Download pdf - Le Galassie: popolazioni stellari - INAFmarconi/Lezioni/FisGal14-15/Lezione03.pdf · A. Marconi Fisica delle Galassie 2014/2015 Popolazioni Stellari E’ possibile analizzare lo spettro

Le Galassie: popolazioni stellari

Lezione 3

A. Marconi Fisica delle Galassie 2014/2015

La funzione di luminosità delle galassie ϕ(L) è definita da dN = ϕ(L) dLdN è il numero di galassie per unità di volume con luminosità tra L e L+dL.ϕ(L) si misura di solito in h-3 Mpc-3; h-3 serve per togliere la dipendenza dalla costante di Hubble H0 = 100 h km/s/Mpc (h=0.72).La forma funzionale che meglio descrive la funzione di luminosità è la cosiddetta funzione di Schechter:

!

ϕ✶ normalizzazione, α pendenza a basse L e L✶ luminosità caratteristica (L>0.1 L✶ “bright” galaxy).

La densità totale di galassie è:

La densità di luminosità totale è:

Funzione di Luminosità delle galassie

2

L ~ Lα

L ~ exp(-L/L✶)

log ϕ

(L) [

Mpc

-3]

log L [L⊙]L✶

ϕ✶

�(L)dL

L⇥= �⇥

�L

L⇥

⇥��

exp(�L/L⇥)dL

L⇥

nTot =�

⇥(L)dL = ⇥��(� + 1)

⇥L =�

L⇥(L)dL = L�⇥��(� + 2)

nTot =�

⇥(L)dL = ⇥��(� + 1)

⇥L =�

L⇥(L)dL = L�⇥��(� + 2)

�(L)dL = �

?

✓L

L?

◆↵

exp(�L/L?)

dL

L?

Funzione di Luminosità delle galassieα = 0.46;L✶ ≈9×109 h-2 L⊙ L✶ ≈2×1010 L⊙ (h=0.7)(~Milky Way);ϕ✶ ≈0.02 h3 Mpc-3

ϕ✶ ≈0.007 Mpc-3 (h=0.7); la densità di luminosità locale è (in B e K):ρL(B) ≈2×108 h L⊙ Mpc-3 ρL(K) ≈6×108 h L⊙ Mpc-3

ρ(L) ~L ϕ(L)

ϕ(L)

Supponiamo l’universo abbia densità media pari alla densità critica (ΩM=1)

⇢c⇢L

=

✓M

L

◆= 1475h

✓M�L�

◆

B

= 492h

✓M�L�

◆

K

Allora M/L in unità solari è

⇢C =3H2

0

8⇡G= 2.0⇥ 10�26h2 kgm�3 = 2.95⇥ 1011h2 M� Mpc�3

>> dei valori osservati in galassie e ammassi (anche con materia oscura): galassie e ammassi non sono sufficienti a chiudere l’universo!


Abbiamo visto le proprietà “globali” delle galassie ellittiche e spirali ma non abbiamo ancora considerato le proprietà delle stelle che costituiscono una galassia.

Lo spettro di una galassia è dato dalla somma degli spettri delle singole stelle costituenti, ma anche dalla somma degli spettri degli altri componenti come regioni HII, nucleo attivo ecc.

Popolazioni Stellari

4

Stelle

Regione HII



5

E

S0

SbSc

Starburst

O5

A1

G2

K0

M5


Popolazioni StellariE’ possibile analizzare lo spettro di una galassia considerandolo come sovrapposizione di varie popolazioni stellari.

