Chapter 6: Stellar Evolution (part 1)

Transcription

1 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.

2 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

3 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

4 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.

5 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 3kTM M = µm A 0 GM r dm r = 3 r 5 (GM2 /R) = 3 5 ( ) 1/3 4πρ GM 5/3 (1) 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: M > M J ( ) 1/2 ( ) 3/2 3 5kT. 4πρ Gµm A For a typical temperature and density of a large molecular cloud, M J 10 5 M with a collapse time scale of 3

6 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 3kTM M = µm A 0 GM r dm r = 3 r 5 (GM2 /R) = 3 5 ( ) 1/3 4πρ GM 5/3 (1) 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: M > M J ( ) 1/2 ( ) 3/2 3 5kT. 4πρ Gµm A For a typical temperature and density of a large molecular cloud, M J 10 5 M with a collapse time scale of t ff (Gρ) 1/2. 3

7 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).

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

9 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.

10 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. We cannot yet theoretically determine the initial mass function (IMF) of stars. The IMF may be determined empirically and may be expressed in forms such as dn/dm = φ(m) = C(M/0.5M ) x 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.

11 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).

12 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.

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

14 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

15 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.

16 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 ( yrs)(m/m ) 2 (L/L ) 1 (R/R ) 1.

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

18 Observational signatures and classification Observational signatures of YSOs: emission lines from the disk and/or outflow more infrared luminosity due to dust emission variability on hours and days due to temperature irregularities on both the stellar surface and disk high level of magnetic field triggered activities (flares, spots, corona ejection, etc) due to fast rotation and convection 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. Class 0 protostars are highly obscured and have short time scales (corresponding to the stage 2); few are known. 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.

19 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: 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. The energy released by the core (now obeying the virial theorem) is absorbed by the envelope and radiated away as infrared radiation.

20 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. The steady increase of the central temperature causes the dissociation of the H 2, 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. The maximum initial radius of a YSO R i max R (M/M ).

21 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: T eff (Z /0.02) 4/51 µ 13/51 (M/M ) 7/51 (L/L ) 1/102. 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 T eff. 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.

22 Pre-main sequence stage In this PMS stage, the YSO has formed a radiative core, though still growing with time. The star is no longer fully convective. 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.

23 Pre-main sequence stage In this PMS stage, the YSO has formed a radiative core, though still growing with time. The star is no longer fully convective. 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. When the temperature in the core reaches the order of 10 6 K, deuterium is transformed into 3 He 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.

24 Pre-main sequence stage In this PMS stage, the YSO has formed a radiative core, though still growing with time. The star is no longer fully convective. 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. When the temperature in the core reaches the order of 10 6 K, deuterium is transformed into 3 He 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 10 6 K with masses M ), may still be called stars.

25 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

26 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 and how fast they can run, and how they end their lives. Mass Cut diagram showing the fate of single stars in various mass classes.

27 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 M η ), based on dimensional analysis, using the hydrostatic equilibrium state and the EoS of the ideal gas and assuming the radiative heat transfer equation,

28 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 M η ), 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 1 R 4 T 4, where κ is assumed to be constant in the nuclear burning region, while the hydrostatic equilibrium condition as P M2 R 4. Considering the EoS, one then find 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)

29 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 M η ), 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 1 R 4 T 4, where κ is assumed to be constant in the nuclear burning region, while the hydrostatic equilibrium condition as P M2 R 4. Considering the EoS, one then find 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) η = 1 if the radiation pressure dominates (for more massive stars; T P 1/4 M 1/2 /R).

30 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 M η ), 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 1 R 4 T 4, where κ is assumed to be constant in the nuclear burning region, while the hydrostatic equilibrium condition as P M2 R 4. Considering the EoS, one then find 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) η = 1 if the radiation pressure dominates (for more massive stars; T P 1/4 M 1/2 /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.

31 How does R depend on M? We know L ɛm M 2+ν R (ν+3), assuming ɛ = ɛ 0 ρt ν, and T M/R. Equating this to L M 3, as an example, we have R M (ν 1)/(ν+3). (2) For example, ν = 18 for CNO. Then R = M Replacing R in L R 2 T 4 eff with Eq. 2 and M L 1/3, we then obtain ( Teff T eff, ) ( L = L ) (ν+11) ( ) 12(ν+3) 0.12 L =. L This insensitivity of T eff 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.

32 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 M η, then τ MS = yrs(m/m ) 1 η Clearly, the MS lifetime of a star is a strong function of its mass. 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. 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. 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. 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.

33 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.

34 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. The metallicity chiefly affects the opacity, or the amount of bound-free absorption, which is dominated by metals. The smaller opacity allows the energy to escape more easily (so the star appears bluer). The lower opacity also reduces the pressure; hence the luminosity of the star needs to be increased to balance its gravity. Illustration of the effects of varying Y and Z on the shape and position predicted for the 14 Gyr isochrone: Z = (heaviest line, (intermediate), and (lightest). The solid lines are for Y = 0.2 and the dotted lines are for Y = 0.3 [From the calculation of van den Berg & Bell (1985)]

35 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 L RT 4 κρ. To replace R and T in the above relation, we can use R (M/ρ) 1/3 and T µm 2/3 ρ 1/3, 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 L M16/3 ρ 1/6 µ 15/2 κ 0. Here the mass of the star is fixed, while κ 0 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) L(0) [ ] 15/2 µ(t). (3) µ(0)

36 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 We can then get dµ dt µ = = 5 dx µ2 4 dt X = 5 4 µ2 L MQ, where Q = ergs g 1 is the energy released from converting every gram of hydrogen to helium. This equation, together with Eq. 3, gives with solution dl(t) dt [ L(t) = L(0) = 75 8 µ(0)l 1+17/15 (t) MQL 1+17/15 (0), ] 15/17 µ(0)l(0) MQ t.

37 For our sun, expressing the luminosity in units of the present value L and assuming the present age of years, and letting µ(0) 0.6, we then have L(t) L = L(0) [ L(0) ] 15/17 t. L L t So the luminosity of the sun on the ZAMS must be L(0) /17 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)].

38 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?

39 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? 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. The result would then be a compression of the core with a corresponding increase in density. The virial theorem (e.g., Eq. 1) gives T µρ 1/3 M 2/3. 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 T eff.

40 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 ( M ), 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.

41 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 ( M ), 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.

42 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

43 Post-Main Sequence Evolution At this stage, it is useful to make a division based on the stellar mass: 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. 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. 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.

44 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. 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. Simultaneously, the smaller stellar radius translates into a hotter effective temperature a general trend seen in stars at this evolutionary stage. For higher mass stars, the mass fraction of the convective core begins to shrink rapidly.

45 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. 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. Simultaneously, the smaller stellar radius translates into a hotter effective temperature a general trend seen in stars at this evolutionary stage. For higher mass stars, the mass fraction of the convective core begins to shrink rapidly. 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.

46 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. 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). This thick shell phase continues with the shell moving outward in mass, until the core contains 10% of the stellar mass (point 4). This is the Schönberg-Chandrasekhar limit. Stars with larger masses will reach this point faster than stars with low messes.

47 The Schönberg-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.

48 The Schönberg-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 M c /M t, which can be qualitatively understood as follows: For a star in hydrostatic equilibrium, 2K + Ω = 0, where Ω = V GM r dm r /r and 2K = 3 V PdV = 2K c + 2K e, 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 2K e = 3 PdV = 3P 0 V c 3 (dp/dr)(4π/3)r 3 dr, e e where P 0 is the pressure at the boundary between the core and envelope.

49 The Schönberg-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 M c /M t, which can be qualitatively understood as follows: For a star in hydrostatic equilibrium, 2K + Ω = 0, where Ω = V GM r dm r /r and 2K = 3 V PdV = 2K c + 2K e, 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 2K e = 3 PdV = 3P 0 V c 3 (dp/dr)(4π/3)r 3 dr, e e where P 0 is the pressure at the boundary between the core and envelope. Assuming hydrostatic equilibrium, the above equation becomes 2K e = 3P 0 V c Ω e.

