# THE $a$-NUMBER OF $y^{q^{2}+q+1}=x^{q^{2}+1}+x^{q}$ OVER FINITE FIELD

Vahid Nourozi and Behrooz Mosallaei  The Klipsch School of Electrical and
Computer Engineering, New Mexico State University, Las Cruces, NM 88003 USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we compute a formula for the $a$-number of curve $\mathcal{X}$
given by the equation $y^{q^{2}+q+1}=x^{q^{2}+1}-x^{q}$ over the finite field
$\mathbb{F}_{q^{2}}$. The $a$-number is an invariant of the isomorphism class
of the $p$-torsion group scheme. In this paper, we compute a closed formula
for the $a$-number of $\mathcal{X}$ using the action of the Cartier operator
on $H^{0}(\mathcal{X},\Omega^{1})$.

##  1\. Introduction

Let $k$ be an algebraically closed field of characteristic $p>0$. Let
$\mathcal{A}$ be an abelian variety defined over $k$. Let $\alpha_{p}$ be the
group scheme $\text{Spec}(k[X]/(X^{p}))$ with co-multiplication given by

$X\rightarrow 1\otimes X+X\otimes 1.$

The group $\text{Hom}(\alpha_{p},\mathcal{A})$ can be considered as a
$k$-vector space since $\text{End}(\alpha_{p})=k$. The $a$-number
$a(\mathcal{A})$ is defined to be the dimension of the vector space
$\text{Hom}(\alpha_{p},\mathcal{A})$.

Let $\mathcal{X}$ be a (non-singular, projective, geometrically irreducible,
algebraic) curve defined over $k$. The $a$-number $a(\mathcal{X})$ of
$\mathcal{X}$ can be defined as the $a$-number of its Jacobian variety
$\mathcal{J}_{X}$. As a matter of fact, the $a$-number of a curve is a
birational invariant which can be defined as the dimension of the space of
exact holomorphic differentials.

The $a$-number of Hermitian curves was computed by Gross in [9], and for
Fermat and Hurwitz curves, it was computed by Montanucci and Speziali [13].
Some results on the rank of the Cartier operator (especially $a$-number) of
curves were introduced by Kodama and Washio [11], González [8], Pries and Weir
[26], and Yui [25].

In this work, we consider the hyperelliptic curve $\mathcal{X}$ given by the
equation

$y^{q^{2}+q+1}=x^{q^{2}+1}+x^{q}$

over $k$, where $q=p^{s}$ for $s\geq 1$, and $\mathcal{X}$ has genus

$g(\mathcal{X})=\frac{q^{4}+q^{2}}{2}.$

Consider the multiplication by $p$-morphism $[p]:X\rightarrow X$ which is a
finite flat morphism of degree $p^{2g}$. It factors as $[p]=V\circ F$. Here,
$F:X\rightarrow X^{(p)}$ is the relative Frobenius morphism coming from the
$p$-power map on the structure sheaf, and the Verschiebung morphism
$V:X^{(p)}\rightarrow X$ is the dual of $F$. The kernel of multiplication-
by-$p$ on $X$ is defined by the group of $X[p]$. The important invariant is
the $a$-number $a(\mathcal{X})$ of curve $\mathcal{X}$ defined by

$a(\mathcal{X})=\dim_{\overline{F}_{p}}\text{Hom}(\alpha_{p},X[p]).$

Where $\alpha_{p}$ is the kernel of the Frobenius endomorphism on the group
scheme $\text{Spec}(k[X]/(X^{p}))$. Another definition for the $a$-number is

$a(\mathcal{X})=\dim_{\mathbb{F}_{p}}(\text{Ker}(F)\cap\text{Ker}(V)).$

A few results on the rank of the Cartier operator (especially $a$-number) of
curves are introduced by Kodama and Washio [11], González [8], Pries and Weir
[26], Yui [25], and Montanucci and Speziali [13]. Besides that, Vahid talked
about the rank of the Cartier of the maximal curves in [19], the hyperelliptic
curve in [16], the maximal function fields of $\mathbb{F}_{q^{2}}$ in [15],
the Picard Curve in [14] and, the Artin-Schreier curve and application of the
Cartier operator in coding theory [20]. Vahid also wrote about these topics in
their [18] dissertation.

##  2\. The Cartier operator

Let $k$ be an algebraically closed field of characteristic $p>0$. Let
$\mathcal{X}$ be a curve defined over $k$. The Cartier operator is a
$1/p$-linear operator acting on the sheaf
$\Omega^{1}:=\Omega^{1}_{\mathcal{X}}$ of differential forms on $\mathcal{X}$
in positive characteristic $p>0$.

Let $K=k(\mathcal{X})$ be the function field of the curve $\mathcal{X}$ of
genus $g$ defined over $k$. A separating variable for $K$ is an element $x\in
K\setminus K^{p}$.

######  Definition 2.1.

(The Cartier operator). Let $\omega\in\Omega_{K/K_{q}}$. There exist
$f_{0},\ldots,f_{p-1}$ such that
$\omega=(f_{0}^{p}+f_{1}^{p}x+\ldots+f_{p-1}^{p}x^{p-1})\,dx$. The Cartier
operator $\mathfrak{C}$ is defined by

$\mathfrak{C}(\omega):=f_{p-1}\,dx.$

The definition does not depend on the choice of $x$ (see [22], Proposition 1).

We refer the reader to [22, 2, 3, 24] for the proofs of the following
statements.

######  Proposition 2.2.

Global Properties of $\mathfrak{C}$). For all $\omega\in\Omega_{K/K_{q}}$ and
all $f\in K$:

  * •

$\mathfrak{C}(f^{p}\omega)=f\mathfrak{C}(\omega)$;

  * •

$\mathfrak{C}(\omega)=0\Rightarrow\exists h\in K,\omega=dh$;

  * •

$\mathfrak{C}(\omega)=\omega\Rightarrow\exists h\in K,\omega=dh/h$.

If $\text{div}(\omega)$ is effective, then differential $\omega$ is
holomorphic. The set $H^{0}(\mathcal{X},\Omega^{1})$ of holomorphic
differentials is a $g$-dimensional $k$-vector subspace of $\Omega^{1}$ such
that $\mathfrak{X}(H^{0}(\mathcal{X},\Omega^{1}))\subseteq
H^{0}(\mathcal{X},\Omega^{1})$. If $\mathcal{X}$ is a curve, then the
$a$-number of $\mathcal{X}$ equals the dimension of the kernel of the Cartier
operator $H^{0}(\mathcal{X},\Omega^{1})$ (or equivalently, the dimension of
the space of exact holomorphic differentials on $\mathcal{X}$) (see [12],
5.2.8).

The following theorem is due to Gorenstein; see [4], Theorem 12.

######  Theorem 2.3.

A differential $\omega\in\Omega^{1}$ is holomorphic if and only if it is of
the form $\left(h(x,y)/F_{y}\right)\,dx$, where $H:h(X,Y)=0$ is a canonical
adjoint.

######  Theorem 2.4.

[13] With the above assumptions,

$\mathfrak{C}(h\frac{dx}{F_{y}})=(\frac{\partial^{2p-2}}{\partial
x^{p-1}\partial y^{p-1}}(F^{p-1}h))^{\frac{1}{p}}\frac{dx}{F_{y}}$

For any $h\in K(\mathcal{X})$.

The differential operator $\nabla$ is defined by

$\nabla=\frac{\partial^{2p-2}}{\partial x^{p-1}\partial y^{p-1}},$

has the property

$\nabla(\sum_{i,j}c_{i,j}X^{i}Y^{j})=\sum_{i,j}c_{ip+p-1,jp+p-1}X^{ip}Y^{jp}.$
(2.1)

##  3\. The $a$-number of Curve $\mathcal{X}$

In this section, we consider the curve $\mathcal{X}$ given by the equation
$y^{q^{2}+q+1}=x^{q^{2}+1}+x^{q}$ of genus $g(\mathcal{X})=\frac{q^{4}+q}{2}$,
with $q=p^{s}\mathbb{F}_{q^{2}}$. From Theorem 2.3, one can find a basis for
the space $H^{0}(\mathcal{X},\Omega^{1})$ of holomorphic differentials on
$\mathcal{X}$, namely

$B_{h}=\Bigg{\\{}\frac{y^{q^{2}+q-i-h-1}}{x^{q^{2}-j}}\,|\,hq\leq
i\leq(h+1)q,h\leq j\leq i-1\Bigg{\\}}$ (3.1)

For $1\leq h\leq q-1$. Then the basis of the Riemann-Roch space is given by
$\mathcal{B}=\bigcup_{h=1}^{q-1}\mathcal{B}_{h}$.