Una singola popolazione stellare è un insieme di stelle caratterizzate da:

storia di formazione stellare SFR(t) ovvero il numero di masse solari convertite in stelle per unità di tempo [M☉/yr] in funzione del tempo: dMgas/dt. Per esempio “burst istantaneo” (ovvero SFR(t) = S0 δ(t-t0) cioè all’istante t0 si formano stelle per una massa totale di M) o “burst continuo” (ovvero SFR(t) = cost. ovvero si convertono continuamente varie M☉/yr in stelle)

initial mass function ovvero data massa M in stelle che si formano, quante sono le stelle che si formano ad una data massa m?ϕ(m)dm è il numero di stelle che si formano tra m e m+dm.La più nota è la IMF di Salpeter ϕ(m) ~ m-2.35 tipicamente mL~0.1 M☉ (H burning limit), mU~100 M☉ (oltre non osservate)

Z ovvero le abbondanze “iniziali” degli elementi pesanti. L’abbondanza poi varia a seguito della produzione di elementi pesanti.

6

Z mU

mL

m�(m) dm = M


Popolazioni StellariPer una data popolazione stellare i modelli di evoluzione stellare forniscono le isocrone nel diagramma HR (costituite dai punti delle tracce evolutive con M, Z allo stesso tempo t).!!I modelli di atmosfere forniscono lo spettro di una stella con M, L e Te.Sommando gli spettri di tutte le stelle con varie masse [pesate per ϕ(m)] è possibile ottenere lo spettro e la luminosità della popolazione stellare in funzione del tempo. Le stelle giovani creano regioni HII di cui bisogna tener conto nel modello.

7

Stellar Tracks (per Z dato)

Massa [M☉]


N (L, Te, ⌧) =

Z ⌧

0

Z mU

mL

T[L,Te](M, ⌧ � t)SFR(t)�(M) dM dt


8

Stelle formate tra t, t+dt e per M, M+dM:

F� = F�(L, Te)Modelli di atmosfere stellari:

F�(�) =� �

L,Te

F�(L, Te)N (L, Te, �) dL dTeSpettro totale:

dN = SFR(t)�(M) dM dt

T[L,Te](M, t) dLdTe

Track evolutivi sul diagramma HR:probabilità che una stella di massa M abbia L, Te dopo un tempo t dalla nascita

Numero totale di stelle tra L,L+dL e Te,Te+dTe al tempo τ sarà perciò

dN = N (L, Te, ⌧) dLdTe



9

138

3. The World of Galaxies

Fig. 3.50. Spectra of gal-axies of different types,where the spectral flux isplotted logarithmically inarbitrary units. The spec-tra are ordered accordingto the Hubble sequence,with early types at the bot-tom and late-type spectraat the top

according to the characteristic age of their stellar popu-lation or according to their star-formation rate. Ellipticaland S0 galaxies essentially have no star-formation ac-tivity, which renders their spectral energy distributiondominated by red stars. Furthermore, in these galaxiesthere are no HII regions where emission lines could begenerated. The old stellar population produces a pro-nounced 4000-Å break, which corresponds to a jumpby a factor of ∼ 2 in the spectra of early-type galaxies.It should be noted that the spectra of ellipticals and S0galaxies are quite similar.

By contrast, Sc spirals and irregular galaxies havea spectrum which is dominated by emission lines, wherethe Balmer lines of hydrogen as well as nitrogen andoxygen lines are most pronounced. The relative strengthof these emission lines are characteristic for HII regions,implying that most of this line emission is produced inthe ionized regions surrounding young stars. For irregu-lar galaxies, the spectrum is nearly totally dominated bythe stellar continuum light of hot stars and the emissionlines from HII regions, whereas clear contributions bycooler stars can be identified in the spectra of Sc spiralgalaxies.

The spectra of Sa and Sb galaxies form a kind oftransition between those of early-type galaxies and Sc

galaxies. Their spectra can be described as a super-position of an old stellar population generating a redcontinuum and a young population with its blue con-tinuum and its emission lines. This can be seen inconnection with the decreasing contribution of thebulge to the galaxy luminosity towards later spiraltypes.

The properties of the spectral light distribution ofdifferent galaxy types, as briefly discussed here, is de-scribed and interpreted in the framework of populationsynthesis. This gives us a detailed understanding ofstellar populations as a function of the galaxy type. Ex-tending these studies to spectra of high-redshift galaxiesallows us to draw conclusions about the evolutionaryhistory of their stellar populations.

