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

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3. Possible detection

At [FORMULA], for the bulk of high-energy electrons in the region ahead of the wind front the characteristic time of their synchrotron energy losses is much less than the GRB duration. In this case, the luminosity per unit area of the wind front in [FORMULA]-rays is [FORMULA] while the same luminosity in low-frequency waves is [FORMULA]. Using these, the ratio of the luminosity in low-frequency waves and the [FORMULA]-ray luminosity is

[EQUATION]

From Table 1, we can see that the GRB light curves in both [FORMULA]-rays and low-frequency waves have maximum when [FORMULA] is about 0.4, and the maximum flux in low-frequency waves is about two times smaller than the maximum flux in [FORMULA]-rays ([FORMULA]). At [FORMULA] the value of [FORMULA] decreases with increasing [FORMULA]. Therefore, we expect that the undispersed duration [FORMULA] of the low-frequency pulse to be somewhat smaller than the GRB duration, and the energy fluence in low-frequency waves to be roughly an order of magnitude smaller than the energy fluence in [FORMULA]-rays. The rise time of the radio pulse is very short because that the value of [FORMULA] increases very fast when [FORMULA] changes from 0.3 to 0.4 (see Table 1).

From Eq. (7), taking into account the Doppler effect, in the observer's frame the spectral maximum for low-frequency waves is expected to be at the frequency

[EQUATION]

where z is the cosmological redshift. For typical parameters of cosmological GRBs, [FORMULA] G and [FORMULA], we have [FORMULA] MHz. Unfortunately, the bulk of the low-frequency waves is at low frequencies, and cannot be observed. However, their high-frequency tail may be detected.

At high frequencies, [FORMULA], the spectrum of low-frequency waves may be fitted by a power law (Smolsky & Usov 2000):

[EQUATION]

where [FORMULA] (see Fig. 1). In the simulations of the wind-ambient gas interaction (Smolsky & Usov 1996, 2000; Usov & Smolsky 1998) both the total numbers of particles of the ambient gas and the sizes of spatial grid cells are restricted by computational reasons, so that the spectrum (10) is measured reliably only at [FORMULA]. The amplitudes of the computed oscillations with [FORMULA] are so small ([FORMULA]) that they cannot be distinguished from computational noise (Smolsky & Usov 1996). Future calculations with greater computational resources may alleviate this problem.

The value of [FORMULA] depends on many parameters of both the GRB bursters and the ambient gas around them, and its estimate, [FORMULA] MHz, is uncertain within a factor of 2-3 or so. In the most extreme case in which [FORMULA] is as high as a few MHz, the high-frequency tail of low-frequency waves may be continued up to [FORMULA] MHz where ground-based radio observations may be performed. In this case, the energy fluence in a pulse of radio emission at [FORMULA] MHz may be as high as a few percent of the GRB energy fluence in [FORMULA]-rays.

A pulse of low-frequency radio emission is strongly affected by intergalactic plasma dispersion in the process of its propagation. At the frequency [FORMULA], the radio pulse retardation time with respect to a GRB is

[EQUATION]

where [FORMULA] is the intergalactic dispersion measure in electrons/cm2, [FORMULA] is the group velocity of radio emission, [FORMULA] is the refractive index and [FORMULA] in Hz. From Eq. (11), for the plausible parameters of [FORMULA] cm-3 and a distance of [FORMULA] cm, at [FORMULA] MHz we have [FORMULA] s. This is time enough to steer a radio telescope for the radio pulse detection. In Eq. (11), we neglected the radio pulse retardation time in our Galaxy, which is typically one or two orders of magnitude less than that in the intergalactic gas.

The observed duration of the low-frequency pulse at the frequency [FORMULA] in the bandwidth [FORMULA] is

[EQUATION]

For plausible parameters, [FORMULA] MHz, [FORMULA] MHz, [FORMULA] s and [FORMULA] s, we have [FORMULA] s; the observed duration of low-frequency radio pulses is determined by intergalactic plasma dispersion, except for extremely long GRBs.

It is now possible to estimate, given assumed values for the magnetic field, the amplitude of the signal produced. We also assume that the plasma field couples efficiently to the free space radiation field. For a radio fluence [FORMULA] and a radio fluence spectral density

[EQUATION]

the radio spectral flux density is

[EQUATION]

where [FORMULA] is the GRB fluence in [FORMULA]-rays and [FORMULA] is the mean value of [FORMULA]. For the latter (dispersion-limited) case with the parameters [FORMULA] erg cm-2, [FORMULA], [FORMULA] MHz, [FORMULA] MHz, [FORMULA] MHz, [FORMULA] s we find [FORMULA] Jy.

The appropriate value of [FORMULA] is very uncertain. In some models it may be [FORMULA], but in internal shock models for GRB with sharp subpulses its value is limited by the requirement that the magnetic stresses not disrupt the thinness of the colliding shells (Katz 1997). For subpulses of width [FORMULA] of the GRB width this suggests [FORMULA]; typical estimates are [FORMULA] and [FORMULA], leading to [FORMULA] Jy.

These large values of [FORMULA] may be readily detectable, although the assumed values of [FORMULA] are very uncertain. There are additional uncertainties. We have assumed that the radio pulse spectrum (13) is valid up to a frequency [FORMULA] MHz that may be hundreds of times higher than [FORMULA]. As discussed above, the spectrum of low-frequency waves is calculated directly only at [FORMULA]. At [FORMULA], the spectrum must be extrapolated, with unknown confidence, from the calculations. The radio spectral flux density at [FORMULA] MHz may therefore be less than the preceding estimates. However, even in this case the very high sensitivity of measurements at radio frequencies may permit the detection of coherent low-frequency radio emission from GRBs.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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