# Andromeda XXV - a dwarf galaxy with a low central dark matter density

Emily J. E. Charles,1 Michelle L. M. Collins,1 R. Michael Rich,2 Justin I.
Read,1 Stacy Y. Kim,1 Rodrigo A. Ibata3 Nicolas F. Martin,3 Scott C. Chapman,4
Eduardo Balbinot,5 Daniel R. Weisz 6
1Physics Department, University of Surrey, Guildford, GU2 7XH, UK
2Department of Physics and Astronomy, University of California at Los Angeles,
Los Angeles, CA 90095, USA
3Observatoire astronomique de Strasbourg, Universié de Strasbourg, CNRS, UMR
7550, 11 rue de l’Université, F-67000 Strasbourg, France
4Department of Physics and Atmospheric Science, Dalhousie University, Coburg
Road, Halifax B3H1A6, Canada
5Kapteyn Astronomical Institute, University of Groningen, Landleven 12,
NL-9747 AD Groningen, the Netherlands
6 Department of Astronomy, University of California Berkeley, Berkeley, CA
94720, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Andromeda (And) XXV has previously been reported as a dwarf spheroidal galaxy
(dSph) with little-to-no dark matter. However, the uncertainties on this
result were significant. In this study, we double the number of member stars
and re-derive the kinematics and mass of And XXV. We find that And XXV has a
systemic velocity of $\nu_{\rm r}=-107.7\pm 1.0$ kms-1 and a velocity
dispersion of $\sigma_{\rm\nu}=4.5\pm 1.0$ kms-1. With this better constrained
velocity dispersion, we derive a mass contained within the half-light radius
of M(r$<\rm r\textsubscript{h})=6.9^{+3.2}_{-2.8}\times 10^{6}$ M$\odot$. This
mass corresponds to a mass-to-light ratio of $\text{[M/L]}_{\rm
r_{h}}=37^{+17}_{-15}$ M⊙/L⊙, demonstrating, for the first time, that And XXV
has an unambiguous dark matter component. We also measure the metallicity of
And XXV to be $\rm[Fe/H]=-1.9\pm 0.1$ dex, which is in agreement with previous
results. Finally, we extend the analysis of And XXV to include mass modelling
using GravSphere. We find that And XXV has a low central dark matter density,
$\rho_{\rm DM}$(150 pc)= 2.7${}^{+1.8}_{-1.6}\times$107 M⊙ kpc-3, making And
XXV a clear outlier when compared to other Local Group (LG) dSphs of the
similar stellar mass. In a companion paper, we will explore whether some
combination of dark matter cusp-core transformations and/or tides can explain
And XXV’s low density.

###### keywords:

galaxies: dwarf – galaxies: kinematics and dynamics – galaxies: haloes – dark
matter

††pubyear: 2022††pagerange: Andromeda XXV - a dwarf galaxy with a low central
dark matter density–A

##  1 Introduction

Dwarf galaxies are typically dark matter dominated systems, with mass-to-light
ratios in the range 10-1000s, making them ideal systems for testing dark
matter physics and cosmology (e.g. Mateo, 1998; Tolstoy et al., 2009; Simon,
2019). The most successful cosmological model to date is Lambda Cold Dark
Matter ($\Lambda$CDM), which explains the cosmic microwave background
radiation (e.g. Peebles, 1982; Komatsu et al., 2009; Aubourg et al., 2015;
Planck Collaboration et al., 2016, 2020), nucleosynthesis (e.g. Walker et al.,
1991; Smith et al., 1993; Burles et al., 1999; Cyburt, 2004; Steigman, 2007;
Fields, 2011) and structure formation on large scales (e.g. Bond et al., 1996;
Springel et al., 2005; Gao et al., 2005a; Gao et al., 2005b; Tegmark et al.,
2006; Springel et al., 2006), albeit by invoking three mysterious components –
dark matter, dark energy and inflation (Planck Collaboration et al., 2016,
2020). However, on galactic scales, particularly dwarf galaxies, there have
been a number of long-standing tensions between $\Lambda$CDM predictions and
observations (e.g. Bullock & Boylan-Kolchin, 2017; Buckley & Peter, 2018;
Sales et al., 2022). One such tension is the so-called ‘cusp-core’ problem
(Moore, 1994; Flores & Primack, 1994; de Blok, 2010; Navarro et al., 2010;
Walker & Peñarrubia, 2011; Read et al., 2017; Genina et al., 2018; Read et
al., 2019), which arises when comparing the dark matter density profiles of
dwarf galaxies from dark matter only simulations in a $\Lambda$CDM cosmology
to observations of dwarf galaxies. $\Lambda$CDM predicts that the dark matter
inside dwarf galaxies should follow a universal ‘cusped’ profile that has a
highly dense centre that steeply decreases with radius, such that
$\rho_{\rm{DM}}\propto r^{-1}$ (Dubinski & Carlberg, 1991; Navarro et al.,
1996b, 1997; Moore et al., 1999). Instead, a growing number of dwarfs have
been observed with ‘cored’ dark matter profiles described by a shallow central
density that remains roughly constant in the centre, such that
$\rho_{\rm{DM}}\sim\rm{constant}$ (Flores & Primack, 1994; Moore, 1994; de
Blok et al., 2001a, b; de Blok, 2010; Marchesini et al., 2002; Simon et al.,
2005; Battaglia et al., 2008; Walker et al., 2010; Agnello & Evans, 2012;
Amorisco & Evans, 2012; Newman et al., 2013; Read et al., 2017; Read et al.,
2019).

For many of the cored dwarfs observed, a plausible solution to the cusp-core
problem within $\Lambda$CDM paradigm is dark matter heating. This is the
process of sudden gas removal due to stellar winds, dynamical friction or
supernovae feedback which results in gravitational fluctuations, causing the
dark matter halo to expand. When this process is repeated across many cycles
of star formation the dark matter halo expands irreversibly, reducing the
central dark matter density and flattening the cusped profile into a core over
time (Navarro et al., 1996a; Gnedin & Zhao, 2002; Read & Gilmore, 2005;
Mashchenko et al., 2008; Pontzen & Governato, 2012; Zolotov et al., 2012;
Brooks & Zolotov, 2014; Oñorbe et al., 2015; Tollet et al., 2016; Read et al.,
2016, 2019). However, dark matter heating is only proposed to be effective in
galaxies with extended star formation (Read et al., 2016, 2019) that are above
a stellar-mass-to-halo-mass ratio threshold of $\rm{M}_{*}/\rm{M_{200}}\sim
5\times 10^{-4}$ (Di Cintio et al., 2014, but see Orkney et al. (2021)).

Puzzlingly, there is a subset of dwarfs within the low surface brightness
population of the LG with $\rm{M}_{*}/\rm{M_{200}}$ below this threshold with
unusually low central densities that are unlikely to be explained by dark
matter heating alone. Around the Milky Way (MW) two such dwarfs have been
found, Crater II (Torrealba et al., 2016) and Antlia II (Torrealba et al.,
2019). Similarly around Andromeda (M31) two more have been observed, Andromeda
XIX (McConnachie et al., 2008; Collins et al., 2020) and Andromeda XXI (Martin
et al., 2009; Collins et al., 2021). These systems have a diffuse nature,
described by a large half-light radius and very low surface brightness.
Furthermore, the mass contained within the half-light radius for these
systems, determined from velocity dispersions measurements, is far lower than
expected for their size or brightness (Caldwell et al., 2017; Fu et al., 2019;
Torrealba et al., 2019; Collins et al., 2020, 2021), meaning that it is likely
that these systems reside in low mass and low density dark matter halos. While
detailed star formation histories for these systems are not currently
available to completely rule out dark matter heating as the cause, even under
the (unlikely) assumption of highly efficient dark matter heating, their low
densities cannot be reproduced (Torrealba et al., 2019; Collins et al., 2021).
Instead, tidal interactions with the host (MW or M31) are the suspected
culprit causing the low density dark matter halos of these systems.

In this paper, we investigate another potential outlier, Andromeda XXV (And
XXV), a dSph satellite galaxy of M31 first discovered as part of the Pan-
Andromeda Archaeological Survey (Richardson et al., 2011). And XXV was
previously identified as a potential outlier within the LG in a 2013 study
(Collins et al. (2013), hereafter known as C13) that performed a kinematic
analysis of numerous M31 dSph satellites. And XXV was found to have a low
velocity dispersion, $\sigma_{\rm v}$=3.0${}^{+1.2}_{-1.0}$ kms-1, which
results in a mass-to-light ratio of [M/L]half=10.3${}^{+7.0}_{-6.7}$ M⊙/L⊙
consistent with a simple stellar system with no appreciable dark matter
component, albeit with large uncertainties due to the small number of member
stars (26) with stellar kinematics in their sample. As such, and as C13 point
out, more stellar velocities are needed to confirm or rule out the presence of
dark matter in this system. We present a revised kinematic analysis, using the
spectroscopic data set outlined in C13 combined with new observations to
provide a larger sample size, more than double that used in C13, allowing us
to critically reassess if And XXV really is a dSph with no appreciable dark
matter. We will also extend the analysis to investigate the metallicity of And
XXV. Finally we use the dynamical mass modelling tool BINULATOR + GravSphere
(Read & Steger, 2017; Read et al., 2018, 2019; Gregory et al., 2019; Genina et
al., 2020b; Collins et al., 2021; Read et al., 2021), to gain an insight into
the dark matter content of And XXV.

The outline of the paper is as follows: in $\S$2 we detail the photometric and
spectroscopic observations used in this study. We detail the contaminant
removal procedure used to remove MW and M31 contaminant stars from our sample
in $\S$3 and discuss our kinematic analysis of And XXV in $\S$4. In $\S$5 we
investigate the metallicity of And XXV. Next, we outline the dynamical mass
modelling results for And XXV $\S$6. In $\S$7 we discuss our results and
finally we conclude in $\S$8.

##  2 Observations

###  2.1 DEIMOS spectroscopy

The spectroscopic data for And XXV were obtained using the Deep Extragalactic
Imaging Multi-Object Spectrograph (DEIMOS) Faber et al. (2003); Cooper et al.
(2012) mounted on the Keck II telescope. The observations are comprised of two
masks. The first was taken in September 2010 (previously presented in C13),
and the second mask was observed on 18th August 2018. The instrumental setup
was the same for both masks, using a 1200 line mm-1 grating with a resolution
of 1.3 Å. To determine the velocity and metallicity of each member star, we
use the calcium triplet (Ca(II)) lines. The Ca(II) lines are present in the
region around $\sim$8500 Å. As such our observations targeted the wavelength
range of $\sim$5600 - 9800 Å with a central wavelength of 7800 Å to resolve
the Ca(II) lines. Each mask was split into 3$\times$20 minute exposures,
combining to give a total exposure time of 3600 seconds per mask. The average
seeing was $\sim 0.8^{\prime\prime}$ for both masks, which resulted in an
average signal-to-noise ratio (S/N) of $\sim$5 per pixel.

The data were reduced using a custom pipeline, described in detail in Ibata et
al. (2011) and C13. In short, the pipeline detects and removes cosmic rays,
then corrects for scattered light, slit function, illumination and fringing.
To account for pixel-to-pixel variations flat-fielding was performed. Next
wavelength calibrations of each pixel were conducted using arc-lamp exposures.
Then, the sky was subtracted from the 2-dimensional spectra. Finally, each
spectrum was extracted (without resampling) from a small spatial region around
each target. The velocities and corresponding uncertainties for all stars were
derived using the strong Ca(II) triplet feature found in the spectra. The non-
resampled data was compared to a template Ca(II) spectrum and a most-likely
velocity and uncertainty for each star were derived using a Markov Chain Monte
Carlo (MCMC) routine. The final value for the velocity uncertainty comprises
of the uncertainty from the MCMC posterior distribution combined with the
uncertainty inherent to DEIMOS which was taken to be ${2.2}$ kms-1 (Simon &
Geha, 2007). The results from each mask were combined into a single catalogue.

Misalignments of the slitmasks can cause velocity shifts of up to 15 km/s. We
correct for these by comparing atmospheric models to the telluric absorption
lines in the spectrum to shift the spectra to the correct frame as described
in C13. To further confirm if any significant misalignment had occurred we
check for a velocity error gradient across the mask. None were found in either
mask. Finally, there were a few stars present both masks. The matching stars
were compared to ensure the velocity uncertainties were well measured.

###  2.2 LBT LBC photometry

The photometric data for And XXV were obtained using the Large Binocular
Cameras (LBC) mounted on the Large Binocular Telescope (LBT) in the
$V\textsubscript{Bessell}$\- and $i\textsubscript{SLOAN}$-band. Observations
were conducted on the nights of 28th October and 8th November, 2011 with a
seeing that was ranging from 0.7" to 1.2". With its 4i say CCDs, the cameras
both cover an area of about $23\times 25$ arcmin2. In total, 29 exposures of
360 s were taken with each filter, for a total exposure time of 2.9 h per
band.

The raw photometry was reduced using the CASU pipeline where the images are
debiased, flat-fielded, trimmed, and gain-corrected (Irwin & Lewis, 2001). The
reduced photometric data was made into a catalogue. Each datum was then
morphologically categorised to distinguish between stellar, non-stellar (e.g.
background galaxies) or noise-like objects. Only stellar objects were
considered for further analysis. Finally, the data were extinction corrected
using the dust maps from Schlafly & Finkbeiner (2011), using 0.271
($V\textsubscript{Bessell}$) and 0.171 ($i\textsubscript{SLOAN}$).

The photometric and spectroscopic observations were combined by cross-matching
the on-sky position of each star within an allowed tolerance of $\pm$1
arcsecond.

##  3 Selecting members of Andromeda XXV

Before we can perform the kinematic analysis of And XXV we must identify the
most probable members of And XXV and remove any contaminants from the MW or
M31 halo. It is difficult to differentiate between member and contaminant
stars using velocity information alone. This is especially true for foreground
MW stars as the two systems have similar systemic velocities. Instead, to
determine likely member stars, we use a ‘triple-threat’ probabilistic approach
which was first outlined in Tollerud et al. (2012) and C13, then further
developed in Collins et al. (2020, 2021) and Gregory et al. (2019). The method
assigns each star a probability of membership using three probability
criteria: (1) the star’s position on the sky with relation to the centre of
And XXV, $P\textsubscript{dist}$; (2) the star’s position on a colour
magnitude diagram (CMD) of And XXV, $P\textsubscript{CMD}$; and (3) the
velocity of the star, $P\textsubscript{vel}$. We discuss each of these
criteria in more detail below.

###  3.1 Distance probability

Pdist is determined using an exponential radial surface brightness profile
modelled as:

$P\textsubscript{dist}=\exp{\left[-\left(\frac{r^{2}}{2{\eta\textsubscript{dist}}r\textsubscript{p}^{2}}\right)\right]}$
(1)

where $r$ is the radial distance of the star from the centre of And XXV (taken
to be 0h 30m 9.9s \- RA, 46${}^{\text{\textdegree}}$ 51${}^{{}^{\prime}}$ 41"
\- Dec Martin et al. (2016)) and $\eta\textsubscript{dist}$ is a free
parameter used to scale the exponential profile to the size of And XXV.
$\eta\textsubscript{dist}$ was extensively tested for any potential biases or
dependencies and a final value of $\eta\textsubscript{dist}=2.5$ was used.
$r_{p}$ is a Plummer profile (Plummer, 1911) used to modify the half-light
radius, $r\textsubscript{h}$, to account for any ellipticity, described by:

$r\textsubscript{p}=\frac{r\textsubscript{h}(1-\epsilon)}{1+{\epsilon}\cos{(\theta)}}$
(2)

where $r\textsubscript{h}=2.7$ arcmin (Savino et al., 2022)), $\epsilon$ is
the ellipticity ($\epsilon$ = 0.03) and $\theta$ is the stars angular position
with respect to the dwarfs major axis ($\theta=-16$ °) (Martin et al., 2016).

###  3.2 Colour-magnitude diagram probability

PCMD is determined using the colour-magnitude diagram of And XXV. A by-eye
best-fit isochrone was overlaid onto the CMD to identify the stars most likely
to be red giant branch (RGB) stars of And XXV. The isochrone used was an old,
metal-poor isochrone ($\rm[Fe/H]=-1.9$ dex, $\rm[\alpha/Fe]=0.0$ dex, $\rm
age=12$ Gyr) obtained from the DARTMOUTH stellar evolutionary models Dotter et
al. (2008) shifted to the distance modulus of And XXV, $m-M=24.38$ (Savino et
al., 2022)). The CMD of And XXV and the by-eye best-fit isochrone are shown in
the right panel of Figure. 1. The minimum distance, $d\textsubscript{min}$, of
each star to the isochrone was converted into a probability of membership
using:

$P\textsubscript{CMD}=\exp{\left[-\left(\frac{d\textsubscript{min}^{2}}{2\eta\textsubscript{CMD}^{2}}\right)\right]}$
(3)

where $\eta\textsubscript{CMD}$ is another free parameter, this time used to
account for the scatter of stars around the best-fit isochrone. Again, this
free parameter was tested for any potential biases or dependencies and a final
value of $\eta\textsubscript{CMD}=0.2$ was chosen.

###  3.3 Velocity probability

Pvel is determined by simultaneously fitting the velocities for all of the
stars in the spectroscopic observations by assuming that these stars inhabit a
profile of four dynamically distinct peaks. Four Gaussians are used to
describe the different peaks which correspond to And XXV stars (PAnd XXV),
then M31 halo contaminant stars (PM31) and two peaks for the MW contaminant
stars (PMW1, PMW2). The MW velocity profile is often assumed to be a single
distribution for contaminant removal purposes. However, due to the similarity
in systemic velocity for And XXV and the MW, for our study, this assumption is
not valid. Instead, we need to include this complexity in our model by
modelling the MW as two velocity distributions (e.g. Gilbert et al., 2006).
Each Gaussian is defined by a systemic velocity ($\nu_{\rm r}$) and a velocity
dispersion ($\sigma_{\rm\nu}$), such that the probability of each star
belonging to each peak (PAnd XXV, PM31, PMW1 and PMW2) is:

$P\textsubscript{peak}=\frac{1}{\sqrt{2\pi}\sqrt{{\sigma_{\nu_{\rm
peak}}^{2}}+\nu\textsubscript{err,i}^{2}}}\times\exp{\left(-\frac{1}{2}\left[\frac{\nu\textsubscript{peak}-\nu_{i}}{\sqrt{\sigma_{\nu_{\rm
peak}^{2}}+\nu\textsubscript{err,i}^{2}}}\right]^{2}\right)}$ (4)

where $\nu\textsubscript{i}$ and $\nu\textsubscript{err,i}$ are the velocity
and velocity uncertainty of a given star respectively. The overall log-
likelihood function is therefore described by:

$\log(\mathcal{L})=\sum_{i=1}^{N}(\alpha P\textsubscript{And~{}XXV}+\beta
P\textsubscript{M31}+\gamma P\textsubscript{MW1}+\delta P\textsubscript{MW2})$
(5)

where $\alpha$, $\beta$, $\gamma$ and $\delta$ are constants describing the
proportion of stars belonging to And XXV, M31, MW1 and MW2 respectively and
are normalised, such that $\alpha+\beta+\gamma+\delta=1$. The components of
this likelihood function were found using emcee, a python Markov Chain Monte
Carlo (MCMC) package (Foreman-Mackey et al., 2013). The routine used 200
walkers, over 5000 iterations with a burn-in stage of 1550 and uniform flat
priors were introduced for each parameter, see Table. 1 for a summary of the
priors and results for the MCMC analysis.

It is important to note that, while the velocity and velocity dispersion
values will likely resemble the final values, this is not the final kinematic
result for And XXV. It is only used to determine Pvel. This is because it does
not consider the impact of any potential M31 or MW contamination in the wings
of the And XXV Gaussian, hence why the other two probability filters are
important and the combination of all three filters is used to weight the
likelihood function in the final kinematic analysis. The probability
distribution described by the Gaussian for each peak can then be combined to
give $P\textsubscript{vel}$, using:

$P\textsubscript{vel}=\frac{P\textsubscript{AndXXV}}{P\textsubscript{MW1}+P\textsubscript{MW2}+P\textsubscript{M31}+P\textsubscript{AndXXV}}$
(6)

Table 1: Prior values and results for the variables used in our emcee analysis for Pvel. [*] Note: This is not the final systemic velocity and velocity dispersion value for And XXV - it is only used to determine Pvel Priors: |  |
---|---|---
Peak |  $\nu_{\rm r}$ (kms-1) |  $\sigma_{\rm\nu}$ (kms-1)
PAnd XXV | $-130<\nu_{\rm r}<-90$ | $0<\sigma_{\rm\nu}<50$
PM31 | $-400<\nu_{\rm r}<-130$ | $0<\sigma_{\rm\nu}<200$
PMW1 | $-60<\nu_{\rm r}<50$ | $0<\sigma_{\rm\nu}<100$
PMW2 | $-90<\nu_{\rm r}<-50$ | $0<\sigma_{\rm\nu}<100$
Results: |  |
Peak |  $\nu_{\rm r}$ (kms-1) |  $\sigma_{\rm\nu}$ (kms-1)
PAnd XXV | $-107.6\pm{1.4}$ |  $5.9^{+2.1}_{-1.7}$ [*]
PM31 | $-268.2^{+49.5}_{-48.0}$ | $132.6^{+29.8}_{-27.0}$
PMW1 | $-45.7^{+15.5}_{-9.3}$ | $28.3^{+13.2}_{-9.8}$
PMW2 | $-66.7^{+10.8}_{-14.2}$ | $36.9^{+16.0}_{-8.3}$

The final probability of membership to And XXV for each star is the product of
the three probability cuts, such that:

$P\textsubscript{member}=P\textsubscript{dist}\times
P\textsubscript{CMD}\times P\textsubscript{vel}$ (7)

Stars with a probability of $P\textsubscript{member}>0.15$ were considered
member stars. The probability of membership cut-off is kept intentionally low
as we use the velocity dispersion for the kinematic analysis. Hence, we do not
want to artificially decrease the velocity dispersion by removing potential
candidates with a probability cut that is too strict. Furthermore, all non-
members have a probability of membership significantly below this cut-off
point. A total of 53 members were identified (as shown in Fig. 1). This data
contains more than double the number of members compared to that used in C13.

