Astronomía Extragaláctica y Cosmología ObservacionalDepto. de Astronomía (UGto)

Lecture 11 Groups and Clusters of Galaxies – III (DM)

Virial Mass EstimatesX-rays Mass Estimate

plasma temperature mass profile scale relations

Mass from Gravitational Lenses Einstein ring deflection angle and the mass types of lenses magnification and ellipticities

Mass Content and Mass Function

Cluster Masses from the Virial Theorem

Assumptions:• the orbits are periodic

• there is no mass variation

• ergodic condition: the system is gravitationally bound / dynamically relaxed / in steady state (neither expanding nor contracting) (tcross < tH, Emec < 0)

• spherical symmetry and isolation (boundary)

• the galaxy velocities are isotropic [σx

2 = σy2 = σz

2 = σ2/3 σ2 = 3σLOS2

RH = (π/2) RHP]

½ dt2 Ijk – ½ dt Jjk = 2 Kjk + Ujk

Jjk = 0 2 <Kij> + <Uij> = 0

<dt2 Ijk> = 0 (time mean)

2 K + U = 0 (at any time)

Mvir = (1/G) V2 RH

Mvir = (3π/2G) σLOS2 RHP

Cluster Masses from the Virial Theorem

But usually, clusters of galaxies have no quite finished forming and equilibrating (merging of subclusters) Jjk ≠ 0; <K> ≠ K; <U> ≠ U

consequently they are neither spherically symmetrical nor isolated

in the external part of the clusters, radial orbits are predominant

observationally, it is difficult to identify interlopers (galaxies passing by the cluster, infalling galaxies that have not yet entered into virial equilibrium and galaxies seen in projection)

So some corrections may be applied

typically the uncertainty in the Virial mass estimate is below 50% for regular (bound) clusters [Small et al. 1998, ApJ 492, 45]

Cluster Masses from the Virial Theorem

Projected Mass Estimator Heisler, Tremaine & Bahcall 1985 [ApJ 298, 8] proposed an expression for the Virial mass estimation with corrections for orbital shapes and for the fact that center must be determined from the (projected) galaxy positions (and not the exact mass distribution)

where k depends on the typical orbital shape (32/π for purely isotropic, and 64/π for purely radial), α ~ 1.5 is the factor accounting for the center difference, vi is the radial velocity of the galaxy with respect to cluster mean, and Ri is the projected distance from the cluster center

Mproj = 1 k Σ vi2 Ri

G (N-α)

Surface Term Correction Carlberg et al. 1996 [ApJ 462, 32] proposed a correction on virial mass estimates to account for the fact that the system is not entirely enclosed in the observational sample, called surface pressure term correction such correction depends on the anisotropic component of galaxy motions, which can be measured from the VDP [Carlberg et al. 1997, ApJ 478, 462; Girardi & Mezzeti 2001, ApJ 548, 79]:

• decreasing/radial VDP – median 45%• flat/isotropic VDP – median 20%• increasing/rotational VDP – median 14%

Cluster Masses from the Virial Theorem

Dark Matter F. Zwicky [1933, Helv. Phys. Acta 6, 110; 1937, ApJ 86, 217] was the pioneer on applying the Virial Theorem to estimate the mass of a galaxy cluster, Coma, and found that the calculated mass was far greater than the observed mass in stars S. Smith [1936, ApJ 83, 23] showed that the same was true for the Virgo Cluster this became known as the missing mass problem

(to be true it is the missing light problem), and this additional matter as dark matter

Coma (A1656)Coma (A1656)

Cluster Masses from X-ray Emission

Gas Temperature Since the temperature of the plasma, in a hydrostatic equilibrium regime, is simply a way to support the attraction by the overall gravitational potential, this temperature is closely related to the total mass of the cluster The hydrostatic equilibrium may be described by the Euler Equation:

Assuming:• spherical symmetry

• “ideal gas” state equation (P V = N kB T) and ρ = μ mH N / V, where μ is the mean molecular weight (μ ≈ 0.6 for a primordial ionized gas)

• (1/ρT) d(ρT)/dr = (dlnρ/dr) + (dlnT/dr), and d/dr = –F(r)/m = G M(r) / r2

kB T dlnρ + dlnT = – G M(r) μ mH dr dr r2

(1/ρ) P = –

1 dP(r) = – d(r)ρ(r) dr dr

P = n kB T = (ρ / μ mH) kB T 1 kB d(ρT) = – d ρ μ mH dr dr

M(r) = – kB T r2 dlnρ + dlnT GμmH dr dr

Cluster Masses from X-ray Emission

Mass profile Thus, the mass distribution within the cluster can be determined if the variation of the gas density and temperature with radius are known

the density distribution is directly associated to the surface brightness (luminosity) profile the temperature may be measured from the X-ray spectra (thermal bremsstrahlung), along many lines of sight through the cluster

also, measuring temperatures requires higher quality data than luminosity measurements, because the photons must be divided among multiple energy bins...

furthermore, even with the highest quality data, the derived mass is still slightly model dependent because T(r) and ρ(r) must be determined by deprojecting the surface brightness information

Cluster Masses from X-ray Emission

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Cluster Masses from Gravitational Lensing

A. Einstein [1915, K. Preuss. Akad. Wiss. 1, 142] showed that his General Theory of Relativity predicted the deflection of a light ray passing close to a mass concentration due to the curvature produced by this mass in the space-time his prediction was first verified in 1919 [Dyson et al. 1920, Phil. Trans. of Royal Society 220, 291], with the observation of apparent position variations of stars close to the Sun during an eclipse, in Sobral (CE, Brasil)

however, the first evidence of extragalactic gravitational lensing was obtained only in 1979 [Walsh et al., Nature 279, 381], with the discovery of multiple images of QSOs

gravitational lensing arcs were first observed in 1987 [Soucail et al., A&A 172, 14], for the cluster A370