######  Proposition 3.1.

The rank of the Cartier operator $\mathfrak{C}$ on the curve $\mathcal{X}$
equals the number of pairs $(i,j)$ with $i,j$ satisfying the equation 3.1 such
that the system of congruence’s mod $p$

$\Bigg{\\{}\begin{array}[]{c}(q^{2}+1)(p-1)+q(h-k)+i\equiv 0,\\\
(q^{2}+q+1)(p-1-h)+j\equiv p-1,\\\ \end{array}$ (3.2)

has a solution $(h,k)$ for $0\leq h\leq p-1$, $0\leq k\leq h$.

###### Proof.

By Theorem 2.4,
$\mathfrak{C}\left(\left(x^{i}y^{j}/F_{y}\right)dx\right)=\left(\nabla\left(F^{p-1}x^{i}y^{j}\right)\right)^{1/p}dx/F_{y}$.
So, we apply the differential operator $\nabla$ to

$\begin{array}[]{cccc}(y^{q^{2}+q+1}-x^{q^{2}+1}-x^{q})^{p-1}x^{i}y^{j}&=&\sum^{p-1}_{h=0}\sum^{h}_{k=0}\binom{p-1}{h}\binom{h}{k}(-1)^{h-k}y^{(q^{2}+q+1)(p-1-h)+j}\\\
&&\hskip 130.88284ptx^{(q^{2}+1)(p-1)+q(h-k)+i}\\\ \end{array}$ (3.3)

for each i, j.

From the Formula (2.1),
$\nabla(y^{q^{2}+q+1}-x^{q^{2}+1}-x^{q})^{p-1}x^{i}y^{j}\neq 0$ if and only if
for some $(h,k)$, with $0\leq h\leq p-1$ and $0\leq k\leq h$, satisfies both
the following congruences mod $p$:

$\Bigg{\\{}\begin{array}[]{c}(q^{2}+1)(p-1)+q(h-k)+i\equiv 0,\\\
(q^{2}+q+1)(p-1-h)+j\equiv p-1,\\\ \end{array}$ (3.4)

Take $(i,j)\neq(i_{0},j_{0})$ in this situation, both
$\nabla\left((y^{q^{2}+q+1}-x^{q^{2}+1}-x^{q})^{p-1}x^{i}y^{j}\right)$ and
$\nabla\left((y^{q^{2}+q+1}-x^{q^{2}+1}-x^{q})^{p-1}x^{i_{0}}y^{j_{0}}\right)$
are nonzero. We claim that they are linearly independent over $k$. To show
independence, we prove that, for each $(h,k)$ with $0\leq h\leq p-1$ and
$0\leq k\leq h$, there is no $(h_{0},k_{0})$ with $0\leq h_{0}\leq p-1$ and
$0\leq k_{0}\leq h_{0}$ such that

$\Bigg{\\{}\begin{array}[]{c}(q^{2}+1)(p-1)+q(h-k)+i=(q^{2}+1)(p-1)+q(h_{0}-k_{0})+i_{0},\\\
(q^{2}+q+1)(p-1-h)+j=(q^{2}+q+1)(p-1-h_{0})+j_{0},\\\ \end{array}$ (3.5)

If $h=h_{0}$, then $i\neq i_{0}$ and $j\neq j_{0}$ from the second equation;
therefore, $k\neq k_{0}$. We may assume $k>k_{0}$. Then
$i-i_{0}=q(k-k_{0})>q$, a contradiction as $i-i_{0}\leq q-1$. Similarly, if
$k=k_{0}$, then $h\neq h_{0}$ by $(i,j)\neq(i_{0},j_{0})$. We assume that
$h>h_{0}$.

Then $j-j_{0}=(q^{2}+q+1)(h-h_{0})>(q^{2}+q+1)$, a contradiction as by
definition on $i,j$.

∎

For the rest in this Section, $A_{s}:=A(\mathcal{X})$ denotes the matrix
representing the $p$-th power of the Cartier operator $\mathfrak{C}$ on the
curve $\mathcal{X}$ with respect to the basis $\mathcal{B}$, where $q=p^{s}$.
Now we are able to compute the $a$-number of curve $\mathcal{X}$.

######  Theorem 3.2.

If $q=p^{s}$ for $s\geq 1$, then the $a$-number of the curve $\mathcal{X}$
equals

$\frac{p^{s}(p-1)\Bigg{(}p^{s-1}(3p^{2s}+1)+(p^{s}-3)\Bigg{)}}{12}.$

###### Proof.

First, we prove that if $q=p^{s}$, $s\geq 1$, then

$\text{rank}(A_{s})=\frac{p^{s}(p+1)(3p^{3s-1}+3-p^{s+1}+2p^{s}+p^{s-1})}{12}.$

In this case, $hq\leq i\leq(h+1)q$, $h\leq j\leq i-1$, and System (3.2) mod
$p$ reads

$\Bigg{\\{}\begin{array}[]{c}i\equiv 0,\\\ -h-1+j\equiv p-1,\\\ \end{array}$
(3.6)

First, assume that $s=1$. For $q=p$, we have

$hq\leq i\leq(h+1)q$

, $h\leq j\leq i-1$, and from equation 3.1, System (3.6) becomes

$\Bigg{\\{}\begin{array}[]{c}i=np+t,\\\ j+h=mp+l,\\\ \end{array}$

with $h+c\leq n\leq 2+c$ and $c\leq m\leq m-c-1$, where $0\leq l,t\leq p-1$
and $c\in\\{0,1\\}$.

In this case, $\text{rank}(A_{1})$ is equal to the cardinality of the set

$\Bigg{(}(np+t,mp+l-h)\in\mathbb{N}^{2}:h+c\leq n\leq 2+c\text{ and }c\leq
m\leq m-c-1\Bigg{)}.$

Thus, $\text{rank}(A_{1})=\frac{p(p+1)(p^{2}+p+4)}{6}$.

For $s\geq 3$, $\text{rank}(A_{s})$ equals $\text{rank}(A_{s-1})$ plus the
number of pairs $(i,j)$ with $h(p^{s-1})+c\leq n\leq(s+1)p^{s-1}+c$ and $c\leq
m\leq m-c-1$, where $0\leq l,t\leq p-1$ and $0\leq c\leq p^{s-1}$, such that
the system mod $p$

$\Bigg{\\{}\begin{array}[]{c}i\equiv 0,\\\ -h-1+j\equiv p-1.\\\ \end{array}$
(3.7)

has a solution. With our usual conventions on $n$, $m$, a computation shows
that such pairs $(i,j)$ are obtained for

$\bigg{(}(np+t,mp+l-h)\in\mathbb{N}^{2}:hp^{s-1}+c\leq
n\leq(s+1)p^{s-1}+c\text{ and }c\leq m\leq m-c-1\bigg{)}.$

In this case we have

$\frac{p^{s-1}(p+1)}{12}\left(3p^{3s-4}(p^{4}-1)-p^{s-2}(p^{4}+2p^{3}-2p^{2}-2p-1)-3\right)$

choices for $(i,j)$. Therefore, we get

$\text{rank}(A_{s})=\text{rank}(A_{s-1})+\frac{p^{s-1}(p+1)}{12}(3p^{3s-4}(p^{4}-1)-p^{s-2}(p^{4}+2p^{3}-2p^{2}-2p-1)-3)$

Now, our claim on the rank of $A^{s}$ follows by induction on $s$. Hence,

$\begin{array}[]{ccccccc}a(\mathcal{X})&=&\dfrac{q^{4}-q}{2}-\dfrac{p^{s-1}(p+1)(3p^{3s-1}+3-p^{s+1}+2p^{s}+p^{s-1})}{12}\\\
&=&\dfrac{p^{s}(p-1)\left(p^{s-1}(3p^{2s}+1)+(p^{s}-3)\right)}{12}\\\
\end{array}$

∎

### Acknowledgements.

This paper was written while Vahid Nourozi was visiting Unicamp (Universidade
Estadual de Campinas), supported by TWAS/Cnpq (Brazil) with fellowship number
$314966/2018-8$.

