11institutetext: Yunnan Observatories, Chinese Academy of Sciences, Kunming
650216, China
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext: University
of Chinese Academy of Sciences, Beijing 100049, China

# Mixing fraction in classical novae

Yunlang Guo 1122 Chengyuan Wu 1122 Bo Wang 1122

(Received xxx, aaaa; accepted yyy, aaaa)

###### Abstract

Context. Classical novae are powered by thermonuclear runaways occurring on
the surface of accreting white dwarfs (WDs). In the observations, the
enrichments of heavy elements in nova ejecta have been detected, indicating
the existence of a mixing process between the accreted matter and the matter
from the outer layers of the underlying WDs prior to nova outbursts. However,
the mixing fraction in classical novae is still uncertain.

Aims. The purpose of this article is to investigate some elemental abundance
ratios during nova outbursts, which can be used to estimate the WD-mixing
fraction in classical novae.

Methods. By considering different WD-mixing fractions with the stellar
evolution code Modules for Experiments in Stellar Astrophysics (MESA), we
carried out a series of simulations of nova outbursts, in which the initial CO
WD masses range from $0.7-1.0\,M_{\odot}$.

Results. We identified four elemental abundance ratios (i.e. $\rm(H+He)/\sum
CNO$, $\rm(H+He)/Ne$, $\rm\sum CNO/Mg$ and $\rm\sum CNO/Si$) that satisfy the
conditions for determining the WD-mixing fraction, in which $\rm(H+He)/\sum
CNO$ is the most suitable mixing meter. We also estimated the WD-mixing
fraction in some representative classical novae. Additionally, we found that a
higher metallicity (i.e. higher WD-mixing fraction) prefers to be accompanied
by a longer $t_{\rm 2}$ (the time of decline by two magnitudes from peak
luminosity) during nova outbursts. The present results can be used to
constrain the mixing process in classical novae.

###### Key Words.:

binaries: close - stars: evolution - novae, cataclysmic variables - white
dwarfs

##  1 Introduction

It is generally believed that classical novae are powered by thermonuclear
runaways occurring on the surface of white dwarfs (WDs) in close binaries, in
which the WDs accrete matter from a main-sequence companion through the inner
Lagrangian point when the companion fills its Roche lobe (e.g. José 2016;
Chomiuk et al. 2021). Classical novae are suggested as the noteworthy
contributors to the Galactic chemical evolution, providing about $0.3\%$ of
the interstellar matter in the Galaxy. The nuclides ${}^{13}\rm C$,
${}^{15}\rm N$ and ${}^{17}\rm O$ in the interstellar medium are thought to be
mainly synthesized by classical novae (e.g. José & Hernanz 1998; Denissenkov
et al. 2014).

Observationally, the enrichments of intermediate-mass elements (e.g. C, N, O,
Ne, Na, Mg and Al) have been detected in nova ejecta, indicating that the
accreted shell is mixed with the outer layers of the underlying WD (e.g. Livio
& Truran 1994; Gehrz et al. 1998). Two types of WDs in nova systems can be
identified by the spectroscopic studies for nova ejecta, i.e. carbon-oxygen
(CO) WDs and oxygen-neon (ONe) WDs. Some mechanisms from one-dimensional
models have been proposed to explain the mixing process, including diffusion-
induced convection (e.g. Prialnik & Kovetz 1984; Fujimoto & Iben 1992), shear
mixing (e.g. Durisen 1977; Kippenhahn & Thomas 1978; MacDonald 1983; Kutter &
Sparks 1987; Fujimoto 1988), and convective overshoot-induced flame
propagation (see, e.g. Woosley 1986). For multidimensional models, including
mixing by resonant gravity waves (e.g. Rosner et al. 2001; Alexakis et al.
2004), and Kelvin-Helmholtz instabilities (e.g. Glasner & Livne 1995; Glasner
et al. 1997, 2005, 2007, 2012; Casanova et al. 2010, 2011, 2016, 2018).

The chemical abundances in nova ejecta have been proposed to reveal the
characteristics of the nova outbursts. José & Hernanz (1998) studied five
classical novae by simulating 14 evolutionary sequences of CO/ONe nova models,
span a range of WD masses and WD-mixing fractions. Their results indicate that
the significant ${}^{19}\rm F$, ${}^{35}\rm Cl$, and low O/N, C/N ratios can
be reproduced in the nova systems with a massive ONe WD. Downen et al. (2013)
found some useful elemental abundance ratios strongly correlated with the peak
temperatures during nova outbursts (e.g. $\rm N/O$, $\rm N/Al$ and $\rm O/S$,
etc.), providing a way to determine the WD mass. Kelly et al. (2013)
identified some elemental abundance ratios (e.g. $\rm\sum CNO/H$, $\rm Ne/H$,
$\rm Mg/H$, $\rm Al/H$ and $\rm Si/H$) to estimate the mixing fraction for ONe
novae, providing some constraints on the study of the WD-mixing process. These
ratios show a strong dependence on the mixing fraction, however, they are not
significantly affected by the WD mass.

The enrichments of He have also been detected in classical novae (e.g. GQ Mus,
HR Del and RR Pic, etc.), indicating that there could be a mixing process
between the He-shell and the accreted material (e.g. Truran & Livio 1986;
Gehrz et al. 1998). Starrfield et al. (1998) suggested that the accreted
material may have mixed with the He-layer by diffusion or accretion-driven
shear mixing before the dredge-up of the WD material. Besides, it is also
possible that the WD material has enriched in the He-layer due to the previous
evolutionary history, which means that the mixing of the accreted material and
the He-layer can naturally increase the abundance of He and WD matter. By
assuming the mixing of the He-shell and the accreted material in
multidimensional simulations, Glasner et al. (2012) estimated that the He-
mixing fraction is about $20\%$. Mixing with different chemical substrates has
also been reported in Casanova et al. (2016, 2018). Guo et al. (2021b)
recently reproduced the chemical abundances in some novae with He-enrichments
by considering the He- and WD-mixing process. They found that the He-mixing
level in classical novae is less than $30\%$.

The purpose of this article is to study several elemental abundance ratios
that can be used to estimate the WD-mixing fraction in CO novae. These ratios
should be sensitive to the WD-mixing degree but not affected by the WD
mass.111 Although the WD-mixing fraction would be affected by the WD mass
(e.g. Casanova et al. 2018), we focus on the measurement of the WD-mixing
degree, thus the mixing meters must only be sensitive to the mixing degree.
The accuracy of the measurement will be reduced if the WD mass has an effect
on the mixing meters. In addition, it is worth noting that these ratios cannot
be affected by the He-mixing that could occur in CO novae, since the
abundances of H and He in nova ejecta are sensitive to the He-mixing fractions
(see Guo et al. 2021b). This paper is organized as follows. In Section 2, the
basic assumptions and methods for numerical calculations are introduced. In
Section 3, we identified some WD-mixing meters, and estimated the WD-mixing
fraction in several representative CO novae. Besides, we showed the
relationship between $t_{\rm 2}$ (the time of decline by two magnitudes from
peak luminosity) and the metallicity in nova ejecta. Finally, we presented a
discussion in Section 4, and a summary in Section 5.

##  2 Numerical methods and assumptions

###  2.1 Nova models

The stellar evolution code used in our simulations is the Modules for
Experiments in Stellar Astrophysics (MESA, version 10398; see Paxton et al.
2011, 2013, 2015, 2018). It is widely accepted that the properties of
classical novae (e.g. light curves, times scales, nucleosynthesis and ejecta
mass) depend on the WD mass ($M_{\rm WD}$), the mass-accretion rate
($\dot{M}_{\rm acc}$), the initial luminosity of WDs and the composition of
accreted matter (e.g. Yaron et al. 2005; Wolf et al. 2013; Rukeya et al. 2017;
Wang 2018). In the present work, we created the CO WD models with the initial
mass range of $0.7-1.0\,M_{\odot}$.222 The WDs with mass $\lesssim
0.6\,M_{\odot}$ are considered to be hybrid HeCO WDs with a CO-rich core
surrounded by a $\sim 0.1\,M_{\odot}$ He-rich mantle (e.g. Dan et al. 2012;
Yungelson & Kuranov 2017; Liu et al. 2017). According to the current stellar
evolution theories, the composition of massive WDs is ONe instead of CO if the
WD mass is larger than $\sim 1.05\,M_{\odot}$ (see Siess 2007).  Following Guo
et al. (2021b), our initial WD models have a luminosity of
$10^{-2}\,L_{\odot}$, and the accretion rate ($\dot{M}_{\rm acc}$) in the nova
models is $2\times 10^{-9}\,M_{\odot}\rm yr^{-1}$. The nuclear reaction
network is coupled by 86 isotopes from ${}^{1}\rm H-^{48}\rm Ca$, including
794 nuclear reactions. Besides, we did not consider the influence of rotation
and convective overshooting in this work.

