# Sodium enhancement in evolved cataclysmic variables

Natsuko Yamaguchi,1 Kareem El-Badry1,2, Antonio C. Rodriguez1, Maude Gull3,
Benjamin R. Roulston1, Zachary P. Vanderbosch1
1 California Institute of Technology, Department of Astronomy, 1200 E.
California Blvd, Pasadena, CA, 91125, USA
2Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
3 Department of Astronomy, University of California Berkeley, Berkeley, CA
94720, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present follow-up spectroscopy of 21 cataclysmic variables (CVs) with
evolved secondaries and ongoing or recently-terminated mass transfer.
Evolutionary models predict that the secondaries should have anomalous surface
abundances owing to nuclear burning in their cores during their main-sequence
evolution and subsequent envelope stripping by their companion white dwarfs.
To test these models, we measure sodium (Na) abundances of the donors from the
Fraunhofer "D" doublet. Accounting for interstellar absorption, we find that
all objects in our sample have enhanced Na abundances. We measure 0.3
$\lesssim$ [Na/H] $\lesssim$ 1.5 dex across the sample, with a median [Na/H] =
0.956 dex, i.e., about an order of magnitude enhancement over solar values. To
interpret these values, we run MESA binary evolution models of CVs in which
mass transfer begins just as the donor leaves the main sequence. These
generically predict Na enhancement in donors with initial donor masses
$\gtrsim 1\,M_{\odot}$, consistent with our observations. In the models, Na
enrichment occurs in the donors’ cores via the NeNa cycle near the end of
their main-sequence evolution. Na-enhanced material is exposed when the
binaries reach orbital periods of a few hours. Donors with higher initial
masses are predicted to have higher Na abundances at fixed orbital period
owing to their higher core temperatures during main-sequence evolution. The
observed [Na/H] values are on average $\approx$0.3 dex higher than predicted
by the models. Surface abundances of evolved CV donors provide a unique
opportunity to study nuclear burning products in the cores of intermediate-
mass stars.

###### keywords:

binaries: close – white dwarfs – novae, cataclysmic variables – binaries:
spectroscopic

††pubyear: 2023††pagerange: Sodium enhancement in evolved cataclysmic
variables–F

##  1 Introduction

Cataclysmic variables (CVs) are binary systems in which a non-degenerate star
(the “donor” or “secondary”) transfers mass to a white dwarf (WD) companion
through stable Roche lobe overflow (see Warner, 1995, for a review). CV
evolution is fundamentally governed by angular momentum loss (AML), which
shrinks CV orbits and drives mass transfer. The strength and dominant
mechanism of AML in CVs is still uncertain, and varies with orbital period. In
the standard evolutionary model (e.g. Knigge et al., 2011), AML at orbital
periods $P_{\rm orb}\gtrsim 3$ h is driven primarily by magnetic braking,
while gravitational wave radiation dominates at shorter periods. Weakening of
magnetic braking at $P_{\rm orb}\lesssim 3$ h is proposed to give rise to the
CV “period gap”; i.e. the observed lack of CVs with orbital periods between 2
and 3 hours (e.g. Rappaport et al., 1983; Knigge, 2006; Knigge et al., 2011;
Inight et al., 2021; Pala et al., 2022).

The vast majority of CV donors fall on a tight “donor sequence”, in which
physical parameters such as mass, radius, temperature, and spectral type
depend primarily on orbital period (e.g. Beuermann et al., 1998; Smith &
Dhillon, 1998; Knigge, 2006; Abrahams et al., 2020). In the standard
evolutionary model, CV donors evolve along this sequence from long to short
periods as their donors are whittled down by mass transfer. Objects on the
donor sequence have temperatures and radii similar to main-sequence stars of
the same mass. Although mass transfer causes some radius inflation relative to
main-sequence stars, this inflation is modest because objects on the donor
sequence are only mildly out of thermal equilibrium (e.g. Knigge, 2006; Knigge
et al., 2011).

A number of CVs have been discovered over the years with physical parameters
that differ significantly from the standard donor sequence. Observationally,
these systems stood out from the bulk of the CV population for having
unusually warm and luminous donors compared to normal CVs at the same orbital
period (e.g. Thorstensen et al., 2002a, b; Thorstensen, 2013; Rebassa-
Mansergas et al., 2014). Several of the same systems also have infrequent
outbursts and faint disks, suggesting low mass transfer rates. Evolutionary
models predict that such “evolved CVs” can form if mass transfer began when
the secondary star was near the end of its main sequence lifetime and thus had
a core enhanced in helium (e.g. Podsiadlowski et al., 2003). At the short
periods where they are observed, the donors in these systems have semi-
degenerate helium cores and thick hydrogen-burning envelopes. This leads them
to have quite different physical parameters from ordinary CV donors, which
consist mostly of hydrogen.

Evolved CVs are predicted to follow qualitatively different evolutionary
pathways from ordinary CVs. Their donor’s cores remain radiative (as opposed
to donors in normal CVs which become fully convective) so they are not
expected to detach at $P_{\rm orb}\approx 3$ hours as a result of weakened
magnetic braking. Instead, this can occur at a wide range of orbital periods
when the donors lose enough of their convective envelopes at $T_{\rm
eff}\gtrsim 6500\,\rm K$. These donors are predicted to detach from their
Roche lobes and be observable as extremely low mass (ELM) WDs in close
binaries (Sun & Arras, 2018; Li et al., 2019; El-Badry et al., 2021b).

Evolved CVs provide one possible formation channel for the AM Canum
Venaticorum binaries (AM CVns; Tutukov et al., 1985; Podsiadlowski et al.,
2003; Kalomeni et al., 2016), which are ultra-compact binaries with helium
donors and $P_{\rm orb}\lesssim 70$ minutes, below the minimum period of
normal CVs. AM CVns are an especially interesting outcome of evolved CV
evolution since they will be among the loudest sources of gravitational waves
for LISA (Breivik et al., 2018; Kupfer et al., 2018). At longer periods, CVs
with evolved donors can have higher mass transfer rates than ordinary CVs and
can be observed as supersoft X-ray sources (e.g. Li & van den Heuvel, 1997;
Schenker et al., 2002). These high accretion rates can lead to stable burning
on the surface of the accretion WD and may provide one channel for Type Ia (or
.Ia) supernovae (Livne, 1990; Bildsten et al., 2007; Nomoto et al., 2007; Wolf
et al., 2013; Brooks et al., 2015).

Because evolved CV donors completed most of their main-sequence evolution
before the onset of mass transfer, their photospheres – which contain material
previously inside their donors’ convective cores – are expected to have
unusual surface abundances. The first observed evolved CVs, EI Psc and QZ Ser,
bore out this prediction, with optical spectra that hinted at helium (He) and
sodium (Na) enhancement (Thorstensen et al., 2002b, a). Ultraviolet
spectroscopy soon showed that the same systems were enhanced in nitrogen and
deficient in carbon, as expected if material on the donors’ surface was
previously processed by CNO burning (Haswell et al., 2002; Gänsicke et al.,
2003; Toloza et al., 2022). Studies based on infrared spectroscopy have
similarly reported carbon deficits (Harrison, 2016, 2018).

A large sample of CVs with evolved donors was presented by El-Badry et al.
(2021b, hereafter E21). Using light curves from the Zwicky Transient Facility
(ZTF; Bellm et al. 2019), they identified 21 binaries with $P_{\rm orb}<6$ h
exhibiting large-amplitude ellipsoidal variability and falling below the main-
sequence in the color-magnitude diagram. The donors in these systems have
comparable temperatures and surface gravities to main-sequence A and F stars,
but significantly lower masses and smaller radii. Using low-resolution follow-
up spectroscopy, they showed that some objects in the sample have ongoing mass
transfer, while others likely just recently became detached. They noted hints
of excess Na absorption for all donors with $T_{\rm eff}\lesssim$ 7000 K,
suggesting potential Na enhancement (at higher temperatures, more of the Na is
ionised so neutral lines in the optical become weaker).

A literature search indeed reveals several other scattered reports of Na
enhancement in the donors of evolved CVs (e.g. Thorstensen et al., 2002b;
Thorstensen, 2013; Harrison, 2018; Green et al., 2020; Zhang et al., 2022).
There has, however, been little systematic study or quantitative evolutionary
predictions of the expected enhancement. In contrast to CNO lines, Na has
strong optical lines, namely the 5900Å doublet (i.e. Fraunhofer D-lines, at
5890 and 5896Å) and the 8200Å doublet (at 8183 and 8195Å), making it readily
accessible in evolved CVs (where the donor, rather than the disk, dominates in
the optical). Na is produced in advanced H burning processes which can only
take place in the high-temperature interiors of stars with masses $M\gtrsim 1\
M_{\odot}$, making it a potential diagnostic of the initial masses of the
donors in evolved CVs.

In this paper, we follow up on the work of E21 and obtain higher-resolution
and higher-SNR spectra of their targets to look for and quantify potential
excesses in Na absorption lines compared to model spectra. We also use the
Modules for Experiments in Stellar Astrophysics (MESA) to model binary
evolutionary scenarios that give rise to evolved CVs and test whether they are
able to predict the observed abundances. In Section 2, we describe our
spectroscopic observations as well as the subsequent data reduction and
calculations of several parameters of our objects using the spectra. In
Section 3, we explain the process of generating a grid of model stellar
spectra with different chemical abundances. By comparing equivalent widths of
Na lines in the model spectra to our data, we infer the Na abundances of our
objects. We then describe and present the results of our MESA models in
Section 4.

##  2 Spectroscopic observations

We observed all 21 evolved CVs from E21 across 4 nights. A detailed observing
log can be found in Appendix A.

