stars within the lower right quadrant of Fig. 17 among the chemically selected
accreted MSTO stars. These stars would otherwise allow us to study if there is
an age gradient in this plane. Our prediction for future studies with reliable
stellar ages is that stars (born during the merger) with lower $\sqrt{J_{R}}$
should not only be more enriched in [Mg/Fe], but also younger, as their birth
material was likely mixed with the $\alpha$-enhanced material of the Milky Way
(contrary to their older accreted siblings) or exhibited a burst of star
formation (Gilmore & Wyse, 1991). Finding such a gradient would allow us to
estimate how much mixing happened over the time-scale of the merger and also
put limits on the time-scale of the merger. Comparisons of their ages with
those of the high-$\alpha$ disk and halo would also aid the necessary
estimation of false-positive chemical selection (contamination) of disk stars
as accreted stars.

###  5.3 Towards a chemodynamical selection of the GSE

We have identified that the chemical selection extends significantly outside
the clean dynamical one in dynamical space and vice-versa in chemical space.
Only $(29\pm 1)\%$ of the stars of our chemical selection were found within
the clean dynamical selection box (Sec. 4.1 and Fig. 10d). There are three
avenues to solve this disagreement: (i) loosen constraints on the chemical
abundances for the chemical selection (to also include the high [Na/Fe] stars
of the GSE), (ii) loosen constraints on the dynamical selection (to also
include the low $\sqrt{J_{R}}$ stars found for the GSE), or (iii) combine less
strict constraints of the chemical and dynamical selections.

Concerning option (i), this would lead to adding stars from the intermediate
Gaussian component (see blue component in Fig. 7) to our selection. As we are
not able to separate the contamination by in-situ stars within this blue
component, it is however beyond the scope of this paper to estimate the
contamination and we resort to the other options here.

Figure 18:  Mean eccentricity (panel a) and [Fe/H] (panel b) in different
regions of the [Na/Fe] vs. [Mg/Mn] plane for stars of GALAH+ DR3. Only bins
with more than 5 stars have been populated. Density contours correspond to
those from Fig. 10a) with the chemically selected accreted stars (orange) and
all stars of GALAH+ DR3 (black).  Figure 19:  Distribution of eccentricity as
a function of different abundances of GALAH+ DR3 (black contours) and the
dynamically selected stars (red contours). Panel a) as a function of [Fe/H].
Panel b) as a function of an adjusted difference between [$\alpha$/Fe] and
[Fe/H] as suggested by Naidu et al. (2020). Panel c) as a function of [Na/Fe].
Panel d) as a function of [Na/Fe] with additional contours indicating our
chemically selected accreted (orange) and intermediate (blue) components. Red
dashed lines indicates the $e$ limited as suggested by Naidu et al. (2020).

Option (ii) was already tested by Feuillet et al. (2020) and suggested that
below $\sqrt{J_{R}/\,\mathrm{kpc\,km\,s^{-1}}}<30$ a significant contamination
by the disk is starting to dominate a dynamical selection.

Finally, we are interested in combining less-strict chemical with informative
dynamical properties towards a chemodynamical selection. The literature is
already rich in suggested selections (see Sec. A). Inspired by the promising
analysis of eccentricity $e$ by Mackereth et al. (2019) and Naidu et al.
(2020), we assess the possible combination of this orbit parameter with
chemical abundances. In Fig. 18, we plot our chemical selection plane [Na/Fe]
vs. [Mg/Mn], but coloured by mean eccentricities (panel a) and coloured by
mean [Fe/H] (panel b). We see a very sharp transition between typically low
eccentricity stars (red colours in Fig. 18a) for the disk stars (black
contours) and high eccentricities ($e>0.6$) in the upper left quartile. From
this figure, it also becomes evident that we are only selecting the low
[Na/Fe] stars (orange contours showing the chemically selected stars) of the
high eccentricity stars. We remind ourselves that Mackereth et al. (2019)
found $\sim 2/3$ of nearby halo stars have $e>0.8$, and Naidu et al. (2020)
selected stars based on their high eccentricities with $e>0.7$.

Which values of eccentricity should now be favoured? Whilst we stress that
finding the best chemodynamic selection is beyond the scope of this paper, a
first look at the distribution of chemical parameters as a function of
eccentricity $e$, see Fig. 19, can inform future studies. Here, we see clear
overdensities around $e\ll 0.5$ and $e\gg 0.8$. The latter also coincides with
the position of our dynamically identified GSE stars (red contours). To allow
the comparison with different chemical abundances, we plot [Fe/H], the
selection of low- and high-$\alpha$ stars by Naidu et al. (2020) \- similar to
the cut by Nissen & Schuster (2010) \- and [Na/Fe] in the different panels.
Depending on what a certain survey is able to measure, it thus could be
explored to combine eccentricity with any of these combinations. Fig. 19c also
suggests, that already upper limits on [Na/Fe], like $\mathrm{[Na/Fe]}\not>0$,
could suffice to select accreted stars and overcome detection limits. We thus
suggest to further assess the combination of abundance limits - such as
$\mathrm{[Na/Fe]}\not>0$ or $\mathrm{[Al/Fe]}\not>0$ \- with orbit limits -
such as $e>0.7$ as suggested by Mackereth et al. (2019) and Naidu et al.
(2020).

###  5.4 Timescales of star formation and accretion

Stellar ages help us to limit accretion timescales (also of the GSE) and trace
back, which events shaped the formation of the Milky Way, including the
merging history of our Galaxy (Wyse, 2001). Multiple studies (e.g. Jofré et
al., 2010; Schuster et al., 2012; Hawkins et al., 2014; Gallart et al., 2019;
Das et al., 2020; Montalbán et al., 2021) have delivered estimates of stellar
ages of accreted/GSE stars. As we are still unable to both estimate very
reliable ages and further disentangle the MW halo from the disk reliably, the
age estimates of different samples are in disagreement on several fronts.

Looking at the kinematic halo stars, Gallart et al. (2019) found that the
accreted stars, selected as the blue sequence (in photometric colours) of the
kinematic halo were coeval with their redder counterpart, and both
significantly older than the average MW high-$\alpha$ (thick) disk star. While
Schuster et al. (2012) and Hawkins et al. (2014) also found the metal-poor
accreted stars to be coeval with the old high-$\alpha$ halo and disk stars,
they identified the metal-rich end of the accreted population to be younger
than the majority of the high-$\alpha$ halo/disk.

With the help of asteroseismically aided observations, Chaplin et al. (2020)
and later Montalbán et al. (2021) were able to estimate some of the most
precise ages of GSE stars to date and confirmed that the average GSE star is
likely slightly younger (or coeval within uncertainties) than the average
high-$\alpha$ star with robust asteroseismically aided age estimates. They
thus concluded that a significant part of the MW high-$\alpha$ disk was
already in place before the infall of the GSE at around $10\,\mathrm{Gyr}$,
echoing the conclusions of several other earlier papers, based on more limited
data (e.g. Wyse, 2001, and references therein).

Our average ages of $11.3_{-3.1}^{+0.8}$$\,\mathrm{Gyr}$ and
$11.6_{-2.4}^{+0.6}$$\,\mathrm{Gyr}$, respectively, agree well with the
asteroseismically aided constraint of $11.0\pm 0.7\text{(stat)}\pm
0.8\text{(sys)}\,\mathrm{Gyr}$ for the GSE stars by Chaplin et al. (2020), but
appears to be older (although consistent within uncertainties) than the
average of the distribution found by Montalbán et al. (2021) at $9.7\pm
0.6\,\mathrm{Gyr}$. As the accuracy of stellar age estimation is subject to
several complex factors, such as atomic diffusion (see e.g. Jofré & Weiss,
2011), a discussion of this disagreement, including the contentious ages of
young $\alpha$-enhanced stars with large masses (e.g. Chiappini et al., 2015;
Zhang et al., 2021), is beyond the scope of this paper.

Looking at the lower end of stellar ages, Bonaca et al. (2020) found both the
star formation rate of the high-$\alpha$ disk and in-situ halo stars to
truncate $8.3\pm 0.1\,\mathrm{Gyr}$ ago ($z\simeq 1$), whereas they find the
star formation of accreted stellar halo to truncate
$10.2_{-0.1}^{+0.2}\,\mathrm{Gyr}$ ago ($z\simeq 2$). While small in size, 75%
(9/12) and 80% (89/112) of the MSTO stars of our chemically and dynamically
identified accreted stars also show ages above $10\,\mathrm{Gyr}$ (Fig. 14).