We adopted the super-Eddington wind as the mass loss mechanism during nova
outbursts, in which the super-Eddington luminosity ($L_{\rm Edd}$) and the
wind mass-loss rate ($\dot{M}$) can be expressed as follows (e.g. Denissenkov
et al. 2013; Wang et al. 2015; Wu et al. 2017; Guo et al. 2021a):

$L_{\rm Edd}=\frac{4\pi GcM_{\rm WD}}{\tau},$ (1) $\dot{M}=\frac{2\eta_{\rm
Edd}(L-L_{\rm Edd})}{\upsilon_{\rm esc}^{2}},$ (2)

where $G$, $c$ and $\tau$ are the gravitational constant, vacuum speed of
light and Rosseland mean opacity, $\eta_{\rm Edd}$ and $\upsilon_{\rm esc}$
are the super-Eddington wind factor and escape velocity, respectively. The
super-Eddington wind is triggered when the WD luminosity exceeds the Eddington
luminosity, and all the radiant energy exceeding Eddington luminosity is used
to eject matter (i.e. $\eta_{\rm Edd}=1$).

###  2.2 White dwarf mixing

We adopted the pre-mixed model described in Politano et al. (1995), in which
the material accreted by WDs is assumed to be a mixture of the companion star
material (solar abundances) and the outermost layers of the underlying CO WDs.
Different WD-mixing degrees can be modified by artificially giving the various
abundance of the WD material in the mixture. In our simulations, we assumed
that the WD material mixed with the accreted layer is composed of $X(^{12}\rm
C)$ = 0.495, $X(^{16}\rm O)$ = 0.495, $X(^{22}\rm Ne)$ = 0.01 (e.g. José &
Hernanz 1998; Guo et al. 2021b). In order to understand the dependence of
chemical abundances in nova ejecta on the WD-mixing level, we carried out a
series of calculations for our nova models with different WD-mixing fractions
($f_{\rm WD}$) $\leq 40\%$.

###  2.3 Helium mixing

In order to test whether the He-mixing has an effect on the WD-mixing meters,
we calculated some nova models, including both the He- and WD-mixing (see the
second paragraph of Sect. 3.1.1 and Fig. 2 for details). Following the studies
of Guo et al. (2021b), we mixed the accreted material with a given mass
fraction of He-layer and WD material. Similar to the method of the WD-mixing,
the He-mixing degrees can be modified by considering the various abundance of
the He-rich material in the mixture. If we only consider the He-mixing during
the outbursts, the He mass fraction in the already mixed envelope ($X_{\rm
He}$) is contributed by the mixed He-shell and the accreted material. We
suppose the mass ratio of the mixed He-shell to the already mixed envelope as
the He-mixing fraction ($f_{\rm He}$), and set the He abundance in accreted
material and He-shell to be $28\%$ and $100\%$, respectively. Thus, the He
mass fraction can be calculated by $X_{\rm He}=f_{\rm He}+0.28\times(1-f_{\rm
He})$, in which $0.28\times(1-f_{\rm He})$ represents the He abundance
contributed by the accreted material. Similarly, if we further consider both
He-mixing and WD-mixing during the outbursts, then $X_{\rm He}=[f_{\rm
He}+0.28\times(1-f_{\rm He})]\times(1-f_{\rm WD})$, in which $f_{\rm WD}$ is
the given WD-mixing fraction.333 In the present work, we assume that the
accreted material mixes first with the He-shell and then with the WD material
(e.g. Starrfield et al. 1998).

Figure 1: Elemental abundances (mass fraction) in nova ejecta vs. atomic
number, in which we set the WD-mixing fraction $\leq 40\%$ and the WD mass to
be $0.9\,M_{\odot}$. Different colored lines represent different WD-mixing
fractions.

##  3 Numerical results

###  3.1 The WD-mixing meters

We performed a series of simulations for nova models with different WD masses
and WD-mixing degrees. Table 1 lists the main properties of initial nova
models computed in this work, as well as the characteristics of the outbursts.
Fig. 1 presents some chemical abundances (i.e. H-Ar) in nova ejecta changing
with different WD-mixing fractions, in which we set the initial WD mass to be
$0.9\,M_{\odot}$. Obviously, C, N, O, F and Ne exhibit a strong monotonic
increase with the WD-mixing levels, while other elements show an opposite
trend. Note that the abundance of some elements are relatively low (e.g. F, P
and Cl) or rarely detected in CO nova ejecta (e.g. Al, S and Ar). Thus, these
elements cannot be used as the WD-mixing meters. In the present work, we
studied the elemental abundance ratios that satisfy the conditions for
determining the WD-mixing degree, e.g. $\rm(H+He)/\sum CNO$, $\rm(H+He)/Ne$,
$\rm\sum CNO/Mg$ and $\rm\sum CNO/Si$.

Table 1: Initial parameters of nova models and the characteristics of the outbursts: the WD-mixing fraction ($f_{\rm WD}$) and WD mass ($M_{\rm WD}$); $T_{\rm peak}$ is the peak temperature at the burning shell; $M_{\rm ej}$ and $L_{\rm max}$ are the ejected mass and the peak luminosity during nova outbursts, respectively; $Z$ is the metallicity in nova ejecta. Mixing fraction |  | $M_{\rm WD}$ | $T_{\rm peak}$ | $M_{\rm ej}$ | $L_{\rm max}$ | $Z$
---|---|---|---|---|---|---
($f_{\rm WD}$) |  | ($M_{\odot}$) | (GK) | $(10^{-5}\,M_{\odot})$ | $(10^{4}\,L_{\odot})$ |
|  | 0.7 | 0.113 | 4.89 | 1.82 | 0.1204
$10\%$ |  | 0.8 | 0.122 | 3.83 | 2.57 | 0.1201
|  | 0.9 | 0.134 | 3.23 | 2.89 | 0.1196
|  | 1.0 | 0.148 | 2.20 | 3.09 | 0.1192
|  | 0.7 | 0.114 | 5.75 | 2.02 | 0.2264
$20\%$ |  | 0.8 | 0.124 | 3.84 | 2.54 | 0.2268
|  | 0.9 | 0.135 | 2.44 | 3.00 | 0.2261
|  | 1.0 | 0.147 | 1.60 | 3.70 | 0.2256
|  | 0.7 | 0.114 | 5.22 | 2.04 | 0.3285
$30\%$ |  | 0.8 | 0.124 | 3.43 | 2.63 | 0.3320
|  | 0.9 | 0.135 | 2.20 | 3.06 | 0.3331
|  | 1.0 | 0.145 | 1.41 | 3.98 | 0.3327
|  | 0.7 | 0.114 | 4.66 | 2.20 | 0.4274
$40\%$ |  | 0.8 | 0.124 | 3.10 | 2.85 | 0.4239
|  | 0.9 | 0.136 | 2.14 | 3.45 | 0.4368
|  | 1.0 | 0.144 | 1.28 | 4.27 | 0.4386

####  3.1.1 $\rm(H+He)/\sum CNO$

Previous studies suggested that the peak temperature attained in H burning
shell would not exceed $4\times 10^{8}$K during nova outbursts (e.g. Livio &
Truran 1994; Denissenkov et al. 2013). The sum abundance of C, N and O
($\rm\sum CNO$) would not change significantly in this temperature regime
(e.g. Truran & Livio 1986; José & Hernanz 1998). For this reason, Kelly et al.
(2013) used $\rm H/\sum CNO$ to measure the WD-mixing degree in ONe novae.

