Optical Studies of Massive X-Ray Binaries" Door E.J

C 7A OPTICAL STUDIES OF SIVE m

EJ. ZUIDERWIJK optical studies of massive X- ray binaries

ACADEMISCH PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE WISKUNDE EN NATUURWETENSCHAPPEN AAN DE UNIVERSITEIT VAN AMSTERDAM, OP GEZAG VAN DE RECTOR MAGNIFICUS, DR. J. BRUYN, HOOGLERAAR IN DE FACULTEIT DER LETTEREN, IN HET OPENBAAR TE VERDEDIGEN IN DE AULA DER UNIVERSITEIT (TIJDELIJK IN DE LUTHERSE KERK, INGANG SINGEL 411, HOEK SPUI) OP WOENSDAG 20 JUN11979 DES NAMIDDAGS TE 15.00 UUR PRECIES

DOOR

EDUARDUSJOSEPHUSZUIDERWIJK

GEBOREN TE HAARLEM

PROMOTOR : Prof. Dr. E.P.J. van den Heave 1 COREFERENT : Dr. H.G L.M. Lamers

The work described in this thesis was supported by the "Netherlands Organi- sation for the advancement of Pure Re- search" (ZWO) and by the University of of Amsterdam.

Voor mijn Iet en onze Ouders

Na het voltooien van mijn proefschrift wil ik graag allen bedanken die aan het tot stand komen ervan hebben bijgedragen. Allereerst dank ik mijn ouders, die mij de gelegenheid gaven een aca- demische opleiding te volgen. Bijzonder erkentelijk ben ik mijn Promotor Prof. Dr. E.P.J. van den Heuvel, die het voor mij mogelijk maakte het onderzoek uit te voeren. Hij is daarbij altijd een steun geweest door de interesse getoond in mijn werk, die onder andere tot uiting kwam in langdurige en diepgaande gedach- tenwisselingen. De coreferent Dr. H.J.G.L.M. Lamers dank ik voor het kritisch lezen van het proefschrift. Het grootste deel van de in mijn proefschrift verwerkte publicaties zijn tot stand gekomen in nauwe samenwerking met mijn collega's van het Sterrenkundig Instituut van de universiteit van Amsterdam en van het As- trophysisch Instituut van de Vrije Universiteit van Brussel. Door hun kenr' nis en ervaring hebben zij ieder op zijn of haar eigen wijze bijgedragen aan dit proefschrift, waarvoor ik hen zeer erkentelijk ben. Bijzondere dank ben ik verschuldigd aan Jan van Paradijs, die de aanzet gaf tot de theoretische berekeningen aan lichtkrommen en die veel waarnemingen uitwerkte, aan Godelieve Hammerschlag-Hensberge voor haar steun bij het waarnemen en bij reductie werk en aan Roelf Takens voor o.a zijn adviezen bij het programmeren. Met veel genoegen denk ik terug aan öe gezelligheid en de goede werk- sfeer in het Sterrenkundig Instituut te Amsterdam. De vaak diepgaande (en soms verhitte) discussies bij de koffie zullen door mij gemist worden. De algemeen direkteur van de ESO, Prof. Dr. L. woltjer, ben ik zeer erkentelijk omdat hij mij de gelegenheid gaf mijn proefschrift te voltooien. Hans de Ruiter bedank ik voor het kritisch lezen van enkele delen van het manuscript. Mijn schoonvader, de heer J.M.M. vanLith, bedank ik voor het verzorgen van de lay-out, de omslag en de fraaie illustraties, waarin hij zijn geheel eigen visie op het onderwerp van mijn proefschrift verbeeldde. Drukkerij Trepico te Hooglanderveen, in het bijzonder de heer P.C.v.d. Kletersteeg, dank ik voor de prettige wijze van samenwerken bij het druk- ken van dit proefschrift. Tenslotte, maar niet op de minste plaats, bedank ik mijn lieve echt- genote. Niet alleen verzorgde je een groot deel van het typewerk, Julie, maar zonder jouw voortdurende steun en zonder het door jou opgebrachte geduld was dit proefschrift er hoogstwaarschijnlijk nooit gekomen.

CONTENTS

PAGE 11 INLEIDING 21 SAMENVATTING 25 INTRODUCTION AND SUMMARY

31 CHAPTER I ANALYSIS OF PHOTOMETRIC OBSERVATIONS OF MASSIVE X-RAY BINARIES 33 IA Theoretical Light Curves and Radial Velocity Orbits for Massive X-ray Binaries 33 1. introduction 34 2. Basic Assumptions of the Computational Model 41 3. The Computer Program and Results 59 4. Limitations of the Assumptions used in the Computational Model 70 IB Four Colour Photometric Observations of the X-ray Binary Star HD 77581 (Vela X-l) I 75 ic Four Colour Photometric Observations of the X-ray Binary Star HD 77581 (Vela X-l) II 82 ID Four Colour Photometric Observations of the X-ray Binary HD 153919 (3U1700-37) 86 IE study of the Light Curve of the Of star HD 153919 95 IF Photometric Variations of Wray977 (3U1223-62 ? ) 98 IG Evidence for an Accretion Disk in SMC X-l 103 CHAPTER II THE RADIAL VELOCITY ORBIT OF HD77581

105 HA Mass Determination for the X-ray Binary System Vela X-l 107 IIB systematic Distortions of the Radial Velocity Curve of HD 77581 (Vela X-l) due to Tidal Deformation 'I4 IIC The Spectroscopie Orbit and the Masses of the Components of the Binary X-ray Source 3U0900-40/HD 77581

131 CHAPTER III SPECTROSCOPIC STUDIES OF MASSIVE X-RAY BINAIRES

133 UIA Spectroscopie Observations of the Early Type B-Supergiant Wray977 (401223-62): Description of the Spectrum and Classification I45 IIIB The Spectrum of HD77581 (Vela X-l): Variations in the Profiles of H and Hell A 4685 8 Po 169 CURRICULUM VITAE

INLEIDING

RONTGENBRONNEN

Ongeveer 16 jaar geleden werd het eerste hemellichaam ontdekt dat zijn energie voor het grootste deel uitzendt in dr vorm van rontgenstraling. Evenals licht en radiostraling bestaat rontgenstraling uit electromag- —8 netische golven, maar dan met een zeer korte golflengte (kleiner dan 10 meter). Omdat deze straling in de aardse dampkring wordt geabsorbeerd moeten waarnemingen van rontgenstraling van hemellichamen worden uitgevoerd op een hoogte van minstens 200 kilometer, waar men zich boven 99,9% van de dampkring bevindt. Op 12 juni 1962 werd door een groep natuurkundigen onder leiding van R. Giacconi een sondeerraket gelanceerd met instrumenten aan boord om röntgen-fluorescentie straling van de maan waar te nemen. De raket be- reikte een hoogte van 230 kilometer en gedurende ongeveer 350 seconden kon worden waargenomen. Veel straling afkomstig van de maan werd niet gemeten, maar wel werd, volkomen onverwacht, een sterke bron van rontgenstraling ontdekt die zich bevond in" de richting van het sterrenbeeld Schorpioen (Scorpio). Tijdens twee volgende raketvluchten, in oktober 1962 en juni 1963, werd deze röntgenbron opnieuw waargenomen, waarbij . werd vastgesteld dat deze niet van plaats veranderde ten opzichte van de "vaste" sterren. Dit betekende dat dit object zich op aanzienlijke afstand van het zonnestelsel moest bevinden, een afstand minstens vergelijkbaar met die van de ons omringende sterren. Er werden ook twee zwakkere röntgenbronnen ontdekt in de richting van de sterrenbeelden Stier (Taurus) en Zwaan (Cygnus). De pas ontdekte bronnen werden al spoedig aangeduid als resp. Sco X-I, Tau X-l en Cyg X-l, waarbij de "X" staat voor "X-ray" (= rontgenstraling).

