# The Capacity of Wedge Sum of Spheres of Different Dimensions

Mojtaba Mohareri<EMAIL_ADDRESS>Behrooz Mashayekhy<EMAIL_ADDRESS>Hanieh Mirebrahimi<EMAIL_ADDRESS>Department of Pure Mathematics,
Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University
of Mashhad,
P.O.Box 1159-91775, Mashhad, Iran.

###### Abstract

K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the
concept of the capacity of a compactum and raised some interesting questions
about it. In this paper, during computing the capacity of wedge sum of
finitely many spheres of different dimensions and the complex projective
plane, we give a negative answer to a question of Borsuk whether the capacity
of a compactum determined by its homology properties.

###### keywords:

Homotopy domination, Homotopy type , Moore space, Polyhedron , CW-complex ,
Compactum.

###### MSC:

[2010]55P15, 55P55, 55P20,54E30, 55Q20.

††journal:

##  1 Introduction and Motivation

K. Borsuk in [2], introduced the concept of the capacity of a compactum
(compact metric space) as follows: the capacity $C(A)$ of a compactum $A$ is
the cardinality of the set of all shapes of compacta $X$ for which
$\mathcal{S}h(X)\leqslant\mathcal{S}h(A)$ (for more details, see [10]).

For polyhedra, the notions shape and shape domination in the above definition
can be replaced by the notions homotopy type and homotopy domination,
respectively. Indeed, by some known results in shape theory one can conclude
that for any polyhedron $P$, there is a one to one functorial correspondence
between the shapes of compacta shape dominated by $P$ and the homotopy types
of CW-complexes (not necessarily finite) homotopy dominated by $P$ (for both
pointed and unpointed polyhedra) [6].

S. Mather in [11] proved that every polyhedron dominates only countably many
different homotopy types (hence shapes). Note that the capacity of a
topological space is a homotopy invariant, i.e., for topological spaces $X$
and $Y$ with the same homotopy type, $C(X)=C(Y)$. Hence it seems interesting
to find topological spaces with finite capacity and compute the capacity of
some of their well-known spaces. Borsuk in [2] asked a question: “ Is it true
that the capacity of every finite polyhedron is finite? ”. D. Kolodziejczyk in
[9] gave a negative answer to this question. Also, she investigated some
conditions for polyhedra to have finite capacity ([5, 6, 7, 8]). For instance,
a polyhedron $Q$ with finite fundamental group $\pi_{1}(Q)$ and a polyhedron
$P$ with abelian fundamental group $\pi_{1}(P)$ and finitely generated
homology groups $H_{i}(\tilde{P})$, for $i\geq 2$ where $\tilde{P}$ is the
universal cover of $P$, have finite capacities.

Borsuk in [2] mentioned that the capacity of $\bigvee_{k}\mathbb{S}^{1}$ and
$\mathbb{S}^{n}$ equals to $k+1$ and 2, respectively. The authors in [12]
computed the capacity of Moore spaces $M(A,n)$ and Eilenberg-MacLane spaces
$K(G,n)$. In fact, we showed that the capacities of a Moore space $M(A,n)$ and
an Eilenberg-MacLane space $K(G,n)$ equal to the number of direct summands of
$A$ and semidirect factors of $G$, respectively, up to isomorphism. Also, we
computed the capacity of wedge sum of finitely many Moore spaces of different
degrees and the capacity of the product of finitely many Eilenberg-MacLane
spaces of different homotopy types. In particular, we showed that the capacity
of $\bigvee_{n\in I}(\vee_{i_{n}}\mathbb{S}^{n})$ equals to $\prod_{n\in
I}(i_{n}+1)$ where $\vee_{i_{n}}\mathbb{S}^{n}$ denotes the wedge sum of
$i_{n}$ copies of $\mathbb{S}^{n}$, $I$ is a finite subset of
$\mathbb{N}\setminus\\{1\\}$ and $i_{n}\in\mathbb{N}$.

In Section 2, we will state some basic facts that we need for the rest of the
paper. In Section 3, we will compute the capacity of wedge sum of finitely
many spheres of different dimensions (containing spheres of dimension 1). As a
consequence, we show that two dimensional CW-complexes with free fundamental
groups have finite capacities.

Borsuk in [2] stated some questions concerning properties of the capacity of
compacta. One of them is as follow:

“Is the capacity $C(A)$ determined by the homology properties of $A$?”

In Section 4, we give a negative answer to this question, in general. However,
the authors in [12] showed that the capacity of Moore spaces determined by
their homology properties.

##  2 Preliminaries

In this paper, every CW-complex is assumed to be finite and connected and
every map between two CW-complexes is assumed to be cellular. We assume that
the reader is familiar with the basic notions and facts of homotopy theory. We
need the following results and definitions for the rest of the paper.

######  Definition 2.1.

[16]. Let $\phi:K\longrightarrow X$ be a map between CW-complexes with mapping
cylinder $M=X\bigcup_{\phi}(K\times I)$. Denote $\pi_{n}(M,K\times\\{1\\})$ by
$\pi_{n}(\phi)$. The map $\phi$ is called $n$-connected if $K$ and $X$ are
connected and $\pi_{i}(\phi)=0$ for $1\leq i\leq n$.

######  Theorem 2.2.

[4, Theorem 4.37]. If $(X,A)$ is an $(n-1)$-connected pair of path-connected
spaces with $n\geq 2$ and $A\neq\emptyset$, then the Hurewicz homomorphism
$h_{n}:\pi_{n}(X,A)\longrightarrow H_{n}(X,A)$ is an isomorphism and
$H_{i}(X,A)=0$ for $i<n$.

Recall that a pair $(X,A)$ of topological spaces is called $n$-connected
$(n\geq 0)$ if $\pi_{i}(X,A)=0$ for $0\leq i\leq n$.

######  Theorem 2.3.

[4, Proposition 4.1]. A covering projection
$p:(\tilde{X},\tilde{x}_{0})\longrightarrow(X,x_{0})$ induces isomorphisms
$p_{*}:\pi_{n}(\tilde{X},\tilde{x}_{0})\longrightarrow\pi_{n}(X,x_{0})$ for
all $n\geq 2$.

######  Definition 2.4.

[16]. Let $X$ be a CW-complex. Then conditions $\mathcal{F}_{i}$ and
$\mathcal{D}_{i}$ on $X$ are defined inductively as follows:

$\mathcal{F}_{1}$: the group $\pi_{1}(X)$ is finitely generated.

$\mathcal{F}_{2}$: the group $\pi_{1}(X)$ is finitely presented, and for any
2-dimensional finite CW-complex $K$ and any map $\phi:K\longrightarrow X$
inducing an isomorphism of fundamental groups, $\pi_{2}(\phi)$is a finitely
generated module over $\mathbb{Z}\pi_{1}(X)$.

$\mathcal{F}_{n}$: the condition $\mathcal{F}_{n-1}$ holds, and for any
$(n-1)$-dimensional finite CW-complex $K$ and any $(n-1)$-connected map
$\phi:K\longrightarrow X$, $\pi_{n}(\phi)$ is a finitely generated
$\mathbb{Z}\pi_{1}(X)$-module.

$\mathcal{D}_{n}$: $H_{i}(\tilde{X})=0$ for $i>n$, and
$H^{n+1}(X;\mathcal{B})=0$ for all coefficient bundles $\mathcal{B}$ (for more
detalis, see [15]).

######  Theorem 2.5.

[16, Theorem F]. A CW-complex $X$ is dominated by a finite CW-complex of
dimension $n\geq 1$ if and only if $X$ satisfies $\mathcal{D}_{n}$ and
$\mathcal{F}_{n}$.

######  Definition 2.6.

[13]. A group $G$ is called Hopfian if every epimorphism $f:G\longrightarrow
G$ is an automorphism (equivalently, $N=1$ is the only normal subgroup of $G$
for which $G/N\cong G$).

######  Definition 2.7.

[1]. A Moore space of degree $n$ $(n\geq 2)$ is a simply connected
$CW$-complex $X$ with a single non-vanishing homology group of degree $n$,
that is $\tilde{H}_{i}(X,\mathbb{Z})=0$ for $i\neq n$. A Moore space is
denoted by $M(A,n)$ where $A\cong\tilde{H}_{n}(X,\mathbb{Z})$.

Note that for $n=1$, the Moore space $M(A,1)$ can not be defined because of
some problems in existence and uniqueness of the space (for more details see
[4]).

######  Theorem 2.8.

[1]. The homotopy type of a Moore space $M(A,n)$ is uniquely determined by
$A$ and $n$ ($n>1$).

######  Theorem 2.9.

[4]. If a map $f:X\longrightarrow Y$ between connected CW-complexes induces
isomorphisms $f_{*}:\pi_{n}(X)\longrightarrow \pi_{n}(Y)$ for all $n\geq 1$,
then $f$ is a homotopy equivalence.

######  Theorem 2.10.

[4]. A map $f:X\longrightarrow Y$ between simply-connected CW-complexes is a
homotopy equivalence if $f_{*}:H_{n}(X)\longrightarrow H_{n}(Y)$ is an
isomorphism for all $n\geq 2$.

######  Theorem 2.11.

[1]. 1) A connected CW-complex $X$ is contractible if and only if all its
homotopy groups $\pi_{n}(X)$ ($n\geq 1$) are trivial.

2) A simply connected CW-complex $X$ is contractible if and only if all its
homology groups $H_{n}(X)$ ($n\geq 2$) are trivial.

######  Theorem 2.12.

[14]

$H_{p}(\mathbb{CP}^{n})=\begin{cases}\mathbb{Z}&p=0,2,4,\cdots,2n\\\
0&otherwise.\end{cases}$

######  Theorem 2.13.

[3, Theorem 3.15]. Let $(X,E,\Phi)$ be a CW-complex and $(\tilde{X},p)$ be its
universal covering. For each cell $e_{\alpha}$ of $X$, let an specific
characteristic map $\phi_{\alpha}:I^{n}\longrightarrow X$ ($n=n(\alpha)$) and
an specific lift $\tilde{\phi}_{\alpha}:I^{n}\longrightarrow\tilde{X}$ of
$\phi_{\alpha}$ be chosen. Then
$\\{\langle\tilde{\phi}_{\alpha}\rangle\;|\;e_{\alpha}\in X\\}$ is a basis for
$C(\tilde{X})$ as a $\mathbb{Z}\pi_{1}(X)$-module.

