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Home , Blue giant, Hertzsprung gap, Horizontal branch, Main sequence

Post evolution

Contents

1 Introduction1

2 Main sequence evolution and lifetime2 2.1 Main sequence duration...... 2 2.2 Evolution along the main sequence...... 3

3 Evolutionary tracks and isochrones5

4 phase5 4.1 The turn-off point...... 5 4.2 From H burning to He burning: overview...... 6 4.3 Giant : red or blue...... 7 4.4 shell burning and core growth...... 7 4.5 The Schonberg-Chandrasekhar¨ limit and the Hertzprung gap...... 9 4.6 Low- stars: degenerate core and the Helium flash...... 10 4.7 : low-mass. core helium burning, stars...... 10 4.8 Sub Giant Branch...... 10 4.9...... 11

5 Cepheids 11

6 The 11

1 Introduction

The density of stars in the HR diagram (or simply HRD) directly tells us where stars spend significant amounts of time. After the main sequence, which is the densest part of the HRD, high density of points in the HRD are found in regions above the MS, namely where the radius is increased. These stars are in the giant region which, as we will see, can be divived into the red giant branch (RGB), the horizontal branch (HB), and the asymptotic giant branch (AGB). The evolution of stars beyond the main sequence is much more complex than that on the main sequence, essentially because the timescale separation not always applies. Nevertheless, some of the basic features of the evolutionary path in the HRD can still be understood using 2 4 simple equations, one of the most useful being L = 4πR σTeff . The post-MS evolution de- pends primarily on the mass of the , and one can define simple categories such as low, intermediate, and high-mass, stars. However, the boundaries between these categories may vary, depending on the dominant process at the particular stage of evolution under consideration. In practice, a given star may be considered a low-mass star with respect to one process (e.g. de- generacy) but not with respect to another (pp or CNO cycle). This makes the whole story a non-linear one, and I have decided to not follow a chronological order which would have the consequence that some of the story would have to be repeated several times.

1 Figure 1: The HR diagram of a several open clusters of different ages (color coded), as obtained from the ESA/Gaia mission. Open clusters are associations of ∼10-100 stars, formed relatively recently. Taken from Gaia Collaboration et al.(2018).

2 Main sequence evolution and lifetime

2.1 Main sequence duration The duration of the main sequence, or core hydrogen burning, phase, depends on the mass. This is a direct consequence of the mass- relation,

L ∝ M 3

(indeed, observations rather lead to M 3.5). The is thus

−2 tnucl ∝ M/L ∝ M .

This may be seen in Fig.1 which compares the HRDs of several open clusters of different ages, containing coeval (i.e. formed at the same time) stars of different mass. The most massive stars will leave the main sequence first. Assuming that these different clusters contain stars covering a similar range of mass, we thus expect that the HRD of the oldest clusters contain more stars away from the main sequence. This is confirmed observationnaly, as can be seen of Fig.1. In addition, stars do not stay on the main sequence until they have consumed all of their hydrogen atoms: actually, they will leave it well before that. And, as we will see, which fraction of the nuclear timescale a star will effectively spend on the main sequence also depends on its mass. Low-mass stars, which are powered by the pp chain, have a radiative core and will only be able to fuse hydrogen atoms into helium within a very central region. In contrast, higher mass stars which have a convective, hence more homogeneous, core, will remain on the main sequence until all the hydrogen in the convective region has been consumed. The limit between the two is at M ≈ 1.5M , where the pp chain and CNO cycles have equal specific production rates. Therefore, stars with M < 1.5 will have an effective main-sequence lifetime shorter than the nuclear timescale. The tMS/tnucl ratio increases steadily with mass, for stars with mass larger than 0.6M . Note that mass loss can be important for massive stars even on the main sequence, which complicate the above general picture.