Sinds 1963 zijn er talloze waarnemingsvluchten gemaakt met sondeerra- ketten. Door de steeds toenemende gevoeligheid van de meet-apparatuur werd het waarnemen van steeds zwakkere röntgenbronnen mogelijk. Eind 1967 waren een dertigtal bronnen bekend, ondanks de beperkte hoeveelheid waar- neemtijd per vlucht (557 minuten). In 1970 kwam voor de röntgen-astronomie de grote doorbraak. In december van dat jaar werd de UUURU (SAS-A) kunstmaan gelanceerd, de eerste die 12 geheel gebouwd was voor uitsluitend het waarnemen van rontgenbronnen. In de loop van 4 jaar is hiermee de gehele "röntgen-hemel" in.kaart gebracht; de 4e (en- laatste) UHURU catalogus bevat gegevetis over 339 rontgenbronnen (het "4U" nummer van een bron geeft de positie aan de hemel weer), Doordat veel van deze bronnen meerdere keren zijn waargenomen is hun positie aan de hemel vrij goed bekend. Bij elke rontgenbron wordt een klein gebiedje aan de hemel opgegeven, de zogenaamde "error box", waarbinnen de betref- fende bron zich met grote waarschijnlijkheid bevindt. Na UHURU zijn verschillende andere kunstmanen gelanceerd waarmee rontgenbronnen werden en nog worden waargenomen, zoals Copernicus, ANS, Ariel, COS-B, SAS-3 en HEAO-A. Het aantal bekende "vaste" bronnen wordt regelmatig groter en de posities aan de hemel worden steeds nauwkeuriger bepaald. Dat laatste vooral door de waarnemingen met de SAS-3 die een buitengewoon preciese positiebepaling mogelijk maken. Voor de toekomst wordt vooral veel verwacht van de HEAO-B die eind 1978 is gelanceerd. Vanwege de grote gevoeligheid van de waarneeminstrumenten en het hoge ruir.telijk oplossend vermogen kunnen met HEAO-B zelfs bronnen in sterrenstelsels op grote afstand van ons "eigen" melkwegstelsel worden waargenomen.

RONTGEN DUBBELSTERREN EN RONTGEN PULSARS

Een van de belangrijkste ontdekkingen die met UHURU gedaan werden was die van de röntgendubbelsterren: de rontgenbron bevindt zich hier in een dubbelstersysteem. Een dubbelster bestaat uit twee sterren die door de wederzijdse zwaartekrachtsaantrekking bijeen gehouden worden en een.baanbeweging om elkaar uitvoeren (eigenlijk: om hun gemeenschappelijke zwaar- tepunt) . Minstens de helft van alle sterren maakt deel uit van dubbel- stersystemen. Een röntgendubbelster bestaat uit een "normale" optisch waarneembare ster en een niet zichtbare compacte ster, meestal een neutronenster, die de bron is van de röntgenstraling. Bij de zware röntgen- dubbelsterren is de optisch -waarneembare component verreweg de zwaarste van de twee; deze ster heeft een massa ("gewicht!') en ook een diameter van enkele tientallen keren die van de zon - respectievelijk 2.10 kg en 1.4 miljoen kilometer - terwijl het steroppervlak een temperatuur heeft van 25000 tot 35000 graden. Daardoor is deze reuzenster zeer helder en straalt iedere seconde enkele honderduizendén keren meer energie uit dan de zon. 13

Een neutronenster heeft een massa vergelijkbaar met die van de zon, maar ziin diameter daarentegen is slechts ongeveer 20 kilometer. De materie in 18 3 zo'n ster is sterk samengeperst; de dichtheid bedraagt ongeveer 10 kg/m , wat van dezelfde orde van grootte is als de dichtheid in atoomkernen. Een ster opgebouwd uit materie met zo'n grote dichtheid bestaat dan ook voor het grootste deel uit neutronen. De rontgenstraling ontstaat wanneer gas (afkomstig van de reuzenster) onder invloed van de zwaartekracht naar het oppervlak van de neutronenster valt. Hierbij worden valsnelheden bereikt van meer dan honderdduizend kilometer per seconde. Het gas wordt door het zeer sterke magneetveld van 12 de neutronenster (10 Gauss) gekanaliseerd in twee stromen die op het oppervlak botsen bij de zich diametraal tegenover elkaar bevindende magnetische polen. Daar wordt het gas - doordat de bewegingsenergie wordt om- gezet in warmte - verhit tot een temperatuur van meer dan tien miljoen graden waardoor het een grote hoeveelheid rontgenstraling uitzendt. De zo gevormde hete gebieden beslaan elk een oppervlakte van slechts ongeveer één vierkante kilometer.

Omdat de neutronenster een baan beschrijft om de reuzenster (figuur 1) kan het gebeuren dat vanaf de aarde gezien de röntgenbron ( = neutronenster) zich gedurende enige tijd daarachter bevindt; er wordt dan geen rontgenstraling waargenomen. Zo'n röntgen-eclips vindt tijdens iedere omloop van de neutronenster plaats en de röntgenbron wordt daarom met grote regelmaat verduisterd. Het tijdsverschil tussen twee opeenvolgende eclip- sen is gelijk aan de tijd die de neutronenster nodig heeft om één omloop te voltooien. Deze zogenaamde dubbelsterperiode van de röntgenbron Vela X-l is 8.96 dagen; die van Centaurus X-3 is 2.087 dagen. De rontgenstraling zelf heeft bij een aantal bronnen een regelmatig ge- pulst karakter: de uitstraling in de richting van de aarde geschiedt niet gelijkmatig maar varieert in intensiteit volgens een zich steeds herhalend patroon (figuur 2). De bron is dan een röntgenpulsar. Dit gedrag wordt veroorzaakt door de rotatie van de neutronenster om zijn as. Doordat de magnetische as in het algemeen niet samenvalt met de rotatieas van de ster verandert door de draaiing steeds de hoek waaronder de hete gebieden bij de magnetische polen worden waargenomen en daarmee de ontvangen hoeveelheid rontgenstraling. De pulsperioden, dus de rotatietijden van de betref- fende neutronensterren, verschillen van bron tot bron en zijn van de orde van seconden tot minuten. De pulsar Vela X-l bijvoorbeeld heeft een puls- b.J. JLiUlJLSJCifVVVlJ.IV

«1 TIJD

X X

Jf *

* K

'13 '17 jan 1971 1DAG

Figuur 1; (A) Model voor een zware röntgendubbelster. De neutronenster N beschrijft een baan om de reuzenster R. (B) De intensiteit 1% van de waargenomen röntgenstraling als . functie van de tijd (schematisch). De tijdstippen tj t/m tg komen overeen met de posities 1 t/m 6 in figuur 1&. Tussen t$ en tg bevindt de rOntgenbron (=neutronenster) zich achter de reuzenster; dan wordt geen röntgenstraling waargenomen. Het midden van de rbntgenecïips is aangegeven met tg.. (C) De door UHURU gemeten rpntgenhelderheid van SMC X-l in 1971. Duidelijk zijn de eclvgsen te- zien op 13 en op 17 januari. De dubbelsterperiode bedraagt3.89 dagen. 15

