Interaction of Type Ia Supernovae With The Circumstellar Environment

Type Ia Supernovae and their Environment: Theory & Applications to SN 2014J

Paul Dragulin11affiliation: Department of Physics, Florida State University, Tallahassee, FL 32306, USA; pd09@my.fsu.edu , Peter Hoeflich22affiliation: Department of Physics, Florida State University, Tallahassee, FL 32306, USA; phoeflich77@gmail.com
Abstract

We present theoretical semi-analytic models for the interaction of stellar winds with the interstellar medium (ISM) or prior mass loss implemented in our code SPICE 111Supernovae Progenitor Interaction Calculator for parameterized Environments, available on request, assuming spherical symmetry and power-law ambient density profiles and using the -theorem. This allows us to test a wide variety of configurations, their functional dependencies, and to find classes of solutions for given observations.

Here, we study Type Ia Supernova (SN Ia) surroundings of single and double degenerate systems, and their observational signatures. Winds may originate from the progenitor prior to the white dwarf (WD) stage, the WD, a donor star, or an accretion disk (AD). For explosions, the AD wind dominates and produces a low-density void several light years across surrounded by a dense shell. The bubble explains the lack of observed interaction in late time SN light curves for, at least, several years. The shell produces narrow ISM lines Doppler shifted by 10-100 , and equivalent widths of and in case of ambient environments with constant density and produced by prior mass loss, respectively. For SN 2014J, both mergers and mass explosions have been suggested based on radio and narrow lines. As a consistent and most likely solution, we find an AD wind running into an environment produced by the RG wind of the progenitor during the pre-WD stage, and a short delay, to , between the WD formation and the explosion. Our framework may be applied more generally to stellar winds and star-formation feedback in large scale galactic evolution simulations.

Subject headings:
Supernovae: Type Ia: Circumstellar: Environment: Interaction : SN2014J
slugcomment: Accepted for publication by the Astrophysical Journal

1. Introduction

Type Ia supernovae (SNe Ia) allow us to study the Universe at large and have proven invaluable in cosmological studies, the understanding of the origin of elements, and they are laboratories to study physics such as: hydrodynamics, radiation transport, non-equilibrium systems, nuclear and high energy physics. The consensus picture is that SNe Ia result from a degenerate C/O white dwarf (WD) undergoing a thermonuclear runaway (Hoyle & Fowler, 1960), and that they originate from close binary stellar systems. Potential progenitor systems may either consist of two WD, the so called double degenerate system (DD), or a single WD along with a Main Sequence (MS) or Red Giant (RG) star, the so called single degenerate system (SD).

Within this general picture for progenitors, four classes of explosion scenarios are discussed which are distinguished by the mechanism which triggers the thermonuclear explosion: (1) Within the Chandraskhar mass models, the explosion is triggered by compressional heat close to the center when the WD approaches . The accretion material may originate either from a RG or MS companion via Roche-lobe overflow, or from tidal disruption of another WD in an DD system. (2) In a second class, the explosion is triggered by heat released during the dynamical merging (DM) of two WDs of a DD system. (In some cases, the combination may result in what is known as an accretion induced collapse). For overviews, see Branch et al. (1995); Nomoto et al. (2003); Di Stefano et al. (2011); Di Stefano & Kilic (2012); Wang & Han (2012); Hoeflich et al. (2013). (3) Recently, a third trigger mechanism has been revived known as double detonation scenario or He-detonation in a sub- mass WD (Nomoto, 1982a; Livne, 1990; Woosley & Weaver, 1994; Hoeflich & Khokhlov, 1996; Kromer et al., 2010; Woosley & Kasen, 2011). In this picture, a C/O WD star accretes He-rich material at a low rate to prevent burning. Explosions are triggered from ignition of the surface He layer with masses of about a few hundredths to 0.1 . The resulting strong shock wave may trigger a detonation of underlaying C/O. Previous calculations produced a few of which is inconsistent with the early time spectra when the photosphere is formed within (Höflich et al., 1997). Modern recalculations have utilized a smaller He shell mass and obtain better agreement with observations (Kromer et al., 2010; Woosley & Kasen, 2011), though, the problem with the outer layers still persists. Recent studies of helium detonations including curvature and expansion effects may be in better agreement with the observations (Moore et al., 2013; Ruiter et al., 2014; Zhou et al., 2014). For this class of explosions, we may expect accretion disk winds similar to those in scenarios as discussed below. (4) Another explosion path, also recently set forth by Kashi & Soker (2011), is known as the core-degenerate (CD) scenario. They suggested that the merger between a WD and the hot core of an AGB star could take place within a common-envelope (CE) phase of binary evolution. The rotation of the core slows down by emitting magnetic dipole radiation until the angular momentum is insufficient to prevent collapse and susequent SN Ia. Binary/envelope interaction is expected to produce wind during a very short phase, similar to a planetary nebula (PN) mass loss phase. It is not obvious that the properties would resemble a typical SNe Ia. Depending on the delay time from merging to explosion, the merged pair might explode within the PN shell or in a lower density ISM (Tsebrenko & Soker, 2015).