##  3 The Capacity of Wedge Sum of Spheres

Borsuk in [9] mentioned that the capacity of $\bigvee_{k}\mathbb{S}^{1}$
equals to $k+1$ and $C(\mathbb{S}^{n})=2$ for $n\geq 1$. The authors in [12]
showed that the capacity of a Moore space $M(A,n)$ equals to the number of
direct summands of $A$, up to isomorphism. Also, we computed the capacity of
the wedge sum of finitely many Moore spaces of different degrees. In
particular, we showed that the capacity of $\bigvee_{n\in
I}(\vee_{i_{n}}\mathbb{S}^{n})$ equals to $\prod_{n\in I}(i_{n}+1)$ where
$\vee_{i_{n}}\mathbb{S}^{n}$ denotes the wedge sum of $i_{n}$ copies of
$\mathbb{S}^{n}$, $I$ is a finite subset of $\mathbb{N}\setminus\\{1\\}$ and
$i_{n}\in\mathbb{N}$. As a special case, the capacity of
$\mathbb{S}^{m}\vee\mathbb{S}^{n}$ $(m,n\geq 2,m\neq n)$ equals to 4.

In this section, we compute the capacity of wedge sum of finitely many spheres
of different dimensions (containing spheres of dimension 1). As a consequence,
we prove that two dimensional CW-complexes with free fundamental groups have
finite capacities.

######  Remark 3.1.

[16]. Suppose that the fundamental group $\pi_{1}(X)$ of a CW-complex $X$ is a
free group of rank $r$. One can find a map $\phi:K\longrightarrow X$ inducing
an isomorphism of fundamental groups, where $K$ is wedge sum of $r$ copies of
$\mathbb{S}^{1}$.

######  Lemma 3.2.

Suppose that the fundamental group $\pi_{1}(X)$ of a CW-complex $X$ is a free
group of rank $r$ and $\phi$ is as in Remark 3.1. Also, assume that
$\pi_{i}(\phi)$ is a free $\mathbb{Z}\pi_{1}(X)$-module of rank $r_{i}$ for
$i\geq 2$ and $H_{k}(\tilde{X})=0$ for $k>n$. Then $X$ has the homotopy type
of wedge sum
$\bigvee_{r}\mathbb{S}^{1}\vee\big{(}\bigvee_{i=2}^{n}(\vee_{r_{i}}\mathbb{S}^{i})\big{)}$
where $\vee_{r_{i}}\mathbb{S}^{i}$ denotes the wedge sum of $r_{i}$ copies of
$\mathbb{S}^{i}$.

###### Proof.

It is easy to see that $\phi$ is 1-connected ($\phi$ induces an isomorphism of
fundamental groups). Using the construction introduced in [16, p. 59] and by
induction on $i$, since by the hypothesis $\pi_{i}(\phi)$ is a free
$\mathbb{Z}\pi_{1}(X)$-module, we can attach finitely many $i$-cells to $K$,
obtaining $K^{(i)}$ say, necessarily with trivial attaching maps to make
$\phi$ $i$-connected. By Theorem 2.9, $K^{(n)}$ is homotopy equivalent to $X$.
By hypothesis $C_{k}(\tilde{X})=H_{k}(\tilde{X})=0$ for $k>n$
($B_{k}(\tilde{X})=0$ because of the shape of $\tilde{X}$) and hence by
Theorem 2.13, $C_{k}(X)=0$ for $k>n$. Thus $K^{(n)}$ is the wedge sum
$\bigvee_{r}\mathbb{S}^{1}\vee\big{(}\bigvee_{i=2}^{n}(\vee_{r_{i}}\mathbb{S}^{i})\big{)}$.
∎

Note that Lemma 3.2 is a generalization of the following result:

######  Theorem 3.3.

[16, Proposition 3.3]. If a CW-complex $X$ satisfies $\mathcal{D}_{2}$ and
$\mathcal{F}_{2}$ and $\pi_{1}(X)$ is a free group, then $X$ has the homotopy
type of a finite bouquet of 1-spheres and 2-shperes.

######  Theorem 3.4.

The capacity of $\bigvee_{n\in I}(\vee_{i_{n}}\mathbb{S}^{n})$ equals to
$\prod_{n\in I}(i_{n}+1)$ where $\vee_{i_{n}}\mathbb{S}^{n}$ denotes the wedge
sum of $i_{n}$ copies of $\mathbb{S}^{n}$, $I$ is a finite subset of
$\mathbb{N}$ and $i_{n}\in\mathbb{N}$.

###### Proof.

If $I\subset\mathbb{N}\setminus\\{1\\}$, the result has been proved by the
authors in [12]. Hence we can suppose that
$X=\bigvee_{r}\mathbb{S}^{1}\vee\big{(}\bigvee_{n\in
I\setminus\\{1\\}}(\vee_{i_{n}}\mathbb{S}^{n})\big{)}$. Assume that $A$ is
homotopy dominated by $X$ and $\pi_{1}(A)$ is of rank $s$ where $0\leq s\leq
r$. By Remark 3.1, one can find a map $\phi:K\longrightarrow A$ induces an
isomorphism of fundamental groups, where $K$ is wedge sum of $s$ copies of
$\mathbb{S}^{1}$ . By Theorems 2.3 and 2.2, we have
$\pi_{n}(\phi)\cong\pi_{n}(\tilde{\phi})\cong
H_{n}(\tilde{A},\tilde{K})\cong\frac{C_{n}(\tilde{A})}{B_{n}(\tilde{A})}$.
Note that $B_{n}(\tilde{A})\leqslant B_{n}(\tilde{X})$ and
$B_{n}(\tilde{X})=0$ since $\tilde{X}$ is a tree with a $\bigvee_{n\in
I\setminus\\{1\\}}(\vee_{i_{n}}\mathbb{S}^{n})$ on each of its vertex, hence
$B_{n}(\tilde{A})=0$. This shows that $\pi_{n}(\phi)\cong C_{n}(\tilde{A})$ as
$\mathbb{Z}\pi_{1}(A)$-module. By Theorem 2.13, the rank of $C_{n}(\tilde{A})$
as $\mathbb{Z}\pi_{1}(A)$-module equals to the rank of $C_{n}(A)$ as
$\mathbb{Z}$-module (which is a submodule of $C_{n}(X)$). Therefore
$\pi_{n}(\phi)$ is a free $\mathbb{Z}\pi_{1}(A)$-module of rank $j_{n}$
($0\leq j_{n}\leq i_{n}$). Hence by Lemma 3.2, $A$ is homotopy equivalent to
$\bigvee_{s}\mathbb{S}^{1}\vee\big{(}\bigvee_{n\in
I\setminus\\{1\\}}(\vee_{j_{n}}\mathbb{S}^{n})\big{)}$. ∎

######  Corollary 3.5.

The capacity of $\mathbb{S}^{1}\vee\mathbb{S}^{n}$ equals to 4, for every
$n\geq 2$.

######  Remark 3.6.

Kolodziejczyk in [6] asked the following question:

“Does every polyhedron P with the abelian fundamental group $\pi_{1}(P)$
dominate only finitely many different homotopy types?”

She proved that two large classes of polyhedra, polyhedra $Q$ with finite
fundamental groups $\pi_{1}(Q)$, and polyhedra $P$ with abelian fundamental
groups and finitely generated homology groups $H_{i}(\tilde{P})$ $(i\geq 2)$,
have finite capacities, where $\tilde{P}$ is the universal covering of $P$
(see [7],[6]). Note that the wedge sum $\mathbb{S}^{1}\vee\mathbb{S}^{2}$ is a
simple example of a polyhedron $P$ with infinite abelian fundamental group
$\pi_{1}(P)$ and infinitely generated homology group
$H_{2}(\tilde{P};\mathbb{Z})$ which is not wedge sum of Moore spaces. So its
capacity can not determined by neither the results of Kolodziejczyk nor the
results of [12].

######  Corollary 3.7.

Every 2-dimensional CW-complex $X$ with free fundamental group $\pi_{1}(X)$
has finite capacity. Moreover, if the ranks of $\pi_{1}(X)$ and $H_{2}(X)$ are
$r$ and $s$, respectively, then the capacity of $X$ equals to
$(r+1)\times(s+1)$.

###### Proof.

It can be concluded from Theorems 2.5, 3.3 and 3.4. ∎

##  4 A Negative Answer to a Question of Borsuk

Borsuk in [2] stated some questions concerning properties of the capacity of
compacta. One of them is as follow:

Is the capacity $C(A)$ determined by the homology properties of $A$?

In this section, we give a negative answer to this question, in general.
However, the authors in [12] showed that the capacity of Moore spaces
determined by their homology properties. In fact, they proved that the
capacity of a Moore space $M(A,n)$ equals to the number of direct summands of
$A$, up to isomorphism.

######  Lemma 4.1.

[4, Example 2.43]. Let $X$ be the Eilenberg-MacLane space
$K(\mathbb{Z}_{m},1)$. Then $H_{n}(X)$ is $\mathbb{Z}_{m}$ for odd $n$ and $0$
for even $n>0$.

######  Lemma 4.2.

The capacity of projective plane $\mathbb{CP}^{2}$ equals to 2.

###### Proof.

Suppose that $A$ is homotopy dominated by $\mathbb{CP}^{2}$. Then $A$ is
simply connected and $H_{n}(A)$ is isomorphic to a direct summand of
$H_{n}(\mathbb{CP}^{2})$. Since $H_{n}(\mathbb{CP}^{2})\cong\mathbb{Z}$ for
$n=0,2,4$ and $H_{n}(\mathbb{CP}^{2})=0$ otherwise, we have the following
cases:

  1. a)

If $H_{2}(A),H_{4}(A)=0$, then by part (2) of Theorem 2.11, $A$ is homotopy
equivalent to a one-point space.

