Download Probing Galaxy Evolution with Environment: Ram Pressure Stripping and Major Mergers in Group Environments

Probing Galaxy Evolution with Environment: Ram Pressure
Stripping and Major Mergers in Group Environments
Janice A. Hester
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: David N. Spergel
September, 2008
c Copyright 2008 by Janice A. Hester.
All rights reserved.
Abstract
Environment provides a potentially powerful, and under exploited, test for models of galaxy
evolution. This thesis explores the use of environment through the study of two physical
processes, ram pressure stripping and major mergers. As a satellite orbits through the hot
inter-cluster or inter-group medium, its cold gas can be stripped. The effectiveness of ram
pressure stripping depends on the group environment, making it an ideal candidate for an
environmental study. To examine this dependence an analytical model is developed which
can be applied to a wide range of host group and satellite masses and can therefore easily
be placed in a cosmological context. The implications of this model are explored, and the
model is then confronted with observations of galaxy group members in the Sloan Digital
Sky Survey. A strong correlation is observed between galaxy color and group mass. This
correlation is not predicted by other models of galaxy evolution, but can be understood as
the effect of ram pressure stripping.
Current cosmological models predict that mergers between galaxies are common. Major
mergers are an extreme class of merger in which the morphology of the galaxies involved
is severely disrupted. Correlations between the major merger rate and environment are
explored using the Millennium Simulation, a cosmological gravitational N-body simulation.
The Millennium Simulation combines a large volume, which includes the full range of environments, with the mass resolution necessary to track the merger histories of the dark
matter halos that typically host galaxies. The simulation results guide a physical understanding of the major merger rate as arising from the interplay between cosmology, which
sets the accretion rate of halos, and the interactions of bound substructures, or subhalos,
iii
with the host halo and with each other. Inelastic scattering between subhalos plays a strong
role in determining the merger rate by driving subhalos into the center of the host. The
observational implications of the simulation results are briefly considered. The goal of this
study is to provide the theoretical background necessary for using environment to test major
merger driven models of galaxy evolution.
iv
Acknowledgements
I’d like to thank my husband, Amin Nasr, for his patience and understanding; my family,
for supporting me in every possible way; my fellow graduate students, for helping me keep
my perspective and sanity; and my advisor, Prof. David N. Spergel, for keeping me more
or less on the right track.
The work presented in Chapters 2 and 3 was funded by NASA Grant Award #NNG04GK55G.
I’d like to thank A. Berlind, M. Blanton, and D. Hogg for the use of the SDSS group catalog
and for useful discussions.
Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P.
Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese
Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins
University, the Korean Scientist Group, Los Alamos National Laboratory, the Max-PlanckInstitute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New
Mexico State University, University of Pittsburgh, University of Portsmouth, Princeton
University, the United States Naval Observatory, and the University of Washington.
The work presented in Chapter 4 was partially funded by NSF grant 0707731. The
Millennium Simulation databases and the web application providing online access to them
were constructed as part of the activities of the German Astrophysical Virtual Observatory.
v
The particular structure of the database design which allows efficient querying for merger
trees is described in: Lemson G. & Springel V. 2006, Astronomical Data Analysis Software
and Systems XV, ASP Conference Series, Vol. 351, C. Gabriel, C. Arviset, D. Ponz and E.
Solano, eds.
vi
Contents
Abstract
iii
Acknowledgements
v
Contents
vii
List of Figures
x
List of Tables
xvii
1 Introduction
1
1.1
Background Cosmology & Hierarchical Structure Formation . . . . . . . . .
1
1.2
Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3
Galaxy Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4
Galaxies in a Cosmological Context . . . . . . . . . . . . . . . . . . . . . . .
16
1.4.1
Halo Occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.4.2
N-body Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2 An Analytical Model of Ram Pressure Stripping
23
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.2
The Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.4
Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
vii
2.5
2.6
2.4.1
ICM Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.4.2
Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.4.3
Inclination
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
2.4.4
Concentration of the Cluster . . . . . . . . . . . . . . . . . . . . . .
37
2.4.5
Concentration of the Satellite . . . . . . . . . . . . . . . . . . . . . .
37
2.4.6
Length Ratios, λi . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.4.7
Mass Fractions, mi . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.5.1
Comparison with Observations and Simulations . . . . . . . . . . . .
40
2.5.2
Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3 Ram Pressure Stripping in Groups - Confronting Theory with Observations
50
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.2
Ram Pressure Stripping . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
3.2.1
Ram Pressure Stripping and the SFR . . . . . . . . . . . . . . . . .
56
3.2.2
Model Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.3
The Group Catalog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.4
Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.4.1
Comparison to Previous Results . . . . . . . . . . . . . . . . . . . .
69
3.4.2
Disk-like Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
4 Major Mergers Between Dark Matter Halos in the Millennium Simulation
4.1
88
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
4.1.1
89
Major Mergers, AGN, & Morphology . . . . . . . . . . . . . . . . . .
viii
4.1.2
4.2
4.3
4.4
4.5
Mergers & Environment - Previous Results . . . . . . . . . . . . . .
93
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4.2.1
Simulation & Numerical Issues . . . . . . . . . . . . . . . . . . . . .
97
4.2.2
Merger Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
4.3.1
Evolution of the Merger Rate . . . . . . . . . . . . . . . . . . . . . .
103
4.3.2
Mergers Between Subhalos and Their Host Halos . . . . . . . . . . .
104
4.3.3
Mergers Between Subhalos . . . . . . . . . . . . . . . . . . . . . . .
108
4.3.4
Merger Rate vs Local Halo Density . . . . . . . . . . . . . . . . . . .
110
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
4.4.1
Evolution of the Merger Rate . . . . . . . . . . . . . . . . . . . . . .
113
4.4.2
Groups and Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
4.4.3
Major Merger Rate and the Assembly Bias . . . . . . . . . . . . . .
123
4.4.4
Subhalo Completion . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
134
A NFW Profile
151
B Orbits in an NFW Profile
153
C Hot Galactic Halo
155
References
157
ix
List of Figures
2.1
Mass at which a satellite’s gas disk is ram pressure stripped versus Rstr /Rd .
Short-dashed line: sorbit = .25, solid : sorbit = .35, long-dash: sorbit = .5,
dash-dot: sorbit = .75, dash-triple-dot: sorbit = 1. Top-left: Spiral galaxy
orbiting in a 1015 M cluster. Top-right: Spiral galaxy orbiting in a 1014 M
cluster. Bottom-left: Spiral galaxy orbiting in a 1013 M group. Bottomright: Dwarf galaxy orbiting in a 1013 M group. . . . . . . . . . . . . . . .
2.2
46
Mass at which a satellite’s disk is ram pressure stripped versus the mass
fraction of gas disk that is stripped. Lines are the same as in Fig. 1. Top-left:
Spiral galaxy orbiting in a 1015 M cluster. Top right: Spiral galaxy orbiting
in a 1014 M cluster. Bottom left: Spiral galaxy orbiting in a 1013 M group.
Bottom right: Dwarf galaxy orbiting in a 1013 M group. . . . . . . . . . . .
2.3
47
Mass at which a satellite’s gas disk is ram pressure stripped versus the deficiency in a 3.5 × 1014 M cluster. Lines are the same as in Fig. 1. . . . . . .
x
48
3.1
Fraction of galaxies that belong to the red sequence, fr , and the fraction with
ni > 2.5, f> , vs Mr . Stars/solid : Central galaxies Diamonds/dash-doubledot: Satellite galaxies in groups with log(Mgr (M )) < 13. Squares/dash:
13 < log(Mgr (M )) < 14. Triangles/dash-dot: 14 < log(Mgr (M )) < 15.
See the text for a description of the groups and the selection of central and
satellite galaxies. (a): fr for all galaxies. (b)((c)): fr for galaxies with
ni < 2.5 (ni > 2.5). (d): f> for all galaxies. (e)((f)): f> for galaxies
with blue (red) colors, separated using an Mr dependent color cut. Note
that galaxies in groups have both a higher fr and a higher f> than isolated
galaxies, in agreement with previous studies of the relationship between color
and ni and environment. Among disk-like satellite galaxies, that is satellites
with ni < 2.5, the correlations between fr , Mgr , and Mr are all consistent
with inflow within the outer gas disk plus ram pressure stripping of the outer
gas disk determining star formation rates. . . . . . . . . . . . . . . . . . . .
3.2
86
Difference between satellite galaxies and central galaxies in the red fraction,
fr,gr − fr,is , and in the fraction of galaxies with ni > 2.5, f>,gr − f>,is . Symbols, line styles, and panels are the same as in Figure 3.6. Note that an
excess of red galaxies is observed among the satellite galaxies in both of the
ni selected sub-populations, but no excess of satellites with ni > 2.5 is seen in
either the blue or red sub-populations. This asymmetry has been observed
before in studies based on local over-densities and conditional correlation
functions, and indicates that some interaction between satellite galaxies and
the IGM is suppressing star formation among disk-like satellite galaxies. This
figure highlights this asymmetry. Panel (b) also highlights the decrease with
luminosity among the disk-like satellites in the fraction of galaxies that transition from the blue to the red sequence as satellites. . . . . . . . . . . . . .
xi
87
4.1
Evolution of the merger rate. The left hand panel displays the evolution of
the specific merger rate, the merger rate per halo, for all halos. The forward
looking rate,R+ , and the backward looking rate, R− are both plotted, where
R+ is the fraction of all halos with Vmax > 175km/s that will become the most
massive progenitor of a major merger in the next Gyr and R− the fraction
of all halos with Vmax > 175km/s that are a remnant of a major merger
that occurred in the last Gyr. These two definitions correspond to merger
rates determined using counts of close pairs and morphologically identified
merger remnants respectively. The right hand panel displays the physical
number density of major mergers per Gyr resulting in remnants with Vmax >
175km/s. The red solid line corresponds to all halos. The blue and purple
lines correspond to mergers occurring in ‘groups’ and ‘rich groups’ (See the
text for group halo definitions). The solid lines include both mergers between
two subhalos and mergers between a subhalo and the host halo. The dashed
lines include only mergers between two subhalos. . . . . . . . . . . . . . . .
xii
142
4.2
Tidal stripping of subhalos at z=0. Logarithmically spaced contours of subhalo number density in the log[np/np(Vmax )] versus log(ro /rvh ) plane, where
np/np(Vmax ) is a well defined surrogate for mts /mvs , as discussed in the text,
and ro /rvh is the subhalo’s distance from the center of the host halo in units of
the host halo’s virial radius. In the three different panels, contours are shown
for subhalos residing in hosts with Vh values corresponding to galaxy-like host
halos, group-like host halos, and rich group and cluster-like host halos. The
black contours are for all subhalos with Vs > 120km/s. The colored contours
are for subhalos whose merger with the host would be counted as a major
merger. Contours are shown for Vs /Vh > 0.7 and Vs /Vh > 0.94, roughly
corresponding to pre-tidal stripping mass ratios of 1:3 and 1:1.2. With the
exception of some subhalos with high Vs /Vh , all subhalos are similarly tidally
stripped. The thin diagonal lines show the predicted relationships between
mts /mvs and rp /rvh for NFW halos for group-like (center panel) and clusterlike (right panel) hosts from Mamon (2000), where rp is the peri-center of
the subhalo’s orbit. The right-hand line has been shifted to higher ro /rvh by
0.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
143
4.3
Tidal stripping of subhalos at z=1. The same as Figure 4.5, but for z = 1. .
144
4.4
Distribution of Vlmp /Vmmp for major mergers at z = 0, 1, and 2. The three
panels show the distribution split by merger remnant Vmax . The same ranges
are used as in Figures 4.5 and 4.5. The solid lines are histograms of the
fraction of major mergers in each Vlmp /Vmmp bin and the dashed lines are
the cumulative probability distribution of Vlmp /Vmmp values. Mergers are
heavily dominated by lower values of Vlmp /Vmmp , and it is safe to assume that
the vast majority of major mergers are true sub-host mergers in which the
subhalo merger partner has been tidally stripped before completely merging
with the host halo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
145
4.5
The effect of interactions between subhalos on the major merger rate. The
left panel shows R+ for sub-host mergers in hosts with 345 < Vh (km/s) < 455
split by the number of subhalos in the host. The dark blue long-dashed line
shows R+ for mergers between a host and a lone subhalo. The light blue dotdashed line shows R+ for sub-host mergers of hosts with two subhalos, and
the green dotted line shows R+ for sub-host mergers of hosts with at least
two subhalos. Introducing a second subhalo increases R+ by a factor of ≈ 10,
rather than ≈ 2, as would be expected if interactions between subhalos had
no effect on the merger rate. The right panel examines how the increased
merger rate in hosts with multiple subhalos, due to interactions between
subhalos, affects the frequency of merger remnants among halos with grouplike Vmax , 380 < Vmax (km/s) < 500. It is merger remnant frequency rather
than close pair counts that is easiest to interpret in group-like environments.
The dark blue dashed line shows the merger remnant frequency for halos
with no subhalos with Vs > 120km/s. These remnants would be the result of
mergers between the host and a single bright subhalo. The green line shows
the frequency of merger remnants among group-like halos with at least one
bright subhalo and the red line shows the frequency among all group-like
halos. Subhalo interactions clearly play a strong role in driving the overall
merger rate as without them the expected remnant frequency would be only
≈ 2 times higher than the dark blue line. The observational implications are
that merger remnants should be more likely than average galaxies to have a
least one fainter close companion and that when merger remnants are cross
correlated with a sample of fainter galaxies they should show an enhanced
correlation on small scales. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
146
4.6
The evolution of R− for ‘group’ and ‘rich group’ members is compared to that
for all halos. Groups and rich groups are defined as host halos with Vh values
in the indicated ranges and at least two subhalos within their virial radius,
indicated by ‘r/rv < 1’ or two non-central halos within their FOF group,
indicated by ‘all’. Group ‘members’ includes both the central host and either
all subhalos, ‘r/rv < 1’, or all non-central halos, ‘all’. These correspond to
observed groups with at least three members which are either conservatively,
‘r/rv < 1’, or generously, ‘all’, defined. Contrary to previous expectations,
the specific merger rate in groups is lower than in the field. As is discussed in
the text, while mergers are dominated by sub-host mergers, the halo count
in groups is dominated by subhalos. . . . . . . . . . . . . . . . . . . . . . .
4.7
147
Merger rate versus local halo number density and Vmax . The left panel displays R− versus the local halo number density, measured by counting all halos
with Vmax > 120km/s within a 2h−1 Mpc comoving sphere, for all halos with
Vmax > 175km/s for z = 0, 1, and 2. The right panel shows R− versus Vmax
for halos in all environments. Insets are for orientation. The left inset shows
the distribution of local densities for z = 1 for three Vmax ranges, galaxy-like
(175 < Vmax (km/s) < 380), group-like (380 < Vmax (km/s) < 500), and rich
group-like (500 < Vmax (km/s) < 950). This inset clearly illustrates halo biasing. The right inset shows the distribution of Vmax in three environment
ranges, 0-4, 5-9, and 10-14 halos within 2h−1 Mpc comoving, displaying the
counterpart of halo biasing. The merger rate is independent of Vmax and, for
environments typical of most halos, of environment. . . . . . . . . . . . . .
xv
148
4.8
Merger rate versus local halo number density for halos grouped by Vmax . In
both panels, R− is plotted versus the local density at z = 0, 1, and 2 for
halos in three Vmax ranges, as shown in the legend. The middle Vmax range,
the green dotted line, evolves from moderately biased at z = 2 to unbiased
at z = 0. The left panel includes both central halos and non-central halos
that are not within the virial radius of the central halo of their dark matter
FOF group. Linear theory predicts no dependence of R− on environment
for these halos. In the left panel, halos with low Vmax values display a R−
that decreases with local density while halos with high Vmax values display
a R− that increases with local density. This is similar to other ‘assembly’
biases that have been recently observed in large N-body simulations. The
right panel includes only halos that are the central halo of their dark matter
FOF group. The left panel is sensitive to the influence of the central halo
of an FOF group on the halos surrounding it, the right panel is not. For
halos with low Vmax , R− is observed to decrease with local density in the
right panel, but not in the left. The right panel is particularly sensitive to
the effect that environment on 2h−1 Mpc scales has on halos which dominate
their immediate environment. At z = 2, the intermediate Vmax range displays
a hill-like behavior in the left panel, but increases continuously with local
environment in the right panel. At z = 0 and 1, halos with intermediate
values of Vmax behave like those with low values of Vmax . Comparing the two
panels provides insight into the possible mechanisms responsible for these
trends and supports the hypothesis that they arise from the extension of
the effects of tidal stripping and unbound inelastic collisions between halos
beyond the virial radius of the central/host halo. . . . . . . . . . . . . . . .
xvi
149
List of Tables
2.1
Parameter Values and Scatter . . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
49
Chapter 1
Introduction
The linear growth of structure from the CMB to the large scale structure at z ∼ 0 is
successfully described by ΛCDM. The next step in following the evolution of the universe
is to determine how galaxies form and evolve within a ΛCDM universe. Environment is
potentially one of the most useful tools for studying galaxy evolution, especially as the
advent of large galaxy redshift surveys has allowed trends between structural and stellar
properties and environment to be measured with statistical significance. Making the best
use of these surveys requires physics-based models that predict environmental dependence
and evolution of both dark matter dynamical and gas-dynamical processes.
1.1
Background Cosmology & Hierarchical Structure Formation
The determination of a concordance cosmological model has opened up the possibility of
studying galaxy evolution through observations of large populations of galaxies. The predictions of ΛCDM include a picture of hierarchical structure formation in which small objects
undergo gravitational collapse first and become the building blocks of successively larger
objects. The range of typical formation histories of collapsed objects are extremely sensitive
to cosmology, as are the range of environments in which typical halos reside. Formation
1
2 Introduction
history and environment are thought to be the two dominant factors that determine galaxy
morphology. Therefore, studying populations of galaxies relies on an accurate cosmological
model.
In the concordance ΛCDM cosmological model, the universe is flat, 24% of the massenergy density is in the form of matter, Ωm = 0.24, and 76% is in the form of an ill
understood ‘dark energy’, ΩΛ = 0.76 (Spergel et al., 2007). Here Ωi designates the fraction
of the critical density, or the density required for flatness, that a given component provides at the current epoch, z = 0. The mass is divided into normal, baryonic, matter and
‘dark matter’. Dark matter may be weakly interacting, but is not electromagnetically or
strongly interacting. The matter component is dominated by dark matter, Ωbaryon = 0.04.
Observations of the CMB alone, most notably by the Wilkinson Microwave Anisotropy
Probe (WMAP; Page et al., 2003), constrain the basic cosmological model. A variety of
small-scale CMB experiments and non-CMB cosmological observations are all consistent
with the parameters derived from the CMB and can be used to further constrain these
parameters (Spergel et al., 2007). Hence the current model is truly a concordance cosmological model. Observations of galaxy clustering, cluster abundances, weak lensing, and the
lyman alpha forest all constrain the shape and normalization of the matter power spectrum as well as Ωm . The matter power spectrum characterizes fluctuations in the matter
density field, and is therefore extremely important to studies of structure formation and
galaxy formation and evolution. The variety of constraints on the power spectrum probe a
wide range of scales and redshifts. Observations of the luminosity-brightness relationship
versus redshift for supernova indicate that the expansion of the universe is accelerating
and require a repulsive component, termed ‘dark energy’. These observations constrain the
density of this component, ΩΛ . The density of baryons, which plays an important role in
modeling galaxy formation, is further constrained by big bang nucleosynthesis predictions
of the abundances of the lightest elements, which are created early in the universe and not
by stellar nucleosynthesis. A thorough discussion of this topic, with references, is given
in Spergel et al. (2007).
1.1 Background Cosmology & Hierarchical Structure Formation 3
The early universe is extremely uniform, however density fluctuations exist on all scales.
These fluctuations appear as temperature variations in the CMB and grow to become galaxies and large scale structure. At the time of recombination, the height of these fluctuations
is of order a part in 100. The power in the density fluctuations is distributed approximately uniformly over all scales and the modes are initially decoupled (Komatsu et al.,
2003). Initially, the evolution of the universe can be described using linear theory, in which
each density mode is allowed to evolve independently. Structure formation occurs on all
scales as regions that are over dense undergo gravitational collapse. As the universe evolves,
modes become coupled and deviations from linear theory become increasingly important.
The growth of large scale structure in this framework can be nicely visualized using the
Zeldovich approximation (Zel’Dovich, 1970). Over dense regions are not spherically symmetric, but instead have three axes ranging from a shortest axis to a longest. Over dense
regions therefore collapse first along the short axis, forming sheets or walls, then along the
middle axis, forming filaments, and finally along the last axis to form spherical dark matter
halos. This scenario is supported by observations of the lyman alpha forest (see review in
Claude-André et al., 2008). On scales of 10-20 Mpc, the large scale structure at z = 0 is
characterized by long filaments which intersect in massive, 1014 to 1015 M , galaxy clusters (Bond, 2008). Accordingly large scale structure is often described as a ‘cosmic web’.
Galaxies form in smaller dark matter halos within the cosmic web, which in turn grew from
over densities on smaller scales.
On the smaller scales at which galaxy formation occurs, the height of the typical initial
density peak which has just finished collapsing at a given redshift can be estimated by
considering the evolution of a spherically symmetric region (Gunn and Gott, 1972). Such a
region begins by following the Hubble expansion, which expansion is slowed and reversed by
the excess gravitational attraction of the over density. The density of the over dense region is
the only factor controlling the speed of the collapse; density peaks on different scales take the
same time to collapse if they have the same average density. Denser regions collapse faster.
At every redshift there is a threshold density contrast above which spherically symmetric
4 Introduction
over dense regions have collapsed and below which they are still collapsing. This threshold
decreases with time. Rather than collapsing to a point, dark matter collapses to form three
dimensional virialized halos. Weakly interacting dark matter cannot effectively cool and
therefore the final halo preserves the energy of the initial expanding region. Gravitational
interactions are assumed to virialize the halo, that is energy is evenly shared between
the internal kinetic energy and the gravitational potential energy of the halo (Binney and
Tremaine, 1988). Dark matter halos that form in cosmological N-body simulations, and that
therefore form from non-spherical over densities, have triaxial shapes and acquire internal
angular momentum, they have a spin (Bett et al., 2007, and references therein).
The growth and spatial distribution of dark matter halos can be modeled analytically by
combining the spherical collapse model with an initial density power spectrum (Press and
Schechter, 1974). At each redshift, all regions with an average density above the threshold
for spherical collapse are assumed to have collapsed into virialized halos. Over densities on
different scales correspond to halos with different masses, the larger the scale the greater
the mass. Given an initial density power spectrum, density peaks that exceed the threshold
for collapse can be identified. These models provide a framework for understanding the
evolution of dark matter halos. They predict the mass function of halos, the hierarchical
nature of structure formation, and the clustering of dark matter halos. A review of the
following is presented in Peacock (1999).
The distribution of halo masses evolves with time. At every redshift, a small scale
density peak is more likely to exceed the threshold for collapse than a large scale peak;
the distribution of halo masses is heavily weighted towards low masses. The likelihood of
a density peak exceeding the threshold for collapse increases with time as the threshold
decreases. The typical mass of a dark matter halo and the mass of the most massive
halos both increase with time as larger scale peaks become increasingly likely to exceed the
threshold (Press and Schechter, 1974; Jenkins et al., 2001). Accordingly, the most massive
halos are also on average the youngest halos, having collapsed the most recently. Lower
mass halos also continue to form in regions with successively lower large scale densities.
1.1 Background Cosmology & Hierarchical Structure Formation 5
In addition to collapsing to form increasingly massive young halos, large scale density
peaks are the site of the earliest forming lower mass halos. A peak is more likely to exceed
the density contrast for collapse if it resides on top of a larger scale peak. Massive halos are
therefore formed by assembling many older lower mass halos. This hierarchical structure
formation occurs on all scales. Dark matter halos with masses that typically host galaxies
are assembled from halos that hosted dwarf galaxies. Groups of galaxies are hosted in
massive dark matter halos assembled from halos that once hosted the individual galaxies.
Clusters of galaxies form by combining the halos of multiple galaxy groups. Accreted halos
do not immediately completely merge with or disintegrate into the more massive forming
halo. Instead the cores of accreted halos are retained as substructure for some time. The
evolution of substructure, including any dependence of substructure evolution on halo mass,
remains an open field of study.
While in fourier space the matter density distribution contains power on all scales, in
real space, the dark matter is clustered on all scales. The clustering of dark matter, dark
matter halos, and of the galaxies that halos host, is measured using the correlation function,
ξ(r). The correlation function is the fourier transform of the power spectrum and, for a
discreet distribution, measures the excess of pairs that are separated by a given distance
compared to a random distribution. Specifically
dP = n2 [1 + ξ(r)] dV1 dV2 ,
where dP is the probability of finding an object in the volume dV1 and in the volume dV2 ,
which volumes are separated by r, and n is the number density of objects. Dark matter
halos and the galaxies they host can be more strongly clustered than the dark matter itself;
they are biased.
A framework for understanding dark matter halo clustering is again provided by considering the initial density distribution. Density fluctuations around a peak can be split
into small scales, below which collapse has occurred, and large scales, which represent the
environment of the collapsed object. Halos that form on top of a large scale peak, as do the
earliest forming halos of a given mass and the most massive halos at a given redshift, form
6 Introduction
in denser than average environments. These halos are strongly clustered and are biased
relative to the dark matter. As the density threshold decreases, halos of a given mass can
collapse in successively less dense environments and become less biased. Today the most
massive collapsed halos host clusters of galaxies and have masses of ≈ 1014−15 M . These
halos are young, biased, and reside in dense environments. Halos that host groups of galaxies and have masses of ≈ 1013−14 M are less clustered than clusters of galaxies but are still
biased compared to the dark matter. Before the evolving density modes begin coupling to
each other, that is in the linear regime, small scale density perturbations are not correlated
with large scale density perturbations. In this case, at fixed mass, the accretion history or
assembly history of a collapsed object is not correlated to its environment (White, 1996).
Note that halo mass and environment clearly are correlated; massive halos preferentially
reside in dense environments and dense environment preferentially host massive halos.
1.2
Galaxies
Galaxies are collections of gas and stars that reside in the cores of dark matter halos. They
typically fall into two classes, late-type or spiral galaxies and early-type or elliptical galaxies.
Late type galaxies consist of both a stellar disk and a cold gas disk. Star formation is ongoing
in these galaxies, giving them the blue colors typical of massive, and therefore young, stars.
Star formation frequently occurs in large spiral arms. The resulting bright blue arms of
young massive stars give these galaxies their characteristic appearance. Late-type galaxies
are rotationally supported, as attested to by their shapes. Early type galaxies are pressure
supported, with triaxial shapes, and consist of an old, red stellar population. Gas in these
galaxies is generally hot and diffuse. A galaxy that forms from a slowly cooling cloud of gas
will generically take on a disky shape. The formation of elliptical galaxies likely depends
on a faster, more violent, mechanism.
The canonical model of galaxy formation postulates a shock heated spherical halo of hot
gas that cools from the inside out forming a growing disk of cold gas (White and Rees, 1978;
Dalcanton et al., 1997; Mo et al., 1998). Modifications to this model account for different
1.2 Galaxies
7
modes of gas accretion and for the continual accretion of smaller objects. The simplest
model of galaxy formation postulates first that accreted gas is shock heated to the virial
temperature of the dark matter halo at the halo’s virial radius. The virial radius is the outermost radius to which a halo has collapsed and virialized, and the virial temperature is the
temperature of an isothermal gas in hydrostatic equilibrium in the gravitational potential
of the dark matter halo. The model assumes that as gas cools it forms an exponential gas
disk at the center of the dark matter halo. The hot gas halo is assumed to be in hydrostatic
equilibrium, and the core of the gas halo is therefore the densest and cools the fastest.
Cooling times become increasingly longer at larger radii. The shock heated halo therefore
cools from the inside out and the gas disk grows larger and denser (Mo et al., 1998). A
version of this model in which the gas and dark matter share the same initial specific angular momentum distribution, there is no angular momentum transport, and there are no
torques after the collapse begins produces exponential disks (Dalcanton et al., 1997). Star
formation then occurs within the cold gas disk. Star formation within disks is observed to
be proportional to the density of the cold gas disk and to only occur over a threshold density (Kennicutt, 1998). Assuming such a relationship, star formation within the exponential
gas disk naturally forms an exponential stellar disk. The most visually striking features of
late type galaxies, the bright blue spiral arms, are the result of star formation induced by
spiral density waves in the self gravitating gas and stellar disks (Lin and Shu, 1964; Binney
and Merrifield, 1998).
Several modifications of this simplified picture have been introduced. Accreting gas
may not shock at the virial radius of the halo. If the halo is not massive enough, then
it may not be able to support a shock at large radii and cool gas may flow directly into
the core of the halo (Dekel and Birnbiom, 2006). This gas likely shocks at smaller radii,
where cooling times are short. In contrast, halos above a cut off mass can support a shock.
Halos in a transition region may experience both forms of gas accretion, with cold flows
occurring along filaments and hot flow occurring elsewhere. Forming halos also accrete
less massive halos which host their own galaxies. The gas in these galaxies is accreted
8 Introduction
cold, but is likely stripped and heated by the hot gaseous halo surrounding the accreting
galaxy (Bland-Howthorn et al., 2007; Dekel and Birnbiom, 2008). The cold gas and stellar
disks in late-type galaxies are both observed to have exponential radial density profiles.
The source of this profile is unknown. The hot halo may cool to form an exponential
disk directly, which then naturally grows an exponential stellar disk (e.g. Dalcanton et al.,
1997). Alternatively, viscosity within the disk may drive inflow. Combined with a density
dependent star formation rate, inflow can be invoked to grow exponential gas and stellar
disks (Lin, 1987a,b; Olivier et al., 1991; Bell, 2002). Models that rely on viscous inflow are
relatively insensitive to the initial conditions of the gas disk (Olivier et al., 1991). This may
be an advantage in the messy context of hierarchical structure formation with both cold
and hot flows. A suitable source of the viscosity has not been identified however.
Like halos, galaxies continually accrete smaller objects. Numerous dwarf galaxies and
tidal streams in the Milky Way attest to this ongoing accretion (Siegal-Gaskins and Valluri, 2008, and references therein). The fate of disrupted dwarf galaxies may be to form a
thick stellar disk (Yoachim and Dalcanton, 2008). Counter-rotating gas and stars, observed
in some spirals, may also be a relic of hierarchical structure formation (Kannappan and
Fabricant, 2001). Thick disks themselves may be counter-rotating, lending support to hierarchical disk building scenarios (Yoachim and Dalcanton, 2005). Mergers between equal
mass galaxies likely have significantly more disruptive effects on galaxy morphology.
The second class of galaxies, early-type or elliptical galaxies, are distinct from latetype galaxies. The stellar components have triaxial shapes and are primarily pressure
supported. The complex stellar orbits that give these galaxies their shapes have little net
angular momentum. The stellar component may have some rotational component, and in
detail their shapes may be described as either ‘disky’ or ‘boxy’ depending on the ratio of
the maximum rotation velocity along the major axis to the velocity dispersion in the center
of the galaxy (Davies et al., 1983; Cappellari et al., 2007). Here ‘disky’ and ‘boxy’ refer to
the shape of the galaxies isophotes, with ‘disky’ isophotes corresponding to galaxies with
higher maximum rotation velocities. The stellar populations are old and therefore red.
1.2 Galaxies
9
All sort-lived massive blue stars have left the main sequence, leaving only the long-lived
low mass red stars. Both the stellar densities and the light profiles of elliptical galaxies
are more centrally concentrated than those of disk galaxies. The light profiles of elliptical
galaxies are characterized by a de Vaucouleurs profile, I(R) ∝ I0 exp(−kR1/4 ), in contrast
to the exponential profiles of disk galaxies (de Vaucouleurs, 1948). Gas in these galaxies is
generally hot and diffuse and is observed in the X-ray (for example, Fabbiano et al., 1989).
While the cooling times in the outskirts of this hot halo may be longer than a hubble time,
in the inner core they are not. The lack of cool gas in the cores of these galaxies is often
explained by invoking feedback from either low levels of star formation or low luminosity
AGN. Faint blue cores and LINERs are observed in the cores of some elliptical galaxies,
and the energetics of these objects are roughly consistent with the feedback scenario (Best
et al., 2001; Schwinski et al., 2007). Red bulges resembling small elliptical galaxies are often
observed in the centers of spiral galaxies. At the center of every spheroid, a generic term for
either bulges or elliptical galaxies, resides a super massive black hole, capable of becoming
an AGN in the presence of fresh fuel (Richstone et al., 1998, and references therein). The
masses of these black holes are tightly correlated with the dynamical properties of the
spheroids that host them (Magorrian et al., 1998; Ferrarese and Merritt, 2000; Gebhardt
et al., 2000; McLure and Dunlop, 2002; Tremaine et al., 2002; Marconi and Hunt, 2003).
Early and late type galaxies were distinguished early in the history of galaxy observations (Hubble, 1936). With modern galaxy surveys the distinction between two fundamentally different galaxy types can be well motivated. The two classes of galaxies can
be separated using either their star formation histories or their stellar structures. When
a large population, 10s or 100s of thousands of galaxies, are plotted on the optical color
versus luminosity plane, a clear bimodality is seen in the optical colors (Strateva et al.,
2001; Blanton et al., 2003b; Kauffmann et al., 2003a). Galaxies either reside on a narrow
‘red sequence’ or in a broader ‘blue cloud’. The structures of galaxies can be quantified by
measuring their light profiles, either by measuring the concentration of the light or by fitting
a light profile with a flexible exponent, I(R) = I0 exp(−kR1/n ), where n is call the sersic
10 Introduction
index. A pure exponential disk has n = 1 while for a de Vaucouleurs profile n = 4. Like the
optical colors, the light profiles of galaxies are bimodal (Blanton et al., 2003b; Kauffmann
et al., 2003a). Further, optical colors and light profiles are highly correlated. Most galaxies
are either blue with low concentrations or red with high concentrations (Blanton et al.,
2005a). The exception is the case of red disk galaxies, which occur in small but significant
numbers. These galaxies were also observed by Hubble and are classes as S0 galaxies or as
early-type disk galaxies. These are disk galaxies that have lost or consumed their gas and
are no longer forming stars.
The two classes of galaxies, and the clear distinction between them, must be accounted
for by scenarios of galaxy formation and evolution. The formation of gas rich disk galaxies
was discussed above. Slow and prolonged star formation always occurs in a disk as conservation of angular momentum dictates that cooling gas collapses to form a disk. Dark matter
halos and the galaxies they host not only undergo minor mergers in which a more massive
halo accretes a significantly less massive halo, they can also experience major mergers in
which the two halos are of similar masses. A major merger between two disk galaxies can
be quite dramatic. As the two stellar populations collide and mix a pressure supported
remnant can be created (Toomre and Toomre, 1972; Mihos and Hernquist, 1996; Springel,
2000; T. and Burkert, 2003; Barnes, 2004; Springel et al., 2005a; Cox et al., 2006). In
addition tidal torques during the merger drive a significant fraction of the gas in the two
colliding galaxies into their respective centers (Barnes and Hernquist, 1992). An intense
burst of star formation can occur in this gas and the resulting stars form a dense core in
the eventual remnant (Mihos and Hernquist, 1996). This gas may also fuel luminous AGN
in the centers of the galaxies and in the center of the remnant. Feedback from the star
burst and the AGN can heat and expel any gas that has not been consumed, terminating
both star formation and AGN activity (Silk and Rees, 1998; Fabian, 1999; Di Matteo et al.,
2005; Springel et al., 2005b,a; Hopkins et al., 2006a). After the newly formed core dims
and reddens, the remnant comes to resemble an elliptical galaxy. Observational evidence
for this scenario is presented in 4.1.1. If the remnant accretes a significant mass of gas after
1.3 Galaxy Populations 11
the merger, then it may re-grow a gas disk and the remnant becomes the bulge of a new
spiral galaxy (Steinmetz and Navarro, 2002). An alternative formation scenario has been
advanced in which elliptical galaxies form early through monolithic collapse. These models
may be able to accurately reproduce the color magnitude relationship and metallicities of
elliptical galaxies, but do not address their structural properties (for example Larson, 1974;
Yoshii and Arimoto, 1987; Bressan et al., 1994).
1.3
Galaxy Populations
Galaxy formation and evolution can be influenced by many factors, both environmental and
internal. The history of an individual galaxy is likely to be complex and difficult to interpret.
