The Double Detonation of a Double-degenerate System, from Type Ia Supernova Explosion to its Supernova Remnant

Supernovae (SNe) mark the end of the life of stars, and are one key step in the life cycle of elements in the Galaxy. Type Ia SNe have attracted a lot of attention, for enabling the discovery of cosmic acceleration (see Garnavich 2017, for a review). SNe Ia are thought to be the thermonuclear explosion of a white dwarf (WD), although they come in many types (Jha et al. 2019) and their progenitor system(s) are still unclear (Ruiter 2020). Many scenarios have been proposed for the explosion (Hillebrandt et al. 2013), a basic idea being that the WD is part of a binary system so that it can increase its mass via accretion or merger.

There is no consensus on the nature of companion stars of exploding WDs. The companion star may be a nondegenerate star—the single degenerate scenario (Whelan & Iben 1973; Nomoto 1982), another WD—the double-degenerate scenario (Iben & Tutukov 1984; Webbink 1984), or the core of an asymptotic giant star—the core degenerate scenario (Kashi & Soker 2011). Recent observations have put some constraints on the single degenerate scenario. No red-giant star has been found in the pre-explosion images of SN 2011fe and 2014J (Li et al. 2011; Kelly et al. 2014, respectively), which are the nearest type Ia SNe observed in the past decade. No surviving stellar companions were convincingly detected in the remnants of nearby Ia SNe (see Ruiz-Lapuente 2019, for a review). On the other hand, some type Ia SNe may indicate the presence of nondegenerate companion stars, such as PTF11kx (Dilday et al. 2012), iPTF14atg (Cao et al. 2015), and SN 2012cg (Marion et al. 2016). These observational results suggest that type Ia SNe may have multiple origins (Hillebrandt et al. 2013).

The double-degenerate scenario is one of the promising scenarios for type Ia SNe. In this scenario, the exploding WD may be a near-Chandrasekhar-mass explosion with high central density, or a sub-Chandrasekhar-mass explosion with low central density. Near-Chandrasekhar-mass explosion has been thought to be difficult theoretically, because such a WD converts its composition from carbon–oxygen to oxygen–neon–magnesium (Saio & Nomoto 1985; Schwab et al. 2012; Ji et al. 2013), and collapses to a neutron star rather than exploding as a type Ia SN (Nomoto & Kondo 1991). Various sub-Chandrasekhar-mass explosion models have been suggested. Although the violent merger model can successfully produce type Ia SNe (Pakmor et al. 2011, 2012; Tanikawa et al. 2015), the total mass of two WDs should be larger than the Chandrasekhar mass (Sato et al. 2015, 2016), which means that the event rate is smaller than the type Ia SN rate (Maoz et al. 2014). Another model, called the "helium-ignited violent merger" or "dynamically driven double-degenerate double-detonation" (hereafter, D⁶) allows the total mass of the two WDs to be less than the Chandrasekhar mass (Guillochon et al. 2010; Pakmor et al. 2013). The D⁶ scenario is supported by the recent discovery of a hypervelocity WD by Shen et al. (2018b), which is expected to be left as a by-product of the explosion. Thus, the D⁶ model has become a promising model for type Ia SNe.

This has motivated many researchers to examine the double detonation of a sub-Chandrasekhar-mass WD with respect to its explodability, nucleosynthesis, and observability (e.g., Shen et al. 2018a, 2021; Gronow et al. 2020, 2021; Polin et al. 2019, 2021; Pakmor et al. 2021). These studies have mainly focused on the exploding WD, not the surviving WD. Papish et al. (2015) and Tanikawa et al. (2018, 2019) have shown that D⁶ SN ejecta exhibit nonspherical features because of the presence of the surviving WD. If such nonspherical features can survive until the remnant phase, it would be inconsistent with spherically symmetric objects. On the other hand, SN ejecta may become spherical through interaction with the interstellar medium (ISM). In addition, the chemical compositions of D⁶ ejecta would be different from those of SN Ia ejecta with a nondegenerate companion star. In the D⁶ scenario, ejecta can contain a significant amount of carbon and oxygen, stripped from the companion WD (Tanikawa et al. 2018, 2019). If the companion star is nondegenerate, the ejecta may contain hydrogen stripped from a main-sequence or red-giant star, or helium in the case the companion is a helium star (Marietta et al. 2000; Pan et al. 2012; Liu et al. 2013; Boehner et al. 2017). It is unknown whether this difference is observable or not after SN ejecta interact with the ISM.

