THERMONUCLEAR SUPERNOVAE: PROBING MAGNETIC FIELDS BY POSITRONS AND LATE-TIME IR LINE PROFILES

Several previous investigators have studied the transport of energy by high-energy photons through SN envelopes (Ambwani & Sutherland 1988; Burrows & The 1990; Chan & Lingenfelter 1991; Hoeflich et al. 1994; Hoeflich 2002). As part of his PhD thesis, Penney (2011) implemented new bound–free and bound–bound transitions that allow us to calculate X-ray spectra. This transport module has been further extended to incorporate positron transport in the presence of magnetic fields.

Several authors have addressed the transport of positrons (Bussard et al. 1979; Brown & Leventhal 1987). Early studies by Colgate et al. (1980) and Chan & Lingenfelter (1993) used one-dimensional approximations and assumed that the magnetic field was either chaotically twisted at a scale small enough to trap the positrons in place or radially combed by the expansion, restricting the positrons to move radially. They calculated the energy deposition by positrons at late times and demonstrated the extent to which positron escape would lead to LCs declining faster than the rates of the radioactive decay driving them. Milne et al. (1999) presented calculations using cross sections and a Monte Carlo technique more similar to ours, but continued to use either a radial magnetic field or ones that are chaotic and sufficiently large to trap all positrons. As part of the thesis of Penney (2011), LC results have been compared to those published Milne et al. (2004). As discussed below, we found a generally good agreement, but with a slightly reduced positron escape. In this paper, we make use of line profiles to measure positron transport effects and their dependency on B. Profiles can be expected to be more sensitive and cleaner than LCs because they measure directly the change in the redistribution function of the energy deposition from radioactive decay. In contrast, LCs rely on one quantity, the total energy deposition in the envelope that powers the bolometric LC. The reconstruction of the bolometric LC depends, in turn, on the photon redistribution in frequency space, and after a few years, other energy sources may become important, such as interaction between the SN and the interstellar material.

The primary source of positrons is the β⁺ channel of the ⁵⁶Co → ⁵⁶Fe*, which accounts for about 18% of all ⁵⁶Co decays. About 1.4 MeV of the total excitation energy of ⁵⁶Fe* is available to be split between neutrinos and positrons, with the resulting positron energy spectrum (Nadyozhin 1994) given by

$$\begin{equation} N(E)=Cp^2(E_o-E)^2(2\pi \eta (1-\exp (-2\pi \eta))^{-1})\protect, \protect \end{equation} \tag{ 1 }$$

where C is a scale factor and η is the charge of the nucleus times ℏ over the velocity of the electron. The mean energy of the spectrum is 0.632 MeV.

A small number of relatively high energy positrons are also contributed by pair production from the high-energy gamma-ray lines of ⁵⁶Co decay. As most γ rays escape without interaction at late times, these contribute only a few percent to the total positron production by 300 days, but their production and transport are included. The cross section for this channel of positron production has been discussed by Ambwani & Sutherland (1988b).

For the Monte Carlo transport, we take into account three dominant processes of interaction: scattering off atomic electrons as the positron moves through the media, the similar excitation of free electrons in a plasma, and direct annihilation with electrons.

The annihilation cross section is given by Equation (2) (Lang 1999):

$$\begin{equation} \hspace{-10pt}\sigma \,{=}\, \frac{\pi r_o}{\gamma \,{+}\,1}\left[\frac{\gamma ^2\,{+}\,4\gamma \,{+}\,1}{\gamma ^2-1}ln(\gamma +\sqrt{\gamma ^2-1})-\frac{\gamma +3}{\sqrt{\gamma ^2-1}}\right]\protect, \protect \end{equation} \tag{ 2 }$$

where r_o = e²/mc² and γ is the Lorentz factor. In undergoing this interaction, the positron is destroyed, its kinetic energy is deposited into the surrounding material, and its rest-mass energy is released as gamma rays.

Following Chan & Lingenfelter (1993) and Gould (1971), the energy loss by interaction with charged particles is given by

$$\begin{equation} \frac{dE}{dx} = \frac{-4\pi r_o^2m_ec^2q}{AM_n\beta ^2}\left(qln(\frac{\sqrt{\gamma -1}\gamma \beta m_ec^2}{b_{{\rm max}}}\right)\Pi (\gamma)\protect. \protect \end{equation} \tag{ 3 }$$

Π(γ) is a relativistic correction given by

$$\begin{equation} \Pi (\gamma) = \frac{\beta ^2}{12}\left[\frac{23}{2}+\frac{7}{\gamma +1}+\frac{5}{(\gamma +1)^2}+\frac{2}{(\gamma +1)^3}\right]\protect, \protect \end{equation} \tag{ 4 }$$

where q is the relative charge of the particles in the media. For atomic electron interactions, q is the atomic number Z for the atom. For plasma scattering, q is the ionization fraction. Parameter b_max is the maximum impact parameter, equivalent to the maximum amount of energy the positron can lose in one interaction. For atomic electron scattering, it is the ionization potential. Since the dependence on this parameter is weak, we simplify by using an average ionization potential for the medium. For the case of interactions with electrons in the plasma, the maximum impact parameter is ℏω because the impact with a free electron sets up a disturbance in the plasma that has an energy proportional to the frequency. Note that the ionization stage depends on time and requires ionization models that, in turn, will depend on the energy deposition by positrons. In this study, we assume single ionization. Other sources of continuous energy loss, those of bremsstrahlung and synchrotron radiation, were considered but found to be minor contributions and, for computational efficiency, have been omitted in this study.

The relative strengths of the three types of cross sections are shown in Figure 1. Annihilation is a very small part of the cross section throughout most of the energy range of the positron spectrum. As the positrons scatter and continuously lose energy, though, they will either fall into the range where direct annihilation dominates and be destroyed or slow down until their energies become similar to the binding energy of atomic electrons, form positronium, and be annihilated.

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Between 1 keV and the binding energy of nearby electrons, the positron can undergo charge exchange, stripping an electron off nearby atoms to form positronium, which has a negligible lifetime before undergoing either e⁺e⁻ → 2γ or e⁺e⁻ → 3γ for para- and ortho-positronium, respectively. We assume that the para-state and ortho-state of positronium are in statistical equilibrium for the creation of γ photons. The cross section of this process for elements other then H is not well known but is generally proportional to the Bohr radius and thus much larger than the direct annihilation cross section.

This uncertainty in the cross section for annihilation at low energies will hardly influence the location of the energy deposition because the energy loss is large (see Equation (3)). Even at day 1000 after the explosion, the range of a 1 keV positron is less than 0.1% of the radius of the envelope. Thus, we assume that once a positron has fallen below 1 keV, it deposits its kinetic energy locally and annihilates via positronium formation. Chan & Lingenfelter (1993) found that some small fraction of thermalized positrons (≈0.1% in models similar to ours) will avoid annihilation, surviving as "slow" positrons as the envelope density becomes low enough for the lifetime before positronium formation to become very long. However, these positrons will have already deposited their kinetic energy in the ejecta, and their mass energy would not contribute to the low energy light curves and spectra whether they eventually annihilate or not. At late times, γ rays from annihilation are almost guaranteed to escape the envelope without interaction.

The transport is solved via a Monte Carlo method very similar to the photon transport (Hoeflich et al. 1994; Hoeflich 2002). A background model is discretized, and the radiation transport is performed in individual computational cells. Upon interaction, the process is chosen randomly, weighted by the size of the individual cross sections.

The motion of positrons depends on the presence, size, and morphology of the magnetic field B imposing a Lorentz force on the positron. The positron is gyrating around the magnetic field line. The scale imposed by B is the Larmor radius

$$\begin{equation} r_L = \frac{p_{\perp }}{qB} {(p_{\perp} the momentum perpendicular to the field.)} \end{equation} \tag{ 5 }$$

The significance of the gyro motion depends on the size of r_L relative to the scale of variations of the explosion model r_M, such as the density scale height or the scale of the B-field changes.

For computational efficiency, we distinguish three different implementations: (1) r_L ≫ r_M, (2) r_L ≪ r_M, and (3) the case in between.

In Case 3, the path of the positron must be explicitly integrated by breaking the path length into segments sufficiently small so that the B-field changes little along individual segments. A positron propagates along a circular arc on that path (the radius determined by Equation (5)) until it is absorbed, scatters, or escapes the envelope.

Cases 1 and 2 avoid the need to reconstruct the path for individual positrons; thus, they allow for a higher computational efficiency by a factor of more than ≈10.

For Case 1, small magnetic fields, the path of the positron is nearly straight and we integrate along linear rays until the positron is absorbed, is scattered, or escapes.

