# Finding the unusual red giant remnants of cataclysmic variable mergers

Nicholas Z. Rui TAPIR, California Institute of Technology, Pasadena, CA 91125,
USA  Jim Fuller TAPIR, California Institute of Technology, Pasadena, CA 91125,
USA

###### Abstract

Mergers between helium white dwarfs and main-sequence stars are likely common,
producing red giant-like remnants making up roughly a few percent of all low-
mass ($\lesssim 2M_{\odot}$) red giants. Through detailed modelling, we show
that these merger remnants possess distinctive photometric, asteroseismic, and
surface abundance signatures through which they may be identified. During
hydrogen shell burning, merger remnants reach higher luminosities and possess
pulsations which depart from the usual degenerate sequence on the
asteroseismic $\Delta\nu$–$\Delta\Pi$ diagram for red giant branch stars. For
sufficiently massive helium white dwarfs, merger remnants undergo especially
violent helium flashes which can dredge up a large amount of core material (up
to $\sim 0.1M_{\odot}$) into the envelope. Such post-dredge-up remnants are
more luminous than normal red clump stars, surface carbon-rich, and possess a
wider range of asteroseismic g-mode period spacings and mixed-mode couplings.
Recently discovered low-mass ($\lesssim 0.8M_{\odot}$) red clump stars may be
core helium-burning remnants of mergers involving lower-mass helium white
dwarfs.

stellar mergers, cataclysmic variables, red giant stars, asteroseismology

††software: Modules for Experiments in Stellar Astrophysics (Paxton et al.,
2010, 2013, 2015, 2018)

##  1 Introduction

Growing evidence suggests that mergers between main-sequence (MS) stars and
low-mass white dwarfs should be fairly common. The likely outcome of such a
merger is to accrete the MS star onto the white dwarf, igniting hydrogen shell
burning and creating an unusual red giant (RG) star. These RG remnants of
those mergers should exist within the stellar population, and may be
identifiable through a combination of their photometry, pulsations, and
surface abundances.

Cataclysmic variables, or CVs, are stably mass-transferring systems with white
dwarf (WD) accretors and MS donors. Typically, the progenitor binaries of CVs
are post-common-envelope systems (Paczynski et al., 1976) which have been
tightened by magnetic braking and gravitational waves until the onset of Roche
lobe overflow (Knigge, 2011).

For a while, the research field has been dogged by a WD mass problem (e.g.,
Zorotovic & Schreiber, 2020):

  1. 1.

the typical accretor mass in CVs is $\simeq~{}0.8M_{\odot}$, greater than the
typical mass of carbon–oxygen white dwarfs (CO WDs) $\simeq~{}0.6M_{\odot}$,
and

  2. 2.

no helium(-core) white dwarf (hereafter He WD) accretors have ever been
discovered in CVs, even though they should be frequent outcomes of
conventional binary stellar evolution (Zorotovic et al., 2011; Zorotovic &
Schreiber, 2020).

Where are all of the low-mass CV accretors?

In the past decade, it was realized that the WD mass problem can naturally be
solved by an extra, accretor mass-dependent angular momentum loss mechanism
(Schreiber et al., 2015). This so-called consequential angular momentum loss
would preferentially destabilize the mass transfer of CV-like systems with
lower-mass white dwarfs, causing them to quickly merge before they can be
observed (Belloni et al., 2018). Physically, this mechanism may be a
frictional effect associated with nova events (Shen, 2015), which are expected
to be much longer-duration for lower-mass white dwarfs (Shara et al., 1993),
although the subsequent analysis is agnostic to the details.

If CVs with low accretor masses are missing because they merge, it is obvious
to ask whether their remnants can be observed and identified. In the case
where the progenitor system is a close He WD–MS binary (the scenario of focus
in this study), the remnant is expected to evolve along highly modified
versions of the red giant branch (RGB) and possibly core helium burning (CHeB,
or the red clump) phases of isolated, low-mass ($M~{}\lesssim~{}2M_{\odot}$)
stars.

More recently, in only the last few years, the community has realized that
stellar interactions may produce RGs with conspicuous and lasting
asteroseismic signatures. Rui & Fuller (2021) and Deheuvels et al. (2022) show
that many first-ascent RGs which gain hydrogen mass from companions (either
through merger or stable mass transfer) possess unusually low gravity-mode
period spacings. Li et al. (2022) and Matteuzzi et al. (2023) similarly
identify anomalously undermassive CHeB stars through asteroseismology. Because
asteroseismology probes internal structures, it constrains separate
information from traditional techniques which probe the surface properties of
the star. It may thus play a critical role in identifying the remnants of
cataclysmic variable mergers.

In this work, we investigate observable signatures of He WD–MS merger
remnants. The physical picture we advance is shown schematically in Figure 1.
We construct and evolve merger remnant models (described in Section 2), and
demonstrate a significant number of telltale signs (photometric,
asteroseismic, and surface compositional) which are complex downstream
consequences of a highly cold and degenerate core during hydrogen shell
burning. Sections 3 and 4 discuss merger remnants during hydrogen shell
burning (RGB) and helium core burning (CHeB), respectively. In Section 5, we
identify some merger remnant candidates in existing observations. Section 6
discusses other possible signatures, and contextualizes this work in the
broader nascent field of binary interaction asteroseismology. Section 7
summarizes our key findings.

Figure 1: A summary diagram of the behavior of the He WD–MS merger remnants
investigated in this work. On the RGB, merger remnants have cold and slightly
more compact cores, which inflates their radii and changes the radiative
structures of their cores to which asteroseismology is sensitive (Section 3).
Sufficiently massive merger remnants ignite helium in abnormally energetic
helium flashes. For massive enough progenitor He WDs, these events can dredge
up significant amounts of helium into merger remnants’ envelopes (Section 4),
changing their surface abundances, and greatly modifying both their structure
and evolution. Less massive remnants become horizontal branch stars (Section
4.5), or even hot subdwarfs or He WDs (not explored in this work).

##  2 Stellar models

By the time of merger, the He WD component of a close He WD–MS binary has
undergone an extended phase of radiative cooling lasting potentially up to
gigayears. When the binary subsequently merges, the MS component (now the
remnant’s envelope) quickly ignites hydrogen in a burning shell around the He
WD (now the remnant’s core) and sets up thermal equilibrium on a short
envelope thermal timescale:

$\begin{split}\tau_{\mathrm{th,env}}&=\frac{GM^{2}}{RL}\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\approx
0.1\,\mathrm{Myr}\times\left(\frac{M}{M_{\odot}}\right)^{2}\left(\frac{R}{10R_{\odot}}\right)^{-1}\left(\frac{L}{30L_{\odot}}\right)^{-1}\end{split}$
(1)

In contrast, the degenerate core only thermalizes (via electron-mediated
conduction) with the hydrogen burning shell on a much longer timescale
$\sim\\!10$–$100\,\mathrm{Myr}$ comparable to the duration of the RGB phase
(see Appendix B). Therefore, as previously noticed by Zhang et al. (2017), the
low-entropy core can persist for long enough to influence the long-term
evolution of the remnant.

Using Modules for Experiments in Stellar Astrophysics (MESA, version r10398;
Paxton et al., 2010, 2013, 2015, 2018), we create and evolve evolutionary
models of merger remnants, consisting of cold helium cores surrounded by
hydrogen envelopes111Inlists and other files required to reproduce our results
can be found at the following Zenodo link:
https://zenodo.org/records/10828187.

We do this in the following stages:

  1. 1.

An isolated star with $M=1.2M_{\odot}$ is evolved through the MS and part of
the RGB until it attains the desired initial core mass of the remnant,
$M_{\mathrm{WD}}$. This initial mass is chosen to be close to typical RG
masses from Kepler (e.g., Yu et al., 2018). Because RGs with $M\lesssim
2M_{\odot}$ all obey the same core mass–luminosity relation, the subsequent
analysis is insensitive to this choice.

  2. 2.

The hydrogen envelope is removed by applying a high mass-loss rate (using
relax_initial_mass with $|\dot{M}|=10^{-3}M_{\odot}\,\mathrm{yr}^{-1}$). To
simulate a realistic white dwarf atmosphere, we retain an additional
$M_{\mathrm{atm}}=10^{-4}M_{\odot}$ of material from the base of the envelope,
on top of the core mass $M_{\mathrm{WD}}$.

  3. 3.

The resulting object (which quickly relaxes into a He WD) is evolved through a
cooling track to $\log(L_{\mathrm{WD}}/L_{\odot})=-4.0$.

  4. 4.

To stellar-engineer the merger remnant, we start with a scaffold RG model with
core mass $M_{\mathrm{WD}}$, and replace shells with mass coordinates $m\leq
M_{\mathrm{WD}}$ with the He WD model. To build the scaffold model, we start
with the original RG model from above. We remove envelope mass from the
scaffold model via relaxation with
$|\dot{M}|=10^{-3}M_{\odot}\,\mathrm{yr}^{-1}$ with burning disabled, until
the RG attains the desired final envelope mass $M_{\mathrm{MS}}$ (physically
equal to the mass of the MS star participating in the merger, assuming no mass
lost). This produces a merger remnant with total mass
$M=M_{\mathrm{MS}}+M_{\mathrm{WD}}$. For most merger remnant models in this
work, we arbitrarily fix $M_{\mathrm{MS}}$ such that $M=0.8M_{\odot}$, to
resemble fairly typical low-mass RGs which may result from CV mergers.

  5. 5.

To ensure numerical convergence, the resulting remnant is evolved for
$2\,\mathrm{kyr}$ without burning and gold tolerances disabled. Nuclear
reactions and gold tolerances are re-enabled at $2\,\mathrm{kyr}$ and
$4\,\mathrm{kyr}$, respectively. Evolution through the helium flash involves
disabling gold tolerances again, and the models are terminated after helium-
burning when the central helium fraction drops to $Y_{c}\leq 10^{-3}$.

In post-processing, we truncate the first $1\,\mathrm{Myr}$ (a few envelope
thermal times) of our merger remnant models in order to avoid possibly
unphysical transient behavior closely following the merger, which is not
modelled accurately.

The relatively low He WD luminosity $L_{\mathrm{WD}}=10^{-4.0}L_{\odot}$ is
chosen to highlight the effects of a highly degenerate core in the limiting
case. In Appendix A, we show that, once the core has thermalized sufficiently
long on the RGB, the effect of increasing $L_{\mathrm{WD}}$ is very similar to
that of decreasing $M_{\mathrm{WD}}$. This is because these changes ultimately
affect the entropy of the merger remnants’ cores in the same direction, and
subtler differences in the remnants’ core temperature profiles are erased by
thermal conduction on the core’s thermal time (see Appendix B).

Our models include the predictive mixing scheme described in Section 2.1 of
Paxton et al. (2018) to account for near-core mixing, and to suppress
numerical instability associated with definition of the convective core
boundary, especially during the CHeB phase. We omit winds in order to avoid
sensitivity to the wind prescription, which is highly uncertain. The effect of
winds at the tip of the RGB (or mass loss during the merger itself) is
primarily to increase the value of $M_{\mathrm{MS}}$ required for a given
merger remnant mass during CHeB. Stellar models are initialized to solar
metallicity, using the metal mass fractions given by Grevesse & Sauval (1998).
We use the built-in pp_cno_extras_o18_ne22 network for nuclear reactions.