If one is concerned less with the evolution of a single galaxy and more with understanding
the physical mechanisms that are important for driving galaxy evolution in general, then
the study of entire populations of galaxies can be more rewarding. Galaxy populations can
be characterized in a variety of ways. The distribution of individual galaxy properties, the
correlation between two or more properties, the clustering and environments of galaxies, and
correlations between galaxy morphology and environment can all be studied. The advent
of large galaxy redshift surveys, for example the 2 degree Field of View Galaxy Redshift
Survey (2dFGRS) (Colless et al., 2001) and the Sloan Digital Sky Survey (SDSS) (York
et al., 2000), allow such studies to be carried out with meaningful statistical significance.
Two popular ways of quantifying galaxy populations are the galaxy luminosity function,
Φ(L), and the 2 point autocorrelation function, ξ(r). These allow the relationship between
galaxies and dark matter halos to be probed. The luminosity function is defined such that
Φ(L)dL gives the number density of galaxies with luminosities in the range (L, L + dL).
The galaxy luminosity function decreases monotonically with luminosity, is approximately
exponential at low luminosities, and declines steeply above a characteristic luminosity. It
can be reasonably fit using a Schechter function,
Φ(L) = (Φ∗ /L∗ )(L/L∗ )α exp(−L/L∗ ).
12 Introduction
Here L∗ is the characteristic luminosity, α is the faint end slope, and Φ∗ sets the overall
galaxy density (Schechter, 1976). The galaxy luminosity function can be compared with the
dark matter halo mass function found either analytically or using large gravitational N-body
simulations. The galaxy luminosity function is inconsistent with a constant mass to light
ratio at all halo masses. Instead, the mass to light ratio appears to increase at both faint and
bright luminosities. Attempts to reconcile the two include invoking photoionization heating
and supernova feedback in low-mass systems and feedback from supernova or AGN in highmass systems (White and Rees, 1978; White and Frenk, 1991; Efstathiou, 1992; Benson
et al., 2003; Croton et al., 2006). Clustering provides a second way to map galaxies onto
dark matter halos. While the clustering strength of dark matter halos increases with halo
mass, for galaxies it increases with luminosity (Norberg et al., 2001; Zehavi et al., 2005).
Galaxies are strongly clustered on small scales and their correlation function generally
declines as a power law with separation, that is ξ(r) = (r/r0 )−γ , where r0 is the correlation
length and γ the slope of the correlation function (Norberg et al., 2001; Zehavi, 2004).
The correlation function for dark matter halos does not have a power law shape (Cooray
and Sheth, 2002). The two can be reconciled by allowing massive halos to host multiple
galaxies (see for example Seljak, 2002). Clustering can be used more generally to measure
the typical halo masses of any class of objects. Any random sampling of halos will have
the same clustering strength as the original halo population. Using either the luminosity
function or the correlation function to probe dark matter halo properties requires modeling
the underlying dark matter halo mass function and clustering properties. Weak lensing can
be used to probe the relationship between halos and galaxies directly. A typical L∗ galaxy is
hosted in a dark matter halo of 1.4 × 1012 M (Mandelbaum et al., 2006). The same lensing
study also confirmed that mass to light ratios increase with mass for high mass halos and
that halos can host multiple galaxies.
Large redshift surveys are ideal for correlating multiple galaxy properties and have
been used to verify that there are two fundamental classes of galaxies. The most striking
correlation is the bimodality in the optical color - luminosity relationship (Strateva et al.,
1.3 Galaxy Populations 13
2001; Blanton et al., 2003b; Kauffmann et al., 2003a). Optical color is closely related to
the star formation history of a galaxy. While blue galaxies are currently star forming or
have formed stars recently, red galaxies have not formed stars in the last Gyr or more. The
bimodality correlates with the galaxy luminosity. The fraction of red galaxies increases
with luminosity and the brightest galaxies are almost exclusively red (Kauffmann et al.,
2003a; Blanton et al., 2005a). The average color in the tight red sequence also increases
with luminosity. Similar trends have also been observed in related properties such as the
star formation rate and the strength of the 4000Å break (Kauffmann et al., 2003b; Baldry
et al., 2004; Kelm et al., 2005; Weinmann et al., 006a). A similar bimodality exists for
galaxy light profiles. The fraction of galaxies with elliptical-like light profiles increases
with luminosity (Blanton et al., 2005a). Optical colors and light profiles are themselves
correlated. The overwhelming majority of galaxies with elliptical-like light profiles are red,
while a small but significant fraction of red galaxies with disk-like light profiles are red. The
majority of disk like galaxies are blue (Blanton et al., 2005a).
The existence of a relationship between galaxy morphology and local environment has
been known for some time. Galaxies in dense regions corresponding to groups and clusters of
galaxies are more likely to be red and to be elliptical galaxies (Dressler, 1980, and references
therein). This basic trend has been confirmed in large galaxy redshift surveys. Galaxies in
observationally constructed groups and clusters are redder and have lower star formation
rates (Martinez et al., 2002; Gómez et al., 2003; Goto et al., 2006; Weinmann et al., 006a).
As the local density increases, where local density is measured by counting galaxies within
a cylinder in redshift space, the fraction of red, concentrated galaxies increases (Kauffmann
et al., 2004), and bright, red, and concentrated galaxies are all found preferentially in dense
environments (Hogg et al., 2003b; Blanton et al., 2005a). Red galaxies and galaxies with
highly concentrated light profiles are more strongly clustered than blue galaxies and low
concentration galaxies (Norberg et al., 2002; Li et al., 2006). These observations are all
highly related. Galaxies that reside preferentially in groups and clusters share the environments and clustering properties of these more massive objects. The observed enhanced
14 Introduction
clustering of red and early-type galaxies up to scales of several Mpc may be due entirely to
the preference of these objects to reside in groups and clusters (Collister and Lahav, 2005;
Blaton et al., 2006).
All of the above observations can be used to confront theoretical models of galaxy
formation and evolution. One popular approach to this problem is to use Semi-Analytic
models (SAMs; for example see Kauffmann et al., 1999; Croton et al., 2006; De Lucia et al.,
2006). These models use simple, physically motivated, analytical models to map galaxies
onto dark matter halos in cosmological N-body simulations. The structural morphology
and the star formation history of each galaxy is determined by the dark matter accretion
history and environment of the dark matter halo that hosts it. This approach is tempting
for several reasons. The environment of every ‘galaxy’ is included in the model output,
many observational trends can be matched simultaneously, and the interplay between the
various physical mechanisms can be captured. Unfortunately, while the input parameters
of SAMs can be well constrained with observations, it is unclear whether SAMs ca be used
to directly confront the input physics.
Semi-Analytic models include several input parameters which are set by matching a
range of observations. As an example, the SAM presented in Croton et al. (2006) includes
11 input parameters. The model matches the field luminosity-color distribution, the stellar
mass - age relation, the Tully-Fisher relation, the cold gas fraction and gas phase metallicity
of spiral galaxies, the color-magnitude relation for elliptical galaxies, the volume averaged
cosmic star formation history, and the cosmic black hole formation history. At first glance,
producing such a range of observations with only 11 input parameters is impressive. However, many of the observations are highly interrelated and the number of truly independent
observables that have been used to constrain the model is obscured. It is likely that the
model is not heavily constrained and that the input physics itself it not being adequately
confronted. When studying galaxy evolution it is quite important that such a model be
heavily constrained because it is often the case that several physical mechanisms can be
advanced that produce qualitatively similar behavior.
1.3 Galaxy Populations 15
The work presented in this thesis takes an alternate approach to confronting models of
galaxy evolution. Rather than modeling many processes simultaneously, an attempt is made
to isolate a single physical process. Two physical processes are considered, ram pressure
stripping and major mergers. Each is placed in a cosmological context and predictions are
made regarding correlations between relevant galaxy properties and environment. Focusing
on correlations with environment is physically motivated. Both processes, and many of
the processes that are considered important drivers of galaxy evolution, correlate with
environment. Ram pressure stripping is an interaction between a satellite galaxy and the
inter-group or inter-cluster medium and as such occurs only in groups and clusters. The
rate of major mergers is also assumed to depend on the local halo density and velocity
distribution. The goal is to find one or more correlations with environment that isolate an
individual process and that are not sensitive to the details of the physical model. The focus
is not on constraining the parameters of a particular model, but on determining whether a
particular piece of physics is correct and important. By confronting a predicted correlation
with observations, one can determine whether a given process has a significant impact on
galaxy evolution. Once a suite of physical processes has been tested in this manner, it might
then be rewarding to consider a SAM-like model in order to study the interplay between
processes.
In this approach, the interplay between different physical processes must still be accounted for in some manner. This is not done by trying to model multiple processes. As a
first step, some observed correlations both between galaxy properties and between galaxy
properties and environment are accepted as inputs into the model. This can be done either directly through the model or by accounting for these correlations when confronting
observations. These correlations are presumably set by physical processes other than the
one under study. The second step is to rely on a large, uniformly observed, population of
galaxies, such as large redshift surveys make available. The effect of some physical processes
is assumed only to introduce noise into the predicted correlation. With a large enough population of galaxies, an attempt can be made to average over all of these processes. This
16 Introduction
can be done as long as they don’t share a related correlation with the environment. In
this approach, the size of this noise must be considered as it determines the strength of the
correlation that one can expect to observe. If the noise is too large, or the galaxy sample
too small, then an underlying correlation may not be observed. An attempt must be made
to ensure that this is not occurring. If all physical processes other than the one under study
can be split between those with related correlations, that must be considered directly, and
those without, that introduce noise, then the process under study can be isolated. It is an
open challenge to this approach to identify competing physical processes that predict the
same correlations as the process under consideration.
1.4
Galaxies in a Cosmological Context
In order to use large galaxy surveys to confront models of galaxy evolution, these models
must be placed in a cosmological context. This can be done in at least two ways. To
first order, all dark matter halos of the same mass are statistically equivalent. This allows
a halo occupation approach to the problem for any process which is isolated to groups
and clusters. First a model must predict how a process of interest depends on the host
halo and satellite halo masses. The model must then be used to predict the level of noise
that variations in host halo and satellite galaxy properties introduce at fixed host halo and
satellite mass. That is how strongly does the effectiveness of the process vary at fixed host
halo and satellite mass. Once this is done, all the relevant correlations with environment
and clustering strengths can be predicted. In practice, however, such models are easiest to
confront with observationally identified group and cluster catalogs.
Cosmology can also be included in a model directly by using cosmologically large gravitational N-body simulations. This is useful both when studying dynamical processes and
when studying deviations from the assumption that all halos of the same mass are statistically equivalent. Individual dynamical processes do interact heavily with each other.
Therefore, truly isolating a dynamical process, in the sense of constructing an analytical
model describing the correlation between an individual process and environment, is difficult.
1.4 Galaxies in a Cosmological Context 17
Such a model would have to treat all relevant processes simultaneously and would likely
treat the interactions between processes in an average sense, which may not be appropriate. N-body simulations overcome this challenge by tracking all gravitational interactions
simultaneously. Simulations have their own limitations however, as they are subject to numerical issues and resolution constraints. Simulations are likely most useful for informing
an analytical understanding of the dynamics under consideration. Simulations also have a
role to play in halo occupation treatments as they can be used to track accretion histories
as a function of halo mass.
1.4.1
Halo Occupation
Many of the physical process that are thought to drive galaxy evolution operate only in
groups and clusters. For example, interactions with the intra-cluster medium (ICM) can
heat and strip gas from satellite galaxies (Gunn and Gott, 1972; Larson et al., 1980).
Subhalos are tidally stripped and are dynamically heated both by interactions with the host
potential and by unbound interactions with other subhalos (Moore et al., 1998; Mamon,
2000; Peñarrubia and Benson, 2005; Knebe et al., 2006). In addition, linear theory predicts
that the accretion history of dark matter halos is independent of their environment (White,
1996). The previous provide theoretical support for the basic assumption of halo occupation
models. This is that any halo property, and therefore all halo properties that may be relevant
to a given model of galaxy evolution, can be described as a distribution at fixed halo mass,
with no residual dependence on environment. Halo properties may, of course, correlate with
each other, but a complete codependent distribution of all halo properties is specified by the
halo mass alone. In essence halo occupation models extend the prediction of linear theory
to include all relevant gastrophysics.
Halo occupation models were first used to match the galaxy correlation function to the
dark matter halo correlation function. Each dark matter halo may host several galaxies,
usually divided between a central galaxy and satellite galaxies which are hosted by bound
subhalos orbiting within the main host halo. Halo occupation models make the simple
18 Introduction
assumption that the distribution of the number of satellite galaxies is determined by the host
halo mass alone. That is, there is no residual dependence on environment on a larger scale.
A halo occupation distribution (HOD) is then determined by specifying P (Ngal |Mhalo ).
This distribution provides a map between the dark matter halo correlation function and
the galaxy correlation function (see review in Cooray and Sheth, 2002). One success of these
models is recovering the power spectrum form of the galaxy cross-correlation (for example
see Seljak, 2002). Halo occupation models can also be extended to include the luminosities
and colors of both the central and satellite galaxies (for example see Guzik and Seljak,
2002).
A halo occupation distribution is a convenient tool for describing the relationship between dark matter halos and galaxies. The theoretical support for this treatment, however,
is only suggestive. Fortunately for this class of models, there is strong observational support
for the assumption that halo mass alone determines the distribution of galaxy properties.
As stated previously, the clustering of galaxies selected on color and morphology is consistent with these properties being determined by halo mass alone (Collister and Lahav, 2005;
Blaton et al., 2006). This analysis includes a healthy contribution from satellite galaxies
and extends well beyond the virial radii of the host halos. A supporting analysis was performed by Berlind et al. (2006b) who explicitly test whether the clustering of galaxy groups
depends on any group property other than mass. They find a weak tendency for groups with
the reddest central galaxies to be slightly less clustered, but no other residual dependence.
Taken together these studies show that the assumption of halo occupation models is good
to first order.
The HOD approach lends itself well to attempts to use environment to isolate the effects
of individual physical mechanisms, at least in the case of processes that are confined to
groups and clusters. This approach explicitly splits halo properties into a mass dependence,
plus a distribution at fixed mass. In these cases, the most direct approach is to study how
a given process depends on host halo mass and satellite mass. Correlations which are set
by other physical processes then fall into two categories; correlations between morphology
1.4 Galaxies in a Cosmological Context 19
and mass and correlations between morphological properties. These two types exist for
both the host and the satellite. For example, more massive groups tend to host brighter
satellites. This trend is set by the accretion history of the groups, is predicted by models of
structure formation, and needs to be considered when correlating satellite galaxy properties
with group mass. For galaxies, the fractions of red and elliptical galaxies increase with
luminosity. If this is not related to the physics under study, then this correlation to will
need to be accounted for. Correlations between galaxy properties, such as between color and
stellar structure, may need to be considered, for instance if the transition between the two
classes of galaxy objects is not under consideration. In addition to any relevant correlations,
the distribution of properties at fixed group mass or satellite luminosity, which explicitly
introduce noise into any underlying correlation, must be considered. At fixed group mass,
the number and luminosity of satellite galaxies can vary. This noise can be quantified
in order to determine the number of groups that must be observed in order to measure
any underlying correlations. Similarly, variations in galaxy properties at fixed luminosity;
such as variations in surface brightness, gas fraction, star formation rate, etc..., if these
are not related to the galaxy property of interest; can be averaged over by observing large
number of galaxies. It should be noted that satellite galaxies are being treated in a similar
fashion to host groups. The same split is being made between properties that correlate
with satellite luminosity, correlations between satellite properties, and unrelated variations
in satellite properties. Before they were accreted by their host group, satellite galaxies were
the central galaxies of their own halos and had properties that could be characterized by a
mass dependence plus a distribution at fixed mass.
One aspect of the HOD based approach may need to be considered in an explicitly
theoretical fashion. The accretion history of host halos may depend on the mass of the
host. That is, the typical time since a satellite was accreted may vary with both group
mass and satellite luminosity. This process can be considered within the HOD framework,
assuming that the distribution in accretion times is set by host halo mass and satellite
luminosity, recalling that the distribution of satellite luminosity is itself determined by host
20 Introduction
halo mass. Unfortunately, such a correlation could not be observed directly, but would be
captured in an N-body simulation. Any such correlation will likely need to be considered,
and will have to be drawn from simulations. This is an exception to modeling only the
physical process under consideration and letting observations themselves capture all other
physical processes.
1.4.2
N-body Simulations
Cosmologically large N-body simulations have several important roles to play in the study
of galaxy evolution. N-body simulations confirm analytical treatments of structure formation and capture any deviations from these treatments that are introduced by non-linear
effects. This includes capturing any deviations from the assumption that dark matter halo
properties are completely characterized by their mass. Structure formation forms the background to gastrophysical processes that drive galaxy formation and evolutions. Simulations
are also crucial for studying dynamics, particularly for subhalos within more massive host
halos. Dynamical processes can effect the morphologies of galaxies. Minor mergers between
galaxies and tidal interactions are likely responsible for determining many of the details of
galaxy morphologies. Major mergers may be the mechanisms where by elliptical galaxies
and bulges in spiral galaxies are created.
Analytical models of subhalo dynamics provide valuable insight into the fate of subhalos, but can easily fail to capture the interplay between different dynamical mechanisms.
Interactions are generally considered in an average sense, rather than by tracking individual
subhalos. In addition, how two or more processes that operate on similar time scales interact may be quite difficult to predict analytically. Subhalos are tidally stripped and shocked
by the host dark matter potential, they experience some degree of dynamical friction which
degrades and circularizes their orbits, they undergo inelastic unbound collisions which each
other, trading orbital energy and angular momentum and further heating the subhalos, and
subhalos may also occasionally merge with each other (Mamon, 2000; Peñarrubia and Benson, 2005; Boylan-Kilchin et al., 2008; Knebe et al., 2006). The effect of each interaction
1.4 Galaxies in a Cosmological Context 21
on an individual subhalo is determined by that subhalo’s orbit, which is in turn set by its
orbit at the time of accretion, modified by all subsequent interactions. The eventual fate of
subhalos is to merge with the core of the host halo, but the time scale on which this takes
place can clearly be heavily affected by all of the interactions that the subhalo experiences.
By considering the interplay between all of these processes, the overwhelming challenge
that the analytical modeler faces becomes clear. N-body simulations overcome this obstacle
by explicitly tracking all gravitational interactions. Simulations, however, have their own
challenges.
Cosmological N-body simulations track the evolution of matter under the influence of
gravity. Initial conditions are set at high redshifts, before structure formation begins to
be non-linear. The majority of matter in the universe is dark, and therefore primarily
interacts gravitationally. Pure N-body simulations assume that, after the redshift at which
the initial conditions are set, the effects of baryonic physics on the evolution of the dark
matter is weak. Such simulations are expensive, and the computer time needed to run
them increases with the number of dark matter particles involved. There is therefore a
trade off between expense, size, and both mass and space resolution. The density of the
universe is set by cosmology. Hence there is a very explicit trade off between size and mass
resolution. Typical cosmological simulations are quite large, containing tens of millions to
billions of equal mass dark matter particles. This is necessary to simultaneously probe
a cosmologically interesting volume and to resolve the halos that typically host galaxies.
The analysis of N-body simulations is therefore typically pushed to their absolute limits.
This is particularly true when analyzing subhalos. If there is no physical understanding
of the dynamics of subhalos, then it is possible to misidentify the limits of a dark matter
simulation. Hence, simulations and analytical treatments must be used simultaneously to
probe the evolution of subhalos.
The results of studying of halo and subhalo dynamics could be placed in a HOD context.
There is however little motivation to do so. Any correlations between dynamical history and
halo mass, host halo mass, or any other measure of environment can be made directly in the
22 Introduction
simulation. Correlation functions are also easy to measure directly in cubical simulation
boxes. In addition, recent studies of particularly large N-body simulations indicate that
there may be residual dependences of dark matter halo assembly histories on local environment (Wechsler et al., 2006; Croton et al., 2007; Gao and White, 2007). This dependence
is captured in simulations, but not by the HOD treatment. Correlations between galaxy
properties and local environment that cannot be reduced to correlations with host halo
mass may expand the range of interesting observational tests of galaxy evolution scenarios.
Chapter 2
An Analytical Model of Ram
Pressure Stripping
The contents of this chapter have been previously published as Hester (2006a)
2.1
Introduction
Ram pressure stripping was first proposed by Gunn and Gott (1972) to explain the observed
absence of gas-rich galaxies in clusters. They noted that galaxies falling into clusters feel
an intracluster medium (ICM) wind. If this wind can overcome the gravitational attraction
between the stellar and gas disks, then the gas disk will be blown away. They introduced a
simple analytical condition to determine when gas is lost:
2
ρICM vsat
< 2πGσ∗ σgas .
(2.1)
The right-hand side, 2πGσ∗ σgas , is a gravitational restoring pressure, ρICM is the density
of the ICM, and vsat is the orbital speed of the satellite galaxy. Using this condition they
concluded that spirals should lose their gas disks when they pass through the centers of
clusters.
Galaxies near cluster centers have earlier type morphologies, are redder, and form fewer
stars than galaxies in the field of similar luminosity (Gómez et al., 2003; Goto et al., 2006;
23
24 An Analytical Model of Ram Pressure Stripping
Hogg et al., 2003b,a). Stripped galaxies have low H i masses for their size and morphology.
Such H i deficiency correlates with lower star formation rates (SFRs) (Gavazzi et al., 2006),
and lowered SFRs lead to redder colors. Therefore, ram pressure stripping, in conjunction
with processes like harassment that can alter the structure of a galaxy, may play a role in
transforming galaxy morphologies.
Several sets of observations indicate that stripping occurs to spirals in clusters. Giovanelli and Haynes (1983) define a deficiency parameter that compares a galaxy’s observed
H i mass to the expected H i mass in a field galaxy with the same morphology and optical
size. Galaxies in Virgo, (Giovanelli and Haynes, 1983), Coma, (Bravo-Alfaro et al., 2000),
and other nearby clusters, (Solanes et al., 2001) are observed to be H i deficient. Solanes
et al. (2001) observe that the average deficiency increases with decreasing distance to the
clusters’ centers and that deficient galaxies have, on average, more eccentric orbits. Cayatte
et al. (1994) observe indications of stripping in the H i distributions of individual galaxies in
the Virgo cluster. Compared to an average profile for a field galaxy of similar morphology
and optical extent, these galaxies have normal gas densities in their inner disks but are
sharply truncated. More detailed observations of several Virgo spirals have since been carried out. NGC 4522 is observed to have an undisturbed optical disk, a severely truncated H
i disk, and extra planar Hα and H i on one side of the disk (Kenney and Koopmann, 1999;
Kenney et al., 2004b). IC 3392, NGC 4402, NGC 4419, and NGC 4388 all have a truncated
H i disk, and the first three have extraplanar H i (Kenney et al., 2004a). NGC 4569 has
a truncated H i disk and extra planar Hα (Kenney et al., 2004a). Observing asymmetric,
extraplanar gas in combination with an undisturbed optical disk is a strong indication that
the galaxy’s gas is interacting with the ICM.
Detailed simulations of ram pressure stripping confirm that spirals in clusters should be
stripped. For some simulations the final radii of the galaxies are compared to the predictions
of the Gunn and Gott condition. In addition, the morphologies of simulated galaxies have
been compared to observations of spirals in Virgo. Simulations by Abadi et al. (1999) show
that galaxies in a face-on wind are stripped to radii near those predicted by the Gunn and
2.1 Introduction
25
Gott condition, but galaxies in an edge-on wind are only mildly stripped. In contrast, other
simulations show a two-step stripping process that strips galaxies at all inclination angles.
Ram pressure stripping first quickly strips gas from the outer disk, and then the remaining
disk is slowly viscously stripped (Marcolini et al., 2003; Quilis et al., 2000; Roediger and
Hensler, 2005; Schulz and Struck, 2001). Marcolini et al. (2003) point out that, unlike ram
pressure stripping, viscous stripping is more effective in an edge-on wind. They find that the
final radii of the face-on galaxies match those predicted using the Gunn and Gott condition
and that edge-on galaxies are stripped to slightly larger radii. Roediger and Hensler (2005)
run a series of simulations to test the dependence of the stripping radius on the speed of the
ICM wind, the mach number, and the vertical structure of the gas disk. They find that the
most important parameters are the ram pressure and surface density of the gas disk and
that the effects of varying the mach number and vertical structure are minor. They find
that the original Gunn and Gott condition does a fair job predicting the stripping radius.
However, when they use an adjusted ram pressure, they find that using the thermodynamic
pressure in the central plane of the disk more accurately predicts the stripping radius.
Simulations of ram pressure stripping can also match the range of morphologies of
stripped galaxies. Vollmer et al. (2000, 2004) track the density, velocity, and velocity
dispersion of a simulated galaxy’s cold disk gas. In NGC 4522, the kinetically continuous
extraplanar gas matches simulations of a galaxy that is currently being stripped. However,
the gas in NGC 4569 is less pronounced, not kinetically continuous with the disk gas, and
has a large velocity dispersion, more like simulations of a galaxy that has been stripped
in the past. In their simulations Schulz and Struck (2001) observe a process they term
”annealing”. The ICM wind compresses the gas disk and triggers the formation of large
spiral arms. The interaction of these arms with the wind leads to the contraction of the
galaxy and the formation of a dense truncated gas disk. A burst of star formation may
occur at the edge of the truncated disk. A ring of star formation is seen at the truncation
radius of the disk in IC 3392.
Some groups have studied stripping in groups and cluster outskirts where the ram pres-
26 An Analytical Model of Ram Pressure Stripping
sure is lower than in cluster centers. While simulations and observations both support the
idea that large spirals in clusters are stripped of their H i disks, it is not clear that stripping
occurs in poorer environments. Two types of stripping may occur in groups. The ram pressure in groups is lower than in clusters. However, dwarf galaxies have lower masses than
large spirals and therefore lower restoring pressures, and dwarf galaxies may be stripped
of most of their H i. In addition, large spirals have restoring pressures that decrease with
radius and may be stripped of their outer disks. Marcolini et al. (2003) run simulations o
dwarf-like galaxies in winds typical of the ram pressure in groups and find that the simulated dwarfs are stripped. The Roediger and Hensler (2005) simulations include runs with
low ram pressure. Their large and medium-sized model spirals are stripped of some of their
outer H i in these runs. The observational evidence for stripping in low-mass systems is circumstantial. The gas content and star formation histories of Local Group dwarfs correlate
with their distances from the dominant spirals (Blitz and Robishaw, 2000; Grebel, 2002).
This project aims to use the clear observations of stripping in clusters to predict when
stripping occurs in groups. This is done by examining how the ram and restoring pressures
depend on the masses of the cluster or group and the satellite galaxy. Two ways in which
the masses of the host system and satellite galaxy affect the ram and restoring pressures
are identified. Satellites’ orbital speeds are set by the depth of the cluster’s potential, and
the restoring force on the gas disk is determined by the depth of the satellite’s potential.
In addition, both cluster and galaxy morphologies change with mass. A model is developed
and used to study both types of mass dependence. This is done in the following steps.
In the next section a model for the mass and gas distributions is presented both in terms
of physical parameters and in terms of scale-free parameters. In § 2.3 the dependence of
ram pressure stripping on the masses of the satellite and cluster is presented assuming that
the scale-free parameters are constant. In §2.4 the dependence of the scale-free parameters
on the masses of the satellite and cluster and the sensitivity of the model to these changes
are explored. In §2.5 the results are discussed and the model is compared to observations
of stripping. Section 2.6 concludes.
2.2 The Model
2.2
27
The Model
The analytical model of ram pressure stripping developed here uses the Gunn and Gott
condition to predict the extent to which galaxies are stripped. This condition is based on
a simplified picture of stripping that ignores the details of the hydrodynamical processes
and balances forces on large scales. However, simulations of galaxies that are stripped by
a face-on wind result in stripping radii that match the predictions of the Gunn and Gott
condition (Abadi et al., 1999; Marcolini et al., 2003; Roediger and Hensler, 2005). In the
simulations galaxies in an edge on wind are eventually viscously stripped to radii similar to
those of face-on, ram pressure stripped galaxies. This is discussed further in section 2.4.3.
In light of the general agreement between Gunn and Gott’s condition and simulations of
ram pressure stripping, it is assumed that a model based on this condition can be used to
make general statements about where and to what extent galaxies are stripped.
The model for the group or cluster has two components, a gravitational potential in
which the satellite orbits and an ICM against which the satellite shocks. The gravitational
potential is modeled using an NFW profile with mass Mv,gr , radius, rv,gr , and concentration,
cg . The scaled radius s ≡ r/rv,gr .
φhalo =
−GMv,gr
ln(1 + cg s)
g(cg )
rv,gr
s
(2.2)
The NFW profile is discussed in more detail in appendix A. The ICM is modeled using a
β profile with scale length rc , and central density, ρ0 .
(−3/2)β
r2
ρ = ρ0 1 + 2
rc
(2.3)
The β profile can be rewritten as
αvρ ρ0c
ρ=
3
1+
s2
(rc /rv,gr )2
(−3/2)β
(2.4)
The parameter α relates ρ0 to the characteristic density of the NFW profile, and vρ ρ0c is
the average dark matter density within rv,gr .
The gravitational potential of the satellite and the surface density of the gas disk are
needed to model the restoring pressure. The gravitational potential combines a dark matter
28 An Analytical Model of Ram Pressure Stripping
halo, a stellar disk, and a stellar bulge. The dark matter halo is modeled as an NFW profile
with mass Mv,sat and concentration cs . The disk, of mass Md , is described using a Miyamoto
and Nagai potential (Miyamoto and Nagai, 1975).
−GMd
2
√
R2 + Rd + z 2 + h2
φdisk = r
(2.5)
This potential is easily differentiated, and with the appropriate choice of disk scale length,
Rd , and height, h, many of the properties of the potential of an exponential disk can be
matched (Johnston et al., 1995). This potential can be rewritten as
φdisk =
−GMd
1
r
q
2
Rd
λ2d S 2 + 1 + λ2d zs2 + λ2h
(2.6)
where λd ≡ rv,sat /Rd , λh ≡ h/Rd , S ≡ R/rv,sat , and zs ≡ z/rv,sat .
The bulge, of mass Mb , is described using a Hernquist potential (Hernquist, 1990).
−GMb
r + rb
(2.7)
1
−GMb
rb λb ssat + 1
(2.8)
φbulge =
This can be written as
φbulge =
where λb ≡ rv,sat /rb , ssat ≡ rsat /rv , and rsat is the distance from the center of the satellite.
By introducing mds ≡ Md /Mv,sat and mb ≡ Mb /Mv,sat , the full gravitational potential of
the satellite can be written in the form
φf ull =
−GMv,sat
f (s~sat , mi , λi , cs )
rv,sat
(2.9)
Gas in galaxies can be found in three spatial components: an exponential molecular
gas disk with a scale length comparable to the stellar disk, a nearly flat atomic disk that
extends beyond the stellar disk and shows a sharp cutoff, and a hot galactic halo. Beyond
the disk cutoff, the atomic disk may continue as an ionized gas disk (Binney and Merrifield,
1998).
2.2 The Model
29
The H i is modeled as a thin flat disk with mass Mg , surface density, σg , and a sharp
cutoff at radius Rg
σg =



σ
0
if R < Rg


0,
if R > Rg
Mg
Mv,sat mdg λ2g
σ0 ≡
= 2
πRg2
π
rv,sat
(2.10)
(2.11)
where mdg ≡ Mg /Mv,sat is the fractional mass of the gas and λg ≡ rv,sat /Rg is the scaled
size of the disk.
The inner gas disk is dominated by the H2 disk. The molecular disk is significantly
more difficult to strip than the H i disk both because the H2 disk is more compact and
because the H2 is found in molecular clouds. These clouds do not feel the effect of stripping
as strongly as the diffuse H i. How stripping of H2 occurs and how the H2 clouds and
diffuse H i in the inner disk interact are not known. The two phases may be tied together
by magnetic fields; in which case the inner H i disk will remain until the ram pressure can
remove the entire inner disk. It may also be possible for the wind to remove the H i from
the inner disk while leaving the H2 behind. Because of this uncertainty, the stripping of
the inner disk will not be studied here. The model is only used to discuss stripping beyond
1.5 stellar scale lengths. If the ram pressure is not strong enough to affect the H2 , the H2
will contribute to the restoring pressure in the outer disk. Therefore, the mass of the H2
disk is added to the stellar mass.
The emphasis of this project is to study the stripping of the H i disk. However, the
fate of the hot galactic halo can also influence the evolution of the stripped galaxy. It is
expected that the hot halo will be easily stripped both because it is diffuse and because
the restoring potential of the satellite is strongest in the disk. To check this a hot halo is
included in the model galaxy.
Mori and Burkert (2000) give an analytic condition for the complete stripping of a hot
galactic halo.
2
ρICM vsat
> P0,th =
ρ0,sat kB T
µmp
(2.12)
30 An Analytical Model of Ram Pressure Stripping
where P0,th and ρ0,sat are the central thermal pressure and density of the hot galactic halo,
kB is the Boltzmann constant, T is the temperature of the galactic halo, and µmp is the
average molecular mass. The thermal pressure replaces the gravitational restoring pressure
in the original stripping condition. In their simulations, this condition does a fair job of
predicting the mass of a galaxy that can be completely stripped. This condition is adapted
for the current model and all gas outside the radius where Pram = Pth is assumed to be
stripped.
The hot galactic halo is modeled by assuming that the gas is at the virial temperature,
Tv , of the satellite and is in hydrostatic equilibrium in the satellite’s dark matter potential.
The self-gravity of the gas is ignored. The density of the hot halo is
ρ(ssat ) =
vρ ρc0
3
msg j(ssat , cs )
(2.13)
where scaled density profile j(ssat , cs ) is defined in appendix C and msg is the ratio of the
mass of gas in the hot galactic halo to Mv,sat .
A dimensionless temperature, t(cs ) is also introduced in appendix C. It is defined by
t(cs ) ≡
kB Tv
µmp
rv,sat
GMv,sat
(2.14)
It is a function of only cs .
2.3
Results
In this section the dependence of stripping on the masses of the large and satellite halos
and on the model’s parameters is explicitly determined.
The maximum ram pressure, the ram pressure at the pericenter of the galaxy’s orbit, is
given by
Pram,max
GMv,gr b2s αvρ ρ0c
=
rv,gr s20 v 3
1+
s20
(rc /rv,gr )2
(−3/2)β
(2.15)
where s0 is the distance of closest approach, is the orbital energy per unit mass, v ≡
2 ), and l is the orbital angular momentum per unit mass.
−GMv,gr /rv,gr , bs ≡ l2 /(||rv,gr
2.3 Results
31
Both bs and /v are expected to be independent of the masses of the satellite and the
cluster.
At any point in the orbit, sorbit , the ram pressure is
Pram (sorbit ) =
2GMv,gr αvρ ρ0c
p(sorbit , cg )
rv,gr
3
(2.16)
where dimensionless pressure, p(s, cg ), is defined.
(−3/2)β
ln(1 + cg s)
s2
p(s, cg ) ≡ g(cg )
−
1+
s
v
(rc /rv,gr )2
(2.17)
Equations for the orbital speeds are derived in appendix B. Assuming that α, β, rc /rv,gr ,
and cg do not depend on the mass of the halo, the Mv,gr dependence of both Pram (s0 ) and
2/3
Pram (sorbit ) is Pram ∝ Mv,gr .
For a potential of the form in eq. 2.9, the force per unit gas mass in the z direction is
found as follows.
fz =
GMv,sat ∂
f (s~sat , mi , λi , cs )
rv,sat ∂z
(2.18)
fz =
GMv,sat ∂
f (s~sat , mi , λi , cs )
2
∂zs
rv,sat
(2.19)
Assuming that the mi and λi introduced in section 2.2 and cs are constant with mass, the
1/3
mass dependence of the restoring force per unit gas mass is given by fz ∝ Mv,sat .
For the gas disk, if the fractional mass of the gas, mdg , and the scaled size of the disk,
λg , are both constant, then
1/3
2
σg ∝ Mv,sat /rv,sat
∝ Mv,sat
(2.20)
2/3
(2.21)
Combining the above,
Prest,max = σg fz,max ∝ Mv,sat
For any radius along the disk, Rstr , there is a maximum restoring pressure. If the
maximum restoring pressure is greater than the ram pressure, the satellite holds the gas at
this radius. The condition for the satellite holding its gas can be written as
∂
2
2/3
Mv,sat ∂zs Rstr f (~ssat , mi , λi , cs ) 2mdg λg
Prest,max
=
>1
2/3
Pram,max
αp(sorbit , cg )
Mv,gr
(2.22)
32 An Analytical Model of Ram Pressure Stripping
3/2

Mv,sat 
> 
Mv,gr
αp(sorbit , cg )
∂
∂zs
Rstr
f
(s~sat , mi , λi , cs ) 2mdg λ2g
(2.23)


The restoring pressure of the galactic hot halo is
GMv,sat
vρ ρc0
Prest =
t(cs )msg j(ssat , cs )
rv,sat
3
(2.24)
The condition for retaining this gas within ssat is
Mv,sat
t(cs )msg j(ssat , cs ) 3/2
>
Mv,gr
2αp(sorbit , cg )
(2.25)
If the model’s parameters: cg , cs , /v , bs , β, rc /rv,gr , α, the mi , and the λi ; are all
independent of the mass of the satellite and cluster, then the fraction of gas that is stripped
3/2
from a satellite depends on the ratio Mv,sat /Mv,gr . In this case, Pram ∝ Mv,gr and for both
3/2
the gas disk and hot galactic halo Prest ∝ Mv,sat .
