LUM INOSITY FU NCTIO NS FOR OLD STELLAR SYSTEM S
byP eter A nthony Bergbusch L> ^ B.Sc., University of Saskatchewan, 1974 rACULTY 0 f GRADUATE S T o td - M.Sc., University of Regina, 1984
„JL D issertation su bm itted in partial fulfillment f l r ' ^ DEAN of the requirem ents for th e degree of
P D O C TO R OF PHILOSO PHY
in th e D epartm ent of Physics and Astronom y We accept this dissertation as conforming
to the required standard
Dr. D.A. VandenBerg, Supervisor (D epartm ent of P ’.ysics and Astronomy)
Dr. F.D .A , Hartwick, D epartm ental M ember (Dept, of Physics and Astronomy)
Th'. O .J. P ritch et, D epartm ental M em ber (D epartm ent of Physics and Astronomy)
Dr. R.D. M cClure, O utside M ember (Dominion Astrophysical Observatory)
Dr. F.P. Robinsojv-Qutside M em ber (D epartm ent of C hem istry)
Dr. II. Srivastava, O utside M em ber (D epartm ent of M athem atics)
“ " / ■ —y — • r
---Dr. (i.Ci. Fahlm an, E xternal Exam iner (U niversity of B ritish Columbia) © P E T E R ANTHONY BERGBUSCH, 1992
University of V ictoria Septem ber 1992
A ll rights reserved. This dissertation m a y not be reproduced, in whole o r in part, by m imeograph o r other means,
11
Supervisor: Professor Don A, VandenBerg
ABSTRACT
T he potential for lum inosity functions (LFs) of post-turnoff stars to constrain basic cluster param eters such as age, metallicity, and helium abundance is exam ined in this di, sertation. A review of the published LFs for th e globular cluster (GC) M92 suggests th a t the morphology of th e transition from th e m ain sequence to th e red giant branch (ltG B ) is sensitive to these param eters. In particular, a small bum p in this region m ay provide an im p ortant age discrim inant for GCs. A significant deficiency in the num ber of stars over a 2 mag interval, ju s t below th e turnoff, remains unexplained.
A m ethod of interpolating isochrones and LFs accurately from evolutionary se quences, from th e lower main sequence to th e RGB tip, i j discussed. T h e interpolation schf me is based on prim ary interpolation points which are identified by th e behaviour of the derivative d (lo g T e# )/d (lo g t) along an evolutionary sequence.
New BV CCD observations, calibrated w ith Landolt and G raham stan d ard stars, for the old open cluster NGC 2243 and for th e bright stars in the GCs NGC 288 and NGC 7099 are presented. The colour m agnitude diagram (CMD) of NGC 2243 contains a strong binary sta r component. Comparisons w ith th e fiducial sequences of the G C 47 Tuc (Hesser et al. 1987) indicate th a t t! two clusters have similar abundances, while comparisons with the new oxygen-enhanced isochrones (Bergbusch & VandeuBerg 1992) suggest th a t NGC 2243 has an age of 4-5 Gyr, and a m etallicity [Fe/H] - 0.65. T h e morphology of both th e CMD and the LF through th e turnoff region cannot be a ttrib u te d to the merging of the binary and single s ta r sequences, but convective overshooting works m the correct sense to account for th e differences between th e isochrones and th e CMD.
For NGC 288 and NGC 7099, excellent overall consistency among th e Zero Age Horizontal Branch, isochrone, and LF fits is obtained for cluster ages of 14-16 Gyr. The m anifestation of the transition bum p in NGC 288’s LF provides a particularly
I l l
strong constraint 011 the age, since this feature becomes more prominent as the m etal licity increases, /t'-method helium abundance estim ates give V ~ 0.23 for NGC 288 and F w 0.31 for NGC 7099. The 2nd param eter problem is discussed in light of these results. T he RGB bum p, present in canonical LFs, is only weakly identified in the cum ulative LF (C’LF) of NGC 288, and may not be present at all in NGC 7099’s CLF. However, th e brightest RGB stars in both clusters are found within as 0.2 mag of the RGB tip predicted by the oxygen-enhanced models.
Exam iners: Dr. D.A. VandenBere Dr. F.D .A . Hartwick D k C .J~ P ritc h et Dr. R.D. M cClure Dr. F.P. Robinson Dr. H. Srivastava / Dr. G .6 . Fahlmafi
iv
Table o f C on ten ts
A b str a c t
ii
T ab le o f C o n ten ts
iv
L ist o f T ab les
viii
L ist o f F ig u res
ix
A c k n o w le d g e m e n ts
xvii
C h a p ter
1
Introduction
1
1.1 The Relevance of Luminosity Functions 1
1.2 The Helium A bundance 9
1.3 M92: An Illustration 13
1.3.1 T he Age Lum inosity Relations 14
1.3.2 T h e Luminosity Function Near th e Turnoff 17 1.3.3 T he G iant Branch Luminosity Function 22
1.3 4 Discussion 23
1.3.5 Conclusions 28
1.4 Scope of the Work 30
C h a p ter 2
T he Construction of M odel LFs and Isochrones
31
2.1 Introduction 31
2.2 T he M athem atical Formalism 32
2.3 Equivalent Evolutionary Phases (EE Ps) 36
2.3.1 T he Zero-Age Main-Sequence (ZAMS) 37
V
2.3.3 T h e Blue Hook (BH ) 40
2.3.4 Post-M ain-Sequence EEPs 41
2.4 Joining th e G iant Branch to the M ain Sequence Track 45
2.4.1 Idealized Upper G iant Branches 47
2.5 Tests of Interpolation Accuracy 48
C h a p ter 3
D ata Acquisition and Reduction
55
3.1 Observations 55
3.2 S tandard Svars 56
3.2.1 C luster Photoelectric Sequences 64
3.3 Profile-Fitting P hotom etry of C luster Fields 67
3.4 Artificial S tar Tests 70
3.5 Rectification of th e LF 71
C h a p ter 4
The Old Open Cluster NGC 2243
77
4.1 Introduction 77
4.2 Cluster and Background Fields 79
4.3 Artificial S tar Tests 83
4.4 C luster M embers: T he Location of the G iant Branch 94
4.5 T he Color-M agnitude D iagram 95
4.5.1 C om parison w ith 47 Tuc 98
4.5.2 C om parison w ith Isochrones 101
4.6 T he Lum inosity Function 104
vi
4.8 Sum m ary 114
C h a p ter 5
The Globular Cluster NG C 288
116
5.1 ( ’luster param eters 116
5.2 Observations 119
5.2.1 Comparison w ith O ther C luster P hotom etry 123
5.3 Artificial S tar Tests 125
5.4 Analysis of the CMD 133
5.5 T he Lum inosity Function 142
5.6 The Helium Abundance 157
5.7 Discussion 159
C h a p ter
6
The Globular Cluster NGC 7099
162
6.1 Cluster P aram eters 162
6.2 Observations 164
6.2.1 Comparisons w ith O ther C luster P h otom etry 167
6.3 Artificial S tar Tests 171
6.4 Analysis of th e CMD 179
6.5 T he Lum inosity Function 185
6.6 The Helium A bundance 193
6.7 Discussion 197
C h a p ter 7
Conclusions and Ftature Work
198
R efer en ces
203
A. NGC 2243 B. NGC 288 C. NGC 7099 VH 211 216 263
List o f T ables
vm
Table 1-1 A pparent Distance Moduli for Various Ages and Compositions 15
Table 3-1(a) Tem poral Coefficients 61
Table 3-1 (b) Zero-Points and Zenith Extinctions 61
Table 3-2 NGC 2243 Photoelectric Sequences 66
Table 3-3 NGC288 Photoelectric Sequence 66
Table 3-4 NGC 7099 Photoelectric Sequence 66
Table 4-1 O bserving Log (NGC 2243) 81
Table 4-2 Artificial S tar P hotom etric Accuracy (NGC 2243) 89 Table 4-3 Artificial S tar Completeness Fractions (NGC 2243) 108 Tabic 4-4 Rectified Luminosity Function (N GC 2243) 108
Table 5-1 O bserving Log for NGC 288 120
Table 5-2 Fiducial Sequences for NGC 288 127
Table 5-3 Artificial S tar P hotom etric A ccuracy (NGC 288) 131 Table 5-4 Artificicl S tar Completeness Fractions (NGC 288) 147 Table 5-5 Rectified Lum inosity Function (N G C 288) 149
Table 6-1 O bserving Log for NGC 7099 164
Table 6-2 Fiducial Sequences for NGC 7099 174
Table 6-3 Artificial S tar P hotom etric Accuracy (NGC 7099) 176 Table 6-4 Artificial star Completeness Fractions (NGC 7099) 190 Table 6-5 Rectified Luminosity Function (N GC 7099) 192
List o f F igures
Figure 1-1 Figure 1-2 Figure 1-3 Figure 1-4 Figure 1-5 Figure 1-6 Figure 1-7 Figure 1-8 Figure 1-9 Figure 1-10T h e effects of age, helium abundance, and m etallicity on model LFs through th e turnoff region.
T h e effects of age, helium abundance, ;.nd m etallicity on RGB LFs.
T h e effects of age, helium abundance, and m etallicity on RGB CLFs.
V -m agnitude as a function of m etallicity for the brightest RGB stars in 33 globular clusters.
Age-luminosity relations a t th e turnoff for selected m etal-poor LFs.
A com posite LF for th e turnoff region of M92, based on d a ta from th e literature.
T heoretical LFs through th e turnoff region, normalized a t Afy = 2.
T heoretical LFs through th e turnoff region, superim posed on th e com posite LF for M92.
T heoretical RGB LFs superim posed on Hai vick’s observed LF for M92.
Figure 2-1 Figure 2-2{a) Figure 2-2(b) Figure 2-3 Figure 2-4 Figure 2-5 Figure 2-6(a) Figure 2-6(b) Figure 3-1 Figure 3-2(a) T h e functional relation L — L ( M , t ) in th e L-M.-1 coordinate frame.
T h e identification of prim ary E E P s on th e tem p eratu re and lum inosity derivatives.
Evolutionary sequences w ith th e prim ary E E Ps, as identified in Fig. 2-2(a), indicated.
T h e interpolation scheme, based on th e prim ary EEPs identified in Fig. 2-2.
Comparisons between idealized RGBs and th e original sequences com puted with th e Eggleton code.
Isochrones interpolated from evolutionary sequences separated by 0.3A4q, com pared w ith those interpolated from th e 0.1 M ® grid to illustrate th e linearity of th e interpolation scheme. Evolutionary sequences and isochrones with approxim ately the sam e spacing in th e L — Teg plane.
Evolutionary sequences, recovered from the isochrones in 2-6(a) a,’e com pared w ith th e original sequences through the turnoff region.
Correlation between th e tem poral coefficients «3 and 64 in th e photom etric transform ation equations.
Differences between the observed and standard m agnitudes and colours as a function of tim e.
xi
Figure 3-2(b) Differences between the observed and standard m agnitudes
and colours as a function of airiness. 62
Figure 3-3(a) Differences between th e observed and standard m agnitudes
and colours as a function of V -m agnitude. 6? Figure 3-3(b) Differences between th e observed and standard m agnitudes
and colours as a function of B — V. 62
Figure 3-4 C um ulative distributions of th e X and Y coordinates for
NGC 7099. 72
Figure 4-1 A finder chart for th e observed fields in NGC 2243. 80 Figure 4-2 The CMD of th e observed fields in NGC 2243. 84 Figure 4-3 The CMD of a field « 15' n o rth of NGC 2243. 84 Figure 4-4 C um ulative coordinate distributions used to assign positions
in X (a) and Y (b) to the artificial stars. 85 Figure 4-4(c) V -m agnitude cum ulative distribution used to assign m agnitudes
to th e artificial stars. 86
Figure 4-5 Fiducial sequences for NGC 2243. 88
Figure 4-6 Differences between th e assigned and recovered m agnitudes (a)
and colours (b) of th e artificial stars. 90 Figure 4-7 T he artificial star CM D, showing the locations of th e in pu t and
recovered stars. 93
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 4-9 The cleaned CMD of NGC 2243.
4-10 T he fiducial sequences of 47 Tuc, together with a semi-em pirical m ain sequence superim posed on th e cleaned NGC 2243 CMD.
4-11 [Fe/H] = —0.65 oxygen-enhanced isochrones for various ages, together with a corresponding synthetic HB, superim posed on th e cleaned NGC 2243 CMD.
4-12 The sam e as Fig. 4-11, b u t for [Fe/H] = —0.47
4-13 P robability distribution param eters, estim ated from the artificial scar tests.
4-14(a) Two initial estim ates of th e tru e LF for NGC 2243, superim posed on th e observed LF,
4-14(b) A com parison of th e convergence achieved w ith th e two models illustrated in (a).
4-15 Rectification factors as a function of V -m agnitude.
4-16 A single-star model LF for [Fe/H] = —0.65, superim posed on th e rectified LF of NGC 2243.
4-17 A model LF containing a binary s ta r contribution,
for th e sam e metallicity, superim posed on th e rectified LF for NGC 2243.
5-1 A finder chart for th e fields observed in NGC 2S8. 5-2 The CMD of the th e fields observed in NGC 288.
Figure 5-3 Figure 5-4 Figure 5-5 Figure 5-6 Figure 5-7 Figure 5-8 Figure 5-9 Figure 5-10 Figure 5-11 Figure 5-12 xin
Comparisons between B olte’s photom etry and th a t
presented in this study. 124
Fiducial sequences for NGC 288, superim posed on th e
observations. 128
Comparisons among th e observed, model, and adopted
CLFs used to assign m agnitudes to th e artificial stars. 129 The recovered artificial star CMD, superim posed on th e input
artificial star CMD for NGC 288. 130
Differences between th e assigned and recovered m agnitudes (a) and colours (b) of th e artificial stars. 90 Fiducial points for NGC 288 together with th e HB
observations. 134
D orm an’s synthetic HBs superim posed on th e observations
for various distance m oduli and reddenings. 136 D orm an’s synthetic HBs for various m etallicities and distance
moduli superim posed on th e observations. 138 Isochrones and ZAHBs for various m etallicities and distance
m oduli superim posed on th e fiducial points and the observed
HB of Fig. 5-8. 140
Isochrones for various ages and m etallicities superim posed on
xiv
Figure 5-13 External and in tern a1 errors as a functiion of observed
m agnitude, estim ated from th e artificial star tests. 143
Figure 5-14 T h e cleaned CMD for NGC 288. 145
Figure 5-15 Probability distribution param eters, estim ated from th e
artificial sta r tests for NGC 288. 146
Figure 5-16 Rectification factors for the LF of NGC 288 as a function
of th e observed V -m agnitude. 148
Figure 5-17 14 Gyr m odel LFs for [Fe/H] = —1.26 and various power law mass spectra, superim posed on th e rectified LF
through th e turnoff region cf NGC 288. 151 Figure 5-18 14 Gyr m odel LFs for various m etallicities superim posed
on th e rectified LF of NGC 288. 153
Figure 5-19 Model LFs for various ages and m etallicities superim posed
on th e turnoff region of NGC 288’s rectified LF. 155 Figure 5-20 Model CLFs for various ages and m etallicities superim posed
on th e upper RGB portion of NGC 288’s rectified CLF. 156 Figure 6-1 A finder ch art for th e fields observed in NGC 7099. 165 Figure 6-2 T he CMD of th e th e fields observed in NGC 7099. 166 Figure 6-3 Comparisons between B olte’s photom etry and th a t
presented in this study. 168
Figure 6-4 Com parisons between th e photom etry of Richer et ah and
Figure 6-5 Figure 6-6 Figure 6-7 Figure 6-8 Figure 6-9 Figure 6-10 Figure 6-11 Figure 6-12 Figure 6-13 Figure 6-14
Fiducial sequences for NGC 7099. superim posed on the observations.
