Now derive both sides of above equation with respect to ψ: (known as Abel integral equation) the above equation is invertible thanks to the identity:

Transcription

1 Change of variable in the integral: Now derive both sides of above equation with respect to ψ: (known as Abel integral equation) the above equation is invertible thanks to the identity: so that one gets

2 Eddington s equation input: density profile output: velocity distribution function suitable for numerical integration! the trick is getting ρ(ψ) Consistency check: given a density profile ρ(r) the necessary condition F(ε)>0 is not guaranteed (not every density profile is actually consistent to a steady-state solution)!

3 The simplest possible application of Eddington s procedure (and by far the most used!): the isothermal sphere model a flat rotational curve implies the density profile: from Poisson s equation: R 0 =distance of the Earth from Galactic Center ρ 0 Earth s position the potential is: so that we can get ρ(ψ): with σ=constant

4 Plugging ρ(ψ) into Eddington s equation: suitable change of variable: Maxwellian!

5 Maxwellian r.m.s velocity: so that the potential is properly normalized: and from the 2 equations:

6 PRD71,043516,2005

7 Too many models: some classification is needed

8 Class ρ (density profile) σ r (velocity dispersion) A spherical isotropic B C spherical axisymmetric non isotropic non isotropic D triaxial non isotropic

9 [Belli, Cerulli, Fornengo, Scopel, PRD66(2002)043503]

10 CLASS A: spherical ρ, isotropic σ DF depends on phase space only through the energy, which is an integral of motion: F=F(ε) starting from a given ρ(r), one gets the potential ψ(r) and thtough a Laplace inversion the DF: (Eddington s formula) consistency check: F>0

11 Radially symmetric density profiles considered: logarithmic model (isothermal sphere with a core radius) power-law models (spherical limit of a class of known axisymmetric solutions, Evans, Mon. Not. R. Astron. Soc. 267, 333 (1994) ) profiles suggested by numerical simulations (Navarro- Frank- White, Kravtsov, Moore).,

12 One complication: F does not depend only on the DM distribution, but also on the visible one However, the visible parts of the Galaxy may be modeled as: a bulge, given by a spheroidal density distribution within ~ 2 kpc from the center a disk, with an exponential distribution which dies away at ~ 4 kpc (R 0 =8.5 kpc) So we can assume that the only contribution of non halo components is through the rotational velocity:

13 Two possibilities: a maximal halo: M vis <<M halo a non maximal halo All the dependence on visible components is contained in M vis From the observation: ρ 0 max M vis <<M halo (v rot 100 =v rot (r=100 kpc), flatness condition) For each model an interval in ρ 0 can be obtained (ρ 0 <ρ 0 max )

14 v 0 =220 km/sec

15 Allowed intervals for ρ GeV/cm 3 <ρ 0 <1.7 GeV/cm 3

16 CLASS B: spherical ρ, non-isotropic σ same models as before, but now the DF depends also on L L : F=F(ε,L) Osipkov-Merrit solution: 2 L F( ε, L) = F( Q= ε 2 2 r a is related to the degree of anisotropy β 0 of the velocity dispersion tensor evaluated at the Earth s position (β 0 <0.4): r a ) F(Q) is derived from Eddington s equation with ε Q

17 CLASS C: axisymmetric ρ In the case of axial symmetry the DF depends in general on E and L z. The DF can be written as the sum of an even (F + ) and an odd (F - ) contribution with respect to L z : the density ρ DM turns out to depend only on the even part F +, so that, by inverting it, the DF may be determined up to an arbitrary odd part F - Known analytical solution for F + for a particular class of models (Evans) for which: N.B.: these solutions are valid in the hypothesis that the halo potential is dominant over the other components maximal halo

18 Corotation of the halo The DF for an axisymmetric model is known up to an arbitrary odd component F -. The case F - (ε,l Z ) 0 corresponds to the case of a rotating halo, where the number of particles moving clockwise around the axis of symmetry is different from that in the opposite sense. Explicit example for F - : with: F right(left) describe configurations with maximal v φ at fixed ρ 0

19 A DF with an intermediate value of v φ can be obtained as a linear combination of F right and F left : 0.36 < η < 0.64, related to the dimensionless spin parameter of the Galaxy [S. Warren, P. J. Quinn, J. K. Salmon and W. H. Zurek, Astrophys. J. 399, 405 (1992); S. Cole and C. Lacey, Mon. Not. R. Astron. Soc. 281, 7126 (1996)]

20 CLASS D: triaxial ρ in this case the problem to determine F is non trivial. numerical integrations known to be unstable, 2 additional non-classical effective integral of motions needed for selfconsistent equilibrium configurations simplest case: logarithmic ellipsoidal potential: Given the difficulty of the task, only the lowest order velocity moments of the distribution function (related directly to observable properties) are calculated via the Jeans equations ( stellar hydrodynamical equations). However, Jeans equations requires assumptions regarding either the shape or the orientation of the velocity ellipsoid. allignment to conical coordinates, <σ i σ j >=0,i j (Evans, Carollo, de Zeeuw, Mon. Not. R. Astron.Soc. 318,1131(2000))

21 CLASS D: triaxial ρ conical coordinates (r,µ,ν): µ and ν solutions to the quadratic equation in τ: at large radii ellipsoidal coordinates reduce to conical coordinates most natural generalization of axisymmetric case

22 CLASS D: triaxial ρ in practice, use a Maxwellian with the following velocity dispersions (obtained solving Jean s equations): Earth s position on major axis Earth s on intermediate axis and δ is a free parameter that in the spherical limit (p=q) quantifies the amount of anisotropy in the velocity dispersion: