Supernova Remnant


The exact solution of point explosion is called as Sedov-Taylor self similar solution and well known as a basic model on the evolution of supernova explosion.
Cas A (Chandra) CAS A Radio Cas A Optical
Above figures are the photographs of the typical supernova remnant CasA in X-ray, non-thermal radio, and optical radiations from left to right, respectively. This 'cloud' has about 10 light years diameter and is regarded as a figure of spherical shock wave generated by the supernova explosion took place about 300 years ago and being expanded into the inter stellar medium.
From the observed X-ray spectrum, the gas emitted this X-ray has temperature as high as 50 million K. From its intensity distribution, high temperature gas is considered to distribute spherically. The outer most surface is the shock wave front and the scene where the shock wave heats up the inter stellar gas entrained by the expanding shock wave is seen. On the otherhand, the strong polarized non-thermal radio emission is observed in radio (VLA) observation, this is the result of synchrotron emission emitted from relativistic electrons circulating around magnetic filed lines. It is considered that electrons are accelerated up to very high energy by Fermi acceleration at the shock surface.

Simulation Results


To show initial states, please place the mouse on the figures.

High temperature and high pressure gas at the center is expanding along with swept up surrounding low temperature gas (adiabatic expansion). Please refer detailed explanation for the simulation results in detail.

Basic Equations

Spherically symmetric Hydrodynamic equations.

Characteristics of Exact Solution

Free Expansion Phase

Initially, envelope ejected by the supernova explosion is expanding freely into the interstellar medium. Then this expanding envelope compresses interstellar gas around it, creates a shock wave inside of the interstellar gas because of high expanding velosity, and sweep up the inter stellar gas along with its expansion. During this initial phase, the amount of gas swept up is much smaller than the original exploded envelope, the expantion of the envelope is not affected by the outer interstellar gas and kept initial speed and energy.

Adiabatic Expansion Phase

When the amount of gas swept up becomes larger than the original exploded envelope, Kinetic energy of the original exploded envelope is transfered to the swept up gas, the shock wave is propagating into interstellar gas, and the swept up gas is heated up by the shock wave despite of the details of explosion. The period during when energy is not released by the radiation from the swept up gas (when energy is conserved) is called adiabatic expansion phase. The evolution during this phase is determined only by the energy of explosion E0, the density of interstellar gas [roh], and the elapsed time from the explosion t. Then a self similar solution exists as shown in the next section, the thermodynamic quantities of hot gas inside the shock front, e.g., density, pressure, and temperature of the gas, and the distribution of the expansion velocity is well described by the Sedov-Taylor self similar solution. The way to derive Sedov-Taylor self similar solution is found in "Fluid Mechanics" by Landau and Lifshitz, "Similarity Methods in Mechanics" by L. I. Sedov, and/ or "Uchu Ryutai Rikigaku (in Japanese)" by Sakashita and Ikeuchi.

Constant Temperature Expanding Phase

The expansion velocity is decreasing with time. As a result, when the expansion proceeds, radiation cooling behind the shock front becomes effective. Because the energy loss per unit volume and unit time by radiation cooling (radiation cooling rate) becomes larger along with the decrease in temperature (decceleration of shock wave). When radiation cooling time of the gas becomes shorter than the expansion time, the evolution deviates from self similar one. Then radiation cooled interstellar gas distributes spherically (shperical shell) behind the shock wave, hot rarefied gas fills inside it. This phase of supernova remnant (SNR) is called in constant temperature expanding phase. In this phase, the SNR evolves under rather conserving momentum of the cooled down high density shell (spherical shell) of inter stellar gas than conserving energy. In this phase, the radius of the shell expands in proportion to 1/4 power of the elapsed time since the explosion.

Brief Explanation on Sedov-Taylor Solution

As an initial condition, energy E0 is released at the center of isotropic gas with constant density [roh]. It is too much complicated to describe exeact solution completely, we will write down characteristic physical quantities of the exact solution here. Time evolution of the radius R and expanding velocity D of the shock front are described as
Here, [Xi] is a dimensionless constant determined by specific heat ratio [gamma]. Typical values are listed in the Table below.
At just behind the shock wave (r=R), time evolution of physical quantities are described as follows.
Please refer THIS document (PDF file in Japanese) for detailed explanation.


"Fluid Mechanics (Course of Theoretical Physics, Vol. 6)", Landau and Lifshitz, 1959, Read Edu. & Prof. Publ. Ltd.
"Similarity and Dimensional Methods in Mechanics", L. I. Sedov, 1982, Mir Publ., Moscow.
"Uchu Ryutai Rikigaku (in Japanese)", Sakashita and Ikeuchi, Baifukan.