Morphology of relaxed and merging galaxy clusters. Analytical models for monolithic Minkowski functionals - arXiv

MNRAS 000, 1–12 (2019) Preprint 1 February 2021 Compiled using MNRAS LATEX style file v3.0

 Morphology of relaxed and merging galaxy clusters.
 Analytical models for monolithic Minkowski functionals

 1
 C. Schimd,1★ M. Sereno2,3
 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
 2 INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Piero Gobetti 93/3, I-40129 Bologna, Italy
 3 INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
arXiv:2101.12634v1 [astro-ph.CO] 29 Jan 2021

 Accepted 2021 January 25. Received 2021 January 19; in original form 2020 July 23

 ABSTRACT
 Galaxy clusters exhibit a rich morphology during the early and intermediate stages of mass
 assembly, especially beyond their boundary. A classification scheme based on shapefinders
 deduced from the Minkowski functionals is examined to fully account for the morphological
 diversity of galaxy clusters, including relaxed and merging clusters, clusters fed by filamen-
 tary structures, and cluster-pair bridges. These configurations are conveniently treated with
 idealised geometric models and analytical formulae, some of which are novel. Examples from
 CLASH and LC2 clusters and observed cluster-pair bridges are discussed.
 Key words: galaxies: clusters: general – cosmology: observations

 1 INTRODUCTION outskirts of Abell 2744 probed by XMM-Newton X-ray data (Eck-
 ert et al. 2015); the proto-clusters revealed by Herschel-SPIRE from
 Morphology of galaxy clusters is an indicator of their state of relax-
 Planck candidates (Greenslade et al. 2018) or combining VUDS and
 ation and can be used to infer their formation history and evolution.
 zCOSMOS-Deep data (Cucciati et al. 2018); the filaments bridg-
 As result of the gravitational dynamics of dark and luminous matter,
 ing the cluster systems A399-A401 and A3016-A3017, detected
 relaxed galaxy clusters and their hosting dark matter haloes have a
 combining Planck data with ROSAT (Planck Collaboration et al.
 triaxial shape (Limousin et al. 2013), with tendency to prolateness
 2013) or Chandra (Chon et al. 2019); the gaseous and dusty bridge
 over oblateness especially in their final stage of evolution as as-
 IRDC G333.73+0.37 (Veena et al. 2018); the molecular filamentary
 sessed by high-resolution -body simulations (e.g. Bett et al. 2007;
 structures around Centaurus, Abell S1101, and RXJ1539.5 probed
 Macciò et al. 2007; Despali et al. 2014; Bonamigo et al. 2015) and
 by ALMA and MUSE (Olivares et al. 2019); the multiple filaments
 confirmed by X-ray, optical, Sunyaev-Zel’dovich (SZ), and weak-
 within the SSA22 protocluster (Umehata et al. 2019). Weak gravita-
 lensing measurements (Cooray 2000; De Filippis et al. 2005; Sereno
 tional lensing analyses have been successful in detecting the dense
 et al. 2006, 2018b). The persistence of this trend in the outskirts of
 environment and the correlated dark matter around the main cluster
 clusters depends on their mass (Prada et al. 2006), mass accretion
 halo (Sereno et al. 2018a).
 rate (Diemer & Kravtsov 2014), and assembly history (Dalal et al.
 2008; Faltenbacher & White 2010; More et al. 2016). The increasingly large samples of haloes detected in optical
 The three-dimensional shape of these structures is normally (Rykoff et al. 2014; Oguri et al. 2018; Maturi et al. 2019), X-ray
 described by the eigenvalues of the mass distribution or inertia ten- (Pierre et al. 2016), or SZ surveys (Bleem et al. 2015; Planck Col-
 sors and related parameters such as sphericity, elongation, ellipticity, laboration et al. 2016) demand for flexible and reliable indicators of
 prolateness, and triaxiality (Springel et al. 2004). These statistics morphology that can be applied to the full zoo of galaxy clusters. A
 are well-suited for dynamically evolved or poorly resolved clusters, number of statistics, such as halo concentration, peak-centroid shift,
 however they cannot account for the rich morphology of unrelaxed power ratio, axial ratio, and position angle, have been considered
 structures or beyond the virial radius shown by high-quality imag- to quantify the degree of regularity and symmetry of these struc-
 ing and spectroscopy. New instruments are opening indeed a golden tures (Donahue et al. 2016; Lovisari et al. 2017). However, these
 age for a multi-wavelenth study of protoclusters, merging clusters indicators can fail for very irregular systems. A cluster progenitor
 and their filamentary environment at both low and high redshift. experiences very different shapes during the merger history and the
 The clearest example in the local universe is the Virgo cluster with configuration of satellite halos and local environment dramatically
 its different substructures identified using GUViCS (Boselli et al. changes. Major mergers can be followed by slow accretion along
 2014) and HyperLeda (Kim et al. 2016) data. At intermediate and filaments until the cluster ends up in a relatively viralized final phase
 high redshift, some spectacular illustration of rich structures are: the with a nearly regular and spherical shape. We aim at finding a small
 set of morphological parameters that can in principle describes all
 the different phases of the merging accretion history.
 ★ E-mail: carlo.schimd@lam.fr In this paper, we propose to use the three non-trivial Minkowski

 © 2019 The Authors

2 C. Schimd, M. Sereno
functionals to fully characterise the morphology of spatial struc-
tures (Mecke et al. 1994; Mecke 2000). We show that very different
morphologies, namely major mergers, multiple mergers, and fil-
amentary structures can be suitably described by a single set of
geometrically motivated parameters. We calculate analytical ex-
pressions for the triaxial ellipsoid, a -fused balls model accounting
for non-relaxed clusters undergoing merging, a spiky model with
 cylindrical branches radially connected to a central ball possibly
accounting for filaments of matter feeding a central halo, and a
dumbbell model describing axially-symmetric cluster-pair bridges
(§2; details of the calculations reported in the Appendices). These
systems are then classified using the so-called shapefinders deduced
from the Minkowski functionals (§3). Conclusions are addressed in
§4.