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# Differential reddening in the direction of 56 Galactic globular clusters

M. V. Legnardi1, A. P. Milone1,2, G. Cordoni1, E. P. Lagioia1, E. Dondoglio1,
A. F. Marino2,3, S. Jang4, A. Mohandasan1, and T. Ziliotto1
1 Dipartimento di Fisica e Astronomia “Galileo Galilei”, Univ. di Padova,
Vicolo dell’Osservatorio 3, Padova, IT-35122
2 Istituto Nazionale di Astrofisica - Osservatorio Astronomico di Padova,
Vicolo dell’Osservatorio 5, Padova, IT-35122
3 Istituto Nazionale di Astrofisica - Osservatorio Astrofisico di Arcetri,
Largo Enrico Fermi, 5, Firenze, IT-50125
4 Center for Galaxy Evolution Research and Department of Astronomy, Yonsei
University, Seoul 03722, Korea E-mail<EMAIL_ADDRESS>
(Accepted 2023 April 05. Received 2023 April 04; in original form 2022
December 23)

###### Abstract

The presence of differential reddening in the direction of Galactic globular
clusters (GCs) has proven to be a serious limitation in the traditional
colour-magnitude diagram (CMD) analysis. Here, we estimate local reddening
variations in the direction of 56 Galactic GCs. To do that, we use the public
catalogs derived as part of the Hubble Space Telescope UV Legacy Survey of
Galactic Globular Clusters, which include photometry in the F275W, F336W,
F438W, F606W, and F814W filters. We correct photometry for differential
reddening finding that for 21 out of 56 GCs the adopted correction procedure
significantly improves the CMDs. Moreover, we measure the reddening law in the
direction of these clusters finding that $R_{V}$ exhibits a high level of
variability within the Galaxy, ranging from $\sim 2.0$ to $\sim 4.0$. The
updated values of $R_{V}$ have been used to improve the determination of local
reddening variations and derive high-resolution reddening maps in the
direction of the 21 highly-reddened targets within our sample. To compare the
results of the different clusters, we compute the 68th percentile of the
differential-reddening distribution, $\sigma_{\Delta A_{\rm F814W}}$. This
quantity ranges from 0.003 mag to 0.030 mag and exhibits a significant anti-
correlation with the absolute module of the Galactic latitude and a strong
correlation with the average reddening in the direction of each cluster.
Therefore, highly-reddened GCs located in the proximity of the Galactic plane
typically show higher differential-reddening variations across their field of
view.

###### keywords:

globular clusters: general, Hertzsprung–Russell and colour–magnitude diagrams,
techniques: photometry

††pubyear: 2023††pagerange: Differential reddening in the direction of 56
Galactic globular clusters–14

##  1 Introduction

Galactic globular clusters (GCs) are remarkable fossils from the early
Universe. Being among the most ancient objects of the Galaxy, they provide
unique insights into nucleosynthesis, star formation, and chemical enrichment
at the earliest stages of the Galaxy’s formation. One of the most serious
challenges in the study of many GCs, especially the ones in the direction of
the Galactic Center, is the presence of spatially-variable extinction, better
known as differential reddening, caused by variation of dust column density
across the field of view (FoV). By introducing a significant broadening of all
evolutionary sequences in colour-magnitude diagrams (CMDs), differential
reddening complicates the traditional CMD analysis preventing an accurate
determination of some fundamental parameters, such as age, distance, and
metallicity (e.g. Bonatto et al., 2013). Moreover, differential reddening
hinders the detection and characterization of the various stellar sequences
that compose the CMDs, including binaries and multiple stellar populations in
GCs (e.g. Milone et al., 2012). Therefore, correcting the photometry for the
effects of differential reddening is a crucial step for a correct analysis of
the CMD.

Over the years, several methods have been developed to model and correct for
spatially variable extinction in the direction of Galactic GCs. A first
approach consists of dividing the FoV across the cluster in a regular cell
grid, extracting the CMD of the stars included in each cell, and computing the
differential reddening of each cell as the average colour distance, calculated
across the reddening line, between the extracted CMD and the bluest ones (e.g.
Kaluzny & Krzeminski, 1993; Heitsch & Richtler, 1999; Piotto et al., 1999; von
Braun & Mateo, 2001; Sarajedini et al., 2007; McWilliam & Zoccali, 2010;
Gonzalez et al., 2011; Bonatto et al., 2013). An alternative method has been
introduced by Milone et al. (2012) and then extensively used to investigate
star clusters in the Milky Way and in Magellanic Clouds (e.g. Lagioia et al.,
2014; Milone, 2015; Bellini et al., 2017; Li et al., 2017; Pallanca et al.,
2021; Dondoglio et al., 2022; Legnardi et al., 2022). In a nutshell, they
estimated the differential reddening of each star in a cluster by defining a
reference line for the upper main sequence (MS), the sub-giant branch (SGB),
and the red-giant branch (RGB) and then calculating the colour distance of
each cluster star from it along the reddening vector. The final differential-
reddening value associated with each star corresponds to the median colour
displacement of the spatially closest objects.

Recently, our group has undertaken an extensive investigation of differential
reddening in the FoV of more than a hundred star clusters in the Magellanic
Clouds by means of Hubble Space Telescope (HST) images (Milone et al., 2022).
Moreover, we investigated reddening variations across the FoV of 43 Galactic
GCs (Jang et al., 2022), based on ground-based wide-field photometry from
Stetson et al. (2019). Here, we constrain the amount of differential reddening
in the central regions of 56 Galactic GCs by using high-precision HST
photometry and astrometry by Nardiello et al. (2018).

Milone et al. (2012) and Bonatto et al. (2013) investigated local reddening
variations across the FoV for a large sample of Galactic GCs (59 and 66,
respectively) by using HST data. Both works are based on the data of the
GO-10775 program alone (PI: A. Sarajedini; Sarajedini et al. 2007), which is a
HST Treasury project where 66 GCs have been homogeneously observed through the
F606W and F814W filters of the Wide Field Camera of the Advanced Camera for
Surveys (WFC/ACS). Recently, this dataset has been integrated through the
images taken as part of the GO-13227 program (PI: G. Piotto; Piotto et al.
2015), a UV-initiative proposal designed to observe a large sample of clusters
through the UV/blue F275W, F336W, and F438W bands of the Ultraviolet and
Visual Channel of the Wide Field Camera 3 (UVIS/WFC3) on board HST.

Based on multi-band photometry, we estimate the differential reddening
associated with each source in the catalog and derive a high-resolution and
high-precision reddening map in the directions of 56 Galactic GCs. To do that,
we adapt to the available catalogs (Nardiello et al., 2018) the approach by
Milone et al. (2012). The main difference is that our dataset comprises
magnitudes in five filters, namely F275W, F336W, F438W, F606W, and F814W,
whereas the work by Milone and collaborators is based only on the F606W and
F814W bands. Our procedure, combining information from five magnitudes,
provides more accurate differential-reddening corrections than the previous
one. Moreover, we take advantage of stellar proper motions that allow us to
separate the bulk of cluster members from field stars in all GCs. The
resulting differential-reddening catalogs and the reddening maps are publicly
released to the astronomical community.

This paper is organized as follows. Section 2 presents the GC sample and
describes the procedures used to identify those stars that provide information
on differential reddening. In Section 3 we explain in detail the method to
calculate differential reddening, and we estimate the impact of changing the
reddening law on the differential-reddening determination. In Section 4 we
describe the procedure to constrain the reddening law in the direction of the
highly-reddened clusters and we construct high-resolution reddening maps.
Finally, Section 5 briefly overviews the results.

Figure 1: This figure illustrates the procedure to select well-measured stars
in NGC 6171. The top and middle panels present the selection based on the qfit
(top) and RADXS (middle) parameters. Red lines separate well-measured stars
(black points) from those that have poorer photometry (gray points). The
$m_{\rm F336W}$ versus $m_{\rm F336W}-m_{\rm F814W}$ CMDs for well-measured
stars and for the stars that we rejected are displayed in the bottom-left and
the bottom-right panel, respectively.

##  2 Data and data analysis

To estimate the amount of differential reddening we used the catalogs by
Nardiello et al. (2018) that provide astrometric positions, high-precision
multi-band photometry, and cluster membership from proper motions in the
central $\sim 2.7^{\prime}\times 2.7^{\prime}$ FoV for a large sample of 56
Galactic GCs. All the clusters were observed with WFC/ACS camera in F606W and
F814W filters as part of the GO-10775 program (PI: A. Sarajedini; Sarajedini
et al., 2007) and with UVIS/WFC3 in F275W, F336W, and F438W filters mostly
within GO-13227 (PI: G. Piotto; Piotto et al., 2015).

The data reduction has been carried out with the computer program KS2
developed by Jay Anderson as an evolution of kitchen_sync, written initially
to reduce the images from GO-10775 (Anderson et al., 2008). KS2 derives three
distinct astro-photometric catalogs for each cluster obtained through three
different methods. Method one provides accurate measurements of bright,
unsaturated stars, whereas the photometry of faint stars is best measured with
methods two and three (see Sabbi et al., 2016; Bellini et al., 2017; Nardiello
et al., 2018; Milone et al., 2022, for details).

In the following, we use photometry from method one. Indeed, the regions of
the CMD where the colour broadening is most sensitive to differential
reddening correspond to the upper MS, SGB, and the lower part of the RGB.

Figure 2: This figure presents the method used to calculate the differential
reddening of the target star marked with the large blue starred symbol. In the
left panel, we plot the $m_{\rm F814W}$ versus $m_{\rm F336W}-m_{\rm F814W}$
CMD, whereas in the right panel, we show the same stars but in the rotated
reference frame. In both diagrams, the red arrow indicates the reddening
direction, and black dots, included between the two dotted lines, identify
reference stars, among which the 35 closest neighbours of the target star are
represented with cyan crosses. The red line is the fiducial of reference stars
calculated in the rotated reference frame. Finally, the right-panel inset
shows the position of the 35 neighbouring reference stars zoomed around the
target star. See Section 3 for details on the employed procedure.

