On Presolar Stardust Grains from CO Classical Novae

We compute the nucleosynthesis using a reaction network consisting of 213 nuclides, ranging from p, n, ⁴He, to ⁵⁵Cr. These nuclides are linked by 2373 nuclear interactions (proton and α-particle captures, β-decays, light-particle reactions, etc.). Thermonuclear reaction rates are adopted from STARLIB v6.5 (09/2017).⁸ This library has a tabular format and contains reaction rates and rate probability density functions on a grid of temperatures between 10 MK and 10 GK (Sallaska et al. 2013). The probability densities can be used to derive statistically meaningful reaction rate uncertainties at any desired temperature. Many of the reaction rates important for the present work that are listed in STARLIB have been computed using a Monte Carlo method, which randomly samples all experimental nuclear physics input parameters (Longland et al. 2010). Most of the reaction rates important for studying hydrogen burning in CO novae are based on experimental nuclear physics information and provide a reliable foundation for robust predictions. For some reactions of interest to classical novae (Iliadis 2015), however, experimental rates are not available yet, and the rates included in STARLIB are adopted from nuclear statistical model calculations using the code TALYS (Goriely et al. 2008). In such cases, a reaction rate uncertainty factor of 10 is assumed.

Stellar weak interaction rates, which depend on both temperature and density, for all species in our network are adopted from Oda et al. (1994) and, if not listed there, from Fuller et al. (1982). The stellar weak decay constants are tabulated at temperatures from T = 10 MK to 30 GK, and densities of $\rho {Y}_{e}=10\mbox{--}{10}^{11}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, where Y_e denotes the electron mole fraction. For all stellar weak interaction rates, we assumed a factor of 2 uncertainty. Short-lived nuclides (e.g., ¹³N (T_1/2 = 10 min), ¹⁴O (${T}_{1/2}=71\,{\rm{s}}$), ¹⁵O (${T}_{1/2}=122\,{\rm{s}}$), ¹⁷F (${T}_{1/2}=64\,{\rm{s}}$), and ¹⁸F (${T}_{1/2}=110\,\min$)) present at the end of a network calculation were assumed to decay to their stable daughter nuclides.

To explore the effects of thermonuclear reaction rate uncertainties, we perform some of our calculations by randomly sampling all rates simultaneously using the rate probability densities provided by STARLIB. This method is discussed in detail in Iliadis et al. (2015) and was recently applied to explain abundance anomalies in globular clusters (Iliadis et al. 2016). It suffices to mention here that we adopt a lognormal distribution for the nuclear rates, given by

$$\begin{eqnarray}&&f(x)=\displaystyle \frac{1}{\sigma \sqrt{2\pi }}\displaystyle \frac{1}{x}{e}^{-{(\mathrm{ln}x-\mu )}^{2}/(2{\sigma }^{2})},\end{eqnarray} \tag{ 1 }$$

where the lognormal parameters μ and σ determine the location and the width, respectively, of the distribution. For a lognormal probability density, samples, i, of a nuclear rate, y, are computed from

$$\begin{eqnarray}&&{y}_{i}={y}_{\mathrm{med}}{(f.u.)}^{{p}_{i}},\end{eqnarray} \tag{ 2 }$$

where y_med and $f.u.$ are the median value and the factor uncertainty, respectively, which are both provided by STARLIB. The quantity p_i is a random variable that is normally distributed (i.e., according to a Gaussian distribution with an expectation value of zero and a standard deviation of unity). We emphasize that the factor uncertainty of experimental Monte Carlo reaction rates depends explicitly on stellar temperature (Iliadis et al. 2010; Longland 2012).

We adopt a simple, one-zone analytical parameterization for the thermodynamic trajectories of the explosion,

$$\begin{eqnarray}&&T(t)={T}_{\mathrm{peak}}{e}^{-t/{\tau }_{T}}\,,\,\rho (t)={\rho }_{\mathrm{peak}}{e}^{-t/{\tau }_{\rho }},\end{eqnarray} \tag{ 3 }$$

where t ≥ 0 is the time since peak temperature, T_peak, or peak density, ρ_peak; τ_T and τ_ρ are the times at which temperature and density, respectively, have fallen to 1/e of their peak values. Notice that we do not assume an adiabatic expansion, since we treat τ_T and τ_ρ as independent parameters. This is consistent with the results of one-dimensional hydrodynamic nova simulations, which predict non-adiabatic T–ρ evolutions.

It is necessary to demonstrate that our simple simulation has some predictive power in regard to nova nucleosynthesis. In a first step, we generated a hydrodynamic CO nova model using the one-dimensional code SHIVA (José & Hernanz 1998), assuming a white dwarf mass and initial luminosity of M_WD = 1.0 M_⊙ and ${L}_{\mathrm{WD}}={10}^{-2}\,{L}_{\odot }$, respectively, and accretion of solar-like material at a rate of ${\dot{M}}_{\mathrm{acc}}=2\times {10}^{-10}\,{M}_{\odot }$ yr⁻¹. The composition of the nuclear fuel was obtained by pre-mixing equal amounts of solar-like matter with carbon–oxygen white dwarf matter (assumed to be 50% ¹²C and 50% ¹⁶O, by mass). The model included 45 envelope zones containing all material involved in the thermonuclear runaway. The deepest zone achieved a peak temperature of 179 MK, while the innermost ejected zone reached a peak temperature of 163 MK. Final isotopic abundances for matter that exceeds escape velocity (i.e., the fraction of the envelope ejected) are determined 1 hr after peak temperature is achieved, and the abundance of each nuclide is mass-averaged over all ejected zones.

