# Hooke and Coulomb energy of tripod spiders

Giorgi Khimshiashvili  Ilia State University, Tbilisi, Georgia  and  Dirk
Siersma  Mathematisch Instituut, Universiteit Utrecht

###### Abstract.

Tripod spiders are the simplest examples of arachnoid mechanisms. Their
workspaces and configuration spaces are well known. For Hooke potential, we
give a complete description of the Morse theory and treat the robust control
of the spider. For the Coulomb energy, we use stationary charges and the
trapping domain to study the robust control of spiders. We show that, for a
regular triangle and positive charges, the domain of robust control is non-
void. This relates to questions about the Maxwell conjecture about point
charges. We end with several natural problems and research perspectives
suggested by our results.

###### Key words and phrases:

Spider linkage, workspace, Hooke energy, Coulomb energy, stationary charges,
robust constrol

###### 2020 Mathematics Subject Classification:

58K05, 70B15

##  1\. Introduction

The geometry and topology of mechanical linkages play an important and
increasing role in applied problems. Most of the previous studies were
concerned with the workspace and the topology of the configuration space,
which is only known in a few cases summarized in [KM], [Oh], [Mo]. The aim of
this paper is to extend the known results to new classes of linkages and
enrich them by considering potential functions on the workspace, with
aplications to control of the linkages considered. Our approach is based on
Morse theory which yields an explicit connection between the topology of
configuration space and the critical points of potential.

We will be basically concerned with the so-called arachnoid mechanisms, the
topology of which is a largely unexplored topic. Nowadays the same type of
objects is often called ”spidery linkage”. Detailed information on the
topology of arachnoid linkages is important for the design and control of
certain types of spider robots. More concretely, we study the simplest
arachnoid mechanism, the $3$-leg spider also known as tripod spider [Oh],
[Mo].

In Section 2 we use Hooke energy as a potential function and describe in
detail the workspace and the critical point theory. This implies, in
particular, that a weighted version of Hooke energy can be used to control
such a linkage.

In section 3 we study the Coulomb potential of point charges placed at the
foots of a $3$-leg spider linkage. We deal with the so-called stationary
charges of the spider’s center (the common point of legs) and the so-called
trapping domain of stationary charges determined in [GK1], [GK2]. This enables
us to determine the domain in the workspace of spider, where the position of
spider’s center can be robustly controlled by the values of stationary charges
using the Coulomb control scenario developed in [KPS], [GK1]. For a symmetric
spider based on a regular triangle and having a contractible workspace, we
show that the domain of robust Coulomb control is a non-void open subset of
the workspace containing the center of the reference triangle.

In the sequel we use Morse theory for manifolds with boundaries and corners.
Most of it is ”folklore” hidden in the literature. We mention here [JJT] and
[GM] as the most general reference. For critical points of functions on
manifolds with boundary and corners we refer to [Si]. For brevity, we refer to
criteria of such critical points and corresponding topological changes in
level surfaces as ”standard rules”.

In conclusion we mention several related problems and perspectives suggested
by our results. In general, this paper may be considered as a first step in
applying our approach to spider linkages and creating a paradigm for further
research in this direction.

The results of this paper were obtained and written up in the framework of a
”Research in Residence” project at the ”Centre des Rencontres Mathematiques”
(CIRM, Luminy, France) realized in November of 2022. It is our pleasure to
acknowledge the support and excellent working conditions at CIRM which largely
facilitated our research.

##  2\. Hooke Energy as a potential function of tripod spider

We will consider the Hooke potential for three points in several situations.
First without any constraint, next with constraints on maximal an minimal
distances, and finally for a 3-leg spider.

###  2.1. No constraints

We start with the following simplified situation. Given $3$ points $A,B,C$ (in
vector notation $a,b,c$), the foot points of the spider, and its center
(joint) the point $X$. The legs $AX$, $BX$, $CX$ are completely flexible
around the foot, their length is allowed to change.

The Hooke Energy is defined by

$H(x)=||x-a||^{2}+||x-b||^{2}+||x-c||^{2}$

The stationary points are determined by: $\nabla H=6x-2(a+b+c)=0$, so
$x=\frac{1}{3}(a+b+c)$, the center of gravity. There are no other stationary
points. Note that $H(X)=||x-z||^{2}+K,$ where $z=\frac{1}{3}(a+b+c)$, the
center of gravity $Z$ and $K=||a||^{2}+||b^{|}|^{2}+||c||^{2}-||z||^{2}$. All
level curves are circles.

###  2.2. Maximal length constraints

We require

$|XA|\leq R_{A}\;,\;|XB|\leq R_{B}\;,\;|XC|\leq R_{C}.$

The configuration space is the intersection of 3 discs $D_{A},D_{B},D_{C}$
with centers resp $A,B,C$ and radii $R_{A},R_{B},R_{C}$. We study the boundary
extrema of $H$.

######  Lemma 1.

The map $H$ restricted to the boundary of $D_{A}$ has an extrema in $Y$ iff
$AY$ has the direction of $AB+AC$, i.e the two intersection points of the line
trough $A$ and $Z$ with the boundary of $D_{A}$, a minimum and a maximum.

###### Proof.

Suppose $A$ is the origin. Let $R_{A}e_{\phi}$ a point on $\delta D_{A}$. Use
Lagrange multiplyers:

$\nabla H=6R_{A}e_{\phi}-2(b+c)=\lambda e_{\phi}$

Therefore

$(-\lambda+6r_{A})e_{\phi}=2(b+c)$

∎

######  Proposition 1.

Let the workspace $W=D_{A}\cap D_{B}\cap D_{C}$ contain an open neighborhood
of $Z$ then $Z$ is the only stationary point of $H$, an absolute minimum.
There are no boundary singularities.

###### Proof.

The center of gravity $Z$ is clearly a minimum. Potential other critical
points are the critical points of the restiction of $H$ to the boundary
circles (intersections of $ZA$ with the circle around $A$, etc) and also the 3
‘corner points ‘, where 2 boundary circles intersect. The statement follows
now from the standaard rules . For boundary points, they are here as follows:

type of restriction | direction normal | type | cell attaching
---|---|---|---
minimum | inward | no change | no
minimum | outward | minimum | 0-cell
maximum | inward | no change | no
maximum | outward | saddle | 1-cell

With no change we mean, that the topological type of the lower level sets
don’t change.
For corner points there are similar rules. They give in our case: no change.
Due to the special (circular) form of level curves and boundaries one can
obtain the same result by ‘inspection of pictures’. ∎

###  2.3. Two sided length constraints

We require

$0<R_{A}^{-}\leq|XA|\leq R_{A}\;,\;0<R_{B}^{-}\leq|XB|\leq
R_{B}\;,\;0<R_{C}^{-}\leq|XC|<R_{C}.$

The configuration space is the intersection of 3 discs with centers $A,B,C$
and radii $R_{A},R_{B},R_{C}$, where some smaller (open) discs have been taken
out. From the many different posibilties, we consider here the case that these
3 discs have a relatively small radii and the workspace $W$ is a disc with 3
holes, containing an open neighborhood of the gravity center $Z$.

######  Proposition 2.

In this situation the point $Z$ is an absolute minumum of $H$ on $W$ ,
moreover there are 3 saddle points on the (outer) intersection points of the
small discs with $ZA$, $ZB$, $ZC$.

###### Proof.

The center of gravity $Z$ is an absolute minimum. Other potential critical
points are the critical points of the restiction of $H$ to the boundary
circle, the 3 small circles and also the 3 ‘corner points ‘, where 2 boundary
circles intersect. The statement follows now from standard rules for boundary
and corner singularities as explained above.

∎

######  Remark 1.

The phase portrait of the gradient of $H$ consists of straight lines to $Z$,
except for those lines that intersect one of the small discs. Their
trajectories follow from the moment that they intersect these discs a part of
the boundary circle until they become ‘visible’ from $Z$ and then continue as
a straight line. There are 3 intervals between the outer and the inner
circles, which are conflict strata for the gradient flow.

######  Remark 2.

The Morse theory in the 3 cases above is as follows:
In 2.1: $b_{0}=1,b_{i}=0\;(i\geq 1)$ and $\mu_{1}=1,\mu_{i}=0\;(i\geq 1)$,
In 2.2: $b_{0}=1,b_{i}=0\;(i\geq 1)$ and $\mu_{1}=1,\mu_{i}=0\;(i\geq 1)$,
In 2.3: $b_{0}=1,b_{2}=3,b_{i}=0\;(i\geq 2)$ and
$\mu_{1}=1,\mu_{1}=3,\mu_{i}=0\;(i\geq 2)$.
In these three cases, $H$ is a perfect Morse function.

###  2.4. Robust control

A similar study can be made for the weighted Hooke Energy:

$H_{\alpha,\beta,\gamma}=\alpha||x-a||^{2}+\beta||x-b||^{2}+\gamma||x-c||^{2}$

Assume $\alpha>0,\beta>0,\gamma>0$. This potential function has the point
$Z=(\alpha:\beta:\gamma)$ (barycentric coordinates) as an absolute minimum.
The level curves are circles with the center at this point $Z$. The critical
point theory is similar to the case of $H$, which corresponds to
$(\alpha:\beta:\gamma)=(1:1:1)$.

A proper choice of the controls $(\alpha,\beta,\gamma)$ can be used to move
the 3-leg spider to any point in the triangle $ABC$ via minimum points of
Hooke energy. This procedure yields a robust control of the spider.

###  2.5. The 3-leg spider

In this case the telescopic connections are replaced by 2-arms with fixed arm
lengths and flexible turning point. We assume that the two parts of the arm
have different lengths. So $AX$ is replaced by the arm $AP\cdot PX$, $BX$ by
$BQ\cdot QX$ and $CX$ by $CR\cdot RX$. The configuration space $\mathcal{C}$
is an 8-fold cover of the workspace $W$ with certain identifications at the
boundaries. The topology has been studied in full generality by P. Mounod
[Mo]. He showed that an $n$-leg spider with generic arm lengths has a smooth
two-dimensional configuration space, and gave a formula for its Euler
characteristic. In his paper he also solved some questions of J. O’hara [Oh].

The configuration space of 3-leg spider projects now clearly to the workspace
$W$ from 2.3 (with the 3 small holes). $R_{A}^{-}$ is equal to the difference
of arm-lengths in the $AP\cdot PX$ and $R_{A}$ is equal to their sum. Similar
for the points $B$ and $C$. We assume here that the arm lengths are such, that
they give the workspace mentioned above.

According to the formula of [Mo] the Euler characteristic of $\mathcal{C}$ is
-22. So we have a smooth surface with genus 12.

We consider the quadratic distance function $H$ (see above).

######  Proposition 3.

The Hooke energy $H$ on the configuration space $\mathcal{C}$ of the 3-leg
spider has:

  * •

8 minima

  * •

36 saddles

  * •

6 maxima

###### Proof.

The potential $H$ is defined on the configuration space via projection to the
work space. We will use the results from Propostion 2 and use (branched)
covering arguments. The covering is 8-fold above the open part of the
workspace. The branching takes place at the boundaries and corners. The
potential critical points are just the preimages of the special points, which
have been studied in Proposition 2.

We find in this way $8$ critical pre-images of the center of gravity $Z$,
these are all absolute minima (with the same value). The $36$ saddles come
from the $9$ special points on boundary circles, where we have covering degree
$4$. Their types can be deduced by topological study of the gluing of the two
boundary pieces. Finally, we have $6$ maxima, which corespond to the three
corner points, where the covering is $2$-fold and at each such point such four
pieces are glued together.

There is is a dictionary between the Morse theory on the workspace treated
above and the configuration space. Compare the level curves in both cases via
gluing and this results in the Morse indices mentioned in the proposition. ∎

##  3\. Coulomb Energy as a potential function of tripod spider

From the viewpoint of control theory, it is also interesting and practically
important to consider the Coulomb potential of point charges which are placed
at the fixed foot points of a spider and can be varied in order to change the
position of its body endowed with a fixed charge. A similar scenario was
studied in big detail in the papers [GK1], [GK2] and our exposition in this
section relies on the constructions and results given in those two papers.

We begin with recalling several concepts and constructions in the form needed
in the sequel. Recall that the Coulomb potential $E=E(Q@A)$ of $n$ point
charges of non-zero magnitudes $q_{i}$ placed at $n$ points $A_{i}$ is a
function of point $X$ in the complement of points $A_{i}$ by the formula

$E=\sum\frac{q_{i}}{d_{i}},$

where $d_{i}=d(X,A_{i})$ is the Euclidean distance between the points $P$ and
$A_{i}$. As usual a point $X$ is called a stationary point (or equilibrium
point) of potential $E$ if its gradient $\nabla E$ vanishes at $X$. There
exists a huge number of results and problems concerned with the Coulomb
potential and equilibrium points of point charges. For us it is important to
mention that, as was proven by M. Morse himself, for a generic configuration
and generic values of point charges Coulomb potential is a Morse function
[CM]. This suggests that Coulomb potential is a reasonable candidate for
developing Morse theory for a spider linkage along the same lines as in the
case of Hooke’s energy considered above. The aim of this section is to present
a number of results in this direction and present an application in the spirit
of robust Coulomb control discussed in [KPS], [GK1], [GK2]. Another relevant
topic in our context is the so-called Maxwell’s conjecture on point charges
stating that the number of isolated equilibrium points of $n$ point charges in
$2$-dimensional Euclidean space does not exceed $(n-1)^{2}$ (see, e.g., a
recent review in [GNS]).

We next focus on the case of 3 point charges. Given a triangle $\triangle$
with vertices $A,B,C$, referred to as a reference triangle (or base triangle),
and a point $X$ in the same plane, the triple of normalized stationary charges
$Q(X)=\\{q_{i}(X;\triangle)\\}$ is defined as a triple of non-zero real
numbers $q_{i}$ such that $\nabla E(X,Q)=0$. As was proven in [GK1] these
charges are uniquely determined by the ratio’s

$q_{1}:q_{2}:q_{3}=d_{1}^{3}\mathcal{A}_{1}:d_{2}^{3}\mathcal{A}_{2}:d_{3}^{2}\mathcal{A}_{3},$

where $\mathcal{A}_{1}$ is the area of $\triangle BCX$, $\mathcal{A}_{2}$ of
$\triangle CAX$, $\mathcal{A}_{3}$ of $\triangle ABX$. We denote the angles at
X of these triangles by $\alpha_{1},\alpha_{2},\alpha_{3}$ respectively.

In [GK1] , [GK2] the Coulomb trapping domain $T(\triangle)$ is defined as the
set of minimum points of the potential induced by the charges $Q(X)$. The
trapping domain is given by the inequality $h(X)>0$, where $h$ is the Hessian
of $E(X,Q(X))$, which we call the  trapping Hessian. We have in geometric
terms the following formula:

$h(X)=-2\mathcal{A}+9(\prod_{i}^{3}\sin\alpha_{i})(\sum_{i=1}^{3}d_{i}^{2}\mathcal{A}_{i}).$

It was conjectured in [GK2] that $T(\triangle)$ is a non-empty subset
containing the incenter of $\triangle$. This conjecture has been verified in a
number of cases, including the regular triangle.

As was mentioned, a complete description of equilibria (stationary points) of
$E$ and their Morse indices is not known even in the case of three charges.
However rather complete results for special configurations of three point
charges have been obtained in [Ts]. In particular, the Maxwell’s conjecture
was proven for arbitrary three non-zero point charges at the vertices of a
regular triangle and the Morse indices are computed.

For this reason, in the rest of the text we always assume that the reference
triangle is regular since it is the most important case for applications and,
at the same time, in this case all necessary background results are rigorously
proven in [GK2].

Further, given a $3$-leg spider $S$ with the center at (moving) point $X$,
(fixed) foot-points $A,B,C$ and identical legs with the links of lengths $a$
(thigh length) and $b$ (foot length), we denote by $W(S)$ its workspace
considered above. To simplify the discussion we assume that $R>a>b>0$, in
which case the whole workspace of the spider is the intersection of three
circular annuli with centers $A,B,C$.

######  Proposition 4.

The domain of robust Coulomb control of spider $S$ in the above scenario is
equal to the intersection $D(S)=W(S)\cap T(\triangle)$.

As said before the topology of the workspace depends on the triangle and the
arm lengths. This influences the intersection which could even be void. For
the regular triangle, we make the choice that the workspace becomes a
contractible region bounded by three circular arcs and containing the
incenter. Also we assume that all charges are positive. The trapping domain
$T(\triangle)$ is explicitly known as the interior of a compact convex region
bounded by $h=0$ and containing the incenter of the triangle. Under these
conditions this yields the following conclusion.

######  Corollary 1.

The domain of robust Coulomb control of spider $S$ based on a regular triangle
is non-void.

For a spider based on the regular triangle one can also carry out a complete
Morse analysis in a similar way as for the Hooke potential. One knows that for
given positive charges there is a single minimum of E in the trapping area
$T(\triangle)$ and $3$ saddle points in $\triangle$ ouside $T(\triangle)$. The
poles of $E$ are not in the workspace and therefore not in the configuration
space, which is an $8$-fold cover of the workspace branched (glued) over
boundary components. After obtaining information about the boundary
singularities one gets a complete Morse analysis.

##  4\. Concluding remarks

There are several natural problems and research perspectives suggested by our
results. We mention some of them which seem most interesting and feasible.

First of all, note that all the concepts used in our paper make sense for
$n$-leg spiders with arbitrary base $n$-gon and arbitrary lengths of the
links. So a natural next step is to generalize the above results to Hooke and
Coulomb potential of an $n$-leg spider.

