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THE PLEIADES an astrometric and photometric study of an open cluster
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FLOOR VAN LEEUWEN THE PLEIADES an astrometric and photometric study of an open cluster THE PLEIADES an astrometric and photometric study of an open cluster
proefschrift
ter verkrijging van de graad van Doktor in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit te Leiden, op gezag van de Rector Magnificus Dr. A.A.H. Kassenaar, hoogleraar in de Faculteit der Geneeskunde, volgens het besluit van het College van Dekanen te verdedigen op donderdag 26 mei 1983 te klokke 15.15 uur
door
Floor van Leeuwen geboren te Amsterdam in 1952
Sterrewacht Leiden
Royal Greenwich Observatory, U.K.
1983 Beugelsdijk Leiden B.V. De beoordelingscommissie bestond uit de volgende leden: Professor Dr. A. Blaauw, promotor Dr. J. Lub, referent Professor Dr. J.H, van der Waals voorzitter van de Subfaculteit Sterrenkunde en Natuurkunde Professor Dr. H.C, van de Hulst Professor Dr. W.B. Burton Professor Dr. C, Zwaan Errata page tU Reference 3 read '18UU' instead of '1W 16 Reference 23 read '19«6' instead of '19U51 70 2nd par., 5th line read 'surface gravity' instead of 'temperature' 80 Tabled add: 1531. m=1?.5B, obs: 1931. per .a!' ."^hr, flares: ? 112 Equ.16 read 'n(n»nCi)' instead of 'n(l)1 116 Equ.32 read ' (x(0.i)a+ yC'l.if )' instead of • (x(o,i)+y(n,i))' STELLINGEN behorende bij het proefschrift De PLEIADEN, An Astrometrie and Photometric Study of an Open Cluster
1. De onzekerheid in de ruimtelijke snelheden zoals die door O.J.Eggen in 1963 (Astronomical Journal Vol 68, p 697) zijn bepaald voor heldere A type sterren bedraagt ten minste ^7 km/sec. 2. De details welke Eggen(1963) in de verdeling van ruimtelijke snelheden onderscheidt zijn significant kleiner dan 7 km/sec en daardoor statistisch niet te rechtvaardigen. 3. Het bestaan van een "Pleiades Moving Group" is door Eggen(1963) geenszins aangetoond, H, Discrepanties tussen cluster leeftijden zoals afgeleid van pre en post hoofdreeks ster evolutie modellen duiden niet noodzakelijk enkel op een spreiding in leeftijden voor cluster leden maar kunnen ook deels het gevolg zijn van onvolledigheid in de beschrijving van het sterevolutie proces. (R.Stauffer, 1979: Astron.Journal Vol 85, p 1341)
5. De klassieke astronomische waarneem technieken, zoals astrometrie en fotometrie, bieden nog volop mogelijkheden tot het verkrijgen van nieuwe gezichtspunten in de sterrenkunde. 6. Het onderzoek naar variabele sterren dient gestimuleerd te worden. 7. In het kader van de Brits-Nederlandse samenwerking met betrekking tot de sterrenwacht op La Palma dient een eenvoudige vergelijk fotometer ontwikkeld te worden.
8. De variabele sterren welke geklassificeerd worden als BY Dra, RS CVn, CC Eri, FK Com en de K sterren in de Pleiaden zijn allen in eerste instantie variabel door een interactie tussen de hoge hoeksnelheden en de convectie zones die al deze sterren gemeen hebben. (sectie 3.3 van dit proefschrift) 9. Door bezuinigingen op posities wordt veel wetenschappelijk werk sterk vertraagd in afronding en publikatie. 10. Het classificeren van nog niet verklaarde visuele en radar waarnemingen als UFO onttrekt mogelijk interessante waarnemingen aan wetenschappelijk onderzoek. 11. Het verdient aanbeveling de onderverdeling van het genus Aloe te herzien en daarbij, in navolging van het onderscheid met Haworthia's en Gasteria's, de nadruk te leggen op de verschillende bloeiwijzen eerder dan de verschillen in groeivorm. (G.W.Reynolds, The Aloes of South Africa, 1976, A.A.Balkema, Rotterdam) 12. De plantengroei en zijn aanpassing aan het heersende klimaat kan veel informatie verschaffen over de langere termijn betrouwbaarheid van de klimaats omstandigheden rond een astronomisch waarneem station. aan Gtoèka en Magda Contents
Historical development leading to the present work 8 1.1 The Pleiades as observed from 1829 to 1950 8 1.3 Outline of the present work 13 Notes to the presentation 13 References 14
The astrometric measurements and their reduction 17 2.0 Summary 17 2.1 Introduction 18 2.2 The astrometric measurements 21 2.2.1 The ASTROSGAN as astrometric measuring device 21 2.2.2 The photometric and positional parameters 21 2.2.3 The interpretation of the photometric parameters 26 2.2.1 The positional accuracies 30 2.3 Construction of the reference frame of positions 31 2.3.1 The preparatory catalogue of star positions 31 2.3.2 The ideal picture 33 2.3.3 The transformations 35 2.3.3.1 Transformations between linear pictures for different plate centres 35 2.3.3.2 Transformations between linear and ideal pictures 36 2.3.4 The influence of the earth atmosphere 36 2.3.5 Aberrations 40 2.3.6 The central 3 by 3 degrees field 41 2.3.7 Construction of the catalogue of 1978 positions 43 2.4 Determination of the proper motions 48 2.4.1 Reducing old epoch plates to new epoch reference positions 48 2.4.2 Definition of the proper motion system in the central region 48 2.4.3 The proper motions in the outlying regions 50 Appendix 1 Definition of the proper motions 51 Appendix 2 Updating mechanism for proper motions 51 References 52
The photometric observations, differential reddening and the variable K-stars in the Pleiades 53 3.0 Summary 53 3.1 VBLUW photometry of the Pleiades and a determination of its distance 54 3.1.0 Summary 54 3.1.1 Introduction 55 3.1.2 Selection of stars 56 3.1.3 The measurements 57 3.1.4 Identification of members 58 3.1.5 The distance to the Pleiades 59 3.2 Differential reddening in the Pleiades 67 3.2.0 Summary 67 3.2.1 Introduction 67 3.2.2 Empirical reddening corrections 68 3.2.3 Theoretical grid reddening corrections 69 3,3 Variable K-type stars in the Pleiades 73 3.3.0 Summary 73 3.3.1 Introduction 73 3.3.2 The photometric measurements 75 3.3.2.1 1980 measurements 75 3.3.2.2 1981 measurements 78 3.3.3 The photometric data and their interpretation 79 3.3.3.1 The Hghtourves and periods 83 3.3.3.2 The colour indices 79 3.3.H Spectroscopie observations 86 3.3.5 Rotational modulation 86 3.3.6 The angular momentum 89 3.3.7 Future prospects 90 References 91
4. On the distribution of masses and motions in the Pleiades 95 4,0 Summary 95 1 Introduction 96 4.2 The density distributions 98 4.2.1 The counts 98 4.2.2 Surface and space density distributions 100 4.2.3 The noise level of the density distribution 104 4.3 The occurrence of hard binaries 105 4.4 The luminosity and mass functions 109 4.4.1 The observed luminosity function 109 4.4.2 The fainter extension 111 4.5 The internal motions 113 4.5.1 The influence of plate reductions on proper motions 113 4.5.2 The influences of the motion of the Sun relative to the cluster 115 4.5.2.1 The relative radial velocity 115 4.5.2.2 The relative secular parallax 117 4.5.2.3 On the detection of real rotation and expansion 119 4.5.3 The internal velocities 119 4.6 The potential energy distribution and the total mass of the Pleiades 121 4.7 The Galactic tidal force and mass segregation 125 4.8 Conclusions 127 References 129
Nederlandse samenvatting 132
Studie overzicht 137
Dankwoord 138 1. Historical development leading to the present work 1.1 The Pleiades as observed from 1829 to 1950 The Pleiades are among the oldest observed objects In the sky. Refer- ences to them date back more than 4500 years. Also positional research on the cluster is relatively old. It started with the mapping of the cluster by Galilei (1); in 1653 the first map based on his telescope observations was published. The first reasonably accurate relative positions of stars in the Pleiades were provided In 1755 by Le Monnter (2). All positional work those days was performed using meridian telescopes and had as Its main purpose the provision of a reference frame for the determination of the positions of the Moon and Planets and for navigation. The positions of the Pleiades were especially used to determine the parallax and angular diameter of the moon, as they could provide many reference points In a small area, (at the occasion of occultation).
In 1829, Bessel (3) started observing the Pleiades at the Observatory of Ktfnigsberg with the Heliometer built by Fraunhofer. This Instrument, of which the basic Idea was developed by 01e Romer in 1675, is a refractor of which the objective has been cut in two equal halves. By shifting one half with respect to the other, two shifted images are observed of the same object. Distances between stars can be measured accurately using a Helio- meter, by first aligning the shifting direction with the two stars and then shifting one half of the objective till one of the images of one star co- incides with one of the other star. The angular distance of the stars is then determined by the shift of the objectives and the focal distance of the telescope»
In the report of his observations and those performed later by Plantamour and Schluter, Bessel added a new purpose to positional astronomy, with which he Initiated astrophysical research on the Pleiades. Beasel wrote:
Endlich habe Ich einer mSglichst slcheren Bestlmmung der relativen Orter dieser Sterne elnen for sich selbst bestehenden Werth beigelegt, indem ich glaube, dass eine Zeit kommen wird, fur welche die Frage nach den Inneren Bewegungen dieses merkwurdigen Sternhaufens ein Interesse gewinnen wird, welches, bei aussersten Kleinheit dieser Bewegungen, nur durch friihere Beobachtungen befriedigt werden kann, wenn sle durch die Anwendung der dazu geeignetsten Mittel erlangt worden sind und daher die grösste erreichbare Genauigkeit besltzen. For astrometric purposes these measurements with the Kttnigsberg Heliometer are no longer of value, but up to the beginning of the 20th cen- tury they served as very useful first epoch measurements. The first to use them as such was G. Wolf (4) in 1874. He took up the programme of Bessel and measured new relative positions for 79 stars, using a micrometer behind a refractor. He identified and listed 476 stars in the field and presented a map of 625 stars down to the 13th magnitude. The accuracies of Wolf's measures were however, low, and the proper motions he derived from a com- parison with Bessel's observations were therefore spurious. He realized that the relative proper motions of the stars In the cluster are small, but got a wrong impression of the real moving direction of the cluster, which Is be- tween South East and South: Si nous considérons les dix étoiles bien dêterminées par Bessel, nous trouvons que, sauf pour 26s_, Ie mouvement relatif est extrèmement petit, sa direction est vers 1'est avec une incli naison trêis faible vers Ie nord, C. Pritchard (5) tried in a similar way to determine proper motions in 1879, using micrometer and meridian observations obtained in Oxford, but his results were of lower quality than those of Wolf. A more successful attempt was made by W. t. Elkin (6) in 1885, using the new Heliometer of Yale Observa- tory The first paragraph of the introduction to his report is characteristic for much of the research on the Pleiades! On taking charge of the new Heliometer of the Yale Observatory the first piece of work which I planned, and of which the execution is presented In the following pages, was a redetermination of the rela- tive places of the principal stars forming the group of the Pleiades. As is well known, this was one of the first problems attacked by Besael with his new Kttnigsberg Heliometer, and which occupied his attention during some twelve years until it was finally completed in 1841. The desirability of repeating such work from time to time is obvious, and apart from the value of the evidence of the results for the knowledge of the group and its motions there is presented so favourable an opportunity for a thorough test of the instrument a? to make such investigation especially suitable for an initial piece of work. Elkin extensively studied systematic errors in Bessel's work and finally de- rived differences between the Yale and Ktfaigsberg data for which he claimed a probable accuracy of 0V2, i.e. proper motions with an accuracy of 5" per millennium He also compared his data with those of Wolf and Pritchard and found large systematic differences especially in Right Ascension. Twenty years earlier, in 1865, Rutherford started photographing the sky and in particular some star clusters. Measures of Rutherford's plates are presented by Gould (7) and Jacoby (8). Other early photographic measures are presented by Turner (9) based on Oxford plates and Olsson (10) from plates taken in Stockholm. The photographic measures provided much improved relative positions, especially when good measuring equipment started to be developed» By the end of the 19th century the photographic plates became the main instru- mentation for the determination of relative positions. In 1903, J. Lagrula (10) from the Observatory of Lyon, summarized the results obtained In the 19th century by means of Heliometers, Micrometers, meridian observations and photographic plates. He carefully checked all measures on systematic errors and calculated proper motions. With respect to the in- ternal dispersion in these proper motions, and results obtained earlier by Wolf, Pritchard and Elkin, Lagrula wrote: Les mouvements propres relatifs de Pritchard different tellement de ceux que nous avons obtenus, ainsi que de ceux qui ré*sultent des mesures hêliomêtriques de ResseX et d'Elkin, qu'ils ne semhlent plus acfuellement devoir être pris en consideration, tes mouvements de Wolf, quoique meilleur, doivent être êgalement abandoneest on s'explique, en effect, que les erreurs systé - mattques relevées en ascension droite dans ces mesures affectent beaucoup les mouvements eonclus pour cent ans, d'autant plus que l'intervalle de temps sêparant les observations de Wolf et de Pritchard cte celles de Bessel est relativement court. D*un autre cofcé", la concordance de nos résultats avec ceux dédults spêcialement de mesures héliomêtrtques, et les faibles erreurs moyennes qu'ils comportent, paraisseat démontrer que, malgré1 leur exiguïté, les mouvements relatifs des Pleiades ne peuvent plus être mis en doute a notre époque. Although Lagrula was wrong in this respect, as hls "internal motions" were still due to errors of measurement his next statement was of grew1: importance: Enfin la petitessa de ces mouvements relatifs par rapport au mouvement general d'entrainement du groupe en meme temps que 1'agglomeration des étoiles brillantes dans cette portion restreintre du ciel montrent, avec une grande probabilité , que l'amas des Plefades est un systême physique. Meanwhile, also the photometric properties of stars in the Pleiades field became a subject of research. In 1900, Gaultier (12) presented a list of photographic magnitudes of 300 of these stars. This list has served for a long time as a reference. It was examined in 1907 by Hertzsprung (13) together with similar observations by Dugan and Schiller (14). Hertzsprung determined and applied corrections for systematic differences and compared these and measures by himself with visual photometric magnitudes derived in Potsdam by Muller and Kempf (15). Hertzsprung noticed systematic differences between photographic (mc) and photometric visual (mp) magnitudes for cluster members: Werden nur die Sterne betrachtet, fur welche eine mit Alcyone gemeinsame Eigenbewegung gefunden wurde, so zeigt sich ein ausgeprSgter Gang in den Dtfferenzen m(--mp. DLeser Gang entspricht aber einer Rnderung der Farbe mit der Stern grosse, welche man nach den Spektralangaben erwarten kan. The relation found by Hertzsprnng was further investigated In 1914 by Lau (16) and led to what is known today as the Hertzsprung-Russel diagram. From the beginning of the 20th century the Pleiades were established as a star cluster, a dynamically bound &roup of stars. The main, interest from that time on was to find more members, especially among the fainter stairs and in outer regions, and to datermine more accurate proper motions and photometric parameters. The main role in obtaining these data was played by four astrono- mers, viz, Robert Triimpler, John Titus, Ejnar Hertzsprung and Leendert Binnendijk.
Triiiroler (17) started in 1914 in Zurich with a reexamination of Lagrula's work. Using only those stars already observed by Bessal between 1829 and 1840,
10 together with more recent, corrected observations, he derived an upper limit for the internal notions of 6" per millennium He showed a proper motion diagram for the Cluster and the field stars, from which he separated cluster members- As an additional selection criterion, Trilmpler used the colour magnitude relation for cluster stars found by Hertzsprung:
E& von denen Sternen, deren relative Eigenbewegung gegen n Tauri nach Lagrula kleiner ist als 2" im Jahrhundert, keiner tnit seinen Farbindex stark von der GesetzmSssigkeit abweicht, sind dieselbe alle als physische Mitglieder der Plejaden Gruppe anzusehen.
With respect, to the internal motions, Tritmpler wrote;
Die relative Eigenbewegungen der 11 hallsten Plejadensterne sind sehr klein, sie liegen unterhalb der von Lagrula, Kromm (18) und Boss (19) for ihre Eigenbewegungen erreichten Genauichkeit,
After moving to the USA, Triimpler (20,21) started an extensive search for stars in the Pleiades Cluster, using almost all available old observations and some new photographic plates taken at the Allegheny and Zurich Obser- vatories. His work resulted in a list of 246 stars, which according to him, could be members of the cluster. The list contains stars brighter than mpg • 9 within 3.5 degrees and brighter than mpg « 13.5 within 1 degree from the cluster centre. Using photographic magnitudes determined by himself and colour indices measured by Hertzsprung, he was able to establish a colour magnitude relation for cluster members. He also determined a luminosity law fcr the cluster and found it to be different for different areas:
It is to be noted that the luminosity law of the outlying region is not the same as that of the central field; the former does not con- tain any stars between the third and sixth magnitude, and the lumi- nosity function rises more rapidly for the fainter stars. This effect is produced by what seems to be a general characteristic of star clusters that the brightest members are more concentrated to- wards the centre, while the fainter stars are scattered over a larger field.
TruMpler compared the average brightness of Pleiades stars with those of nearby stars of the same spectral type and derived a cluster parallax of 0'.'008. Although he derived proper motions accurate enough to distinguish members from non-members, the actual proper motions and their accuracy were never published, which decreased the value of his work.
In 1938, Titus (22) published his measurements based on a direct compari- son between 26 exposures on 15 plates taken by Rutherford at Columbia Uni- versity, New York, between 1868 October and 1874 November, and 26 new plates taken with the same telescope between 1935 October and 1936 September. The average time interval used was 64 years. He was able to bring the accuracy for the proper motions of cluster members brighter than mp„ = 9.0 at "5 per mil- lennium, a factor 10 better than all that was achieved before. Because of this high accuracy he was able to distinguish an internal proper motion dispersion of '.'9 per millennium, a value which is close to more recent estimates. After correcting for parallactic motion and assuming a parallax of 0V011 he derived an internal velocity dispersion of 600 m sec"'1-.
11
l r Titus was, however, not the first to derive a reliable value for the internal proper motion dispersion. Hertzsprung (23) had published his estimate (almost the same value) in 1934. It was based on preliminary results of an extensive research programme concerning the Pleiades, and carried out between 1924 and 1944 at the Observatory of Leiden. In his Darwin Lecture on 1929 May 10, Hertzsprung (24) explained what kind of plates he used and why those plates were available. The apparent diameter of the Pleiades is about 2 degrees. That is to say, they just cover a normal Carte du Ciel plate. Therefore the Pleiades, which show both bright and faint stars all over the plate, have formed a favoured test object for refractors of the Carte du Ciel type. As soon as a refractor had been erected, the Pleiades were among the first objects to be photographed. And then Che nebulae present in the group are just bright enough to show on plates taken with the aperture ratio of the Carte du Ciel refractor, 1:10, with an exposure time of a few hours. This circumstance may also have some bearing on the fact, that at different observatories there are old, rather long exposed plates of the Pleiades present, which, when repeated now, will enable us to determine fairly accurate values of the relative motions of the fainter stars.
Hertzsprung borrowed a total of 161 plates from 15 observatories. Algiers, Bonn, Bordeaux, K«5benhavn, Greenwich, Harvard, Helsinki, Leiden, Oxford (Uni- versity), Oxford (Radcliffe), Paris, Potsdam, Poulkova, Tachkent and Vatican. They were measured in 80 pairs by many of the Leiden Observatory staff, com- bining as often as possible old and new plates from the same observatory. Preliminary publications came from Kreiken (25) in 1924, Raimond (26) in 1928 and Hertzspruig (23) in 1934. The final publication on the proper motions came in 1947 (27), and yielded a catalogue of positions for 3259 stars and proper motions for 2920 of these. Many faint members of the Pleiades were detected, as this study provided proper motions for all stars brighter than mpg =15.5 in the field, 2 magnitudes deeper than the study by Trtimpler. Also a few members of the Hyades were found. Parallel to the astrometric research, Binnendijk (28) performed a photo- metric study of 1276 stars in the same field. He presented a discussion of both the astrometric and photometric data, involving the distance of the cluster, differential reddening of cluster members, the internal proper motion dispersion for different magnitude intervals and a discussion of the properties of the field stars. Only part of the astrometric data has been improved since 1947, the major improvements being achieved for photometric and spectroscopie data.
References to relevant papers published after 1950 will be given in the appro- priate sections.
12 1.2 Outline of the present work The main objective of the preaent work is to determine proper motions in an area of 8..' by 8.2 degrees around the cluster centre. Although all necessary measurements and reductions have been accomplished, there was no time and money left to entirely finish this part of the work in the context of this thesis. The astrometric measurements and their reduction to the extent that the work was completed, are described in section 2. Fortunately, a survey by Pels, although at much lower accuracy, was able to provide several possible members' in the outlying regions of the cluster. Photometric meaaurementa on these stars are presented in section 3.1. In section 3.2 the differential reddening of the £ and A stars is discussed. In section 3.3 the K stars, and in particular their variations are described. In section 4 the astrometric and photometric data on the cluster members are combined and examined. Their surface and space density distributions are derived as well as the luminosity and mass functions of the cluster, These data are compared with models for open clusters, and, together with the models, with internal velocity dispersion data derived from proper motions.
Notes to the presentation Sections 2, 3.1, 3.2, 3.3 and H have been written as individual papers and will therefore sometimes partly overlap in their contents, in particular with respect to introductions. In the notation I often used the symbols * and ** for multiplication and powers of 10; 10"8 = 108 . A cross reference tabel for star numbers used for the brighter members of the Pleiades can be found on page 73 and maps of the cluster on pages 55 and 64
13 References 1. Petti Gasaendi, 1653, Instituto astronomica, ... cui accesserunt Galilei Nuntius et Johanues Kepleri. DLoptice, secunda editio priori correctior, Londini, typis Jacob! Flesher, page 32 (reference from (4)) 2. M. l'abbé Outhitr, 1755, Tome II des Mémoires présente's & 1'Academie royale des Sciences par dives Savants, page 607 (reference from (4)) 3. Friedrich Wilhelm Bessel, 1841, Astronomische Untersuchungen, Ersterband. V: Beobachtungen verschiedener Sterne der Plejaden, page 209 till 238 quotation from page 210. 4. C. Wolf, 1877, Annales de l'observatoire de Paris, Tome XIV, deuxieme partle, page Al till A81: Description du groupe des Pleiades et mesure micrometrique des positions relatives des principales étoiles qui le composent. quotation from page A79. 5. C, Pritchard, 1884, Memoirs of the Royal Astronomical Society, Vol XLVIII, Part III page 225. On the Relative Proper Motions of 40 Stars in the Pleiades, determined from Micrometric and Meridional Observations. 6. William L. Elkin, 1887, Transactions of the Astronomical Observatory of Yale University, pi. Researches with the Heliometer; Determination of the Principal Stars in the group of the Pleiades, quotation from page 5. 7. Benjamin Ap^horp Gould, 1897, Cordoba Photographs, Photographic Observations of Star Clusters from Impressions Made at the Argentine National Observatory, page 50 till 66. 8. H. Jacoby, 1892, Contributions from the observatory of Columbia Colege New York, No.3: The Rutherford photographic measures of the group of the Pleiades. 9. H. H. Turner, 1894, Monthly Notices Royal Astronomical Society, page 489 till 504. Some Measures of Photographs of the Pleiades at the Oxford University Observatory. 10. K. G. Olsson, 1898, Astronomiska Iakttagelser och undersökningar austalda p9 Stockholms Observatorium, Vol. VII No 3: Photographische Aussmessung der Piejades. 11. J. Lagrula, 1903, Traveaux de l'Observatoire de Lyon, III, page 1 till 152: ïtude sur les occultations d'araas d1étoiles par la lune avec un catalogue normal des Pleiades, quotations from Page 88 till 90.
14 12, E. CR. Gaultier, 1900, Bulletin de la Société astvonomique de France, p. 441 till 444: Catalogue annuel des grandeurs photographiques de 300 étoiles das Pleiades* 13, E. Hertzsprung, 1907, Astronomische Nachrichten: Band 176, Nr 4204 p.49 till 58: Züv Bestiniraung der photographischen Sterngröbe. quotation from page 53. 14, R. S, Dugan, 1504, Publ. Astroph. Inst. Konigstuhl-Heidelberg, Vol II, page 129: Helligkeit und Or ter von 359 sternen der Plejaden gruppe. K. Schiller, 1905, idem page 133, Photographische Helligkeiten und mittlere Orter von 251 Sternen der Piejade Gruppe. 15. G. Muller and ?. Kempf, 1899, Astronomische Nachrichten, Band 150, Nr.3587. page 193 to 216: Bestimmung der Helligkelt von 96 Plejaden sternen. 16. H. E. Lau» 1912, Astronomische Nachrichten. Band 191, Nr. 4583, page 409 to 420: Die FarbentSnung der Piejadensterne. 17. R. Trvimpler, 1914, Astronomische Nachrichten. Band 200, Nr.4790, page 213 to 230. Die relativen Eigenbewegungen der Plejaden sterne. quotations from p.222 and 18. F. Kromm, 1911, Annales de l'observatoire de Bordeaux 14. lre partie: page 173: Mouvements propres de 160 étoiles de la region des Pleiades. 19. L. Bosa, 1910, Preliminary General Catalogue 1900.0 20. R. Trümpler, 1920, Publications of the Astronomical Society of the Pacific. Vol.32, p 43-49. A study of the Pleiades Cluster. 21. R. Trtfmpler, 1921, Lick Observatory Bulletin number 333, p 110-119: The physical members of the Pleiades group, citation from p 115. 22. J. Titus, 1938, The Astronomical Journal, Vol 47, No 4 page 25 to 35: The internal motions and mass of the Pleiades cluster. 23. E. Hertzsprung, 1934, Bulletin of the Astronomical Institutes of the Netherlands, Vol. 7, No 258, page 187 and 188: Provisional search for internal motions in the group of the Pleiades. 24. E. Hertzsprung, 1929, Monthly Notices of the Royal Astronomical Society, Vol. 79.8 pages 660 till 678: The Pleiades (George Darwin Lecture) quotation from page 661. 25. E. A. Krelken, 1924, Bulletin of the Astronomical Institute of the Netherlands, Vol. 12, No 49, page 55 till 59: Proper motions of stars belonging to the Pleiades.
15 26. J» J. Raimond Jr., 1928, Bulletin of the Astronomical Institutes of the Netherlands, Vol,A, no 143, page 135 till 137: Proper motions of 12 bright Pleiades. 27. E. Hertssprung, 1947, Annalen van de Sterrewacht te Leiden, Iteel XIX, eerste stuk A: Catalogue de 3259 etoiles dans les Pleiades. 28. L. Blnnendijk, 1945, Annalen van de Sterrewacht te Leiden. Eteel XIX, tweede stuk; A study of stars In the Pleiades region based on photographic magnitudes, colour-equivalents, spectral types and proper motions.
16 2. The Astrometric Measurements and their Reduction
2.0 Summary Astrometric measurements and their reduction are presented, aimed at a detection of internal motions in the central field of the Pleiades and the detection of members in the outlying region of this cluster. The area investigated measures 8,2 by 8.2 degrees and is centred on the brightest star in the cluster, Alcyone. Plates of different astrographs have been used, showing a spread in epoch of up to 85 years. Different problems connected with the use of overlapping plates in the definition of positions and proper motions are described. The position of the tangential point and the optical distortion of the Leiden Astrograph are determined by means of internal consistency of overlapping plates. A final reduction of the 400000 star positions measured on 719 different exposures has not been possible in the available time, but is still in progress.
17 1
2.1 Introduction An astrometric study of the Pleiades Cluster can provide two types of information on the dynamics of the cluster, viz, the distribution of stars and the distribution of velocities in the cluster. The mass function of an open cluster, giving the number of stars per mass interval, is of vital importance in understanding its dynamics. A proper motion study should therefore cover as large a magnitude interval as reasonably possible, Furthermore, the area over which the cluster extends, and how stars of different mass are represented in it, is essential information in the determination of the total mass of the cluster. Studies that only cover the central region of an open cluster can only provide biased information on those quantities. Members of the Pleiades can be selected with a high degree of reliability when proper motions are obtained with accuracies of 1 arcsec per millennium or better, while for information on the internal velocity dispersions the accuracy needs to be at least 0.3 arcsec per millennium. The astrometric studies presently available cover the requirements set above only partly. They are, in chronological order: 1. Hertzsprung (1947): A catalogue of proper motions and positions for 2920 and 3259 stars respectively, covering a field of 2.5 by 2.5 degrees and a magnitude range from 2.8 to 16. The accuracies in the proper motions range from 0.8 arcsec per millennium for stars brighter than phot.magn. 12.5 and increasing to 8 arcsec per millennium for the faintest stars. 2. Artyukhina and Kalinina (1970): Proper motions for 4228 stars in a region extending up to 1.5 degrees from the cluster centre. The accuracies claimed are 6 arcsec per millennium, although 8 arcsec per millennium seems more likely from a comparison with paper 1. The magnitude coverage is inhomogeneous, and varies between 6.5 to 12.0 and 6.5 to 14.0. Old epoch positions are those of the Astrographic Catalogue. 3. Pels (unpublished, a list cf probable and possible members selected by Pel3 is presented here in section 3.1). This study is basically the same as paper 2, only different new epoch plates with a scale of 40 arcsec per mm instead of 90 arcsec per mm, have been used. In addition, three Carte du Ciel plates were measured, providing proper motions for some fainter stars.
4. Jones (1970): Proper motions for 88 cluster members in the central region. Coverage in field and in magnitude range is inhomogeneous. The proper motion accuracy ranges from 0.2 to 1.8 arcsec per millennium, for stars between magn. 4.3 and 15.3.
5. Vasil<*vskis, Van Leeuwen, Nicholson and Hurray (1979): This project led to our much more extensive present program. It produced relative positions and proper motions for 146 stars in the central region of the cluster, covering a magnitude range from 2.8 to 12.5. The proper motion accuracy varies from 0.15 to 0.3 arcsec per millennium for cluster members, and up to 0.4 arcsec per millennium for non members. The information contained in these papers only partly covers the requirements set above for a study of the dynamics of the cluster. In what follows, the presence and possibilities of old epoch photographic plates will be investigated in order to see what improvements are possible to these studies. In estimating the possibilities of the old epoch exposures, they are compared with hypothetical infinitely good 1980,0 positions, because of which the errors in the resulting proper motions are only due to the standard deviations in the old positions. Later on, the extent to which these possibilities can be achieved is estimated, on basis of which the new epoch plates have been taken. Members of the Pleiades Cluster are found up to H.H degrees from the cluster centre, while the tidal radius of the cluster is probably at least 6 degrees or 12 pc (see Van Leeuwen, 1980). It was therefore decided to investigate a square region of 8.2 by 8.2 degrees, thus covering most of the cluster and providing a possibility of detecting a larger extent in the corners of the field. A covering of such an area by old epoch plates is only provided in surveys and the oldest of them is the Astrographic Catalogue. The area is covered by two zones, the Oxford zone for declinations 25 to 28 and the Paris zone for declinations 20 to 24. The original plates used for the Catalogue still exist and were made available by the Royal Greenwich Observatory and the Observatoire de Paris respectively.
The Astrographic Catalogue plates have a scale of 60 arcsec per mm. Assuming a standard deviation of relative star positions on these plates of 1.5 jlm, they yield positions with accuracies of 0'.'09 (the accuracies used here drop rapidly when stars are only marginally visible above the plate background). Each star is recorded on two different plates, while the brighter stars appear twice on each plate. The average epoch of these plates is 1895, which can at present provide proper motions with an accuracy of 0.5 arcsec per millennium for the brighter stars, for which four positions are available, and 0.7 arcsec per millennium for the fainter ones. The Astrographic Catalogue plates cover a magnitude range down to 12 for most, and down to 13 or 11 for some. There is especially a difference between the Paris and the Oxford plates, where the latter normally do not show images fainter than 12th magnitude, while the first generally reach 13th magnitude.
The possibilities to extend this investigation to fainter stars are limited. A few of the Paris Carte du Ciel plates were made available, each showing three images of all stars brighter than magn. 15. These plates were taken around '1900 and are of the same scale as the Astrographic Catalogue plates. They can provide proper motions for the faint stars, between magn. 11 and 15• with accuracies of about 0.7 arcsec per millennium. The images of the brighter stars overlap each other and require special pattern recognition algorithms. One other source of old epoch plates is available, although of much lower accuracy. It consists of four plates taken with the Franklin Adams refractor, at a scale of 200 arcsec per mm, and each covering a 9 by 9 degrees field. Their epoch is 1930. If the star positions recorded on these plates can be reproduced within an accuracy of 1.5 Jim, which is doubtful because of serious image deformations towards the plate edge (probably caused by bad alignment of the three objective elements), one may achieve proper motions for all stars in the field, brighter than magn. 14.5, with an accuracy of 3 arcsec per millennium.
19 The material presented above can provide a significant improvement in membership selection for the extended field. Detection of internal velocity dispersion is, however, only possible for the limited central region of 3 by 3 degrees. Here, deep exposures at old epochs exist, specially centred on the Pleiades. Five such exposures, taken with the Paris Astrograph around 1930, and several taken with the Leiden Astrograph, have been available for the present work. They can provide proper motions with an accuracy of 0.3 arcsec per millennium in the central 1,8 by 1,8 degrees region and 0,5 arcsec per millennium for the region extending to 3 by 3 degrees, New epoch exposures have been made with the Leiden Astrograph, at a scale of MC arcsec per mm, such that the possibilities of the old epoch plates could be used optimally and no loss larger than 15 percent in possible proper motion accuracy would occur. In order to estimate the number of new epoch exposures needed to reach this aim, four quantities are introduced, viz: T1, the difference in epoch between the new and old exposures 0, the standard deviation of positions on the old exposures ni, the weight-ratio between all new and all old exposures
I the standard deviation of the proper motion obtained from a least squares solution The last quantity is related to the first three by (see App.1):
ALL = ~ C1 + —) (1) T1 n1 From (1) it will be clear that, when ni goes to infinity, the situation described above for the old epoch plates is reached. Here we want to keep 1/n1 less than 15 percent, or n1 larger that: 7. As n1 is a ratio in weight and not in numbers, we still need to find the ratio in weight between one old Astrographic Catalogue exposure and one new Leiden exposure. This ratio is approximately given by the square of the ratio of their scales, which is (1.5)2. As will be shown in section 2.2.3, the real ratio is probably slightly smaller due to the scaling of the seeing, If one takes it at (LU)2, then two Astrographic Catalogue exposures have th<> same weight as one Leiden exposure. Accordingly, for ni exceeding 7 we require eight exposures at the new epoch for every two exposures at the old epoch.
New epoch exposures have been taken such as to cover the extended field eight times, with two exposures per plate. Each star appears twice on four differently centred plates, providing a four-fold overlapping system. The central 1.8 by 1.8 degrees region is covered by 60 exposures, while the region out to 3 by 3 degrees is covered by 36 exposures, except for the corners, where 20 exposures are generally available.
20 A total of 719 exposures on 471 plates have been measured, using the ASTROSCAN measuring device of Leiden Observatory. This device, the measurements and their reduction to positions and photometric parameters, are described in section 2.2. The construction of the new epoch reference frame, a task which could not be completed in the context of this thesis, is described in section 2.3, while in section 2.4 the reduction of old epoch exposures to the new epoch reference frame and the determination of the proper motions are described. Due to several problems in the performance of the ASTROSCAN, causing unacceptable positional scatter, the final measurements could not be started until the end of 1980. In consequence, the positional data became available rather late, which left only limited time for a proper reduction of the new epoch reference frame. In addition, a deformation arose in building the new epoch reference frause, possibly due to asymmetric errors, of the order of a few tenths of a micron, in the projection of the sky on a Leiden photographic plate. Such errors are hard to detect on a single plate, but cause rapidly increasing deformation in plate overlap procedures. The astrometric work presented in this section is therefore not entirely complete and in this respect the resulting proper motions, for stars brighter than magn. 1;t,5 in a 1.8 by 1.8 degrees region, are of a preliminary nature.
