Chapter 6: Stellar Evolution (Part 1)

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

Chapter 6: Stellar Evolution (part 1)

With the understanding of the basic physical processes in stars, we now proceed to study their evolution. In particular, we will focus on discussing how such processes are related to key characteristics seen in the HRD. The CD-ROM that came with the text book (HKT) contains some nice and informative description and movies from stellar evolution modeling (e.g., CD-ROM/StellarEvolnDemo/index.html). They cover the Main Sequence (MS) and evolved stages (although some of the movies are missing). The same programs may also be obtained from http://astro.if.ufrgs.br/evol/evolve/hansen/index.htm. Outline

Star Formation

Young Stellar Objects

The Main Sequence Dependence on stellar mass Dependence on chemical composition

Post-Main Sequence Evolution Leaving the MS The red giant branch The helium burning phase The asymptotic giant branch

Final evolution stages of high-mass stars Consider a medium of uniform density and temperature, ρ and T . From the virial theorem, 2E = −Ω, we have

Z M 1/3 3kTM GMr 3 2 3 4πρ 5/3 = dMr = (GM /R) = GM (1) µmA 0 r 5 5 3

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

3 1/2 5kT 3/2 M > MJ ≡ . 4πρ GµmA

For a typical temperature and density of a large molecular cloud, 5 −1/2 MJ ∼ 10 M with a collapse time scale of tff ≈ (Gρ) .

Star Formation Here we brieﬂy discuss the gravitational instability, Jeans mass, fragmentation of gas clouds, as well as the resultant initial mass function (IMF) of stars. −1/2 tff ≈ (Gρ) .

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

Z M 1/3 3kTM GMr 3 2 3 4πρ 5/3 = dMr = (GM /R) = GM (1) µmA 0 r 5 5 3

3 1/2 5kT 3/2 M > MJ ≡ . 4πρ GµmA

For a typical temperature and density of a large molecular cloud, 5 MJ ∼ 10 M with a collapse time scale of Star Formation Here we brieﬂy discuss the gravitational instability, Jeans mass, fragmentation of gas clouds, as well as the resultant initial mass function (IMF) of stars. Consider a medium of uniform density and temperature, ρ and T . From the virial theorem, 2E = −Ω, we have

Z M 1/3 3kTM GMr 3 2 3 4πρ 5/3 = dMr = (GM /R) = GM (1) µmA 0 r 5 5 3

3 1/2 5kT 3/2 M > MJ ≡ . 4πρ GµmA

For a typical temperature and density of a large molecular cloud, 5 −1/2 MJ ∼ 10 M with a collapse time scale of tff ≈ (Gρ) . What exactly happens during the collapse depends very much on the temperature evolution of the cloud.

I Initially, the cooling processes (due to molecular and dust radiation) are very efﬁcient. If the cooling time scale tcool is much shorter than tff , the collapse is approximately isothermal. −1/2 I As MJ ∝ ρ decreases, inhomogeneities with mass larger than the actual MJ will collapse by themselves with their local tff . This fragmentation process will continue as long as the local tcol is shorter than the local tff , producing increasingly smaller collapsing subunits. I Eventually the density of subunits becomes so large that they become optically thick and the evolution becomes adiabatic (i.e., 2/3 1/2 T ∝ ρ for an ideal gas), then MJ ∝ ρ . I As the density has to increase, the evolution will always reach a point when M = MJ , when a subunit reaches approximately hydrostatic equilibrium. We assume that a stellar object is born.

Cloud fragmentation Such mass clouds may be formed in spiral density waves and other density perturbations (e.g., caused by the expansion of a supernova remnant or superbubble). Cloud fragmentation Such mass clouds may be formed in spiral density waves and other density perturbations (e.g., caused by the expansion of a supernova remnant or superbubble). What exactly happens during the collapse depends very much on the temperature evolution of the cloud.

−x dn/dm = φ(M) = C(M/0.5M )

where x = 2.35 (Salpeter’s law), valid for M/M ≥ 0.5, and x = 1.3 for 0.1 ≤ M/M < 0.5 in the solar neighborhood.

The Initial Mass Function

This way a giant molecular cloud can form a group of stars with their mass distribution being determined by the fragmentation process. The process depends on the physical and chemical properties of the cloud (ambient pressure, magnetic ﬁeld, rotation, composition, dust fraction, stellar feedback, etc.). Much of the process is yet to be understood. The Initial Mass Function

−x dn/dm = φ(M) = C(M/0.5M )

where x = 2.35 (Salpeter’s law), valid for M/M ≥ 0.5, and x = 1.3 for 0.1 ≤ M/M < 0.5 in the solar neighborhood. Probably the strongest evidence for a top heavy IMF comes from the Galactic 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 Orion nebula, the observed total diffuse X-ray luminosities from massive young stellar clusters suggest that the number of low-mass YSOs are a factor of ∼ 10 smaller than what would be expected from the standard IMF and the massive star populations observed in the clusters. If conﬁrmed, this has strong implications for understanding the star formation at high z, the mass to light ratio, etc.

Is the IMF universal?

While the IMF in galactic disks of the MW and nearby galaxies seem to be quite consistent, there are good reasons and even lines of evidence 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 the Galactic 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 Orion nebula, the observed total diffuse X-ray luminosities from massive young stellar clusters suggest that the number of low-mass YSOs are a factor of ∼ 10 smaller than what would be expected from the standard IMF and the massive star populations observed in the clusters. Is the IMF universal?

