Vol.0 (20xx) No.0, 000–000

11institutetext: Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS>
Received 20xx month day; accepted 20xx month day

# Pre-explosion helium shell flash in type Ia supernovae

Noam Soker 11

###### Abstract

I study the possibility that within the frame of the core degenerate (CD)
scenario for type Ia supernovae (SNe Ia) the merger process of the core of the
asymptotic giant branch (AGB) star and the white dwarf (WD) maintains an
envelope mass of $\approx 0.03M_{\odot}$ that causes a later helium shell
flash. I estimate the number of pre-explosion helium shell flash events to be
less than few per cents of all CD scenario SNe Ia. A helium shell flash while
the star moves to the left on the HR diagram as a post-AGB star (late thermal
pulse—LTP) or along the WD cooling track (very-LTP) causes the star to expand
and become a ‘born again’ AGB star. Merger remnants exploding while still on
the AGB form peculiar SNe Ia or peculiar SNe II (if hydrogen is detected),
while an explosion inside an inflated born-again star results in an early flux
excess in the light curve of the SN Ia. The fraction of systems that might
show an early flux excess due to LTP/VLTP is $<{\rm few}\times 10^{-4}$ of all
SNe Ia, much below the observed fraction. In the frame of the CD scenario SNe
Ia with early flux excess result from SN ejecta collision with planetary
nebula fallback gas, or from mixing of 56Ni to the outer regions of the SN
ejecta. Ongoing sky surveys might find about one case per year where LTP/VLTP
influences the SN light curve.

###### keywords:

(stars:) white dwarfs – (stars:) supernovae: general – (stars:) binaries:
close

##  1 INTRODUCTION

There are several theoretical scenarios for the ignition of carbon oxygen
white dwarfs (CO WDs) as type Ia supernovae (SN Ia; e.g., Hoeflich 2017; Livio
& Mazzali 2018; Soker 2018; Wang 2018; Jha et al. 2019; Ruiz-Lapuente 2019;
Soker 2019a; Ruiter 2020 for less than five years old reviews). I classify
these scenarios as follows (see Soker 2019a for a comparison table of the six
scenarios).

  1. 1.

The core-degenerate (CD) scenario involves the merger of a CO, or HeCO, WD
with the core of a massive asymptotic giant branch (AGB) star. The merger
remnant is a WD close to the Chandrasekhar mass limit that explodes at a later
time. The time to explosion is the merger to explosion (MED) delay time
$t_{\rm MED}$ (e.g., Kashi & Soker 2011; Ilkov & Soker 2013; Aznar-Siguán et
al. 2015). The possibility that the CO WD merger remnant maintains a very
small mass of helium and the possible consequences are the subject of the
present study.

  2. 2.

The double degenerate (DD) scenarios and the DD-MED scenario involve the
merger of two WDs (e.g., Webbink 1984; Iben & Tutukov 1984) that lose energy
to gravitational waves. Helium might play a crucial role in igniting the
merging WDs (e.g., Yungelson, & Kuranov 2017; Zenati et al. 2019; Perets et
al. 2019). The explosion might take place during the merger itself (e.g.,
Pakmor et al. 2011; Liu et al. 2016; Ablimit, Maeda, & Li 2016), or at a MED
time (e.g., Lorén-Aguilar et al. 2009; van Kerkwijk et al. 2010; Pakmor et al.
2013; Levanon, Soker, & García-Berro 2015; Levanon & Soker 2019; Neopane et
al. 2022).

  3. 3.

In the double-detonation (DDet) scenario ignition of helium that a CO-rich WD
accretes from a companion triggers the WD explosion (e.g., Woosley & Weaver
1994; Livne & Arnett 1995; Shen et al. 2018).

  4. 4.

In the single degenerate (SD) scenario a CO WD accretes hydrogen-rich gas from
a non-degenerate companion (e.g., Whelan & Iben 1973; Han & Podsiadlowski
2004; Orio 2006; Wang et al. 2009; Meng, & Podsiadlowski 2018). The WD might
explode as soon as it approaches the Chandrasekhar-mass, or it might explode
after a long delay (long MED) after losing angular momentum (e.g., Piersanti
et al. 2003; Di Stefano et al. 2011; Justham 2011).

  5. 5.

In the WD-WD collision (WWC) scenario the two WDs explode as they collide at
about their free fall velocity (e.g., Raskin et al. 2009; Rosswog et al. 2009;
Kushnir et al. 2013; Aznar-Siguán et al. 2014). Studies (e.g., Toonen et al.
2018; Hallakoun & Maoz 2019; Hamers & Thompson 2019) found that the WWC
scenario contributes $<1\%$ of all SNe Ia.

