A Robust Approximation to
a Lambert-Type Function
——————————————————————–

Ken Roberts111Physics and Astronomy Department, Western University, London,
Canada<EMAIL_ADDRESS>
April 8, 2015

###### Abstract

The function $y=g(x)=\mathrm{log}\big{(}W(e^{x})\big{)}$, where $W()$ denotes
the Lambert W function, is the solution to the equation $y+e^{y}=x$. It
appears in various problem situations, for instance the calculation of
current-voltage curves for solar cells. A direct calculation of $g(x)$ may be
inaccurate because of arithmetic underflow or overflow. We present a simple
algorithm for calculating $g(x)$ that is robust, in that it will work for
almost all $x$ values which are representable in the arithmetic of one’s
chosen computer language. The algorithm does not assume that the chosen
computer language implements the Lambert W function.

## 1 Introduction

The Lambert W function $w=f(z)$ is the solution of $w\,e^{w}=z$, for complex
$w$ and $z$. It can be considered as the multi-branch inverse of the conformal
map $w\to w\,e^{w}=z$ between the complex $w$-plane and the complex $z$-plane.
When $w$ and $z$ are restricted to having real values, the graph of the
Lambert W function is as shown in figure 1. For further background regarding
the Lambert W function, see [1, 2].

Some problem situations, for instance the modeling of current and voltage in
diodes or solar cells, reduce to an implicit equation which can be solved
explicitly by means of the Lambert W function. As a simple example, consider
the implicit equation

$a\,e^{bU}+bU=V$ (1)

where $a,b$ are positive real numbers, parameters of the model, and $U,V$ are
real variables.222 Equation (1) is a considerable simplification, in order to
have an example in mind. The typical solar cell model has four or five
parameters. See [4], page 14, for instance. The corresponding explicit
solution for $U$ as a function of $V$ is

$U=\frac{1}{b}\,\mathrm{log}\Big{(}\frac{1}{a}\,W(a\,e^{V})\Big{)}$ (2)

Here $\mathrm{log}()$ denotes the principal branch of the natural logarithm
function, and $W()$ denotes the principal branch of the Lambert W function.

A typical task might be, given values of the model parameters, to draw a graph
of $U$ as a function of $V$. For that task, it is computationally efficient to
have, instead of the implicit equation (1), the explicit solution (2) for $U$
in terms of $V$ and the model parameters.

Another task might be to estimate the model parameter values $a,b$ which best
fit experimental observations of $U,V$ pairs. For that task, one wants to have
an understanding of how varying the model parameters will affect the $U$-$V$
curves.

Still another task might be to determine the relationships among the model
parameters which correspond to having an extremum of a function of $U$ or $V$.
This task also requires an understanding of how varying the model parameters
will affect the $U$-$V$ curves, but it will also be helpful if the formula
utilized has partial derivatives with respect to all the parameters.

The expression in equation (2) is analytic, so is well-behaved with respect to
its argument and the model parameters. It can be repeatedly differentiated,
its extrema lie at stationary points, and so on. Because one is working with
real values for $U$ and $V$ and the positive real model parameters, the
argument for the Lambert W function evaluation in equation (2) is positive, so
the principal branch of the Lambert W function is being used. Moreover, one is
using the principal branch of the natural logarithm function. Everything is
single valued in this expression.

However, there can be a numerical difficulty in performing computations with
equation (2). Making the substitutions (coordinate changes)
$y=bU+\mathrm{log}(a)$ and $x=V+\mathrm{log}(a)$, the computations involve an
evaluation of the function

$y=g(x)=\mathrm{log}\Big{(}W(e^{x})\Big{)}.$ (3)

One difficulty which can arise, depending upon the computer programming
language used, is numerical underflow or overflow, related to the evaluation
of an exponential of $x$ where $x$ (negative or positive) has a large
magnitude. The value of $x$ must therefore be restricted to the set of
logarithms of floating point numbers whose exponents can be accurately
represented in the arithmetic facility of the computer language. A second
difficulty which can arise is that the best computer programming language for
the rest of one’s problem solution may not have a built-in Lambert W function
evaluator.

The purpose of this note is to address those two difficulties. We will
describe a simple procedure, which can be implemented in any programming
language with floating point arithmetic, for the robust calculation of the
function $y=\mathrm{log}(W(\mathrm{exp}(x)))$. The procedure is valid for
essentially any real value of $x$ which is representable in the programming
language.

Figure 1: Graph of $y=LambertW(x)$ for both $x$ and $y$ real.
The graph shows all pairs of real values of $x,y$
which satisfy $y=W(x)$; that is, satisfy $x=y\,e^{y}$
Necessarily $x\geq-1/e$.
If $-1/e<x<0$, then there are two possible $y$ values
corresponding to the two branches of Lambert W function.
The upper branch, with $y\geq-1$, is the principal branch.
If $x$ is real and positive, the only real value of $y=W(x)$
is the positive value of $y$ on the principal branch.

## 2 Description of the Function $y=g(x)=\mathrm{log}(W(\mathrm{exp}(x)))$

We may consider the function $y=g(x)=\mathrm{log}(W(\mathrm{exp}(x)))$ as a
transformation of the Lambert W function, with a change of representation or
coordinate space. For clarity, we will restrict to real arguments. We can
think of $W(u)$, the principal branch of the Lambert W function, as a mapping
of the positive real line to itself. The function $u=h(x)=\mathrm{exp}(x)$
maps the whole real line to the positive real line, and its inverse
$h^{-1}(u)=\mathrm{log}(u)$, the principal branch of the natural logarithm,
maps the positive real line to the whole real line. In this interpretation,
the function $y=g(x)=\mathrm{log}(W(\mathrm{exp}(x)))$ is the composition of
functions $g=h^{-1}\circ W\circ h$, and it maps the whole real line to the
whole real line.

Suppose that $y=\mathrm{log}(W(\mathrm{exp}(x)))$. Taking exponentials (ie,
applying the function $h$ to both sides of the equation) gives

$e^{y}=W(e^{x}).$

That is, using the definition of the Lambert W function,

$e^{y}\,e^{e^{y}}=e^{x}$

or

$e^{y+e^{y}}=e^{x}.$

Taking logarithms (ie, applying the function $h^{-1}$ to both sides) gives

$y+e^{y}=x.$ (4)

Equation (4) is a simple equation structure, as simple as the Lambert W
defining equation structure

$w\,e^{w}=z.$ (5)

In fact, equation (4) is just equation (5) in another coordinate system.

When we are evaluating $y=g(x)=\mathrm{log}(W(\mathrm{exp}(x)))$ as the
solution to equation (4), we are just evaluating the Lambert W function. There
is an important difference, however: The evaluation of $g(x)$ does not involve
much risk of underflow or overflow in the numerical representation of the
computer language.

Since $y=g(x)$ satisfies $g(x)+e^{g(x)}=x$, the first derivative of $g(x)$
satisfies

$g^{\prime}(x)+e^{g(x)}\,g^{\prime}(x)=1$

or

$g^{\prime}(x)=\frac{1}{1+e^{g(x)}}>0$

The second derivative of $g(x)$ satisfies

$g^{\prime\prime}(x)=-\frac{e^{g(x)}\,g^{\prime}(x)}{(1+e^{g(x)})^{2}}<0$

Figure 2 shows the function $y=g(x)$ for moderate values of the argument $x$,
that is, for $-4<x<4$. Figure 3 shows the same function for larger values of
the argument $x$, that is, for $-1000<x<1000$.

Figure 2:  Graph of $y=g(x)=\mathrm{log}(W(\mathrm{exp}(x)))$
for moderate magnitudes of the argument.

Figure 3:  Graph of $y=g(x)=\mathrm{log}(W(\mathrm{exp}(x)))$
for large magnitudes of the argument.