2.2 The Astrometric Measurements and Their Reduction
2.2.1 The ASTRQSCAN as Astrometric Measuring Device The ASTROSCAN of Leiden Observatory has been developed in order to measure densities on a photographic plate at accurately defined positions. The device consists of a mechanical part, which is basically a David Mann Comparator with two perpendicular screws driven by stepping motors, and an optical part, which consists of a stabilized 150 W Tungsten halide lamp, an illumination system and a lens which focusses the plate on a Reticon photo diode array RL 128 EC. The optimization of the optical system and its use for surface photometry have been described by Swaans (1981). Here, only some remarks will be made with respect to the astrometric use of the ASTROSCAN.
The mechanical part of the ASTROSCAN is shown schematically in Fig.1. It consists of a Table (A) to which a stepping motor, driving a screw, and a magnetic ruler are connected. The table has two polished flat ways on which the x-frame (B) rests by means of four roller bearings and two sliding graphite blocks. This frame is connected to the screw by a long nut. The readout of the magnetic ruler is connected to the frame. The nut is pulled on the screw by a preload mechanism. On the x-frame a second stepping motor is connected, driving the second screw. This screw as well as one "-shaped and one flat scraped way which are part of the frame, are perpendicular to the previous ones. Parallel to them a second magnetic ruler is connected. The scraped ways on the x-frame carry the y-frame (C), which is again connected to the screw by a long nut and a preload system. To the y-table another readout-head for the second magnetic ruler is connected. The stepping motors with their screws and the magnetic rulers provide two independent means of
21 measuring differentially the positions of the frames. One of these, the screw position, is recorded for each complete measure together with the positional difference with respect to the magnetic ruler. If this difference is too large, the measures are halted and. have to be restarted manually. The positional repeatability of the screw and ruler positions are 0.8 and 1.6 respectively. The overall linearity of the screws lies also within 1 JJ,m, ut the rulers show semi-regular deviations with amplitudes up to 2 u,m and repeating over intervals of 100 |j.m. The repeatability is, however, not regular enough to correct for. It was therefore decided to use the screw positions as reference for the positions of the frames.
S f I . ffH ' f I*N / S > •ii 11 j 111 ui i B 1111 in ii ii i [ ir. s /
Figure 1: Figure 2: A schematic picture of the mechanical An idealized measuring pattern part of the ASTEOSCAN. The different for a photographic plate, devided symbols have the following meaning: in 8 x 8 squares, and each square containing one star. A : Table r : magnetic ruler B : x - frame s : screw C : y - frame m : stepping motor n : nut g : gliding strips The circle with cross in the centre is the opening in the Table (A) for the illumination of the plate.
22 A photographic plate, which is situated in a plateholder fixed to the y-frame, is measured according to a file, giving for each star which is expected to be shown on the plate a preliminary (x,y) position on that plate and a matrix size to be measured at that position. A transformation from the (x,y) coordinates in the file to those on the plate in the ASTRO^CAN is obtained from a manual identification of two stars from the file on the plate, and performing a transformation between their file coordinates and the ASTROSCAN coordinates. The file has been arranged in diagonal ordar, 3hown in Fig.2 and Fig.3 in a simplified way. Each file has been divided into 16 x 16 squares of 1 x 1 cm, The sequence in which stars, positioned in these squares, have been measured, is based on the principle shown in Fig.2, By measuring this way, one avoids locally warming up the screws. The minimal rate of displacement along the screws is about 1 cm/min, compared with a thermal time constant of about 15 minutes.
At each file position the density distribution is measured in a matrix of which the size, determined by the magnitude of the star, is given in the file. This is done by scanning the plate in the y - direction in steps of 10 (im. At each step the array of 128 diodes records the intensity of the lamp transmitted through the plate in squares of 10 by 10 |J.m each. Only those covering the matrix requested are read out and recorded. Matrices broader than the diode array (1.28 mm) are covered by more scans, where the number of scans is always odd for symmetry reasons. The intensities are converted on-line into densities, based on calibrations carried out before the actual measures. Before storage on tape, the resolution of a matrix has been brought back to 20 by 20 LLm, by combining 4 elements each time. This can be justified by the fact that the time resolution of the emulsions used for the photographic plates is approximately 25 p,m. In addition, the decrease by a factor 4 of the data + X stream significantly decreases the computer time * needed for the reduction to positional and * + photometric parameters of the star from the 1 I II I I I I I density matrix.
Figure 3: The x (x) and y(+) positions as function of time in the case where a measuring pattern like that presented in Fig.2 is followed. Due to the diagonal measuring, local warming up of screws i3 avoided. (At s the x and y coordinates coincide)
23 2.2.2 The Photometric and Positional Parameters In this section the matrices of densities are reduced to positional and photometric parameters of the stars they cover. First the star in a matrix is located and a preliminary position is determined. Then, in an iterative loop, photometric parameters and the position of the star in the matrix are established. Finally, a listing is presented of all the different quantities that are to be stored on magnetic tapes, to be used later on. The positional information of a stellar image is contained only in its slopes. The first step has therefore been to determine the derivatives in x and y over the matrix given by 'd'. The presence of systematic gradients over the matrix indicates the presence of a star. The derivatives have been determined ovsr squares of 3 by 3 points and assigned to the central point of such a submatrix. They are defined as follows:
gy(i,j) = I (d(k,j+1) - d(k.j-D) (2) gji.j) - Z (d(i+1 l) - d(i-1,l>) (3) y ki3 t The preliminary position of the star can now be determined in many different ways. Here a method is chosen in which gx and g are summed up at constant i and j respectively:
Gx(i) =1
Gy(j> = I gy(i.j) (5)
The maxima and minima of Gx and Gy and their accompanying i and j values provide the provisional position of the star. A comparison of the maxima and minima shows the symmetry, and thus the quality of the stellar image. All four are part of the output data file created for each star. Before the density distribution of the stellar image is further investigated, the mean sky-level is subtracted. This level, and its standard deviation, are determined from the densities at the edge of a matrix. They are both part of the star's data file. A closer look at the mean sky-level and its standard deviation can reveal the presence of another image near to or at the edge of the matrix, which may disturb the photometric and positional parameters <.f a star. Such effects are corrected for, if possible, while the presence of a second star is recorded in the data file of the star. As the next step the mean gradient of the slope and the width of the stellar image are determined. First, weights are assigned to all points in the matrix, based on the derivatives in x and y:
2 w(i,j) = g2(i,j) + g2(i,j) Next, the distance r(i,j) to the provisional image centre is calculated for all points with w(i,j) above a certain value, thus selecting only points on the slope of the image. For these points, equations of condition are formed: d'(i.j) = T + S.r(i.j) (6) whei > d'(i,j) are the densities corrected for the mean sky-level. A least
24 squares solution is performed over these equations, all given equal weight, resulting in values for T and S„ These two parameters describe a cone of height T and slope S, which is transformed to a width at base level, W and a slope S, the first two photometric parameters. The next step involves a positional fit of a cone determined by (W,S) to the stellar image. This is cone by means of a least squares solution over small positional corrections AX and AYt which provide an updated centre for the star. Here, the equations of condition are given a weight w(i,j>, according to the coordinntes of the point used. Using the updated centre positions a new determination of W and S is performed, followed by a new determination of the positional correction. This iterative procedure is continued till the positional corrections are smaller than 0.5 ^.m. or at maximum three times. As the last step, two more photometric parameters are determined, viz, the mean density at the centre of the image, T, and the curvature of the density distribution, C, near to the centre. Both parameters result from a least squares fit of a paraboloid to the image top. The.fit is made „*or all points within a distance W/3 from the image centre. The data file for each star now contains the following items:
1. Star name 2. Position in x and y 3. Accuracy of position, 6* and 6y 4. Difference between screws and ruler position 5. Width and top density with their accuracies, W and T, ÖW and ÖT 6. Slope and curvature with their accuracies, S and C, fis and Qc 7. Mean sky and its standard deviation. Sky and ÖSky
8. Maxima and minima of Gx and Gy 9. Duplicity information 10. Positional system : a. screws + stepping motors or b. magnetic rulers
These data are stored on tape, ^receded by a record containing information on the exposure, such as epoch, plate number, exposure time, observer, and also the total number of stars identified on the exposure and contained in the file. Each exposure makes one file, stored on its own label on magnetic tape, thus making easy access possible to the data of any exposure. The star-datafiles are sorted on the number or name of the stars, to faciliate cross identifications between different exposures.
25 2.2.3 Interpretation of the Photometric Parameters The four photometric parameters derived in the previous section all contain information, over a limited range, about the magnitude of the stars. The most useful of them, the width W, has already been known for a very long time, and the almost linear relation between the logarithm of W and the magnitude of a star was already referred to by Gaultier (1900) as a commonly known relation. It is based on the assumption that a stellar image on a photographic plate is basically a distribution in density of which any cross-section is Gaussian, and of which the volume is proportional to the intensity of the image. When the dispersion of the Gaussian is no longer significantly influenced by seeing, it becomes linearly dependent on the image amplitude. Then, the volume is also proportional to the radius of the image, which explains the relation between the logarithm of the radius and the logarithm of the intensity. It is shown in Fig.1 for the cluster members of an older Leiden Astrograph exposure, which will be used also in later
B (magn) 11 13 15
Figure »l: The relation between W (plotted on logarithmic scale) and the Walraven B magnitudes for 131 clustermembers, as measured on plate LE05464 of the Leiden file. The different symbols indicate the distance of a star from the plate centre: (.) 0 - 10 ran; (+) 40 - 60 ran; (x) 60 - 80 mm; (o) > 80 mm. The bars indicate the accuracy of W at the given magnitude. An important influence on the shape of the stellar image, and thus on the four photometric paremeters, is the seeing. It will influence mainly the fainter images, and contributes to the width V of the image by a constant W(S) in first approximation a3: W2= (W(R)2+ W(S)2) (7) where W(R) would be the real width of the image without the presence of seeing. For unsaturated and slightly saturated images, the total "density-volume" of the image must remain the same, independent of the seeing. With worse seeing, the top height T and slope S, as well as the curvature C will decrease, and thus also the positional information in the image. The effect is best observed from diagrams of S and C against log(W), as shown in Fig.5 and 6. Both diagrams show an increase of S and C at low W, followed by a maximum, called here S2 and S1 respectively, after which S decreases slowly and C rapidly, the latter soon becoming equal to zero. At S1 the curvature of the top has stopped increasing due to saturation, while at S2 the same happens to the slopes. With increasing bad seeing, saturation will be delayed and occur at larger W values, while also S and C will be lower at S2 and SI. Thus, the seeing at the time of an exposure can be read off from the coordinates of Si and S2,
1.0 + + Figure 5: XXX The relation between X •• X ++ the slope S ft* ft < (density/arcsec) and the width W. The 08 maximum in S is called S2 and is a powerful tool to estimate the quality of an exposure. Symbols as in Fig.1. 0.6
0.4-
0.2
0.0 1" W 2" 4" 8" 16"
27 8 1 1 i Figure 6: C X *#W The relation between 6 * m • — the curvature C •• • + x (desity/arcsec2) and the width W. The maximum in C is called .4 • - S1, and indicates when * 0 saturation starts to x * x* a affect the top of the .2 stellar image. Symbols as in Fig.4. .0 — i 1 1 1 1" 2" 8" 16" 1 W l
2.S I 1 1 X Figure 7: The relation between T * * • "••-* • x '° ** the top density T and dens, •4 .** x the width W. The saturation effects 2.0 -** become visible for T >1.5 and cause an 1 increasing spread in T.
* X Symbols as in Fig.4. X
«X 1.5 ~* X -
. i *
1.0 -
• +
0.5 - -
00 1 1 1 1 8" 16"
28 Bad guiding will also influence SI and S2. It has been observed, however, that this affects mainly W, by increasing this parameter with only slight decrease in S and C. The positions of SI and S2 have been noted in the reduction log-book, in order to select later good quality exposures for the construction of the reference frame for new exposures. Figure 7 shows the relation between the top density, from which the sky level has been subtracted, against log(W). It clearly reflects the influences of saturation, as the top densities tend to go to a maximum. In addition, beyond W s 8?, the resJout of the ASTROSCAN gets saturated (densities >3.0). For the large T values, the linear subtraction of the sky level is no longer allowed, due to severe non linearities in the blackening of the emulsion and later in the readout of the ASTROSCAN. As a result, saturated stars at positions on a plate with different sky background level show a spread in top densities at equal W values. The spread shown in Fig.7 at large width originates from sky variations between 0.15 and 0.68. Near to complete saturation of the ASTROSCAN, the effective sky level subtraction should be reduced almost to zero. Top densities for saturated stars at different sky levels purely reflect these differences in T. A proper correction before saturation of the ASTROSCAN, however, involves a detailed description of the saturation curve for each emulsion. Saturation also influences the values obtained for the slope S.
When a proper correction for saturation effects has been applied, the parameters T and S can provide independent additional information about the magnitude of a star. Both contain more information on the magnitudes of the faintest stars than W, as this last parameter is too much determined by seeing for these stars and hardly changes as function of magnitude. The parameter S can also provide magnitude information for the brighter stars, as can be seen in Fig.5. The curvature C will hardly ever be able to add any significant magnitude information to the combination of W, T and S. The significance and the possibilities of the different photometric parameters are finally shown in a different way in Fig.8, as the growth of a stellar image, characterized by W, T, S and C.
0" o 2" 4" 6" 8" 10" 12" Figure 8: The growth of a stellar image, characterized by the photometric parameters W, T, S and C. The images are superimposed on the sky level, at 0.5 in density, indicated by a broken line. The numbers next to each profile represent B magnitudes connected to the images following Fig.4.
29 2.2.4 The Positional Accuracies The positional accuracies, obtained from the least squares positional fit described in section 2.2,2, show how well or how badly a cone can be fitted to the stellar image. It is determined by the tot;al amount of information present in the image, i.e. the integrated intensity over the exposure time, that ':ws used to build up the image. It is also influenced by seeing, especially for faint images and by non radial symmetry of images, either due 'co guiding errors or to focussing errors near plate edges (see section 2.3.5).
Figure 9 shows how the accuracies in x and y depend on log(W) and on the distance to the plate centre. A level of 0.7 }J,m i3 established as soon as the real width of the image starts to influence the parameter W, which occurs here at W = 2V'. For smaller W values the accuracies rapidly decrease and the positional standard deviation gets to values of + 2.0 |J,m and larger. The comparison of (6x,6y) values at similar W values is a strong tool in discriminating faulty images. The (6x,6y) values are to be used as weights (given by (0.8/Öx)2 and (0.&7§y)3) in the plate reductions described in sections 2.3 and 2.1.
l 1 1 1 2 "™ + Ay (jinn) — « ° X 1 + o o
0 1 i i i 1 I i I 2 "" + AX — + ° (/uml X + 1 - ^, °0"tf * «*•• ^^Bc"** • '•» \ « * n 1 I i 1 1" w 2" 8" 16"
Figure 9: The accuracies in x and y of the cone fits to stellar images. The influence of distance of a star from the plate centre becomes important for stars that are more than 80 mm away from this centre. The average accuracy soon reaches 0.7 Jim, which means that the positional fit with a cone does not contribute a significant error to the relative positional accuracies of the images, which vary from 1.2 to 2p.m. Symbols as in Fig.4.
30 2.3 Construction of the reference frame of positions
2.3.1 The preparatory catalogue of star positions A field of 8.2 by 8.2 degrees, approximately centred on the brightest star in the Pleiades, Alcyone, has been extensively covered by photographic exposures taken with the Leiden photographic refractor during the period 1977 to 1980, In this field, 121 plate centres have been defined, that form a rectangular grid in such a way that almost every part of the field is covered by four platss centred at different positions on the sky. The plate centres and their identification codes are given in Table 1. Figure 10 shows the zones (indicated in Table 1) in which the plates are grouped,
The central field (G7) and the first zone have been covered by 10 or more exposures on each of the nine plate centres. For all other zones at least one plate with exposures of 10 and 15 minutes has been taken at each plate centre. Thus a minimum of eight exposures is available for nearly every star in the field.
Table 1: Plate centres with their codes, zone numbers and numbers of exposures covered with the 1978 - 1980 survey of the Pleiades with the Leiden Photographic Refractor.
Codr • (197» d ion* nr Cods • (1975) d lorn nr Cod» a (197B) d tona nr • IP ••» • IB
1 B 1 <"* 5 27* 32' 5 2 1 2 3"««?« 27* 32- 5 2 1 2 3»36' 2 27' 32 5 2 B > 01. a 26 50 5 3 1 3 3 «a.a 26 50 « a i 2 3 36. 1 26 5n a 2 B a 11 01. j 26 OK ! 2 1r a 3 an.a 26 OB 3 a i 3 36. 3 26 OB 3 2 B I a m.2 25 26 5 2 1 5 3 aa.» 25 26 2 2 1 3 36. « 2J 26 3 ft B S a 01. 1 2a «« 5 2 1t A 3 •a.a 2« •a 1 10 1 3 36. • 2« • • 3 a B T a 01. 0 2a 02 < a p 7 3 aa.a 2a 02 1 11 1 3 36. 5 2« 02 3 a B B a on 9 23 20 ; 2 ' a 3 aa.7 23 20 1 11 1 3 36. 5 23 20 3 2 B 9 « 00 9 22 3» 5 2 r 9 3 «6.7 22 38 2 2 36. 6 22 3B 3 2 BIO a 00 A 21 56 i s no 3
31 Table 2: Plates used for the preparatory catalogue, giving the first two digits of the star numbers assigned B C D e F G H i K L to stars' in these fields (Nr), the 1 lit it 2 code as given in Table 1, the Leiden Photographic Refractor plate i file number, the approximate number 4 of stars identified in each field 5 and the estimated limiting magnitudes. - Nr Cede PIICc n 7 D "H "1 H f> IC 13092 202 It 5 8 02 F ft LtC 11045 265 \ H "3 F » LE13O9* 202 1* 5 9 nu H 8 LEI 3099 125 1* 5 01 It 8 LEI 3275 622 ,, 10 06 it e LEI337» «87 1* f7 IC « Lt1313T «13 lt 5 08 H « LE1J321 627 It 5 11 n9 F 4 LC1332] Aoa 15 0 10 V « LE1332Ï M? 15 0 tl 0 ( itnw 567 12 iii i i i 12 p ft LE1M79 5*3 It 13 010 LE13t70 920 15 0 Ift no LC1J35» TOO 11 5 15 H10 LEI 3357 «9* ik 1) 16 110 LC133M 5*0 it S
17 K10 LC13351 T10 it .5 11 K a LE133<6 too it .0 Figure 10: 19 H 6 LE13588 800 it ,5 20 H « Lt13«19 1000 15 0 The covering of the 8.2 by 8.2 degrees 21 H 2 LE13«66 • 30 It .5 22 It 2 LC13M2 1050 15 .0 field by a central field and five zones. H 2 LE<3*07 1200 15 .0 24 F 2 LEI 3*09 1250 15 .0 25 0 2 LE13t7t 1500 15 .5 26 B 2 LE13*75 1250 15 .0 27 a a LC11327 900 II .0 28 B 6 LE13269 500 It .5 29 B 8 LEI3*75 1100 15 .0 30 B10 LE13376 700 It ,5 31 B12 LE13172 1000 15 .0 32 D12 LE13*71 750 It .5 33 F12 LE13*6Q 780 ^5 HIJ I.E13«9 «0 11 45 35 K12 LE13*67 «50 It .0 16 M12 LE13152 650 11 .5
All stars that are to be measured in the whole field had to be identified and measured manually once. This has been done using 36 plates which together cover the field with only small overlap. For each of these plates a three tim«3 enlarged paper copy ha3 been made. On these copies stars are identified making use of the two exposures in order to avoid fake images. The stars are named by a 6 digit number, of which the first two digits identify the field (the number indicated in column one of Table 2) and the last four the star in the field. Table 2 gives for each of the 36 plates used the field identification, field number, the plate used, the approximate number of identified stars, and the estimated limiting magnitude.
The 36 plates used for the preparation of the catalogue are not all equally good. Unfortunately, some can only be measured down to m-osi1», while others easily reach the target of m =14.5. Because of extrapolation of the magnitude calibrations for stars fainter than mpg=i4.5 (see section 2.2.3) the estimates beyond this limit can be wrong by up to 0.5 magnitudes.
32 Only images shown on both the 15 and the 10 minutes exposures have been identified as stars. For each of the 36 plates the positions of all stars identified have been measured by hand, using the KOMESS Zeiss two-coordinate machine of the Leiden Observatory. This instrument has been equipped with magnetic rulers, that make an automatic readout of positions possible in increments of 10 microns. The positions are recorded on a disk file of the Huygens Laboratorium FDP 11/45 computer, together with manually provided information regarding star number and image size.
The files thus created have been converted into ASTROSCAN preparation files and using these files each of the 36 plates has been measured on the ASTROSCAN, To assure the identifications, also the second exposures have been measured using the same files. By means of the small overlaps between the plates the 36 files of positions that resulted from the Astroscan measures have been put together provisionally, in a way described in section 2.3.3. Thus a catalogue of all stars in the field has been provided, from which a file of positions can be prepared for the measurement of each exposure taken anywhere in the field,
During 1978, 1979 and 1980 a total of 373 exposures have been taken at the 121 plate centres, comprising some 2H2000 positions of 23000 stars. All these positions have to be combined to form a catalogue good enough to use it as reference frame for tha reduction of the old epoch plates. As these old plates were mainly taken with refractors other than the Leiden refractor, the constraints on the internal accuracy of the catalogue need to be very high. To fulfil such requirements a thorough investigation of the imaging behaviour of photographic refractors has been made, which is reported here first.
2«3«2 The ideal picture In the following the projection through ideal optics, given by a negligibly small objective, is discussed for two ^aaes. First, the projection on a tangential plane is derived, which is the idealized projection of the sky on a photographic plate. We shall refer to such a projection as the ideal picture. Next, we will refer to a linear picture, which is formed by a projection on a sphere with its centre at the negligibly small objective or entrance pupil, and which will be used to describe transformations between overlapping plates. In the ideal picture, the plane of projection is called the tangential plane, and the point on this plane Figure 11: where the distance to the entrance The projection on a tangential pupil is shortest is called the plane through ideal optics. tangential point, while the distance pp: projection pupil itself is given by T and usually tp: tangential point
33 referred to as tha focal distance. Pointing our ideal optics to the sky, a point (<&0),6(J>) on the sky is projected at the tangential point. A star at angular distance p radians from (a(0),6(0)) is projected at a distance r(i) from the tangential point, given by (see Fig.11):
r(i) s T,tanp (8a) Similarly, the linear picture, defined on a sphere with radius T, is defined as: r(l) s T.p (8b) where r(l) is measured along the sphere. The differences between ths ideal and the linear picture are thus given by: T.p3/3 (9) which amounts to up to 12 p,m for the Leiden Astrograph at the corners of a photographic plate of 16 by 16 cm and a focal distance of 524.3 cm (as derived from our measures). This far experts the measuring accuracy of 1.2 to 2.0 microns; it illustrates the necessity for considering deviations from linearity in the relations between partly overlapping plates. On the tangential plane (x,y) coordinates are defined such that the coordinate centre is at the tangential point. The y-coordinate corresponds with the direction to the celestial pole as seen from (Ct(0),6(0)). Seen from the tangential point and the corresponding point on the sky, the orientation angle ^ of a star with respect to ths direction of the celestial pole remains the same. In the (x,y) coordinates defined above one can thus write : x = T.tanp.sin-d* (10) y = T.tanp.cos-d' (11) Usually the positions of stars on the sky are expressed in right ascension and declination, (01,6). By means of the expressions for spherical triangles, equations 10 and 11 are written as (see Fig. 11) (A(X=(X-0t(0)):
sin(p).sinC9-) sin (& a) .cos (6) cos(p) sin(6).sin(6(0))+cos(ö).cos(6(0)).cos(ACt>
cos(p) .sinOfr) sin(6) .eos(6(0))-cos(6) .sin(6(0)) .oos( AOt) cos (p) 3in (6). sin (6 (0)) -t-cos (6). cos (6( 0)). cos (& a) These are the familiar projection formulae for a plate taken with a photographic refractor, which can be found in many text books (see e.g. Van de Kamp, 1967). They describe the geometric distortion in the ideal case of a projection through ideal optics.
34 2.3.3. The Transformations When plates are taken at different positions on the sky, transformations should be found between (x,y) coordinates that result from different plate centres. This transformation is described in three steps, of which the first and the last describe the transformation to and from the linear picture (r(l)s T.p). The second step describes the transformation between linear pictures.
2.3.3.1 Transformations between Linear Pictures for different Plate Centres We now consider the relation between coordinates on the sphere with respect to (Cl( 0,1),8(0,1)), corresponding to a tangential point ti, with those with respect to a different direction, (
35 2.3.3.2. Transformations between ijuuijl ttiul The transformation of the (x,y) coordinates with respect to ti to tli..»- with respect to t2, needs in addition to the above given spherical relations, the transformations from (x(1),y(1)) to (u(1)9v(t),w(D) and from (u(2).v(2)»w(2)) to (x(2),y(2)J. The first of these ia given by:
1'2 (15) u(1) = x(1> ,w(1) / T (16) v(1) = y(1) ,w(1) / T (17) and the second by: x(2) s T.(u(2)/w(2)) (18) y(2) s T,(v(3)/w(2)) (19) The idealized transformation described above has been used for the provisional plate connections needed for the preparatory catalogue described in section 2,3.1. For the final reference catalogue a more precise description is needed. It involves influences of the atmosphere and the performance of the telescope objectives,
2.3.4 The influence of the earth'3 atmosphere Because of refraction by the earth's atmosphere stars are seen at a higher elevation than where they actually stand. This refraction also influences the positions on a photographic plate. It causes a change in angular scale for distances measured in the direction to the zenith as well as perpendicular to this direction. Here we approximate the earth's atmosphere by concentric Homogeneous shells (see Fig.12). If one imagines a vacuum concentric shell at telescope level, then the radiation passing through the atmosphere to this shell does not change direction. Connes (1979) has shown that deviations from mean refraction as described by this approximation, known as anomalous refractions, are sufficiently cancelled for exposures of the order of 1000 seconds or more. The refractive behaviour of the atmosphere can accordingly be described as the effect of only a prism of air in front of the telescope, bounded by the vacuum layer and the telescope objective (see Fig.12). The upper surface of this prism is still curved with a radius equal to its distance from the centre of the earth. That curvature is, however, for practical purposes negligible. A classical prism with flat sides is all that is left (see Fig.13). The refraction resulting from this prism, expressed in zenith distance z, can now easily be calculated. It causes stars to be seen at a higher elevation by an amount:
be s (-Az) = (n-i).tan(z) (20) where n is the refractive index of the prism and thus of the atmosphere at telescope level. zenith star
if plote
Figure 12: Figure 13: The influence of the atmosphere The refraction through a prism of air on inconming radiation, A vacuum in front of the telescope objective. layer is imagined at telescope level. Radiation passing through the atmosphere to the vacuum layer does not change direction. The influence of the atmosphere as seen through the telescope is given by the influence of a prism of air (aee Fig.13).
On a photographic plate covering only a small part of the sky (not more than a few degrees) only differential refraction matters, that is, the change of b,z over the plate. This differential refraction is given by: dAz/dz = -(n-D/cos^z (21)
In order to see whether scale variations over a plate can be significant, equation 21 is expanded in a Taylor series in z, around z(0):
2 (d&z/dz)z.20= -((n-1)/cos z(0)).(1 + 2.dz.tan(z(0))) (22) The second term in equation 22 is small and can usually be neglected. In case of the Leiden photographic refractor it becomes significant, with respect to the measuring accuracy, at zenith distances beyond i»5 degrees. This has therefore been used as a limit for the plates that were taken, so that here only the first term in (22) needs to be considered.