When evolving the non-merged and merger remnant models, we run MESA in the
hydrodynamical mode (evolving the radial velocity variable $v_{r}$
explicitly), in order to stably evolve our models through the helium flash. We
find that this is particularly necessary in the merger remnant models, where
the helium flash is abnormally violent (see Section 4). We are able to model
merger remnants involving He WD masses as high as
$M_{\mathrm{WD}}=0.38M_{\odot}$. Higher He WD masses cause numerical problems
during the He flash.

This scheme for producing He WD–MS merger remnants produces very similar
models to those of Zhang et al. (2017), who instead manually add the hydrogen
envelope back onto the He WD using a large, time-dependent mass gain rate. We
find that our method more reliably produces a model which MESA can evolve
without case-by-case human intervention, and allows for the successful
evolution of remnants with more degenerate cores.

In order to demonstrate the effects of the enhanced core degeneracy in our
merger remnant models, we also run a model (hereafter the non-merged model) of
a RG whose envelope has only partially been removed on the RGB (so that it has
a total mass $M\\!=\\!0.80M_{\odot}$). Besides having possibly a low-mass
envelope, this model otherwise behaves like a normal RG and should be thought
of as representing standard stellar evolution.

##  3 Red giant branch

Soon after merger, a merger remnant quickly relaxes into a RG which behaves
similarly to a normal star on the RGB. Specifically, it is composed of an
inert degenerate helium core surrounded by a tenuous hydrogen envelope which
is inflated to a large radius by hydrogen burning in a thin shell at its base.
However, merger remnants differ from single RGs because their cores are cold
(owing to the potentially long cooling phase of the progenitor He WD), rather
than being almost isothermal with the burning shell.

Soon after their formation, merger remnants, especially those involving lower-
mass He WDs, are out of thermal equilibrium and temporarily shrink in radius.
The duration of this phase is a decreasing function with $M_{\mathrm{WD}}$,
lasting $\simeq 10\,\mathrm{Myr}$ for $M_{\mathrm{WD}}=0.20M_{\odot}$ and
dropping to $\lesssim 1\,\mathrm{Myr}$ by $M_{\mathrm{WD}}=0.27M_{\odot}$. The
degree of this secular dimming also drops off strongly with $M_{\mathrm{WD}}$.
However, after this short-lived period, the remnant evolves monotonically up a
modified version of the RGB.

Overall, this cold core has three consequences on the RGB:

  1. 1.

At fixed helium core mass, the star is brighter and has a larger radius
(Section 3.1). Because the helium core of a remnant is colder and thus
slightly more compact, the overlying hydrogen-burning shell has a
significantly higher luminosity.

  2. 2.

The degenerate core alters the structure of the gravity-mode (g-mode) cavity
(Section 3.2), modifying the propagation of g modes in an observable way.

  3. 3.

The helium flash, which occurs when some shell in the core reaches a
sufficient temperature $T\simeq 10^{8}\,\mathrm{K}$, is slightly delayed
(Section 3.3). This allows merger remnants to exceed the maximum luminosity
attainable by a single RG at the tip of the RGB. Additionally, the more
violent helium flash can have significant consequences for the CHeB stage
(Section 4).

###  3.1 More compact cores inflate the radius

In a typical RG, the total luminosity $L$ is essentially entirely determined
by its core mass $M_{c}$ (Refsdal & Weigert, 1970), with very little
sensitivity to the mass of the envelope. This correspondence is called the
core mass–luminosity relation. Photometrically, this means that single RGs
with similar core masses will appear essentially indistinguishable on a
Hertzsprung–Russell diagram, even if their envelopes differ significantly in
mass. Asteroseismically, it results in a tight relationship between the large
frequency spacing $\Delta\nu$ and the g-mode period spacing $\Delta\Pi$
(defined and interpreted below) for degenerate-core RGs with $M\lesssim
2M_{\odot}$ (e.g., Deheuvels et al., 2022).

Crucial to this relation is that the radius of the burning shell (which is
similar to the radius of the core, $R_{c}$) is a strict function of $M_{c}$
(for a degenerate core, roughly $R_{c}\propto M_{c}^{-1/3}$). In merger
remnants, the core is cooler than the hydrogen-burning shell and therefore
slightly more compact (Althaus et al., 2005). In this case, the luminosity is
no longer fixed by $M_{c}$ alone, although it is still determined by the
environment around the burning shell.

Figure 2: The trajectories on the Hertzsprung–Russell diagram of merger
remnant models with varying He WD mass $M_{\mathrm{WD}}$ and total mass
$M=0.80M_{\odot}$. The red dashed line indicates a non-merged model with
$M=0.80M_{\odot}$. The top-left and bottom-left panels zoom into the tRGB and
CHeB stages, respectively. The red-outlined star symbols on the top-left panel
indicate the location of the helium flash. On the bottom-left panel, only
evolution $\gtrsim 5\,\mathrm{Myr}$ following the helium flash is shown, and
points are distributed $10\,\mathrm{Myr}$ apart. We have excised times near
the helium flash, when the envelope sometimes becomes hydrodynamical.

In addition to modifying its placement on the Hertzsprung–Russell diagram
(Figure 2), the inflated radius significantly modifies the observable pressure
(p) modes in the envelope. Asteroseismically, RGs are solar-like oscillators
(Chaplin & Miglio, 2013), for which a frequency of maximum power
$\nu_{\mathrm{max}}$ and large (p-mode) frequency spacing $\Delta\nu$ can be
measured. These observables are approximately related to the mass $M$ and
radius $R$ of the RG as

$\nu_{\mathrm{max}}\propto g/T_{\mathrm{eff}}^{1/2}\propto
M\,R^{-2}\,T_{\mathrm{eff}}^{-1/2}$ (2)

and

$\Delta\nu\simeq\sqrt{G\bar{\rho}}\propto M^{1/2}\,R^{-3/2}$ (3)

(Ulrich, 1986; Brown et al., 1991).

In the absence of additional information, $M_{c}$ and $R_{c}$ are not known.
While $L$, $\nu_{\mathrm{max}}$, and $\Delta\nu$ are all significantly
different in merger remnants, this is only a direct consequence of their
inflated radii. Therefore, these observables alone cannot distinguish merger
remnants except near the tip of the RGB (Section 3.3). However, independent
probes of the core, most notably the asteroseismic g-mode period spacing
(Section 3.2) can help distinguish merger remnants from normal stars.

###  3.2 Asteroseismic signatures on the red giant branch

Figure 3: The Brunt–Väisälä frequency $N$ (top) and temperature $T$ (bottom)
shown for the $M_{\mathrm{MS}}=0.55M_{\odot}$, $M_{\mathrm{WD}}=0.25M_{\odot}$
merger remnant (solid gray-blue) and non-merged model of $M=0.80M_{\odot}$
(red dashed), at various stages of evolution. The two models are compared at
equal core masses $M_{c}$. The shaded region denotes helium added to the
helium core of the merger remnant model during its evolution.

Asteroseismically, RGs which are sufficiently low on the RGB (roughly before
the red bump; Pinçon et al., 2020) have strong enough mixed-mode coupling such
that the g-mode period spacing $\Delta\Pi$ can be measured:

$\Delta\Pi\approx\frac{\sqrt{2}\pi^{2}}{\int_{\mathcal{R}}(N/r)\,\mathrm{d}r}\mathrm{,}$
(4)

defined here specifically for the dipole ($\ell=1$) modes. This quantity
probes the radiative core of the star. The integral in Equation 4 is taken
over $\mathcal{R}$, the g-mode cavity of the RG, i.e., the region where the
mode frequency $\omega<N$, where $N$ is the Brunt–Väisälä frequency. At
present, $\Delta\Pi$ has been measured for a few thousand RGs (e.g., Vrard et
al., 2016), observationally constraining the structures of their radiative
cores.

By Equation 4, more stratified radiative zones with larger integrals over
$N\,\mathrm{d}\ln r$ have lower period spacings. In Appendix C, we show that
the Brunt–Väisälä frequency in the degenerate part of the radiative core is
given by

$N^{2}\approx
N_{0}^{2}\frac{k_{B}T}{ZE_{F}}\left(1-\frac{5}{2}\nabla\right)\propto\frac{T}{ZE_{F}}\left(1-\frac{5}{2}\nabla\right)\mathrm{.}$
(5)

where $E_{F}$ is the Fermi energy of the core, $\nabla=d\ln T/d\ln P$, and the
normalization $N_{0}^{2}$ is comparable to the dynamical frequency of the
core, and is given by

$N_{0}^{2}=\frac{\rho g^{2}}{p}\sim\sqrt{GM_{c}/R_{c}^{3}}\mathrm{.}$ (6)

Both a typical RG core and cooling WD are approximately isothermal ($\nabla\ll
1$). Then, ignoring the temperature dependence of the stellar structure for
the moment, we approximately expect

$N\propto\sqrt{T}\mathrm{,}$ (7)

A “cold,” highly degenerate isothermal core will have a significantly smaller
$N$, and therefore significantly larger $\Delta\Pi$, than a “warm” one
(Bildsten & Cutler, 1995). However, if there is a temperature gradient, it may
significantly affect $\Delta\Pi$.

The evolution of the buoyancy profile of a merger remnant with
$M_{\mathrm{WD}}=0.25M_{\odot}$ and $M_{\mathrm{MS}}=0.55M_{\odot}$ (together
with a non-merged model of equal mass) is shown in the top panels of Figure 3.
In He WD–MS merger remnants, the period spacing is modified by two effects
which act roughly in opposite directions with comparable magnitudes.

First, at early times (left panels of Figure 3), the core of a merger remnant
is roughly isothermal at its initial temperature $T\ll T_{\mathrm{shell}}$. In
comparison, the core of a single RG is roughly isothermal with the hydrogen-
burning shell ($T\approx T_{\mathrm{shell}}$). During this time period, the
Brunt–Väisälä frequency profile in the core is suppressed from that of a
single star by a factor $\sqrt{T_{\mathrm{shell}}/T}$. This effect tends to
increase the period spacing.

However, in a short time $<\tau_{\mathrm{th,core}}$, electronic thermal
conduction (approximately obeying the nonlinear law in Equation B5) sets up a
temperature gradient $\nabla$. Curiously, we find that this temperature
gradient happens to mostly cancel out the factor $\sqrt{T_{\mathrm{shell}}/T}$
(see Equation 5) to cause the $N$ profiles of the single and merger remnant
models to be similar (top center panels of Figure 3). Equality between the
Brunt–Väisälä frequency profiles occurs long before the interior of the core
has thermalized to the shell temperature. Therefore, in the relatively short
time that it takes heat conduction to set up a temperature gradient throughout
the core, the lower temperature of the merger remnant core no longer works to
increase $\Delta\Pi$. The reason for this “coincidental” cancellation remains
mysterious, but may be related to some properties of long-lived pre-
thermalized solutions of nonlinear heat diffusion. These solutions (called
“intermediate asymptotics”; Barenblatt, 1996) exhibit self-similar behavior in
many nonlinear heat diffusion problems (often in the context of diffusion in a
porous medium), and have been studied extensively in applied mathematics (by,
e.g., Witelski & Bernoff, 1998; Galaktionov et al., 2004; Hayek, 2014).

The second effect is that, because $N$ scales with the dynamical frequency of
the core, it is higher in a more compact core, and is therefore larger in a
merger remnant’s core than in a single star’s core. In contrast with the first
effect, the increased Brunt–Väisälä frequency due to this effect persists
until the core heats up enough for the difference in $R_{c}$ between the
merger remnant and single stars to be erased (which occurs on the thermal
timescale $\tau_{\mathrm{th,core}}$). This effect tends to reduce $\Delta\Pi$
in the merger remnant compared to a single RG.