###  2.1 Selection of well-measured cluster stars

The main impact of differential reddening on the CMD is a colour broadening of
the evolutionary sequences. Specifically, the stars that are affected by a
larger amount of reddening than the average value exhibit red colours and
faint magnitudes relative to the fiducial line derived from all cluster
members in the CMD. Conversely, stars with a small amount of reddening are
shifted to the bright-blue side of the fiducial.

To estimate the amount of differential reddening associated with each star in
the catalog, we need to calculate the shift along the reddening direction
relative to the colour and magnitude of a sample of nearby cluster members.
However, not all the stars provide information on differential reddening. In
the following, we describe the method to identify the probable cluster members
with high-quality photometry.

####  2.1.1 Selection of probable cluster members

The FoVs of GCs are contaminated by a variable number of background/foreground
sources. To separate cluster members and field stars, we used the cluster
membership probabilities provided by Nardiello et al. (2018, see their Section
4 for details), which corresponds to the probability that a star is part of
the cluster based on proper motions. Specifically, we considered probable
cluster members all stars with a membership probability greater than 90%.

####  2.1.2 Selection of stars with high-quality photometry

Similarly to differential reddening, photometric uncertainties spread the
stars around the fiducial line of the CMD. Hence, it would be challenging to
disentangle the effect of reddening and errors from stars with large
photometric uncertainties. For this reason, selecting a sample of well-
measured stars is mandatory to derive accurate reddening estimates.

For all clusters, our selection criteria rely on the quality fit (qfit) and
the RADXS diagnostic parameters, which are provided in the Nardiello et al.
(2018) catalogs. Specifically, the qfit estimates the goodness of the PSF-
fitting procedure, whereas the RADXS measures the amount of flux that exceeds
the predictions from the best-fitting PSF. It is a powerful tool to
distinguish cosmic rays and PSF artifacts, which have negative RADXS values,
and to identify background galaxies, which have $\rm RADXS>0$.

Fig. 1 shows an example of the selection procedure for NGC 6171. In the top
and middle panels, we plot the qfit (top) and the RADXS (middle) parameters as
a function of magnitudes in the F336W (left) and F814W (right) bands. Red
lines are drawn by hand with the criteria for separating well-measured stars
(black points) from those that have poorer photometry (gray points). The
results of the selection are illustrated in the bottom-left and -right panel,
where we show the $m_{\rm F336W}$ versus $m_{\rm F336W}-m_{\rm F814W}$ CMD for
well-measured and rejected stars, respectively.

##  3 Differential reddening determination

We dealt with differential reddening by adapting to our dataset the method
introduced by Milone et al. (2012) to correct the photometry of 59 Galactic
GCs for differential reddening. Our dataset, including magnitudes in the
F275W, F336W, F438W, F606W, and F814W bands, allowed us to obtain improved
differential-reddening determinations compared to the original ones by Milone
et al. (2012), based only on the F606W and F814W magnitudes.

To this aim, we exploited the $m_{\rm F814W}$ versus $m_{\rm X}-m_{\rm F814W}$
CMDs, where X=F275W, F336W, F438W, and F606W, and derived four distinct
estimates of the amount of differential reddening suffered by each star. We
then averaged together information from the different colour combinations to
obtain a more accurate differential-reddening determination. In the following
we summarize the adopted procedure, which is illustrated in Fig. 2 for the
$m_{\rm F814W}$ versus $m_{\rm F336W}-m_{\rm F814W}$ CMD of NGC 6171 that we
used as a test case.

  * •

The reddening vector’s direction, which is indicated by the red arrow in the
left panel of Fig. 2, has a non-zero slope with respect to CMD axes. To
simplify the correction process, we started by defining a new reference frame
in which the abscissa (x’) and the ordinate (y’) are parallel and orthogonal
to the reddening direction, respectively. To this aim, we arbitrarily
identified a rotation point near the MS turn-off (red dot) and rotated the CMD
counterclockwise by an angle $\theta$, defined as:

$\theta=\arctan\frac{A_{\rm F814W}}{A_{\rm X}-A_{\rm F814W}}$ (1)

where AX and AF814W are the absorption coefficients in the X and F814W
filters, respectively (Dotter et al., 2008). A zoom around the SGB of the
resulting rotated CMD is presented in the right panel of Fig. 2, where for
completeness we also show the rotated direction of the reddening vector (red
arrow) and the rotation point. The reddening vector is now parallel to the x’
axis.

  * •

We then selected a sample of reference stars in the CMD region that includes
the upper MS, the SGB, and the lower portion of the RGB. The reference stars
distribute along sequences where there is a wide angular variation between the
direction of the fiducial line and that of the reddening vector. Hence, they
allow us to disentangle the contribution of photometric errors and
differential reddening to the colour broadening. As an example, the selected
reference stars of NGC 6171 are marked as black points in the $m_{\rm F814W}$
versus $m_{\rm F336W}-m_{\rm F814W}$ CMD and in the corresponding rotated
diagram of Fig. 2.

  * •

We calculated the red fiducial line plotted in the right panel of Fig. 2. To
do that, we divided the sample of reference stars into y’ bins and we
calculated the median values of x’ and y’ for each of them. We derived the
fiducial line of the selected reference stars by linearly interpolating these
median points.

  * •

As a final step, we computed the distance from the fiducial line along the
reddening direction ($\Delta\,x^{\prime}$) by subtracting from the pseudo-
colour (x’) of each reference star that of the fiducial line at the same y’
level.

In summary, this procedure allows us to measure the value of
$\Delta\,x^{\prime}$ for each reference star. These quantities will be used in
Section 3.1, and 4.1 to correct the photometry for the effects of differential
reddening and to investigate the reddening in front of the GCs, respectively.

Figure 3: Panel a-b. Comparison between the original $m_{\rm F814W}$ versus
$m_{\rm F336W}-m_{\rm F814W}$ CMD of NGC 6171 and the corresponding
photometric diagram corrected for differential reddening. In the inset, we
plotted a zoom around the SGB where the correction is more evident. Panel c.
Differential-reddening map in the direction of NGC 6171. The levels of red are
proportional to the reddening variation as indicated by the associated colour
bar on the right. Panel d. Variation of the resolution of the differential-
reddening correction within the observed FoV.

###  3.1 Correcting the photometry for differential-reddening effects

To derive the amount of differential reddening associated with each star in
the catalog, we used an iterative procedure that consists of the following
steps:

  1. 1.

For each star in the catalog, we defined a local sample of $N_{\rm
ref}=30$-100 nearby reference stars. As an example, in Fig. 2 we mark with
cyan crosses the 35 closest reference stars used to estimate the differential
reddening suffered by the star represented by the blue starred symbol. As
illustrated in the inset of the right panel, all the reference stars lie in a
circular region centered on the target star. The radius of this circle,
$\rho$, corresponds to the distance of the farthest reference star with
respect to the target one, which is indicative of the map resolution in that
specific position. Clearly, the value of $\rho$ changes from one star to
another. It is typically small in the central cluster regions, where the
density of reference stars is high, and will increase towards the external
regions.

  2. 2.

For each star, we calculated the median $\Delta\,x^{\prime}$ value of the
neighbouring reference-star sample excluding the target from the computation.
We estimated the associated error as the root mean scatter of
$\Delta\,x^{\prime}$ divided by the square root of $N-1$, where $N$ is the
number of selected nearby reference stars. To correct the photometry for the
effects of differential reddening we subtracted the derived quantity from the
$x^{\prime}$ value of each star in the rotated reference frame. We then used
the new diagram to select reference stars with more accuracy and therefore
obtain an improved fiducial line.

  3. 3.

We repeated all steps above until the procedure converges. Typically, this
happens after three iterations. At this point, we converted the corrected
$x^{\prime}$ and $y^{\prime}$ into $m_{\rm X}$ and $m_{\rm F814W}$ magnitudes
and we used the corresponding absorption coefficients to derive the
differential-reddening value associated with each star.

The best differential-reddening estimate for each star is obtained by
combining the information from the different $m_{\rm F814W}$ versus $m_{\rm
X}-m_{\rm F814W}$ CMDs. Specifically, our final estimate of $\Delta A_{\rm
F814W}$ corresponds to the weighted average of the four distinct differential-
reddening determinations. The corresponding uncertainties are given by the
error of the weighted mean.

We derived various determinations of differential reddening by assuming
different values of $N_{\rm ref}$ ranging from 30 to 100 in step of 5. For
each differential-reddening determination, we calculated the residuals between
the corrected $x^{\prime}$ values and the fiducial line in each CMD. The best
determination of differential reddening is provided by the value of $N_{\rm
ref}$ that gives the minimum values of the r.m.s of these residuals.