Figure 2: The elemental abundance (mass fraction) ratios that may be
candidates for the WD-mixing meters vary with different He-mixing degrees, in
which we set the WD mass to be $0.7\,M_{\odot}$ and the WD-mixing fraction to
be $20\%$. Panel (a): the results of $\rm(H+He)/\sum CNO$ and $\rm H/\sum
CNO$. Panel (b): the results of $\rm(H+He)/Ne$ and $\rm H/Ne$. Panel (c): the
results of $\rm\sum CNO/Mg$ and $\rm\sum CNO/Si$.

However, it is not negligible to verify whether the WD-mixing meters are
affected by the He-mixing that may occur in CO novae. We then performed a
series of calculations for nova models with He-mixing fraction ranging from
$0-20\%$, where we set the WD mass to be $0.7\,M_{\odot}$ and the WD-mixing
fraction to be $20\%$. Fig. 2 shows the variation of elemental abundance (mass
fraction) ratios with different He-mixing degrees. We can see that the value
of $\rm H/\sum CNO$ is affected by the He-mixing fraction significantly, while
the effect can be eliminated by choosing $\rm(H+He)/\sum CNO$ (see Fig. 2a).
This is because, for a given WD-mixing degree, the abundance of H+He and heavy
elements in nova ejecta is hardly changed by the He-mixing process (see Table
3 in Guo et al. 2021b).

Table 2: The comparison of neon abundance between José & Hernanz (1998) and this work. | $M_{\rm WD}$ |  | $f_{\rm WD}$ |  | Ne |  | $Z$ |  | $\rm Ne/Z$
---|---|---|---|---|---|---|---|---|---
| ($M_{\odot}$) |  | ($\%$) |  |  |  |  |  |
| 0.80 |  | 25 |  | 3.80e-3 |  | 0.280 |  | 0.014
| 0.80 |  | 50 |  | 5.82e-3 |  | 0.530 |  | 0.011
José & Hernanz (1998) | 1.00 |  | 50 |  | 5.85e-3 |  | 0.530 |  | 0.011
| 1.15 |  | 25 |  | 3.60e-3 |  | 0.280 |  | 0.013
| 1.15 |  | 50 |  | 5.77e-3 |  | 0.540 |  | 0.011
| 1.15 |  | 75 |  | 7.82e-3 |  | 0.790 |  | 0.010
| 0.70 |  | 20 |  | 2.75e-3 |  | 0.227 |  | 0.012
| 0.90 |  | 10 |  | 2.11e-3 |  | 0.120 |  | 0.018
This work | 0.90 |  | 20 |  | 2.36e-3 |  | 0.226 |  | 0.010
| 0.90 |  | 30 |  | 3.14e-3 |  | 0.333 |  | 0.010
| 0.90 |  | 40 |  | 4.10e-3 |  | 0.437 |  | 0.010
| 1.00 |  | 20 |  | 2.39e-3 |  | 0.226 |  | 0.011

Fig. 3 represents the changes of four elemental abundance ratios with
different WD-mixing fractions and peak temperatures, where the peak
temperature corresponds to the WD mass in the range of $0.7-1.0\,M_{\odot}$.
As shown in Fig. 3a, $\rm(H+He)/\sum CNO$ is weakly affected by the peak
temperature but strongly depends on the WD-mixing fraction, indicating that
$\rm(H+He)/\sum CNO$ can be used to determine the WD-mixing fraction.

Figure 3: Peak temperature vs. the elemental abundance ratios that can be used
to determine the WD-mixing fraction, where the peak temperature corresponds to
the WD mass in the range of $0.7-1.0\,M_{\odot}$. Different colors represent
different WD-mixing fractions. Panel (a): the results of $\rm(H+He)/\sum CNO$.
Panel (b): the results of $\rm(H+He)/Ne$. Panel (c): the results of $\rm\sum
CNO/Mg$. Panel (d): the results of $\rm\sum CNO/Si$. Figure 4: WD-mixing
fractions in some representative classical novae determined by different
elemental abundance ratios. Different marks represent the WD-mixing degree
from $10\%-40\%$. The black stars refer to the observed data. The error bar
represents the effect of peak temperature on the elemental abundance ratios.
Panel (a): the results of GQ Mus (Morisset & Pequignot 1996). Panel (b): the
results of PW Vul (Andreä et al. 1994). Panel (c): the results of V842 Cen
(Andreä et al. 1994). Panel (d): the results of V827 Her (Andreä et al. 1994).
Panel (e): the results of V723 Cas (Iijima 2006). Panel (f): the results of
V2214 Oph (Andreä et al. 1994).

####  3.1.2 $\rm(H+He)/Ne$

Table 2 lists the Ne abundance in different nova models, including the results
of José & Hernanz (1998) and this work. These simulations show that the
abundance of Ne hardly changes with the WD mass for a given WD-mixing
fraction. As shown in Fig. 2b, $\rm(H+He)/Ne$ is more suitable for the WD-
mixing meter than $\rm H/Ne$ because of the weak dependence of the former on
the He-mixing fraction. Meanwhile, $\rm(H+He)/Ne$ shows a weak dependence on
the peak temperature, whereas sensitive to the WD-mixing fraction (see Fig.
3b). Thus, $\rm(H+He)/Ne$ can also be a WD-mixing meter.

####  3.1.3 $\rm\sum CNO/Mg$ and $\rm\sum CNO/Si$

Downen et al. (2013) studied the ratio of final to initial elemental
abundances during nova outbursts changing with different WD masses. Their
results show that the abundances of Mg and Si are hardly affected by the WD
mass. Fig. 2c shows that $\rm\sum CNO/Mg$ and $\rm\sum CNO/Si$ are not
affected by the He-mixing fraction. Meanwhile, we represented the changes of
$\rm\sum CNO/Mg$ and $\rm\sum CNO/Si$ with peak temperature in Fig. 3c-d,
respectively. It is obvious that these two abundance ratios are weakly
dependent on the WD mass but strongly depend on the WD-mixing degree. This
means that $\rm\sum CNO/Mg$ and $\rm\sum CNO/Si$ meet the conditions for
determining the WD-mixing degree.

###  3.2 Comparison to observations

Fig. 4 shows the values of these four ratios with different WD-mixing degrees,
as well as the estimation of WD-mixing levels in six representative novae,
i.e. GQ Mus, PW Vul, V842 Cen, V827 Her, V723 Cas and V2214 Oph. GQ Mus is a
classical nova, showing signs of the He and metal enrichments. According to
the studies of Guo et al. (2021b), the WD- and He-mixing level in this nova
are $20\%$ and $30\%$, respectively. Similarly, the simulations in this work
show that the WD-mixing fraction in GQ Mus is about $20\%$. For nova PW Vul,
$\rm(H+He)/\sum CNO$ and $\rm(H+He)/Ne$ imply that the WD-mixing fraction is
about $30\%$, which is consistent with the results of José & Hernanz (1998).

In the case of nova V842 Cen, the WD-mixing fraction in the range of
$30\%-40\%$ is estimated by $\rm(H+He)/\sum CNO$, while $\rm(H+He)/Ne$ implies
a value less than $10\%$. For nova V827 Her, $\rm(H+He)/\sum CNO$ indicates a
WD-mixing fraction of $30\%$, while $\rm(H+He)/Ne$ shows the value less than
$10\%$. Nova V723 Cas shows a WD-mixing fraction below $10\%$ by the
estimation of $\rm(H+He)/\sum CNO$, while a value far greater than $40\%$ is
suggested by $\rm(H+He)/Ne$. V2214 Oph is also a classical nova showing the
He-enrichments, and the He-mixing fraction is about $13\%$ (see Guo et al.
2021b). For this nova, $\rm(H+He)/\sum CNO$ implies a WD-mixing degree in the
range of $30\%-40\%$, while $\rm(H+He)/Ne$ indicates that the value is lager
than $40\%$. In addition, DQ Her exhibits a high enrichment of CNO, but only
H, He, C, N and O were determined by Petitjean et al. (1990). According to our
simulations, we speculated that the WD-mixing fraction of this nova is greater
than $40\%$.

Fig. 4 also represents the WD-mixing fractions measured by $\rm\sum CNO/Mg$
and $\rm\sum CNO/Si$. We note that the measurement results of these two ratios
are not in good agreement with $\rm(H+He)/\sum CNO$ and $\rm(H+He)/Ne$ (for
more discussions, see Sect. 4). From this figure, we can see that the WD-
mixing degree in classical novae spans a relatively wide range.

