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A search for faint variable stars in the globular cluster M71 Hodder, Philip Jeremy Crichton 1990

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A S E A R C H F O R F A I N T V A R I A B L E S T A R S IN T H E G L O B U L A R C L U S T E R M71 by P H I L I P J E R E M Y C R I C H T O N H O D D E R B . S c , University of St. Andrews, 1988 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Department of Geophysics and Astronomy We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A September 1990 © P h i l i p Jeremy Crichton Hodder, 1990. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Geophysics and Astronomy The University of British Columbia Vancouver, Canada Date 20 September 1990  DE-6 (2/88) Abstract A 67" x 104" area of the metal-rich globular cluster M71 was searched for variable stars using 73 C C D frames. Using mean B and V values a colour-magnitude diagram down to V FH 22 is constructed. Four variables were discovered, with two more stars classed as possible candidates for variability. Phase diagrams and real time light curves are presented for al l variables. One variable blue straggler (or S X Phe star) has been discovered with a period of 0.d05181. Values for the mass depend on the pulsation mode assumed for this star - (0.90 db 0.13).A4© for the first overtone mode, and (1.57 ± 0.22)A4© for the fundamental mode. A second variable, of similar period (0.d06053), but with a magnitude l . m 75 below the main sequence turn off was also found but it may be a field star. Two candidate eclipsing binary systems were found. The most likely period of one is 0.d37244. This value, and the shape of the light curve, suggest it may be a W U M a type variable. Its position on the C M D suggests that it too may be a field star. No period was obtainable for the other candidate binary due to a lack of phase coverage. Further data is needed to confirm and strengthen these claims. n Contents Abstract ii List of Tables v List of Figures vi Acknowledgements viii 1 Introduction 1 1.1 Globular Clusters and Variable Stars 1 1.2 The Globular Cluster M71 ( N G C 6838) 5 2 Description of the Data 8 3 Data Reduction 13 3.1 D A O P H O T 13 3.1.1 S K Y , F I N D and P H O T 13 3.1.2 P S F 14 3.1.3 G R O U P , N S T A R and S U B S T A R 14 3.1.4 Comments on the data reduction 15 3.2 Matching the frames 15 3.2.1 A U T O M M A T C H 15 3.3 Calibrat ion 17 i n 3.3.1 Pr imary Standards and Colour Coefficients 18 3.3.2 Secondary Standards and Zero Points 22 3.3.3 A Discussion of Errors 26 4 Results and Discussion 29 4.1 Photometry 29 4.2 A Colour - Magnitude Diagram for M71 29 4.2.1 A M a i n Sequence Ridge Line for M71 30 4.3 Variable stars 33 4.4 Period Analysis 35 4.5 Description of Variable Stars 36 4.5.1 Variable H I 36 4.5.2 Variable H2 43 4.5.3 Variable H3 48 4.5.4 Variable H4 52 4.5.5 The Candidate Variables H5 and H6 52 4.5.6 Comparison Stars and Photometry 54 4.5.7 A Finding Chart 54 5 Conclusions 65 References 67 i v List of Tables 1.1 Da ta for the globular cluster M71 ( N G C 6838) 7 2.1 Log of observations for 1985 Ju ly 12/13 9 2.2 Log of observations for 1985 July 13/14 10 2.3 Log of observations for 1985 July 14/15 11 2.4 Log of observations for 1985 July 15/16 12 3.1 Pr imary Standards 19 3.2 Secondary standards 24 4.1 F iduc ia l Ridge Line for M71 32 4.2 Summary of Variable Star Properties 55 4.3 Photometry for H I , H2 , H3 and H4 56 4.3 Photometry for H I , H2 , H3 and H4 57 4.3 Photometry for H I , H2, H3 and H4 58 v List of Figures 3.1 Pr imary standard calibration 20 3.2 Secondary standard (zero point) calibration 25 3.3 Plot of e and a against V magnitude 28 4.1 M71 Colour - Magnitude Diagram 31 4.2 Plot of sv against V 34 4.3 Real time light curve a,nd amplitude spectrum for H I 38 4.4 Ampl i tude spectrum and phased light curve for H I . ' 39 4.5 Window function and residual for H I 40 4.6 P - L relation for cluster S X Phe stars 42 4.7 Real time light curve and amplitude spectrum for H2 44 4.8 Ampl i tude spectrum and phase diagram for H2 45 4.9 Window function and residual for H2 46 4.10 Real time light curve and amplitude spectrum for H3 49 4.11 Ampl i tude spectrum and phase diagram for H3 50 4.12 Window function and residual for H3 51 4.13 Phase diagrams for H3 53 4.14 Real time light curve for H4 59 4.15 Real time light curve for H5 and H6 60 4.16 Real time light curves for C l and C2 61 4.17 Real time light curves for C3 and C4 62 vi 4.18 Real time light curves for C5 and C6 4.19 A finding chart for variables H I to H6 Acknowledgments This thesis would not have been possible without the help of many people. I would like to thank Dr . J . M . Nemec and Dr . H . B . Richer, my supervisors for this thesis, for their patient guidance and continued interest in what I was doing. Together with Dr . Richer, I would also like to thank Dr . G . G . Fahlman for obtaining the data and letting me use it (and the matching program). Gerry Grieve and Jaymie Matthews (Drs. both) provided invaluable advice on al l manner of subjects. Thanks also to Dr . Jason "Big Boots" A u m a n for reading this "milk toast" thesis. Thanks to B r a d and Ted (Stooges 2 and 1) and to everyone else in the Deptartment (see Brad's thesis for a complete list!). Last ly I would like to thank my parents, and the rest of my family, for their continued love and support, without which it would have al l been very much harder. cat the-words > / d e v / n u l l v i i Chapter 1 Introduction 1.1 G l o b u l a r C l u s t e r s a n d V a r i a b l e S t a r s The study of globular clusters is intimately concerned with the creation and fate of the Universe. Globular clusters are amongst the oldest objects in the Universe - age estimates range from 12 to 18 Gyr - and, due to the relative proximity of our Galaxy's population of globular clusters, are among the best studied objects outside the Galactic disc. The study of globular clusters mainly confines itself to the production of colour - magnitude diagrams ( C M D s ) , dynamical modelling, production of mass and luminosity functions and the study of variable stars in the cluster. The study of colour - magnitude diagrams has benefited enormously over the past few years by the introduction of C C D detectors and the development of the computer software required to properly analyse the data. As a result, age estimates, as determined by isochrone fitting, have become much more certain. Such fitting can now be attempted down to approximately 2 3 r d magnitude wi th ease, compared to the more time consuming and less accurate methods of photographic and photoelectric photometry. Dynamical modelling of globular clusters is an extremely involved subject. There are usually only a few data points in the observed surface brightness and velocity dispersion profiles, which must then be fitted by models which involve many unknown parameters. Despite this "uniqueness" problem, a great deal of progress has been made in this field, 1 ini t ial ly using K i n g models ( K i n g 1966), and more recently using the Fokker - Planck equation (e.g. Spitzer 1987, Drukier et al. 1991). The production of luminosity functions, and hence of mass functions, can be accom-plished by counting stars in various magnitude bins and then applying various geometrical and statistical corrections. Such studies (e.g. Fahlman et al. 1989) seem to indicate the presence of "brown dwarfs", one of the prime candidates for galactic dark matter. The study of variable stars in globular clusters goes back as far as 1890 (Pickering 1890), followed rapidly by the discovery of many more variables and also by increased insight into the nature of stellar pulsations in general. Perhaps one of the most signifi-cant developments in this subject was the discovery of the period - luminosity (P - L) relationship for the S Cephei class of variables (Leavitt 1912) followed later by the recog-nit ion of two populations of Cepheids (see Baade 1952) leading to the doubling of the extra-galactic distance scale. The original direction of the present work was to systematically examine some blue stragglers (BSs) in the globular cluster M71 for variability. A short discussion of blue stragglers, and of variability i n them is therefore in order. The discovery of the first "blue stragglers" (a term coined by Burbidge and Sandage 1959) in a globular cluster was due to Sandage (1953), when he obtained a C M D for the globular cluster M 3 . One group of stars on the diagram appeared to be lying above the main sequence turn off - that is, they had not evolved with the rest of the cluster. Many other blue stragglers have since been discovered (see Table 1 of Nemec 1989) and large numbers have been found in the nearby dwarf spheroidal galaxies in Draco (Carney and Seitzer 1986), Sculptor (Da Costa 1984), Ursa Minor (Olszewski and Aaronson 1985) and possibly, Carina, (Mould and Aaronson 1983). One of the first explanations put forward for the presence of blue stragglers was that they are indeed stars that have been formed comparatively recently, well after the majority 2 of the cluster members. However the idea of continuing episodes of star formation in globular clusters is hard to support, principally owing to the fact that any residual gas left over from the first batch would have been swept out of the cluster by stellar winds from any rapidly evolving, massive stars. Eggen and Iben (1989) conclude that while it may be possible to have bursts of star formation in old disk and halo objects, there are more viable alternatives. For young disk population objects a "delayed formation" hypothesis is quite tenable (Eggen and Iben 1988). One of the first alternate suggestions was that blue stragglers have extended main se-quence lifetimes because of mixing (as mentioned in M c C r e a 1964), either due to magnetic fields, rotation (Saio and Wheeler 1980, A b t 1985) or just strong convection. Hydrogen from the envelope would be brought down into the core to extend the period of hydrogen burning. A significant problem wi th this kind of internal mixing is that it would have to be rather arbitrarily applied to only a small fraction of the stars in the cluster. M c C r e a (1964) mentions mixing almost as an adjunct to the hypothesis that blue stragglers are, in fact, relatively massive binary systems in which an appreciable amount of mass transfer has occurred from one component to the other. Renzini , Mengel and Sweigart (1977) demonstrate that the formation of the so called "anomalous Cepheids" - that is, Cepheids that are too bright for their periods - can occur via mass transfer in binary systems, if the stars remain within certain mass l imits. They show that the interaction of the binary system wi th the rest of the cluster may eventually cause the two stars to merge together and predicts that the relative numbers of blue stragglers to anomalous Cepheids should be greater than the ratio of their respective lifetimes. This kind of scenario can also explain the existence of blue stragglers, as has been demon-strated by, for example, Webbink (1976), Mateo et al. (1990),and Eggen and Iben (1989). However a suitably efficient mechanism for the transport of angular momentum out of the system needs to be present. In this theory a blue straggler can form if a detached binary 3 system wi th a period of from 2 to 5 days becomes, v ia angular momentum loss, a contact binary. From there it can merge into a single star. The most probable mechanism for this loss of angular momentum is a magnetic stellar wind, which implies that stars more massive than 2 to 3A4@ (and which do not have a deep enough convection zone to support a magnetic stellar wind) cannot merge by this process unless the companion does all the work (and is of low enough mass). Systems with periods greater than about 5 days are unlikely to merge due to the time scales involved (Eggen and Iben 1988). This provides an important observational test of this theory. Renzini , Mengel and Sweigart (1977) demonstrate that blue stragglers should be found preferentially in envi-ronments of low stellar density (because systems of a high stellar density wi l l disrupt the binary before merging can occur) - a fact that seems to have been confirmed by Nemec and Harris (1987), though the data is slightly incomplete due to image crowding. Nemec and Harris have also computed the mean mass of the blue stragglers in N G C 5466 to be ~ 1.2 A4Q. Nemec and Cohen (1989) later found that the more luminous blue stragglers of N G C 5053 are more centrally concentrated than the less luminous ones. Approximately 30% of the blue stragglers found in N G C 5466, N G C 5053 and u> Cen have been shown to exhibit some kind of photometric variability (see Nemec 1989). Most of this sample (about 20 stars) are S X Phe variables (that is, population II dwarf Cepheids) wi th periods of between 0.03 and 0.06 days. In every respect they are similar to the S X Phe stars in the general field of the Galaxy (such as S X Phe itself, and, for example, B L C a m and K Z Hya). It also appears that fundamental and first overtone modes of pulsation in S X Phe stars are identifiable from different P-L relations and light curve morphologies. The derived masses of a l l known S X Phe stars are in the range ~ 1.0 to ~ 1 .2A^0 which is consistent with the coalescence of a binary system. However different formation mechanisms may be at work in different clusters, or even in the same cluster. In addition to the S X Phe stars, three contact binaries have been found amongst the 4 blue stragglers in N G C 5466 (Mateo et al. 1990) and a wide ( l . d 4 period) eclipsing binary in the blue straggler region of the iv Cen colour - magnitude diagram has recently been shown to be a member of that cluster (Jensen and J0rgensen 1985, Margon and Cannon 1989). These observations of binary systems among the blue stragglers provide direct evidence for the binary hypothesis. In addition the similarity between the N G C 5466 contact ( W U M a type) binaries and those found in M67 and N G C 188 (Kaluzny and Shara 1987) suggests that variable stars might also be present among the upper main sequence stars in open clusters. 1.2 T h e G l o b u l a r C l u s t e r M 7 1 ( N G C 6 8 3 8 ) The first major modern work on M71 is that of A r p and Hartwick (1971), conducted using photoelectric and photographic observations. They note the existence of a small number (less than 10) of blue stragglers, and derive an age estimate, reddening, metallicity and distance. A more modern survey of the cluster is that by Cudworth (1985), who sup-plies photographic photometry and membership probabilities for most apparent members brighter than V ~ 17. Cudworth's magnitudes and colours were used to calibrate the data for the present work. Richer and Fahlman (1988) used deep C C D photometry over a large (6' x 4') area to identify both a sequence of D B white dwarfs and over 50 blue stragglers; as well they derived better estimates of the foreground reddening and distance modulus (see Table 1.1). A further study (Richer and Fahlman 1989) investigated both the mass and luminosity functions of M71 using the same data, and concluded that this cluster may contain a substantial number of low mass stars (between 50% and 90% may be less massive than O . 3 3 A 4 0 ) ; that the global mass function cannot be well fitted by a power-law; and that multimass K i n g models can model the inner part of the cluster very well indeed. Al len and Martos (1988) show, by performing numerical integrations of the cluster's 5 orbit backwards in time, that M71 has a very similar galactic orbit to the open cluster M67. However it should be born in mind that the errors in the cluster proper motion derived by Cudworth (1985) are quite large, almost at the 50% level. In fact, in plots of metal abundance versus apocentric distance, pericentric distance and orbital eccentricity, M71 always groups with the galactic clusters in their sample. This tends to strengthen the idea that M71 is a disk population object i n some respects. Sawyer Hogg (1973) catalogues four variable stars in M71 , including an eclipsing bi-nary. However Lil ler and Tokarz (1981) show, on the basis of its radial velocity, that the latter object is a field star. Cudworth (1985) notes that two other variables may also be present using data from that study in addition a study by Welty (private communication to Cudworth) and one by Frogel et al. (1979) who cite Sawyer Hogg (private communica-tion to Frogel et ah). A l l these variables lie near the tip of the red giant branch, and none are within the field studied in this survey. Table 1.1 (below) summarizes some pertinent information about M71 . The ini t ia l objective of this study, the preliminary results of which have been discussed in Hodder et al. (1990), was to search for variable blue stragglers in M 7 1 . However, the method employed in the data analysis enabled any type of variable with a period from approximately 30 minutes to 7 hours to be searched for. Chapter 2 contains a detailed discussion of the observations themselves. Chapter 3 is concerned with the data analysis and calibration procedures employed. Chapter 4 discusses the results of the study and presents a table of photometry for four of the variables discovered, as well as a finding chart for them. Chapter 5 makes some concluding remarks about the work. 6 Table 1.1: Data for the globular cluster M71 ( N G C 6838). Quantity Value Reference* Right Ascension, a i 9 5 0 1 9 h 5 1 m 3 3 s 1 Declination, ^ 950 +18° 38'47" 1 Galact ic longitude, £ 56°7 2 Galactic latitude, b — 4°5 2 Distance modulus, (rn - M)v 13.m70 3 Foreground reddening, E(B - V) 0.m28 3 Integrated absolute magnitude, My — 5.m60 2 Galactocentric distance (kpc), R 7.6 2 Radia l velocity, vr (km s - 1 ) —19 5 Metall ici ty, [Fe/H] - 0 . 6 to - 1 . 0 6 T i d a l radius (arcmin), logr* 1.1 2 Central concentration, c 1.1 4 a) (1) Cudworth (1985), (2) Harris and Racine (1979), (3) Richer and Fahlman (1988), (4) Richer and Fahlman (1989), (5) Webbink (1981), (6) Leep, Oke and Wallerstein (1987) 7 Chapter 2 Description of the Data The data upon which this work is based were taken during four nights (1985 July 12 to 15 (U.T.) ) at the Canada - France - Hawaii telescope by H . B . Richer and G . G . Fahlman. The R C A 1 chip was used at the Cassegrain focus (f/8). The R C A 1 chip is a 320 x 512, 3 phase R C A SID 501, backside il luminated buried channel C C D wi th 30 /xm square pixels. A t f/8 it therefore has a plate scale of 0."21 per pixel. A total of 87 frames of approximately the same field of M71 were taken over that period, of which 75 were reduced for this analysis. The others were not reduced owing to the poor seeing on those frames. A condensed form of the observing log is given in Tables 2.1 to 2.4. In these tables, column (1) is the name of the image frame, columns (2), (3) and (4) are the Universal T ime at the start of the exposure, the exposure time (in seconds) and the Heliocentric Jul ian Date at mid-exposure, respectively. Column (5) shows the filter used for that frame and column (6) lists the object taken on the frame. After preprocessing, the frame size was reduced to 317 x 497 pixels, or 67" x 104". Each frame was centered (to within approximately 5") on the same field, 30" N and 55" E of the centre of M 7 1 , given as a = 19 h 51 m 32. s 77, S = +18038'47."2 (1950) by Cudworth (1985). O n the third night of observation, which was photometric, frames of six Landolt (1973) standard stars were also taken. The log of these observations is included in Table 2.3. 8 Table 2.1: Log of observations for 1985 July 12/13. Frame U . T . E x p . H J D b Fil ter t ime a 2446200+ (1) (2) (3) (4) (5) 27M71 08:34 300 58.86377 V 28M71 08:42 300 58.86768 V 29M71 08:48 300 58.87158 V 30M71 08:54 300 58.87549 V 31M71 08:59 300 58.87939 V 32M71 09:05 300 58.88330 V 33M71 09:11 300 58.88721 V 34M71 09:18 300 58.89502 V 35M71 09:24 300 58.89893 V 36M71 09:30 300 58.90283 V 37M71 09:36 300 58.90674 V 38M71 09:43 300 58.91064 V 39M71 09:49 300 58.91455 V 40M71 09:55 300 58.91846 V 41M71 10:00 300 58.92236 V 42M71 10:07 300 58.92627 V 43M71 10:16 300 58.93408 V 44M71 10:46 300 58.95361 V 45M71 10:52 300 58.95752 V 46M71 10:58 300 58.96143 V 47M71 11:04 300 58.96533 V 48M71 11:11 300 58.97315 V 49M71 11:18 300 58.97705 V 50M71 12:36 600 59.03174 B 51M71 12:49 600 59.04346 B 52M71 13:00 600 59.05127 B 53M71 13:12 600 59.05909 B 54M71 13:24 600 59.06690 B 55M71 13:53 600 59.08643 B 56M71 14:04 600 59.09424 B 57M71 14:16 600 59.10206 B 58M71 14:27 600 59.10987 B Object (6) M71 Standard field M71 Standard field a) in seconds b) Heliocentric Julian Date of mid exposure 9 Table 2.2: Log of observations for 1985 July 13/14. Frame U . T . Exp . H J D a Fil ter Object t ime b 2446200+ (1) (2) (3) (4) (5) (6) 60M71 08:24 600 59.85598 B M71 Standard field 61M71 08:36 600 59.86379 B 62M71 08:57 600 59.87942 B 63M71 09:08 600 59.88723 B 64M71 09:19 600 59.89504 B 65M71 09:31 600 59.90286 B 66M71 09:42 600 59.91067 B 67M71 09:53 600 59.91848 B 68M71 10:04 600 59.92629 B 69M71 10:16 600 59.93410 B 70M71 10:28 600 59.94583 B 71M71 13:56 300 60.08645 V 72M71 14:03 300 60.09036 V M71 Standard field a) in seconds b) Heliocentric Julian Date of mid exposure 10 Table 2.3: Log of observations for 1985 July 14/15. Frame U . T . E x p . H J D b Fil ter Object t ime a 2446200+ (1) (2) (3) (4) • (5) • (6) SA108_1 05:43 20 60.74272 V SA108-719,727,728 SA108.2 05:46 20 60.74272 V SA108.3 05:49 20 60.74662 V SA108.4 05:53 20 60.75053 V SA108-5 05:55 20 60.75053 B SA108-6 05:58 20 60.75444 B SA108-7 06:03 60 60.75444 U SA108-8 06:08 300 60.75834 u SA108-719,727,728 74M71 08:15 7200 60.89116 u M71 Standard field 75M71 10:25 7200 60.98100 u 76M71 12:51 600 61.04350 B 77M71 13:02 600 61.05132 B 78M71 13:19 300 61.05913 B 79M71 13:27 300 61.06694 B M71 Standard field SA114_1 14:16 300 61.09819 U SA114-548 SA114.2 14:22 30 61.10210 B SA114.3 14:24 30 61.10601 B SA114-4 14:26 10 61.10601 V SA114-548 SA115.1 14:30 20 61.10991 V SA115-486 SA115.2 14:31 30 61.10991 B SA115.3 14:33 300 61.10991 U SA115-486 GD246.1 14:44 120 61.11772 u GD246 GD246-2 14:48 120 61.12163 u GD246.3 14:54 30 61.12554 B GD246.4 14:56 30 61.12554 V GD246 a) in seconds b) Heliocentric Julian Date of mid exposure 11 Table 2.4: Log of observations for 1985 July 15/16. Frame U . T . E x p . H J D b Fil ter Object t ime a 2446200+ (1) (2) (3) (4) (5) (6) 80M71 10:11 300 61.93024 V M71 Standard field 81M71 10:23 300 61.93805 V 82M71 10:29 300 61.94196 V 83M71 10:35 300 61.94587 V 84M71 10:43 300 61.95368 V 85M71 10:49 300 61.95759 V 86M71 10:55 300 61.96149 V 87M71 11:01 300 61.96540 V 88M71 11:07 300 61.96931 V 89M71 11:13 300 61.97321 V 90M71 11:19 300 61.97712 V 91M71 11:25 300 61.98102 V 92M71 11:34 300 61.98884 V 93M71 11:39 300 61.99274 V 94M71 11:45 300 61.99665 V 95M71 12:24 600 62.02399 B 96M71 12:34 600 62.03181 B 97M71 12:44 600 62.03962 B 98M71 12:57 600 62.04743 B 99M71 13:08 600 62.05524 B 100M71 13:19 600 62.06306 B 101M71 13:30 600 62.07087 B 102M71 13:41 600 62.07868 B 103M71 13:53 600 62.08649 B M71 Standard field a) in seconds b) Heliocentric Julian Date of mid exposure 12 Chapter 3 Data Reduction 3 . 