Presentation on theme: "Waves and Solitons in Multi-component Self-gravitating Systems"— Presentation transcript:
1 Waves and Solitons in Multi-component Self-gravitating Systems Kinwah WuMullard Space Science LaboratoryUniversity College LondonCurtis Saxton (MSSL, UCL)Ignacio Ferreras (P&A, UCL)
2 OutlineCollective oscillations, non-linear waves and solitons (a brief overview)Multi-component self-gravitating systemsTwo studies: results and some thoughts- “tsunami” & “quakes” in galaxy clusters- solitons in self-gravitating sheets and 1Dinfinite media
3 Collective phenonmona in multi-component systems - Coupled oscillators:different oscillation normal modes and dissipation processesnon-linear mode couplingresonance- Two-stream instability- Landau dampingcf. electron-ion plasmawave-wave interactionparticle-particle interactionwave-particle interaction
4 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 likedLinear superposition often not applicablePropagation speed proportional to pulse heightAn example soliton pulse profile:pulse heightpropagation speed
5 Situations that give rise to solitons - fluids with surface tension- ion acoustic oscillations in plasmas- etcKadomstev-Petviasgvili (KP) EquationKorteweg-de Vries (KdV) Equationmathematical methods developed to solve various soliton equationse.g. Baecklund transformation, inverse scattering method,Zakharov-Shabat method ……
6 Mutiple solitonsdevelopment of “soliton trains” (Zabusky & Kruskal 1965)kdV equation with a cosine-like function as initial conditioncomponent solitions with larger pulse height travel fastera train of solitions lined up with the tallest leading the way=> “big” solitions are more likely to find each otherdepending on dimension- resonance and phase shiftbefore collisionresonant states: momentum exchangephase shifttimeafter collision
7 Multi-component self-gravitating systems The fluid description (non-rotational case)Conservation equations:Poisson equation:
9 Galaxy clusters as “spherical” 2-component coupled oscillators DM - momentum carrierGas - coolant (dissipater)inflowgas coolingSolve to obtain the stationary structure
10 Perturbative analysis Lagrange perturbation for the DM and gas components:the perturbed Poisson equationplus 6 coupled perturbed hydrodynamic equationsDiagonise the matrixDefine the boundary conditionsNumerically integrate the DEsalgebraic functions of hydrodynamic variables
11 Normal mode oscillations (the DM component) mass inflow rate = 100 M_sun per year(size ~ 1 Mpc; kT_max ~ 10 keV)eigen-mode- oscillation modes depend on BCs- high order modes not damped- different stability properties foreven and odd modes in some casessimilar eigen-planes can be generatedfor the gas component
12 Quakes and tsunami in galaxy clusters: dark-matter oscillations and gas dissipation - close proximity between clusters--> excitation of DM oscillations, ie. cluster quakeshigh-order modes are also fast growing--> oscillations may occur in a wide range of scalesoscillations in DM coupled with oscillations in gascooler 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 tsunamimode cascade--> inducing turbulence and hence heating of the cluster throughoutcf. “original” cluster tsunami model of Fujita, Suzaki & Wada (2004) andFujita, Matsumoto & Wada (2005)stationary DM provide background potential (ie. no quake), waves in gaspiled up when propagating inward (“self-excited” tsunami only)
13 Solitons in self-gravitating sheets and 1D infinite media
14 Planar systems: two component infinite self-gravitating sheets Suppose: 1. inertia of one component is unimportant2. the component is approximately isothermal3. polytropic EOS“quasi-1D” approximation
15 Perturbative expansion constant yet tobe determinedUse two new variables:
16 Formation of solitons KdV equation --> soliton solution effective sound speedrescaling variablesKdV equation --> soliton solutioncf. Solitons in single-component self-gravitating systems(Semekin et al. 2001)
17 Some thoughts and questions: 2 colliding DM solitonsresonant stateSuppose the resonant half lifeQ1: Are ridge solitons manifested as filaments in cosmicsheets?Q2: Can soliton collisions make globular clusters?
18 Summary Collective and non-linear oscillations, which may not be present in single component self-gravitating systems,could be important in multi-component systems.DM and gas play different roles in exciting and sustainingoscillations in astrophysical systems.Galaxy clusters can be considered as couple oscillatorswith DM as the mode resonant medium and gas as theenergy dissipater. Gas in clusters can be compressionheated by acoustic coupling with the DM oscillations.Solitons can be excited in DM/gas sheets and infiniteself-gravitating systems, and they could lead to “bright”structure formation, provided that certain dynamicalconditions are satisfied.
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