$R_{\mathrm{tc}}$ it ends and here, $\sigma(R_{\mathrm{tc}})=0$. With this
assumption we get:

$\sigma^{2}(r)=\frac{\mathrm{G}\,M}{6\,\varepsilon}(1+r^{2}/\varepsilon^{2})^{5/2}\left[\frac{1}{(1+r^{2}/\varepsilon^{2})^{3}}-\frac{1}{(1+R_{\mathrm{tc}}^{2}/\varepsilon^{2})^{3}}\right].$
(81)

The radial dependence of the velocity dispersion for the truncated and the
infinite galaxy are compared in Fig. 43.

####  17.4 Velocity dispersion in a double Plummer sphere

For a galaxy modeled as two Plummer spheres – one for the luminous component
and another one for the dark halo – the situation with the velocity dispersion
is more complex. The presence of one component influences the dispersion in
the other one and vice versa. Eq. (79) changes to

$\sigma_{1}^{2}(r)=\frac{1}{\rho_{1}(r)}\intop_{r}^{\infty}\rho_{1}(r^{\prime})\frac{\mathrm{d}\left[\phi_{1}(r^{\prime})+\phi_{2}(r^{\prime})\right]}{\mathrm{d}r^{\prime}}\mathrm{d}r^{\prime}.$
(82)

Using Eq. (74) and Eq. (73) and after a partial integration, we obtain

$\sigma_{1}^{2}(r)=\frac{\mathrm{G}\,M_{1}}{6\,\sqrt{\varepsilon_{1}^{2}+r^{2}}}+\frac{\mathrm{G}\,M_{2}}{\varepsilon_{2}^{3}}\left(1+r^{2}/\varepsilon_{1}^{2}\right)^{5/2}I(r,\varepsilon_{1},\varepsilon_{2}),$
(83)

where the first term is identical to the dispersion of the first component
without in the absence of the second one. The integral
$I(r,\varepsilon_{1},\varepsilon_{2})$ is solved as follows

$I(r,\varepsilon_{1},\varepsilon_{2})=\intop_{r}^{\infty}\frac{r^{\prime}}{\left(1+r^{\prime
2}/\varepsilon_{1}^{2}\right)^{5/2}\left(1+r^{\prime
2}/\varepsilon_{2}^{2}\right)^{3/2}}\mathrm{d}r^{\prime}=$ (84)

$=\frac{1}{3\left(\varepsilon_{2}^{2}-\varepsilon_{1}^{2}\right)}\left[\frac{1}{\left(r^{2}+\varepsilon_{1}^{2}\right)^{3/2}\left(r^{2}+\varepsilon_{2}^{2}\right)^{1/2}}+\frac{4}{\left(\varepsilon_{2}^{2}-\varepsilon_{1}^{2}\right)^{2}}\left(2-\sqrt{\frac{r^{2}+\varepsilon_{2}^{2}}{r^{2}+\varepsilon_{1}^{2}}}-\sqrt{\frac{r^{2}+\varepsilon_{1}^{2}}{r^{2}+\varepsilon_{2}^{2}}}\right)\right].$

Fig. 44 illustrates the effect of the presence of the other component on the
velocity dispersion of a Plummer sphere.

?figurename? 44: An illustration of the effect of a second component on the
dispersion of a Plummer sphere ($M_{*}=3.2\times 10^{11}$ M⊙ ,
$\varepsilon_{*}=7$ kpc). Red: the dispersion of the isolated sphere, green:
additional dispersion caused by the presence of a second component of large
mass ($M_{\mathrm{DM}}=6.4\times 10^{12}$ M⊙ ) and large Plummer radius
($\varepsilon_{\mathrm{DM}}=60$ kpc), blue: the sum of the two. The dispersion
is normalized so that the dispersion in the center of the first component in
the absence of the second one (181 km$/$s) equals 1.

####  17.5 Standard set of parameters

For the future reference, let us define the standard set of parameters for
simulations (used in this Part) as the following set of values:

The mass of the primary galaxy: $M_{\mathrm{p}}=3.2\times 10^{11}$ M⊙

The secondary to primary mass ratio: 0.02

Plummer radius of the primary galaxy: $\varepsilon_{\mathrm{p}}=20$ kpc

The cut-off diameter for the primary galaxy: $R_{\mathrm{tc}}=200$ kpc

Plummer radius of the secondary galaxy: $\varepsilon_{\mathrm{s}}=2$ kpc

The initial radial distance of the secondary galaxy: 180 kpc

The initial velocity of the secondary galaxy: $125$ km$/$s, the escape
velocity for the initial distance

These values are used as the usual setup of the presented simulations and we
will refer to them often, so we do not have to repeat them.

Let us only remark that the escape velocity, $v_{\mathrm{esc}}$, is computed
only approximately, on the same grounds as the force between the galaxy (see
Eq. 77), i.e. we put

$v_{\mathrm{esc}}=\sqrt{\frac{2\,G\,(M_{\mathrm{p}}+M_{\mathrm{s}})}{\sqrt{r^{2}+\varepsilon_{\mathrm{common}}^{2}}}.}$
(85)

The results of our simulation show that, in the relevant range of radii, its
difference from reality is negligible.

###  18 Dynamical friction

Dynamical friction is a braking force of gravitational origin acting on a body
that moves through the field of stars or any other matter. We will be
interested in the dynamical friction incurred on the secondary galaxy by the
stars and dark matter of the primary galaxy. We encourage the reader to
consult Appendix D – Introduction to dynamical friction – which is a modified
chapter from Ebrová (2007). It explains in detail the nature of this
phenomenon and it is likely to be of interest even to a reader already
familiar with the topic.

Appendix D also contains a derivation of the Chandrasekhar formula (Sect.
D.2). Chandrasekhar formula is an analytical expression derived by
Chandrasekhar (1943) that is still often used to calculate the dynamic
friction. The formula is a good approximation for the dynamical friction and
is easy to use in test-particle simulations.

There are several different simplifications done during its derivation (see
Sect. E.1). One is the assumption of homogeneity of the stellar field around
the braked body (both density and velocity dispersion are taken as constants).
This leads to a relatively simple expression that contains the so-called
Coulomb logarithm. The exact value of this logarithm is unknown and is usually
roughly estimated and taken as a constant.

In Ebrová (2007), we have devised an alternative way to calculate the
dynamical friction in radial mergers (that are the most likely to produce
shell structures). We call it our modification of the Chandrasekhar formula
and a detailed description and derivation can be found in Appendix E. Here we
summarize only the main ideas.

The homogeneity of density and velocity dispersion is not assumed during the
derivation of the Chandrasekhar formula. Instead, a more realistic stellar
distribution function is used, varying both the density and velocity
dispersion based on the chosen model of the host galaxy. Using the radial
symmetry, the originally 5-dimensional problem is reduced to a 2-dimensional
one, Eq. (104), which is analytically insolvable and so numerical integration
is used to calculate the final result for the dynamical friction. In this
approach, no estimated values are needed as an input, only the distribution
function chosen for the galaxy determines the friction.

In Sect. E.2, we compare the result of Eq. (104) to the Chandrasekhar formula.
It is shown that using a constant as the Coulomb logarithm is completely
inadequate for the problem at hand.

###  19 Multiple Three-Body Algorithm (MTBA)

We now investigate another alternative method to calculate the dynamical
friction in radial minor merger. The method is described in the paper Seguin
and Dupraz (1994) and it is also suitable for test-particle simulations.

####  19.1 Principle and characteristics

Seguin and Dupraz (1994) used restricted tree-body simulations to examine
dynamical friction in head-on encounter. They adopted the Multiple Three-Body
Algorithm which was originally proposed by Borne (1984). The basis of the
method is to calculate the motion of the satellite galaxy from the
gravitational influence of the particles in the primary galaxy. However, it is
not a self-consistent simulation, as the particles are otherwise treated as
test particles – their motion is calculated as the motion of massless
particles in the sum of the gravitational potentials of both galaxies, in the
same manner as in our simulations of the creation of the shell structure
(Sect. 13 and Sect. 22). In the case of the MTBA, the particles are generated
so that they follow the distribution function of the primary galaxy. Only when
the motion of the secondary galaxy is calculated, these particles are used as
if each of them had a mass of $m=M_{\mathrm{p}}/N$, where $M_{\mathrm{p}}$ is
the total mass of the primary galaxy and $N$ is the total number of particles
used. The force/acceleration acting upon the secondary galaxy in each step is
fully determined by the action of all particles in the primary galaxy upon a
chosen smooth potential of the secondary galaxy. Having also the potential of
the satellite act on these particles naturally perturbs their trajectories and
from their force exerted back on the satellite galaxy the dynamical friction
naturally arises.

To summarize, this method to calculate the dynamical friction requires a model
for the potentials of the primary and the secondary galaxy and the use of
particles in the primary galaxy. The particles are treated in two different
ways: as massless when their motion is calculated and as massive when the
motion of the secondary galaxy is calculated.

Seguin and Dupraz (1994) have directly compared the results of a MTBA
simulation with the coupled solution of the linearized Poison and
collisionless Boltzmann equations for the first passage of the satellite. They
found MTBA to be equivalent to the analytical method. Compared to their
analytical method, the MTBA has the advantage of easier and faster
calculation. Moreover the MTBA is more flexible so it can follow the whole
process until a complete merger. Both these methods show that the dynamical
friction in radial merger is not strictly proportional to the local density –
contrary to what is assumed in the Chandrasekhar formula. Moreover, it is a
time-dependent process which depends on the full past history of the merger,
contrary to a satellite on a circular orbit in the co-rotating frame. This
observation cannot be reproduced in any modification of the Chandrasekhar
formula (including ours) which is fundamentally local.

In Seguin and Dupraz (1996) the MTBA has been compared with a self-consistent
Particle-Mesh simulation. The MTBA gives an accurate estimate of the decay
rate of orbital energy of the satellite, within 10% of the $N$-body simulation
during the first orbit. But it fails to reproduce the ultimate phase of the
merger.

####  19.2 Merger parameters

To compare different methods for the calculation of the dynamical friction, we
have modeled the secondary as a point mass (eventually with a very small
softening – 0.01 kpc) and have chosen the following parameters of the
collision:

The mass of the primary galaxy: $M_{\mathrm{p}}=10^{12}$ M⊙

The secondary to primary mass ratio: 0.01

Plummer radius of the primary galaxy: $\varepsilon_{\mathrm{p}}=10$ kpc

The cut-off diameter for the primary galaxy: $R=200$ kpc

The initial radial distance of the secondary galaxy: 100 kpc

The initial velocity of the secondary galaxy: 0 km$/$s

####  19.3 Results of simulations

It turns out that for a successful application of the MTBA it is necessary to
use a high enough number of particles in the primary galaxy and a small enough
time step of integration. The simulation for the chosen set of parameters
(Sect. 19.2) stabilizes for about 100,000 particles with time step of 0.01
Myr, but even then there are noticeable differences mainly in the later part
of the merger as we further increase the number of particles and decrease the
time step, see Fig. 45 and Fig. 46. On the other hand, the introduction of the
slight softening in the interaction of the secondary does not influence the
results provided that enough particles and a small enough time step are used.

?figurename? 45: (a) Distance of the secondary from the center of the primary
galaxy; (b) energy of the secondary. The motion was calculated using the MTBA
with 100,000 particles for time steps of 0.001–1 Myr. Parameters of the
collision are given in Sect. 19.2.

?figurename? 46: (a) Distance of the secondary from the center of the primary
galaxy; (b) energy of the secondary.The motion was calculated using the MTBA
with time step 0.01 Myr for 1,000-1,000,000 particles. Parameters of the
collision are given in Sect. 19.2.

###  20 Comparison with self-consistent simulations

To compare the calculation of the dynamical friction using the methods
mentioned earlier (Appendix E and Sect. 19) with the self-consistent
simulations, we use the simulations performed by Kateřina Bartošková using
GADGET-2. GADGET-2 is free software, distributed under the GNU General Public
License. The code can be used for studies of isolated systems, or for
simulations that include the cosmological expansion of space. It computes
gravitational forces with a hierarchical tree algorithm (optionally in
combination with a particle-mesh scheme for long-range gravitational forces)
and represents fluids by means of smoothed particle hydrodynamics (SPH). Both
the force computation and the time stepping are fully adaptive. The code is
written in highly portable C and uses a spatial domain decomposition to map
different parts of the computational domain to individual processors. GADGET-2
was publicly released in 2005 (Springel, 2005) and presently is the most
widely employed code for the cosmic structure formation.