Figure 5: The relationship between $t_{\rm 2}$ and the metallicity in nova
ejecta ($Z$), in which these novae are listed in Gehrz et al. (1998). Note
that we only selected classical novae that show the significant signs of the
WD-mixing ($Z\textgreater 0.1$). $t_{\rm 2}$ is the time of decline by two
magnitudes from peak brightness, which can be directly derived from the
observations.

###  3.3 Relationship between $t_{\rm 2}$ and the metallicity in CO novae

In order to explore the physical parameters that affect the WD-mixing
fraction, we summarized the observed data of CO novae listed in Gehrz et al.
(1998), in which the selected novae show the obvious signs of the WD-mixing
(i.e. the metallicity $Z\textgreater 0.1$). Fig. 5 shows the relationship
between $t_{\rm 2}$ and $Z$ in these classical novae, where $t_{\rm 2}$ is the
time of decline by two magnitudes from peak brightness. We used $t_{\rm 2}$
instead of WD mass or accretion rate because $t_{\rm 2}$ can be obtained
directly from the observations, and the metallicity was chosen because it can
reflect the WD-mixing fraction in classical novae.

As shown in Fig. 5, longer $t_{\rm 2}$ time tends to correspond to the higher
metallicities in classical novae. It is generally believed that a faster-
declining nova which has a shorter $t_{\rm 2}$ implies a more massive WD in
nova system.444Note that other parameters (e.g. the accretion rate, the WD
initial luminosity and the composition of the accreted envelope) also affect
$t_{\rm 2}$, but the WD mass is the key role affecting the main trend (e.g.
Yaron et al. 2005; Hachisu & Kato 2016; Hachisu et al. 2020). Thus, the trend
illustrated in Fig. 5 indicates that high metallicity (i.e. high WD-mixing
fraction) may exist in the nova systems with less massive WDs, except for a
particular object nova T Aur, i.e. a low metallicity but with $t_{\rm 2}$
close to that of DQ Her.

The WD mass would affect the mixing time, and in turn affects the metallicity
in classical novae. It has been suggested that the temperature at the envelope
base ($T_{\rm base}$) is about $10^{8}$ K when the convection has already
extended throughout the whole envelope (e.g. José et al. 2016). Meanwhile, in
the multidimensional simulations, Casanova et al. (2011) mapped the 1D nova
model on the 2D or 3D Cartesian grid when $T_{\rm base}$ reached $10^{8}$ K by
considering the multidimensional mixing (see also Casanova et al. 2018). José
& Hernanz (1998) studied the effect of the WD mass on the time required from
$10^{8}$ K to the peak temperature ($t_{\rm max}$). They found that $t_{\rm
max}$ decreases with the WD mass, which means that shorter mixing time exist
in the nova systems with massive WDs, resulting in a low WD-mixing degree.

##  4 Discussion

As shown in Fig. 4, the WD-mixing degree measured by different mixing meters
may be inconsistent. It has been suggested that the elements C and O are the
main components in the underlying CO WDs based on the studies of stellar
evolution and asteroseismology (e.g. Umeda et al. 1999; Giammichele et al.
2018). Consequently, the WD material mixed with the accreted envelope is
mainly composed of C and O, indicating that $\rm(H+He)/\sum CNO$ is the most
suitable elemental abundance ratio to estimate the WD-mixing level. On the
other hand, the abundances of Mg and Si in CO novae are relatively low, hence
they are not good candidates for the WD-mixing meters.

According to the stellar evolution theories, ${}^{22}\rm Ne$ is produced by
${}^{14}\rm N(\alpha,\gamma)^{18}\rm F$ $(e^{+}v)$ ${}^{18}\rm
O(\alpha,\gamma)^{22}\rm Ne$ when stars evolve to the He-burning phase, making
${}^{22}\rm Ne$ become the dominant neon isotope in CO WDs. In this work, we
set the mass fraction of ${}^{22}\rm Ne$ in the WD material to be 0.01.
However, the ${}^{22}\rm Ne$ abundance in CO WDs may be different from our
assumptions due to the different metallicity environments (Umeda et al. 1999),
which will affect the WD-mixing degrees measured by $\rm(H+He)/Ne$. For
example, V723 Cas has a Ne abundance as high as those of ONe novae, but was
identified as a CO nova (see Iijima 2006). Such high abundance of Ne is due to
the high concentration of ${}^{22}\rm Ne$ in the underlying CO WD (Livio &
Truran 1994; Hachisu & Kato 2016), resulting in that the WD-mixing degree
measured by $\rm(H+He)/Ne$ is much higher than $40\%$ (see Fig. 4e).

Figure 6: Relation between $t_{\rm 2}$ and the value of $\rm Ne/Z$ obtained
from some classical novae, in which these novae show obvious signs of WD
mixing ($Z\textgreater 0.1$).

In Table 2, we can see that the ratio of Ne abundance to the metallicity ($\rm
Ne/Z$) in nova ejecta shows a weak dependence on the WD mass and the WD-mixing
degree, and its value is consistent with the Ne abundance set in the CO WD
model (i.e. $X(^{22}\rm Ne)$ = 0.01). Thus, we expect that the Ne abundance in
WDs ($X(^{22}\rm Ne)$) can be represented by the value of $\rm Ne/Z$ in nova
ejecta. Fig. 6 shows the relation between $t_{\rm 2}$ and $\rm Ne/Z$ obtained
from six classical novae, where these novae show obvious signs of WD-mixing
($Z\textgreater 0.1$). Some other novae (e.g. V1668 Cyg, DQ Her and T Aur) are
not presented here because of no information about the Ne abundance in the
references (e.g. Gallagher et al. 1980; Petitjean et al. 1990; Andreä et al.
1994). As shown in Fig. 6, lower $\rm Ne/Z$ (i.e. $X(^{22}\rm Ne)$) prefers to
be detected in the faster nova systems (i.e. shorter $t_{\rm 2}$)
corresponding to the massive WDs. Previous studies suggested that as the
stellar population metallicity increases, the ${}^{22}\rm Ne$ abundance
increases while the WD mass decreases (see, e.g. Umeda et al. 1999). Thus, we
speculate that these novae shown in Fig. 6 may have different stellar
population metallicities.

By assuming the mixing process driven by Kelvin-Helmholtz instabilities in the
multidimensional simulations, Casanova et al. (2018) suggested that the mixing
fraction in nova systems increases with the WD mass due to the higher base
density and peak temperatures. However, in their simulations, the
metallicities in novae with less massive WDs are lower than that of the
observed values, e.g. $Z=0.0268$ for $0.8\,M_{\odot}$, and $Z=0.0772$ for
$1\,M_{\odot}$ in CO novae. We note that the nova models calculated in their
work involve a limited number of combinations of WD masses and accretion
rates. Thus, the influence of WD mass and other parameters such as the mass-
accretion rate on the WD-mixing deserves further analysis.

The observed He-enrichments in classical novae imply that helium can be
accumulated on the WD surface. We speculate that the WD mass could affect the
thickness of He-shell, in which the heavier He-shell is expected to accumulate
on the surface of low-mass WDs owing to the weaker outbursts caused by the
relatively low degeneracy of the H-shell. Meanwhile, Chen et al. (2019)
studied the characteristics of nova outburst by considering a wide range of WD
masses and mass-accretion rates. They found that low-mass WDs can lead to
relatively high accreted masses and mass-accumulative efficiencies for a given
mass-accretion rate. In addition, Glasner et al. (2012) show that the He-
mixing level in nova systems is about $20\%$, indicating that the mixed He-
shell mass is $\sim 20\%$ of the critical mass.

##  5 Summary

In the present work, by employing the stellar evolution code MESA, we
investigated the elemental abundance ratios during nova outbursts, which can
be used to estimate the WD-mixing fraction in CO novae. These ratios show a
weak dependence on the WD mass and the He-mixing, but they are sensitive to
the WD-mixing fraction. We identified the elemental abundance ratios
$\rm(H+He)/\sum CNO$, $\rm(H+He)/Ne$, $\rm\sum CNO/Mg$ and $\rm\sum CNO/Si$
that can meet the conditions for determining the WD-mixing degree, where
$\rm(H+He)/\sum CNO$ is the most useful mixing meter. In addition, we
estimated the WD-mixing fraction in some representative classical novae.
Moreover, we found that a higher metallicity (i.e. higher WD-mixing fraction)
prefers to be accompanied by a longer $t_{\rm 2}$ time in classical novae. The
present work can be used to provide some constraints on the mixing theories in
nova systems. In order to better understand the mixing process, more observed
samples and theoretical simulations are needed.