As an example, the results of this procedure are illustrated in panels a and b
of Fig. 3 for NGC 6171, where we compare the original $m_{\rm F814W}$ versus
$m_{\rm F336W}-m_{\rm F814W}$ CMD and the corresponding diagram corrected for
differential reddening. Noticeably, the evolutionary sequences in the
corrected CMD are consistently narrower than in the original one. In panel c
of Fig. 3 we provide the resulting high-resolution differential-reddening map,
whereas in panel d we show the variation of the resolution in the observed FoV
of the cluster. As expected, the highest resolution ($\rho=5.5$ arcsec) is
achieved in the central regions of the cluster where the density of the
reference stars is higher.

For 21 out of 56 GCs, the photometry corrected for differential reddening
provides improved CMDs, thus demonstrating the efficiency of the correction.
Conversely, the correction for differential reddening does not improve the
photometry of the remaining 35 GCs. This fact is not surprising. Indeed these
clusters have $E(B-V)\lesssim 0.01$ mag and the amount of differential
reddening is likely comparable to or smaller than the uncertainties of
$E(B-V)$ determinations.

In Table 1 we provide for each cluster the minimum, maximum, and median value
of $\rho$ used to derive the differential reddening of cluster members
($\rho_{min}$, $\rho_{max}$, and $\rho_{med}$). The average values of
$\rho_{min}$, $\rho_{max}$, and $\rho_{med}$ for all clusters are 5.6, 41.9,
and 11.6 arcsec, respectively.

Figure 4: Upper panel. Extinction law for three different assumptions of the
$R_{V}$ coefficient, namely the standard value for diffuse interstellar medium
(3.1, gray line), the value commonly assumed in the direction of the Galactic
Center (2.5, red line) and the value measured for the $\rho$ Ophiuchi cloud
(4.1, blue line). The vertical dashed lines are drawn corresponding to the
central wavelengths of the five photometric filters used in this work. Bottom
panel. Transmission curves of the five filters used in this work to correct
photometry for differential reddening. Figure 5: Local reddening variations in
the direction of NGC 6171 as a function of the X coordinates calculated by
assuming $R_{V}=2.5$ (upper-left panel), $R_{V}=4.1$ (upper-right panel), and
$R_{V}=3.1$ (middle panels). Bottom panels illustrate the residuals, derived
by comparing $\Delta A_{F814W}$ calculated by assuming $R_{V}=2.5$ (left) and
$R_{V}=4.1$ (right), respectively, with the original $\Delta A_{F814W}$,
obtained by assuming $R_{V}=3.1$. The dashed red line marks the average
difference, whereas the dispersion of the residuals is reported in the upper-
right corner.

Figure 6: $m_{\rm F606W}+m_{\rm F814W}-km_{\rm F438W}$ versus $m_{\rm
F275W}-m_{\rm F438W}$ pseudo-CMD of NGC 3201 (panel a). $\Delta(m_{\rm
F438W}-m_{\rm F814W})$ versus $\Delta(m_{\rm F275W}-m_{\rm F438W})$ diagrams
for NGC 3201 (b1), NGC 6121 (b2), and NGC 6144 (b3). Figure 7: Spatial
distribution of the 56 GCs studied in this work in Galactic coordinates. The
size of each point is proportional to the average reddening of the cluster.
The 21 GCs for which the corrected photometry provides improved CMDs are
colour-coded according to the measured $R_{V}$ in their FoV.

Figure 8: Differential-reddening maps in the direction of twenty Galactic GCs.
The colour bar in the bottom panel of each plot codifies the $\Delta A_{\rm
F814W}$ values ranging from less extinct regions (lighter colours) to more
extinct areas (redder colours). Figure 9: $m_{\rm F814W}$ versus $m_{\rm
F336W}-m_{\rm F814W}$ CMDs for four GCs before (left panel) and after (right
panel) applying the procedure to correct for differential reddening. Figure
10: Same as in Fig. 9 but for NGC 6352, NGC 6656, NGC 6779, and NGC 6838.

###  3.2 The impact of the reddening law on differential-reddening
determination

The interstellar reddening law, namely the relation between the absorption
coefficient $A_{\lambda}$ and the wavelength, has been expressed in several
ways by different authors in the literature (e.g., Cardelli et al., 1989;
Fitzpatrick & Massa, 1990; O’Donnell, 1994). All of these formulations depend
on a single parameter, $R_{V}$=$A_{V}/E(B-V)$, which is usually set to 3.1,
the standard value for the diffuse interstellar medium (e.g., Sneden et al.,
1978). However, it is well known that the extinction law is not uniform in the
Galaxy, and the value of $R_{V}$ changes from place to place. For example,
several authors found that $R_{V}\sim 2.5$ is an appropriate value to describe
the reddening law toward the Galactic Center (e.g., Udalski, 2003; Nataf et
al., 2013, 2016). In addition, $R_{V}$ can change significantly also in nearby
regions, as in the case of the $\rho$ Ophiuchi cloud, where $R_{V}=4.1$
(Cardelli et al., 1989; Hendricks et al., 2012). Similarly, the investigation
of the various regions of recent star formation in the Large Magellanic Cloud
revealed that the extinction curve is systematically flatter (in logarithmic
units) than in the diffuse interstellar medium (De Marchi & Panagia, 2014,
2019; De Marchi et al., 2016; De Marchi et al., 2020, 2021).

In the upper panel of Fig. 4 we show three curves corresponding to the
extinction laws derived assuming the O’Donnell (1994) parametrization and
three different values of $R_{V}$, namely 2.5 (red), 3.1 (gray), and 4.1
(blue). The bottom panel of Fig. 4 illustrates the transmission curves of the
five filters used in this work to estimate differential reddening.

To verify whether different values of $R_{V}$ can significantly alter the
outcomes of the procedure to correct photometry, we corrected the CMDs of NGC
6171 with the same procedure described in Section 3 by assuming a non-
universal reddening law. Specifically, we calculated differential reddening in
the case of $R_{V}=2.5$ and $R_{V}=4.1$, and then we compared the results with
our previous determination, derived assuming $R_{V}=3.1$. To do that, we first
determined the absorption coefficients in the UVIS/WFC3 F275W, F336W, and
F438W filters and in the WFC/ACS F606W and F814W bands. We used a synthetic
spectrum of a star with effective temperature $T_{\rm eff}=5900\,K$, gravity
$\log g=4.5$, and metallicity $\rm[Fe/H]=-1.5$ as reference. Then, we derived
an absorbed spectrum by convolving the flux of the reference spectrum with the
extinction law by O’Donnell (1994). To this aim, we assumed $E(B-V)=0.33$
(from the 2010 version of Harris, 1996, catalog) and $R_{V}=2.5$ and 4.1,
respectively. The two synthetic spectra have been integrated over the
transmission curves of the filters used in this paper to derive the
corresponding magnitudes. Finally, by comparing the magnitudes of the absorbed
and the reference spectrum we found the absorption coefficients that we used
to derive the direction of the reddening vector.

Results are illustrated in Fig. 5, where we show the local reddening
variations as a function of the X coordinate derived assuming $R_{V}=2.5$,
$R_{V}=4.1$, and $R_{V}=3.1$. The bottom panels show the residuals calculated
comparing $\Delta A_{\rm F814W}$ derived assuming $R_{V}=2.5$ (left) and
$R_{V}=4.1$ (right), with the original $\Delta A_{\rm F814W}$ distribution,
derived assuming $R_{V}=3.1$. In both cases, we obtained an average difference
close to zero and a dispersion, $\sigma\sim 0.01$ mag. We found similar
results for NGC 6441, which is the GC that shows the highest variations of
reddening in the FoV.

We conclude that changing $R_{V}$ has a moderate impact on the differential-
reddening map of highly-reddened clusters. The effect is negligible for the
GCs that exhibit small reddening variations in their FoV, which include most
of the investigated objects. In the following Section, we constrain the
reddening law in the direction of the 21 GCs that are significantly affected
by differential reddening and improve their differential-reddening map.

##  4 The reddening law in the direction of 21 clusters

For 21 out of 56 GCs, the photometry corrected for differential reddening
provides improved CMDs. These clusters, which are significantly affected by
differential reddening, offer us the possibility to constrain the reddening
law within the Galaxy in their directions.

To do that we adopted the following iterative procedure. We first constructed
the $m_{\rm F606W}+m_{\rm F814W}-km_{\rm F438W}$ versus $m_{\rm F275W}-m_{\rm
F438W}$ and $m_{\rm F606W}+m_{\rm F814W}-km_{\rm F438W}$ versus $m_{\rm
F438W}-m_{\rm F814W}$ pseudo-CMDs. The constant $k$ is chosen in such a way
that the combination of magnitudes in the y-axis is reddening-free so that the
differential reddening affects only the colour on the x-axis. Similar to the
values of absorption in the F438W, F606W, and F814W bands, $k$ depends on the
reddening law. At the first iteration, we assumed the O’Donnell (1994)
reddening law with $R_{V}=3.1$, which corresponds to $k=1.19$. As an example,
we show the $m_{\rm F606W}+m_{\rm F814W}-km_{\rm F438W}$ versus $m_{\rm
F275W}-m_{\rm F438W}$ diagram of NGC 3201 in panel a of Fig. 6, where we use a
red arrow to indicate the reddening direction.