The stellar environment will shed light on the evolutionary history of the progenitor, supernovae light curves (LCs), and spectra, with X-rays and radio emission being the probes (e.g. Chugai & Danziger (1994); Dwarkadas & Chevalier (1998); Chevalier & Irwin (2012); Chandra et al. (2012); Fransson et al. (2014)). As discussed below, the density limits for the environment of a typical SNe Ia are well below those of the solar neighborhood and one of the goals is to probe whether SD and/or DD systems may create this environment.

In the case of DD progenitors, we may expect long evolutionary time scales after the formation of the WDs compared to the accretion phase in and the double detonation scenarios. The time scales depend on the unknown initial separation and mass of the binary WDs and the decay of the orbits due to gravitational waves (possibly modified during a common envelope phase). The time scale of angular momentum loss by gravitational waves scales with the fourth power of the separation (Landau & Lifshitz, 1971). For example, the orbits of two 1 WDs at will decay in representing a period with no or little mass loss. However, we may expect wind just prior to the explosion when the WDs fill their Roche lobe. The size of the Roche lobe corresponds to a separation of (Eggleton, 1983) which translates to mass loss at most some months prior to the merging (Han & Webbink, 1999), and material close to the system which will be quickly overrun by the SN material. In DD, therefore, we expect no ongoing wind with the exception of a brief period just prior to the dynamical merging. Thus, the environment of a DD system may be dominated by the ISM the system has moved into which depends on its peculiar velocity and the delay between WD formation and explosion. Mannucci et al. (2006) argued that the observed evidence of SNe Ia rates favors a bimodial distribution of the delay times between star formation and explosion with about 50 % of all explosions take place after and , respectively. Using the star formation rates and assuming that all SNe Ia originate from DD systems, Piersanti et al. (2009) concluded that 50 % of all DD systems explode within , with a long tail to about . Recent studies show that the distribution of delay times is more continuous (see Maoz et al. (2014) and references therein). It is likely that the DD system moved far away from its birthplace and that the explosion happens in the (constant density) ISM . In most cases, we may expect a low density environment consistent with observations.

The environment of SD systems can be expected to consist of three main components: 1) Some matter bound in the progenitor system at the time of the explosion that may originate from the accretion disk or be shed from the donor star; 2) the wind from the WD, accretion disk or donor star; and 3) the interstellar medium (ISM).

Within the scenario of WD explosions, hydrodynamic calculations have shown that the expanding supernova ejecta wraps around the companion star and may pull off several tenths of a solar mass of material in case of a RG donor (Marietta et al., 2000; Kasen, 2010). Besides the donor star, another source of matter is the accretion disk material (Gerardy et al., 2003) lifted during a pulsational phase during the explosion, or debris from the merging of two WDs (Hoeflich & Khokhlov, 1996; Quimby et al., 2007a). There has been some observed evidence for interaction between the explosion and the immediate environment. Although H-lines like in SN 2002ic are rare, a common feature is a high-velocity line which, first, was prominently seen in events like SN1995D, SN 2001el, SN2003du, and SN 2000cx, a feature present in almost all SNe Ia (Hatano et al., 2000; Fisher, 2000; Wang et al., 2003; Folatelli et al., 2013; Silverman et al., 2015). This line may be attributed to the material even of solar metallicity bound or in close proximity to the progenitor system 222 emission is suppressed by collisions, charge exchange and low ionization of hydrogen in absense of a strong, ongoing interaction with a Red Giant wind, see also Sect. 4.. (Gerardy et al., 2004; Quimby et al., 2007b).