####  20.1 Altering GADGET-2 computational setting

The parameters of the collision have been set the same as in the previous
case, Sect. 19.2, but with no cut-off diameter. $10^{5}$ particles have been
used to represent the primary galaxy. The results differ for different
settings of computational parameters in GADGET-2. Here we present results of
five simulations that differ in settings for three chosen parameters and in
the accuracy of variables during the calculation.

During the calculation of the gravitation force, spline softening is used.
$SoftPar$ is the magnitude of the softening used for mutual interactions of
the particles of the primary galaxy. $SoftSec$ is the softening for the
secondary and in an interaction between the secondary and a particle of the
primary galaxy, the larger value from $SoftPar$ and $SoftSec$ is used. $ETIA$
(ErrorTolIntAccuracy) influences the accuracy of the integration method. It is
used in the estimation of the adaptive integration step $\Delta t$

$\Delta t=\sqrt{\frac{2\,ETIA\,SoftPar}{a}},$ (86)

where $a$ is the amount of acceleration the particle has been subjected to in
the previous step. Thus the smaller $ETIA$ we choose, the shorter will be the
time step. $Precision$ refers to the type of the floating-point precision used
during numerical calculations.

The values we have used in the five different simulations and the labels of
the simulations are shown in Table 6. The orbital decay of the satellite for
all the runs is shown in Fig. 47. Run D has been calculated with the highest
precision and we thus use it as a reference in the following section.

run | $ETIA$ | $SoftPar$ | $SoftSec$ | $Precision$
---|---|---|---|---
|  | kpc | kpc |
A | 0.002 | 0.21 | 0.05 | Single
B | 0.008 | 0.05 | 0.05 | Single
C | 0.04 | 0.01 | 0.01 | Single
D | 0.04 | 0.01 | 0.01 | Double
E | 0.002 | 0.05 | 0.05 | Single
?tablename? 6: The settings for the GADGET-2 simulations. The meaning of the
parameters is explained in Sect. 20.1.

?figurename? 47: (a) Distance of the secondary from the center of the primary
galaxy; (b) energy of the secondary.The motion has been calculated using
GADGET-2. The parameters of the collision are given in Sect. 19.2, the
settings for each simulation in Table 6.

####  20.2 Comparison of methods

Fig. 48 shows the orbital decay of the secondary in the merger with parameters
given in Sect. 19.2 for three different methods of calculation of dynamical
friction. Our modification of Chandrasekhar formula adds to the equations of
motion of the secondary the dynamical friction calculated using a numerically
integrated analytical formula as described in Appendix E. The MTBA method
(Sect. 19) is represented by a simulation with 100,000 particles and time step
of 0.01 Myr. From the self-consistent simulation with GADGET-2 we show run D
(see Sect. 20.1).

?figurename? 48: (a) Distance of the secondary from the center of the primary
galaxy; (b) energy of the secondary in three different methods: our
modification of Chandrasekhar formula (Sect. E.1, red curve); inconsistent
simulation with GADGET-2 (Sect. 20.1, green curve); and MTBA (Sect. 19.1, blue
curve). Parameters of the collision are given in Sect. 19.2.

Our modification of Chandrasekhar formula gives by far the fastest loss of the
orbital energy of the satellite, but even the MTBA gives a significantly
larger value of the dynamical friction than the self-consistent simulation.

In Sect. 22 we will however use our modification of Chandrasekhar formula for
the calculation of the dynamical friction, as we have conducted a sizable
number of simulation using this method before we became familiar with the
MTBA. The MTBA is also more computationally demanding. It requires a small
enough time step and the inclusion of test particles in the primary galaxy,
which are otherwise of no interest for us. Our modification of Chandrasekhar
formula, on the contrary, gives the same results for the motion of the
secondary galaxy for the time step of 1 Myr as for any shorter step.

Doing self-consistent simulations is not an option because of the number of
different simulations required for this study (most of which we do not show
explicitly in this thesis). Because it seems that our modification of
Chandrasekhar formula significantly overestimates the real value of the
friction, the results have to be considered an upper bound for the influence
of the dynamical friction on the shell structure. At the end, it turns out
that the differences in the shell structure related to the choice of a method
to calculate the dynamical friction is smaller than the uncertainty in the
models of the tidal decay of the secondary galaxy (Sect. 21).

###  21 Tidal disruption

Together with the dynamical friction, the tidal disruption is another effect
that is important for the galactic merger. The tidal disruption gradually
lowers the mass of the cannibalized galaxy and thus mitigates the effect of
the dynamical friction. During shell formation, it is of particular
importance, because the gradual release of stars from the secondary galaxy has
an important effect on the growing shell structure. The introduction of the
tidal disruption into test-particle simulation is nevertheless a difficult
task.

####  21.1 Massloss of the secondary

In the context of the tidal disruption of an object in the gravitation field
of another body, the notion of the _tidal radius_ is frequently introduced.
This is an approximative approach to the tidal forces, assuming that under the
tidal radius the matter is still bound to the disrupted body, but it is not
the case anymore outside the tidal radius. The reader may find more details on
the concept in Appendix F. Here we will only show how we used it in our test-
particle simulations.

?figurename? 49: The purely analytical approach to the decay of the secondary
galaxy during the first passage for the standard set of parameters (Sect.
17.5). Left: the evolution of the mass of the secondary galaxy. Rights:
Distance of the secondary from the center of the primary galaxy (blue curve)
and tidal radius of the secondary (red curve).

First we have implemented a purely analytical approach, where we calculate the
current tidal radius in every step using Eq. (107) and update the mass of the
secondary galaxy accordingly to the mass of a Plummer sphere with the original
parameters of the secondary galaxy but restricted to the tidal radius. But
this leads to us only lowering the satellite mass during the first passage
through the center of the primary galaxy, see Fig. 49. Particles are released
in limited amount also during further passages, but this mechanism obviously
does not reflect the real situation for multiple passages.

?figurename? 50: Gradual decay of the secondary galaxy calculated using test
particles. Top: distance between the centers of the primary and the secondary
galaxy. Bottom: the number of particles bound to the secondary galaxy. Blue
curves show the development for the simulation where we consider as bound
particles those inside the sphere of the tidal radius, the red curves
correspond to keeping particles with lower than escape velocity. Both
simulations are carried out for the standard set of parameters (Sect. 17.5),
the dynamical friction is calculated using our modification of the
Chandrasekhar formula (Appendix E).

To describe the decay of the satellite during further passages, we have
included in its calculation the test particles of the secondary galaxy. We
count particles that we still consider bound with the satellite galaxy. The
ratio between their number and the number of particles that we have put in the
secondary galaxy at the beginning of the simulation determines its current
mass. As a criterion for bound particles we consider that 1) the distance of
the particle from the center of the secondary galaxy is lower than the current
tidal radius; 2) the velocity of the particle with respect to the secondary
galaxy does not exceed the escape velocity for its given distance from the
center of the secondary galaxy. Fig. 50 shows how these two approaches differ
for otherwise identical initial conditions.

The use of the tidal radius causes large fluctuations of the number of bound
particles near the passage of the secondary galaxy through the center of the
primary galaxy, when many particles suddenly find themselves outside the tidal
radius. When later the secondary galaxy retreats from the center of the
primary, the tidal radius quickly increases and more particles are included.
Some of them eventually escape before the secondary reaches its apocenter, but
still more particles stay bound to the secondary than there were during its
passage through the center of the primary. In the other simulation the loss of
particles is more monotonous, the orbital decay slightly faster, and more
particles are caught in the center of the host galaxy.

?figurename? 51: Development of the distance between the primary and the
secondary galaxy (top) and the Plummer radius of the secondary galaxy
(bottom). The simulations carried out for the standard set of parameters
(Sect. 17.5), the dynamical friction is calculated using our modification of
the Chandrasekhar formula (Appendix E). The radial density of the secondary
galaxy at the beginning of the simulation and in 5 Gyr is shown Fig. 52.

The use of the two different methods to model the tidal disruption of the
secondary does not have a dramatic impact on the merer. Nevertheless, the
times of the passages of the secondary through the center of the host galaxy
and the volume of particles released in each passage differ between the two
models, mainly in the later phases of the merger. This may have a noticeable
impact on the appearance of the shell system in different time, that is the
positions of the shells, their number, brightness, opening angle and so forth.

The problem is that we have no hint as to which of the methods is a better
approximation for the true decay of the secondary galaxy. If we were to
compare the results with self-consistent simulations, we would likely get
different results depending mainly on the configuration of the merger. Thus we
compare the test-particle simulations done with different methods for the
tidal disruption of the secondary galaxy and focus on features of the shell
system that are independent of the method used (Sect. 22.1).

####  21.2 Deformation of the secondary galaxy

Another thing going on during the merger that is difficult to reproduce in
test-particle simulations is the deformation if the cannibalized galaxy. We
model components of galaxies with spherically symmetric Plummer spheres. Thus
we have tried at least to change the profile of the sphere of the secondary
galaxy during the simulation.

The mean value of the radial distance of a particle $\left\langle
r\right\rangle$ in a Plummer sphere is given as

$\left\langle r\right\rangle=\frac{\intop_{0}^{R_{\mathrm{tc}}}r^{\prime
3}\rho(r^{\prime})dr^{\prime}}{\intop_{0}^{R_{\mathrm{tc}}}r^{\prime\prime
2}\rho(r^{\prime\prime})dr^{\prime\prime}},$ (87)

where $\rho(r^{\prime})$ is the density of the Plummer sphere Eq. (74) and we
express the cut-off in multiplies of the Plummer radius
$R_{\mathrm{tc}}=p\varepsilon$. The mean value of the radial distance is then

$\left\langle
r\right\rangle=\varepsilon\frac{2\left(1+p^{2}\right)^{3/2}-2-3p^{2}}{p^{3}}.$
(88)

Thus if we calculate the mean value radial distance from the center of the
secondary galaxy for the particles that we consider bound to it in the
simulation

$\left\langle r\right\rangle=\sum_{i=1}^{N}r_{i}/N,$ (89)

we can easily convert it to a new Plummer radius for the secondary galaxy
$\varepsilon_{\mathrm{s}}$. Fig. 51 shows the development of the Plummer
radius of the secondary galaxy in a simulation with the standard set of
parameters (Sect. 17.5). The Plummer radius is calculated using Eq. (88),
where $\left\langle r\right\rangle$ is the mean radial distance of particles
under the current tidal radius. The radial density of the secondary galaxy at
the beginning of the simulation and in 5 Gyr is shown in Fig. 52. It is
important to keep in mind that the density is calculated only from radial
distances from the center of the satellite even though the spherical symmetry
was surely broken during the simulation.

?figurename? 52: The radial density of the secondary galaxy at the beginning
of the simulation and in 5 Gyr for the standard set of parameters (Sect.
17.5). In blue is the density calculated from the test particles of the
secondary galaxy, in green the model of the secondary chosen at the start of
the simulation and in red the density of the Plummer sphere that corresponds
to the changing Plummer radius which is calculated from the distribution of
the test particles. The density is normalized so that the central density of
the initially chosen Plummer sphere of the secondary galaxy is one.
?figurename? 53: Snapshots of simulations. For description of all runs see
text in Sect. 22.1. Time 0 is defined as the moment then the secondary galaxy
reaches the center of the primary galaxy for the first time, which is (for all
three runs) almost exactly 1 Gyr after it has been released from the distance
of 180 kpc with escape velocity. Only the surface density of particles
originally belonging to the satellite galaxy is displayed corresponding to the
subtraction of the profile of the primary galaxy. Each box, centered on the
host galaxy, shows 300$\times$300 kpc. Radial histogram of particles in 5 Gyr
is shown in Fig. 54.