While these observations suggest that the GSE has not significantly influenced
the formation of the thick disk, it has fuelled the hypothesis that the last
major merger of the GSE is chronologically not only consistent with the
decrease of star formation in the high-$\alpha$ disk (Bonaca et al., 2020) and
beginning of star formation of the low-$\alpha$ disk, but that there is
actually a causal connection (see e.g. Buck, 2020). If true, this allows us to
put constraints on the merger timescale and subsequent onset of the
low-$\alpha$ disk formation (e.g. Wyse, 2001; Di Matteo et al., 2019;
Belokurov et al., 2020).

The jury is still out as to whether the GSE is responsible for the decrease in
thick disk star formation. But it seems that at least the timing is plausible
in this eventful Galactic epoch. Future studies of the chrono-chemodynamic
properties of the accreted and in-situ stars promise to shed light on the
circumstances of drastic changes in the Galactic structure.

##  6 Conclusions

In this study we set out to identify which elements are best used to identify
accreted stars in our Milky Way based on elemental abundances from the third
data release of the GALAH Survey (Buder et al., 2021) and to compare this
chemical selection with dynamical ones from the literature. The key findings
of our study are:

  * •

To identify the best set of elements for this task, we follow the approach by
Nissen & Schuster (2010) to identify accreted stars (in their paper called
low-$\upalpha$ halo stars) via their high total velocities and low [Mg/Fe]
compared to the (thick) disk stars. We find several elements showing a
significant separation in GALAH+ DR3 between abundances of the accreted stars
and the disk (or in-situ stars), including Mg, Si, Na, Al, Mn, Ni, and Cu.
Their detection rates as a function of [Fe/H] vary strongly, and we find the
best compromise of significance of separation and detection rates for Mg, Mn,
and Na - ranking them as the best telltale elements of accretion based on
GALAH+ DR3 (see discussion in Sec. 5.1.1).

  * •

We test the identification of accreted stars based on these elements in
different abundance planes via Gaussian Mixture Models and find the best
results from [Na/Fe] vs. [Mg/Mn], similar to the selection via [Al/Fe] vs.
[Mg/Mn] used by Hawkins et al. (2015) and Das et al. (2020) for data from the
APOGEE Survey.

  * •

We compare the chrono-chemodynamic properties of stars identified via this
chemical selection with those of the accreted stars of the GSE, selected via a
box in $L_{Z}$ vs. $\sqrt{J_{R}}$ space as suggested by Feuillet et al.
(2021). We discuss the implications for chemical tagging of accreted stars as
well as how we can interpret the difference between chemical and dynamical
selections. Our main points are: Firstly, values of [Na/Fe] of stars
identified through dynamical selection are typically $0.1\,\mathrm{dex}$
higher than those of the chemical selection and secondly, the radial actions
$J_{R}$ of the chemical selection extend well below the clean selection in
dynamical space suggested by Feuillet et al. (2021). In particular, only
$(29\pm 1)\%$ of the chemically selected stars fall within the clean dynamical
selection box.

  * •

We discuss the reliability and contamination of our selection in Sec. 5.1.2,
finding that our chemical selection is possibly - but insignificantly -
contaminated by the IGS/Heracles (7%) and other accreted structures.

  * •

We find 20% of the chemically selected accreted stars on prograde orbits
$L_{Z}>500\,\mathrm{kpc\,km\,s^{-1}}$, that is, overlapping with the hot disk.
More follow-up is necessary to identify if this is caused by contamination or
an actual overlap in dynamical space, thus suggesting changes of orbits of
accreted stars (see our discussion in Sec. 5.2.2).

  * •

If one is interested in the chemical properties of the GSE it is favourable to
use the quantities estimated based on the dynamical selection. To analyse the
dynamical properties of the GSE, however, those estimated from the chemical
selection should be preferred. Again, we caution that we expect a small
contamination by the IGS/Heracles (7%) as well as possibly the dynamically hot
stellar disk. 20% of our chemically selected stars exhibit
$L_{Z}>500\,\mathrm{kpc\,km\,s^{-1}}$. This is an upper limit of our
contamination, as it could be either caused by contamination of our selection
or changed orbits of truly accreted stars. We do not suggest to use the
overlap of both selections, as we have identified significant differences due
to the overlap of accreted and in-situ stars in [Na/Fe], which prohibit us to
distinguish the accreted stars with high [Na/Fe] from the in-situ ones within
our uncertainties. In particular, 72% of the stars of the dynamically selected
GSE have [Na/Fe] above the $84^{\text{th}}$ percentile of the chemically
selected ones.

  * •

We therefore also discuss how we can find a better selection of accreted stars
in chemodynamical space. In Sec. 5.3, we thus show how the previously
suggested orbit eccentricity (see e.g. Mackereth et al., 2019; Naidu et al.,
2020) in combination with chemistry can help future studies to find
appropriate selections.

  * •

Finally, we use age estimates of MSTO stars to find typical ages of
$11.3_{-3.1}^{+0.8}$$\,\mathrm{Gyr}$ (chemically selected) and
$11.6_{-2.4}^{+0.6}$$\,\mathrm{Gyr}$ (dynamically selected) for the accreted
stars in GALAH+ DR3. We caution though that these are estimated from only 12
and 112 MSTO stars, respectively, and prone to outliers. When neglecting the
few stars below $10\,\mathrm{Gyr}$, our distribution suggests the truncation
of star formation of accreted stars around $10\,\mathrm{Gyr}$ as found by
Bonaca et al. (2020).

##  7 Outlook

With the ongoing development of new instruments and the beginning of the era
of large-scale stellar surveys (see Nissen & Gustafsson, 2018; Jofré et al.,
2019, for reviews), the bulge and halo have now also come into reach and we
start to see streams and substructures in the halo (see e.g. Helmi, 2020, for
a review), which are evidence of ongoing and past accretion events. How
significant these events were is still under investigation: How many mergers
happened? Where are their remnants now? How (dis-)similar are their properties
to the in-situ stars that were already in the Galaxy? Which of these were
major mergers? How much (primordial) gas did they bring into the Galaxy? What
is the connection between mergers with the pause in star formation and
different chemistry that we observe between the high- and low-$\alpha$ disk?
Helmi (2020) concludes that, to be able to interpret various structures, we
need detailed chemical abundances of stars with full phase-space information,
which in-turn motivates the continuing efforts of ongoing and upcoming
surveys.

With the availability of astrometric information from the Gaia satellite
mission and its Data Releases 1 (Gaia Collaboration et al., 2016b), DR2 (Gaia
Collaboration et al., 2018a), and eDR3 (Gaia Collaboration et al., 2021) as
well as the industrial revolution of stellar spectroscopic surveys, delivering
millions of chemical abundance measurements (for a review see Jofré et al.,
2019), our selection techniques of accreted stars start to shift from
kinematic towards chemodynamic or even purely chemical properties.

We are, however, just at the beginning of truly understanding the interplay of
kinematics/dynamics, chemistry, and ages of the different substructures. We
know that, when it comes to these different properties, kinematic properties
change on short timescales, whereas dynamic properties (in a slowly evolving
potential) are conserved for a longer period. But we hope that chemical
abundances, as locked in the stellar atmosphere at birth, do not change
significantly over cosmic time for individual stars, and are furthermore
significantly different for different Galactic and extra-galactic birth
places. Stellar ages (which are difficult to extract from our observables) are
our best hope to narrow down the formation scenarios of our Galaxy.

Two major question that need to be answered in more holistic studies are: When
can we actually identify a star as accreted? And how can we tell it apart from
other accreted stars? In this study, we have been able to answer this question
in terms of the most extreme cases, that is the significantly different
enrichment of some stars for example in Na (and Al). However, we clearly are
struggling at the overlap of accreted structures themselves (e.g. GSE and
IGS/Heracles or GSE vs. Arjuna) as well as accreted stars and in-situ ones.
When do we actually call stars in-situ, given that the Milky Way itself likely
started from several smaller structures that then kept accreting and star
formation also takes place during mergers from material of both in-situ and
accreted material? More research is needed to push our understanding of the
underlying (accreted) structure of our Milky Way and its building blocks.
Possible clues might also be found by studying the spatial distribution of
these stars compared to the older GSE stars. Due to the low number of
identified stars, this should, however, be done by combining the stars
identified by the various different and complementary surveys.