We observed 20 targets with the Echellette Spectrograph and Imager (ESI;
Sheinis et al. 2002) on the Keck II telescope in echellette mode. For the
majority of these observations, we used the 0.5” slit with a 600s exposure
which yielded spectral resolution $R\sim 9300$ and a typical SNR $\sim$ 20 per
pixel. A few targets were observed with other ESI slits; see Table 4 for
details. All ESI spectra were reduced using the MAuna Kea Echelle Extraction
(MAKEE) pipeline, which performs bias-subtraction, flat fielding, wavelength
calibration, sky subtraction, and a flexure correction to the wavelength
solution using sky lines.

Due to poor weather conditions affecting several observing runs and
observability constraints during the year, we observed one object, P_2.74a,
with the Low Resolution Imaging Spectrometer (LRIS; Oke et al. 1995) on the
Keck I telescope. We observed the binary over a full orbit, with 63 (red) and
78 (blue) $\times$ 90 second exposures and a 1.0” slit (see Appendix A), but
most of the abundance analysis focuses on a single exposure, taken at
quadrature. The red and blue arms covered a wavelength range of $\sim$ 3150 to
5600Å and 5400 to 10290Å, respectively. The data was reduced using LPipe
(Perley, 2019). This provided a significantly lower resolution of $R\sim 1540$
compared to the ESI spectra but as we are primarily concerned with the
calculation of equivalent widths, this should be acceptable to complete the
sample.

Cutouts of the spectra for all objects can be found in Figures 1, 3, and 11,
which respectively show spectral regions containing the Mg I triplet, the Na D
doublet, and the H$\alpha$ line.

The phases at which the spectra were taken could be deduced from the recorded
mid-exposure time and the orbital ephemeris obtained from light curve fitting,
summarized in Table 4 of E21.

E21 also inferred the effective temperatures of all the targets from SED
fitting of UV-to-IR photometry from Pan-STARRS, WISE, and 2MASS, using the
results from fitting their low-resolution spectra as a weak prior. These
temperatures, as well as several other relevant parameters from the same
paper, are summarized in Table 1.

ID | Gaia eDR3 ID |  $P_{\rm orb}$ [days] |  $T_{\rm eff}$ [K] |  $R_{\rm donor}$ [$R_{\odot}$] |  $i$ [deg]
---|---|---|---|---|---
P_2.00a | 4393660804037754752 | 0.08331301(2) | 7734 $\pm$ 73.0 | 0.17${}^{+0.01}_{-0.01}$ | 55.49${}^{+14.14}_{-5.33}$
P_2.74a | 861540207303947776 | 0.11414278(4) | 4726 $\pm$ 52.0 | 0.23${}^{+0.01}_{-0.01}$ | 79.76${}^{+7.0}_{-6.75}$
P_3.03a | 1030236970683510784 | 0.12642666(6) | 5910 $\pm$ 77.0 | 0.27${}^{+0.03}_{-0.03}$ | 73.10${}^{+11.29}_{-9.92}$
P_3.06a | 2133938077063469696 | 0.1274832(2) | 5862 $\pm$ 70.0 | 0.27${}^{+0.03}_{-0.02}$ | 79.49${}^{+7.09}_{-6.61}$
P_3.13a | 4228735155086295552 | 0.13056512(8) | 6662 $\pm$ 54.0 | 0.28${}^{+0.01}_{-0.01}$ | 83.66${}^{+4.65}_{-2.75}$
P_3.21a | 45968897530496384 | 0.13375228(7) | 7022 $\pm$ 72.0 | 0.31${}^{+0.02}_{-0.02}$ | 71.56${}^{+7.29}_{-3.18}$
P_3.43a | 184406722957533440 | 0.1429875(2) | 6193 $\pm$ 83.0 | 0.26${}^{+0.01}_{-0.01}$ | 79.21${}^{+7.31}_{-6.44}$
P_3.48a | 4002359459116052864 | 0.14479504(6) | 6444 $\pm$ 62.0 | 0.26${}^{+0.02}_{-0.02}$ | 73.90${}^{+11.05}_{-10.57}$
P_3.53a | 1965375973804679296 | 0.146883(3) | 5324 $\pm$ 66.0 | 0.32${}^{+0.01}_{-0.01}$ | 79.66${}^{+6.96}_{-6.67}$
P_3.81a | 373857386785825408 | 0.1586339(7) | 6689 $\pm$ 72.0 | 0.35${}^{+0.07}_{-0.05}$ | 78.34${}^{+7.76}_{-7.71}$
P_3.88a | 1077511538271752192 | 0.1615714(3) | 5875 $\pm$ 102.0 | 0.28${}^{+0.02}_{-0.01}$ | 79.64${}^{+6.96}_{-6.6}$
P_3.90a | 3053571840222008192 | 0.1624549(1) | 7442 $\pm$ 87.0 | 0.33${}^{+0.01}_{-0.01}$ | 64.85${}^{+10.19}_{-4.42}$
P_3.98a | 896438328413086336 | 0.16582294(9) | 5122 $\pm$ 53.0 | 0.40${}^{+0.03}_{-0.03}$ | 70.36${}^{+13.16}_{-12.32}$
P_4.06a | 4358250649810243584 | 0.16898413(4) | 7587 $\pm$ 72.0 | 0.28${}^{+0.01}_{-0.01}$ | 59.48${}^{+12.19}_{-5.3}$
P_4.10a | 3064766376117338880 | 0.1708102(4) | 4846 $\pm$ 38.0 | 0.37${}^{+0.01}_{-0.01}$ | 79.51${}^{+7.05}_{-6.58}$
P_4.36a | 2126361067562200320 | 0.181743(2) | 5998 $\pm$ 80.0 | 0.39${}^{+0.02}_{-0.02}$ | 74.15${}^{+10.74}_{-10.16}$
P_4.41a | 2171644870571247872 | 0.1838691(2) | 7013 $\pm$ 105.0 | 0.35${}^{+0.01}_{-0.01}$ | 64.02${}^{+10.6}_{-3.9}$
P_4.47a | 1315840437462118400 | 0.18604299(9) | 7171 $\pm$ 108.0 | 0.36${}^{+0.07}_{-0.05}$ | 73.03${}^{+6.83}_{-2.77}$
P_4.73a | 2006232382792027904 | 0.1969063(2) | 7619 $\pm$ 85.0 | 0.38${}^{+0.01}_{-0.01}$ | 57.45${}^{+12.3}_{-5.32}$
P_5.17a | 4382957882974327040 | 0.2153735(2) | 6187 $\pm$ 61.0 | 0.37${}^{+0.03}_{-0.03}$ | 67.57${}^{+14.9}_{-11.87}$
P_5.42a | 286337708620373632 | 0.2258108(1) | 7326 $\pm$ 78.0 | 0.40${}^{+0.01}_{-0.01}$ | 68.34${}^{+8.65}_{-4.01}$
Table 1: Physical parameters of the 21 objects, combining information from
Tables 1, 2, and 5 of E21. The “ID”s are from E21, with the numbers
corresponding to the orbital period in hours. $P_{\rm orb}$ is the orbital
period in days from ZTF light curves and confirmed by spectral fitting;
$T_{\rm eff}$ is the effective temperature of the donor from SED fitting;
$M_{\rm donor}$, $R_{\rm donor}$, and $i$ is the mass of the donor, its
radius, and the minimum inclination respectively, obtained from a combined
fitting using the Markov chain Monte Carlo (MCMC) method for properties of
both the donor and WD companion.

###  2.1 Spectra and radial velocities

For each spectrum, the radial velocity was obtained using the cross-
correlation function (CCF) method (Tonry & Davis, 1979), which we implemented
as described in Appendix A of Zhang et al. (2021). This involved shifting the
spectra in log space of wavelength with a range of radial velocities
corresponding to -600 and 600 km/s and finding the shift that maximized the
CCF when compared to a standard stellar model spectrum. For measuring RVs, we
used the wavelength range 4900-5400Å, which includes the MgI triplet (5167,
5173, 5184Å), shown for all objects in Figure 1. For the model spectra, we
used BOSZ Kurucz models (Bohlin et al., 2017) with solar abundances and log(g)
and $T_{\rm eff}$ values close to those of the target. We retrieved models
with $R=50,000$ and broadenend them to the resolution of each observed
spectrum. Instrumental broadening was carried out using convolution with a 1D
Gaussian filter kernel (Astropy Collaboration et al., 2022) with FWHM obtained
from a gaussian fit to the [OI]5577Å line in the sky spectra.

We inferred projected rotation velocities $v\sin i$ for the donor in each
system from the observed spectra (Section 2.2), but at this initial stage, we
simply set $v\sin i=(2\pi R/P_{\rm orb})\sin(i)$ using values from Table 1.
Rotational broadening was implemented using the rotBroad function (as
originally described by Gray, 1992) from PyAstronomy (Czesla et al.,
2019),with linear limb-darkening coefficients taken from Claret & Bloemen
(2011). RV errors were obtained by calculating chi-square
($\chi^{2}=\sum\left(F_{\rm obs}-F_{\rm model}\right)^{2}/{\sigma_{\rm
obs}^{2}}$) values between the observed and model spectra at each shift (for
the wavelength range considered) and identifying the points at which
$\chi^{2}$ increased by 1 from the minimum value. We note that while the
radial velocity that minimizes $\chi^{2}$ is not precisely the same as that
which maximizes the CCF, they typically only differ by $\lesssim 1$ km/s and
the calculated errors should still provide a reasonable estimate of the
statistical errors.

Figure 1: The Mg I triplet for all objects, plotted with their corresponding
model spectra, used in the cross-correlation to obtain RVs. For four objects
(P_3.13a, P_3.21a, P_3.43a, P_3.90a), the triplet is noticeably deeper than
the standard solar abundance models and fits better to those with He mass
fraction $Y=0.868$. These are used later in the calculation of abundances.

