## Appendix## A.1. Lagrangian form of hydrodynamical equations and shock conditionsWe 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 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 frontEqs. (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 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
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-ordinatesThe 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 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 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 where © European Southern Observatory (ESO) 1999 Online publication: March 10, 1999 |