2 MORPHOLOGY BY MINKOWSKI FUNCTIONALS:
 MODELS
The Minkowski functionals are a complete set of morphological Figure 1. Minkowski functionals iso-contours for an ellipsoid, E , as
descriptors that characterise the geometry and topology of a contin- function of the intermediate-to-major and minor-to-major axis ratio, =
 / and = / . Volume (solid lines; = 0), surface (dashed;
uous body. In three dimensions they are its volume ( ), surface area
 = 1), and integrated mean curvature (dotted; = 2) are shown
( ), integral mean curvature ( ) and integral Gaussian curvature
 for values ranging from (0.05,0.7,0.1) and increasing in steps Δ =
( ) of the surface, the latter being linearly related to the Euler char- {0.5, 0.25, 0.01} ℎ −3 Mpc3− . The underlying density plot represents the
acteristic that counts the number of connected components minus triaxiality . Points with error bars are CLASH clusters from Sereno et al.
the number of tunnels plus the number of cavities (Mecke 2000). (2018b, colour coded by redshift; point size proportional to the mass) and
According to a characterisation theorem, the Minkowski functionals Chiu et al. (2018, black), softly following the median prolatness-ellipticity
are the only valuations invariant under rotations and translations and relation of Despali et al. (2014, white long-dashed line).
preserving additivity and continuity (Hadwiger 1957). These prop-
erties along with the Steiner formula allow the calculation of 0 ≡ ,
 1 ≡ /6, and 2 ≡ /3 , the fourth functional 3 ≡ = 1 being et al. (1999), one has
trivial for isolated bodies with no tunnels and cavities as here.
 4 3
 0E∗ = , (2a)
 3
   
 2 1
2.1 Ellipsoidal model: relaxed clusters 1E∗ = 1+ , (2b)
 3 
 2 
Nearly viriliazed clusters can be conveniently described as ellip- 2E∗ = [ + ( )] , (2c)
soidal haloes. For a triaxial ellipsoid E with principal semi-axes 3
 √
 > > , with defining the polar axis and ( , ) the equa- where ( ) = (arccos )/ 1 − 2 , and { = , = 1/ , ( ) =
torial plane, namely with ≡ / and ≡ / respectively the ( )} for prolate ellipsoids (E∗ = E P ), { = , = , ( ) =
intermediate-to-major and minor-to-major axis ratio, the non-trivial −1 ( −1 ) for oblate ellipsoids (E∗ = E O ).1
Minkowski functionals are Equations (1-2) reduce to the familiar expressions for a sphere
 4 3 S ( = = ), viz. 0S = 4 3 /3, 1S = 2 2 /3, and 2S = 4 /3.
 0E = , (1a) As illustrated in Figure 1, the surface area 1 and the integrated
 3
   mean curvature 2 are nearly degenerate with the volume 0 for
 2 2 
 1E = 1 + ( , ) + 2 ( , ) , (1b) nearly prolate shapes or orthogonal for nearly oblate structures. They
 3 follow the trend of the triaxiality parameter = (1 − 2 )/(1 − 2 ),
 
 2E = ( + ), (1c) which distinguishes oblate ( & 0) from prolate ( . 1) structures
 3 1 2 and can be used to define three broad morphological classes (Chua
 √ et al. 2019, long dashed lines). Coloured points (size proportional
in which = 1 − 2 , = −1 [1 − ( / ) 2 ], ( , ) and ( , )
 to the mass, colour-coded by redshift) have been obtained for the
are elliptic integrals of first and second kind with sin = 
 Cluster Lensing and Supernova Survey with Hubble (CLASH) clus-
(Abramowitz & Stegun 1970), and 1,2 are dimensionless integrals
 ters (Sereno et al. 2018b). Their 3D shape are constrained with a
that we evaluate numerically in the general case; see equations (A8)
 multi-wavelength analysis combining the surface mass density as
in Appendix A.
 determined by gravitational lensing, which probes the size in the
 Analytic limits of the previous equations exist for prolate el-
 plane of the sky, and X-ray and SZ data, to infer the radial extent
lipsoids of revolution about axis ( > = , so that = , = 0,
 (Sereno 2007). With convenient priors, some less strong constraints
 = = arcsin ), which could account for virialised clusters, and
 on the 3D shape can be still determined based on lensing alone
for oblate ellipsoids of revolution about axes ( = > , so that
 = 1, = 1, = arctanh , = ), which could account for an in-
termediate stage of merging. Following the notation in Schmalzing 1 Our results slightly differ from Schmalzing et al. (1999).

 MNRAS 000, 1–12 (2019)

Morphology of relaxed and merging galaxy clusters 3
(Chiu et al. 2018, black points). These points softly follows the
median prolatness-ellipticity relation of Despali et al. (2014, white
long-dashed line) that fits ΛCDM -body simulations.

2.2 Fused-balls model: merging clusters
Clusters are usually neither relaxed nor isolated. The complex mor-
phology of merging clusters, or a central cluster with satellite haloes
can be conveniently pictured as a group of partially overlapping
balls. In this subsection, we consider first the case of a major merger
( = 2 balls) and then the case of satellite halos ( > 2 balls).

Major mergers. The Minkowski functionals of two merged balls
2 M = ∪ cannot be calculated like for the ellipsoid because
 1 2
the surface is not regular enough to uniquely define the fundamental
form. Instead, they can be calculated using additivity, ( 1 ∪ 2 ) =
 ( 1 ) + ( 2 ) − ( 1 ∩ 2 ). For two merged balls with unequal
radii and 6 and centres at distance 6 + , the volume and
surface area are trivial (see also Gibson & Scheraga 1987), while
 Figure 2. Minkowski functionals iso-contours for major mergers, M
the integrated mean curvature can be calculated using the Steiner (Eqs. 3), as function of the smaller ball radius and distance from the ma-
formula; see Appendix B. One finally obtains jor ball with radius . Volume (solid lines), surface (dashed), and integrated
 2 3
 
 1
 
 mean curvature (dotted) levels increase by 10% moving top-right from the
 0M = + 3 − 3 + ( 2 + 2 ) + ( 2 − 2 ) 2 , lower values attained for a single ball, S (for 2M , also the contours corre-
 3 8 2 4 
 sponding to 93 and 96 % of 2S are shown). The lower (‘total embedding’)
 (3a)
 and upper (‘no overlap’) triangular regions account for trivial morphologies
 
 1M = ( 2 + 2 ) + ( + ) + ( − )( 2 − 2 ), (3b) of one and two isolated balls, respectively. Points indicate major mergers
 3 6 6 from the LC2 cluster catalogue (Sereno 2015, colour-coded by redshift as
 √︄
 2 in figure 1, size proportional to 200c ). Filled (empty) symbols designate a
 2 + 2
  2
 2 − 2
 2M = ( + + ) − 2 − 1 − , (3c) merging subclump at the pericenter (splashback) for a system at = 0.3;
 3 3 2 2 see Table 1.