Using the RV ephemerides from E21, we can also calculate the expected RV of
the donor at the orbital phase of our observed spectra. We compare these to
the measured RVs in Table 2. The typical disagreement between measured and
predicted RVs is about 30 km/s: small compared to the RV semi-amplitudes, but
large compared to the formal uncertainties. We suspect that the main reason
for disagreement between the measured and predicted RVs is that the RV
zeropoint of the low-resolution spectra obtained by E21 was only stable at the
$\sim 50$ km/s level (see their paper for details), leading to a systematic
uncertainty in the binaries’ center-of-mass RVs. The RV zeropoint for the ESI
spectra analyzed in this work is stable at the few km/s level, so we expect
our RVs to be more reliable than the predictions from E21.

ID | RVspec [km/s] | RVphase,E21 [km/s]
---|---|---
P_2.00a | -326.9${}^{+5.2}_{-5.3}$ | -397.4 $\pm$ 10.8
P_2.74a | -281.7${}^{+3.3}_{-2.8}$ | -395.9 $\pm$ 15.6
P_3.03a | -102.0${}^{+2.6}_{-2.0}$ | -64.7 $\pm$ 10.4
P_3.06a | 9.0${}^{+1.4}_{-1.3}$ | 14.8 $\pm$ 9.0
P_3.13a | -358.6${}^{+5.7}_{-5.8}$ | -318.4 $\pm$ 18.4
P_3.21a | -216.8${}^{+1.8}_{-2.1}$ | -294.8 $\pm$ 12.5
P_3.43a | -279.7${}^{+2.1}_{-2.0}$ | -236.5 $\pm$ 9.2
P_3.48a | 273.0${}^{+1.8}_{-1.0}$ | 244.4 $\pm$ 5.4
P_3.53a | 216.0${}^{+8.1}_{-8.2}$ | 160.0 $\pm$ 29.2
P_3.81a | -201.9${}^{+2.4}_{-2.0}$ | -229.1 $\pm$ 5.1
P_3.88a | 134.0${}^{+1.3}_{-1.2}$ | 174.2 $\pm$ 8.0
P_3.90a | -85.0${}^{+1.8}_{-2.5}$ | -133.6 $\pm$ 3.4
P_3.98a | 35.0${}^{+3.9}_{-3.4}$ | -17.6 $\pm$ 8.3
P_4.06a | -205.9${}^{+0.7}_{-1.5}$ | -225.9 $\pm$ 8.6
P_4.10a | 208.0${}^{+4.1}_{-4.6}$ | 197.9 $\pm$ 24.9
P_4.36a | 235.0${}^{+2.5}_{-3.2}$ | 196.9 $\pm$ 7.5
P_4.41a | -34.0${}^{+2.6}_{-2.3}$ | -101.8 $\pm$ 3.0
P_4.47a | 214.3${}^{+30.3}_{-21.1}$ | 199.5 $\pm$ 17.5
P_4.73a | -313.7${}^{+1.9}_{-2.7}$ | -329.6 $\pm$ 6.7
P_5.17a | -165.9${}^{+1.1}_{-1.5}$ | -193.3 $\pm$ 11.0
P_5.42a | 107.0${}^{+4.1}_{-3.7}$ | 112.2 $\pm$ 6.3
Table 2: Radial velocities of all objects. RVspec are the results from CCF
maximization using the spectra obtained in this work; RVphase,E21 are those
calculated using the best fit orbital solutions from Table 3 of E21. We find
typical disagreements between the two RVs of $\sim 30$ km/s which can likely
be attributed to the improved stability in the RV zeropoints of the ESI
spectra compared to the lower resolution spectra from E21.

###  2.2 Projected Rotational Velocity

As mentioned in 2.1, we can get an expected value for the projected rotational
velocity by using the values of radius, period, and inclination from E21.
However, this is only an estimate. In particular, the radii are obtained using
fits to the spectral energy distributions (SEDs) while neglecting possible
contamination from the companion or the accretion disk. Though comparisons to
model spectra do not suggest significant contamination for most targets,
strictly speaking, this makes the derived radii upper limits.

Therefore, we also measured best-fit $v\sin i$ values directly from the
rotational broadening of the observed spectra. For each target, this was done
by first calculating radial velocities using model spectra rotationally
broadened with a range of $v\sin i$ values from 50 to 150 km/s and shifting
the target spectra with the derived RV. Then, the $\chi^{2}$ value was
calculated by comparison with each model and the $v\sin i$ of the
corresponding model that minimized the $\chi^{2}$ was identified. The errors
were obtained by finding the location for which $\chi^{2}$ increased by one.
Note that some of the parameters from E21 used to calculate the $v\sin i$
values have asymmetric errors. The errors on these have been approximated
through standard error propagation using averages of the upper and lower
values. The differences in the $v\sin i$ values from the two calculations are
shown in Figure 2.

We find that for three objects, the chi-square method did not result in clear
local minima, instead monotonically decreasing towards larger $v\sin i$. These
are marked in red in the figure, placed at the zero line (as the calculated
differences have little meaning), and given arbitrarily large error bars for
emphasis. This result is not unexpected given the particularly weak lines of
P_2.00a and P_4.47a as well as the broad lines of P_2.74a due to a lower
resolution spectrum, all of which can be seen in Figure 1. Meanwhile, the rest
of the objects have $v\sin i$ values that are in agreement to within about
$\pm$ 10 km/s on average, suggesting that the radius and inclination
constraints from from E21 are reliable.

Figure 2: Difference in the $v\sin i$ calculated from $2\pi R/Psin(i)$ using
the values derived in E21 and that calculated with $\chi^{2}$ minimization.
The points in red indicate objects for which no good solution was found using
the $\chi^{2}$ method - they have been placed at 0 and their error bars have
been arbitrarily extended to emphasize this. We see that most objects show an
agreement of the two values to $\sim$ 30 km/s, suggesting that orbital
parameters from E21 are reasonable estimates.

###  2.3 Equivalent Widths

In order to measure the Na abundance in a donor, we first calculate the
equivalent width (EW) of its absorption lines in the spectrum, in particular
the 5900Å doublet. Later, we repeat this process for many model spectra
generated over a grid of abundances and make comparisons between the EWs of
the models and the data (see Section 3).

To estimate the local continuum, the lines themselves were masked out, then a
range 60Å above and below the center of the doublet (120Å for the one LRIS
spectrum) was selected and fitted with a first order polynomial. The integral
under the lines, $\pm$15Å around the center ($\pm$30Å for LRIS), was
calculated numerically using the trapezoidal rule and subtracted from the area
under the continuum over the same wavelength range to get the equivalent width
(EW). Figure 3 shows these regions around the doublet for all of our objects.
It is already clear by eye that for most targets, the observed line is
significantly deeper and broader than that of the corresponding model spectrum
with solar abundance.

Figure 3: The Na 5900Å doublet for all targets (for one of the two orders in
which it is found). The region of the spectra over which the continuum is
calculated is in gray, with the resulting first order polynomial fit in the
black dashed line. The orange is the region considered to be occupied by the
doublet and that under which the integrals, and thus EW, is calculated. A
model spectra with $T_{\rm eff}$ matching each target (with [Na/H] = 0.0 dex)
is also shown in blue for reference. The main takeaway from this figure is
that the observed lines are noticeably deeper than those of the solar
abundance model, indicating enhancement.

The error on the EW for each target was obtained by calculating the EWs of the
same line in 100 Monte Carlo realizations of the observed spectra and taking
their standard deviation. These were generated using the normalized flux of
the original spectrum and its error as the standard deviation of a Gaussian
distribution from which a random sample was drawn at each wavelength.

There are overlaps in the wavelength coverage towards the edge of each order
for the ESI spectra. This means that some lines may be found in two orders,
which is the case for the 5900Å doublet. This allowed us to make the same
calculations for the two orders and take their inverse-variance weighted
average. For the one LRIS spectra (P_2.74a), the EW was calculated just using
the blue side where the line is located and no average was taken. Figure 4
plots the EWs of the 5900Å doublet for all objects against their IDs and
temperatures, for each of the two orders as well as their average. We see that
there is a clear downward trend with increasing temperature: the lines become
weaker at high $T_{\rm eff}$, where most of the Na is ionized. The EW values
are also listed on Table 3, along with the derived Na abundances from
comparison with models, described in Section 3.3.

####  2.3.1 Accounting for interstellar or circumbinary absorption

For many of the spectra, there were narrow and deep interstellar lines
superimposed on top of the broad Na lines from the donors. Though they do not
occupy a large area in most cases, to minimize their effect on our EW
calculations, we cut out any data points $\pm$ 0.6Å around the rest
wavelengths of the doublet (as the interstellar lines have very small RV
shifts) and replaced them with a median of several points above and below this
range. This process effectively blocks out any obvious interstellar Na
contribution which are present for a few objects (See Appendix C). This was
not done for the LRIS spectrum because the resolution was too low that the
doublet itself, and the interstellar lines, were not resolved and there were
no distinguishable features on the single broad absorption line.

In addition to absorption from the interstellar medium (ISM), it is in
principle possible that there is additional Na absorption due to circumbinary
material in the immediate vicinity of the targets. We explore this possibility
– and conclude that it is unlikely, because all the observed Na lines are
broad and RV variable, tracking the donor – in Appendix D.

Figure 4: EWs of the Na 5900Å doublet for all objects, plotted against their
IDs (i.e. their orbital periods, _left_) and their temperatures (_right_). The
gray points with the circle and triangle markers are the EWs of the line in
each of the two orders while the red points with the star markers takes their
weighted average. The solid, dash-dot, and dotted blues lines on the figure to
the right plots the EW against temperature of model spectra with [Na/H] = 0.0,
1.0, and 1.3 dex respectively. All objects lie above the solar abundance line,
meaning that they all show some level of enhancement, while more than half
show $\gtrsim$ an order-of-magnitude enhancement.

