Superluminal neutrinos?

Transcription

1 Superluminal neutrinos? Measurement of the neutrino velocity with the OPERA detector in the CNGS beam, T. Adam et al. 2011, arxiv: v1: v > c by 2.48 x 10-5

2 Superluminal neutrinos? Neutrino velocity from the supernova SN1987A: distance of the LMC ~ l.y. => delta_t = x 2.48 x 10-5 ~ 4 years!

3 Potential theory

4

5 Si ρ=0, (Gauss theorem) Potential energy: Part of the system already in place, with ρ(x) and Φ(x); work to bring δm from to x is δmφ(x) ; then,

6 (2.14) because ; the divergence theorem then implies: 0

7 Spherical systems Newton s first theorem: A body that is inside a spherical shell of matter experiences no net gravitational force from that shell. Newton s second theorem: The gravitational force on a body that lies outside a spherical shell of matter is the same as it would be if all the shell s matter were concentrated into a point at its center. Corollary of Newton s first theorem: The gravitational potential inside an empty spherical shell is constant, because. One can then choose Φ using its definition, as = G R d 3 x'ρ(x') = GM R R=radius of the spherical shell Consequence of both theorems: The gravitational attraction of a spherical density distribution ρ(r ) on a unit mass at radius r is entirely determined by the mass interior to r: with

8 Spherical systems (cont.) The potential is the sum of the potentials due to each spherical shell: Circular speed: centripetal acceleration is v c2 /r Circular frequency: Escape speed:

9 Spherical systems (cont.) Potentials of some simple systems (cont.): a) Point mass: (keplerian potential) b) Homogeneous sphere: Orbital period: => r = f(ρ), indep. from r free fall time: Harmonic oscillator with ω=2π/t ; if dropped at distance r, the star reaches the center after T/4 = which is very close to

10 Circular velocity in the nucleus of M31: a behaviour close to that expected for a homogeneous sphere Source: ; adapted from Kormendy & Bender 1999, ApJ 522, 772

11 Spherical systems (cont.) Potentials of some simple systems (cont.): b) Homogeneous sphere (cont.): A significant fraction of the orbit is completed in a time ~ (Gρ) -1/2, whatever the size & shape of the orbit. True also of inhomogeneous systems, if the average ρ is considered: c) Plummer model: the simplest of all, that does not diverge at r = 0 Caveat: no analytical formula for the orbits in this potential!

12 Dark halo Stars Plummer density profiles fitted to hydrodynamical N-body/Tree-SPH simulations of dsph galaxies (source: Revaz et al. 2009, A&A 501, 189)

13 Spherical systems (cont.) Potentials of some simple systems (cont.): d) Isochrone potential: orbits can be described analytically. The orbital period depends on E, not on L (like in the keplerian case) GM/r for r >> b e) Two-power density models: 2 power laws for the density, one near the center, the other at large radii: Dehnen models: all models with β = 4 Hernquist models: (α, β) = 1, 4 elliptical galaxies Jaffe models: (α, β) = 2, 4 NFW (Navarro, Frenk & White 1995): (α, β) = 1,3 DM halos NFW: ρ 0 and a parameters not independent (simulations) => 1 effective param.: r 200, such that ρ(r 200 ) = 200 ρ c (t) = 200 (3H 2 (t)/8πg)

14 Fit of a Jaffe profile (α, β = 2, 4) to the elliptical galaxy NGC 3379 (source: astr553/topic07/t7_jaffe_fit.html) Source: heavensgloryobservatory.com

15 Spherical systems (cont.) Potentials of some simple systems (cont.): e) Two-power density models (cont.): NFW : (α, β) = 1,3 Concentration parameter: c = r 200 /a ; c = 16 for M ~ M " 6 for M ~ M Mass within r: M const if r M const if r M if r

16 Spherical systems (cont.) Potentials of some simple systems (cont.): f) Sérsic 3D (or Einasto): - 3D analogue of the Sérsic model originally designed to fit projected profiles of elliptical galaxies - Better than NFW to fit N-body cosmological simulations (Navarro et al. 2004): a) finite central density b) finite total mass with m~6 Sérsic 2D index n vs integrated absolute magnitude (Kormendy et al. 2009, ApJS 182, 216, Fig. 33)

17 Quelques profils de densité (source: G. Mamon, Dynamique gravitationnelle des systèmes à N corps, )

18 Flattened systems Kuzmin and Myamoto-Nagai models: a) Kuzmin (1956): (also called «Toomre s model 1») model of a razor-thin disk - at points with z < 0, Φ K = potential of a point mass M located at (R, z) = (0, a) - at points with z > 0, Φ K = potential of a point mass M located at (R, z) = (0, -a) => 2 Φ K and ρ = 0 everywhere, except at z = 0 Surface density

19 Flattened systems (cont.) Kuzmin and Myamoto-Nagai models (cont.): b) Myamoto & Nagai (1975): generalization of Plummer s and Kuzmin s models - Identical to Plummer for a = 0 (r 2 = R 2 +z 2 ) - Identical to Kuzmin for b = 0 - More and more spherical with increasing b/a

20 Flattened systems (cont.) Kuzmin and Myamoto-Nagai models (cont.): Example: Plummer s and Kuzmin s models used to interpret the IR surface brightness profile of the quasar host 1 Zw 1 Scharwächter et al. 2003, A&A 405, 959