O CM •e in

TUD 283 sec

Figuur 2: De neutronenster als röntgenpulsar (boven). De invallende materie kan het oppervlak alleen bereiken bij de magnetische polen (M) van het zeer sterke magneetveld; daar wordt het gas sterk verhit. Door de draaiing van de neutronenster om zijn as (R) varieert de door de waarnemer ontvangen straling voortdurend in intensiteit volgens een steeds herhalend patroon. Omdat de twee "hete plekken" bij de magnetische polen beurtelings in beeld komen ziet men vaak twee maxima en twee minima per pulsperiode. Dit is duidelijk zichtbaar in een serie pulsen (onder) van de bron 3U0900-40 (Vela X-l), waargenomen met SAS-3 in juli 1975. 16 periode van 283 seconden, die van Cen X-3 is 4.84 seconden en de pulsar SMC X-l heeft een pulsperiode van 0.71 seconden (SMC=Kleine Magelhaese Wolk). De snelheid van de neutronenster ten opzichte van de aarde - de radiële snelheid - verandert voortdurend als gevolg van de baanbeweging; hierdoor varieert de waargenomen pulsperióde (de intrinsieke pulsperiode - de rota- tietijd van de neutronenster - verandert niet). Dit zogenaamde Doppler- effeet stelt ons in staat om de baansnelheid van de neutronenster te bepalen: die van Vela X-l bedraagt 270 km/s, die van Cen X-3 is 415.1 km/s. Uit de grootte van de baansnelheid van de neutronenster kan worden berekend dat de reuzenster in Vela X-l minstens 18 en die in Cen X-3 minstens 16 keer zo zwaar is als de zon. (Op veel langere tijdschaal gezien neemt de pulsperiode langzaam af. Deze verandering is intrinsiek - de rotatie- snelheid van de neutronenster neemt langzaam tóe - en wordt veroorzaakt door de naar het oppervlak vallende materie die draaiimpuls overdraagt op de neutronenster. Uit de sc«r.--id waarmee de rotatie toeneemt - die bij de meeste pulsars tot een halvering van de pulsperiode leidt in enkele duizenden jaren - kan worden afgeleid dat de röntgenbron zó compact is dat het alleen een neutronenster kan zijn). Hoewel de neutronenster in een zware röntgendubbelster optisch niet waar- neembaar is wordt zijn aanwezigheid duidelijk kenbaar door de getij denwer- king die de veel zwaardere en wel zichtbare reuzenster ervan ondervindt. Die bevindt zich immers in het zwaartekrachtsveld van de neutronenster en door de getijkrachten die daardoor op de reuzenster werken heeft deze niet meer een zuivere bolvorm, maar is enigszins uitgerekt in de richting van de neutronenster. Deze vervorming volgt de baanbeweging van de neutronenster en draait dus eenmaal per dubbelsterperiode rond. Dit heeft tot gevolg dat de grootte van het voor ons zichtbare steroppervlak steeds verandert, waardoor de waargenomen helderheid van de reuzenster regelmatig varieert. Per dubbelsterperiode worden twee helderheidsmaxima waargenomen gescheiden door twee helderheidsminima (figuur 3); we spreken van ellipsoïdale helderheidsvariaties. Een ander gevolg van de vervorming is dat de temperatuur van het oppervlak van de reuzenster niet overal dezelfde is: de door getijkrachten veroorzaakte "punt" die zich het dichtst bij de neutronenster bevindt is enkele duizenden graden koeler dan de rest van de ster. Dit effect ver- groot de helderheidsvariaties en heeft ook tot gevolg dat de twee helderheidsminima enigszins ongelijk zijn; het diepste minimum wordt waargenomen als de neutronenster zich tussen ons en de reuzenster bevindt. Behalve de 17 »f

« * r *

0.0 0.4 0.6 0.8 1.0 XRRY PHflSE

Figuur 3: De ellipsoidale helderheidsvariaties van een door getijkrachten vervormde ster. (A) Door de draaiing van de vervormde reuzenster (als gevolg van de baanbeweging van de neutronenster) verandert de positie van de "punt" voortdurend ten opzichte van de waarnemer3 waardoor de grootte van het voor hem zichtbare oppervlak varieert. In de richtingen a en c wordt een helderheidsminimvm en in de richtingen b en d een helderheidsmaximum waargenomen. (B) De helderheidsvariaties van HD153919 (4U1700-37). Alle waarnemingen zijn getekend als functie van de fase, dat is de fractie van de dubbelsterperiode die op het tijdstip van elke waarneming is verlopen sinds de laatste rOntgeneclips; rechts van fase 1.0 en linke van fase 0.0 wordt de tekening herhaald. Duidelijk zichtbaar zijn de maxima bij fase 0.25 en fase 0.75 (overeenkomend met de posities b en d van de waarnemer) en de minima bij fase 0 en 0.5 (posities a en c). 1 l 18 helderheidsvariaties vertoont de reuzenster ook radiële snelheidsveran- deringeri als gevolg van de baanbeweging. Deze komen tot uiting in kleine veranderingen in de golflengte van absorptielijnen in het waargenomen spectrum van de ster. Ook dit is een vorm van het Dopplereffect en kan worden gebruikt om de baansneïheid van de reuzenster te'bepalen. Indertijd werd de ster HD77581 geidentificeerd met de röntgenbron Vela X-l (AU0900-40). De ster bevindt zich binnen de "error box" van de rönt- genbron en vertoont helderheidsveranderingen van ongeveer 10% met een periode van 8.96 dagen, precies de dubbelsterperiode die eerder uit de rontgeneclipswaarnemingen was afgeleid. Bovendien varieert de snelheid van de ster ten opzichte van de aarde met dezelfde periode; de baansneïheid is ongeveer 19 km/s, dus veel kleiner dan die van de neutronenster (270 km/s). Dat komt doordat de reuzenster een veel grotere massa heeft.

MASSABEPALING

Dubbelsterren zijn van buitengewoon groot belang voor het astrofysisch . onderzoek, omdat zij als de enige hemellichamen de mogelijkheid bieden om de massa's van sterren te meten. De massa's van neutronensterren in het bijzonder kunnen alleen worden bepaald als deze sterren deel uit maken, van een dubbelster, zoals bij de röntgendubbelsterren het geval is. Het kennen van de preciese massa's van sterren stelt ons in staat om theoretische berekeningen over de inwendige structuur en de ontwikkeling van sterren, die de laatste jaren door het beschikbaar zijn van snelle rekenautomaten mogelijk werden, op hun juistheid te testen. Deze berekeningen worden uitgevoerd aan de hand van zogenaamde stermodellen, waarin een groot aantal uit de natuurkunde bekende-wetmatigheden zijn verwerkt over het gedrag van straling en materie bij de omstandigheden die in zo'n ster voorkomen (zoals hoge temperaturen en hoge dichtheden). De massa van een ster blijkt de belangrijkste grootheid te zijn die de structuur van de ster vastlegt en de ontwikkeling ervan in de loop van miljoenen jaren bepaalt. Voor een ster waarvan de massa bekend is kunnen met zo'n stermodel waarneembare grootheden worden berekend , zoals de afmetingen van de ster, de temperatuur van het oppervlak en de totale hoeveelheid uitgezonden energie; deze kunnen dan worden vergeleken met de waarnemingen van die ster.

