ON SILICON GROUP ELEMENTS EJECTED BY SUPERNOVAE TYPE IA

We first select a system consisting of the major SiG and FeG elements to trace out the most useful equations connecting the individual abundances and their relationship to Y_e. We choose ²⁸Si, ³²S, and ⁴⁰Ca from the SiG isotopes, and ⁵⁸Ni and ⁵⁴Fe from the FeG isotopes. As Figure 1 suggests, these are the dominant isotopes under QNSE conditions. Conservation of mass and charge can therefore be expressed as

$$\begin{eqnarray} &&Y_{\rm n}+ Y_{\rm p}+ 28Y_{{\rm 28Si}}+ 32Y_{{\rm 32S}}+ 40Y_{{\rm 40Ca}}+ 54Y_{{\rm 54Fe}}+ 58Y_{{\rm 58Ni}}=1\nonumber\\ \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &&Y_{\rm p}+ 14Y_{{\rm 28Si}}+ 16Y_{{\rm 32S}}+ 20Y_{{\rm 40Ca}}+ 26Y_{{\rm 54Fe}}+ 28Y_{{\rm 58Ni}}= Y_{\rm e}.\nonumber\\ \end{eqnarray} \tag{ 2 }$$

From minimization of the Helmholtz free energy there follows the fundamental QNSE relations (Bodansky et al. 1968; Hix & Thielemann 1996; Meyer et al. 1998; Iliadis 2007)

$$\begin{eqnarray} \frac{Y_{{A,Z}}}{Y_{{A^{\prime },Z^{\prime }}}} &=& f(\rho,T)Y_{\rm p}^{{ Z-Z^{\prime }}} Y_{\rm n}^{{ A-A^{\prime }-(Z-Z^{\prime })}}, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} f(\rho,T) &=& \frac{G_{{ A,Z}}}{G_{{ A^{\prime },Z^{\prime }}}}\left(\frac{\rho N_{\mathrm{A}}}{\theta }\right)^{A-A^{\prime }}\exp \left(\frac{B-B^{\prime }}{k_{\mathrm{B}}T}\right), \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} \theta &=& \left(\frac{m_{\mathrm{u}}k_{\mathrm{B}}T}{2\pi \hbar ^{2}}\right)^{\frac{3}{2}}. \end{eqnarray} \tag{ 5 }$$

Here T is the temperature, ρ is the baryonic mass density, G_{A, Z} is the temperature-dependent partition function, B is the nuclear binding energy, N_A is the Avogadro constant, k_B is the Boltzmann constant, and m_u is the atomic mass unit. The molar abundances are the local abundances that correspond to a region of the star associated with a specific ρ and T. The nuclei are treated as an ideal gas and we ignore screening corrections, both of which are justifiable assumptions under the thermodynamic conditions of interest. Specification of T, ρ, Y_e, the aggregate molar abundance of the SiG isotopes Y_SiG, and the aggregate molar abundance of the FeG isotopes Y_FeG is sufficient to solve for all the abundances in a two-cluster QNSE environment. At a given ρ and T, we use Equation (3) to write Y_32S and Y_40Ca in terms of Y_28Si, Y_p, and Y_n. Similarly Y_58Ni is written in terms of Y_54Fe, Y_p, and Y_n. This leaves us with four unknowns, Y_p, Y_n, Y_28Si, and Y_54Fe, and, for a known Y_e, four constraints: Equations (1) and (2), and the sums Y_SiG = Y_28Si + Y_32S + Y_40Ca and Y_FeG = Y_54Fe + Y_58Ni, which are both specified externally to the solution of the QNSE.

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Measurement of four quantities Y_28Si, Y_32S/Y_28Si, Y_40Ca/Y_32S, and Y_54Fe/Y_28Si is an equally sufficient basis from which to solve for all the abundances in the silicon-rich region of SNIa. For our choice of isotopes, Equation (3) leads to

