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Structures, Oscillations, Waves and Solitons in Multi-component Self-gravitating SystemsPowerPoint Presentation

Structures, Oscillations, Waves and Solitons in Multi-component Self-gravitating Systems

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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 Multi-component Self-gravitating Systems

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: Multi-component Self-gravitating SystemsSome 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 Multi-component Self-gravitating Systems

- 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 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: Multi-component Self-gravitating SystemsThe components and their roles

Dark matter -

unknown number of species

Trapped baryons

(stars and galaxies)

Dominant momentum carriers

Main energy reservoir

dynamically unimportant

Hot ionized gas (ICM)

Magnetic field ?

Cosmic rays ?

…..

Radiative coolant

Galaxy clusters: Multi-component Self-gravitating SystemsGeneralised self-gravitating “fluid”

Poisson equation

Dark matter -

unknown number of species

Dominant momentum carriers

Main energy reservoir

Generalised equations of states

Hot ionized gas (ICM)

velocity dispersion

(“temperature”)

Radiative coolant

entropy

degree of freedom

Galaxy clusters: Multi-component Self-gravitating SystemsMulti-component formulation

Mass continuity equation

Momentum conservation equation

gravitational force

Entropy equation (energy conservation equation)

energy injection

radiative loss

stationary situations:

Galaxy clusters: Multi-component Self-gravitating SystemsStationary structures

After some rearrangements, we have

gas cooling

inflow

Inversion of the matrix

integration over the radial coordinate

+ boundary conditions

Profiles pf density and other variables

Galaxy clusters: Multi-component Self-gravitating SystemsProjected 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: Multi-component Self-gravitating SystemsDensity and temperature profiles

Saxton and Wu (2008a)

Galaxy clusters: Multi-component Self-gravitating SystemsSpatial resolved X-ray spectra

Top row:

Bottom row:

Saxton and Wu (2008a)

Galaxy clusters: Multi-component Self-gravitating SystemsX-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: Multi-component Self-gravitating SystemsLocal Jean lengths

Saxton and Wu (2008a)

Galaxy clusters: Multi-component Self-gravitating SystemsDark 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: Multi-component Self-gravitating SystemsStability analysis

Lagrange perturbation:

hydrodynamic

equations

a set of coupled linear

differential equations

+ appropriate B.C.

dimensionless eigen value

“eigen-value problem”

numerical shooting method

(for details, see Chevalier and Imamura 1982, Saxton and Wu 1999, 2008b)

Galaxy clusters: Multi-component Self-gravitating SystemsWave excitations and mode stability

Spacing of the modes depends on the B.C.; stability of the modes depends on the energy transport processes

red: damped modes

black: growth modes

Saxton and Wu (2008b)

Galaxy Clusters: Multi-component Self-gravitating SystemsCould this be ….. ?

(ATCA radio spectral image of Abell 3667 provided by R Hunstead, U Sydney)

Galaxy clusters: Multi-component Self-gravitating SystemsGas 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: Multi-component Self-gravitating SystemsCluster 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: Multi-component Self-gravitating SystemsTwo-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: Multi-component Self-gravitating SystemsQuasi-1D Newtonian treatment

dark matter

gas

quasi-1D approximation

Cosmic walls: Multi-component Self-gravitating SystemsNon-linear perturbative expansion

a constant yet to

be determined

Consider two new variables:

Cosmic walls: Multi-component Self-gravitating SystemsSoliton formation in dark matter

rescaling the

variables

Korteweg - de Vries (KdV) Equation

soliton solution

Wu (2005); Wu and Younsi (2008)

Solitons in astrophysical systems: Multi-component Self-gravitating Systems1D 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: Multi-component Self-gravitating SystemsTrain solitons

Zabusky and Kruskal (1965)

Younsi (2008)

Solitons in astrophysical systems: Multi-component Self-gravitating SystemsHigher 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: Multi-component Self-gravitating SystemsHigher dimension solitions

Single rational soliton obtained by Zakharov-Manakov method:

Younsi and Wu (2008)

Solitons in astrophysical systems: Multi-component Self-gravitating SystemsPropagation of solitons in 3D

Younsi and Wu (2008)

Solitons in astrophysical systems: Multi-component Self-gravitating SystemsResonance 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: Multi-component Self-gravitating SystemsStability 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: Multi-component Self-gravitating SystemsResonance, 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 Multi-component Self-gravitating Systems

Collison and resonant interaction of two small-amplitude solitons on a beach in Oregon in USA (from Dauxois and Peyrard 2006).

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