Astronomía Extragaláctica y Cosmología Observacional

1. Depto. de Astronomía (UGto) Astronomía Extragaláctica y Cosmología Observacional Lecture 5 Masses of Galaxies • Dynamical masses • virial theorem • virial masses of ellipticals • rotation curves of spirals • hot gas in ellipticals • Mass/Luminosity ratios • definition • dark matter: Milky Way • dark matter: other galaxies • dark matter: other scales

2. Virial Masses • All direct methods of measuring masses in Astronomy are dynamical! • Virial theorem: • Introduced by Clausius (1879) and first applied in Astronomy by Eddington (1915) for star clusters • Premise: star clusters, galaxies and clusters of galaxies can be considered to be steady, gravitationally bound configurations, meaning that the objects of which they are composed have come into dynamical equilibrium under gravity. • Tests for boundedness: • crossing times → tcross < tsys tsys: age of the system • tcross = R / <v> R: size of the system • <v>: typical vel. or vel. dispersion • Ex: Milky Way R = 8.5 kpc • <v> = 220 km s-1 • trot = 2π R/<v> ≈ 2.5108 a << tMW 1010 a • mechanical energy → E < 0 K: kinetic energy • E = K + U U: gravitational potential energy

3. Virial Masses • Virial theorem - deduction: • Generically, virial theorems are moments of the motion equations, obtained by multiplying these eqs by powers of x and summing through all the galaxies. dt (m v) = F = – m Ф = – G m Σj mj (r – rj) / |r – rj|3 dt = d/dt Σkxj dt (mk vk) = – G ΣkΣj≠k mk mjxj / (xk – xj)2 → dt2 (Σk mk xj xk) = dt [dt(Σk mk xj xk)] = dt [Σkdt(mk) xj xk + Σk mkdt(xj) xk + Σk mk xjdt(xk)] = dt [Σk dt(mk) xj xk] + dt[Σk mk (vj xk + xjvk)] = dt [Σk dt(mk) xj xk] + dt[2Σk mk(xj vk)] = dt [Σk dt(mk) xj xk] + 2[Σkdt(xj) mk vk + Σk xjdt(mk vk)]  ½ dt2(Σk mk xj xk) – ½ dt [Σk dt(mk) xj xk] – Σk mk vj vk = Σk xj dt(mk vk) → ΣkΣj 1 / (xk – xj) = ΣkΣj (xk – xj) / (xk – xj)2 = ΣkΣjxk / (xk – xj)2 + ΣkΣj(–xj) / (xk – xj)2 = 2ΣkΣj xj / (xk – xj)2  ½ ΣkΣj 1 / (xk – xj) = ΣkΣj xj / (xk – xj)2 ½ dt2(Σk mk xj xk) – ½ dt [Σk dt(mk) xj xk] – Σk mk vj vk = – ½ G ΣkΣj≠k mk mj / (xk – xj)

4. Virial theorem - deduction: ½ dt2(Σk mk xj xk) – ½ dt [Σk dt(mk) xj xk] – Σk mk vj vk = – ½ G ΣkΣj≠k mk mj / (xk – xj) IjkJjk 2KjkUjk inertia tensor mass variation kinetic energy potential energy tensor tensor tensor ½ dt2Ijk – ½ dtJjk = 2 Kjk + Ujk • If there is no variation on mass: Jjk = 0 • If the orbits are periodic: <dt2Ijk> = 0 (time mean) 2 <Kij> + <Uij> = 0 • If the system is dynamically relaxed: 2 K + U = 0 (at any time) • Virial Masses

5. Mass determination: 2K = – U Σk mk vk2 = ½ G ΣkΣj≠k mk mj / (xk – xj) Σk mk vk2 = G ½ ΣkΣj≠k mk mj(Σk mk) Σk mk(xk – xj)(Σk mk)2 quadratic mean vel, harmonic radius = mean weighted by mass separation weighted by mass V2 = G Mvir / RH Mvir = (1/G) V2 RH • no assumption is made about the distribution of the particles and of their orbits! • if the system is not “virialized” (dynamical equilibrium), M will be overestimated by Mvir • Virial Masses

6. Virial Masses • Virial masses of Ellipticals: • since Re includes half of the luminosity of the galaxy, on the assumption that it also includes half of the mass, one can expect that RH Re • for a roughly spheric symmetrical distribution of mass, RH  3 Re • the mean velocity of an E is given by its velocity dispersion (σ) • σ can be measured from the width of absorption line profiles (Doppler broadening) • usually only the central velocity dispersion can be measured – the relation between σ and σ0 depends on the geometry and the shape of the vel distribution • on the assumption of isotropy, σx2 = σy2 = σz2 = σ2/3, so σ2 = 3σLOS2 • thus: Mvir ~ (9/G) σLOS2 Re  • Virial masses of Spirals: • for spirals, the virial theorem can be recovered from the balance between gravitational and centrifugal forces: G M(r) / r2 = Vrot2(r) / r M(r) = (1/G) Vrot2(r) r • axial symmetry is assumed, so Vrot is a circular velocity (and corrections to inclination may be applied) • measuring the M(r) in a range of r produces the circular rotation curve of the galaxy • the total mass may be obtained by measuring the total radius and the respective M(r)