In a second step, we adjusted the parameters of our simple one-zone simulation by trial and error to see if we can approximately reproduce the final isotopic ratios of the multi-zone hydrodynamic model, assuming exactly the same initial abundances in both calculations. The resulting isotopic ratios for the most important elements (C, N, O, Si, and S) are shown in Figure 2. The solid lines correspond to the time evolutions predicted by the simple one-zone simulation and were obtained with the following parameter values: T_peak = 177 MK, ${\rho }_{\mathrm{peak}}=200\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, τ_T = 2500 s, τ_ρ = 38 s. The total time was 10,000 s, but the results are not sensitive to this parameter once peak temperature and density have significantly declined from their peak values. The dotted line in each panel indicates the mass-zone-averaged ejected final abundance ratios predicted by the multi-zone hydrodynamic calculation. An interesting finding is that the one-zone simulation reproduces the results of the multi-zone hydrodynamic calculation within a factor of 2. We repeated the test for other CO white dwarf masses, and even for models of ONe novae, and again obtained agreement within a factor of 2.

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This level of agreement may be at first surprising, considering that our simulation, unlike the hydrodynamic model, follows a single zone only, and disregards accretion, convection, and ejection of matter. However, recall that we are mainly interested in isotopic ratios instead of absolute isotopic or elemental abundances, which are very likely more sensitive to such effects. We emphasize that the parameters (T_peak, ρ_peak, τ_T, τ_ρ) derived from the simple simulation correspond neither to the averages over different mass zones, nor to a given zone, of the hydrodynamic simulation. They nevertheless provide, albeit crude, approximations of the physical conditions during the nuclear burning, mainly because thermonuclear reaction rates are highly sensitive to the plasma temperature.

It is also interesting to note that the value ${\rho }_{\mathrm{peak}}\,=200\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ of the exponentially decaying density profile in Figure 2 does not correspond to any maximum density in the one-dimensional hydrodynamic simulation, but is approximately equal to the density at maximum temperature in the innermost zones of the hydrodynamic calculation. In the latter simulation, the density declines from a maximum value, which typically amounts to a few thousand grams per cubic centimeter and is achieved well before the temperature peaks, to a very small value.

A factor of 2 agreement is sufficient for the purposes of the present work, as will be shown below. The advantage of our simple procedure is that we can independently sample over the parameters and repeat the simulation many times. If a given stardust grain has indeed a nova paternity, we would expect that certain combinations of parameter values (T_peak, ρ_peak, τ_T, τ_ρ) approximately reproduce the measured isotopic ratios, with magnitudes near the ranges typical for classical novae.

Before we can discuss the presolar stardust grain data, however, we need to introduce three more parameters that will be important for our study: the ¹²C/¹⁶O ratio of the outer white dwarf core, the mixing of matter at the interface of the white dwarf and the envelope, and the dilution of the ejecta by mixing with solar-like matter.

3.3.1. The White Dwarf Composition

3.3.2. Mixing between White Dwarf and Accreted Matter during the Flash

Spectroscopic observations show that CNO elements are considerably enriched relative to hydrogen in many nova ejecta (Gehrz et al. 1998). This enrichment plays a critical role for the dynamic ejection of a portion of the envelope and presumably results from mixing of the outer core white dwarf matter with the accreted matter (red region in Figure 1(b)). Recent two- and three-dimensional simulations now yield encouraging results by demonstrating that Kelvin–Helmholtz instabilities during the thermonuclear runaway can lead to an enrichment of the accreted envelope with material from the underlying white dwarf at levels that approximately agree with observations (Glasner & Livne 1995; Casanova et al. 2011, 2016; Glasner et al. 2012).

So far, multi-dimensional nova simulations incorporate very small nuclear reaction networks (≈30 nuclides only), and are not suitable for studying the nucleosynthesis in detail. Consequently, one-dimensional simulations are indispensable for this purpose, but cannot account self-consistently for the mixing at the interface between the outer white dwarf core and the accreted matter. Most one-dimensional simulations work around this problem by artificially enriching the envelope with outer core matter to a predetermined degree. The enriched matter is then accreted and its history is followed through the nuclear burning and mass ejection stages.

Frequently, one-dimensional nova simulations assume that the accreted matter from the companion and the white dwarf core material pre-mix with equal mass fractions. Recent work, albeit in the context of ONe novae, hinted at a pre-mixing fraction of 25% white dwarf matter and 75% accreted matter (Kelly et al. 2013), which provides a significantly better fit to the measured elemental abundances in several, but not all, nova ejecta.

While these are encouraging first steps, we are far from being able to predict the degree of element mixing in a given observed nova. In particular, the pre-mixing parameter, f_pre, is poorly constrained at present. We define it by mixing one part of outer white dwarf core matter with f_pre parts of accreted solar-like matter,

$$\begin{eqnarray}&&{X}_{\mathrm{pre}}\equiv \displaystyle \frac{{X}_{\mathrm{WD}}+{f}_{\mathrm{pre}}{X}_{\mathrm{acc}}}{1+{f}_{\mathrm{pre}}},\end{eqnarray} \tag{ 4 }$$

where X_i denotes mass fraction. We will randomly sample this parameter over a reasonable range (0.5 ≤ f_pre ≤ 5.0) to see which values best reproduce the stardust grain data. The outer bounds of this interval correspond to white dwarf admixtures of 67% and 17%, respectively.