Assuming that this set of parameters is constant is equivalent to assuming that the physical parameters scale with mass in the most obvious way. It assumes that component masses
scale as Mv , lengths scale as rv , and the central densities of the ICM and the satellite’s hot
galactic halo are constant. For any model in which these scalings hold, the result that the
extent of stripping depends on Mv,sat /Mv,gr will hold. For any cluster mass, the scaled
radius will set the ICM density and the orbital velocity will be proportional to Mv,gr /rv,gr .
The restoring pressure depends on the depth of the satellite galaxy’s gravitational potential
well and the density of the gas disk. For a generic potential, φ ∝ Mv,sat /rv,sat and the
1/3
restoring force per unit gas mass is proportional to Mv,sat . If the mass of the gas disk scales
1/3
with Mv,sat and the disk length with rv,sat , then the density of the gas disk scales as Mv,sat
2/3
and the restoring pressure as Mv,sat .
2.4
Model Parameters
In this section a reference set of values for the model’s parameters is introduced and the effect
of varying these parameters is discussed. The reference parameters determine the values of
Mv,sat /Mv,gr at which stripping occurs. They are taken from observations, where possible,
2.4 Model Parameters 33
and λCDM simulations, otherwise. Two types of parameter variations are discussed. First,
several of the model parameters vary systematically with Mv,sat or Mv,gr . Second, at fixed
masses the parameters have scatter.
The scatter in the parameters is discussed for several reasons. By combining an estimate
for the scatter in each parameter with the model, the parameters that have the most effect
on the radius to which an individual galaxy is stripped can be identified. Second, it is useful
to know the range of ratios across which stripping occurs. Finally, if the overall scatter is
not too large, the relationship between Rstr and Mv,sat /Mv,gr could be used to search for
signs of stripping in a large galaxy survey. The results of this section are discussed in the
terms above in section 2.5.2.
In Figures 2.6 and 2.6, the Mv,sat at which a satellite is stripped is plotted versus
Rstr /Rg and Mstr /Mg for a satellite orbiting at a variety of sorbit values in 1015 , 1014 , and
1013 M clusters. The mass of the stripped gas is denoted as Mstr . Table 1 lists the model
parameters used for three models, a large satellite in a 1015 M cluster, the reference model;
a large satellite in a 1014 M cluster, the middle-mass model; and a small satellite in a
1013 M group, the low-mass model. The large satellite model is based on the Milky Way.
Physical length scales and masses for the stellar disk and bulge are from Johnston et al.
(1995). The stellar disk is modeled using Mds = 1011 M , Rd = 6.4 kpc, and h = .26
kpc. The equivalent Rd for an exponential disk is 5 kpc. The bulge is modeled using
Mb = 3.4 × 1010 M and rb = .7 kpc. A flat disk with a radius of 30 kpc gives Rg /Rd
similar to that seen in observations. The H i gas mass is chosen to result in a disk density
of 5 M pc−2 . The virial mass is chosen such that Mv,sat /Mbaryon ≈ 12. Setting Mv,sat
also sets rv,sat . The concentrations used are cs = 12 and cg = 5.5, 7.5, and 10. The
concentrations are based on Bullock et al. (2001) and assume an average over-density within
rv , vρ , of 337. The assumed over-density affects the concentrations, which are measured
using vρ = 337, and α, because it relates the ICM and dark matter densities. Note that
vρ does not appear in equation 2.23 or 2.25. The ICM parameters, α, β, and rv,gr /rc , are
set by X-ray observations of clusters and groups, and the orbital parameters are taken from
34 An Analytical Model of Ram Pressure Stripping
simulations. Both are discussed below. Table 1 also lists the assumed scatter in each model
parameter and resulting scatter in the mass at which a satellite is stripped.
2.4.1
ICM Profile
The profile of the ICM is determined by the parameters α, rv,gr /rc , and β (eq. 2.4). Sanderson and Ponman (2003) compile a set of average ICM profiles for clusters with temperatures
ranging from 0.3 to 17 keV. They find that the ICM of groups and poor clusters is both
less dense and more extended than in rich clusters. Mohr et al. (1999) present X-ray observations of a set of clusters and Helsdon and Ponman (2000) and Osmond and Ponman
(2004) present X-ray observations of groups. The three groups present β and rc (in units
of kpc) for a best-fit β profile and the X-ray temperature, TX , for each group or cluster.
The average β and rv,gr /rc for the cluster observations are approximately 0.67 and 12. The
averages for the group observations are approximately 0.45 and 170. To convert rc (in kpc)
to rc /rv,gr , virial masses for the groups and clusters are found using the M200 (TX ) relationship from Popesso et al. (2005). The mass M200 is the mass contained within the radius
at which the average over-density reaches 200. It is converted to Mv using equations A.7
and A.6. The profiles in Sanderson and Ponman (2003) are scaled using r200 as determined in Sanderson et al. (2003). The differences in the rc /rv,gr determined using the two
M200 (TX ) relationships are within the intergroup scatter in rc /rv,gr . The average profiles
for the hottest and coolest groups from Sanderson and Ponman (2003) are well matched
by using α = 6.5 and 20 respectively and the appropriate β and rv,gr /rc . In Figures 2.6
and 2.6, plots are made for stripping in 1013 , 1014 , and 1015 M clusters. A cluster with
Mv,gr = 1015 M falls into the hottest temperature bin from Sanderson and Ponman (2003)
while a group with Mv,gr = 1013 M falls into the coolest. The ICM parameters for the
1014 M cluster are α = 5.5, β = .57, and rv,gr /rc = 25.
Scatter in these parameters is estimated using the three sets of observations. The
observed scatter in β is ±0.08. The majority of clusters have rv,gr /rc between 7 and 20, and
groups have rv,gr /rc between 75 and 270. For both groups and clusters, α is assumed to
2.4 Model Parameters 35
vary by a factor of
√
6 in either direction. Varying β varies the ram pressure at sorbit = .35
by a factor of 2 for the clusters and a factor of 7.4 for the groups. The ram pressure varies
by a factor of 6 in the clusters and 5.4 in the groups when rv,gr /rc is varied. Varying α by
a factor of 6 results in an increase in the ram pressure by the same factor.
2.4.2
Orbits
The orbital parameters are important for determining both satellites’ pericenters and orbital
speeds. In theory, the pericenter of a satellite’s orbit is determined by /v and bs (Eq. B.2).
However, Gill et al. (2004) present pericenter distributions for subhalos at z = 0 in clustersized dark matter halos with Mv,gr = 1 − 3 × 1014 M . They find that the s0 distribution
can be fit with a Gaussian with s¯0 = .35 and σs0 = .12. This distribution shows little
variation with either Mv,gr or Mv,sat . The maximum ram pressure in a cluster with ICM
parameters corresponding to a few 1014 M and cg = 7.5 increases by a factor of 5.4 when
the pericenter decreases from s0 = 0.47 0.23.
Some satellites have not yet passed through the pericenter of their orbits, and the maximum ram pressure they have experienced is determined by their current sorbit . Increasing
sorbit from 0.35 to 1 results in a decrease of the mass at which a satellite can be stripped by
two orders of magnitude. Therefore, unless it is known that a group of satellites have all
passed through their pericenters, the effect of varying sorbit cannot be treated like the scatter in the other parameters. Instead, when comparing satellites in different environments
or with different masses, sorbit must be specified.
At a given sorbit , the ram pressure is dependent on /v , but not l/lv . In simulations
Vitvitska et al. (2002) find that the speeds of satellites entering dark halos can be fit with a
2
Maxwell-Boltzmann distribution with vrms
≈ 1.15vc2 , vc2 ≡ GMv,gr /rv,gr . The total energy
per unit mass of an object at the virial radius with this speed is ≈ v , and is a weak
function of cg . This distribution has a large scatter, σv = .6vc , corresponding to /v
between 0.2 and 2.8. At sorbit = .35, when vsat (sorbit = 1)/vc is varied between 0.55 and
1.75, the Mv,sat /Mv,gr needed for stripping vary by a factor of ≈ 1.6. Vitvitska et al. (2002)
36 An Analytical Model of Ram Pressure Stripping
also find that for Mv,sat < .4Mv,gr the angular momenta of satellites entering halos can
be fit with a Maxwell-Boltzmann distribution with lrms /lv = .7, lv ≡ vc rv . As expected,
the distributions of /v and bs do not depend on Mv,sat and Mv,gr . However, unless the
correlation between vsat (sorbit = 1)/vc and l/lv is understood, these parameters cannot be
used to determine the distribution of s0 .
2.4.3
Inclination
The Gunn and Gott condition assumes that all galaxies orbit in a face-on wind. In reality, galaxies orbit at all inclinations. Simulations show that galaxies are stripping at all
inclinations, but edge-on galaxies are stripped to larger radii than face-on galaxies.
Marcolini et al. (2003) point out that while galaxies that are hit face-on by the ICM
wind experience the most ram pressure stripping, galaxies in an edge-on wind experience the
most viscous stripping. They derive an expression for the ram pressure needed to viscously
strip an edge-on galaxy to radius R and treat this pressure as a restoring pressure for the
edge-on case. When they compared the new condition to the Gunn and Gott condition,
they find that any ram pressure that can completely strip a face-on galaxy will viscously
strip an edge-on galaxy and that any ram pressure that cannot strip a face-on galaxy will
not strip an edge-on galaxy. The two models differ when a galaxy is partially stripped.
In this case edge-on galaxies are still stripped, but to larger radii than face-on galaxies.
They run simulations at a variety of inclination angles, i, and find that the final stripping
radii match the predictions of the two conditions. When galaxies at intermediate angles are
partially stripped, they are stripped to intermediate radii.
When Marcolini et al. plot the two restoring pressures for three galaxy masses, 0.76,
7.4, and 77.2 ×109 M , their Figure 3, the fractional difference between the two restoring
pressures is close to constant throughout the outer disk but increases with the mass of the
galaxy. For the largest galaxy the restoring force in the edge-on case is ≈ 4.5 times larger.
2.4 Model Parameters 37
2.4.4
Concentration of the Cluster
Small dark matter halos tend to be more concentrated than large dark matter halos. Bullock
et al. (2001) use an analytical model of the evolution of dark matter halos to describe this
dependency. For the standard λCDM model, near Mv = M ∗ ≈ 1.5 × 1013 h−1 M they find
c(z = 0) = 9
Mv
M∗
−.13
(2.26)
The cg used for the cluster and group models are determined using this relation. Bullock
et al. (2001) also see a scatter that is as large as the evolution of the concentration over the
range 0.01M∗ < Mv < 100M∗ . To explore the scatter introduced through cg , cg is varied
between 5 and 16.
The cluster concentration affects both s0 and the ram pressure at a given sorbit (eqs. B.2
2
2 (s
and B.6). When cg is increased from 5 to 16, for vsat
orbit = 1) = 1.15vc and l/lv = 0.7, the
pericenter decreases from s0 = 0.53 to 0.5. This is smaller than the scatter in the pericenter.
At a given sorbit , the orbital speed increases slightly as cg increases. This difference increases
as sorbit decreases. For sorbit = 0.75, the change is not noticeable. For sorbit = 0.35, the
Mv,sat /Mv,gr at which a satellite is stripped increases by a factor of 1.2 when cg is increased
from 5 to 16.
2.4.5
Concentration of the Satellite
In the outer H i disk the restoring pressure of the dark matter halo makes a non-negligible
contribution to the restoring pressure. More concentrated satellites have higher restoring
pressures than other satellites of the same mass. Varying cs between 12 and 20 varies the
Mv,sat /Mv,gr at which stripping occurs by a factor of ≈ 1.7.
The concentration also alters how the hot galactic halo is stripped. In a more concentrated dark matter halo the gas is denser in the center and more diffuse elsewhere. Stripping
can therefore occur down to a smaller radius, but a smaller fraction of the gas is lost.
38 An Analytical Model of Ram Pressure Stripping
2.4.6
Length Ratios, λi
How disk scale lengths vary with Mv,sat is an open question. Theory suggests that when
cold gas disks form in dark matter halos, their initial radii are proportional to the virial radii
of the halos they form in (Mo et al., 1998). There is a large scatter in this relationship that
is due to a large scatter in the disk angular momenta. However, cold disks are composed of
a combination of stars and gas, and how these components arise from the original cold disk
is not understood.
The fraction of the H i that is found in the outer disk, where the restoring force from
the stellar disk is weakest, is set by the ratio λd /λg (eqs. 2.6 and 2.11). Swaters et al. (2002)
observed the H i disks of 73 local dwarfs. They found λd /λg = 6 ± 2.5 for the dwarfs. For
late-type spirals λd /λg = 5.5 ± 1.6. These observations suggest that λd /λg contributes to
the scatter in Mv,sat /Mv,gr , but not to the mass dependence. Varying λd /λg changes both
the size and the shape of the restoring force. When λg is held at 5.7 and λd /λg is decreased
to 3.5, the restoring force decreases within R/Rg = 0.45 and increases at larger R. The
fractional change in Prest is less than 1.1 throughout the disk. Holding λg constant and
increasing λd /λg to 8.5, results in decreasing the restoring force outside of R/Rg = 0.3 by
a factor of less than 1.13.
Varying λg while holding λd /λg constant changes the density of both the gas and stellar
disks. At a given R/Rg , the density of both disks ∝ λ2g and the restoring pressure ∝ λ4g . The
scatter in λg and how λg evolves with mass are not known. However, because of the heavy
dependence of the restoring pressure on it, the scatter in λg is likely to make an important
contribution to the scatter in the Mv,sat /Mv,gr at which stripping occurs. Increasing λg
from 0.75λg to 1.25λg decreases the Mv,sat /Mv,gr needed for stripping by a factor of 20.
The presence of molecular gas complicates the physics of stripping in the inner disk.
Therefore, this model is only applied to disk radii beyond 1.5 stellar disk scale lengths. At
these radii, R > 10rb , the bulge acts as a point mass, and changing λb (Eq. 2.8) has no
effect. In particular, the radius of the bulge changes with the mass of the bulge. Below, the
1/3
mass of the bulge is allowed to vary by 40%. Assuming that rb ∝ Mb , this corresponds
2.4 Model Parameters 39
to allowing rb to vary by 12%. This is not noticeable in the outer H i disk.
2.4.7
Mass Fractions, mi
Both mb and mdg (eqs. 2.8 and 2.11) are correlated with the mass of the satellite. While
the majority of large spiral galaxies have a significant fraction of their stars in a stellar
bulge, low-mass late-type galaxies do not. Very low mass spiral galaxies exist that do not
contain any bulge (Matthews and Gallagher, 1997) and dwarf galaxies are observed to be
either dE/dSph or dIrr. In addition, low-mass late-type galaxies have a higher fraction of
their mass in the gas disk than large spirals (Swaters et al., 2002). Decreasing mb reduces
the restoring pressure in the outer disk. In contrast, increasing mdg increases the restoring
pressure.
In the bottom right panels of Figures 2.6 and 2.6, the satellite is bulgeless, the gas
mass fraction, mdg , is doubled, and the stellar mass fraction, mds , is reduced by 20%. The
original model corresponds to a large late-type spiral, while the bulgeless model is closer
to a late-type dwarf galaxy. Because the restoring force due to the bulge is small, the net
effect is to increase the restoring pressure. Most galaxies will lie between the two plotted
models.
The mi can be split between those that set the depth of the potential well, mds + mH2
and mb , and mdg which sets the gas density. In late-type galaxies, bulges can contribute up
to 50% of the stellar light (Binney and Merrifield, 1998). In the reference model, 25% of
the stars are in the bulge. Varying mb between 15% and 35% of the satellite’s stars, varies
the restoring force in the outer disk by a factor of ≈ 1.1. Neither the mass dependence nor
the scatter in mds + mH2 are known. Varying mds + mH2 by 20%, varies the restoring force
by a factor of ≈ 1.3. The effect of scatter in the gas density parameter, mdg , is straight
−3/2
forward. The Mv,sat /Mv,gr at which stripping occurs varies as mdg
. Varying mdg by a
factor of 1.5, similar to the scatter in MHi /LR in Figure 9 of Swaters et. al., varies the
Mv,sat /Mv,gr needed for stripping by a factor of 1.8.
Variations in the mi values should be correlated because the variation in the mass in
40 An Analytical Model of Ram Pressure Stripping
baryons in a satellite is likely smaller than the variation in the mass of a given component.
The correlation is such that it will reduce the overall scatter.
2.5
2.5.1
Discussion
Comparison with Observations and Simulations
In this section, the Mv,sat /Mv,gr values at which the model predicts stripping should occur
are compared to observations of H i disks from Solanes et al. (2001). This sample is
useful for making comparisons between the model and observations because the number of
spirals is large and the observations span several clusters. The average temperature of the
clusters is 3.5keV. This corresponds to Mv,gr = 3 × 1014 M . The average gas profile for
this temperature can be described using the middle mass cluster model. In Figure 4 from
Solanes et al. the fraction of spirals with def > 0.3 and the average deficiency are plotted
versus r/RA , where RA is the Abel radius. In Figure 2.6 of the present paper Mv,sat is
plotted versus the deficiency in a 3 × 1014 M cluster at several sorbit . The deficiency is
defined as def ≡ log(M̄ /Mobs ), where Mobs is the observed H i mass and M̄ is the average
H i mass for field spirals with the same morphology and optical diameter. For the model
def = log(Mg /(Mg − Mstr )).
The observed fraction of spirals with def > 0.3 reaches 0.3 at r/RA ≈ 1. and 0.45
at r/RA ≈ 0.5. The model predicts that almost all galaxies within sorbit = 0.5 have
def > 0.3. However, only ≈ 30% of the galaxies viewed within r/rv,gr = 0.5 actually
reside within r/rv,gr = 0.5. This estimate assumes that the galaxy distribution follows an
NFW profile and that all galaxies within r/rv,gr = 2 are included in the projected cluster.
At sorbit = 1, the model predicts that an average galaxy traveling face-on to the wind is
stripped to def > 0.3 for Mv,sat < 1011.5 M . This estimate can be roughly adjusted for
inclination by multiplying by 10−.5 , Mv,sat < 1011 M . Approximately 45% of the galaxies
seen within sorbit = 1 actually reside within sorbit = 1. Some galaxies beyond the sorbit under
consideration will be stripped to def > 0.3. Therefore, the observed fraction of spirals with
2.5 Discussion
41
def > 0.3 should be greater than that predicted by multiplying the volume fraction with
the fraction of stripped galaxies within sorbit . This condition is met for sorbit = 1 if at least
a third of the observed spirals have Mv,sat > 1011 M , corresponding to Md ≈ 8 × 109 M .
For sorbit = 0.5, the volume fraction is lower than the observed fraction and this condition
is easily met. The difference between the observed value and product of the volume fraction
and stripped fraction should decrease as sorbit increases. A 1011 M spiral traveling face-on
to the wind is stripped to def ≈ 1 at sorbit = 0.8. A 1012 M spiral traveling face-on to
the wind is stripped to def ≈ 1 at sorbit = 0.4. Spirals with def ≈ 1 are not observed in
the Solanes et al. (2001) sample outside of 1RA .
The model does a good job of matching the observations of H i disks from Solanes et al.
(2001). If the model over-predicted the degree of stripping, then it would over-predict the
fraction of spirals with def > 0.3 and predict deficiencies at large radii greater than those
observed. If the model under-predicted the degree of stripping, then it might not predict
that spirals in the center of the cluster can have def ≈ 1 and would require that a larger
fraction of the observed spirals reside in the cluster center.
Both Marcolini et al. (2003) and Roediger and Hensler (2005) run simulations of disk
galaxies in winds typical of the outskirts of a group, equivalent to sorbit = 1 and 0.7, and the
center of a group, sorbit = 0.2 and 0.25. In Marcolini et al. the two lowest mass simulated
dwarfs are completely stripped in the high ram pressure. In the other cases, the dwarfs
are partially stripped to a range of radii. When the gas surface density in the current
model is adjusted to the exponential gas density in the dwarf models, the current model
and the simulations match well for both sorbit . The only disparity is that for sorbit = 0.2
the high-mass model is stripped to Rstr /Rg = 0.1 in the simulations and to Rstr /Rg = 0.45
in the model. This is probably due to differing dark matter halo profiles. The high-mass,
M > 1011 M , galaxy model of Roedinger and Hensler is stripped to Rstr /Rg = 0.8 in the
lower ram pressure and Rstr /Rg = 0.6 in the higher ram pressure. After the gas density is
adjusted, the stripping radii of their high-mass model in the two winds are matched well by
the model.
42 An Analytical Model of Ram Pressure Stripping
2.5.2
Predictions
In this model there exist three stripping regimes, stripping of the hot galactic halo, the flat
H i disk, and the inner gas disk. The focus is on stripping of the flat H i disk. However, the
model also includes a hot galactic halo. This hot gas is easily stripped in all environments.
A satellite galaxy that is orbiting at sorbit = 1. and has a hot galactic halo consisting of
0.5% of its mass, is stripped of at least 40% of its hot halo. At sorbit = 0.35, a typical
pericenter, all satellite galaxies are stripped of practically their entire halo in both groups
and clusters. The stripping of the inner disk has not been studied. However, this gas should
be difficult to strip and any spiral that has not been stripped of its entire outer H i disk
should retain this gas. The rest of this section focuses on the outer disk.
The model does a reasonable job of matching H i observations in large cluster spirals.
Therefore, it can be used to predict the extent of stripping at different Mv,sat and Mv,gr . The
model’s basic prediction, that the stripping radius is determined by the ratio Mv,sat /Mv,gr
is modified by the mass dependence of the model’s parameters. Therefore, the usefulness
of the model’s predictions depends in part on how well this dependence is understood.
Fortunately, the mass dependence of many of the parameters is based on observations. The
model parameters that most alter Mstr (Mv,sat /Mv,gr ) in low-mass systems are the ICM
profile parameters and the distribution of mass in the galaxy. The scaled ram pressure,
2/3
Pram /Mv,gr , that a satellite experiences decreases with Mv,gr because the ICM density
2/3
throughout most of the cluster decreases. The scaled restoring pressure, Prest /Mv,sat , is
larger in low-mass disks because a higher fraction of the disk mass is found in the H i disk.
The changes in both the scaled ram and restoring pressures act to decrease the Mv,sat /Mv,gr
at which stripping occurs in low-mass systems. For example, the Local Group dwarfs have
Mv,sat /Mv,gr smaller than the bright spirals in Virgo but should experience similar amounts
of stripping.
The model makes three basic predictions. Dwarf galaxies are stripped of their entire
outer H i disks in clusters and lose varying fractions of their outer disk in groups. Massive
spirals can also be stripped of significant fractions of their H i disk in groups if they travel
2.5 Discussion
43
to small sorbit . In a 1013 M group, a 1010 M dwarf is stripped of 50% of its H i disk at
sorbit = 0.7, a 1011 M satellite is similarly stripped at either sorbit = 0.5 or sorbit = 0.35,
depending on the satellite model used, and a large 5 × 1011 M satellite is similarly stripped
at sorbit = 0.3. The corresponding orbits from stripping 80% of the H i disk are sorbit = 0.55,
0.35, 0.25, and 0.2. It should be kept in mind that these numbers are for an average satellite
traveling face-on to the ICM wind.
The degree of stripping that a particular satellite of mass Mv,sat experiences orbiting in
a cluster of mass Mv,gr can vary greatly. The parameters that are responsible for most of
the variation in Mstr are the ICM parameters, α, β, rv,gr /rc , the galaxy’s inclination, i, the
pericenter, s0 , and the extent of the disk, λg . The variation in the stellar and gas surface
densities is largely contained in λg . The ICM parameters are set by the choice of cluster, but
clusters contain galaxies with a variety of orbits, morphologies, and inclinations. Within
the same cluster, a galaxy of mass Mv,sat = M with a low surface density on a radial orbit
traveling face-on to the wind can be as severely stripped as a galaxy with Mv,sat ≈ M/25
that has a high surface density and is traveling edge-on. At the same time, the ram pressure
can vary significantly between clusters with the same mass because of variations in the ICM
profile. Two identical satellite galaxies on identical orbits in different clusters of the same
mass can experience ram pressures that vary by a factor of ≈ 10.
An effective mass, Mef f , can be defined for each individual galaxy. This is the mass of
a satellite that the model predicts is stripped to the same Rstr as the given galaxy. The
effect of the scatter in the parameters can then be discussed in terms of the scatter in Mef f
at a given Mv,sat . The source of scatter in Mef f can be grouped into three types, that due
to the ICM parameters, α, β, rv,gr /rc , the orbital parameters, /v , i, cg , and the restoring
pressure parameters, cs , mi , λi . The overall scatter in the high- and low-mass models, the
result of varying all of the model parameters, are σMe f f,h ≈ 0.75 dex and σMe f f,l ≈ 0.85 dex,
respectively. These can be broken down into σMef f ,ICM,h = 0.7 dex, σMef f ,ICM,l = 0.8 dex,
σMef f ,orbit = 0.5 dex, and σMef f ,rest = 0.65 dex. The discussion in the paragraph above is
based on these estimates.
44 An Analytical Model of Ram Pressure Stripping
2.6
Conclusions
The model developed here relates the degree of ram pressure stripping a satellite galaxy
experiences to the galaxy and cluster masses and can be used to quickly determine the extent
to which a galaxy is likely to be stripped. In clusters galaxies at moderate mass ratios,
Mv,sat /Mv,gr ≈ 10−2 , are moderately stripped, Rstr /Rd ≈ 0.6, at intermediate distances,
0.5 < sorbit < 1, from the cluster center and severely stripped closer to the cluster center.
Satellites with lower Mv,sat /Mv,gr are severely stripped even at intermediate distances.
Stripping also occurs in groups. However, the same degree of stripping occurs for lower
Mv,sat /Mv,gr in groups than in clusters. Dwarf galaxies are moderately to severely stripped
at intermediate distances in groups, and large spiral galaxies can be moderately stripped if
they travel to small sorbit .
The model is simple and motivated by observations. Dark matter profiles are well
matched to NFW profiles outside a possible core, observations of X-ray gas in clusters can
be fit using β profiles, and the average gravitational potential of most galaxies should be
well matched by the model potential. The model has a large number of parameters, but
the values for many can be taken from observations. However, the model assumes that
clusters are static. From an evolutionary standpoint, it is only valid after the group or
cluster has acquired an ICM. On the other hand, in dynamic clusters bulk ICM motions
may cause more stripping to occur than is predicted by the model. This is observed in the
Virgo cluster (Kenney et al., 2004b).
Ram pressure stripping is not the only way to remove gas from galaxies. In groups
and clusters tidal stripping of both gas and stars can occur (e.g., (Bureau et al., 2004;
Patterson and Thuan, 1992)). For dwarf galaxies it has been proposed that supernova
winds associated with bursts of star formation may expel gas (Ferrara and Tolstoy, 2000;
Silich and Tenorio-Tagle, 2001), and the efficiency of this mechanism may depend on the
environment (Murakami and Babul, 1999). However, ram pressure stripping is capable of
removing gas from galaxies across a large range of galaxy masses and environments. In
particular ram pressure stripping can act when supernova-driven ejection is likely to be
2.6 Conclusions
45
inefficient and can remove gas from galaxies that are either not tidally interacting or that
are experiencing only weak tidal interactions.
The tidal radius, rt , at the physical orbital radius, rorbit , as estimated using rt =
(Mv,sat /3Menc )rorbit , can be compared to the stripping radius, Rstr , as determined using the model. Here Menc is the cluster mass enclosed within rorbit . In scaled coordinates,
rt /Rd = λg sorbit [3m(sorbit )]−1/3 , where m(sorbit ) = Menc /Mv,gr . The H i fraction lost to
tidal stripping depends only on the scaled size of the gas disk and sorbit and not on either
mass. Because the gas disk resides in the center of the satellite’s dark halo, tidal stripping
rarely effects the H i disk and practically never competes with the effect RPS. For example,
in order for the H i disk to be tidally stripped to rt /Rg = .8, a satellite must travel to
sorbit ≈ .1. By this sorbit the entire outer H i disk has been ram pressure stripped for almost
all satellites.
The effect of varying the model’s parameters was studied both to identify the parameters
that most effect the extent to which an individual galaxy is stripped and to determine the
range of satellite and cluster masses that result in the same Rstr . The extent to which an
individual galaxy is stripped depends most strongly on the galaxy’s sorbit or s0 , inclination,
i, and disk scale length, λg , and on the cluster’s ICM profile. Galaxies that reach smaller
sorbit , are face-on to the wind, have denser disks, or orbit through a denser ICM are more
effectively stripped. Galaxies in the same cluster with Mv,sat that differ by as much as a
factor of 25 can be stripped to the same Rstr . In different clusters, galaxies on identical
orbits with identical morphologies, identical mi , λi , cs , can be stripped to the same Rstr
with Mv,sat values that differ by as much as a factor of 30.
46 An Analytical Model of Ram Pressure Stripping
Figure 2.1: Mass at which a satellite’s gas disk is ram pressure stripped versus Rstr /Rd .
Short-dashed line: sorbit = .25, solid : sorbit = .35, long-dash: sorbit = .5, dash-dot: sorbit =
.75, dash-triple-dot: sorbit = 1. Top-left: Spiral galaxy orbiting in a 1015 M cluster. Topright: Spiral galaxy orbiting in a 1014 M cluster. Bottom-left: Spiral galaxy orbiting in a
1013 M group. Bottom-right: Dwarf galaxy orbiting in a 1013 M group.
2.6 Conclusions
47
Figure 2.2: Mass at which a satellite’s disk is ram pressure stripped versus the mass fraction
of gas disk that is stripped. Lines are the same as in Fig. 1. Top-left: Spiral galaxy orbiting
in a 1015 M cluster. Top right: Spiral galaxy orbiting in a 1014 M cluster. Bottom left:
Spiral galaxy orbiting in a 1013 M group. Bottom right: Dwarf galaxy orbiting in a 1013 M
group.
48 An Analytical Model of Ram Pressure Stripping
Figure 2.3: Mass at which a satellite’s gas disk is ram pressure stripped versus the deficiency
in a 3.5 × 1014 M cluster. Lines are the same as in Fig. 1.
Eq. 2.4
Eq. 2.4
–
–
Eq. 2.4
–
–
§2.4.3
Eq. B.6
Eq. 2.2
Eq. 2.9
Eq. 2.11
Eq. 2.6
Eq. 2.6
Eq. 2.8
Eq. 2.11
Eq. 2.6
Eq. 2.8
Eq. 2.13
Eq. B.2
α
βb
–
–
rv,gr /rc b
–
–
i
vsat (sorbit = 1)/vc
cg
cs
λg
λd /λg
λh
λb
mdg
mds + mH2
mb
msg
s0
6.5
0.67
–
–
12
–
–
0◦
1.15
5.5
12
5.7
6
0.04
180
0.008
0.05
0.017
0.005
0.35
reference model
(3)
5.5
–
0.57
–
–
25
–
–
–
7.5
–
–
–
–
–
–
–
–
–
–
mid mass model
(4)
20
–
–
0.45
–
–
170
–
–
10
20
–
–
–
na
0.016
0.04
0
–
–
low mass model
(5)
×6
±.08
–
±.08
7 - 20
–
75 - 270
0◦ - 90◦
±.6
5-16
12-20
±1.4
±2.5
–
–
×1.5
±.01
0.012-0.024
×1.5
±.12
scatter
(6)
6
2
–
7.4
6
–
5.4
4.5
1.4
1.1
1.4
8
1.2
–
–
1.5
1.3
1.1
1.5
5.4
∆P a
(7)
15
3
–
20
16
–
13
10
1.6
1.2
1.7
20
1.4
–
–
1.8
1.4
1.2
1.8
13
∆M a
(8)
∆P and ∆M columns list the scatter in Pram or Prest and Mv,sat /Mv,gr . The scatter is given as the fractional change
when the parameter of interest is varied over its entire range. Therefore, log(∆M ) ≈ 2σlog (Mv,sat /Mv,gr ).
b The first row shows affects of scatter for the large mass model and the third row for the small mass model. Columns 7
and 8 show the fractional increase in Pram at sorbit = 1.
a
ref in text
(2)
parameter
(1)
Table 2.1: Parameter Values and Scatter
2.6 Conclusions
49
Chapter 3
Ram Pressure Stripping in Groups
- Confronting Theory with
Observations
The contents of this chapter have been submitted to the Astrophysical Journal.
3.1
Introduction
Galaxies in clusters have earlier type morphologies, redder colors, and lower star formation
rates (SFRs) than galaxies in the field of similar luminosity (Gómez et al., 2003; Goto et al.,
2006; Dressler, 1980). Galaxies in groups and clusters formed both earlier and in denser
environments than their field counterparts. Cluster galaxies may therefore naturally appear
older than field galaxies. In addition, group and cluster galaxies are more likely to have
undergone major mergers (Gottlober et al., 2001) and are susceptible to harassment (Moore
et al., 1998), both of which may transform blue, late-type galaxies into red, early-type
galaxies. Furthermore, galaxies in groups and clusters no longer accrete fresh gas and their
galactic halo gas is easily heated and stripped, together termed ‘strangulation’ (Larson
et al., 1980; Balogh et al., 2000). They are also subject to ram pressure striping of their gas
50 Ram Pressure Stripping in Groups - Confronting Theory with Observations
3.1 Introduction
51
disks (Gunn and Gott, 1972). These interactions with the IGM are capable of suppressing
star formation in group and cluster members, resulting in redder colors, but do not alter
their structural morphology.
Analysis of large galaxy redshift surveys has revealed strong bimodalities in both stellar
properties; color, SFR, 4000Å break; and structural properties; concentration, sersic index,
central surface brightness, (Baldry et al., 2004; Blanton et al., 2003b; Kauffmann et al.,
2003a; Li et al., 2006; Strateva et al., 2001). The stellar and structural properties of galaxies
correlate with each other and with luminosity. Galaxies show a strong tendency to be either
blue, with disk-like light profiles (low sersic index and central surface brightness), or red,
with elliptical-like light profiles (high sersic index and central surface brightness) (Blanton
et al., 2005a). In addition, the relative frequency of late and early type galaxies correlates
with luminosity and with stellar mass. The fraction of galaxies with red colors and ellipticallike profiles increases with i-band absolute magnitude (Blanton et al., 2005a). The relative
fraction of galaxies with both red colors and low star formation rates increases with rband absolute magnitude (Weinmann et al., 006a). The average 4000 Å break, g-r color,
concentration, and µ0 all monotonically increase with stellar mass (Kauffmann et al., 2003b).
(See also Baldry et al. (2004); Balogh et al. (2004a); Kelm et al. (2005).) An analysis of the
effect of environment on galaxy evolution should take these correlations into account and
study the environmental dependence of both the distribution of galaxy properties and the
relationships between them.
Several recent observational studies have explored the relationship between galaxy properties and environment. In the SDSS, bright, red, and concentrated galaxies are found preferentially in dense environments (Hogg et al., 2003b; Blanton et al., 2005a). As the local
density increases the fraction of galaxies found in the red and high concentration modes
increases (Kauffmann et al., 2004), and galaxies in the red or high concentration modes are
more clustered than galaxies in the blue or low concentration modes (Li et al., 2006). In
the 2dFGRS, Norberg et al. (2002) found that galaxies with early-type stellar properties
are more clustered that those with late-type stellar properties. For satellite galaxies, cor-
52 Ram Pressure Stripping in Groups - Confronting Theory with Observations
relations between galaxy types and group mass are particularly interesting. Observations
made at intermediate redshifts indicate that significant galaxy evolution occurs in grouplike environments (Cooper et al., 2006, 2007; Gerke et al., 2007). Previous studies using
group catalogs in the SDSS and 2dFGRS have found that the fraction of passive red galaxies increases with group mass (Weinmann et al., 006a; Martinez et al., 2002), though some
studies have disagreed with this conclusion, (for example Balogh et al. (2004a); Tanaka
et al. (2004)).
The various correlations between environment and morphology are not all independent.
For example, correlations between galaxy properties and local environment, that is environment measured on scales less than 1-2 Mpc, can give rise to correlations on larger scales.
