Pulsar magnetospheres: Variation principle, singularities, and estimate of power

We formulate a variation principle for the force-free magnetosphere of an inclined pulsar: ε+ Ω · M (where ε and M are electromagnetic energy and angular momentum and Ω is the angular velocity of the star) is stationary under isotopological variations of the magnetic field and arbitrary variations of the electric field. The variation principle gives the reason for the existence and proves the local stability of singular current layers along magnetic separatrices. Magnetic field lines of inclined pulsar magnetospheres 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 the inner separatrix terminates and the outer separatrix emanates. The electromagnetic field should have an inverse square root singularity near the singular line inside the inner magnetic separatrix. The large-distance asymptotic solution is calculated and used to estimate the pulsar power, L ≈ c^-3μ²Ω⁴ for spin-dipole inclinations ≲30°.