Energy of Tycho's Supernova Remnant is increasing with time

Abstract

It is shown, using the Zeldovich integral relations, that the energy of Tycho's Supernova Remnant is strongly growing with time, approximately as t^1/3. This growth can be attributed to the exothermic reactions going inside the remnant. The use of the assumption of the adiabaticity of the motion inside of the shock front, and no losses or gain of energy at the front, seems therefore unjustified.

Danish astronomer Tycho Brahe observed in 1572 and studied the explosion of a star that became known as Tycho's Supernova. Modern observations of Tycho's Supernova Remnant (SNR) showed that it forms an expanding sphere filled by high-temperature debris. The most accurate measurements of the expansion of the SNR are, as I was informed by R. D. Blandford, those presented in the article by E. M. Reynoso et al. (1). In particular, in this work, the power-law index (expansion parameter) ν = d ln R/d ln t = 0.471, where R is the radius of the shock front and t is time, was evaluated by processing the experimental data. The authors emphasize in ref. 1 that this value is in excess of the value ν = 0.4 that was obtained theoretically under the assumption of adiabaticity of the flow inside of the shock wave and no radiation losses at the front. I remind the reader that the value ν = 0.4 was obtained by G. I. Taylor (2, 3), J. von Neumann (4, 5), and L. I. Sedov (6, 7) for a very intense and concentrated explosion in a medium without radial density gradient; i.e., with constant density ρ₀ ahead of the shock front and adiabatic flow behind the strong shock wave. Here, “strong” means that the pressure at the shock front is much larger than the initial pressure. Under these assumptions, the energy inside of the shock front remains constant.

The positive excess of the expansion parameter ν over the value of 0.4 obtained in the paper by Reynoso et al. (1) shows that the energy inside of the shock front is increasing with time because of some exothermic processes. We demonstrate below, using the similarity method and integral relations derived by Ya. B. Zeldovich in the foreword to the author's book (ref. 8; reproduced also in ref. 9), that the results presented in ref. 1 allow us to evaluate the law of the energy E growth with time, which appears to be close to E ∼ t^1/3.

Basic Assumptions and the Model

We assume that both the energy variation rate and the expansion rate depend only on the instantaneous values of energy and radius. By the energy, we mean, naturally, the gas-dynamic part of energy with exclusion of the nuclear form. To operate with kinematic quantities, we introduce, instead of the energy E, the kinematic energy ℰ = E/ρ₀, where ρ₀ is the initial density. Therefore, dℰ/dt = f₁(ℰ, R) and dR/dt = f₂(ℰ, R).

The dimensions of the quantities ℰ and R are equal, obviously, to [ℰ] = L⁵T⁻², [R] = L, where L and T are, respectively, the dimensions of length and time. Dimensional analysis [see, e.g., the author's books (8–10)] gives

The equations shown here as Eqs. 1 and 2 were proposed by Zeldovich in his foreword to the author's book (ref. 8; also reproduced in ref. 9, pp. XVIII–XIX). He was motivated by the intention to give a simplified presentation (“for pedestrians,” as he said) of the work by the present author and G. I. Sivashinsky (11) concerning a model of the influence of the radiation losses at the shock front in a concentrated, very intense explosion (also presented in ref. 8). In the equation presented here as Eq. 1, Zeldovich introduced a minus sign because the energy loss, not gain, was the subject of his analysis. Zeldovich's equations (here, Eqs. 1 and 2) will be the basis for our model.

The Evaluation of the Energy Evolution with Time

According to ref. 1, the observed law of the SNR expansion can be represented in the form

where δ > 0 is the increase in the expansion parameter ν over the adiabatic value 0.4. We substitute Eq. 3 into Eq. 2 and obtain, easily,

Furthermore, substituting Eq. 3 into Eq. 1, we obtain the relation between α, β, and δ:

Eq. 6 shows that at small δ, α is proportional to δ, so if α = 0, ℰ is constant, and we return to the adiabatic law of R ∼ t^2/5. In the case of SNR expansion, the situation is different, δ = 0.071; therefore, the final expression for the energy takes the form

So, it is important that δ is multiplied in the exponent of time in Eq. 4 by a large factor, 5.

Discussion

As we see, at the stage reported in ref. 1, the energy in the SNR is substantially growing with time. It can be explained by exothermic reactions taking place in the SNR. No transition to the adiabatic stage can be seen from the results presented in ref. 1.

In the recent paper by Cassam-Chenaï et al. (12), the role of radiation at the shock wave is, in particular, emphasized. Apparently, the exothermic reactions accelerating the shock front are also concentrated near the front, so that the scheme of the adiabatic flow inside of the shock wave with energy gain or losses at the front can also be an explanation.

It can be anticipated that, at a later stage, when radiation losses will outweigh energy generation, the front will decelerate and δ will become negative, so that the total energy of the SNR will decay, after passing through the adiabatic stage. However, at the present time, the use of the assumption of the adiabaticity of the motion inside of the shock front, and no gain or losses of energy at the front, seems unjustified.