GALAXIES

Prof Steve Phillipps – Physics Room 4.12

Level H – Spring 2009

Galaxies in the Universe

Galaxies are basically large systems of stars. The Sun and all (∼ 2000) visible stars are in one suchsystem - the Galaxy. A pale band crossing the sky is also visible from a dark site - the Milky Way, madeup of vast numbers of faint stars. It appears as a band around the sky because stars are distributed ina flattened disc; in directions in the plane of this disc we see many stars along the same line of sight,out of the plane relatively few.

From the southern hemisphere, you can see separate patches away from the Milky Way – theLarge and Small Magellanic Clouds, small companion galaxies of our Galaxy. The Andromeda Nebula(M31) looks even smaller, but is the nearest external giant galaxy. If it is similar in size to our Galaxy(d ' 20 kpc; see below), then its angular size θ ' 1 degree (0.02 radians) implies a distance

D = d/θ ∼ 1000 kpc = 1 Mpc.

Many more nebulae were discovered in the 18th and 19th centuries, but proof that they were externalgalaxies, outside our own, only came in the 1920s.

Distance Measurements

The problem was determining distances. Inside the Galaxy, we can measure distances to stars bytrigonometric parallax, the change in apparent direction of a star when viewed from two differentpositions. For a large baseline, we utilise the movement of the Earth around the Sun; observations sixmonths apart give a baseline of 2AU (1 Astronomical Unit is the mean Earth - Sun distance, 149.6million km). A measurement with a baseline of 1AU is a star’s ‘annual parallax’ p (in arcsec).

The first stellar parallax was obtained by Bessel (1838), who measured p = 0.29 ′′ for the star 61Cygni. He chose this star as it has a large ‘proper motion’, i.e. it moves across the sky (relative tothe positions of neighbouring stars) at a faster rate (µ ' 5′′ per year) than almost any other star.

Defining a parsec as the distance at which a star has p = 1′′ (i.e. 206265 AU, 3.086 × 1016m),implied a distance D = 1/p ' 3.5 pc for 61 Cygni. (Proxima Centauri is the closest star, at 1.3 pc).

William Herschel (in the 1790s) pre-empted the measurement of stellar distances by a simpleargument, viz. if he assumed that other stars were of similar intrinsic luminosity to the Sun, he couldestimate how much further away they must be in order to look as faint as they do.

Given the distance D to the Sun and the measured flux of sunlight at the Earth (F = 1.37 kW/m2),the luminosity of the Sun (i.e. its power output, L) is given by

F = L/4πD2.

This gives L = 3.86 × 1026 W. From the fluxes of the brightest stars and assuming L ' L, thebrightest stars should be ∼ 2× 105 times further away than the Sun, i.e. about 1 pc.

Going a step further, if stars were ∼ 1 pc apart, then from the implied volume density, and thetotal number of stars he could see, Herschel deduced that the overall size of the Galaxy was about800 pc (in the plane) by 150pc (perpendicular to the plane).

The key breakthrough in distance measurement came in 1908 with Henrietta Leavitt’s study ofCepheid variable stars in the Magellanic Clouds. Cepheids have a characteristic variation in brightnesswith time, and Leavitt discovered a relationship between the period of the variations and the apparentbrightness of the star. Since all the Cepheids in one of the Clouds are at essentially the same distance,

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this implies a relationship between their periods (P ) and luminosities (L). To calibrate it, we needthe true L of one or more Cepheids, which requires their distances. No Cepheid was near enough fora trigonometric parallax to be obtained, so Harlow Shapley used a method called ‘statistical parallax’(using stars’ velocities) to obtain their distances.

Once it was calibrated, Shapley could use the Cepheid P − L relation to determine L for anyCepheid with an observed P and then deduce its distance from its apparent brightness. In 1915, heobtained distances up to 50 kpc for a number of globular clusters (GCs), densely packed clusters of∼ 106 stars, a few pc across, which contained Cepheids.

GCs are distributed all around the sky but with a preponderance in one direction. Assuming theyare symmetrically placed with respect to the centre of the Galaxy, this observation is easily explainedif we are viewing them from an off-centre position. Since Shapley had distances to many GCs, he wasable to estimate that the Sun must be ∼ 10 kpc from the Galactic Centre.

In 1923, Edwin Hubble used the new 100′′ telescope at Mount Wilson to find Cepheids in M31.With the known P − L relation he showed that M31 must be at least 300 kpc away (and have asize comparable to our own Galaxy). Observations of Cepheids in other spiral and irregular nebulae(published in a paper by Hubble in 1925) confirmed the existence of external galaxies.

Redshifts, Distances and Dynamics

Spectroscopy had been used in astronomy from the 1860s and provided a way of measuring stars’velocities - the Doppler Effect. For a source moving away from an observer at speed v, successivewave crests, emitted a time ∆tem apart, have an extra distance v∆tem to travel so take an extra time(v/c)∆tem to arrive. The observer sees wave crests separated by time intervals

∆tobs = (1 + v/c)∆tem,

so sees the wavelength of the light stretched by the factor 1 + v/c. If we define redshift z via

1 + z = λobs/λem

where λobs and λem are the observed and emitted wavelengths, then

z = ∆λ/λem = v/c

where ∆λ = λobs − λem is the change in wavelength seen for lines in the spectrum of a source movingat speed v. (We ignore any relativistic effects here).

Radial velocities of stars are ∼ 10 − 100 km s−1. Starting around 1912, observations showed thatspiral nebulae were nearly all moving away from us, at velocities reaching 2000 km s−1. These are toolarge for the nebulae to be Galactic objects, as they exceed the escape velocity from the Galaxy.

Redshifts are also used for the dynamics of objects, e.g., in a rotating disc galaxy seen edge on,one side will be approaching the observer (blueshifted) and the other receding (redshifted). Opik (in1922, prior to Hubble) showed that the observed rotation velocity for M31 can be used to estimateits distance. If stars at the edge of its visible disc move in circular orbits at velocity V around a massM, then the gravitational force must match the centripetal acceleration

GM(θD)2

=V 2

θD

2

where the physical radius is the measured angular radius θ times the unknown distance D.On average, to produce 1 L we require about 3 solar masses (3 M) of stars (since most are

smaller than the Sun and less efficient at power generation), i.e. the mass-to-light ratio M/L ' 3 insolar units. The luminosity of M31 is obtained from its flux F as L = 4πD2F , so combining all theseresults

D =V 2θ

4πGF (M/L).

Using the observed values, Opik deduced a distance of about 450 kpc.

Expansion of the Universe

Using Cepheids in nearby galaxies and then taking the brightest stars as ‘secondary distance indi-cators’ (i.e. assuming that the most luminous stars in any galaxy are always physically the same),Hubble estimated distances to 18 galaxies with measured redshifts and in 1929 demonstrated a linearrelationship between recession velocity cz and distance D.

Friedmann and Lemaitre had already found solutions of Einstein’s equations of General Relativitywhich allowed uniform expansion of the universe and Hubble’s observational result was immediatelyassociated with such a general expansion, as it required velocity proportional to distance.

The key application of ‘Hubble’s law’ is in assigning a distance to any galaxy for which z canbe obtained, regardless of any actual distance measurement (via Cepheids etc.). The constant ofproportionality H0 in the law

cz = H0D

is known as Hubble’s constant, conventionally written with units of km s−1/Mpc.

The Distance Scale

Distance determinations rapidly become insecure once trigonometric parallaxes become impossible.Until the 1990s this limit was at tens of parsecs because of the practical limitation on measuringangles < 0.1 arcsec (the angle subtended by a 200m diameter crater on the Moon). Satellite basedobservations, particular from Hipparcos, have extended these measurements to stars at distances upto 1 kpc. Lacking precise measurements, astronomers concocted a whole series of alternatives, usefulfor different distant ranges, forming the ‘cosmic distance ladder’.

The ‘moving cluster’ method is used to determine the distance to nearby star clusters. If allthe cluster stars are moving together with parallel (3-dimensional) velocity vectors, their motion willappear to converge to some point on the sky. If the angular distance of a star from this convergentpoint is θ, its radial and tangential velocities are related by

vt = vrtan θ.

But vt is also measured by the proper motion µ times the distance D, so

D = vrtan θ/4.74µ

where the 4.74 comes from the translation of µD from arcsec/year × pc, i.e. AU/year, to km/sec.The statistical parallax method mentioned earlier is related to this in that we use the velocities

of a whole set of stars. Once the overall Galactic rotation is allowed for, on average the randomcomponents of vr and vt (or µD) should be about the same. Thus we can deduce the mean distanceto the sample stars.

Once we have distances to some clusters, we can use the global characteristics of stellar populations.If we plot the temperatures of stars (from their colours) against their luminosities, only certain regionsof the ‘Hertzsprung-Russell (H-R) diagram’ are populated. The majority of stars occupy a swathe -the ‘main sequence’ (MS) - from bright and blue (hot) to faint and red (cool).

Once we have calibrated the relationship between the colour and luminosity of stars, from a clusterat known distance, we can use it as a distance indicator; we deduce L for a star from its colour anduse the flux F and the inverse square law. This is known as ‘spectroscopic parallax’.

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Astronomers normally use magnitudes rather than fluxes and luminosities. The apparent magni-tude

m = −2.5 log10 F + constant,

where the constant is chosen so that the star Vega has m = 0.0, and the absolute magnitude

M = −2.5 log10 L + constant.

is defined so that it matches the apparent magnitude for an object exactly 10 pc away. Thus (for Din pc) the ‘distance modulus’

m−M = 5 log10(D/10).

If we look at other star clusters, the standard MS will just be shifted in magnitude by the distancemodulus to the cluster. This method of distance determination is known as ‘MS fitting’. If the clustercontains a ‘useful’ sort of star (e.g. variable star such as an RR Lyrae), we can calibrate its luminosityand use that as another ‘standard candle’ (i.e. source of known brightness) to determine yet moredistances. As well as the MS, the H-R diagram of a typical cluster contains a ‘giant branch’ of redgiant stars which have evolved off the MS, increasing in size and luminosity. The stars at the tip ofthe red giant branch have quite characteristic luminosities, so this ‘TRGB’ provides another distanceindicator for any star cluster (or galaxy) in which individual stars can be resolved.

Galactic Extinction

Uncertainty is introduced into distance measurements by the fact that spiral galaxies like ours containan ‘interstellar medium’ (ISM), of gas and dust acting like a ‘fog’ between the stars. This makes starsor external galaxies look dimmer than they would on the basis of their distance alone, so complicatesall distance estimates based on apparent brightnesses and the inverse square law.

Foreground absorption by dust in our Galaxy depends on direction in the sky; absorption is greaternearer to the Galactic Plane, as we are looking through a greater depth of ISM. Approximating ourGalaxy by a uniform slab with the Sun in the central plane, the line of sight length varies as cosec b,where b is called the Galactic latitude.

Each element of length removes a certain fraction of the light, so if I(x) is the intensity of the lightafter it has passed through a length x of the ISM from its source, then after a further distance ∆x

I(x + ∆x) = I(x)−∆I = I(x)[1− κ∆x]

where κ represents the rate at which the light is absorbed (the absorption coefficient). Thus

dI

dx= −κI

which has the solutionI = Ic e−κx

if Ic was the original intensity at x = 0. This is usually written

I = Ic e−τ

where τ is called the optical depth. Translating into magnitudes,

m = mc − 2.5 log(e−τ ) = mc + 1.086τ = mc + A

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where mc is the magnitude we would see in the absence of absorption and A is the (total) ‘extinction’.The ‘selective’ extinction, also called ‘reddening’ as it changes the apparent colour of obscured objects,is the difference in extinction at two different wavelengths, e.g. in the B and V bands, written E(B−V ).The ratio RV = AV /E(B − V ) ' 3.1 for the Galaxy.

A rough estimate of the absorption in the blue band when looking all the way out of the Galaxy isAB ' 0.2 cosec b magnitudes. In reality, the Galaxy is not uniform and the Sun is not at the centre,so we expect some variation with angle around the Plane – the Galactic longitude ` (measured from0o towards the Galactic Centre). As the absorption is due to dust, which then re-emits at longerwavelengths, we can deduce AB(`, b) by looking at FIR emission.

Hubble’s Constant

Beyond the distance where individual stars could be seen, Hubble and his successors had to rely on evenless direct means, such as the size of the largest HII regions (luminous regions of ionized hydrogen) ina galaxy, or the appearance of the whole galaxy. To reach the greatest distances we can assume thatthe brightest galaxy in any cluster of galaxies always has approximately the same absolute magnitude.More recently, there has been considerable success in using supernovae as standard candles.

The large variety (and dubious precision) of the distance indicators led to uncertainty over thevalue of H0 for several decades, but with the Hubble Space Telescope (HST) it is possible to detectmuch more distant Cepheids, allowing direct distance estimates to many more galaxies. Working inthe near infra-red to reduce the absorption by intervening interstellar dust, the P − L relation forCepheids in terms of I-band magnitudes is

MI = −1.3− 3.5 log P

for P in days. A Cepheid with a period of 100 days will have MI = −8.3 and be visible with HST(which can easily reach mI = 25) out to a distance modulus m−M ≥ 33, i.e. a distance of 40 Mpc,well beyond the Virgo Cluster, the nearest large grouping of galaxies (at D ' 20 Mpc). The HST ‘keyproject’ on the distance scale obtained

H0 = 70± 2 km s−1Mpc−1.

which we assume hereafter.

Galaxy Types

There are three main ‘morphological types’ of galaxy; spirals like M31 and our Galaxy, irregulars likethe Magellanic Clouds, and ellipticals, which are regular in shape and free of substructure (e.g. M87in Virgo). The appearance of elliptical galaxies depends just on the stars they contain; as they emitmost light at longer wavelengths they must contain mostly red stars. Spirals contain gas and dustbetween the stars and the spiral patterns are delineated by both dark dust lanes and bright regionswhere new stars are forming from the gas. As a population of young stars will contain bright blueones, spirals are also bluer than ellipticals.

Hubble produced the basic classification system; the ‘tuning fork’ diagram. The handle containselliptical galaxies, from circular (as projected on the sky) E0s to more and more elongated E1s to E7s.

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The numeral (0 to 7) represents the shape of the galaxy’s image via the quantity 10(1 − b/a), whereb/a is the ratio of the short (minor) to long (major) axis lengths, e.g. an E3 has an axis ratio of 0.7:1.

The prongs contain spiral galaxies. Elliptical galaxies are ellipsoidal in 3-D, but spirals are basicallyflat discs. They can look very elongated if seen edge-on, but often also have a central spheroidal‘bulge’. Sa galaxies have tightly wound spiral arms, while the pattern in Sb, Sc and Sd galaxiesbecomes progressively more open. In step with this, spirals show a decrease in the importance of theirspheroidal component, i.e. Sa s have large bulges while Sc s have small bulges and bulges are almostnon-existent in Sd galaxies.

The reason for the two prongs to the fork is that spirals separate into two sequences dependingon whether the arms start from the central bulge or from the ends of a further component, a central‘bar’ – the types SBa, SBb, etc.

At the end of the sequence, Hubble placed the Irregulars, which are again ‘flat’ but with chaoticpatterns of bright regions. Intermediate are the Sm s, with fragmentary arm-like structures, andIrregulars are often now referred to as Im galaxies (the ‘m’ in each case standing for Magellanic).

Hubble added S0 (or lenticular) galaxies, where the prongs join the handle. These have disccomponents, but with no sign of any spiral pattern, and large bulges. E and S0 galaxies are referredto jointly as ‘early type’ galaxies, and Sa s as early type spirals, while Sc s etc. are late type spirals.

More elaborate schemes fill in intermediate types such as Sab or Sbc, and the family of weaklybarred systems, SABa and so on. (In this system the original unbarred galaxies are SAa etc.). deVaucouleurs introduced a numerical sequence to represent the main morphological types. These runfrom T = −5 to 0 for ellipticals and S0s, to 1 for Sa galaxies and so on up to 5 for Sc and 9 for Sm(and 10 for irregulars), i.e. one numerical class for each Hubble sub-class.

van den Bergh’s scheme attempts to indicate the luminosity of a galaxy; more luminous spiralshave well defined continuous arms (‘grand design’), while low luminosity ones have weak, patchy arms(‘flocculent’). A luminous Sc galaxy with very clearly defined arms is an ScI and a less bright onewith indistinct arms an ScIII, the Roman numerals denoting the ‘luminosity class’. Classes IV and Vare used for Im galaxies with low surface brightness (SB), i.e. small flux per unit area.

Galaxies at Other Wavelengths

Some galaxies are more prominent at non-optical wavelengths. Spirals have radio emission associatedwith star formation and interstellar matter, but ‘radio galaxies’, as such, are giant E galaxies. Theseare powered by ‘central engines’ containing massive black holes and often exhibit jets and twin lobesof emission on either side, in some cases extending for hundreds of kpc.

At millimetre, sub-millimetre and far infra-red (FIR) wavelengths, we see thermal re-emissionfrom dust at ∼ 10 − 100K. Observations by IRAS (the Infra-Red Astronomical Satellite) led to thediscovery of ultra-luminous infra-red galaxies (ULIRGs) powered by starbursts, the rapid formationof large numbers of stars over a short time period. The starburst regions are enshrouded in thick dustlayers and ULIRGs have LFIR >> Lopt.

X-rays are produced thermally in galaxies by material at temperatures T > 106K. They can arisefrom X-ray binary stars or hot gas in or between galaxies, but the most impressive source are ActiveGalactic Nuclei (AGN) harbouring black holes, where the emission is due to accretion of gas onto thecentral object. The brightest AGN are quasars (QSOs), which can outshine their host galaxy by afactor 10 or more. (See the HEA course for details on non-optical emission from galaxies).

Luminosities

Astronomers seldom work in conventional ‘physics’ units of Watts, preferring Solar luminosities, L,or working in magnitudes. Our Galaxy contains ∼ 1011 stars and has L ' 2 × 1010L. Given theSun’s (blue) absolute magnitude of 5.48, this implies a total absolute magnitude for the Galaxy ofMB ' −20.3. M31 has MB ' −20.8 (i.e. it is about 1.5 times as luminous). It was originally thoughtthat the luminosity range of galaxies was quite small, with the Magellanic Clouds representing thefaint end of the distribution. (They have MB ' −18 and −16.5.)

However, this is a ‘selection effect’. Very luminous galaxies really are rare, but we will be biasedagainst including low L systems in our samples, as they must be nearby to look bright. High L galaxieswill be visible throughout a large volume of space. Exactly this problem is encountered with stars;

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apparently bright stars like Vega and Rigel are mostly distant stars intrinsically much brighter thanthe Sun, but in a representative volume of space most stars are less luminous than the Sun.

Both our Galaxy and M31 have a number of small companions and are part of the ‘Local Group’of 30-40 galaxies. There are only 3 giant galaxies (the Galaxy, M31 and M33), the rest are irregulars(mostly dwarf irregulars, dIs), dwarf ellipticals (dEs) and dwarf spheroidals (dSphs).

M31’s companions M32 and NGC205 are prototype dEs. They are ∼ 100 times less luminous thanM31 – the dividing line for dwarfs is at about MB = −18 or 3×109L – and look like small versions ofa normal (giant) E galaxy, ellipsoidal in shape with little or no internal structure (‘nucleated’ dE,N scontain bright central star clusters). dIs have similar L to dEs, but have very clumpy structures.

Dwarf spheroidals (dSph) are even lower luminosity versions of dEs; the first one discovered,Sculptor, was 1/100 of the brightness of previously known systems. The lowest L companions to ourGalaxy are the Draco and Ursa Minor dwarfs, while M31’s companion And IX is marginally fainter.Each is about 10−5 of the luminosity of M31 (i.e. a few ×105L). At MB ' −8, they have magnitudessimilar to GCs and even very bright single stars. Even fainter systems have recently been detectedaround M31, but it is not clear if these are genuine galaxies or just star clusters. No dwarf types,most of which have low SB, are included properly in the tuning fork.

At the other extreme, the brightest giant Es in galaxy clusters, called cD galaxies, range up toabout MB ' −24, 25 times brighter than our Galaxy.

The Luminosity Function

The number of galaxies of different luminosities is quantified by the ‘luminosity function’, φ, thenumber of galaxies per unit volume (in practice per Mpc3) per unit luminosity interval. The mostcommon form of the LF is that due to Schechter,

dn

dL= φ(L) =

φ∗L∗

e−L/L∗

(

L

L∗

)α

.

φ∗ is a normalisation factor (roughly the number density of bright galaxies) and L∗ is a characteristicluminosity where the numbers level off from the steep exponential cut-off at the bright end to thepower-law of slope α at the faint end.

Alternatively, this can be written as the number density of galaxies per unit volume in a onemagnitude interval,

φ(M) = 0.4 loge10 φ∗ exp(−10−0.4(M−M∗))10−0.4(α+1)(M−M∗)

where M∗ is the magnitude corresponding to luminosity L∗. Here we have used the usual identity

M −M∗ = −2.5 log(L/L∗).

and the fact that we must haveφ(L)dL = φ(M)dM

if dM is the magnitude range corresponding to luminosity interval dL.Current estimates in the B band give

L∗ ' 2× 1010L or M∗ ' −20.6, φ∗ ' 0.0055 Mpc−3, α ' −1.2.

Again using the large redshift surveys, but with the galaxies differentiated by morphological type,the Schechter function parameters for Es and S0s are

MB∗ ' −20.4, φ∗ ' 0.0034, α ' −0.5.

M∗ is marginally fainter for Es and S0s than for all galaxies combined (MB∗ ' −20.6). The volumedensity is obviously lower, but φ∗ is only a factor 1.6 less than that for the overall population. Thisdoes not mean that ' 60% of all galaxies are early type giants. This is approximately true aroundMB∗ (and at very bright MB nearly all the galaxies are early types), but the faint end slope parameteris less negative, so the numbers per magnitude interval do not increase towards faint MB like those ofgalaxies in general. Thus the overall fraction of galaxies which are giant Es and S0s is small.

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Spirals have Schechter function parameters

MB∗ ' −20.6, φ∗ ' 0.004, α ' −1.2.

Splitting this further, late types have fainter M∗ but steeper α than earlier types; a typical Sd issignificantly less luminous than an average Sa. The fractions of galaxies in each spiral type arerelatively poorly defined, but amongst brighter spirals there are similar numbers of Sa and Sb galaxiescombined as there are of Sc s. At the fainter end Sd s and Sm s dominate.

A slope of −1.2 implies that the number of galaxies per magnitude interval rises slowly towardsfainter objects: if α is exactly −1, there are the same number of galaxies in each magnitude bin at thefaint end. However, if we consider the dwarfs separately, the slope steepens to α ' −1.5. The slopecan not be as steep as α = −2, or the total amount of light from the galaxy population would diverge.

Writing L for the total luminosity density,

L =

∫ ∞

0Lφ(L)dL,

and for a Schechter functionL = φ∗L∗Γ(2 + α).

Γ(x) is the gamma function, which can not be written in simpler terms unless x is an integer, whenit is the factorial function (x − 1)!. If α = −1, Γ(2 + α) = Γ(1) = 1, so L = φ∗L∗. With the moreaccurate observed values

L ' 1.4 × 108 L/Mpc3.

The total number of galaxies per unit volume is

n = φ∗Γ(1 + α),

which is generally a divergent function. In reality galaxies do not exist with arbitrarily small lumi-nosities; the number brighter than some minimum brightness Lmin is

n(> Lmin) = φ∗Γ(1 + α,Lmin/L∗)

where Γ is now the ‘incomplete gamma function’.

Redshift Surveys

To measure the LF, we need L for a large number of galaxies, so the key is obtaining distances.Fortunately, from Hubble’s Law, we can use redshift as a substitute for distance. Over the last twodecades technological advances have meant that ‘redshift surveys’ have risen from samples of ∼ 100galaxies up to 105−106 objects in the 2dF galaxy redshift survey (2dFGRS), on the Anglo-AustralianTelescope, and the US Sloan Digital Sky Survey (SDSS).

Suppose we have a complete sample of galaxies, all with measured z and m, from which we candeduce each M . The distribution of M values is still not the LF, as more luminous galaxies are visibleat greater distances. To allow for this, we weight each galaxy by the inverse of the volume in which

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it could lie and still appear bright enough to get into our sample. For an apparent magnitude limitmlim, the maximum z at which a galaxy of given M could be seen, zmax(M), follows from

mlim −M = 5 logD(zmax) + 25 + k(zmax) + A(`, b).

where the distance to the galaxy is now measured in Mpc (rather than pc as in the usual definition)and k(z) is the k-correction which accounts for the fact that we observe a different range of rest framewavelengths through a given filter if the galaxy spectrum is redshifted.

Specifically, if Fλ(λ) is the flux from a galaxy at wavelength λ – the spectral energy distributionor SED – then the ratio of the fluxes in the redshifted and un-redshifted bandpasses will be

K =Fλ(λB/(1 + z))∆λB/(1 + z)

Fλ(λB)∆λB

where λB is the central wavelength of the B passband (in this case) and ∆λB is the bandwidth. Thek-correction in magnitudes is then

k(z) = −2.5 logK = 2.5 log(1 + z)− 2.5 logFλ(λB/(1 + z))

Fλ(λB)

Even if Fλ is constant, there is still a k-correction 2.5 log(1 + z), since a narrower range of restframewavelengths contributes to the observed flux from the redshifted galaxy.

In general we may need to account for the curvature of space when calculating distances andvolumes, but if not the available volume will be

Vmax(M) = (Ω/3)D3(zmax) = (Ω/3)(czmax/H0)3,

where Ω is the size of the survey area on the sky in steradians.We then obtain the LF by summing (1/Vmax) for all galaxies in each magnitude bin. We still see

far fewer dwarfs than giants in any sample, so there will be much greater statistical uncertainty at thefaint end of the LF than at the bright end. This uncertainty is increased by the fact that many lowL galaxies are also low SB and not all surveys are equally sensitive to such objects. This has led toconsiderable argument as to the value of the faint end slope α and whether a single Schechter functionremains a good fit to the LF over its whole range.

Galaxy Distributions

The LF gives the average number density of galaxies, but galaxies are not uniformly distributed. Evenin the Local Group, dEs and dSphs only occur as satellites of larger galaxies, while dIs can also be‘free flying’ and fill the overall volume, a region of radius 1.5 Mpc around the mid-point between M31and the Galaxy. The Local Group is rather flat, with most galaxies close to a single plane.

Other nearby galaxies lie in similar structures, e.g. the Sculptor Group, at a distance of a fewMpc. About 5 Mpc away is the Cen A group, containing Centaurus A, the nearest giant E galaxyand one of the brightest radio sources in the sky. Groups tend to line up, making larger filamentarystructures. Few galaxies are completely isolated.

On larger scales there are the Virgo and Fornax Clusters, ∼ 20 Mpc away. Clusters containhundreds to thousands of galaxies but are much rarer than groups. The Virgo Cluster is also thecentre of an even larger structure, the Virgo Supercluster, of which the Local Group is an outlier.

Environment influences general galaxy properties; the denser the region, the greater the fractionof early type galaxies. This ‘morphology - density relation’ shows that in small groups and other lowdensity regions (the ‘field’), the fraction of bright galaxies which are spirals is around 80%, with about20% S0s and very few Es. Moving in from the outskirts of clusters, the fraction of spirals decreasessteadily (to near zero by the time we reach the densest regions), while the S0 fraction rises steadily(up to ∼ 50%) but is rapidly caught up by the E fraction at high densities. Dwarf galaxies alsoshow a dependance on density, with greater numbers (relative to giants) in clusters; dEs and dSphdominate towards the centres, dIs in the outskirts. The galaxy LF must then be different in differentenvironments, the increased fraction of dwarfs in clusters being equivalent to a steeper faint end ofthe LF in clusters than the field.

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Sizes

Sizes of galaxies are ill defined; they do not have obvious ‘edges’ but fade away gradually. To describethe physical size of a galaxy, we need the stars’ (or other constituents’) distribution as a function ofdistance R from the galaxy’s centre. For stars, the radial distribution is seen in the intensity I, thelight emitted per unit area, or surface brightness (SB). Different galaxy types each have characteristicprofiles, I(R), but a general measure of the size of a galaxy is its ‘half-light’ or ‘effective’ radius Re.This is the radius of the circle (projected on the sky) which encloses half the total light. Large spiralsand ellipticals have Re up to 10 kpc, while cD galaxies, with large extended envelopes, have Re up to∼ 100 kpc. Dwarf galaxies are much smaller, down to ∼ 200 pc or less.

The part of a galaxy we can see depends on its contrast against the night sky and the ‘noise’ inthe observations. We are always limited by detecting a finite number of photons; measurements of anarea of a galaxy which emits ng photons can never be more accurate than the ‘Poisson noise’

√ng.

In addition there is noise from the number of photons received from the underlying sky, ns, so the‘signal-to-noise ratio’ will be

S

N=

ng

(ng + ns)1/2.

When the number of galaxy photons in a given area (e.g. detector pixel) drops to the point whereS/N is small (say 3), it cannot be distinguished from the background. This is the isophotal detectionthreshold, a minimum intensity level Imin, typically a few percent of the sky background level, or acorresponding limiting SB, µlim. The size of the galaxy out to this limit is its ‘isophotal diameter’. Weare not sensitive to the remaining light outside this isophote, so a more directly measureable quantitythan the total luminosity LT is the isophotal luminosity Liso, which only integrates the light out tothe isophotal radius.

Surface Brightness

High L galaxies also have high SB (in their central regions). Dwarfs are usually low SB, hence diffuseand hard to see against the foreground glow from our atmosphere (the main reason for their relativelyrecent discovery). Physically, low SB is due to a low surface density of stars; whether this correspondsto a low volume density depends on how ‘thick’ the galaxy is along the line of sight.

SB is not usually quoted in Wm−2, but in L/pc2 or magnitudes per square arc second (µ). Totranslate between these, 1L = 3.86 × 1026W and 1 parsec = 3.086 × 1016m so

1L/pc2 = 4.05 × 10−7Wm−2.

For the Sun MB = 5.48, and as 1 pc subtends 0.1 radians (20626 arc sec) at the standard distance of10 pc used to calculate absolute magnitudes, 1L/pc2 corresponds to

µB = MB + 5 log(20626) = 27.05 B magnitudes per sq arc sec (Bµ).

The mean SB inside the effective radius (L/2πR2e) is about the same for both giant ellipticals and

spirals, ∼ 100L/pc2 (' 22Bµ; similar to the brightness of the night sky). Es have I(r) rising moresteeply towards the centre than do spiral discs (see below), so the central values differ widely; a few102L/pc2 for discs, ∼ 104 for Es and bulges.

Dwarf galaxies are typically a factor 10 lower in average SB than giants, and some are even fainter– the class of ‘low surface brightness galaxies’ or LSBGs. These include the fainter dI galaxies, thedE and dSph galaxies and a class of large but low SB disc galaxies. Their prototype, Malin 1, is∼ 100 kpc in extent, but with central SB ' 2L/pc2. Not all dwarf galaxies have low SB, ‘bluecompact dwarfs’ or BCDs, contain small, blue, high SB central regions, while ultra-compact dwarfs(UCDs) have properties between normal dwarfs and GCs.

Surface Brightness Profiles

Spiral galaxy discs seen face-on have a simple azimuthally averaged intensity distribution,

I(R) = I0 exp(−R/a)

10

where I0 is the central intensity and a is the scale length. In magnitudes

µ(R) = µ0 + 1.086R/a.

The 1.086 factor (= 2.5 log10e) arises from the logarithmic definition of magnitudes.

Ellipticals are traditionally fitted by the de Vaucouleurs or ‘R1/4’ law

I(R) = I0 exp(−(R/a)1/4) or µ(R) = µ0 + 1.086(R/a)1/4 .

The bulge components of spiral galaxies are also often assumed to follow an R1/4 law, but dE,dSph and dI galaxies generally have profiles close to exponentials. The very luminous cD galaxieshave an excess above the standard R1/4 profile, extending out to large radii.

The total luminosity is

LT =

∫ ∞

02πR I(R)dR

(assuming the image of the galaxy is circular; ‘face-on’ to the observer). More generally, the ‘curve ofgrowth’ gives the luminosity within a circle of radius R projected on the sky,

L(R) =

∫ R

02πwI(w)dw.

For exponential profilesL(R) = 2πa2I0(1− (1 + x)e−x),

where x = R/a, andLT = 2πa2I0.

In magnitudes (with a in arcsec)

m = −2.5 log(2π)− 5 log a + µ0 = µ0 − 5 log a− 2.00.

Since the effective radius contains half of the total light, L(Re) = 0.5LT , so (1 + x)e−x = 0.5,implying x = 1.69. Thus

Re = 1.69 a.

The SB at Re (Ie or µe) can be used as an alternative to I0 or µ0. From above

Ie = I0 e−1.69 = 0.184 I0, µe = µ0 + 1.83

andLT = 3.81πR2

eIe.

For ellipticals (or bulges) the integrals are lengthier to compute, giving

LT = 8!πa2I0, L(R) = LT

(

1− e−x7∑

k=0

xk/k!

)

11

where now R/a = x4. In this case

Re = 7.674a = 3461 a, Ie = e−7.67 I0, µe = µ0 + 8.33

soI(R) = I0 exp(−7.67(R/Re)

1/4) = Ie exp(−7.67[(R/Re)1/4 − 1])

andLT = 7.22πR2

eIe.

A general spiral will have an overall radial profile

I(R) = I0Dexp(−R/aD) + I0Bexp(−(R/aB)1/4)

where the subscripts ‘B’ and ‘D’ refer to the bulge and disc components. (The SB in magnitudes cannot be written in any simple form). The bulge-to-disc ratio is then

B

D= 3.5

R2eBIeB

a2DI0D

.

We can also generalise the de Vaucouleurs law to the Sersic law

I0 exp(−(R/a)1/n) = Ie exp(−b[(R/Re)1/n − 1])

with b ' 2n− 1/3. Lower L Es (and bright dEs) have n ' 2− 4.

Structure of Ellipticals

Although numerical simulations of the gravitational interactions between large numbers of stars seemto lead to the de Vaucouleurs distribution from many possible starting points, there is no theoreticaljustification for the R1/4 law for E galaxies. Also, the SB is a projection onto the sky of the true 3-Ddistribution of stars and there is no analytic mathematical function for the density which projectsexactly to the de Vaucouleurs form.

Starting with a simple form of the density, a power law in true 3-D radius, ρ ∝ r−γ , the surfacedensity or SB as a function of projected radius R will be

I(R) ∝∫ ∞

0(h2 + R2)−γ/2dh

where h is measured along the line of sight through the galaxy. With a change of variables to g = h/R,

∫ ∞

0

R dg

Rγ(g2 + 1)γ/2= R−γ+1G(γ)

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where

G(γ) =

∫ ∞

0(g2 + 1)−γ/2dg

is a number which depends only on γ. Thus a power law volume density of slope −γ projects to asurface density of slope −γ + 1 so an observed profile of slope −δ must be the projection of a 3-Ddensity distribution r−δ−1.

There are some simple forms which give reasonable approximations to E galaxy profiles, e.g.

I(R) = I0(1 + R/R0)−2

which approximates a power law of slope −2 for the SB in the outer parts, but flattens in the centre.A useful density function is Hernquist’s

ρ(r) =M2π

rc

r(rc + r)3

where rc is the core radius. Calculating the same integral as before we obtain a projection which isvery close to the de Vaucouleurs law over a wide range of radii.

Kormendy Relation

The luminosity of an elliptical galaxy correlates tightly with the scale length a. Since L ∝ I0a2 (or

IeR2e), this implies a correlation between L and SB. Brighter Es are also larger, but the increase in

a more than accounts for the increase in L, so the SB goes down; the ‘Kormendy relation’. Dwarfellipticals with MB > −18 are quite different, as their L−SB relation goes in the opposite direction.A good fit to Kormendy’s original relation for giant Es is (for our value of H0)

µe = 20.2 + 3.0 logRe

or the equivalentMB = −19.3 − 2.0 logRe.

Note that these implyL ∝ R0.8

e and Ie ∝ R−1.2e ∝ L−1.5.

Since a typical E has Re ∼ a few kpc, the scale length a ∼ 1 pc. The region where the intensitydrops by the first few factors of e in a de Vaucouleurs profile will subtend a tiny angle even for nearbygalaxies. The HST resolution of 0.05′′ makes the exploration of the central regions feasible. Veryluminous ellipticals are generally found to have a ‘core’, a region where I is nearly constant, but inless luminous ellipticals the SB continues to rise steeply to a ‘cusp’ at the centre. For a cuspy profile,the 3-D density must be rising faster than r−1. The luminosity density right at the centre of cuspyellipticals can be ∼ 106L/pc3.

Shapes

If we assume that Es have regular isodensity surfaces, there are three possible 3-D shapes. They couldbe oblate spheroids, squashed down at the poles, or prolate spheroids, with one long and two shortaxes, or they could have all three principal axes of different lengths, a tri-axial ellipsoid. The oblatecase would seem the most reasonable, with the galaxies flattened by rotation. An oblate spheroidalways looks less (or at most equally) flattened in 2-D projection than it is in 3-D, so we need a rangeof true flattenings at least up to the 0.3:1 axis ratio seen for E7 galaxies. (This is true for prolatespheroids, too). From most viewing angles, oblate spheroids should appear nearly circular, so therelatively small number of E0 galaxies implies that Es are not (usually) oblate.

It is likely that most luminous Es are tri-axial. Though it is difficult to picture, the projection ofgeneral tri-axial isodensity surfaces in 3-D (even if all aligned with the same axes) can give rise, formany viewing angles, to elliptical isophotes which are not aligned, an effect seen in some galaxies as‘twisting isophotes’.

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Even in this case the isophotes themselves are still perfect ellipses, but not all real Es are exactlyelliptical! Two characteristic deviations are seen, either ‘boxy’ or ‘disky’. Disky isophotes are moreelongated along the major axis (pointier) than a true ellipse, whereas boxy isophotes are compressedalong the axes (more rectangular).

Other properties of ellipticals tend to correlate with the isophote shapes, e.g. very luminous Esare usually boxy. Boxy Es are also more likely to be radio and X-ray emitters. Medium luminosityEs are more likely to be disky, and also to be faster rotating. Disky Es can be seen as intermediatesbetween other Es and S0s. S0s themselves resemble Es as smooth ensembles of stars, but with a largedisc contribution. Typically the bulge contributes about 50-60% of the light, but the disc to bulgeratios vary from about 0.1 to 2.

Structure of Spirals

Spirals come in a variety of forms, characterised by (relatively) thin, (more or less) flat discs and(usually) a central bulge. The bulge-to-disc luminosity ratio B/D spans a large range. The earliesttype spirals, and the S0s, have the largest B/D, ∼ 1, with B/D then decreasing to ∼ 0.3 for Sbgalaxies (e.g. M31) and < 0.1 for later type spirals (like M33). Sd and Sm galaxies (e.g. LMC) haveno real bulge (but can have a central star cluster).

Bulges have physical sizes Re ' 100pc to several kpc. They can be fairly round spheroids, quiteflattened or even prolate tri-axial ellipsoids. Seen edge-on, most bulges have elliptical isophotes, butsome are ‘peanut shaped’, with the isophotes showing a dip at the minor axis. The bulge in our Galaxyis best viewed in the IR where the dust extinction in the intervening disc is minimised. It containsabout 20% of the stellar mass of the Galaxy and is somewhat elongated.

Roughly half of all spirals and S0s have a bar. In early type barred spirals, the bar co-exists withthe bulge. Bars can have axis ratios up to 5 and can contain as much as 30% of the total galaxy light.They are also flat, like the disc component.

Spiral Arms

The arms of Sa galaxies are very tightly wound, but in later types the arms open out. The shape ofthe pattern can be represented by a logarithmic spiral. In polar co-ordinates

ln(R/R0) = kθ

where R0 is the ‘radius’ of the spiral as it crosses the axis at θ = 0. A pure mathematical spiral isinfinitely long and has infinitely many crossings of θ = 0; for a spiral galaxy, the arms start from theedge of the bulge or the ends of the bar and complete no more than 2 - 3 turns around the galaxy.

The ‘pitch angle’, i, measures the angle between the direction of the arm and the tangent to acircle at the same R, so

tan i =1

R

dR

dθ

and for a logarithmic spirali = tan−1k.

For Sa galaxies i ∼ 5o, for mid-type spirals it is ' 10o to 12o and for late types i ∼ 20o.In two-armed spirals, at any radius R there are two brightness peaks 180o apart, but spirals can

have multiple arms or have numerous branching sub-arms.The arms are superimposed on the exponential disc of starlight and despite appearances are a

relatively minor component in terms of the total number of stars. A cross-section through a spiralgalaxy in the infra-red, which is insensitive to the presence of a few very bright blue stars, shows onlya small intensity enhancement as we cross the arms.

Disc Surface Brightnesses

Disc components of bright spirals have scale lengths of a few kpc; M31 has a ' 4.5kpc. For exponen-tials, Re = 1.69a, so half light radii of discs range up to ∼ 10kpc. The observable extent can be muchlarger than this. If we characterise it by the point where the SB is 25Bµ (the de Vaucouleurs radius),

14

it is typically ' 3a. At fainter limits, the Holmberg radius (at the 26.5Bµ isophote) is generallyaround 4− 5a or ∼ 25kpc. Some discs appear to end quite suddenly here, rather than just fade intothe background. (26.5Bµ is only ' 2% of the sky background, so the contrast of the galaxy imageagainst the sky is low).

Even if the central disc regions are hidden by the bulge, we can extrapolate the radial profilefrom further out to estimate the disc brightness at the centre, the extrapolated central SB, µ0. Manybright spirals have values of µ0 ' 21.65Bµ (just over 100L/pc2). This is known as ‘Freeman’s law’.However, it may be another selection effect.

Consider a set of discs with the same LT but different µ0. A galaxy of very high SB will have avery small scale size and its isophotal radius will also be small. If the SB is very low, only the centralregion is above the limiting isophote and again the isophotal radius is small. There will then be apreference for galaxies of some intermediate SB which lends a galaxy its maximum apparent size.

The isophotal radius Rlim at some Ilim is given by

Ilim = I0 exp(−Rlim/a) ⇒ Rlim = a ln(I0/Ilim) = 0.92a(µlim − µ0).

But also LT = 2πa2I0, so for fixed LT ,

Rlim ∝ I−1/20 ln(I0/Ilim) ∝ (µlim − µ0)10

0.2(µ0−µlim).

This is a peaked function of µ0, with the peak around 2.2 magnitudes brighter than µlim. With olderphotographic imaging (µlim ' 24Bµ) this closely coincided with the Freeman’s law value.

This bias can affect the inclusion of galaxies in samples if we only include objects with images abovesome minimum angular diameter, since galaxies with large physical isophotal sizes will be included outto much larger distances and hence in relatively large numbers. Galaxies with high or low SB exceedthe angular diameter limit only if quite local. Very low SB galaxies may not rise above a survey’s SBthreshold at all. Similarly for isophotal magnitudes, low SB galaxies have a smaller fraction (possiblynone) of their total light inside the limiting isophote, so are less likely to be included. Galaxies withlow SB and low L (i.e. dwarfs) are doubly selected against.

Taking this into account, the decline in numbers on the bright side of Freeman’s peak is genuine(though some high SB discs do exist), but on the other side, there is a whole population of low surfacebrightness galaxies (LSBGs), uncovered by deeper imaging. The distribution of µ0 for all discs (plusdwarfs) is quite flat from ' 21.5Bµ down to the faintest surveyed levels at ' 25Bµ, though luminousspirals do have a peaked distribution of µ0.

From the joint distribution of µ0 and a or µ0 and L, the majority of discs are quite small and ofmoderate SB. However, the large, higher SB discs have higher LT , so the large galaxies emit most ofthe light.

Vertical Structure

Edge-on, spiral discs show a flattening of up to a factor 10, i.e. disc thicknesses ∼ 10% of their radii.The radial surface density profile of face-on discs is exponential, and the minor axis profile of edge-ongalaxies shows the same form in the vertical (z-) direction. In cylindrical polars, the (3-D) densitystructure is then

ρ(R, z) = ρ0 exp(−R/a) exp(−|z|/h),

with the scale height h ∼ a/10. Note that as ρ is separable in R and z (unlike for ellipsoidal distri-butions), projection onto the sky leaves the exponential form unchanged. (The only exception is themajor axis profile in the edge-on case, since we are integrating across different ranges of R at differentpoints along the axis).

In our Galaxy the vertical structure is found via star counts. For a particular type of star, e.g.MS K stars (most stars fainter than mV = 14, are K or M ‘dwarfs’; see below), all will have similar L(' 0.5L for K dwarfs). But if they all have the same MV (' 6), then the number of stars counted ina given patch of sky at a given m will correspond to the number at some particular distance. Detailedspectroscopy or accurate colours allow us to determine L, and hence distance D, for individual stars.Alternatively, if we just count all the stars at a given m, we get a convolution of the density variation

15

along the line of sight, ρ(D), with the stellar luminosity function φ(M), since stars at a given m cancome from any suitable combination of M and D, i.e.

n(m) =

∫

φ(M = m− 5log(D/10) + A(`, b,D)) ρ(D)ΩD2dD,

where A is the absorption out to distance D in direction (`, b) and Ω is the solid angle surveyed.

Thin Disc, Thick Disc and Halo

Observations towards the Galactic Poles show that the vertical distribution of K dwarfs near theGalactic Plane is close to exponential, with h ' 350pc. For A stars h ' 200pc and the youngest, veryluminous, stars lie in an even thinner layer. (Older stars have had time to be scattered to greaterheights by gravitational interactions). For the K dwarfs, there is a ‘tail’ to higher z distances, thesignature of the ‘thick disc’, with a scale height ' 1350pc. Further out still, another componentdominates, the halo, a diffuse spheroidal extension of the bulge, with stellar density fall-off ∼ r−3.In the volume near the Sun, called the Solar Neighbourhood, around 98% of stars belong to the thindisc, ' 2% to the thick disc and ∼ 0.1% are halo stars.

Also associated (mostly) with the halo are the GCs. Other spirals and ellipticals have systems ofGCs like those around our own Galaxy. The distribution of luminosities of globulars (i.e. the GCLF),between ∼ 104 and ∼ 106L, is virtually identical, irrespective of the galaxy they surround. Theuniversal, roughly gaussian, GCLF shape makes the V magnitude of its peak, MGC ' −8.5, into auseful distance indicator.

The number of GCs around a galaxy varies widely. Our Galaxy has ' 130, while NGC1399, thebrightest galaxy in the Fornax Cluster, has ' 1600. The numbers go up with the luminosity of thegalaxy, but we can define a ‘specific frequency’ SN as the number per unit galaxy light (usually per108L). Most spiral galaxies have SN ' 0.4, but Es generally have larger specific frequencies, ' 2.The largest ellipticals in clusters have even larger SN .

Stars in Galaxies

Stars are divided into spectral types, based on the strength of the absorption lines of various elements.The stars with the strongest Balmer lines of hydrogen (Hα etc.) were assigned to spectral type Aand those with the next strongest to type B. A more useful sequence is based on the stars’ coloursor surface temperatures, as in the H-R diagram. Main sequence (MS) stars are ordered by decreasingtemperature O, B, A, F, G, K and M. The MS is a mass sequence from the most massive, hot, blue,high luminosity O and B stars, down to low mass, cool, red, low luminosity M stars. The generalshape of the spectra roughly matches that of black bodies with T > 30 000K for O stars, through6000K for G stars like the Sun, to ≤ 3000K for M stars.

In the spectra, again working from O to M, we see first primarily the Balmer lines of neutralhydrogen. These are not particularly strong in O stars as the hydrogen is almost all ionized, butincrease to a maximum for A stars. Recall that Balmer lines are from transitions between the firstexcited state and higher energy states of the electron. Any photons which are energetic enough (ofshort enough wavelength), can remove the electron entirely, ionizing the atom. This causes the ‘BalmerLimit’ at about 360 nm where the spectrum drops off sharply.

Reducing T further, the Balmer lines get weaker again, but absorption lines due to other elementsbecome more prominent. In G stars the most important are the Sodium D lines (' 590 nm), theMagnesium b feature (' 520 nm), the G band of the CH radical (around 430 nm) and the H andK lines of singly ionized calcium, CaII (at 400 nm). Note that optical astronomers traditionally useAngstroms (A) rather than nm, where 10 A= 1nm. The combined effect of the Balmer limit and theCaII H and K lines sharply cutting off the spectrum is therefore known as the ‘4000 A break’. It isa key feature of the spectra of all intermediate temperature stars. In cooler stars, molecules such asTiO provide characteristic absorption bands at the red end of the spectrum.

For a galaxy as the sum of many stars, the spectral shape and features reflect the mix of stars thegalaxy contains. The hottest stars dominate the blue end of the spectrum so the lines there shouldbe characteristic of O, B or A stars, if present. Likewise the red light should be produced mostly by

16

K and M stars (assuming there are enough of them), with typical features at long wavelengths. Wecan then reproduce the shape of the galaxy spectrum by adding the correct relative number of starsof different types and masses, using a computer code for ‘population synthesis’.

Stellar Lifetimes

The luminosity L of a MS star is related to its mass M approximately as

L ∝Mα

with α ' 3 for M < 0.5 M and α ' 4 above that. Above ' 10 M the slope flattens out again toα ' 2. Massive stars generate more light per unit mass than smaller stars. O and B stars – massesaround 40M and 5M – have L in the range ∼ 106 down to 100L.

Assuming a fixed fraction (∼ 10%) of a star’s original H is burned to He at the core, the time itremains on the MS

τms ∝ML∝M1−α.

With the α values above, high mass stars have very short lifetimes relative to stars like the Sun. TheSun will stay on the MS for ' 1010 years, while a 15M star will exhaust its fuel in 107 years. Thusa substantial contribution from luminous blue stars in a galaxy’s spectrum implies that stars haveformed in the galaxy in the past 107 years. If we see no sign of such stars, e.g. E galaxy spectra, nostar formation has taken place in the recent past.