At intermediate distances of up to several light years, in the case of explosions, the environment may be dominated by the wind from the donor star, the accretion disk or, for high accretion rates, the wind from the WD, or the interstellar material (ISM). A number of possible interaction signatures has been studied, including X-rays, Radio, and narrow H and He lines, but no evidence has been found, with an upper limit of for the mass loss (Chugai, 1986; Schlegel & Petre, 1993; Schlegel, 1995; Cumming et al., 1996; Chomiuk et al., 2012a). In late-time light curves, interaction should result in excess luminosity but, in general, is not seen. No sign of an interaction has been found even in SN1991T, which has been observed up to day 1000. This implies particle densities less than (Schmidt et al., 1994).

At large distances, from few tenths to several light years, the environment is determined by the ISM. It is known that Type Ia SNe generally explode away from star forming regions (Wang et al., 1997). This can be partly attributed to the long stellar evolutionary lifetimes of the low-mass stars in the progenitor systems, allowing them sufficient time to move away from their place of birth. It is also known that SNe Ia occur in elliptical and spiral galaxies, including galactic disks, the bulge and the halo. One may expect the explosion to occur in ISM particle densities of (Ferrière, 2001). Light echos from SNe Ia have been used to probe their environments, and showed that many SNe Ia have circumstellar dust shells at distances ranging from a few up to several hundred parsecs (Hamuy et al., 2000; Rest et al., 2005; Aldering et al., 2006; Patat et al., 2007b; Crotts & Yourdon, 2008; Wang et al., 2008; Rest et al., 2008).

Most evidence for a link between SNe Ia and their environment comes from the observations of narrow, time-dependent, blue-shifted NaID and KI absorption lines which, for a significant fraction of all SNe Ia, indicates outflows (D’Odorico et al., 1989; Patat et al., 2007a; Blondin et al., 2009; Foley et al., 2012; Sternberg et al., 2011; Phillips et al., 2013). In addition, extinction laws derived from SNe Ia seem to be different from the interstellar medium in our galaxy, suggesting a component linked to the environment of SNe Ia rather than the general host galaxy (Cardelli et al., 1989; Krisciunas et al., 2000; Elias-Rosa et al., 2006; Nordin et al., 2011; Pastorello et al., 2011). Possibly, the hydrodynamical impact of the SN ejecta will produce additional emission and may modify the outer structure of the envelope and, thus, the Doppler shift of spectra features. Light emitted from the photosphere of the supernovae may heat up matter in the environment, which, in turn may change the ionization balance or the dust properties (Raymond et al., 2013; Krügel, 2015; Slavin et al., 2015; Patat et al., 2015a). We note that this effect for different dust formation in the host galaxy may lead to extinction laws different from the Milky Way as commonly observed in SNe Ia (Goobar, 2008; Krisciunas et al., 2003; Folatelli et al., 2010; Kawara et al., 2011; Burns et al., 2014). The dust properties may effect the light echoes which could in turn change the extinction laws (Wang, 2014).

The following picture of the environment emerges: SNe Ia are surrounded by a cocoon with a much lower environment than the ISM. Sometimes, narrow ISM lines indicate clumps or surrounding shells. However, the diversity of supernova and progenitor channels leaves open a huge parameter space which cannot be covered by numerical simulations. To cover the parameter space, we use a semi-analytic approach similar to those developed by Weaver et al. (1977) for stellar wind/ISM interaction, Chevalier & Imamura (1983) for stellar wind/stellar wind interaction, and Chevalier (1982) for supernova remnants. In our study, we make use of the theorem (Buckingham, 1914; Sedov, 1959) to study the classes of self-similar solutions for the environment of SNe Ia.

The current state of the research leaves some important questions unresolved. How can we understand the ubiquitously low density environment, their general structure, and their link to the progenitor systems? Do SNe Ia all originate from merging WDs? Which of the wide variety of progenitor systems are compatible with the observations and the range of parameters? What other possible signatures might be seen due to the interaction of the explosion within the possible progenitor systems? For SN 2014J, can we find a class of progenitor systems which is consistent with the lack of X-rays and radio (Margutti et al., 2014; Pérez-Torres et al., 2014), which favor dynamical merging scenarios, and the narrow ISM lines, which favor mass explosions (Graham et al., 2015b)?

