24P153-P
A new self-similar solution for shock break-out of super-novae
超新星爆発Shock break-outに関する新自己相似解
Tatsushi Kobashi, Masakatsu Murakami
小橋 龍知, 村上匡且
Institute of Laser Engineering, Osaka University, Suita, Osaka 565-0871, Japan
大阪大学 レーザーエネルギー研究所 〒565-0871 吹田市山田丘
Although many researches are focused on the shock wave accompanying supernova explosion, the self-similar solutionof the self-consistent explosion which taking into account both self-gravity and radiant heat conduction has never beenaddressed. A new self-similarity solution was found in terms of a novel with the help of the Lie-group method. This
scheme is expected to be also applied to different problems with multiple heat conduction mechanism.

1. Introduction
Self-similar solutions play a crucial role in any
branches of physics, in particular, for such fields
as hydrodynamic phenomena in astrophysics. The
self-similar solutions under different assumptions
have indeed widened their applicability andenriched our qualitative understanding of physics.
A new self-similar solution describing spherical
explosions of a gaseous sphere under bothself-gravity and radiative diffusion is investigatedin detail, where the diffusion is modeled by a powerlaw with respect to density and temperature. The
reduced dimensional eigenvalue problem is solved
to show that there is a unique quantitative relation
between the physical effects for self-similardynamics. The resultant spatial and temporal
behaviors are also determined uniquely, once theopacity is specified.
2. Basic equation and similarity ansatz
The one-dimensional spherical gas-dynamical
equations with both self-gravity and diffusivity arevelocity, and is the gravitational potential.
We assume an ideal gas equation of state in
the form(5)

where kB is the Boltzmann constant, is the mean
atomic mass, andis the specific heat ratio.
Equation (4), described by the one-temperaturemodel, includes the nonlinear heat diffusion termon the right-hand side, where we assume a powerlaw dependence for the diffusion coefficient,(6)

with 0, m, and n being constants. For normal
physical values, n > 0 and m > 0 are assumed.
To find a self-similar solution, we here introduce
the following group transformation;(1)(2)

(7)

(3)

(4)

where p is the pressure, is the density,isthe specific internal energy, u is the flowwhere a, b, c, d, and e is the constant and R(t) is the
temporal characteristic scale length of the system; Aand B are positive constants defining the scale of
the radius and the density, respectively. These
constants can be uniquely determined bysubstituting equation (1) - (4) such that the(8)
(9)

102
101
τ

100
10-1

g

10-2v

10-310(10)642velocity v

ξdensity g temperature τtransformed system is the kept symmetric and thus
has the same structure as the original one based on
Lie’s idea. Then, equation (1), (2), and (4) are
reduced, respectively, to the following ordinary
differential equations:

0-4

10-1

100

ξ

101

102

Fig.1. the self-similar solution for reference case,m=1, n=13/2, =1.3, and K1=0.01

Since equation (3) is automatically satisfied, its
reduced form does not appear in the above set ofequations. As can be seen in equations (18) and(19), the present system is characterized by two
positive dimensionless parameters, K1 and K2.
Equations (8)-(9) are a second order ordinary
equation system for g, , v, and the obvious
boundary conditions at = 0 are(11)The last relation physically means that there is nopressure gradient at the center.
3. The self-similar solution
At a glance, the ODE system, together withboundary condition (11), seems to be closed
mathematically. However one can easily find thatnumerical integration of the system produced aphysically unacceptable picture under an arbitrary
set of the values of K1, K2, and S , where S is theshock front parameter, such that the temperaturesuddenly drops to zero with the density being kept
nonzero at a finite radius. Since the physicallyquantities are expected to change smoothly in space,
it is conjectured that some special values of K1, K2,
and S which are still unknown, can give such aphysically acceptable picture. Therefore the present
system is supposed to be a two-dimensional
eigenvalue problems treated in previous work.
To determine a unique set of parameter, we need
two more physical conditions. The first one is thatthe temperature should converge to zerosimultaneously with increasing radius. Note that
self-similar models based on the isothermal or

adiabatic assumptions do not need energy equation(4), and the system is described in terms of only asingle unknown constant, S, which is determinedby this first condition. The second one is that thevelocity should converge to zero at∞.
Although we can determine the value of S, and K2
by using both condition, K1 is not still unknown.
Figure 1 shows the eigenstructure of the self-similarsolution for the reference case, m=1, n=13/2, =1.3,
and K1=0.01.

4. Conclusion
Under the appropriate similarity ansatz andvariable transformations, the hydrodynamic system
is reduced to the novel two-dimensional eigenvalue
problem. The physical implication is that a uniquequantitative relation between the gravity and theheat diffusivity indwells in the self-similardynamics.
References[1] Landau, L. D., & Lifshitz, E.M 1959, Fluid
Mechanics(London: Pergament)[2] Sedov, L. I. 1959, Similarity and Dimensional
Methods in mechanics(New York: Academic)[3] Zeldvich, Ya. B., & Raizer, Yu. P. 1960, Physics ofShock Waves and High Temperature Hydrodynamic
Phenomena(New York: Academic)[4] G. Guderley, Luftfahrforschung 19, 302 (1942)[5] M. Murakami, K. Nishihara, and T. Hanawa,Astrophys. J. 607, 879 (2004)