Not all stars are on the MS and in terms of generating light, it is the red giants that are mostimportant in Es. These are stars which have evolved off the MS (due to running out of fuel in theircentres), becoming much larger, and thereby more luminous despite their low temperatures.

Stellar Population Evolution

The stellar population in a galaxy is determined by when the stars formed – the star formation historyor SFH – and the distribution of masses of the stars which formed – the initial mass function or IMF.Although the IMF may vary in extreme environments or at very early cosmic times, a universal IMFis a reasonable approximation. The most often used is the Salpeter mass function dNi/dM∝M−2.35.A better approximation may be a double power law, with slopes ' −2.3 for massive stars and ' −1.3below about 0.5M. The overall range of stellar masses is often taken to be 0.1 to 100 M.

For a single generation of stars, all the stars initially populate the MS, but after a short time themost massive (bluest) stars will move off to become giants. The MS will thus become truncated atthe bright end. As time goes on, stars of lower M also evolve into red giants and the ‘main sequenceturn-off’ moves gradually downwards as seen in the H-R diagrams of star clusters of different ages.The colour of the overall stellar population becomes redder as we replace blue MS stars by red giants,and the red colours of Es imply that they (like GCs) have very old stellar populations, with starsformed ' 1010 years ago. The strengths of some absorption features in the spectra, e.g. the Balmerlines, are also age indicators.

In stellar populations with no recent SF, the red giants dominate the total light (since they aremuch more luminous than red dwarfs). This gives us a way to calculate how the total luminosity ofan E galaxy should change with time. L will be proportional to the number of red giants presentat any time, t, after the stars formed. This is the same as the number of stars with MS lifetimesbetween t − ∆trg and t, where ∆trg is the time that stars remain as red giants. MS lifetimes arefunctions of mass, so the mass of a star which has a MS lifetime τms is M(τms). The given range ofages corresponds to a range of masses, M(τms) to M(τms −∆trg) = M(tms) + ∆M. So, if we writethe IMF as the number dNi of stars formed with masses between M and M+ dM, then the numberof red giants at time t will be

Nrg =dNi

dM ×∆M =dNi

dM|dM||dτms|

∆trg

For an IMF, dNi/dM∝M−1−x, with x ' 1.35, and lifetimes τms ∝M−β with β ' 3

L ∝ Nrg ∝ t−1+x/β ' t−0.6.

17

Thus E galaxies were more luminous in the past. We expect to see this effect if we study galaxiesat high redshift, because of the long light-travel or ‘look-back’ time involved. Detailed calculations ofE galaxy evolution – made in the same way as for population synthesis models, but now including theknown evolution of each mass of star – give similar results to our approximate one.

Metallicity

Nuclear processing in stars leads to the creation of heavier elements (known collectively as ‘metals’)from H and He. By convention, the fraction by mass of H in a star is written as X, that of He as Y(universally ' 0.25) and that of metals (everything else) as Z. About 2% of the Sun is made up ofheavy elements, i.e. Z = 0.02. Old stars shed their outer layers, either quietly via stellar winds andthe production of planetary nebulae or spectacularly in supernova explosions, so processed material isreturned to the ISM to be incorporated into the next generation of SF. As a galaxy evolves it producesstars of higher Z. Since the atoms or ions of these metals absorb photons (e.g. the Mg and Ca lines),the more heavy elements present the more light is absorbed. The absorption occurs mainly at theblue end of the spectrum, so stars with higher Z look redder than those of low Z. A galaxy made upof metal rich stars will be redder than one composed of metal poor stars. But the age of a populationof stars also changes its colour, leading to an ‘age-metallicity degeneracy’; an early type galaxy whichlooks bluer than others might be younger or more metal poor.

While Es and S0s are generically red, there is a very tight correlation between absolute magnitudeand precise colour such as U−V ; lower L Es are slightly bluer than bright ones. Detailed spectroscopyshows they have metal poor stars, which implies a correlation between Z and M for E galaxies. Alower mass galaxy has a lower escape velocity, so it is easier for heavy elements produced in onegeneration of stars to be lost from the galaxy, via ‘galactic winds’, before they can be incorporatedinto later generations, keeping Z down. The correlation carries on from giant Es down to dEs, whileLocal Group dSph have even lower Z. The tightness of the colour - magnitude relation was originallytaken to imply that E galaxy formation was ‘co-eval’, a once only event in the history of the universe.Strictly though, it says that all the stars in Es formed at about the same time; they might haveassembled into large galaxies later. Not all stars in a galaxy need have the same Z and even in Es thestars at the centre tend to be more metal rich (up to Z ' 2Z) than those further out.

Note that metallicity is also given in terms of the number of atoms of a given element relative toH. For example 12 + log (O/H) is the logarithm of the number of O atoms for every 1012 H atoms.For the Sun 12 + log(O/H) = 8.93. [Fe/H] means the logarithm of the ratio of Fe to H atoms in anobject, compared to what it is in the Sun; [Fe/H] = log (NFe/NH) − log (NFe/NH).

Stellar Populations in Spirals

The disc, and especially arms, of a spiral are blue while the bulge is yellow, suggesting they containdifferent sorts of stars. In the 1940s, Baade observed that the brightest stars in the bulge of M31 werered giants, but those in the disc were blue supergiants and proposed the existence of two separatestellar populations. Population I stars occur in the disc and arms and Population II stars in the bulge,GCs and stellar halo.

Studies of the H-R diagram for stars in GCs in our Galaxy show that they are generally oldsystems, with MS turn-off points indicating ages > 10 Gyr. The presence of variable stars knownas RR Lyraes confirm these ages. They are lowish mass post-MS core He burning stars with closelysimilar luminosities, ' 50L, making them useful distance indicators. Stars take at least 8 Gyr toreach this stage of their evolution.

Using precise star colours, most Galactic GCs appear very metal poor with [Fe/H] ' −2, thoughsome reach [Fe/H] ' −0.6 (Z ∼ 1/100 to 1/4 solar). Halo stars in the outer parts of the Galaxyhave similar ages and Zs. (There are some GCs, closer-in, with somewhat higher Z and younger ages,though still several Gyr).

Bulges of external galaxies appear photometrically to be an inwards continuation of the stellarhalo, but the evidence from our Galaxy is that the halo and bulge differ in make-up. Although allbulge stars are old, their Zs show a wide range and are typically ' 0.5Z, compared to < 0.01Zin the halo. (The recently discovered HE 0107-5420 holds the record as the metal poorest halo starknown, with [Fe/H] ' −5.3, 1/200000 Z).

18

Population II stars are identified with those formed early in the histories of the galaxies. Theuniverse has a well determined age close to 13.7 Gyr, so the galaxies cannot be older than about13 Gyr. At the other extreme, ‘open clusters’ in the Galaxy (much less populous than GCs, with∼ 102 − 104 stars) still contain bright MS stars, so must be young. The Pleiades stars are only about100 Myr old and very few open clusters have ages > 1 Gyr. The older ones are in the outer partsof the Galaxy; in the inner parts, gravitational interactions lead to the dispersal of clusters on Gyrtime scales. Their H-R diagrams indicate metal abundances not very different from the Sun’s. Theseare classic Population I stars. Population II stars thus formed before Population I. Any even earliergeneration of stars, not now visible, was Population III.

Since bulge stars are older and therefore redder on average than disc stars, the overall colour ofspirals changes systematically with B/D along the Hubble sequence. Early type spirals are red, likeS0s (B − V ∼ 0.9), while late types are blue (B − V ∼ 0.5). The spectra of early type spirals andS0s are similar to K stars, with Ca H and K, the G-band etc. Sc s have much more energy at shortwavelengths from O and B stars, as well as emission lines from regions of SF.

The Stellar Mass Function

Considering just MS stars, we can determine the LF – e.g. for the Solar Neighbourhood – in exactlythe same way as for galaxies. This differs from the IMF, dNi/dM, in two ways; the translation betweenMS luminosity and stellar mass (via models of stellar structure) and the fact that the high M starsobserved must have formed in the last few Myr, whereas low M stars have been accumulating sincethe Galaxy formed ∼ 1010 years ago. The present day (t = t0) number density of MS stars of mass M

N0(M)dM = dNi(M)

∫ t0

t1(M)Ψ(t)dt

where Ψ(t) is the total star formation rate (see later). The interval t0 − t1(M) is the MS lifetime ofstars of mass M, given empirically (in years) by

log τms(M) ' 10.0− 3.42 log (M/M) + 0.88 (log (M/M))2.

(For near solar mass stars, τms ' 1010M−3.4 yr). For individual young clusters such as the Pleiades,we can assume that all the stars formed in a single SF event. Clusters have the advantage of bettersampling the very faint objects, to extend the IMF down to red dwarf and even brown dwarf stars.

A consequence of the form of the IMF is the rate of production of supernovae (SNe). For a SalpeterIMF, we expect 1 star above 8M – able to produce a Type II (core collapse) SN – for every ' 135M

turned into stars.

Gas in Galaxies

The interstellar medium (ISM) contains a variety of components depending on the local conditions,viz. neutral H, ionized H, molecular H, He and dust. The relative importance of the ISM is quantifiedby the H mass to optical light ratio MH/LB (in M/LB). For an early type spiral this may be< 0.1, rising to ' 0.8 for late types. For average spirals the mass in gas, compared to stars, is ' 0.1 to0.2. The rare gas rich galaxies, with ∼ 1/2 their mass in gas, are frequently LSBGs, suggesting thatthey have been inefficient at forming stars.

HI

In 1944, van de Hulst predicted that neutral hydrogen (HI) would emit in the radio at 21.1 cm(1420 MHz) due to a ‘hyperfine’ transition. Besides the quantum numbers representing its ‘orbit’, theelectron has a ‘spin’ quantum number. If this spin is in the same direction as that of the proton, thenthe atom has a slightly different energy than if the two spins are opposed. If the electron ‘flips’ fromone spin state to the other then the excess energy (∼ 6× 10−6eV) is released as a 21 cm photon.

Although the transition is quite unlikely – an electron will flip only once in 3.5 × 1014 s – there isso much HI in spiral galaxies that it produces by far the most important radio line and is the key to

19

studies of the ISM in our Galaxy and other spirals. The HI mass of our Galaxy is about 6× 109M,corresponding to 1066 hydrogen atoms, 3× 1051 of which will flip each second.

Until recently, HI surveys targeted optically catalogued galaxies, generally large galaxies at knownz in order to ‘tune’ the radio observations to the correct frequencies, so any optically inconspicuousgalaxies with large HI contents were underrepresented.

Technical advances with ‘multi-beam’ receivers made it possible to search large areas of sky.HIPASS, at Parkes, and HIJASS, at Jodrell Bank, have covered the whole local volume out tov ' 10000 km s−1, D ' 140Mpc. The detection limit is MHI/M ' 106D2, i.e. they can de-tect 106M of HI at D = 1Mpc, but need 1010M for a detection at D = 100Mpc.

Observed HI masses span 106.8 to 1010.6M with a peak at about 109.5M. Allowing for thesmaller volumes surveyed for smaller MHI produces an HI mass function (HIMF) with a similarshape to the optical LF. The low mass end has a power law slope ' −1.3 but is steeper for latertypes. Radio detections need to be made over several frequency channels to distinguish them frominterference so narrow lines will not be detected. This biases against low mass galaxies with lowrotation velocities, and also against nearly face-on galaxies.

Brightness Temperature

A cloud of HI will both emit and absorb radiation along our line of sight, so in passing through somethickness dx, the resultant intensity I must vary according to

dI

ds= ε− κI

where ε is the emissivity and κ is the absorption coefficient. If we define the ‘source function’ S to bethe ratio of emission to absorption, ε/κ, and use the optical depth

τ = −∫

κdx ,

the radiation transfer equation isdI

dτ= I − S.

If S is constant across a medium of total optical depth τν at frequency ν, then, neglecting stimulatedemission, the observed intensity outside the cloud will be

Iν = S(1− e−τν ).

The hyperfine levels are separated by an energy hν = kT for T = 0.07K, very much less than typicalHI temperatures of 100K, implying perfect LTE. Hence the source function is the Planck function

Bν(Ts) =2hν3/c2

ehν/kTs − 1

where the ‘spin temperature’ Ts is essentially equal to the kinetic temperature of the atoms.Since hν << kTs we take the Rayleigh-Jeans limit

Bν(Ts) =2kTsν

2

c2

which implies

Iν =2kTsν

2

c2

(

1− e−τν)

.

Radio astronomers define ‘brightness temperature’ Tb such that

Iν =2kTbν

2

c2,

so in these termsTb = Ts

(

1− e−τν)

.

20

HI Observations

For an optically thick source (large τν) Tb = Ts, so we can say nothing about the total HI in the lineof sight. However, when τν is small Tb ' τνTs, so the brightness temperature is proportional to theoptical depth and hence the HI column density NH . Numerically,

τν =CNHφν

Ts

with C = 2.57 × 10−15 for T in Kelvin, ν in Hz and column densities in atoms/cm2. φν describeshow ‘spread out’ the emission is in frequency; φνdν is the fraction of atoms which can emit or absorbphotons of frequencies ν to ν + dν. Inverting this and integrating

NH = 3.88 × 1014∫

Tsτνdν = 3.88 × 1014∫

Tb(ν)dν.

For a given line the observed ν is equivalent to a Doppler velocity v (in km s−1), so alternatively

NH = 1.82 × 1018∫

Tb(v)dv.

For a galaxy at distance D, subtending solid angle Ω, we can integrate over its physical area D2Ω toobtain the total number of H atoms

NT = 1.82× 1018D2∫

dv

∫

Tb dΩ.

If D is in Mpc and we revert from Tb to observed flux Fν – measured in units of Janskys, where 1 Jy= 10−26 W m−2 Hz−1 – we obtain the total mass

MHI = 2.36 × 105D2∫

Fνdν M.

Even ignoring any rotation, turbulent velocities in the gas will generate a Doppler width ' 10 km s−1,corresponding to δν ' 5 × 104 Hz. For a hydrogen density of 0.5 atoms/cm3, say, we get a columndensity NH ' 1021cm−2 for a pathlength of 2× 1021cm or 700 pc. If φν is a gaussian of width δν, φν

takes the value 1/(√

πδν) ' 10−5 at the centre of the line. Thus with Ts ' 125K, τν ' 0.2, so despitethe huge amounts of HI in a galaxy, along most sight lines the HI is optically thin.

Since each photon has energy hν, the rate at which photons will be received, in some bandwidth∆ν, from a source of angular size Ω (or with a radio telescope of beam size or resolution Ω) will be

nν = 2τνkTs

hν

ν2

c2∆ν Ω m−2s−1.

A 100m dish with a beam size ' 1 square arc minute and a receiver with resolution ' 1 kHz observinga cloud with Ts ∼ 100K should therefore collect about 100 21cm line photons per second per channel.These will have a total energy of 10−3 eV.

HII Regions

Luminous young stars emit ionizing photons, so large parts of the ISM are in the form of HII. A hotstar (surface temperature Ts > 20000K) inside a H cloud of reasonably uniform density will ionize allthe gas in a region around the star known as a Stromgren Sphere, making an HII region.

In equilibrium, the ionization rate must equal the rate at which electrons recombine with protons.If the star emits NUV ionizing photons per second this will ionize the atoms in a volume of radius rs

such that

NUV =4π

3r3snenHα2

21

or

rs =

(

3NUV

4πnenHα2

)1/3

where ne and nH are the electron and hydrogen densities and α2 is the recombination coefficient forelectrons not falling directly to the ground state. (Those that do produce another ionizing photon, sohave no net effect). The captures to the n = 2 state or above result in a further non-ionizing photon asthey drop to the ground state, most of them eventually producing Balmer lines, especially the 656.3 nmHα line, so we can use Hα emission to measure ionizing radiation. For an O5 star, NUV ' 5 × 1049

s−1, and for a typical cloud ne ' nH ' 109m−3, while at ' 8000K α2 ' 3 × 10−19m3s−1. Thus weobtain a Stromgren radius ' 1 pc. HII regions are usually sharply ‘ionization bounded’, because werun out of ionizing photons, rather than atoms to ionize (the density bounded case).

HII regions also produce ‘free-free’ emission in the radio when an electron loses energy by beingscattered by a proton, but not captured. This emission is characterised by the mean energy of theelectrons (hence is ‘thermal’ radiation) and its intensity depends on nenH , integrated along the lineof sight. Since the change of energy in a scattering can take any value, free-free emission leads to aradio continuum, with a spectrum Fν ∝ ν−α and α ' 0.1, except at low frequencies where it turnsover to ν2 (as for black body radiation) due to ‘self-absorption’, i.e. the medium becomes opticallythick to its own radiation.