3.3.3. Dilution of the Ejecta

We already mentioned that previous classical nova simulations result in much more anomalous isotopic ratios compared to the values measured in nova candidate grains. To explain the observations, an additional mixing episode, after the outburst, has been postulated (which we will term "post-mixing"), whereby the ejecta processed by nuclear burning mix with more than 10 times the amount of unprocessed, solar-like matter before grain condensation (Amari et al. 2001). However, the source and mechanism of this potential dilution is not understood.

The grains we are considering here condensed at temperatures well over 1000 K, and therefore it is difficult to explain the formation of SiC or olivine grains by mixing of nova ejecta with the interstellar medium.

One idea was pursued by Figueira et al. (2017), who studied the collision of the nova ejecta initially with the accretion disk and subsequently with the companion. They found that the matter escaping from the binary system is predominantly composed of the ejecta (i.e., the contribution of the accretion desk or the companion is small). Under these conditions, we expect on average only minor post-mixing, although enhanced dilution may perhaps occur in local regions. If grains can condense in this environment, we nevertheless expect that only a small fraction will show signatures of significant post-mixing.

More studies of this important issue are needed, since we neither know the source of the solar-like matter for post-mixing, nor if any post-mixing took place at all. At this time, we conclude the following. If we consider two measured stardust grains, and for one grain all data agree with CO nova simulations without the need for any post-mixing, while the other grain requires dilution to match data to the CO nova simulations, then the former grain is more likely to originate from a CO nova. We will return to this argument in Section 4.2.

Since the post-mixing process is poorly constrained, our simulations will account for it using a post-mixing parameter, f_post, defined by

$$\begin{eqnarray}&&{X}_{\mathrm{post}}\equiv \displaystyle \frac{{X}_{\mathrm{proc}}+{f}_{\mathrm{post}}{X}_{\mathrm{pris}}}{1+{f}_{\mathrm{post}}},\end{eqnarray} \tag{ 5 }$$

where X_proc and X_pris denote the mass fractions from the reaction network output (i.e., processed matter) and the pristine matter (i.e., solar-like), respectively. We will sample this parameter over a range of 0 ≤ f_post ≤ 10⁴ to see which values best reproduce the stardust grain data.

Isotopic data for all presolar stardust grains that have been suggested over the years to originate from novae are compiled in Table 2. The data are separated according to grain chemistry (SiC, silicate, graphite, and oxide). As already noted, we have no unambiguous evidence linking any of these grains to a nova paternity. Neither can we exclude unambiguously a nova paternity for many other grains among the thousands of stardust samples measured so far. Our goal is to investigate the conditions, if any, that could give rise to the measured isotopic anomalies.