Star Formation

Young Stellar Objects

The Main Sequence Dependence on stellar mass Dependence on chemical composition

Post-Main Sequence Evolution Leaving the MS The red giant branch The helium burning phase The asymptotic giant branch

Final evolution stages of high-mass stars The proto-star stages have the KH time scale 7 2 −1 −1 ∼ (2 × 10 yrs)(M/M ) (L/L ) (R/R ) .

Young Stellar Objects

Objects that are on the way to become stars, but extract energy primarily from gravitational contraction are called young stellar objects (YSOs) here. They represent the entire stellar system throughout all pre-main sequence (MS) evolutionary phases. Theoretically, the formation and evolution of a YSO may be divided into four stages: 1. proto-star core formation; 2. protostar star builds up from inside out, forming a disk around (core still contracts and is optically thick); 3. bipolar outflows; 4. surrounding nebula swept away. The proto-star stages have the KH time scale 7 2 −1 −1 ∼ (2 × 10 yrs)(M/M ) (L/L ) (R/R ) . YSOs are classified into classes 0, 1, and 2, according to the ratio of infrared to optical, amount of molecular gas around, inflow/outflow, etc. I Class 0 protostars are highly obscured and have short time scales (corresponding to the stage 2); few are known. I Class 1 or 2 protostars are already living partly on nuclear energy (3 and 4); but the total luminosity is still dominated by gravitational energy. The low-mass YSO prototype is T Tauri. We still know little about high-mass YSOs, which evolve very fast and interact strongly with their environments.

Observational signatures and classification Observational signatures of YSOs: I emission lines from the disk and/or outflow I more infrared luminosity due to dust emission I variability on hours and days due to temperature irregularities on both the stellar surface and disk I high level of magnetic field triggered activities (flares, spots, corona ejection, etc) due to fast rotation and convection I strong X-ray emission from hot corona. Observational signatures and classification Observational signatures of YSOs: I emission lines from the disk and/or outflow I more infrared luminosity due to dust emission I variability on hours and days due to temperature irregularities on both the stellar surface and disk I high level of magnetic field triggered activities (flares, spots, corona ejection, etc) due to fast rotation and convection I strong X-ray emission from hot corona.

YSOs are classified into classes 0, 1, and 2, according to the ratio of infrared to optical, amount of molecular gas around, inflow/outflow, etc. I Class 0 protostars are highly obscured and have short time scales (corresponding to the stage 2); few are known. I Class 1 or 2 protostars are already living partly on nuclear energy (3 and 4); but the total luminosity is still dominated by gravitational energy. The low-mass YSO prototype is T Tauri. We still know little about high-mass YSOs, which evolve very fast and interact strongly with their environments. Hayashi tracks

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

I During the collapse the density increases inwards. The optically thick phase is reached ﬁrst in the central region, which leads to the formation of a more-or-less hydrostatic core with free falling gas surrounding it. I The energy released by the core (now obeying the virial theorem) is absorbed by the envelope and radiated away as infrared radiation. I Because of the heavy obscuration by the surrounding dusty gas, stars in this stage cannot be directly observed in optical and probably even in near-IR. I The steady increase of the central temperature causes the dissociation of the H2, then the ionization of H, and the ﬁrst and second ionization of He.

The sum of the energy involved in all these processes has to be at most equal to the energy available to the star through the virial theorem. The luminosity can be very large and hence usually requires convection. i The maximum initial radius of a YSO Rmax ≈ 50 − 100R (M/M ). If the effects due to the accretion of matter to the forming star may be neglected, the object follows a path on the HRD with the effective temperature similar to that given by the early expression:

−4/51 13/51 7/51 1/102 Teff ∝ (Z/0.02) µ (M/M ) (L/L ) .

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

The increase of the metallicity (that causes an increase of the opacity) shifts the track to lower Teff . The increase of the metallicity has little effect on µ. An increase of the helium abundance at constant metallicity has the opposite (and less relevant) effect, due to the increase of µ. Hayashi tracks of a 0.8 solar mass star with helium mass fraction 0.245, for 3 different metallicities. 6 I When the temperature in the core reaches the order of 10 K, deuterium is transformed into 3He by proton captures. The exact location when this happens depends on the stellar mass. In any case, the energy generation of this burning is comparably low and does not signiﬁcantly change the evolution track. Brown dwarfs, which are only able to burn deuterium (at T ∼ 106 K with masses ∼ 0.05 − 0.1M ), may still be called stars.

Pre-main sequence stage

I In this PMS stage, the YSO has formed a radiative core, though still growing with time. I The star is no longer fully convective. I Its evolution has to depart from its Hayashi track, which forms the rightmost boundary to the evolution of stars in the HRD. As the center temperature increases due to the virial theorem, the path is almost horizontal on the HRD. Brown dwarfs, which are only able to burn deuterium (at T ∼ 106 K with masses ∼ 0.05 − 0.1M ), may still be called stars.

Pre-main sequence stage

6 I When the temperature in the core reaches the order of 10 K, deuterium is transformed into 3He by proton captures. The exact location when this happens depends on the stellar mass. In any case, the energy generation of this burning is comparably low and does not signiﬁcantly change the evolution track. Pre-main sequence stage

Star Formation

Young Stellar Objects

The Main Sequence Dependence on stellar mass Dependence on chemical composition

Post-Main Sequence Evolution Leaving the MS The red giant branch The helium burning phase The asymptotic giant branch

Final evolution stages of high-mass stars The Main Sequence

A star spend the bulk of its lifetime in the MS, where it burns hydrogen in the core. We consider how the basic stellar properties depend on the mass and chemical composition of a star. The mass — a deciding parameter in the stellar evolution — determines what the central temperature can reach, hence what nuclear reactions can occur ”Mass” Cut” diagram showing the fate of and how fast they can run, and single stars in various mass classes. how they end their lives. which can be written as L ∝ M−1R4T 4, where κ is assumed to be constant in the nuclear burning region, while the hydrostatic M2 equilibrium condition as P ∝ R4 . Considering the EoS, one then ﬁnd I η = 3 if the pressure is primarily due to the ideal gas (i.e., for stars with masses lower than ∼ 10M ; T ∝ P/ρ ∝ M/R) I η = 1 if the radiation pressure dominates (for more massive stars; 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 difference is due to the structure change caused by the convection, which makes 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 can roughly estimate the mass dependence of the luminosity (L ∝ Mη), based on dimensional analysis, using the hydrostatic equilibrium state and the EoS of the ideal gas and assuming the radiative heat transfer equation, which can be written as L ∝ M−1R4T 4, where κ is assumed to be constant in the nuclear burning region, while the hydrostatic M2 equilibrium condition as P ∝ R4 . Considering the EoS, one then find I η = 3 if the pressure is primarily due to the ideal gas (i.e., for stars with masses lower than ∼ 10M ; T ∝ P/ρ ∝ M/R) I η = 1 if the radiation pressure dominates (for more massive stars; 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 difference is due to the structure change caused by the convection, which makes the nuclear burning more efficient. How does R depend on M? We know L ∝ M ∝ M2+ν R−(ν+3), ν 3 assuming = 0ρT , and T ∝ M/R. Equating this to L ∝ M , as an example, we have R ∝ M(ν−1)/(ν+3). (2) For example, ν = 18 for CNO. Then R = M0.81. 2 4 Replacing R in L ∝ R Teff with Eq. 2 and M ∝ L1/3, we then obtain

(ν+11) 0.12 T L 12(ν+3) L eff = = . Teff , L L

This insensitivity of Teff to L is due to the strong temperature dependence of the CNO cycle. Nevertheless, the exponent, 0.12, is still a factor of ∼ 10 larger than that for the Hayashi track or the RGB and AGB. The right ﬁgure shows pre-MS evolutionary tracks adopted by Stahler (1988) from various sources, as well as the locations of a number of T Tauri stars. Since a star’s luminosity on the MS does not change much, we can estimate its MS lifetime from simple timescale arguments and the mass-luminosity relation. If L ∝ Mη, then

10 1−η τMS = 10 yrs(M/M )

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 to the age of the Universe. Thus we are primarily concerned with stars more massive than this. I While practically most of relatively low-mass stars are close to Zero-Age Main Sequence stars (ZAMSs), massive stars burn hydrogen much faster, especially via the CNO cycle. I Because of the much steeper temperature dependence, the CNO cycle occurs in a much smaller region than do the pp-chains. The requirement for fast energy transport drives convection in the stellar core. I While we focus here on the evolution of isolated stars, it should be noted that if a star is in a close binary then the story can change drastically. I The metallicity chieﬂy affects the opacity, or the amount of bound-free absorption, which is dominated by metals. I The smaller opacity allows the energy to escape more easily (so the star appears bluer). Illustration of the effects of varying Y and Z on I The lower opacity also reduces the shape and position predicted for the 14 Gyr the pressure; hence the isochrone: Z = 0.006 (heaviest line, 0.001 luminosity of the star needs to be (intermediate), and 0.0001 (lightest). The solid increased to balance its gravity. lines are for Y = 0.2 and the dotted lines are for Y = 0.3 [From the calculation of van den Berg & Bell (1985)]

Dependence on chemical composition Now we consider how the metallicity of a star affects the color and luminosity of a star. We first briefly consider the effect on the ZAMS, which are those stars who arrived at the MS recently. Dependence on chemical composition Now we consider how the metallicity of a star affects the color and luminosity 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 the opacity, or the amount of bound-free absorption, which is dominated by metals. I The smaller opacity allows the energy to escape more easily (so the star appears bluer). Illustration of the effects of varying Y and Z on I The lower opacity also reduces the shape and position predicted for the 14 Gyr the pressure; hence the isochrone: Z = 0.006 (heaviest line, 0.001 luminosity of the star needs to be (intermediate), and 0.0001 (lightest). The solid increased to balance its gravity. lines are for Y = 0.2 and the dotted lines are for Y = 0.3 [From the calculation of van den Berg & Bell (1985)] Why does the luminosity increase with time? The nuclear burning changes the abundances of elements and hence the molecular weight (µ). Here we use the sun as an example of the luminosity evolution. for an ideal gas star. If we assume that radiative diffusion controls the energy ﬂow, then RT 4 L ∝ . 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 in the core of the sun), then we have M16/3ρ1/6µ15/2 L ∝ . κ0

Here the mass of the star is ﬁxed, while κ0 does not vary strongly with the abundances. Neglecting the weak dependence on ρ, the above relation can be written in time-dependent form L(t) µ(t) 15/2 ≈ . (3) L(0) µ(0) To see how µ varies with time, we assume that the bulk of the stellar interior is completely ionized and neglect the metal content that is small compared to hydrogen and helium. Then we have

4 µ = 3 + 5X

We can then get

dµ 5 dX 5 L = − µ2 = µ2 , dt 4 dt 4 MQ where Q = 6 × 1018 ergs g−1 is the energy released from converting every gram of hydrogen to helium. This equation, together with Eq. 3, gives dL(t) 75 µ(0)L1+17/15(t) = , dt 8 MQL−1+17/15(0) with solution

85 µ(0)L(0) −15/17 L(t) = L(0) 1 − t . 8 MQ For our sun, expressing the luminosity in units of the present value L and assuming the present age of 4.6 × 109 years, and letting µ(0) ≈ 0.6, we then have

L(t) L(0) L(0) t −15/17 = 1 − 0.3 . L L L t

So the luminosity of the sun on the ZAMS 15/17 must be L(0) ≈ 0.7 L = 0.79L from this solution [by setting L(t ) = L ], which is very close to the value, 0.73, from the numerical model quoted above. The model shows that the core of the sun is indeed radiative and that the convection zone occupies only the outer 30% of the radius (but only 2% of the mass). The right ﬁgure shows the representative theoretical evolutionary tracks for stars of different masses [Iben (1967)]. I If P ∝ ρT /µ, then the increase in µ must be compensated by an increase in ρT to maintain the hydrostatic equilibrium of the star. I The result would then be a compression of the core with a corresponding increase in density. I The virial theorem (e.g., Eq. 1) gives T ∝ µρ1/3M2/3. Therefore, the increase of µ and ρ must lead to an increase in T , hence the energy generation rate and the total luminosity. But this increase of T in the core does not necessarily reﬂected by an increase of Teff .

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

I If P ∝ ρT /µ, then the increase in µ must be compensated by an increase in ρT to maintain the hydrostatic equilibrium of the star. I The result would then be a compression of the core with a corresponding increase in density. I The virial theorem (e.g., Eq. 1) gives T ∝ µρ1/3M2/3. Therefore, the increase of µ and ρ must lead to an increase in T , hence the energy generation rate and the total luminosity. But this increase of T in the core does not necessarily reﬂected by an increase of Teff . When the mass fraction of hydrogen in a stellar core declines to X ∼ 0.05 (point 2 on the evolutionary track), the MS phase has ended, and the star begins to undergo rapid changes.

Chemical proﬁles in the MS Due to the relatively weak temperature dependence of the p-p chain, H-burning involves a relatively large mass fraction in a 1 M star, for example. In contrast, for more massive upper MS stars (& 1.2 − 1.3M ), CNO cycle becomes the dominant energy production mechanism. Its strong temperature dependence results in a more centrally concentrated nuclear burning process and in a convective core.

Chemical proﬁles of hydrogen in 1 M (left panel) and 5M (right) stars at different stages during the core hydrogen burning phase. Chemical proﬁles in the MS Due to the relatively weak temperature dependence of the p-p chain, H-burning involves a relatively large mass fraction in a 1 M star, for example. In contrast, for more massive upper MS stars (& 1.2 − 1.3M ), CNO cycle becomes the dominant energy production mechanism. Its strong temperature dependence results in a more centrally concentrated nuclear burning process and in a convective core.

Chemical proﬁles of hydrogen in 1 M (left panel) and 5M (right) stars at different stages during the core hydrogen burning phase. When the mass fraction of hydrogen in a stellar core declines to X ∼ 0.05 (point 2 on the evolutionary track), the MS phase has ended, and the star begins to undergo rapid changes. Outline

Star Formation

Young Stellar Objects

The Main Sequence Dependence on stellar mass Dependence on chemical composition

Post-Main Sequence Evolution Leaving the MS The red giant branch The helium burning phase The asymptotic giant branch

Final evolution stages of high-mass stars 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 He burning ignited stably in a non-degenerate core and ends up as a degenerate carbon-oxygen (CO) WD. I Massive stars (& 8M ). Such a star also ignites carbon in a non-degenerate core. Stars with masses & 11M can have nuclear burning all the way to Fe and then collapse to form neutron stars or BHs. At point 3, the hydrogen is essentially exhausted in the core, which becomes nearly isothermal. While this is happening, the hydrogen rich material around the core is drawn inward and eventually ignites in a thick shell, containing ∼ 5% of the star’s mass.

Leaving the MS From point 2 to 3 (overall contracting phase): As X becomes less than 0.05, the nuclear energy generated is not sufﬁcient to maintain the hydrostatic equilibrium, the entire star begins to contract.

I The increasing gravity due to the contraction is balanced by the heat or the luminosity due to the conversion of gravitational energy to thermal energy. I Simultaneously, the smaller stellar radius translates into a hotter effective temperature — a general trend seen in stars at this evolutionary stage. I For higher mass stars, the mass fraction of the convective core begins to shrink rapidly. Leaving the MS From point 2 to 3 (overall contracting phase): As X becomes less than 0.05, the nuclear energy generated is not sufﬁcient to maintain the hydrostatic equilibrium, the entire star begins to contract.

At point 3, the hydrogen is essentially exhausted in the core, which becomes nearly isothermal. While this is happening, the hydrogen rich material around the core is drawn inward and eventually ignites in a thick shell, containing ∼ 5% of the star’s mass. Leaving the MS

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

I Gradually, the envelope approaches the thermal equilibrium again (i.e., the rate of energy received is roughly equal to that released at the star’s surface). I This thick shell phase continues with the shell moving outward in mass, until the core contains ∼ 10% of the stellar mass (point 4). I This is the Schonberg-Chandrasekhar¨ limit. Stars with larger masses will reach this point faster than stars with low messes. There exists an upper limit to the ratio Mc/Mt , which can be qualitatively understood as follows: For a star in hydrostatic equilibrium, 2K + Ω = 0, where R R Ω = − 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 partial integration (assuming the P = 0 at the outer radius R) gives Z Z 3 2Ke = 3 PdV = −3P0Vc − 3 (dP/dr)(4π/3)r dr, e e

where P0 is the pressure at the boundary between the core and envelope. Assuming hydrostatic equilibrium, the above equation becomes 2Ke = −3P0Vc − Ωe.

The Schonberg-Chandrasekhar¨ limit

At the exhaustion of central H, a star is left with a He core surrounded by a H-burning shell and then an H rich envelope. Given that there is no nuclear burning in the core, its thermal stratiﬁcation is nearly isothermal. There exists an upper limit to the ratio Mc/Mt , which can be qualitatively understood as follows: For a star in hydrostatic equilibrium, 2K + Ω = 0, where R R Ω = − 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 partial integration (assuming the P = 0 at the outer radius R) gives Z Z 3 2Ke = 3 PdV = −3P0Vc − 3 (dP/dr)(4π/3)r dr, e e

where P0 is the pressure at the boundary between the core and envelope. The Schonberg-Chandrasekhar¨ limit

where P0 is the pressure at the boundary between the core and envelope. Assuming hydrostatic equilibrium, the above equation becomes 2Ke = −3P0Vc − Ωe. Assuming that the core contains only a small fraction of the total 2 4 stellar mass Mt so that we can roughly approximate Pe ∝ Mt /R (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. 4 2 Hence at the interface,, Pe ∝ Tc /Mt . Therefore, the condition Pe ≤ P0,m dictates the existence of an upper limit to Mc/Mt .

Putting all these together, we have 2K + Ω = 2Kc + Ωc − 3P0Vc = 0, or 2 McTc Mc P0 = K1 3 − K2 4 , Rc Rc where the K1 and K2 are constants. For given values of Mc and Tc, P0 attains a maximum value 4 Tc P0,m = K3 2 when the core radius Rc = K4Mc/Tc (where K3 and K4 Mc are constants). For the star to be in equilibrium, P0,m must be larger than, or at least equal to, the pressure Pe exerted by the envelope on the interface with the core. Assuming that the core contains only a small fraction of the total 2 4 stellar mass Mt so that we can roughly approximate Pe ∝ Mt /R (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. 4 2 Hence at the interface,, Pe ∝ Tc /Mt . Putting all these together, we have 2K + Ω = 2Kc + Ωc − 3P0Vc = 0, or 2 McTc Mc P0 = K1 3 − K2 4 , Rc Rc where the K1 and K2 are constants. For given values of Mc and Tc, P0 attains a maximum value 4 Tc P0,m = K3 2 when the core radius Rc = K4Mc/Tc (where K3 and K4 Mc are constants). For the star to be in equilibrium, P0,m must be larger than, or at least equal to, the pressure Pe exerted by the envelope on the interface with the core. Assuming that the core contains only a small fraction of the total 2 4 stellar mass Mt so that we can roughly approximate Pe ∝ Mt /R (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. 4 2 Hence at the interface,, Pe ∝ Tc /Mt . Therefore, the condition Pe ≤ P0,m dictates the existence of an upper limit to Mc/Mt . Physically, this is because as the mass of the core increases, its gravity increases, while the the thermal energy of the core is provided chieﬂy by the hydrogen-burning in the shell, which is set to hold the envelope in balance.

I The exact value of the Schonberg-Chandrasekhar¨ limit depends on the ratio between the mean molecular weight in the envelope and in the isothermal core:

M µ 2 c = 0.37 e . Mt SC µc

I At the end of the MS phase of a solar chemical composition object, µe ∼ 0.6 and µc ∼ 1.3 (the core is essentially made of pure 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 have a ratio equal to the limit and will then contract on the Kelvin-Helmholtz timescale. For a star with . 3M , the core mass is below the Schonberg¨ -Chandrasekhar limit, and the contraction phase takes much longer time. The Hertzsprung gap effectively disappears. Thus, many stars in 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 the limiting Hayashi line).

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 the temperature increase up to ∼ 108 K, when He fusion is ignited. I The core (for a star with mass . 2M ) can also reach sufﬁcient densities that the effect of degeneracy pressure comes into play. I As the envelope cools due to expansion, the opacity in the envelope increases (due to the Kramers opacity). The thermal energy trapped by this opacity causes 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 longer time. The Hertzsprung gap effectively disappears. Thus, many stars in 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 the limiting Hayashi line). The red giant branch As the star cools further, the surface opacity becomes less (H− opacity dominates near the surface: κ ∝ ρ1/2T 9). The energy blanketed by the atmosphere eventually escapes, and the luminosity of the star increases. The evolution of low-mass stars with degenerate cores is almost independent of the total stellar masses. I In such a star, a very strong density contrast has developed between the core and the envelope, which is so extended that it exerts very little weight on the compact core. I For a low mass star, the core is degenerate. Its structure is independent of its thermal properties (temperature) and only depends on its mass. I Therefore the structure of a low-mass red giant is essentially a function of its core mass. I The large pressure gradient across the hydrogen-burning shell determines the luminosity of the star. Empirically, from models, and later analytically1, the energy generation in a thin shell of ideal gas around a degenerate core is z L = KMc (4) Replacing L in Eq. 5 with that in Eq. 4 with z ∼ 8 for CNO burning and then integrating leads to  −1/(z−1) shells (z ∼ 15 for helium (z − 1)K (t − t ) M = M 1 − 0 burning shells). The shell c c,0  (1−z)  burning continually XQMc,0 −z/(z−1) increases the mass of the 1/z ! (z − 1)K (t − t0) core with L = L0 1 − , XQL(1−z)/z dM L 0 c = , (5) dt XQ where Mc,0 and L0 are the core mass and where X is the mass luminosity when t = t0. This luminosity fraction of the fuel and Q is increases sharply when the star ascends the energy yield. the RGB.

Why does the RGB luminosity increase sharply?

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

1http://adsabs.harvard.edu/abs/1983ApJ...268..356T Replacing L in Eq. 5 with that in Eq. 4 and then integrating leads to  −1/(z−1) (z − 1)K (t − t ) M = M 1 − 0 c c,0  (1−z)  XQMc,0 !−z/(z−1) (z − 1)K 1/z (t − t ) L = L 1 − 0 , 0 (1−z)/z XQL0

where Mc,0 and L0 are the core mass and luminosity when t = t0. This luminosity increases sharply when the star ascends the RGB.

Why does the RGB luminosity increase sharply?

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 its gravitational potential. This makes the temperature in the shell higher which gives a greater luminosity by the CNO cycle. Empirically, from models, and later analytically1, the energy generation in a thin shell of ideal gas around a degenerate core is z L = KMc (4) Replacing L in Eq. 5 with that in Eq. 4 with z ∼ 8 for CNO burning and then integrating leads to  −1/(z−1) shells (z ∼ 15 for helium (z − 1)K (t − t ) M = M 1 − 0 burning shells). The shell c c,0  (1−z)  burning continually XQMc,0 −z/(z−1) increases the mass of the 1/z ! (z − 1)K (t − t0) core with L = L0 1 − , XQL(1−z)/z dM L 0 c = , (5) dt XQ where Mc,0 and L0 are the core mass and where X is the mass luminosity when t = t0. This luminosity fraction of the fuel and Q is increases sharply when the star ascends the energy yield. the RGB.

1http://adsabs.harvard.edu/abs/1983ApJ...268..356T The ﬁrst dredge-up

I As the star ascends the RGB, the decrease in envelope temperature due to expansion guarantees that energy transport will be by convection. The convective envelope continues to grow, until it almost reaches down to the hydrogen burning shell. The outer convective envelope base I In stars with mass 1.5M , the core & and the He core mass as a function of decreased in size during MS evolution, time for a 0.8 M star. leaving behind processed CNO. I As a result, the surface abundance of 14N grows at the expense of 12C, as the processed material gets mixed onto the surface. I This is called the first dredge-up. Typically, this dredge-up will change the surface CNO ratio from 1/2 : 1/6 : 1 to 1/3 : 1/3 : 1; this result is roughly Hydrogen abundance profile within a independent of stellar mass. 0.8 M star, after the first dredge up. The exact luminosity position of the RGB bump is a function of metal abundance, helium abundance, and stellar mass (and hence stellar age), as well as any additional parameters that determine the maximum inward extent of the convection envelope or the position of the H-burning shell.

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

I This behavior is due to the change in the H abundance after the ﬁrst dredge-up and hence the decrease of the mean molecular weight (L ∝ µ15/2 as discussed earlier). I After the shell has crossed the discontinuity, the surface luminosity grows again, monotonically with The HRD of a 0.8 M star and the trend increasing core mass. corresponding to the RGB bump (inset). The open circle marks the ﬁrst dredge up. RGB bump RGB bump was theoretically predicted by Thomas (1967) and Iben (1968) as a region in which evolution through the RGB is stalled for a time when the H-burning shell passes the H abundance inhomogeneity envelope.

I This behavior is due to the change in the H abundance after the ﬁrst dredge-up and hence the decrease of the mean molecular weight (L ∝ µ15/2 as discussed earlier). I After the shell has crossed the discontinuity, the surface luminosity grows again, monotonically with The HRD of a 0.8 M star and the trend increasing core mass. corresponding 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 metal abundance, helium abundance, and stellar mass (and hence stellar age), as well as any additional parameters that determine the maximum inward extent of the convection envelope or the position of the H-burning shell. Mass loss by red giants

I Stars on the RGB undergo mass loss in the form of a slow (between 5 and 30 km s−1) wind. I Mass loss rates for stars of ALMA image of R Sculptoris, a RGB star. ∼ 1M can reach −8 −1 ∼ 10 M yr at the tip of the RGB. I The total amount of mass lost during the RGB phase can be ∼ 0.2M . I This rate is high enough so that a star at the tip of the RGB (TRGB) will be surrounded by a circum-stellar shell, which can redden and extinct the 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 shell decreases to ≈ 0.001M near the tip of the RGB. I At the same time, the energy generation rate per unit mass increases strongly, which means the temperature within the burning shell also increases. With it, the temperature in the degenerate helium core increases. I To have the helium burning, we need much higher temperature and density than for the hydrogen burning, because the large Coulomb 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 the scaling: the higher the mass, the lower the density. For M & 2M : temperature wins, He-burning is triggered gently. The luminosity increase is small, because of little core contraction.

He ﬂash

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

I Until the degeneracy is overcome, the He burning increases the temperature, but without reducing the density, leading to a “He 3.4 ﬂash” (when the luminosity of the star reaches ∼ 10 L ) and then to the HB.

I The mass of the core at this time is ∼ 0.5M . This, together with the tight relation between the luminosity and core mass, determines the TRGB luminosity. 11 I The flash luminosity ∼ 10 L is all absorbed in the expansion of the non-degenerate outer layers. I As the flash proceeds, the degeneracy in the core is removed, and the core expands. He flash

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

For M & 2M : temperature wins, He-burning is triggered gently. The luminosity increase is small, because of little core contraction. Horizontal Branch Lower mass stars which do undergo the He ﬂash quickly change their structure and land on the stable ZAHB on a Kelvin-Helmholtz timescale.

I These stars burn helium to carbon in their core and also have a hydrogen-burning shell. I As the core becomes slightly larger, the resultant pressure drop at the hydrogen burning shell reduces the luminosity of the star from its pre-helium ﬂash luminosity to ∼ 100L . I Because of the large luminosity associated with helium burning, the central regions of horizontal branch (HB) stars are convective. A star with a helium core of ∼ 0.5M , for example, will have a convective helium burning core of ∼ 0.1M . Horizontal Branch

I The effective temperature of a ZAHB star depends principally on its envelope mass (especially when the mass of the envelope is small). Stars with low mass envelopes will be extremely blue, with log Teff ≥ 4.3. Stars with large envelope masses (∼ 0.4M ) appear near the base of the RGB. I On average, a star spends about 108 years of the life time on the HB Cluster M3 H-R diagram. The gap (about 10% of the lifetime in the in the horizontal branch is due to RGB phase) and has a luminosity of the instability strip, where stars, ∼ 102 times the MS counterpart. known as RR Lyrae variable stars with periods of up to 1.2 days, are I While QHB ∼ 0.1QMS, much of the luminosity of a HB star arises from typically not included in such the H-shell burning (up to ∼ 80%; diagrams. i.e., the bulk of the H-burning occurs after the MS for a low-mass star). Red clump stars

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

I Red clump stars are the (relatively metal-rich and/or young population I counterparts to HB stars (which belong to population II). I They have metallicity greater than about 10% solar. Above this value, red clump location in a CMD is fairly insensitive to the metallicity. I They look redder because opacity rises with Z. They nestle up against the RGB in HR diagrams for old, open cluster, making Hipparcos CMD. The box outlines a clump of dots of similar luminosity. the redclump region. Observing red clump stars in near-IR

Sample CMD in the ﬁeld of the Galactic center (from Hui Dong). The solid, 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 of the stars with initial mass of 5, 10, 20 and 25 M . However, at a given metallicity, clusters of apparently the same age show 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 could result from different mass-loss laws, which may depend on stellar rotation or dynamic interaction within the clusters.

HB color and second parameter problem

The metallicity is the main (’the ﬁrst’) parameter controlling the color distribution of HB stars, due mainly to the envelope opacity and secondarily 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 a measurement of the age. However, at a given metallicity, clusters of apparently the same age show 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 could result from different mass-loss laws, which may depend on stellar rotation or dynamic interaction within the clusters. The Asymptotic giant branch This is a period of stellar evolution undertaken by all low- to intermediate-mass stars (1 - 10 solar masses) late in their lives. The AGB phase is divided into two parts, the early AGB (E-AGB) and the thermally pulsing AGB (TP-AGB). I During the E-AGB phase, the main source of energy is helium fusion in a shell around a core, consisting mostly of carbon and oxygen. The luminosity from this shell will cause the region outside of it to expand. I After the helium shell runs out of fuel, the TP-AGB starts. Now the star derives its energy from fusion of hydrogen in a thin shell. I The Helium shell builds up and eventually ignites explosively, a process known as a helium shell ﬂash, causing the star to expand and cool, which shuts off the hydrogen shell burning and causes convection in the zone between the two shells. I When the helium shell burning nears the base of the hydrogen shell, the increased temperature reignites hydrogen fusion and the cycle begins again. I The luminosity during AGB phase is largely determined by the L 4 Mc CO core mass. For Mc > 0.5M , = 6 × 10 − 0.5 , L M 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 the −4 −1 order of 10 M yr . I As a consequence of the superwind, stars of initial mass in the range 1 M . M . 10 M are left with C-O cores of mass between 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 a result, AGB stars may show S-process elements in their spectra and strong dredge-ups can lead to the formation of carbon stars. Core growth, population, and mass loss of AGB stars Using the conversion factor for hydrogen to carbon, ˙ −11 −1 L Mc = L/Q = 1.2 × 10 M yr , L the lifetime can be estimated as 6 Mc − 0.5M τAGB = (1.4 × 10 yr)ln . Mc,0 − 0.5M