The community is far from a consensus on the leading SN Ia scenario(s).
Therefore, studies continue to explore each one of these stellar binary
scenarios (e.g., some examples from the last two years, Cui et al. 2020; Zou,
Zhou, & Huang 2020; Blondin et al. 2021; Clark et al. 2021; Ablimit 2021;
Chandra et al. 2021; Liu et al. 2021; Meng & Luo 2021; Michaely 2021; Wang et
al. 2021; Zeng et al. 2021; Acharova, Sharina, & Kazakov 2022; Chu et al.
2022; Cui & Li 2022; Dimitriadis et al. 2022; Ferrand et al. 2022; Lach et al.
2022; Liu, Roepke, & Zeng 2022; Livneh & Katz 2022; Mazzali et al. 2022;
Pakmor et al. 2022; Patra et al. 2022; Piersanti et al. 2022; Sharon & Kushnir
2022; Shingles et al. 2022).

My view (see table in Soker 2019a) is that the CD scenario and DD-MED scenario
account for most normal SNe Ia. The main differences between the core-WD
merger in the CD scenario and the WD-WD merger in the DD scenario are the hot
core and the hydrogen-rich envelope at merger in the CD scenario. In the
present study I propose that in some cases the core-WD merger remnant in the
CD scenario might maintain sufficient amount of helium to experience a late
helium shell flash, i.e., a late thermal pulse, (LTP), or a very late thermal
pulse (VLTP).

In single star evolution a post-AGB star might experience a LTP, i.e., a
helium shell flash while it is still burning hydrogen and moves leftward
horizontally on the HR diagram, or it might experience a VLTP, i.e., a helium
shall flash after hydrogen burning has ceased and the post-AGB is now on the
WD cooling track. The parameters that determine whether a post-AGB star
experiences a LTP, a VLTP, or none, are the mass of the helium layer when it
leaves the AGB and where in the thermal pulse cycle on the AGB the star
departs the AGB (e.g., Iben 1987; Lawlor & MacDonald 2006). The massive WD
that we deal with here is formed from the merger of the core with a WD, and
therefore it is not straightforward to deduce on its evolution by using
single-star evolutionary tracks. I will rather use some general properties of
single-star evolution to speculate on possible outcomes for the merger
product.

In section 2 I discuss the conditions for a pre-explosion LTP or VLTP. In
section 3 I explore the way by which the core-WD merger remnant retains some
helium, and in section 4 I refer to the relevant timescales of the scenario.
In section 5 I estimate the rate of SNe Ia where LTP/VLTP might influence the
light curve, and in section 6 I discuss possible consequences. I discuss and
summarize this somewhat speculative study in section 7.

##  2 The conditions for a pre-explosion (very) late thermal pulse (LTP/VLTP)

The occurrence of a pre-explosion LTP/VLTP requires the following ingredients.
It is the first condition (section 2.1) that I expect to be rare. The second
condition (section 2.2) might hold for a large fraction of SNe Ia.

###  2.1 Retaining helium in the merger process

In low-mass single-star evolution an LTP (when hydrogen still burns in a thin
shell) or a VLTP (after the hydrogen burning has ceased) take place when the
helium mass on top of the CO core when the star leaves the AGB is $\Delta
M_{\rm He}\simeq 0.01M_{\odot}$ (e.g., Iben 1987; Lawlor & MacDonald 2006). On
the other hand, more massive stars, super-AGB stars that form oxygen-neon
cores, have weak thermal pulses with short time intervals between pulses
(e.g., Siess 2010; see discussion of thermal pulses by, e.g., Gautschy &
Althaus 2007). This implies that only a small hydrogen mass is required to
lead to the next pulse. Siess (2010) finds that for a core of about
$1.3M_{\odot}$ the hydrogen mass advances by only $5\times 10^{-5}M_{\odot}$
between pulses. Two comments related to these values are as follows. (1) The
young merger product is more extended than an isolated massive WD of the same
mass (Neopane et al. 2022), and so the required mass for a LTP/VLTP is larger
than for a single star of the same mass. (2) The relevant point for this study
is that if the LTP/VLTP is very weak, lasts for a short time, and does not
contain much mass, then it has only a minor effect on the light curve of the
SN Ia.

Following the above discussion I will scale quantities with the requirement of
massive helium shell, but note that the scaling might overestimate the mass
required for the LTP/LTP to take place, and it is possible that in all merger
remnants the LTP/VLTP are very weak. In any case, I will conclude that strong
LTP/VLTP must be very rare (less than few per cents) among all SNe Ia that
result from the CD scenario.

In the single-star simulations the mass of the post-AGB star that is evolving
to become a WD is $\simeq 0.6-0.9M_{\odot}$, and the WD evolves as a slowly
rotating single star. In the scenario I study here the WD is a merger remnant
that is rapidly rotating and its mass is $\simeq 1.4M_{\odot}$. I take the
condition on the remaining helium mass in the merger remnant to be

$\Delta M_{\rm He,TP}\approx 0.01M_{\odot};\qquad{\rm Condition~{}1}.$ (1)

As stated above, because of the high mass of the merger remnant, this mass
limit can be smaller even.

