# Where are the missing symbiotic stars? Uncovering hidden Symbiotic Stars in
public catalogues

Stavros Akras ${}^{1\href https://orcid.org/0000-0003-1351-7204{}}$
1Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing,
National Observatory of Athens, GR 15236 Penteli, Greece
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Theoretical predictions of the population of Galactic symbiotic stars (SySts)
are highly inconsistent with the current known population. Despite intense
effort over the past decades, observations are still far below the
predictions. The majority of known SySts so far are identified based on
selection criteria established in the optical regime. The recent discovery of
SU Lyn with very faint optical emission lines uncloaked a subgroup of SySts
with accreting-only white dwarfs. In this particular case, the luminous red
giant may overshadow the dimmed white dwarf companion. A new approach to
search for this subgroup of SySts is presented, employing GALEX UV and
2MASS/AllWISE IR photometry. The FUV-NUV colour index is an indicator, direct
or indirect, for the presence of hot compact companions. The cross-match of
the Catalogue of Variable Stars III obtained from the All-Sky Automated Survey
for SuperNovae with the GALEX, 2MASS and AllWISE catalogues result in a sample
of 814 potential SySt candidates. From them, 105 sources have photometric
measurements from both FUV and NUV bands and 35 exhibit FUV-NUV$<$1, similar
to what it is expected from known SySts. Five known SySts are recovered, while
two new genuine SySts are discovered in spectroscopic follow-up observations
after the detection of the typical emission lines.

###### keywords:

Astronomical data bases: catalogues – (stars:) binaries: general – (stars:)
binaries: symbiotic – stars: late-type – stars: variables: general – (stars:)
white dwarfs

††pubyear: 2022††pagerange: Where are the missing symbiotic stars? Uncovering
hidden Symbiotic Stars in public catalogues–Where are the missing symbiotic
stars? Uncovering hidden Symbiotic Stars in public catalogues

##  1 Introduction

Theoretical predictions of the population of symbiotic stars (SySts) in the
Milky Way are strongly contradicted by the observations of an order of 2–3
magnitudes due to the assumptions considered for the evolution of binary
systems, for which very little was known 20–30 years ago.

The first estimation of the SySts’s population in the Milky Way was
4$\times$103 (Kenyon, 1986). In 1992, the total number of SySts was computed
around 3$\times$105 by Munari & Renzini (1992), and a year later Kenyon et al.
(1993) reduced it to 3.3$\times$104 SySts111The number of 3.3$\times$104
corresponds only to S-type SySts, while the D-type with a Mira companion are
$\sim$0.6$\times$104.. Magrini et al. (2003) recalculated the population of
Galactic SySts of the order of 4$\times$105 assuming that a 0.5 percent of the
population of red giants (RGs) and AGB stars should be in orbit around white
dwarfs (WDs). More recently, Lü et al. (2006) refined the Galactic population
of SySts between 1.2$\times$103 and 15$\times$103 employing a population
synthesis model. Discrepancies among theoretical studies are associated to the
different assumptions on the SySts’s lifetime and/or the evolutionary paths of
binary systems that eventually become SySts.

From the comparison of these theoretical predictions with the most recent
census of known SySts in the Milky Way (Akras et al., 2019a; Merc et al.,
2019, and references therein) and the latest discoveries over the past few
years (Merc et al., 2020; Akras et al., 2021; Merc et al., 2021b; Merc et al.,
2021a; Merc et al., 2022; Munari et al., 2021, 2022; De et al., 2022; Petit et
al., 2023), it is evident that many Galactic SySts are still missing and
waiting to be discovered. Thus far, the vast majority of the known SySts if
not all of them, have been discovered and identified following a number of
criteria based on optical observations: (i) the detection of some typical
emission lines (e.g., [Fe vii] $\lambda\lambda$5727,6087), H$\alpha$, [O
iii]$\lambda$5007, and He ii $\lambda$4686, (ii) the absorption molecular
features such a TiO, VO, and CN that ensure the presence of a red giant,
and/or (iii) the detection of the O vi Raman-scattered
$\lambda\lambda$6830,7088 lines (Mikolajewska et al., 1997; Belczyński et al.,
2000; Akras et al., 2019a).

Figure 1: Left panel: 2MASS J-H versus H-Ks diagnostic diagram of all the
sources (29,237) in the ASAS-SN/2MASS/AllWISE cross-matched list. The locus of
S and D-type SySts from Corradi et al. (2008); Rodríguez-Flores et al. (2014);
Akras et al. (2019b) are also plotted. All sources are concentrated in the
regimes of S-type SySts. Right panel: Gaia MG versus GBP-GRP HR diagram for
the same ASAS-SN/2MASS/AllWISE list. The vast majority has GBP-G${}_{\rm
RP}>$2 and M${}_{\rm G}>$5 suggesting evolved stars in the red-giant branch.
The smaller group with GBP-G${}_{\rm RP}<$2.5 and M${}_{\rm G}<$2 likely
presents some M-dwarfs and main-sequence stars.

Although, Mukai et al. (2016) argued for the discovery of the first SySt (SU
Lyn) with very weak H$\alpha$ line, whilst other typical emission lines such
as He ii, [Fe vii], [O iii] were absent. The optical spectrum of SU Lyn
resembled that of a typical M-type giant star with a weak H$\alpha$ emission
line, but its hard X-ray emission and UV variability could not be explained
without the presence of an accreting hot companion. Note that SU Lyn satisfies
the newly infrared selection criteria of S-type SySts proposed by Akras et al.
(2019b, 2021), providing an additional verification of its symbiotic nature.
Follow-up X-ray, UV, optical and near-IR observations of SU Lyn have
unquestionably confirmed the symbiotic nature of this accreting-only SySt with
M=0.7-0.85 M$\odot$ and Teff=37000$\pm$15000 K (Lopes de Oliveira et al.,
2018; Kumar et al., 2021). According to the X-ray spectral classification
scheme (the $\alpha$, $\beta$, $\gamma$, and $\delta$ scheme, Luna et al.,
2013, and references therein), SU Lyn is classified as $\delta$-type symbiotic
star. An interesting relation between the X-ray spectral types of SySts and
the O vi $\lambda$6830 Raman-scattered line has been reported by Akras et al.
(2019a). In particular, O vi $\lambda$6830 line is only detected in $\alpha$
and $\beta$-type SySts for which the X-ray emission is associated with shell-
burning WDs resulting in highly ionized gas and strong optical emission lines.

Based on the intriguing discovery of SU Lyn, Mukai et al. (2016) argued that
the current catalogues of SySts are biased to those with more luminous WDs
capable of producing strong optical emission lines and a highly ionized
circumstellar envelope. Where are the missing SySts with low luminosity WD
companions? It is very likely that many low-luminosity SySts are misidentified
and may be listed in catalogues of typical red giants or even in catalogues of
variable stars or emission line sources.

A new accretion-only SySt, THA 15-31, has recently been discovered by Munari
et al. (2022). THA 15-31 displays only Balmer lines in its optical spectrum,
molecular TiO bands and strong UV-excess, but no X-ray emission. Note that
this newly discovered SySt also satisfies the IR-selection criteria of S-type
SySts (Akras et al., 2019b).

Besides SySts, the family of WD + RG binary systems also includes Barium and
Technetium-poor extrinsic S stars or more general chemically-peculiar stars
(hereafter CPSs). They do not show symbiotic activity (Jorissen, 2002, and
reference therein) and do not satisfy the optical criteria of SySts.
Nevertheless, it may be possible that SySts with low luminosity WDs or feeble
symbiotic activity are hidden in these classes of binary systems. The
parameters responsible for triggering the symbiotic activity or s-process
elements enhancement due to the mass transfer in these binary systems are
discussed by Jorissen (2002). Should both CPSs and SySts be included in the
general population of WD + RG binary systems when they are compared with the
theoretical predictions in the Milky Way?

In this paper, I search for potential SySt candidates camouflaged as regular
stars without any prior information coming from optical spectroscopy, but
combining information from ultraviolet (GALEX, Bianchi et al., 2011b; Bianchi
et al., 2017), near-infrared (2MASS, Cutri et al., 2003) and mid-infrared
(WISE, Wright et al., 2010; Cutri et al., 2021). Information from the All-Sky
Automated Survey for Super-Novae (ASAS-SN, Shappee et al., 2014) and Gaia
(Gaia Collaboration et al., 2016, 2018) are also combined in order to better
constrain the list of SySt candidates. The paper is organized as follows:
Section 2 hosts the methodology followed to search for new candidates which
display the representative characteristics of the presence of red giants and
hot WDs or an accretion disk around it. The results of the GALEX and
2MASS/AllWISE analysis as well as the follow-up spectroscopic discovery of two
new genuine SySts are presented in Sections 3 and 4, respectively. The
difference in FUV-NUV colour index between SySts and single dwarfs, red
giants, white dwarfs, Barium and Technetium-poor stars is discussed in Section
5. I end up with the conclusions in Section 6.

Figure 2: The distribution of GALEX UV photometry of the sources in the
GALEX/ASAS-SN (upper left panel) and the GALEX/Known-SySts (upper right panel)
lists. The lower left panel displays the distribution of the FUV-NUV colour
index of the sources in the GALEX/ASAS-SN and GALEX/Known-SySts lists. The
lower right panel presents the distribution only for those source with FUV-
NUV$<1$.

##  2 Methodology

The recent discovery of SU Lyn with weak H$\alpha$ emission, absent [O iii]
and He ii lines, and obvious UV-excess (Mukai et al., 2016) in conjunction
with the most recent discovery of THA 15-31 (Munari et al., 2022) have
inspired me to search for those red giants that display strong UV-excess and
at the same time satisfy the IR selection criteria of S-type SySts (Akras et
al., 2019b, 2021). The methodology is divided into two steps. First, it
identifies the stars that exhibit characteristics of red giants satisfying the
IR selection criteria and then those stars that show strong evidence for the
presence of a hot component, either directly due to the photospheric emission
from WDs or indirectly from the emission of an accretion disk around WDs. Each
step is described in more details in the following subsections.

