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

Appendix

A.1. Lagrangian form of hydrodynamical equations and shock conditions

We use the set of hydrodynamical equations for the case of one- dimensional adiabatic motion of nonviscous perfect gas in Lagrangian form (Klimishin 1984)

Here gas pressure , its density , velocity and Eulerian co-ordinate are functions of Lagrangian co-ordinate, i.e., initial gas particle position a and time t, is the adiabatic sound velocity, is the adiabatic index. Subscripts indicate partial derivatives with respect to corresponding variables, is the initial density distribution.

The continuity equation (A4) may be written in the following form

or

At front of a strong shock with trajectory the following conditions are satisfied

where is the shock velocity, superscript "s" corresponds to values of parameters at the shock front .

A.2. Derivatives of functions of parameter distribution at the shock front

Eqs. (A1)-(A5) and shock conditions (A6)-(A9) allow to find the values of arbitrary order partial derivatives of hydrodynamical functions at the shock front using the law of shock motion (Gaffet 1978).

To find first derivatives of functions and u at front, we use Eqs. (A1)-(A3), written for , and add three equations, resulting from differentiation of boundary conditions (A6)-(A8) along the shock trajectory by operator . Solving the obtained set of six equations for six unknown partial derivatives at the shock front, we obtain expressions for (Hnatyk & Petruk 1996).

From Eq. (A5), we have now

where .

To find second derivatives of hydrodynamical functions at the shock front we differentiate equations (A1)-(A3) separately with respect to a and at the shock front (a=R), what gives us six equations for nine unknown derivatives. Additional three equations are given by differentiation of shock conditions (A6)-(A8) by the operator

Solving the system of nine equations with nine unknowns, we obtain expressions for (Hnatyk & Petruk 1996). From (A5) we obtain

where .

The time derivative , and for other ones we have

where

Derivatives from B and Q with respect to time are

In the self-similar case , .

According to the approximate formula for shock velocity (1) in the common (non-similar) case

In the work of Hnatyk & Petruk 1996, the rest derivatives are presented.

A.3. Approximation of the connection between Lagrangian and Eulerian co-ordinates

The algorithm considered above allows to calculate partial derivatives of arbitrary order. We consider the case when an expansion of density, pressure and velocity into series are restricted by second order at shock front and first in the central region. Namely, if at time t the shock position is , we approximate a connection between Eulerian co-ordinate and the Lagrangian one a in the following way

where , , and for each sector parameters are choosen from the condition that partial derivatives at shock front and in center of explosion correspond to their exact values:

In case of a uniform medium from the Sedov self-similar solution, it follows that in the central region at the dependence is and the factor C is connected with central pressure in the following way:

thus for in case (plane shock) , and in case (spherical shock) .

In the more general case of anisotropic explosion with direction-dependent energy release, and shock trajectory from (7) one obtains generalized condition for factor C:

where

© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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