with cos = ( 2 + 2 − 2 )/2 . These equations are defined for
non-trivial merging, i.e. as long as the two spheres overlap with no
total embedding ( 1 ∩ 2 ≠ ∅ and 2 * 1 , i.e. − 6 ); for Table 1. LC2 merging clusters (Sereno 2015): parameters of two-balls model
non-overlapping spheres the correct expression is recovered setting ( ≡ 200c , flat ΛCDM cosmology). For comparison (lower part of the
 = + . table), indicative values for the splashback and pericenter phase of some
 The results are illustrated in Figure 2 as function of the radius major merger. Lengths in ℎ −1 Mpc.
of the smaller ball and separation between the centres, both nor-
 Name redshift 
malised to the radius of the larger ball. Note that for major ( . )
and advanced (  ) mergers, 0 and 1 are nearly degenerate. Abell 1750 0.0678 0.98 ± 0.19 0.85 ± 0.20 0.57 ± 0.06
As reference, the values for the major-mergers ( ∼ ) from the Abell 901 0.16 0.88 ± 0.15 0.77 ± 0.19 0.85 ± 0.08
LC2 catalogue (Sereno 2015) calculated assuming a flat ΛCDM Abell 115 0.197 1.13 ± 0.10 1.07 ± 0.11 0.63 ± 0.06
 Zw Cl2341 0.27 0.87 ± 0.15 0.86 ± 0.15 0.73 ± 0.07
cosmology with Ωm = 0.3 and ℎ = 0.7 and ≡ 200c are shown,
 Abell 1758 0.28 0.95 ± 0.26 0.68 ± 0.17 1.46 ± 0.14
along with the characteristic splashback (Diemer et al. 2017) and Bullet Cluster 0.296 1.91 ± 0.09 1.66 ± 0.09 0.50 ± 0.05
pericenter values estimated for binary systems at redshift = 0.3 MACS J0025 0.5842 1.15 ± 0.28 1.06 ± 0.23 0.45 ± 0.05
with main halo mass 200c = 1014 ℎ−1 M and secondary halo with CLJ0102-4915 0.87 1.10 ± 0.05 0.96 ± 0.05 0.52 ± 0.05
3 or 10 times smaller mass;3 see Table 1. 1 = 1014 ℎ −1 M , 2 = 1 /10:
 For balls with equal radius ( = ) the Minkowski functionals splashback O 0.3 1.47 0.68 2.03
of the resulting body M P are well-known, pericenter H 0.3 1.47 0.68 0.20
 1 = 1014 ℎ −1 M , 2 = 1 /3:
 1 3
  
 4 3 3 
 0M P = 1+ − , (4a) splashback M 0.3 1.47 0.68 2.03
 3 4 16 3 pericenter N 0.3 1.47 0.68 0.20
  
 2 2 
 1M P = 1+ , (4b)
 3 2 with limits 0S = 4 3 /3, 1S = 2 2 /3, and 2S = 4 /3 when
 √︄
 4 
 
 
 
 2 2
 
 2
  = 0, i.e. when M P becomes a sphere, S.
 2M P = 1+ − 1− arccos 1 − . (4c)
 3 2 3 4 2 2 2
 Multiple mergers. Equations (3) can be extended to the easiest
 configuration for multiple merging, i.e. a central ball (halo) of
2 We denote ≡ [ , ] the -th ball centred in with radius , radius intersecting smaller balls (satellites) of radius ,
omitted for clarity. with centres at distance from the centre of and not mutually
3 
 200c denotes the mass enclosed within a sphere of radius 200c with intersecting ( ∩ = ∅; , = 1, . . . , ; = +1). The Minkowski
mean over-density 200 times the critical density. functionals of the simply-connected resulting body M = ∪

MNRAS 000, 1–12 (2019)

4 C. Schimd, M. Sereno

Table 2. Observed cluster-pair bridges of single (upper table) or stacked through thick filaments (e.g. Werner et al. 2008; Dietrich et al. 2012;
(lower table) systems. The compilation is restricted to clusters pairs with Planck Collaboration et al. 2013; Bonjean et al. 2018, Table 2, rows
similar radius ≈ = 200 , separated by and with filament radius . 1-3), also detected by stacking techniques (Clampitt et al. 2016;
Lengths in ℎ −1 Mpc. Epps & Hudson 2017; Tanimura et al. 2019; de Graaff et al. 2019,
 Table 2, rows 4-7).
 Cluster/data set redshift The morphology of an axially-symmetric body defined by
 A222 - A223 (•) 0.21 1.2 15 ± 3 0.6 two balls connected by a cylinder, D = 1 ∪ 2 ∪ , can be
 A399 - A401 (N) 0.073 1.70 3 1.52 ± 0.09 deduced from the previous equations using additivity and noting
 A21 - PSZ2 G114.9 (H) 0.094 1.36 4.2 0.92 that the sum of the Minkowski functionals of the two spherical
 caps chopped by the cylinder bases, L1,2 = 1,2 ∩ , are equiv-
 SDSS/DR17 (LRG) 0.2–0.5 0.5–1 6–14 &1
 BOSS + CFHTLenS 0.3–0.6 1.25 7.1 ± 1 1.25
 alent to the Minkowski functionals of the lens L = L1 ∪ L2 as
 SDSS/DR12 + Planck 0–0.4 1.35 6–10 6 0.5 reported in Appendix B. After some algebra and recognising the
 CMASS + Planck ∼ 0.55 0.5–1 6–14 6 2.5 two-fused balls model, one obtain (D) = M + ( ). The ex-
 act though cumbersome mathematical expression combines Equa-
 tions (3) for two balls of radius and separated by an effective
 distance = ( 2 − 2 ) 1/2 + ( 2 − 2 ) 1/2 , and the well-known
Ð 
 =1 (so M 1 ≡ M) are trivially obtained by additivity; see
Appendix C. The two top-line panels in Figure 3 illustrate the results Minkowski functionals of a cylinder with circular basis of radius 
for = 3 and 5. and height − , with the actual distance between the centres
 of 1 and 2 . An illustrative example of Minkowski functionals
 iso-contours for balls with same radius is shown in Figure 4 as
2.3 Spiky model: filaments feeding clusters function of length and radius of the cylindric bridging filament. For
Massive halos form at the highest density nodes of the cosmic relatively small bridge lengths ( ∼ 2 ), the functionals are quite
web. Even in absence of major mergers, dark matter is continuously degenerate. For larger radii, degeneracy is broken. A compilation
accreting along filaments connecting the nodes (e.g. Eckert et al. of systems that can be approximated by this geometry are reported
2015; Connor et al. 2018). We can approximate such spiky geometry for comparison; see Table 2.
by a ball of radius attached to distinct i.e. not mutually A more advanced configuration is obtained by replacing the
intersecting cylinders ( = 1, . . . , ) radially joined to , each cylinder by a truncated cone with circular bases of radius 1 and
with length ℓ and basis with radius lying on the surface of 2 and height ℎ = − ( 2 − 12 ) 1/2 − ( 2 − 22 ) 1/2 ; see Appendix D.
 . Using additivity, the Steiner formula, and Equation (B2) the The Minkowski functionals are (D ) = ( 1 ) + ( 2 ) +
Minkowski functionals of the resulting body S̄ = ∪ =1
 Ð 
 are ( ) − (L1 ) − (L2 ), with the functionals for obtained from
 h i Equations (D1-D3). It is not difficult to further generalise this model
 ∑︁ 
 0S̄ = (2 − ) 3 + 2 ℓ + (3 2 − 2 − 2 ) , by adding two additional cylindric filaments that protrude from the
 3 3 two haloes in opposite directions. These two haloes can be regarded
 
 (5a) as local clumps of matter embedded in a single, bent cosmic filament
 ∑︁ 2 similar to the A3016-A3017 system (Chon et al. 2019). Finally, note
 1S̄ = (2 − ) 2 + ( + 2 ℓ − 2 ), (5b) that a pile of truncated cones with matching bases can describe
 3 6
 axially-symmetric filaments with varying thickness, well-suited for
 2 1 ∑︁  
 2S̄ = (2 − ) + ℓ + 2 + arcsin , (5c) systems such as the one recently reported by Umehata et al. (2019)
 3 3 and Herenz et al. (2020).
 