###  22 Simulations of shell structure

Now we finally show the combined effect that the inclusion of both the
dynamical friction and gradual decay of the secondary galaxy in the
simulations has on the shell formation. The simulations are carried out using
the method described in Sect. 17, i.e. millions of test particles were
generated so that they follow the distribution function of the secondary
galaxy at the beginning of the simulation. The particles then move according
to the sum of the gravitational potentials of both galaxies that are both
represented by a smooth potential. The galaxies move with respect to each
other as dictated by their masses, shape of potentials, positions and
velocities.

The dynamical friction, when included, is calculated using our modification of
the Chandrasekhar formula, see Appendix E, and the gradual decay of the
secondary galaxy, when included, is calculated using some of the methods from
Sect. 21.1. In Sect. 22.2, we have added the dark halo to the primary galaxy
and Sect. 22.3 shows the shell formation in a self-consistent simulations. All
the outputs are oriented so that the secondary originally approached the
primary galaxy from the right hand side.

####  22.1 Dynamical friction and tidal disruption

We have compared three simulations, all of them for the standard set of
parameters (Sect. 17.5).

  * •

Run 1 – without dynamical friction and with instant disruption of the
secondary.

  * •

Run 2 – dynamical friction is calculated using our modification of the
Chandrasekhar formula and the tidal disruption using the analytical approach
based on the tidal radius as described at the beginning Sect. 21.1.

  * •

Run 3 – dynamical friction is again calculated using our modification of the
Chandrasekhar formula, the tidal disruption is based on the counting of
particles inside/outside the current tidal radius. Additionally, the Plummer
radius of the secondary galaxy is constantly recalculated as described in
Sect. 21.2.

Snapshot from all the runs for two different times are shown in Fig. 53,
radial histograms of particles in Fig. 54. Video from run 1 and run 2 is part
of the electronic attachment. For the description of the video, see Appendix H
point 4.

We compare a simple simulation (Run 1) with a pair of simulations (Runs 2 &
3), where the tidal decay of the secondary galaxy is modeled using two
different methods. However, we can see a qualitative shift in the same
direction between both Runs 2 & 3 and the simple simulation. The result of
both Runs 2 & 3 is a multi-generation shell system, whereas Run 1 can in
principle give rise only to one generation of shells.

For both Runs 2 & 3 there were more particles trapped in the gravitational
field of the host galaxy and a large part of them has been transported to the
vicinity of the center of the host galaxy. The outer shells (of the first
generation) are more diffuse and significantly less luminous when compared to
Run 1, whereas their positions remain essentially the same. On the other hand,
in the later generations, there are brighter shells, some of which can overlap
with the first-generation shells. The shells of the later generations appear
on smaller radii and are often bright, whereas in Run 1 shells at small radii
are completely missing. The evolution of the shell brightness in Run 1 is
somehow calmer, whereas the shells of the later generations in Runs 2 & 3 have
a tendency to reach very high brightness in certain small range of radii.

?figurename? 54: Radial histogram of stars of the secondary galaxy, centered
on the primary 5 Gyr after the first passage of the secondary galaxy through
the center of the primary galaxy for the three different simulations – run 1
(red), run 2 (green) and run 3 (blue). For description of all runs, see text
in Sect. 22.1.

Runs 2 & 3 are more consistent with observations (Sect. 3) in the sense that
their contain shells on both small and large radii. An important thing to
notice is that within our model, any subsequent passage of the secondary
galaxy through the center of the primary galaxy does not lead to a complete
destruction of the shells from the previous passages. Towards the center of
the host galaxy, we find shells with larger surface brightness, also a feature
found in real shell galaxies. At the same time, in Runs 2 & 3 we can find
faint shells surrounded by brighter ones from both sides, another effect
observed in real galaxies and impossible to reproduce in a simple simulation.

The main difference between Run 2 and Run 3 lies in the positions of the
shells from the later generations – those shells that dominate the system in
later times thanks to their brightness. The timing of the second passage of
the secondary galaxy through the center of the host galaxy is very similar for
Run 2 and Run 3 but the difference in energy, mass and decay of the secondary
galaxy is sufficient to produce shells at different radii. Run 3 also differs
significantly from Run 2 (and also Run 1) in that a bright shells system
persist even a long time after the first approach of the secondary galaxy (7
Gyr). However, we cannot say whether it is Run 2 or Run 3 that better
describes the real merger of two galaxies under given initial conditions. This
indicates that quantitative modeling of a shell system using test-particle
simulation is very difficult or even impossible.

In spite of the difficulties, we dare to state qualitative conclusions
independently on the method chosen for the tidal decay of the secondary
galaxy: the introduction of the dynamical friction and the gradual decay to
our simulations dramatically changes the appearance of shell structures. Only
the outermost shell of the first generation is not overlayed by later,
brighter generations of shells added during next passages of the satellite
through the center of the primary. While the position of the outermost shell
is not much affected by the dynamical friction, its brightness is rapidly
lowered due to the many particles staying trapped in the weakened but
remaining potential of the small galaxy.

####  22.2 Dark halo

To be even more realistic, we present a two-component model of the galaxy – a
luminous component with a dark halo. The velocity dispersion of each component
is under the influence of the other (Sect. 17.4). The velocity dispersion is
an important parameter of the dynamical friction a thus values of the friction
induced by each component slightly differ (the amount depends on parameters)
from the values we get when the component is isolated (Sect. E.1).

We performed three simulations with parameters listed in Table 7. In all the
cases, the mass of the secondary galaxy is 0.02 of the total mass of the
primary; and the secondary approaches with escape velocity. Dynamical friction
is calculated using our modification of the Chandrasekhar formula (Appendix
E). The mass of the secondary galaxy was gradually lowered during the
simulation according to the number of test particles under the current tidal
radius (Sect. 21.1) and its Plummer radius was being adjusted according to the
method described in Sect. 21.2.

run | $\varepsilon_{*}$ | $M_{*}$ | $\varepsilon_{\mathrm{DM}}$ | $M_{\mathrm{DM}}$ | $\varepsilon_{\mathrm{s}}$ | $M_{\mathrm{s}}$ | $D_{\mathrm{ini}}$ | $v_{\mathrm{ini}}$
---|---|---|---|---|---|---|---|---
| kpc | M⊙ | kpc | M⊙ | kpc | M⊙ | kpc | km$/$s
M0B0 | 7 | $3.2\times 10^{11}$ | - | - | 2 | $6.4\times 10^{9}$ | 180 | 125
M2B6 | 7 | $3.2\times 10^{11}$ | 60 | $6.4\times 10^{12}$ | 2 | $1.344\times 10^{11}$ | 300 | 443
M6B10 | 7 | $3.2\times 10^{11}$ | 100 | $1.92\times 10^{13}$ | 2 | $3.904\times 10^{11}$ | 300 | 756
?tablename? 7: Parameters of simulations. The potentials of the galaxies are
modeled as a single Plummer sphere for the secondary galaxy in all runs and
the primary galaxy in the run M0B0; and as a double Plummer sphere for the
primary in runs M2B6 and M6B10. Indices *, DM and S refer to the luminous and
dark components of the primary galaxy and the secondary galaxy, respectively.
$\varepsilon$ is Plummer radius, $M$ total mass of the Plummer sphere,
$D_{\mathrm{ini}}$ initial distance between centers of the secondary and
primary galaxies and $v_{\mathrm{ini}}$ their mutual velocity.

?figurename? 55: Evolution of the merger for three different configurations of
the dark halo of the primary galaxy – distance between galaxies, number of
particles bound to the secondary galaxy and its Plummer radius. For the
parameters of the mergers, see Table 7.  ?figurename? 56: Snapshots from three
simulations, for the parameters of the mergers, see Table 7. Time stamps refer
to the time elapsed since the first passage of the secondary galaxy through
the center of the primary galaxy. Each panel covers 300$\times$300 kpc and is
centered on the host galaxy. Only the surface density of particles originally
belonging to the satellite galaxy is displayed. The density scale varies
between frames, so that the respective range of densities is optimally
covered.

Fig. 55 illustrates the evolution of the distance between the galaxies and the
gradual decay of the secondary galaxy. Time stamps of each run have been
shifted so that in each case the secondary galaxy reaches the center of the
primary galaxy at time 0\. In the first case (run M0B0 without any halo), the
secondary galaxy lost all particles during the first passage and this
simulation is rather equivalent to simulations with instant disruption. In the
configurations that include the halo (runs M2B6 and M6B10), the velocity is
such on the other hand that the primary galaxy catches only very few particles
in the first passage and a significant growth of the shell structure is
observed only in later phases of the merger.

Snapshots for three different times are shown in Fig. 56 and radial histograms
for time 3 Gyr in Fig. 57. For the simulation with a heavy halo (run M6B10)
the particles cover the largest span of energies (apocenters) and in both
simulations with a halo (runs M2B6 and M6B10), new shells on lower radii are
created in further passages of the secondary through the center of the primary
galaxy and many particles end up being caught in the center of the primary. In
the simulation without a halo (run M0B0) the secondary decays in the first
passage, but the particles have mostly sizable energies at that time and thus
have apocenters at larger galactocentric radii or outright escape the system.
The positions of the shells in a given time are obviously different for
different potentials of the primary galaxy.

?figurename? 57: Radial histogram of stars of the secondary galaxy, centered
on the primary 3 Gyr after the first passage of the secondary galaxy through
the center of the primary for three different simulations. The meaning of
colors is the same as in Fig. 55 (red: M0B0 – without halo, green: M2B6 – halo
20 times more massive than the luminous component, blue: M6B10 – 60 times more
massive). For the parameters of the mergers, see Table 7.

The main effect of the halo on the shell system is probably in that its
presence (through the increased mass of the primary galaxy) allows for a
faster development of shells at larger radii, despite the secondary releasing
in our case only a small part of its stars during its first passage through
the center of the host galaxy. Meanwhile, there are additional shells created
in the following passages, creating the high radial range of shells observed
in some galaxies which has continuously proven difficult to reproduce in
simulations.

The increased total mass of the host galaxy is apparently more important than
the difference in the dynamical friction caused by the differences in local
density and velocity dispersion for different halo configurations. The more
massive halo accelerates the secondary galaxy more, reducing the loss of its
energy via the dynamical friction and increasing the time before a subsequent
return of the secondary galaxy. But the higher energy/velocity of the
secondary galaxy allows the existence of shells at larger radii - while it is
important to note that in our simulations, we see shells at 200 to 300 kpc
from the center of the host galaxy, which is a distance where noone ever
observed (or even looked for) shells in real galaxies.

?figurename? 58: Comparison of histograms of radial distances of shells‘
particles in the self-consistent (green) and test-particle (red) simulations
at two different time steps.

####  22.3 Self-consistent versus test-particle simulations

In this section, we compare two simulations with the same initial conditions,
one conducted in a self-consistent manner using GADGET-2 by Kateřina
Bartošková, the other one with test particles. Originally we intended to keep
the parameters of the primary galaxy, but a two-component (luminous+dark
matter) Plummer sphere is not a consistent system for an arbitrary choice of
parameters, particularly for those we have used so far. The system is
consistent when each physically distinct component has a positive distribution
function (Ciotti, 1996). Thus we have chosen the following parameters for the
merger:

The potential of the primary galaxy is a double Plummer sphere with respective
masses $M_{*}=2\times 10^{11}$ M⊙ and $M_{\mathrm{DM}}=8\times 10^{12}$ M⊙ ,
and Plummer radii $\varepsilon_{*}=8$ kpc and $\varepsilon_{\mathrm{DM}}=20$
kpc for the luminous component and the dark halo, respectively. The potential
of the secondary galaxy is chosen to be a single Plummer sphere with the total
mass $M=2\times 10^{10}$ M⊙ and Plummer radius $\varepsilon_{*}=2$ kpc. The
cannibalized galaxy is released from the distance of 200 kpc from the center
of the host galaxy with the initial velocity 102 km$/$s in the radial
direction (as always).

Snapshots from several times for both of the simulations are shown in Fig. 59,
radial histograms for the chosen times in Fig. 58. Video from the self-
consistent simulation is part of the electronic attachment. For the
description, see Appendix H point 5.