As a follow-up study we also strongly propose to attempt to associate the
substructures detected in dynamical space that are not overlapping with the
clean GSE box by Feuillet et al. (2021) in detail. This would be an
application of the methodology similar to the one by Naidu et al. (2020), as
already done for the Helmi streams for GALAH+ DR3 by Limberg et al. (2021),
but is beyond the scope of this study. Along a similar line of thought, we
also suggest to continue the efforts of searching for associations between
accreted structures with other substructure. Such studies include the search
for associations between globular clusters and accreted structures (Massari et
al., 2019), stellar streams with globular clusters as potential progenitors
(Bonaca et al., 2021) and moving groups with accreted structures (Schuler et
al., 2021). These studies made use of the data from different surveys,
including data provided by Gaia Collaboration et al. (2018c) in combination
with the work by Vasiliev (2019) and the H3 Survey (Conroy et al., 2019). The
data from the GALAH survey is very complementary to these surveys, as it
probes different regions of the Galaxy and/or adds the high-dimensional
chemical perspective and thus allows to confirm found accreted associations
even stronger.

In the future it will be vital to continue the effort of comparing present-day
observations with both higher redshift observations as well as potential
formation scenarios. High redshift observations may allow us to observe major
mergers as they happen and inform us on their importance. Was the MW
high-$\alpha$ (thick) disk for example heated up by major mergers like those
of the GSE, or was it already born hot as the correlation of higher gas
velocity dispersions at higher redshifts would suggest (Wisnioski et al.,
2015; Leaman et al., 2017)? Can we find a consistent story over all redshifts?

Favouring or excluding formation scenarios will need to go hand-in-hand with
the comparison to (cosmological hydrodynamical) simulations (e.g. Mackereth &
Bovy, 2018; Bonaca et al., 2017; Wu et al., 2021), which allow us to time
accretion events and trace accreted stars spatially, dynamically, and now also
chemically. Much progress has been made in recent years through studies of the
in-situ and ex-situ fractions (e.g. Pillepich et al., 2015), the influence of
mergers on the $\alpha$-enhancement (e.g. Zolotov et al., 2010; Grand et al.,
2020; Buck, 2020; Renaud et al., 2021a), the estimation of infall scenarios
and parameters of the GSE (e.g. Villalobos & Helmi, 2008; Koppelman et al.,
2021; Naidu et al., 2021), including the amount and importance of gas-rich and
gas-poor mergers (e.g. Fensch et al., 2017; Renaud et al., 2021b) and telling
apart different components of simulated galaxies (e.g. Obreja et al., 2019).
We expect great progress here and an iterative convergence on deciphering the
origin of the elements, as elemental abundance measurements - especially of
environments different than the already well-studied disk - inform the
constraints on chemical enrichment processes and yields (e.g. Fernández-Alvar
et al., 2018; Vincenzo et al., 2019; Eitner et al., 2020; Sanders et al.,
2021; Ishigaki et al., 2021).

## Acknowledgements

We acknowledge the traditional owners of the land on which the AAT and ANU
stand, the Gamilaraay, the Ngunnawal and Ngambri people. We pay our respects
to elders past, present, and emerging and are proud to continue their
tradition of surveying the night sky in the Southern hemisphere. This work was
supported by the Australian Research Council Centre of Excellence for All Sky
Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.
KL acknowledges funds from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 852977). TB acknowledges support from the European Research
Council under ERC-CoG grant CRAGSMAN-646955. JK and TZ acknowledge financial
support of the Slovenian Research Agency (research core funding No. P1-0188)
and the European Space Agency (PRODEX Experiment Arrangement No. C4000127986).
KCF acknowledges support from the the Australian Research Council under award
number DP160103747. CK acknowledges funding from the UK Science and Technology
Facility Council (STFC) through grant ST/R000905/1 and ST/V000632/1, and the
Stromlo Distinguished Visitorship at the ANU. YST acknowledges financial
support from the Australian Research Council through DECRA Fellowship
DE220101520. We thank Rohan Naidu for sharing the [Fe/H] values from Naidu et
al. (2020). Hyperlink figures to code access are inspired by Rodrigo Luger.

## Facilities

AAT with 2df-HERMES at Siding Spring Observatory: The GALAH Survey is based
data acquired through the Australian Astronomical Observatory, under programs:
A/2013B/13 (The GALAH pilot survey); A/2014A/25, A/2015A/19, A2017A/18 (The
GALAH survey phase 1), A2018 A/18 (Open clusters with HERMES), A2019A/1
(Hierarchical star formation in Ori OB1), A2019A/15 (The GALAH survey phase
2), A/2015B/19, A/2016A/22, A/2016B/10, A/2017B/16, A/2018B/15 (The HERMES-
TESS program), and A/2015A/3, A/2015B/1, A/2015B/19, A/2016A/22, A/2016B/12,
A/2017A/14, (The HERMES K2-follow-up program). This paper includes data that
has been provided by AAO Data Central (datacentral.aao.gov.au). Gaia:  This
work has made use of data from the European Space Agency (ESA) mission Gaia
(http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. Other facilities: This
publication makes use of data products from the Two Micron All Sky Survey
(Skrutskie et al., 2006) and the CDS VizieR catalogue access tool (Ochsenbein
et al., 2000).

## Software

The research for this publication was coded in python (version 3.7.4) and
included its packages astropy (v. 3.2.2; Astropy Collaboration et al., 2013,
2018), corner (v. 2.0.1; Foreman-Mackey, 2016), galpy (version 1.6.0; Bovy,
2015), IPython (v. 7.8.0; Pérez & Granger, 2007), matplotlib (v. 3.1.3;
Hunter, 2007), NumPy (v. 1.17.2; Walt et al., 2011), scipy (version 1.3.1;
Virtanen et al., 2020), sklearn (v. 0.21.3; Pedregosa et al., 2011),
statsmodels (v. 0.10.1), xdgmm (v. 1.1). We further made use of topcat
(version 4.7; Taylor, 2005);

## Data Availability

The data used for this study is published by Buder et al. (2021) and can be
accessed publicly via https://docs.datacentral.org.au/galah/dr3/overview/. We
provide full tables for Tables 3 and 5 in the supplementary material. All code
to reproduce the analysis and figures can be accessed via
https://github.com/svenbuder/Accreted-stars-in-GALAH-DR3 and is also marked
behind each figure with a link icon to this repository.

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1Research School of Astronomy & Astrophysics, Australian National University,
Canberra, ACT 2611, Australia
2ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
Australia
3Department of Astronomy, Stockholm University, AlbaNova University Centre,
SE-106 91 Stockholm, Sweden
4Department of Astronomy, Columbia University, Pupin Physics Laboratories, New
York, NY 10027, USA
5Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue,
New York, NY 10010, USA
6Lund Observatory, Department of Astronomy & Theoretical Physics, Box 43,
SE-221 00 Lund, Sweden
7Astrophysics Research Institute, Liverpool John Moores University, 146
Brownlow Hill, Liverpool L3 5RF, UK
8Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482
Potsdam, Germany
9Sydney Institute for Astronomy, School of Physics, A28, The University of
Sydney, NSW 2006, Australia
10Monash Centre for Astrophysics, Monash University, Australia
11School of Physics & Astronomy, Monash University, Australia
12Department of Physics & Astronomy, Macquarie University, Sydney, NSW 2109,
Australia
13Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Padova,
vicolo dell’Osservatorio 5, 35122, Padova, Italy
14Faculty of Mathematics & Physics, University of Ljubljana, Jadranska 19,
1000 Ljubljana, Slovenia
15School of Physics, UNSW, Sydney, NSW 2052, Australia
16Stellar Astrophysics Centre, Department of Physics & Astronomy, Aarhus
University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
17Macquarie University Research Centre for Astronomy, Astrophysics &
Astrophotonics, Sydney, NSW 2109, Australia
18Mullard Space Science Laboratory, University College London, Holmbury St.
Mary, Dorking, Surrey, RH5 6NT, UK
19Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD
4350, Australia
20Centre for Astrophysics Research, Department of Physics, Astronomy &
Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK
21Center for Astrophysical Sciences & Department of Physics & Astronomy, The
Johns Hopkins University, Baltimore, MD 21218

