b a = 1 n 10. Surface brightness profile of most elliptical galaxies can be fit well by the R 1/4 (or de Vaucouleurs) law, (1 ɛ) 2 a 2 = 1.

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1 7 Elliptical Galaxies Basic properties of elliptical galaxies Formation of elliptical galaxies 7.1 Photometric Properties Isophotes of elliptical galaxies are usually fitted by ellipses: Major axis a; minor axis b, ellipticity ɛ = 1 b/a. Types of ellipticals: E n (n = 0,1,,7) if b a = 1 n 10. Surface brightness profile of most elliptical galaxies can be fit well by the R 1/4 (or de Vaucouleurs) law, ( I(R) = I 0 exp ξ 1/4) with ξ R/R, R : a characteristic radius, which scales with the azimuthal angle φ (relative to the major axis) as R [(1 ɛ) 2 1]cos 2 φ. Proof: x 2 a 2 + y 2 (1 ɛ) 2 a 2 = 1. r 2 = x 2 + y 2 = x 2 + (a 2 x 2 )(1 ɛ) 2 = a 2 (1 ɛ) 2 + r 2 cos 2 φ[1 (1 ɛ) 2 ]. r 2 = b 2 /[1 + ((1 ɛ) 2 1)cos 2 φ]. If R is taken to be the effective radius R eff within which half of the total luminosity is contained: R eff /R 0 ζ exp( ζ 1/4 )dζ = ζ exp( ζ 1/4 )dζ, then [ ( )] I(R) = I eff exp 7.67 ξ 1/4 1 with ξ R/R eff, where I eff is the surface brightness at R eff. The value of R eff is usually quoted as that of the semi-major axis. 1

2 2 Fig The ratio between ordered motion and randon motion. The curve in (a) shows flattening due to rotation. The open symbols in (a) are for giant ellipticals, while the filled dots are for relative small ellipticals. 7.2 Kinematic Properties Velocity dispersion versus ordered motion Big ellipticals not supported by angular momentum, but by random motion. Importance measured in terms of the ratio between the maximum line-ofsight velocity v m (relative to the mean velocity of all stars in the galaxy) and the mean velocity dispersion σ Flattening caused by rotation As another application of virial theorem, consider an axisymmetric system. Let the axis of symmetry be the z-axis. Virial theorem applied to z and x directions are Thus, 2K zz + W zz = 0; 2K xx + W x = 0. W xx = K xx V rot/2 2 + σx 2 W zz K zz σz 2. Here we have used the fact that half of the rotation energy is in x component. For a oblate body with axis ratio b/a, it can be shown that W zz W xx (b/a) 0.9 = (1 ɛ) 0.9. Thus, for isotropic velocity dispersion, i.e. σ x = σ z = σ, we have V rot σ 2(1 ɛ) (1 ɛ) 1 1 = ɛ/(1 ɛ). This is a relation between the flattening (ɛ) and the rotation (V rot ) that causes it. Observations show that giant ellipticals are NOT flattened by rotation, but by anisotropic velocity dispersion.

3 Elliptical Galaxies 3 Fig The fundamental plane of elliptical galaxies. 7.3 Gas Content and stellar population Little cold gas and dust; but extended X-ray halos of hot gas. Not much current star formation. Giant ellipticals contain mainly old (Population II) stars. Most of the stars in elliptical galaxies form early. 7.4 Scaling Relations Basic properties: R eff ; L (or I eff ), and σ 0 (central velocity dispersion). Each of these parameters covers a large range, but they are tightly correlated: The fundamental plane relation: log R eff = alog σ 0 blog I eff + constant. where a 1.24, b 0.82 in the optical, and a 1.53, b 0.79 in near infrared. The Faber-Jackson relation: L σ 4 The D n -σ 0 relation: D n σ 1.2, where D n is the physical radius at some isophotal level. Both are projections of the fundamental plane 7.5 Physical explanation of the fundamental plane The fundamental-plane relation is interpreted in terms of the the virial theorem: GM R = k E v 2 2,