Figure 1: Left Top: The velocity histogram from And XXV. Light grey represents
non-members likely contamination from foreground MW stars or M31 halo stars.
Dark grey represents the 53 identified member stars of And XXV. The black
dashed line represents the systemic velocity of And XXV determined in this
study. Left Bottom: Radial distance from the centre of And XXV as a function
of the line of sight velocity. The smaller light grey circles are the non-
member stars from the spectroscopic data. The larger coloured circles are the
member stars, colour-coded by the probability of membership (see colour bar on
the right) and the error bars are the 1$\sigma$ uncertainties. The horizontal
grey dashed lines represent 1$\times$ and 2$\times$ rh from the bottom up
respectively. Right:  The colour magnitude diagram for And XXV. Again the
smaller light grey circles are non-member stars within 1.5$\times$ rh and the
larger coloured circles are the member stars colour-coded by the probability
of membership. The black solid line is a by-eye best-fit isochrone
($\rm[Fe/H]=-1.9$ dex, $\rm[\alpha/Fe]=0.0$ dex, $\rm age=12$ Gyr) for And XXV
taken from the DARTMOUTH stellar evolutionary models Dotter et al. (2008) and
shifted to the distance modulus of And XXV, $m-M=24.38$ (Savino et al.,
2022)).

##  4 Kinematic Analysis of Andromeda XXV

We used another emcee routine to determine the velocity ($\nu$) and velocity
dispersion ($\sigma_{\rm\nu}$) of And XXV. This time, the routine fits a
single Gaussian for the 53 member stars, with the likelihood of each star
weighted by the star’s respective probability, such that the log-likelihood
function is:

$\log(\mathcal{L})=\sum_{i=1}^{N}\log(P\textsubscript{{member}\textsubscript{i}}P\textsubscript{{AndXXV}\textsubscript{i}})$
(8)

The routine used 500 walkers over 5000 iterations with a burn-in stage of
3750. The values of $\nu$ and $\sigma_{\rm\nu}$ for And XXV from the
$P\textsubscript{vel}$ analysis were used as initial guesses and the same flat
priors used for $P\textsubscript{vel}$ were implemented. The resulting
posterior distribution can be seen in Fig. 2.

Figure 2: A corner plot showing the results of our kinematic analysis using
EMCEE. The velocity and velocity dispersion are well resolved, giving
$\nu_{r}=-107.7\pm 1.0$ km s-1 and $\sigma_{\nu}=4.5\pm 1.0$ kms-1

###  4.1 Kinematic results

The velocity and velocity dispersion are well resolved, giving
$\nu=-107.7\pm{1.0}$ km s-1 and $\sigma_{\rm\nu}=4.5\pm{1.0}$ km s-1, the
uncertainties are the 1$\sigma$ uncertainty intervals from the posterior
distributions. The properties derived for And XXV are listed in Table 2. The
velocity and velocity dispersion are consistent with those derived in C13
($\nu=-107.8\pm{1.0}$ km s-1, $\sigma_{\rm\nu}=3.0^{+1.2}_{-1.0}$ km s-1)
although our velocity dispersion is higher. This is likely due to the
targeting of the C13 study. The first mask preferentially targeted stars close
to the centre of And XXV. As seen in Fig. 1, the central stars are
kinematically colder than those in the outskirts (velocity dispersion of
$\sim$3.3 km s-1 in the centre vs $\sim$5.7 km s-1 on the outskirts). The
outermost stars included in the new spectroscopic data set increase the
average dispersion measured in this study. It is important to note that at the
outskirts of And XXV we are more susceptible to contamination from non-member
stars, especially due to the similarity in velocity between And XXV and the
foreground MW stars. We extensively tested out membership selection, however,
it is possible inclusion of contaminates could potentially contribute to the
velocity flaring at larger radii.

Table 2: The properties of And XXV. Sources a: Martin et al. (2016), b: Savino et al. (2022), c: this work. Property | Value | Source
---|---|---
$\rm\alpha$,$\rm\delta$ (J2000) | 00:30:09.9 +46:51:41 | a
mV | 15.3${}^{+0.3}_{-0.2}$ | a
MV | -9.1${}^{+0.3}_{-0.2}$ | b
D (kpc) | 751.6${}^{+25}_{-21}$ | b
rh (arcmin) | 2.7${}^{+0.4}_{-0.3}$ | b
rh (pc) | 590${}^{+90}_{-47}$ | b
L (L⊙) | 3.7${}^{+1.4}_{-0.5}\times$105 | b
$\nu$ (kms-1) | -107.7$\pm$1.0 | c
$\sigma_{\nu}$ (kms-1) | 4.5$\pm$1.0 | c
M(r<rh) (M⊙) | 6.9${}^{+3.2}_{-2.8}\times$106 | c
[M/L]${}_{\rm r_{h}}$ (M⊙/L⊙) | 37${}^{+17}_{-15}$ | c
[Fe/H] (dex) | -1.9$\pm 0.1$ | c
$\rho_{\rm DM}$(150 pc) (M⊙ kpc-3) | 2.7${}^{+1.8}_{-1.6}\times$107 | c

Using the now better constrained velocity dispersion, we can measure the mass,
M($<r_{\textsubscript{h}}$), and mass-to-light ratio, $\text{[M/L]}_{\rm
r_{h}}$, contained within the half-light radius of And XXV. Assuming a flat
velocity dispersion profile, $M\textsubscript{r\textsubscript{h}}$ can be
calculated using Walker et al. (2009):

$M\textsubscript{r\textsubscript{h}}=580\,r\textsubscript{h}\,\sigma_{\nu}^{2}$
(9)

The mass of And XXV was determined to be
$M\textsubscript{r\textsubscript{h}}=6.9^{+3.2}_{-2.9}\times 10^{6}$ M$\odot$,
which is in agreement with the C13 study within the 1$\sigma$ uncertainties.
Using the updated absolute magnitude value of And XXV,
$M\textsubscript{V}=-9.1$ (Savino et al., 2022), which is equal to a
luminosity of $L=3.7^{+1.4}_{-0.5}\times 10^{5}$ L$\odot$, the mass-to-light
ratio contained within the half-light radius is
$[M/L]\textsubscript{r\textsubscript{h}}=37^{+17}_{-15}$ M$\odot$/L$\odot$.
The mass-to-light ratio agrees with the value from C13 within the 2$\sigma$
uncertainties. If we instead use the previous luminosity value ($L=6.8\times
10^{5}$ L$\odot$) with the updated velocity dispersion we derive a mass-to-
light ratio of $[M/L]\textsubscript{r\textsubscript{h}}=21^{+10}_{-9}$
M$\odot$/L$\odot$, which is consistent within 1$\sigma$. This demonstrates
that the moderate increase in the mass-to-light ratio is predominantly due to
the updated luminosity value that is almost half the previous value that was
used in C13.

###  4.2 Comparison to Kinematics of Local Group dSphs

Figure 3: Left: Mass-to-light ratios of dSphs in the LG. The diagonal black
lines illustrate a mass-to-light ratio of 10, 100 and 1000 from right to left
respectively. Right:  The mass contained within the half-light radius for the
LG dSphs. The solid black line is the best-fitting NFW mass profile for the LG
population (Collins et al., 2014), with the 1$\sigma$ uncertainties for this
relationship shown by the blue shaded region. Both: The light grey triangles
are MW dSphs and the light grey circles are M31 dSphs. The C13 result is
illustrated by the pink diamond and the updated value from this study for And
XXV is shown by the purple star. For all data points, the error bars show the
1$\sigma$ uncertainties.

Dwarf galaxies are typically dark matter dominated systems at all radii and as
such we would expect them to have mass-to-light ratios of
$[M/L]\textsubscript{r\textsubscript{h}}>10$ M$\odot$/L$\odot$. The left panel
of Fig. 3 shows the mass-to-light ratios of LG dSphs, the light grey circles
and triangles are M31 and MW dSphs respectively, taken from Tollerud et al.
(2012); Collins et al. (2013); Collins et al. (2017); Collins et al. (2020,
2021); Walker et al. (2007, 2009); Simon & Geha (2007); Simon et al. (2011,
2015); Martin et al. (2007); Martin et al. (2013a, b, 2014); Ho et al. (2012);
Kirby et al. (2015); Kirby et al. (2017). The pink diamond shows the C13
result. The mass-to-light ratio is indicative of a simple stellar system with
no appreciable dark matter content. The purple star shows the mass-to-light
ratio of And XXV derived from this study. We can see that with the combination
of the slightly increased velocity dispersion and updated luminosity used in
this study, the mass-to-light ratio of And XXV moves in line with other LG
dSphs. We have shown for the first time that And XXV has an unambiguous dark
matter component and is indicative of a dark matter dominated dwarf galaxy.

The right panel of Fig. 3 shows the mass contained within the half-light
radius as a function of the half-light radius for the LG dSphs. Again the
light grey circles and triangles are M31 and MW dSphs respectively, taken from
the same sources. With the updated velocity dispersion the half-light radius
mass moves more in line with what we would expect (compare the pink diamond -
C13 to the purple star -this study). Despite this, we can see that the mass of
And XXV is still significantly lower when compared to the best-fitting NFW
mass profile for the LG population (Collins et al., 2014), illustrated by the
solid black line and the light blue shaded region is the 1$\sigma$
uncertainty. This extended radial behaviour with significantly less mass than
expected for its size is also shown by the other four outliers And XXI
(Collins et al., 2021), And XIX (Collins et al., 2020), Crater II (Torrealba
et al., 2016) and Antlia II (Torrealba et al., 2019), as highlighted in Fig.
3. For these systems, it is suspected that tides have caused the low masses.
Furthermore, C13 noted that tidal interactions could have acted to lower the
central density of And XXV, and the increase in velocity dispersion with
radius could indicate a tidal influence.

To investigate if we can observe the impact of tides in the kinematics, we
altered the Gaussian component of the likelihood function described in Equ. 4
to include a velocity gradient component that would indicate the presence of
tidal streams, following the methodology outlined in Martin & Jin (2010). No
statistically significant velocity gradient was found. It is important to note
that this does not mean And XXV has not experienced tidal influences,
especially as we potentially run into the same small number statistics issues
reported by the C13 as we are trying to fit more parameters than the simple
Gaussian outlined in Equ. 4. We return to the potential impact of tides on And
XXV in $\S$ 7.

