For the well observed 16 globular clusters with known metal abundance (Z), the helium abundances (Y) and ages are determined by various methods, and the relations between Y, Z and age are examined. The luminosity L R R of RR Lyrae stars is known to be dependent of evolutionary models and pulsation theory in the sense that the pulsation theory and horizontal branch (HB) models yield the anticorrelation between L R R and Z whereas main sequence (MS) and red giant branch (RGB) models yield the direct correlation between them. Similarly the anticorrelation between Y and Z is obtained from the HB models and pulsation theory whereas the direct correlation between them is obtained when the RGB model is applied. The current evolutionary models yield the anticorrelation between Z and age of clusters whenever the direct correlation between Y and Z holds. However when the anticorrelation between Y and Z is applied for age determination, the similar age of clusters is obtained as shown by Sandage (1982b). The ages, which are determined by the fitting of C-M diagrams to isochrones in the ( M v , B-V)-plane, suggest the two different chemical enrichment processes, which could be accounted for by the disk-halo model for the chemical evolution of the Galaxy (Lee and Ann 1981). Also it is known that the R-method is very useful for Y-determination and the derived Y's show the increasing rate of Δ Y Δ Z ≃ 0.5 which is comparable to the observed value of Δ Y Δ Z ≃ 0.3 from HII regions and planetary nebulae by Peimbert and Torres-Peimbert (1976). In this case, the age-metallicity relation of globular clusters could be explained by the disk-halo model.

The well observed 8 open clusters, NGC 6530, 2264, 654, 129, 2168, Pleiades, Praesepe and Hyades were selected on the basis of photometric observation and proper motion study. The luminosity functions (LF) and mass functions (MF) of these clusters are different with cluster age and they could be divided into three age groups (t< 10 7 yrs, 10 7 10 8 yrs, 10 8 10 9 yrs). From these LF's and MF's, the mean LF and MF of the open clusters are derived and these functions suggest the time-dependent initial mass function (IMF) and the variation of observed MF with cluster age.

According to the star formation rate and metal enrichment rate given by the disk-halo model of Lee and Ann (1981), the two different forms of time-dependent initial mass function (IMF) and the present day mass function (PDMF) of nearby stars have been examined. It was shown that the constraint for the initial rapid metal enrichment requires the time-dependence of IMF at the very early phase ( t ≲ 5 × 10 8 yrs) of the solar neighborhood. The computed PDMF's show that the PDMF is nearly independent of any specific functional form of IMF as long as the latter includes a Gaussian distribution of log m. This result is due to the very small fractional mass ( × 5 of stars formed at the very early period during which the IMF is time-dependent. The computed PDMF suggests the presence of more numerous low mass stars than shown in Miller and Scalo's (1979) PDMF, supporting the possibility of the existence of low-velocity M dwarfs. According to the number distribution of stars with respect to [Fe/H], the mean age of these low mass star must be very old so as to yield the mean metal abundance ¯ [ F e / H ] ≈ − 0.15 for the stars in the solar neighborhood.

For ∼ 240 nearby stars their age and mass were determined and kinematic parameters determined for 362 stars, applying Woolley's three-dimensional potential. Metallicity and kinematic parameters of these stars were correlated with their age, suggesting the slow collapse ( t ≳ a few billion years) of the Galaxy and the initial rapid enrichment in metal abundance ( Δ Z ≈ 1 / 3 Z 1 (present) for ∼ 4 × 10 8 yrs). The late slow enrichment rate is given by d ( Z / Z ⊙ ) / d t = 5.9 ∼ 7.0 ± 3.4 per Gyr.

Defining a metal parameter ( S p ) c , which is related to the morphological parameters of C-M diagrams, we have estimated metal abundances for 97 globular clusters in our Galaxy. A correlation between absolute magnitude of the horizontal branch and metal abundance is derived, which is used for the determining distances to globular clusters whose visual magnitudes of the horizontal branch are known. The space distribution of globular clusters and the chemical evolution of the halo are examined. Our analysis suggests an initial mean gradient of metallicity to be d[Fe/H]/ d r G = -0.06 k p c − 1 for the halo in galactocentric distance, r G <20 kpc. Our findings also imply a slow collapse of protogalaxy.