$$\begin{eqnarray} \frac{Y_{{\rm 28Si}}}{Y_{{\rm 32S}}} & \approx & \left(\frac{\rho N_{\mathrm{A}}}{\theta }\right)^{-4} Y_{\rm p}^{-2} Y_{\rm n}^{-2} \exp \left(\frac{B_{\mathrm{28Si}}-B_{\mathrm{32S}}}{k_{\mathrm{B}}T}\right) \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \frac{Y_{{\rm 54Fe}}}{Y_{{\rm 58Ni}}} & \approx & \left(\frac{\rho N_{\mathrm{A}}}{\theta }\right)^{-4} Y_{\rm p}^{-2} Y_{\rm n}^{-2} \exp \left(\frac{B_{\mathrm{54Fe}}-B_{\mathrm{58Ni}}}{k_{\mathrm{B}}T}\right) \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \frac{Y_{{\rm 32S}}}{Y_{{\rm 40Ca}}} & \approx & \left(\frac{\rho N_{\mathrm{A}}}{\theta }\right)^{-8} Y_{\rm p}^{-4} Y_{\rm n}^{-4} \exp \left(\frac{B_{\mathrm{32S}}-B_{\mathrm{40Ca}}}{k_{\mathrm{B}}T}\right). \end{eqnarray} \tag{ 8 }$$

Here we assume all ratios of nuclear partition functions are unity. This is justifiable, as at typical QNSE temperatures the nuclei are mostly in their ground state, and all of these nuclei have zero spin.

Using Equations (6) and (8), consider the local SiG element ratio

$$\begin{eqnarray} \Phi (T) &=&\frac{Y_{{\rm 28Si}}}{Y_{{\rm 32S}}} \left(\frac{Y_{{\rm 40Ca}}}{Y_{{\rm 32S}}} \right)^{1/2}\nonumber \\ &\approx & \exp \left({\frac{B_{{\rm 28Si}}-B_{{\rm 32S}}-0.5(B_{{\rm 32S}}-B_{{\rm 40Ca}})}{k_{\mathrm{B}}T}} \right)\nonumber \\ &=& \exp \left(\frac{-1.25}{T_{9}}\right), \end{eqnarray} \tag{ 9 }$$

where T₉ is the temperature in units of 10⁹ K. Typical temperatures in the QNSE regions where the SiG elements are formed are (3.4–4.0) × 10⁹ K (a range of 15% in temperature); over this range Φ varies by 6%, from 0.73 to 0.69. Measuring Φ at a single epoch from the abundance ratios Y_28Si/Y_32S and Y_40Ca/Y_32S allows a test of whether the SiG material was produced in a QNSE state. With sufficient precision, measurement of Φ allows a determination of the temperature in the QNSE region before freeze-out. We assume for the remainder of this paper that such precision is available and that the QNSE temperature is a known quantity. Measuring Φ at multiple epochs when silicon features dominate the SNIa spectrum allows trends in the QNSE temperature to be assessed.

More generally, a double ratio of the form

$$\begin{eqnarray} K &=& \frac{Y_{Z-2,A-4}}{Y_{Z,A}}\frac{Y_{Z^{\prime }+2,A^{\prime }+2}}{Y_{Z^{\prime },A^{\prime }}}\nonumber\\ &\approx& \exp \left[\frac{(B_{Z-2,A-2}-B_{Z,A}) - (B_{Z^{\prime },A^{\prime }} -B_{Z^{\prime }+2,A^{\prime }+4})}{k_{\mathrm{B}}T}\right]\quad \end{eqnarray} \tag{ 10 }$$

is independent of ρ, θ, Y_p, and Y_n. Additionally, if the isotopes in this ratio are major constituents of clusters in NQSE, then the argument of the exponential will be of order unity and K will not vary strongly over the narrow range of temperature for which NQSE conditions attain. A relatively precise value of K can then be specified from the ratio Φ. Such a quasi-constant can also be defined for the FeG elements. Equations (6) and (7) imply that

$$\begin{equation} \Psi \equiv \frac{Y_{{\rm 58Ni}}}{Y_{{\rm 54Fe}}}\frac{Y_{{\rm 28Si}}}{Y_{{\rm 32S}}} \approx \exp \left(\frac{6.36}{T_{9}}\right), \end{equation} \tag{ 11 }$$

where we again assume all ratios of nuclear partition functions are unity, and, by construction, Ψ is independent of ρ and θ. Over the range of QNSE temperatures, Ψ varies by 28%: from Ψ = 6.5 at T₉ = 3.4 to Ψ = 4.9 at T₉ = 4.0.

