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Astron. Astrophys. 344, L37-L40 (1999)

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2. GRB jet model

In our model GRBs develop in a pre-existing jet. We consider a binary system formed by a neutron star and an O/B/WR companion in which the energy of the GRB is due to the accretion-induced collapse of the neutron star to a black hole.

The high speed in the energy flow in AGN-jets is generally believed to be initiated by strong magnetic field coupling to the accretion disk of the black hole (e.g. Falcke & Biermann 1995, Romanova & Lovelace 1997, Romanova et al. 1997). We use the same mechanism here. We assume that in this transition a large amount of energy is anisotropically released as Poynting flux along the polar axis. One may think of this process as a violent and rapid twist, such as occasionally debated as a possible cause for supernova explosions (Kardashev 1970, Bisnovatyi-Kogan 1970, Le Blanc & Wilson 1970, and more recently Biermann 1993). This energy release naturally initiates an ultrarelativistic shock wave in the pre-existing jet. The emission microphysics are as in existing fireball models.

To fix the jet parameters, we use the basic ideas of the jet-disk symbiosis model by Falcke & Biermann (1995, 1999). In this model the direction of the magnetic field is mostly perpendicular to the axis of the jet and the values of the total particle number [FORMULA] (relativistic electrons + thermal protons) and magnetic field [FORMULA] are calculated using the equipartition between magnetic field energy density and kinetic plasma energy density in the umperturbed jet. Both, [FORMULA] and [FORMULA], are a function of the jet ejection rate [FORMULA], the equipartition parameter [FORMULA] and the bulk velocity of the jet [FORMULA], where [FORMULA] is the bulk Lorentz factor of the jet prior to the burst. The values of the particle number density and the magnetic field in the unperturbed jet are then given as

[EQUATION]

[EQUATION]

where [FORMULA] is the minimum electron Lorentz factor [FORMULA] of the relativistic electrons (assumed to be in a power law distribution), [FORMULA] is the opening angle of the jet, and [FORMULA] is the distance along the jet. z is the redshift.

The advantages of using the propagation of an ultrarelativistic shock wave inside a jet can be summarized in the following two points

  • The initiation of the shock by magnetic fields does not necessarily involve considerable matter. It implies a low amount of baryonic matter, of order [FORMULA], in the jet.

  • The initial amount of energy deposited in the jet is of the order [FORMULA], and we can obtain an apparent isotropic energy as high as [FORMULA] [FORMULA]. This fits well the requirements of GRB971214.

The evolution of the shock Lorentz factor [FORMULA] with distance [FORMULA] along the jet axis can be obtained using the conservation of the unperturbed jet gas energy in the shock rest frame, [FORMULA], that is

[EQUATION]

The characteristic time scale to see the emission across the entire region when the shock has reached a distance [FORMULA] along the axis of the jet is given by [FORMULA]. Substituting this into the formula for [FORMULA], we get the time evolution of the ultrarelativistic shock front

[EQUATION]

[EQUATION]

where [FORMULA].

A change in the emission properties occurs when the opening angle of our jet passes the relation [FORMULA]. Using the formula of the bulk Lorentz factor as a function of time, we can see at what time [FORMULA] this relation holds: [FORMULA] s. Hence, after about 6 hours this limit is reached. Prior to this, the observed emission is limited by the Lorentz boost to a conical section of the shock front of angle [FORMULA]. For the following calculation of the afterglow emission several hours after the burst we can therefore consider [FORMULA].

In the flow behind a steady shock front, relativistic particles are usually accelerated and magnetic fields can be amplified. After the shock, [FORMULA] and [FORMULA], relative to the jet axis. The resulting magnetic field in the shocked plasma will have a strength of roughly [FORMULA] and the shock wave compresses the magnetic field component perpendicular to the jet axis. The jump conditions for the density particle number give [FORMULA], (De Hoffmann and Teller 1950, Marscher and Gear 1985). We use here the approximation for an ultrarelativistic shock so that [FORMULA], where 1 and 2 are related to the zone before and after the shock. This allows a straight and simple limit to nonrelativistic shocks. In our model, this corresponds to

[EQUATION]

[EQUATION]

where (sf) shows the quantities in the shock frame and (ob) the corresponding values in the observer frame.

In our jet we consider an electron power law distribution [FORMULA] where [FORMULA], with a cutoff at a constant minimum electron Lorentz factor [FORMULA]. In fact, if p-p collisions inject a population of electrons in the unperturbed jet, then one would expect that in the unperturbed jet the electron Lorentz factor [FORMULA] goes from [FORMULA] up to some large value (Falcke & Biermann 1995).

The equation for [FORMULA] has been obtained considering that in the shock frame the powerlaw electron energy density is taken as a fraction [FORMULA] of the pre-existing relativistic electron energy density and using the value p=2

[EQUATION]

If we were to relate to the proton energy density in the shock, [FORMULA].

To calculate the boosting factor of the transition from the shock frame to observer frame, we assume the angle between the jet-axis and the line of sight to the observer is [FORMULA] some time during the [FORMULA]-ray burst. The critical synchrotron frequency after the shock front in the observer frame is given by [FORMULA], where we consider the minimum electron Lorentz factor evolves in phase space with the bulk Lorentz factor [FORMULA]:

[EQUATION]

We assume isotropic emission in the shock rest frame. Only when we transform the radiation emitted into the observer frame, the photons are concentrated in the forward direction, lying within a cone of half-angle [FORMULA]. For the afterglow, however, the solid angle is determined by the actual opening angle of the jet.

At the critical frequency, [FORMULA], in our model the synchrotron cooling time is less than the dynamical time, [FORMULA]. In fact, in the shock frame

[EQUATION]

so the synchrotron time scale is shorter than the dynamical time scale for our standard parameters.

Considering the cooling evolution of our electron power law spectrum, to calculate the flux we use the formula [FORMULA], where [FORMULA] is the total emission power per unit volume per unit frequency for an electron power law distribution with an index [FORMULA], (e.g. Sect. 6.4 of Rybicki & Lightman, 1979). In our model this corresponds to a flux in the observer frame given by

[EQUATION]

Here, the quantity [FORMULA] represents the thickness of the radiating shock front, extending behind the shock front to a point where the local [FORMULA] drops below [FORMULA]

[EQUATION]

where [FORMULA]. In both equations (11) and (12) [FORMULA] is in the shock frame.

Substituting the values of the equations (4), (5), (7), (8) and (12) in (11), and with [FORMULA] in the observer frame

[EQUATION]

where D is the luminosity distance, [FORMULA] corresponds to a redshift of about [FORMULA], using [FORMULA] and [FORMULA].

The flux at frequencies [FORMULA] below the critical frequency, [FORMULA], can be obtained using the flux [FORMULA] at frequencies [FORMULA], calculated at the frequency [FORMULA], that is [FORMULA]. The corresponding flux is given by

[EQUATION]

For optically thin emission, [FORMULA] is equal 1/3 and the time dependence is [FORMULA]. This suggests a gentle optical rise.

Another important check is to calculate the synchrotron self-absorption coefficient [FORMULA] in our model (Sect. 6.8 of Rybicki & Lightman, 1979). Integrating it over an isotropic distribution of particles and using the value obtained for the coefficient [FORMULA], we find [FORMULA]. The optical depth along the path of a travelling ray is equal to the product of the absorption coefficient [FORMULA] times the width of the shell (equation 12), and it will be equal to unity at the time

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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