Molecules

Molecules radiate via rotational and vibrational transitions of the whole molecule. H2 will be themost abundant molecule, but is hard to observe since it is symmetrical and has no dipole moment.Only weak quadrupolar transitions are possible and there are no useful optical or radio emission lines.Observers use measurements of the much rarer but more amenable CO molecule, though this requiresthe translation factor between the mass in CO and that in H2. The value is still uncertain and maybe different for metal rich and metal poor galaxies, but a reasonable conversion is

NH2' 1.2× 104NCO

where the CO (strictly 12CO) is measured via rotational transitions at 1.3 and 2.6 mm. These areparticularly strong for molecular densities 109 − 1010 m−3 and transitions are excited even at verycold interstellar temperatures, 10 - 20K. Our Galaxy has a molecular mass ' 3× 109M. Lines fromthe isotope 13CO or other rarer molecules with large dipole moments (e.g. NH3, HCN) can be usedat higher densities; the centres of Giant Molecular Clouds (GMCs) can reach ∼ 1012molecules/m3.

Gas Distributions

The atomic and ionized ISM in most spiral galaxies is in a thin layer, with scale height ' 150 pc. Itsradial extent is often much greater than that of the stars. The gas layer often ‘flares’ out, far from thecentre, and it may also ‘warp’ away from the central plane (as defined by the stars). All these effectsare seen for our Galaxy, where the extent of the HI is at least 25 kpc.

Many spirals have their peak HI column density some distance from the centre; M31 has a hole inthe distribution inside a few kpc. Overall HI density varies relatively little across discs. Peak valuesare ≤ 1021 atoms/cm2, while the radius at the 1020 atoms/cm2 level is on average 2R25. This is adrop of a factor ∼ 10 in NH while the optical SB falls by about 6 magnitudes, a factor 250.

HI size correlates with optical size, while HI mass correlates with both, indicating similar mean HI

surface densities in most spirals. This standard value is probably determined by optical depth effects;a layer with NH > 4× 1020cm−2 (3M/pc2) is optically thick to UV photons which could dissociateH2 molecules. Denser layers will self-shield their inner parts, allowing molecules to form from HI.

The molecular gas, mostly in distinct clouds with densities > 2×108m−3, is more tightly compressedinto the plane than the HI, with half-thickness < 100 pc. It is mostly in the inner regions of spirals,often concentrated in a ring; our Galaxy has a high molecular density ' 4 kpc from the centre.

Most of the densest (and coldest) gas in spiral galaxies is in the arms. This is further concentratedinto the GMCs (our Galaxy contains ∼ 103) with M > 105M and diameters > 10 pc . MolecularClouds also exist outside the arms but are much smaller, ∼ 2 pc across with M < 100M.

22

GMCs have even denser cores where new stars are formed, so the SF regions and HII regionsappear along the arms. The dense material contains dust, so we see dust lanes along the inner edgeof the arms. In our Galaxy we can follow the spiral arm pattern with the same tracers, but we needto determine distances to individual GMCs, OB associations, open clusters or HII regions.

The multi-phase structure of the ISM is in approximate pressure equilibrium, with cool densematerial, including the molecular gas, at T < 100K and n > 107 atoms/m3, a mixture of neutral andionized warm interstellar material (WIM) at a few 103 K and n ∼ 105 m−3, and hot ionized plasma atT ∼ 106K and n ∼ 103 m−3. Most of the gas mass is in the cool component, but the hot diffuse phasefills most of the volume. The WIM extends to heights 1-2 kpc above the plane and the hot plasmaforms an extensive halo around the warm clouds. The different phases are in continual motion. Coolgas can be heated by SN explosions and rise up out of the plane into the halo before cooling andfalling back in, making a ‘galactic fountain’.

The Sun is near the outer edge of the Sagittarius-Carina Arm. Slightly further out than the Sunwe see a short filamentary arm segment, the Orion Spur, but the next major arm is the Perseus Arm' 2 kpc beyond the Sun. The Sun is surrounded by the Local Bubble of hot gas about 300 pc across,shown up by X-ray and extreme UV observations. Inside the bubble, the density is < 2 × 104 m−3

and T ' 106K. On a smaller scale (' 8 pc) there is the mostly neutral (T ∼ 7000K) Local InterstellarCloud, with the Sun near one edge.

In the outer parts of discs there is insufficient NH to absorb all the ionizing photons in generalintergalactic space (emitted by other galaxies and quasars). Thus below NH ∼ 1019 cm−2 we expectthe HI to drop sharply as any hydrogen present will be ionized.

Large halos of ionized gas around galaxies can be detected by their absorption of light from distantquasars, producing QSO absorption line systems (QSOALS). Optical spectra of quasars typically showthe lines of the MgII doublet at 280 nm or, at higher (absorber) redshifts, the UV lines of CIV at155 nm. The large number of QSOALS seen per unit redshift interval (i.e. distance range) and theknown number density of largish galaxies imply that the absorbers should have extents 40 - 80 kpc.Far UV observations from satellites such as IUE (the International Ultraviolet Explorer), show thesame absorption lines in the spectra of stars, due to ionized gas in our Galaxy.

Gas in Ellipticals

In present day spirals mass lost as stars evolve mixes in with the existing ISM, providing fuel forfurther SF. The same must have happened in the evolution of early type galaxies to increase Z to theobserved levels. Currently, though, most early types contain negligible amounts of HI, so where hasthe mass lost since SF ceased gone? For a standard IMF the stars in a 1010L E will collectively lose1-2 M/year, so over several Gyr ∼ 1010M of gas should be cycled back into the ISM.

It was finally observed by X-ray satellite observatories, as giant Es were found to contain hot X-rayemitting plasma. If the KE per unit mass of the material lost from stars is roughly the same as thatof the stars themselves, then

3

2kT ' 1

2m <v2 >

where m is the average particle mass and <v2 > is the mean square speed of the stars. Since very hotgas will be completely ionized, the mean particle mass is around half the proton mass so

T =mp <v2 >

6k' 2× 106 <v2 >

(500 km s−1)2K.

The X-ray emission is ‘free-free’, or ‘thermal bremsstrahlung’, radiation from electrons scatteringoff protons, so

LX ∝ nenHΛ(T ) ∝ n2HΛ(T ) ∝ n2

HT 1/2

where Λ (∝ T 1/2 for bremsstrahlung) is a measure of the cooling rate. The ‘cooling time’ required toradiate away the energy of the gas

tcool =3

2(ne + nH)kT/nenHΛ(T ) ' 3kT

nHΛ(T )> 1Gyr

at typical particle densities ≤ 10−8 m−3. [See HEA notes for details.]

23

Dust

Individual dusty nebulae such as the Coal Sack obscure the stars behind them, so are seen as darkpatches within the Milky Way. Dust can also result in bright ‘reflection nebulae’, seen via scatteringof light from nearby bright stars.

In the disc of the Galaxy, about 1% of the ISM mass is in dust with mean density ng ' 10−6

grains/m3 or about 1 dust grain per 1012 atoms in the ISM. The optical depth along a sightlineof length x is τλ = σgngx, where σg = πa2

gQλ is the effective cross-section of a grain of radius ag.(Qλ is the relative extinction coefficient, which measures how efficiently a grain removes photons ofwavelength λ). Observationally, the dust absorption follows

AB ' NH/(2 × 1021 cm−2).

The variation of extinction with λ, the extinction curve, depends on the make up of the interstellargrains. The Galactic curve is roughly τ ∝ λ−1, but shows a bump at about 220 nm due to very smallparticles, ' 0.01µm in size, probably made of graphite. The UV extinction in general is thought tobe produced by even smaller silicate grains, while the visible part of the curve may result from icygrains ' 0.1µm across. There is also a contribution from PAHs (polycyclic aromatic hydrocarbons),consisting of 20 to 100 C atoms and with a size ' 1 nm. The extinction curves for the LMC and SMCare different to the Galactic one as a result of the lower Z of their ISM. Since different elements aremore ‘depleted’ on to grains than others, some gas phase ISM abundances differ substantially from‘standard’ cosmic (or solar) values.

Optical Depths

The observed SB of a galaxy disc will change as we tilt it relative to our line of sight. The pathlength through the stellar distribution must increase, thereby increasing the surface intensity, but onthe other hand, for dust filled discs, obscuration must also increase as we look through longer pathlengths. In a realistic model, the absorbing dust will be mixed in with the stars, or in a thinner layerin the middle of the stars, so doesn’t act as a simple absorbing screen.

Consider a uniformly mixed slab of physical thickness X containing stars of volume emissivity εand dust of total optical depth τ (i.e. optical depth per unit path length τ/X). In the face-on casethe observed intensity from an element of thickness dx a distance x into the slab will be

dI(0) = εdx e−τx/X

so integrating through the slab

I(0) =εX

τ(1− e−τ ) = I0

(

1− e−τ

τ

)

where I0 would be the intensity in the absence of dust. Thus the face-on absorption is

A = −2.5 log

(

1− e−τ

τ

)

not simply −2.5 log e τ .If we tilt the galaxy by an angle i we change the effective depth by a factor sec i and

I(i) =εX

τ[1− e−τ sec i] = I(0)

[

1− e−τ sec i

1− e−τ

]

.

For the optically thin case τ << 1I(i) = I(0) sec i

while for the optically thick case, τ >> 1,

I(i) = I(0)

24

because we see just the top layer of stars, ∼ 1 optical depth thick, regardless of inclination. Noticethat the area of the galaxy image will decrease as cos i, so in the optically thin case the total flux (orapparent magnitude m) is unchanged, while for τ > 1 m will be fainter for inclined galaxies.

The light absorbed in the optical is re-emitted in the FIR. LFIR/LB depends both on the mean τand on how clumpy the absorbing dust is, e.g. around star forming regions with the rest of the discrelatively clear. For our Galaxy we know that τ through the disc at the solar position ' 0.5 (AB ' 0.2magnitudes towards the poles) and should be higher, τ ≥ 1, nearer to the centre. This may be typicalof spiral discs.

Star Formation

As we observe HII regions and bright blue (short lived) stars in the spiral arms, these are importantareas for SF. SF regions show features of different ages, detectable in different ways. When molecularclouds collapse, their dust content heats up to ' 30-50K leading to FIR emission at 100µm. As proto-stars form, dust near them reaches T ' 1000K, thus emitting in the near infra-red at 3µm, while thesurrounding layers are still at only 100K. When a star reaches the MS (assuming it is massive andtherefore hot enough) it will ionize its surroundings, forming a compact HII region detectable at radiowavelengths. As the hot ionized region expands, the dust cools back to FIR emitting temperaturesbefore dissipating, leaving the stars visible at optical wavelengths as an OB association or open cluster.Once the SF has taken place, the stars start to feed back energy and material into the ISM, allowingthe process to repeat. Feedback occcurs both via the stars’ radiation (heating and ionizing photons)and mechanically from mass loss as the stars age or more spectacularly in SN explosions which caneach deposit ∼ 1044J into the surrounding medium. Shock waves propagate ahead into the molecularclouds, compressing the gas and instigating gravitational collapse and the formation of further newstars on a time scale ∼ 1 Myr (‘sequential star formation’).

The nearby Orion Nebula region shows all these features. OMC1 is a dense molecular core ofseveral solar masses behind the main nebula and is the least evolved (youngest) component. There arenumerous infra-red sources, including the Kleinman-Low and Becklin-Neugebauer objects in the coreof OMC1, the latter apparently an obscured 10 M B0 star just arriving on the main sequence. Othermolecular material exists in parsec scale clumps containing 10-100 M. The stars of the Trapeziumrepresent a 106 yr old OB association, the brightest of several hundred stars within a 0.3 pc radiusinside an HII region. These are now dissociating the H2 via their UV radiation. The Trapezium isthe youngest (and smallest) of the OB associations in Orion and adjoins what is left of the moleculargas, which may all be gone in a few 107 yrs.

The Jeans Mass

In a molecular cloud of size r and density ρ at temperature T , hydrostatic balance between inwardgravitational force and outward pressure force requires

dP

dr= −GM(r)ρ

r2.

At the edge of the cloud M = 4πr3ρ/3, so approximating dP/dr by P/r,

P ' 4π

3Gρ2r2.

Also, for a perfect gas

P =ρkT

µmp= ρc2

s

where µ is the mean molecular weight (' 2 if the cloud is almost all H2) and cs is the sound speed.Matching these expressions

r '(

3kT

8πmpGρ

)1/2

' 107(

T

ρ

)1/2

m

25

for T in Kelvin and ρ in kg m−3. This is the Jeans radius. For a typical GMC, T ' 10 and ρ ' 10−15,so r ' 1015m ∼ 0.1 pc. The mass contained in such a region is the Jeans mass

MJ =

(

3

4πρ

)1/2(

kT

2Gmp

)3/2

' 1030kg ∼ 1M.

If the same mass is squashed into a smaller space the gravitational term exceeds the pressure termand the clump will continue to get smaller and denser. Thus the mass scale on which clumps aresuceptible to collapse is of the right order for forming stars. It is much less than the mass of a wholeGMC. A more sophisticated treatment, using the equations of gas dynamics and Poisson’s equationgives essentially the same Jeans criterion.

Global Star Formation

The total rate at which galaxies are forming stars can be estimated from a variety of observables.Foremost is Hα emission, which measures essentially the number of ionizing photons from youngO and B stars. Assuming a standard Salpeter IMF (between 0.1 and 100M), we can translate thenumber of massive stars to the total mass of gas turned into stars. Thus Hα luminosity is proportionalto the current star formation rate (SFR). Numerically,

SFR =L(Hα)

1.3× 1034WM/yr.

Typical early type spirals have SFRs ∼ 0.1 to a few M/yr. Mid-type spirals have SFR ∼5M/yr, rising to 10-20M/yr for Sc s. ‘Starburst’ galaxies, undergoing a sudden galaxy-wide burstof SF can have SFRs > 100M/yr. For an L∗ galaxy and a typical M/L, we expect a stellar massa few ×1010M. This can be produced in ∼ 1010yr by mid to late types operating at their currentrates, but early types must have had higher SFRs in the past.

Larger spirals have ‘more of everything’, including more SF, so it is useful to normalize the SFR.We can do this by dividing the derived SFR by the optical area, to give rates inM/yr/kpc2 (typically' 0.01), or we can utilise the spectra and use the ‘equivalent width’ of the Hα line. This is definedsuch that if the total flux in a line is F (line), and the intensity (per A) in the continuum around theline is I(cont), then the equivalent width (in A) is

EW =F (line)

I(cont).

In other words, a piece of continuum of width EW generates the same flux as the line. Spirals typicallyhave Hα EWs from a few up to ∼ 60 A (or 100 A for starbursts). Using EWs is essentially the sameas normalizing by the red luminosity, which is equivalent to comparing the current SFR to the totalpast SFR (which built up the population of red stars). SF regions are dusty, so we should allow forthe extinction. Typically a factor ' 2.8 is applied, but it may vary widely as the SF and dust areclumped. In principle we can obtain the extinction from the ‘Balmer decrement’. The ratio of thefluxes of the Balmer lines Hα and Hβ has a theoretically calculable value, so if the observed ratiois different, we can deduce the extra extinction at the wavelength of Hβ compared to Hα and hencethe total extinction. To make Hα based SFR estimates agree with those from longer wavelengthindicators (below), we need to increase the extinction corrections systematically as the SFR increases.This increases the upper limit for ordinary spirals to ' 100M/yr. Integrating over all star forminggalaxies, we obtain a present day SF density ' 0.025M/yr/Mpc3.

Other Star Formation Indicators

Other emission lines can be used as star formation indicators, e.g the lines [OII]λ3727 and [OIII]λ5007.Usefully, the OII line is shifted into the observed optical spectra of distant galaxies when the Hα lineis redshifted out into the IR. For typical spirals, EW[OII] ' 1/3EW(Hα).

26

In the FIR (∼ 10 to 1000µm) there are contributions to the emission from dust heated by thegeneral interstellar radiation field – called ‘infrared cirrus’ in our Galaxy – and from dust in SF regions.This correlates linearly with the SFR, with

SFR ' L(FIR)

2.2 × 1036WM/yr.

In the radio, HII regions and general ionized gas emit thermal continuum radiation, while SFregions give rise to supernovae that accelerate the electrons which produce synchrotron emission.Observationally there is a tight linear correlation between the FIR flux and the radio flux for all kindsof star forming galaxies, so we can also use the latter to determine the SFR. For starburst galaxies,including ULIRGs, the dust extinction is particularly large (2 - 3 magnitudes), so it is preferable touse the infrared or radio indicators.

Another alternative for dusty galaxies is their X-ray emission. X-ray emission is produced by massflow from a donor star onto a companion compact object. In high mass X-ray binaries (HMXBs)the donors are ≥ 8M, so the lifetime is short and HMXBs (with LX ' 6 × 1030W) trace currentSF. (Low mass X-ray binaries – LMXBs – trace the overall stellar mass). Our Galaxy has about 50HMXBs and a SFR ' 3M/yr so

SFR ' LX(HMXB)

1032WM/yr.