To address the questions, we developed a parameterized model in chapter 2 using fluid mechanics. In chapter 3, we present the application of semi-analytic models and our code SPICE as an analysis tool in the framework of environment of SNe Ia. We evaluate the imprint of different environments and wind properties. In chapter 4, we apply the framework to SN 2014J as an example (see Fig. 17) and discuss the results. In chapter 5, final discussions and conclusions are presented.

2. Theory and Assumptions

We develop here a model for wind-environment interaction from basic fluid mechanics. In the case of spherical symmetry and adiabatic flows, the hydrodynamic equations take the form:

(1)
(2)
(3)

where is the fluid velocity, is the mass density, is the pressure, and is the adiabatic index. The structures have four characteristic regions (Fig. 1): I) undisturbed wind emanating from the source between and an inner shock front , II) the inner shocked region of accumulated wind matter between and the contact discontinuity , III) an outer region of swept-up interstellar gas between and the outer shock , and IV) the outermost, undisturbed, ambient medium.

Figure 1.— Velocity, pressure and density structure of a typical model shows four distinct regions dominated by the wind (I), the reversed shock (II), a shell (III), and the environment (IV) both for constant density environments (left) and environments produced by prior mass loss (right). In the figures of this work, density is shown in particle density , where is Avogadro’s number and is the mean molecular weight. In this work, is set equal to 1. For different , in all equations.

The solutions for regions I and IV are trivial. The solution for region III was found by Parker (1963) who implimented a self-similar ansatz and the following transformations:

(4)
(5)
(6)
(7)

where for the outer density. The similarity exponent can be found by dimensional analysis.

(8)

for and for (See Eqs. 16 and 24 for exact expressions of ). Substituting these into equations 1, 2 and 3, making the substitution , and re-arranging gives us the following:

(9)
(10)
(11)

The outer boundary conditions are obtained from the Rankine-Hugoniot jump conditions for large Mach numbers:

(12)
(13)
(14)

Integration is carried out with respect to from the outer shock where to the contact discontinuity where the velocity is equal to and hence .

The solution for region III is self-similar because the only relevant parameters are the mechanical luminosity of the wind emanating from the origin and the outer density constant . There is no way to obtain a parameter with dimension of length or time using those parameters. This is not the case in region II, where the relevant parameters are , as well as and , individually. Therefore, in the case of a self-similar solution is not possible in region II, as was shown by Weaver et al. (1977). They did however obtain useful analytic relations directly from the hydrodynamic equations by assuming that region II was isobaric. Their results for the locations of the inner shock, contact discontinuity, outer shock, the velocity, pressure, and density are the following:

(15)
(16)
(17)
(18)
(19)
(20)

after correcting a small typographical error in their given expression for the velocity. This is the solution for the structure of the interaction region for the (constant IS density) case as the Mach number goes to infinity.

One notable feature about these structures is that the density goes to zero as approaches from above but diverges to infinity upon approach from below. Pressure and fluid velocity are finite and continuous across the boundary. Using the analytic expression for (equation 20) we may define a characteristic width of the density peak in the inner region by , where is the inverse of equation as given in 20. The width of region III is given by the integration of the equations 9 and 10; it is . The temperature varies with the inverse of the density. In reality the extreme temperature discontinuity will smooth out due to finite heat conduction.

2.1. Self-similar solutions for s=2

The case for allows us self-similar solutions because all characteristic scales are proportional to (Eq. 8) and, thus, the time-dependence cancels out. Chevalier & Imamura (1983) found self similar solutions for the interaction regions of colliding winds. Their work is similar to what we do here. The density in region IV is of the form assuming it is of a prior stellar wind with parameters and . The boundary conditions become

(21)
(22)
(23)

where the subscript i is either 1 or 2, referring to the outer and inner shock front boundaries, respectively (). Using the Buckingham Pi theorem (Buckingham, 1914; Sedov, 1959), we obtain the following expression for :

(24)
(25)
(26)

with

(27)
(28)

are functions to be determined numerically. By requiring pressure continuity across , we acquire an analytic expression for the inner shock radius as a function of time:

(29)

The outer shock radius can be given by

(30)

and likewise

(31)

The quantities , , and are found from integrating equations 9, 10 and 11 in either region III () or II (). However, in order to calculate them, initial guesses of , and are required. A consistent solution is obtained by damped fixed-point iteration. As Fig. 1 shows, the structures are qualitatively different for than for when . The density goes to infinity as one approaches from either side while the pressure is finite. Formally, the temperature therefore goes to zero.