##  3 Analysis

###  3.1 Kurucz model spectra

To infer abundances from the observed spectra, we calculated a grid of 1D LTE
model atmospheres and synthetic spectra for a range of effective temperatures
and Na abundances, which we subsequently compared to the observed spectra. We
used ATLAS 12 (Kurucz, 1970, 1979, 1992) to compute the atmosphere structure
and SYNTHE (Kurucz, 1993) for the radiative transfer calculations, self-
consistently re-computing the atmosphere structure for each model spectra. We
use the linelist maintained by R.
Kurucz111http://kurucz.harvard.edu/linelists.html and assumed a microturbulent
velocity of $2\,\rm km\,s^{-1}$. We generated spectra at resolution
$R=300,000$ and applied instrumental and rotational broadening to match the
observed data.

We assumed a surface gravity $\log\left[g/\left({\rm
cm\,s^{-2}}\right)\right]=4.8$ dex for all models, as the expected value for
all the binaries in our sample is within 0.2 dex of this value. We generated
models with a spacing of 100 K in $T_{\rm eff}$, ranging from 4500 to 7700 K,
and a spacing of 0.1 dex in [Na/H], ranging from -0.1 to 1.3 dex. Our fiducial
calculations assumed the solar abundance pattern for elements besides Na. As
described below, we also generate several grid with He enrichment to explore
its effect on the Na lines.

###  3.2 Helium enhanced models

From Figure 1, we see that for four objects in particular - P_3.13a, P_3.21a,
P_3.43a, P_3.90a - the observed Mg lines are noticeably deeper than those of
the model with solar abundances. The same is true for other metal lines in
these objects.

One possible explanation for this is that the Teff values inferred from SED
fitting are overestimated. A higher Teff means more of the Mg will be ionized,
thus weakening its neutral lines. To explore this possibility, we compared the
observed spectra with several models with lower Teff and found that models
cooler by $\approx 750$ K better matched the depth of the observed Mg
triplets. However, these lower temperatures (with a corresponding increase in
radius) resulted in a much worse fit to the broadband SEDs (Figure 9 in E21),
even when plausible light contributions from the accreting WD and/or disk were
taken into account. We thus consider it unlikely that a too-cool assumed
$T_{\rm eff}$ is the reason for the deeper metal lines in these objects.

An alternative explanation for the deeper-than-expected Mg lines in these
sources is surface helium enhancement of the donors. Evidence of enhancement
has been observed in the donors of some other evolved CVs (e.g. Harrison,
2018), and some enhancement is also predicted by evolutionary models (Section
4). Lending support to the helium enhancement hypothesis, the same objects
that have deeper-than-expected Mg lines also have deeper He I lines at 5876Å
than predicted by the fiducial spectral models (Figure 3).

To analyze these objects and assess the sensitivity of Na abundance
measurements to the assumed helium abundance, we generated several grids of
models (with the same spacing in $[Na/H]$ and $T_{\rm eff}$ as before) with
surface He mass fractions, $Y$, of 0.569 and 0.868 (compared to the solar
value of 0.254). We plot the models with $Y=0.868$ in Figure 1 for the four
objects and we see that these have deeper Mg I triplet lines than the model
with $Y=0.254$, in better agreement with the observed spectra. Thus, we carry
out the calculation of Na abundances for these objects in Section 3.3 using
these He enhanced models.

For the remaining objects, the good agreement of the observed spectra and
models with solar helium abundances rules out strong helium enhancement, so we
use the solar models. However, it is possible that some modest enhancement
could go unrecognized. We explore how unaccounted-for He enhancement would
affect our inferred Na abundances in Appendix E, where we conclude that the
uncertainty in our inferred Na abundances associated with possible He
enhancement is typically about 0.1 dex.

###  3.3 Abundance calculations

We calculate the EWs of the same Na lines for all model spectra with different
abundances at each $T_{\rm eff}$. This is plotted on Figure 5. At all $T_{\rm
eff}$, we find that with more Na (i.e. higher [Na/H]), the EW of its line is
greater. However, the sensitivity of this relation depends strongly on the
$T_{\rm eff}$ whereby the overall slope gets less steep at higher temperatures
as more of the Na is ionized.

For each object, we derive the relation between EW and abundance at its
temperature $T_{\rm eff,0}$ using interpolation between the relations for two
models lying above and below $T_{\rm eff,0}$. The corresponding abundance is
then obtained at the EW of the line measured from its spectra. The error is
calculated by propagating the errors in the EW and $T_{\rm eff}$. We note that
the uncertainty in $T_{\rm eff}$ obtained from SED fitting (listed on Table 2
of E21, ranging from $\sim 50-100$ K) is likely underestimated due to various
systematics so we instead take it to be $100$ K for all objects.

The resulting plots are shown for all objects in Figure 6. Green plots
indicate that that the objects are being compared to He enhanced models with
$Y=0.868$, as discussed in Section 3.2. The abundances with their errors are
also summarized in Table 3. We see that all objects show some level of Na
enhancement with [Na/H] values ranging from $\sim$ 0.4 - 1.6 dex, with a
median value at a significant 1.024 dex i.e. about an order of magnitude
enhancement compared to solar value.

Figure 5: EWs of the Na 5900Å doublet against the Na abundances for model
spectra with a range of effective temperatures. At a given $T_{\rm eff}$, we
see that EW increases monotonically with [Na/H]. The slope gets shallower at
higher $T_{\rm eff}$ as more Na gets ionized. Figure 6: Plots of EW against
[Na/H] obtained by interpolating those in Figure 5 to the temperature of each
object. The red lines indicate extrapolation, where the calculated EW of the
object is above the maximum value that was generated. The blue and green lines
corresponds to models with solar and enhanced He mass fractions. Most objects
have [Na/H] ranging from $\sim$ 0.4 - 1.3 dex, with the exception of P_3.88
with a highly extrapolated value of 1.576 dex.

###  3.4 Non-LTE corrections

In generating the curves of growth (Figures 5 and 6) used to calculate the Na
abundances of our objects, we neglected deviations from the standard
assumption of local thermodynamic equilibrium (LTE) used in the model
atmospheres. It has been found that the inclusion of non-LTE (NLTE) effects
can increase the EWs of Na lines at fixed [Na/H] (e.g. Mashonkina et al.,
2000; Lind et al., 2011; Lind et al., 2022). To avoid overestimating the
abundances, we apply NTLE corrections to our abundances using the results from
Lind et al. (2022).

At the measured EWs of all objects (sum of the 5890 and 5896 Å lines), we
calculate the difference between the predicted NLTE and LTE abundances,
interpolated to their effective temperatures. We used values for [Fe/H] = 0.0
dex, log(g) = 5.0, and microturbulence of 2 km/s. The thus-inferred NLTE
corrections are shown as a function of temperature in Figure 7. The mean
correction is $-0.126$ dex, and the correction is more important at higher
temperatures. We add these corrections to our previously measured abundances
in Section 3.3. These “NLTE-corrected" abundances can be found on Table 3.
These values range from $\sim$ 0.3 - 1.5 dex, with a median value of 0.956
dex, and will be used when referring to [Na/H] in the rest of the paper.

We emphasize that these are approximate corrections, done to avoid overstating
the Na enhancements. The LTE curves of growth from Lind et al. (2022) are
similar but not identical to those shown in Figure 5 as the synthetic spectra
were not calculated with the same code. For several objects, indicated with
blue stars on Figure 7, our measured EWs exceeded the maximum value calculated
by Lind et al. (2022) (which corresponded to [Na/H]=0.7) so we simply used the
NLTE correction at the maximum value. However, as the absolute differences get
smaller towards larger EWs, this only results in more conservative values for
the final abundances. Lastly, it should be noted that as discussed in Section
3.2, some of our objects were modelled with He enhancement but this was not
taken into account in calculating the NLTE corrections.