###### Acknowledgements.

We thank the anonymous referee for valuable comments that help to improve the
paper. BW is supported by the National Natural Science Foundation of China (No
11873085), the science research grants from the China Manned Space Project
(Nos CMS-CSST-2021-A13/B07), the Western Light Youth Project of CAS, and the
Yunnan Fundamental Research Projects (Nos 2019FJ001 and 202001AS070029). CW is
supported by the National Natural Science Foundation of China (No. 12003013).
The authors gratefully acknowledge the “PHOENIX Supercomputing Platform”
jointly operated by the Binary Population Synthesis Group and the Stellar
Astrophysics Group at Yunnan Observatories, Chinese Academy of Sciences.

## References

  * Alexakis et al. (2004) Alexakis, A., Calder, A. C., Heger, A., et al. 2004, ApJ, 602, 931
  * Andreä, Drechsel, & Starrfield (1994) Andreä J., Drechsel H., Starrfield S., 1994, A&A, 291, 869
  * Casanova et al. (2010) Casanova, J., José, J., García-Berro, E., Calder, A., & Shore, S. N. 2010, A&A, 513, L5
  * Casanova et al. (2011) Casanova, J., José, J., García-Berro, E., Calder, A., & Shore, S. N. 2011b, Nature, 478, 490
  * Casanova et al. (2016) Casanova, J., José, J., García-Berro, E., & Shore, S. N. 2016, A&A, 595, A28
  * Casanova et al. (2018) Casanova, J., José, J., & Shore, S. N. 2018, A&A, 619, A121
  * Chen et al. (2019) Chen, H.-L., Woods, T. E., Yungelson, L. R., et al. 2019, MNRAS, 490, 1678
  * Chomiuk, Metzger, & Shen (2021) Chomiuk, L., Metzger, B. D., & Shen, K. J. 2020, arXiv e-prints, arXiv:2011.08751
  * Dan et al. (2012) Dan, M., Rosswog, S., Guillochon, J., & Ramirez-Ruiz, E. 2012, MNRAS, 422, 2417
  * Denissenkov et al. (2013) Denissenkov, P. A., Herwig, F., Bildsten, L., & Paxton, B. 2012, The Astrophysical Journal, 762, 8
  * Denissenkov et al. (2014) Denissenkov, P. A., Truran, J. W., Pignatari, M., et al. 2014, MNRAS, 442, 2058
  * Downen et al. (2013) Downen L. N., Iliadis C., José J., & Starrfield S., 2013, ApJ, 762, 105
  * Durisen (1977) Durisen R. H., 1977, ApJ, 213, 145
  * Fujimoto (1988) Fujimoto, M. Y., 1988, A&A, 198, 163
  * Fujimoto & Iben (1992) Fujimoto, M. & Iben, Icko, J. 1992, ApJ, 399, 646
  * Gallagher et al. (1980) Gallagher, J. S., Hege, E. K., Kopriva, D. A., Williams, R. E., & Butcher, H. R. 1980, ApJ, 237, 55
  * Gehrz et al. (1998) Gehrz, R. D., Truran, J. W., Williams, R. E., & Starrfield, S. 1998, PASP, 110, 3
  * Giammichele et al. (2018) Giammichele, N., Charpinet, S., Fontaine, G., et al. 2018, Nature, 554, 73
  * Glasner & Livne (1995) Glasner, S. A. & Livne, E. 1995, ApJ, 445, L149
  * Glasner, Livne, & Truran (1997) Glasner, S. A., Livne, E., & Truran, J. W. 1997, ApJ, 475, 754
  * Glasner, Livne, & Truran (2005) Glasner, S. A., Livne, E., & Truran, J. W. 2005, ApJ, 625, 347
  * Glasner, Livne, & Truran (2007) Glasner, S. A., Livne, E., & Truran, J. W. 2007, ApJ, 665, 1321
  * Glasner, Livne, & Truran (2012) Glasner, S. A., Livne, E., & Truran, J. W. 2012, MNRAS, 427, 2411
  * Guo et al. (2021) Guo, Y., Liu, D., Wu, C., & Wang, B. 2021a, Research in Astronomy and Astrophysics, 21, 034
  * Guo, Wu, & Wang (2021) Guo, Y., Wu, C., & Wang, B. 2021b, MNRAS, 505, 2975
  * Hachisu & Kato (2016) Hachisu, I. & Kato, M. 2016, ApJ, 816, 26
  * Hachisu et al. (2020) Hachisu, I., Saio, H., Kato, M., Henze, M., & Shafter, A. W. 2020, ApJ, 902, 91
  * Iijima (2006) Iijima, T., 2006, A&A, 451, 563
  * Jose & Hernanz (1998) José, J. & Hernanz, M. 1998, ApJ, 494, 680
  * Jose, Hernanz, & Iliadis (2006) José, J., Hernanz, M., & Iliadis, C. 2006, Nucl. Phys. A, 777, 550
  * Jose (2016) José, J., Halabi, G. M., & El Eid, M. F. 2016, A&A, 593, A54
  * Jose (2016) José, J. 2016, Stellar Explosions: Hydrodynamics and Nucleosynthesis (Boca Raton, FL: CRC/Taylor and Francis)
  * Jose, Shore, & Casanova (2020) José, J., Shore, S. N., & Casanova, J. 2020, A&A, 634, A5
  * Kelly et al. (2013) Kelly, K. J., Iliadis, C., Downen, L., José, J., & Champagne, A. 2013, ApJ, 777, 130
  * Kippenhahn & Thomas (1978) Kippenhahn, R. & Thomas, H. C. 1978, A&A, 63, 265
  * Kutter & Sparks (1987) Kutter, G. S. & Sparks, W. M. 1987, ApJ, 321, 386
  * Liu et al. (2017) Liu, D., Wang, B., Wu, C., & Han, Z. 2017, A&A, 606, A136
  * Livio & Truran (1994) Livio, M. & Truran, J. W. 1994, ApJ, 425, 797
  * MacDonald (1983) MacDonald, J. 1983, ApJ, 273, 289
  * Morisset & Pequignot (1996) Morisset, C. & Pequignot, D. 1996, A&A, 312, 135
  * Paxton et al. (2011) Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3
  * Paxton et al. (2013) Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4
  * Paxton et al. (2015) Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15
  * Paxton et al. (2018) Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34
  * Petitjean, Boisson, & Pequignot (1990) Petitjean, P., Boisson, C., & Pequignot, D. 1990, A&A, 240, 433
  * Politano et al. (1995) Politano, M., Starrfield, S., Truran, J. W., Weiss, A., & Sparks, W. M. 1995, ApJ, 448, 807
  * Prialnik & Kovetz (1984) Prialnik, D. & Kovetz, A. 1984, ApJ, 281, 367
  * Rosner et al. (2001) Rosner, R., Alexakis, A., Young, Y. N., Truran, J. W., & Hillebrandt, W. 2001, ApJ, 562, L177
  * Rukeya et al. (2017) Rukeya, R., Lü, G., Wang, Z., & Zhu, C. 2017, PASP, 129, 074201
  * Shara et al. (2018) Shara, M. M., Prialnik, D., Hillman, Y., & Kovetz, A. 2018, ApJ, 860, 110
  * Siess (2007) Siess, L. 2007, A&A, 476, 893
  * Starrfield et al. (1998) Starrfield, S., Truran, J. W., Wiescher, M. C., & Sparks, W. M. 1998, MNRAS, 296, 502
  * Truran & Livio (1986) Truran, J. W. & Livio, M. 1986, ApJ, 308, 721
  * Tylenda (1978) Tylenda, R. 1978, Acta Astron., 28, 333
  * Umeda et al. (1999) Umeda, H., Nomoto, K., Yamaoka, H., & Wanajo, S. 1999, ApJ, 513, 861
  * Wang (2015) Wang, B., Li, Y., Ma, X., et al. 2015, A&A, 584, A37
  * Wang (2018) Wang, B. 2018, Research in Astronomy and Astrophysics, 18, 049
  * Wolf et al. (2013) Wolf, W. M., Bildsten, L., Brooks, J., & Paxton, B. 2013, ApJ, 777, 136
  * Woosley (1986) Woosley, S. E. 1986, in Saas-Fee Advanced Course 16: Nucleosynthesis and Chemical Evolution, ed. J. Audouze, C. Chiosi, & S. E. Woosley, 1
  * Wu et al. (2017) Wu, C., Wang, B., Liu, D., & Han, Z. 2017, A&A, 604, A31
  * Yaron et al. (2005) Yaron, O., Prialnik, D., Shara, M. M., & Kovetz, A. 2005, ApJ, 623, 398
  * Yungelson & Kuranov (2017) Yungelson, L. R., & Kuranov, A. G. 2017, MNRAS, 464, 1607


