Linear acceleration emission: 2 Power spectrum

The theory of linear acceleration emission is developed for a large amplitude electrostatic wave in which all particles become highly relativistic in much less than a wave period. An Airy integral approximation is shown to apply near the phases where the electric field passes through zero and the Lorentz factors of all particles have their maxima. The emissivity is derived for an individual particle and is integrated over frequency and solid angle to find the power radiated per particle. The result is different from that implied by the generalized Larmor formula which, we argue, is not valid in this case. We also discuss a mathematical inconsistency that arises when one evaluates the power spectrum by integrating the emissivity over solid angle. The correct power spectrum increases as the 4/3rd power of the frequency at low frequencies, and falls off exponentially above a characteristic frequency. We discuss application of linear acceleration emission to the emission of high frequency photons in an oscillating model for pulsars. We conclude that it cannot account for gamma-ray emission, but can play a role in secondary pair creation.Comment: 25 pages; Accepted for publication in Ap

An index theorem of Callias type for pseudodifferential operators

We prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which have the form D + i \Phi, where D is elliptic pseudodifferential with Hermitian symbols, and \Phi is a Hermitian bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index of such operators is completely determined by the symbolic data over the boundary. We use the scattering calculus of R. Melrose in order to prove our results using methods of topological K-theory, and we devote special attention to the case in which D is a family of Dirac operators, in which case our theorem specializes to give families versions of the previously known index formulas.Comment: 18 pages, 1 figure. Revisions incorporated as suggested by referee report. To appear in the Journal of K-theor

Generic model for magnetic explosions applied to solar flares

An accepted model for magnetospheric substorms is proposed as the basis for a generic model for magnetic explosions, and is applied to solar flares. The model involves widely separated energy-release and particle-acceleration regions, with energy transported Alfv\'enically between them. On a global scale, these regions are coupled by a large-scale current that is set up during the explosion by redirection of pre-existing current associated with the stored magnetic energy. The explosion-related current is driven by an electromotive force (EMF) due to the changing magnetic flux enclosed by this current. The current path and the EMF are identified for an idealized quadrupolar model for a flare

Conservation of both current and helicity in a quadrupolar model for solar flares

A model for a solar flare, involving magnetic reconnection transferring flux and current between current-carrying magnetic loops connecting two pairs of footpoints, is generalized to include conservation of magnetic helicity during reconnection, as well as conservation of current at all four footpoints. For a set of force-free loops, with the $i$th loop having flux $F_i$ and current $I_i$, the self and mutual helicities are proportional to the self and mutual inductances with the constant of proportionality determined by $\alpha_i=F_i/\mu_0I_i$. In a constant-$\alpha$ model, the change in magnetic energy is proportional to the change in helicity, and conservation of helicity implies conservation of magnetic energy, so that a flare cannot occur. In a quadrupolar model, with $\alpha_1>\alpha_2$ initially, $\alpha_1$ increases and $\alpha_2$ decreases when flux and current are transferred from loops~1 and~2 to loops~3 and~4. A model that conserves both current and helicity is constructed; it depends on the initial $\alpha$s, and otherwise is somewhat simpler than when helicity is neglected.Comment: 14 pages, 1 figure. Solar Physics (in press

Faraday rotation: effect of magnetic field reversals

The standard formula for the rotation measure, RM, which determines the position angle, $\psi={\rm RM}\lambda^2$, due to Faraday rotation, includes contributions only from the portions of the ray path where the natural modes of the plasma are circularly polarized. In small regions of the ray path where the projection of the magnetic field on the ray path reverses sign (called QT regions) the modes are nearly linearly polarized. The neglect of QT regions in estimating RM is not well justified at frequencies below a transition frequency where mode coupling changes from strong to weak. By integrating the polarization transfer equation across a QT region in the latter limit, I estimate the additional contribution $\Delta\psi$ needed to correct this omission. In contrast with a result proposed by \cite{BB10}, $\Delta\psi$ is small and probably unobservable. I identify a new source of circular polarization, due to mode coupling in an asymmetric QT region. I also identify a new circular-polarization-dependent correction to the dispersion measure at low frequencies.Comment: 25 pages 1 figure, accepted for publication in The Astrophysical Journa

Magnetic explosions: role of the inductive electric field

Inclusion of the inductive electric field, ${\bf E}_{\rm ind}$ due to the temporally changing ${\bf B}$, in magnetic explosions is discussed, with emphasis on solar flares. Several roles played by ${\bf E}_{\rm ind}$ are identified: on a global scale, ${\bf E}_{\rm ind}$ produces the EMF that drives the explosion; the associated ${\bf E}_{\rm ind}\times{\bf B}$ drift is identified with the inflow of magnetic field lines into a reconnection region; the polarization current, associated with $\partial{\bf E}_{\rm ind}/\partial t$, implies a ${\bf J}\times{\bf B}$ force that accelerates this inflow; and the component of ${\bf E}_{\rm ind}$ parallel to ${\bf B}$ accelerates the energetic electrons that cause hard X-ray emission and type III radio bursts. Some simple models that describe these effects are presented. A resolution of the long-standing "number problem" in solar flares is suggested