Figure 7: The difference in the Na abundances implied from EWs of the Na 5900 Å doublet of synthetic spectra with and without NLTE effects, as a function of effective temperature for all objects. The blue stars indicate objects whose EWs were greater than the maximum value provided by Lind et al. (2022) for which the difference plotted is that at the maximum value. In all cases, we see that inclusion of NTLE effects leads to a reduction in the [Na/H], with a mean value of -0.126 dex, so neglecting them will result in an overestimate of the Na enhancements. ID | EW [Å] | [Na/H] | [Na/H]NLTE
---|---|---|---
P_2.00a | 0.808 $\pm$ 0.429 | 1.029 $\pm$ 0.526 | 0.777 $\pm$ 0.526
P_2.74a | 16.176 $\pm$ 0.251 | 1.335 $\pm$ 0.197 | 1.296 $\pm$ 0.197
P_3.03a | 4.043 $\pm$ 0.231 | 1.090 $\pm$ 0.105 | 1.016 $\pm$ 0.105
P_3.06a | 5.014 $\pm$ 0.174 | 1.275 $\pm$ 0.104 | 1.201 $\pm$ 0.104
P_3.13a | 4.177 $\pm$ 0.078 | 1.136 $\pm$ 0.076 * | 1.016 $\pm$ 0.076 *
P_3.21a | 3.074 $\pm$ 0.258 | 1.114 $\pm$ 0.108 * | 0.964 $\pm$ 0.108 *
P_3.43a | 3.723 $\pm$ 0.138 | 0.676 $\pm$ 0.085 * | 0.583 $\pm$ 0.085 *
P_3.48a | 1.936 $\pm$ 0.111 | 0.843 $\pm$ 0.096 | 0.736 $\pm$ 0.096
P_3.53a | 6.566 $\pm$ 0.098 | 1.014 $\pm$ 0.127 | 0.970 $\pm$ 0.127
P_3.81a | 2.102 $\pm$ 0.366 | 1.118 $\pm$ 0.177 | 0.995 $\pm$ 0.177
P_3.88a | 6.251 $\pm$ 0.166 | 1.576 $\pm$ 0.116 | 1.502 $\pm$ 0.116
P_3.90a | 1.941 $\pm$ 0.055 | 1.024 $\pm$ 0.078 * | 0.822 $\pm$ 0.078 *
P_3.98a | 6.298 $\pm$ 0.168 | 0.699 $\pm$ 0.142 | 0.653 $\pm$ 0.142
P_4.06a | 0.581 $\pm$ 0.142 | 0.608 $\pm$ 0.265 | 0.328 $\pm$ 0.265
P_4.10a | 7.424 $\pm$ 0.073 | 0.475 $\pm$ 0.162 | 0.442 $\pm$ 0.162
P_4.36a | 4.260 $\pm$ 0.169 | 1.220 $\pm$ 0.098 | 1.143 $\pm$ 0.098
P_4.41a | 0.834 $\pm$ 0.111 | 0.565 $\pm$ 0.141 | 0.397 $\pm$ 0.141
P_4.47a | 1.692 $\pm$ 0.990 | 1.316 $\pm$ 0.506 | 1.152 $\pm$ 0.506
P_4.73a | 0.833 $\pm$ 0.110 | 0.987 $\pm$ 0.143 | 0.756 $\pm$ 0.143
P_5.17a | 3.054 $\pm$ 0.259 | 1.048 $\pm$ 0.115 | 0.956 $\pm$ 0.115
P_5.42a | 0.908 $\pm$ 0.057 | 0.869 $\pm$ 0.091 | 0.668 $\pm$ 0.091
Table 3: Table of the EWs of the 5900Å doublet with the derived [Na/H] from
Figure 6. The [Na/H]NLTE has been corrected for NTLE effects using values from
Figure 7, calculated using curves of growth provided in Lind et al. (2022).
[Na/H] values were calculated using solar He abundance (Y = 0.254) models,
except those marked with an asterisk (*) which were calculated using He
enhanced (Y = 0.868) models. [Na/H] of objects range from $\sim$ 0.4 - 1.6
dex, with a median value of 1.026 dex. Meanwhile, [Na/H]NLTE range from $\sim$
0.3 - 1.5 dex, with the median at 0.956 dex.

##  4 MESA evolutionary models

We ran MESA models (Paxton et al., 2011; Paxton et al., 2013, 2015, 2018,
2019) to study the evolutionary history of the evolved CVs and to test whether
they predict the observed overabundance of Na. Similar to the models used in
E21 (described in greater detail in El-Badry et al. 2021a and summarized
here), we only follow the evolution of the donor/secondary and model the WD
companion/primary as a point mass. For the mass transfer rates, we use those
of optically thick Roche lobe overflow described by Kolb & Ritter (1990). We
consider fully non-conservative mass transfer meaning that all of the mass
transferred to the primary is instantaneously ejected from its vicinity as
fast winds, described by the $\beta$ parameter being set to 1 (note that MESA
uses $\beta$ as defined by Tauris & van den Heuvel 2006 which is 1-$\beta$ as
defined earlier by Rappaport et al. 1982). This is a reasonable assumption in
the case of CVs where WDs periodically lose accreted mass by classical novae
explosions on timescales short to those over which the orbit evolves. We
include the torque applied by magnetic braking as described by equation 36 of
Rappaport et al. (1983) and set the gamma exponent to 3. The strength of
magnetic braking in CVs at the periods of our sample are uncertain e.g. (e.g.
El-Badry et al., 2022), but the adopted magnetic braking law primarily affects
how quickly CVs evolve, not their abundances at a fixed evolutionary state.

We made several modifications compared to the MESA models described by E21.
Firstly, we used a newer version of MESA (r22.05.1) which called for changes
to the options in the ‘star’ module of the secondary. To initialize it as a
ZAMS star with solar abundances from Grevesse & Sauval (1998), we used the
set_uniform_initial_composition option and manually defined the initial mass
fractions of H, He, and metals (0.70, 0.28, 0.02 respectively). We also turned
on rotation with the surface velocity set by tidal synchronization (which
applies at the short orbital periods of our objects). Most importantly, MESA’s
default nuclear reaction network (basic.net) does not include Na and thus we
instead use the network sagb_NeNa_MgAl.net which follows 22 isotopes from 1H
to 27Al, including 21-23Na. It uses rates provided by the JINA reaclib
database V2.0 (Farmer et al., 2015). The transitions between the CNO, NeNa,
and MgAl cycles are depicted in Figure 1 of Boeltzig et al. (2016). The step
producing 23Na is proton capture by 22Ne or in shorthand notation,
22Ne(p,$\gamma$)23Na. The NeNa and MgAl cycles are often discussed in the case
of AGB stars undergoing Hot Bottom Burning (HBB) where they are activated at
the high temperatures reached at the base of the convective layer (Izzard et
al., 2007) as well as in novae outbursts (Jose et al., 1999), but 23Na is
generically produced as long as sufficiently high temperatures are reached. If
the NeNa cycle produces excess Na in the cores of CV donors during their main-
sequence evolution, the resulting Na enhancement can be observed on the stars’
surfaces after their outer layers are stripped.

We calculated models with initial donor masses of $1.0\,M_{\odot}$ and
$1.5\,M_{\odot}$, assuming a WD mass $M_{\rm WD}=0.7\,M_{\odot}$ in all cases.
For each $M_{\rm donor}$, we ran models over a grid of initial periods,
$p_{0}$ (ranging from $\sim 0.5-5$ days), to find the initial periods which
would produce systems with properties close to those of the evolved CVs in our
sample. We did not consider lower initial donor masses here because such
donors would take more than a Hubble time to complete their main-sequence
evolution and become evolved CVs. Meanwhile, donors with higher initial masses
would undergo a phase of thermal-timescale mass transfer shortly after initial
Roche lobe overflow, resulting in a high mass transfer rate and stable H
burning on the surface of the accreting WD (e.g. Schenker et al., 2002). This
would lead to an increase in the WD’s mass that would not be reliably followed
by our calculations, which assume mass transfer is fully non-conservative on
long timescales.

Figure 8: MESA calculations for initial donor masses of $1.0\ M_{\odot}$
(Left) and $1.5\ M_{\odot}$ (Right), for a range of initial periods, $p_{0}$.
The figures plot the mass, effective temperature, Roche-Lobe filling factor,
mass transfer rate, and remaining hydrogen mass of the donor/secondary star
against the orbital period. We also add the tracks for a normal CV with an
initial donor mass of $0.4\ M_{\odot}$ and $p_{0}=0.3$ days (black dashed
line). The effective temperatures of our objects are also plotted over the
tracks for reference (blue and green points). We see that these temperatures
are significantly higher than that of a normal CV. For the $1.0M_{\odot}$
model, the $p_{0}$ range which result in the temperatures of our objects
($\sim 4500-7500$ K) at their orbital periods ($\sim 2-6$ h) is $\sim 3.0-3.2$
days. For the $1.5\ M_{\odot}$ model, this range is $\sim 0.7-0.8$ days.
Figure 9: Surface abundances of 4He, 12C, 16O, 14N, and 23Na against orbital
period for the MESA models in Figure 8. The points in the plots of 23Na
abundance are those of the objects from this paper ([Na/H]NLTE from Table 3)
to serve as comparison to those predicted by the models (the blue and green
points indicate abundances calculating using solar He abundance (Y = 0.254)
and He enhanced (Y = 0.868) models, respectively). It is seen that while the
MESA models do predict significant Na enhancements, they generally
underpredict them compared to observed values.

Figure 8 shows the evolution of several basic parameters of the donor - the
mass, effective temperature, Roche lobe filling factor, mass loss rate, and
total hydrogen mass - as a function of the orbital period for a range of
initial periods. For reference, we also add tracks for a "normal" (i.e. not
evolved) CV (black dashed lines) with initial with $M_{\rm donor}=0.4\
M_{\odot}$ and $p_{0}=0.3$ days. With increasing $p_{0}$, the donor takes
longer to become Roche lobe filling so that it is more evolved at the onset of
mass transfer (although this figure is restricted to the short orbital periods
of our objects where most models are already mass-transferring i.e.
R/R${}_{\rm RocheLobe}\sim$ 1 and log $\dot{M}_{\rm donor}<$ 0, plotted on the
third and fourth panels respectively). Therefore, systems with a longer
$p_{0}$ end up reaching higher effective temperatures at short periods
$\lesssim 5$ h, as seen on the second panels. The last panels show that they
are also left with smaller total masses of remaining hydrogen as they have
undergone more H core burning while on the main sequence.

For initial $M_{\rm donor}=1.0\ M_{\odot}$, models with $p_{0}\gtrsim 2.5$
days remain mass transferring across the period gap ($P_{\rm orb}\approx 2-3$
hours, where the normal CV (black, dashed) detach due to the donor becoming
fully convective and magnetic braking weakening). This reflects the fact that
these donors are sufficiently evolved for their helium-enriched cores to
remain radiative so that they do not experience a disruption of magnetic
braking at $P_{\rm orb}\approx 3$ hours and thus have continuous mass
transfer. These objects also reach very short periods $\lesssim 1$ h where
they would be observed as AM CVn binaries (e.g. Podsiadlowski et al. 2003).
The model with $p_{0}=2.25$ days has a somewhat less evolved donor that does
detach from its Roche lobe when it becomes fully convective, but this only
occurs at $P_{\rm orb}\sim 2$ h – shorter than the 3 hour period where normal
CVs with fully unevolved donors detach. We see that while a larger range of
initial periods remain in contact across the gap, only the models with
$3.0\lesssim p_{0}\lesssim 3.2$ days reach the range of effective temperatures
at the observed periods of our sample, also plotted on the figure ($T_{\rm
eff}\sim 4500-7500$ K at $P_{\rm orb}\sim 2-6$). Beyond $p_{0}\approx 3.2$
days, the donor leaves the main sequence before mass transfer starts
(described further below for the $1.5\ M_{\odot}$, $p_{0}=0.85$ case).

