Thermonuclear Explosions of Chandrasekhar Mass White Dwarfs

Abstract

The thermonuclear explosion of a Chandrasekhar mass white dwarf is an important class of supernovae which can attribute to various subclasses of Type Ia supernovae and accretion induced collapse. Type Ia supernovae are not only essential as their roles of standard candle in the discovery of dark energy, but also robust sources of iron-peak elements for the galactic chemical evolution. In this chapter we discuss the physics of the explosion mechanisms of the Chandrasekhar mass white dwarf. First we review the possible evolutionary paths for the accreting white dwarf to increase its mass to the Chandrasekhar mass in the binary systems. When the white dwarf’s mass reaches near the Chandrasekhar limit, carbon burning is ignited and grows into deflagration in the central region. We review the principle component of deflagration physics and how it is implemented in supernova simulations. We then review the physics of detonation by examining its structure. We also discuss how the detonation is triggered physically and computationally. At last, we describe how these components are applied to various explosion mechanisms, including the deflagration-detonation transition, pure deflagration, and gravitationally confined detonation. Their typical behaviour, nucleosynthesis, and applications to the galactic chemical evolution and observed supernovae are examined.

Appendices

Appendix: A Short Review of Detonation Physics

1.1 Deflagration to Dedonation Transition

In the previous section why the deflagration phase is necessary and how to implement the flame physics in SNe Ia simulation are discussed. In this section the motivation of including detonation in the framework, its background physics, and the implementation technique are further discussed.

The multidimentional turbulent flame model attempts to introduce a flame-acceleration scheme which allows the flame to burn more material before the expansion of the WD quenches the flame. It is found that the multidimentional PTD model predicts a significant amount of unburnt carbon and oxygen in the central region. As a result, the explosion energy inclines to the weaker side of the observed explosion.

If the transition of deflagration to detonation occurs, the above problem might be solved. Detonation wave is the propagation of burning through shock compression. Unlike the deflagration counterpart, the supersonic motion of detonation can ensure that all necessary material in WD is burnt before the density becomes too low to sustain nuclear burning. This also bypasses the inconsistency of pure detonation model which tends to overproduce⁵⁸Ni and⁵⁴Fe due to electron capture (see Figs. 9 vs. 10, and Figs. 16 vs. 14) because in DDT model, deflagration part is confined in small mass region and the detonation mostly burns the outer material, which has too low density for electron capture to take place.

1.2 Physics of Detonation and Transition

The detonation in general consists of three parts, the pre-shock region, the reaction zone, and the post-reaction zone region. To study the detonation structure, one usually solve the eigenstate(s) for the steady-state detonation wave structure equations (Sharpe 1999).

By assuming the matter remains in thermodynamics equilibrium, that

$$\displaystyle{ \Delta \varepsilon = \frac{\partial \varepsilon } {\partial \rho }\vert _{T,X_{i}} + \frac{\partial \varepsilon } {\partial T}\vert _{\rho,Y _{i}} +\sum _{i} \frac{\partial \varepsilon } {\partial Y _{i}}\vert _{\rho,T}, }$$

(4)

the steady-state Euler equation can be written as

$$\displaystyle\begin{array}{rcl} \frac{d\rho } {dx}& =& -\frac{\rho a_{f}^{2}} {v} \frac{\boldsymbol{\sigma }\cdot \mathbf{R}} {\iota },{}\end{array}$$

(5)

$$\displaystyle\begin{array}{rcl} \frac{dT} {dx}& =& \left ( \frac{\partial p} {\partial T}\right )_{\rho,Y }^{-1}\left \{\left [u^{2} -\left (\frac{\partial p} {\partial \rho } \right )_{T,Y }\right ] \frac{d\rho } {dx} -\sum _{i=1}^{N}\left ( \frac{\partial p} {\partial X_{i}}\right )_{\rho,T,Y _{j\neq i}}\frac{dY _{i}} {dx} \right \},{}\end{array}$$

(6)

$$\displaystyle\begin{array}{rcl} \frac{dY } {dx} & =& \frac{R} {v},{}\end{array}$$

(7)

where

$$\displaystyle{ \eta = a_{f}^{2} - v^{2} }$$

(8)

is the sonic parameter,

$$\displaystyle{ a_{f}^{2} = \left (\frac{\partial p} {\partial \rho } \right )_{T,Y } + \left [\frac{p} {\rho ^{2}} -\left (\frac{\partial \varepsilon } {\partial \rho }\right )_{T,Y }\right ]\left ( \frac{\partial p} {\partial T}\right )_{\rho,T}\left ( \frac{\partial \varepsilon } {\partial T}\right )_{\rho,Y }^{-1} }$$

(9)

is the sound speed of constant composition (also known as frozen sound speed in the literature of detonation), and

$$\displaystyle\begin{array}{rcl} \sigma _{i} = \frac{1} {\rho a_{f}^{2}}\left \{\left ( \frac{\partial p} {\partial Y _{i}}\right )_{\rho,T,Y _{j\neq i}} -\left ( \frac{\partial p} {\partial T}\right )_{\rho,Y }\left ( \frac{\partial \varepsilon } {\partial T}\right )_{\rho,Y }^{-1}\right.& & \\ \left.\left [\left ( \frac{\partial \varepsilon } {\partial Y _{i}}\right )_{\rho,T,Y _{j\neq,i}} -\left ( \frac{\partial q} {\partial Y _{i}}\right )_{Y _{j\neq i}}\right ]\right \}& &{}\end{array}$$

(10)

is the thermicity constant, such that $\boldsymbol{\sigma }\cdot \mathbf{R}$ is the thermicity.