Table 2.  Measured Isotopic Ratios in Nova Candidate Presolar Stardust Grains

Grain	12C/13C	14N/15N	17O/16O	18O/16O	${\delta }^{25}\mathrm{Mg}{/}^{24}\mathrm{Mg}$	${\delta }^{29}\mathrm{Si}{/}^{28}\mathrm{Si}$	${\delta }^{30}\mathrm{Si}{/}^{28}\mathrm{Si}$	${\delta }^{33}{\rm{S}}{/}^{32}{\rm{S}}$	${\delta }^{34}{\rm{S}}{/}^{32}{\rm{S}}$
			(×10−4)	(×10−3)
SiC Grains
AF15bB-429-3 (1)	9.4 ± 0.2	⋯	⋯	⋯	⋯	28 ± 30	1118 ± 44	⋯	⋯
AF15bC-126-3 (1)	6.8 ± 0.2	5.22 ± 0.11	⋯	⋯	⋯	−105 ± 17	237 ± 20	⋯	⋯
Ag2 (2)	2.5 ± 0.1	7.0 ± 0.1	⋯	⋯	⋯	−304 ± 26	319 ± 38	−92 ± 222	162 ± 106
Ag2_6 (2)	16.0 ± 0.4	9.0 ± 0.1	⋯	⋯	⋯	−340 ± 57	263 ± 82	48 ± 334	−394 ± 106
KJC112 (1)	4.0 ± 0.2	6.7 ± 0.3	⋯	⋯	⋯	⋯	⋯	⋯	⋯
KJGM4C-100-3 (1)	5.1 ± 0.1	19.7 ± 0.3	⋯	⋯	⋯	55 ± 5	119 ± 6	⋯	⋯
KJGM4C-311-6 (1)	8.4 ± 0.1	13.7 ± 0.1	⋯	⋯	⋯	−4 ± 5	149 ± 6	⋯	⋯
G1614 (2)	9.2 ± 0.07	35.0 ± 0.7	⋯	⋯	⋯	34 ± 5	121 ± 6	⋯	⋯
G1697 (2)	2.5 ± 0.01	33.0 ± 0.8	⋯	⋯	⋯	−42 ± 12	40 ± 15	⋯	⋯
G1748 (2)	5.4 ± 0.02	19.0 ± 0.2	⋯	⋯	⋯	21 ± 4	83 ± 5	⋯	⋯
G270_2 (2)	11.0 ± 0.3	13.0 ± 0.3	⋯	⋯	⋯	−282 ± 101	−3 ± 131	−615 ± 385	−542 ± 175
G283 (2)	12.0 ± 0.1	41.0 ± 0.5	⋯	⋯	⋯	−15 ± 3	75 ± 4	⋯	⋯
G278 (2)	1.90 ± 0.03	7.0 ± 0.2	⋯	⋯	⋯	1570 ± 112	1673 ± 138	⋯	⋯
G1342 (2)	6.40 ± 0.08	7.00 ± 0.14	⋯	⋯	⋯	445 ± 34	513 ± 43	⋯	⋯
GAB (2)	1.60 ± 0.02	13.0 ± 0.2	⋯	⋯	⋯	230 ± 6	426 ± 7	−82 ± 279	−6 ± 122
G240-1 (2)	1.00 ± 0.01	7.0 ± 0.1	⋯	⋯	⋯	138 ± 14	313 ± 23	⋯	⋯
M11-151-4 (3)a	4.02 ± 0.07	11.6 ± 0.1	⋯	⋯	⋯	−438 ± 9	510 ± 18	⋯	⋯
M11-334-2 (3)b	6.48 ± 0.08	15.8 ± 0.2	⋯	⋯	⋯	−489 ± 9	−491 ± 18	⋯	⋯
M11-347-4 (3)	5.59 ± 0.13	6.8 ± 0.2	⋯	⋯	⋯	−166 ± 12	927 ± 30	⋯	⋯
M26a-53-8 (4)	4.75 ± 0.23	⋯	⋯	⋯	⋯	10 ± 13	222 ± 25	⋯	⋯
Silicate Grains
1_07 (5)	⋯	⋯	49.1 ± 3.6	1.36 ± 0.19	⋯	⋯	⋯	⋯	⋯
4_2 (6)	⋯	⋯	128.0 ± 1.4	1.74 ± 0.05	1025 ± 29	24 ± 40	134 ± 52	⋯	⋯
4_7 (6)	⋯	⋯	149.0 ± 2.0	1.30 ± 0.06	213 ± 56	136 ± 46	−49 ± 80	⋯	⋯
A094_TS6 (7)	⋯	⋯	95.4 ± 1.1	1.50 ± 0.01	⋯	29 ± 43	43 ± 54	⋯	⋯
AH-106a (8)	⋯	⋯	50.1 ± 2.2	1.78 ± 0.07	⋯	15 ± 59	80 ± 67	⋯	⋯
B2-7 (9)	⋯	⋯	133.0 ± 1.0	1.43 ± 0.04	⋯	21 ± 56	57 ± 69	⋯	⋯
GR95_13_29 (10)	⋯	⋯	62.5 ± 2.5	1.96 ± 0.14	79 ± 21	−16 ± 63	379 ± 92	⋯	⋯
Graphite Grains
KFB1a-161 (4), (11)	3.8 ± 0.1	312 ± 43	⋯	⋯	−28 ± 62	−133 ± 81	37 ± 87	⋯	⋯
KFC1a-551 (4)c	8.46 ± 0.04	273 ± 8	⋯	⋯	−157 ± 443	84 ± 54	761 ± 72	⋯	⋯
LAP-149 (12)	1.41 ± 0.01	941 ± 81	3.86 ± 0.34	1.94 ± 0.07	⋯	−8 ± 24	−23 ± 29	−23 ± 143	6 ± 70
Oxide Grains
12-20-10 (13)	⋯	⋯	88.0 ± 3.0	1.18 ± 0.11	⋯	⋯	⋯	⋯	⋯
8-9-3 (13)	⋯	⋯	51.4 ± 1.1	1.89 ± 0.07	−66 ± 21	⋯	⋯	⋯	⋯
C4-8 (13)	⋯	⋯	440.4 ± 1.2	1.10 ± 0.02	949 ± 8	⋯	⋯	⋯	⋯
KC23 (14)	⋯	⋯	58.5 ± 1.8	2.19 ± 0.06	45 ± 35	⋯	⋯	⋯	⋯
KC33 (14)	⋯	⋯	82.2 ± 0.6	0.68 ± 0.08	⋯	⋯	⋯	⋯	⋯
MCG67 (10)	⋯	⋯	47.3 ± 1.4	1.77 ± 0.03	⋯	⋯	⋯	⋯	⋯
MCG68 (10)	⋯	⋯	62.6 ± 1.1	1.89 ± 0.02	⋯	⋯	⋯	⋯	⋯
S-C6087 (15)	⋯	⋯	75.2 ± 0.3	2.18 ± 0.03	36 ± 22	⋯	⋯	⋯	⋯
T54 (16)	⋯	⋯	141 ± 5	0.5 ± 0.2	⋯	⋯	⋯	⋯	⋯
Solar	89	272	3.8	2.0	0.0	0.0	0.0	0.0	0.0

Notes. Presented errors are 1σ. Some ratios are presented as deviations from solar abundances in permil, ${\delta }^{i}X{/}^{j}X\equiv [{(}^{i}X{/}^{j}X)/{({}^{i}X{/}^{j}X)}_{\odot }-1]\times 1000$. ¹²C/¹³C and ¹⁴N/¹⁵N (ratio in air) are from José et al. (2004) and references therein; ¹⁷O/¹⁶O and ¹⁸O/¹⁶O solar values are from Leitner et al. (2012).