One can see from these graphs that the function $g(x)$ behaves like $x$ when
$x$ is much less than 0, and behaves like $\mathrm{log}(x)$ when $x$ is much
more than 0. For values of $x$ around 0, there is a smooth blend between the
two behaviors, with $g(1)=0$. The function $g(x)$ is strictly monotonic
increasing, as it has a positive first derivative. It curves downward, as it
has a negative second derivative.

One can further see, from figure 3, that when the range of the argument $x$ is
large, the graph of $y=g(x)$ looks like it has a sharp corner at the origin.
Actually, as figure 2 illustrates, the graph does not really have a sharp
corner. Nonetheless, at a suitable distance (large scale), one has in the
function $g$ a useful smooth function for representing a function which has a
step in its derivative.

## 3 Algorithm for Calculating $y=g(x)=\mathrm{log}(W(\mathrm{exp}(x)))$

In the terminology of H. Kuki ([3] page 23), the function $g(x)$ is
contracting. That is, $|\Delta g(x)|\leq|\Delta x|$, or $|g^{\prime}(x)|\leq
1$, for all $x$ values in its argument domain. That means the task of finding
an estimate for $g(x)$ given $x$ is relatively stable. A slight change in $x$
(noise in the input) will produce only a slight change in $g(x)$. The only
challenges in developing a formula to estimate $g(x)$, given $x$, are finding
an appropriate algorithm, coding the sequence of calculations to avoid
unwanted cancellation, and being reasonably efficient in the number of
computations performed.

A suitable algorithm can be an initial estimate, followed by some number of
iterations of a refinement. Halley’s method is used to perform refinements
because it has cubic convergence, and the derivatives involved can be
calculated efficiently.

Given any fixed real number $x$, we wish to find a real number $y$ such that

$h(y)=y+e^{y}-x$

is zero. The first and second derivatives of $h(y)$ are needed for Halley’s
method. They are

$\displaystyle h^{\prime}(y)$ $\displaystyle=$ $\displaystyle 1+e^{y}$
$\displaystyle h^{\prime\prime}(y)$ $\displaystyle=$ $\displaystyle e^{y}$

and hence are particularly easy to calculate. Once one has $e^{y}$ from the
calculation of $h(y)$, the derivatives are also at hand. It is also necessary,
in order to use Halley’s method, that the first derivative is non-zero; that
is the case for $h^{\prime}(y)=1+e^{y}$.

As an initial estimate $y_{0}$, we choose to use $y_{0}=x$ for $x\leq-e$, and
$y_{0}=\mathrm{log}(x)$ for $x\geq e$. For $-e<x<e$, we linearly interpolate
between the two values $-e$ and 1. This is an extraordinarily crude initial
estimate, but it is sufficient, since Halley’s method is very robust and
rapidly convergent in this application.

The general iteration formula for Halley’s method is

$y_{n+1}=y_{n}-\frac{2h(y_{n})h^{\prime}(y_{n})}{2h^{\prime}(y_{n})^{2}-h(y_{n})h^{\prime\prime}(y_{n})}$

In this particular case $h(y)=y+e^{y}-x$ and the iteration formula becomes

$y_{n+1}=y_{n}-\frac{2(y_{n}+e^{y_{n}}-x)(1+e^{y_{n}})}{2(1+e^{y_{n}})^{2}-(y_{n}+e^{y_{n}}-x)e^{y_{n}}}.$
(6)

The details of coding depend upon the computer language. It will be efficient
to evaluate $e^{y_{n}}$ only once per iteration. All other computations are
straightforward arithmetic. When evaluating the denominator of the adjustment
in the iteration equation (6), there is little risk of cancellation resulting
from the subtraction, as the first term in the denominator is larger than the
second term.

In practice, just a few iterations suffice to give a good result. For
arguments in $-10^{6}\leq x\leq 10^{6}$, four iterations of Halley’s method
reduce the absolute error to less than $10^{-80}$. The actual coding can use a
convergence criterion, based upon the desired maximum error in the estimate of
function value, to determine how many iterations to perform. Alternatively, if
the precision is fixed by the computer language’s arithmetic representation or
by the needs of the application situation, then one can determine how many
iterations of the refinement will suffice, and perform only that number,
omitting the final redundant iteration which verifies the convergence. This
technique, due to H. Kuki as seen in his algorithms for computing square root
(see [3], pages 49-50 and 135-136), probably deserves a name. Perhaps it
should be called “Kuki’s convergence non-test”.

## 4 Acknowledgments

The author had the good fortune and honor to work for Hirondo Kuki in 1968-69,
and thanks him for his guidance, support and friendship. He also thanks S. R.
Valluri for an introduction to the Lambert W function and the many interesting
problems associated with its properties, and in particular for a stimulating
discussion of the topic of this note. He thanks Mark Campanelli for suggesting
Halley’s method for iterative refinement in solar cell calculations.

## References

  * [1] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, “On the Lambert W function”, Advances in Computational Mathematics, vol 5 (1996), pp 329-359.
  * [2] S. R. Valluri, D. J. Jeffrey, R. M. Corless, “Some applications of the Lambert W function to physics”, Canadian Journal of Physics, vol 78, no 9 (September 2000), pp 823-831.
  * [3] Hirondo Kuki, Mathematical Functions, University of Chicago Computation Center Report, February 1966.
  * [4] Jenny Nelson, The Physics of Solar Cells, Imperial College Press, 2003.


#  ${S}^{5}$: The destruction of a bright dwarf galaxy as revealed by the
chemistry of the Indus stellar stream

Terese T. Hansen George P. and Cynthia Woods Mitchell Institute for
Fundamental Physics and Astronomy, and Department of Physics and Astronomy,
Texas A&M University, College Station, TX 77843, USA  Department of Physics
and Astronomy, Texas A&M University, College Station, TX 77843, USA  Alexander
P. Ji Observatories of the Carnegie Institution for Science, 813 Santa Barbara
St., Pasadena, CA 91101, USA  Gary S. Da Costa Research School of Astronomy
and Astrophysics, Australian National University, Canberra, ACT 2611,
Australia  Ting S. Li NHFP Einstein Fellow  Observatories of the Carnegie
Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA
Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544, USA  Andrew R. Casey Center of Excellence for Astrophysics in Three
Dimensions (ASTRO-3D), Australia  School of Physics & Astronomy, Monash
University, Wellington Rd, Clayton 3800, Victoria, Australia  Andrew B. Pace
McWilliams Center for Cosmology, Carnegie Mellon University, 5000 Forbes Ave,
Pittsburgh, PA 15213, USA  Lara R. Cullinane Research School of Astronomy and
Astrophysics, Australian National University, Canberra, ACT 2611, Australia
Denis Erkal Department of Physics, University of Surrey, Guildford GU2 7XH, UK
Sergey E. Koposov Institute for Astronomy, University of Edinburgh, Royal
Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK  Institute of Astronomy,
University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK  Kavli
Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK  Kyler Kuehn Lowell Observatory, 1400 W Mars Hill Rd, Flagstaff,
AZ 86001, USA  Australian Astronomical Optics, Faculty of Science and
Engineering, Macquarie University, Macquarie Park, NSW 2113, Australia
Geraint F. Lewis Sydney Institute for Astronomy, School of Physics, A28, The
University of Sydney, NSW 2006, Australia  Dougal Mackey Research School of
Astronomy and Astrophysics, Australian National University, Canberra, ACT
2611, Australia  Jeffrey D. Simpson School of Physics, UNSW, Sydney, NSW 2052,
Australia  Nora Shipp Department of Astronomy & Astrophysics, University of
Chicago, 5640 S Ellis Avenue, Chicago, IL 60637, USA  Kavli Institute for
Cosmological Physics, University of Chicago, Chicago, IL 60637, USA  Fermi
National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510, USA  Daniel
B. Zucker Department of Physics and Astronomy, Macquarie University, Sydney,
NSW 2109, Australia  Macquarie University Research Centre for Astronomy,
Astrophysics & Astrophotonics, Sydney, NSW 2109, Australia  Joss Bland-
Hawthorn Sydney Institute for Astronomy, School of Physics, A28, The
University of Sydney, NSW 2006, Australia  Centre of Excellence for All-Sky
Astrophysics in Three Dimensions (ASTRO 3D), Australia T. T. Hansen
<EMAIL_ADDRESS>
###### Abstract