Note that condition 1 (equation 1) does not require that immediately after the
merger process ends this is the helium mass. It rather allows also for an
increase of the helium mass by hydrogen burning after the merger ends (section
3.2).

###  2.2 A merger to explosion delay (MED) time

In single-star evolution typical LTP/VLTPs occur at about a thousand to tens
of thousands of years after the star has left the AGB (e.g., Iben et al. 1983;
Blöcker 2001). In the case of a massive WD that is a remnant of a core-WD
merger the time scale can be shorter even. I will return to this timescale in
section 4. Here I only point to the requirement of a MED.

The requirement that, at least some, progenitors of SNe Ia have a MED, i.e., a
time delay from merger (or from a mass transfer event) to explosion comes from
the observational findings that many SNe Ia have no close circumstellar matter
(CSM; see review by Soker 2018) at explosion time, and that many SNe Ia have a
more or less spherical explosion (see review by Soker 2019a).

Studies have considered a MED of tens of years (e.g., Neopane et al. 2022 for
a recent paper on the DD scenario) to tens of thousands of years (e.g., Soker
2022 for a recent paper on the CD scenario). For the VLTP to have any role on
the light curve of the SN, the LTP/VLTP cannot occur a long time before the
explosion. The typical time of the LTP/VLTP after the AGB phase sets an upper
limit on the MED of $\approx 10^{5}{~{}\rm yr}$. Overall, for a pre-explosion
LTP/VLTP to occur and play a role I require the MED time to be

$t_{\rm med}\la 10^{5}{~{}\rm yr};\qquad{\rm Condition~{}2}.$ (2)

Condition 2 (equation 2) does not impose a strong limit on pre-explosion
LTP/VLTP if we consider that, crudely, $\approx 50\%$ of SNe Ia might obey
this condition (Soker 2022). It is Condition 1 (equation 1) that strongly
constrains the number of pre-explosion LTP/VLTP. I therefore turn to study the
possibility of the merger remnant to retain some helium.

##  3 Retaining low helium mass

###  3.1 The CD scenario versus the DD scenario

Critical to the occurrence of a LTP/VLTP is that some helium survives the
merger process. When two WDs merge in the DD scenario the helium, which one or
two of them have, easily burns because of the high temperatures that the outer
zones achieve (e.g., Dan et al. 2012; Pakmor et al. 2013; Perets et al. 2019;
Pakmor et al. 2021). Therefore, it is not clear at all whether any helium can
survive the merger of two WDs. The answer will come from future 3D
simulations.

I therefore consider only the CD scenario where the merger takes place at the
end of the CEE (Kashi & Soker 2011). In this case the helium on the outer zone
of the core is already very hot, $\simeq 10^{8}{~{}\rm K}$. It is possible
therefore that this helium is not compressed to high enough densities to be
ignited. On the other hand, the violent merger process likely mixes helium
with carbon, a mixture that burns more easily. Again, only 3D numerical
simulations can determine whether some helium from the core survives the core-
WD merger in the CD scenario. Therefore, I first consider the case where
hydrogen and helium from the envelope survive.

###  3.2 Surviving envelope helium

To have a core-WD merger in the CD scenario the mass of the envelope of the
AGB star at the time it engulfs the WD secondary should be $M_{\rm AGB,env}\ga
3M_{\odot}$ (e.g., Soker et al. 2013; Canals, Torres, & Soker 2018).

I use the Reimers mass loss rate (Reimers 1975) in the form $\dot{M}_{\rm
wind}\simeq 3\times 10^{-13}LM^{-1}RM_{\odot}{~{}\rm yr}^{-1}$, where the
stellar luminosity $L$, mass $M$, and radius $R$ are in solar units. I note
that other expressions are also possible (e.g., Schröder & Cuntz 2007). For
this mass loss rate, which is linear with the stellar luminosity, the ratio of
the rate at which envelope burning adds helium to the core, $\dot{M}_{\rm
He}$, to the wind mass loss rate, $\dot{M}_{\rm wind}$, is

$\frac{\dot{M}_{\rm He}}{\dot{M}_{\rm wind}}\simeq 0.3\left(\frac{M_{\rm
rem}}{1.4M_{\odot}}\right)\left(\frac{R_{\rm g}}{200R_{\odot}}\right)^{-1},$
(3)

where $M_{\rm rem}$ is the mass of the merger remnant, and for the giant
radius $R_{\rm g}$ I take a moderate radius because the envelope mass is low.
To retain a helium mass that allows a LTP/VLTP, i.e., $\Delta M_{\rm
He,TP}\approx 0.01M_{\odot}$ (equation 1), the merger process should leave an
envelope mass of

$M_{\rm rem,env}\approx 0.03M_{\odot}\simeq 0.01M_{\rm AGB,env}.$ (4)

Kashi & Soker (2011) estimated that to facilitate a core-WD merger in the CD
scenario a fraction of $\eta_{\rm CBD}\simeq 0.01-0.1$ of the common envelope
remains bound in a circumbinary disk at the end of the CEE. Here I require
that either in addition to the circumbinary disk there is a low-mass envelope
that survives the CEE, or that a fraction of the circumbinary disk remains
bound after merger and it is inflated to form a low-mass envelope.

The conclusion is that to have a LTP/VLTP the merging process should leave
about one per cent or more of the initial AGB envelope mass bound to the
merger remnant. I expect that only a minority, if at all, of SN Ia progenitors
in the CD scenario will leave such an envelope, but that nonetheless it is
possible.

###  3.3 Surviving core helium

I here estimate that not enough helium to lead to a LTP/VLTP from the core
survives the merger.

As I mentioned above (section 3.1) it is not clear at all that the merger
process leaves any helium from the core. In the core-WD merger process either
the WD or the core stay intact, and the other object is tidally destroyed. The
helium, being on the outer part of the core, is mixed with the CO of the
destroyed object, being the WD or the core. Therefore, if the process leaves
any helium, the helium is mixed with the carbon and oxygen in the outer parts
of the merger remnants. At the outer regions the densities are too low for the
pycnonuclear reactions that Chiosi et al. (2015) studied to take place. Chiosi
et al. (2015) proposed an isolated sub-Chandrasekhar WD scenario for SNe Ia.
In their scenario pycnonuclear reactions between carbon and tiny amounts of
hydrogen or helium during the crystallisation of the WD ignite a SN Ia
explosion.

If some helium is mixed with the CO in the outer zone of the WD, it diffuses
out. I very crudely estimate the diffusion coefficient of helium through CO
matter based on expression from, e.g., Paquette et al. (1986) and Caplan,
Bauer, & Freeman (2022), to be $D\approx 0.1(T/10^{8}{~{}\rm
K})^{3/2}(\rho/10^{6}{~{}\rm g}{~{}\rm cm}^{-1})^{-2/3}{~{}\rm cm}^{2}{~{}\rm
s}^{-1}$. The helium is spread in a layer of width $\Delta r\approx
1000{~{}\rm km}$. The diffusion time will be therefore $\tau_{\rm
diff}\approx(\Delta r)^{2}/D$. For the above parameters, and since the WD
cools over this long time scale, I estimate that $\tau_{\rm
diff}>10^{9}{~{}\rm yr}$. (I get the same crude timescales from scaling the
tables that Koester 2009 give). By that time the WD has cooled. I conclude
that if the helium mixture with carbon and oxygen in the hot WD merger remnant
did not ignite the helium (but I do expect ignition of helium), the floating
of helium on the surface of the WD is unlikely to ignite a VLTP.

##  4 Relevant timescales

In single-star evolution a LTP (a post-AGB helium shell flash while hydrogen
still burns) takes place hundreds to thousands of years after departure from
the AGB (e.g., Blöcker 2001). For a VLTP (when hydrogen burning has already
ceased) the star needs to fade first to a luminosity of $L_{\rm
VLTP}\approx{\rm few}\times 100L_{\odot}$ along the WD cooling track (e.g.,
Iben et al. 1983; Blöcker 2001). This implies that VLTPs take place tens of
thousands of years after departure from the AGB, with longer timescales for
more massive stars (e.g., Bloecker & Schoenberner 1990; Bloecker 1995, for
evolutionary times on the WD cooling track). Overall, I crudely take the
LTP/VLTP of the merger remnant to occur sometime in the post-AGB age of

$100{~{}\rm yr}\la t_{\rm AGB-TP}\la 10^{5}{~{}\rm yr}.$ (5)

This is the reason for condition 2 (equation 2). I do note that the merger
remnant is expected to be highly magnetized (e.g., Pelisoli et al. 2022).
Future studies should examine the role of the high magnetic fields in the
LTP/VLTP.