Evolution calculations show that a relation exists between the initial core mass and the initial mass of the star M0, of the form Mc,0 = a + bM0, where a and b are constants. Hence τAGB is essentially a function of the initial stellar mass. The number of AGB stars is proportional to the lifetime of stars in this stage. So if the IMF is assumed, we know the relative population of AGB stars. Compared with the observed, one can conclude that the core mass can grow by only about 0.1 M , indicating the mass loss must be very intense. Core growth, population, and mass loss of AGB stars Using the conversion factor for hydrogen to carbon, ˙ −11 −1 L Mc = L/Q = 1.2 × 10 M yr , L the lifetime can be estimated as 6 Mc − 0.5M τAGB = (1.4 × 10 yr)ln . Mc,0 − 0.5M

I The core of a star at the end of its AGB phase is surrounded by an extended shell, planetary nebula, illuminated by intense UV radiation from the contracting central star. I The central star, or PN nucleus, initially moves toward higher temperatures, powered by nuclear burning in the thin X-ray/optical composite image shell still left on top of the C-O core. of the Cat’s Eye Nebula. I When the mass of the shell decreases below a critical mass of the order of −3 −4 10 M to 10 M , the shell can no longer maintain the high temperature for the nuclear burning and the luminosity of the star drops. I At the same time, the nebula, expanding at a rate of ∼ 10 km s−1, gradually HST image of the PN, NGC disperses, after some 104 − 105 yrs. 6326, with a binary central star. Outline

Star Formation

Young Stellar Objects

The Main Sequence Dependence on stellar mass Dependence on chemical composition

Post-Main Sequence Evolution Leaving the MS The red giant branch The helium burning phase The asymptotic giant branch

Final evolution stages of high-mass stars Final evolution stages of high-mass stars

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

I A star with mass above ∼ 11M will ignite and burn fuels heavier than carbon until an Fe core is formed which collapses and causes a supernova explosion. I For a star with mass & 15M , mass loss by the stellar wind becomes important during all evolution phases, including the MS. Kippenhahn Diagram When the degenerate core’s mass surpasses the Chandrasekhar limit (or close to it), the core contracts rapidly. No further source of nuclear energy 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 stars

For stars with masses & 30M , I The mass loss time scale is shorter than the MS timescale. The MS evolutionary paths of such stars converge toward that of a 30M star. I Mass-loss from Wolf-Rayet stars leads to CNO products (helium and nitrogen) exposed. I The evolutionary track in the H-R diagram becomes nearly horizontal, since the luminosity is already close to the Eddington limit. I Electrons do not become degenerate until 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 nuclear energy in the iron core, the temperature rises from the contraction, but not fast enough. It collapses on a time scale of seconds! ˙ I How could M and vw be measured? I In general, mass-loss rates during all evolution phases increase with stellar mass, resulting in timescales for mass loss that are less that the nuclear Kippenhahn diagram of the evolution of

Mass loss of high-mass stars Mass loss plays an essential role in regulating the evolution of very massive stars.

I WR stars are examples, following the ˙ 1/2 correlation: log[Mv∞R ] ∝ log[L]. I In general, mass-loss rates during all evolution phases increase with stellar mass, resulting in timescales for mass loss that are less that the nuclear Kippenhahn diagram of the evolution of

Mass loss of high-mass stars Mass loss plays an essential role in regulating the evolution of very massive stars.

I WR stars are examples, following the ˙ 1/2 correlation: log[Mv∞R ] ∝ log[L]. ˙ I How could M and vw be measured? Mass loss of high-mass stars Mass loss plays an essential role in regulating the evolution of very massive stars.

I WR stars are examples, following the ˙ 1/2 correlation: log[Mv∞R ] ∝ log[L]. ˙ I How could M and vw be measured? I In general, mass-loss rates during all evolution phases increase with stellar mass, resulting in timescales for mass loss that are less that the nuclear Kippenhahn diagram of the evolution of

timescale for M & 30M . As a result, a 60 M star at Z = 0.02 with mass there is a convergence of the final loss. Cross-hatched areas indicate (pre-supernova) masses to ∼ 5 − 10M . where nuclear burning occurs, and I However, this effect is much diminished curly symbols indicate convective for metal-poor stars because the regions. See text for details. Figure mass-loss rates are generally lower at from Maeder & Meynet (1987). low metallicity. Review Key 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 on the initial mass of the star? What are the relatively time durations that the star spend on different evolutionary stages? 7. The location of the Hayashi track, the MS, or the red-giant branch is sensitive to the chemical composition of the stars. Does Teff increase or decrease with the increase of the metallicity or the decrease of the He abundance? Why? 8. How does the metallicity of ZAMS stars affect their color and luminosity? Review cont. 9. How does the luminosity of a MS star depend on its mass? Can you characterize this dependence from a simple dimensional analysis, assuming that the radiative heat transfer dominates? 10. What is the first dredge-up? How does it affect the observed surface abundances of the elements? 11. Why do some stars undergo He-flash, while others don’t? Has He-flash actually been observed? 12. What is the horizontal branch? Why does it tend to have a narrow 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 in these 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?