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4. Elliptical Galaxies

Before late 1970’s, elliptical galaxies were regarded as simple systems. They were thought to

• be flattened by rotation,

• contain little amount of gas,

• be made up of old stars only.

Much progress has been made since then. The old paradigm is now largely gone, people find that

• massive ellipticals are not flattened by rotation, but they are supported by anisotropicvelocity dispersions.

• many massive ellipticals actually contain decoupled (e.g. counter-rotating) centres (see Fig.14).

• ellipticals contain significant amount of hot gas that emits in the X-ray.

• Some ellipticals contain weak stellar disks and/or counter-rotating cores.

Fig. 1 shows M87 – the giant elliptical in the Virgo cluster. It appears to be quite simple in theoptical (at least in this image), but the galaxy appears dramatically different in the X-ray andradio. Fig. 2 shows two elliptical galaxies with shells and dust lanes, illustrating that ellipticalshave more complex appearances and can also contain gas and dust.

Fig. 1.— M87 in the optical (left), X-ray (middle) and radio (right). All three images are 8.7arcminute by 13.7 arcminute.

There are still many open questions, for example

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• What is the star formation history of ellipticals?

• How do elliptical forms? Is merging of disk galaxies the only formation route?

• How does the X-ray gas evolve and form in ellipticals?

• How does the fundamental plane of ellipticals arise?

• Why do the core properties such as the central black hole mass and more global propertiescorrelate so well?

We will touch upon some of these open questions later in this chapter.

Fig. 2.— Images of NGC 1344 and NGC 1316. Notice the shells and dust lanes in these twogalaxies, respectively.

4.1. Photometry

4.1.1. Surface Brightness Profile

A few models (e.g., King profile, Jaffe sphere) have been proposed to describe the light profilein ellipticals. A popular model is the de Vaucouleurs law or R1/4 law:

I(R) = Iee−7.67[(R/Re)1/4−1], (1)

where Re is the effective radius within which half of the light is contained, Ie is the surfacebrightness at Re and R is the projected radius. Fig. 3 shows a R1/4 fit to the surface brightnessprofile of NGC 1700.

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The typical ranges for Ie, Re and MB are 10 − 104L�/ pc2, 0.5 − 40 kpc, and 108 − 1012L�,respectively.

Fig. 3.— Surface photometry for NGC 1700. The solid line is the best R1/4-law fit to the data(indicated as solid points). The B-band surface brightness µB is in units of magnitude per squarearcsecond (a larger value of µB indicates a fainter surface brightness).

4.1.2. Cores of elliptical galaxies

The de Vaucouleurs profile cannot fit the inner-most (core) light profiles. Using the HubbleSpace Telescope photometry, Faber et al. (1997) find that the inner light profiles can be dividedinto two types:

• power-law inner profiles. The surface brightness profile keeps on rising steeply in the innerpart. Such profiles are found in lower-luminosity ellipticals and spiral bulges.

• cored profiles. The surface brightness profile rises much more slowly in the inner part. Suchprofiles are found in more luminous ellipticals.

Fig. 4 one example for these two types of profiles.

4.2. Isophotal shapes

4.2.1. Three-dimensional shapes

Elliptical isophotes are approximately elliptical. This implies that the three-dimensionalshapes can be

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Fig. 4.— Surface brightness profiles for the central light distribution in two elliptical galaxies:NGC 596 (power-law profile, upper curve at small mean radius) and NGC1399 (cored profile, lowercurve at small mean radius). The V -band surface brightness µV is in units of magnitude per squarearcsecond (a larger value of µV indicates a fainter surface brightness).

• oblate (a = b > c, e.g., flying saucers), prolate (a > b = c, e.g., cigars) or triaxial (a 6= b 6= c)

• from the observed projected ellipticity distribution, one infers that a : b : c ≈ 1 : 0.95 : 0.7.Ellipticals are hence mildly triaxial but close to oblate.

Isophotal twists are observed in many ellipticals; one example is shown in Fig. 5.

4.2.2. Shapes of isophotes: disky and boxy ellipticals

Isophotes of ellipticals are not exactly ellipses. Their deviations from perfect ellipses canclassify ellipticals into two types: disky or boxy.

Suppose we fit an ellipse Re(φ) to an isophote, where φ is the polar angle. For eachangle, we can measure the difference between the observed radius and the fitted radiusδ(φ) = Robs(φ)−Re(φ). We can expand δ(φ) in a Fourier series

δ(φ) = δ0 +∞∑

n=1

an cos(nφ) +∞∑

n=1

bn sin(nφ), (2)

where δ0 is the average deviation from a perfect ellipse. It turns out that a4 is the most interestingterm in classifying the isophotes. If a4 < 0, the isophotes are said to be boxy; if a4 > 0, disky. Fig.6 illustrates disky and boxy elliptical galaxies.

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Fig. 5.— Isophotes of the elliptical galaxy NGC 5831 (classified as E3). Notice that the isophotaltwists from 4 arcseconds to 40 arcseconds (their major-axes are indicated in the plot).

Ellipticals that have disky isophotes have more rotational support, and they lack X-ray andradio powers; boxy ellipticals rotate more slowly, and they are more luminous in the X-ray andradio.

