the surrounding of neutron stars is one of the key questions in pulsar magnetospheric physics. A very extensive literature exists about the assumption of a dipolar magnetic ﬁeld but very little progress has been made in attempts to include multipolar components in a self-consistent way. In this paper, we study the effect of multipolar electromagnetic ﬁelds anchored in the star. We give exact analytical solutions in closed form for any order l and apply them to the retarded point quadrupole ( l = 2), hexapole ( l = 3) and octopole ( l = 4), a generalization of the retarded point dipole ( l = 1). We also compare the Poynting ﬂux from each multipole and show that the spin-down luminosity depends on the ratio R / r L , R being the neutron star radius and r L the light-cylinder radius. Therefore the braking index also depends on R / r L . As such multipole ﬁelds possess very different topology, most importantly smaller length scales compared to the dipolar ﬁeld, especially close to the neutron star, we investigate the deformation of the polar caps induced by these multipolar ﬁelds. Such ﬁelds could have a strong impact on the interpretation of the pulsed radio emission suspected to emanate from these polar caps as well as on the inferred geometry deduced from the high-energy light-curve ﬁtting and on the magnetic ﬁeld strength. Discrepancies between the two-pole caustic model and our new multipole caustic model are emphasized with the quadrupole ﬁeld. To this respect, we demonstrate that working with only a dipole ﬁeld can be very misleading. with the electric and magnetic ﬁeld. The general framework in general relativity has been given in P´etri (2013). Here we specialize the discussion to Minkowski space–time (P´etri 2012), avoiding frame-dragging effects which would generate higher multipoles from lower ones. All the details about the derivation of the partial differential equations recalled below and the associated expansion method are taken from chapter 9 of Jackson (2001) dealing with radiating systems and multipolar ﬁelds. However, our deﬁnitions do not strictly follow the usual conventions taken for quantum mechanics. We give exact analytical solutions for any multipole moment, specifying boundary conditions on the star being treated as a perfect conductor.