For our simplified system consisting of the a few major SiG and FeG elements, Equation (2) may be written as

$$\begin{eqnarray} Y_{\rm e} &=& Y_{{\rm 28Si}}\left[ 14 + 16 \frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} + 20 \frac{Y_{{\rm 40Ca}}}{Y_{{\rm 32S}}}\frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} + 26\frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}}\right. \nonumber\\ &&\left. + 28 \frac{Y_{{\rm 58Ni}}}{Y_{{\rm 28Si}}} + \frac{Y_{\rm p}}{Y_{{\rm 28Si}}} \right]. \end{eqnarray} \tag{ 12 }$$

We factor out Y_28Si because, as we show in Section 3, the silicon yield is the least sensitive to changes in ρ, T, and Y_e in QNSE material. We may also drop Y_p since it is much smaller (Y_p < 10⁻⁴) than the other abundances.

The first step in reconstructing Y_e is to determine from observations the Y_28Si/Y_32S and Y_40Ca/Y_32S abundance ratios from strata with similar velocities. Measurement of these ratios determines the second and third terms of Equation (12). Their ratio also forms Φ (Equation (9)), which if near unity verifies that the SiG elements were synthesized in a QNSE environment. This relation may also be inverted to determine the temperature of the QNSE environment when the SiG elements were synthesized.

The next step is to measure the Y_54Fe/Y_28Si abundance ratio. Usually it is difficult to extract the ⁵⁴Fe abundance from the iron lines. However, ⁵⁴Fe is the only iron isotope that is abundant in the regime where both ²⁸Si and ³²S are also abundant, in the absence of significant mixing of the QNSE material with core material. The reason for this is that in NSE, where most of the mass is in the iron group, the requirement that Z ≈ A forces ⁵⁶Ni to be the dominant abundance. In contrast, for QNSE, most of the mass is in the Si-group isotopes and this charge/mass constraint is lifted, so that the greater binding energy of the slightly neutron-rich ⁵⁴Fe results in ⁵⁴Fe having a larger abundance than ⁵⁶Ni (Hix & Thielemann 1996; Meyer et al. 1998; Iliadis 2007). In the absence of large-scale mixing, the ⁵⁴Fe produced in the silicon-rich regions is physically separated from the ⁵⁶Ni produced deeper in the core, so that signatures of iron at early times from ⁵⁴Fe do not depend on the ⁵⁶Ni decay chain. Therefore, if iron features are detected in the early time spectra (≈8 days) at the same expansion velocities where SiG elements dominate the spectral features, they are produced by ⁵⁴Fe. This result is due to material being in QNSE and is not dependent on any particular SNIa model.

The final step is to determine the Y_58Ni/Y_28Si abundance ratio. Using Equation (11) we write

$$\begin{equation} \frac{Y_{{\rm 58Ni}}}{Y_{{\rm 28Si}}} = \frac{Y_{{\rm 58Ni}}}{Y_{{\rm 54Fe}}}\frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}} = \Psi \frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}}\frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}}. \end{equation} \tag{ 13 }$$

Thus, the last two terms of Equation (12) are determined and may be rewritten as

$$\begin{eqnarray} Y_{\rm e}&=& Y_{{\rm 28Si}} \left[ 14 + 16 \frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} + 20 \frac{Y_{{\rm 40Ca}}}{Y_{{\rm 32S}}}\frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} + 26\frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}}\right. \nonumber\\ &&\left. + 28 \Psi \frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} \frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}} \right]. \end{eqnarray} \tag{ 14 }$$

Equation (14) is our principal result. Highly accurate abundance determinations of the four quantities Y_28Si, Y_32S/Y_28Si, Y_40Ca/Y_32S, and Y_54Fe/Y_28Si under the relevant temperatures is sufficient to determine Y_e to within 6% because these abundance dominate the QNSE composition.