For a surviving envelope mass of $M_{\rm rem,env}$, and using the same mass
loss rate as in section 3.2 and for a remnant mass of $M_{\rm
rem}=1.4M_{\odot}$, the time from the merger event to the termination of the
AGB by the merger remnant is

$\begin{split}&\Delta t_{\rm mer-AGB}\approx 1.1\times
10^{4}\left(\frac{M_{\rm rem,env}}{0.03M_{\odot}}\right)\\\
&\times\left(\frac{L_{\rm rem}}{5\times
10^{4}L_{\odot}}\right)^{-1}\left(\frac{R_{\rm
g}}{200R_{\odot}}\right)^{-1}{~{}\rm yr},\end{split}$ (6)

Another process that determines the post-merger evolution is angular momentum
loss through the magnetic dipole radiation torque. This timescale might set
the MED time in the CD scenario (e.g., Ilkov & Soker 2012) and in the DD
scenario (e.g., Neopane et al. 2022). Several parameters determine the angular
momentum loss timescale. Some of them are more secure, like the moment of
inertia of the WD. Assuming that the merger remnant is rapidly rotating and
has a mass of $\simeq 1.4M_{\odot}$, there are two undetermined parameters
that influence the angular momentum loss timescale the most. These are the
dipole magnetic field of the remnant $B$, and the angle between the magnetic
axis and the rotation axis of the WD remnant, $\delta$. The angular momentum
loss timescale dependence on these two parameters is (for more details see
Ilkov & Soker 2012 and Neopane et al. 2022)

$\begin{split}\tau_{\rm B}\approx 10^{4}\left(\frac{B}{10^{9}{~{}\rm
G}}\right)^{-2}\sin^{2}\delta{~{}\rm yr}.\end{split}$ (7)

Note that Ilkov & Soker (2012) scale (their equation 10) by $B=10^{8}{~{}\rm
G}$ and by $\sin\delta=0.1$, while Neopane et al. (2022) scale (their equation
10) by $B=10^{10}{~{}\rm G}$ and by $\sin\delta=1$. If the angular momentum
loss timescale determines the time to the WD explosion then $t_{\rm
MED}\simeq\tau_{\rm B}$.

If indeed condition 2 (equation 2) holds, then the explosion takes place
before the nebula that the CEE (that leads to the merger) ejected disperses to
the ISM, although in might be at a large distance from the explosion. This
nebula is a planetary nebula, as the WD remnant ionises the nebula. The
explosion of a SN inside a planetary nebula is termed SNIP. In (Soker 2022) I
crudely deduced by analysing observations that SNIPs account for $\approx
50\%$ of all normal SNe Ia (i.e., not including peculiar SNe Ia).

##  5 The rate of the rare LPT/VLTP SNIPs

Equation (6) shows that if the merger remnant maintains enough envelope mass
to experience a strong LTP/VLTP, then its envelope mass dictates that it stays
an AGB star for $\Delta_{\rm mer-AGB}\approx 10^{4}{~{}\rm yr}$ after merger.
In Soker (2022) I crudely estimated that about half of all SNe Ia are SNIPs
that take place within a time scale of $t_{\rm SNIP}\approx 10^{4}{~{}\rm yr}$
after the end of the CEE of the WD and the AGB star. The equality $\Delta_{\rm
mer-AGB}\approx t_{\rm SNIP}$ implies that about half the merger remnants that
maintain a large enough envelope mass to experience a helium shell flash will
explode while the merger remnants is still an AGB star. This holds whether the
helium shell flash triggers (ignites) the core explosion or not. The ignition
might result from a last helium shell flash during this final AGB phase that
ignites the WD. Else, the loss of angular momentum might lead to the core
explosion (see section 6.1).

The outcome of an explosion of the core while there is a hydrogen-rich
envelope of radius $\approx 100R_{\odot}$ and a mass of $\approx
0.01M_{\odot}$ is a peculiar SN Ia if the hydrogen is not detected. It is
peculiar because the relatively massive envelope, which in turn might lead to
an early flux excess much higher than in normal SNe Ia with early flux excess
(section 6.3). If the hydrogen is detected, then it is a peculiar SN II. It is
a peculiar SN II because of the extremely low hydrogen mass and the large 56Ni
that the explosion yields relative to typical SNe II.

Because I am not aware of such peculiar SNe Ia and SN II, I conclude that the
number of cases, $N_{\rm TP,SN}$, where the WD-core merger remnant maintains a
large enough envelope to stay an AGB and experience a helium shell flash on
the AGB or experience a LTP/VLTP are very rare, at most few per cents of the
number of all SNe Ia, $N_{\rm Is}$. Namely,

$N_{\rm TP,SN}<{\rm few}\times 0.01N_{\rm Ia}.$ (8)

If and when such peculiar SNe are identified, we might be able to better
constrain this fraction.

I emphasise again that the mass limit for LTP/VLTP of the massive merger
remnant might be much smaller than what condition 1 (equation 1) gives. In
that case the merger remnant stays a short time on the AGB, and even if
explosion does take place during the AGB phase the effect of the very low mass
envelope is small. Therefore, weak LTP/VLTP, those that result from a very low
helium mass of $\Delta M_{\rm He,TP}\ll 10^{-2}M_{\odot}$, might be more
common. This should be explored in future studies.