2.2. Self-similar Solutions for s=0 and Boundary Conditions

As discussed above, the self-similar solution depends on the ambient density, , and the kinetic energy flux, at the inner boundary. For constant density ISM (s=0), the solutions are not valid for and . This is because physical assumptions break down and the results are unphysical solutions.

For small times, this can be seen as follows: As shown in Table 1, and . This implies the velocity of the reverse shock would go against infinity. No interaction is possible if the wind cannot overrun , therefore the description becomes unphysical. Also, the reverse shock is greater than the contact discontinuity for small enough . Therefore, no self-similar solutions exist for and , i.e. times shorter than . For our application to SN environments at times greater than when interaction takes place within the progenitor system, this hardly poses a limitation. For our reference model in the accretion disk wind case, the MS wind, and the RG-like wind (see tables 3,4 & 5) the critical times are , , and , respectively. The times where no self-similar solutions exist are short compared to the duration of the winds from the progenitor system.

Table 1For constant ISM and , the relations for the radius of the contact discontinuity , the forward and reverse shock, and , the fluid velocity at the shell , its mass column density , and the density of the inner void, , all as a funciton of the density of the environment , the mass loss , its wind velocity , and duration . For , we assumed (see Eq. 20). Asymptotic () values of the reverse shock , and the particle density are obtained from the FAP model and given below. is the ambient pressure.

For large times, the solution depends on the outer boundary condition, namely the pressure of the ambient medium. In the following, we want to consider the validity of solutions at large times, and develop approximations which allow us to study environmental properties. We will compare solutions with and without ambient pressure. We will refer to those as zero-ambient pressure (ZAP) and finite-ambient pressure models (FAP), respectively.

For , goes out indefinitely according to the self-similar solution as external pressure is neglected. In reality, the outer pressure will increasingly confine the expansion of the structure and, thus, . In the self-similar solution without ambient pressure, decreases with (Eq. 39) and, eventually, it will drop below the ambient pressure of the physical medium. As reference, we define the pressure-equilibration time as the time at which the pressure just inside equals the ambient (constant) pressure, . It is given by

(32)

In Fig. 2, we give the evolution of the basic physical quantities as a function of time for models with parameters typically for AD-, RG-like and MS-star winds. For our reference models (see Tables 3, 4 and 6), assuming an ideal gas ambient pressure given by temperature , we obtain in the AD wind case, the RG-like wind, and the MS wind , , , respectively.

Figure 2.— Structure feature comparisons of ZAP and FAP models for given sets of parameters as a function of for the reference models of the AD-, RG-like and MS wind (right to left). Dependence of the radii of the outer and inner shock , of the contact discontinuity , and the ram pressure are shown as a function of the duration of the wind normalized to the time at which the inner and outer pressure are equal. We show the functions for the zero ambient and finite pressure model indicated by the small and large symbols, respectively.

As an extreme case and benchmark for modifications, we use a MS wind with parameters similar to the Sun and an evolutionary time (see reference model in Table 6). is some 124 years only, i.e. smaller than by a factor of . In the ZAP model, the contact discontinuity and the location of the reversed shock are about 23 and 20 ly, respectively. The solar wind has similar properties but the termination shock is at about 75 to 95 AU based on Voyager 1 (Shiga, 2007). The discrepancies can be understood due to the ambient pressure not being taken into account. Moreover, for times much larger than , where is the ambient sound speed. We would therefore expect turbulent instabities which results in mixing. Ignoring ambient pressure for the MS (solar) model results in the contact discontinuity overrunning the the heliopause in about 10 years. Therefore, it is imperative that we consider how to account for finite ambient density in order to get realistic solutions.

A first order estimate for the solution may be obtained by stopping the time integration at for a model without ambient pressure, and neglecting the further evolution. For and and with this crude approximation, we obtain the right order of magnitude with compared to the solar value of 75 to 95 AU.

In the following, we will construct physically motivated boundary conditions for moderate 1 … 3, and discuss the uncertainties estimated by a comparison between the finite ambient pressure model (FAP) and zero ambient pressure model (ZAP).

