Waves and Solitons in Multi-component Self-gravitating

Systems

Kinwah Wu Mullard Space Science Laboratory

University College London

Curtis Saxton (MSSL, UCL)Ignacio Ferreras (P&A, UCL)

Outline

• Collective oscillations, non-linear waves and solitons (a brief overview)

• Multi-component self-gravitating systems• Two studies: results and some thoughts - “tsunami” & “quakes” in galaxy clusters - solitons in self-gravitating sheets and 1D infinite media

Collective phenonmona in multi-component systems

- Coupled oscillators: different oscillation normal modes and dissipation

processes non-linear mode coupling resonance - Two-stream instability - Landau damping

cf. electron-ion plasma wave-wave interaction particle-particle interaction wave-particle interaction

Solitons as non-linear wave packets

• Non-linear, non-dispersive waves: The nonlinearity which 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• Propagation speed proportional to pulse height • An example soliton pulse profile:

pulse height propagation speed

Situations that give rise to solitons

- fluids with surface tension - ion acoustic oscillations in plasmas - etc

Korteweg-de Vries (KdV) Equation

Kadomstev-Petviasgvili (KP) Equation

mathematical methods developed to solve various soliton equations e.g. Baecklund transformation, inverse scattering method, Zakharov-Shabat method ……

Mutiple solitons

- development of “soliton trains” (Zabusky & Kruskal 1965) kdV equation with a cosine-like function as initial condition component solitions with larger pulse height travel faster a train of solitions lined up with the tallest leading the way

=> “big” solitions are more likely to find each other

depending on dimension

- resonance and phase shift

time

before collision

after collision

resonant states: momentum exchange phase shift

Multi-component self-gravitating systems

The fluid description (non-rotational case)

Conservation equations:

Poisson equation:

Quakes and Tsunami in Galaxy Clusters

Galaxy clusters as “spherical”2-component coupled oscillators

• DM - momentum carrier • Gas - coolant (dissipater)

inflowgas cooling

Solve to obtain the stationary structure

Perturbative analysis

Lagrange perturbation for the DM and gas components:

plus 6 coupled perturbed hydrodynamic equations

the perturbed Poisson equation

Diagonise the matrixDefine the boundary conditions Numerically integrate the DEs

algebraic functions of hydrodynamic variables

Normal mode oscillations (the DM component)

- oscillation modes depend on BCs- high order modes not damped - different stability properties for even and odd modes in some cases

eigen-mode

- similar eigen-planes can be generated for the gas component

mass inflow rate = 100 M_sun per year(size ~ 1 Mpc; kT_max ~ 10 keV)

Quakes and tsunami in galaxy clusters: dark-matter oscillations and gas

dissipation- close proximity between clusters --> excitation of DM oscillations, ie. cluster quakes - high-order modes are also fast growing --> oscillations may occur in a wide range of scales - oscillations in DM coupled with oscillations in gas - cooler interior of gas (due to radiative loss) --> slower sound speeds in the inner cluster region (“cooling flow” core) --> waves piled up when propagating inward, ie. cluster tsunami - mode cascade --> inducing turbulence and hence heating of the cluster throughout

cf. “original” cluster tsunami model of Fujita, Suzaki & Wada (2004) and Fujita, Matsumoto & Wada (2005) stationary DM provide background potential (ie. no quake), waves in gas piled up when propagating inward (“self-excited” tsunami only)

Solitons in self-gravitating sheets and 1D infinite media

Planar systems: two component infinite self-gravitating sheets

Suppose: 1. inertia of one component is unimportant 2. the component is approximately isothermal 3. polytropic EOS

“quasi-1D” approximation

Perturbative expansion

Use two new variables:

constant yet to be determined

Formation of solitons

KdV equation --> soliton solution

rescaling variables

effective sound speed

cf. Solitons in single-component self-gravitating systems (Semekin et al. 2001)

Some thoughts and questions:

2 colliding DM solitons

resonant state

Suppose the resonant half life

Q1: Are ridge solitons manifested as filaments in cosmic sheets? Q2: Can soliton collisions make globular clusters?

Summary

1. Collective and non-linear oscillations, which may not be present in single component self-gravitating systems, could be important in multi-component systems. 2. DM and gas play different roles in exciting and sustaining oscillations in astrophysical systems. 3. Galaxy clusters can be considered as couple oscillators with DM as the mode resonant medium and gas as the energy dissipater. Gas in clusters can be compression heated by acoustic coupling with the DM oscillations. 4. Solitons can be excited in DM/gas sheets and infinite self-gravitating systems, and they could lead to “bright” structure formation, provided that certain dynamical conditions are satisfied.