11institutetext:  Rosseland Centre for Solar Physics, University of Oslo, P.O.
Box 1029 Blindern, NO-0315 Oslo, Norway
11email<EMAIL_ADDRESS>22institutetext: Institute of Theoretical
Astrophysics, University of Oslo, P.O. Box 1029 Blindern, NO-0315 Oslo, Norway
33institutetext: Dep. of Space, Earth and Environment, Chalmers University of
Technology, Onsala Space Observatory, 43992 Onsala, Sweden 44institutetext:
Grupo de Astrofísica Molecular, Instituto de Física Fundamental, IFF-CSIC, C/
Serrano, 123, 28006, Madrid (Spain)

# First detection of AlF line emission towards M-type AGB stars

M. Saberi First detection of AlF line emission towards M-type AGB starsFirst
detection of AlF line emission towards M-type AGB starsFirst detection of AlF
line emission towards M-type AGB starsFirst detection of AlF line emission
towards M-type AGB stars T. Khouri First detection of AlF line emission
towards M-type AGB starsFirst detection of AlF line emission towards M-type
AGB stars L. Velilla-Prieto First detection of AlF line emission towards
M-type AGB starsFirst detection of AlF line emission towards M-type AGB stars
J. P. Fonfría First detection of AlF line emission towards M-type AGB
starsFirst detection of AlF line emission towards M-type AGB stars W. H. T.
Vlemmings First detection of AlF line emission towards M-type AGB starsFirst
detection of AlF line emission towards M-type AGB stars S. Wedemeyer First
detection of AlF line emission towards M-type AGB starsFirst detection of AlF
line emission towards M-type AGB starsFirst detection of AlF line emission
towards M-type AGB starsFirst detection of AlF line emission towards M-type
AGB stars

The nucleosynthesis production of fluorine (F) is still a matter of debate.
Asymptotic giant branch (AGB) stars are one of the main candidates for F
production. However, their contribution to the total F budget is not fully
known due to the lack of observations. In this paper, we report the detection
of aluminium monofluoride (AlF) line emission, one of the two main carriers of
F in the gas-phase in the outflow of evolved stars, towards five nearby
oxygen-rich (M-type) AGB stars. We studied the Atacama large millimetre/sub-
millimetre array (ALMA) observations of AlF ($v=0$, $J$ = 4–3, 9–8, 10–9, and
15–14) and ($v=1$, $J$ = 7–6) line emission towards $o$ Ceti, and ($v=0$, $J$
= 7–6 and 15–14) lines towards R Leo. We also report a tentative detection of
AlF ($v=0$, $J$ = 7–6) line in IK Tau, ($v=0$, $J$ = 15–14) line towards R
Dor, and ($v=0$, $J$ = 7–6 and $J$ = 15–14) lines in W Hya. From spatially
resolved observations, we estimated the AlF emitting region with a radius
$\sim 11R_{\star}$ for $o$ Ceti and $\sim 9R_{\star}$ for R Leo. From
population diagram analysis, we report the AlF column densities of $\sim
5.8\times 10^{15}$ cm-2 and $\sim 3\times 10^{15}$ cm-2 for $o$ Ceti and R
Leo, respectively, within these regions. For $o$ Ceti, we used the C18O
($v=0$, $J$ = 3–2) observations to estimate the H2 column density of the
emitting region. We found a fractional abundance of $f_{\rm
AlF/H_{2}}\sim(2.5\pm 1.7)\times 10^{-8}$. This gives a lower limit on the F
budget in $o$ Ceti and is compatible with the solar F budget $f_{\rm
F/H_{2}}=(5\pm 2)\times 10^{-8}$. For R Leo, a fractional abundance $f_{\rm
AlF/H_{2}}=(1.2\pm 0.5)\times 10^{-8}$ is estimated. For other sources, we
cannot precisely determine the emitting region based on the available data.
Assuming an emitting region with a radius of $\sim 11R_{\star}$ and the
rotational temperatures derived for $o$ Ceti and R Leo, we crudely
approximated the AlF column density to be $\sim(1.2-1.5)\times 10^{15}$ cm-2
in W Hya, $\sim(2.5-3.0)\times 10^{14}$ cm-2 in R Dor, and
$\sim(0.6-1.0)\times 10^{16}$ cm-2 in IK Tau. These result in fractional
abundances within a range of $f_{\rm AlF/H_{2}}\sim(0.1-4)\times 10^{-8}$ in W
Hya, R Dor, and IK Tau.

###### Key Words.:

Stars: abundances – Stars: AGB and post-AGB – Stars: circumstellar matter

##  1 Introduction

Fluorine (F) is among the few elements whose cosmic origin is still the
subject of debate. It has only one stable isotope ($\rm{}^{19}F$), which can
be easily destroyed by proton, neutron, and alpha particle capture reactions
in stellar interiors (e.g. Ziurys et al. 1994; Abia et al. 2015). F is the
most electronegative element, so it is extremely chemically reactive and can
strongly bond to electron donors such as metals (Ziurys et al. 1994).

There are several scenarios to explain the cosmic F production: (i) He-burning
shell flashes in asymptotic giant branch (AGB) stars with initial masses of
$\sim$2–4 $M_{\odot,}$ subsequent thermal pulses, and third dredge-up; (ii)
neutrino process occurring during supernova explosions; (iii) mergers between
helium and carbon-oxygen white dwarfs; (iv) He-burning phase in Wolf-Rayet
(WR) stars; (v) rapidly rotating massive stars (e.g. Woosley & Weaver 1995;
Meynet & Arnould 2000; Karakas 2010; Longland et al. 2011; Abia et al. 2015;
Jönsson et al. 2017; Limongi & Chieffi 2018; Ryde et al. 2020). A detailed
overview of the F production sites and its role on the Galactic chemical
evolution can be found in Grisoni et al. (2020, and references therein). The
relative contributions of the aforementioned sites must be constrained by
observations (e.g. Timmes et al. 1995; Spitoni et al. 2018; Olive & Vangioni
2019). Among all these candidates, AGB stars are the only sites of F
production that have been observationally confirmed (Jorissen et al. 1992;
Federman et al. 2005; Werner et al. 2005; Abia et al. 2015, 2019).

The AGB phase is a late phase of evolution for stars with an initial mass of
1–8 $M_{\odot}$. AGB stars play a significant role in the Galactic chemical
evolution by ejecting of newly synthesised elements to the interstellar medium
through strong stellar winds.

Models of Lugaro et al. (2004) and Karakas (2010) predict that the F
enrichment is a strong function of the initial stellar mass and metallicity.
For solar metallicity models, they predict the highest F formation in AGB
stars with initial masses to be between 2 and 4 $M_{\odot}$ and for them to be
maximal in the 3–3.5 $M_{\odot}$ mass range.

Jorissen et al. (1992) reported a fluorine over-abundance up to 30 times the
solar value in AGB stars using the infrared vibration-rotation lines of
hydrogen fluoride (HF). They also found that the fluorine enrichment is
correlated with the enrichment of atomic carbon. An over-abundance ranging
from 10-250 times the solar abundance are reported in a number of hot post-AGB
stars from far-UV observations of F V and F IV (Werner et al. 2005). This
significant over-abundance of F is believed to be the F synthesised during the
preceding AGB phase, which is brought to the surface of the post-AGB star.

Previous determinations of F abundance in AGB stars are mainly based on near
infrared observations of HF lines. However, significant contamination of these
spectral lines with telluric lines in this wavelength region prevents accurate
determination of the F abundance (e.g. Abia et al. 2009, 2010, 2015).