in which = ( 2 − 2 ) 1/2 is the distance from the centre of 
to the -th spherical cap bounded by the cylinder . The condition 2.5 Mass assembly history and morphology
 ∩ = ∅ is possible if approximately 2 . 4 2 .
 Í
 During the late stage of evolution before virialisation, satellite haloes
 Equations (5) are simpler but still keep the essential informa-
 are closer to the main halo. This tends to accrete mass at merging
tion if the free heads of cylinders are not flat but spherical caps with
 rate and with time-scale depending on the epoch, initial mass and
the same curvature radius as the central ball, i.e. L = ∩ , so
 Í statistics of the primordial density field (Bond et al. 1991), mass
that (S ) = ( ) + ( ). The Minkowski functionals are
 and kinematic of sub-haloes (e.g. Zhao et al. 2003), and tidal forces
then
 (Lapi & Cavaliere 2011), which are possibly conditioned by dark-
 4 3 ∑︁
 energy (Pace et al. 2019). The filamentary structures feeding clusters
 0S = + 2 ℓ , (6a)
 3 tend to become shorter and thinner (e.g. Cautun et al. 2014), and the
 
 2 2 ∑︁ connectivity of the more massive hence largest and latest formed
 1S = + ℓ , (6b) haloes decreases over time (Choi et al. 2010; Codis et al. 2018;
 3 3
 Kraljic et al. 2020). Since Minkowski functionals account for the
 2 1 ∑︁ non-trivial geometrical and topological content of fused bodies de-
 2S = + ℓ . (6c)
 3 3 spite their evolutionary stage, relaxed or not, one expects that they
 
 correlate with the dynamical state of galaxy clusters. This claim is
The bottom rows of Figure 3 illustrate results for = 1, 2, 4. supported by the results we obtained with idealised models.
 As shown in Figure 3 (top panels), while the volume of merg-
 ing models M is mainly sensitive to the relative size (radius) of
2.4 Dumbbell model: cluster-pair bridge
 the satellites, area and integrated mean curvature strongly depend
Cluster of galaxies may reside in superclusters still not in equi- also on their relative distance from the main halo, lifting the degen-
librium. In the simpler configuration, major haloes are connected eracies. Overall, late-time structures are more compact i.e. occupy

 MNRAS 000, 1–12 (2019)

Morphology of relaxed and merging galaxy clusters 5

Figure 3. Minkowski functionals of merging models M with = 3, 5 satellites (rows 1-2 from the top) and spiky models S with = 1, 2, 4 filaments
(rows 3-4-5), illustrated by topologically equivalent bodies in the top left corner of the right panels. Left-to-right: volume, surface, and integrated curvature,
normalised to the values of the central ball. In M the satellite balls have different radius and are at distance ∝ from the central ball (see legends);
lines with increasing thickness would represent subsequent stage of merging, with satellites going closer to . In S the cylindric filaments have bases of
radius and length ℓ ∝ ℓ (see legends); thicker lines represent later stages of gravitational evolution. For the S models 2 does not depend on the radius of
filaments but only on their length. Lengths are in units of the central ball radius .

MNRAS 000, 1–12 (2019)

6 C. Schimd, M. Sereno
 The dependence of shapefinders on the parameters of models
 is illustrated only for the two-fused ball model, M, as prototype
 for major mergers and fully accounted for by two parameters only,
 i.e. the ratio of balls radii / and the distance between balls in
 units of the major ball radius, / . Figure 5 (left panel) suggests
 that major-merging clusters from the LC2 catalogue, which share
 similar geometric scales or , can be distinguished by the values
 of 1 , 2 , and 3 , whose iso-contours are markedly orthogonal
 in different part of the ( , ) parameter space. For reference, the
 two major-merging systems with secondary haloes orbiting at the
 splashback radius of the main halo and having 3 and 10 times
 smaller mass (upper and lower empty triangles, respectively) differ
 by about 16, 12, and 6 percent in volume, surface, integrated mean
 curvature, corresponding to a 4, 16, and 6 percent difference in 1 ,
 2 , and 3 , respectively.
 The so-called planarity = ( 2 − 1 )/( 2 + 1 ) and filamen-
 tarity = ( 3 − 2 )/( 3 + 2 ) are less suitable shapefinders to
 classify the systems considered in this study, especially the ‘stellar’
 models M and S ; here the words ‘planarity’ and ‘filamentarity’
 are equivocal. Nonetheless, for the two-fused ball model and 
Figure 4. Minkowski functionals iso-contours for the dumbbell model D can differ by about ±0.15 from zero, which is the value for a ball. A
with balls of same radius as function of distance between the balls’ Blaschke diagram based on ( , ) (see e.g. Schmalzing et al. 1999)
centres and of radius of the cylindric bridge. Volume (solid lines; = 0), could be therefore an alternative interesting diagnostic to classify
surface (dashed; = 1), and integrated mean curvature (dotted; = 2) more realistic systems.
are shown for values ranging from the smaller as indicated and in steps The geometrical models presented in Section 2 illustrate the po-
Δ = {2, 0.5, 0.5} Mpc3− moving rightward. Data points are described tentiality of the classification scheme based on ( 1 , 2 , 3 ), which
in Table 2. can be applied to clusters of galaxies or any astrophysical or physical
 system with non-trivial geometry. Figure 6 shows the three projec-
 tions of the ( 1 , 2 , 3 ) parameter space populated with triaxial el-
smaller volume, cover smaller area, and have smaller intrinsic curva- lipsoids E (with axes , , ∈ [1, 5] ℎ−1 Mpc), merging models M 
ture than at early time (later stages of the gravitational evolution are with = 1, 2, 3 satellites (balls with radius ∈ [ /2, ] at distance
represented by thicker lines and by solid-dashed-dotted sequence). ∈ [ , 2 ] from the central ball with radius = 1, 2ℎ−1 Mpc),
 The morphology captured by Minkowski functionals for spiky spiky models S with = 1, 2, 3 filaments (cylinders with radius
models S (Figure 3, bottom rows) is similar to merging models: ∈ [ /4, 3 /4] and length ℓ ∈ [1, 5]ℎ−1 Mpc, feeding a cen-
regardless the number of filaments attached to the central ball, the tral ball with radius = 1, 2 ℎ−1 Mpc), and dumbbell models Dℎ
volume primarily depends on the thickness (radius) of filaments, (major ball with radius = 1, 2 ℎ−1 Mpc, minor ball with radius
the area is likewise sensitive to the relative length of filaments, ℓ , ∈ [ /2, ], cylindric bridge with radius ∈ [ /4, 3 /4] and
while integrated mean curvature only depends on ℓ . Again, the height ℎ = 5, 10 ℎ−1 Mpc).4
overall amplitude of Minkowski functionals decreases for late-time The triaxial ellipsoids E (shown only in the left panels) extend
morphologies, converging towards the values of the central main over the broadest region of the parameter space, quite well-separated
cluster. from the other models. For fixed width 2 , the maximum thickness
 A numerical study based on -body simulations is needed 1 and minimum length 3 is achieved for prolate and oblate ellip-
to quantitatively assess the correlation between the full set of soids, which are almost superposed. Instead, as shown in Figure 1,
Minkowski functionals and the relaxation state of these structures. the same value of the Minkowski functionals corresponds to differ-
It will be of interest to evaluate the ability of these statistics to dis- ent values of the triaxiality parameter, viz. are orthogonal to 
tinguish between ‘stalled’ and ‘accreting’ haloes, which are located so bringing less discriminating power than the shapefinders.
at the nodes of a network of respectively thin and thick filaments All but the E models are approximately centred around a
feeding them (Borzyszkowski et al. 2017). value of 1 that is equal to the radius of the central or major
 ball. Disregarding the unavoidable degeneracies between models,
 for fixed value of (see right panels, showing only models with
3 CLASSIFICATION BY SHAPEFINDERS = 2ℎ−1 Mpc) there is a clear trend of 2 that increases with the
Sahni et al. (1998) introduced the thickness 1 = 0 /2 1 , width number of satellites in merging models M , while it only mildly
 2 = 2 1 / 2 , and length 3 = 3 2 /4 of isodensity contours, decreases with the number of cylindric filaments (connectivity)
dubbed shapefinders, to investigate in a non-parametric way the of spiky models S . The connectivity is instead more evident in
size and shape of the matter density field above or below a given the ( 2 , 3 ) plane, increasing on average with 3 . A similar trend
threshold on large scales. Shandarin et al. (2004) used these statistics occurs for the M models, with larger integrated mean curvature
for voids and superclusters. We employ the shapefinders to attempt a or 3 occurring for systems with more satellites.
classification of the morphology of galaxy clusters. The shapefind- The Minkowski functionals support these conclusions. As sug-
ers are geometrically and physically motivated and can characterise
all the different phases of the merging accretion history, or, in a
complementary view, all the halo configurations that populate the 4 Length units are here irrelevant since only ratios do matter; in Figure 6
universe at a single cosmic time. we adopt ℎ −1 Mpc as common practice in cosmology.