3.10 Chemical Evolution of GalaxiesDuring its evolution, the chemical composition of a gal-axy changes. Thus the observed metallicity yieldsinformation about the galaxy’s star-formation history.We expect the metallicity Z to increase with star-formation rate, integrated over the lifetime of the galaxy.We will now discuss a simple model of the chemical evo-

Spettri di galassie di diversi tipi morfologici lungo la sequenza di Hubble, con gli early types in fondo, ed i late types in cima.


134


Fig. 3.46. a) Evolutionary tracks in the HRD for stars of dif-ferent masses, as indicated by the numbers near the tracks (inunits of M⊙). The ZAMS (zero age main sequence) is theplace of birth in the HRD; evolution moves stars away fromthe main sequence. Depending on the mass, they explode asa core-collapse SN (for M ≥ 8M⊙) or end as a white dwarf

(WD). Prior to this, they move along the red giant branch(RGB) and the asymptotic giant branch (AGB). b) Isochronesat different times, indicated in units of 109 years. The uppermain sequence is quickly depopulated by the rapid evolutionof massive stars, whereas the red giant branch is populatedover time

Fig. 3.47. a) Comparison of the spectrum of a main-sequencestar with a blackbody spectrum of equal effective temperature.The opacity of the stellar atmosphere causes clear deviationsfrom the Planck spectrum in the UV/optical. b) Spectrum

of a stellar population with solar metallicity that was in-stantaneously born a time t ago; t is given in units of 109

years

t [Gyr]

Popolazioni StellariInizialmente dominano le stelle più massicce (emissione UV)dopo ~107 yr, flusso <1000Å diminuisce fino a sparire; cresce NIR per evoluzione stelle massicce a (super)giganti;108 - 109 yr forte emissione NIR resta ma diminuisce ancora quella ottica UV;dopo 109 yr giganti rosse determinano quasi tutta emissione NIR;dopo 3×109 yr, UV cresce per stelle blu nella fase di ramo orizzontale e per nane bianchetra 4-13 Gyr evolve molto poco

10



11

Notare l’enorme evoluzione in luminosità e l’arrossamento dello spettro

Evoluzione dello spettro di una popolazione stellare (burst di 100 Myr)



12

Notare anche l’apparire di un prominente break a 4000 Å (inizialmente è il Balmer break, poi dovuto ad assorbimento combinato di Ca, Mg ed altre righe metalliche).D4000 ~ F(4000+)/F(4000-) Caratteristica importante degli spettri e può essere usato per stimare l’età della popolazione stellare dagli spettri.

Balmer break (4000 Å)

134


Fig. 3.46. a) Evolutionary tracks in the HRD for stars of dif-ferent masses, as indicated by the numbers near the tracks (inunits of M⊙). The ZAMS (zero age main sequence) is theplace of birth in the HRD; evolution moves stars away fromthe main sequence. Depending on the mass, they explode asa core-collapse SN (for M ≥ 8M⊙) or end as a white dwarf

(WD). Prior to this, they move along the red giant branch(RGB) and the asymptotic giant branch (AGB). b) Isochronesat different times, indicated in units of 109 years. The uppermain sequence is quickly depopulated by the rapid evolutionof massive stars, whereas the red giant branch is populatedover time

Fig. 3.47. a) Comparison of the spectrum of a main-sequencestar with a blackbody spectrum of equal effective temperature.The opacity of the stellar atmosphere causes clear deviationsfrom the Planck spectrum in the UV/optical. b) Spectrum

of a stellar population with solar metallicity that was in-stantaneously born a time t ago; t is given in units of 109

years



13

3.9 Population Synthesis

135photometric properties – so-called photometric redshiftestimates.

3.9.3 Color EvolutionDetailed spectra of galaxies are often not available.Instead we have photometric images in different broad-band filters, since the observing time required for spec-troscopy is substantially larger than for photometry. Inaddition, modern wide-field cameras can obtain photo-metric data of numerous galaxies simultaneously. Fromthe theory of population synthesis we can derive pho-tometric magnitudes by multiplying model spectra withthe filter functions, i.e., the transmission curves of thecolor filters used in observations, and then integratingover wavelength (A.25). Hence the spectral evolutionimplies a color evolution, as is illustrated in Fig. 3.48(a).