For large magnetic fields, r_L is small compared to the structure of the explosion model. The positron can be assumed to follow the field line on a spiral trajectory mean-free path. The general motion of a positron is along the field line with its path length increased by a factor of 1/sin (θ_p), where θ_p is the pitch angle the positrons momentum vector makes with the field line. Effects of gradients in B are implemented following Chen (2003). We divide these into two forces, one parallel to the field lines and one perpendicular. The former causes the "magnetic mirror" effect. Because the first adiabatic invariant is constant, a change in the B-field strength along the direction of travel causes a change in the pitch angle. In the code, we average the derivative of the B-field over the zone in the direction of the field lines and use this to calculate an average gyro-radius for that zone. The result is that positron diffusion is discouraged in the direction of increasing magnetic field and encouraged in the opposite direction, as an increasing pitch angle will increase the path length. The effect of the perpendicular component of the B gradient forces the mean positron path to not follow the field line due to gradients perpendicular to the B-field. We add this velocity to the motion of the particle in each zone using the following equation:

$$\begin{equation} v=\frac{1}{2}v_{\perp }r_L\frac{B \times \nabla {B}}{B^2}\protect. \protect \end{equation} \tag{ 6 }$$

For illustration, Table 1 shows the evolution of r_L for a typical SN Ia model as a function of time. In practice, we use Case 3 or Case 1 if r_L is smaller than the resolution of the background model or larger than twice the model grid, respectively. Cases 1 and 3 have been tested against the general Case 2. With time and lower magnetic fields, the range of the positron transport increases. For B-fields larger than 10⁹ G, positrons follow the field lines. For B-fields less than 10³ G and about 1 year, the bending of the path of the positrons remains small and they travel along rays. We note that turbulence will produce structures of about 1%–5% of the envelope structure. Thus, the energy deposition by positrons requires full transport of positrons for all B-fields less than B ≈ 10⁶ G when the transport effects are important.

Radioactive decay of ⁵⁶Ni → ⁵⁶Co → ⁵⁶Fe is the dominant energy source that powers the LCs and spectra of SNe Ia. The total energy input depends on the total mass of the initial ⁵⁶Ni and the redistribution by transport effects. ⁵⁶Ni decays by electron capture, which leads to the emission of high-energy photons when the excited ⁵⁶Co transitions to its ground state. Besides decay via electron capture, 18% of all decays of ⁵⁶Co are β⁺ and lead to the emission of a positron.

Escape Probabilities: First, we want to discuss the role of the escape probabilities and their implications for LCs (Figure 3). The small cross sections for γ rays result in rapidly increasing escape fractions. Already at maximum light, ≈20 days after the explosion, 15% of all γ rays escape. This fraction increases to about 85% and 98% by day 100 and day 300, respectively. In contrast, the large positron cross sections cause almost complete local annihilation of positrons up to about 200 days. The length of the positron path depends on the size of B, which causes a strong dependence of the positron escape fraction on B at later times. However, as the kinetic energy of positrons only accounts for some 3% of the total energy produced by ⁵⁶Co decay, the total energy input is dominated by γ rays until around days 150–200 (Figure 3). Thus, studying the B-field via the influence of energy deposition on optical/IR wavelengths inherently requires late-time observations.

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Light Curves: Previously, detailed simulations for positron transport effects on the bolometric LCs have been published by (Milne et al. 1999) for a wide range of models based on the deflagration model W7 (Nomoto 1984), the delayed-detonation models (Yamaoka et al. 1992; Hoeflich & Khokhlov 1996; Hoeflich et al. 1998), helium detonations (Ruiz-Lapuente et al. 1993; Hoeflich & Khokhlov 1996), and pulsating-delayed detonation and merger models (Hoeflich et al. 1995; Hoeflich & Khokhlov 1996). We have compared the energy escape by positrons of our reference model and dipole fields with model DD23C (Hoeflich et al. 1998) and radial magnetic fields used in the sample of Milne et al. (1999). We find that our escape fractions agree with Milne et al. (1999) within ≈20%. For high B-fields, our escape fractions are slightly smaller because we include nonradial fields whereas Milne et al. (1999) consider the radial component only.

For a brief discussion, we use the V band (V) as a proxy for the bolometric LC (Arnett 1980). LCs are shown in Figure 4 for low and high B for dipole and turbulent fields. The scale of the turbulent field has been taken to be one pressure scale height in the progenitor WD. The formation of early-time optical and IR LCs has been previously discussed (Hoeflich et al. 1998; Hoeflich et al. 2002; Dessart et al. 2014).