Fig. 6.— Disky and boxy isophotes and two example galaxies. Taken from Bender’s lecture notes.

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4.2.3. Colour gradient in ellipticals

Elliptical galaxies are redder near their centers than further out. This indicates systematictrends in the age and metallicities of stars as a function of radius. But due to the so-calledage-metallicity degeneracy, the precise meaning of the colour-gradient in ellipticals is notcompletely clear.

4.3. Dynamical properties

Fig. 7 shows a typical spectrum for elliptical galaxies. From detailed analyses of the spectrum,we can infer (not an easy task!) the redshift, circular velocity, velocity dispersion and absorptionline strengths as a function of radius in elliptical galaxies (see Fig. 8). In particular, we can derivethe velocity dispersion along different lines of sight.

Fig. 7.— A typical spectrum for elliptical galaxies. From Simon Driver’s talk. Notice that thisgalaxy is at redshift of z ≈ 0.07, so the emission lines (from Mg, Na etc.) have been slightlyredshifted.

There are many interesting correlations between the kinematics and photometric propertiesof elliptical galaxies. We discuss some of these in turn.

4.3.1. Central massive black holes

Most elliptical galaxies and bulges of spirals (these two together are referred to as spheroidals)are found to host massive black holes. For example, the Milky Way hosts a central massive blackhole with M ≈ 2.6× 106M�. The presence of the black hole is inferred from the sharp rise in therotation speeds or velocity dispersions as a function of radius. This rise is interpreted due to thepresence of central black holes (see Fig. 9).

Fig. 10 shows the correlation of the black hole mass with the total luminosity and with thebulge/spheroidal luminosity. The total luminosity for a disk galaxy includes contributions fromboth the bulge and disk. Clearly the correlation with the total luminosity is not as good as thatwith the spheroidal luminosity. It shows that black holes are more physically connected with thespheroidal component.

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Fig. 8.— Velocity, dispersion and line strengths as a function of radius r inferred for the ellipticalgalaxy NGC 4621.

Fig. 9.— The rotation curve of the Milky Way, M31 and NGC 4258. Notice the sharp rise in theinner part due to the presence of a central black hole.

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Fig. 10.— The correlation between the mass of black holes with the total bulge/spheroidalluminosity (left panel) and the total luminosity (right panel).

The best correlation is, however, found between the central velocity dispersion and the blackhole mass, which has small scatters (cf. Fig. 11).

MBH = 108.13M�

(σ

200km/s

)4.02

(3)

This correlation is puzzling as it indicates the black holes which are at the very centre of galaxiescorrelate well with galaxy properties on much larger scales. The Schwarzschild radius of a blackhole is given by

rSch =2GM

c2= 3× 108 km

M

108M�≈ 10−5 pc

M

108M�. (4)

The radius at which the Keplerian motion induced by the black hole is equal to the velocitydispersion of a galaxy (induced by the galaxy as a whole) is given by

GM

r≈ σ2 −→ r =

GM

σ2= 10pc

(σ

300 km s−1

)2 M

108M�. (5)

Both scales are much much smaller than typical scales involved in elliptical galaxies. We return tothis point in §4.4.

4.3.2. Fundamental plane

Fig. 12 shows the fundamental plane and its projections on various planes. In particular, thetop right panel shows the correlation between the luminosity and velocity dispersion L ∝ σ4. Therelation is called the Faber-Jackson relation.

An edge-on view of the fundamental plane is shown in the bottom right panel. It shows atight correlation between the size, surface brightness and velocity dispersion in elliptical galaxies:

re ∝ σ1.40 Σ−0.85

e , (6)

where σ0 is the central velocity dispersion and Σe is the mean surface brightness within re.

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Fig. 11.— The correlation between the mass of black holes and velocity dispersion.

Let us use dimensional analysis to see what correlation we naively expect for re, σ0 and Σe.From the definition of Σe, we have

Σe =L/2πr2

e

. (7)

andGM

re= kσ2

0, (8)

where k is a fudge factor that contains all the unknown details of ellipticals. Combining theprevious two equations we find that

re =k

2πG

(M

L

)−1

σ20Σ

−1e . (9)

If k and the mass-to-light ratio (M/L) do not change from elliptical to elliptical, the aboveequation clearly defines a plane in the three-dimensional space of (re, σ0,Σe). Surprisingly, such aplane exists with very small deviations. The shape of the plane (see eq. 6) deviates slightly fromthe above equation. One can easily show that if k−1(M/L) ∝ M0.2 ∝ L0.25 then the observedfundamental plane is reproduced (see Problem set 4).

It shows that ellipticals may be complex in details, their global properties are, however, quitewell regulated. This is a vital piece of information that constrains the formation of ellipticalgalaxies.

One important application of the fundamental plane is as a distance indicator. We can observeσ0 (from spectroscopy) and Σe (from photometry), we can hence infer the physical half-light radiusre. From the observed angular size of the half-light radius, we can then determine the distancefrom d = re/θe.

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Fig. 12.— This figure shows the fundamental plane (lower-right panel) and its three projections.〈µe〉 is related to Σe. Galaxies with larger 〈µ〉e values are fainter in surface brightness.

4.3.3. Rotation in elliptical galaxies

It is common to use a parameter (V/σ)∗ to indicate the importance of rotation in ellipticals.If this parameter is equal to zero, then there is no rotation. If this parameter is close to 1, then thesystem is significantly affected by rotation (rotationally flattened). Fig. 13 shows the correlationbetween the (V/σ)∗ parameter with luminosity and the a4 parameter. It is clear that faint anddisky ellipticals are rotationally flattened, while bright and boxy ellipticals are more supported byan-isotropic random motions.

A more detailed examination of the rotation velocity as a function of radius revealscounter-rotating cores and peculiar rotation patterns in some elliptical galaxies (Fig. 14). Thesepeculiar kinematics indicate the formation of ellipticals is not simple: it is perhaps suggestive ofphysical processes such as merging of galaxies.

4.4. Dynamics and formation of elliptical galaxies

The existence of the fundamental plane of ellipticals indicates that the formation of ellipticalsprobably involves some fairly homogeneous processes – ellipticals may be some sort of equilibriumstate that stellar systems relax to.

For a typical elliptical galaxy, the gravitational force is long-range, but the mean-free pathof stars is much larger than the typical size of a galaxy, and so the relaxation due to two-bodyinteractions is very long. So stars move in the smoothed overall potential provided by numerousstars (∼ 1010). The energy of a star is largely conserved (due to the lack of collisions which change

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Fig. 13.— The left panel shows the correlation between the rotation parameter (V/σ)∗ and thetotal luminosity; the right panel shows the correlation between (V/σ)∗ and the a4 parameter.

Fig. 14.— The bottom panels indicate the variation of the rotation velocity as a function of radius.Notice that the velocity changes sign for NGC 4365 around 10 arcseconds, indicating counter-rotating stars in this galaxy at different radii. The rotation velocity is not monotonic as a functionof radius in NGC 4494.

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the energy of stars). We next discuss how stars move in the overall potential, and how ellipticalmay be formed through merging of disk galaxies.

4.4.1. Orbits of stars

In a Keplerian potential, all orbits with negative energy are closed ellipses. However, in morecomplex potentials, orbits are not closed. Fig. 15 shows the box and loop orbits in a logarithmictwo-dimensional potential with an axial ratio of 0.9. Notice that the box orbits come close to thecentre but the loop orbits never do so.

Fig. 15.— planar orbits in a two-dimensional potential. See Binney & Tremaine (1987) for moredetails.

Fig. 16 shows the orbits in a triaxial potential. The box and tube orbits are similar to thebox and loop orbits in Fig. 15.

4.4.2. Violent relaxation

Two-body collisions between stars are a very slow process to establish an equilibrium state.Ellipticals may be formed through some faster relaxation processes.

One such (collective) relaxation process was discovered by Lynden-Bell in 1967. Ellipticalgalaxies may be formed by the merger of two galaxies that have comparable masses. In this case,the potential itself is rapidly changing during the merger. This in turn induces rapid changes inthe energy of individual stars.

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Fig. 16.— Stable orbit families in tri-axial potentials, see Binney & Tremaine (1987) for moredetails. The top left is the box orbits while the other three are tube orbits.

To see this, we study the rate of the energy change

dE

dt=

d

dt

[12v2 + φ(~x(t), t)

](10)

Using the chain rule we find that

dE

dt= v

dv

dt+

∂φ

∂t+∇φ · d~x

dt= ~v ·

(d~v

dt+∇φ

)+

∂φ

∂t=

∂φ

∂t, (11)

where we have used ~v = d~x/dt and the Newtonian equation: d~v/dt = −∇φ. As the potentialchanges with time, the particle energy changes as well.

• Violent relaxation is efficient in re-distributing energy. As the potential rapidly fluctuates,the orbits of stars are randomly scattered, which randomises the energy of particles and thesystem quickly relaxes to some equilibrium state.

• Computer simulations show that violent relaxation in merging galaxies can lead to a lightprofile that is similar to the R1/4 law, in agreement with the observed light distribution inelliptical galaxies.

• However, violent relaxation is often incomplete, leaving behind peculiar kinematics orphotometric signatures, such as counter-rotating cores (Fig. 14) and shells (Fig. 2).

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Problem Set 4

1. Show that the luminosity L(R) that is predicted by the R1/4 law to lie inside projectedradius R is

L(R) = L(∞)

[1− exp(−y)

(1 + y +

y2

2!+

y3

3!+

y4

4!+

y5

5!+

y6

6!+

y7

7!

)]

where y = 7.67(R/Re)1/4.

2. (2003 exam question) What is the fundamental plane relation for elliptical galaxies? Explainhow it can be used as a distance indicator. [5 marks]

3. In the lecture notes, we have shown (from dimensional analysis ground) that

Re ∝ k(M/L)−1σ20Σ

−1e

Show that if k−1(M/L) ∝ L0.25 then we have

Re ∝ Σ−0.83σ1.330 ,

very similar to the observed fundamental plane.