The supernova remnant (SNR) is the phase following the SN, made from the interaction of the ejecta with the circumstellar and/or interstellar matter (Reynolds 2017). In a young, ejecta-dominated SNR, the shell of shocked matter is bounded by two shocks: the forward shock (FS) developing ahead of the supersonic ejecta, and the reverse shock (RS) forming inside the ejecta as they are decelerated. The shocked matter (shocked ejecta and shocked ISM) is heated to X-ray-emitting temperatures, for thousands of years (Vink 2017). The interface between the ejecta and the ISM, a contact discontinuity (CD), is unstable and subject to the Rayleigh–Taylor instability (RTI). This shapes the SNR independently of its initial conditions (the explosion) and boundary conditions (the ISM).

In this work we are specifically investigating the impact of the initial explosion on the SNR morphology. A D⁶ SN is rather symmetric, except for the marked imprint from the companion WD. In the study of thermonuclear explosions in binary systems, there has been a number of works that considered the interaction of the ejecta with the companion star, in terms of the impact on the SN emission, or the fate of the companion (e.g., Marietta et al. 2000; Pakmor et al. 2008; Kasen 2010; Liu et al. 2013; Maeda et al. 2014; Boehner et al. 2017; Dessart et al. 2020; Zeng et al. 2020; Liu & Zeng 2021). There have also been several searches for surviving companions in SNRs (Ruiz-Lapuente 2019), but looking for normal stars—an exception being Kerzendorf et al. (2018), who looked for a WD too, motivated by the theoretical study of Shen & Schwab (2017). A feature of the D⁶ scenario is that the surviving companion is itself a WD. ⁷ In contrast only a few works have looked into the signature of a companion on the SNR—and all assuming normal stars for the companion (Lu et al. 2011; Vigh et al. 2011; García-Senz et al. 2012; Gray et al. 2016). In this paper we investigate for the first time the imprints on the SNR with a WD companion and follow the evolution up to when all the ejecta have been shocked.

Our approach is to run numerical simulations from the 3D SN to the 3D SNR. We already applied it to the canonical model for a Type Ia SN: the explosion of a Chandrasekhar-mass WD (Ferrand et al. 2019, hereafter Paper I), examining the impact of the ignition and of the propagation of the flame (Ferrand et al. 2021, hereafter Paper II). In Paper I we showed that, assuming a uniform ambient medium, the impact of the SN on the SNR may still be visible after hundreds of years. Furthermore, in Paper II we showed that the details of the explosion mechanism matter: the more asymmetric ignition setups produce more asymmetric remnants, and in a way that is different for deflagrations versus detonations. In this paper we are investigating a different kind of Type Ia model: the double detonation of a WD in a double-degenerate system.

The paper is organized as follows. In Section 2 we summarize the methods for carrying out the numerical simulations and analyzing their results. In Section 3 we present the results on the SNR morphology, using a selection of representative maps. In Section 4 we discuss implications for observations of young, nearby SNRs. In Section 5 we conclude and outline our perspectives with the D⁶ model.

Our simulation is done in two steps, the SN explosion per se, already published, and the subsequent SNR evolution, presented in this paper.