For a young stellar population the color evolution israpid and the population becomes redder, again becausethe hot blue stars have a higher mass and thus evolve

Fig. 3.48. a) For the same stellar population as in Fig. 3.47(b),the upper two graphs show the colors B − V and V − K asa function of age. The lower two graphs show the mass-to-light ratio M/L in two color bands in Solar units. The solidcurves show the total M/L (i.e., including the mass that is

later returned into the ISM), whereas the dashed curves showthe M/L of the stars itself. b) The fraction of B- (top) andK -luminosity (bottom) contributed by stars in their differentphases of stellar evolution (CHeB: core helium burning stars;SGB: subgiant branch)

quickly in the HRD. For the same reason, the evolutionis faster in B − V than in V − K . It should be mentionedthat this color evolution is also observed in star clustersof different ages. The mass-to-light ratio M/L also in-creases with time because M remains constant while Ldecreases.

As shown in Fig. 3.48(b), the blue light of a stel-lar population is always dominated by main-sequencestars, although at later stages a noticeable contributionalso comes from horizontal branch stars. The NIR ra-diation is first dominated by stars burning helium intheir center (this class includes the supergiant phase ofmassive stars), later by AGB stars, and after ∼ 109 yrby red giants. Main sequence stars never contributemore than 20% of the light in the K-band. The factthat M/L K varies only little with time implies that theNIR luminosity is a good indicator for the total stellarmass: the NIR mass-to-light ratio is much less depen-dent on the age of the stellar population than that forbluer filters.

Evoluzione del colore e del rapporto M/L. Rapida evoluzione in colore per popolazioni giovani (< 3 Gyr) guidata dall’evoluzione rapida delle stelle massicce che dominano emissione.M/L cresce nel tempo perchè M costante ed L diminuisce.M/LK varia meno ed indica che emissione K (NIR) è miglior tracciante della massa in stelle.



14

Colori più rossi (e M/L più elevati)sembrano indicare popolazioni stellari più vecchie;ma questo è anche interpretabile con metallicità più elevata(le atmosfere di stelle piùmetalliche sono più fredde)aumento di età di fattore X è quasi equivalente ad un aumento di un fattore 0.65X della metallicità.Questa degenerazione può essere rimossa tenendo conto, ad esempio, dell’informazione spettroscopica (EW delle righe, D4000 ecc.)


137

Fig. 3.49. a) Evolution of colors between 0 ≤ t ≤ 17×109 yrfor a stellar population with star-formation rate given by(3.69), for five different values of the characteristic time-scaleτ (τ = ∞ is the limiting case for a constant star-formationrate) –Galactic center see solid curves. The typical colors forfour different morphological types of galaxies are plotted. Foreach τ , the evolution begins at the lower left, i.e., as a bluepopulation in both color indices. In the case of constant starformation, the population never becomes redder than Irr’s; to

achieve redder colors, τ has to be smaller. The dashed lineconnects points of t = 1010 yr on the different curves. Here,a Salpeter IMF and Solar metallicity was assumed. The shiftin color obtained by doubling the metallicity is indicated byan arrow, as well as that due to an extinction coefficient ofE(B − V ) = 0.1; both effects will make galaxies appear red-der. b) The dependence of colors and M/L on the metallicityof the population

• A simple model of star-formation history reproducesthe colors of today’s galaxies fairly well.

• (Most of) the stars in elliptical and S0 galaxies areold – the earlier the Hubble type, the older the stellarpopulation.

• Detailed models of population synthesis provideinformation about the star-formation history, andpredictions by the models can be compared withobservations of galaxies at high redshift (and thussmaller age).