For initial $M_{\rm donor}=1.5\ M_{\odot}$, the range of initial periods
forming evolved CVs that reach the temperatures of our objects is shorter and
narrower, from $\sim 0.7-0.8$ days. This is expected because more massive
stars are hotter and have thinner convective envelopes. This means that
magnetic braking is weaker so the stars must be initially closer together to
begin mass transfer as the donors leave the main sequence. Furthermore, they
have shorter lifetimes and evolve more quickly off the main sequence. The
$p_{0}=0.6$ model is a case where the donor is hardly evolved and thus its
evolution closely follows that of a normal CV. Meanwhile, the $p_{0}=0.85$
model represents a system that is above the“bifurcation limit” where the donor
has had enough time for it to ascend the giant branch by the time it begins
mass transfer (Podsiadlowski et al., 2003). By the time the system reaches the
period range of our objects, the donor has evolved into a WD and is on the
cooling track, with gravitational radiation slowing bringing it closer to its
companion until they are brought back in contact at very short periods. E21
also tested the effect of altering other initial parameters including the
metallicity and the exponential overshooting parameter but found that, given
the same degree of donor evolution at the onset of mass transfer, these
parameters only modestly change the mass of the donor and qualitative
evolutionary tracks at short periods.

Figure 9 shows surface abundances of the donor for several elements against
the orbital period. Once again, we add the plots for a normal CV in all panels
which remain approximately constant at 0.0 dex (i.e. solar) for all elements.
For the [Na/H] panel, we also plot points for the measured abundances of our
objects on top of the models.

We see that 4He, 14N, and 23Na are enhanced while 12C and 16O are depleted at
the observed periods. The He enhancement can be explained simply by the fact
that we are seeing the core of an evolved donor that has undergone significant
H burning and whose envelope has been stripped away as a result of the mass
transfer. This also supports the possibility that enhanced He may be what is
responsible in deepening of the magnesium lines (Section 3.2; we note that
24Mg is one stable element in the MgAl cycle - while its abundance was traced
in several models, there were no significant changes from the initial solar
values as a result of this process). The abundances of C, N, and O can be
explained by looking at the reactions in the CNO cycle of which
14N(p,$\gamma$)15O is the slowest. This results in a build up of 14N and a
corresponding exhaustion of C (and O to a lesser extent). As described
earlier, the Na enhancement is likely the result of advanced H burning which
only occur in the high temperatures of evolved stars. From the plots of
[Na/H], we see that the abundance at $P_{\rm orb}\sim 2-6$ h is sensitive to
the initial period. Thus, for a given initial donor mass, we are only able to
obtain a maximum predicted value of [Na/H] (corresponding to the longest
$p_{0}$ i.e. the most evolved model) at a specified orbital period. For
$M_{\rm donor}=1\ M_{\odot}$, this is at [Na/H] $\sim 0.6$ dex and for $1.5\
M_{\odot}$, this is greater at [Na/H] $\sim 0.7$ dex at $P_{\rm orb}\sim 4$ h.
Thus, a system with an initially more massive secondary (within the mass range
that leads to stable mass transfer) results in a greater Na enhancement at the
observed periods as it reaches higher temperatures and thus undergoes more
nuclear burning which produce more Na.

We see that a majority of the measured abundances of our objects lie above the
maximum values, so while we do expect a significant Na enhancement based on
the evolved CV models, they under-predict its magnitude compared to
observations. There are several possible reasons for this discrepancy.
Firstly, the rate of the 22Ne(p,$\gamma$)23Na reaction have particularly large
uncertainties compared to others in the NeNa cycle, stemming from
contributions from possible low-energy resonances whose existence and
properties have been debated in the literature (for further discussions on
this topic, refer to Izzard et al. 2007; Iliadis et al. 2010; Kelly et al.
2017). JINA reaclib V2.0 uses rates from Iliadis et al. (2010) (see their
Table B.20). Fortunately, since the dependence of the yield of 23Na on the
reaction rate is not linear and becomes relatively flat at fast rates as the
22Ne fuel gets used up, the magnitude of the uncertainty in the rate is not
directly carried over to the yield (see Izzard et al. 2007 for a more detailed
explanation). Nevertheless, the 23Na abundance calculated by MESA and
discussed throughout should only be taken to be accurate to within a factor of
a few.

Also, it is seen that irrespective of initial mass, the Na abundance plateaus
at $P_{\rm orb}\lesssim 2$ h. This is likely to be the result of a nuclear
statistical equilibrium being reached which prevents more Na from being
produced in the outer layers than is initially produced at the core when it
reaches the necessary temperature for the NeNa cycle to begin. This is further
discussed in Appendix F by looking at radial profiles of the abundances at
several points in the evolution.

Lastly, it is possible that our measured abundances are overestimated, which
can occur if the donor temperatures are underestimated or if there is He
enhancement that was unaccounted for in some objects, as discussed in Section
3.2 and Appendix E. Another possibility is that there are other physics that
we have neglected that furthers the Na enhancement of evolved CVs. We explore
one such process, namely rotational mixing, in the following section.

###  4.1 Rotational Mixing

Rotation effects in massive stars have been studied extensively in the
literature (see Maeder 1998 for an overview of the various physical processes
considered in models of rotating massive stars). Meanwhile, there has been
comparatively less work done on this topic in lower mass stars. One reason for
this is that the original treatment of mixing in rotating stars, the so-called
Eddington-Sweet circulation, has a characteristic timescale $\propto\tau_{\rm
KH}/\epsilon$ where $\tau_{\rm KH}$ is the Kelvin-Helmholtz timescale and
$\epsilon$ is the ratio of centrifugal to gravitational acceleration
(Eddington, 1925; Mestel & Moss, 1986). This timescale is short and thus this
process is very important for fast-rotating massive stars but less so for
solar-type stars. However, more recent studies have shown that its effects may
be significant even for lower mass stars (Palacios et al., 2003; Chatzopoulos
et al., 2012; Istrate et al., 2016). In particular, mixing may change stellar
abundances by allowing for advanced nuclear processing in a greater portion of
the star, or by counteracting the effect of gravitational settling so that
more heavier elements can be found in the outer layers (e.g. Istrate et al.
2016 used MESA models to study this effect in proto-ELM WD binaries and found
that it is important in their early stages). Furthermore, there can be
contributions from other types of mixing such as the Spruit-Taylor magnetic
diffusion (fluid motion resulting from toroidal magnetic field instabilities
in differentially rotating stars (Tayler, 1973; Spruit, 1999); e.g.
Chatzopoulos et al. 2012 considered this mechanism in explaining carbon
deficits observed in some compact object - solar-type binaries).

Therefore, we also ran a few models including rotational mixing. This is
implemented in MESA with am_D_mix_factor (i.e. the rotational diffusion
coefficient) which we set to the commonly used value of 1/30 from Heger et al.
(2000). We also set D_ES_factor and D_ST_factor to one, corresponding to the
inclusion of both the Eddington-Sweet and Spruit-Taylor mixing processes. As
an in-depth exploration of rotational effects is beyond the scope of this
paper, we just focus on a single comparison of models with mixing turned on
and off. Because the timescale of main sequence evolution is highly sensitive
to any changes in the internal structure of the stars, the inclusion of
rotational mixing has a significant effect on how evolved the donor is when
mass transfer begins. Thus, rather than comparing models with the same initial
periods, with and without mixing, we should compare models that have similar
trends in the effective temperature (i.e. are at similar evolutionary stages
at fixed orbital period).

Figure 10 plots the evolutionary tracks of reference (i.e. no mixing) models
with initial periods $p_{0}=2.9948$ and $3.15$ (also plotted on Figures 8 and
9), as well as a model again with $p_{0}=2.9948$ but with mixing. From the
panels on the left showing several basic donor parameters, we find that for a
given initial period, the inclusion of mixing pushes the donor towards
becoming more evolved in a shorter amount of time. In other words, the $T_{\rm
eff}$ curves of the $p_{0}=2.9948$ model is raised upwards to almost overlap
that of the $p_{0}=3.15$ model.

Now comparing the plots of surface abundances to the right of the reference
$p_{0}=3.15$ model against the $p_{0}=2.9948$ model with mixing, we see that
there is little difference between them across all elements. In particular,
there is no consistent rise in the Na abundance. Any small differences can be
explained by the fact that they are not identically evolved and as they are
smaller than most of the error bars in the measured abundances of our objects,
such differences are not enough to account for the discrepancy between the
models and observations. Thus, we find that given donors at a similar
evolutionary stage, mixing does not have a significant effect on the surface
abundances at the orbital periods of our interest.

Moreover, we note that we did not include element diffusion processes (e.g.
gravitational settling) which, as explored by Istrate et al. (2016), work
against rotational mixing, meaning that any impact on the abundances found
here are likely optimistic. For completeness, we did run several models with
element diffusion (implemented on MESA with do_element_diffusion) but found
that once again, given donors that are similarly evolved, it has little effect
on the predicted surface abundances.

