Sternberg Astronomical Institute Moscow University

1. Astronomical Distancesor Measuring the Universe(Chapters 5 & 6)by Rastorguev Alexey,professor of the Moscow State University and Sternberg Astronomical Institute, Russia Sternberg Astronomical Institute Moscow University

2. Content • Chapter Five: Main-Sequence Fitting, or the distance scale of star clusters • Chapter Six:Statistical parallaxes

3. Chapter Five Main-Sequence Fitting, or the distance scale of star clusters • Open clusters • Globular clusters

4. Main idea: to use the advantages of measuring photometric parallax of a whole stellar sample, i.e. close group of stars of common nature: of the same • age, • chemical composition, • interstellar extinction, but of different initial masses

5. Advantages of using star clusters as the “standard candles” - 1 • (a) Large statistics (N~100-1000 stars) reduce random errors as ~N-1/2. All derived parameters are more accurate than for single star • (b) All stars are of the same age. Star clusters are the only objects that enable direct age estimate, study of the galactic evolution and the star-formation history • (c) All stars have nearly the same chemical composition, and the differences in the metallicity between the stars play no role

6. Advantages of using star clusters as the “standard candles” - 2 • (d) Simplify the identification of stellar populations seen on HRD • (e) Large statistics also enables reliable extinction measurements • (f) Can be distinguished and studied even at large (5-6 kpc, for open clusters) distances from the Sun • (g) Enable secondary luminosity calibration of some stars populated star clusters – Cepheids, Novae and other variables

7. DataBase on open clusters: W.Dias, J.Lepine, B.Alessi (Brasilia) • Latest Statistics - Version 2.9 (13/apr/2008): • Number of clusters: 1776 • Size: 1774 (99.89%) • Distance: 1082 (60.92%) • Extinction: 1061 (59.74%) • Age: 949 (53.43%) • Distance, extinction and age: 936 (52.70%) • Proper motion (PM): 890 (50.11%) • Radial velocity (VR): 447 (25.17%) • Proper motion and radial velocity: 432 (24.32%) • Distance, age, PM and VR:379 (21.34%) • Chemical composition [Fe/H]:158 ( 8.90%) • “These incomplete results point out to the observers that a large effort is still needed to improve the data in the catalog” (W.Dias)

8. Astrophysical backgrounds of “isochrone fitting” technique: • (a) Distance measurements: photometric parallax, or magnitude difference (m-M) • (b) Extinction measurements: color change, or “reddening” • (c) Age measurements: different evolution rate for different masses, declared itself by the turn-off point color and luminosity ----------------------------------------------- • Common solution can be found on the basis of stellar evolution theory, i.e. on the evolutional interpretation of the CMD

9. Difference with single-stars method: • Instead of luminosity calibrations of single stars, we have to make luminosity calibration of all Main Sequence as a whole: ZAMS (Zero-Age Main Sequence), and isochrones of different ages (loci of stars of different initial masses but of the same age and metallicity)

10. Important note: Theoretical evolutionary tracks and theoretical isochrones are calculated in lg Teff – Mbol variables • Prior to compare directly evolution calculations with observations of open clusters, we have to transform Teff to observed colors, (B-V) etc., and bolometric luminosities lg L/LSun and magnitudes Mbol to absolute magnitudes MV etc. in UBV… broad-band photometric system (or others)

11. Important and necessary step: the empirical (or semi-empirical) calibration of “color-temperature” and “bolometric correction-temperature” relations from data of spectroscopically well-studied stars of • (a) different colors • (b) different chemical compositions • (c) different luminosities with accurately measured spectral energy distributions (SED), or calibration based on the principles of the “synthetic photometry”

12. Bolometric magnitudes and bolometric corrections • Bolometric Magnitude, Mbol, specifies total energy output of the star (to some constant): • Bolometric Correction,BCV, is defined as the correction to V magnitude: >1 BCV≤ 0 By definition,Mbol = MV + BCV

13. Example:BCV vs lg Teff:unique relation for all luminosities From P.Flower (ApJ V.469, P.355, 1996)

14. Note: Maximum BCV ~0 at lgTeff~3.8-4.0 (for F3-F5 stars), when maximum of SED coincides with the maximum of V-band sensitivity curve • Obviously, the bolometric corrections can be calculated to the absolute magnitude defined in each band

15. For modern color-temperature and BC-temperature calibrations see papers by: • P. Flower (ApJ V.469, P.355, 1996): lgTeff - BCV – (B-V) from observations • T. Lejeune et al. (A&AS V.130, P.65, 1998): Multicolor synthetic-photometry approach; lgTeff–BCV–(U-B)-(B-V)-(V-I)-(V-K)-…-(K-L), for dwarf and giants with [Fe/H]=+1…-3 (with step 0.5 in [Fe/H])

16. lgTeff – (B-V) • for different luminosities; based on observations • (from P.Flower, ApJ V.469, P.355, 1996) • Shifted down by Δ lgTeff = 0.3 relative to next more luminous class for the sake of convenience

17. T.Lejeune et al. (A&AS V.130, P.65, 1998): • Colors from UV to NIR vs Teff(theory and empirical corrections)

18. Before HIPPARCOS mission, astronomers used Hyades “convergent-point” distance as most reliable zero-point of the ZAMS calibration and the base of the distance scale of all open clusters • Recently, the situation has changed, but Hyades, along with other ~10 well-studied nearby open clusters, still play important role in the calibration of isochrones via their accurate distances

19. Revised HIPPARCOS parallaxes of nearby open clusters (van Leeuwen, 2007)

20. Pleiades problem: HST gives smaller parallax (by ~8%) ΔMHp≈ -0.17m • Combined MHp – (V-I) HRD for 8 nearby open clusters constructed by revised HIPPARCOS parallaxes of individual stars (from van Leeuwen, 2007) and corrected for small light extinction • Hyades MS shift (red squares) is due to • Larger [Fe/H] • Larger age ~650 Myr • Bottom envelope (----) can be treated as an observed ZAMS MHp (V-I)

21. (a) Observed ZAMS (in absolute magnitudes) can be derived as the bottom envelope of composite CMD, constructed for well-studied open clusters of different ages but similar chemical composition • (b) Isochrones of different ages are appended to ZAMS and “calibrated”

22. Primary empirical calibration of the Hyades MS & isochrone for different colors, by HIPPARCOS parallaxes(M.Pinsonneault et al. ApJ V.600, P.946, 2004) Solid line: theoretical isochrone with Lejeune et al. (A&AS V.130, P.65, 1998) color-temperature calibrations MV

23. ZAMS and Hyades isochrones: sensitivity to the age for 650±100 Myr (from Y.Lebreton, 2001) • Fitting color of the turn-off point ZAMS

24. Best library of isochrones recommended to calculate cluster distances, ages and extinctions: • L.Girardi et al. “Theoretical isochrones in several photometric systems I. (A&A V.391, P.195, 2002) • Theoretical background: • (a) Evolution tracks calculations for different initial stellar masses (0.15-7MSun) and metallicities (-2.5…+0.5) (also including α-element enhanced models and overshooting) • (b) Synthetic spectra by Kurucz ATLAS9 • (c) Synthetic photometry (bolometric corrections and color-temperature relations) calibrated by well-studied spectroscopic standards

25. L.Girardi et al. “Theoretical isochrones in several photometric systems I. (A&A V.391, P.195, 2002) • Distribution of spectra in Padova library on lg Teff – lg g plane for [Fe/H] from -2.5 to +0.5 • Wide variety of stellar models, from giants to dwarfs and from hot to cool stars, to compare with observations in a set of popular photometric bands: • UBVRIJHK (Johnson-Cousins-Glass), WFPC2 (HST), … Giants

26. Ages of open clusters vary from few Myr to ~8-10 Gyr, age of the disk • For most clusters, [Fe/H] varies approximately from -0.5 to +0.5 • Necessary step in the distance and age determination – account for differences in metallicity ([Fe/H] or Z)

27. Metallicity effects on isochrones:modelling variables, Mbol - Teff Turn-off point

28. Metallicity effects on isochrones: optics Turn-off point

29. Metallicity effects on isochrones: NIR Turn-off point

30. The corrections ΔM and ΔCI (CI –Color Index) vs Δ[Fe/H] or ΔZ to isochrones, taken for solar abundance, can be found either • from theoretical calculations, • or empirically, by comparing multicolor photometric data for clusters with different abundances and with very accurate trigonometric distances

31. Metallicity differences can be taken into account by • (a) Adding the corrections to absolute magnitudes ΔM and to colors ΔCI to ZAMS and isochrone of solar composition. These corrections can follow both from observations and theory. • (b) Direct fitting of observed CMD by ZAMS and isochrone of the appropriate Z – now most common used technique • These methods are completely equivalent