The recently discovered Indus stellar stream exhibits a diverse chemical
signature compared to what is found for most other streams due to the
abundances of two outlier stars, Indus$\\_$0 and Indus$\\_$13\. Indus$\\_$13,
exhibits an extreme enhancement in rapid neutron-capture ($r$-)process
elements with $\mathrm{[Eu/Fe]}=+1.81$. It thus provides direct evidence of
the accreted nature of $r$-process enhanced stars. In this paper we present a
detailed chemical analysis of the neutron-capture elements in Indus$\\_$13,
revealing the star to be slightly actinide poor. The other outlier,
Indus$\\_0$, displays a globular cluster-like signature with high N, Na, and
Al abundances, while the rest of the Indus stars show abundances compatible
with a dwarf galaxy origin. Hence, Indus$\\_0$ provides the first chemical
evidence of a fully disrupted dwarf containing a globular cluster. We use the
chemical signature of the Indus stars to discuss the nature of the stream
progenitor which was likely a chemically evolved system, with a mass somewhere
in the range from Ursa Minor to Fornax.

Stellar abundances (1577), Dwarf galaxies (416), Milky Way stellar halo
(1060),Globular star clusters (656)

††software: MOOG (Sneden, 1973; Sobeck et al., 2011), IRAF (Tody, 1986, 1993),
ATLAS9 (Castelli & Kurucz, 2003), linemake
(https://github.com/vmplacco/linemake), NumPy (van der Walt et al., 2011),
Matplotlib (Hunter, 2007), AstroPy (Astropy Collaboration et al., 2013, 2018),
CarPy (Kelson, 2003), SMHR (Casey, 2014)

##  1 Introduction

The detection of numerous stellar streams in large photometric surveys like
the Sloan Digital Sky Survey (SDSS; York et al. 2000; Stoughton et al. 2002)
and the Dark Energy Survey (DES; Abbott et al. 2018) has made it clear that
the Milky Way (MW) halo contains remnants from numerous accreted globular
clusters and dwarf galaxies. Stellar streams form when satellite systems fall
into the MW halo and are slowly disrupted and mixed into the MW stellar
population. The streams thus provide a unique window to study the hierarchical
build up of the Galaxy. With the recent addition of all sky proper motions
from the $Gaia$ mission (Gaia Collaboration et al., 2016, 2018) a full
chemodynamical study of the systems can be performed providing strong
diagnostics of the stream progenitor systems.

In particular, the chemically distinct pattern of stars formed in globular
clusters has been used to link streams to their parent object (Simpson et al.,
2020; Hansen et al., 2020) and further explore the chemical enrichment
channels of these systems (Casey et al., in prep). Although detailed abundance
analysis of disrupted dwarf galaxy stars is still sparse, chemical
characterization of some systems has revealed these to contain stars
exhibiting enhancements in rapid neutron-capture ($r$-process) elements.
Specifically, Aguado et al. (2020) found an overall enhancement of
$\mathrm{[Eu/Fe]}\sim+0.65$ in their nine $Gaia$ Sausage/Sequoia stars
analysed and Gull et al. (2021) detected a range of
$-0.5<\mathrm{[Eu/Fe]}<+1.3$ in stars belonging to the Helmi debris and tails
streams and the $\omega$ Centauri progenitor stream. These results suggest
that the progenitors of these streams are likely relatively massive systems
which experienced multiple $r$-process enrichment events such as neutron star
mergers. In addition, the search for dynamical substructure in data from
spectroscopic surveys such as the HK, Hamburg/ESO and LAMOST surveys, which
includes a large number of stars with stellar abundances derived, have also
revealed several dynamically tied groups (Yuan et al., 2020; Limberg et al.,
2021)

In this paper we investigate the chemical signature of the stellar stream
Indus, which contains both an $r$-process enhanced star, Indus$\\_$13, and a
globular cluster like star, Indus$\\_$0\. The Indus stream was discovered as a
stellar overdensity in DES data (Shipp et al., 2018) and further characterized
with radial-velocity and metallicity measurements of member stars through the
Southern Stellar Streams Spectroscopic Survey ($S^{5}$) project (Li et al.,
2019). Seven of the Indus stream members were selected for high-resolution
follow-up and their abundances are presented in Ji et al. (2020). In this
paper we present new abundances, as well as further explore those from Ji et
al. (2020). Section 2 describes the observations of Indus$\\_$0 and
Indus$\\_$13\. A detailed analysis of neutron-capture elements in Indus$\\_$13
is presented in Section 3 and the results given in Section 4. Section 5
discusses the implications of the abundance patterns of Indus$\\_$0 and
Indus$\\_$13 and the general chemical signature of the stream. A summary is
provided in Section 6.

Table 1:  Observing log for Indus$\\_$0 and Indus$\\_$13 Obs Date | MJD | $t_{\text{exp}}$ | SNR | SNR | $v_{\text{hel}}$ | $\sigma(v)$ | $N_{\text{ord}}$
---|---|---|---|---|---|---|---
|  | (min) | 4500Å | 6500Å | (kms-1) | (kms-1) |
Indus$\\_$0
2019-07-27 | 58691.16 | 50 | 25 | 47 | $-28.8$ | 0.2 | 34
Indus$\\_$13
2019-07-25 | 58689.23 | 86 | 23 | 43 | $-58.0$ | 0.3 | 35
2020-11-17 | 59170.60 | 92 | 12 | 35 | $-59.2$ | 0.6 | 37

Table 2: Basic Data for Indus$\\_$0 and Indus$\\_$13

Quantity | Indus$\\_$0 | Indus$\\_$13 | Unit | Reference
---|---|---|---|---
Gaia DR2 id | 6390575508661401216 | 6412626111276193920 |  |
RA | 23:24:01.75 | 22:05:30.97 | hh:mm:ss.ss |
Dec |  $-$64:02:20.8 |  $-$56:30:53.4 | dd:mm:ss.s |
$g$ | 16.20 | 17.05 | mag | Abbott et al. (2018)
$r$ | 15.60 | 16.46 | mag | Abbott et al. (2018)
Teff | 5040$\pm$47 | 5063$\pm$58 | K | Ji et al. (2020)
$\log g$ | 1.93$\pm$0.16 | 2.29$\pm$0.16 | cgs | Ji et al. (2020)
$\xi$ | 1.73$\pm$0.23 | 1.59$\pm$0.16 | kms-1 | Ji et al. (2020)
$\mathrm{[M/H]}$ | -2.41 |  $-$1.91 |  | Ji et al. (2020)

Table 3: Atomic Data and Abundances for Individual Neutron-Capture Element
lines

El. | $\lambda$ | $\chi$ | $\log gf$ | $\log\epsilon$ | $\sigma_{\log\epsilon}$ | ref
---|---|---|---|---|---|---
| (Å) |  |  |  | (dex) |
Sr II | 4077.71 | 0.00 |  $+$0.15 |  $+$1.68 | 0.06 | 1
Sr II | 4161.80 | 2.94 |  $-$0.47 |  $+$1.92 | 0.18 | 1
Sr II | 4215.52 | 0.00 |  $-$0.17 |  $+$1.62 | 0.08 | 1
Y II | 3950.35 | 0.10 |  $-$0.73 |  $+$1.17 | 0.51 | 2
Y II | 4124.90 | 0.41 |  $-$1.38 |  $+$0.71 | 0.28 | 2
Y II | 4358.72 | 0.10 |  $-$1.15 |  $+$0.85 | 0.40 | 2

