ASTR 505 Galactic Astronomy

Course Notes

HCG59. NASA/ESA HST

Paul Hickson

The University of British Columbia, Department of Physics and Astronomy

January 2016

c Paul Hickson. Not to be copied, used, or revised without explicit written permission from the copyright owner. Galactic Astronomy 2016

1 Introduction

Figure 1.1: NGC 3370, a spiral galaxy that resembles the Milky Way. NASA, Hubble Heritage Project.

1.1 Why study galaxies?

Galaxies are the largest stellar systems in the Universe. They contain the vast majority of luminous matter. Galaxies are the primary sites of star formation activity and element production. They trace the large-scale structure of the Universe, and cosmic history over „ 13 Gyr. They are the site of a wide variety of interesting phenomena, including supernovae, gamma-ray bursts, supermassive black holes, and relativistic jets to name a few.

There are many questions. Some of the biggest are: Why are there galaxies? Why do they have such varied structure? Why are there such large variations in mass and size? How do they form and evolve? What do they tell us about the Universe?

At the same time one must ask the question, “What is a galaxy?” They come in such a wide range of shapes, sizes, masses and luminosities that one may wonder what distinguishes a galaxy from its satellites, or what distinguishes a galaxy from a star cluster? We shall approach this by examining the properties of these objects in some detail.

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1.2 A brief history

Here we provide just a very brief summary. For more-detailed accounts see for example • https://ned.ipac.caltech.edu/level5/March02/Gordon/Gordon2.html • http://apod.nasa.gov/diamond jubilee/papers/trimble.html • Berendzen, Hart and Seeley, Man Discovers the Galaxies, Columbia Univ. Press 1984.

The brightest galaxies were seen as “diffuse nebulae” by 18th-century observers such as Thomas Wright and Sir William Herschel. The Andromeda galaxy (M31) can even be seen with the naked eye, but only as a fuzzy luminous patch of light.

The nature of these objects was long a subject of debate. The philosopher Immanuel Kant first suggested that they were “island universes”. Herschel, on the contrary, was led to the conclusion that they were small object within the Milky Way since some “planetary nebulae” were clearly associated with stars.

Spiral structure was first noted (in M51) in 1845 by William Parsons (the 3rd Earl of Rosse) observing with his 72-inch telescope. Although large for the time, its performance, and that of earlier reflecting telescopes, was limited by the low efficiency of its primary mirror made from “speculum” metal.

Figure 1.2: Lord Rosse’s 72-inch telescope (the “Leviathan of Parsonstown”), used by Rosse and Dreyer to observe spiral nebulae.

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Figure 1.3: Early photograph of M31 taken by Issac Roberts (from A Selection of Photographs of Stars, Star-clusters and Nebuae, Volume II, The Universal Press, London, 1899)

The introduction of photography in the mid 19th century provided a powerful new tool. The first photograph of a nebula (the Orion Nebula) was obtained in 1880 by the American doctor and amateur astronomer Henry Draper using an 11-inch refractor built by Alvan Clark. Early photographs by Welsh engineer and amateur astronomer Issac Roberts and American astronomer James Keeler allowed classification into knotty (M33) and smooth (M31) types. Keeler and Heber Curtis, working at Lick Observatory, made extensive photographic studies of nebulae starting around 1900 using the 36-inch Crossley reflector.

(a) (b)

Figure 1.4: (a) The 40-inch Alvan Clark refractor at Yerkes Observatory. (b) The 36-inch Crossley reflector at Lick Observatory circa 1905 (Mary Lee Shane Archives).

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Evidence was mounting for the extragalactic nature of at least some nebulae, which clearly resembled a miniature version of the Milky Way. However, the central surface brighness of some of these exceeded that of the centre of the Milky Way, which argued against this hypothesis. (We now know that the bright centre of our galaxy is obscured by dust.)

Another breakthrough came with the application of photographic spectroscopy. Spectroscopy of M31 showed it to have a continuous spectrum, unlike the emission spectrum seen for gaseous nebula. In 1914 Vesto Slipher measured the radial velocity of M31 and Wolf measured internal rotational velocities in M81. It soon became evident that the radial velocities of some nebulae are too large for them to be bound to the Milky Way. However their distances were still unknown.

Measurements made by van Mannen of photographic plates taken with the Hale 60-inch tele- scope at Mount Wilson seemed to indicate motion in M81. He interpreted this as either rotation or expansion, which would support the local hypothesis. However, independent mea- surements of his plates by Lundmark found proper motions that were an order of magnitude smaller!

(a) (b)

Figure 1.5: The 60-inch Hale reflector (a) and 100-inch Hooker telescope (b) at Mt. Wilson.

Measuring thousands of stars in the Large Magellanic Cloud (LMC), Henrietta Leavitt found a relationship between the period and luminosity of a certain type of pulsating star called

Page 5 Galactic Astronomy 2016 a Cepheid variable. Published in 1908, the period-lumnosity relation formed the basis for distance measurements using the inverse square flux law.

In 1917 George Ritchey discovered a nova in M31, supporting the extragalactic theory. More novae discoveries followed. Debate over the nature of these objects reached a peak in 1920, with the Curtiss-Shapley debate.

The spiral nebulae were firmly identified as extragalactic with the discovery of Cepheid vari- ables in M31 by Hubble in 1925. New plates taken with the Mt. Wilson 100-inch Hooker telescope also ruled out van Mannan’s proper motions in M81.

1.3 Catalogues

A necessary starting point for the study of galaxies is a sample of objects. Many catalogues have been compiled over nearly three centuries. Some of the most significant are listed below.

It is important to realize that all of these catalogues are largely heterogeneous samples of ob- jects. Some contain other objects in addition to galaxies, and none have uniform quantitative selection criteria.

In fact, the very nature of galaxies makes identification difficult. Low-surface-brightness (LSB) galaxies may be lost in the noise of the light of the night sky. Typically one can detect and measure only to an intensity level that is about 1% of the sky “background” intensity.

High-redshift galaxies are subject to numerous biases, due to cosmological surface-brightness dimming, small angular size, redshift of their spectra, etc.

Nevertheless, the catalogs form a starting point for many studies, and also serve to identify the brightest galaxies. Many such galaxies have Messier or NGC identifications that are frequently used.

• Messier Catalogue (Messier, circa 1758) contains 110 nebulae, galaxies and star clus- ters.

• General Catalog of Nebulae (GC: John Herschel 1864) contains 5079 non-stellar objects.

• New General Catalog (NGC: Dreyer, circa 1880) was compiled from the GC and results of many observers. Dreyer observed many of these objects using Lord Rosses 72 reflector. The catalogue lists 13,226 non-stellar objects, including those in the Index Catalogs.

• Revised NGC corrected many errors in the original NGC.

• Index Catalogs (IC) are two expansions to the NGC (IC I and IC II).

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• Shapley-Ames Catalog (SA: Shapley & Ames, 1932) contains 1200 galaxies brighter than 13th magnitude.

• Revised Shapley-Ames Catalog (RSA: Sandage & Tammann 1981) provides data on magnitudes, types and redshifts for the SA. It also added some additional galaxies.

• Palomar Observatory Sky Survey (POSS) consists of complete photographic imag- ing of the northern sky to „ 30 deg declination in red and blue wavelength bands. The limiting magnitude is „ 19. Digitized images available at http://archive.stsci.edu/cgi- bin/dss form

(a) (b)

Figure 1.6: (a) Edwin Hubble with the 48-inch Schmidt telescope at Mt. Palomar. (b) The elliptical galaxy M87, from the Digitized Palomar Observatory Sky Survey.

• Reference Catalog of Bright Galaxies (de Vaucouleurs and collaborators). There are three editions. The RC (1964) provided photometric data for 2594 galaxies. The RC2 (1976) lists 4362 galaxies, and the RC3 (1993) lists 23012 galaxies.

• Zwicky Catalogue (Zwicky et al 1961-1967) is a catalogue of galaxies and clusters of galaxies (CGCG) that contains „ 20, 000 galaxies found on Palomar 18-inch Schmidt plates, to a limiting magnitude of „ 15.5. From this galaxy sample, 1931 clusters of galaxies were identified. Zwicky also published a Catalog of Compact Galaxies (1971)

• Updated Zwicky Catalogue (Falco et al 1999) provides better positions and redshifts for Zwicky catalogue.

• Uppsala General Catalogue (UGC: Nilson 1973) is an essentially complete catalogue of „ 13, 000 galaxies to limiting diameter of 1.0 arcmin and limiting magnitude of 14.5

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on blue POSS plates.

• Third Cambridge Catalogue of radio sources (2C: Bennett et al. 1962) contains 328 radio sources.

• CFA Redshift Survey (Centre for Astrophysics) (1982 - 1987) obtained reshifts for about 18,000 galaxies to a magnitude of 15.5.

• IRAS (Infrared Astronomy Satellite) all-sky mid-infrared survey (1982).

• Automatic Plate Measuring Machine (APM) survey of more than 2 million galax- ies (1990) found on POSS and UK Schmidt plates.

• Two-Micron All Sky Survey (2MASS, 2003) infrared all-sky survey

• Two Degree Field Redshift Survey (2DF, 2002) provided spectra and redshifts for more than 200,000 galaxies over 1500 square degrees of sky.

• Sloan Digital Sky Survey (SDSS, 2003-2013) provides photometry for 500 million objects and spectra for a million objects.

1.4 Electronic archives

Electronic archives have become indispensable tools in astronomy. Here are some of the most useful:

• Many catalogues are available in digital form from the CDS Data Centre in Strasbourg, cdsweb.u-strasbg.fr

• Digital images of any region of the sky can be downloaded from the Digitized Sky Survey at archive.stsci.edu/cgi-bin/dss form

• The NASA Extragalactic Database (NED), maintained at Caltech’s Infrared Processing and Analysis Center (IPAC) has extensive data and tools, ned.ipac.caltech.edu

• The Lyon Extragalactic Data Archive (LEDA) has an extensive galaxy database and useful search tool, HyperLeda, leda.univ-lyon1.fr

• The most extensive and useful archive of scientific literature is the NASA Astrophysics Data System (ADS) and search engine, adsabs.harvard.edu

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2 Galaxy morphology and classification

Galaxy classification is an important first step towards a physical understanding of the nature of these objects. For a detailed description of classification systems and their implications, a good reference is the book Galaxy Morphology and Classification by Sidney van den Bergh (Cambridge University Press, 1998).

One can divide galaxies into four general categories: • normal galaxies – no gross peculiarities • active galaxies – strong nuclear activity • starburst galaxies – intense star-formation activity • interacting galaxies – disturbed by recent or current gravitational encounter

We begin by considering the morphology of normal galaxies.

The first classification schemes predate an understanding of the nature of spiral nebulae. An early example Wolf’s system, described by Sandage in Stars and Stellar Systems, Volume 9.

Figure 2.1: Wolf’s classification scheme (from Stars and Stellar Systems, Volume 9).

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2.1 The Hubble sequence

The basic classification scheme still employed today is based on tuning-fork diagram devised by Hubble (1936). Comprehensive discussions of the system and its application have been given by Sandage (1961, 1975).

Figure 2.2: The Hubble tuning fork classification (www.astro.princeton.edu/„frei/ Gcat htm/cat ug 1.htm#galaxies).

Hubble (mistakenly) thought galaxies might evolve from left to right in the diagram, hence the designation early and late type galaxies.

2.2 Elliptical galaxies

Elliptical galaxies are characterized by a smooth, symmetric distribution of stars. There is little or no, evidence of dust, gas, or star formation. The stars in these galaxies are old, typically „ 10 ´ 13 Gyr.

The isophotes (lines of constant intensity) are elliptical, generally to within a few percent accuracy. However, most show significant variation in ellipticity and position angle with radius.

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Figure 2.3: Messier 87, a giant elliptical galaxy at the centre of the Virgo Cluster (Anglo- Australian Observatory, David Malin).

Ellipticity ε is defined by ε “ 1 ´ b{a, (2.1) where a and b are the semi-major and semi-minor axes of the ellipse.

No elliptical galaxies are observed to have ε ą 0.7.

Elliptical galaxies are classified according to apparent flattening. The designation is En where n “ 10ε.

2.3 S0 galaxies

Also called lenticular galaxies (due to their lens-like appearance when see edge-on), S0 galaxies are more flattened than elliptical galaxies, and have a noticeable disk. However, they show no evidence of gas, spiral arms, star formation or young stellar populations.

Many S0 galaxies have visible dust. The subclasses S01, S02, S03 designate increasing amounts of dust.

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(a) (b)

Figure 2.4: (a) NGC1316, an S0 galaxy showing dust lanes. This galaxy, also know as Fornax A, is a powerful source of radio emission (NASA/ESA Hubble Heritage Project). (b) NGC 3115, an S0 galaxies seen edge on (J. Kormendy, CFHT).

2.4 Spiral galaxies

Spiral galaxies are characterized by the presence of a distinct disk with abundant gas, dust and ongoing star formation. Spiral arms, traced by young luminous stars, highlight regions of active star formation triggered by density waves propagating in the disk,

As the sequence progresses from Sa through Sc, (from early- to late-type spirals) the spiral arms increase in prominence and become less tightly would and more irregular. The diffuse central bulge becomes less prominent.

A bar (a radial density enhancement) may be present. The bar may terminate in a ring. Spiral arms often attach to the bar or ring.

2.5 Irregular galaxies

Hubble defined two classes of irregular galaxies: • Irr-I are normal low-luminosity star-forming galaxies. • Irr-II are peculiar systems, such as M82 (Figure 2.9).

The amorphous appearance of M82 is due to a dust-enshrouded burst of star formation thought to have been triggered by a close encounter with M81.

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Figure 2.5: Messier 104, the Sombrero Galaxy, is an Sa seen nearly edge on. Note the prominent stellar bulge and the thin symmetric disk (NASA/ESA Hubble Heritage project).

Figure 2.6: NGC 3949 displays the typical characteristics of an Sc galaxy prominent patchy open spiral arms and a small bulge (NASA Hubble Heritage project).

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(a) (b)

Figure 2.7: (a) Barred spirals M83, an SBc galaxy (W. Keel, U. Alabama, KPNO 4m tele- scope), and (b) NGC 613, an SBc galaxy (M. Neeser (Univ.-Sternwarte Mnchen), P. Barthel (Kapteyn Astron. Institute), H. Heyer, H. Boffin (ESO), ESO).

Figure 2.8: The Large Magellanic Cloud, classified as SBm by de Vaucouleurs, exhibits irreg- ular, patchy star formation with little evidence of spiral structure (AAO).

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Figure 2.9: Messier 82, the prototypical example of an Irr-II galaxy (P. Challis, CfA, Whipple Observatory).

2.6 de Vaucouleurs extensions de Vaucouleurs (1959, 1963) defined extensions to the Hubble system. He introduced inter- mediate types (eg. Sab) and additional types: Sd, Sm, Im which have increasingly-irregular light distributions. de Vaucouleurs introduced a more detailed classification of features: (r) means a ring, (s) means S-shaped, SA denotes a normal spiral and SB a barred spiral. He also recognized intermediate types, SAB. de Vaucouleurs’ Im is equivalent to Hubble’s Irr-I .

2.7 Yerkes system

Morgan (1970) introduced a galaxy classification system based primarily on the degree of cen- tral concentration of the light. He found that the integrated spectrum of the galaxy correlates strongly with central concentration (Elliptical galaxies have an integrated spectrum resem- bling those of K giant stars, whereas Sc galaxies, which have small bulges, have integrated spectra resembling those of A stars). This is the first classification scheme based on physical properties rather than merely on appearance.

Morgan’s classification system uses a letter (see Table), followed by one or two letters (a, af, f, fg, g, gk, k) indicating the approximate spectral type of the integrated spectrum and finally a number in the range 1 ´ 7 representing elongation or inclination. For example, Egk3.

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Table 2.1: Morgan’s classification codes

Shape code Description B barred spiral D rotational symmetry but not a normal spiral or elliptical galaxy cD large D galaxy in a cluster E elliptical Ep elliptical with dust I irregular L low surface brightness N small bright nucleus S spiral

2.8 DDO system van den Bergh (1960a,b, 1976), working at the David Dunlap Observatory, noticed that S0 galaxies have a range of central concentrations, as do spirals. He proposed a trident classification in which S0s are placed in a sequence parallel to that of spirals. S0a, S0b, S0c refer to disk galaxies that resemble Sa, Sb, Sc in shape but lack gas.

He also introduced a sequence Aa, Ab, Ac, of anemic galaxies that are intermediate between spirals and S0s. The presence of bars is indicated by a B, as in ABc for instance. van den Bergh noted that luminous spiral galaxies have more distinct and regular spiral arms that do low-luminosity spirals and introduced a luminosity classification system (I to V) based on the appearance of the arms:

Class I (eg M101 – type Sc I), with well developed structure, are the most luminous spiral galaxies Class V are the least luminous, with little spiral structure.

2.9 Giant elliptical galaxies

Morgan (1958, 1959) defined new categories of luminous elliptical galaxies. D – a giant elliptical cD – a supergiant elliptical.

Both have extended luminous envelopes, and are found in clusters. cD galaxies are the most massive and luminous galaxies known. It is believed that their unusual properties are a result of their location at the centre of rich clusters, where they have accreted debris stripped from other cluster galaxies by gravitational interactions.

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Figure 2.10: NGC 6166, the large galaxy in the centre, is a cD galaxy (Adam Block/NOAO/AURA/NSF).

2.10 Dwarf elliptical galaxies

For a review see Ferguson & Binggeli (1994). There are two distinct categories: Dwarf ellipticals (dE) - Small galaxies, typically „ 10´3 of the mass of a normal elliptical. having a smooth distribution of stars and little or no gas. Dwarf spheroidal (dSph) - Low-luminosity very diffuse spheroidal systems of stars

The Andromeda galaxy (M31) has three dE companions (NGC 147, 185 and 205 see Figure 2.11).

7 Dwarf spheroidal galaxies have masses of order 10 Md. They are very common – nine are known to orbit the Milky Way. They show no signs of recent star formation or gas, but their stellar populations exhibit a range of ages indicating an extended period of star formation.

2.11 Structural components

It is useful to consider galaxies as being made up of one or more distinct structural components. The morphological type of a galaxy is then a statement of a more fundamental property the types of components that comprise the galaxy.

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Figure 2.11: NGC 205 (lower left), a dE companion to the Andromeda Galaxy (Messier 31 upper right) (J-C Cuillandre, CFHT).

Figure 2.12: The dwarf spheroidal galaxy Leo I is believed to be a distant satellite of the Milky Way (David Malin, AAO).

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The spheroid is the smooth elliptical distribution of stars found in elliptical galaxies. It is composed primarily of an old, metal-poor, population of stars typically having ages „ 12 Gyr or more. The spheroid is thought to be among the first stellar components to form.

The stellar halo is a diffuse, roughly spherical, system of stars and globular clusters that surrounds most large galaxies. It is comprised mostly of old, metal-poor, stars. The halo has little or no net angular momentum. It is thought to have formed, at least in part, by the accretion of small satellite galaxies.

The dark halo is the dark matter component that envelopes all galaxies. It extends well beyond the visible extents of the galaxy. The ,mass density decreases with radius roughly as where measurable. The nature of the particles that comprise this component is not known, but there is strong evidence that the particles are non-relativistic (and therefore have mass).

The gaseous halo refers to the hot gas that envelopes some galaxies. It can be detected by the soft X-rays that are emitted from thermal bremsstrahlung. The typical gas temperature is found to be „ 106 K. This component includes cosmic rays and the galactic magnetic field.

Figure 2.13: X-ray image of the elliptical galaxy M87. A diffuse envelope, due to thermal emission from hot gas, is punctuated by filaments emanating from the nucleus.

The bulge is a smooth distribution of old stars that surrounds the nucleus of many spiral galaxies. Large bulges are similar in appearance to the spheroids of elliptical galaxies. Many galaxies have small bulges or none at all. Small bulges have light distributions that more resemble the disk, and may be rotating.

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The thin disk is a flat distribution of stars, gas and dust, having a typical vertical scale height of about 300 pc. In spiral galaxies, it contains stars with a wide range of ages. The thin disk is the home of spiral arms, molecular clouds, star formation, etc. It is thought to have formed by dissipation during the collapse of the protogalactic gas cloud.

The thick disk is a fainter component seen in some galaxies (see Figure 2.13). It has a typical scale height of „ 1 kpc, and is comprised of a population of old, metal-poor, stars. The thick disk is thought to arise soon after the formation of the thin disk as a result of heating by accretion of satellite galaxies.

Figure 2.14: NGC 4762 (lower left). This edge-on galaxy shows a prominent thick-disk com- ponent (NASA/ESA HST).

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3 Photometric properties

3.1 Intensity distribution

See Kormendy and Djorgovski (1989) for a review of surface photometry of elliptical galaxies and bulges.

An important characteristic of a galaxy is its intensity profile, Iprq which is a plot of intensity vs. radius r. This can be measured either along a particular radial axis (eg. the major or minor axis of the galaxy), or an appropriate average intensity can be used.

An isophote is defined as line of constant intensity in a two-dimensional image.

For elliptical isophotes, one can define the radius to be that of a circle having the same area A as that enclosed by the isophote, ? r “ A{π “ ab (3.1)

We will normally use physical distances (measureda in kpc for example) rather than angular distances (measured in arcsec).

The measured intensity profile is affected by diffraction and/or atmospheric seeing, which affect small radii, and by errors in sky subtraction, which affect large radii

The flux of the galaxy, within radius r, can be determined by integrating the intensity profile, 2π r F prq “ Ipxqxdx. (3.2) d2 ż0 Here d is the distance to the galaxy, introduced to convert the radius to angular units.

The total flux F is defined by F “ F p8q.

3.2 Relationship to the density distribution

The emission coefficient jprq is the power emitted per unit volume, per steradian, over some specified range of wavelength, as a function three-dimensional of position r within the galaxy. Integrating over solid angle gives the luminosity density ρLprq. For isotropic emission (independent of direction) we have

ρLprq “ jprqdΩ, (3.3) ż “ 4πjprq. (3.4)

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The emission coefficient is related to the intensity I through the equation of radiative transfer dI “ ´αI ` j, (3.5) ds where α is the absorption coefficient. Integrating this along the line of sight through the galaxy at projected position r, assuming spherical symmetry and no absorption, 8 Iprq “ jpsqds. (3.6) ż0 Substituting s2 “ x2 ´ r2, we get, 8 xdx Iprq “ 2 jpxq? . (3.7) 2 2 r x ´ r ż

This is an Abel integral equation for jprq. The solution is 1 8 dIpxq dx jprq “ ´ ? . (3.8) 2 2 π r dx x ´ r ż

Note that when we write Iprq is a two-dimensional (projected) radius but when we write jprq, r refers to a three-dimensional radius.

For example, consider a power-law Iprq “ βr´α, where α and β are constants. Then αβ 8 dx jprq “ x´pα`1q ? . (3.9) 2 2 π r x ´ r ż

Substitute u “ x2{r2 ´ 1 to extract r from the integral, αβr´pα`1q 8 jprq “ u´1{2p1 ` uq´pα{2`1qdu. (3.10) 2π ż0 We recognize the integral as a beta function, 8 ΓpxqΓpyq Bpx, yq “ tx´1p1 ` tq´px`yqdt “ , (3.11) Γpx ` yq ż0 where Γpxq is the gamma function 8 Γpxq “ tx´1e´tdt. (3.12) ż0

Thus, αβr´pα`1q jprq “ Br1{2, pα ` 1q{2s, 2π αβΓrpα ` 1q{2s “ ? r´pα`1q. (3.13) 2 πΓpα{2q

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This shows that a luminosity distribution that is a power law gives rise to an intensity distri- bution that is also a power law, but with a flatter slope,

Iprq 9 r´α, jprq 9 r´pα`1q. (3.14)

It is important to point out that a pure power law is not physical. Iprq diverges at large r if α ă 0 and at small r if α ą 0. F prq diverges at large r if α ă 2 and at small r if α ą 2.

3.3 Empirical profiles

The Hubble - Reynolds profile (invented by Reynolds in 1913, but popularized by Hubble) is ´2 Iprq “ I0p1 ` r{r0q (3.15) where I0 is the central intensity and r0 is a characteristic length scale. It is simple, but the flux diverges and the corresponding jprq is cumbersome.

The Modified Hubble profile (Rood et al. 1972) is similar to Hubble-Reynolds, but has a simpler density profile,

2 ´1 Iprq “ 2j0r0r1 ` pr{r0q s , (3.16) 2 ´3{2 jprq “ j0r1 ` pr{r0q s . (3.17)

The fact that this fits galaxies reasonably-well indicates that the luminosity density must resemble the above jprq, which falls off as r´3, over a large range of radius. However, the luminosity diverges at large radius, so some kind of cutoff is required.

The de Vaucouleurs (1948, 1953) profile is

1{4 Iprq “ Ie expt´7.66925rpr{req ´ 1su, (3.18) where re is the effective radius and Ie “ Ipreq is the intensity of the corresponding isophote. The numerical constant is chosen so that re contains one half of the total flux.

The total flux is 2 F “ 22.6652Iere , (3.19) which is finite.

The de Vaucouleurs profile is a good fit to most large elliptical galaxies (see Figure 3.1). This profile corresponds to a straight line in a plot of surface brightness µ “ ´2.5 logpIq ` const vs. r1{4.

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Figure 3.1: Composite B-band surface brightness profile of the elliptical galaxy NGC 3379. The flattening near the centre is a result of atmospheric seeing. Outside this region, the profile is fit quite well by a de Vaucouleurs law, which corresponds to a straight line in this diagram (Capaccioli et al. 1990).

The S´ersicprofile (S´ersic1963) is a 3-parameter function 1{n Iprq “ Ie expt´b rpr{req ´ 1su. (3.20) The flux within radius r can be found analytically and is given by neb F prq “ 2πI r2 γr2n, bpr{r q1{ns (3.21) e e b2n e where x γpn, xq “ xn´1e´xdx (3.22) ż0 is the incomplete gamma function.

The total flux is neb F “ 2πI r2 Γp2nq. (3.23) e e b2n

The fraction of the total light contained within radius r is therefore F prq γr2n, bpr{req1{ns “ (3.24) F Γp2nq

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The constant b is determined from the requirement that F preq “ F {2, ie. by solving the equation 2γp2n, bq “ Γp2nq. (3.25)

Note that the choice n “ 4 gives the de Vaucouleurs profile.

The exponential profile Iprq “ I0 expp´r{r0q (3.26) is equivalent to a S´ersicfunction having n “ 1,

Iprq “ Ie expr´1.67835pr{re ´ 1qs (3.27) where I0 “ 5.35671Ie and r0 “ 0.59582re.