The supernova—Our D⁶ SN model is described in detail in Tanikawa et al. (2018). The main model parameters are summarized in Table 1. The simulation was made with a smoothed-particle hydrodynamics (SPH) code, using a Helmholtz equation of state (Timmes & Swesty 2000), and coupled with the nuclear reaction network Approx13 (Timmes et al. 2000). As for the initial conditions, we considered a binary system with WDs of mass 1.0M_⊙ for the primary and 0.6M_⊙ for the secondary. According to Shen et al. (2018a; see their Figure 5) a 1.0M_⊙ WD produces about the mass of ⁵⁶Ni required to power a typical SN Ia, while lower masses would produce underluminous events and higher masses would produce overluminous events. As for the mass of the secondary, the way it impacts the explodability of the primary is not settled, but it should not affect the result of the explosion of the primary. The primary is made of 95% CO in its core and 5% He in its shell, while the secondary is pure CO with equal C and O, in mass (there is the possibility that the secondary also has an He layer). Although the exact composition of the primary WD does affect the yields somewhat, it does not alter the total kinetic energy of the ejecta (Sim et al. 2010), which controls the dynamics of the remnant phase. The two WDs orbit each other with a semimajor axis of 1.6 × 10⁴ km. The first detonation is ignited by adding a hot spot on the surface of the primary. The hot spot is located in the orbital plane of the binary system, and in the propagating direction of the primary WD. ⁸ The first detonation develops over the He layer, and folds over on itself in t = 1.25 s, making a splash at the opposite point. The shock is then channeled into the WD, and it converges at the center at t = 1.625 s, triggering the second detonation, the thermonuclear explosion that disrupts the primary WD. The explosion generates a kinetic energy of ≃10⁵¹ erg and synthesizes a mass of ⁵⁶Ni of 0.54 M_⊙.

Table 1. Model Parameters

primary's mass	M1 = 1.0 M⊙ (95% CO + 5% He)
secondary's mass	M2 = 0.6 M⊙ (100% CO)
orbital separation	d12 = 1.6 × 104 km
primary's orbital velocity	v1 = 1100 km s−1
secondary's orbital velocity	v2 = 1800 km s−1
angular size of secondary seen from primary	2θ2 = 44°, Ω2 = 0.46 sr
ejecta mass	Mej = 0.97 M⊙
total kinetic energy	${E}_{\mathrm{SN}}=1.11\times {10}^{51}$ erg
ISM density	ρISM = 0.1 mp .cm−3

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The interaction of the primary WD's ejecta with the secondary WD produces distinctive features in otherwise spherically symmetric ejecta, as seen in Figure 5 in Tanikawa et al. (2018) and our annotated Figure 1. The main feature is a wide shadow in the ejecta (solid angle Ω = 1.8 sr, of conical shape with half-opening angle θ = 445), a region of lower density and more irregular material, which was cast by the companion. A second feature is a narrow stream (Ω = 0.21 sr, θ = 148) of material that was stripped from the companion; in particular, low-velocity carbon and oxygen moving at ≃3000 km s⁻¹. Such low-velocity oxygen can be seen in the nebular phase of SN 2010lp (Kromer et al. 2013; Taubenberger et al. 2013) and iPTF14atg (Kromer et al. 2016), two SN 2002es-like subluminous type Ia SNe. Also visible in Figure 1 in the outer, less dense region, is an extended tail, along the line where the first detonation folded over on itself on the surface of the primary WD. Another feature of the D⁶ explosion is that the ejecta have a velocity shift of ∼1000 km s⁻¹. In our simulation the secondary WD survives the explosion of the primary (it may or may not explode; see Tanikawa et al. 2019). It has a velocity of ≃1700 km s⁻¹, that is, traveling 1 pc every 575 yr, in a direction perpendicular to the axis of the conical shadow, and it has a peculiar composition from ejecta contamination. An interactive 3D model of the ejecta is available online, ⁹ made from isocontours of the mass density, which shows the conical shadow, as well as the tail.

