The Boltzmann moment equation approach for

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The Boltzmann moment equation approach for
the
dynamics of galactic stellar disks.
Eduard Vorobyov The University of
Western Ontario
and Christian Theis Institut fur Astronomie,
Universitat Wien
HST image of NGC 4414
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The Boltzmann Moment Equation Approach
On galactic scales, stars can be considered as
point sources that move virtually without
collisions on orbits determined by their common
gravitational potential. Hence, we can use the
collisionless Boltzmann equation to describe the
dynamics of stars.
where f f(t,r,v) is the distribution function
of stars and includes all accelerations
due to gravitational forces and curvilinear
terms.
The collisionless Boltzmann equation is not valid
for an arbitrarily long time.
Two-body stellar encounters ? perturbations to
stellar orbits ? alter the DF
The time over which the memory about the original
DF is completely lost is called the
relaxation time
Trel.
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Direct solution of the collisionless Boltzmann
equation in the 6D position-velocity space is
numerically intractable!
Owing to the large number of stars (1011), it is
possible to describe their
motion as a fluid.
This approach involves taking moments of the
collisionless BE in velocity space to obtain a
set of equations for densities, mean velocities,
velocity dispersions, heat fluxes, etc. of stars.
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Boltzmann moment equations up to second
order in cylindrical
coordinates (Vorobyov Theis, MNRAS, 2006)
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Boltzmann moment equations need to be closed by
some sort of a closure relation. Equation for
the n-th moment involves n-th 1
moments. The zero-heat-flux approximation is
assumed.
We have ten equations for ten unknowns
density r, mean velocities ur, uj, uz
Velocity dispersion tensor (symmetric!)
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Why is this quantity called the heat flux?
All quantities r Qijk in the dispersion
equations can be convolved to the divergence
of a rank-three tensor
In classic hydrodynamics, there is a similar
quantity divergence of a heat flux
By neglecting all terms r Qijk, we consider an
ideal ANISOTROPIC fluid of stars
Are heat fluxes important for the dynamics of
stars in galactic stellar disks?
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The thin-disk approximation and the BEADS-2D code
A substantial fraction of stars in spiral
galaxies are concentrated in the disk. Hence,
to a first approximation, stellar disks in spiral
galaxies may be regarded as having a
zero thickness. We assumed the thin-disk
approximation and neglected all vertical
motions and vertical structure.
Observationally justified, Binney
Merrifield, Galactic Astronomy
The equations are simplified greatly and we have
only six equations for six
variables S , ur , uf , srr , sff , and srf
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The BEADS-2D code solves the Boltzmann moment
equations up to second order in the thin-disk
approximation on a polar grid (r,f) using the
method of finite differences. The gravitational
potential of a thin stellar disk is computed
using the convolution theorem (Binney
Tremaine, Galactic Dynamics).
resolution 512 x 512 grid cells
The artificial viscosity stress tensor
is introduces to smooth out the shocks
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Initial setup
Initial conditions
We consider an initially axisymmetric stellar
disk surrounded by a spherical dark matter halo.
The initial surface density of stars is
distributed exponentially according to
, where S01000 M8
pc-2 r0 4 kpc. The initial Toomre parameter
of the disk is Q1.3
At t0, initial random perturbation to the
stellar surface density is introduced. The
maximum relative amplitude of the perturbation is
10-5.
Spherical dark matter halo
thin stellar disk
Initial rotation curve
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The birth of a spiral structure in the Q1.3
stellar disk
The movie below shows the positive stellar
density perturbation in the disk (log units)
Excellent agreement is found with the results of
linear stability analysis by Polyachenko et al.
(1997). Thin stellar disks with a flat rotation
curve are stable for Q 3.0 (Vorobyov Theis,
MNRAS, 2006).
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The shape of stellar velocity ellipsoids in
spiral galaxies
The local properties of stars in the disk plane
are determined by the velocity
dispersion tensor
srr velocity dispersion in the radial
direction sff velocity dispersion in the
tangential direction srf mixed velocity
dispersion
The local velocity dispersion tensor can be
diagonalized by rotating
the local basis (r,f) at an angle lv
s1, s2 principal velocity dispersions lv
vertex deviation
The principal velocity dispersions form an
imaginary surface that is called the velocity
ellipsoid (velocity ellipse in 2D case). The
ratio s1s2 determines the shape of the velocity
ellipsoid.
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Spatial distribution of the positive stellar
density perturbation (relative to the initially
axisymmetric distribution) at t1.6 Gyr
Ratio s1s2 of the smallest versus largest
axes of the stellar velocity ellipsoids within
the disk plane at t1.6 Gyr
s1s2 can be as small as 0.25
The stellar velocity ellipsoid may have quite an
elongated shape in specific regions
of a spiral galaxy!
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The shape of the velocity ellipsoid along a
radial cut at f90o
Solid line s1s2 Dotted line solar
neighborhood value for s1s2
Stellar spiral arm
The s1s2 ratio attains deep local minima at the
outer (convex) edges of the spiral arms. This
property of the stellar velocity ellipsoid can
potentially be used to trace the STELLAR spiral
arm in spiral galaxies.
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The epicycle approximation
In the epicycle approximation (i.e. small
perturbations to stellar circular orbits), the
tangential and radial stellar velocity
dispersions can be related through the circular
speed uc as
Binney Tremaine Galactic Dynamics
It is difficult to measure the circular speed
(i.e. the speed of a hypothetical star in a
circular orbit determined exclusively by the
gravitational potential) since it requires the
independent knowledge of the gravitational
potential in a galaxy.
The local tangential velocity of stars uf is
often used as a proxy
for the circular speed.
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We check the validity of the epicycle
approximation in spiral galaxies by calculating
the relative error
Solid line is the radial profiles of x(r,f) along
a radial cut at the azimuthal angle f 90o.
Shaded area shows the positive stellar density
perturbation in the disk at f 90o
Stellar spiral arms
The epicycle approximation is severely broken
near the outer (convex) edges of spiral arms!
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The residual velocity field and spiral structure
The residual velocities are obtained by
subtracting the rotational velocities due to the
axisymmetric gravitational field from the proper
velocities of stars. Residual velocities show
the perturbations to stellar orbits introduced
by the stellar spiral arms and the bar
Note this retrograde stream of stars. We believe
it might be responsible for the peculiar shape of
the stellar velocity ellipsoids and the failure
of the epicycle approximation.
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Conclusions