A comparison between th e observed and model CLFs used to assign m agnitudes to th e artificial stars.
The recovered artificial sta r CMD, superim posed on th e input artificial sta r CMD for NGC 7099.
Differences between th e assigned and recovered m agnitudes (a) and colours (b) of th e artificial stars. Fiducial points for NGC 7099 together with th e HB observations.
D orm an’s [Fe/H] -= -2 .0 3 synthetic HB ;* superim posed on the observations for various distance moduli and reddenings.
D orm an’s synthetic HBs for various m etallicities and distance moduli superim posed on the observations. Isochrones and ZAHBs for various m etallicities and distance moduli superim posed on th e fiducial points and th e observed HB of Fig. 6-9.
Isochrones for various ages and m etallicities super
imposed on the fiducial points through th e turnoff region. E xternal and internal errors as a function of observed m agnitude, estim ated from the artificial star tests.
xvi Figure 6-15 Figure 6-16 Figure 6-17 Figure 6-18 Figure 6-19 T he cleaned CMD for NGC 7099. 188
Probability distribution param eters, estim ated from the
artificial star tests for N GC 7C39. 189
Rectification factors for th e LF of NGC 7099 as a
function of th e observed V -m agnitude. 191 14 and 16 G yr model LFs for [Fe/H] = —2.03 and x = 0.0,
superim posed on the rectified LF through th e turnoff
region of NGC 7099. 194
14 and 16 G yr model LFs and CLFs for [Fe/H] = —2.03 superim posed on the u p p er RGB portion of NGC 7099’s
A ck n ow led gem en ts
xvn
It is a pleasure finally to be in a position to recognize and thank all of those people who have encouraged, helped, and supported m e while this work was in progress. First of all, I would like to thank Dr. Ishrat Naqvi and Dr. Len Greenberg of the University of R egina for encouraging me to undertake gradu ate studies. T he University of Regina also provided support by granting m e an extended educational leave of absence.
P eter Stetson, of th e Dominion Astrophysical Observatory, generously m ade his expertise, knowledge, and software for the reduction of CCD d a ta available to me before much of it entered the public domain. I hope th a t I was as useful guinea pig for him as he was a guru for me!
Very special thanks are due to Don VandenBerg, who has tolerated m e as a stu den t through all my ups and downs, who seems never to have doubted th a t I had the “right s tu f ” (if he did, he kept it quiet), who let m e choose my own path through the work, and who found ways to support m e financially during my stay in V ictoria. Needless to say, his superb oxygen-enhanced evolutionary sequences provided the startin g point for all of th e work presented in this dissertation.
M any family members have supported me, both em otionally and financially. In particular, my m other M ary and my brother E rn est contributed funds when necessary — b u t all of m y brothers and sisters lovingly prodded m e when I needed it.T he greatest credit belongs to my wife Jean, and to m y children Ju lia and Tom, who lived with th e financial and em otional sacrifices, and who deserved much m ore of m e than I was able to give over th e past few years.
C hapter 1
In trod u ction
1.1
T h e R e le v a n c e o f L u m in o sity F u n ctio n s
Observed lum inosity functions (LFs) have not yet had th e expected im pact on stellar astrophysics, in spite of a num ber of theoretical studies (e.g., Sim oda & Iben 1970; Ratcliff 1987) and impressive observational efforts (e.g., Sandage 1957; Hartwick 1970; D a Costa 1982). As em phasized by Renzini & Fusi Pecci (1988), th e L F is a much m ore critical te st of stellar evolution theory th a n th e fitting of isochrones to observed color-m agnitude diagram s (CM Ds), because th e num bers of stars in different evolutionary phases is a direct reflection of th e relative lifetimes in these phases. M oreover, it has th e advantage (see Paczynski 1984) th a t it is com pletely independent of predicted and observed colors and is sensitive to m odel tem p eratures only through the bolom etric correction scale.
T h e inform ation th a t can be derived from an observed L F depends on which por tion of th e CMD has been observed. T h e stan d ard power-law form of th e in itial m ass function (IM F) is <j>(M)dM oc w here <f>(M)dM is th e num ber of stars in th e m ass range M , M + d M . If it is assum ed th a t th e stars in a cluster are coeval and th a t mass-loss is not significant for th e evolutionary phases considered, th e n th e num ber of stars per u nit V -m agnitude interval is sim ply $ (V ) = T he exponent x is derived from th e lower m ain sequence, w here a statistically significant num ber of stars, spread over a reasonable range in m ass, can be obtained. T his has been th e m ain th ru s t of recent CCD-based LF studies (e.g., Richer et al., 1991).
From the turnoff point to th e tip of th e giant branch, th e LF is insensitive to the IM F and to th e effects of mass segregation because th e range in mass is very small ( < 0.03jM© from model calculations for reasonable cluster ages). However, as illustrated in Figure 1-1, th e morphology of th e LF, in the transition from the turnoff to th e giant branch, is sensitive to age, helium abundance, and metallicity. The location of this transition, together w ith th e bum p evident near th e base of th e giant branch, shifts to fainter m agnitudes as th e age is increased. Moreover, th e size of the bum p decreases w ith increasing age. T h e effect of increasing the helium abundance is to depress th e height of th e tran sition , while increasing th e m etal abundance serves to steepen the slope and to accentuate th e size of th e bum p. (See Sim oda & Iben 1970, and Ratcliff, 1987 for a m ore detailed discussion regarding th e transition from th e turnoff to th e giant branch.) However, th e slope of th e LF between the base of th e gian t branch and the location of th e evolutionary pause, where th e H-burning shell contacts th e com position discontinuity produced when th e convective envelope a ttain s its deepest p enetratio n into th e interior, is insensitive to any of the input model param eters, and so serves as a good region over which to norm alize different m odel predictions for com parison to observation.
T h e bum p in th e LF near th e base of the giant branch is m ainly a reflection of th e m orphology of th e subgiant branch: th e flatter th e transitio n from the turnoff to th e RGB, th e larger the num ber of stars in th e m agnitude interval centered on it. Post-turnoff stars evolve rapidly in tem perature, while their lum inosity evolu tion m ay actually slow down (Fig. 2-2(a) illustrates this for a 1.25.M© sta r with [Fe/H] = —0.65). Such tem p eratu re evolution occurs m ore rapidly as the stellar
lo
g
$
i i i i i i i i i r
T J(a) Age
12 Gyr -18 Gyr ---[F e/H ], Y, [O /F e] -2.26,0.235,+ 0.75(b) Helium
Y, [F e/H ], [0/Fe] 0.20, -2 .2 7 , 0.0 — 0.30, -2 .2 1 , 0.0 - - 0.235,-2.26,+ 0.75 Age = 16 Gyr(c) [F e/H ]
.1 .1 _l L ..1 1, J L [F e/H ], Y. [ 0 /F e ] -2 .26,0.235,+ 0.75 ■ —0.65,0.241,+0.30 Age = 16 Gyr .1— 1.-1- L4
M y6
F IG U R E 1-1 The effects of (a) age, (b) helium abundance, and (c) metallicity on model LFs through the turnoff region, normalized at M v = +2.0, for the parameters indicated.
