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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
Lagrange identity, Jacobi criterion and virial theorem Lagrange identity & Jacobi criterion: Moment of inertia: =>
the kinetic energy and the potential energy of the system,
If W is a k-homogeneous function, i.e. if , then
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
∑ ∂W ∂ x i
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
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