  2. b)

If $H_{2}(A)\cong\mathbb{Z}$ and $H_{4}(A)=0$, then $A$ is the Moore space
$M(\mathbb{Z},2)$ and so, by Theorem 2.8, it is homotopy equivalent to
$\mathbb{S}^{2}$. But $\mathbb{S}^{2}$ can not be homotopy dominated by
$\mathbb{CP}^{2}$. Since if $\mathbb{S}^{2}\leqslant\mathbb{CP}^{2}$, then
$\pi_{3}(\mathbb{S}^{2})\cong\mathbb{Z}$ is isomorphic to a subgroup of
$\pi_{3}(\mathbb{CP}^{2})\cong\pi_{3}(\mathbb{S}^{5})=0$ which is a
contradiction (note that by [14, Exercise 11.47],
$\pi_{q}(\mathbb{CP}^{n})\cong\pi_{q}(\mathbb{S}^{2n+1})$ for all $q\geq 3$).

  3. c)

If $H_{2}(A)=0$ and $H_{4}(A)\cong\mathbb{Z}$, then $A$ is the Moore space
$M(\mathbb{Z},4)$ and so, by Theorem 2.8, it is homotopy equivalent to
$\mathbb{S}^{4}$. Similar to the case (b), $\mathbb{S}^{4}$ can not be
homotopy dominated by $\mathbb{CP}^{2}$ since
$\pi_{4}(\mathbb{S}^{4})\cong\mathbb{Z}$ is not isomorphic to a subgroup of
$\pi_{4}(\mathbb{CP}^{2})\cong\pi_{4}(\mathbb{S}^{5})=0$.

  4. d)

Let $H_{2}(A)\cong\mathbb{Z}$ and $H_{4}(A)\cong\mathbb{Z}$. Suppose
$g:\mathbb{CP}^{2}\longrightarrow A$ is the domination map. It is easy to see
that $g_{*}:H_{n}(\mathbb{CP}^{2})\longrightarrow H_{n}(A)$ is an epimorphism
between two isomorphic Hopfian groups for all $n\geq 2$ which implies that
$g_{*}$ is isomorphism for all $n\geq 2$. Therefore by Theorem 2.10, $g$ is a
homotopy equivalence and hence, $A$ and $\mathbb{CP}^{2}$ have the same
homotopy type.

Thus $A$ can only be homotopy equivalent to either one-point space or
$\mathbb{CP}^{2}$ and so, $C(\mathbb{CP}^{2})=2$. ∎

Now, we are in a position to give a negative answer to the question of Borsuk.

######  Theorem 4.3.

The capacity of a compactum is not determined by its homology properties, in
general.

###### Proof.

Consider $X=\mathbb{S}^{2}\vee\mathbb{S}^{4}$ and $Y=\mathbb{CP}^{2}$. It is
easy to see that $H_{n}(X)\cong H_{n}(Y)$ for all $n\geq 0$ ($H_{n}(X)\cong
H_{n}(Y)\cong\mathbb{Z}$ for $n=0,2,4$ and $H_{n}(X)=H_{n}(Y)=0$ otherwise).
But by Theorem 3.4, $C(X)=4$ and by Lemma 4.2, $C(Y)=2$. ∎

######  Remark 4.4.

Note that the capacity of a compactum is not also determined by its homotopy
groups. Consider $X=\mathbb{S}^{2}$ and $Y=\mathbb{S}^{3}\times
K(\mathbb{Z},2)$. For these simply connected spaces, we have
$\pi_{i}(X)\cong\pi_{i}(Y)$ for all $i\geq 2$, however $C(X)=2$ and $C(Y)\geq
4$ since $\\{*\\}$, $\mathbb{S}^{3}$, $K(\mathbb{Z},2)$ and $Y$ are homotopy
dominated by $Y$ which are not homotopy equivalent to each other.

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# Measuring Dwarf Galaxy Intrinsic Abundance Scatter with Mid-resolution
Spectroscopic Surveys:
Calibrating APOGEE Abundance Errors

Jennifer Mead Department of Astronomy, Columbia University, New York, NY
10027, USA  Melissa Ness Research School of Astronomy & Astrophysics,
Australian National University, Canberra ACT 2611, Australia  Department of
Astronomy, Columbia University, New York, NY 10027, USA  Eric Andersson
Department of Astrophysics, American Museum of Natural History, New York, NY
10024, USA  Emily J. Griffith Center for Astrophysics and Space Astronomy,
Department of Astrophysical and Planetary Sciences, University of Colorado,
389 UCB, Boulder, CO 80309-0389, USA  NSF Astronomy and Astrophysics
Postdoctoral Fellow  Danny Horta Center for Computational Astrophysics,
Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA

###### Abstract

The first generations of stars left their chemical fingerprints on metal-poor
stars in the Milky Way and its surrounding dwarf galaxies. While instantaneous
and homogeneous enrichment implies that groups of co-natal stars should have
the same element abundances, small amplitudes of abundance scatter are seen at
fixed [Fe/H]. Measurements of intrinsic abundance scatter have been made with
small, high-resolution spectroscopic datasets where measurement uncertainty is
small compared to this scatter. In this work, we present a method to use mid-
resolution survey data, which has larger errors, to make this measurement.
Using APOGEE DR17, we calculate the intrinsic scatter of Al, O, Mg, Si, Ti,
Ni, and Mn relative to Fe for 333 metal-poor stars across 6 classical dwarf
galaxies around the Milky Way, and 1604 stars across 19 globular clusters. We
first calibrate the reported abundance errors in bins of signal-to-noise and
[Fe/H] using a high-fidelity halo dataset. We then apply these calibrated
errors to the APOGEE data, and find small amplitudes of average intrinsic
abundance scatter in dwarf galaxies ranging from 0.032 $-$ 0.14 dex with a
median value of 0.043 dex. For the globular clusters, we find intrinsic
scatters ranging from 0.018 $-$ 0.21 dex, with particularly high scatter for
Al and O. Our measurements of intrinsic abundance scatter place important
upper limits on the intrinsic scatter in these systems, as well as constraints
on their underlying star formation history and mixing, that we can look to
simulations to interpret.

Galactic chemical evolution – Chemical abundances – Metal-poor stars –
Intrinsic abundance scatter – Dwarf galaxies – Globular clusters – Stellar
halo

##  1 Introduction

Quenched dwarf galaxies that are bound to and inhabit our stellar halo serve
as laboratories for studying snapshots of the early chemical evolution of the
universe. With average stellar metallicities typically $\rm{[Fe/H]}<-1$ (Kirby
et al., 2008, 2013; Geha et al., 2013; Li et al., 2018; Fu et al., 2023),
their oldest, most metal-poor stars contain the chemical fingerprints of the
very first generations of stars, including Population III (Frebel & Bromm,
2012; Ji et al., 2015; Hartwig et al., 2018, 2019). An understanding of the
contribution to chemical enrichment from Population II stars is also necessary
for placing constraints on the chemical evolution of the early universe. In
all, to decode the signatures of the chemical ancestry of dwarf galaxies, it
is essential to understand the observational effects of the propagation of
elements from various populations of stars and from a variety of
nucleosynthetic sources throughout the ISM (Ji et al., 2015).

The chemical history of a stellar population is observed through measurements
of element abundances, as well as their intrinsic abundance scatter – the true
scatter of the data, accounting for measurement uncertainties – which
necessitates careful consideration of uncertainties on abundances. Previous
studies have uncovered overall element abundance scatter in numerous
abundances, both within and between dwarf galaxies (Hill et al., 2019; Ji et
al., 2020, 2022), but there remains a need to resolve the intrinsic scatters
of chemical abundances above the level of observational errors; a difficult
task for even small high-resolution spectroscopic studies. Mid-resolution
spectroscopic surveys [e.g. APOGEE (Majewski et al., 2017), GALAH (De Silva et
al., 2015), Gaia RVS (Gaia Collaboration et al., 2016, 2023) and LAMOST (Cui
et al., 2012a, b)] afford the community with copious amounts of data, and
target numerous stellar systems. Nonetheless, measurements of the intrinsic
abundance scatter are made even more challenging by larger measurement
uncertainties than higher signal-to-noise (SNR), higher resolution data, as
well as potentially inaccurate estimates of measurement uncertainty in some
parameter spaces. An important question to ask, however, is if we can use
large mid-resolution surveys, to test the the intrinsic scatter of metal-poor
systems that have been targeted and furthermore if we can validate or if
necessary, calibrate, the uncertainty estimates so that these are accurate.

From a theoretical perspective, the intrinsic scatter of element abundances in
any system of stars is dependent on a number of properties. This includes the
efficiency of gas mixing in the ISM (Krumholz & Ting, 2018; Emerick et al.,
2020a; Ji et al., 2023), metal-retention and recycling through inflows and
outflows (Emerick et al., 2018, Mead et al. in prep), stochastic star
formation (Welsh et al., 2021; Pan & Kravtsov, 2023; Griffith et al., 2023),
and mass and metallicity dependence of nucleosynthetic sources (Muley et al.,
2021). The assessment of the scatter of individual elements is one indirect
approach to access these environmental parameters.

The interpretation of the intrinsic scatter for an ensemble of elements is a
powerful link to the details of the formation and enrichment histories of
dwarf galaxies, star clusters, and stellar halos. For example, discoveries in
the last few decades have found multiple populations of stars in globular
clusters, and thus larger intrinsic scatter in some abundance dimensions
(Bastian & Lardo, 2018, and references therein), hinting at a more complex
formation pathway than previously thought. Similarly, in dwarf galaxies the
magnitude of the intrinsic scatter provides clues to the mixing of chemical
yields and stochasticity of star formation, particularly in the early
Universe. In the halo, which is thought to be composed of globular clusters
and disrupted dwarfs, the intrinsic scatter may provide information on where
the stellar populations of the halo originated from.

