Astron. Astrophys. 364, 655-659 (2000)

2. Low-frequency electromagnetic waves generated at the wind front

Our mechanism for production of short pulses of low-frequency radio emission from relativistic, strongly magnetized wind-generated cosmological GRBs applies very generally. For the sake of concreteness, we consider wind parameters that are natural in a GRB model that involves a fast rotating compact object like a millisecond pulsar or dense transient accretion disc with a surface magnetic field G (Usov 1992; Blackman et al. 1996; Katz 1997; Kluzniak & Ruderman 1998; Spruit 1999; Wheeler 1999; Woosley 1999; Ruderman, Tao, & Kluzniak 2000).

In this model the rotational energy of compact objects is the energy source of cosmological GRBs. The electromagnetic torque transfers this energy on a time scale of seconds to the energy of a Poynting flux-dominated wind that flows away from the object at relativistic speeds, (e.g., Usov 1994). The wind structure at a time is similar to a shell with radius and thickness of , where is the characteristic deceleration time of the compact object's rotation, or the dissipation time of a transient accretion disc.

The strength of the magnetic field at the front of the wind may be as high as

where cm is the radius of the compact object,  s^-1 is the angular velocity at the moment of its formation and s cm is the radius of the light cylinder. Eq. (1) gives the real value of B at the wind front if both the magnetic field of the compact object is strictly dipolar and the thickness of the wind shell does not increase essentially in the process of the shell outflow.

The distance at which deceleration of the wind due to its interaction with an ambient gas becomes important is (Mészáros & Rees 1992; Piran 1999)

where n is the density of the ambient gas and is the initial kinetic energy of the outflowing wind. Eq. (2) assumes spherical symmetry; for beamed flows is times the wind energy per steradian. At , the main part of is lost by the wind in the process of its inelastic interaction with the ambient medium, and the GRB radiation is generated.

Substituting for r into Eq. (1), we have the following estimate for the magnetic field at the wind front at :

where we have introduced a parameter which gives the fraction of the wind power remaining in the magnetic field at the deceleration radius. For plausible parameters of cosmological -ray bursters,  G,  s^-1,  ergs, and cm^-3, from Eq. (3) we have  G.

For consideration of the interaction between a relativistic magnetized wind and an ambient gas, it is convenient to switch to the co-moving frame of the outflowing plasma (the wind frame). While changing the frame, the magnetic and electric fields in the wind are reduced from B and in the frame of the -ray burster to and in the wind frame. This is analogous to the the well-known transformation of the Coulomb field of a point charge: purely electrostatic in the frame of the charge, but with in a frame in which the charge moves relativistically. Using this and Eq. (3), for typical parameters of cosmological -ray bursters we have  G at .

In the wind frame, the ambient gas moves to the wind front with the Lorentz factor and interacts with it. The main parameter which describes the wind-ambient gas interaction is the ratio of the energy densities of the ambient gas and the magnetic field, , of the wind

where is the density of the ambient gas in the wind frame and is the proton mass.

At the initial stage of the wind outflow, , is , but it increases in the process of the wind expansion as decreases. At , when is more than , the interaction between the wind and the ambient gas is strongly nonstationary, and effective acceleration of electrons and generation of low-frequency waves at the wind front both begin (Smolsky & Usov 1996, 2000; Usov & Smolsky 1998). For , the mean Lorentz factor of outflowing high-energy electrons accelerated at the wind front and the mean field of low-frequency waves weakly depend on (see Table 1) and are approximately given by

to within a factor of 2, where and are the magnetic and electric field components of the waves. The mechanism of generation of these waves is coherent and consists of the following: At the wind front there is a surface current that separates the wind matter with a very strong magnetic field and the ambient gas where the magnetic field strength is negligible. This current varies in time because of nonstationarity of the wind-ambient gas interaction and generates low-frequency waves.

Table 1. Derived parameters of simulations for both high-energy electron Lorentz factor and low-frequency electromagnetic wave amplitudes and their power ratio in the region ahead of the wind front

Fig. 1 shows a typical spectrum of low-frequency waves generated at the wind front in the wind frame. This spectrum has a maximum at the frequency which is about three times higher than the proton gyrofrequency in the wind field :

Fig. 1. Power spectrum of low-frequency electromagnetic waves generated at the front of the wind in the wind frame in a simulation with G, , and . The spectrum is fitted by a power law (dashed line).

© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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