4 maso increases, which accounts, in p a rt, for the increasing size of the bum p at younger ages. T h e effect is fu rther enhanced in th e T -m agnitude LF by the increasing effect of m etallicity on stellar tem peratures, which, through th e bolom etric corrections, re sults in large effects on M y . Consequently, when th e I F is partitioned into m agnitude bins, those bins w hich sam ple th e subgiant portion may contain stars in significantly different evolutionary stages, whereas th e rest ol' th e bins contain stars which are in essentially the sair ’ stage of evolution.
Stellar evolution theory (Sweigart & Gross 1978) predicts th a t two features of the red-giant branch (R G B ) LF are affected by m etallicity, helium abundance, and age; they are th e location of th e evolutionary pause, which also m anifests itself as a bum p in th e L F , and th e m agnitude of the RG B tip. As illu strated in Figure 1-2, differences in the tip m agnitude and in the location of th e bum p are m ost obvious for changes in m etallicity. A gain, the effect a t the tip is due to th e large bolom etric corrections at cooler tem peratu res (see, for exam ple, VandenBerg 1992), associated w ith the line blanketing and increasing metallicity. T he shift in position of th e bum p to fainter m agnitudes -as th e m etallicity increases results from th e combined effects of a real reduction in lum inosity and th e bolom etric corrections. However, th e location of the bum p is also sensitive to th e trea tm e n t of convection, particularly to th e am ount of overshoot a t the b o tto m of th e convective envelope (e.g., see Alongi et al. 1991). For this reason, th e identification of the RGB bum p in observed LFs m ay prove to be a m ore useful constraint on th e trea tm e n t of convection th an on th e basic cluster param eters.
lo
g
$
5
r n - 7 i j=
(a) Age
I I 1 T I-
(b) Helium
0 —=
(c) [F e/H ]
\ _ 0 L 1 1 _.1... I I I I I I 1 I- 2
- 1 0 t I i r * I__ LF IG U R E 1*2 The effects of (a) age, (b) helium abundance, and (c) metallicity on model LFs along the giant branch. The parameters for each Une type are the same as in Figure 1-1.
6
Because of th e difficulty in obtaining large samples of bright RGB stars, Rood & Crocker (1985) have argued th a t the best way to locate th e RGB bum p is through the logarithm of th e cum ulative lum inosity function (CLF). Figure 1-3 illustrates th a t the b u m p m anifests itself in the CLF as a small break in th e slope, approxim ately 2-2.5 m ag below th e RGB tip . Even though the break in the slope is a subtle feature, it is due entirely to th e contribution of th e bum p stars to th e CLF. Moreover, th e effect on th e slope of th e C LF below the bum p persists over several m agnitudes, which provides an additional constraint when com parisons are m ade between m odel and observed CLFs. (O f course, th e shape of th e CLF a t the bright end is not affected by th e presence of th e b u m p at all.) Recently, Fusi Pecci et al. (1990) have identied the b u m p in 11 clusters by locating th e break in th e slope of observed CLFs, and have dem onstrated th e po tential of th e bum p as a stand ard candle.
In addition to th e indications given by model calculations, observational evidence (e.g., Frogel et al. 1981; Frogel et al. 1983; VandenBerg &: Durrel! 1S90) suggests that th e R G B tip m agnitude has potential as a stan dard candle because of its relative insensitivity to age. In Figure 1-4, th e data, compiled by Frogel et al. (1983) are p lo tted together w ith th e RGB tip loci for th e ages 12 and 18 Gyr derived from the oxygen-enhanced isochrones of Bergbusch & VandenBerg (1992). T h e two relevant points to be m ade from this diagram are: 1) a t a given m etallicity, th e models predict a difference of no m ore th a n 0.03 m ag over th e 6 G yr age difference, and 2) the model loci seem to form an approxim ate upper envelope to the d ata. T he m ain observational difficulties appear to be: 1) obtaining a sufficiently large sam ple of bright stars near th e R G B tip , to ensure th a t the brightest RGB star is observed; and 2) discrim inating
7
3
w " &0 O(a) Age
2
1 0(a) Helium
2
1 0(a) [F e/H ]
o 1 02
1 0 1My
F IG U R E 1-3 The effects of (a) age, (b) helium abundance, and (c) met»llicity on cumulative LFs along the giant branch. The parameters for each line type are the same as in Figure 1-1.
3
0
[ F e /H ]
F IG U R E 1-4 The brightest GB star data for 33 globular clusters compiled by Frogel ti al. (1983) together with the model RGB tip luminosities derived from the Bergbusch Sc VandenBerg (1992) isochrones for 12 Gyr (solid line) and 18 Gyr (dotted line). Cluster metallicities based on the infrared measures of Frogel ti al. have been adopted, and the symbols plotted match those of their Figure 6; open circles indicate that the brightest cluster star may not have been observed; crosses indicate clusters for which the effects of crowding were severe.
9
between stars on th e RGB and those on th e asym ptotic giant branch (AGB). Through intercomparisons w ith m odel CLFs, the shape of observed CLFs m ay provide a way to estim ate th e RG B tip lum inosity to somewhat b e tte r precision.
1.2
T h e H e liu m A b u n d a n ce
It is difficult to establish helium abundances in cluster stars because th e spectral lines due to helium only becom e (relatively) strong in stars w ith spectral types earlier than AO. In globular clusters, this observational constraint restricts th e stellar sam ple to hot, blue horizontal branch (H B) stars. Even so, it is not certain th a t helium abundances derived from such observations reflect th e helium content when th e stars were formed. For one thing, model calculations show th a t th e dredge-up phase on the RGB may serve to enhance th e surface abundance of helium , depending on th e significance of diffusion (cf. Proffitt &: VandenBerg 1992) through m ain sequence and giant branch evolution. On th e other hand, the observational evidence (e.g., H eber et al. 1986, and Glaspey et al. 1989) indicates th a t blue HB stars are helium poor as the result of diffusion.
T h e R -m ethod, first elucidated by Iben (1968a), combines th e results of stellar evolution theory w ith observed lum inosity functions to deduce Y . T he ratio R is defined by R — Ih b/ Irgb = • ^h b/ ^r g b, where tRB and Irgb are th e predicted lifetimes of stars on the horizontal branch and on th e region of th e RGB above th e HB, which may b e deduced from theory; Nr b and Nrgb are th e observed num bers of staxs in the corresponding regions of th e CMD.
10 T h e sensitivity of these ratios to th e helium abundance can be understood by com paring stellar models at th e RGB tip with those along th e HB. F irst of all, the helium flash, which signals the end of th e first ascent of the giant branch, is initiated in a highly degenerate helium core, where th e tem p eratu re and density are of th e order 106 g / c m 3 and 8 x 107 K respectively (Renzini 1977). According to Iben ’s red giant models, th e m ass of this inert helium core (A fc) is m ost strongly a function of th e helium abundance (d M c/ d Y « — 0.4Af©), b ut it is only a weak function of the m etallicity ( d M cj d(log Z ) « O.OlAd©) and of the to ta l stellar mass ( d M c/ d M « -0 .0 6 ).