##  5 Metallicity of Andromeda XXV

We determined the metallicity of And XXV by measuring the equivalent widths of
the calcium II triplet lines from the spectroscopic observations. These
absorption lines are a good proxy for iron abundance [Fe/H] (Armandroff & Da
Costa, 1991). We only include member stars with a good signal-to-noise ratio
(S/N). For our data set, this was determined to be $S/N>5$ per pixel, which
resulted in a sample that comprised of 41 stars. To calculate the equivalent
widths, we followed the methodology outlined in C13. Firstly, we apply a
Doppler correction to each star to ensure the spectra are measured in the rest
frame. The spectra were then interpolated onto a common framework before co-
adding the S/N weighted spectra. Next, the spectra were normalised, such that
the mean continuum was equal to one. Finally, we simultaneously fit the
continuum and three Gaussian peaks to each of the three Ca II triplet lines to
obtain the equivalent widths.

Figure 4: The luminosity-metallicity relation for LG dSphs. MW dSphs are
represented by light grey triangles whereas the M31 dSphs are the light grey
circles. The pink diamond shows the results for And XXV from the C13 study and
the purple star shows the result for And XXV from this study derived from the
coadd spectra. The black dash line is the best fit luminosity-metallicity
relation from Kirby et al. (2013) with the light blue band illustrating the
1$\sigma$ scatter. And XXV is in perfect agreement with this relation.

The metallicity was determined from the equivalent widths using the approach
described by Starkenburg et al. (2010) such that:

$\text{[Fe/H]}=-2.87+0.195~{}M+0.48~{}\Sigma Ca-0.913~{}\Sigma
Ca^{-1.5}+0.00155~{}\Sigma{Ca}~{}M$ (10)

where $\Sigma\rm Ca=$
$0.5\text{EW}_{8498}+\text{EW}_{8542}+0.6\text{EW}_{8662}$ and $M$ is the
absolute magnitude of each star given by:

$M=V-5\times\log_{10}(D_{\odot})+5$ (11)

where $V$ is the V-band magnitude of the star and $D_{\odot}$ is the
heliocentric distance of the star which, for all stars, was assumed to be the
heliocentric distance of And XXV, 751.6 kpc (Savino et al., 2022). This
resulted in a metallicity for And XXV of $\rm[Fe/H]=-1.9\pm{0.1}$ dex. This
value is in direct agreement with the result from the C13 study. Furthermore,
this result is in agreement with the luminosity-metallicity relation (Kirby et
al., 2013) for LG dwarfs, as shown in Fig. 4 with data taken from the sources
mentioned for Fig. 3 in addition to Ho et al. (2015) and Wojno et al. (2020).

Figure 5: Top: The metallicity distribution function derived from the
individual spectra of And XXV member stars with a S/N > 5 per pixel. Bottom: A
corner plot showing the mean and spread of metallicity for And XXV. We find a
mean metallicity of $\rm{[Fe/H]}=-2.02\pm 0.02$ dex and a spread of
$\sigma_{\rm{[Fe/H]}}=0.08^{+0.02}_{-0.01}$ dex.

With high S/N spectra, it is possible to obtain metallicities for individual
stars. To do this we performed the same fitting procedure described above but
this time for each of the 41 spectra separately. The fitting procedure was
successful for 36 of the 41 spectra. The top panel in Fig. 5 shows the
metallicity distribution function (MDF) for And XXV. An MCMC routine fitting a
Gaussian to the MDF found the mean metallicity to be $\rm[Fe/H]=-2.02\pm 0.02$
dex and the metallicity dispersion to be
$\sigma\textsubscript{[Fe/H]}=0.08^{+0.02}_{-0.01}$ dex, see the corner plot
Fig. 5. The values obtained from the MDF are also in agreement with the
results from the coadded spectra, the C13 study and the luminosity-metallicity
relation (Kirby et al., 2013).

##  6 Mass Modelling of Andromeda XXV

###  6.1 BINULATOR \+ GravSphere \- mass modeling tool

GravSphere111The updated GravSphere code along with the new BINULATOR binning
method, described in this paper and Collins et al. (2021), is available to
download from https://github.com/justinread/gravsphere. pyGravSphere, a free
form mass version of the GravSphere code (Genina et al., 2020b), is available
to download from https://github.com/AnnaGenina/pygravsphere. is the dynamical
mass modelling tool we used to measure the dark matter density profile of And
XXV and is described described in detail in Read & Steger (2017); Read et al.
(2018, 2021); Genina et al. (2020b); Collins et al. (2021). In this section,
we briefly review the GravSphere methodology and describe its application to
And XXV. GravSphere solves the projected spherical Jeans equations (Jeans,
1922; Binney & Mamon, 1982) for a set of tracers, in this instance the 53
member stars of And XXV, to determine the dark matter density profile,
assuming that it is a spherical, non-rotating system. These equations are
known to have a mass-velocity anisotropy degeneracy for which a wide range of
solution combinations can satisfy the model (Merrifield & Kent, 1990;
Wilkinson et al., 2002; Łokas & Mamon, 2003; de Lorenzi et al., 2009). Several
methods have been proposed to break this degeneracy (see e.g. Read & Steger
(2017). GravSphere addresses it by fitting two higher order “Virial Shape
Parameters” (VSPs), first proposed by Merrifield & Kent (1990, see also ()).
The velocity anisotropy profile is difficult to constrain observationally.
However, using VSPs we only need line of sight velocities (easily obtained
from spectroscopic observations) to place meaningful constraints on the
velocity anisotropy profile, hence breaking the mass-velocity anisotropy
degeneracy. GravSphere uses a symmetric version of the velocity anisotropy
profile, $\tilde{\beta}$, is used to avoid issues with infinite values (Read
et al., 2006), where $\tilde{\beta}=0$ describes an isotropic velocity
distribution, $\tilde{\beta}=-1$ a complete tangential and $\tilde{\beta}=1$ a
complete radial distribution. Finally, in line with previous studies (Read et
al., 2018; Collins et al., 2021), we adopt the CORENFWTIDES model to describe
the dark matter distribution. This mass profile includes within its
parameterisation the cusped Navarro-Frenk-White profile (NFW; Navarro et al.
1996). The NFW profile gives a good fit to dark matter density profiles in
pure dark matter simulations and is described by two parameters: a virial mass
$M_{200}$ and concentration parameter $c_{200}$. The CORENFWTIDES model adds
four new parameters to this: $n$, $r_{c}$, $r_{t}$ and $\delta$. The first two
control how “cored” or “cusped” the dark matter profile is inside $r_{c}$,
where $n=1$ corresponds to a constant density flat core, $n=0$ corresponds to
an $r^{-1}$ cusp, as in the NFW profile, and $n=-1$ corresponds to an even
steeper $r^{-2}$ cusp. The second two parameters model the effect of tidal
forces from a larger host galaxy stripping some of the outer mass away, where
$r_{t}$ is the tidal radius beyond which the density falls off as
$r^{-\delta}$.

GravSphere has been rigorously tested on a wide variety of mock data (Read &
Steger, 2017; Read et al., 2018, 2021; Genina et al., 2020b; Collins et al.,
2021) and stands up well when compared to other dynamical mass modeling tools
(Read et al., 2021). However, for systems with a small number of stars, and/or
where the velocity uncertainty is large, the binning method in the previous
version of GravSphere can become slightly biased towards cusped profiles
(Gregory et al., 2019; Zoutendijk et al., 2021; Collins et al., 2021). To
resolve this issue GravSphere was updated to include a new separate binning
routing, BINULATOR, first introduced and outlined in detail in Collins et al.
(2021). This update reduces the aforementioned biases by fitting a generalised
Gaussian probability distribution to each bin, providing a robust estimate of
the mean, variance, kurtosis and corresponding uncertainties for each bin,
even for systems with an extremely small sample size - such as And XXV. These
estimates are then used as input to GravSphere. Detailed tests of this updated
version of BINULATOR+GravSphere can be found in Collins et al. (2021)
Appendix. A.

###  6.2 Implementing BINULATOR \+ GravSphere for Andromeda XXV

The surface brightness profile for And XXV was constructed from photometric
data obtained through the LBT imaging described in $\S$2.2. We included point
sources from the entire data set out to $\sim 5\times r_{h}$. For each star
the radial distance from the centre of And XXV was measured and a probability
of membership was assigned using the $P_{\text{{dist}}}$ and
$P_{\text{{iso}}}$ method outlined in Equ. 1 and Equ. 3, albeit with a looser
probability constraint (P${}_{\text{member}}>0.01$) and larger $\eta$
parameters to ensure a representative and complete surface brightness profile.
The velocity dispersion profile was constructed from the spectroscopic data
for the 53 identified members of And XXV, using the velocity, velocity
uncertainty, radial distance from the centre and the probability of membership
determined using Equ. 7 for each member star. For both data sets the
probability of membership is summed to give the total number of ‘effective’
tracers:

$N\textsubscript{eff}=\sum^{N\textsubscript{mem}}_{i=0}P\textsubscript{mem,i}$
(12)

giving $N\textsubscript{eff}=2450$ for the photometric data and
$N\textsubscript{eff}=32$ for the velocity data, which were then split into 98
bins of 25 and 4 bins of 8 respectively. GravSphere fits the surface
brightness profile and radial velocity profile from the kinematic and
photometric data input using an emcee routine (see Appendix A). For the
parameters in the CORENFWTIDES profile we implemented the following priors:
$7.5<\log_{10}(M_{200}/\text{M}_{\odot})<11.5$, $7<c_{200}<53$,
$-2<\log_{10}(r_{c}/\text{kpc})<10$, $1<\log_{10}(r_{t}/\text{kpc})<20$,
$3<\delta<5$ and $-1<n<1$ . For the symmetric velocity anisotropy,
$\tilde{\beta}$, the priors were: $-0.1<\tilde{\beta}_{\infty}<1$,
$-2<\log_{10}(r_{0}/\text{kpc})<0$ and $1<q<3$. Finally, for the stellar mass
of And XXV we convert the updated luminosity value from Savino et al. (2022)
using the assumption of a stellar mass-to-light ratio of 1, in line with
McConnachie (2012). This gives a stellar mass of
$M_{*}=3.7^{+0.3}_{-0.4}\times 10^{5}$ M⊙.

Figure 6: The dark matter density profile for And XXV. The dark grey and light
grey shaded regions are the 1$\times$ and 2$\times~{}\sigma$ uncertainties
respectively and the purple dashed line is the literature half-light radius of
And XXV (Savino et al., 2022). The blue dotted line corresponds to 150 pc, the
radial distance at which we determine the central dark matter density (Read et
al., 2018). And XXV has a low central dark matter density of $\rho_{\rm
DM}$(150 pc)= 2.7${}^{+1.8}_{-1.6}\times$107 M⊙ kpc-3.