See, for example, Blanton et al. (2006). In addition, more massive halos are biased, having a
greater correlation length and amplitude. The preference of early-type galaxies, particularly
faint early type galaxies, to reside in more massive halos can enhance the clustering of these
galaxies relative to their late-type counter parts of scales up to several Mpc (Collister and
Lahav, 2005; Blaton et al., 2006). Taken together these studies indicate that processes that
are effective in group-like environments are responsible for many of the observed correlations
between morphology and environment. Many of the mechanisms which are thought to be
responsible for driving galaxy evolution also operate within virialized groups and clusters.
Therefore, it is particularly interesting to study correlations between galaxy properties and
group properties such as virial mass.
The corollary to the biasing of massive halos is that the luminosity function is dependent
on environment, with higher density environments or higher mass halos tending to include
brighter galaxies or host brighter satellite galaxies (Yang et al., 2003, 2005; Croton et al.,
2005; Zheng et al., 2005). In combination with the correlation between galaxy properties
and luminosity, the correlation between environment and luminosity dictates that when
analyzing the dependence of galaxy properties on group mass it is essential to separate
galaxies by luminosity, or stellar mass, and group mass simultaneously.
Several studies have concluded that the relationship between stellar properties and envi-
3.1 Introduction
53
ronment is stronger than that between structural properties and environment. Kauffmann
et al. (2003b) show that the relationship between concentration and stellar mass depends
on the local environment only weakly and only at low stellar masses, M∗ < 3 × 1010 M . In
contrast the relationship between the 4000Å break or the SFR and the stellar mass shows
a strong dependence on the local environment. Li et al. (2006) study the correlation function, in bins of luminosity, for galaxies belonging to the different galaxy modes. Galaxies
that are red, have large 4000Å breaks, high concentrations, or bright µ all have enhanced
correlation functions on scales less than 5Mpc. However, for galaxies that are red and have
large breaks, the enhancement is larger and extends to larger scales. Blanton et al. (2005a)
demonstrate that in the SDSS the observed correlation between the i-band sersic index, ni ,
and environment can be reproduced by assigning galaxies new local densities based on their
g − r colors and absolute i-band magnitudes, Mi , alone. However, the observed correlation
between color and environment cannot be reproduced by assigning local densities based on
ni and Mi . This asymmetry indicates that at least one process frequently occurs in group
like environments that affects star formation, but not structure. Strangulation and ram
pressure stripping are two such processes.
It is a generic prediction of most models of ram pressure stripping that the extent of
stripping increases with the mass of the host group or cluster and decreases with the mass
of the satellite galaxy. In contrast, strangulation occurs to all satellite galaxies. When the
two scenarios in their most generic forms are compared, they make qualitatively different
predictions for the correlations between star formation rates, group mass, and satellite
mass. In this paper a ram pressure stripping scenario is introduced which focuses on the
effect of stripping in groups, where ram pressures are low. Moderately bright satellites
in groups are not stripped of gas from within their stellar disks. They can, however, be
stripped of gas from the gas disk that extends beyond the stellar disk, where restoring
pressures are also low. If this disk is viscous, either normally or magnetically, then inflow
from it may supply enough gas to increase the time after accretion for which a satellite
can continue to form stars. If this gas is stripped, then it cannot continue to fuel star
54 Ram Pressure Stripping in Groups - Confronting Theory with Observations
formation. Introducing this scenario also makes one quantitative prediction. Introducing
an additional supply of gas increases the time for which a satellite galaxy can form stars,
hence increasing the predicted number of blue satellite galaxies. This is something of a
post-diction as semi-analytic models that do not consider inflow from the outer disk under
predict the blue fraction in groups (Weinmann et al., 2006). When comparing the stripping
and pure strangulation scenarios, it is most interesting to study to correlations between
the fraction of satellites which are red, group mass, and satellite luminosity among the
population of disk-like satellite galaxies. It is only among these galaxies that the stripping
scenario and the traditional strangulation scenario make different predictions. Therefore,
towards the end of distinguishing between the two scenarios of star formation fueling and
suppression in groups new results are presented here examining these correlations among
disk-like satellite galaxies.
Hester (2006a) (Hereafter, Paper 1) presents a specific analytical model of ram pressure
stripping that predicts the fraction of the gas that is stripped from the outer H i disk of a
satellite galaxy. The outer H i disk indicates the gas disk that extends beyond the stellar
disk. In the analytic model, the fraction of the outer H i disk that a galaxy retains depends primarily on the ratio of the galaxy’s mass to the mass of the group or cluster in
which it orbits, a quite generic prediction. The model also introduces several descriptive
model parameters which allow the model to be calibrated and modify the generic prediction
somewhat. When confronting the hypothesis that ram pressure stripping is an important
quencher of star formation in groups with observations, a specific model, like that presented
in Paper 1, has several uses. First, it can be used to predict whether the extent of ram pressure stripping is expected to vary across the group masses and luminosities of an observed
group catalog. That it does vary is the first condition for observing an trend with group
mass or luminosity. If the galaxies in the catalog either all experience slight stripping or all
experience extreme stripping, then no correlation will be observed. In addition, a specific
model can be used to examine whether the number of observed galaxies and the dynamical
range in luminosity and group mass are adequate.
3.2 Ram Pressure Stripping 55
This paper combines theory and observations to examine the importance of ram pressing
stripping in driving star formation rates in groups. In § 3.2.1 the scenario in which inflow
from the outer H i disk plus ram pressure stripping of this disk drives star formation rates
in group is introduced. The analytic model is reviewed in § 3.2.2 and used to verify that
the SDSS group catalog is appropriate for this project. The group catalog used here was
assembled by A. Berlind and is presented in Berlind et al. (2006a). It is reviewed briefly in
§ 3.3. In § 3.4 the results of the observational study are presented. The role of ram pressure
stripping in driving galaxy evolution in groups is discussed in § 3.5. § 3.6 concludes.
3.2
Ram Pressure Stripping
Gunn and Gott (1972) proposed ram pressure striping to explain the observed absence of
gas rich galaxies in clusters. Galaxies in clusters feel an intracluster medium (ICM) wind
that can overcome the gravitational attraction between the stellar and gas disks and strip
the gas disk. They introduced the following condition to estimate when this occurs;
2
ρICM vsat
< 2πGσ∗ σgas .
(3.1)
The left-hand side is a ram pressure, where ρICM is the density of the ICM and vsat is the
orbital speed of the satellite. The right-hand side is a gravitational restoring pressure where
σ∗ and σgas are the surface densities of the stellar and gas disks respectively. Using this
condition they concluded that spirals should lose their gas disks when they pass through
the centers of clusters.
Galaxies in nearby clusters are observed to be deficient in H i and to have truncated gas
disks when compared to field galaxies of similar morphology and optical size (Bravo-Alfaro
et al., 2000; Cayatte et al., 1994; Giovanelli and Haynes, 1983; Solanes et al., 2001). In
addition, asymmetric extra-planar gas that appears to have been pushed out of the disk
is observed in several Virgo spirals (Kenney et al., 2004a; Kenney and Koopmann, 1999;
Kenney et al., 2004b). These observations can be explained by ram pressure stripping,
and observing an undisturbed stellar disks accompanied by extra-planar gas is a strong
56 Ram Pressure Stripping in Groups - Confronting Theory with Observations
indication that the gas disk is interacting with the ICM. Ram pressure stripping has also
been repeatedly observed in simulations of disk galaxies in an ICM wind (Abadi et al., 1999;
Marcolini et al., 2003; Quilis et al., 2000; Roediger and Hensler, 2005; Schulz and Struck,
2001).
Paper 1 uses an analytical model of ram pressure stripping to explore the range of
environments in which stripping can occur and the galaxy masses that are susceptible to
stripping in each environment. It focuses on the H i disk beyond the stellar disk, and
stripping of gas from within the stellar disk is not modeled. The gravitational restoring
pressure is found by placing a flat H i disk in a gravitational potential consisting of a dark
matter halo, a stellar disk, and a stellar bulge. The ram pressure is determined by letting the
satellite orbit in an NFW potential through a β profile ICM. The gas fraction that a galaxy is
striped of is found to depend on the ratio of the satellite mass to the group mass, Msat /Mgr ,
and the values of several descriptive parameters. Observations of stripped spirals in clusters
compare well with the model’s predictions for large Mgr and Msat . Paper 1 concludes that
many galaxies, particularly low-mass galaxies, can be stripped of a substantial fraction of
their outer H i disks in a wide range of environments. In the next subsections, the possible
effects of stripping the outer H i are discussed and the analytical model is used to predict
the effects of ram pressure stripping on the galaxies in the SDSS catalog.
3.2.1
Ram Pressure Stripping and the SFR
The model of ram pressure stripping presented in Paper 1 addresses the stripping of the
outer H i disk. Galaxies can be stripped of their inner gas disk. However, this likely happens
only to the smallest galaxies or in the highest density environments. As is seen in Paper 1
and reviewed above, ram pressure stripping of the outer H i disk, that is the gas disk at radii
beyond the stellar disk, occurs for a wide range of satellite masses and environments. This
section briefly introduces a scenario in which the outer gas disk can fuel star formation. It
is the combination of this fueling scenario with model of stripping which is being confronted
with observations. If the predicted trends are not observed, then stripping of the outer H i
3.2 Ram Pressure Stripping 57
disk may still be occurring without affecting star formation.
While disk galaxies are currently forming stars from the gas in their inner disks, they
cannot continue to form stars at their current rate for longer than a few Gyr unless this gas is
replenished. Outside of groups star formation can be sustained by in-fall of new gas into and
the continual cooling of the hot galactic halo. Upon accretion satellite galaxies are stripped
of their hot galactic halo gas and experience no new in-fall. The standard ‘strangulation’
scenario assumes that the fuel for star formation is supplied by gas within the stellar disk,
galactic halo cooling, and in-fall. When a galaxy is accreted by a group the later two are
shut off and every satellite galaxy experiences a slowly declining SFR as it consumes its
inner gas disk. If star formation is only fueled by these sources, then the effectiveness of
star formation truncation would not vary with Mgr or luminosity. The scenario presented
here introduces a fourth supply of gas for star formation in the form of inflow from the
outer H i disk. Ram pressure stripping of this gas modifies the amount available to fuel
post-accretion star formation. This scenario therefore introduces correlations between postaccretion star formation rates, Mgr and satellite luminosity. By including a further source
of fuel for star formation, this scenario increases the time for which satellites can typically
form stars; hence predicting a higher fraction of blue, star forming, galaxies in groups than
the traditional strangulation scenario.
In addition to cooling and in-fall, the inner gas disk, and hence star formation, may be
fed by the inflow of gas from the outer H i disk. Inflow is expected if there is any viscosity
in the disk. In-falling gas should often join the outer H i disk as it is both physically larger
and may be better matched in angular moment to the in-falling gas. Hence a mechanism
for transporting gas inward through the disk is in some sense often implicitly assumed.
Models which feed star formation with a viscous disk can be used to form exponential
stellar profiles (Bell, 2002; Lin, 1987a,b). If star formation is fed by gas in the outer gas
disk, then galaxies in groups that retain some or all of this gas will be able to continue
forming stars for a longer time after in-fall than traditional strangulation models would
predict. In cases of extreme stripping, when gas is in fact stripped from the inner disk,
58 Ram Pressure Stripping in Groups - Confronting Theory with Observations
star formation must be truncated abruptly. In contrast, when the gas disk is stripped
down to the radius at which star formation is occurring, the time scale over which star
formation declines is determined by the rate at which star formation consumes the inner
gas disk. When the outer disk is not fully stripped, it may fuel star formation, possibly at
a declining rate, for some time longer before the inner disk again slowly consumes itself.
Therefore, like the traditional strangulation scenario, the stripping scenario presented here
predicts that the star formation rate will decline slowly. In contrast, it predicts that the
time after accretion for which a satellite can continue to form stars is dependent on Mgr
and luminosity.
In this scenario, the effectiveness of ram pressure stripping can be related to the red
fraction within a sample of satellite galaxies in a straight forward manner. Once a galaxy is
accreted, its supply of fresh gas is shut-off and its galactic halo is heated and stripped. All
subsequent star formation relies on gas which the galaxy retains within the group environment, that is gas in the inner and outer gas disk. Galaxies which retain a greater fraction
of their gas continue forming stars for a longer period of time after accretion. Once star
formation is quenched, a galaxy joins the red sequence. Therefore, the more effective ram
pressure stripping is for a given group mass and galaxy luminosity, the less time, on average, the galaxies will continue to form stars and the greater will be the fraction of galaxies
which have migrated onto the red sequence. The fraction of galaxies that transition from the
blue cloud onto the red sequence should therefore shadow the effectiveness of ram pressure
stripping, increasing with group mass and decreasing with satellite luminosity. This simple
scenario ignores the effect of dynamics within the group, assuming that the average time
since accretion is independent of Mgr and satellite luminosity. Consideration of dynamical
effects will be postponed until §3.5
Studying the fraction of blue galaxies that transition to the red sequence also has observational advantages. While within the red sequence and blue cloud the exact colors of
galaxies are influenced by factors such as metallically, the separation between the two is a
relatively clean indicator of the presence, or absence, of recent star formation. Within the
3.2 Ram Pressure Stripping 59
red sequence and the blue cloud the average colors of galaxies are only weakly correlated
with either host group mass or local environment (Balogh et al., 2004a; Weinmann et al.,
006a). Hence, all of the relevant information is captured in the fraction of galaxies which
reside on the red sequence, hereafter the red fraction or fr . Focusing on the red fraction also
simplifies comparisons with related previous related studies that make use of the bimodality
in galaxy colors.
3.2.2
Model Predictions
Any generic model of ram pressure stripping which is based on the Gunn and Gott condition
will predict that the severity of ram pressure stripping will increase as the mass of the group
increases and the mass of the satellite decreases. In the first case, the ram pressure will
increase, and in the second the restoring pressure will decrease. The analytical model
presented in Paper 1 predicts that if the morphologies of both groups and galaxies are
mass independent, then the extent of stripping should depend on the ratio Msat /Mgr . This
simplest assumption must be modified somewhat primarily because the average IGM density
of groups decreases as Mgr decreases, and the average gas fraction in the disk increases as
Msat decreases (Sanderson and Ponman, 2003; Swaters et al., 2002). The generic prediction
holds; if ram pressure stripping is an important driver of star formation suppression in
groups, then two trends should be observed. First, at fixed luminosity, typical star formation
rates should decrease with group mass. Second, at fixed group mass, typical star formation
rates should increase with luminosity. Assuming that this second trend will be seen in this
simple form, however, ignores other processes which are at work. Among central galaxies,
which one can assume reflect the properties of in-falling satellite galaxies, the fraction of
galaxies with red colors increases with luminosity. Galaxies that have red colors upon in-fall
are not affected by ram pressure stripping. Therefore, rather than focusing on the absolute
number of red galaxies at each luminosity, the change in the fraction of red galaxies when
comparing central and satellite galaxies must be considered.
While these generic predictions are model independent, a simple, reasonably calibrated,
60 Ram Pressure Stripping in Groups - Confronting Theory with Observations
model of ram pressure stripping is needed in order to determine whether a given group catalog is appropriate for testing the importance of ram pressure stripping. If an inappropriate
catalog is used, then one, or both, of the predicted trends may not be observed. In this
case, the theory that ram pressure stripping drives star formation quenching in groups will
not have been appropriately tested. To test the importance of stripping, three conditions
must be meet. First, the extent of stripping must vary across the galaxy catalog. Second,
the number of groups and galaxies must be adequate. Finally, the dynamic range of the
catalog, in both group mass and galaxy luminosity, must be wide enough. The analytic
model developed in Paper 1 will be used to determine whether the SDSS groups are appropriate for testing the importance of ram pressure stripping. The model is reasonably
calibrated in that it matches observations of stripped galaxies for bright galaxies in rich
groups. It should be emphasized that the model will not be used to make specific predictions about average colors or the red fraction as a function of group mass and luminosity,
as is sometimes done with more ambitious models. Rather, a simple and generic prediction
exists which is capable of distinguishing the effect of ram pressure stripping. The model
is only used to determine whether the SDSS group catalog is an appropriate sample with
which to test this generic prediction.
Observing a color or SFR trend due to ram pressure stripping requires a group catalog
with an appropriate range of group and galaxy masses. A useful group catalog is one in
which the effectiveness of stripping varies across the luminosities and group masses present
in the sample. If the galaxies are either all only mildly stripped or all severely stripped,
then the predicted correlation between stellar properties and Mgr and luminosity will not be
observed. The groups in the catalog have masses between 1012 and 1015 M and the galaxies
have −19 > Mr > −22. To test whether these ranges are appropriate for this work, the
analytical model is used to predict the extent of stripping these galaxies experience. To
do this Mr must be converted into an estimate of the pre-infall virial mass of the satellite
galaxy. For this conversion mass to light ratios ranging between 40 - 90 are considered.
These are a factor 10-15 higher than the stellar mass to light ratios of 4-6 found for the
3.2 Ram Pressure Stripping 61
R-band by Maraston (1998). The exact model parameters used for this test are given in
Paper 1. The galaxy parameters are those for a large spiral; with a dark matter halo
mass, stellar mass, disk size, and neutral gas surface density corresponding to a MW like
galaxy. The IGM parameters for the lowest mass galaxy groups, Mgr ≈ 1013 M , match the
observed properties of poor groups with observed X-ray emission (Helsdon and Ponman,
2000; Sanderson and Ponman, 2003; Osmond and Ponman, 2004). The ICM parameters for
the higher mass groups correspond to rich groups and poor clusters (Mohr et al., 1999). In
Paper 1 these are referred to as the ‘low-mass group model’ and the ‘middle-mass cluster
model’ respectively. With these parameters, the predictions of the model indicate that
the high-mass groups should contain many severely and moderately stripped galaxies while
the lowest mass groups should be capable of stripping few of the galaxies in the sample.
Few if any of these galaxies are stripped from within their stellar disks; here a ‘severely’
stripped satellite has been stripped down to the edge of the stellar disk. The range in galaxy
mass present in the sample is not great, the galaxy masses likely vary by a factor of ≈ 10.
However, the dimmest galaxies are, on average, stripped of greater fractions of their initial
gas than the brightest. The group masses and galaxy luminosities which make up the SDSS
group catalog are therefore appropriate.
In Paper 1, an analytic model is also presented which predicts the extent of stripping
of the hot galactic halo surrounding a galaxy. The galactic halo is modeled by placing gas
at the virial temperature of the galaxy’s dark matter halo in hydrostatic equilibrium with
an NFW potential. It is assumed that the galactic halo is stripped down to the radius at
which the ram pressure equals the thermal pressure. The galaxies in this sample can only
maintain ≤ 0.5% of their mass in the galactic halo without the gas cooling rapidly. Using
this mass fraction, the model predicts that the galaxies in the group sample have more than
70% of this gas stripped in a 1013.5 M group and the entire halo is stripped in any group
with a mass above 1014 M . This is an upper limit on the hot halo gas these galaxies can
retain in the absence of fresh in-falling gas. For lower gas densities, galaxies in low mass
groups retain even less of their halo gas and there is less variation in the fraction of this gas
62 Ram Pressure Stripping in Groups - Confronting Theory with Observations
which is stripped. Heating of the galactic halo by the IGM will also tend to remove this
gas, again resulting in less variation. Therefore, in contrast to the outer H i disk, it is likely
that all of the galaxies in this sample are stripped of the entirety of their galactic halo.
We now turn our attention to the question of whether the dynamic range and the
numbers of groups and galaxies in the group catalog are sufficient. These are related
questions. Given a greater dynamic range, a smaller sample will suffice, and vice versa.
The dynamic range in satellite masses, a factor of ≈ 10, is significantly smaller than the
range of group masses, a factor of ≈ 103 . Therefore, we will consider whether the group
catalog contains enough galaxies to detect a trend with luminosity. The size of the galaxy
catalog needed to detect this trend across a given range is determined first by the scatter
in the relationship between the fraction of the H i disk mass that is stripped, fstr , and the
satellite galaxy mass, Msat , and second by the sensitivity of the red fraction to the average
fstr . The first of these issues can be addressed using the analytical model. Galaxies with the
same Msat are stripped of different gas fractions due to differences in morphology and orbit.
The scatter can be thought of in terms of a distribution of effective masses, Mef f , at each
physical satellite mass Msat . The effective mass is defined such that all satellites with the
same effective mass are stripped of the same fraction of their H i disk. The first condition
on measuring a difference in fr is that, given the sample size, it is theoretically possible
to measure a difference in the average effective mass of two samples of galaxies. We will
effectively be attempting to do so through fr . The larger the scatter in the effective mass
at fixed luminosity, the greater the number of galaxies a sample must contain to measure
this difference.
The scatter between Msat and Mef f is substantial. As shown in Paper 1, a 1011 M
galaxy can be stripped as much as a 1010 M galaxy or as little as a 1012 M galaxy. The
scatter is mainly due to differences in the galaxies’ orbits, stellar and H i disk scale lengths,
and in the density and extent of the groups’ IGM. The scatter is split equally between
factors affecting the ram pressure and those affecting the restoring pressure. Galaxies on
radial orbits, orbiting face-on, with low surface brightnesses or low gas fractions are more
3.2 Ram Pressure Stripping 63
susceptible to ram pressure stripping than those on circular orbits orbits, orbiting edge-on,
with dense stellar and gas disks. Similarly, variations in the density and concentration
of the IGM across groups increase the scatter. Galaxies on average orbits experience less
striping if the IGM is less dense or more concentrated. The parameters of the model of
Paper 1 are all descriptive parameters which correspond directly to observables such as
surface brightness, gas disk density, IGM profile, etc.... By introducing variations in these
parameters similar to that observed for real galaxies and groups, the scatter between Msat
and Mef f can be estimated. As stated above, this scatter is roughly and order of magnitude.
The scatter between Msat and Mef f is similar to the dynamic range of Msat in the sample.
Therefore, a catalog must contain many galaxies in order for there to exist a statistically
significant change in the average Mef f across the luminosity range. Similarly, one expects
that a substantial number of groups and galaxies will be required to observe a variation in
the red fraction with luminosity. In contrast, the dynamic range in Mgr is larger than the
scatter, and a change in the red fraction with Mgr should be easier to measure.
To take this analysis further using theory alone would require modeling the sensitivity
of the red fraction to changes in the effective mass. This is beyond the scope of this simple
model. We can instead use the observations themselves to constrain the necessary number
of galaxies. The question of import is, if no correlation between luminosity and the red
fraction is observed, then can we conclude that ram pressure stripping is not important?
This question can be answered through comparison with the correlation between Mgr and
the red fraction. Such a trend has been previously observed and we will confirm that it
exists in this sample. The analytical model predicts that the effectiveness of stripping
depends on Msat and Mgr as Msat /Mgr , with some variation due to correlations between
morphology and mass. These correlations are such that the ram pressure increases slightly
steeper with Mgr than the restoring pressure does with Msat . The IGM density increases
with Mgr while the gas fraction in the disk decreases with Msat . Therefore, across the same
dynamic range, say a factor of 10 in Mgr and Msat , we expect that the change in the red
fraction with Mgr should be slightly larger than the change in the red fraction with Msat ,
64 Ram Pressure Stripping in Groups - Confronting Theory with Observations
or luminosity. There is some uncertainty in even this estimate unless we carefully consider
the distribution of Mgr and Msat in our Mgr and luminosity bins. However, we can make
the following approximation. If a subset of groups is considered in which the dynamic range
in Mgr matches that in Msat , then the difference in the red fraction across the Mgr range
should be similar to that across the luminosity range. If no correlation between luminosity
and the red fraction is observed, and the measurement errors are in fact inconsistent with a
difference similar to that measured for Mgr , then ram pressure stripping is not a dominant
driver of star formation suppression in groups. If no correlation is observed, but a similar
difference is consistent with the measurement errors, then a larger sample is required. If
a similar correlation is observed, then this is of course consistent with the ram pressure
stripping model of star formation suppression in groups.
With a similar group catalog, also drawn from the SDSS, Weinmann et al. (006a) find
that the red fraction increases a few tenths from group masses of 1012 M to 1015 M . Therefore, to estimate whether the size of the group catalog used for this project is appropriate,
the number of galaxies need to observe a difference in the red fraction of 0.1 is now determined. As errors in the colors are small in comparison, it is assumed that the scatter in the
observed red fraction is given by σf2r = frt (1 − frt )/N , where frt is the true red fraction, as
is appropriate for an essentially binomial distribution. This error estimate assumes that all
galaxies can be correctly assigned to either the red sequence or blue fraction and that the
error in the measured red fraction is due to statistics alone. In the case that two frt are close
to 0.5, approximately 400 galaxies are needed in each sub-sample to observe a difference in
the observed red fractions between them of 0.1 with 3σ confidence. The scatter between
Msat and Mef f is due to both variations in orbital and satellite parameter values, which
vary within a single group, and to variations in the IGM parameter values, which vary across
groups. Therefore, this estimate is only valid for a sample that includes a large number of
groups. The SDSS group catalog contains 2700 groups with at least three members, 18250
central galaxies, and 19688 satellite galaxies. It is large enough to place 400 galaxies into
reasonable bins in Mr and Mgr and has an adequate number of individual groups. In other
3.3 The Group Catalog 65
words, the catalog is large enough to detect differences in the red fraction as small as 0.1
with reasonable significance. There are two caveats to this conclusion. First, it should be
recalled that due to the color luminosity relation the change in the fraction of red galaxies
between central and satellite galaxies will be measured rather than the red fraction itself.
Second, the galaxies will be split by their light profiles in order to isolate galaxies which are
structurally disk-like. This will reduce the number of galaxies and increase the measurement
errors.
The catalog of SDSS groups used for this project covers a range of Mgr and Mr in which
stripping should occur and across which the degree of stripping should vary. The fraction of
galaxies that belong to the red sequence will be focused on rather than average galaxy colors.
This choice is made because within a sample the effectiveness of ram pressure stripping and
the relative increase in the red fraction should be simply related. The relative red fraction
should shadow the effectiveness of stripping, increasing with Mgr and decreasing with Msat .
Using the red fraction will also simplify comparisons between this project and others that
use the galaxy color bimodality. Following the discussion above, we expect that the group
catalog should be large enough to observe a correlation between the red fraction and both
Mgr and luminosity.
3.3
The Group Catalog
The Sloan Digital Sky Survey (SDSS) (York et al., 2000) is conducting an imaging and
photometric survey of Π sr in the northern hemisphere as well as three thin slices in the
southern hemisphere. Observing is done using a dedicated 2.5m telescope in Apache Point,
NM. The telescope operates in drift scan mode and observes in five bandpasses (Fukugita
et al., 1996). Magnitude calibration is carried out using a network of standard stars (Smith
et al., 2002). Three sets of spectroscopic targets are selected automatically, the main galaxy
sample, the luminous red galaxy sample, and the quasar sample (Strauss et al., 2002;
Eisenstein et al., 2001; Richards, 2002). Objects in the main galaxy sample have Petrosian
magnitudes r0 < 17.7 and are classified as extended. Magnitudes are corrected for galactic
66 Ram Pressure Stripping in Groups - Confronting Theory with Observations
extinction using the reddening maps of Schlegel et al. (1998) prior to selection. Spectroscopy
is taken using a pair of fiber-red spectrographs, and targets are assigned to fibers using an
adaptive tiling algorithm (Blanton et al., 2003d). Data reduction for the SDSS is done using
a series of automated pipelines (Hogg et al., 2001; Ivezic et al., 2004; Lupton et al., 2001;
Pier et al., 2003; Smith et al., 2002).
The catalog used here is a volume limited sample drawn from the NYU Value Added
Galaxy Catalog (NYU-VAGC) (Blanton et al., 2005b). The sample goes down to Mr <
−19.0 and has a redshift range of 0.015-0.068. The group finding algorithm is described
in detail in Berlind et al. (2006a). The group finder is a friends of friends algorithm with
two linking lengths, one for projected distances and one in red shift space. Linking lengths
are chosen such that the multiplicity function, richness, and projected size of recovered
groups from simulations projected into redshift space are unbiased measures. The linking
lengths are also chosen to maximize the number of groups recovered and minimize the
number of spurious groups. Fiber collisions are treated by assigning each ‘collided’ galaxy
the redshift of its nearest neighbor. The velocity dispersions of the recovered groups are
systematically low because the group finder does not link the fastest moving group members
to the group. However, as the fastest group members are also those most likely to be
stripped, not including these galaxies biases against observing the signature of ram pressure
stripping.
Virial masses and radii for the groups are determined by assuming a monotonic relationship between group mass and total group luminosity and matching a ΛCDM mass function
to the measured group luminosity function (Berlind et al., 2006a). The group finder is tuned
to recover the multiplicity function of groups, and should therefore also recover the group
luminosity function. This matching assumes no scatter in the relationship between mass
and richness. Individual mass measurements are therefore noisy. In this paper, galaxies are
placed into wide bins in group mass, each of which contains hundreds of groups. Groups
that scatter within their bin do not affect the results presented here. It is only scatter
between bins which is important. This scatter will tend to decrease the difference in Mgr
3.3 The Group Catalog 67
between the bins, and hence works against the signal we are attempting to observe. Group
mass and group luminosity are correlated, however, and the large number of groups in the
catalog should ensure significant differences in the distribution of Mgr between the bins.
Both ram pressure stripping and strangulation only affect the satellite galaxies within
a group or cluster. Therefore, the group members are split into ‘satellite’ galaxies and
‘central’ galaxies. This was done in two ways. First, for groups with more than 1 member,
‘central’ galaxies were selected by requiring that a central galaxy have an absolute r-band
magnitude, Mr , at least 0.5 brighter than the next brightest galaxy in the group. Thus
central galaxies were conservatively selected and not all all groups were required to have
a central galaxy. All galaxies that weren’t assigned to the sample of central galaxies were
assigned to the satellite sample. Second, a distance criteria was introduced and all groups
were required to have a central galaxy. A galaxy which was 0.5 Mr brighter than the next
brightest galaxy was still selected as the central galaxy. When no such dominant galaxy was
identified, among galaxies less than 0.5 Mr dimmer than the brightest galaxy the galaxy
nearest the group center was selected. In both cases ‘groups’ with only one member were
assigned to the central galaxy sample, and pairs, groups with only two members, were either
both assigned to the central sample, if ∆Mr between them was less than 0.5, or the dimmer
of the pair was assigned to the satellite sample otherwise. Practically identical results were
obtained with both separation criteria. The results presented here use the first.
The absolute magnitudes in the NYU-VAGC are k-corrected and corrected for passive
evolution to z = 0.1 (Blanton et al., 2003a,c). The measured Mr and g − r colors used
here are based on these k-corrected magnitudes. Membership in the red sequence is defined
using the Mr dependent color cut presented in Li et al. (2006) for the SDSS; g − r >
−0.788 − 0.078Mr . The i-band Sersic index, ni , is defined as Ii (r) = A exp −(r/r0 )(1/ni )
and its measurement is discussed in Blanton et al. (2003b). Galaxies with de Vaucouleurs
profiles are assigned ni = 4, galaxies with exponential disk profiles are assigned ni = 1,
and late and early type galaxies can be separated using ni ≈ 2.5. As discussed in Blanton
et al. (2003b), the sersic index is not biased by seeing, which is an improvement over
68 Ram Pressure Stripping in Groups - Confronting Theory with Observations
other measures of the light profile such as the concentration. Both seeing and the nonaxisymmetric shapes of real galaxies introduce scatter into the measurement of ni .
While stellar masses are available for most of the galaxies in the group catalog, this
work will use Mr as an indicator of the typical restoring pressure of a galaxy. That is,
we will separate galaxies according to the mass of the group in which they reside, Mgr ,
and their luminosity, as measured by Mr . This is done for two reasons. First, the volume
limited sample used to identify the groups was luminosity selected. Using Mr therefore
avoids completion issues. Second, recent work comparing N-body simulations to galaxies in
the SDSS indicates that luminosity may in fact be a fair indicator of the maximum circular
velocity of the dark matter halos (Tasitsiomi et al., 2004). The maximum circular velocity
is directly related to the depth of the gravitational well in the halo, and thus to the restoring
pressure on the disk. Completion issues aside, luminosity may be a better indicator of a
galaxies resistance to stripping than the stellar mass.
3.4
Results and Analysis
This section presents an analysis of the fraction of galaxies on the red sequence and the
fraction of galaxies with early-type light profiles as a function of group mass and luminosity.
In Figure 3.6a, the fraction of galaxies residing on the red sequence, fr , is plotted versus
the r-band absolute magnitude, Mr . The four lines correspond to all central galaxies and to
satellite galaxies in groups of log(Mgr (M )) < 13, 13 - 14, and 13.5 - 14.5. In Figure 3.6d,
the the fraction of galaxies with an i-band sersic index, ni , greater than 2.5, f> , is plotted
in the same manner. Panels b and c of Figure 3.6 show the same plots of fr versus Mr , but
only include galaxies with ni < 2.5 and ni > 2.5, respectively. Figures 3.6(e) and 3.6(f) plot
f> versus Mr for galaxies with blue (red) colors, as determined using the Mr dependent.
Galaxies are binned in Mr such that in the intermediate Mgr bin each Mr bin has an equal
number of blue galaxies, resulting in uniform measurement errors across the Mr range. The
errors shown in Figure 3.6are given by σ 2 = f (1 − f )/N , as discussed in §3.2.2. These errors
were propagated to determine the errors for Figure 3.6.
3.4 Results and Analysis 69
3.4.1
Comparison to Previous Results
The results presented in Figure 3.6 agree well with previous observations. Both fr and
f> are higher for satellite galaxies than for central galaxies, Figures 3.6(a) and 3.6(d), in
agreement with previous observations of the morphology density relation. The higher fr
and f> found for satellite galaxies is reflected in the correlation between color and structure
and local environment, as well as in the enhanced clustering of both red and concentrated
galaxies. In addition to being enhanced for satellites, fr increases with Mgr , as observed
previously by Weinmann et al. (006a) and Martinez et al. (2002). Note that f> also increases
with Mgr . For central galaxies fr increases with luminosity, as has often been observed.
However, among satellite galaxies, fr appears to be independent of Mr . This was also noted
previously by Weinmann et al. (006a).
The asymmetry between color and structure which was previously observed is apparent
in Figure 3.6. The difference in fr between central and satellite galaxies is larger than the
difference in f> . This is reflected in the greater strength of the environmental dependence
and clustering of red galaxies versus concentrated galaxies (Kauffmann et al., 2003b; Li
et al., 2006). The true strength of the asymmetry is seen clearly when Figures 3.6(b) and
3.6(c) are compared. Among satellite galaxies with ni < 2.5 there is a large difference in fr
between central and satellite galaxies, Figure 3.6(b). However, among blue galaxies, there
is no difference in f> between central and satellite galaxies, Figure 3.6(e). Additionally,
among galaxies with ni > 2.5, fr is higher for satellite galaxies while for red galaxies f> is
independent of group environment, Figure 3.6(c) and 3.6(f). This is the perfect asymmetry
observed by Blanton et al. (2005a) and Quintero et al. (2005); the distribution of ni among
galaxies with similar colors is unaffected by environment while galaxies with all ni are redder
in dense environments such as groups.
3.4.2
Disk-like Satellites
Ram pressure stripping and strangulation both affect star formation rates and colors without
affecting structure. Accordingly, the results presented here aim at differentiating between
70 Ram Pressure Stripping in Groups - Confronting Theory with Observations
these two scenarios by attempting to answer the following question. Among galaxies with
ni < 2.5, that is among galaxies that remain disk-like, does the fraction of blue galaxies that
transition from the blue cloud to the red sequence increase with group mass and decrease
with satellite galaxy luminosity, as predicted by the ram pressure scenario presented above.
This question is approached by examining fr and f> in some detail.
As a supplement to Figure 3.6, Figure 3.6 shows the differences in fr and f> between
the central and satellite galaxies. In both Figures 3.6(a) and 3.6(b), the difference in fr
between the central and satellite galaxies clearly increases with Mgr . This is seen just as
clearly in Figure 3.6. When all galaxies are considered, Figure 3.6(a), fr is flat across the
Mr range for satellite galaxies in all Mgr bins, while for the central galaxies fr increases with
luminosity. When only galaxies with ni < 2.5 are considered, fr decreases with luminosity
for satellite galaxies in the highest Mgr bin while for the central galaxies fr is independent
of luminosity. In the middle Mgr , fr for galaxies with ni < 2.5 may decrease slightly, and
in the lowest Mgr bin fr is flat. In Figure 3.6 these comparisons appear as a decrease of
the difference in fr between the central and satellite galaxies with increasing luminosity.
The change in the difference in fr between the central and satellite galaxies across the Mr
range in the catalog is small but entirely consistent with the expectations of the stripping
model discussed in §3.2.2. The observed trends between fr , Mgr , and Mr , among galaxies
with ni < 2.5 are consistent with the ram pressure stripping scenario for color evolution
in groups, and these trends are reflected in the full galaxy sample. Before accepting this
conclusion, the affects of separating galaxies using ni must be considered.