50 Putting all these together, we have 2K + Ω = 2K c + Ω c 3P 0 V c = 0, or P 0 = K 1 M c T c R 3 c Mc 2 K 2 Rc 4, where the K 1 and K 2 are constants. For given values of M c and T c, P 0 attains a maximum value T P 0,m = K 4 c 3 when the core radius R Mc 2 c = K 4 M c /T c (where K 3 and K 4 are constants). For the star to be in equilibrium, P 0,m must be larger than, or at least equal to, the pressure P e exerted by the envelope on the interface with the core.

51 Putting all these together, we have 2K + Ω = 2K c + Ω c 3P 0 V c = 0, or P 0 = K 1 M c T c R 3 c Mc 2 K 2 Rc 4, where the K 1 and K 2 are constants. For given values of M c and T c, P 0 attains a maximum value T P 0,m = K 4 c 3 when the core radius R Mc 2 c = K 4 M c /T c (where K 3 and K 4 are constants). For the star to be in equilibrium, P 0,m must be larger than, or at least equal to, the pressure P e exerted by the envelope on the interface with the core. Assuming that the core contains only a small fraction of the total stellar mass M t so that we can roughly approximate P e Mt 2/R4 (from the hydrostatic equilibrium of the entire star) and T c M t /R (from the virial theorem), where R is the total radius of the star. Hence at the interface,, P e Tc 4 /Mt 2.

52 Putting all these together, we have 2K + Ω = 2K c + Ω c 3P 0 V c = 0, or P 0 = K 1 M c T c R 3 c Mc 2 K 2 Rc 4, where the K 1 and K 2 are constants. For given values of M c and T c, P 0 attains a maximum value T P 0,m = K 4 c 3 when the core radius R Mc 2 c = K 4 M c /T c (where K 3 and K 4 are constants). For the star to be in equilibrium, P 0,m must be larger than, or at least equal to, the pressure P e exerted by the envelope on the interface with the core. Assuming that the core contains only a small fraction of the total stellar mass M t so that we can roughly approximate P e Mt 2/R4 (from the hydrostatic equilibrium of the entire star) and T c M t /R (from the virial theorem), where R is the total radius of the star. Hence at the interface,, P e Tc 4 /Mt 2. Therefore, the condition P e P 0,m dictates the existence of an upper limit to M c /M t.

53 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. The exact value of the Schönberg-Chandrasekhar limit depends on the ratio between the mean molecular weight in the envelope and in the isothermal core: ( Mc M t ) SC = 0.37 ( µe µ c ) 2. At the end of the MS phase of a solar chemical composition object, µ e 0.6 and µ c 1.3 (the core is essentially made of pure helium). The limit is thus equal to (M c /M t ) SC 0.1. 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.

54 From point 4 to 5: This region in the HRD corresponds to the Hertzsprung Gap because of the short (KH) evolution time scale. The contraction leads to the temperature increase up to 10 8 K, when He fusion is ignited. The core (for a star with mass 2M ) can also reach sufficient densities that the effect of degeneracy pressure comes into play. 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.

55 From point 4 to 5: This region in the HRD corresponds to the Hertzsprung Gap because of the short (KH) evolution time scale. The contraction leads to the temperature increase up to 10 8 K, when He fusion is ignited. The core (for a star with mass 2M ) can also reach sufficient densities that the effect of degeneracy pressure comes into play. 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 Schönberg -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.

56 From point 4 to 5: This region in the HRD corresponds to the Hertzsprung Gap because of the short (KH) evolution time scale. The contraction leads to the temperature increase up to 10 8 K, when He fusion is ignited. The core (for a star with mass 2M ) can also reach sufficient densities that the effect of degeneracy pressure comes into play. 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 Schönberg -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).

57 The red giant branch As the star cools further, the surface opacity becomes less (H opacity dominates near the surface: κ ρ 1/2 T 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. 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. For a low mass star, the core is degenerate. Its structure is independent of its thermal properties (temperature) and only depends on its mass. Therefore the structure of a low-mass red giant is essentially a function of its core mass. The large pressure gradient across the hydrogen-burning shell determines the luminosity of the star.