###  6.3 Dark matter density profile of Andromeda XXV

The resulting dark matter density profile is shown in Fig. 6. The dark grey
and light grey shaded regions are the 1 and 2$~{}\sigma$ uncertainties
respectively and the purple dashed line is the half-light radius of And XXV.
Within the uncertainties, it is not possible to distinguish between a cusped
or cored profile. Instead, we turn our attention to the central dark matter
density, $\rho_{\rm{DM}}(150\rm{pc})$, illustrated by the blue dotted line.
This value is used because this is the key region where core formation is
expected to reduce dark matter densities, compared to CDM predictions (Read et
al., 2018; Genina et al., 2020a). For And XXV we get a central dark matter
density of $\rho_{\rm DM}$(150 pc)= 2.7${}^{+1.8}_{-1.6}\times$107 M⊙ kpc-3.
This is low when compared to other LG dSphs, as shown in Fig. 7, with data
taken from Read et al. (2019). Here dSphs are illustrated as circular data
points and dIrrs are triangles. Quiescent (non-star forming) dwarfs are purple
whereas currently star-forming dwarfs are blue. The sample of dSph galaxies in
Read et al. (2019) was chosen to be tidally isolated in order to measure the
impact of dark matter heating on $\rho_{\rm DM}$(150 pc), independently of
tidal effects. However, while the dSphs in Read et al. (2019) are known to be
tidally isolated today (based on their orbits, measured using proper motion
data from Gaia+HST (Gaia Collaboration et al., 2016, 2018), this does not mean
that they have been tidally isolated for their whole history. As shown in
Genina et al. (2020b), some dwarfs can have past interactions with one another
and/or other infalling structure that lowers their inner densities while
leaving them on apparently benign orbits today. While it is not possible to
control for this for the dSphs in the Read et al. (2019) sample, like Fornax,
such effects cannot explain the similarly low density of the isolated dIrrs in
the Read et al. (2019) sample. Fornax is illustrated by the partially filled
purple circular data point. Fornax is an interesting case that has a density
and star formation history similar to dIrrs, but is a dSph. As such, its low
density could owe to star formation and/or tides (Genina et al., 2022). A star
formation (SF) induced dark matter core provides the best fit to the kinematic
data for Fornax, but tides could still have had a major influence even though
Fornax’s orbit looks to be tidally benign today.

Figure 7: Top:  The central dark matter density as a function of pre-infall
mass, M200. The dark grey band corresponds to the inner DM densities of
completely cuspy profiles, whereas the light grey band corresponds to a fully
cored coreNFW profile (Read et al., 2016). The width of the bands corresponds
to the 1$\sigma$ scatter in DM halo concentrations (Dutton & Macciò, 2014).
Bottom: Central dark matter density as a function of the stellar mass to halo
mass ratio, M${}_{\text{*}}$/M200. The vertical grey dashed line illustrates
the M${}_{\text{*}}$/M200 ratio above which cusp-core transformations should
become efficient (Di Cintio et al., 2014). Both:  Circular data points are
dSphs, while triangles are dIrrs. Dwarfs that are currently star forming are
light blue whereas dwarfs that are quiescent are purple. Fornax is illustrated
by the partially filled purple circular data point due to its unusual star
formation history atypical of dSphs. The error bars are the $1\sigma$
uncertainties. And XXV and And XXI are labelled and have thicker error bars
with caps. Data for And XXI were obtained from Collins et al. (2021) and from
Read et al. (2019) for the other dwarfs.

When comparing And XXV to the rest of the LG dwarfs we can see just how low
And XXV’s central dark matter density is, at approximately an order of
magnitude smaller than expected for a quiescent dSph, $\rho_{\rm
DM}(150\rm{pc})>10^{8}$ M⊙ kpc-3. Instead, And XXV has a central dark matter
density more in line with currently star forming dIrrs,
$\rho_{\rm{DM}}(150\rm{pc})<10^{8}$ M⊙ kpc-3, which are expected to have lower
central dark matter densities due to the dark matter heating effects of their
continued star formation. Even when compared to the unusual Fornax, with even
lower central density, And XXV is still a significant outlier. Furthermore, in
the top panel in Fig. 7, the dark grey band corresponds to a fully cusped
profile whereas the light grey band corresponds to a fully cored profile. The
width of the bands corresponds to the 1$\sigma$ scatter in DM halo
concentrations (Dutton & Macciò, 2014). Here we can see that And XXV appears
to reside in the light grey region which corresponds to a fully cored profile.
This behaviour is also reflected in And XXI, another documented M31 outlier
for which we have dynamical mass modelling (Collins et al., 2021), as labelled
in Fig. 6

##  7 Discussion

Dark matter heating is one way in which dwarfs can lower the central dark
matter densities. Could the unusually low central density observed in And XXV
be due to dark matter heating caused by extended star formation? From the
shallow horizontal branch star formation history, we can see that And XXV
formed 50 per cent of its total stars 8.7 Gyrs ago and formed 90 per cent of
stars (and thus likely finished star forming) 5.8 Gyrs ago (Weisz et al.,
2019). It is possible that And XXV had a very short burst of star formation
$\sim$1-2 Gyrs ago, although within the uncertainties this is not significant
and could be an artefact of the method used as many of the dwarfs in their
sample display this short burst. Deeper star formation data is required to
confirm the nature of this burst. Dark matter heating is more effective in
galaxies with extended star formation (Read et al., 2016, 2019), hence the
relatively short, isolated bursts of star formation history that And XXV
undergoes means it is unlikely that star formation alone could explain the
extremely low central density observed in And XXV. The same conclusion was
reached for the other outliers with low central densities (Torrealba et al.,
2019; Collins et al., 2021). However, in the bottom panel of Fig. 7 we can see
that And XXV lies below the dashed grey line which indicates the stellar mass-
halo mass ratio for which dark matter heating can become viable (Di Cintio et
al., 2014). However, a recent study by Orkney et al. (2021) showed that
gravitational fluctuations due to late minor mergers can also instigate slight
core formation, even in dwarfs below this stellar mass-halo mass ratio.
Therefore, it is possible that And XXV may have undergone a slight cusp-core
transformation, although none of the simulated, isolated, dwarfs reported in
Orkney et al. (2021) reach the low density of And XXV.

Another process that can lower the central density is tidal interactions.
Tidal interactions lower densities at all radii, unlike dark matter heating
which only lowers the central density. Tidal interactions can be split into
two different processes, tidal stripping and tidal shocking. Tidal stripping
occurs when the gravitational force from the host galaxy (in this case M31)
exceeds the gravitational force from the dwarf allowing matter (dark
matter/stars) to become unbound from the dwarf. Tidal stripping preferentially
removes matter from the outer radii working inwards. As such the central dark
matter density is only noticeably reduced after significant mass loss. For
cusped profiles $\gtrsim 99$ per cent mass of the original mass would need to
be lost before the central dark matter density would decrease (Peñarrubia et
al., 2008; Peñarrubia et al., 2010; Errani et al., 2018; Errani & Peñarrubia,
2020; Errani & Navarro, 2021). However, cored profiles are less efficient at
protecting their central densities and require less extreme mass loss for a
noticeable effect (Read et al., 2006; Peñarrubia et al., 2010; Brooks &
Zolotov, 2014). Tidal shocking is another type of tidal interaction. It occurs
for satellites on highly eccentric orbits moving in and out of the
gravitational potential of the host. If the resulting gravitational
fluctuations occur on shorter time scales than the dynamical time of the
interaction, it will dynamically heat the stars and dark matter. The effect of
tidal shocking is most pronounced at the pericentre of the orbit, as this is
where the gravitational field changes most rapidly. Furthermore, tidal
shocking is only effective at lowering central densities for cored systems
(Read et al., 2006; Errani et al., 2017, 2018; Errani & Peñarrubia, 2020; van
den Bosch & Ogiya, 2018) or for systems that reside in a low concentration
dark matter halo (Amorisco, 2019).

At a 3D-project distance of 85.2 kpc (Savino et al., 2022), And XXV is one of
the closest M31 satellites, meaning it may potentially have had a previous
encounter with M31 resulting in tidal forces. However, without proper motions,
it is difficult to place meaningful constraints on the orbital history to
ascertain if And XXV has had any close approaches with M31. Even armed with
hypothetical proper motions, it is important to remember that the current
orbital properties are not always a robust indicator of possible past close
interactions (Lux et al., 2010; Genina et al., 2020a). Without orbital history
information, we turn our attention to other potential indicators of tidal
interaction. Firstly, we turn to the photometry to see if we can find
indications of tidal interactions. No obvious tidal tails or other tidal
substructure can be seen in the photometry of And XXV. The photometry used is
shallow, as such tidal features, which tend to have low surface brightnesses,
could readily be missed. Without dedicated deep imaging out to the very
outskirts of And XXV, such tidal substructure would be difficult to detect
(e.g. Shipp et al., 2022). Recent studies have shown that dSphs can undergo
tidal interactions and show no observable tidal substructure (Read et al.,
2006; Peñarrubia et al., 2009; Genina et al., 2020a), this is especially true
for dwarfs on highly eccentric orbits, since the episodes of shocking are
restricted to only the pericentric passages. As such, the dwarfs can relax
back to equilibrium on a dynamical timescale, quickly erasing any signs of the
interaction from the main body of the dwarf. Moreover, as the stellar
component is strongly embedded in the dark matter halo, dSphs can be
excessively stripped by tidal forces preferentially removing dark matter and
as such, not demonstrate any observable tidal features in the stars until very
extreme mass loss ($\gtrsim 90$ per cent Peñarrubia et al., 2008; Peñarrubia
et al., 2010; Errani et al., 2018). Secondly, in the kinematic analysis, we
observed a kinematically colder centre in And XXV with the velocity dispersion
increasing with radial distance. This may demonstrate that And XXV is not in
dynamical equilibrium which could be due to previous tidal interactions.
Interestingly a similar kinematic profile is observed in Antlia 2, a system
for which tides are the suspected culprit (Torrealba et al., 2019). Moreover,
the unusually low mass contained within the half-light radius measured for And
XXV could be explained by tidal stripping/shocking, which would reduce the
mass of the system over time. This is especially true for cored systems which
are able to maintain their original radius while losing mass (Peñarrubia et
al., 2010), resulting in the extended radial behaviour observed. Finally,
dwarfs undergoing strong tidal interactions, which result in stellar mass
loss, are expected to be outliers on the luminosity-metallicity relation (e.g.
Tucana III, see Simon et al., 2017). From Fig. 4 we can see that And XXV
agrees with this relation which suggests it has not lost much stellar mass
through tides. However, this does not mean that tides are not important as
significant dark matter mass loss can lower the inner density, as can tidal
shocking, without any associated stellar mass loss ((e.g. Read et al., 2006)).

