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Pulsar Electrodynamics Current Density
by Rai Yuen Supervisor: Don Melrose
and


Contents
What is this talk about? First briefly outline two important models for pulsar electrodynamics: Vacuum and Magnetospheric models. Two models, two electric fields: inductive and potential. Screening of the induced E and produced co-rotation E by the screening current density. What is this talk NOT about? Observational development / data. Complete picture for pulsar electrodynamics. Interpretations of results.


Motivation
Seems nothing mention about this screening current density or the inductive electric field in the literature. Current Density: GJ out-flowing at c. Pairs created in magnetosphere. Global scale. These are not the screening current density we needed. Screening of E: Parallel E along B lines. The screening is usually discussed in the context of e+/e- pairs - gaps. But not the inductive electric field we concern here. Not Even in the classical works by Mestel (1995), Michel (1982) and his book (1991) ...


History of Pulsars
First widely accepted prediction of an ultra-dense star was given by Zwicky and Baade (1934). Proposed the idea that a supernova represents the transition of an ordinary star into a neutron star, consisting mainly of neutrons. Pulsar was discovered by Bell and Hewish in 1967 in the radio band. Gold [2] and Pacini (1968) proposed that pulsars = neutron stars. So rotating neutron stars. Typical pulsed periods range from ~1 ms to ~5s, with surface magnetic field ~1012 G.


Model of Pulsar
Rapidly rotating magnetised neutron stars. Dominated by electromagnetic fields, so can be modelled using the classical theory of electrodynamics.


Mathematical Dipole
To see the problem, start from simple model. Rotating Magnetic Dipole: Time-dependent magnetic moment. Gives E and B at x at retarded time tretarded = t ­ r/c.


The Vacuum Model
Given the rotating dipole, how to form a star?
Add a boundary (star surface). Inside contains well-conducting medium with the magnetic dipole at the centre. Outside is vacuum. Angular velocity, . Well-conducting medium implies co-rotation. By Ohm's law, E in the co-rotating frame vanishes, and E in the lab frame is,

We have a vacuum model.


The Magnetospheric Model
Proposed by Goldreich and Julian (1969). Concepts of light cylinder, open and closed magnetic field lines, co-rotation, etc . . . Assumptions: A well-conducting plasma filled magnetosphere Co-rotating with pulsar. Free to specify B, but assume as given previously. Force-free condition: Co-rotation E: GJ charge density:


Two Electric Fields
EI from the Vacuum model is inductive in nature, E from the magnetospheric model is potential and inductive.

The two electric fields are not compatible with each other. ER is required by other theoretical models. How to screen the induced EI, and at the same time, produced the co-rotation ER?


The Screening Current Density
Screening of potential E requires charge density, Screening of induced E requires current density. Consider Maxwell-Ampere's Law:

It is satisfied by the vacuum model: J = 0. E and B given in the model.


The Screening Current Desnity
In the Magnetospheric model: Co-rotation requires E be given by ­( x x) x B. Free to choose B. If vector field, B, in a co-rotating frame is time-independent, the evolution of B is given by (Melrose 1967)


The Screening Current Density
The screening current density, JS, in coordinate-free form is,

Approximate with dipole magnetic field, and consider lower-order terms in spherical coordinates,

Usual definition for r, , .


Parallel Flow of J
Consider flow of JS. Project JS along the dipole field: parallel and perpendicular components. From orbit theory, flow of charge particles is due to electromagnetic field. Parallel component: Due to the EI // B.

S


Perpendicular Flow of JS
Due to inertial drift: Curvature + polarization drifts

Curvature Drift, vc: From parallel flow. Particles travel along curved B induces curvature drift. Perpendicular to radius and B.


Perpendicular Flow of JS
Polarization Drift, vp: ER is dependent on B. B is dependent on t varying ER induce polarization drift. Particles move in direction of dER /dt, which is B.

Total perpendicular drift, vT = vc + v

p


Properties of J
For aligned case, // m or = 0, there is no variation in the electromagnetic field, JS = 0.

S

At the two poles:

At the equatorial plane:

Circulates around the field lines and forms closed circuit. Maximum magnitude at = 90o, but ceases its effects at = 0o.


How Significant is JS
JS has characteristics different from other currents: Co-rotation current, vGJ. Goldreich-Julian current density, JGJ = cGJ. Compare JS with JGJ JGJ sets a limit for magnetospheric processes, E.g, j JGJ.


How Significant is JS
Perpendicular rotator. P = 1s, = 85o, = 0 Consider the ratio:
o

|JS| is comparable to |JGJ| close to the light cylinder. Ignore JS in the past, but seems unjustified.


Conclusion and Future Work
Outlined the two electric fields from the Vacuum and Magnetospheric models. The two electric fields are not compatible with each other. Screening of the induced E requires a current density. JS becomes important close to LC. Still continuing: Properties JS at the light cylinder. Some other implications from JS.


References
Baade, W. & Zwicky, F.: "Cosmic Rays From SuperNovae", Proc. Natl. Acad. Sci. 20, 259, 1934 Gold, T.: Nature, 218, 731, 1968 Goldreich, P., Julian, W.H.: ApJ, 157, 869, 1969 Melrose, D.B.: Planet. Space Sci., 15, 381, 1967 Mestel, L. J.: Astrophys. Astr., 16, 119, 1995 Michel, F. C.: Rev. Mod. Phys., Vol. 54, No.1, 1982