While separating galaxies according to their light profiles using the sersic index is potentially a powerful way of distinguishing between the effects of processes that do and do not
alter structure, care must be taken when attempting this. The measurement errors on the
colors are quite small and likely don’t affect these results. Scatter in the measured ni , due
to both seeing and the non-axisymmetric shapes of real galaxies, may however. Observed
trends not only in f> itself, but also in fr within ni selected sub-samples can be biased.
The measured f> will tend to be less extreme than the true f> . When f> is high, more
3.4 Results and Analysis 71
true ni > 2.5 galaxies will scatter out of the sample with measured ni > 2.5 than into it,
and vice versa. Furthermore, as the red fraction in the ni > 2.5 population is clearly higher
than in the ni < 2.5 population, scattering of galaxies between these two sub-populations
will tend to make the measured fr (ni < 2.5), frm< , greater than the true value, frt< . A
consequence of this is that a change in fn2.5 can cause a change in frm< while frt< remains
unchanged or can can cause frm< to appear flat while frt< is changing.
In order to examine these effects the following definitions are made. Let nit and nim be
the true and measured ni of a galaxy, respectively. Let n be the percentage of galaxies with
nit > 2.5 that scatter into the nim < 2.5 population,
n≡
N (nim < 2.5|nit > 2.5)
N (nit > 2.5)
Let t be the percentage of galaxies with nit < 2.5 that scatter in the opposite direction. Let
f>t be the fraction of galaxies with nit > 2.5 and f>m the fraction with nim > 2.5. Then
f>m and f>t are related by
f>m = (1 − n)f>t + t(1 − f>t ).
(3.2)
or
f>t =
f>m − t
.
1 − (n + t)
(3.3)
Let frt<(>) be the true fr in the ni < (>)2.5 population and frm<(>) be the measured fr .
With these definitions
frm< =
Nr<m
frt< (1 − f> )(1 − t) + frt> f> n
=
N<m
(1 − f> )(1 − t) + f> n
(3.4)
or
frt< = frm< −
n
f>t
(frt> − frm< ).
(1 − t) (1 − f>t )
(3.5)
Introducing the corresponding equations for frt> followed by some algebra gives
frt<
= frm< −
n
f>t
nt
(frm> − frm< ) −
frm<
(1 − t) (1 − f>t )
(1 − n)(1 − t)
1+
nt
(1 − n)(1 − t)
(3.6)
−1
.
72 Ram Pressure Stripping in Groups - Confronting Theory with Observations
By combining the above with Equation 3.3, frt< can be estimated for any n and t. Given
the relatively small errors in color, changes in frm in the full galaxy sample reflect true
differences, galaxies can be cleanly separated by color, and changes in frt do not induce
changes in f>m .
The effect of scatter on f>m is straight forward. Equation 3.2 can be written as
f>m − f>t = −(n + t)f>t + t
(3.7)
t − f>m (n + t)
.
1 − (n + t)
(3.8)
or
f>m − f>t =
If n = t, then f>m = f>t at 0.5, and at values above and below this f>m is less than f>t
or greater than f>t , respectively. When comparing two f> , the effect is always to make
the measured difference less than the true difference. Therefore, f>t is more sensitive to
both luminosity and Mgr than Figure 3.6(d) indicates. Previous conclusions about the
relative strengths of the correlations between color and environment and structure and
environment do still hold. The true values of n and t are unknown. Assuming that n = t,
however, the data can be used to constrain them. In order for fr among galaxies with
ni > 2.5 not to exceed 1, n < 0.065. With this constraint and using Equation 3.8, the
true increase in f> with increasing Mgr is still smaller than the measured increase in fr . A
further characteristic of Equation 3.7 is that when comparing two f> , the difference between
the true difference, f>t,1 − f>t,2 , and the measured difference, f>m,1 − f>m,2 , decreases as
f>t,1 − f>t,2 decreases. In addition, galaxies can be relatively cleanly separated by color.
Therefore, that no difference in f>m with Mgr is observed among red or blue galaxies
indicated that there is no significant difference in f>t among these populations. A nearly
perfect asymmetry exists between color and structure.
The correlation between fr< and Mgr is now considered. At fixed luminosity, n and t
should be independent of Mgr . Assuming that n = t = 0.065, the largest value consistent
with the observations, Equation 3.6 can be used to check whether the observed fr< are
consistent with no increase with Mgr . The observed differences are not consistent with no
3.4 Results and Analysis 73
increase, the true fr among the population of satellite galaxies with ni < 2.5 does increase
with Mgr . This was perhaps never in doubt given that the observed increase in fr among
all satellite galaxies is higher than the increase in f> .
To estimate the size of the induced change in frm< with Mr changes in f> , n, and t
must be considered concurrently. In general, scatter between the populations tends to induce
∆frm< which mimic f> . Contrary to this expectation, while f> increases with luminosity,
Figure 3.6d, frm< in the highest Mgr bin decreases while frm< in the intermediate and
lower Mgr bins is flat, Figure 3.6(b). As luminosity increases, however, the scatter between
the two populations may decrease, tending to decrease the induced ∆frm< , potentially even
inducing a decline in frm< with luminosity. The most extreme possible case would be to
assume that n = t = 0 in the brightest bin, that is there is no scatter, while in the lowest
luminosity bin n = t = 0.065. With these assumptions, frt< in the highest bin still decreases
with luminosity while frt< does not correlate with Mr in the middle and low Mgr bins. Such
extreme assumptions about the luminosity dependence of the scatter are not valid. Each Mr
bin mixes galaxies with different observed mr , with varied shapes, observed under varying
conditions. It is likely a better assumption to use a constant n and t across the luminosity
range. In the case of constant n and t, the scatter does tend to make frm< mimic f> . Given
the steep increase in f> with luminosity, introducing a small scatter between the two ni subsamples can effectively hide a correlation between frt< and Mr . Assuming n = t = 0.065
for all Mr , frt< decreases with luminosity in all three Mgr bins. Assuming n = t = 0.02,
frt< decreases with luminosity in the two higher Mgr bins but not in the lowest Mgr bin. In
each case in which a correlation is observed, the measurement errors are inconsistent with
no correlation. In at least the highest mass groups frt< does decrease with luminosity, as
observed in Figure 3.6. In the middle mass bin, a true correlation may exist which is being
hidden by scatter in the measured ni . There may or may not be a true correlation in the
lowest Mgr bin, depending on the true values of n and t.
Ram pressure stripping affects only galaxies that are blue and star forming when they
are accreted. Therefore, the clearest prediction to be drawn from the ram pressure stripping
74 Ram Pressure Stripping in Groups - Confronting Theory with Observations
scenario is that the fraction of accreted galaxies that are blue upon in-fall and subsequently
join the red sequence increases with Mgr and decreases with luminosity. This fraction can
be estimated by dividing the observed difference in fr< between the satellite and central
galaxies by the blue fraction among the central galaxies; that is (fr<,sat − fr<,cent )/(1 −
fr<,cent ). This assumes that when they were accreted the satellite galaxies had the same fr
as the current central galaxies. This normalized difference is therefore only a rough estimate
of the number of blue accreted galaxies that have since transitioned to the blue sequence.
The normalized difference clearly increases with Mgr as in each luminosity bin (1 − fr<,cent )
is the same for all Mgr .
Whether the normalized difference decreases with luminosity depends on the slopes of
the correlations between frt< and Mr among both the satellite and central galaxies. While
there appears to be no correlation between fr< and Mr among the central galaxies, the
measurement is subject to the same issues with scatter in nim . Much like the lowest Mgr
bin, frt< may be flat or may decrease somewhat with luminosity, depending on the true
values of n and t. It appears that the slope of any decrease in frt<,cent with luminosity is
shallower than the slope among satellite galaxies. In this case, frt<,sat − frt<,cent and the
normalized difference decrease with luminosity, in agreement with the predictions of the
ram pressure stripping scenario.
This conclusion is unaffected by scatter in the measured ni . Comparing the slopes of
the correlations between frt< and Mr is remarkably robust to assumptions about n and t.
One can rely on the relatively straight forward assumption that n and t depend only on
luminosity and are independent of Mgr . In each luminosity bin,the strength of the induced
∆frm< increases with f> . The highest Mgr bin is therefore always affected the strongest by
induced ∆frt< and all of the satellites are more effected than the central galaxies. Recalling
that the effect of the induced ∆frt< is to decrease the slope of the correlation between
frm< and Mr , observing slopes among the satellite galaxies which are steeper than among
the central galaxies implies the same holds for the true slopes. This applies equally to the
relative slopes of the different Mgr bins. The slope of the correlation between frm< truly
3.5 Discussion
75
declines with Mgr . An interesting observation which will be taken up again later.
The trends predicted by the ram pressure stripping scenario are both seen in the groups
and are both robust trends, they exist in the true ni < 2.5 population and are not induced by
scatter in measuring ni . The estimated fraction of blue accreted galaxies that transition to
the red sequence as a satellite galaxy increases with Mgr and decreases with luminosity. The
stripping scenario also predicts that, given the greater range in Mgr present in the sample
the increase fr< with Mgr is larger than with Mr . This is consistent with the observed
frm< . Verifying this observation is difficult, however, given that the change with Mgr is
overestimated due to induced ∆frm< , while the change with Mr is underestimated. The
results presented here are therefore consistent with ram pressure stripping, plus viscous flow
from the outer to inner H i disk, being an important driver of star formation suppression
in groups. As discussed in the next section, the results may indicate that disk stripping
and an expanded definition of strangulation work together to drive star formation. Ram
pressure stripping, of course, only drives stellar evolution. Other processes, such as mergers
and harassment, must account for the observed structural evolution, which is generally
accompanied by stellar evolution.
3.5
Discussion
Several mechanisms may be responsible for driving galaxy evolution in groups. The remnants of major mergers may preferentially find themselves within groups and clusters, harassment is effective in the dense environments of groups, and processes such as strangulation and ram pressure stripping only occur within virialized groups. A previously observed
abundance of red disk galaxies in group environments indicates that some combination of
these latter two processes does play a role. Interactions with the IGM can suppress star
formation without affecting structure. This paper focuses on distinguishing between the
affects of stripping and strangulation. A new stripping scenario is first presented in which
star formation can be fueled by inflow from the outer H i disk and ram pressure stripping
modifies this source of fuel. New observational results are also presented showing correla-
76 Ram Pressure Stripping in Groups - Confronting Theory with Observations
tions between the red fraction, group mass, and satellite luminosity within the population
of disk-like satellites, defined as galaxies with ni < 2.5. In this section, the predictions of
the strangulation and stripping scenarios are contrasted and then compared to observations
of the SDSS groups. The two scenarios are first compared without considering structure,
and then the new observational results are considered.
Across the Mgr and Msat present in the group catalog, all satellite galaxies experience
strangulation. In-fall is truncated and the hot galactic halo stripped. The simplest prediction based on this would be that fr for satellite galaxies should be independent of both Mgr
and Mr . This generic prediction, however, may be modified by tracking the in-fall histories
of satellites as, for example, the time since accretion may depend on Mgr or Msat . This has
been done using Semi-Analytic Models (SAMs), and the results of such a model have been
compared to observations of a similar set of groups drawn from the SDSS in Weinmann et al.
(2006). The results of their study were presented in similar fashion to the results presented
here. Galaxies are binned by Mgr and Mr and trends in the blue fraction, fb , are presented.
Structural information is not considered, hence the two scenarios will first be compared
without regard to structure. When structure is not considered, among observed satellite
galaxies with Mr < −19 fr increases with Mgr and is independent of Mr . These results are
seen in the observed SDSS groups discussed here and those discussed in Weinmann et al.
(2006).
Across much of the Mgr and Mr range considered, the SAM results predict fb is independent of Mgr . In the highest luminosity bins, however, an upturn in fb is predicted.
The strength of this upturn increases as Mgr decreases. There is an important distinction
between the SAM result and the observed trends. The luminosity at which the SAM results deviate from this independence increases with Mgr ; for each Mgr , it is only among
the most luminous satellite galaxies of these groups that fb increases. In contrast, in the
SDSS groups fr depends on Mgr and Mr across the full dynamic range. Satellite galaxies
that are significantly dimmer, and correspondingly less massive, than these most luminous
satellites have observed blue fractions significantly higher than predicted by the SAM. The
3.5 Discussion
77
deficiency of blue galaxies in the SAM, when compared to the observations, increases as
Mgr decreases.
The upturn in the SAM results may be capturing several effects. The brightest satellite
galaxies at a given Mgr are more likely to be new arrivals, which tend to be bluer and
therefore brighter. Strangulation is a slow process, with time these brightest galaxies will
become not only redder, but dimmer, possibly leaving the brightest luminosity bins. The
brightest bins are also capturing satellites that have the highest pre-accretion masses. These
satellites may therefore spend less time on average within the group before merging with
the central halo. Regardless of the reason for this upturn, it is possible to examine the sense
in which the SAM prediction would be affected by including the outer H i disk as a further
source of star formation fueling.
Introducing an additional source of gas will tend to decrease the predicted fr , increase fb ,
in groups. If this source decreases with Mgr , then the decrease in fr will decline with Mgr .
To facilitate comparing the correlation of fr with Mr between the two scenarios, satellites
can be split into three Msat /Mgr regimes, where Msat is the satellite’s pre-accretion virial
mass. At the smallest Msat /Mgr , stripping is extremely effective, and all satellites are
stripped of their outer H i disk. In this case there is no difference between the stripping and
strangulation scenarios. At the largest Msat /Mgr stripping is ineffective, and all satellites
retain their gas disk and continue to form stars for longer than the SAM would predict. At
intermediate Msat /Mgr , stripping is effective, with an effectiveness that correlates with Mr .
These three regimes occupy different locations within the Mgr vs Mr plane. The satellite
galaxies in the SDSS groups occupy the later two of these regimes.
Both scenarios predict that satellites which were accreted quite recently remain star
forming and blue; fb among satellite galaxies is not zero in the SAM predictions. Whatever
the origin of the blue SAM satellites, introducing any additional supply of gas will leave these
galaxies blue and affect only satellites which the SAM predicts are red. If a constant fraction
of the red galaxies in the SAM predictions were instead blue, then the change in fb would
increase as fb decreased. Therefore, if an additional supply of gas was introduced, then the
78 Ram Pressure Stripping in Groups - Confronting Theory with Observations
generic effect would be to decrease the strength of any predicted correlation between fr and
Mr . In the case of the steep up-turn in fb at high luminosity in low mass groups, introducing
any additional gas supply, or increasing the time scale over which gas is consumed, would
tend to bring the SAM predictions into line with the observations. These satellites are in the
Msat /Mgr regimes in which the inflow plus stripping scenario does introduce an additional
source of gas. In the case of the highest Mgr ; where fr is large, the percentage change in fr
with Mr is small, and the SAM predicts that fr is increasing with Mr ; bringing the SAM
predictions into line with the observations may require that the additional supply of gas
increase with luminosity, as it does in the stripping scenario. Note that either introducing a
new supply of fuel for star formation or increasing the time scale over which gas is consumed
can flatten the correlation between fr and Mr , but cannot account for the increase in fr
with Mgr .
Adding the outer H i disk as a source of fuel for star formation will modify the SAM
predictions by decreasing fr , increasing fb , in groups, particularly in low mass groups where
stripping is ineffective. The decrease in fr , from the SAM to the stripping scenario, will
increase as Mgr decreases, tending to bring the predictions into agreement with the observations of SDSS groups. On this merit alone, the stripping scenario introduced here deserves
further consideration. Introducing an additional supply of gas will also tend to bring the
predicted correlation between fr and Mr into line with the observations. That this gas
supply increase with luminosity may be required. The inflow from the outer H i disk plus
ram pressure stripping scenario meets these requirements. The new observational results
are now considered.
The first relevant result was to confirm that an excess of red, ni < 2.5 galaxies are found
in the satellite population. That this is so can be clearly seen in Figures 3.6 and 3.6. Within
the population of satellite galaxies with nim < 2.5, fr is significantly higher among satellite
galaxies than among central galaxies. The dependence of fr on Mgr and Mr within this
population was next examined. Doing so required taking care to consider the effects of the
significant scatter between the measured nim and the ‘true’ nit for individual galaxies. In
3.5 Discussion
79
concert with the strong tendency for galaxies with ni > 2.5 to reside on the red sequence,
this scatter can bias the observed results. This scatter was considered, and the results can
be summarized as follows. Among the population of satellite galaxies with nit < 2.5, fr
increases with Mgr and decreases with r-band luminosity. Among central galaxies with
nit < 2.5, fr may also decrease with luminosity. The absolute slopes of the correlations
between fr and Mr cannot be measured without a better characterization of the errors in
and distribution of ni , but the slopes can be meaningfully compared. As Mgr decreases,
the strength of the correlation between fr and Mr decreases, and the weakest correlation is
found among the central galaxies.
If the average time since accretion is independent of Msat and Mgr , then the inflow
plus stripping scenario predicts that the fraction of satellite galaxies that were blue when
accreted that have since migrated to the red sequence increases with Mgr and decreases
with luminosity, as discussed in §3.2.1. The time scale for dynamical friction to merge a
satellite with its host group is naively proportional to Msat /Mgr , indicating that the average
time since accretion, and hence fr , may tend to increase as Mgr increases and luminosity
decreases. Dynamical friction is not the only dynamical process at work in groups however.
Satellite galaxies are quickly tidally stripped, decreasing the effectiveness of dynamical
friction and possibly resulting in satellites which survive for many Gyr, as for example
in Boylan-Kilchin et al. (2008). When multiple satellites are present, satellites experience
unbound collisions with each other in which angular momentum is exchanged. This process
may become quite important in the case of long-lived satellite galaxies. The combined
actions of tidal stripping and unbound collisions tend to decrease the correlation between
the average time since accretion and Msat /Mgr . These dynamical effects are captured by the
SAM predictions, which nevertheless fail to predict the appropriate correlations. Therefore,
any underlying dynamical correlation between the time since accretion and Msat /Mgr is too
weak to alone account for the observed trends. Introducing inflow within the gas disk plus
ram pressure stripping can greatly accentuate correlations between fr , Mgr , and Mr within
the population of satellites with ni < 2.5, in agreement with the observed trends.
80 Ram Pressure Stripping in Groups - Confronting Theory with Observations
If inflow plus stripping is to be invoked as a means of reconciling the strangulation
based SAM predictions with observations of real satellite galaxies, then the two must be
reconciled by correlations within the population of disk-like galaxies. That is, it must be
among the disk-like galaxies that the red fraction increases with Mgr and decreases with
luminosity as these are the satellite galaxies which are affected by stripping. This is indeed
what is observed. Furthermore, the observational results indicate that trends in fr among
the full satellite sample are strongly influenced by trends within the ni < 2.5 sub-sample.
While f> among the satellites increases with both Mgr and luminosity, fr shows a greater
increase with Mgr and does not correlate with luminosity. Brief considerations of dynamics
within groups indicate that introducing inflow plus stripping either introduces or enhances
correlations between fr , Mr , and Mgr such as those observed in the population of disk-like
satellite galaxies. Taken together, the results presented here indicate that the inflow plus
stripping scenario is an extremely promising candidate for bringing predictions of galaxy
evolution closer to observations.
When considering the effectiveness of ram pressure stripping, the ram pressure is determined by the group and orbital properties alone and the restoring pressure by the satellite’s
properties alone. Therefore the average restoring pressure as a function of Mr is independent of Mgr , and we might expect the slope of the correlation between fr and Mr would
be the same in all Mgr bins. This is contrary to the observation that within the nit < 2.5
population the slope of this correlation actually decreases with Mgr . This could be due to
either accretion histories or gas physics. While the restoring pressure is determined by the
satellite properties alone, Mgr plays a role in setting the dynamics within a group. The
differences in the slope of the correlation between fr and Mr could also be hinting at the
underlying gas physics.
The stripping scenario presented here could be considered as an extension to the traditional strangulation scenario. An extra supply of fuel for star formation is introduced in the
form of the outer H i disk. Inflow levels in this disk may be low and likely cannot sustain
the star formation levels necessary to maintain a satellite in the blue cloud indefinitely.
3.6 Conclusions
81
If they can sustain star formation for ‘long enough’ however, then when this reservoir is
modified by ram pressure stripping, trends like those observed in the SDSS groups would
occur. The observations may be showing a hint as to what ‘long enough’ is. In the lowest Mgr bin the observations are consistent with no correlation between fr and Mr among
galaxies with nit < 2.5. This is a more traditional strangulation result, with lower than
predicted fr . The analytical model predicts that little to no stripping should be occurring
to the observed satellite galaxies in these groups. This gives rise to the question of what
occurs in this new stripping scenario when stripping is not effective. In this case, all disks,
regardless of their Mr , have access to their outer H i disk. In these galaxies star formation
is sustained by inflow for some average, unspecified, length of time, before is begins slowly
declining. This may lead to fr which are independent of Mr , but lower than predicted by
traditional strangulation models, matching the observations. As Mgr increases, the effectiveness of stripping increases, and the contribution of stripping to determining the slope
of the correlation between fr and Mr may increase.
3.6
Conclusions
This paper introduces a new scenario for star formation fueling in groups in which the star
formation rate is driven by ram pressure stripping of the outer H i disk. In this scenario
traditional modes of star formation fueling are extended by including an additional supply of
fuel for star formation in the form of inflow from the outer H i disk. While the ram pressure
in groups is too low to strip gas from within the stellar disk, the restoring pressures in the
outer disk are equally low. Within groups the ram pressure can strip gas from the outer H i
disk of luminous galaxies. If there is inflow within this disk, then it may extend the length
of time for with a satellite galaxy can continue to form stars after is has been accreted.
In this scenario, satellites that are not effectively stripped may continue forming stars for
some time, until inflow is no longer capable of maintaining the star formation rates required
to retain a galaxy in the blue cloud. These satellites will then slowly transition to the red
sequence. Satellites that are stripped down to their stellar disks experience an immediate,
82 Ram Pressure Stripping in Groups - Confronting Theory with Observations
but slow, decline in their star formation rates, as in the traditional strangulation scenario.
Satellites which are stripped of intermediate gas fractions will continue to form stars for
intermediate lengths of time before they too make a slow transition onto the red sequence.
Introducing inflow from the outer disk plus ram pressure stripping in groups tends to
bring the predictions of semi-analytic models that consider only traditional strangulation
into better agreement with observations. By making an additional supply of gas available
for star formation, the length of time for which satellite galaxies can continue forming stars
is increased, and so too the blue fraction in groups. Introducing a new supply of gas also
softens the correlation between the average time since accretion and the red or blue fraction,
smoothing the sharp up-turn in fb seen among the most luminous galaxies of the low mass
groups in the SAM predictions of Weinmann et al. (2006). An analytical model of stripping
predicts that in the SDSS group catalog the effectiveness of stripping increases with group
mass. The difference between the predictions of the stripping scenario and the traditional
strangulation scenario therefore decreases as Mgr increases. All of these effects work to
improve the predicted correlations.
These results point towards a generic argument for fueling star formation in groups
with inflow from the outer H i disk which does not depend on the predictions of a specific
semi-analytic model. This argument for inflow rests on comparing the average time since
accretion with the time for which a galaxy can continue to form stars from the gas within the
stellar disk alone. At the star formation rates and gas densities typical of spiral galaxies, star
formation will consume the inner gas disk within a couple Gyr. If typical satellite galaxies
were accreted longer than a few Gyr ago, then all models that neglect inflow will predict
an abundance of red satellite galaxies. This comparison is made implicitly in the SAM,
which does predict a large red fraction among satellite galaxies. While dynamical friction
arguments predict that satellites typically survive for only a few Gyr, these predictions
ignore tidal stripping. The time scale for tidal stripping is significantly shorter than the
time scale for dynamical friction. Therefore, the tidal stripping typical of groups and
clusters can lead to satellite lifetimes a few times longer than dynamical friction alone
3.6 Conclusions
83
would predict (see for example Boylan-Kilchin et al. (2008)). If typical satellite lifetimes
are several times longer than the 1-2 Gyr for which these galaxies can continue to form stars
from their inner gas disks, then traditional strangulation models generically predict a high
red fraction among satellite galaxies. The relative abundance of blue satellite galaxies in
observed groups is therefore a challenge to this generic class of models and supports models
that include inflow.
If inflow plus stripping is to be invoked to improve the predictions of fr in groups, then
the relevant trends must be observed within the disk-like population of satellite galaxies. It
is only for these galaxies that the predictions of the stripping and strangulation scenarios
differ. The correlations among these galaxies must be consistent with the stripping scenario
and must be strong enough to influence the correlations among all satellite galaxies. In the
absence of correlations between Mgr , satellite luminosity, and the typical time since accretion, the stripping scenario predicts that fraction of accreted blue galaxies that transition
to the red sequence increases with Mgr and decreases with luminosity. A brief consideration
of the dynamics within groups in § 3.5 indicated that dynamics may enhance the strength
of these predictions, but they do not reverse them. To test these predictions, new observational results were presented of the conditional red fraction for central and satellite galaxies
with disk-like light profiles, as determined using the i-band sersic index, in a group catalog
drawn from the SDSS.
The observational results presented here are in agreement with earlier studies. Both
the fraction of red galaxies, fr , and the fraction of galaxies with ni > 2.5, f> , are higher
among satellite galaxies than central galaxies, and both increase with group mass, Mgr .
The excess of red galaxies in the groups is greater than the excess of galaxies with ni > 2.5,
indicating that a population of red disk-like galaxies can be found in the groups. Among
disk-like satellite galaxies, fr increases with Mgr and decreases with luminosity. Comparing
the relative slopes of the correlation between fr and luminosity indicates that the fraction
of accreted blue galaxies which have since transitioned to the red sequence decreases with
luminosity. These results are consistent with inflow plus stripping driving star formation
84 Ram Pressure Stripping in Groups - Confronting Theory with Observations
rates among the disk-like galaxies. The correlations among the disk-like satellites are strong
enough to affect the observed red fraction among the entire satellite population. The increase with Mgr in the fraction of satellite galaxies with red color is greater than that in the
fraction of galaxies with ni > 2.5. In addition, while the number of satellites with ni > 2.5
increases with luminosity, the fraction with red colors is relatively constant. Both of these
demonstrate the effect of correlations within the population of disk-like satellites on the full
satellite population. The new observational results are entirely consistent with the scenario
in which inflow within the gas disk plus ram pressure stripping determines star formation
rates in groups.
Making use of measures like ni for studies such as the one presented here requires some
caution. While colors can be measured with little noise, structural measures can be quite
noisy. In the catalog use here, a significant fraction of galaxies likely scatter across the
ni = 2.5 line. This can have several consequences. First, as f> deviates from 0.5, the
numbers of galaxies scattering in each direction will not be equal. Thus, this scatter works
against observing extreme values for f> . Second, galaxies with ni > 2.5 are substantially
more likely to be red. Among central galaxies, the measured fr for galaxies with ni < 2.5
is 0.2 while for galaxies with ni > 2.5 it reaches as high as 0.8. Therefore, this scatter
has a strong tendency to increase the red fraction among galaxies with measured ni < 2.5.
Finally, the size of the induced increase in fr increases as either f> increases or the scatter
increases. The induced increase in the red fraction among galaxies with ni < 2.5 will tend to
mimic f> . In theory, variations in the scatter between the two populations can work against
this tendency. In practice, however, it seems to hold. These issues were all considered in
some detail when arriving at the conclusions discussed in the previous paragraph.
The results presented here make a strong case for introducing inflow within the gas
disk plus ram pressure stripping into studies of galaxy evolution in groups. In order to
determine the relative importance of inflow and ram pressure stripping on the star formation
and color evolution of satellite galaxies several tracks should be followed. Primarily, the
mechanisms by which galaxies can fuel their star formation must be understood in more
3.6 Conclusions
85
detail. Specifically, the ability of galaxies to sustain star formation via disk inflow within
group-like environments should be studied both theoretically and observationally. Second,
the effects of hierarchical structure formation on the stripping scenario should be considered
in more detail than is done here. Finally, further observational tests should be considered
which can distinguish between the stripping scenario presented here and the traditional
strangulation model.
86 Ram Pressure Stripping in Groups - Confronting Theory with Observations
Figure 3.1: Fraction of galaxies that belong to the red sequence, fr , and the fraction with
ni > 2.5, f> , vs Mr . Stars/solid : Central galaxies Diamonds/dash-double-dot: Satellite
galaxies in groups with log(Mgr (M )) < 13. Squares/dash: 13 < log(Mgr (M )) < 14.
Triangles/dash-dot: 14 < log(Mgr (M )) < 15. See the text for a description of the groups
and the selection of central and satellite galaxies. (a): fr for all galaxies. (b)((c)): fr for
galaxies with ni < 2.5 (ni > 2.5). (d): f> for all galaxies. (e)((f)): f> for galaxies with
blue (red) colors, separated using an Mr dependent color cut. Note that galaxies in groups
have both a higher fr and a higher f> than isolated galaxies, in agreement with previous
studies of the relationship between color and ni and environment. Among disk-like satellite
galaxies, that is satellites with ni < 2.5, the correlations between fr , Mgr , and Mr are all
consistent with inflow within the outer gas disk plus ram pressure stripping of the outer gas
disk determining star formation rates.
3.6 Conclusions
87
Figure 3.2: Difference between satellite galaxies and central galaxies in the red fraction,
fr,gr − fr,is , and in the fraction of galaxies with ni > 2.5, f>,gr − f>,is . Symbols, line
styles, and panels are the same as in Figure 3.6. Note that an excess of red galaxies is
observed among the satellite galaxies in both of the ni selected sub-populations, but no
excess of satellites with ni > 2.5 is seen in either the blue or red sub-populations. This
asymmetry has been observed before in studies based on local over-densities and conditional
correlation functions, and indicates that some interaction between satellite galaxies and the
IGM is suppressing star formation among disk-like satellite galaxies. This figure highlights
this asymmetry. Panel (b) also highlights the decrease with luminosity among the disk-like
satellites in the fraction of galaxies that transition from the blue to the red sequence as
satellites.
Chapter 4
Major Mergers Between Dark
Matter Halos in the Millennium
Simulation
The contents of this chapter represent work done with A. Tasitsiomi and will be submitted
to the Astrophysical Journal.
4.1
Introduction
The ongoing mergers of both dark matter halos and the galaxies they contain is an inevitable
component of hierarchical structure formation. The potential impacts of these mergers on
galaxy evolution are both varied and heavily debated. Mergers can be roughly divided into
two classes; minor mergers, in which a small halo is accreted by a substantially larger halo,
and major mergers, in which the two halos are of roughly similar mass. The division between
the two is usually placed near a mass ratio of 3:1. Minor mergers contribute both stars and
gas to forming galaxies, and are important for understanding the detailed morphologies
of galaxies, particularly spiral galaxies. Tidal forces during a minor merger may heat the
thin stellar disk and drive bar instabilities, thick disks may also represent the remnants
88 Major Mergers Between Dark Matter Halos in the Millennium Simulation
4.1 Introduction
89
of disrupted satellites (Steinmetz and Navarro, 2002; Yoachim and Dalcanton, 2008). The
effects of a major merger are likely more dramatic. Major mergers between two gas rich
spiral galaxies are a popular mechanism for creating elliptical galaxies, an idea dating back
to Toomre and Toomre (1972). More recently, they have been invoked as a means of fueling
intense starbursts and luminous AGN (Mihos and Hernquist, 1996; Di Matteo et al., 2005;
Springel et al., 2005b,a, among others).
Theories connecting major mergers, starbursts, AGN, and the creation of elliptical galaxies have not been suitably tested observationally. Theoretical models can reproduce the
luminosity function and number density evolution of AGN while correctly recovering correlations between black hole masses and spheroid dynamics (Kauffmann and Haehnelt, 2000;
Wyithe and Loeb, 2003a,b; Hopkins and Hernquist, 2006). These models include assumptions about AGN and starburst lifetimes and efficiencies which tend to be under constrained
by the observational data. Additional constraints on these models are needed, as are model
independent tests of major merger driven evolutionary scenarios. Environment is a potentially powerful probe which has been under exploited in previous work on major mergers.
To make use of environment, the environmental dependences of the major merger rate must
be well understood. This work therefore explores the environments of major mergers in the
Millennium Simulation, a large N-body simulation with a box length of 500h−1 Mpc and
a particle mass below 109 M (Springel et al., 2005c). This simulation is large enough to
probe the full range of environments and has a fine enough mass resolution to follow galaxylike halos and subhalos. The goal of this project is to provide the theoretical groundwork
necessary to use environment as a probe of merger driven galaxy evolution scenarios.
4.1.1
Major Mergers, AGN, & Morphology
The link between major mergers, starburst and AGN fueling, and galaxy morphology has
been extensively studied using simulations, and there is strong circumstantial observational
evidence that these populations are correlated. Simulations of individual major mergers
involve imbedding a stellar and gaseous disk into each of two dark matter halos, placing them
90 Major Mergers Between Dark Matter Halos in the Millennium Simulation
on a collision course, and observing the stages of the simulated merger. Such models vary
considerably in complexity, ranging from including only stellar disks to modeling multiphase
gas disks. They also vary in prescriptions for AGN and star formation fueling. These
simulations capture the physics behind the major merger scenario and can predict the
appearance of the different stages of a major merger (for examples see Mihos and Hernquist,
1996; Springel, 2000; T. and Burkert, 2003; Barnes, 2004; Springel et al., 2005a; Cox et al.,
2006). Complimentary simulations invoke semi-analytic prescriptions to introduce simplified
gas physics into large dark matter simulations. These allow the entire population of AGN,
starbursts, and galaxy morphologies to be studied (Kauffmann and Haehnelt, 2000; Wyithe
and Loeb, 2003a,b).
The results of previous theoretical studies of major mergers can be briefly summarized
as follows. The tightly bound stellar components remain in the center of their dark matter
halos, loose angular momentum through gravitational interactions with their own dark
matter halos, and remain at the center of the merger remnant. The two stellar components
encounter each other with a tightly bound orbit, quickly merging after the cores of the
dark matter halos have merged. The resulting stellar remnant is pressure supported rather
than rotationally supported. When no gas component is included, the central densities of
simulated major merger remnants are lower than the observed cores of elliptical galaxies.
When gas is included, tidal forces funnel the gas into the centers of the merging galaxies
prior to the galaxies merging. This gas then settles to the center of the merger remnant.
Of the initial gas disk, ≥ 50% ends up in the central component of the merger remnant (see
review in Barnes and Hernquist, 1992). Allowing this gas to form stars builds a stellar core
in the remnant, similar to that seen in elliptical galaxies (Mihos and Hernquist, 1996). Gas
that is funneled to the centers of the merging galaxies can also fuel AGN activity both in
the merging galaxies and in the remnant. When black hole growth and star formation in
the merger remnants both occur with efficiencies that are proportional to the gas fraction in
the central region, semi-analytic models can reproduce observed correlations between black
hole masses and spheroid masses (Kauffmann and Haehnelt, 2000). Self regulated models of
4.1 Introduction
91
black hole growth go a step further. With abundant fuel, the black hole at the center of the
remnant grows exponentially both in mass and in luminosity. Feedback from the growing
AGN heats and drives winds in the surrounding gas and can become strong enough to expel
gas from the center of the remnant (Silk and Rees, 1998; Fabian, 1999), terminating both
star formation and AGN activity. Self regulated models of black hole growth predict that
the ultimate mass of the black hole is strongly correlated with the depth of the central
gravitational potential (Di Matteo et al., 2005; Springel et al., 2005b,a). Furthermore,
nuclear star formation and black hole growth are both regulated by the same process (e.g..
Hopkins and Hernquist, 2006). The resulting major merger remnant has a stellar profile
which resembles an elliptical galaxy, hosts a central super massive black hole with a mass
that correlates both with the central potential and the central stellar density, is gas poor,
and has a diffuse halo of hot gas. If this gaseous halo is prevented from cooling and forming
stars, the post-starburst remnant will fade to become ‘red and dead’. Feedback between
cooling in the nucleus and low level AGN activity is often invoked to prevent further star
formation in the remnant (Best et al., 2001; Fabian et al., 2006).
There is observational evidence to support this merger driven evolutionary scenario.
Ultraluminous infrared galaxies (ULIRGS) represent powerful starbursts. Observations of
ULIRG morphologies indicate that they are ongoing or recent major mergers (see reviews
in Sanders and Mirabel, 1996; S., 2005). Radio observations of these galaxies indicate that
they have central concentrations of dense cool gas (Scoville et al., 1986; Sargent et al., 1987,
1989). Powerful IR selected starbursts are also known to host obscured or low luminosity
AGN (Komossa et al., 2003; Gerssen et al., 2004; Alexander et al., 2005; Borys et al., 2005).