58 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. 1

59 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 analytically 1, the energy generation in a thin shell of ideal gas around a degenerate core is L = KM z c (4) with z 8 for CNO burning shells (z 15 for helium burning shells). The shell burning continually increases the mass of the core with dm c dt = L XQ, (5) where X is the mass fraction of the fuel and Q is the energy yield. 1

60 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 analytically 1, the energy generation in a thin shell of ideal gas around a degenerate core is L = KM z c (4) with z 8 for CNO burning shells (z 15 for helium burning shells). The shell burning continually increases the mass of the core with dm c dt = L XQ, (5) where X is the mass fraction of the fuel and Q is the energy yield. Replacing L in Eq. 5 with that in Eq. 4 and then integrating leads to 1/(z 1) M c = M c,0 1 (z 1)K (t t 0) XQM (1 z) c,0 ( ) L = L 0 1 (z 1)K z/(z 1) 1/z (t t 0 ) XQL (1 z)/z, 0 where M c,0 and L 0 are the core mass and luminosity when t = t 0. This luminosity increases sharply when the star ascends the RGB. 1

61 The first dredge-up 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. In stars with mass 1.5M, the core decreased in size during MS evolution, leaving behind processed CNO. As a result, the surface abundance of 14 N grows at the expense of 12 C, as the processed material gets mixed onto the surface. 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 independent of stellar mass. The outer convective envelope base and the He core mass as a function of time for a 0.8 M star. Hydrogen abundance profile within a 0.8 M star, after the first dredge up.

62 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. 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 µ 15/2 as discussed earlier). After the shell has crossed the discontinuity, the surface luminosity grows again, monotonically with increasing core mass. The HRD of a 0.8 M star and the trend corresponding to the RGB bump (inset). The open circle marks the first dredge up.

63 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. 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 µ 15/2 as discussed earlier). 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.

64 Mass loss by red giants Stars on the RGB undergo mass loss in the form of a slow (between 5 and 30 km s 1 ) wind. Mass loss rates for stars of 1M can reach 10 8 M yr 1 at the tip of the RGB. The total amount of mass lost during the RGB phase can be 0.2M. 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. ALMA image of R Sculptoris, a RGB star. HST image of the Hourglass Nebula.

65 Toward the helium burning phase A star with mass 0.5M will eventually ignite helium in its core. 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. 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. 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 s; 8 Be is not stable. The path depends on the race between the temperature (from 10 7 K to 10 8 K, ignition temperature of helium) and density ( 10 2 to 10 6 cm 3 the degeneracy density). This race depends on the (initial) mass again. Remembering the scaling: the higher the mass, the lower the density.

66 He flash For low-mass stars, the temperature is reached after the partial degeneracy in the He core. partial degeneracy in the He core. Until the degeneracy is overcome, the He burning increases the temperature, but without reducing the density, leading to a He flash (when the luminosity of the star reaches L ) and then to the HB. 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. The flash luminosity L is all absorbed in the expansion of the non-degenerate outer layers. As the flash proceeds, the degeneracy in the core is removed, and the core expands.

67 He flash For low-mass stars, the temperature is reached after the partial degeneracy in the He core. partial degeneracy in the He core. Until the degeneracy is overcome, the He burning increases the temperature, but without reducing the density, leading to a He flash (when the luminosity of the star reaches L ) and then to the HB. 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. The flash luminosity L is all absorbed in the expansion of the non-degenerate outer layers. 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. The luminosity increase is small, because of little core contraction.

68 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. These stars burn helium to carbon in their core and also have a hydrogen-burning shell. 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. 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.

69 Horizontal Branch 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 T eff 4.3. Stars with large envelope masses ( 0.4M ) appear near the base of the RGB. On average, a star spends about 10 8 years of the life time on the HB (about 10% of the lifetime in the RGB phase) and has a luminosity of 10 2 times the MS counterpart. While Q HB 0.1Q MS, much of the luminosity of a HB star arises from the H-shell burning (up to 80%; i.e., the bulk of the H-burning occurs after the MS for a low-mass star). Cluster M3 H-R diagram. The gap in the horizontal branch is due to the instability strip, where stars, known as RR Lyrae variable stars with periods of up to 1.2 days, are typically not included in such diagrams.

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

71 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 (A K = 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.

72 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.

73 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.

74 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.

75 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). 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. 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. 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.

76 When the helium shell burning nears the base of the hydrogen shell, the increased temperature reignites hydrogen fusion and the cycle begins again. The luminosity during AGB phase is largely determined ( by the ) L CO core mass. For M c > 0.5M, = Mc 0.5, L M which is of the order of the Eddington luminosity. 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 order of 10 4 M yr 1. 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. 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.

77 Core growth, population, and mass loss of AGB stars Using the conversion factor for hydrogen to carbon, Ṁ c = L/Q = M yr 1 L, L the lifetime can be estimated as ( ) τ AGB = ( Mc 0.5M yr)ln. M c,0 0.5M Evolution calculations show that a relation exists between the initial core mass and the initial mass of the star M 0, of the form M c,0 = a + bm 0, where a and b are constants. Hence τ AGB 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.

78 Core growth, population, and mass loss of AGB stars Using the conversion factor for hydrogen to carbon, Ṁ c = L/Q = M yr 1 L, L the lifetime can be estimated as ( ) τ AGB = ( Mc 0.5M yr)ln. M c,0 0.5M Evolution calculations show that a relation exists between the initial core mass and the initial mass of the star M 0, of the form M c,0 = a + bm 0, where a and b are constants. Hence τ AGB 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.

79 Planetary nebulae 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. The central star, or PN nucleus, initially moves toward higher temperatures, powered by nuclear burning in the thin shell still left on top of the C-O core. When the mass of the shell decreases below a critical mass of the order of 10 3 M to 10 4 M, the shell can no longer maintain the high temperature for the nuclear burning and the luminosity of the star drops. At the same time, the nebula, expanding at a rate of 10 km s 1, gradually disperses, after some yrs. X-ray/optical composite image of the Cat s Eye Nebula. HST image of the PN, NGC 6326, with a binary central star.

80 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

81 Final evolution stages of high-mass stars 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. 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. For a star with mass 15M, mass loss by the stellar wind becomes important during all evolution phases, including the MS.

82 Kippenhahn Diagram

83 Mass-loss of high-mass stars For stars with masses 30M, 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. Mass-loss from Wolf-Rayet stars leads to CNO products (helium and nitrogen) exposed. The evolutionary track in the H-R diagram becomes nearly horizontal, since the luminosity is already close to the Eddington limit. Electrons do not become degenerate until the core consists of iron.

84 Mass-loss of high-mass stars For stars with masses 30M, 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. Mass-loss from Wolf-Rayet stars leads to CNO products (helium and nitrogen) exposed. The evolutionary track in the H-R diagram becomes nearly horizontal, since the luminosity is already close to the Eddington limit. 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!

85 Mass loss of high-mass stars Mass loss plays an essential role in regulating the evolution of very massive stars. WR stars are examples, following the correlation: log[ṁv R 1/2 ] log[l].

86 Mass loss of high-mass stars Mass loss plays an essential role in regulating the evolution of very massive stars. WR stars are examples, following the correlation: log[ṁv R 1/2 ] log[l]. How could Ṁ and v w be measured?

87 Mass loss of high-mass stars Mass loss plays an essential role in regulating the evolution of very massive stars. WR stars are examples, following the correlation: log[ṁv R 1/2 ] log[l]. How could Ṁ and v w be measured? 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 timescale for M 30M. As a result, there is a convergence of the final (pre-supernova) masses to 5 10M. However, this effect is much diminished for metal-poor stars because the mass-loss rates are generally lower at low metallicity. Kippenhahn diagram of the evolution of a 60 M star at Z = 0.02 with mass loss. Cross-hatched areas indicate where nuclear burning occurs, and curly symbols indicate convective regions. See text for details. Figure from Maeder & Meynet (1987).