##  Appendix A Selection and visualisation techniques of halo/accreted stars

Table 6:  A compilation of different techniques to identify major accretion structures. The list includes photometric information used in colour-magnitude diagrams (CMD), stellar kinematic properties such as Galactic longitude $l$ and latitude $b$, radial velocity $v_{\text{rad}}$, tangential velocity ($V_{T}$), total velocity ($V_{\text{tot}}$), Galactocentric Cartesian velocities ($V_{X}$, $V_{Y}$, and $V_{Z}$), Galactocentric cylindrical velocities ($V_{R}$, $V_{\phi}$, and $V_{Z}$), stellar dynamic properties such as maximum Galactocentric radius ($R_{\text{max}}$), actions ($J_{R}$, $J_{\phi}=L_{Z}$, $J_{Z}$, and total $J_{\text{tot}}$), eccentricity $e$, orbit energy $E$, as well as stellar chemical information such as the iron abundances relative to hydrogen [Fe/H], and element abundances of element X relative to iron [X/Fe]. $k$-means and Gaussian Mixture Models (GMM) are scikit-learn clustering algorithms (Pedregosa et al., 2011), whereas StarGo is a neutral-network-based clustering method Yuan et al. (2018). We note that the references are not necessarily the first ones finding these properties, but examples of their application. In the case of [Na/Fe] vs. [Ni/Fe] for stars with high $V_{\text{tot}}$, the correlation has e.g. found by Nissen et al. (1997); Nissen & Schuster (2010) and discussed by Venn et al. (2004) before being applied explicitly by Bensby et al. (2014).  Category | Properties | Example Reference(s)
---|---|---
Stellar photometry |  $m_{i}$ and/or $m_{i}-m_{j}$ | Belokurov et al. (2006)
Stellar kinematics |  $V_{X}$, $V_{Y}$, $V_{Z}$, and $\sqrt{V_{X}^{2}+V_{Z}^{2}}$ | Koppelman et al. (2018)
|  $V_{R}$, $V_{\phi}$, $V_{Z}$ ellipsoid membership probability | Carollo et al. (2010)
| … | Ishigaki et al. (2012, 2013)
| two-point velocity correlation function | Re Fiorentin et al. (2015)
| Neural-network based classification with Gaia DR2 6D | Ostdiek et al. (2020)
| (same as the previous) | Necib et al. (2020)
Stellar dynamics |  $V_{\phi}$ and $R_{\text{max}}$ | Gratton et al. (2003)
|  $J_{Z}$ and $J_{\perp}=\sqrt{J_{X}^{2}+J_{Y}^{2}}$ | Helmi et al. (1999)
|  $L_{Z}$ and $E$ | Helmi et al. (2017, 2018)
|  $L_{Z}$, $E$, and $L_{Z}/|L_{Z,\text{circ}}|$ | Koppelman et al. (2019)
|  $J_{\phi}/J_{\text{tot}}$ and $(J_{Z}-J_{R})/J_{\text{tot}}$ | Myeong et al. (2019)
|  $L_{Z}$ and $J_{R}$ | Feuillet et al. (2020)
|  $E$, $L$, $\theta=\arccos{L_{Z}/L}$, $\phi=\arctan{L_{X}/L_{Y}}$ via StarGo | Yuan et al. (2020a)
Stellar kinematics and photometry |  $v_{\text{rad}}$ and $m_{i}$ | Ibata et al. (1994)
|  $l$, $b$, $\mu_{l}$, $\mu_{b}$, $m_{i}$ via streamfinder | Malhan & Ibata (2018)
|  $V_{T}$ and sequences in the CMD | Gaia Collaboration et al. (2018b)
| (same as the previous) | Haywood et al. (2018b)
| (same as the previous) | Gallart et al. (2019)
Stellar chemokinematics |  $V_{\text{tot}}$, [Fe/H], and [Mg/Fe] | Nissen & Schuster (2010)
| … | Navarro et al. (2011)
|  $V_{\text{tot}}$, [Na/Fe], [Ni/Fe] | Bensby et al. (2014)
|  $l$, $b$, $v_{\text{rad}}$, [Fe/H], and [$\alpha$/Fe] | Hawkins et al. (2015)
|  $V_{\phi}$ and [Fe/H] | Belokurov et al. (2020)
| (same as the previous) | An & Beers (2021)
| Galactocentric spherical $V_{\rho}$, $V_{\phi}$, and [Fe/H] | Belokurov et al. (2018)
| Galactocentric spherical $V_{\rho}$, $V_{\phi}$, $V_{\theta}$ and [Fe/H] via GMM | Myeong et al. (2018a)
|  $V_{X}$, $V_{Y}$, $V_{Z}$, and [Fe/H] via GMM | Nikakhtar et al. (2021)
Stellar chemodynamics |  $e$, [Fe/H], [Mg/Fe], [Al/Fe], [Ni/Fe] via $k$-means | Mackereth et al. (2019)
| [Fe/H], $J_{R}$, $L_{Z}$, and $J_{Z}$ | Myeong et al. (2018b)
|  $e$, [Fe/H], and [$\alpha$/Fe] | Naidu et al. (2020)
|  $E$ and $e$ informed by [Al/Fe] and [Mg/Mn] | Horta et al. (2021)
Stellar chemistry | [Fe/H] and [Mg/Fe] | Di Matteo et al. (2019); Di Matteo et al. (2020)
| [Fe/H] and [$\alpha$/Fe] | Carollo & Chiba (2021)
| [Al/Fe], [Mg/Mn] via GMM | Das et al. (2020)
| [Al/Fe], [Mg/H] | Feuillet et al. (2021)
| [Fe/H], [X/Fe] for (C+N), O, Mg, Al, Si, K, Ca, Cr, Mn, and Ni via $k$-means | Hayes et al. (2018)

Many selections of major substructures in the halo via spatial, photometric,
kinematic, dynamic, and chemical properties (or a combination of those) exist
- we list several of them, categorised by the used selection properties in
Table 6.

We acknowledge that this list is incomplete and not all studies that used the
listed techniques are included. We hope this list is of use for giving the
reader an overview of the existing techniques to uncover major merger events
in the early history of the Milky Way (see Helmi, 2020, for a review).


# Anomalous coupling in radiation mediated shocks

Amir Levinson1<EMAIL_ADDRESS>Alon Granot 1  Arno Vanthieghem 2,3
Jens Mahlmann 3  1School of physics and Astronomy, Tel Aviv University, Tel
Aviv 2 International Research Collaboration Center, National Institutes of
Natural Sciences, Tokyo 105-0001, Japan 3 Department of Astrophysical
Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA

###### Abstract

We summarize recent attempts to unravel the role of plasma kinetic effects in
radiation mediated shocks. Such shocks form in all strong stellar explosions
and are responsible for the early electromagnetic emission released from these
events. A key issue that has been overlooked in all previous works is the
nature of the coupling between the charged leptons, that mediate the radiation
force, and the ions, which are the dominant carriers of the shock energy. Our
preliminary investigation indicates that in the case of relativistic shocks,
as well as Newtonian shocks in multi-ion plasma, this coupling is driven by
either, transverse magnetic fields of a sufficiently magnetized upstream
medium, or plasma micro-turbulence if strong enough magnetic fields are
absent. We discuss the implications for the shock breakout signal, as well as
abundance evolution and kilonova emission in binary neutron star mergers.

##  1 Introduction

Astrophysical shocks are commonly divided into two main types: collisionless
shocks, which form in optically thin regions, and are mediated by collective
plasma instabilities on skin depth scales (Blandford & Eichler, 1987;
Spitkovsky, 2005), and radiation mediated shocks (RMS), which form in
optically thick regions and are mediated by Compton scattering and, under
certain conditions, pair creation on scales of the order of the Thomson length
(see Waxman & Katz 2017; Levinson & Nakar 2020, and references therein).