De modellen van neutronensterren voorspellen dat er een grenswaarde I —4 I

bestaat voor de massa van zo'n neutronenster. Dat wil zeggen dat een neutronenster met een massa groter dan die grenswaarde niet stabiel is en door zijn eigen gewicht ineenstort tot een zwart gat (daar is de zwaartekracht zo sterk dat zelfs licht niet meer van de ster kan ontsnappen). De omstandigheden in neutronensterren zijn echter dermate extreem, dat in de modellen verwerkte natuurkundige wetmatigheden vrijwel uitsluitend langs theoretische weg kunnen worden afgeleid, o.a. met behulp van de quan- tum-mechanica. Afhankelijk van de daarbij gekozen uitgangspunten volgt voor de bovengrens aan de massa van een neutronenster bij verschillende modellen een waarde van tussen 1.4 en 2.7 zonnemassa's. Door nu de massa's van nautroiVGristerren te meten kan in beginsel de werkelijke, door de na- tuur bepaalde, grenswaarde worden vastgesteld en kan zo een keuze worden gemaakt welk neutronenstermodel het juiste is. Indirect wordt daardoor ook informatie verkregen over het gedrag van materie bij de in neutronensterren voorkomende temperaturen en dichtheden.

De massa's van de twee componenten in een dubbelster kunnen worden berekend als voor beide sterren de afmetingen bekend zijn van de banen waarin ze om elkaar bewegen. De baan van elke component afzonderlijk wordt vastgesteld door de radiële snelheid van de ster te meten in de loop van de dubbelsterperiode; als gevolg van de baanbeweging verandert deze per riodiek; uit de snelheidsveranderingen kunnen dan zowel de vorm als de grootte van de baan worden afgeleid, mits ook de inclinatie - dat is de hoek die het baanvlak maakt met de hemelbol - bekend is. Deze inclinatie is een onmisbaar gegeven. Uit de radiële snelheidsmetingen wordt alleen de beweging van de ster ten opzichte van de aarde bepaald; we meten alleen de snelheidscomponent langs de gezichtslijn. Veronderstel eens dat we precies loodrecht op het baanvlak kijken: de (baan)beweging van de ster is dan altijd loodrecht op de gezichtslijn, hetgeen betekent dat we geen radiële snelheidsveranderingen zien hoe groot de baansnelheid van de ster ook moge zijn. Bevinden wij ons als waarnemers daarentegen in het baanvlak zelf - weliswaar op grote afstand van de dubbelster - dan zijn de radiële snelheidsvariaties maximaal. Bij een waarneempositie tussen deze twee uitersten nemen we in de vorm van de radiële snelheid de baansnelheid van de ster dus verkleind waar, waarbij de mate van verkleining afhankelijk is van de preciese waarde van de inclinatie. De gemeten radiële snelheden moeten daarom gecorrigeerd worden om de werkelijke baansnelheden te vinden (de factor ertussen is gelijk aan de sinus van de inclinatie).

De radiële snelheidsvariaties van de neutronenster en de reuzenster in n * 20

een röntgendubbelstersysteem worden bepaald met behulp van het Doppler- effect uit de regelmatige veranderingen van respectievelijk de röntgen- pulsperiode en de golflengte van absorptielijnen in het sterspectrum. De inclinatie van het baanvlak kan bij sommige röntgendubbelsterren worden afgeleid uit de duur van de röntgeneclips, maar in de meeste gevallen wordt de inclinatie bepaald uit de grootte van de ellipsóïdale helderheidsvariaties van de reuzenster. Daarvoor geldt namelijk een zelfde soort re- denering als voor de radiële snelheden: als we loodrecht op het baanvlak kijken zien we altijd hetzelfde deel van het steroppervlak waardoor de helderheid van de ster konstant is; bevinden we ons in hef. baanvlak, dan zijn de helderheidsvariaties maximaal. De grootse van de waargenomen el- lipsóïdale helderheidsvariaties kan theoretisch worden berekend o.a. als functie van inclinatie. Door de waarnemingen met deze berekeningen te ver- gelijken kan de inclinatie worden bepaald of kunnen grenzen worden afgeleid voor de waarde ervan. 21

SAMENVATTING

Dit proefschrift beschrijft een fotometrisch en spectroscopisch onderzoek aan de zware röntgendubbelsterren HD77581/4O0900-40 (Vela X-l), HD153919/ 4U1700-37, Skl60/4O0115-74 (SMC X-l) en Wray 977/4U1223-62. Het doel van dit onderzoek was: zo nauwkeurig mogelijk de massa van één of meer neutronensterren te bepalen; voor Vela X-l werd dit doel verwezen- lijkt. Bovendien werd de invloed bestudeerd die de neutronenster uitoefent op de reuzenster in een röntgendubbelster. Het waarnemingsmateriaal werd verzameld tijdens een groot aantal waarneem- perioden tussen 1973 en eind 1976, door medewerkers van het Sterrenkundig Instituut van de Universiteit van Amsterdam en van het Astrofysisch Insti- tuut van de Vrije Universiteit van Brussel; door mijzelf werden zes waar- neemprogramma's uitgevoerd. De fotometrische waarnemingen (helderheidsme- tingen) werden uitgevoerd op de Europese Zuidelijke Sterrenwacht (ESO) te Chili met de 50 cm ESO telescoop en de 50 cm Deense telescoop en op het Leidse Zuidelijke Station (Zuid-Afrika) met de 90 cm telescoop. De spectrogrammen werden opgenomen met de Coudé en Echelle spectrografen van de 152 cm telescoop van de ESO. Met de Faul Coradi volautomatische microphotometer van het Sterrenkundig Instituut te Utrecht werden gedigitiseerde densiteitsregistraties van deze spectra verkregen. De positie van absorptielijnen in de spectra werden gemeten met de Grant comperateur van het Sterrenkundig Laboratorium Kapteyn te Groningen.

Het eerste hoofdstuk is gewijd aan de studie van helderheidsvariaties van de reuzenstarren HD77581, HD153919, Skl60 en Wray 977. Er wordt een computerprogramma beschreven dat ontwikkeld werd om de lichtkrommen van zware röntgendubbelsterren theoretisch te berekenen. Deze berekeningen zijn gebaseerd op het Roche model voor een door rotatie en getijde- krachten vervormde ster; om de bijdrage van ieder zichtbaar oppervlakte element aan de totale helderheid van de ster in een bepaalde fotometrische band te bepalen wordt gebruik gemaakt van de steratmosfeermodellen van Kurucz, Peytremann en Avrett (1974). De resultaten verkregen met dit programma komen goed overeen met die van soortgelijke programma's waarin echter min pf meer ruwe benaderingen voor de modelatmosferen werden gebruikt; deze programma's werden onderling vergeleken door Wickramasinghe en Whelan (1975).

1 22

HD77581 werd in 1975 gedurende 46 nachten waargenomen in het Stfömgren uvby vier-kleuren systeem. De ellipsoïdale heldérheidsvariaties hebben een amplitude van ongeveer 0.10 magnitude (d.w.z. 10% variatie). Van de ene dubbel- steromloop tot de andere komen onregelmatige variaties voor; bovendien zijn de afwijkingen van de gemiddelde lichtkromme vaak gekorreleerd; dit wordt mogelijk veroorzaakt door een opgedrongen pulsatie van HD77581 als gevolg van de excentrische baan van de neutronenster. De gemiddelde lichtkromme van HD77581 werd vergeleken met theoretisch berekende lichtkrommen; het was hierdoor mogelijk om een ondergrens voor de inclinatie van het systeem vast te stellen: deze ligt tussen 70 en 90-. Dit resultaat is van groot belang voor de bepaling van de massa van de neutronenster (hoofdstuk II) en is geheel, in overeenstemming met de waarde van i > 74 die Avni en Bahcall (1975) vinden uit röntgenwaarneminge»-'. Bovendien werd gevonden dat de ster het kritische (Roche) oppervlak vrijwel volledig vult. De waargenomen variatie van dé Ström- gren cl kleurindex is consistent met de modelberekeningen.. De Of ster HD153919 werd waargenomen in het Stromgren systeem en in het Wal- raven VBLUW vijf kleuren systeem. De ster vertoont ellipsoïdale heldérheids- variaties met een amplitude van 0.04 magnitude (d.w.z. 4% variatie). Uit een vergelijking van de gemiddelde lichtkromme met de rontgenwaarnemingen werd gevonden dat het midden van de röntgeneclips niet precies samenvalt met het secundaire helderheidsrainimum, maar een faseverschuiving van 0.06 vertoont.' Het eerste deel van de röntgeneclips lijkt het gevolg te zijn van verduiste- ring door de zeer dichte sterrewind van deze ster, die meer dan 10 zonnemassa's per jaar verliest (Hensberge,1974). Het blijkt evenwel niet mogelijk te zijn de lichtkromme van deze ster bevredigend te verklaren met het Roche mo- • del voor dubbelsterren dat de basis van de theoretische berekeningen vormt. Een verklaring hiervoor kan mogelijk worden gevonden in de zeer hoge temperatuur die heerst in de atmosfeer van HD153919 (35000-40000K); de (Thomson) verstrooiing aan vrije electronen speelt daardoor een belangrijke rol bij het stralingstransport, waardoor de in het Roche-model toegepaste methode om de verdeling van de oppervlaktehelderheid te bepalen minder nauwkeurig is. De Bi la superreus Wray 977 werd door waarnemingen met de SAS-3 kunstmaan (Bradt e.a., 1976) geïdentificeerd met de röntgenbron 4U1223-62. Deze bron is een röfitgenpulsar, maar hij vertoont geen röntgeneclipsen, wat het gevolg is van de kleine inclinatie van het baanvlak (i < 60°). Uit onze fotometrische waarnemingen werd een dubbelsterperiode van 23 dagen afgeleid; dit resultaat werd onlangs bevestigd door Pakull (1978), die een periode van 22.6 dagen vond aan de hand van waarnemingen uit vier achtereenvolgende jaren. 23

Het laatste artikel van dit hoofdstuk beschrijft de analyse van de lichtkromme van Skl60, een 13e magnitude ster in de Kleine Magelhaese Wolk. Qndat zowel de massa' s van de twee componenten als de inclinatie van het baanvlak bekend zijn uit röntgen- en spectroscopische waarnemingen (Primini e.a., 1977; Hutchings e.a., 1977) kon een nauwkeurige theoretische lichtkromme worden berekend. De waargenomen lichtkromme (in het VBLUW systeem) wijkt daarvan echter sterk af; in het bijzonder de amplitude van de helderheidsvariaties is veel groter dan op grond van de massaverhouding en de inclinatie verwacht kan worden. Het bleek evenwel mogelijk te zijn om de lichtkromme te verklaren door aan te nemen dat in dit dubbelstersysteem de neutronenster zich in het middelpunt bevindt van een schijfvormige gaswolk, die ongeveer vijf procent bijdraagt aan de totale helderheid van het dubbelstersysteem. De aanwezigheid van zo 'n gasschijf duidt erop dat de massaoverdracht hier plaatsvindt doordat de reuzenster Skl60 iets buiten zijn kritische Lagrange oppervlak puilt; de uitpuilende materie stroomt naar de neutronenster en vormt eerst een gasschijf alvorens op de neutronenster zelf te vallen.

In het tweede hoofdstuk wordt een uitgebreide studie gepresenteerd van de radiële snelheidsvariaties van HD77581. De baan die de röntgenpulsar Vela X-l (P=283 ) om deze reuzenster beschrijft werd met grote precisie bepaald door Rappaport en McClintock (1975). De nauwkeurigheid van de massabepaling van deze neutronenster is daarom geheel afhankelijk van de nauwkeurigheid waarmee de baan van HD77581 wordt bepaald. Om een juiste interpretatie te kunnen maken van de snelheidsmetingen werd de invioed bestudeerd van de ver- von.iiijg van de reuzenster op de profielen van absorptielijnen in het waargenomen spectrum. Het al eerder genoemde computerprogramma werd uitgebreid om ook (absorptie)lijnprofielen te kunnen berekenen; de bijdrage daaraan van elk oppervlakte elementje van de ster (afhankelijk van de temperatuur en de versnelling van de zwaartekracht ter plaatse) wordt bepaald met de KPA steratmosfeermodellen. Het blijkt uit deze berekeningen dat er aanzienlijke verschillen kunnen bestaan tussen de werkelijke (baan)snelheid van de reuzenster en de snelheid die wordt afgeleid uit de positie van een absorp- tielijn in het spectrum; het theoretisch voorspelde gedrag van spectraallijnen van verschillende ionen (Hel, Oil, SilV) is echter zeer verschillend, waardoor het in principe mogelijk is verstoringen van deze soort te herken- nen. Deze studie leverde voor de massa van de neutronenster Vela X-l een

waarde van 1.74 M ; de massa van HD77581 bedraagt 21.3 M . o 'o In het derde hoofdstuk worden de eerste resultaten gepresenteerd van een gedetailleerde studie van de spectra van Wray 977 en HD77581. 24

Bij de reductie van gedigitiseerde densiteitsregistraties van de Echelle en Coudë spectrogrammen werd gebruik gemaakt van speciaal hiertoe door mij ontwikkelde computerprogramma's. Het is hiermee onder andere mogelijk om een gemiddeld spectrum te berekenen uitgaande van de optelling van een aantal afzonderlijke densiteitsregistraties. Op deze manier wordt een aanzienlijke verbetering van de signaal/ruisverhouding (t.o.v. de individuele spectra) verkregen, waardoor bijvoorbeeld spectraallijnen en de variaties daarin veel duidelijker zichtbaar worden. Gemiddelde spectra van Wray 977 werden gebruikt om een betrouwbare spec- trale classificatie te maken van deze ster (Blla). De spectraallijnen Ho P en H zijn duidelijk variabel en verdienen verdere studie; opvallend in dit spectrum zijn de zeer sterke interstellaire absorptielijnen. Een dergelijke studie van het spectrum van HD77581 leverde als eerste resultaten, interessante variaties in de spectraallijnen H en Hell X4686 A

