Pulsar Magnetosphere: Variation Priciple, Singularities, Estimate of Power

Abstract

We formulate variation principle for force-free magnetosphere of an inclined pulsar: ${\cal E} +{\bf \Omega}\cdot {\bf M}$ ($\cal E$, ${\bf M}$ are electromagnetic energy and angular momentum, ${\bf \Omega}$ is the angular velocity of a star) is stationary under isotopological variations of magnetic field and arbitrary variations of electric field. The variation principle gives the reason for existence and proves local stability of current singular layers along magnetic separatrices. Magnetic field lines of inclined pulsar magnetosphere lie on magnetic surfaces, and do have magnetic separatrices. In the framework of the isotopological variation principle, inclined magnetospheres are expected to be simple deformations of the axisymmetric pulsar magnetosphere. A singular line should exist on the light cylinder, where inner separatrix terminates and outer separatrix emanates. The electromagnetic field should have an inverse square root singularity near the singular line inside the inner magnetic separatrix. Large distance asymptotic solution is calculated, and used to estimate the pulsar power, $L\approx c^{-3}\mu ^2\Omega^4$ for spin-dipole inclinations $\lesssim 30^{\circ}$ .