##  6 Possible consequences of LTP/VLTP

I speculate here on three possible consequences of a LTP/VLTP. As I stated in
section 4, because all explosions occur at a short time ($\la 10^{5}{~{}\rm
yr}$) after the AGB phase for the LTP/VLTP to play a role, all these SNe Ia
are SNIPs, i.e., SNe Ia inside planetary nebulae.

###  6.1 Triggering WD remnant explosion

In single star evolution the WD mass does not reach a mass larger than about
$1.1M_{\odot}$. Calculations show that a LTP/VLTP does not ignite the WD
(otherwise we would have a much larger number of SNe Ia than observed).
However, a LTP/VLTP on a merger remnant that is about the Chandrasekhar mass,
$M_{\rm rem}\simeq 1.4M_{\odot}$, might ignite the WD, similar to the ignition
by a thin helium shell in some channels of the DDet scenario (e.g., Shen &
Moore 2014). Shen & Moore (2014) find that a pure helium shell mass of
$\simeq{\rm few}\times 0.001M_{\odot}$ can ignite a WD of mass $\ga
1.3M_{\odot}$. A small mixture of CO into the helium ease the ignition of
helium. Since I take the helium mass to be $\Delta M_{\rm He,TP}\ga
0.01M_{\odot}$, which is condition 1 (equation 1), the LTP/VLTP might ignite
the WD. During the LTP/VLTP the star is hydrogen-deficient with total hydrogen
mass of $\la 10^{-5}M_{\odot}$ (e.g., Herwig et al. 1999), and so the
explosion will be indeed classified as a SN Ia (or a peculiar SN Ia).

Previous studies of the CD scenario (e.g., Ilkov & Soker 2012) assume that the
loss of angular momentum leads to the explosion of the Chandrasekhar-mass
merger remnant, with a time scale as given by equation (7). In addition to
this possibility, I raise here the possibility that in rare cases (equation 8)
LTP/VLTP ignites the WD.

Chandrasekhar-mass WDs that explode in the SD scenario by detonation alone,
without deflagration to detonation transition, produce too much 56Ni to be
compatible with observations of normal SNe Ia. However, there are some super-
Chandrasekhar-mass WDs that do have larger than usual 56Ni masses (e.g., Deb,
Mukhopadhyay, & Weber 2022; Dimitriadis et al. 2022 for recent studies).
Moreover, in a recent study Neopane et al. (2022) argue that a Chandrasekhar-
mass merger product that explodes within a short time after merger, namely, a
short MED time, might have a different density structure than a Chandrasekhar
WD in the SD scenario (be more extended), and therefore has different
properties at explosion. Specifically, Neopane et al. (2022) find that their
calculated synthetic spectra of a merger product that explodes by detonation
(with no deflagration to detonation transition) with $t_{\rm MED}=\simeq
100{~{}\rm yr}$ in the DD scenario matches normal SN spectra. Future studies
should examine the properties of detonation of Chandrasekhar-mass WDs with MED
times of up to $t_{\rm EMD}\simeq 10^{4}-10^{5}{~{}\rm yr}$, and determine
whether they produce normal or peculiar SNe Ia.

Yet another possibility is that a LTP/VLTP ignites sub-Chandrasekhar WD
remnants, but these are more massive than what single-star evolution gives.
Roy et al. (2022) find that helium detonation does not ignite a WD of $\simeq
1M_{\odot}$ in the DDet scenario. They further suggest that helium detonation
might ignite the most massive CO WDs. Namely, LTP/LTP events, which are
similar in many respects to helium detonation in the DDet scenario, might
ignite WD merger remnants in the mass range of

$1.2M_{\odot}\la M_{\rm WD,mer}({\rm LTP/VLTP})<1.4M_{\odot}.$ (9)

Even in this case the WD needs to be more extended than a cold WD (as Neopane
et al. 2022 find) to be a normal SN Ia, because cold WDs that explode in the
DDet scenario have to be of mass $\simeq 1M_{\odot}$ to account for normal SNe
Ia (e.g., Shen et al. 2018).

###  6.2 WD explosion inside a born-again AGB star

Different simulations show that the born-again star can stay inflated, i.e.,
$R\gg 1R_{\odot}$ (e.g., Schoenberner 1979) for hundreds of years (e.g.,
Althaus et al. 2005; Guerrero et al. 2018; Lawlor 2021). However, more massive
stars at the time of LTP/VLTP evolve faster (e.g., Bloecker 1995; Bloecker &
Schoenberner 1997). On top of these, the time of evolution depends on the
uncertain efficiency of convective mixing of elements (e.g., Herwig 2002;
Althaus et al. 2005), and the evolution time seems to be short (e.g., Herwig
2002; Lechner & Kimeswenger 2004), down to several years (e.g., Clayton et al.
2006). Overall, I scale the WD-inflated phase after a LTP/VLTP with a time
scale of $\tau_{BA}\approx 100{~{}\rm yr}$, but it might be shorter even. This
implies that only $\la 1\%$ of the SNe Ia that explode within
$10^{4}-10^{5}{~{}\rm yr}$ from the termination of the AGB will take place
inside a born-again envelope. With equation (8) I conclude that the fraction
of SN Ia inside an inflated born-again envelope is extremely low