We have $[f_p,[\bar{f},\partial_{\bar{u}^2}]]=\Ad_{\bar{f}}^2\partial_{\bar{u}^2}+\bar{u}^1_1[\partial_{\bar{u}^1},[\bar{f},\partial_{\bar{u}^2}]]+\bar{u}^1_2[\partial_{\bar{u}^1_1},[\bar{f},\partial_{\bar{u}^2}]]=\Ad_{\bar{f}}^2\partial_{\bar{u}^2}+\bar{u}^1_1[\partial_{\bar{u}^1},[\bar{f},\partial_{\bar{u}^2}]]\Mod\Delta_1$. The involutivity of $\Delta_2$ implies that $\Ad_{\bar{f}}^2\partial_{\bar{u}^2}$ and $[\partial_{\bar{u}^1},[\bar{f},\partial_{\bar{u}^2}]]$ are collinear modulo $\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1},\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}]}$. In case of $k_1\geq 2$, we certainly have $\Ad_{\bar{f}}^2\partial_{\bar{u}^2}\notin\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1},\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}]}$. Indeed, assume that $k_1\geq 2$ and $\Ad_{\bar{f}}^2\partial_{\bar{u}^2}\in\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1},\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}]}$. We then have

\begin{align*}
\Delta_2&=\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1},\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}],\underbrace{[\partial_{\bar{u}^1},[\bar{f},\partial_{\bar{u}^2}]]}_{\in D_1}}
\end{align*}

(where $[\partial_{\bar{u}^1},[\bar{f},\partial_{\bar{u}^2}]]\in D_1$ due to the involutivity of $D_1$). Thus, either $\Delta_2=\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1}}+D_1$ or $\Delta_2=\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1}}+\Span{\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}]}$. It follows that the first case would lead to $\Delta_i=\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1}}+D_{i-1}$ which for $i=k_1+1$ would imply that $D_{k_1}$ is involutive, contradicting with $D_{k_1}$ being non-involutive. The second case, $\Delta_2=\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1}}+\Span{\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}]}$ would lead to $\Delta_3=\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1}}+D_1$ and in turn $\Delta_i=\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1}}+D_{i-2}$ which for $i=k_1+2$ would imply that $D_{k_1}$ is involutive, again contradicting with $D_{k_1}$ being non-involutive. Thus, in case of $k_1\geq 2$, we always have $\Ad_{\bar{f}}^2\partial_{\bar{u}^2}\notin\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1},\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}]}$, and thus

\begin{align}\label{eq:delta2case1}
\begin{aligned}
\Delta_2&=\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1},\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}],\Ad_{\bar{f}}^2\partial_{\bar{u}^2}}\,.
\end{aligned}
\end{align}

The form (<ref>) then easily follows from the involutivity of the distributions $\Delta_2,\Delta_3,\ldots$ and the fact that $f_p$ and $\bar{f}$ coincide modulo $\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1}}\subset\Delta_2,\Delta_3,\ldots$

However, for $k_1=1$, it can indeed happen that $\Ad_{\bar{f}}^2\partial_{\bar{u}^2}\in\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1},\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}]}$ and thus

\begin{align}\label{eq:delta2case2}
\begin{aligned}
\Delta_2&=\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1},\partial_{\bar{u}^1},\partial_{\bar{u}^2},[\bar{f},\partial_{\bar{u}^2}],[\partial_{\bar{u}^1},[\bar{f},\partial_{\bar{u}^2}]]}\,
\end{aligned}
\end{align}

(in which case we necessarily have $[\partial_{\bar{u}^1},[\bar{f},\partial_{\bar{u}^2}]]\notin D_1$). In this case, due to the involutivity of the distributions $\Delta_2,\Delta_3,\ldots$ and the fact that $f_p$ and $\bar{f}$ coincide modulo $\Span{\partial_{\bar{u}^1_2},\partial_{\bar{u}^1_1}}\subset\Delta_2,\Delta_3,\ldots$ the form (<ref>) follows.

§.§ Proof of Lemma <ref>

Under the assumption $d=2$ with $D_{k_1-1}\subset\C{D_{k_1}}$, the distribution $\mathcal{I}_{k_1}=\Delta_{k_1+1}\cap D_{k_1}$ is involutive, which can be shown by contradiction. Assume that $\mathcal{I}_{k_1}=\Delta_{k_1+1}\cap D_{k_1}$ is non-involutive. From (<ref>), it follows that we have[It is immediate that $\Ad_{\bar{f}}^{k_1+1}\notin D_{k_1}$, as it would either lead to $\Delta_{k_1+1}\cap\TXU=D_{k_1}$ or $\Delta_{k_1+2}\cap\TXU=D_{k_1}$, which contradicts with $D_{k_1}$ being non-involutive.]

\begin{align*}
\begin{aligned}
\mathcal{I}_{k_1}=\Delta_{k_1+1}\cap D_{k_1}=D_{k_1-1}+\Span{\Ad_{\bar{f}}^{k_1}\partial_{\bar{u}^2}}\,.
\end{aligned}
\end{align*}

Since $\mathcal{I}_{k_1}\subset\Delta_{k_1+1}\cap\TXU$ and

\begin{align*}
\begin{aligned}
\Delta_{k_1+1}\cap\TXU&=D_{k_1-1}+\Span{\Ad_{\bar{f}}^{k_1}\partial_{\bar{u}^2},\Ad_{\bar{f}}^{k_1+1}\partial_{\bar{u}^2}}\,,
\end{aligned}
\end{align*}

is involutive, we then have $\overline{\mathcal{I}}_{k_1}=\Delta_{k_1+1}\cap\TXU$, $\mathcal{I}_{k_1}\underset{1}{\subset}\overline{\mathcal{I}}_{k_1}$. We have $\mathcal{I}_{k_1}\underset{1}{\subset}D_{k_1}$ and since $D_{k_1-1}\subset\C{D_{k_1}}$, we have $[D_{k_1-1},\mathcal{I}_{k_1}]\subset D_{k_1}$. However, since $\mathcal{I}_{k_1}$ is assumed to be non-involutive, we necessarily have $[D_{k_1-1},\mathcal{I}_{k_1}]\not\subset\mathcal{I}_{k_1}$ and thus, the direction which completes $\mathcal{I}_{k_1}$ to its involutive closure is contained in $D_{k_1}$, which would lead to $\overline{\mathcal{I}}_{k_1}=D_{k_1}$ and contradict with $D_{k_1}$ being non-involutive. Thus, $\mathcal{I}_{k_1}$ is indeed involutive.

