Timescales for Galactic Collisions
Sarah Ragan, Drake University
Advisor: Dr. Curt Struck, Iowa State University
ISU Summer REU Program
1. Introduction
Galaxies, which were once thought to be isolated, unchanging universes, are
now believed to have an interactive history. In fact, interactions are thought to be a key
factor in galactic evolution [1]. Galactic interactions observed today range in magnitude
from tidal interactions to nuclear coalescence. Interactions between galaxies will
change the structure and activity of the galaxies involved.
Most galaxies in the local universe contain supermassive black holes, which are
on the order of 107 – 1010 solar masses [2]. Throughout its lifetime, a galaxy grows in
mass through the hierarchical merging of smaller progenitors [3].
Merging galaxies usually also involve the merger of the central black holes. For
instance, if two galaxies, which each host a supermassive black hole, collide, the black
holes form a binary system, whose lifetime is characterized by several stages before
their probable coalescence.
1.1 Dynamical Friction Stage: S. Chandrasekhar discovered the dynamical friction
phenomenon, and published his results in 1943 [4]. Chandrasekhar’s dynamical friction
model describes a massive body (m1) moving through an infinite, homogeneous
population of smaller objects (individual mass m2, m2<m1). As m1 moves, it will
gravitationally interact with m2’s, pulling them towards itself as it moves, which creates
an overdense wake [1]. Only stars moving slower than m1 contribute to the force,
which, like friction, opposes the motion of m1 [5]. The wake then has a significant
gravitational effect on m1, which will decelerate the massive object.
Not until the 1970’s was Chandrasekhar’s formula applied to the problem of galactic
collisions [1]. Toomre was a pioneer in his claim that this was the chief agent in major
mergers [see 6]. In galaxy collisions, dynamical friction causes each black hole to sink
towards the center of the common gravitational potential [7].
The form of Chandrasekhar’s dynamical friction force formula that we used is
given by the following:
4π ln ΛG 2 (m1 + m2 )m1 ρ
=−
m1 ⋅
dt
v m21
dv m1

2X −X 2 
e .
erf ( X ) −
π


(1)
In the above formula, ρ is the mass density of the background stars, vm1 is the velocity
of m1, X = v m1
2σ (σ = velocity dispersion), and ln (Λ) is known as the Coulomb
logarithm, which is approximated as a constant in our calculations [5].
1.2 Non-hard Binary Stage: When m2 shrinks to a radius at which the mass of the
stars interior to it is approximately equal to the total mass of the binary system (m1 +
m2), the binary becomes “bound.” In this situation, the gravitational force on m2 is
dominated by m1 rather than the surrounding stars [7, 8]. The binary will shrink, and the
effectiveness of dynamical friction will decrease due to the increase of the black hole’s
velocity and reduction of orbital period. During this stage, the binary expels stars that
interact with it. Further evolution of the black hole binary involves two-body relaxation
between stars [9].
1.3 Hard Binary Stage: The black holes continue to spiral together under dynamical
friction. The binary becomes “hard” when the binding energy of the system exceeds the
typical particle kinetic energy. Generally, the binary separation must be
ah ≡
Gm2
4σ c2
(2)
where G is the gravitational constant and σc is the one-dimensional velocity dispersion
of the core [7]. Depending on the mass ratio of the two black holes in the system, the
evolution may proceed in slightly different ways. Compared with a binary black hole
system with equal mass black holes, a system with a low mass ratio (m1>>m2) becomes
hard at smaller separations.
1.4 Gravitational Radiation Stage: As the binary black hole continues to harden, the
separation decreases to the sub-parsec range, and gravitational radiation becomes the
dominant dissipative force [7]. Gravitational radiation is emitted when massive bodies
accelerate, much like electromagnetic radiation is emitted when charged particles move.
With the plans for a gravitational radiation detector currently in the works, astronomers
hope to detect significant signals from a merger in progress.
1.5 Applications: When two galaxies of comparable mass merge, the gas compresses
and a significant starburst is often seen. Galactic mergers are also believed to be to
blame for the existence of counter-rotation of material in galaxies, as well as ultraluminous infrared galaxies (ULIRGs). Counter-rotating galaxies are relatively old
merger remnants that have an inner stellar disk that rotates in the opposite direction as
the rest of the galaxy. ULIRGs are younger, multiple core systems with bolometric
luminosity of order 1012 Lsolar related to nuclear starburst activity. Ninety percent of
ULIRGs can be linked to galactic merger activity [10]. Still another possible state of a
merger remnant is a “double core.” Currently, the belief holds that only in major
mergers (when the masses of the two galactic nuclei are comparable) is the
coalescence of the black holes expected.
1.5 Hypothesis/Purpose: The main focus of this study was to explore the stages of a
galactic merger, the evolution of the black hole binary, and the associated timescales.
After reviewing the literature, we proceeded to explore in detail the first of these stages.
Using simple modeling techniques, we computed orbital timescales for simple cases of
galaxy collisions.
2. Methods
Due to a rapid increase in interest in this topic over the past few decades, a
thorough review of the literature was in order. Upon investigation, it appeared that
some processes had been studied in more detail than others. In light of this fact, the
focus of our study became the early stages of binary black hole evolution.
2.1 Modeling: In order to produce realistic simulations, an appropriate model of a
typical galaxy is necessary. Dehnen and Binney’s [11] mass models of the Milky Way
galaxy were most suitable for our purposes. Dehnen and Binney’s model allowed us to
effectively vary a number of parameters to suit our desired mass model. In accordance
with model 4 in the article, the density of the disk is given by
r 
r 
r 