Because different nucleosynthetic channels contribute to the enrichment of
different elements, intrinsic scatter is also a metric by which we can study
the contribution of different nucleosynthetic sources to the chemical
evolution of different stellar systems, as well as the mass and metallicity
dependence of yields. For example, light elements such as C, N, and F are
primarily produced by low and intermediate mass stars on the asymptotic giant
branch (AGB) (Karakas, 2010; Kobayashi et al., 2011). The $\alpha$-elements
(e.g. O111O is also produced through the CNO cycle, and so not always
considered an $\alpha$-element, but this contribution is subdominant for low-
metallicity stars., Mg, Si, Ca, Ti) are primarily produced by massive stars
and ejected by core-collapse (Type II) supernovae (SNe) (Timmes et al., 1995;
Kobayashi et al., 2006). About half of Fe-peak elements (e.g. Cr, Mn, Fe, Ni,
Co) are produced in Type Ia supernovae (Kobayashi & Nomoto, 2009). The well-
known [$\alpha$/Fe]-[Fe/H] relationship, which shows the delay in enrichment
in Fe from Type Ia SNe versus the short-timescale production of
$\alpha$-elements by Type II SNe, can be used to characterize the star
formation rates in galaxies (e.g. Tinsley, 1980; Matteucci & Greggio, 1986;
Hayden et al., 2015).

Other elements, such as the odd-Z elements (e.g. Al, P) and neutron-capture
elements (s- and r-process elements) depend on the excess of neutrons
available. Odd-Z elements depend on the metallicity of the progenitor star as
their formation requires a surplus of neutrons made from 22Ne during He-
burning (Kobayashi et al., 2011). Strong s-process elements (Sr to Pb) are
produced in low-mass AGB stars (Busso et al., 1999; Herwig, 2005; Karakas &
Lattanzio, 2014), whereas weak s-process elements (between Fe and Sr) are
produced in massive stars near solar metallicity (Pignatari et al., 2010). The
formation mechanism for r-process elements is debated but neutron star mergers
are a likely scenario (Lattimer & Schramm, 1974; Rosswog et al., 1999; Goriely
et al., 2011; Wanajo et al., 2014).

Kobayashi et al. (2020) developed galactic chemical evolution models to place
constraints on the origin of elements as a function of time and galactic
environment. Models such as this, along with an understanding of gas mixing,
the ratio of nucleosynthetic sources, and stochasticity of star formation can
be used to predict the origin and amount of scatter in dwarf galaxy
observations. High-resolution star-by-star models with individual stellar
feedback (e.g. Brauer et al. in prep; Andersson et al. in prep) are uniquely
poised to explore these physical drivers in the context of intrinsic abundance
scatter.

Figure 1: Measurements of element abundances relative to Fe for stars in our
sample with SNR $\geq$ 25\. Points are colored by $\rm{T_{eff}}$ to
demonstrate there are no systematic trends in [X/Fe]. The pink line is the
best fit second order polynomial that minimizes the residuals with respect to
the data. The blue shaded region represents 1$\sigma$ around the mean in bins
of 0.5 dex in [Fe/H] from -2.5 to -1.

Measurements of the intrinsic scatter have been successfully made for
different systems and structures. Intrinsic abundance scatters have been
calculated in both the globular cluster systems of the stellar halo, and open
cluster systems in the disk. Metal-rich open star clusters, which are thought
to be the building blocks of the disk, and born together from the same
molecular cloud, are near chemically homogeneous. Studies measure small
amplitudes of open cluster element abundance intrinsic scatter, on the order
of $<$ 0.03 dex (e.g. Bovy, 2016; Ness et al., 2018; Poovelil et al., 2020).
In the disk itself, scatter in [O/Fe] has been shown to be 0.03-0.04 dex in
the high- and low-$\alpha$ disk (Bertran de Lis et al., 2016). Several studies
also identify non-zero intrinsic abundance scatters in globular clusters, in
particular, this is most significant for light elements (for example, He has a
scatter of up to 0.1 dex) (Milone et al., 2018; Mészáros et al., 2020), but is
also seen to a lesser extent in heavy elements (Gratton, 2020).

In the solar neighborhood, Griffith et al. (2023, hereafter G23) measured
element abundances and intrinsic abundance scatters for metal-poor stars for
12 elements ranging from 0.04 for [Cr/Fe] to 0.16 for [Na/Fe] dex. Because of
their low metallicity, these stars are likely field halo stars. They notably
show that these scatters can be produced by N$\sim$50 core-collapse
supernovae, which is a number too small to accurately represent an IMF.

Quantifying the scatter in co-natal (similar [Fe/H]) stellar populations
serves as a baseline for understanding the temporal, chemical, and mechanical
assembly of these systems. In the case of dwarf galaxies, this accesses the
earliest modes of star formation, as well as the earliest epochs of Milky Way
assembly. For stellar clusters, this quantification has implications for the
feasibility of chemical tagging (Freeman & Bland-Hawthorn, 2002), or the use
of chemical abundance patterns to identify common birth sites of stars.

Figure 2: Same as Figure 1 with best fit line subtracted to remove scatter
dependence on [Fe/H].
$\rm{[X/Fe]^{*}}=\rm{[X/Fe]}-\rm{\langle{[X/Fe]}\rangle}$.

In our study, we use the SDSS (Blanton et al., 2017) Apache Point Observatory
Galactic Evolution Experiment (APOGEE) survey (Majewski et al., 2017) DR17
(Abdurro’uf et al., 2022) to study the intrinsic abundance scatter of metal-
poor ([Fe/H] $\leq$ -1) stars in dwarf galaxies surrounding the Milky Way.
Previous works have studied the abundances of the Milky Way satellites with
APOGEE data (e.g. Hasselquist et al., 2021; Fernandes et al., 2023). The
APOGEE survey has IR spectra (R=22,500) for nearly 734,000 stars in and around
the Milky Way, including 7 targeted dwarf spheroidal galaxies (dSph): Draco,
Ursa Minor, Bootes I, Sextans, Fornax, Sculptor, and Carina (total 718 stars).
The spectra were fit for stellar parameters and chemical abundances using the
APOGEE Stellar Parameter and Chemical Abundances Pipeline (ASPCAP) (García
Pérez et al., 2016). In this work, we focus on elements from the $\alpha$, Fe-
peak, and light nucleosynthetic families.

The layout of this paper is as follows: in Section 2 we discuss the sample of
APOGEE data that we use, along with checks applied to the errors and the
spectra themselves to estimate the uncertainty of their quality. In Section 3,
we present a method for calibrating the uncertainties on the APOGEE abundance
data using intrinsic abundance scatters derived from high fidelity data.
Section 4 contains measurements and results of intrinsic abundance scatter
using the calibrated APOGEE abundance uncertainties. Lastly, we conclude in
Section 5 and discuss the implications of our results.

##  2 Data

In our work, we use APOGEE DR17 to study metal-poor red giant stars in three
types of objects: dSphs, globular clusters (GC), and the Milky Way stellar
halo.

###  2.1 Data Cuts

For all objects, we implement a metallicity cut of $-2.1\leq$ [Fe/H] $\leq-1$
to obtain the metal-poor stars whose scatter we are interested in. For all but
the halo stars, we identify likely members of each system using the assigned
APOGEE MEMBERFLAG. Halo stars are identified as metal-poor stars with distance
$d\leq 2500\rm{pc}$. As in G23, these stars are from the solar neighborhood,
but due to their low-metallicity, are likely halo stars. We remove stars in
known substructure including Gaia-Sausage-Enceladus (GSE) (Horta et al.,
2023a) and globular clusters (Schiavon et al., 2024) from this sample. We also
tested a stricter cut on GSE stars by eliminating high eccentricity (e $>$
0.7) stars, but this had no effect on our results.

To ensure the quality of the data, we make additional data cuts. We remove all
stars which have been flagged with warnings generated by ASPCAP for their
stellar parameters (STAR_BAD) or abundance measurements (M_H_BAD; ELEMFLAG).
To exclude very low-quality spectra, we only use stars with a $\rm{S/N}\geq
25$ for our analysis, with the final data cut for dSphs having a median
$\rm{S/N}=61.22$. Lastly, we only include stars with $4000\leq\rm{T_{eff}}\leq
5000K$ (median $\rm{T_{eff}}=4434\,\rm{K}$ for dSphs) and
$0.0\leq\rm{log}g\leq 3.0$ (median $\rm{log}g=0.99$ for dSphs), keeping the
range wide enough to contain as much of the data as possible but narrow enough
to avoid systematic variations of these quantities with [X/Fe] (e.g. Griffith
et al., 2021; Weinberg et al., 2022). Table 3 in the Appendix includes a
summary of the resulting number of stars in our sample for each object we
study. Because Bootes I has so few metal-poor stars at the required SNR, we
exclude Bootes I from analysis.

###  2.2 Data Summary

We use chemical abundances derived by ASPCAP for the stars in our sample
(García Pérez et al., 2016). We focus on the following elements: Al (light
elements); O, Mg, Si, Ti ($\alpha$-elements); and Ni and Mn (Fe-peak
elements). Figure 1 shows the abundance [X/Fe] and metallicity [Fe/H]
measurements for stars in our dSph sample. Data points in Figure 1 are colored
by $\rm{T_{eff}}$ to demonstrate that there are no trends in [X/Fe] with
$\rm{T_{eff}}$ and that measured scatter is independent of stellar parameters.
Generally, we see negative trends in [X/Fe] with [Fe/H]. As we are interested
in processes that influence scatter in abundances for stars at the same
metallicity, or in other words, the scatter around the trend line, we fit and
subtract a second order model to the data in the [Fe/H]-[X/Fe] plane. In
Figure 2, we show the resulting residuals for the dSphs, revealing the scatter
of our sample with [Fe/H]. We similarly do this for the Milky Way stellar halo
and GC stars in our sample. This trend-subtracted data is used in all further
analysis.

Figure 3: Example line windows for star 2M02383346-3428246 in Fornax. The
combined spectrum is shown in black with the ASPCAP model in red. The line
window is indicated by the green shading.