^aAdditional isotopic ratios for Grain M11-151-4 are given in Nittler & Hoppe (2005): ²⁶Al/²⁷Al = 0.27 ± 0.05, ${\delta }^{46}\mathrm{Ti}/{\delta }^{48}\mathrm{Ti}=28\pm 59$, ${\delta }^{47}\mathrm{Ti}/{\delta }^{48}\mathrm{Ti}=215\pm 57$, ${\delta }^{49}\mathrm{Ti}/{\delta }^{48}\mathrm{Ti}=82\pm 55$, ${\delta }^{50}\mathrm{Ti}/{\delta }^{48}\mathrm{Ti}=-100\pm 123$. ^bAdditional isotopic ratios for Grain M11-334-2 are given in Nittler & Hoppe (2005): ²⁶Al/²⁷Al = 0.39 ± 0.06, ${\delta }^{42}\mathrm{Ca}/{\delta }^{40}\mathrm{Ca}=-70\pm 200$, ${\delta }^{44}\mathrm{Ca}/{\delta }^{40}\mathrm{Ca}=535\pm 150$, ${\delta }^{46}\mathrm{Ti}/{\delta }^{48}\mathrm{Ti}=-61\pm 33$, ${\delta }^{47}\mathrm{Ti}/{\delta }^{48}\mathrm{Ti}=-5\pm 36$, ${\delta }^{49}\mathrm{Ti}/{\delta }^{48}\mathrm{Ti}=380\pm 47$, ${\delta }^{50}\mathrm{Ti}/{\delta }^{48}\mathrm{Ti}=-20\pm 59$. ^cMultiple values exist for the ¹⁶O/¹⁸O (or ¹⁸O/¹⁶O) ratio of Grain KFC1a-551 and are not given here (see Amari et al. 1995; M. Bose 2017, private communication; and http://presolar.wustl.edu/∼pgd/welcome.html).

References. (1) Amari et al. (2001), (2) Liu et al. (2016), (3) Nittler & Hoppe (2005), (4) Nittler & Alexander (2003), (5) Vollmer et al. (2007), (6) Nguyen & Messenger (2014), (7) Nguyen et al. (2007), (8) Nguyen et al. (2010), (9) Bose et al. (2010), (10) Leitner et al. (2012), (11) José & Hernanz (2007), (12) Haenecour et al. (2016), (13) Gyngard et al. (2010), (14) Nittler et al. (2008), (15) Choi et al. (1999), (16) Nittler et al. (1997).

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The values listed for C, N, and O represent isotopic number abundance ratios, while for Mg, Si, and S, the data correspond to parts-per-thousand deviations from solar matter, for example,

$$\begin{eqnarray}&&\delta ({}^{25}\mathrm{Mg}{/}^{24}\mathrm{Mg})\equiv {\delta }^{25}\mathrm{Mg}\\ &&\quad \equiv \,\left[\displaystyle \frac{{({}^{25}\mathrm{Mg}{/}^{24}\mathrm{Mg})}_{\exp }}{{({}^{25}\mathrm{Mg}{/}^{24}\mathrm{Mg})}_{\odot }}-1\right]\times 1000,\end{eqnarray} \tag{ 6 }$$

where the most abundant isotope of the element (i.e., of even mass number) appears in the denominator.

We chose to exclude consideration of δ²⁶Mg values and inferred ²⁶Al/²⁷Al ratios. First, for grains with significant amounts of both Mg and Al, some of the observed ²⁶Mg may have condensed originally as ²⁶Al. In such cases, we cannot simply add the simulated ²⁶Mg and ²⁶Al abundances and compare the total to the measured δ²⁶Mg values because the Al and Mg may have condensed in a particular grain with different efficiencies. Although, in principle, this can be taken into account if the Al/Mg ratio of a grain is known (e.g., for MgAl₂O₄ grains; Zinner et al. 2005), this ratio has not been reported for many of the grains. Second, even for grains like SiC with high Al/Mg ratios and hence purely radiogenic ²⁶Mg, Al contamination has been shown to have impacted nearly all reported isotopic measurements (Zinner & Jadhav 2013; Groopman et al. 2015), leading to an underestimate of the original ²⁶Al/²⁷Al ratios by up to a factor of 2. The magnitude of such contamination can be estimated for a given grain through detailed analysis of the original measurement data (Groopman et al. 2015), but this information has not been reported for the nova candidate grains of this study.

When comparing grain measurements to results from nucleosynthesis simulations, two important issues need to be addressed. First, we cannot reasonably expect that a simulation will precisely reproduce the grain measurements, since there are too many approximations involved in any nucleosynthesis model. If the simulation results are "close" to the data, say, within some factor, we may accept the computed results as a possible solution. Second, we need to account for the systematic bias in the grain measurements, in addition to the statistical uncertainty that is included in the reported error. Systematic effects arise from contamination (e.g., from sampling the meteorite material surrounding the grain), or from sample preparation. It is important to emphasize that this bias could move a data point into the direction of less anomalous values only (i.e., contamination will not make a grain appear more anomalous than it really is).