A370

Cl2244

Cluster Masses from Gravitational Lensing

η is called “collision parameter”/DOL = tg θ ≈ θ

α is the deflection angle DOS, DOL and DLS are “angular diameter” distances, in the cosmological sense

DOLDLS

DOS

Observer SourceLens

α

θ

Apparentposition

η

β

Gravitational lensing geometry (general case)

Cluster Masses from Gravitational Lensing

Einstein Ring (the simplest case) Chwolson [1924, Astr. Nachrichten 221, 329] and Einstein [1936, Science 84, 506] realized that, if the background object were precisely aligned with the deflecting point of the lens, the gravitational deflection of the light rays would result in a circular ring, centered upon the lens

DOLDLS

DOS

Observer SourceLens

α

Apparentposition

ηθE δ

α = θE + δη/DOL = tg θE ≈ θE η/DLS = tg δ ≈ δ

η = θE DOL

δ = θE DOL / DLS

α = θE + θE(DOL/DLS)α = θE [(DLS + DOL)/DLS]

θE = α (DLS/DOS)

Cluster Masses from Gravitational Lensing

Einstein Rings

MG1131+0456 (VLA)

B1938+666 (Merlin)

B1938+666 (HST)

Cluster Masses from Gravitational Lensing

Dependence of the deflection angle on the mass

where is the 3D gravitational potential and Ψ is projected potential. for a point lens, in cylindrical coordinates,

for a flat and homogeneous distribution of matter, with projected density ΣL (from the 2D Poisson equation),

to estimate the cluster mass using the images produced by the gravitational lensing effect one may fit a projected mass profile (or potential) and try to reproduce the image(s) of the background object(s) that are subject to the effect

α = 2/c2 ∫ (,z) dz = 1/c2 Ψ()

2 Ψ = 8πG ΣL α = [4 G M(<)] / ( c2)

= –(GM) / √(2+z2) α = (4GM) / (c2)

Cluster Masses from Gravitational Lensing

Types of lenses by replacing the lens equation (Einstein ring), M() = π2ΣL, and =DOLθ, we can derive a critical surface density:

a lens with ΣL = Σcrit perfectly focuses the image in a well defined point – usually this is not the case and the lens produces aberrations in the image (distortions and multiple images).

α = θ DOS = 4 G π 2 ΣL = 4 G π (DOLθ) ΣL

DLS c2 c2

Σcrit = c2 DOS = c2 . 4 G π DLS DOL 4 π G D

Cl1358+62

Gravitational Lensing

the lensing effect can be classified in 4 types depending on θE and Σcrit

• when the source and the lens are aligned, an Einstein ring is produced at θE

• when ΣL > Σcrit (usually inside θE), the effect is called strong lensing, and multiple images are always formed• when ΣL < Σcrit (usually outside θE), the lensing is weak, and only one image is produced, stretched tangentially to θ, producing an arclet• when θ >> θE the weak lensing effect distorts only slightly the galaxy images, making them more elongated – this effect is called shear

Einstein´s Cross Cl0024+16

A1689

Cluster Masses from Gravitational Lensing

Types of lenses:

α

Cluster Masses from Gravitational Lensing

Magnification A fundamental property of gravitational lenses is to preserve surface brightness: since the source image size is changed, the observed flux will also vary the image size is usually enlarged, so the flux is magnified! (more light will arrive to the observer due to the convergence power of the lens)

Measuring scales With strong lensing it is possible to measure the mass on the central region of the clusters, while from arclets and shear one can measure the mass in larger radii strong lensing is easily identified, but weak lensing, specially the shear may be only found by statistics of orientations of galaxies in the field: galaxies have arbitrary orientations but with the shear their ellipticities will be coherently oriented with minor semi-axis pointing to the lens a complex ellipticity can be defined:

where a and b are the major and minor axis of the image, and φ is the position angle of a in the weak lensing regime

where γ is the shear and κ is the convergence

ε = (a–b)/(a+b) e2iφ

<ε> = <γ / (1–κ)>

Cluster Masses from Gravitational Lensing

Cluster Mass Content

Typical galaxy: ~ 1011 stars 1011 M in stars Typical cluster: ~ 102 – 103 galaxies 1013 – 1014 M in stars ICM: ~ 2-20 × 1013 M

Cluster masses (Virial, X-rays, Lenses): 1014 – 1015 M

Stars: < 10%

ICM: 10-25%

DM: 70-90%

References for the images of Extragalactic Astronomy Part

Astronomy Picture of the Day -- http://apod.nasa.gov/apod/astropix.html Digitized Sky Survey -- http://archive.stsci.edu/cgi-bin/dss_form Hubble Space Telescope Gallery -- http://hubblesite.org/gallery/album/ Sloan Digital Sky Survey -- http://www.sdss.org/ Chandra X-ray Observatory -- http://chandra.harvard.edu/photo/ Astronomia Extragalática (Notas de Aula by Gastão Lima Neto, IAG/USP, Brasil) Anglo Australian Observatory -- http://www.aao.gov.au/images/ NASA Extragalactic Database -- http://nedwww.ipac.caltech.edu/ Two-Micron All Sky Survey -- http://www.ipac.caltech.edu/2mass/