Literatuur

Avni, Y., Bahcall, J.N. (1975) -. Ap. J. (Letters) 202, L131 Bradt, H.V., Apparao, K.M.V., Clark, G.W., Dower, R., Doxsey, R., Hearn, D.R., Jernigan, J.G., Joss, P.C., Mayer, W., McClintock, J., Walker, F. (1977): Nature 269_, 21 Hensberge, G. (1974): Astron. & Astroph. 36_, 295 Hutchings, J.B., Crampton, D., Cowley, A.P., Osmer, P.S. (1977): Ap. J. 217^, 186 Kurucz, R., Peytremann, E., Avrett, E. (1974): "Blanketed Model Atmos- pheres for Early Type Stars", Washington D.C.: Smithonian Institute Pakull, M. (1978): IAUC 3317 Primini, F., Rappaport, S., Joss, P.C. (1977): Ap. J. 217, 543 Rappaport, S., Joss, P.C., McClintock, J.E. (1976): Ap. J. (Letters) 206_, L103 Wickramasinghe, D.T., Whelan, J. (1975): M.N.R.A.S. 172, 175 25

INTRODUCTION AND SUMMARY

One of the most Important recent developments in astrophysics was the discovery of several neutron stars in close binary systems. The existence of neutron stars had been predicted theoretically by Landau in 1933 and was confirmed by the discovery of the radio pulsars in 1967. These objects, however, - with the sole exception of the recently discovered binary radio pulsar PSR1913+16 (Taylor et al., 1976) - are always single ra- pidly rotating neutron stars and therefore no information about their masses - an important parameter in theoretical neutron star models - can be obtained directly. X-ray observations with UHURU, the first satellite entirely designed for the detection of X-rays from celestial objects, have revealed since its launch in December 1970 several hundreds of X-ray sources; the final "4U"- catalog lists 339 sources (Forman et al., 1978). More than ten of them turned out to be members of close binary systems. The binary character is - in most cases - firmly established from the regular occurrance of X-ray er.lipses. Moreover, in various binary systems the X-ray source is a short period X-ray pulsar and the orbital motion is reflected by small Doppler changes in the observed pulse period. The short time variability (down to milliseconds) as well as the hardness of the X-ray spectrum indicate that the emitting region is very small and hot, and that the X-ray sources are very compact objects, such as neutron stars. The ac- curately known orbital periods (derived from the X-ray eclipse timing) made it possible to identify the optical companions of about a dozen of these sources. These stars usually show variations in brightness or radial velocity - or both - with the same periodicity as the X-ray source.

Tiie optical counterparts in massive X-ray binaries are in most cases massive luminous early-type supergiants. The currently accepted model of such massive X-ray binary systems consists of an optically invisible neutron star orbiting the visible supergiant:. The X-ray source is powered by mass transfer from the normal primary star tb the neutron star either by means of a stellar wind or by Roche-lobe overflow (Davidson and Ostriker, 1973; van den Heuvel, 1975). The extremely compact neutron star constitutes a very deep gravitational potential well and the infalling material gains on its way down an amount of kinetic energy equivalent to some ten percent of its restmass energy. The very strong magnetic field of the neutron star _1

12 (10 Gauss) channels the inflow of the matter towards the magnetic poles. There the kinetic energy is converted into heat which leads to the for- mation of two hot X-ray emitting regions with an area of about one square kilometer each. The pulsed character of the observed X-ray flux is explained "by the rotation of the neutron star: the non-alignment of the magnetic and rotational axes causes for an outside observer a periodically varying aspect of the "hot spots" near the magnetic poles, resulting in the pulsed modula- lation of the observed flux (Davidson and Ostriker, 1973; Lamb, 1977). X-ray binaries are of great importance for astrophysics, because they constitute - sofar - the only possibility to measure the masses of neutron stars. Current theoretical models predict the existence of a limiting mass for these compact objects, above which no stable configuration can exist; a neutron star more massive than this upper limit is expected to continue its collapse, presumably to become a black hole. Depending on the particular choice of the equation of state in various models the derived values of the limiting mass range from 1.4M up to 2,7 M (Canuto, 1977). In order . 0 Q to distinguish between the various equations of state, the real limiting value has to be estimated from observations. Mass determinations of neutron stars can in principle provide us with a firmly established lower boundary, value for the limiting mass. In a binary system the masses of the components are determined by esta- blishing the actual dimensions of both orbits around the common- centre of gravity. This is performed by estimating the inclination of the orbital, plane and by measuring the varying radial velocity (due to orbital motion) of each star individually throughout the binary period. For the neutron star in an X-ray binary this radial-velocity orbit is determined from delay measurements of the X-ray pulse-arrival times; the radial velocity orbit of the normal primary can be estimated in the classical way from the Doppler shifts of absorption lines in its spectrum. Knowledge of the inclination angle is required to correct the radial velocity measurements for the fact that these refer only to the motion with respect to the observer. In most cases the photometric variations of the primary, where possible in combination with the X-ray eclipse durations, are used to determine - or to set constraints on - the value of the inclination. These photometric variations result from the rotational and tidal distortions of the primary by the neutron star. Because in most X-ray binaries the contribution from the neutron star to the total optical luminosity is negligible, these systems also- offer an unique opportunity to study the effects of these dis-

40 27

tortions on the photometric and spectroscopie characteristics* of a binary component.

In this thesis photometric and spectroscopie studies of several optical counterparts of massive X-ray binaries are presented. Subjects of study were the binary systems:HD77581/4U0900-40 (Vela X-l), HD153919/4U1700-37, Wray 977/4U1223-62 and Skl60/4U0115-74 (=SMC X-l). The observational material was collected during many observing runs by several observers from the Astronomical Institute of the University of Amsterdam and from the Astrophysical Institute of the Vrije Universiteit at Brussels. Six observing runs at the European Southern Observatory in Chile were carried out by the author. Strömgren (uvby) four-colour photometric observations .of HD77581, HD153919 and Wray 977 were performed at the European Southern Observatory, using the ESO 50 cm telescope and the Danish 50 cm telescope with four channel photometer of the University of Kopenhagen. Walraven (VBLUW) five colour observations of Skl60 and HD153919 were collected with the 90 cm telescope and five channel photometer at the for- mer Leiden Southern Observatory in South-Africa. Spectrograms of HD77581 and Wray 977. were recorded with the Coudé and Echelle spectrographs of the 1.52 m spectroscopie telescope of the European Southern Observatory. Density tracings of theae spectra were obtained with the Faul Coradi Microphotometer of the Astronomical Institute at Utrecht. The positions of absorption lines in the Coudé spectra were measured with the Grant comparator of the Kapteyn Astronomical Laboratory of the Univer- sity of Groningen.