The supernova remnant—Results of the SN simulation at 50 s were mapped to a Cartesian grid of physical size 6 × 10¹¹ cm with spatial resolution 256³, and loaded into our custom version of the Eulerian hydrodynamics code Ramses (Teyssier 2002) to conduct the SNR simulations. The ability of the code to follow SNR dynamics was demonstrated in our earlier works focused on particle acceleration (Ferrand et al. 2010, 2012, 2014). For D⁶, the companion itself is not included in the SNR simulation (its size is ≲2 × 10⁹ cm, about the size of a single grid cell); only its impact on the ejecta is considered (the direct interaction between the companion WD and the ejecta is finished by that time). The method is the same as in Papers I and II. Considering the initial quasi self-similarity, we first scale up the hydro profiles from 50 s to 1 day, and start the SNR simulation at 1 day. We follow the evolution up to 4000 yr after the explosion, which is after the RS has shocked all of the ejecta and reached the SNR center. This is a more advanced age than in previous papers, because we observed longer-lasting features with D⁶. The SNR remains in an adiabatic phase of evolution, with our parameters the radiative snowplow phase would start at about 50,000 yr (Cioffi et al. 1988). The SNR simulation is performed in a comoving grid: factoring out the global expansion of the SNR over that period of time allows us to focus numerical resolution on the dynamics of the shocked ejecta. ¹⁰ For reference the box size is 4.5 × 10⁻⁴ pc at 1 day, 5.6 pc at 100 yr, 40 pc at 4000 yr. In order to separate the effects of the explosion and of any structure in the ambient medium, we assume a homogeneous ISM of number density 0.1 cm⁻³ to have dynamics roughly similar to Tycho's SNR. The overall hydrodynamic evolution of the SNR is controlled by three characteristic scales for the radius, velocity, and time (Dwarkadas & Chevalier 1998; Warren & Blondin 2013):

$$\begin{eqnarray}&&{r}_{\mathrm{ch}}={\left(\displaystyle \frac{3{M}_{\mathrm{ej}}}{4\pi {\rho }_{\mathrm{ISM}}}\right)}^{1/3}\simeq 4.54\,\mathrm{pc},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{u}_{\mathrm{ch}}={\left(\displaystyle \frac{2{E}_{\mathrm{SN}}}{{M}_{\mathrm{ej}}}\right)}^{1/2}\simeq 10\,700\,\mathrm{km}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{t}_{\mathrm{ch}}=\displaystyle \frac{{r}_{\mathrm{ch}}}{{u}_{\mathrm{ch}}}\simeq 412\,\mathrm{yr},\end{eqnarray} \tag{ 3 }$$

where M_ej is the mass of the ejecta, E_SN is the (kinetic) explosion energy, and ρ_ISM is the mass density of the ISM.

The analysis of the SNR morphology is performed as explained in the previous papers. At each step we track three surfaces of particular interest: RS, CD, FS. At run time we extract in 3D the surface of each wave front and record its radius from the explosion center. Treating the latter as a function on a sphere, we expand its relative variations in spherical harmonics and compute the angular power spectrum. The simulation was also done with smooth initial conditions (labeled 1Di, versus 3Di for the full initial conditions), made by averaging the mass density over all angles, in order to assess what happens from the SNR phase only.

Our simulation shows the expected dynamical evolution for a young SNR: the formation of a shell of dense and hot matter bounded by the RS and the FS. This is the region where thermal emission is produced in the X-ray band, which can be observed with space-borne telescopes. In Figure 2 we show a 3D view of the SNR shell, at an age of 500 yr for illustration. In this plot only the shocked matter is shown, the ejecta shocked by the RS (green hues), and the ISM shocked by the FS (purple hue). We use both isocontours and a volume rendering of the mass density. Part of the shell has been removed to show its interior. The typical field of Rayleigh–Taylor fingers is visible all around. The shell is globally symmetric except for a flat edge of the RS (at the bottom of the plot), with reduced RTI between the shocks. As we will see more clearly in the subsequent figures, this is the imprint from the shadow cast by the companion WD in the ejecta of the exploded WD. We also note a region with more marked fingers (toward the bottom right corner of the plot); this corresponds to the tail where the first detonation folded over on itself. Another 3D model of the SNR is available online, ¹¹ made from isocontours of the ejecta tracer, which shows the morphology of the RT fingers in the shell of shocked matter, and is interactive.