Figure 10: Results of MESA models for initial $M_{\rm donor}=1.0M_{\odot}$
with (blue dashed) and without (blue solid) rotational mixing for
$p_{0}=2.9948$, as well as for $p_{0}=3.15$ without mixing (black solid). We
see that the blue dashed and black solid lines have almost overlapping Teff
curves meaning that they are similarly evolved. Thus, at a given initial
period, mixing causes the donor to become more evolved in a shorter period of
time. We also find that there is very little difference in the black solid and
blue dashed lines on the plot of [Na/H] on the right. Therefore, at a given
evolutionary stage of the donor, mixing does not have a significant effect on
the predicted Na abundance.

###  4.2 Conclusions

We have obtained follow-up high resolution spectroscopy of the 21 CVs with
evolved donors first studied by E21. The donors in these systems first
overflowed their Roche lobes near the end of their main-sequence evolution and
are thus predicted to have helium-enriched cores and anomalous surface
abundances. Our goal was to measure the surface Na abundances of the donors to
test for enhancement in Na from nuclear burning, which occured when the
material now in the donors’ photospheres was inside their convective cores. We
also ran evolutionary models to find the range of initial parameters that lead
to the formation of evolved CVs and test if they are able to explain the
observed abundances. Our main findings are as follows:

  1. 1.

Na enhancement: By measuring equivalent widths (EWs) of the Na 5900 Å doublet
from the observed spectra and comparing them to those of synthetic spectra
generated over a grid of Na abundances (Figure 3), we calculated [Na/H] values
for the objects. As described in Section 3.4, we also corrected for non-LTE
effects. We find $0.3\lesssim$ [Na/H] $\lesssim 1.5$ dex with a median value
of 0.95 dex (i.e. nearly an order of magnitude enhancement compared to solar;
Table 3). Our study thus shows that surface Na enhancement, which has
previously been detected in several CVs, is generically found in CVs with
evolved donors.

  2. 2.

Evolutionary models for evolved CVs: We ran binary models using MESA to
investigate the initial binary parameters that produce evolved CVs similar to
those in the observed sample. We modelled the WD as a point mass of 0.7
$M_{\odot}$ and tested initial donor masses of 1.0 and 1.5 $M_{\odot}$ over a
range of initial periods (Section 4). We found that for initial $M_{\rm
donor}=1.0\ M_{\odot}$, initial periods of $3.0\lesssim p_{0}<3.2$ days
produced evolved donors with effective temperatures similar to the observed
systems. For $M_{\rm donor}=1.5M_{\odot}$, the required period range was
$0.7\lesssim p_{0}<0.9$ days, which is shorter and narrower (Figure 8).

  3. 3.

Na enhancement predicted by MESA models: Models of evolved CVs predict a
significant enhancement in the surface Na abundance, consistent with our
observations. At the orbital periods of the observed systems, the predicted
amount of enhancement is sensitive to both the initial period and initial
donor mass. The ubiquitous Na enhancement of evolved CV donors can thus likely
be understood as a consequence of nuclear processing in the donors during
their main-sequence evolution. Models with higher initial donor masses
generally predict stronger Na enhancement, while models for “normal” unevolved
CVs predict no enhancement (Figure 9).

  4. 4.

Connection to Future Modeling: A majority of the observed donors have higher
inferred Na abundances than predicted by the MESA models. Possible sources of
this discrepancy include the large uncertainty in the reaction rate of the
step producing 23Na in the NeNa cycle, or missing physical processes (mixing
and diffusion are described below). These may be worth further investigation
when constructing future models. The discrepancy could also be due to an
overestimation of the calculated abundances which could result from an
underestimation of the donor effective temperature or unaccounted for He
enhancement (see Section 3.2 and Appendix E).

  5. 5.

Effects of rotational mixing: We explored the possibility of additional
enhancement through rotational mixing by turning on Eddington-Sweet
circulation and Spruit-Taylor instabilities in our MESA models. We found that
at a given initial donor mass, mixing pushed the donor to become evolved more
quickly. However, given donors at the same evolutionary stage (i.e. similar
curves of effective temperature at a given period), mixing had very little
effect on the predicted Na abundances (Section 4.1). Therefore, while mixing
does have an effect on the evolutionary path of a system given the same
initial conditions, it is unlikely to be able to account for the extra Na
enhancement seen in our objects.

## Acknowledgements

We thank Shrinivas Kulkarni for help acquiring observational data, and John
Thorstensen, Tom Marsh, and the ZTF stellar variables group for helpful
discussions. We also thank the anonymous referee for the detailed feedback.

The data presented herein were obtained at the W. M. Keck Observatory, which
is operated as a scientific partnership among the California Institute of
Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W. M. Keck Foundation.

The authors wish to recognize and acknowledge the very significant cultural
role and reverence that the summit of Maunakea has always had within the
indigenous Hawaiian community. We are most fortunate to have the opportunity
to conduct observations from this mountain.

Based on observations obtained with the Samuel Oschin Telescope 48-inch and
the 60-inch Telescope at the Palomar Observatory as part of the Zwicky
Transient Facility project. ZTF is supported by the National Science
Foundation under Grant No. AST-2034437 and a collaboration including Caltech,
IPAC, the Weizmann Institute for Science, the Oskar Klein Center at Stockholm
University, the University of Maryland, Deutsches Elektronen-Synchrotron and
Humboldt University, the TANGO Consortium of Taiwan, the University of
Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National
Laboratories, and IN2P3, France. Operations are conducted by COO, IPAC, and
UW.

## Data availability

The data underlying this article are available upon reasonable request to the
corresponding author.

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##  Appendix A Observing log

Table A summarizes the observing dates and instrument specifications for the
spectra collected for all objects. Note that the “ID”s used here are internal
to E21 and this work – the “P” stands for period, followed by a number which
is the period in hours, and the letter can be used to differentiate objects
with the same period to two decimal places.

We observed 20 (out of 21) systems with the Echellette Spectrograph and Imager
(ESI) in Echellette mode on the Keck II telescope. The majority (18) of these
were observed on 09/02/22 and 10/24/22. For these, the wavelength coverage was
between 4000 - 10150 Å, divided across 10 orders. During the first night, 300s
exposures were taken using the 0.3" slit, which resulted in a median
resolution of R $\sim$ 11700\. The resolutions were calculated using the
[OI]5577Å sky line by taking the ratio of the central wavelength to the FWHM
of the line obtained from Gaussian fitting. Signal-to-noise ratios (SNRs) were
calculated over 20 pixels centered around 5850 and 5890 Å surrounding the Na
doublet, by taking the ratio of the mean flux over the mean flux error. This
gave us a median SNR (averaged over the two regions) of $\sim 6$ (the values
for each spectra can be found in Table 4). Based off of these SNRs, and
checking the effect of orbital smearing in determining the rotational velocity
was minimal over longer exposure times, it was decided that for the next
night, 600s exposures with a 0.5" slit would be used instead, which gave us R
$\sim$ 9300 and SNR $\sim$ 21\. The remaining two objects with ESI spectra
were observed by a collaborator as backup targets on 03/06/22 and multiple
300s exposures were taken with the 1.0" slit, resulting in R $\sim$ 4800\. To
improve the SNR, these spectra were co-added after correcting for each of
their radial velocities which gave us SNRs $\sim$ 17.

As described in Section 2, one object (P_2.74a) was observed using LRIS with
90s exposures over its complete orbit.

ID | Date [UTC] | MJD | Exposure [s] | Slit width ["] | Resolution | SNR | Instrument
---|---|---|---|---|---|---|---
P_2.00a | 09/02/2022 | 59732.26710000 | 300 | 0.3 | 11720 | 5.77 | ESI
P_2.74a | 06/01/2022 | 59877.60220858 | 63x(r), 78x(b) 90 | 1.0 | 1543 | 34.92 | LRIS
P_3.03a | 10/24/2022 | 59877.21684573 | 600 | 0.5 | 9352 | 11.26 | ESI
P_3.06a | 10/24/2022 | 59825.40700344 | 600 | 0.5 | 9332 | 13.25 | ESI
P_3.13a | 10/24/2022 | 59877.31297247 | 600 | 0.5 | 9321 | 28.81 | ESI
P_3.21a | 10/24/2022 | 59825.53532983 | 600 | 0.5 | 9299 | 9.20 | ESI
P_3.43a | 10/24/2022 | 59877.43533115 | 600 | 0.5 | 9473 | 18.65 | ESI
P_3.48a | 03/06/2022 | 59825.54595483 | 3x300 | 1.0 | 4845 | 21.03 | ESI
P_3.53a | 10/24/2022 | 59877.54255511 | 600 | 0.5 | 9360 | 22.71 | ESI
P_3.81a | 09/02/2022 | 59644.55657553 | 300 | 0.3 | 11611 | 6.50 | ESI
P_3.88a | 03/06/2022 | 59877.32205094 | 3x300 | 1.0 | 4825 | 13.73 | ESI
P_3.90a | 10/24/2022 | 59825.44070830 | 600 | 0.5 | 9321 | 38.95 | ESI
P_3.98a | 10/24/2022 | 59644.57136847 | 600 | 0.5 | 9342 | 14.60 | ESI
P_4.06a | 09/02/2022 | 59877.61201310 | 300 | 0.3 | 11459 | 16.39 | ESI
P_4.10a | 10/24/2022 | 59877.55295580 | 600 | 0.5 | 9321 | 28.79 | ESI
P_4.36a | 10/24/2022 | 59877.63229504 | 600 | 0.5 | 9277 | 13.32 | ESI
P_4.41a | 10/24/2022 | 59877.30323879 | 600 | 0.5 | 9322 | 21.04 | ESI
P_4.47a | 09/02/2022 | 59825.45260830 | 300 | 0.3 | 12214 | 2.38 | ESI
P_4.73a | 10/24/2022 | 59877.33081067 | 600 | 0.5 | 9299 | 23.49 | ESI
P_5.17a | 09/02/2022 | 59877.33934018 | 300 | 0.3 | 11725 | 9.73 | ESI
P_5.42a | 10/24/2022 | 59877.53413636 | 600 | 0.5 | 9318 | 38.58 | ESI
Table 4: An observing log for all objects. For clarity, the date provided is
the date in UTC at the start of the observing run. The resolutions were
calculated using the FWHM from the Gaussian fitting of the [OI]5577Å line in
the sky spectra. The signal-to-noise ratio (SNR) values provided are the
averages between the values calculated over 20 pixels around 5890 and 5850 Å,
above and below the Na doublet (for ESI spectra, the values listed here are of
the sixth order - those of the fifth order are comparable). For all ESI
observations, Echellete mode was used and for LRIS, both the red and blue
spectra were taken.