Figure 4: An asteroseismic spacing diagram ($\Delta\Pi$ versus $\Delta\nu$)
showing merger remnant evolutionary sequences and a non-merged model (red
dashed) all with $M=0.80M_{\odot}$. The top-right panel shows the entire
parameter space, and the left and bottom panels zoom in on CHeB stars and the
RGB, respectively. Roughly speaking, both RGB and CHeB stars evolve from right
to left on this diagram (i.e., towards decreasing $\Delta\nu$, or increasing
$R$). The black points are observations taken from the catalog of Gaulme et
al. (2020). In the top-right panel, colored triangles denote the last
$5\,\mathrm{Myr}$ of the CHeB phase.

As a merger remnant evolves, $\Delta\Pi$ typically evolves from being larger
than that of a single star (at fixed $M_{c}$) to smaller (Figure 4), although
it may not achieve the latter regime by the helium flash if the merger
involves a sufficiently massive WD. In our models, departures of $\Delta\Pi$
from the non-merged case can reach up to $\approx\\!10\,\mathrm{s}$ at a given
core mass. However, we again caution that $M_{c}$ is not directly known, and
diagnosis of a past merger requires another observable, such as the radius or
$\Delta\nu$.

Asteroseismic measurements $\Delta\nu$ and $\Delta\Pi$ of RGs are typically
represented on a spacing diagram as shown in Figure 4. Most stars on the RGB
(zoomed in on the bottom panel) cluster around a tight sequence which
ultimately arises from the core mass–luminosity relation (e.g., Deheuvels et
al., 2022). This is also essentially the path followed by our $M=0.8M_{\odot}$
non-merged model, modulo a small order-unity factor owing to a weak dependence
on total mass ($\Delta\nu\propto M^{1/2}$). In this space, it is clear merger
remnants usually lie above the degenerate sequence, before slowly evolving
back towards it. The position of merger remnants above that of normal RGB
stars is dominated by the larger radii of these objects at the same core mass.
Since $\Delta\nu\propto R^{-3/2}$, this shifts their positions on the diagram
to the left. Merger remnants may also pass slightly below the degenerate
sequence during their evolution (due to sufficiently small values of
$\Delta\Pi$). This effect is however subtle. Stars in this region of
$\Delta\nu$–$\Delta\Pi$ space may more naturally be explained by RG–MS mergers
(Deheuvels et al., 2022), particularly if they lie far below the degenerate
sequence.

Much of the phase where merger remnants depart from the degenerate sequence
takes place during the contraction phase soon after the formation of the
remnant. This is only observable for remnants sufficiently low on the RGB
where g modes are detectable (i.e., below the RGB bump, which is the “hook”
feature in Figure 4). Hence, this sort of asteroseismic diagnosis may only be
possible for sufficiently recent mergers involving sufficiently low-mass He
WDs.

###  3.3 Overbright tip of the RGB

During the RGB phase of a single star, the helium core grows in mass and
contracts over time, as the envelope expands. Simultaneously, the core heats
up gradually until the temperature is high enough for helium burning through
the triple-$\alpha$ process ($\simeq 10^{8}\,\mathrm{K}$), at which time the
star has reached the tip of the RGB (tRGB).

In our most extreme model with $M_{\mathrm{WD}}=0.38M_{\odot}$, the tRGB
surface luminosity exceeds that of a single star by a factor of $2$ (Figure
2). This factor is likely to be even larger for more massive values of
$M_{\mathrm{WD}}$ than we can run (but which are still physical). At the tRGB,
our merger remnant models outshine the tRGB of the single star model for
between $140\,\mathrm{kyr}$ ($M_{\mathrm{WD}}=0.20M_{\odot}$) and
$1.8\,\mathrm{Myr}$ ($M_{\mathrm{WD}}=0.38M_{\odot}$). Since RG luminosities
are essentially totally determined by near-core properties, even merger
remnants of low mass (involving small values of $M_{\mathrm{MS}}$) are still
expected to outshine single RGs at the tRGB.

The ostensibly well-known luminosity of the tRGB is often leveraged to measure
cosmological distances (Bellazzini et al., 2001, 2004). We point out that
overbright merger remnants near the tRGB may affect the tRGB of a stellar
population’s role as a standard candle. However, this effect is probably
minor, since merger remnants of this type likely make up no more than a few
percent of all RGs.

Figure 5: Top: The maximum helium-burning luminosities $L_{\mathrm{He,flash}}$
attained during the helium flash in our merger remnant models with total mass
$M=M_{\mathrm{MS}}+M_{\mathrm{WD}}=0.80M_{\odot}$. Bottom: For the merger
remnant models, we show core masses $M_{\mathrm{c@flash}}$ at the helium flash
(purple circles), mass coordinates $M_{\mathrm{flash}}$ where the helium flash
begins (magenta stars), and initial (turquoise triangles) and final (green
squares) core masses $M_{\mathrm{CHeB,ci}}$ and $M_{\mathrm{CHeB,cf}}$ during
the CHeB phase. In both panels, dashed lines denote values for the
$M=0.80M_{\odot}$ non-merged model.

##  4 Core helium-burning phase

When the core reaches a sufficient temperature, helium burning begins. In
single RGs, it is well known that helium ignites off-center due to a slight
temperature inversion caused by neutrino cooling (Thomas, 1967). The peak
burning rate reached during the subsequent helium flash depends sensitively on
the density where helium ignites (Salpeter, 1957), and therefore on the mass
of the helium core. Subsequent intermittent burning events occur in a series
of subflashes (Thomas, 1967). In a normal RG, these subflashes propagate
inwards over the course of a few megayears until they reach the core, fully
lifting the degeneracy of the helium core and burning $\approx\\!4\%$ of its
mass into carbon. Once this occurs, RGs quickly contract on a thermal time
until they have radii $\approx\\!11R_{\odot}$ (the CHeB phase), after which
they are supported by a combination of helium core- and hydrogen shell
burning, which generally contribute in comparable amounts to the stellar
luminosity.

Figure 6: Kippenhahn diagrams for the non-merged $M=0.80M_{\odot}$ and
selected merger remnant models, during the helium flash (left) and CHeB phase
(right). Convective (radiative) zones are shown in blue (yellow), and burning
regions fusing more vigorously than
$\epsilon=10^{3}\,\mathrm{erg}\,\mathrm{g}^{-1}\,\mathrm{s}^{-1}$ are shown in
red (at this scale, the CHeB region is highly localized to the center of the
star and cannot be seen). The black dashed line indicates the helium core mass
$M_{c}$. Merger remnant models with $M_{\mathrm{WD}}\geq 0.27M_{\odot}$
experience core dredge-up events during the helium flash, and subsequently
develop thinner convective envelopes by mass during CHeB. If $M_{\mathrm{MS}}$
is small enough, the convective envelope disappears entirely, and the star
enters the horizontal branch.

In merger remnants, the helium flash occurs similarly, although, as previously
pointed out by Zhang et al. (2017), the flash occurs farther off-center
(Figure 5), since the core temperature inversion is now dominated by the low
temperature of the WD progenitor rather than the weaker effect of neutrino
cooling (compare the dashed red and solid blue curves of the lower-right panel
of Figure 3). Additionally, the subflashes occur closer together in time, and
are more energetic in general.

In our models, the maximum helium-burning luminosity attained by a merger
remnant is roughly a linear function of $M_{\mathrm{WD}}$, with the most
extreme model attaining a factor of $\approx\\!5$ higher helium-burning
luminosity than attained by a non-merged star (see the top panel of Figure 5).
The energy production rate can be a whopping $\approx\\!10^{10}L_{\odot}$ at
the peak of the helium flash. Since mergers involving more massive WDs than
those on our model grid are possible, even larger helium-burning luminosities
may occur in nature. The most vigorous helium flash in our merger models also
burns a larger fraction of the core into carbon, up to $\simeq 7\%$ by mass.

We verify in our models that the helium flash, though abnormally energetic in
merger remnants, still does not result in hydrodynamical burning (which might
result in detonation). Specifically, following Shen et al. (2010), we verify
that convection is always efficient enough to flatten temperature gradients
created by helium burning, and also that the hierarchy $|v_{r}|\ll
v_{\mathrm{conv}}\ll c_{s}$ is maintained throughout the helium core (though
not necessarily in the hydrogen envelope). The properties of this dredge-up
concern physics in the helium core only, and is essentially independent of the
envelope mass. Similar core dredge-up events during the helium flash have also
been predicted in low-metallicity stars which have undergone extreme mass loss
(Sweigart, 1997; Cassisi et al., 2002).

The CHeB-phase evolution of our merger remnant models fits into two regimes,
sorted by $M_{\mathrm{WD}}$:

  1. 1.

For $M_{\mathrm{WD}}\leq 0.26M_{\odot}$, the flash is delayed slightly, and
occurs when the core is slightly more massive. As a result, the CHeB phase of
these remnants begins with a slightly higher helium core mass. Specifically,
our single non-merged model ignites helium at a core mass $M_{c}\approx
0.46M_{\odot}$, and our merger remnant model with
$M_{\mathrm{WD}}=0.26M_{\odot}$ undergoes its flash at a slightly higher core
mass $M_{c}\approx 0.50M_{\odot}$.

  2. 2.

For $M_{\mathrm{WD}}\geq 0.27M_{\odot}$, the flash produces so much heat that
it causes the convective envelope to deepen significantly, dredging up both
the hydrogen-burning shell and the outer layers of the helium core (see Figure
6). For our most extreme model ($M_{\mathrm{WD}}=0.38M_{\odot}$),
$\approx\\!20\%$ of the core ($\approx 0.1M_{\odot}$ of helium) is dredged up
into the envelope, significantly modifying its mean molecular weight.

Figure 7: Time evolution of the CHeB-phase luminosity $L$, hydrogen- and
helium-burning luminosities $L_{\mathrm{H}}$ and $L_{\mathrm{He}}$, effective
temperature $T_{\mathrm{eff}}$, radius $R$, and helium core mass $M_{c}$ for
merger remnant models and a non-merged model (red dashed) all with
$M=0.80M_{\odot}$. The left and right panels show merger remnant models with
$M_{\mathrm{WD}}\leq 0.26M_{\odot}$ and $M_{\mathrm{WD}}\geq 0.27M_{\odot}$,
respectively. The latter models have helium flashes which are vigorous enough
to dredge up some of the helium core into the envelope.

As a result of this, merger remnants involving lower-mass He WDs
$M_{\mathrm{WD}}\leq 0.26M_{\odot}$ evolve very similarly to normal red clump
stars, appearing almost identical to them in both photometric (bottom-left
panel of Figure 2) and asteroseismic (top-left panel of Figure 4 and left
panel of Figure 9) observables. One minor difference is that the merger
remnants have systematically shorter CHeB lifetimes, by up to $\lesssim 20\%$.

In contrast, merger remnants involving more massive He WDs
$M_{\mathrm{WD}}\geq 0.27M_{\odot}$ evolve and appear very differently from
their single counterparts. The following sections focus on observational
signatures in this latter case (hereafter post-dredge-up merger remnants). The
dredge-up event ultimately affects the luminosity (Section 4.1),
asteroseismology (Sections 4.2 and 4.3), and surface abundances (Section 4.4)
significantly.

###  4.1 Over-luminous red clump stars

During the CHeB phase, post-dredge-up remnants have significantly modified
photometric properties:

  1. 1.