References. — (1) Kramida et al. (2018); (2) Biémont et al. (2011); (3) Ljung
et al. (2006); (4) Wickliffe et al. (1994); (5) Kramida et al. (2018), using
HFS/IS from McWilliam (1998); (6) Lawler et al. (2001), using HFS from Ivans
et al. (2006); (7) Lawler et al. (2009); (8) Li et al. (2007), using HFS from
Sneden et al. (2009); (9) Den Hartog et al. (2003), using HFS/IS from Roederer
et al. (2008); (10) Lawler et al. (2006), using HFS/IS from Roederer et al.
(2008); (11) Lawler et al. (2001), using HFS/IS from Ivans et al. (2006); (12)
Den Hartog et al. (2006); (13) Lawler et al. (2001), using HFS from Lawler et
al. (2001); (14) Wickliffe et al. (2000); (15) Lawler et al. (2004), using HFS
from Lawler et al. (2009); (16) Lawler et al. (2008); (17) Wickliffe & Lawler
(1997); (18) Lawler et al. (2009) for $\log gf$ values and HFS; (19) Lawler et
al. (2007); (20) Quinet et al. (2006); (21) Xu et al. (2007), using HFS/IS
from Cowan et al. (2005); (22) Biémont et al. (2000), using HFS/IS from
Roederer et al. (2012); (23) Nilsson et al. (2002)

Note. — The complete version of this Table is available online only. A short
version is shown here to illustrate its form and content.

##  2 Observations

Indus$\\_$0 and Indus$\\_$13 were observed in July 2019 with the Magellan/MIKE
spectrograph (Bernstein et al., 2003) as part of the $S^{5}$ high-resolution
follow-up program. General background on the $S^{5}$ project can be found in
Li et al. (2019), and information on membership probability calculations and
target selection for the high-resolution follow-up are described in Li et al.
(2019) and Ji et al. (2020). Indus$\\_$13 was again observed in November 2020.
All observations were executed using the 0.7” slit and 2x2 CCD binning,
yielding a typical resolution of $R\sim$ 35000/28000 in the blue and red arms,
respectively. The data were reduced with the CarPy MIKE pipeline (Kelson,
2003) and coadded. Table 1 provides an observing log for the observations
listing the signal-to-noise ratio (SNR) per pixel, heliocentric radial
velocities and associated uncertainty, and the number of orders
($N_{\text{ord}}$) used for the cross-correlation of the spectra.

##  3 Stellar Parameter Determination and Abundance Analysis

Ji et al. (2020) presented abundances for $\sim$25 elements in both
Indus$\\_$0 and Indus$\\_$13 as part of their analysis of stars across
multiple stellar streams. For completeness, we list the Ji et al. (2020)
measurements for Indus$\\_$0 including errors in Table 3. This star exhibits
an abundance pattern consistent with it having formed in a globular cluster,
and we discuss the implications of this in Section 5. Indus$\\_$13 exhibits
significant $r$-process enhancement, such that we are able to derive
abundances for 24 neutron-capture elements in addition to the measurements
presented in Ji et al. (2020). These new abundances are determined following
the analysis in Ji et al. (2020) and using the parameters for Indus$\\_$13
determined in that paper (see Table 2). We provide here a brief description of
the procedure and refer the reader to Ji et al. (2020) for details. Effective
temperatures were determined from photometry using de-reddened $g-r$ colors
from the DES survey (Abbott et al., 2018) and color-temperature relations
derived from Dartmouth isochrones (Dotter et al., 2008). Surface gravities
were determined using the distance modulus from Shipp et al. (2018) in
combination with DES $g$-band magnitudes. Finally, metallicities and
mictroturbulences were determined from equivalent width measurements of Fe II
lines in the stellar spectra. All abundances presented here are derived via
spectral synthesis using the analysis code SMHR
111https://github.com/andycasey/smhr to run the 2017 version of the 1D LTE
radiative transfer code MOOG 222https://github.com/alexji/moog17scat.
$\alpha$-enhanced ($\mathrm{[\alpha/Fe]=+0.4}$) ATLAS model atmospheres
(Castelli & Kurucz, 2003) were used as input, with line lists generated from
linemake 333https://github.com/vmplacco/linemake updated with CH lines from
Masseron et al. (2014)
444https://nextcloud.lupm.in2p3.fr/s/r8pXijD39YLzw5T?path=%2FCH. Solar
abundances were taken from Asplund et al. (2009). Our synthesis also includes
isotopic and hyperfine structure broadening, when applicable, employing the
$r$-process isotope ratios from Sneden et al. (2008). Values and references
for the oscillator strengths ($\log gf$) employed are listed in Table 1 along
with wavelengths, and the excitation potential ($\xi$) values for the lines
used in the analysis. Finally, Ji et al. (2020) introduced a thorough
systematic and statistical abundance error analysis, which we also employ
here.

Figure 1:  Synthesis of Mo I, Os I, and Th II lines in Indus$\\_$13\. The top
panels display the best fit synthesis (solid lines) to the observed spectrum
(black squares) along with $\pm$0.2 dex uncertainty (shaded region). Residuals
between the observed spectrum and the various synthesis are shown in the
bottom panel. For the Th II line blended features included in the synthesis
are labeled.