The physical number density of luminous AGN rises with redshift, peaking around z ∼
2−3 (Boyle et al., 2000; Fan et al., 2001), similar to the major merger rate of massive halos in
N-body simulations. Black hole growth and spheroidal growth appear to be closely related.
Black hole masses correlate both with the mass and the velocity dispersion of the spheroids,
either elliptical galaxies or spiral bulges, that host them (Magorrian et al., 1998; Ferrarese
and Merritt, 2000; Gebhardt et al., 2000; McLure and Dunlop, 2002; Tremaine et al., 2002;
92 Major Mergers Between Dark Matter Halos in the Millennium Simulation
Marconi and Hunt, 2003). Post-starburst E+A (or K+A) galaxies have high central surface
densities, kinematically hot older stellar populations, and will fade to resemble elliptical
galaxies in the absence of additional star formation (Norton et al., 2001; Yang et al., 2004;
Goto, 2005). E+A galaxies also frequently display disturbed morphologies, indicative of
mergers or tidal interactions (Yang et al., 2004; Goto, 2005; Owers et al., 2007). Elliptical
galaxies are found preferentially in group and cluster environments (Dressler, 1980; Norberg
et al., 2002; Hogg et al., 2003b; Kauffmann et al., 2004; Blanton et al., 2005a), and simulated
dark matter halos in equivalent environments preferentially experienced major mergers in
their pasts (Gottlober et al., 2001).
We intend to use a dark matter simulation to develop environmental diagnostics of
major merger populations that can be used to test the merger driven galaxy evolution
scenario. By doing so we are assuming both a one to one correspondence between dark
matter mergers and galaxy mergers and that the relevant dynamics are dominated by the
dark matter. As discussed above, when the cores of two dark matter halos that each host a
galaxy merge, the galaxy merger is immanent. The final stages of the galaxy merger occur
quickly; Cox et al. (2006) find that the final galaxy merger in a 1:1 merger takes ≈ 200Myr.
While there has been some discussion of ‘dark halos’ which do not host galaxies (Maccio
et al., 2006, and references therein), this occurs at Vmax well below those considered here.
Similarly, their is no observational evidence for orphan galaxies (Mandelbaum et al., 2006),
and truncated dark matter halos have been observed around galaxies in clusters (Natarajan
et al., 2007). A one to one correspondence between halo mergers and galaxy mergers is
therefore a reasonable assumption. While the inclusion of baryons might affect some of the
relevant dynamics, dark matter constitutes a strong majority of the matter. Hence, the
dynamics is dominated by the dark matter, with possible refinements to be introduced by
including baryons. Exploring this issue is a potential topic for future work.
4.1 Introduction
4.1.2
93
Mergers & Environment - Previous Results
The earliest, and simplest, theoretical studies of the merger rate were based on extended
Press Schechter theory, which was in turn based on linear theory plus spherical collapse
models (Press and Schechter, 1974; Bond et al., 1991; Lacey and Cole, 1993). Mergers
between halos were assumed to occur on the time scale of dynamical friction and mergers
between subhalos were neglected. With these assumptions, merger trees can be built for all
halos existing today (Kauffmann and White, 1993; Somerville and Kolatt, 1999). Extensions
of this treatment first considered the merger rate between subhalos, concluding that the
sub-sub merger rate could be quite high within group mass hosts. Further work began to
include dynamical effects within the host halo such as tidal stripping. This was done both
analytically and with N-body simulations (Mamon, 2000; Peñarrubia and Benson, 2005;
Boylan-Kilchin et al., 2008). Assuming that linear theory is modified only by dynamics
within virialized halos, that is that linear theory correctly describes accretion histories, all
non-linear effects are confined to within host halos. In this scenario, these effects are entirely
captured by correlating both the merger rate between subhalos and the time-scale for subhost mergers with the relevant halo masses. In this treatment, correlations between the
merger rate and environment on scales beyond the virial radii of the hosts, measured either
directly or through clustering, arise from convolving any correlations between merger rate
and host mass with the clustering of the hosts. The final refinement is to use cosmological
N-body simulations to test the linear theory and to search for correlations between the
merger rate, or accretion histories in general, and environment which are not derivative. In
general, linear theory breaks down when tidal forces become important, so some deviations
should be expected.
Previous treatments of the merger rate between subhalos have found that the specific
merger rate, that is the merger rate per halo or galaxy, should be enhanced in groups and
suppressed in clusters. Within a bound virialized halo, where it may be possible to assume
relative subhalo velocities are random, the specific major merger rate between subhalos can
94 Major Mergers Between Dark Matter Halos in the Millennium Simulation
be expressed as
Rm = nh hσm vi
where σm is the merger cross section, v is the relative velocity of the halos, and nh is the
number density of potential interaction or merger partners. Generically, the merger cross
section must increase with subhalo mass and be highly dependent on the relative velocities of
the subhalos. In clusters, where relative subhalo velocities substantially exceed the internal
velocities of the subhalos, merger rates between subhalos are likely suppressed relative to
distinct, or non-sub, halos. Groups however appear to represent an environment which
combines higher than average halo densities with relative velocities that are conducive to
merging. Merger cross sections based on simulated encounters between distinct halos of
equal mass peak near the internal velocity of the halos (Makino and Hut, 1997). Using
these cross sections and assuming a Maxwellian velocity distribution for the subhalos that
scales appropriately with host halo mass, Makino and Hut (1997) find that merger rates
are enhanced in groups and decline with the mass of the host halo, becoming suppressed
in clusters. In this treatment the subhalo merger rate also increases with the mass of the
subhalos, as the merger cross section increases.
Several studies, both analytical and numerical, have found that tidal stripping occurs on
a time scale which is shorter than the dynamical friction time scale; that is newly accreted
subhalos are tidally truncated before dynamical friction has time to act (Mamon, 2000;
Peñarrubia and Benson, 2005; Boylan-Kilchin et al., 2008). Tidal stripping has one of two
effects on the merger rate in groups. In a mass defined subhalo sample, tidal stripping
reduces the subhalo number densities by removing subhalos from the sample. In a subhalo
sample based on pre-accretion masses, tidal stripping can drastically reduce the subhalo
merger cross section. Previous work has taken the first approach. Mamon (2000) presents
a detailed analytical estimate of the merger rate in groups and clusters as a function of
group or cluster internal velocity distribution, Vh , and subhalo mass, ms , which account for
tidal stripping. When subhalo populations are selected using post tidal stripping masses,
the specific merger rate is extremely sensitive to the host halo mass and is still enhanced
4.1 Introduction
95
in groups and suppressed in clusters. An analytical estimate of this dependence is given
in Mamon (2000); Rm ∝ nG2 m2s /Vh3 .
Numerical simulations support these analytical results. Ghinga et al. (1998) simulate
a large cluster with a high resolution in order to observe subhalos. They find that halos
cease merging once they enter the cluster. De Lucia et al. (2004) also find that the merger
rate drops after sub-halos are accreted by a cluster. Gottlober et al. (2001) make a complimentary measurement in a cosmological N-body simulation. They measure the merger
rate of the most massive progenitors of the halos identified at z = 0, therefore not counting
major mergers that result in remnants that later merge with a more massive halo. They
present the merger rate of the most massive progenitors of halos at z = 0 versus redshift
as a function of environment. Halos that reside in clusters at z = 0 have the lowest merger
rates near z = 0, but had merger rates higher than the progenitors of isolated halos in the
past. The merger rates for halos that reside in groups at z = 0 are higher than for isolated
galaxies.
The consideration that subhalos are tidally stripped has implications for the decay of
subhalo orbits via dynamical friction. By reducing the bound mass of the subhalos, tidal
stripping increases the dynamical friction time scale and mergers between the subhalo and
the core of the host halo do not occur as rapidly as is generally assumed. Simulations
offer the opportunity to measure the times time scales on which mergers occur between
an accreted halo and its new host halo. Boylan-Kilchin et al. (2008) run high resolution
simulations in which a single subhalo is accreted by a host with no other substructure and
find that the subhalos merge with their hosts on time scales that are several times longer
than predicted by dynamical friction alone.
Studying the merger rate as a function of host halo mass is attractive because it has direct observation consequences. Doing so also likely captures most of the non-linear physics
affecting the merger rate. Under the assumptions of linear theory, once a halo mass is
specified, in this case the mass of the host halo, the accretion history is independent of environment (White, 1996). The non-linear physics can thus be divided into dynamics within
96 Major Mergers Between Dark Matter Halos in the Millennium Simulation
the host halo, which do not violate this assumption, and non-linear correlations between
accretion history and local environment. Several recent studies of halo properties have indicated that accretion history does have a residual dependence on local environment (Wechsler
et al., 2006; Croton et al., 2007; Gao and White, 2007). There may be a similar dependence
between the major merger rate and local environment. This has not yet been studied.
We are interested not only in the dynamics that drive major mergers but in laying
the groundwork to craft observational tests both of the simulations themselves and for
identifying major merger populations. To this end we will not rely on a mass defined halo
sample as has been done in most previous work, with some exceptions (e.g.. Gottlober
et al., 2001). The luminosity of the galaxy that is hosted by a halo, which is the relevant
observable quantity, correlates more strongly with the maximum circular velocity of the
halo, Vmax , than the mass of the halo (Kravtsov et al., 2004a; Tasitsiomi et al., 2004, 2008).
This is particularly true for subhalos. In the case of isolated halos, Vmax and halo mass
are closely related. For a mass defined subhalo catalog, tidal stripping both destroys any
correlation between halo mass and galaxy luminosity and begins removing halos from the
catalog. A Vmax selected subhalo catalos is less susceptible to these effects. Because many
previous results rely on mass defined halo catalogs, they must be carefully compared to our
results.
While the environments of major mergers are clearly a widely studied topic, the Millennium Simulation should allow us to make an important advancement. The Millennium
Simulation, with its superb combination of size and resolution, allows the study of all of the
above issues in concert. We will use a common, well defined, language to study these issues
and, in addition, should be able to see the effects of the interplay between them. Finally,
we will focus not only studying the dynamics of major mergers, but on using definitions of
environment and the major merger rate that have clear observational counterparts. This is
essential as the ultimate goal is to use the results of this work to craft observational tests
that are capable of identifying merger populations.
4.2 Methods 97
4.2
4.2.1
Methods
Simulation & Numerical Issues
Our study is performed using the Millennium Simulation (MS) (Springel et al., 2005c) run
using a version of GADGET2 (Springel, 2005). The MS is a cosmological N-body simulation
of the ΛCDM universe that follows the evolution of more than 10 billion particles in a box of
500h−1 Mpc comoving on a side. The particle mass is 8.6×108 h−1 M , and particle-particle
gravitational interactions are softened on scales smaller than 5h−1 kpc. The simulation uses
parameters in agreement with the WMAP1 results (Spergel et al., 2003): Ωm = 0.25,
ΩΛ = 0.75, h = 0.73, n = 1 and σ8 = 0.9.
We use halos drawn from the MS halo catalog. The first step in halo identification is
a friends-of-friends (FOF) group finder which is built into the simulation code. Particles
separated by less than 0.2 times the mean particle separation are grouped together in
a FOF group. This combines particles into groups with a mean over-density which is
somewhat lower than the expected overdensity of virialized halos at low z and approaches
the expected overdensity as z increases (Bullock et al., 2001). In post-processing, bound
halos are identified within these FOF groups using an improved version of the SUBFIND
algorithm (Springel et al., 2001). In each FOF group both a dominant central halo and
its subhalos are identified. SUBFIND first computes the smoothed density field at the
positions of all the particles. It then defines as possible halos all regions centered around
locally overdense points that are bounded by the first isodensity contour to traverse a
saddle point in the density. Each halo candidate is subjected to a gravitational unbinding
procedure. Structures that retain greater than 20 particles are kept in the halo catalog
and their basic properties are determined. At low redshifts the FOF groups includes halos
within a few virial radii of the central halo, which will be made use of later.
In this study we characterize halos predominantly by the maximum of their rotation
velocity curve, Vmax , rather than by mass. The reasons for not using halo mass, particularly
in the case of subhalos, are twofold. First, defining mass in the case of subhalos can be
98 Major Mergers Between Dark Matter Halos in the Millennium Simulation
problematic. It is not clear that SUBFIND provides reliable subhalo masses, see discussion
in Natarajan et al. (2007), and, regardless of the specifics of the halo finder, subhalo mass is
itself a relatively ill-defined concept. Second, as we are most interested in results that can
be compared to observations, we are concerned primarily with the hypothetical luminosity
of the galaxy hosted by a subhalo and not with the bound mass of the subhalo. The ‘correct’
halo measure for our purposes is therefore the quantity that is most closely correlated with
the luminosity of the galaxy that would have formed at the center of the halo or subhalo.
Even a well defined ‘tidal mass’ is therefore not a suitable measure. In the general scenario
galaxies form within distinct halos which may then be accreted and become a subhalo.
After accretion the subhalo can undergo a substantial degree of tidal stripping achieving a
tidal mass that depends on several structural and orbital parameters. While in the majority
of cases, the central galaxy remains intact, the surrounding dark matter halo decreases in
mass. Any initial correlation between halo mass and the luminosity of the hosted galaxy is
thus destroyed.
Using Vmax as a proxy for galaxy luminosity is both well motivated and does not suffer
as strongly from the issues mentioned above. The use of Vmax is strongly justified by an
array of successful comparisons between observations and collision-less simulations in which
halos are assigned a galaxy luminosity by associating luminosity and Vmax (e.g., Kravtsov
et al., 2004a; Tasitsiomi et al., 2004, 2008). These studies make a case for the hypothesis
that, as a measure of the central halo potential, Vmax quantifies the ability of a halo to allow
baryons to cool and from stars. Note that while Vmax may not be the only such quantity,
these studies indicate that it is the dominant quantity. As a measure of the central halo
potential, Vmax is also significantly better defined for subhalos, thus overcoming a technical
issue associated with using tidal masses. Of course Vmax itself is not immune to the affects
of tidal stripping. However, it has been shown that it responds to tidal stripping much
slower than mass (e.g., Kravtsov et al., 2004b).
The completion of the halo catalog, and in particular of subhalos, is important for this
project. The first concern is that distinct, that is non-sub, halos are complete down to the
4.2 Methods 99
necessary Vmax . The second concern is that subhalos are not lost to numerical issues. The
halo catalog which we worked with directly for this project included halos with Vmax >
120km/s, though our mergers may have progenitor halos with Vmax < 120km/s. The larger
complete halo catalog from the Millennium Simulation includes all halos with greater than
20 particles, or a bound mass greater than 1.7 × 1010 M . There is no guarantee, however,
that the halo catalog is complete down to this limit. The standard test for completion,
applied most commonly to mass selected halo catalogs, is to check for a flattening of the
mass function at low halo masses. We do the same for the Vmax selected catalog and find
that no flattening is observed down to Vmax = 120km/s at z = 0 or at z = 4. Merger
remnants with Vmax = 120km/s typically have a least massive merger progenitor with
Vmax > 100km/s, and we would ideally be able to check for flattening down to this limit.
At redshift z = 0, an average distinct halo with Vmax = 100km/s has a virial mass Mv ≈
1.2 × 1011 M and includes np ≈ 150 bound particles. The virial mass scales approximately
3 , as expected. The relationship between V
as Vmax
max and Mv in the simulation evolves with
redshift in the sense that as redshift increases Mv (Vmax ) decreases. At z = 4, an average
distinct halo with Vmax = 100km/s has a virial mass Mv ≈ 3.3 × 1010 M and includes
np ≈ 38 bound particles. This introduces an interesting aspect of this work. At z = 0, we
can be reasonably sure that distinct halos are complete down to Vmax = 100km/s, given
the relatively large particle number and the lack of flattening at higher z. As the redshift
increases however, the halo catalog becomes less complete. At z = 4 completion of even
distinct halos may begin to be an issue. Each trend we will present below relies on halos
down to a different Vmax , and each will therefore only be reliable up to a given redshift.
In the case of the MS, the particle mass resolution is fine enough to correctly describe
distinct halos down the smallest Vmax halos of interest. However, subhalos with Vmax values
greater than the minimum reliable Vmax , as determined via a convergence test, may suffer
from over-merging if inadequate mass and force resolution allow subhalos to be prematurely
and artificially disrupted. Mass resolution limits the reliable mass, or Vmax , for subhalos
in that as the subhalo is stripped and the number of bound particles decreases, two-body
100 Major Mergers Between Dark Matter Halos in the Millennium Simulation
interactions can be being to operate, causing the subhalo to artificially evaporate. Similarly,
inadequate force resolution results in subhalos with a large artificial core. Such halos are
easily disrupted and are artificially lost. Increased force resolution results in smaller cores
and more robust halos.
These and similar issues have initiated multiple numerical studies. While ideally one
would rerun the MS with different mass and spatial resolution in order to estimate the
impact of these effects, this route is prohibitively expensive. As a feasible alternative we
will rely on the results of Klypin et al. (1999), which agree with other, similar, numerical
studies. Klypin et al. (1999) provide a pair of criteria that must be fulfilled in order for overmerging of subhalos not to be important. The authors motivate their conditions analytically
before verifying them with ART simulations. In principle they are therefore independent of
the specifics of the simulation. They find that at least 20-30 particles are required in order
for two-body evaporation to be negligible and that the tidal radius of the subhalo must be
at least as large as a couple of spatial resolution lengths. Note that the 20-30 particles refers
to the number of bound particles post-tidal stripping. In general, subhalo orbits are slowly
degraded and the tidal mass and radius of a subhalo are monotonically related to its orbital
radial position. These conditions can therefore be expressed as a minimum orbital radius
to which a subhalo can survive numerical effects. When verifying that these conditions are
met, one can begin by asking which of these two effects first becomes important for the
subhalos of interest and conclude by determining whether the orbital radius to which the
halo survives is sufficiently small. That is, at sufficiently small orbital radii we can assume
that a merger between the subhalo and its host is imminent and are no longer concerned
with the survival of the subhalo. At a given orbital radius, subhalos with higher values
of Vmax will contain more bound particles and will have a larger tidal radius. Ultimately,
therefore, these considerations dictate a minimum Vmax to which subhalos can reliably be
studied. These limits are not exact, but they highlight the relevant concerns and will be used
to indicate approximately when subhalo completion becomes an issue. The possible effects
of subhalo completion on our individual results are discussed in detail in Section 4.4.4.
4.2 Methods
101
The final simulation issue which must be considered is the spacing of the simulation
outputs for which the halo finder is run. There are 64 saved outputs, most of which are
equally spaced in log(1 + z) between z=20 and z=0. The typical time elapsed between two
consecutive outputs is 2-400 Myr. Typical time scales for an accreted halo to merger with
its host halo are a few to 10 Gyr (Boylan-Kilchin et al., 2008), though we find that in some
cases this time scale may be shortened by approximately a factor of 10. Therefore, except
in some extreme cases, accreted halos are resolved as subhalos in multiple outputs before a
merger with the host occurs. This temporal resolution distinguishes this work from some
past studies (e.g., Klypin et al., 1999).
4.2.2
Merger Trees
Every halo in the MS is associated with a merger tree that contains all of the progenitors of
that halo during each previous simulation output. The merger tree therefore also contains
the merger history of each halo. Mergers trees are constructed after halos are identified in
the 64 saved outputs. The spacing of these outputs is dense enough to reliably identify the
descendant of every halo in the following timestep. Halos are required to have a unique
descendant in either the following timestep or, in a small number of cases, the time step
after. Identifying all descendants at each timestep defines the merger trees. Descendants are
identified by tracking particles, each of which is labeled. Every halo in the next timestep
that shares particles with the halo under consideration is identified. These particles are
weighted according to how bound they are to the halo under consideration, with tightly
bound particles receiving a higher weight. The halo in the following timestep with the
highest weighted sum is marked as the unique descendant. This scheme aims at tracking
the inner cores of the halos which are less vulnerable to mergers and tidal stripping. The
construction of the merger trees is described in the supplementary information to Springel
et al. (2005c).
The merger trees are used to identify major mergers. While all halos have a unique
descendant, when a merger has occurred halos have two or more progenitors in the previous
102 Major Mergers Between Dark Matter Halos in the Millennium Simulation
timestep. For this work, major mergers are defined using the Vmax ratio of two merger
progenitors. The progenitor with the higher Vmax is referred to as the mmp, or most
massive progenitor, and that with the lower Vmax as the lmp, or least massive progenitor.
For halos that are not the subhalo of a larger halo, Vmax correlates closely with mass. Major
mergers are defined as having Vmmp /Vlmp > 0.7 (0.6933). In the case of halos that are not
subhalos, this is the equivalent of mergers of with less than a 3:1 mass ratio. In the case of
subhalos, the mass ratio of the halos may vary considerably, but the luminosity ratio of the
galaxies typically hosted in the halos will not.
Two definitions of a specific merger rate are used in this paper. A specific merger rate is
a merger rate per Gyr normalized by the number of halos, as opposed to a physical number
density of mergers per Gyr. The first rate is a backwards looking merger rate, R− . It is
defined as the number of major mergers in the last timestep that resulted in remnants with
Vmax divided by the number of halos with the same Vmax in the current simulation output,
R− ≡ Nm− (Vmax )/Nh (Vmax ). The observational equivalent of the is definition is a merger
rate measured using morphologically identified merger remnants. It is also the appropriate
rate to compare with potential post-merger populations such as starbursts and luminous
AGN. A forward looking specific merger rate, R+ , is also used. The forward looking merger
rate is defined as the number of halos in the Vmax range of interest that will experience a
major merger in the next timestep divided by the number of halos in the same Vmax range
in the current timestep, R+ ≡ Nm+ (Vmax )/Nh (Vmax ). The observational equivalent of R+
is a merger rate measured using pair counts. In these definitions Nm− and Nm+ distinguish
mergers that happened in the previous Gyr from those that will happen in the next Gyr.
4.3
Results
In this section results are presented for several different measurements of the evolution
of and environmental dependence of the major merger rate. The physical motivation for
exploring each of these measurements is presented in this section, but a full discussion of the
physical significance of each measurement is postponed until §4.4. In what follows, several
4.3 Results
103
different classes of dark matter halos are discussed. A ‘distinct’ halo is any halo that is not
a subhalo. A ‘subhalo’ is bound to and resides within the virial radius of a distinct host
halo. A distinct halo may host subhalos, and therefore be a ‘host’ halo, but it cannot itself
be a subhalo. While Vmax may be used to designate the maximum circular velocity of any
halo, Vh is used specifically for either distinct or host halos and Vs specifically for subhalos.
4.3.1
Evolution of the Merger Rate
Figure 4.5 presents the evolution of the merger rate for all halos with Vmax > 175km/s.
The right panel displays the evolution of both specific merger rates, R+ and R− . Again
R+ is a forward looking merger rate which measures the fraction halos that will become
the most massive progenitor (MMP) of a major merger in the next Gyr, and R− is a
backward looking merger rate which measures the fraction of halos that are a remnant of a
merger that occurred in the last Gyr. Comparing R+ and R− demonstrates the differences
one might expect when comparing the evolution of the merger rate as determined using
different observational techniques, such as identifying merger remnants morphologically
and immanent mergers using pair counts. The evolution of the forward looking rate has
a shallower slope than the backward looking rate. The backward merger rate, Rm− ∝
(1 + z)2.2±0.05 and the forward merger rate, Rm+ ∝ (1 + z)1.82±0.01 . Due to completion
issues at high z the actual slopes may be somewhat steeper. Mergers are lost to completion
issues before halos, suppressing the specific merger rate at high z. This is due to loosing
the least massive progenitor halos.
The left panel of Figure 4.5 shows the evolution of the physical density of mergers.
The physical number density of merger remnants with Vmax > 175km/s increases with
redshift like nm ∝ (1 + z)5.44±0.02 . For comparison the physical number density of halos
with Vmax > 175km/s evolves as nh ∝ (1 + z)3.13±0.03 . Both of these results differ from their
mass selected counterparts. In the case of a mass defined halo sample, the evolution of the
halo number density above a lower mass limit is shallower than (1 + z)3 as halos are built
up over time and the mass function evolves towards higher masses. Similarly, the peak in
104 Major Mergers Between Dark Matter Halos in the Millennium Simulation
the physical density of mergers at intermediate or high redshift seen in mass selected halo
samples is not observed for a Vmax selected halo sample. The decline in the density of mass
selected mergers at high z is due to the increasing scarcity of halos above a given mass as
z increases. The differences between these two types of samples can be reconciled. The
relationship between Vmax and halo mass is evolving. As the redshift increases Mv (Vmax )
decreases. The halo density and merger rate of lower mass halos are being measured at
high z.
4.3.2
Mergers Between Subhalos and Their Host Halos
When a halo is accreted by a more massive host halo and becomes a subhalo, it is potentially
subject to tidal stripping, dynamical friction, unbound and bound collisions with other
subhalos, and may eventually merge with the host halo. The results presented in the next
two subsections examine these issues in the light of their influence on the major merger rate.
Figures 4.5 and 4.5 begin by demonstrating that tidal stripping of subhalos is occurring
in the Millennium Simulation. These figures show subhalo number density contours in the
log[np/np(Vmax )] versus ro /rvh plane at z = 0 and z = 1, where r0 /rvh is the current orbital
radius of the subhalo divided by the virial radius of the host halo. Contours are shown for all
subhalos, and for subhalos with Vs /Vh > 0.7 and 0.94, corresponding to approximate, preaccretion, mass ratios, mvs /Mvh , of 3:1 and nearly 1:1 (1:1.2), where mvs is the virial mass
of the subhalo and Mvh the virial mass of the host halo. Here log[np/np(Vmax )] is a measure
of the tidal stripping a subhalo has experienced. Specifically, np is the number of particles
that remain bound to the subhalo and np(Vmax ) is the typical number of bound particles in
a distinct halo with the same Vmax . Distinct halos exhibit a well defined correlation between
np and Vmax with small but measurable scatter, as expected. Therefore, log[np/np(Vmax )] is
a well defined substitute for log(mts /mvs ), where mts is tidal mass of the subhalo. The three
panels correspond to subhalos residing in galaxy-like host halos, 175 < Vh (km/s) < 380,
group-like host halos, 380 < Vh (km/s) < 500, and a combination of rich group-like host
halos, 500 < Vh (km/s) < 950 and cluster-like host halos, 950 < Vh (km/s). Later ‘group-
4.3 Results
105
like’ and ‘cluster-like’ host halos will be distinguished from halos that actually host groups
and clusters.
In Figures 4.5 and 4.5 it can be clearly seen that even relatively massive subhalos,
subhalos with Vs /Vh values that for distinct halos would corresponding to 3:1 and lower mass
ratios, are accreted and effectively tidally stripped by the larger, host, halo. The contours for
these relatively high Vmax subhalos lie above the densest contours for all subhalos, consistent
with these subhalos as a random subset of all halos. Relatively high Vmax subhalos are
subject to the same tidal stripping as all subhalos. The exception to this is the case of
subhalos with Vs /Vh values near 1:1 which show a spur at high log[np/np(Vmax )] within the
virial radius of the host. Subhalos with high Vs /Vh values are found primarily at large radii,
which is consistent with these subhalos as a sparsely sampled random set of all subhalos.
The number of subhalos increases with ro /rvh , as expected, and subhalos with high Vs /Vh
are therefore found preferentially at these radii. There is no significant difference in the
distribution of Vs /Vh at different ro /rvh .
Figure 4.5 shows the distribution of Vlmp /Vmmp for all major mergers resulting in galaxylike, group-like, and rich group and cluster-like halos. The Vmax for the most massive
of the two merger progenitors is designated as Vmmp , and Vlmp designates Vmax for the
least massive progenitor. The panels display both the distribution and the cumulative
distribution of Vlmp /Vmmp . Major mergers are dominated by the Vmax equivalent of mass
defined 2:1 and 3:1 mergers. As can be seen in Figure 4.5, only a few percent of mergers are
close to 1:1 mergers (Vlmp /Vmmp ≈ 0.95). The percentage decreases with the Vmax of the
merger remnant and increases with redshift. Comparing Figure 4.5 with Figures 4.5 and 4.5
demonstrates that most mergers occur in the regime where the least massive progenitor
(LMP) is first accreted as a subhalo, and tidally stripped, before it merges with the MMP,
its host halo. It is therefore assumed in what follows that the vast majority of mergers can
be cleanly separated into sub-host and sub-sub mergers.
Figure 4.5 presents both the forward looking, left panel, and backward looking, right
panel, merger rate for sub-host mergers resulting in group-like halos. As is discussed in the
106 Major Mergers Between Dark Matter Halos in the Millennium Simulation
introduction, dynamical friction is not effective at degrading the orbits of tidally stripped
subhalos. The time-scale for dynamical friction scales as Mvh /mvs , the less massive the
stripped subhalo, the longer the time-scale for dynamical friction. Tidal stripping is clearly
occurring in the simulation. Despite this the R+ for sub-host mergers can reach values as
high as a merger every one to a few Gyr, as can be seen in the left panel of Figure 4.5. This
suggests that the dynamics of these systems are not completely captured by considering a
combination tidal stripping and dynamical friction alone. In addition to interacting with
their host, subhalos also interact with each other, motivating us to examine whether the
presence of multiple subhalos affects the sub-host merger rate.
In the left panel of Figure 4.5, R+ is plotted for host halos with 345 < Vmax (km/s) <
455 and 1, 2, and greater than 1 subhalo, ns = 1 ,2, and > 1. The subhalo counts
include all subhalos with Vmax > 120km/s. To compute this rate, at each time-step the
number of major merger remnants in the next time-step with a most massive progenitor
with 345 < Vmax (km/s) < 455 and with one fewer subhalo than the ns of interest, is
divided by the number of halos in the current time-step with 345 < Vmax (km/s) < 455
and the desired ns. This assumes, for example, that all merger remnants with no subhalos
are the result of a merger between a host and its single subhalo. The time-step is small
enough that the occurrence of a sub-host merger and the accretion of a new subhalo should
be rare enough not to effect these results. When this does occur it will bias against the
result observed. The right panel of Figure 4.5 shows the number of merger remnants with
380 < Vmax (km/s) < 500 divided by the number of halos in this Vmax range in the same
timestep. This Vmax range roughly corresponds to the remnants of the mergers plotted in
the left-hand panel. In this panel the number of merger remnants with the ns of interest is
divided by the number of all halos with the same ns. While the left hand panel best captures
any dynamics, the right hand panel is easiest to compare with observations. Particularly
in groups, it is a backward looking rate, a rate calculated using mergers that have already
happened, that is easiest to observe. A forward looking merger rate necessarily relies on
pair counts.
4.3 Results
107
The specific forward looking merger rate in host halos that have two subhalos is approximately 10 times larger than the specific merger rate in host halos with only one subhalo
pre-merger. The distribution of Vs /Vh for host halos within this narrow range in Vh does
not depend on ns; a host halo with two subhalos is twice a likely to host a subhalo with
Vs > 0.7Vh and only twice as likely. If the existence of multiple subhalos did not affect the
dynamics of sub-host mergers, then host halos with two subhalos would experience major
mergers twice as often. Interactions between subhalos appear to be enhancing the rate of
sub-host mergers substantially. Including these interactions is necessary to correctly connect the sub-host merger rate to the accretion rate onto host halos. These interactions are
examined in the next section and a mechanism behind this observation is advanced.
The dynamics captured when considering R+ is reflected in the backwards looking
merger rate, shown in the right panel of Figure 4.5. The frequency of merger remnants
among host halos with ns > 0 is several times larger than the frequency among host halos
without a subhalo with Vmax > 120km/s. Subhalo interactions play an important role in determining the overall merger rate. At low redshifts, where the effects of subhalo completion
are less important, a second subhalo is present for 80-90% of the sub-host mergers in grouplike halos. These halos typically host 2-3 subhalos. Were subhalo interactions unimportant,
the over all merger rate for all group-like halos would be a similar fraction higher than the
merger rate for only group-like halos with one subhalo. At low redshifts, R− among grouplike halos is considerably more enhanced than this, demonstrating the universal importance
of subhalo interactions. The converse to this observation is that group-like merger remnants
preferentially host a subhalo. Differences between the two panels of Figure 4.5 are mainly
due to the ratios Nh (ns = 1)/Nh (ns = 0) and Nh (ns = 2)/Nh (ns = 1). Mergers of hosts
with lone subhalos are now normalized by the number of halos with no subhalos rather than
the number of halos with one subhalo. These ratios evolve with redshift, accounting for the
differences in the evolution of the merger rate seen between the two panels.
108 Major Mergers Between Dark Matter Halos in the Millennium Simulation
4.3.3
Mergers Between Subhalos
Observationally, galaxies can be separated into ‘isolated’ galaxies and members of groups
or clusters. Previous work has suggested that the specific merger rate may be enhanced
in groups, as groups combine high galaxy densities with moderate relative velocities; that
is, it is postulated that group members are more likely than isolated galaxies to undergo
major mergers. This hypothesis does not consider the effects of tidal stripping, which can
reduce the merger cross section considerably. Less massive halos must approach each other
at slower velocities and shorter distances in order to become bound and merge. Tidal
stripping therefore lowers the specific merger rate between subhalos. More sophisticated
treatments, for example Mamon (2000), consider tidal stripping, but use mass defined subhalo samples which are less likely to correspond to observed subhalos. In these treatments,
tidally stripped subhalos can drop below the lower mass limit and are no longer considered.
These treatments therefore share the conclusion that the specific merger rate is enhanced in
groups. The next step is to use a Vmax defined subhalo sample, include tidal stripping, and
determine if the specific merger rate is indeed higher in groups than for isolated galaxies.
Observational groups are constructed by identifying over-densities in the projected
galaxy density within a narrow redshift range. Observational group finding algorithms
seek to identify bound, virialized objects. The simulation equivalent to an observed group
is a host halo with its subhalos. Accordingly, groups in the simulation are defined as grouplike halos that host at least two subhalos with Vmax > 120km/s. Rich groups are defined
as rich group-like halos with at least two subhalos. The observational equivalent to these
simulation groups would be a catalog of groups containing at least three member galaxies, the ‘central’ galaxy and two satellites, which is constructed using a volume limited
galaxy catalog. In addition to the evolution of the physical number density of all halos,
the right panel of Figure 4.5 displays the physical number density of mergers of all group
‘members’, the host halo and its subhalos. Figure 4.5 also includes the physical number
density of mergers between two subhalos, or sub-sub mergers. Specifically, Figure 4.5 shows
the number of merger remnants that are subhalos of group hosts and of rich group hosts.
4.3 Results
109
In these environments, these are likely the remnants of true sub-sub mergers rather than
recently accreted merger remnants. Mergers within groups account for a small percentage
of all mergers. Sub-sub mergers make up roughly 10% of the mergers that occur in group
environments and a fraction of a percent of all mergers. The simulation groups host 2-3
subhalos on average, depending on the redshift, and the specific merger rate for subhalos
is therefore 20-30 times lower than the specific merger rate for the host halos of groups. In
host halos with Vh < 320km/s, it is somewhat more common for a subhalo to be a merger
remnant. This occurs most frequently for subhalos with Vs /Vh ≈ 1 which does not correspond to a true sub-sub merger, but rather a sub-host merger in disguise. Not only are
mergers between subhalos rare in an absolute sense, the specific merger rate of subhalos, or
the frequency of merger remnants among subhalos, is low. Major mergers are dominated
by sub-host mergers.
Together the sub-host merger rate and sub-sub merger rate determine the specific merger
rate among group members. In Figure 4.5, R− is plotted versus redshift for all halos,
‘group’ members, and members of ‘rich groups’. The merger rate includes all halos with
Vmax > 120km/s. ‘Groups’ are group-like host halos, 380 < Vh (km/s) < 500, that contain
two subhalos within their virial radius. Group members include the host halo and its
subhalos and are designated as ‘Group Members, r/rv < 1’ in Figure 4.5. An alternate
definition of groups requires the host halo to have two ‘subhalos’ within the dark matter
FOF group surrounding the central host halo. Under this definition group members include
the host halo and all subhalos within the dark matter FOF group; this is plotted as ‘Group
Members, all’. In a similar fashion, observational group finding algorithms vary in how
conservatively they define groups and group members. ‘Rich Group Members, r/rv < 1’
follows the first definition but includes systems with 500 < Vh (km/s) < 950. As can be seen
in Figure 4.5, the specific merger rate among group members is in fact lower than for all, or
for ‘isolated’, halos in a Vmax selected halo sample. It is also likely to be suppressed in groups
constructed from luminosity selected galaxy samples. This results combines the results of
both this and the previous sub-section. The specific merger rate of a distinct halo that hosts
110 Major Mergers Between Dark Matter Halos in the Millennium Simulation
multiple subhalos is higher than for a distinct halo that only hosts one subhalo. The subhost merger rate in group halos is enhanced relative to group-like halos. While mergers in
these systems are dominated by sub-host mergers, however, the halo count is dominated by
the subhalos, which have a greatly suppressed specific merger rate. In addition, the group
host halos are significantly more massive, and therefore less numerous, than the average
halo in the sample of halos with Vmax > 175km/s. Mergers in groups therefore make a
small contribution to the total merger rate, as can be seen in Figure 4.5
Previous analytical work has shown that for mass selected subhalos the sub-sub merger
rate decreases drastically with host halo mass while increasing with subhalo mass. For
example following Mamon (2000), R+ ∝ nG2 m2s /Vh3 . These are generic scalings that arise
for a range of specific analytical cross sections. It is interesting to check whether similar
scalings occur for Vmax selected subhalos in order to determine if one should expect to
observe these trends for real galaxies. Somewhat surprisingly, no such trend is observed for
the Vmax selected subhalos. The forward looking merger rate, R+ , for sub-sub mergers is
significantly suppressed relative to the sub-host merger rate and shows no dependence on
either Vh or Vs . This is taken up in §4.4.