Type II SNe also arise from massive, short lived stars, so the SN rate traces the SFR. For mid-typespirals, it is ' 1− 2 SNU, where a ‘supernova unit’ is 1 SN per 100 years per 1010L of starlight. Forthe Galaxy this is about 1 SN every 30 years. The lack of observed SN in the 400 years since Kepler’sSupernova can be explained by interstellar extinction which limits us to seeing ≤ 10% of the Galacticdisc. Many supernova remnants (SNR) can be observed in the radio, for instance Cas A, about 3 kpcaway and only 250 years old, judged by its size and the expansion speed of the shell.

Densities and Time Scales

The SF in a galaxy should depend on the available fuel, and this is quantified by the ‘Schmidt law’

SFR ∝ ΣnH

where ΣH is the gas surface mass density (∼ NH). It was originally proposed that n ' 2, reflectingthe collision rate between pairs of interstellar clouds, but other considerations lead to n ' 1, i.e. moregas allows more stars to form. Observations suggest n ' 1.4 and the SFR per unit area then followsthe ‘Kennicutt law’

ΣSFR

M/yr/kpc2 ' 2.5× 10−4(

ΣH

M/pc2

)1.4

.

Even starbursts follow the above relation, their very high SFRs matched by high gas surface densities∼ 102 − 105M/pc2 compared to 1− 100M/pc2 for normal spiral discs.The average SFR near theSun (over a 109yr timescale)is ≤ 1 solar mass star per pc2 per Gyr. Given the disc scale height, thisis ' 2 stars per 103 pc3 per Gyr. Even modest sized galaxies can contain regions with high SFR,though; the central 10 pc core of the 30 Doradus region of the LMC has 104M of young stars and aSFR (over this small area) equivalent to ∼ 100M/yr/kpc2.

SFR correlates better with total gas density (HI and H2) than just H2, even though the SF occursin the molecular gas. The SFR - ΣH relation also provides a timescale – the Roberts time – on whichthe existing gas will be used up by SF at the current rate. This is frequently much less than 1010 yrs.

The power, ' 3/2, can be understood if SFR depends not only on the gas density ρ, but also onthe time scale. This should be of order the ‘free fall time’, tff , the time taken for a perturbation tocollapse. Imagine a particle falling from the edge of a perturbation of size r (and mass M = 4πρr3/3)to the centre along an infinitely thin ‘ellipse’ with semi-major axis a = r/2. From Kepler’s third law,a is related to the period of the orbit, P , by

P 2 =4π2

GMa3.

27

Since the time to fall to the centre is P/2 we have

tff =

(

3π

32Gρ

)1/2

(of order the Jeans radius divided by the sound speed), so

SFR ∝ ρ/tff ∝ ρ3/2.

Within a given galaxy, SF appears not to occur below a column density ' 4M/pc2. Above that,there is a Schmidt type relation

ΣSFR(R) ∝ ΣH(R)1.3.

The threshold value can be understood from the stability of a gas disc (the Toomre criterion). Also, ifwe set the dynamical time at any radius as simply 2πR/V (R) = 2π/Ω, where V is the rotation speedand Ω the angular velocity, then the radial variation in SFR closely follows ΣH/tdyn, with

ΣSFR ' 0.017ΣHΩ.

This suggests that ' 10% of the gas is turned to stars per orbital period.The environment of a galaxy can affect its SF. Strongly interacting galaxies have very high SFRs

but spirals in clusters like Virgo have lower SFRs – and lower gas masses – than galaxies of the sametype in the field. These ‘HI deficient’ galaxies suggest that gas consumption was faster in the clusterenvironment, possibly as a result of spirals ‘falling in’ to the cluster from the periphery. The largeredshift surveys have shown that SFR decreases with increasing density of the local environment.

Galaxy Dynamics

Besides the stellar populations, and interstellar matter, they contain, the observed structure of galaxiesis related to the dynamics of these constituents. Given their different forms, we might expect these todiffer between ellipticals and spirals.

Dynamics of Ellipticals

Most E galaxies appear flattened. The obvious interpretation of this is rotation, as with the flatteningof the Earth, but the obvious turns out to be wrong! Rotation is usually measured via the absorptionlines in the spectrum of a galaxy as we should see a differential Doppler shift from one side of thegalaxy to the other. Such observations are traditionally done by using a ‘long slit’ on the spectrograph,so that we simultaneously take in light from a slice along the major axis of the image. This resultsin a 2-D image on the spectrograph detector, where one dimension – the dispersion direction – showsthe different wavelengths and the other – the spatial direction – represents positions along the slit.The signature of a rotating galaxy is the presence of tilted spectral lines, indicating changing Dopplervelocities with position. Measurements for giant Es show them to be much too slowly rotating tocause significant flattening.

In a non-rotating system, random kinetic motions of the particles (the ‘temperature’ of the system)support it against gravity. From conservation of energy, the faster a particle is fired outwards in agravitational field, the further it can go before being brought to a halt and falling back.

If we consider the components of the particle’s (i.e. star’s) original outward velocity in the x−,y− and z−directions, then generally speaking it will go furthest in the direction in which it has thegreatest velocity. The same will be true if we consider three separate stars, each moving outwardsalong a principle axis with a different speed, and we can then extend this to a whole ensemble of stars.If the mean square velocity (velocity dispersion) in the z−direction, σ2

z , is less than σ2x and σ2

y, forinstance, then the stars will typically travel less far in the z−direction than the others. Thus our ballof stars will be less extended in the z−direction than in x and y and the system will look flattened.Different values for all the σ2

x, σ2y and σ2

z will give a tri-axial shape. (The same general considerationsare still true if the mass is not all concentrated at the centre, as we have implicitly assumed).

28

In E galaxies velocity dispersions are not measured from the spread of the velocities of individualstars, as they can not be observed separately. Instead we observe the width of the galaxy’s spectrallines. For a given star, a spectral line will be at a specific wavelength, but because all the stars aremoving at different velocities the galaxy spectrum – the sum of those of all its constituent stars –will show a line broadened to an extent dependent on the velocity dispersion. Notice that this isthe one-dimensional velocity dispersion radially along the line of sight, σ2

r . In general the velocitydispersion can vary with position within the galaxy – if not, we have an isothermal distribution (sametemperature everywhere) and in practice galaxies are not too far from this.

Rotation of Ellipticals

Consider a galaxy rotating about the z−axis with rotational speed V , and with isotropic velocitydispersion. The distance the stars can go in each direction scale with the KEs, Kx, Ky, Kz, associatedwith the respective velocity components. Viewing the system ‘side-on’, with y in the radial direction,the axis ratio should follow roughly

b

a' Kz

Kx' σ2

z

V 2 + σ2x

since in the plane there is a contribution from the rotation. In this orientation σ2y(= σ2

z = σ2x) = σ2

r ,the observed radial velocity dispersion, so

b

a' σ2

r

V 2 + σ2r

.

Inverting this(

V

σr

)2

'(

a

b− 1

)

=ε

1− ε

where ε = 1 − b/a is the observed ellipticity. A more sophisticated treatment gives essentially thesame result and applies even for inclined systems.

A flattening b/a = 0.5 (an E5) requires V/σr ' 1; even b/a = 0.8 implies V/σr ' 0.5. A typicalgiant E has σr ≥ 250 km s−1, so we require rotation speeds of the same order, close to those ofspirals. Much lower rotation velocities are usually seen, giving V/σr ∼ 0.2. Thus Es can not beoblate rotating ellipsoids with isotropic velocity dispersions – they must have the anisotropic velocitydispersions discussed above.

Some ellipticals do rotate fast enough to be rotationally flattened, generally the lower luminosityones and almost always disky rather than boxy ones. They probably have ‘buried’ rotating discswithin the main body of their stars, making them more like S0 galaxies.

The Virial Theorem

Exactly as for any other isolated dynamical system, the stars in a galaxy satisfy the virial theorem

2K + U = 0

providing the density distribution is not changing with time. The total KE for a collection of individualstars (labelled by index i) is

K =∑

i

miv2i /2.

If the stars’ velocities are uncorrelated with their masses, this reduces to

2K =M <v2 > =Mσ2

where M is the total mass.The total gravitational PE is the sum of contributions from all pairs of particles (at positions

ri, rj), so

U =∑

i>j

−Gmimj

|ri − rj|

29

(the condition i > j prevents each pair being counted twice). Assuming no separation of the stars bymass,

U = −GM2

rg

where 1/rg is some weighted average of the 1/ri so represents a characteristic extent of the system.For a spheroidal system

U = −αGM2

Re

where Re is the usual (projected) effective radius and α is a constant (for galaxies of a given radialdensity profile) of order unity.

In the limit, the sum becomes an integral, so building up the galaxy by successively adding thinshells of density ρ(r) and mass dM = 4πρ(r)r2dr on top of an existing mass M(r),

U = −G

∫ ∞

0

M(r)dMr

.

For a uniform density sphere of radius rs, M(r) = 4πr3ρ/3 and dM = 4πr2ρdr, so

U = −G

∫ rs

0

16π2

3ρ2r4dr = −16

15π2Gρ2r5

s .

The total mass M = 4πr3sρ/3 and we can calculate that Re = (1− 2−2/3)1/2rs = 0.608rs, so

U = −0.365GM2

Re,

the standard form with α ' 1/3.Returning to the virial theorem

Mσ2 = αGM2

Re

i.e.

M =1

α

Reσ2

G.

Here σ2 is the 3-dimensional velocity dispersion, so for a near isotropic system σ2 ' 3σ2r , giving

M' 3σ2rRe

αG.

If the mass-to-light ratio is M/L then

L ' 3σ2rRe

αG(M/L).

Indeed, from dimensional considerations, we must always have

L ∝ σ2rRe (M/L)−1

and since L ∝ IeR2e we obtain

L ∝ σ2(L/Ie)1/2(M/L)−1

orL ∝ σ4I−1

e (M/L)−2.

30

The Faber-Jackson Relation and the Fundamental Plane

In its original form, the observational Faber-Jackson (F-J) relation follows the simplest predictionL ∝ σ4 or more quantitatively

L ' 2× 1010(

σr

200 km s−1

)4

L

over the range roughly 50 ≤ σr ≤ 500 km s−1. Other estimates suggest that the power of σ might benearer 3 than 4 and depend on the wavelength at which L is measured.

Earlier we had a relationship between an E’s SB (or size) and its luminosity. We can combine thisand the F-J relation in the three dimensional parameter space of L, I and σ. The points all lie close tothe so-called ‘Fundamental Plane’ (FP), with the three parameters linked by a relation approximately

L ∝ I−2/3e σ5/2.

or equivalentlyRe ∝ σ5/4I−5/6

e

We can think of the Kormendy relation (between L and Ie) and the F-J relation (between L and σ)as projections of this tilted plane in 3-D onto the respective 2-D slices.

Ignoring any variation due to different density distributions (called ‘non-homology’) hidden in theparameter α and in the proportionality constant between L and IeR

2e, the virial theorem result implies

ML∝ σ2Re

σ5/2I−2/3e

∝ σ−1/2Re(σ5/2R−2

e )2/5 ∝ σ1/2R1/5e .

Thus more massive Es have higher mass-to-light ratios than smaller ones. This is consistent with moreluminous Es being redder, since redder populations generate less light per unit mass.

Given correlations between L and one or more distance independent observables, we can deduceL from the measured Ie and σ, and then the distance from the observed flux. Thus the F-J and FPrelations provide distance indicators in the range where the properties of whole galaxies are used.

They are often used to determine the ‘peculiar velocities’ of galaxies, relative to the overall Hubbleexpansion. If we know the distance D and the observed recession velocity cz then (to first order) thepeculiar velocity

vpec = cz −H0D.

(Contrary to appearances, this is not dependent on using the correct H0, as it cancels out with thevalue built into the luminosities used in obtaining distances). Investigating velocity fields and galaxy‘flows’ in this way is important in the study of the large scale distribution of mass in the universe asthe peculiar velocities are generated by the gravitational effect of surrounding masses.

Elliptical Galaxy Masses

From the virial theorem, we can deduce a typical M/L ' 5 in the central regions of Es. For nearby Esin which velocities of individual stars (or planetary nebulae) can be measured, the overall M/L ≥ 10.But GCs containing similar old stars have σ2 indicating M/L ' 2. In addition, if we calculate thelight ouput for a certain total mass in stars with a realistic IMF, an old galaxy should have M/L ' 2- 3. Thus Es seem to have more mass than expected.

If the galaxy has satellites, either GCs or dwarf galaxies, then we can use their dynamics todetermine the enclosed mass out to large scales. As a minimum, we require the galaxy to be sufficientlymassive that the satellites do not exceed the local escape velocity at the distance Rsat

v2esc '

2GM(Rsat)

Rsat.

The masses required to bind satellites at radial distances ' 100 kpc imply M/L ∼ 50.

31

We can also use the X-ray emitting gas to measure the gravitational potential via the standardequation of hydrostatic equilibrium

dP

dr= −ρ(r)

GM(r)

r2

where P is the pressure and ρ the density of the gas at radius r. For an isothermal gas with an averageparticle mass µmp, where mp is the proton mass (for completely ionized H, plus He, µ ' 0.6),

P = nkT =ρ

µmpkT.

For a particle density n ∝ r−β, the emission per unit volume ε ∝ n2 ∝ r−2β. Projecting this gives anX-ray SB IX ∝ R1−2β, so we can work back from the observed X-ray profile to n(r) and hence ρ(r).We can also relax the assumption of isothermality if we have a spatially resolved X-ray spectrum togive T (r). Finally, inverting the hydrostatic equation,

M(r) =k

Gµmp

r2

ρ

(

−d(ρT )

dr

)

.

For the isothermal case and an observed power law SB profile, this reduces to

M(r) =βkT

Gµmpr.

Large central cluster ellipticals can contain ∼ 1011M of X-ray gas (' their mass in stars), but thetotal dynamical mass, out to the limits of the X-ray observations, is typically ∼ 1012M.

The general increase of M/L with r, and the fact that M/L is far higher than expected for anypopulation of stars, are manifestations of the ‘missing mass problem’, i.e. the inferred presence ofgravitating mass not accounted for by visible (stellar plus gaseous) matter.

Massive Black Holes

Similar methods are used to study the mass in the nuclear region of Es. Observations at high resolutionshow that σ increases as we approach the centre. Keeping these rapidly moving stars close to the centrerequires a deep potential well. The LG (dwarf) elliptical M32 has a central mass of around 106M

on pc scales. In giant Es like M87, σ2 implies a mass a few ×109M inside the central 10 pc, i.e.ρ ≥ 106M pc−3. M87 also contains a very small central gas disc, observable with HST, whoserotation velocity corroborates the large central mass. This is so concentrated that it is virtuallycertain that it must be a supermassive black hole (SMBH).

The inferred mass, MBH , correlates with the luminosity of the elliptical host (or bulge luminosityin other galaxy types), or with galaxy velocity dispersion, e.g.

MBH ' 1.4× 108(

σ

200 km s−1

)3.5

M.

This apparently universal relationship (the ‘Magorrian relation’) is evidence for a close connectionbetween the growth of the black hole and the formation of the galaxy.

Some ellipticals are strong radio sources and their radio emission is almost certainly powered byaccretion onto a central black hole. However, there is little correlation between MBH and radio powerexcept that radio emission requires a minimum MBH ' 109M.

Dynamics of Spirals

The main dynamical feature of discs is their rotation, which is easy to measure in external galaxies(especially if nearly edge-on) from the relative Doppler shift of lines in the galaxy’s spectrum atdifferent points along the major axis. The rotation speed rises fairly quickly in the inner galaxy beforelevelling off. Early type spirals with the largest bulges have the most rapid rise.

32

For motion in a circle

V 2 =GM(R)

R

and if the mass is centrally concentrated like the light, at large R we can approximate M(R) by the

total mass MT to get ‘Keplerian’ orbits with V ∝ R−1/2. There is no sign of this decline in velocityin optical galaxy spectra, though these are limited to fairly small radii because of the fall in SB withR.

To measure V (R) for our Galaxy, consider a star at Galactic longitude `, a distance D from theSun and at Galacto-centric radius R. Ignoring non-circular motions, suppose the region of the Galaxynear the Sun orbits with speed V0. (The Sun moves relative to this average local motion – the LocalStandard of Rest – at a few km s−1). The star will have a radial velocity

vr = V cos α− V0 sin `

where α is the angle between the line of sight and the tangent to the circular motion of the star. Fromthe sine rule (see diagram)

sin `

R=

sin(π/2 + α)

R0

so

vr = R0 sin `

(

V

R− V0

R0

)

.

For solid body rotation (V ∝ R) there is no relative velocity vr, but outside the central regions,discs show differential rotation, stars further out taking longer to orbit.