ρ d (r ) = 6.70 exp − 25.625 −

 + 0.00134 exp − 1 −
 + 1.042 exp − 5.56 −
6400 
3200 
3200 



(3)
where r is the radius in parsecs. The three components of the above expression
correspond to the density of the interstellar medium, thin disk, and thick disk,
respectively. The spherical density distribution for the bulge and halo is given by
 r 2 + 2.78 × 10 6
ρ b ,h (r ) = 

1000





−1.8
 r 2 + 2.78 × 10 6
exp −
3.61 × 10 6

  r 2 + 1.56 × 10 6
 +
1899
 




−1.629
2
6

1 + r + 1.56 × 10

1899





−0.538
(4)
The above distribution is slightly different that what the Dehnen and Binney originally
gave. For simplicity, our models placed a higher emphasis on the density of the halo
and bulge such that the mass distribution is dominated by the spherical component.
Figure 1 is a plot of the resulting mass function that was used in some of the
integrations.
Combination Disk and Bulge MW M(r)
1.E+11
1.E+10
1.E+09
Enclosed Mass (solar)
1.E+08
1.E+07
1.E+06
1.E+05
1.E+04
1.E+03
1.E+02
1.E+01
1.E+00
0
200
400
600
800
1000
1200
1400
1600
Radius (pc)
z = 100
z = 300
z = 500
z = 800
z = 1000
Power (z = 500)
1800
2000
y = 1E+06x1.1821
R2 = 0.8256
Figure 1: Mass model used in our orbit calculations, based on
Dehnen and Binney’s [11] model of the Milky Way. The different colored marks represent
Mass values with different z coordinates (cylindrical).
2.2 Simulations: Unfortunately, much time was spent attempting to integrate orbits
using numerical methods that were not stable enough for our problem. For example, an
early idea was to use the leapfrog algorithm. This algorithm utilizes initial values for
radius and velocity, and uses a constant time step. The velocity is calculated at each
half time step, and that value is used to calculate the radius at the next time step. Alas,
this algorithm, along with others, gave unreasonable results.
The Adams-Bashforth numerical method seemed to produce results that were
more reasonable. However, this algorithm proved to be slightly less but still rather
dependent on the choice of the timestep. Due to limited available time, we more or less
abandoned these numerical methods, and our results are based upon early estimations.
3. Results
Our estimates show a wide range of possible timescales for the merger of black
holes at the center of merging galaxies. Based on Binney and Tremaine’s [5] estimates
of the decay of a satellite globular cluster near a much more massive galaxy (This
estimation is comparable to our situation, because our model considered a point-like
satellite in orbit around a more massive host galaxy.), the general trend that we see is
that more massive companion (secondary) black holes (~108 – 109 Msolar) take much
less time compared to less massive companions (~105 – 106 Msolar) (see Figure 2).
Speaking relatively, when the mass ratio (Msecondary/Mprimary) is close to 1, the merger
timescale is shorter. Also, the greater the initial relative velocity, the more time it will
take for dynamical friction to bring the two nuclei together. For instance, a 108 Msolar
companion black hole interacting with the primary mass distribution (~ 1010 Msolar at 1