In the following, in order to explain the morphological evolution we use three different kinds of plots: (i) 2D slices of the mass density, of all the matter, to reveal the inner structure of the SNR (Figure 3); (ii) 2D projections along a line of sight of the density squared, which serves as a proxy of the thermal X-ray emission (Figure 4); and (iii) spherical maps and angular spectra, which show the surface of the wave fronts and quantify their angular content (Figures 5, 6, 7). To make it easier to identify the features from the explosion, we generated color-coded masks that show the shadow cone from the WD companion (green), and the tail where the first detonation converges at the surface of the exploding WD (blue), versus the regular ejecta (red). For each of the three series of figures listed above, the masks are generated in the same way as the maps: a slice (not time-dependent, as it is purely geometrical), a sum in projection weighted by the quantity of interest (slightly time-dependent, as the quantity may be evolving), the intersection with the wave fronts (slowly time-dependent, as the waves are moving). For the slice and projection maps, on each figure we show the results along three directions (the three principal axes x, y, z of our simulation box), so as to grasp the entire morphology of the SNR. By design the spherical maps show the entire surface of the SNR at once.

Slices—Slices of the mass density are shown in Figure 3 at select times, and a movie from 1 yr to 4000 yr by steps of 1 yr is available online. The projected position of the companion WD, assuming ballistic motion, is indicated by a white cross; it is still inside the SNR by the end of the simulation. Over time, the characteristic shell structure develops, bounded by the RS and FS. The RTI grows on top of the irregular ejecta boundary. While the instability is in the linear phase, the fingers grow exponentially in time, the resolution of our simulation does not allow us to probe this phase in detail. According to Schulreich & Breitschwerdt (2022), the instability saturates and enters the nonlinear phase by the time the extent of the perturbations has reached half their wavelength, which in our simulation happens within the first few years of the SNR evolution. For hundreds of years after, we see that the evolution of the RTI fingers is essentially self-similar. After about a thousand years, we see that the fingers start merging with one another.

Looking at features from the explosion, we note the persistence of a dense region in the center of the remnant. The shell is off-center, as a result of the initial velocity shift—so the geometric center of the remnant is not the explosion point. There are two other prominent features that stem from the explosion. First, the tail from the first detonation is visible past the average shock surface. At early times (around a hundred years after the explosion) it looks like a protrusion, being ahead of the average FS surface, but actually it is present from the beginning and is being overtaken by the bulk of the expanding ejecta. This feature is mostly erased after a few hundred years. Second, the conical shadow from the companion WD is visible as a deformed RS. In this underdense region the RS is traveling faster, the net result after a few hundred years is that the shock front looks segmented, with a straight edge across the shadow. Also we note that the edges of the cone have enhanced RT fingers growth.

The RS moves inward inside the ejecta, ¹² reaching the center at about t = 2690 yr. This time is expected to scale as Equation (3), that is, ${E}_{\mathrm{SN}}^{-1/2}{M}_{\mathrm{ej}}^{5/6}{\rho }_{\mathrm{ISM}}^{-1/3}$. In particular a lower/higher ambient density would mean a slower/faster evolution. After the RS has converged at the center, there is evidence of a rebound. This feature, which is robust in 3D (Petruk et al. 2021), will be studied in more detail in future work. The peculiar shape of the RS described above is visible all the way from its birth to its disappearance.

Projections—Projections of the density squared of the shocked ejecta, a proxy for their broadband thermal X-ray emission, are shown in Figure 4 at select times, and a movie from 1 yr to 4000 yr by steps of 1 yr is available online. Most of the small-scale structures visible on these maps come from the RTI. Over time the distance between the CD and the FS is increasing, and so in a comoving frame the ejecta appear to be shrinking in size. Only the shocked ejecta are shown, and so the central density peak is visible only when the RS reaches the center of the SNR. In projection, the tail from the first detonation is more difficult to see. The shadow from the companion is well visible; it appears as a dark disk surrounded by a brighter ring. The dark disk corresponds to an underdense region, while the bright ring corresponds to enhanced RTI. These features are actually getting more marked over time. They are visible until the rebound of the RS. As the shadow is a localized feature, the SNR morphology is different depending on the direction of observation. Along the x-axis one is looking almost along the conical axis and so one sees the entire darker disk, along the y-axis one is looking at some angle and sees a bright ellipse, and along the z-axis one is looking sideways and sees a bright bar on the edge.