Table 4: Abundance Summary for Indus$\\_$0

El. | N | $\log\epsilon$ | $\mathrm{[X/H]}$ | $\sigma_{\mathrm{[X/H]}}$ | $\mathrm{[X/Fe]}$ | $\sigma_{\mathrm{[X/Fe]}}$ | $\Delta_{T_{\rm eff}}$ | $\Delta_{\log g}$ | $\Delta_{\xi}$ | $\Delta_{\mathrm{[M/H]}}$ | $s_{X}$
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | (dex) |  | (dex) | (dex) | (dex) | (dex) | (dex) | (dex)
C-H | 2 |  $+$6.16 |  $-$2.27 | 0.08 |  $+$0.20 | 0.09 | 0.09 |  $-$0.06 | 0.01 | 0.04 | 0.03
C-N | 1 |  $+$6.53 |  $-$1.30 | 0.47 |  $+$1.16 | 0.47 | 0.13 |  $-$0.05 | 0.01 | 0.05 | 0.00
O I | 1 |  $<+$7.81 |  $-$0.88 |  |  $+$1.58 |  |  |  |  |  |
Na I | 2 |  $+$4.72 |  $-$1.52 | 0.08 |  $+$0.94 | 0.08 | 0.07 |  $-$0.07 |  $-$0.07 |  $-$0.01 | 0.06
Mg I | 7 |  $+$5.54 |  $-$2.06 | 0.06 |  $+$0.40 | 0.07 | 0.04 |  $-$0.04 |  $-$0.04 | 0.00 | 0.09
Al I | 2 |  $+$4.31 |  $-$2.14 | 0.25 |  $+$0.32 | 0.25 | 0.04 |  $-$0.09 |  $-$0.09 |  $-$0.06 | 0.30
Si I | 2 |  $+$5.83 |  $-$1.68 | 0.08 |  $+$0.78 | 0.09 | 0.07 |  $-$0.05 |  $-$0.03 | 0.01 | 0.00
K I | 2 |  $+$3.42 |  $-$1.61 | 0.10 |  $+$0.85 | 0.10 | 0.04 |  $-$0.01 |  $-$0.05 |  $-$0.01 | 0.07
Ca I | 15 |  $+$4.28 |  $-$2.06 | 0.06 |  $+$0.40 | 0.07 | 0.03 |  $-$0.00 |  $-$0.02 | 0.00 | 0.10
Sc II | 8 |  $+$0.82 |  $-$2.33 | 0.12 |  $+$0.12 | 0.12 |  $-$0.00 | 0.05 |  $-$0.03 | 0.01 | 0.19
Ti I | 17 |  $+$2.73 |  $-$2.22 | 0.08 |  $+$0.24 | 0.08 | 0.06 |  $-$0.01 |  $-$0.00 | 0.01 | 0.12
Ti II | 31 |  $+$2.79 |  $-$2.16 | 0.07 |  $+$0.29 | 0.09 | 0.02 | 0.06 | 0.04 | 0.02 | 0.14
V I | 2 |  $+$1.49 |  $-$2.44 | 0.11 |  $+$0.02 | 0.12 | 0.02 | 0.02 | 0.01 |  $-$0.01 | 0.11
V II | 2 |  $+$1.72 |  $-$2.21 | 0.10 |  $+$0.24 | 0.11 |  $-$0.01 | 0.06 | 0.01 | 0.01 | 0.00
Cr I | 7 |  $+$3.18 |  $-$2.46 | 0.09 |  $-$0.00 | 0.09 | 0.05 |  $-$0.01 |  $-$0.02 | 0.00 | 0.14
Cr II | 3 |  $+$3.27 |  $-$2.37 | 0.07 |  $+$0.08 | 0.08 |  $-$0.01 | 0.06 |  $-$0.01 | 0.01 | 0.00
Mn I | 5 |  $+$2.60 |  $-$2.83 | 0.08 |  $-$0.37 | 0.09 | 0.03 |  $-$0.01 |  $-$0.01 |  $-$0.00 | 0.12
Fe I | 118 |  $+$5.04 |  $-$2.46 | 0.05 |  $+$0.00 | 0.00 | 0.04 | 0.00 | 0.02 | 0.01 | 0.23
Fe II | 18 |  $+$5.06 |  $-$2.44 | 0.09 |  $+$0.00 | 0.00 | 0.00 | 0.06 | 0.00 | 0.01 | 0.16
Co I | 4 |  $+$2.98 |  $-$2.01 | 0.09 |  $+$0.46 | 0.09 | 0.05 | 0.01 |  $-$0.00 | 0.00 | 0.04
Ni I | 9 |  $+$3.89 |  $-$2.33 | 0.08 |  $+$0.14 | 0.08 | 0.05 |  $-$0.00 | 0.00 | 0.00 | 0.13
Cu I | 1 |  $<+$2.49 |  $-$1.70 |  |  $+$0.76 |  |  |  |  |  |
Zn I | 2 |  $+$2.22 |  $-$2.34 | 0.11 |  $+$0.12 | 0.11 | 0.02 | 0.02 |  $-$0.01 | 0.01 | 0.10
Sr II | 2 |  $+$0.68 |  $-$2.19 | 0.22 |  $+$0.25 | 0.19 | 0.04 | 0.03 |  $-$0.12 | 0.02 | 0.00
Y II | 4 |  $-$0.30 |  $-$2.51 | 0.09 |  $-$0.07 | 0.10 | 0.01 | 0.04 |  $-$0.01 | 0.01 | 0.06
Zr II | 1 |  $+$0.36 |  $-$2.21 | 0.14 |  $+$0.23 | 0.15 |  $-$0.08 | 0.07 |  $-$0.05 |  $-$0.05 | 0.00
Ba II | 5 |  $-$0.11 |  $-$2.29 | 0.16 |  $+$0.15 | 0.14 | 0.02 | 0.05 |  $-$0.07 | 0.01 | 0.15
La II | 3 |  $<-$0.91 |  $-$2.01 |  |  $+$0.43 |  |  |  |  |  |
Eu II | 2 |  $-$1.50 |  $-$2.02 | 0.10 |  $+$0.42 | 0.09 | 0.02 | 0.07 | 0.01 | 0.02 | 0.00
Dy II | 1 |  $-$0.76 |  $-$1.85 | 0.16 |  $+$0.59 | 0.16 | 0.01 | 0.06 |  $-$0.02 | 0.02 | 0.00

Note. — Abundances are taken directly from Ji et al. (2020).

Table 5: Abundance Summary for Indus$\\_$13

**footnotetext: NLTE corrected: $\Delta_{\rm NLTE}$ = +0.41 dex.

Note. — Abundances C to Zn are taken directly from Ji et al. (2020)

##  4 Indus$\\_$13 Abundance Results

We have derived abundances for 24 neutron-capture elements, including lead
(Pb) and thorium (Th), in Indus$\\_$13\. Synthesis of selected elements (Mo I,
Os I, and Th II) are shown in Figure 1 and final abundances including errors
are listed in Table 3, where $\Delta_{\rm X}$ are the abundance errors due to
stellar parameter errors and $s_{X}$ is the systematic error. In addition,
individual line abundances are listed in Table 2. For completeness we also
include abundances derived for Indus$\\_$13 from Ji et al. (2020) in Table 3.
All abundances are derived assuming local thermodynamic equilibrium. For the
majority of the neutron-capture elements, abundances are derived from the
ionized states of the species, for which effects due to departures from local
thermodynamic equilibrium (NLTE) are expected to be minimal. For the range of
neutron-capture elements analysed here, to our knowledge, NLTE corrections
only exist for Ba and Pb. For Ba, Korotin et al. (2015) find average
corrections of $\pm$0.1 dex for the three red Ba II lines at 5853, 6141, and
6496Å, two of which are used in this study. However, with a Ba abundance of
$\mathrm{[Ba/Fe]}=+1.29$, Indus$\\_$13 falls outside the grid of Ba abundances
for which Korotin et al. (2015) provide corrections. We therefore use the
uncorrected Ba abundance. For Pb, Roederer et al. (2020) derived abundances
using the 2203Å Pb II line in the UV spectra of three metal-poor stars,
yielding higher Pb abundances than results from the more commonly used Pb I
line at 4057Å for all three stars. This confirms that Pb I abundances are
likely affected by NLTE effects. We therefore correct our Pb abundance using
the grid from Mashonkina et al. (2012), which predicts a correction of $+0.41$
dex for the star with the best-matched parameters to Indus 13 (HD 6755; Teff =
5100 K, $\log g$ = 2.93, and $\mathrm{[Fe/H]}=-1.68$), resulting in
$\log_{\epsilon}$(Pb) = +1.52 for Indus$\\_$13.

Figure 2:  Top panel: neutron-capture element abundances derived for
Indus$\\_$13 (dots) along with scaled Solar-System $r$-process (black line)
and $s$-process (gray line) abundance patterns (Sneden et al., 2008). Bottom
panel: residuals between observed abundances and scaled Solar-System
$r$-process abundance pattern.

##  5 Discussion

###  5.1 $R$-Process Signature in Indus

Figure 2 displays the derived neutron-capture element abundances for
Indus$\\_$13 (blue dots) along with the scaled Solar-System $r$-process (black
line) and $s$-process (gray line) abundance patterns (Sneden et al., 2008). An
average of the abundances from La to Hf has been used for the scaling (Ba was
excluded due to possible NLTE effects as described above). It is clear that
for the heavy elements (La - Hf), the abundance pattern of Indus$\\_$13
displays a good match to the scaled Solar-System $r$-process abundance
pattern. Thus Indus$\\_$13 was enriched by an $r$-process nucleosynthesis
event and, with an Eu abundance of $\mathrm{[Eu/Fe]}=+1.81$, can be classified
as an $r$-II star according to the classification scheme of Holmbeck et al.
(2020). The other six Indus stream stars exhibit milder enhancements in Eu
($+0.40<\mathrm{[Eu/Fe]}<+0.67$) with all having $\mathrm{[Ba/Eu]}<+0.0$ (Ji
et al., 2020, see also Figure 3) consistent with enrichment from an
$r$-process event. They can thus be labelled $r$-I stars.

