Stellar Structure Section 6: Introduction to Stellar Evolution Lecture 15 – Cluster HR diagrams Main-sequence lifetime Isochrones Evolution during H burning Mass dependence: 5 and 1 solar masses Convective overshooting Evolution after central H exhaustion Formation and development of isothermal core Schönberg-Chandrasekhar limit

Cluster HR diagrams (see blackboard sketches) Clusters: groups of stars that formed together => same age and same chemical composition – ideal observational test Galactic clusters – young to middle-aged: show main sequence of variable length, some giants and gap between: ‘Hertzsprung gap’ Globular clusters – old: show short main sequence, well developed giant branch, and horizontal branch Ages from luminosity at turn-off point: L  M 4, while fuel supply  M, so MS lifetime  M/L  M -3 Hence lower-mass stars take much longer to exhaust their fuel supply

Isochrones – lines of constant age Stars in a cluster all have approximately the same age For comparison with observation, need to construct model HR diagrams for stars of the same age, but different evolutionary stages Proceed by computing the evolution of stars of range of mass, then drawing isochrones that join stars of different mass but the same age (see blackboard sketch) Details of evolution depend on mass – choose two examples:  5 M  : relevant to HR diagrams of galactic clusters  1 M  : relevant to HR diagrams of globular clusters Stars > 10 M  behave a bit differently, but rare – so omit here

Main sequence evolution Common feature – all stars stay close to MS until hydrogen exhausted at centre: X c = 0 Low mass stars – no convective core, no other mixing, composition changes localised: Handout 12 (top) Higher mass stars – convective cores, fresh fuel mixed in from further out If convective core shrinks, composition changes uniformly throughout convective core, and radiative zone develops at edge: Handout 12 (middle) (for stars with M < 10 M  ) [Stars > 10 M  : convective core grows, and get semi-convection – see Kippenhahn & Weigert pp.284-5; more complicated, but general features of evolution remain the same, so omit here]

Definition of convective core – ‘over- shooting’ Where is ‘edge’ of convective core? Where  rad =  ad ? But – rising elements don’t stop abruptly there: acceleration changes sign, but elements keep rising, now decelerating This is ‘over-shooting’ – and means that some mixing still occurs in convectively stable regions Need some kind of mixing-length theory to describe behaviour, even if  =  ad inside the convective core (Handout 12, foot) (see Kippenhahn & Weigert, pp , for a good discussion) Effect: more fuel mixed into core, and lifetime for nuclear burning is longer (by as much as a third) Calibrate theory against observations of clusters

Structure at end of MS evolution (mass range about M  ) Core-envelope structure Envelope – original chemical composition Core – higher mean molecular weight (helium-rich) Early models of red giant stars showed large radius for such structure, even before evolutionary calculations made

Evolution after central H exhaustion After X c = 0, H can still burn in shell around centre Shell burns outwards, feeding He onto growing inert core To keep shell burning fast enough to balance luminosity, temperature in shell must rise Leads to slow, almost homologous contraction of core, on nuclear timescale Low-mass stars – shell burning starts at once: just a continuation of core burning, with the centre omitted Higher-mass stars – H exhausted over entire convective core: pause in H-burning until shell temperature high enough Overall contraction of whole star, on thermal timescale, between X c = 0 and onset of shell burning

Development of isothermal core Slow core contraction → H-burning in shell  straightaway for ~ 1 M   after a delay for ~ 5 M  (overall contraction first) Both cases – core has no (nuclear) energy source, and rapidly becomes isothermal (see blackboard) Shell burning → slow increase in mass of core For stars in 2-6 M  range, there is a maximum mass for the core Arises from fact that core is both isothermal and self-gravitating, and has overlying layers pressing down upon it

Schönberg-Chandrasekhar instability (really failure of equilibrium) Core is contracting to keep shell temperature high enough So core grows in mass, but shrinks in radius Outer layers sink deeper into potential well of core, and increase in weight – i.e. external pressure on core increases Initially, core’s internal pressure also increases: Boyle’s law, PV = constant, => P  1/R 3. As R decreases, self-gravity of core grows, providing effective negative pressure,  -1/R 4, increasing faster than Boyle’s law Hence maximum internal pressure (see blackboard) and core collapses when this is reached

Schönberg-Chandrasekhar limiting mass fraction Virial theorem treatment (including external pressure) allows estimate of core mass at which instability occurs Can be shown (e.g. Kippenhahn & Weigert, pp ) that instability occurs when core mass reaches a certain fraction of the total mass of the star (see blackboard) This fractional mass is known as the Schönberg- Chandrasekhar limit [NOT to be confused with the Chandrasekhar limit for the mass of a white dwarf (see later)] Core initially collapses on dynamical timescale (pressure balance lost), but structure rapidly ceases to be isothermal, and contraction slows to thermal timescale How does star respond? Next lecture!