###  2.3 Spectra Quality Check

Metal-poor stars have weak metal lines which make abundance measurements
uncertain. Therefore, we directly examine the spectra of stars in our sample
to assess the quality of the ASPCAP fit for spectral lines of elements we are
using. We use the strongest spectral lines from the line list (Smith et al.,
2021) used in the APOGEE abundance pipeline. Ultimately, we find that Al, O,
Mg, Si, Ti222Ti is labeled as ‘deviant’ from literature expectations in DR17,
however visual inspection shows the lines are well-fit for the stars in our
sample., and Ni show absorption features, which are generally well-fit by the
model. We conclude that it is reasonable to proceed with our analysis using
these abundance measurements. We also include Mn in our analysis, but note
that these lines were not as well-fit, and that these abundance measurements
should be treated with caution and assumed to be upper limits. Figure 3 shows
a typical example of ASPCAP line fits for the stars and elements included in
our sample. In addition to a visual inspection, we implement a quality cut
whereby we calculate the $\chi^{2}$ for the ASPCAP best fit model (i.e.
$\chi^{2}=\Sigma\frac{(\rm{x-y})^{2}}{\rm{\delta x}^{2}}$) and remove
individual abundance measurements with a reduced $\chi^{2}>6$ from
consideration in the analysis.

##  3 Calibration of APOGEE Abundance Uncertainties

There are two sources of variability in our abundance measurements that we
define: $\delta x$, the internal measurement uncertainty, and $s$, the
intrinsic scatter. The intrinsic scatter of a data set quantifies the "true"
scatter among observed stars generated by physical processes. Equation 1 is a
basic calculation of intrinsic scatter that assumes the data are Gaussian with
standard deviation $\sigma$ and have the same uncertainties, $\delta x$. From
Equation 1, the inverse relationship between intrinsic scatter and measured
error is apparent.

$s=\sqrt{\sigma^{2}-\overline{\delta x}^{2}}$ (1)

We perform our analysis taking into account each star’s individual errors
using a maximum likelihood approach to estimate the mean abundance of each
element, $\bar{x_{i}}$, and intrinsic scatter, $s_{i}$, of the stars in each
object given their individual measurement uncertainties, $\delta x_{i,n}$. The
joint probability function (Walker et al., 2006; Ness et al., 2018; Li et al.,
2018) is:

$P(x_{i,n}^{o}|\bar{x_{i}},s_{i},\delta x_{i,n})=\\\
\prod_{n=1}^{N}1/\sqrt{2\pi(\delta
x^{2}_{i,n}+s^{2}_{i})}\cdot\rm{exp}(-\frac{(x^{o}_{i,n}-\bar{x_{i}})^{2}}{2(\delta
x^{2}_{i,n}+s^{2}_{i})})$ (2)

We use the log of the probability function to determine the posterior
distribution of the mean and intrinsic scatter using emcee (Foreman-Mackey et
al., 2013). Values for the intrinsic scatter are cited as the mean of the
posterior distribution with errors that are the 1$\sigma$ standard deviation
of the distribution. All MCMC calculations were required to have a minimum of
3 stars in the sample.

Figure 4: Comparison of the average intrinsic scatters and 1$\sigma$
dispersions of uncalibrated APOGEE dSphs (black) to the average intrinsic
scatter of dSphs from Kirby et al. (2018) (K18) (pink). Intrinsic scatter of
individual dSphs in APOGEE are in purple. Average APOGEE intrinsic scatters
are consistently higher than the K18 intrinsic scatters for elements reported
in K18.

###  3.1 APOGEE Data Intrinsic Scatter

In Figure 4 we report intrinsic scatters for individual APOGEE dSphs (purple)
using the reported APOGEE abundance uncertainties (henceforth referred to as
uncalibrated data). These range from 0.08 to 0.14 for [Al/Fe], 0.09 to 0.19
for [O/Fe], 0.08 to 0.14 for [Mg/Fe], 0.11 to 0.18 for [Si/Fe], 0.11 to 0.22
for [Ti/Fe], 0.11 to 0.18 for [Ni/Fe], and 0.12 to 0.22 for [Mn/Fe]. For each
abundance ratio (in black), we also report the average, uncalibrated,
intrinsic abundance scatter of all dSphs and the errorbars are the 1$\sigma$
standard deviation on the mean intrinsic scatters of individual dSphs. The
average scatter ranges from a minimum of 0.11 dex for [Mg/Fe] to a maximum of
0.16 dex for [Mn/Fe].

APOGEE has errors on the level of intrinsic scatter reported in other
literature, therefore we compare the average scatters calculated with the
APOGEE data used in our work to average scatters of other data for dwarfs in
the literature, namely, the Kirby et al. (2018, hereafter K18) dataset which
carefully accounts for the random and systematic uncertainties. From this data
set, we only consider dSphs with at least 10 observed stars. This dataset
overlaps partially with the APOGEE dataset (Draco, Fornax, Sculptor, Sextans,
and Ursa Minor) and includes 3 additional dSphs (Canes Venatici I, Leo I, and
Leo II). Figure 4 also shows that the uncalibrated data for the dSphs in the
APOGEE dataset have higher average intrinsic scatter than the average for
those in K18. This difference is substantial, and ranges from 0.02 dex to 0.1
dex.

Similarly high intrinsic abundance scatters can be shown for field stars in
the APOGEE halo data when compared to intrinsic abundance scatters calculated
in the same way from G23 data. In Figure 5, we report the intrinsic abundance
scatters in the halo for the same elements. These range from a minimum of 0.09
dex for [Ni/Fe] to a maximum of 0.20 dex for [Al/Fe]). We also show that these
values range from 0.02 up to 0.09 dex higher than intrinsic scatters for halo
stars in G23. These differences call into question the accuracy of the APOGEE
uncertainty estimates for the metal-poor stars.

Figure 5: Comparison of intrinsic scatters of uncalibrated APOGEE halo data to
intrinsic scatters of data from G23 recalculated using Equation 2. The
recalculated G23 scatters are consistent with those reported in G23.

###  3.2 Uncertainty Estimates and Error Calibration

####  3.2.1 Reported APOGEE Uncertainties

ASPCAP uses FERRE (García Pérez et al., 2016), a $\chi^{2}$ minimization
optimization algorithm, to compare the APOGEE observed spectrum to a grid of
synthetic spectra that uses the flux errors computed during data reduction as
the weights, and identify the atmospheric parameters and abundances that yield
the best fit spectrum. A first pass of FERRE derives the atmospheric
parameters of each star, [$\alpha$/Fe], and [Fe/H], and then a second pass
treats the values of these parameters as fixed and derives the abundance
measurements.

Uncertainties on the derived parameters are estimated from the inverse of the
curvature matrix which depends on the flux error and partial derivatives of
the synthetic spectra with the different parameters estimated. These
uncertainties are likely underestimated. Indeed, estimates of abundance
scatter in star clusters, yield substantially larger errors than the internal
FERRE uncertainty estimates (Holtzman et al., 2015). These uncertainties were
adjusted in recent data releases using repeat observations, but abundance
measurements may still have systematic errors for metal-poor stars as the data
is sparse in this regime and at low SNR (Jönsson et al., 2020). Repeat
observations provide an empirical estimate of uncertainty as a function of
Teff, [M/H] and SNR (Holtzman et al., in prep). These empirically derived
uncertainties are reported for each abundance in the data release table which
we used to calculate the intrinsic scatters in Figures 4 and 5.

As it is unlikely that the intrinsic scatter the APOGEE element abundances are
substantially larger than those of K18 as shown in Figure 4, we conclude that
underestimated errors in APOGEE are the source of inflated $s$ as would be the
result of Equation 1. The importance of having accurate errors in calculating
a meaningful intrinsic scatter motivates us to test and calibrate the
uncertainty estimates reported for our stars using an independent approach we
develop for this work.

####  3.2.2 High-Fidelity G23 Data Sample

We use the high-fidelity data presented in G23 to compare to and subsequently
calibrate the APOGEE halo element abundance intrinsic scatters. The G23 sample
contains abundance measurements of 12 elements (Na, Mg, Si, Ca, Sc, Ti, V, Cr,
Mn, Fe, Co, Ni) for 86 metal-poor ($-2.1\leq\rm{[Fe/H]}\leq-1$) subgiants in
the solar neighborhood ($d\leq 2500\,\rm{pc}$) observed with the Potsdam
Echelle Polarimetric and Spectroscopic Instrument (PEPSI) (Strassmeier et al.,
2015) on the Large Binocular Telescope (LBT). Spectra were obtained with
resolution $R=50,000$, and $S/N>125$ and $S/N>236$ respectively in the
wavelength ranges of 4260–4800 and 5440–6270 Å.

The primary goal of the survey was to measure the intrinsic abundance scatter.
To do so, they selected a stellar sample with tight Teff and logg constraints
to minimize systematic abundance trends and undertook a careful accounting of
measurement errors. G23 assume that their data has negligible differential
systematic scatter, and account only for the photon-noise scatter. Lacking
abundances measurements from repeat observations of the same star, they use
the error spectrum calculated in the PEPSI reduction pipeline to simulate 10
synthetic spectra for each star. They vary the flux at each wavelength by an
amount drawn from a Gaussian with a standard deviation equivalent to the
error. The new realizations of the spectra are used to recalculate the
abundances and the standard deviation is adopted as the photon-noise. G23 find
that the median photon-noise uncertainty is $\rm{\sigma_{phot,med}}<0.04$ dex.
They also consider NLTE effects, but conclude that while the shape of the
abundance pattern is influenced by NTLE effects, the abundance scatter is not.

Table 1: Number stars in each calibration bin. | -2.1 < [Fe/H] < -1.55 | -1.55 < [Fe/H] < -1
---|---|---
25 < S/N < 125 | 21 | 84
125 < S/N < 225 | 35 | 134
225 < S/N < 325 | 27 | 109
325 < S/N < 425 | 17 | 52
425 < S/N | 36 | 113
Figure 6: Added error terms for all elemental abundances, binned by SNR and
[Fe/H]. Within each elemental abundance, added error terms generally decrease
with increase SNR and [Fe/H].

G23 find intrinsic scatters ranging from $0.04-0.08$ dex in [X/Fe] for element
abundances in our set (Mg, Si, Ti, Ni, Mn). This is greater than the photon-
noise uncertainty by a factor of 2 for most elements. Overall, the errors that
G23 calculate for their data range from an average of 0.048 dex on [Si/Fe] to
0.068 dex on [Ti/Fe]. Because the uncertainties are significantly smaller
compared to the scatter, small inaccuracies in the abundance uncertainty will
have a small impact on the intrinsic scatter. Therefore, the method provides
robust upper limits on the intrinsic scatter of the element abundances for
this population.