Next, one can also consider other geometrically or physically meaningful
potentials of $n$-leg spider like Riesz energies or the oriented area of the
polygon formed by the moving joints of spider.

Further, one can formulate several extremal problems related to the design of
spider linkages with desirable properties like the area or shape of the
workspace.

Moreover, one can try to extend our results to the case of $n$-leg spiders
with a moving $n$-gon platform instead of the center point.

Finally, it is natural to extend our discussion to spatial spiders where the
legs can move in a three-dimensional Euclidean space.

It is also possible to consider ”legs” with more than two links or with
flexible non-extendible tether links. Each of the mentioned possibilities
deserves a closer look and careful exploration which we intend to undertake in
the future research.

Returning to $3$-leg spiders we add that an urgent topic is to clarify what
happens with the domain of robust Coulomb control for arbitrary configuration
of foots. In this case, there are several feasible problems concerned with the
optimization of workspace with given restrictions on the sum of the lengths of
links and circumradius of the base triangle. For example, there is good
evidence that the maximal area of $W(S)$ arises for the most symmetric spider
with equal links and regular configuration of foots.

## References

  * [CM] S. Cairns, M. Morse, _Critical point theory in global analysis and differential topology_ , Academic Press, 1969.
  * [JJT] H. Th. Jongen, P. Jonker, F. Twilt, _Nonlinear Optimization in Finite Dimensions_ , Kluwer, Dordrecht, 2000.
  * [GK1] G. Giorgadze, G. Khimshiashvili, _Triangles and electrostatic ion traps_ , J. Math.Phys. 62, 053501 (2021)
  * [GK2] G. Giorgadze, G. Khimshiashvili, _Incenter of triangle as a stationary point_ , Georgian Math. J. ,29 (2022), 515-525.
  * [GM] M. Goreski, R. MacPherson, _Stratified Morse theory_ Springer Verlag, 1988.
  * [GNS] A. Gabrielov, D. Novikov, B. Shapiro, _Mystery of point charges_ , J. Lond. Math. Soc. 2007.
  * [KM]
  * [KPS] G. Khimshiashvili, G. Panina, D. Siersma, _Coulomb Control of Polygonal Linkages_ , Journal of Dynamical and Control Systems 20 (2014), 491–501.
  * [Mo] P. Mounoud, _Sur l’éspace des configurations du aragnee_ Osaka J. Math. 48 (2011), 149-178.
  * [Oh] J.-I. O’Hara, _The configuration space of planar spider linkages_ , Top. Appl. 154 (2007), 502-526.
  * [Si] D. Siersma _Singularities on boundaries, corners, etc_ , Quarterly Journal of Mathematics, 32, (1981), 19–127
  * [Ts] Y.-L. Tsai, _Special cases of three point charges_ , Nonlinearity 24 (2011), no.12, 3299-321.


# Effect of AGN on the morphological properties of their host galaxies in the
local Universe

Tilahun Getachew-Woreta1,2,3, Mirjana Pović1,4, Josefa Masegosa4, Jaime
Perea4, Zeleke Beyoro-Amado1,2,5, and Isabel Márquez4
1Astronomy and Astrophysics Research and Development Department, Entoto
Observatory and Research Center (EORC), Ethiopian Space
Science and Technology Institute (ESSTI), P.O.Box 33679, Addis Ababa, Ethiopia
2Addis Ababa University (AAU), P.O.Box 1176, Addis Ababa, Ethiopia
3Physics Department, Bule Hora University (BHU), P.O.Box 144, Bule Hora,
Ethiopia
4Instituto de Astrofísica de Andalucía (IAA-CSIC), 18008, Granada, Spain
5Physics Department, Kotebe Metropolitan University (KMU), P.O.Box 31248,
Addis Ababa, Ethiopia<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The morphological classification of active galaxies may be affected by the
presence of active galactic nuclei (AGN). In this paper, we provide the most
detailed analysis on how different AGN contributions, from 5% to 75%, to the
total optical light may affect six commonly used morphological parameters and
the final classification of AGN host galaxies at z $\sim$ 0\. We used a local
sample of $>$ 2000 visually classified non-active galaxies, to quantify how
the contribution of a bright nuclear point source of different intensity could
affect morphological parameters such as: asymmetry, Abraham concentration
index, Gini, M20 moment of light, smoothness, and Concelice-Bershady
concentration index. We found that most of the morphological parameters are
affected by AGN contributions above 25%, with late-type galaxies being more
affected than early-types. We found that Gini, Abraham concentration index and
asymmetry are the most stable parameters even for AGN contributions above 25%,
in comparison to Concelice-Bershady concentration index and M20 moment of
light. Smoothness parameter shall be avoided when classifying AGN host
galaxies, or at least it shall be used simultaneously in combination with
several other parameters.

###### keywords:

Galaxies; active galactic nuclei; morphological parameters; morphological
classification.

††pubyear: 2021††pagerange: Effect of AGN on the morphological properties of
their host galaxies in the local Universe–LABEL:lastpage

##  1 Introduction

Morphology is a key parameter used to study the properties of galaxies hosting
Active Galactic Nuclei (AGN), their connection with AGN, and their formation
and evolution (e.g., Kauffmann et al., 2003; Pierce et al., 2010; Pović et
al., 2012; Leslie et al., 2016; Dubois et al., 2016; Gu et al., 2018).
Morphology is one of the fundamental parameters for understanding some of the
open questions related to the effect of AGN on their host galaxy and vice-
versa, the origin of accretion material, the triggering mechanisms that
initiate an active phase in a galaxy, the duration of nuclear active phase,
etc. Therefore, the study of the morphology of active galaxies can place
important constraints on models of supermassive black hole (SMBH) formation
and growth, as well as on the formation and evolution of galaxies across the
cosmic time (e.g., Mahmood et al., 2005; Dubois et al., 2016; Fontanot et al.,
2020; Setoguchi et al., 2021). It has been suggested that AGN may play an
important role in the morphological transformation of galaxies from late- to
early-types, being responsible for quenching (e.g., Nandra et al., 2007; Pović
et al., 2012; Leslie et al., 2016; Dubois et al., 2016; Fontanot et al.,
2020), but also enhancing (e.g., Mahoro et al., 2017, 2019) star formation.
The dependence of the environment on AGN host galaxies in combination with
morphology has been extensively studied to better understand the AGN
triggering mechanisms. It has been suggested that minor and major mergers do
not (e.g., Sharma et al., 2021; Silva et al., 2021, and references therein)
and do (e.g., Cisternas et al., 2011; Ellison et al., 2019; Kim et al., 2020;
Gao et al., 2020) contribute significantly to the nuclear activity in
galaxies. In particular, the interpretation of morphology in terms of galaxy
evolution still remains a problem. Since the determination of the morphology
depends strongly on image resolution, morphological classification in deep
surveys remains difficult, especially when dealing with faint and high
redshift sources (Pović et al., 2015).

Different methods have been used to classify galaxies in terms of their
morphology. These include visual classification (e.g., Hubble, 1926; de
Vaucouleurs, 1959; Sandage, 1961; Buta, 2013; Lintott et al., 2008; Nair &
Abraham, 2010; Simmons et al., 2017; Willett et al., 2017;
$Z^{\textbullet}$ywucka et al., 2020) and parametric methods that assume an
analytic model for fitting galaxy surface brightness based on its physical and
mathematical parameters (e.g., Peng, 2010; Simard et al., 2011; , 2012;
Jiménez-Teja et al., 2012). Non-parametric methods have been also commonly
used, by measuring different galaxy properties that correlate with
morphological types, such as colour, light distribution, galaxy shape, etc.
(e.g., Abraham et al., 1994, 1996; Conselice et al., 2000, 2003; Huertas-
Company et al., 2008; Tasca et al., 2009; Pović et al., 2013; Mahoro et al.,
2019), being very often mixed with the machine and deep learning for giving
galaxy a probability to be early- or late-type (e.g., Huertas-Company et al.,
2008; Pović et al., 2012, 2013, 2015; Pintos-Castro et al., 2016; , 2019),
and/or purely based machine and deep learning methods (e.g., Kuminski et al.,
2014; Huertas-Company et al., 2015; Aniyan et al., 2017; Domínguez et al.,
2018; Lukic et al., 2018; de Diego et al., 2020). Each method has its
advantages and disadvantages. Visual and parametric methods are more used for
nearby and well-resolved galaxies, providing a more detailed information about
galaxy structure, while non-parametric and machine/deep learning methods are
less time-consuming and showed to be useful when dealing with large datasets,
and higher redshift sources.