In the first chapter the photometric variations of HD77581, HD153919, Wray 977 and Skl60 are discussed. A computer program is described that has been developed to calculate theoretical light curves of X-ray binaries. These calculations are based on the Roche model for a tidally and rotationally distorted star; this model is extensively discussed. A set of model atmospheres given by Kurucz, Peytremann and Avrett (1974) was used to estimate the contribution of each visible surface element of the primary to the total luminosity (in a specified photometric passband). The light curves computed with this program agree well with results from similar programs'discussed by Wickramasinghe and Whelan (1975).. I UHtilM V V * W MMUG

The BO.5Ib supergiant HD77581 was observed during 46 nights in 1975. The (uvby) ellipsoidal brightness variations have an amplitude of about 0.1 magnitude. We compared this light curve with theoretically computed curves; it turned out to be possible to set constraints on the inclination of the orbital plane, which is in the range 70 -90 . This result is in agreement with the constraint i > 74° derived by Avni and Bahcall (1975) from X-ray observations. We also found that the primary is on the edge of filling its critical e^uipotential surface. The observed periodic variation of the cl colour index is also consistent with the model calculations. Deviations of individual observations (from the same night) from the mean light curve are often correlated. This may be due to some type of forced pulsation of HD77581, which is subject to the time dependent gravitational attraction of the secondary as a result of the orbital eccentricity. The ellipsoidal brightness variations (uvby and VBLUW), of the 06.5f star HD153919 have an amplitude of about 0.04 magnitude. A comparison of the light curve with X-ray observations revealed a phase shift of about 0.06 between the moment of mid X-ray eclipse and the secondary minimum. This shift is interpreted as due toJ the very dense wind of this star, which is losing more than 10 M /yr (Hensberge, 1974); in fact the first part of the eclipse is due to absorptions of X-rays in the asymmetrically .out- flowing atmosphere. It appears to be impossible to reproduce the mean observed light curve by theoretical model calculations. A possible explana-. tion of this result may be found in the very high temperature (35OOOK - 40000K) in the atmosphere of HD153919. Due to this, Thomson scattering contributes considerably to the total continuum opacity which makes the. use of van Zeipel's law of gravity darkening in the Roche model less accurate. Therefore the light curve of the primary cannot be used to set any constraint on the mass of the X-ray source 4U1700-37.

The Blla supergiant Wray 977 has been the optical candidate star for the X-ray source 4U1223-62 for a long time, but only recently this identification has been confirmed by high precision X-ray observations with SAS-3 (Bradt et al., .1977). Our uvby observations suggest an ellipsoidal light curve with a period of about 23 days. This result has been confirmed recently by Pakull (1978). The final paper of this chapter deals with VBLUW observations of the. BOla supergiant Skl6O. An accurate theoretical light curve could be computed because both the masses of the two components in this X-ray binary and

42 29 the inclination of the orbital plane were known from X-ray- and spectroscopie observations (Primini et al., 1977; Hutchings et al., 1977). The observed light curve strongly deviates from the theoretical one; especially the amplitude of the brightness variations is much larger than expected. However, it turned out to be possible to explain the light curve of Skl60 by assuming the existence of a gaseous disk around the neutron star, which contributes some five percent to the total luminosity of the binary system. The existence of such a disk is a strong indication that the mass transfer from the primary to the neutron star is due to the Roche lobe overflow of Skl60. An extended radial velocity study of HD77581 is presented in the second g chapter. The orbit of the X-ray pulsar, Vela X-l (P = 283 ) was determined with great precision from pulse time delay measurements by Rappaport, Joss and McClintock (1975) . Therefore, the accuracy of the mass determination for this neutron star depends critically on the precision of the estimated radial velocity orbit of HD77581. In order to eliminate non-orbital contributions to the radial velocity variations we investigated the effects of the deformation of the primary on the observed profiles of absorption lines. The light curve synthesis computer program was extended to enable the calculation of absorption line profiles; the contribution to the line profile from each surface element of the primary (dependent on the local temperature and gravity) was calculated by using the KPA model atmospheres. It turned out that large discrepances can exist between the true orbital velocity of the center of gravity of the star and the "velocity" derived from the measured position of an absorption line in the spectrum. Lines of various ions (Hel, OH, SilV), however, have a completely different behaviour, which makes it in principle possible to recognize this type of distortions in the radial velocity orbit. The result of this study was the determination of the mass of Vela X-l which is 1.74 ' M ; + 1~1 0 the mass of the supergiant HD77581 was estimated to be 21.3 ." M , — 1 .8 0 (the errors are 1 o limits). The first results of spectroscopie studies of Wray 977 and HD77581 are presented in the last chapter. The software for the reduction of the di- gitized density registrations of the spectra was written and developed by the author. With these programs it is possible (among other things) to compute averaged spectra from several individual density tracings, which results in a much better signal to noise ratio compared with those of the individual tracings. 30

Averaged spectra of Wray 977 were used to determine the precise spectral classification of the star (Blla) and to study the time variations rfc of Bo and H . The diffuse interstellar absorption lines are very strong P Y in the spectrum, consistent with the reddening, • A similar study of the spectrum of HD77581 reveals interesting variations of H. and Hell \ 4686 A* . p

References

Avni, Y., Bahcall, J.N. (1975): Ap- J. (Letters) 202, L131

Bradt, B.V., Apparao, K.M.V., Clark, G.w., Dower, R., Doxsey, R.f Beam, D.R., Jernigan, J.G., Joss, P.C., Mayer, W., McClintock, J., Walker, F. (1977): Nature 269, 21 Canuto, V. (1977) in: Proc. Eighth Texas Symposium on Relativistic Astrophysics, Ann. N.Y. Acad.Sci. 302^, 514 Davidson, K., Ostriker,J.P. (1973): Ap. J. 17£, 598 Forman, W., Jones, C., Cominsky, L., Julien, P., Murray, S., Peters, G., Tananbaum, H., Giacconi, R. (1978): Ap. J. (Suppl.Series) 38_, 357 Bensberge, G. (1974): Astiyn. & Astroph. 36_, 295 van den Beuvel, E.P.J. (1975): Ap. J. (Letters) 198_, L109 Butchings, J.B., Crampton, D., Cowley, A.P., Osmer, P.S. (1977): Ap. J. 217, 186 Kurucz, R., Peytremann, E., Avrett, E. (1974): "Blanketed Model Atmos- pheres for Early-Type Stars", Washington D.C.: Smithonian Institute Lamb, F.K. (1977) in: Proc. Eighth Texas Symposium on Relativistic Astrophysics, Ann. N.Y. Acad. Sci. 302, 482 Landau, L. (1933): Nature 14a, 333 Pakull, M. (1978): IAUC 3317 Primini, F., Rappaport, S., Joss, P.C. (1977): Ap. J. 217, 543 Rappaport, S., Joss, P.C., McClintock, J.E. (1976): Ap. J. (Letters) 206, L103 Taylor, J.B., Bulse, R.A., Fowler, L.A. (1976): Ap. J. (Letters) 206_, L53 Wickramasinghe, D.T., Whelan, J. (1975): M.N.R.A.S. .172, 175

| (overeenkomend met de posities b en d van de waarnemer) en de minima bij fase O en 0.5 (posities a en o).

