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

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

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

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  1. 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)

  2. 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

  3. 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

  4. 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.

  5. 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

  6. Galaxy clusters: The 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

  7. Galaxy clusters: Generalised 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

  8. Galaxy clusters: Multi-component formulation Mass continuity equation Momentum conservation equation gravitational force Entropy equation (energy conservation equation) energy injection radiative loss stationary situations:

  9. Galaxy clusters: Stationary 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

  10. 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)

  11. Galaxy clusters:Density and temperature profiles Saxton and Wu (2008a)

  12. Galaxy clusters:Spatial resolved X-ray spectra Top row: Bottom row: Saxton and Wu (2008a)

  13. 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)

  14. Galaxy clusters:Local Jean lengths Saxton and Wu (2008a)

  15. 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)

  16. Galaxy clusters:Stability 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)

  17. Galaxy clusters: Wave 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)

  18. Galaxy Clusters: Could this be ….. ? (ATCA radio spectral image of Abell 3667 provided by R Hunstead, U Sydney)

  19. 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)

  20. 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)

  21. 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 ……..

  22. Cosmic walls:Quasi-1D Newtonian treatment dark matter gas quasi-1D approximation

  23. Cosmic walls:Non-linear perturbative expansion a constant yet to be determined Consider two new variables:

  24. Cosmic walls:Soliton formation in dark matter rescaling the variables Korteweg - de Vries (KdV) Equation soliton solution Wu (2005); Wu and Younsi (2008)

  25. 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

  26. Solitons in astrophysical systems: Train solitons Zabusky and Kruskal (1965) Younsi (2008)

  27. 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

  28. Solitons in astrophysical systems:Higher dimension solitions Single rational soliton obtained by Zakharov-Manakov method: Younsi and Wu (2008)

  29. Solitons in astrophysical systems:Propagation of solitons in 3D Younsi and Wu (2008)

  30. 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)

  31. 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.

  32. 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.

  33. End 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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