There are still many inconsistencies between the results obtained regarding
the morphology of active galaxies (e.g., Kauffmann et al., 2003; Pierce et
al., 2007; Georgakakis et al., 2008; Choi et al., 2009; Gabor et al., 2009;
Pović et al., 2009, 2012; Mahoro et al., 2019; Ramón-Pérez et al., 2019), and
how AGN may affect the morphological classification of their host galaxies
(e.g., Gabor et al., 2009; Cardamone et al., 2010; Pierce et al., 2010). Many
previous studies found that AGN host galaxies usually reside in spheroids or
bulge-dominated galaxies (e.g, Kauffmann et al., 2003; Pierce et al., 2007,
2010; Pović et al., 2009, 2012; Dubois et al., 2016), although some studies
find a higher concentration of later-types (e.g., Choi et al., 2009; Gabor et
al., 2009; Mahoro et al., 2019). It has been also shown that morphological
properties of AGN host galaxies depend on the type of detection, where most
radio detected AGN are hosted by red sequence galaxies (e.g., Hickox et al.,
2009), X-ray detected AGN hosts occupy all areas of the colour-magnitude
diagram, but mostly green valley (e.g., Nandra et al., 2007; Georgakakis et
al., 2008; Hickox et al., 2009; Pović et al., 2009, 2012), while many of the
infrared detected AGN hosts reside in slightly bluer galaxies and occupy
mainly the blue cloud region with a significant fraction of peculiar galaxies,
with clear signs of interactions and mergers (e.g., Hickox et al., 2009;
García-Bernete et al., 2015; Mahoro et al., 2019). Although several works
commented how the AGN contribution may affect the morphological classification
of their host galaxies, finding insignificant contributions in most cases
(e.g., Gabor et al., 2009; Cardamone et al., 2010; Pierce et al., 2010; Trump
et al., 2015; Wang et al., 2017), we are still lacking detailed analysis of
how different morphological parameters, commonly used in galaxy
classification, are affected by the AGN contribution and therefore how the
final classification of galaxies will be affected.

In this work, we go a step further and study how different AGN contributions
may affect the six morphological parameters that are commonly used in non-
parametric methods for galaxy classification. We carried out all the analysis
by using a sample of a number of local, non-active galaxies with known visual
morphological classification (Nair & Abraham, 2010). In this paper, we present
the results obtained for the original sample at z $\sim$ 0\. The simulations
at fainter magnitudes and higher redshifts will be presented in a forthcoming
paper (Getachew et al., in prep.).

The paper is organized as follows: in section 2 we describe the used data,
along with a brief description of the generated source catalogue. The applied
methodology is described in section 3, while all analysis are given in section
4. Section 5 gives the results and discussion of our work, and finally, we
draw our conclusions in section 6. We assumed cosmological model with
$\Omega_{\Lambda}$=0.7, $\Omega_{M}=0.3$, and $H_{0}$ = 70 kms-1Mpc-1. All
magnitudes in this paper are given in the $AB$ system (Oke, 2015).

##  2 Data

To study the effect of the AGN contribution on the morphological parameters of
their host galaxies in the local Universe, we used an initial sample of 8000
local galaxies visually classified in the g-band by Nair & Abraham (2010). The
authors used the data from the Sloan Digital Sky
Survey111https://www.sdss.org/ (SDSS; York et al. 2000, Stoughton et al. 2002)
Data Release 4 (DR4) down to an extinction-corrected apparent magnitude of
$g<16$ and in the redshift range of $0.01\leq z\leq 0.1$ (with a mean redshift
of $0.04$). The galaxies were selected randomly out of $\sim 14000$ sources
contained in the Nair & Abraham (2010) catalogue, as explained in detail in
Pović et al. (2013). To make sure that the selected sub-sample of local
galaxies is a fair representation of the parent sample, we compared the two
samples in terms of their general properties. In Fig. 1 we show the
distributions for both samples for the absolute magnitude in g-band,
morphology, star formation rate (SFR), and stellar mass, where the last two
parameters have been obtained from the MPA-JHU222https://wwwmpa.mpa-
garching.mpg.de/SDSS/ SDSS catalogue (Brinchmann et al., 2004; Salim et al.,
2007). The Kolmogorov-Smirnov (KS) test for all comparison plots indicates
that the maximum deviation (D) between the two distributions is $\sim$ 0,
indicating that the two samples are consistent.

Figure 1: The comparison between the properties of the full Nair & Abraham
(2010) sample (red lines) and the sub-sample of 8000 galaxies (blue lines)
used in this work, including absolute magnitude in g-band (top left),
morphology (top right), SFR (middle to left), stellar mass (middle to right),
and size (bottom). The KS test maximum deviation D parameter between the two
distributions is given in all cases.

.

Taking into account the nature of this work, we need a sample of non-active
galaxies for our analysis. To eliminate AGN hosts, we cross-matched our sample
of 8000 galaxies with the MPA-JHU SDSS DR7 emission-line catalogue of $\sim$
920000 sources, using a cross-matching radius of 2 arcsec. We obtained a total
of 7940 counterparts with available redshifts and emission lines with signal-
to-noise ratio S/N $>$ 3\. For these galaxies, we plotted the Baldwin,
Phillips & Terlevich (1981) BPT-NII diagram as shown in Fig. 2, using the
emission-line ratios of $[NII]\lambda 6584/H\alpha$ and $[OIII]\lambda
5007/H\beta$. To separate star-forming (non-AGN) galaxies from composite and
AGN galaxies we used the separation limits given by Kauffmann et al. (2003)
and Kewley et al. (2006). We also checked the number of Seyfert 2 and LINER
sources using the separation limit of Schawinski et al. (2007). We found in
total 2744 (35%) star-forming galaxies, 1918 (24%) composites, 594 (7%)
Seyfert 2s, and 2684 (34%) LINERs.

After selecting these 2744 non-AGN galaxies, we went through the visual
inspection of their images and we further removed those galaxies with
superposed foreground bright stars, and also galaxies with signs of
interactions and mergers. Our final sample of non-active galaxies contains
2301 sources. Using the visual morphological classification of Nair & Abraham
(2010) and their T-type classification scheme (see their Table 1), we have 471
early-type galaxies (elliptical and lenticular, with T-type $\leq$ 0), 891
early-spirals (0 $>$ T-type $\leq$ 4), 889 late-spirals (4 $>$ T-type $\leq$
8), and 50 irregular and peculiar galaxies (T-type $>$ 8). We inspected
visually the images of those galaxies with T-type $>$ 8 and found that the
majority ($>80\%$) of them are not irregular galaxies, but rather disturbed
galaxies with clear signs of interactions, and therefore along this paper we
will call these galaxies peculiar. An example of 10 galaxies included under
this class is given in Fig. 3.

Figure 2: The diagnostic BPT-NII diagram separating the selected sample of
local galaxies into: star-forming (red symbols), composite (in blue), Seyfert
2 (in indigo) and LINERs (in green). Used separation limits between four
classes (from left-to-right) correspond to: Kauffmann et al. (2003, dashed
line), Kewley et al. (2006, solid line), and Schawinski et al. (2007, dotted
line). Figure 3: An example of peculiar galaxies analysed in this paper, with
clear signs of disturbed structures.

##  3 Methodology

In this section, we analysed six of the most used parameters in the non-
parametric morphological classification of galaxies, and how sensitive they
are on AGN contribution when dealing with the classification of active
galaxies. We measured the six parameters for the sample of 2744 non-AGN
galaxies. We then artificially added in their center different AGN
contributions (from 5% - 75%), and we again measured the same parameters in a
consistent way.

###  3.1 Morphological parameters of the local sample of non-active galaxies

The six morphological parameters analysed in this work have been used in many
previous studies to distinguish between early- (elliptical and lenticular) and
late-type (spiral and irregular) galaxies (e.g., Scarlata et al., 2007;
Huertas-Company et al., 2008, 2009; Pović et al., 2012, 2013, 2015; Licitra et
al., 2016; Pintos-Castro et al., 2016; Tarsitano et al., 2018; , 2019; Mahoro
et al., 2019; Baes et al., 2020; Bignone et al., 2020). These six parameters
are:

Asymmetry index (hereafter ASYM; Abraham et al. 1994). This parameter
describes the level of asymmetry in the shape of one galaxy. It is calculated
by comparing an object flux in every pixel using an original image and the
rotated one by $180^{o}$, and taking into account the background. It can be
expressed as:

$ASYM=\frac{\Sigma{|I(i,j)-I_{180}(i,j)|/2}}{I(i,j)}-\frac{\Sigma{|B(i,j)-B_{180}(i,j)|/2}}{I(i,j)},$
(1)

where I$(i,j)$ and I${}_{180}(i,j)$ are the fluxes in the $(i,j)$ pixel
position on the original and rotated image, while B$(i,j)$ and
B${}_{180}(i,j)$ are the fluxes of the background in the original and rotated
image, respectively. Highly symmetrical sources, such as undisturbed
elliptical or lenticular galaxies, will have minimal residuals and small
values of ASYM, while the significant residuals of spiral, irregular, and
peculiar galaxies will correspond to higher values of ASYM. In general, the
allowed range of this parameter is 0 $<$ ASYM $<$ 1\.