We used each pseudo-CMD to derive the amount of reddening that affects the
$m_{\rm F275W}-m_{\rm F438W}$ and $m_{\rm F438W}-m_{\rm F814W}$ of each star
($\Delta\,(m_{\rm F275W}-m_{\rm F438W})$ and $\Delta\,(m_{\rm F438W}-m_{\rm
F814W})$) by using the procedure of Section 3.

To constrain the value of $R_{V}$ in the direction of the investigated targets
we compared the slopes of the observed points in the $\Delta\,(m_{\rm
F438W}-m_{\rm F814W})$ versus $\Delta\,(m_{\rm F275W}-m_{\rm F438W})$ diagram
with the values inferred from simulated photometry. The latter comprises the
same number of stars, observational errors, and $\Delta\,(m_{\rm F438W}-m_{\rm
F814W})$ distribution as derived from the observed CMD. We assumed the
O’Donnell (1994) reddening law and various values of $R_{V}$ ranging from 1.5
to 5.0 in steps of 0.1. For each value of $R_{V}$, we calculated the slope of
the least-squares best-fit line in the $\Delta\,(m_{\rm F438W}-m_{\rm F814W})$
versus $\Delta\,(m_{\rm F275W}-m_{\rm F438W})$ plane. The value of $R_{V}$
that provides the smallest difference between the slopes of the line derived
from simulated and observed points corresponds to the best determination of
$R_{V}$. This ends one iteration. At this stage, we repeated the procedure
above by using the updated value of $R_{V}$.

The results for NGC 3201 ($R_{V}=3.19\pm 0.12$) are illustrated in the panel
b1, whereas panels b2 and b3 compare the results obtained in the directions of
NGC 6121 (red, $R_{V}=3.85\pm 0.13$) and NGC 6144 (blue, $R_{V}=1.93\pm 0.25$)
with NGC 3201. Clearly, the distributions of points in the $\Delta\,(m_{\rm
F438W}-m_{\rm F814W})$ versus $\Delta\,(m_{\rm F275W}-m_{\rm F438W})$ plane
are consistent with lines that have different slopes.

The resulting values of $R_{V}$ are listed in Table 2 and range from $\sim
2.0$ to $\sim 4.0$ with a high level of variability within the Galaxy. In Fig.
7 we plot the spatial distribution of the studied GCs in Galactic coordinates,
colour-coded according to the measured values of $R_{V}$. Typically, the
clusters that are close to the Galactic plane and in proximity to the Galactic
center exhibit smaller values of $R_{V}$ than the standard one.

We used the updated values of $R_{V}$ to calculate the direction of the
reddening vector and then provide a more precise determination of local
reddening variations in the direction of the 21 highly-reddened targets within
our sample. The reddening maps in the direction of these clusters are provided
in Fig. 8 (the map for NGC 6171 is already shown in panel c of Fig. 3),
whereas in Fig. 9 and 10 we show the original (left panels) and corrected
(right panels) $m_{\rm F814W}$ versus $m_{\rm F336W}-m_{\rm F814W}$ CMDs for
eight GCs. These plots point out the effectiveness of the adopted method, as
in all presented GCs the CMD region including the upper MS, the SGB, and the
lower part of the RGB appears sharper and better defined.

For each of the 21 clusters, we publicly release the photometry of all stars
in the catalogs corrected for differential reddening, together with the
corresponding high-resolution reddening map. Specifically, for each star, we
provide the ID (from Nardiello et al., 2018), the amount of differential
reddening with the corresponding uncertainty, and the value of $\rho$.

###  4.1 Reddening in the directions of 56 Globular Clusters

The procedure presented in Section 3.1 provides high-precision differential
reddening estimates that allowed us to significantly improve the photometry of
21 GCs. However, the radii of the regions used to derive the amount of
differential reddening associated with each star change from one star to
another, thus making it challenging to compare the results from the different
clusters.

To further quantify the amount of differential reddening and better compare
the reddening variation in the direction of each cluster, we divided the FoV
into a regular grid of $\sim$7 $\times\,$7 arcsec2 cells. We calculated the
median $\Delta\,x^{\prime}$ of reference stars inside each cell and converted
this quantity into the corresponding differential-reddening value using the
absorption coefficients. We estimated the uncertainty as the ratio between the
dispersion of the differential-reddening values of the stars in each cell
divided by the square root on $N-1$, where $N$ is the number of stars in that
cell. We then derived the weighted average of the four distinct estimates of
$\Delta A_{\rm F814W}$ obtained from the $m_{\rm F814W}$ versus $m_{\rm
X}-m_{\rm F814W}$ CMDs, where X=F275W, F336W, F438W, and F606W.

Table 1 provides $\sigma_{\Delta A_{\rm F814W}}$ and $\Delta A_{\rm F814W,{\it
max}}$, respectively the 68.27th percentile and the difference between the
98th and the 2nd percentile of the differential-reddening distribution. We
estimated the corresponding errors by means of bootstrapping statistics.
Specifically, we generated 1,000 samples and for each extraction, we used the
method presented above to obtain an estimate of $\sigma_{\Delta A_{\rm
F814W}}$ and $\Delta A_{\rm F814W,{\it max}}$. The uncertainties associated
with these two quantities correspond to the random mean scatter of the 1,000
determinations.

Since the FoVs of all studied clusters span a similar area, the results from
our analysis allow us to compare the reddening along the directions of the 56
GCs. We used $\sigma_{\Delta A_{\rm F814W}}$ and $\Delta A_{\rm F814W,{\it
max}}$ to quantify the variation of extinction across the observed FoV.
Specifically, we assumed $\sigma_{\Delta A_{\rm F814W}}$ and $\Delta A_{\rm
F814W,{\it max}}$ as a proxy of the average and maximum differential-reddening
variation, respectively.

The values of $\sigma_{\Delta A_{\rm F814W}}$ range from 0.003 mag to 0.030
mag, whereas $\Delta A_{\rm F814W,{\it max}}$ varies from a minimum of 0.012
mag to a maximum of 0.120 mag. As shown in Fig. 11, where we plot the spatial
distribution of the studied GCs in Galactic coordinates (l,b), the GCs with
large values of $\sigma_{\Delta A_{\rm F814W}}$ are located in the proximity
of the Galactic plane. Specifically, $\sigma_{\Delta A_{\rm F814W}}$ exhibits
a significant anti-correlation with the absolute value of the Galactic
latitude (left panel of Fig. 12), and a strong correlation with the average
reddening of the host cluster (from the 2010 version of the Harris, 1996,
catalog, right panel of Fig. 12).

Our results corroborate previous findings that highly-reddened GCs typically
show higher differential-reddening variations across their FoV (Bonatto et
al., 2013; Jang et al., 2022). However, we note large variations of the
$\sigma_{\Delta A_{\rm F814W}}$ value among clusters with similar reddening. A
remarkable example is provided by the two clusters with the highest $E(B-V)$
values, NGC 6366 and NGC 6304, which are characterized by low values of
$\sigma_{\Delta A_{\rm F814W}}$ with respect to the common trend. These facts
would suggest that in addition to $E(B-V)$, another parameter is responsible
for the value of $\sigma_{\Delta A_{\rm F814W}}$. Similar conclusions can be
extended to $\Delta A_{\rm F814W,{\it max}}$.

Figure 11: Spatial distribution of the 56 GCs studied in this work in Galactic
coordinates. The size of each point is proportional to the average reddening
of the cluster, whereas the levels of red are indicative of the 68th
percentile of the differential-reddening variation, as indicated by the colour
bar on the bottom. Figure 12: Left panel. 68th percentile of the differential-
reddening variation, $\sigma_{\Delta A_{\rm F814W}}$, against the absolute
value of the Galactic latitude. Right panel. $\sigma_{\Delta A_{\rm F814W}}$
as a function of the average reddening calculated in the direction of the
cluster. Red dots represent the 21 GCs for which the photometry corrected for
differential reddening provides improved CMDs. The Spearman’s rank correlation
coefficient and the associated p-value are quoted in the upper-right corner.

##  5 Summary and conclusions

Our group has undertaken an extensive investigation of differential reddening
in the directions of the Galactic and Magellanic Cloud star clusters based on
multi-band photometry from HST and wide-field ground-based telescopes (Jang et
al., 2022; Milone et al., 2022). In the present work, we estimated the amount
of differential reddening in the direction of 56 Galactic GCs based on HST
photometry. To do that, we adapted the method by Milone et al. (2012) to
archival data uniformly observed through the F275W, F336W, F438W of UVIS/WFC3,
and the F606W, and F814W filters of WFC/ACS (Nardiello et al., 2018). We
derived the amount of differential reddening associated with all stars in the
catalogs with a typical spatial resolution of $\sim$12 arcsec.