To be consistent with the calculated APOGEE intrinsic scatters, we use
Equation 2 to calculate the intrinsic scatter of the G23 data for the 86 stars
in their sample, showing the result in Figure 5. We report a range of
intrinsic scatters in [X/Fe] of $0.04-0.08$ dex for elements that overlap with
APOGEE (Mg, Si, Ti, Ni, Mn). The calculated intrinsic scatters for G23 are
within (Mg, Si, Ti) or slightly lower (Ni, Mn) than the reported intrinsic
scatters relative to the two- and three-parameter models reported in Table 4
of G23.

####  3.2.3 Calibration of APOGEE Uncertainties

To accurately calibrate the APOGEE uncertainties, we construct a comparable
sample of halo stars from APOGEE as described in Section 2, and calculate the
intrinsic scatter to compare to the G23 intrinsic scatter. Figure 5 highlights
the discrepancy between the intrinsic scatter of a comparable APOGEE halo
sample and the G23 sample.

Due to the robustness of intrinsic scatter from the G23 data, we take these
measurements as the "ground truth" intrinsic scatter of halo field stars and
calibrate the uncertainties on the APOGEE data such that the calculated
intrinsic scatter of the APOGEE halo data matches that of G23. In practice, we
calculate an additional term, $a$, every abundance and add it in quadrature to
the reported uncertainty on abundance measurements in APOGEE.

$P(x_{i,n}^{o}|\bar{x_{i}},s_{i},\delta x_{i,n})=\\\
\prod_{n=1}^{N}1/\sqrt{2\pi(\delta
x^{2}_{i,n}+a^{2}+s^{2}_{i})}\cdot\rm{exp}(-\frac{(x^{o}_{i,n}-\bar{x_{i}})^{2}}{2(\delta
x^{2}_{i,n}+a^{2}+s^{2}_{i})})$ (3)

Using equation 3 we fix the intrinsic scatter, $s$, to the calculated value
from the G23 data and maximize the likelihood over $a$. We bin over the APOGEE
halo data in [Fe/H] and SNR as we expect that the total uncertainties are
primarily dependent on these quantities. Due to the small number of high SNR
stars at low metallicity, we divide our [Fe/H] range into two bins from $-2.1$
dex to $-1.55$ dex and $-1.55$ dex to $-1$ dex. We then bin the SNR in ranges
of 100, from an SNR of 25 up to 425. The final bin contains all stars with SNR
higher than 425 as there are very few stars in this range and SNRs beyond this
range are thought to be unphysically high due to persistence effects (Holtzman
et al., 2018), though the data does still have very high SNR. We note that
finer binning would result in more precise added error terms on the data, but
that we are limited by the APOGEE halo sample size. The number of stars in
each bin is listed in Table 1.

We extrapolate added error terms for the remaining abundances not in the G23
dataset as all the abundances in our sample have approximately the same
average uncertainty, enabling us model the added error terms for abundances
for which we calculate $a$ as a Gaussian. The extrapolated terms are the mean
of this Gaussian distribution333We also tested a uniform model using the
maximum and minimum as the bounds on the range. The difference between models
was insignificant.. Our results for $a$ are reported in Table 2.

Table 2: Average added error terms, $a$, for each S/N and [Fe/H] bin for each abundance. Abundances for which $a$ was extrapolated are in italics. |  | 25 < S/N < 125 | 125 < S/N < 225 | 225 < S/N < 325 | 325 < S/N < 425 | 425 < S/N
---|---|---|---|---|---|---
[Al/Fe], [O/Fe] | -2.1 < [Fe/H] < -1.55 | 0.17 | 0.14 | 0.16 | 0.09 | 0.08
-1.55 < [Fe/H] < -1 | 0.22 | 0.10 | 0.03 | 0.03 | 0.04
[Mg/Fe] | -2.1 < [Fe/H] < -1.55 | 0.33 | 0.09 | 0.00 | 0.00 | 0.00
-1.55 < [Fe/H] < -1 | 0.24 | 0.10 | 0.00 | 0.00 | 0.00
[Si/Fe] | -2.1 < [Fe/H] < -1.55 | 0.27 | 0.18 | 0.19 | 0.00 | 0.00
-1.55 < [Fe/H] < -1 | 0.25 | 0.09 | 0.00 | 0.00 | 0.06
[Ti/Fe] | -2.1 < [Fe/H] < -1.55 | 0.12 | 0.11 | 0.29 | 0.27 | 0.19
-1.55 < [Fe/H] < -1 | 0.20 | 0.15 | 0.05 | 0.00 | 0.06
[Ni/Fe] | -2.1 < [Fe/H] < -1.55 | 0.01 | 0.21 | 0.14 | 0.10 | 0.07
-1.55 < [Fe/H] < -1 | 0.17 | 0.07 | 0.00 | 0.00 | 0.00
[Mn/Fe] | -2.1 < [Fe/H] < -1.55 | 0.13 | 0.11 | 0.17 | 0.05 | 0.13
-1.55 < [Fe/H] < -1 | 0.22 | 0.09 | 0.12 | 0.12 | 0.08

Figure 6 shows the added error terms for all abundances explored with the
APOGEE data. There is a clear trend of decreasing added error terms with
higher SNR and [Fe/H]. This is expected as higher SNR and higher metallicity
data have stronger metal lines. Furthermore, we note that the size of the
added error terms at low SNR are on the order of the uncalibrated intrinsic
scatter. This implies the intrinsic scatters are very sensitive to the
estimated uncertainty which is a limitation of the data quality, and that the
intrinsic scatters reported in this study are upper bounds. As we construct
the G23 halo and APOGEE samples to be drawn from nearly the same underlying
population, this is a robust data-driven method to obtain accurate
uncertainties that we can propagate for our analysis of the dSph systems and
provides a benchmark for comparison for further studies.

To obtain calibrated uncertainties on the data, the added error terms are
applied in quadrature to the reported uncertainty from ASPCAP for each element
by drawing from a random Gaussian distribution defined by the posteriors on
the added error. The joint probability is again maximized for the intrinsic
scatter as in equation 2. Intrinsic scatters for the APOGEE data reported from
this point forward use this calibrated data.

##  4 Intrinsic Scatter from Calibrated APOGEE Data

In this section, we apply the derived added error terms for the individual
abundances to the APOGEE data for the Milky Way stellar halo, dSphs, and Milky
Way globular clusters, and compare the calibrated and uncalibrated intrinsic
scatters. A table of the calibrated intrinsic scatters as well as object
properties is provided in the Appendix.

###  4.1 Milky Way Stellar Halo

To validate our calibrated uncertainty estimates, we first compare the
calibrated intrinsic scatter of the APOGEE halo sample to the G23 halo sample
as shown in Figure 8. Calibrated intrinsic scatters for all abundances
decrease by at least 0.02 dex compared to the uncalibrated intrinsic scatters.
For those abundances overlapping with the G23 data, intrinsic scatters for the
calibrated APOGEE data are approximately the same as the intrinsic scatters
derived from the G23 data, as expected. Specifically, we report intrinsic
scatters of 0.07 dex for [Mg/Fe], 0.06 dex for [Si/Fe], 0.06 dex for [Ti/Fe],
and 0.05 dex for [Ni/Fe], which are all within 0.015 dex of G23. We report an
intrinsic scatter of 0.11 for [Mn/Fe], notably 0.03 dex higher than the G23
value, but this is perhaps due to the added error terms for [Mn/Fe] having
greater variation particularly in the high-metallicity bin. This suggests that
the data may need to be more finely binned to get an accurate estimate of the
added error for each star, which would require a larger sample of low-
metallicity stars. In general, the differences compared to G23 are likely due
to the coarse binning for the added error term. In addition, since the
maximization of the joint probability function accounts for the individual
errors on stars, and the added error terms are not calibrated for individual
stars; this likely introduces some small differences between the intrinsic
scatter in the G23 data and the calibrated APOGEE data. Nevertheless, these
differences are slight and the comparison in Figure 8 validates our procedure.

###  4.2 Milky Way dSphs

With the robustness of our calibration confirmed, we now turn to an analysis
of the dSphs. Figure 7 shows the comparison between the calibrated and
uncalibrated average dSph intrinsic scatter, calibrated intrinsic scatters for
individual dSphs in APOGEE, and the average intrinsic scatters from the K18
sample. The calibrated average intrinsic scatter of dSphs is reduced by up to
0.11 dex overt the uncalibrated data, making the average scatter of dSphs in
the APOGEE data comparable to the average scatter of dSphs in K18.

The data also shows that generally, the dSphs have lower intrinsic scatter and
dispersion in the light and $\alpha$-elements (average $0.04\pm 0.02$ dex)
than in the Fe-peak elements (average $0.11\pm 0.05$ dex). This variation may
arise from the difference in time-scales for core-collapse versus Type Ia SNe.
More common core-collapse SNe occur relatively frequently and on short-
timescales which could result in greater mixing over time and lower impact
from any single event, leading to lower intrinsic scatter. Conversely, less
frequent longer-timescale Type Ia SNe could result in less mixing before star
formation episodes, thereby increasing intrinsic scatter.

###  4.3 Milky Way Globular Clusters

For comparison with the Milky Way stellar halo and dSphs, we apply our method
to the GCs available in APOGEE DR17. These old, gravitationally bound stellar
systems have traditionally been thought to form as a single object, which
should give them small intrinsic abundance scatter; however, this scenario is
made complicated by the presence of multiple stellar populations in most GCs
(Bedin et al., 2004; Bellini et al., 2010; Piotto et al., 2012, 2015; Bastian
& Lardo, 2018; Milone et al., 2018; Milone & Marino, 2022). Furthermore, these
populations are only pronounced in specific elements, typically associated
with high-temperature H-burning (Bastian & Lardo, 2018). Of the abundances
examined in this work, Al and O are the only ones that shows multiple
populations in GCs.