?figurename? 59: Snapshots from a test-particle simulation (left) and from the
corresponding self-consistent simulation (right). Time equal zero corresponds
to the passage of the secondary galaxy through the center of the primary
galaxy. Each panel covers 400$\times$400 kpc and is centered on the host
galaxy. Only the surface density of particles originally belonging to the
satellite galaxy is displayed. The density scale varies between frames, so
that the respective range of densities is optimally covered.

Unfortunately it turns out that for this choice of parameters, our method of
including the gradual decay of the secondary galaxy (Sect. 21.1) does not lead
to a very gradual decay at all. In the case of simulation with test particles,
the secondary galaxy loses all its particles near its first passage through
the center of the primary galaxy. Thus we use the model with instant
disruption of the secondary instead. To make the comparison even worse, the
self-consistent simulations behaves in yet another way: the core of the
secondary galaxy survives the first two passages through the center of the
primary galaxy and for some reason dissolves close to its apocenter.

However, despite these significant differences, the results are surprisingly
similar. Most importantly, the radii of the outermost shells differ by less
than 10%. In comparison with our enhanced test-particle simulations (e.g.,
Sect. 22.1 – Runs 2 & 3), the self-consistent simulation does not show a
significant transport of particles of the secondary galaxy to the area around
the center of the host galaxy, neither does it produce shells at low radii.
Where on the other hand the self-consistent simulation resembles more the
enhanced test-particle simulations than the simple test-particle simulation
(with instantaneous disruption of the secondary galaxy and no dynamical
friction) is the dramatic decrease of the brightness of the outermost shell on
large radii (compare with Sect. 22.1 – Fig. 54).

###  23 Discussion

Our goal in this Part of the work was to include the dynamical friction and
the gradual decay of the secondary galaxy in the test-particle simulations. It
has been previously pointed out that coupling of these phenomena is a key
effect in the shell structure formation but it was never modeled in much
detail so far. Using these simulations, we aimed to asses the plausibility of
timing the shell-creating merger using the outermost observed shell in a shell
galaxy.

For the dynamical friction we used our own modification of the Chandrasekhar
formula for radial trajectories, Appendix E, which is more faithful to the
true stellar distribution function of the host galaxy. The dynamical friction
calculated in this way is fully determined by the distribution function of the
host galaxy and the mass and velocity of the secondary, thus is contains no
free parameters. Comparison between our modification and the commonly used
form of the Chandrasekhar formula, Sect. E.2, shows that the use of a constant
Coulomb logarithm is completely inappropriate for radial mergers. But when
compared with the self-consistent simulations, our method is found to
significantly overestimate the friction, Sect. 20.2. In reality, the dynamical
friction in a radial merger depends on the whole merger history and thus can
be hardly reproduced by any modification of the Chandrasekhar formula, Sect.
19.1. Our simulations thus have to be understood as the upper estimate on the
true effect of the dynamical friction on the shell formation.

Including the tidal disruption of the secondary galaxy in test-particle
simulation is even more complicated. We have tried several methods, Sect.
21.1, and none of them is a priori better than any other. Moreover we tried to
reflect on the change of the shape of the gravitational potential of the
cannibalized galaxy during the merger using a variable Plummer radius, Sect.
21.2. We have carried out several simulations using different methods for the
decay of the secondary galaxy, focusing on qualitative effects in which these
simulations differ from simple simulations that assume instantaneous breakdown
of the secondary galaxy and no dynamical friction. We can believe that effects
that are independent of the method used are more likely to participate in
shell forming process in reality.

One such effect is that while the position of the outermost shells of the
first generation is not much affected by the inclusion of the gradual decay
and dynamical friction in the simulations, its brightness is drastically
lowered. The same effect is observed in our self-consistent simulation, Sect.
22.3. Even easily inferring the age of the collision is rendered impossible
(as already pointed out by Dupraz and Combes, 1987). The shell systems in Fig.
54 (Sect. 22.1), all having the outermost shell at $+150$ kpc, are seen 5 Gyr
after the first passage of the cannibalized galaxy through the center of the
host galaxy. If we observationally identify the leftmost shell (around $-80$
kpc in Fig. 54) as being the outermost one, we would mistakenly estimate the
merger age to be only $\sim 2.5$ Gyr. We would also wrongly determine the
direction from which the secondary galaxy came: assuming the classical picture
(based on simulations without friction and with instantaneous disruption), the
outermost shell would be located on the side from which the satellite came, so
we would conclude it went from the left while the opposite is true.

Furthermore, with respect to the simple simulation, in the simulations with
gradual decay of the secondary, we observe the creation of new generations of
shell during every passage of the remnant of the secondary galaxy through the
center of the primary. In the consecutive generations, shells are created at
lower radii and with higher brightness. It is important to see that, in our
simulations, the subsequent passages of the secondary galaxy do not
significantly disturb the existing shell structure of the previous generation
and thus a shell system with a large range of radii is created. The radial
range of shells observed in some real shell galaxies is truly impressive and
it is impossible to reproduce in a simple simulation. It is also worth noting
that in simulations with the gradual decay of the secondary, a large part of
the mass of the secondary ends up in the proximity of the center of the
primary galaxy.

Presence of a dark matter halo in the primary galaxy, Sect. 22.2, changes not
only the dependence of the period of radial oscillations on radius (Sect. 9),
but also the range of stellar energies through the change of the velocity of
the accreted satellite. The halo allows for a faster development of shells at
larger radii. A more massive halo creates a larger range of shell radii in our
simulations than a less massive one. The increased total mass of the host
galaxy is more important than the difference in the dynamical friction caused
by the differences in local density and velocity dispersion for different halo
configurations. The more massive halo accelerates the secondary galaxy more,
reducing the loss of its energy via the dynamical friction and increasing the
time before a subsequent return of the secondary galaxy. The higher velocity
of the secondary galaxy also means that the primary galaxy catches only very
few particles in the first passage and a significant growth of the shell
structure is observed only in later phases of the merger.

In general, it seems that test-particle simulations are not suitable for a
quantitative reproduction of observed shell systems. There is no reliable
(semi-)analytical method to calculate the dynamical friction in radial and
close-to-radial minor mergers. Apparently even more importantly, there is no
universal method to model the tidal decay of the cannibalized galaxy in test-
particle simulations. Unfortunately, it turns out that it is exactly the
details of the decay of the secondary galaxy that affect significantly the
overall shell structure. In two simulations, with apparently small differences
in the loss of mass and energy of the secondary galaxy during the first
passage and the time of the second passage, shells of the second generations
were created at different radii with respect to the shells from the first
generation (which are otherwise very similar between the simulations).
Moreover, the brightness of these shells differs and with each farther passage
of the secondary galaxy, the difference in the appearance of the shell system
increases and the observability of shells in the host galaxy changes by whole
gigayears. Overall, an accurate reproduction of a shell galaxy is a very
delicate matter, as in practice we do not know an exact distribution of mass
in the host galaxy, the original trajectory of the secondary galaxy, nor its
own mass distribution and our simulations suggest that the shell structure is
very sensitive even to small details in these quantities.

Nevertheless even despite the simplicity of the models we used, it turned out
that our test-particle simulations with gradual disruption and dynamical
friction of the secondary galaxy do better than the simple simulations in
reproducing observed features in real shell galaxies. We thus conclude that
also in real galaxies, these features are the result of combined effects of
the gradual decay and dynamical friction.

At the end, we shall stress that while all these details have a large effect
on the overall appearance of the shell system, they are not very important for
the application of the method to measure the host galaxy potential from
kinematical data that we have introduced in Part II. This method relies only
on the assumption that the stars that form one particular shell are moving
along radial trajectories and were released in the center of the primary
galaxy together at some moment in the past. Within the framework the radial-
minor-merger model, neither the gradual decay of the secondary galaxy nor the
dynamical friction do not in principle have a large influence on the radiality
of the stellar trajectories. Also, even when these effects are present, stars
are being released in short time intervals when the secondary galaxy passes
through the center of the primary galaxy, however these intervals are slightly
larger than zero, which would be the case for the instantaneous decay of the
secondary galaxy. This fact causes the shells to be slightly more diffuse and
can interfere with an effort to determine the positions of the spectral peaks
and the shell edge. Nevertheless, in principle the measurement of the
potential should be still possible.

##  ?partname? IV Conclusions

In Part I we have summarized observational and theoretical knowledge about the
shell galaxies according to the available literature. Shell galaxies are
mostly elliptical galaxies containing fine structures which are made of stars
and form open, concentric arcs that do not cross each other. The most
prominent observational characteristics of shells are summarized in 22 points
in Sect. 4. In Sect. 5, we introduce all proposed scenarios of origin of shell
galaxies. The most widely accepted theory, supported by a multitude of
observational evidence, is the close-to-radial minor merger of galaxies
introduced by Quinn (1984). In the framework of this model, Merrifield and
Kuijken (1998) suggested using shell kinematics to measure the potential of
the host galaxy. The issue of the determination of the overall potential and
distribution of the dark matter in galaxies is among the most prominent in
galactic astrophysics since the most successful theory of the evolution of the
Universe so far seems to be the theory of the hierarchical formation based on
the assumption of the existence of cold dark matter, significantly dominating
the baryonic one. Thus, independent measurement of the dark matter content in
galaxies is highly desirable. Measurement of galactic potential is
particularly difficult in elliptical galaxies at large distances from the
center of the galaxy. Incidentally, shells are found mainly in elliptical
galaxies and they do occur in distances up to 100 kpc from the center.

The method of Merrifield and Kuijken (1998) is based on the approximation of a
stationary shell. Using positions of peaks in the line-of-sight velocity
distribution (LOSVD), it allows the calculation of the gradient of the
potential near the shell edge. We have developed this method further in Part
II assuming validity of the radial-minor-merger model and spherical symmetry
of the host galaxy. Using both analytical calculations and test-particle
simulations, we have shown that the LOSVD has a quadruple shape in this
situation. Assuming a constant shell phase velocity and a constant radial
acceleration in the host galaxy potential for each shell, we have developed
three different analytical and semi-analytical approaches (Sect. 11.6) for
obtaining the circular velocity in the host galaxy and the current shell phase
velocity from the positions of the peaks of the maxima of the LOSVD.

The applicability of our different approaches varies with the character of
measured data. As obtaining suitable data is at the very limit of current
observational tools and thus no such data is yet available for analysis, we
have applied our methods to results of a simulation of a radial minor merger.
We were able to reproduce the circular velocity at shell radii to within $\sim
1$ % from the actual value. Applying the method of Merrifield and Kuijken
(1998) to the simulated data, we have derived a circular velocity larger by
40–50% than the true value.

All our approaches, however, derive the shell phase velocity systematically
larger, 7–30%, than the real velocity is. That can be caused by nonradial
trajectories of the stars of the cannibalized galaxy or by poor definition of
the shell radius in the simulation. The method of Merrifield and Kuijken
(1998) does not allow to derive the shell phase velocity at all since it is
based on the approximation of a stationary shell.

In the case of spherical symmetry, the value of the circular velocity directly
determines the amount of mass enclosed under the given radius, thus
determining the dark matter content of the galaxy. On the other hand, the
shell velocity depends on the serial number of the shell and on the whole
potential from the center of the galaxy up to the shell radius and thus its
interpretation is less straightforward. A comparison of its measured velocity
to theoretical predictions is possible only for a given model of the potential
of the host galaxy and the presumed serial number of the observed shells. In
such a case, however, it can be used to exclude some parameters or models of
the potential that would otherwise fit the observed circular velocity.
Moreover, the measurement of shell velocities can theoretically decide whether
the outermost observed shell is the first one created; determine the time from
the merger and the impact direction of the cannibalized galaxy; and reveal the
shells from different generations, which can be present in a shell galaxy
(Bartošková et al., 2011).

In Part II we have examined effects of the gradual decay and dynamical
friction of the cannibalized (secondary) galaxy on the appearance of the shell
structure. Our goal was to asses the plausibility of timing the shell-creating
merger using the outermost observed shell in a shell galaxy. Attempts to date
a merger from observed positions of shells, using simple test-particle
simulations, have been made in previous work of Canalizo et al. (2007)
supporting a potential causal connection between the merger, the post-
starburst ages in nuclear stellar populations, and the quasar.