As can be seen from Figure 2, the Th abundance of Indus$\\_$13 falls +0.33 dex
below what is predicted from the Sneden et al. (2008) Solar-System $r$-process
abundance pattern. We find $\log\epsilon(\rm Th/Eu)=-0.66$, while the Sneden
et al. (2008) pattern predicts $\log\epsilon(\rm Th/Eu)=-0.33$. This means
that Indus$\\_$13 joins the sub-group of actinide-poor $r$-process enhanced
stars. This group also includes the two other $r$-II stars in dwarf galaxies
for which a Th abundance has been derived: COS 82 in UMi for which Aoki et al.
(2007) found $\log\epsilon(\rm Th/Eu)=-0.59$ and DES J033523$-$54040 in
Reticulum II (Ret II) for which Ji & Frebel (2018) found the extremely low
ratio $\log\epsilon(\rm Th/Eu)=-0.84$. The other extreme is the so-called
actinide boost stars, where the actinides are over-produced relative to the
rare-earth elements. Examples include CS 31082$-$001 with $\log\epsilon(\rm
Th/Eu)=-0.22$ (Hill et al., 2002) and 2MASS J09544277$+$5246414 with
$\log\epsilon(\rm Th/Eu)=-0.12$ (Holmbeck et al., 2018). This abundance
signature have been known to exist for some time in MW halo stars, and has
also recently been detected in stars associated with the Helmi debris stream
(Gull et al., 2021), but actinide boost stars have yet to be discovered in any
present day surviving dwarf galaxy.

The actinide elements, Th and U, are exclusively produced via the $r$-process,
and modelers have struggled to explain the variations seen in these elements
in $r$-process enhanced metal-poor stars (Mashonkina et al., 2014). However,
recent results from $r$-process nucleosynthesis modeling suggest that the
abundance patterns seen in actinide-poor and actinide-boosted stars can be
produced by varying the electron fraction ($Y_{e}$) distribution within the
same $r$-process site (Holmbeck et al., 2019). The Holmbeck et al. (2019)
analysis was performed for the $r$-process nucleosynthesis in neutron-star
mergers, which is currently the only $r$-process element production site that
has been directly observed (Drout et al., 2017). A similar analysis of
actinide element production in other proposed $r$-process elements production
sites such as magneto-rationally driven supernovae and collapsars (Winteler et
al., 2012; Siegel et al., 2019) may help to better identify the source of
$r$-process elements in systems like Indus.

Another way to investigate the source of $r$-process elements in Indus is to
calculate the fraction of lanthanide elements to lighter neutron-capture
elements - the so-called lanthanide fraction ($X_{\rm LA}$) - for
Indus$\\_$13, which can be directly compared to the lanthanide fraction found
for the 2017 neutron-star merger GW170817 (Ji et al., 2019). We find $\log
X_{\rm LA}$= -1.33$\pm$0.2 for Indus$\\_$13 (the error is dominated by
systematics). This is somewhat higher than the $\log X_{\rm LA}$= -2.2$\pm
0.5$ found for GW170817 but similar to the $\log X_{\rm LA}$=-1.44 found for
the highly $r$-process enhanced stars in the MW halo. This discrepancy between
the $\log X_{\rm LA}$ for GW170817 and the highly $r$-process enhanced MW halo
stars led Ji et al. (2019) to conclude that neutron-star mergers with more
lanthanide rich ejecta are needed for neutron-star mergers to remain viable as
the dominant $r$-process element production site. It is likely that many of
the stars included in the highly $r$-process enhanced MW halo sample used for
the calculation in Ji et al. (2019) are accreted dwarf galaxy stars similar to
Indus$\\_$13\. With the separation of the halo $r$-process enhanced stars into
dynamically tied groups, a comparison $X_{\rm LA}$ can be done on a system-by-
system basis, allowing us to better characterize the range of lanthanide
fractions captured in metal-poor stars.

A further interesting feature of Indus$\\_$13, in connection to its
$r$-process enrichment, is its relatively high metallicity. The stars in the
Indus stream span a range in metallicity from $\mathrm{[Fe/H]}=-2.44$ to
$-1.69$ (Ji et al., 2020), where Indus$\\_$13, with a metallicity of
$\mathrm{[Fe/H]}=-1.92$, joins the metal-rich end of this range. With its
relatively high metallicity, Indus$\\_$13 adds to the sample of $r$-II stars
in the MW with $\mathrm{[Fe/H]}>-2$ that has emerged in recent years (e.g.
Roederer et al., 2018; Hansen et al., 2018; Sakari et al., 2018). Early
exploration and surveys for $r$-II stars in the MW halo found $r$-II stars
only in a fairly narrow metallicity range from $\mathrm{[Fe/H]}=-3.19$ to
$-2.58$, suggesting that these stars are preferentially found at the lowest
metallicities. On the other hand, the first $r$-I stars were discovered over a
wider metallicity range, extending to metallicities of
$\mathrm{[Fe/H]}\sim-1.5$ (Barklem et al., 2005). However, the recent
extensive search for $r$-process enhanced stars initiated by the $R$-Process
Alliance (RPA) (Hansen et al., 2018) has found a number of $r$-II stars with
$\mathrm{[Fe/H]}>-2$. In fact, of the 74 $r$-II stars discovered in the four
RPA data release papers, 21 have $\mathrm{[Fe/H]}>-2$, corresponding to 28%.
This percentage is very similar to that found for the 242 $r$-I stars
discovered, where 70 have $\mathrm{[Fe/H]}>-2$, corresponding to 29% (Hansen
et al., 2018; Sakari et al., 2018; Ezzeddine et al., 2020; Holmbeck et al.,
2020). These results strongly suggest that the metallicity distribution
function is the same for $r$-I and and $r$-II stars in the halo. Thus the high
metallicity of Indus$\\_$13 is not unexpected or unusual since $r$-II stars,
though rare, are equally likely to be found at high metallicities as $r$-I
stars.

Apart from the MW, $r$-process enhanced stars have also been discovered in
several classical dwarf galaxies (e.g. Cohen & Huang, 2009, 2010) as well as
in ultra-faint dwarf (UFD) galaxies. The discovery of $r$-process enhanced
stars in the UFD galaxies Ret II (Ji et al., 2016; Roederer et al., 2016),
Tucana III (Hansen et al., 2017; Marshall et al., 2019), and Grus II (Hansen
et al., 2020) supports the idea that $r$-process enhanced stars in the MW halo
likely found their way into the halo through the accretion and disruption of
such smaller systems (Brauer et al., 2019). With the release of kinematic data
from the $Gaia$ satellite, this theory has been tested: first on the smaller
sample in Roederer et al. (2018), and most recently by Gudin et al. (2020) on
the full literature sample that includes the large number of $r$-process
enhanced stars discovered as part of the RPA. Both of these find that a number
of $r$-process stars cluster in groups sharing kinematic features, suggesting
that the stars were likely stripped from the same parent object. Indus$\\_$13,
as the first highly $r$-process enhanced star to be discovered in a disrupted
dwarf galaxy stream, along with the recent discovery of $r$-process enhanced
stars in the Helmi debris, Helmi tail, and $\omega$ Centauri progenitor
streams (Gull et al., 2021), thus provides a vital observational link
supporting the accretion origin for $r$-process enhanced stars. In addition,
as is pointed out by Gull et al. (2021), many $r$-process enhanced stars in
the MW halo likely come from relatively massive galaxies that have experienced
multiple $r$-process events and not just from UFD galaxies. As a consequence,
we should expect to find stronger kinematic clustering among these stars from
larger galaxies, compared with the smaller number of stars accreted from small
galaxies.

Figure 3:  Light and neutron-capture element abundances for the seven Indus
stars and MW halo stars from JINA base (Abohalima & Frebel, 2018) highlighting
the separation from the rest of the stars of Indus$\\_$0 for the light
elements N, Na, and Al, and of Indus$\\_$13 for the neutron-capture elements
Eu and Ba. NLTE corrected abundances for Na and Al are also shown in the top
row displaying and general shift for all the Indus stars. Dotted line in the
$\mathrm{[Ba/Eu]}$ plot (bottom middle) marks a pure $r$-process
$\mathrm{[Ba/Eu]}$ ratio while the arrows mark a dominant contribution to the
neutron-capture elements being from either the $s$\- or $r$-process (Sneden et
al., 2008).