We will frequently refer to results from populationsynthesis in the following chapters. For example, wewill use them to interpret the colors of galaxies athigh redshifts and the different spatial distributions ofearly-type and late-type galaxies (see Chap. 6). Also,we will present a method of estimating the redshift ofgalaxies from their broad-band colors (photometric red-shifts). As a special case of this method, we will discussthe efficient selection of galaxies at very high redshift(Lyman-break galaxies, LBGs, see Chap. 9). Becausethe color and luminosity of a galaxy are changing even

when no star formation is taking place, tracing backsuch a passive evolution allows us to distinguish thispassive aging process from episodes of star formationand other processes.

3.9.7 The Spectra of GalaxiesAt the end of this section we shall consider the typicalspectra of different galaxy types. They are displayedfor six galaxies of different Hubble types in Fig. 3.50.To make it easier to compare them, they are all plottedin a single diagram where the logarithmic flux scaleis arbitrarily normalized (since this normalization doesnot affect the shape of the spectra).

It is easy to recognize the general trends in these spec-tra: the later the Hubble type, (1) the bluer the overallspectral distribution, (2) the stronger the emission lines,(3) the weaker the absorption lines, and (4) the smallerthe 4000-Å break in the spectra. From the above discus-sion, we would also expect these trends if the Hubblesequence is considered an ordering of galaxy types


Popolazioni StellariUn’altra origine dell’arrossamento è l’estinzione da parte della polvere.

15


137

Fig. 3.49. a) Evolution of colors between 0 ≤ t ≤ 17×109 yrfor a stellar population with star-formation rate given by(3.69), for five different values of the characteristic time-scaleτ (τ = ∞ is the limiting case for a constant star-formationrate) –Galactic center see solid curves. The typical colors forfour different morphological types of galaxies are plotted. Foreach τ , the evolution begins at the lower left, i.e., as a bluepopulation in both color indices. In the case of constant starformation, the population never becomes redder than Irr’s; to

achieve redder colors, τ has to be smaller. The dashed lineconnects points of t = 1010 yr on the different curves. Here,a Salpeter IMF and Solar metallicity was assumed. The shiftin color obtained by doubling the metallicity is indicated byan arrow, as well as that due to an extinction coefficient ofE(B − V ) = 0.1; both effects will make galaxies appear red-der. b) The dependence of colors and M/L on the metallicityof the population

• A simple model of star-formation history reproducesthe colors of today’s galaxies fairly well.

• (Most of) the stars in elliptical and S0 galaxies areold – the earlier the Hubble type, the older the stellarpopulation.

• Detailed models of population synthesis provideinformation about the star-formation history, andpredictions by the models can be compared withobservations of galaxies at high redshift (and thussmaller age).

We will frequently refer to results from populationsynthesis in the following chapters. For example, wewill use them to interpret the colors of galaxies athigh redshifts and the different spatial distributions ofearly-type and late-type galaxies (see Chap. 6). Also,we will present a method of estimating the redshift ofgalaxies from their broad-band colors (photometric red-shifts). As a special case of this method, we will discussthe efficient selection of galaxies at very high redshift(Lyman-break galaxies, LBGs, see Chap. 9). Becausethe color and luminosity of a galaxy are changing even

when no star formation is taking place, tracing backsuch a passive evolution allows us to distinguish thispassive aging process from episodes of star formationand other processes.

3.9.7 The Spectra of GalaxiesAt the end of this section we shall consider the typicalspectra of different galaxy types. They are displayedfor six galaxies of different Hubble types in Fig. 3.50.To make it easier to compare them, they are all plottedin a single diagram where the logarithmic flux scaleis arbitrarily normalized (since this normalization doesnot affect the shape of the spectra).