 MNRAS 000, 1–12 (2019)

Morphology of relaxed and merging galaxy clusters 7

 thickness H1, width H2, length H3 planarity P, filamentarity F

Figure 5. Shapefinders isocontours for merger model M, where two balls with radius and are separated by . Left: 1 , 2 , 3 isocontours in units of
the values attained for a unit ball S, i.e. 1S = 2S = 3S = 1, increasing by Δ / S = (0.025, 0.1, 0.05) rightward. 1 and 2 are minimum when
the two balls do not overlap, respectively for / ∼ 0.89 and 0.41; 3 is minimum for non-trivial overlap, at ( / , / ) ≈ (0.54, 0.84). Right: planarity
(filamentarity) isocontours range in [−0.025, 0.125] ([−0.15, 0.3]) in steps of 0.025 (0.05), crossing the vanishing value valid for a ball or through total
embedding. Symbols as in Figure 2.

gested by Figure 3 (rows 1-2), while the volume ( 0 ) and, to smaller etc.), which are well-suited for relaxed or poorly resolved systems
extent, the surface ( 1 ) of merging systems M mainly inform but less appropriate to describe merging clusters or their filamentary
about the size of the central ball and that of the largest satellite, environment.
the integrated mean curvature ( 2 ) is sensitive also to the smaller The usual morphological parameters can be impractical for
satellites even when is small, catching both their size and distance diverse samples. Axial ratios and inertia eigenvectors provide a
from the central ball. For S models (rows 3-5), 0 and 1 equally very accurate and physically motivated description of regular and
respond to the thickness and lengths of the filaments, while the slope approximately triaxial haloes, but they can fail to properly describe
of 2 as function of the typical length increase on average with the major mergers or bridges and filaments. In some sense, classic
connectivity ; in this case the classification in the ( 2 , 3 ) plane morphological schemes usually adopted in cluster astronomy can
seems more selective. be properly used only after the shape of the halo as been already
 Dumbbell models Dℎ attain the largest value of 1 , which assessed. One first determines the class which the cluster belongs
increases with the length ℎ of the bridge. Consistently with to and then adopts the relevant shape classifier.
(see Figure 4), the width 2 is mainly sensitive to the radius of the
 The shapefinders based on the Minkowski functionals, intro-
smaller ball, while the length 3 is strongly responsive to the bridge
 duced by Sahni et al. (1998) to investigate the morphology of the
length almost regardless the other scales of the dumbbell.
 large scale structure and dubbed thickness ( 1 ), width ( 2 ), and
 Figure 7 illustrates the ability of shapefinders to separate the
 length ( 3 ), provide instead a small set of parameters that can prop-
observed systems presented in the precedent sections. Triaxial el-
 erly describe very diverse morphologies, possibly correlating with
lipsoids by Sereno et al. (2018b, disks), major mergers of the LC2
 the entire accretion history of the halo. This study assesses the ca-
catalogue Sereno (2015, squares), and cluster-pair bridges (see Ta-
 pability of Minkowski functionals and shapefinders to discriminate
ble 2, symbols and large squares) nicely occupy different positions
 between ellipsoidal, merging, spiky, and dumbbell morphologies,
in the parameter space.
 providing explicit formulae for simplified geometries.
 Equations (1) for triaxial ellipsoids E and (3) for two-fused
4 DISCUSSION AND CONCLUSIONS balls M are the main analytical result of this study; to our knowledge
The forthcoming generation of imaging and spectroscopic surveys the formulas for their integrated mean curvature, E and M , are
carried out with DESI, WEAVE, 4MOST, Rubin Observatory, Eu- new in the literature. Using the additivity of Minkowski functionals
clid, Roman Space Telescope, eROSITA, or SKA will collect thou- and the Steiner formula, one can generalise the model to merging
sands of new galaxy clusters and proto-clusters at low and high balls (satellites), M . Analytical expression for axially-symmetric
redshift and with a considerable spatial resolution, allowing us to filaments with varying thickness, Equations (D1-D3), are pivotal
establish a firm relationship between their complex morphology and to build spiky geometries S accounting for filaments feeding a
the mass assembly history. The recent exquisite observations oper- central halo or cluster-pair bridges D.
ated by CFHT/Megacam, VLT/MUSE or ALMA already support It is important to remind that the (scalar) Minkowski function-
the introduction of new spatial statistics besides the traditional ones als for the merger and spiky models, M and S , in which the differ-
calculated from the mass or inertia tensors (ellipticity, triaxiality, ent satellites or branches do not mutually overlap, do not supply any

MNRAS 000, 1–12 (2019)

8 C. Schimd, M. Sereno

Figure 6. Classification by shapefinders ( 1 , 2 , 3 ) of ellipsoidal triaxial, merging, spiky, and dumbbell models in two projections (left and right panels;
see § 3 for values). Top: merging, spiky, and dumbbell models have central or major ball with radius = 1 or 2ℎ −1 Mpc. Correspondingly, they have 1 ≈ 1
or 2ℎ −1 Mpc and 2 & 1 or 2ℎ −1 Mpc. Bottom: zoom on models with major ball with = 2ℎ −1 Mpc.