They are brighter and slightly cooler, with the effect being stronger for
higher values of $M_{\mathrm{WD}}$ (bottom-left panel of Figure 2 and top-
right panel of Figure 7). For our most extreme model
($M_{\mathrm{WD}}=0.38M_{\odot}$), the luminosity is roughly tripled.

  2. 2.

Their luminosities evolve much more significantly over the CHeB phase. Again,
this effect is stronger for higher values of $M_{\mathrm{WD}}$. In comparison,
normal CHeB stars have essentially fixed luminosities for almost the entirety
of the helium-burning phase.

As in normal CHeB stars, nuclear energy production has two main contributions:

  1. 1.

Core helium burning. The now non-degenerate helium core burns helium through
the triple-$\alpha$ process in a convective core. Structurally, the core is
similar to normal, intermediate-mass MS stars in that they have convective
core burning surrounded by a radiative envelope. Analogously, the helium-
burning luminosity is essentially set by the helium core mass.

  2. 2.

Hydrogen shell burning. Hydrogen continues to burn in a shell around the
helium core through the CNO cycle. The luminosity of the hydrogen burning is a
sensitive function of the environment around the burning shell.

The luminosities of these burning regions are highly coupled to each other.
For example, the mass of the helium core determines the helium-burning
luminosity and, thus, the radius of the core. This in turn strongly affects
the hydrogen-burning luminosity and thus the total luminosity of the star. In
turn, the luminosity of the hydrogen-burning shell determines the growth rate
of the helium core, therefore feeding back onto the time dependence of both
contributions to the luminosity.

As can be seen in the right panels of Figure 7, post-dredge-up merger remnants
can have significantly larger hydrogen-burning luminosities $L_{\mathrm{H}}$.
While normal clump stars have comparable hydrogen- and helium-burning
luminosities ($\approx\\!20L_{\odot}$ in both cases), our most extreme merger
remnant model with $M_{\mathrm{WD}}=0.38M_{\odot}$ has $L_{\mathrm{H}}\simeq
100L_{\odot}$ (compared to $L_{\mathrm{He}}\lesssim 40L_{\odot}$).

The dominant factor setting $L_{\mathrm{H}}$ in a post-dredge-up remnant is
the mean molecular weight $\mu$ at the hydrogen-burning shell, which is
significantly enhanced by the helium dredge-up event. Single star models have
a near-solar helium mass fraction $Y\approx 30\%$. In contrast, the
$M_{\mathrm{WD}}=0.38M_{\odot}$ remnant model (with total mass
$M=0.80M_{\odot}$) has a much larger value $Y\approx 46\%$ owing to the
$\approx\\!0.1M_{\odot}$ of helium added to a pre-existing, solar-composition
envelope with mass $\approx 0.3M_{\odot}$. These values of $Y$ correspond
approximately to mean molecular weights $\mu\approx 0.62$ and $\mu\approx
0.70$, respectively. Refsdal & Weigert (1970) show that the hydrogen-burning
luminosity exhibits a steep scaling with $\mu$: $L\propto\mu^{7\text{--}8}$
for CNO-cycle burning, corresponding to increases in $L_{\mathrm{H}}$ by a
factor $\simeq 2.5$, which is roughly consistent with the behavior of
$L_{\mathrm{H}}$ in our models.

Because $L_{\mathrm{H}}$ is very sensitive to $\mu$, the appearance and
evolution of the remnant is now acutely sensitive to the initial envelope mass
of the remnant. This is contrary to single stars, in which $L$ and
$T_{\mathrm{eff}}$ during the CHeB phase are nearly independent of the
envelope mass. Since $\mu$ is set by the final mass fraction of the envelope
after the dredge-up event, larger pre-existing envelopes dilute the added
helium and reduce the mean molecular weight enhancement. We note in passing
that structurally important increases in $\mu$ during a core dredge-up event
also appear in some RG–He WD merger models of Zhang & Jeffery (2013).

As can be seen in Figure 7, the helium-burning luminosity $L_{\mathrm{He}}$ is
also different in post-dredge-up models. $L_{\mathrm{He}}$ essentially tracks
the helium core mass $M_{c}$ until significant core helium depletion at the
end of the CHeB phase. The high value of $L_{\mathrm{H}}$ translates to a
fast-growing helium core, and, in turn, an up to a factor of a few increase in
$L_{\mathrm{He}}$ over the CHeB phase. In sum, the higher $L_{\mathrm{H}}$ and
$L_{\mathrm{He}}$ naturally translate to an overbright CHeB phase with
significant time evolution in luminosity.

The increased luminosity causes merger remnants to have abnormally large
radii, up to $R\approx 20R_{\odot}$ in our models (for
$M_{\mathrm{WD}}=0.38M_{\odot}$). In addition to affecting the photometry,
these large radii can also be directly measured asteroseismically via
$\nu_{\mathrm{max}}$ and $\Delta\nu$, together with the scaling relations in
Equations 2 and 3. However, in order to distinguish merger remnants from
ordinary stars beginning to ascend the asymptotic giant branch, it may also be
necessary to measure mixed mode period spacings or surface abundances.

###  4.2 Wider range of asteroseismic g-mode period spacings

Figure 8: CHeB-phase asteroseismic propagation diagrams for a non-merged model
(left) and merger remnants with $M_{\mathrm{WD}}=0.26M_{\odot}$ (center) and
$M_{\mathrm{WD}}=0.38M_{\odot}$ (right). The Brunt–Väisälä ($N$), dipole Lamb
($L_{1}$), and maximum power ($2\pi\nu_{\mathrm{max}}$) frequencies are shown
as the dashed blue, dotted red, and solid purple curves, respectively. Color
coding of areas denotes the p-mode propagating regions (pink), g-mode
propagating regions (light blue), evanescent regions (white), and the exterior
of the star (yellow). Top panels show propagation diagrams near the beginning
of the CHeB phase, and bottom panels show the same farther along the CHeB
phase, when the models attain maxima in the asteroseismic g-mode period
spacing $\Delta\Pi$. All models have total mass $M=0.80M_{\odot}$. Figure 9:
Time evolution of the CHeB-phase asteroseismic g-mode period spacing
$\Delta\Pi$ (top), and $e^{-\mathcal{I}_{1}}$ (bottom), which is a simple
proxy for the dipole mixed-mode coupling factor. We plot merger remnant models
and a non-merged model (red dashed) all with $M=0.80M_{\odot}$. As in Figure
7, the left and right panels show merger remnant models with
$M_{\mathrm{WD}}\leq 0.26M_{\odot}$ and $M_{\mathrm{WD}}\geq 0.27M_{\odot}$,
respectively.

During the CHeB phase, post-dredge-up mergers remnant models have
substantially different structures which directly modify asteroseismic
observables such as $\Delta\Pi$. This can be seen by comparing the left and
right panels of Figure 8, which show the propagation diagrams of the non-
merged and $M_{\mathrm{WD}}=0.38M_{\odot}$ merger remnant models at two
selected times. On the other hand, merger remnants which do not experience
significant dredge-up of the helium core ($M_{\mathrm{WD}}\leq 0.26M_{\odot}$)
have essentially identical structures (compare the left and center panels of
Figure 8).

Because the nuclear burning luminosities are substantially modified in post-
dredge-up models, the g-mode period spacing $\Delta\Pi$ of merger remnants
spans a wider range of values over the CHeB phase (see the top-right panel of
Figure 9, as well as the top-left panel of Figure 4). While $\Delta\Pi$
(Equation 4) depends on the integral of $N$ within the radiative core, its
value is dominated by the regions with the largest $N$. As can be seen in
Figure 8, $N$ is maximal underneath the hydrogen-burning shell (i.e., below
the compositional spike in $N$ at radii $\lesssim 0.1R_{\odot}$).

At the beginning of the CHeB phase (the top panels of Figure 8), post-dredge-
up merger remnants have smaller helium core masses, resulting in smaller
$L_{\mathrm{He}}$. This results in a smaller convective core (slightly
extending the bottom boundary of the g-mode cavity) as well as a slightly
higher core dynamical frequency (resulting in slightly higher values of $N$
overall). Both of these effects tend to decrease $\Delta\Pi$.

In contrast, near the end of CHeB (the bottom panels of Figure 8), the helium
core of merger remnants has grown substantially due to the large hydrogen
burning rate. They therefore evolve to have larger $L_{\mathrm{He}}$, which
ultimately results in an increased convective core size and overall lower $N$
profile, and therefore larger $\Delta\Pi$. Because they also have inflated
radii (Section 4.1), they have smaller values of $\Delta\nu$, and will occupy
a region to the left of normal clump stars on a $\Delta\nu$–$\Delta\Pi$
spacing diagram (top panels of Figure 4).

###  4.3 Asteroseismic mixed-mode coupling

In RGs, pulsations can probe not only the g-mode cavity (through $\Delta\Pi$)
but also the evanescent zone between the p- and g-mode cavities. The extent of
coupling between the two cavities is usually described by a coupling factor
$q$ (Unno et al., 1989). In addition to determining the visibility of mixed
modes, $q$ itself is an independent observable which probes a different
internal structural feature than does $\Delta\Pi$ (Mosser et al., 2017;
Dhanpal et al., 2023). The recent discovery that low-mass or low-metallicity
RGs have preferentially high $q$ (Matteuzzi et al., 2023; Kuszlewicz et al.,
2023) has created revitalized demand for physical interpretations of $q$.

Computing $q$ is mathematically nontrivial and involves detailed solution of a
wave transmission problem (e.g., Takata, 2016a, b, 2018) or fitting a model
spectrum directly (e.g., Jiang & Christensen-Dalsgaard, 2014). Takata (2016a)
show that the coupling factor is related to the transmission coefficient $T$
by

$q=\frac{1-\sqrt{1-T^{2}}}{1+\sqrt{1-T^{2}}}\mathrm{.}$ (8)

Takata (2016b) further write the transmission coefficient as

$T=e^{-\pi(X_{I}+X_{R})}\mathrm{.}$ (9)

Here, $X_{I}$ is defined as an integral over the evanescent zone $\mathcal{E}$
of the radial wavenumber with respect to the asymptotic dispersion relation,
i.e., for dipole modes,

$\begin{split}\mathcal{I}_{1}\equiv\pi
X_{I}&=\int_{\mathcal{E}}|k_{r}|\,\mathrm{d}r\\\
&=\int_{\mathcal{E}}\frac{\omega}{c_{s}}\sqrt{\left(1-N^{2}/\omega^{2}\right)\left(L_{1}^{2}/\omega^{2}-1\right)}\,\mathrm{d}r\end{split}$
(10)

where, for simplicity, we have applied the Cowling approximation. $X_{R}$ is a
remainder term which can be specified analytically in the limit of a thin
evanescent zone (Takata, 2016b), where it is most important.

We focus on the value of $e^{-\mathcal{I}_{1}}$ as a rough proxy for the
mixed-mode coupling. Although CHeB stars typically have strong coupling such
that $X_{R}$ is likely to be important, calculation of this contribution
requires more care (see, e.g., van Rossem, 2023), and we therefore defer a
detailed calculation of $q$ in these merger remnants to a potential future
investigation.

From the propagation diagrams (Figure 8), it can be seen that the evanescent
zone evaluated at $\nu_{\mathrm{max}}$ is initially substantially wider in the
$M_{\mathrm{WD}}=0.38M_{\odot}$ post-dredge-up merger remnant than in the non-
merged model. This decreases the value of $e^{-\mathcal{I}_{1}}$ and, in turn,
the g-mode coupling factor $q$ at early phases of CHeB evolution.

