[Paper Review] Ohmic Power of Ideal Pulsars

Andrei Gruzinov|arXiv (Cornell University)|Jan 31, 2011

Pulsars and Gravitational Waves Research3 citations

TL;DR

This paper proposes a new boundary condition at the equatorial current layer in ideal axisymmetric pulsar magnetospheres—requiring $B^2 - E^2 = 0$—which leads to finite Ohmic power dissipation. Using this condition in the force-free magnetosphere equation, the study finds that approximately 50% of the Poynting flux is dissipated via Ohmic heating outside the light cylinder, with 10% occurring between 1 and 1.5 light cylinder radii, implying significant energy release into radiation, pair production, and particle acceleration.

ABSTRACT

Ideal axisymmetric pulsar magnetosphere is calculated from the standard stationary force-free equation but with a new boundary condition at the equator. The new solution predicts Ohmic heating. About 50% of the Poynting power is dissipated in the equatorial current layer outside the light cylinder, with about 10% dissipated between 1 and 1.5 light cylinder radii. The Ohmic heat presumably goes into radiation, pair production, and acceleration of charges -- in an unknown proportion.

Motivation & Objective

• To resolve the long-standing issue of energy dissipation in ideal, force-free pulsar magnetospheres, which are expected to be dissipationless but show unexplained power loss.
• To identify a physically motivated boundary condition at the equatorial current layer that enables finite Ohmic power dissipation in the ideal force-free limit.
• To quantify the spatial distribution and magnitude of Ohmic heating in the pulsar magnetosphere, particularly near and beyond the light cylinder.
• To explore the phenomenological consequences of this Ohmic dissipation for pulsar radiation, pair production, and particle acceleration.

Proposed method

• Derives the stationary, axisymmetric force-free magnetosphere equation using cylindrical coordinates and scalar potentials $\psi$, $\phi$, and $A(\psi)$.
• Imposes a new Lorentz-invariant boundary condition at the equatorial current layer: $B^2 - E^2 = 0$, which ensures the field becomes electric-like to drive large currents.
• Adapts the CKF relaxation method to solve the Grad-Shafranov-like equation for $\psi$, with boundary conditions at the star surface, infinity, and the light cylinder.
• Uses a trial function $g(r)$ to approximate the equatorial $\psi(r,0)$ profile, adjusting it to minimize the field invariant $I(r) = B^2 - E^2$ to zero for $r > 1$.
• Employs numerical feedback to enforce the condition $I(r) \approx 0$ across $r > 1$, with $g(r) = 0.52 + 0.48/r$ achieving near-perfect nulling.
• Calculates Poynting flux $L$ and Ohmic power $L_{\rm Ohm}$ by integrating over field lines crossing the equatorial current layer.

Experimental results

Research questions

• RQ1 Can finite Ohmic power dissipation occur in an ideal, force-free pulsar magnetosphere without introducing physical viscosity or resistivity?
• RQ2 What boundary condition at the equatorial current layer outside the light cylinder leads to consistent, finite Ohmic heating in the force-free limit?
• RQ3 How much of the total Poynting flux is dissipated via Ohmic processes, and where is this dissipation localized in space?
• RQ4 What are the implications of this Ohmic dissipation for pulsar phenomenology, including radiation, pair production, and particle acceleration?

Key findings

• Approximately 50% of the total spin-down power (Poynting flux) is dissipated via Ohmic heating in the equatorial current layer outside the light cylinder.
• About 10% of the spin-down power is dissipated between 1 and 1.5 light cylinder radii, indicating significant energy deposition in a narrow radial zone.
• The field invariant $I = B^2 - E^2$ is nullified to numerical accuracy ($\sim 10^{-5}$) across $r > 1$ using the boundary condition $\psi(r,0) = \psi_0(0.52 + 0.48/r)$.
• The proposed boundary condition $B^2 - E^2 = 0$ at the equator is Lorentz-invariant and physically meaningful, as it ensures the field becomes electric-like to sustain large currents.
• The Ohmic power is not due to numerical artifacts but arises from a consistent solution of the force-free magnetosphere equation with the new boundary condition.
• The result implies that a large fraction of pulsar energy is converted into radiation, pair production, and particle acceleration near the light cylinder, with direct consequences for pulsar emission models.