We have searched for a method to include the gradual decay and dynamical
friction of the secondary galaxy into the test-particle simulations. While
these effects are (along with many other physical processes) naturally
included in self-consistent simulations, using these has also some serious
drawbacks when compared to test-particle simulations. For example, some
effects seen in self-consistent simulations are difficult or outright
impossible to reproduce by analytical or semi-analytical methods. At the same
time, their manifestation in self-consistent simulations is difficult to
separate and sometimes they may even be confused with non-physical outcomes of
used methods. Moreover, self-consistent simulations with high resolution
necessary to analyze delicate tidal structures such as the shells are
demanding on computation time. This demand is even larger if we want to
explore a significant part of the parameter space.

For the dynamical friction we used our own modification of the Chandrasekhar
formula for radial trajectories, Appendix E. The dynamical friction calculated
in this way is fully determined by the distribution function of the host
galaxy and the mass and velocity of the secondary, thus is contains no free
parameters. But when compared with the self-consistent simulations, our method
is found to significantly overestimate the friction, Sect. 20.2. Our
simulations thus have to be understood as the upper estimate on the true
effect of the dynamical friction on the shell formation.

We have tried several methods for including the tidal disruption and
deformation of the secondary galaxy, Sect. 21, and none of them is a priori
better than any other. In our simulations it turns out that the resulting
shell system is very sensitive to small differences during the decay of the
cannibalized galaxy and thus the test-particle simulations are not suitable
for a quantitative reproduction of observed shell systems. We have thus
focused on qualitative effects in which our enhanced simulations differ from
simple simulations that assume instantaneous breakdown of the secondary galaxy
and no dynamical friction. It turned out that these enhanced test-particle
simulations do better than the simple simulations in reproducing observed
features in real galaxies, including features that the simple simulations
cannot show at all. We thus conclude that also in real galaxies, these
features are the result of combined effects of the gradual decay and dynamical
friction.

One effect found commonly in all the enhanced test-particle simulations is
that while the position of the outermost shells of the first generation is not
much affected by the inclusion of the gradual decay and dynamical friction in
the simulations, its brightness is drastically lowered. The same effect is
observed in our self-consistent simulation, Sect. 22.3. Even just inferring
the age of the collision is thus tricky: if we observationally miss the
weakened outermost shell, which should be clearly visible according to simple
simulations, we would underestimate the merger age by a factor of 2. At the
same time, we would also wrongly determine the direction from which the
secondary galaxy came.

Ideally, for systems with multiple shells we would like to combine
measurements of shell kinematics and their radial distribution, possibly also
with measurements of surface brightness profile (Sect. 14.4). The kinematical
measurements supply us with the magnitude of acceleration at the shell edge
and an estimate of the phase shell velocity, which allows us to separate the
shells in different generations, if these are present. Simulations with the
dynamical friction and gradual decay of the secondary galaxies that reproduce
the kinematic and photometric data will then constrain other parameters of the
merger such as its age and the trajectory and nature of the satellite galaxy.
A similar result has been obtained for M31, Fardal et al. (2007, 2008, 2012),
whereas for the other shell galaxies, obtaining the kinematical data is a
great challenge for the next generation of astronomical instruments.

##  ?partname? V Appendix

###  ?appendixname? A Units and conversions

When dealing with galaxies, we need to describe objects and time spans
incommensurable with our daily experience that defines the standard sets of
units, such as SI. Throughout the text we thus use a set of units adapted for
this task – we measure the mass in M⊙ the length in kpc and the time in Myr.
Although their meaning is clear, they sometimes give rise to rather awkward
derived units. We will briefly list the most prominent of them (together with
the basic ones) and give their relation to the SI and cgs units.

  * Time:

1 Myr = $10^{6}$ yr = $3.156\times 10^{13}$ s

  * Distance:

1 kpc = 3 262 ly = $3.086\times 10^{19}$ m = $3.086\times 10^{21}$ cm

  * Mass:

1 M⊙ = $1.989\times 10^{30}$ kg = $1.989\times 10^{33}$ g

  * Velocity:

1 kpc$/$Myr = 977.8 km$/$s = $9.778\times 10^{7}$ cm$/$s (the roundness of
this value allows for an easy conversion for most of our plots)

  * Acceleration:

1 kpc$/$Myr2= $3.098\times 10^{-8}$ m$/$s2 = $3.098\times 10^{-6}$ cm$/$s2

  * Density:

1 M⊙$/$kpc3= $6.768\times 10^{-29}$ kg$/$m3 = $6.768\times 10^{-32}$ g$/$cm3

  * Grav. unit:

1 kpc3$/$Myr2$/$M⊙ = 14.83 m3$/$s2$/$kg = 14 830 cm3$/$s2$/$g –
thus G = $6.674\times 10^{-11}$ m3$/$s2$/$kg = $4.500\times 10^{-12}$
kpc3$/$Myr2$/$M⊙

###  ?appendixname? B List of abbreviations

AU

arbitrary unit, a relative placeholder unit for when the actual value of a
measurement is unknown or unimportant

DM

dark matter

FWHM

full width at half maximum, parameter of Gaussian function

GADGET-2

free software used for self-consistent simulations, see Sect. 20

KDC

kinematically distinct/decoupled cores of galaxies, see Sect. 3.7

LOS

line-of-sight

LOSVD

line-of-sight velocity distribution

MK98

paper about measuring gravitational potential using shell kinematics
Merrifield and Kuijken (1998)

MTBA

Multiple Three-Body Algorithm, a method used by Seguin and Dupraz (1994) to
study dynamical friction in head-on galaxy collisions, see Sect. 19

S$/$N

signal-to-noise ratio

WIM

Weak Interaction Model of origin of shell galaxies by Thomson and Wright
(1990), see Sect. 5.2

###  ?appendixname? C Initial velocity distribution

The shell-edge density distribution,
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$, is defined by Eq. (13).
Note that, since, in the model of the radial oscillations, all stars at the
shell edge have the same energy, the function $N\left(r_{\mathbf{s}}\right)$
determines the distribution of stellar apocenters, the radial dependence of
which differs just slightly from
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)$.

Let $f(r_{\mathrm{ac}})$ and $g(v_{0})$ be the distribution function of the
stellar apocenters and the initial velocities, respectively, then

$g(v_{0})=f(r_{\mathrm{ac}})\frac{\mathrm{d}r_{\mathrm{ac}}}{\mathrm{d}v_{0}}.$
(90)

In almost all cases in the thesis
$\sigma_{\mathrm{sph}}\left(r_{\mathbf{s}}\right)\propto
1/r_{\mathbf{s}}^{2}$, so the distribution function of the stellar apocenters
is a constant function $f(r_{\mathrm{ac}})=A$.

Initially, all stars are at the center of the host galaxy, so

$v_{0}=\sqrt{-2\left[\phi(r_{\mathrm{ac}})-\phi(0)\right]},$ (91)

where $\phi(r)$ is the spherically symmetric potential of the host galaxy.
Then

$g(v_{0})=A\left.\frac{\mathrm{d}\phi(v)}{\mathrm{d}v}\right|_{v=v_{0}}v_{0},$
(92)

where $\phi(v)$ is inverse function to $v_{0}\left(\phi\right)$ given by Eq.
(91).

The correspondence between the shell-edge density distribution and initial
velocity distribution is one-to-one, unlike for example the one between the
spatial (or projected) density and the shell-edge density distribution, Eq.
(14), as the density at one radius receives contributions from particles with
two distinct velocities.

###  ?appendixname? D Introduction to dynamical friction

Appendix D was, with some adjustments, adapted from the master thesis Ebrová
(2007).

####  D.1 A thermodynamic meditation

The dynamical friction is a braking force of gravitational origin, caused by
the sole fact that the area, through which the secondary galaxy (or, in
general, any object passing through a galaxy or another extended object) flies
is not an empty space filled with a smooth potential, but a large sea of
individual stars.

Thinking deeper, we can easily see that some slowdown of the secondary galaxy
is inevitable. Every system, where energy transfer is possible tends to
temperature equilibrium. In a system of at least three gravitating bodies such
a transfer is indeed possible and frequently happens. The relatively fast and
heavy secondary galaxy possesses a decent amount of kinetic energy and as such
it is just a hot piece thrown into a colder sea of the stars of the primary
galaxy. The slowdown of the intruder that cannot be accounted for in the
fixed-potential model, is the way of leveling the temperatures. The kinetic
energy transfers to the primary’s stars – the same effect causes the heating
of the cold disk population in the week interaction model, as mentioned in
Sect. 5.2.

The reality of this process can be grasped from a different point of view. The
relatively massive secondary galaxy attracts the primary’s stars and thus
creates an area of a higher density of stars behind itself. The passing galaxy
is attracted backwards by this condensation, lowering its speed towards the
primary.

####  D.2 Chandrasekhar formula

An analytical derivation of such a braking force is based on the following
thought: In a distant encounter with just one star, the velocity of an object
cannot be changed, instead it is only deflected from the original direction
and thus enriched with a component of speed perpendicular to the original
direction. For a very massive body, as our secondary galaxy is, the magnitude
of this perpendicular component will not be large, neither will be the loss of
the velocity in the original direction. But when it undergoes many such
encounters, the contributions add. The contributions in the perpendicular
directions will have randomly scattered azimuthal angles and thus add to zero
(except for the overall action of the smooth potential). On the other hand,
the contributions to the original direction of the velocity will always be
opposite to it, resulting in the braking of the galaxy.

The Chandrasekhar formula was originally derived by Chandrasekhar (1943). Here
we present a short version of the presentation of the chapter 7.1 in the bible
of the galactic astronomy, “Galactic Dynamics” by Binney and Tremaine (1987).

To start, let us imagine the encounter of our object of interest with a single
star. When two bodies meet, energy is not transferred, but the direction of
velocity of our object changes. It is a matter of a simple mechanics and as a
result, the change of the component of velocity parallel to its original
direction, $\mid\mathbf{\bigtriangleup v}_{M\parallel}\mid$ between the times
$t=-\infty$ and $t=\infty$ is given by (Eq. 7-10b in Binney and Tremaine,
1987; see its derivation there):

$\mid\bigtriangleup\mathbf{v}_{M\parallel}\mid=\frac{2\,m\,V_{0}}{M+m}\left[1+\frac{b^{2}V_{0}^{4}}{G^{2}(M+m)^{2}}\right]^{-1},$
(93)

where $M$ is our object’s (the secondary galaxy) mass, $m$ is the mass of the
star, $b$ the impact parameter (the length of $\mathbf{b}$, the vector
indicating the position of the star in a plane perpendicular to the original
velocity of the galaxy) and $\mathbf{V}_{0}$ is the difference between the
original velocity of the star $\mathbf{v}_{m}$ and velocity of our object
$\mathbf{v}_{M}$, so $\mathbf{V}_{0}=\mathbf{v}_{m}-\mathbf{v}_{M}$. The bold
typeface indicates vectors, and their length is indicated by the same symbol
in normal type.