The abundance of any α-chain SiG element, which can be used to improve the accuracy of the Y_e determination, can be recovered using Equation (3) and Y_32S/Y_28Si. For example, the abundance ratio Y_32S/Y_36Ar is related to Y_28Si/Y_32S by

$$\begin{equation} \frac{Y_{{\rm 28Si}}}{Y_{{\rm 32S}}} = K_2 \frac{Y_{{\rm 32S}}}{Y_{{\rm 36Ar}}}, \end{equation} \tag{ 15 }$$

with K₂ ≈ exp  (− 3.56/T₉). As the temperature ranges from T₉ = 3.4 to 4.0, K₂ varies from 0.35 to 0.41, a variation of 16%. Including Y_36Ar in the sum for Y_e in Equation (12), adds the term 36K₂(Y_32S/Y_28Si)² to the right hand side of Equation (14). Other α-chain non-major elements may be added in a similar manner.

Deviation of Y_e from 0.5 is primarily due to the major element Y_54Fe with contributions from other non-major SiG and FeG elements. For a more accurate determination of Y_e, one can use the mass conservation, Equation (1), to determine the abundances of these non-major isotopes. For example, consider the case when Y_56Ni and Y_30Si are to be included. Treating Y_p and Y_n as trace abundance in the QNSE regions, Equation (1) becomes

$$\begin{eqnarray} &&Y_{{\rm 28Si}} \left[ 28 + 32\frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} + 40\frac{Y_{{\rm 40Ca}}}{Y_{{\rm 28Si}}} + 54\frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}} + 58\frac{Y_{{\rm 58Ni}}}{Y_{{\rm 28Si}}} \right]\nonumber\\ &&\quad + Y_{{\rm 56Ni}}\left[ 56 + 30\frac{Y_{{\rm 30Si}}}{Y_{{\rm 56Ni}}} \right] = 1, \end{eqnarray} \tag{ 16 }$$

which we are going to solve for Y_56Ni in the QNSE region. From Equation (3) the abundance ratio Y_30Si/Y_56Ni may be written as

$$\begin{equation} \frac{Y_{{\rm 30Si}}}{Y_{{\rm 56Ni}}}=K_{3}\frac{Y_{{\rm 28Si}}}{Y_{{\rm 56Ni}}}\frac{Y_{{\rm 58Ni}}}{Y_{{\rm 56Ni}}} = K_{3} \left(\frac{Y_{{\rm 28Si}}}{Y_{{\rm 56Ni}}} \right)^2 \frac{Y_{{\rm 58Ni}}}{Y_{{\rm 28Si}}}, \end{equation} \tag{ 17 }$$

where K₃ ≈ exp  (− 39.3/T₉). Substituting Equation (17) into Equation (16) and using previous relations gives

$$\begin{eqnarray} && 56 Y_{{\rm 56Ni}}+ 30 K_{3} \frac{Y_{{\rm 28Si}}^{2}}{Y_{{\rm 56Ni}}}\left(\Psi \frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} \frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}} \right) = 1-Y_{{\rm 28Si}}\nonumber\\ &&\times \left[ 28 + 32\frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} + 40 \frac{Y_{{\rm 40Ca}}}{Y_{{\rm 32S}}}\frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} + 54\frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}} + 58\Psi \frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} \frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}} \right].\nonumber\\ \end{eqnarray} \tag{ 18 }$$

Multiplying Equation (18) by Y_56Ni thus yields a simple quadratic equation, $56 Y_{{\rm 56Ni}}^2 - b Y_{{\rm 56Ni}}+ c = 0$, with $c = 30 K_{3}Y_{{\rm 28Si}}^2 \ Y_{{\rm 58Ni}}/Y_{{\rm 28Si}}$ and b being the right hand side of Equation (18). Taking Y_56Ni as the positive root and substituting it into Equation (17) then gives Y_30Si. The derived Y_56Ni and Y_30Si abundances may then be used to refine the estimate for Y_e by adding the now known terms to Equation (14):

$$\begin{eqnarray} Y_{\rm e}&=& Y_{{\rm 28Si}}\left[ 14 + 16 \frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} + 20 \frac{Y_{{\rm 40Ca}}}{Y_{{\rm 32S}}}\frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} + 26\frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}}\right. \nonumber \\ &&\left. + 28 \Psi \frac{Y_{{\rm 32S}}}{Y_{{\rm 28Si}}} \frac{Y_{{\rm 54Fe}}}{Y_{{\rm 28Si}}} \right] + Y_{{\rm 56Ni}}\left[ 28 + 14\frac{Y_{{\rm 30Si}}}{Y_{{\rm 56Ni}}} \right]. \end{eqnarray} \tag{ 19 }$$

Abundances of other non-major elements may be added in a similar manner to the example given above to improve the determination of Y_e.

