Galactic Astronomy Chapter 4....................................................................................................................................................2

Evolution of stars and stellar populations ..............................................................................................2 Stellar evolution and CM diagram.........................................................................................................2

Stellar models ....................................................................................................................................2 Uncertainties in the models ...............................................................................................................3

1) Convection.................................................................................................................................3 2) Mass loss ...................................................................................................................................4 3) Metal diffusion ..........................................................................................................................4 Degenerate objects .........................................................................................................................5

Interpretation of CM diagram............................................................................................................6 Characteristic initial masses ............................................................................................................10 Igniting He in degenerate core.........................................................................................................12 Remnant-mass- initial-mass relation ................................................................................................16 Bounding curves in CM diagram.....................................................................................................17 Dependence of CM diagram on metallicity.....................................................................................18

Order of magnitude estimates ..............................................................................................................22 Pulsating stars ......................................................................................................................................23

Types of variable stars in instability strip ........................................................................................24 Classical Cepheids.......................................................................................................................24 Mira variables .............................................................................................................................26 W Virginis variables – Population II Cepheids ...........................................................................26 RR Lyrae ......................................................................................................................................26

Synthesis of the chemical elements .....................................................................................................31 Production of metals ........................................................................................................................38 Cosmic Helium abundance ..............................................................................................................42

Measuring cosmologicalY in unevolved stars ......................................................................................42 First method .............................................................................................................................42 Second method.........................................................................................................................42 Results......................................................................................................................................42

Star formation......................................................................................................................................43 Initial mass function (IMF)..............................................................................................................45 Prototypical IMF..............................................................................................................................47

2

Chapter 4

Evolution of stars and stellar populations

• Goals of stellar evolutionary theory:

o Explain arrangement of stars in CM diagram; o Reproducing regularities such as MS mass-luminosity relation.

Stellar evolution and CM diagram

Stellar models

• Assumptions made by stellar models: o Spherical symmetry; o Hydrostatic equilibrium; o Description of structure ( ),R T :

§ Four non- linear partial differential equations: ( )RM , ( )L R , ( )P R , ( )T R ; § Equation of state for the gas.

o Luminosity generated at the center

• Structure of stars determine by mass + metallicity;

• Nature of energy: thermonuclear fusion ==> MS = episode of conversion of H in He;

o p-p chain reaction: 2

2 3

3 4 7

3 3 4 7 7

7 4

2 or

2

p p H e

H p He

He He Be

He He He p Be e Li

Li p He

ν

γ

γ

ν

+

−

+ → + +

+ → +

+ → +

+ → + + → + + →

3

o CNO cycle : catalyse reaction 12 13

13 13

13 14

14 15

15 15

15 12 4

C p N

N C e

C p N

N p O

O N e

N p C He

γ

ν

γ

γ

ν

+

+

+ → +

→ + +

+ → +

+ → +

→ + +

+ → +

• Speed of reaction proportional to 2ρ and T ; • CNO cycle sensitive to T ==> dominant process in hottest (massive) stars; • MS sequence: zero-age, chemically homogeneous, H-burning stars of different masses; • Changes of metallicity ==> different families of MS; • As stars evolve ==> changes in chemical composition and physical properties due to successive

episodes of thermonuclear reactions;

Uncertainties in the models

1) Convection

• Schwarzschild criterion: adiabaticT T∇ > ∇ o Gradient of temperature at r for the entire heat flux to be carried by the outward

diffusion of photons is greater than the gradient of temperature if specific entropy of gas was constant in the neighborhood;

o Cell of hot gas move upwards and replaced by cells of cooler gas that fall downwards; o Gas transport energy more efficiently; o Evolutionary calculation shows that the minimum gradient = adiabaticT∇ .

• Mixing length theory o All rising and falling convective cells travel one mixing length l before dissolving into

the ambient medium;

o Dimensionless parameter: l

Hα ≡ , where H local scale height

1lnd PH

dr

− =

, the

distance over which the pressure declines by factor 2.7e = ;

4

o α determines the speed with which convection erodes gradient in chemical composition;

o Convection regions ==> top + bottom ==> boundaries = region of strong chemical

inhomogeneities; o Convection overshoot: boundaries of convection regions where turbulence – rising and

falling cells – spill over into adjacent non convective regions; size of overshoot region β (unit of H) play an important role in the evolution of stars;

o Not obvious if α and β are the same for stars of every spectral type; o MS characterized by small convective zones, except at extreme ends; o Subgiants and giants ==> enormous convective zones

2) Mass loss

• Stellar wind: roughly symmetric flow of plasma from the surface of the star out into ISM; More important for evolved stars – but very difficult to determine theoretically

3) Metal diffusion

• Rate at which elements heavier than H settle into the center of the star; • Internal equilibrium ==> stratification of heavier elements at the center and lightest floating at

the surface; • Helium diffusion: rate at which Helium sink towards the center;

o Originally a primordial element homogeneously mixed with H; o Downward motion of He reduces fraction of H in the core and brings forward the

moment at which nuclear burning exhausts the supply in H;

5

Degenerate objects

Ø Consider a box filled with N non- interacting Fermions ==> N different single states must be occupied (Pauli exclusion principle);

Ø At 0T = system is in lowest-energy eigenstate;

Ø If linear scale L changes, momentum increases ==> 1 3pL

ρ∝ ∝h

;

Ø Pressure exerted by material: 3

EP E

Lρ∝ ∝ ;

Ø Non relativist ==> 2 2 3 5 3E p Pρ ρ∝ ∝ ⇒ ∝ ==> adiabatic compression ==> ratio of

specific heat 53

γ = ;

Ø Relativist ==> 4 3E p P ρ∝ ⇒ ∝ ==> ratio of specific heat 43

γ = ;

Ø Mass of fully relativistic degenerate object as a function of density 0ρ : as 0ρ → ∞ ,

25.87chan µ −→ = eM M M , where µ is the number of atomic mass units per relativist

fermions; for He WD, 2 1.47chanµ = ⇒ = eM M ; Ø In many stars, volume of atom < isolated one ==> e- no longer bound if star contract ==>

wavefunction shrink and energy increases (same for neutron stars).