Oort’s Constants

If we could observe stars of known ` and D throughout the disc we could determine the entire V (R)directly. However, the disc is full of obscuring dust, especially towards the Galactic Centre, so therotation curve can only be determined for regions close to the Sun. In this case, D << R0, so

R = R0 −Dcos ` = R0 + ∆R.

The radial velocity

vr = R0 sin `d

dR

(

V

R

)

∆R = −R0D

2sin 2`

d

dR

(

V

R

)

R0

= −R0D

2sin 2`

[

1

R

dV

dR− V

R2

]

R0

=D

2sin 2`

[

V0

R0−(

dV

dR

)

R0

]

.

Now define ‘Oort’s constant A’ via

A = −R0

2

d

dR

(

V

R

)

R0

=1

2

(

V

R− dV

dR

)

R0

33

sovr = AD sin 2`.

A is a measure of the gradient of the angular velocity Ω(= V/R) near the Sun and from observationis ' 14 km s−1kpc−1.

Similarly, the tangential velocity of the star, measured from its proper motion, is

vt = −R0D

2cos 2`

d

dR

(

V

R

)

− D

2R0

d

dR(RV )

= D(A cos 2` + B),

where Oort’s constant B is

B = −1

2

[

1

R

d

dR(RV )

]

R0

= −1

2

(

dV

dR+

V

R

)

R0

.

B is negative and is ' −12 km s−1kpc−1. It is a measure of the angular momentum gradient. Noticethat

A + B = −(

dV

dR

)

R0

and

A−B =V0

R0.

The measured A and B imply that V is decreasing slightly with radius near the Sun; a ‘flat’ rotationcurve with constant V has B = −A. Also, assuming that the Sun is about 8.5kpc from the GalacticCentre, we must have V0 ' 220 km s−1.

Vertical Velocities and Surface Densities

The Sun has a vertical velocity (‘upwards’ towards the North Galactic Pole) ' 7 km s−1. Many thindisc stars have vz ' 20 km s−1. Thick disc stars have vz ' 40 km s−1. Comparing the verticalvelocities to the vertical extent, we can estimate the surface mass density; stars will move out togreater distances for smaller restoring forces. If we treat some class of stars as tracer particles, withnumber density n, then from the collisionless Boltzmann equation

d

dz

(

n(z)σ2z

)

+∂Φ

∂zn(z) = 0.

Φ is the gravitational potential and therefore ∂Φ/∂z represents the vertical force, often called Kz. Wealso have Poisson’s equation relating the mass density ρ to the potential,

4πGρ(R, z) = ∇2Φ(R, z) =∂2Φ

∂z2+

1

R

∂

∂R

(

R∂Φ

∂R

)

,

while matching the radial force to the centripetal acceleration,

R∂Φ

∂R= V 2(R).

As V is nearly constant in the outer parts of most spiral discs, ∂V /∂R is small and

4πGρ(R, z) ' d

dz

[

− 1

n(z)

d

dz

(

n(z)σ2z

)

]

.

Integrating ρ over all z gives the surface density Σ(R).Oort first used this method for determining Σ near the Sun in 1932. Allowing for a gradual increase

in σ2z with height, from ' 20 km s−1 to 30 km s−1, the observed n(z) implies Σ ' 60 ± 10 M/pc2

34

(integrated out to ' 1kpc; above that we would be including the halo). This is greater than thedensity in visible stars. However, adding faint white dwarfs and low mass red and brown dwarfs, thestellar contribution ∼ 40 M/pc2 and there is a contribution of ' 12 M/pc2 from gas in the ISM,so ∼ 50 M/pc2 is directly accounted for. (This translates to about 100 grams per square metre, thesame surface density as a sheet of paper!). The amount of dark matter in the disc is quite small.

Approximating the mass distribution by an infinite thin sheet of uniform density, Φ(z) = 2πGΣz,so for fixed σ2

zd

dz

(

n(z)σ2z

)

= σ2z

dn

dz= −n

∂Φ

∂z= −2πGΣn

implyingn = n0 exp(−2πGΣz/σ2

z )

i.e. an exponential distribution with scale height h = σ2z/2πGΣ. For the above densities and velocities,

we obtain h ∼ 200pc, consistent with observation.

Rotation of Gas in the Galaxy

To extend the rotation curve to larger radial distances we need another tracer. This is supplied by thegas disc. For our Galaxy, observing from the inside complicates the determination of the HI rotationcurve since we have no direct way of deciding how far away any particular 21cm emitting gas actuallyis. However, if the HI is continuously distributed throughout the disc, we can use the ‘tangent pointmethod’ for the inner Galaxy.

Consider some Galactic longitude `. Gas close to the Sun will have its velocity mainly across thesky, as will distant gas on the far side of the Galaxy. However, at the point where ` is tangent to thecircular motion we see entirely radial motion and hence the highest vr. At the tangent point

R = R0 sin `

and in generalvr = V (R)− V0 sin `

henceV (R0 sin `) = vmax

r (`) + V0 sin `.

Once we have V (R) we can map the HI in the Galaxy by finding the distance from us, in direction`, at which vr takes a particular observed value.

If we look specifically at ` near 90o (or 2700), so that the tangent point is close to the Sun, thenwith the earlier approximations, the distance ∆R = R−R0 = R0 sin `−R0 so

vmaxr = V (R)− V0sin ` = V0 + ∆R

dV

dR− V0sin ` =

[

V −R

(

dV

dR

)]

R0

(1− sin `) = 2AR0(1− sin `)

where A is again the Oort constant. Thus measuring vmaxr at given ` gives us the value of AR0 and

hence the distance to the Galactic Centre if A is known. We can also obtain the distance to the centrein other ways, e.g. by using stellar standard candles such as RR Lyraes. The value usually assumedis 8.5 kpc, though more recent estimates suggest it may be nearer 8 kpc.

Rotation Curves

For external galaxies it is trivial to measure the redshift of the radio 21 cm line at different points acrossa galaxy to determine V (R). Although the spatial resolution of the observations is not usually as goodas in the optical (because of the diffraction limit ∼ λ/D for radiation of wavelength λ observed witha telescope of diameter D), HI is frequently detectable far from galaxy centres. Deep HI observationscan reach column densities ' 1019 atoms/cm2 or 0.1M/pc2, typically at 2 - 3 optical radii.

Even beyond the visible stellar disc the rotational velocity still fails to drop in Keplerian fashion.The rotation curves generally remain flat to the limits of the observations. From the simple V 2 ∝M/R, this requires M(R) to increase linearly with R even where there are few stars. As the gas itself

35

provides only a small contribution to the required mass, such observations provided the main impetusbehind serious theoretical consideration of dark matter in the 1980s.

If we have reasonable estimates ofM/L for the stars in the bulge and disc of a spiral, and allow forthe ISM mass, then the shortfall compared to what is needed to generate the observed V (R) must bedark matter. Even with a ‘maximum disc model’ (choosing the largest M/L that does not overpredictthe rotation curve near the centre), a ‘dark halo’ with a density distribution similar to an isothermalsphere is required to match the rotation curve.

For most spirals, 50 - 90% of the mass out to large radii (tens of kpc) must be dark and the overallM/L ratios rise to 10 - 20 as we move outwards. The mass of our Galaxy follows

M(R) ' 1011(R/10 kpc) M

from ∼ 3 to at least 20 kpc. A singular isothermal sphere (SIS) has the required ρ ∝ 1/r2 to generateM(R) ∝ R, but unfortunately an infinite density at the centre and divergent total mass. Computer‘N-body’ simulations of the mutual gravitational attractions between large numbers of dark matter‘particles’ suggest that dark halos should instead have the NFW (Navarro, Frenk and White) profile

ρNFW

∝ 1

(r/rs)(1 + r/rs)2

where rs is a core size. This is a flatter than a SIS profile near the centre and steeper at large r.

The Tully-Fisher Relation

The overall rotation velocities for distant spiral galaxies can be obtained without spatially resolving thevelocity curve, since radio observations directly give the flux at any frequency (the radiation is detectedin separate ‘channels’ of the radio receiver). For an unresolved spiral galaxy entirely within the radiotelescope ‘beam’, we can measure the ‘velocity width’ from the range of frequencies at which we detect21 cm emisssion. This width ∆v is essentially twice the (maximum) rotation velocity, modulated bythe inclination angle, i.e. ∆v ' 2Vmax sin i, and the radio spectrum shows a characteristic ‘twinhorned’ profile from the approaching and receding material.

The Tully-Fisher relation between luminosity and linewidth is the analogue for spirals of theFaber-Jackson relation for ellipticals and in its basic version has the same form.

L ∝ ∆v4.

If we assume that ∆v represents the rotation velocity in the outer parts of the galaxy (radius R),so ∆v2 ' GM/R, and that spirals all have roughly the same mass-to-light ratio M/L and surfacebrightness I, then

∆v4 ∝ M2

R2∝MM

R2∝(M

L

)2

I L

implies the standard form. One might object that ∆v is measuring the mass of the dominant darkmatter halo and L relates only to the visible stars, but nevertheless the relation still holds.

Recent studies in the IR H band (to minimise the effects of dust) suggest

LH ' 3× 1010LH

(

Vmax

200 km s−1

)3.8

(1)

while in the optical B band the exponent is nearer to 3. Surprisingly, given the simple justificationabove, low SB spirals follow the same relation as high SB objects, requiring some trade off betweenM/L and I. Vmax is usually in the range 100 to 300 km s−1.

Like the F-J relation, the T-F relation gives L in terms of a distance independent velocity mea-surement, so we can use the apparent brightness and the inverse square law to deduce distances forlarge numbers of spiral galaxies. It has therefore been one of the major methods in distance scale andHubble constant work. In addition it is a key observable which any successful theory of spiral galaxyformation must be able to match, since it links the dark halo to the stars.

36

Bulges, Bars and Arms

The other components of spiral galaxies have their own dynamics. Bulges share many dynamical prop-erties with lower L Es, with large velocity dispersions (' 110 km s−1 for the Galaxy), and significantrotation (' 100 km s−1), so V/σ ∼ 1; bulge shapes are consistent with rotational flattening.

Bars are quite flat, like the disc component, and rotate as solid bodies, with a ‘pattern speed’ Ωbar,i.e. the entire bar will point in different directions at different times. However, the stars (and gas)orbit along the bar (i.e. they remain within the bar shape). The bar typically ends where Ωbar = Ω(R),where the bar stars corotate with the disc stars.

The formation of the bar itself is probably due to an instability of the rotating disc, if the surfacedensity is too high. Theoretically, stellar discs should be so unstable that they would all have turnedinto bars unless surrounded by a massive (dark) halo. Bars are also important in ferrying materialtowards the centres of galaxies, as the non-central gravitational forces allow angular momentum to belost.

If spiral arms were ‘material’ arms, differential rotation of the disc (Ω 6= constant) would windthem up in a few rotation periods (< 109 yrs). Some spiral structure, e.g. in flocculent spirals, maybe short lived and (re)generated by the shearing of SF regions by the differential rotation, but as mostdiscs show spiral structure, in most cases the spiral shape must be maintained. Different particles(stars and gas) must be in the arms at different times, suggesting that the spiral is some sort of wave.(For ordinary water waves, the crests move across a lake but individual water molecules do not).

‘Density wave theory’ suggests that the gravitational attraction between stars at different radiicounteracts the tendency of a spiral to wind up and reinforces a pattern which rotates with a patternspeed Ωp. Except around corotation, where Ω(R) = Ωp, the speed at which material enters the armR(Ω(R) − Ωp), is supersonic. This drives a shock wave at the front edge of the arm, increasing thedensity and instigating SF. OB stars have lifetimes of a few Myr so their positions closely track theshock front.

The Galactic Halo

The rotation curve of the Galaxy implies a mass ' 2× 1011M inside 20 kpc. Beyond this, the orbitsof satellite galaxies suggest a total mass ' 1012M out to 100 kpc, implying M/L ' 50. Althoughmuch of the missing mass must be in non-baryonic form, it is possible that some is supplied by invisiblebut macroscopic objects known as MACHOs (for Massive Astrophysical Compact Halo Objects). TheMACHOs could be small black holes, planet sized bodies or dim stars such as brown dwarfs.

These might be detectable by their gravitational effects. According to General Relativity, lightpassing close to a large mass will be deflected. The bending of the light ray means that the mass actslike a lens, amplifying the light of background sources.

Given a large number of background stars – e.g. in the Magellanic Clouds – a MACHO in ourGalaxy’s halo will sometimes move across the line of sight to one of the stars, briefly causing it tobrighten. In the 1990s several teams monitored millions of stars looking for such random brightenings.They found a number of convincing events, which can be differentiated from variable stars becausethey vary by the same amount at all wavelengths, i.e. achromatically. The duration of a lensing eventreflects the lensing mass and the results suggested that up to 20% of the halo mass might be in dimobjects of mass around 0.5M, presumably white dwarfs. However, other considerations, such as starcounts, proper motions and the amount of heavy elements that would be produced by the white dwarfprogenitors, probably limit the actual contribution to ∼ 2%.

The Galactic Centre

The Galaxy contains a bulge (probably of bar-like shape)' 2 kpc in radius and with a mass' 1010M.On a much smaller scale, there is a nuclear star cluster of mass ' 107M and size ∼ 3 pc. The nuclearregion contains old metal rich stars, but also very young blue and red supergiants (the central parseccontains around 25 30-100M blue supergiants), along with' 106M of molecular material, probablyin a toroidal shaped region.

The centre region is obscured in the optical, so we must use infra-red or radio observations. Radioemission at cm wavelengths reveals two main regions, Sagittarius (Sgr) A and B containing a string

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of radio sources aligned along the Plane in a region about 250 pc across. Their thermal radio spectraimply these are HII regions. In the mid infra-red, at 10µm, we see emission from dust, possibly fromregions around forming O stars.

Sgr A is very close to the Galactic Centre itself. At high resolution it breaks up into at least 3components, East, West and A*, a few parsecs apart. Sgr A East is a non-thermal source, probablya supernova remnant, while Sgr A West appears to be an HII region, also visible as a dusty infra-redsource. Sgr A* lies within or behind Sgr A West and is a non-thermal point source (at most a fewAU in diameter) which almost certainly marks the actual Galactic Centre. Ionized gas in the centralregions is rotating at several hundred km s−1, indicating a mass of a few ×106M inside 0.2 pc.

IR observations from the ESO Very Large Telescope’s adaptive optics system with a resolution of10s of milli-arc seconds (mas) probe down to scales ∼ 100 AU and it is found that the stellar densityis still rising, approximately as R−1.4; σ2 also rises, up to ' 1000 km s−1 1′′ from the centre, implyinga huge mass in a very small volume.

Some spirals (e.g. Seyfert galaxies) have Active Galactic Nuclei in their centres, like those in Egalaxies, so although our Galaxy is rather inactive (the luminosity of Sgr A* is only 103L), we canreasonably assume that this mass is a black hole. Observations taken over a span of about 10 yearsshow that star S2 is orbiting Sgr A* with a period of 15.2 years and at pericentre was just 120AUfrom Sgr A* and moving at 8000 km s−1. Allowing for the known stellar density (' 108M/pc3), theobservational data are well fitted by a model with a (2.9 ± 0.1) × 106M black hole in the centre.

The other LG spirals have differing central regions. M31 has a double nucleus; one componentappears to contain a 106M black hole, the other, about 2pc away, to be a compact star cluster. Thedynamics of the nuclear region of M33 shows no evidence for a black hole, though it does have a strongcentral X-ray source and a nuclear star cluster, comparable to a large globular cluster (luminosity∼ 2 × 106L, but M/L ∼ 0.4). Nuclear clusters appear to be common in late type spirals and alsoappear in some dwarf galaxies. The low M/L may be associated with episodes of star formation.Abundant gas in the central 100 pc is a common feature of many spirals.

Dynamical Evolution

Although the stars in Es are very old, it is likely that many Es, arrived at their current form quiterecently. A few, like Cen A, show significant departures from the standard smooth appearance. CenA itself has a huge dust lane across its centre and 10% of Es have significant dust, contrary to thestandard picture with no cold ISM. In addition, a few show the cold gas directly as a ‘polar ring’ ata large angle to the main plane, suggesting an external origin.

Other Es contain ‘shells’ within the main body of stars, which can be seen by ‘unsharp masking’the image (i.e. by subtracting off a smoothed version of the image to leave any sharp edged features).We also observe close pairs of galaxies with ‘tails’ and other non-axisymmetric features. ULIRGS andQSO hosts often have close neighbours and distorted appearances.

All these observations suggest that galaxies interact or even merge with one another. The processcan be seen taking place in the “Antennae”, where the two main galaxies are in contact and tidalstreams of debris are stretched out behind them. Later on, the merging galaxies will look like oneirregular object and the tails will be less prominent.