Chemical equilibrium models of F-bearing species predict a considerable amount
of F to be locked into aluminium monofluoride (AlF) and HF in the outflow of
AGB stars. The metal-containing molecules such as AlOF, CaF, and CaF2 are also
expected to be abundant at radii larger than $10R_{\star}$ in M-type AGB stars
(Agúndez et al. 2020). However, these species have not been detected in the
outflow of AGB stars yet.

The lines of rotational transitions of AlF and HF have been previously
reported in the outflow around the carbon-rich AGB star IRC+10216 (Ziurys et
al. 1994; Agúndez et al. 2011, 2012). They estimated relative abundances of
$f_{\rm AlF/H_{2}}\sim 1\times 10^{-8}$ and $f_{\rm HF/H_{2}}\sim 8\times
10^{-9,}$ which results in a lower limit of $f_{\rm F/H_{2}}\sim 2\times
10^{-8}$ for the total F budget in IRC+10216. This is consistent with the
solar value of $f_{\rm AlF/H_{2}}=(5\pm 2)\times 10^{-8}$ reported by Asplund
et al. (2021). Moreover, Danilovich et al. (2021) recently reported the
detection of rotational lines of AlF ($v=0,J=7-6$) towards the S-type AGB
star, W Aql. They estimated a fractional abundance of $f_{\rm
AlF/H_{2}}=7.2\times 10^{-8}-1.0\times 10^{-7}$ in the inner CSE within a
radius of $\sim 15R_{\star}$ and $f_{\rm AlF/H_{2}}=4.0\times 10^{-8}$ within
$\sim 90R_{\star}$ using radiative transfer modelling.

Observations of multiple transitions of F-bearing species at sub-millimetre
wavelengths can provide a more accurate determination of the total F budget in
AGB stars. In this paper, we report the first detection of rotational lines of
AlF towards M-type AGB stars, $o$ Ceti and R Leo, with the Atacama large
millimetre/sub-millimetre array (ALMA). We also report tentative detection of
AlF lines towards W Hya, R Dor, and IK Tau.

Table 1: M-type AGB stars with detected AlF line emission.

Star | $\dot{M}$ | d | $V_{\rm LSR}$ | $V_{\rm exp}$ | $R_{\star}$ | $T_{\star}$ | P | $M_{\star}$
---|---|---|---|---|---|---|---|---
| ($10^{-7}M_{\odot}yr^{-1}$) | (pc) | (km s-1) | (km s-1) | (mas) | (K) | (days) | ($M_{\odot}$)
$o$ Ceti | $1$ | 102 | 47 | 3 | 15 | 2800 | 332 | 1.0
R Leo | 1 | 130 | 0.5 | 6 | 13.5 | 2800 | 310 | 1.5
W Hya | 1 | 104 | 41 | 7 | 20 | 2950 | 388 | 1.0
R Dor | 1 | 45 | 6.9 | 6 | 27.5 | 2400 | 175† | 1.0-1.3
IK Tau | 80 | 265 | 34 | 18.5 | 6 | 2100 | 470 | 1.1-1.5
111Columns 2 to 8, for the first four sources, are taken from Vlemmings et al.
(2019) and the stellar masses for W Hya, R Dor, and IK Tau are from Danilovich
et al. (2017). For $o$ Ceti and R Leo, they are estimated based on the 17O/18O
ratio presented by De Nutte et al. (2017) and the O isotopic ratios are taken
from Hinkle et al. (2016). For IK Tau, all parameters are taken from Velilla
Prieto et al. (2017, and references therein). We note that $R_{\star}$ listed
here are the measured radii in the near-infrared. †Period of R Dor varies
between 175 and 362 days.

##  2 Observations

We used observations of the ALMA main arrays in Bands 4, 6, 7, and 8 from
projects 2016.1.00004.S, 2017.1.00191.S, 2018.1.00749.S, and 2018.1.00649.S
for $o$ Ceti, projects 2016.1.01202.S and 2017.1.00862.S for R Leo, and
project 2016.1.00119.S for IK Tau. In addition, we used ALMA compact array
(ACA) observations in Band 8 from project 2018.1.01440.S for $o$ Ceti, R Leo,
R Dor, and W Hya and project 2016.2.00025.S for W Hya.

Table 1 lists the observed sources and their physical properties. The observed
lines, their spectroscopic parameters, date of observations, and measured
intensities are listed in Table 2. We also investigated any possible blends
from the Cologne database for molecular spectroscopy (CDMS) (Müller et al.
2001, 2005) and jet propulsion laboratory (JPL) spectroscopic databases
(Pickett et al. 1998) (see notes in Table 2).

The data calibration was performed following the standard ALMA procedures with
the Common Astronomy Software Application (CASA) (McMullin et al. 2007).
Details of the data processing can be found in Khouri et al. (2018); Vlemmings
et al. (2019); Fonfría et al. (2019). The calibration uncertainty will depend
on the flux calibrator used and typically ranges from 5-20$\%$ (Francis et al.
2020). In this paper, we assumed a typical $10\%$ uncertainty for the flux
calibration. The integrated flux uncertainties that are listed in column 7 in
Table 2 include the uncertainties from flux calibration and from Gaussian
fitting on the lines.

Table 2: Detected AlF emission lines towards M-type AGB stars by ALMA
observations.

Source | $\nu$ | Transition | $E_{\rm u}$ | $g_{u}$ | $A_{ul}$ | Flux | FWHM | $\rm V_{c}$ | Ang. res. | M.R.S. | Apr. | Obs. Date | $\phi$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | (GHz) |  | (K) |  | ($10^{-3}s^{-1}$) | (Jy km s-1) | (km s-1) | (km s-1) | (arcsec) | (arcsec) | (arcsec) |
$o$ Ceti | 131.8988 | $v=0,J=4-3$ | 16 | 9 | 0.025 | $0.12\pm 0.04$ | 5.9 | 47.8 | 0.077 | 2.897 | 0.25 | 21-09-17 | 0.9
| 228.7165 | $v=1,J=7-6$ | 1184 | 15 | 0.147 | $0.46\pm 0.14$ | 9.7 | 48.5 | 0.049 | 1.267 | 0.5 | 22-09-17 | 0.9
| 296.6988 | $v=0,J=9-8$ | 71 | 19 | 0.299 | $1.84\pm 0.23$ | 6.6 | 47.7 | 0.286 | 9.238 | 0.5 | 19-11-18 | 0.1
| 329.6416∗ | $v=0,J=10-9$ | 87 | 21 | 0.412 | $4.39\pm 0.59$ | 8.7 | 47.2 | 0.022 | 0.570 | 0.5 | 09-11-17 | 0.8
| 494.2268 | $v=0,J=15-14$ | 190 | 31 | 1.41 | $7.60\pm 0.84$ | 6.2 | 47.4 | 0.172 | 11.962 | 0.5 | 28-11-18 | 0.1
R Leo | 230.7938 | $v=0,J=7-6$ | 44 | 15 | 0.138 | $0.24\pm 0.06$ | 7.2 | -1.0 | 0.133 | 1.941 | 1 | 01-10-16 | 0.5
|  |  |  |  |  | $0.26\pm 0.04$ | 6.4 | 0.3 | 1.275 | 13.166 | 5 | 22-03-17 | 0.6
|  |  |  |  |  | $0.22\pm 0.03$ | 5.8 | 0.2 | 0.365 | 5.521 | 2 | 03-05-17 | 0.0
|  |  |  |  |  | $0.15\pm 0.04$ | 6.2 | -2.3 | 0.028 | 0.695 | 0.3 | 21-09-17 | 0.7
|  |  |  |  |  | $0.24\pm 0.05$ | 4.3 | -0.6 | 0.022 | 0.431 | 0.3 | 03-10-17 | 0.7
|  |  |  |  |  | $0.19\pm 0.05$ | 5.7 | -0.6 | 0.025 | 0.597 | 0.3 | 27-10-17 | 0.9
| 494.2268 | $v=0,J=15-14$ | 190 | 31 | 1.41 | $3.16\pm 0.49$ | 5.7 | 0 | 2.570 | 15.118 | CP | 24-03-19 | 0.2
W Hya | 230.7938∗∗ | $v=0,J=7-6$ | 44 | 15 | 0.138 | $0.49\pm 0.15$ | 11 | 39.0 | 5.089 | 29.537 | CP | 21-05-17 | 0.5
| 494.2268 | $v=0,J=15-14$ | 190 | 31 | 1.41 | $4.29\pm 1.22$ | 7.3 | 41.1 | 2.039 | 13.634 | CP | 08-06-19 | 0.8
R Dor | 494.2268 | $v=0,J=15-14$ | 190 | 31 | 1.41 | $2.13\pm 0.72$ | 7.7 | 5.5 | 2.135 | 13.812 | CP | 08-06-19 | 0.3
IK Tau | 230.7938∗∗ | $v=0,J=7-6$ | 44 | 15 | 0.138 | $0.23\pm 0.04$ | 13.7 | 36.0 | 0.752 | 8.412 | 3 | 13-04-17 | -