1 D A O P H O T Each frame was analysed to give positions and magnitudes for each star using the DAOPHOT computer code (Stetson 1987). There follows a brief description of DAOPHOT for those unaquainted wi th its use. The reader is directed to the DAOPHOT User Manual , and to the aforementioned paper for a much more rigourous description. 3.1.1 SKY, FIND and P H O T After a median value of the sky "background" has been calculated (for use in various DAOPHOT routines), a list of approximate centroids of objects on the frame is obtained by convolving the image with a Gaussian function of approximately the same F W H M as the individual stellar images. Areas brighter than some threshold (supplied by the user) have shape parameters calculated for them. F I N D calculates two fitting parameters for each object by fitting Gaussians in the x and y directions to the stellar profile. These are sharp (the height of the best-fitting delta function divided by the height of the best-fitting Gaussian function) and round (the difference between the heights of the two one dimensional Gaussian divided by the average of those heights). A n y objects with these parameters outside of preset limits are not classed as stars and are rejected from the list. Concentric aperture photometry (usually, at this stage in the reduction, with just one 13 small aperture) is then performed on the star to get an approximation of the instrumental magnitude. 3.1.2 PSF It is then necessary to determine the point spread function (PSF) for the frame. P S F s can, and do, vary from frame to frame taken during a particular night, and sometimes vary across one frame as well. To calculate the P S F , a list of fairly bright but unsaturated stars is assembled. Stars with close neighbours are rejected from the list. For each star in the list the program determines the deviation between a Gaussian profile and the actual stellar image profile. As it progresses through the list of P S F stars a mean of the deviations is built up. This is then assumed to apply to a l l stars on the frame. The actual procedure is iterative, involving the subtraction of neighbouring stars to isolate the P S F star, and then the redetermination of the P S F . 3.1.3 GROUP, NSTAR and SUBSTAR Once a P S F has been constructed, the stars on the original list (made by the F I N D routine) are arranged into groups of (usually) less than 60. The purpose of this is so that the N S T A R routine can then perform simultaneous profile fitting of al l the stars in the group. This system has the advantage of automatically accounting for the intrusion of light from other, close by, stars. DAOPHOT is thus quite good at getting accurate magnitudes in fairly crowded fields. The N S T A R routine also produces a % 2 statistic for each star to indicate how well it has been fitted to the "Gaussian plus P S F " model. Once a list of more accurate centroids and magnitudes has been produced by the profile fitting routine N S T A R , each star on the list can be subtracted out of the frame to produce a frame where faint stars, which might have been hidden in the wings of a brighter star's profile, can be identified. This process of N S T A R - S U B S T A R - F I N D can be repeated as often as desired, in order to measure al l the stars on the frame. 14 3.1.4 Comments on the data reduction A l l the frames were reduced using the aforementioned procedure. Typica l seeing discs for each of the first, second and fourth nights were (in the V and B filters respectively) 0."7 and l."2, l."0 and 0."6, and 0."5 and 0."9. O n the third night the seeing in the B and U filters was 0."6 and 0."8. It should be noted that the B data on the first night of observation -was quite poor. P S F variations have been noted in the B filter for the C C D used i n this study (Richer and Fahlman 1988). The number of stars measured on each frame ranged from ~ 1800 (best frame) to ~ 700 (worst frame). 3.2 M a t c h i n g t h e f r a m e s In order for mean magnitudes to be computed, and for real time light curves to be obtained for each star, it was necessary to match together the lists of N S T A R magnitudes and positions for each frame acquired from DAOPHOT. The problem of matching together many lists, each containing a large number of stars is an interesting one in its own right, and, considering the increase in the use of C C D photometry in the last few years (and consequently colour - magnitude and colour - colour diagrams based on matching such lists together), is one that warrants some discussion. The maximum positional deviation between two frames was about 20 pixels in the East - West direction and about 6 pixels in the Nor th - South direction. Some stars could not, therefore, be matched on all frames. 3.2.1 A U T O M M A T C H A program named AUTOMMATCH was supplied by G . G . Fahlman for this purpose. The input into this program the lists of stars (sorted by increasing X co-ordinate), the position offsets between each frame, and a critical match radius (in pixels). Two objects on different 15 lists are considered to be the same star if their centroids are wi th in this distance. These stars are then considered to have been "matched" together. Posit ion offsets were obtained, relative to the first frame (27M71 on Table 2.1), by manually matching the first 20 brightest stars on each list, and computing the mean X and Y offset for six of those stars. The largest standard deviation of this mean was 0.5 pixels. (The offset did not appear to vary across individual frames.) This led to the selection of a match radius of 1.0 pixels. The program reads in the lists and applies the position offsets. It then matches the first list to each subsequent list, generating an array of pointers between the lists. It then matches the second list wi th each subsequent list, and also wi th the first list. This continues unti l every list has been matched to every other list, and each list now has its own array of pointers for each star indicating the ID number of that star on every other list. If every list has matched with every other list perfectly there should be a "closed loop" of pointers, in that it should be possible to go from one list to the next (using the pointers) and return to the starting position. The program checks this loop and generates a warning message if it is not closed at that point in the list. When there is a conflict between two possible targets for a match, the program can be set to follow an automatic conflict resolution criteria by assuming that the star with the lower photometric error is the correct one. Mismatches occur more often in crowded fields, as might be expected. The principle effect of mismatching stars, for the purpose of this work, depends mainly on the number of such occurrences for a particular star. When a mismatch occurs, or a conflict resolution is in error, a unmatched star is left over, which is later written to a file wi th the matched stars. If this has only occurred once or twice for a particular star these can be filtered out of the database generated by AUTOMMATCH by only considering stars wi th a certain number of measurements (say 10% of the total number of frames 16 taken). A large number of mismatches can usually be spotted by examination of the standard deviation of the mean magnitude of the star. A high standard deviation of a stars' mean magnitude is usually the result of inherent variability or of measurements being interchanged between two stars. Due to the sheer volume of data processed the incidence of mismatching was quite high, however this was a problem mainly for faint stars. Satisfactory matching was obtained for approximately 1600 stars in total. 3.3 C a l i b r a t i o n To transform the instrumental (or N S T A R ) system magnitudes to a standard system (in this case the Johnson U B V system) i t is usually necessary to apply transformations of the form V-v = av + & (B-V) {B-V) = ctbv + /3bv (b-v) (U-B) = aub + f3ub (u-b) where U, B and V are the standard system magnitudes and u, b and v are the instru-mental magnitudes. A primary calibration is usually performed on several standard stars observed on the same night as the program object to derive the /? coefficients. Bright, but unsaturated stars, on the program frame can then be used as secondary standards to find the zero point of the transformation, and thereby bring a l l the stellar magnitudes on the program frame onto the standard photometric system. Unfortunately, on the third night of the observing run, when the primary standard frames were taken, an error prevented the acquisition of program frames for M71 in the V filter. The 6 coefficients could be calculated using the Landolt (1973) standards shown in Table 3.1. However, in order to obtain zero points for the transformation equations, it was necessary to adopt as secondary standards those stars observed by Cudworth (1985) which could be identified on the program frames (Table 3.2). 17 3.3.1 Primary Standards and Colour Coefficients Although the zero points for the calibration could not be calculated independently, the colour coefficients could be so derived using the primary standards observed on the third night (1985 July 14/15). Using the P H O T routine in the DAOPHOT program, concentric aperture photometry was performed for a variety of apertures on each primary standard, extending to a radius of 22 pixels in most cases. A plot of magnitude versus radius was made for each star on each frame. The magnitude used in the following calculations was the average of the last two to three points on this "curve of growth" where the curve had approached, or was approaching, its l imit ing magnitude. Table 3.1 lists the primary standards, their magni-tudes and colours from Landolt (1973) (V and B — V ) , together wi th the instrumental magnitudes and colours. The latter [vo and (6 — v)0] have been corrected for atmospheric extinction and have been normalised to an exposure time of one second. In cases where two or more observations of the same star had been made in the same filter the values were averaged together. The errors quoted by Landolt are the sigma of the mean magnitude and colour of a particular star - hence the error in B — V can be less than the error in V. The maximum of oy and OB-V was chosen in order to have an estimate of the error in the B magnitude. The following equations were then fit to this data: V-v0 = av + flv (B-V) B-b0 = ab + fa (B-V) {B-V) = abv + (b-v)0. Determination of the colour coefficients was achieved using a maximum likelihood tech-nique which took into account errors in both variables (see Stetson 1990). W i t h this approach a straight line was fit to the data to obtain the colour coefficients. The advan-tage of this was that the fit is "invertible", that is, it makes no difference which variable is taken to be the "independent" one. This enables the last (the B — V) transformation 18 Table 3.1: Pr imary Standards. Landolt ID V ( B - V ) 0~(B-V) v 0 (b - v ) 0 0"(b-v) G D 246 13.100 0.029 -0.320 0.025 15.673 0.004 -0 .573 0.004 SA115-486 12.480 0.013 +0.490 0.024 15.119 0.006 +0.050 0.008 SA114-548 11.600 0.015 + 1.370 0.014 14.256 0.006 +0.743 0.012 SA108-719 12.710 0.025 +1.000 0.029 15.276 0.009 +0.471 0.013 SA108-727 12.730 0.028 +0.700 0.028 15.332 0.006 +0.219 0.020 SA108-728 13.650 0.026 +0.960 0.023 16.260 0.003 +0.425 0.028 J : 1 | 1 i i i | i i i — i — | — i — i — i — i — | — i — i — i — i — | — r J — I — i — i — i — i — L _ J — i — i — i I i i i i I i i i i I i -- 0 . 