4.3.4
Merger Rate vs Local Halo Density
Figure 4.5 shows R− versus local density for all halos with Vmax > 175km/s in the left
panel, and in the right panel displays R− versus Vmax for all halos in all environments.
Both are shown for z = 0, 1, and 2. The local density is the count of all halos, both distinct
and sub, with Vmax > 120km/s within a sphere of radius 2h−1 Mpc comoving. The halo
sample is complete to lower values of Vmax than the merger sample, simply because both
progenitor halos must be resolved in order to identify a merger. A lower Vmax is therefore
used to measure the environment than is used to measure R− . Using a larger halo sample
has the advantage of sampling the environment more accurately. The insets in Figure 4.5
are provided for orientation. The inset in the left panel displays the distribution of halo
environments at z = 1 for three Vmax ranges; galaxy-like (175 < Vmax (km/s) < 380), group-
4.3 Results
111
like (380 < Vmax (km/s) < 500), and rich group-like (500 < Vmax (km/s) < 950). The biasing
of high Vmax halos can clearly be seen in the inset. The distribution of the local comoving
halo number density evolves little between z = 2 and z = 0. What evolution there is of
the sense that halos become less biased with time. The inset in the right panel displays
the distribution of Vmax at z=1 for halos in residing in three local density bins, 0-4, 5-9,
and 10-14 halos within 2h−1 Mpc comoving. Between z = 2 and z = 0, these distributions
evolve slightly towards higher values of Vmax .
A correlation between the merger rate and either local environment or Vmax would be
straight forward to compare with observations. Unfortunately, as can be seen in Figure 4.5,
there is little dependence of the merger rate on environment or Vmax independently. The
merger rate does decline slightly above local densities of 15-20 at z = 0 and 1, and at
slightly lower densities at z = 2. As can be seen by referring to the insets, however, the
vast majority of halos reside in environments with lower densities, where the merger rate is
constant with local density.
Recent work has shown that codependent correlations between halo properties such as
concentration or age and both Vmax , or mass, and local environment do exist in large N-body
simulations (Wechsler et al., 2006; Croton et al., 2007; Gao and White, 2007). In Figure 4.5
R− is plotted versus the local environment for halos in three Vmax ranges for z = 0, 1, and
2. The Vmax ranges used are 175 < Vmax (km/s) < 280, 280 < Vmax (km/s) < 380, and
380 < Vmax (km/s) < 900. These are different than the Vmax ranges used previously and
have been selected to highlight the evolution of the trends observed in Figure 4.5. Halos in
the intermediate Vmax range evolve from moderately biased at z = 2 to relatively unbiased
at z = 0. The left panel includes all halos that are either the central halo of their dark
matter FOF group or are beyond the virial radius of their central halo. These halos are
selected because linear theory predicts that, when true subhalos are excluded, R− at fixed
Vmax should be independent of local environment. In contrast, in the simulation the halos
with the lowest values of Vmax show a slight trend of decreasing R− with local density, and
the halos with the highest values of Vmax show a slight trend of increasing R− with local
112 Major Mergers Between Dark Matter Halos in the Millennium Simulation
density. This is similar in sense and size to the trends seen in other halo properties.
Halos in the vicinity of a more massive halo, that is halos under the dynamical influence
of a nearby massive halo, may be subject to dynamics similar to those occurring within the
virial radius of a host halo. In order to gain insight into the dynamics beyond the virial
radius, the right panel of Figure 4.5 shows R− versus the local density for only halos that are
the central halo of their FOF group. By comparing the right and left panels of Figure 4.5
the contribution to the trends seen in the left panel of halos which dominate their local
dynamics and those that are under the influence of a more massive halo can be separated.
Only the two lower Vmax ranges are plotted in the right panel. This panel excludes the effect
of any influence of the central halo of a FOF group on other ‘distinct’ halos that are within
the FOF group but beyond the virial radius of the central halo. Conversely, it emphasizes
the interaction between central halos, which dominate their immediate environment, and
environment on intermediate scales. There are two points of note when comparing the two
panels of Figure 4.5. First, for the lowest Vmax range R− versus local density is significantly
flatter in the right panel. Second, in the right panel the trend for the mid Vmax range evolves
from a R− that increases with local density at z = 2 to an absence of a trend at z = 0.
This is in contrast to a relatively flat relation between R− and environment at all redshifts
for the same Vmax range in the left panel.
For Figures 4.5 and 4.5, it is appropriate to include both distinct halos and subhalos
in the local environment definition for a variety of reasons. Observationally, the galaxy
equivalent of distinct and isolated halos cannot be cleanly separated. Therefore, the trends
shown in Figure 4.5 and in the left panel of Figure 4.5 are the most useful for comparing to
observations. Including subhalos also makes sense theoretically. Ideally, one would plot the
trends shown in Figure 4.5 for dark matter over-density rather than halo number density.
This ideal is best approached by including subhalos as the local halo count is thereby
implicitly weighted by both the mass and proximity of nearby halos. A massive halo near
the edge of the 2h−1 Mpc sphere contributes to the local density by as many of it’s subhalos
as lie within the sphere. As was observed in Section 4.3.2, however, the sub-host merger
4.4 Discussion
113
rate increases with the number of subhalos so it is prudent to examine how this correlation
affects the results presented here.
The correlation between the number of subhalos and the merger rate may bias the
results in the sense that increasing the number of subhalos will increase both the merger
rate and the local halo count. This biases against observing the trend observed for halos in
the low Vmax range, but has the same sense as the trend observed for the high Vmax range.
The number of subhalos typical for the majority of halos in even the high Vmax range is
smaller than the range of environments plotted. This is not surprising given that the virial
radius of these halos is smaller than 2h−1 Mpc. The question of import must therefore be
whether the number of subhalos and the local halo count, excluding the hosts own subhalos,
are correlated. If they are not correlated then at high local densities, where non-subhalos
dominate the halo counts, the correlation between subhalo count and the merger rate may
actually work to smooth any underlying trend between local environment and the merger
rate by introducing a large scatter into the merger rate. On the other hand, if the number
of subhalos rises with the local halo count, again excluding the subhalos themselves, then
this correlation may actually drive the observed trend between the merger rate and the
local environment. It is indeed the case that the number of subhalos in a host increases
with the local halo density, which is consistent with the results of Wechsler et al. (2006).
This is another hint at the physical mechanism driving the correlations between the merger
rate and the environment on intermediate scales.
4.4
4.4.1
Discussion
Evolution of the Merger Rate
There are two main points to be drawn from the evolution of the halo major merger rate, as
presented in Figure 4.5. First, the evolution of both R− and R+ is significantly flatter than
that of nh . While nh ∝ (1+z)3.13±0.03 , R− ∝ (1+z)2.2±0.05 and R+ ∝ (1+z)1.82±0.01 . Mergers involving distinct halos, that is sub-host mergers, are not mergers between halos with
114 Major Mergers Between Dark Matter Halos in the Millennium Simulation
random motions; the merger rate of distinct halos cannot be modeled by Rm = nh hσm vi.
Rather major mergers occur at a rate determined primarily by the typical accretion rate
of halos. This rate does increase with redshift, but it is not constrained to evolve like nh
or a−3 . It is the growth of structure, set by cosmology and the initial matter distribution,
that sets the merger rate.
The merger rate is primarily driven by the backward looking accretion rate. The evolution of this rate is not mass independent. There is a characteristic halo mass, which
evolves with redshift, that corresponds to a typical forming halo. The typical backward
looking accretion rate is highest near this characteristic mass. Therefore, the evolution of
the merger rate is likely to depend on the Vmax limit used to define the halo catalog. Similarly, different rates of evolution may be measured for observational catalogs with different
selection criteria. For the halo sample used here, this effect is not large. As can be seen in
Figure 4.5, at fixed redshift R− is relatively independent of Vmax . The lower Vmax limit used
in this sample corresponds to moderately bright galaxies, slightly dimmer than the Milky
Way. These results predict that the specific merger rate for galaxies of moderate to bright
luminosity evolves with a shallower slope than their number density. In order to directly
compare the evolution of an observed merger rate to a simulated merger rate, that is to
directly compare the measured slopes, the luminosity range of the observed sample must
be matched to the Vmax range of the simulated halos. Any evolution in the relationship
between Vmax and luminosity will complicate such a comparison.
The second point is that care must be taken when comparing the merger rate measured
using differing definitions, either in simulations or with observations. As can be seen in the
left panel of Figure 4.5, the slope of the evolution of the merger rate depends on whether R−
or R+ , corresponding to morphologically identified mergers or to close pair counts, is used.
The differences between R− and R+ should be kept in mind when comparing the evolution of
the merger rate derived using different observational definitions. In a galaxy sample, defined
by a lower luminosity limit, the merger rate as measured using close pair counts and using
morphologically identified mergers will evolve differently within that same sample. When
4.4 Discussion
115
the effects of changing the luminosity limit of a sample are included, comparing the merger
rate determined using different techniques across two galaxy samples should be undertaken
with caution.
The difference between the slopes of the two measures of the merger rate, R− and R+ ,
is to first order set by the evolution of the mass, or Vmax , function. As seen in Figure 4.5,
at fixed redshift the merger rate is relatively independent of Vmax . In addition, the merger
rate and number of halos change only slightly between consecutive simulation outputs. It
follows that
Nh (Vmmp )
R+ (t − dt) Nh (Vmmp , t − dt)
R−
≈
≈
R+
R+ (t)
Nh (Vrem , t)
Nh (Vrem )
where Vmmp is the typical Vmax for the most massive progenitor of merger remnants with
Vrem . This ratio is always greater than one, Vmmp < Vrem . It increases with redshift as the
Vmax function shifts towards lower values of Vmax and the local slope of the Vmax function
becomes steeper. This is consistent with the relationship between R− and R+ observed in
Figure 4.5.
A similar effect likely occurs when comparing the evolution of R− for halo samples with
different Vmax limits, particularly at high Vmax . The major merger rate depends on the
accretion rate and the Vs /Vh distribution of halos that are being accreted; both of which
evolve. For halos near the break in the Vmax function, Vs /Vh evolves towards higher values
as the local slope of the Vmax function near the host Vmax flattens. In contrast to halos well
below the break, where the slope evolves little, the slope of the evolution of the merger rate
for the high Vmax halos may be flatter. This is yet another way in which galaxy selection
can affect the observed evolution of the merger rate.
Gottlober et al. (2001) consider a third measure of the evolution of the merger rate in
which only halos that belong to a MMP chain of a halo identified at z = 0 are considered.
They find that this rate evolves as (1 + z)∼3 , which is steeper than our result for all major
mergers. The MMP chain is the trunk of a merger tree, consisting at each redshift of the
most massive progenitor of the z = 0 halo. A halo that is part of a branch of a tree that
becomes the least massive progenitor (LMP) of any merger is therefore not included in this
116 Major Mergers Between Dark Matter Halos in the Millennium Simulation
merger rate. As it captures the merger history of halos that exist at z = 0, this merger
rate is intrinsically interesting. However, it should not be compared directly to an observed
merger rate. Naively, one would expect the merger rate of the MMP chain to evolve slower
than R− or R+ , contrary to what is found. As z increases, the fraction of halos that are fated
to become a LMP of a future merger must increase. If current merger rate is unrelated to
future survival, then the number of mergers that occur, but result in a halo that is not part
of an MMP chain, should increase with redshift resulting in a shallower measured merger
rate. That Gottlober et al. (2001) instead measure a steeper rate suggests that halos which
undergo major mergers are more likely to survive as part of a MMP chain. This dependence
is only suggested by these results. While the merger rate measured here includes halos with
the same Vmax at all redshifts, the average Vmax of the progenitor halos in a MMP chain
decreases with redshift. This comparison emphasizes the importance of both measuring the
merger rate as Gottlober et al. (2001) have in order to compare morphologies at z = 0 to
merger history and also measuring the merger rate as is done here in order to compare to
the observed evolution of the merger rate. It is not correct either to integrate the merger
rate observed here in order to compare with morphologies at z = 0 or to compare observed
merger rates to the evolution measured by Gottlober et al. (2001).
Previous observations of the evolution of the major merger rate, using both pair counts
and morphologies, have found a wide range of slopes when the evolution is characterized
as Rm ∝ (1 + z)m . Most are flatter than m = 3, with some that are actually consistent
with no evolution. For both methods, previous results can be roughly split between groups
that find m = 2 − 4 (Le Fèvre et al., 2000; Patton et al., 2002; Conselice et al., 2003;
Cassata et al., 2005; Kampczyk et al., 2007) and m < 1 (Carlberg et al., 2000; Lin et al.,
2004; Lotz et al., 2008). There are many differences between these studies including sample
selection, merger selection, pair selection, and whether the evolution is measured within
a single sample or by comparing two or more samples. In general, stricter dynamical
requirements on pair counts result in steeper evolution. Berrier et al. (2006) find that
pair counts in simulations show little evolution because the decline in the merger rate is
4.4 Discussion
117
counterbalanced by the build up of groups and clusters. Measurements at high redshift rely
on morphologically identified mergers which can be complicated by resolution, loss of low
surface brightness features, and by observing galaxies at non-ideal rest frame wavelengths.
Automated morphological measurements may also have difficultly separating major mergers
from galaxies with asymmetric star formation. While individual groups do attempt to
quantify and correct for these effects, the disparity between different measurements indicates
there is more work to be done. When comparing the current work to observations it should
be kept in mind that we explicitly consider not mergers between distinct halos, but between
either two-sub halos or between a sub-halo and the core of its host halo. This rate is
therefore meant to be compared directly to the galaxy merger rate. We predict that the
specific galaxy merger rate evolves with m ≈ 2, shallower than the evolution of the number
density of galaxies.
4.4.2
Groups and Clusters
The merger rate in groups and clusters reflects an interplay between tidal stripping, dynamical friction, and unbound interactions between subhalos. In this section the results
presented above will be discussed in terms of these interactions. Two kinds of mergers are
occurring in groups and clusters, sub-host mergers and sub-sub mergers. This separation
between merger types is motivated by our results. In the MS, relatively massive subhalos
and less massive subhalos are tidally stripped of similar mass fractions. The least massive
progenitor of a sub-host major merger therefore spends a considerable length of time as a
subhalo before merging with the central host halo. The exception to this is 1:1 mergers. As
can be seen in Figure 4.5, these mergers are quite rare.
A simple model of tidal stripping, which assumes that subhalo orbits are nearly circular
and that the host-halo and the newly accreted subhalos share a self similar density profile, for
example an NFW profile, predicts that the fractional mass loss of a subhalo is determined by
ro /rvh alone. Specifically, rts /rvs = ro /rvh or mts /mvs = Mh (ro )/Mvh , where the subscript
vh designates the viral value of the host, vs the pre-accretion virial value of the subhalo,
118 Major Mergers Between Dark Matter Halos in the Millennium Simulation
ts tidal value for the subhalo, and ro the orbital radius of the subhalo within the host. A
linear, monotonic relationship between the fractional mass loss and ro /rvh does appear in
Figure 4.5. Scatter about this relationship can be introduced by scatter in the relationship
between Vmax and np, deviations from circular orbits, and scatter in the concentrations of the
host-halo and subhalo density profiles. Subhalos that lie in the bulge to the lower left of the
contour plot are likely subhalos on radial orbits that have passed through their peri-center
at least once. Mamon (2000) presents an analytical estimate of the tidal mass of a subhalo
with an NFW profile orbiting in an NFW host in which the correlation between mass and
concentration and the typical orbits of subhalos in NFW hosts have been considered (their
equation 8). In Figure 4.5 the diagonal lines in the left and right panels correspond to this
analytical estimate in groups and clusters respectively. The left line is simply the estimate
given in Mamon (2000), and the right line is this estimate shifted to higher log(ro /rvh ) by
0.3. As long as some subhalos have made one peri-centric passage, the average peri-centric
radius, rp /rvh of subhalos found at each ro /rvh is lower than ro /rvh itself. The contours
presented in Figures 4.5 and 4.5 are clearly consistent with representing tidal stripping of
subhalos in the MS, that is the MS appears to be correctly capturing tidal stripping of
subhalos.
Figures 4.5 and 4.5 demonstrate that even relatively massive subhalos, subhalos that
are potential major merger partners, are accreted and then tidally stripped. Before a
major merger can occur, these subhalos must lose their orbital energy and, perhaps more
importantly, angular momentum. These tidally stripped subhalos retain only 10-30% or
their original mass, resulting in long dynamical friction time-scales. This effect is made
particularly strong by the continual tidal stripping of the subhalo should it manage to spiral
in towards the center of its host. Boylan-Kilchin et al. (2008) consider exactly this problem
by running high resolution simulations of a single halo as it is accreted by and eventually
merges with a larger host halo. They consider a variety of mvs /Mvh and orbital parameters
and find that merger time-scales are considerably longer than models of dynamical friction
that neglect tidal stripping would predict. Their results, including both their simulations
4.4 Discussion
119
and a fitting formula based on the simulations, indicate that the merger time-scale for major
mergers likely ranges from ≈ 3.5 to 7 Gyr, with longer time-scales for lower mvs /Mvh and
higher circularities. Major mergers are dominated by mergers with initial mass ratios closer
to 3:1 than 1:1, and are therefore expected to have the extremely long merger time-scales
closer to the long end of this range. The forward looking merger rate for hosts with a
single subhalo, shown in the right panel of Figure 4.5, is roughly consistent with the results
of Boylan-Kilchin et al. (2008). Assuming that all host halos with a single subhalo at z = 1
whose single subhalo has Vs /Vh > 0.7 experience a merger at z = 0 correctly predicts the
merger rate at z = 0. This is of course a very rough consistency check, but it indicates
that the MS is correctly capturing sub-host mergers in the case of a host with a single
subhalo. The relatively high rate of sub-host mergers when all hosts halos are considered
indicates that further dynamics is at work, which is confirmed by the studying the effect of
the number of subhalos in a host on the sub-host merger rate.
Mergers, or bound collisions, between subhalos are rare, likely due to the small merger
cross section of tidally stripped halos. Unbound collisions between subhalos in groups and
clusters are likely quite common. Unbound collisions between subhalos with opposing angular momentum can reduce the angular momentum of the individual subhalos and drive
them towards the center. Loosing angular momentum in this fashion could potentially be
much more effective than dynamical friction. Assuming elastic unbound collisions, angular
momentum would be effectively scattered between subhalos, but not effectively cancelled.
An analytical treatment of this problem that assumes inelastic collisions finds no enhancement for the sub-host merger rate from sub-sub interactions (Peñarrubia and Benson, 2005).
In reality, however, sub-sub interactions are extremely inelastic, as indicated by studying
the mass loss induced by these interactions (Knebe et al., 2006). In the case of inelastic
collisions, subhalos with opposing angular momenta can effectively dump both orbital energy and angular momentum when they experience an unbound collision. Given inelastic
collisions between subhalos, the presence of another, or several, other subhalos of comparable mass or Vmax may considerably shorten the time it takes a subhalo to merge with its
120 Major Mergers Between Dark Matter Halos in the Millennium Simulation
host. This is consistent with the results presented above; hosts with 2 subhalos are ≈ 10
times more likely to experience a major merger than hosts with 1 subhalo. That this is
occurring requires further study. It is, however, both physically plausible and consistent
with the results.
While mergers between two subhalos are rare, they do occur. In contrast to the major
merger rate between subhalos in a mass selected subhalo sample, which previous analytical
work has found declines with increasing Mvh or decreasing mvs , R+ for Vs selected subhalos
does not depend on either Vh or Vs . The absence of a trend with Vh is perhaps the most
startling. In cluster-like host halos the relative velocities of subhalos are higher than in
group-like host halos. Therefore, one would expect the specific merger rate to decline. The
key to this puzzle may be the relatively large masses of the subhalos studied. The analytic
results assume that the number of subhalos scales with Mvh and that the number density
of subhalos within the host is therefore constant. In this case the relative velocities of
the subhalos is the only relevant factor. This assumption may not be appropriate for the
subhalos studied here. While the average number of subhalos with Vmax > 175km/s does
increase approximately as Vh3 , these host halos occupy a regime in which the average number
of these somewhat massive subhalos ranges from < 1 to a few. A sub-sub major merger can
only occur if two relatively large subhalos exist within the same host halo. For group-like
host halos there are multiple hosts that host only one such subhalo for every host with two
or more such halos. The number of halos that host multiple large subhalos increases with
Vh , off setting the increase in relative velocities between subhalos. A simulation with a finer
mass resolution, which could be used to detect subhalo merger remnants down to lower
Vmax may reveal a correlation between R+ and Vh .
It is less surprising that no trend is observed between R+ for sub-sub mergers and Vs .
There are two types of mixing involved in transitioning from a mass-based major merger rate
to a Vmax based rate which can significantly weaken this dependence. First, as can be seen in
Figures 4.5 and 4.5, after tidal stripping is considered, subhalos with the same Vs correspond
to a wide range of bound masses. While sub-sub mergers may still be more likely for higher
4.4 Discussion
121
post-tidal stripping subhalo masses, and subhalo mass and Vs are loosely correlated, there
is significant scatter in the mass-Vs relation of both subhalos and therefore of major merger
partners. Second, major mergers are defined differently in the two scenarios. For Vs defined
subhalos, the case of two merging subhalos with similar Vs is marked as a major merger
even if one subhalo is newly arrived and the other significantly tidally stripped. Conversely,
a merger of two subhalos with similar masses and different degrees of stripping will not be
considered a major merger if the Vs are significantly different. This is the reverse of the
mass based major merger definition. The previously predicted dependence of the sub-sub
merger rate on subhalo mass may simply be washed out when considering a Vs defined
specific merger rate. As a final consideration, the relatively low number of mergers between
subhalos results in poor statistics, and a weak potential dependence on Vh or Vs would
therefore not be observed. This leaves open the possibility that a weak trend observed in a
large survey would not be inconsistent with the simulation results.
Observed galaxies can be separated into ‘isolated’ galaxies and group and cluster members. While previous results have suggested that the specific merger rate may be enhanced
in groups, the results presented here find the opposite effect. In the simulation equivalent of
groups the specific merger rate is suppressed relative to that for distinct halos. This result
reflects a combination of sub-host and sub-sub mergers. While the sub-host merger rate
is enhanced in ‘groups’, defined as halos with 380 < V max(km/s) < 500 hosting multiple
subhalos, the extremely low specific merger rate of the subhalo group members more than
offsets this gain. In more massive systems with multiple subhalos the specific merger rate
of ‘group members’ does decline. This reflects only the increasing numeric dominance of
subhalos over the host halos. When attempting to compare an observed specific merger
rate in groups with the rate in the simulations it should be kept in mind that the predicted
rate may depend quite heavily on the construction of the group catalog. As can be seen by
comparing ‘Group Members, all’, and ‘Group Members, r/rv < 1’ in Figure 4.5, the specific
merger rate differs between a conservatively defined and a generously defined set of group
members. Generously defined groups, groups that include ‘subhalos’ beyond the true virial
122 Major Mergers Between Dark Matter Halos in the Millennium Simulation
radius of the host display a lower specific merger rate.
The merger rate in groups is not enhanced relative to the field, as has sometimes been
predicted. There are however, several predicted trends that can be confronted with observations. First, we find that mergers are dominated by sub-host mergers and that the
specific merger rate of sub-host mergers is 20-30 times larger than for sub-sub mergers. Second, there are observational tests that can be made arising from the strong enhancement
of the merger rate in halos hosting multiple sub-halos. The simulation results predict that
merger remnants should be more likely than an average galaxy to have a fainter near-by
companion, which is likely bound to the merger remnant. Similarly, when observationally
identified mergers are cross correlated with a fainter galaxy sample, mergers are predicted
to show a significantly enhanced correlation on small scales, that is scales corresponding to
the one-halo term of the correlation function.
McIntosh et al. (2007) study the major merger rate in groups by identifying morphologically disturbed close pairs of galaxies. They find that a slight majority of mergers involve
the central galaxy of a group and that the specific merger rate for central galaxies is ≈ 3%
and for satellite galaxies is less than 1%. Their observations agree qualitatively, but not
quantitatively, with the simulation results presented here. When translating the simulation
results into observational predictions, any scatter between the Vmax of the host halo and
either luminosity or stellar mass, as well as any uncertainties in the observations, will tend
to result in observational results that are less extreme than the simulation results. That is,
while the difference between the specific merger rate of host halos and of sub-halos may differ by a factor or 20-30 in the simulations, a factor of 10 or less difference in the observations
may be consistent. McIntosh et al. (2007) also find that the sub-sub specific merger rate
declines with the mass of the host group. Their groups have an average of ≈ 5 members and
all have multiple sub-halos. They therefore avoid the contribution of group-like halos with
only one sub-halo and appear to capture the effect of increasing relative sub-halo velocities
with group mass.
Previous theoretical and observational results have indicated that starbursts and AGN
4.4 Discussion
123
show an excess of companion galaxies on small scales. Thacker et al. (2006) study the AGNgalaxy cross correlation in a SPH simulation in which AGN are fueled by major mergers
and find an excess at small scales compared to the galaxy-galaxy correlation. Goto (2005)
cross correlate 266 E+A galaxies in the SDSS with the SDSS imaging catalog and find that
E+A galaxies have an excess of companions on scales < 150kpc when compared to normal
galaxies. Serber et al. (2006) preform a similar analysis for AGN and find an excess of
companions on similar scales. Previous studies of companion frequencies have suggested
that tidal interactions with the companion trigger activity. We offer an alternative scenario
in which activity was triggered by a major merger which was facilitated by the companion.
The companion galaxies must also have lost considerable angular momentum, hence their
small distances from the active galaxy.
4.4.3
Major Merger Rate and the Assembly Bias
The major merger rate shows no independent correlation with either local environment
or Vmax for the Vmax range considered. At fixed Vmax however, R− does correlate with
the local environment. The sense and size of these correlations are similar to so called
‘assembly biases’ seen before for related halo properties such as concentration and age. The
results presented in §4.3.4 provide insight into the potential physical mechanism driving
this correlation.
Despite the codependent correlations between the merger rate, Vmax and local environment, no independent correlation on local environment or Vmax is observed. The absence of
a correlation between the merger rate and local environment for a mass selected halo sample
was observed previously by Kauffmann and Haehnelt (2000). A slight correlation between
the merger rate and the halo mass has been observed previously (Fakhouri and Ma, 2008).
The absence of this trend in the results presented here may be due to the scatter between
halo mass and Vmax . Lower Vmax halos are found on average in lower density environments.
The corollary is also true; the average Vmax increases with local over density. These trends
appear to conspire to hide the underlying codependent correlations when the independent
124 Major Mergers Between Dark Matter Halos in the Millennium Simulation
correlations are considered. If the merger rate for low Vmax decreased and for high Vmax
increased with increasing local halo density, without also changing the relative weights of
these populations, then the merger rate would decline with environment. There is no theoretical reason that the major merger rate should not depend on environment, and it appears
to be a coincidence that no correlation is observed in this sample. It should therefore be
kept in mind that a correlation may be observed for a halo sample with a significantly lower
Vmax limit. Indeed, one may predict that the merger rate would decline with a similarly
defined local environment for such a sample.
As seen in Figure 4.5, the backward looking merger rate, R− , correlates with the local
halo density, defined as the count of halos with Vmax > 120km/s within 2h−1 Mpc, when
halos in a limited Vmax range are considered. For group-like halos the major merger rate
increases with local density while for galaxy-like halos it decreases. In dense environments
all halos have a greater number of potential major merger partners. This is particularly true
for more massive halos, or halos with higher values of Vmax , because the typical halo mass
increases with the local halo number density. The relative velocities in dense environments
are also higher, suppressing the accretion rate and hence the merger rate. This has a smaller
effect on massive halos. One might therefore expect the observed trends to appear in the
case of a forward looking merger rate, or R+ . Indeed, massive halos are more likely to
form in dense environments; they are biased. Conversely, in linear theory the formation
history of a halo of fixed mass does not depend on environment. Initial over-densities on
small scales, which set the merger history of a halo, are uncorrelated with the initial overdensities on larger scales, which set the environment of a halo (White, 1996). Therefore,
linear theory predicts that a backward looking merger rate, R− , at fixed halo mass should
be independent of the local over-density. Assuming that the relationship between mass and
Vmax is independent of environment, which linear theory would also predict, then R− at
fixed Vmax should be independent of the local over-density. This is not what is observed in
the Millennium Simulation.
Similar deviations from the expectations of linear theory have recently been observed in
4.4 Discussion
125
simulations for other halo properties such as halo formation time, concentration, number
of subhalos, subhalo mass function, and halo angular momentum (Wechsler et al., 2006;
Croton et al., 2007; Gao and White, 2007). The correlation between the likelihood of
having experienced a recent major merger and the local environment may be yet another
facet of what has been termed the ‘assembly bias’. These correlations are all closely related
through halo’s assembly histories. Younger halos are also less concentrated (Navarro et al.,
1997; Wechsler et al., 2002; Zhao et al., 2003), and a halo that has recently experienced a
major merger is younger. Halos that host a higher than average number of subhalos are
more likely to experience, and to have experienced, a major merger. The results of Wechsler
et al. (2006) indicate that at the Vmax values characteristic of groups, host halos with greater
numbers of subhalos are more clustered. In the Millennium Simulation, at group-like values
of Vmax , halos in dense environments host more subhalos than is typical for their Vmax . It
seems likely that this trend drives the trend in R− for group-like halos. Previous studies
have all found that the size of the assembly bias is small, and the results here are consistent
with this. For the range of local environments in which the majority of halos find themselves,
the merger rate is relatively independent of environment. At z = 0, the merger rate per
halo for galaxy-like halos decreases by only 20% up to local halo counts of 10.
Linear theory is based on the spherical collapse model, in which the radial evolution of
shells of dark matter are determined by the mass contained within the shell. Tidal forces are
neglected, as are the orbits and interactions of previously virialized halos within these shells.
The role of tidal forces has been previously acknowledged; the halo mass function can be
predicted correctly only if tidal forces are included in an average sense (Sheth and Tormen,
1999). Tidal forces, halo orbits, and unbound interactions between halos must contribute to
the non-linear trends observed in larger simulations. These interactions are similar to those
occurring within virialized halos, but extend beyond the virial radius into the surrounding
regions. Extending the analysis of the dynamics within halos beyond the virial radius may
therefore provide some insight into the assembly biases. Fortuitously, the peculiarities of
the Millennium Simulation halo catalog facilitate such an analysis. Simulation particles
126 Major Mergers Between Dark Matter Halos in the Millennium Simulation
are first grouped into FOF groups before halos are found, and information about these
FOF groups is retained in the final catalog. All non-central halos within a FOF group
are flagged as a subhalo of the central halo. Central halos are the dominant halo in the
immediate vicinity; there is no more massive halo within several of the central halo’s own
virial radii, rvc . Conversely, non-central, non-sub halos do have a more massive halo in their
immediate vicinity; they are within a few rvc of a more massive, central, halo.
Tidal stripping within FOF groups is occurring beyond the virial radii of the FOF central
halos. Figures 4.5 and 4.5 include all FOF non-central halos. The relationship between
tidal mass and ro /rvh extends beyond the viral radius of the FOF central halo to radii of a
few rvc . Note that this is actually predicted by the analytical estimate of Mamon (2000).
When a halo gets within a few viral radii of a more massive halo, tidal forces can both
suppress accretion onto and eventually strip the halo. The tidal forces in the outskirts are
weak, but likely suppress the major merger rate of these halos by decreasing accretion onto
and thereby the typical number of subhalos within them.
Halos which are destined to be accreted by a larger halo have in some sense always been
bound to the larger halo. That is, their orbital energy with respect to the center of mass of
the host to be has always been negative. In contrast to the spherical collapse model, however,
their orbital angular momentum is not zero. Their orbit cannot be simplified as solely radial
expansion followed by turn around and collapse. Major mergers in N-body simulations are
observed to have a range of initial orbital parameters, with the distribution of orbital
circularities, j/jc (E), peaking near 0.5 (Khochfar and Burkert, 2006). Any mechanism
which causes the subhalo-to-be to loose orbital energy and angular momentum before it
is accreted will cause it to be prematurely accreted by the central host-to-be. The same
mechanisms for energy and angular moment loss are available to these halos as are to
subhalos that have been accreted by the host; dynamical friction and unbound interactions
with other halos in the vicinity. While these halos are only mildly tidally stripped, the dark
matter density beyond the virial radius is much lower than within it and dynamical friction
is not effective. On the other hand, unbound collisions between halos in the outskirts of a
4.4 Discussion
127
forming host may be an effective means of loosing orbital energy and angular momentum.
In the outskirts of a forming host, subhalos-to-be retain most of their mass and have slower
relative velocities, though lower halo number densities are also found beyond rvc . If the
accretion rates of galaxy-like halos into group-like and rich-group like halos are enhanced
by unbound interactions between halos in the vicinity of group-like halos, then group-like
halos in denser environments will experience higher accretion rates, have higher than average
subhalo counts, and higher major merger rates. This of course assumes that the group-like
halo is not itself in the vicinity of a cluster-like halo, in which case accretion onto the
group-like halo will be terminated.
The suppression of the accretion rate in the vicinity of a more massive halo and the
enhancement of the accretion rate for central galaxies in dense environments by unbound
collisions between surrounding halos may work together to set the major merger rate assembly bias. This hypothesis can be tested indirectly by using the dark matter FOF groups.
In the case that tides play a vital role in decreasing the merger rate for galaxy-like halos
in dense environments, the trend of decreasing R− with environment for these halos will
not be observed when only central galaxy-like halos are considered. This is exactly what is
observed in the right panel of Figure 4.5 for z = 0 and 1. At z = 2 a trend remains, but it
is considerably reduced. A similar comparison can be made to test the hypothesis that unbound collisions between non-central halos drives the increase in R− with environment for
group-like halos. This comparison is best made with halos in the intermediate Vmax range
in Figure 4.5, that is halos with 280 < Vmax (km/s) < 380. The highest Vmax range consists
almost entirely of central halos, and no useful comparison can be made for this range. For
the lowest Vmax range, 2h− 1Mpc measures the local environment on scales far larger than
a few virial radii. At z = 2, for high local densities the merger rate for the intermediate
Vmax halos transitions from decreasing, for all halos, to increasing, for central halos only.
While poor statistics prevent this trend from being measured with clear significance, the
difference between the panels is suggestive. When considering all halos with intermediate
Vmax , halos in the densest environments tend to be in the vicinity of a more massive halo.
128 Major Mergers Between Dark Matter Halos in the Millennium Simulation
As the local density increases, the likelihood of the halo being a central halo decreases and
the probable proximity and mass of the nearby central halo increases, resulting in stronger
tidal forces. For central halos, as the local density increases, the number of halos between
one and a few rv increases, possibly resulting in increased unbound collisions between these
halos and an enhanced accretion rate. At these local halo densities, the halo counts for
halos in the intermediate Vmax range are dominated by halos beyond rv .
Observing the trends between R− and local halo density may be possible either directly
or through clustering. It can be predicted that, when cross correlated with a fainter galaxy
sample, galaxy populations corresponding to the brightest merger remnants will show enhanced clustering on scales larger than the typical virial radius of the remnants. Cross
correlating with a fainter galaxy sample allows the environment on scales of a few virial
radii to be adequately sampled. If this correlation is driven by unbound halo interactions
beyond the virial radius of the halo, then the enhancement in the correlation function will
decline beyond scales of a few virial radii. The trend will persist somewhat due to correlations between environment measured on different scales.
Comparing observations to instances of assembly biases identified previously is difficult both because the correlation is small and because the relevant halo properties may
only loosely correlate with galaxy properties. One exception, the number of subhalos
a group hosts, has its own difficulties due to constraints imposed by estimating group
masses (Berlind et al., 2006b). The major merger rate may therefore be an interesting
way to directly probe the assembly biases as a class. Further, the picture presented here
may contribute to a physical understanding of any assembly type bias that is observed.