7. Rotation curves of Spirals • Differential rotation measurements: • inclination in the optical 2D spectra – differential shifts on stellar absorption lines (central regions) or HII regions emission lines (outer regions of galaxy) • radio observations of 21cm line (HI gas) profile   

8. kinematical minor axis (systemic velocity) region of solid body rotation [V(r)  r] kinematical major axis (usually ~ // to apparent major axis) closed contour (elongated along the apparent major axis) means decline in rotation curve maximum rotation velocity (Vmax) • Rotation curves of Spirals Position-velocity diagram Differential rotation curves: NGC 2742 Spider diagram (contours of constant LOS velocity = isovelocities) NGC 1744

9. Rotation curves of Spirals

10. Rotation curves of Spirals • Differential rotation curves: • in general, rotation velocities of the stars (absorption lines) and the gas (HI) do not differ by more than 30 km/s (about the measurement errors) in the discs of S • the observed circular speed curves very rarely show a significant decline at the largest observed radii, that is they remain flat through large distances from the centre (in the most extreme cases to almost 100 kpc!) • the form of the rotation curve is fairly constant for distinct spiral types, with the amplitude (Vmax) changing from Sa to Sc – the Vmax has median values of 299, 222 and 175 km/s respectively for Sa, Sb and Sc[Rubin et al. 1985, ApJ 289, 81] • amongst the same type, the more luminous galaxies have higher Vmax (quantified by the Tully-Fisher relation)

11. Other mass estimates • Masses from X-ray emission of hot gas in Ellipticals: • most E have no gas, but others have some • more frequently than cold gas (HI), hot gas (T  106 K) of H and He fully ionized is found • the heating source is usually supernovae explosions • hot gas emits in X-ray band by bremsstrahlung (free e- scattered by ions) and bound-bound transitions of ions • since the gas is assumed to be in hydrostatic equilibrium in the potential well of the galaxy, the total mass of the galaxy (necessary to confine the gas) can be estimated  NGC 720

12. Mass/Luminosity ratios • Definition: • since evidences of a “missing mass” are found, in general from dynamical measures, it is conventional to estimate the mass/luminosity ratio of the galaxies: M /LЧ= M / M = dex[0.4(MЧ – MЧ)] M /M(Ч band) LЧ / LЧ • Dark matter: • Jan Oort[1932, Bull. Astron. Inst. Neth. 6, 249], analyzing velocities of stars near the Sun, concluded that visible stars can supply only 30-50% of the amount of gravitating matter implyed by their velocity • Fritz Zwicky[1933, Helw. Phys. Acta 6, 110], by measuring velocity dispersions of rich clusters, found that about 10 to 100 times more mass than the one in visible galaxies were needed to keep them bound • Ostriker, Yahil & Peebles [1974, ApJ L 193, L1] and Einasto, Kraasik & Saar [1974, Nature 250, 309] measured galaxies masses as a function of radius (from rotation curves) and found that masses increase linearly with r out to at least 100 kpc, and that normal S and E have masses ~ 1012M

13. Dark matter: Milky Way • local discM/L is estimated to be about 3 tracers: uncertainties: F and K giants evolution, metallicity, distance young population of thin disc little evidence of DM • MW rotation curve (total disc, to ~ 2 R) • inner disc → 5-10 • outer disc → 15-20 [Clemens 1985, ApJ 295, 422] tracers: stars, planetary nebulae, HI gas, HII regions

14. Dark matter: Milky Way • halo (to 5-10 R) → M/L ~ 10-30 tracers: uncertainties: RRLyrae runaway stars? Globular Clusters circular orbits at large radius? satellites dwarf galaxies which of them are really bound to the MW? • since we do not know how the invisible component is distributed, usually the M/L ratios are estimated using the spherical hypothesis for the sake of simplicity • there are also theoretical reasons to the DM haloes of S: Ostriker & Peebles [1973, ApJ 186, 467] showed (and it was latter confirmed by detailed computation) that DM haloes can stabilize the disc of S galaxies • since the rotation curve is flat, V2(r)  r, and so mass grows with r2 • some authors suggest that M/L grows with r other uncertainties: outer gas in noncircular orbits (owing to effects of recent arrival, p.e.) luminosity at large radius may be underestimated because sky background brightness is overestimated [Faber & Gallagher 1979, ARAA 17, 135*]

15. Dark matter: other galaxies • Globular Clusters have larger σ than the field stars at the same radius (at least in the most studied GC systems: M87, NGC 5128 and NGC 720) • gE at the center of clusters may trace cluster DM halo (M87, p.e.) • typical M/L of E gals are about 10-20 • bulges of S show M/L similar to that of E, about 10-20 • over the visible regions, M/L of S is usually less or about 10, but this value usually increase at larger radii • for relatively bright Irr, HI rotation curves extend far enough to demonstrate the existence of large and rising M/L outside the optical region • dwarf galaxies are found to have proportionally more DM than massive galaxies M87 NGC 6744 NGC 2403

16. Dark matter: other scales • SM/L may grow up to 100h at r ~ 200 kpc • EM/L are larger, reaching 400h at r ~ 200 kpc • on larger scales, M/L grows to 200-400h • most of the DM may reside on galactic haloes [Bahcall et al. 1995, ApJ 447, L81] 