Figure 9 shows the average intrinsic scatters of 1604 stars across 19 GCs
(median 79 stars per GC) that were targeted in APOGEE DR17 (see the Appendix
for a list of GCs). We separately analyze Omega Centauri and M54, as they are
suspected cores of disrupted dwarf galaxies (Neumayer et al., 2020). The GCs
have a median SNR of 157, and [Fe/H] of -1.1. The average intrinsic scatter is
$0.21$ dex for [Al/Fe] and $0.08$ for [O/Fe]; higher than the
$\alpha$-elements which have an average scatter of $0.04$ dex and the Fe-peak
elements which have an average scatter of 0.05 dex. This is consistent with
the multiple stellar populations mentioned previously.

Mészáros et al. (2015) presented a similar study of GCs in APOGEE DR10,
comparing the observed scatter of the abundances with their uncertainty
estimates. Similar to our work, they find significant scatter in [Al/Fe],
attributing this to the light-element multiple populations. Furthermore, they
highlight that the scatter decreases with increasing [Fe/H]. We find similar
trends in [Al/Fe], [Mn/Fe], and possibly [Ni/Fe].

Figure 7: Comparison of average calibrated APOGEE intrinsic scatters for dSphs
(purple circles) with average intrinsic scatter from K18 dSphs (pink stars).
Individual calibrated APOGEE dwarfs are in grey. Purple squares are the
average uncalibrated APOGEE intrinsic scatter for dSphs. The calibrated
scatters are generally comparable to the K18 scatters.

##  5 Discussion and Conclusions

###  5.1 Summary

In this work we introduce a methodology for calibrating abundance
uncertainties derived from mid-resolution surveys by leveraging smaller, high-
fidelity, high-resolution data sets. Data-driven calibration or label-transfer
efforts have previously been employed to garner greater value from low SNR
data (e.g. Casey et al., 2017; Ho et al., 2017; Rice & Brewer, 2020; Xiang et
al., 2021, 2022). However, these typically rely on stars in common between the
surveys. Our approach does not rely on this, but instead utilizes population
statistics. We apply our method to calibrate the measurement errors and
calculate the intrinsic abundance scatter of metal-poor stars for elements Al,
O, Mg, Si, Ti, Ni, and Mn in dSphs, GCs, and the Milky Way stellar halo. Our
calculation relies on the calibration of the reported APOGEE abundance
uncertainties and is specifically for APOGEE DR17 metal-poor red giant stars
($-2.1\leq$[Fe/H]$\leq-1$) and in the temperature range $4000\,\rm{K}\,\leq
T\leq\,5000\,\rm{K}$.

Making measurements of the intrinsic abundance scatter of dSphs, GCs, and the
halo with the APOGEE data, particularly for metal-poor stars, is important for
providing constraints for simulations and linking to physical drivers. Doing
this requires using data at lower metallicities and SNR than the majority of
APOGEE spectra, in a regime where there is little previous work. As the
magnitude of the uncertainties impact the measurement of intrinsic scatter, it
is vital to test the accuracy of the APOGEE reported uncertainty. In doing so,
we found them to be underestimated and using the method introduced in this
paper, we derive the complete uncertainties on the APOGEE data in this regime.

Figure 8: Calibrated intrinsic scatters for the APOGEE halo compared to G23.
Calibrated APOGEE intrinsic scatters are consistent with the recalculated G23
scatters.  Figure 9: Comparison of average APOGEE intrinsic scatters for
globular clusters before (blue squares) and after (blue circles) calibration.
Individual objects are in grey.

To do this, we use a high fidelity reference set of halo stars (G23) to
provide ground truth information about the intrinsic scatter of element
abundances in the stellar halo. We construct a comparable sample of field halo
stars from the APOGEE data using the same distance and metallicity criteria as
G23. Our primary assumption in this work is that the underlying population of
stars in the reference set (G23) is comparable to the halo population in the
calibration set we use from APOGEE. Comparing the intrinsic scatters between
the two datasets, we find that the intrinsic scatters of the uncalibrated
APOGEE halo data are substantially higher than G23, indicating the APOGEE
uncertainties are underestimated. This is unsurprising as the reference
objects used in APOGEE for providing empirical uncertainty estimates are
sparse in the metal-poor and low SNR regime (SNR $\sim$ 50) (Jönsson et al.,
2020). We note that the G23 uncertainties themselves are very small so their
intrinsic scatters, which we take as ground truth, are minimally impacted by
any inaccuracy in these. However, strictly, any underestimation in the G23
uncertainties would result in lower intrinsic scatter, and so these data set
an upper limit on the ground truth intrinsic scatter and therefore a lower
limit on the complete uncertainties for APOGEE.

Figure 10: Comparison of the average and dispersion of the observed intrinsic
scatters between the stellar halo, dSphs, and GCs.

To calibrate the APOGEE uncertainties, we determine the uncertainty on the
APOGEE halo data required to match the intrinsic abundance scatter of the halo
population from G23. We subsequently calculate an added error term for every
abundance as a function of [Fe/H] binned across a range of -2.1 to -1 and SNR
binned across a range of 25 to 1420, which we can apply to stars in this
parameter space.

After calibrating the uncertainties for each of the stars in the dSphs, we
find that on average, dSphs have comparatively low intrinsic scatter in the
light and $\alpha$-elements (on average, 0.04 dex) and higher intrinsic
scatter in the Fe-peak elements (on average, 0.11 dex). We also undertake a
similar analysis for GCs, but show that in contrast to the dSphs, GCs have
high intrinsic scatter in the light elements (Al and O with 0.21 and 0.08 dex
respectively), and no clear pattern in intrinsic scatter among the $\alpha$\-
and Fe-peak elements. This may suggest that for elements ejected through
higher-energy processes, mixing is only efficient on small scales.

###  5.2 Comparison of dSphs, MW Halo, and GCs

Comparing the average intrinsic abundance scatters of dSphs, GCs, and the
stellar halo enables us to set constraints on the origin of scatter and the
formation histories of different structures. Figure 10 shows that the
intrinsic scatter of the halo is more similar to the average scatter of GCs
than it is to the average scatter of dSphs. However, this is not to say that
the halo and GCs are formed of the same populations of stars, rather, that we
distinctly see the signatures of multiple populations in both. This may be an
indicator of the existence of stars from destroyed GCs in the halo.

Additionally, that the classical dSphs have a different chemical composition
than the halo has been shown both in observational (Unavane et al., 1996; Venn
et al., 2004) and simulated data (Robertson et al., 2005; Font et al., 2006;
Johnston et al., 2008; Cunningham et al., 2022; Horta et al., 2023b). Much of
the difference is attributed to the short star formation histories of accreted
dwarfs compared to longer-lived classical dSphs. The higher scatter seen in
the halo could be interpreted as a result of early quenching of star formation
in dwarfs destroyed by the halo, where metals do not have time to mix and end
up in stars before star formation ceases. In comparison, the classical dSphs
have more extended star formation histories which allow for more complete
mixing, resulting in lower intrinsic scatters.

###  5.3 Future Prospects for Observations and Simulations

The ability to account for the complete uncertainties in the metal-poor regime
of survey data is critical in order to fully exploit these data. Large-scale
surveys also offer unique coverage of wider areas of dSphs than smaller
studies, which typically focus on the core. Broader area coverage in general
enables studies of element abundance scatter across the Galaxy, as well as
spatially dependent scatter. Currently, available survey data of dSphs stars
are generally low SNR ($<70$ median in APOGEE), are limited to the brightest
stars, and element abundance measurement uncertainties are nearly as high as
the total abundance scatter. In order to make high-fidelity measurements of
the intrinsic scatter of element abundances of low-metallicity dSph systems,
reasonable sample sizes of high resolution (e.g. R $>$ 50,000), high SNR
spectra (e.g. $>$ 100) are required.

However, our efforts to calibrate large medium-resolution survey data using
small high resolution, high SNR studies makes the most use of the full
parameter space of the available data, as well as provides a better
understanding of the unaccounted uncertainties in the metal-poor regime in
APOGEE, which is relevant to an ensemble of studies. Our results subsequently
serve as a baseline for the next generation of data. High-resolution, high SNR
spectra can be obtained with upcoming 30-meter class telescope programs such
as the Giant Magellan Telescope (GMT), European Extremely Large Telescope
(E-ELT), and Thirty Meter Telescope (TMT). Additionally, 30-meter class
telescopes are expected to reach up to the 25th magnitude in the V-band, or 2
magnitudes deeper than existing observations of resolved stars in dSphs (Ji et
al., 2019). The intrinsic scatter in element abundances as we showcase here
can serve as an important test and comparison for future ELT studies where
measurement uncertainties will undoubtedly be smaller, and will serve as
another source of calibration data for the uncertainty estimates used in this
work.

Intrinsic abundance scatter of dSphs and other galactic systems also provides
important constraints on theoretical simulations and models for studying the
physical origins of abundance scatter and formation histories of dwarf
galaxies, halos, and star clusters. In particular, it constrains the amount of
gas mixing, the stochasticity of star formation, the variability in and ratio
between different nucleosynthetic sources, and the chemical signatures
produced by the first stars. A better understanding of all of these processes
can assist us in interpreting the element abundance scatter we see in metal-
poor stars today. An in-depth analysis of high-resolution simulations will
help us to interpret the trends we have found in the APOGEE data.

Of particular interest are star-by-star models (e.g. Æos, Brauer et al., in
prep.; GRIFFIN, Lahén et al. 2020; INFERNO, Andersson et al. 2023; LYRA,
Gutcke et al. 2021), which allow for detailed chemical yield models linked to
specific stellar types (e.g. Emerick et al., 2020a, b; Lahén et al., 2023).
The details of when and where stars inject enriched material likely plays a
role for abundance features like the intrinsic scatter. Furthermore, star-by-
star models sample an IMF as opposed to treating stars as unresolved mono-age
populations. This latter feature is important in modeling the early universe
to resolve low-mass stellar systems due to the formation of only handfuls of
massive stars that inject enriched material into each halo. Future work with
simulations, including analyzing how supernovae ejecta mix in the interstellar
and intergalactic mediums both spatially and temporally, and how the intrinsic
abundance scatters of stars in each halo evolves as these elements mix, will
be anchored by the measurements we provide.

##  6 Acknowledgments

The authors thank Professor Alexander Ji for extensive discussions on
methodology for calculating intrinsic scatter with the ASPCAP measurements of
APOGEE data, Professors Mary Putman and Kathryn Johnston, and Dr. Emily
Cunningham for discussions about early results, and Drs. Christian Hayes and
Rachael Beaton for their time and insight on the usability of dwarf galaxy
abundances at low metallicities.