It is easy to recognize the general trends in these spec-tra: the later the Hubble type, (1) the bluer the overallspectral distribution, (2) the stronger the emission lines,(3) the weaker the absorption lines, and (4) the smallerthe 4000-Å break in the spectra. From the above discus-sion, we would also expect these trends if the Hubblesequence is considered an ordering of galaxy types

Tracce evolutive per 0<t<17 Gyr di una popolazione stellare con SFR(t) ~ exp[-(t-tf)/τ] IMF di Salpeter e Z=Z⊙

Per SFR costante (τ=∞) i colori non diventano mai più rossi di una galassia Irregolare;la riga tratteggiata è il luogo dove t = 1010 yr;le frecce sono i vettori spostamento per variazione di metallicità (Z) e estinzione E(B-V)


SSFR =SFR

Mstars=

1

�

b =SFR

hSFRipast=

SFR

Mstars/tHubble

Tasso di Formazione stellare

16

Più che lo SFR assoluto interessa lo SFR relativo allo SFR medio nel passato della galassia oppure relativo alla massa in stelle della galassia

“birth rate parameter”

“specific SFR”

τ = tempo richiesto per formare Mstar al rate osservato attualmente

Galassie quiescenti (e.g. ellittiche) hanno τ ≫ tHubble o SFR≪〈SFR〉past

→ Hanno formato il grosso delle stelle in passato con SFR molto maggiore Galassie a spirale hanno τ >~ tHubble o SFR~<〈SFR〉past

→ Hanno avuto una storia di formazione stellare abbastanza costante Galassie “starburst” hanno τ ≪ tHubble o SFR≫〈SFR〉past

→ Si trovano in un periodo di attività di SF eccezionalmente elevato


Popolazioni StellariCombinando varie popolazioni stellari diverse (eventualmente tenendo conto dell’estinzione da parte della polvere per ciascuna di esse) è possibile ricostruire lo spettro della galassia:

G(λ) = Σi Pi(λ) exp[-τi(λ)] dove Pi(λ) è lo spettro della popolazione stellare i-esima e τi(λ) è la profondità ottica della polvere tra noi e Pi.

Problemi: ci sono molti parametri liberi tra loro degeneri. Per esempio età, metallicità ed estinzione da polvere sono degeneri (galassia vecchia e ricca di metalli o galassia giovane arrossata hanno distribuzioni spettrali fotometriche all’apparenza simili).

17

Popolazione “vecchia”

Popolazione “giovane”

Spettro sintetico totale

Spettro osservato

New galaxy classificationWith the huge number of galaxies observed by SDSS a new classification scheme for galaxies has emerged beyond the Hubble one.Early and late type galaxies form two sequences (seen in many properties):

the red sequence made of non-star-forming, high mass spheroidal galaxies, (old, red and dead galaxies)the blue sequence (blue cloud) consists of star-forming, low mass galaxies which are disc-dominated.

3.9 The Red and Blue Sequences 89

type galaxies are found to form two distinct sequences which are known as the redand blue sequences. In summary:

– The red sequence consists of non-star-forming, high mass spheroidal galaxies,or, more colloquially ‘old, red and dead’ galaxies.

– The blue sequence consists of star-forming, low mass galaxies which are disc-dominated.

These two sequences are defined by a number of the characteristic properties whichhave already been introduced.

3.9.1 Colour Versus Absolute Magnitude

Perhaps the most striking distinction between the two sequences appears in theplot of the colour 0.1(g − r) against absolute magnitude M. Figure 3.20a showsthe distribution of these properties for 144,000 galaxies from the SDSS catalogue(Blanton et al., 2003). Superimposed on the diagram are isodensity contours, thebulk of the galaxies lying within the heavy white contours. The separation into twosequences is clearly defined, the oval region at the top of the diagram being the redsequence and the broader region towards the bottom right the blue sequence.