However, late in the CHeB phase of post-dredge-up merger remnants involving
especially massive He WDs with $M_{\mathrm{WD}}\geq 0.33M_{\odot}$, the
evanescent region becomes exceedingly small because of a larger radiative
core. This temporarily causes $e^{-\mathcal{I}_{1}}$ and the mixed-mode
coupling to become larger than in typical CHeB stars, giving their mixed modes
high visibility.

In the last $\lesssim 20\,\mathrm{Myr}$, both $\Delta\Pi$ and
$e^{-\mathcal{I}_{1}}$ dive sharply as the CHeB star begins to enter the
asymptotic giant branch phase. During this time, the star expands especially
quickly, and it will lie even farther to the left of most merger remnants on
the asteroseismic spacing diagram in Figure 4.

Figure 10: Selected surface abundances $\mathrm{A}(\mathrm{X})$ (defined in
Equation 11) during the CHeB phase. Red points denote the $M=0.80M_{\odot}$
non-merged model, and starred points denote post-dredge-up merger remnants.
For ${}^{7}\mathrm{Li}$, filled symbols denote lithium enrichment due to the
helium flash only (described in Section 4.4), whereas hollow symbols denote
lithium enrichment due to both the helium flash and merger itself (which may
be inaccurately modelled).

###  4.4 Abundance anomalies

Merger remnants that experience a core dredge-up at the flash naturally
display unusual surface abundances. While dredge-up of the helium core most
obviously produces an enhancement in ${}^{4}\mathrm{He}$, direct spectroscopic
measurement of helium abundance in late-type stars is likely infeasible.
Fortunately, dredge-up probably enriches the surface with other elements that
can be directly probed spectroscopically. Figure 10 shows surface abundances
for selected species in terms of the index $\mathrm{A}(\mathrm{X})$, defined
for a given species $\mathrm{X}$ as

$\mathrm{A}(\mathrm{X})\equiv\log(n_{\mathrm{X}}/n_{\mathrm{H}})+12$ (11)

where $n_{\mathrm{X}}$ and $n_{\mathrm{H}}$ are the surface number densities
of $\mathrm{X}$ and hydrogen, respectively.

We find that post-dredge-up remnants exhibit significant 12C surface
enrichment, up to $\simeq 1\,\mathrm{dex}$. This enhancement arises from
dredge-up of core material which has been partially fused during the helium
flash into carbon. Post-dredge-up remnants thus possess increased values of
$\mathrm{C}/\mathrm{Fe}$ and decreased values of
${}^{13}\mathrm{C}/{}^{12}\mathrm{C}$. Additionally, our post-dredge-up
remnant models also possess significant surface enhancements in
${}^{18}\mathrm{O}$ and ${}^{22}\mathrm{Ne}$, which are created by successive
$\alpha$ captures of ${}^{14}\mathrm{N}$ (e.g., Clayton, 2003) during the
helium flash. While not included in our reaction network, it is also probable
that a significant amount of ${}^{26}\mathrm{Mg}$ is formed through $\alpha$
capture of ${}^{22}\mathrm{Ne}$ in the abnormally hot helium flash (as in Shen
et al., 2023). The surface abundance of ${}^{26}\mathrm{Mg}$ is therefore also
likely to be enhanced.

The lithium abundance is also important, because it can be created by the
burning of 3He or destroyed by burning at temperatures comparable to those
required for hydrogen burning. Lithium-rich giants have previously been
suggested to have been formed via binary interactions or mergers (e.g., Zhang
& Jeffery, 2013; Casey et al., 2019). Our merger remnant models become highly
lithium-rich very soon after merger, owing to a brief dredge-up event which
occurs when hydrogen burning is first turned on (similar to the Cameron–Fowler
mechanism; Cameron & Fowler, 1971). While such a merger-era lithium
enhancement seems plausible (see, e.g., observations of lithium enrichment in
luminous red novae; Kamiński et al., 2023), our treatment of the merger
process itself is highly artificial, and the evolution of the remnant is
unreliable for post-merger ages less than a thermal time,
$\tau_{\mathrm{th,env}}$. Modification to the surface abundances of other
species during this (possibly artificial) dredge-up event are minor, since
they primordially occur in much higher abundance than lithium.

If lithium is not enhanced during the merger, we investigate whether core
dredge-up during the helium flash may be able to create lithium-rich giants
anyway. To test this, we remove all of the lithium from our merger remnant
models (with total $M=0.8M_{\odot}$) just prior to the tRGB, and evolve them
through the helium flash and CHeB phase. As shown in Figure 10, most models do
not become lithium-rich according to the standard criterion
$\mathrm{A}(\mathrm{Li})\geq 1.5$ (e.g., Deepak & Lambert, 2021), with the
exception of the $M_{\mathrm{WD}}=0.38M_{\odot}$ model. It is possible that
merger remnants with higher values of $M_{\mathrm{WD}}$ or lower values of
$M_{\mathrm{MS}}$ may attain stronger lithium enhancements. Moreover, extra
mixing during the helium flash suggested by recent evidence may increase the
amount of lithium surfaced (Kumar et al., 2020; Martell et al., 2021; Schwab,
2020). Overall, we conclude that, unless lithium enrichment occurs at merger
or non-canonical mixing processes operate during the helium flash, He WD–MS
merger remnants are unlikely to become lithium-rich.

###  4.5 Populating the horizontal branch with merger remnants

Neglecting mass loss during merger or through winds, $M_{\mathrm{MS}}$ sets
the envelope mass of the merger remnant. In the preceding discussion, we have
focused on varying $M_{\mathrm{WD}}$ and fixed $M_{\mathrm{MS}}$ so that the
total mass of the merger remnant models is
$M=M_{\mathrm{WD}}+M_{\mathrm{MS}}=0.80M_{\odot}$. However, in principle,
$M_{\mathrm{MS}}$ could span a wide range of masses, including very small ones
(if the hydrogen-rich component is a brown dwarf or if there is significant
mass loss, e.g., Metzger et al. 2021), up to a few solar masses for mergers
with intermediate-mass MS stars. In this section, we explore the behavior of
merger remnants under varying $M_{\mathrm{MS}}$.

Variation of $M_{\mathrm{MS}}$ may significantly change the behavior of the
remnant in the following ways:

  1. 1.

If $M_{\mathrm{MS}}$ is sufficiently small, merger remnants may only ignite
helium with scant hydrogen envelopes (starting CHeB as subdwarf B-type stars;
sdBs), or may fail to ignite helium altogether (fizzling out into He WDs). For
reasons of scope, we do not investigate the He WD or sdB outcomes (but see the
detailed modelling of Zhang et al. 2017 and Zhang et al. 2023). Alternatively,
if the envelope mass drops below $M_{\mathrm{env}}\simeq 0.1M_{\odot}$ during
CHeB but can still sustain hydrogen shell burning, the remnant can evolve onto
the horizontal branch (Catelan, 2009).

  2. 2.

In post-dredge-up remnants (where a helium flash mixes a fixed amount of core
helium into the envelope), the helium fraction of the envelope during the CHeB
phase is set by $M_{\mathrm{MS}}$ (larger hydrogen-rich envelopes during this
stage more effectively dilute this additional helium). For merger remnants
massive enough to enter a CHeB phase with a hydrogen-burning shell, this
significantly affects their hydrogen-burning (and, thus, total) luminosity (as
in Section 4).

Fixing $M_{\mathrm{WD}}=0.30M_{\odot}$, we present in Figure 11 the CHeB-phase
evolution of merger remnants with
$M_{\mathrm{MS}}/M_{\odot}\in[0.30,0.35,0.40,0.50,0.70]$. All of these models
are massive enough to reach the tRGB and undergo a helium flash, which in this
case is energetic enough to dredge up $\approx 0.06M_{\odot}$ of helium.
However, the envelopes of models with lower $M_{\mathrm{MS}}$ possess much
more helium-enriched envelopes: envelope helium mass fractions during the CHeB
phase for these models range from $Y=0.36$ (for
$M_{\mathrm{MS}}=0.70M_{\odot}$) to $Y=0.48$ (for
$M_{\mathrm{MS}}=0.30M_{\odot}$).

At the zero-age CHeB, all of these models possess a convective envelope. As
expected, models with lower $M_{\mathrm{MS}}$ (and higher envelope $Y$) have
higher hydrogen shell-burning luminosities at the beginning of the CHeB phase
(as can be seen in the $L_{\mathrm{H}}$ panel in Figure 11). Models with
higher $M_{\mathrm{MS}}=0.40$, $0.50$, and $0.70M_{\odot}$ retain these
convective envelopes and behave similarly to the remnants discussed in Section
4.

The lower-mass $M_{\mathrm{MS}}=0.30$ and $0.35M_{\odot}$ models display
significantly different behavior. These models burn most of their remaining
hydrogen during the CHeB phase such that their envelope mass drops below
$0.1\,M_{\odot}$. The outer layers of these models become completely radiative
(at $t-t_{\mathrm{flash}}\approx 25\,\mathrm{Myr}$ and $55\,\mathrm{Myr}$,
respectively), and the remnants behave like horizontal branch stars. When this
occurs, the stars become very blue, roughly reaching respective effective
temperatures $T_{\mathrm{eff}}\approx 20000\,\mathrm{K}$ and $\approx
12000\,\mathrm{K}$. While these models continue to sustain hydrogen-shell
burning to some extent, $L_{\mathrm{H}}$ significantly drops during this
horizontal branch stage (decreasing by factors $\simeq 10$ and $\gtrsim 2$ for
the $M_{\mathrm{MS}}=0.30M_{\odot}$ and $M_{\mathrm{MS}}=0.35M_{\odot}$
models). In the $M_{\mathrm{MS}}=0.30M_{\odot}$ model, this extreme drop in
$L_{\mathrm{H}}$ precipitates a significantly lower total luminosity, which is
readily apparent on a Hertzsprung–Russell diagram (top-left panel of Figure
11). We confirm for these two cases that the inclusion of gravitational
settling does not change the results. A Reimers wind (Reimers, 1975, 1977)
scaled as in Reimers (1977) suggests that winds in these objects are small
($\dot{M}<10^{-10}\,M_{\odot}\,\mathrm{yr}^{-1}$) and may be ignored.

Figure 11: Evolution during the CHeB phase for merger remnants with
$M_{\mathrm{WD}}=0.30M_{\odot}$ but varying $M_{\mathrm{MS}}$ (Section 4.5).
We plot a Hertzsprung–Russell diagram (left, with zoomed-in inset on the top
right) and the time evolution of the hydrogen-rich mass
$M_{\mathrm{env}}=M-M_{c}$ and hydrogen shell-burning luminosity
$L_{\mathrm{H}}$ (bottom right). A non-merged model with $M=0.80M_{\odot}$ is
also shown for comparison (red dashed line).

##  5 Candidate merger remnants

###  5.1 Undermassive red clump stars may be merger remnants

Figure 12: Measured masses and radii of observed CHeB stars, measured using
asteroseismology. Blue and green squares denote the very low-mass (VLM) sample
of Li et al. (2022) and red horizontal branch (rHB) stars of Matteuzzi et al.
(2023). Gray data points are taken from the catalog of Yu et al. (2018).
Orange stars denote CHeB-phase merger remnant models with
$M_{\mathrm{WD}}=0.20M_{\odot}$ and varying $M_{\mathrm{MS}}$. Red triangles
denote a CHeB-phase non-merged model with $M=0.8M_{\odot}$, for comparison.
Model track points are sampled $10\,\mathrm{Myr}$ apart.