Besides simply truncating the solution at , there is a way to approximately incorporate the ambient pressure in a way that retains the self similar solution at any time, although with a modified ambient pressure profile. In order to see how, we first notice the Rankine-Hugoniot jump conditions (in shock rest frame):

(33)
(34)
(35)

where the 0 subscripts denote pre-shock and the primed variables are post-shock quantities. After applying the substitutions in 4, 5, and 6, we see that the boundary conditions can remain constant with respect to space and time if follows a spacial power law with (or, equivalently, a time power-law of . This is because .). An effective pressure power law environment can be defined by requiring that, at a certain final time , the thermal energy contained within in our effective environment is equal to the thermal energy in the physical, constant ambient density at the same radius. We find it by volume integration along : . Although can be thought of as a constant parameter used to define the environment, in practice and . However, we note that, as the solution advances forward in time, this means that and the boundary condition will vary as well, meaning the solution will not be truly “self-similar,” ie the morphology changes with time. The solution obtained in this way is, in fact, a series of snapshots of self-similar solutions where is equal to the instantaneous time . Using this parameterisation, the Buckingham theorem gives us the following:

(36)
(37)
(38)

where the s are to be determined numerically and the ideal gas relation was used: is the gas constant divided by the mean molecular weight and is the physical ambient, constant temperature. will be taken to be typical for ISM gas (Osterbrock & Ferland, 2006). Note that, using Eqs 38 and 32 (for ), we have .

The outer boundary conditions of the ODEs are then given as:

(39)
(40)
(41)

where the Mach number is given by:

(42)

An initial guess of is required in order to numerically solve the ODEs and obtain the structure, therefore iteration is necessary in order to obtain a consistent solution. Following the method of Weaver et al. (1977) using with our modified boundary conditions, we obtain the following expressions for the radii and inner structure profile:

(43)
(44)
(45)
(46)
(47)
(48)

where , , and and are evaluated at the contact discontinuity. Comparison of eqs. 44, 45 with eqs. 37, 36 give and . We can then define a proportionality for the reverse shock:

(49)

where

(50)

For , , , , and eqns 15-20 from the last section are reproduced. For the reference models for an AD-, RG-like and MS-wind, a comparison of the basic properties between the ZAP and FAP models as a function of is shown in Fig. 2. Qualitatively, the main differences are as follows: For FAP models, and are smaller and is larger than for ZAPs, and goes to a constant value for large . For the range shown and for the RG-like wind, becomes larger than at about marking the regime of “unphysical” solutions already discussed above for the ZAP model. The functional relations appear to be similar and, in fact, they are identical as a consequence of the theorem. The relative shifts between the various quantities are given by proportionality factors which, in turn, depend only on the basic parameters, namely , and . For the FAP model, the proportionality constants have to be determined numerically (Fig. 3). A further consequence of the theorem is that the differences are only a function of and do not depend individually on , and (Fig. 4). For a constant density medium, the characteristic parameters can be directly obtained using Fig. 3.

Figure 3.— Proportionality factors for the characteritic distances as a function of for constant density environments and the FAP model. Note that all factors are constant for ZAP models with values corresponding to large (see Sect.2).

Figure 4.— Fractional difference in between models of zero ambient pressure () and finite pressure models () in the parameter space of the mass loss , the wind velocity , and the environment density as a function of . The plots visualize the -theorem as discussed in Sect. 3: The difference depends on only. The ratios between scale-free variables are constant throughout the parameter space. The -theorem applies also to . In the lower right, we show the fractional difference of (red), (blue) and (magenta) between the ZAP and FAP models as a function of .

The detailed solutions for our reference models are shown in Figs. 5 & 6. The morphology of the envelopes does not change for a wide range of parameters and time. As discussed above in case of the MS star wind, however, we must expect strong mixing for in FAP models. The solution becomes “un-physical” for in the regime of a weak shock.

Figure 5.— Structure of model 3 for AD wind (see Table 3). We show the ZAP (left) with and at (middle), and the FAP model at . The overall structure is similar within the parameter range.

Figure 6.— Same as Fig. 5 but for RG-like winds (see Table 4).

2.3. Existence of Solutions

Here, we want to provide the range for which self-similar solutions exists using the theorem.

Case I (s=0) : For FAP and s=0, there is one group given by Eq 38. Solutions do not exist for Mach numbers less than 1.3849.