6

Interpretation of CM diagram

• Requires conversion of T and Lbol in color and M; 1. Fastest approach

o Assemble samples of nearby stars (stars with metallicity comparable to the Sun) for which effT , bolM , z and colors have been observationally determined;

o Assign to theoretical model nearest values of effT and bolM ;

o In particular, interpolating in tables of values of effT vs. colors ==> estimate of color of theoretical stars;

2. Second approach (metal richer or poorer stars) ==> requires sophisticated stellar-atmosphere

codes + precise mass of atomic and molecular data o Tuning stellar atmosphere code to stars bolM and effT ; o Determining colors observing model stars; o Complicated for cooler or metal rich stars ==> lots of lines to fit!

• Life of stars rule by simple thermodynamic principles: heat flows from inside to outside ==> from hotter to colder;

• Heavier nuclei form in core: ...H He C O Si Fe→ → → → → → ; • Energy released in formation of given nuclear species ==> stop compression for a while ==>

exhaustion of nuclear fuel implies further compression; • Each period of steady nuclear burning = concentration of stars in CM diagram;

o Quantitatively, number of stars at each burning stage varies roughly as τ , the time it takes to exhaust nuclear fuel burnt at that stage;

• Most important stage: Main Sequence (MS) ==> transformation of H into He; the longest

period – because relative fuel-economy – more energy is released out of MS; out of MS luminosity increase ==> energy liberated increases;

o Position on MS ==> Mass – more massive ==> bluer and more luminous ==> upper left of MS;

o Locus of MS ==> metallicity – metal poor ==> below (less luminous) and left (bluer- hotter) than Solar MS;

• Massive stars 8≥ eM that leave MS: moves to the right, in a strip where stars pulsate –

Instability strip + SN explosion (in a short period), running almost vertically (increases in luminosity);

o Evolutionary track of massive stars intersect instability strip ==> Cepheid Variables;

7

• Stars of mass 8< eM spend years moving up the Red Giant Branch (RGB); o H shell burning stage (rather than in core) + fully convective envelop;

• The base of the RGB joins the MS: Sub Giant Branch (SGB) – which is short and slope form

lower left to upper right in CM diagram; o H burning stage + no convective envelop; o In Solar neighborhood SGB is sparsely populated – Hertzsprung Gap;

• Important concentration of stars: Horizontal Branch – stars that burn He in their core; o Locus of stars of different masses that have just started core-He-burning: Zero-Age HB

(ZAHB); o During time on HB, the luminosity slowly increases ==> non-zero vertical extent with

ZAHB bounding it below; o Stars of metal rich clusters congregate at the red end of the HB – Red Clump (RC); o HB of metal-poor cluster intersect instability strip ==> RR Lyrae Variables;

8

• Once stars run out of He in core move – rather quickly – away from HB up to Asymptotic

Giant Branch (AGB); o Run parallel to RGB but slightly bluer; o Burn He in shell + C in core; o As they move up the AGB – variability in light + mass loss;

• Dredge-up: very common phenomenon – important to bring C nuclei from below He-

burning shell to the surface; Carbon-star made of M Giants; • Mass loss: increases with age (luminosity); by the time stars reach HB, 20% of mass is

returned to ISM; many stars return most of their masses to ISM;

• Approaching the AGB, oscillations develop because of He burning shell at small radii + H-

burning shell around it; for 90% of stars, H-burning shell more important source of energy; occasionally, extinguished by He-burning shell;

• Mass- loss increases approaching top of AGB ==> OH/IR stars ;

o Stars near top of AGB = luminous in IR due to dusty gas ejected envelop; o Soon after reaching the top of AGB stars eject their remaining H ==> become

extremely blue (moving leftward in CM diagram)

9

• Planetary nebulae : H envelop fluoresce due to intense ionization by remaining very hot WD;

• As He-burning ceases, stars start to cool moving down the WD cooling sequence;

o For masses ~ 0.55-0.6 eM : degenerate gas in the core stop the compression;

10

Characteristic initial masses

• Least massive stars: 0.08H e;M M o Objects less massive never get hot enough to initiate H burning – supported by electron

degeneracy – Black dwarfs (Ex. Jupiter 0.001 e;M M ); o If they burn D or Li ==> Brown dwarfs;

• Next significant mass: 1.1conv e;M M o i conv<M M radiative core; o i conv>M M convective core, because of strong dependence of CNO cycle to T, which is

more important in massive stars; o Actually, size of convective regions grows gradually starting at convM and engulfed the

core at 1.5 e; M ;

11

• Further important mass: 1.8 2.2HeF − e;M M , depending on chemical composition (ignition of

He = point 6 in Fig 5.2);

• i HeF<M M explosive ignition of helium core as the star reaches top of RGB - helium flash; § Luminosity of HB independent of mass, because He ignites explosively when

core reaches 0.45 eM in He and luminosity determined by core;

12

Igniting He in degenerate core

o In degenerate objects the energy of highest occupied state Fermi energy = e

FE ;

o Object with temperature eFE

Tk

= are degenerate;

o Thermal energy of electron is independent of T, because few quantum state

available for excitation by energy kT ; o In degenerate objects, the pressure is provided by electrons with energy e

FE ;

o When heated, the energy of electrons changes only when eF

FE

Tk

→ and the object

cease to be completely degenerated;

o Since for a non relativist object, 1

Fpar

Em

∝ , the nuclei become non-degenerate for

310 FT T−; , hence for a large range in temperature the velocity dispersion of the nuclei is Maxwellian while that of the electrons is independent of T;

o According to Maxwell-Boltzmann law, as the nuclear reactions heat the objects, the

reaction rate increases, because the number of nuclei capable of penetrating the Coloumb barrier increase;

o However, little pressure is generated ==> negligible expansion; o Since the reaction rate is proportional to 2ρ , and is scarcely moderated by the

expansion ==> increases exponentially ==> explosion;

13

o i HeF>M M helium ignites quiescently, because their core is not electron degenerate § Do not burn He on HB or in red clump ==> star above HB during core helium

burning; § At the point of ignition, the mass of the He core increases with iM ==>

luminosity increases with iM ;

14

§ For 20i ≤ eM M , point of He ignition correspond to point 6 in Fig. 5.2; § For 3i > eM M , most of He is burnt near the bottom of the track, down from

point 6 up to tip of AGB above 6; § For 6i > eM M , significant horizontal displacement due to cessation of

convection in the star’s outer envelope;

o 8i up> e;M M M C ignites quiescently in the core; o In all but most massive stars ( 30i ≥ eM M ) nuclear burning proceeds until core contain

iron peak nuclei ==> no more energy available ==> core contract and heats;

o Two processes lead to catastrophic gravitational collapse: 1) Capture of electrons by the nuclei; 2) Shattering of nuclei into α -particles by energetic photons; • Both process diminish ambient pressure by removing energy from photons and

electrons fields which sustain the core against gravity ==> runaway contraction; • The smaller the nuclei, the higher is e

FE and the faster is processes 1 and 2.