4 4 R : average radius such that the l.h.s. is the mean specific potential energy, v 2 is so defined that v 2 /2 is the mean specific kinetic energy, k E is equals to 2 for a system in equilibrium. Expressing R and v 2 in terms of observables R e (e.g. R eff ), and σ e (e.g. σ 0 ): R e = k R R ; σ e = k V v 2, writing L = M/(M/L), L = k L I e R 2 e (k R, k V and k L are dimensionless quantities), we obtain ( ) M 1 R e = C R σei 2 e 1, C R L k E 2Gk R k L k 2 V ( ) M 2 ; L = C L σei 4 e 1, C L = k L CR 2. L If elliptical galaxies are homologous (i.e. they have the same density and velocity profiles so that C R and C L are the same for all of them), and if they all have the same (M/L), then the virial relation defines a plane in the log R e -log σ e -log I e space, or in the log L-log σ e -log I e space. In the more general case where (M/L), C R and C L are power laws of I e, σ e and R e (or L), the virial theorem still defines a plane in the parameter space, but the plane will be tilted with respect to the one with constant (M/L) and C R (or C L ). The observations on the FP can therefore provide important information on the formation and evolution of elliptical galaxies. Assume that ellipticals are homologous, the observed FP implies a change of (M/L) with L and I eff as (M/L) L (2 a)/2a I 1/2 (1+2b)/a eff. The observed values of a and b implies that (M/L) L 0.31 I 0.02 eff (in optical), (M/L) L 0.15 I 0.12 eff (in near-infrared). Other possibilities: elliptical galaxies may not be homologous in their structures, and so k R and k V may depend on σ e and I e. 7.6 The Morphology-Density Relation Given that galaxies have different intrinsic properties, one natural question is whether these properties are correlated with the environments. The answer is yes for some properties. An important example is the morphology-density relation. 7.7 Elliptical Galaxies as Collisionless Stellar Systems Since elliptical galaxies consist mainly of stars, one may hope to understand their structural and kinematic properties by treating them as self-gravitating stellar systems.

5 Elliptical Galaxies 5 Fig The morphology-density relation Simple Dynamical Models The structure and kinematics given by the phase-space distribution function f(x,v,t): ρ (x) = m f(x,v)d 3 v, ρ(x) = (M/L)ρ (x). v i (x) = m ρ (x) v i f(x,v)d 3 v, v i v j (x) = m ρ (x) v i v j f(x,v)d 3 v. Observables: surface density, mean velocity along a line-of-sight, and mean velocity dispersion, at a position (x,y) Σ = ρ (x)dz, v z = 1 ρ (x) v z (x)dz, σz 2 = 1 ρ (x) vz 2 (x)dz v z 2, Σ Σ The Isothermal Sphere The distribution function: f(e) = (ρ ( ) 0/m) E (2πσ 2 exp ) 3/2 σ 2 where E v 2 /2 Φ, and σ 2 is a constant. The density is ρ(r) = ρ 0 exp( Φ/σ 2 ), and Poisson s equation: ( 1 d r 2 r 2dΦ ) = 4πGρ 0 exp ( Φ ) dr dr σ 2. With the boundary condition ρ(0) = ρ 0, dφ/dr = 0, the above equation can be integrated. The resulted density profile is characterized by a King,

6 6 radius: r 0 = 3σ 4πGρ0. ρ(r) ρ 0 [1 + (r/r 0 ) 2 ] 3/2(for r < 2r 0), ρ(r) = σ 2 /(2πGr 2 )(for r > 10r 0 ) Note that the total mass involved is infinite! King Model { ( (ρ0 /m)(2πσ f(e) = 2 ) 3/2 e E/σ2 1) (E > 0) 0 (E 0) Note that no particles have E 0. The density is ρ(r) = 4π f(e)v 2 dv ( ) ( Φ 4Φ = ρ 0 e Φ/σ2 erf σ πσ 2 1 2Φ ) 3σ 2, (7.1) which in Poisson s equation gives an equation for Φ. This equation for Φ can be integrated numerically with the boundary conditions Φ = Φ 0 and (dφ/dr) = 0 at r = 0. The observed light distribution in bright ellipticals can be fit by King models with Φ(0)/σ Models with r 4 outer profiles ρ(r) = (3 γ)m 4π a r γ (r + a) 4 γ, For γ = 1 (the Hernquist model), the [ f(ẽ) = M 3sin 1 q + q ] 1 q 2 (1 2q 2 )(8q 4 8q 2 3) 8 2π 3 a 3 vg 3(1, q2 ) 5/2 q GM Ẽ and v g a. 7.8 Formation Scenarios for Elliptical Galaxies The fact that the collapse of an N-body system with cold and clumpy initial configuration generally leads to the formation of an elliptical-like remnant with R 1/4 -profile suggests that ellipticals may have formed from the collapses of cold and clumpy stellar systems. To understand the formation of elliptical galaxies, it is important to understand how such initial conditions are generated. Three possibilities.

7 Elliptical Galaxies 7 (i) The formation of an elliptical by a major merger of two disk galaxies (perhaps with bulges) of comparable masses. In this case, most stars in the elliptical galaxy are formed in its progenitor disks. (ii) The formation of an elliptical galaxy by a sequence of mergers of galaxies with individual masses much smaller than the remnant elliptical. As in the major-merging scenario, most stars in the remnant are formed in its projenitors. (iii) The formation of an elliptical through the collapse of a large chunk of gas cloud. In this model, stars are formed as the cloud contracts and fragments. The last possibility is usually called the monolithic-collapse scenario, while the first two are called the merging scenarios.