.

Using asteroseismology, Li et al. (2022) discovered a population of
undermassive CHeB stars with masses $\lesssim 0.8M_{\odot}$ (their “very low-
mass” sample). Since single stars of these masses could not have evolved off
of the MS in the age of the universe, these undermassive stars must have
undergone non-standard evolution, such as stripping by a companion or binary
assembly. Although Li et al. (2022) argue that these undermassive giants are
the product of partial envelope stripping by close companions, most of these
objects do not exhibit the expected radial velocity variability between APOGEE
and Gaia (Kareem El-Badry, private communication), disfavoring this formation
channel. Matteuzzi et al. (2023) identify several more such objects (referred
to as “red horizontal branch” stars), further demonstrating their extremely
strong mixed-mode coupling.

Observed undermassive clump stars likely have typical helium core masses and
only stand out due to their low envelope masses. He WD–MS mergers naturally
explain these objects: as long as $M_{\mathrm{WD}}$ is small enough that no
dredge-up occurs at the helium flash, merger remnants have core masses which
are basically normal. Furthermore, a merger remnant’s envelope mass is simply
set by $M_{\mathrm{MS}}$ (modulo merger or tRGB mass loss), which can be
arbitrarily low. Finally, this binary scenario does not leave any companion
behind, explaining why most undermassive CHeB stars are consistent with being
single at present. While other mechanisms may exist for forming single
undermassive CHeB stars (e.g., mass loss during failed common-envelope
events), He WD–MS mergers appear to be highly promising.

To demonstrate this possibility, we run additional merger remnant models,
fixing $M_{\mathrm{WD}}=0.20M_{\odot}$ and varying $M_{\mathrm{MS}}$ between
$0.35M_{\odot}$ and $0.60M_{\odot}$ (total masses $M$ between $0.55M_{\odot}$
and $0.80M_{\odot}$). Lower-mass merger remnant models have masses and radii
which are consistent with the observed very low-mass sample of Li et al.
(2022) (Figure 12). Because the He WD–MS merger channel produces CHeB stars
with essentially normal cores, our models behave almost identically to models
performed by Li et al. (2022) of normal CHeB stars with artificial envelope
stripping to mimic mass loss from an initially standard RG.

###  5.2 Zvrk: a possible post-dredge-up merger remnant?

Using asteroseismology, Ong et al. (2024) recently discovered a peculiar RG
(“Zvrk”) with the following features:

  1. 1.

The oscillation spectrum is complex, superficially resembling typical spectra
of CHeB giants (which are dense due to their strong mixed-mode coupling, e.g.,
Mosser et al., 2017; Dhanpal et al., 2023).

  2. 2.

The asteroseismic scaling relations imply a radius $R\approx 24R_{\odot}$, a
factor of two larger than that of a typical CHeB star.

  3. 3.

The star is highly lithium-rich ($\mathrm{A(Li)}>3$), and also slightly
carbon-enhanced relative to typical first-ascent RGs, or indeed CHeB stars
(e.g., Bufanda et al., 2023).

  4. 4.

Photometric modulation and asteroseismic rotational splittings (assuming a
pure p-mode spectrum) are consistent with a fast-spinning envelope with period
$P\approx 100\,\mathrm{d}$.

Mainly on the basis of its radius, Ong et al. (2024) conclude that Zvrk cannot
be in the CHeB phase and must instead be a first-ascent RG. Because the
expected mixed-mode coupling for an RG of this size would be weak, they argue
that the spectrum is actually composed of pure p modes (rather than mixed
modes), with the complexity of the spectrum instead coming from large
rotational splittings from its fast rotation rate.

We suggest instead the possibility that Zvrk is a post-dredge-up He WD–MS
merger remnant which is on the CHeB phase, not the RGB. This addresses the
aforementioned observations in the following ways:

  1. 1.

If Zvrk were a CHeB star, it would naturally have a large mixed-mode coupling
and, thus, a CHeB-like spectrum (see also Section 4.3).

  2. 2.

Our models of post-dredge-up CHeB remnants (Section 4.1), like Zvrk, have
radii approximately twice as large as those of typical CHeB stars.

  3. 3.

As demonstrated in Section 4.4, post-dredge-up remnants are expected to be
carbon-rich, similar to Zvrk. While the core dredge-up event at the helium
flash is unlikely to match the measured value of $\mathrm{A}(\mathrm{Li})$,
lithium enrichment may occur at merger (see Section 4.4).

  4. 4.

As discussed in Section 6.3, merger remnants are expected to be rapidly
rotating.

It is unclear at present whether He WD–MS mergers can reproduce these effects
in the correct combination to match observations. The mass $M\approx
1.2M_{\odot}$ inferred by scaling relations is larger than those of our
fiducial models ($M=0.8M_{\odot}$), possibly requiring even stronger core
dredge-ups which probably occur for larger values of $M_{\mathrm{WD}}$ than we
can model. We point out that an extreme dredge-up event of this type probably
modifies $\nu_{\mathrm{max}}$ (in a still-contested manner; Viani et al.,
2017; Zhou et al., 2024) and, thus, the accuracy of scaling relation-based
values of $M$ and $R$. The similarly behaving remnants of RG–He WD mergers
Zhang & Jeffery (e.g., 2013) may also possibly reproduce the properties of
Zvrk.

Further complicating the picture, Ong et al. (2024) point out that a naïve
identification of the double-ridged feature in Zvrk’s échelle diagram with the
usual $\ell=0$ and $2$ degrees implies an unusually small p-mode offset
$\epsilon_{p}\sim 0.25$. This is too low to be consistent with a low-mass RG
of any canonical evolutionary state: none of the observed stars in Figure 10
of Kallinger et al. (2012) have $\epsilon_{p}<0.4$ (though CHeB stars do have
lower values of $\epsilon_{p}$ than do first-ascent RGs with comparable
$\Delta\nu$). Of course, if Zvrk is a post-dredge-up remnant, it may well be
possible that it attains an unusual value of $\epsilon_{p}$. While not
theoretically characterized in this work, $\epsilon_{p}$ may turn out to be
another observational diagnostic for He WD–MS remnants.

While it is beyond the scope of the present work, we encourage a more detailed
investigation to determine whether this hypothesis can explain Zvrk’s large
radius, oscillation spectrum, surface abundances, and rapid rotation in a
quantitative and self-consistent way.

###  5.3 Other potential post-dredge-up merger remnants

On the $\Delta\nu$–$\Delta\Pi$ diagram, Mosser et al. (2014) identify several
RGs which lie near, but slightly leftward, of the red clump (see their Figure
1). While Mosser et al. (2014) argue that these stars have recently undergone
helium subflashes, we suggest that they might be post-dredge-up remnants. As
we show in Section 4.2, post-dredge-up remnants are also expected to have
values of $\Delta\Pi$ comparable to those of normal CHeB stars, but smaller
values of $\Delta\nu$ on account of their larger radii.

Recently, a small number of highly carbon-deficient giants has been discovered
and characterized as a distinct class (Bidelman & MacConnell, 1973; Bond,
2019; Maben et al., 2023). These objects are almost all in the CHeB phase, and
many are also lithium-rich and overluminous compared to the usual red clump
(Maben et al., 2023). However, the He WD–MS merger scenario predicts a carbon
enrichment and thus fails to explain the abnormally low carbon abundances in
these stars.

##  6 Discussion and future prospects

###  6.1 Progenitors and rates

While no CVs with He WD accretors have been discovered (to our knowledge), CV
progenitors (known as pre-CVs) have been, some shown in Figure 13. As their
name suggests, such systems are expected to eventually initiate (possibly
unstable) mass transfer after a combination of magnetic braking and
gravitational radiation tighten their orbits sufficiently.

Zorotovic et al. (2011) compile a catalog of post-common envelope binaries
(PCEBs), WD–MS binary systems, some of which may contain He WD components,
identified in this work as those with $M_{\mathrm{WD}}<0.5M_{\odot}$ (see
Figure 13). Most of the MS components are of relatively low mass, with
$M_{\mathrm{MS}}\simeq 0.3M_{\odot}$. However, many of these systems contain
fairly massive He WDs and could, upon merging, display fairly extreme versions
of the asteroseismic and photometric merger remnant signatures we propose.

In later years, Maxted et al. (2014) identified a separate class of close
detached binaries involving AF-type MS stars orbiting low-mass ($\simeq
0.2M_{\odot}$) proto-He WDs, with van Roestel et al. (2018) later measuring
$M_{\mathrm{WD}}$ and $M_{\mathrm{MS}}$ for $36$ such systems (see Figure 13).
Due to the low masses of the proto-WD components, EL CVns are likely formed
via stable mass transfer rather than common-envelope events (which would
likely result in merger; Chen et al., 2017). The EL CVns in Figure 13 will
likely merge when the stellar component is either a MS star or a subgiant
(Lagos et al., 2020). In the He WD–subgiant case, the He WD is expected to
merge with the subgiant core and produce a similar, low-entropy-core remnant
as in the He WD–MS case explored in this work. However, He WD–subgiant mergers
may also have their own distinctive signatures, and a detailed investigation
of the associated remnants is probably warranted.

Figure 13: Parameters for a collection of known pre-CVs which may contain He
WDs ($M_{\mathrm{WD}}\leq 0.5M_{\odot}$) taken from Zorotovic et al. (2011)
for PCEBs and van Roestel et al. (2018) for EL CVns. Color indicates the
orbital periods of each system.

Based on MESA models of merger remnants (Zhang et al., 2017), Zhang et al.
(2018) perform a population synthesis calculation to determine how much these
sorts of merger remnants contribute to the population of single He WDs, which
are not a natural outcome of isolated stellar evolution. Assuming a star-
formation rate of $5M_{\odot}\,\mathrm{yr}^{-1}$, their calculation implies a
Galactic formation rate of merger remnants $\sim 0.02\,\mathrm{yr}^{-1}$
(about half of which fizzle out into He WDs before starting helium burning),
with a factor-of-a-few uncertainty when the Reimers wind parameter is varied
(Reimers, 1975, 1977). If merger remnants continue to appear as RGs for
$100T_{\mathrm{m}8}\,\mathrm{Myr}$, the Milky Way should contain $\sim
2T_{\mathrm{m}8}\times 10^{6}$ remnants at a given time. Put another way, for
a Kroupa initial mass function (Kroupa, 2001), single low-mass RGs in the
range $0.8M_{\odot}$ to $2.0M_{\odot}$ form at a rate $\sim
0.3M_{\odot}\,\mathrm{yr}^{-1}$. If isolated RGs have lifetimes
$100T_{\mathrm{s}8}\,\mathrm{Myr}$, He WD–MS merger remnants should make up
roughly $\sim 7\%\times(T_{\mathrm{m}8}/T_{\mathrm{s}8})$ of all low-mass RGs.

###  6.2 Merger transients and mass retention

The observable signatures of He WD–MS merger remnants described in this paper
are applicable long (at least a thermal time) after the merger. However, the
merger itself as well as any preceding novae events should produce observable
transients and other emission, whose rates should be consistent with the
population of merger remnants.

