Introduction

The Magellanic Clouds, and the numerous rich stellar clusters located within them, are a largely untapped avenue of research. At the time of commencement of this thesis, relatively few regions of the Magellanic Clouds had been studied in depth with regard to finding variable stars. In particular, virtually all the Cepheid variables known in the Magellanic Clouds were discovered from photographic plates taken in the early part of this century. This technique of discovery is restricted to finding variables of relatively large amplitude ( $\raisebox{-0.6ex}{$\,\stackrel {\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}$0.5 magnitudes), those which are bright and blue, and lying in the less crowded areas of the Magellanic Clouds (Payne-Gaposchkin and Gaposchkin 1966, Payne-Gaposchkin 1974). The more recent wide-area surveys for Long-Period Variables (LPVs) (e.g. Hughes 1989) were also carried out using photographic plates. Although large numbers of variables have been found with such surveys by virtue of the large areas of sky covered, it is expected that they are seriously incomplete at fainter magnitudes, lower amplitudes, and in areas which are extremely crowded, such as near to the rich clusters of the Magellanic Clouds. Modern CCD photometry, although limited in the area of sky able to be covered, is capable of extending the completeness down to much lower luminosities and amplitudes, and is much better able to probe the crowded environs of the Magellanic Cloud (MC) Clusters. If variable stars associated with MC Clusters can be found, many questions associated with various aspects of stellar evolution may be addressed.

Observations of MC clusters offer numerous advantages over observations of galactic clusters. There is both a large number and a wide variety of clusters in the Magellanic Clouds, with ages ranging from 10⁷ years (NGC 346, NGC 2004) to around 10¹⁰ years (NGC 121, NGC 1841, NGC 2257). The young MC clusters are much richer than galactic clusters of the same age. This allows better populated colour-magnitude diagrams to be constructed, especially in the vitally important evolved regions such as the Cepheid instability strip and the giant branches. The stars in the MC clusters are located at a distance which is reasonably well known, yet far enough away so as to make depth effects small. They are still close enough so the clusters are relatively bright, and accurate photometry may be comfortably performed with modest equipment. Both the Magellanic Clouds fortuitously lie well away from the obscuring matter near the Milky Way, so reddening corrections are much smaller and more certain than for the young clusters of our galaxy, the majority of which lie at low galactic latitudes.

Several outstanding problems in stellar astronomy can be investigated through the study of the rich clusters of the Magellanic Clouds. The initial aims of this thesis were to address two such problems: to investigate the Cepheid mass discrepancy, and to study the initial masses of Long-Period Variables (LPVs) of various periods.

The Cepheid mass discrepancy is the disagreement between Cepheids masses based on evolutionary considerations and masses derived from pulsation calculations (Section 1.1.3). The main cause of this discrepancy is thought to be uncertainties in the evolutionary models, relating to the treatment of convective core overshoot during the main-sequence evolution of the Cepheids. This is described in Section 1.1.2.

The masses of LPVs are not well-determined, in part because there are very few LPVs known to be associated with clusters, and also because there are few LPVs with reliable distance measurements. The discovery of LPVs associated with the MC Clusters would allow ages and masses to be derived. This is described in Section 1.2.

Cepheid Variables

Pulsation Masses

The observed properties of any discovered Cepheids (period, luminosity, colour) may be used to determine the pulsation mass M_puls. Pulsation theory states that, to first order, the so-called pulsation constant Q should be the same for all Cepheids, where Q is given by

$$\sqrt{{\frac{\overline{\rho}}{\rho_\odot}}} \sqrt{{\frac{(M/M_\odot)} {(R\;/\;R_\odot)^{3}}}}$$ (1.1)

From the definition of T_eff, the luminosity of a star is

L = 4πσR²T_eff⁴. (1.2)

Substituting for R in ( ) from () yields the P-L-M-T relation:

M∝L^3/2T^-6P^-2. (1.3)

In practice, the exponents in equation (1.3) are not the canonical values shown. Appropriate values are determined by making many linear pulsation models of a given type of star, say Cepheids, with varying M, P and T.

In terms of observational parameters, where T_eff may be converted to colour (V-I), and L is determined from the observed magnitude, equation (1.3) may be rewritten as M_puls = M(M_V, P,  V-I). The usual form of this equation is:

$${\frac{{M\;\;}}{{M_{\odot}}}}$$ (1.4)

where A, B, C and D are constants. The pulsation mass is independent of any assumptions regarding the amount of convective core overshoot during main-sequence evolution. It should be noted that for commonly used passbands (eg. B-V, V-I), the magnitude of the C coefficient in equation (1.4) is large enough (typically ∼1.5) that the derived pulsation masses are fairly sensitive to colour.

