12 Hyperluminal CBs and radio scintillations

The radio AGs of GRBs often show temporal variations of a factor of two or more on a time scale of hours at early times and on a time scale of days at later times; e.g. GRB 000926 in Figs. 15, 16, GRB 991216 in Fig. 22, GRB 991208 in Figs. 30, 31, GRB 000418 in Fig. 36, GRB 990123 in Fig. 35 and GRB 970508 in Figs. 47, 48.

Similar variations have not been seen in the optical and X-ray bands. The intensity variations of GRB radio AGs are very reminiscent of the ones seen in radio signals from pulsars in our galaxy, interpreted as scintillations due to the motion of the line of sight through the refractive, diffractive and dispersive ISM of the Galaxy (see, e.g., Lyne & Smith 1982). Some very compact active galactic nuclei also show an intraday variability that has been the subject of much debate (e.g., Wagner & Witzel 1995 and references therein). At least in one case - the variations in the radio intensity of the quasar J1819+3845, the most extremely variable AGN known at radio wavelengths (Dennett-Thorpe & de Bruyn 2000) - it was shown unambiguously that the variations are scintillations caused by the ISM (Dennet-Thorpe & Bruyn 2002).

The (de)coherence properties in time and frequency of the radio scintillations have been used to measure the transverse speed of pulsars (e.g., Lyne & Smith 1982). Gupta (1995) has demonstrated for a sample of 59 pulsars that their transverse speed, $V_{\rm iss}$, measured from their inter-stellar scintillations, agrees well with their transverse speed, $V_{\rm pm}$, measured from their proper motion (see also Nicastro et al. 2001).

The movement of the line of sight to pulsars is in most cases dominated by their proper motion at a transverse velocity $V_{\rm pm}$ larger than the turbulent speeds in the ISM, or of the sun relative to the ISM, or of the Earth around the sun. The mean $V_{\rm pm}$ of Gupta's 59 pulsars is 311 $\rm km\, s^{-1}$ and their mean distance is estimated to be 1.96 kpc. Their angular speeds are within an order of magnitude of a central value:

$$\dot \omega_{ps}\sim {\langle V_{\rm pm}\rangle \over \langle D\rangle} \simeq 5.1\times 10^{-15}\, \rm rad\, s^{-1}.$$ (34)

Travelling with a characteristic $\gamma\sim 10^3$ and viewed at typical angles $\theta$ of milliradians, CBs have apparent superluminal velocities, $V_{_{\rm CB}}$ of Eq. (30), that are so high (a few hundred times the speed of light) that they deserve to be called hyperluminal. The angular speed in the sky is:

$$\dot\omega_{_{\rm CB}}(t)= {V_{_{\rm CB}}(t)\over D_A}\simeq {\gamma(t)\, \delta(t)\, c\, \theta\over (1+z)\, D_A}\, \cdot$$ (35)

For the reference values $\theta\sim 1$ mrad and an initial $\gamma_0\sim\delta_0\sim 10^3$, the initial angular speed of a CB at redshift z=1 is $\rm\dot\omega_{_{CB}}(0)\sim 2.7\times 10^{-15}$ rad s^-1, in the very same range as that of Galactic pulsars. The CBs' angular velocity $\dot\omega_{_{\rm CB}}(t)$ and the resulting (inverse) coherence time of the scintillations should decline as $\gamma(t)\,\delta(t)$ does. Both pulsars and CBs are pointlike from the point of view of their radio scintillations. Thus, all conditions are met to expect pulsar-like scintillations in the radio signals from CBs.

The deviations from a smooth behaviour of the radio signals in the case of GRB 980425, as can be seen in Figs. 51, 52, are chromatic, but correlated in time over a much longer period than for the other GRBs. Because GRB 980425 is so close (z=0.0085) and is viewed at the unusually large angle of $\sim$8 mrad (Table 3 and DDD 2001), its apparent angular velocity, Eq. (35), is much larger than for other GRBs. The line of sight to this GRB swept a much bigger region of galactic ISM than for other GRBs or, for that matter, pulsars. Thus, we have no independent information on the ISM irregularities causing scintillations on this large scale.

The analysis of CB scintillations could result in a measurement of their hyperluminal speeds and a decisive test of the cannonball model (fireballs do not have relativistic proper motions, firecones stop moving close to their progenitors and, unlike CBs, have an increasing size that should rapidly quench their scintillations with time). The presently available information (or the current information in its published form) is insufficient for us to attempt at the moment to extract conclusions about CB hyperluminal velocities from the observed scintillation patterns in the radio AG of GRBs.