## 2. Hydrodynamical modellingAs proposed by Hnatyk & Petruk (1996) the hydrodynamical description includes two steps: the calculation of shock front dynamics (shape of SNR) and the calculation of the state of the plasma inside the SNR. ## 2.1. Calculation of shock front motionKlimishin & Hnatyk (1981), Hnatyk (1987) have shown that the motion of a strong one-dimensional adiabatic shock wave in a medium with an arbitrary distribution of density is described with high accuracy by the approximate formula:
Formulae (1)-(2) generalize two basic features of shock motion in non-uniform media: -
*a deceleration*in a medium with increasing , constant or slowly decreasing density ; then the parameter*k*is close to ; -
*an acceleration*if the density is decreasing fastly enough as ; then the parameter*k*is close to .
Since in all realistic cases ,
i.e. the initial stage of the motion of the shock from a point
explosion is always described by the self-similar Sedov solution for a
where is the self-similar constant for a uniform medium (, ), is the energy of explosion. If the density distribution is such that a region of acceleration with exists and begins at some distance where , then an approximate formula for shock velocity is: Here, is determined from (3). If the density distribution is such that there is a transition from a region of acceleration with into a region of deceleration where , then another feature appears. At the beginning, the shock deceleration will be analogous to the motion of an external (forward) shock as described by Nadyezhyn (1985) and Chevalier (1982). The external shock decelerates with and, when its velocity equals velocity from (3) the further deceleration is described by . Therefore, a general formula for the shock velocity which takes into account the three cases is (Hnatyk 1987): where ;
are zeros of the function
(the points of changing of regimes
of shock motion) in increasing order of It must be emphasized that the majority of models interesting for astrophysics are described more simply by formulae (3)-(4). The equation of the trajectory of the shock motion is When the density distribution departs from spherical symmetry, the calculation should be carried out by sectors. The 3D region is divided into the necessary quantity of sectors and Eq. (6) is integrated for each of them. Such an approach allows also to consider an anisotropic energy release replacing by for . ## 2.2. Calculation of the plasma characteristics inside SNRThe second step concerns the determination of gas parameters inside
the SNR. It uses the shock motion law by sector discribed in the
previous step. We will use the Lagrangian coordinates
( In this way, we approximatively take into account a redistribution of energy between the sectors in the cases of an anisotropic explosion and/or a non-uniform medium. The calculation of is presented in Appendixes A.1-A.3. The other parameters (pressure , density , velocity ) are exactly derived from in case of adiabatic motion of gas behind the shock front (Hnatyk 1987): ## 2.3. Testing the methodThe proposed method exactly describes the shock trajectory in self-similar (Sedov) cases with including in particular the uniform () medium. The space distributions of the gas parameters: pressure density temperature and velocity inside the SNR are accurate within 3% in the case (Hnatyk & Petruk 1996). The result of the calculation for a point explosion in medium with an exponential density distribution where
Fig. 2 presents the results of a direct 2D numerical calculation together with results of different approximate approaches (Kestenboim et al. 1974). As we see, for the largest time the proposed method is accurate within 7%, i.e., of the same order of accuracy as the numerical method. Even a numerical calculation by sector (1D numerical calculation in each sector) yields a considerably lower accuracy, only about 20%. Another important quantity useful to check the accuracy the shock acceleration in exponential atmosphere is the breakout time when shock velocity The combination of the 2D numerical solutions for with the self-similar solution for for yields (Kestenboim et al. 1974) while approximation (5) gives The accuracy of the density and pressure calculations in the proposed method is illustrated by Fig. 3. Additional results of testing of the method for more complex density distributions, such as a density discontinuity are presented by Hnatyk (1987). All these results reveal that the proposed method has high enough accuracy in all cases in the sector aproximation. It fails however in the case of shock interaction with a small dense cloud. © European Southern Observatory (ESO) 1999 Online publication: March 10, 1999 |