Case II (s=2): Two groups exist and, thus, possible solutions are a combination of groups with given by equations 27 and 28. For ambient density profiles, the shell velocity range is sufficient to determine the relation between and for the prior mass loss. In Figs. 7, 8 & 9, we show and as a function of the wind parameters covering the entire range discussed in this paper.

Regime I: For high mass loss rates, we have no power law relation between the wind and environmental parameters, and the values of and need to be interpolated in the figures or can be calculated by SPICE.

Regime II: If a low mass loss wind runs into a high mass loss wind, , hardly vary with the ratio . Thus, the contour lines in Figs. 7, 8 & 9 are horizontal. Their value can be approximated therefore as a function of only the relative wind velocities. In Fig. 10, the variation of the cuts for various . and can be well described by single functions. needs two descriptions valid at low and high ratios of separated at . The resulting power law dependencies are given in Table 2.

Figure 7.— Value of as a function of and log where and . is close to constant in the regime of low mass winds running into high mass loss wind, i.e. .
Figure 8.— Value of as a function of and log where and . is close to constant in the regime of low mass winds running into high mass loss wind, i.e. .
Figure 9.— Value of as a function of and log where and . is close to constant in the regime of low mass winds running into high mass loss wind, i.e. .

Figure 10.— as a function of the ratio between inner and outer wind velocity described by as obtained by cuts at of indicated by progressively thicker lines. The exact solutions are given in red. We give the solutions for in comparison to the fits. The approximations are given (magenta and blue, dotted).
Table 2Same as Table 1 but for an environment produced by a prior wind (s=2). The index 1 corresponds to the prior wind. The relations are valid for high velocity winds running into environments produced by low velocity winds (see Fig. 7). For , , and , the relations are valid for . For and , it is valid for and between and . is calculated by formal integraion in SPICE.

3. Applications: Environments of Type Ia Supernovae Progenitors

We will explore winds emanating from the progenitor system and interacting with the ISM of mass loss of the system prior to the supernova explosion. We consider winds from each source separately, and we address the question of which component is mostly responsible for the formation of the environment, and the typical structure to be expected. Subsequently, we discuss the link between observables and progenitor systems, and analyze SN 2014J.

We employ our spherical, semi-analytical models constructed by piecewise, scale-free analytic solutions. Scales enter the system via the equation of state, the boundary, and the jump conditions. The free parameters are: 1) The velocity , 2) mass loss rate from the central object, and the 3) or a mass loss rate with , i.e. , and 4) the duration of the wind interaction . As a result, we obtain the density, velocity and pressure as a function of time, namely and which can be linked to observables. We use typical parameters to discuss the different regimes which may occur in nature. For actual fits of observations, appropriate solutions can be constructed by tuning these parameters with SPICE.

The wind may originate from the AD, the donor star which may be a MS or a RG-, horizontal- and asymptotic-branch star, the WD during a phase of over-Eddington accretion, or a combination of AD with a RG-like wind. As shown below, the time scale for the accretion and, thus, the progenitor, is an important factor in formation of the environment. The time scales vary widely depending on the scenario and chemical composition of the accreted material and the initial mass of the progenitor (e.g. (Sugimoto & Nomoto, 1980; Piersanti et al., 2003a; Wang & Han, 2012) and reviews cited in the introduction). To reach , about 0.2 to 0.8 of material needs to be accreted. For hydrogen accretion, the rates for stable hydrogen burning are between depending on the metallicity (Nomoto, 1982b; Hachisu et al., 2010). The upper and lower limits for are set by the Eddington limit for the luminosity and the minimum amount of fuel needed for steady burning, respectively. However, it is under discussion whether and at which accretion rates steady H-burning can continue until the WD approaches . It depends on the chemical composition, rotation of the WD, and details of the approximations used (Nomoto, 1982b; Starrfield et al., 1985; van den Heuvel et al., 1992; Hachisu et al., 1999; Piersanti et al., 2003a; Yaron et al., 2005; Sako et al., 2008; Hachisu et al., 2012; Bours et al., 2013). For a recent review, see Maoz et al. (2014). In this study, we use the wide range of accretion rates to avoid restricting possible solutions. Thus, we consider time scales between and years. Larger rates of mass overflow result in over-Eddington luminosity and a strong wind from the progenitor WD with properties typical of RG winds (Hachisu et al., 1996, 2008). Subsequently, we refer to the high-density, low velocity winds as “RG-like”. Accretion of He and C/O -rich matter allow much shorter timescales down to the dynamical times of merging WDs.