2 Figure 2: The HR diagram (HRD) of globular clusters. Globular clusters are old, gravitationally bound, clusters of stars formed at the same moment. The HRD thus contains the evolution of stars of all formed originally in the cluster. Because they were formed early in the history of the universe, they contain little amount of heavy atoms and have low (here measured as the abundance of iron, [Fe/H], with respect to the (see Appendix)). Left: the M3 (NGC 5272) cluster, with an age of 12.6 Gyr. The main evolution stages have been written on top of the plot. Right: a collection of 14 globular clusters, with color-coded metallicity with ages from 11.5 to 13.5 Gyr. The MS has different length in different clusters, which reflects differences in their population (in terms of mass) and age: the bluest (not a true color, but in the color code of this Figure) clusters, which have the lowest metallicity ([Fe/H]< −2, or less than a 1/100th that of the Sun) are also the oldest (M30, M92, 13.25 Gyr) and thus have the longest MS evolution. One can see that different clusters have TO points at different locations along the MS. Also, notice the horizontal branch (HB) which corresponds to low-mass stars with core helium burning. From Renzini & Fusi Pecci(1988) and Gaia Collaboration et al.(2018).

2.2 Evolution along the main sequence Regardless mass loss complications, the basic observational fact is that, during the core hydro- gen burning phase, stars move on the main sequence. In particular, their luminosity increases. This is due to the evolution of the chemical composition. To see how, we will refer to the Ed- dington model (which we did not see), which shows that the luminosity of a star on the main sequence is given by1:  3 4 4 M L = 0.003LEddµ β (1) M where

3 −2 −1 LEdd = 4πcGM/κ = 1.3 ×10 L (M/M )(κ/1 g cm ) (2) is the —that is, the maximum luminosity a star can have before radiative pressure disperses it—and κ is the surface . Numerically, this gives:

L  M 3 = 5.4 β4 (κ/1 g cm−2)(µ/0.61)4 (3) L M For low-mass stars, β ≈ 1, while for very massive stars, µβ is constant which means that L is constant2. But for most stars, L will thus increase with µ. Since µ increases as hydrogen

1 3 4 3 The so-called Eddingtong quartic equation is 1 − β = 0.003(M/M ) µ β , where β = Pgas/(Prad + Pgas). 2This is true because mass loss is negligible on the main sequence.

3 Figure 3: The evolution of a 1.3M star from the main sequence. During steps A-C, hydrogen burning takes place in the core. The luminosity increases as a consequence of the increase of the mean weight per particle µ (see Eq.1). From Thomas(1967). Dotted regions have varying hydrogen composition. C-D is the phase (SGB). Beyond point D (RGB phase), takes place in the envelope at times larger than 7 Gyr. is converted into helium, the luminosity increases. This can be seen in the first part of the evolutionary tracks shown in Fig.3 and Fig.6, which correspond to stars of 1.3 and 5 M respectively. Numerical calculations show that a 1M star, with intially X = 0.70 and Y = 0.28, would have X = 0.4 after 4.2 Gyr, hence Y ≈ 0.58. For a 5M star, this would be achieved after ≈ 50 Myr. Correspondingly, the mean molecular weight of the gas increases. Rememeber that the mean weight per particle is given by

1/µ = 1/µI + 1/µe (4)

1/µI ≈ X + Y/4 + Z/hAi (5)

1/µe ≈ (1 + X)/2. (6) P P For the Sun (see Fig.4), where the average weight of the metals is hAi = i nimi/ i ni ≈ 17.5 (adopting the present-day abundances). Assuming hAi does not change much (it actually does very little since the main change is in H and He), the mean mass per particle was µ = 0.61 amu when it entered the ZAMS: this is what is the value in the present-day (see Fig.4) because there is no convection except in the very outer parts. In the present-day core of the Sun, we see from Fig.4 that µ = 0.85. If all hydrogen were consumed in the core (and we know it will not be the case), X=0, Y=0.98, and we would have µ = 1.34. We thus see that, in the nuclear burning core, which is where the luminosity is generated, the mass per particle increases, hence L increases with time. Another consequence of the fusion of hydrogen into helium is to reduce the number of free particles and hence the pressure: the core contracts, while the and density increase. Therefore, the temperature gradient dT/dr increases and, since L ∝ dT/dr, the energy flow makes the envelope of the star to expand: the radius increases, hence the luminosity. The envelope radius also increases, and in such way that the decreases 2 4 (L ∝ R Teff ): the star moves towards the upper right in the HRD and becomes a red giant.

4 Figure 4: Numerical calculations for the present-day Sun. Taken from Kevin France, Johns Hopkins University, based on Standard Solar Model data.

3 Evolutionary tracks and isochrones

Before we get into a more detailed view of , we start with a look at what is called an evolutionary track: this is the time evolution of a single star of a given mass (e.g. Fig.9). These tracks are the result of complex numerical calculations, which integrate the differential equations we have established, including the calculation of the opacities, nuclear energy produc- tion rates, and chemical composition evolution. Nevertheless, there are observational equivalents of such evolutionary tracks, which are provided by the HRD of star clusters. Clusters come in two types: open clusters are younger than globular clusters (typically older than 10 Gyr), but both types of clusters have a very interesting property: stars in a given cluster are coeval, and they span a range of masses, which is given by the . Since stars of different mass evolve at different pace, they will distribute themselves at different locations in the HRD, and they will, altogether, build up an evolutionary track. To convince yourself, compare the very recent HRD of clusters from the GAIA mission, in Fig.2, and numerical calculations for a 1 M star Fig.3.

4 Red giant phase

• Main sequence: core hydrogen burning, envelope, photosphere

• Red Giant: inert helium core followed by helium core burning, hydrogen shell burning, envelope, photosphere

•

4.1 The turn-off point Once hydrogen has been fused into He, the pp chain in the core will stop. In the course of hy- drogen fusion, the composition of the star has changed in the core which has become dominated by helium. Eventually, one obtains an inert core of helium, surrounded by a relatively thin shell of burning hydrogen and, beyond, the non-burning envelope. Having an inert core of helium surrounded by a hydrogen shell burning is not a stable situa- tion however, and this will lead to the first major main change during the of a star, namely its evolution away from the main sequence, and the beginning of the red giant phase. The turn-off point (TO) marks the location, in the HRD, where the star leaves the main sequence (see Fig.2). Soon (for massive stars) or later (for low mass stars), the next step will be helium burning. We will see in the following what makes hydrogen burning to stop well before all of it has been converted into helium. But let us begin with an overview of what will happen.

5 Figure 5: Pre-MS (Hayashi and Henyey tracks, left and right panels) followed by the MS and early post-MS evolution (right panel) of a 1M star. The three phases outlined by filled areas correspond to: 1/ core hydrogen burning on the MS (10 Gyr); the luminosity increases because µ increases; 2/ hydrogen shell burning on the Sub Giant Branch (0.5 Gyr) and building of a electron-degenerate helium core, followed by 3/ expanding envelope around a nearly isothermal core (0.5 Gyr) which drives the star along the Red Giant Branch, which is parallel to the . Adapted from Iben(2013).

4.2 From H burning to He burning: overview The red giant phase is what happens after core hydrogen, and which ends when core helium burning stops. During the red giant phase, hydrogen burning takes place in a shell which pro- gressively moves outwards at hydrogen is being consumed. The turn-off point is the position of the star when it leaves the main sequence. The fate of the star after the TO point depends on its mass. Stars with a mass lower than 0.5M will evolve directly to helium dwarfs, withough going through the red giant phase. After the turn-off (TO) point: SGB, RGB, Helium flash, HB, AGB. • RGB: hydrogen-shell burning; envelope expansion; temperature in the core increases; M < 2M : core supported by degenerate pressure; M > 2M , core collapses; Tc increases leading the 3α process;

• if M < 2M , 3α ignition is explosive: Helium flash • HB: core helium burning (also called Helium Main Sequence); duration ∼ 100 Myr; • AGB: core helium fuel exhausted; - core; mass loss (); fate beyond AGB depends on the mass: white dwarfs (M < 8M ) or core-collapse supernova.