For example, Berlind et al. (2006b) find that groups whose central galaxies are bluer than
average, or rather less red than average, are biased on large scales compared to average
groups. While bluer galaxies are typically considered ‘younger’, blue colors suggest the
central galaxy has recently received either gas or young stars. This is physically distinct
from an enhanced dark matter accretion rate onto the group’s host halo. In the present
work, we find not only that groups with an enhanced merger rate are biased, but that in
4.4 Discussion
129
these groups the average time between accretion and the sub-host merger is shorter than
average. Therefore, sub-host mergers, both major and likely minor, are more likely to bring
young stars, and possibly gas, to the central galaxy. This may be the physical mechanism
behind the observation of Berlind et al. (2006b).
When measuring these trends, the scale on which the environment is sampled plays a
role in determining what is observed. This is discussed here in halo terms, but it could be
converted to galaxy terms by converting Vmax to luminosity. In order to observe an increase
in R− with environment, it is best to measure the environment on scales corresponding to
a few rv for the Vmax of interest. Measuring environment on larger scales will introduce
scatter into the relationship between the measured environment and the density of noncentral halos that are under the dynamical influence of the halo sample of interest. In
order to observe a decrease in R− , it is best to measure the environment on scales of
a few times the virial radius of the largest halo typically found in dense environments.
Using a smaller radius will increase the scatter between the measured environment and
the local tidal forces. Using a scale of 2h−1 Mpc is optimized for observing the increase in
R− for rich group-like halos and a decrease in R− for galaxy-like halos. It is a somewhat
awkward scale to use for the intermediate Vmax range. In the left panel of Figure 4.5, R−
for these halos first increases with environment up to halo counts around 5 or 6 and then
decreases with environment. For these halos and the particular environment measurement
used, halos with environment counts below ≈ 5 are central halos with a lower than average
number of companions. For these cases increasing environment corresponds to an increasing
number density of non-central FOF members and an increased R− . Note that this initial
increase is very similar to the initial increase observed in the right panel when only central
halos are included. At higher environments increasing halo counts reflects a more complex
combination of decreasing fraction of central halos and increasing environment both within
and beyond the FOF group. Even when central halos are isolated, the measurement of
environment on scales greater than a few rv for the intermediate Vmax halos introduces
noise between the measured environment and the number density of non-central halos. If
130 Major Mergers Between Dark Matter Halos in the Millennium Simulation
environment were measured using a halo sample with a lower Vmax cut off and on a smaller
scale, an initial increase in R− with environment would likely also be observed for halos in
the lower Vmax range, 175 < Vmax (km/s) < 280, and an increase with environment might
be observed across a larger range of environments once central halos were isolated.
The evolution of the correlations between R− , Vmax , and local environment can also
be discussed in terms of the picture presented above. As structure evolves, halos with
increasing values of Vmax form. When the first halos with a given Vmax form they are
universally dynamically dominant halos, akin to the central halos of the dark matter FOF
groups. In all environments R− increases with local density, which measures or at least
correlates with the density of halos which are under the dynamical influence of these new
halos. At any redshift, the youngest, most massive halos will display R− that increases
with local environment in all environments. As the growth of structure advances halos with
increasingly higher values of Vmax form, initially residing in dense environments. When
considering smaller halos, as the local environment increases the chances of being in the
vicinity of one of these newly formed large halos increases. While in less dense environments
R− for the smaller halos may still increase with environment, in rich environments it will
decline. As time advances and these larger halos become less biased, the smaller halos will
find themselves in the vicinity of more massive halos in less extreme environments. Once
the more massive halos become unbiased, in all local environments increasing environment
corresponds to an increase in the mass and proximity of the nearest massive halo, and
hence to an increase in the tidal forces. At this stage R− decreases with environment in
all environments. As halos with increasing values of Vmax are formed this entire sequence
marches towards higher Vmax and larger scales.
This discussion of the evolution of these effects is meant to consider the issues involved
and suggest an aspect of this picture that it may be rewarding to explore further. It is
possible that studying evolution of these trends would be cleaner using the correlation
function as changes in scale with Vmax and redshift can be more naturally accommodated.
One general characteristic of this scheme is that the Vmax at which the dependence of R−
4.4 Discussion
131
on environment transitions from increasing to decreasing with increasing density evolves
towards higher values. This transition must be typical of any trend that is driven by the
proximity of a more massive halo, and is typical of ‘assembly’ biases.
4.4.4
Subhalo Completion
In order to examine the ways in which our results may be impacted by the artificial disruption of subhalos, we now apply the criteria of Klypin et al. (1999) to the relevant subhalos.
Due to the evolving relationship between Vmax and the halo mass, we expect to find a maximum redshift out to which each of our results is valid. We will begin by determining whether
our results are limited by mass resolution or force/spatial resolution. For the MS, a tidal radius of two times the spatial resolution corresponds to 10h−1 kpc. Assuming, for simplicity,
a static subhalo density profile, mass resolution is the limiting factor for subhalos. That is,
for the particle mass and spatial resolution of the MS, subhalos drop below the 30 particle
limit before their tidal radius reaches 10h−1 kpc. A typical halo with Vmax = 100km/s,
the lowest relevant Vmax , that has been stripped to the 30 particle limit has a tidal radius
significantly larger than 10h−1 kpc, so this conclusion likely holds in the realistic case that
stripping alters the subhalo density profile. To check this conclusion we follow subhalos
in the bound particle number versus orbital radius plane and find the redshift at which
subhalos with Vmax = 120km/s, the lower limit of our catalog, begin dropping below the 30
particle limit. We do not appear to be loosing any subhalos before this point. For each of
the minimum Vmax values considered below, the same result applies; the limiting condition
is that a typical subhalo retain at least 30 bound particles. When relevant subhalos begin
dropping below the 30 particle limit before reaching orbital radii at which a merger with
the host is presumed imminent, subhalo retention has become an issue.
From here, examining the effects of subhalo retention becomes a question of identifying
if and when subhalos with a Vmax of interest drop below the 30 particle limit. To do this
we use the analytical estimate of Mamon (2000) for tidal stripping of NFW halos;
mts ≈ mvs ar (ro /rvh )br
132 Major Mergers Between Dark Matter Halos in the Millennium Simulation
where mts /mvs is the tidal mass of the subhalos as a fraction of the initial virial mass, ro /rvs
the orbital radius scaled by the virial radius of the host halo, and ar and br are chosen to
reflect departures from self similarity. As can be seen in Figure 4.5, this provides a fair
description of tidal stripping in the MS. This relation can be used to estimate the number
of bound particles which an in-falling subhalo must possess in order to retain 30 bound
particles to a desired ro /rvh . The correlation, measured in the simulation, between Vmax
and bound particle number for distinct halos can then be used to estimate the minimum
Vmax to which subhalos are complete at a given redshift. Recalling that this correlation
evolves with redshift, one can also determine the maximum redshift at which a typical halo
with a given Vmax survives to a desired ro /rvh .
The results of this completion study are most concisely stated by determining the maximum redshift to which subhalos are complete for several Vmax of interest. The pertinent
Vmax are 100 km/s for sub-sub mergers with remnant Vmax > 175km/s, 120 km/s for subhalo counts, and 210 km/s for sub-host mergers in ‘group-like’ halos with Vmax > 380km/s.
In the simulation, 90% of the least massive progenitors of major mergers have Vmax values
greater than 0.6 times that of the merger remnant. Hence the values of 100 km/s and 210
km/s. Typical subhalos with Vmax = 210km/s survive numerical effects to ro /rvh = 0.1 out
to z > 4. subhalos with Vmax = 120km/s survive to ro /rvh < 0.1 at z = 0, ro /rvh ≈ 0.2
at z = 1, and ro /rvh ≈ 0.4 at z = 2. Typical subhalos with Vmax = 100km/s survive to
ro /rvh ≈ 0.25 at z = 0 and ro /rvh ≈ 0.5 at z = 1.
The estimates of the redshifts at which relevant subhalos are complete can in turn be
used to examine the probable effects of completion on the results presented here. Clearly
we are pushing the ability to track subhalos in the MS. Most of the important results,
however, are quite robust against the effects of subhalo completion. The largest exception
is the case of mergers between subhalos. Subhalo mergers are more likely to happen before
subhalos are significantly tidally stripped, which mitigates the effects of loosing potential
least massive merger progenitors to artificial evaporation at moderate radii. That said, the
result that subhalo mergers are rare must rest solely on results from redshifts near z = 0.
4.4 Discussion
133
The most robust result is that the presence of multiple subhalos drastically reduces the
time-scale for sub-host mergers. It is the presence of bound structures in the simulation that
drives the merger rate. It is therefore appropriate to separate halos based on the presence or
absence of such a structure, regardless of whether a subhalo would exist in the hypothetical
case of better mass resolution. The related result that an ‘assembly’-type bias is observed
in the major merger rate is somewhat less robust as it depends on identifying the particular
halos in which subhalos counts and the sub-host merger rate are enhanced. The fidelity of
this identification starts fading near z ≈ 1 as subhalos with Vmax ≈ 120km/s begin to be
subject to artificial evaporation. In the absence of a correlation between environment and
the probability of a subhalo evaporating, however, this effect only introduces noise. This
may degrade the observed trend but cannot create it. To the extent that environment and
subhalo loss may be correlated, the initial bound particle counts of fresh subhalos are likely
suppressed in over-dense environments, biasing against the observed result.
Finally, the evolution of the merger rate for halos with Vmax > 175km/s in both all
environments and group environments is affected by subhalo incompletion. Completion
affects both the measured evolution of R+ and R− in all environments and the comparison
between environments. In all environments, the first subhalos lost are not the least massive
progenitor halos of the sub-host mergers, which dominate the merger rate. Rather they are
the lower Vmax subhalos which do play a role in facilitating the sub-host major mergers.
Again, these halos are not only not identified but are artificially dissolved. In grouplike halos this results in a major merger rate that is underestimated at high redshifts.
This effect may occur to some extent at all redshifts as there is always a lowest Vmax to
which subhalos are resolved, but the strength of the effect increases with redshift. Another
effect is eventually felt by galaxy-like halos when the least massive progenitors of sub-host
major mergers are themselves artificially dissolved. When this occurs a major merger is
prematurely recorded. The combined effect on the total merger rate for all halos with Vmax >
175km/s is difficult to predict. At intermediate z, the merger rate may be suppressed, while
at the highest z it may be enhanced, yet in Figure 4.5, the merger rate for all halos shows a
134 Major Mergers Between Dark Matter Halos in the Millennium Simulation
smooth evolution. Subhalos also play a role in group identification. As completion begins
to affect subhalos with Vmax ≈ 120km/s, the requirement that a ‘group’ halo have at least
two subhalos becomes increasingly stringent. As some ‘real groups’ are not identified, the
physical density of mergers in groups is underestimated at high redshift and the measured
slope of the evolution is too shallow. The flattening observed in the right panel of Figure 4.5
must be due in part to completion, but may also be due to the evolution of the Vmax function,
as discussed above. Compared to all environments, the specific merger rate in groups also
shows a flatter evolution, Figure 4.5. The average number of subhalos in these groups
declines with redshift, either due to completion or to the evolution of the Vmax function and
the growth of structure. This in-turn suppresses the merger rate. The relative shallowness
in the evolution of the merger rate in group environments may be either artificial or physical.
4.5
Conclusions
We have made a detailed study of the environments of major mergers in a the Millennium
Simulation (Springel et al., 2005c). Our goal in doing so is to provide a theoretical background for observational tests both to confront the dark matter simulations directly and
to identify populations of merger remnants through their environments. To conclude, the
results of this paper will be considered directly in this light. Comparing simulation results
with observations correctly requires some caution. These comparisons can be placed into
three categories. The most robust observational tests rely on physically motivated comparisons between actual observations. Such tests are likely to be remarkably insensitive to
the details of the simulation. Comparisons between actual observations that are motivated
by the simulations, but which are not physically motivated may be interesting, but their
interpretation will likely be difficult. They may be useful for testing the accuracy of numerical simulations when subtle resolution effects are involved. For example, in this work
we find that the major merger rate, which is dominated by sub-host mergers, depends on
resolving subhalos with Vmax below that of the merger progenitors themselves. Comparing
simulation results to observations directly is tempting, but perilous. When trying to make
4.5 Conclusions
135
such a comparison extreme care must be taken to understand sample selection, both in the
simulations and the observations, and to understand the effects of halo completion, with all
subtleties included.
To explore the kinds of comparisons that are likely to be rewarding, the several measures
of the evolution of the merger rate presented here can be taken as an example. We find that
the evolution of the specific merger rate is flatter than the evolution of the number density
of halos. This comparison can be made directly in the observations, and has interesting
physical implications because it probes the growth of structure. It is also possible to confront
the simulation results by making such a comparison. In this case, halo completion issues
affect the evolution of the merger rate, but not the evolution of the number density of
halos. Completion effects likely cannot account for the discrepancy between the two slopes
however. A second comparison that the simulation results motivate is to compare the slope
of the evolution of the merger rate for galaxy samples with different luminosity limits. The
simulations suggest brighter galaxies may display a shallower slope. This trend can be
weakly physically motivated by appealing to the evolution of the Vmax function, but it is
also affected by completion. This comparison may be a good check against the simulations
as it probes the effects of subhalo retention. Finally, comparing the slope of the evolution
of the major merger rate as measured in the simulations directly to a slope measured using
observations is likely to be unrewarding. The measured slope is affected by completion,
it may depend on halo selection, and comparing it to observations requires assumptions
about the evolution of the Vmax – luminosity relation. In what follows we will focus on
observational tests that are physically motivated and that do not require a direct comparison
between observations and the simulations.
Three different measures of the merger rate can be considered, each with its own physical
counterpart. The forward looking specific merger rate measures mergers that will happen,
similar to pair counts. The backward looking specific merger rate measures mergers that
have happened, similar to morphologically identified merger remnants and other potential
merger remnants such as starbursts and AGN. These are the two merger rates measured
136 Major Mergers Between Dark Matter Halos in the Millennium Simulation
in the MS. A look-back merger rate, such as used in Gottlober et al. (2001), measures
the major merger history of halos identified at z = 0, and has implications for galaxies’
morphologies and star formation histories. These three rates are not interchangeable. For
a Vmax selected halo catalog, or a luminosity selected galaxy catalog, R− will always be
lower, and will evolve slower, than R+ . The forward looking merger rate measures more
massive mergers, but normalizes the merger rate using the same number of halos or galaxies.
Comparing observationally determined R+ and R− would require mapping between pre- and
post-merger luminosities, which is not a trivial task. The look-back merger rate measures
a very different quantity than either R− or R+ . At any given redshift, every halo identified
at z = 0 has a dominant progenitor with the highest mass, or highest Vmax , of all of the
halos that will become the z = 0 halo. Only mergers involving these halos are considered
in the look-back major merger rate. Any correlation between the major merger rate and
the probability of being this dominant halo will result in differences between the look-back
major merger rate and R− or R+ . In addition, the typical Vmax of the progenitors of today’s
halos decreases with redshift. There are two observational consequences of this difference.
First, the evolution of the major merger rate measured by Gottlober et al. (2001) cannot be
compared to observed merger rates. Rather these observations should be compared to R− or
R+ as appropriate. Second, one cannot integrate an observed R− or R+ to draw conclusions
about the relationship between the merger rate and observed galaxy morphologies. Rather,
a study like that presented in Gottlober et al. (2001) is required.
Due to the fine time resolution in the MS, all mergers are either sub-host mergers or subsub mergers. That is the least massive progenitor of a major merger between two distinct
halos is identified as a subhalo for at least one, and usually several, time-steps before the
merger occurs. Subhalos are quickly tidally stripped after they are accreted by their host,
which has important implications for both sub-sub and sub-host merger rates. The tidal
stripping of subhalos drastically reduces their cross section for sub-sub mergers, resulting in
specific merger rates between subhalos that are 20-30 times lower than the sub-host merger
rate in the same hosts. Mergers in group environments are therefore dominated by sub-
4.5 Conclusions
137
host mergers while halo counts are dominated by subhalos. Therefore, the specific merger
rate among group members is actually lower than among the entire halo catalog. It is a
physically motivated prediction of this work that the observed specific major merger rate,
that is the merger rate per galaxy, among members of galaxy groups will be lower than in
the field. This is different from previous predictions based on mass selected halo samples.
Here again the specific merger rate measured in the simulation should not be compared
directly to a rate measured observationally as group selection can have a significant effect
on the specific merger rate measured.
The tidal stripping of subhalos occurs for all accreted halos, even those that have Vmax
similar to that of their host. That is, the least massive progenitor of a major merger is accreted and tidally stripped before it merges with the most massive progenitor. Tidal stripping drastically increases the dynamic friction time scale, lengthening the time separating
accretion from the merger. This effect has been observed in high resolution simulations for
host halos that accrete a single subhalo (Boylan-Kilchin et al., 2008). In the MS, extremely
low specific major merger rates are observed in single subhalo systems, consistent with the
previous result. In hosts with two or more subhalos, however, the merger rate is enhanced
by more than a factor of 10. This observation can be dynamically motivated. Subhalos
experience substantial inelastic collisions with each other (Knebe et al., 2006). When two
subhalos approach each other with opposing orbital angular momentum, their orbital angular momentum can be effectively cancelled while orbital energy is lost to motions internal
to the subhalos. Subhalo interactions therefore facilitate sub-host mergers. Observing this
trend in the simulation is also quite robust to the effects of completion. It is driven by the
presence of bound objects in the simulation. A subhalo that is is artificially dissolved is
not counted as a subhalo and does not contribute to this effect. This result leads to what
may be the most useful of our observational predictions. Merger remnants are substantially
more likely that the average halo to host a subhalo. Hence, the frequency of faint near-by
companions should be enhanced for merger remnants. Similarly, when cross-correlated with
a fainter galaxy sample, merger remnants should show an enhanced correlation on scales
138 Major Mergers Between Dark Matter Halos in the Millennium Simulation
less than the virial radius of the remnants.
The dependence of the merger rate on Vmax and on the local environment, defined as
the count of halos with Vmax > 120km/s within 2h−1 Mpc, was also studied. No dependence
on either Vmax or local environment was identified when each was studied independently.
There is no physical motivation for this result, and it may not hold down to lower Vmax .
At fixed Vmax however, for distinct halos the backward looking merger rate R− depends
weakly on environment. This is contrary to the prediction of linear theory that, at fixed
halo mass, environment and merger history are uncorrelated. The observed correlations are
similar in both sense and in size to similar deviations from linear theory observed in large
simulations, the so called ‘assembly biases’ (Wechsler et al., 2006; Croton et al., 2007; Gao
and White, 2007). For galaxy-like halos R− decreases with increasing halo density. For
group-like halos it increases. A physical mechanism behind this observations was advanced
in §4.4. Both tidal effects and interactions between halos extend beyond the virial radius of
a massive host halo. Tidal effects beyond the virial radius are in particular known to occur.
They are required to correctly predict the halo mass distribution Sheth and Tormen (1999).
Tidal stripping is predicted to extend up to several rv (Mamon, 2000), which is observed
in the MS (Figures 4.5 & 4.5). Tidal effects may set the major merger rate assembly
bias for galaxy-like halos by suppressing accretion onto galaxy-like halos in the vicinity of a
dynamically dominant massive halo. As the local halo density increases, the likely proximity,
mass, distance from a massive halo, and hence the typical strength of tidal effects increase.
Inelastic interactions between bound halos within a few rv of a group-like halo may result
in the loss of orbital energy and angular momentum and the premature accretion of these
halos. As the local halo density surrounding a group-like halo increases, the rate of these
interactions also increases. That this is responsible for the group-like half of the major
merger assembly bias is supported by the observation that subhalo counts correlate with
the local density and that groups with subhalo counts that are higher than average are more
strongly clustered (Wechsler et al., 2006). For assembly biases in general, the typical Vmax ,
or mass, at which the cross over from galaxy-like to group-like behavior occurs increases
4.5 Conclusions
139
with time, being smaller at higher redshift. This is as expected if the cross over is one
from halos that are found in the vicinity of more massive halos to halos that dominate the
local dynamics. The observational consequences of this are that merger remnants which
are the central galaxies of groups should show enhanced clustering on scales beyond the
virial radius of the group while galaxies that are not group or cluster members should show
suppressed clustering on scales beyond their own virial radii. A physical understanding of
the dynamics driving ‘assembly biases’ may help in interpreting observations. For example,
group-like halos in dense environments likely experienced an enhanced merger rate for both
major and minor mergers. The subhalos involved in these mergers spend less time on
average orbiting within the host halo and may be morphologically younger when the subhost merger occurs. This may have implications for the morphology of the central galaxy.
The central galaxies of groups with bluer, or at least less red, colors do show enhanced
clustering (Berlind et al., 2006b).
The physically motivated predictions presented above that involve comparisons between
observed populations, generally major merger remnants versus non-remnants, can all be
used to confront the simulation results with observations provided that major merger remnants are conservatively morphologically identified. Using them to identify other merger
remnants populations, or to determine whether a given population are merger remnants,
will be complicated by the physics of these objects. Most least massive progenitors spend
considerable time as a subhalo prior to merging. In the case of mergers between galaxy-like
halos, this may not affect the galaxy residing within the subhalo. In the case of group-like
halos, however, gas physics may play a role. Three populations of interest are starbursts,
K+A galaxies, and AGN which are considered possible stages of the evolution from two
merging blue spiral galaxies to a red elliptical. Whether a dark matter merger can be
identified with these populations depends on whether substantial amounts of gas are also
involved in the merger. One can consider first whether the subhalo likely brings any gas
to the merger and second whether the host halo has gas of its own. Again, in the case of
mergers between isolated galaxies, one or both of the galaxies is quite likely to bring gas to
140 Major Mergers Between Dark Matter Halos in the Millennium Simulation
the merger. In the case that the mmp is the central galaxy of a group, the group may host a
hot halo which can remove gas from the lmp while it orbits as a subhalo. In the case that a
group hosts multiple subhalos, the lmp will spend less time as a subhalo but undergo more
interactions with other subhalos. The lmp may or may not bring any gas to the merger. Not
all groups host a hot inter-group medium however. In these groups the lmp may retain its
gas until the merger with the central galaxy. Such groups appear not be be representative,
but instead tend to host blue disk galaxies at their centers (Osmond and Ponman, 2004;
Brough et al., 2006). These groups have had calmer than average merger histories and may
therefore reside in regions of low halo density and have low subhalo counts for their masses.
With these complicating factors in mind, the frequency of a fainter companion may prove
to be more useful for identifying populations of merger remnants than cross correlation
studies.
Studying the frequency of fainter companions has clear advantages over clustering studies. The predictions for the correlation function can be split into the one-halo term, which
should be enhanced for all mergers, and the two-halo term, which should be suppressed
for galaxy-like remnants and enhanced for group-like remnants. Observing suppressed or
enhanced clustering on scales beyond the virial radius would require that the observed potential merger population represent a fair sampling of the underlying merger population.
For remnant populations that require gas, this assumption is not fair. For the one-halo
term, assuming a cross-correlation between galaxies with luminosities typical of the central
galaxies of groups and a fainter galaxy population, every central galaxy will contribute like
ns, the number of subhalos it hosts. If the likelihood of being able to fuel a starburst or
AGN declines with ns, then enhanced clustering on small scales may or may not be seen,
depending on the, unknown, strength of this correlation. Using the frequency of a fainter
companion does not require assuming a fair sampling of the underlying merger population
and every bright galaxy is weighted equally, regardless of the number of fainter companions.
If a potential merger remnant population shows an increased companion frequency, when
compared to a large non-merger galaxy sample of comparable mass, then that population
4.5 Conclusions
141
likely consists of merger remnants.
When studying the environments of AGN further physics comes into play. AGN display
a complex relationship between host mass or Vmax and AGN luminosity which may depend
on both environment and redshift (Hopkins and Hernquist, 2006; Hopkins et al., 2006b).
The distribution of Vmax for luminous AGN will be different than that of the underlying
merger population if the AGN lifetime correlates with Vmax . This will trivially affect the
clustering of these AGN, even in the absence of a correlation between environment and
the merger rate. This will also complicate constructing a comparison sample for studies of
companion frequencies. Determining whether AGN are mergers may require modeling AGN
fueling and attempting to over constrain such models with several observational measures
of environment, number density, and evolution. Additionally, some AGN may be triggered
by by alternate mechanisms, though merger driven AGN likely dominate at high redshift
and bright luminosity (Hopkins et al., 2006c, and references therein).
Designing specific observational tests based on the results presented here is a topic for
future work.
142 Major Mergers Between Dark Matter Halos in the Millennium Simulation
Figure 4.1: Evolution of the merger rate. The left hand panel displays the evolution of the
specific merger rate, the merger rate per halo, for all halos. The forward looking rate,R+ ,
and the backward looking rate, R− are both plotted, where R+ is the fraction of all halos
with Vmax > 175km/s that will become the most massive progenitor of a major merger in
the next Gyr and R− the fraction of all halos with Vmax > 175km/s that are a remnant of
a major merger that occurred in the last Gyr. These two definitions correspond to merger
rates determined using counts of close pairs and morphologically identified merger remnants
respectively. The right hand panel displays the physical number density of major mergers
per Gyr resulting in remnants with Vmax > 175km/s. The red solid line corresponds to
all halos. The blue and purple lines correspond to mergers occurring in ‘groups’ and ‘rich
groups’ (See the text for group halo definitions). The solid lines include both mergers
between two subhalos and mergers between a subhalo and the host halo. The dashed lines
include only mergers between two subhalos.
4.5 Conclusions
143
Figure 4.2: Tidal stripping of subhalos at z=0. Logarithmically spaced contours of subhalo
number density in the log[np/np(Vmax )] versus log(ro /rvh ) plane, where np/np(Vmax ) is a
well defined surrogate for mts /mvs , as discussed in the text, and ro /rvh is the subhalo’s
distance from the center of the host halo in units of the host halo’s virial radius. In the
three different panels, contours are shown for subhalos residing in hosts with Vh values
corresponding to galaxy-like host halos, group-like host halos, and rich group and clusterlike host halos. The black contours are for all subhalos with Vs > 120km/s. The colored
contours are for subhalos whose merger with the host would be counted as a major merger.
Contours are shown for Vs /Vh > 0.7 and Vs /Vh > 0.94, roughly corresponding to pretidal stripping mass ratios of 1:3 and 1:1.2. With the exception of some subhalos with
high Vs /Vh , all subhalos are similarly tidally stripped. The thin diagonal lines show the
predicted relationships between mts /mvs and rp /rvh for NFW halos for group-like (center
panel) and cluster-like (right panel) hosts from Mamon (2000), where rp is the peri-center
of the subhalo’s orbit. The right-hand line has been shifted to higher ro /rvh by 0.3.
144 Major Mergers Between Dark Matter Halos in the Millennium Simulation
Figure 4.3: Tidal stripping of subhalos at z=1. The same as Figure 4.5, but for z = 1.
4.5 Conclusions
145
Figure 4.4: Distribution of Vlmp /Vmmp for major mergers at z = 0, 1, and 2. The three
panels show the distribution split by merger remnant Vmax . The same ranges are used as
in Figures 4.5 and 4.5. The solid lines are histograms of the fraction of major mergers
in each Vlmp /Vmmp bin and the dashed lines are the cumulative probability distribution of
Vlmp /Vmmp values. Mergers are heavily dominated by lower values of Vlmp /Vmmp , and it is
safe to assume that the vast majority of major mergers are true sub-host mergers in which
the subhalo merger partner has been tidally stripped before completely merging with the
host halo.
146 Major Mergers Between Dark Matter Halos in the Millennium Simulation
Figure 4.5: The effect of interactions between subhalos on the major merger rate. The
left panel shows R+ for sub-host mergers in hosts with 345 < Vh (km/s) < 455 split by
the number of subhalos in the host. The dark blue long-dashed line shows R+ for mergers
between a host and a lone subhalo. The light blue dot-dashed line shows R+ for subhost mergers of hosts with two subhalos, and the green dotted line shows R+ for sub-host
mergers of hosts with at least two subhalos. Introducing a second subhalo increases R+ by
a factor of ≈ 10, rather than ≈ 2, as would be expected if interactions between subhalos had
no effect on the merger rate. The right panel examines how the increased merger rate in
hosts with multiple subhalos, due to interactions between subhalos, affects the frequency of
merger remnants among halos with group-like Vmax , 380 < Vmax (km/s) < 500. It is merger
remnant frequency rather than close pair counts that is easiest to interpret in group-like
environments. The dark blue dashed line shows the merger remnant frequency for halos
with no subhalos with Vs > 120km/s. These remnants would be the result of mergers
between the host and a single bright subhalo. The green line shows the frequency of merger
remnants among group-like halos with at least one bright subhalo and the red line shows
the frequency among all group-like halos. Subhalo interactions clearly play a strong role
in driving the overall merger rate as without them the expected remnant frequency would
be only ≈ 2 times higher than the dark blue line. The observational implications are that
merger remnants should be more likely than average galaxies to have a least one fainter close
companion and that when merger remnants are cross correlated with a sample of fainter
galaxies they should show an enhanced correlation on small scales.
4.5 Conclusions
147
Figure 4.6: The evolution of R− for ‘group’ and ‘rich group’ members is compared to that for
all halos. Groups and rich groups are defined as host halos with Vh values in the indicated
ranges and at least two subhalos within their virial radius, indicated by ‘r/rv < 1’ or two
non-central halos within their FOF group, indicated by ‘all’. Group ‘members’ includes both
the central host and either all subhalos, ‘r/rv < 1’, or all non-central halos, ‘all’. These
correspond to observed groups with at least three members which are either conservatively,
‘r/rv < 1’, or generously, ‘all’, defined. Contrary to previous expectations, the specific
merger rate in groups is lower than in the field. As is discussed in the text, while mergers
are dominated by sub-host mergers, the halo count in groups is dominated by subhalos.
148 Major Mergers Between Dark Matter Halos in the Millennium Simulation
Figure 4.7: Merger rate versus local halo number density and Vmax . The left panel displays
R− versus the local halo number density, measured by counting all halos with Vmax >
120km/s within a 2h−1 Mpc comoving sphere, for all halos with Vmax > 175km/s for z = 0,
1, and 2. The right panel shows R− versus Vmax for halos in all environments. Insets are for
orientation. The left inset shows the distribution of local densities for z = 1 for three Vmax
ranges, galaxy-like (175 < Vmax (km/s) < 380), group-like (380 < Vmax (km/s) < 500), and
rich group-like (500 < Vmax (km/s) < 950). This inset clearly illustrates halo biasing. The
right inset shows the distribution of Vmax in three environment ranges, 0-4, 5-9, and 10-14
halos within 2h−1 Mpc comoving, displaying the counterpart of halo biasing. The merger
rate is independent of Vmax and, for environments typical of most halos, of environment.
4.5 Conclusions
149
Figure 4.8: Merger rate versus local halo number density for halos grouped by Vmax . In
both panels, R− is plotted versus the local density at z = 0, 1, and 2 for halos in three Vmax
ranges, as shown in the legend. The middle Vmax range, the green dotted line, evolves from
moderately biased at z = 2 to unbiased at z = 0. The left panel includes both central halos
and non-central halos that are not within the virial radius of the central halo of their dark
matter FOF group. Linear theory predicts no dependence of R− on environment for these
halos. In the left panel, halos with low Vmax values display a R− that decreases with local
density while halos with high Vmax values display a R− that increases with local density.
This is similar to other ‘assembly’ biases that have been recently observed in large N-body
simulations. The right panel includes only halos that are the central halo of their dark
matter FOF group. The left panel is sensitive to the influence of the central halo of an
FOF group on the halos surrounding it, the right panel is not. For halos with low Vmax ,
R− is observed to decrease with local density in the right panel, but not in the left. The
right panel is particularly sensitive to the effect that environment on 2h−1 Mpc scales has
on halos which dominate their immediate environment. At z = 2, the intermediate Vmax
range displays a hill-like behavior in the left panel, but increases continuously with local
environment in the right panel. At z = 0 and 1, halos with intermediate values of Vmax
behave like those with low values of Vmax . Comparing the two panels provides insight into
the possible mechanisms responsible for these trends and supports the hypothesis that they
arise from the extension of the effects of tidal stripping and unbound inelastic collisions
between halos beyond the virial radius of the central/host halo.
Appendix A
NFW Profile
When simulations are run that allow cold, non-interacting particles to evolve gravitationally
from a flat power spectrum at high redshift, the dark matter halos that emerge at low z are
well fit by a universal density profile over many orders of magnitude in mass. This profile
was originally parameterized by Navarro et al. (1997). Both the satellite and group/cluster
dark matter halos are described using an NFW profile, the relevant properties of which are
summarized here.
The NFW profile is
ρ(r)
δchar
=
ρ0c
(r/rs )(1 + r/rs )2
(A.1)
where ρ0c is the average density of the universe at z = 0, rs is an inner scale radius, and
δchar is a characteristic over-density. It can be rewritten in terms of a concentration, c, an
overdensity, vρ , and a scaled radius, s ≡ r/rv
vρ c2 g(c)
ρ(s)
=
0
ρc
3s(1 + cs)2
c≡
rv
rs
vρ c3 g(c)
3
1
g(c) =
ln(1 + c) − c/(1 + c)
δchar =
151
(A.2)
(A.3)
(A.4)
(A.5)
152 NFW Profile
The overdensity, virial mass, Mv , and virial radius, rv , are related by
4
Mv = πrv3 vρ ρ0c
3
(A.6)
1/3
Note that the virial radius scales as Mv . The mass contained within s is
M (s) = g(c)Mv ln(1 + cs) −
cs
1 + cs
(A.7)
and the gravitational potential is
Φ(s) = −
GMv g(c) ln(1 + cs)
rv
s
(A.8)
The above can be found in (Cole and Lacey, 1996; Lokas and Mamon, 2001) and (Navarro
et al., 1997).
Appendix B
Orbits in an NFW Profile
Bound orbits in a central force field travel between two radial extremes. These are found
by solving.
1
2 [ − Φ(r)]
=
2
r
l2
(B.1)
where and l are the energy and angular momentum per unit mass (Binney and Tremaine,
1988).
The scaled radial extremes, s0 , for an NFW profile are given by
1
−2
v
ln(1 + cs0 )
= 2 1 − g(c)
bs
s0
s20
(B.2)
where v and bs are defined as
v ≡ −
b2s ≡
GMv
rv
l2
b2
=
||rv2
rv2
(B.3)
(B.4)
The parameter b would be the impact parameter for an unbound orbit.
Conserving angular momentum and using Eq. B.2, the orbital speed at the pericenter,
v0 , is
v02 =
1/3
It is proportional to Mv
b2s GMv
s20 v rv
and depends on the concentration through s0 (c).
153
(B.5)
154 Orbits in an NFW Profile
At any point in the orbit, the orbital speed, vsat , is
2
vsat
2GMv
ln(1 + cs)
=
g(c)
−
rv
s
v
(B.6)
Appendix C
Hot Galactic Halo
The virial temperature, Tv , of a gravitational potential is defined by
3kB Tv
|W |
≡ hv 2 i =
µmp
Mv
(C.1)
where W is the work done by gravity in forming the halo, kB is the Boltzmann constant,
mp is the mass of the proton, and µ is the mean molecular weight (Binney and Tremaine,
1988).
Using eqs. A.2, A.6, and A.7, the virial temperature of an NFW profile is
|W |
GMv 2 2
=
c g (c)
Mv
rv
≡
1
Z
0
ln(1 + cs) − cs/(1 + cs)
sds
s(1 + cs)2
GMv 2
c g(c)I1 (c)
rv
(C.2)
(C.3)
A dimensionless temperature, t, is defined.
kB Tv
rv
t(c) ≡
µmp
GMv
2
c g(c)I1 (c)
t(c) =
3
(C.4)
(C.5)
For a gas in hydrostatic equilibrium in potential φ, ignoring self-gravity
ρ(r) = ρ0 exp
µmp
[φ(r) − φ(0)]
kB T
155
(C.6)
156 Hot Galactic Halo
Combining eqs. C.3 and C.6, the gas density profile is
ρg (s) = ρ0 exp
3
ln(1 + cs)
−1
cI1 (c)
cs
(C.7)
Letting the gas mass within rv equal a fraction msg of the virial mass results in a central
density of
ρ0 =
ρc0 vρ msg
3I2 (c)
(C.8)
ln(1 + cs)
3
−1
cI1 (c)
cs
(C.9)
The integral, I2 (c), is
Z
1
2
s ds exp
I2 (c) =
The scaled density profile j is defined by
j(s, c) ≡
1
exp
I2 (c)
ρ(s) =
3
ln(cs)
−1
cI1 (c)
cs
vρ ρc0
3
(C.10)
msg j(s, c)
(C.11)
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