##  Appendix B H-alpha lines

We plot H-$\alpha$ lines for all objects in Figure 11, along with the Kurucz
model spectra with solar abundances at the corresponding temperatures. It is
seen that the model provides a relatively good fit to the data for around half
of the objects. Meanwhile, several of them have noticeably shallower observed
absorption lines compared to their models (e.g. P_3.03a, P_3.06a, P_3.43a,
P_3.53a) while others have little to no absorption or instead show emission
(e.g. P_2.74a, P_3.98, P_4.10a). The same features were identified for many of
the same objects in E21 and can most likely be explained by the contribution
from an accretion in systems with ongoing accretion.

Figure 11: The spectra at the H$\alpha$ transition for all objects, plotted
with their corresponding model spectra. It is seen that for about half of the
objects, the line of the observed spectra is visibly shallower than that of
the model or is an emission instead of an absorption. These objects are likely
mass-transferring with contribution from an accretion disk.

##  Appendix C Interstellar lines

The plots in Figure 12 show interstellar Na 5900Å doublet lines highlighted in
blue which are removed in the calculation of equivalent widths, using the
spectra in orange. This can be done because interstellar lines are not shifted
by the radial velocity of the star and so are observed very closely to their
rest wavelengths. While for many objects, they are relatively narrow and
indistinguishable from noise, in cases like P_4.41a, their contribution can be
significant and thus important to account for.

Figure 12: The Na 5900 Å doublet for several representative objects showing
the removal of interstellar (IS) lines. The IS lines can be distinguished from
the donor lines as they are not broadened and observed very closely to their
rest wavelengths (i.e. not shifted with the RV of the donor).

##  Appendix D Contribution from a circumbinary disk

While we have masked out narrow interstellar lines, one may reasonably ask
whether there could potentially be contribution from a circumbinary disk.
Lines from such a disk would be broadened due to its own orbital velocity
around the binary system and thus would be more challenging to distinguish
from the lines of the donor.

One way to check whether or not this is the case is to obtain multiple spectra
over the course of a complete orbit and trace radial velocity shifts of the
line. This was done for one object in our sample, P_2.74a, for which we
obtained 90s exposures over 2.75 hours using LRIS (Section 2 and Table A) .
Calculating the orbital period using the MJD at mid-exposure for each
spectrum, stacking all of the normalized spectra, and displaying them as an
image, we get a trailed spectrum as shown in Figure 13, zoomed into the Na
doublet (note that the two lines cannot be resolved with LRIS so they are seen
as a single dark band on the plot). Qualitatively, it is seen that the line
moves back and forth along with any neighbouring features as it traces the
donor’s orbital motion. Thus, there is no clear sign of contamination from ex.
a circumbinary disk which will add a separate component to the radial velocity
of the donor and cause deviation from a sinusoidal evolution over the orbit.
While this was only done for a single object, it does show that significant Na
enhancement can be present without evidence of a circumbinary disk.

Figure 13: A trailed spectrum showing the Na doublet (unresolved at this
resolution) for P_2.74a collected over one period. An eclipse of the WD
accretor occurs at phase = 0.5. The line is close to sinusoidal and follows
the trend of neighbouring features, meaning it is tracing the donor without
significant contribution from an external source such as a circumbinary disk.

##  Appendix E Abundances with He enhancement

If He enhancement is unaccounted for, it generally results in an
overestimation of the Na abundance. Thus we calculate the inferred [Na/H] of
all objects by comparison to models with a He enhanced mass fraction of
$Y=0.569$. The results are summarized in Table 5, where we see that with the
exception of P_2.74a, all other objects see a decrease in the calculated
[Na/H] compared to those from Table 3 (at the low temperature of P_2.74a, the
He enhanced model has a steeper curve of growth so that it predicts lower Na
abundances compared to the solar model for small EWs but eventually crosses
over). In this case, [Na/H] ranges $\sim$ 0.2 to 1.5 dex with a median value
of 0.826 dex. Therefore, while there is a decrease in the inferred Na
enhancement if He enhancement is present for all objects, it remains quite
significant.

ID | [Na/H] (Y = 0.254) | [Na/H] (Y = 0.569)
---|---|---
P_2.00a | 0.693 $\pm$ 0.517 | 0.597 $\pm$ 0.506
P_2.74a | 1.296 $\pm$ 0.197 | 1.455 $\pm$ 0.179
P_3.03a | 1.016 $\pm$ 0.105 | 0.852 $\pm$ 0.105
P_3.06a | 1.201 $\pm$ 0.104 | 1.033 $\pm$ 0.099
P_3.13a * | 1.016 $\pm$ 0.076 | 1.016 $\pm$ 0.076
P_3.21a * | 0.964 $\pm$ 0.108 | 0.964 $\pm$ 0.108
P_3.43a * | 0.583 $\pm$ 0.085 | 0.583 $\pm$ 0.085
P_3.48a | 0.736 $\pm$ 0.096 | 0.227 $\pm$ 0.086
P_3.53a | 0.970 $\pm$ 0.127 | 0.792 $\pm$ 0.127
P_3.81a | 0.995 $\pm$ 0.177 | 0.826 $\pm$ 0.173
P_3.88a | 1.502 $\pm$ 0.116 | 1.305 $\pm$ 0.106
P_3.90a * | 0.822 $\pm$ 0.078 | 0.822 $\pm$ 0.078
P_3.98a | 0.653 $\pm$ 0.142 | 0.475 $\pm$ 0.141
P_4.06a | 0.328 $\pm$ 0.265 | 0.181 $\pm$ 0.251
P_4.10a | 0.442 $\pm$ 0.162 | 0.260 $\pm$ 0.162
P_4.36a | 1.143 $\pm$ 0.098 | 0.978 $\pm$ 0.095
P_4.41a | 0.397 $\pm$ 0.141 | 0.237 $\pm$ 0.137
P_4.47a | 1.152 $\pm$ 0.506 | 0.973 $\pm$ 0.525
P_4.73a | 0.744 $\pm$ 0.143 | 0.595 $\pm$ 0.139
P_5.17a | 0.956 $\pm$ 0.115 | 0.795 $\pm$ 0.113
P_5.42a | 0.668 $\pm$ 0.091 | 0.505 $\pm$ 0.089
Table 5: Na abundances calculated using He enhanced models with $Y=0.569$. For
ease of comparison, we also add a column from Table 3 using solar abundance
($Y=0.256$) models of He. The objects marked with an asterisk (*) are those
whose abundances are calculated using $Y=0.868$ models as discussed in Section
3.2. All abundances here have been corrected for NLTE effects (Section 3.4).
With the exception of P_2.74a, [Na/H] for all other objects decrease with
increasing He abundance. The values now range from $\sim$ 0.2 - 1.5 dex with
the median being 0.826 dex which is however, still a significant Na
enhancement.

##  Appendix F MESA profile plots

Figure 14 show several profiles (plots as a function of the radial coordinate
of the donor at some snapshots during the evolution) of the chemical
abundances for the model with initial donor mass of $1.0M_{\odot}$ and initial
orbital period of 3.15 days.

The plots on the left show profiles at three orbital periods $\lesssim 10$
hours. As expected, the radius of the donor falls as the period decreases and
more of its mass is transferred to the WD. More importantly, the chemical
abundances as a function of the fraction of radius evolves over time. In
particular, it is seen that at the longest period (green), there is a core
rich in Na but its abundance falls off rapidly to solar values at the surface.
Therefore, if the donor is observed at this period, no Na enhancement will be
observed. As the donor continues to evolve, it loses its outer envelope and
undergoes further burning so that we begin to see a rise in Na abundance at
the outer layers. If the donor is observed at the shortest period plotted
(blue), we will observe a significant Na enhancement of $\sim$ 0.5 dex. Note
that the abundances at the central region ($\lesssim 0.1R_{\odot}$) plateaus
to a constant value at all periods and in particular for Na, the abundance at
the outer layers closely approach but never exceeds this value, placing a
limit onto the achievable enhancement. This points to an equilibrium being
reached between the two regions.

To identify when the core abundance was established, we also provide profiles
spanning a much greater length of time, from $\sim$ 10 Myr - 10 Gyr, on the
right. We see that there is a large increase in the central Na abundance
between 1 Gyr to 10 Gyr as it has had a long time to build up, where it
increases from $\sim$ 0.05 to the maximum of 0.7 dex. The small rise in
central temperature between these two times may also contribute slightly as it
can result in a significant increase in the reaction rate of the step
producing the Na.

Figure 14: Profiles of the MESA models for initial parameters $M_{\rm
donor}=1.0M_{\odot}$ and $p_{0}=3.15$ days, showing the abundances of several
elements at several different orbital periods (left) and star ages (right).
The plots on the left show that as the donor evolves towards shorter periods,
its radius decreases as it sheds more of its envelope and the surface Na
abundance increases. Meanwhile, from the plots on the right (spanning a much
greater length of time), we find that the central Na abundance increases
significantly (from $\sim$ 0.05 to 0.7 dex) between $10^{9}$ to $10^{10}$
years where it is maintained thereafter.