In the direction of the zenith, as seen from the plate centre, z (as given in equ.21) equals the zenith distance. This direction is indicated by s in Fig.13. In the perpendicular direction, t, z equals zero. If we now define a rescaling factor in the direction s of (1-q) and in the perpendicular direction t of (1-p), we can write for p and q, using (22):
p = (n-1) (23) q = (n-1) / coszz (2*0
37 The scale corrections derived above are for angular distances, and can thus be applied directly to the (u.v.w) coordinates. As long as the field size is not more than 5 by 5 degrees, they can also be applied to the (x,y) coordinates. In case of larger fields serious interference can occur between the projection and the refractive rescaling, which makes the use of additional terms necessary. The coordinate system in which the rescaling, described by (23,2*1), takes place is rotated over an angle ^) with respect to the (x,y) coordinates as defined is. equations 10 and 11. The coordinates in which the refraction is described are orientated in the direction of the zenith. By means of the (Qti6) coordinates of the plate centre, the mean siderlal time 'ST' of the exposure and the geographic latitude 'b' of the telescope the zenith distance 'z' and the orientation angle \p' of the zenith with respect to the direction to the pole, also called the parallactio angle, can be calculated:
cos(z)=sin(b),sin(6)+oos(b).eos(Ö) .cos(ST-O) (25) 3in(q»=sln(ST-GQ.co3(b)/sin(z) (26) To correct the (x,y)-coordinates for the differential refraction, the coordinates need to be rotated first, over an angle ip after which scale corrections can be applied. By rotating back over (-vp) a corrected system of (x,y) coordinates is obtained,, This operation can be represented as a multiplication of the vector (x,y) by the matrix S:
\D) + (1+q).sinz(vp) ((1+p)-(1+q)).sin(Ü)).eos(afl i s = I , -I i ((1+p)-(1+q)).sin(ip).cos(ij}> (1+p).sin2(vp) + (1+q).cosz(^p) i (27) By means of equation 23 and 24 matrix S can be written as: (n-1) .(cos2(*p)+sin2(q))/cos2(z)) -(n-1) .tan2(z) .sinfcp) .cosftp) S = ! -(n-1).tan2(z).sin( The transformation described by S is not orthogonal. It rotatea the x and y axes by a difference of E described by: sin(E)=-(n-1).sin(2vp).tan2(z) (29) The difference in rescaling amounts to: 2 sx -sy = -(n-1).cos(2vp).tan (z), (30) where (1+sx) and (1+Sy) reprosent the rescaling factors over the x and y coordinates respectively. 38 In case one uses for plate transformations independent linear constraints for the x and y coordinates, the effects described above are fully accounted for. This can be done in the case of a small field covered by one plate, as described by Vasilevskis et al (1979) for a study of the internal motions in the central field of the Pleiades cluster. The values of E and (sx -sy ) have been calculated from the linear transformation constants that are found in that study for different plates. They are compared with values predicted on basis of equations 29 and 30. Figure 14 shows that the general behaviour is as expected, but the observed dispersion is about a factor two larger than expected based on the mean errors found. Probably the anomalous refraction (Connes, 1979) was not completely cancelled for most of the exposures, of which the exposure times were generally about 600 seconds. The relation has been used to reconstruct the hour angles with an accuracy of 10 minutes of some of the old exposures, for which the exposure data were partly lost. Corrections for differential refraction have only been applied for the zonewise plate connections described in sections 2.3.6 and 2,3.7t in order to avoid distortion of the field. Mi- o o r _2h 1 Figure 14: The relation between the hour angle of an exposure and the non orthogonality (E) and scale differences of the x - and y - axes, due to refraction and the direction of the zenith. The points and circles represent the observed (E) and (sx-Sy) values obtained from the reductions of several leiden Astrographic plates of the central region, described by Vasilevskis ^t £l (1979). L 2,3.5 Aberrations The objective of a telescope is not point shaped as was assumed in section 2.3*2, but has instead a definite size which has to be accounted for. The lens or lenses that form an objective are made in such a way that they approach ideal optics in imaging characteristics. For a real objective, however, the rays passing through the objective centre and those passing through the edges do not necessarily cross the tangential plane at the same position. The deviations, which are called aberrations, cause defocussed images of which the positions of the brightness centres do not always coincide with the ideal image described in section 2,3.2. This ideal image is, in first approximation only, described by the rays passing through the centre of the objective. Figure 15; The diffraction of radiation through a real objective (left) on a tangential plane (right). T: tangential point. The projection of a central ray and of a ray passing through an area o.do.djj) are shown, where the projection of the latter differs by dr and dt from the first. It has been assumed here that the optical axis coincides with the tangential point. Here we will assume that the lenses that form the objective are axially symmetric. The aberrations are then described with respect to the symmetry axis of the objective, the so called optical axis, which does not necessarily coincide with the direction to the tangential point. With respect to this optical axis we define a point (r,0) on the tangential plane and an area o.do.dlj) on the objective, as illustrated in Fig.15. If (r,0) is the position where the central ray crosses the tangential plane, then the aberrations from (r,0) caused by the size and shape of the objective are given by (dr.dt), where dt is directed perpendicular to dr. The size of dr and dt will depend on r, o, and (Jj . Here only aberrations of the third order in r and o will be considered. The coefficients of these aberrations are according to Van Heel (1933): idri i a4.cos(i)j) a3.(2+cos(aj))) (2a1+a5).cos((J)) a2 i j o3i idti ! aU.sinO))) a3.sin(2({j) a5.sin(t|j) ! • I ! O i io2r', i i !or2' Only the second and fourth coefficients of dr cause deviations in position, all others only cause defocussing. The effects described by the second coefficients of dr and dt are generally known as coma; they can cause a linear change in scale. The fourth coefficient of dr describes deviations known as pincushion or barrel shaped distortion. It is this coefficient which needs to be known in order to make transformations between positions on plates with different centres possible. For some objectives, for which a3 and a2 are large, one may expect colour dependence for scale and distortion. In that case a more refined description is needed and in particular knowledge of colours of stars will be essential. Diminishing the 3izes of the coefficients a3 and a2 is thus very important for objectives used in astrometry. In order to make transformations possible between the (x,y) and (u,v,w) coordinates, the amount of optical distortion, given by a2, and thus also the position of the optical axis need to be known. One also has to determine the amount of colour dependence of the coefficient a3. This has been done by means of a number cf partly overlapping plates taken in a 3 by 3 degrees area, covering the central field and the first zone. 2.3.6 The central 3_ by 3. degrees field The positions of the tangential point and the optical axis of the Leiden Astrograph have never been determined before. Also the amount of optical distortion still has to be found, as well as the amount of coma. The best way to determine all these parameters would be by means of a well defined reference frame. Such a reference frame with a sufficient number of stars does not exist at the required accuracy (approximately 0*101). Another possibility for finding these parameters is by means of an iterative procedure with different values, followed by a plate overlap solution as described in sections 2.3.2 and 2.3.3. This way the parameters can be found by means of internal consistency of the overlaps. The iterative overlapping plate reductions have been performed using 30 exposures, taken at approximately the same epoch on the 9 different field centres in the zones zero and one, covering a region of 3 by 3 degrees. Data on these exposures are presented in Tables 4 and 5, where they are indicated by an asterisk in tha last column,. The positions of the tangential point and the optical axis have been assumed fixed for all these exposures at one and the same position. This assumption finds its justification in the very rigid construction of the tube of the Leiden Astrograph, whil? no refocussing or other adjustmens to the camera took place in the period when these exposures were taken. In the overlap reduction, all exposures are simultaneously reduced to one reference field. To make this possible, the coordinates for the stars on each exposure and on the reference field are transformed to coordinates describing the star positions on spheres, the (u,v,w) coordinates defined in section 2.3.3. This transformation assumes positions for the tangential point and the optical axis, as well as the optical distortion. In the iterations, different values for these quantities are assumed, such as to minimize the final standard deviation that results from the least squares solution. 41 The transformations between different exposures in the system of the spherical surfaces have been performed with respect to one of the central field plates, LE13192. Each star appearing on one of the other exposures as well as on LE13192 provides an equation of condition describing the transformation of the plates to the reference field. Stars appearing on two or more plates, but not on LE13192, describe an equation of condition by the difference of the transformations to the central field for both plates. Thus n-1 equations for each star appearing on n plates are constructed, independent of the position of the star. The 30 plates used yield a total of 15000 equations of condition based on positions of 1900 different stars, These equations have been solved in a simultaneous least squares solution. In order to avoid distortion of the spherical reference field thus constructed, the transformations are performed while keeping the coordinate system of each plate orthogonal. This is only possible if the main source for non-orthogonality of the coordinates, the influence of atmospheric refraction, has been removed based on external information by multiplying (x,y) coordinates by the matrix S given by equation 28. Very small non orthogonalities of the ASTROSCAN are not important, as these are reflected the same way for all exposures. Then the transformations can be orthogonal and are described by four coefficients; ) + B(j) .v(j,i) + C(j) .u(j,i) (31a) where the index i indicates a star and the index j a plate; j = 0 if the reference field. In addition, two linear constraints, describing a possible magnitude dependence of the positions, are added, which provide a total of 6 free parameters for the reduction of each plate. u(0,i) r + I(j) .magn(j.i) (31b) v(0,i) = + K(j) .magn(j.i) The 15000 equations of condition describing the transformations of 29 plates to LE13192 thus yield a total of 29 times 6 = 17t unknowns. In the first step the position of the tangential point is sought. It has by far the strongest influence on the stellar positions on a plate. In this step the amount of optical distortion is assumed negligible. A total of 12 test points finally yielded a position of the tangential point situated 2.2 cm West and 2.3 cm North from the plate centre. The geometric distortion caused by the tangential point is pictured in Fig.16. It shows the camera to be badly out of adjustment. In the second step both the optical distortion and optical axis have been determined. This is done in an iterative way, in which first the optical distortion is determined at a fixed position for the optical axis, followed by a determination of the optical axis position using the optical distortion. The optical distortion was found to be barrel shaped and rather small, which makes the position of the optical axis uncertain. The maximum corrections needed are just below one micron. The final overlap reduction provides a preliminary catalogue of positions for all stars in the 3 by 3 degrees field. 42 Figure 16; The position of the tangential point on plates taken in the period 1977 till 1980 with the Leiden Astrograph. The circles indicate lin?s of equal geometric distortion that result from the tangential projection, at values of 1, 3, 5, 10, 15, 20, 25 and 30 micron. The size of the field is 15 by 15 cm. The orientation: North up and West right. 2.3.7 Construction of the catalogue of epoch 1978 positions All the new exposures positioned in the central field for the first zone have been used for the final definition of positions in the 3 by 3 degrees field. Each of these exposures, including the 30 exposures used in section 2.3.6, has been reduced to the positions defined in the overlap solution made for this field. The reductions are again carried out in the (u,v,w) domain. The coordinates of the stars in the preliminary catalogue have also been expressed in the (u,v,w) domain, which makes it possible to describe a linear transformation from exposure to catalogue coordinates: (32) v(0,i) = DU).u(j.i) + E(j).v(j,i) + F(j).W(j,i) where the index j stands for the exposure, 0 for the catalogue and i for a star. In case the position of the tangential point is not known with sufficient accuracy, additional coefficients are needed; this will be the case for the old epoch plates, but this can also be the ca3e for a new epoch plate if e.g. a glass splinter in the plate holder causes the plate to be slightly tilted. A splinter of only a quarter of a millimeter thick could cause a deviation of 7.5 mm in the position of the tangential point. In the case of the Leiden photographic refractor, where the tangential point is situated 13 cm from one 43 of the plate corners, this could give rise to deviations in position up to 5 microns. To correct for the influence of a deviating tangential point in the transformation from (x,y) to (u,v,w), two coefficients are added to (32), vis,: (32a) These coefficients are similar to the usual tilt coefficients for transformations in the (x,y) coordinates (see Vasilevskis ^t al, 1979). They can be applied for shifts up to a few centimetres. For larger shifts the coefficients are no longer equal for u and v, and additional terms have to be added. In order to correct for possible magnitude dependence of positions, caused usually by incorrect guiding or instability of the telescope at moments of strong wind, four terms were added to (23)t viz: u(0,i) ; + I(j).magn(j,i) + J(j).magn(j.i),magn(j,i) (32b) ) = + K(j).magn(j,i) + L(j).magn(j,i).magn(j,i) where 'magn' stands for the magnitude or equivalent parameter of the star. In these reductions only cluster stars are used as reference points. This makes it possible to investigate also the colour dependence of the scale, the influence of the coma described in section 2.3.5. Instead of the colour, a magnitude parameter could be used, as for unevolved stars in a cluster the magnitudes are more or less proportional to the (B-V) colour index. In such a situation one can add another term to (32), viz: (32O In case of the Leiden photographic refractor this coefficient was found to be less than 10-6, which causes stars at the edge of the plate with (B-V)=0 to be projected less than 1 micron (0.04 arcseconds) closer to the optical axi3 than those with (B-V)=1. By means of least squares solutions of the equations of condition ' given under (32,a,b,c) all exposures of the zones 0 and 1 are reduced to the preliminary catalogue formed in section 2.3.6. All of them are presented in Tables 4 and 5. Thus a catalogue containing positions in a 3 by 3 degrees field was formed, which will be called catalogue A. The reduction described in section 2.3*6 will be called the phase one reduction, and the reduction presented here phase two. 44 All new epoch exposures bearing zone number 2 (see Fig,10) are added to catalogue A to form catalogue B, First a phase one reduction was made in which catalogue A served as reference field around which a ring of 16 exposures has been reduced. All of the telescope parameters, viz. the tangential point, the optical axis and the optical distortion, were kept fixed at the values found in section 2.3.6. A phase two reduction should have been made for all available exposures towards the preliminary catalogue B in order to form the final version of this catalogue. At this stage, however, a distortion in the field was noticed from an unjustifiable increase in residuals as can be seen in Table 6. It made further reductions useless. The distortion is most likely not axially symmetric, as axially symmetric distortions have been accounted for. It is likely to amount to at most a few tenths of a micron on a single plate, which makes it hard to detect. A careful study of residuals left after the least squares reduction of overlapping plates could make it possible to detect the source and cure it. Table 4: Reduction data for the exposures covering zone 0. (H.A,: hour angle; Ext: exposure time; Nr: number of stars identified; 3.d.: standard deviation of positions obtained after transformation to reference frame; • : exposure used for initial reference frame, phase one reduction.) Fld 1«(H !•oordinatea II.» Ei t (»••!•1 Fp • rti Nr a .4. u« LE11063. G 7 h„I ?S« ii:s ., !?7 moo 1977 .777 578 ï LE13064. G 7 ai 29 ja n .6 -0 .7a ?atin 1977 777 174 .51 LE1106T. 0 7 1 aa 16 ;a 11 .1 0 .75 900 1977 .777 579 .68 LE11069. G 7 a? aa ?a ?*>. ^ 1.ai 900 1977 .777 527 .67 LEU071. G 7 3 aa 17 ?« 1.1 2.05 900 1977 .777 403 .7? LEIJ076. G 7 i ai 10 11 3a .1 -2 .«3 1?00 1977 .7(13 571 .67 LE13O77. G 7 1 aa IR 2a 11 .1 -1 .99 1200 1977 .7B1 5B5 .aa LE1307B. G 7 1 aj a? ;a 13 ,a .1 .5a 1200 1977 .783 551 .on IEI1O79. G 7 1 aj u? ?a ?.6 -1 .?6 1200 1977 .783 54? .5? LFiioan, G 7 ) aaIR 2a 2 .a .0 .31 1200 1977 .713 572 .19 LEI30S1, G 7 1 aa 19 2a ?a _j 0 .ia 1200 1977 .7(13 5B8 .53 LE13082. G 7 3 aj 33 ja 2a ,a 0 .59 1200 1977 .7B3 55a .66 UE130BJ. G 7 3 ai 32 2a ?a .« 0 .97 Ifflfl 1977 .783 571 .IB LEI30Ba, G 7 1 aa 19 2a 2a .3 1 .36 390 1977 .783 36a .90 LE1J0B7. G 7 3 «3 30 24 13 .5 -2 .21 600 1977 .802 577 .30 LE13107. 1 G 7 3 43 31 2a 13 .7 _? .73 600 1977 .805 5ao .58 LE131O9. 1 0 7 3 a3 10 ?a 13 .5 -2 .12 600 1977 .R5R 553 .29 UE1111B. 1 G 7 J «3 19 21 13 .? -0 .39 2460 1977 .919 571 ,4' LE111H6. 1 G 7 3 43 3a 2a ia .n -n .aa 3780 1977 .928 585 .61 LE13192. 1 G 7 3 «1 22 ?a ia .3 -1 .17 600 1977 .936 570 1.37 45 Table 5: As Table 4, for plates covering zone 1. h H 7 An' ?«* V.* ,n M i?nn 1177. W, 1 "cntuli H 7 3 »n w R.n • \'.?f Ann 1971. V"i .1' ltUllfl.1 H 77 1 «0 1? ?« ?.9 n,«? Ann 1177. Run «8 • I7 LII1136.1 H 7 l to IA ?• ?,9 • ?.O8 Ann 1977. 117 ««7 ,7R wiim.i H 7 1 to 11 ?t 13.9 -1 ,AA Ann 1977. 919 «81 .11 icuiti.i H 7 1 ti 1 11.9 1.9? (ion 1977. 919 tm .7? LC111S2.1 H 7 1 ii I 21 11.1 t\,91 ftnn 1977. 9?? 178 .7? 1(1116.1,1 1 tO 11.3 1. A? 1?00 1977, «2? «88 ,r,n H 7 r LU3I69.1 H 7 1 19 18 21 II.1 -0,<19 Mir, 1977. 92S • 79 • iA 1.(11190.1 H 7 1 10 ID 21 11.a .1,115 600 1977, 916 «An .AT 1.113093.1 H 6 1 to IS 21 St. 5 .0,17 Ann 1977. ftq? 1B7 .01 L(1109*.1 H t 3 10 »1 21 St.7 O.«7 too 1977. »n? 113 . ?1 LE11108.1 H 6 1 to • 1 it 51.7 .1,06 s?n 1977. 801 1?n .AA « a 3 «0 tT 21 13.1 -0.20 600 1977. 818 1?' .11 1113135.1 H 6 1 to IT 21 St.l 0.01 aoo 1977. 919 576 .91 1.(13111.1 H 6 3 tl 3 21 53.6 1,82 aoo 1977. 919 5«5. .11 LI131S3.1 H a 1 to to 21 Si.I .1.65 eoo 19TT, 922 t«a 1.65 LH3156.1 H a 3 10 13 21 53.5 0.2B 1200 1977. 921 sit ,66 1.113170,1 H a 3 10 1 21 St.l 4.23 too 1977. 925 SIS 1.IS LI13191.1 H i 3 to tl 21 53.» -1.10 too 1977. 936 195 1.79 1,113165.1 0 t 3 1328 21 50.9 0,01 900 1977. 777 Sit 1,69 LtlSOIS.I 0 a 3 13 29 it St.9 .1.8* 110 1977. 802 312 1.S1 LH30II.2 0 a 3 13 31 21 St.t -i.ta 615 I9TT. «02 «93 l.*0 LI13101.1 0 a 3 13 31 It 53.6 -i.]« aoo 1977. 80S 636 1.8? LI1311S.1 a t 3 «3 11 21 5S.T 0.11 aoo 197». asB 560 l.li 1,113127,1 a t 3 »3 tt 21 S3.) -1.78 too 1917. 917 620 1.50 L(l3U«.t a a 3 «3 21 21 55.2 -1.02 aoo 1911 919 611 i.ts LI131I5.1 S 6 3 13 sa J» 53.5 2.05 aoo 1977 919 602 1.T0 111315».! G a 3 »1 28 21 53.7 -1 .12 aoo 1917 922 595 3.61 LI111S9.1 c a 3 13 IS 21 53.6 0.62 1200 19TT 925 115 1.81 LI13171.1 0 a 3 12 10 21 51.3 0.15 aoo 1917 925 822 1.67 LI1118O.1 e s 3 «1 «0 21 S3.5 -2.78 aoo 1977 928 5B3 1.96 1(13095.1 a 3 ta 1 21 5I.S 0.08 630 1917 802 72t 1.75 H11106.1 a, 3 «a 10 21 53.6 -1.62 aoo 19TT 80S Toe 1.22 H13112.1 a 3 ta 11 It 53.4 -0.96 aso 1977 858 ati 1.52 H1312B.1 a 3 ta 1 2t S3.9 -1.5» «00 1977 917 asi 1.86 U1H17.1 « 1 tt 5 21 5». t -o.to aoo 191T 919 701 1.72 H13U6.1 3 «4 23 21 53.7 2.29 eoo 1977 919 as9 1.71 LI13I55.1 t 3 ta 5 21 53,1 -1.21 1200 1971 922 6S3 1.5S 1113161.1 ' a 3 ts 51 It 53.7 -1.T1 aoo 1971 915 651 1.68 111)111.1 t 3 ta 16 21 S3.B -2,13 too 197T 928 ata 2.56 LI13193.1 ' a 3 ta t 11 53.6 -0.90 aoo 1971 93a 601 1.63 LI13090.1 1 3 ta t 21 13,8 -1.23 aoo 19TT «02 630 1.95 Hl 1094.1 • ^ 3 ts SB 21 13.S 0.28 aoo 1911 eoj 1.79 LI13105.1 r 1 3 ta 11 21 11.6 -1.91 aoo 1977 80S 613 1.28 LI13U7.1 F 1 3.1a I 21 13.6 2.23 sas 1977 .858 558 i.aa LI13U9.1 F T 3 ta 2 21 H.3 -1.1T «00 1977 917 629 1.63 LI13I39.1 r 7 3 ta IB 21 12.t 0.35 aoo 1977 .919 621 1.1* LdlUT.I F T 3 •« 22 21 13.3 2.51 aoo 1977 .919 551 2.29 LI13160.1 r 7 3 ta 10 21 it.T 1.09 1200 1977 .922 ai9 1.71 111)165.1 F 7 3 ts 52 21 11.0 -1.31 aoo 1977 .925 638 1.59 II111B2.1 r 1 3 ia21 21 11.0 -2, IT aoo 19T7 .921 551 2.12 1.11319».1 F 7 3 ta 21 13.a -0,66 too 1977 .936 575 1.51 Lt1309t.l F 1 3 16 I 23 31.3 -0.17 ais 1977 .«02 609 1.90 utiigg.i F a 3 IS sa 23 31.2 1.25 aoo 1977 .802 579 2.01 1.(1110».! F 1 3 16 7 23 30.S -1.21 600 1977 .«05 568 1.31 LI13114.1 F 1 3 ta a 23 3t.a 1.87 600 1917 .B5B 505 1.(1 1113130.1 F I 3 ta 3 23 31.1 -o.aa «00 1977 .917 5S3 1.36 LI131I0.1 F ( 3 ta IB 23 31.7 0.62 aoo 1977 .919 576 2.85 LI131M.1 F 1 3 «4 23 23 2.80 aoo 1977 .919 513 1.38 1.111156.1 F ! 3 te 1 23 3tls -0.77 1200 1977 .922 579 2.2B Limaa.i F ( 3 IS 52 23 32.9 -0.01 aoo 1971 .925 SB! 1.76 LI13195.1 F ( 3 it a 23 31.3 -0.11 aoo 1977 .93a 582 1.92 LI13089.1 0 1 3 13 30 23 31.0 -1.39 «00 1977 .«02 555 1.72 LI11103-1 a 1 3 13 31 23 30.5 -2.17 aoo 1977 .805 SIS 1.B3 LI13113.1 a i 3 «3 27 23 32.1 -0.55 600 1977 .858 537 1.16 Lt13131.1 0 11 3 13 19 23 3».7 -o.ta 600 1977 .919 532 1.85 LI13U1.1 a ) 3 13 35 23 31.6 0.91 600 1977 .919 558 I.tl LI11162.1 e 1 3 1131 2] 14.t 2.09 1100 1917 .922 536 1,16 LI.1167.1 s 1 3 13 12 23 12.a -0.71 600 1917 .915 513 1.91 LI131BI.1 a 1 3 «333 23 3S.0 -1.33 1200 1977 .921 SIB 1.87 LI131B9.1 0 1 3 1336 23 15. t 600 1911 .938 102 1.8S LEIJ201.1 0 1 3 13 21 23 3».8 •olit 600 1911 .958 187 1.B8 LI13093,1 H 1 3 10 IB 23 31.0 -0.38 600 1977 .802 «60 1.11 LH3O99.1 H 1 3 to tt 2! 31.0 1.22 60u 1977 .802 «59 1.63 LE13110.1 H 1 3 10 •8 2< J2.7 -1.79 600 1977 .858 «01 LH3125.1 K I 3 «0 14 2J 33.« -2.33 600 1977 .917 132 1I5C LI13133.1 H 1 3 to17 23 33.6 -2.05 600 1977 .919 115 1.52 LII3M2.1 H ! 3 11 3 2: 31.6 1.25 600 1977 .919 117 1.R1 LEI3151.1 H ! 1 10 It 23 3t.7 -2.25 600 1977 .912 116 1.69 LE131S7.1 H 1 1 10ta 21 It.t -0.13 600 1971 .922 • 18 2.27 LE1316B.I H ! 3 10 37 23 32.8 -O.»2 600 1971 .925 I3« 1.75 LCinas.i H II 3 «0 58 13 11.9 -0.81 820 1977 .928 • 19 i.m LE11196.1 H I 3 «0«8 23 -o.aa too 1977 .9JA 1.46 46 Table 6: As Table 4, for plates covering zone 2. Pin.-.KI Fin ( UW 1 coordmntra II. h. F.xt(see) F.poch Hr 3.il. H* T n 4» 31* 30'. 6 A i? 900 1978, 831 381 1.97 !FM?7(I! ? I B 1 17 1 15. 2 79 600 1978, 833 319 1.78 T R 1 17 4 0.98 900 1979. 810 387 2.58 I 7 1 16 53 1?. 4 1.69 900 1979, 808 «66 2.45 -F. 11111'.? I 7 1 16 53 34 9. 9 1,80 600 1979. 808 404 2.7« LEI1457, 1 I 7 1 3d 57 24 12. 6 54 900 1979, 958 «67 2,62 t 7 1 36 57 24 9. 0 28 600 1979, 958 «21 2.53 t 6 1 15 41 24 52, J 2. 11 900 1978. 777 398 2.08 LKIJ?S7,'? I 6 1 15 40 24 57. 0 2. 40 600 1978. 777 32* 1.99 LF.Utin, 1 6 1 16 47 34 55. 0 1, 12 901 1979. 808 «90 2.75 ! 6 3 16 48 24 51. 1 1,40 600 1979. 808 «10 2,65 «13350. 1 t 5 3 36 60 25 37. 2 -1, 82 900 1978. 907 580 2,16 L61135O,2 l 3 17 0 35 42. 0 -1. 55 600 1978. 907 494 2.29 «11417. 1 I 5 16 35 25 3». 2 -1. 41 600 1979. 818 5«5 2.58 LE11417.3 l 5 3 36 35 35 30. 2 -1. 69 90(1 1979. 818 «21 2.19 «13351. 1 H 5 3 40 1 35 37.5 -1, 38 900 1978. 966 60S 2.17 «13151.2 H 5 «ft 1 29 42. 3 • 1.12 600 1978. 966 303 2.02 «13433. 1 H 5 3 39 56 25 32. -1. 27 900 1979 810 «47 2.41 «114J1 2 H 5 3 39 58 25 35. 8 .1 00 600 1979 810 582 2.43 LE13297 1 G 5 3 43 12 25 «3. 9 0 58 600 1978 907 603 1.95 «13297 2 G 5 3 «3 12 25 38. 7 0 79 900 1978 907 575 2.10 «13461 1 G 5 3 42 56 25 36 8 0 17 900 1980 030 581 2.60 «13161 2 G 5 3 42 57 25 32 8 0 45 600 1980 030 «03 1.96 «1329R 1 F 5 3 45 38 25 «1 7 1 04 900 1978 907 703 1.78 LEI 3298 2 F 5 3 45 38 25 36 7 1 22 900 1978 907 655 1.81 Lt13299 1 E 5 3 49 22 25 38 7 1 49 600 1978 907 SOI 2.26 LE13299 2 E 5 3 49 22 25 43 6 1 68 900 1978 907 644 2.09 LE13439 1 E 5 49 S 25 .13 7 -0 5.1 900 1979 818 566 LE13439 1 E 5 3 «9 5 25 36 4 -0 S3 900 1979 818 738 Ü85 LE1345B 1 E 5 3 48 55 25 37 3 1 .89 900 1979 966 8*5 2.19 «13458 .2 E 5 3 48 55 25 33 9 2 .22 600 1979 .966 761 2.31 LE13266 1 E 6 3 49 13 24 55 5 .2 .85 900 1978 ,810' «S2 2.59 «13459 .1 E 6 3 48 56 24 52 3 2 .50 600 1979 .966 720 2.48 «13159 .2 e 6 3 48 56 24 56 3 2 .75 900 1979 .966 684 2.29 LE13243 .1 E 7 3 49 26 24 12 .0 1.36 900 1978 .766 599 2.27 «13243 .2 7 3 49 25 24 16 .2 1.65 600 1978 .766 405 1.71 «13441 .1 E 7 3 49 3 24 14 .0 0 .83 900 1977 .818 61* 3.62 LE13463 .1 E 7 3 48 6 24 13 .5 1.29 900 1980 .030 624 1.90 LE13463 .2 E 7 3 48 7 24 8 .9 1.55 600 1980 .030 160 1.79 LE13277 .1 E 8 3 49 6 23 31 .7 0 .83 900 1978 .833 56» 2.59 «13377 ,2 E ft 3 49 7 23 26 .2 1.12 600 1978 .833 462 1.96 «13310 .1 E 9 3 49 8 22 50 .6 .15 900 19T8 .928 «24 2.7« «13310 .2 E 9 • 49 8 22 45 .7 -0 .88 600 1978 .928 338 3.07 LE1344D .1 E 9 49 12 22 45 .3 -0 .24 900 1979 .821 351 1.54 E 9 3 «9 12 22 «7 .7 0 .07 600 1979 .821 «20 2.24 LE13465 II E 9 48 8 22 48 .6 2 .29 900 1980 .030 364 3.18 LE11465 .2 E 9 48 8 22 «4 .5 i .58 600 1980 .030 284 2.67 LE133O9 .1 F 9 3 46 22 49 .6 -1 .63 900 1978 .928 402 2.12 LE13309 .2 F 9 3 46 5 22 54 ,5 -ï .37 600 1978 .928 403 2.16 «HJOR G 9 «3 5 22 49 .9 -i.10 900 1978 .928 305 2.04 LE1310R !? G 9 43 5 22 45 .7 -1 .82 600 1978 .928 275 1,92 LE13107 ,1 H 9 I 39 59 ?.2 4R .0 -ï .63 600 1978 • 922 202 3.96 IE 11107 .? H 9 39 59 32 53 .2 -i .45 540 1978 .92? 2.06 «1129? .1 I 9 \ 37 4 22 «7 .8 i.68 900 1978 .969 418 2.96 LK1129? .? t 9 1 17 ?2 «4 .0 i.98 600 1978 .969 351 2.74 47 2.4 Determination of the Proper Motions 2.4.1 Reducing Old Epoch Exposures to New Epoch Reference Positions Proper motions are determined by means of a comparison of positions recorded at different epochs. Here, the comparison is made between positions in the new epoch reference catalogue and those on old epoch exposures. In order to make such a comparison possible, all external influences on the old epoch positions have to be removed. These external influences, however, can only be found exactly in the case where the old epoch positions are already known, i.e. if next to the new epoch also the absolute proper motions are known. In the situation described here, only the new epoch reference positions can be used as reference frame. This reference frame provides only positions at an epoch that differs by HO to 80 years from those of the old epoch exposures. When using new epoch positions as reference frame for the reduction of old epoch exposures, the residuals found after the least squares solution are no longer only due to positional scatter, but also to the uncorrected proper motions. In fact, the least squares solution tends to spread the influences of the proper motions of tha reference stars over the parameters used in the solution, in a way described in detail in section 4.5.1. Every different selection of stars used as reference points will therefore put different constraints on the least squares solution and will lead to a different set of parameters needed to transform the old epoch exposures. Thus, also the resulting positions of the old epoch exposure will depend on the selection of reference stars, as well as the derived proper motions. The problem described above can be avoided to a large extent by constructing a consistent proper motion system in the central region, described in the next section, followed by reducing exposures in the outlying region under constraints described in section 2.4.3. 2.4.2 Definition of the Proper Motion System in the Central Region In order to avoid the creation of a different proper motion system for each individual old epoch exposure, a number of measures have been taken, of which the first is described here. Following Vasilevskis et £»1 (1979), a system of relative proper motions has been defined, based on the reduction of a selection of old' epoch exposures, all taken with the Leiden Astrograph. These exposures have been reduced directly to the new epoch reference frame, using for all one and the same selection of cluster members as reference points. The selected stars cover a large magnitude range and are well distributed over the field. By choosing only cluster members as reference points one diminishes the influences of proper motions, as only the relative proper motions, which are very small for these stars, can disturb the reference frame over large epoch differences. Thus, the influences of the relative proper motions are small, and are distributed for all exposures used in exactly the same way. The reduced positions thus obtained can be combined for each star, providing a reference system of proper motions defined by the selection of reference 48 stars. The reduction of the selected exposures has been performed using transformation equations as presented in section 2.3-7, equ. 32 a, b, c, but without using the quadratic magnitude term of 32 b and the coma term of 32 c. The first has been left out in order to avoid magnitude dependent distortions of the proper motion field and the second because no coma has been detected for the new epoch exposures taken with the same objective. The second step in the definition of the proper motion system involves an improvement of the quality of the proper motions, 'Phis is done by using all available old epoch exposures of the central field, taken with the Leiden Astrograph, and including those used in the previous step. All cluster stars are used as possible reference points, as long as a relative proper motion in the system defined above is known. The old epoch exposures are reduced to positions of the new epoch catalogue, corrected to their cwn epoch by means of the relative proper motions. In the reduction, equations of condition have been used as given in equ. 32 a, b. The system of proper motions remained unchanged. The steps that follow have been carried out but the results are of little or no value due to the distortion in the new epoch reference frame. Four sets of exposures taken also with the Leiden Astrograph, and covering the fields North (G6), South (G8), East (F7) and West (H7> of the central field (G7), have been reduced next. Here, the reduction, again based on only cluster members, involves two types of equations of condition. The first type is the same as described above, and is used for stars of which the proper motion has been determined. The second type, for stars of which the proper motion has not yet been established, describes a connection between two old exposures such that the proper motion of a star is accounted for without it being known or calculated. This is done in a uay very similar to that described for the overlapping plate reductions in zones described in section 2.3.6. First, the equations of condition are normalized, i.e. the difference between the resulting old epoch position and the new epoch position is divided by the epoch difference, which is for convenience expressed in units of 50 years. Thus normalized, a difference of these equations of condition can connect the reductions of two or more exposures outside the region where proper motions are knowtt, still taking into account all influences of proper motions and still maintaining the constraints of the reference system of proper motions from which we started. The last step in the central region proper motion reduction involves the reduction of five long exposures (30 to 90 minutes) taken with the Paris Astrograph, Their reduction is basically the same as described in the previous step. Here, however, special care has to be taken with respect to optical distortions, which can now be read off from residuals left after the least squares solution. As the new epoch reference system is still distorted, the optical distortions have not yet been deduced. 49 In order to combine all old and new epoch positions properly, a least squares updating mechanism is used, due to which only a minimum amount of data (nine quantities) needs to be stored in order to use all available information (The method is shown in Appendix 2). This also implies that the final catalogue with postion and proper motion data, can be updated any time with new or old exposures, still using all available data. In order to control faulty positions, a record is kept of the last five data points added to the star's data file, which can still be subtracted from the normal equations. All data points added in the normal equations are checked, as far as possible, for faults. 2.4.3 The Proper Motions of the Outlying Region The reductions of the old epoch exposures that cover the outlying region of the cluster are basically the same as the reductions described above. The only difference, however, is in the necessity of using other than cluster stars as reference points, as simply less and less cluster stars are available in those regions. Still, the proper motions are defined in the same system, only disturbed by possible influences of colours on the positions of the reference stars. From the reduction of the central region, however, it seems that for Leiden Astrograph plates these colour effects contribute a standard deviation of 0.3 arcsec per millennium or less to the proper motions, which is significantly lower than the expected accuracies for proper motions in the outer region. These effects may be somewhat smaller or larger for other Astrographs. The reductions are to be performed in zones, thus systematically extending the proper notion field outwards . Due to the problems described at the end of section 2.3.T, no results can yet be presented. Acknowl edgemevtts It is a pleasure to thank Prof Blaauw for initiating the present astrometric research of the Pleiades and Prof Vasilevskis for hi3 support and advice during his stay as guest professor at the Leiden Observatory. I would also like to thank R.S.LePoole for stimulating discussions and A.A.Schoenmaker, R.Rijf and A.de Jong for all the work done with respect to the ASTROSCAN measuring device. W.Brokaar made the prints used for the preparatory catalogus. Prof.Pagel red the manuscript and provided many corrections to the original text. Last but not least I would like to thank the many students of the Leiden Observatory, who spent often cold nights making the new photographic exposures used in the present work. 50 Appendiidix !_1 Given rposition„ s x(j,i) at epoch T(j), the proper motion |I(x,l) is determine•mined byby:: which leads to normal equations i II I(T(j)-T(O)) ! fx(O.i)! llx(j.i) ! I * i ! = !_. I A.2 I Choosing T(0) such that 2E(T(j)- T(0)) = 0, the accuracy of |A(x,i) is given by: 6 I2I2)l/2 A.3 Using ni new exposures at epoch T1 and one old exposure at epoch zero, and taking the standard deviation of Ax(j,i) as 0, one arrives at equ. (1). Appendix £ The updating of the normal equations, given in A.2, involves the knowledge of: n, T(0), I(T(j)-T(O)), JTx(j.i), Zx(j,i).(T(j)~T(O)) as well as*. Zy(j.i) and ry(j»i).(T(j)-T(O)). In order to update the estimate of the dispersion and the standard deviations of x(0,i), |X(x,i), y(O,i) and pCi), the first part of equ. A.3 needs to be examined: )2= ICx(j.i) - x(0,i> -^Cx,i).(T(j)-T(O)>)2 A.U With Z(T(j)- T(0)) = 0, we can write for x(0,i) and^i(x.i): x(0,i) = £x(j,i) / II A.5 |I(x,i) = Ix(j,i).(T(j)-T(O)) /I (T(j)-T(O))2 A.6 From A.4, A.5 and A.6 the following relation can be deduced: _. (I(x(j,i)) (Ix(j,i).(T(j)T(O))) = Ix(j,i) ~- A.7 I I2 Thus, by also keeping a record of Ex(j,i)2and Sy(j.i)2, also the standard deviations can be updated, which means that nine quantities altogether are needed to maintain all information used in a least squares solution. 51 Artyukhina, N.M. and Kalinina, E.P., 1970; Trudy Gos.Astron^Shternberga, 39, 111 Connes, P., 1979: in Modern Astrometry, IAU coll.18, 339; ed. F.V.Prochazka and R.H.Tucker faultier, E.Ch., 190Q: Bull.de la Soc.Astron.de France, Hertzsprung, E,, 1917: Ann.Stèrrew.Leiden, Vol XIX, part one Jones, B.F., 1970: Astron.J., 75, 563 Swaans, L.W.J.G., 1981: Implementations of Electronographic Pictorial Photometry, Thesis Leiden University. Van de Kamp, P,, 19(>?