Abraham concentration index (hereafter CABR; Abraham et al. 1996). This is one
of the most used light concentration parameters. It is measured as the ratio
between the flux at 30% $(F_{30})$ of the Petrosian radius (Petrosian, 1976)
and the total flux $(F_{tot})$ of the galaxy:

$CABR=F_{30}/F_{tot}.$ (2)

The typical range of values of this parameter is $0<CABR<1$, where galaxies
with higher light concentration in their centers have higher values of CABR.

Gini coefficient (hereafter GINI; Abraham et al. 2003; Lotz et al. 2004). This
parameter has been adopted from the economic statistics (Gini, 1912) and is
now used in astronomy to describe the distribution of light among the pixels
associated with a galaxy. If $n$ is the number of pixels assigned to a galaxy,
$f_{i}$ the flux from pixel $i$ (ordered in such a way that $f_{i}$ increases
with the pixel index), and $\bar{f}$ the mean values of the pixel, GINI
parameter can be defined as:

$GINI=\frac{1}{|\bar{f}|n(n-1)}{\Sigma^{n}_{i}}{(2i-n-1)|f_{i}|}.$ (3)

The values of GINI cover the range $0<GINI<1$, where low values characterise
an equal distribution of light across all galaxy pixels, while high values
point to extreme inequality in light distribution contained within a small
number of pixels (e.g., in a single bright nucleus of a galaxy or a galaxy
with multiple bright nuclei). This parameter has been suggested to be the
efficient one for morphological measurements of strongly lensed galaxies
(Florian et al., 2016a).

M20 moment of light (hereafter M20; Lotz et al. 2004). The value of this
parameter depends on the spatial distribution of the light in a galaxy
relative to its center, summed over the 20% brightest pixels and normalised by
the total second-order moment of light (Mtot):

${M20=log_{10}\left({\frac{\Sigma_{i}{M_{i}}}{M_{tot}}}\right),}$ (4)

while the flux $f_{i}$ in each pixel in comparison to the total flux $f_{tot}$
is $\Sigma_{i}{f_{i}<0.2f_{tot}}$,
and where

$M_{tot}=\Sigma^{n}_{i}{M_{i}}=\Sigma^{n}_{i}{f_{i}[(x_{i}-x_{c})^{2}+(y_{i}-y_{c})^{2}]},$
(5)

with $(x_{c},y_{c})$ standing for the galaxy center obtained by minimising the
total light (Huertas-Company et al., 2008). This parameter in general has
values in the range $-3<M20<0$. It is very sensitive to the spatial
distribution of light. For undisturbed early-type galaxies when the brightest
20% of the light may easily be positioned close to the galactic center, M20
approaches values of -2.5 or -3.0. On the other hand, the presence of multiple
bright sources of light that are spatially separated from the galaxy center
(e.g., bars, spiral arms, strong star-forming regions, etc.) increase the
value of M20 toward 0 (Conselice et al., 2014).

Smoothness or clumpiness (hereafter SMOOTH; Conselice et al. 2000). This
parameter is sensitive to the small-scale structures in galaxies. To measure
it, the original galaxy image is smoothed with a boxcar of a given width and
subtracted from the original image. SMOOTH is then obtained as:

$SMOOTH=\frac{\Sigma{|I(i,j)-I_{s}(i,j)|/2}}{I(i,j)}-\frac{\Sigma{|B(i,j)-B_{s}(i,j)|/2}}{I(i,j)},$
(6)

where $I(i,j)$, $B(i,j)$ are the flux of the galaxy and background,
respectively, in the pixel position $(i,j)$ of the original image, while
$I_{s}(i,j)$ and $B_{s}(i,j)$ represent the same in the case of the smoothed
image. This parameter has values $0<SMOOTH<1.0$, where higher values indicate
a larger number of small-scale structures in a galaxy.

Conselice-Bershady concentration index (hereafter CCON; Bershady et al. 2000).
Similar to CABR and GINI, this parameter gives the level of light
concentration within a galaxy. It is measured as the logarithm of the ratio of
the circular radii containing 80% and 20% of the total flux:

$CCON=5log(r_{80}/r_{20}).$ (7)

In general, lower CCON values correspond to lower fractions of light in the
central region.

We have written an in-house IDL code for measuring all of the six described
parameters of the selected sample of the non-AGN galaxies. To obtain all
needed input parameters we first run SExtractor (Bertin & Arnouts, 1996) using
the SDSS g-band images. When necessary, the galaxy centre is determined by
minimizing ASYM (Abraham et al., 1996), while the total flux is defined as the
one contained within 1.5 times the Petrosian radius (Huertas-Company et al.,
2008) as measured by SExtractor.

###  3.2 Morphological parameters of the local sample with added AGN
contributions

For analysing how the presence of an AGN may affect the different
morphological parameters and hence the final morphological classification of
active galaxies, we used the selected sample of star-forming (non-AGN)
galaxies and simulated AGN hosts by adding in their centers different AGN
contributions. For simulating the AGN contribution, for each galaxy we used
its corresponding point spread function (PSF) image in the g-band obtained
from the SDSS database (see Pović et al., 2013, for more details). To model
the PSF we used the Moffat function (Moffat, 1969), which showed to be its
good analytical approximation (e.g., Molina et al., 1992; Trujillo et al.,
2001). For each galaxy we constructed five simulated images using different
fractions of AGN contribution to the total galaxy flux: 5%, 10%, 25%, 50%, and
75%. The total galaxy flux was measured by running SExtractor code on the
g-band images. Once we obtained simulated images of galaxies with the
different AGN contributions added we repeated the steps from Sec. 3.1. We
first run SExtractor on all simulated images to obtain the different input
parameters needed for measuring the morphological parameters. Secondly, for
each AGN contribution of 5% - 75% we measured the six morphological parameters
following the procedure in Sec. 3.1.

Figure 4: The example of surface brightness profiles of early-type (top left), early-spiral (top-right), late-spiral (bottom left), and peculiar (bottom right) galaxy when using the original g-band image (blue line) and simulated once with AGN contributions of 5% (yellow line), 10% (green line), 25% (purple line), 50% (red line), and 75% (brown line). The black dashed and solid vertical lines indicate the PSF and galaxy Petrosian half-light radius. Table 1: Additional information of galaxies showed in Fig. 4, including: object ID, right ascension and declination, Petrosian radius at 50% and 90% of the total galaxy flux, total galaxy flux, morphology, Petrosian radius at 50% and 90% of the PSF flux, and PSF flux, respectively. Object ID | RA | DEC |  $R_{50}$% |  $R_{90}$% | Total-flux | Morphology | PSF $R_{50}$% | PSF $R_{90}$% | PSF Total-flux
---|---|---|---|---|---|---|---|---|---
| (deg) | (deg) | (pix) | (pix) | (counts) |  | (pix) | (pix) | (counts)
535-51999-573 | 218.32646 | 2.7014122 | 5.69 | 14.30 | 387.95 | early-type | 1.73 | 3.88 | 347.51
420-51871-506 | 13.712063 | 15.278116 | 12.51 | 26.31 | 407.49 | early-spiral | 1.74 | 3.76 | 357.42
476-52314-66 | 144.95767 | -0.56153035 | 17.74 | 27.96 | 362.99 | late-spiral | 1.71 | 3.90 | 326.41
615-52347-59 | 233.80957 | 57.514824 | 11.90 | 26.03 | 382.77 | peculiar | 1.95 | 4.13 | 432.98

Figure 4 shows the surface brightness profiles using the original image (blue
line) and the simulated images with 5% to 75% of AGN contribution, in the case
of an early-type galaxy (top left), an early-spiral (top right), a late-spiral
(bottom left), and a peculiar galaxy (bottom right). The two vertical lines in
each plot correspond to the Petrosian half-light radius (R50%) of the PSF
(dashed line) and galaxy (solid line). The size of each galaxy at 50% and 90%,
its total flux and corresponding PSF values are given in Table 1 as an
example.

Finally, as a result of the work described in Sec. 3.1 and here, for each
galaxy and each morphological parameter we obtained six measurements: the real
one using original images, and simulated ones when having 5% to 75% of AGN
contribution in comparison to the total galaxy flux.

##  4 Analysis and results

In this section, we measure the effect of the AGN contribution on the
morphological parameters when considering the total sample of selected
galaxies and when taking into account their morphological type. We also test
the effect of the AGN on some of the most used morphological diagrams when
dealing with the classification of galaxies.

###  4.1 Effect of the AGN on the morphological parameters of the total sample

Using the total sample of the 2301 selected sources described in Sec. 2, we
analysed in detail the effect of AGN on the six morphological parameters
described in Sec. 3.1. For each parameter obtained using original and
simulated images (with 5 different added AGN contributions), we measured their
median values, first quarter (Q1333Q1 corresponds to 25% of sample.), and
third quarter (Q3444Q3 corresponds to 75% of sample.), where the range Q1-Q3
corresponds to 50% of the sample. We used the median values of samples plus
the change in the [Q1-Q3] range as indicators of the change (or scatter) in
samples with AGN contribution.

