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Post Main Sequence Evolution

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Post Main Sequence Evolution Post Main Sequence 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 Red giant phase5 4.1 The turn-off point.................................5 4.2 From H burning to He burning: overview....................6 4.3 Giant stars: red or blue..............................7 4.4 Hydrogen shell burning and helium core growth.................7 4.5 The Schonberg-Chandrasekhar¨ limit and the Hertzprung gap..........9 4.6 Low-mass stars: degenerate core and the Helium flash.............. 10 4.7 Horizontal branch: low-mass. core helium burning, stars............ 10 4.8 Sub Giant Branch................................. 10 4.9.......................................... 11 5 Cepheids 11 6 The Asymptotic Giant Branch 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 star, 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-luminosity relation, L / M 3 (indeed, observations rather lead to M 3:5). The nuclear timescale 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 energy 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 masses 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 metallicity (here measured as the abundance of iron, [Fe/H], with respect to the Sun (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 Eddington luminosity—that is, the maximum luminosity a star can have before radiative pressure disperses it—and κ is the surface opacity. 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 subgiant phase (SGB). Beyond point D (RGB phase), convection 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) photosphere 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 temperature 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.
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