15

o Supernovae: energy released by the core implosion • Depending on the progenitor mass, remnant = Black Hole (BH) or neutron

star; • Mass of neutron star limited to 1.4 eM ;

o Star with 60i ee> e;M M M becomes unstable before forming iron-peak elements; at the point of instability their core contain O;

• At temperature 2

722 10e

em c

T Kk

≡ ×; , thermal density of positrons becomes

significant ==> pair-production instability; rapid increase in the number of e e+ − pairs with T ==> specific heat of vacuum becomes unusually large and the ratio of specific heat γ in the star drops below 4 3γ = ==> star becomes unstable;

• If 120i > eM M a few violent pulsations ensue as O-burning reverse the

gravitational collapse, but at the end the core collapse and the envelop may be completely ejected;

• For 120 300i< <e

MM O-burning completely disrupt the star while even more

massive stars collapse into a BH without pulsation; • Before they become unstable, stars more massive than 60 eM will have ejected

their H envelopes ==> SN of type Ib (no H lines in their spectra);

16

Remnant-mass-initial-mass relation ( )r iM M

• Stars eventually return most of their envelop to the ISM ==> problem = predicting rM from iM ; NOTE interest only for 0.8i > eM M , because no lower-mass stars

have had time to complete their evolution;

• For low values of iM , rM climbs the possible values for WD; • For 8i up= = eM M M up to 60i = eM M , remnant is neutron star and rM clusters

around 1.4 eM ; • Some less massive stars may turn into BH; but not much more massive than 1.4 eM ; • More massive BH ==> 60i > eM M ; X-ray astronomy revealed 3BH ≥ eM M ;

possibly, rM increases from 1.4 eM at 60i = eM M to 3 eM at 90i = eM M ;

17

Bounding curves in CM diagram

Ø Dot lines in figure 5.1 bound a region of different pulsating stars (instability strip):

o RR Lyrae stars – fairly metal poor HB stars; o Cepheid variables – SG o W virginis stars; o δ Scuti – pulsating MS stars o Pulsating WD;

Ø De Jager limit (1984): forms the upper edge of region occupied by variable stars; slope

downwards from left to right ==> blue stars ~ 6 times brighter than red ones; o Massive stars evolve from MS at constant luminosity ==> evolution carries them across

the de Jager limit; o Largest mass- loss rates occur in the vicinity of de Jager limit; o No stars appear above and to the right of Jager limit ==> mass loss stop redward

movement; o Alternatively, stars evolve so quickly once they reach the de Jager limit that probability

to observed them is consequently low; Ø To the right of the populated CM diagram lies a forbidden region: where stars cannot found

equilibrium; the boundary between forbidden and populated regions = Hayashi line - Locus of fully convective stars in CM diagram;

o At lowest setting of gas consumption, heat produced escape radiatively from surface ==> L and T are small ==> as L increases, T also increases;

o At critical setting, same part of cloud becomes convective (at higher setting = completely convective) ==> T nearly independent of L – at this point clouds swell sufficiently rapidly that additional luminosity can be radiated from surface without changing temperature;

18

Dependence of CM diagram on metallicity

• Structure of CM diagram is the same but the precise location of various branches depend on ( ), ,X Y Z ;

• Figure 5.8 shows the evolution for moderately metal poor stars: 0.23, 0.0004Y Z= = (extreme

case); while most He produced by Big Bang, significant quantity of He produced by stars alongside of heavy elements ==> lower both Y and Z;

• Comparing figure 5.2 with figure 5.8 ==> lowering Z by 1.7 dex affects individual stellar

tracks; • In general, stars becomes brighter and hotter;

o Stars with 0.6 eM becomes brighter by factor 2.3 and hotter by a factor 1.2; o Above 4 eM stars are mostly hotter; 1.3 brighter and 1.3 hotter;

• In general, ZAMS moves down by just under 1 mag at a given color (shift of 0.06 in log effT and 0.32V K− −; );

19

• Subdwarfs: extremely metal poor MS stars ==> lie below the ZAMS of the Hyades cluster ==> can be detected based on UV-excess;

o ( ) ( ) ( )

star MS HyadesU B U B U Bδ

−− = − − − for same value of B V− ;

o At fixed metallicity, ( )U Bδ − decreasing function of B V− ==> larger for blue stars

due to line blanketing; o Color index 0.6δ (Sandage 1969): UV-excess corrected by a factor to make it

independent of B V− at fixed metallicity;

o Correlation with metallicity (Laird, Carney & Latham 1988):

[ ]( )1 2

0.6 0.0776 0.01191 0.05353 Fe Hδ − + −; ;

o Vertical offset VMδ between ZAMS and Subdwarfs of given 0.6δ :

( )2 30.6 0.6 0.60.862 0.6888 53.14 97.004VMδ δ δ δ= − + − .