Formation of a RMS occurs when, (i) the photon diffusion time across the shock
is comparable to the shock crossing time of the flow111This is also the
deceleration length of the flow. To see this note that the mean force acting
on a baryon (assuming tight coupling between ions and electrons) is
$-m_{p}d\beta/dz=\sigma_{T}e_{rad}$, where $e_{rad}$ is the local energy
density of the radiation. Energy conservation yields $e_{\gamma}\approx
n_{u}m_{p}c^{2}\beta_{u}^{2}$ in the immediate post shock region, where
$\beta_{u}$ is the shock velocity and $n_{u}$ is the plasma density upstream,
with which one obtains
$-d\beta/dz\sim\beta_{u}/L_{dec}\sim\sigma_{T}n_{u}\beta_{u}^{2}$. Thus,
$L_{dec}\simeq(n_{u}\sigma_{T}\beta_{u})^{-1}$ equals the photon diffusion
length (see Levinson & Nakar 2020). A similar result can be derived for
relativistic shocks (Granot et al., 2018; Vanthieghem et al., 2022)., and (ii)
the shock is radiation dominated (e.g., Weaver, 1976; Katz et al., 2010).
Condition (i) requires the optical depth ahead of the shock to exceed
$\beta_{u}^{-1}$ (roughly the shock width), where $\beta_{u}$ is the shock
velocity in units of $c$ 222In relativistic RMS, this scaling is altered by
Klein-Nishina effects and excessive pair production.. Condition (ii) requires
the shock to be fast enough, $\beta_{u}>2\times 10^{-4}(n_{u}/10^{15}\,{\rm
cm^{-3}})^{1/6}$, here $n_{u}$ is the upstream density. These two conditions
pertain in essentially all strong stellar explosions, including various types
of supernovae, low luminosity GRBs, regular GRBs and binary neutron star (BNS)
mergers (Levinson & Nakar, 2020). Consequently, RMS are inherent features of
stellar explosions. The RMS structure and dynamics depend on the progenitor
type, the explosion energy, and the explosion geometry. In particular, the
shock velocity at breakout can range from sub-relativistic to mildly
relativistic, and in extreme cases even ultra-relativistic.

The prime motivation to explore the physics of RMS is the desire to predict
the properties of the radiation emitted during the breakout of the shock from
the opaque envelope surrounding the explosion center. This breakout emission -
the first electromagnetic signal a distant observer can detect - and the
subsequent cooling emission, carry valuable information regarding the
explosion mechanism and the progenitor type, which is otherwise inaccessible.
Moreover, as explained in §3.3 (see granot2023 for a detailed analysis), in
case of BNS mergers the RMS can trigger nuclear transmutations prior to its
breakout, that can significantly impact the abundance evolution of r-process
material and the resultant kilonova emission.

The physics of RMS involves two disparate scales; the Thomson scattering
length,

$\lambda=(\sigma_{T}n_{u})^{-1}\approx
10^{9}\left(\frac{n_{u}}{10^{15}\,cm^{-3}}\right)^{-1}\quad\rm cm,$ (1)

that dictates the global shock dynamics and structure, and the (proton) skin
depth,

$l_{p}=\frac{c}{\omega_{p}}\approx
0.5\left(\frac{n_{u}}{10^{15}\,cm^{-3}}\right)^{-1/2}\quad\rm cm,$ (2)

on which coupling between ions, electrons and, in relativistic RMS, positrons
is anticipated to occur. Here $\omega_{p}=(4\pi e^{2}n_{u}/m_{\rm p})^{1/2}$
is the (proton) plasma frequency. As seen, under typical astrophysical
conditions the separation between kinetic and radiation scales is huge. This
is presumably the reason why the issue of ion-electron coupling in RMS has
been overlooked until very recently. In what follows, we outline recent
attempts to investigate the role of plasma kinetic effects in RMS, and discuss
potential implications for observations of stellar explosions.

The interaction of intense radiation field with plasma has been studied also
in other astrophysical contexts. In particular, the dynamics of Compton-driven
plasma wakefields generated by the interaction of GRB prompt emission with the
circumburst medium has been studied in Frederiksen (2008) and Del Gaudio et
al. (2020). A kinetic study on the development of plasma instabilities in
bright GRBs is presented in Martinez et al. (2021).

##  2 The role of plasma turbulence in the coupling of pairs and ions

All models previously developed to compute the structure and emission of RMS
invoke the single-fluid approach, which implicitly assumes infinitely strong
coupling between all plasma constituents. However, since the cross section for
Compton scattering off protons is smaller by a factor $(m_{\rm p}/m_{\rm
e})^{2}$ than that for electrons (and positrons when present), the radiation
force acting on the ions is completely negligible. This raises the question of
how the radiation force is mediated to the ions. The conventional wisdom has
been that a tiny charge separation, induced by the radiation force experienced
by the charged leptons, generates electrostatic field that decelerates the
ions, and apart from providing a mediator of the radiation force this coupling
has no further consequences for the shock properties beyond the single fluid
models. Indeed, in case of a Newtonian RMS that propagate in a pure hydrogen
gas, a velocity separation of

$\Delta\beta/\beta_{u}\approx(l_{p}/\lambda)^{2}\lesssim 10^{-19}$ (3)

is sufficient to generate the required electric field that decelerates the
protons (Levinson, 2020). However, as shown below, in the presence of
positrons and/or ion species with different charge-to-mass ratio electrostatic
coupling fails, and a different mechanism is needed to couple the various
plasma constituents.

###  2.1 Relativistic RMS

In sufficiently fast RMS ($\beta_{u}\gtrsim 0.5$) the temperature is high
enough to allow rapid creation of $e^{+}e^{-}$ pairs inside the shock
(Levinson & Bromberg, 2008; Budnik et al., 2010; Katz et al., 2010; Ito et
al., 2020a, b). For $\gamma_{u}>1$ the pair multiplicity approaches
$\gamma_{u}m_{\rm p}/m_{\rm e}$, completely dominating the shock opacity.
Under such conditions electrostatic forces cannot provide coupling between
pairs and ions since the electric field required to decelerate the ions exerts
opposite forces on electrons and positrons. To illustrate this, a multi-fluid
model for unmagnetized, relativistic RMS (RRMS) propagating in a pure hydrogen
plasma has been developed recently (Levinson, 2020). The analysis indicates
that once the density of newly created positrons approaches the baryon
density, which in RRMS occurs at the onset of the shock transition layer, the
charge density, and ultimately the electric field, reverse sign. This leads to
decoupling of the different species early on, and to the development of
relative drifts between the different beams. The presence of a strong enough
background magnetic field perpendicular to the shock velocity can lead to
tight coupling, preventing velocity spreads (Mahlmann et al., 2023).

The velocity separation between the different plasma constituents imposed by
the radiation force in unmagnetized RRMS, is expected to induce a rapid growth
of plasma instabilities. Linear stability analysis (Vanthieghem et al., 2022)
indicates a growth of various plasma modes, which ultimately become dominated
by a current filamentation instability driven by the relative drift between
the ions and the pairs. Particle-in-cell simulations (Vanthieghem et al.,
2022) validate these results and further probe the nonlinear regime of the
instabilities, elucidating the pair-ion coupling by the microturbulent
electromagnetic field. These simulations are local, in the sense that they
encompass kinetic-scale region inside the shock transition layer. The
radiation force is modeled as a prescribed force acting solely on the pairs.
The relevancy of such simulations to realistic RMS stems from the huge scale
separation between radiation and kinetic physics.

An interesting result found in Vanthieghem et al. (2022) is a dependence of
the coupling length on the pair multiplicity $\mathcal{M}$ (it scales roughly
as $\mathcal{M}^{1/2}$). For the large multiplicity anticipated well inside
the shock, $\mathcal{M}\approx\gamma_{u}m_{\rm p}/m_{\rm e}$, it becomes
macroscopic, $\sim 10^{6}l_{p}$. This is still much smaller than the width of
an infinite, planar RRMS, but might exceed the shock width during the breakout
phase (see §3.2 for further discussion).

Figure 1:  Structure of the RMS precursor for fiducial parameters described in
the text. We determine the position $x\,=\,0$ as the tip of the photon beam.
(Top) Transverse magnetic field profile; (Middle) Density of the ions;
(Bottom) Density profile of the injected positrons in log-scale. As seen, the
combined lepton deceleration and pair loading lead to progressive current
filamentation of the background, followed by anomalous coupling of the
different species.