Figure 7. Example of shapefinders classification of observed systems (projections as in Figure 6): triaxial haloes (Sereno et al. 2018b, disks, colour-coded by
redshift as in Figure 1), major mergers (LC2 catalogue by Sereno 2015, squares, colour-coded by redshift as in Figure 2), dumbbell systems (black symbols
and shaded regions, see Table 2 and Figure 4). For reference, major merging models M are shown, all having main halo with radius = 0.1 − 3ℎ −1 Mpc
and secondary halo with radius located at distance such that ( , ) = ( , 0.3 ) (i.e. close haloes with same radius; solid line), (0.5 , ) (dashed),
and ( , 1.5 ) (i.e. distant haloes with same radius; dot-dashed). Triaxial haloes by Sereno et al. (2018b) nicely fit the ellipsoidal prolate model with
( , , ) = (1 ℎ −1 Mpc, , ), ∈ [0.1, 1] ℎ −1 Mpc in the ( 2 , 3 ) plane (dotted line) but not in the ( 1 , 2 ) plane.

 MNRAS 000, 1–12 (2019)

Morphology of relaxed and merging galaxy clusters 9
information about the relative orientation of the substructures. The ACKNOWLEDGMENTS
morphology of anisotropic bodies can be instead distinguished by
 We thank G. Covone, K. Kraljic, and E. Sarpa for discussions and
the vector and tensor-valued Minkowski functionals (e.g. Beisbart
 a critical reading of the manuscript, and the anonymous referee for
et al. 2001, 2002), which can be interpreted as generalisation of the
 the fruitful suggestions that largely improved the illustration of our
moment of inertia of the body. Consistently, the so-called planarity
 results. This work has been partially supported by the Programme
and filamentarity shapefinders deduced from ( 1 , 2 , 3 ) would
 National Cosmology et Galaxies (PNCG) of CNRS/INSU with INP
be misleading for the simplified models considered here, thus not
 and IN2P3, co-funded by CEA and CNES, and Labex OCEVU
used for the classification.
 (ANR-11-LABX-0060). MS acknowledges financial contribution
 Not surprisingly, as shown in Figure 3, the Minkowski func- from contracts ASI-INAF n.2017-14-H.0 and INAF mainstream
tionals respond to the distance and size of satellites, to the connec- project 1.05.01.86.10. MS acknowledges LAM for hospitality.
tivity of central haloes, and to the thickness of feeding filaments.
These geometrical and topological properties trace the growth of
structures and have an impact on the physical properties of galax- DATA AVAILABILITY
ies in the nodes of the cosmic-web (Choi et al. 2010; Codis et al.
2018; Kraljic et al. 2018, 2020). Reasonably enough, Minkowski The data underlying this article are available in the article and in its
functionals and shapefinders are therefore correlated with the mass online supplementary material.
assembly history both in the dark and gaseous components and
could serve as diagnostics to investigate the relationship between
local morphology and global dynamics. REFERENCES
 Figures 6 and 7 summarise this study. They show how a simple Abramowitz M., Stegun I. A., 1970, Handbook of mathematical functions :
three-dimensional parameter space is adequate to describe the full with formulas, graphs, and mathematical tables
variety of cluster. Thought not fully lifting the degeneracies neces- Beisbart C., Valdarnini R., Buchert T., 2001, A&A, 379, 412
sarily occurring between very different morphologies, ( 1 , 2 , 3 ) Beisbart C., Dahlke R., Mecke K., Wagner H., 2002, in Mecke K., Stoyan D.,
 eds, Lecture Notes in Physics, Berlin Springer Verlag Vol. 600, Morphol-
can be effectively used as classifiers provided at least some effective
 ogy of Condensed Matter. pp 238–260 (arXiv:physics/0203072)
radius of the major structure is estimated.
 Bett P., Eke V., Frenk C. S., Jenkins A., Helly J., Navarro J., 2007, MNRAS,
 This study is a proof-of-concept to illustrate the potential of 376, 215
Minkowski functionals and shapefinders for clusters studies. The Bleem L. E., et al., 2015, ApJS, 216, 27
 Bonamigo M., Despali G., Limousin M., Angulo R., Giocoli C., Soucail G.,
full practical potential of this approach in cluster morphological
 2015, MNRAS, 449, 3171
analysis has still to be assessed. The toy models we considered can Bond J. R., Cole S., Efstathiou G., Kaiser N., 1991, ApJ, 379, 440
capture some of the main features of a diverse sample of clusters Bonjean V., Aghanim N., Salomé P., Douspis M., Beelen A., 2018, A&A,
but likely fail to describe more complex configurations that shows 609, A49
up in the observed sky or in numerical simulations. In these case, Borzyszkowski M., Porciani C., Romano-Díaz E., Garaldi E., 2017, MN-
alternative computation techniques shall be used. Depending on RAS, 469, 594
the discrete or continuous nature of the mass tracers, the under- Boselli A., et al., 2014, A&A, 570, A69
lying Minkowski functionals can be estimated using the so-called Cautun M., van de Weygaert R., Jones B. J. T., Frenk C. S., 2014, MNRAS,
germ-grain or excursion set models (Mecke et al. 1994; Schmalzing 441, 2923
et al. 1999), i.e. by dressing the point-processes (e.g. galaxies or Chiu I.-N., Umetsu K., Sereno M., Ettori S., Meneghetti M., Merten J.,
 Sayers J., Zitrin A., 2018, ApJ, 860, 126
subhaloes) with balls of fixed radius, whose union forms the con-
 Choi E., Bond N. A., Strauss M. A., Coil A. L., Davis M., Willmer C. N. A.,
tinuous body, or considering the iso-contours of suitably smoothed 2010, MNRAS, 406, 320
random field (e.g. the density or temperature of the cluster), respec- Chon G., Böhringer H., Dasadia S., Kluge M., Sun M., Forman W. R., Jones
tively using the radius of balls or the threshold value defining the C., 2019, A&A, 621, A77
iso-contours as diagnostic parameter. Chua K. T. E., Pillepich A., Vogelsberger M., Hernquist L., 2019, MNRAS,
 484, 476
 We showed the potential of shapefinders as morphological Clampitt J., Miyatake H., Jain B., Takada M., 2016, MNRAS, 457, 2391
classifier in 3D. The three-dimensional shape of galaxy clusters Codis S., Pogosyan D., Pichon C., 2018, MNRAS, 479, 973
can be constrained with joint multi-wavelength analyses combining Connor T., et al., 2018, ApJ, 867, 25
lensing, X-ray, and SZ (Sereno et al. 2018b) or deep spectroscopic Cooray A. R., 2000, MNRAS, 313, 783
campaigns (Rosati et al. 2014; Finoguenov et al. 2019; Kuchner Cucciati O., et al., 2018, A&A, 619, A49
et al. 2020), which unveil the third dimension orthogonal to the Dalal N., White M., Bond J. R., Shirokov A., 2008, ApJ, 687, 12
projected sky. The data sets required by these analyses can be very De Filippis E., Sereno M., Bautz M. W., Longo G., 2005, ApJ, 625, 108
expansive and the full 3D analysis of galaxy clusters is usually not Despali G., Giocoli C., Tormen G., 2014, MNRAS, 443, 3208
 Diemer B., Kravtsov A. V., 2014, ApJ, 789, 1
feasible for most of the known halos. Even though this situation can
 Diemer B., Mansfield P., Kravtsov A. V., More S., 2017, ApJ, 843, 140
change with the next generation surveys and instruments, it might
 Dietrich J. P., Werner N., Clowe D., Finoguenov A., Kitching T., Miller L.,
be useful to consider 2D shapefinder classes in the projected space. Simionescu A., 2012, Nature, 487, 202
This could be more useful in the context of large surveys. Donahue M., et al., 2016, ApJ, 819, 36
 Eckert D., et al., 2015, Nature, 528, 105
 Finally, it is worth to stress that morphology alone cannot un-
 Epps S. D., Hudson M. J., 2017, MNRAS, 468, 2605
ambiguously determine the degree of equilibrium of a halo. Appar- Faltenbacher A., White S. D. M., 2010, ApJ, 708, 469
ently, morphological regular clusters can be unrelaxed (Meneghetti Finoguenov A., et al., 2019, The Messenger, 175, 39
et al. 2014). Any possible correlation between shapefinders and Gibson K. D., Scheraga H. A., 1987, Mol. Phys., 62, 1247
dynamical state of the clusters shall require a more accurate inves- Greenslade J., et al., 2018, MNRAS, 476, 3336
tigation, also based on -body simulations.
MNRAS 000, 1–12 (2019)

