Astronomía Extragaláctica y Cosmología ObservacionalDepto. de Astronomía (UGto)

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

Virial Masses

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> ≈ 2.5108 a << tMW 1010 a

• mechanical energy → E < 0 K: kinetic energyE = K + U U: gravitational potential energy

All direct methods of measuring masses in Astronomy are dynamical!

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/dtΣk xj dt (mk vk) = – G Σk Σj≠k mk mj xj / (xk – xj)2

→ dt2 (Σk mk xj xk) = dt [dt (Σk mk xj xk)]

= dt [Σk dt(mk) xj xk + Σk mk dt(xj) xk + Σk mk xj dt(xk)]= dt [Σk dt(mk) xj xk] + dt[Σk mk (vj xk + xj vk)]= dt [Σk dt(mk) xj xk] + dt[2 Σk mk (xj vk)]= dt [Σk dt(mk) xj xk] + 2[Σk dt(xj) mk vk + Σk xj dt(mk vk)]

½ dt2 (Σk mk xj xk) – ½ dt [Σk dt(mk) xj xk] – Σk mk vj vk = Σk xj dt(mk vk)

→ Σk Σj 1 / (xk – xj) = Σk Σj (xk – xj) / (xk – xj)2

= Σk Σj xk / (xk – xj)2 + Σk Σj (–xj) / (xk – xj)2

= 2 Σk Σj xj / (xk – xj)2

½ Σk Σj 1 / (xk – xj) = Σk Σj xj / (xk – xj)2

½ dt2 (Σk mk xj xk) – ½ dt [Σk dt(mk) xj xk] – Σk mk vj vk = – ½ G Σk Σj≠k mk mj / (xk – xj)

Virial Masses

Virial theorem - deduction:

½ dt2 (Σk mk xj xk) – ½ dt [Σk dt(mk) xj xk] – Σk mk vj vk = – ½ G Σk Σj≠k mk mj / (xk – xj)

Ijk Jjk 2Kjk Ujk

inertia tensor mass variation kinetic energy potential energy tensor tensor tensor

½ dt2 Ijk – ½ dt Jjk = 2 Kjk + Ujk

If there is no variation on mass: Jjk = 0 If the orbits are periodic: <dt

2 Ijk> = 0 (time mean)

2 <Kij> + <Uij> = 0

If the system is dynamically relaxed: 2 K + U = 0 (at any time)

Virial Masses

Mass determination:

2K = – UΣk mk vk

2 = ½ G Σk Σj≠k mk mj / (xk – xj)

Σk mk vk2 = G ½ Σk Σj≠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

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, σx

2 = σy2 = σz

2 = σ2/3, so σ2 = 3σLOS2

thus:Mvir ~ (9/G) σLOS

2 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)

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

Rotation curves of Spirals

Differential rotation curves:

Spider diagram(contours of constant LOS velocity = isovelocities)

Position-velocity diagram

region of solid body rotation [V(r) r]

kinematical minor axis(systemic velocity)

kinematical major axis(usually ~ // to apparent major axis)

closed contour (elongated along theapparent major axis) means

decline in rotation curve

maximum rotation velocity(Vmax)

NGC 1744

NGC 2742

Rotation curves of Spirals

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)

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

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 ~ 1012 M

Dark matter: Milky Way

local disc M/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

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

[Faber & Gallagher 1979, ARAA 17, 135*]

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

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

NGC 6744

NGC 2403

M87

Dark matter: other scales

S M/L may grow up to 100h at r ~ 200 kpc E M/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]