###  2.1 Red giant companion

In this first step, we make use of the Catalogue of Variable Stars III333The
catalogue was downloaded in the beginning of 2020. Since then, three more
catalogues have been published. The most recent update was made on 08/10/2021,
and it lists almost 71,000 more variable objects. (Jayasinghe et al., 2018,
2019) from the All-Sky Automated Survey for SuperNovae222https://asas-
sn.osu.edu/variables (ASAS-SN, Shappee et al., 2014). The ASAS-SN survey
covers the entire visible sky with telescopes in both hemispheres to a depth
of $<$17 mag in the V band and saturation level of $\sim$10-11 mag. Each field
of the ASAS-SN survey has 200–600 epochs of observations. The catalogue
contains $\sim$412,000 variable stars classified as $\delta$ Scuti, RR Lyrae,
Cepheids, rotational variables, eclipsing binaries, semiregular and irregular
variables and Mira variables. The semiregular and irregular variables classes
include red giants, and they are of particular interest.

The IR (2MASS/AllWISE) selection criteria of S-type SySts (Akras et al.,
2019b, 2021) were applied to the Catalogue of Variable Stars III resulting in
a sample of 96,808 variable sources. To further filter the sample, we used the
IR criteria between SySts and K/M-type single giants from Akras et al. (2019b)
which reduced the sample to 64,730 stars. According to Gromadzki et al.
(2013), SySts stars display short-term variabilities between 50 and 200 days,
besides the long-term and orbital variations, associated with the stellar
pulsations of the giant companions. A variability period cut-off was applied
to our sample in order to keep only those sources with period $>$50 days. This
resulted in our final ASAS-SN/2MASS/AllWISE cross-matched list of 29,237
stars.

In Figure 1, we present the distribution of the sources in the ASAS-
SN/2MASS/AllWISE list in the typical J – H versus J – Ks diagnostic colour-
colour diagram (DCCD, left panel) (Corradi et al., 2008; Rodríguez-Flores et
al., 2014; Akras et al., 2019b) and in the Gaia MG versus GBP-GRP
Hertzsprung–Russell diagram (right panel). The vast majority of the sources
are well-placed in the locus occupied by the S-type SySts in the near-IR DCCD.
Yet, several other classes of stars such as single K/M type giants, Mira
stars, T Tauri, young stellar objects (YSO), may exhibit the same colour
indices (see fig. 1 in Akras et al., 2019b). Nevertheless, the application of
the IR selection criteria is expected to have reduced substantially the
contamination from these classes.

Figure 3: Upper panels:FUV-NUV colour index distribution of M-type giants
(left panel) and DA white dwarfs (right panel). Lower Panel: FUV/NUV
photometry (left panel) and FUV-NUV colour index (right panel) distributions
of M-type dwarfs

It should be noted that M-type dwarfs stars were excluded from the training
sample in Akras et al. (2019b) because of their weak H$\alpha$ emission and
the violation of the IPHAS criterion of SySts (Corradi et al., 2008). In view
of the fact that we do not take into account the IPHAS criterion or any
H$\alpha$-criterion, variable M dwarfs may be still present in our list.

According to the Gaia MG versus GBR-GRP HR diagram, the majority of the
sources in the ASAS-SN/2MASS/AllWISE list are evolved stars in the red giant
branch phase, GBP-G${}_{\rm RP}>$2 and M${}_{\rm G}>$5, with very few
contaminants from T Tauri, YSO, dwarfs stars or main sequence stars,
GBP-G${}_{\rm RP}<$2.5 and M${}_{\rm G}<$2 (see also fig. 22 in Jayasinghe et
al. (2019)).

###  2.2 White dwarf companion

The analysis in the previous section ensures that the majority of the sources
exhibit signatures of red giant stars, and thus they are potential cool
companions in binary systems. In the next step, we search for those stars from
the previous step that display either direct signatures for the presence of a
hot WD companion or indirect signatures from the UV emission of an accretion
disk around the WD companion. For this exercise, we make use of the revised
catalogue of GALEX UV sources (GUVcat_AIS, Bianchi et al., 2017).

The GALEX survey mapped an area of $\sim$25000 square degrees with two bands.
The far-UV band (FUV) has $\lambda_{\rm eff}\sim$1258Å covering the spectral
range from 1340 to 1806Å, and the near-UV band (NUV) has $\lambda_{\rm
eff}\sim$2310Å with a spectral range from 1693 to 3007Å. GALEX has
successfully been used to find new single WD or in binary systems (e.g.
Bianchi et al., 2011b). Therefore, the unique photometry of GALEX will be
valuable for the mining of potential WD companions, either directly or
indirectly. The GUVcat_AIS GALEX catalogue lists 82,992,086 unique UV sources,
and it is constructed based on the observations from the All Sky Survey (AIS)
with both detectors (FUV and NUV) on. This means that UV sources in the
GUVcat_AIS catalogue without available FUV photometry were below the detection
limit (Bianchi et al., 2017). It should also be noted that AIS observations do
not cover the Galactic disk, and thus the majority of the Galactic stellar
sources are not observed. The results of this work represent only a part of
the stellar population in the Milky Way.

The cross-matching of the ASAS-SN/2MASS/AllWISE list with the GUVcat_AIS GALEX
catalogue resulted in a list of 814 matches, considering a radius of 5 arcsec
due to the FWHM of the GALEX PSF of 4.2 and 5.3 arcsec for the FUV and NUV
filters.

ASASSN-Va | Other name | RA | Dec. | FUV | NUV |  MG |  GBP-GRP | SIMBAD | Refs. |  P${}_{\rm ASASSN}^{a}$ |  P${}_{\rm VSX}^{b}$ |  Distancec
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (J2000.0) | (J2000.0) | (mag) | (mag) | (mag) | (mag) |  |  | (days) | (days) | kpc
J103437.09-401940.4 |  | 158.654896 | -40.327779 |  19.226$\pm$0.090 |  18.543$\pm$0.045 | -0.024 | 3.577 |  |  | 215.9039612 | 65.4 |  1.12${}^{+0.16}_{-0.12}$
J180148.62+813925.5 | 2MASS J18014909+8139252 | 270.455019 | +81.656976 |  20.350$\pm$0.192 |  19.793$\pm$0.084 | -2.509 | 2.931 | LPV |  | 548.0866926 |  |  3.47${}^{+0.53}_{-0.42}$
J080553.51-065032.6 | IRAS 08034-0641 | 121.473753 | -6.84334 |  21.062$\pm$0.318 |  21.621$\pm$0.336 | 0.589 | 3.838 | star |  | 630.6335752 |  |  0.92${}^{+0.13}_{-0.11}$
J231839.50+182539.0 | IRAS 23161+1809 | 349.663968 | 18.426775 |  22.008$\pm$0.396 |  21.817$\pm$0.279 | -1.226 | 3.699 | star |  | 270.9739143 |  |  1.94${}^{+0.55}_{-0.38}$
J115151.88-422751.3 | V422 Cen | 177.966346 | -42.464231 |  16.343$\pm$0.036 |  16.477$\pm$0.023 | -1.533 | 4.098 | VSM |  | 415.8376321 | 193 |  0.83${}^{+0.08}_{-0.07}$
J085048.16-124517.9 | IRAS 08484-1234 | 132.700944 | -12.755286 |  21.814$\pm$0.349 |  21.316$\pm$0.218 | -1.309 | 3.515 | star |  | 671.0213235 | 953 |  1.67${}^{+0.31}_{-0.23}$
J203557.17+201127.2 | LT Del | 308.988768 | +20.191100 |  15.937$\pm$0.021 |  16.381$\pm$0.017 | -1.262 | 1.662 | SySt | 1 | 530.3445585 | 476 |  5.81${}^{+1.72}_{-1.20}$
J194225.42-680735.5 | V399 Pav | 295.606744 | -68.126459 |  13.805$\pm$0.007 |  14.662$\pm$0.006 | -2.954 | 2.838 | SySt | 1 | 590.8957458 | 565 |  3.71${}^{+0.99}_{-0.68}$
J210006.29-423843.7 | DD Mic | 315.026121 | -42.645440 |  14.038$\pm$0.008 |  14.144$\pm$0.006 | -3.333 | 1.693 | SySt | 1 | 418.2553265 | 399 |  6.53${}^{+1.59}_{-1.15}$
J155930.27+255511.9 | T CrB | 239.875575 | +25.920248 |  16.199$\pm$0.043 |  15.245$\pm$0.009 | -0.735 | 2.419 | SySt | 1 | 883.577155 | 227.6 |  0.81${}^{+0.03}_{-0.03}$
J060910.32-833528.6 | TYC 9493-59-1 | 092.294499 | -83.591200 |  19.528$\pm$0.155 |  19.906$\pm$0.136 | -0.677 | 1.982 | star |  | 138.39 | 138 |  11.0${}^{+1.69}_{-1.39}$
J172133.84+015011.7 | SS 305 | 260.391045 | +01.836472 |  15.776$\pm$0.021 |  16.405$\pm$0.014 | -2.063 | 2.642 | ELS | 2 | 459.11 | 459 |  7.74${}^{+2.63}_{-1.79}$
J165002.92-230517.0 | THA 23-30 | 252.512028 | -23.088131 |  18.379$\pm$0.113 |  18.591$\pm$0.076 | -1.282 | 2.887 | ELS | 3 | 334.37 | 334 |  7.85${}^{+3.44}_{-2.17}$
J165319.30+330958.2 | KO Her | 253.330200 | +33.166149 |  22.285$\pm$0.386 |  22.951$\pm$0.412 | -1.113 | 3.171 | LPV |  | 661.6351211 | 61 |  2.27${}^{+0.27}_{-0.22}$
J115539.67+123447.8 | LEE 107 | 178.915616 | +12.580103 |  19.702$\pm$0.154 |  19.141$\pm$0.078 | -2.245 | 1.948 | Pe |  | 294.0876492 | 57.556521 |  3.23${}^{+0.65}_{-0.49}$
J184639.07-552112.7 | AN Tel | 281.662882 | -55.352938 |  20.783$\pm$0.305 |  19.863$\pm$0.151 | -2.121 | 1.924 | LPV |  | 213.0895841 |  |  2.93${}^{+0.38}_{-0.30}$
J022508.61-132400.6 | BD-14 450 | 036.285797 | -13.400167 |  16.477$\pm$0.042 |  15.873$\pm$0.018 | -2.357 | 2.151 | Pe |  | 674.3792712 | 73.8 |  3.03${}^{+0.42}_{-0.34}$
J105143.65-270433.1 |  | 162.931781 | -27.075852 |  21.589$\pm$0.286 |  22.765$\pm$0.439 | -1.344 | 2.233 |  |  | 673.7733942 | 128.50 |  3.79${}^{+0.93}_{-0.68}$
J210350.55-641008.5 | UCAC2 3609393 | 315.960620 | -64.168942 |  20.733$\pm$0.194 |  20.159$\pm$0.100 | -0.619 | 3.145 | star |  | 272.0216292 | 202.101852 |  2.76${}^{+0.79}_{-0.53}$
J175754.82+312529.4 | OQ Her | 269.476839 | +31.424950 |  21.172$\pm$0.301 |  20.724$\pm$0.176 | -1.762 | 2.609 | VSM |  | 371.1020341 |  |  2.52${}^{+0.22}_{-0.18}$
J204427.66+191440.5 |  | 311.115194 | +19.244580 |  21.561$\pm$0.343 |  20.831$\pm$0.188 | -0.679 | 3.517 |  |  | 622.4804518 | 194.061707 |  1.94${}^{+0.51}_{-0.34}$
J083048.08+412223.1 | IRAS 08274+4132 | 127.700378 | +41.373186 |  22.014$\pm$0.488 |  21.409$\pm$0.312 | -1.604 | 2.973 | star |  | 442.7656954 |  |  2.07${}^{+0.35}_{-0.27}$
J073506.43+113737.3 |  $[$WWV2004$]$ J0735065+113736 | 113.777990 | +11.626260 |  20.901$\pm$0.202 |  20.310$\pm$0.125 | -0.341 | 2.941 | LPV |  | 992.69815 |  |  1.72${}^{+0.47}_{-0.31}$
J104440.47+192523.0 | EW Leo | 161.168463 | +19.423048 |  17.977$\pm$0.085 |  18.283$\pm$0.037 | -2.168 | 3.785 | LPV |  | 551.4496933 | 90.895804 |  0.95${}^{+0.10}_{-0.08}$
J165236.05+165046.2 | TYC 1521-203-1 | 253.150389 | +16.847076 |  17.618$\pm$0.049 |  16.989$\pm$0.023 | -2.785 | 2.446 | CVS |  | 399.3786493 | 17.147459 |  4.63${}^{+1.00}_{-0.73}$
J213439.96-022057.0 | TYC 5212-477-1 | 323.666496 | -02.348910 |  21.842$\pm$0.447 |  20.965$\pm$0.187 | -1.382 | 3.445 | star |  | 530.4592668 | 66.093857 |  1.38${}^{+0.24}_{-0.18}$
J065955.32-615041.1 |  | 104.980970 | -61.845086 |  21.583$\pm$0.442 |  22.292$\pm$0.381 | -1.643 | 3.053 |  |  | 517.2080886 | 52.97 |  2.69${}^{+0.38}_{-0.30}$
J190955.52-764421.4 | UCAC2 614990 | 287.478400 | -76.738797 |  21.465$\pm$0.294 |  21.302$\pm$0.218 | -2.312 | 3.135 | star |  | 540.9608249 | 189.87352 |  6.47${}^{+1.73}_{-1.25}$
J194456.44-230115.4 | UCAC2 22584087 | 296.234872 | -23.021048 |  20.187$\pm$0.161 |  19.897$\pm$0.089 | -1.577 | 3.121 | star |  | 416.2300275 | 64.05056 |  3.33${}^{+1.98}_{-1.00}$
J005904.44+455222.0 | V520 And | 014.768515 | +45.872855 |  18.779$\pm$0.077 |  17.751$\pm$0.032 | -0.329 | 3.652 | LPV |  | 73.9352712 | 70.0 |  1.91${}^{+0.44}_{-0.31}$
170633.94+200805.9 | V455 Her | 256.642073 | +20.134955 |  20.604$\pm$0.242 |  19.914$\pm$0.119 | -0.871 | 3.418 | LPV |  | 62.9524961 | 57.201015 |  1.89${}^{+0.21}_{-0.23}$
J195948.16-825237.0 | Hen 3-1768 | 299.949838 | -82.877005 |  14.589$\pm$0.010 |  14.905$\pm$0.009 | -3.252 | 1.818 | ELS/SySt | 4 | 90.4840311 | 52.67 |  7.07${}^{+1.25}_{-0.97}$
J200938.65-052808.4d | C* 2863 | 302.411034 | -05.468843 |  20.643$\pm$0.178 |  21.063$\pm$0.092 | -1.678 | 1.975 | carbon |  | 55.8698306 |  |  4.29${}^{+1.33}_{-0.87}$
J131436.41-312510.3e | V957 Cen | 198.651872 | -31.419652 |  20.511$\pm$0.304 |  18.749$\pm$0.037 | 0.067 | 3.014 | Candidate SySt | 1 | 89.5268432 | 54.545456 |  1.14${}^{+0.27}_{-0.19}$
J122432.81-261408.1 | KV Hya | 186.136647 | -26.235779 |  16.145$\pm$0.029 |  15.598$\pm$0.015 | -2.600 | 2.053 | carbon |  | 90.7332789 | 97.4 |  4.99${}^{+1.18}_{-0.89}$
J085541.52-200603.7 | BD-19 2567 | 133.923212 | -20.100973 |  21.311$\pm$0.391 |  20.577$\pm$0.121 | -2.015 | 3.459 | star |  | 93.834604 | 331 |  1.08${}^{+0.09}_{-0.07}$

a The Catalogue of Variable Stars III (Jayasinghe et al., 2018, 2019) from the
All-Sky Automated Survey for SuperNovae (ASAS-SN, Shappee et al. (2014))

b AAVSO International Variable Star Index (VSX, Watson et al., 2006).

c Geometric distances from (Bailer-Jones et al., 2018) based on Gaia DR2
parallaxes. The indices correspond to the upper and lower bounds on the
confidence interval of the estimated distance.

d This object has received the variable star designation, V2011 Aql, in the
GCVS5.1.

e This candidate SySt has a FUV-NUV=1.7608 mag.

LPV: long period variable, VSM: variable star of Mira Cet type, ELS:emission-
line star, Pe: Peculiar star, CVS: Cepheid variable Star

References:(1) Akras et al. (2019a) and references therein, (2) Stephenson &
Sanduleak (1977), (3) The & Lim (1964), (4) Lucy et al. (2018)

Figure 4: Low resolution long-slit spectra of the emission lines sources SS
305 and THA 23-30. The upper panels present the full spectra whilst the lower
panels zoom-in the characteristic optical emission lines of SySts.

The majority of AGNs and QSOs have already been excluded from the ASAS-
SN/2MASS/AllWISE list because of the W1-W2 WISE selection criterion. Based on
the work by Akras et al. (2019b), SySts have W1-W2$<$0.1 while QSOs and AGNs
have W1-W2 of 1.20 (standard deviation of 0.16) and 0.51 (standard deviation
of 0.26), respectively (e.g. Nikutta et al., 2014). QSO are also characterized
by FUV-NUV$>$0 (or $>$0.5 for red-shift$>$0.5-0.6) (Bianchi et al., 2011a;
Deharveng et al., 2019). The contamination of our list with background ANGs
and QSOs is expected to be small.

From these 814 matches, only 105 have measurements from both UV GALEX bands,
so the FUV-NUV colour index can be explored. Hereafter, we will refer to this
list as the GALEX/ASAS-SN list. Note that the FUV-NUV colour index has not
been corrected for interstellar extinction, given that the GALEX colour is
nearly independent for Milky Way dust (Bianchi et al., 2011a). As for the
remaining 709 candidates without FUV measurements, they are very faint UV
sources to be detected in the FUV band, which implies high FUV-NUV colour for
these sources. Moreover, the cross-matching of the list of known Galactic
SySts from Akras et al. (2019a) with GALEX returned 30 matches (hereafter
GALEX/Known-SySts list).

The distributions of the GALEX UV photometry as well as the FUV-NUV colour
index for both lists, GALEX/ASAS-SN and GALEX/Known-SySts, are presented in
Figure 2. The FUV and NUV magnitudes in the GALEX/ASAS-SN list vary from 14 up
to 22 mag with peaks at 21 and 19 mag, respectively (upper left panel). On the
other hand, the known SySts (GALEX/Known-SySts list) are found to exhibit UV
photometry between 12 and 21 mag with a peak around 16 mag for both bands
(upper right panel). This analysis demonstrates that most of the known SySts,
discovered employing the traditional optical criteria, are relatively bright
UV sources and certainly brighter than our GALEX/ASAS-SN candidates. Although,
there are some cases of genuine SySts being relatively faint in UV such as
StHA 164 (FUV=21.4694$\pm$0.3946, NUV=21.0719$\pm$0.2503) or GH Gem (FUV=not
measured, NUV=19.6457$\pm$0.1433) implying that UV-faint SySts ($>$19 mag)
cannot be ruled out.

A comparison of the FUV-NUN colour index between the GALEX/ASAS-SN and
GALEX/Known-SySts lists shows that there is a number of GALEX/ASAS-SN
candidates with FUV-NUN colour index comparable to that of genuine SySts (FUV-
NUV$<$1, see bottom left panel in Figure 2). This suggests the presence of a
potential hot, compact companion in these candidates. The distribution of FUV-
NUV index of the GALEX/ASAS-SN sources with UV colour index less than 1 is
displayed in the bottom right panel of Figure 2.

Beside the bluer GALEX/ASAS-SN sources in the FUV-NUV colour, there are also
70 redder sources with 1$<$FUV-NUV$<$5, which may be either SySts like o Ceti
(FUV=15.2035$\pm$0.0200, NUV=12.9255$\pm$0.0044; FUV-NUV=2.278) or even single
M-type giant stars with strong chromospheric activity. The upper panels in
Figure 3 display the distribution of the FUV-NUV colour for a sample of single
red giants and single WDs from the Catalogue of Stellar Spectral
Classifications (Skiff, 2014). The former show a peak at around 3.5 mag while
the latter are clearly bluer with a peak at -0.5 mag. The higher the FUV-NUV
colour index, the lower the probability of being a genuine SySts or more
general, a WD+RG binary system, but not negligible.

##  3 Results: GALEX/ASAN-SN sources

In Table 2.2, we list all the 35 GALEX/ASAS-SN sources with FUV-NUV$<$1, and
they are considered as potential candidates of genuine SySts or WD + RG binary
systems. The first and second columns list the ASAS-SN ID number and other
names of the sources. The third and fourth columns give the coordinates (R.A.
and Decl.) in the J2000.0 equinox. The GALEX FUV and NUV photometry as well as
the Gaia (G) photometry and the GBP-GRP colour index of each source are listed
in the fifth, sixth, seventh and eight columns, respectively. Their
classification in SIMBAD and references are given in the ninth and tenth
columns. Finally, the variability period of each source provided by the ASAS-
SN (Shappee et al., 2014; Jayasinghe et al., 2018, 2019) and the International
Variable Star Index (VSX) database available at the AAVSO (Watson et al.,
2006) are also listed in the last two columns.

Our final GALEX/ASAS-SN list contains long period variables, variable star of
Mira Cet type, emission-line star, peculiar star, Cepheid variable Star,
carbon stars, symbiotic stars and some sources without a spectral
classification yet. Moreover, significantly different variably periods are
provided by ASAS-SN and VSX.

SS 305 and THA 23-30 are of particular interest as they have been classified
as emission line stars (Stephenson & Sanduleak, 1977; The & Lim, 1964). This
means that their symbiotic nature can be verified with optical spectroscopic
data.Their periods provided by ASASSN and VSX are in excellent agreement:
459.11/459 days (SS 305) and 334.37/334 days (THA 23-30). In addition to these
emission line sources, four known (T CrB, Hen 3-1761, LT Del, CD-43 14304) and
one candidate SySt (V957 Cen) are also recovered from the ASAS-SN catalogue.
All four genuine SySts are among the brightest UV sources with GALEX
photometry $<$16.5 mag and FUV-NUV indices of 0.9545, -0.8567, -0.4443, and
-0.1066 mag, respectively.

The SySt candidate, V957 Cen (Akras et al., 2019a) is a much fainter UV source
with FUV=20.5107 mag and NUV=18.7499 mag. Its FUV-NUV colour index (1.7608) is
found to be significantly higher than the known SySts, and therefore it is
less likely to be a genuine SySt.

Hen 3-1768 is also included in the list of emission line stars (Henize, 1976).
It is a very bright UV source (FUV=14.5894, NUV=14.9054) with FUV-NUV=-0.316,
indicative of a potential symbiotic star if we take into account that it
satisfies the near-IR selection criteria of S-type SySts (Akras et al., 2019b,
2021). The characteristic O vi Raman-scattered line of SySts was detected in
this source during the validation phase of the RAMan Search for Extragalactic
Symbiotic Stars (RAMSES II) project (Angeloni et al., 2019) and
spectroscopically by Lucy et al. (2018) providing the necessary confirmation
of its symbiotic nature.

##  4 Follow-up spectroscopy of the emission line sources

The previous classification of SS 305 and THA 23-30 as H$\alpha$-emitters
prompted me to obtain their first optical spectroscopic data and unveil their
true nature. Low resolution long-slit spectra were obtained on 3rd and 7th of
August 2019 using the Goodman High Throughput Spectrograph (Clemens et al.,
2004) on the 4.1 m telescope at the Southern Astrophysical Research (SOAR)
observatory in Chile.

The configuration of the observations covers the wavelength range from 3000 to
9050 Å in two parts: the blue M1 arm from 3000 to 7050 Å and the red M2 arm
from 5000 to 9050 Å (see also, Akras et al., 2021). This wavelength range
covers all the characteristic lines (e.g. H$\alpha$, He i, [O iii], [Fe iii],
[Fe vii], O vi) as well as the TiO and VO molecular bands needed for their
classification as genuine SySts.

The optical spectra of SS 305 and THA 23-30 are shown in Figure 4. Both
emission line sources are found to be unquestionable genuine SySts. Hydrogen
recombination lines are detected, as expected based on their previous
classification as emission line sources (Stephenson & Sanduleak, 1977; The &
Lim, 1964) together with high excitation lines such as He ii, [O iii], [Fe
iii], [Ne iii], C iv indicating the presence of a hot companion. The
characteristic O vi Raman-scattered line detected mainly in SySts (Allen,
1980; Akras et al., 2019a) is also observed in one of the candidates (THA
23-30). The molecular bands associated with presence of a cold companion are
also detected in both sources. Hence, these newly discoveries of SySts
demonstrate that combining information/data from the ultraviolet and infrared
spectral ranges can also lead to interesting results.

It is worth mentioning that these two H$\alpha$-emitters have significantly
lower FUV-NUV colour indices (-0.63 and -0.21, respectively) compared to the
colour of SU Lyn (0.32). This can explain the optical emission-line spectra of
SS 305 and THA 23-30. This suggests the presence of shell-burning WDs in these
newly discovered SySts in contrast with the accreting-only SU Lyn and THA
15-31.

It should also be noted that three more candidates, which are not classified
as emission line sources (AN Tel, C* 2863 and UCAC2 614990), were observed
without any emission line detected. The FUV-NUV colour index of these three
sources is 0.92, -0.42 and 0.16 mag, respectively. The study of their ASAS-SN
and GALEX light curves is required to bring more insights on their
classification as accreting-only SySts.

Overall, the proposed methodology provides us with some good SySt candidates
for follow-up optical spectroscopic surveys. However, the rate of true
positives may not be very high. In this work, five candidates with FUV-NUV$<$1
were observed and only two of them turned out to be genuine SySts (40 per
cent). If we restrict the FUV-NUV criterion to $<$0, then 2 out of 3
candidates (66 percent) were turned out to be genuine SySts.

It has to be pointed out that there are several hundreds of common sources
between ASAS-SN and GALEX for which FUV measurements are missing. Considering
that within the 105 ASAS-SN and GALEX matches with available FUV and NUV
photometry, there are 35 that satisfy the FUV-NUV$<$1 criterion (see Table 2),
and within these 35 there are seven genuine SySts (five previously known and
two newly discovered), we reckon that nearly 50 potential genuine SySts,
depending on their FUV-NUV colour index, may exist in the list of 814 ASAS-
SN/GALEX matches.

An additional tool that can also be very useful for the identification of new
SySts, and it will be presented in a forthcoming paper, is the investigation
of the GALEX light curves of the sources, seeking for evidence of interaction
between the two components in potential binary systems. This approach may also
reveal potential SySts with low UV WDs and faint or even absent emission
lines.

##  5 Discussion

###  5.1 SySts versus single dwarfs, red giants and white dwarfs.

Bianchi et al. (2007) pointed out that GALEX sources bluer in the FUV-NUV
colour and redder in the optical are likely binary systems with a hot WD and
an M-dwarf type star (see also Rebassa-Mansergas et al., 2010) or a giant star
(see also Li et al., 2018). Smith et al. (2014) carried out a study on the so-
called UV-excess sources which appear to be bluer in the FUV-NUV colour index
relative to their spectral type, and possible scenarios, e.g. WDs, subdwarfs,
binary systems, and chromospheric activity, for the origin of the UV-excess
are discussed by the authors. Only a small part of the GALEX sources with UV-
excess can be explained ($\leq$2 per cent are compact sources; WDs, subdwarfs,
$\sim$9 percent are old and young stars with chromospheric activity and
$\leq$4 percent are close active binaries).

To ensure that the FUV-NUV colour cut-off results in GALEX detections for
which the UV emission is attributed to the presence of WDs (either from an
accretion disc or a stellar wind irradiated by the hot WD photosphere) rather
than the chromospheric activity due to magnetic fields on the surface of
sinlge giants and dwarfs (e.g. Simon & Drake, 1989; Konstantinova-Antova et
al., 2013; Smith, 2018), the GALEX photometry of a sample of single M-type
giants (986 giants) from the Catalogue of Stellar Spectral Classifications
(Skiff, 2014) and a sample of single M-type dwarf (762) selected from the All
sky catalogue of bright M dwarfs (Lépine & Gaidos, 2011) were explored. The
upper panels in Figure 3 display the FUV-NUV distribution of M-type giants and
DA-type WDs and the lower, right panel the distribution of M-type dwarf stars.

The mean FUV-NUV colour index of M-type giants is 3.271 mag with a standard
deviation of 0.703 mag, which is consistent with previous studies (see also
fig. 1 in Smith et al., 2014). This analysis also shows that late M-type
giants are described by lower FUV-NUV colour index than the earlier types (see
fig. 1 in Smith et al., 2014). As for M-type dwarfs, the mean FUV-NUV colour
index is 1.76 mag with a standard deviation of 0.776 mag. A recent study on
the chromospheric activity of cold dwarfs has shown that late type dwarfs,
with low effective temperature, are characterized by a weaker chromospheric
activity (Boro Saikia et al., 2018). Nearly 100 M dwarfs or 13 percent are
found to exhibit FUV-NUV$<$1\. This number becomes significantly smaller in
case a more restricted FUV-NUV$<$0.5 cut-off is applied (23 sources or 3
percent).

On the other hand, the single DA WDs display a mean FUV-NUV=0.064 mag with a
standard deviation of 0.719 mag (14652 WDs) being significantly bluer than
both giants and dwarfs. From the sample of DA WDs, we find that 91 percent
exhibit FUV-NUV$<$1, while the same percentage for M-giants is only 1.5
percent. This does not necessarily mean that there are no WDs in binary
systems with FUV-NUV$>$1\. Overall, the identification of binary systems with
a WD companion is not feasible by employing only the FUV-NUV colour, as it can
also be seen in the figures 5 and 6 in Bianchi & Shiao (2020). The FUV-NUV$<$1
colour cut-off is valid for excluding the vast majority of single M-type giant
and dwarf stars, but additional data are needed. We also deduce that SySts
with weak UV emission ($>$19 mag) are likely misidentified. Based on the
sample of known SySts with available GALEX measurements, genuine SySts can
exhibit UV photometry down to 22 mag (right panel in Figure 2). Recently, a
new classical SySt (Hen 3-860), previously classified simply as H$\alpha$
emitter, was discovered in the ASAS-SN catalogue (Merc et al., 2022). This new
SySt is not recovered in this work, simply because there is no available GALEX
photometric data.

It should also be noted that there are two sources (BY CVn and V934 Her) with
FUV-NUV$<$1 in the list of M-type giants used for this analysis, and I
searched for further information in public catalogues. V934 Her is a known
SySt (Akras et al., 2019a; Merc et al., 2019) while BY CVn is classified as
MIII-type long-period variable star in various catalogues (All-sky Compiled
Catalogue of 2.5 million stars (Kharchenko, 2001), JMMC Stellar Diameters
Catalogue (JSDC) Version 2 (Bourgés et al., 2014), the Tycho-2 Spectral Type
Catalogue (Wright et al., 2003), 2nd Catalogues of Radial Velocities with
Astrometric Data (Kharchenko et al., 2007), Extended Hipparcos Compilation
(XHIP) (Anderson & Francis, 2012), All-sky spectrally matched Tycho2 stars
(Pickles & Depagne, 2010), Catalogue of Stellar Spectral Classifications
(Skiff, 2014), General Catalogue of Variable Stars (Samus’ et al., 2017),
AKARI/HIP and AKARI/2MASS samples (Ita et al., 2010)).

BY CVn is also included in the 9th Catalogue of Spectroscopic Binary Orbits
(SB9, Pourbaix et al., 2004). It is certainly a binary system with an orbital
period of 496.7 days (Famaey et al., 2009). A binary model of this object
implies a1$\times$sini=6.62759e+07 km with a standard deviation (SD) of 2e+06
km and mass function f(m)=0.0471997 M$\odot$ with SD=0.0043 M$\odot$. These
parameters are comparable with those obtained from known Systs (Fekel et al.,
2000) or even binary systems with WD companion. We, hence, consider BY CVn as
a good SySt candidate for further investigation.

Figure 5: FUV/NUV photometry (left panel) and FUV-NUV colour index (right
panel) distributions of Barium stars.

###  5.2 SySts versus Barium and Technetium-poor stars

Besides SySts, our GALEX/ASAS-SN list may also contain other families of WD +
RG binary systems such as Barium stars and Technetium-poor (Tc-poor) extrinsic
S stars444For more information about these families of WD + RG binary systems,
the reader should refer to McClure et al. (1980); Han et al. (1995); McClure &
Woodsworth (1990); Jorissen et al. (2019)). These two groups of stars exhibit
peculiar abundances of s-process elements like Ba, Sr or Tc that cannot be
explained from the evolutionary status of single stars. The process of mass
transfer between the companions in binary systems has been proposed to explain
these peculiarities McClure et al. (1980); Han et al. (1995); McClure &
Woodsworth (1990); Jorissen et al. (2019).

Barium stars have been found to be either dwarf or giant stars (Escorza et
al., 2017). A UV-excess in Barium stars has been reported relative to normal
giants (Böhm-Vitense et al., 2000; Gray et al., 2011) indicative of a hot
compact companion. Most of them have been proved to be in binary systems
(McClure et al., 1980), in which the more massive companion evolved faster, it
produced the s-process elements in their interior during the Asymptotic Giant
Branch (AGB) phase, brought them to the surface through the 2nd and 3rd
dredges up processes, and finally contaminated the less massive and less
evolved companion with the s-process elements through mass transfer. When the
secondary is finally evolved to the giant phase, the primary has become a
white dwarf.

The analysis of the IR and UV criteria in this work ensure the presence of a
giant and WD companions, respectively. Therefore, Barium and extrinsic Tc-poor
S stars may also be included in our GALEX/ASAS-SN list. To probe this
hypothesis, we gathered a sample of Barium and extrinsic Tc-poor S stars from
the literature in order to explore their distributions in IR and UV colours.
Our lists contain 327 Barium and 50 Tc-poor S stars (a list of 30 Tc-rich
intrinsic S stars is also used as a test sample).

Barium stars exhibit a mean 2MASS J-H colour index equal to 0.51 (SD=0.14) mag
which is clearly smaller than the lower limit of SySts (J-H$>$0.78) and they
can be easily distinguished. From the list of 327 Ba stars, only 14 (4
percent) satisfy the IR criteria of SySts (Akras et al., 2019b).
Interestingly, all Barium stars violate the criterion that separates SySts
from single red giants (K-W3$>$0.27, see fig. A5 in Akras et al., 2019b).
Barium stars have a mean K-W3 colour equal to 0.06 with SD=0.21. Regarding the
Tc-poor S stars, 41 out of 50 Tc-poor S stars are found to satisfy all the IR
criterion of SySts, but they also violate the criterion between SySts/RGs
((K-W3)mean=0.16, SD=0.22). The remaining nine Tc-poor S stars have J-H$<$0.78
mag.

From the analysis above, it is clear that the possibility of finding a Barium
or Tc-poor S stars in the GALEX/ASAS-SN list of candidates is nearly
negligible. Both are well separated from SySts based on their infrared
2MASS/WISE colour indices, despite that they are all members of the WD + RG
binary systems. SySts are characterized by higher J-H, K-W3 and W3-W4 colour
indices compared to Barium and Tc-poor S stars, indicative of higher mass-loss
rates. This might be one of the reasons that symbiotic activity is not
triggered in most of the Barium and Tc-poor S stars.

GALEX UV photometry has also been gathered for our lists of Barium stars (101
have available measurements from both GALEX bands) and Tc-poor S stars (18).
Figure 5 displays the distribution of the FUV and NUV photometry, as well as
the FUV-NUV colour index for our sample of Barium stars. All but three Barium
stars and all Tc-poor S stars display FUV-NUV$>$1\. The mean of FUV-NUV colour
index of Barium and Tc-poor S stars is calculated 4.8 mag (SD=1.7 mag) and
2.38 mag(SD=0.85 mag), respectively. Hence, both classes of WD + RG binary
systems exhibit a distinct difference in the GALEX colour index from SySts.
This is the second main difference, besides the IR colour indices, between
SySts and Barium/Tc-poor S stars.

It is worth mentioning that among the three Barium stars with FUV-NUV$<$1,
there are two known SySts which also show the Barium phenomenon (BD-21 3873:
FUV-NUV=-0.562 mag, LT Del: FUV-NUV=-0.443 mag) and a variable star (29 Dra:
FUV-NUV=0.312 mag). 29 Dra is a well-known WD+RG binary system (K0 III, Fekel
et al., 1993; Zboril & Messina, 2009; Bilíková et al., 2010; Pickles &
Depagne, 2010) with an orbital period of 903.8 days (Fekel et al., 1993) and a
rapidly rotating giant with strong chromospheric activity (strong Ca II H and
K emission lines). The giant star in this system is characterized by non-
chemical abundances peculiarities (e.g., Fekel & Simon, 1985; Zacs et al.,
1997; Barisevičius et al., 2010; Merle et al., 2016) and positive FUV-NUV
index. Moreover, it only marginally passes the criteria of S-type SySts
(J-H=0.705$\pm$0.504, K-W3=-0.004$\pm$0.423, W1-W2=0.338$\pm$0.466,
W1-W4=0.234$\pm$0.305) because of the large photometric error. The WD
component has M=0.55 M$\odot$, slightly larger than the threshold of 0.5
M$\odot$ (Merle et al., 2016), Teff=30000 K and logg=8 (Fekel & Simon, 1985).
It’s FUV-NUV colour index of 0.312 mag is comparable with the colour index of
SU Lyn 0.323 mag. It is also characterized by hard X-ray emission (e.g.,
Bilíková et al., 2010) like the $\delta$-type no shell-burning (accreting-
only) SU Lyn. Therefore, any potential symbiotic activity in this binary
system may not be detectable through the typical optical emission lines
criteria.

The mean FUV-NUV colour index was also computed for the small sample of 9 Tc
rich stars (Jorissen et al., 2016, 2019), 22 carbon-enhanced, metal-poor stars
(8 CEMP-no, 7 CEMP-rs, and 7 CEMP-s, Beers & Christlieb (2005)), and 9 CH
stars with available measurements in both GALEX filters. None of these types
of stars is characterized by a low FUV-NUV colour index. Despite the small
populations of our samples, their mean FUV-NUV colour indices are clearly
higher than 1: Tc rich: 3.02(SD=1.13) mag, CEMP-no: 4.62(SD=1.07) mag, CEMP-
rs: 4.48(SD=1.71) mag, CEMP-s: 4.41(SD=1.69) mag, and CH: 6.17(SD=0.96) mag.

All these classes of sources have been proposed to be members of binary
systems with a WD companion (McClure et al., 1980; McClure & Woodsworth, 1990;
Han et al., 1995; Jorissen et al., 2016, 2019). Their high FUV-NUV colour
indices, e.g. Barium and Tc-poor S stars, distinguish them from SySts. Their
WD companions have significantly different stellar parameters than those in
SySts or no accretion disk. For the connection between SySts and peculiar
stars as well as the conditions necessary to provoke symbiotic activity and/or
barium enhancement, the reader should refer to the review by Jorissen (2002).

##  6 Conclusions

In this work, I searched for new candidate SySts hidden in the ASAS-SN
catalogue of variable stars III, combining photometric data from GALEX, Gaia,
2MASS and AllWISE. The discovery of the accreting-only SU Lyn and THA 15-31
SySts with weak or absent optical emission, intrigued me to search for SySts
without take into account information from the visible.

GALEX UV photometry was useful to recover this hidden population of SySts
unveiling the presence of the WDs in the spectral regime where red giants are
not the dominant source. By applying the IR selection criteria of S-type SySts
(Akras et al., 2019b) to the ASAS-SN catalogue of variables stars, I obtained
a list of potential SySts with RG companions (29,237) as it can be seen from
their MG magnitude and GBP-GRP colour indices. The cross-match of that list
with the revised catalogue of GALEX UV sources (Bianchi et al., 2017) provided
us with 814 matches, from which only 105 have available photometry from both
GALEX bands. The remaining 709 sources have only NUV photometry and were
considered only as potential candidate SySts.

Based on the FUV/NUV photometry as well as the FUV-NUV colour index of known
SySts, it was found that the FUV-NUV$<$1 criterion is a reliable indicator for
the presence of WDs. Moreover, the comparison of the GALEX photometry of WDs,
M-dwarfs and M-giants demonstrated that the FUV-NUV$<$1 criterion yields
mainly WDs with very low contamination from giant stars. The higher the FUV-
NUV colour index, the higher the contamination from M-dwarfs and thus
M-giant+M-dwarfs binary systems.

The presence of two emission line stars (SS 305 and THA 23-30) in our list of
GALEX/ASAS-SN candidates verified the method with follow-up optical
spectroscopy. The symbiotic nature of both candidates is confirmed with the
detection of the molecular bands and the strong H$\alpha$, He ii
$\lambda$4686, O vi $\lambda$6380 Raman-scattered lines, among others.

These new discoveries strongly supported the combination of the IR, GALEX and
Gaia information for searching potential SySt candidates. We also argued that
many more SySts are likely hidden in our list of GALEX/ASAS-SN candidates
(814) with only NUV photometry available. Considering that there are at least
7 genuine SySts in our list of 105 GALEX/ASAS-SN candidates with available FUV
and NUV bands photometry, we reckoned that around 50 more SySts may be
included in our list of GALEX/ASAS-SN candidates. Two more interesting sources
with FUV-NUV$<$1 were found in the lists of M-type giants (BY CVn) and Barium
stars (29 Dra).

## Acknowledgements

I would like to thank the anonymous Referee for the constructive comments that
improved the paper. I would also like to thank Denise Rocha Gonçalves and
Gerardo Juan Manuel Luna for their comments and suggestions on an earlier
draft of this paper. This paper makes use of data from the ASAS-SN catalogue
of Variable Stars III, from the Two Micron All-Sky Survey which is a joint
project of the University of Massachusetts and the Infrared Processing and
Analysis Centre/California Institute of Technology, funded by the NASA and the
National Science Foundation (NSF), and from the Wide-field Infrared Survey
Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the NASA. This work has also made use of data from the European
Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed
by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. SIMBAD database, operated at
CDS, Strasbourg, France has also been used. Finally, the following software
packages in Python were employed: Matplotlib (Hunter, 2007), NumPy (Van der
Walt et al., 2011), SciPy (Virtanen et al., 2020) and AstroPy Python (Astropy
Collaboration et al., 2013, 2018).

## DATA AVAILABILITY

The data and results of this article will be shared on reasonable request to
the corresponding author.

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#  Kinetic rate coefficients for electron-driven collisions with CH+:
dissociative recombination and rovibronic excitation.

Joshua Forer1,2 , Dávid Hvizdoš3, Mehdi Ayouz4, Chris H. Greene3, Viatcheslav
Kokoouline1
1Department of Physics, University of Central Florida, 32816, Florida, USA
2Institut des Sciences Moléculaires, Université de Bordeaux, CNRS UMR 5255,
33405, Talence Cedex, France
3Department of Physics and Astronomy and Purdue Quantum Science and
Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA
4Université Paris-Saclay, CentraleSupélec, Laboratoire de Génie des Procédés
et Matériaux, 91190, Gif-sur-Yvette, France Email<EMAIL_ADDRESS>
###### Abstract

Cross sections and rate coefficients for rovibronic excitation of the CH+ ion
by electron impact and dissociative recombination of CH+ with electrons are
evaluated using a theoretical approach combining an R-matrix method and
molecular quantum defect theory. The method has been developed and tested,
comparing the theoretical results with the data from the recent Cryogenic
Storage Ring experiment. The obtained cross sections and rate coefficients
evaluated for temperatures from 1 K to 10,000 K could be used for plasma
modeling in interpretation of astrophysical observations and also in
technological applications where molecular hydrocarbon plasma is present.

###### keywords:

molecular processes – plasmas – scattering – astrochemistry – ISM: clouds

††pubyear: 2023††pagerange: Kinetic rate coefficients for electron-driven
collisions with CH+: dissociative recombination and rovibronic
excitation.–Supplementary Material

##  1 Introduction

Hydrides and their ions are often used to trace various characteristics of the
interstellar medium (ISM). Understanding their formation and destruction
mechanisms is therefore necessary in using them as accurate tracers. The CH+
ion, discovered in the interstellar medium by Douglas and Herzberg (1941) and
Adams (1941), was originally thought to be formed primarily through the
reaction

$\ce{C++H2->CH++H}.$ (1)

The reaction is endothermic, requiring 0.398 eV ($\sim$4620 K) to proceed
(Hierl et al., 1997), but typical kinetic temperatures of diffuse clouds are
roughly between 40 and 130 K (Shull et al., 2021). Hence, the reaction cannot
explain the observed abundance of CH+ in diffuse clouds (Godard and
Cernicharo, 2013). Observations of CH+ in diffuse clouds have motivated many
theoretical and experimental studies on the structure and reactivity of the
ion. The structure of the ion has been well established for several decades,
while there is still need for cross sections for processes involving
collisions of CH+ with electrons. In particular, the knowledge of cross
sections for excitation of the ion by electron impact is important for
interpretations of astrophysical observations. For example, rotational
excitation of CH+ by electron impact was found to be the dominant process
producing CH+ in the ISM Godard and Cernicharo (2013).

Besides astronomical applications, cross sections and rate coefficients for e-
+ CH+ collisions are important for the interpretation and modelling of
hydrocarbon plasma behavior. Other than rotational excitation, mentioned
above, one needs the data on vibrational and electronic excitation and
dissociative recombination (DR). CH+ is also a suitable candidate for
benchmark theoretical studies of such processes. On one hand, this is because
there are several experimental measurements (Amitay et al., 1996; Paul et al.,
2022; Kálosi et al., 2022) available – of DR in particular. On the other hand,
the ion has a relatively complex electronic structure, such that theoretical
methods could be tested on this system and, if successful, be applied to other
similar problems.

Our recent theoretical study (Forer et al., 2023) has demonstrated that theory
can now accurately describe the DR process in CH+. The theoretical approach
developed in that study (originated from several previous works (Hamilton and
Greene, 2002; Kokoouline and Greene, 2003; Čurík et al., 2020)) with small
modifications can also be used to obtain cross sections for rotational,
vibrational, and even electronic excitation of the CH+ ion by electron impact.
Because the theoretical method was validated comparing the DR results with the
experimental data, it is expected to provide reliable data on the electron-
impact excitation processes as well. This study is devoted to the theoretical
evaluation of cross sections and rate coefficients for rovibronic excitation
of CH+ ion by electron impact. The only available experimental result on
excitation of CH+ is from a recent study in the Cryogenic Storage Ring (CSR)
(Kálosi et al., 2022), where the rate coefficient for rotational excitation
from the ground rovibrational level $v=0,j=0$ of CH+ to the first excited
$v=0,j=1$ was measured. However, there have been several theoretical studies
on rotational excitation using the Coulomb-Born approximation (Chu and
Dalgarno, 1974; Dickinson and Munoz, 1977; Lim et al., 1999) and a semi-
classical method (Flower, 1979). More recently, vibronic excitation of CH+ was
studied by Jiang et al. (2019) using a combination of an R-matrix approach
(Carr et al., 2012) and molecular quantum defect theory (Seaton, 1983; Aymar
et al., 1996).

The present study is based on a fully quantum description of e- + CH+
collisions and considers rotational and vibrational degrees of freedom of the
target ion, as well as its electronic structure, including the three lowest
electronic states and corresponding Rydberg resonances appearing in the CH+ +
e- collisional spectrum. The article is organized in the following way:
section 2 gives an overview of the present theoretical approach and the
differences between its application for DR and rovibronic excitation, section
3 presents our DR rate coefficients, section 4 describes the Coulomb-Born
approximation and its application to the present rotational excitation
calculations, section 5 is devoted to the discussion of obtained results on
rovibronic excitation, and section 6 concludes our findings.

##  2 Theoretical Approach

The method described in this section combines fixed-nuclei electron-scattering
calculations with the R-matrix method, rovibrational frame transformation, and
multichannel quantum defect theory (MQDT). Only the main elements of the
approach will be presented here. We use the same method to calculate
scattering matrices as in our previous study of the dissociative recombination
(DR) of CH+ (Forer et al., 2023). The theory and computational details have
only two main differences: the formula for obtaining the rovibronic excitation
(RVE) cross sections (instead of DR cross sections) and the vibrational
Hamiltonian used in the vibrational frame transformation of the electronic
S-matrix.

We perform electron-scattering calculations with the R-Matrix method
implemented in UKRMol (Carr et al., 2012; Tennyson, 2010), accessed via the
Quantemol-N interface (Tennyson et al., 2007). The K-matrices for the e- + CH+
system are represented in a basis of electronic scattering channels, indexed
by $n$, $l$, and $\lambda$. The quantum numbers $l$ and $\lambda$ correspond
to the magnitude of the orbital angular momentum of the incident electron and
its projection on the molecular axis, respectively. The electronic states of
the target, $\ce{CH+}$, are indexed by $n$. K-matrices are obtained for
several values of $R$, the internuclear distance of CH+. We then transform the
K-matrices into the S-matrix, via the intermediary matrix of phase shifts,
$\underline{\delta}(R)$. The matrices are formally related by

$\underline{K}(R)=\tan\underline{\delta}(R),\quad\underline{S}(R)=(I+iK)(I-iK)^{-1}=e^{\underline{\delta}(R)}.$
(2)

The reasons for this transformation are twofold: the S-matrix is a smooth
function of $R$, which is necessary to have a more accurate vibrational
integral in the frame transformation, and the S-matrix is used in the formula
to compute the cross sections.

After the electronic S-matrices are obtained for each value of $R$, we proceed
with the vibrational frame transformation. The first difference in this
approach with our study of DR (Forer et al., 2023) is that the vibrational
Hamiltonian is Hermitian. To study DR, we used a complex absorbing potential
to represent absorption by discretizing the continuum. Here, we only need to
obtain vibrational wave functions for bound states. The vibrational frame
transformation proceeds as

$S^{\Lambda}_{n^{\prime}v^{\prime}l^{\prime}\lambda^{\prime},nvl\lambda}=\int
dR\;\phi_{n^{\prime}v^{\prime}}(R)\;S^{\Lambda}_{n^{\prime}l^{\prime}\lambda^{\prime},nl\lambda}\;\phi_{nv}(R),$
(3)

where $v$ indexes a vibrational level. The superscript $\Lambda$ indicates
that the S-matrices are block diagonal with respect to $\Lambda$, the
projection of the total angular momentum on the molecular axis.

The Hermitian Hamiltonian implies, of course, real eigenvalues. The channel
energies in the case of DR are complex, with a nonzero imaginary part for
continuum states. Here, all channel energies and vibrational wave functions
$\phi_{nv}(R)$ are real-valued. The S-matrix on the left-hand-side of (3) is
the vibronic S-matrix and, unlike in the case of DR, unitarity is defined with
the usual spectral norm, i.e., $S^{J}S^{J\dagger}=I$. Following the
vibrational frame transformation, we perform the rotational frame
transformation on the vibronic S-matrix to obtain the rovibronic S-matrix,
i.e.,

$S^{J}_{n^{\prime}v^{\prime}j^{\prime}\mu^{\prime}l^{\prime},nvj{\mu}l}=\sum\limits_{\Lambda}\sum\limits_{\lambda\lambda^{\prime}}(-1)^{l^{\prime}+l+\lambda^{\prime}+\lambda}C^{j^{\prime}\mu^{\prime}}_{l^{\prime}-\lambda^{\prime},J\Lambda}S^{\Lambda}_{n^{\prime}v^{\prime}l^{\prime}\lambda^{\prime},nvl\lambda}C^{j\mu}_{l-\lambda,J\Lambda},$
(4)

where the total angular momentum of the ion-electron system is
$\overset{\rightharpoonup}{J}=\overset{\rightharpoonup}{j}+\overset{\rightharpoonup}{l}$,
$\overset{\rightharpoonup}{j}$ is the total angular momentum of the ion, and
$\mu$ is the projection of $j$ on the molecular axis. The S-matrix, now
expressed in a basis of rovibronic channels, block diagonal over $J$.

For each scattering energy, we then partition $S^{J}$ into blocks
corresponding to open ($o$) and closed ($c$) channels and construct the
diagonal matrix $\underline{\beta}$ for closed channels,

$\underline{S}^{J}=\begin{pmatrix}\underline{S}_{oo}&\underline{S}_{oc}\\\
\underline{S}_{co}&\underline{S}_{cc}\end{pmatrix},\quad\beta_{i^{\prime}i}(E_{\text{tot}})=\frac{\pi}{\sqrt{2(E_{i}-E_{\text{tot}})}}\delta_{i^{\prime}i},$
(5)

where $E_{i}$ is the energy of the $i^{\text{th}}$ channel and is real, and
$E_{\text{tot}}$ is the total energy of the ion-electron system. We proceed
with the closed-channel elimination procedure, borrowed from MQDT, to reduce
the S-matrix to only open channels.

$\underline{S}^{J,phys}(E_{\text{tot}})=\underline{S}_{oo}-\underline{S}_{oc}\left(\underline{S}_{cc}-e^{-2i\underline{\beta}}\right)\underline{S}_{co}.$
(6)

The physical S-matrix, $S^{J,phys}$, is then used to calculate the total RVE
cross section from some initial channel $\ket{nvj\mu}$ to some final channel
$\ket{n^{\prime}v^{\prime}j^{\prime}\mu^{\prime}}$,

$\sigma_{n^{\prime}v^{\prime}j^{\prime}\mu^{\prime}\leftarrow
nvj\mu}(E_{el})=\frac{\pi}{2m_{e}E_{el}}\sum\limits_{J}\frac{2J+1}{2j+1}\sum\limits_{ll^{\prime}}\left\lvert
S^{J,phys}_{n^{\prime}v^{\prime}j^{\prime}\mu^{\prime}l^{\prime},nvj\mu
l}\right\rvert^{2},$ (7)

where $m_{e}$ is the mass of an electron and $E_{el}$ is the incident electron
energy. It is also possible to calculate vibronic excitation (VE) cross
sections, i.e., not including the rotational structure, by simply skipping the
rotational frame transformation (4). The closed-channel elimination procedure
remains identical, except that the S-matrices are block diagonal over
$\Lambda$ and not $J$. The total VE cross section from some initial channel
$\ket{nv}$ to some final channel $\ket{n^{\prime}v^{\prime}}$ is then

$\sigma_{n^{\prime}v^{\prime}\leftarrow
nv}(E_{el})=\frac{\pi}{2m_{e}E_{el}}\sum\limits_{ll^{\prime}}\sum\limits_{\lambda\lambda^{\prime}}\left\lvert
S^{\Lambda,phys}_{n^{\prime}v^{\prime}l^{\prime}\lambda^{\prime},nvl\lambda}\right\rvert^{2}.$
(8)

The cross sections obtained from (7) and (8) only describe a single scattering
event. To better describe conditions in the ISM, kinetic rate coefficients are
needed, which rely on the above cross sections. State-selected kinetic rate
coefficients, for DR, RVE, or VE, are obtained following

$\alpha_{k}(T)=\frac{\int\limits_{0}^{\infty}\sigma(E_{el})\sqrt{2E_{el}/m_{e}}\sqrt{E_{el}}e^{-E_{el}/kT}dE_{el}}{\int\limits_{0}^{\infty}\sqrt{E_{el}}e^{-E_{el}/kT}dE_{el}},$
(9)

where $k$ is the Boltzmann factor and $\sigma$ is a cross section. In
practice, these integrals are carried out numerically. Additionally, one can
average the state-selected rate coefficients obtained from (9) by

$\overline{\alpha}_{k}(T)=\frac{\sum\limits_{i}\alpha_{k}^{i}(T)(2j_{i}+1)e^{-E_{i}/kT}}{\sum\limits_{i}(2j_{i}+1)e^{-E_{i}/kT}},$
(10)

where $i$ indexes starting channels where the total angular momentum quantum
number of the ion is $j_{i}$. The rate coefficient (DR or (R)VE), starting
from some channel indexed by $i$ and obtained with (9), is given by
$\alpha_{k}^{i}(T)$. If $\alpha_{k}^{i}(T)$ is a rate coefficient without
rotational resolution, $j_{i}$ can be taken to be zero in (10).

The precision of theoretical cross sections are only limited by the numerical
precision of the calculations. Experimental measurements have much larger
uncertainties, so comparisons are often made by convolving theoretical results
with experimental parameters. The convolution function differs for every
experimental setup, but Gaussian functions are fairly common:

$\displaystyle\overset{\sim}{\sigma}(E)$ $\displaystyle=\frac{\int
dE\sigma(E_{el})e^{-\left(E_{el}-E\right)^{2}/(2\gamma^{2})}}{\int
dEe^{-\left(E_{el}-E\right)^{2}/(2\gamma^{2})}}$ (11)
$\displaystyle\overset{\sim}{\sigma}(E)$
$\displaystyle=\frac{1}{\gamma\sqrt{2\pi}}\int
dE\sigma(E_{el})e^{-\left(E_{el}-E\right)^{2}/(2\gamma^{2})}.$ (12)

The parameter $\gamma$ is the convolution width (in the same energy units as
the electron energy grid). The prefactor in (12) is the analytic expression of
the denominator in (11). Because calculations are performed numerically on
predetermined grids of scattering energies, (12) and (11) may give different
results near the endpoints.

##  3 Rate Coefficients for Dissociative Recombination

Fig. 1 compares state-selected kinetic DR rate coefficients obtained with the
present method (Forer et al., 2023). DR and RVE are related in the sense that
they are competing processes. During a collision between an ion and an
electron, if the initial channel is not the only open channel, both processes
may take place. This is why we take the DR probability to be the probability
that no RVE occurs. Therefore, DR results exhibiting a certain level of
agreement with experimental results would suggest that RVE is described,
overall, with similar accuracy.

Figure 1:  State-selected DR cross sections from the ground vibronic state of
CH+. The solid line was obtained with (9) using the theoretical DR cross
sections obtained with the current method (Forer et al., 2023). The dashed
line with an error-curve represents recent experimental data (Paul et al.,
2022). The dot-dashed line represents previous calculations (Mezei et al.,
2019).

The present method produces much more accurate kinetic DR rate coefficients
than the previous theoretical results of Mezei et al. (2019) when compared to
recent measurements made at the Cryogenic Storage Ring (Paul et al., 2022)
over the astrophysically relevant temperature range for diffuse clouds
($\sim$40 K – 130 K (Shull et al., 2021)).