10 C. Schimd, M. Sereno

Figure A1. Local mean curvature loc of triaxial ellipsoids (inset) with axes ratio ( / , / ) = (0.6, 0.6) (left), (0.6, 0.4) (centre), and (0.6, 0.2) (right) as
function of the angles measured from the centre of the body and spanning a quarter of the equatorial plane (solid line) and of the two perpendicular meridian
planes (dashed, dotted; same ). For the axially symmetric, prolate ellipsoid (left) two directions are equivalent. Note the different range of values of loc .

Hadwiger H., 1957, Vorlesungen über Inhalt, Oberflache und Isoperimetrie. Veena V. S., Vig S., Mookerjea B., Sánchez-Monge Á., Tej A., Ishwara-
 Die Grundlehren der mathematischen Wissenschaften, Springer Chandra C. H., 2018, ApJ, 852, 93
Herenz E. C., Hayes M., Scarlata C., 2020, arXiv e-prints, p. Werner N., Finoguenov A., Kaastra J. S., Simionescu A., Dietrich J. P., Vink
 arXiv:2001.03699 J., Böhringer H., 2008, A&A, 482, L29
Kim S., et al., 2016, ApJ, 833, 207 Zhao D. H., Mo H. J., Jing Y. P., Börner G., 2003, MNRAS, 339, 12
Kraljic K., et al., 2018, MNRAS, 474, 547 de Graaff A., Cai Y.-C., Heymans C., Peacock J. A., 2019, A&A, 624, A48
Kraljic K., et al., 2020, MNRAS, 491, 4294
Kuchner U., et al., 2020, MNRAS, 494, 5473
Lapi A., Cavaliere A., 2011, ApJ, 743, 127
 APPENDIX A: INTEGRATED MEAN CURVATURE OF AN
Limousin M., Morandi A., Sereno M., Meneghetti M., Ettori S., Bartelmann
 M., Verdugo T., 2013, Space Sci. Rev., 177, 155 ELLIPSOID
Lovisari L., et al., 2017, ApJ, 846, 51 Following Poelaert et al. (2011), an ellipsoid described by
Macciò A. V., Dutton A. A., van den Bosch F. C., Moore B., Potter D.,
 Stadel J., 2007, MNRAS, 378, 55 2 2 2
 + + = 1, (A1)
Maturi M., Bellagamba F., Radovich M., Roncarelli M., Sereno M., Moscar- 2 2 2
 dini L., et al. 2019, MNRAS, 485, 498
 with principal semi-axes > > and central Cartesian coordi-
Mecke K. R., 2000, in Mecke K. R., Stoyan D., eds, Lecture Notes in
 nates
 Physics, Berlin Springer Verlag Vol. 554, Statistical Physics and Spatial
 Statistics. The Art of Analyzing and Modeling Spatial Structures and = cos , (A2)
 Pattern Formation. p. 111
 = sin cos , (A3)
Mecke K. R., Buchert T., Wagner H., 1994, A&A, 288, 697
Meneghetti M., et al., 2014, ApJ, 797, 34 = sin sin , (A4)
More S., et al., 2016, ApJ, 825, 39
 written in terms of the eccentric anomalies and , has local mean
Oguri M., et al., 2018, PASJ, 70, S20
 curvature
Olivares V., et al., 2019, A&A, 631, A22
Pace F., Schimd C., Mota D. F., Del Popolo A., 2019, J. Cosmology As- ℎ3 ( 2 + 2 + 2 − 2 )
 tropart. Phys., 2019, 060
 loc ( , ) = , (A5)
 2 2 2 2
Pierre M., et al., 2016, A&A, 592, A1
 with
Planck Collaboration et al., 2013, A&A, 550, A134
Planck Collaboration et al., 2016, A&A, 594, A27 
 ℎ = √︃ (A6)
Poelaert D., Schniewind J., Janssens F., 2011, arXiv e-prints, p.
 2 2 cos2 + 2 ( 2 cos2 + 2 sin2 ) sin2 
 arXiv:1104.5145
Prada F., Klypin A. A., Simonneau E., Betancort-Rijo J., Patiri S., Gottlöber being the shortest distance (‘height’) from the centre to the tan-
 S., Sanchez-Conde M. A., 2006, ApJ, 645, 1001 gent plane to the ellipsoid at the point considered and =
 √
Rosati P., et al., 2014, The Messenger, 158, 48 + 2 + 2 the radius to this point upon the ellipsoid surface
 2
Rykoff E. S., et al., 2014, ApJ, 785, 104
 (see Figure A1). The local Gaussian curvature is loc = ℎ4 / 2 2 2 .
Sahni V., Sathyaprakash B. S., Shandarin S., 1998, ApJ, 495, L5
Schmalzing J., Buchert T., Melott A. L., Sahni V., Sathyaprakash B. S.,
 The mean curvature integrated over the surface is
 ∫ 2 
 Shandarin S. F., 1999, ApJ, 526, 568
 ∫ 
 sin 
Sereno M., 2007, MNRAS, 380, 1207 ≡ d d loc = 2 ( 1 + 2 ), (A7)
 0 0 ℎ 
Sereno M., 2015, MNRAS, 450, 3665
Sereno M., De Filippis E., Longo G., Bautz M. W., 2006, ApJ, 645, 170 with
 ∫ 2 2
Sereno M., et al., 2018a, Nature Astronomy, 2, 744 + 2 arctanh 
Sereno M., Umetsu K., Ettori S., Sayers J., Chiu I.-N., Meneghetti M., 1 = d , (A8a)
 0 2 
 Vega-Ferrero J., Zitrin A., 2018b, ApJ, 860, L4 ∫ 2 2
 − 2 − 2 − 2 1
  
Shandarin S. F., Sheth J. V., Sahni V., 2004, MNRAS, 353, 162 arctanh 
 2 = − d , (A8b)
Springel V., White S. D. M., Hernquist L., 2004, in Ryder S., Pisano D., 0 2 2 3
 Walker M., Freeman K., eds, IAU Symposium Vol. 220, Dark Matter in 2 2
 Galaxies. p. 421 where 2 = 1 − 2 2 ≤ 1 and 2 = 2 sin2 + 2 cos2 . The
Tanimura H., et al., 2019, MNRAS, 483, 223 dimensionless integrals (A8) are evaluated numerically. Analytic
Umehata H., et al., 2019, Science, 366, 97 limits exists for prolate and oblate ellipsoids (see main text).