For an object flying through a field of stars with the phase-space number
density of stars $f(\mathbf{v}_{m},\mathbf{b})$, the change in the parallel
component of velocity $\mathrm{d}\mathbf{v}_{M\parallel}$ in an infinitesimal
time $\mathbf{\mathrm{d}}t$ will be given by the integration of Eq. (93)
multiplied by the density $f(\mathbf{v}_{m},\mathbf{b})$ over the plane of
$\mathbf{b}$ and the space $\mathbf{v}_{m}$. For $\mathbf{b}$ is measured from
a given point in a plane, it is advantageous to use the polar coordinates
$(b,\varphi)$:

$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}=\intop\intop\intop
f(\mathbf{v}_{m},b,\varphi)\frac{2m\,V_{0}(\mathbf{v}_{m})\,\mathbf{V}_{0}(\mathbf{v}_{m})}{(M+m)\left[1+\frac{b^{2}V_{0}^{4}(\mathbf{v}_{m})}{G^{2}(M+m)^{2}}\right]}\mathrm{d}^{3}\mathbf{v}_{m}\,b\mathrm{d}b\,\mathrm{d}\varphi.$
(94)

To derive the Chandrasekhar formula we further assume the homogeneity of the
field of stars, so as the distribution function of the stars does not depend
on $\mathbf{b}$. The remaining $\mathbf{b}$-dependent part is of the following
form a can be easily integrated from 0 to some $b_{\mathrm{max}}$:

$\intop_{0}^{b_{\mathrm{max}}}\frac{b\mathrm{d}b}{1+c^{2}b^{2}}=\left[\frac{\ln(1+c^{2}b^{2})}{2\,c^{2}}\right]_{b=0}^{b=b_{\mathrm{max}}},$
(95)

where in our case $c=V_{0}^{2}/[G(M+m)]$. It is conventional to introduce the
notation

$\Lambda=\frac{b_{\mathrm{max}}V_{0}^{2}}{G(M+m)}.$ (96)

A typical value of $\Lambda$ would be of the order of $10^{3}$, thus we can
neglect the one and put $\frac{1}{2}\ln(1+\Lambda^{2})\cong\ln(\Lambda)$. This
factor is often called the Coulomb logarithm. Furthermore we assume that we do
not err too much when replacing $V_{0}$ in $\Lambda$ by $v_{\mathrm{typ}}$, a
typical speed. Then the Coulomb logarithm does not depend on $\mathbf{v}_{m}$,
and still $V_{0}=\mid\mathbf{v}_{m}-\mathbf{v}_{M}\mid$ and the whole Eq. (94)
goes to

$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}=4\pi\ln(\Lambda)G^{2}m(M+m)\intop
f(\mathbf{v}_{m})\frac{\mathbf{v}_{m}-\mathbf{v}_{M}}{\mid\mathbf{v}_{m}-\mathbf{v}_{M}\mid^{3}}\mathrm{d}^{3}\mathbf{v}_{m}.$
(97)

The integral is of exactly the same form as in the Newton’s law of gravity and
if the stars move isotropically, the density distribution is spherical and by
Newton‘s first theorem (see Binney and Tremaine, 1987; chapter 2), the total
acceleration of our object by dynamical friction is:

$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}=-16\pi^{2}\ln(\Lambda)G^{2}m(M+m)\frac{\intop_{0}^{v_{M}}f(v_{m})v_{m}\mathrm{d}v_{m}}{v_{M}^{3}}\mathbf{v}_{M}$
(98)

i.e., only stars moving slower then our object contribute to the force and
this force always opposes the motion. Eq. (98) is usually called the
Chandrasekhar dynamical friction formula.

If $f(v_{m})$ is Maxwellian with dispersion $\sigma$

$f=\frac{n_{0}}{(2\pi\sigma^{2})^{3/2}}\exp(-\frac{1}{2}v^{2}/\sigma^{2}),$
(99)

we can integrate Eq. (98). The density of the stars is $\rho_{0}=n_{0}\,m$ and
for $M\gg m$, what happens to be our case, we can put $(M+m)\cong M$, and then
Eq. (98) reads:

$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}=-\frac{4\pi\ln(\Lambda)G^{2}\rho_{0}M}{v_{M}^{3}}\left[\mathrm{erf}(X)-\frac{2\,X}{\sqrt{\pi}}\mathrm{e}^{-X^{2}}\right]\mathbf{v}_{M},$
(100)

where $\Lambda$ is given by Eq. (96), $X\equiv v_{M}/(\sigma\sqrt{2})$ and
erf($X$) is the error function given by

$\mathrm{erf}(X)\equiv\frac{2}{\sqrt{\pi}}\intop_{0}^{X}\mathrm{e}^{-t^{2}}\mathrm{d}t$
(101)

for which we can obtain tabulated values, or we can pre-generate them
numerically with an arbitrary precision.

####  D.3 What a wonderful universe

Giving it a deeper thought, one can consider the validity of the Chandrasekhar
formula almost a miracle. We have by the way disclosed that it works, at least
approximately – the confrontation with numerical simulations of flybys through
a galaxy or a cluster has been carried out by e.g. Lin and Tremaine (1983);
Bontekoe and van Albada (1987), who proved that the analytical solution (given
by the Chandrasekhar formula) is in a good agreement with the simulations in a
relatively wide range of situations. The analytical solutions has some freedom
in the Coulomb logarithm which is not completely well-defined. Its correct
choice can help to better reproduce the numerical results and compensate other
drawbacks of the formula – anyway, the freedom is small when we demand the
Coulomb logarithm to stay constant.

?figurename? 60: The path and velocity changes of the objects undergoing
encounters with individual stars. The absolute value of the velocity remains
unchanged.

But back to our astonishment. When the secondary galaxy deviates from its
course, its speed in the original direction is reduced. But after meeting
another star that compensates the deviation, it also gets back the original
velocity in this direction, as is shown in Fig. 60.

The point is that the Chandrasekhar formula evaluates the change of the
parallel component of the velocity after the flyby from infinity to infinity
for every single star with the same initial conditions and then adds these
changes and applies them to the secondary galaxy in one moment, the moment of
the closest approach with these stars, see Fig. 61. The change of the parallel
component of the velocity and the compensation of the changes in the
perpendicular direction then happen somehow at the same time, although the
magnitude of their effect is calculated as if they happen consecutively – and
by some wonder, it works.

?figurename? 61: A schematic depiction of the change of the velocity of the
secondary galaxy after three steps. In every moment, only the influence of the
stars lying in one plane perpendicular to the motion of the galaxy is taken
into account.

Let us just remark that the fact that we account for the influence of the
stars in the moment of the closest approach is not so strong neglection.
During an encounter of two bodies, roughly one half of the velocity change
takes place around the point of the closest approach on the scale of the
impact parameter. For the encounter of the galaxy with two stars, it is
confirmed in the right panel of Fig. 62.

####  D.4 Why does it work?

We can see the mechanism of the dynamical friction in action even in a simple
model of a “galaxy” interacting with two “stars”, results of which are seen in
Fig. 62. Although the model is a very simple one, it allows us to see in
practice that yet in the system of three bodies (in contrary to two) the
permanent energy and momentum transfer is possible. The symmetry of the
configuration ensures that the galaxy will keep a straight line and thus any
change of velocity it undergoes will be a change in the magnitude of the
velocity. According to the idea of an infinite sea of stars, we take into
account only the interaction between the stars and the galaxy, not mutually
between the stars.

?figurename? 62: The result of the simulation. A large body (with the mass of
200 M⊙, straight black line) moves in the direction of the $x$-axis (with the
velocity of only 100 m$/$s – this and the other unrealistic values have been
chosen only to make the picture more illustrative in a linear and uniform
scale) and encounters a pair of stars (2 M⊙ each) that are initially place
symmetrically with respect to its track (0.1 pc from the track). The mutual
gravitational attraction of the stars is neglected. The right panel shows the
development of the velocity of the large body during the closest approach. The
blue line represents its original velocity, thus its path if the stars were
not present.

It is clear that due to the galaxy’s gravity, the stars begin to move towards
its track (meanwhile also moving towards the galaxy along the track, but let
us not care for a moment). While the stars move towards the track, the
attraction accumulates and they gather speed. When they cross the galaxy’s
track, the galaxy starts pulling them back (at least when we speak about the
perpendicular component of the velocity) and they slow down. Anyway, thanks to
the fact that they cross the track _after_ the galaxy’s passage, they spend
more time in the phase where their perpendicular velocity component is
increased than otherwise and finally they retain some speed in this direction.
But it means they have gathered kinetic energy, what must be at the expense of
the galaxy’s kinetic energy and so the speed of the galaxy must have decreased
(that is the dynamical friction) – even though it has moved much faster than
before during the closest approach of the encounter. In reality, the situation
is a little more complex, because apart from the energy, the momentum has to
be also conserved – the momentum of the galaxy has decreased and so the stars
must have also a non-zero parallel component of the velocity, to maintain this
component of momentum.

In accordance with the derivation of the Chandrasekhar formula, we use Eq.
(93) just multiplied by two to derive the analytical formula for the change of
the galaxy’s velocity. For the impact parameter $b$ we obviously put the
original distance between the stars and the galaxy’s track. Our numerical
tests for various values of parameters (masses, initial velocity of the
galaxy, impact parameter) show that the analytical results obtained this way
tend to overestimate the decrease in the velocity, typically by about 15 per
cent.

It could be anticipated that the numerical and analytical results will differ,
as the analytical formula counts with two separated encounters from infinity
to infinity. In such a case the galaxy follows a curved trajectory and thus
its interaction with the star is slightly different than when both encounters
happen at the same time and the galaxy is forced to stay on a straight line.
Let us remark that we have tested the model by removing one of the stars and
then the results for the change of the parallel component of the velocity
differ from the prediction in fractions of per mille.

In reality, the situation is even more complex, there are many stars in the
game and they also mutually interact and undergo the influence of all the
surrounding stars that do not take part in the dynamical friction directly.

###  ?appendixname? E Our method

In Ebrová (2007) we have introduced our method to calculate the dynamical
friction in restricted $N$-body simulations during the radial merger. In this
section we remind the reader of its characteristics and derivation as
introduced in the master thesis.

####  E.1 Avoiding some approximations

The Chandrasekhar formula contains two kinds of inaccuracies. The first of
them is the principal one, namely the fact that the change in the parallel
component of the velocity from any individual star is added instantaneously at
the point of the closest approach (of the secondary galaxy) to it. We have
already shown that it is not too wrong, but what is worse, the influence of
the star is taken to be such as if the galaxy passed it from infinity to
infinity and there was nothing in the universe but the star and the galaxy.
Sects. D.2–D.4 for details.

The second source of inaccuracy lies in all the approximation that have been
done when passing from Eq. (94) to Eq. (98). These will concern us in this
section, leaving aside the assumptions of the Maxwellian velocity distribution
and the negligence of the masses of the stars compared to that of the
secondary galaxy, that led us from Eq. (98) to Eq. (100), which we use in the
simulations and keeping the “principal inaccuracy” mentioned above.

The first approximations that allowed us to integrate Eq. (94) over the plane
of the impact parameter was the assumed homogeneity of the star field, i.e.
that the distribution function does not depend on position. Then we have taken
the Coulomb logarithm to be independent of velocity of the stars
$\mathbf{v}_{m}$ (it obviously isn’t, but it varies slowly) and this has
allowed us to simplify the $\mathbf{v}_{m}$-integral and given a suitable
choice of the distribution function we could even carry out the integration
(see Sect. D.2). Both steps are only approximate even in the simple case of
the spherical galaxy with the Plummer profile, as both the density – Eq. (74)
and the velocity dispersion – Eq. (80) of the Plummer sphere do depend on the
radius.

If we wish to avoid these simplification, we have to turn back to Eq. (94) and
put in e.g. the Maxwellian distribution, Eq. (99), for
$f(\mathbf{v}_{m},b,\varphi)$, together with putting $n_{0}m=\rho$, where
$\rho$ is the density of the primary at a given point – keeping in mind that
the radius _$r$_ (the distance of a point from the center of the primary
galaxy) on which the formulae depend is a function of $b$_,_ $\varphi$ and in
fact also of the direction of motion of the braked body (the secondary
galaxy). When dealing with the radial mergers, this direction points towards
the center of the primary galaxy and $r$ becomes a particularly simple
function of $b$:

$r=\sqrt{d^{2}+b^{2}},$ (102)

_where_ $d$ is (also in the following) the distance between the centers of the
primary and the secondary galaxy. There is no $\varphi$-dependence in the
radial case and the integration gives a trivial factor of $2\pi$. For
simplicity, we put the Eq. (80) for the velocity dispersion, as the friction
is essentially negligible for both the simple and the cut-off dispersion in
the areas where they significantly differ (see Fig. 43). Furthermore, during
the multiple passages that occur in the simulations (where the friction
becomes significant) the secondary galaxy does not reach these areas at all.
Using Eq. (80) for the cut-off velocity dispersion would thus unnecessarily
complicate the already complex formulae.