20

• Lower metallicity stars make longer excursions towards higher temperatures and bluer colors during He burning;

• Blueward excursion significant for stars with 2≥ eM M ==> cross instability strip;

• Typical evolutionary phases = shorter for low metallicity stars (Table 5.3); • RGB is also much more vertical ==> colors depend less strongly on masses of stars;

21

• AGB runs nearly vertical and less blueward ==> half way up AGB star hotter by 0.1dex;

• HB slightly lower: ( ) [ ]0.17 0.83VM HB Fe H= + (Lee, Demarque & Zinn 1990; but this is still

controversial see: Bounanno, Corsi & Fusi Pecci 1989; Chaboyer, Demarque & Sarajedini 1996);

• On HB a 0.7 eM lies significantly to the left of a 1.1 eM stars and slightly below a 1.5 eM

one; • For solar metallicity stars, the less massive the further to the right on HB + total extent small

==> formation of redclump;

22

Order of magnitude estimates

• Relation Mass-Luminosity on the ZAMS (Bressan et al. 1993):

2.14

3.5

4.8

81 ; 20

1.78 ; 2 20

0.75 ; 2

MSLL

≥ ∝ < <

≤

ee

e e e

ee

M M MM

M MM M

M M MM

• MS lifetime MSτ : length of time the luminosity can be supported by the conversion of H into

He o Fusion of 4 26.7Mevp He E+ → ⇒ ; ; 25 Mev injected into burning region and the rest

radiated by neutrino; o Conversion of mass ∆M of H releases energy: 20.0067E c= ∆M ; o If fraction α of total mass can be converted: 20.0067MS c Lτ α= M ; o Detailed computation ==> when 1/10 of stellar mass consumed, stars move away from

MS rapidly ==> 1

10 GyrMSLL

τ−

e e

∼ MM ;

• HB lifetime HBτ : length of time the luminosity can be supported by the conversion of He into

C+O on the Horizontal branch o For i HeF<M M , luminosity on HB 50HBL Le∼ ; o Stars remain there until core of He is burnt 0.45He = eM M ; o Typically ½ converted in C and ½ converted in O ==> 1 1

2 2He C O∆ → +M with energy

released 4 27.2 10 c−× ∆M ;

o Horizontal branch lifetime estimate: 4 27.2 10 0.45

0.1 Gyr50HB

cL

τ−× × e

e; ∼M

;

• SG lifetime SGτ : length of time passed as Supergiant:

o For i up>M <M , 1.4 eM is turned into Fe before exploding into a SN;

o Conversion of mass into energy: 20.0085 c∆M ; o Luminosity 3

3 10L L= e

o 2

3

0.0085 1.40.18 GyrSG

cL

τ×

= e ;M

23

• Cooling lifetime of WD in CM diagram:

o Radius stay constant as it cools, because degenerate pressure independent of T; o 4

effL T∝ or 10log constantbol effM T= − + ==> constant-radius lines run roughly parallel to MS (Cool, Piotto & King 1996);

Pulsating stars

• Oscillation in stars ==> variation in luminosity • Usually very small – asteroseismology (Brown & Gilligand 1994); • Frequency of oscillations = normal modes;

o p-mode: stellar atmosphere heathing in and out, associated with substantial fluctuations in pressure (explains the p);

§ Radially propagating sound wave confined within the star with 12

Rλ = ;

§ Large amplitude + asymmetrical form; o g-modes:

§ fundamental: all points of atmosphere move together and no point where pressure fluctuation vanishes throughout oscillation;

§ first overtone: some points move inward while other move outward – nodes =

point where the pressure vanishes; 32

Rλ = and lightcurve is sinusoidal;

• Most large amplitude pulsation excited by instability produced by κ-mechanism (Eddington):

Opacity, κ , of material increase rather than decrease when heated ==> atmosphere function as a heat engine: radiation = steam, opacity = valve, layer = piston;

o At one point of cycle, a layer of material loses support and falls downward ==> compression ==> heating ==> higher opacity;

o Heats buildup below the layer ==> pressure rise and push the layer up; o Expansion ==> cooling ==> lower opacity ==> material loses support and falls

downward ==> cycle repeats;

24

• Order of magnitude of period analysis: the pulsation period, P, should be of the order of the time required for a sound wave to move through the star:

o 1 2

P Qρρ

=

e, where ρ = mean density and Q is the pulsation constant

( )1 21 hrGρe∼ ;

o At constant effT , L increases when ρ decreases ==> Period-Luminosity (PL) relation,

for star with similar effT ; o More luminous stars have longer pulsation Period Ex. Cepheids (Leavitt 1912);

• In the CM diagram, the instability strip is related to the κ-mechanism, because ionization of Helium 2He He+ +→ causes opacity to increase, with temperature at radii strategically placed for excitation of fundamental and/or for overtone modes (Cacciari & Clementini 1990; Gautschy & Saio 1995, 1996);

Types of variable stars in instability strip

Classical Cepheids

• Stars with 3≥ eM M ==> make excursion to the left of CM diagram after reaching the RGB;

• Stars with 5≥ eM M ==> pass into the instability strip

• Instability strip becomes wider for high absolute magnitude;

25

• Cepheids are massive young stars in the spiral arm of our galaxies; • Their light curves are strictly periodic:

o B-band ==> abrupt rise ~ 1 mag (20% of the period) followed by a slower decline (50% of period);

o IR-band ==> smaller amplitude + sinusoidal variations; o At maximum light (phase 0.0) ==> highest temperature + earliest spectral type +

greatest outward radial velocity; o At minimum light (phase 0.75) ==> lowest temperature + latest spectral type +

greatest inward radial velocity; o Reach maximum radius (phase 0.4) on descending part of lightcurve and minimum

shortly after next lightrise begins; o 6 2VM− ≤ ≤ − o Period several days to hundred of days

• Modes = fundamental + overtone ==> two independent mass estimates: CM location +

pulsation theories ( coincident only if accurate opacities are used); • PL relation ==> reflects variation with mass of the frequency of the fundamental mode; • PL relation depends on metallicity (color varies with T and Z) ==> PLC = slope in figure

5.13; o Problems:

1) Significant IS absorption ==> important reddening correction; 2) Full PLC nonlinear in P, MV, and ( )0

B V− (Coulson 1986);

• PL relation in IR ==> reddening correction negligible + PL relation steeper in K (but higher uncertainties due to the small amplitudes of the pulsations);

• Optimal = multicolor PL relation ==> accurate periods in B and V and apparent magnitude

in K (Jacoby et al. 1992);

• Feast & Walker 1987: 2.78log 4.1310dV

PM = − −

with a scatter of ~ 0.3 mag;

• Madore & Friedmann 1991:

( )2.43 0.14 log 3.50 0.0610dB

PM = − ± − ±

with a scatter of ~ 0.36 mag

( )2.76 0.11 log 4.16 0.0510dV

PM = − ± − ±

with a scatter of ~ 0.27 mag

( )2.94 0.09 log 4.52 0.0410dR

PM = − ± − ±

with a scatter of ~ 0.22 mag

( )3.06 0.07 log 4.87 0.0310dI

PM = − ± − ±

with a scatter of ~ 0.18 mag

( )3.42 0.09 log 5.70 0.0410dK

PM = − ± − ±

with a scatter of ~ 0.13 mag

26

Mira variables

• Stars that approach the AGB; • Long period 80≥ days; • Large amplitude of pulsation; • κ-mechanism within zones of H and He+; • Because the atmosphere is highly convective there are no theory yet on their mode of

pulsation; in particular, unknown if fundamental or overtone (Gautscy & Saio 1996); • P increases with L : Feast et al. 1989 PL relation in LMC:

( )3.57 0.16 log 1.21 0.391dK

PM = − ± − ±

• In MW, P correlated to kinematics ==> stars with longer P confined to plane (Jura & Kleinmann 1992);

W Virginis variables – Population II Cepheids

• Low mass, metal poor stars - evolve away of HB on exhaustion of He in their core; • Halo stars ==> high galactic latitude or GC or center of MW; • More closely related to RR Lyrae variables than to Classical Cepheids; • 0.8d 30dP≤ ≤ • PL relation not as tight as for Classical Cepheid;

RR Lyrae

• HB stars with spectral type centered on A; • 0.8 e∼M M while original mass 1.0i e∼M M • Old, relatively metal poor; • 0.5dP ; ; • RR Lyrae with 0.5dP ≥ and 5S∆ ≥ found in GC + halo + center of MW; • RR Lyrae with 0.4dP ≤ and 5S∆ ≤ found in MW plane ==> old disk stars; • Much fainter than Cepheids;

27

• Peak of RR Lyrae at definite magnitude ==> easy to detect; • Used to determine Sun distance from center of MW; • Used in interstellar density analysis of spheroid component; • Good distance estimator for cluster:

o 2-3 magnitude fainter than brightest stars ==> easy to observe; o Easily identified; o Well defined absolute magnitude;

• Bailey types: a, b, c and d, according to shapes of lightcurve: o abRR (majority):

§ Asymmetric lightcurve; § 0.4dabP ≥ ;

§ 0.55dabP ≈ ; § Large amplitude variations: 0.5 1.5m≤ ∆ ≤ ; § Pulsate in fundamental mode; § Halo stars;

o cRR (50% of remaining ones): § symmetric (sinusoidal) lightcurves; § small amplitudes 0.5m∆ ≤ ; § short period 0.4dcP ≤ ;

§ 0.3dcP ≈ ; § Pulsate in first overtone; § Old disk stars;

28

o dRR (50% of remaining ones): § Pulsate in fundamental and first overtone mode; § Halo stars;

• Because most RR Lyrae stars are metal poor their metallic lines spectral type tend to be

classified earlier than spectral type appropriate for hydrogen lines;

==> Preston 1959: ( ) ( )10 S spectraltype H lines spectraltype CaII K lines ∆ ≡ − , where

the spectral type is measured at minimum light, in 1/10 of spectral class;

S∆ correlates with metal content: 0S∆ = for solar metallicity & 10 to 12 for most metal poor; ==> Butler 1975:[ ] 0.16 0.23Fe H S≈ − ∆ − , can be used to estimate cluster’s metallicity;

• Estimating VM - 5 methods:

1. Statistical parallax of field stars (Layden et al. 1996); o Suffer from errors in proper motion + intrinsic spread produced by various metal

contents

2. Calibration of GC CM diagrams by MS fitting; 3. Combining apparent magnitude of stars in Magellanic clouds with known distance of

clouds; o Assume RR Lyrae stars are the same than in the MW; o Distance to LMC and SMC must be determined by independent means;

4. Matching observation to prediction of complete stellar model; o Error in stellar structure calculations;

5. Baade-Wesselink method (see details below);

o Error in radial velocities – due to non uniform expansion; o Non radial expansion; o Stellar color not well correlated with temperatures;

29

Ø Proposed by Baade (1926) – developed by Wesselink (1946); o Principle: Doppler velocities allow to measure change in linear size of the source

and measurement of color and apparent magnitude provide a measure of fractional change in size and hence absolute luminosity and distance;

o Starting with the radius of the star R and from the optical color or spectral shape

effT 2 44 effL R Tπ σ⇒ = or 10log 5logbol effM T R C= − − + , where C is a known constant;

o Determining the radius: lines in spectrum show Doppler shifts that vary cyclically

with the same period as the star’s brightness variations ==> global expansion and contraction of envelope;

o Amount by which size changes between time 0t and 1t obtained by integrating the

line of sight velocity: ( )1

01

t

lostr p v t dt∆ = − ∫ ;

* The negative sign is because 0losv < for expansion (blueshift); * Factor p is correction for expansion out of line of sight (outer limb ==> losv transverse ==> observed mean lower than true value);

o Starting with unknown 0r at 0t , by time 1t , size changes by measurable amount

0 1r r+ ∆ ==> change in apparent magnitude:

( )1 0 1 0 0 1 0 1 05 log log 10 log logbol bol bol bol eff effm m M M r r r T T − = − = − + ∆ − − − ;

==> If we can determine effT we can then determine 0r ;

o Easy solution = select moments when 1 0eff effT T= ==> 10 1

H rr

H∆

=−

, where

( )1 0 510 bol bolm mH −≡ ; o Complete method:

§ Obtain spectra throughout period of oscillation and measuring Doppler shift in spectral line ==> ( )losv t ;

§ Series of measures of apparent magnitude in several band-passes ==> ( )bolM t∆ , corrected for extinction;

§ Selecting data from 2 parts of oscillation cycle at which stars has identical color ==> same efficient temperature;

§ Calculation of r∆ and M∆ ==> R ; § Result: series of R varying along the cycle.

30

• Absolute magnitude of RR Lyrae stars = 0.5 1VM≤ ≤ , with average = 0.6VM ∼ (obtained by averaging flux over a period);

• Correlation with metallicity (Jones et al. 1992): ( ) [ ]0.16 0.03 1.02 0.03VM Fe H= ± + ±

==> if metallicity can be determined from spectra, VM can be determined to 10%; • Dependence of PL on metallicity is less important in NIR:

( )2.3 0.2 log 0.88 0.061dK

PM = − ± − ±

;

• PLZ relation: ( ) ( ) [ ]2.0 0.3 log 0.06 0.04 0.7 0.11dK

PM Fe H = − ± − ± − ±

;

31

Synthesis of the chemical elements

• Atomic Nuclei = bound state of Z protons + N neutrons ==> nuclide = points (Z, N) in Z-N plane;

• Atomic number: A ≡ Z + N;

Radioactive decay: β ± decay (emission of e− or e+ ) α decay (emission of He nucleus)

32

• In Z-N plane stable nuclide trace a stability band; parallel and centered just above the N = Z;

• Binding energy per proton and neutron ε varies systematically within Z-N plane: as one moves

across stability band along a line A = Cte, ε increases then decreases; if we imagine that

nuclide lie a distance ε below the Z-N plane ==> stability valley, with slopes from each end

towards the middle ==> deepest point = 56Fe (8.79 Mev below H and 1.21 Mev below 238U );

• Stars derive much of the energy they radiate by shifting their protons and neutrons down the valley from H to He forward to Fe;

33

Groups of nuclides: • Primary nuclides: synthesized in star that started with only H and He ==> Amount produced,

independent of metallicity of stars; • Secondary nuclides: requires the presence of primary nuclides synthesized in previous

generation of stars;

v α nuclides: 20 24 28 32 36 40Ne, Mg, Si, S, Ar, Ca can be formed by adding 2, 3, 4, … α particles to 16 O ;

o Formed by C and O burning: § 12 12 20 4C C Ne He+ → + ; § 16 16 28 4O O Si He+ → + ; § Capture of α particles: 20 4 24Ne He Mg ?+ → + ;

o More abundant than their immediate neighbors on Z-N plane; o Abundance decreases smoothly with increasing atomic number;

v 23 Na and 27 Al = sole stable isotopes with odd-Z; o related to α nuclides through C and Ne burning in hot stars: 12 12 23C C Na n+ → + ==>

primary; o But can also be produced when He and 16 O is present ==> secondary;

34

v Iron peak nuclides: 40 < A < 65 ==> Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu o Floor of stability valley here is flat ==> stable isotopes have similar biding energies; o Form late in the evolution of most massive stars – with exceedingly hot cores:

§ Temperature so great that nuclei are constantly forming and splitting apart ==> thermal equilibrium ==> probability of each configuration E kTe−∝ (Boltzmann factor) where E is the energy of the configuration;

§ Flatness of floor + thermal equilibrium ==> many nuclides are present in

significant number;

v s-process nuclides (slow process): o Lie along the valley floor up towards higher A from the iron peak;

88 89 90 138 139 140 141 208 209Sr, Y, Zr, Ba, La, Ce, Pr, Pb, Bi ; o Certain reactions release free neutrons: Ex. 12 12 23C C Na n+ → +

§ Free neutrons are easily absorbed by nucleus: Ex. 56 57Fe n Fe+ → § Further similar absorption leads to 59Fe which is unstable with ½ life 45 days; § β − decay ==> 59Co ; § with further neutron absorption ==> 60Co again which is unstable with ½ life

330 days; § β − decay ==> 60 Ni ; § Proceeding in this way, the original 56Fe moves ever further up the periodic

table;

35

o At any given time within nuclear burning regions there are nuclei at different level of transmutation chain – time spent at given level is inversely proportional to cross-section for absorption of neutron ==> when the neutron production ceases the number of nuclei at each point in chain is inversely proportional local cross-section for absorption of neutron;

o Multiplying the abundance of s-process nuclides by the cross-section for absorption of

neutron ==> not independent of A, but decreases smoothly with increasing A; o s-process = secondary elements

§ They require elements from iron-peak to be present in a C or O burning zones; § Since these elements form after C and O, they must have been synthesized in

previous generation of stars;

o Naively, the abundance of secondary nuclides would be expected to be proportional to square of the abundance of primary elements:

§ Assume the production rate of primary element to be constant: 1primda

cdt

= ;

§ The rate of secondary elements: sec2 prim

dac a

dt= ;

§ Integrating ==> 2 22sec 1 2

1

12 2 prim

ca c c t a

c= = ;

§ In fact observed slope of mean relation between log abundance of secondary vs primary elements smaller than two; reason = density of neutron poisons ( 3He or 14 N , which absorb neutrons) increase with metallicity ==> diminishes the

36

number of neutrons and the rate of production of s-process nuclides, which is proportional to the multiplication of density of heavy elements per neutron flux;

v r-process nuclides (rapid process): o Lie on neutron rich side of the va lley floor just above the s-process elements;

80 81 84 128,130 127 192 193 196,198Se, Br, Kr, Te, I, Os, Ir, Pt ; o Form by rapid capture of neutron; o Principle: depends if the first unstable nuclide that is encountered on moving vertically

up the Z-N plane from the stability valley have time to β -decay to stable nuclide of higher Z before a further neutron is absorbed; § Yes ==> nucleus shuffles along the chain of s-process nuclides that lie near the

bottom of the stability valley; § No ==> nucleus is driven up the side of valley where nuclides are more unstable

until the local ½ life to β -decay becomes comparable to mean time between neutron captures; then jogs along the length of the valley;

o When neutron-generating ceases each nucleus slides down towards the valley floor by

β -decay coming to rest as an r-process nuclide; o Abundance is not expected to be inversely proportional to their neutron-capture cross-

sections ==> distinguish between r and s process; o but in fact, distinction is not sharp because many nuclides can be synthesized by both

processes;

37

o Distinction sharpest when several stable nuclides lie on the neutron-rich side of the valley of stability and are isolated from the floor by unstable nuclides;

o Figure 5.17 shows the valley in the region of Sb where 124Sn and 128Te (two pure r-

process) form islands insuring that 124Te and 128Xe are pure s-process nuclides; o Nearly all elements heavier than Fe have isotopes that can be synthesized by s-process

==> small abundances due to large neutron-absorption cross-sections; o Still unclear if r-process elements are primary or secondary; not established at what

stage in star evolution material might be exposed to large flux of neutrons;

38

Production of metals

• 4 process contribute to the production of the metals:

o Straightforward He, C and O-burning ==> 12C , 16 O and α nuclides; o Iron-peak nuclides: form when the nuclei achieve approximate equilibrium in hot cores

of evolved stars; o s-process elements: form from slow neutron irradiation of heavy nuclides synthesized in

an earlier generation of stars; o r-process elements: form from rapid neutron irradiation of heavy nuclides in an primary

or secondary manner;

• Important questions to resolve: o At what stage of the stellar evolution the r-process really occurs? o How and when is the metal injected into the ISM?

