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

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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)

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

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

- 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.

- the universe
- superclusters
- galaxy clusters, groups
- galaxies
- young star clusters
- giant molecular clouds
- ……

Dark Matter

Baryons - hot gas

galaxies and stars

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

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

Mass continuity equation

Momentum conservation equation

gravitational force

Entropy equation (energy conservation equation)

energy injection

radiative loss

stationary situations:

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

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)

Saxton and Wu (2008a)

Top row:

Bottom row:

Saxton and Wu (2008a)

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)

Saxton and Wu (2008a)

Constraints set by by the allowed mass of the “massive object” at the centre of the cluster

Saxton and Wu (2008a)

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)

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)

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

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)

- - 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)

Suppose that

- the equations of state of

both the dark matter and

gas are polytropic;

- the inter-cluster gas is

roughly isothermal.

Then ……..

dark matter

gas

quasi-1D approximation

a constant yet to

be determined

Consider two new variables:

rescaling the

variables

Korteweg - de Vries (KdV) Equation

soliton solution

Wu (2005); Wu and Younsi (2008)

- 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

Zabusky and Kruskal (1965)

Younsi (2008)

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

Single rational soliton obtained by Zakharov-Manakov method:

Younsi and Wu (2008)

Younsi and Wu (2008)

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)

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.

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.

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