Figure 5 shows the effect of the AGN contribution on the six morphological
parameters used in this work: ASYM (top left), CABR (middle left), GINI
(bottom left), M20 (top right), SMOOTH (middle right), and CCON (bottom
right). We represent how the median values of these parameters (solid lines)
change from the values measured on the original images (without AGN being
added, indicated in Fig. 5 as "Org") when 5%, 10%, 25%, 50% and 75% of AGN
contribution is added. We also represent the corresponding Q1 and Q3
measurements (dashed lines), that can be considered as an indicator of the
error.

Most of the parameters change with the added AGN contribution in comparison to
real values, with gradually larger differences being observed as the AGN
contribution to the total flux increases. In Table 2, the effect of the AGN
has been measured using the median values of each parameter and each sample
(total and for different morphological types), measured on original images
(param_orig) and after adding different AGN contributions (param_AGN), such
as: (param_AGN - param_orig)/param_orig. A total number of sources with
measured parameters under each sample is also listed in the table. CCON is one
of the most sensitive parameters to AGN contribution, facing 22% and 46% of
change (toward higher values) in case of 10% and 25% of added AGN
contribution, respectively. On the other side, ASYM is one of the parameters
that almost doesn’t change independently on AGN contribution. As can be seen
in Fig. 5 it takes median values between 0 and 0.13, and therefore we are not
listing any values for this parameter in Table 2. This parameter has been
measured for 72%, 91%, 71%, 61%, and 66% of the total sample, early-type,
early-spiral, late-spiral, and peculiar galaxies, respectively. After ASYM,
GINI is the least affected parameter changing only 8%, 16%, 27%, and 37% for
10% 25%, 50%, and 75% of AGN contribution, respectively. Up to 10% of AGN,
CABR and M20 show 10-15% of change. Similar to CABR and M20, SMOOTH is also
showing a low level of contamination below 25% of AGN contribution, however,
this parameter is very sensitive to noise, as shown previously (e.g., Huertas-
Company et al., 2008; Pović et al., 2013, 2015), and it could be measured
properly for a very small sub-sample (44%) of the total number of galaxies
(see Table 2). In general, we can observe that AGN contribution to the total
flux of 25% and above produce significant changes in the parameters of
$\gtrsim$ 30% (or $\gtrsim$ 20% in the case of GINI and M20), except in the
case of ASYM.

Figure 5: Distribution of median values (black solid lines) of ASYM (top left), CABR (middle left), GINI (bottom left), M20 (top right), SMOOTH (middle right) and CCON (bottom right) in relation to AGN contribution to the total flux from 0% up to 75%. Dashed lines correspond to Q1-Q3 range where 50% of the sample is located. Table 2: Quantification of the effect of 5% - 75% of AGN on the five morphological parameters in the case of the total sample, early-type, spiral early-type, spiral late-type, and peculiar galaxies. ASYM parameter doesn’t vary much with AGN contribution (see Fig. 5 and the text), and therefore, it is not listed in this table. Fraction of galaxies with good measurements of analysed morphological parameters of each sample is given in column 2, and in the text in the case of ASYM. Measured parameter | Number of sources | 5% | 10% | 25% | 50% | 75%
---|---|---|---|---|---|---
(sample) |  | AGN | AGN | AGN | AGN | AGN
CABR (total sample) | 2292 (100%) | 8% $\pm$ 2% | 13% $\pm$ 2% | 31% $\pm$ 5% | 49% $\pm$ 7% | 60% $\pm$ 8%
CABR (early-type) | 470 (100%) | 4% $\pm$ 1% | 7% $\pm$ 1% | 18% $\pm$ 2% | 27% $\pm$ 3% | 35% $\pm$ 4%
CABR (early-spiral) | 887 (100%) | 7% $\pm$ 1% | 13% $\pm$ 2% | 30% $\pm$ 3% | 49% $\pm$ 6% | 66% $\pm$ 9%
CABR (late-spiral) | 885 (100%) | 9% $\pm$ 2% | 18% $\pm$ 3% | 35% $\pm$ 2% | 55% $\pm$ 6% | 73% $\pm$ 9%
CABR (peculiar) | 50 (100%) | 3% $\pm$ 2% | 8% $\pm$ 2% | 24% $\pm$ 3% | 42% $\pm$ 10% | 58% $\pm$ 13%
GINI (total sample) | 2294 (100%) | 4% $\pm$ 1% | 8% $\pm$ 2% | 16% $\pm$ 5% | 27% $\pm$ 9% | 37% $\pm$ 12%
GINI (early-type) | 470 (100%) | 3% $\pm$ 1% | 5% $\pm$ 1% | 11% $\pm$ 3% | 20% $\pm$ 4% | 25% $\pm$ 5%
GINI (early-spiral) | 888 (100%) | 4% $\pm$ 1% | 8% $\pm$ 2% | 16% $\pm$ 5% | 27% $\pm$ 9% | 37% $\pm$ 11%
GINI (late-spiral) | 886 (100%) | 5% $\pm$ 2% | 9% $\pm$ 2% | 18% $\pm$ 4% | 33% $\pm$ 7% | 42% $\pm$ 9%
GINI (peculiar) | 50 (100%) | 2% $\pm$ 1% | 6% $\pm$ 2% | 12% $\pm$ 2% | 20% $\pm$ 3% | 30% $\pm$ 4%
M20 (total sample) | 2185 (95%) | 6% $\pm$ 1% | 11% $\pm$ 1% | 20% $\pm$ 2% | 21% $\pm$ 3% | 19% $\pm$ 3%
M20 (early-type) | 447 (96%) | 2% $\pm$ 1% | 4% $\pm$ 1% | 6% $\pm$ 1% | 4% $\pm$ 1% | 2% $\pm$ 2%
M20 (early-spiral) | 846 (95%) | 5% $\pm$ 2% | 11% $\pm$ 2% | 19% $\pm$ 2% | 21% $\pm$ 2% | 20% $\pm$ 2%
M20 (late-spiral) | 836 (94%) | 14% $\pm$ 2% | 27% $\pm$ 4% | 44% $\pm$ 7% | 47% $\pm$ 5% | 46% $\pm$ 5%
M20 (peculiar) | 48 (96%) | 3% $\pm$ 1% | 7% $\pm$ 1% | 19% $\pm$ 3% | 18% $\pm$ 2% | 17% $\pm$ 4%
SMOOTH (total sample) | 1023 (44%) | 4% $\pm$ 1% | 8% $\pm$ 1% | 29% $\pm$ 7% | 58% $\pm$ 15% | 87% $\pm$ 24%
SMOOTH (early-type) | 352 (75%) | 6% $\pm$ 5% | 16% $\pm$ 6% | 32% $\pm$ 15% | 64% $\pm$ 29% | 84% $\pm$ 33%
SMOOTH (early-spiral) | 347 (39%) | 6% $\pm$ 4% | 12% $\pm$ 4% | 24% $\pm$ 6% | 58% $\pm$ 5% | 87% $\pm$ 13%
SMOOTH (late-spiral) | 291 (33%) | 3% $\pm$ 2% | 9% $\pm$ 3% | 26% $\pm$ 2% | 78% $\pm$ 8% | 113% $\pm$ 9%
SMOOTH (peculiar) | 30 (60%) | 8% $\pm$ 4% | 13% $\pm$ 4% | 30% $\pm$ 5% | 65% $\pm$ 8% | 113% $\pm$ 29%
CCON (total sample) | 2186 (95%) | 10% $\pm$ 2% | 22% $\pm$ 6% | 46% $\pm$ 11% | 66% $\pm$ 14% | 67% $\pm$ 13%
CCON (early-type) | 457 (97%) | 5% $\pm$ 1% | 9% $\pm$ 1% | 17% $\pm$ 2% | 18% $\pm$ 2% | 18% $\pm$1%
CCON (early-spiral) | 855 (96%) | 10% $\pm$ 1% | 22% $\pm$ 3% | 53% $\pm$ 6% | 72% $\pm$ 7% | 73% $\pm$ 6%
CCON (late-spiral) | 836 (94%) | 11% $\pm$ 3% | 23% $\pm$ 3% | 50% $\pm$ 5% | 76% $\pm$ 8% | 78% $\pm$ 9%
CCON (peculiar) | 49 (98%) | 10% $\pm$ 3% | 19% $\pm$ 4% | 35% $\pm$ 5% | 51% $\pm$ 5% | 68% $\pm$ 8%

