## 4. SolutionTo develop the solution we have to make a short excursus into the Lie group analysis of differential equations. A reader not familiar with this theory is referred to the textbooks (Ovsiannikov 1982; Ibragimov 1983; Olver 1986). It is known that the system of These operators constitute the linear space with the definite properties (commutator of any pair of operators can be expressed as a linear combination of the entire set) which is called the Lie algebra. The existence of a group of symmetry of the system (1)-(3) in other words means that the system is invariant under the action of infinitesimal transformations given by (7)-(13). It is worthwile to note that a group generally defines the global transformations given by finite algebraic equations. In case of the finite transformations, finding a group of symmetry is not an easy problem since this procedure involves solving the nonlinear equations. It happened to be more convenient to deal with the infinitesimal transformations because they lead to linear determining equations which are always solvable. In this stage the appearance of algebra which determines the structure of a group is natural. A Lie group and its associated Lie algebra are intimately related so that any one of these allows to recover the other one and vice versa. In what follows we make no difference between them unless it is specially emphasized. Each transformation we are dealing here with can be assigned a
conventional physical meaning. In particular, the operators (7)-(8)
describe the invariance of Eqs. (1)-(3) under translations in
time and in space, the operators (9) describe invariance under
rotations, the operators (10) present the Galilean transformations.
The transformations allow to breed solutions once a particular one is
known and, what is more important, to construct new classes of
particular, so-called invariant, solutions reflecting the
corresponding particular symmetrical properties of equations. The
invariant solution meeting to If , the group is extended by an additional projective symmetry first found by Ovsiannikov (1958) The invariant solution corresponding to Obviously, the solution we are interested in must coincide with the
standard Sedov solution at early stages ( with some constants We are looking for the spherically symmetric solution. In spherical
coordinates The invariant solution is derived as follows. First we find the
invariants Eq. (17) is a linear first order partial differential equation that can be solved with the help of the method of characteristics. It has four functionally independent solutions To draw a parallel between the invariant and self-similar solutions
it is worthwile to turn from the invariants Here is the dimensional constant to be determined later. Any function of invariant is also invariant, so we can construct
invariant solution suggesting that where we have introduced the sound velocity after which Eqs. (23)-(26) finally transform to The variable in (30)-(33) plays the role of the independent invariant variable. The constant is traditionally defined in such a way that at the shock front; its rigorous definition is deferred until Eq. (60). Hence the range of the invariant variable is , where is either a positive constant in case of a shell-like solution or zero if a solution extends up to the center of symmetry. After the substitution of Eqs. (30)-(33) into the system (1)-(3) the latter one is reduced to the system of ordinary differential equations which completely coincide with the system derived by Sedov for the
self-similar solution (1959). This is not surprising because at small
times The boundary conditions for the problem are the standard conservation laws at the strong adiabatic shock jump where the subscript 1 denotes the values taken immediately behind the shock front. The shock front velocity can be found by setting in Eq. (30) and differentiating this with respect to time: Substituting , the preshock and postshock values taken from Eqs. (4), (6) and (31)-(33) into (37)-(39) we find that the boundary conditions are compatible with the invariant form and can be eventually written as We again find that the boundary conditions coincide with those of Sedov for (Sedov 1959). Since both the dynamic equations and the boundary conditions are the same as in the Sedov case we conclude that the solution of the problem (34)-(36), (40)-(42) is exactly the Sedov one and hence we may draw on the known solution from (Sedov 1959): Here the variable In case Altogether, the invariant solution is given by the formulas (30)-(33), (43)-(53). An inspection of expressions (31), (33) shows that the profiles of
density and pressure remain similar throughout the whole lifetime of
the blast wave, their form coincides with the profiles of the Sedov
self-similar solution. At the same time according to Eq. (32) the
profile of velocity changes with time (Figs. 1 and 2) and tends
to the homologous one,
It is important that in the matter reference frame the blast wave decelerates and eventually comes to rest: the relative jump of velocity and the mass flux through the discontinuity asymptotically vanish. This means that the blast wave asymptotically encloses the finite mass However, at any finite moment Let us consider how the energy of explosion At the same time the total energy is combined as the kinetic,
At the initial moment One may wonder how Eqs. (54) and (56) containing the terms
with different time dependencies can hold. It would not be difficult
to show that this is the case. Substituting expressions (55), (57) and
(58) into Eqs. (54) and (56), equating (54) to (56) and
multiplying this by gives us an equation
quadratic in time. This equation must be an identity in where In case of the uniform medium, , Eq. (60) is reduced to the well-known (Sedov 1959) the subscript hereafter denotes parameters of the ambient fluid taken at the moment of explosion. The remaining two equations for hold automatically. © European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |