chapter 6: stellar evolution (part 1) · chapter 6: stellar evolution (part 1) with the...

Chapter 6: Stellar Evolution (part 1)

With the understanding of the basic physical processes in stars, wenow proceed to study their evolution. In particular, we will focus ondiscussing how such processes are related to key characteristicsseen in the HRD.

The CD-ROM that came with the text book (HKT) contains some niceand informative description and movies from stellar evolutionmodeling (e.g., CD-ROM/StellarEvolnDemo/index.html). They coverthe Main Sequence (MS) and evolved stages (although some of themovies are missing). The same programs may also be obtained fromhttp://astro.if.ufrgs.br/evol/evolve/hansen/index.htm.

Outline

Star Formation

Young Stellar Objects

The Main SequenceDependence on stellar massDependence on chemical composition

Post-Main Sequence EvolutionLeaving the MSThe red giant branchThe helium burning phaseThe asymptotic giant branch

Final evolution stages of high-mass stars

Star FormationHere we briefly discuss the gravitational instability, Jeans mass,fragmentation of gas clouds, as well as the resultant initial massfunction (IMF) of stars.

Consider a medium of uniform density and temperature, ρ and T .From the virial theorem, 2E = −Ω, we have

3kTMµmA

=

∫ M

0

GMr

rdMr =

35

(GM2/R) =35

(4πρ

3

)1/3

GM5/3 (1)

for the hydrostatic equilibrium of a sphere with a total mass M. If theleft side is instead smaller than the right side, the cloud wouldcollapse. For the given chemical composition, this criterion gives theminimum mass (called Jeans mass) of the cloud to undergo agravitational collapse:

M > MJ ≡(

34πρ

)1/2( 5kTGµmA

)3/2

.

For a typical temperature and density of a large molecular cloud,MJ ∼ 105M with a collapse time scale of tff ≈ (Gρ)−1/2.

Star FormationHere we briefly discuss the gravitational instability, Jeans mass,fragmentation of gas clouds, as well as the resultant initial massfunction (IMF) of stars.Consider a medium of uniform density and temperature, ρ and T .From the virial theorem, 2E = −Ω, we have

3kTMµmA

=

∫ M

0

GMr

rdMr =

35

(GM2/R) =35

(4πρ

3

)1/3

GM5/3 (1)


M > MJ ≡(

34πρ

)1/2( 5kTGµmA

)3/2

.

For a typical temperature and density of a large molecular cloud,MJ ∼ 105M with a collapse time scale of

tff ≈ (Gρ)−1/2.

Star FormationHere we briefly discuss the gravitational instability, Jeans mass,fragmentation of gas clouds, as well as the resultant initial massfunction (IMF) of stars.Consider a medium of uniform density and temperature, ρ and T .From the virial theorem, 2E = −Ω, we have

3kTMµmA

=

∫ M

0

GMr

rdMr =

35

(GM2/R) =35

(4πρ

3

)1/3

GM5/3 (1)


M > MJ ≡(

34πρ

)1/2( 5kTGµmA

)3/2

.

For a typical temperature and density of a large molecular cloud,MJ ∼ 105M with a collapse time scale of tff ≈ (Gρ)−1/2.

Cloud fragmentationSuch mass clouds may be formed in spiral density waves and otherdensity perturbations (e.g., caused by the expansion of a supernovaremnant or superbubble).

What exactly happens during the collapse depends very much on thetemperature evolution of the cloud.

I Initially, the cooling processes (due to molecular and dustradiation) are very efficient. If the cooling time scale tcool is muchshorter than tff , the collapse is approximately isothermal.

I As MJ ∝ ρ−1/2 decreases, inhomogeneities with mass largerthan the actual MJ will collapse by themselves with their local tff .This fragmentation process will continue as long as the local tcolis shorter than the local tff , producing increasingly smallercollapsing subunits.

I Eventually the density of subunits becomes so large that theybecome optically thick and the evolution becomes adiabatic (i.e.,T ∝ ρ2/3 for an ideal gas), then MJ ∝ ρ1/2.

I As the density has to increase, the evolution will always reach apoint when M = MJ , when a subunit reaches approximatelyhydrostatic equilibrium. We assume that a stellar object is born.

Cloud fragmentationSuch mass clouds may be formed in spiral density waves and otherdensity perturbations (e.g., caused by the expansion of a supernovaremnant or superbubble).What exactly happens during the collapse depends very much on thetemperature evolution of the cloud.

I Initially, the cooling processes (due to molecular and dustradiation) are very efficient. If the cooling time scale tcool is muchshorter than tff , the collapse is approximately isothermal.

I As MJ ∝ ρ−1/2 decreases, inhomogeneities with mass largerthan the actual MJ will collapse by themselves with their local tff .This fragmentation process will continue as long as the local tcolis shorter than the local tff , producing increasingly smallercollapsing subunits.

I Eventually the density of subunits becomes so large that theybecome optically thick and the evolution becomes adiabatic (i.e.,T ∝ ρ2/3 for an ideal gas), then MJ ∝ ρ1/2.

I As the density has to increase, the evolution will always reach apoint when M = MJ , when a subunit reaches approximatelyhydrostatic equilibrium. We assume that a stellar object is born.

The Initial Mass Function

This way a giant molecular cloud can form a group of stars with theirmass distribution being determined by the fragmentation process.The process depends on the physical and chemical properties of thecloud (ambient pressure, magnetic field, rotation, composition, dustfraction, stellar feedback, etc.). Much of the process is yet to beunderstood.

We cannot yet theoretically determine theinitial mass function (IMF) of stars.The IMF may be determined empiricallyand may be expressed in forms such as

dn/dm = φ(M) = C(M/0.5M)−x

where x = 2.35 (Salpeter’s law), valid forM/M ≥ 0.5, and x = 1.3 for0.1 ≤ M/M < 0.5 in the solarneighborhood.

Is the IMF universal?

While the IMF in galactic disks of the MW and nearby galaxies seemto be quite consistent, there are good reasons and even lines ofevidence suggesting different IMFs in more extreme environments(e.g., bottom-light in the Galactic center and top-cutoff in outer disks;Krumholz, M. R. & McKee, C. F. 2008, Nature, 451,1082).

Probably the strongest evidence for a top heavy IMF comes from theGalactic center stellar clusters (Sunyaev & Churazov 1998, MNRAS,297, 1279; Wang et al. 2006, MNRAS, 371, 38).Compared with the X-ray emission from young stars in the Orionnebula, the observed total diffuse X-ray luminosities from massiveyoung stellar clusters suggest that the number of low-mass YSOs area factor of ∼ 10 smaller than what would be expected from thestandard IMF and the massive star populations observed in theclusters.

If confirmed, this has strong implications for understanding the starformation at high z, the mass to light ratio, etc.

Outline

Star Formation






Objects that are on the way to become stars, but extract energyprimarily from gravitational contraction are called young stellarobjects (YSOs) here. They represent the entire stellar systemthroughout all pre-main sequence (MS) evolutionary phases.

Theoretically, the formation andevolution of a YSO may be dividedinto four stages:

1. proto-star core formation;2. protostar star builds up from

inside out, forming a disk around(core still contracts and isoptically thick);

3. bipolar outflows;4. surrounding nebula swept away.

The proto-star stages have the KH time scale∼ (2× 107 yrs)(M/M)2(L/L)−1(R/R)−1.

Observational signatures and classificationObservational signatures of YSOs:

I emission lines from the disk and/or outflowI more infrared luminosity due to dust emissionI variability on hours and days due to temperature irregularities on

both the stellar surface and diskI high level of magnetic field triggered activities (flares, spots,

corona ejection, etc) due to fast rotation and convectionI strong X-ray emission from hot corona.

YSOs are classified into classes 0, 1, and 2, according to the ratio ofinfrared to optical, amount of molecular gas around, inflow/outflow,etc.

I Class 0 protostars are highly obscured and have short timescales (corresponding to the stage 2); few are known.

I Class 1 or 2 protostars are already living partly on nuclearenergy (3 and 4); but the total luminosity is still dominated bygravitational energy.

The low-mass YSO prototype is T Tauri. We still know little abouthigh-mass YSOs, which evolve very fast and interact strongly withtheir environments.

Hayashi tracks

The structure of a YSO changes with its evolution. During theso-called protostar evolutionary stage, the optically thick stellar coregrows during the accretion phase.The YSOs are fully convective and are thus homogeneous,chemically. They evolve along the so-called Hayashi track in the HRdiagram:

I During the collapse the densityincreases inwards. The optically thickphase is reached first in the centralregion, which leads to the formation of amore-or-less hydrostatic core with freefalling gas surrounding it.

I The energy released by the core (nowobeying the virial theorem) is absorbedby the envelope and radiated away asinfrared radiation.

I Because of the heavy obscuration bythe surrounding dusty gas, stars in thisstage cannot be directly observed inoptical and probably even in near-IR.

I The steady increase of the centraltemperature causes the dissociation ofthe H2, then the ionization of H, and thefirst and second ionization of He.

The sum of the energy involved in all these processes has to be atmost equal to the energy available to the star through the virialtheorem. The luminosity can be very large and hence usuallyrequires convection.The maximum initial radius of a YSO R i

max ≈ 50− 100R(M/M).

If the effects due to the accretion of matter to the forming star may beneglected, the object follows a path on the HRD with the effectivetemperature similar to that given by the early expression:

Teff ∝ (Z/0.02)−4/51µ13/51(M/M)7/51(L/L)1/102.

Notice the very weak dependence on L and M. The effect of thechemical composition is reflected by the values of both µ and Z(metal abundance).

The increase of the metallicity (that causes anincrease of the opacity) shifts the track tolower Teff . The increase of the metallicity haslittle effect on µ.

An increase of the helium abundance atconstant metallicity has the opposite (and lessrelevant) effect, due to the increase of µ. Hayashi tracks of a 0.8 solar mass

star with helium mass fraction0.245, for 3 different metallicities.

Pre-main sequence stageI In this PMS stage, the YSO has formed a

radiative core, though still growing withtime.

I The star is no longer fully convective.I Its evolution has to depart from its Hayashi

track, which forms the rightmost boundaryto the evolution of stars in the HRD. As thecenter temperature increases due to thevirial theorem, the path is almost horizontalon the HRD.

I When the temperature in the core reaches the order of 106 K,deuterium is transformed into 3He by proton captures. The exactlocation when this happens depends on the stellar mass. In anycase, the energy generation of this burning is comparably lowand does not significantly change the evolution track.

Brown dwarfs, which are only able to burn deuterium (at T ∼ 106 Kwith masses ∼ 0.05− 0.1M), may still be called stars.

Outline

Star Formation





The Main Sequence

A star spend the bulk of itslifetime in the MS, where it burnshydrogen in the core. Weconsider how the basic stellarproperties depend on the massand chemical composition of astar.

The mass — a decidingparameter in the stellar evolution— determines what the centraltemperature can reach, hencewhat nuclear reactions can occurand how fast they can run, andhow they end their lives.

”Mass” Cut” diagram showing the fate ofsingle stars in various mass classes.

Dependence on stellar mass

We first check how L and R are related to the mass of a star. We canroughly estimate the mass dependence of the luminosity (L ∝ Mη),based on dimensional analysis, using the hydrostatic equilibrium stateand the EoS of the ideal gas and assuming the radiative heat transferequation,

which can be written as L ∝ M−1R4T 4, where κ is assumedto be constant in the nuclear burning region, while the hydrostaticequilibrium condition as P ∝ M2

R4 . Considering the EoS, one then find

I η = 3 if the pressure is primarily due to the ideal gas (i.e., forstars with masses lower than ∼ 10M; T ∝ P/ρ ∝ M/R)

I η = 1 if the radiation pressure dominates (for more massivestars; T ∝ P1/4 ∝ M1/2/R).

These exponents are close to the empirical measurements (e.g.,η ∼ 3.5 for stars of a few solar masses; Ch. 1). The small differenceis due to the structure change caused by the convection, whichmakes the nuclear burning more efficient.

Dependence on stellar mass

We first check how L and R are related to the mass of a star. We canroughly estimate the mass dependence of the luminosity (L ∝ Mη),based on dimensional analysis, using the hydrostatic equilibrium stateand the EoS of the ideal gas and assuming the radiative heat transferequation, which can be written as L ∝ M−1R4T 4, where κ is assumedto be constant in the nuclear burning region, while the hydrostaticequilibrium condition as P ∝ M2

R4 . Considering the EoS, one then find

I η = 3 if the pressure is primarily due to the ideal gas (i.e., forstars with masses lower than ∼ 10M; T ∝ P/ρ ∝ M/R)

I η = 1 if the radiation pressure dominates (for more massivestars; T ∝ P1/4 ∝ M1/2/R).

These exponents are close to the empirical measurements (e.g.,η ∼ 3.5 for stars of a few solar masses; Ch. 1). The small differenceis due to the structure change caused by the convection, whichmakes the nuclear burning more efficient.

How does R depend on M? We know L ∝ εM ∝ M2+νR−(ν+3),assuming ε = ε0ρT ν , and T ∝ M/R. Equating this to L ∝ M3, as anexample, we have

R ∝ M(ν−1)/(ν+3). (2)For example, ν = 18 for CNO. Then R = M0.81.

Replacing R in L ∝ R2T 4eff with Eq. 2 and

M ∝ L1/3, we then obtain(Teff

Teff ,

)=

(L

L

) (ν+11)12(ν+3)

=

(L

L

)0.12

.

This insensitivity of Teff to L is due to the strongtemperature dependence of the CNO cycle.Nevertheless, the exponent, 0.12, is still afactor of ∼ 10 larger than that for the Hayashitrack or the RGB and AGB.

The right figure shows pre-MS evolutionarytracks adopted by Stahler (1988) from varioussources, as well as the locations of a numberof T Tauri stars.

Since a star’s luminosity on the MS does not change much, we canestimate its MS lifetime from simple timescale arguments and themass-luminosity relation. If L ∝ Mη, then

τMS = 1010 yrs(M/M)1−η

Clearly, the MS lifetime of a star is a strong function of its mass.

I The MS lifetime of a star with a mass of 0.8 M is comparable tothe age of the Universe. Thus we are primarily concerned withstars more massive than this.

I While practically most of relatively low-mass stars are close toZero-Age Main Sequence stars (ZAMSs), massive stars burnhydrogen much faster, especially via the CNO cycle.

I Because of the much steeper temperature dependence, theCNO cycle occurs in a much smaller region than do thepp-chains. The requirement for fast energy transport drivesconvection in the stellar core.

I While we focus here on the evolution of isolated stars, it shouldbe noted that if a star is in a close binary then the story canchange drastically.

Dependence on chemical compositionNow we consider how the metallicity of a star affects the color andluminosity of a star. We first briefly consider the effect on the ZAMS,which are those stars who arrived at the MS recently.

I The metallicity chiefly affects theopacity, or the amount ofbound-free absorption, which isdominated by metals.

I The smaller opacity allows theenergy to escape more easily (sothe star appears bluer).

I The lower opacity also reducesthe pressure; hence theluminosity of the star needs to beincreased to balance its gravity.

Illustration of the effects of varying Y and Z onthe shape and position predicted for the 14 Gyrisochrone: Z = 0.006 (heaviest line, 0.001(intermediate), and 0.0001 (lightest). The solidlines are for Y = 0.2 and the dotted lines arefor Y = 0.3 [From the calculation of van denBerg & Bell (1985)]

Why does the luminosity increase with time?The nuclear burning changes the abundances of elements and hencethe molecular weight (µ). Here we use the sun as an example of theluminosity evolution. for an ideal gas star. If we assume that radiativediffusion controls the energy flow, then

L ∝ RT 4

κρ.

To replace R and T in the above relation, we can use R ∝ (M/ρ)1/3

and T ∝ µM2/3ρ1/3, which is inferred from the virial theorem (Eq. 1).If Kramers’ is the dominant opacity (e.g., due to f-f transitions as inthe core of the sun), then we have

L ∝ M16/3ρ1/6µ15/2

κ0.

Here the mass of the star is fixed, while κ0 does not vary strongly withthe abundances.Neglecting the weak dependence on ρ, the above relation can bewritten in time-dependent form

L(t)L(0)

≈[µ(t)µ(0)

]15/2

. (3)

To see how µ varies with time, we assume that the bulk of the stellarinterior is completely ionized and neglect the metal content that issmall compared to hydrogen and helium. Then we have

µ =4

3 + 5X

We can then get

dµdt

= −54µ2 dX

dt=

54µ2 L

MQ,

where Q = 6× 1018 ergs g−1 is the energy released from convertingevery gram of hydrogen to helium. This equation, together with Eq. 3,gives

dL(t)dt

=758

µ(0)L1+17/15(t)MQL−1+17/15(0)

,

with solution

L(t) = L(0)

[1− 85

8µ(0)L(0)

MQt]−15/17

.

For our sun, expressing the luminosity in units of the present value Land assuming the present age of 4.6× 109 years, and lettingµ(0) ≈ 0.6, we then have

L(t)L

=L(0)

L

[1− 0.3

L(0)

Lt

t

]−15/17

.

So the luminosity of the sun on the ZAMSmust be L(0) ≈ 0.715/17L = 0.79L fromthis solution [by setting L(t) = L], which isvery close to the value, 0.73, from thenumerical model quoted above.The model shows that the core of the sun isindeed radiative and that the convection zoneoccupies only the outer 30% of the radius(but only 2% of the mass).

The right figure shows the representativetheoretical evolutionary tracks for stars ofdifferent masses [Iben (1967)].

We may understand the above by considering what happens as µincreases with time in the hydrogen-burning core. Does T have toincrease when µ increases?

I If P ∝ ρT/µ, then the increase in µmust be compensated by an increase inρT to maintain the hydrostaticequilibrium of the star.

I The result would then be a compressionof the core with a correspondingincrease in density.

I The virial theorem (e.g., Eq. 1) givesT ∝ µρ1/3M2/3. Therefore, the increaseof µ and ρ must lead to an increase in T ,hence the energy generation rate andthe total luminosity. But this increase ofT in the core does not necessarilyreflected by an increase of Teff .

Chemical profiles in the MSDue to the relatively weak temperature dependence of the p-p chain,H-burning involves a relatively large mass fraction in a 1 M star, forexample.In contrast, for more massive upper MS stars (& 1.2− 1.3M), CNOcycle becomes the dominant energy production mechanism. Itsstrong temperature dependence results in a more centrallyconcentrated nuclear burning process and in a convective core.

Chemical profiles of hydrogen in 1 M (left panel) and 5M (right) stars at differentstages during the core hydrogen burning phase.

When the mass fraction of hydrogen in a stellar core declines toX ∼ 0.05 (point 2 on the evolutionary track), the MS phase hasended, and the star begins to undergo rapid changes.

Outline

Star Formation





Post-Main Sequence Evolution

At this stage, it is useful to make a division based on the stellar mass:I Low-mass stars (0.8− 2M). Such a star develop a degenerate

helium core after the MS, leading to a relatively long-lived RGB(RGB) phase and to the ignition of He in a so-called helium flash.

I Intermediate-mass stars (2− 8M). Such a star has its Heburning ignited stably in a non-degenerate core and ends up asa degenerate carbon-oxygen (CO) WD.

I Massive stars (& 8M). Such a star also ignites carbon in anon-degenerate core. Stars with masses & 11M can havenuclear burning all the way to Fe and then collapse to formneutron stars or BHs.

Leaving the MSFrom point 2 to 3 (overall contracting phase): As X becomes lessthan 0.05, the nuclear energy generated is not sufficient to maintainthe hydrostatic equilibrium, the entire star begins to contract.

I The increasing gravity due to thecontraction is balanced by the heat orthe luminosity due to the conversion ofgravitational energy to thermal energy.

I Simultaneously, the smaller stellarradius translates into a hotter effectivetemperature — a general trend seen instars at this evolutionary stage.

I For higher mass stars, the mass fractionof the convective core begins to shrinkrapidly.

At point 3, the hydrogen is essentially exhausted in the core, whichbecomes nearly isothermal. While this is happening, the hydrogenrich material around the core is drawn inward and eventually ignites ina thick shell, containing ∼ 5% of the star’s mass.

Leaving the MS

From point 3 to 4 (thick shell phase): Much of the energy fromshell burning now goes into pushing matter away in both directions.As a result, the luminosity of the star does not increase; instead theouter part of the star expands.

I Gradually, the envelope approaches thethermal equilibrium again (i.e., the rateof energy received is roughly equal tothat released at the star’s surface).

I This thick shell phase continues with theshell moving outward in mass, until thecore contains ∼ 10% of the stellar mass(point 4).

I This is the Schonberg-Chandrasekharlimit. Stars with larger masses will reachthis point faster than stars with lowmesses.

The Schonberg-Chandrasekhar limit

At the exhaustion of central H, a star is left with a He core surroundedby a H-burning shell and then an H rich envelope. Given that there isno nuclear burning in the core, its thermal stratification is nearlyisothermal.

There exists an upper limit to the ratio Mc/Mt , which can bequalitatively understood as follows:For a star in hydrostatic equilibrium, 2K + Ω = 0, whereΩ = −

∫V GMr dMr/r and 2K = 3

∫V PdV = 2Kc + 2Ke, where the

subscripts c and e stand for the core and shell of the star. A partialintegration (assuming the P = 0 at the outer radius R) gives

2Ke = 3∫

ePdV = −3P0Vc − 3

∫e(dP/dr)(4π/3)r3dr ,

where P0 is the pressure at the boundary between the core andenvelope.Assuming hydrostatic equilibrium, the above equation becomes2Ke = −3P0Vc − Ωe.


At the exhaustion of central H, a star is left with a He core surroundedby a H-burning shell and then an H rich envelope. Given that there isno nuclear burning in the core, its thermal stratification is nearlyisothermal.There exists an upper limit to the ratio Mc/Mt , which can bequalitatively understood as follows:For a star in hydrostatic equilibrium, 2K + Ω = 0, whereΩ = −




2Ke = 3∫



where P0 is the pressure at the boundary between the core andenvelope.

Assuming hydrostatic equilibrium, the above equation becomes2Ke = −3P0Vc − Ωe.


At the exhaustion of central H, a star is left with a He core surroundedby a H-burning shell and then an H rich envelope. Given that there isno nuclear burning in the core, its thermal stratification is nearlyisothermal.There exists an upper limit to the ratio Mc/Mt , which can bequalitatively understood as follows:For a star in hydrostatic equilibrium, 2K + Ω = 0, whereΩ = −




2Ke = 3∫



where P0 is the pressure at the boundary between the core andenvelope.Assuming hydrostatic equilibrium, the above equation becomes2Ke = −3P0Vc − Ωe.

Putting all these together, we have 2K + Ω = 2Kc + Ωc − 3P0Vc = 0,or

P0 = K1McTc

R3c− K2

M2c

R4c,

where the K1 and K2 are constants.For given values of Mc and Tc , P0 attains a maximum valueP0,m = K3

T 4c

M2c

when the core radius Rc = K4Mc/Tc (where K3 and K4

are constants).For the star to be in equilibrium, P0,m must be larger than, or at leastequal to, the pressure Pe exerted by the envelope on the interfacewith the core.

Assuming that the core contains only a small fraction of the totalstellar mass Mt so that we can roughly approximate Pe ∝ M2

t /R4

(from the hydrostatic equilibrium of the entire star) and Tc ∝ Mt/R(from the virial theorem), where R is the total radius of the star.Hence at the interface,, Pe ∝ T 4

c /M2t .

Therefore, the condition Pe ≤ P0,m dictates the existence of an upperlimit to Mc/Mt .

Putting all these together, we have 2K + Ω = 2Kc + Ωc − 3P0Vc = 0,or

P0 = K1McTc

R3c− K2

M2c

R4c,

where the K1 and K2 are constants.For given values of Mc and Tc , P0 attains a maximum valueP0,m = K3

T 4c

M2c

when the core radius Rc = K4Mc/Tc (where K3 and K4

are constants).For the star to be in equilibrium, P0,m must be larger than, or at leastequal to, the pressure Pe exerted by the envelope on the interfacewith the core.Assuming that the core contains only a small fraction of the totalstellar mass Mt so that we can roughly approximate Pe ∝ M2

t /R4

(from the hydrostatic equilibrium of the entire star) and Tc ∝ Mt/R(from the virial theorem), where R is the total radius of the star.Hence at the interface,, Pe ∝ T 4

c /M2t .

Therefore, the condition Pe ≤ P0,m dictates the existence of an upperlimit to Mc/Mt .

Physically, this is because as the mass of the core increases, itsgravity increases, while the the thermal energy of the core is providedchiefly by the hydrogen-burning in the shell, which is set to hold theenvelope in balance.

I The exact value of the Schonberg-Chandrasekhar limit dependson the ratio between the mean molecular weight in the envelopeand in the isothermal core:(

Mc

Mt

)SC

= 0.37(µe

µc

)2

.

I At the end of the MS phase of a solar chemical compositionobject, µe ∼ 0.6 and µc ∼ 1.3 (the core is essentially made ofpure helium). The limit is thus equal to (Mc/Mt )SC ≈ 0.1.

I A star with the total mass larger than ∼ 3M will evolve to havea ratio equal to the limit and will then contract on theKelvin-Helmholtz timescale.

From point 4 to 5: This region in the HRD corresponds to the“Hertzsprung Gap” because of the short (KH) evolution time scale.

I The contraction leads to thetemperature increase up to ∼ 108 K,when He fusion is ignited.

I The core (for a star with mass . 2M)can also reach sufficient densities thatthe effect of degeneracy pressurecomes into play.

I As the envelope cools due to expansion,the opacity in the envelope increases(due to the Kramers opacity). Thethermal energy trapped by this opacitycauses the star to further expand.

For a star with . 3M, the core mass is below the Schonberg-Chandrasekhar limit, and the contraction phase takes much longertime. The Hertzsprung gap effectively disappears. Thus, many starsin an old stellar cluster may be found in this sub-giant phase.By the time the star reaches the base of the RGB (point 5),convection dominates energy transport (similar to YSOs at thelimiting Hayashi line).





For a star with . 3M, the core mass is below the Schonberg-Chandrasekhar limit, and the contraction phase takes much longertime. The Hertzsprung gap effectively disappears. Thus, many starsin an old stellar cluster may be found in this sub-giant phase.

By the time the star reaches the base of the RGB (point 5),convection dominates energy transport (similar to YSOs at thelimiting Hayashi line).





For a star with . 3M, the core mass is below the Schonberg-Chandrasekhar limit, and the contraction phase takes much longertime. The Hertzsprung gap effectively disappears. Thus, many starsin an old stellar cluster may be found in this sub-giant phase.By the time the star reaches the base of the RGB (point 5),convection dominates energy transport (similar to YSOs at thelimiting Hayashi line).

The red giant branchAs the star cools further, the surface opacity becomes less (H−

opacity dominates near the surface: κ ∝ ρ1/2T 9). The energyblanketed by the atmosphere eventually escapes, and the luminosityof the star increases. The evolution of low-mass stars withdegenerate cores is almost independent of the total stellar masses.

I In such a star, a very strong density contrasthas developed between the core and theenvelope, which is so extended that it exertsvery little weight on the compact core.

I For a low mass star, the core is degenerate.Its structure is independent of its thermalproperties (temperature) and only dependson its mass.

I Therefore the structure of a low-mass redgiant is essentially a function of its coremass.

I The large pressure gradient across thehydrogen-burning shell determines theluminosity of the star.

Why does the RGB luminosity increase sharply?

The more massive the core, the smaller its radius and stronger itsgravitational potential. This makes the temperature in the shell higherwhich gives a greater luminosity by the CNO cycle.

Empirically, from models, and later analytically1, the energygeneration in a thin shell of ideal gas around a degenerate core is

L = KMzc (4)

with z ∼ 8 for CNO burningshells (z ∼ 15 for heliumburning shells). The shellburning continuallyincreases the mass of thecore with

dMc

dt=

LXQ

, (5)

where X is the massfraction of the fuel and Q isthe energy yield.

Replacing L in Eq. 5 with that in Eq. 4and then integrating leads to

Mc = Mc,0

1 −(z − 1)K (t − t0)

XQM(1−z)c,0

−1/(z−1)

L = L0

(1 −

(z − 1)K 1/z(t − t0)

XQL(1−z)/z0

)−z/(z−1)

,

where Mc,0 and L0 are the core mass andluminosity when t = t0. This luminosityincreases sharply when the star ascendsthe RGB.

1http://adsabs.harvard.edu/abs/1983ApJ...268..356T

Why does the RGB luminosity increase sharply?

The more massive the core, the smaller its radius and stronger itsgravitational potential. This makes the temperature in the shell higherwhich gives a greater luminosity by the CNO cycle.Empirically, from models, and later analytically1, the energygeneration in a thin shell of ideal gas around a degenerate core is

L = KMzc (4)

with z ∼ 8 for CNO burningshells (z ∼ 15 for heliumburning shells). The shellburning continuallyincreases the mass of thecore with

dMc

dt=

LXQ

, (5)

where X is the massfraction of the fuel and Q isthe energy yield.

Replacing L in Eq. 5 with that in Eq. 4and then integrating leads to

Mc = Mc,0

1 −(z − 1)K (t − t0)

XQM(1−z)c,0

−1/(z−1)

L = L0

(1 −

(z − 1)K 1/z(t − t0)

XQL(1−z)/z0

)−z/(z−1)

,

where Mc,0 and L0 are the core mass andluminosity when t = t0. This luminosityincreases sharply when the star ascendsthe RGB.

1http://adsabs.harvard.edu/abs/1983ApJ...268..356T

The first dredge-upI As the star ascends the RGB, the

decrease in envelope temperature dueto expansion guarantees that energytransport will be by convection. Theconvective envelope continues to grow,until it almost reaches down to thehydrogen burning shell.

I In stars with mass & 1.5M, the coredecreased in size during MS evolution,leaving behind processed CNO.

I As a result, the surface abundance of14N grows at the expense of 12C, as theprocessed material gets mixed onto thesurface.

I This is called the first dredge-up.Typically, this dredge-up will change thesurface CNO ratio from 1/2 : 1/6 : 1 to1/3 : 1/3 : 1; this result is roughlyindependent of stellar mass.

The outer convective envelope baseand the He core mass as a function oftime for a 0.8 M star.

Hydrogen abundance profile within a0.8 M star, after the first dredge up.

RGB bumpRGB bump was theoretically predicted by Thomas (1967) and Iben(1968) as a region in which evolution through the RGB is stalled for atime when the H-burning shell passes the H abundanceinhomogeneity envelope.

I This behavior is due to the change inthe H abundance after the firstdredge-up and hence the decrease ofthe mean molecular weight (L ∝ µ15/2

as discussed earlier).I After the shell has crossed the

discontinuity, the surface luminositygrows again, monotonically withincreasing core mass.

The HRD of a 0.8 M star and the trendcorresponding to the RGB bump (inset).The open circle marks the first dredge up.

The exact luminosity position of the RGB bump is a function of metalabundance, helium abundance, and stellar mass (and hence stellarage), as well as any additional parameters that determine themaximum inward extent of the convection envelope or the position ofthe H-burning shell.

Mass loss by red giants

I Stars on the RGB undergomass loss in the form of aslow (between 5 and30 km s−1) wind.

I Mass loss rates for stars of∼ 1M can reach∼ 10−8M yr−1 at the tip ofthe RGB.

I The total amount of mass lostduring the RGB phase can be∼ 0.2M.

I This rate is high enough sothat a star at the tip of theRGB (TRGB) will besurrounded by acircum-stellar shell, whichcan redden and extinct thestar.

ALMA image of R Sculptoris, a RGB star.

HST image of the Hourglass Nebula.

Toward the helium burning phase

I A star with mass & 0.5M will eventually ignite helium in its core.I As the density contrast between the helium core and its

hydrogen envelope increases, the mass within the burning shelldecreases to ≈ 0.001M near the tip of the RGB.

I At the same time, the energy generation rate per unit massincreases strongly, which means the temperature within theburning shell also increases. With it, the temperature in thedegenerate helium core increases.

I To have the helium burning, we need much higher temperatureand density than for the hydrogen burning, because the largeCoulomb barrier and three-bodies to come together in 10−16 s;8Be is not stable.

I The path depends on the race between the temperature (from∼ 107 K to ∼ 108 K, ignition temperature of helium) and density(∼ 102 to ∼ 106 cm−3 – the degeneracy density).

I This race depends on the (initial) mass again. Remembering thescaling: the higher the mass, the lower the density.

He flash

For low-mass stars, the temperature is reached after the partialdegeneracy in the He core. partial degeneracy in the He core.

I Until the degeneracy is overcome, the He burning increases thetemperature, but without reducing the density, leading to a “Heflash” (when the luminosity of the star reaches ∼ 103.4L) andthen to the HB.