We will adopt the following procedure for determining approximate agreement between simulation and measurement ("acceptable solutions"). The simulated mass fraction of a nuclide with atomic number Z and mass number A, denoted by ${X}_{\mathrm{sim}}{(}^{A}Z)$, is divided and multiplied by a systematic uncertainty factor, n_sim, according to

$$\begin{eqnarray}&&{X}_{\mathrm{sim}}{(}^{A}Z)/{n}_{\mathrm{sim}}\leqslant {X}_{\mathrm{sim}}{(}^{A}Z)\\ &&\quad \leqslant \,{X}_{\mathrm{sim}}{(}^{A}Z)\times {n}_{\mathrm{sim}}.\end{eqnarray} \tag{ 7 }$$

This range is then transformed into a range of simulated δ-values, ${\delta }^{A}{Z}_{\mathrm{sim}}$,

$$\begin{eqnarray}&&[{\delta }^{A}{Z}_{\mathrm{sim}}^{\mathrm{low}},{\delta }^{A}{Z}_{\mathrm{sim}}^{\mathrm{high}}].\end{eqnarray} \tag{ 8 }$$

Next, using an experimental uncertainty factor, n_exp, we define a range for a measured value of ${\delta }^{A}{Z}_{\exp }^{\mathrm{mean}}\pm {\delta }^{A}{Z}_{\exp }^{\mathrm{err}}$,

$$\begin{eqnarray}&&[{n}_{\exp }^{(1-\pi )/2}\times {\delta }^{A}{Z}_{\exp }^{\mathrm{mean}}-{n}_{\exp }\times {\delta }^{A}{Z}_{\exp }^{\mathrm{err}},\\ &&{n}_{\exp }^{(1+\pi )/2}\times {\delta }^{A}{Z}_{\exp }^{\mathrm{mean}}+{n}_{\exp }\times {\delta }^{A}{Z}_{\exp }^{\mathrm{err}}],\end{eqnarray} \tag{ 9 }$$

where $\pi =\mathrm{sign}({\delta }^{A}{Z}_{\exp }^{\mathrm{mean}})=\pm 1$ denotes the sign of the experimental mean value. We define an acceptable solution if the two regions given by Equations (8) and (9) overlap. For the two factors containing the effects of the simulation and measurement bias, we adopt values of n_sim = 1.7 and n_exp = 2. The former value implies an uncertainty factor for a simulated abundance ratio of 1.7² ≈ 3. It was chosen to exceed the factor of 2 within which our one-zone simulations reproduce the results of the multi-zone hydrodynamic calculation (Section 3.2). As already pointed out in Section 2, we will only accept solutions for which simulated and measured δ-values overlap for all measured isotopic ratios.

To gain a better understanding of this procedure, consider the following numerical example. Suppose a value of ${\delta }^{13}{C}_{\exp }^{\mathrm{mean}}\pm {\delta }^{13}{C}_{\exp }^{\mathrm{err}}=12734\pm 167$ has been measured for a hypothetical grain. Accounting for both statistical and systematic effects, we translate this experimental result into an experimental interval of [12400, 25802], according to Equation (9). Furthermore, suppose that one among many reaction network runs results in final mass fractions of ${X}_{\mathrm{sim}}{(}^{12}C)=0.203$ and ${X}_{\mathrm{sim}}{(}^{13}C)=0.0193$. According to Equation (7), these values are translated to ranges of $0.1194\leqslant {X}_{\mathrm{sim}}{(}^{12}C)\leqslant 0.3451$ and $0.011353\leqslant {X}_{\mathrm{sim}}{(}^{13}C)\leqslant 0.03281$. The interval for the corresponding ${X}_{\mathrm{sim}}{(}^{13}C)/{X}_{\mathrm{sim}}{(}^{12}C)$ ratios is then given by [0.03289, 0.2748], resulting in a range of simulated ${\delta }^{13}{C}_{\mathrm{sim}}$ values of [1701, 21568] . In this case, the experimental and simulation ranges overlap. Only if the same applies to all other isotopic ratios measured for this particular grain do we retain the solutions (i.e., the model parameters, such as peak temperatures and densities, mixing parameters, etc.) of this particular reaction network calculation.

The nova candidate graphite grain LAP-149 has a diameter of about 1 μm and exhibits one of the lowest ¹²C/¹³C ratios ever measured (Table 2). The ¹⁴N/¹⁵N ratio is high, but the oxygen, silicon, and sulfur isotopic ratios are close to solar within experimental uncertainties. Haenecour et al. (2016) found that the C, N, Si, and S isotopic ratios could be reproduced by a CO nova model involving a 0.6 M_⊙ white dwarf with 50% pre-mixing (f_pre = 1; Equation (4)) without assuming any post-mixing. However, the measured and simulated ¹⁷O/¹⁶O and ¹⁸O/¹⁶O ratios disagreed by orders of magnitude. The other CO nova models, for white dwarf masses in the range of M_WD = 0.8–1.15 M_⊙, did not provide a match to any of the measured isotopic ratios. Notice that LAP-149 is the only nova candidate grain with eight measured isotopic ratios (see Table 2).

