Self similar solutions in phase space

Transcription

1 Self similar solutions in phase space x F (x, v,t )=F (, v, t) f (λ x, λ v, λt t )=λ f ( x, v, t ) No dependence on scale for the phase space density solution F α 3 4 Infinitesimal variations of λ implies solution for F x v f (x, v, t )=t F α, α t t α0 ( )

2 The n-dimensional Vlasov equation : f f f Φ + v =0 t x x v The same equation in reduced variables : F F F F Φ (+n α )F +(+α ) x +α v + =0 x v x x v α 0= n α x x = α t v v= α t α =+ α ()

3 How to find a solution to Eq ()? Self similarity is a simplification and a reduction of the general problem, But Eq () is still very complex Is finding a solution hopeless in itself? Some clues on the solution : It is a series of folds in phase space We consider an adiabatic regime

4 The clues ) Series of folds : Need some kind of polar coordinates Adiabatic The potential variations are small During an orbit : thus a fold is close to a line of constant energy

5 Let find the solution for a general power law potential + Φ=k x v + E=k x + h(r,θ)=c 0 θ=θ+ π If θ is large the angular variation during a fold is small and the fold θ=θ+ π Equation of a spiral, a fold implies Corresponds to R=Constant=E Constant density, k=/, E= (x + v ) Idenfication with polar coordinates, R =E General case : change variable, u=x η, tan (θ)= v x E=u+ v

6 Introducing the appropriate change in variable η η= + u= x G=ln (F ) G G G G + +α +(+α ) η u+α v sgn( x )η v u u =0 u v u v Polar coordinates: ( ) R= u +v tan (ψ)= v u H H H η sin ψcos ψ(+α ) + R ( cos ψ (+α )η +α ) +η R cos ψ ) + +α +=0 ( ψ ψ R η=η α α + The nearly adiabatic regime: interesting cases ) Stationary solution ) very close to origin=large number of folds

7 Stationary solution ρ(t )=t ρ (t)=t x =t x α t ( ) α = η =0 The vlasov self similar equation is transformed to: H H R α +η R cos ψ ) + +α +=0 ( ψ R This equation has a general solution α + +α α α H (R, ψ)= α log R +Q R + α cos ψ d ψ ( The iso contours of the solution are spirals )

8 Small radius approximation Equation is reduced to dominant terms: H η R cos ψ )+ + α +=0 ( ψ With general solution: (+ α)(+α ) α α H (R, ψ)= R cos ψ d ψ+ H (R ) α The iso-contours are also spirals consistent with the stationary case

9 Solution iso-contours: geometry of the spirals ( R Q α ) + I (ψ) =R α+ α Eq. of the iso-contour F0 Q=log Q α I (ψ)=(+ α ) cos ψ d ψ ψ=ψ0 + π=ψ0+ ϵd ψ A fold implies: R=R 0+ ϵdr Requiring that all functions are power laws implies that: dr α R0 R0

10 Series of numerical simulations with power law initial conditions 0 Initial density: ρ0 x + α Fold spacing prediction: dr R 0 R 0+ 0 Numerical estimation: dx dr R 0γ

11 Results: measured versus predicted exponents +0 γ 0

12 Re-scaling of potential of power law force field F F F F (+n α ) F +(+α ) x +α v +k x =0 x v x v + Φ=k x <<0 <, x =λ x, v =λ v, k x = k>0 attraction k<0 repulsion k>0 repulsion k<0 attraction λ =λ =k

13 Consequence of changing force sign on the solution contours λ = Changes of sign in the equation R R + + cos ψ d ψ=c 0 + d ψ R x + + +v dψ v Small ψ d ψ tan ψ= x ( ) x (+) + λ = x v + λ ( λ x v ) λ (+) λ v ( ) +v = x + (+) v x v + ( ) x + (+)

14 Illustration on the effect of force sign =, k >0 x +v =, k <0 x +v =

15 Symmetrically spherical system with Angular momentum f f J Φ f + v+ =0 t r r r 3 r v r ( ) f (r, v r, J,t)=g(r, v r, t)δ(j J 0) J 0 Φ g g g + v+ =0 t r r r 3 r v r ( ) The problem is very similar to the D problem J 0 r force dominated by r force dominated by Φ r Two asymptotic regimes r 3 term = 4