5 0.0 0.5 1.0 1.5 B - V ( b - v ) 0 Figure 3.1: Pr imary standard calibration, (a) F i t to V - v0 = av + BV(B — V). (b) F i t to B - b0 = ab + pb(B - V). (c) F i t to (B - V) = abv + Pbv(b - v)0. The a and /? coefficients resulting from each fit are given in the top left hand corner of each panel. Open circles show points not considered in the final fit. 20 to be written in the form shown above. , In the first pass of the calibration calculations it was noted that the star SA108-709 deviated from the fit, but by less than the 3<7 level. Landolt (1973) notes that this star is "marginally useable as a standard". Hurley (1989) uses the same standard frames as in the present study and notes the poor fit of this star in the calibration. The data reductions for this study are independent and seem to strengthen the conclusion that this star may not be a good standard. Accordingly, SA108-709 was dropped from the list of standards and the calibration was recalculated. The following colour coefficients were derived: 0V = -0 .036 ± 0 . 0 2 0 Ph = 0.177 ± 0 . 0 1 5 pbv = 1.282 ± 0 . 0 0 5 . The method used to perform the above fitting procedure also returned an estimate of the scatter inherent in the data. This is the mean error of unit weight (m.e.l) . It is basically the root mean square value of the observational errors. It is defined, following Stetson (1990), as m.e . l = / V £ \ , N-2 2 where e, is the residual of the (AQ observed points wi th errors o~x and ay, m being the slope of the fit. A n m.e . l of unity would indicate that the observational errors had been estimated almost exactly. The mean errors of unit weight for the V, B and B — V fits were 1.168, 0.774 and 0.200 respectively. Figure 3.1 shows these transformations in a graphical manner. It was then assumed that these coefficients had not changed significantly from night to night. This allowed their use in determining the zero points for the reference frames on the fourth night (1985 Ju ly 15/16). 21 3.3.2 Secondary Standards and Zero Points Cudworth (1985) obtained photographic and photoelectric photometry for stars in M71 down to V « 16. There was no difficulty in identifying a set of 20 stars from Cudworths ' list, to act as secondary standards. Once this list was obtained stars whose mean x 2 value (from N S T A R fits on the program frames) was more than 2.5 were removed from the list. This left 14 stars, whose properties from Cudworths' study are shown in Table 3.2. A n error of 0.04, 0.06 and 0.07 was assumed for the Cudworth V , B magnitudes and the (B — V) colour index respectively (as discussed in Cudworths ' paper). The values of t>o and (b — V)Q are the N S T A R magnitudes in the reference frames discussed below. The errors quoted for these quantities are those returned by N S T A R . A completely independent calibration is therefore not obtained, and the results dis-cussed in the following sections should be considered with this in mind. In order to determine the zero point corrections, two reference frames were chosen, one in B (95M71) and one in V (80M71). The zero points were found by calculating, for each of the secondary standards, the appropriate value for a, viz. av = V-v0 - A, (B-V) ab = B.-bo - p\ (B-V) ctbv = (B-V) - pbv (b-v)0. These values for a were then averaged over al l the secondary standards. Stars KC-336 and KC-371 were found to deviate by more than Sa from the mean in the ctbv calculation. KC-336 also deviated by more than 3a in the ab calculation. The case for av was not as clear cut - six of the stars (both of the former two stars, together wi th KC-254, 1-39, KC-367 and 1-34) deviated by more than Za from the mean. Because of the large scatter in the V fit (see Figure 3.2), and because rejection of al l these six stars would have nearly halved the number of data points, it was decided to remove only KC-336 and KC-371 from all future calculations and keep the other stars in. The mean value of a for each zero point was then recalculated. The following results, which are displayed graphically 22 in Figure 3.2, were obtained. av = -2 .903 ± 0 . 0 2 3 ab = -3 .066 ± 0 . 0 2 7 abv = -0.219 ± 0 . 0 1 4 . The following transformation equations were therefore used to transform the B and V magnitudes of al l stars found on both the B and V reference frames (948 in total) to the Johnson U B V system: (B-V) = -0.219 + 1.282 (b-v)0 B-b0 = -3 .066 + 0.177 (B-V) V-v0 = -2 .903 - 0.036 (B-V). Once this had been accomplished, magnitude offsets, relative to the reference frames, were calculated and applied to al l other frames. Each frame was considered in turn, B and V frames being dealt wi th separately. Only stars which had been found on both of the reference frames were considered in the calculation. The difference in magnitude of those stars, between the reference and program frame, was computed. These differences were sorted by increasing value and the lower 10% and upper 20% of the measurements were discarded to remove outliers. The mean of the remaining differences was then found, weighted by the a of each difference (which came from the error in each magnitude, as returned by DAOPHOT, and added in quadrature). This mean offset was then applied to the measurements of a l l stars on that frame. When each frame had been placed on the same magnitude system the mean values of the B and V magnitudes were computed, together with mean values of the X and Y position and the x 2 statistic. 23 Table 3.2: Secondary standards. Cudworth ID V B - V v 0 (b - v ) 0 <7(b-v)0 KC-336 15.97 0.45 19.053 1.024 0.012 0.012 KC-371 16.03 1.02 19.093 0.844 0.006 0.007 KC-369 15.92 1.40 18.836 1.209 0.004 0.005 KC-255 15.64 1.09 18.601 1.029 0.010 . 0.010 KC-254 16.05 0.77 19.166 0.764 0.008 0.010 1-40 14.69 1.21 17.701 1.121 0.011 0.011 KC-368 15.96 1.00 18.926 1.021 0.004 0.004 1-39 14.52 1.10 17.381 1.020 0.008 0.008 KC-367 16.65 1.02 19.477 1.012 0.007 0.007 KC-252 15.38 0.69 18.356 0.748 0.010 0.010 1-34 14.45 1.05 17.318 0.975 0.007 0.007 1-32 14.46 1.07 17.373 1.005 0.006 0.006 KC-250 15.19 1.25 18.104 1.090 0.006 0.006 KC-249 14.97 1.25 17.938 1.117 0.010 0.010 24 -2 .8 -2 .9 -3 .0 -3.1 ~t 1 1 r (a) <crT> = -2.903 ± 0.023 J 1_ I L 0.5 1.0 B-V -3 .0 -3.5 "i i i r n 1 1 r (b) <ab> = -3.066 ± 0.02? _i i i i_ j i i i_ 0.5 1.0 B-V 0.0 -0 .2 a b v -0 .4 -0 .6 -0 .8 p 1 1 1 1 1 r (c) <abT> = -0.219 ± 0.014 _i i i i_ J i i i_ 1.5 1.5 0.5 1.0 1.5 B-V Figure 3.2: Secondary standard (zero point) calibration, (a): fit to av — V—v0—/3v(B—V). (b): F i t to ctb = B —bo - p\(B - V). (c): F i t to abv = {B-V)~ /3bv(b - v)0. The mean a resulting from each calculation are given in the top right hand corner of each panel. Dotted lines represent the 3<r level. Open circles denote points not used in the final calculation. 25 3.3.3 A Discussion of Errors A n explanation is required of the various photometric errors that have been quoted and that w i l l be quoted in later sections. The errors that DAOPHOT returns along wi th its magnitude calculations for stars are derived from the off diagonal elements of the matrix used to calculate the magnitude (amongst other parameters) in a "least squares" type of fitting process. These elements are really the correlation 'coefficients for the profile fit of a certain brightness, at a certain position, that is performed by the NSTA-R routine (Stetson 1987). The errors quoted for the zero points are the values of the sigma of the (unweighted) mean, taken over each individual value. The errors in the colour coefficients are also derived from the off diagonal elements of the matrix used in the iterative maximum likelihood method used to calculate them. Because the magnitude of each star on the appropriate reference frame was transformed to the Johnson U B V system, the error in that transformed magnitude was calculated by a standard propagation of errors procedure (see, for example, Bevington 1969). As mentioned above the next step i n the calibration procedure was to apply a magni-tude offset to al l frames (taken in a given filter). This was a weighted mean over a number of stars of the difference in the magnitude in the reference frame and that in the frame in question. The sigma of this weighted mean (where the weight was equal to 1/cr2) was also calculated and was typically of order 10~ 3 to 1 0 - 4 . This error was therefore not included in further calculations. Once this error had been applied, any error of the new magnitude if calculated by normal error propagation methods would be very close to the original DAOPHOT error: the larger, and more realistic error in the reference frame magnitude would be "washed out" by the averaging process used to determine the offset. This does not give an accurate representation of the true error in a stellar magnitude on a particular frame, since any frame could, in principle, have been chosen as the reference frame. To 26 arrive at an estimate of the true error, the reference frame error was added in quadrature with the instrumental (DAOPHOT) error. When the (unweighted) mean magnitude was calculated both the sigma of the mean (cr), and the standard deviation of the individual values (s) were derived. The former quantity was, due once more to the averaging effect, considerably lower than the individual errors. The latter quantity was used to search for variable stars, as described below. Figure 3.3 demonstrates this situation. 27 0.5 0.4 0.3 0.2 0.1 0.0 t "I I I 1 T" n i I i i I I i 1 1 1 1 1 r ~1 I T" (a) 12.0 J i . . ~l 14.0 16.0 18.0 v r e f 20.0 22.0 24.0 0"v 0.5 0.4 0.3 0.2 0.1 0.0 " i i i I i i r "| i 1 1 1 1 1 r (b) 12.0 14.0 16.0 18.0 20.0 22.0 24.0 Figure 3.3: (a) Plot of e (the error in the transformed V magnitude of the reference frame) against V magnitude, (b) Plot of a (the error in the mean magnitude after cali-bration)DAOPHOT against V magnitude. 