Astron. Astrophys. 332, L21-L24 (1998)

2. Coherent curvature radiation

Curvature radiation can be described in terms of emission by a relativistic particle moving around the arc of a circle chosen such that the actual acceleration corresponds to centripetal acceleration. A relativistic electron with Lorentz factor , constrained to follow a path with radius of curvature , radiates similarly to an electron in a circular orbit with frequency . The critical frequency where most of the radiation is emitted is approximately given by (e.g. Zheleznyakov 1996, p. 231)

Or if we observe a particular frequency we find the corresponding Lorentz factor via

The curvature radius of the field lines in a magnetic dipole field can be expressed as (Smirnow 1973, p. 211)

where is the collateral angle of a point on the field line starting at at the pulsar surface. A reasonable approximation up to is given by

where is the curvature radius at the pulsar surface.

The total power radiated by a particle of energy along a curved field line is ( denotes the charge)

The classical Goldreich-Julian charge density in the pulsar magnetosphere (Goldreich and Julian 1969)

provides us with a typical particle density of if P=0.5 s and G. It enables us to estimate the typical size of the emission region for incoherent curvature emission. For the lowest luminosity (0655+64 (Taylor et al. 1993) we find . Using Eqs. (1b), (4) and (5) together with the simplifying assumption that the emission region extends across the line of sight as well as along it, we find that the predicted profile widths for incoherent curvature emission turn out to be within

For frequencies below we predict profile widths of roughly 180 degrees and only when we approach () can we expect widths of around 25 degrees. Typical pulsar profile widths at radio frequencies are however of the order of 5-10 degrees (e.g. Izvekova et. al. 1994, Seiradakis et. al. 1995). Because of the narrow profiles of the pulsed emission we must rule out any consideration of incoherent curvature emission as the source of the observed pulsar radio luminosities.

As incoherent emission is obviously insufficient it might be worthwhile to turn or attention to coherent curvature emission which has been suggested as an emission mechanism (see references above). Coherence exists in volumes of the order of (Melrose 1992). The power output of such a coherence volume is then given by:

with being the number of particles within a coherence volume. To account for the observed luminosity L one requires of these volumes, which will give us an estimate of the total number of particles involved

and the minimal size of the emission region

Energy conservation alone provides us with an important constraint that will enable us to restrict the free parameter and to check the consistency of the model.

The energy loss of a charge within the emission region is now

which must never exceed the energy of the particle itself.

The bunching of particles requires an external electric force which confines the particles in a coherence volume. Although electrostatic instabilities have been criticized for several reasons, the principal argument, that electrostatic plasma wave can be responsible for the bunching of particles has not been questioned, only processes that excite the waves are under discussion. The Doppler shifted plasma waves oscillate with the local plasma frequency of the particles and therefore their electrostatic fields present the ideal bunching force. When the particles are in resonance with the wave they experience the wave electric field most intensively and they are confined (bunched) in the coherence volume . The same resonance condition holds for the propagation of electromagnetic waves with frequency (where ), which defines a certain radius in the pulsar magnetosphere, at which the coherent radiation can escape from the plasma. Particle bunching and wave propagation is determined by the same condition (e.g. Melrose 1992). Since the bunching leads to a very fast and efficient emission the place at which the condition is fulfilled is also the place of emission, i.e. the emission height. For larger radii only lower frequencies can be emitted (see Eqs. (1a) and (3)). An increase in density corresponds to an increase in frequency at a given radius (Eq. (5)). With the determination of the minimum emission height from using Eq. (1b) to eliminate

we can express the ratio of emitted energy over the total energy as

which cannot exceed unity. From the condition we obtain a condition for the luminosity by inserting Eqs. (1-9) into Eq. (10), leaving

© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998
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