We used these differential-reddening values to correct photometry for the
effects of absorption. For 21 out of 56 GCs with reddening variations
$\sigma_{\Delta A_{\rm F814W}}\gtrsim 0.01$ we obtain improved CMDs, as all
evolutionary sequences become thinner and better defined. In addition, we
constrained the reddening law in the direction of these clusters. To do that,
we compared the slopes of the observed points in the $\Delta\,(m_{\rm
F275W}-m_{\rm F438W})$ versus $\Delta\,(m_{\rm F438W}-m_{\rm F814W})$ diagram
with the values inferred from simulated photometry derived assuming various
values of $R_{V}$. We found that $R_{V}$ exhibits a high level of variability
within the Galaxy, with values ranging from $\sim 2.0$ to $\sim 4.0$. The
updated values of $R_{V}$ have been used to calculate the direction of the
reddening vector and then derive a more precise determination of local
reddening variations within the FoV of the 21 highly-reddened targets. For
these clusters, we publicly release the photometric catalogs corrected for
differential reddening, together with the corresponding reddening map.

To quantify the variation of reddening across the observed FoV and compare the
results from the different clusters, we computed the 68.27th percentile
($\sigma_{\Delta A_{\rm F814W}}$) and the difference between the 98th and the
2nd percentile ($\Delta A_{\rm F814W,{\it max}}$) of the differential-
reddening distribution. We found that $\sigma_{\Delta A_{\rm F814W}}$ ranges
from 0.003 to 0.030 mag, whereas $\Delta A_{\rm F814W,{\it max}}$ varies from
0.012 to 0.120 mag. The 68.27th percentile of the differential-reddening
distribution exhibits a strong correlation with the average reddening of the
host cluster. In addition, GCs with large values of $\sigma_{\Delta A_{\rm
F814W}}$ are located in the proximity of the Galactic plane, as we found that
there is a significant anti-correlation between $\sigma_{\Delta A_{\rm
F814W}}$ and the absolute module of the Galactic latitude. Similar results
have been found for $\Delta A_{\rm F814W,{\it max}}$, thus corroborating the
conclusion that GCs with high values of average reddening are affected by
large amounts of differential reddening.

## Acknowledgements

We thank the anonymous referee for various suggestions that improved the
quality of the manuscript. This work has received support from the European
Research Council (ERC) under the European Union’s Horizon 2020 research
innovation programme (Grant Agreement ERC-StG 2016, No 716082 ’GALFOR’, PI:
Milone, http://progetti.dfa.unipd.it/GALFOR). APM and ED acknowledge support
from MIUR through the FARE project R164RM93XW SEMPLICE (PI: Milone) and from
MIUR under PRIN program 2017Z2HSMF (PI: Bedin).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

Table 1: This table lists for each cluster the average reddening (from the 2010 version of the Harris, 1996, catalog), the 68th percentile of the differential-reddening distribution, the maximum variation of differential reddening across the FoV of the cluster (difference between the 98th and the 2nd percentile of the differential-reddening distribution), the minimum, maximum, and median radius of the regions used to estimate differential reddening in the observed FoV. Cluster ID |  $E(B-V)$ |  $\sigma_{\Delta A_{\rm F814W}}$ |  $\Delta A_{\rm F814W,{\it max}}$ |  $\rho_{min}$ |  $\rho_{max}$ |  $\rho_{med}$
---|---|---|---|---|---|---
|  (mag) |  (mag) |  (mag) |  (arcsec) |  (arcsec) |  (arcsec)
NGC 0104 |  0.04 |  $0.008\pm 0.001$ |  $0.031\pm 0.004$ |  4.1 |  66.2 |  8.4
NGC 0288 |  0.03 |  $0.003\pm 0.001$ |  $0.012\pm 0.003$ |  9.1 |  39.1 |  13.6
NGC 0362 |  0.05 |  $0.003\pm 0.001$ |  $0.013\pm 0.001$ |  4.1 |  48.9 |  6.9
NGC 1261 |  0.01 |  $0.005\pm 0.001$ |  $0.016\pm 0.002$ |  3.8 |  40.1 |  9.0
NGC 1851 |  0.02 |  $0.006\pm 0.001$ |  $0.022\pm 0.002$ |  3.8 |  32.9 |  7.2
NGC 2298 |  0.14 |  $0.029\pm 0.002$ |  $0.098\pm 0.008$ |  5.2 |  57.6 |  16.2
NGC 2808 |  0.22 |  $0.006\pm 0.001$ |  $0.021\pm 0.001$ |  5.5 |  23.8 |  8.0
NGC 3201 |  0.24 |  $0.022\pm 0.002$ |  $0.078\pm 0.005$ |  6.6 |  32.2 |  12.7
NGC 4590 |  0.05 |  $0.006\pm 0.001$ |  $0.020\pm 0.002$ |  4.1 |  38.6 |  10.6
NGC 4833 |  0.32 |  $0.011\pm 0.001$ |  $0.044\pm 0.006$ |  4.3 |  21.0 |  8.1
NGC 5024 |  0.02 |  $0.004\pm 0.001$ |  $0.014\pm 0.002$ |  4.8 |  35.7 |  8.2
NGC 5053 |  0.01 |  $0.004\pm 0.001$ |  $0.015\pm 0.002$ |  12.9 |  50.3 |  20.2
NGC 5272 |  0.01 |  $0.006\pm 0.001$ |  $0.020\pm 0.001$ |  4.8 |  56.6 |  8.0
NGC 5286 |  0.24 |  $0.006\pm 0.001$ |  $0.022\pm 0.004$ |  5.2 |  40.5 |  9.5
NGC 5466 |  0.00 |  $0.003\pm 0.001$ |  $0.013\pm 0.002$ |  9.8 |  39.1 |  15.3
NGC 5897 |  0.09 |  $0.003\pm 0.001$ |  $0.012\pm 0.001$ |  7.5 |  30.4 |  12.2
NGC 5904 |  0.03 |  $0.005\pm 0.001$ |  $0.016\pm 0.001$ |  4.0 |  71.4 |  9.0
NGC 5927 |  0.45 |  $0.019\pm 0.003$ |  $0.076\pm 0.012$ |  5.7 |  43.3 |  12.5
NGC 5986 |  0.28 |  $0.025\pm 0.001$ |  $0.086\pm 0.004$ |  3.9 |  37.1 |  8.0
NGC 6093 |  0.18 |  $0.020\pm 0.001$ |  $0.066\pm 0.005$ |  5.4 |  52.1 |  10.1
NGC 6101 |  0.05 |  $0.004\pm 0.001$ |  $0.014\pm 0.001$ |  5.7 |  36.4 |  11.5
NGC 6121 |  0.35 |  $0.020\pm 0.002$ |  $0.082\pm 0.012$ |  11.3 |  43.1 |  19.8
NGC 6144 |  0.36 |  $0.008\pm 0.001$ |  $0.028\pm 0.002$ |  6.4 |  32.7 |  11.7
NGC 6171 |  0.33 |  $0.030\pm 0.002$ |  $0.100\pm 0.004$ |  5.5 |  39.2 |  11.9
NGC 6205 |  0.02 |  $0.006\pm 0.001$ |  $0.021\pm 0.001$ |  4.3 |  35.8 |  7.5
NGC 6218 |  0.19 |  $0.006\pm 0.001$ |  $0.022\pm 0.001$ |  6.6 |  28.8 |  12.1
NGC 6254 |  0.28 |  $0.016\pm 0.001$ |  $0.052\pm 0.003$ |  4.1 |  27.2 |  8.7
NGC 6304 |  0.54 |  $0.016\pm 0.002$ |  $0.061\pm 0.007$ |  5.5 |  47.7 |  18.3
NGC 6341 |  0.02 |  $0.007\pm 0.001$ |  $0.022\pm 0.001$ |  4.2 |  30.5 |  8.5
NGC 6352 |  0.22 |  $0.017\pm 0.002$ |  $0.070\pm 0.010$ |  7.3 |  40.2 |  14.4
NGC 6362 |  0.09 |  $0.003\pm 0.001$ |  $0.013\pm 0.001$ |  7.3 |  36.3 |  12.3
NGC 6366 |  0.71 |  $0.020\pm 0.002$ |  $0.078\pm 0.006$ |  12.4 |  46.0 |  20.8
NGC 6388 |  0.37 |  $0.010\pm 0.001$ |  $0.038\pm 0.005$ |  5.5 |  39.4 |  8.1
NGC 6397 |  0.18 |  $0.007\pm 0.001$ |  $0.025\pm 0.003$ |  3.1 |  40.6 |  13.2
NGC 6441 |  0.47 |  $0.029\pm 0.003$ |  $0.120\pm 0.013$ |  6.2 |  39.6 |  9.1
NGC 6496 |  0.15 |  $0.019\pm 0.002$ |  $0.072\pm 0.007$ |  5.6 |  43.6 |  13.0
NGC 6535 |  0.34 |  $0.011\pm 0.001$ |  $0.043\pm 0.006$ |  6.6 |  81.9 |  25.9
NGC 6541 |  0.14 |  $0.012\pm 0.001$ |  $0.048\pm 0.006$ |  4.5 |  34.9 |  10.4
NGC 6584 |  0.10 |  $0.005\pm 0.001$ |  $0.019\pm 0.002$ |  3.1 |  41.2 |  8.1
NGC 6624 |  0.28 |  $0.010\pm 0.001$ |  $0.042\pm 0.008$ |  3.8 |  43.5 |  11.7
NGC 6637 |  0.18 |  $0.007\pm 0.001$ |  $0.025\pm 0.002$ |  2.8 |  34.7 |  7.3
NGC 6652 |  0.09 |  $0.005\pm 0.001$ |  $0.020\pm 0.003$ |  3.0 |  50.8 |  12.9
NGC 6656 |  0.34 |  $0.012\pm 0.001$ |  $0.047\pm 0.002$ |  7.3 |  30.9 |  12.4
NGC 6681 |  0.07 |  $0.003\pm 0.001$ |  $0.013\pm 0.001$ |  2.9 |  37.4 |  9.9
NGC 6715 |  0.15 |  $0.006\pm 0.001$ |  $0.022\pm 0.002$ |  6.2 |  70.0 |  11.2
NGC 6717 |  0.22 |  $0.004\pm 0.001$ |  $0.017\pm 0.003$ |  4.6 |  54.4 |  23.3
NGC 6723 |  0.05 |  $0.004\pm 0.001$ |  $0.017\pm 0.002$ |  3.4 |  24.2 |  7.2
NGC 6752 |  0.04 |  $0.004\pm 0.001$ |  $0.015\pm 0.002$ |  2.6 |  85.7 |  10.0
NGC 6779 |  0.26 |  $0.017\pm 0.001$ |  $0.057\pm 0.003$ |  2.9 |  29.4 |  7.6
NGC 6809 |  0.08 |  $0.004\pm 0.001$ |  $0.014\pm 0.001$ |  8.6 |  31.9 |  14.0
NGC 6838 |  0.25 |  $0.011\pm 0.001$ |  $0.039\pm 0.002$ |  11.9 |  44.6 |  19.2
NGC 6934 |  0.10 |  $0.010\pm 0.001$ |  $0.034\pm 0.005$ |  4.1 |  44.5 |  9.4
NGC 6981 |  0.05 |  $0.004\pm 0.001$ |  $0.015\pm 0.001$ |  3.6 |  42.1 |  9.5
NGC 7078 |  0.10 |  $0.008\pm 0.001$ |  $0.029\pm 0.002$ |  5.2 |  33.3 |  8.3
NGC 7089 |  0.06 |  $0.007\pm 0.001$ |  $0.025\pm 0.001$ |  5.5 |  34.7 |  8.3
NGC 7099 |  0.03 |  $0.005\pm 0.001$ |  $0.018\pm 0.003$ |  2.4 |  29.8 |  8.2
Cluster ID | $R_{V}$ | Cluster ID | $R_{V}$
---|---|---|---
NGC 2298 | $3.11\pm 0.11$ | NGC 6352 | $2.36\pm 0.20$
NGC 3201 | $3.19\pm 0.12$ | NGC 6366 | $3.70\pm 0.43$
NGC 5286 | $2.04\pm 0.28$ | NGC 6388 | $1.92\pm 0.24$
NGC 5927 | $2.43\pm 0.16$ | NGC 6441 | $2.60\pm 0.43$
NGC 5986 | $4.15\pm 0.44$ | NGC 6496 | $3.53\pm 0.23$
NGC 6093 | $3.93\pm 0.18$ | NGC 6535 | $2.32\pm 0.18$
NGC 6121 | $3.85\pm 0.13$ | NGC 6541 | $3.40\pm 0.19$
NGC 6144 | $1.93\pm 0.25$ | NGC 6656 | $2.53\pm 0.11$
NGC 6171 | $3.01\pm 0.11$ | NGC 6779 | $2.87\pm 0.19$
NGC 6254 | $3.35\pm 0.27$ | NGC 6838 | $2.36\pm 0.20$
NGC 6304 | $2.64\pm 0.16$ |  |
Table 2: This table lists for each of the 21 highly-reddened GCs the new
values of $R_{V}$ with the corresponding error calculated in Section 4.