From the Jacobi identity

\begin{align*}
\begin{aligned}
\underbrace{[\Ad_{\bar{f}}^{k_1-1}\partial_{\bar{u}^1},[\bar{f},\Ad_{\bar{f}}^{k_1}\partial_{\bar{u}^2}]]}_{\in\Delta_{k_1+1}\cap\TXU\subset D_{k_1}+\Span{\Ad_{\bar{f}}^{k_1+1}\partial_{\bar{u}^2}}}+\underbrace{[\Ad_{\bar{f}}^{k_1}\partial_{\bar{u}^2},[\Ad_{\bar{f}}^{k_1-1}\partial_{\bar{u}^1},\bar{f}]]}_{\in D^{(1)}_{k_1}}+\underbrace{[\bar{f},\overbrace{[\Ad_{\bar{f}}^{k_1}\partial_{\bar{u}^2},\Ad_{\bar{f}}^{k_1-1}\partial_{\bar{u}^1}]}^{\in\mathcal{I}_{k_1}}]}_{\in D_{k_1}+\Span{\Ad_{\bar{f}}^{k_1+1}\partial_{\bar{u}^2}}}&=0\,,
\end{aligned}
\end{align*}

where we have used the involutivity of $\mathcal{I}_{k_1}$ for the third term, it follows that the vector field $[\Ad_{\bar{f}}^{k_1}\partial_{\bar{u}^1},\Ad_{\bar{f}}^{k_1}\partial_{\bar{u}^2}]$, which completes $D_{k_1}$ to $D_{k_1}^{(1)}$, is contained in $D_{k_1}+\Span{\Ad_{\bar{f}}^{k_1+1}\partial_{\bar{u}^2}}$, and thus $D^{(1)}_{k_1}=D_{k_1}+\Span{\Ad_{\bar{f}}^{k_1+1}\partial_{\bar{u}^2}}$, as desired.

§.§ Proof of Lemma <ref>

The vector field $\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2}\in E_{k_2-1}$ is a Cauchy characteristic vector field of $E_{k_2}$. Indeed, we have (see (<ref>) and (<ref>))

\begin{align}\label{eq:Ek2}
\begin{aligned}
\end{aligned}
\end{align}

From the Jacobi identity

\begin{align*}
\begin{aligned}
[\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2},[\bar{f},\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1}]]+\underbrace{[\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1},[\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2},\bar{f}]]}_{\in\Delta_{k_2}\cap\TXU\subset E_{k_2}}+\underbrace{[\bar{f},\overbrace{[\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2}]}^{\in E_{k_2-1}}]}_{\in E_{k_2}}&=0\,,
\end{aligned}
\end{align*}

it follows that $[\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^1}]\in E_{k_2}$, where we have used the fact that $[\bar{f},E_{k_2-1}]\subset E_{k_2}$ and that $\Delta_{k_2}\cap\TXU$ and $E_{k_2-1}$ are involutive. The involutivity of $\Delta_{k_2}\cap\TXU$ together with $[\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^1}]\in E_{k_2}$ implies that $\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2}\in\C{E_{k_2}}$.

The distribution $E_{k_2}$ is by assumption non-involutive. Since $E_{k_2}\subset\Delta_{k_2+1}\cap\TXU$ and $\Delta_{k_2+1}\cap\TXU$ is involutive, it follows that $\overline{E}_{k_2}=\Delta_{k_2+1}\cap\TXU=E_{k_2}+\Span{\Ad_{\bar{f}}^{k_2+1}\partial_{\bar{u}^2}}$.

In case of $k_2\geq k_1+2$, it immediately follows from the Jacobi identity and the way the distributions $E_i$ are constructed that $E_{k_2-2}\subset\C{E_{k_2}}$. We have $E_{k_2-1}=E_{k_2-2}+\Span{\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2}}$ and we have just shown that $\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2}\in\C{E_{k_2}}$. Since by assumption $E_{k_2-1}\not\subset\C{E_{k_2}}$, it follows that $\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1}\notin\C{E_{k_2}}$. Since $\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1}$ is contained in the involutive subdistribution $\Delta_{k_2}\cap\TXU\underset{1}{\subset}E_{k_2}$ (see (<ref>)), it follows that the vector field $[\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^1}]$ cannot be contained in $E_{k_2}$ and thus, $\overline{E}_{k_2}=E_{k_2}+\Span{[\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^1}]}$, which in turn implies that the vector fields $[\Ad_{\bar{f}}^{k_2}\partial_{\bar{u}^2},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^1}]$ and $[\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^1}]$ are collinear modulo $E_{k_2}$, $[\Ad_{\bar{f}}^{k_2}\partial_{\bar{u}^2},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^1}]=\lambda [\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^1}]\Mod E_{k_2}$. From the latter, it follows that $[\Ad_{\bar{f}}^{k_2}\partial_{\bar{u}^2}-\lambda\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1},\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^1}]\in E_{k_2}$, which because of $\Ad_{\bar{f}}^{k_2}\partial_{\bar{u}^2}-\lambda\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1}\in\Delta_{k_2}\cap\TXU$ implies that $\Ad_{\bar{f}}^{k_2}\partial_{\bar{u}^2}-\lambda\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1}\in\C{E_{k_2}}$. It follows that $\C{E_{k_2}}=E_{k_2-2}+\Span{\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2},\Ad_{\bar{f}}^{k_2}\partial_{\bar{u}^2}-\lambda\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1}}\underset{2}{\subset}E_{k_2}$, further Cauchy characteristic vector fields which are linearly independent of those we have already found cannot exist, otherwise $E_{k_2}$ would be involutive.

In case of $k_2=k_1+1$, the distribution $E_{k_2-2}$ does not exist. However, we have the involutive distribution $\Delta_{k_1}\cap D_{k_1-1}=D_{k_1-2}+\Span{\Ad_{\bar{f}}^{k_1-1}\partial_{\bar{u}^2}}$. Recall that we have $E_{k_1}=D_{k_1-1}+\Span{\Ad_{\bar{f}}^{k_1}\partial_{\bar{u}^2}}$. Thus, it follows that $[\bar{f},\Delta_{k_1}\cap D_{k_1-1}]\subset E_{k_1}$. Based on the latter, by applying the Jacobi identity
, it can be shown that $\Delta_{k_1}\cap D_{k_1-1}\subset\C{E_{k_2}}$. Repeating the above proof with $E_{k_2-2}$ replaced by $D_{k_1-1}\cap\Delta_{k_1}$ gives the desired result that in case of $k_2=k_1+1$ we have $\C{E_{k_2}}=D_{k_1-1}\cap\Delta_{k_1}+\Span{\Ad_{\bar{f}}^{k_2-1}\partial_{\bar{u}^2},\Ad_{\bar{f}}^{k_2}\partial_{\bar{u}^2}-\lambda\Ad_{\bar{f}}^{k_2-2}\partial_{\bar{u}^1}}$ .

§.§ Proof of Lemma <ref>

By assumption, we have $G_{k-1}\subset\C{G_k}$, which actually implies $G_{k-1}=\C{G_k}$. Indeed, since $G_k$ is non-involutive, we necessarily have $\Dim{\C{G_k}}\leq\Dim{G_k}-2$. Because of $G_{k-1}\subset\C{G_k}$ and $G_{k-1}\underset{2}{\subset}G_k$, we thus have $G_{k-1}=\C{G_k}$. Apply a change of coordinates $(z_3,z_2,z_1)=\Phi(x,u)$ such that $G_{k-1}$ and $\overline{G}_k$ get straightened out simultaneously, such that in the new coordinates we have

\begin{align*}
\overline{G}_k&=\Span{\partial_{z_1},\partial_{z_2}}\,.
\end{align*}

Since $G_{k-1}=\C{G_k}$, in these coordinates, there exists a basis for $G_k$ of the form $G_k=G_{k-1}+\Span{v_1,v_2}$ with the two vector fields $v_j$ being of the form $v_j=v_j^{i_2}(z_3,z_2)\partial_{z_2^{i_2}}$, $i_2=1,\ldots,\Dim{z_2}$ (where $\Dim{z_2}=3$).