16 Numerical simulations: Thierry Sousbie, IAP Spherical collapse with angular momentum

17 Folds with angular momentum Φ r regime J 0 r 3 regime

18 Small r force dominated by angular momentum repulsive term dr r +, = 4 dr r 3

19 General statistical properties of solutions near equilibrium Why do we observe some universal laws in numerical simulations? For smooth phase space density the expectation are all power laws Fold in phase space of cold initial conditions corresponds to un-smooth density Smoothing the phase space density breaks the auto similarity: no power law

20 Some auto similar properties are conserved in a smoothing process Fold thickness is proportional to fold distance: a key property R=s( ψ,.., ψn ) Generalized spiral In a given dimension associated with A variation of the initial condition ψk π d ψk ψk a fold corresponds to ψk =ψk + π corresponding to the fold thickness Linearization δ( π) δ(d ψk ) Fold distance and fold thickness are proportional ψk =ψk +d ψk

21 f fs δr ΔR δr =Constant ΔR δr ΔR Constant scaling between values and occurrence of f and fs

22 Histogram of f and fs are identical except for constant scaling factors N Self similarity of the distribution of the values of f are preserved In the smoothing operation f Consequence: the expectation of smoothed quantities derived from the probability distribution of f is a power law

23 Quantities derived from the smoothed probability distribution k pσ 3 Q n= P (f ) f df 3 p σ f d v σ n Qn=t n α0 r tα γ ( ) r n ρ Q σ p σ f d v 3 r σ 3 pσ 3 Simulation data from: Wojtak etal; Q, Q, Q,Q 4 4

24 n 3 5 (n m) Qn f d v 8 Rnm = = =r m 3 Qm f d v 3 3 R, R, R, R Simulation data from: Wojtak etal;

25 DM self similarity induced by baryons At the begining of galaxy formation the initial DM halo is affected by the baryonic feedback, Supernovae winds, AGN,...resulting in a re-shaping of the halo The cuspy halo center is destroyed and replaced with a core In this starburst episode the momentum driven winds impose an equilibrium condition The violent re-shaping of the halo and the equilibrium condition is consistent with a new self similar class This new baryonic induced self similar model of DM halo explains and is consistent With a number of observational facts.

26 Equilibrium condition for the gas (Murray, Qataert & Thompson, 005) Optically thick gas radiation momentum equal gravity: G M MG r B LM = c Total momentum proportional to number of star and to gas mass LM M G GM =constant rb Universal acceleration at a scale radius Constant DM acceleration new similarity class α = constant baryon acceleration Former similarity class, infall on a seed mass Bertschinger, α = 9

27 Constant DM and baryons acceleration at a galactic scale radius Acceleration at a galactic scale For DM (Donato etal 009) Acceleration at a galactic scale For baryons (Gentile etal 009)

28 The scaling of DM and baryons distribution must be similar r B r DM rb r DM Donato etal. 004 demonstrate that it is the case for a sample of about 40 galaxies

29 Velocity rotation curve VM Maximum of the velocity curve situated at large distances rm

30 V M =c 0 rm Self similarity GMB r B B =a 0 & r B r DM r M 4 V M G M B Deep MOND regime Transition between Newtonian & Mond regime at fixed acceleration also But the gas equilibrium condition does not apply to clusters: unlike MOND the self similar model model that the similarity relation for velocities break beyond the scale of galaxies.

31 Conclusion The analytic self similar solutions of the Vlasov equation are generalized folds in phase space. The predicted structure of these folds is observed in numerical simulations The statistical properties of the solutions are predictable and also observed In numerical simulations. The baryons feedback introduce a specific similarity class, the properties of The associated solutions explains and connect a number of facts: ) universal accelerations ) proportionality between DM and baryons scale length 3) MOND like properties and the break of these properties at the scale of Clusters of galaxies.