In addition to the impact of AGN on measured morphological parameters, we also
checked the effect of AGN in relation to the galaxy size. In Fig. 6, we showed
the median effective radius measured on original and simulated (with 5-75% of
AGN) images for a total sample (top panel) and for two different ranges of
absolute magnitude (g-band Mab $<$ -25 (middle panel) and Mab $>$ -25 (bottom
panel)). Just as an example, we showed also the relative change of CABR
parameter. We can observe that in all three cases for a total sample and the
two samples with different g-band absolute magnitude, the effective radius
starts to be more affected when AGN contribution is above 25%, indicating that
smaller galaxies will be more affected. In all three cases, independently on
brightness, the change in the effective radius is $<$ 10% up to 10% of AGN
contribution and becomes of the order of 20%, 40%, and 60% for larger AGN
contributions of 25%, 50%, and 75%. The median values of the effective radius
obtained when adding 50% and 75% of AGN, correspond to only 7% and 1% of
sources in the original sample, respectively. We also checked the effect of
AGN on the Petrosian radius, finding insignificant effect of $<$ 4% up to 25%
of AGN contribution, increasing to 13% and 28% of change for 50% and 75% of
AGN contribution, respectively. Finally, we also checked the effect of size on
morphological parameters without having an AGN. In general, we do not observe
any significant effect of size on ASYM, SMOOTH, M20 (except a small effect of
$<$ 20% for ET galaxies), and CCON parameters. A slight change in CABR with
size has been observed for a total sample, where more compact galaxies (with
CABR $\sim$ 0.5) have $\sim$ 25% smaller sizes in comparison to less compact
galaxies (CABR $\sim$ 0.2 - 0.3). This effect is a bit stronger in the case of
the GINI parameter, where more compact galaxies with GINI $\sim$ 0.7 have
$\sim$ 40% smaller sizes in comparison to less compact galaxies (GINI $\sim$
0.35).

Figure 6: The effective radius (Re) as a function of AGN contribution for a
total sample (top), and sources with the g-band absolute magnitude of Mab $<$
-25 (middle) and Mab $>$ -25 (bottom). The colour bar shows the relative
change of CABR morphological parameter as an example. The effective radius is
represented through median values, while the vertical lines indicate the Q1
and Q3 quartiles.

###  4.2 Effect of AGN on the studied parameters for the different
morphologies

In this section, we present a similar analysis as above, but now taking into
account morphology. We use the four broad morphological classes as described
in Sec. 2.

Figures 7 to 10 show how the values of the six morphological parameters change
when including an AGN contribution from 0% to 75% of early-type, early-spiral,
late-spiral, and peculiar galaxies, respectively. Table 2 provides the
quantification of the change in these parameters for each morphological type
depending on the AGN contribution.

Figure 7: Same as Fig. 5, but only for early-type galaxies. Figure 8: Same as
Fig. 5, but only for early-spiral galaxies. Figure 9: Same as Fig. 5, but only
for late-spiral galaxies. Figure 10: Same as Fig. 5, but only for peculiar
galaxies.

Table 2 shows that the change in all morphological parameters with the AGN
contribution is different for different morphological types. In general, when
dealing with early-type galaxies, light concentration parameters such as CABR,
GINI, M20, and CCON are less affected by the AGN contribution than in the case
of late-type galaxies (both early- and late-spirals and peculiar), while
SMOOTH is more sensitive to the AGN contribution when measured for early-type
galaxies than for late-types. Similar as in the case of the total sample, ASYM
parameter doesn’t change much with AGN contribution, independently on
morphology. Besides ASYM, GINI is the most stable parameter to be used when
dealing with morphologies of active galaxies, independently on the
morphological type. In addition to this, M20, CABR, and CCON are also quite
stable for active early-type galaxies for AGN contributions up to 25%. In
general, the morphological parameters of late-type galaxies, in particular
late-spirals and peculiar, are more affected by the AGN, especially for AGN
contribution of 25% and above.
Finally, to show the behaviour of the different morphological types in
relation to AGN contribution, all median values are plotted in Fig. 11. It can
be seen the distribution of early-type, early-spiral, late-spiral, and
peculiar galaxies showed previously in Figs. 7 to 10. This figure can help us
to put constraints on each of the parameters regarding their use in
morphological classification of active galaxies, that will be discussed in
more detail in Sec. 5.

Figure 11: Comparison of the effect of 5% - 75% of AGN contribution on early-
type (in red), early-spiral (in blue), late-spiral (in green), and peculiar
(in magneta) galaxies. Median values of each sample are used to represent each
distribution, as shown in Figs. 7 to 10.

###  4.3 Effect of the AGN contribution on the morphological diagrams

In this section, we analyse the effect of the AGN contribution on four of the
most commonly used morphological diagnostic diagrams, that we selected since
they have been broadly used in previous studies to separate between different
morphological types (e.g., Abraham et al., 1994, 1996, 2003; Conselice et al.,
2000, 2003; Lotz et al., 2004; Yagi et al., 2006; Scarlata et al., 2007;
Pierce et al., 2010; Pović et al., 2013, 2015; Pintos-Castro et al., 2016;
Tarsitano et al., 2018; , 2019; Mahoro et al., 2019). Figures 12 to 15 show
the following diagrams: ASYM vs. CABR, CABR vs. GINI, CCON vs. CABR, and CCON
vs. M20, respectively. In each figure, we represent the original local sample
of non-active galaxies (top-left plot), and simulated active galaxies with
added 5% (top-right), 10% (middle-left), 25% (middle-right), 50% (bottom-
left), and 75% (bottom-right) of AGN contribution to the total flux. In all
plots we represent the location of early-type (red), early-spiral (blue),
late-spiral (green), and peculiar (magenta) galaxies. The distribution of each
parameter for the four morphological types are also represented as attached
histograms on the top and to the right axes of each diagram. In line with the
results obtained in the previous two sections, in general, it can be observed
that for up to 10% of AGN contribution the distribution of simulated galaxies
does not change significantly in comparison to the original samples. However,
starting from 25% of AGN contribution larger contamination can be observed
between different morphological types. Mix between early- and late-types
becomes significantly larger when having strong AGN contribution of 50% and
75% to the total galaxy flux, making difficult the separation between these
two types as discussed more in Sec. 5. For easier comparison with the original
sample, in all histograms, as a reference we introduced the vertical lines
that correspond to the median values of parameters of different morphological
types. Quantification for each parameter is given in Table 2.

Figure 12: Relation between ASYM and CABR morphological parameters of the
original sample of non-active galaxies (top-left plot), and simulated active
galaxies with added 5% (top-right), 10% (middle-left), 25% (middle-right), 50%
(bottom-left), and 75% (bottom-right) of AGN contribution to the total flux.
Top and right histograms show the distribution of CABR and ASYM parameters,
respectively, of different morphological types. In all, central plot and
histograms we represent the distribution of early-type (red), early-spiral
(blue), late-spiral (green), and peculiar (magenta) galaxies. In all
histograms, as a reference, the red, blue, green, and magenta vertical lines
indicate the median values of morphological parameters of the original sample
of early-type, early-spiral, late-spiral, and peculiar galaxies, respectively.

Figure 13: Same as in Fig. 12, but showing the relation between CABR and GINI
parameters.

Figure 14: Same as in Fig. 12, but showing the relation between CCON and CABR
parameters.

Figure 15: Same as in Fig. 12, but showing the relation between CCON and
$M_{20}$ parameters.