Deprojection of the de Vaucouleurs or S´ersic formulae is non trivial. Numerical tables for jprq are given by Young (1976) for de Vaucouleurs’ law.

3.4 King models

King (1966) proposed a family of models based on the dynamics of self-gravitating collisionless stellar systems. Unlike the previous fitting formulae, King models are based on physical principles – they result from specifying a phase space distribution function. The details of the physics will be discussed later.

King models have three free parameters:

I0 – central intensity rc – core radius rt – tidal radius

No exact closed-form analytic expression exists, but approximations have been suggested by Liu (1996) and Natarajan & Linden-Bell (1997).

Apart from a trivial scaling in intensity and radius, the King models form a one-parameter family of curves (Fig. 3.2) defined by the concentration index

c “ logprt{rcq (3.28)

Globular clusters are well fit by King models with c „ 0.75 ´ 1.75 . Elliptical galaxies are best fit by models with c „ 2.2.

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Figure 3.2: The King models: logpI{I0q is plotted against r{rc . Individual curves are labeled by the concentration index (from King 1966).

Figure 3.3: Comparison of de Vaucouleurs r1{4 law (dashed line) with a King model having c “ 2.2 (from King 1966).

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3.5 Cusps and cores

High-resolution images from the Hubble Space Telescope (HST) have allowed the intensity profiles of the nuclear regions of galaxies to be studied. One generally finds two types of profiles: cores – Iprq Ñ const as r Ñ 0 cusps – Iprq Ñ 8 as r Ñ 0

Hubble, de Vaucouleurs, King models all have central cores since the central intensity is finite.

However, many galaxies have cusps: as r Ñ 0, d ln I{d ln r Ñ const (Lauer et al 1992, Crane & Stiavelli 1993, Faber et al 1997, see Figure 3.4).

Figure 3.4: HST observations of the central regions of 55 elliptical galaxies and bulges. High- luminosity galaxies tend to have cores, lower luminosity galaxies have cusps (from Faber et al. 1997).

Some galaxies are found to have have a massive central star cluster (Kormandy & Djorgovski 1989).

Jaffe (1983) and Hernquist (1990) each proposed physical models with a central cusp that resemble a de Vaucouleurs profile in the outer part of the galaxy.

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Dehnen (1993) models start with the density profile

p3 ´ γqL r j r 0 . (3.29) p q “ γ 4´γ 4π r pr ` r0q There are three free parameters: L – total intensity r0 – characteristic radius γ – logarithmic slope of the density cusp

The corresponding intensity profile is

8 p3 ´ γqLr0 xdx Iprq “ ? . (3.30) γ 4 γ 2 2 2π r x px ` r q ´ x ´ r ż 0 This can be expressed in terms of elementary functions for integer γ and in terms of elliptic integrals for half-integer γ. Jaffe’s model corresponds to γ “ 1 , Hernquist’s to γ “ 2. The profile most closely resembles de Vaucouleurs profile when γ “ 3{2.

Even Dehnen’s model is not completely adequate to fit the inner regions of galaxies. Lauer et al. (1995) therefore introduced the 5-parameter empirical “nuker” profile:

pβ´γq{α γ α pγ´βqα Iprq “ Ib2 prb{rq r1 ` pr{rbq s . (3.31)

3.6 Profiles of disk galaxies

´r{r The outer portions of disk galaxies are well-fit by exponential profiles I0e 0 (Figure 3.5).

The inner region sometimes shows a dip (Freeman’s Type II profile – see Figure 3.6). It is not known if this dip is the result of a central “hole” or if the outer part of the disk is brighter than usual (Talbot, Jenson & Dufour 1979).

Freeman (1970) found that I0 has roughly the same value for all disk galaxies: µB “ 21.65, ´2 which corresponds to „ 145Ld pc . This is possibly due to selection effects (Disney 1976), but Bosma and Freeman (1981, unpublished) disagree. However, if one removes the light due to the spheroid component this changes the slope of the exponential, altering I0 (Kormendy 1977, Boroson 1981). It seems unlikely that the disk and bulge have conspired to give a constant sum.

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Figure 3.5: Radial intensity profiles of six spiral galaxies: a) spiral arms in the B band, b) the disk in the O (similar to V) band. The solar symbol in (b) denotes the estimated surface brightness of the Milky Way at the radius of the Sun (from Schweizer 1976).

3.7 Azimuthal intensity distributions

The shape of any isophote can be represented, in polar coordinates pr, φq by the Fourier expansion 8 r “ a0 ` ak cospkφq ` bk sinpkφq (3.32) k“1 ÿ The k “ 1 terms should be absent as they represent a centering error. The k “ 2 terms represent an ellipse. Nonzero a3, b3 indicate “egg-shaped” isophotes:

If a4 ă 0, the isophote is boxy If a4 ą 0, the isophote is disky

Elliptical galaxies have elliptical isophotes, with small deviations |a4{a0| „ 0.01 (Kormendy & Djorgovski 1989). However, galaxies often show a rotation of the position angle of the isophotes with radius.

Intrinsically-twisted spheroids are unlikely to be dynamically stable. Such twists cannot arise from the projection of purely oblate or prolate ellipsoids. Therefore, this is interpreted as evidence for triaxial ellipsoids (Kormendy 1982).

Bulges, particularly small ones, are often boxy or even peanut shaped.

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Figure 3.6: Freeman’s (1970) Type 1 (left) and Type 2 disk-galaxy profiles. NGC 4459 (Type I) is classified as Sa, NGC 5236 is SAB(s)c.

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4 Kinematics

4.1 Kinematics of galactic rotation

The rotation of the Milky Way can be mapped by observations of radial velocities and proper motions of stars.. One assumes circular motion with rotation speed vprq. The general relations for the radial and tangential velocity components of a star are

vr “ pω ´ ωdqrd sin l, (4.1)

vt “ pω ´ ωdqrd cos l ´ ωd, (4.2) where ω “ v{r is the angular velocity at the radius r of the star, ωd “ vd{rd is the angular velocity at the radius of the Sun, rd is the distance from the Sun to the galactic centre, l is the galactic longitude of the star and d is the distance of the star from the Sun.

For stars near the Sun, a good approximation can be made by expanding vprq in a Taylor series, keeping only the first-order term. This gives Oorts formulae (Oort 1927)

vr “ Ad sin 2l, (4.3)

vt “ dpA cos 2l ` Bq, (4.4) where the Oort constants are defined by 1 A “ ´ r ω1pr q, (4.5) 2 d d B “ A ´ ωd, (4.6)

Currently adopted IAU values of these are A “ 14 km s´1 kpc´1 , B “ ´12 km s´1 kpc´1 , ´1 rd “ 8.5 kpc, vd “ 220 kms .

For external galaxies, rotation can be mapped either by 21 cm observations, molecular lines, or by means of optical spectroscopy (Figure 4.1). The radial velocity field of a differentially- rotating disk, with spin axis inclined at an angle i to the line of sight, is given by 2 2 ´1{2 vrprK, φKq “ vprq sin ir1 ` tan φK sec is , (4.7) 2 2 2 1{2 r “ rKrcos φK ` sin φK sec is . (4.8)

For a rigidly-rotation disk, v “ rω so this becomes

vrprK, φKq “ ω sin i rK cos φK, (4.9) which shows that the isovelocity contours (lines of constant radial velocity) are straight lines parallel to the minor axis.

For a constant rotation velocity, v “ v0 we have 2 2 ´1{2 vr “ v0 sin i 1 ` tan φK sec i (4.10) so the isovelocity contours are radial lines.“ ‰

Deviations of the isovelocity contours can indicated warps or other kinematic peculiarities.

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(a) (b)

Figure 4.1: 21-cm observations of IC 342. Left: HII map, contours are at 0.4, 0.8, 1.2, 1.6 and 2.4 Jy beam´1 km s´1. Right: radial velocity contours at 10 km s´1 intervals. Solid lines indicate positive velocities, dotted lines negative (Crosthwaite, Turner & Ho 2000).

4.2 Rotation curves

A plot of the rotation velocity as a function of radius, for a disk galaxy, is called the rotation curve (see Sofue & Rubin 2001 for a review of rotation curves of spiral galaxies). One finds that the rotation velocity of the Milky Way is approximately constant, away from the central black hole, v » 200 km s?1 (see Figure 4.2).

Rotation curves of other galaxies are also roughly constant to as large a radius as can be measured. Typically v „ 150 ´ 260 km s´1 (see Figure 4.3).

For a spherical mass distribution, the orbital velocity is related to the mass within the orbit,

GMprq 1{2 vprq “ . (4.11) r „  Therefore, a constant rotational velocity requires that Mprq 9 r, ρprq 9 r´2.

Elliptical galaxies have little or no net rotation, so one measures instead the velocity dispersion σv along the line of sight 2 1{2 σv “ pv ´ hviq . (4.12) ´1 ´1 Typically σv ranges from ă 50 km s in dwarfs to „ 400 km s in large elliptical galaxies (di Nella et al 1995).

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Figure 4.2: Rotation curve of the Milky Way, NGC 4258 and M31. The rotation speed is nearly constant over „ 5 orders of magnitude in radius (from Sofue & Rubin 2001).

Figure 4.3: Rotation curves of spiral galaxies (from Sofue & Rubin 2001).

4.3 Random motions

Stars have, to varying degrees, a component of random velocity. This component is well- represented by a Gaussian distribution

2 2 2 1 1 v vy v fpvq “ exp ´ x ` ` z , (4.13) 3{2 2 2 2 p2πq σxσyσz « 2 ˜σx σy σz ¸ff

Page 33 Galactic Astronomy 2016 where x, y and z are orthogonal spatial directions.

´3{2 ´1{2 ´1 In velocity space, the surface f “ p2πq e pσxσyσzq is an ellipsoid, called the velocity ellipsoid. The principal axes of the ellipsoid coincide with the x, y and z axes and have semi-lengths equal to σx, σy and σz. The angle between the major axis of the ellipsoid and the direction to the galactic centre is called the vertex deviation.

The various stellar populations in the Galaxy have different velocity ellipsoid parameters, population II stars having higher velocities that population I.

4.4 Elliptical galaxies

Elliptical galaxies generally do not show any significant systematic rotation. Their shapes must therefore be determined by random internal stellar motions. If σx “ σy “ σz, a non- rotating galaxy would be spherical. Most Elliptical galaxies therefore require anisotropic velocity dispersion – two of the velocity σx, σy and σz for a prolate or oblate shape, and all must differ for a triaxial shape.

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5 Stellar populations

5.1 The disk

The thin disk of the Milky Way contains many relatively young stars with high metallicity, referred to as population 1. The ages of disk galactic clusters and individual stars (from isochrone fitting) range from 0.1 to „ 10 Gyr (eg. McClure & Twarog, 1977). The Galactic disk has a radial abundance gradient – the inner disk has rFe{Hs » 0, the outer disk has rFe{Hs » ´0.3 (Figure 5.1).

Figure 5.1: Abundance gradient exhibited by galactic clusters in the Galactic disk (from Freeman & Bland-Hawthorne 2002).

There is no clear evidence for an age-metalicity relation (eg. Friel 1995, see Figure 5.2), however, the velocity dispersion increases with age (Figure 5.3).

5.2 The bulge

The bulge is generally assumed to be old – the presence of RR Lyrae stars and ages derived from colour-magnitude diagrams (CMDs) support this. However, the Galactic bulge is metal rich, rFe{Hs » ´0.25, which is closer to the disk than to the halo.

How can this be if metallicity increases with time? The bulge has short dynamical time so metal content can increase more rapidly.

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Figure 5.2: Abundance vs. age for Galactic clusters. The radial metalicity gradient has been removed. There is no evidence for an age-metalicity relation (from Freeman & Bland- Hawthorn 2002).

5.3 The stellar halo

The stellar halo of the Milky Way consists primarily of old stars referred to as population 2 that have metallicities as low as rFe{Hs » ´5.

There are two populations of globular clusters in the Galaxy, a metal-poor (rFe{Hs ă ´0.8) halo population and a metal-rich (rFe{Hs ą ´0.8) disk population in rapid rotation.

The range of metallicity of galactic globular clusters is comparable to that of stars in the thick disk ´2.2 ď rFe{Hs ď ´0.5. However, globular clusters have ages up to „ 13 Gyr, older than the oldest disk stars, Old age does not mean low metallicity! Young globular clusters have been detected in interacting galaxies, so at least some globular clusters formed recently.

Kinematic streams of stars have been identified in the halo which presumably are due to capture and disruption of dwarf galaxies (eg. Saggitarius Dwarf).

5.4 Spheroids

The light from elliptical galaxies is dominated by red giants, there are no young hot stars. The determination of age and metallicity is complicated by age-metalicity degeneracy which affects integrated colours (Figure 5.4).

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Figure 5.3: Components of velocity dispersion for Galactic disk stars as a function of age. Stars with ages between 2 and 10 Gyr belong to the thin disk. Stars with ages of 10 Gyr or more belong to the thick disk and have higher velocity dispersion (Freeman & Bland- Hawthorne 2002).

Spectroscopic studies indicate that elliptical galaxies have a range of metallicities with ´1.2 ď rFe{Hs ď 0.3.

Elliptical galaxies are redder in the centre, due to higher metalicity (metals absorb blue light) Metalicity is found to decrease with radius, by roughly 0.2 per decade (Kormendy & Djorgovski 1989).

On the other hand, dSph galaxies tend to be bluer in the centre (Vader et al 1988).

Observations and population synthesis studies indicate that at least some dSph have an ex- tended star formation history (eg. Gallart 1999 )

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Figure 5.4: Age-metalicity degeneracy. Variations in age and metallicity affect stellar isochrones in similar ways (Worthey 1993).

Figure 5.5: Synthetic colour-magnitude diagrams for the Leo 1 dSph galaxy. Different colours indicate different ages. Left: no noise, right: noise added to simulate HST observations (Gallart et al. 1999).

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(a) (b)

Figure 5.6: Two possible star formation histories for the Leo 1 dSph galaxy. The data are inconsistent with a single burst of star formation at any past epoch (Gallart et al. 1999).

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6 Integrated properties

6.1 Luminosities

The Luminosity function ΦpLq is the number of galaxies per unit volume per unit luminosity interval (see Binggeli, Sandage & Tammann 1988 for a review). The number of galaxies with luminosity in the range pL, L ` dLq is ΦpLqdL. Equivalently, the number of galaxies per unit volume with absolute magnitude in the range pM,M ` dMq is ΦM pMqdM. Since the two must be equal, we have

dM ´1 Φ pMq “ ΦpLq , M dL ˇ ˇ ˇ ˇ “ 0.921Lˇ ΦpLˇq (6.1) ˇ ˇ where the constant is more precisely 0.4 lnp10q.

A convenient empirical fitting formula is that of Schechter (1976)

α ´x ˚ Φpxq “ Φ0x e , x ” L{L . (6.2)

The Schechter function has three free parameters,

Φ0 – characteristic number density L˚ – characteristic luminosity

α – slope parameter

The number density of galaxies with luminosity L or greater is

8 npLq “ ΦpLqdL, L ż 8 α ´x “ Φ0 x e dx, ˚ żL{L ˚ “ Φ0Γpα ` 1,L{L q. (6.3)

The luminosity density of galaxies with luminosity L or greater is

8 ρLpLq “ ΦpLqLdL, L ż 8 α`1 ´x “ Φ0 x e dx, ˚ żL{L ˚ “ Φ0Γpα ` 2,L{L q. (6.4)

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Note that npLq diverges at low luminosity if α ď ´1, and ρLpLq diverges at low luminosity if α ď ´2.

For the overall population one finds M ˚ “ ´20.3 ˘ 0.7, L˚ » 2.0 ˆ 1010L , a » ´1.25, BT B Bd ´3 Φ0 » 0.004 Mpc . From this one concludes that the mean blue luminosity density of the 8 ´3 Universe is ρLB “ p1.1 ˘ 0.1q ˆ 10 LBd Mpc .

Different morphological types have different luminosity functions (Figure 6.1)

Figure 6.1: Luminosity functions for individual morphological types in field and cluster envi- ronments (from Binggeli, Sandage & Tammann 1988).

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6.2 Diameters

Galaxies do not have distinct diameters. Several definitions have been suggested. The Holm- berg radius is the length of the semimajor axis of the µpg “ 26.5 isophote. The de Vau- couleurs radius is the radius that the µpg “ 25.0 isophote would have if the galaxy were seen face on and unobscured by dust.

The standard diameter pD0q is twice the de Vaucouleurs radius. It ranges from ă 1 kpc for dwarfs to „ 40 kpc for large elliptical galaxies and is „ 23.8 kpc for Milky Way. It is 90 kpc for NGC 4489 (a D galaxy in the Coma cluster) and 220 kpc for A1153+23 (a cD galaxy in the cluster Abell 1413).

6.3 Spectrophotometric properties

There is a tight correlation of spectrum with Hubble type. Spectra vary from K giant for elliptical galaxies to A for Im (see Figure 6.2). Also, there is an increasing prominence of nebular emission lines, particularly Hα, in later-type galaxies.

Multi-wavelength SEDs of galaxies are shown in Figure 6.3. The energy output peaks in the ´α visible and far infrared (FIR). In the radio-FIR spectral range, Fν 9 ν with α „ 2.3.

6.4 Star formation rate

The star formation rate (SFR) can be measured from the UV spectrum, which is dominated by young hot, stars. The best spectral region is around 125 nm, for which Lνpνq is nearly constant.

The Kennicutt relation (Kennicutt 1998) relates the star-formation rate to the UV luminosity,

´1 ´21 ´1 SFRpMd yr q “ 1.4 ˆ 10 Lνp W Hz q. (6.5)

One can also estimate the SFR from nebular line strength (Kennicutt 1992)

´1 ´31 SFRpMd yr q “ 7.9 ˆ 10 LHαp Wq. (6.6) ´1 ´21 SFRpMd yr q “ p1.4 ˘ 0.4q ˆ 10 LrOIIsp Wq. (6.7)

Equation (6.5) is the most accurate relation. The observed luminosities must be corrected for extinction both within the Milky Way and the target galaxy.

The star-formation rate can also be estimated from the far-infrared luminosity (FIR: 8 – 1000 um). The connection occurs because radiation from young luminous stars is absorbed and

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Figure 6.2: Spectral energy distributions of E, Sa, Sc and Im type galaxies. Note the flattening of the SED, due to increasing numbers of hot stars, and increasing prominence of emission lines (from Kennicutt 1998). reradiated in the FIR by dust. If the dust opacity is high, as in starburst galaxies, the FIR luminosity reflects the SFR (Kennicutt 1998),

´1 ´37 SFRpMd yr q “ 4.5 ˆ 10 LFIRp Wq. (6.8)

´1 Global star formation rates in normal galaxies range from 0 in E to „ 20Md yr in gas-rich spirals (Kennicutt 1998). Star formation rates as high as 1000Md yr´1 are found in luminous starburst galaxies.

The star formation rate is found to correlate strongly with local gas density (Schmidt 1959, see Figure 6.4). The Schmidt law relates the star formation rate per unit area in the disk to the gas surface density,

1.4˘0.15 Σgas Σ “ p2.5 ˘ 0.7q ˆ 10´4 M yr´1 kpc´2. (6.9) SFR M pc´2 d ˆ d ˙

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Figure 6.3: Multi-wavelength spectral energy distributions of galaxies. Normal elliptical and spiral galaxies are shown in the top two graphs (from Schmitt et al 1997).

6.5 Masses

Masses are estimated dynamically. For spirals, we have seen that the rotation curve gives the mass interior to the largest radius at which the velocity is measured.

For ellipticals, the mass can be estimated from the velocity dispersion using the virial theorem

2K ` U “ 0, (6.10)

Page 44 Galactic Astronomy 2016 where K is the mean kinetic energy per unit mass and U is the mean potential energy per unit mass. This gives

2 GM “ 3σvR, (6.11) 4π 8 R´1 “ jprqLprqrdr, (6.12) L2 ż0 Lprq “ 4π rjprqr2dr. (6.13) ż0 (6.14)

?Applying this to individual galaxies gives mass-to-light ratios M{LB „ 10 for spirals and M{LB „ 30 for ellipticals (solar units).

The total mass can be probed to much greater radii by observations of satellite galaxies. From the 2dF redshift survey, Brainerd & Specian (2003) have estimated the masses interior to 370 kpc for 809 galaxies. Their results are:

ellipticals :M{LB » 190 ˘ 18, (6.15) ˚ spirals :M{LB » 400pLB{LBq. (6.16)

The surprising result here is that the mass of elliptical galaxies is proportional to the lumi- nosity, but the mass of bright spiral galaxies appears to be independent of the luminosity!

Figure 6.4: Global Schmidt law. Star formation rate correlates with local gas density over five orders of magnitude (from Kennicutt 1998).

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6.6 Correlations between parameters

For elliptical galaxies, one finds that Ie, re and σv are correlated,

1.4˘0.05 ´0.9˘0.1 re 9 σv Ie . (6.17)

As a result, elliptical galaxies are mostly confined to a two-dimensional plane in three di- mensional (Ie, re and σv) parameters space. This is called the fundamental plane (Figure 6.5)

Projections of this correlation result in the Kormendy relation,

´0.85 re 9 Ie (6.18) and the Faber-Jackson relation 4 L 9 σv. (6.19)

Figure 6.5: Projections of the fundamental plane of elliptical galaxies. The first two graphs show the Kormendy and Faber-Jackson relations respectively. The fourth shows a projection parallel to the plane (Kormendy & Djorgovski 1989).

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Dwarf spheroidal galaxies do not obey these relations. They have a larger radius than pre- dicted and are thus not the same as elliptical galaxies. The current view is that they have lost mass from expulsion of gas by supernova-driven winds.

The colours of E and S0 galaxies correlate with luminosity (Baum 1959, see Figure 6.6). This is believed to be primarily a metallicity effect (Faber 1977).

Figure 6.6: Variation of spectral energy distributions of Virgo cluster elliptical galaxies with magnitude. More luminous galaxies are redder (Visvanathan & Sandage 1977).

The metallicity of elliptical galaxies correlates with velocity dispersion (Bender, Burstein & Faber 1993, see Figure 6.7). A possible explanation is that more massive galaxies are able to retain a higher fraction of the metals produced by supernovae.

For spiral galaxies, the maximum rotation velocity correlates with the luminosity (Tully & Fisher 1977, see Figure 6.8) 2.5˘0.3 L 9 vmax (6.20) where vmax is measured from the width of the 21-cm line of atomic hydrogen.

The Tully-Fisher relation was originally used as a distance indicator for spiral galaxies. How- ever, it can also be used to probe galaxy evolution (eg. Ziegler et al. 2002 find evidence that moderate- luminosity galaxies are 1 ´ 2 magnitudes more luminous at a redshift z „ 1 than predicted by their rotation velocities).

The scatter is reduced if near-infrared (eg H-band) luminosity is used because this is less- affected by star formation and extinction.

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6.7 Intrinsic shapes

Spiral galaxies seen edge-on give a direct indication of the intrinsic shapes of the disk and bulge.

Figure 6.7: Magnesium line index is plotted vs velocity dispersion for elliptical galaxies (Ben- der, Burstein & Faber 1993).

Figure 6.8: Tully-Fisher relation for nearby spiral galaxies (Tully & Fisher 1977).

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The observed shapes of elliptical galaxies are related to the intrinsic shapes through (unknown) projection angles. However, from the statistics of large samples, one can in principle learn about the distribution of intrinsic shapes. For ellipsoids with rotational symmetry, viewed at an angle θ from the axis of symmetry, one has the following relationships between the intrinsic axial ratio β and the apparent axial ratio q :

β2 sin2 θ ` cos2 θ oblate, q2 “ (6.21) pβ2 sin2 θ ` cos2 θq´1 prolate. " Assuming that galaxies are randomly oriented, this leads to the following integral equations relating the frequency functions of intrinsic and apparent axial ratios (Mihalas & Binney 1981)

q Npβqdβ fpqq “ q oblate, (6.22) 2 2 2 0 p1 ´ β qpq ´ β q ż 1 q Npβqβ2dβ fpqq “ a prolate. (6.23) 2 2 2 2 q 0 p1 ´ β qpq ´ β q ż Statistically, large elliptical galaxiesa are not well fit by prolate or oblate shapes, but require some degree of triaxiality (see Figure 6.9 and Figure 6.10).

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Figure 6.9: Distribution of observed axial ratios (left) with derived distributions of intrinsic axial ratios for oblate (centre) and prolate (right) spheroids. The rows (a) to (d) illustrate the effect of reducing the amount of smoothing applied to the observational data. Dashed lines denote 90% confidence intervals (Tremblay & Merritt 1995).

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Figure 6.10: Derived distributions of intrinsic axial ratios for triaxial shapes. The number in the top left of each graph is the value of Z “ p1 ´ β1qp1 ´ β2q . Satisfactory solutions exist for 0.3 ď Z ď 0.8 (Tremblay & Merritt 1995).

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7 Gas in galaxies

Much information about physical conditions in galaxies comes from observations of the gaseous component. In addition to tracing the kinematics, spectral analysis provides information about density, temperature and composition of the interstellar medium.

At low temperatures, gas is found in molecular form. At higher temperatures, atomic gas predominates. At higher temperatures still (several thousand degrees) and in the vicinity of hot stars, much of the gas is ionized.

Molecules can be excited to any of an infinite number of discrete vibrational and rotational states. Atoms have only electronic states. Ionized gas has a continuum of (unbound) electronic states.

The states can be populated by collisions and also by the emission and absorption of photons. This latter process is of particular importance as the photons provide key information about the density, temperature and ionization state of the gas.

We begin by reviewing briefly the most relevant physics for these processes.

7.1 Fundamentals

Absorption and emission of radiation passing through a medium is governed by the equation of radiative transfer dIν “ ´αI ` j (7.1) ds ν ν

The absorption and emission coefficients (α and jν) are generally functions of position and wavelength.

In terms of the optical depth τ “ αds, (7.2) ż this equation can be written as dIν jν “ ´I ` (7.3) dτ ν α which has the solution τ jν I “ I p0qe´τ ` ex´τ dx. (7.4) ν ν α ż0

In thermal equilibrium at temperature T , dIν{ds “ 0 and we have Kirchoff’s law,

jν “ B pT q, (7.5) α ν

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where BνpT q is the Planck function 2hν3 1 BνpT q “ . (7.6) c2 ehν{kT ´ 1

At low frequencies (ν ! kT {h) this becomes the Rayleigh-Jeans law 2kT ν2 B pT q » . (7.7) ν c2 This motivates the definition of the brightness temperature

2 Iνc T ” . (7.8) B 2kν2 If the cloud has a constant temperature, Eq (7.4) simplifies to

´τ ´τ TB “ TBp0qe ` T p1 ´ e q (7.9) showing that the brightness temperature becomes the physical temperature in the optically- thick limit (τ Ñ 8).