T he dependence of M c on th e to tal stellar mass occurs because th e tem perature and density conditions, m entioned above, are obtained sooner in m ore massive stars, due to th e constraint im posed by hydrostatic equilibrium . Consequently, helium ignition occurs sooner. In th e case of stars w ith M > 2.2A4®1, helium burning may be in itiated w ithout th e flash, because th e interior tem peratures become high enough before degeneracy sets in. T he direct effect of increasing th e helium abundance in a star of a given mass, is to increase its m ean m olecular weight. Through th e equation of state, th is results in higher tem peratures (at eqivalent evolutionary stages) throughout its interior, thereby producing an earlier onset of th e flash conditions. T hus, an increased helium abundance serves to reduce both th e tim e spent on th e RGB and th e lum inosity a t th e RG B tip.
1 This is th e lim it in th e canonical models for RGB evolution. If convective over shoot is im p o rtan t (c/. M aeder h M eynet 1989), th en the transition mass can be significantly lower.
1 1
A nother consideration is th a t th e lum inosity of th e horizontal branch is effectively set by M c, because H e-burning in th e core is th e dom inant source o f lum inosity in this evolutionary phase. Furtherm ore, M c is essentially a constant for stars arriving on th e HB (see, for exam ple, VandenBerg 1992), despite th e fact th a t to tal stellar mass increases from blue to red. (T he size of th e o u ter envelope increases tow ards the red end of the HB. T he difference in mass is thought to arise from variable am ounts of m ass loss which could occur along th e RGB a n d /o r during th e transition from th e RGB tip to th e zero-age horizontal branch.) Since th e HB consists of all stars in the helium core burning phase, and since all HB stars have nearly the sam e luminosity, they all evolve a t approxim ately th e sam e rate. Consequently, th e num ber of stars on the HB is proportional to th e lifetime of an HB star, and therefore o n Y . Moreover, since b o th th e HB and th e RGB tip lum inosity are controlled by M c, th e num ber of stars seen on th e RGB betw een th e HB and th e tip is also predom inantly a function of Y .
T h e chief lim itation of th e R -m ethod is th a t it is m odel dependent through the theoretically determ ined ratio of th e lifetimes. T h e original calibration by Iben (1968a) lead to the conclusion (Iben 1968b) th a t th e initial helium abundance was Y « 0.33, which a t th a t tim e agreed reasonably well w ith th e results of h o t big- bang calculations. However, im provem ents in th e trea tm e n t of convection, including convective overshooting and semiconvection (R obertson & Faulkner 1972; Sweigart & Gross 1974, 1976) have produced sub stantial increases in estim ated HB lifetimes. Recalibrations o f th e m ethod by Buzzoni st al. (1983) and C aputo et al. (1987) gave mean values of Y = 0.23 ± 0 .0 2 and Y = 0.24 ± 0 .0 1 respectively, again in rem arkable
12 agreem ent w ith th e m ore current predictions (e.g., Yang et al. 1984; see also Denegri et al. 1990) ot th e big-bang calculations. D orm an et al. (1989) obtained similar results for th e globular cluster 47 Tuc, independent of th e iZ-method estim ates, by comparing m odel horizontal oranch sequences to th e observations of Hesser et al. (1987) in the CM D. 2
A nother problem w ith the iZ-method is th a t it is difficult to separate red HB stars and AGB stars from th e RGB stars since th e colour differences between them can be q u ite small. To account for th e AGB contribution, Buzzoni et al. derived another ratio, R ' = Nh b/ ( Nrc;b + Na g b), and arrived at the sam e helium content as they
inferred from th e iZ-method estim ates. However, it is preferable to use only RGB and HB stars, since evolution theory is b e tte r understood for these stars th an for those on th e AGB.
N either Buzzoni et al. nor C aputo et al. were able to rule out th e possibility of a variation in Y from cluster to cluster, possibly correlated w ith m etallicity. This m ay b e a ttrib u te d to th e scatter in th e d a ta and th e large uncertainty in each datum due to poor statistics. However, a significant variation in Y could also be obtained if M . c varies am ong th e clusters for some other reason. (For exam ple, calculations by M engel & Gross (1975) suggest th a t rotatio n in giant branch stars can induce
2 A furth er increase in theoretical horizontal branch lifetimes m ay be required (C astellani et al., 1985) by th e occurrence of core-breathing pulses (convective in stabilities), which bring fresh helium in to the core, thus extending th e helium core burning phase. This would necessitate a fu rther reduction in th e estim ates of Y via th e iZ-method.
13 variations in M c, in th e sense th a t a higher ro tation rate results in a greater core mass.) In th e absence of compelling evidence for such a random variation, a sizeable sam ple of clusters w ith good statistics and accurate photom etry, covering a wide range in m etallicity should m ake it possible to discern any significant cluster-to-cluster variation in F , and w hether it is system atic or not.
1.3
M 92: A n Illu str a tio n
Among th e globular clusters in the Galaxy, M92 is th e only one for which repeated observations of th e LF in the turnoff region have been made. T h e first lum inosity function of M92 was derived by Tayler (1954), who m anaged to obtain s ta r counts down to abo ut one m agnitude below th e turnoff. Hartwick (1970) produced a lu m inosity function th a t reached « 1 m ag deeper th an this, and separated horizontal branch stars from giant branch stars on th e basis of th e ir B — V colours. A dditional LFs covering different portions of the CMD from th e subgiant region to th e lower m ain sequence have been obtained by van den Bergh (1975), Fukuoka & Sim oda (1976), and Sandage h K atem (1983). All of these LFs were obtained from sta r counts on photographic plates and therefore suffer from incompleteness a t some m agnitude level, due to crowding of stellar images and to th e inability to d etect faint images.
Most recently, Stetson &; H arris (1988) have produced a CCD stu dy of th e cluster which reaches from th e base of th e giant branch to th e lower m ain sequence ( M v « 8). They derived an app aren t distance m odulus (m — M ) aj>p « 14.6 by com paring th e ir fiducial m ain sequence w ith th e local P opulation II subdw arf stan dard s, a n d obtained a good m atch to th e observations w ith an isochrone for Y = 0.24, [Fe/H] = —2.03,
14 [O/Fe] = +0.7, and an age of 16-17 Gy. For th e lower m ain sequence, a power law exponent x « 0.5 fit th e LF well, although a t th e bright end, th e inferred mass spectrum appeared to have a higher slope.3
In w hat follows, a com posite lum inosity function for th e turnoff region of M92 is derived by com bining th e published LFs m entioned above. This com posite LF is com pared w ith a variety of m odel LFs derived from evolutionary tracks (VandenBerg & Bell 1985; Bergbusch & V andenBerg 1992) to see if th e lum inosity function can constrain any of th e model inp ut param eters, the age a n d /o r the distance modulus, and to shed some light on th e direction fu tu re observations should t~ke.
1.3.1
T h e A g e -L u m in o sity R e la tio n s
T h e age-lum inosity relations for th e m ain sequence turnoff shown in Figure 1-5 were derived from evolutionary tracks, where th e adopted turnoff point for each track was taken as th e location where th e tem p eratu re derivative w ith respect to tim e, d(\ogT eff)/d (\o g t) changed sign. T h e m otivation for using tracks ra th e r than isochrones is th a t th e m ain sequence turnoff point cau be obtained m ore readily from an evolutionary sequence, an d as can be seen from th e figure, th e technique does 3 In a sam ple of nine globular clusters, ranging in m etallicity from [Fe/H] — —2.1 to —0.7, M cClure et al. (1986) found th a t th e exponent for th e IM F varies with th e m etallicity of th e cluster, w ith th e m ost m etal poor clusters requiring x « 2.5, while th e m ost m etal rich clusters require x « —0.5. However, O rtolani et al. (1989) have noted th a t th e mass functions of M30 and NGC 6397 (bo th m etal-poor clusters) have much lower slopes th a n would be expected from th e M cClure et al. results. M92 is sim ilar to these two clusters in this respect.