In this section we seek a simple analytic relation between the abundances of the major QNSE elements, Y_28Si, Y_32S, and Y_40Ca, with respect to Y_e. We begin by re-writing Equation (3) as

$$\begin{eqnarray} &&\frac{1}{f(\rho,T)}\frac{Y_{AZ}}{Y_{A^{\prime }Z^{\prime }}}= Y_{\rm n}^{A-A^{\prime }-(Z-Z^{\prime })}Y_{\rm p}^{Z-Z^{\prime }}. \end{eqnarray} \tag{ 20 }$$

We may assume without loss of generality that A > A' and Z > Z'. Now let

$$\begin{eqnarray} &&w = \sum _{i \ne {\rm protons}} Z_{i} Y_{A_{i}Z_{i}}, \end{eqnarray} \tag{ 21 }$$

which is identical to the definition of the electron fraction Y_e but without the free protons. We define $v=Y_{A^{\prime }Z^{\prime }} f(\rho,T)$. Therefore, with this notation Equation (20) becomes

$$\begin{equation} Y_{A_iZ_i} = v(1-Y_{\rm e}- w)^{A_i-A^{\prime }-(Z_i-Z^{\prime })}(Y_{\rm e}- w)^{Z_i-Z^{\prime }}. \end{equation} \tag{ 22 }$$

Multiplying by Z_i and summing,

$$\begin{eqnarray} w &=& \sum _{i \ne {\rm protons}} Z_i Y_{A_i,Z_i}\nonumber \\ & =&\sum _{i \ne {\rm protons}} Z_{i}v_{i} (1-Y_{\rm e}- w)^{(A_i-A^{\prime })-(Z_i-Z^{\prime })}(Y_{\rm e}- w)^{Z_i-Z^{\prime }}.\nonumber \\ \end{eqnarray} \tag{ 23 }$$

Since Y_e → 0.5, and Y_e < 0.5, therefore, 0 < (1 − Y_e − w) < 1 and also that 0 < (Y_e − w) < 1. This implies that the right hand side of Equation (22) has the most contribution from terms with the smallest values of (Z_i − Z') and (A_i − A') − (Z_i − Z'). Note that we have chosen Z_i > Z'. For most major QNSE elements we may then choose A' = 2Z' amd A_i = 2Z_i. For the SiG group pair ²⁸Si and ³²S, Equation (23) becomes

$$\begin{eqnarray} &&w = v Z_{\rm 32S}(1 - Y_{\rm e}- w)^2 (Y_{\rm e}- w)^2, \end{eqnarray} \tag{ 24 }$$

which is quartic in w. To order $Y_{\rm e}^2$, Equation (24) has the solution

$$\begin{equation} w= \frac{1}{2v Z_{\rm 32S}} [ \left(2Y_{\rm e}+1\right)\pm \sqrt{\left(4Y_{\rm e}v Z_{\rm 32S}+1 \right)} ]. \end{equation} \tag{ 25 }$$

Substituting Z_32S = 16 and expanding the square root leads to a zeroth order term that is a constant and a first order term that is linear in Y_e. From the expression for w in Equation (23), the largest contribution comes from Z_28SiY_28Si. We thus identify Y_28Si as the constant term and the linear term in Y_e with Y_32S. Finally, we identify Y_40Ca, keeping the higher order terms in Y_e in Equation (24).

We begin with the W7 model (Nomoto et al. 1984; Thielemann et al. 1986; Iwamoto et al. 1999) because the synthetic light curves and spectra from W7 models have been extensively analyzed (e.g., Nugent et al. 1997; Hachinger et al. 2009; Jack et al. 2011; van Rossum 2012). W7 is a 1D explosion model with a parameterized flame speed that captures the stratified ejecta observed in Branch normal SNIa (Branch et al. 1993). For our purposes, a model is a series of temperature and density snapshots from ignition at time = 0 s to homologous expansion at time = 4.1 s. The W7 model assumes a solar ²²Ne mass fraction uniformly distributed throughout the white dwarf. Our W7-like models change the assumed value of the uniformly distributed ²²Ne mass fraction. Specifically, we control the value of the electron fraction Y_e, which is set by the original composition and the pre-explosive convective simmering (Timmes et al. 2003; Piro & Bildsten 2008; Chamulak et al. 2008; Townsley et al. 2009; Walker et al. 2012), by setting the mass fraction of ²²Ne: Y_e = 0.5 − X(²²Ne)/22 = 0.5 − Q · X(²²Ne)_☉/22, where Q is a multiplier on the solar ²²Ne mass fraction X(²²Ne).