J.M. acknowledges support from the National Science Foundation Graduate
Research Fellowship under grant DGE-2036197. E.A. acknowledges support from
the US National Science Foundation under grants AST18-15461 and AST23-07950.
E.J.G. is supported by an NSF Astronomy and Astrophysics Postdoctoral
Fellowship under award AST-2202135.

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Table 3: (1) Object Name; (2) Total number of stars that meet initial data cuts; (3) V-band Magnitude555From McConnachie (2012) for dSphs and Harris (1996) (2010 edition) for GCs; (4) Average [Fe/H] of stars in cut; (5) Median SNR of stars in cut; (6-12) Intrinsic scatter $\pm$ standard deviation of [X/Fe] for stars in cut. Object | Stars | $M_{V}$ | $\overline{[Fe/H]}$ | S/N | $\sigma_{[Al/Fe]}$ | $\sigma_{[O/Fe]}$ | $\sigma_{[Mg/Fe]}$ | $\sigma_{[Si/Fe]}$ | $\sigma_{[Ti/Fe]}$ | $\sigma_{[Ni/Fe]}$ | $\sigma_{[Mn/Fe]}$
---|---|---|---|---|---|---|---|---|---|---|---
halo |  |  |  |  |  |  |  |  |  |  |
MW Halo | 492 | N/A | 1.33 | 251.4 | $0.18\pm 0.01$ | $0.12\pm 0.01$ | $0.07\pm 0.00$ | $0.06\pm 0.00$ | $0.06\pm 0.01$ | $0.05\pm 0.00$ | $0.11\pm 0.01$
dSph |  |  |  |  |  |  |  |  |  |  |
Draco | 31 | -8.8 | -1.74 | 78.4 | $0.05\pm 0.04$ | $0.06\pm 0.04$ | $0.10\pm 0.06$ | $0.05\pm 0.04$ | $0.06\pm 0.05$ | $0.18\pm 0.04$ | $0.07\pm 0.05$
Ursa Minor | 40 | -8.8 | -1.84 | 48.9 | $0.05\pm 0.04$ | $0.09\pm 0.05$ | $0.05\pm 0.03$ | $0.05\pm 0.04$ | $0.06\pm 0.05$ | $0.19\pm 0.04$ | $0.19\pm 0.09$
Sextans | 27 | -9.3 | -1.74 | 54.3 | $0.05\pm 0.04$ | $0.06\pm 0.05$ | $0.05\pm 0.04$ | $0.06\pm 0.04$ | $0.11\pm 0.08$ | $0.12\pm 0.04$ | $0.09\pm 0.07$
Fornax | 62 | -13.4 | -1.28 | 57.1 | $0.04\pm 0.03$ | $0.03\pm 0.02$ | $0.03\pm 0.02$ | $0.03\pm 0.02$ | $0.05\pm 0.03$ | $0.12\pm 0.02$ | $0.03\pm 0.02$
Sculptor | 126 | -11.1 | -1.57 | 57.5 | $0.04\pm 0.02$ | $0.02\pm 0.02$ | $0.02\pm 0.02$ | $0.02\pm 0.02$ | $0.04\pm 0.03$ | $0.15\pm 0.02$ | $0.15\pm 0.03$
Carina | 47 | -9.1 | -1.59 | 68.6 | $0.05\pm 0.04$ | $0.04\pm 0.03$ | $0.04\pm 0.03$ | $0.04\pm 0.03$ | $0.04\pm 0.03$ | $0.11\pm 0.02$ | $0.04\pm 0.03$
Undetermined |  |  |  |  |  |  |  |  |  |  |
OmegaCen | 1211 | -10.26 | -1.63 | 153.0 | $0.42\pm 0.01$ | $0.18\pm 0.01$ | $0.18\pm 0.00$ | $0.04\pm 0.00$ | $0.08\pm 0.01$ | $0.06\pm 0.00$ | $0.15\pm 0.01$
M54 | 40 | -9.98 | -1.46 | 88.6 | $0.30\pm 0.04$ | $0.06\pm 0.02$ | $0.07\pm 0.02$ | $0.03\pm 0.02$ | $0.04\pm 0.03$ | $0.02\pm 0.02$ | $0.03\pm 0.02$
GC |  |  |  |  |  |  |  |  |  |  |
M53 | 34 | -8.71 | -1.89 | 131.9 | $0.40\pm 0.06$ | $0.03\pm 0.03$ | $0.07\pm 0.02$ | $0.02\pm 0.02$ | $0.05\pm 0.04$ | $0.11\pm 0.03$ | $0.20\pm 0.06$
N5466 | 14 | -6.98 | -1.81 | 71.2 | $0.22\pm 0.10$ | $0.06\pm 0.05$ | $0.06\pm 0.05$ | $0.06\pm 0.05$ | $0.08\pm 0.06$ | $0.20\pm 0.06$ | $0.07\pm 0.06$
M2 | 28 | -9.03 | -1.48 | 137.7 | $0.26\pm 0.04$ | $0.06\pm 0.02$ | $0.03\pm 0.02$ | $0.02\pm 0.01$ | $0.03\pm 0.02$ | $0.02\pm 0.02$ | $0.03\pm 0.02$
M13 | 90 | -8.55 | -1.48 | 134.4 | $0.33\pm 0.03$ | $0.10\pm 0.02$ | $0.08\pm 0.01$ | $0.03\pm 0.01$ | $0.03\pm 0.02$ | $0.04\pm 0.02$ | $0.12\pm 0.02$
M3 | 236 | -8.88 | -1.42 | 148.0 | $0.29\pm 0.01$ | $0.08\pm 0.01$ | $0.06\pm 0.01$ | $0.03\pm 0.01$ | $0.25\pm 0.02$ | $0.04\pm 0.01$ | $0.11\pm 0.01$
M5 | 147 | -8.81 | -1.22 | 166.3 | $0.23\pm 0.02$ | $0.15\pm 0.01$ | $0.05\pm 0.01$ | $0.02\pm 0.01$ | $0.01\pm 0.01$ | $0.04\pm 0.01$ | $0.05\pm 0.02$
M12 | 75 | -7.31 | -1.28 | 125.5 | $0.09\pm 0.01$ | $0.04\pm 0.01$ | $0.04\pm 0.01$ | $0.01\pm 0.01$ | $0.02\pm 0.02$ | $0.01\pm 0.01$ | $0.13\pm 0.02$
N6397 | 56 | -6.64 | -2.00 | 360.4 | $0.40\pm 0.04$ | $0.08\pm 0.02$ | $0.05\pm 0.01$ | $0.01\pm 0.01$ | $0.05\pm 0.04$ | $0.02\pm 0.01$ | $0.07\pm 0.04$
M55 | 83 | -7.57 | -1.77 | 181.3 | $0.28\pm 0.03$ | $0.03\pm 0.02$ | $0.02\pm 0.01$ | $0.01\pm 0.01$ | $0.03\pm 0.02$ | $0.03\pm 0.02$ | $0.06\pm 0.03$
M22 | 231 | -8.50 | -1.69 | 140.1 | $0.29\pm 0.02$ | $0.09\pm 0.01$ | $0.06\pm 0.01$ | $0.03\pm 0.01$ | $0.13\pm 0.03$ | $0.09\pm 0.01$ | $0.17\pm 0.02$
M79 | 31 | -7.86 | -1.51 | 166.3 | $0.23\pm 0.04$ | $0.04\pm 0.02$ | $0.03\pm 0.02$ | $0.02\pm 0.01$ | $0.03\pm 0.02$ | $0.17\pm 0.04$ | $0.14\pm 0.03$
N3201 | 101 | -7.45 | -1.35 | 192.9 | $0.27\pm 0.02$ | $0.14\pm 0.01$ | $0.03\pm 0.01$ | $0.01\pm 0.01$ | $0.04\pm 0.01$ | $0.01\pm 0.01$ | $0.07\pm 0.02$
M10 | 83 | -7.48 | -1.51 | 176.3 | $0.39\pm 0.03$ | $0.14\pm 0.02$ | $0.07\pm 0.01$ | $0.01\pm 0.01$ | $0.02\pm 0.01$ | $0.02\pm 0.01$ | $0.09\pm 0.02$
N6752 | 79 | -7.73 | -1.49 | 202.0 | $0.32\pm 0.03$ | $0.18\pm 0.02$ | $0.07\pm 0.01$ | $0.02\pm 0.01$ | $0.06\pm 0.02$ | $0.01\pm 0.01$ | $0.13\pm 0.02$
N288 | 35 | -6.75 | -1.27 | 150.0 | $0.09\pm 0.02$ | $0.02\pm 0.01$ | $0.02\pm 0.01$ | $0.02\pm 0.01$ | $0.02\pm 0.02$ | $0.01\pm 0.01$ | $0.03\pm 0.03$
N362 | 58 | -8.43 | -1.11 | 130.9 | $0.18\pm 0.02$ | $0.03\pm 0.01$ | $0.05\pm 0.01$ | $0.02\pm 0.01$ | $0.05\pm 0.01$ | $0.01\pm 0.01$ | $0.02\pm 0.02$
N1851 | 34 | -8.33 | -1.11 | 128.3 | $0.12\pm 0.02$ | $0.06\pm 0.01$ | $0.02\pm 0.01$ | $0.02\pm 0.01$ | $0.03\pm 0.02$ | $0.02\pm 0.01$ | $0.02\pm 0.02$
M4 | 109 | -7.19 | -1.05 | 222.4 | $0.06\pm 0.01$ | $0.04\pm 0.01$ | $0.03\pm 0.00$ | $0.01\pm 0.01$ | $0.01\pm 0.01$ | $0.00\pm 0.00$ | $0.01\pm 0.01$
N2808 | 80 | -9.39 | -1.11 | 148.2 | $0.31\pm 0.03$ | $0.09\pm 0.01$ | $0.09\pm 0.01$ | $0.01\pm 0.01$ | $0.07\pm 0.01$ | $0.01\pm 0.01$ | $0.01\pm 0.01$