$N_{\rm BA,SN}<{\rm few}\times 10^{-4}N_{\rm Ia}.$ (10)

The gas that the born-again AGB star ejects might be H-poor and He-poor (e.g.,
Montoro-Molina et al. 2022). The inflated envelope of the born-again star in
these cases is H-poor and He-poor. However, calculations show that the
inflated envelope can be H-rich and He-rich as well (e.g., Althaus, Córsico, &
Miller Bertolami 2007). The envelope can be also He-rich and H-poor, as with
V605 Aql, the central star of the planetary nebula A58 (e.g., Clayton et al.
2006).

An explosion inside the very low mass, $\approx 10^{-3}M_{\odot}$, H-poor or
H/He-poor envelope would still be classified as a SN Ia. However, the very hot
thin envelope will cool in the first hours to days to give an early flux
excess, similar to that when the ejecta collides with circumstellar matter
(section 6.3). The outcome is therefore a SN Ia with early flux excess in the
frame of the CD scenario.

###  6.3 Explosion into a born-again ejecta

A born-again star, which following a LTP/VLTP becomes an AGB star again,
ejects a mass of $\approx 10^{-4}-10^{-3}$ (but the mass can be larger even)
in a clumpy morphology (e.g., Jacoby, Hillwig, & Jones 2020 for a discussion
and references). Fang et al. (2014) model the born-again phase with a duration
of 20 years, a wind velocity of $20{~{}\rm km}{~{}\rm s}^{-1}$, and a mass
loss rate of $10^{-4}M_{\odot}{~{}\rm yr}^{-1}$. Following these studies and
others, I take a typical ejecta of a born-again star to be clumpy (e.g.,
Kerber et al. 2009; Hinkle et al. 2020; Rodríguez-González et al. 2022), to
contain a mass of $\approx 10^{-4}-10^{-3}M_{\odot}$, and to expand at ${\rm
few}\times 10{~{}\rm km}{~{}\rm s}^{-1}$. However, for the inner and slowest
parts of the ejecta I take a velocity of $\simeq 10{~{}\rm km}{~{}\rm
s}^{-1}$.

The born-again ejecta adds up to the planetary nebula shell and does not play
a prominent additional role at late times. The born-again ejecta can lead to
an early flux excess of the SN Ia only if the SN ejecta collides with the
born-again ejecta in the first few days, before the SN Ia becomes bright by
radioactive decay. The process is like the early flux excess that might result
from the collision of the SN Ia ejecta with the circumstellar matter in some
other SN Ia scenarios (e.g., Raskin & Kasen 2013; Levanon, Soker, & García-
Berro 2015; Kromer et al. 2016; Piro & Morozova 2016).

For the fastest supernova ejecta at $v_{SN}\simeq 20,000{~{}\rm km}{~{}\rm
s}^{-1}$ to collide with the slow born-again ejecta moving at $10{~{}\rm
km}{~{}\rm s}^{-1}$ within three days after explosion, the SN explosion should
take place with a time delay after the LTP/VLTP of only $\Delta t_{TP-SN}\la
16{~{}\rm yr}$. The fraction of such systems that lead to early flux excess
(ELE) is less even than SN Ia inside a born-again envelope, and it is only

$N_{\rm ELE,SN}<{\rm few}\times 10^{-5}N_{\rm Ia}.$ (11)

Overall, the fraction of systems that explode inside an extended born-again
envelope or a decade after the star contracts back, is about $\la 1\%$ of all
cases with LTP/VLTP. These SNe Ia will have an early flux excess. But
equations (10) and (11) such systems can account for no more than a fraction
of ${\rm few}\times 10^{-4}$ of ann SNe Ia.

##  7 Discussion and Summary

In this study I raise the (speculative) possibility that within the frame of
the CD scenario (section 1) in rare cases the merger process of the core and
the WD maintains an envelope mass (equation 4) that is sufficiently massive to
cause a later helium shell flash. The helium shell flash might take place
while the merger remnant is still on the AGB (section 5), while the star moves
to the left on the HR diagram as a post-AGB star (this is a LTP), or along the
WD cooling tack (VLTP).

Although the merger remnant is massive, it is hot and more extended than a WD
of the same mass that evolves as a single-star or grows by slow mass accretion
(Neopane et al. 2022). In addition the merger remnant is rapidly rotating and
sustains a strong magnetic field. Future studies should determine the helium
mass that can drive LTP/VLTP on such complicated WD-core merger remnants. In
this study I scaled the required helium mass for an LTP/VLTP as in equation
(1), but one should keep in mind that the required mass might be much smaller.