7.2 Thermal equilibrium

Fundamental equation of statistical mechanics – in thermal equilibrium at temperature T , a system that can be in any of n energy levels Ei (i “ 1, 2, ¨ ¨ ¨ , n) has the probability

gk P pE q “ e´Ek{kT (7.10) k Z to be found in a particular energy level Ek. Here Z is the partition function n Ei{kT Z “ gie (7.11) j“1 ÿ and the statistical weight gk is the number of distinct quantum states that have the same energy Ek.

Accordingly, the distribution of excitation states j of an atom or molecule is given by the Boltzmann equation njpXrq grj “ e´pErj ´Erkq{kT , (7.12) nkpXrq grk where njpXrq is the number per unit volume of species X in ionization state r and excitation state j.

The fraction of atoms excited to a particular energy level is therefore

njpXrq grj “ e´Erj {kT . (7.13) npXrq Zr

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The gas is often not in thermal equilibrium. The occupation of state k is then some fraction bk of the equilibrium value.

It also follows from the fundamental equation that the equilibrium distribution of atoms over ionization levels is given by the Saha equation

npXr 1qne Zr 1Ze ` “ ` , (7.14) npXrq Zr where ne is the number density of free electrons. Ze is the partition function of free electrons, per unit electron density. It can be found by integrating over all possible momentum states. The result is 3{2 2πmekT Z “ 2 » 4.83 ˆ 1021 T pKq3{2. (7.15) e h2 ˆ ˙ The Saha equation then becomes

3{2 npXr 1q 2 Zr 1 2πmekT ` “ ` e´χr{kT (7.16) npX q n Z h2 r e r ˆ ˙ where χr “ Er`1,1 ´ Er,1 is the ionization potential from the ground state.

7.3 Collisions

Collisions are an important mechanism maintaining the distribution of atoms in various ex- cited states. The probability that a particle moving with speed v experiences a collision with a field particle f, in time dt is Cpvqdt where

Cpvq “ nf vσpvq (7.17) and σpvq is the collision cross section.

The probability per unit time of collisional excitation from level j to level k is γjknf where γjk is the rate coefficient obtained by averaging Cpvq over velocities.

For a Maxwell-Boltzmann velocity distribution,

2 2 fpvqdv “ π´3{2l3e´l v d3v 2 2 “ 4π´1{2l3v2e´l v dv (7.18) where mmf l2 ” , (7.19) 2pm ` mf qkT one has 8 4 2 2 γ “ hvσpvqi “ ? l3 v3σ pvqe´l v dv. (7.20) jk π jk ż0

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The probability per unit time of collisional de-excitation from level k to level j is γkjnf , which is obtained by averaging the de-excitation cross section σkjpvq over velocity.

The number of excitations produced by particles with velocities in the range v to v ` dv is njnf fpvqvdvjσjkpvq. Similarly, the number of de-excitations produced by particles with velocities in the range v to v ` dv is nknf fpvqvdvσkjpvq.

In thermal equilibrium, the number of excitations between any two states j and k must equal the number of de-excitations between those same states. Therefore,

njnf fpvqσjkpvqvdv “ nknf fpvkqσkjpvqvdv (7.21) ż ż Which is the same as njnf hvσjki “ nknf hvσkji , (7.22) thus, njγjk “ nkγkj. (7.23)

Using the Boltzmann equation we therefore obtain a relation between the coefficients of exci- tation and de-excitation, ´Ejk{kT gjγjk “ gkγkje , (7.24) where Ejk “ Ek ´ Ej is the energy difference between the two states.

Electrons are usually the most important particles causing collisional excitation and de- excitation of atoms. The rate of these processes is therefore proportional to the electron density.

7.4 Emission and absorption

Consider transitions between a lower level j and an upper level k. The energy emitted, per cubic meter, per second, per steradian, due to spontaneous transitions is

hνjknkAkj j dν “ , (7.25) ν 4π ż where Akj, the Einstein A coefficient for spontaneous emission, is the rate of spontaneous decay from level k to level j for a single atom.

The energy absorbed, per cubic meter, per second, per steradian, is

hνjk I αdν “ ρ pn B ´ n B q, (7.26) ν 4π ν j jk k kj ż

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where Bjk and Bkj are the Einstein B coefficients. These represent the rate, per atom, for stimulated absorption and emission, respectively, per unit energy density ρν of the radiation field.

The energy density is related to the mean intensity Jν by 1 4π ρ “ I dΩ “ J (7.27) ν c ν c ν ż where 1 J “ I dΩ. (7.28) ν 4π ν ż

If the radiation field is isotropic, Iν “ Jν and we have

hνjk αdν “ pn B ´ n B q. (7.29) c j jk k kj ż

In thermal equilibrium, Iν “ Bν and Kirchoff’s law applies. Eq (7.26) becomes

Bν αdν “ jνdν, ż ż hνjknkAkj “ , 4π hνjkBν “ pn B ´ n B q. (7.30) c j jk k kj Using the Boltzmann equation, we find

4π nj A “ B pT q B ´ B , kj c ν n jk kj ˆ k ˙ 3 ´hν {kT 8πhνjk gjBjke jk {gk ´ Bkj “ . (7.31) c3 e´hνjk{kT ´ 1

Requiring this purely atomic relation to be independent of temperature gives the following relation between the Einstein coefficients,

3 gj c Bkj “ Bjk “ 3 Akj. (7.32) gk 8πhνjk

The Einstein coefficients are related to the oscillator strengths fjk and fkj.

gk mehνjk fjk “ ´ fkj “ 2 Bjk. (7.33) gj πe

It is common to see the B coefficients defined as transition rates per unit mean intensity Jν, rather than per unit energy density ρν. Those coefficients are equal to ours multiplied by a factor of 4π{c.

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7.5 Kinetic equilibrium

The rate at which atoms or molecules are excited to various states, by all processes, must equal the rate at which they radiate or are de-excited from these states. This is embodied in the equations of detailed balance

nj Bjkρν ` Ajk ` ne γjk “ nkBkjρν ` nkAkj ` ne nkγkj. (7.34) » fi k kăj k k kąj k ÿ ÿ ÿ ÿ ÿ ÿ – fl The left side of this equation describes the rate at which transitions occur out of the j state, while the right side is the rate of transitions into the j state.

If there are m separate levels, there are m ´ 1 such equations that then determine the m ´ 1 ratios of the population densities.

For example, for a 2-level atom in a weak radiation field, the ratio b2{b1 is given by

b2 1 » . (7.35) b1 1 ` A21{neγ21

4 ´6 8 Typically, in the interstellar medium, ne „ 10 , γ21 „ 10 and A21 » 10 for a permitted line. Therefore the upper level is greatly depleted compared to the thermal equilibrium case.

This is not necessarily the case for forbidden lines (those for which electric dipole transitions are not allowed quantum mechanically).

7.6 Emission and absorption lines

The equivalent width of a line is defined by

W “ p1 ´ Iν{Iνcqdν, (7.36) ż where Iνc is the intensity of the adjacent continuum.

As an example, consider the hydrogen 21-cm line. This is a hyperfine transition (spin flip of electron) in neutral hydrogen at a frequency ν “ 1420.406 MHz. The spontaneous transition ´15 ´1 probability is Akj “ 2.868 ˆ 10 sec and the power emitted per unit volume is given by

hνjkAkj j dν “ n , ν 4π k ż hνjkAkj gk » nHI, 4π gk ` gj 3hνjkAkj “ n . (7.37) 16π HI

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If the optical depth is small, the observed intensity is

I “ Iνdν “ jνdνds, ż ż ż 3hνjkAkj “ n ds. (7.38) 16π HI ż

The column density, integrated over the line, is therefore

NHI “ nHIds, ż 16π “ Iνdν, 3hνjkAkj 2 ż 32πkνjk “ T pvqdv, 3hc3A B kj ż 15 ´2 ´1 ´1 » 1.82 ˆ 10 TBpvqdv m K (km/s) . (7.39) ż

Prominent optical emission lines include Hα,Hβ,Hγ, [O III] at 500.69 & 495.89 nm, [N II] at 654.81 & 658.36 nm, [O II] at 372.61 & 372.88 nm (see Figure 7.1).

Figure 7.1: Optical spectrum of a compact galaxy nucleus exhibiting emissions lines of HII regions (from Sarajedini et al. 1999).

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At optical wavelengths, hydrogen line emission is proportional to the emission measure

2 Em ” neds. (7.40) ż From the Boltzmann and Saha equations,

´E {kT grke rk nkHI “ nHI ZHI ´E {kT g e rk nenp “ rk ZpZe

´Erk{kT 2 “ grke ne{Ze (7.41)

(Zp = 1 since the proton has no excited states). Therefore,

hνjkAkj I “ j dνds “ n ds 9 n2ds. (7.42) ν 4π k e ż ż ż ż Ratios of intensities of lines from the same ion probe temperature and electron density. [O II] 3729/3726 is sensitive to ne (which determines the rate of collisional excitations). [O III] 5007/4363 is sensitive to T .

Line ratios can be used to distinguish active galaxies from normal galaxies (Figure 7.2)

7.7 Structure of the interstellar medium

The ISM has several components. The two-phase model (Field, Goldsmith & Habing 1969) postulated a cool neutral phase in equilibrium with a warm mostly-ionized phase. The cool phase consists of atomic and molecular gas with temperature T ă 300 K and density n „ 3 ˆ 107 atoms m´3. The hot phase has a temperature T „ 104 K and density n „ 3 ˆ 105 atoms m´3.

The warm phase is found in HII regions, where the gas is photoionized by UV radiation from hot stars. The theory was first developed by Str¨omgren(1937), so the ionized region is called a Str¨omgrensphere. Within this region the photoionization rate exceeds the recombination rate. However, the density of ionizing photons decreases with distance from the star. When the ionization rate drops below the recombination rate the gas is primarily neutral.

The neutral gas is an effective absorber of UV photons. Thus the opacity of the IGM, at wavelengths less than Lyman-α (121.5 nm) is typically very high over distances greater than a few hundred pc.

McKee & Ostriker (1977) introduced a three-phase model which includes high-temperature T „ 106K gas. This gas is primarily heated by shock waves generated by supernovae. It can be detected by X-ray telescopes, which reveal bubbles surrounding hot stars, and superbubbles surrounding star-forming regions.

The ISM is turbulent, containing structure on a wide range of scales. The dense cores of molecular clouds are sites of star formation.

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Figure 7.2: Line ratio diagnostic diagram. Active galaxies occupy the upper right region of the figure above the solid line. Dashed lines indicate theoretical H II region models for a range of temperatures (from Veilleux & Osterbrock 1987).

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8 Interstellar dust

The interstellar medium contains particulate matter grains normally referred to as dust. These may be silicates, ices, metals, or combinations. The grains scatter or absorb incident radiation. They also radiate in the infrared region of the spectrum.

8.1 interstellar extinction

Observations of stars in the Milky Way reveal that many appear to be underluminous for their spectral type. This is attributed to absorption and scattering of light by interstellar dust grains. One defines the extinction A as the difference between the expected and observed magnitude. Normally this is given for the V band, and denoted AV . It can also be defined more generally as a function of wavelength Apλq.

These same stars show a shift of their colours towards the red. Here colour is defined to be the difference between two magnitudes, for example, B ´ V . The reddening EpB ´ V q is defined as the difference between the expected and observed colour.

The ratio of extinction to reddening, RV “ AV {EpB ´ V q is found to vary in different regions of the galaxy (Figure 8.1). An average value for the galaxy is RV » 3.1.

At visible and infrared wavelengths, one finds that the extinction is approximately inversely proportional to wavelength, Apλq 9 λ´1.

8.2 Scattering and absorption

The absorption efficiency factor is defined to be the ratio between the optical cross section for absorption σa and the geometric cross section of the grain σg.

σa Qa “ . (8.1) σg The scattering efficiency factor is defined in a similar manner

σs Qs “ . (8.2) σg

The extinction efficiency factor is the sum of these Qe “ Qa ` Qs.

The ratio Qs{Qe is the fraction of light scattered rather than absorbed and is called the albedo. The extinction is

AV ” ´2.5 logrIν{Iνp0qs » 1.086NgQeσg (8.3) where Ng is the column density of grains.

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Figure 8.1: Interstellar extinction curves. Extinction is plotted as a function of wavelength, normalized by AV , the extinction in the V band. RV denotes the ratio AV {EpB ´ V q where EpB ´ V q is the reddening (ned.ipac.caltech.edu).

The emission coefficient for scattering is given by

1 1 jνpkq “ ngQsσg Iνpk qF pk ´ k qdΩ (8.4) ż where k points in the direction of propagation and F pk ´ k1q is a phase function equal to the fraction of radiation scattered by angle φ (from direction k to k1q.

In Mie scattering theory the grains are modelled as dielectric spheres of radius a and complex index of refraction n. One finds that Qe depends on the ratio of particle size to wavelength, x ” 2πa{λ,

4 For x ! 1, Qs „ x ,

For x „ 1, Qs „ 3,

For x " 1, Qs “ 2 (Babinet’s principle).

If n is real, the grains scatter but do not absorb, Qa “ 0.

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Rayleigh scattering describes scattering by particles that are much smaller than the wave- length. In this case 2 2 8 4 n ´ 1 4 Qs “ x 9 ν . (8.5) 3 n2 ` 2 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ By comparing the wavelength dependence of Qs for various models with observed reddening curves, one can estimate the properties of the grains. These are found to have diameters 0.1 to 1 µm and a total mass density „ 1% of the gas density.

8.3 Emission by grains

In HII regions, collisional heating of the grains is important. Outside these regions, radiative heating dominates. In that case, the power radiated per unit volume in the infrared must be equal to that absorbed

1 4πj “ σgngQa Iνpk qdνdΩ, ż “ σgngQaργc, (8.6) where ργ is the radiation energy density.

4 For spherical grains, this must equal 4σgngσBT , where σB is the Stefan-Boltzmann constant, hence 4σ T 4 ρ “ B “ Q ρ . (8.7) IR c a γ which states that the energy density in the radiated infrared photons is a fraction Qa of the total radiation field.

The temperature of the grains is then given by

1{4 Qaργc T “ (8.8) 4σ „ B  and is typically, „ 10 to 20 K.

Radiative processes are particularly important in that they affect the dynamics of the gas. Thermal pressure prevents the gas from reaching high densities during gravitational contrac- tion or in shocks. Star formation (and galaxy formation) requires efficient radiative processes to effectively cool the gas.

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9 Star formation

Stars form when molecular clouds become unstable and undergo gravitational collapse. As the cloud contracts, fragmentation occurs resulting in many small stellar mass cores contracting to form individual stars. The result is a star cluster. Galactic clusters in the Milky Way are examples of this process.

9.1 Gravitational free-fall time

It is easy to show that, if there is no internal pressure, a homogeneous gas sphere will collapse in the free-fall time 3π t “ . (9.1) ff 32Gρ c However, clouds are not pressure free. Contraction can only occur if the attractive force exceeds the internal pressure force within the cloud. This was quantified in 1902 by James Jeans.

9.2 The Jeans instability

A simple way to derive Jeans’ result is to consider a self-gravitating sphere of gas having mass M, radius R, uniform density ρ and temperature T .

The total gravitational energy is given by

3 GM 2 U “ ´ . (9.2) 5 R The thermal energy is M 3kT K “ . (9.3) µmH 2 According to the virial theorem of classical mechanics, if the cloud is stationary (neither expanding or collapsing) we must have

2K ` U “ 0. (9.4)

It follows that contraction will occur if 2K ă ´U. Thus the condition for contraction is kT GM ă (9.5) µmH 5R

Writing M “ 4πρR3{3 and introducing the sound speed

dP kT cs “ “ , (9.6) d dρ dµmH

Page 64 Galactic Astronomy 2016 this becomes cs 15 R ą R “ , (9.7) J 2 πGρ c where RJ is the Jeans radius.

We conclude that the cloud will contract if its mass exceeds the Jeans mass 4πρR3 5c3 15 M ” J “ s J 3 2G πGρ c T 3{2 ρ ´1{2 » 2.2Md . (9.8) 10 K 3 ˆ 10´15 kg m´3 ˆ ˙ ˆ ˙ Jeans obtained this esult from a perturbation analysis of the equations of hydrodynamics. The Jeans length π λ “ c (9.9) J s Gρ c is the maximum wavelength for stable oscillations of waves in the gas. The amplitudes of wavelengths that are longer than the Jeans length grow exponentially due to gravitational instability.

We see that the Jeans length corresponds roughly to the distance that a sound wave can travel in the free-fall time. On scales larger than this, a gas cloud will collapse before a pressure wave can resist it.

9.3 Contraction and fragmentation

Contraction can only continue if the cloud is able to radiate energy. It does this via infrared emission from the dust grains. The luminosity is

3M 2 4 3σB 4 LIR “ 3 4πa σBT “ MT 4πρga ρga If the cloud contracts slowly, remaining near virial equilibrium, its total energy will be 3MkT E “ K ` U » ´K “ ´ . 2µmH Equating the rate of change of this energy to the infrared luminosity, 3kM dT 3σ “ ´ B MT 4 2µmH dt ρga so 1 dT 2µmH σB 4 “ ´ . T dt kρga

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Integraton gives the time dependence of temperature,

´3 ´3 8µmH σB T ptq “ T0 ` pt ´ t0q kρga

Here T0 is the initial temperature and t0 the initial time. This was first worked out by Kelvin and Helmholz (1856), and is called Kelvin-Helmholtz contraction.

Typically, during the collapse, the cloud fragments into smaller cores, which have masses comparable to those of stars. These cores continue to contract individually.

When the temperature inside a core reaches T „ 107 K, nuclear reactions become important. The energy released increases the pressure, stopping the collapse.

After several million years, stellar winds clear residual gas and dust and we have a galactic star cluster.

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10 Potential theory

The study of dynamics is fundamental to any understanding of galaxies. The following few chapters provide a brief summary of the most important results. More details can be found in Binney and Tremaine (1987).

For the topics discussed here, Newtonian gravitation theory is sufficient. Only in regions close to black holes found in the centres of galaxies are relativistic effects important.

Because galaxies contain such a large number of stars, the gravitational potential may gen- erally be regarded as smooth and constant. A large part of the stellar dynamics problem is then the determination the potential and the orbits of stars in that potential. The exception occurs during the collapse phase, or during encounters with other galaxies, when the potential can change on a timescale comparable to the orbital period.

10.1 Gravitational potential

The gravitational force vector can be written as the gradient of a scalar function U. This results in considerable simplification of dynamical equations.

For a continuous distribution of mass ρpxq, Newtons law gives a force per unit mass

x1 ´ x F pxq “ Gρpx1qd3x1, (10.1) |x1 ´ x|3 ż “ ´∇Φpxq. (10.2) where G is Newton’s constant and Gρpx1q Φpxq “ ´ d3x1. (10.3) |x1 ´ x| ż is the scalar gravitational potential. Differentiating the potential and applying Gausss theorem gives Poisson’s equation ∇2Φpxq “ 4πGρpxq. (10.4) In empty space ρ “ 0 and this becomes Laplace’s equation

∇2Φpxq “ 0. (10.5)

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10.2 Potential energy of a mass distribution

The fact that the gravitational force can be written as the gradient of a potential means that it is conservative force – the energy expended by a test particle moving from point A to point B does not depend on the path taken. In particular, the energy required to bring a small mass δm from infinity to x is x dW “ ´δm F ¨ ds, 8 xż “ δm ∇Φ ¨ ds, ż8 x dΦpsq “ δm ds, ds ż8 “ δmΦpxq. (10.6)

(which of course is negative). Therefore, to create a change δρpxq in density by bringing in matter from infinity requires energy

δW “ Φpxqδρpxqd3x. (10.7) ż This produces a change in the potential δΦpxq that satisfies Poisson’s equation, ∇2δΦpxq “ 4πGδρpxq, hence 1 δW “ Φ∇2δΦd3x, 4πG 1 ż “ r∇ ¨ pΦ∇δΦq ´ ∇Φ ¨ ∇δΦs d3x, 4πG ż 1 1 “ ∇ ¨ pΦ∇δΦq ´ δp∇Φ ¨ ∇Φq d3x. 4πG 2 ż „  (10.8)

The first term is a perfect divergence and so can be converted to a surface integral by Gauss’s theorem. If the volume of integration extends to infinity and the mass distribution is localized, the potential goes to zero on the surface and this term vanishes. We then have 1 W “ ´ |∇Φ|2 d3x. (10.9) 8πG ż To obtain a second form, apply the divergence theorem and Poisson’s equation, 1 W “ ´ ∇pΦ∇Φq ´ φ∇2Φ d3x, 8πG ż 1 ˇ ‰ “ ρΦd3ˇx. (10.10) 2 ż The Chandrasekhar potential energy tensor is defined by BΦ W j “ ´ ρxj d3x. (10.11) k Bxk ż

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This tensor is symmetric and its trace is the potential energy,

j j W “ W j ” W j, j ÿ “ ´ ρx ¨ ∇Φd3x. (10.12) ż For a spherically symmetric body, the potential energy tensor is diagonal and

8 W “ ´4πG ρMprqrdr, (10.13) ż0 where Mprq is the mass interior to radius r.

10.3 Newton’s theorems

Two elementary theorems are easily proved: 1. A body inside a spherical shell of matter experiences no net gravitational force. 2. A body outside a spherical shell of matter experiences the same force as it would if all the shell’s mass were concentrated into a point at its centre.

These imply that the effect of a spherically-symmetric mass distribution is the same as if the mass interior to r were concentrated into a point at the centre.

10.4 Thin disk

For a thin axisymmetric disk,

B2Φ 1 B BΦ “ 4πGρpR, zq ´ R , Bz2 R BR BR » 4πGρpR, zq. (10.14) where the second term on the right, the divergence of the radial force, is much smaller than the first. Therefore the vertical force on a particle above the disk is

BΦ Fz “ ´ » ´4πG ρpR, zqdz, Bz ż » ´4πGΣpRq, (10.15) where ΣpRq is the mass surface density of the disk.

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10.5 Potential-density pairs

Here are some examples of potential-density pairs:

Point mass M GM Φprq “ ´ , (10.16) r ρpxq “ Mδ3pxq (10.17)

Homogeneous sphere of radius a and density ρ 2πG ´ ρp3a2 ´ r2q r ď a 3 Φprq “ (10.18) $ 4πGa3 ’´ ρ r ą a & 3r ’ρ r ď a ρprq “ % (10.19) #0 r ą a

Plummer potential GM Φprq “ ´? , (10.20) r2 ` b2 3M r2 ´3{2 ρprq “ 1 ` . (10.21) 4πb3 b2 ˆ ˙ Compare this with the luminosity distribution that produces the Modified Hubble intensity profile.

Kuzmin potential GM ΦpR, zq “ ´ (10.22) R2 ` pa ` |z|q2 aM ρpR, zq “ apR2 ` a2q´3{2δpzq “ ΣpRqδpzq. (10.23) 2π This potential, created by a thin disk, mimics that of a point mass M located a distance a along the axis on the opposite side of the disk.

Miyamoto-Nagai potential GM ΦpR, zq “ ´ ? , (10.24) R2 ` pa ` z2 ` b2q2 ? ? b2bM aR2 ` pa ` 3 z2 ` b2qpa ` z2 ` b2q2 ρpR, zq “ ? . (10.25) 4π rR2 ` pa ` z2 ` b2q2s5{2pz2 ` b2q3{2

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This potential can be varied continuously between the Plummer and Kuzmin potentials. It can therefore describe a range of flattened systems.

Multipole expansion

A general solution that gives the potential for any mass distribution ρ is provided by an expansion in spherical harmonics and radial polynomials,

Y mpΩq 1 r 8 Φpr, Ωq “ ´4πG l ρ pxqxl`2dx ` rl ρ pxqx1´ldx , (10.26) 2l ` 1 rl`1 lm lm l,m 0 r ÿ „ ż ż  m˚ ρlm “ ρpr, ΩqYl pΩqdΩ. (10.27) ż where Ω ” pθ, φq and dΩ “ sinpθqdθdφ.

The dipole term (l “ 1) vanishes if the origin is taken at the centre of mass.

m The spherical harmonics, Yl pΩq obey the following relations

m˚ m1 2 m Yl pΩqYl1 pΩqdΩ “ δll1 δmm1 , ∇ Yl pΩq “ 0. (10.28) ż

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11 Stellar orbits

11.1 Symmetries and integrals of motion

Noether’s theorem states that any continuous symmetry of the system implies a conserved quantity. Here are some examples:

Axial symmetry – conservation of angular momentum about the symmetry axis,

BΦ dLz “ 0 Ñ “ 0. (11.1) Bφ dt

Spherical symmetry – conservation of angular momentum about all axes,

BΦ BΦ dL “ “ 0 Ñ “ 0. (11.2) Bθ Bφ dt

Stationarity – conservation of energy,

BΦ dE “ 0 Ñ “ 0. (11.3) Bt dt

A constant of motion is a any function of the phase space coordinates and time px, v, tq conserved along the orbit. There are 6 independent constants of motion (eg. the position and velocity at some particular time).

An integral of motion is any function of the phase space coordinates px, vq that is conserved along the orbit. An example is the energy per unit mass: E “ v2{2 ` Φ.

An integral of motion is a constant of motion but a constant of motion is not necessarily an integral (because it may involve t). An integral is a constraint on the phase space coordinates and thus reduces the dimension occupied by the orbit by 1.

An isolating integral confines the orbit to some region of phase space. For example, energy, and angular momentum. A non-isolating integral does not.

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11.2 Spherical potential

If the potential is a function only of r, F “ ´∇Φ is directed radially – ie. it is a central force. Because of this, the orbital angular momentum (per unit mass) L of any particular star is conserved, dL d “ pr ˆ r9 q “ r9 ˆ r9 ` r ˆ r: “ r ˆ F “ 0. (11.4) dt dt Because L is perpendicular to r and v, constant L requires that the orbit lie in a fixed plane. To find the form of the orbit in this plane, use polar coordinates (r, φ). The equations of motion are

r: ´ rφ92 “ ´Φ1prq. (11.5) 2r9φ9 ` rφ: “ 0. (11.6)

These equations can be integrated to give two conserved integrals, the angular momentum and energy,

L “ r2φ,9 (11.7) 1 E “ pr92 ` r2φ92q ` Φ. (11.8) 2 Write u “ 1{r and rearrange the above to obtain

du 2 2 “ rE ´ Φpuqs ´ u2. (11.9) dφ L2 ˆ ˙

Setting du{dφ “ 0 generally gives two solutions u1 and u2 that bound u from above and below (Figure 11.1). The star oscillates between inner radius r1 “ 1{u1 and outer radius r2 “ 1{u2 as it orbits.

Generally, the orbit is a rosette that eventually covers the entire region between r1 and r2.

There are two special cases in which the orbit closes: Point mass: ellipse with one focus at the centre Homogeneous sphere: centered ellipse

11.3 Axial symmetry

Few galaxies are spherically symmetric, but many have at least approximate axial symmetry. For these, we use cylindrical coordinates (R, φ, z). Symmetry demands conservation of

2 Lz “ R φ.9 (11.10)

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Figure 11.1: Graph showing bounding points of the orbit. The two points where the curves cross correspond to du{dφ “ 0 and therefore limit the range of the radius r “ 1{u. If the energy E is positive, there is only one crossing point, which sets a lower limit on the radius.

The problem can be reduced to a 2-dimensional orbit in a rotating plane (the meridional plane) which contains the z axis, as follows

BΦ L2 BΦ R: “ Rφ92 ´ “ z ´ , BR R3 BR BΦ “ ´ eff , (11.11) BR BΦ z: “ ´ eff (11.12) Bz where L2 Φ pR, zq “ ΦpR, zq ` z (11.13) eff 2R2 is an effective potential.

The meridional plane rotates at angular velocity φ9 which is generally not a constant, but changes as R varies in order to conserve angular momentum,

Lz φ9 “ . (11.14) R2 The shapes of orbits in the meridional plane depend on the potential and on the energy of the star. Generally, the orbits are bounded both in r and in z.

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The energy per unit mass of a star is 1 E “ pR9 2 ` R2φ92 ` z92q ` Φ, 2 1 L2 “ pR9 2 ` z92q ` Φ ` z , 2 2R2 1 “ pR9 2 ` z92q ` Φ . (11.15) 2 eff We see that Φeff represents the sum of the potential energy and the kinetic energy of motion in the φ direction.

Figure 11.2: Sketch of the effective potential for a typical galaxy. The minimum of the potential corresponds to the radius Rg of a stable circular orbit. Stellar orbits are bounded in radius by the intersection points of the curve with the dashed horizontal line, determined by the total energy E and the energy associated with the z motion.

To examine the nature of small departures from a circular orbit, we expand Φeff in a Taylor series about the point pRg, 0q keeping only the quadratic terms (the linear terms vanish by symmetry), 1 1 Φ “ Φ pR , 0q ` κ2x2 ` ν2z2 ` ¨ ¨ ¨ , (11.16) eff eff g 2 2 where 2 2 B Φeff κ “ pRg, 0q, (11.17) BR2 2 2 B Φeff ν “ pRg, 0q, (11.18) Bz2 x “ R ´ Rg. (11.19) This is the epicyclic approximation. The solutions are simple harmonic motion in R and z. x “ X cospκt ` ξq, (11.20) z “ X cospνt ` ζq. (11.21)

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Inserting the radial solution into the angular momentum equation and integrating gives (to first order) y ” Rpφ ´ ωtq “ ´Y sinpκt ` ξq (11.22) where ω is the angular speed of a star in a circular orbit at radius Rg.

The star executes simple harmonic motion in the z direction with vertical frequency ν. In a frame rotating with angular speed ω, the star follows an elliptical path with frequency κ, in the direction opposite to ω (it moves faster when the radius decreases and slower when the radius increases, to conserve angular momentum).

The elliptical path is called an epicycle and κ is the epicyclic frequency. The epicyclic frequency is related to the Oort constants

κ2 “ ´4Bω “ ´4BpA ´ Bq, (11.23) κ ´B “ 2 . (11.24) ω A B c ´ The lengths of the semimajor and semiminor axes of the epicycle have the ratio

X κ ´B “ “ . (11.25) Y 2ω A B c ´

Some special cases are: point mass, κ “ ω, homogenous sphere, κ “ 2 ω, Milky Way (Sun’s orbit), κ » 1.3 ω.

11.4 Non-axisymmetric potentials

In non-axisymmetric potentials, such as the triaxial potentials of many elliptical galaxies, one typically finds two types of orbits:

Stars in tube orbits rotate about the galaxy, while oscillating in the radial and z direc- tions. They are confined in both of these directions, eventually covering completely a kind of distorted torus or tube.

Stars in box orbits reverse their direction, and never completely circle the galaxy. They are confined in azimuth as well as radius and z.

Figure 11.3 shows some examples.

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Figure 11.3: Examples of box and tube orbits in a triaxial potential (Statler 1986).

11.5 Rotating non-axisymmetric potentials

Suppose that the potential is rotating about the z axis with constant frequency ωb. An example is a galaxy with a rotating bar. Let r be the position vector of a star in the rotating frame. The equation of motion is

r: “ ´∇Φ ´ 2pωb ˆ r9 q ´ ωb ˆ pωb ˆ rq. (11.26) Take the dot product of this equation with r9 and rearrange to obtain a conserved quantity, the Jacobi integral EJ , dE J “ 0. (11.27) dt where 1 1 E “ r92 ´ |ω ˆ r|2 ` Φ, J 2 2 b 1 1 “ r92 ´ ω2R2 ` Φ. (11.28) 2 2 b After some manipulation (see Binney & Tremaine 1987) one obtains

EJ “ E ´ ωb ¨ L. (11.29)

L “ r ˆ r9 ` r ˆ pωb ˆ rq. (11.30)

Page 77 Galactic Astronomy 2016 showing that neither energy nor angular momentum alone is conserved.

Define the effective potential 1 Φ “ Φ ´ ω2R2. (11.31) eff 2 b then 1 E “ r92 ` Φ . (11.32) J 2 eff

Stellar orbits are bounded by the zero-velocity surface EJ “ Φeff.

In the plane z = 0, there are five stationary points at which

BΦ BΦ eff “ eff “ 0. (11.33) BR Bφ

These are called Lagrange points (Figure 11.4). The points L1 and L2 are saddle points and are therefore unstable. The point L3 is a minimum of the effective potential and is therefore stable. The points L4 and L5 correspond to local maxima of the potential. However, they can be stable for certain potentials (stars near these points librate – slowly oscillate around them).

Figure 11.4: Plot of the effective potential, in the z “ 0 plane, of a rotating bar. The Lagrange points are indicated. L3 is located at the central minimum, coinciding with the centre of the galaxy. The potential is maximum at L4 and L5 and then decreases with increasing radius (Buta & Combes 1996).

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Resonances occur when a natural frequency of the star’s orbit coincides with the angular frequency ωb of the rotating potential. This pumps energy into the orbit, increasing the amplitude of the oscillations.

Corotation is a resonance that occurs when the pattern frequency coincide, ω “ ωb.

Lindblad resonances occur when the azimuthal epicyclic frequency is a multiple of the orbital frequency in the rotating frame, mpω ´ωbq “ ˘κ, where m “ 1, 2, 3, ¨ ¨ ¨ . Galaxies may have multiple Lindblad resonances, corresponding to different values of m. These resonances are of particular importance in the theory of spiral structure.

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12 Collisionless systems

In galaxies, the distances between stars is so vast that stellar collisions, or close encounters, rarely happen. It is then possible to represent a galaxy as a system of collisionless particles. But first, let’s estimate how long it takes for stellar encounters to significantly change stellar orbits. This is called the relaxation time.

12.1 Relaxation time

If a star passes in the vicinity of another star, it feels an acceleration both parallel to its velocity, and perpendicular to it. If the encounter is not a close one, the change in velocity is small, and almost entirely in the perpendicular direction (the change in the parallel direction is positive approaching the star and negative when receding from it, which cancels).

The change in the perpendicular velocity component vK due to a weak encounter with a star of mass m is given by

8 Gm cos θ ds δv “ v9 dt » , K K s2 ` b2 v ż ż´8 Gm 8 “ bps2 ` b2q´3{2ds, v ż´8 Gm 8 “ p1 ` x2q´3{2dx, vb ż´8 2Gm “ . (12.1) vb The cross section for an encounter with impact parameter b P pb, b ` dbq is 2πbdb . Multiply this by the average surface density of stars N{πR2, in a galaxy of N stars having characteristic radius R, and integrate over b. The change in mean square velocity after one pass through the galaxy is

N bmax 2Gm 2 ∆v2 » 2πbdb, K πR2 vb żbmin ˆ ˙ 8N Gm 2 “ ln Λ, (12.2) R2 v ˆ ˙ where ln Λ “ lnpbmax{bminq is called the Coulomb logarithm.

Take Gm b “ pfor this value, |δv | » vq, (12.3) min v2 K bmax “ R. (12.4)

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By the virial theorem, v2 » GM{R “ NGm{R, therefore,

bmax » N. (12.5) bmin The number of crossings required to change the velocity of a star by an amount comparable to its initial velocity is therefore

R2v2 v 2 n » , relax 8N ln N Gm N ´ ¯ “ . (12.6) 8 ln N 12 Since N „ 10 for a large galaxy and the crossing time tcr “ R{v is typically of order 8 20 10 yr, the relaxation time trelax “ nrelaxtcr „ 10 yr. This justifies the neglect of stellar encounters in galaxies. However, encounters cannot be neglected in smaller systems such as globular clusters.

12.2 Collisionless Boltzmann equation

Consider now not just a single star, but a large number of stars moving in a smooth potential Φ. At any given time, the position and velocity of each star can be represented by a point in 6-dimensional phase space w “ pr, vq. The number of stars with phase space coordinates within a small volume d6w “ d3xd3v centered at w is

dN “ fpw, tqd6w, (12.7) where fpw, tq is called the distribution function or phase space density. A specification of fpw, tq gives a complete statistical description of the configuration of the system.

In the absence of encounters, stars move smoothly through phase space as they follow their or- bits. If stars are not created or destroyed, the phase space density therefore obeys a continuity equation Bf ` ∇w ¨ pfw9 q “ 0. (12.8) Bt This states that the rate of change of the density in a small volume of phase space is equal to the (negative) divergence of the flow of stars.

Expanding the continuity equation we get Bf Bf ` ∇r ¨ pfr9 q ` ∇v ¨ pfv9 q “ ` r9 ¨ ∇rf ` v9 ¨ ∇vf ` f∇r ¨ v ´ f∇v ¨ p∇rΦq Bt Bt “ 0. (12.9)

The second last term vanishes because v and r are independent variables and the last term vanishes because the potential does not depend on velocity. Therefore we have df Bf ” ` w9 ¨ ∇wf “ 0. (12.10) dt Bt

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This is called the collisionless Boltzmann equation (sometimes called the Vlasov equa- tion).

By analogy with fluid dynamics, df{dt is the convective derivative (or Lagrangian deriva- tive) of the flow – the sum of the explicit time derivative plus the change due to the motion of the flow. The collisionless Boltzmann equation is therefore a statement that the phase space density around any star remains constant.

One can equally well interpret f as proportional to the probability of finding a star in some phase space volume. This is useful for systems with a small number of stars.

The collisionless Boltzmann equation must hold for subsets of stars. In particular we can divide the stellar population into a very large number of subsets of different mass. Multiplying the distribution function of each subset by the respective mass and adding gives a distribution function for the mass density in phase space. So, we can also interpret f as a mass density in phase space.

12.3 The Jeans equations

Now consider moments of the distribution function. Integrating the distribution function over velocities gives the stellar number density,

npr, tq “ fpr, v, tqd3v, (12.11) ż and the first moment of the velocity is the mean stellar velocity, 1 v¯pr, tq “ fvd3v. (12.12) n ż Integrating the collisionless Boltzmann equation over velocity therefore gives

Bn 3 3 Bn 3 3 ` v ¨ ∇rfd v ` v9 ¨ ∇vfd v “ ` ∇r ¨ vfd v ` v9 ¨ ∇vfd v “ 0. (12.13) Bt Bt ż ż ż ż The last term vanishes by virtue of the divergence theorem and the requirement that f Ñ 0 as v Ñ 8 (there are no stars with infinite velocity). This gives the first Jeans equation – a continuity equation for the stellar number density,

Bn ` ∇r ¨ pnv¯q “ 0. (12.14) Bt Now take the first velocity moment of the collisionless Boltzmann equation,

B 3 3 pnv¯q ` vpv ¨ ∇rqfd v ` vpv9 ¨ ∇vqfd v “ 0. (12.15) Bt ż ż

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In component notation this equation is

B B 3 BΦ Bf 3 pnv¯jq ` vivjfd v ´ vj d v Bt Bx Bx Bv i ż i ż i B B BΦ B 3 BΦ 3 “ pnv¯jq ` pnvivjq ´ pvjfqd v ` δijfd v “ 0. (12.16) Bt Bx Bx Bv Bx i i ż i i ż Again, the divergence theorem can be used to eliminate the third term. This gives

B B BΦ pnv¯jq ` pnvivjq ` n “ 0. (12.17) Bt Bxi Bxj

Now expand the first term and substitute from Eq. (12.14),

Bv¯j B BΦ B n ` pnvivjq “ ´ n ` v¯j pnv¯iq. (12.18) Bt Bxi Bxj Bxi This can be written in the form

2 dv¯j Bv¯j Bv¯j BΦ 1 Bpnσijq ” ` v¯i “ ´ ´ , (12.19) dt Bt Bxi Bxj n Bxi

2 where σij “ pvi ´ v¯iqpvj ´ v¯jq “ vivj ´ v¯iv¯j is the velocity dispersion tensor. In vector notation this becomes dv¯ Bv¯ 1 ” ` v¯ ¨ ∇v¯ “ ´∇Φ ´ ∇ ¨ pnσ2q. (12.20) dt Bt n

This is the second Jeans equation. It is the equivalent of the Euler equation of fluid dynamics. The left side is the convective derivative of the mean velocity. The right side consists of the gravitational force and a “pressure gradient” term.

The velocity dispersion tensor is symmetric and so can be diagonalized by a suitable change of coordinates. The ellipsoid with principal axes along the diagonalizing coordinates and semi- 2 axis lengths equal to the eigenvalues of σij is the velocity ellipsoid introduced in Section 4.3.

The Jeans equations were first derived by Maxwell, but were first applied to stellar dynamics by Jeans.

12.4 The virial equations

Multiply the second Jeans equation by xi and integrate over the spatial coordinates to get

B 3 B 3 BΦ 3 xipρv¯jqd x “ ´ xi pρv vjqd x ´ ρxi d x (12.21) Bt Bx k Bx ż ż k ż j

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The second term on the right side of this equation is the potential energy tensor Wij. The first term on the right side can be integrated by parts, with the surface term vanishing,

B 3 B 3 3 xi pρv vjqd x “ pρxiv vjqd x ´ δ ρv vjd x “ ´2Kij, (12.22) Bx k Bx k ik k ż k ż k ż where 1 K “ ρv v d3x (12.23) ij 2 i j ż is the kinetic energy tensor. The kinetic energy tensor can be split into systematic and random components 1 K “ T ` Π , (12.24) ij ij 2 ij where 1 T “ ρv¯ v¯ d3x, (12.25) ij 2 i j ż 2 3 Πij “ ρσijd x. (12.26) ż With these definitions, Eq. (12.22) becomes

B 3 xipρv¯jqd x “ 2Tij “ Πij ` Wij. (12.27) Bt ż Since the right side is symmetric, the left side must be also, so we may write

1 B 3 ρpxiv¯j ` xjv¯iqqd x “ 2Tij “ Πij ` Wij. (12.28) 2 Bt ż The moment of inertia tensor is defined by

3 Iijptq “ ρpx, tqxixjd x. (12.29) ż Differentiating with respect to time, and using the continuity equation we find

1 dIij 1 Bρ 3 “ xixjd x, 2 dt 2 Bt ż 1 Bρv¯k 3 “ ´ xixjd x, 2 Bx ż k 1 B 3 1 3 “ ´ pρv¯kxixjqd x ` ρv¯kpδikxj ` xiδjkqd x 2 Bxk 2 1 ż ż “ ρpv¯ x ` v¯ x qd3x, (12.30) 2 i j j i ż which is the same integral as appears in the left side of Eq. (12.28).

Differentiating again with respect to time and substituting in (3.70) gives the tensor virial theorem, 2 1 d Iij “ 2T ` Π ` W . (12.31) 2 dt2 ij ij ij

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If we take the trace of Eq. (12.31), and suppose that the system is in a steady state so that the left side is zero, we get the scalar virial theorem,

2K ` W “ 0, (12.32) where K “ Tr Kij is the total kinetic energy.

Since the kinetic energy of a system of mass M is K “ M v2 {2, where v2 is the mean square stellar velocity, we have GM v2 “ (12.33) rg where the gravitational radius is defined by

GM 2 r ““ . (12.34) g W

Since f can be interpreted as a mass density in phase space, we can replace the stellar number density n by the mass density ρ in all the equations above.

12.5 Jeans’ theorem and spherical systems

A fundamental theorem due to Jeans states that any steady-state solution of the collisionless Boltzmann equation depends on the phase space coordinates only through integrals of motion, and any function of the integrals yields a steady-state solution of the collisionless Boltzmann equation.

The first part of Jeans’ theorem follows because f is an integral of motion – it is constant along the orbit and it a function only of the phase space coordinates (for a steady-state solution). The second follows because any function of the integrals is necessarily constant along the orbit and therefore satisfies the collisionless Boltzmann equation.

In a spherical system, f can be expressed as a function of E and L (it cannot depend on the direction of L), f “ fpE,Lq. The simplest models have f “ fpEq.

A simple model having an isotropic velocity dispersions is the isothermal sphere. Start with the distribution function proportional to expp´E{σ2q,

2 ρ0 Φ ` v {2 fpEq “ exp ´ . (12.35) p2πσ2q3{2 σ2 ˆ ˙ The density is

8 2 4πρ0 Φ ` v {2 ρ “ fd3v “ exp ´ v2dv, p2πσ2q3{2 σ2 ż ż0 ˆ ˙ ´Φ{σ2 “ ρ0e , (12.36)

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8 2 1 4πρ0 Φ ` v {2 P “ fv2d3v “ exp ´ v4dv, 3 3p2πσ2q3{2 σ2 ż ż0 ˆ ˙ “ σ2ρ. (12.37) Compare this with the equation of state an ideal isothermal gas at temperature T ρk T P “ B . (12.38) m The two are the same if k T σ2 “ B . (12.39) m Inserting Eq. (12.36) into Poissons equation gives 1 d d ln ρ 4πg r2 “ ´ ρ. (12.40) r2 dr dr σ2 One solution is the singular isothermal sphere, σ2 ρ “ . (12.41) 2πGr2 However, this solution is infinite at the centre. A second isothermal sphere solution can be found numerically by integrating out from the centre with boundary condition ρ1p0q “ 0. This solution is finite at the centre and approaches the singular solution at large radii.

The isothermal sphere has infinite mass and stars extending to infinite radius. As this is not physically realistic, King (1966) introduced the “lowered” distribution function that falls to zero at finite radius and velocity

ρ1p0q “ 0, ρ 0 exp E Φ σ2 1 E Φ 2 3 2 r´p ´ 0q{ s ´ ă 0 fpEq “ p2πσ q { (12.42) $ ( &0 E ě Φ0. % To find the density of the King model, integrate this over velocity 8 ρprq “ 4π fpEqv2dv, ż0 2 4Ψ 2Ψ “ ρ eΨ{σ erfp Ψ{σ2q ´ 1 ` , (12.43) o πσ2 3σ2 « c ˆ ˙ff a where Ψprq “ Φ0 ´ Φprq.

Now substitute this into Poisson’s equation to get a differential equation for Ψ.

1 d dΨ 2 ? 4Ψ 2Ψ r2 “ 4πGρ eψ{σ erfp Φ{σq ´ 1 ` . (12.44) r2 dr dr 0 πσ2 3σ2 « c ˆ ˙ff

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This equation can be integrated numerically to give Ψprq . Substitution into then gives ρprq. Finally, if one assumes a constant mass-to-light ratio, Eq. (3.7) gives the intensity profile.

As this equation is second-order, two boundary conditions are needed. The first of these, dΨ{drpr “ 0q “ 0 is obvious by symmetry. To find the second, we note that as Ψ Ñ 0, ρ Ñ 0. The radius rt at which ρprtq “ 0 is called the tidal radius. The second boundary condition, the value of Ψp0q, implicitly determines the tidal radius.

The mean square velocity 1 v2 “ fv2d3v (12.45) ρ ż decreases steadily outward from the centre and reaches zero at rt.

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13 Stability

13.1 Jeans instability for collisionless systems

Collisionless stellar systems are described by the collisionless Boltzmann and Poisson equations

Bf ` v ¨ ∇rf ´ p∇rΦq ¨ p∇vfq “ 0, (13.1) Bt 2 3 ∇rΦ “ 4πG fd v. (13.2) ż In equilibrium, the distribution function and potential are independent of time.

Now consider a small departure from equilibrium

fpr, v, tq “ f0pr, vq ` εf1pr, v, tq, (13.3)

Φpr, tq “ Φ0prq ` εΦ1pr, tq. (13.4)

Substituting this in the above and neglecting terms of order ε2 and higher leads to linearized collisionless Boltzmann equation and Poisson equations for the perturbations,

Bf1 ` v ¨ ∇rf1 ´ p∇rΦ0q ¨ p∇vf1q ´ p∇rΦ1q ¨ p∇vf0q “ 0, (13.5) Bt 2 3 ∇rΦ1 “ 4πG f1d v. (13.6) ż Now consider departures from a homogeneous state f0 “ f0pvq. In this case we can set Φ0 “ 0 (this is called the Jeans swindle – see Binney & Tremaine 1987 for a discussion),

Bf1 ` v ¨ ∇rf1 ´ p∇rΦ1q ¨ p∇vf0q “ 0, (13.7) Bt 2 3 ∇rΦ1 “ 4πG f1d v. (13.8) ż We now look for solutions of the form

f1 “ fa expripk ¨ r ´ ωtqs, (13.9)

Φ1 “ Φa expripk ¨ r ´ ωtqs. (13.10)

Substituting this in Eqns. (13.7) and (13.8) gives

pk ¨ v ´ ωqfa ´ Φak ¨ p∇vf0q “ 0, (13.11)

2 3 ´k Φa “ 4πG fad v. (13.12) ż hence k ¨ ∇vf0 k2 “ ´4πG d3v. (13.13) k ¨ v ´ ω ż

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This is the dispersion relation connecting k and ω.

If ω2 ą 0, the system is stable – only oscillations occur. If ω2 ă 0, the system is unstable and small perturbations grow exponentially with time. The critical wave number kJ is that for which ω “ 0, k ¨ ∇vf0 k2 “ ´4πG d3v. (13.14) J k ¨ v ż For example, a homogeneous system with a Maxwellian velocity dispersion has

ρ0 ´v2{2σ2 f0 “ exp , (13.15) p2πσ2q3{2 8 df0 k2 “ ´16π2G vdv, J dv ż0 4πGρ0 “ . (13.16) σ2 This is analogous to the Jeans instability for a homogeneous fluid in which the fluid is 2 2 unstable to collapse on scales larger than the Jeans length λJ “ 2π{kJ where k “ 4πGρ{cs and cs is the sound speed.

13.2 Stability of rotating disks

Consider an infinitely-thin homogeneous sheet of gas with surface density Σ rotating with constant angular velocity Ω. In the plane of the sheet, in rotating coordinates R “ pr, φq, the equations are (continuity, Euler, and Poisson) BΣ ` ∇R ¨ pΣvq “ 0, (13.17) Bt Bv 1 2 ` pv ¨ ∇Rqv “ ´ ∇RP ´ ∇RΦ ´ 2Ω ˆ v ` Ω R, (13.18) Bt Σ ∇2Φ “ 4πGΣδpzq. (13.19)

A perturbation analysis, similar to that of the previous section, leads to the dispersion equa- tion 2 2 2 2 ω “ 4Ω ´ 2πGΣ0|k| ` k cs, (13.20) 2 where cs “ dP {dΣ is the sound speed. Stability requires that ω ě 0. a If Ω “ 0, the sheet is stabilized by pressure on small scales,

2πGΣ0 k ą kJ ” 2 . (13.21) cs

2 More generally, the right side is quadratic in k, and has a minimum at k “ πGΣ0{cs. For stability, we require that ω2 ě 0 at this value of k. This gives

2 2 2 2 2 2 2 2π G Σ0 π G Σ0 4Ω ´ 2 ` 2 ě 0. (13.22) cs cs

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This simplifies to csΩ π ě » 1.5708, (13.23) GΣ0 2 which is independent of wave number.

While the above analysis applies to a gaseous disk, a similar relation holds for stellar systems. Toomre (1964) obtains σΩ ě» 1.68, (13.24) GΣ0 where σ is the one-dimensional velocity dispersion.

The stability of a more-realistic uniformly-rotating isothermal disk, having vertical extent, has been investigated by Goldreich and Lynden-Bell (1965) who find

csΩ ě 1.06, (13.25) GΣ0

For a differentially-rotating gaseous disk, the dispersion relation is

2 2 2 2 pmΩ ´ ωq “ κ ´ 2πGΣ0|k| ` k cs, (13.26) where m is the azimuthal mode number of the perturbation pexpripmφ ´ ωtqs) and κ is the epicyclic frequency.

The uniformly-rotating disk is a special case of this with m “ 0, κ “ 2Ω. The stability condition is csκ Q ” ě 1. (13.27) πGΣ0 This is a local stability criterion, where it has been assumed that the spatial scale of pertur- bations is significantly smaller than the size of the disk. Q is essentially the ratio of random velocity cs to the orbital velocity πGΣ0{κ „ ΩR.

For a differentially-rotating stellar disk a similar relation holds, called the Toomre stability criterion (Toomre 1964), σκ Q ” ě 1. (13.28) 3.36GΣ0 For the solar neighbourhood, one finds that Q „ 1.7, so the disk is locally stable.

A disk may be locally stable but have a global instability. Theoretical analysis (see Binney & Tremaine 1987, Section 5.3.3) and numerical simulations (Figure 3.4) indicate that if a disk rotates too rapidly it is unstable and forms a bar. The stability condition has the form Ω ă 0.507, (13.29) Ω0

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Figure 13.1: Numerical simulations of a rotating disk of 100,000 stars. Although stable on small scales, the system is globally unstable, creating a rotating bar after a few rotation times (Hohl 1971).

where Ω0 ” π GΣp0q{2a is a characteristic frequency for a rotation disk of radius a and central surface density Σp0q. a

Ostriker & Peebles (1973) confirmed Hohl’s results and found that their N-body models were stable only if T ă 0.14 ˘ 0.02, (13.30) |W | where T is the kinetic energy of rotation and W is the potential energy. This is the Ostriker- Peebles criterion.

In terms of the kinetic energy in random motions Π “ ´W ´2T , the Ostriker-Peebles criterion becomes Π ą 5.1. (13.31) T For stars in the solar neighbourhood one finds Π{T » 0.15 which is much less than the value needed for stability. Ostriker and Peebles therefore proposed that the galaxy must have an unseen component, more massive than the disk, which would stabilize it. This is the dark halo.

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13.3 Density waves

Lin and Shu (1964, 1966) proposed that spiral arms result from the passage of stars and gas through a density enhancement propagating as a wave in the disk. Gas encountering the density wave is compressed, triggering star formation. The H-II regions and hot blue stars that result outline the wave. By the time the stars leave the wave, the most luminous have already died. Observations support this idea, showing a progression of stellar ages across the spiral arms.

For density waves to persist, amplification and feedback mechanisms are required. These are:

Lindblad resonance – if the frequency with which a star encounters a density wave is an integer multiple of the epicyclic frequency, a resonance will occur that can feed energy into the density wave. If the pattern rotates at constant angular speed ΩP , the condition for resonance is Ω ´ ΩP “ ˘κ{m (13.32) Swing amplifier – Goldreich & Lynden-Bell (1965) and Toomre (1981) discovered that a leading spiral pattern in a disk unwinds to form a trailing spiral pattern, and in the process is greatly amplified (Fig. 13.2). The amplification occurs because the pattern of the unwinding wave roughly matches the epicyclic motions of stars, creating a resonance.

Swing amplification can boost random noise to form transient spiral patterns. Feedback can occur when a density wave propagates inward and passes across the centre of the galaxy. An ingoing trailing spiral emerges as an outgoing leading spiral. A problem with this arises if the galaxy has an inner Lindblad resonance, which can absorb ingoing spiral waves.

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Figure 13.2: Evolution of a leading spiral density wave in a differentially rotating stellar disk. Contours indicate fractional density enhancement. The time interval between diagrams is half a rotation period at the radius of corotation. From Toomre (1981).

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14 Encounters

Until now we have neglected stellar encounters as being unimportant in galaxies. While this is true for stars, it is not true for encounters of galaxies with one another.

14.1 Dynamical Friction

Dynamical friction is a drag force that slows a massive object moving through a background of lower-mass objects. If a particle of mass M encounters a particle of mass m, with initial relative velocity v “ vM ´ vm and impact parameter b, the change in its velocity components perpendicular and parallel to v are (see Binney & Tremaine 1987, Section 7.1, for a derivation)

2mbv3 b2v4 ´1 ∆vM “ 1 ` , (14.1) K GpM ` mq2 G2pM ` mq2 „  2mv b2v4 ´1 ∆v “ ´ 1 ` . (14.2) Mk M ` m G2pM ` mq2 „  The change in kinetic energy of the mass M is M ∆K “ rpv ` ∆v q2 ` ∆v2 ´ v2 s, 2 M Mk MK M » MvM ∆vMk. (14.3)

Hence, 2 4 ´1 ∆K ∆vMk 4mv b v » 2 “ ´ 1 ` , (14.4) K v v pM ` mq G2pM ` mq2 M M „  and the particle looses energy in the encounter.

3 The rate at which M encounters stars that have velocities in d vm and impact parameters 3 b P pb, b ` dbq is p2πbdbqpvqpfpvmqd vmq so the rate of energy loss is

bmax 2 4 ´1 1 dK 4m 2 3 b v “ ´ v fpvmqd vm 1 ` 2πbdb, K dt v pM ` mq G2pM ` mq2 M ż ż0 „  2 2 ´1 ´2 3 “ ´4π lnp1 ` Λ qG mpM ` mqvM v fpvmqd vm, ż 2 ´1 ´2 3 » ´8π lnpΛqG mpM ` mqvM |vM ´ vm| fpvmqd vm. (14.5) ż where 2 bmaxv Λ “ . (14.6) GpM ` mq For an isotropic velocity distribution, this becomes Chandrasekhar’s dynamical friction formula 1 dK vM » ´32π2 lnpΛqG2mpM ` mqv´3 fpv qv2 dv . (14.7) K dt M m m m ż0

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If the mass M is moving much slower than the typical particle velocity, vM ! vm, the integral becomes vM 1 fpv qv2 dv » fp0qv3 (14.8) m m m 3 M ż0 and d ln K{dt is independent of velocity, as in Stokes formula for a particle moving through a viscous fluid.

If the mass M is moving much faster than the typical particle velocity, the integral converges to the number density of “background” particles

vM nm fpv qv2 dv » (14.9) m m m 4π ż0 In this case, 1 dK » ´8π lnpΛqG2pM ` mqρ v´3. (14.10) K dt m M In the limit that M " m, the dynamical friction time becomes

1 dK ´1 v3 t “ ´ » M . (14.11) df K dt 8π lnpΛqG2Mρ „  m which is inversely proportional to M and to the background density, and is independent of the mass of the background particles.

Insight can be gained from the following physical explanation. The mass M attracts the background particles, crating a “wake” of excess density. The gravitational attraction of the wake creates a drag force on the mass.

Chandrasekhar’s formula fails if the mass M is so large as to be comparable to that of the background interior to its orbit.

Dynamical friction affects satellites orbiting a galaxy. Globular clusters within „ 3 kpc of the centre of an L˚ galaxy should spiral into the nucleus in less than a Hubble time. The Magellanic clouds will be consumed by the galaxy on a similar timescale (Binney & Tremaine 1987, Section 7.1).

Galaxies in dense groups suffer orbital decay as they move through a sea of dark matter stripped from the individual galaxies. This results in a merging instability that can destroy the group in less than a Hubble time.

Suppose that a group contains N galaxies and that the baryonic matter constitutes a fraction f of the total mass, the remainder being in light dark-matter particles. Let R be the radius of the group and MT “ NM{f be its total mass. If the group is dense, interactions will strip the dark matter halos, producing a envelope of dark matter in which the baryonic cores of the galaxies orbit.

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Consider the motion of a single galaxy through this envelope. The average dark matter density 3 is ρ “ 3MT p1 ´ fq{4πR , so the dynamical friction timescale is

v3 t “ , df 8π lnpΛqG2Mρ v4R2 R “ 2 , 6 lnpΛqG MMT p1 ´ fq v 1 MT “ tcr, 5 lnpΛq M N “ tcr, (14.12) 6 lnpΛqfp1 ´ fq

2 where we have used the virial theorem v R » GMT and the definition of the crossing time tcr “ R{v.

´1 ´1 For a typical compact group, n “ 4, f „ 0.1, ln Λ „ 3 and tcr „ 0.04H0 , hence tdf „ 0.1H0 , showing that the galaxy orbits should decay in a few tenths of a Hubble time.

14.2 Kinetic theory

The Kinetic theory of gasses contains powerful concepts that can be applied to other “parti- cles” such as stars, or dark matter particles. We represent the entire system of N particles by a single point in 6N dimensional Γ-space. This is called the microstate. Now consider all possible realizations of the system that have the same macroscopic properties. The den- sity of such microstates in Γ-space is the N-particle distribution function f pNq. Effectively, pNq 6 6 6 f pw1, w2, ¨ ¨ ¨ wN qd w1d w2 ¨ ¨ ¨ d wN is the probability that the system will be found in the 6 6 6 6N-dimensional Γ-space volume d w1d w2 ¨ ¨ ¨ d wN centered at pw1, w2, ¨ ¨ ¨ wN q. f pNq obeys a continuity equation, and by the same process that lead to the collisionless Boltzmann equation, we obtain Liouville’s equation,

pNq pNq N df Bf pNq ” ` w9 a∇w f “ 0, (14.13) dt Bt a a“0 ÿ which states that the density of Γ-points is conserved. This is Liouville’s theorem.

Liouville’s theorem was actually first derived by Gibbs, two years after Liouville died.

Integrating f pNq over all particles but one gives the single-particle distribution function f

p1q pNq 6 6 f “ f pw1, tq ” f pw1, w2, ¨ ¨ ¨ , wN , tqd w2 ¨ ¨ ¨ d wN . (14.14) ż

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Integrating Liouville’s equation over all particles but one yields the collisionless Boltzmann equation if, and only if, the particles are uncorrelated, ie. if and only if the 2-particle distri- bution function is separable,

p1q pNq 6 6 f “ f pw1, bsw2, tq ” f pw1, w2, ¨ ¨ ¨ , wN , tqd w3 ¨ ¨ ¨ d wN , ż “ fpw1, tqfpw2, tq. (14.15) Encounters change the phase space density. Scattering moves some stars out of the phase space volume and other stars into the phase space volume around a star. Represent this by adding a collision term Cptq to the Boltzmann equation, df “ Γptq. (14.16) dt To obtain an explicit form for the collision term, let Ψpw, ∆wqd6∆w be the rate at which stars are scattered from phase space coordinates w to a volume d6∆w centered at w ` ∆w. Then we have df “ rΨpw ´ ∆w, ∆wqfpw ´ ∆wq ´ Ψpw, ∆wqfpwqsd6∆w, (14.17) dt ż which is called the master equation. The first term on the right corresponds to stars scattered into the phase space volume, the second terms corresponds to stars scattered out of this volume.

The master equation is an integro-differential equation. To simplify it, expand Ψf in a Taylor series, keeping first and second order terms, B Ψpw ´ ∆w, ∆wqfpw ´ ∆wq “ Ψpw, ∆wqfpwq ´ ∆wi rΨpw, ∆wqfpwqs Bwi 1 B2 ` ∆wi∆wj rΨpw, ∆wqfpwqs ` ¨ ¨ ¨ . (14.18) 2 BwiBwj Substituting this in the master equation gives the Fokker-Planck equation df B 1 B2 “ ´ rfDp∆wiqs ` rfDp∆wi∆wjqs, (14.19) dt Bwi 2 BwiBwj where the all the details of the interaction are embodied in the diffusion coefficients

6 Dp∆wiq ” ∆wiΨpw, ∆wqd ∆w, (14.20) ż 6 Dp∆wi∆wjq ” ∆wi∆wjqΨpw, ∆wqd ∆w. (14.21) ż Once the diffusion coefficients are estimated, the Fokker-Plank equation becomes a purely differential equation that is amenable to numerical solution.

One could include higher-order terms but these are usually much smaller than the linear and quadratic terms, which are comparable in magnitude.

The Fokker-Planck equation is widely used in studies of systems in which collisional effects are important, such as globular clusters. See Binney & Tremaine (1987, Section 8.3) for more detail and examples.

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14.3 Violent relaxation

If the relaxation time in elliptical galaxies is much longer than the age of the Universe, why do these galaxies have such a smooth distribution of light? One might expect inhomogeneities to be present, relics from galaxy formation.

A resolution was provided by Lynden-Bell in 1967. He whosed (Lynden-Bell 1967) that rapid fluctuation of the gravitational potential can effectively mix stellar orbits, erasing any pre- existing structure. Such fluctuations will occur, for example, if two galaxies merge. This is called violent relaxation.

Clearly, if the potential is time dependent, the energies of stellar orbits are not conserved. However, the interaction remains collisionless, so the phase space density surrounding any star is conserved (as per the Collisionless Boltzmann Equation). What happens is that regions of low phase-space density are mixed with regions of high phase-space density. Thus if an average is taken over a reasonably-large phase-space volume, the result approaches an equilibrium distribution. This process is a kind of phase mixing.

Numerical simulations confirm that when galaxies collide, the resulting merger quickly equi- librates, resembling an elliptical galaxy after several crossing times.

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15 Active galaxies - phenomenology

Much of the material for this section is based on Osterbrock (1991).

The first active galaxy was discovered over one hundred years ago when Edward Fath working at Lick Observatory noticed that the spiral galaxy NGC 1068 had six emission lines in its spectrum Fath (1909). Subsequently, Hubble (1926) called attention to NGC 1068, 4051 and 4151 as having spectra like planetary nebulae. These galaxies, and others like them were studied by Carl Seyfert (1943) who observed that their emission lines are broader than those seen in many normal galaxies.

Many of these galaxies were found to have an unusually-bright star-like nucleus (Fig. 15.1). Some galaxies, called Type 1, or Seyfert 1, have very broad permitted lines, with Doppler widths „ 104 km/s, as well as less-broad forbidden lines. Seyfert 2 galaxies show only the forbidden lines.

Figure 15.1: Seyfert galaxies NGC 1068 (left) and NGC 4151 (University of Alabama, W. Keel; HST).

Twelve years later, Walter Baade and Rudolph Minkowski (1954a) identified the strong ra- dio source Cygnus A with a distant cD galaxy at redshift 0.057 (Fig. 15.2). This galaxy also showed strong high-ionization emission lines like the Seyfert galaxies. Surprisingly, the strongest radio emission was found not to originate from the galaxy, but from two extended lobes on either side.

Some radio sources appeared to not be associated with galaxies. When a lunar occultation allowed the positions of 3C273 to be precisely determined, it coincided with a star-like object with a faint jet (Fig 15.3). The spectrum showed broad emission lines at unusual positions.

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Figure 15.2: Top: 5 GHz radio map of Cygnus A obtained with the VLA; left: HST optical image of the central 5 arcsec, showing an elliptical-like galaxy containing dust; right: Infrared image taken with the Keck telescope adaptive optics system revealing an intense star-like nucleus.

Maartin Schmidt identified the brightest of these as hydrogen lines at redshift 0.158 (Schmidt 1963). Shortly after, Jesse Greenstein and Thomas Matthews identified 3C48 as a quasi- stellar source (QSS) with a redshift of 0.367 (Greenstein & Matthews 1963).

Similar star-like objects with broad emission lines were found which were not strong radio sources. These were called quasi-stellar objects (QSOs). Later, the term quasar was adopted to refer to both QSOs and QSSs.

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Figure 15.3: 3C273 and its jet (montage created by W. Keel, University of Alabama).

Some radio galaxies were found to have a bright star-like nucleus. These were called N galaxies. SOme radio galaxies show emission lines in the spectra of their nuclei. Broad-line radio galaxies (BLRG) have broad lines like Seyfert 1 and quasars. Narrow-line radio galaxies (NLRG) have less-broad emission lines, like Seyfert 1.

In 1968 John Schmidt matched the positon of the radio source VRO 44.22.01 with the variable “start” BL Lacertae. He obtained a spectrum of this object which showed no absorption or emission lines at all. Similar objects were soon found. By 1976, 30 BL Lac objects were known.

In 1974, M. Disney obtained images of the BL Lac object PKS 0548-322 which revealed it to be the nucleus of an elliptical galaxy having a redshift of 0.069.

Many BL Lac objects exhibit extreme variability. Their flux can bary by as much as 50% in a single day. These are called optically-violent variables (OVVs). Collectively, BL Lac objects and OVVs are now given the name blazars.

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(a) (b)

Figure 15.4: (a) Spectrum a the BL Lac object PG 1553+113. The absorption features are due to oxygen in the Earth’s atmosphere (ESO). (b) Image of the BL Lac object H 0323 022, showing what appears to be an elliptical galaxy with a redshift of 0.147 (ESO).

15.1 Classification and properties

Let’s summarize some basic observational results: Seyfert galaxies are spiral. radio galaxies are elliptical. N-Galaxies are usually BLRGs. QSO and QSS have similar optical spectra. Seyferts are the most numerous active galaxies (about 2% of all spirals). blazars have the brightest nuclei, and the most extreme variability.

15.2 Emission spectra

Typical spectra are shown in Figure 15.6.

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Figure 15.5: Seyfert galaxy NGC 5548 (left) compared with a normal galaxy NCG 3277 with similar distance and Hubble type. Note the diffraction spikes from the bright nucleus (HST, University of Alabama, W. Keel).

Emission lines have typical linewidths (FWHM) „ 200 ´ 700 km/s. This is much broader than those seen in normal galaxies.

Table 15.1: Classes of active galaxies Name Defining property Seyfert galaxy Spiral galaxy with bright star-like nucleus Seyfert 1 Permitted lines have very broad component Seyfert 2 Broad component is missing Radio galaxy Galaxy with strong radio emission Quasi-stellar object (QSO) Star-like object with broad emission lines Quasi-stellar source (QSS) Star-like object with radio emission Quasar A QSO or QSS BL Lac object Star-like object with no spectral features Optically-violent variable (OVV) Highly-variable star-like object Blazar BL Lac or OVV N-Galaxy Elliptical galaxy with bright star-like nucleus Broad-line radio galaxy (BLRG) Radio galaxy with spectrum like Seyfert 1 Narrow-line radio galaxy (NLRG) Radio galaxy with spectrum like Seyfert 2 Low-ionization nuclear emission region Galaxy with lower-ionization lines than Seyfert (LINER)

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Figure 15.6: Typical spectra of active galaxies (University of Alabama, W. Keel).

Permitted lines may have a very broad component with maximum widths „ 5000 ´ 30000 km/s.

In the ultraviolet, the strongest lines are Ly-α λ1216 A,˚ and next, C IV λ1549. These lines are very prominent in high-redshift quasars.

15.3 Narrow-line region

The narrow emission lines arise in a region within „ 102 pc of the nucleus called the narrow- line region (NLR).

The distribution of ionized gas is clumpy, often showing filaments (Figure 15.8). Line ratios 4 9 ´3 indicate T „ 1.5 ˆ 10 K and ne „ 10 m .

The size of the NLR can be estimated from the line flux. For Hβ, we have 4π L “ R3fn n α hν , (15.1) Hβ 3 e p Hβ Hβ

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Figure 15.7: Broad-band spectrum of the Seyfert Galaxy NGC 4151 (University of Alabama, W. Keel).

where αHβ is the Hβ recombination coefficient the number of recombinations that produce an Hβ photon per unit time per unit electron density and f is the filling factor – the fractional volume occupied by the gas clouds.

Typically ne » np ` 1.5nHe (about half the Helium is ionized to He II and half to He III).

8 The most luminous Seyfert 2 galaxies have LHβ » 2 ˆ 10 Ld, which gives

´1{3 10 ´3 ´2{3 RNLR » 20f pne{10 m q pc, (15.2) 5 10 ´3 ´1 MNLR » 7 ˆ 10 pne{10 m q Md. (15.3)

10 ´3 ´2 6 For ne „ 10 m and f „ 10 , one has RNLR „ 90 pc and MNLR „ 10 Md.

The relatively-low electron temperature, combined with the high state of ionization indicates that photoionization is the primary energy source. (Other mechanisms such as shock waves from cloud collisions, and relativistic particle beams, produce a much higher temperature for the same degree of ionization.)

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Table 15.2: Principal optical emission lines found in active galaxies

Forbidden lines Permitted lines Line Wavelength (A)˚ Line Wavelength (A)˚ [O I] 6300, 6364 Fe II [O II] 3727 HI 1217, 6563 [O III] 4363, 4959, 5007 He II [Ne III] 3869 He I [S II] 6716, 6731 C III] 1909 [Fe VII] 6087 C IV 1549 [Fe X] 6375 Mg I

Figure 15.8: Seyfert galaxy NGC 1068 and nuclear region showing ionized gas in the narrow- line region (KPNO, NASA).

Since the number of recombinations (each of which produce an emission line photon) is propor- tional to the number of ionizing photons, this implies that the ratio of line flux to continuum flux should be constant, which is confirmed by observations.

The ionizing flux required is harder (more high-energy photons) than that produced by stars.

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This is consistent with the source of ionization being the power-law continuum.

The high-energy photons in the ionizing spectrum penetrate the gas, producing an extended region of partial ionization not seen in normal galaxies. This accounts for the strength of the low-ionization lines.

15.4 Broad-line region

Forbidden lines are never seen with broad components. This means that they must be colli- sionally de-excited. Therefore the broad-line emitting region must have a density of at least 14 ´3 ne „ 10 m .

The presence of semi-forbidden C III] λ 1909 in the broad line region indicates that the density 15 ´3 cannot be much higher than ne „ 3 ˆ 10 m .

Seyfert 1 have a broad component, Seyfert 2 do not. The belief is that this is because the broad line region (BLR) is obscured by dust in the latter. There is evidence to support this – images taken in polarized light reveal weak broad lines in some Seyfert 2, which is interpreted as scattered light from the BLR.

9 The most luminous Seyfert 1 have LHβ » 10 Ld. Equation 5.2 then gives

´1{3 15 ´3 ´2{3 RBLR » 0.015f pne{10 m q pc, (15.4) 15 ´3 ´1 MBLR » 36pne{10 m q Md. (15.5)

15 ´3 ´2 For ne „ 10 m and f „ 10 , one has RBLR „ 0.07 pc and MBLR „ 40Md. it is generally presumed that photoionization is the dominant energy input mechanism. The lines strengths are observed to be proportional to the continuum flux, but this alone is not conclusive evidence.

For examples of theoretical models of the BLR see Kwan (1984) and Rees et al (1989).

Some AGN show variability in the BLR. This should correlate with variations in the optical continuum as the photon flux from the central source propagates across the BLR. Cross correlation of variations in various lines and the continuum can give information on the size and geometry of the BLR (see Peterson 1988).

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15.5 Continuous spectrum

´α AGN often show a power law component, Fν 9 ν , where the spectral index α is typically of order unity, that extends from the infrared to soft-X-ray frequencies. This is presumed to arise from synchrotron radiation.

Superimposed on this is a broad component extending from the optical to ultraviolet frequen- cies called the blue bump.

The radiation is presumed to arise from accretion onto a massive black hole at the centre of the galaxy. The blue bump is believed to be due to thermal emission from the accretion disk (Shields 1978).

The blue bump spectrum is not black body as a temperature variation is expected in the disk. The highest-energy photons would come from the inner edge of the accretion disk, where the Keplerian velocity, and temperature, are the greatest. Relativistic effects are important. Many models exist (eg. Laor & Netzer 1989), but none are in complete agreement with the data.

A useful landmark is the Eddington luminosity LE, at which radiation pressure prevents gravitational accretion in a spherically symmetric system, L GMm E σ “ H , 4πr2 T r2 4πGmH 4 LE “ M » 3.29 ˆ 10 pM{MdqLd, (15.6) σT

2 ´29 ´2 where σT “ 8πre {3 » 6.652 ˆ 10 m is the Thompson cross section. Here re “ 3 2 ´15 e {p4π0mec q “ 2.82 ˆ 10 m is the classical electron radius.

The spectrum depends on the mass of the black hole, the accretion rate and the inclination of the accretion disk. The data are consistent with black hole masses in the range M „ 7.5 9.5 8 9.5 10 ´ 10 Md. Masses in the upper part of this range, M „ 10 ´ 10 Md are found in quasars. Smaller masses are found in Seyferts.

Accretion rates and luminosities range from near Eddington in luminous quasars to a few percent of Eddington in Seyferts.

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Figure 15.9: Broadband spectrum of BL Lac object Mrk 501. (A.E. Wehrle/M.A. Catanese/J.H. Buckley/Whipple Collaboration).

15.6 Radio morphology

The most luminous radio sources known are quasars and radio galaxies. However, the entire radio luminosity is just a few percent of the total luminosity of these objects.

Edge-brightened double lobe sources, called Fanaroff-Riley class II, have luminosities 26 ´1 L128MHz ą 2 ˆ 10 W Hz (Fanaroff & Riley 1974).

Fanaroff-Riley class I, brightest at the inner part of the lobe, have luminosities L128MHz ă 2 ˆ 1026 W Hz´1.

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Figure 15.10: Broadband spectrum of BL Lac object Mrk 501. Curves represent synchrotron and inverse Compton models (E.D. Bloom/J. Kataoka et al.).

The radio emission has a synchrotron spectrum, with a power law index of α „ 0.7 ´ 0.8.

Radio galaxies are always cD, D or E type. Seyfert galaxies are usually Sa or Sb, never E.

Most radio galaxies have emission-line optical spectra, either BLRG or NLRG.

Radio galaxies show a linear relationship between [O III] luminosity and total radio power (Rawlings et al 1989).

Seyfert galaxies are weak radio sources. Seyfert 2 and Seyfert 1 are indistinguishable in the radio (Giuricin et al 1990, Ulvestad & Wilson 1989).

The radio emission in Seyfert galaxies is confined to jets within a few hundred pc of the nucleus. These jets are not aligned with the major or minor axes, indicating that they are neither perpendicular nor parallel to the rotation axis of the galaxy.

It seems likely that both Seyfert galaxies and radio galaxies produce jets, but the lack of an interstellar medium in elliptical galaxies allows the jet to escape, producing the radio lobes.

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Figure 15.11: Broadband spectrum of 3C 273. curves represent synchrotron and inverse Compton models, plus a thermal component (M. A. Catanese, Iowa State Univ).

15.7 Infrared flux

Seyfert galaxies and quasars emit much of their power in the infrared. In addition to the power law continuum and blue bump, there is an infrared bump extending from 2 to 103 um, due to thermal emission from dust grains. The derived dust temperatures are in the range 40 ´ 80 K.

2 4 The typical dust mass is of order 10 Md, corresponding to a gas mass of order 10 Md which is comparable to that of the NLR.

The power-law continuum peaks around 80 um and decreases at longer wavelengths. This turnover is interpreted as synchrotron self-absorption.

Often one observes an excess flux around 5 um wavelength. This 5-um excess can be fit by thermal emission from dust near the nucleus, perhaps in the form of a torus, heated by radiation from the accretion disk.

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Figure 15.12: Broadband spectrum of BL Lac object Mrk 421. Curves are as in Figure 15.12. (M. A. Catanese, Iowa State Univ).

15.8 X-ray and gamma ray flux

Seyfert galaxies and quasars are found to be strong X-ray emitters. In the 2 ´ 20 keV band the spectrum is usually well described by a power law with index α „ 0.5 ´ 1.

The BLR Hα flux is found to be proportional to the X-ray flux.

The most plausible source of the radiation is the synchrotron power-law seen in the optical and IR, augmented by synchrotron self-Compton radiation (inverse Compton scattering of synchrotron photons) that produces most high-energy X-rays. The models predict a lowering of the spectral index by about 0.5 around 1 keV (Odell et al 1987). High-quality spectra reveal a more complex situation requiring multiple components (Band & Malkan 1989).

Variations in X-ray flux have been seen on a timescale of hours. This is consistent with 7 emission from the inner part of an accretion disk at a few Schwarzchild radii from a 10 Md black hole (Turner & Pounds 1988).

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(a) (b)

Figure 15.13: Examples of Fanaroff-Riley class I (3C31) and II (3C175 and PKS 2356-61 radio galaxies (left: VLA, right: Koekemoer, Schilizzi, Bicknell & Eckers/ATNF).

Gamma rays have been observed from some AGN, particularly blazers, which show a double peak in the spectrum. The first peak, in the X-ray region is due to Synchrotron radiation and the second peak at higher energy results from inverse-Compton scattering.

15.9 Jets

Narrow, well-collimated jets are seen emerging from the nuclei in many AGN. In radio galaxies, these jets transport energy from the nucleus to the radio lobes. Radio galaxies usually have two jets, emerging in opposite directions.

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Often, the jets appear to be of unequal brightness, and sometimes only one jet is visible. This asymmetry in the flux ratio of the jets is presumed to arise from relativistic effects, which boost the apparent power of an approaching jet compared to that of a receding jet (discussed in the next section).

Condensations in the jets are observed, with radio interferometers, moving outward from the nucleus. Sometimes the apparent speed exceeds the speed of light. This apparent superluminal motion can also be understood in terms of relativistic effects in a jet approaching us in a direction close to the line of sight.

(a) (b)

Figure 15.14: (a) HST Optical image of the jet in M87. Note the presence of gas filaments near the nucleus. (b) VLA radio images of inner region of the jet, showing apparent superluminal motion (John Biretta, Space Telescope Science Institute).

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16 Active galaxies - theory

16.1 Relativistic beaming

Relativistic effects provide an explanation both for superluminal motion, and the observed brightness asymmetry of jets.

Suppose that a blob of luminous matter is approaching us, moving with speed v (in units of the speed of light) at an angle θ to the line of sight. We see the emitted radiation boosted by the relativistic Doppler effect ´1 ν “ ν0rγp1 ´ v cos θqs (16.1) where γ “ p1 ´ v2q´1{2 is the Lorentz factor.

3 Specific intensity Iν divided by ν is a Lorentz scalar, being proportional to fγ, the number density of photons in phase space, Iν “ h4cf , (16.2) ν3 γ where h is Planck’s constant.

From this, we see that ´3 ´3 Iν 9 γ p1 ´ v cos θq . (16.3) For the total intensity, ´4 ´4 I “ Iνdν 9 γ p1 ´ v cos θq . (16.4) ż The ratio of intensities between the approaching and receding jets is therefore

I` 1 ` v cos θ 4 “ , (16.5) I´ 1 ´ v cos θ ˆ ˙ This is a very large boost. Only 0.1% of the emitting fluid needs to move towards us at γ „ 2.8 in order for it to dominate the emission! This easily accounts for the asymmetry seen in jets.

To convert intensities to fluxes, we must integrate over the angular size of the source. The directions transverse to the velocity do not undergo Lorentz contraction, and contraction in the longitudinal direction is independent of the sign of the velocity. Therefore the approaching and receding jets have the same amount of contraction, and the flux ratio is the same as the intensity ratio, F ` 1 ` v cos θ 4 “ , (16.6) F ´ 1 ´ v cos θ ˆ ˙ ´α For a power-law spectrum, Fν 9 ν , we have

F ` 1 ` v cos θ 3´α ν “ , (16.7) F ´ 1 ´ v cos θ ν ˆ ˙

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The spectral index appears because the observed frequency corresponds to different frequencies in the rest frames of the approaching and receding jets.

The apparent transverse velocity is v sin θ v “ (16.8) K 1 ´ v cos θ which has a maximum value vK “ γv when cos θ “ v. This velocity can easily be superluminal even for modest Lorentz factors.

Variability is observed on a shorter timescale than in the rest frame of the jet. The observed timescale is smaller by a factor „ γ´1,

t ν0 1 ´ v cos θ 1 “ “ γp1 ´ v cos θq “ » . (16.9) t0 ν p1 ´ vqp1 ` vq 2γ a 16.2 Theoretical models

In recent years, a plausible framework has been developed in which all AGN phenomena arise as a result of accretion of matter by a central black hole. Differences between types result from different black hole masses, accretion rates, gas content, viewing angles, environment, etc.

The general picture is illustrated in Figure 16.1. The principal components are • a massive black hole • a gaseous accretion disk • a dusty torus • gas clouds forming the BLR • gas clouds forming the NLR • opposed jets emerging perpendicular to the accretion disk

Unified models propose the following: QSOs are the most luminous examples of a continuum of gas-rich active galaxies that include Seyfert galaxies and LINERS QSSs are the most luminous examples of a continuum of gas-poor active galaxies that include radio galaxies of various kinds Blazars are active galaxies in which one of the jets points almost directly towards us. Their extreme properties are due mostly to relativistic beaming effects. The only difference between Seyfert 1 and Seyfert 2 (and between BLRG and NLRG) is that in the latter, the orientation of the galaxy prevents us from seeing into the BLR, which is obscured by a dusty torus.

Before discussing physical models, it will be useful to summarize some of the most important observed quantities (Table 16.1).

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Figure 16.1: Schematic diagram of an active galactic nucleus (not to scale) (C.M. Urry and P. Padovani).

Table 16.1: Observational properties of active galaxies

Total luminosity* 1036 ´ 1041 W Emission line luminosity* 1037 ´ 1039 W Radio luminosity* 1032 ´ 1039 W Optical/IR continuum luminosity* ď 1040 W X-ray luminosity ď 1040 W Optical-IR spectral index 0 ´ 2 Energy content of radio lobes 1045 ´ 1054 J BLR linewidth 7 ˆ 103 ´ 3 ˆ 104 km s´1 BLR temperature „ 104 K BLR gas density 1014 ´ 1017 m´3 Radio variability timescale years Optical variability timescale weeks – months X-ray variability timescale hours – days * Luminosities in the table assume that the emission is isotropic. The luminosity will be lower if there is significant relativistic beaming.

16.3 Nature of the central object

The mass of the central object can be estimated in several ways:

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• Eddington limit – In the case of spherical accretion, a lower limit on the mass can be found from the total luminosity

σ L L M ě T » 7.95 ˆ 109 M . (16.10) 4πGm 1042 W d H ˆ ˙ This mass will be an overestimate if the observed radiation is highly beamed or the luminosity is above the Eddington limit. • Velocity width of the broad lines

v2R v 2 R M » » 2.3 ˆ 1910 M . (16.11) G 104 km s´1 pc d ˆ ˙ ˆ ˙ This mass will be an overestimate if the orbital radius of the cloud has been overesti- mated or the line broadening is not due to orbital motion (eg. radiative acceleration). • Velocity width of the narrow lines

v2R v 2 R M » » 2.3 ˆ 109 M . (16.12) G 200 km s´1 kpc d ˆ ˙ ˆ ˙ • Model fitting of the continuum radiation

7 9 M „ 3 ˆ 10 ´ 3 ˆ 10 Md. (16.13)

This mass will be inaccurate if the model is incorrect.

9 From the above it seems likely that masses of up to „ 10 Md are required. This leads 3 inevitably to a massive black hole because supermassive stars of 10 Md or more are believed to be unstable and a sufficiently-dense star cluster would be unstable due to collisions.

9 The conclusion is that the central object is a black hole of as much as 10 Md. We now have two immediate problems: 1) how to create this black hole and 2) how to produce the luminosity seen in active galaxies.

16.4 Black hole basics

Non-rotating (and electrically neutral) black holes are described by Schwarzschild’s solution to Einsteins equations. An event horizon exists at the Schwarzschild radius

2GM M r “ » 2.94 km. (16.14) S c2 M ˆ d ˙ Rotating (and electrically neutral) black holes are described by the Kerr solution to Einstein’s equations. An event horizon exists at radius r “ rS and a region exists, called the ergosphere, outside the horizon, in which particles are dragged around around the black hole.

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There is a limit to the amount of angular momentum that a black hole can possess,

GM 2 J “ . (16.15) max c We can writing the angular momentum as

J “ aJmax, (16.16) where the spin parameter a P r0, 1s.

Objects having a ą 1 cannot collapse to form a black hole unless they can first transfer some of their angular momentum elsewhere.

There is a minimum stable orbit around a black hole. For a Schwarzschild (non-rotating) black hole, the radius of this minimum stable orbital is rmin “ 3rS.

Quantum mechanical effects cause black holes to radiate as if the horizon were a black body at the Hawking temperature.

3 ~c ~c TH “ “ . (16.17) 4πkBrS 8πkBGM

The Hawking luminosity is therefore

M 2 L “ 4πr2 σ T 4 » 3.96 ˆ 10´29 d W. (16.18) H S B H M ˆ ˙

16.5 Accretion of stars

Hawking radiation is negligible for a massive black hole, so the luminosity most likely comes from gravitational energy released by infalling matter. The most obvious source of matter is the stars in the galactic centre. The required rate of consumption of matter is

L 16.7 L M9 “ » M yr´1. (16.19) ηc2 η 1041 W d ˆ ˙ where η is the fraction of the rest-mass energy that is converted to radiation.

If the black hole swallows a star whole, the energy released is negligible. Therefore, for this to work the star must be tidally disrupted.

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As a rough approximation, star of mass M˚ and radius R˚, at a distance r from the black hole, will be disrupted if the tidal force exceeds the gravitational binding force

2GMR‹ GM˚ 3 ě 2 , r R˚ ρ 1{3 r ď 21{3 S r , (16.20) ρ S ˆ ˚ ˙ 3 3 where ρ˚ “ 3M˚{4πR˚ and ρS “ 3M{4πrS are the mean densities of the star and the black hole.

This ignores centrifugal force and distortion of the star. A more accurate calculation gives the Roche limit, ρ 1{3 r ď 2.423 S r . (16.21) ρ S ˆ ˚ ˙ This shows that the black hole is incapable of disrupting stars if its density

3c6 M ´2 ρ “ » 8.2 ˆ 1018 kg m´3, (16.22) S 32πG3M 2 M ˆ d ˙ is less than 0.07 of that of typical stars,

M R ´3 ρ » 3.8 ˆ 104 ˚ ˚ kg m´3, (16.23) ˚ M R ˆ d ˙ ˆ d ˙ 7 which means that black holes having mass exceeding the Laplace-Hills limit M „ 10 Md or more will swallow most stars whole (Hills 1975).

A possible solution to this problem is that stars in the vicinity of a black hole will likely collide with each other, leading to disruption (Frank & Rees 1976).

16.6 Accretion of gas

A plausible alternative is that interstellar gas is accreted directly. As a simple model, consider spherical accretion by a mass M placed in an infinite homogeneous perfect gas whose density, pressure and temperature far from the mass are ρ8, P8 and T8. We ignore the small increase in M due to the accretion and assume that the gravitational field is static.

The dynamics of the gas is determined by the continuity and Euler equations, Bρ ` ∇ ¨ pρvq “ 0, (16.24) Bt Bv 1 ` pv ¨ ∇qv ` ∇P ´ ∇Φ “ 0. (16.25) Bt ρ We assume that the flow is in steady state, so the time derivatives vanish.

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Integration of the continuity equation gives the mass accretion rate

M9 “ 4πr2ρv “ const. (16.26)

Integration of the Euler equation gives the Bernoulli equation

v2 P dP GM ` ´ “ 0 (16.27) 2 ρ r żP8

For an equation of state P 9 ργ, (16.28) the Bernoulli equation becomes

v2 γ P ρ γ´1 GM ` 8 ´ 1 “ . (16.29) 2 γ ´ 1 ρ ρ r 8 «ˆ 8 ˙ ff If γ “ 1, the accretion is isothermal; if γ “ 5{3 it is adiabatic. The simultaneous solution of equations (16.26) and (16.29) leads to a upper limit on the mass flow given by the Bondi accretion rate (Bondi 1951) 9 4πλ 2 M ď 3 pGMq ρ8, (16.30) cs where cs is the (adiabatic) sound speed far from the mass

2 P8 kT cs “ γ “ γ (16.31) ρ8 m and λpγq equals e3{2{4 for the isothermal case and 1{4 for the adiabatic case.

Combining (16.30) and (16.31) gives

M 2 T ´3{2 n f M9 87 e M yr´1. (16.32) ď 8 4 9 ´3 d 10 Md 10 K 10 m 0.01 ˆ ˙ ˆ ˙ ´ ¯ ˆ ˙

That this is at all close to the estimated accretion rate required to power a luminous quasar is encouraging. However...

The Bondi accretion rate ignores radiation pressure, which will cut off the flow when the lumi- nosity reaches the Eddington limit, and the assumption of spherical symmetry is unrealistic. Angular momentum will prevent the gas from reaching the black hole. Instead, an accretion disk will form. The rate of accretion will be determined by the efficiency with which viscous forces can transfer angular momentum outward in the disk.

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16.7 The accretion disk

Accretion of matter with significant angular momentum will spin up the black hole to near its maximum rate.

Gas having specific angular momentum (angular momentum per unit mass) ? j ą 3rS (16.33) cannot cross the event horizon unless it can first shed angular momentum via viscous or magnetic torques.

In the process, orbital energy is converted to heat, or magnetic energy, and ultimately radiated away. If this radiative cooling is effective, the accretion disk will be thin.

Angular momentum can be transported by viscous drag (Carter 1979). Because the orbital velocity increases as radius r decreases, viscous forces within the accretion disk will act to transfer angular momentum outward. At the same time, there must be a net flux of mass inward, to feed the black hole.

Lets examine the effects of viscous drag alone. Electromagnetic effects will be considered later.

We assume that the disk is thin, with surface density Σ, stationary and has circular symmetry. In cylindrical coordinates r, φ, z, the continuity equation becomes

BΣ ` ∇ ¨ pΣvq “ 0. (16.34) Bt The first term vanishes because the system is stationary. The second term gives 1 d rΣv “ 0. (16.35) r dr r From this we see that the mass flux

M9 “ ´2πrΣvr (16.36) is constant.

The flux of angular momentum passing inward through radius r is the product of the mass flux and the specific angular momentum of the gas at that radius, Mrv9 φ. The angular momentum transferred outward across r by viscous forces is 2πr2T , where T is the viscous force per unit length. The difference between these is the net flux of angular momentum. It must be constant, since angular momentum is conserved,

2 J9 “ 2πr T ´ Mrv9 φ “ const. (16.37)

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Thus M9 vφ J9 T “ ` . (16.38) 2π r 2πr2 2 2 2 For a Keplerian disk, vφ “ r ω “ GM{r, where M is the mass of the black hole, so this becomes M9 pGMq1{2 J9 T “ ` . (16.39) 2π r3{2 2πr2

The viscous force must vanish at the inside edge of the disk, where r “ rmin, so

9 9 1{2 1{2 J “ ´MpGMq rmin, (16.40) hence 1{2 M9 pGMq rmin 1{2 T “ 1 ´ . (16.41) 2π r3{2 r „ ´ ¯  As the gas spirals inward, The virial theorem requires that 2K`U “ 0, so E “ K`U “ ´U{2. The energy liberated per unit time between r and r ` dr is therefore

1 GMM9 dL “ d ´ ´ 2πrvφT , « 2 r ff

1 GMM9 GMM9 rmin 1{2 “ d ´ ´ 1 ´ , « 2 r r r ff " ´ ¯ * 3 GMM9 GMMr9 1{2 d min , (16.42) “ ´ ` 3 2 « 2 r r { ff so in the outer part of the disk, the rate of energy release will be three times the change of potential energy of the matter crossing r per unit time.

The total luminosity will be (recalling that rmin “ 3rS)

8 L » dL, rmin ż 8 3 GMM9 GMMr9 1{2 “ ´ ` min , 2 r r3{2 « ffrmin 1 GMM9 “ , 2 rmin 1 » M,9 (16.43) 12 showing that this can be a relatively efficient source of energy (η „ 0.1).

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16.8 Thermal radiation from the disk

Suppose now that all the energy released in the accretion disk is radiated as thermal energy, and that the disk is optically thick. From Eqn. (16.42) we have

3 GMM9 3 GMMr9 1{2 dL min dr, “ 2 ´ 5 2 «2 r 2 r { ff 4 “ 2σBT p2πrdrq, (16.44) hence the temperature is given by

1{4 3GMM9 rmin 1{2 T “ 3 1 ´ . (16.45) « 8πσBr r ff " ´ ¯ *

The maximum temperature occurs at r “ 7rmin{6 and is given by

1{4 3GMM9 6 1{2 T “ 1 ´ , max 8πσ r3 7 « B # ˆ ˙ +ff 1{4 3GMM9 » 0.522 3 , «4πσrmin ff 1{4 Mc9 6 » 0.522 2 2 . (16.46) «288πσBG M ff

Setting L “ M9 {12 “ LE, the Eddington luminosity, we get

L c4 1{4 T » 0.522 E , max 24πσ ηG2M 2 „ B  m c5 1{4 » 0.522 H , 6σ ηGM „ B  M ´1{4 » 1.7 ˆ 107 K. (16.47) M ˆ d ˙ 9 For an accreting solar mass black hole, this temperature will generate X-rays, but for a 10 Md black hole, the temperature is only „ 105 K » 9 eV, not hot enough to account for the observed X-ray and gamma-ray emission.

Thermal emission from the accretion disk might account for the blue bump seen in the optical- UV as it has about the right temperature. We need to look elsewhere for the source of the X-ray and gamma radiation. Possibilities include instabilities in the disk and the release of magnetic energy.

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16.9 Electrodynamics of the accretion disk

It is likely that electric and magnetic fields will play an important role, both in the transfer of angular momentum in the accretion disk, and in the transfer of energy to charged particles and electromagnetic radiation. Here we outline a possible mechanism described in detail by Blandford (1976).

We adopt “natural” units in which ε0 “ µ0 “ c “ 1. Maxwell’s equations are

∇ ¨ B “ 0, (16.48) BB ∇ ˆ E ` “ 0, (16.49) Bt ∇ ¨ E “ ρe, (16.50) BE ∇ ˆ B ´ “ J, (16.51) Bt where ρe is the electric charge and J is the current density.

Figure 16.2: Magnetosphere of the accretion region. The accretion disk, seen edge on, is denoted by D; the black hole by C. The spin axis is vertical. The dashed line indicates (for a Keplerian disk) the position of the light surface, where the rotation speed of the field lines equals the speed of light. By this point the field lines are no longer in a vertical plane, but spiral backwards in the toroidal direction. (Blandford 1976).

At the temperatures inferred for the accretion disk, the gas will be highly ionized and will be strongly coupled to the magnetic field. As the plasma orbits the black hole, the magnetic field will be dragged along.

In the inertial (non-rotating) frame, we see an electric field

E “ γv ˆ B, (16.52)

Page 125 Galactic Astronomy 2016 which is directed inward toward the axis.

In the magnetosphere above the disk, charges move freely and electromagnetic forces dominate gravity and pressure. The electromagnetic field quickly relaxes to a configuration in which the total electromagnetic force is approximately zero

ρeE ` J ˆ B “ 0. (16.53)

The charge density satisfies Eqn. (16.50) and the magnetic field lines are equipotentials.

Each field line has a fixed angular velocity ω, so its rotational velocity is rω, where r is the radial coordinate in the cylindrical coordinate system pr, φ, zq. At a certain radius, this velocity equals the speed of light and charged particles can no longer keep up. The field line must therefore be swept back, forcing the particles to travel outwards along the field line at speeds „ c.

The electromagnetic and particle energy and angular momentum lost across the light surface must come from the accretion disk, so there is a magnetic drag that can be the main source of torque on the disk.

To be specific, suppose that the rotating accretion disk is infinitely thin and has a current density of the form ´α Jφ 9 r (16.54) By Ampere’s law, this current generates a generates a poloidal magnetic field. It can be shown that (Blandford 1976) ´α Bz 9 Jφ 9 r . (16.55) The induced electric field near the disk is

1´α Er » ωrBz 9 ωr . (16.56)

In order to make the field lines sweep back at the light surface by, say, 45˝, there must be a poloidal current that will generate a toroidal magnetic field component equal to the poloidal field there. According to Maxwell’s fourth equation (ignoring the displacement current) this current is given by the curl of the toroidal field 1 B 1 ω J “ B » ωB » B . (16.57) pls 4π Br φls 4π φls 4π pls

Flux and charge conservation require that Jpls{Bpls „ Jzd{Bzd, so the current leaving the disk is ´α Jz 9 ωr . (16.58) This current must be supplied by a radial current flowing in the disk

1´α Jr 9 rJz 9 ωr . (16.59)

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This results in a torque per unit area of τ “ r ˆ pJ ˆ Bq, hence 2p1´αq τ » rJrBz 9 ωr . (16.60) The torque must equal the rate of transport of angular momentum, therefore

2 B 2 1 1 τ “ Σv ¨ ∇pr ωq “ Σvr pr ωq » Σvrrω “ ωM.9 (16.61) Br 2 4π where vr is the radial velocity in the disk and Σ is the mass surface density. From this it follows that 2p1´αq 2πΣrvr 9 r . (16.62) The LHS of Eqn. (16.62) is equal to M9 , which is independent of radius, so the model will be self-consistent if α “ 1.

The power transferred to the magnetosphere, per unit area, is 1 F “ ω ¨ τ » ω2M.9 (16.63) 4π For a Keplerian disk, ω2 “ GM{r3, so the total power is 1 8 P » GMM9 r´32πrdr, 4π żrmin GMM9 1 “ » M.9 (16.64) 2rmin 12 We see that the efficiency can be quite high (η » 1{12).

16.10 Formation of the jet

The electric field induced by this process has a typical strength

E „ ωrBz „ Bz, (16.65) and Bz, the strength of the poloidal magnetic field at the disk, is related to the mass flux through the requirement that it produce sufficient torque to transfer the angular momentum. This gives (Blandford 1976) 1{2 M9 r ´1 L ´1 M ´1 B » » 0.3η´1{2 T, (16.66) z 8πr2 r 1039 W 109M ˜ ¸ ˆ S ˙ ˆ ˙ ˆ d ˙ which gives an electric field strength of order a volt per metre.

In this field, particles will be accelerated to TeV energies after traversing a distance, „ 1011 m, that is a fraction of a Schwarzschild radius. At these energies pair creation will dominate and a relativistic particle/electromagnetic jet will develop along the axis of symmetry.

Instabilities and shocks that develop in the beam will convert kinetic energy to radiative energy, giving rise to fluctuating gamma-ray, X-ray and optical emission.

Relativistic effects will amplify the variability and reduce the timescale for observers whose line of sight is within an angle „ γ´1 of the axis of the jet.

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Figure 16.3: Inner region of an AGN jet. The accretion disk and black hole is shown at the left. Charged particles are accelerated by an intense electric field producing a collimated outflow. X and gamma ray photons are emitted by the most energetic electrons in the inner jet, and from local regions heated by shocks that form in the supersonic flow (A. Marscher, Boston Univ).

Figure 16.4: Artists rendering of a dusty torus surrounding an accretion disk and black hole. The light blue emission from the inner walls of the torus is due to the fluorescence of iron atoms excited by X-ray emission from the hot gas (CXC/M. Weiss).

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17 Starburst and infrared-luminous galaxies

Some galaxies exhibit unusually-strong infrared emission. This results from heating of gas and dust, either by intense star formation, or by radiation from an active nucleus.

17.1 Classification

Starburst galaxies are gas-rich galaxies undergoing intense star formation. The cause of this activity is usually an encounter with another galaxy. Tidal, or pressure, forces compress the gas, triggering rapid, global star formation.

Well-known examples are M82 (Figure 2.9) and Arp 220 (Figure 17.1), which have star for- ´1 ´1 mation rates of „ 10Md yr and „ 100Md yr respectively. These are one to two orders of magnitude higher than the star formation rates found in normal spirals.

UV radiation, primarily from O and B stars produced in the burst, heats the dust giving rise to intense thermal emission in the infrared.

Figure 17.1: Arp 220 in the infrared (left) and X-ray (right). Classified as peculiar, it is the prototype ultra-luminous infrared galaxy. With a redshift of 0.018, This object is believed to be the collision of two spiral galaxies (comparable to the Milky Way). Deep optical images reveal a galaxy shrouded in dust, with faint tidal plumes indicative of an interaction or merger. The high IR luminosity comes from a massive starburst. The X-ray emission reveals high temperature gas outflow driven by supernovae. A compact source, possibly a massive black hole, is coincident with the nucleus of one of the galaxies, with a second nearby. (R. Thompson et al. U. Arizona, NASA HST NICMOS; NASA/SAO/CXC/J.McDowell) .

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The IRAS satellite opened up this field, providing fluxes in 4 wavelength bands (12 um, 25 um, 60 um and 100 um) for a large sample of galaxies. From these, luminosities are calculated according to Table 17.1.

???????? Table 17.1: Flux and luminosity definitions from the IRAS Point Source Catalog

´14 ´2 Ffir 1.26 ˆ 10 t2.58f60 ` f100u [W m ] 2 Lfir Lp40 ´ 500µmq “ 4πDLCFfirrLds ´14 ´2 Fir 1.8 ˆ 10 t13.48f12 ` 5.16f25 ` 2.58f60 ` f100u [W m ] 2 Lir Lp8 ´ 1000µmq “ 4πDLCFirrLds Lir{LB Fir{νfνp0.44µmq 11 LIG Luminous Infrared Galaxy, Lir ą 10 Ld 12 ULIG UltraLuminous Infrared Galaxy, Lir ą 10 Ld 13 HyLIG HyperLuminous Infrared Galaxy, Lir ą 10 Ld Luminosities are based on a Hubble constant of 74 km s´1 Mpc´1. The bolometric luminosity of the Sun is 3.83 ˆ 1026 W. (From Sanders & Mirabel 1996)

Most high-luminosity galaxies emit the bulk of their energy in the IR (5-500 um).

11 Luminous Infrared Galaxies (LIRGs or LIGs) have LIR ą 10 Ld. They emit more radiation in the IR than at all other wavelengths combined (Figure 17.3).

12 Ultraluminous Infrared Galaxies (ULIRGs or ULIGs) have LIR ą 10 Ld. They are comparatively rare. Arp 220 is the closest example.

17.2 Luminosity function

Galaxy samples selected by IR flux have a flatter luminosity function (Figure 17.2)

ULIRGs appear to be „ 2 times more common than optically-selected QSOs with comparable luminosities.

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17.3 Spectral properties

ULIRGs have spectral energy distributions that peak between 50 and 100 um (Figure 17.3 ). About 15% of ULIRGs have warm spectral energy distributions The excess flux in the 1-10 um region in these objects is believed to be thermal radiation from a dusty torus surrounding an active nucleus.

Spectra show a high degree of reddening (eg. large Balmer decrement, etc).

The fraction of objects showing AGN spectral characteristics increases with luminosity (Figure 17.4).

ULIRGs have submillimetre flux that is consistent with thermal emission from dust at „ 60 K.

There is a strong correlation between infrared and radio emission

F p3.75 ˆ 1012 Hz q ” log FIR » 2.35 ˘ 0.2. (17.1) F p1.49 GHzq „ ν  Most radio galaxies and quasars have low q values because of the presence of strong compact radio components and extended radio lobes that do not appear in the IR.

Xray emission from ULIGs is consistent with emission from compact objects (X-ray binaries) and „ 107 K winds associated with the starburst. AGN are sometimes detected.

17.4 Morphology

9 Nearly all E and S0 have LIR ă 10 Ld. Nearly all LIRGS are Sb or Sc galaxies.

Nearly all ULIGS are interacting/merging systems (Figure 17.5 and Figure 17.6). The fraction 10 of objects that are interacting or mergers increases from 10% at LIR 10 Ld to 100% at 12 LIR ą 10 Ld.

10 The most luminous ULIRGs contain large central concentrations („ 10 Md) of molecular gas.

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Figure 17.2: Luminosity function of infrared galaxies compared to optically-selected normal galaxies. From Sanders & Mirabel (1996).

17.5 Formation and Evolution

The high fraction of interactions and mergers found in the highest luminosity LIRGs points directly to the probable cause.

During the interaction, gas looses angular momentum and flows inward toward the centre, creating a large concentration of molecular gas. The gas fuels the intense star-formation activity that produces the infrared flux. If significant quantities of gas reach the central black holes AGN activity develops. Eventually the nuclei (and black holes) of the two galaxies merge. The strongest infrared emission appears to develop at about the same time as the nuclei merge.

Examples of well-known mergers are shown in Figure 17.7. These systems are usually found in small groups of galaxies, not large clusters. In clusters the encounter velocities are too high and most galaxies have already lost their gas by ram-pressure stripping.

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Figure 17.3: Optical-IR spectral energy distributions for a sample of 60 um-selected ultra- luminous infrared galaxies. About 15% have warm SEDs, as shown in the insert (Sanders & Mirabel 1996).

Figure 17.4: Distribution of spectral types as a function of infrared luminosity (Sanders & Mirabel 1996).

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Figure 17.5: Optical contours of 10 ULIRGs. All show distinctive features of interactions or mergers (Sanders & Mirabel 1996).

Figure 17.6: Optical images of three IR-luminous QSOs: a) Mrk 1014, b) 4C 12.50 , c) 3C 48 (Sanders & Mirabel 1996).

It is possible that ULIRGs evolve into QSOs or radio galaxies as gas and dust are consumed by the starburst. Eventually they may resemble elliptical galaxies. About 50% of merger remnants have K-band profiles that are well-fit by de Vaucouleurs law (Wright et al 1990, Stanford & Bushouse 1991, Kim 1995).

Gravitational lensing may be important for high-redshift galaxies. The extremely-luminous 14 galaxy IRAS 10214+4724 (z “ 2.286, LIR „ 2 ˆ 10 Ld) appears to be lensed (Figure 17.10).

?

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Figure 17.7: Examples of merging galaxies: a) Arp 244, the “Antennae”, b) Arp 226, “Atoms for Peace”, c) IRAS 19254-7248, d) Arp 220. H I 21-cm contours are superimposed on deep r-band images. Inserts show K-band images for all but Arp 220 and an r-band image for the latter. White contours delineate CO emission. The bar indicates 20 kpc (from Sanders & Mirabel 1996).

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Figure 17.8: Optical image of the central region of the Antennae. The two nuclei are bridged by gas and dust lanes. Blue stars highlight H II regions and areas of rapid star formation. (NASA HST).

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Figure 17.9: Optical and X-ray image of the ultraluminous infrared galaxy NGC 6240. The Chandra image reveals X-rays emanating from two massive black holes near the centre of the galaxy. (MPE, W. Keel).

Figure 17.10: IRAS 10214-4724, a ULIRG with z “ 2.286. The small CO linewidth (250 km/s) for such an apparently luminous object, and the arclike geometry suggest that this object has been lensed by a foreground galaxy or group (Sanders & Mirabel 1996).

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18 Groups and Clusters of galaxies

Few galaxies are truly isolated. Most are found in small loose associations of galaxies such as the Local Group. These system have a low mean density but are gravitationally bound. Their dynamical times are comparable to the Hubble time. Occasionally, subgroups form that can have very high density and short dynamical times. These dense groups are the primary sites of galaxy interactions and mergers.

One also finds clusters of hundreds or even thousands of galaxies. Rich clusters are massive bound systems in which galaxies orbit at high relative speeds. These systems are the typical home of elliptical galaxies and S0s, which have lost their gas to the intra-cluster medium by a variety of physical processes.

Figure 18.1: Abell 2218, a rich cluster containing a central cD galaxy and numerous gravita- tionally lensed arcs. (NASA HST).

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18.1 Morphology and taxonomy

See Abell’s article in Stars & Stellar Systems (Abell 1975). Also, see the extensive set of review articles at nedwww.ipac.caltech.edu/level5/cog.html.

18.1.1 Cluster identification

In the early 1900s large associations of nebulae such as those found in the Coma, Fornax and Ursa Major constellations were identified. Hubble established the nature of these objects as galaxies in 1924.

The distribution of bright galaxies in the Shapley-Ames catalog (1932) revealed at least four clusters. Shapley (1933) later catalogued 25.

Hubble (1934) studied the distribution of galaxies over the sky and estimated that there was at least one great cluster for every 50 square degrees of sky. If each contain on the average 500 galaxies, this would make up „ 1% of the galaxy population.

Zwicky (1937) noted that the distribution of galaxies in the Coma cluster resembled the distribution of stars in a globular cluster. Reasoning that these were relaxed and stationary systems, he applied the virial theorem to estimate the mass of the Coma cluster.

The first major photographic sky survey was the Lick Astrographic Survey conducted with the 20-inch telescope. Galaxy counts on the survey plates were used to determine galaxy surface densities (eg. Shane & Wirtanen 1954, 1967). Statistical analysis detected clustering on scales as large as 4˝. They concluded that most clusters contain „ 102 galaxies have have diameters of a few Mpc (Neyman, Scott & Shane, 1953, 1954). Limber (1953, 1954) and Layzer (1956) also performed independent analyses of the Lick counts using different techniques.

The Palomar Observatory Sky Survey (POSS) provided an extensive resource with which to study the distribution of galaxies. Abell (1958) catalogued 2712 of the richest clusters found on the POSS plates.

Subsequently, Zwicky and collaborators compiled a very extensive Catalog of Galaxies and Clusters of Galaxies based on the POSS plates (Zwicky, Herzog, Wild, Karpowicz & Kowal 1961-1968)

The ACO catalogue (Abell, Corwin and Olowin 1989), based on the POSS and the UK Schmidt photographic surveys, contains 4073 rich clusters of galaxies over the entire sky.

Redshift surveys (eg. Geller & Huchra 1983, Tully 1987) identified bound systems of galaxies in 3-d. These studies indicate that most galaxies occur in small groups

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18.1.2 Classification schemes

Zwicky et al (1961-67) define three classes:

• compact contains a pronounced concentration of 10 or more galaxies that appear to be in contact (when seen in projection) • medium compact contains a central concentration of galaxies that appear separated by just a few galaxy diameters • open no pronounced central concentration of galaxies

Morgan (1961):

• Clusters containing many galaxies having minor central concentration of light (late-type spirals, irregulars, etc) • Clusters containing few or no such galaxies

Abell (1965, 1975):

• regular rich clusters of „ 103 or more galaxies (within 6 mag of the brightest) with a pronounced central concentration and spherical symmetry. These clusters contain few late-type galaxies. • irregular range from poor groups to large aggregates of galaxies like the Virgo cluster, that have an irregular shape, sometimes with multiple concentrations of galaxies. These clusters contain many late-type galaxies.

Bautz & Morgan (1970):

• Type I clusters containing a central cD galaxy (eg A2199) • Type II clusters containing one or more D galaxies (eg Coma) • Type III clusters containing no dominant galaxies

Rood & Sastry (1971) tuning fork classification of rich clusters:

• cD contain a central cD galaxy (eg. A2199) • B contain a pair of D galaxies (eg. Coma) • L have a linear distribution of bright galaxies (eg. Perseus) • C show a core-halo distribution of faint galaxies around a central concentration (eg. Corona Borealis) • F a flattened distribution of galaxies (eg. A397) • I an irregular distribution of galaxies (eg. Hercules)

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Figure 18.2: Rood and Sastrys (1971) Tuning fork classification of rich clusters, based largely on the configuration of the 10 brightest galaxies.

18.2 X-ray observations

See Mulchaey (2000) for a review.

Figure 18.3: Rosat image showing diffuse X-ray emission in the NGC 2300 group of galaxies, superimposed on an optical image (NASA).

The advent of X-ray satellites, particularly Einstein, Rosat, ASCA, XMM-Newton and Chan- dra, opened up a new window on clusters of galaxies by detecting directly their hot baryonic

Page 141 Galactic Astronomy 2016 component.

The Rosat all-sky survey revealed that „ 1{2 of nearby groups and clusters contain diffuse X-ray emission from hot intra-cluster gas.

Groups that contain elliptical galaxies are more likely to have diffuse X-ray emission. The radial extent of the X-ray emission is typically 10 ´ 50% of the virial radius of the cluster (0.1 ´ 1 Mpc).

The gas temperature ranges from kT „ 0.3 keV to 2 keV (106 ´ 107 K). The gas metalicity ranges from „ 5% solar in small groups to „ 30% solar in rich clusters

Figure 18.4: Rosat image of the Virgo clusters. The X-ray emission arises from a compact core centered on M87 and an irregular extended region (Bohringer et al. 1994).

18.3 Compact groups

For a review see Hickson (1997).

Compact groups are the densest galactic systems in the Universe. They typically contain four to 10 bright galaxies within a few galaxy radii of each other. The first examples were found by accident (eg. Stefan 1877, Seyfert 1948).

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Figure 18.5: Stephans Quintet. Discovered by E. M. Stephan (1888), it is the first compact group known. The largest galaxy towards the lower right has a much lower redshift than the others and is presumably in the foreground.

Spectroscopic studies (Burbidge & Burbidge 1959, 1961) revealed velocity dispersions that were much higher than would be expected from the mass of the galaxies alone. It was pro- posed by some that they are transient systems either expanding or collapsing, or just chance projections of galaxies.

The relative velocities of galaxies in these groups is comparable to their internal stellar veloc- ities making them particularly prone to strong tidal interactions and mergers.

Hickson (1982) searched the POSS prints for systems similar to the classical compact groups, but using specific selection criteria, and identified a sample of 100 such groups. Since the se- lection criteria are known, biases can be quantified, making this sample suitable for statistical analysis. This approach has since been extended to the Southern Hemisphere (Iovino 2002)

18.4 Structure and galaxy content

The surface density of galaxies in rich regular clusters can be adequately represented by King models, or de Vaucouleurs profiles. There is a tendency for the very brightest galaxies to be

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Figure 18.6: Seyferts Sextet. Discovered in 1948, this system of 4 interacting galaxies is one of the densest known (Seyfert 1951). The small spiral has a much higher redshift and is presumably a background object. (NOAO)). found preferentially in a central core. Other than this there is little or no evidence for general luminosity segregation.

Clusters contain fewer spiral galaxies than do samples of isolated galaxies or loose groups. About 80% of such field galaxies are spirals.

Spiral galaxies are common in irregular clusters but rare in regular ones. Dressler (1980) has quantified this as a correlation between galaxy type and local space density called the morphology-density relation (Figure 18.8).

Loose groups of galaxies are spiral-rich, with a spiral fraction comparable to that of field galaxies. Compact groups typically contain fewer spirals. One finds that on average 50% of the galaxies are E or S0 types.

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Figure 18.7: VV 172. Optical image of the chain of galaxies discovered by Vorontsov- Velyaminov (1959, 1972). The second galaxy from the top has a redshift much higher than the other members.

Compact groups do not show a significant morphology-density relation, but instead exhibit a strong correlation between galaxy morphology and the group velocity dispersion (Hickson et al. 1988)

The majority of galaxies in compact groups are interacting, showing morphological peculiar- ities such as tidal tails, distorted isophotes, kinematic peculiarities, etc.

Elliptical galaxies in compact groups tend to lie off the fundamental plane. They have lower internal velocity dispersions that do other ellipticals with similar effective radii, magnitudes and colours. They are also more likely to have irregular isophotes. Almost all of them show nuclear emission lines.

The seemingly-high fraction of discordant redshifts found in compact groups is in fact con- sistent with chance projections of foreground and background galaxies. Approximately 3% of the blue luminosity density of the Universe (ie. starlight) is found in galaxies in compact groups.

Active or starburst galaxies are often found in compact groups. For example, HCG 16 contains a Seyfert 2 galaxy, two LINERs and three starburst galaxies. These types of galaxies are rarely found in rich clusters.

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The radial velocity dispersions range from ă 100 km s´1 in small groups to „ 103 km s´1 in rich clusters. Velocity dispersions as high as 500 km s´1 are found in compact groups.

Figure 18.8: The morphology density relation in a sample of 55 rich clusters. The fractions of E and S0 galaxies increases with the local surface density (Dressler 1980).

X-ray observations reveal that diffuse hot gas permeates rich clusters and many groups of galaxies. This X-ray emission results primarily from thermal bremsstrahlung.

The gas is assumed to be in hydrostatic equilibrium and isothermal. This is probably a good assumption for regular clusters, but not for irregular ones. Also, there is evidence that the gas temperature varies significantly in groups.

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Figure 18.9: The morphology-velocity correlation in compact groups. Curves show the cumu- lative distribution of velocity dispersion for spiral-rich (fs ą 0.5) and spiral-poor (fs ă 0.5) groups. The latter have about twice the median velocity dispersion as the former (Hickson 1997).

Since the gas and the galaxies share the same gravitational potential, the equations of hydro- static equilibrium are

dPgas dΦ “ ´ρ , (18.1) dr gas dr dPgal dΦ “ ´ρ . (18.2) dr gal dr

The equations of state for gas and galaxies are

ρgaskT Pgas “ , (18.3) µmH 1 P “ ρ v2 “ ρ σ2, (18.4) gal 3 gal gal where ν is the mean molecular weight and σ is the 1-dimensional galaxy velocity dispersion.

Combining these two sets of equations gives

kT d ln ρgas d ln ρgal “ σ2 . (18.5) µmH dr dr

Integrating, we get β ρgasprq ρgalprq “ (18.6) ρ p0q ρ p0q gas „ gal 

Page 147 Galactic Astronomy 2016 where the parameter β is the ratio of the specific energy in galaxies to the specific energy in hot gas, µm σ2 β “ H (18.7) kT The galaxy distribution is well-represented by King’s (1962) approximation to the isothermal sphere (same as the modified Hubble profile),

2 ´3{2 ρgal “ ρgalp0q 1 ` pr{rcq (18.8) where rc is the core radius. The coresponding“ gas density‰ is therefore 2 ´3β{2 ρgas “ ρgasp0q 1 ` pr{rcq . (18.9) The X-ray emission is proportional to the square“ of the‰ gas density, so 2 ´3β jprq “ jp0q 1 ` pr{rcq (18.10) Projecting this to two dimensions gives the X-ray“ inensity‰ profile 2 ´3β`1{2 Iprq “ Ip0q 1 ` pr{rcq (18.11) This is called the beta model. “ ‰ Beta-model fits to the X-ray data (eg. Jones & Forman 1984) give estimates of the gas mass and temperature. Temperatures range from 0.3 to 2 keV, increasing with the richness of the cluster (18.10).

Strong correlations are found between X-ray temperature, luminosity and cluster velocity dispersion (eg. Figs. 18.11 and 18.12). However, there is at present no consensus about the exact form of these correlations.

The total cluster mass can be estimated as follows. Hydrostatic equilibrium gives ∇P “ ´ρ∇Φ. (18.12) Taking the divergence of this and using the gas equation of state and Poisson’s equation, we get 1 ∇ ¨ ∇P “ ´∇2Φ “ ´4πGρ (18.13) ρ tot where ρtot is the total mass density (gas, galaxies, dark matter...). If we now assume spherical symmetry, the total mass contained within radius r is r 2 Mprq “ 4π ρtotr dr, ż0 1 r 1 d r2 dP “ ´ r2 dr, G r2 dr ρ dr ż0 „  k r r2 dpρT q “ ´ d , Gµm ρ dr H ż0 „  kr2 1 dpρT q “ ´ , GµmH ρ dr krT rdρ rdT “ ´ ` , (18.14) Gµm ρdr T dr H „ 

Page 148 Galactic Astronomy 2016 so we finally obtain kT prqr d log ρ d log T Mprq “ ´ ` . (18.15) Gµm d log r d log r H „ 

These mass estimates agree with those obtained by applying the virial theorem to the velocity dispersion of the galaxies (Fabricant, Rybicki & Gorenstein1984).

One finds that groups and clusters contain a gas mass that is at least as large as the mass of galaxies in groups, and several times the mass of galaxies in rich clusters.

Figure 18.10: Gas temperature vs radial extent of X-ray emission. Triangles represent rich clusters and circles represent galaxy groups (Mulchaey 2000). The total baryonic mass (gas plus galaxies) is of order 10 ´ 20% of the total mass estimated from the virial theorem, or from the X-ray data.

18.5 Evolution of galaxies in clusters and groups

A number of dynamical processes affect galaxies in groups and clusters:

Ram pressure stripping

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Figure 18.11: Correlation between gas temperature and galaxy velocity dispersion in groups (circles) and clusters (triangles). This is expected as both parameters increase with mass (Mulchaey 2000).

Ram pressure stripping is a dynamical effect that occurs primarily in rich clusters. Galaxies moving through the hot intracluster gas feel a wind which removes their intergalactic gas. This process can convert spiral galaxies in anaemic galaxies, or even into S0s. It is likely that this is responsible for the high fraction of S0 galaxies in rich clusters.

2 The dynamic pressure exerted by the intracluster gas is P “ ρigmv {2, where v is the galaxy velocity and ρigm is the mass density of the intra-cluster gas. If this exceeds the gravitational force, per unit area, binding an interstellar gas cloud, gas will be swept out of the galaxy. Thus the condition for ram pressure stripping is 1 ρ v3 ą ρ Lv ¨ ∇Φ, (18.16) 2 igm ism where ρism is the density, and L a characteristic length, of an interstellar cloud.

Ram pressure stripping is believed to be the mechanism that removes gas from spiral galaxies as they fall into a cluster, converting them to S0 galaxies. An example of this effect can be

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Figure 18.12: Correlation between X-ray luminosity and galaxy velocity dispersion in groups (circles) and clusters (triangles). This is expected as both parameters increase with mass (Mulchaey 2000). seen in NGC 4522, which is falling into the Virgo cluster. Multicolour imaging (Sivanandam, Rieke & Rieke, 2014), shows a displacement of thermal emission and a dust trail behind the galaxy (Figure 18.13).

Induced star formation

This process occurs in clusters and groups. Galaxy-galaxy interactions and possible inter- action with the intracluster medium can trigger star formation. This star formation may eventually deplete the gas in the galaxy converting it into an S0. This is a major effect in compact groups. There is some evidence that this may be occurring on the outskirts of clusters

Galaxy harassment

Repeated encounters with small and/or fast galaxies can strip the stars from the outer parts of galaxies causing them to become more compact. Energy can be added or removed from stellar orbits, changing the structure of the galaxy

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Figure 18.13: Three-color composite image of NGC 4522. North is up and east is left. Blue is HST F435W data, green is HST F814W data, and red is IRAC 8 um excess emission data. This image clearly shows that there is a significant amount of dust stripping occurring in this galaxy. There are a few extraplanar 8 um excess sources that are either offset from their optical counterparts in the direction of the ICM wind or with no optical counterparts at all in the stripped dust trail immediately west of the southern end of the galaxy. (Sivanandam, Rieke & Rieke 2014).

Tidal distortion

Close and/or slow encounters can cause severe distortion of galaxies as stars are pulled out into tidal tails. Interactions can shock the gas, causing star formation. Gas can loose angular momentum in the interaction and flow towards the centre of the galaxy. Most important in compact groups. Less frequent in clusters

Mergers

Close binary encounters can result in merging of the two galaxies. This typically requires several orbital periods. The galaxy nuclei loose energy via dynamical friction and spiral inward, eventually merging. This is most important in compact groups and at the centre of rich clusters.

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Gas in the galaxies is consumed in a massive starburst. As the newly-formed stars age, the relic begins to resemble an elliptical galaxy. The initially-blue colours fade as the stars evolve. Such galaxies may exhibit spectral features of hot stars in their spectrum, (Figure 18.14). After approximately 109 yrs, the SED looks like that of a normal elliptical galaxy.This process may account for the fact that elliptical galaxies are found predominantly in high-density environments.

In compact groups, the process can lead to isolated, or nearly isolated elliptical galaxies. If the group contained a significant amount of hot gas, one could end up with an elliptical galaxy surrounded by an X-ray halo (Figure 18.15). Several such objects have been found (Ponman et al 1994, Jones et al. 2003).

Figure 18.14: Spectrum of the post-starburst galaxy SDSS J101345.39+011613.06, showing an “E+A” spectrum indicating the presence of both old and youg stellar populations (Swinbank et al. 2005).

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Figure 18.15: RXJ1331.5+1108. An isolated elliptical galaxy embedded in a diffuse X-ray halo (Jones et al. 2003).

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Dynamical evolution of clusters and groups

Clusters of galaxies are so large that their dynamics needs to be considered within the context of the overall dynamics of the Universe. Initially, the gas, dark matter and eventually galaxies are participating in the overall expansion of the Universe. Eventually, their excess density stops their expansion and they collapse. Much can be learned about this general scenario by considering a tractable special case the collapse of a homogenous sphere within a uniform background.

18.6 Spherical collapse

Consider a homogeneous sphere of density ρptq and radius Rptq that is initially expanding with the Universe. The time evolution of the radius of this sphere is governed by the equation GMpRq R: “ ´ , (18.17) R2 where M “ 4πR3ρ{3 is the mass contained within the sphere. Multiplying this equation by 2R9 and integrating, we obtain 2GM R9 2 “ ´ k, (18.18) R where k is a constant of integration. Substituting for M and dividing by R2 gives the Fried- mann equation R9 2 8πG k H2 ” “ ρ ´ , (18.19) R2 3 R2 This analysis ignores the effects of a cosmological constant. The implications of this assump- tion will be discussed later.

2 If the density is equal to the critical density ρc ” 3H {8πG, it follows from Equation (18.42) that k = 0. If the density is dominated by non-relativistic matter with pressure P ! ρ, the first law of thermodynamics requires that ρ 9 R´3 if the evolution is adiabatic.

Equation (18.42) can then be written as 8πG RR9 2 “ ρ R3 ” A2. (18.20) 3 0 0 where ρ0 and R0 are the density and radius at some early time t0. This has the solution

3 2{3 R “ At (18.21) 2 ˆ ˙ where t is measured from the Big Bang.

Therefore 2 Hptq “ (18.22) 3t

Page 155 Galactic Astronomy 2016 and the critical density evolves as 3H2 1 ρ “ “ . (18.23) c 8πG 6πGt2 Now suppose that the sphere has density slightly greater than critical density

ρ “ ρcr1 ` δptqs (18.24)

From the Friedmann equation 8πGR2 k “ ρ r1 ` δs ´ H2R2 “ H2R2δ (18.25) 3 c which is clearly positive, k ą 0. The Friedmann equation can now be written

RR9 2 ` kR “ A2. (18.26)

This equation has the parametric solution A2 R “ p1 ´ cos ηq, (18.27) 2k A2 t “ pη ´ sin ηq. (18.28) 2k3{2 which is the equation of a cycloid.

2 2 3{2 We see that the sphere expands to a maximum radius Rmax “ A {k at time tmax “ πA {k and collapses to high density an equal time later. The density is

3 2 3 ρ0r 3A 2k ρ “ 0 “ , R3 8π A2p1 ´ cos ηq „  3k3 “ . (18.29) πA4p1 ´ cos ηq3 Now we also have 1 ` δ ρ “ ρ p1 ` δq “ , c 6πGt2 2k3p1 ` δq “ . (18.30) 3πGA4pη ´ sin ηq2 Therefore 9 pη ´ sin ηq2 1 ` δ “ . (18.31) 2 p1 ´ cos ηq3

At some point in the expansion, galaxies form from the matter, so the expanding sphere will cluster of galaxies when it collapses. At maximum expansion, η “ π. From Equation (18.31), the density ratio between the sphere and the background is then 9π2 1 ` δ “ » 5.55. (18.32) 16

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If δ is small, so is η and we can expand as follows

9 η ´ η ` η3{6 ´ η5{120q2 p1 ´ η2{20q2 1 ` δ » “ , 2 1 ´ 1 ` η2{2 ´ η4{24q3 p1 ´ η2{12q3 » p1 ´ η2{10p1 ` η2{4q » 1 ` 3η2{20. (18.33)

Therefore, for small δ, 3 δ » η2. (18.34) 20

The time of collapse of a sphere whose overdensity at some early time t0 is δ0 ! 1 is then given by

5{2 2π 12π 3 π ´3{2 tc “ t0 » t0 » δ t0, η ´ sin η η3 2 ¨ 53{2 0 ´3{2 » 2.19 δ0 t, (18.35) which shows that regions of high overdensity collapse sooner than regions of low overdensity.

18.7 Virialization

At maximum expansion, the sphere has no kinetic energy. The total energy is therefore equal to the potential energy 3GM 2 Umax “ (18.36) 5Rmax

During collapse, the potential fluctuates rapidly, resulting in a redistribution of energy among the galaxies by “violent relaxation” (Lynden-Bell 1967). The cluster relaxes to a smooth distribution resembling an isothermal sphere.

After relaxation, the cluster shape is roughly stationary so the virial theorem holds. We see that

2K ` U “ 0, K “ ´U{2, E “ K ` U “ U{2, (18.37) so the potential energy after virialization is

U » 2Umax. (18.38)

This shows that the radius of the cluster after collapse is about half its radius at maximum expansion, R » Rmax{2.

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18.8 Post-collapse infall

If the cluster contains total mass M, and the background has critical density, then a particle outside the sphere, at radius r sees a mean density interior to it of 3M ρ “ ρ ` . (18.39) c 4πr3 The overdensity is therefore 3M 2MG ´3 δprq » 3 “ 2 r . (18.40) 4πr ρc H A thin shell of matter at this radius will therefore fall into the cluster at a time given by Equation (18.35). From this we see that there is a continual rain of matter into the cluster.

The material close to it (for which δ is greatest) falls in first, followed by matter at larger radius, etc. This process slowly increases the mass, and richness of the cluster.

If the material surrounding the cluster is also clustered, we have a situation in which large clusters consume smaller ones that have formed near it. Small clusters grow by accreting smaller neighbours. This is the hierarchical clustering scenario.

18.9 Effect of a cosmological constant

The presence of a positive cosmological constant, also called dark energy, indicated by recent observations of distant supernovae, has a profound effect on gravitational clustering.

A cosmological constant Λ is equivalent to a constant vacuum energy density Λ ρ “ . (18.41) Λ 8πG Including this in Equation (18.42 gives 8πG k Λ H2 “ ρ ´ ` , (18.42) 3 R2 3 When the mean density drops below the vacuum density, ρ ă ρΛ, Friedmann’s equation approaches the limit R9 Λ 1{2 » , (18.43) R 3 ˆ ˙ which has the exponentially-growing solution R 9 expp Λ{3tq (18.44)

Observations indicate that at the present time,aρΛ „ 2.7ρ. The matter density is related to redshift z by ρ 9 p1`zq3, so the Λ-dominated epoch began at a redshift of z „ 2.71{3´1 „ 0.39. At z ą 1, ρ ą 2ρΛ and we expect that the cosmological constant (dark energy) had little effect on the dynamics.

Beginning around z „ 0.4, the cosmological constant begins to dominate, delaying or prevent- ing the collapse of over-dense regions. At the present time, for example, an over-density of δ „ 2 would be required just to equal the cosmological-constant term. Regions having mean density less than „ 3 times the critical density will not collapse at all.

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18.10 Clustering hierarchy

It is apparent from the observations that there is a continuous range of sizes and populations of clusters, ranging from just a few galaxies to thousands. It is therefore useful to seek a scale- free description of clustering. These are provided by power spectral analysis and covariance analysis.

Suppose that we locate ourselves on a galaxy, then ask the question What is the probability of finding another galaxy with a distance in the range pr, r ` drq. This probability can be written as dP “ 4πnr¯ 2drr1 ` ξprqs (18.45) wheren ¯ is the mean number of galaxies per unit volume. The first term on the right hand side corresponds to the Poisson probability for a uniform (unclustered) random distribution of galaxies. The second term is an additional contribution arising from clustering.

The function ξprq is called the spatial 2-point correlation function of the galaxy distri- bution.

The observational data are well fit by a power-law form

r ´γ ξprq “ , (18.46) r ˆ 0 ˙ with γ » 1.77. The parameter r0 is the distance at which the probability is twice the Poisson probability. Typically, r0 „ 4 Mpc.

In a similar way, one can define an angular 2-point correlation function, wpθq, which describes the excess probability of finding a galaxy, seen projected on the sky, at a given angular distance θ from another. This is easier to measure than ξprq as redshifts are not required (Figure 18.16).

The angular and spatial covariance functions are related by Limber’s equation (Peebles 1980). For the power law of Equation (18.46), one has

θ 1´γ wpθq “ (18.47) θ ˆ 0 ˙

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Figure 18.16: Angular 2-point correlation function for galaxies in the SDSS within the mag- nitude range 18 ă r1 ă 22 (Connolly et al. 2002).

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Galaxy Formation

This section provides a very brief overview of the growth of structure in the Universe. Unless otherwise indicated, we adopt units in which the speed of light c “ 1.

18.11 Cosmological model

One assumes that at early times the Universe was very nearly homogeneous and isotropic. Good evidence for this comes from the isotropy of the cosmic microwave background radiation. The growth of structure can then be examined by considering small perturbations to the homogeneous background density.

It is convenient to distinguish distances from coordinates. The latter are fixed to the matter and the distance between them increases with time. This is represented by the relation

ds “ aptqdx (18.48) where ds is the distance corresponding to an infinitesimal coordinate difference dx. The parameter aptq is called the scale factor and the coordinates are called comoving coordinates.

One can show (Friedmann, 1922; Roberson 1935, 1936; Walker 1936) that the most general form for the metric of a homogeneous isotropic three-dimensional space is

dr2 ds2 “ a2ptq ` r2dΩ2 , where (18.49) 1 ´ kr2 ˆ ˙ dΩ2 “ sin2 θdθ2 ` dφ2, (18.50) which is called the Friedmann-Robertson-Walker line element (FRW). Here k is a constant that represents the curvature of the space (it equals the Gaussian curvature when a(t) = 1).

In four dimensions, the line element becomes

dτ 2 “ dt2 ´ ds2 (18.51) where τ is proper time (the time measured by clocks that are locally at rest and therefore have constant comoving coordinates). This implicitly defines the components of the metric tensor gij via 2 i j dτ “ gijdx dx , (18.52) where ~x “ pt, r, θ, φq. (Note: we are using a signature p`, ´, ´, ´q for the metric tensor. The alterative p´, `, `, `q, also commonly used, results in some sign changes in the equations for Tij and Rij below.)

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Matter and energy are represented by the energy-momentum tensor for a perfect fluid having density ρ and pressure P in its rest frame.

Tij “ pρ ` P quiuj ´ P gij. (18.53)

Here ui are covariant components of the four velocity dxj u “ g . (18.54) i ij dτ The gravitational dynamics is governed by Einstein’s equations 1 R ´ Rg ´ Λg “ 8πGT . (18.55) ij 2 ij ij ij

Rij is the Ricci tensor, a geometrical quantity related to the curvature. It is a function of the metric tensor, which serves as a generalized gravitational potential. The scalar curvature ij R “ g Rij is the trace of the Ricci tensor.

Substituting the FRW metric, and the above energy-momentum tensor, into Einstein’s equa- tions gives two Friedmann equations,

a9 2 8πG k Λ “ ρ ´ ` , (18.56) a2 3 a2 3 a: 4πG Λ “ ´ pρ ` 3P q ` . (18.57) a 3 3

The mass/energy density ρ can be separated into nonrelativistic matter, ρm and radiation (relativistic matter) ρr, ρ “ ρm ` ρr. (18.58)

The Hubble parameter is defined by a9 H “ (18.59) a and its present value H0 is called the Hubble constant.

One can define dimensionless density parameters,

8πGρm Ω “ , (18.60) m 3H2 8πGρr Ω “ , (18.61) r 3H2 k Ω “ ´ , (18.62) k 3H2 Λ Ω “ , (18.63) Λ 3H2 (18.64)

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In terms of these, the first Friedmann equation takes the simple form

Ωm ` Ωr ` Ωk ` ΩΛ “ 1. (18.65)

We adopt the so-called ΛCDM model, with the following parameters taken from the 7-year Wilkinson Microwave Anisotropy Probe (WMAP) data combined with supernova and baryon acoustic oscillation results (Jarosik et al. 2011),

´1 ´1 H0 “ 70.4 km s Mpc , (18.66)

T0 “ 2.73 K, (18.67)

Ωm0 “ 0.27, (18.68)

Ωλ0 “ 0.73, (18.69)

Ωk0 “ 0.00, (18.70) (18.71)

The subscript ‘0’ denotes the present value of a time-varying parameter.

The radiation density is related to the temperature by the black body relation,

4 ρr “ 4σBT , (18.72) where σB is the Stefan-Boltzmann radiation constant.

The cosmological redshift is related to the scale factor by

a0 1 ` z “ . (18.73) a The radiation and matter densities obey the scaling relations

3 ρm “ ρm0p1 ` zq , (18.74) 4 ρr “ ρr0p1 ` zq , (18.75) provided that the number of particle species does not change. (The extra factor of 1 ` z in the radiation density is due to the energy loss per particle from the redshift.)

From Equations (18.72) and (18.75) we see that

T “ T0p1 ` zq. (18.76)

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18.12 Radiation dominated era

Because ρr has the strongest dependence on 1 ` z, the radiation term will initially dominate the others on the RHS of the Friedmann equation. In this radiation dominated era, the Friedmann equation becomes

a9 2 8πG “ ρ a4 a´4 (18.77) a 3 r0 0 ˆ ˙ ˆ ˙ which is readily solved to give

1{4 a 32πGρr0 “ t1{2, (18.78) a 3 0 ˆ ˙ 1 3 1{2 t “ “ p1 ` zq´2. (18.79) 2H 32πGρ ˆ r0 ˙

We have implicitly assumed that photons are the only relativistic particles. This is not the case in the early Universe, because neutrinos are relativistic. The energy density of each species of neutrino is equal to 7/8 of the number density of photons. This factor arises because neutrinos obey Fermi-Dirac statistics and photons obey Bose-Einstein statistics.

At very early times, neutrinos were in equilibrium with photons, but as the Universe expanded reaction rates fell (due to decreasing density) and the neutrinos decoupled from the photons. When the temperature dropped below the rest-mass energy of the electron, electrons and positrons annihilated. This increased the number of photons, boosting the energy density of the photons, and therefore the temperature.

Before annihilation, the entropy per photon is 2ˆ7{8`1 “ 11{4 (corresponding to an electron, positron and photon). The annihilation reaction reaction is reversible and therefore entropy is conserved. After annihilation, the entropy per photon is increased by a factor of 11{4. Since entropy is proportional to T 3, the temperature of the photons is now higher than that of the neutrinos by a factor of p11{4q1{3.

The total energy density is obtained by adding the energy density of the neutrinos to that of the photons. Since there are three neutrino species, the energy density becomes

7 4 4{3 ρ “ 1 ` 3 ˆ ρ p1 ` zq4, (18.80) r 8 11 r0 « ˆ ˙ ff 4 » 1.68ρr0p1 ` zq . (18.81)

Therefore in the radiation era, 1{4 T » 1.68 T0p1 ` zq. (18.82)

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18.13 Matter dominated era

As the expansion proceeds, the radiation density decreases faster than the matter density, by a factor of 1 ` z. The two become equal at time teq. One can estimate the corresponding redshift found from Equation (18.80),

4{3 ´1 21 4 ρm0 1 ` z “ 1 ` , (18.83) 8 11 ρ « ˆ ˙ ff r0 ´1 4{3 2 21 4 3H Ωm0 “ 1 ` 0 , (18.84) 8 11 32πGσ T 4 « ˆ ˙ ff B 0 » 3200. (18.85)

The matter-dominated era begins at teq, and continues until the cosmological term takes over, at a redshift z „ 0.4. If we ignore the radiation and curvature terms in the Friedman equation, but include the cosmological term, the Friedmann equation is

a9 2 8πG Λ “ ρ a3 a´3 ` . (18.86) a 3 m0 0 3 ˆ ˙ ˆ ˙ This equation has the solution

3 2 ´1 p1 ´ Ωm0qa t “ ? sinh 3 (18.87) 3H0 1 ´ Ω0 «d Ωm0a0 ff Thus we have a relation between time and redshift,

2 ´1 1 ´ Ωm0 tpzq “ ? sinh 3 . (18.88) 3H0 1 ´ Ω0 «dΩm0p1 ` zq ff

This equation provides a convenient way of relating cosmic time tpzq, and lookback time 5 t0 ´ tpzq, to redshift, for all but the first 10 years or so of cosmic time.

18.14 Recombination

When the Universe has cooled to a temperature of „ 3000 K, electrons and protons combine to form neutral hydrogen. This period of “recombination” occured at a redshift of z „ 3000{2.73 ´ 1 » 1100, which corresponds to a time (from Equation 18.88) of tr » 500, 000 yr.

Prior to recombination, the mean free path of photons was small, due to electron scattering. At recombination, the Universe became transparent and photons propagated freely. We see them today as the cosmic microwave background radiation (CMB).

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18.15 Density fluctuations

Small fluctuations in the density of matter in the early Universe grow and eventually form the galaxies, and dark matter halos, that we see today. In the early Universe, fluctuations occur on all spatial scales and the density contrast increases with time.

The expansion stretches the spatial scales of all fluctuations, so it is convenient to use co- moving coordinates x “ px, y, zq, in which the scales do not change.

Most theories predict that density fluctuations will be Gaussian. This means that the distri- bution of fluctuation amplitudes ρpxq δpxq “ ´ 1. (18.89) hρi has a Gaussian distribution. Exceptions to this are theories that contain topological defects, such as cosmic strings, which give rise to non-Gaussian fluctuations.

18.16 Density power spectrum

In the early universe, density fluctuations form a random field. Although the locations and amplitudes of individual fluctuations cannot be predicted, statistical properties can. If the fluctuations are Gaussian (ie. a Gaussian distribution of amplitudes at any given time), they are uniquely defined by their mean and power spectrum. The spatial Fourier transform of the density field, with respect to the co-moving coordinates, gives the spectral content. Define the Fourier component of the density fluctuation by

´ik¨x 3 δkpkq “ δpxqe d x. (18.90) ż where k is the co-moving wave vector. We shall not worry about whether the integral con- verges. More rigour can be found in text books such as Peebles (1993).

The reciprocal relationship is 1 δpxq “ δ pxqeik¨xd3k. (18.91) p2πq3 k ż For Gaussian fluctuations, different spectral modes are independent, so the ensemble average of two Fourier components has the form

˚ 3 3 1 hδkδk1 i “ p2πq P pkqδ pk ´ k q (18.92) where the delta on the RHS is the three-dimensional Dirac delta function, 1 δ3pkq “ eik¨xd3x. (18.93) p2πq3 ż

Page 166 Galactic Astronomy 2016 and P pkq is the power spectrum.

There are no preferred scales in the early universe, so one expects that density fluctuations will have a power-law form with some index n,

P pkq 9 kn. (18.94)

We shall see that physical effects, during the radiation-dominated era, modify the spectrum, reducing the power on small scales. The shape can be accurately predicted, within the context of a given model such as cold dark matter (CDM).

The density autocorrelation function is defined by

ξprq “ hδpxqδpx ` rqi . (18.95)

Here the brackets represent an ensemble average. Since the Universe is homogeneous on large scales, this is equivalent to averaging over volume (the ergodic assumption)

The autocorrelation function is related to the spatial 2-point correlation function of galax- ies, discussed in the previous section. If we assume that galaxies trace the mass (including dark matter), we can use the observed 2-point correlation function to estimate the density autocorrelation function.

The power spectrum is directly related to the autocorrelation function via the Wiener- Khinchine theorem. Replacing δpxq by its Fourier integral, and using the fact that a real function is equal to its complex conjugate, we see that

1 ˚ ipk´k1q¨x ik¨r 3 3 1 ξprq “ hδ δ 1 i e e d kd k , p2πq6 k k ż ż 1 1 “ P pkqδ3pk ´ k1qeipk´k q¨xeik¨rd3kd3k1, p2πq3 ż ż 1 “ P pkqeik¨rd3k. (18.96) p2πq3 ż This shows that the autocorrelation function is the Fourier transform of the power spectrum, and vice-versa, P pkq “ ξprqe´ik¨rd3r. (18.97) ż If the clustering is isotropic, ξprq can depend only on the distance r, and the power spectrum is a function only of k. The integrals in Equations (18.96) and (18.97) simplify to

1 8 sinpkrq ξprq “ P pkq k2dk, (18.98) 2π2 kr ż0 8 sinpkrq P pkq “ 4π ξprq r2dr. (18.99) kr ż0 (18.100)

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18.17 Curvature fluctuations

The power spectrum can be put in dimensionless form. One defines

k3 ∆2pkq ” P 9 kn`3, (18.101) 2π2 k which represents the power per logarithmic interval of k.

Fluctuations in the matter generate fluctuations δΦ in the gravitational potential. In Ein- stein’s theory, this corresponds to spacetime curvature. From Poisson’s equation,

2 ∇ δΦ “ 4πGρcδ, (18.102) therefore 2 δΦ ” pδΦqk “ ´4πGρcδk{k . (18.103) From this we see that 2 ´4 2 n´1 ∆Φpkq 9 k ∆ pkq 9 k . (18.104)

If n ‰ 1, the amplitude of the curvature fluctuations increase without limit at either small or large scales leading to large nonlinearities in the geometry of spacetime. Since we see no evidence for this, Zeldovich, Harrison and Peebles have argued that the primordial spectrum must have n “ 1.

If n “ 1, we have equal perturbations in the spacetime curvature on all scales. This is called the scale-invariant spectrum. The scale-invariant spectrum agrees well with observations, A small deviation from n “ 1 is predicted by inflationary theories.

18.18 Growth of fluctuations

Lets now consider the growth of matter fluctuations in the early Universe. We consider small perturbations to critical density and write

ρ “ ρcp1 ` δq, (18.105)

a “ acp1 ` ζq, (18.106) where δ and ζ are both small. The Friedman equation gives 8πG a9 2 “ ρa2 ´ k, (18.107) 3 and for the unperturbed background we have 8πG a9 2 “ ρ a2. (18.108) c 3 c c

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We have dropped the cosmological term as it is unimportant in the early Universe, but we must keep the curvature term in the perturbation, since the perturbed density is not critical.

Substituting Equations (18.105) and (18.106) into Equation (18.107), keeping terms to first order and subtracting Equation (18.108) we obtain

8πG 2a9 pa9 ζ ` a ζ9q “ ρ a2pδ ` 2ζq ´ k. (18.109) c c c 3 c c

The perturbed density is related to the perturbed scale factor ρ 9 a´β, where β “ 3 if matter dominates and β “ 4 if radiation dominates. Therefore

δ “ ´βζ (18.110) and Equation (18.109) simplifies to

9 βk 2δ ` βHδ “ 2 . (18.111) Hac

Here H “ a9 c{ac is the Hubble parameter in the unperturbed background. In the radiation 1{2 dominated era, β “ 4, H “ 1{2t and ac 9 t . Equation (18.111) becomes δ δ9 ` “ const, (18.112) t and we see that δ 9 t.

2{3 In the matter dominated era, H “ 2{3t and ac 9 t , so Equation (18.111) becomes δ δ9 ` 9 t´1{3, (18.113) t and we see that δ 9 t2{3.

In summary, we have

a 9 t1{2, δ 9 t, radiation dominated, a 9 t2{3, δ 9 t2{3, matter dominated. (18.114)

As the Universe expands, particles that are further apart than ct cannot communicate. This is the horizon size, and it grows linearly with time.

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18.19 Cosmic transfer function

Fluctuations that are larger than the horizon grow freely, according to Equation (18.114). However when the horizon size grows larger than the scale of the fluctuation, pressure exerted by the radiation becomes important. While the Universe is radiation dominated (ρr ą ρm), fluctuations that are smaller than the horizon oscillate, like sound waves, and cannot grow.

Let tk be the time when a Fourier component δk enters the horizon. The physical length corresponding to wavenumber k is 2πaptq{a0k “ 2π{kp1 ` zq. Setting this equal to ctk, and using (18.114) we find ´2 tk 9 k . (18.115)

Now a fluctuation that enters the horizon at time tk has been growing for a time tk. So, while the Universe is radiation dominated,

´2 δk 9 tk 9 k . (18.116) As the Universe expands, the radiation energy density drops faster than the matter density and the Universe becomes matter dominated at a time t “ teq. Radiation pressure soon becomes 2{3 unimportant and the dark matter fluctuations are free to grow, so δk 9 t , independent of scale.

Figure 18.17: Measurements of the mass density power spectrum from fluctuations in the cosmic microwave background (WMAP) and galaxy counts (2dF). Here h » 0.7 is the Hub- ble constant in units of 100 km/s/Mpc (Spark & Galagher). The solid line is the ΛCDM prediction.

Because small-scale fluctuations have had less time to grow, during the radiation-dominated era, than large-scale fluctuations, the primordial power spectrum is modified. On small scales, P pkq 9 knpk´2q2 “ kn´4. (18.117)

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On large scales, which did not enter the horizon during the radiation-dominated era, the primordial spectrum is preserved. The wavenumber at which the break in the spectrum occurs corresponds to the size of the horizon at t “ teq. Thus

n k , k ă keq, P k p q 9 n´4 #k , k ą keq. where keq “ 2π{teq. Taking n “ 1, the cold-dark-matter model predicts that the power spectrum will have a slope of 1 for small k and ´3 for large k. This agrees with observations (Figure 18.17) although the slope is slightly flatter than 1 on large scales, as is predicted by inflationary models. Also, on sub-galaxy scales CDM predicts more power than is observed.

18.20 Baryon Acoustic Oscillations

When the Universe becomes matter dominated, the dark matter fluctuations are free to grow under the influence of gravity alone. This is not the case for the baryonic matter (the gas) which still interacts strongly with the photons. Because the gas is highly ionized, the electrons are coupled to the photons by Thompson scattering. And, the protons and nuclei are coupled to the electrons by electrostatic forces.

The photons, exert a pressure that resists the gravitational force acting on the gas. The result is acoustic oscillations. Sound waves travel with a speed

dP cs “ (18.118) d dρ

The pressure is due to the photons, so P “ ρr{3. The density includes both the gas (ρg) and photon energy density (ρr), which scale differently with temperature. Thus

1 dρr{dT cs “ ? , 3ddpρg ` ρrq{dT

1 ´4ρr{T “ ? , 3dp´3ρg ´ 4ρrq{T 1 1 “ ? . (18.119) 3 1 ` 3ρg{4ρr

At high redshift, this is close to the speeda of light, so the Jeans length (the distance that sound can travel in a collapse time) is comparable to the size of the horizon.

The oscillation frequency for co-moving wavenumber k is

ω “ kcsp1 ` zq. (18.120)

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Maximum compression or expansion will occur at times πn t “ (18.121) n ω where n “ 0, 1, ¨ ¨ ¨ . If this equals the time of recombination, the fluctuation will have a maximum (positive or negative) when the CMB photons last scatter, and will generate CMB anisotropy. The wave numbers of the density maxima (odd n) and minima (even n) will be πn kn “ . (18.122) tncsp1 ` zq This equation is approximate as we have ignored the variation in sound speed with time, due the the implicit dependence on the scale factor in Equation (18.118).

We can estimate the position of the first peak by noting that it corresponds to the wavenumber that just reached maximum contraction at the time of recombination. Thus the physical scale is on the order of the sound horizon at recombination 2π l1 » „ cstr. (18.123) p1 ` zqk1

At recombination, the sound horizon cstr » 130 kpc, which corresponds to a comoving scale of p1 ` zqcstr » 146 Mpc (referred to the present time t0).

Although dark matter does not participate in these oscillations, it is affected by the gravita- tional potential of the baryons. Thus there is a weak imprint in the matter power spectrum, leading to a preferred scale for galaxy clustering.

Large scale surveys of galaxies show a peak in the galaxy power spectrum that can be identified with this scale (Figure 18.18). The observed angular scale of the peak is the ratio of l1 to the mean distance of the galaxies. Since l1 is known, this gives a distance-redshift relation that can be used to study the equation of state of dark energy. (The cosmological constant is a special case having P “ ´ρ.)

18.21 Cosmic background radiation anisotropy

Acoustic oscillations produce temperature fluctuations in the CMB from the following effects:

• Photons emitted from a higher density regions have a higher temperature since these regions are hotter

• Photons that are last scattered in a potential well receive a gravitational redshift - the Sachs-Wolfe effect (Sachs & Wolfe 1967).

• Photons scattered from moving matter receive a Doppler shift

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Figure 18.18: Baryon acoustic oscillation peak seen in the galaxy 2-point correlation function, from SDSS data. (Eisenstein et al. 2005).

The resulting CMB temperature fluctuation is given by (eg. Tegmark 1996), ∆T 1 1 “ δ ` ∆Φ ´ v . (18.124) T 3 3 r The higher order peaks are damped due to dissipation of the oscillations, so the first peak is the most prominent.

Odd numbered peaks correspond to positive (high density) and even peaks to negative density fluctuations. Since dark matter does not oscillate, but only collapses, it enhances the odd peaks and reduces the even peaks.

Current models are based on numerical integration of the relativistic equations of photon transport with Compton scattering in an expanding Universe (eg. Bond and Efstathiou 1984).

In a flat universe the angular size corresponding to the first peak is

l1 θ1 » p1 ` zrq, t0 cstr „ p1 ` zrq, t0 „ 1˝. (18.125) The position of the peak is sensitive to the curvature of space (due to focussing of light), so CMB anisotropy measurements provide a strong constraint on the curvature constant.

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Figure 18.19: Temperature (TT) and temperature-polarization (TE) power spectra for the seven-year WMAP data set. The solid lines show the predicted spectrum for the best-fit ΛCDM model (WMAP collaboration).

18.22 First stars and galaxies

At recombination, the baryons decouple from the photons and the acoustic oscillations stop. The gas is now free to fall into the potential wells created (primarily) by the dark matter.

The mean square density fluctuation within a spherical volume VR having co-moving radius R can be found by integrating the power spectrum,

8 2 2 1 2 2 σR ” δ R “ 2 P pkqW pkRq k dk, 2π 0 8 ż “ ∆2pkqW pkRq2dk{k. (18.126) ż0 Here W pxq is a window function that describes the smoothing produced by convolving the fluctuation field with a function that is equal to 3{4π within the sphere and zero outside. It

Page 174 Galactic Astronomy 2016 is the Fourier transform of this “tophat” filter, x 3 sinprq 2 W pxq “ 3 r dr, x 0 r 3 ż “ rsinpxq ´ x cospxqs. (18.127) x3 To get an order-of-magnitude estimate of the amplitude of the dark matter fluctuations at the time of recombination, note that W pxq equals unity at low frequencies and cuts off when x » 2π, so 2π{R 2 2 σR » ∆ pkqdk{k. (18.128) ż0 n n´4 Recall that P pkq 9 k when k ă keq and P pkq 9 k when k ă keq, and that n » 1. A simple approximation that has the correct asymptotic behaviour is 4 2 Apk{keqq ∆ pkq » 4 , (18.129) 1 ` pk{keqq where A is a dimensionless constant.

Inserting this into Equation (18.128) and doing the integration, we get

teq{R x3 σ2 » A dx, R 1 ` x4 ż0 A “ lnr1 ` pt {Rq4s. (18.130) 4 eq

Measurements of large-scale structure indicate that at the present epoch, the rms density ´1 fluctuation on a co-moving scale of 8 h Mpc, σ8 » 0.8, and the co-moving horizon size at ´1 ´1 equality is teq » 120 Mpc (h ” H0{100 km s Mpc ). Inserting these values into Equation (18.130), we find A » 0.24.

Since fluctuations grow in proportion to the scale factor in the matter-dominated regime, the ´3 observed value of σ8 corresponds to a fluctuation δ „ 10 at recombination.

Smaller scales have a higher density contrast, and so will collapse on shorter timescales. From Equation (18.35), we expect that the collapse time for a (positive) fluctuation δ (at the time of recombination) is ´3{2 tc » 2.19 δ tr. (18.131) After recombination, the matter is free to fall into the potential wells created by the dark matter. However, the matter is impeded by gas pressure. The critical Jeans length is on the order of the sound speed times the gravitational free-fall time. When the photons decouple, ´1 the sound speed drops to cs “ kBT {µmH » 6500 km s . The Jeans mass becomes

a 1 15k 3{2 M “ ? B T 3{2ρ´1{2, J 6 π µm G ˆ H ˙ 5 » 10 Md. (18.132)

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This is about equal to the mass of a globular cluster. The first stars likely formed from clouds of this size that fragmented to form stars.

A key factor in this scenario is the ability of the clouds to radiate away thermal energy during the contraction. If they cannot cool, they cannot contract. With no dust grains to radiate, most of the cooling likely comes from radiation by molecular hydrogen.

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