Ï Principles of Astrometry, W,H.Freeman and Co, London Van Heel, A.C.S., 1933: in Leerboek der Optiek, by C.Swikker, P.Noordhof N.V., Groningen, Netherlands Van Leeuwen, F., 1980: in Star Clusters, IAÜ Symp.85, 157; ed. J.E.Hesser, Reydel Publ., Dordrecht, Netherlands Vasilev3kis, S., Van Leeuwen, F., Nicholson, W. and Murray, C.A., 1979: Astron.Astroph.Suppl., J7, 333 3. The Photometric Observations. Differential Reddening and the Variable K Stars in the Pleiades 3.0 Summary Photometric observations of stars in the Pleiades cluster are presented. First, in section 3.1, a photometric survey is described aimed at confirming indications for membership of 193 stars selected by G.Pels and a determination of the distance to the cluster by means of nearby stars. The data for the brighter stars have been used in section 3.2 for a determination of the differential reddening distribution in the Pleiades cluster. The survey described in sections 3.1 showed the first indications that many late G and K stars in the Pleiades are variable. New observations, which are described in section 3.3, confirmed this and showed that most or all of these stars are variable due to rotational modulation. The surface flux distributions of these stars are found to be modulated with an amplitude that is increased with increasing angular velocity. Moreover, the angular momentum contained in each of these stars, which are probably just arriving on the main sequence, is found to be compatable with that of the total solar system. There are indications from much slower rotating G stars in the Pleiades that these stars are rapidly losing their angular momentum, a process which may presently be observed from the K stars in the Pleiades. 53 3.1 VBLUW Photometry of the Pleiades and JJ Determination £f their Pistand's 3.1.0 Summary We present measurements in the Walraven VBLUW system of 391 known or suspected members of the Pleiades cluster. Of these, 62 stars are found not to be members. The remaining stars range from spectral type B8 to K5 and cover the main sequence over an interval of 6 magnitudes. They are used here together with similar measurements of nearby stars (distances < 8 pc) to measure the cluster distance. This distance is determined as 130+5 pc (M-m = -5.57), where the main contribution to the uncertainty is due to the inaccuracy of the parallaxes. 3.1.1 Introduction Photometric surveys of the Pleiades cluster are published on the Johnson UBV system by Nicolet(1978) for stars earlier than late K. In the Geneva system Rufener(1981) published results for stars earlier than spectral type GO and in the StrBmgren system Hauck and Mermilliod(1980) compiled measurements of Pleiades stars earlier than spectral type F. For all these studies stars were chosen from the Catalogue of the Pleiades by Hertzsprung(1947) and from a selection made by Trumpier(1921). Our photometric study in the VBLUW system of known and suspected members was planned for various purposes. One of these was to establish membership for a selection of 193 probable and possible members by Pels. These stars were refered to by Pels, Oort and Pels Kluyver(1975) and are presented here. The second was to determine the distance to the Pleiades independent of the distance to the Hyades. For this purpose 13 stars with parallaxes ranging from 0.12 to 0.30 arcsec were included in the photometric measurements. The paper is organized as follows. Section 3.1.2 describes the selection of the stars and section 3.1.3 the reduction of their photometric measurements. Section 3.1.4 gives a discussion on the membership of the stars in the Pleiades and the last section, 3.1.5 describes the distance determination. 3.1.2 Selection of Stars Three groups of stars were selected. The first group consists of 198 stars brighter than photographic magnitude 14.5 from possible members selected by Hertzsprung(1947). Most of these stars are well determined cluster members, based on accurate proper motions. The membership is uncertain only for. a few stars near to the edge of the measured field and stars with proper motions only approximately near to the cluster proper motion. The second group consists of the 193 stars outside the central region, that were selected by Pels. This selection is part of an extensive proper motion survey set up to find members of the Hyades cluster (Pels et al(1975). (The North-Western region of the Hyades lies in front of the Pleiades cluster.) Pels determined proper motions by means of a comparison of positions published in the Astrographic Catalogues of Oxford and Paris with positions 54 measured on new plates taken with the Leiden Photographic Refractor. This refractor has a scale of 39*.'35 per mm so that the Astrographic Catalogue fields with a scale of 60V per mm (and the same 16 by 16 cm plate size) were not completely coverd. The coverage of the measured field is shown in Fig.1, as well as the field measured by Hertzsprung. Figure 1: The surface distribution of plates used by G.Pels to derive proper motions is shown together with the area studied by HertzsprungdW) (broken line), superimposed on a Franklin Adams plate taken in 1933. The partly dotted line on the edge of the field indicates the approximate border of the astrometric study described in section 2. 55 The faintest stars measured on these plates were usually of the 13th photographic magnitude. For some fields the original Carte du Ciel plates of the Paris zone could be used as reference, in which case this limit was at 15th magnitude. The proper motions measured by Pels have an average accuracy of O','OO8 per annum. With this accuracy a considerable number of non-members are included among the possible members, especially for the fainter ones, In his selection Pels compiled a list of 118 probable and 75 possible members, shown in Tables 1a and 1b respectively. Two stars turned out to be listed twice. The selection criterion was based on the coincidence of the proper motion of a star with the cluster proper motion. In both tables the stars selected by means of the Astrographic Catalogue are listed first, followed by the selection from the Carte du Ciel plates. Search maps for all these stars are in preparation. Table 2 provides a cross reference for stars with HD numbers and for those selected as possible members by Trumpler(1921). Table 2 Table 3 Cross reference for star numbers Data on selected nearby stars. of Pels, Trumpler(19ai) and HD pi = 1000*parallax in arcsec Pals HD Trunpler Pels HD Trumplar No 11921) No 11921) J-No HD P' e m sp 1989-cQordlnates Name 3 22444 57 23664 S115 e s* 58 23433 S84 5562 216803 123 8 6.5 K5 22 55 14 -31 49.5 i» 22637 S26 61 23498 S93 5563 216899 159 5 8.5 M2 22 55 45 •16 27.6 Ross 671 13 22987 S37 67 2395» S149 5565 216956 144 7 1.3 A3 22 56 33 -29 43.7 Alpha PsA 16 22578 SJ3 70 24382 Rfifl 5763 169 7 9.2 Ml 23 48 96 + 2 18.7 5« 1581 134 8 4.3 F8 19 91 -65 ue.8 Zetta Tuc 17 2268» 73 24836 21 22614 S25 74 J4J79 S16S 156 4628 145 5 5.8 K2 9 47 14 • 5 12.3 26 23290 82 24711 S185 352 19369,1 148 7 6.9 05 1 39 99 -56 17.9 p Erl •27 2Ï977 S39 125 22491 S29 356 11476 133 6 5.3 C5 1 41 28 •29 11.4 107 Psc 32 23133 127 22702 S29 36S 1(799 275 5 3.6 KB 1 43 99 -16 92.6 eau Cet 528 16169 147 5 5.9 KB 2 34 49 • 6 46.2 35 S42 134 23820 48 21792 S127 149 23312 S6t 59» 17925 1J7 E E.I C5 2 51 31 -12 51.1 49 23388 S76 159 23935 S144y 783 29794 156 8 4.3 C5 3 19 96 -43 98.8 82 Erl 52 23492 S7C 163 24899 S194 742 22949 393 4 3.9 K2 3 31 59 - 9 31.5 epsil. Eri 54 23852 SI 37 164 25U8 1»77 39652 125 5 3.3 FS 4 48 45 • e 55.6 1 Orl P!3) The group of nearby stars with well known parallaxes was selected from the General Catalogue of Trigonometric Parallaxes by Jenkins(1952, 1963). I1* stars were selected of which one is double. All have parallaxes larger than QÏ12. They cover a spectral range from A3 to M2 and are presented in Table 3. 3.1.3 The Measurements On 28 nights during 1979 October, November and December we performed photometry of the selected stars, using the VBLUW photometer (Walraven and Walraven, I960) on the Dutch 91 cm telescope at La Silla, ESO. With this photometer the incomming light is recorded simultaneously in five channels, ranging from the visual to the ultra violet. A description of the telescope and the photometer was given by Lub(1979) and a more detailed description of the calibration of the system by Lub and Pel(1977). Other references can be found in these papers. 56 Table la Table 1b Probable members selected by Pels. Possible members selected by Pels Numbers 106 to 118 originate from Numbers 184 to 195 originate from Carte du del Plates. Carte du del Plates. 3 as 21 •II.I -41.4 I 1 2t Ul \1 \ M 1 «14 H,t 11,* •)«.! f««,l i* ii *JI II,a •lltl -Sl.S 1 1 1* «1.1 111 1HN «11 11,1 *,1 «II,* -«, t 14 IS «11 ll.t «tl.I -•*.• * »at «aa <•* >MII «*i M,i u«* •!«.* -*7.« at 14 tl2 ll.S % i It «44 M III I *M «,t If,I «*•** -13.1 )* I* til SI,I 1* «44 3 31 I «14 11.7 «11,4 -ll.* •14 -If il i\ «ia ».* 1* •M 1 II 47 «IS SS.l «tltl -49,7 It 44 •!! S.I H •11 i at ii «it s,« «31,3 t*l,4 11 11 «If 11,1 31 «31 1 14 17 ill U.I •>*«,* « 41 «IS I.I 11 '4t 1H1I *i\ 41,1 S':5 •11 -43 H II 'J* tl.t 11 «11 1 1* H «It 14,1 It «1* tl.t * 7.» .(1,J 11 111 1 11 11 *11 lltl • 1,1 -«1.1 ) It 11 til 11.7 '5:5 «» -H 11 «11 171 « 41 «11 11,1 IJ.4 1 11 11 «11 t.i ll U *V It,I U.I in» «a_a in 11 \\ til 44,* II.I •II -11 11 It *ll ]i,C til.I i*7,« «a* -ii It ]* «It VI,S I 1111 «31 «11.4 • ».• 1 itit •)» «14.4 -43,1 11,1 1 11 «11.7 311 li,4 I 11 I 14 4* tl* 14, i II, «U.I W'i tl.a 114 t «14 H.S It «14 lit i ii ii «as u.t II 1 ill 11.» tlti «ll.l -ll.I *i ~ "i «IS 27,7 1 H « II II «11 U.I « 1,1 -11.4 1 H » «1* 11,1 44 It ill 11.1 * V,« -17.1 «ii at s i ii «17 H.I I* 1* «It ll.l «II,* -«1.1 14t u «it n,i «H.4 4, a 144 1 4* •11 41 H tl* 41,1 Ü 1 » I «11 U.I «1S.1 -41.1 i4S 4* «11 44 |t «1* 4t.l •17,1 -»S.4 1H» til 14.1 «13,I -41*1 14* . 44 •It 4» i n* ai.t •IS.S -JJ.l 1 » II «11 14.1 •lt,S -«1.1 117 : 14 •12 41 * «11 «1,4 * «.* -1S.1 ii4 ni It* 1 44 Ml 4.1 ii m n,< •IS,] -1J.4 i 4i ai *ii 17.* •IS.» ««*,* It* I 44 «It It SI «IS 11,1 1 M H *M M,7 •«.* -»,« •33 -3t «1 41 «IS 11,1 •I:? «14.9 -11,* SI > tjS Sl.t 1 41 II «It lltl U.t 1 41 4* «11 ll,« til.» -M.t «9 41 t» 11,1 «ll.t •)t,4 t I* «14 lift inn tit u,% «11.1 «41.4 •11 -44 n H u* u.i «14.T -ll.l «11 41.1 ti.» «11,1 -M.I M 11•11 U,l 11.4 «M>4 -11.1 •IÏ ll\ 1 4* 14 «11 11.4 •11.3 -4*, I IT » it.' 11*3 • *.* -H.I 11.1 •11.» -«1,3 11 II 11.* «IV.» -li.H tl.i u.* • ll.t •,!«,* 41 11:« It.l • It.l -M.I iU «31.4 -IS.t 21.1 • t.t -11.4 3 41 44 «IS M.S U.I •it.* -«*.s 41 1* •II 11 I It.t •IÏ.2 -M.7 1 J* «16 3 «I IS *1S 11,4 II. I «aj.i -42.1 44 14 •11 44 7 II.I • tt,i -«.I 1 J* tit 3 4* 4* «14 M.t ii.a «11.4 -«1.1 SS 11 •23 M 4 1.1 •14.1 -41.* 1 » «1* 1 41 41 «IS 41.1 •;•.* -«•.* 12 •!« Ill SI I •21 IS * t.l « t.a -II.I i 3* «a* 14i 7 «a* i.i «U.t -31.1 s: «a* us ll.t «ll.S -11.1 1 41 14 tl* II.* •IS.» -13.3 Si *« 17f 5* J* •2* 11 * 11.3 •11.7 -ll.t i ia » «a* at.t «ll.l -41.1 • 1.4 -41, t 3 14 41 «31 M.« •31.1 -«,* si «at ai* •11.9 -11.7 J «J 11 «31 4.* «33.* -SI.1 Si It «It ll.l 12.1 •lt.l »M.« It* til 1411 4* H «II ll.t «IS.t -31.1 •21.S -11.» "» II «II H.I » t «II iT 3 13.3 •It.* -M.I 1 41 •11 11.4 4t St «ll.l -SI.I 11 l*.l 14 «11 111 *!),* -)!,! 11 IT* t 4* «11 .1 S til • 1,1 -ll.l 11* «ii i4* i ib «i m i u *:i in « 1 «at 14 4 4.4 • *.* -41.1 1 44 V , •It.* -ll.l i ai «12 )• I ll.i 1 44 41•?i U.I «11,1 -41.4 in 4 i «ii an i ii •ii II ) U.t «It.t -ll.S 1 41 » «14,4 ,W.J 11* 4 I «11 US it si •ii 11 • tt.4 • t.t -44.2 •IS 14.1 •11.3 -ll.t t 4t S •IS 11.7 • ll.t -U.I lit 4 t *» lit 1 44 H » »4 •I*.* -«I.i 111 4 • «It 1*1 •ll.S -41.1 •13.4 -41,* * H •IS 14.1 •31.1 -It.l 4« •as *n 4i. t 1 II OS 14.1 • ll.S -14.3 44 •i* «it n.4 •ii.a -si.! ttl 4 I *12 111 U It •11 1.1 •at.i -is.* •It.l -41.S 44 •22 *» ai.t • 11.1 -M.I 44 •22 «la a.» •lt.t -ll.l 1 H 31 43.1 13.4 •12. -19.4 tt •23 «tl.t -44.2 1 11as 49.7 13.3 •19. •ll.l tt •22 «a».« -4S.I •11 -44 It «12 311i at is•I:5kf 13.4 11.* •XI. -Sl.t «« •22 •31. t -M.J 31 *13 lilt i iiit •ii 14.2 14.1 44 •22 *2I ».* 1J.>•«.4 -3*.* » *H 1132i ii u•ii ll.l 11.» •22. -ll.t 14 «34 «14 14.1 1.4 •14.1 -11.« •31 -*1 32 •13 Ul.t 13.1 •43. 1 44 •14 14.14 1 »3f.« ill at •24 ll.l 14.1 •3*. 44 •1* 1* 1 |i IS «IS U.t «114.4 1 -43.1 1 91 17 •21 ll.l ll.l «11, i -4*.l 44 •1* *s i it * «is at.t v.i It.It I -44.1 3 11 M •ai 17.C 11.2 •11. 14 •2* is* i i* s* «as u,i i.i K.I -41.1 «4 •i* Ms 3 si at «it aa.t ll.t 34.3 -31.4 3 SI «ai ti.« 11.2 •11. 44 •a* 11 1 II 11 •It 4*.l ll.S •as. • -41.3 i ai !! •It U.I U.I •It. «4 «It «1 1 41 21«27 41.1 ll.l «ii.• -41.7 41 «i *n. » -Jï.i II •at SI 1 12 14•at Si:! ll.l *ii. 9 -41.4 II •i.' 14 1 SI 11*n ll.l -H.S W.S •14.2 «t •23 1 S3 14 «* •23 3 SI H •as 1 SI It «t *2t 1 S3 31 «a •21 3 SI 41 S3 «a* •Iff 11.1 •31.4 -41.1 »* •3* 1 S3 1ft•It 3S.I •34.3 '34,9 92 «23 1 U SS •ai 41.4 •ll.l -SI.» 93. «23 3 SI It •aa «.7 •ai.i -M,4 H •12 ) SI SI«ai ll.t •11.7 -34.1 SI •1* «is t».t •ïs.ï -n.s 13 •1* •at H.S •tt.t -«t.i Mi •1* •II 13.4 13,3 «11,1 -U.4 S* «It •1* 14.1 11.1 *U.» -It.* u «31 •1* 41.3 «It.* -49.1 H •2J H •31 st •IS I* 4 • II tit 4 I «11 1*2 4 V «1* lit I 1 1 «11 S.I •14.1 -11.1 1» *» S.t «ll.t -43.1 1*4 4 I «2* 114 4 3 It *1* 11.1 1 *11 lif 43i tin 1i lia II *lt 41.1 1S.4 *».* -41 2 32 14 «24 1.1 14.1 •!•.* -41 •24 at •11 H.4 14.1 •11 .1 -«I.I » •14 31 3 SI It •13 41.4 U.I •17 .1 -43.7 111 ii M •24 2S.I 11.? •2*.* -44.1 •23 ai.4 11.7 •11.7 -44.1 •34 24.* •a* •1* tn 1 tt II •11 ll.t 13.1 •19 -19.4 «14 mi 3 u tt •I* 11.3 U.I •33 -13.1 Ü -Jf.l 57 The telescope and the photometer moved in 1979 from the Leiden station on the SAAO annex near Hartbsespoortdam in South Africa to the ESO observatory at La Silla. A careful! cheek of the photometer was made at that time, which resulted in some minor changes in the performance of the V, B, L and W channels (Lub, private communication). After installing the photometer there were still some problems with the U and W channels, These problems were solved in March 1980 which again changed the performance of some channels, in particular the Vf. Because of all these changes, which were partly unforeseen, one has to distinguish three photometric systems. The first is the 1973-1975 system as described by Lub and Pel(1977), which was probably maintained till 1978. The second is the 1979 system, used form March 1979 till March 1980, and the third is the 1980 system, used from March 1980 onwards. All data presented in this paper are in the 197? system, The transformations to the 1973-1975 system, as based on Pleiades measurements performed in 1977 and those described in the present paper, are given in Table J». Transformations to the 1980 system are not yet available. Table 4 Transformations between the Walraven 1977 and 1979 measurements of stars in the Pleiades cluster. The accuracies are primarily determined by the low measuring accuracies of the 1977 data. 78-indax 79 indices accuracy interval V • V .817 < In the Walraven system the calibrations are made with respect to a system of 20 standard stars. These stars are calibrated internally with respect to each other over a whole season, as described by Lub and Pel(1977). Such a calibration was also performed for the 1979 system by Pel and Van Leeuwen (1981, Roden internal report). The results obtained after the calibration are presented in Table 5. All data are expressed in log(I) values as is the custom in the Walraven system. Multiplication by -2.5 provides a relative scale in magnitudes. (V-B) (B-U) (U-WI (B-L) Table 5 5731 1.1298 -.1515 .1497 .1348 .1241 17»B1 1.8567 -.1428 .2131 .13B5 .1541 The 1979 calibration 24315. 1.1116 -.1135 .3571 .1739 .126] 16511 1.9118 -.089» -.1924 -.1413 -.1428 of the primary 18446 1.6899 -.1932 -.1981 -.1475 -.8495 standards of the 41534 1.4977 -.1671 .1514 -.1124 .1115 am -1.5S69 -.1961 -.1351 -.1482 -.1624 VBLUW system cues 0.4762 -.1561 -.1191 -.1217 -.1122 74575 1.2B5! -.1559 -.11107 -.1190 -.1184 87514 1.9167 -.1291 .2964 .1597 .1821 aam 1.198» -.1493 .1114 • 1B71 .1229 114337 1.6474 -.1662 .0115 -.1229 -.1135 121847 1.6747 -.1292 .2414 .1426 .1714 112981 1.1167 -.1698 .1394 -.1267 .aac4 144471 1.1799 .ties .1152 -.««43 .1135 163955 «.8598 -.11(1 .4156 .1162 .1132 umi 1.7739 .1121 -.1257 .6055 -.1135 191B91 -1.1536 -.1167 .3327 .B677 .1112 198111 1.1451 .1141! .4477 .1224 .1417 ss In addition to the standard stars a local standard in the Pleiades cluster, HD 23109 (V=7«85, spectral type=A3V), was frequently measured. This star, which is a well established cluster member, will be refered to by its Hertzsprung(19t?) number, viz. Hz801, It was calibrated to the standard stars and found to be sufficiently stable. Each of the selected stars was measured on two to four different nights. Each measurement consisted of two to four integrations of 16 to 61 seconds. During a night either the 1VJ5 or 16V5 diaphragm was used. The quality of the measurements using the 1H'3 diaphragm turned out to be inferior, especially for the ultra violet channels. For six nights the (U-W) index could not be used at all. They involved mainly measurements of the brightest cluster members. The calibration of the parallax stars has been done directly with respect to the standard stars. The Pleiades were calibrated with respect to their local standard Hz 801. Tables 6, 7 and 8 give the averaged results for the parallax stars, the stars selected by Hertzsprung and those selected by Pels respectively. The indicated spread in measurements is rather uncertain as only very few measurements were used to calculate them. The mean standard deviations, which range in V from 0.001 for the brighter stars to 0.003 in log(i) for the faintest members, give a more reliable picture of the accuracies obtained. Some of the stars show signs of variability, in particular among the late G and K stars. Eight of these were investigated further during 1980, which resulted in the discovery of variability for Hz 31. Hz 129, Hz 686, Hz 716, Hz 879, Hz 1121, Hz 1220 and Hz 1883, as well as the F type stars Hz 727 and Hz 1797 (see Alphenaar and Van Leeuwen, 1981). Another set of observations carried out in 1981 resulted in the discovery of variability for Hz 451, Hz 625, Hz882, Hz 883, Hz 1039, Hz 1332, Hz 1531, Hz 2031, Hz 3019 and Hz 3163 (see Meys, Alphenaar and Van Leeuwen, 1982). Repeated measurements during 1980 and 1981 for most of the members of the central region showed probable or possible variability (AV>0.006 in log(D) for those indicated by 'v' or 'v?' in Tables 7 and 8. A description of the variabilities of the Pleiades K stars is presented in section 3.3. Table 6 Combined photometric data with standard deviations for 15 nearby stars. 1581 ».2« 1.051 1 .253 1 .289 « .20» 3 .237 1 4 VCR' 5.76 o.««« ,&(]« 2 .«5 1 .271 2 .458 Ü 2 10360 5.91 O.3B5 .107 8 .«61 7 .273 6 .QRQ 5 3 10361 5.79 O.«34 .391 6 .mo it .260 3 .436 2 3 10476 5.26 O.5»5 .377 4 .«36 2 .239 7 .«3? 1 1 10700 3.52 1.315 .326 u .329 4 .247 3 .119 1 3 16*60 s.a« O.«ll .««3 1 .533 2 .319 ?. .528 1 2 17925 6.05 0.329 .19» 0 .155 2 .255 Ï .440 1 ? 20791 ».?8 1.038 .318 2 .330 2 .25? 3 .315 2 3 92009 3.75 1.552 .400 1 .«56 1 .>•» 4 .452 3 1 30652 3.20 l.«7« .197 1 .292 3 .1X1 2 ,?10 1 3 216801 6.50 0.116 -•513 P .607 ? .237 3 .575 2 5 216899 B.71 -0.719 > .701 q .ft«2 t .780 15 .524 q 5 216956 1.15 2.204 0 .00*» 0 .«29 3 .128 .191 1 2 J 5763 9.03 -0.E6S .690 .593 II .62? 1ft .«93 u U 59 Table 7 Combined photometric data for 199 stars from the Hertzsprung catalogue. The indications of variability are based on the 1979 and the 1980 and 1981 measurements if such were available. See section 3.1.4 for comments, H K I 131 9.«W -1,011 1 .m 1 .31» • ,19T .323 1 1 4fl 9»t 1 .3» t •in .3» 3 1 .37T 3 ,«7> 11 4,193 5 .«« 1 .«51 II .169 9 it'aii •4»tW tl ;«; i .«• < .m« 3 )!«« .161 .231 1 11 ,4W < ,«15 1) ,355 .311 5 1139 -1,916 1 .311 3 U.Ml 1161 .170 1 |T ,565 31 ,301 .514 11 -i.m 1 !m 1 .«« 1 ,3°T i 1200 9.9» -1.2H 1 .319 2 .113 .33» 1 I» ift.Mi 131? .3M ,!H ,275 t .ia 1 ; I19T 19.51] •1.W 0 •NO 1 .311 1 306 .379 3 m W 1X1 -1.990 f .3tt 0 .3» II .111 3 .301 II ,37» 3 .147 3 1215 19.951 -l.W 3 •M3 0 .319 0 .316 .315 6 -MM : ,1«M ]1 12» -1,991 t ,17131 ,»!> 3? •339 «T i« 10..7W .30* .20* ,391 11,131 • UT « '«*« ,137 3 .200 1 1.111 .91] 1 .009 9 .369 3 .OW 1 .116 1 1» "«.HI ' ,«4 1 '• .3*9 •309 M« ,1H 1 '.191 I .169 1 .315 1 3 I15T 11.31? -1,910 .OU .«09 1 .197 .313 0 \ I1U . 5ft 1 ,15» 1 ,191 1 .175 ,311 1 15* 1 22T •,34* 3 .«05 5 Ï*H5 .301 3 ,169 ,311 1 IÜ591 •1*151 3 .361 1 ,111 It .151 39 •t.OTJ 2 ; ,179 2 .171 ,301 1 IQ'.U» -I.fti» 1 •384 \ ,2«6 5 ,111 3 III* I.3U tl» 1 ni .931 5 .171 11.401 .157 1" 1291 -2.I5« 1 ,M« 3 .lit il t7i -.1.8M30 .375 4 iiH 12 ,111 IK -1,474 .359 .3» .137 • 3t« 1309 9.H1 •1,050 1 ,30» l .199 8 ,310 3 .515 10 193 ii.jn -1,7*1 4 ,3»! » ,221 15 ,159 t D12 12 51.3 -l.ltt 1 ,»7« 5 .961 !• .157 ,w • ,»lt 1 .1» 1 ,31« 1 1311 J!?M -,W j ,301 1 •33t i .191 .Hl 0 m «.'#79 ium .2?» 3 .131 3 .1» .131 2 12.7!» ,516 1 ,603 19 • 393 .5»»« [41 -1,61,5» .295 ( .131 1 .231 .311 I 12,731 -1.393 5 ,«99 > .560 5 .201 ,957 11 W* li.Oltf -1.A9C ,3« .«11 1 .153 31 .375 11 1311 !J.5»7 -3.«79 ' .75» .571 31 J» «*.4*T 1 r .»» .991 •• .133 •IH 1< 151« 10.577 .39» 1 .131 t .111 .719 • MM 4,094 .141 .!«« 1 .1*5 .215 1 1511 13.511 -3,49! 35 ,513 7 .571 » ,w< .«99 11 -.5» > .ne 1 '.3M 1 .700 J 1553 -1.197 : ,«» 5 .511 7 .1«5 17 .•73 35 3*^ 11.7» -1.9M ' .144 .«on 3 .211 17 .380 3 13<3 1K192 -1.T35 3 .3«5 1 .376 IS .23« 15 .195 3 »T ïl.174 f .5*2 l .•9t «« 1613 9.191 -1.111 ! .210 I .199 • .30T .333 2 11.293 ï .W .C«t 12 .?** .5W IT 1621 -2.855 • •ï«t .179 19 •2W •306 30 'tQ5 4,P4S -t.I9S 1 ,?3Ï .30? 2 .197 ,337 1 16«5 I1.3U -1.111 .326 > .«60 • .361 11 .710 1 1M3S -1.111 .335 17 ,371 11 1738 9.311 -.961 .235 .316 1 .116 .336 1 447 5,4*? -5?0 : -t.010 ,7«6 5 ,(*! 7 1762 1.275 .152 .369 1 .190 .209 0 Ml 1766 -.913 .303 46ft 1.704 l.?Tt ! -'.(11? > .725 15 .«7 « .m« 7 1776 m'.vn •1.111 ï .3» .11? 5 .333 3 .32* 1 «70 B.9» -.94? 1 ,V? > .ja o .176 .701 1 179» 10,100 -t.itr .373 .137 0 .706 .71(1 6 -1.594 ,121 •> .101 2 1797 -1.3C5 .305 1 .192 .2»ti 0 -,1.415 1 .321 7 «33 5,»»9 .570 -.011 .711 »1 .073 3 ,M1 1? 414 10,747 «1» .396 1156 10.019 -1,273 .306 0 .191 V? 11.911 -9?.f>ll 1 .tlS • !«8T 10 .310 TT .«IA « l«7t 6.M3 -.033 .IW7 .«37 31 .12» < .199 1 130 -.441 t) .1(7 1 .11» » .171 .107 1 mui 12.107 -3.OT 1 .«71 1 .533 70 .197 .531 «9 1 .»** 1 .«13 7 .17» .319 1 (912 9.0» -.UI .71* .315 3 .111 .32» 1 5.M? ? -Ml .IH6 > .071 7 10.151 -1.399 .78H .313 1 .313 .271 5 13,3W -Ï.4W -> .4JT 1 .!«! 1993 1.362 -.999 ! .131 1 .399 0 .169 .217 0 *.C5« 1 !MT 10.«95 .1.619 1 .379 .117 1 .«> I .«03 1 57» 11,33? -1.750 9 .«« « .117 1 .271 t7 .152 1 201» 12.619 -1.119 1 .«7 .««7 7 ."«1 .<«9 17 AM 9.010 1 .111 1 .115 0 .ie? .317 0 11.577 -1.819 .117 .«36 9 .301 16 .«05 < 1 .578 7 .«91 26 .171 2176 Ü.715 -1.996 « .«53 13 .22» 11 .«16 3 «?T 9.PB9 -».13> 1 .221 7 ,1M 1 • 22« 7 21«7 in.905 -1.119 .371 1 .393 3 .269 29 .351 2 MA t?,5t? -?.2f4 7 ,«T7 t ,55« 35 .«96 37 7111 J.626 1.102 1 -.K3 ? .3«9 70 .011 3 .051 3 «I? (t,M5 -.471 0 uw 1 .«50 7 .1M .21« 7 2IT3 111.«71 -I.U7 1 .777 ? .315 1 .303 .311 I 6V) 12^01- -3.06S i :«i3 « ,«23 7 ,77« 10 !l62 11 2111 9.215 2 -.OOf 3 .379 « .117 19 .013 3 6T6 7 .577 2t .472 «7 .319 3195 «.120 -.901 ; .090 .«07 1 .199 .705 t 11.414 7 .920 5 ,'31 3320 7.518 -.260 1 .019 2 .«33 3 .132 1 • 116 7 C97 «.«13 -.700 0 .150 1 .1«3 0 .115 .709 3 33«k 12.911 -1.293 2 .«97 t .173 « .37? .«13 17 •*AA 1ft.13» -1.11» 1 .2(11 n .122 1 .301 .211 1 2?«1 1.509 .til 3 -.006 ! .165 15 .011 • .117 1 717 T.ITi -.121 1 .«17 3 .125 I .1H1 n 3275 11.tol -1.901 .350 ,«05 .151 7?7 -1.149 " .715 1 .100 1 .705 3371 10,901 -1.610 1tl .319 1 .«76 T .735 21 .«06 « 733 i?,ir* -2.1!") f .513 5 .515 10 .1*9 .«'11 1! 321« 11.119 -1.715 1 .157 3 .196 5 .!!» 9 .376 1 719 9.«47 -t.r*T« 1 .2T5 0 .119 5 .?'# 2319 7.959 -.«37 1 ."73 1 .«10 1 .117 t .21* 2 7»r> 1 .«93 1 .919 T .7«9 .103 79 3311 11.«If -1.I23 I .316 1 .«36 77 .30118 .199 10 W 1 .331 7 .116 1 .335 1 -t.l*? » .317 1 .150 7 .161 12 .111 5 -1.1-P « .«1 3 .115 t[t .2.17 17 .199 11 3.U5 9.t25 -.909 1 .11» 1 .107 1 .tfl .303 II -1.167 1 .1» • .799 5 2366 11.517 -1.115 1 ..171 2 .«II 5 .290 21 •39« 1 7S 21 2«B« 7.510 -.165 1 .011 3 .«33 1 .133 5 .190 1 579 i?."Jl 2 .«99 1 .939 tl .51" 11 25» 10,290 -1.3T1 3 .151 1 .301 9 .199 .250 3 8Sj 0 .«17 3 .«93 10 .311 2507 t.Tl* •0*3 * ,031 3 .!«> 6 .10] 1 • 1«5 1 7 .91» i3.0*tV 1 .803 9 .315 '.«tl 33 -1.997 3 .551 1 .110 II .51« 1« 1 .«85 1 -11! » .362 • .«19 5 SÏÏ !?:«5 ..1.613 1 .310 « .319 1 'hi 10 .Hl « Sift 11.«6 -l.R» 1 .37» « .«1? 10 .117 33 .«aa 10 3663 ll'«tT -1.133 5 .311 1 .106 16 .31*» 4?3 UUM .1,318 « .377 1 ,331 • .229 .173 1 27>1 13.737 -1.199 5 ,1» 1 .535 II .361 .538 «0 4*5 1J.T23 •Ï.W» 2 .«'7 7 ,919 10 .110 .551 71 3711 10.3II -1.312 3 .360 1 .301 9 .201 .351 3 -.72ft 1 .759 3 .122 » .778 0 HU 6.933 -.«5 0 .031 1 ..199 3 .113 2 .I7Q 3 T.9t» -.411 7 .!!« *56 1 ,37« 1 .119 1 .307 1 3170 tl.529 -3.1TI 3 .176 3 .551 10 .219 .519 10 ..157 .«00 -MM .37» .399 3110 1I.69* -1.936 3 .312 5 .«» • .353 K .«09 1 •.1»? l.BT« »o o -.et] 2 .22] 1» .069 0 .080 2 3U1 11.93» -1.173 1 .«29 » .«15 T •M5 '1 .«91 II tft.161 «.I.4U 3 .331 5 .701 .311 1 5 .«SU 1005 .Ml " .911 31 .510 33 29t« -1.235 3 .«70 1015 10^573 12.«1» 4 .5*1 11 .331 3» .511 19 2 '.m 2 !« 7 .212 3 3019 J.857 1B .961 .517 17 ion 7.J52 -*193 1 .011 3 .«3 3 5 .«21 27 .595 ;»9 3 .197 3 3011 1.190 -.799 1 .1(6 2 .KI 2 .111 .19» 3 1032 -1.749 7 .311 > .366 3 .20t 1] .313 « 1019 3U> 11.633 -1.911 I .339 3 .393 J .in 13 ,209 2 1!:J5 1 .372 2 .279 3096 5 .«29 IO«« 3 .151 1 .511 3 12.105 -1.101 » .«7» 6 .211 15 .««7 5 .110 9 .136 1 I9.U9 -Ml] 1 .319 3 .359 1 .331 11 .315 9 1099 1KM3 -j!oi2 5 .«11 3 .«U 5 .21] >t mr .«69 8 3122 11.917 -2.0H 5 .311 1 .!!« 6 .1» » .306 ( ttoc -*.t*2 2 .«3 3 .5K19 .321» .501 13 3163 12.73» -3.Ml M .157 5 .506 33 .79» .«57 9 not -t.3t3 2 .«. 1 .lot .153 t 3179 19.067 1 .119 0 .31» 6 1110 13*3J1 • .571 -I.K! •213 .350 « 2 S. 3117 5 .9W .517 1117 10.244 2 .322 1 .361 1 11.139 -1.510 1 .UI» .215 .337 fl 3197 13.102 4.139 3 .199 « .516 T .399 35 .111 !« 1122 1.306 -tra ' .199 0 .301 6 .1» .211 1 31«l 13.U7 -2.7H « .210 6 .311 39 .175 .301 71 112 60 Table 8 Combined photometric data for 193 possible and probable members selected by Pels. Comments like in Table 7. 1 10.«I -1.14) 1 .1)9 I •71 1 ,1U • ,«1 2 1 191 1.134 -I.22J 2 ,193 1 .371 0 .210 ,n\ z 2 ft I 11,131 4,191 I .13) S 330 < •lifts .315 10 1 M 11.999 -1.19» 5 .260 7 .«1 » .734 12 3 H 3 9.19» -,9)1 1 .121 1 joi i •in ' .115 1 1 101 11.159 4,0» 1 ,0H 2 .371 6 .019 .IM 2 « n 4 II.«5 -l.U» 1 ,3M 2 9I> 5 .ui it .in « i 104 9.841 si.lOO 15 .301 19 ,Wt M .131 .17* it i i 10,«71 .1,1» 4 ,]in o 3)1 9 ,210 « .lit S 3 in 11.03» -1,171 1 .555 0 .MO 1 .31] .367 1 * n « 9.HT »1.0« It .219 1 90» 1 .117 • .2» 2 ] IM 15.3« -J.I1» 2 .999 T T l(l.41t -I.I4T I ,M« 2 in 3 ,22» 10 •in i i 14.»» -3,112 11 !<•] 9 -.093 11 .311 JO 1 H I I9.T9» i1.5»1 3 .199 i »» « .1» 1 .211 2 3 101 11.70» 4.TII 2 .Ill T .3» tt .W H ! n 9 12,493 4,2«0 ft , II 11.«4 ' -1.8»! 1 .111 1 «1» 1 ,181 11 .!»« « 1 ttt 11.109 4.501 1 .It» »*1* «ff 7 ,1M 1 371 J .111 21 ,331 t 3 111 14,22» 4.9)1 10 .11» .«1 9.1t] ."14 I 115 1 .179 1 ,20] 2 1 113 I1.H4 4.5» 1 .Hit 3 (fi0J tt 3 T 11.91) 4.01) 2 •«15 3 «73 9 ,191 11 1 114 12.107 4.33« 11 .«92 1 .nu a* .91» !« 3 «.on -I.2W » .249 I .113 • .Ml 4 ,2» 3 1 US 11.6» 4.112 1 .901 1 .MQ3] .W19 •W 13 1 ». II 11.118 -1.797 2 .10» I .12» 10 .MI 32 .21) 1 « 113 9.219 -.9«9 ( .SOO .««! 3 1.319 -•51» 2 .112 1 .127 1 .113 « .210 1 2 11.117 4.6)9 2 .«70 } .516 10 .nw .513 " 11.092 -1.194 I .320 3 .319 ' .217 IT .217 5 2 1*1151 11.111 4.525 0 .311 .10» 10 .443 .916 10 I n 1t.5»t -1.195 I •111 • .519 22 .219 U .50» 9 i lit 11.1'S -3.117 3 .027 ,?.»: in E n 9.79» -I.I7« I .2» I .311 2 .211 • .25» 2 2 117 11.11) 4.H7 1 ',1» n .21» 9.391 -I.OIt 1 .215 1 .295 1 .lit 1 .211 0 2 111 1I.4U -3.019 2 .397 14 .390 31 .430 > n 9.091 -.177 2 .912 I .111 0 .JM20 .51» 1 2 119 11.295 4.177 5 .112 4 .901 9 .393 1 .4» 1 I2.IM 4,327 « .172 « .«2 10 • Hi 22 .377 10 3 190 -3.160 7 .9)5 12 .936 33 ! " 11.11» -1.90» I •n i .115 5 .25» 27 .111 3 3 191 io:9 9» 12.311 4.11» 0 .313 1 .«5 31 .177 23 .«17 1 2 97 11.13» 4.2)5 1 .197 2 .«• 11 .211 • .399 » 3 9f 11.701 4.1W 1 .315 1 .1» < •Kit! •272 12 3 11.19* -1.73» 5 .111 3 .311 1 .130 It .3» 2 i IK 13.509 4.1». 2 .iw .!M 7 .20» 2» .197 21 3 6f 3.1,4 Identification of Members In the determination of membership those stars for which the proper motions are determined by HertzsprungCW?) (with a few exceptions) were used as a reference frame in the V-(V-B) diagram. The position in this diagram of each of the stars selected by Pels was compared with the reference frame. This method of membership determination may cause problems for unresolved binaries of comparable brightness. So some of these stars may have been assigned erroneously as probable members. The stars that are obviously further away C&V >0.06 in log(D) or nearer to us (^V <-(M0 in log(D) than the Pleiades are indicated in Tables 7 and 8 by 'n'. On the other hand, the stars for which the membership is obvious (0.04 >&V >-0.06) are not indicated. The indication 'd' shows that a star is situated in the region where unresolved binaries can be expected (-0.06 > AV >-Q,4Q), while a 'D' indicates that the star was listed by Batten et al(1978) as a spectroscopie binary. The indication '?' applies for stars that are only slightly furthtir away than the cluster (0.06 >AV > 0.01). The membership for these stars and for those assigned as 'd' remains uncertain until proper motions are available with accuries better than 0V001 per annum. The photometric membership determination for stars fainter than V=-2.2 is uncertain due to an increased intrinsic scatter in their V-(V-B) relation. M 1V-BI — 11 02 03 U US Figure 2: The V-(V-B) diagram for cluster members (units are in log(I)>. (.): members selected by Hertzsprung(19M7); Co): members selected by Pels. The left part of the diagram shows the brighter stars and the right part the fainter, shifted by 2.0 in log(I) upwards at constant (V-B). 62 30 stars from the 118 probable members selected by Pels were found not to be members. For two stars the identification was uncertain and seven stars are possibly double stars. From the selection of 75 possible members 33 were found not to be members, while one was uncertain and two possibly double. The selection by Hertzsprung contained nine stars which are not members. All of these have proper motions which are badly determined or barely coincide with the cluster proper motion. In addition, one of the measured stars, Hz 2275, was known from its proper motion not to be a member, 25 3tars were found to be probable unresolved binaries and one star, Hz 303 (indicated by 'm' in Table 7) looks like being double and in addition considerably reddened. The proper motion of this star as measured by Vasi.Levskis et aL(1979) gives no reason to doubt its membership. One other star is known to be reddened considerably, Hz 10S1 and is indicated by 'r'. It is probably situate» behind interstellar dust connected with the reflection nebulae around Merope, Hz 980. A comparison between the relative numbers of possible unresolved binaries among the stars selected by Hertzsprung and those selected by Pels shows that the number of erroneously identified members among the latter selection will be small. The distribution of unresolved binaries will be described in section 4.3. Figures 2, 3 and 4 show the results for V, (B-U) and (B-L) against (V-B) for all stare not marked by 'n*. The stars from the selections by Hertzsprung and Pels are indicated by different symbols. Figure 5 shows a map of members of the Pleiades cluster that was presented earlier by Van Leeuwen(1980). 1 1 t 1 • 1 • 1 _• Hi - s 1 • 03 : • % Ot • •* *, ' 4 • * . r . . 85 - '" V • . " • Ot - • 07 1 1 _. „ o.o tv-a) -» oi oi 03 Figure 3: The (V-B) versus (B-U) diagram (units in log(D) 63 l 1 t 1 1 I I ai • - I * • • 9 • •'• • • V... - V* « Jr*.«$- ». •.-.,.* * 03 •'•'t.' *m m * • «A • * "*• v m Figure 4; like Fig,3, • • for (V-B) as - • • * • versus {. (B-U. *• * • •••• • 1 1 • 1 * 1 oo IV-BI — ai Figure 5: A map of all cluster members brighter than phot .magi.. 12.5, the approximate limit of completeness of the survey by Pels. The stars are indicated by circles, of which the size increases linearly with "•"— decreasing magnitude. Coordinates are of epoch 1900. ." P . o • O .. o O 3.1.5 The distance to the Pleiades Most of the recent determinations of the distance to the Pleiades are based on a comparison with the Hyades. Such a comparison involves an uncertain correction for a difference in metal abundance, which itself is uncertain (see Cayrel de Strobel, 1980). Moreover the distance moduli of the Hyade3 as derived by different authors still show differences up to 0.2 magnitudes (see e.g. Hanson,1980). Here we try to make an independent calibration of the distance of the Pleiades stars, using the photometric data of 11 of the 15 parallax stars presented in Table 8. These stars, which have parallaxes larger than 0V12, have (V-B) indices less than 0.41, Beyond this limit the intrinsic luminosities of the Pleiades stars can be different due to pre-main-sequence evolution (see section 3.3)• The intrinsic accuracy of the photometric data presented here is at least 10 times better than that of the distance moduli of the parallax stars. The comparisons are made here with respect to a dereddened single star or main sequence fit. HD 216956 is of spectral type A3V and agrees well in all colours with the dereddened colours of Hz 804 (presented in section 3.2), which is also a A3V star (see Table 9). Table 9 Differences in V and the colour indices between HD 216956 and Hz 804, expressed in log(I). dV d(V-B) d(B-U) d(U-W) d(B-L) 2,573 0.001 -0.002 -0.008 0.006 The other 10 stars are compared with the dereddened main sequence calibration for the F and G stars, covering the interval 0.17 < (V-B) < 0.41. This calibration, described in section 4.3, takes into account the duplicity effects, the spread in distance moduli and differential reddening. It is reddening corrected except for - possible total reddening of the entire cluster. The main uncertainty left in comparing the parallax stars with the calibration grid is due to possible differences in metal abundance. Using Kurucz(1979) models of stellar atmospheres it was found that such differences are revealed by deviatons in the (B-U) and (B-L) colour indices. No systematic differences are observed in these indices and we therefore conclude that there are no large systematic differences in abundance between the parallax and the Pleiades stars that could influence V and (V-B). The calibration, derived in section 4.3 for F and G stars, has to be corrected for an average differential reddening of 0.033. Thus, the calibration is given by (see section 4.3): V = 0.499 - 9.26(V-B) + 9.32(V-B)2 -4.4(V-B)3 Using this relation, the distance ratios given in Table 10 are found. These ratios have been compared with the parallaxes of the stars as measured at different observatories. Their differences are large and often more than the accuracies claimed. Here the parallaxes of Allegheny have been corrected for a systematic magnitude effect following Lutz et al(1981). The R.A. effect in the Yale parallaxes, which was noted by Lutz(1981) but for which no correction 65 was given, is possibly present in our data but has not been corrected for. Table 10 Distance ratios and parallaxes for 11 parallax stars, and the distance to the Pleiades for each of the observatories. HB r(V) parallax (in 0.001 arcsec units) Yale Cape Allegh HoCor Sproul 1581 15.1 130 10 116 5 1628 17.0 136 7 139 28 155 8 10360 16.1 160 12 133 10361 15.9 160 13 133 10176 16.8 126 16 113 8 10700 33.1 277 7 288 312 5 257 5 17925 m,2 113 7 118 133 10 2079« 22.1 162 10 151 32019 «2.3 297 10 306 291 28 302 312 28 30652 17. ft 110 12 112 20 151 216803 19.3 133 10 157 10 distance 130 128 112 122 133 po Pleiades +12 +13 +20 • 16 •11 The parallax of the Pleiades has been calculated for each observatory independently, providing 5 different parallaxes for the Pleiades cluster that agree well within their errors. From these five determinations the distance to the Pleiades is estimated as 130 + 5 pc, a distance modulus of 5.57 + 0.08 magnitudes. If the entire cluster is~Veddened by 0.012 magnitudes in V (see section 3.2) then the cluster distance decreases to 128 po. These values agree well with the distance of 129 pc derived by Turner(1979), based on a comparison between the Pleiades and the Hyades, and somewhat less with the distance of 13** pc derived from a similar comparison with the Hyades (but using different assumptions for the distance and metal abundance of the Hyades) by Jones(1981). Only a small fraction(about 10 percent) of the useful stars within 8 pc has been used here: By measuring all these stars the uncertainty of + 5 po may be reduced to + 3 pc. After a successful HIPPARCOS mission a 1 percent accuracy of the Pleiades distyance becomes feasible. Acknowled gement s It is a pleasure to thank J.Brand, who assisted in performing the observations. I would also like to thank J.Lub and J.W.Pel for the definition of the 1979 system. P.Alphenaar took care of the final reduction and prepared Tables 7 and 8. 66 3.2 Differential Reddening in the Pleiades 3.2.0 Summary A preliminary report is presented describing the differential reddening for B and A stars in the Pleiades as determined from VBLUW measurements. 