The sufficiently massive envelope according to condition 1 (equation 1)
implies that the merger remnant spends $\approx 10^{4}{~{}\rm yr}$ on the AGB
(equation 6). I estimated that in the frame of the CD scenario the fraction of
merger remnants that maintain sufficiently massive envelope to have a LTP/VLTP
is at most few per cents of all SNe Ia (equation 8). Otherwise there will be a
too large number of WD explosions inside a thin AGB envelope. However, if the
required helium mass is smaller because of the massive remnant, i.e., $\Delta
M_{\rm He,TP}\ll 10^{-2}M_{\odot}$ (see discussion in section 2.1 and 5), then
the merger remnant stays on the AGB for a much shorter time and the effect of
the very low mass envelope on the SN Ia light curve is small. Therefore, weak
LTP/VLTP that results from these low mass helium shells might be more common.

In cases the WD merger remnant explodes while the merger remnant is still on
the AGB the outcome is a peculiar SN Ia if no hydrogen or helium are detected,
or a peculiar SN II if hydrogen is detected (section 5). If the WD merger-
remnant explodes inside an inflated born-again star, or within $<20{~{}\rm
yr}$ after the born-again star has contracted back, then the outcome is an
early flux excess in the light curve of the SN Ia. The fraction of systems
that might show an early flux excess due to this process is the combined rates
of equations (10) and (11), a fraction of $<{\rm few}\times 10^{-4}$ of all
SNe Ia.

In recent studies Deckers et al. (2022) estimate that SNe Ia with early flux
excess, such as SN 2015bq (Li et al. 2022), amount to $\simeq 18\pm 11\%$ of
all SNe Ia at redshift of $z<0.07$, and Magee et al. (2022) estimate this
fraction to be $\simeq 28\pm 11\%$. My conclusion is that the inflated
envelope and mass ejection by LTP/VLTP events might at most account for a very
small fraction of SNe Ia with early flux excess.

Deckers et al. (2022) also find that SNe Ia with early flux excess originate
from younger stellar populations. This, and the early flux excess in some
super-Chandrasekhar SNe Ia (e.g., SN 2020hvf; Jiang et al. 2021), might
suggest that some early flux excess SNe Ia come from the CD scenario. The CD
scenario can account for super-Chandrasekhar SNe Ia and can explain a short
CEE to explosion delay time ($t_{\rm CEED}$; Soker 2022). Therefore, it is
possible that the CD scenario might account for early flux excess in some SNe
Ia, but not by LTP/VLTP. Either the early flux excess results from ejecta-CSM
collision like in other SN Ia scenarios (e.g., Raskin & Kasen 2013; Levanon,
Soker, & García-Berro 2015; Kromer et al. 2016; Piro & Morozova 2016), or the
explosion mixes 56Ni to the outer regions of the SN ejecta as in some other SN
Ia scenarios (e.g., Piro & Morozova 2016; Jiang et al. 2018). The CSM might be
gas from the planetary nebula that falls back towards the remnant. The
fallback process requires further study.

The helium burning during the core-WD merger in the CD scenario, and in rare
cases in an LTP/VLTP, nucleosynthesises the radioactive isotope 44Ti, like
helium burning in the DDet scenario (e.g., Roy et al. 2022). In most SNe Ia
according to the CD scenario, as I found above, LTP/VLTP does not take place.
As well, most explosions take place more than a thousand years after merger.
Because the radioactive isotope 44Ti has a half-life of 60 years I do not
expect the presence of more 44Ti than what the explosion by deflagration to
detonation transition predicts (see comparison between models by, e.g, Troja
et al. 2014). Recent studies suggest that most SNe Ia contain small masses of
44Ti, as the deflagration to detonation transition predicts or less (e.g.,
Lopez et al. 2015; Weinberger et al. 2020). In rare cases the explosion takes
place hundreds of years after merger. Then there might be some extra 44Ti. In
these cases there is a close CSM because of the short delay from the CEE to
explosion (short $t_{\rm CEED}$). In the very rare cases of LTP/VLTP that
triggers a SN Ia or take place a short time before the SN Ia explosion, I
expect to find a somewhat larger 44Ti mass than what the deflagration to
detonation transition predicts.

Despite the very low fraction of SNe Ia with pre-explosion LTP/VLTP, the large
number of SNe Ia (thousands per year) that ongoing sky surveys will find in
coming years might lead to about one case per year where LTP/VLTP might have
an influence on the light curve.

## Acknowledgments

I thank Robert Fisher for enlightenment discussions and Marcelo M. Miller
Bertolami and Yossef Zenati for useful comments. This research was supported
by a grant from the Israel Science Foundation (769/20).

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