No one piece of evidence is enough to decisively confirm or omit the
possibility of either dark matter heating or tidal interactions as the cause
of the low central density. However, the above would imply that we may
potentially explain And XXV’s extremely low central density due to a
combination of both factors. Meaning And XXV could have undergone a slight
cusp-core transformation from dark matter heating, which made And XXV more
susceptible to tidal stripping and shocking which further reduced the central
dark matter density, with tidal interactions being the significant
contributing factor. This conclusion is in line with the results from studies
investigating the other anomalous dwarfs (Torrealba et al., 2019; Collins et
al., 2021). Although, detailed analysis in combination with additional data,
such as proper motions, is required to consolidate this conclusion. And XXV
joins the small but growing list of unusual LG ’puffy’ dwarfs with low central
densities. Around the M31 we now see three such systems (And XXV, And XIX and
And XXI). A detailed study (Charles et al, in prep) into potential formation
and evolution pathways resulting in the low central densities observed will
improve our understanding of the nature of dark matter and potentially place
constraints on different cosmological models.

##  8 Conclusions

We present an updated kinematic analysis for And XXV. And XXV was previously
identified as a Local Group outlier. Using previous spectroscopic observations
in combination with a new data set, providing 53 member stars, more than
double that in the previous study, we were sufficiently able to determine the
presence of dark matter within And XXV, for the first time. In addition, we
dynamically mass model And XXV using BINULATOR \+ GravSphere to constrain its
dark matter density profile. Our key findings are as follows:

  * •

We measure the a systemic velocity for And XXV of $\nu=-107.7\pm{1.0}$ km s-1
and a velocity dispersion of $\sigma_{\rm\nu}=4.5\pm{1.0}$ km s-1. These
values are consistent with the results derived by C13, which had a much
smaller sample size.

  * •

We observe that the most central stars have a low velocity dispersion, which
increases with increasing radial distance, potentially indicating And XXV is
not in dynamical equilibrium and may have undergone tidal interactions.

  * •

Assuming dynamical equilibrium, we determine the mass contained within the
half-light radius to be M(r$<\rm r\textsubscript{h})=6.9^{+3.2}_{-2.8}\times
10^{6}$ M$\odot$, which is lower than we would expect for the size of And XXV.
This mass value corresponds to a mass-to-light ratio of $\text{[M/L]}_{\rm
r_{h}}=37^{+17}_{-15}$ M⊙/L⊙, which, for the first time, indicates And XXV is
a dark matter dominated system.

  * •

For stars with a $S/N>5$ we measure the metallicity of And XXV to be
$\rm[Fe/H]=-1.9\pm{0.1}$ dex from the coadded spectra. We also model the
metallicity distribution function from which we are able to resolve a mean
metallicity ($\rm[Fe/H]=-2.02\pm 0.02$ dex) and metallicity dispersion
($\sigma\textsubscript{[Fe/H]}=0.08^{+0.02}_{-0.01}$ dex). These results are
perfect in agreement with C13 and with the luminosity-metallicity relation for
low-mass LG dwarfs (Kirby et al., 2013).

  * •

Using the dynamical mass modelling tool BINULATOR \+ GravSphere we measure a
low central dark matter density of $\rho_{\rm DM}$(150 pc)=
2.7${}^{+1.8}_{-1.6}\times$107 M⊙ kpc-3. From the dark matter density profile
alone we cannot distinguish between a cusped or cored halo. However, when
compared to other quiescent star forming dSphs we find the And XXV has a
central dark matter density approximately one order of magnitude smaller than
we would expect, more similar to the central dark matter density of isolated
star forming dIrrs.

  * •

In a companion paper (Charles et al in prep.), we will consider whether dark
matter heating, tides, or some combination of these can explain And XXV’s low
density in the context of LCDM. We will also explore its implications for
alternative dark matter models.

## Acknowledgements

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

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

The LBT is an international collaboration among institutions in the United
States, Italy and Germany. LBT Corporation partners are: The University of
Arizona on behalf of the Arizona Board of Regents; Istituto Nazionale di
Astrofisica, Italy; LBT Beteiligungsgesellschaft, Germany, representing the
Max-Planck Society, The Leibniz Institute for Astrophysics Potsdam, and
Heidelberg University; The Ohio State University, representing OSU, University
of Notre Dame, University of Minnesota and University of Virginia.

EB acknowledges financial support from a Vici grant from the Netherlands
Organisation for Scientific Research (NWO). RI and NM acknowledge funding from
the European Research Council (ERC) under the European Unions Horizon 2020
research and innovation programme (grant agreement No. 834148).

## Data Availability

The raw spectra obtained with DEIMOS are available via the Keck archive. Fully
reduced 1D spectra and photometry will be made available upon reasonable
request to the lead author. The updated GravSphere code, along with the new
BINULATOR binning method is, available to download from
https://github.com/justinread/gravsphere.

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##  Appendix A Radial Profiles from GravSphere

Here we include the radial profile fits from GravSphere, see Fig. 8. First,
the top panel shows the well defined surface brightness profile,
$\Sigma_{\text{*}}$. The blue data points are the binned photometry from the
LBT imaging data out to four effective half light radii. Second, in the bottom
panel is the radial velocity dispersion, $\sigma_{\text{LOS}}$. The blue data
points are the binned velocity dispersion data taken from our probability
weighted member stars. For both panels, the black line shows the fit from
GravSphere with the dark and light gray shaded regions showing the 1 and
2$\sigma$ uncertainty intervals respectively. The vertical purple dashed line
is the half-light radius of And XXV. We see that the observed surface
brightness profile is well reproduced by GravSphere. We note that the same
velocity dispersion increasing with increasing radius behaviour is observed in
the binned kinematic data that is explained in $\S$ 4.2 and $\S$ 7. We
reiterate that this behaviour may indicate that And XXV is not in dynamical
equilibrium and could be indicative of And XXV having undergone tidal
interactions. The velocity dispersion profile inferred by GravSphere within
the 1$\sigma$ uncertainties for all bins. However, from visual inspection it
seems possible that GravSphere has artificially biased the velocity dispersion
to higher values at small radii in an attempt to fit this flaring feature.
Although, this potential bias would not affect the conclusion that And XXV has
an unusually low central dark matter density. A lower velocity dispersion
profile would result in a lower density. As such, the value from this study
can be considered an upper bound of the density.

Figure 8: Top: The surface brightness profile, $\Sigma_{\text{*}}$. The blue
data points are the binned photometry from the LBT imaging data. Bottom:  The
radial velocity dispersion, $\sigma_{\text{LOS}}$. The blue data points are
the binned velocity dispersions from our probability weighted member stars.
Both: The black line shows the fit from GravSphere. The 1 and 2$\sigma$
uncertainty intervals are shown by the dark grey and light grey shaded regions
respectively. The vertical purple dashed line is the half-light radius of And
XXV.


# Forbidden ”ortho”–”para” electric dipole transitions in H${}_{2}^{+}$ ion.

Vladimir I. Korobov  Bogoliubov Laboratory of Theoretical Physics, Joint
Institute for Nuclear Research, Dubna 141980, Russia  D. Bakalov  Institute
for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,
blvd. Tsarigradsko ch. 72, Sofia 1142, Bulgaria

###### Abstract

We present the first systematic calculation of the electric dipole forbidden
transitions in the homonuclear molecular ion H${}_{2}^{+}$. We get that the
transition rate from the ground ”ortho” $(v\\!=\\!0,N\\!=\\!1,J\\!=\\!1/2)$
state to the ground ”para” $(v\\!=\\!0,N\\!=\\!0,J\\!=\\!1/2)$ state is
$4.9\\!\times\\!10^{-14}$ s-1 that corresponds to the lifetime of
$6.4\\!\times\\!10^{5}$ yr. The Einstein coefficient $A_{nn^{\prime}}$ for the
$(v\\!=\\!1,N\\!=\\!1,J\\!=\\!1/2)\\!\to\\!(v\\!=\\!0,N\\!=\\!0,J\\!=\\!1/2)$
transition is equal to $0.12\\!\times\\!10^{-9}$ s-1, which is of ”comparable”
order with the values for quadrupole transitions in H${}_{2}^{+}$. It gives
hope that the ”ortho”–”para” transitions in H${}_{2}^{+}$ can be induced by
the laser spectroscopy.

##  I Introduction

Forbidden E1 transitions between ”ortho” (triplet in the proton spins, $I=1$)
and ”para” (singlet, $I=0$) states in the hydrogen molecular ion H${}_{2}^{+}$
are of significant interest. So far, only transitions between near-threshold
high vibrational $v\sim 18\\!-\\!19$ states were studied numerically [1] and
experimentally [2], where it was discovered that due to strong $g/u$ state
mixing (gerade/ungerade electronic symmetry states) near threshold the
transition rate is high enough for spectroscopic measurements. For example,
the rate of the $(v=19,N=1)\to(v=18,N=0)$ transition is $9.6\times 10^{-6}$
s-1.

In the case of hydrogen molecule H2, Wigner gave a qualitative estimate for
the E1 radiative decay rate from ”ortho” to ”para” hydrogen as $R_{\rm
est}=10^{-14}~{}\mbox{s}^{-1}$ (see [3]). First calculations by Raich and Good
[4] showed that the decay rate of the $X^{1}\Sigma^{+}_{g}(N\\!=\\!1)$ lowest
vibrational ”ortho” state is about $\sim\\!2\\!\times\\!10^{-13}$ yr-1, many
orders of magnitude smaller than the nominal rate due to strong cancellations.
Dodelson [3] reanalyzed the calculations of Raich and Good using a fully
quantum electrodynamical approach, introduced by Feinberg and Sucher [5] for
calculations in the helium atom, and derived additional terms, which change
the previous result by about 20%. Later Pachucki and Komasa [6] extended the
calculations by taking into account both E1 and M2 transitions, they’ve got
the E1 decay rate of the $(v\\!=\\!0,N\\!=\\!1,J\\!=\\!1)$ ”ortho” state equal
to $1.68\\!\times\\!10^{-13}$ yr-1 (or $5.3\\!\times\\!10^{-21}$ s-1). The
radiative lifetimes of the $1s\sigma_{g}$ $(v\\!=\\!0,N\\!=\\!1,J\\!=\\!1/2)$
”ortho” state of the H${}_{2}^{+}$ molecular ion have been only roughly
estimated in [1]. Our calculation shows that this estimate was incorrect.

In the present work we intend to perfom calculations of the E1 forbidden
transitions for the hydrogen molecular ion H${}_{2}^{+}$ at low $v$ and $N$,
which takes into account all relativistic corrections of order
$(m_{e}/m_{p})(Z\alpha)^{2}$ to the transition amplitude. We intend to show
that in the case of the E1 decay, transition rates are much higher than for
the neutral molecule and are in agreement with Wigner’s estimate for this
quantity.

In what follows we assume atomic units: $\hbar=|e|=m_{e}=1$.