3.2.1 Introduction The Pleiades cluster is situated in an area of the sky that is densily populated with small dust concentrations. The presence of dust near to the Pleiades was first inferred from reflection of light emitted by the brightest members of the cluster. The bright small reflection nebula near HD 23180 (Merope) was optically discovered in Venice in 1859. October 19. by Wilhelm Tempel(i860, 1875). The large scale structure was first seen by means of a three hours exposure made by Paul and Prosper Henry(1885) in Paris in 1885, November 16, Two copies of this exposure, present in the Leiden Observatory Photographic plate archive, clearly show the structured reflecting nebulae. Unfortunatly, the original exposure no longer exists (Dëbarbat, jjrivate communication). Binnendijk(W6) noted that the distribution of background stars in the field of the Pleiades indicates the existence of cloud concentrations. As similar concentrations could be deduced from the reddening distribution of cluster members, which moreover coincides with some parts of the reflection nebulae, it seems that at least some of the clouds are situated within the area occupied by the cluster. The scale size of the clouds as deduced from the reflection nebulae is 0.1 pc, while fine structure can be seen on a scale of 0.01 pc and less. The area occupied by the B and A stars is much larger, nearly 16 pc in diameter (see section H.2). Light received from different cluster members is therefore differently influerscd by the dust and consequently reddening corrections have to be determined for each star individually. This section is a preliminary report on a more extensive study of the physical parameters of the B and A stars in the Pleiades, the way they are influenced by rotation and evolution and a determination of the age of the cluster. This report will therefore be only a brief one, giving mainly results obtained so far with respect to reddening corrections. It has been included as such, to provide the distribution of reddening in the cluster which plays an important role in the deduction of the number of nearly equal mass unresolved binary F and G stars, described in section H.3. The determination of differential interstellar reddening has been done by' means of the VBLUW photometric data presented in section 3.1. All data will be expressed in log(I) units as is the custom in this system. Multiplying by -2.5 provides a relative scale in magnitudes. 67 3.2.2 Empirical Reddening Corrections The amount to which the light of a star as measured on the Earth is influenced by interstellar extinction can be determined by comparing its colour indices with a reddening free calibration grid. Two types of grids can be distinguished, empirical and theoretical. The empirical calibration grids are generally based on measures of colours of nearby stars, from which it is assumed that they are only slightly influenced by interstellar extinction. The brightest stars are usually used for this purpose. This method suffers from systematic errors due to the fact that massive stars are less common than light stars, because of which these massive stars are on average further away than the light stars and more influenced by interstellar reddening. In order to avoid such problems, we have used the cluster stars themselves as reference frame in the determination of their extinction. The empirical calibration grids have been obtained in an iterative way. First, blue envelopes are determined in the various two colour diagrams. Reddening corrections towards these envelopes are compared and combined according to the accuracy of their determination. This accuracy strongly depends on the relation between the slope of the calibration grid and the reddening direction in a two colour diagram. After applying the combined reddening corrections to the data new blue envelopes are determined and the procedure is repeated. The reddening corrections have been determined in the (V-B) colour index of the VBLUW system. The other colour indices, (B-U), (B-L) and (U-W) have been transformed to reddening free, so called 'Q' indices defined in Table 1. This simplifies the procedure described above without losing information. Table 1 Reddening corrected or 'Q' [B-U] = (B-U) - 0.610 « (V-B) indices in the VBLUW system. [B-L] = (B-L) - 0.390 « (V-B) [U-W] = (U-W) - 0.450 * (V-B) In the determination of the calibration grids the stars have been devided into four groups, to which blue envelopes have been fitted with second degree polynomials. The selection criteria used are as much as possible in accordance with spectroscopie data. The first group consists of the six brightest and evolved stars. The second group covers the other B stars, the A0 and A1 and some of the A2 stars. When measured by [B-U], the Balmer discontinuity is largest for A2 and A3 stars. The third group covers some of the A2 and all A3 and A4 stars. No stars of spectral types A5, A6 or A7 occur in the spectral classification by Abt and Levato(1978), a discontinuity which i3 also shown as a change in performance of the index measuring the depth of the Balmer lines, [B-L]. Finally, the fourth group covers the spectral types A8, A9 and F0. Table 2 presents the final reddening corrections obtained for the stars in those groups. Figure 1 presents a histogram of the distribution of the reddening indices found. Table 4 provides a cross reference between the HD numbers and other numbers that have been used currently in refering to the stars shown in Table 2. 68 1 15 N JB] all 101 • 51L • 0 SSOOSXXM sa i 10 N Table 2 Reddening corrections obtained with the empirical calibration. Evolv. (1 stnra B to «2 stars Ki and »» stars AB till FO stars Stir nr E(V-B) StHr nr ECV-n> St.w nr E(V-R) Star nr F.(V-R) HB 23630 0.00? HD 83751 0.007 HD 234*9 0.024 HD 23157 0.028 HD 23850 ft.005 HD 93288 0.018 HD 23155 -0.001 •ID 23479 0.009 HD 23302 0.000 •ID 2332a 0.005 HD 23872 0.007 HD ?3863 0.000 HD 23408 0.015 HD 83432 0.014 HD 24178 0.029 HD 23610 0.011 HD 23480 0.023 HD 23950 0.036 HD 23628 0.010 HD 23246 0.000 HD 23338 0.000 HD 23923 0.003 HD 23852 0.014 HD 23156 0.015 HD 53659 0.010 HD 23388 0.008 HD 23607 0.004 HD 234111 0.012 HD 23643 0.007 HD 23733 0.015 HD 23873 0.004 HD 23402 0.020 HD 23567 0.035 HD 82578 0.005 HD 23409 0.042 HI) 24711 0.098 HD 23961 0.032 HD 238B6 0.010 HD 23791 0.016 HD 23568 0.020 HD 23430 0.014 HD 23585 0.003 HD 23642 0.033 HD 23361 0.040 HD 23792 0.028 HD 23410 0.047 HD 23194 0.015 HD 23325 0.037 HD 24076 0.041 HD 23924 0.00G HD 23375 0.020 HD 23763 0.054 HD 23512 0.152 HD 23290 0.003 HD 23632 0.013 HD 23664 0.024 HD 23488 0.033 HD 23913 0.019 HD 22702 0.00R HD 2261« 0.007 HD 23387 0.071 HD 24899 0.014 HD 23631 0.022 HD 22637 0.034 HD 23943 0.018 3.2.3 Theoretical Grid Reddening Corrections Jh«°»;etical calibration grids, based on the model stellar atmospheres as £?y ïürUCZf]975) haVe been obfcained for the Walraven system by Lub ZLltl^ l\ The grids have been updated with the 1979 passbands and the improved models (Kurucz 1979) by J.Tinbergen at Leiden Observatory. The nt y WeU the chara SS'SÜE? nT <*eristics of stars with temperatures above 9500 K. Between 9500 and 7500 K they are not sufficiently accurate for passbands covering the Balmer discontinuity, such as the L band in the VBLUW system. A sudden change in the models between 9000 and 8000 K is shown in 69 I Fig.2 for the (B-L) index. This change is probably related to the introduction of convection in the models of these stars. It is not shown in observations of stars in that temperature range. Models for stars with temperatures below 7500 K are again in agreement with the ob servations. Figure 2: The (B-L) index as a function of T(eff) and log(g) according to the Kurucz(1979) model stellar atmospheres. A change in the models is shown in the temperature region 9000 to 8000 K, probably due to the introduction of micro convection. 5 20 The calibration grids as obtained from folding the models with the photometric passbands are a function of effective temperature, surface gravity and for cooler stars metal abundance. In a small passband system like the VBLUW system it is possible to determine simultaneously the reddening corrections and the physical parameters, assuming that all reddening corrected indices should indicate the same effective temperature and surface gravity. In the following these parameters are derived for 30 B and early A stars, assuming that their determination is not significantly influenced by spectral anomalies and rapid rotation. In the final paper these influences will be investigated too. The VBLUW system offers very favourable possibilities for calibrating B and early A type stars, due to the unique properties of the (U-W) index. In combination with the (B-L) index it offers an almost rectangular grid of temperature against surface gravity, in which the reddening direction alignes with lines of equal temperature (see Fig.3). It provides the possibility to derive surface gravity estimates without accurate knowledge of reddenings. Using the estimated reddening and surface gravity, the index (B-U) can provide a temperature estimate for a star (see Fig.4), which together with the surface gravity provides an expected (V-B) value. The difference with the observed (V-B) provides an updated reddening correction. By iterating through these steps, values for E(V-B) are obtained, as well as log(g) and T(eff) values. All of these will, however, be influenced more or less by the initial assumptions made, in particular with respect tc neglecting the influences of spectral anomalies and rapid rotation. The results obtained from the theoretical grid calibration are presented in Table 3. Figure 5 presents a comparison between the empirical and the theoretical grid calibrations, showing a systematic difference of 0.005 in log(I) in E(V-B). This difference is found to depend on (V-B), such that it equals 0.01 for the brightest B stars and 0.00 for the A2 stars. As both calibration grids suffer from difficulties in their determination near to A2 stars, it is not clear which of them is most in error. A comparison with StrBmgren and Geneva photometric data will be presented in the final paper. 70 Figure 3ï The (U-W) versus (B-L) diagram with the theoretical calibration grid and an isoctrone of 10"*8 years (Maeder and Nermilliod, 1980). Each data point represents a reddening corrected star, The six brightest, evolved stars are indicated by crosses. 0.0 Figure 4: The (B-U) - T(eff) diagram. As in Fig.3, the data points are corrected for reddening. t>-Ulo login 71 i i r Table 3 E(B-V) (•1 Unictni V Physical parameters and reddening «•pïr corrections obtained from the .iv theoretical grid calibration HD log 6 log(T eff) E(V-B Fig. 4 Fig. 5 «3630 3.02 4,069 4,072 0.011 23850 3.33 4.10R 4.101 0.019 2310ft 3.13 4.098 4.106 o.ooe 23108 3.21 4.105 4.096 0,024 23180 3-35 4,081 4.086 0.023 002 23338 3,54 4.119 4.119 0.011 23862 33753 3,74 4.066 4.077 0.015 23288 3.88 4.100 4,100 0.G29 23321 3.91 4.099 4.098 0.015 _i I i 0.M tlB-VI 9.02 th«r, 0.04 OOt 23132 4.06 4,066 4.076 0.023 23950 3.96 4.100 4.097 0.047 23923 3.86 4.048 4.049 0.012 23629 4.21 3.993 4.002 0,008 Figure 5: 23111 4.19 4.037 4.047 0.017 A comparison between the empirical and theoretical reddening 23873 1.32 4.013 4.025 0.004 corrections, showing a systematic 22578 4.18 3.998 4.006 0.007 difference of 0.005 in E(V-B) that 23961 4.20 4.027 4.021 0.039 may be due to a general reddening 23568 4.21 4.020 4.027 0.024 of all cluster stars. 23642 4.26 4.008 4.003 0.035 23*) 10 4.24 3.993 3.997 0.018 21076 4.28 4.007 4.002 0.045 23763 4.29 3.973 3.972 0.046 23632 4.16 3.975 3.978 0.003 23913 4.21 4.003 4.006 0.021 22611 4.23 3.973 3.974 0.006 23387 4.26 4.008 4.000 0.074 24899 4.20 3-991 3.992 0.016 23631 4.23 3.987 3.985 0.024 22637 tt.15 u.nn? 0.040 Acknowledgements It is a pleasure to thank Prof Blaauw and P.T.de Zeeuw for stimulating and participating in the present research. 72 J Table U Cross reference far different numbers used for the brighter Pleiades stars HD Hzl HzII Trum Pels Gaul HD Hzl HzII Truni Pels Gaul 22578 S 23 16 23585 457 1284 365 121 226 14 S 25 21 23607 501 1362 290 132 22637 S 26 10 23610 526 1407 S108 22702 S 29 127 23628 513 1384 399 135 23155 153 47 23629 508 1375 395 133 23156 28 158 51 1 236 JO 542 1432 414 144 23157 27 157 50 23631 520 1397 402 137 23197 43 232 74 8 23632 510 1380 397 134 23216 92 344 121 27 23642 540 1431 413 145 23288 117 447 139 34 23643 534 1425 410 141 23290 26 23664 S115 57 23302 126 468 148 38 23733 693 1762 493 177 23324 150 541 166 46 23753 722 1823 506 185 23325 146 531 162 44 23763 742 1876 518 191 23338 156 563 170 49 23791 792 1993 551 204 23361 187 652 195 56 23792 S127 48 23375 206 697 2Q8 59 23850 870 2168 594 219 23387 216 717 215 63 23852 S137 54 23338 S 76 49 23862 878 2181 602 224 23*02 S 78 52 23863 885 2195 607 227 23408 242 785 231 71 23872 891 2220 613 231 23409 251 804 235 73 23873 910 2263 622 236 23410 248 801 234 23886 924 2289 629 239 23430 S 84 58 23913 S142 55 23432 255 817 240 76 23923 977 2425 671 252 23441 265 859 247 78 23924 975 2415 670 251 23479 313 956 281b 86 23948 996 2488 688 263 23480 323 980 286 90 23950 S149 67 23488 S 93 61 23964 1003 2507 697 267 23489 341 1028 295 94 24076 1129 2866 791 295 23512 371 1084 311 99 24178 S165 74 23567 447 1266 359b 118 24711 S185 82 23568 436 1234 354 113 24899 S194 163 Bessel HD HzII Bessel HD HzII 16 Tau Celaeno 23288 477 22 Tau Asterope.2 23441 859 17 Tau Electra 23302 468 23 Tau Merope 23480 980 19 Tau Taygeta 23338 563 •n Tau Alcyone 23630 1432 20 Tau Maia 23408 785 27 Tau Atlas 23850 2168 21 Tau Asterope.1 23432 817 28 Tau Pleione 23862 2181 Hzl : Hertzsprung(1923) Pels : Section 3.1 HzIIi Hertzsprung(1947) Gaul : Gaultier(1900) Trum: Trumpler(1921) 73 3.3 Variable K-Type Stars in the Pleiades 3.3.0 Summary Photometric measurements of 19 G and K type stars in the Pleiades cluster are presented. Of these, 18 are found to be variable, while the remaining one is probably variable too. For 12 stars lightcurves have have been obtained, which are characteristic for many fast rotating stars with convective envelopes, such as BY Dra, RS CVn and FK Com stars. All these stars are thought to vary due to rotational modulation. Spectroscopie observations for two of the Pleiades stars confirm this and show the highest angular velocities for K type stars near to the main sequence ever found; 50 to 100 times faster than the Sun. The rotational modulation and the observation that all Pleiades K stars observed so far are variable imply that all of these stars have a non-axial-symmetric surface flux distribution. Variations in the colour indices show that the Pleiades K stars get redder when getting fainter. We discovered in addition the existence of a relation between the periods and the amplitudes of the lightcurves of these stars. This relation shows that the amplitude of the non-axial-symmetric modulation of the surface flux increases with increasing angular velocity, Using the relation between the photometric and rotational periods we conclude that the Pleiades K stars still contain a high amount of angular momentum, compatable to that of the total solar system. The much lower angular velocities of the earlier and slightly further evolved G type stars indicate that these stars lose their angular momentum on a short tine scale at or soon after reaching the main-sequence; this process can at present be observed by means of the G and K stars in the Pleiades cluster. 3.3.1 Introduction Many young late K and M stars show photometric variations classified as BY Dra, which means that they have semiregular lightcurves with periods between one and ten days, and amplitudes up to 0.35 magnitudes (see e.g. Kunkel, 1975; Hall, 1980). These stars show emission in the Call H and K lines and are often known for flare activity (Busko and Torres 1978). Some, but certainly not all, are binaries (Bopp and Fekel 1977). The lightcurves are characterized by round minima and maxima and almost straight rising and descending branches with slightly different slopes. Secondary maxima do also occur. A typical example of such a lightcurve is shown by AU Mic as observed in 1971 by Torres et al (1972). Figure 1 shows AU Mic based on their data, with an asymmetric ligthtcurve and an amplitude of 0.35 magnitudes. There is evidence that the variations of BY Dra stars can change on timescales of weeks to years. Observations of AU Mic by Torres et jal (1972), Hoffmann(1981) and Van Leeuwen jet al (1983) show significant changes in amplitude and shape of its lightcurve during the past decade. Phillips and Hartmann (1978) found variations for BY Dra and CC Eri on a timescale of years, whereas variations of BY Dra on a timescale of months were shown by Melkonyan(i98D. 74 At the IAU Symposium 67, Kunkel(1975) suggested that the BY Dra 8.6 '+ ' ' ' syndrome might be present among the many flare stars in the Pleiades cluster. 8.7 These stars are of spectral types KO and later. Robinson and Kraft (1974) found V • variabilities for K-type members of the 8.8 Pleiades cluster and assumed them to be of the BY Dra type. They found 8.9 — fractions of a lightcurve for Hz 347 (Hz : Hertzsprung 1917) indicating a period of less than 34 hours and detected variations for 7 of the other 8 0,9 „U-B O stars they investigated, BY Dra type variability in this spectral range in 1.0 the Pleiades was established for five * * • • • • • stars in 1980 October and November at La ; Silla, ESO (Alphenaar and Van Leeuwen 1.1 — • • • * * * * ». • 1981, paper I). Seven others were found in 1981 October and November at the same 1.4 ^B-V site (Meys, Alphenaar and Van Leeuwen 1982, paper II). In addition 1.5 spectroscopie observations of two stars 0.2 -V-r were performed by M.F.Walker at Lick • Observatory. A preliminary discussion • of the combined, photometric and 0.3 -Ï V.» • .....' spectroscopie data has been presented by i r * i * i l i Van Leeuwen and Alphenaar (1982). The 0.6 0.8 0 0.2 0.4 0.6 present paper describes the observations in more detail. Figure 1: Lightcurves of AU Mic in the Johnson UBV and r bands as observed by Torres £t al(1972). Notice the clear resemblance with the lightcurve of Hz 1883 as shown in Fig.2a. The paper is organized as follows. The photometric measurements obtained in 1980 and 1981 are described in section 3.3.2 and investigated in section 3.3.3. The spectroscopie observations and their implications are described in section 3.3.4, while in section 3.3.5 the implications of rotational modulation are investigated. In section 3.3.6 the angular momentum problem is described and in section 3.3.7 some suggestions for future research are made. The star numbers used throughout this paper are those given in the Catalogue of the Pleiades by Hertzsprung (1947) and will be indicated by 'Hz '. All Pleiades stars described in the present paper have well determined proper motions and all are very probable members. This applies also for those stars used as local standard in the cluster or as direct comparison stars for the 1980 observations. 75 3.3.2 The Photometric Measurements 3.3.2.1 1980 Measurements In 1979 November, Van Leeuwen and Brand (see section 3.1) performed a photometric survey of the Pleiades cluster. All known or suspected cluster members brighter than photographic magnitude 14.5 (spectral type K5) were measured two to four times on different nights using the VBLUW photometer behind the Dutch telescope on La Silla, ESO, Many of the stars with (V-B) larger than 0.35 (spectral type later than G8) showed standard deviations in their measurements higher and sometimes much higher than what could be expected (0.0Q5 magn in V). They showed clear indications of being variable. According to Iben(1965) stars of spectral types G and K reach the main-sequence at an age of 10**8 years or later. The age of the Pleiades is also estimated to be 10**8 years (see Golay and Mauron 1982). It is thus likely that the G and K type stars in the Pleiades are completing their contraction towards the main-sequence (see also Stauffer, 1979). Little is known about this phase in stellar evolution, and because variabilities can provide valuable information on -he physical state of a star a more detailed study of the late G and K stars of the Pleiades cluster was initiated. In 1980 a random selection of 8 late G and early K-type cluster members was made, evenly covering the magnitude interval between photographic magnitudes 12.5 and 14.5. Only two other selection criteria were used: the stars had to be well situated on the empirical main sequence and should have an F type cluster member nearby to be used as comparison stars. It was aimed at detecting a possible relation between spectral type and variability. Using the photometric data obtained in 1979 we tried to avoid spectroscopie binaries of comparable mass, i.e. stars that seem to be too bright for their (V-B) index. The photometric measurements were performed during 1980 October and November, by Van Leeuwen and Alphenaar, using the VBLUW photometer on the Dutch 91 cm telescope on La Silla, ESO. A 16.5 arcsecs diafragm was used throughout. The telescope and photometer are described by Walraven and Walraven (I960), the data-aquisition system by Rijf et al (1969), the system calibration by Lub and Pel (1977) and the installation of the telescope and the photometer on La Silla by Lub (1979). The incoming light is recorded by this photometer simultaneously in five channels, called VBLUW, ranging from visual to near ultra-violet. All data obtained with this photometer will be expressed in log(I) units, as is the custom in the VBLUW system. Multiplying by -2.5 gives a relative scale in magnitudes. The measurements have been organized as follows. For each suspected variable a nearby cluster member of spectral type F was selected as comparison star. Due to the rather bright reflection nebulosity in the Pleiades cluster the choice of the position of a sky measurement is critical. We therefore used for each 3tar a fixed sky position as reference point throughout our observing period. These positions are given in Table 1, along with other information o;> „;.,e selected 3tars. The sequence in which the suspected variables and >\/ifeir comparison stars were measured is shown in Table 2. 76 Table 1 Table 2 Data on the stars measured in 1980, The sequence in which the (S, W, N, E: South, West etc) suspected variables and their comparison stars have been measured. Kr M (7000) DEC V-B sky Uog(I>> pos loeiX standard measurement integration 804 3 45 51 +21 02 26 7.85 0.082 30" 3 time atar* program star 25 3 «2 55 «24 29 11 9,49 0.208 30" S 4 • 64 sec ISO 3 13 42 •23 35 47 9.55 0,214 30» II 3 45 «0 «24 37 44 9.T3 S.?3« 30' 3 sky near program atar 2 * 61 see 745 3 15 «I •24 17 85 9.4» 0,221 30» H 1122 3 « 39 •24 Hi 18 9.31 0.199 30» 3 1132 3 "6 38 •22 55 18 9.44 0.208 30" S comparison star 3 * 33 sec 1797 3 18 IT •23 38 IB 0.243 30» J sky near comparison star a » 32 sec atari (•ot»ptrison atar) program star H • 6tt sets 74& (• 745) 3 4$ 41 •24 25 59 11.28 0.351 30» 3 12} (• 16») 3 43 34 •23 45 49 11,45 0.372 30» S 1220 (•1132) 3 44 53 «22 52 58 11.84 0.372 30» S sky near program star 2 • 61» aec 34 (• ZS> 3 41 03 •24 40 17 12.08 0,429 30» S 1124 («1122) 3 40 39 •24 01 53 12.32 0.441 30" II 1881 (•179T) 3 48 28 •23 18 09 12.51 O.47T 30» 3 879 (• 727) 3 48 08 •24 34 01 12.83 O.4}9 30» * «« <• 745) 3 15 33 •2» 18 IT 13.44 0.520 30' V Hz 804 (HD 23409, mtf = 7.86, spectral type A3V) has been used as local standard for the determinations of the atmospheric extinctions. This star had proven to be sufficiently stable in 1979, when it was used for the same purpose. In order to check its stability once more, some of its measurements were followed by measurements of Hz 652 (HD 23361, my = 8.05, spectral type A3V). During four successive nights 26 pairs were measured, showing a stability of better than 0.0015 in log(I) in V. The mean differences and their standard deviations in V and the colour indices are presented in Table 3« along with a similar test performed in 1981. Table 3 Differences between Hz 625 and Hz 804 (the local standard) as measured in 1980 and 1981. V M »~U U-V B-L 652-804 -0.0790 0.0124 -O.0OO8 0.0078 0.0038 (1980) a.d. «0.0017 •O.OOtO «O.0015 «9.0028 «0,0009 852-804 -0.0788 0.0125 -0.0005 0.0083 0.0036 (1981) i.d. «0,0030 «0.0010 «p.OOIT «0.0027 «O.OOtO Several stars of the VBLUW standard system were measured before and after the five hours spent each night on measuring Pleiades stars. In addition, during some very good nights, some standards were measured during the culmination of the Pleiades. The latter were used to calibrate Hz 804 to the standard system. The others were used mainly as a check on the stability of the atmospheric extinction and the instrument. 77 The local standard, Hz 804, was measured seven times each night, with intervals of half an hour to one hour, providing an even distribution of sec(z) values between 1.6 and 2.3. The extinction determination consisted of a least squares fit to the measurements of Hz 804 with sec(z) and the time as variables. Most of the nights had well determined extinctions, while they showed only small variations from one night to the other. Usually a small correction was needed for a change of zeropoint during the night, possibly due to the influence of temperature changes on the high voltage supply. The extinction determinations showed that during good nights on La Silla differential photometry is possible over a field of 2 degrees up to 65 degrees out of the zenith. This provided a maximum of five hours of observing on the Pleiades per night. The comparison stars were calibrated relative to Hz 804, Two of them, Hz 727 and Hz 1797, turned out to be variable (see Paper I). The mean values and standard deviations for all seven comparison stars are presented in Table 4, Because the use of these stars hardly improved the quality of the final data, we decided to reduce all stars directly with respect to Hz 804. Table 4 Photometric data on the comparison stars and the local standard. The first standard deviation with the local standard Hz 804 has been obtained from the calibration of 9 of its measurements to the VBLUW standard system, while the second indicates the standard deviation with respect to the extinction determinations, like is also the case for the standard deviations of the comparison stars. The last column provides the total number of measurements used to calculate the means and the standard deviations. Jtair V V-B B-U U-U B-L nr ... log(I) • • • Hz 804 -0.3948 0.0855 0.4435 0.1472 0.2104 9 +0.0013 •0.0019 +0.0012 +0.0033 +0.0030 9 +0.0015 •0.0007 +0.0012 +0,0018 +0.0009 135 Hz 25 -1.0490 0.2127 0.3057 0.1888 0.2073 25 +0.0014 +0.0008 +0.0012 +0.0046 +0.0012 Hz 164 -1.0759 0.2207 0.2961 0.1902 0.2078 9 +0.0011 •0.0017 +0.0020 +0.0041 +0.0021 Hz 727 -1.1538 0.2417 0.2926 0.1970 0.2226 30 •0.0053 +0.0017 +0.0028 +0.0039 +0.0023 Hz 745 -1.0432 0.2295 0.3258 0.1986 0.2184 60 •0.0015 •0.0011 +0.0021 +0.0034 +0.0020 Hz 1122 -0.9783 0.2040 0.2980 0.1854 0.2066 22 •0.0009 +0.0010 +0.0021 +0.0032 +0.0012 Hz 1132 -1.0333 0.2145 0.3069 0.1932 0.2104 11 +0.0010 +0.0011 +0.0019 •0.0026 •0.0015 Hz 1797 -1.3105 0.2496 0.2902 0.2019 0.2272 41 +0.0034 +0.0014 •0.0035 +0.0073 +0.0026 78 AU eight selected stars were found to be variable, all showing variations larger than 0.02 magnitudes. For the faintest five of them (Hz 34, Hz 686, Hz 879, Hz 1124 and Hz 1883) periods anfl lightcurves could be obtained (see Paper I). The other three (Hz 129, Hz 746 and Hz 1220) show variations with small amplitudes and on a timescale longer than days. The pc iod3 and amplitudes of the variations are shown in Table 6, together with the 1981 results, The five faintest stars were remeasured a year later in order to investigate their variations on a longer timescale. 3.3,2,2 1981 Measurements During 1981 October and November a second series of measurements was performed by Alphenaar and Meys, In addition to the five stars mentioned above, a new selection of 11 stars was investigated using the same instrumentation as before. Because of our experience with comparison stars mentioned in section 3.3.2.1, all measurements were reduced directly to Hz 804 of which a renewed stability check is shown in Table 3. Information on additional stars selected for the 1981 measurements is presented in Table 5. These additional stars were not chosen at random, but instead a selection was made based on the 1979 and 1980 measurements, nine stars were selected because of large deviations, up to 0.04 in log(I) in V. while another two (Hz 883 and Hz 1332) were selected because of small deviations, 0.002 in log(I) in V. These last two star3 were chosen in order to see whether non variable K stars in the Pleiades exist. Of these, Hz 1332 was found to be variable and regular; Hz 883 still may or may not be variable. Table 5 Like Table 1, for the additional stars selected in 1981. Hz i(A (2000) DEC ra V-B sky V log(I) pos 451 3 44 50 +24 54 46 13.43 0.570 30» N 625 3 45 SI +23 43 45 12.66 0.526 30» S 659 3 45 26 +23 25 55 12.02 0.413 30" N 882 3 46 04 +23 24 26 12.90 0.487 30" S 883 3 46 07 +24 33 52 13.05 0.534 30» M 1039 3 46 27 +23 35 40 13.05 0.572 30» S 1332 3 47 13 +23 42 58 12.52 0.474 30" S 1531 3 47 41 +23 58 25 13.58 0.563 30» N 2034 3 48 49 +23 58 44 12.65 0.447 30" S 3019 3 51 24 +24 05 20 13.49 0.567 30" S 3163 3 51 .53 +24 23 19 12.73 0.457 30» S Because the weather on La Silla in 1981 has been much better than in 1980, the observations could be carried out according to plan. The observing period of three weeks wa3 devided in three equal parts. The first week was 79 used to check the new selection for variability and to compare the lightcurves of the old selection with the previous measurements. The second week was spent on obtaining fractions of lightcurves and determining preliminary periods. The third week was used to adjust periods and to evenly cover the lightcurves. To make this plan feasible all measurements were to be reduced within 24 hours after they had been obtained. All available information could thus be used in the planning for the coming night. Tn 1980 we used for this purpose an HPM1C pocket calculator and in 1981 the reduction program for VBLUW photometry on the ESO computing facilities at La Silla could be used. The final reduction of the measurements was carried out at the Leiden Observatory in the way described in section 3.3.2.1, Periods and lighteurves, presented in Table 6, were found for seven of the eleven selected stars. Three of the other four stars are variable and one probably. All individual measurements obtained in 1980 and 1981 will be published in the near future. Table 6 The compiled information on variations, periods and lightcurves, The last column gives the total number of flares observed for these stars (Mirzoyan, 1981). Nr ID Observed Period V-ampl. flares •80 '81 (hours) vmagn) observed 3« 12.06 • • 28.116 0.05 139 11.15 0 >0.02 «51 13.15 (*) • ? >0.05 1 625 12.66 r 10.26 0.12 659 12.02 ? >0.05 686 13.11 • • 9.5282 0.10 3 7M6 11.38 * ? >0.02 879 12.83 21.086 0.07 382 12.90 13.89 0.12 883 13.05 Ml. 02 1 1039 13.05 < 29.2 >0.02 1 112'! 12.31 • i 20.550 0.07 1220 ii.au • ? >0.02 1332 12.52 27.3 0.05 1883 12.61 • i 5.61*98 0.20 2 2031 12.65 i 8.53 0.07 2 3019 13.19 i ? >0.07 5 3163 12.73 i 10.28 0.10 3.3-3 The Photometric Data and their Interpretation 3.3^3.1 The Lightcurves and Periods Lightcurves have been obtained for a total of 12 stars. All of these show a clear resemblance with those of BY Dra stars. The periods found range from 5.5 to 30 hours, which is considerably shorter than what is usually found for BY Dra 3tars. This difference in periods may be related to the difference in spectral type: the BY Dra field stars are late K and M type stars, whereas 80 the Pleiades stars are early K types. According to Haro (1976) and Mirzoyan (1981) eight of the 19 variables show flare activity (see Table 6). This includes Hz 883 for which the variations are very small or not existing. During a total of 12.5 hours of integration time spent on all 19 stars shown in Table 6, no flares have been observed. Lightcurves may be distinguished into three groups, according to their shapes: the 'v' shaped, shown in Figs.2a-c, the 'u' shaped, shown in Fig.2d and the 'n' shaped, shown in Fig.2e. They are sorted on period and all shifted to minimum light at zero phasr, Most of the lightcurves show different slopes for the rising and descending branches, but no preference has been found for either the rising or the descending branches to be steepest. The intensities as measured for all these stars change continually as a function of time. -i—i . , r- "i—'—r—"—i—•— • Ml 01.24 * 1 ' 1 • T <—T r—| 1 =! 1i • • t1 • • •• _ * • i • % •. 1*1 1 2» • m •• •.'•: • .* • ft - V. * • • • • *.'. .«» •ZM * + • • • * + + + ++ + . •* Ui OUd au os» t • • * •• • , • • . • • • * • •IU • ' . • * - • *". * • ' ' . '• " _ * •• • _ i . i , * *• •* 1.1. I.I.I 1 . l . t 0? 01 00 02 Figure 2a Figure 2b Figure 2: a,b and c: The 'v1 shaped lightcurves; d: the 'u' shaped lightcurves; e: the 'n' shaped lightcurves. Dots indicate the 1980 observations and open circles the 1981 observations. The lightcurves are all shifted to minimum light at zero phase and are plotted in log(I) in the V band of the VBLUW system. 81 L * TOOSM». i ' r ' 1 1 r 1 "1 ..• • • • . " . 2034 0 *U T • "" . . • .. * * * • * 233 * • -1...V • • •• . + •)- + + •. + + -13» • + • 606 Q4Qd t • 261 • • ; * + + • : • • • S + + » ' V." . ; : ,. * • '+ ++ + H + • 1I2I> Olid * •• ,; • + + + '. •. 233 '• 1039:12211 i - -J4 . ' * " ' "" • 1 1,1. l . 1 1 1 «a 02 ot oi no o; 00 02 04 04 01 00 0.2 «MX pta» Figure 2c Figure 2e Figure 2d 82 Figure 2 indicates the existence of a relation between period and amplitude. Which is shown in Fig,3. Stars with 'v' or 'u' shaped lightcurves show larger amplitudes at shorter photometric periods. Those with 'n' shaped lightQurves generally show smaller amplitudes. A relation like this is not known for field stars of the BY Dra type, which may be due to the fact that, unlike the field stars, the Pleiades K stars are all approximately of the same age and mass. i .01 - IMfll (logt) Figure 3: QA The relation between period and amplitude for the stars presented in Fig.2. The 'v', 'u1 or 'n' no indicates the shape of the "o lightcurves, A relation between „n periods and amplitudes clearly .02 exists for the 'v' and 'u' shaped "o lightcurves. .00 i . i 10 K 20 KH100 (boon) Apart from the absence of pronounced variations for the brighter stars, those of spectral type G, no relation has yet been found between the photometric variations and the mean brightness of the stars. Moreover, one of the comparison stars used in 1980, viz. Hz 727 (m =9.73, spectral type F7.5v), shows similar variations, as can be seen in Fig.1». Its amplitude is small and it has a possible period of 27.4 hours. Figure 1: The lightcurve of Hz 727 (spectral type F7.5 V), which was initially taken as comparison stsr. -0768 phi» The five stars that were measured both in 1980 and in 1981 showed no changes in their periods. Changes in the lighteurves, however, have been observed for two of them, while three of these stars showed changes in mean brightness. The 1980 and 1981 data in Fig.2 are indicated by different symbols. In the three cases where the shapes of the lightcurves remained the same, both the data of 1980 and 1981 have been plotted in the same frame, to show to what extent the lightcurves were reproduced. Pronounced changes are shown by Hz 1124, which decreased in mean brightness by 0.01 in log(I) in V from 1980 to 1981. This change was accompanied by a change in lighteurve from 'nf to 'vf shape. The same star showed an increasing depth of the minimum in 1980. An increase in brightness has been observed for Hz 31, by 0.008 in log(I) in V between 1980 and 1981. Hz 686 showed a relatively noisy lightcurve in 1980 and a change from 'v' to 'n' between 1980 and 1981. 3.3.3.2 The Colour Indices The simultaneous recording of light in five channels by the VBLUW photometer permits a description of the colour indices (V-B, B-U, U-W and B-L) on the same basis as the description of the V observations. The information recorded in the near ultra-violet channels is, however, for K-type stars ra' .ter noisy due to the lack of photons. A detailed description of the behaviour in the colour indices is therefore only possible for Hz 1883, as this star is relatively bright and shows the largest amplitude in its variations. Figure 5 shows the lightcurves of Hz 1883 I 1 I 1 I I in V and the colour indices, The variations in V are v in (V-B) but reflected tog(i) not in the other indices. The relation between V and (V-B) is linear as can be seen in Fig,6. There are no -2.30 differences found in this relation between the rising and descending branches of the lightcurve. Figure 7 shows the size and direction of the variations of Hz 1883 H—I—I—I—I—I—r—I—I—h—r- in a V - (V-B) diagram, together with the data 0.46 of other cluster IV-Bl members of that \t 1 spectral region. loglll Similar relations between V and (V-B) are found for nine other I I I I I I .1 I I I I I I stars (see Table 7), o.t while the noise level on the (V-B) data for (B-L .*>'. the others is too high • • • •• -. • . with respect to their 0.S amplitudes to make such a detection possible. H—I—I—h—{-H—I—I—I—h—I- (B-U! «.so .• .* . Figure 5: The lightcurves of login Hz 1883 in the VBLUW system V channel and (V-B), (B-L) and (B-U) 0.5! I I I I I I colour indices. 0.0 ph,« 02 0.4 0.6 0.8 1.0 0.2 84 -2,25 - log (I) -2.201- -2.35 O',6 (V-B1W log(l) 0.47 0.48 Figure 6: The V-(V-B) relat,-, a for Hz 1883 as derived from the 1980 (.) and 1981 (o) measurements. Table 7 A(V-B)/AV for the Pleiades K-type stars, indicating that all of these get redder when getting fainter. Hz A(V-B)/AV Hz A(V-B)/AV 34 -.28 +.05 625 -.24 +.07 686 -.36 +.11 879 -.13 +.08 882 -.23 +.06 1124 -.17 +.05 1531 -.29 +.12 1883 -.19 +.02 2034 -.18 +.09 31B3 -.12 +.05 Figure 7: The V-(V-B) relation for the K-type stars of the Pleiades cluster. Open circles represent the stars of Tables 1 and 5, while crosses indicate other members. The size and direction of the variations of Hz 1883 are indicated by arrows. 85 The variations of Hz 1883 and the V - (V-B) relation set by the cluster members of that spectral region both show the slope: (V-B) = -0,20*V + const +0,016 This relation probably indicates that the variations in V are related to variations in surface temperature, A simple comparison with UBV data (Nicolet, 1977) and (B-V)(Johnson) - T(eff) relations given by Allen (1976) shows an average surface temperature for Hz 1883 of 4450 K, with a difference between the minimum and maximum of 9Q K, The expected brightness variations that result from a^ch temperature changes are, however, smaller than those observed. This means that the variations ox1* this star do not follow lines of equal radius, which for a simple B'lackbody approximation are given by: (V-B) = -0.26»V +const The effective surface at maximum light is 6.0+1.2 percent larger than at minimum light. Figure 8 pictures the calculations given above. 1 1 ' 1 t 1 <*" ^ uu / / tuo - / \ KKI / I I I I i 1 1 1 1 1 1 1 1 Figure 8: The changes in temperature as Ui - • derived from the measurements of / Hz 1883 are shown in the top graph. The bottom graph shows — how the observed intensity 109 variations (full line) differ l/l. from those derived from the . '/ temperature variations alone on - y/ (line with open circles). • , 1 I 1 1 | f f | Op».' .2 The variations in V are not reflected in (B-U) and (B-L). In the Blackbody approximation a A(B-U) of -0.23*AV would be expected. The variations in the U and L channels can be smaller than expected if these channels also receive light from an other source, not or less participating in the variations. In that case the relative changes in U and L intensities would be smaller. This may also change the amplitude in B and bring the V - (V-B) relation closer to the Blackbody relation. A likely candidate for a (variable) source of near ultra violet light is the chromospheric activity. Although the chromosphere is probably active, no fit could be made to explain 86 the observed behaviour in U and L. A similar colour dependence of photometric amplitudes has been observed for stars in the solar neighbourhood by Vrba and Rydgren (1983). 3.3,4 Spectroscopie observations In order to verify whether the photometric variations of Hz 1883 could be the result of duplicity, four spectra were kindly taken by M.F.Walker: three on 1980, November 28 and one on November 29. They are single trail time resolved spectra of which the first three cover the complete period. The observation? will be described in detail by M.F.Walker in the near future. The spectra look quite normal, the only peculiarity being the presence of emission cores in the Call H and K lines. There are no double lines observed, while changes in radial velocity are smaller than +10 km/sec. We therefore conclude that Hz 1883 is most likely a single starT Line profile measurements were performed in 1981, October and Dfcember by M.F.Walker for two stars, viz Hz 1883 and Hz 3163. The spectrum of the first showed H(Xweakly in emission with the impression of being double. The line profiles were compared with similar measurements of four field stars with known rotational velocities and line broadenings as given by Herbig '-d Spalding (1955). V.sin(i) values of 146 and 75 km/sec were found for Hz 1133 and Hz 3163 respectively, with an uncertainty of approximately 16 km/sec (Walker, private con.munication). The ratio between the photometric periods and the rotational periods as derived from v.sin(i) for Hz 1883 and Hz 3163 is almost the same. In addition, the rotational periods can be equal to the photometric, periods for reasonable assumptions of the stellar radius and inclination (R(Hz 1883)=0.72R(Sun)/sin(i), R(Hz 3163)=0.67R(Sun)/sin(i)>. It seems therefore reasonable to assume the photometric and rotational periods to be equal, in which case the photometric variations can be attributed to rotational modulation. This has also been observed for other fast rotating stars with convective envelopes that show similar variations, e.g. those classified as BY Dra, RS CVn and FK Com stars. 3.3.5 Rotational Modulation The photometric and spectroscopie data are combined in order to gain inside in the background of the photometric variations. It will be shown that probably all Pleiades K stars have a non axial symmetric surface flux distribution. This is found to be due to the influence of their high angular velocies on their convective envelopes. No relation is found between the longitudonal modulation of the surface fluxes and flare activity. Photometric variations can be the result of rotational modulation in two different situations: (1) with a geometric deformation of the outer layers of the star; (2) wi'.h a non axial symmetric flux distribution over the stellar surface. The first solution is hard to maintain, as the lightcurves simply cannot be explained as due to geometric distortion only. The most likely deformations that could cause light variations are cigar or egg shaped. Both of these, however, would give rise to two distinguishable minima and maxima 87 rather than one minimum and maximum. Thus, although deformations of the stellar surface may be present and are even likely considering the high angular velocities, they are unlikely the direct reason for the variations. It will therefore be assumed in the following that the variations are primarely due to an inhomogeneous flux distribution over the stellar surface. The present paper described measurements of 19 late G and early K type members of the Pleiades which were all found to be variable. Robinson and Kraft(1974) investigated 8 other Pleiades K stars and found all of these to be variable too. In selecting the stars for the present paper indications of variability were either no criterion or were compensated by also selecting some stars that gave a first impression of not being variable. It therefore seems justified, under the assumption of rotational modulation, to state that most, if not all Pleiades late G and K type stars are variable. This implies for all these stars a non-axial symmetric flux distribution on their surfaces, The lightcurves that have been observed can very accuratly be represented by a first and a second harmonic, which are only slightly shifted in phase. Higher harmonics do not contribute significantly to the lightcurves. Even though the amplitudes of higher harmonics on the stellar surface get decreased due to projection in the lightcurves, they are still too small to be of importance on the stellar surface. The lightcu'-ves do not show discontinuities in their first derivative that could indicate discontinuities in the flux distribution on the stellar surface as a function of longitude. The lightcurves as observed for stars like AU Mie (see Fig.1), the FK Com type stars (Bopp, 1983) and the RS CVn stars (Catalano, 1983) are all very similar to those of the Pleiades K stars. They indicate that the characteristics of the non-axial symmetry of the surface flux distributions are for all of these stars very similar. The variations in flux distribution are accompanied by colour variations, which are linearly related to the flux variations. If these colour variations are interpreted as due to temperature variations, then they will account for most of the flux variations. Similar effects have also been observed by e.g. Vogt(1983) and in particular for AU Mic (Van Leeuwen, Meys and Alphenaar 1983) and HD 18131, HD 18407 and TO Ps A (Meys,Van Leeuwen and Alphenaar 1983). Of these, the colour variations of HD 18134 were found to be larger than needed to explain the V variations. These observations indicate that the main feature observed is an overall temperature gradient over the surfaces of these stars. From the observations of the Pleiades stars it is found that such temperature gradients can remain stable over long time intervals (>1 year or 1550 rotations) but may also change in periods of weeks (20 rotations). The relation that was found to exist between the photometric periods of the Pleiades K stars and their photometric amplitudes can be translated to one between the amplitude of the modulation of the surface flux distributions and the angular velocities. It clearly shows that these modulations are due to the high angular velocities and therefore explains why all the Pleiades K stars ate found to be variable: because of their high angular velocities. The variations found for these stars are also typical for other fast rotating stars with convective envelopes (young stars, such as those described in the present paper, corotating late type binaries such as BY Dra and giants such as the RS CVn stars). It is therefore likely that the longitudonal modulation of their surface fluxes is primarely due to an interaction between the angular velocity and the structure of the convection zone. 8B Some of the K-type stars in the Pleiades cluster are also known for chromospheric (McCarthy 1974) and flare activity. There was, however, no relation found between flare activity and photometric amplitudes or periods (see Tablvi 6). The flare activity of the Pleiades K stars is rather low compared to some field stars showing similar variations, such as AU Mie for which two flares were observed in 1.5 hours of total integration time (Van Leeuwen, Meys and Alphenaar 1983). There seems to be no unique relation between photometric amplitudes or periods and flare activity, neither in the Pleiades, nor with stars outside the cluster, A non uniform light distribution on a stellar surface as described above is usually explained as due to an analogon of Sun spots on the stellar surface. It was first suggested by Kron(1952) and Chugainov(1966) and further investigated by Krzeminski(1969). Evans(1971) and Torres and Ferraz Mello(1972) applied it to CC Eri and AU Mic respectively. It was further investigated with respect to BY Dra by Vogt(1975) and Oskanyan(1977). They all found ways to reproduce the observed lightcurves in one channel but some encountered problems when also colour variations had to be explained. It should be realized that any regular Hghtcurve can be explained by means of an arbitrary inclination angle and latitude and temperature distribution of spots. In the models of 'v'-shaped lightcurves the inclination angle is assumed to be rather small, with spots positioned near to one pole. This way no fraction of the lightcurve is at constant light, and the 'v' shape of the lightcurve becomes mainly the result of projection effects. Here we prefere to leave open the question whether the actual blocking of the surface flux is due to sunspots. It is, according to us, more important to find out why all the Pleiades K stars show a non axial symmetric surface flux distribution which is very similar from one star to the other, as well as similar to those found for other fast rotating stars with conveetive envelopes. It is important to find out how the non axial symmetry of these surface fluxes is related to the high rotational velocities and their interaction with the convective envelopes and the magnetic fields that are likely to develope in such envelopes. Such a study is likely to reduce considerably the degrees of freedom in applying a sunspot or any other model and would therefore add much to the value of to their application. The question asked by Hall(198O): 'why are these stars variable?' can no longer be answered by 'otherwise we would not observe them'. It should be replaced by the following question: why do so many fast rotating stars with convective envelopes have non-axial symmetric surface flux distributions? 3.3.6 The Angular Momentum The one to one relation between the photometric and rotational periods derived in section 3.3.^ implies that the angular velocities of the Pleiades K stars are 50 to 100 times higher than that of the Sun, Also their angular momentum will be larger by approximatly the same ratio. Differences between the internal structure of main-sequence and pre-main-sequence stars may lead to a slightly different ratio. Figure 9 shows those Pleiades K stars for which periods have been found in a diagram like presented by McNally (1965), in which the logarithm of the angular momentum per unit mass is plotted against the logaritnm of the mas3. A similar. diagram showing the same features has more recently be presented by Gray(1982). As McNally pointed 89 out, there exist two relations between angular momentum and mass. The first is set by O, B and A stars and may represent the initial angular momentum distribution. The second is set by the F and C stars and may represent a stability criterium: stars above this relation are unable to maintain their initial amount of angular momentum, The cross point of both relations coincides with the onset of the convection zone. It is therefore now-adays generally believed that the second relation is due to magnetic braking. As the onset of sonveetion causes magnetic fields to be developed, it is thought that stellar winds streaming out along these fields slow down rotation, The Pleiades K stars are found on the relation set by the 0, B and A stars. We therefore conclude that this process had not much effect on these stars yet; they confirm the impression that this relation is due to an initial angular momentum distribution and show that it initially also applies for K-type stars. 1 1 J»D5 18 ïéo "" logA PM 17 -18B3/7 — Figure 9: "i f The relation between the logarithm of the angular momentum per unit mass (A) and the logarithm of the total mass as 5 &F5 derived for main-sequence stars by McNally(1965). The diagram also shows oio the positions of the Sun, the solar system and the Pleiades K stars for which periods are found. Note that the fastest rotating K stars as well 15 as the solar system follow the ~ /oSUN relation set by the 0, B and A stars. 1 r 33 34 logH The angular velocities of the Pleiades K stars are much higher than those of the G-type stars as measured by Mayor (private communication) with the CORAVEL radial velocity detector. This is also indicated by the rapid increase in the timescale and decrease in the amplitudes of the photometric variations for these stars as described in section 3.3.2.1. The G stars in the Pleiades are only slightly further evolved. Their low angular velocities show that the slowing down of these stars is taking place on a short timescale, a process which can at present be observed i'rom e.g. photometric and spectroscopie observations of the G and K stars in the Pleiades. In the situation described in section 3.3.5 the magnetic field would be beamed due to the non axial symmetry of the convective envelope. If this beam, causing the slowing down of the angular velocity, is pointed not perpendicular to the stellar rotation axis, then this slowing down of the stellar rotation would be accompanied by a precession of the star. This precession may be observable 90 from changes in the Hghtcurves. One further observation is of interest with respect to MeNally's diagram. It concerns the positions of the Sun and the solar system. The first follows the relation set by other G stars, while the latter coincides with the initial relation set by the 0, B and A stars, and has a total angular momentum compatable to the K stars in the Pleiades. McNally pointed out, that this was an indication of how the initial angular momentum of the Sun has been redistributed. If the total angular momentum of the solar system is not accidentally following the relation set by the 0, B, A and Pleiades K stars, but is also a reminder of the initial angular momentum distribution, then the following question could be posed: Is the rapid slowing down of the G and K stars, which takes place at the time of arrival on the main sequence under the influence of a probably beamed magnetic field, related to the formation of planetary systems? 3.3.7 Future prospects The observation and determination of periods and lightcurvep for a large number of stars, as well as the determination of their rotational velocities, is a time consuming program. Still we hope to be able to continue it for a few seasons in order to improve the statistical value of the data. In addition, observations of similar stars in other, younger and older open clusters have to be obtained. These may provide period-amplitude relations similar to that observed in the Pleiades, from which the mass and age dependence of this relation could be found. Detailed line profil» measurements may provide in the near future a possibility to obtain latitude resolution of the flux distributions on the surfaces of the K stars in the Pleiades. Acknowledgements It is a pleasure to thank P.Alphenaar and J.J.M.Meys for their enthousiastic cooperation in the observations and the reductions. 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Vasilevskis, S., Van Leeuwen, F., Nicholson, W. and Murray, C.A., 1979: Astron.Astroph.Suppl., _37, 333 Vogt, S.S., 1975: Astroph.J. _199, 418 Vogt, S.S., 1983: in Activity in Red-Dwarf Stars, IAU coll.71 Vrba, F.J., and Rydgren, A.E., 1983: in Activity in Red-Dwarf stars, IAU coll.71 Walraven, Th. and Walraven, J.H., 1960: Bull.Astron.Inst. of the Netherlands 15, 67 4. On the Distribution of Masses and Motions in the Pleiades 4.0 Summary Astrometric and photometric data on stars in the Pleiades cluster are used to determine the distribution of masses and motions of these stars. Surface and space density profiles are calculated and it is shown that the density of low mass stars in the cluster centre is very low. This is interpreted as due to the influence of the galactic tidal field, which can increase the angular momentum dispersion of stars in the cluster halo. Many low mass stars that are ejected to the cluster halo will therefore lose their interaction with the cluster core. When the contribution of the cluster halo is taken into account, no significant difference is found between the luminosity function of the cluster and that of the solar neighbourhood over the range from 4 to 0.4 solar masses. Information on stars with lower masses, which are all pre-main sequence objects, is incomplete. The occurrence of unresolved binaries is investigated by means of the luminosity distribution of the F and G type clustermembers. It is found that at least 15 percent of these stars are unresolved binaries of comparable mass and brightness. A comparison between their projected distribution and that of the A and B type members clearly indicates that the suspected binaries are indeed more massive than single F and G stars. Information on the distribution of motions in the cluster is obtained from the proper motions that were presented by Vasilevskis et ^1 (1979). A procedure is given to separate from these and similar differential proper motions the external influences. The internal proper motion dispersion thus derived is O'.'OOI per annum in each coordinate projected on the centre. The tangential component decreases rapidly going outwards. A model of the potential energy profile of the cluster is determined from the spatial distribution of its stars. Using the luminosity function and the space density distribution of G stars, it is extrapolated such as to fit with the observed velocity dispersion and space distribution for the brighter stars. A total cluster mass of 1000 to 1600 solar masses is found this way, indicating the existence of a large number of low mass stars. A comparison with numerical and analytical cluster simulations shows that only few use realistic input parameters. In particular the evaporation could be less than predicted and core collapse could disappear if the more realistic parameters are used. The modeling of the Pleiades cluster is probably best done with a numerically calculated core evolution and a statistically or analytically treated halo and halo core interaction. 95 4.1 Introduction Attempts to determine the internal proper motion dispersion of the Pleiades are as old as the determinations of the proper motions themselves. The first, although unsuccessful, of these were made by Wolf(1877) and Pritchard(1884). Their proper motions were, however, not good enough to distinguish members properly. The proper motions as determined by Elkind887) and later by Lagrula(1903) were the first to make a proper distinction between members and field stars in the central region of the cluster. Elkin combined his heliometer observations with those obtained by Bessel(1844) while Lagrula used practically all heliometric, micrometric and photographic positions obtained by several observers between 1829 and 1900. TrumpleKW1!) reinvestigated these data and derived an upper limit of 6 arcsec per millennium for the internal proper motion dispersion of the cluster members. A significant improvement of the accuracies of the proper motions in the Pleiades region was obtained when Hertzsprung(1934) presented preliminary results of an extensive photographic study of the central region of the cluster. This study was based mainly on photographic exposures taken with Astrographs built for the Astrographic Catalogue and Carte du Ciel surveys. A few years later, Titus(1938) presented similar results based on a comparison between the original Rutherford plates taken between 1865 and 1874 and a repeated set taken with the same equipment. Both Hertzsprung and Titus found a dispersion in the proper motions of cluster members of 0.9 arcsec per millennium, from which Titus derived an internal velocity dispersion of 600 m per sec. This value is close to more recent determinations by Binnendijk(1946), JonesC1970b) and Van Leeuwen(1980). The distribution of stars in the Pleiades has first been investigated by Trumpler(1920), who showed that the massive, bright stars are much more concentrated to the centre than the faint, low mass stars. This observation has since been confirmed by Artyukhina(1968) and Van Leeuwen(1980). Problems arise in the density determinations for stars with masses less than 1.5 solar masses in the cluster centre. These densities are so low that the statistical accuracy of their determination is very low too. This gave rise to completely opposite results obtained for the central space densities by Mirzoyan and Mnatsakanian(197D and Kholopov(1971b). Estimates of the total mass of the Pleiades have been obtained by Titus(1938) and Jones(1970b), who found masses of 260 and 690 solar masses respectively by applying the Virial theorem to the internal velocity dispersion in the cluster centre. The conditions in an open cluster, and in particular in its halo, however, violate the basic assumptions of this theorem and such estimates are therefore only very crude approximations. A mass estimate based on the potential energy distribution, the central velocity dispersion and the space density distribution of the massive stars indicates a significantly higher mass (see Van Leeuwen 1980 and this paper) The luminosity function of the Pleiades has been investigated by Trumpler(1921), Jones(1970b), Van Leeuwen(198O) and Jones(198D. Differences of opinion are shown with respect to the position where the luminosity function has its maximum and from where it flattens or decreases towards fainter stars. These differences are mainly due to a different interpretation of the observations of flare stars in the cluster. An early flattening can 96 justify the low mass estimate obtained by Jones, while a late flattening is needed for the high mass estimate obtained by Van Leeuwen. The relevant data will be reviewed in the present paper. The occurrence of binaries in open clusters has major impacts on their dynamical evolution, as they can act as energy reservoirs. Bettis(1977) observed an abundance of binary F stars in the Pleiades comparable to that of the solar neighbourhood. Following Heggie(1975) we define hard binaries as those that have binding energies higher than the local mean kinetic energy in the cluster centre, which implies that in the Pleiades all F and G type binaries with separations less than 5" are hard, and will tend to get harder in interactions with single stars (see e.g. Heggie 1975, Dokuchaev and Ozernoy 1982). This is due to the fact that these systems are unable to obtain enough energy from a single interaction to be disrupted. The occurrence of such stars in the Pleiades will be investigated in the present paper, using a virtually complete sample of all F and G members. The total number of stars in the Pleiades cluster is too high for a complete numerical simulation, but it is also too small for the statistical approach of Monte Carlo simulations and analytical methods. The influence of the galactic tidal field, however, causes segregation of a core with a few massive stars and a halo with a high number of low mass stars (see also Terlevich 1980). This makes a combined approach possible where only the core is described numerically. In the present paper the available astrometrie and photometric data have been combined in order to provide an empirical model of an open cluster in as much detail as possible. It is expected that this model will add to the development of theoretical and numerical models, which may teach us about the dynamical evolution of open clusters. The paper has been organized as follows. In section 1.2 the surface densities of B, A, F and G stars are examined, and the spatial densities derived. Some remarks are made with respect to the significance of these determinations. An estimate of the occurrence of binary stars is obtained in section 1.3, using the luminosity distribution of F and G stars. The suspected, unresolved and therefore hard binaries are compared in their projected distribution with B and A and F and G stars. Section 4.1 deals with the luminosity and mass functions of the cluster, which are compared with those derived for the solar neighbourhood and from other open clusters. A cluster mass estimate is obtained from an extrapolation of the observed luminosity function. In section 1.5 the very precise proper motions obtained in the central region of the cluster by Vasilevskis, Van Leeuwen, Nicholson and Murray(1979) are used to determine the dispersion in the internal motions as a function of the distance to the cluster centre. First, however, the external influences on relative proper motions are described, such as the relative motion of the cluster with respect to the Sun. Some remarks are made with respect to the possibilities of detecting rotation and expansion. In section 1.6 a model of the potential energy distribution in the cluster is obtained, and in section 1.7 the influence of the galactic tidal field on the tangential velocity dispersion is derived. Finally, in section 1.8, the empirical cluster model is compared with existing numerical and analytical models. 97 Throughout this paper it will be assumed that the Pleiades cluster is spherically symmetric. Such an assumption is necessary in order to ma 3 a three dimensional interpretation of two dimensional data possible. It can be checked only for the projected surface density distribution, which is done in section 4.2. Non spherical symmetry can, in principle, also be checked by means of the distribution of distance moduli and by combining proper motion data with radial velocities. Both checks, however, are still of too low accuracy or too much disturbed by other influences. 4.2 The Density Distributions The determination of surface and space densities for open clusters is hampered by the low number of stars they contain. The outlying region of an open cluster, to which we shall refer as the halo, is not visible against the background of field stars. At the outer edge of the Pleiades, at a projected radius of 4.5 degrees from the centre, the projected field star density down to phot.mag 13.0 outnumbers that of the cluster stars by more than a factor 100. Statistical methods are thus useless for the detection of open cluster halos, as was also shown in practice by Artyukhina and Kholopov(1966). Only by means of well determined cluster members can one determine the content and distribution of stars in the halo of an open cluster. The cluster members used in the present paper come from the selection made in section 3.1 and consists of only stars with high probability of membership, up to phot.mag 14.5 for the centre and phot.mag 12.5 for the halo. Photometric data that are used come from section 3.1, while the astrometric data are provided by Hertzsprung (1947) and Vasilevskis et al(1979). 4.2.1 The Counts The surface density as observed for an open cluster is the result of the projection of the space density along the line of sight. The space density, p(R), is assumed to be spherically symmetric; the surface density, 0(r), will therefore be circular symmetric. Still another quantity will be introduced here, \(z), which represents the space density as projected on the line of sight, and which provides information on the distribution of distance moduli in the cluster. It will be used as such in section 4.3. (r.z^p) are cylindrical coordinates, where z is defined along the cylinder axis, coinciding with the line of sight through the cluster centre, and r,tp are the radius and orientation in the cylinder. The relations between (Jt X and p, expressed in cylindrical coordinates, are given by (see e.g. Smart, 1938): /"* RR p(R)) O(r) = 2|~- dR (1) = 2 I 2 J r> (R - r . T X(z) = 2T\J p(R)RdR (2) 98 Both 0(r) and \(z) can in principle be observed. In the Pleiades cluster, however, \(z), as determined from photometric data, is disturbed by duplicity, differential reddening, possibly by "cosmic scatter" and by photometric inaccuracies. Some of these effects will be described in section 4.3. The function CKr) can be determined as soon as the cluster centre is known. Trumpler(1921) gave for it a position 6' West of Alcyone, based mainly on the stars of the central field. A reexamination of the centre position by using also cluster members of the halo reveals some new aspects. This has been done in two ways, once by taking the minimum of the sum of distances to the centre, given by SI, and once by taking the minimum of the sum of squared distances to the centre, given by S2. The latter is equivalent to the least squares mean position in rectangular coordinates. S1 and S2 will show different positions if the surface distribution is not circular symmetric, or if the selection of members is not complete in some outlying regions. Table 1 shows the minima as obtained for three groups, viz. the B and A, F and G stars in S1 and S2. For all groups the S2 minima, which are more influenced by the halo, are approximately 7' South of the S1 minima, while the F and G stars are found more to the West. The S1 centre for the B and A stars agrees well with the centre as found by Trumpler, as could be expected. Assuming that there may well be some incompleteness among Northern halo stars, as these are derived from the low quality Oxford Astrographic Catalogue positions, a centre position based on the S1 minimum for all stars seems to be the most realistic. It puts the cluster centre 7'50" North and 6 sec East of Merope. Figure 1 shows this position and the others of Table 1 on a chart of the brightest Pleiades. Figure 2 shows the distribution of position angles (measured from North through East to South) with respect to the adopted centre. There is an indication of flattening of the cluster perpendicular to the galactic plane (like found in numerical models), which will, however, be ignored in further calculations. Figure 1: A chart of the brightest Pleiades with the centre positions as determined for the B & A, the F and the G stars, (x): most weight on stars in the cluster centre; (+): equal weight for all stars. 99 Table 1 Centre positions with respect to Merope A & B F G All 51 +40s +9'15" -39s +5'39 -31s +6'00" +6s +7'50" 52 +39s -0'20" -75s -1'02" +7s +1'12" -7s 0'00" Figure 2: Distribution of the position ».A itirt angles with respect to the mean S1, (x), position of all stars, showing a possible flattening of the cluster perpendicular to the Galactic plane. 4.2.2 Surface and Space Density Distributions Due to the low number of stars in an open cluster, its surface density is noisy. In order to see how noisy it is, and to obtain smoothed surface densities, a function £(M) is introduced, giving the cumulative number of stars up to a projected radius r1: ?1 I(r1) = 2TC J0(r) r dr (3) This function is shown in Fig.3 for B and A, F and G stars as well as for suspected F and G binaries (see section 4.3). 100 I i i i Figure 3: The distribution function Z(r1) 20 - for B & A, F and G stars and F F.C-bin. and G binaries. The vertical £(r1 bars indicate the radii within 10 - which 10, 25, 50 and 75 percent of the stars are found . • " ' B.A respectively. 0 _ 30 - . 1 - .- ' .••••' F 0 „il 1 _ 30 ƒ | . - _ :' i ..••"•' o 0 30 - t _ ' i n .• , i 1 1 1 Statistical and numerical models of clusters predict the development of a halo, where the space density drops by B~1, with q between 3 and 3.5 (see e.g. Spitzer 1975; King 1966). From equ.1 it can easily be shown that in that region the surface density drops by r-q+1. The function Z(r1) for the halo can accordingly be given by: Z(r1) = a b r1-q+ (r1>rh), q not = a b In r1 (r1>rh), q r 3 (4) , where rh indicates the inner radius of the halo. By plotting £(r1) as a function of rri+3 for different values of q the extent of the halo can be determined. Table 2 shows a, b and q as obtained by means of a least squares solution for the halos of the distributions of B and A, F and G stars. They indicate a decreasing slope of the halo towards lighter stars. The values of 00 and po (the surface and space densities of the halo extrapolated to 1 pc distance from the centre) can be derived directly from equations 1, 3 and 4: 00 = b (3-q)/ 2TC (5) po =00/ 2 I (co&9)('"zd'& (6) 0 where cos&= r/R has been substituted in equ.1 to obtain equ.6. 101 By integrating 0(r) and p(R) over the projected and real halo distributions, and comparing the numbers of stars thus obtained with the total numbers of stars observed, one can easily find the total numbers of stars in the projected and real core respectively. These numbers, and the corresponding densities, can also be found in Table 2. The uncertainties have been derived from the above mentioned I 1 least squares solution for a, b and q. The data of Table 2 are shown in Fig.4 B.A for the surface densities and in Fig.5 logo-/* for the space densities. They show \ how massive stars are more («^1/prt concentrated to the cluster centre F IK than light stars. There is also a strong indication that for the lighter G 1W stars the density decreases towards 0 the centre of the cluster. A similar feature has been observed for the distribution of flare stars (mainly K type members of the Pleiades) by Mirzoyan and Mnatsakanian(1971), but their results were questioned by Kholopov(1971a,b). From the -1 distributions of F and G stars as obtained here it seems, however, that the central cavity as claimed by \\ Mirzoyan and Mnatsakanian could be I real. In section 4.7 it will be i described how such a cavity can be the I 1 result of an interaction of the -1 logr/r. o Jr„=ipc) 1 galactic tidal field with the orbits of the halo stars. Figure 4: The surface density distribution for the B & A, F and G stars as derived from the function £(r1). Table 2 Surface and space density distributions, (see equ.4, 5 and 6) A 4 B q -3.73+. 05 -3.55+.04 -3.35+.04 extend 1.0-8 .0 1.4-10. 1.5-10. PC halo 00 6.9+. 5 8.5+.6 8.0+.9 stars/pc i* po 4.2+. 3 4.8+.4 4.3+.6 stars/pe^ N(halo) 56+4 60+7 65+9 stars N(proj) 19 13 8 stars core N(space) 10+4 6+7 3+9 stars Pc 3.3+1 .3 0.5+0.6 0.2+0.6 stars/pc^ 102 Figure 5: The space density distributions for the B & A, F and G type stars as derived from£(r1). The dashed lines indicate the uncertainty levels (10) of the central space density determinations. logR/R (Ro=1pcl 4.2.3 The Noise Level of the Density Distributions A least squares solution of equ.4, using the best value of q to represent the halo slope, provides not only values for a and b, but also a standard deviation of Z(r1) over the halo. These standard deviations are found to be +0.5, i.e. £(r1) fits a straight line with a scatter of only +0.5 stars. In this section the noise level on £(r1) is further investigated. First an estimate is made of the expected noise level in the unrealistic case that the stars are only governed by the average space density distribution O(R). In other words, the average space density distribution for all stars as observed now is also the space density distribution averaged over a sufficiently long time interval for a single star. The same applies in that case for 0(r), the surface density. The chance of finding n stars in a ring between rO and r1 around the cluster centre is given by a binomial distribution (see e.g. Menzel 1960): f(n) = ( [j) pn(1-p)N-n (7) where N equals the total number of stars, and p is the fraction of N that one expects to find in the ring as based on the smoothed O(r): 103 r1 p r N-12TE J0(r) r dr (8) rO The average number of stars expected in this ring, If one now takes rO = 0, then p = £(r1)/N and n = X(r1). Thus the expected noise level onZ(r1) is given by: s(I(r1)> = (I(r1).