##  5 Results and Discussion

As shown in Sec. 4., in general, when the light coming from an AGN is $\geq$
25% of the total galaxy light, most of the used morphological parameters and
therefore the morphological classification of AGN host galaxies can be
significantly affected. We found that the classification of early-type
galaxies (elliptical and lenticular) hosting an AGN will be more accurate than
when dealing with spiral and peculiar hosts. Difficulties in classifying late-
type galaxies in comparison to early-types have been already observed in the
past, e.g., when studying the effect of survey resolution and depth on the
morphological classification of galaxies (e.g, Pović et al., 2015, and
references therein).
In general, when introducing the different AGN contributions, ASYM parameter
varies very little. However, taking into account previous studies, this
parameter is sensitive to noise (Pović et al., 2013) and is measured for a
smaller sample of galaxies, and therefore in galaxy classification it shall be
used in combination with other parameters. In addition to ASYM, we found GINI
to be one of the most stable parameters in the case of both early- and late-
types, showing variations $<$ 20% up to the 25% of AGN, which still permits
the separation between early- and late-types. GINI has been also suggested to
trace well sub-structures in galaxies (e.g., Lisker et al., 2006), mergers and
interacting galaxies (e.g., Lotz et al., 2004, 2008a, 2008b), strong-lensed
galaxies (e.g., Florian et al., 2016a, b), and to correlate with stellar mass
tracing the overall structure of a galaxy better than any other morphological
parameter (e.g., Zamojski et al., 2007). However, it has been also shown that
GINI depends strongly on the signal-to-noise ratio and the choice of the
aperture within which it is measured (e.g., Lisker et al., 2008; Pović et al.,
2015). In addition to this, we saw that light concentration parameters in
general (including CABR, CCON, and M20) are less affected by AGN in early-type
than in late-type galaxies, and more stable than SMOOTH parameter which shows
higher changes when an AGN is present (although lower when dealing with late-
type sources), as shown in Table 2.
Previous findings refer to the observed variation in parameters when different
AGN contributions are taken into account. However, not necessarily a stronger
variation means that a parameter is less suitable for morphological
classification. We can observe this in Fig. 11. ASYM parameter shows to have
well separated median values of all four morphological types even up to 50% of
the AGN contribution. CABR and GINI also shown to be efficient for separating
early- and late-type (early-spiral, late-spiral, and peculiar) galaxies
independently on AGN contribution, where GINI shows better separation between
different late-types up to 25% of AGN. CCON can be also used in classifying
early- and late-type AGN hosts up to 25% of AGN. M20 shows strong mix between
morphological types, starting from 10% of AGN contribution, while SMOOTH
parameter shows to be inefficient in classifying active galaxies. These
findings with median distributions of parameters can be compared with those in
Figs. 12 to 15 where the whole samples are represented and similar trends can
be observed. Morphological diagrams based on ASYM vs. CABR (Fig. 12) and CABR
vs. GINI (Fig. 13) parameters remain much more stable on AGN contribution,
reaching levels of contamination of early-types by late-types of $\sim$ 30 -
35% for 25% of AGN contribution, with a slight increment of 5-10% when dealing
with stronger AGN. We can see in both Fig. 12 and 13 that even at 50% and 75%
of AGN contribution to the total optical light, these diagrams still maintain
different distributions of early- and late-type galaxies. On the other hand,
when using CCON and/or M20 parameters in Figs. 14 and 15 contamination is $>$
40% for 25% of AGN contribution, while for stronger AGN, distributions of
early- and late-type galaxies almost totally overlap. Therefore, taking all
previous into account, we recommend GINI, CABR, and ASYM to be more suitable
to be used in non-parametric methods of morphological classification of AGN
host galaxies, even if combined with machine learning. However, we remind once
again that for stronger AGN (above 25% of contribution to the total light),
special attention should be taken into account for a reliable classification
of AGN hosts at z $\sim$ 0, considering the possibility to quantify the AGN
contribution to the total optical light of a galaxy and to remove the AGN
contribution before going through any detailed morphological classification.
To our knowledge, only a few of the previous studies considered the effect of
the AGN contribution on the morphological classification of their host
galaxies. Cardamone et al. (2010) studied the effect of AGN on the rest-frame
colours using a sample of X-ray detected obscured and unobscured AGN in the
Extended Chandra Deep Field-South (ECDF-S) at 0.8 $\leq$ z $\leq$ 1.2. They
found an insignificant effect of the AGN on the galaxy colour in the case of
moderate luminosity, obscured AGN, with a more significant effect for very
blue, very luminous AGN (sources out of the blue cloud being classified as
quasars, see their Fig. 1). A similar study considering the effect of AGN on
the host colours has been carried out by Wang et al. (2017) for AGN of
moderate X-ray luminosity up to the redshift of z $\sim$ 2.5; again
significant difference is found in the distribution of U - V rest-frame colour
between active and non-active galaxies. These results go in line with several
others showing that in moderate luminosity AGN, the AGN contribution to the
total optical light is not significant (e.g., Silverman et al., 2008; Xue et
al., 2010; Simmons et al., 2011).
Gabor et al. (2009) studied the effect of AGN on two morphological parameters,
asymmetry, and concentration. They used the same asymmetry parameter as
defined in our work, finding an insignificant difference between asymmetries
of active and non-active galaxies. However, in their concentration index,
defined similarly to CCON, but using $r_{90}$ instead of $r_{80}$, the authors
found much larger values of the concentration index when an AGN is present,
that is in line with our findings and Fig. 11. The most similar work to ours
is that by Pierce et al. (2010), where they used a sample of $\sim$ 600 non-
active galaxies from the AEGIS survey (Davis et al., 2007) at 0.2 $<$ z $<$
0.6, and analysed the effect of an AGN on four morphological parameters: Gini,
asymmetry, Concelice concentration index, and M20 moment of light. They
concluded that AGN contributions above 20% have an impact on most of the
morphology measurements that depend on the morphological characteristics of
the original galaxies. In comparison to Pierce et al. (2010), we provide a
more detailed analysis, considering a larger sample of testing non-active
galaxies, with precise known visual morphologies, testing five different AGN
contributions, and adding two more parameters (CABR and SMOOTH) that were not
analysed previously, and that very often are used simultaneously with other
parameters in the morphological classification of galaxies (e.g., Huertas-
Company et al., 2008, 2009; Pović et al., 2013; Pintos-Castro et al., 2016; ,
2019; Mahoro et al., 2019)
Finally, we recall that the results obtained in this paper refer to the AGN
effect on the morphological properties of their host galaxies at z $\sim$ 0\.
When going to higher redshifts and fainter magnitudes, and due to the effect
that both the survey depth and resolution have on the morphological
classification of galaxies, the study will be more complex, as mentioned above
and analysed in detail in Pović et al. (2015). We plan to proceed with this
approach in a forthcoming paper (Getachew et al., in prep.).

##  6 Summary and conclusions

In this paper, we provide a detailed study regarding the effect of the AGN on
the morphological properties of their host galaxies and the corresponding
morphological classification. We studied six parameters commonly used in non-
parametric methods, often in combination with machine and deep learning, to
classify galaxies in terms of their morphology. These parameters are: Abraham
concentration index (CABR), Gini coefficient (GINI), Conselice-Bershady
concentration index (CCON), $M_{20}$ moment of light (M20), asymmetry index
(ASYM), and smoothness (SMOOTH). We used the sample of 8000 local galaxies
with visual morphological classification in g-band (Nair & Abraham, 2010) to
select a sample of 2301 visually classified non-active local galaxies. We
simulated their AGN counterparts by adding in their centers an AGN
contribution of 5%, 10%, 25%, 50%, and 75% of the total galaxy flux. For the
AGN construction for each galaxy we used its corresponding PSF, described by
the Moffat function. We then measured in a consistent way the six
morphological parameters in both the original images and all simulated images.
We finally studied how the presence and relative contribution of the AGN may
affect each of the morphological parameters in relation to galaxy morphology,
and the final classification of active galaxies. Our main findings are the
following:

  * •

When observing the total sample of galaxies, independently on their
morphology, we observe that significant change of $\sim$ 30% in parameters
corresponds to AGN contributions to the total optical flux of 25% and above,
except in ASYM which almost doesn’t vary when AGN is added.

  * •

An impact of AGN on different morphological parameters vary depending on
galaxy morphology. Light concentration parameters (e.g., CABR, GINI, M20, and
CCON) are less affected by AGN in early-type galaxies, while SMOOTH is less
affected in late-types. However, in general, spiral and peculiar galaxies will
be more affected by AGN contribution than early-type galaxies.

  * •

GINI and ASYM have been found to be some of the most stable parameters when
dealing with morphological classification of active galaxies, in the case of
both early- and late-types, in combination with CABR. CCON and M20 can be used
for classifying both early- and late-type AGN hosts up to 10% - 25% of AGN
contribution to the total optical light, but not higher. We suggest avoiding
SMOOTH parameter when classifying active galaxies, or to use it at least in
combination with other parameters.

  * •

In line with previous, morphological diagrams based on the combination of
ASYM, CABR, and GINI parameters are much more stable, and even when dealing
with 50% and 75% of AGN contribution, a significant fraction of $\sim$ 50% of
early- and late-type galaxies can still be classified. This is not the case
when either CCON and/or M20 parameters are included, when $>$ 40%
contamination between early- and late-types has been obtained for 25% of AGN
contribution, and $>$ 60% in case of 50% and 75% of AGN contribution to the
total optical light.

## Acknowledgments

We thank the anonymous referee for accepting to review this paper, giving us
constructive comments that improved our manuscript. TGW acknowledges the
support from Bule Hora University under the Ministry of Science and Higher
Education. TGW, MP, and ZBA acknowledge financial support from the Ethiopian
Space Science and Technology Institute (ESSTI) under the Ethiopian Ministry of
Innovation and Technology (MInT). TGW, JM, and MP acknowledge support given
through the grant CSIC I-COOP 2017, COOPA20168. MP, JM, JP, and IMP
acknowledge the support from the Spanish Ministerio de Ciencia e Innovación -
Agencia Estatal de Investigación through projects PID2019-106027GB-C41 and
AYA2016-76682C3-1-P, and the State Agency for Research of the Spanish MCIU
through the Center of Excellence Severo Ochoa award to the Instituto de
Astrofísica de Andalucía (SEV-2017-0709). In this work, we made use of Virtual
Observatory Tool for OPerations on Catalogues And Tables (TOPCAT).

## DATA AVAILABILITY

In support of this study, no new data were generated or analyzed. The data
used in this article can be obtained from the public sources cited in the
article (or references therein).

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