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).