28 Chapter 4 Results and Discussion 4.1 P h o t o m e t r y The C C D frames used in this study were not ideally suited for the purpose of searching for variable stars. The seeing, while generally quite good in most respects, was often bad enough to blend the wings of the majority of bright stars together, and to ensure that most faint stars were ini t ial ly lost in the wings of the brighter ones. The calibration described in Chapter 3 is not independent - it relies on Cudworth's (1985) photometry for its zero points. Cudworth estimates that his photometry is accurate to ± 0 . m 0 2 for the brighter stars and to ± 0 . m 0 4 for stars of approximately 16 t h magnitude. Ult imately it is this, and the error in the zero points, that determines the accuracy of the final calibration. However the photometry performed by DAOPHOT is quite precise. Thus when light curves are presented in the following sections, the DAOPHOT errors w i l l be used for the error bars, but when, for example, masses are derived from a P - L relation the error in the reference frame wi l l be used, since it is this that determines the absolute error. 4.2 A C o l o u r - M a g n i t u d e D i a g r a m f o r M 7 1 Having obtained calibrated mean V and B magnitudes for each star found on the reference frames, a colour - magnitude diagram ( C M D ) was constructed, as is shown in Figure 4.1. Variables and candidate variables (see §4.3) are also shown on this figure. Several selection 29 criteria were applied to the overall list of stars in order to include in the C M D only those stars with the most reliable photometry. These criteria were: 1. Each star was measured on both the B and the V reference frame. 2. Each star was measured on a total of 10 B and 10 V frames. 3. The l imit on the mean value of the N S T A R x 2 parameter was set at 3.0. 4. The l imit of the sigma of the mean B and V values for a star to be included was set at 0.05. A total of 577 stars appear in the figure. As can be seen by inspection of Figure 4.1, the width of the main sequence widens appreciably after 2 0 t h magnitude - this is due to the increase in photometric error as magnitudes become fainter (that is, it becomes more difficult for N S T A R to arrive at a good profile fit to a star as the star becomes progressively fainter). The C M D also shows substantial scatter throughout the main sequence and red giant branch. This is thought to be due principally to P S F variations across the frame, a known problem with the Cassegrain focus at C F H . However M71 is at a low galactic latitude (see Table 1.1) so there is a definite possibility of some field star contamination and differential reddening across the cluster. 4.2.1 A Main Sequence Ridge Line for M71 The following methods were employed to derive a ridge fine for the main sequence shown in Figure 4.1. For the range V = 17.75 to 20.25 the following method was used to establish this curve. The data were divided into "strips" in magnitude across the C M D , each strip was 0.m5 in depth (a thinner strip would have reduced the number of points available for the fit described below). Each strip was then binned in colour, each bin being 0.m01 wide. A histogram of the number of stars in a colour bin was generated and a Gaussian curve was fit to this histogram. 30 T 12.0 14.0 V 16.0 18.0 20.0 22.0 H1 H4 H2 H6 • • .: • • . • ••• -I I I L. 0.5 1.0 B - V 1.5 Figure 4.1: The colour - magnitude diagram derived for M71. Variable stars and candi-date variables are shown as open circles and are labelled. (See §4.3 for a description of the properties of each one.) Reddening corrections have not been applied. The ridge line in Table 4.1 has been included, and the horizontal branch at V = 14.35 is also shown. 31 Table 4.1: F iducia l ridge line for M 7 1 . V ( B - V ) C T ( B - V ) Reference* (1) (2) (3) (4) 12.00 1.85 -— 1 12.50 1.67 — 1 13.00 1.51 — 1 13.50 1.40 — 1 14.00 1.32 — 1 14.50 1.25 — 1 15.00 1.19 — 1 15.50 1.13 — 1 16.00 1.07 — 1 16.50 1.04 — 2 17.00 1.03 — O 17.50 0.84 •— 2 18.00 0.83 0.05 2 18.50 0.84 0.03 2 19.00 0.89 0.08 2 19.50 0.92 0.06 2 20.00 0.97 0.08 2 a) (1) Cudworth (1985), (2) the present work. For the range V — 16.75 to 17.75 the line was estimated by eye from Figure 4.1.For brighter regions of the CMD (V = 12 to 16), where data were not available in the present study due to the comparatively small area of the survey, an estimate of the fiducial sequence was made by eye from Cudworth's (1985) Figure 2 and hence no FWHM is given for these data. Table 4.1 lists the centre of each magnitude strip (column 1), together with the corresponding central point of the Gaussian (column 2) and its full width at half maximum (FWHM) (column 3). Below 20 t h magnitude the photometric errors become too large for this method to be meaningful. The FWHM given in Table 4.1 are not very indicative of the true spread in the main sequence photometry; however, they are compatible with the photometric errors discussed in Chapter 3. 4.3 V a r i a b l e s t a r s The principle purpose of this work was to search for variable stars. Due to the large amount of data (typically 70 measurements of approximately 1000 stars) an automated approach was adopted in order to carry out the initial phases of this search. Due to the generally superior nature of the V data (that is, the larger number and good quality of the frames), it was this that was used in the search procedure. The method of searching for variable stars was to plot the mean V magnitude against the standard deviation of that magnitude, as shown in Figure 4.2. As might be expected the standard deviation generally increases with decreasing brightness. The data were binned in magnitude, and the mean standard deviation in that bin was calculated. To reject outliers from the calculation of the mean the process was repeated, this time reject-ing points lying more than 3.0 standard deviations (of the mean of the bin) away from the original mean (of the standard deviations). The stars could then be grouped into classes, depending on how far (in terms of the standard deviation of the mean) the star's "sigma of mean" was from the mean of the bin. Figure 4.2 shows these levels as lines of different 33 1 1 1 1 ' 1 • 1 | I I I • « I I j 1 1 1 • ! 4 • • • • • • • • • « « -...J • m \ ~ i i • • • • • • t • • ««. r ~ • • • r • ; i— n • • — I I 1 »I j • • I % • • • 0 • *• • V 1 t 1 1 l -12.0 14.0 16.0 . 18.0 20.0 22.0 24.0 V Figure 4.2: Plot of sy against V. Solid lines are the mean of each bin. Dashed lines are the la level above the mean. Dotted lines are the Za level above the mean. (See text for a fuller description.) 34 types. Stars more than 3.5 standard deviations form the mean were classed "variable", those between 3.5 and 2.0 were classed "probably variable", and those below 2.0 times the mean were considered "constant" stars. However, although this is a good method in theory, it does not function correctly for all stars because of the possibility of mismatching stars. A mismatch, as mentioned in §3, causes a star's mean magnitude to have a much higher standard deviation. The end result was that the real time lightcurve of each and every star was viewed by the author, in order to pick out the variable stars. This leads to the possibility of having missed some variables, however the identification was attempted three times (using the data for the first, second and fourth nights) and was cross checked between each set of data. A total of six variables and candidate variables were discovered. Each of these wi l l be discussed i n turn, following a brief description of the method used to analyse the periods. 4.4 P e r i o d A n a l y s i s Several of the variables discovered had sufficient phase coverage to allow the use of a Fourier analysis technique to determine the periods. The approach used followed that of Deeming (1975) and modified by K u r t z (1985) and Matthews and Wehlau (1985). This latter modification uses trigonometric identities to reduce computing time when calculating the Fourier components. The program used for this analysis was supplied by J . M . Matthews, and subtracts the mean of the data set so as to remove any long term trends (as noted in Deemings' (1975) paper) before the analysis commences. The program then calculates the Fourier components of the real time light curve of the data set over a given frequency range at a given frequency interval. The peaks in the resulting amplitude spectrum give the most likely frequencies (or periods) of any periodicities existing in the data. Natural ly this procedure wi l l also pick out any aliasing periods, most notably in this 35 study, the one day aliasing expected from the manner in which the data were obtained. The usual approach adopted for this work was to perform an ini t ia l analysis over a fairly wide range of frequencies (say, 0 to 100 cycles per day (c/d) with a grid of 2000 frequencies), then to narrow down onto the most prominent peaks and perform a more detailed analysis (using the same number of frequencies, but over a much narrower range, usually around 20 c /d . There is no guarantee that the maximum peak is at the true frequency, so the data were phased for periods given by several frequencies adjacent to the maximum. Once a period had been determined a window function could be generated. In order to do this an amplitude spectrum was generated using as input, instead of the real data, a sine wave whose period, phase and amplitude was obtained from the analysis of the star in question. If the window function resembles the amplitude spectrum, then it is quite likely that al l aliasing effects have been accounted for. Residuals (in the sense amplitude spectrum — window function) were generated, first by normalizing the window function to the amplitude spectrum and then subtracting. Note that when referring to a variation i n magnitude i n the following discussion "amplitude" refers (unless stated otherwise) to half of the peak to peak amplitude of the light curve. 4.5 Description of Variable Stars Each variable and candidate variable w i l l now be discussed in turn. A table summarizing the properties of each star is given at the end of this section as are tables of photometry for a l l of the variables and a finding chart. 