Initially the shocked ejecta dominate the overall shocked mass and thus the thermal X-ray emission, but eventually the shocked ISM dominates in mass. ¹³ When adding the contribution for the shocked ISM using the same proxy of density squared summed along the line of sight, the D⁶ signature features are still visible at 500 yr, but become hidden at about 1000 yr. So one needs to separate the emission of the shocked ejecta and of the shocked ISM, which is possible using spatially resolved X-ray spectroscopy, as they have different compositions and thermodynamic states. In a subsequent paper focused on the thermal X-ray emission, we will present the separate contributions of the different media and of the different elements they contain. Furthermore, when particle acceleration happens, the induced nonthermal (synchrotron) emission traces the outline of the FS at X-ray wavelengths.

Wave fronts—The SNR shell is bounded by two shocks, the RS and the FS, while the ejecta and the ISM are separated by one interface, the CD. Spherical maps and angular spectra of the location of CD, RS, FS are shown in Figures 5, 6, 7, respectively, at select times, and movies from 1 yr to 4000 yr by steps of 1 yr are available online. ¹⁴ The spherical maps allow us to see at a glance the entire surface considered. The angular spectra, as a function of the angular wavenumber ℓ, allow us to quantify which angular scales contribute to the overall morphology of the SNR. For each wave front we observe SN modes at large scales (small ℓ) that decay quickly in time, except for a permanent dipole (ℓ = 1, meaning two-sided), which comes from the velocity shift from the binary motion. For the CD, we see that the RTI is growing from the smallest scales (largest ℓ) to larger scales (smaller ℓ), as expected. In the first part of the simulation, the RTI spectral distribution is fairly symmetric, with a central peak slowly shifting to lower ℓ. After about 1000 yr, the distribution gets increasingly skewed toward lower ℓ, as finger growth is now driven by mergers. Maps for the RS and the FS are simpler than the ones for the CD, although they bear some imprint of the RTI as well: the feet of the fingers for the RS, the tips of the fingers for the FS. The tail from the first detonation is visible at the beginning. Then the ring from the companion shadow appears, seen on the spectra in the low-ℓ modes.

As a final step, we aim to quantify which of the initial conditions (progenitor system and explosion mechanism) and of the ejecta–ISM interaction (growth of the RTI) shape the SNR at a given age. The summed angular power in the wave front deformations is shown in Figure 8 as a function of time. The top panel shows the evolution until the simulation end time 4000 yr, for each of CD, RS, FS, for both 1Di and 3Di cases. The 1Di case uses smooth initial conditions. For the FS and RS nothing is expected to happen and so this is merely a test of the numerical precision; for the CD this case shows the effect of purely the RTI. The 3Di case uses the actual initial profiles, and shows the imprint of the explosion. On the angular spectra, we see that the SN modes (at large scales) and the SNR modes (at small scales) can be separated for the first few hundred years around ℓ = 10; the power in these two separate ℓ bands is plotted in the bottom panel, for the CD only, up to 500 yr. At small scales (excluding the permanent dipole, thin lines) the evolution is similar in the 1Di and 3Di cases, which comforts us in the fact that it corresponds to RTI growth. At large scales (thick lines), in 1Di nothing happens as expected, while in 3Di a decay is visible, strong initially but not reaching zero level. Looking at the 3Di case (solid lines), the low-ℓ and high-ℓ curves for the CD intersect at 85 yr. It means that, after that time, most of the angular power in the ejecta morphology is coming from the SNR RTI, which is in its nonlinear phase of self-similar growth. However signatures of the explosion are still visible on the maps at much later ages, because they are localized in space, stable in time features. This is in contrast with our previous studies in Papers I and II with the N series of models, for which signatures were seen as statistical deviations across the entire SNR surface.