From B ¨ o h m -Vitense's atmospheric model calculations, the relations, [ T e , (B-V)] and [B.C, (B-V)] with respect to heavy element abundance were obtained. Using these relations and evolutionary model calculations of Rood, and Sweigart and Gross, analytic expressions for some physical parameters relating to the C-M diagrams of globular clusters were derived, and they were applied to 21 globular clusters with observed transition periods of RR Lyrae variables. More than 20 different parameters were examined for each globular cluster. The derived ranges of some basic parameters are as follows; Y = 0.21 ∼ 0.33 , Z = 1.5 × 10 − 4 ∼ 4.5 × 10 − 3 , a g e , t = 9.5 ∼ 19 × 10 9 years, mass for red giants, m R G = 0.74 m ⊙ ∼ 0.91 m ⊙ , mass for RR Lyrae stars, m R R = 0.59 m ⊙ ∼ 0.75 m ⊙ , the visual magnitude difference between the turnoff point and the horizontal branch (HB), ${\Delta}V_{to}=3.1{\sim}3.4(<{\Delta}V_{to}>=3.32)$, the color of the blue edge of RR Lyrae gap, $(B-V)_{BE}=0.17{\sim}0.21=(<(B-V)_{BE}>=0.18),\;[\frac{m}{L}]_{RR}=-1.7{\sim}-1.9$, mass difference of m R R relative to m R G , ( m R G − m R R ) / m R G = 0.0 ∼ 0.39 . It was found that the ranges of derived parameters agree reasonably well with the observed ones and those estimated by others. Some important results obtained herein can be summarized as follows; (i) There are considerable variations in the initial helium abundance and in age of globular clusters. (ii) The radial gradient of heavy element abundance does exist for globular clusters as shown by Janes for field stars and open clusters. (iii) The helium abundance seems to have been increased with age by massive star evolution after a considerable amount (Y>0.2) of helium had been attained by the Big-Bang nucleosynthesis, but there is not seen a radial gradient of helium abundance. (iv) A considerable amount of heavy elements ( Z ∼ 10 − 3 ) might have been formed in the inner halo ( r G C <10 kpc) from the earliest galactic co1lapse, and then the heavy element abundance has been slowly enriched towards the galactic center and disk, establishing the radial gradient of heavy element abundance. (v) The final galactic disk formation might have taken much longer by about a half of the galactic age than the halo formation, supporting a slow, inhomogeneous co1lapse model of Larson. (vi) Of the three principal parameters controlling the morphology of C-M diagrams, it was found that the first parameter is heavy clement abundance, the second age and the third helium abundance. (vii) The globular clusters can be divided into three different groups, AI, BI and CII according to Z, Y an d age as well as Dickens' HB types. BI group clusters of HB types 4 and 5 like M 3 and NGC 7006 are the oldest and have the lowest helium abundance of the three groups. And also they appear in the inner halo. On the other hand, the youngest AI clusters have the highest Z and Y, and appear in the innermost halo region and in the disk. (viii) From the result of the clean separations of the clusters into three groups, a three dimensional classification with three parameters, Z, Y and age is prsented. (ix) The anomalous C-M diagrams can be expalined in terms of the three principal parameters. That is, the anomaly of NGC 362 and NGC 7006 is accounted for by the smaller age of the order of 1 ∼ 2 × 10 9 years rather than by the helium abundance difference, compared with M 3. (x) The difference in two Oosterhoff types I and II can be explained in terms of the mean mass difference of RR Lyrae variables rather than in terms of the helium abundance difference as suggested by Stobie. The mean mass of the variables in Oosterhoff type I clusters is smaller by 0.074 m ⊙ which is exactly consistent with Rood's estimate. Since it was found that the mean mass of RR Lyrae stars increases with decreasing Z, the two Oosterhoff types can be explained substantially by the metal abundance difference; the type II has Z< 3.4 × 10 − 4 , and the type I has higher Z than the type II.

The four dimensional classification of globular dusters with the parameters, Z, Y, age and HB type is presented defining two new parameters. ( B − V ) 1 / 2 a n d S 3 / 2 which are shown to be tightly correlated with Kinman's spectral types and the helium abundances obtained from the R-method, respectively. The Z- and Y- abundances are derived from ( B − V ) 1 / 2 a n d S 3 / 2 , respectively, and the latter parameters determine the age class of clusters with help of Dickens' HB type, which is a function of Z. Y and age. For the examined forty two globular clusters the computed range at Z and Y are 1.5×10 -4 ≤ Z ≤ 4.5×10 -2 and 0.23 ≤ Y ≤ 0.41. The age difference between the oldest (HB type 1) and the youngest (HB type 7) clusters is roughly estimated to be 2 − 4 × 10 9 years. Using these four parameters the known anomalous C-M diagrams seem to be reasonably interpreted without taking into account some complicate parameters such as unusually overabundant heavy elements, mass loss and mass spread, etc. The four dimensional scheme strongly suggests the slow successive collapses of the proto-Galaxy rather than a single fast collapse, and by this slow collapse model the inversion of chemical abundance gradient in the Galaxy can be explained. It is also shown that the clump position along the RGB near the HB level removes down to the fainter magnitude as the Z(Y)- abundance increases (decreases).

To reduce the instrumental and calibration errors in the photoelectric photometry as much as possible it is necessary to select the optimum photocell voltage and energy attenaution and to observe as many standard stars as possible over the wide range of color, spectral type and air mass.