15 produce a very sm ooth set of relations. It should be noted, however, th a t had the age-luminosity relations been derived from isochrones, slightly different results would necessarily have been obtained. On an isochrone, th e turnoff point is identified as the location of th e bluest stars near a t th e top of th e m ain sequence. Such stars m ay already be in th e thick hydrogen-burning shell stage of evolution, whereas th e location of th e turnoff point on an evolutionary track, defined by th e te m p eratu re derivative, corresponds very closely to th e point of core hydrogen exhaustion. In w hat follows, such differences m ay be neglected, since th e derived relations will be used only in the differential sense.
Table 1-1. A pparent D istance M oduli for Various Ages and Com positions
Age A /£° (m — M ) [Fe/H] [O/Fe] Y
3.95 14.86 - 2.21 0.0 0.30 3.80 15.01 -2 .2 7 0.0 0.20 14.0 Gyr 4.00 14.81 -1 .7 7 0.0 0.20 4.01 14.80 -2 .2 6 +0.75 0.235 4.08 14.73 -2 .0 3 +0.70 0.235 4.09 14.72 - 2.21 0.0 0.30 3.95 14.87 -2 .2 7 0.0 0.20 16.0 Gyr 4.13 14.68 -1 .7 7 0.0 0.20 4.14 14.67 -2 .2 6 +0.75 0.235 4.21 14.6 -2 .0 3 + 0.70 0.235 4.21 14.60 - 2.21 0.0 0.30 4.07 14.74 - 2 .2 7 0.0 0.20 18.0 Gyr 4.24 14.57 - 1 .7 7 0.0 0.20 4.25 14.46 - 2 .2 6 +0.75 0.235 4.32 14.49 -2 .0 3 +0.70 0.235
T he age-luminosity relation for th e com position Y = 0.24, |Fe/H ] = —2.03, [O/Fe] = +0.7, together w ith th e Stetson & H arris estim ates of age (16 Gyr) and distance m odulus (14.6), fix the transform ation of th e m odel LFs to th e observer’s plane. T h e distance m oduli given in Table 1-1 th en follow from th e age-lum inosity
16
3
3.5
4
4.5
20
10
Age (Gyr)
F IG U R E 1*5 Age-Luminosity relations at the main sequence turnoff point derived from the evolutionary sequences of VandenBerg & Bell (1983) and VandenBerg (1992). The composition parameters corresponding to the plotted curves are: a [Fe/H] = —2.27, Y = 0.20, [O/Fe] = 0.0; a [Fe/H] = -1.77, Y = 0.20, [O/Fe] = 0.0; o [Fe/H] = -2.21, K = 0.30, [O/Fe] = 0.0; * [Fe/H] =
r
17 relations o f Fig. 1-3, for th e compositions and ages listed. T he d istance m odulus (m - M ) app = 14.74 for an age of 18 G yr w ith Y — 0.20, [Fe/H] = —2.27, and [O/Fe] = 0.0 is in good agreem ent w ith (m — M ) app — 14.72, o btained by Heasley & C hristian (1986) for th e sam e model param eters.
1 .3 .2
T h e L u m in o sity F u n ctio n N ea r t h e T u rn off
T he various observed LFs were norm alized to H artw ick’s (1970) LF so as to have th e same num ber of stars in th e m agnitude range over which th ey overlapped and over which they were thought to be com plete. H artw ick’s L F was th e only one used to define th e giant branch LF because M92’s horizontal branch reaches down to V « 17, and none of th e o th er LFs discrim inate between horizontal branch stars and giants. T he com posite LF is shown in Figure 1-6. A t V = 17, th e Poisson error, <r($), in th e Hartwick LF am ounts to « 0.1 dex in lo g $ . T he app aren t discrepancy of van den Bergh’s LF at th is m agnitude is probably due to th e fact th a t his bins have very few stars in th em , giving <r(log $ ) « 0.3 dex. A t th e faint end, th e Poisson errors are typically < 0 . 1 dex. A part from th e large am ount of scatter in th e d a ta , there is a pronounced dip in th e LF between 19 < V < 20, and a t V = 18, th e re appears to be a sm all plateau adjacent to a step of about 0.2-0.3 dex in log $ a t V = 18.2.
T h e morphology of th e plateau and step in th e LF near V — 18 is subtle, and given th e un certain ty in th e data, requires some justification. However, th e individual LFs spanning 17.9 < V < 18.3 w ith bin w idths narrow enough to resolve sm all features (i.e., those by H artw ick, van den Bergh, and Fukuoka & Sim oda), regardless of how well they agree on th e slope of th e L F through this region, show a step of « 0.3 dex
18 & <aO O
M92
3
x AC t2
o o o o16
18
20
F IG U R E 1-6 The composite luminosity function for M92. The data are from o Tayler (1954), o Hartwick (1970), * van den Bergh (1975), Fukuoka Sc Simoda (1976), v Sandage Sc Katem (1983), and * Stetson Sc Harris (1988). The Poisson errors at V = 17 in the Hartwick data amount to ps 0.1 dex in log$. Near V = 18, the LFs of Fukuoka Sc Simoda and of van den Bergh show a small plateau, and at V — 18.2 they have a step amounting to ts 0.3 dex which is also repeated in Hartwick’s LF.
19 near V = 13.2. Moreover, these three LFs show a hesitation im m ediately adjacent to this step over the next two brighter bins. T h e evidence for th e existence of th e bum p in the LF is certainly not conclusive, b u t it is suggestive. F urtherm ore, m odel LFs using th e current best estim ates of the helium abundance an d reasonable estim ates of cluster m etallicity do have a bum p a t this location for a w ide range of possible cluster ages.
Model LFs, binned in 0.2 m agnitude intervals, were co nstructed using th e tech niques described in C hapter 2 and in Bergbusch & V andenBerg (1992). For th e model loci shown in Fig’”-e 1-7, th e exponent adopted for th e IM F was x = 1.0. This is roughly th e m ean of th e Stetson & Harris results — b u t, as n o ted earlier, th e choice of x makes little difference for this region of th e LF. T h e 16 G yr LF for Y = 0.24, [Fe/H] = —2.03, and [O/Fe] = +0.70 was normalized to th e H artw ick LF between +0.4 < M y < +2.6, w ith (m — M ) — 14.6, to give a good fit to th e giant branch LF. T he ap propriate distance m odulus obtained from the age-lum inosity relations was then applied to each of th e other model LFs, which were then norm alized to m atch along th e giant branch. A part from th e gross morphology of th e break in th e LF above th e tu-noff point, th e only o th er feature ap parent in th e m odel LFs is a small bum p near V = 18 which develops m ore strongly w ith decreasing age. N either the location nor the size of this bum p is very com position sensitive for the sm all range in m etallicities shown, b u t th e V = 0.30 L F (indicated by th e long-dashed curve) is distinct from the o th er cases.
20 tax)
o
2
—14 Gyr
16 Gyr
[F e/H ], Y, [O /F e] -2 .2 6 ,0 .2 3 5 ,+ 0 .7 5 - 1 .7 7 , 0.20, 0.0 - 2 .2 7 , 0.20, 0.0 2.21, 0.30, 0.0 —18 Gyr
17
18
19
V
20
21
F IG U R E 1*7 Theoretical luminosity functions normalized to match along the lower giant branch. The distance moduli given in Table 1-1 have been applied.
lo
g
$
2 1I I I I | I I I
14 Gyr
rf-H-HH
16 Gyr
18 Gyr
■ l - L g l . t i i I i i i i 1 ■ ■ ■19
V
20
21
F IG U R E 1-8 Comparisons between theoretical LFs and the composite LF for M92. The model curves are '^entified as in Fig. 1-7; the observed points are identified as in Fig. 1-6.
T he com parisons between theory and observation are shown in Figure 1-8. None of th e m odel LFs can be singled out as fitting the d a ta substantially b e tte r in this m agnitude range th a n any of the others. T h e dip feature already alluded to (19 < V < 20) is best m atched by th e LFs w ith Y = 0.3, b u t between 18 < V < 19, th e observed LF is located about 0.2 m ag brighter. T his portion of th e LF can be m atched w ith th e Y — 0.3 models, only if a distance modulus inconsistent with the S tetson &: Harris age estim ate and distance modulus is used. It is possible th a t a feature corresponding to the bum p in th e model LFs is present in th e com posite LF at V = 18, b u t th e error bars in th e d a ta are sufficiently large to make this identification uncertain. Moreover, th e observed LFs were binned over different m agnitude intervals (ie. roughly 0.5 m ag bins for Tayler’s LF, and 0.2 m ag bins for H artw ick’s LF), so the resolution and exact location of this feature varies between them . However, if the bum p is real, th e Y = 0.3 m odel LF is ruled o ut for all ages and for th e others (i.e., V = 0.20,0.24), the 16-18 G yr model LFs appear to m atch it best.
1.3.3
T h e G ia n t B ran ch L u m in o sity F u n ction
H artw ick’s giant branch lum inosity function is shown in Figure 1-9, together with th e m odel LFs for th e inp ut param eters described in th e previous section. There is a distin ct transition in th e sm oothness of th e observed LF near V = 14.6, which corresponds w ithin « 0.2 m ag of the predicted location of the RGB bum p. T h e CLF shown in Figure 1-10 does show a discontinuity a t th e position of th e transition, but ra th e r th a n steepening, as predicted by th e models, th e slope of th e CLF flattens after th e discontinuity. (To reiterate, th e break in th e slope of the m odel C LFs is due
lo
g
$
2314 Gyr
1 016 Gyr
1 0 —©18 Gyr
10
— ©15
12
13
14
16
F IG U R E 1-9 Comparisons between theoretical LFs and the Hartwick’s RGB LF for M92. The identifications of the model curves are as given for Figure 1-7.
2
1 02
1 02
1 0 1-10 (o)L 2414 Gyr
16 Gyr
[F e/H ], Y. [O /F e] -2 .2 6 ,0 .2 3 5 ,+ 0 .7 5 1.77, 0 .2 J, 0.0 2.27, 0.20, 0.0 - 2 .2 1 , 0.30, 0.0-18 Gyr
12
13
14
V
15
16
Comparisons between theoretical CLFs and the observed CLF for M92, based on '. Crosses (x) represent the CLF when the brightest star is removed.
2 5
entirely to th e presence of th e giant branch bum p.) This discrepancy could be due to the accidental inclusion of one or two AGB stars a t th e b righ t end. As illu strated in Fig. 1-10, the location of the R G B tip would m atch th e m odel CLFs very well if the brightest star were remove rrom the sample.
1 .3 .4
D isc u ssio n
M 92’s LF in th e region of th e main-sequence turnofF point appears to offer at least one tantalizing feature, nam ely th e bum p a t V = 18. T h e theoretical LFs indicate, over th e small range in m etallicity explored here, th a t th e location of the bum p is sensitive both to th e helium content and to th e oxygen abundance ratio, while th e size of th e bum p is sensitive to age. Increasing th e helium abundance delays th e appearance of th e b u m p to la ter evolutionary phases, while increasing th e oxygen abundance ratio causes th e bum p to appear earlier. T he current view is th a t [O/Fe] > 0 for m etal-poor stars (e.g., VandenBerg 1992), so th a t [O/Fe] = 0.75 at [Fe/H] = —2.26 places an upper lim it Y < 0.24 on th e helium abundance. A lthough not shown, comparisons have been m ade for 10 and 12 G yr m odel LFs. Based on the size of th e bum p, an age o f 10 G yr can be ruled o u t — b u t 12 G yr rem ains a possibility, particularly for th e Y = 0.20, [Fe/H] = - 2 .2 7 and Y = 0.24, [Fe/H] = - 2 .2 6 models, because th e 0.2 mag bins sm ear th e feature enough to depress th e height of th e bum p.
T h e situation for th e giant branch LF is ra th e r m ore discouraging. W hile th e character of th e observed LF does change near th e expected location of th e RGB bum p, it is hard to claim th a t th e bum p itself can be recognized. However, if the identification of th e RG B bum p w ith th e location of th e break in th e slope of the
26 observed CLF is correct, th e n th e Y = 0.30 case again appears to be ruled out. W hen both th e location of the bum p and th e RGB tip are considered, th e best overall fits are obtained w ith th e [Fe/H] = —2.26, Y = 0.235, [O/Fe] = +0.75 models for 16- 18 Gyr. A m ore populous sam ple of bright RGB stars, clearly free of contam ination by AGB stars, would certainly provide a m ore rigourous te st of th e models.
A lthough th e observations currently available do not yield th e model param eters unam biguously, they do show th a t th e LF can b e used to constrain b oth th e age and the helium abundance even w hen th e m etallicity is uncertain by a few ten ths of a dex. T he m ajo r difficulty from an observational point of view lies in obtaining a sufficiently large sam ple of stars w ith good photom etric accuracy to give both the statistical co ntrast and th e binning resolution required to determ ine th e location and th e size of b o th th e turnoff and RGB bum ps m ore precisely.
An estim ate of th e num ber of stars above th e m ain sequence turnoff point can be m ade if th e to ta l mass of a cluster and th e slope of its m ass function are known. For exam ple, from th e m ass-to-light ratios given by Illingworth & King (1977), the estim ated mass of M15 is fa 106A 4q. T he slope of the m ass function according to Fahlm an et al. (1985) could be as large as * = 2.0, and if all of th e mass in th e cluster is contained in stars of 0.1-0.8.M©, th en fa 5,000 of th em lie above th e turnoff. A pproxim ately 80% of these stars will be found in th e one m agnitude interval above th e turnoff point, so th e potential to im prove th e observed LFs in th e turnoff region certainly exists.
O n th e basis of M onte Carlo sim ulations, Rood & Crocker (1085) have claimed th a t m ore th a n 1000 stars are required in th e upper 3.5 mag of th e RG B to provide a
2 7
statistically significant detection of the bum p. Using th e param eters for M15 quoted above, no m ore th a n « 250 such stars can be expected in th e entire cluster. A ppar ently, the C LF is likely to be of more use th an th ' differential LF in constraining cluster param eters — particularly the distance m odulus — through its p oten tial to define both th e location of th e RG B bum p and th e tip luminosity.
T he dip feature between 19 < V < 20 rem ains a problem for in terpretation because v irtually all of th e observed LFs agree on its existence, while none of our standard m odels of stellar evolution predict it. T h e com parison between th e model LFs and th e observations in Figure 1-8, suggests th a t it m ay reach as faint as V = 21. In this m agnitude range th e observed LFs do not suffer from incompleteness. As can be seen from Figure 1-6, the observed LFs are in good agreem ent as faint as V = 21, where th e S tetson & H arris LF is known to be com plete. Regardless of its extent, a significant fraction of th e stars predicted by stan d ard evolutionary calculations is missing from this p a rt of th e LF.
