1. Stellar Structure Section 6: Introduction to Stellar Evolution Lecture 14 – Main-sequence stellar structure: … mass dependence of energy generation, opacity, convection zones, density profile … mass limits … effects of composition changes Post-main-sequence evolution: … calculations and observational tests

2. Mass dependence of opacity and energy generation • Central (and mean) temperature of a MS star increases with mass, though not strongly (T ~ M/R, R ~ M0.6-0.8) • Bound-free opacity (Kramers’) decreases with temperature, while electron-scattering stays constant, thus becoming relatively more important • Energy generation increases with temperature – and CNO cycle much more sensitive than pp chain, so becomes more important • Roughly speaking: • M > 1.5 M: CNO cycle, Thomson scattering dominate • M < 1.5 M: pp chain, bound-free absorption dominate • Transition actually occurs gradually, at slightly different masses

3. Mass dependence of convection • CNO cycle has much stronger T-dependence than pp chain, so central temperature gradient steeper in more massive stars • Steep gradient unstable to convection → convective core (radiative core in less massive stars with pp chain) • Less massive stars have cooler surfaces => ionisation zones in surface regions → convective envelopes (more massive stars ionised right to surface, so have radiative envelopes) • Thus convective envelopes are deep at low mass, and shrink to nothing as mass increases, while convective cores grow with mass, from zero at about 1.1 M (Handout 11, top) • Note strong mass-dependence of the concentration of mass to the centre (50% of R contains ~95% M at Sun, <50% at 0.4 M)

4. Mass dependence of central conditions • Seen already that T generally increases slowly with mass – Handout 11 (foot) shows the detail – note the dramatic change of T with density for masses around the Sun • Density decreases as mass increases (contributing to decrease of bound-free opacity) – but note almost constant central density (~100 g cm-3) over mass range 0.3 to 1.3 M • Ratio of central to mean density at Sun ~100 • Radiation pressure increases gradually towards higher mass, and degeneracy towards lower mass

5. Mass limits (see http://www.sheffield.ac.uk/mediacentre/2010/1713.html for details of R136a1) • Lower mass limit for MS: ~0.08 M, caused by T being too low at centre for H fusion (some D fusion pre-main-sequence) • Less massive stars simply cool slowly; seen as brown dwarfs (with degenerate cores, and surface molecules, such as methane); below ~17 Jupiter masses, usually classed as planets • High mass limit less clear: cores of 50-100 M have luminosities large enough for radiation pressure to stop further accretion, so this often taken as upper limit (also: pulsationally unstable) • Recent VLT observations suggest more massive stars exist: • eclipsing binary NGC 3603-A1, masses 116 and 89 M - consistent • R136a1, mass ~265 M, birth mass ~320 M ! R136a Sun

6. Summary of basic picture • For solar composition, model HR diagram agrees satisfactorily with observations, remembering that models are zero age and observed stars have range of ages – see blackboard sketch • Changes in assumed composition of models cause small shifts in ZAMS, but little change in shape – see blackboard sketch, which implies giants cannot remain well-mixed as they evolve • Theoretical M-L relation agrees reasonably with observation (Handout 2), despite small number of well-determined masses • Switchover from convective to radiative envelopes seems to occur at about the right effective temperature, if we are interpreting spectra correctly

7. Post-main-sequence evolution • Better understood than pre-MS evolution • Pioneering calculations in 1950s and 1960s by 3 main groups: • Icko Iben (USA) • Rudolf Kippenhahn and collaborators (Germany) • Bohdan Paczynski and collaborators (Poland) • Results agreed well, despite 3 independent computer codes • Many other groups now active, from Switzerland to Japan to USA – results differ from each other only in detail: slightly different assumptions about equation of state, opacity, nuclear reaction rates, treatment of convection etc

8. Observational tests give additional confidence • Many observational tests available for MS stars => firm foundation for post-MS studies • MS lifetime >> pre-MS timescale => much more data available (even though star formation studies now observation-led) • Post-MS timescale also nuclear (except for a few phases) – so again much more data than for pre-MS studies • Two kinds of observational constraint • Statistical studies of large numbers of field stars (problem: selection effects, e.g. more luminous stars dominate sample) • Look at star clusters: stars all at ~same distance, and probably all of ~same age. See next lecture