Stars with a mass 0.5 < M < 8M will also evolve towards white dwarfs, but after having lost most of their envelope during the red giant phase: these will become carbon-oxygen (or Ne-Mg) white dwarfs. Stars more massive than 8M will go through successive burning stages involving heavier and heavier elements, and will build an onion-like structure, with heavier elements towards the center; eventually, in the latest stages, the core of these high-mass stars will become gravitationally unstable and will produce core-collape . It is important to keep in mind that the above limit of 2M is not a sharp limit. Indeed, the maximum initial mass, below which degeneracy takes place before reaching the grav- itational instability threshold (see the SC limit below), is not known. It lies somewhere

6 between 1.7 and 2.5M , depending on the metallicity and details of the convection, and to a lesser extent on the chemical composition. Let us now look at these different phases in a little more details.

4.3 Giant stars: red or blue What we have described before are red giants. They have a fully convective envelope, thus reaching the Hayashi track in the HRD, with effective temperature Teff = 2000 − 4000 K. But not all giants are red. There are also blue giants, with photospheric effective temperature larger ≈ 1 − 3 ×104 K. Whether a star will become a red or a depends on its precise radial chemical composition. Giant stars have a high-density core surrounded by a more diffuse envelope, with lower surface temperature than it had while on the MS. The core-envelope structure is characteristic of giants stars. They can be modeled as a combination of two , with different K constant. The different values of K in the core and in the envelope stems from the difference in their chemical composition, hence of entropy. The jump in chemical composition between the core and the envelope also gives rise to a jump in pressure, and very generally, we have that the core pressure is much larger than in the envelope and in the photosphere: Pc  Penv  Pph. In giant stars, the core is a self-gravitating sphere, and can be viewed as a star surrounded by an envelope which represents the boundary conditions, while the boundary conditions for the envelope are provided by the photosphere. The properties of the core actually depends on the mass of the star: it can consist of a electron-degenerate helium gas, where the degeneracy pressure is able to counteract the . Or it can be a non-degenerate, self-gravitating, helium gas which has collapsed under gravity, as we will see briefly below.

4.4 Hydrogen shell burning and helium core growth At the end of their main-sequence life, stars possess an inert core made of helium, which is too cold for the 3α process to take place. This core does not (yet) generate energy, and from the temperature gradient equation (e.g. in the radiative diffusion case, dT/dr ∝ L), must be (nearly) isothermal and radiative. The hydrogen around the core may be at the appropriate conditions for fusion to take place, hence leading the hydrogen shell burning. As the material surrounding the core consumes hydrogen, the hydrogen burning shell progressively moves outwards, while the mass of the inter helium core grows. In giant stars (low and high mass), the hydrogen shell burning proceeds through the CNO cycle. In both low (M < 2M ) and high (M > 2M ) mass stars, the core contracts because it does not generate heat, and thus also gets hotter: Tc ∝ GMc/Rc from the . The contraction of the core will stop when the radiative loss from the core are balanced by energy input from the hydrogen burning shell above it. This happens when the burning rate is sufficiently large which requires that it is due to the CNO cycle instead of the pp chain, at a temperature which is 1.5 ×107 K. That H burning proceeds through the CNO cycle which makes the temperature gradient very steep; the luminosity varies fast with radius (dL/dr ∝ r−ν−4), and hydrogen burning takes place in thin (both in radius and mass) shell. Surprisingly, the properties of the shell depend on those of the cores and not on those of the envelope. That the shell is thin is linked to the high density in the core which makes its gravity very strong. In Lagrangian form, the translates into dP/dm ∝ m/r4 which implies that the pressure varies very fast with radius above the core. To look into more details what happens after core hydrogen burning, we need to split our discussion according to the mass of the star.

7 Figure 6: The evolution of a 5M star. Panel a): during core hydrogen burning by the CNO cycle (≈ 79 Myr) and early post-MS evolution: overall contraction (from 79 to 80.9 Myr, or ≈ 2 Myr; core plus envelope contraction), followed by hydrogen shell burning. Mcc refers to the mass of the convective core. Note that the abundance of hydrogen in the convective core goes to zero at the end of the overall contraction phase, when Mcc tends to zero. Panel b): from hydrogen shell burning to core helium burning. The region covered in panel a) is outlined. Notice that core expansion is accompanied by envelope expansion, which makes the star moving to the right in the HRD. Panel c): evolution of the internal structure. Note the change of scale for the abscissa to account for the accelerated evolution. The onset of helium burning (end of panel a), point E) occurs at T ∼ 108 K. From Iben(2013) and Kippenhahn et al.(2012). 8 Figure 7: Evolutionary tracks, in the HRD, of stars with masses M = 3 to 10M . The evolution of such stars is similar, and is characterized by the of an helium core which is surrounded by an expanding, cooling, envelope making these stars red giants. The dashed lines delineate the region of the Cepheids (known as the ). From Kippenhahn et al.(2012).

4.5 The Schonberg-Chandrasekhar¨ limit and the Hertzprung gap

For stars with M > 2M (intermediate- and high-mass stars), there is an upper limit the mass of the helium core can reach. Actually, core growth must stop well before all the star has been converted into helium. Indeed, this is why the main sequence timescale corresponds to the fusion of 10% of the initial hydrogen mass. The demonstration of this is very similar to that of the stability of an isothermal, pressure confined, self-gravitating, sphere. It can be done with the virial theorem after including the surface pressure term. One obtains that, for equilibrium configurations to exist, the surface pres- 4 4 2 sure must be lower than some maximum value, Psurf,max ∝ Tc /(µc Mc ), where the subscript c refers to quantities in the core. Very much like what was found in the study of pressure con- fined Bonnor-Ebert spheres, when the external pressure is increased (while keeping the mass of −3 the core constant), the radius shrinks, with Psurf ∝ Rc , and self-gravity increases up to that point where internal thermal pressure can no longer oppose the sum of external pressure plus self-gravity: the core must collapse. The condition for stability can be written in terms of the contrast between the mean weight per particle between the surrounding envelope and the core:

M µ 2 c ≤ 0.37 e M µc where M is the total mass of the star (M  Mc), and µc and µe are the mean weight per particles in the core and in the envelope respectively. When the mass of the core becomes larger than the critical mass, it becomes unstable and collapses. If we take µe = 0.6 and µc = 1.3, we find that Mcrit ≈ 0.10M. This limit is known as the Schonberg-Chandraskehar¨ limit (1942). The core collapse takes place on the -Helmoltz timescale, which is ≈ 30 Myr for a 5M star. At the same time, the layers above the core have expanded, making the stellar radius to increase (Fig. ??) by a factor ≈ 10: the star has become a red giant. Because the evolution during the core collapse is so fast, the probability of finding stars along this path is

9 small, and there is a gap in the HRD between the main sequence and the red giant branch for stars with M > 2M . This is called the . It corresponds to the empty portion between the main sequence and the RGB, below the Horizontal branch (see e.g. Fig.2). As the core contracts, it heats up, and reaches the helium burning ignition temperature, at an age of 83 Myr for our 5M star. Core helium burning will last 16 Myr (see Fig.6 and Fig.9). During this phase, the nuclear energy produced by helium burning serves primarily to stabilize the core and bring the whole star to near thermal equilibrium, while the luminosity of the star is essentially due to hydrogen shell burning. Actually, the helium core can be seen as a pure , with particular boundary conditions. As is clearly seen in Fig.7 and Fig.9, once helium burning has been ignited, the track makes loops, going back and forth from the Hayashi line. And these loops are more pronouced (larger variation of Teff ) as the increases.

4.6 Low-mass stars: degenerate core and the Helium flash

During the core contraction of stars with M < 2M , the density in the core increases faster than the temperature, which leads to an increase of the degeneracy of the electrons. The result is that an electron-degenerate helium core will form before Mc reaches the 0.1M critical mass. As a consequence, the SC limit does not apply to such stars, which possess an nearly isothermal, de- generate, helium core, which is still growing due to the helium ashes produced by the hydrogen burning in the shell above. Another consequence, is that these low-mass stars do not go throw fast (KH timescale) contraction of their self-gravitating core and there is no such gap in the HRD: the subgiant branch (SGB) is mostly populated by low-mass stars. Also, since there is no contraction of the core, there is no heating source in the core. The temperature of the core will be that of the hydrogen burning shell above it, much lower than that of helium ignition (≈ 108 K), and it will be so for a long time (∼ tnucl): we thus expect to see these stars in the HRD. As hydrogen is 2 4 consumed in the envelope, the latter grows: but since L ∝ R Teff , either L increases or Teff decreases. But, since the star can not cross the Hayashi line, and because low-mass stars begin their life close to the Hayashi line (contrary to massive ones), the luminosity must increase. During the growth of the inert (0.15 < Mc < 0.45M ), degenerate, helium core surrounded by a thin H burning shell, the mass-luminosity relation is approximately:

 7.6 Mc L ≈ 200L 0.3M

4.7 Horizontal branch: low-mass. core helium burning, stars When a low-mass star has reached the tip of the RGB, a series of helium shell flashes begins, which propagate inwards. When they reach the helium core, these flashes stop and the star enters the horizontal branch (HB). Stars on the HB are low-mass stars converting helium into carbon and oxygen, at the base of a convective core, whose mass increases due to shell hydrogen burning above the core. Stars with low metallicity are located at the red end (low Teff ) of the HB, while stars rich in heavy elements are located at the blue end (high Teff ). The diversity of the shape of the HB is exemplified in a spectacular way with the ESA/Gaia data shown in Fig.8.

4.8 Sub Giant Branch Once they have left the main sequence, hydrogen burning takes place in a shell surrounding the isothermal, helium, core.

10 Figure 8: The horizontal branch, as seen in four different globular clusters spanning a broad range of (hence age). The age (in Gyr) of the clusters range from 12.75 to 13.50 Gyr.

2 The timescale of core collapse is the KH timescale: tKH ∼ GM /RL ∼ 1/(RM). We thus have little chance to see a massive star on the SGB, because they cross it too rapidly. For low mass stars, the core is supported by degenerate pressure and it takes a longer time to reach the 3α ignition temperature. The KH timescale is, as we know, longer than the dynamical timescale, but significantly shorter than the nuclear burning one. On the SGB, stars move towards the upper right of the HRD. However, we know there is a forbidden region in the HRD, which is delimited by the Hayashi tracks, which correspond to fully convective stars.

4.9 Intermediate mass stars: no electron-degenerate helium core before helium ignition, but develop an electron-degenerate carbon-oxygen core after exhausting central helium (Iben 1991).

5 Cepheids

6 The Asymptotic Giant Branch

After core helium burning, and after shell hydrogen burning, stars enter the Asymptotic Giant Branch (AGB).

11 Figure 9: The evolution of a 5M star away from the main sequence. From Iben(1991).

12 References

Gaia Collaboration, Babusiaux, C., van Leeuwen, F., et al. 2018, ArXiv e-prints

Iben, Jr., I. 1991, ApJS, 76, 55

Iben, Jr., I. 2013, Stellar Evolution Physics, Volume 1: Physical Processes in Stellar Interiors

Kippenhahn, R., Weigert, A., & Weiss, A. 2012, and Evolution

Renzini, A. & Fusi Pecci, F. 1988, \araa, 26, 199

Thomas, H.-C. 1967, Z. Astrophy., 67, 420

13 Index

Bonnor-Ebert,8 clusters,5 coeval,1 globular,3,5 open,2,5 core-collape,6

Eddington luminosity,2 giants asymptotic giant branch,1 blue giants,6 horizontal branch,1 red giant branch,1

Hertzsprung gap,9 horizontal branch, 10

Initial Mass Function,5 nuclear burning core,2 shell,8 timescale Kelvin-Helmoltz,9 main sequence,8 nuclear,1 turn-off point,6 white dwarfs,6

14