Evolution Masses

The Cepheid evolution mass is derived from stellar evolution models, and is simply the mass of a model star as it passes through the same region of the Cepheid instability strip as the real Cepheid. Naturally, the mass derived is dependent upon the input physics of evolutionary models used.

If a Cepheid is known to be a member of a cluster, then isochrones may be fitted to the cluster CMD, and the Cepheid mass may then be determined from the cluster turn-off mass. If the Cepheid is not a cluster member, but an isolated field Cepheid, then the evolution mass may be inferred by assuming an appropriate distance modulus, and finding an isochrone of appropriate mass which lies at the same luminosity as the Cepheid during the core-helium burning phase of evolution. It is when the star is passing along these core-helium burning loops that it passes through the Cepheid instability strip.

Cepheid evolution masses are probably much less well-determined than pulsation masses. The uncertainties are mainly due to a poor knowledge of the amount of convective overshoot which occurs at the edge of the convective core during main-sequence evolution of stars. Such overshoot only occurs in stars with a mass greater than ∼1.5M_$\odot$. For a star of a given mass, an increase in the amount of overshoot increases the luminosity at which the core helium burning loops occur. Equivalently, increasing the degree of convective core overshoot in the models leads to lower derived evolution masses for a given Cepheid luminosity. Thus, the evolution mass derived for a star of a given luminosity is dependent on the degree of convective overshoot. It should also be noted that the degree of convective core overshoot itself may not be constant for all Cepheids, and may be a function of mass. The study of Cepheids of a wide range of masses is required to shed light on this possibility.

The Cepheid Mass Discrepancy

Cepheids which are members of the rich clusters of the Magellanic Clouds may be used to help address the so-called mass discrepancy problem of Cepheids (Cox 1980), namely that Cepheid masses as derived from pulsation models (pulsation masses) are systematically different from masses derived from evolutionary models (evolution masses). Historically, the pulsation masses derived have been up to 50% smaller than the evolutionary masses, although the size of the discrepancy depends on the input physics. The advent of new stellar opacities has reduced, but not eliminated this discrepancy.

The observational properties of a Cepheid at a known distance (the LMC or SMC distance in the present case) uniquely determine the pulsation and evolution masses. Thus by comparing the pulsation mass to evolution masses obtained from models using differing amounts of convective core overshoot, it should be possible to determine the degree of convective core overshoot required to equate the two masses. Some work on this problem has already been done by Mateo et al. (1990a), Welch et al. (1991) and Chiosi et al. (1992).

Once the amount of convective core overshoot is known, some useful theoretical predictions can be made. In the past, the theoretical prediction of the Cepheid P-L and P-L-C relations have not been possible, due to the uncertainties in the Cepheid Mass-Luminosity relation caused by the unknown amount of convective core overshoot in the evolutionary models. If the amount of convective core overshoot can be determined, it becomes possible to derive these relations purely from theory. This would be a comforting result, as these relations are a cornerstone for the astronomical distance scale.

LPVs

The Long-Period Variables (LPVs) are another group of stars that could benefit from the study of Magellanic Cloud clusters. Relatively little is known about the precursors of LPVs, and the masses and luminosities are not well determined. This is due to the fact that virtually all galactic LPVs are isolated members of the general field. Studies of galactic kinematics of LPVs (Feast 1963) show that they evolve from stars with masses of typically 1.2M_$\odot$. A few short-period LPVs are known to be members of old Galactic globular clusters, and a few supergiant examples are known in very young Galactic open clusters. However, a literature search has revealed no large-amplitude LPVs known to belong to clusters with a turn-off mass in the range 1 - 9M_$\odot$. It is this range of stellar masses that mainly gives rise to the high-mass-loss-rate AGB stars such as Mira variables, OH/IR stars and dusty carbon stars. These stars are major contributors to mass-return back to the interstellar medium (Schild 1989).

