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• 1

Bienvenue au

Cours d’astrophysique III : Dynamique stellaire et galactique

Semestre automne 2011

Dr. Pierre North Laboratoire d’astrophysique

Ecole Polytechnique Fédérale de Lausanne Observatoire de Sauverny

CH – 1290 Versoix

http://lastro.epfl.ch

• Lagrange identity, Jacobi criterion and virial theorem Lagrange identity & Jacobi criterion: Moment of inertia: =>

where

the kinetic energy and the potential energy of the system,

such that:

If W is a k-homogeneous function, i.e. if , then

Lagrange identity

Gravitation: k = -1, W ≤ 0 =>

Integrating on time: ⇒ Jacobi criterion: if E > 0, I(t) → ∞ for t → ∞ : mean instability if E < 0, system remains confined ∀t : necessary but insuff.

K = 1 2

mi i ∑  ˙ x i2

W = −G 2

mim j |  x j −

 x i |j ∑

i≠ j ∑

= ∂W ∂  x i

= +G mim j

|  x i −  x j |

3 j≠ i ∑  x i −

 x j( ) = mi ∂Φ ∂  x i ( x i)

 x i i=1

N

∑ ∂W ∂  x i

= kW

• Lagrange identity, Jacobi criterion and virial theorem (cont.) Virial theorem: Lagrange identity ; time average:

=> ; stationary system: .

Gravitation: k = -1, =>

A more general form of the virial theorem:

Confinement provided by gravitation and external pressure only!

Applications of the virial theorem: ,

Where and assuming that => Bound system => E < 0 =>

• Discovery by Zwicky of dark matter in galaxy clusters (application of the virial theorem; his first suspicion dates back to 1933)

• Fritz Zwicky (1898-1974): a colourful character. Predicted the neutron stars as remnants of core-collapse supernovae (to which he also attributed the origin of cosmic rays), discovered the dark matter in galaxy clusters and predicted the gravitational lensing effects of the galaxy clusters.

• Lagrange identity, Jacobi criterion and virial theorem (cont.) Applications of the virial theorem (cont.): Systems undergoing uniform expansion or contraction: but

Spiral galaxies: assuming a spherical distribution of matter, ;

same result from the virial:

If vrot ≈ const., M(r) ∝ r → ∞ if r → ∞ ; finite visible mass => DM!

Elliptical galaxies: velocity dispersion

Energy ; no gas => difficult to measure σ far from the center

Galaxy clusters: intergalactic hot gas => Kinetic energy of the gas very small. Gas mass ~ 20-30% of gravitational mass → still 70-80% of DM!

˙ I ≠ 0

˙ I = cste ⇒ ˙ ̇ I = 0

• Lagrange identity, Jacobi criterion and virial theorem (cont.)

Applications of the virial theorem (cont.) Globular clusters: no significant DM Example: Palomar 13 has σ = 0.6-0.9 ± 0.3 km/s => M/LV = 3-7 (Blecha, Meylan et al. 2004, A&A 419, 533)

dSph galaxies: small to large amounts of DM; problems: - tidal tails => σ ➚ - unrecognized binaries => σ ➚ but insufficient tu rule out DM.

Example: Sculptor; Mdyn = 3 × 108 Msol (within 1.8 kpc)

=> M/L ~ 160 (Tolstoy et al. 2009, ARAA 47, 371)

• DM in dSph galaxies

Source: Walker M. G. 2011, in A universe of dwarf galaxies, eds. M. Koleva, Ph. Prugniel and I. Vauglin, EAS Pub. Series 48, 425

• Projection and deprojection Objects with spherical symmetry Density profile: projected density given by

Σ is the Abell transform of ρ. Solution:

Any uncertainty on Σ(R) is amplified in ρ(r)!

Kinematic deprojection: 1) spherical isotropic systems:

• Projection and deprojection (cont.) Kinematic deprojection: 2) spherical anisotropic systems:

Because the non-diagonal terms of the anisotropy tensor are 0, and with β the anisotropy tensor of the velocities: β = 1 for radial dispersion β = 0 for purely isotropic dispersion β → -∞ for purely tangential dispersion

In general, β is unknown : how to eliminate it? 2nd Jeans equation stationary and in spherical coord:

but , so if ,

• Projection and deprojection (cont.) Kinematic deprojection: 2) spherical anisotropic systems (cont.):

=>

=>

If M(r), ρ(r), Σ(R) and σldv(R) known, the right-hand side term is known. Inversion of the integral equation not easy: Binney & Mamon (1982), Tonry (1983), Bicknell et al. (1989), Solanes & Salvador-Solé (1990) etc. (« anisotropy inversion »). Other way round: assume β(r), deproject, then use Jeans equation to determine M(r) (« mass inversion », Mamon & Boué 2010, MNRAS 401, 2433)

• Stability of collisionless systems All theoretical models are not necessarily stable, e.g. a thin disk of stars is highly unstable => necessary to test the stability with respect to retroactive effects (≠ stability of orbits in a given, fixed potential). One neglects the effects of differential rotation => static systems, or systems in solid-body rotation.

Analogy between stellar systems and 1)  Autogravitating fluids: p  2)  Electrostatic plasmas: in rarefied plasmas, average field > 2-body

interactions; but plasma neutral at large scales

Linear response theory: Equilibrium system with density forced by an external gravitation field with ε response ∝ perturbation ε ; induced density perturbation such that

with the response function.

• Stability of collisionless systems (cont.) Linear response theory (cont.): Causality => for . The system’s dynamics depend on the total perturbing potential corresponding to the density . The polarization function is

with for

R function: describes the ρ response to an external perturbing force P function: describes the ρ response to a TOTAL perturbing force, including

the contribution from the self-gravity of the ρ response.

Temporal FT of a function y(t) that vanishes for t < 0:

With c real, > 0 and large enough that converges. Then