222Spectroscopic data are taken from the CDMS. Integrated flux, the full width
at half maximum (FWHM), and the line central velocity (Vc) are from Gaussian
fitting. Flux uncertainties ($\sigma_{Flux}$) are a summation of Gaussian
fitting uncertainty and $10\%$ of the total flux due to calibration
uncertainty of ALMA data. MRS stands for maximum recoverable scale. Apr.
denotes circular aperture used to extract the spectra, and CP denotes where
the spectra are extracted from the central pixel ($6.6^{\prime\prime}\times
4.4^{\prime\prime}$) for the ACA observations. $\phi$ denotes the stellar
variability phase at visual wavelengths where $\phi_{min}=1$ and
$\phi_{max}=0$. ∗ and ∗∗ lines are likely blended with an SO2 line at 329.6459
GHz and a 50TiO2 line at 230.7931 GHz, respectively.

##  3 Excitation analysis and results

Figure 1: Integrated emission of the spatially resolved lines of AlF towards
$o$ Ceti. The lines are integrated in the 131.873-131.883 GHz (41-54 km s-1)
range for the $v$ =0 , $J$ = 4–3 line, 228.675-228.685 GHz (36-59 km s-1) for
$v$ =1 , $J$ = 7–6, and 329.606-329.577 GHz (32-59 km s-1) for the $v$ =0 ,
$J$ = 10–9 line. The scale is given in Jy km/s/beam. The lines shown are
indicated at the top of each panel. The white contours show the 50% stellar
continuum emission level at the corresponding frequencies and the dashed
yellow contours mark the 3-$\sigma$ level of the line emission. The filled
white ellipses indicate the beam size in each observation. In the right panel,
the regions over which the lines were integrated (see text) are indicated by
the dashed blue ellipse (peak AlF emission), dotted magenta circle (CO
$v=1,J=3-2$ line emission region), and dotted-dashed white circle (all AlF
emission lines).

###  3.1 $o$ Ceti

####  3.1.1 AlF population diagram

We identify four rotational lines of AlF within the ground vibrational state
$v$ = 0, $J$ = 4–3, 9–8, 10–9, and 15–14, and the vibrationally excited line
$v$ = 1, $J$ = 7–6, which are listed in Table 2. Rotational transition
frequencies of AlF are taken from CDMS (Wyse et al. 1970; Hoeft et al. 1970).
Figure 1 presents the integrated flux density of spatially resolved AlF lines
towards $o$ Ceti. From the $J$ = 10–9 line, the emitting region is estimated
to have a diameter of $\sim 0.34^{\prime\prime}\sim 34$ au $\sim 22R_{\star}$
based on the $3\sigma$ emission region, where $R_{\star}$ refers to the
stellar radius measured in infrared.

We derived the column density of AlF towards these three regions: 1: the
region with a diameter of $\sim 0.34^{\prime\prime}(\sim 22R_{\star})$ centred
on the star shown in Fig. 1, rightmost panel, shown by a dotted-dashed white
circle, encompassing all five observed AlF lines; 2: a circular region centred
on the star with a diameter of $0.168^{\prime\prime}(\sim 11R_{\star})$ shown
in Fig. 1, rightmost panel, represented by a dotted magenta circle, where we
use the three spatially resolved lines ($v$ =0 , $J$ = 4–3, 10–9, and $v$ =1 ,
$J$ = 7–6); 3: a small elliptical region at north-east of the star with a
diameter of $0.055^{\prime\prime}\times 0.07^{\prime\prime}(\sim 4R_{\star})$
shown in Fig. 1, rightmost panel, represented by a dashed blue ellipse, where
we could only use the AlF ($v$ =0 , $J$ = 10–9, and $v$ =1 , $J$ = 7–6) lines.
The $v$ =0 , $J$ = 4–3 extracted from this region was too weak to be used for
the analysis in region 3.

For regions 1 and 2, we used the population diagram method (Goldsmith & Langer
1999) to estimate the AlF rotational temperature ($T_{\rm rot}$) and mean
column density ($N_{\rm AlF}$). This is a reasonable approximation since AlF
lines are optically thin, as we show later, and AlF arises from a relatively
small region ($<11R_{\star}$) in the inner part. Our analysis also shows that
the rotational temperature does not vary significantly between regions 1 and
2. The rotational temperature describes the excitation of the rotational
levels of the molecule, and in the case of local thermodynamic equilibrium
(LTE), it also represents the kinetic temperature of the gas. For region 3, we
were only able to use the $J$ = 10–9 line in the ground vibrational state, and
a population diagram analysis was not possible.

The population level of a given molecule follows

$ln(\frac{N_{u}}{g_{u}})=ln(\frac{N_{\rm AlF}}{Z})-\frac{E_{u}}{k_{\rm
B}T_{exc}},$ (1)

where $N_{u}$ and $g_{u}$ are the column density and statistical weight of the
upper level, respectively, $N_{\rm AlF}$ is the total column density of the
AlF molecules, $Z$ is the partition function, $E_{\rm u}$ is the energy of the
upper level, $k_{\rm B}$ is the Boltzmann constant, and $T_{\rm exc}$ is the
excitation temperature.

We assumed optically thin emission in the calculation of $N_{\rm u}$, which is
estimated using

$N_{u}=\left(\frac{4\pi d^{2}}{\pi
r_{e}^{2}}\right)\left(\frac{W}{A_{ul}h\nu}\right),$ (2)

where the first term accounts for the geometrical dilution in which $d$ is the
distance to the source and $r_{e}$ is the radius of the emitting region. $W$
is the flux of the line in units (W m-2), and for each line it is individually
extracted from the regions that are listed above. $A_{\rm ul}$ represents the
Einstein-A coefficients, which express the probability of the spontaneous
emission from the upper level $u$ to the lower level $l$.

From the population diagram shown in Fig. 2, we find $T_{\rm rot}=145\pm 40$ K
and $T_{\rm rot}=320$ K for regions 1 and 2, respectively. The accuracy of the
latter results is limited as we only have two points in the population diagram
that correspond to rotational lines in $v$ =0 . Using
$Z(145\leavevmode\nobreak\ {\rm K})=183.57$ and $Z(320\leavevmode\nobreak\
{\rm K})=404.73$, we find mean column densities $N_{\rm AlF}=(3.6\pm
1.1)\times 10^{15}$ cm-2 and $N_{\rm AlF}=2.1\times 10^{16}$ cm-2 for regions
1 and 2, respectively, as listed in Table 3 (Model A).

To test our assumption of optically thin emission, we estimated the optical
depth at the line centre of all lines from all selected regions using

$\tau_{0}=\frac{c^{2}A_{ul}N_{u}}{8\pi\nu^{2}\Delta\nu\sqrt{\pi}/2\sqrt{ln2}}\left(\exp\left(\frac{h\nu}{k_{\rm
B}T_{\rm exc}}\right)-1\right).$ (3)

We find optical depths in the $0.09-0.5$ range in Region 1 and $0.1-0.5$ in
Region 2, confirming our assumptions of emission that is not optically thick.
To investigate the accuracy of the excitation temperature and column density
from the population diagram, we calculated the flux density ($S_{\nu}$) of the
lines using

$S_{v}=N_{\rm AlF}(\pi r_{e}^{2})\frac{g_{u}\exp(\frac{-E_{u}}{k_{B}T_{\rm
rot}})}{Z_{\rm rot}(T_{\rm rot})}\frac{A_{\rm ul}h\nu}{4\pi
d^{2}}\>\>\phi_{\nu},$ (4)

where
$\phi_{\nu}=(exp(-(\nu-\nu_{0})^{2}/\Delta\nu^{2})/(\Delta\nu\sqrt{\pi})$ is
the line profile where $\Delta\nu=\nu_{0}\Delta v/c,$ and we assumed $\Delta
v=4$ km s-1, which fits the width of the observed AlF lines. Using the derived
column densities and rotational temperatures from the population diagrams in
Eq. 4 to reproduce the synthetic spectra in both regions, we significantly
under-predicted the line strength of the vibrationally excited $v$ =1 line
(see dashed blue lines in Fig. 3 for region 1 and upper panel of Fig. 4 for
region 2). As can be seen, the derived temperature and column density only
characterise the populations of the rotational levels, which also indicates
the quasi-thermal excitation discussed above. This can be caused by different
excitation mechanisms dominating the excitation of the $v$ =0 and $v$ =1
levels. The ground state excitation is most likely dominated by molecular
collision, while the vibartionally excited state is mostly populated by the
radiation from the central star and therefore require a higher excitation
temperature. Hence, we considered excitation temperatures for the rotational
levels, $T_{\rm rot}$, and vibrational states, $T_{\rm vib}$, which are
independent. In this way, the number of molecules in a given $v$ state and $J$
level is given by

$N_{v,J}=N_{\rm AlF}(\pi r_{e}^{2})\frac{g_{J}\exp(\frac{-E_{J}}{k_{B}T_{\rm
rot}})}{Z_{\rm rot}(T_{\rm rot})}\>\>\frac{g_{v}\exp(\frac{-E_{v}}{k_{B}T_{\rm
vib}})}{Z_{\rm vib}(T_{\rm vib})},$ (5)

where $g_{\rm J}=(2J+1)$ and $g_{v}=1$ are the rotational and vibrational
degeneracy, $E_{J}$ and $E_{\nu}$ are the rotational and vibrational
excitation energies, and $Z_{\rm rot}$ and $Z_{\rm vib}$ are the rotational
and vibrational partition functions and are given by

$Z_{\rm rot}(T_{\rm rot})=\sum_{J}(2J+1)e^{-J(J+1)B/k_{B}T_{\rm rot}},$ (6)

where we considered that $J$ runs from 0 to 94 and B=16488.355 MHz, which is
taken from CDMS (Yousefi & Bernath 2018).

$Z_{\rm vib}(T_{\rm vib})=\sum_{v}e^{-E_{v}/k_{B}T_{vib}};$ (7)

here, the vibrational state $v$ runs from 0 to 5, which are the levels
available in CDMS. We varied $N_{\rm AlF}$ and $T_{\rm vib}$ in Eq. 5 to
obtain the best fit model to all observed lines in regions 1 and 2. We find
that a vibrational temperature of $T_{\rm vib}=1300\pm 500$ K with associated
$Z_{\rm vib}$=1.71 and column densities of AlF molecules $N_{\rm AlF}=(5.8\pm
2.0)\times 10^{15}$ cm-2 and $N_{\rm AlF}=(3.0\pm 0.7)\times 10^{16}$ cm-2
reproduces the flux density of all observed line in regions 1 and 2,
respectively (see dashed red lines in Figs. 3 and 4). These best models are
selected based on a chi-square analysis and the results are summarised in
Table 3 (Model C).

We note that if we include the $v=1$ data point in the calculation of the
excitation temperature and column density in the population diagram shown in
Fig. 2, this results in an excitation temperature of $T_{\rm rot}=1332\pm 794$
K and a column density of $N_{\rm AlF}=(2.7\pm 0.7)\times 10^{16}$ cm-2 for
region 1. The derived temperature is in agreement with the vibrational
temperature that we derived from separating the rotational and vibrational
temperature above; however, the column density is higher by a factor of four
assuming the same emitting region, which results in an overestimation of the
flux density of all observed lines. For region two, it results in $T_{\rm
rot}=956\pm 100$ K and a column density of $N_{\rm AlF}=(7.5\pm 0.4)\times
10^{16}$ cm-2 for region two which also results to an overestimation of the
flux density of observed lines (Model B in Table. 3).

The AlF lines in region 3 are also not very optically thick, with optical
depths at the line centre of 0.8 and 0.02 for $v=0$ and 1 lines, respectively,
calculated using Eq. 3. For region 3, we assumed the same rotational and
vibrational temperatures of $T_{\rm rot}=320$ K and $T_{\rm vib}=1300\pm 500$
K as we found in region 2. This results in an AlF column density of $N_{\rm
AlF}=(5\pm 2)\times 10^{16}$ cm-2. The model results overlaid with the
observed spectra are presented in the lower panel of Fig. 4. The summary of
the derived column densities in all regions are listed in Table 3.