4.5.1 Variable H I Analysis The star modestly (!) designated here as H I is in the blue straggler region of the C M D (see Figure 4.1). It has a mean magnitude and colour of V ~ 16.19 and B — V 0.63., 36 Figure 4.3(a) shows the real time light curve for this star for the first, second and fourth nights of observations. Although the B data appear to show regular variations they were found during the analysis to produce very poorly defined peaks with substantial aliasing problems. Therefore it was decided to utilize only the V data in the analysis. Attention is drawn to the outlying point at H J D 2446258.93408, which is ringed on the aforementioned figure. Removal of this point did not change the following analysis. Figure 4.3(b) shows the resultant amplitude spectrum from 0 to 100 c /d . There is a significant peak at around 19 c / d . A second amplitude spectrum was generated, this time in the region 10 to 30 c /d wi th a frequency spacing of 0.01 c /d . This is shown in Figure 4.4(a). The maximum peak is at 19.3002 c / d (i.e. a period of 0.d05181), with an amplitude of 0.m0511. The error in the period - i f this is indeed the true period - is due to the spacing of the grid; for this star it is 0.d00003. The phasing produced by this period is very good (see Figure 4.4(b)), significantly better than that for any other peak. The epoch for the phasing was chosen to be H J D 2446261.96540, the heliocentric Jul ian date of the brightest magnitude in the sequence. The window function shows no major differences from the amplitude spectrum - it, and the residuals, are shown in Figure 4.5(a) and 4.5b. Discussion Variable H I appears to be a variable blue straggler of the S X Phe type. Its period is towards the upper end of the range discussed in Chapter 1. Nemec (private communica-tion) has supplied a period - luminosity diagram for field and cluster S X Phe stars. The distance modulus of 13.70 ± 0.20 (Richer and Fahlman 1988) and the mean V magnitude of 16.186 ± 0.027 leads to an Mv = 2.49 ± 0.20. On the period - luminosity relation in Figure 4.6, this star is probably pulsating in the first overtone mode. It should be noted however, that the l imit ing factor in determining these sorts of relationships is, of course, 37 16.0 L 1 ] (a) . | i i i i 1 i i i 1 1 1 1 i i 1 i i i i 1 i i 1 16.4 V 16.4 ••••• 1 I . , I , , i r 1 i i i t * * A - A 1 A A * A 1 1 1 1 16.8 17.2 B ~ 58.90 58.95 59.00 59.05 59.10 16.4 16.0 r V 16.4 T < I , 1 , , i i I i i i i A « A A A , 1 , ' , l r 1 i r l 1 1 1 1 16.8 17.2 B 59.80 59.85 59.90 59.95 60.00 16.4 16.0 : V 16.4 t i , 1 , ." i i 1 i i i ... , 1 , • " A A A * " ' 1 1 1 1 1 i 16.8 17.2 B 61.90 61.95 62.00 62.05 62.10 HJD - 2446200.00 • i i i i i i i i i i i i i i - 1 i i i i i i i i ' i r i i i i i i i i I ' ^ i ' i i i i i TW i i i i i i 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 frequency (c/d) Figure 4.3: (a) Real time light curve for H i . Filled circles denote V data; filled triangles, B data, (b) Amplitude spectrum for HI. Frequencies range from 0 to 100 c/d, with a frequency interval of 0.05 c/d. 38 0.10 0.0B - (a) ~i i i i I i i I 1 1 1 1 1 1 1 1 1 1 r 10.0 15.0 20.0 25.0 frequency (c/d) 30.0 16.0 16.1 V 16.2 • f1* 16.3 -16.4 ~1 1 1 1 1 1 1 j 1 1 1 1 1 1 1 1 1—T (b) i ~ i — i — I — i — i — r i i i I i i t i i i t 1.00 0.00 0.20 0.40 0.60 0.80 Phase Figure 4.4: (a) Amplitude spectrum for H i . Frequencies range from 10 to 30 c/d, with a frequency interval of 0.01 c/d. (b) Light curve for HI. The data have been phased using a period of 0.d05181. The epoch was chosen to be HJD 2446261.96540 (maximum light). The error bars are the DAOPHOT errors. 39 0.10 - (a) 0.08 ca fi 0.06 • 1 i i I i i 1 r "| i i 1 r ~i 1 v 0.00 1 1 1 1 1 10.0 15.0 20.0 25.0 30.0 frequency (c /d) 0.04 5? 0.02 CO cu xi 0.00 3 fi -0.02 -0.04 ~i i i i I i 1 1 1 1 1 1—~r " | 1 1 1 r (b) .j i i_ 10.0 J — i — i — i — i i ' ' i i i i i i 15.0 20.0 25.0 30.0 frequency (c /d) Figure 4.5: (a) The window function generated for the H I data using the period and am-plitude given in the text, (b) The residuals (in the sense amplitude spectrum — window function) for H I . 40 the error in the luminosity of the stars (see Nemec and Mateo 1991). In the above figure, one can see considerable scatter of the points about the supposed best fit line. Note that the variable in M3 (DaCosta 1988) has not been included in the fit because it is still unconfirmed. Figure 4.6 should, therefore, be taken as indicative. Knowing the absolute magnitude and the pulsation period, the mass of HI can be derived from the period - mean density relation (e.g. Collins 1989): P^-p = Q(M,L,Te,X,Y) (4.1) where P is the period and p is the mean density. Q is the pulsation constant and depends on the mass, effective temperature and luminosity of the star (A4,Te,andL), and the composition (X and Y). Nemec and Mateo (1991) give a more useful variant of this equation, which can be written as: l o g - ^ - = 2 1 o g % - 0 . 6 M b o i - 6 1 o g T e + 25.40 (4.2) where the subscript m refers to the first overtone mode and M b o i is the bolometric mag-nitude. The pulsation constant Q was taken to be 0.033 ± 0.05 for the fundamental mode and 0.025 db 0.05 for the first overtone mode (Petersen and Jorgensen 1972). The error in the period of 0.d00003 was derived from the spacing of the grid used in the period search. Values for the effective temperature (of 7900±400 K) and the bolometric correction (B.C.) (—0.mll ± 0.05) were obtained by interpolation of Table 66 of Lang (1986). The errors in the quantities calculated from tables of data are "worst case" estimates. Assuming a first overtone pulsation mode a mass of Mm = (0-90 ± O.52)AT0 was derived for H I . The error in the mass was calculated using standard error propagation techniques. By far the most dominant error is that in My- The resulting mass is somewhat 41 1.0 i i i i I i i i i I i i i i I i i T i I i i i i I i i i — r 2.0 3.0 4.0 M71 -fr NGC 5053 D NGC 5466" Field O CJ Cen x M3 A 5.0 1 , 1 1 J i i i i i i i i i i i i i i i i ' I i i i i 1.6 -1.5 1.4 -1.3 -1.2 Log P (days) - l . l 1.0 Figure 4.6: A period - luminosity relation for field and cluster SX Phe stars. Data for stars from the clusters NGC 5053, NGC 5466, LO Cen and M3 are shown. HI is represented by the asterisk. The upper line shows the first overtone pulsation mode; the lower one depicts the fundamental mode. The field star SX Phe itself is known to pulsate in both modes. This is indicated by the bar joining two of the points. 42 low considering that, as described in Chapter 1, the mean mass of most SX Phe stars is around 1.2MQ. However this figure is well within the error quoted for the mass of HI. Note that if the star is pulsating in the fundamental mode the following mass results: Mm = (1.57 ± 1 . 0 2 ) ^ 0 . The most likely explanation for the seemingly low mass for the first overtone pulsation mode is an error in the zero point of the calibration that was not discovered and included in the error for My. For example, shifting the absolute magnitude to 2.3 (which is within the error of the distance modulus) would result in a mass of ~ (1.2 ± 0.52)^© (see Nemec and Mateo 1991). Note that even if the calibration procedure described in §3.3 is changed, for example, to include the stars rejected in the fit, or to use weighted rather than unweighted means, then the zero points do not change by more than ~ 0.m02. 4.5.2 Variable H2 Analysis From its position on the CMD (Figure 4.1), H2 is a very blue star. It is quite faint, lying approximately l.m75 below the main sequence turn off. For the period analysis for this star both the B and the V data sets were used, at first independently and then combined to improve the phase coverage. The real time light curve for H2 is shown in Figure 4.7(a). Four points (ringed in Figures 4.7a and 4.8(b)) in the B light curve were found on a preliminary analysis to be outliers in the phase diagram and were excluded from further analysis. The amplitude spectrum generated for the V data only revealed several strong peaks around 17 c/d, five of which gave reasonably good phase diagrams (these were at 16.53026, 16.85027, 17.18028, 17.51029 and 17.84029 c/d). In order to try and increase the phase coverage the B data was normalized to the V data by subtracting the colour (B — V). The amplitude spectrum thus generated is shown in Figure 4.7(b). A second spectrum 43 19.2 i i 1(a) i 1 i i i 1 j 1 1 1 i 1 i i i i 1 i i i i 1 i i 1 19.6 V 19.4 -- « « * A 4 19.8 B 19.6 t i I 1 i > i i 1 i i > , 1 , , t 1 1 1 1 1 i 20.0 19.2 58.90 58.95 59.00 59.05 59.10 19.6 V 19.4 * * A * A ® 19.8 B 19.6 • i 1 1 1 1 1 1 1 1 1 1 1 1 1 l 1 1 i 1 1 l 1 1 20.0 19.2 59.80 59.85 59.90 59.95 60.00 19.6 • » A * A ® * B V 19.4 - « * 4 * 19.8 19.6 1 1 1 1 1 i I i i i , 1 , , 1 1 1 1 1 1 1 1 1 I 1 20.0 61.90 61.95 62.00 62.05 62.10 HJD - 2446200.00 i i i i i i i i i i i i i i i i i i i i - i i i i 1 i i m i M i r ^ n - i n i ' i l , i n ' ! i i i I V I 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 frequency (c/d) Figure 4.7: (a) Real time light curve for H2. Fi l led circles denote V data; filled triangles, B data, (b) Ampli tude spectrum for H2. Frequencies range from 0 to 100 c /d , with a frequency interval of 0.05 c /d . The B data was normalized to the V data to produce this diagram. See §4.5.2 for further explanation. 44 i i i i I i i 1 1 1 1 1 1 1 1 1 1 1 r 5.0 10.0 15.0 20.0 25.0 frequency (c/d) 19.2 i i i | i :(b) I i | i I 1 | 1 1 1 | 1 1 1 | t i i i 1 r-: 19.3 V ( B ) 19.4 19.5 — 19.6 i i i 1 i i i i i i i i i i i i i i i i i i i 1 i i i 0.00 0.20 0.40 0.60 0.80 1.00 Phase Figure 4.8: (a) Ampl i tude spectrum for H2 . Frequencies range from 10 to 30 c /d , with a frequency interval of 0.01 c /d . Aga in the B data has been normalized to the V data, (b) Phased light curve for H2 . The data have been phased using a period of 0.d06050. The epoch for V was chosen to be 2446258.89502. The epoch for B was 2446262.03962. Error bars denote DAOPHOT errors. 45 T i ' | i i i I | I I 1 1 1 1 1 1 r 5.0 10.0 15.0 20.0 25.0 frequency (c/d) i i i i | i i i i | i i i i 1 1 1 1 r 0.04 - (b) 2? °-02h fi -0.02 -co 0.04 -1 1 1—_i 1 1 i i i I i i ' ' I i ' ' i 5.0 10.0 15.0 20.0 25.0 frequency (c/d) Figure 4.9: (a) The window function generated for the H2 data using the period and am-plitude given in the text, (b) The residuals (in the sense amplitude spectrum — window function) for H2. 46 obtained using a finer frequency grid (an interval of 0.01 c / d over a range 5 to 25 c/d). This is shown in Figure 4.8(a). The three main peaks are at frequencies of 15.54024, 16.53026 and 17.51029 c /d . (Note the usual alias frequency of around 1 c /d . This may be expected given the observing schedule.) The strongest peak, with an amplitude of 0.m056 and a period of 0.d06050 was found to produce the best phase diagram in that it had the least dispersion about a smooth curve. The B data were then examined independently - the strongest peak in its amplitude spectrum was at a frequency of 16.51026 c /d , i.e. a period of 0.d06057. Figure 4.8(b) shows the phase diagram for H2, using the V and normalized B data (shown as filled triangles). Both sets of data were phased independently, both with a period of 0 d06050 but wi th an epoch of H J D 2446258.89502 in V and 2446262.03962 in B (the brightest magnitudes in each filter). Figure 4.9(a) and b shows the window function and the residuals for the data set described above. The window function was generated as described in §4.5.1. Note that the amplitude spectra, window functions and residual plots are a l l to the same vertical scale - the residuals for H2 are very small (less than 0.m005). Discussion As mentioned earlier H2 lies blueward of the main sequence, and below the turn off point. The photometry of Richer and Fahlman (1988) verifies the blue colour of this star. If this star is indeed a cluster member then its position in the C M D is quite unusual. Kaluzny (1989) has recently demonstrated the existence of a sequence of stars in N G C 188 that appears to be an extension of the blue straggler sequence downward and to the left of the turnoff point. That one of these kinds of stars (if a member of M71) is variable is an intriguing prospect. M71 does not, from simple examination of Richer and Fahlman's Figure 4, appear to have a significant number of such stars, however. Close examination of Figure 4.8(b) reveals a slight phase difference between the B and 47 the V data - that is, there is a periodic temperature variation. This is to be expected if the star is undergoing radial pulsation, as appears likely from the shape of the light curve. With the relatively few number of data points it is not possible to be more specific. 4 . 5 . 