i i Laura Greggio and Alvio Renzini: Stellar Populations — Chap. renzini9181c01 — 2011/7/12 — page 1 — le-tex i i 1 1 Firm and Less Firm Outcomes of Stellar Evolution Theory Stellar evolution theory is a mature science which provides the backbone to the theory of stellar populations and the construction of their synthetic luminosities, colors and spectra. In turn, synthetic stellar populations offer the virtually unique possibility to estimate the ages of stellar systems from star clusters to whole galax- ies, their star formation rates and star formation histories, and their mass in stars. This chapter is no substitute to a whole textbook on stellar evolution, of which there are many. It rather offers an introduction to stellar evolution, meant to familiarize the reader with the pertinent nomenclature and a few key concepts. Particular at- tention is devoted to highlight the strengths and weaknesses of current methods, assumptions and results of stellar evolution theory, in particular concerning their impact on synthetic stellar populations. Further readings on the treated issues are listed at the end of this chapter. 1.1 A Brief Journey through Stellar Evolution 1.1.1 A9Mˇ Star Figure 1.1 shows the calculated evolutionary path on the Hertzsprung–Russell di- agram (HRD) of a 9 Mˇ star of solar composition. Letters mark critical points in thecourseofevolutionandTable1.1givesthe corresponding times to reach them. Specifically, the various points mark the following events: A: Beginning of steady hydrogen burning, zero age main sequence (ZAMS). C-C0: Exhaustion of hydrogen in the core, and ignition of hydrogen burning in a shell surrounding the hydrogen-exhausted core. E: Arrival on the Hayashi line, that is, the envelope is (almost) fully convective. F: Ignition of helium burning in the center. J: Exhaustion of helium in the core. JK: Ignition of helium burning in a shell surrounding the helium-exhausted core. Stellar Populations, First Edition. Laura Greggio and Alivio Renzini © 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA. i i i i i i Laura Greggio and Alvio Renzini: Stellar Populations — Chap. renzini9181c01 — 2011/7/12 — page 2 — le-tex i i 2 1 Firm and Less Firm Outcomes of Stellar Evolution Theory Table 1.1 Evolutionary times for the 9 Mˇ sequence. Point Time (106 yr) Point Time (106 yr) A 0.0000 G 24.3046 B 21.8130 H 24.8211 C 22.5220 I 26.1765 0 C 22.5378 J 26.7813 D 22.6435 K 26.8740 E 22.7752 L 26.9380 F 22.8126 M 26.9488 Figure 1.1 The evolutionary track in the along the sequence (A–M) is discussed in Hertzsprung–Russel diagram of a 9 Mˇ star the text, while times to reach such points are with solar composition from the ZAMS all the reported in Table 1.1 (source: Renzini et al. way to the asymptotic giant branch phase (hy- (1992, Astrophys. J., 400, 280)) (Reproduced by drogen and helium burning in two separate permission of the AAS.). shells). What happens at the labeled points L: Back to the Hayashi line (fully convective envelope). LM: Early asymptotic giant branch phase (E-AGB). Before disclosing what happens at points BDGHI and K (not mentioned in the previous list) it is necessary to introduce a few concepts in the language of stellar model makers. The core is the innermost part of the star, where nuclear reactions have greatly altered the original composition. By envelope one means the outer part i i i i i i Laura Greggio and Alvio Renzini: Stellar Populations — Chap. renzini9181c01 — 2011/7/12 — page 3 — le-tex i i 1.1 A Brief Journey through Stellar Evolution 3 of the star, over (most of) which nuclear burning is negligible and whose composi- tion may still be close to the original one. Next is the concept of thermal equilibrium. A star is said to be in thermal equilib- rium (TE) when the total rate of nuclear energy generation (LN)isalmostperfectly equal to its surface luminosity (LS). TE means that the envelope is able to transfer outsideandradiateintospacequitepreciselythesameamountofenergywhich per unit time is produced by nuclear reactions in the core. If LS ' LN the star is in TE and its evolution proceeds on a nuclear timescale. Conversely, if LS ¤ LN the star is out of TE, and its evolution proceeds on a thermal timescale, which usu- ally is much shorter than the nuclear timescale. TE is often broken when the core runs out of fuel or when a new fuel is ignited. A dramatic example of breaking TE is the helium ignition in a degenerate core, called a helium flash. At flash peak 10 LN ' 10 Lˇ while LS ' 2000 Lˇ (and decreases!). The energy that does not escape the star is used to expand the helium core, hence relieving its degeneracy. Evolutionary phases in TE and those out of TE can be easily identified by plotting LS versus LN, or, equivalently, versus the luminosity impinging at the base of the envelope (LB), as shown in Figure 1.2 relative to the evolution of the same 9 Mˇ star model whose HRD is shown in Figure 1.1. Letter labels in this figure mark the Figure 1.2 The luminosity radiated from the along the sequence are the same as in Fig- star’s surface as a function of the luminosity ure 1.1 (source: Renzini et al. (1992, Astro- released by the stellar core and entering the phys. J., 400, 280)) (Reproduced by permission envelope from its base, for the same track of the AAS.). shown in Figure 1.1. Labels of relevant points i i i i i i Laura Greggio and Alvio Renzini: Stellar Populations — Chap. renzini9181c01 — 2011/7/12 — page 4 — le-tex i i 4 1 Firm and Less Firm Outcomes of Stellar Evolution Theory same events as in Figure 1.1, so that one can identify those phases that are in TE and those which are out of it. Thus, phase AB (which is most of the main sequence phase) proceeds in strict TE, but as hydrogen approaches exhaustion in the core, nuclear burning starts to fall short of keeping in pace with the rate at which energy is being radiated away, the star starts departing from TE and begins to contract (point B). The virial theorem may help in understanding why the star contracts. If ΔL D LB LS ¤ 0itmeans that the envelope is either gaining or losing energy, that is: env env C env dE d(E Egrav) ΔL D L L D D th , (1.1) B S dt dt where the energy of the envelope is split in its thermal and gravitational parts, with Z M E env DG r dM . (1.2) grav r r env From the virial theorem: env C env ' 2Eth Egrav 0, (1.3) env hence, using this virial relation to eliminate Eth from Eq. (1.1), we obtain: env dEgrav D 2ΔL . (1.4) dt Therefore, if ΔL > 0 the envelope gains energy, its gravitational energy increases, that is, it becomes less gravitationally bound, and expands. Conversely, if ΔL < 0 the envelope loses more energy from the surface than it receives from the bottom, its gravitational energy decreases, it becomes more bound, hence it contracts. This means that the star contracts in those evolutionary phases that lie in the upper half of Figure 1.2, and expands in those that lie in the lower half, and in both cases does so on a thermal timescale, if they are noticeably away from the LS D LB TE line. All this is quite rigorous, but mere physical intuition should also suffice: if the envelope emits less energy than it receives from the bottom it inflates, if it emits more than it receives it deflates. Notice that only half of the energy needed to expand the envelope comes from the trapped energy flux, the other half comes from the thermal energy of the envelope itself. This is indeed the case starting at point B: as the core is running out of fuel the envelope starts losing more energy than it receives, and contracts until at point C hydrogen is effectively exhausted over the central regions, and nuclear energy gen- eration quickly shifts to a shell surrounding a hydrogen-exhausted helium core. The quick readjustment from core to shell burning leaves the star at point C0, somewhat out of TE, but then the structure tends to approach again TE, until at point D this tendency is reverted and a major excursion away from TE begins. In stars a little less massive than the one considered here, TE is actually well restored shortly after point C0, and yet – as here at point D – TE is broken and stars under- go an extensive loop in the LS LB diagram, as shown in Figure 1.2. The journey i i i i i i Laura Greggio and Alvio Renzini: Stellar Populations — Chap. renzini9181c01 — 2011/7/12 — page 5 — le-tex i i 1.1 A Brief Journey through Stellar Evolution 5 across the HRD from point D to point E takes place on a thermal timescale, and the star expands to red giant dimensions. Clearly, a thermal instability erupts at point D. This is indeed a quite severe thermal instability, suffice to say from Figure 1.2 that at the peak of the instability the core releases 7000 Lˇ but the envelope radiates away only 4000 Lˇ,and 3000 Lˇ are absorbed for its expansion. The physical origin of this thermal instability is actually quite easy to understand.