3. The quasi-analytical simplified model based on the constant-thickness approximation
The PEM pulse expansion in the absence of baryonic matter has been discussed in a previous paper (Ruffini et al. 1999) where the quasi-analytical approach of an expanding shell of constant thickness in the laboratory frame was adopted and validated by comparison with the numerical integration of the general relativistic hydrodynamical equations. We here generalize these results by examining the collision of the PEM pulse with baryonic matter and adopting the constant-thickness approximation both for the description of the collision and the further expansion of the PEM pulse by a simplified approach to the system of equations outlined in the previous paragraph.
We first recall the main results of the PEM pulse expanding in vacuum: we indicate by the four-velocity of the slab and by the constant width of the slab in the laboratory frame of the plasma fluid, the average bulk relativistic gamma-factor is,
The evolution of the slab is governed by the total energy and entropy conservations, which are cast into the following equations as a function of the coordinate volume of the plasma fluid expanding from to ,
where the proper volume V of the plasma fluid and the thermal index Eq. (20), a slowly-varying function of the state with values around 4/3, has been approximately assumed to be constant. The coordinate number density of -pairs in equilibrium is and the coordinate number density of -pairs . These equations have already been numerically integrated (Ruffini et al. 1999).
The baryonic matter remnant is assumed to be distributed well outside the dyadosphere in a shell of thickness between an inner radius and an outer radius at a distance from the EMBH at which the original PEM pulse expanding in vacuum has not yet reached transparency,
As the PEM pulse reaches the region , it interacts with the baryonic matter which is assumed to be at rest. In our simplified quasi-analytic model we make the following assumptions to describe this interaction:
These assumptions are valid if: (i) the total energy of the PEM pulse is much larger than the total mass-energy of baryonic matter , , and (ii) the comoving number density ratio of pairs and baryons at the moment of collision is extremely high (e.g., , and (iii) the PEM pulse has a large value of Lorentz factor ().
In the collision process between the PEM pulse and the baryonic matter at , we impose the total energy and momentum conservations. We consider the collision process between two radii , satisfying and . The amount of baryonic mass acquired by the PEM pulse is
These equations determine the gamma-factor and the internal energy density in the capture process of baryonic matter by the PEM pulse. After collision (), the further adiabatic expansion of PEM pulse is described by the total baryon number, energy and entropy conservations, i.e., the following hydrodynamical equations which generalize those derived in our previous paper (Ruffini et al. 1999) with ,
In these equations () the comoving baryonic mass- and number densities are and , where V is the comoving volume of the PEM pulse. The integration is continued untill the transparency condition in Eq. (34) is reached.
© European Southern Observatory (ESO) 2000
Online publication: July 13, 2000