Putting all this together, we get

$\displaystyle\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}$
$\displaystyle=$ $\displaystyle\frac{3^{5/2}\varepsilon_{\mathrm{p}}^{2}}{(\pi
G)^{3/2}M_{\mathrm{p}}^{1/2}M_{\mathrm{s}}}\intop\intop\frac{\mid\mathbf{v}_{m}-\mathbf{v}_{M}\mid(\mathbf{v}_{m}-\mathbf{v}_{M})}{(b^{2}+d^{2}+\varepsilon_{\mathrm{p}}^{2})^{7/4}}\times$
(103)

$\times\left[1+\frac{b^{2}(\mathbf{v}_{m}-\mathbf{v}_{M})^{4}}{G^{2}M_{\mathrm{s}}^{2}}\right]^{-1}\mathrm{\exp\left[-\frac{3\mathbf{v}_{\mathit{m}}}{\mathit{GM_{\mathrm{p}}}}\sqrt{\mathit{b}^{2}+\mathit{d}^{2}+\varepsilon_{\mathrm{p}}^{2}}\right]}b\mathrm{d}b\,\mathbf{\mathrm{d^{3}}v}_{m},$

where the meaning of the variables is the same as when we derived the
Chandrasekhar formula in Sect. D.2. The indexes $p$ and $s$ again stand for
the parameters of the primary and the secondary galaxy, respectively.

First, we shift the integration variable to
$\mathbf{v}_{m}^{\prime}=\mathbf{v}_{m}-\mathbf{v}_{M}$ and immediately rename
it back $\mathbf{v}_{m}^{\prime}\rightarrow\mathbf{v}_{m}$. We then perform
the scalar product with the unit vector $\mathbf{v}_{M}/v_{M}$ on both sides,
getting the projection of the friction acceleration to the direction of the
velocity of the secondary galaxy. This is by symmetry its only nonzero
component in the radial case and it will be advantageous to deal with a
scalar-valued integral. The negative value means that the friction acts in the
direction opposite to the motion of the braked body, what is the only feasible
situation in any setup with an isotropic velocity distribution in the primary
galaxy.

Transforming to the spherical coordinates (taking the $z$-axis parallel with
the velocity of the secondary galaxy), we have
$\mathbf{v}_{m}\cdot\mathbf{v}_{M}=v_{m}v_{M}\cos\theta$ and again no
dependence on the azimuthal angle, leaving us with the obligatory factor of
$2\pi$. The $\theta$-integral then can be carried out in the form that could
be with some effort put on mere three lines:

$\frac{\mathrm{d}\mathbf{v}_{M\parallel}}{\mathrm{d}t}\cdot\frac{\mathbf{v}_{M}}{v_{M}}=\frac{\sqrt{3\,M_{\mathrm{p}}}G^{3/2}M_{\mathrm{s}}\varepsilon_{\mathrm{p}}^{2}}{2\sqrt{\pi}v_{M}^{2}}\intop_{0}^{\sqrt{R^{2}-d^{2}}}\intop_{0}^{\infty}\frac{b\mathrm{d}b\,v_{m}^{2}\mathrm{d}v_{m}}{\left(\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}\right)^{11/4}\left(G^{2}M_{\mathrm{s}}^{2}+b^{2}v_{m}^{4}\right)}\times$
(104)

$\times$[$\mathrm{e}^{-3\,\frac{\sqrt{\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}}\left(v_{m}-v_{M}\right)^{2}}{GM_{\mathrm{p}}}}\left(\mathrm{G}M_{\mathrm{p}}-6\,v_{m}v_{M}\sqrt{\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}}\right)-$

$-\mathrm{e}^{-3\,\frac{\sqrt{\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}}\left(v_{M}+v_{m}\right)^{2}}{GM_{\mathrm{p}}}}\left(\mathrm{G}M_{\mathrm{p}}+6\,v_{m}v_{M}\sqrt{\varepsilon_{\mathrm{p}}^{2}+d^{2}+b^{2}}\right)$]
,

where $R$ is the considered cut-off of the primary galaxy. We cannot proceed
analytically with the integration (not even in one of the variables), instead
we have solved it numerically in Maple for chosen values of the parameters.

We have come to a formula for the dynamical friction Eq. (104) that is
physically more accurate than the Chandrasekhar formula, but it is valid only
for a radially moving body in the Plummer sphere. It is also only more
accurate in the sense of avoiding the approximation used between Eq. (98) and
Eq. (100) but it is still built atop the “principal inaccuracies” described
above.

?figurename? 63: The value of the integrand (including all the constants) from
Eq. (104) in the dependence on the integration variables (the impact parameter
and the relative velocity between the secondary galaxy and the stars) for the
standard set of parameters (see __ Sect.  _17.5_) and the distance of the
braked body (the secondary galaxy) from the center of the primary of 70 kpc.
The velocity of the body is taken to be 0.2 kpc$/$Myr (1 kpc$/$Myr $\doteq$
1000 km$/$s, see Appendix A). This value is also indicated in the graph by a
red marker – it not surprising to find it near the peak, because there is a
strong contribution from the stars that are in rest with respect to the center
of the primary galaxy, as the Maxwellian distribution peaks in zero.

The reader who considers a formula to be the best figure can enjoy Eq. (104)
and who considers a figure to be the best formula can explore Fig. 63, where
the integrand of Eq. (104) is shown in dependence of both integration
variables for a chosen set of parameters. It is clear that far most of the
acceleration comes from a close neighborhood of the braked body both in the
plane of the impact parameter and the velocity space. However, the maximum of
the integrand does not exactly coincide with the actual speed of the body, as
there is no reason for it to be so, but it is very close.

For a primary galaxy made of two Plummer spheres – one for the luminous
component and one for the dark matter halo – the equivalent of Eq. (104)
becomes much more complicated. It can be obtain in much the same manner as
described in this chapter, only using Eq. (83) instead of Eq. (80) for the
velocity dispersion. But the angular integration is not possible to
analytically, and the resulting three-dimensional integral cannot be written
in a couple of lines’ worth of space. A numerical solution is necessary for
specific values of parameters.

####  E.2 Back to Chandrasekhar formula

We have examined how the braking force according to Eq. (104) differs from
that calculated using the Chandrasekhar formula. The Coulomb logarithm is in
some sense a free parameter of the formula, thus we have adjusted it to
maximize the agreement between the two methods of calculation of the friction.
For further details, see Ebrová (2007).

?figurename? 64: The logarithmic and linear plots of the time dependence of
the dynamical friction for multiple passages of the secondary galaxy for the
standard set of parameters (Sect. 17.5), using $b_{\mathrm{max}}=10$ kpc
together with the lower limit of the Coulomb logarithm
$\ln\Lambda_{\mathrm{crit}}=2$. Red values are computed in the model, blue
values are numerical solution of Eq. (104).

Using a constant value of the Coulomb logarithm we did not obtain a good
agreement between the friction calculated using the Chandrasekhar formula and
using our method. The best option seems to be to calculate the value of the
Coulomb logarithm in every step from the actual value of the velocity of the
secondary galaxy. The $V_{0}$ in the definition Eq. (96) for $\Lambda$ is the
difference between the velocities of the stars and the secondary galaxy. As
the stellar velocities are isotropic, the average value is just the velocity
of the secondary galaxy with respect to the center of the primary.

There is a uncertainty in the parameter $b_{\mathrm{max}}$ in the same
equation – it should be theoretically equal to the distance between the center
of the secondary and the outer boundary of the primary measured in the plane
perpendicular to the motion of the secondary. But Eq. (100) assumes a
homogeneous field of stars across all this distance, what is obviously not
true. As the plane of the impact parameter is the plane perpendicular to the
radial motion of the secondary galaxy, the density of the primary galaxy is
always the highest in its center and decreases outwards. Thus it may seem that
the $b_{\mathrm{max}}$ should be smaller than the normal distance to the edge
of the primary galaxy, but the approximation of the $V_{0}$ with the velocity
of the galaxy and other circumstances make the situation more complex. The
value of $b_{\mathrm{max}}$ must be chosen in a trial-and-error method for the
chosen parameters of collision so that the magnitude of the friction agrees
best with the numerical solution of the integral Eq. (104).

The adaptive version of the Coulomb logarithm with a suitable chosen
$b_{\mathrm{max}}$ fits nicely in the high velocity regime. The problem
appears when the satellite gets close to its apocenter and also mainly in the
late parts of the merger when the velocity of the satellite is much lower than
during its first passage through the center of the primary galaxy. Here the
adaptive version of the Coulomb logarithm with the chosen $b_{\mathrm{max}}$
significantly underestimates the friction when compared with the numerical
solution of the integral Eq. (104). So we use the adaptive Coulomb logarithm
until its value drops under a certain limit $\ln\Lambda_{\mathrm{crit}}$, then
we put this limit for the Coulomb logarithm instead. With this modification of
the Chandrasekhar formula, we can achieve a reasonable agreement, see Fig. 64.
$b_{\mathrm{max}}$ and the lower limit for the Coulomb logarithm are free
parameters and they depend on the parameters of the radial merger – the
initial mutual velocity of the galaxies, their masses and Plummer radii.

####  E.3 Incorporation of the friction in the simulation

The question of incorporation of the dynamical friction in the simulations of
the shell formation is tricky. In a fully self-consistent simulation, the
dynamical friction would be automatically included, but such a simulation
would be extremely demanding on resources – for the friction to be really well
simulated, the number of particles of primary galaxy should not be several
orders of magnitude smaller than the true amount of stars in the galaxies.
Joining the stars in a smaller amount of more massive objects systematically
overcounts the friction. Peñarrubia et al. (2004) remarked that Prugniel and
Combes (1992); Wahde and Donner (1996) have indeed shown that the dynamical
friction is artificially increased if the particle number is small. Using the
analytical formula for the friction is not devoid of problems, but in some
respects it could be more accurate than some of the self-consistent
simulations.

On the other hand, the number of the particles of the secondary is an
important quantity for the visibility of the shells in the simulations. And
for the large number of required test particles $(\sim 10^{6})$ that represent
just the secondary galaxy, even our “simple” simulations take hours of
computation on a contemporary desktop computer. Furthermore, to explore the
parameter space we have to run a lot of simulations, so we can really use a
handy (semi-)analytical formula. We can easily add the acceleration calculated
by Eq. (104) into the equation of motion of the galaxies.

It is worth mentioning that we departed in two aspects from the potential that
we chose to model the merging galaxies. We assumed Maxwell velocity
distribution, Eq. (99). This is not exactly true for the Plummer sphere, but
the difference is small and the true velocity distribution in real galaxies is
not known, so we cannot do much better, or say exactly how big mistake do we
make.

The secondary galaxy is here treated as a point mass what artificially
increases the friction, because the extended character of the galaxy softens
the force (Sect. 17.2). Specially the stars with a small impact parameter with
respect to the center of the secondary galaxy fly straight through it and
their effect is significantly reduced compared to the Chandrasekhar formula
for the point mass. The overestimation of the dynamical friction is not a
crucial problem as we want to estimate how much the shell system is influenced
by it – we can assume that the reality is not worse than our results and we
get the upper bound on the effect.

###  ?appendixname? F Tidal radius

For starters, let us remind the reader of the derivation of the tidal radius,
as presented in Ebrová (2007). The tidal forces acting on an object are often
derived using the following picture: A massive body (secondary galaxy) as a
whole follows the force acting on it in its center of mass. But the force
acting on outer parts of the body is different, as it is at different
distances of the source (the primary galaxy). If this difference is larger
than the binding force with the secondary for a given star, it is stripped
off.

The tidal radius $r_{\mathrm{tidal}}$ is then defined as the distance (from
the center of the secondary), where the difference of the force of the primary
from its force in the center of mass of the secondary is just equal to the
force from the secondary:

$F_{\mathrm{p}}(d-r_{\mathrm{tidal}})-F_{\mathrm{p}}(d)=F_{\mathrm{s}}(r_{\mathrm{tidal}}),$
(105)

where _$d$_ is the separation between the centers of the galaxies and
$F_{\mathrm{p}}(r)$ and $F_{\mathrm{s}}(r)$ is the force from the primary and
the secondary for a given test particle (its mass is immediately canceled out
from the equation).

