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Structures, Oscillations, Waves and Solitons in Multi-component Self-gravitating 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
Hot ionized gas (ICM)
Trapped baryons (stars and galaxies)
Dominant momentum carriersMain energy reservoir
Radiative coolant
dynamically unimportant
Magnetic field ? Cosmic rays ? …..
Galaxy clusters: Generalised self-gravitating “fluid”
Dark matter - unknown number of species
Hot ionized gas (ICM)
Dominant momentum carriersMain energy reservoir
Radiative coolant
Poisson equation
Generalised equations of states
velocity dispersion(“temperature”)
entropy
degree of freedom
Galaxy clusters: Multi-component formulation
Mass continuity equation
Momentum conservation equation
Entropy equation (energy conservation equation)
gravitational force
energy injectionradiative loss
stationary situations:
Galaxy clusters: Stationary structures
gas cooling inflow
Inversion of the matrix integration over the radial coordinate + boundary conditions
After some rearrangements, we have
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
Saxton and Wu (2008a)
Top row: Bottom row:
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
a set of coupled linear differential equations
+ appropriate B.C.
“eigen-value problem”
numerical shooting method
dimensionless eigen value
(for details, see Chevalier and Imamura 1982, Saxton and Wu 1999, 2008b)
Galaxy clusters: Wave excitations and mode stability
Saxton and Wu (2008b)
red: damped modesblack: growth modes
Spacing of the modes depends on the B.C.; stability of the modes depends on the energy transport processes
Galaxy Clusters: Could this be ….. ?
(ATCA radio spectral image of Abell 3667 provided by R Hunstead, U Sydney)
Galaxy clusters:Gas tsunami
Fujita et al. (2004, 2005)
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
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
Consider two new variables:
a constant yet to be determined
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
Top: 2-soliton interaction Bottom: 3-soliton interaction
- preserve identities - linear superposition not applicable - phase shift
Methods for solutions:- Baecklund transformation - inverse scattering - Zakharov method ……
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
At resonance, the amplitude can be twice the sum of the amplitudes of the two incoming solitons.
evolving two spherical rational solitons to collide and resonate
Younsi and Wu (2008)
Solitons in astrophysical systems:Stability of solitons
spherical soliton shell
transverse perturbation
longitudinal 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).