Our results for LAP-149 are shown in Figure 3, which was obtained after computing 25,000 network samples. The top row displays the measured isotopic ratios (red) together with the simulations. Only those simulation results are displayed that simultaneously agree with all data, according to Equations (8) and (9). The different colors for the simulation results correspond to different peak temperature values (blue: T_peak < 0.20 GK, green: T_peak ≥ 0.20 GK). The corresponding sampled model parameters are shown in the bottom row.

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We obtain acceptable solutions for a wide range of pre-mixing fractions, between f_pre = 1 and f_pre = 5 (first bottom panel), corresponding to outer white dwarf core admixtures between 50% and 16%, respectively. Post-mixing fractions are in the range of f_post = 30–100, implying a significant admixture of solar-like matter after the explosion. In particular, no acceptable solutions are obtained without post-mixing, in agreement with the findings of Haenecour et al. (2016). Acceptable peak temperature and peak density values (second bottom panel) scatter throughout the sampled ranges (150 MK ≤ T_peak ≤ 250 MK, $5\,{\rm{g}}\,{\mathrm{cm}}^{-3}\,\leqslant {\rho }_{\mathrm{peak}}\leqslant 5\times {10}^{3}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$). The 1/e exponential decay times for temperature and density (third bottom panel) scatter within the range of ${10}^{3}\,{\rm{s}}\leqslant {\tau }_{T}\leqslant {10}^{4}\,{\rm{s}}$ and ${10}^{2}\,{\rm{s}}\leqslant {\tau }_{\rho }\leqslant {10}^{4}\,{\rm{s}}$, respectively. Solutions are obtained for the entire range of sampled outer white dwarf core composition, $0.6\leqslant {X}_{\mathrm{WD}}{(}^{12}C)\leqslant 0.9$, $0.1\,\leqslant {X}_{\mathrm{WD}}{(}^{16}O)\leqslant 0.4$ (fourth bottom panel). Lastly, the ratios of elemental carbon to oxygen, after post-mixing, are in the range of X(C)/X(O) = 0.6–1.0 (fifth bottom panel).

So far, we used in the simulations for all rates of thermonuclear reactions and weak interactions the recommended (i.e., median) values provided by STARLIB. However, the nuclear rates have uncertainties, either derived from experimental nuclear physics input or from theoretical models (Section 3.1). For this reason, we repeated the above Monte Carlo procedure of computing 25,000 network samples, but this time included the random sampling of the nuclear rates according to their probability densities contained in STARLIB (Section 3). As a result, the scatter of the simulation points (black, blue, green) in Figure 3 increased slightly, but all relevant features discussed above are preserved. In other words, current reaction rate uncertainties have only a small impact on our results for this particular grain.

The above results do not prove a CO nova origin for LAP-149, because we are not considering here the production in sites other than novae (e.g., supernovae or AGB stars). But we can conclude that this grain could only have been produced by the temperature and density conditions, and compositions, typical of CO novae if the ejecta mixed with a large fraction of solar-like matter (i.e., 1 part of ejecta with at least 30 parts of solar-like matter).

We carefully repeated the random sampling using different random number seeds and total numbers of simulations. All results shown in this work, including Figure 3, are robust, in the sense that they are reproducible apart from small statistical fluctuations.

Similar to grain LAP-149 discussed in the previous section, we investigated for each of the 39 grains listed in Table 2 if our simulations can reproduce the measured isotopic ratios according to Equations (8) and (9). Three points need to be considered for the following discussion.

First, the number of measured isotopic ratios will strongly impact the likelihood that a given grain originates from a CO nova. In other words, if simulations reproduce to a similar degree the data for grains A and B, and only two isotopic ratios have been measured in grain A compared to six ratios in grain B, then the latter grain is more likely to be of CO nova paternity. Clearly, the more isotopic ratios measured, the tighter the constraints on the grain origin.

Second, the relative number of network runs ("acceptable solutions") that provide simultaneous solutions for all measured isotopic ratios of a given grain is also important in this regard. We cannot claim that the absolute number of acceptable solutions reflects the probability of a CO nova paternity. But we can conclude that a low number of solutions indicates a fine-tuning of model parameters, while a high number of solutions results from model parameter combinations that occupy a larger volume of the parameter space. In other words, the relative number of acceptable solutions reflects the likelihood of a CO nova paternity.

Third, we discussed in Section 3.3.3 that the origin and mechanism for a possible dilution of the ejecta by solar-like matter ("post-mixing") is not well understood, and that it is likely that a fraction of CO nova grains condense in the ejecta with little, if any, post-mixing. For this reason, we assume that a given grain has a higher chance of a CO nova paternity if its measured isotopic ratios can be simulated without any post-mixing.

If we allow for post-mixing of various degrees, we find acceptable solutions for almost all grains listed in Table 2. The only exceptions are grains 8-9-3 and KFC1a-551. For these, not a single acceptable solution is obtained, and thus a CO nova paternity is highly unlikely. Also, it would be a fortuitous coincidence if all the other 37 stardust grains listed in Table 2 would be of CO nova origin. Therefore, we conclude that only a weak case can be made for a CO nova paternity if significant post-mixing must be invoked to match observed and simulated isotopic ratios.