##  4 On the Coulomb-Born Approximation for Rotational and vibrational
(De-)Excitation

The dipole moment of CH+ — present in the molecular center-of-mass about which
the molecule rotates — couples partial waves of $\Delta l=\pm 1$, which
reduces the accuracy of our partial wave basis ($l$=0–2) for such a long-range
process as rotational excitation. We include the effect of higher partial
waves in the Coulomb-Born approximation (Boikova and Ob’edkov, 1968; Gailitis,
1976; Chu and Dalgarno, 1974), similar to the method described in the work of
Rabadán and Tennyson (1998), by calculating three different cross sections:
cross sections obtained from our R-matrix method ($\sigma^{\text{R-mat}}$,
calculated according to 7), total cross sections obtained in the Coulomb-Born
approximation representing the contribution of all partial waves
($\sigma^{\text{TCB}}$, 16), and partial cross sections obtained in the
Coulomb-Born approximation representing the contribution of the partial waves
included in our basis ($\sigma^{\text{PCB}}$, 14). The final rovibrational
excitation cross sections are then a sum of the R-matrix cross sections and
the difference between the total and partial Coulomb-Born cross sections,
i.e.,

$\sigma^{\text{RVE}}=\sigma^{\text{R-mat}}+\sigma^{\text{TCB}}-\sigma^{\text{PCB}},.$
(13)

Lower partial-wave scattering is typically not well described by the Coulomb-
Born approximation because the electron is too close to the molecule for the
dipole interaction to be considered a perturbation. This is especially true
for $s$-wave scattering. Therefore, we replace the $l=0$ – 2 partial wave
contribution from the Coulomb-Born approximation with those in our R-matrix
calculations. The partial Coulomb-Born cross sections are given by

$\displaystyle\sigma^{\text{PCB}}_{j^{\prime}v^{\prime}\leftarrow jv}$
$\displaystyle=16\pi\frac{k^{\prime}}{k}\left\lvert\braket{v^{\prime}}{Q_{\xi}(R)}{v}\right\rvert^{2}\frac{2j^{\prime}+1}{2\xi+1}{\small\begin{pmatrix}j&j^{\prime}&\xi\\\
0&0&0\end{pmatrix}}^{2}$ (14)
$\displaystyle\times(2j+1)(2j^{\prime}+1)\sum\limits_{ll^{\prime}}^{l_{\text{max}}}{\small\begin{pmatrix}l&l^{\prime}&\xi\\\
0&0&0\end{pmatrix}}^{2}\left\lvert M_{ll^{\prime}}^{\xi}\right\rvert^{2},$

where $l_{\text{max}}=2$ because our R-matrix calculations only include up to
$l=2$ partial waves. The dipole moment function is given by $Q_{\xi}(R)$ and
the matrix elements $M^{\xi}_{ll^{\prime}}$ are given by

$M^{\xi}_{ll^{\prime}}=\frac{1}{kk^{\prime}}\int\limits_{0}^{\infty}drF_{l}(\eta,r)r^{-\xi-1}F_{l^{\prime}}(\eta^{\prime},r),$
(15)

where $F_{l}(\eta,r)$ is the regular radial Coulomb function, $\eta=-1/k$, and
$\eta^{\prime}=-1/k^{\prime}$.

For an approach that does not treat vibration, the integral
$\braket{v^{\prime}}{Q_{\xi}(R)}{v}$ in (14) can be replaced with the dipole
moment at the equilibrium geometry of the ion. Considering the dipolar
coupling ($\xi=1$), the partial Coulomb-Born cross sections (14) converge to
the following as $l_{\text{max}}\to\infty$:

$\sigma^{\text{TCB}}_{j^{\prime}v^{\prime}\leftarrow
jv}=\frac{8}{3}\frac{\pi^{3}}{k^{2}}\left\lvert\braket{v^{\prime}}{Q_{\xi}(R)}{v}\right\rvert^{2}(2j^{\prime}+1){\small\begin{pmatrix}j&j^{\prime}&1\\\
0&0&0\end{pmatrix}}^{2}f(\eta,\eta^{\prime}),$ (16)

where

$\begin{gathered}f(\eta,\eta^{\prime})=\frac{e^{2\pi\eta}}{(e^{2\pi\eta}-1)(e^{2\pi\eta^{\prime}}-1)}\chi_{0}\frac{d}{d\chi_{0}}\left\lvert{}_{2}F_{1}(-i\eta,-i\eta^{\prime};1;\chi_{0})\right\rvert^{2},\\\
\zeta=\eta^{\prime}-\eta,\quad\chi_{0}=-4\eta\eta^{\prime}/\zeta^{2}.\end{gathered}$
(17)

The function ${}_{2}F_{1}(a,b;c;z)$ is the Gaussian hypergeometric function
defined as:

${}_{2}F_{1}(a,b;c;z)=\sum\limits_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}},\quad(q)_{n}=\frac{\Gamma(q+n)}{\Gamma(q)}.$
(18)

It should be noted that we only include the Coulomb-Born correction to $\Delta
j=\pm 1$ transitions; our $\Delta j=\pm 2$ transitions are obtained purely
from our R-matrix method. For further detail in computing (17), we invite the
reader to read the work of Chu and Dalgarno (1974). Additionally, the work of
Rabadán and Tennyson (1998) contains minor errors (non-squared Wigner 3-$j$
symbols) in their equations (3) and (4), which are corrected here.

##  5 Rate Coefficients for Rovibronic (De-)Excitation

Fig. 2 shows the VE rate coefficients obtained by the present method. The
corresponding state-selected kinetic rate coefficients agree well with those
of Jiang et al. (2019), as shown in Fig. 2. In Fig. 3, we compare the present
VE rate coefficients to the same work, starting from the ground vibronic state
to the first excited electronic state. The agreement is worse for excitation
between electronic states, possibly due to our improved treatment of channels
attached to the excited electronic states.

Figure 2:  State-selected kinetic VE rate coefficients within the ground
electronic state of CH+. Solid lines represent rate coefficients from the
present calculations, dashed lines are taken from a previous calculation
(Jiang et al., 2019). The cross sections are obtained according to (8) and the
kinetic rates are obtained according to (9).  Figure 3:  State-selected
kinetic VE rate coefficients from the ground electronic state of CH+ to the
first excited state of CH+. Solid lines represent rate coefficients from the
present calculations, dashed lines are taken from a previous calculation
(Jiang et al., 2019). The cross sections are obtained according to (8) and the
kinetic rates are obtained according to (9).

Fig. 4 shows the present RE cross sections convolved with a Gaussian
distribution according to (12,11) with widths ($\gamma$) of 1 meV and 5 meV to
demonstrate the smoothing out of resonances and to compare their overall
magnitude. The $\Delta j=\pm 1$ transition is the largest over all displayed
electron energies ($<1$ eV), as expected for such a strongly dipolar system as
CH+. Fig. 5 shows the individual cross sections of (13), which show an
increasing correction as a function of the incident electron energy. The
$\sigma^{\text{RVE}}$ and $\sigma^{\text{R-mat}}$ cross sections are not
convolved, unlike Fig. 4.

Figure 4:  Rotational excitation cross sections within the ground vibronic
state of CH+. The cross sections are obtained according to (7), and convolved
with a Gaussian as per (11) with $\gamma=1$ meV (thin lines) and $\gamma=5$meV
(thick lines).

Fig. 6 compares the present rotational excitation rate coefficients with those
obtained by Hamilton et al. (2015), who included the Coulomb-Born correction
for $\Delta j=\pm 1,\pm 2$ transitions. We only include this correction for
$\Delta j=\pm 1$ transitions, but the agreement between the results is good
overall. Hamilton et al. (2015) also use an R-matrix approach, but use the
adiabatic-nuclei-rotation approximation to obtain rotational excitation (RE)
cross sections and rate coefficients, while we use a frame transformation (4)
to describe the rotational structure of the ion. Fig. 7 illustrates the
difference between the state-selected RE rates with and without the Coulomb-
Born approximation, compared to the experimentally determined RE rate
coefficient at the CSR (Kálosi et al., 2022) and again to the results of
Hamilton et al. (2015). Our present results with the Coulomb-Born correction
show the best overall agreement with the CSR measurements, although all
theoretical rates are within the provided 1-$\sigma$ uncertainty for most of
the kinetic temperatures shown (10–140 K).

Figure 5:  Comparison of cross sections for rotational excitation $j=0\to
j^{\prime}=1$ obtained using the R-matrix approach with $s$, $p$, and $d$
partial waves ($\sigma^{\text{R-mat}}$) and the closed-form total Coulomb-Born
approximation ($\sigma^{TCB}$). The figure shows also the partial Coulomb-Born
cross section obtained with $s$, $p$, and $d$ partial waves
($\sigma^{\text{PCB}}$) and the cross section where the R matrix data is
combined with the total Coulomb-Born cross section accounting for partial wave
with $l>2$ ($\sigma^{\text{RVE}}$).  Figure 6:  State-selected kinetic RE rate
coefficients within the ground vibronic state of CH+. Solid lines represent
the present theory, dashed lines represent the results of Hamilton et al.
(2015).  Figure 7:  State-selected kinetic RE rate coefficients from $j=0$ to
$j^{\prime}=1$ within the ground vibronic state of CH+. The measured rates
coefficients from the CSR (Kálosi et al., 2022) are compared to the results of
Hamilton et al. (2015) and our kinetic rate coefficients with
($\alpha^{\text{RVE}}$) and without ($\alpha^{\text{R-mat}}$) the Coulomb-Born
correction.

##  6 Conclusions

This paper presents kinetic state-selected DR, VE, and RE rate coefficients
obtained with our DR method (Forer et al., 2023), which also allows us to
calculate RVE cross sections and rate coefficients with little extra effort.
The DR rate coefficients agree well overall with recent experimental
measurements made at the Cryogenic Storage Ring (Paul et al., 2022) and better
than previous theoretical treatments. Our VE rate coefficients, compared to
the work of Jiang et al. (2019), agree well for vibrational excitation within
the ground electronic state of CH+. However, the results differ by up to an
order of magnitude for vibronic excitation to the first excited state of CH+,
which we attribute to a more accurate description of channels attached to
excited electronic states (Forer et al., 2023).

We also obtain RE rate coefficients within the ground electronic state of CH+,
which we compare to the work of Hamilton et al. (2015) which is an R-matrix
method that describes rotational excitation with the adiabatic-nuclei-rotation
approximation and does not treat vibration. They correct their $\Delta j=\pm
1,\pm 2$ transitions with the Coulomb-Born approximation, while we only do so
for $\Delta j=\pm 1$ transitions. Results between our approaches agree well
over the presented kinetic temperatures. Compared to $j=0\to j^{\prime}=1$
rate coefficients recently measured at the CSR (Kálosi et al., 2022), our
theoretical results using the Coulomb-Born correction agree better over all
plotted kinetic temperatures than our theoretical results without the Coulomb-
Born correction, and slightly better than the recent theoretically determined
rate coefficients of Hamilton et al. (2015) over most kinetic temperatures
between 10 K and 140 K. However, all theoretically determined rates under 100
K are within the experimental uncertainty.

## Acknowledgements

We are thankful for the support from the National Science Foundation, Grant
Nos.2110279 (UCF) and 2102187 (Purdue), the Fulbright-University of Bordeaux
Doctoral Research Award, and the program “Accueil des chercheurs étrangers” of
CentraleSupélec.

## Data Availability

State-selected kinetic rate coefficients from the present calculations are
included in the supplementary materials.

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## Supplementary Material

Supplementary Material is available in the online version of this article