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 1
The Nobel Prize in Physics 2011 Saul Perlmutter (Berkeley, CA) Brian P. Schmidt (Weston Creek, Australia) Adam G. Riess (STScI, Baltimore, MD) The Nobel Prize in Physics 2011 was awarded "for the discovery of the accelerating expansion of the Universe through observations of distant supernovae" with one half to Saul Perlmutter and the other half jointly to Brian P. Schmidt and Adam G. Riess.
Source: Knop et al. 2003, ApJ 598, 102 (Perlmutter s group)
Euclid: Mapping the geometry of the dark Universe Theme: How did the Universe originate and what is it made of? Primary Goal: To understand the nature of dark energy and dark matter by accurate measurement of the accelerated expansion of the Universe through different independent methods. Targets: Galaxies and clusters of galaxies out to z~2, in a wide extragalactic survey covering 15 000 deg², plus a deep survey covering an area of 40 deg² Wavelength: Visible and near-infrared Telescope: 1.2 m Korsch Orbit: Second Sun-Earth Lagrange point, L2 Lifetime: 6 years Type: M-class mission
Laplacian in cylindrical coordinates: Poisson s equation in very flattened systems => Poisson s equation with F(R) = - Φ/ R Myamoto-Nagai potential: But F(R) - Φ K / R remains well behaved => 1/b ~ when b 0, for z=0 Valid potential : the vertical variation of Φ at a given R depends only on the ρ distribution at that R Solution of Poisson s equation in 2 steps: 1) Disk = surface density layer of zero thickness; determine Φ(R,0) 2) Solve the «vertical» Poisson eq. to find Φ(R,z) => 0 if disk symmetrical around the equatorial plane
The potentials of disks - One considers first oblate spheroidals : small axis = axis of symmetry - small volume limited by 2 similar spheroïdal surfaces = thin homoeoid - spheroid = superposition of thin homoeoids - axisymmetric disk = very flattened spheroid Find the surface density Σ(R) as a function of a distribution of central surface densities Σ 0 (a), then the potential Φ(R,0) and the circular velocity v c2 (R) To get a thin disk, q 0 but with = const.
The potentials of disk (cont.) If Σ(R) is known, the disk can be built by combining homoeoids whose summed surface density equals Σ(R) R find the function Σ 0 (a) that satisfies the Abell integral equation Solution: Potential: Homoeoid theorem: The exterior equipotential surfaces of a thin homoeoid are the spheroids that are confocal with the shell itself. Inside the shell the potential is constant. Newton s third theorem: A mass that is inside a homoeoid experiences no net gravitational force from the homoeoid. => The radial force felt at R depends not only on the mass interior to R, but also on that exterior to R
The potentials of disk (cont.) Potentials of spheroidal shells: Oblate spheroidal coordinates: (R, z) (u, v) with semiellipses u=const. and hyperbolae v=const. R = Δ cosh(u) sin(v) z = Δ sinh(u) cos(v) with u 0, 0 v π Source: Binney & Tremaine 2008, Galactic dynamics, 2nd ed.
The potentials of disk (cont.) Potential of a thin homoeoid of eccentricity e = (1-c 2 /a 2 ) 1/2 : remembering, the potential outside the homoeoid is, for e = 1: Tedious calculation leads to Which becomes, after integrating by parts and putting z = 0 Putting z = 0 in the previous equation and differentiating with respect to R:
Two examples: a) the Mestel disk (flat rotation curve) The potentials of disk (cont.) \ / V cosh -1 (R max /a) \ / V R max 2 a R max a 2 R max 1 a = constant! arcsin(a/r) 0 R => v 0 2 = G M(R)/R spherical case!
The potentials of disk (cont.) Two examples (cont.): b) the exponential disk \ / V = Σ 0 a K 1 (a/r d ) ; K 1 = modified Bessel function => where y R/(2R d ) and I ν, K ν are modified Bessel functions
V c of the exponential disk (full curve), of a point with the same total mass (dotted) and of the spherical system with the same M(R) law (dashed) Source: Binney & Tremaine 2008, Galactic dynamics, 2nd ed.
The potential of our Galaxy Several components: - disk hypothesis: each component has its own, constant M/L ratio - bulge - stellar halo (not modelled: negligible contribution to the potential) - dark halo Available data to constrain the model: 1. v c (R), from v 0 = v c (R 0 ) and HI clouds 2. Values of the Oort constants A, B 3. Surface density within 1.1 kpc of the Galactic plane near the Sun, Σ 1.1 (R 0 ) 4. Velocity dispersion of bulge stars in Baade s window, σ = 117±15 km/s 5. Total mass within 100 kpc from the Galactic Center 6. Distance R 0 = 8 kpc of the Sun to the G.C. Density law of each component: 0) SMBH a) Bulge : with Oblate spheroid truncated at r b = 1.9 kpc, with α b = 1.8, q b = 0.6, a b = 1 kpc
The potential of our Galaxy (cont.) Density law of each component (cont): b) Dark halo: Dehnen s model (generalized to oblate spheroids) with h But q h very poorly constrained => one assumes q h = 0.8! c) Stellar disk: 2 components, the thin disk & the thick disk with α 0 +α 1 = 1 Thin disk: z 0 ~ 0.3 kpc Thick disk: z 1 ~1 kpc d) ISM disk: there is a hole for R < 4 kpc! Hole at R < 4 kpc With R m = 4 kpc z g = 80 pc R g = 2 R d the gas extends much beyond the stars! Contribution of Σ g : 25% of Σ d
Queues de marée entourant NGC 5907 qui ne sont que les débris d une galaxie naine après collision
Distribution of the interstellar matter (ISM) HI distribution in M81 Source: Yun et al. 1994, Nature 372, 530 HI distribution in M31 (lower panel) compared to the optical picture (upper panel) Source: Brinks & Bajaja 1986, A&A 169, 14
Source: Binney & Tremaine 2008, Galactic dynamics, 2nd ed.
Model I potential Model II Source: Binney & Tremaine 2008, Galactic dynamics, 2nd ed.
Model I Model II Source: Binney & Tremaine 2008, Galactic dynamics, 2nd ed.
Force perpendicular to the disk (continuous line) with the contribution from the disk and from the bulge+halo Source: Binney & Tremaine 2008, Galactic dynamics, 2nd ed.
«Boxy» bulge in NGC 4710 (NASA & ESA photo release, 18 Nov. 2009) «Boxy» bulge in NGC 5746 (Bureau & Freeman 1999, AJ 118, 126)
Orientation of the bar of the Galaxy Source: Binney & Tremaine 2008, Galactic dynamics, 2nd ed.
Orientation de la barre de la Galaxie Source: F. Combes, Pour la Science No 337, nov. 2005
Equipotentials of an axisymmetric bulge (dotted lines) and of the bar (continuous lines) Source: Binney & Tremaine 2008, Galactic dynamics, 2nd ed.