Table 3 lists all grains for which we found acceptable solutions without assuming any post-mixing. For each grain we show the mineralogy, the number of measured isotopic ratios, and the measured elements. The last column shows the total number of network runs that provide simultaneous solutions for all measured isotopic ratios of a given grain according to Equations (8) and (9). The grains are rank ordered, from top to bottom, according to the plausibility of a CO nova paternity. As discussed above, we ranked the grains according to the number of measured isotopic ratios (column 3) and the number of acceptable simulations (column 5).

We find that six grains, all of them of the SiC variety, have a high plausibility of a CO nova origin (from G270-2 to Ag2_6 in Table 3). These grains will be discussed in more detail in Section 4.3. In this case, we obtain acceptable solutions without any post-mixing, and we have several measured isotopic ratios (≥4) or many acceptable solutions (≥10).

The next group consists of 12 grains (from Ag2 to 1_07 in Table 3) that do not require any post-mixing. These have a medium plausibility of a CO nova paternity. We rank them below the top group because they either have a small number of measured isotopic ratios (i.e., fewer experimental constraints) or a small number of acceptable simulations.

Figure 4 shows the measured C, N, O, Si, and S isotopic ratios of all grains listed in Table 2. The colors red and green indicate grains of high and medium plausibility, respectively, of a CO nova paternity. For these two groups, acceptable solutions are obtained without assuming any post-mixing of the ejecta. Grains shown in blue require post-mixing to match the measured isotopic ratios and correspond to a low plausibility of a CO nova paternity. For the two grains shown in black, no solutions are obtained with or without post-mixing of the ejecta, and thus they most likely do not originate from CO novae.

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The SiC grains G270-2, G278, Ag2_6 (Liu et al. 2016), and M11-334-2, M11-347-4, M11-151-4 (Nittler & Hoppe 2005), shown in boldface in Table 3, have the highest plausibility of a CO nova paternity. For these grains, between four and six isotopic ratios of the elements C, N, Si, and S have been measured, and our simulations sampling the CO nova parameter space provide simultaneous solutions to all data without requiring any dilution of the ejecta with solar-like matter.

Interestingly, most of these have been argued to originate from supernovae rather than novae on the basis of their isotopic signatures. For example, M11-334-2 has ²⁸Si, ⁴⁴Ca, and ⁴⁹Ti excesses and a very high inferred initial ²⁶Al abundance, similar to those seen in type X SiC grains from supernovae, and M11-151-4 has an unexplained ⁴⁷Ti anomaly. The ³²S excesses seen in G270-2 and AG2_6 and the strong ²⁸Si depletion seen in G278 have not been predicted by previous nova models.

We will now consider the simulated CO nova peak temperature and peak density conditions that are obtained for these grains, assuming no post-mixing. They are shown in Figure 5, using the same color scheme that was employed in Figure 3 (black, blue, and green for T_peak ≤ 0.15 GK, 0.15 GK < T_peak < 0.20 GK, and T_peak ≥ 0.20 GK, respectively). The simulation results are not uniformly spread over the T_peak–ρ_peak plane. Instead, the solutions occupy select regions. Black simulation points are not apparent, except for a small number of points for grains G278 and M11-334-2. This indicates that all six grains likely originate in nova explosions with peak temperatures in excess of 150 MK, involving higher-mass CO white dwarfs. For grains G270_2 and Ag2_6, the most likely peak temperature exceeds 200 MK, as can be seen from the relative number of green simulation points. Figure 6 shows for the same six grains the exponential 1/e decay timescale of the density profile versus the exponential 1/e decay timescale of the temperature profile. The simulation points occupy a parameter space typical for CO nova conditions.

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It is interesting to consider the measured elemental carbon and oxygen abundances in CO nova ejecta, and compare the observations to the simulations. The observational results are summarized in Table 1. The carbon-to-oxygen mass fraction ratios range from 0.084 (for GQ Mus) to 5.4 (for V827 Her). Dust has been directly observed in PW Vul, QV Vul, V827 Her, V842 Cen, and V1668 Cyg (column 5). At least two of these, QV Vul and V842 Cen, have produced SiC dust. The simulated elemental oxygen versus carbon abundances (by mass) are shown in Figure 7, without assuming any post-mixing. Most of the simulation results scatter about the dotted line, corresponding to equal carbon and oxygen mass fractions. A few simulation points, for grain G278 only, exhibit ratios of X(O)/X(C) ≲ 0.3 (i.e., the points on the far left in the second top panel), which would be unfavorable for the condensation of SiC grains. For all solutions shown in Figure 7, the median of the elemental silicon mass fraction amounts to ${X}_{\mathrm{med}}(\mathrm{Si})\approx 6\times {10}^{-4}$, which is close to the solar value, ${X}_{\odot }(\mathrm{Si})\approx 8\times {10}^{-4}$.

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Finally, we repeated the simulations under exactly the same conditions, except that we included this time thermonuclear reaction rate variations. As mentioned in Section 3.1, all reaction rates in the network were sampled simultaneously according to their rate probability densities given by STARLIB. The results for the peak density versus peak temperature are shown in Figure 8. Comparison to Figure 5, which was obtained without thermonuclear rate variations, shows that the thermonuclear rate uncertainties increase the scatter of the simulation points (black, blue, green) noticeably. A detailed analysis of which nuclear reaction rate variations have the largest impact on the scatter is beyond the scope of the present work and will be presented in a forthcoming publication. Nevertheless, all relevant features discussed above are preserved, even when taking into account thermonuclear rate variations.

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