The nucleosynthesis of these W7-like models (see Figure 1) is calculated by integrating a 489 isotope nuclear reaction network (Timmes 1999) over the thermodynamic trajectories of each Lagrangian mass shell. This post-processing calculation is not precisely self-consistent because the W7 thermodynamic trajectories have the built-in assumption of an energy release from the original W7 model, carbon+oxygen material complimented with a solar ²²Ne mass fraction. While changes to the abundance of ²²Ne, hence Y_e, slightly influence the energy generation rate (Hix & Thielemann 1996, 1999), burned material still reaches QNSE conditions and our analysis should still hold.

First we test the validity of QNSE in the abundances synthesized in our W7-like SNIa models. We use local values of ρ and T from these models to construct the QNSE predicted abundances.

Figure 2 shows the final mass fractions of the SiG elements ²⁸Si, ³²S, and ⁴⁰Ca between 0.8 M_☉ and 1.2 M_☉. Two cases for each element are shown, Q = 1.0 and 2.0, representing 1.0 and 2.0 times the solar ²²Ne abundance, respectively. Solid lines represent the results from post-processed W7-like models and the symbols represent the results from our QNSE solutions. Figure 3 shows the final mass fractions of FeG elements ⁵⁴Fe and ⁵⁶Ni over the same mass range and white dwarf metallicities as in Figure 2. These figures show ⁴⁰Ca and ⁵⁴Fe have the largest systematic changes (up to a factor of two depending on the mass shell) within the silicon-rich region as the electron fraction varies. Also, a major conclusion from Figure 3 is that ⁵⁴Fe is the only iron isotope and the most abundant FeG element in the QNSE regime bound by regimes enclosing 0.8 M_☉ and 1.1 M_☉. This relation is also supported by Mazzali et al. (2014).

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A weakly varying Φ (see Equation (9)) implies dΦ/dT  ≃  0. It follows that the constituent ratios Y_28Si/Y_32S and Y_32S/Y_40Ca reach an extremum at the same temperature. This property is evident in Figure 2: when there is a deflection in Y_28Si/Y_32S with mass coordinate there is a corresponding variation in Y_32S/Y_40Ca. In addition, the individual abundances Y_28Si, Y_32S, and Y_40Ca reach an extremum at the same (ρ,T) point where the ratios reach an extremum. In general, isotopes with A = 2Z will have a QNSE abundance Y_AZ that scales as (Y_pY_n)ⁿ, where n is a positive or negative integer. These isotopes can be expressed as a function of Y_28Si/Y_32S (Equation (9) is one example) and will reach an extremum at the same (ρ,T) point as Y_28Si. The only assumption in deriving these properties is that the system achieves QNSE conditions, where most of the SiG elements are synthesized. The values of the ratios Y_28Si/Y_32S and Y_32S/Y_40Ca at the extremum will, of course, depend on the details of the explosion. Treating Ψ (see Equation (11)) as a quasi-constant and following arguments similar for the SiG elements, one finds Y_54Fe, Y_58Ni, and Y_28Si also reach an extremum at the same (ρ,T) point, as shown in Figure 3.

Next we explore the global abundances of the SiG and FeG elements with Y_e when our W7-like models reach homologous expansion at t = 4.1 s. Figure 4 shows the total ²⁸Si, ³²S, and ⁴⁰Ca molar abundances ejected as a function of the Y_e. The curves correspond to post-processing the W7-like thermodynamic trajectories and the symbols are the results from our analytical QNSE model. Both the post-processing and the model independent QNSE results suggest a nearly constant ²⁸Si yield with respect to Y_e, a systematic quasi-linear ³²S yield with respect to Y_e, and a more complex trend for the global abundance of ⁴⁰Ca with Y_e. Among the SiG elements, ⁴⁰Ca has the largest sensitivity to the electron fraction, in agreement with the trends seen in the local abundances. These results are in accord with trends explored in Section 2.4.