• For i up<M M remnant = WD ==> locks C, O and α elements ==> these stars enriched ISM

only in He (and N); o Importance of Dredge-up ==> some heavier elements can reach ISM; Ex. Carbon stars; o Analysis of PN ==> abundance of He, N, O and C in ejecta;

§ Problem = to determine mass of element synthesized (quite small), need distinguishing between abundance of ejecta and original ISM;

§ Easier in metal-poor ISM systems (Ex. SMC) ==> (Monk, Barlow & Clegg 1988): [ ]N H larger in PN than in HII regions by 1 dex; differences fall to 0.8 and 0.4 in the LMC and MW PN, respectively; For other elements, the difference is still smaller Ex. [ ]O H in PN only 0.3 dex higher than in HII regions;

§ Since the enhancement of N is thought to be due to free-neutron captured by C ==> one expect other s-process elements to formed by PN;

o Heavy-element production by small mass stars plays an important role in the chemical

evolution (especially when the ISM is metal poor);

39

• For i up>M M : SN = prime source of iron-peak + r-process elements o Classification:

§ Type II: spectra show H lines;

• Luminosity 9 90.4 10 4 10BLL

× ≤ ≤ ×e

(Tammann & Schröder 1990);

• Do not occur in early-type galaxies – mostly in spiral arms; • Progenitor = massive stars.

§ Type I: spectra show no H lines; • Ia: absorption due to Si+ (some authors require it to also show He lines,

otherwise it is classified as Ic); o Luminosity: 99.6 10BL L= × e ; o Occur in all types of galaxies – anywhere; o Progenitor = WD in binary system;

• Ib: show He lines (more similar to type II than type Ia): o Differences with Type Ia established only from 1980’s; o Occur in spirals near spiral arms; o Progenitor = massive stars;

o SN rate in MW = 1 per 40 years and only 15% are of type Ia (Tammann, Löffler &

Schröder 1994); o Physical Interpretation (Wosley 1990):

§ Type II and Ib = high mass stars suffering a core-collapse (core-collapse SN); § Type Ib more massive than type II have lost their H envelope through stellar

wind or lost it to companion (in this case mass argument is weak); § Type Ia = thermonuclear explosion of C/O at the surface of a WD triggered by

accretion of matter from a companion;

• Metal production by core-collapse SN: o Model = structure of massive star at the moment of core-collapse;

§ Difficulty #1: Amount of mass blew away by implosion? • 99% of the energy is released in a blast of neutrinos; • Only a small fraction transferred mechanically to the envelop; • Some implosion may not produced detectable electromagnetic radiation;

§ Difficulty #2: Changes in chemical composition?

40

o 1970’s mass ejection model = mass-cut; § the mass-cut define a radius within pre-collapse star separating the ejected from

the remnant matter; § A shock wave induces a violent burst of nuclear reactions in departing matter

==> Si burnt to iron-peak nuclides including 56 Ni ; § During nuclear flash, iron-peak elements are intensely bombarded by neutron

==> r-process selements; § Further out layers of materials that had long ago been processed by the star to C,

O, α -elements and s-process elements are ejected by the shock; § Core-collapse SN contribute every kind of nuclides to ISM: H, He, C, O, α -

elements, s-process and r-process elements;

§ Table 5.4 shows the mass ejecta expected for stars with different initial mass;

• Column #1: mass of He Core at the onset of He burning; • Column #2: initial mass (no wind); • Other columns = masses of various elements lying outside the mass-cut;

• Ex. SN 1987A (50kpc in LMC – 23 feb 87):

o Neutrino burst was observed (Arnett et al. 1989; McCray 1993); o Progenitor ignite He when the core contained 6 eM of He o Progenitor = Blue Giant – MK type = B3I ==> suffered significant mass loss prior to the

explosion; o From Table 5.4, 20i ≥ eM M ; o After 100 days of the explosion, SN light emission was powered by radioactive decay of

0.069 eM of 56 Ni (1/2 life of 8.8d) to 56Co then 56Fe ; ==> 100d-300d light curve

41

declining exponentially with time constant that perfectly fits the 111.3d mean life of 56Co and 0.069 eM of 56 Ni ;

o After 300d, the light curve decline faster because the ejecta becomes too rarified to

degrade into optical photons all the gamma rays that carry much of the energy released by the decaying Co ==> from 175d, 837Kev and 1240 Kev of gamma ray lines of 56Co were detected;

o Observed spectra ==> total kinetic energy 491.5 10 J×∼ in the ejecta; o P-Cygni profile ==> outermost envelop ejected at 20000 km/s≥ , soon slowed to 2500

km/s; o Structure of the envelop highly complex due to hydrodynamical instabilities ==>

metallicities of the envelop not well determined;

• Metal production by Type Ia SN: o Progenitor = C/O WD achieving 1.38 eM through gas accretion from a companion; o Electron in core highly degenerate ==> star’s nuclear fuel liable to ignite explosively; o Model: a deflagrating wave of burning sweeps through the star ==> ½ of C and O

converted to iron-peak elements in about a second; o Predicted lightcurve and mixture of nuclides depend sensitively on the speed of the

wave ~as fast as sound speed in material; o Explosion releases 1% of the energy of a SN type II, a large fraction channeled into the

kinetic energy of the ejecta 4410 J∼ ==> energy concentrated in only 1.4 eM instead of