###  5.2 A Globular Cluster Star in Indus

The Indus stream contains a second chemically distinct star, namely
Indus$\\_$0, for which Ji et al. (2020) found $\mathrm{[N/Fe]}=+1.16$,
$\mathrm{[Na/Fe]}=+0.94$, and $\mathrm{[Al/Fe]}=+0.32$, clearly separating it
from the other Indus stars for these elements, as can be seen in Figure 3. The
abundances for other elements in this star, however, are very similar to those
for the rest of Indus sample. Since both Na and Al are known to be affected by
NLTE effects we have also plotted the NLTE corrected abundances for Na and Al
for the Indus stars in Figure 3 using corrections from Lind et al. (2011) and
Nordlander & Lind (2017), respectively. These corrections mainly result in an
overall shift of the Na and Al abundances of the Indus stars and do not change
the status of Indus$\\_$0 as an outlier.

With high N, Na and Al abundances, Indus$\\_$0 is similar to the abundance
pattern found in so-called ‘second generation’ stars in globular clusters.
Such stars show distinct enhancements in N, Na and Al abundances coupled with
depletions in C, O and Mg relative to the abundances of ‘first generation’ or
‘primordial’ stars in the cluster. While the enrichment process is not
understood, it is clear that nuclear-burning cycles, like the CNO, Ne-Na and
Mg-Al cycles, are involved in generating the variations in the light element
abundance pattern of the stars (e.g. Gratton et al., 2019). The presence of
these light element abundance variations is a characteristic of globular
clusters, and there is no evidence of such correlated light-element abundance
variations occurring in dSph or UFD field stars (e.g. Shetrone et al., 2003;
Norris et al., 2017; Salgado et al., 2019). It should be noted that, with a
metallicity of $\mathrm{[FeII/H]}=-2.44$, a globular cluster progenitor of
Indus$\\_$0 would be among the most metal-poor globular clusters known
(Beasley et al., 2019). However, Letarte et al. (2006) have shown from high
dispersion analyses of a small number of stars that similar abundance
anomalies are found in the Fornax globular clusters that are only slightly
more metal-rich (see discussion below).

The spread in metallicity and the velocity dispersion of the Indus stream
firmly point to it being a disrupted dwarf galaxy (Li et al., 2019; Ji et al.,
2020). However, given that Indus$\\_$0 is in the observed sample, and given
the apparent lack of globular cluster second generation abundance pattern
stars in the field populations of other dSph and UFD systems, the inevitable
conclusion from our data is that the Indus progenitor possessed a globular
cluster in which Indus$\\_$0 formed.

Excluding the most massive MW satellites (Magellanic Clouds and Sagittarius)
globular clusters have been detected in association with only two MW dwarf
galaxies; Fornax and Eridanus 2 (Huang & Koposov, 2021). Specifically, one
faint (M${}_{\rm V}\approx-3.5$) cluster has been detected in the ultra faint
system Eridanus 2 (Crnojević et al., 2016; Koposov et al., 2015), while Fornax
has six globular clusters (Wang et al., 2019). The Fornax globular clusters
have long been known to be old and metal-poor (e.g. Zinn & Persson, 1981;
Buonanno et al., 1998, 1999; Strader et al., 2003), and Larsen et al. (2012)
found that the 4 most metal-poor clusters, which have
$-2.5<\mathrm{[Fe/H]}<-2$, match the metal-poor tail of the Fornax field star
abundance distribution. Further, Larsen et al. (2014) have estimated the
nitrogen abundances of red giants in these four clusters using multi-band HST
imaging. They find a range in N abundance of $\Delta\mathrm{[N/Fe]}\sim+2$
dex, with roughly equal numbers of N-normal and N-enhanced stars in each
cluster. Together with the small sample of Fornax globular cluster red giants
analyzed at high dispersion discussed in Letarte et al. (2006), these results
point to the existence of multiple stellar populations in the Fornax globular
clusters, with properties very similar to those seen in MW globular clusters.
The inference that can be drawn from the Indus stars is then a similar picture
to Fornax, with Indus$\\_$0 originating in a metal-poor globular cluster in
which multiple stellar populations existed, and which was associated with the
Indus progenitor.

Aside from the on-going accretion and disruption of the Sgr dwarf, which
possesses at least six associated globular clusters (e.g. Da Costa &
Armandroff, 1995; Law & Majewski, 2001; Massari et al., 2019; Vasiliev, 2019),
our abundance results for the Indus stream members provide the first evidence
for the accretion of a dwarf galaxy that contained at least one globular
cluster. Other kinematic studies have found evidence for potentially similar
events. For example, Malhan et al. (2019) found a cocoon of stellar material
associated with the globular cluster stream GD-1, which they interpret as the
remnants of the dwarf galaxy with which GD-1 was accreted. Similar analysis of
the Jhelum stream has revealed a two-dimensional structure for this stream
with a thin dense component and a more diffuse component (Bonaca et al.,
2019). These components may represent distinct but related progenitors (Bonaca
et al., 2019). In the context of the Indus stream results, this scenario is
intriguing as it has been suggested that the Indus and Jhelum streams are two
wraps of the same stream (Shipp et al., 2018; Bonaca et al., 2019). None of
the Jhelum stars analyzed, however, display abundance patterns compatible with
a globular cluster origin (Ji et al., 2020). We note for completeness that no
current Galactic globular cluster has been found to have orbital
characteristics similar to those of the Indus stream in the study of Riley &
Strigari (2020).

###  5.3 The Nature of the Indus Stream Progenitor

Apart from clues from various nucleosynthesis channels, the chemical signature
and specific abundance trends of a stream also provide some information on the
nature of the stream progenitor. For example, streams originating solely from
globular clusters can be separated from dwarf galaxy streams using properties
such as the presence or absence of iron abundance spreads, stream width and
velocity dispersion (e.g. Fig. 4 of Ji et al., 2020). For dwarf galaxy
streams, the mean metallicity and the iron abundance of the ‘knee’ in the
([$\alpha$/Fe], [Fe/H]) relation can also yield information on the stellar
mass or luminosity of the progenitor (Kirby et al., 2011). We now use our
information and that in Ji et al. (2020) to place constraints on the Indus
stream progenitor.

Of the present-day intact dwarf spheroidal galaxies, a mixture of $r$-I and
$r$-II stars has been discovered in three systems: Ursa Minor (UMi), Draco,
and Fornax (Shetrone et al., 2001; Cohen & Huang, 2009; Letarte et al., 2010;
Lemasle et al., 2014). $r$-I stars are also known to be present in Carina
(Shetrone et al., 2003; Venn et al., 2012) and Sculptor (Kirby & Cohen, 2012).
Figure 4 shows the $\mathrm{[Eu/Fe]}$ abundances as a function of metallicity
for the $r$-I and $r$-II stars in Draco, UMi, Fornax, and Indus. For the dwarf
galaxies, the stellar sample is that compiled in Hansen et al. (2017), where
$r$-process enhanced stars in dwarf galaxies were selected to have
$\mathrm{[Eu/Fe]}>+0.3$ and $\mathrm{[Ba/Eu]}<-0.4$. The latter criterion is
to ensure the stars were predominantly enriched by an $r$-process event as
opposed to the slow neutron-capture process ($s$-process), which also produces
both Ba and Eu but in a different ratio (Sneden et al., 2008). In Figure 4 we
also include three of the recently re-analysed Fornax stars from Reichert et
al. (2021) satisfying the above criteria. In metallicity, the Indus stream
stars mostly overlap with the $r$-process enhanced stars found in Draco and
UMi, whereas the $r$-I and $r$-II stars in the more massive Fornax galaxy
generally have higher metallicities, and those in the small ultra-faint, Ret
II galaxy are predominantly found at lower metallicities. This suggests that
the Indus’s progenitor was a relatively massive galaxy with efficient chemical
enrichment, similar to UMi. It should be noted, however, that the Indus stars
selected for high-resolution follow-up, and thus included in this study, were
preferentially selected to be metal-poor. Ongoing analysis of medium-
resolution spectra for a larger sample of Indus stream stars finds a mean
metallicity for the stream of $\mathrm{[Fe/H]}=-1.97$ (Pace et al., in prep).
Consequently, a similar analysis of a larger sample of Indus stars would
likely increase the overlap with the Fornax sample.