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We have presented for the first time the evolution and morphology of a D⁶ SNR, up to 4000 yr after the explosion, which with our chosen ISM density is after all the ejecta have been shocked. In this section we discuss implications for the interpretation of observations of SNRs.

4.1. Other Effects

4.2. Comparison with Previous Works

4.3. Possible Target SNRs

In this paper, for the first time we have followed a "helium-ignited violent merger" or "dynamically driven double-degenerate double-detonation" (D⁶) SN model into the SNR phase, up to 4000 yr after the explosion. We have analyzed the structure of the SNR using a variety of representations: 2D slices, 2D projections, 3D contours, and their angular variations. We have found that a D⁶ progenitor system and explosion leave clear signatures on the SNR:

• 1.  
  the first detonation produces an ejecta tail, which at early times looks like a protrusion from the shell;
• 2.  
  the second detonation leaves a central density peak, which is revealed in X-rays when the RS reaches the center;
• 3.  
  because of the initial velocity shift, the SNR shell is off-center at all times, which shows as a strong dipole component in the angular spectra;
• 4.  
  the companion star generates a conical shadow in the ejecta, which is visible in projection as a dark patch surrounded by a bright ring.

Basically we found that the specific 3D structure of the explosion is preserved in the SNR phase. The features from the first detonation and from the companion are localized, and so the way they look depends on the direction of observation, producing various SNR morphologies. However, as we see all the shocked material in projection in X-rays, they should be visible to some degree along any orientation. The features from the shadow are long-lasting; they could in principle be detected in the shocked ejecta up to the time when the RS rebounds at the center of the SNR, which happens just short of 2700 yr after the explosion for an assumed density of 0.1 cm⁻³. The conical shadow is visible on the SNR shell as a brighter ring encompassing a darker disk. The ring is a region of overgrowth of the RTI, while the disk is a region of undergrowth of it. This rather unusual SNR morphology is obtained even in a uniform ambient medium. So we point out that observing an irregular Type Ia SNR does not necessarily implies an inhomogeneous medium.

We emphasize that we have only one realization of the D⁶ model. The size of the shadow may depend on the strength of the initial interaction between the ejecta and the companion, which may depend on the masses and explosion energy. The relative orientation between the tail from the first detonation and the conical shadow from the companion may also vary. For instance, the tail may happen to be aligned with the shadow, which may affect its evolution. Our work, which shows that the initial configuration matters in latter phases, should entice modelers to do more exhaustive studies of the D⁶ scenario.

In a follow-up paper, we will conduct a more advanced study of the thermodynamical state of the plasma and its thermal emission, in order to allow for more precise comparisons of our simulation results with spatially and spectrally resolved X-ray observations. Our additional perspectives with D⁶ modeling include, for the early phase, to study the nebular emission by means of radiative transfer calculations, and for the later phase, to study the visibility of the RS at and after its rebound. The later point warrants further study regarding its observability in SNRs, not just for D⁶. Finally we would like to encourage searches for WDs in SNRs; D⁶ is an example of a thermonuclear explosion with a surviving companion that is not a normal star.

This research has been supported by Grants-in-Aid for Scientific Research (16K17656, 19K03907). SN acknowledges the support by JSPS Grants-in-Aid for Scientific Research "KAKENHI" (A: grant No. JP19H00693) and Pioneering Program of RIKEN for Evolution of Matter in the Universe (r-EMU). This work was funded in part by the Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS, https://ithems.riken.jp) program at RIKEN. S.S.H. acknowledges support by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canadian Space Agency. We thank the anonymous referee for comments that helped clarify the manuscript.

Facilities: Simulation for the SN explosion was done on Oakforest–PACS at Joint Center for Advanced High Performance Computing and on Cray XC50 at Center for Computational Astrophysics - , National Astronomical Observatory of Japan. Simulations for the SNR evolution were performed on the iTHEMS clusters at RIKEN. -

Software: HEALPix (Gorski et al. 2005), SciPy (Virtanen et al. 2020), Matplotlib (Hunter 2007), VisIt (Childs et al. 2012).