R otation and diffusion are am ong th e physical processes commonly believed to play a role in stellar evolution, b u t are not included in stan d ard models. R o tatio n has th e effect of increasing m ain sequence lifetimes slightly, b u t evolutionary rates (which would affect LF morphology) rem ain virtually unchanged through th e m ain sequence and turnoff regions. (See Deliyannis et al. (1989), who argue from quite a sophis ticated study of ro tation , th a t it cannot be of much significance for globular cluster stars.) A dispersion in rotational velocities am ong th e cluster stars would effect th e the intrinsic w idth of th e cluster GMD along th e m ain sequence and turnoff regions. So far, the observed w idths of CMDs in m any globular cluster studies have been a t
28 trib u ted to photom etric errors (see Renzini & Fusi Pecci (19S8) for further discussion of these points). The m ain effect of helium diffusion on stellar evolution, according to th e theoretical c a lc u la tio n s of Stringfellow et al. (1983), is to speed up evolution along th e m ain sequence and to slow it down during the H-shell thinning phase. How ever, it is hard to understand how diffusion could produce th e dip seen in th e LF, for which th e m ass range is « 0.05-0.1.M®, or why it would b e so sharply focussed on such a small p a r t of an isochrone. Moreover, LFs com puted from stellar models th a t include diffusion (Proffitt & VandenBerg 1991) are alm ost indistinguishable from LFs derived from canonical models.
A possible answer to this dilem m a is th a t th e IM F cannot be represented by a sim ple power law, as Stetson & H arris (1988) have already suggested for their m ain sequence LF. Such an explanation would severely reduce th e usefulness of the LF through th e turnoff region in constraining any of th e cluster param eters. From th e perspective of stellar astrophysics, it would be m ore palatable to incorporate some, as yet unaccounted for, physical process(es) into th e models to reconcile the observations'. For exam ple, th e occurence of a sm all isotherm al core in main sequence stars (V andenBerg & Stetson 1991) can account for much of th e morphology in M92’s LF th rough th e turnoff region.
1.3.5
C o n c lu sio n s
A sm all p lateau near V = 18 in th e observed LF for M92 has been identified ten tativ ely w ith th e bum p in th e subgiant region predicted by stan d ard models of stellar evolution. T he size of the plateau suggests an age 16-18 G yr for th e cluster.
29 A pronounced dip in th e observed LF between 19 < V < 20 has been identified, but th is feature has no counterpart in th e theoretical LFs. A lthough th e dip has been a ttrib u te d to variations in th e com monly accepted power law behaviour of th e IM F, it m ay instead b e an indication th a t canonical models are missing som ething. T he available observed LFs for th e turnoff region of M92 are not good enough to constrain any of th e astrophysical param eters used in th e construction of m odel LFs as accurately as one would like, except possibly the helium content, for which 0.20 < Y < 0.24 is preferred when 0.0 < [O/Fe] < 0.75 are the corresponding oxygen abundance ratios.
A sim ilar situation holds for th e giant branch lum inosity function. T he LF is too sparse to clearly define th e RG B bum p predicted by th e stan d ard models. B oth the differential and cum ulative lum inosity functions suggest th a t th e bum p occurs near V = 14.6, which again seems to exclude th e Y — 0.30 models w ith scaled solar abundances. T h e oxygen-enhanced models provide th e best over-all fit.
CCD observations have th e potential to m ake great im provem ents over th e old photographic work because profile-fitting photom etry can be done w ith relative accu racy in crowded fields, and q u an titativ e estim ates of th e com pleteness of th e sample, as well as of th e photom etric accuracy, can be derived. A lthough m ost of th e available CCDs cover only small areas of a globular cluster a t a tim e, th e newer, large form at detectors do provide th e op po rtunity to cover significant fractions of th e cluster on a single fram e. Nevertheless, as will be shown in C h apter 5, th e reduction process is com puter intensive and tim e consuming.
30
1.4
S c o p e o f t h e W ork
In th is chapter, the lum inosity function has been presented as a potentially pow erful diagnostic tool for confronting th e current models of stellar stru ctu re and evolu tion. T h e basic problem , b o th from th e theoretical and observational points of view, is to generate precise and accurate LFs. To this end, in C hapter 2 a new, accurate m eth od of generating m odel LFs from evolutionary sequences is described. C hap te r 3 presents som e of th e problem s and the m ethods employed in th e acquisition and reduction of CCD data, and together w ith a form ulation for th e analysis of artifi cial s ta r tests based on Bayes’ Theorem for conditional probabilities. In C hapter 4, ph otom etry reaching « 4 m ag below the turnoff of th e old open cluster NGC 2^43 is presented and analysed. In such clusters, b oth the CMD and th e LF through the turnoff region can potentially constrain th e degree of convective overshooting in stel lar models. Furtherm ore, th e analysis presented in C hapter 4 also provides th e first confrontation between th e isochrones and LFs derived w ith th e techniques described in C h ap ter 2 and CCD observations. In C hapters 5 & 6 , photom etry of th e evolved stars in th e core regions of th e globular clusters NGC 288 and NGC 7099 (M30) is presented. Comparisons between th e observed LFs, rectified according to the m eth ods outlined in C hapter 3, an d th e oxygen-enhanced isochrones and LFs of Bergbusch & VandenBerg (1992) are m ade. Finally, in C hapter 7, th e results of th e preceding chapters are sum m arized, an d some proposals for fu rth er studies are sugge&ted.
31
C hapter 2
T h e C on stru ction o f M o d el LFs an d Isoch ron es
2.1
In tr o d u c tio n
T he most im p o rtan t consideration in generating m odel lum inosity functions and isochrones from evolutionary sequences, is th a t th e interpolated quantities should produce the m orphology evident in those sequences as accurately as possible. The Revised Yale Isochrones and Luminosity Functions (Green et al. 1987), often show fea tures (bum ps an d wiggles) which have n o t been a ttrib u te d to evolutionary processes — presum ably, they are m anifestations of numerical noise in th e original evolutionary sequences, or th ey have been produced by the interpolation scheme. In some cases, the m agnitude of these apparently spurious features rivals th a t of real features such as th e RGB bum p.
As has been illustrated in C hapter 1, th e detailed m orphology of th e lum inosity function in th e region n ear th e m ain sequence turnoff shows sub tle differences as the basic in pu t param eters are varied. If it is to be used successfully to constrain th e input param eters, th en particular care will have to be taken to produce th e best possible models, and to obtain observations with high statistical significance. In this chapter, a m ethod of accurately interpolating isochrones and m odel L F s from evolutionary sequences will b e described.
32
2 .2
T h e M a th e m a tic a l F orm alism
Deep CCD studies have revealed th a t th e CMDs of th e G alactic globular clusters te n d to b e exceedingly tight through th e turnoff region, w ith a photom etric scatter th a t is fully consistent w ith (small) observational uncertainties. This result provides very strong su p p o rt for th e basic assum ptions th a t are m ade ab ou t the stars in most globular clusters; namely, th a t ( 1) they are coeval, and (2) th e m aterial out of which th ey were formed was essentially chemically homogeneous. These assum ptions, to gether w ith th e Vogt-Russell Theorem , imply th a t a t any given epoch, the num ber of stars, <f>(L)dL, in th e lum inosity interval (L, L + dL) is equal to th e num ber of stars, N ( M ) d M ) in th e corresponding mass interval ( M , M + d M ) . T he quantity <f>(L) is called th e luminosity function (LF), and the quantity N { M ) is called the initial
mass function (IM F)1. Formally, the equality is expressed as
<f>(L)dL = N ( M ) d M , (2 - 1)
w here L = L ( M , t ) , and N ( M ) oc is th e form usually adopted for th e IM F. T h e LF m ay th en be expressed as
=
(2 - 2)
an d it rem ains to find a suitable expression for th e derivative d M f d L which can be evaluated from th e theoretical evolutionary tracks and isochrones.
1 O bservations will sam ple th e present mass function (P M F ). How well it m atches th e IM F depends o n th e dynam ical evolution of th e cluster.