Figure 2:  Population diagram of observed AlF rotational transitions towards
$o$ Ceti that are listed in Table 2. Blue points correspond to flux densities
that are extracted from region 1 and green points correspond to those from
region 2 (see Sect. 3.1 for explanations). The vibrationally excited $v=1$
lines in both regions are also shown by x symbols, but they were not
considered in the calculations of the rotational temperatures. The vertical
error bars are calculated from the uncertainties of the integrated fluxes
(uncertainty from the Gaussian fitting plus 10$\%$ of the total flux from the
calibration of the data) and are considered in the fitting process. Figure 3:
AlF lines observed by ALMA towards $o$ Ceti that are listed in Table 2
extracted from region 1 (black solid lines) overlaid with model results
(dashed lines). Models indicate that two separate rotational and vibrational
temperatures are need to reproduce all observed lines, which is discussed in
Sect. 3.1. The line rest frequencies and transitions are marked in each panel.
The grey dashed lines show the rms level. Figure 4:  AlF lines observed by
ALMA towards $o$ Ceti extracted from region 2 with a diameter of
$0.168^{\prime\prime}\sim 11R_{\star}$ (upper panel) and region 3 with
diameter of $0.055^{\prime\prime}\times 0.07^{\prime\prime}\sim 4R_{\star}$
(lower panel) shown with black solid lines overlaid with model results (dashed
lines). Models indicate that two separate rotational and vibrational
temperatures are needed to reproduce all observed lines, which is discussed in
Sect. 3.1. The line rest frequencies and transitions are marked in each panel.
The grey dashed lines show the rms level. We note that the asymmetric feature
in the right wing of the $J=10-9$ line in region 3 might be due to a blend
with a high excitation SO2 line lying at 329.6459 GHz that becomes more
prominent in the innermost regions of the circumstellar envelope. Table 3:
Results of excitation analysis of AlF towards $O$ Ceti in three selected
regions shown in Fig. 1.

region | diameter | $T_{\rm rot}$ | $T_{\rm vib}$ | $N_{\rm AlF/H_{2}}$ | $N_{\rm H_{2}}^{*}$ | $f_{\rm AlF/H_{2}}$ | Model
---|---|---|---|---|---|---|---
| ($R_{\star}$) | (K) | (K) | ($\rm cm^{-2}$) | ($\rm cm^{-2}$) |  |
1 | 22 | $145\pm 40$ | $145\pm 40$ | $(3.6\pm 1.1)\times 10^{15}$ |  |  | A
1 | 22 | $1332\pm 794$ | $1332\pm 794$ | $(2.7\pm 0.7)\times 10^{16}$ |  |  | B
1 | 22 | $145\pm 40$ | $1300\pm 500$ | $(5.8\pm 2.0)\times 10^{15}$ | $2.4\times 10^{23}$ | $(2.5\pm 1.7)\times 10^{-8}$ | C
2 | 11 | 320 | 320 | $2.1\times 10^{16}$ |  |  | A
2 | 11 | $956\pm 100$ | $956\pm 100$ | $(7.5\pm 0.4)\times 10^{16}$ |  |  | B
2 | 11 | 320 | $1300\pm 500$ | $(3.0\pm 0.7)\times 10^{16}$ | $2.0\times 10^{24}$ | $(1.5\pm 0.8)\times 10^{-8}$ | C
3 | 4 | 320 | $1300\pm 500$ | $(5.0\pm 2.0)\times 10^{16}$ | $6.1\times 10^{24}$ | $(0.8\pm 0.5)\times 10^{-8}$ | C

333For regions 1 and 2: models A include $v$ =0 lines in the population
diagram (PD) to calculate the excitation temperature and column density which
result to underestimation of the flux density of $v$ =1 lines as shown by blue
dashed lines in Figs. 3 and 4, models B include both $v$ =0 and $v$ =1 lines
in the PD, which results in the overestimation of the flux densities of all
lines, and model C separates $T_{\rm rot}$ and $T_{\rm vib}$ and results in
the best model outcomes, as shown in Figs. 3 and 4 with red and green dashed
lines and discussed in Sect. 3.1.1. ∗ We note that $N_{\rm H_{2}}$ is
uncertain by a factor of two, as discussed in Sect. 3.1.2.

####  3.1.2 AlF/H2 fractional abundance

Estimating the AlF/H2 fractional abundance is subject to a relatively larger
uncertainty because the gas density and temperature distributions in the
extended atmospheres and wind-acceleration regions is variable and complex
with regard to time.

Moreover, Mira is in a binary system including Mira A ($o$ Ceti) and a white
dwarf (Mira B) at a distance of $\sim 0.5^{\prime\prime}$ (Ramstedt et al.
2014). It is well known that the binary companion affects the gas density at
large scales ($5^{\prime\prime}-10^{\prime\prime}$) Ramstedt et al. (2014).
However, the gravitational field is expected to be dominated by Mira A up to
$\sim 0.3^{\prime\prime}\sim 20R_{\star}$ in the inner most region Mohamed &
Podsiadlowski (2012). Therefore, the binary companion is not expected to
influence the gas density in the regions we study here.

The radiation field is likely strongly affected by Mira B in the UV spectral
region, even close to Mira A as suggested by the relative UV brightness of the
two sources reported by Karovska et al. (1997). This probably affects the
molecular abundances in the inner CSE, but how the different parameters of
binary systems influence the abundances of specific molecules is not yet
established. This is a topic of ongoing research (e.g. Saberi et al. 2018,
2019; Van de Sande & Millar 2022). We speculate that the asymmetry seen in the
AlF molecular distribution ($J=10-9$) seen in Fig. 1 could be due to the
gravitational influence of Mira B, but investigating this is beyond the scope
of this paper.

To constraining the H2 gas column density, we used the spatially resolved
C18O($v$ = 0, $J$ = 3–2) line observed with ALMA. We calculated the H2 column
density towards the three different regions described in Section 3.1.1. To
convert the C18O column density to H2 column density, we assumed the typically
used CO fractional abundance $f_{\rm C^{16}O/H_{2}}=4\times 10^{-4}$ for
M-type AGB stars (e.g. Khouri et al. 2018), and a ratio
$\rm{}^{16}O/^{18}O=282\pm 100$ reported for $o$ Ceti by Hinkle et al. (2016).
Together, these imply a fractional abundance of C18O, $f_{\rm
C^{18}O/H_{2}}=(1.4\pm 0.6)\times 10^{-6}$.

For the excitation analysis of the C18O line, we used the
RADEX444http://var.sron.nl/radex/radex.php radiative transfer code. We
considered a line width of 6 km s-1 and varied the excitation temperature and
C18O column density to reproduce the observed C18O flux from observations.
Calculations of the $\rm H_{2}$ gas density in the three selected regions are
given below.

In region 1, the brightness temperature of the C18O line peaks at 13 K,
corresponding to a flux density of 90 mJy as shown in Fig. 7. We varied the
C18O excitation temperature and column density for C18O to reproduce the
observed flux density. Assuming an excitation temperature of $700\pm 300$ K,
we found that a column density $\rm N_{C^{18}O}=(3.3\pm 1.3)\times 10^{17}$
cm-2 reproduces the observed line. Considering $f_{\rm C^{18}O/H_{2}}=(1.4\pm
0.6)\times 10^{-6}$ gives the H2 column densities $N_{\rm H_{2}}=2.4\times
10^{23}$ cm-2. The derived $N_{\rm H_{2}}$ has an uncertainty of a factor of
2.1 considering the uncertainty on the $\rm{}^{16}O/^{18}O$ isotopic ratio and
excitation temperature. Assuming the AlF column density of $N_{\rm
AlF}=(5.8\pm 2)\times 10^{15}$ cm-2 that we describe in the previous section,
we find an AlF fractional abundance of $f_{\rm AlF/H_{2}}\sim(2.5\pm
1.7)\times 10^{-8}$ in region 1.

In region 2, the brightness temperature of the C18O line peaks at 30 K,
corresponding to a flux density of 55 mJy. Similarly to region 1, we varied
the excitation temperature and the column density of C18O to reproduce the
line flux extracted from region 2. We assumed an excitation temperature of
$1500\pm 500$ K and found a column density of $\rm N_{C^{18}O}=(1.5\pm
0.5)\times 10^{18}$ cm-2 that reproduces the C18O line. These yield a H2
column density of $N_{\rm H_{2}}=1.1\times 10^{24}$ cm-2 with an uncertainty
of a factor of 1.8 due to the uncertainties on the isotopic ratio and the
excitation temperature. For the same region, Khouri et al. (2018) reported
$N_{\rm H_{2}}=3.4\times 10^{24}$ cm-2 based on the radiative transfer
modelling of the CO ($v$ = 1, $J$ = 3–2) and 13CO ($v$ = 0, $J$ = 3–2) line.
Thus, a column density of $N_{\rm H_{2}}\sim 2\times 10^{24}$ would be
consistent with both the value determined by us and those by Khouri et al.
(2018) given the intrinsic uncertainties and different approaches, and is
probably a better estimate of the real column density. The best model from
Sect. 3.1.1 gave an AlF column density $N_{\rm AlF}=(3.0\pm 0.7)\times
10^{16}$ cm-2. Therefore, the estimated H2 and AlF column densities result to
a fractional abundance of $f_{\rm AlF/H_{2}}\sim(1.5\pm 0.8)\times 10^{-8}$ in
region 2.

In region 3, the C18O brightness temperature peaks at 160 K corresponding to a
flux density of 38 mJy. We assumed an excitation temperature of $1500\pm 500$
K and found a column density of $(8.5\pm 2.5)\times 10^{18}$ cm-2 for C18O to
reproduce the observed brightness temperature in this region. This implies a
H2 column density of $N_{\rm H_{2}}=6.1\times 10^{24}$ cm-2 with an
uncertainty of a factor of 2.0. From Sect. 3.1.1, we find an AlF column
density of $\rm N_{AlF}=(5\pm 2)\times 10^{16}$ cm-2 for the best model. These
translate to a fractional abundance of $f_{\rm AlF/H_{2}}\sim(0.8\pm
0.5)\times 10^{-8,}$ and in region 3 this is within $4R_{\star}$. The derived
fractional abundances in the three regions are listed in Table 3 (Model C).

Our derived AlF fractional abundances in the three regions are consistent with
chemical models of Agúndez et al. (2020, see Fig. A.4,) which also predict a
mean AlF fractional abundance of $\sim 10^{-8}$ within $\sim 9R_{\star}$ and
also a lower abundance in a range of $\sim 10^{-11}-10^{-8}$ in the innermost
region with a radius of $1R_{\star}<R<3R_{\star}$. Considering the initial
mass of $o$ Ceti $\sim 1M_{\odot}$, our results are also in agreement with the
stellar yield models by Lugaro et al. (2004); Karakas (2010) and are
consistent with the Solar F budget of F/H${}_{2}=(5\pm 2)\times 10^{-8}$
(Asplund et al. 2021).

####  3.1.3 The excitation of AlF

The derived rotational temperatures of 320 K and 145 K from the population
diagram in the inner CSE are rather low and seem to indicate sub-thermal
excitation of AlF. A detailed study of the excitation of AlF is necessary to
understand the distribution of the level populations. However, such an
analysis is complicated by the fact that the radiation field as a function of
position is poorly constrained at the relevant wavelengths (because of, for
example, dust absorption and emission). We do not expect the binary companion
to have a significant effect on the radiation field close to Mira A at the
relevant wavelengths.

Although, the three-dimensional gas density distribution in the inner region
of $o$ Ceti is only constrained by one-dimensional models. Nonetheless, we
were able to estimate the relative effects of collisions and the radiation
field on the excitation of AlF.

The radiative pumping of AlF from $v=0$ to $v=1$ takes place through an
infrared band at 12.48 $\mu$m. Assuming the effective temperature and
effective near-infrared (IR) radius as determined by Wittkowski et al. (2016),
we can estimate the 12.48 $\mu$m flux density for a 2450 K black body, and we
find a flux density of $\sim 500$ Jy. This does not include the contribution
from diffuse emission from dust and the binary companion and, hence, is a
lower limit to the actual radiation field. For comparison, this naked-star
estimate corresponds to 0.23 times the flux taken directly from the infrared
space observatory (ISO)
observations555https://irsa.ipac.caltech.edu/data/SWS/.

To compute the mean intensity at 12.48 $\mu$m from a naked star with the
considered radius and effective temperature, we calculated the dilution of the
radiation field over the solid angle of region 1 ($\Omega=2.134\times
10^{-12}$ sr). We find a mean intensity of $J_{\nu}=F_{\nu}/\Omega=1.04\times
10^{-8}\rm ergs\>s^{-1}$ cm-2 Hz-1 sr-1. The IR pumping rate can be written as
$\rho=A_{\rm vib}J_{\nu}/(2hc(\nu/c)^{3})=0.1{\rm s}^{-1}$, where $A_{\rm
vib}\sim 8.62$ s-1 is the spontaneous-emission coefficient of AlF for a
$v=1-0$ transition. Considering collisional rates of $\sim 2\times 10^{-10}$
cm3 s-1 (Danilovich et al. 2021), we find a upper limit for the H2 densities
below which IR pumping dominates by 109 cm-3. If the radiation field is
stronger than our naked-star estimate, the IR pumping will be efficient at
larger densities. Given the observed ISO spectrum integrated over a much
larger area than region 1, the mean intensity is likely at most a factor of a
few larger than our estimate.

Assuming depths for regions 1 and 2 along the line of sight similar to their
radial extent on the sky ($\sim 5\times 10^{14}$ cm and $\sim 3\times 10^{14}$
cm, respectively), the H2 column densities we obtained imply average densities
in these regions of $\sim 5\times 10^{8}$ cm-3 and $\sim 4\times 10^{9}$ cm-3.
Hence, the radiation field is expected to dominate the excitation through
vibrational pumping in a large fraction of region 1 but not as much in region
2. Interestingly, IR pumping would be expected to help increase the rotational
temperature of the $v=0$ levels for low gas densities, making the low values
we derive puzzling. In order to study the excitation of AlF in detail,
radiative transfer models including the three-dimensional gas-density
distribution and mean intensity at relevant frequencies is necessary.
Nonetheless, the rotational and vibrational temperatures we derive provide an
empirical description of the average excitation of AlF and can be used to
infer column densities as above.

####  3.1.4 HF

As discussed in Sect. 1, HF is among the two most abundant F-bearing species
in the outflow of AGB stars (Agúndez et al. 2020). We investigated the archive
HF data available for $o$ Ceti, aiming to determine the HF fractional
abundance and the total F budget in the gas phase. There is a tentative
detection of HF ($J$ = 1–0) at 1232.476 GHz observed with Herschel/SPIRE.
There is also potential detection of the HF lines ($J$ = 2–1, 3–2, and 4–3)
observed by PACS in Herschel. However, all these lines are blended with H2O
lines, making it difficult to determine the HF fractional abundance based on
low spectral resolution of Herschel data.

###  3.2 R Leo

In R Leo, the AlF ($v$ = 0, $J$ = 7–6) line is detected at five different
epochs with ALMA at various pulsation phases. The data are taken over a period
of one year at a visual phase of $\phi=$ 0.0, 0.5, 0.6, 0.7, and 0.9. One
extra line of AlF ($v$ = 0, $J$ = 15–14) is covered by the ACA observations.
All the covered lines with their spectroscopic parameters and measured
intensities are listed in Table 2. The observations show that the emission
region is barely resolved by the $\sim 0.13^{\prime\prime}$ beam (Fig. 8). The
line flux extracted from the higher angular-resolution images (beam $\sim
0.02^{\prime\prime}$) increases up to apertures of $\sim 0.25^{\prime\prime}$
. Hence, we considered the emission region to have a diameter of $\sim
0.25^{\prime\prime}\sim 33$ au $\sim 18R_{\star}$.

Figure 5 presents the multi-epoch observations of AlF at 230.79 GHz. Our
observations with the highest angular resolution at $\phi=$0.7 and 0.9 are
subject to flux loss due most likely to the limited maximum recoverable scale
of $0.4-0.6^{\prime\prime}$ shown in Fig. 5, left panel, with magenta and cyan
profiles. The flux variation seen in the AlF ($J$ = 7–6) line is most likely
due to low surface brightness sensitivity of the long baseline observations
and the added limitations related to imaging emission with an extent similar
to the maximum recoverable scale. Possible small calibration uncertainties on
the shortest baselines and changes in antenna configuration between the
observations makes a direct comparison between the highest angular resolution
observations uncertain. Therefore, based on the current observations, we
cannot confirm any flux variation due to the stellar variability.

For the population diagram, we used the weighted mean value of the first three
$J$ = 7–6 data points that are listed in Table 2. We removed the three
observations with MRS $<0.6^{\prime\prime}$ that are likely subject to flux
loss. As shown in Fig. 6, we derived the rotational temperature of $T_{\rm
rot}\sim 300$ K and the column density of $N_{\rm AlF}\sim 3.0\times 10^{15}$
cm-2. The accuracy of the results is limited since we only have two points in
the population diagram. We used the column density and the rotational
temperature derived from the population diagram to reproduce the observations.
The observed spectra and the model results are presented in Fig. 5.

Since R Leo and $o$ Ceti have similar mass-loss rates and pulsation periods,
we assumed the same H2 column density of $(2.4\pm 1.4)\times 10^{23}$ cm-2 as
we derived for a region with a radius of 17 au in $o$ Ceti (region 1). We also
crudely estimated an AlF fractional abundance of $f_{\rm AlF/H_{2}}\sim(1.2\pm
0.5)\times 10^{-8}$ for R Leo.

Figure 5:  ALMA observations of AlF line emission towards R Leo (solid lines)
and the results of an LTE model (dashed black lines). Stellar variability
phases ($\phi$) are indicated in the upper-right, and the line transitions are
indicated in the upper-left panels. The grey dashed lines show the rms level.
Figure 6:  Population diagram of observed AlF rotational lines towards R Leo.
The data point with $E_{u}=44$ K is the mean of the first three observations
that are listed in Table 2 (see Sect. 3.2 for explanations).

###  3.3 R Dor, W Hya, and IK Tau

We report a tentative detection of AlF ($v$ = 0, $J$ = 15–14) line emission at
494.223 GHz towards W Hya and R Dor. We also report tentative detection of AlF
($v$ = 0, $J$ = 7–6) line emission at 230.79 GHz towards IK Tau and W Hya. The
observed lines with their properties extracted from Gaussian fits are listed
in Table 2. We assumed an emitting region of $11R_{\star}$, where $R_{\star}$
refers to the stellar radius measured in infrared as listed in Table. 1, for
these sources similar to what we derived from observations of $o$ Ceti. This
results in an emitting region with a radius of 22 au for W Hya, 13.5 au for R
Dor, and 17.5 au for IK Tau.

We considered $T_{\rm rot}$ of 145 K from Mira region one and 300 K from Mira
region two and R Leo to estimate the mean column density in these sources. The
results are listed in Table. 4. The model results overlaid on the observed
spectra are presented in Fig. 9.

To estimate the fractional abundances for these sources, we need to know the
H2 column density. Since AlF emission is compact and it comes from a region
that can still be gravitationally bound to the star, part of the gas will be
ejected and some will fall back to the star. Hence, estimating gas densities
in these inner regions from extrapolating models obtained based on the large-
scale envelope will lead to errors in the estimated gas densities. Moreover,
densities from models for the large-scale envelopes are uncertain by a factor
of a few Ramstedt et al. (2008). Finally, the way in which the densities in
the extended atmosphere differ from those predicted by extrapolating the mass-
loss rate inwards may vary between stars because of differences in the
extended atmospheres and wind-acceleration region (e.g. Habing 1996; Höfner &
Olofsson 2018).

To roughly estimate the AlF fractional abundances in these sources, we used
the H2 column density that we derived for $o$ Ceti of $(2.4\pm 1.4)\times
10^{23}$ cm-2 within a radius of $11R_{\star}$ from the central star. This
suggests the $\rm AlF/H_{2}$ fractional abundance is within a range of
$\sim(0.1-4)\times 10^{-8}$ for these sources as listed in Table 4. This is in
agreement with the Solar fractional abundance of F and the stellar yield
models for AGB stars with initial masses in range of $1-2M_{\odot}$. We note
that the approximation we made above is crude. This is even less certain for
IK Tau and R Dor due to different mass-loss rates (in case of IK Tau) and
pulsation periods (in both cases) with respect to $o$ Ceti, which may cause
even larger differences in the H2 gas density and fractional abundances in the
inner region. A proper determination of AlF abundances will require a study of
gas densities in the AlF emission regions.

Table 4: Approximation of AlF column density and fractional abundance in other
sources.

Star |  $T_{\rm rot}=145$ (K) |  $T_{\rm rot}=300$ (K)
---|---|---
|  $N_{\rm AlF}$ ($\rm cm^{-2}$) | $f_{\rm AlF/H_{2}}$ |  $N_{\rm AlF}$ ($\rm cm^{-2}$) | $f_{\rm AlF/H_{2}}$
R Leo |  |  | $3\times 10^{15}$ | $1.2\times 10^{-8}$
R Dor | $2.5\times 10^{14}$ | $0.1\times 10^{-8}$ | $3\times 10^{14}$ | $0.12\times 10^{-8}$
W Hya | $1.2\times 10^{15}$ | $0.5\times 10^{-8}$ | $1.5\times 10^{15}$ | $0.6\times 10^{-8}$
IK Tau | $0.6\times 10^{16}$ | $2.5\times 10^{-8}$ | $1\times 10^{16}$ | $4.2\times 10^{-8}$
666For R Leo, the $T_{\rm rot}$ and $N_{\rm AlF}$ are estimated using the PD
shown in fig. 6. For other sources, we used $T_{\rm rot}$ estimated for $o$
Ceti and R Leo to approximate $N_{\rm AlF}$. For all sources, the H2 density
of $2.4\times 10^{23}\rm cm^{-2}$ is used as derived for $o$ Ceti.

##  4 Discussion and summary

The cosmic origin of F is still uncertain. AGB stars are among the few
candidates to synthesis F in our Galaxy. From stellar yield models, the
efficiency of F synthesis in AGB stars strongly depends on the initial mass
and metallicity (Lugaro et al. 2004; Karakas 2010). For Solar metallicity, the
F synthesis is maximal for stars with an initial mass of $2-4M_{\odot}$. From
chemical models by Agúndez et al. (2020), a significant amount of F is
expected to be locked into AlF and HF in the outflow of all chemical types of
AGB stars. In this paper, we report the first detection of AlF line emission
towards five oxygen-rich AGB stars observed with ALMA: $o$ Ceti, R Leo, W Hya,
R Dor, and IK Tau.

Towards $o$ Ceti, we detected five rotational lines and determined the
fractional abundance of $f_{\rm AlF/H_{2}}\sim(0.8\pm 0.5)\times 10^{-8}$
within a radius of $2R_{\star}$, $\sim(1.5\pm 0.8)\times 10^{-8}$ within a
radius of$5.5R_{\star}$ ,and $\sim(2.5\pm 1.7)\times 10^{-8}$ within a radius
of $11R_{\star}$ from population diagram analysis. The observations are best
reproduced by considering independent rotational and vibrational excitation
temperatures. These derived fractional abundances at various radii from the
star are in agreement with $f_{\rm AlF/H_{2}}$ molecular distribution for an
M-type AGB star from the recent chemical models by Agúndez et al. (2020). This
indicates how spatially resolved observations of several transitions can
verify the accuracy of chemical models on predictions of molecular fractional
abundances as long as the H2 gas density in the emitting region is known.

Towards R Leo, we find a column density of $3\times 10^{15}$ cm-2 for an
emission region with radius $\sim 9R_{\star}$. For other sources, we
considered the rotational temperatures of 145 K and 300 K as derived for $o$
Ceti and R Leo to make a rough estimation of the AlF column density. These
result in $N_{\rm AlF}\sim(1.2-1.5)\times 10^{15}$ cm-2 in W Hya,
$\sim(2.5-3.0)\times 10^{14}$ cm-2 in R Dor, and $\sim(0.6-1.0)\times 10^{16}$
cm-2 in IK Tau within a radius of $11R_{\star}$ from central stars. However,
spatially resolved observations towards these sources are necessary to resolve
the line emitting regions and constrain the column densities. By assuming the
same H2 column density as we derived for $o$ Ceti, we can make a crude
approximation of the AlF fractional abundance $(1.2\pm 0.5)\times 10^{-8}$ in
R Leo and in a range of $(0.1-4)\times 10^{-8}$ for W Hya, R Dor, and IK Tau.

All observed sources in our sample have an initial mass in the range of $M\sim
1-2M_{\odot}$ and are Galactic sources with metallicities probably similar to
solar; thus, they are not expected to efficiently synthesise fluorine from
stellar yield models by Lugaro et al. (2004); Karakas (2010). Our results for
all sources are in a good agreement with both stellar yield models and
chemical models.

Danilovich et al. (2021) recently reported the detection of AlF and HF towards
the S-type AGB star, W Aql. Using radiative transfer analysis, they found
fractional abundances of $f_{\rm AlF/H_{2}}=1\times 10^{-7}$ and $f_{\rm
HF/H_{2}}=1\times 10^{-8}$. Their reported value in the inner part is a higher
than expected AlF abundance for W Aql, which has an initial mass within a
range of $1.2-1.6M_{\odot}$ reported by De Nutte et al. (2017). This can
indicate that either the mass of W Aql is larger, or that models for fluorine
production predict nucleosynthesis at initial masses that are too large.
However, the uncertainty in the radiative transfer analysis based on a single
line observation and uncertainties in the physical parameters in the inner
CSEs can play an important role in the molecular excitation analysis and
abundance derivation.

The estimated $f_{\rm AlF/H_{2}}$ in all M-type AGB stars in our sample are in
agreement with the reported $f_{\rm AlF/H_{2}}$ in the Sun and the C-type AGB
star, IRC+10216, reported by Asplund et al. (2021) and Agúndez et al. (2012),
respectively. We note that dependency of the AlF abundance on the AGB chemical
type is not observationally constrained. This is due to very few observations
that have been done so far to measure the abundances of F-bearing species in
the outflow of evolved stars due to the lack of sensitivity of previous
generations of observational facilities. Chemical models by Agúndez et al.
(2020) have assumed the same photospheric abundance of $f_{\rm AlF/H_{2}}\sim
10^{-8}$ for M-, S-, and C-type AGB stars. Further observations in a larger
sample from all chemical types are still needed to verify this assumption and
also quantify the total F budget in various chemical types.

From chemical models, a significant overabundance of F due to stellar
nucleosynthesis is expected to be seen in both AlF and HF abundances (M.
Agúndez, priv. comm.). We remind the reader that the low spectral resolution
of Herschel PACS/SPIRE data of HF observations make it impossible to
distinguish the HF lines from the H2O lines that are abundant in M-type AGB
stars. Therefore, our study suggests that observations of AlF lines towards
AGB stars with initial masses of $\mbox{2--4}M_{\odot}$ can provide more
reliable observational evidence of the F nucleosynthesis predicted by stellar
yield models of AGB stars. This is important with regard to understanding the
role of AGB stars in the total F production in our Galaxy.

###### Acknowledgements.

We thank John Black for fruitful discussions on the excitation of the AlF and
all his comments on the manuscript. This paper makes use of the following ALMA
data: ADS/JAO.ALMA$\\#$2016.1.00004.S, $\\#$2017.1.00191.S,
$\\#$2018.1.00749.S, $\\#$2018.1.00649.S, $\\#$2016.1.01202.S,
$\\#$2017.1.00862.S, $\\#$2016.1.00119.S, $\\#$2018.1.01440.S, and
$\\#$2016.2.00025.S. ALMA is a partnership of ESO (representing its member
states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and
ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic
of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. MS
and SW acknowledge support by the SolarALMA project, which has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (Grant agreement No. 682462),
and by the Research Council of Norway through its Centres of Excellence
scheme, project number 262622. TK, LVP and WV acknowledge support from the
Swedish Research Council under grants No. 2019-03777 and 2014-05713. LVP
acknowledges the ERC consolidator grant 614264. JPF has received funding
support from the European Research Council under the European Union’s Seventh
Framework Program (FP/2007-2013) / ERC Grant Agreement n. 610256 NANOCOSMOS.

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##  Appendix A CO observation towards $o$ Ceti

Figure 7 presents $\rm C^{18}O$ ($v=0,J=3-2$) line spectra extracted in region
one towards $o$ Ceti. The line has been used to estimate the H2 gas density in
region 1 as detailed in Sect. 3.1.2.

Figure 7:  ALMA observations of $\rm C^{18}O$ line towards $o$ Ceti from
region one. This line has been used for the estimation of H2 density in region
one as discussed in Sect. 3.1.2.

##  Appendix B R Leo

Figure 8 presents the integrated emission of the AlF ($J=7-6$) line observed
with a $0.133^{\prime\prime}$ beam towards R Leo that is discussed in Section
3.2 as an estimator of the size of the emitting region.

Figure 8:  Integrated emission of AlF ($J=7-6$) line towards R Leo. The lines
are integrated in a range of -7 to 5 km/s. The scale is given in Jy km/s/beam.
The white contours show the 50% stellar continuum emission level at the
corresponding frequencies, and the dashed yellow contours mark the 3-$\sigma$
level of the line emission. The filled white ellipses indicate the beam size
($0.133^{\prime\prime}$) in each observation.

##  Appendix C R Dor, W Hya, and IK Tau

Figure 9 presents the observed spectra for R Dor, W Hya, and IK Tau overlaid
with the model results from Section 3.3.

Figure 9:  ALMA observations of AlF lines towards IK Tau, R Dor, and W Hya
(Black solid lines) overlaid with LTE model results using $T_{\rm rot}=145$ K
(red dashed lines) and $T_{\rm rot}=300$ K (green dashed lines) that are
detailed in Sect. 3.3. The line rest frequencies and transition are stated in
each panel. The AlF ($v$ = 0, $J$ = 7–6) line emission at 230.7938 GHz can
potentially be blended with $\rm{}^{50}TiO_{2}$ line at 230.7931 GHz. The grey
dashed lines show the rms level.


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