##  II The Hamiltonian

In this work we adopt the following notation: $v$ is the vibrational quantum
number, $N$ is the total orbital angular momentum of the nonrelativistic wave-
function. The spin part is described by the spin operators of two protons,
$\mathbf{I}_{1}$, $\mathbf{I}_{2}$, and the spin of an electron,
$\mathbf{s}_{e}$, $\mathbf{I}=\mathbf{I}_{1}+\mathbf{I}_{2}$ is the total
nuclear spin, $\mathbf{F}=\mathbf{I}+\mathbf{s}_{e}$ is the total spin of the
H${}_{2}^{+}$ ion, and $\mathbf{J}=\mathbf{F}+\mathbf{N}$ is the total angular
momentum. Thus the ground ”para” state is denoted $(v=0,N=0,I=0,F=1/2,J=1/2)$.
The excited rotational ”ortho” state of the ground vibrational level
$(v=0,N=1)$ has two spin states: $F\\!=\\!1/2$ and $F\\!=\\!3/2$. We will call
the first state as the ground ”ortho” state. While the triplet states with the
total spin $F\\!=\\!3/2$ are not coupled to the subspace of the singlet
nuclear spin states, at least within the first order corrections to the wave
function considered in this work. For this reason we will ignore the
$F\\!=\\!3/2$ states in our studies and will use the shorter notation for
description of a state: $(v,N,J)$.

The Hamiltonian for an ion interacting with the radiation electromagnetic
field, which also includes the leading order Breit-Pauli corrections, is
expressed as follows:

$H=H_{0}+H_{BP}+H_{I},$ (1)

where $H_{0}$ is the zero-order nonrelativistic Hamiltonian of the particles,
$H_{BP}$ is the Breit-Pauli Hamiltonian, and $H_{I}$ is the part which
describes interaction of the ion with the electromagnetic field.

The nonrelativistic Hamiltonian in the center of mass frame may be written as:

$H_{0}=\frac{\mathbf{p}_{1}^{2}}{2M}+\frac{\mathbf{p}_{2}^{2}}{2M}+\frac{\mathbf{p}_{e}^{2}}{2m_{e}}-\frac{Z}{r_{1}}-\frac{Z}{r_{2}}+\frac{Z^{2}}{R},$
(2)

where $\mathbf{r}_{i}=\mathbf{r}_{e}\\!-\\!\mathbf{R}_{i}$ and
$\mathbf{R}=\mathbf{R}_{2}\\!-\\!\mathbf{R}_{1}$ are electron coordinates
relative to nuclei and internuclear position vector in the molecular
coordinate notations, $(\mathbf{r}_{e},\mathbf{R}_{1},\mathbf{R}_{2})$ and
$(\mathbf{p}_{e},\mathbf{p}_{1},\mathbf{p}_{2})$ are the position vectors and
momenta of particles in the center of mass frame, $M=m_{p}$ is the proton
mass, and $Z=1$ is the proton charge. According to the tradition accepted in
the theory of the light atoms we will use $Z$ for the proton charge in order
to distinguish between two scales: $\alpha$, fine structure constant, and
$v/c\approx Z\alpha$, mean particle velocity in a bound system.

The Breit-Pauli Hamiltonian $H_{BP}$ is the leading order relativistic
correction (see, for example, [7]). Here we need only the spin-spin and spin-
orbit interactions responsible for the $g/u$ mixing; we will follow the
notations of [8]. The spin-spin and spin-orbit interactions are expressed,
respectively:

$H_{ss}=\alpha^{2}\frac{(1\\!+\\!\kappa_{e})(1\\!+\\!\kappa_{p})}{m_{e}m_{p}}\frac{8\pi}{3}\left[(\mathbf{s}_{e}\cdot\mathbf{I}_{1})\delta(\mathbf{r}_{1})+(\mathbf{s}_{e}\cdot\mathbf{I}_{2})\delta(\mathbf{r}_{2})\right].$
(3) $\begin{array}[]{@{}l}\displaystyle
H_{so}=\alpha^{2}\Biggl{\\{}\frac{1\\!+\\!2\kappa_{e}}{2m_{e}^{2}}\left[\frac{[\mathbf{r}_{1}\times\mathbf{p}_{e}]}{r_{1}^{3}}+\frac{[\mathbf{r}_{2}\times\mathbf{p}_{e}]}{r_{2}^{3}}\right]\mathbf{s}_{e}-\frac{1\\!+\\!\kappa_{e}}{m_{e}m_{p}}\left[\frac{[\mathbf{r}_{1}\times\mathbf{p}_{1}]}{r_{1}^{3}}+\frac{[\mathbf{r}_{2}\times\mathbf{p}_{2}]}{r_{2}^{3}}\right]\mathbf{s}_{e}\\\\[8.53581pt]
\displaystyle\hskip
42.67912pt-\frac{1\\!+\\!2\kappa_{p}}{2m_{p}^{2}}\left[\frac{[\mathbf{r}_{1}\times\mathbf{p}_{1}]}{r_{1}^{3}}\mathbf{I}_{1}+\frac{[\mathbf{r}_{2}\times\mathbf{p}_{2}]}{r_{2}^{3}}\mathbf{I}_{2}\right]+\frac{1\\!+\\!\kappa_{p}}{m_{e}m_{p}}\left[\frac{[\mathbf{r}_{1}\times\mathbf{p}_{e}]}{r_{1}^{3}}\mathbf{I}_{1}+\frac{[\mathbf{r}_{2}\times\mathbf{p}_{e}]}{r_{2}^{3}}\mathbf{I}_{2}\right]\Biggr{\\}}.\end{array}$
(4)

The operators which connect ortho and para states are

$\begin{array}[]{@{}l}\displaystyle
H_{ss}^{-}=\alpha^{2}\frac{(1\\!+\\!\kappa_{e})(1\\!+\\!\kappa_{p})}{m_{e}m_{p}}\frac{4\pi}{3}\Bigl{\\{}(\mathbf{s}_{e}\cdot\mathbf{I}_{-})\bigl{[}\delta(\mathbf{r}_{1})\\!-\\!\delta(\mathbf{r}_{2})\bigr{]}\Bigr{\\}},\\\\[11.38109pt]
\displaystyle
H_{so}^{-}=\alpha^{2}\Biggl{\\{}\frac{1\\!+\\!\kappa_{p}}{2m_{e}m_{p}}\left(\frac{[\mathbf{r}_{1}\times\mathbf{p}_{e}]}{r_{1}^{3}}-\frac{[\mathbf{r}_{2}\times\mathbf{p}_{e}]}{r_{2}^{3}}\right)\mathbf{I}_{-}-\frac{1\\!+\\!2\kappa_{p}}{4m_{p}^{2}}\left(\frac{[\mathbf{r}_{1}\times\mathbf{p}_{1}]}{r_{1}^{3}}-\frac{[\mathbf{r}_{2}\times\mathbf{p}_{2}]}{r_{2}^{3}}\right)\mathbf{I}_{-}\Biggr{\\}}\end{array}$
(5)

where $\mathbf{I}_{1}$ and $\mathbf{I}_{2}$ are the spin operators of the two
protons, $\mathbf{I}_{-}=(\mathbf{I}_{1}-\mathbf{I}_{2})$, $\kappa_{e}$ and
$\kappa_{p}$ are the magnetic moment anomaly of an electron and proton,
respectively, while $\mu_{p}=1\\!+\\!\kappa_{p}$ is the magnetic moment of the
proton in nuclear magnetons.

In our consideration we use the NRQED formalism [9, 10]. For our needs only
the leading order terms of the Lagrangian [10], Eq. (19), are needed both for
an electron and proton. We use the Coulomb gauge, and the one-photon
interaction Hamiltonian (in the center of mass frame) may be expressed as:

$\begin{array}[]{@{}l}\displaystyle
H_{I}=\sum_{a}Z_{a}\alpha\frac{\mathbf{p}_{a}}{m_{a}}\mathbf{A}_{r}+\sum_{a}\frac{Z_{a}\alpha(1\\!+\\!\kappa_{a})}{2m_{a}}\boldsymbol{\sigma}_{a}\mathbf{B}_{r}-\sum_{a}\frac{Z_{a}\alpha(1\\!+\\!2\kappa_{a})}{8m_{a}^{2}}\>\boldsymbol{\sigma}_{a}\Bigl{(}\mathbf{p}_{a}\\!\times\\!\mathbf{E}_{\perp}-\mathbf{E}_{\perp}\\!\times\\!\mathbf{p}_{a}\Bigr{)}\\\\[8.53581pt]
\displaystyle\hskip
22.76219pt+\sum_{a}\frac{Z_{a}^{2}\alpha^{2}}{2m_{a}}\mathbf{A}\\!\cdot\\!\mathbf{A}_{r}-\sum_{a}\frac{Z_{a}^{2}\alpha^{2}(1\\!+\\!2\kappa_{a})}{4m_{a}^{2}}\>\boldsymbol{\sigma}_{a}\Bigl{(}\mathbf{A}_{r}\\!\times\\!\mathbf{E}_{\parallel}\Bigr{)}+\sum_{a}\frac{Z_{a}^{2}\alpha^{2}(1\\!+\\!2\kappa_{a})}{4m_{a}^{2}}\>\boldsymbol{\sigma}_{a}\Bigl{(}\mathbf{E}_{\perp}\\!\times\\!\mathbf{A}\Bigr{)}+\dots,\end{array}$
(6)

we use $\mathbf{A}_{r}$, $\mathbf{B}_{r}$, and $\mathbf{E}_{\perp}$ to denote
operators of the external electromagnetic radiation field. The transverse
fields $\mathbf{E}_{\perp}$ and $\mathbf{B_{r}}$ depend on $\mathbf{A}_{r}$ as
$\mathbf{E}_{\perp}=-\frac{1}{c}\frac{\partial}{\partial
t}\mathbf{A}_{r}(\mathbf{r},t)$ and
$\mathbf{B}_{r}=\boldsymbol{\nabla}\\!\times\\!\mathbf{A}_{r}(\mathbf{r},t)$.

Operators $\mathbf{E}_{\parallel}$ and $\mathbf{A}$ are the electric-field
strength and the magnetic-field potential, which are induced by the particles
constituting the molecular ion. For our derivations we need only the
expression for the magnetic-field potential produced by the magnetic moment of
the nuclei (corresponds to the $\mathbf{A}\\!\cdot\\!\mathbf{A}_{r}$ term of
the electron line in (6)):

$\mathbf{A}=-\frac{Z\alpha(1\\!+\\!\kappa_{p})}{m_{p}}\left(\frac{\left[\mathbf{r}_{1}\\!\times\\!\mathbf{I}_{1}\right]}{r_{1}^{3}}+\frac{\left[\mathbf{r}_{2}\\!\times\\!\mathbf{I}_{2}\right]}{r_{2}^{3}}\right),$

which results in correction to the nonrelativistic current
$\mathbf{J}^{(0)}=\sum_{a}Z_{a}\alpha\mathbf{p}_{a}/m_{a}$:

$\boldsymbol{\delta}\mathbf{J}=-\frac{Z\alpha^{3}(1\\!+\\!\kappa_{p})}{2m_{e}m_{p}}\left\\{\frac{\left[\mathbf{r}_{1}\\!\times\\!\mathbf{I}_{1}\right]}{r_{1}^{3}}+\frac{\left[\mathbf{r}_{2}\\!\times\\!\mathbf{I}_{2}\right]}{r_{2}^{3}}\right\\}.$
(7)

The other terms in Eq. (6) produce corrections to the electric current, which
either do not contain the proton spin operator, or are of higher order in
$m_{e}/m_{p}$ or $Z\alpha$ expansion and thus will be neglected.