32. Ideally, we should estimate [Fe/H] (or Z) prior to fitting CMD by isochrones • If it is not the case, systematic errors in distances (again errors!) may result • Open question: differences in Helium content (Y). Theoretically, can play important role. As a rule, evolutionary tracks and isochrones of solar Helium abundance (Y=0.27-0.29) are used

33. L.Girardi et al. (2002) database on isochrones and evolutionary tracks is of great value – it provides us with “ready-to-use” multicolor isochrones for a large variety of the parameters involved (age, [Fe/H], [α/Fe], convection,…)

34. Example: Normalized transmission curves for two realizations of popular UBVRIJHK systems as compared to SED (spectral energy distributions) of some stars (from L.Girardi et al., 2002) • See next slides for ZAMS and some isochrones

35. 0.1 1 • Theoretical isochrones (color - MV magnitude diagrams) for solar composition (Z=0.019) and cluster ages 0.1 Gyr, 1 Gyr and 10 Gyr (L.Girardi et al., 2002, green solid lines) 10 Gyr

36. 0.1 • Theoretical isochrones (NIR color-magnitude diagrams) for solar composition (Z=0.019) and cluster ages 0.1 Gyr, 1 Gyr and 10 Gyr (L.Girardi et al., 2002, green solid lines) 1 What are fancy shapes ! 1 Gyr

37. Girardi et al. isochrones in modelling variablesMbol – lg Teff (more detailed age grid) ZAMS

38. Optics NIR • The same but for “standard” multicolor system ZAMS ZAMS

39. How estimate age, extinction and the distance?1st variant • (a) Calculate color-excess CE for cluster stars on two-color diagram like (U-B) – (B-V). Statistically more accurate than for single star. Highly desirable to use a set of two-color diagrams as (U-B) – (B-V)and (B-V) – (V-I) etc., to reduce statistical and systematical errors

40. How estimate age, extinction and the distance?1st variant • (b) If necessary, add corrections for [Fe/H] differences to ZAMS and isochrones family, constructed for solar abundance • (c) Shift observed CMD horizontally, the offset being equal to the color-excess found at (a) step, and then vertically, by ΔM, to fit proper ZAMS isochrone, i.e. cluster turn-off point. Calculate true distance modulus as (V-MV)0 = ΔV - RV∙E(B-V) • (for V–(B-V) CMD)

41. How estimate age, extinction and the distance?2nd variant • (a) If necessary, add corrections for [Fe/H] differences to ZAMS and isochrones family, constructed for solar metallicity • (b) Match observed cluster CMD (color-magnitude diagram) to ZAMS and isochrone trying to fit cluster turn-off point • (c) Calculate horizontal and vertical offsets: H: Δ (color) = CE (color excess) V: (m-M) = (m-M)0 + R· CE (m-M)0 – true distance modulus

42. How estimate age, extinction and the distance?2nd variant • (d) Make the same procedure for all available observations in other photometric bands • (e) Compare all (m-M)0 and CE ratios. For MS fitting performed properly, • distances will be in general agreement, • CE ratios will be in agreement with accepted “standard” extinction law You can start writing paper !

43. MS-fitting example: Pleiades, good case Magnitudes offset gives ΔV=(V-MV)0+RV∙E(B-V) ↨ (m-M)0 = 5.60 E(B-V)=0.04 lg (age) = 8.00 ZAMS G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

44. Young distant cluster, good case (m-M)0=12.55 E(B-V)=0.38 lg (age)=7.15 G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

45. h Per cluster (m-M)0=13.65 E(B-V)=0.56 lg (age)=7.15 RSG (Red Super- Giants) G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

46. RSG (m-M)0=12.10 E(B-V)=0.32 lg (age)=8.22 G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

47. Older and older… (m-M)0=7.88 E(B-V)=0.02 lg (age)=9.25 G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

48. Very old open cluster, M67 (m-M)0=9.60 E(B-V)=0.03 lg (age)=9.60 G.Meynet et al. (A&AS V.98, P.477, 1993) Geneva isochrones

49. Optical data: D.An et al. (ApJ V.671, P.1640, 2007)(Some open clusters populated with Cepheid variables)

50. The same, NIR data: D.An et al. (ApJ V.671, P.1640, 2007)