Baldry and his colleagues have shown that the colour distribution of these galax-ies can be separated into red and blue sequences which can be can be very well-described by Gaussian distributions over the magnitude range −23.5 ≤ Mr ≤−15.75 (Baldry et al., 2004). It is striking how precisely the overall colour distri-bution in each bin of absolute magnitude over this wide magnitude range can bedecomposed into two Gaussian distributions (Fig. 3.21). The red galaxies are the

Fig. 3.20. a Illustrating the bimodality in the distribution of colour 0.1(g − r) of galaxiesas a function of optical absolute magnitude (Blanton et al., 2003). b A plot of Sersic indexagainst colour for 10,095 galaxies selected from the Millennium Galaxy Catalogue (Driveret al., 2006)

Red sequence

Blue cloud

Green valley

Blanton+03 Driver+06

n>2 spheroid dominated (red galaxies)

n<2 disk, pseudobulge dominated (blue galaxies)


Bimodalità Età-Massa

19

Fit degli spettri stellari con modelli di popolazioni stellari per ricavare età della popolazione stellare (principalmente da Hβ - stelle giovani - e dal break a 4000 Å - stelle vecchie).La bi-modalità della distribuzione delle galassie sul piano colore-magnitudine riflette una bi-modalità nella distribuzione età-massa.

Età

Massaluminose con basso SFR (RS)

poco luminose con alto SFR (BC)

giovani e di piccola massa (BC)

vecchie e di grande massa (RS)


Bimodalità Età-MassaLe due distribuzioni posso quindi essere separate in galassie massicce e luminose, vecchie e con poca SF (Red Sequence)galassie poco massicce e luminose, giovani e con molta SF (Blue Cloud)

20

3.8 Galaxies as Gravitational Lenses

121

Fig. 3.33. The density of galaxies in color–magnitude space.The color of ∼ 70 000 galaxies with redshifts 0.01 ≤ z ≤ 0.08from the Sloan Digital Sky Survey is measured by the rest-frame u − r, i.e., after a (small) correction for their redshiftwas applied. The density contours, which were corrected forselection effects like the Malmquist bias, are logarithmicallyspaced, with a factor of

√2 between consecutive contours. In

the left-hand panel, the measured distribution is shown. Ob-viously, two peaks of the galaxy density are clearly visible,one at a red color of u − r ∼ 2.5 and an absolute magnitudeof Mr ∼ −21, the other at a bluer color of u −r ∼ 1.3 andsignificantly fainter magnitudes. The right-hand panel corre-sponds to the modeled galaxy density, as is described in thetext

again is in agreement of what we just learned: for highluminosities, the red galaxies clearly dominate, whereasat small luminosities, the blue galaxies are much moreabundant.

The mass-to-light ratio of a red stellar population islarger than that of a blue population, since the former nolonger contains massive luminous stars. The differencein the peak absolute magnitude between the red and bluegalaxies therefore corresponds to an even larger differ-ence in the stellar mass of these two populations. Redgalaxies in the local Universe have on average a muchhigher stellar mass than blue galaxies. This fact is il-lustrated by the two dotted lines in the right-hand panelof Fig. 3.33 which correspond to lines of constant stel-lar mass of ∼ 2–3×1010 M⊙. This seems to indicatea very characteristic mass scale for the galaxy distribu-tion: most galaxies with a stellar mass larger than thischaracteristic mass scale are red, whereas most of thosewith a lower stellar mass are blue.

Obviously, these statistical properties of the galaxydistribution must have an explanation in terms of theevolution of galaxies; we will come back to this issuein Chap. 9.

3.8 Galaxies as Gravitational Lenses

In Sect. 2.5 the gravitational lens effect was discussed,where we concentrated on the deflection of light bypoint masses. The lensing effect by stars leads to im-age separations too small to be resolved by any existingtelescope. Since the separation angle is proportional tothe square root of the lens mass (2.79), the angular sepa-ration of the images will be about a million times largerif a galaxy acts as a gravitational lens. In this case itshould be observable, as was predicted in 1937 by FritzZwicky. Indeed, multiple images of very distant sourceshave been found, together with the galaxy responsiblefor the image splitting. In this section we will first de-scribe this effect by continuing the discussion we beganin Sect. 2.5.1. Examples of the lens effect and its variousapplications will then be discussed.

3.8.1 The Gravitational Lensing Effect – Part IIThe geometry of a typical gravitational lens system issketched in Fig. 2.21 and again in Fig. 3.34. The phys-