##  III Transition amplitudes

transition | this work | [1]
---|---|---
$(19,0)\\!\to\\!(18,1)$ | 11.5 | 11.5
$(19,0)\\!\to\\!(17,1)$ | 161.4 | 163.7
Table 1: Comparison with calculations by Bunker and Moss [1]. Averaged
transition rates $\bar{A}_{nn^{\prime}}$ (in $10^{-6}$ s-1).

The transition amplitude for the forbidden E1 transitions is expressed

$\begin{array}[]{@{}l}\displaystyle\mathcal{T}_{E1}^{i}=\left\langle\psi_{n}\left|J^{i}\right|\psi_{n^{\prime}}\right\rangle+\left\langle\psi_{n}\left|J^{(0)i}\right|\psi_{n^{\prime}}^{(1)}\right\rangle+\left\langle\psi_{n}^{(1)}\left|J^{(0)i}\right|\psi_{n^{\prime}}\right\rangle=\left\langle\psi_{n}\left|J^{(0)i}\right|\psi_{n^{\prime}}\right\rangle+\left\langle\psi_{n}\left|\delta
J^{i}\right|\psi_{n^{\prime}}\right\rangle+\\\\[8.53581pt] \hskip
34.1433pt\displaystyle+\left\langle\psi_{n}\left|J^{(0)i}Q(E_{n^{\prime}}\\!-\\!H_{0})^{-1}QH_{\text{gu}}\right|\psi_{n^{\prime}}\right\rangle+\left\langle\psi_{n}\left|H_{\text{gu}}Q(E_{n}\\!-\\!H_{0})^{-1}QJ^{(0)i}\right|\psi_{n^{\prime}}\right\rangle,\end{array}$
(8)

where $\mathbf{J}$ is the electric current operator of the system of bound
particles (ion), $\psi_{n}$ and $\psi_{n^{\prime}}$ are the nonrelativistic
solutions of the Schrödinger equation, $\psi_{n}^{(1)}$ and
$\psi_{n^{\prime}}^{(1)}$ are the first order relativistic corrections to the
wave functions, $Q$ is the projection operator on a subspace orthogonal to
$\psi_{n}$ (or $\psi_{n^{\prime}}$), and the operator

$H_{\text{gu}}=H_{so}^{(-)}+H_{ss}^{(-)}$

is the $g/u$ symmetry breaking part of the Breit-Pauli Hamiltonian.

Transition amplitude in the length form may be obtained, using relation:
$[H_{0},\mathbf{r}_{a}]=-i\,\mathbf{p}_{a}/m_{a}$, or
$\mathbf{J}^{(0)}=i\alpha[H_{0},\mathbf{d}]$. Then one gets

$\begin{array}[]{@{}l}\displaystyle\mathcal{T}_{E1}^{i}=\left\langle\psi_{n}\left|\mathcal{D}^{i}\right|\psi_{n^{\prime}}\right\rangle+i\,\alpha
w_{nn^{\prime}}\left\langle\psi_{n}\left|d^{i}Q(E_{n^{\prime}}\\!-\\!H_{0})^{-1}QH_{\text{gu}}\right|\psi_{n^{\prime}}\right\rangle\\\\[8.53581pt]
\displaystyle\hskip 91.04881pt+i\,\alpha
w_{nn^{\prime}}\left\langle\psi_{n}\left|H_{\text{gu}}Q(E_{n}\\!-\\!H_{0})^{-1}Qd^{i}\right|\psi_{n^{\prime}}\right\rangle\end{array}$
(9)

where $d^{i}=\sum_{a}Z_{a}r_{a}^{i}$ is the dipole operator in the length
form, and

$\mathcal{D}^{i}=i\,\alpha w_{nn^{\prime}}d^{i}-i\alpha\left[d^{i},H_{\rm
gu}\right]+\delta J^{i}.$

Here the commutator is

$i\left[\mathbf{d},H_{\rm
gu}\right]=-\frac{\alpha^{2}}{2}\left(1+\frac{1}{m_{p}}\right)\frac{(1\\!+\\!\kappa_{p})}{m_{e}m_{p}}\left(\frac{[\mathbf{r}_{1}\\!\times\\!\mathbf{I}_{1}]}{r_{1}^{3}}+\frac{[\mathbf{r}_{2}\\!\times\\!\mathbf{I}_{2}]}{r^{3}_{2}}\right)$
(10)

and it cancels out the electric current correction term, Eq. (7), in the
leading order in $m_{e}/m_{p}$.

##  IV Results and discussion

Numerical calculations were based on the ”exponential” variational expansion
[12]. Particularly, the nonrelativistic solutions $\psi_{n}$ and the first
order corrections to the wave functions, $\psi_{n}^{(1)}$, were calculated
using this expansion method. By averaging Eq. (9), we obtained the transition
amplitudes $T_{nn^{\prime}}$ and, eventually, the Einstein coefficients
$A_{nn^{\prime}}$ for the spontaneous emission of a photon from the state
$n=(v,N,J)$ to the state $n^{\prime}=(v^{\prime},N^{\prime},J^{\prime})$.

In Table 1 our results are compared with the results of Bunker and Moss [1].
In order to make this comparison, the transition rates are averaged over
$J^{\prime}$,

$\bar{A}_{nn^{\prime}}=\bar{A}_{N=0,N^{\prime}=1}=\frac{2}{3}A_{J=\frac{1}{2},J^{\prime}=\frac{3}{2}}+\frac{1}{3}A_{J=\frac{1}{2},J^{\prime}=\frac{1}{2}}.$
(11)

In [1] only the spin-spin interaction has been taken into account as a source
of the $g/u$ breaking. That is justified for the weakly bound states where the
spin-spin coupling becomes dominant due to proximity of the $2p\sigma_{u}$
states.

Tables 2 and 3 present the results of our numerical calculations for
$N\\!=\\!0\\!\to\\!1$ and $N\\!=\\!1\\!\to\\!2$ transitions, respectively. As
is seen transition rates for $\Delta v=1$ transitions are of ”comparable”
order of magnitude with the rate of quadrupole transitions in H${}_{2}^{+}$
[13, 14]. For example, the E2 transition: $(v\\!=\\!0,N\\!=\\!0)\to(0,2)$, has
$A_{nn^{\prime}}=9.7\\!\times\\!10^{-12}$ s-1, and the E2 transition:
$(v\\!=\\!0,N\\!=\\!0)\to(2,2)$, has $A_{nn^{\prime}}=32.\\!\times\\!10^{-9}$
s-1 [14]. This fact allows to hope that the ”ortho”–”para” transitions may be
studied in laser spectroscopic experiments.

It is interesting to note that the contribution of spin-orbit interactions to
the decay rate of low $v$ states is about 5-10%. It turns out that a simple
approach with only spin-spin coupling should give fairly good results. In this
case, only the reduced matrix element of the transition $N\to N^{\prime}$ can
be calculated, and the spin part of the amplitude can be expressed using
Wigner’s 6-$j$ symbols, as for allowed transitions:

$\left\langle
vFNJ\|\mathbf{d}\|v^{\prime}FN^{\prime}J^{\prime}\right\rangle=(-1)^{J+F+N^{\prime}+1}\sqrt{(2J\\!+\\!1)(2J^{\prime}\\!+\\!1)}\left\\{\begin{matrix}N&1&N^{\prime}\\\
N^{\prime}&F&J\end{matrix}\right\\}\left\langle
vN\|\mathbf{d}\|v^{\prime}N^{\prime}\right\rangle\,.$

The terms, which were derived in the Dodelson work (see Eq. (32), [3]) and
were essential for the molecule, in case of H${}_{2}^{+}$ ion have
contributions of the same magnitude as for the H2. Thus they have a negligible
effect on the transition rate in the H${}_{2}^{+}$ molecular ion.

$v\\!\to\\!v^{\prime}$ |  $J=\frac{1}{2}\\!\to\\!J^{\prime}=\frac{1}{2}$ | $J=\frac{1}{2}\\!\to\\!J^{\prime}=\frac{3}{2}$
---|---|---
$0\\!\to\\!0$ | 0.0558 | 0.0494
$0\\!\to\\!1$ | 123.7 | 128.3
$0\\!\to\\!2$ | 0.1170 | 0.0708
$0\\!\to\\!3$ | 0.0429 | 0.0554
$0\\!\to\\!4$ | 0.0155 | 0.0183
$1\\!\to\\!0$ | 238.7 | 122.7
$1\\!\to\\!1$ | 0.0646 | 0.0585
$1\\!\to\\!2$ | 274.3 | 283.1
$1\\!\to\\!3$ | 0.8526 | 0.6433
$1\\!\to\\!4$ | 0.1131 | 0.1510
Table 2: The Einstein coefficient, $A_{nn^{\prime}}$, (in $10^{-12}$ s-1) for transitions between ”ortho” and ”para” states, $N=0\\!\to\\!N=1$ $v\\!\to\\!v^{\prime}$ |  $J=\frac{3}{2}\\!\to\\!J^{\prime}=\frac{5}{2}$ |  $J=\frac{3}{2}\\!\to\\!J^{\prime}=\frac{3}{2}$ | $J=\frac{1}{2}\\!\to\\!J^{\prime}=\frac{3}{2}$
---|---|---|---
$0\\!\to\\!0$ | 0.5229 | 0.0788 | 0.4186
$0\\!\to\\!1$ | 151.8 | 26.15 | 128.3
$0\\!\to\\!2$ | 0.0817 | 0.0070 | 0.0527
$0\\!\to\\!3$ | 0.0561 | 0.0121 | 0.0523
$0\\!\to\\!4$ | 0.0195 | 0.0039 | 0.0176
$1\\!\to\\!0$ | 211.3 | 23.96 | 236.6
$1\\!\to\\!1$ | 0.6076 | 0.0932 | 0.4899
$1\\!\to\\!2$ | 337.1 | 57.82 | 284.3
$1\\!\to\\!3$ | 0.7216 | 0.0865 | 0.5265
$1\\!\to\\!4$ | 0.1511 | 0.0335 | 0.1426
Table 3: The Einstein coefficient, $A_{nn^{\prime}}$, (in $10^{-12}$ s-1) for
transitions between ”ortho” and ”para” states, $N=1\\!\to\\!N=2$.

## Acknowledgements

The authors acknowledge support from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
Agreement No. 786306 “PREMOL” ERC-2017-AdG).

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