3 Variable H 3 Analysis This star lies to the right of the main sequence. Its very red colour suggests that it may be a field star. The real time light curve is shown in Figure 4.10(a) and appears more like some eclipsing binary variation than a Cepheid like one. An amplitude spectrum was generated from the V data, showing several peaks in the region 6 to 7 c/d. Using only the B data, two peaks of significant amplitude were found at 7 and 9 c/d. The two sets of data were then merged by normalizing the B data to the V data. Since this star was suspected of being a binary system no temperature effects need be accounted for, at least on this level of accuracy. The resulting amplitude spectrum is shown in Figure 4.10(b). Figure 4.11(a) shows a more detailed amplitude spectrum that was generated over the range 0 to 20 c/d. The most noticeable features of this diagram are the peaks indicated at 4.390006 c/d and 5.370028 c/d. It was found that the latter frequency (a period of 0d18622 produced the best phase diagram by a considerable amount, and this is shown in Figure 4.11(b). Again, the B data has been normalized to the V data and is shown as filled triangles. Figures 4.12(a) and (b) show the window function and residuals, respectively, for the second amplitude spectrum. Discussion The shape of the light curve for this star may indicate that it is actually a binary star. If so then the period quoted as 0.d18622 should be doubled to give a 0d37244 period. This is well within the range of established W UMa type binary systems (for example 48 18.6 V 18.8 19.0 18.6 V 18.8 19.0 18.6 18.8 19.0 V ~ i 1 1—I 1 1 1 1 1 1 1 1 r (a) •••• n i i i i i i I i 1 j -• + 1 I I I I I l _ Jl I I I I ' ' i 58.90 58.95 59.00 59.05 59.10 _i I -i i i i i i i i_ _i i _i • i i i 59.80 I I 1 t I 59.85 59.90 59.95 . * * • . A _I I I I I I I I I I I i I I * 60.00 J L 19.8 20.0 B 20.2 19.8 20.0 B 20.2 19.8 20.0 B 20.2 61.90 61.95 62.00 62.05 62.10 H J D - 2446200.00 0.12 0.10 5 0.08 a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 oo 0.00 0.0 10.0 20.0 30.0 40.0 50.0, 60.0 70.0 80.0 90.0 100.0 frequency (c/d) Figure 4.10: (a) Real time light curve for H3 . Fi l led circles denote V data; filled triangles, B data, (b) Ampl i tude spectrum for H3. Frequencies range from 0 to 100 c /d , with a frequency interval of 0.05 c / d . The B data was normalized to the V data to produce this diagram. See §4.5.3 for further explanation. 49 0.12 so '5 0.08 B -i 1 1 r (a) 4.390006 c/d i i I r~ 5.370028 c/d 15.0 20.0 frequency (c/d) V (B) 18.6 18.8 19.0 19.2 T~> i i | i i " | i i — i — | — i — i — i — | — i — i — i — ^ — i — i — i — | — i — i — r L(b) * ti i 0.00 0.20 0.40 0.60 Phase 0.80 1.00 Figure 4.11: (a) Ampli tude spectrum for H3 . Frequencies range from 10 to 30 c /d , with a frequency interval of 0.01 c /d . Again the B data has been normalized to the V data, (b) Phase light curve for H3. The data have been phased using a period of 0.d18622. The epoch was chosen to be H J D 2446261.96540 (maximum light). Fi l led circles are V data; filled triangles, B data. Error bars depict DAOPHOT errors. 50 0.12 I I I I I I I I T" J 1 I L 10.0 frequency (c/d) 0.06 5 0.02 -0.04 -0.06 1 1 1 1 1 1 1 1 1 1 1 1 r 20.0 i 1 1 r 0.0 5.0 10.0 frequency (c/d) 15.0 20.0 Figure 4.12: (a) The window function generated for the H3 data using the period and amplitude given in the text, (b) The residuals (in the sense amplitude spectrum — window function) for H3. 51 the stars NH19 and NH30 in N G C 5466 have periods of approximately 0.d44 and 0.d33, respectively (Nemec and Harris 1987, Mateo et al. 1990)). W i t h this in mind the data were rephased at that period to give Figure 4.13(a). Figure 4.13(b) was produced by phasing the data wi th different epochs (2446258.90674 in V and 2446262.07868 in B ) . As one can see, the data are not completely convincing. Further investigation of this star is required to confirm or refute the above claim. 4.5.4 Variable H4 No period analysis could be done on this star since it appears that only one cycle has been observed (see Figure 4.14). Even so the shape of the light curve is strongly suggestive of an eclipsing binary system. If so this would be a very significant discovery because there is only one other eclipsing binary positively identified in a globular cluster. That star is N J L 5 in u> Cen, discovered by Niss, J0rgensen and Lausten (1978) and confirmed as being a radial velocity member of a; Cen by Jensen and J0rgensen (1985) and Margon and Cannon (1989). However the star H4 seems peculiarly placed on the C M D for it to be a cluster member. Radial velocity information would be required to confirm membership for H4. 4.5.5 The Candidate Variables H5 and H6 The real time light curves of five candidate variables are presented in Figures 4.15(a) and (b). These results are not overwhelmingly convincing as proofs of variability - they are included for completeness sake. H5 seems to have a very short period - so short that it is almost unphysical. H6 seems to be exhibiting some kind of variation but with its very faint magnitude it is hard to tell. 52 V (B) 18.6 IB.8 19.0 1 i i | i i i | i i i | 1 1 1 1 i 1 1 1 1 1 1 1 1 1 r £ 0 0 19.2 -- i * i i i i ^ * * 4 i • • • i -_i i i I i i _i L _i i i I i i i_ 0.00 0.20 0.40 0.60 0.80 1.00 Phase V (B) 18.6 18.8 19.0 19.2 i i i | i i i | i i P | i i—i—| i i — i — | — i — i — i —|— i — i — r £ 0 0 i V • *i t'di ii _i_—i i I i i i I i i i I i i i I i i i I i i i 1 ' i 0.00 0.20 0.40 0.60 0.80 1.00 Phase Figure 4.13: (a) Phase diagram for H3 wi th a period of 0.d37244. V data is shown as filled circles; B data is represented by filled triangles,. The B data has been normalized to the V data, as described in the text, (b) As above but wi th separate epochs for B and V . 53 4.5.6 Comparison Stars and Photometry In order to demonstrate the believability of the variables described here a comparison star of comparable magnitude was chosen from the neighbours of each star. The real time light curves for each of these stars are shown in Figures 4.16, 4.17 and 4.18. The " C " number of each comparison is the same as the variable or candidate variable it is to be compared with. Table 4.2 is a summary table for all the variables mentioned previously. Column (1) is the star's ID number in this work, columns (2) and (3) are the V magnitude and the B — V colour, columns (4) and (5) are the period and amplitude in V (where appropriate) and columns (6) and (7) are the X and Y positions on the finding chart given in Figure 4.19. Table 4.3 contains the photometry for each of the variables H I , H2 , H3 and H4. Column (1) is the H J D of the observation; column (2) is the filter that observation was taken in . Columns (3) to (6) contain the observed magnitudes for H I , H2 , H3 and H4 respectively. The magnitudes in these tables have been calibrated as described in §3.3. 4 .5 .7 A Finding Chart Figure 4.19 is a grey scale rendition of the V reference frame wi th the positions of each of the variables and candidates marked. The X and Y positions i n Table 4.2 correspond to the coordinate system marked upon the figure. North is to the top of the figure; east to the left. The origin of the coordinate system in this work was calculated from the positions of the secondary standards to be at (X = 84."2,y = —25."3) on C u d worth's (1985) system. Note that Cudworth's X values (in arc seconds) increase from West to East, contrary to the values (in pixels) in this work. 54 Table 4.2: Summary of Variable Star Properties Star V B - V Period (days) VAmp X Y (1) (2) (3) (4) (5) (6) (7) H I 16.186 0.631 0.d05181 0.m051 294.35 468.26 H2 19.384 0.374 0.d06053 0.m056 269.65 427.41 H3 18.837 1.193 0.d37244 0.m100 106.98 442.17 H4 17.748 0.553 • — — 229.35 438.41 H5 18.727 1.140 — — 57.47 223.22 H6 20.628 0.601 — — 45.93 148.58 55 Table 4.3: Photometry for H I , H2 , H3 and H4. H J D a Fil ter H I H2 H3 H4 (2446200.0+) (1) (2) (3) (4) (5) (6) 58.86377 V 16.170 19.455 18.649 17.540 58.86768 V — 19.407 18.675 17.550 58.87158 V 16.190 19.391 18.671 17.553 58.87549 V 16.202 19.370 — 17.564 58.87939 V 16.220 19.347 — — 58.88330 V 16.221 19.326 18.665 17:591 58.88721 V . 16.225 19.378 18.662 17.596 58.89502 V 16.216 19.311 18.667 17.605 58.89893 V — 19.327 18.653 17.603 58.90283 V 16.154 19.380 — 17.603 58.90674 V 16.138 19.401 18.673 17.602 58.91064 V — 19.434 18.664 17.608 58.91455 V 16.151 19.420 18.675 17.598 58.91846 V 16.171 19.456 18.666 17.599 58.92236 V — 19.426 18.677 — 58.92627 V 16.232 19.414 18.688 17.567 58.93408 V 16.166 19.330 —• 17.525 58.95361 V 16.142 19.336 18.674 — 58.95752 V 16.120 19.380 18.667 17.508 58.96143 V — 19.365 — 17.513 58.96533 V 16.132 19.431 18.671 17.515 58.97315 V 16.164 19.418 18.653 17.512 58.97705 V 16.179 19.422 — — 59.03174 B 16.797 19.798 19.640 18.248 59.04346 B 16.787 19.843 19.376 18.274 a) Heliocentric Julian Date of mid exposure Table 4.3: Photometry for H I , H2, H3 and H4. H J D a Fil ter H I H2 H3 H4 (2446200.0+) (1) (2) (3) (4) (5) (6) 59.05127 B 16.830 19.597 19.433 — 59.05909 B 16.784 19.756 19.361 18.244 59.06690 B 16.839 19.714 — 18.346 59.08643 B 16.810 19.749 19.384 — 59.09424 B 16.782 19.767 19.406 18.264 59.10206 B — 19.919 19.454 18.288 59.10987 B 16.775 19.904 20.765 18.310 59.85598 B — 19.710 19.508 18.334 59.86379 B 16.789 19.733 — — 59.87942 B 16.767 19.816 19.485 18.331 59.88723 B — 19.839 19.492 18.320 59.89504 B 16.796 19.821 19.492 18.316 59.90286 B 16.790 19.770 19.516 18.302 59.91067 B — 19.727 — — 59.91848 B 16.776 19.715 19.491 18.324 59.92629 B 16.808 19.756 19.487 18.350 59.93410 B 17.109 19.825 19.493 — 59.94583 B — 19.799 19.388 18.368 61.93024 V 16.214 19.397 18.664 17.727 61.93805 V 16.231 19.431 18.653 17.853 61.94196 V 16.236 19.447 18.661 17.945 61.94587 V 16.233 19.455 18.664 18.047 61.95368 V — 19.420 18.671 18.165 61.95759 V 16.154 19.397 18.669 18.202 61.96149 V 16.130 19.350 18.659 — a) Heliocentric Julian Date of mid exposure 57 Table 4.3: Photometry for H I , H2 , H3 and H4. H J D a Fil ter H I H2 H3 H4 (2446200.0+) (1) (2) (3) (4) (5) (6) 61.96540 y 16.115 19.339 18.662 18.221 61.96931 V 16.135 19.347 18.667 18.262 61.97321 V 16.150 19.324 18.652 18.199 61.97712 V — 19.314 18.656 18.133 61.98102 V — 19.339 18.661 18.009 61.98884 V 16.236 19.347 — 17.919 61.99274 V — 19.384 18.653 — 61.99665 V 16.240 — 18.660 17.774 62.02399 B — 19.705 19.403 18.123 62.03181 B 16.789 19.710 19.415 18.282 62.03962 B — 19.685 19.499 18.276 62.04743 B 16.831 19.720 — — 62.05524 B 16.821 19.706 — — 62.06306 B 16.789 19.793 19.420 18.268 62.07087 B 16.773 19.811 19.489 18.431 62.07868 B 16.792 19.770 19.449 18.258 62.08650 B — 19.730 — 18.277 a) Heliocentric Julian Date of mid exposure ~] i i I — i I i i i I I i i i i I i i i i 1 i 1 r 17.6 18.0 B 18.0 — 18.4 17.6 i i i I i i i i I i i i i I i i i i I i i i i I i i i 58.90 58.95 59.00 59.05 59.10 18.0 V 18.0 B 18.4 _i i i i i i i i I i i ' i_ _i i i i i i i i i i i i 17.6 59.80 59.85 59.90 59.95 60.00 18.0 V B 18.0 - T 18.4 1 1 1 I 1 1 I t I I * I I I I t 1 1 I l t l l I l i t 61.90 61.95 62.00 62.05 62.10 HJD - 2446200.00 Figure 4.14: Real time light curve for H4, a possible eclipsing binary. The error bars are the DAOPHOT errors. 59 V 18.4 18.6 18.8 18.4 18.6 18.8 18.4 18.6 18.8 1 1 1 1 1 1 1 1 1 1 1 1 - ( a) i | , i l l | 1 1 l < J 1 1 i - + -> i i l i i i i I i i i , 1 , , 1 1 1 1 l 1 1 1 l l i 58.90 58,95 59.00 59.05 59.10 [ , ,1 » 1 1 1 1 1 1 1 1 1 l 1 1 1 t ( , 1 , , 1 i i 1 i i i i 1 i i i 59.80 59.85 59.90 59.95 60.00 -i i i 1 i i i i 1 li i > • l i t l l 1 l l i i 1 i i t 61.90 61.95 62.00 62.05 HJD - 2446200.00 19.6 19.8 20.0 19.6 19.8 20.0 19.6 19.8 20.0 B B B 62.10 20.2 20.4 V 20.6 20.8 21.0 20.2 20.4 V 20.6 20.8 21.0 -20.2 20.4 V 20.6 -20.8 21.0 T — i — i — I — i — i — i — i — I — i — i — i — i — I — i — i — i — i — | — i — i — i — r (bj 4 _i i i i i i i i J_ _i i i i i i i i ' J_ 58.90 J i i I i i i i i i i i i i i i ' i i i i i i i i i i 58.95 59.00 59.05 59.10 59.80 59.85 I I I I 1 1 I I I I 1 I 4 59.90 t • JL 59.95 _i i i I i i_ 60.00 _L 20.4 - 20.6 B 20.8 20.4 20.6 B 20.8 20.4 20.6 B 20.8 61.90 61.95 62.00 62.05 HJD - 2446200.00 62.10 Figure 4.15: (a) Real time light curve for H5, a candidate variable, (b) Real time light curve for H6, a candidate variable. On both plots the error bars are DAOPHOT errors. 60 p — i — i —i— I — i — i — r V V 17.6 18.0 17.6 18.0 17.6 18.0 : (a) ~i 1 1 1—I 1 1 1 1 1 1 1 1 r 18.4 B 18.8 _i i i i i i - i ' i i i i i • JL 58.90 58.95 59.00 • t i i i i • ' i i i ' i i i i i i 59.05 59.10 _i i I i i i i I i i i 18.4 B 18.8 59.80 59.85 59.90 59.95 1 I 1 I 1 I I I I I I I 61.90 19.2 V 19.4 19.6 19.2 V 19.4 19.6 19.2 V 19.4 19.6 (b) I i • i , I i 61.95 62.00 62.05 HJD - 2446200.00 i—i—i—|—i—r 60.00 J i i I ' i i_ 62.10 18.4 B 18.8 ~i—I—i—i—i—r T i—i—i—|—i—i—r ^ + 4 i i i i ' i i ' i i ' i ' I i i ' 58.90 JL 58.95 59.00 59.05 + 59.10 J I I I I I I I I I ' I I I t ' 59.80 59.85 59.90 59.95 60.00 • + 1 • + • . J I I I I I I I I I I I I I I I I I I I 1 I I I I L 20.0 20.2 20.4 20.6 20.2 20.4 20.6 20.2 20.4 20.6 B B B 61.90 61.95 62.00 62.05 HJD - 2446200.00 62.10 Figure 4.16: (a) Real time light curve for C I , a comparison star for HI . (b) Real time light curve for C2, a comparison star for H2. On both plots the error bars are DAOPHOT errors. 61 18.6 V 18.9 18.8 V 18.9 18.8 V 18.9 19.2 V V V 1 1 1 1 1 : (a) 1 1 1 1 1 1 ' 1 ' • i i 1 i i 1 1 A 1 ' ' A 1 - , , , i , 1 1 1 1 1 1 , 1 , A • A 4 i i i 1 i i 1 1 1 , , 1 58.90 58.95 59.00 59.05 59.10 • i i ! i • « t i 1 i i t * * * A , 1 , * A * 1 , , t I 59.80 59.85 59.90 59.95 60.00 1 1 1 1 1 • i 1 i i i . 1 , l • * * * A t 1 1 1 1 1 1 A • 1 , , ( 61.90 17.7 18.0 18.3 17.7 18.0 18.3 17.7 18.0 E-18.3 rjTl 1 r (b) 61.95 62.00 62.05 62.10 H J D - 2446200.00 ~i i i i | i i i i | i i r~ _j—,—j—j—|—,—|—,—j-i i i i i i i t j i_ _i i i i i i i i i i i i i_ 58.90 58.95 59.00 59.05 59.10 I I I I .j i i i i i_ 59.80 59.85 59.90 59.95 60.00 i t i I i i i i I _i i i i_ _i i i i i i i i i i i_ 19.8 B 20.0 19.8 B 20.0 19.8 B 20.0 18.3 61.90 61.95 62.00 62.05 H J D - 2446200.00 -3 18.6 18.9 1B.2 1B.6 18.9 18.2 18.6 18.9 19.2 B B B 62.10 Figure 4.17: (a) Real time light curve for C 3 , a comparison star for H3. (b) Real time light curve for C4, a comparison star for H4. On both plots the error bars are DAOPHOT errors. 62 17.4 17.6 17.8 18.0 17.4 17.6 17.8 18.0 17.4 l_ ~} T1 1 | i i i i | i i i i | — i r,—i 1 — | — i p 1—i 1 1 1 r : (a) 17.6 17.8 18.0 -- 18.4 H 18.6 18.8 JL _J i i i i i i i i i i i i i i i i 58.90 58.95 59.00 59.05 59.10 .j i i I i i i i I i i ' i J_ _i i i i i i i i i i i i 59.80 59.85 59.90 59.95 60.00 + * * * * + * . . _1 I I 1 I I I I I I I 1 I 1 I ' I I l l l t I l B 61.90 61.95 62.00 62.05 HJD - 2446200.00 19.0 18.4 18.6 18.8 19.0 18.4 18.6 18.8 19.0 B B 62.10 19.8 V 20.1 20.4 19.8 V 20.1 20.4 19.8 V 20.1 -20.4 -~i—i—i—|—i—i—i—i—|—i—n—i—i—|—i—i—i—r—|—i—i—i—i—|—i—i—i—: ( b) 120.8 -_ 21.2 1 I I I I 1 I t I I I I I I I 1 I _L _i i i i i i i i_ 58.90 58.95 59.00 59.05 1 4 • 4 • 4 + * • • * i i i 1 i i i 59.10 _L 59.80 59.85 4 59.90 59.95 60.00 , , ( • «* V \ " t 1 I I I I I ) -J 1 I I I I l_ B 21.6 20.8 21.2 21.6 20.8 21.2 21.6 B B 61.90 61.95 62.00 62.05 HJD - 2446200.00 62.10 Figure 4.18: (a) Real time light curve for C5, a comparison star for H5. (b) Real time light curve for C6, a comparison star for H6. On both plots the error bars are DAOPHOT errors. 63 400 300 goo 100 0 0 100 200 300 X Figure 4.19: A finding chart for variables H I to H6. This is a grey scale plot of the V reference frame (80M71). Coordinates for each variable are given in Table 4.2. 64 Chapter 5 Conclusions A search for variable stars in M71 using extant C C D frames has been described. The reduction and calibration procedures have been discussed, and are dependent on the photometry of Cudworth (1985). However this is almost certainly accurate to at least 0.m05. A n important discovery of this work is the recognition of an S X Phe star (or variable blue straggler) in the cluster, with a period of 0.d05181. This is the first such star to be found in a metal-rich globular cluster. From its position on the P - L diagram obtained for other S X Phe stars, both in the field and in clusters, it is hard to determine its puisation mode exactly. The mass was calculated to be (0.90 ± 0.52)A4© if it pulsates i n the first overtone, and (1.57 ± 1.02).M© if in the fundamental mode. The major uncertainty in this seems to be the uncertainty in the cluster distance. Indeed, the whole P - L relation for these stars seems to hinge on the calibrations for their luminosities. A second variable, similar to the first but about l . m 75 below the main sequence turn off, was also discovered. It has a period of 0.d06050. W i t h such a low luminosity and a period typical of a dwarf Cepheid, it is possible that this is a field star superimposed on the cluster. As mentioned in §4.5.2, Kaluzny (1989) has proven the existence of an "extension" of the blue straggler sequence down to the left of the main sequence in the cluster N G C 188. If the star H2 found in this study is a member of M71 then it may 65 imply an extension of the instabihty strip down to these low luminosities. Of course the star may be a binary of the W U M a type, but, from the shape of the light curve, this does not seem likely. Two variables, which may be binary in nature, were also found. For only one of these could the period be determined - this is most probably 0.d37244. This is typical of a W U M a type binary (e.g. Mateo et al. 1990). Note that its position on the C M D means that it is unlikely to be a cluster member. This is made even more likely by the fact that M71 is at a low galactic latitude. The other binary is almost certainly an eclipsing binary - however wi th no period information it is difficult to comment on it further. More observations on both of these stars are needed to confirm, and to refine the period of variability. Mateo et al. 1990 predict that from "3% to 15% of al l Pop. II blue stragglers should be contact binaries if the merger model is correct". From the work of Richer and Fahlman (1988), one notes that there appear to be approximately 50 blue stragglers in M 7 1 . Thus for this cluster the discovered percentage is (at least) 2%, assuming, of course, that both of these stars are binaries and are cluster members. The discovery of only four variables, with possibly two more as candidates, may seem to be disproportionate compared to the effort put into the data reduction (for 80 frames). It should be born in mind however that the images used in this work were not taken expressly for the purpose of a variable star search. Variable seeing, and the lack of sufficiently extensive phase coverage, has certainly affected the results. 66 References Abt , H . A . , 1985, Ap. J. (Letters), 294, L103. Allen, C. and Martos, M . A . , 1988, Rev. Mexicana Astron. Astrofi, 16, 25. A r p , H . and Hartwick, F . D . A . , 1971, Ap. J., 167, 499. Baade, W . , 1952, in Transactions of the I.A.U., 8, 397. Bevington, P .R. , 1969, Data Reduction and Error Analysis for the Physical Sciences, M c G r a w - H i l l , Inc. Burbidge, E . M . and Sandage, A . , 1958, Astrophys. J., 128, 174. Carney, B . W . and Seitzer, P., 1986, Astron. J., 92, 23. Cudworth, K . M . , 1985, Astron. J., 90, 65. D a Costa, G.S. , 1984, Ap. J., 285, 483. D a Costa, G.S. , 1988, in The Second Conference on Faint Blue Stars, eds. A . G . Davis Ph i l ip , D.S. Hayes and J . W . Liebert, p. 579. Deeming, T . J . , 1975, Astrophys. Space Sci., 36, 137. Drukier, G . , Fahlman, G . G . and Richer, H . B . , 1991, to appear in Formation and Evolution of Star Clusters, ed. K.Janes, (in press). Eggen, O . J . and Iben, I., 1988, Astron. J., 90, 635. Eggen, O . J . and Iben, I., 1989, Astron. J., 97, 431. Fahlman, G . G , Richer, H . B . , Searle, L . and Thompson, I .B. , Ap. J. (Letters), 343, L49. Frogel,J., Persson, S.E. and Cohen, J . , 1979, Ap. J., 227, 499. Harris, W . E . and Racine, R. , 1979, Ann. Rev. Astr. Astrophys., 17, 241. Hodder, P . J .C . , Nemec, J . M . , Richer, H . B . and Fahlman, G . G , 1990, in Formation and Evolution of Star Clusters, A . S . P . Conference Series, ed, K . Janes (in press). 67 Hurley, D . J . C . , 1989, M . S c . thesis, University of Br i t i sh Columbia. Jensen, K . S . and J0rgensen, H . E . , 1985, Astr. Astrop. Supp. Ser., 60, 229. Kaluzny, J . , 1989, to be published i n Acta Astronomica, 40. Kaluzny, J . and Shara, M . , 1987, Ap. J., 314, 585. K i n g , I.R., 1966, Astron. J., 71, 64. Kur t z , D . W . , 1985, M.N.R.A.S., 213, 773. Landolt , A . U . , 1973, Astron. J., 78, 959. Lang, K . R . , 1986, Astrophysical Formulae - A compendium for the Physicist and Astrophysicist, 2 n d ed., Springer-Verlag. Leavitt , H . A . , 1912, Circ. Harvard Obs., No. 173. Leep, E . M , Oke, J . B . and Wallerstein, G . , 1987, Astron. J., 92, 338. Lil ler , M . and Tokarz, S., 1981, Astron. J., 86, 669. Margon, B . and Cannon, R . D . , 1989, Observatory, 109, 82. Mateo, M . , Harris, H . C . , Nemec, J . M . , and Olszewski, E . W . , 1990, Astron. J., 100, 469. Matthews, J . M . and Wehlau, W . H . , 1985, Publ. A.S.P., 97, 841. McCrea , W . H . , 1964, M.N.R.A.S., 128, 147. Mould , J .R . and Aaronson, M . , 1983, Ap. J., 273, 530. Nemec, J . M . , 1989, in The Use of Pulsating Stars in Fundamental Problems of Astronomy, I . A . U . Col loquim N o . l l l , ed. E .G.Schmidt , p.215. Nemec, J . M . and Cohen, J . G . , 1989, Ap J., 336, 780. Nemec, J . M . and Harris, H . C . , 1987, Ap J, 316, 172. Nemec, J . M and Mateo, M . , 1991, to appear in Confrontation between Stellar Pulsation and Evolution, (ed. C.Cacciar i) , A . S . P . Conference Series (in press). Niss, B . , J0rgensen, H . E . and Lausten, S., 1978, Astron. Astrophys. Suppl. Ser., 32, 387. Olszewski, E . and Aaronson, M . , 1985 Astron. J., 90, 2221. Petersen, J .O . and j0rgensen, H . E . , 1972, Astron. Astrophys., 17, 367. Pickering, E . C , 1890, Astr. Nach., 123, 107. 68 Renzini , A . , Mengel, J . G . and Sweigart, A . V . , 1977, Astron. Astrophys., 56, 369. Richer, H . B . and Fahlman, G . G . , 1988, Ap. J., 325, 218. Richer, H . B . and Fahlman, G . G . , 1989, Ap. J., 339, 178. Sawyer Hogg, H . , 1973, Pub l . David Dunlap Obs. 3, no. 6. Saio, H . and Wheeler, J . C . , 1980, Ap. J., 242, 148. Sandage, A . , 1953, Astron. J., 58, 61. Spizter, L . , 1987, Dynamical Evolution of Globular Clusters, (Princeton University Press). Stetson, P . B . , 1987, Publ. A.S.P., 99, 191. Stetson, P . B . , 1990, preprint. Webbink, R . F . , 1976, Ap. J., 209, 829. Webbink, R . F . , 1981, Ap. J. Suppi, 45, 259. 69 

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