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Between 60 and 200 days, the decay of V is steeper than is expected from the decay of ⁵⁶Co because γ-ray photons dominate the energy input and they rapidly increase their escape probability, as shown in Figure 3. In fact, this discrepancy caused early predictions that SNe Ia are not powered by ⁵⁶Ni (Colgate et al. 1962). Note that the decline rate increases at about 100–200 days with decreasing brightness because more centrally concentrated ⁵⁶Ni causes a delay in the γ escape (Hoeflich et al. 1993, 2002).

After 300 days, almost all γ rays escape and the LC is dominated by positrons. As the envelope is now optically thin for optical photons, the anisotropy effects are expected to be small. At ≈300 days, V closely mimics the decay curve of cobalt. The change of slope after this point may be used to estimate B (Milne et al. 1999). At about 300 days, the influence of B remains small, ≈0.05 mag. After about 3 yr, the influence of B grows to about 1 mag. Even for high B, the path of positrons follows the magnetic field line, allowing some positron mobility and, then, positron escape. In contrast, for a turbulent field photons are trapped, which results in local energy deposition (see Figure 4) until the field weakens to the point that the Larmor radius is larger than the size of the turbulence. Despite the significance of both the size and morphology of the B-field on late-time bolometric LCs, there are several limitations in using LCs to probe B. Monochromatic LCs are needed to reconstruct bolometric LC. However, V or UBVRI may not actually represent the bolometric LCs. It has been suggested that emission will shift from the V band to the infrared as the timescales of atomic transitions in the optical range increase relative to those in the infrared owing to falling matter densities (called the IR catastrophe). In models, the IR catastrophe may set in anywhere from 300 to 400 days depending on the details of the atomic model and data and the abundances (Fransson et al. 1996). Moreover, recent observations of SNe Ia identified MIR emission lines as a major cooler starting at about three to four months after the explosion (Gerardy et al. 2007; Li et al. 2014). Finally, other energy sources may contribute to the energetics, such as interaction of the envelope with its surroundings, which will deposit energy in the high-velocity layers. Moreover, the diversity among SNe Ia has been well established, and variations in the ⁵⁶Ni distributions will change the bolometric LCs. These uncertainties and variables make it difficult to extract B-field strengths from just the one quantity provided by LCs.

The Energy Distribution: We want to shift our discussion to the distribution of the energy deposition and the resulting line profiles. The energy depositions by γ rays and positrons and by positrons only are given in Figure 5. Up to about 100–200 days, the contribution from positrons is locally deposited owing to high cross sections. Nonlocality is produced by γ rays. At early times, about a week after the explosion, the optical depth for γ rays is still large and the energy input closely traces the distribution of the radioactive isotopes (Figure 2). We can see the "hole" in the center occupied by stable, electron capture elements. Owing to geometrical dilution, the optical depth for γ photons decreases with time (t) as ≈t⁻². This leads to an increasing energy deposition in the central region and larger heating of the outer layers. By about 100 days, the central heating peaks. With time, positrons take over as the main contributors to the energy deposition, and a central hole reappears.

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We start with the case of positron transport without B. Trapping can be attributed to the opacities only as the positron travels along a ray. The escape fractions and the normalized total and positron energy redistribution functions are given in Figures 3 and 5. Up to about 100 days, positrons continue to be locally confined. At 200 days, the outer layers are transparent, diffuse enough that the positrons escape, lowering the amount of high-velocity material that is heated, but positrons in the inner layers remain locally confined. At 300 days small features like the dip in Ni at 5000 km s⁻¹ are starting to be washed out, and there is some penetration into the core. Note that dip in ⁵⁶Ni is not physical, but an artifact of spherical DDT. However, it is a good indicator to what degree positron transport has become nonlocal. After 500 days, the nonradiating core and the area around it are the only parts of the envelope dense enough to have almost complete positron capture. At 300 to about 400 days, the total energy deposition is at a minimum at the center and subsequently increases as a result of the growing mean free path of positrons.

Magnetic fields determine the path of positrons and thus introduce three more parameters into our discussion: the size of the field, its morphology, and the orientation of the B-field relative to an observer. The main effect of the B-field on the escape and energy redistribution can be understood in terms of the Larmor radius r_L compared to the size of the envelope and any substructures considered. It increases the effective mean free path and influences the direction.

For dipole fields, we see that high magnetic fields delay the overall evolution of the energy distribution by about 1 year. For example, the distribution at 200 days resembles closely that of the high B-field at day 500 (Figure 5).