10 eM the velocity of the ejecta is higher: 310 20 10 km/sejv − ×∼ ; o The star leaves no remnant (completely disrupted); o Entire UV and optical diplay of SN type Ia derives from radioactive decay of 56 Ni to

56Fe ==> produce 5-10 times much 56Fe as SN 1987A 0.7 e∼ M of which 0.45 eM

form by the decay of 56 Ni ; ==> significant source of r-process elements;

42

Cosmic Helium abundance

• Most of He was synthesized in first minutes of Big Bang ==> abundance of oldest + metal poor stars ? cosmic abundance;

Measuring cosmologicalY in unevolved stars

First method 1. Observe hot young stars in dwarf metal-poor galaxies; 2. Plot vs. Y Z ; 3. Extrapolate ( )Y Z to 0Z = ;

Second method 1. metallicity of ISM in metal-poor galaxies;

Results

• 0.228 0.005Y = ± (Pagel et al. 1992); • Standard value: 0.235Y = ;

• Lowering Y at fixed Z raises ZAMS: ,

3eff

bol

Z T

MY

∂ ∂

; (Lebreton 1998);

• Fixing 0.02Z = and lowering 0.23Y = raises ZAMS at constant effT by 0.15 mag; • Possible third method of determining cosmologicalY : correction between Y , Z and ZAMS for

oldest more metal poor stars: (Cayrel 1968) subdwarfs with accurately known distances ==> ZAMS 0.75 0.3 mag± below Hyades MS; confirmation (Eggen 1973) with cooler subdwarfs

comparing VM with ( )R I− ==> 0.23Y = for subdwarfs, compared to 0.28Y = for Solar metallicity stars;

==> subdwarfs ( ) ( ) 0.28N He N H ≈ , compared to 0.10 for Solar metallicity stars;

43

Star formation

• Phase of evolution of star which is very difficult to modelized, because system not in hydrostatic equilibrium + complex nature of ISM + importance of magnetic fields (magnetohydrodynamics + degree of ionization) + interaction with possible companions;

• Clues: young stellar clusters associated with:

1. Dense IS clouds 2. Spiral arms; ==> Ingredients = dense IS gas + large scale shock fronts;

• Possible scenario:

i. Dynamics of IS clouds: Gravity (compression) + turbulence energy (friction + pressure) +

magnetic energy (pressure); ii. At densest points, ionization level is small due to the low level of ionization radiation + low

level of CR ==> coupling to magnetic field reduced ==> magnetic energy dissipates ==> lower ionization + coupling to magnetic field;

iii. Collapse into Protostar ==> gravitational released efficiently through IR radiation ==> luminosity rise, but temperature remains low (10-20K);

iv. Eventually material becomes opaque and temperature in the densest core rise ~ 2000K ==> dissociation of 2 2H H→ ==> produces an energy sink and a free-fall collapse of the core

v. The ionization of H increases the opacity due to the electrons ==> pressure balance achieved stopping core collapse;

vi. As L increases, envelope becomes fully convective ==> protostar now follow Hayashi tracks:

44

• On the Hayashi track, the protostar reach hydrostatic equilibrium ==> moves down on longer time scales; less massive stars slowed by nuclear burning of 2 7D Li+ ;

• For 0.3> eM M , convection stop and radiative core grows ==> tracks turn sharply to the left; • Subsequent evolution: protostar contracts at nearly constant L ==> T increase ==>

thermonuclear fusion start on the MS sequence; • The pre-MS phase evolution of a star is rapid: ~ 75 Myr for 1 eM ;

• Stars with 0.08 0.3≤ ≤e

MM reach the MS with a convective core ==> move vertically in the

CM diagram until the reach the ZAMS;

45

• Objects with 0.08< eM M , run out of fuel once they have burnt 2 7D Li+ and slide down to the right on the MS;

• Many young stars have rotating disk of dust + gas = possibly protoplanetary disks; Ex. T-Tauri

stars; further accretion of matter from this disk ==> increase luminosity or produce jets;

Initial mass function (IMF)

• The Initial Mass Function, ( )ξ M , specified the distribution in mass of freshly formed stars; • In a burst of star formation: ( )0dN N dξ= M M is the number of stars formed with masses in

the range ( ), d+M M M ; • 0N is a normalizing constant that depends on the magnitude of the burst and normalization of

( )ξ M ; • Normalization of ( )ξ M : ( )dξ =∫ eM M M=M ==> 0N is the number of solar masses stars

formed during the burst; • A priori, no reason for the IMF to be universal, but assumption seems to fit observations; • Procedure to determine IMF:

i. Determine luminosity function ( )MΦ for MS in stellar neighborhood; ii. Correction for stellar evolution (not necessary for coeval population):

§ Assume constant SFR: ( ) ( )0

for

1 otherwiseMS MSt t

M Mτ τ <

Φ = Φ ×

; where t is the time

since the burst started and ( )MS Mτ is the MS lifetime of a star of magnitude M; § The correction factor takes into account that only stars of magnitude M formed

in the last fraction MS tτ of the population life;

iii. ( ) ( )0dM

Md

ξ = Φ M MM

;

o where ( )M M specifies the relation between mass and absolute magnitude;

§ theoretically ==> difficult for 0.5< eM M because low-mass stars takes too much time to settle on the MS + spectra too rich in spectral features to be able to constrain continuum + problem of convection for 0.3< eM M ;

§ Observation ==> from binary stars

46

• Problem: ξ and 0Φ are related by the derivative of ( )M M and the quality of the data is insufficient to constrain it;

• Worst, the theory predicts that ( )M M must be complex due to the physics; • Solution = compromised (Kroupa, Tout & Gilmore 1993):

o Assume that ξ is a simple function; because SF is a chaotic process ==> unlikely to bear imprints of any particular length or energy scale ==> ξ must be featureless;

o It is compatible with empirical data: the mass function 0Φ is a distorted reflection of

ξ through the function ( )M M . This function is rich in structures ==> reflected in

0Φ ==> follow the complexity of empirical data.

47

Prototypical IMF

1. Salpeter (1955): ( ) 2.35ξ −∝M M ;

2. Scalo IMF (1986): ( )

2.45

3.27

1.83

, 10

,1 10

, 0.2

ξ

−

−

−

>∝ < < <

e

e

M M MMM M M

M M

;