Metzger et al. (2021) show that the merger itself likely produces a dusty
transient similar to a luminous red nova (e.g., Kulkarni et al., 2007) powered
by recombination of the ejected material, and roughly estimate a Galactic rate
$\sim 0.1\,\mathrm{yr}^{-1}$. This is larger than the rate predicted by Zhang
et al. (2018) by a factor $\sim 5$, although both estimates are subject to
significant uncertainties. From this rate, they estimate that the Milky Way
contains $10^{3}$–$10^{4}$ remnants at the present day, which is much smaller
than the $\sim 10^{6}$ we estimate in Section 6.1. The reason is that their
models predict the merger remnant only retains a small fraction $\sim\\!10\%$
of the hydrogen supplied by an already low-mass MS star, rather than all of
it. This implies a much shorter post-merger lifetime $\sim 10$–$100$ kyr.

However, Metzger et al. (2021) investigates cases where the MS star has a low
mass, $M_{\rm MS}\lesssim 0.5M_{\rm WD}$. As they point out, higher-mass MS
stars will likely produce gravitationally unstable disks that accrete much
more efficiently onto the WD, closer to our assumption of conservative mass
accretion. Ignition of hydrogen burning on the WD during the merger (not
included in their simulations) may also cause the remnant to swell up and gain
more mass from the disk. An independent constraint on the population of He
WD–MS merger remnants in the Milky Way may help distinguish these scenarios.

###  6.3 Rapid rotation

The merger occurs when the MS component of the close binary overflows its
Roche lobe. Very roughly, this occurs at a semimajor axis
$a\\!\sim\\!3\,R_{\mathrm{MS}}\sim
3R_{\odot}(M_{\mathrm{MS}}/M_{\odot})^{0.8}\mathrm{,}$ according to the
traditional mass–radius scaling formula for the MS. At the time of merger, the
total binary has an orbital angular momentum

$L=M_{\mathrm{MS}}M_{\mathrm{WD}}\sqrt{\frac{Ga}{M}}$ (12)

where $M=M_{\mathrm{MS}}+M_{\mathrm{WD}}$ (and we have assumed a circular
orbit). Then, assuming that no mass is lost during the merger, the final
envelope spin rate of the resulting RG remnant is

$\begin{split}\Omega_{\mathrm{env}}&\sim
3.8\,\mu\mathrm{Hz}\times\left(\frac{\kappa}{0.2}\right)^{-1}\left(\frac{M_{\mathrm{MS}}}{0.5M_{\odot}}\right)^{0.4}\left(\frac{M_{\mathrm{WD}}}{0.3M_{\odot}}\right)\\\
\times&\left(\frac{M}{0.8M_{\odot}}\right)^{-1/2}\left(\frac{R}{20R_{\odot}}\right)^{-2}\end{split}$
(13)

where we have scaled the moment of inertia of the merger remnant’s envelope to
$\kappa=I/M_{\mathrm{MS}}R^{2}=0.2$ (as in, e.g., Bear & Soker, 2010).

This corresponds to rotation periods $\sim\\!20\,\mathrm{days}$, more than
half the surface breakup frequency $\Omega_{K}=\sqrt{GM/R^{3}}\sim
6.2\,\mu\mathrm{Hz}$, depending on the inflated radius of the remnant. This is
much faster than the rotation rate of typical RG stars at similar radii: rapid
rotation (without associated radial velocity variability) and associated
magnetic activity can therefore help distinguish merger remnants.

However, Equation 13 gives an estimate of the envelope rotation rate very soon
(roughly a thermal time) after the merger. The subsequent evolution of the
rotation profile is strongly dependent on the physics of magnetic braking
(which saps angular momentum from the system) and angular momentum transport
(which couples the core and envelope rotation rates). Both of these pieces of
physics are not particularly well understood (especially in the context of
fast-rotating RGs), and models which incorporate both of these effects are
necessary for predicting the long-term evolution of core and envelope rotation
rates of merger remnants (Qian et al., in preparation).

###  6.4 Non-asymptotic effects on pulsations

Both Equations 3 and 4 for $\Delta\nu$ and $\Delta\Pi$ rely on the asymptotic
approximation for stellar oscillations, i.e., that the radial wavelength of
the oscillation is much smaller than the scale height of any structural
variable. While the asymptotic approximation is well-justified for most RGs
(see, e.g., the introductory discussion of Ong & Basu, 2020), merger remnants
may possess sharp features in their profiles (glitches) which may cause
departures from the asymptotic formulae. Indeed, such glitches are observed in
a small fraction of CHeB red giants (Vrard et al., 2022).

Although we eschew a comprehensive analysis of non-asymptotic effects in this
work, we point out three possible glitches which may occur in He WD–MS merger
remnants:

  1. 1.

Soon after merger, the Brunt–Väisälä frequency rises sharply with radius at
the interface between the He WD and hydrogen-burning shell. In intermediate-
mass MS stars, a similar spike in $N$ near the convective core can produce
variations in $\Delta\Pi$ versus $P$ whose “period” in this space is a
function of the buoyancy coordinate of the glitch (Miglio et al., 2008;
Pedersen et al., 2018). Given the low-entropy state of the core and the short-
lived nature of the entropy discontinuity, it remains to be seen whether this
buoyancy feature is detectable.

  2. 2.

In merger remnants, more intense helium flashes quickly burn larger fractions
of helium into carbon. This enhances the composition gradient between helium
flash-processed material and outer layers of the helium core. This may produce
an abnormally strong compositional peak in $N$ during the CHeB phase, which
may manifest as an observable buoyancy glitch.

  3. 3.

At the He i and He ii ionization zones, the first adiabatic exponent dips
abruptly, producing sharp features in the sound speed (e.g., Miglio et al.,
2010). Notably, the amplitudes of these acoustic glitches increase with higher
helium mass fraction $Y$ (Houdek & Gough, 2007). During the CHeB phase of our
models, $Y$ can be enhanced to extreme degrees (Section 4), and it is possible
that the effect of these acoustic glitches may be very strong.

Detailed mode calculations are likely required to determine whether these
glitches are observable, what their characteristics are in the oscillation
spectrum, and to what extent they can be used to identify and characterize
merger remnants.

Our predictions for the large frequency spacing $\Delta\nu$ relies on the
scaling relations in Equation 3, which is known to require corrections when
the outer layers deviate from homology to a calibration standard (Belkacem et
al., 2013; Ong & Basu, 2019). This may be slightly modified in merger remnants
by changes in surface composition or rapid rotation, and should be
investigated in the future.

###  6.5 Broader progress in binary interaction asteroseismology

Due to the diversity of stellar interactions expected to occur in the field,
other post-merger stellar structures and their asteroseismic signatures
deserve future investigation, in particular:

  * •

RG–He WD mergers likely produce unusual CHeB giants, which have been
previously explored as a possible channel for producing certain classes of
carbon stars (Izzard et al., 2007; Zhang & Jeffery, 2013; Zhang et al., 2020)
or as a possible progenitor of the 8 UMi planetary system (Hon et al., 2023).
The models of Zhang & Jeffery (2013) suggest that these merger remnants behave
like overluminous CHeB stars, similar to those described in Section 4.1 for
the He WD–MS scenario (compare their Figure 4 to our Figure 2).

  * •

CO WD–MS mergers are likely to result from consequential angular momentum
loss, particularly for lower-mass CO WDs. While existing modelling literature
typically focuses on progenitor systems’ nova eruptions (Iben & Tutukov, 1996;
Shara et al., 2010; Kato et al., 2017) or their role in producing Type Ia
supernovae (Kovetz & Prialnik, 1994; Cassisi et al., 1998; Newsham et al.,
2013; Hillman et al., 2016; Wang, 2018), mergers of such remnants should
become unusual asymptotic giant branch stars with highly degenerate cores
(Cassisi et al., 1998; Piersanti et al., 2000; Wolf et al., 2013), with
possibly observable consequences.

  * •

CO WD–He WD mergers are the likely progenitors of R Coronae Borealis stars (R
Cor Bor stars), which are yellow supergiants consisting of a carbon–oxygen
core surrounded by a helium envelope inflated by shell burning. R Cor Bor
stars are known to pulsate at periods between $40$ and $100\,\mathrm{d}$
(Lawson & Kilkenny, 1996; Karambelkar et al., 2021), making asteroseismology a
promising tool for probing their internal structures (Wong & Bildsten, 2023).

  * •

Last year, Bellinger et al. (2023) identified asteroseismology as a tool for
testing the post-MS merger channel for producing blue supergiants, finding
that the g-mode period spacing $\Delta\Pi$ constrains their formation
channels. In a parallel observational study using TESS, Ma et al. (2023)
discovered peculiar but universal low-frequency ($f\lesssim
2\,\mathrm{d}^{-1}$) photometric power excess, although the physical
interpretation remains unclear.

Binary interaction asteroseismology is a technique at its infancy, with likely
many more fruitful directions.

##  7 Summary

In this work, we presented detailed models of merger remnants of He WD–MS
mergers. Merger remnants quickly initiate hydrogen shell burning and become
unusual giant stars which may hide inside the RG population. However, they
exhibit a number of unique signatures which may be used to distinguish them.
In summary, during hydrogen shell-burning (RGB), merger remnants:

  1. 1.

are over-inflated at a given core mass.

  2. 2.

depart from the standard degenerate sequence on the asteroseismic
$\Delta\nu$–$\Delta\Pi$ diagram. Asteroseismology can thus identify remnants
whose mixed-mode coupling is sufficiently strong.

  3. 3.

undergo delayed helium flashes, and attain higher luminosities at the tRGB
than do single RGs.

During helium core burning (CHeB), remnants of mergers involving lower-mass He
WDs:

  1. 1.

attain core masses which are essentially typical for single CHeB stars.

  2. 2.

are strong candidates for the undermassive red clump stars discovered by Li et
al. (2022) and Matteuzzi et al. (2023).

Remnants of mergers involving higher-mass He WDs dredge up a significant
fraction (up to $\sim 0.1M_{\odot}$) of helium into the envelope. During core
helium burning, these post-dredge-up merger remnants:

  1. 1.

have significantly larger radii and luminosities than single stars on the red
clump.

  2. 2.

exhibit a wider range of asteroseismic g-mode period spacings $\Delta\Pi$ than
do typical stars on the red clump.

  3. 3.

attain abnormally strong degrees of asteroseismic mixed-mode coupling towards
the end of CHeB.

  4. 4.

are enriched in ${}^{12}\mathrm{C}$, as well as ${}^{18}\mathrm{O}$ and
${}^{22}\mathrm{Ne}$.

  5. 5.

may already have been discovered. The rapidly rotating RG discovered by Ong et
al. (2024) (“Zvrk”) has many of the predicted properties of this type of
merger remnant.

Observational probes of these merger remnants can constrain the He WD–MS
merger process at the population level. In turn, this may provide additional
confirmation of the consequential angular momentum loss hypothesis and white
dwarf mass problem for CVs.

We thank Kareem El-Badry, Joel Ong, Marc Hon, and Yaguang Li for their
insightful comments and thorough reading of the manuscript, as well as Lars
Bildsten for useful discussions. We also acknowledge Masao Takata, Andrew
Casey, and Tim Bedding for their helpful remarks. N.Z.R. acknowledges support
from the National Science Foundation Graduate Research Fellowship under Grant
No. DGE‐1745301.

