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Astron. Astrophys. 344, 295-309 (1999)

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2. Hydrodynamical modelling

As 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 motion

Klimishin & 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:

[EQUATION]

where

[EQUATION]

R is the distance from the explosion, [FORMULA] is the initial density distribution of the surrounding medium; [FORMULA]; [FORMULA] for a plane, cylindrical and spherical shock, respectively.

Formulae (1)-(2) generalize two basic features of shock motion in non-uniform media:

  • a deceleration in a medium with increasing [FORMULA], constant [FORMULA] or slowly decreasing density [FORMULA]; then the parameter k is close to [FORMULA];

  • an acceleration if the density is decreasing fastly enough as [FORMULA]; then the parameter k is close to [FORMULA].

Since in all realistic cases [FORMULA], 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 uniform medium, for the deceleration stage of shock motion from Eqs. (1)-(2) we have:

[EQUATION]

where [FORMULA] is the self-similar constant for a uniform medium ([FORMULA], [FORMULA]), [FORMULA] is the energy of explosion.

If the density distribution is such that a region of acceleration with [FORMULA] exists and begins at some distance [FORMULA] where [FORMULA], then an approximate formula for shock velocity is:

[EQUATION]

Here, [FORMULA] is determined from (3).

If the density distribution is such that there is a transition from a region of acceleration with [FORMULA] into a region of deceleration where [FORMULA], 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 [FORMULA] and, when its velocity equals velocity [FORMULA] from (3) the further deceleration is described by [FORMULA].

Therefore, a general formula for the shock velocity which takes into account the three cases is (Hnatyk 1987):

[EQUATION]

where [FORMULA]; [FORMULA] are zeros of the function [FORMULA] (the points of changing of regimes of shock motion) in increasing order of R in the interval [FORMULA].

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

[EQUATION]

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 [FORMULA] replacing [FORMULA] by [FORMULA] for [FORMULA].

2.2. Calculation of the plasma characteristics inside SNR

The 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 [FORMULA] (a is the initial coordinate of the gas element at [FORMULA]) and take as the unknown function the position (Eulerian coordinates) of the gas element [FORMULA] ([FORMULA]) at time t. The function [FORMULA] is expanded into series about the shock front and about the center of explosion. Then these two decompositions are combined. The coefficients of decomposition near the shock front is determined by the motion law in each sector and near the center of explosion assuming a zero pressure gradient (see Appendix A.3). It is important that the value of pressure in the central region be taken equal in all sectors contrary to previous propositions (Laumbach & Probstein 1969, Hnatyk 1987, 1988). The value of pressure around the point of explosion is determined from the condition that the ratio [FORMULA] of pressure in the center of explosion [FORMULA] to the average value of energy density inside SNR [FORMULA] does not change with time and is equal to the self-similar value for a uniform medium:

[EQUATION]

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 [FORMULA] is presented in Appendixes A.1-A.3.

The other parameters (pressure [FORMULA], density [FORMULA], velocity [FORMULA]) are exactly derived from [FORMULA] in case of adiabatic motion of gas behind the shock front (Hnatyk 1987):

  • the density is obtained from the continuity equation

    [EQUATION]

  • the pressure is derived from the equation of adiabaticity [FORMULA] or

    [EQUATION]

  • velocity directly from approximation [FORMULA]

    [EQUATION]

2.3. Testing the method

The proposed method exactly describes the shock trajectory in self-similar (Sedov) cases with [FORMULA] including in particular the uniform ([FORMULA]) medium. The space distributions of the gas parameters: pressure [FORMULA] density [FORMULA] temperature [FORMULA] and velocity [FORMULA] inside the SNR are accurate within 3% in the [FORMULA] case (Hnatyk & Petruk 1996).

The result of the calculation for a point explosion in medium with an exponential density distribution

[EQUATION]

where r is the distance from the center of explosion, [FORMULA] is the angle between the considered direction and the direction opposite to the density gradient, and H is the scale height, are presented in Fig. 2-3. The simulations are carried out for dimensionless parameters [FORMULA] [FORMULA] [FORMULA] [FORMULA] with [FORMULA] [FORMULA] [FORMULA] and [FORMULA] They allow to obtain the solution for any value of the initial parameters [FORMULA] [FORMULA] and [FORMULA]

[FIGURE] Fig. 2. Profiles of a shock front in a non-uniform exponential medium Eq. (11) as a function of dimensionless time [FORMULA] (see text). The coordinates are non-dimensional as defined in the text. Line 1 is the result of the numerical 2D calculation; line 2 is the approximate law (5); line 3 is the result of numerical calculation in the sector approximation; line 4 is the result of the thin layer approximation.

[FIGURE] Fig. 3a-c. Distributions of density a , b and pressure c inside an SNR expanding in a flat exponential medium Eq. (11) for different directions from the place of explosion at the dimensionless time [FORMULA] The solid lines correspond to the present method, the dashed lines are the result of the direct 2D numerical calculations of Kestenboim et al. (1974). Here [FORMULA]

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 [FORMULA] 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 [FORMULA] when shock velocity [FORMULA] The combination of the 2D numerical solutions for [FORMULA] with the self-similar solution for [FORMULA] for [FORMULA] yields [FORMULA] (Kestenboim et al. 1974) while approximation (5) gives [FORMULA]

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.

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© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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