If at least one galaxy is gas rich, the interaction may funnel interstellar material towards the centreof the merger product and drive a burst of SF. ULIRGs can contain 109 − 1010M of gas and dustwhich is so dense that the SF is hidden in the optical and the radiation is reprocessed to give ≥ 1012Lin the FIR. For a standard IMF, their SFR

Ψ ' 5× 10−10L60/ε M/yr

where ε (' 1) is the fraction of the light reemitted in the FIR and L60 is the output at 60µm in L.A by-product of mergers is the creation of new GCs. These are usually thought of as very old

systems, but HST images of merging systems show numerous bright star clusters which should, whentheir stellar populations have aged, end up looking like normal GCs. If many Es formed via mergersthis explains why they have so many GCs per unit galaxy light compared to spirals. (Before the HSTobservations, this was a serious objection to the formation of Es by the merger of spirals).

Even when not actually colliding, satellite galaxies are still subject to gravitational tidal effects.M32, a compact dE, has a central SB 10 magnitudes higher than typical dEs and has the metallicity

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expected of a much more massive galaxy (but no GCs). The dynamics of its central regions showevidence for a supermassive black hole of ' 3 × 106M and the galaxy as a whole shows significantrotation. Hence it may be the remnant of a galaxy, perhaps an early type spiral, which lost its outerregions due to the tidal effects of M31. Its orbital period about M31 is ∼ 800 Myr and its orbitalradius only ∼ 12 kpc so M32 will periodically pass through the outer disc of M31.

Gravitational Interactions

Mergers are best investigated theoretically via computer ‘N-body models’ of the gravitational forcesbetween individual ‘particles’ (each representing thousands or millions of M of stars or gas) as thegalaxies approach one another, but we can make progress with some simple analytic approximations.

To see why galaxies merge, rather than just orbit each other, consider the effect on a star (massm∗) in one galaxy as another galaxy (mass Mg) flies past at velocity v with impact parameter b. Thestar (and galaxy) will gain a small perpendicular velocity due to their gravitational attraction. Thenet effect can be shown to be

∆v⊥ =2Gm∗

bv

for the galaxy, and a corresponding 2GMg/bv for the star. Thus there is a gain in perpendicular KE

∆K⊥ =Mg

2

(

2Gm∗

bv

)2

+m∗

2

(

2GMg

bv

)2

'2G2m∗M2

g

b2v2

with the smaller mass, the star, gaining most of the energy.This energy comes from the original ‘forward’ motion of the galaxy. If the galaxy slows by an

amount ∆v << v, it loses∆K‖ 'Mgv∆v ' ∆K⊥

⇒ ∆v ' ∆K⊥

Mgv' 2G2m∗Mg

b2v3.

To obtain the overall rate of change in forward velocity, we integrate over stars at all b, so

dv

dt= −

∫ bmax

bmin

2G2m∗Mg

b2v3× nv 2πb db = −4πG2m∗Mgn

v2lnΛ

where n is the number density of stars, v dt × 2πb db is the volume of the cylindrical shell at impactparameter b swept out in time dt and lnΛ = ln(bmax/bmin) is called a coulomb logarithm and will beof order unity. Roughly, then,

dv

dt' −4πG2Mgρ∗

v2

where ρ∗ = nm∗ is the mass density in stars. Strictly, this would be for a galaxy travelling through auniform distribution of stars, but it gives a useful first estimate.

The deceleration effect is called ‘dynamical friction’ as it operates like actual frictional forcesin depending on the galaxy’s velocity, and the above is a simplified version of what is called the‘Chandrasekhar formula’. If the deceleration is large enough (requiring small relative velocities and ahigh mass density of stars), it will eventually slow the passing galaxy to such an extent that it will‘fall in’ towards the other galaxy after a number of closer and closer passages. Larger mass galaxiesare slowed more, so a galaxy will ‘swallow’ its larger neighbours first (‘galactic cannibalism’). Thetime it would take to decelerate from velocity v to zero is

tmerge ∼v

|dv/dt| ∼v3

4πG2Mgρ∗.

For v ∼ 200 km s−1, Mg ∼ 1010M and ρ∗ corresponding to ∼ 1011M in a 10 kpc radius spherethis gives ' 3 × 108 years. We implicitly started with the galaxies quite close together (lnΛ ' 1), sothe overall time scale for a merger will be ∼ Gyrs.

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From the N-body models it is found that even for the merger of two disc galaxies the final producteventually looks like an elliptical, with a density distribution close to the de Vaucouleurs law. Anydust lanes, polar rings and so forth are interpreted as the remains of a spiral galaxy which has ‘fallenin’, while features like shells or the so-called ‘kinematically decoupled cores’ (where the central regionsrotate independently of the stars further out) probably indicate where the ‘relaxation’ of the stars inthe merged system to equilibrium is not yet complete.

Timescales

On simple dynamical grounds we might not even expect an isolated elliptical to have reached equi-librium via stellar interactions in the time since it formed, let alone a merger product. In general, acollision is an interaction which is close enough that it completely changes the velocities of the stars(∆v ' v) and causes the same mixing as seen in a gas – providing they occur frequently enough. Suchcollisions require PE gain (relative to when the stars were a long way apart) of the same order as theiroriginal KE, i.e.

Gm2∗

rc>

m∗v2

2

where m∗ and v are the stars’ masses and velocities. For a ' 1M star in a giant E, moving atv ' 500 km s−1, this requires

rc <2Gm∗

v2' 109m ' 3× 10−8pc.

For simplicity consider a star moving straight through the middle of a galaxy for a distance ∼ 2Re,say 15 kpc. This will take one ‘crossing time’

tcross = 2Re/v ' 5× 1020m/5× 105m s−1 ' 1015s ' 3× 107years.

During this time it will encounter all the stars in a cylinder of cross-section πr2c ' 3 × 10−15pc2. If

the column had cross-section 1pc2 it would have a column density – and hence number of stars –corresponding to the central SB of a giant E, ∼ 105L/pc2, i.e. ∼ 105 stars. Thus in our muchthinner column there will be ∼ 3× 10−10 stars so we would expect one close encounter every 3× 109

crossing times, or ∼ 1017 years – much longer than the age of the universe.Distant encounters between stars are much weaker, but there will be many more of them. With

the same notation as before, the force between the stars produces a perpendicular velocity

v⊥ =2Gm∗

bv.

In time t the number of stars passing at a distance between b and b+db is again 2πnvtb db. Integrationover b gives the total change in energy of a star from all other stars passing it,

∆K =∑ 1

2m∗v

2⊥ =

∫ bmax

bmin

1

2m∗nvt

(

2Gm∗

bv

)2

2πb db =4πG2m3

∗nt

vln

(

bmax

bmin

)

.

Relaxation can be taken to occur when the change in energy is equal to the original energy. Thus

trelax =v3

8πG2m2∗n lnΛ

.

If b covers the range between strong encounters (∼ 3 × 10−8 pc, from above) and the overall size ofthe galaxy (15 kpc), then lnΛ = ln(5× 1011) ' 27. The averaged out density n will be of order 1011

stars in a volume of 2× 1012pc3, i.e. about 0.05 pc−3, so

trelax ∼ 2× 1017years.

Distant encounters cannot drive an elliptical towards equilibrium in the age of the universe, either.

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The answer appears to be ‘violent relaxation’. For stars moving in a fixed potential, orbital energyis conserved. However, when the galaxy is first forming by collapse, or is accumulating mass by mergersor accretion, the gravitational potential is not time independent. This will change the stars’ energiesby large amounts, mixing them between various possible orbits and quite rapidly randomizing theirvelocities. This is not due to any sort of ‘two-body relaxation’ between pairs of stars, as in the casesabove, but is a co-operative effect, depending on the changing potential at any given point caused bythe motion, and hence change of position, of all the stars in the galaxy (or pair of galaxies).

Dynamical Evolution of the Galaxy

There are many indications of past interactions in the Galaxy - LMC - SMC system. There is a‘bridge’ of gas (and some young star clusters) between the Clouds, plus a more tenuous trail of gasahead of and behind the SMC in its orbit – the Magellanic Stream. It extends nearly half way aroundthe sky and contains around 2 × 108M of HI (comparable to MH in the Clouds themselves). Theorbits of the Clouds are highly eccentric, bringing them very close to the Galaxy every 2 Gyr. Thelast close approaches were ∼ 200− 400 Myr ago. The SMC has been severely affected by tidal forcesand may now be hardly gravitationally bound. Tidal friction from the Galaxy will bring about thedecay of the Clouds’ orbits and eventually lead to their merger with the Galaxy in a few Gyr.

A system already merging with the Galaxy was discovered as a group of stars in the same directionas the bulge of the Galaxy but with different velocities from bulge stars. These stars are part of asatellite galaxy – the Sagittarius dwarf – plunging through the opposite side of the Galactic Plane tothe Sun, about 16 kpc from the centre of the Galaxy. Stars pulled out of the dwarf are found in astream more than 20o (10 kpc) long. The GC M54 is associated with the Sagittarius dwarf and maybe its nucleus. At least three other GCs also appear to belong to the galaxy, suggesting that it wasonce much more massive than it is now. (Only Fornax of the other LG dSph galaxies has any GCs).The Galaxy may have accumulated other GCs from previously disrupted satellites. Other streams ofstars around the Galaxy (and around M31), suggest that the remains of completely accreted dwarfgalaxies supply a substantial fraction of halo stars.

Galaxy Formation and Evolution

The finite speed of light means that distant galaxies are seen as they were at earlier cosmic times.Roughly, redshifts of order unity probe the universe when it was half its present age (∼ 7 Gyr), whilethe most distant galaxies and quasars at z ∼ 6 are seen as they were ≤ 1 Gyr after the Big Bang.We can therefore make ‘in situ’ observations of the evolution of galaxies as well as studying nearbygalaxies as the endpoints of this evolution.

High Redshift Galaxies

In addition to directly obtaining redshifts for very faint galaxies, there are two key methods for findingvery high z objects. By observing in several bands (preferably ≥ 6, including UV and IR) it is possibleto find the best matching galaxy type and redshift to reproduce all the observed colours, with errorsδz < 0.1. This ‘photometric redshift’ technique has been used widely on HST observations, e.g. theHubble Deep Fields and Ultra Deep Field (reaching about magnitude 30), the deepest ever opticalobservations. Photometric z s, confirmed by spectroscopic follow-up, indicate the presence of galaxiesout to z = 5.6 in the HDF, with most between z = 0.5 and 2. The types of galaxies appear to changeabove z ∼ 1.5; both the optical morphologies and the best fitting spectra indicate that at high z thereis an increasing fraction of irregulars, with a shortage of real spirals.

A related method is the use of ‘drop-outs’, first used to select ‘Lyman break galaxies’ (LBGs) atz ∼ 3. The spectrum of any source drops away sharply bluewards of the Lyman break at 91.2 nm,because shorter wavelength photons are removed by photoionizing any H they encounter. At z ∼ 3this break is redshifted past the U band (i.e. to around 360 nm), so the U−B colour becomes very redor the object is completely undetected at U . At progressively higher z the break moves across longerwavelength bands and objects will drop out of samples selected at B, V or R. In addition, at higherz, the accumulated absorption from the Lyman alpha forest lines (due to intervening HI clouds) cutsout the flux shortward of Lyα in the source galaxy. Thus R band drop-outs occur at z > 4.8 and I

41

band drop-outs are used to search for galaxies at z > 5.6. LBGs’ numbers and clustering propertiesare consistent with them being the direct ancestors of present day giant Es or small galaxies whichwill later merge to form a current epoch type galaxy. The galaxies seen at z ∼ 4−5 in deep HST data(the most luminous in existence at that time) are relatively small compared to present day galaxies,with half-light radii of 0.2′′ to 0.4′′, corresponding to ≤ 3 kpc.

Alternatively, we can search for star forming galaxies via their emission lines, particularly Lyα.Narrow band images (i.e. through filters which transmit only a narrow wavelength range) will detectLyα emitters at specific redshifts even when the continuum (in a normal broad band) is too faintto see. We should note, though, that samples selected in this way can be contaminated by sourcesemitting a different line, say [OII], at lower redshift, while not all galaxies may be strong line emitterseven at early times. The distant galaxy HDF 4-473.0, for instance, appears to have a star formationrate of only 10M/yr, similar to a present day Sc.

In addition, some local starburst galaxies have little Lyα emission because of dust absorption,suggesting that we should also look for high z objects via dust emission redshifted into the sub-mmregion. Increasing z can actually boost the brightness of galaxies in the sub-mm, since the redshiftmoves the observed band closer to the peak of the dust emission spectrum at around 100µm, giving anegative k-correction. In principle, a z = 10 source could be as bright as one at z = 1. Observed highz sub-mm sources are candidates for E galaxies during their main SF phase, with SFRs ∼ 1000M/yr.This would produce all the stars in a giant E in a few ×108 yr.

Evolution Models

In general terms, the evolution of a galaxy consists of three components; the star formation history,the fading (and reddening) of stellar populations (‘passive evolution’) and the increase in mass viamergers. Using observations such as colours and metallicities of galaxies, luminosity or mass functionsetc., we arrive at the following general scheme.

Density perturbations in the early universe (primarily in the dominant dark matter) grow throughgravitational attraction until they are able to collpse despite the overall expansion of the universe.(This stage can be studied computationally via N-body models and tested via the predicted numbersand distribution of dark matter halos of different masses). The baryons within these forming darkmatter halos also collapse and, unlike the dark matter, interact (e.g. via shocks) and dissipate energyto as to create even denser baryon dominated central regions – proto-galaxies. (This requires addinghydrodynamical modelling to the computer codes).

Once a sufficient density is reached, SF begins. As the universe is still almost entirely H and He,the gas has to cool in different ways (via H2 molecular emission) to that in present day SF regions.The metal-free Population III stars are expected to be much more massive (many ≥ 100M) thannormal stars. Hence they have very short lifetimes (so no metal-free stars are seen today) and quicklydeposit heavy elements in the ISM for the next generation of Population II stars. A key ingredientof modelling this (and subsequent star forming phases) is understanding the feedback (of energy andionizing photons, as well as material) into the ISM.

Theoretically, the state of the art are ‘semi-analytic models’, which use the output from thenumerical simulations of the halo growth and then include the numerous likely physical processes –cooling, star formation, feedback etc. – by way of given ‘rules’.

At this stage most systems will be of low mass which subsequently merge to form larger bulge-likesystems containing stars formed early (z > 3− 5) and on a short timescale. This is reflected in theircolours and element abundances; since only SNII will have occurred during the SF phase, producingelements such as Mg and O, there is an ‘α element’ enhancement relative to Fe (which is mostlyproduced on a longer timescale by SNIa). At the same time, gas inflow will fuel (and increase themass of) central BHs, which become visible as quasars (already seen at z > 6). The AGN probablyalso have an important role in feeding energy back into the ISM and hence modulating SF.

These bulges can then accrete further infalling baryons, which form a surrounding disc, becauseof their angular momentum, making a spiral-like galaxy. These may then suffer further mergers tocreate larger Es. Since we know that giant Es contain old stars, their progenitors must have mergedearly on, or have been gas free (‘dry mergers’). Combining measurements of colours and luminosities,it appears that the stellar populations in Es have faded by a factor 2 since z 1, and this has beencounteracted by a similar sized increase in stellar mass.

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Future spirals will have continued to form (Population I) stars from accreted gas over a long timescale, producing element abundances as seen in the Sun. We can track the SF history of the universevia the number of UV (or emission line) emitting galaxies at different z (the ‘Madau plot’). Roughly,the SF per unit volume was constant from high z down to z ' 1, but declined steeply thereafter. Thisis due to a combination of a decrease in SF in individual galaxies and an increase in the fraction ofnon-star forming (red) galaxies, especially in clusters (the ‘Butcher-Oemler effext’). Interestingly, itis the more massive galaxies which ceased SF first, an effect called ‘downsizing’ (i.e. at high z onlymassive galaxies were red).

Use of colours is not limited to bright Es. From stellar population models and a given starformation history, we can calculate the tracks through colour-magnitude or colour-colour diagramswith z for any galaxy type (allowing also for k-corrections). Sophisticated models exist for trackingthe evolution, including the chemical enrichment as star formation proceeds and the related variationin dust opacity, though they are dependent on assumptions such as a universal IMF.

Since the LF and distribution of types are different in clusters to the field, evolution is evidentlyenvironment dependent. Galaxies in clusters may both start differently, e.g. because galaxy sizedperturbations appear earlier in higher density regions of the universe, and develop differently. Clustermembers are subject to extra effects such as greater likelihood of interactions, tidal effects from thecluster potential and sweeping out of gas by the pressure of the intergalactic medium. Galaxies (spiralsand irregulars) may be transformed into early types (with or without a final burst of star formation)when they ‘fall’ into a cluster. (Like galaxies, clusters grow hierarchically via accretion and mergers).The end result of all this is then the diverse galaxy population that we see today.

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