From the previous sections and within the delayed detonation scenario, we can conclude the following: up to about 200–300 days, the energy deposition functions do not depend on magnetic fields because γ-ray transport dominates or positrons are deposited locally regardless of the field. Subsequently, positron transport effects may become important, depending on B.

The goal of this section is to demonstrate the signature of radiation and positron transport on late-time spectra. We discuss in detail the 1.644 μm spectral feature, including its evolution with time and its dependence on the orientation of the observer. In addition, adjoining strong NIR features are considered to demonstrate the impact of line blending and how blending may severely limit a kinematic interpretation of profiles. For the spectral band, we have calculated a full grid of 30 models for B = 10^{0, 3, 4, 6, 9} G at 100, 200, 300, 400, 500, 999 days with counters in five and eight directions in the Eulerian Θ and ϕ coordinates. Here, we show only a few of the resulting 1200 spectra.

Our reference model, though spherical, requires full 3D transport because of the axial symmetry introduced by the magnetic field. Chemical abundances take into account the variability caused by the radioactive decay ⁵⁶Ni → ⁵⁶Co → ⁵⁶Fe. The excitation balance has been calculated using our non-LTE code (Hoeflich et al. 2004) as a function of time. For the ionization, we found a mean ionization of about 1–1.5 electrons per atom and have assumed a value of 1.5. We neglect second-order effects on the excitation and ionization, although the ionization in the central regions will depend on the heating by positrons and, at late times, iron-group elements may be neutral. As discussed in the final section, this approximation can be tested by observations but can be justified as follows: Lower ionization means a slightly larger mean free path, by about 20%–30% (Figure 1). However, a larger mean free path implies more heating, which, in turn, decreases positron transport (see Section 2), which, in turn, reduces the feedback effect. Tests have shown that the total effect on the spectral features remains small.

To discuss line profiles, the continuum fluxes have been subtracted and the maximum fluxes normalized. The actual abundances have been used from the model 5p0z22.25. For the spectral synthesis, we included the forbidden line profiles of [Fe ii] at 1.533, 1.560, 1.644, 1.664, 1.677, 1.712, 1.745 μm and [Co iii] at 1.55, 1.74, 1.76 μm. The continuum flux in the 1.6–1.7 μm band is 0.24, 0.15, 0.12, 0.1, 0.06, and 0.02 at 100, 200, 300, 400, 500, and 999 days, respectively.

The evolution with time is shown in Figure 6 in velocity space corresponding to wavelengths between 1.5 and 1.8 μm. There are three prominent features at about −20, 000, 0, and +18, 000 km s⁻¹.

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Low Magnetic Fields: We start with a discussion of the angle-averaged line profile of [Fe ii] at 0 km s⁻¹ ≡ 1.644 μm because line-blending effects for this feature remain small except in the wings (Hoeflich et al. 2004). By day 100, the profile is peaked because gamma rays are nonlocal and mostly absorbed in the central, high-density region, which has stable ⁵⁴Fe. The [Fe ii] feature is asymmetric and blueshifted by about 2000 km s⁻¹ because the photosphere blocks the redshifted layers of the envelope. By day 200, the blueshift is reduced to about ≈800 km s⁻¹. With time, the feature becomes wider and reaches a maximum width at about 300 days. At around the same time, it develops a flat-top or stumpy appearance in the center. As discussed and shown in Section 5, a caustic profile is produced if ⁵⁶Ni is homogeneously mixed to the center or the explosion of a low-density WD because the emission would follow the density structure (Hoeflich et al. 2004). By about day 200, positron transport results in increasingly wider line wings and a small low-velocity emission component. By day 500, positrons are able to enter and deposit their energy in the core, and the profile becomes increasingly peaked.

Role of Magnetic Fields: Magnetic fields will hardly change line profiles up to about a year after the explosion. At about 300 days, the envelope is mostly optically thin in the quasi-continuum. Observations of the profile of [Fe ii] at 1.644 μm at this time provide an excellent tool to measure the distribution of ⁵⁶Ni in the velocity space. Earlier, optical depth effects in the continuum and the contribution of γ rays are important contributors to the energy input. Later, positrons transport energy into the central regions, and at a rate dependent not just on the ⁵⁶Ni distribution and density structure, but also on the magnitude and structure of the B-field. In practice, early variability in the profiles allows us to separate optical depth and geometry effects, to probe the distribution of the central nonradioactive Fe, and to test for ionization effects.