I The mass of the core at this time is ∼ 0.5M. This, together withthe tight relation between the luminosity and core mass,determines the TRGB luminosity.

I The flash luminosity ∼ 1011L is all absorbed in the expansionof the non-degenerate outer layers.

I As the flash proceeds, the degeneracy in the core is removed,and the core expands.

For M & 2M: temperature wins, He-burning is triggered gently. Theluminosity increase is small, because of little core contraction.

Horizontal BranchLower mass stars which do undergo the He flash quickly change theirstructure and land on the stable ZAHB on a Kelvin-Helmholtztimescale.I These stars burn helium to carbon in

their core and also have ahydrogen-burning shell.

I As the core becomes slightly larger,the resultant pressure drop at thehydrogen burning shell reduces theluminosity of the star from itspre-helium flash luminosity to∼ 100L.

I Because of the large luminosityassociated with helium burning, thecentral regions of horizontal branch(HB) stars are convective. A starwith a helium core of ∼ 0.5M, forexample, will have a convectivehelium burning core of ∼ 0.1M.

Horizontal BranchI The effective temperature of a ZAHB

star depends principally on itsenvelope mass (especially when themass of the envelope is small).Stars with low mass envelopes willbe extremely blue, withlog Teff ≥ 4.3. Stars with largeenvelope masses (∼ 0.4M) appearnear the base of the RGB.

I On average, a star spends about108 years of the life time on the HB(about 10% of the lifetime in theRGB phase) and has a luminosity of∼ 102 times the MS counterpart.

I While QHB ∼ 0.1QMS, much of theluminosity of a HB star arises fromthe H-shell burning (up to ∼ 80%;i.e., the bulk of the H-burning occursafter the MS for a low-mass star).

Cluster M3 H-R diagram. The gapin the horizontal branch is due tothe instability strip, where stars,known as RR Lyrae variable starswith periods of up to 1.2 days, aretypically not included in suchdiagrams.

Red clump stars

While clusters with many blue HB stars are dominated by stars withsmall envelopes, clusters whose HB stars are in a “red clump” havelarge envelope mass stars, evolved from stars with relatively largemasses (& 3M) and metallicity.

I Red clump stars are the (relativelymetal-rich and/or young population Icounterparts to HB stars (which belong topopulation II).

I They have metallicity greater than about10% solar. Above this value, red clumplocation in a CMD is fairly insensitive to themetallicity.

I They look redder because opacity riseswith Z . They nestle up against the RGB inHR diagrams for old, open cluster, makinga clump of dots of similar luminosity.

Hipparcos CMD. The box outlinesthe redclump region.

Observing red clump stars in near-IR

Sample CMD in the field of the Galactic center (from Hui Dong). Thesolid, dotted and dashed lines are the Padova isochrones of 6 Myr,200 Myr and 10 Gyr ages and with a typical extinction (AK = 2.4)toward the center and solar metallicity assumed.The four diamonds on the isochrones of 6 Myr marks the location ofthe stars with initial mass of 5, 10, 20 and 25 M.

HB color and second parameter problem

The metallicity is the main (’the first’) parameter controlling the colordistribution of HB stars, due mainly to the envelope opacity andsecondarily to the expected correlation of the evolving stellar mass(hence more massive stellar envelope) and the metallicity (e.g.,higher metallicity stars evolve more slowly than lower ones).Thus the color distribution of HB stars could in principle be used as ameasurement of the age.

However, at a given metallicity, clusters of apparently the same ageshow different HB colors.This is the origin of the so-called second parameter problem.

The nature of the problem is not clear. But the color difference couldresult from different mass-loss laws, which may depend on stellarrotation or dynamic interaction within the clusters.

The Asymptotic giant branchThis is a period of stellar evolution undertaken by all low- tointermediate-mass stars (1 - 10 solar masses) late in their lives. TheAGB phase is divided into two parts, the early AGB (E-AGB) and thethermally pulsing AGB (TP-AGB).

I During the E-AGB phase, the main sourceof energy is helium fusion in a shell arounda core, consisting mostly of carbon andoxygen. The luminosity from this shell willcause the region outside of it to expand.

I After the helium shell runs out of fuel, theTP-AGB starts. Now the star derives itsenergy from fusion of hydrogen in a thinshell.

I The Helium shell builds up and eventuallyignites explosively, a process known as ahelium shell flash, causing the star toexpand and cool, which shuts off thehydrogen shell burning and causesconvection in the zone between the twoshells.

I When the helium shell burning nears the base of the hydrogenshell, the increased temperature reignites hydrogen fusion andthe cycle begins again.

I The luminosity during AGB phase is largely determined by the

CO core mass. For Mc > 0.5M,L

L= 6× 104

(Mc

M− 0.5

),

which is of the order of the Eddington luminosity.I A strong stellar wind as a result of the high radiation pressure in

the envelope (and the thermal pulses) can reach a rate of theorder of 10−4 M yr−1.

I As a consequence of the superwind, stars of initial mass in therange 1 M . M . 10 M are left with C-O cores of massbetween 0.6 M and 1.1 M.

I During the E-AGB, the so-called second dredge-up may occur,while the third dredge-up happens during TP-AGB phase. As aresult, AGB stars may show S-process elements in their spectraand strong dredge-ups can lead to the formation of carbon stars.

Core growth, population, and mass loss of AGB starsUsing the conversion factor for hydrogen to carbon,

Mc = L/Q = 1.2× 10−11 M yr−1 LL

,

the lifetime can be estimated as

τAGB = (1.4× 106 yr)ln(

Mc − 0.5MMc,0 − 0.5M

).