 MNRAS 000, 1–12 (2019)

Morphology of relaxed and merging galaxy clusters 11

 Figure D1. Section of dumbbell model with axially-symmetric conic fila-
 ment . The intersection of the two balls 1 and 2 with yields two
 “flat-concave lenses” L1 and L2 (light grey).

Figure B1. Section of two fused balls 1 and 2 with centres separated by
a distance and radius 1 and 2 . Their intersection L ≡ 1 ∩ 2 forms
a “bi-concave lens” (light grey). Covering L with balls with radius (only APPENDIX C: MINKOWSKI FUNCTIONALS OF THE
two are shown, dotted) one obtains the parallel lens L (dashed), which MULTIPLE-MERGERS MODEL M∗ 
results in the union of two axially-symmetric dihedra 1 and 2 glued to a
 The Minkowski functionals of M ∗ = ∪ 1 ∪ · · · ∪ for =
wedged torus (sections and 0 in dark gray).
 1, . . . , satellites with radius < , not mutually overlapping, and
 avoiding trivial embedding (| − | 6 ) are
  
APPENDIX B: INTEGRATED MEAN CURVATURE OF ∗ 2 2 ∑︁ 3 1 3
 0S = (2 − ) 3 + − 
TWO MERGED BALLS 3 3 8
 
The integrated mean curvature of M = 1 ∪ 2 is calculated using 2 2 2
 ∑︁ ∑︁ ( − )
 2 ( 1 ∪ 2 ) = 2 ( 1 ) + 2 ( 2 ) − 2 ( 1 ∩ 2 ). The last term + ( 2 + 2 ) + , (C1a)
 2 4 
is derived from the Steiner formula applied to parallel lens L , 
 ∗ ∑︁ 2
namely the uniform coverage of the lens L ≡ 1 ∩ 2 by balls with 1S = (2 − ) 2 − 
radius . As illustrated in Figure B1, L results in the union of 3 3
 
L with two dihedra 1 and 2 of thickness and opening angles ∑︁
 2 2
 ∑︁ ( − )( − )
respectively 1 and 2 , and a wedged torus centred on and with + ( + ) + , (C1b)
 6 6 
cross-section ( − 1 − 2 ) 2 . According to the Steiner formula, 
its volume (L ) ≡ (L) + ( 1 ) + ( 2 ) + ( ) is equal to ∗ 2 2 ∑︁
 2S = + ( + )
 (L) + (L) + (L) 2 + 43 3 (Mecke 2000), i.e. a fourth-order 3 3
 
polynomial in with coefficients proportional to the Minkowski v
 u !2
 2 + 2 2 − 2
 t
functionals; the integrated mean curvature (L) corresponds to 1 ∑︁
the sum of the terms of (L ) proportional to 2 . − 2 −1− , (C1c)
 3
 
 2
 The volume of the spherical dihedron 1 is 

 3 with cos = ( 2 + 2 − 2 )/2 . The condition ∩ = ∅ is
   
 
 ( 1 ) = 2 1 − 12 + 1 2 + , (B1) approximately realised if the surface covered by the basis of the 
 1 3
 balls does not exceed the surface of the central sphere, [( 2 −
 Í
with ≡ 1 = ( 12 − 22 + 2 )/2 the distance from the centre of 2 + 2 )/(2 )] 2 . 4 2 .
 1 and the centroid of the lens and ≡ 1 2 . A similar expression
holds for ( 2 ) replacing 1 by 2 and by − . The volume of
the wedged torus is
 APPENDIX D: MINKOWSKI FUNCTIONALS OF THE
 2 
 ( ) = ( − 1 − 2 ) 2 + (cos 1 + cos 2 ) 3 , (B2) DUMBBELL MODEL D 
 3
 The dumbbell D = 1 ∪ ∪ 2 resulting from the union of two
with = ( 12 − 2 ) 1/2 its major radius, cos 1 = / 1 , and cos 2 =
 balls ( = 1, 2) bridged by a truncated cone (Figure D1) has
( − )/ 2 . The Steiner formula finally yields
 (D ) = ( 1 ) + ( 2 ) + ( ) − (L1 ) − (L2 ). The
 3 Δ4 2 2 3 Minkowski functionals of L ≡ ∩ and are calculated as
 (L) = − − Σ + ( 1 + 23 ) , (B3)
 12 4 2 3 proportional to (Steiner formula) using Equations (B1,B2). The
 non-trivial result for are
 (L) = 2 Δ2 − ( 1 + 2 ) − ( 1 − 2 )Δ2 , (B4)
 
 √︄ ( ) = 12 ℎ − 1 ℎ2 tan + ℎ3 tan2 , (D1)
 Δ4 3
 (L) = 2 ( 1 + 2 − ) + 2Σ2 − 2 − 2 , (B5)
 ( ) = ( 12 + 22 ) + 2 ℎ( 1 cos + ℎ sin ) (D2)
in which cos ≡ cos( − 1 − 2 ) = (Σ2 − 2 )/(2 1 2 ), Σ2 = + ( 22 − 12 ) sin ,
 12 + 22 , and Δ2 = 12 − 22 . All these equations are valid for non + 2 − 1
 ( ) = ℎ cos2 + ( − 2 ) 1 − 2 sin 2 , (D3)
trivial intersection, i.e. overlap with no embedding or | − 1 | 6 2 . 2 2

MNRAS 000, 1–12 (2019)

12 C. Schimd, M. Sereno
where 1 and 2 are the radii of the minor and major circular basis
of , ℎ its height, and tan = ( 2 − 1 )/ℎ. For = 0 one recovers
the known Minkowski functionals for a cylinder with basis ≡
 1 = 2 and height ℎ, i.e. ( ) = 2 ℎ, ( ) = 2 ( + ℎ),
 ( ) = (ℎ + ). For ℎ = 0 the second and third Minkowski
functionals further yield the area and the integrated mean curvature
(or 2/ × perimeter) of a two-faces two-dimensional disk embedded
in a three-dimensional space.

This paper has been typeset from a TEX/LATEX file prepared by the author.

 MNRAS 000, 1–12 (2019)