• We developed the BEADS-2D code, which solves the
  Boltzmann moment equations up to second order in
  the thin-disk approximation on a polar grid. The
  fully 3D version is under development.
• The BEADS code evolves directly the observable
  quantities (such as densities, mean velocities,
  velocity dispersions) , which is an advantage
  over more familiar N-body codes.
• The BEADS-2D (3D) code can be applied to study
  the dynamical properties of
• stellar disks in spiral galaxies, in particular
  the shape of the stellar velocity ellipsoids,
• deviations from the epicycle approximation,
  vertex deviation, Oort constants, etc.
• Stellar velocity ellipsoids at the convex edges
  of stellar spiral arms may have
• a peculiar elongated shape. It is important to
  confirm this effect using the data from Hipparcos
  and GAIA missions.
• The epicycle approximation is severely broken
  near the convex edges of stellar spiral arms and
  hence should be used with an extreme caution.

Help of Sergiy Khan to create animations is
gratefully acknowledged.
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Applications of the BEADS-2D code to study the
stability properties of galactic disks
The stability analysis of a thin stellar disk to
a local axisymmetric perturbation states that
the disk is stable if (Safronov 1960, Toomre 1964)
The Toomre criterion should be modified if local
non-axisymmetric perturbations are considered
(Polyachenko et al. 1997)
where b is a function of the rotation curve. In
the most interesting case of a flat
rotation curve b 1.69 and the modified Toomre
criterion becomes
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Q 1.3 t 1.3 Gyr
Q 2.5 t 7.0 Gyr
(log units)
Q 2.0 t 6.0 Gyr
Q3.15 t8.0 Gyr
At Q 3.15 no regular pattern is seen
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The final fate of the Q1.3 stellar disk the
central nuclear spiral
and the outer resonant ring
Gravitational torques break apart the spirals
1.4 Gyr
1.8 Gyr
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The growth of instabilities by swing amplification
An appealing physical interpretation of the
growth of instabilities in stellar disks is
swing amplification
(e.g. Toomre 1981).
center
Evolution of a packet of leading waves in a
stellar Mestel disk with Q1.5, Toomre (1981)
Trailing waves that propagate through the
center emerge as leading waves
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Ringed galaxies
NGC 3081 (Buta, Byrd, Freeman, AJ, 2004)
HST image
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Snapshot of the inner region (lt10 kpc) of the
Q1.3 stellar disk at 1 Gyr
Positive perturbations in stellar density
1 Gyr
Interference of trailing and leading
short-wavelength waves traveling through
the disk
center.
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It is helpful to introduce the parameter
According to Julian Toomre (1966), the maximum
gain of SA is at 1 lt X lt 3
X-parameter profile of the Q1.3
stellar disk
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Pros and cons of the Boltzmann moment equation
approach
Pros (as compared to the N-body approach)

• Directly evolves observable quantities
• Can be easily (relative to N-body) extended to
  include collisional terms
• (Fokker-Planck-type equations) and phase
  transitions (star formation)
• Numerically affordable (as compared to N-body
  codes) and does not require
• a specialized hardware (like GRAPE systems
  for N-body simulations)
• Better suited for comparisons with the results
  of analytical analysis because it
• uses the same set of basic equations.

Cons (as compared to the N-body approach)

• Boltzmann moment equations have to be closed! We
  use the so-called
• zero-heat-flux approximation (Qijk0)
• There are no publicly available Boltzmann moment
  equation codes (but I am
• working on it!).

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The value for the vertex deviation can be defined
as (Binney Merrifield 1998, p.630)
Incomplete (-45o, 45o) valid in the epicycle
approximation, srrgtsjj
extended (-90o, 90o) valid for any perturbation
(Vorobyov Theis 2006)
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In order to visualize the growth of instabilities
in disks, it is common to use the global Fourier
amplitudes defined as
Global Fourier amplitudes provide a rough measure
to the mean relative density perturbation in the
disk.
We computed the Global Fourier amplitudes for
five models characterized by different values
of the initial Q-parameter and having a
nearly-flat rotation curve.
BEADS-2D code shows excellent agreement with the
linear stability analysis!