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##  Appendix A Dependence on the cooling age of the white dwarf

Pre-merger, all of the merger models discussed in the main text use a He WD
which has been cooled until it achieves a luminosity
$L_{\mathrm{WD}}=10^{-4.0}L_{\odot}$. This relatively low luminosity is chosen
to explore the limiting case of a very degenerate helium core. In our models,
this luminosity corresponds to relatively long He WD cooling ages $5\,{\rm
Gyr}\lesssim t_{\mathrm{cool}}\lesssim 7\,\mathrm{Gyr}$, with a fixed
$L_{\mathrm{WD}}$ corresponding to longer $t_{\mathrm{cool}}$ for He WDs with
higher masses $M_{\mathrm{WD}}$ or more substantial atmospheres. For
comparison, note that the merger remnant models of Zhang et al. (2017) use
$L_{\mathrm{WD}}=10^{-2.0}L_{\odot}$.

In this Appendix, we discuss the effect of varying
$\log(L_{\mathrm{WD}}/L_{\odot})$. Figure 14 shows the evolution of some
selected selected quantities for four models (the first three of which also
appear in the main text):

  1. 1.

A non-merged model, which has a “normal” helium core on the RGB which is close
to isothermal with the hydrogen-burning shell.

  2. 2.

A merger remnant with $\log(L_{\mathrm{WD}}/L_{\odot})=-4.0$ and
$M_{\mathrm{WD}}=0.30M_{\odot}$.

  3. 3.

A merger remnant with $\log(L_{\mathrm{WD}}/L_{\odot})=-4.0$ and a lower mass
$M_{\mathrm{WD}}=0.27M_{\odot}$.

  4. 4.

A merger remnant with a higher pre-merger He WD luminosity
$\log(L_{\mathrm{WD}}/L_{\odot})=-2.5$ and $M_{\mathrm{WD}}=0.30M_{\odot}$.
This value of $\log(L_{\mathrm{WD}}/L_{\odot})$ corresponds to a He WD cooling
age $t_{\mathrm{cool}}\approx 550\,\mathrm{Myr}$.

All models have total masses of $M=0.80M_{\odot}$ and WD atmospheres of mass
$10^{-4}M_{\odot}$, and respectively.

As described in Section 3, merger remnants during the RGB will start with
initially low-entropy cores, but their entropies will gradually grow due to
heat diffusion and deposition of higher-entropy helium resulting from hydrogen
burning. The degree of entropy deficit in the core therefore results from a
combination of the entropy of the original He WD as well as its mass, which
determines its total heat capacity). Because increasing
$\log(L_{\mathrm{WD}}/L_{\odot})$ and decreasing $M_{\mathrm{WD}}$ affect this
core entropy deficit in the same way, merger remnants should be affected by
increases in $\log(L_{\mathrm{WD}}/L_{\odot})$ and decreases in
$M_{\mathrm{WD}}$ in similar ways.

Figure 14 shows the evolution of the models listed above. They all have
comparable radii on the RGB, but $\Delta\Pi$ (reflecting the internal thermal
structure of the core) varies somewhat between them. As can be seen on the
bottom left panel of Figure 14, the core temperature $T_{c}$ of the
$M_{\mathrm{WD}}=0.27M_{\odot}$, $\log(L_{\mathrm{WD}}/L_{\odot})=-4.0$ model
initially behaves very similarly to the $M_{\mathrm{WD}}=0.30M_{\odot}$,
$\log(L_{\mathrm{WD}}/L_{\odot})=-4.0$ model, since not enough time has yet
elapsed for conduction to significantly modify its temperature. Later on,
$T_{c}$ in the $M_{\mathrm{WD}}=0.27M_{\odot}$ model takes a sharp upturn to
more closely resemble the warmer $M_{\mathrm{WD}}=0.30M_{\odot}$,
$\log(L_{\mathrm{WD}}/L_{\odot})=-2.5$ model. By the helium flash, these two
models have very similar $\Delta\Pi$ and temperature profiles.

Once the helium flash occurs, variations in the temperature profile due to a
finite thermal conductivity in the core are erased entirely. During the CHeB
phase, the $M_{\mathrm{WD}}=0.27M_{\odot}$,
$\log(L_{\mathrm{WD}}/L_{\odot})=-4.0$ and $M_{\mathrm{WD}}=0.30M_{\odot}$,
$\log(L_{\mathrm{WD}}/L_{\odot})=-2.5$ models are essentially identical (right
panels of Figure 14). Both models evolve significantly differently than the
$M_{\mathrm{WD}}=0.3M_{\odot}$, $\log(L_{\mathrm{WD}}/L_{\odot})=-4.0$ model
(which had a larger entropy deficit on the RGB).

In summary, decreasing the cooling age of the merging He WD has a very similar
effect to slightly decreasing its mass. Hence, the merger models in the main
text are expected to behave similarly to merger models with younger and
slightly more massive WDs.

Figure 14: The total luminosity $L$, radius $R$, asteroseismic g-mode period
spacing $\Delta\Pi$, and central temperature $T_{c}$ for selected models
during the RGB (left) and CHeB (right) phases. A model with
$M_{\mathrm{WD}}=0.30M_{\odot}$ but a brighter He WD
($\log(L_{\mathrm{WD}}/L_{\odot})=-2.5$) is included.

##  Appendix B Thermal timescale of the core

In this Appendix, we estimate the timescale $\tau_{\mathrm{th,core}}$ for a
cold, highly degenerate remnant core to thermalize with a hot burning shell.
In this environment, electron-mediated heat conduction is the dominant heat
transport mechanism.

The resulting heat flux $\vec{q}$ takes the form

$\vec{q}=-\alpha_{\mathrm{cond}}\rho T\nabla T$ (B1)

where

$\alpha_{\mathrm{cond}}\approx 2.44\times
10^{3}AZ^{-2}\,\mathrm{cm}^{4}\,\mathrm{s}^{-3}\,\mathrm{K}^{-2}\mathrm{,}$
(B2)

see Mestel (1950). The heat capacity per unit volume is dominated by the non-
degenerate ions:

$c_{v}\approx\frac{\varepsilon}{T}\approx\frac{3\rho k_{B}}{2Am_{p}}$ (B3)

In the absence of heat sources, the continuity equation for energy density
$\varepsilon$ within the interior of the WD is thus given by

$\frac{\partial\varepsilon}{\partial t}=-\nabla\cdot\vec{q}$ (B4)

Substituting Equations B1 and B3 into Equation B4 yields the following
nonlinear heat diffusion equation:

$\frac{\partial T}{\partial t}=\frac{\kappa}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}T\frac{\partial T}{\partial r}\right)\mathrm{,}$ (B5)

cf. Mestel (1952). In writing Equation B5, we have assumed spherical symmetry,
and that $\rho$ does not vary with temperature. The diffusion coefficient
$\kappa$ is given by

$\kappa=\frac{2Am_{p}\alpha_{\mathrm{cond}}}{3k_{B}}$ (B6)

At early times, the core is essentially isothermal at a low temperature
$T_{0}\ll T_{\mathrm{shell}}\simeq 3\times 10^{7}\,\mathrm{K}$. Thermal
contact with the hot hydrogen-burning shell at the outer boundary sets up a
temperature gradient $\partial T/\partial r\sim T_{\mathrm{shell}}/R_{c}$. The
natural length scale of the problem is $R_{c}$. By Equation B5, the core
therefore thermalizes on a timescale $\tau_{\mathrm{th,core}}$ given roughly
by

$\frac{1}{\tau_{\mathrm{th,core}}}\sim\frac{\kappa
T_{\mathrm{shell}}}{R_{c}^{2}}$ (B7)

or

$\tau_{\mathrm{th,core}}\sim\frac{3k_{B}}{2Am_{p}\alpha_{\mathrm{cond}}}\frac{R_{c}^{2}}{T_{\mathrm{shell}}}\approx
80\,\mathrm{Myr}\times\left(\frac{R_{c}}{0.035R_{\odot}}\right)^{2}\left(\frac{T_{\mathrm{shell}}}{3\times
10^{7}\,\mathrm{K}}\right)^{-1}\left(\frac{Z}{2}\right)^{2}\left(\frac{A}{4}\right)^{-2}$
(B8)

A similar timescale to $\tau_{\mathrm{th,core}}$ has been previously derived
by Shen et al. (2009). The scaling relation in Equation B8 is normalized to a
typical He WD in thermal contact with a typical hydrogen-burning shell. This
timescale is comparable to the length of the remnant’s RGB phase—the longest
RGB phase leading to helium ignition in our merger remnant models lasts
$\simeq 180\,\mathrm{Myr}$ (for a He WD mass $M_{\mathrm{WD}}=0.20M_{\odot}$).
Heat conduction thus cannot completely destroy the low-entropy state of the
core quickly enough to erase its long-term effects on the appearance and
evolution of the remnant, especially for higher values of $M_{\mathrm{WD}}$.

##  Appendix C Brunt–Väisälä frequency in degenerate helium cores

In this Appendix, we derive the Brunt–Väisälä frequency $N$ in the part of the
g-mode cavity which lies within the helium core of a star on the RGB. This
region is characterized by degenerate electrons which dominate the pressure
support and non-degenerate ions which dominate the heat capacity.

Following Brassard et al. (1991), in the absence of composition gradients, $N$
can be written as

$N^{2}=N_{0}^{2}\frac{\chi_{T}}{\chi_{\rho}}\left(\nabla_{\mathrm{ad}}-\nabla\right)$
(C1)

where

$\displaystyle\chi_{T}=\left(\frac{\partial\ln p}{\partial\ln
T}\right)_{\rho}$ (C2a) $\displaystyle\chi_{\rho}=\left(\frac{\partial\ln
p}{\partial\ln\rho}\right)_{T}$ (C2b)
$\displaystyle\nabla_{\mathrm{ad}}=\left(\frac{\mathrm{d}\ln T}{\mathrm{d}\ln
p}\right)_{\mathrm{ad}}$ (C2c) $\displaystyle\nabla=\frac{\mathrm{d}\ln
T}{\mathrm{d}\ln p}$ (C2d)

and we have defined

$N_{0}^{2}=\frac{\rho g^{2}}{p}$ (C3)

where roughly $N_{0}\\!\simeq\\!\sqrt{GM_{c}/R_{c}^{3}}$ is comparable to the
dynamical frequency of the core.

In degeneracy-supported matter, $\chi_{\rho}\\!\approx\\!5/3$ and
$\nabla_{\mathrm{ad}}\\!\approx\\!0.4$ are basically constant. In WDs, the
pressure has two contributions from the degenerate electrons (which dominate
the total pressure) and non-degenerate ions (which carry the temperature
dependence):

$p=\frac{2}{5}n_{e}E_{F}+n_{i}k_{B}T$ (C4)

where $n_{e}=Z\rho/Am_{p}$ and $n_{i}=\rho/Am_{p}$ are the electron and ion
number densities, respectively, and $E_{F}$ is the electronic Fermi energy.
Roughly

$\displaystyle p\approx\frac{2}{5}\frac{Z\rho E_{F}}{Am_{p}}$ (C5a)
$\displaystyle\left(\frac{\partial p}{\partial
T}\right)_{\rho}\approx\frac{\rho k_{B}}{Am_{p}}$ (C5b)

Then

$\chi_{T}=\frac{T}{p}\left(\frac{\partial p}{\partial
T}\right)_{\rho}=\frac{5}{2Z}\frac{k_{B}T}{E_{F}}\mathrm{.}$ (C6)

Therefore, for mostly degenerate matter,

$N^{2}\approx
N_{0}^{2}\frac{k_{B}T}{ZE_{F}}\left(1-\frac{5}{2}\nabla\right)\mathrm{.}$ (C7)