1. Comprehensive analytic formulae for stellar evolution as a function of mass and metallicity Author ： Jarrod R. Hurley ， Onno R. Pols and Christopher A. Tout Reporter ： Chen Wang 5/27/2014

2. Citation (544)  It is a direction of a rapid evolution code for single stars  Everyone who use the code should cite this paper

3. Star / Twin  Written by Eggleton P.P 1.Eggleton, P. P. (1971) MNRAS,151 ， 351 (550) 2.Eggleton, P. P. (1972) MNRAS,156 ， 361 (343) 3.Eggleton, P. P., Faulkner, J. & Flannery, B. P. (1973) A&A, 23, 325 (340) 4.Eggleton, P. P. (1973) MNRAS, 163, 279 (161) 5.Han, Zh., Podsiadlowski, Ph. & Eggleton, P. P. (1994) MNRAS, 270, 121 (196) 6.Pols, O. R., Tout, C. A., Eggleton, P. P. & Han, Zh. (1995) MNRAS, 274, 964 (360)

4. The author – Cambridge STARS Peter Eggleton 1. Jarrod R. Hurley: Centre for Astrophysics &Supercomputing Swinburne University of Technology (1996-1999 PhD in Cambridge ) 2. Onno R. Pols: University of Utrecht 3. Christopher A. Tout: Cambridge The boss

5. Outline  Introduction  Stellar evolution overview  Stellar models  Procedure  Fitting formulae  Final stages and remnants  Mass-loss and rotation  Discussion

6. What do I concern about  Why we use this kind of code  The stellar evolution path and how do they combine the evolution with the comprehensive formulae  The key point in the code  The accuracy

7. Introduction  The results of detailed stellar evolution calculations are required for applications in many areas of astrophysics. ☆ Model the chemical evolution of galaxies ☆ Determine the ages of star clusters ☆ Simulating the outcomes of stellar collisions ……  Update the results of the calculations

8. Introduction  Tested against observations: reproduce the findings of large-scale star surveys, such as the Hipparcos Catalogue, using population synthesis.  In order to make population synthesis statistically meaningful, it is necessary to evolve a large sample of stars so as to overcome Poisson noise.

9. Introduction  However, detailed evolution codes can take several hours to evolve a model of just one star.  We need to generate a large set of detailed models and present them in some convenient form in which it is relatively simple to utilize the results at a later stage.

10. Introduction  Construct tables (rather large) :interpolate  Approximate the data by a number of formulae as a function of age, mass and metallicity The first approach has been used for many years

11. Introduction  Construct a set of single-star evolution (SSE) formulae.  It is difficult to find analytic approximations than to interpolate in tables.  But the resulting code is very much more compact and adaptable to the requirement of, for example, an N-body code or variable mass-loss.

12. Hertzsprung-Russell diagram (HRD)

13. Stellar evolution overview  The length of a star’s life, its path on the HRD, and its ultimate fate depend critically on its mass  M ≤ 1.1 radiative cores  M ＞ 1.1 convective cores

14. Stellar evolution overview  MS hook : for stars with convective cores, mixing in the core will deplete the fuel over a large region, which leads to a rapid contraction over the inner region at core-hydrogen exhaustion  A bifurcation point

15. Stellar evolution overview Low-mass stars

16. Stellar Models  The fitting formulae are based on the stellar models computed by Pols et al. (1998)  M : 0.5 ~ 50  Z : 0.0001 ， 0.0003 ， 0.001 ， 0.004 ， 0.01 ， 0.02 ， 0.03  Mass-loss from stellar winds was neglected in the detailed stellar models

17. Procedure Assign each evolution phase an integer type, k, where: 0=MS star M<=0.7 deeply or fully convective 1=MS star M>=0.7 2=HG 13=NS 14=BH 15= massless remnant

18. Procedure  Take different features of the evolution in turn, e.g., MS lifetime, ZAHB luminosity, and first try to fit them as for a particular Z  Extend the function to using as a starting point  Fit formulae to the end-point of the various evolutionary phases, as well as to the time-scales.  Fit the behaviour within each phase as

19. Procedure  Fit the L zams and R zams as a function of M and Z (Tout et al. 1996)  L zams accurate to 3 per cent  R zams accurate to 1.2 per cent

20. Fitting formulae  The formulae describe the evolution as a function of mass M and age t  Z-dependence is implicit whenever a coefficient of the form an and bn appears in any of the formulae  There bifurcation points LM: IM: HM

21. Fitting formulae – MS & HG  The base of the giant branch (BGB)  Approximate L and R at the end of the MS & HG, and at the base of the GB

22. Fitting formulae – MS & HG  MS evolution Mimic the hook

23. Fitting formulae – MS & HG  HG evolution  The core mass at the end of the HG Grow linearly

24. Fitting formulae – GB  Power-law  Core He burning  Depend strongly on the mass and metallicity

25. Fitting formulae – CHeB & AGB  Core helium burning: complicated and depends strongly on the mass and metallicity  AGB: determine the CO core mass Radius evolution from the ZAMS to the end of the AGB Excellent performance !

26. Final stages and remnants  Envelope is lost before Mc reaches, the remnant core becomes a WD  Supernova explosion  The supernova leaves either a NS or a BH

27. Stellar remnants

28. Mass-loss and rotation  Mass-loss prescription is independent of the previous formulae and fits observations well.  Use different prescription in different situations  Should be consistent with Eggleton’s code (Nieuwenhuijzen & de Jager, 1990)

29. Mass-loss and rotation  We should follow the evolution of the star’s angular momentum  Start each star with some realistic spin on the ZAMS.

30. Evolution path

31. SSE package  EVOLVE The main routine  ZCNSTS Sets all the constants  STAR  HRDIAG Decide the evolution stages  ZFUNCS Detailed evolution formulae  MLWIND Derives the mass-loss rate

32. Why we choose the SSE package  Fast (It is very useful if we want to know the information derived from the evolution of a large number of stars )  Accuracy (within 5 per cent)  All phases from the zero-age main sequence up to the remnant stages

33. What about Eggleton’s code ◇ It contains a wealth of information detailing the interior structure of each star, information that the formulae simply cannot provide. It is still very useful !

34. Perception  We can observe just a tiny fraction of a population of sources, or cannot resolve single sources at all.  The population synthesis can help to derive or check parameters of the whole set of studied objects, or to predict properties of an unobserved fraction of the sources  SSE/BSE is a useful tool in evolutionary population synthesis

35. Perception  When the whole population can be studied observationally, and also an evolution of its membership can be directly derived from actual data, population modeling becomes unnecessary ! SSE will be an important tool for a long time