(1 -I(rO/N)j/2 (11) This would imply an expected scatter of +3.5 over the halo, significantly more than the number found above. It is thus likely, that the assumption made above concerning the randomness of the distribution is wrong. In an open cluster, and in particularly in its halo, the stars are not moving at random, but instead in well defined and hardly disturbed orbits. The distribution function of an individual star, as defined above, is determined by this orbit and differs entirely from the mean distribution of all stars, as given by p(R). It will show a maximum at its apocentre in the halo, which implies that a star found in the halo is likely to spend most of its time near to the position where it is found. As the chance of finding a star at the place v»tiere it is observed increases, the standard deviation on its detection decreases, as is observed. The distribution function replacing equ.7 is in this case, however, a complex function of the (unknown) distributions of energy and angular momentum in the cluster halo. No new estimate has been made concerning the expected noise level on£(r1). Instead, the observed noise level has been used. The function p(R) can be used to calculate the potential energy distribution in the cluster. In order to make this possible, the occurrence and characteristics of binaries need to be known. To make the extension towards lighter stars possible, the luminosity and mass functions are to be investigated. 4.3 The Occurrence of Hard Binaries As was mentioned in the introduction, all binaries with separations less than 5" in the Pleiades act as hard binaries. These are also all stars that could not be measured separately in the photometric survey in the Walraven system described in section 3.1, which therefore provides a good data base to look for these stars. All photometric data will be expressed in logl values, 104 as is the custom in the Walraven system. Multiplying by -2.5 provides a relative scale in magnitudes. Numerical simulations of open clusters predict the formation of one or a few hard binaries due to occasional three body encounters in the cluster centre (see Van Albada 1968; Aarseth 1975). These systems tend to lose energy when interacting with single stars. The time scale on which the formation takes place is, however, roughly 12tc (tc=crossing time=5.10**7 years in the Pleiades) according to Aarseth (1975). This is about 6 times longer than the age of the Pleiades (10**8 years according to Golay and Mauron 1982). The occurrence of such binaries in the Pleiades is therefore a good indicator of the formation of these systems by processes connected with pre-main sequence evolution or star formation, rather than due to the dynamical evolution of the cluster. Bettis(1977) showed for the F stars in the Pleiades that their binary distribution fits with the one derived for field stars by Abt and Levy(1976). This distribution shows a preference among close binaries for mass ratios close to one. The data presented in section 3.1 for F and G stars are of much higher quality and essentially complete. We expect thus to give a better determination of the binary frequency in the Pleiades. The occurrence of binaries among cluster stars can be examined from their luminosity distribution, i.e. the integrated distribution of AV with respect to the relation V-(V-B) (the colour-magnitude diagram). In the Pleiades the F and G stars are in particular suitable for such an investigation. The 155 stars used here cover the range between (V-B) = 0.15 (F1) and 0.41 (G8). A third order polynomial is needed to fit the V values to the (V-B) values: V = a + b.(V-B) + c.(V-B)2 + d.(V-B)3 (12) This equation has been solved by least squares. First a solution was made for all stars, which made it possible to remove those stars that are, probably due to duplicity, too bright. As the limit for this selection is slightly arbitrary and in addition not much influenced by the exact solution, only one iteration was made. From a second solution the following coefficients were obtained: a = 0.532, b = -9.26, c - 9.32, d = -4.4 +0.001 +0.03 +0.15 +2.1 The full line in Fig.6a shows the distribution in AV with respect to this relation. 105 02 Figure 6: The luminosity distribution of F & G stars. a. full line: The observed distribution in AV with respect to the relation V-(V-B) dashed line: Predicted distribution in AV due to differential reddening and spread in distance moduli. b. The predicted distribution in AV due to differential reddening. c. The same due to spread in distance moduli. The distribution function in AV is mainly composed of three contributions: (1) the distribution of distance moduli, (2) the distribution in differential reddening, (3) the occurrence of unresolved binaries (the influence of measuring errors is with respect to these fully negligable). The first and second of these can be reconstructed, as we know the space density distribution of F and G stars and the reddening distribution for B and A stars (see section 3.2). The distribution of unresolved binaries can thus be deduced. 106 Applying the reddening distribution as derived for B and A stars to the F and G stars is, however, not without ambiguity: the space density distributions for both groups are significantly different. Unfortunately, however, reddening corrections for especially late F and G stars are hard to determine. The reddening distribution for the B and A stars is therefore the best available information. This distribution is given in E(V-B) and has to be converted into a distribution in AV. Due to the nonlinear terms in equ.12 this conversion depends on (V-B) and is given by: AV = [(-3.1-b) - 2c.(V-B) - 3d.(V-B)2 ].E(V-B) (13) for a single star. The distribution in AV has been derived using this relation by deviding the stars used in solving equ.12 into 6 groups, each containing approximately 22 stars and covering a range of 0.045 in logl in (V-B). By applying the average (V-B) of each group to equ.13 and adding the thus obtained distributions in AV, weighted by the number of stars in a group, an overall distribution in AV has been obtained, which is shown in Fig.6b. The distribution of distance moduli has been derived by integrating the space density distributions of F and G stars according to equ.2. It is shown in Fig.6c in intervals of 0.01 in logl in AV. A convolution of this distribution function with tha reddening distribution function in AV is shown as a dashed line in Fig.6a. It has been shifted along AV by 0.033 such that the cross correlation between the observed and predicted distributions is largest. From a comparison between the observed and the predicted luminosity distributions, a total of 20 stars, with AV >0.15, are found to be very likely binary stars. The cumulative surface density of these stars is shown in Fig.3. A comparison between their distribution and those of the B and A and the F and G stars clearly indicates that the 20 suspected binaries have masses similar to single B and A stars. The cumulative surface densities (£(rO) f°r the B & A and F & G binary stars relate as 1 : 1.1, while those for the F 4 G binary and F single stars relate as 1 : 1.1. Thus, not only are these stars brighter than single F and G stars, but they appear also to be more massive, making it very likely for them to be binaries. An indication of the mass ratios involved can be obtained from the observed values of AV for the suspected binaries. Figure 7 shows AV as function of V1-V2 (1: primary, 2: secondary) with respect to V1 and with respect to the relation V - (V-B). Also shown are mass ratios derived from V1-V2, based on the relation log(luminosity) = c + 4.5 log(mass). Counting AV from the peak in the observed distribution (indicated by an arrow in Fig.6a), one derives that the 20 stars most likely to be binaries have AV > 0.17, V1-V2 < 0.76 and M2/M1 >0.66. The total masses of these stars thus lie in the range 1.5 to 3 solar masses and are similar to those of single B and A stars. There is some indication of about 10 binaries with 0.07 < AV < 0.17, 1.16 < V1-V2 < 0.76 and 0.43 < M2/M1 < 0.66 (based on the possible difference between the full line and the dashed line in Fig.6, betweenAV=0.07 and 0.17). This number is smaller than predicted by the relation obtained by Abt and Levy(1976), but considering the uncertainty in'these numbers no conclusion can be drawn from this difference. 107 In any case there is a large number of initially hard binaries among the F and G stars of this young cluster. These systems are most likely not a consequence of the cluster dynamics but must be the result of a star formation process. The fact that most of these stars have comparable masses and are too closely bound to be seen separately (separations less than 0!!5, 65AU) puts restrictions on their formation process. Their binding energies are more than ten times higher than the mean kinetic energy in the cluster centre. In addition, three of these stars are suspected variables, which, if the variability were the result of eclipses, would decrease the average separations and tighten the restrictions on their formation. Table 3. finally, gives the 20 suspected binaries, their V and AV values, as well as indications of variability. Table 3 Suspected binary F and G stars. Hz: Hertzsprung (1947) P : Pels, see section 3.1 Nr V AV var. log(I) M2/M1 1 94 90 .86 »2 .79 .75 72 i) .67 .64 i i i i i i i i i P 48 -0.61 0.18 P 32 -0.69 0.27 0.3 Hz1338 -0.7*» 0.25 AV Hz1912 -0.88 0.18 ft 0 P 3 -0.93 0.17 V 4 Hz1726 -0.97 0.22 log») P 174 -1.12 0.16 0.1 N^ - Hz 739 -1.07 0.33 Hz1117 -1.35 0.28 " » P 53 -1 57 0.19 0 i i i i i i i i i Hz 186 -1.48 0.29 0 V1-V2 °5 log(I) 10 Hz 975 -1.51 0.28 Hz2147 -1.62 0.24 V Hz2027 -1.62 0.25 V Figure 7: Hz 320 -1.69 0.20 AV with respect to V1 (o) and with Hz 173 -1.61 0.29 respect to the relation V-(V-B) Hz 298+ as a function of V1-V2 and H2/M1. 299 -1.63 0.27 Hz2278 -1.62 0.30 V Hz 303 -1.50 0.44 P 39 -1.73 0.20 108 4.4 The Luminosity and Mass Functions The luminosity and mass functions of young open clusters are particularly interesting as they represent still rather undisturbed pictures of the mass spectrum of star formation. The more so because, due to the influence of the galactic tidal field, described in section 4.7, the interaction of halo stars with the cluster core has decreased, and therefore probably also the rate of evaporation. The observed mass function is thus probably very similar to the original one. Due to mass segregation, shown in section 4.2, the luminosity function is a function of the distance to the cluster centre (see also Trumpler 1921). It will show a much steeper slope if the Whole cluster is taken into account than for the projected core only. Van Den Bergh(1957) and Taff(19710 determined average luminosity functions for open clusters. They found a flattening of the number of stars per magnitude interval beyond abs.phot.magn. 4, much earlier than the peak in the luminosity function of the solar neighbourhood (near abs.phot.magn. 16) as observed by Luyten(1968) and Wielen(1974). Recently two papers were published on the luminosity function of the Pleiades: one by Van Leeuwen(1980) arguing that there is no difference from the solar neighbourhood, and one by Jones(1981), stating that there is a difference such as found by Van Den Bergh. The difference in opinion is mainly caused by a different interpretation of flare star observations, which comprise the faintest observed Pleiades members. In the present section all available data are reviewed, and additionally also the influence of binary stars is considered. 4.4.1 The Observable Region By counting the cluster members in magnitude intervals one obtains an observed luminosity function. Such counts can be made for the whole cluster up to vis.mag 11.6 and for the central field up to vis.mag 13.5. Figure 8 shows the counts within a projected distance r1 as a function of apperent magnitude. The luminosity function of the projected cluster centre flattens towards fainter stars like those derived by Van Den Bergh(1957) and Taff(1974), which are also mainly based on cluster centres. The luminosity function of the whole cluster, however, increases over the whole range observed here, similar to that of the solar neighbourhood as observed by Luyten(1968) and Wielen(197lO. There is thus no reason to assume systematic deviations over the range M =0 to My=6.5. The luminosity function can be transformed to a mass function using a mass-luminosity relation (Allen 1976) and a cluster distance of 130 pc. A comparison with other mass functions derived in a similar way is shown in Fig.9. The hard binaries, which are mostly seen resolved for the stars in the solar neighbourhood, can shift the Pleiades luminosity function to a higher number density per mass interval, but will have only little influence on the slope if the binary fraction and the distribution over mass ratios is the same for stars of different mass. 109 • 1 i - r ' I" 1 1 T 1 |—1 j ::1 log dn/ölm«s] : Re 10 pc 3 - 1 tlvt «tars- il ! 1 H : i Re 3 pc 1 R 1175 pc —'• 0 1 04 loglmas») 00 -0.4 0,6 Figure 8: Figure 9: I(r1) as a function of The mass function of the Pleiades (full line) magnitude, showing the compared with that of the solar neighbourhood increasing slope of the as derived by Luyten(1968) (dashed line) and luminosity function with and by Wielen(1974) (dotted line). The increasing distance from luminosity function of the expected total the centre (read r1 for number of flare stars in the Pleiades field R). is also shown. Over the range 0.8 to 2.0 solar masses the mass function shows a slope Q defined by: d n/d(mass) = C*mass«*Q (14) Q = -2.8 +0.1 (15) The brighter Pleiades fit this value of Q if their masses are between 3 and U.6 solar masses, which is in accordance with estimates obtained from photometric data (Van Leeuwen and De Zeeuw, in prep.). Using the more accurately determined Q of -2.74 as given by Taff(1974), based on many open clusters, a zeropoint can be obtained such as to fit the total number of stars found in the mass interval used. This gives for the Pleiades a value of C = 2.39 if all stars are assumed single and 2.48 if 20 percent are assumed double. In the first case the total number of stars and their total mass, 110 covering the range between mass = 4.6 and 0.76 solar masses, equal 221 and 284 solar masses respectively, while in the second case these numbers are 265 and 350 solar masses. By integrating equ.14 for stars with masses higher than those found now-adays in the cluster, one derives that possibly one to ten stars more massive than 4.6 solar masses once were members of the Pleiades. The exact number, however, depends very much on the slope used in equ.15 and on whether the slope is maintained to higher masses. The white dwarf, claimed by Luyten and Herbig(1960) to be a probable cluster member, may be the remnant of one of these stars. The inaccuracy of its proper motion makes it, however, a very uncertain case. This star is in addition not a binary according to Greenstein(1974), which may make it difficult to lose the mass fraction needed to form a white dwarf from a 5 or more solar masses star. 4.4.2 The Fainter Extension. The survey by Pels, presented here in section 3.1. is almost complete for stars brighter than vis.mag 11.6. Little information is available about densities of fainter members in the outlying region. They can however partly be estimated from densities projected on the central region. Hertzsprune(19^7) gave possible members in the central region that are brighter than phot.mag 15.5 (vis.mag 14.0 for cluster members). By assuming that the surface density distribution for those stars equals that of the G stars, one probably obtains a lower limit for their total number. It is likely that the distribution of these fainter stars is even less concentrated to the centre than that of the G stars. The number of photometrically confirmed members of the selection by Hertzsprung for stars fainter than vis.mag 11.6 and brighter than vis.mag 13.2 equals 52 (see section 3.1). According to Allen (1976), these stars have probable masses between 0.77 and 0.60 solar masses. The estimated number of these stars outside the central region is 39, which brings the total to 91 stars (and probably a few more). The number of stars per mass interval thus continues to increase. A local decrease in the luminosity function of the solar neighbourhood as observed by Wielen(1974) in this mass interval is not observed for the Pleiades. Only a limited amount of photometric data is available for the fainter possible members selected by Hertzsprung. His catalogue gives 47 probable members between vis.mag 13.2 and 14.0, a mass range from 0.6 to 0.54 solar masses, which brings the minimum of the total expected number of these stars at 82. This again shows a strong increase in the Pleiades mass function. Thus, over the range 4.6 down to 0.54 solar masses no discrepancy is found between the mass functions of the Pleiades and the solar neighbourhood. The total mass concentrated in these stars is approximatly 400 solar masses, and 500 solar masses if duplicity is taken into account. An extrapolation along Luyten's luminosity function would add at least another 450 solar masses from approximately 2000 stars with masses less than 0.54 solar masses, bringing the total cluster mass to 850 to 1000 solar masses. An extrapolation along the power law luminosity function of equ.15 in down to e.g. 0.08 solar masses stars provides 10500 to 14000 stars and a total mass of 2000 to 2500 solar masses. The only data available for stars lighter than 0.51* solar masses comes from flare star surveys. Many, if not all Pleiades members fainter than vis.mag 11.5 show flare activity. The search for flare stars in the direction of the Pleiades has revealed already 527 of these stars (Mirzoyan 1981, Helikian £t al 198D. Although not all of these stars are members (only about 50 percent are members according to Jones 1981) they still provide the main source of information with respect to the fainter extension of the luminosity function. Another source, by which members may be selected, is the occurrence of emission lines in Call H and K and H(X(see McCarthy 1974 and section 3.3). The detection of flares by means of photographic plates is limited to phot.mag 18, due to which for fainter stars only the largest flares are observed. Beyond this limit the data will be incomplete, i.e. for stars with masses less than 0.35 solar masses approximately. Mirzoyan et al(1977) presented estimated numbers of potential flare stars based on a simple statistical assumption, using the number of stars for which one and the number for which two flares have been observed to estimate the number of stars for which no flares have been observed yet. If the total number of potential flare stars is given by N and those for which one or two flares have been observed by n(1) and n(2) respectively, then if all stars show the same flare frequency the following relation will hold: N = (n(1) /2 n(2)) +In(k) (16) k=1.» Ambartsumian et ail^ (1970) showed that in case of different flare frequencies N will be larger. Thus, the values of Table 4, which were first presented by Mirzoyan et al(1977), are only lower limits. Table 4 Observed and expected numbers of flare stars in phot.magn.intervals. m(pg) 13 14 15 16 17 18 19 20 21 n(k) 5 33 43 79 137 71 54 32 15 n(0) 8 16 23 105 213 80 38 40 30 N 13 49 66 184 350 151 92 72 45 The numbers of stars per mass interval thus obtained show for the brightest flare stars, of which the density is lower than that obtained above, that probably not all the Pleiades members in that mass range are flare stars, or that some have rather low frequencies. The lower mass stars show a density which is rather too high than too low relative to the power law extrapolation of equ.15, indicating that probably only half of these stars are cluster 112 members, as was found also by Jones(1981) on the basis of astrometric measurements. In his estimation of the luminosity function, however, Jones did not take into account the still undetected flare stars, because of which he found a flattening in the luminosity function in the mass range of the flare stars. Still this flattening is considerably later than found by Taff(1974). It has to be realized that the mass-luminosity relation probably does not hold any longer for the flare stars, as they are still in the pre-main sequence phase. Any estimate of the lower extension of the mass function of the Pleiades is therefore liable to error. We can conclude that as far as it is possible to observe no systematic differences are found between the mass functions of the Pleiades and the solar neighbourhood, i.e. over the range 4.6 to 0.35 solar masses. The further extension of the Pleiades luminosity function may be estimated from the total cluster mass, obtained by comparing possible extrapolations of the potential energy distribution with the velocity dispersion in the cluster centre. 4.5 The Internal Motions The dispersions in the internal motions will be assumed spherically symmetric, i.e. they depend only on the distance to the cluster centre. In that case, only two independent velocity dispersion components exist, one along the direction to the centre and the other perpendicular to this direction. Information on both of these can be obtained from high accuracy proper motions, such as provided by Vasilevskis et^ jal(1979). As these are relative proper motions, they still contain influences of the reduction, which will be described in section 4.5.1. They are also influenced by the relative motion of the Sun with respect to the cluster, which is described in section 4.5.2. Finally, after considering the external influences, the information on internal motions in the cluster is derived from the proper motions in section 4.5.3. 4.5.1 The Influence of Plate Reductions on Proper Motions Proper motions are obtained from a comparison of stellar positions on old and new epoch exposures. These positions are influenced by a variety of effects, such as differential atmospheric refraction, the positioning of the photographic plate during the exposure, guiding accuracies and many other small effects (see section 2.3). Most of these can be accounted for by performing a linear transformation to the positions as recorded on a plate, that bring them into accord with an independent reference frame. Such transformations will look like: ) (17) y(0,i) = D(j) + E(j).y(j,i) + F(j).:c(j,i) (18) 113 where (x(O,i),y(O,i)) are coordinates in the reference frame. The index i indicates stars and the index j exposures. Equations 17 and 18 are solved for a selection of stars by least squares and applied to all recorded stars. Those that are selected for the least squares solution will by definition satisfy the following relations: Z(x(0,i)-xt(j,i)> = 0 I(y(O,i)-y'(j,i)) = 0 i i I(x(0,i)-xl(j.i)).x(0,i) = 0 Z(y(O,i)-yf(J,i)).y(O.i) = 0 i i I(x(O,i)-x'(j,i)).y(O,i) = 0 Z(y(0,i)-y'(j.i)).x(O,i) = 0 i i (19) where (x'(j,i),y'(j,i)) are the transformed positions using the least squares solution for coefficients a to f of equ.17 and 18. The transformed positions, obtained at different epoch exposures, are combined to derive proper motions. This involves a least squares solution over the following equations of condition: = A,x(0,i) + |l(x,i) .(T( j)-T(O)) (20) = Ay(0,i) + |l(y,i).(T(j)-T(O)) (21) where (Ax(0,i),Ay(O,i)) are corrections to the reference frame positions, T(j) is the epoch of exposure j and T(0) an arbitrary zero epoch. The latter can be chosen such that Z(T(j)-T(O)) = 0. This does not influence the solution, but only simplifies the expressions. The proper motions are now determined as: ,j,jj (22) j j , £(y(0,i)-y'(j,i)).(T(j)-T(0))/I(T(j)-T(0)r (23) j j From the relations given as equ.19 it follows that the differences (x(0,i)-x'(j,i)) and (y(O,i)-y'(j,i)) are dependent on the selection of stars used to solve equ.17 and 18. It is thus important to choose the same stars as reference frame for each exposure, as was done by Vasilevskis et al(1979). Also the weights for different reference stars have to be the same for different exposures. Then these stars will fulfil the following relations, which are derived from equ.19, 22 and 23: I^(x.i) = 0 I|J.(y,i) = 0 i i I|JUx,i).x(0,i) = 0 IU(y,i).y(0,i) = 0 (24) i i i i I|I(x,i).y(0,i) = 0 I|i(y,i).x(0,i) = 0 114 and thus define the proper motion system. In this system the reference stars as a group show no translation, rotation or expansion from their proper motions. Any other group of stars, however, will show these effects. This also applies for samples used to determine the various external influences. Each sample however can be transformed in their proper motions so as to fulfil relations 24, using the following equations: L) =|l(x,O) + e1.x(0,i) +^pi.y(O,i) (25) D = |X(y.O) + e2.y(0,i) +vp2.x(0,i) (26) where (ejj)) represent the linear expansion and rotation of the reference frame that are to be removed. In this way, the proper motions of the selected stars are brought to a standstill in zero and first orders in x and y. The above described procedure has been applied to 43 cluster member proper motions, all confined to a circle of 0.5 degr. radius around the cluster centre. Within such a circle the external influences as well as a point symmetric internal proper _motion dispersion distribution do not contribute to the parameters e and vj>. Thus, a well defined proper motion reference system is obtained. Also the proper motions of 41 stars outside the circle are transformed to this system and can in some cases be used as additional information. After this treatment the proper motions are ready for investigation of the external influences. 4.5.2 Oin the Influences £f the Motion c>f the Sun Relative to the Cluster 4.5.2.1 The Relative Radial Velocity The radial velocity of the Pleiades relative to the Sun can be derived from radial velocity measurements of its individual stars by Pearce and HÜH1975): V(rad) = 7.5 +0.3 (m.e.) km/sec (27) This radial velocity will cause a perspective linear contraction of the cluster as projected on the sky by a factor: e = -V(rad)/D = -6.1«10*«-8/annum (28) where D is the distance to the cluster centre, for which 130 pc is used. When both e and V(rad) can be determined individually, an estimate of the distance D can be obtained. This is the well known convergent point method used to derive the distance to the Hyades. In what follows we describe the influence of e on the proper motions and investigate whether it is possible to apply the convergent point method to the Pleiades. 115 The previous section described how the proper motions of the cluster stars were made free from linear expansion. By transforming also the proper motions of the field stars to the same system, the contraction of the cluster is reflected as expansion in the field star proper motions. An examination of these proper motions may reveal this expansion. This can be done by solving equ.25 and 26 for the field star proper motions. As we are looking only for linear expansion, they reduce to: |i(x,i) =|!(x,0) + e.x(O.i) (29) ) + e.y(0,i) (30) The accuracy to which e can be determined, solving equ.29 and 30 by least squares, is given by: Öe =6n..(I(x(0,i)2+ y(0,i)2))~1/Z (3D where 6U. is the dispersion in the field star proper motions. From an examination of data presented by Binnendijk(1946), it is found that 6jl equals 07007 per annum for field stars with an apperent phot.magn. of 14+2. By introducing a mean projected density of field stars, Qf, the sum of equ.31 can be approximated with an integral: r I(x(0,i)+y(0,i)) = 2TC|Of.r2.r.dr = 0.5 Of.r^ (32) 0 from which we derive: 1/7 o öe =6|I.(2/TtOf) .r" per annum (33) The Pleiades cluster covers an area of 4.5 degrees radius. If Of is expressed in stars per square degree, then we find: 1/2 6e = 7.66*10*«-8. Of" per annum (34) In order to detect e as given in equ.28 6e should be at least a factor 3 smaller. In order to reach a level of 4 percent to which the distance of the cluster and the radial velocity are known at present, a field star density of 1000 stars per square degree (a total of approximatly 65000 stars) need to be investigated for their proper motions. This density is reached at phot.mag 16. In principle it is thus possible to determine the Pleiades parallax this way. 116 The proper motions in the central region will not be influenced by the perspective contraction, although it is present at a level of 10**-4 arcsec per annum at a distance of 1500 arcsec from the cluster centre. The procedure described in the previous section removed these effects. This is not possible with the other, tangential component of the relative motions of cluster and Sun, which causes a relative secular parallax. 4.5.2.2 The Relative Secular Parallax The relative secular parallax is a perspective effect caused by the tangential motion of the cluster with respect to the Sun. Due to this stars; on the far side of the cluster have a smaller proper motion than those on the near side. The effect is equivalent to a linear rotation of the cluster around an axis perpendicular to the line of sight and the proper motion direction of the cluster. We define, see Fig. 10, (R,«5) as the coordinates of a star in the cluster and |l(cl) as the average proper motion of all cluster stars. Each individual star will show an additional proper motion {i(p) due to projection, of the size: ji(p) = (35) where D is the cluster distance. These additional proper motions will cause a dispersion in proper motions in the direction of |i(cl), which can be determined in two different ways: from the proper motions or from the space density distribution. A comparison between these two determinations may provide a check on the assumption of spherical symmetry. In what follows both determinations are described and their results compared. Figure 10: Projection of positions and velocities in an open cluster. If we asign a quantity A depending on the position of a star in the cluster, then the average of A as projected on the sky, is given by: ]ƒƒApe R)R2cosöaRdöa vp (36) 117 1 Here we take A as the square of the additional proper motion |i(p). From the data presented by Vasilevskis et al we derive |Kel)=0?047 per annum. Equation 36 thus provides the dispersion in JJL(p) as projected on the sky. This dispersion has been calculated as a function of projected distance to the cluster centre, which is shown in Fig. 11. It increases from 0'.'6 per millennium in the centre to 1U4 per millennium at a projected distance of 6pc and then rapidly decreases to zero. Observations of this curve from proper motions can provide information on the density distribution in the cluster along the line of sight. Figure 11: The dispersion in proper motions due to the relative secular parallax as a function of the distance to the cluster centre. 0 r(pc) 2 At present only the central value of Fig.11 can be measured. Only the proper motions in this region, where the dispersion in jJLCp) should be approximately 0'.'6 arcsec per millennium, are accurately enough known. In order to see whether an additional proper motion dispersion is detectable, we first check on the direction of the largest dispersion in the proper motions. This direction can be determined by solving the following equation: tgap(O) = (37) which can be derived by simple mathematics. The position angle vp(O) thus found can be compared with the direction of the cluster proper motions as derived from the cluster stars. The following values are found: tp(O) = 123.3 +12 (cluster stars within 0.5 degrees) J() = 122.7 +1.6 (field stars) The agreement between both values shows the existence of this effect. The dispersion along the cluster proper motion (the \)-direction, see Binnendijk 1946) and the perpendicular direction X are found to be: <|i2>1/2= 1Ï02 +0»11 per millennium <|Jft)2>1/2= °"88 +0Ï09 per millennium 118 The parallactic d5.spersion is related to these values by: The errors are estimated on the assumption of a random distribution, and are thus too high for this non random problem (see also section 4.2.3). The dispersion in |l(p) as found from proper motions agrees well with the expected value of 0V6 per millennium. 4.5.2.3 On the Detection of Real Rotation and Expansion Linear rotation with a rate equal to corotation in the galaxy is of the order of 10**-9 rad per annum and will thus be hard to detect. The detection limit for differential rotation is set by the cluster stars and is, due to their much lower surface density, not better than 10**-7 rad per annum. Such effects are also unlikely to be discovered. Linear expansion of the cluster is impossible without a source of energy in the cluster centre, and linear contraction is also very unlikely. Differential expansion or contraction is only possible to detect by using cluster members and has therefore also a detection limit of 10°-7 per annum. Numerical models show differential expansion at a rate of 10**-8 to 10*#-9 per annum and one has to conclude that also this detection is not yet feasible. 4.5.3 The Internal Velocity Dispersion The velocities in the cluster will be described in polar coordinates, i.e. as one radial and two tangential components. One of the tangential components is situated in the plane through the line of sight, while the other is perpendicular to this plane. The first is seen in projection together with the radial component as the projected centre directed velocity component while the latter is projected directly as the tangential component. This projection, shown in Fig.10, can be described by the following equations: V(t) = V(up) (39) V(r) = V(R).sin(^-) + V(^).cos(^) (40) V(rad) = V(R).cos(£) - V(£) .cosOfr) (41) The dispersion in V(r) and V(t) can be obtained from the proper motions. Using equ.36 we find the following relations: 119 (42) <)JXr)2>1/2= ( No crossterm occurs, as in the spherically symmetrie approximation V(R) and VCd1) are uncorrelated. This approximation also implies that the dispersions in V(«&) and V(O» are indistinguishable. Consequently, the proper motions can in principle provide all the information on the dispersions in the internal motions. Due to projection effects, however, the information on the dispersion in the radial component near to the centre is hard to obtain. This information can be obtained in combination with radial velocity measurements: The accuracy to which the radial velocities are needed is 200 m/sec, which is possible to reach for the F and G type stars. Stars with earlier or later spectral type are rotating too rapidly to make such an accurate determination possible. The dispersions in |l(t) and |J,(r) have been calculated for three concentric rings and are presented in Table 5, which has been presented before by Van Leeuwen(1980). The data are corrected for the internal accuracies and the influence of the relative secular parallax. They clearly indicate a decrease in tangential velocity dispersions at increasing distance from the cluster centre. This confirms the observation by Jones(1970a) that the stellar orbits in the cluster are very excentric. By means of a comparison with a simple model of a velocity field in the cluster (such as presented by Spitzer 1975) Van Leeuwen(1980) derived a central velocity dispersion of 700 m/sec, and a tangential velocity dispersion decreasing as 1/R starting at 0.5 pc from the centre. Table 5 The internal proper motion dispersions. ring n <|Jfr)22>1/2 <|JjCt<|JjCt)) 2>1/2 arcsecconds per millennium 0.0 - 0.5 po 11 0.98+0.26 1.03+0.26 0.21 0.5 - 1.0 pc 22 0.98+0.20 0.86+0.20 0.19 1.0 - 1.5 pc 34 1.04+0.15 0.69+0.19 0.23 120 The Potential Energy Distribution and the Total Mass of the Pleiades An exhaustive discussion of the dynamical aspects of the Pleiades based on the new data presented in the preceding sections could not be pursued in the available time. We nevertheless wish to add in the present and following sections a few remarks which would seem to be relevant. The space density distribution for the B to G stars as derived in section 4.2 and the mass function for the B to K stars derived in section 4.4 provide the possibility to calculate the potential energy distribution (|XR)t provided one can extrapolate both to lower mass stars. Different extrapolations provide different potential energy distributions, which can be compared with the central velocity dispersion of 700 m/sec to provide an estimate of the total mass of the cluster. It will be shown that such an estimate is probably a lower limit. Given a space density distribution for stars with mass M asp(R.M). Then the potential energy per unit mass at a distance R from the cluster centre is given by: R RjM2 2 4TXGJ[J ƒ D(R. ,M).R. . dMdR: (45) 0 0 H1 where M1 and M2 are the lowest and highest masses of individual stars in the cluster respectively. The mass function can be obtained by integrating p(R,M) over the whole cluster. For stars of spectral type B to G (4.6 to 0.76 solar masses) the 1 1 1 i i i i i i function p(R,M) is given in Table 2. 