While small-scale simulations allow us to capture more realistic values of the
effective radiation force acting on leptons, they do not account for the
kinetic and dynamical coupling between microturbulence, pair loading, and
spatially varying radiation force. Here, we extend the previous hybrid
description to investigate this effect, by accounting for the contribution of
Compton scattering and pair production from an idealized photon distribution
as described in Levinson (2020). Specifically, the photon density of the beam
propagating away from the shock provides the magnitude of the lepton recoil
from Compton scattering, while the density of each beam shapes the local
injection rate. In this cold beam approximation, we do not resolve the energy-
momentum tensor of the photon distribution, and hence, the energy of the pairs
at injection is not constrained. As one would expect in a realistic
configuration, we assume that the pairs are injected at rest into the local
lepton frame with an ad hoc temperature. Below, we discuss a first attempt to
perform global hybrid simulations of the precursor of a relativistic
radiation-mediated shock with fluid photons and kinetic electrons, positrons,
and ions.

Figure 1 shows the results of a hybrid simulation performed using the OSIRIS
code (Fonseca et al., 2002, 2013). In this example, plasma is injected into
the right-hand side boundary with a Lorentz factor of $\gamma\,=\,10$, weak
background magnetic field of $eB_{z}\,=\,0.1\,m_{\rm e}\omega_{\rm pe}c$, and
reduced mass ratio of $m_{\rm i}\,=\,25\,m_{\rm e}$. We resolve the plasma
skin depth $\Delta x\,=\,0.7\,c/\omega_{\rm pe}$ where $\omega_{\rm
pe}^{2}\,=\,4\pi\gamma_{\infty}n_{\infty}e^{2}/m_{\rm e}$, $c\Delta
t\,=\,0.5\,\Delta x$ corresponding to the CFL of electromagnetic field solver
introduced in Blinne et al. (2018) and extensively tested in Grošelj et al.
(2022). The lepton recoil is artificially enhanced to observe significant
deceleration over the box size. The combined lepton deceleration and pair
loading lead to progressive current filamentation of the background, followed
by anomalous coupling of the different species.

###  2.2 Shocks propagating in multi-ion plasma

In reality, the medium into which the shock propagates in most of the
aforementioned systems contains multiple ion species with different charge-to-
mass ratio. In such situations, the RMS physics may be considerably altered.
The main point to note is that the deceleration rate of ions inside the shock
depends on the charge-to-mass ratio, and since the charge conservation
condition in a multi-ion plasma is degenerate, a large velocity separation
between the different ions in the post-deceleration zone is, in principle,
allowed, even in sub-relativistic RMS that are devoid of positrons. As in
RRMS, sufficiently strong magnetic fields can couple the ions, however, since
the gyroradius of an ion of mass $Am_{\rm p}$ is larger by a factor $Am_{\rm
p}/Zm_{\rm e}$ than that of an electron (or positron), much stronger magnetic
fields may be needed to couple all ions. Such strong fields may not be present
in most relevant systems.

The structure of a Newtonian, multi-ion RMS has been explored recently using a
semi-analytic, multi-fluid shock model that can incorporate any number of ion
species (Granot, Levinson & Nakar, in preparation). The model invokes the
diffusion approximation for the transfer of radiation through the shock, and
computes the electrostatic coupling between the ions and electrons in a self-
consistent manner, by solving the energy and momentum equations of the
radiation and the multi-fluid plasma, together with Maxwell’s equations,
taking into account the electrostatic force acting on the charged fluids. An
example is shown in Fig. 2 for two ion species. As seen, a substantial
velocity separation is developed inside the shock, and is maintained in the
post deceleration zone, where all forces (radiation and electrostatic) vanish.
In practice, the relative drift between the different ion beams (and the
electrons) is expected to lead to generation of plasma turbulence that will
ultimately couple the ions, transferring their energy to radiation in the
downstream. However, the dominant wave modes should differ from those found in
a single-ion RRMS (Vanthieghem et al., 2022).

In general, the anomalous coupling length is expected to depend on the
threshold drift velocity for the onset of the instability, on the saturation
level of the turbulence, and conceivably other factors. In BNS mergers, where
the plasma density $\gtrsim 10^{25}$ cm-3, the anomalous coupling length may
be large enough to allow full velocity separation, leading to nuclear
transmutations that might have important implications to kilonova emission
(see §3.3 for further discussion). In other systems, where the density is
lower by many orders of magnitudes, anomalous coupling likely occurs on scales
much smaller that the radiation length. In that case, dissipation via
anomalous friction should first lead to formation of collisionless subshocks
before the energy will be converted to radiation. If the growth of the
instability occurs well after decoupling, the ion temperature should be high
enough to allow inelastic ion-ion collisions by random motions. A full kinetic
study is needed to compute the detailed shcok structure and assess whether
nuclear reactions are expected in such systems.

Figure 2: Velocity profiles of electrons (dashed line) and two ion species,
$\alpha 1$ (solid red) and $\alpha 2$ (solid blue) inside a shock moving at a
velocity of $\beta_{u}=0.3$. The charge-to-mass ratio in this example is
$Z_{\alpha_{1}}/A_{\alpha_{1}}=0.47$ and $Z_{\alpha_{2}}/A_{\alpha_{2}}=0.4$,
and the density ratio is
$n_{\alpha_{1}}/n_{\alpha_{2}}=Z_{\alpha_{2}}/Z_{\alpha_{1}}$, where
$Z_{\alpha_{i}}$ and $A_{\alpha_{i}}$ denote the atomic and mass numbers of
species $\alpha_{i}$, respectively. The solid black line gives the velocity
separation, $\beta_{\alpha_{2}}-\beta_{\alpha_{1}}$.

##  3 Implications for stellar explosions

In the following, we highlight potential implications of the results hitherto
discussed for breakout dynamics and emission, as well as changes in
composition.

###  3.1 Generation of nonthermal particle distributions and hard emission

The generation of magnetic turbulence in RRMS can potentially convert a
fraction of the dissipated energy to nonthermal particles with power-law
distributions. Both the increased nonadiabatic heating found in Vanthieghem et
al. (2022) and the possible formation of power-law energy spectra can impact
the shock breakout emission. In particular, a hard spectral component,
extending well beyond current predictions of single-fluid models, may be
present in the breakout signal.

###  3.2 Long range plasma scales at high pair multiplicity

Single-fluid models of finite RRMS (Granot et al., 2018), indicate that during
the shock breakout phase the width of shock transition layer decreases
dramatically, owing to radiative losses. Since the scale over which pairs and
ions couple becomes macroscopic when the pair multiplicity becomes large (see
§2.1), it could be that the coupling length will approach or even exceed the
radiation scale during the shock breakout episode. If this indeed happens, it
means that the RMS structure will be vastly different than predicted by
current RRMS models, which might have profound implications for the observed
shock breakout signal (e.g., in the case of a shock breakout from a stellar
surface a wider shock implies a softer and more energetic breakout radiation).
Under such circumstances, global shock models may be necessary to compute the
breakout dynamics and emission.

###  3.3 Abundance evolution and kilonova emission in BNS mergers

The expulsion of a relativistic jet following neutron star coalescence drives
a fast RMS into the merger ejecta. The breakout of the shock from the high-
velocity tail of the merger ejecta can produce a gamma-ray flash that, even
though much fainter than a typical short GRB, can overwhelm the jet emission
at large enough viewing angles (with respect to the jet axis). According to
one scenario (Kasliwal et al., 2017; Gottlieb et al., 2018; Pozanenko et al.,
2018; Beloborodov et al., 2020), the gamma-ray flash GRB 170817A that
accompanied the gravitational wave signal GW 170818 was produced by such a
process.

As explained in §2.2, the propagation of the RMS through the merger ejecta can
lead to a significant velocity separation of different r-process isotopes just
downstream of the shock transition layer, provided that anomalous friction is
not too effective. An estimate of the scale separation in RMS suggests that,
under conditions anticipated in BNS mergers, the anomalous coupling length is
likely to exceed the shock width (Granot, Levinson & Nakar, in preparation).
In that case, collions of the different ion beams will induce nuclear
transmutations in regions where the shock velocity exceeds the corresponding
activation barriers. Recent analysis (Granot, Levinson & Nakar, in
preparation) indicates that the ion-ion collision length is smaller than the
shock width, and that in regions where $\beta_{u}\gtrsim 0.2$, the collision
energy may be large enough to induce fission and fusion of many elements. This
can significantly alter the composition profile of r-process material behind
the shock and, potentially, the kilonova emission if the change in composition
affects the opacity and/or the radioactive energy deposition in the ejecta.