Evolution calculations show that a relationexists between the initial core mass and theinitial mass of the star M0, of the formMc,0 = a + bM0, where a and b areconstants. Hence τAGB is essentially afunction of the initial stellar mass.

The number of AGB stars is proportional to the lifetime of stars in thisstage. So if the IMF is assumed, we know the relative population ofAGB stars. Compared with the observed, one can conclude that thecore mass can grow by only about 0.1 M, indicating the mass lossmust be very intense.

Planetary nebulaeI The core of a star at the end of its AGB

phase is surrounded by an extendedshell, planetary nebula, illuminated byintense UV radiation from thecontracting central star.

I The central star, or PN nucleus, initiallymoves toward higher temperatures,powered by nuclear burning in the thinshell still left on top of the C-O core.

I When the mass of the shell decreasesbelow a critical mass of the order of10−3 M to 10−4 M, the shell can nolonger maintain the high temperature forthe nuclear burning and the luminosityof the star drops.

I At the same time, the nebula, expandingat a rate of ∼ 10 km s−1, graduallydisperses, after some 104 − 105 yrs.

X-ray/optical composite imageof the Cat’s Eye Nebula.

HST image of the PN, NGC6326, with a binary central star.

Outline

Star Formation






I What do stars in the mass range of ∼ 8− 11M eventuallyevolve to is still somewhat uncertain; they may just developdegenerate O-Ne cores.

I A star with mass above ∼ 11M will ignite and burn fuelsheavier than carbon until an Fe core is formed which collapsesand causes a supernova explosion.

I For a star with mass & 15M, mass loss by the stellar windbecomes important during all evolution phases, including theMS.

Kippenhahn Diagram

Mass-loss of high-mass starsFor stars with masses & 30M,

I The mass loss time scale is shorterthan the MS timescale. The MSevolutionary paths of such starsconverge toward that of a 30M star.

I Mass-loss from Wolf-Rayet stars leadsto CNO products (helium and nitrogen)exposed.

I The evolutionary track in the H-Rdiagram becomes nearly horizontal,since the luminosity is already close tothe Eddington limit.

I Electrons do not become degenerateuntil the core consists of iron.

When the degenerate core’s mass surpasses the Chandrasekhar limit(or close to it), the core contracts rapidly. No further source of nuclearenergy in the iron core, the temperature rises from the contraction,but not fast enough. It collapses on a time scale of seconds!

Mass loss of high-mass starsMass loss plays an essential role inregulating the evolution of very massivestars.

I WR stars are examples, following thecorrelation: log[Mv∞R1/2] ∝ log[L].

I How could M and vw be measured?I In general, mass-loss rates during all

evolution phases increase with stellarmass, resulting in timescales for massloss that are less that the nucleartimescale for M & 30M. As a result,there is a convergence of the final(pre-supernova) masses to ∼ 5− 10M.

I However, this effect is much diminishedfor metal-poor stars because themass-loss rates are generally lower atlow metallicity.

Kippenhahn diagram of the evolution ofa 60 M star at Z = 0.02 with massloss. Cross-hatched areas indicatewhere nuclear burning occurs, andcurly symbols indicate convectiveregions. See text for details. Figurefrom Maeder & Meynet (1987).



I How could M and vw be measured?

I In general, mass-loss rates during allevolution phases increase with stellarmass, resulting in timescales for massloss that are less that the nucleartimescale for M & 30M. As a result,there is a convergence of the final(pre-supernova) masses to ∼ 5− 10M.





I How could M and vw be measured?I In general, mass-loss rates during all

evolution phases increase with stellarmass, resulting in timescales for massloss that are less that the nucleartimescale for M & 30M. As a result,there is a convergence of the final(pre-supernova) masses to ∼ 5− 10M.



ReviewKey Concepts: Jeans mass, initial mass function, Hayashi track,brown dwarfs, ZAMS, red clump, the Schonberg-Chandrasekhar limit,the Eddington limit, Hertzsprung Gap, Wolf-Rayet stars

1. What are the basic signatures of low-mass YSOs?2. What is the basic reason for the mass fragmentation of a

collapsing cloud? When is the fragmentation expected to stop?3. Qualitatively what is the basic structure of a protostar?4. What may be the structure change of a YSO that could end its

evolution along the Hayashi track?5. How does the lifetime of a star depend on its mass?6. Qualitatively describe the track of a star from its protostar stage

to its death in the HR diagram. How does the track depend onthe initial mass of the star? What are the relatively timedurations that the star spend on different evolutionary stages?

7. The location of the Hayashi track, the MS, or the red-giantbranch is sensitive to the chemical composition of the stars.Does Teff increase or decrease with the increase of themetallicity or the decrease of the He abundance? Why?

8. How does the metallicity of ZAMS stars affect their color andluminosity?

Review cont.9. How does the luminosity of a MS star depend on its mass? Can

you characterize this dependence from a simple dimensionalanalysis, assuming that the radiative heat transfer dominates?

10. What is the first dredge-up? How does it affect the observedsurface abundances of the elements?

11. Why do some stars undergo He-flash, while others don’t? HasHe-flash actually been observed?

12. What is the horizontal branch? Why does it tend to have anarrow luminosity range?

13. What are red-clump stars?14. What are the differences between the red-giant and asymptotic

giant branches? What is the main heating transfer mechanism inthese branches? Why?

15. What happens in a thermally pulsing AGB star?16. How would the radius of a star change if its opacity were

increased by a small amount?17. How would the main sequence lifetime of star change if the

mass loss at the surface was enhanced?18. At which evolutionary stage of a star is a planetary nebula

expected to form? What keeps such a nebula bright visually?

chapter 6: stellar evolution (part 1) · chapter 6: stellar evolution (part 1) with the...

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