1000 Mo u These stars comprise 350 solar masses 700M _ althogether for which 2 - o the potential energy distribution is • • ' ' B.A.F-G presented in Fig.12. • . . • ' ' 350 HG e a 1 > Figure 12: A The distribution of / potential energy in the cluster for different n ^— ' 1 i i | i i i I t extrapolations, o R 12t The function O(R,M) for stars with masses below 0.76 solar masses has been assumed equal to that of the G stars. It is, however, likely that the space density distribution of these stars is even less concentrated to the centre. In that case, the total number of these stars needed to obtain the same potential energy difference between the cluster centre and the outer halo would be larger than what is obtained from the present assumption (the total mass within a radius R is smaller in that case, and therefore also the force at that radius and the integral over the force) Figure 12 shows the extrapolations that provide a total mass of 700 solar masses (as proposed by Jones 1971) and 1000 solar masses (as estimated from the luminosity function extrapolation following Luyten 1968). In order to compare the function f(V(R),V^),V(vp))dV = C1 exp(-K(V(&)2+V(vp)2 )-LV(R)2 ) dV (46) C1 = (K2.L/TI?)1/2 dV = dV(R)dV(£)dV By combining V^) and V(^p) into one tangential component V(T)2 = VCJ)-)2* V(^p>2. we obtain: f(V(R),V(T))dV= C2 V(T) exp(-KV(T)2-LV(R?) dV (47) C2 = 2.K. (L/fr)1'2 dV = dV(T)dV(R) 1/2 This function has a maximum at V(T) = 1/(K) . Equation 47 can be expressed in terms of angular momentum per unit mass, J = V(T).R. It can be separated into two distributions: f(V(R))dV(R) = (L/n;)1/2exp(-L.V(R2)dV(R) (48a) f(J)dJ = 2.J.(K/R2).exp(-J2.K/R2)dJ (48b) The angular momentum simulates an additional potential energy barrier for the centre directed velocities, which can be seen from the energy equation expressed in J and V(R) : V(R)2/2 + J /2R + (J)(R) = E (49) 122 where E and J are constants. The effective potential energy in the radial direction (J2/2R -K|)(R)) will show a minimum at a distance R(J) from the centre. Figure 13 shows the potential energy for the 1000 solar masses model and an angular momentum of 0.44 pc.km/sec, a number which will be used later on in a comparison with the space distribution of the B and A stars. Ripe) 2 Figure 13: The effective potential energy distribution (dotted line) for the radial velocity components in a model of 1000 solar masses and a transverse velocity dispersion of 0.44 km/sec at 1 pc. This model provides a minimum in the effective potential energy at the same distance from the centre as the maximum observed in the radial density distribution of the B & A stars, which is shown as a dashed line at the bottom of the graph. The vertical line starting from the minimum of the effective potential energy distribution gives the scale of the potential energy distribution expressed in units of the squared observed velocity dispersion in the cluster centre: 0.49 (km/sec)2. A potential energy distribution as shown in Fig.13 will give rise to a distribution function over R showing a maximum at R(m), where the effective potential energy is lowest. If one assumes, in first approximation, that the maximum in the density distribution for the most likely J value corresponds to the maximum in density distribution for the entire distribution of J values, then this maximum as observed provides an estimate for the angular momentum distribution. In Fig.13 is shown the distribution of B and A stars as a function of R, showing a maximum at R(m)= 1pc. This distance coincides with the minimum of the effective potential energy in the direction of the centre in case K"^= 0.44 km/sec at 1pc. The corresponding values for the F and G stars are 0.85 and 0.94 km/sec at 1pc respectively. 123 If one assumes that the transverse velocity dispersion (K~^2) is proportional to 1/R, then it equals the observed central velocity dispersion of 700 m/sec at a distance of 0.63 pc from the centre. This distance can be interpreted as a mean distance from the cluster centre for the stars that are subject to equipartion of energy, as at that radius the transverse and radial components of the velocity dispersion are no longer distinguishable. For a homogeneously filled sphere of radius R-max this mean radius equals 0.75R-max. Using this approximation, R-max equals 0.63/0.75 pc = 0.85 pc, which is in good agreement with the inner radius of 1pc observed for the halo distribution of the B and A stars. The radius of 0.63 pc fits also well with the radius of 0.5 pc used in the reconstruction of the internal proper motion dispersion as the radius from where the transverse velocity dispersion starts to decrease. The estimate obtained this way for the angular momentum distribution is thus consistent with both the observed proper motion dispersion and the space density distribution. The transverse velocity dispersions found for the F and G stars can be compared with that of the B and A stars. On the assumption of equipartition of energy they should be related by the square root of the mass ratios between those stars. In that case values of 0.51 and 0.66 km/sec can be expected for the F and G stars respectively, which is considerably smaller than the observed values. They both show an additional dispersion of 0.44 km/sec which, as will be shown in the next section, can be attributed to the influence of the galactic tidal field. The model which is based on a total mass of 1000 solar masses provides a potential energy difference between the cluster centre and a shell at 10 pc from the centre equal to 4 times the squared velocity dispersion in the cluster centre. Actually, as here the centre directed component of the velocity is the only one involved, this difference has been measured from the minimum in the effective potential energy, which is indicated in Fig.13 by the scale starting at the minimum of the potential energy distribution. The ratio 4 has been obtained by Chandrasekhar(1912) using the Virial theorem and should be considered as a crude approximation. In case one assumes a uniform Gaussian velocity distribution over the whole cluster, it binds 95 percent of the B and A stars within a radius of 8pc from the centre. This is, however, most likely not the case as the stars move around in orbits through the cluster. This causes stars with high energies to be underabundant in the cluster centre and over abundant in the halo. A simple calculation for orbits in the 1000 solar mass model shows that stars with an energy equal to 1/L in the cluster centre have a density in the cluster centre 50 percent of what could be expected if the energy distribution was everywhere the same. At 4/L this percentage has decreased to 30. This situation can be simulated by multiplying the velocity distribution in the cluster centre by a factor: C = exp(-0.7V(R).l!/2) (50) In that case, however, the observed velocity dispersion in the cluster centre is only 0.7 times the velocity dispersion that would be observed if all stars would be present in the cluster centre. The velocity dispersion for all stars is in that case estimated at (0.7 km/sec)/0.7 = 1.0 km/sec. In order to keep in this situation a potential energy difference of 4/L between the centre and the outer region a total mass of at least 1600 solar masses is needed. This 124 number could still be higher if the low mass stars are less concentrated to the cluster centre than the G stars, which is very likely. In order to achieve a total mass of more than 1600 solar masses from stars only, the luminosity function needs to be extrapolated along the exponential curve further than what is observed for the luminosity function of the solar neighbourhood. Concluding this section we can say that the mass of the Pleiades cluster is at least 1000 solar masses, but is likely to be more than 1600 solar masses, in particular if one assumes the cluster to be in quasi-static equilibrium, i.e. there is only a minor anisotropy in the centre directed velocity distribution. A mass of 700 solar masses is far to low to keep the massive stars together, in particular in case of an isotropic velocity distribution. The total mass of the cluster thus justifies an increasing luminosity function at least up to the maximum observed by Luyten and possibly even further. The Galactic Tidal Force and Mass Segregation It has been 3hown numerically by Terlevich(1980) that the increased mass segregation, leading to a depletion of the cluster centre for low mass stars, is the result of the interaction with the galactic tidal field. Here it will be shown analytically how this field increases the angular momentum dispersion of stars in the halo and thus increases the average minimum distances to the cluster centre of individual orbits. As, due to equipartition of energy, the low mass stars more easily get to the halo, these stars will be most influenced by the mechanism. It will more and more separate the halo population of light stars from the core population of massive stars, and therefore slow down the dynamical evolution of the cluster. The change in angular momentum due to an external force 'Fe' for a particle moving in a central force field like the cluster halo, is given by : J = RAFe (51) Here, the external force is the galactic field and R is the distance of an individual star to the cluster centre. The external force can be separated into two components, one parallel to the line cluster centre-galactic centre and one perpendicular to this line, and can be written as follows: Fe = F0.(R/a)^coS'S'Calong line connecting centres) Fe = F0.(R/a).sin^ (perpendicular to line connecting centres) F0 = (GM/a^) (force per unit mass acting on star in cluster centre) R and •&• are defined as in Fig.10, with the line of sight pointing towards the 125 galactic centre. M is the mass of the Galaxy within a radius 'a', the distance between the cluster and the galactic centre. Accordingly, the change in angular momentum can in first approximation be given by: j = (GM/a2).(R2/a).<3/2)sin2# (52) Integrating over all values of -9" shows that no change in average angular momentum is caused by the galactic tidal field. In case the orbits of the stars are highly elliptical, ft becomes independent of t. In that case j2 will change by the following amount: 7 ' ' V' dJVdt = 2 J.J = 2 J.JJdt (53) 0 The total change over a time interval t2 is given by: Z &J = 2JJ[JJJdt]dt< 1 (54) 0 0 By taking the average over all values of ft ( f (sin2«9T.sin If we take (GM/a2) = (250km/sec)2/10kpc, then ,1/2 In order to obtain the observed increase of 0.44 km/sec at 1pc (see previous section), the cluster should be 1.8.10**8 years old. This age fits well with the evolutionary age derived for the low mass stars by Stauffer(1979) but is a factor two to three higher than the age derived from the B stars by Maeder and Mermilliod(1981) and Golay and Mauron(1982). A study on problems concerned with the derivation of the age of the massive Pleiades by Van Leeuwen and De Zeeuw is in progress. 126 The uncertainty in the dynamical age derived above due to difficulties in determining <£J2>1/2 as well as in the estimate of As the angular momentum of a star increases, the angle -9" becomes more time dependent. This will decrease the efficiency of the above described effect on longer t7ime scales. When the stellar orbits are nearly circular the efficiency goes to zero. Giant interstellar clouds will add also some angular momentum dispersion to the cluster. Their contribution is, however, small compared to that of the galactic tidal field. The increase in angular momentum dispersion due to a cloud of mass 'M' that passes by the cluster at a minimum distance 'a' with a velocity V can be derived from equ.52 to be: 2 1 /2 z <£J y = (3/8) .Cn;? .CGM/a ) .( The velocity of the Pleiades with respect to the interstellar medium can be estimated from its proper motion and radial velocity as well as from the relative velocity of the reflection nebulae in the cluster (Van Leeuwen, in preparation). Both provide a value of approximatly 30 km/sec. If a cloud of 10**5 solar masses is overtaken by the cluster at a minimum distance of 100 pc with the above given velocity, the increase in angular momentum dispersion equals: In order to obtain values near to 0.44 km/sec at 1pc as found above at least 100 clouds need to be overtaken at distances near to 30 pc during an estimated 3 kpc traveled by the cluster relative to the interstellar medium since its birth. Apart from the fact that such high cloud densities are unrealistic, it seems very unlikely that the cluster would still exist in that case. We therefore conclude that the galactic tidal field is likely the main source behind the increase in angular momentum dispersion for a young open cluster. 4.8 Conclusions In this section the results obtained in the previous sections are to be reviewed in the light of different models of open clusters. Special attention will be paid to parameters that can be observed but which are usually not extracted from the models. 127 The first quantities derived in the present paper were the surface and space density distributions. The ^lope of the halo population is similar to that predicted by analytical considerations (Spitzer 1975), numerical simulations (Terlevich 1980) and from velocity distribution models (King 1966). This could indicate that the radial velocity distribution in the cluster halo is to a large extent isotropic, although anisotropy is observable only at the outer edge of the cluster (King 1975). There, however, the density determinations are hampered by statistical problems of low numbers. The observed densities as a function of stellar mass for the cluster core agree only with the numerical models developed by Terlevich(1980), which take the galactic tidal field into account. Other models do predict mass segregation and a relative decrease in core density for low mass stars, but not to the extent observed here. It would be interesting to obtain estimates of the scatter of counts of stars in an open cluster from different models and to compare this scatter with the observations. In particular a diagram of Z(r1) against r1**-(q-3) (see section 4.2.2), which shows in the observations clearly the extent and noise level of the halo, would be an interesting model parameter to compare. It could provide better estimates for the accuracies of counts in an open cluster and therefore increase the reliability of further calculations performed on such counts. The occurrence of a fraction of at least 15 percent of initial binaries of comparable mass is an important parameter in the evolution of the cluster core (see Heggie 1975). The timescale on which they can separate themselves from single stars of the same spectral type can be obtained from numerical models and may provide an additional age estimate of the cluster. The mass function of the cluster found in section 4.4 is rather different from thost; employed in most models. Only Terlevich(198O) has used a realistic mass function. It may, however, be relatively unimportant to take into account the high numbers of low mass stars, as these finally all disappear to the halo and lose the individual interaction with other cluster members. In the early stages of evolution, nevertheless, they will provide an easy means for the massive stars to lose energy and to concentrate to the core more rapidly. Another important property of these stars is their contribution to the total mass of the cluster and therefore its possible extent in space. Velocity information is usually not extracted from numerical cluster models but should be easy to obtain. Next to the space density distributions it can provide one of the most easily comparable parameters between models and observations. The velocity information should preferably be presented as integrated radial and transverse components as presented in equ.42 and 43. In particular a relation between the energy distribution of all stars in the cluster and the velocity dispersion in the centre is an important observation. Numerical simulations, but probably also Monte Carlo simulations, can provide a time scale for 'ühe increase in angular momentum dispersion, which will be a strong tool in obtaining an estimate of the cluster age independent of stellar evolution models. Such simulations can also, much better than described in section 4.6, determine the relation between the space density distribution and the change in angular momentum dispersion. 128 In section 4.7 it was shown how the galactic tidal field can not bring the cluster to corotation and instead hollows it. This provides a rather favorable situation for modeling an open cluster, as it disconnects the halo population from the core. In the halo, interactions between individual stars are very small due to the large interstellar distances, while in the core these interactions are particularly important. It is therefore proposed to describe the cluster core, with approximately 50 to 100 stars, numerically and the cluster halo by either analytical methods or statistical, Monte Carlo simulations. The interactions between both systems can be described as an exchange of energy towards the halo and as the addition or subtraction of a star with given energy and angular momentum towards the core. In this way, the evolution of the whole cluster can be described, while avoiding problems of describing the core evolution statistically or the halo evolution of 2500 stars or more numerically. Acknowled gements It is a pleasure to thank R.S.LePoole, Prof. Blaauw, Prof. Mestel and Prof Oort for stimulating discussions. 129 References Aarseth, S.J., 1975: in Dynamics of Stellar Systems, IAU symp.69, 57 ed. A.Hayli, Reydel Pub1.,Dordrecht, Netherlands Abt.H.A. and Levy.S.G., 1976: Astroph.J.Suppl.Ser. 30, 273 Allen, C.W., 1976: Astrophysical Quantities, Univ.of London, The Athione Press Ambartsumian.V.A., Mirzoyan.L.V.,Parsamian,E.S., Chavushian.O.S. and Erastova.L.K., 1970: Astrofizika £, 7 Artyukhina.N.M. and Kholopov.P.N., 1966: Astron.Zh. _43, 567 Atryukhina.N.M., 1968:Astron.Zh., 45, 1246 Bessel,F.W., 1844: Astronomische üntersuchungen, Erster Band, 209 Bettis.C, 1977: Astroph.J. 214, 106 Binnendijk.L, 1946: Ann.Sterr.Leiden, Vol XIX, part two Chandrasekhar, S., 1942: Principles of Stellar Dynamics, Chicago Dokuchaev.L.M. and Ozernoy.V.I., 1982: Astron.Astroph. 111, 16 Elkin.W.L., 1887: Trans.Astron.Observ.Yale Univ., ±, 1 Golay.M. and Mauron.N., 1982: Astron.Astroph.Suppl.Ser., 47, 547 Greenstein.J.L., 1974: Astron.J., 2£, 964 Heggie,D.C, 1975: in Dynamics of Stellar Systems, IAU Symp.67, p73, Reydel Publ. Dordrecht, Netherlands Hertzsprung.E., 1934: Bull.Astr.Inst.Netherlands, ]_, 187 Hertzsprung,E., 1947: Ann.Sterr.Leiden, Vol XIX, part one Jones,B.F., 1970a: Astroph.J.Letters, 159, L129 Jones,B.F., 1970b: Astron.J., 75., 563 Jones,B.F., 1981: Astron.J., 86_, 290 Kholopov.P.N., 1971a: Inf.Bull.Variable Stars, 566 Kholopov.P.N., 1971b: Astron.Zh., 48, 529 King,I.R., 1966: Astron.J., 71.. 64 King,I.R., 1975: in Dynamics of Stellar Systems, IAU Symp.69, p99, ed. A.Hayli, Reydel Publ., Dordrecht, Netherlands 130 Lagrula.J., 1903: Trav.de l'Obs.de Lyon, III, 1 Luyten.W.J., 1968: Monthly Not.Roy.Astr.Soc., 239, 221 Luyten.W.J. and Herbig.G., 1960: Harvard Announcement Card 1U74 Maeder.A. and Mermilliod.J.C., 1981: Astron.Astroph., 93,, 136 McCarthy,M.F., 1974: Contr.Oss.Astrofis.Asiago No 300-bis, p87 Menzel.D.H., 1960: Fundamental Formulas of Physics, Vol one, Dover Pub1.New York Mirzoyan.L.V., Chavushian.H.S., Erastova.L.K., Ohanian.G.B., Melikian.N.D., Natsulishuili.R.Sh. and Tsvetkov.M.K., 1977, Astrofizika, _13_, 205 Mirzoyan.L.V., 1981: Instationarity and Evolution of Stars, Akad.Nauk Armyanskaj SSR Mirzoyan.L.V. and Mnatsakanian,M.A., 1971: Inf.Bull.Variable Stars, 528 Pearce.J.A. and Hill.G., 1975: Publ.Dom.Astroph.Obs., J4.. No 14 Pritchard.C., 1884: Mem.Roy.Astron.Soc, Jt8, 225 Smart, W.M., 1938: Stellar Dynamics, Cambridge Univ.Press Spitzer,L., 1975: in Dynamics of Stellar Systems, IAU Symp.69, p 1 ed. A.Hayli, Reydel Publ., Dordrecht, Netherlands Stauffer,J.R., 1979: Astron.J, 85, 1341 Taff.L.G., 1974: Astron.J, 79, 1280 Terlevich.E., 1980: in Star Clusters, IAU Symp.85, P165 ed. J.E.Hesser, Reydel Publ., Dordrecht, Netherlands Titus,J., 1938: Astron.J., J+7, 25 Trumpler.R., 1914: Astron.Nachr., Band 200, Nr4790, 213 Trumpler.R., 1920: Publ.Astron.Soc.Pac., J£, 43 Trumpler.R., 1921: Lick Obs.Bull., J333, 110 Van Den Bergh.S., 1957: Astroph.J., JJ25, 445 Van Leeuwen,F., 1980: in Star Clusters, IAU Symp.85, p157, ed. J.E.Hesser, Reydel Publ., Dordrecht, Netherlands v Vasilevskis.S., Van Leeuwen,F., Nicholson,W. and Murray,C.A., 1979: Astron.Astroph.Suppl., ^7, 333 131 Wielen.R., 1971»: in Highlights of Astronomy, Vol 13, ed. Contopoulos, p 395 Wolf.C. 1877: Ann.de l'obs. de Paris, Tome XIV, second part, A1 132 Nederlandse Samenvatting 1. De Pleiaden is een van de meest opvallende ster groepen aan de noordelijke herfst hemel. Als een kleine groep dicht bij elkaar staande vrij heldere sterren vormen ze een unieke constellatie. Er wordt dan ook al meer dan H000 jaar naar ze verwezen in over de gehele wereld verspreide dokumenten, zoals Chinese annalen, de Bijbel, en Maya codices. Naast de Zon, de Maan Halverwege de 19de eeuw begonnen astronomen in Europa en de VS de Pleiaden nader te bestuderen. Dat hield in, in die tijd, dat men zeer nauwkeurig de afstanden mat tussen individuele sterren. Door de metingen over vele jaren te herhalen kon men berekenen hoeveel deze sterren zich jaarlijks ten opzichte van elkaar verplaatsen. Deze verplaatsingen noemt men relatieve eigen bewegingen, relatief omdat ze gemeten worden ten opzichte van andere sterren. Indien de groep sterren die we de Pleiaden noemen al voor langere tijd bij elkaar zijn, zouden ze slechts zeer geringe onderlinge bewegingen vertonen. In geval het een toevallige clustering is, zou men grote onderlinge bewegingen verwachten. De eerste die er in slaagde deze bewegingen voldoende nauwkeurig te bepalen was William L.Elkin van de Yale Sterrewacht in de VS. Hij vergeleek zijn heliometer waarnemingen van 1885 met die welke Friedrich Wilhelm Bessel deed in de periode 1829 tot 1841. Elkin ontdekte dat veel van de sterren die men tussen en rond de zes heldere sterren kan waarnemen ten opzichte van deze weinig of geen beweging vertonen. Een klein aantal, en in het bijzonder onder de minder heldere sterren, vertonen echter een grote en duidelijk meetbare relatieve eigen beweging. Men kwam tot de conclusie dat de Pleiaden, de zes heldere en een groot aantal minder heldere sterren, al voor langere tijd bij elkaar horen, en dat er sprake is van een echte ster cluster. Daar men echter door deze cluster en vele van zijn soortgenoten heen kan kijken, noemde men ze open clusters, als tegenhanger van de veel grotere bol-clusters waar de sterren in het centrum zo dicht opeen zitten dat ze zelfs niet langer individueel onderscheiden kunnen worden. Nieuwe onderzoekingen in de eerste helft van de 20ste eeuw, met name in Leiden onder leiding van Ejnar Hertzsprung, lieten zien dat de onderlinge bewegingen van de sterren in de Pleiaden cluster zeer gering zijn. Ze horen vrijwel zeker al sinds hun geboorte bij elkaar. In de zelfde periode leerde men ook veel meer over individuele sterren. Het vroege onderzoek van Hertzsprung aan de Pleiaden heeft er veel toe bijgedragen dat een dieper inzicht werd verkregen met betrekking tot de relaties tussen de kleur, massa en eigen helderheid van de sterren. Door het werk van Trumpler leerde men dat niet alle scerren een zelfde verdeling in de cluster vertonen; de heldere, zware sterren zijn veel meer naar het centrum geconcentreerd dan de zwakkere lichte sterren. Sinds het Leidse werk van Hertzsprung is de nadruk van het onderzoek aan de Pleiaden komen te liggen op het zeer nauwkeurig meten van de onderlinge bewegingen van de cluster leden, het soort van onderzoek waarvan ook dit proefschrift een directe voortzetting is, en op het fotometrisch 133 bepalen van de kleuren van de leden. Met de eerste soort metingen hoopt men meer te leren over de massa van de cluster, en van de fotometrie hoopt men meer te leren over de leeftijd en de afstand van de Pleiaden. 2. In dit proefschrift wordt eerst een onderzoek beschreven naar de eigen bewegingen van de Pleiaden, deels opgezet om interne bewegingen voor de lichtzwakke sterren in het centrum van de cluster te kunnen meten, en deels om nieuwe leden te zoeken onder de lichtzwakke sterren in een groot veld rond het cluster centrum. Het is als zodanig ook een voortzetting van het werk dat G.Pels rond 1960 in Leiden deed, en waarvan hier ook de resultaten gepubliceerd worden. Door een samenloop van omstandigheden, welke deels samenhangen met de bezuinigingen waartoe ook de RU Leiden gedwongen is, is dit deel van het werk in het zicht van de haven gestrand. Al de metingen en reducties zijn voltooid en alleen de laatste kleine correcties welke essentieel zijn voor de bepaling van de eigen bewegingen moeten nog bepaald en toegepast worden. Hoofdstuk 2 beschrijft dit deel van het onderzoek voor zover het klaar gekomen is. De posities van de sterren in de Pleiaden worden al sinds 120 jaar geregistreerd op fotografische platen. Voor dit onderzoek waren beschikbaar 70 to 90 jaar oude platen van de Sterrewacht van Parijs en van de Sterrewacht in Oxford, waarvan de fotografische refractor en het platen archief nu onder beheer staan van het Royal Greenwich Observatory. Deze platen zijn gemaakt tussen 1885 en 1910 voor de oudste fotografische hemel bedekking, en bedekken een veld van 10 bij 10 graden. Om een zelfde veld te bedekken met nieuwe fotografische platen zijn tussen 1978 en 1980 373 nieuwe opnamen op 121 verschillende centra gemaakt in het Pleiaden veld, gebruik makend van de in 1898 geïnstalleerde fotografische refractor op de Sterrewacht enclave in Leiden. Hierbij is veel assistentie verleend door een grote groep Leidse sterrenkunde studenten. Om op een dergelijke grote hoeveelheid platen de onderlinge posities van sterren te meten werd gebruik gemaakt van de ASTROSCAM meetmachine van de Sterrewacht in Leiden. In sectie 2 wordt beschreven hoe deze metingen plaats vonden evenals de reductie programma's welke ontwikkeld zijn om van de ASTROSCAN metingen positie en helderheids informatie voor de individuele sterren te ontrekken. In totaal werden meer dan 400 000 sterbeeldjes, geregistreerd in 719 belichtingen, door deze programmals verwerkt. De posities van de sterren in het veld moesten echter eerst eenmaal met de hand worden gemeten. Dit is gedaan voor 2M000 sterren in 36 velden, waarbij gebruik werd gemaakt van de KOMESS Zeis twee coördinaten meetmachine met geautomatiseerde uitlezing. De afbeelding van de hemel op een fotografische plaat is die van een sferisch beeld op een plat vlak. Een dergelijke afbeelding is altijd vervormd en het is dan ook niet mogelijk een eenvoudige transformatie te geven voor de ene plaat naar de andere. In sectie 3 wordt beschreven hoe deze platte vervormde afbeeldingen eerst worden terug gebracht naar de oorspronkelijke sferische, en vervolgens wel eenvoudig naar elkaar getransformeerd kunnen worden. Tevens wordt beschreven wat de invloeden van de atmosfeer en het telescoop objectief zijn op dit proces. Het is in het bijzonder dit proces dat nog niet voldoende nauwkeurig beschreven kan worden om de vereiste nauwkeurigheid te bereiken. Afwijkingen, welke waarschijnlijk niet rotatie symmetrisch zijn, en die slechts in tienden van 134 microns gemeten worden, zijn er de oorzaak van dat ontoelaatbare vervormingen van het buiten veld optreden. In sectie 4 word beschreven hoe voor een veld bestaande uit vele overlappende platen gemaakt met verschillende telescopen toch een consistent en homogeen veld van relatieve eigenbewegingen kan worden verkregen. De beschreven programma's zijn toegepast op het centrale veld dat slechts weinig door de vervormingen is beinvloed. Ook hier, echter, konden geen overlappende velden gebruikt worden, waardoor nog geen verbetering van de qualiteit van de relatieve eigen bewegingen is verkregen ten opzichte van vroegere studies. 3. In hoofdstuk drie worden de fotometrische metingen beschreven. Alle metingen zijn gedaan met de VBLUW fotometer, ontwikkeld door Th.Walraven, en de Nederlandse telescoop op de Europese Zuidelijke Sterrewacht op La Silla, Chili. Het omvat drie secties, waarvan de eerste een serie metingen beschrijft welke gedaan werden voor de lidmaatsc»qps bepaling van een selectie van waarschijnlijke en mogelijke leden van de Pleiaden door Pels. De lidmaatschaps bepaling gebeurt door een vergelijking in kleur en helderheid tussen bekende leden en vermoedeljke leden, gebruik makend van het feit dat alle sterren in de cluster een unieke relatie vertonen tussen kleur en helderheid. Door ook een aantal zeer nabij staande sterren te meten werd uit een vergelijking tussen de helderheden en de kleuren van die sterren en de cluster leden een afstand van 130 pc, 3.9*1015 km, bepaald. Foto's van de Pleiaden vertonen vaak een lichtende nevel rond sommige van de heldere sterren. Deze nevel, welke licht refleoteert van die heldere sterren, is deel van een complex van stof en gas wolken dat met vrij grote snelheid door de cluster beweegt. Het veroorzaakt dat sommige sterren wat verduisterd worden en daardoor roder en zwakker lijken dan ze in werkelijkheid zijn. Dit is verschillend voor verschillende leden van de cluster, doordat de wolken sterk onregelmatig in vorm en dichtheid zijn, op een schaal die klein is in vergelijking met de onderlinge afstanden van sterren in de cluster. In sectie 3.2 worden de individuele verkleuringen van de heldere sterren in de cluster bepaald, en een distributie functie voor de verkleuringen afgeleid. Onderzoek aan individuele sterren in de Pleiaden heeft nog maar weinig plaats gevonden. Een dergelijk onderzoek, in dit geval naar de lichte sterren, wordt beschreven in sectie 3.3. Deze lichte sterren hebben veel tijd nodig om van hun initiële gaswolk te contraheren tot een ster. Het moment waarop we deze sterren in de Pleiaden waarnemen is waarschijnlijk in de voltooiing van dit proces. In sectie 3.3 wordt de ontdekking beschreven van de zeer hoge rotatie snelheden van deze sterren en de regelmatige fotometrische variaties die daarmee gepaard gaan. De variaties lijken het gevolg te zijn van variaties in helderheid op het steroppervlak, zoals dit ook voor andere snel roterende sterren met convectieve schillen is waargenomen. Men neemt dan ook aan dat de rotatie en fotometrische perioden aan elkaar gelijk zijn. Een nooit eerder waargenomen relatie tussen de fotometrische perioden en de fotometrische amplituden van deze sterren zowel als het algemene en uniforme optreden van de variaties geven de sterke indruk dat ze het directe gevolg zijn van de hoge rotatie snelheden. De snelst roterende sterren in de Pleiaden hebben een hoekmoment dat vergelijkbaar is met dat van het totale zonnestelsel, 100 maal meer dan dat 135 van de zon. Er zijn indicaties, dat deze sterren snel aan hoekmoment verliezen. De verdeling van hoekmoment zoals hier is waargenomen kan belangrijke implicaties hebben voor de theorie van het ontstaan van het zonnestelsel. 4. In het laatste hoofdstuk worden de fotometrische en astrometrische gegevens gebruikt om een gedetailleerd op waarnemingen gebaseerd model van een open cluster te verkrijgen. Zo wordt allereerst beschreven hoe de ruimtelijke dichtheden van sterren van verschillende massa's bepaald zijn. In het bijzonder de nauwkeurigheid van dit soort berekeningen wordt nader toegelicht. Voor de F en G type sterren wordt in sectie 3 de verdeling van dubbel sterren bepaald, en wordt afgeleid dat zeker 15 procent van deze sterren dubbels zijn van vergelijkbare massa. Sectie 4 beschrijft de massa functie van de Pleiaden in vergelijking met die van de zons omgeving. De massa functie geeft de relatieve aantallen sterren van verschillende massa weer. Voorzover deze functie bepaald kan worden, zijn geen significante verschillen gevonden tussen de Pleiaden en de zons omgeving, hetgeen een indicatie is dat het vormings proces van de Pleiaden niet veel verschilde van dat van veel sterren in de zons omgeving. De interne bewegingen worden in sectie 5 afgeleid uit de relatieve eigen bewegingen. Na eerst gecorrigeerd te hebben voor verschillende interne (reductie) en externe (Zons beweging) invloeden wordt een snelheids verdeling gevonden die duidelijk wijst op sterk elliptische banen voor sterren in de buiten delen van de cluster. Samen met de ruimtelijke verdeling van sterren in de cluster wordt een schatting gemaakt van de totale massa in de cluster. In sectie 7 is beschreven hoe, door een extern krachtveld, de bewegingen van sterren in de cluster verstoord raken. In het bijzonder in geval dit externe krachtveld gevormd wordt door de massa verzameld in de melkweg, wordt aangetoond dat de gemiddelde ellipticiteit van de sterbanen afneemt. Dit leidt ertoe, dat sterren die lang aan deze verstoring onderhevig zijn, gemiddeld minder dicht bij het centrum kunnen terug keren. Daar de sterren in de buiten delen van de cluster hierdoor het meest beinvloed worden, en deze buiten delen voornamelijk door lichte sterren bezocht worden, leidt dit proces tot een versterkte afscheiding tussen de kern met zware sterren en de buiten delen met de lichte sterren. Numerieke experimenten hebben aangetoond dat het proces inderdaad zo werkt en dat dichtheids distributies worden verkregen onder invloed van het getijden veld van de melkweg welke goed vergelijkbaar zijn met die zoals waargenomen voor de Pleiaden. 136 Studie overzicht Na het behalen van het einddiploma HBS B aan het Dalton Lyceum in Den Haag begon ik in september 1971 met mijn studie aan de Rijks Universiteit Leiden. In maart 1974 legde ik het kandidaads examen natuurkunde af en begon mijn doctoraal studie met hoofdvak sterrenkunde. In die periode deed ik een klein onderzoek aan nabij liggende neutrale waterstof concentraties in de Sagittarius arm onder leiding van Dr.W.W.Shane en een groot onderzoek naar interne bewegingen in het centrale deel van de Pleiaden onder leiding van de gasthoogleraar Prof.Dr.S.Vasilevskis. In het kader van dit onderzoek werden in 1976 twee werkbezoeken van twee weken gebracht aan het Royal Greenwich Observatory in Engeland voor het uitmeten van fotografische platen. Van juni tot september 1975 en in januari 1977 heb ik waargenomen met de VBLÜW fotometer te Hartbeespoortdam, Zuid Afrika. In juni 1977 legde ik het doctoraal examen sterrenkunde af aan de Rijks Universiteit van Leiden. In september en oktober 1977 vervulde ik mijn dienstplicht en gedurende november en december 1977 deed ik fotometrische waarnemingen met de VBLUW fotometer te Hartbeespoortdam. Van 1 januari 1978 tot 1 juni 1981 was ik in dienst van de Sterrewacht te Leiden voor een promotie onderzoek met betrekking tot de Pleiaden. In het kader van dit onderzoek bezocht ik gedurende die tijd de volgende conferenties: The Fourth European Regional Meeting in Astronomy, Uppsala, Zweden, augustus 1978 IAU colloquium 51: Modern Astrometry, Wenen, Oostenrijk, september 1978 IAU symposium 85: Star Clusters, Victoria, Canada, augustus 1979 Gedurende die periode en in het kader van het promotie onderzoek werden de volgende waarnemingen gedaan: 1978, 1979 en 1980: vele nachten met de fotografische refractor op de Sterrenwacht enclave in Leiden, november 1978: 10 nachten met de Bologna 1.5 meter telescoop in Loyano, Italië, en de Tinbergen polarimeter. oktober en november 1979: 6 weken met de VBLUW fotometer, ESO, La Silla, Chili oktober en november 1980: 6 weken met de VBLUW fotometer, ESO, La Silla, Chili Verder heb ik colloquia gegeven met betrekking tot het promotie onderzoek aan het Sterrenkundig Instituut van de Universiteit van Amsterdam, de Sterrenwacht van Bonn, W.Duitsland, het Copernicus Instituut in Warszawa, Polen en de Sterrenwacht van Leiden. Tevens werd in januari 1979 een bezoek gebracht aan het Observatoire de Paris voor het verzamelen van fotografische platen. Vanaf 2 november 1981 ben ik in dienst van het Royal Greenwich Observatory in Engeland, waar ik nog gedurende ruim een jaar aan het promotie onderzoek heb kunnen werken. In die periode bezocht ik IAU Colloquium 71, Activity in Red Dwarfs, gehouden in Catania, Italië, in augustus 1982. 137 Dankwoord Hierbij wil ik gaarne Mevr.Debarbat van het Observatoire de Paris en CA.Hurray van het Royal Greenwich Observatory bedanken voor het beschikbaar stellen van fotografische platen. Veel collega's op de Sterrenwacht van Leiden, tijdens het IAU symposium 85 met betrekking tot Ster Clusters en tijdens het IAU colloquium 71 met betrekking tot Activiteit in Rode Dwergen hebben door vragen, suggesties en discussies bijgedragen aan de totstandkoming van dit proefschrift. Bovenal wil ik Peter Alphenaar bedanken, die op velerlei wijze heeft meegewerkt aan de totstandkoming van dit proefschrift en zonder wie de afronding ervan nog zeker een half jaar vertraagd zou zijn. Het Kerkhoven-Bosscha fonds verleende finaciele ondersteuning voor reizen tussen Engeland en Nederland. 138