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##  Appendix A Multiple populations and differential reddening

All GCs studied in this work show multiple stellar populations with different
chemical compositions that define distinct sequences along the entire CMD
(Milone et al., 2017). In this appendix, we investigate whether or not the
presence of multiple populations in a GC affects the differential reddening
determination, described in Section 3.

To do that, we first simulated the $m_{\rm F814W}$ versus $m_{\rm X}-m_{\rm
F814W}$ CMDs, where X=F275W, F336W, F438W, and F606W, to which we have added
the map of differential reddening shown in the left panel of Figure 13. In
close analogy with Milone et al. (2012, see their appendix A), the reddening
variations that we simulated are related to stellar positions (X, Y) by the
following relations:

$\Delta\,E(B-V)=C_{1}(\cos{X^{\prime}}+\sin{Y^{\prime}})$ (2)

where

$\displaystyle X^{\prime}=C_{2}\pi(X-X_{MAX})/(X_{MAX}-X_{MIN}),$
$\displaystyle Y^{\prime}=C_{2}\pi(Y-Y_{MAX})/(Y_{MAX}-Y_{MIN}).$

Specifically, $X_{MIN,MAX}$ and $Y_{MIN,MAX}$ correspond to the minimum and
maximum values of X and Y, respectively, whereas $C_{1}$ and $C_{2}$ are two
free parameters determining how the reddening variations are distributed
across the FoV of the cluster (see Milone et al. 2012 for details). We fixed
$C_{1}=0.012$ and $C_{2}=2.5$ in such a way that the simulated map resembles
the one of NGC 6838. This cluster, which has been widely studied in the
context of multiple populations, hosts first and second-population stars with
different chemical compositions. The stellar populations of NGC 6838 can be
followed continuously along various evolutionary sequences, including the red
horizontal branch, the RGB, the SGB, and the MS (Cordoni et al., 2020;
Dondoglio et al., 2021; Legnardi et al., 2022).

To investigate the impact of multiple populations on the determination of
differential reddening, we simulated two groups of CMDs. We used the same
number of stars and the same F814W luminosity distribution as observed in NGC
6838. We first derived CMDs where stars belong to a single population. We
assumed for all stars the colours and magnitudes inferred from first-
population stars by Legnardi et al. (2022). Then, we simulated CMDs composed
of first- and second-population stars. In this case, second-population stars
comprise 70$\%$ of the total number of stars, which is a typical value for
Galactic GCs (Milone et al., 2017; Dondoglio et al., 2021). We adopted the
colours and magnitudes of the two stellar populations measured by Legnardi et
al. (2022).

We corrected these CMDs with the same procedure described in Section 3
obtaining a reddening map that is similar to the one that we have simulated.
We assumed the differences between the measured differential-reddening values
($\Delta A_{\rm F814W}^{\rm OUT}$) and the simulated ones ($\Delta A_{\rm
F814W}^{\rm IN}$) as an estimate of the effectiveness of the correction
procedure.

Results are shown in Fig. 14 where we plotted this quantity as a function of
the X coordinate for the two investigated cases, namely CMDs hosting a simple
stellar population (SSP, bottom) and two stellar populations (MP, top). In
both cases, we obtained a similar value for the average difference between
$\Delta A_{\rm F814W}^{\rm OUT}$ and $\Delta A_{\rm F814W}^{\rm IN}$, which is
close to zero. However, we note that in the MP simulations we obtain a
dispersion of the $\Delta A_{\rm F814W}^{\rm OUT}-\Delta A_{\rm F814W}^{\rm
IN}$ quantity ($\sigma$) that is $\sim 1.25$ times higher than that derived
from the simple-stellar population CMDs. We conclude that the multiple
populations affect the uncertainties on differential-reddening determinations
but do not introduce significant systematic errors.

Figure 13: Differential-reddening map added to the simulated CMDs. The top and
right panels show $\Delta A_{\rm F814W}$ as a function of the X and Y
coordinate of the horizontal and vertical slices in which we divided the FoV.
Figure 14: Difference between the measured differential-reddening values
($\Delta A_{\rm F814W}^{\rm OUT}$) and the values of differential reddening
used in the simulation ($\Delta A_{\rm F814W}^{\rm IN}$) as a function of the
X coordinates. The black continuous line marks the average difference, whereas
the dashed lines are indicative of the dispersion of the residuals. We show
the results in the case of a simulation of a GC with two stellar populations
(top) and a single stellar population (bottom).