To detect the effects of B, we want to focus on the Late-time evolution (Figure 7). With increasing B, the evolution from flat-topped or stubby to more centrally peaked profiles is delayed by about 1 year for high B-fields, as can be expected from the discussion above. Lines are narrower by about 20% by day 500 compared to models with B > 10⁶ G. Observations at about 1000 days would allow the measurement of ≈10⁹ G.

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Perils of Line Blending: The kinematic interpretation of emission features breaks down if line blending becomes important, as is the case for features in the optical and most features in the NIR. The two features neighboring the 1.644 μm line, at −20, 000 and +18, 000 kms⁻¹ in Figure 6, are illustrations of the perils of apparently clean features. In fact, they are blends. The former feature is produced by transitions of [Fe ii] at 1.533 and 1.560 μm and of [Co iii] at 1.55 μm. The latter feature is produced by transitions of [Fe ii] at 1.712 and 1.745 μm and of [Co iii] at 1.74 and 1.76 μm.

The evolution of both these satellite features is dominated by the decay of radioactive ⁵⁶Co. Up to about day 200, the forbidden ⁵⁶Co lines are responsible for most of the emission. After about day 300, only ⁵⁶Fe is left, as evidenced also by the slowly varying line ratio between the 1.644 μm feature and the satellites.

Early on, the red feature at +18, 000 km s⁻¹ shows a blueshift similar to [Fe ii] at 1.644 μm but it is broader because several line transitions contribute including [Fe ii] at 1.712 and 1.745 μm and [Co iii] at 1.74 and 1.76 μm. One may expect that, with time, the feature would be shifted toward the blue because of the longer wavelength of the [Co iii] transitions. However, we see a redshift because the feature formed on the wing of the 1.644  μm, which produces a sloped quasi-continuum.

The −20, 000 km s⁻¹ feature at ≈1.55 μm dominates the NIR spectra early on. However, owing to its nature, a simple interpretation may be misleading. With time, this feature undergoes a blueshift opposite to the behavior of the [Fe ii] at 1.644 μm. Taking at face value and in a kinematic interpretation, this would mean that we see an offset in ⁵⁶Ni, whereas this shift, at least in the models, is due to the changing importance of [Co iii] versus [Fe ii].

In summary, time evolution allows us to detect and, potentially, to separate line blending, geometry, and B-field effects. The [Fe ii] feature at 1.644 μ is rather unique in the NIR.

Directional Dependence of NIR Features: Large-scale magnetic dipole fields introduce a directional dependence of the observed line profile. The variations in the profiles can be understood in terms of the Larmor radius given in Table 1. We consider only times later than 300 days when positron transport effects may be significant. For low B-fields, the Larmor radius is much larger than the envelope and positrons travel on rays, which results in isotropic profiles.

In Figures 8, 9 and 10, the directional dependence of profiles is shown for high magnetic fields. For B = 10⁹ G, the positrons follow the field lines with an increase in the effective cross section. Variations in the energy deposition and, thus, the energy deposition depend mainly on the mean free path. By day 1000, positrons created in the polar regions can travel to the center regardless of the B-field. More generally, positrons created at more equatorial latitudes are funneled toward the pole. This geometry effect causes maximum directional dependence. From the pole, we see flat-topped profiles with an increased line width. Seen from the equator, late-time profiles are centrally peaked.

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Lowering the B-field increases the mean free path, and the path of the positron is bent but does not exactly follow the field lines. For B = 10⁶ G, the directional dependence shows up earlier. High-precision observations of the time dependence of profiles would allow us to learn about the orientation of the field relative to the observer.

This leads us to the effects of the morphology of the B-fields. We may expect a dipole field in a WD prior to the runaway, but it may be distorted by the turbulent motion during the thermonuclear runaway, or by RT instabilities during the explosion of the WD.

As an example and limiting case, we have calculated the transport for a turbulent field. In calculations of the thermonuclear runaway, the largest turbulent eddies are ≈5% of the WD (Hoeflich & Stein 2002). We note that the energy deposition is close to isotropic.

The results are dominated by the change of the gyro-radius relative to the size of the structure of the B-field (Table 1). B in excess of 10⁶ G results in local trapping of positrons with some transport effects starting at about 500–1000 days. For fields less than ≈10⁴ G, the gyro-radius is much larger than the size of the envelope and, consequently, the results are identical to the angle-averaged profiles in Figures 6 and 7. Probing the magnetic fields requires time series of profiles after about two to three years, or magnetic fields between 10⁴ and 10⁶ G observed beyond 400 days. As discussed above for very high magnetic fields, LCs come close to the case of full trapping.