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Figure 1 shows the local abundances of the major SiG and FeG elements between mass shells 0.7 M_☉ and 1.3 M_☉ at t = 1.125 s (solid colored lines) and t = 4.10 s (dashed black lines) in one of our W7-like models. At t = 1.25 s the burning front has just passed over the 1.28 M_☉ mass coordinate and most of the synthesized ²⁸Si, ³²S, and ⁴⁰Ca have reached their equilibrium abundances. In this mass region, the peak temperatures, (3–5) × 10⁹ K, and peak densities, (2–4) × 10⁷ g cm⁻³, ensure QNSE conditions (Nomoto et al. 1984; Thielemann et al. 1986). The choice of t = 1.125 s is arbitrary and can be replaced by any epoch in any model when the material reaches QNSE conditions. At t = 4.10 s the explosion has entered homologous expansion and synthesis of all the elements has stopped due to the decreasing temperature. Complete freeze-out does not occur for ⁵⁴Fe interior to 0.95 M_☉ at t = 1.25 s due to residual weak reactions. Figure 1 suggests that abundances generated when QNSE conditions apply are preserved during the subsequent freeze-out. The abundance levels at this epoch may be reflected in the observed spectra over subsequent days. Therefore, applying the QNSE equations to recover Y_e from the major SiG and FeG elements as we have done in Section 2 is justified.

The next step to mapping the observed abundances into Y_e at explosion involves estimating the change in flux and luminosity as Y_e changes. It is important to estimate this change which dictates the level of resolution in Y_e mapping accessible from the observed abundances, if determined accurately from the spectra. There are several complicating factors that contribute to estimating accurate abundances from an observed spectra. Below we estimate the change in flux with respect to Y_e using synthetic spectra from radiative transfer modeling.

We use the PHOENIX radiation transfer code (Hauschildt et al. 1997; Jack et al. 2009; Jack 2009; Jack et al. 2011) to produce synthetic spectra from our W7-like models. The thermodynamic profiles of the W7-like models end when the explosion reaches homologous expansion, about 4 s after ignition. The density, velocity, and abundance profiles are then homologously and adiabatically expanded to 5 days after the explosion using analytical expressions that account for the local decay of ⁵⁶Ni and ⁵⁶Co. This is a reasonable assumption for SNIa after the initial break out (Arnett 1982). From day 5 onward, we address LTE radiative transfer through the expanding remnant in 0.5 day increments to about 21 days to calculate synthetic spectra. At each of these 0.5 day increments, we solve an energy equation that includes the contribution of the adiabatic expansion, the energy deposition by γ-rays, and absorption and emission of radiation. As a result we always obtain a model atmosphere that is in radiative equilibrium.

Figure 5 shows the synthetic spectra at day 15, near peak luminosity, for the W7-like models with Q = 0.0, 1.0, 2.0, and 4.0, representing 0.0, 1.0, 2.0, and 4.0 times the solar ²²Ne abundance, respectively. Changes in the synthetic spectra shown in Figure 5 are due to changes in the abundance of a given element. The SiG material has post-explosion homologous expansion velocities ranging between 9000–13, 000 km s⁻¹ (e.g., Nugent et al. 1997; Hachinger et al. 2009; Jack et al. 2011; van Rossum 2012). The continuum optical depth of the SiG material is between 0.2–0.8, which was evaluated from the continuum opacity at 5500 Å at peak luminosity. The wavelength resolution of the PHOENIX calculation is set to 1 Å and the distance for the flux scale shown on the y-axis is the velocity of the outermost expanding shell times the time since the explosion. The PHOENIX model uses complete redistribution, with the same emission and absorption profile function with respect to the frequency spread around the center of the line frequency. Figure 5 suggests that the Ca ii feature changes the most with Y_e when compared to S II and Si ii features. The Si ii line changes the least with variations in Y_e. These trends, Ca being the most sensitive, S having a near linear dependence, and Si the least sensitive, are a reflection of the nucleosynthesis trends shown in Figure 4. We stress the LTE synthetic spectrum of our W7-like models is not compared to observational data, and thus our W7-like models may not accurately model real SNIa.

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