In case of $\overline{G}_k=\TXU$ (we have  $\Dim{z_3}=0$ and thus simply $v_j=v_j^{i_2}(\bar{x}_2)\partial_{\bar{x}_2^{i_2}}$ in this case), choose any function $\psi(z_2)$, $\D\psi\neq 0$ and define the vector field $v_c=(\D\psi\rfloor v_2)v_1-(\D\psi\rfloor v_1)v_2$, which is non-zero due to the non-involutivity of $G_k$ (indeed, $\D\psi\rfloor v_1=\D\psi\rfloor v_2\equiv 0$ would yield $G_k^\perp=\Span{\D\psi}$ which contradicts with $G_k$ being non-involutive). The distribution $H_k=G_{k-1}+\Span{v_c}$ is obviously involutive and we by construction have $G_{k-1}\underset{1}{\subset}H_k\underset{1}{\subset}G_k$. Note that there exist infinitely many valid choices for the function $\psi$ and different choices yield in general different distributions $H_k$.[Note that there is no need to explicitly construct the vector fields $v_j=v_j^{i_2}(\bar{x}_2)\partial_{\bar{x}_2^{i_2}}$, $j=1,2$ for deriving the distribution $H_k$ corresponding to a certain choice $\psi$. Indeed, any vector fields $w_1,w_2$ which complete $G_{k-1}$ to $G_k$ can be written as a linear combination $w_j=\beta_j^1v_1+\beta_j^2v_2\Mod G_{k-1}$ with some functions $\beta_j^i=\beta_j^i(z_2,z_1)$ and $\beta_1^1\beta_2^2-\beta_1^2\beta_2^1\neq 0$. A straight forward calculation shows that $(\D\psi\rfloor w_2)w_1-(\D\psi\rfloor w_1)w_2=(\beta_1^1\beta_2^2-\beta_1^2\beta_2^1)((\D\psi\rfloor v_2)v_1-(\D\psi\rfloor v_1)v_2)\Mod G_{k-1}$ and thus, $\tilde{H}_k=G_{k-1}+\Span{(\D\psi\rfloor w_2)w_1-(\D\psi\rfloor w_1)w_2}=D_{k_1-1}+\Span{(\D\psi\rfloor v_2)v_1-(\D\psi\rfloor v_1)v_2}=H_k$, the particular choice for the vector fields $w_1,w_2$ has no effect as long as they complete $G_{k-1}$ to $G_k$.]

Next, let us consider the case $\overline{G}_k\neq\TXU$. Note that because of $[f,G_{k-1}]\subset G_k\subset\overline{G}_k$, in the above introduced coordinates the vector field $f$ is of the form

\begin{align*}
\end{align*}

We thus have

\begin{align*}
\end{align*}

where $f_3=f_3^{i_3}(z_3,z_2)\partial_{z_3^{i_3}}$. Since by assumption $\Dim{[f,G_k]+\overline{G}_k}=\Dim{\overline{G}_k}+1$, the vector fields $[v_1,f_3]$ and $[v_2,f_3]$ are collinear modulo $\overline{G}_k$. Without loss of generality, we can assume that $[v_1,f_3]\notin\overline{G}_k$, implying that there exists a function $\alpha$ such that $[v_2,f_3]=\alpha[v_1,f_3]\Mod\overline{G}_k$. This function only depends on $\bar{x}_2$ and $\bar{x}_3$ since the vector fields $v_1,v_2$ and $f_3$ only depend on $z_3$ and $z_2$, $\alpha=\alpha(z_3,z_2)$. Define the distribution $H_k=G_{k-1}+\Span{v_2-\alpha v_1}$. It is immediate that this distribution is involutive and since $v_1$ and $v_2$ are independent, we have $G_{k-1}\underset{1}{\subset}H_k\underset{1}{\subset}G_k$.[It follows that the direction of the vector field $v=v_2-\alpha v_1$ is modulo $G_{k-1}$ uniquely determined by the conditions $v\in G_k$, $v\notin G_{k-1}$, $[v,f]\in\overline{G}_k\neq\TXU$ and thus, the distribution $H_k$ is unique in this case. Indeed, assume that there would exist another such vector field $w\in G_k$, $w\notin G_{k-1}$, which is modulo $G_{k-1}$ not collinear with $v$ and still satisfies $[w,f]\in\overline{G}_k$. Then, we would have $G_k=G_{k-1}+\Span{v,w}$ and in turn $[f,G_k]\subset\overline{G}_k$, which is in contradiction with $\Dim{[f,G_k]+\overline{G}_k}=\Dim{\overline{G}_k}+1$.]

So in any of the two cases, $\overline{G}_k=\TXU$ or $\overline{G}_k\neq\TXU$, we have shown that there exists an involutive distribution $H_k=G_{k-1}+\Span{v}$ with some vector field $v\in G_k$, $v\notin G_{k-1}$. What is left to do is to show that $H_k+[f,H_k]=\overline{G}_k$. By construction, we have $[f,H_k]\subset\overline{G}_k$, so we only have to show that $[f,v]\notin G_k$, which can be shown by contradiction. Assume that $[f,v]\in G_k$. Due to the Jacobi identity, for every vector field $w\in G_{k-1}$, we then have

\begin{align*}
\underbrace{[w,\overbrace{[f,v]}^{\in G_k}]}_{\in G_k}+[v,\overbrace{[w,f]}^{\in G_k}]+\underbrace{[f,\overbrace{[v,w]}^{\in H_k}]}_{\in G_k}&=0\,,
\end{align*}

(where we have used that $G_{k-1}\subset\C{G_k}$ and that $[f,v]\in G_k$ implies $[f,H_k]\subset G_k$ as well as the involutivity of $H_k$). However, this implies $[v,[w,f]]\in G_k$ for every $w\in G_{k-1}$, which in turn would imply that $G_k$ is involutive and contradict with $G_k$ being non-involutive.

§.§ Proof of Lemma <ref>

We have $D_{k_1}=D_{k_1-1}+\Span{v_1,v_2}$ and since $D_{k_1-1}\subset\C{D_{k_1}}$, we have $D_{k_1}^{(1)}=D_{k_1-1}+\Span{v_1,v_2,[v_1,v_2]}$. Based on the Jacobi identity, it can easily be shown that $D_{k_1-1}\subset\C{D_{k_1}^{(1)}}$. Since by assumption $\Dim{\overline{D}_{k_1}}=\Dim{D_{k_1}}+2$, the vector fields $[v_1,[v_1,v_2]]$ and $[v_2,[v_1,v_2]]$ are collinear modulo $D_{k_1}^{(1)}$. Without loss of generality, we can assume that $[v_1,[v_1,v_2]]\notin D_{k_1}^{(1)}$ and thus, $[v_2,[v_1,v_2]]=\alpha [v_1,[v_1,v_2]]\Mod D_{k_1}^{(1)}$. The vector field $v=v_2-\alpha v_1$ by construction meets $v\in D_{k_1}$, $v\notin D_{k_1-1}$ and $[v,[v_1,v_2]]=0\Mod D_{k_1}^{(1)}$, implying that $v\in\C{D_{k_1}^{(1)}}$. It follows that $\C{D_{k_1}^{(1)}}=D_{k_1-1}+\Span{v}$ (the existence of further Cauchy characteristic vector fields would contradict with $D_{k_1}^{(1)}$ being non-involutive) and thus $D_{k_1-1}\underset{1}{\subset}\C{D_{k_1}^{(1)}}\underset{1}{\subset}D_{k_1}$.

That $\C{D_{k_1}^{(1)}}+[f,\C{D_{k_1}^{(1)}}]=D_{k_1}^{(1)}$ can be shown as follows. We have $D_{k_1}\underset{1}{\subset}D_{k_1}^{(1)}$ and by assumption $[f,\C{D_{k_1}^{(1)}}]\subset D_{k_1}^{(1)}$, and we just have shown that $\C{D_{k_1}^{(1)}}=D_{k_1-1}+\Span{v}$ with $v\in D_{k_1}$, $v\notin D_{k_1-1}$. Therefore, $\C{D_{k_1}^{(1)}}+[f,\C{D_{k_1}^{(1)}}]=D_{k_1}^{(1)}$ holds if and only if $[f,v]\notin D_{k_1}$, which can be shown by contradiction. Assume that $[f,v]\in D_{k_1}$. Due to the Jacobi identity, for every vector field $w\in D_{k_1-1}$, we then have

\begin{align*}
\begin{aligned}
\underbrace{[w,\overbrace{[f,v]}^{\in D_{k_1}}]}_{\in D_{k_1}}+[v,\overbrace{[w,f]}^{\in D_{k_1}}]+\underbrace{[f,\overbrace{[v,w]}^{\in \C{D_{k_1}^{(1)}}}]}_{\in D_{k_1}}&=0\,,
\end{aligned}
\end{align*}

(where we have used that $D_{k_1-1}\subset\C{D_{k_1}}$ and that $[f,v]\in D_{k_1}$ implies $[f,\C{D_{k_1}^{(1)}}]\subset D_{k_1}$ as well as the involutivity of $\C{D_{k_1}^{(1)}}$). However, this implies that $[v,[w,f]]\in D_{k_1}$ for every $w\in D_{k_1-1}$, which in turn would imply that $D_{k_1}$ is involutive and contradict with $D_{k_1}$ being non-involutive.