Determination of pulsation masses for LPVs is rather uncertain. Firstly, at least in the Galaxy, it is not easy to determine their luminosity due to the lack of a reliable distance estimate. Secondly, effective temperatures for these objects are uncertain, as colour-T_eff relations are not well determined. The colour-T_eff relations are the link between observation and theoretical models. An observationally determined colour must be transformed into an effective temperature in order to determine the theoretical pulsation mass from the P-L-M-T_eff relation. A practical result which would result from finding LPVs in Magellanic Cloud clusters would be the determination of the mass of stars giving rise to LPVs of a given period. A more ambitious project would be a determination of pulsation mass and comparison with turn-off mass. If the pulsation mass and the turn-off mass are in agreement, it would give us confidence that the colour-T_eff relation was correct. For the more evolved (ie, longer period, more dusty) stars, it would be expected that the pulsation mass would be lower than the turn-off mass, due to the high mass loss rate of these stars. This mass difference would provide a direct measure of the accumulated mass loss.

It has turned out that one of the initial aims of this thesis (determination of masses of LPVs) has not been achieved, due to the absence of any LPVs definitely associated with the clusters analysed - all the LPVs discovered appear to be members of the old field population of the LMC. However, during the course of this thesis, a number of low-amplitude and shorter-period LPVs were identified. This unexpected discovery has led to a possible new method for the determination of the pulsation modes of LPVs. This work is described in Chapter 5.

Cluster Selection

In order to maximise the chances of finding cluster Cepheids, the ages of a number of clusters with known Cepheids were compared, with the aim of determining the age at which Cepheids are likely to appear in clusters. For this application, the Elson and Fall ``s'' parameter (Elson and Fall 1985; Elson and Fall 1988) for MC clusters is an ideal measure of the (relative) age of a cluster. The s parameter is an index derived from integrated UBV photometry of a cluster, and it is closely correlated with the cluster age.

Figure 1.1: The number of known Cepheids in various MC clusters. The points are labelled with the NGC number of each cluster. The x-axis is the Elson and Fall ``s'' parameter. The upper panel is the raw number of Cepheids in each cluster, and the lower panel is the number of Cepheids normalised by the visual luminosity L_V of the cluster. The objects plotted as circles are the clusters studied in this thesis.

The upper panel of Figure 1.1 shows the number of Cepheids known to exist in a sample of MC clusters. This figure is based primarily on the data listed in Mateo et al. (1990b), supplemented by data from the literature for a number of other clusters, and includes the clusters analysed in this thesis. It is likely that the sample of clusters in Mateo et al. (1990b) were chosen based on a procedure similar to this.

The upper panel of Figure 1.1 clearly shows a strong peak in the number of Cepheids in clusters around ages s=26-27. The number of Cepheids in a cluster will of course be dependent on the luminosity of the cluster - rich clusters are likely to have more Cepheids than poorer clusters of the same age. The lower panel in Figure 1.1 shows the number of Cepheids in each cluster normalised by the luminosity of the cluster (relative to the luminosity of NGC 1866). The luminosities are the integrated V magnitude, taken from Bica et al. (1996). This figure confirms that there is a peak in the number of Cepheids at ages s=26-27 that is independent of the luminosities of the clusters, but also indicates that clusters of the same age and luminosity can have very different numbers of Cepheids, or even no Cepheids at all (NGC 2025). The true age at which the Cepheid peak occurs is 1.0-2.0×10⁸ years, corresponding to a (turn-off) Cepheid mass of 4.0-5.0M_$\odot$(Bertelli et al. 1993).

For this thesis study, a list of candidate clusters with ages s∼25-27 was made. The integrated cluster luminosity from van den Bergh (1981) was used to refine this list to the more luminous clusters. The Magellanic Cloud fields which have been analysed in this thesis are listed in Table 1.1, and plotted as open circles in Figure 1.1. Note that the smaller clusters NGC 2057, 2059 and 2066 lie in the same field as NGC 2058/65.

Table 1.1: Target Magellanic Cloud clusters

Field	R.A. J2000.0 Dec.		s
NGC 330	00h56m16s	-72o28 .′8	19
NGC 1850	05  08  44	-68 45.5	21
NGC 2058/2065	05  37  21	-70 10.9	26

Thesis outline

Chapter 2 presents the results for NGC 330, one of the brightest young SMC clusters. The results for NGC 1850, a LMC cluster previously known to have a Cepheid in the cluster core, are presented in Chapter 3. Results for the poorly studied field near NGC 2058 and NGC 2065 (lying just 5^′ apart) are presented in Chapter 4. Data have been obtained for several other MC cluster fields but these have not yet been analysed.

In Chapter 5, the optical and IR photometry of the LPVs discovered in Chapters 3 and 4 is combined, and used to investigate the mode of pulsation of the LPVs of the LMC.

Finally, in Chapter 6, a summary of the results of each of the chapters is given, as well as a brief outline of further research required to help clarify the findings of this thesis. Other related avenues for research are also presented.