For two point-like bodies (with masses $M_{\mathrm{p}}$ and $M_{\mathrm{s}}$),
we can write Eq. (105) as:

$\frac{G\,M_{\mathrm{p}}}{d^{2}(1-r_{\mathrm{tidal}}/d)^{2}}-\frac{G\,M_{\mathrm{p}}}{d^{2}}=\frac{G\,M_{\mathrm{s}}}{r_{\mathrm{tidal}}^{2}}.$
(106)

Assuming further $r_{\mathrm{tidal}}\ll d$ we can use the Taylor expansion
$(1-x)^{-2}\cong 1+2x$ for $x=r_{\mathrm{tidal}}/d$ as it is then a small
quantity. under this assumption we get a simple formula for the tidal radius:

$r_{\mathrm{tidal}}=d\sqrt[3]{\frac{M_{\mathrm{s}}}{2\,M_{\mathrm{p}}}}.$
(107)

However, for two point masses we can get an exact result for the tidal radius.
Not making any approximation in Eq. (106) we can cast it as a fourth-order
polynomial

$X^{4}-2\,X^{3}+q\,X^{2}-2\,q\,X+q=0,$ (108)

where $X=r_{\mathrm{tidal}}/d$ and $q=M_{\mathrm{s}}/M_{\mathrm{p}}$. A
polynomial with an order less than five can be always solved. In our case,
where _$q$_ is positive, there are two real roots, from which we take the one
that gives $r_{\mathrm{tidal}}<d$ and thus $X<1$. The second real root
corresponds to a point of the other side of the primary galaxy that is not of
interest for us. The expression for this root does not give much insight, but
an interested reader can find it in Appendix G.

?figurename? 65: Tidal radius for two point masses: the approximate solution,
Eq. (107), is shown in blue, the exact solutions in red (the outer one in
light red, the inner in dark red). The shows y-axis $X=r_{\mathrm{tidal}}/d$,
the $x$-axis shows the secondary-to-primary mass ratio.

Eq. (105) gives the tidal radius for the particles on the line connecting the
centers of the two bodies – we call it the inner tidal radius. Similarly we
can write an equation for the particles on the other side of the secondary
than the center of primary lies:

$F_{\mathrm{p}}(d)-F_{\mathrm{p}}(d+r_{\mathrm{tidal}})=F_{\mathrm{s}}(r_{\mathrm{tidal}}).$
(109)

It again leads to a fourth-order polynomial for which we can obtain the root
that we call the outer tidal radius. The approximate solution Eq. (107) is the
same for both equations, Eq. (105) and Eq. (109). Let us remark that the tidal
radius is in any case just proportional to _$d$_ as there is no other scale in
the problem. Fig. 65 shows the dependence of the three radii on the mass ratio
of the bodies. We can see that for all relevant ratios the approximate formula
is just between the inner and the outer tidal radius.

The tidal radius for a point mass is in some sense an oxymoron, as these
objects have zero proportions by definition. For spherically symmetric bodies
we can write Eq. (106) as

$\frac{G\,M_{\mathrm{p}}(d-r_{\mathrm{tidal}})}{(d-r_{\mathrm{tidal}})^{2}}-\frac{G\,M_{\mathrm{p}}(d)}{d^{2}}=\frac{G\,M_{\mathrm{s}}(r_{\mathrm{tidal}})}{r_{\mathrm{tidal}}^{2}},$
(110)

where $M(r)$ is the mass enclosed in the radius _$r$._ Particularly for the
Plummer sphere we get this value integrating Eq. (74) over the sphere with the
radius _$r$_ :

$M(r)=\frac{M}{(1+\varepsilon^{2}/r^{2})^{3/2}},$ (111)

where _$M$_ is the overall mass of the body and $\varepsilon$ is the Plummer
radius. Unfortunately this makes the equation too complex to be easily solved.
Let us compare graphically the tidal radii for point masses and Plummer
spheres of the same overall masses just for one particular case – Fig. 66.

The figure (or a simple thought) shows that the notion of the tidal radius in
a general potential makes sense only when the force grows with the distance.
Otherwise the tidal force acts in the same direction as the gravitation of the
secondary and thus cannot strip off any mass. In the Plummer potential the
force reaches its maximum in $\sqrt{2}\,\varepsilon/2$, so the tidal radius is
not defined under this radius, whereas for the point masses it is defined
everywhere.

The idea of the tidal radius is just an approximation to the complex processes
during encounters of two extended bodies. It also does not define a sphere
around the center of the secondary galaxy, but as we have seen, it is
different for various locations, with the lowest value towards the center of
the primary galaxy and the highest on the opposite side. For these reasons it
is not really useful to improve its evaluation and so we have used the
approximate Eq. (107) that as we have seen gives the values somewhere in the
middle between the two extreme values of the tidal radius.

?figurename? 66: The outer and inner tidal radii (marked with circles) for the
point masses and Plummer spheres with the secondary-to-primary mass ratio of
0.02. In the Plummer case, the Plummer radius of the primary is 0.5 of the
distance between the bodies and the Plummer radius of the secondary is 0.1 of
the same quantity. Blue lines (light blue for the point mass, dark blue for
the Plummer sphere) show the gravitational force of the primary in arbitrary
units, red lines (light red for the point mass, dark red for the Plummer
sphere) show the difference between the gravitational force of the primary in
a given point and its value in 1, where the center of the secondary is. The
tidal radii are the points of intersection of corresponding curves.

###  ?appendixname? G Expressions for the tidal radius

Here we give the analytical formulae for the tidal radii in the system of two
point masses as discussed in Appendix F. For the inner tidal radius we have:

$\frac{r}{d}=\frac{1}{2}+\frac{\sqrt{3}}{6}\left(\frac{\sqrt{y}}{\sqrt[6]{z}}-\sqrt{6-4\,q-\sqrt[3]{qz}-\sqrt[3]{\frac{q^{5}}{z}}+6\,\sqrt[6]{z}\sqrt{\frac{3}{y}}(q+1)}\,\right),$
(112)

where

$y=(3-2\,q)\sqrt[3]{z}+\sqrt[3]{qz^{2}}+q^{5/3}$ (113)

$z=54+q^{2}+6\,\sqrt{81+3\,q^{2}}$ (114)

and for the outer tidal radius we get similar expressions:

$\frac{r}{d}=\frac{1}{2}+\frac{\sqrt{3}}{6}\left(\frac{\sqrt{u}}{\sqrt[6]{v}}+\sqrt{6+4\,q-\sqrt[3]{qv}-\sqrt[3]{\frac{q^{5}}{v}}+6\,\sqrt[6]{v}\sqrt{\frac{3}{u}}(q-1)}\,\right),$
(115)

where

$u=(3+2\,q)\sqrt[3]{v}+\sqrt[3]{qv^{2}}+q^{5/3}$ (116)

$v=-54-q^{2}+6\,\sqrt{81+3\,q^{2}}$ (117)

and in all the expressions we use

$q=\frac{M_{\mathrm{s}}}{M_{\mathrm{p}}}.$ (118)

###  ?appendixname? H Videos

Several videos are also part of the electronic attachment of the thesis. Here
we present their description. Information on details of the simulation process
can be found in Sect. 17. The videos can be also downloaded at:
galaxy.asu.cas.cz/$\sim$ivaana/phd

  1. 1.

1-shells.avi – Video from a simulation of a shell-producing radial minor
merger from a perspective perpendicular to the axis of the merger. The bottom
three panels show an area of $60\times 60$ kpc centered on the primary which
is the zoomed part of the upper panels of size $300\times 300$ kpc. The first
column shows the surface density of both the primary and the secondary galaxy,
the second only the surface density of the particles originally belonging to
the secondary galaxy (corresponding to the host galaxy subtraction, a
technique used in processing real galaxy images). The third column shows the
surface density of particles originally belonging to the secondary galaxy
divided by the surface density of the primary galaxy (also corresponding to an
observational technique). The parameters of the merger are the following: the
mass of the primary is $3\times 10^{11}$ M⊙ , the secondary-to-primary mass
ratio is 0.02, the Plummer radius of the primary is 7.6 kpc, of the secondary
0.76 kpc. The initial relative velocity of the galaxies was equal to the
escape velocity of the secondary and the separation of their centers was 90
kpc. When the centers of the galaxies pass through each other, the potential
of the secondary is suddenly switched off.

  2. 2.

2-shells.mpg – Video from a simulation of a shell-producing radial minor
merger used in Sect. 13. The top panel ($300\times 300$ kpc centered on
primary) shows the surface density of the particles originally belonging to
the secondary galaxy from a perspective perpendicular to the axis of the
merger; the bottom panel shows the density of the particles originally
belonging to the secondary in the space of radial velocity (vertical axis)
versus galactocentric distance (horizontal axis). The potential of the host
galaxy is the same as the one described in Sect. 8.1. Primary is modeled as a
double Plummer sphere with respective masses $M_{*}=2\times 10^{11}$ M⊙ and
$M_{\mathrm{DM}}=1.2\times 10^{13}$ M⊙ , and Plummer radii $\varepsilon_{*}=5$
kpc and $\varepsilon_{\mathrm{DM}}=100$ kpc for the luminous component and the
dark halo, respectively. The potential of the cannibalized galaxy is chosen to
be a single Plummer sphere with the total mass $M=2\times 10^{10}$ M⊙ and
Plummer radius $\varepsilon_{*}=2$ kpc. The cannibalized galaxy is released
from rest at a distance of 100 kpc from the center of the host galaxy. When it
reaches the center of the host galaxy in 306.4 Myr, its potential is switched
off and its particles begin to oscillate freely in the host galaxy.

  3. 3.

3-projection.mpg – Video shoes the simulation from point 2 (used in Sect. 13)
at the time 2.2 Gyr after the decay of the cannibalized galaxy (2.5 Gyr of the
simulation time) from different perspectives. Angle of 0 degrees corresponds
to the perspective perpendicular to the axis of the merger.

  4. 4.

4-friction.avi – Surface density of the particles originally belonging to the
secondary galaxy from two simulation of a radial minor merger from Sect. 22.1
(run 1 – right panels and run 2 – left panels). The first column corresponds
to the simulation with dynamical friction and gradual decay of the secondary;
the other corresponds to the simulation without friction and with the instant
disruption of the secondary near the center of the primary galaxy. The bottom
panels show an area of $60\times 60$ kpc centered on the primary which is the
zoomed part of the upper panels of size $300\times 300$ kpc. The video covers
8 Gyr since the release of the secondary galaxy from distance of 180 kpc from
the center of the primary with the escape velocity. Both simulations were
executed for the the standard set of parameters (Sect. 17.5): the mass of the
primary is $3.2\times 10^{11}$ M⊙ , the secondary-to-primary mass ratio is
0.02, the Plummer radius of the primary is 20 kpc, of the secondary 2 kpc.

  5. 5.

5-selfconsistent.avi – Video from self-consistent simulation of a radial minor
merger from Sect. 22.3. The bottom panel ($400\times 400$ kpc centered on
primary) shows the surface density of the particles originally belonging to
the secondary galaxy from a perspective perpendicular to the axis of the
merger; the top panel shows the density of the particles originally belonging
to the secondary in the space of radial velocity (vertical axis) versus
galactocentric distance (horizontal axis). The potential of the primary galaxy
is a double Plummer sphere with respective masses $M_{*}=2\times 10^{11}$ M⊙
and $M_{\mathrm{DM}}=8\times 10^{12}$ M⊙ , and Plummer radii
$\varepsilon_{*}=8$ kpc and $\varepsilon_{\mathrm{DM}}=20$ kpc for the
luminous component and the dark halo, respectively. The potential of the
secondary galaxy is chosen to be a single Plummer sphere with the total mass
$M=2\times 10^{10}$ M⊙ and Plummer radius $\varepsilon_{*}=2$ kpc. The
cannibalized galaxy is released from the distance of 200 kpc from the center
of the host galaxy with the initial velocity 102 km$/$s.

Videos 2–4 were made from simulated data by Miroslav Křížek.

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