In addition to inelastic ion collisions, neutron-rich isotopes downstream of
the shock can undergo fission through the capture of free neutrons that cross
the shock. The activation energy for neutron-induced fission ranges from
practically zero for ${}^{235}U$ to a peak of about $40MeV$ for elements of
mass number $A\approx 100$, so a shock velocity of $\beta_{u}\gtrsim 0.1$
should give rise to fission of many elements. Whether free neutrons are
sufficiently abundant at early times (during shock propagation) to
significantly alter the ejecta composition is yet an open issue.

##  4 Summary

A fundamental question in the theory of radiation mediated shocks is: What is
the mechanism that couples the different plasma constituents (ions, electrons
and positrons), and how does it affect the shock thermodynamics and emission?
We have shown, by means of semianalytic approach and PIC simulations, that
this coupling is accomplished through generation of plasma turbulence in
weakly magnetized shocks, or by magnetic fields in sufficiently magnetized,
perpendicular shocks. The generation of plasma microturbulence in weakly
magnetized RMS can lead to particle acceleration inside the shock, which might
alter the characteristics of the shock breakout emission. Moreover, our
simulations indicate that the coupling length increases with increasing pair
multiplicity, becoming macroscopic in regions where the pair multiplicity
saturates. This could considerably affect the shock structure and emission
during the breakout phase, e.g., due to early formation of collsionless
subshocks, since then the shock thickness is expected to shrink dramatically,
owing to radiative losses, ultimately subceeding the coupling length.

We also find that in shocks propagating in multi-ion plasma, a substantial
velocity separation between ions having different charge-to-mass ratio
develops inside the shock. This can lead to inelastic nuclear collisions that
can induce fission (or fusion). In BNS mergers, the capture of free neutrons
that cross the shock by neutron rich isotopes downstream of the shock can also
induce fission. The passage of a fast enough shock in BNS merger ejecta can,
therefore, lead to a considerable evolution of the relative abundance of
r-process elements at early times, which could have important consequences for
the kilonova emission.

## acknowledgment

We thank the referee for useful comments. AL and AG acknowledge support by the
Israel Science Foundation grant 1995/21. AV acknowledges support from the NSF
grant AST-1814708 and the NIFS Collaboration Research Program (NIFS22KIST020).
JFM acknowledges support by the National Science Foundation under grant No.
AST-1909458. This research was supported by the Multimessenger Plasma Physics
Center (MPPC), NSF grant PHY-2206607.

## Declaration of Interests

The authors report no conflict of interest.

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# Transformer-based Flood Scene Segmentation for Developing Countries

Ahan M R
BITS Pilani Goa Campus
<EMAIL_ADDRESS>
Roshan Roy*
BITS Pilani
<EMAIL_ADDRESS>
Shreyas Sunil Kulkarni
BITS Pilani Hyderabad Campus
<EMAIL_ADDRESS>
Vaibhav Soni
MANIT Bhopal
<EMAIL_ADDRESS>
Ashish Chittora
BITS Pilani Goa Campus
<EMAIL_ADDRESS>Equal Contribution

###### Abstract

Floods are large-scale natural disasters that often induce a massive number of
deaths, extensive material damage, and economic turmoil. The effects are more
extensive and longer-lasting in high-population and low-resource developing
countries. Early Warning Systems (EWS) constantly assess water levels and
other factors to forecast floods, to help minimize damage. Post-disaster,
disaster response teams undertake a Post Disaster Needs Assessment (PDSA) to
assess structural damage and determine optimal strategies to respond to highly
affected neighborhoods. However, even today in developing countries, EWS and
PDSA analysis of large volumes of image and video data is largely a manual
process undertaken by first responders and volunteers. We propose
FloodTransformer, which to the best of our knowledge, is the first visual
transformer-based model to detect and segment flooded areas from aerial images
at disaster sites. We also propose a custom metric, Flood Capacity (FC) to
measure the spatial extent of water coverage and quantify the segmented
flooded area for EWS and PDSA analyses. We use the SWOC Flood segmentation
dataset and achieve 0.93 mIoU, outperforming all other methods. We further
show the robustness of this approach by validating across unseen flood images
from other flood data sources.

##  1 Introduction and Context

The Center for Research on the Epidemiology of Disasters, in affiliation with
the World Health Organization (WHO), reported that natural disasters accounted
for 1.3 million deaths and over USD 2 trillion in economic damage — all
between 1998 and 2017 [19]. Flooding related damage is a factor in most of
them [4] and frequent the list of most expensive disasters [17]. Developing
economies of Asia are disproportionately affected and are the worst-hit by
floods, accounting for 44% of all flood disasters from 1987-1997 [18]. India
alone registers 1/5th of global deaths from floods [11]. Rapid urbanization,
global climate change, and rising sea water levels will expose 1.47 billion
more people to flood risk, with 89% of them living in low-middle income
countries [5]. Flood Segmentation technology is instrumental for Disaster
Prediction and Response is critical to save lives and livelihoods.

Flood Response: Typically, disaster management teams complete a Post Disaster
Needs Assessment (PDSA) and rapidly develop infrastructure based on this
report on the collected data [6]. Unmanned Aerial Vehicles (UAVs) are deployed
to collect large volumes of image and video data in affected regions. PDSA is
essential for identification of submerged regions, sanity check of large
building structures, debris identification, and search-and-rescue (S&R)
operations.

Flood Forecasting: Flood segmentation techniques can be critically important
for flooding-related Early Warning Systems (EWS). According to research in
[12], Indians given a flood warning are twice as likely to evacuate safely
than Indians without any notice. which require constant monitoring of river or
sea water levels. Comparison of current levels with historical evidence of
flood-prone water levels can help understand when to trigger warnings
appropriately.

Constraints: Developing countries are plagued by resource and economic
constraints. Failure of macro- and micro- infrastructure planning in Nicaragua
led to re-construction on top of an earthquake faultline [7]. Weak social
safety and insurance policies inflate recovery time [27]. Economic
vulnerability renders countries like Haiti, Ethiopia, Nepal, El Salvador in a
near-permanent state of emergency alert [7]. In these countries, processing
and analysis of large-scale visual data from UAVs for PDSA in Flood response
is a manual process that requires multi-team intervention, which poses a
serious bottleneck in search-and-response speed. Deployment of EWSs is
infeasible because human monitoring of video feeds is too cumbersome and
expensive.

AI Technology: To reduce the burden of manual analysis on crisis responders,
Deep learning is well-suited to scale, automate and expedite these operations.
The last few years have witnessed a tremendous rise in CNN-based image
classification and segmentation research [21]. However, CNNs suffer from a
well-known problem — large inductive biases. Conceretely, CNNs assume locality
and translation equivariance, which hurt the interpretability of pure CNN-
based algorithms. Recently, visual transformers have garnered attention for
image classification, segmentation and object detection tasks [2, 20, 26, 3]
for challenging these assumptions with comparable accuracy.

Contributions: In this work, we propose a hybrid fused CNN-Transformer:
FloodTransformer to tackle flood water segmentation on the Water Segmentation
Open Collection (WSOC) dataset [22]. First, we achieve state-of-the-art
results and are the first work (to the best of our knowledge) to apply new
transformer-driven research to the flood data domain. Second, our approach is
extendable — we demonstrate the ability of our model to generalize well on
unseen data sources. Further local calibration, if required at all, simply
requires weight fine-tuning with previous, region-specific, flood scene data.
Third, our model does not suffer from data scarcity - it only requires image
data input and not complex sensor data which is hard to collect [13]. Last,
the transformer-based encoder applies recent DL innovations to the flood data
domain. Although the hybrid method still uses CNNs in the decoder network, the
aforementioned spatial inductive biases no longer occur throughout the entire
network. Dependencies between patch embeddings are learnt from scratch. This
improves the robustness of our approach.

##  2 Methodology

To achieve the Flood Scene Understanding, we introduce a Deep Learning model
for Flood image segmentation and quantify the impact of flooding with a custom
metric called Flooding Capacity.