It should be noted that at Eq. (7), the denominator η can bring subtlety to the calculation. In Chapman-Jouget detonation, η is always positive that the solution is continuous everywhere. However, for realistic equation of states and network, η can change sign. It corresponds to the point that the fluid velocity equals to the frozen speed of sound. At this point, there are two solutions for the detonation. First, by direct integration, the zone beyond that points has supersonic velocity. This corresponds to self-sustained detonation wave. Second, the reaction zone remains to be subsonic everywhere. This produces cusps in both density and temperature at that point, so that the solution remains continuous while satisfying the above equations.

In general, only the second solution represents the stable detonation wave which occurs in SNe Ia. In Fig. 23 the density and temperature of a typical detonation wave is plotted. After the shock, there is a buffer zone which allows the temperature to increase. Once the matter reaches 4 × 10⁹ K, the burning of carbon and oxygen becomes explosive that the temperature can be doubled within a few 10⁻² cm. At about 0.1 cm, the drop of density has significantly led to a jump in the fluid velocity, due to mass conservation. This makes the wave reach the frozen sound speed. At that point, the solution to the pathological detonation is connected, which ensures the ash propagates subsonic everywhere. The density reaches its equilibrium ∼ 1 cm, while the temperature reaches equilibrium at about 10² cm. In Fig. 24 the abundance profile for the same detonation wave is plotted. Similar to deflagration, at x < 10⁻² cm,¹²C burns to form²⁰Ne and⁴He. At 10⁻² < x < 10⁻¹ cm, both carbon and oxygen burning produce intermediate mass isotopes such as³²S,³⁶Ar,⁴⁰Ca, and⁴⁴Ti. At 10⁻² < x < 10² cm, the matter slowly converts to NSE that iron-peak isotopes, including⁴⁸Cr,⁵²Fe, and⁵⁶Ni form. The matter reaches equilibrium and no net change is observed beyond x > 10² cm.

Fig. 23

Upper panel: The density profile of the detonation wave at density 10⁹ g cm⁻³. The detonation is assumed to start with a post-shock temperature 3. 5 × 10⁹ K with a composition 50 % ¹²C and 50 % ¹⁶O by mass. Lower panel: Same as above, but for the temperature profile

Fig. 24

The chemical profile of the detonation wave for a detonation wave at density 10⁹ g cm⁻³

In SNe Ia simulations, the detonation energy and composition table need to be computed prior to the hydrodynamics simulations. This is because the table includes solving the equations for the detonation wave structure in order to find the energy release, propagation velocity for the pathological detonation, and ash composition as a function of density. In general, it depends on temperature as well. Owing to the electron degeneracy and that the nuclear binding energy change is much larger than the matter internal energy, the exact yield is less sensitive to the choice of temperature than that of density. After that, similar to the deflagration, the front is tracked by some discontinuity tracking scheme. By extracting the geometric properties of the front, corresponding energy and composition of the fluid swept by detonation wave are changed.

One technical difference between deflagration and detonation is that detonation does not start at the beginning of the simulation and requires certain trigger. In practice, detonation is assumed to start when the local Karlovitz number Ka ≥ 1, namely, the ratio between turbulence length scale and the flame width. When it is satisfied, the eddies around the thick flame becomes important to diffuse the heat from the hot ash to the cold fuel and cease the explosive burning. The hot region can carry out carbon burning simultaneously, creating a supersonic pulse and the shock. The shock then develops into detonation and burns the remaining fuel.

To demonstrate the technique in carrying out detonation physics in Type Ia supernova simulations, a two-dimensional hydrodynamics simulation for the explosion phase of a carbon-oxygen core is presented. The configuration is similar to the model in Sect. 4.1. In Fig. 25 the energy evolution is plotted. The phase before DDT that has started is exactly the same as the PTD model since the same initial model and flame physics are used. But once DDT is triggered, the two models deviate. The total energy increases much faster to a much higher equilibrium value. It also leads to the global heating of matter, as shown by the increase of internal energy. Kinetic energy also continues to grow, in contrast to the asymptotic behavior as shown in the PTD counterpart.

Fig. 25

The time evolution of the total, kinetic, internal, and potential energy of the same model as in Fig. 13

1.3 Open Questions in Detonation

It should be noted that there are two outstanding questions in the detonation transition remain unresolved. In one way, detonation transition is shown to be possible in the form of shock compression in a closed system and by turbulent compression in an open system. Certainly, the environment of a WD belongs to the latter one, where turbulent diffusion is relied to generate a smeared hot spot which can undergo supersonic heating. However, in the large-eddy simulations, it is shown that the typical turbulence strength is only marginally strong to diffuse the thermal energy. In terms of power spectrum, the probability of finding a fluid element with the required velocity fluctuation is small. Certainly, in most SNe Ia simulation, subgrid turbulence are used to estimate the turbulent kinetic energy. This points to the uncertainty in the subgrid turbulence model. Future work in how to achieve a robust turbulence model will offer important insight to the feasibility of the DDT model. Second, the exact Karlovitz number for DDT is not exactly known. In SNe Ia simulation in the literature, typical value of Ka ≈ 1. However, in direct numerical simulation of DDT for the H₂-air flame, at least Ka = 100 is required. Certainly, a one-one correspondence between the H₂-air flame and the carbon-oxygen flame cannot be drawn straightforwardly due to the huge differences in the equation of states and reaction channel. Despite that, the terrestrial flame experiment has demonstrated that the detonation transition can be much harder than one has assumed. It is therefore necessary to understand the critical Ka for DDT transition for a carbon-oxygen WD and if it can be achieved in hydrodynamics simulation.

Cross-References

• Electron Capture Supernovae from Super Asymptotic Giant Branch Stars

• Evolution of Accreting White Dwarfs to the Thermonuclear Runaway

• Nucleosynthesis in Thermonuclear Supernovae

• Type Ia Supernovae

• Type Iax Supernovae