Figure 4:  $\mathrm{[Eu/Fe]}$ as a function of metallicity for $r$-I and
$r$-II stars in the Draco, UMi, and Fornax dwarf spheroidal galaxies and the
ultra-faint dwarf galaxy Ret II compared to the Indus stream stars.

The chemical enrichment efficiency of a galaxy, and by proxy its mass, can
also be estimated from the evolution of the $\alpha$-element abundances with
metallicity. The galaxy will initially be enriched by a population of massive
stars. These have short lifetimes that end in Type II supernovae explosions
resulting in an $\alpha$-enhanced enrichment. After this initial enrichment,
lower mass and more slowly evolving stars are formed, and eventually, Type Ia
supernovae will contribute to the chemical signature of the system resulting
in a decrease in the $\alpha$ to Fe ratio of the system, also known as the
$\alpha$-knee (Tinsley, 1979). The metallicity at which this transition in
$\mathrm{[\alpha/Fe]}$ occurs reflects the chemical enrichment efficiency of
the galaxy: in a low-mass galaxy, the $\alpha$-knee will appear at lower
metallicities than in a high-mass galaxy.

All the Indus stream stars for which detailed abundances have been derived are
enhanced in the $\alpha$-elements ($\mathrm{\langle[Mg/Fe]\rangle=+0.40}$,
$\mathrm{\langle[Si/Fe]\rangle=+0.49}$,
$\mathrm{\langle[Ca/Fe]\rangle=+0.43}$), suggesting that the location of the
$\alpha$-knee of the Indus stream progenitor is at higher metallicity than the
most metal-rich star included in Ji et al. (2020), which has
$\mathrm{[FeII/H]}=-1.69$. A recent abundance re-analysis of high-resolution
data for stars in Fornax and UMi has found the $\alpha$-knee in these two
galaxies to be located at $\mathrm{[Fe/H]}=-1.6$ and $\mathrm{[Fe/H]}=-2.2$,
respectively, and predicted the location in Draco to be $\mathrm{[Fe/H]}=-2.1$
(Reichert et al., 2020). Comparing these numbers to the metallicity of the
Indus stars supports the hypothesis from the Eu abundances that the Indus
progenitor was likely a dwarf galaxy whose mass exceeded that of UMi but which
may not have been as massive as Fornax.

We can also use the fact that the Indus progenitor likely possessed at least
one globular cluster to estimate its luminosity by calculating the normalized
globular cluster specific frequency $S_{N}$ (see Harris & van den Bergh (1981)
and Huang & Koposov (2021) for the definition of $S_{N}$). For a system such
as Fornax with $M_{\rm V}=-13.4$ (McConnachie, 2012) one globular cluster
results in $S_{N}$=4.5 while the same number for UMi with $M_{\rm V}=-8.8$
(McConnachie, 2012) is $\approx$300\. An $S_{N}$ of 300 is high compared to
what is generally found for the Local Group (Mackey et al., 2019) and seems
unlikely given the results of Huang & Koposov (2021) on the MW satellites,
thus supporting our contention that the luminosity of the Indus progenitor
likely exceeded that of systems such as UMi. This is also in agreement with
the initial estimate from Shipp et al. (2018) of the Indus progenitor’s
dynamical mass when it was stripping stars to the stream of 6.5$\times$106M⊙
based on the width of the stream. In this context we also note that the
postulated Indus globular cluster was unlikely to have been of low luminosity
or mass. As shown by Lagioia et al. (2019) and Milone et al. (2020) for
example, globular clusters that possess multiple populations tend to have
relatively large initial masses: only a few clusters with initial masses above
$10^{5}$ M☉ lack multiple populations, while those that do not show multiple
populations are generally of low mass (Lagioia et al., 2019; Milone et al.,
2020).

##  6 Summary

In this paper, we have taken a closer look at the chemical signature of the
Indus stream, a disrupted dwarf galaxy recently discovered in the DES data
(Shipp et al., 2018) and further characterised by the $S^{5}$ project (Li et
al., 2019; Ji et al., 2020). In particular, a detailed neutron-capture
elements abundance analysis is presented for the highly $r$-process enhanced
star Indus$\\_$13\. The neutron-capture element abundances in this star
display a good match to scaled Solar-System $r$-process abundance pattern
confirming an $r$-process nucleosynthesis origin for these elements. With a
slightly low abundance of the actinide element Th, Indus$\\_$13 joins the sub-
class of actinide-poor $r$-process enhanced stars, which also includes one of
the $r$-process enhanced stars each in Ret II and UMi (Ji & Frebel, 2018; Aoki
et al., 2007). The metal-rich nature Indus$\\_$13 star
($\mathrm{[Fe/H]}=-1.92$) along with the general $\alpha$ enhancement seen in
all the Indus stars point to the Indus progenitor being a chemically evolved
dwarf galaxy. In addition to Indus$\\_$13, we also discuss the globular
cluster like star Indus$\\_$0 which shows enhanced N, Na and Al abundances. It
is likely that this star formed in a globular cluster associated with the
Indus stream progenitor. Stars with similar abundance patterns have also been
seen in the globular clusters associated with the Fornax dwarf spheroidal
galaxy (Letarte et al., 2006; Larsen et al., 2014). Taking into account all
available information, we conclude that the Indus progenitor was a chemically
evolved dwarf galaxy that contained at least one globular cluster and which
had a mass that likely exceeded that of UMi but which was less than that of
Fornax.

The authors thank the referee for a careful reading of the manuscript. TTH
acknowledges generous support from the George P. and Cynthia Woods Institute
for Fundamental Physics and Astronomy at Texas A&M University. APJ
acknowledges support from a Carnegie Fellowship and the Thacher Research Award
in Astronomy. ABP is supported by NSF grant AST-1813881. TSL is supported by
NASA through Hubble Fellowship grant HST-HF2-51439.001 awarded by the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., for NASA, under contract
NAS5-26555. A.R.C. is supported in part by the Australian Research Council
through a Discovery Early Career Researcher Award (DE190100656). Parts of this
research were supported by the Australian Research Council Centre of
Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through
project number CE170100013. This research made extensive use of the SIMBAD
database operated at CDS, Straasburg, France Wenger et al. (2000), arXiv.org,
and NASA’s Astrophysics Data System for bibliographic information. This work
presents results from the European Space Agency (ESA) space mission Gaia. Gaia
data are being processed by the Gaia Data Processing and Analysis Consortium
(DPAC). Funding for the DPAC is provided by national institutions, in
particular the institutions participating in the Gaia MultiLateral Agreement
(MLA). The Gaia mission website is https://www.cosmos.esa.int/gaia. The Gaia
archive website is https://archives.esac.esa.int/gaia. This project used
public archival data from the Dark Energy Survey (DES). Funding for the DES
Projects has been provided by the U.S. Department of Energy, the U.S. National
Science Foundation, the Ministry of Science and Education of Spain, the
Science and Technology Facilities Council of the United Kingdom, the Higher
Education Funding Council for England, the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign, the Kavli
Institute of Cosmological Physics at the University of Chicago, the Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the
Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho
de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Científico e Tecnológico and the Ministério da Ciência,
Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energéticas, Medioambientales y Tecnológicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the Institut de
Física d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-
Maximilians Universität München and the associated Excellence Cluster
Universe, the University of Michigan, the National Optical Astronomy
Observatory, the University of Nottingham, The Ohio State University, the
OzDES Membership Consortium, the University of Pennsylvania, the University of
Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the
University of Sussex, and Texas A&M University. Based in part on observations
at Cerro Tololo Inter-American Observatory, National Optical Astronomy
Observatory, which is operated by the Association of Universities for Research
in Astronomy (AURA) under a cooperative agreement with the National Science
Foundation. Magellan:Clay

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