###  2.1 Method

Figure 1: FloodTransformer architecture: simplified from Zhang et al. [23] for
representation.

Inspired by Zhang et al. [23], we propose FloodTransformer to solve
segmentation for the flood data domain. It is a fusion architecture of Visual
Transformer [25] and Convolution Neural Networks (CNNs) and its model
architecture is displayed in Figure 1.

Complex flooding imagery may contain heterogeneous objects, flooding patterns
and backgrounds. Using the self-attention module of the visual Transformer
module from [25] and global vector representation learned from the CNN
network, FloodTransformer fuses the trained embeddings to learn long-term
spatial relationships between the aforementioned entities in images of flood
affected areas. Using Hadamard bilinear product [23], the fusion module fuses
information via embeddings from both parallel streams into a dense
representation. The combination of multi-level fusion maps generates the
segmentation output of the model. We summarize each component below, per Zhang
et al. [23].

Transformer Module: We use the encoder-decoder network using Visual
Transformer [25]. The input image $\mathbf{x}\in\mathbb{R}^{H\times W\times
3}$ is sliced into N patches, where
$\mathbf{N}={\frac{H}{F}\times\frac{W}{F}}$ and F is usually set to 16 or 32.
These patches are flattened into a linear projection layer to generate an
image embedding, added to a trainable position embedding, and passed into $L$
layers of the Transformer encoder’s Multi-head Self Attention (MSA) mechanism.
Every layer updates the embedding with $[q,k,v]$ triplets and the projection
matrix. The encoded sequence $z^{L}$ is passed to the Transformer decoder,
which aims to recover spatial context and outputs feature maps $t^{0}$,
$t^{1}$, $t^{2}$ of different scales.

CNN Module: Per [23] and [15], we implement three Res-blocks as a shallow CNN
network; Transformers already provide global context and retain rich local
information and relationships. In parallel to the transformer module, the
three CNN blocks output downsampled feature maps $v_{0}$, $v_{1}$, $v_{2}$ of
different scales.

Fusion Module: Outputs $v_{0},t_{0}\in\frac{H}{8}\times\frac{W}{8}$ ,
$v_{1},t_{1}\in\frac{H}{4}\times\frac{W}{4}$ and
$v_{2},t_{2}\in\frac{H}{2}\times\frac{W}{2}$ from both modules are fused via
the Fusion module from Zhang et al. [23]. The fused vector $f_{i}$ is
generated by implementing the following operations:

where $\odot$ is the Hadamard product as described in [23].

###  2.2 Metric

We introduce a custom metric called Flooding Capacity (FC) to quantify the
impact of flooding in an image. It is the percentage of flooding in an image,
and is calculated from the segmented mask output of the image, as described in
Equation 1.

$FloodingCapacity(FC)=\frac{1}{\sum_{i=0}^{k}\sum_{j=0}^{k}(p_{ij}==0)||(p_{ij}==1)}\sum_{i=0}^{k}\sum_{j=0}^{k}(p_{ij}==1)$
(1)

$p_{ij}$ are the pixel values of the binary mask output and can have values
${0,1}$. We calculate `Flooding area` as the sum of all $p_{ij}==1$ `flooding`
pixels. We divide `flooding area` by `total area`, which is the sum of all
$p_{ij}==0$ `non-flooding` and $p_{ij}==1$ `flooding` pixels.

Flood Scene Segmentation, as described in Section 2.1, used with Flood
Capacity calculation from Equation 1 can be used on images pre/post flooding
for complete Scene understanding. For Flood Response, this helps optimize
operational planning. For Flood Forecasting, this can quantify the water-cap
in a particular region over time separated image inputs.

##  3 Dataset Description

Water Segmentation Open Collection (WSOC) dataset [22] consists of 11900 flood
or water body related images, composed from multiple open-source source
datasets such as COCO [9], Semantic Drone Dataset [10] and other datasets
containing flood-related images [22]. These images were shot from a variety of
sources such as fixed surveillance cameras, drones (UAVs), crowdsourced in-
field observations (on foot, boat, vehicle) and from social media streams.
Some sample images are shown in Figure 2.

Image resolutions vary from 147x150 pixels for low-quality social media
streams to 2448x3264 pixels for sophisticated aerial imagery. Sampling from
disparate sources is the reason for significant variations in visual
appearance, illumination, water body, and geography; this makes water
segmentation challenging. Newly-added social media images undergo a three-step
manual annotation process to generate binary ground truth masks for pixel-wise
classification between water or no-water.

##  4 Results

We follow the WSOC method [22] and perform a 90-10 train-test split of the
data. Qualitative results of output segmentation masks are shown in Figure 2.
We evaluate the performance of our hybrid Flood Transformer model with mean
intersection-over-union (mIOU) and Percentage Accuracy (PA) scores, on WSOC,
as shown in Table 1. We compare our result to that of other benchmark models
on WSOC [22], in Table 1.

As shown, FloodTransformer comfortably achieves state-of-the-art segmentation
mIOU of 0.93. It is based on Zheng et al [23]’s TransFuse architecture, which
comfortably beats traditional benchmarks such as UNet [14]. The Flood Capacity
(FC) from Equation 1 of predicted mask in Fig 2(i)(c) is 0.53 and of mask in
Fig 2(ii)(c) 0.47.

To demonstrate the robustness of our model with further qualitative results,
we perform inference on out-of-dataset images, as shown in Figure 3. The Flood
Capacity (FC) from Equation 1 for the predicted mask in Fig 3(i)(b) is 0.18
and for mask in Fig 3(ii)(b) is 0.26.

Model | Backbone | WSOC Dataset
---|---|---
mIOU [%] | PA [%]
$\mu$ | $\sigma$ | $\mu$ | $\sigma$
Tiramisu [8] | None | 0.38 | 0.17 | 0.73 | 0.24
SegNet [1] | ResNet50 | 0.85 | 0.08 | 0.94 | 0.01
UNet [14] | ResNet50 | 0.88 | 0.06 | 0.94 | 0.03
FCN32 [16] | ResNet50 | 0.79 | 0.10 | 0.87 | 0.08
PSPNet [24] | ResNet50 | 0.83 | 0.08 | 0.90 | 0.07
SegNet [1] | VGG16 | 0.82 | 0.09 | 0.90 | 0.05
UNet [14] | VGG16 | 0.5 | 0.26 | 0.67 | 0.15
FCN32 [16] | VGG16 | 0.76 | 0.13 | 0.91 | 0.02
PSPNet [24] | VGG16 | 0.82 | 0.09 | 0.92 | 0.02
FloodTransformer (Ours) | Transformer | 0.93 | 0.03 | 0.96 | 0.02
Table 1: Average and Standard deviation of Mean Intersection over Union (mIoU)
and Pixel Accuracy (PA) on WSOC dataset Figure 2: Qualitative results on (i)
social media (ii) in-field images in WSOC dataset.
(a) Image (b) Ground Truth (c) Segmented Output Figure 3: Qualitative results
on out-of-dataset, aerial flooding imagery.
(a) Image (b) Segmented Output

##  5 Conclusions and Impact

In this work we propose, to the best of our knowledge, the first hybrid
transformer-CNN model for flood water segmentation: FloodTransformer. We
achieve state-of-the-art segmentation results on the WSOC dataset and
empirically demonstrate the power and robustness of visual transformers,
deviating from pure CNN-based approaches.

We further validate our model’s performance by demonstrating competitive
qualitiative results on out-of-dataset aerial images, indicating competitive
performance on images in the wild. With the increasing number and variability
of flooding-related disasters in developing countries (varying weather
conditions, water flow, geography etc.) and variability of visual data
collected from UAVs (varying mode of capture, camera technology etc.), this
generalization capacity is crucial.

The flood water segmentation methodology adapted in this paper is useful for
Flood Prediction and Response. Given the high incidence of flooding-related
disasters, our method is an important step towards improving flood warning
systems and covering billions of people in high-risk developing countries. It
can also aid in the first step of disaster management: Post Disaster Needs
Assessment (PDSA). In particular, in both cases it reduces the need for manual
analysis; this is crucial in developing countries with constrained resources
and high populations, where speed and efficiency of response is of paramount
importance.

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