kpc) shown in Figure1 starting from a separation of 1 kpc at 300km/s can take ~ 80 Myr
for the separation between the nuclei to become very small.
4. Conclusions and Discussion
Galactic collisions are quite an interesting phenomenon, and they are likely to be
the subject of many studies in the near future. Being a relatively new field, our study
was very general and an elementary first step in answering the countless questions
astronomers have about these events.
Our results show that the timescale for a galactic merger, or, more specifically, a
binary black hole merger, depends heavily on mass and, to a lesser extent, relative
velocity. Dynamical friction is the main agent acting in these interactions, while the
nuclei have a substantial separation. As the two black holes in the system become
closer together, other sources of orbital energy dissipation take over, such as
gravitational radiation.
We have confirmed that in the case of intermediate mass black hole companions
(~107 – 108 Msolar) there is a very wide range of possible timescale requirements. Our
results are consistent with the belief that major mergers are the only ones expected to
show black hole coalescence. At the same time, black holes involved in a minor merger
are not expected to have coalesced.
As was mentioned earlier, there is bound to be a great deal of literature in the
coming years in this area of study. For example, more sophisticated orbit simulations,
allowing for more complex mass models of galaxies (e.g. non-spherical models, random
distribution of mass). Insert gas dynamical effects. Change the orientation of the
companion galaxies approach to the primary galaxy(e.g. prograde and retrograde,
inclination angels, parabolic or elliptical orbits). Do simulations of gas-rich disks, and
study the star formation effects. Consider mergers of more than two galaxies, as
Volonteri et al. [12] suggests in a recent article. There are countless ways to improve
upon the rudimentary work done here. It is, of course, essential to understand the basic
elements of galactic interactions before progress can be made.
time
1´10
9
8´10
8
6´10
8
4´10
8
2´10
8
HL
yr
Dynamical
Friction
time
HL HL
yr
vs. Radius
kpc
radius
2
4
6
8
10
HL
kpc
Figure 2: The above plot shows an approximation of the time required for dynamical
friction to bring the two nuclei of the binary black hole system from a given radius (xaxis) to a very small separation. The color of the curve corresponds to the mass of the
companion (secondary) black hole: black = 106 solar masses, red = 107 solar masses,
green = 108solar masses, and blue = 109solar masses. For each companion mass,
various relative velocities were used. The leftmost (or topmost) curve of one mass set
corresponds to 500 km/s, while the rightmost (or bottommost) corresponds to 100 km/s,
and the intermittant curves are incremented at 100 km/s greater then the previous one.
time
1´10
8
8´10
7
6´10
7
4´10
7
2´10
7
HL
yr
Dynamical
Friction
time
HL HL
yr
vs. Radius
kpc
radius
0.2
0.4
0.6
0.8
1
Figure 3: Above is a plot similar to that of Figure 2, but on a smaller scale.
HL
kpc
References
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