For accretion disk winds the mass loss rate ranges from to solar masses per year; and the wind velocities originating from the disks are believed to be from 2000 to 5000 (Kafka & Honeycutt, 2004).

Mass loss in “RG-like” stars are typically between and with wind velocities between and (Reimers, 1977; Judge & Stencel, 1991; Ramstedt et al., 2009).

Main sequence star winds are similar to the solar wind (Wood et al., 2002). For solar wind the velocity is between and and the mass loss is (Noci et al., 1997; Feldman et al., 2005; Marsch, 2006).

3.1. Parameterized Study

In the following, we will assume typical wind parameters as follows: In the “RG-like” case, we use mass loss rates between of , and , and a wind velocity of 30 . Similar winds can be expected for WDs with high accretion rates. For the case of a MS star donor, we use mass loss rates between , , and , and . For AD winds the mass loss rate ranges from , , and solar masses per year; a typical rate has been measured to be . A wind velocity of 3000 is used. For the duration of the winds, we consider between and years with years for the references.

The outer environment of the system depends on its history including the delay time between the formation of the WD and the onset of the accretion phase. For long delays, we assume an ISM with constant density, i.e. . For short delay times and small peculiar velocities of the system, the outer environment may be created during the final stage of the progenitor evolution, namely the red giant branch (RGB), horizontal giant branch (HGB) and the asymptotic giant branch (AGB) phase.

3.2. Results

3.2.1 Case I: Constant ISM density

We first consider scenarios where the wind blows out into a medium of constant density for a wide range of parameters (Tables 3-5). The structures are characterized by I) an undisturbed, inner layer dominated by the stellar wind, II) an inner, shocked region with almost constant, low density and a velocity declining with distance, III) a slowly expanding shell of high density, swept up material, and IV) the ISM. The overall solution is representative for all cases as has been shown in Sect. 2. For the estimate of the equivalent width of the Na I doublet at 5890/5896 Å, we use solar abundances, . EW is estimated according to Spitzer (1968); Draine (2011). For potassium lines, the corresponding equations apply.

Case Ia: Fast winds from an accretion disk: Table 3 contains calculated results from several cases with different parameters but the same time-scales . Our reference, model 2 of Table 3, is shown in Fig. 11. It has a mass loss of and a wind velocity . For the duration of the wind, we choose a duration of years which, within the SD scenario, corresponds to the evolutionary time for a low mass WD to grow to at an accretion rate of .

Up to about , the environment is dominated by the on-going wind. In this region, the particle density drops below 1(100) at a distance of which will be overrun by the SN ejecta within 5(0.5) days. Particle densities below 100 in this region will hardly affect the light curves or spectra because the swept up mass will be small. Using the same argument, the low densities within are too low to affect the hydrodynamics of the SN envelope. A high density, outer shell expands at a velocity of 11 with a velocity dispersion of . This shell would produce a narrow line Doppler shifted by about 11 . For an interstellar medium, the equivalent width would be about for the NaID line well comparable to values found by (Phillips et al., 2013) who found between 27 and 441 in a sample of some 20 SNe Ia.

Figure 11.— Hydrodynamic profile for a wind typical for an accretion disk (left), and mass-loss rate of , and a RG-like wind (right), 30 and mass-loss rate of , running into a constant interstellar medium density of 1 particle per after a time of 300,000 years. The contact discontinuity is at 21.7 and 5.45 light-years, respectively. Fluid velocity (magenta) is normalized to 100 , pressure (blue) is normalized to the pressure just inside the outer shock, and particle density is unnormalized.
No
() () () () () () () ()
1 0.1 3000 0.3 2.36 33.34 41.94 4.89
2* 1.0 3000 0.3 0.745 20.11 28.53 2.29
3 10.0 3000 0.3 0.236 11.39 22.02 0.98
4 1.0 3000 0.3 0.236 11.39 22.02 0.98
5 1.0 3000 0.3 0.0745 5.80 20.47 0.36
6 1.0 3000 0.15 0.745 13.70 17.84 1.82
7 1.0 3000 0.4 0.745 23.44 35.07 2.50
No