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Structures, Oscillations, Waves and
Solitons in Multi-component Selfgravitating Systems
Kinwah Wu (MSSL, University College London)
Ziri Younsi (P&A, University College London)
Curtis Saxton (MSSL, University College London)
Outline
1. Brief Overview
2. Galaxy clusters as a multi-component systems
- stationary structure
- stability analysis
3. Newtonian self-gravitating cosmic wall
- soliton formation
- soliton interactions
4. Some speculations (applications) in astrophysics
Solitons:
Some characteristics
Non-linear, non-dispersive waves:
- the nonlinearity that leads to wave steeping counteracts
the wave dispersion
Interact with one another so to keep their basic identity
- “particle” liked
Linear superposition often not applicable
- resonances
- phase shift
Propagation speeds proportional to pulse height
Solitons are common
- It is a general class of waves, as much as linear waves and shocks.
- Many mathematics to deal with the solitonary waves were developed
only very recently.
Multi-component self-gravitating
systems
- the universe
- superclusters
- galaxy clusters, groups
- galaxies
- young star clusters
- giant molecular clouds
……
Dark Matter
Baryons - hot gas
galaxies and stars
Galaxy clusters:
The components and their roles
Dark matter unknown number of species
Dominant momentum carriers
Main energy reservoir
Hot ionized gas (ICM)
Radiative coolant
Trapped baryons
(stars and galaxies)
dynamically unimportant
Magnetic field ?
Cosmic rays ?
…..
Galaxy clusters:
Generalised self-gravitating “fluid”
Dark matter unknown number of species
Poisson equation
Dominant momentum carriers
Main energy reservoir
Generalised equations of states
Hot ionized gas (ICM)
Radiative coolant
velocity dispersion
(“temperature”)
entropy
degree of freedom
Galaxy clusters:
Multi-component formulation
Mass continuity equation
Momentum conservation equation
gravitational force
Entropy equation (energy conservation equation)
energy injection
radiative loss
stationary situations:
Galaxy clusters:
Stationary structures
After some rearrangements, we have
gas cooling
Inversion of the matrix
integration over the radial coordinate
+ boundary conditions
inflow
Profiles pf density
and other variables
Galaxy clusters:
Projected density profiles
Projected surface density of model
clusters with various dark-matter
degrees of freedom
Top: clusters with a high mass
inflow rate
Bottom: clusters with a low
mass inflow rates
Saxton and Wu (2008a)
Galaxy clusters:
Density and temperature profiles
Saxton and Wu (2008a)
Galaxy clusters:
Spatial resolved X-ray spectra
Top row:
Bottom row:
Saxton and Wu (2008a)
Galaxy clusters:
X-ray surface brightness
Projected X-ray surface brightness of model clusters with various
dark-mass degrees of freedom
(black: 0.1 - 2.4 keV; gray: 2 - 10 keV)
Saxton and Wu (2008a)
Galaxy clusters:
Local Jean lengths
Saxton and Wu (2008a)
Galaxy clusters:
Dark matter degrees of freedom
Constraints set by by the allowed mass of the “massive object” at
the centre of the cluster
Saxton and Wu (2008a)
Galaxy clusters:
Stability analysis
Lagrange perturbation:
hydrodynamic
equations
dimensionless eigen value
a set of coupled linear
differential equations
+ appropriate B.C.
“eigen-value problem”
numerical shooting method
(for details, see Chevalier and Imamura 1982, Saxton and Wu 1999, 2008b)
Galaxy clusters:
Wave excitations and mode stability
red: damped modes
black: growth modes
Spacing of the modes
depends on the B.C.;
stability of the modes
depends on the energy
transport processes
Saxton and Wu (2008b)
Galaxy Clusters:
Could this be ….. ?
(ATCA radio spectral image of Abell 3667 provided by R Hunstead, U Sydney)
Galaxy clusters:
Gas tsunami
cooler cluster interior
smaller sound speeds
hotter outer cluster rim
larger sound speeds
- subsonic waves propagating from outside becoming supersonic
- waves in gas piled up when propagating inward (tsunami)
- stationary dark matter providing the background potential, i.e.
self-excited tsunami
Fujita et al. (2004, 2005)
Galaxy clusters:
Cluster quakes and tsunami
- close proximity between clusters
excitation of dark-matter oscillations, i.e. cluster quakes
- higher-order modes generally grow faster
oscillations occurring in a wide range of scales
- dark-matter coupled gravitationally with in gas
dark matter oscillations forcing gas to oscillate
- cooler gas (due to radiative loss) implies lower sound speeds in the
cluster cores
waves piled up when propagating inward, i.e. cluster tsunami
- mode cascades
inducing turbulences and hence heating of the cluster throughout
Saxton and Wu (2008b)
Cosmic walls:
Two-component self-gravitating
infinite sheets
Suppose that
- the equations of state of
both the dark matter and
gas are polytropic;
- the inter-cluster gas is
roughly isothermal.
Then ……..
Cosmic walls:
Quasi-1D Newtonian treatment
dark matter
gas
quasi-1D approximation
Cosmic walls:
Non-linear perturbative expansion
a constant yet to
be determined
Consider two new variables:
Cosmic walls:
Soliton formation in dark matter
rescaling the
variables
Korteweg - de Vries (KdV) Equation
soliton solution
Wu (2005); Wu and Younsi (2008)
Solitons in astrophysical systems:
1D multiple soliton interaction
Methods for solutions:
- Baecklund transformation
- inverse scattering
- Zakharov method
……
- preserve identities
- linear superposition not
applicable
- phase shift
Top: 2-soliton interaction
Bottom: 3-soliton interaction
Solitons in astrophysical systems:
Train solitons
Zabusky and Kruskal (1965)
Younsi (2008)
Solitons in astrophysical systems:
Higher dimension solition equations
Relaxing the quasi-1D approximation
2D/3D treatment
Kadomstev-Petviashvili (KP) Equation
Cylindrical and spherical KP Equation
n = 1 for cylindrical; and n = 2 for spherical
Non-linear Schroedinger Equations
Solitons in astrophysical systems:
Higher dimension solitions
Single rational soliton obtained by Zakharov-Manakov method:
Younsi and Wu (2008)
Solitons in astrophysical systems:
Propagation of solitons in 3D
Younsi and Wu (2008)
Solitons in astrophysical systems:
Resonance in 2D soliton collisions
evolving two spherical
rational solitons to
collide and resonate
At resonance, the amplitude can be twice the sum of the amplitudes
of the two incoming solitons.
Younsi and Wu (2008)
Solitons in astrophysical systems:
Stability of solitons
longitudinal perturbation
spherical soliton shell
transverse perturbation
In general, many 3D solitons, particularly,
the Zarhkarov-Manakov rational solitions,
are unstable in longitudinal perturbations,
but can be stabilised in the presence of
transverse perturbations. Ring solitons
are formed.
Solitons in astrophysical systems:
Resonance, density amplification and
a structure formation mechanism
2 colliding solitons with
baryons trapped inside
resonant state
For resonant half life
the baryonic gas trapped by the dark matter soliton resonance will
collapse and condense.
End
Collison and resonant interaction of two small-amplitude solitons on a beach
in Oregon in USA (from Dauxois and Peyrard 2006).