ANALYSIS OF PHOTOMETRIC OBSERVATIONS OF MASSIVE X-RAY BINARIES

I .A: THEORETICAL LIGHT CURVES AND RADIAL VELOCITY ORBITS FOR MASSIVE X-RAY BINARIES

1. Introduction

The observed periodic brightness variations, "light curves" of the luminous primaries in massive X-ray binary systems are of the so-called ellipsoidal type (Avni and Bahcall, 1975). Per orbital cycle they show two maxima of approximately equal brightness, separated by two somewhat un- equal minima. These variations arise from the changing aspect (due to binary revolution) of the primary, which is tidally and rotationally distorted by the optically faint companion. The distribution of the observed surface brightness over the primary is determined by gravity and limb darkening, and in some cases by X-ray heating due to the companion. The distortion of the primary is a function of several binary system parameters, such as the separation of the two components, the mass ratio, the mass of the primary and its size. (The first three parameters fix the binary period; the size of the primary is determined by the evolutionary state of the star.) The appearance of the light curve of the primary in an X-ray binary system and in particular the total amplitude of the brightness variations are fixed by the intrinsic system parameters and by the inclination angle under which the system is observed. This physical "picture" allows one to derive - or set constraints on - some of these parameters by comparing the observed light curve with synthetic light curves calculated with a computer model simulation. Theoretical binary light curves can be obtained by calculating the total flux (in a specified photometric passband) emerging from the primary in the direction of the observer, as a function of binary phase. This calculation is in general carried out by constructing a grid of discrete surface elements distributed approximately uniformly over the stellar surface and by subsequently adding the contributions from the different elements. (Of course, those elements which are situated beyond the hori- zon, are excluded.)

Computer calculations of this type have been performed by several au- 34

thorsj an intercomparison of a number of programs is given by Wickrama - singhe and Whelan (1975). In these models the locally emerging intensity and the limb darkening are treated in a more or less approximate way, i.e. by using Planck's law to compute the monochromatic intensity together with a linear limb-darkening law I,(O,y) ^l-u + uu where u is in the range from 0.4 to 0.6 (Avni and Bahcall, 1975; Wilson, 1972), or by using a grey atmospheric model (Strittmatter et al., 1973) or scaled non- grey model atmospheres based on a T = 20000 K and log g=3.0 model (Wick- ramasinghe and Whelan, 1975).

Instead, we use a set of blanketed LTE atmospheric models given by Kurucz et al. (1974): at each surface element the emerging intensity is obtained by interpolation in tables, calculated beforehand from these models. Several aspects of the computational model and the logical organisation of the program will be discussed in the following sections. The LTE model atmospheres have been used also to compute the shape and the strength of several important spectral absorption lines (in the spectra of early-type stars) in the same way: at each surface element the line profile is constructed from precalculated tables and the "observed" profile is derived by adding the local contributions at a number of wavelengths, taking into account the radial velocity of each surface element. In this way the theoretical radial velocity of the primary as a • function of orbital phase can be estimated by analyzing the shape of the apparent theoretical line profile.

2. Basic Assumptions of the Computational Model

2.1 The Shape of the Primary - the Roche Model

The definition of the shape of the primary in our model is based on the classical centrally-condensed Roche model for a tidally and rotationally distorted star in a binary system. The surface of the star is given by one of the Lagrangian equipotentials, a choica which implies the following assumptions:

• - The actual gravitational potential of each of the components can be

"1 35

approximated by that of a point mass; - The primary is considered to be corotating with the orbital angular velocity, while the axis of rotation is perpendicular to the orbital plane; - The relative orbit of the two components is circular.

The effects of deviations from these purely geometrical conditions will be discussed in section 4.2 . Following Kopal (1959, p.125) a rectangular system of coordinates was chosen, corotating with the orbital angular velocity and having its origin at the center of mass of the primary (figure 1). In this reference . frame the X and Y axes are located in the orbital plane. The X-axis poin- ting towards the secondary which is located at a distance a. The Z-axis is perpendicular to the orbital plane.

Figure 1: The corotating reference frame. F and N denote the •positions of the oenter of mass of the primary and of the neutron start respectively. An outline of the surface of the primary star is indicated. jaren.

The equation of motion for a fluid element at position R (x,y,z) in this rotating frame is (Kruszewski, 1966):

2-y * p —+ 2p3L xS| = - + pV¥(x, y, z) (1) dt2 K dt

where r is the position vector of the fluid element, ü^ is the angular velocity of the frame, which is identical with the Keplerian angular velocity, p is the density and P is the total pressure in the fluid element. The potential ¥ is given by ):

~_ .,2 m (2) (

with: x2 + yZ +

= (x - a)2 + y2 + z2

where m and m are the masses of the primary and secondary respectively, r and r represent the distances of R from the centers of gravity of the components, and G denotes the gravitational constant. The first and second terms on the right-hand side of expression (2) represent the gravitational potentials of the two components; the third term arises from the centripetal force produced by the rotation of the frame around its origin, and the last term is due to the acceleration of the origin of the frame itself, which is located at a. distance am /(in +m ) from the axis of re- s p s revolution of the system. Equipotential surfaces defined by the equation:

¥ = constant

are the "Roche equipotentials" or "Lagrangian surfaces". Because the primary is assumed to be corotating with the frame (as a rigid body) both the acceleration d2r/dt2 and the Coriolis force 2 pin x dr/dt vanish and equation (1) reduces to the equation of hydro- is, static equilibrium:

) We adopted the sign convention of Kopal (1959) and Kruszewski (1966) "1 37

VP - = O (3)

It is now assumed that the surface of the deformed primary can be identi- fied with some equlpotential surface defined by:

y = w (4)

This choice will be discussed later on. .

For computational purposes it is convenient to write expression (2) in dimensionless form by adopting the distance of the two components a^ as unit of length and by changing over from rectangular x, y, z coordinates to the spherical polar coordinates r, 6, (j) defined by (cf. figure 1):

x = r cos 9 =r X y = r sin 6 sin = r ]x (5) z = r sin 9 sin $ = r V

The potential in dimensionless form then becomes (Kopal, 1959, p.127):

^+ q (l-2Xr- - qXr + (6)

m m where: Ü = and q = — 2 m (m +m ) Gm p p s' m

Our definition of X, ]i and V is different from the 'one by Kopal, but is more convenient for the construction of the grid of surface elements. For any given value of the mass ratio q the topological properties of the Lagrangian surfaces clearly depend on the value of the dimensionless potential parameter £2; they are extensively discussed by Kopal (1959, 1972) and Kitamura (1970). A special value £2=?Q exists such that equipotentials defined by Ü > Q are closed convex surfaces around each of the two components (for fi»fic they are close to ellipsoids); those with c ti

ized by il = S2c, the "critical lobe" or "Roche lobe", is, if we restrict ourselves to the primary, the (limiting) convex surface enclosing the 1 J

1 l 2 2 3 3 4 4 5 5 6 CC 6 a ÜJ 7 7 2; 8 O 8 9 9 10 10 -1.0 -o.s o.o 0.5 1.5 2.0 -1.0 -0.5 0.0 0.5 1.5 2.0 X Q=.O76 X Q=.O76

Figure 2: The dimensionless potential Q (af. eq.(6)) along the X-axis of the oorotating system of coordinates. The positions of the primary (B), the secondary (S) and the inner Lagrangian point (L) are indicated. The primary can underfill its critical lobe (left) or overflow it (right). A mass ratio q = 0.076 was adopted.

J 39

maximum possible volume around this star. At the intersection of the critical lobe with the X-axis the gradient of the potential vanishes. This "inner Lagrangian point" is a saddlepoint: the potential Q has a local minimum along the X-axis and a local maximum along a line perpendicular to the X-axis. The value Q depends only on the value of q and can easily . be computed by solving the equation:

- q + ( 1 + q ) x O (7) y,z=O ( 1 - x ) '

followed by substituting the thus found dimensionless coordinates of the inner Lagrangian point L ( x, 0, 0) in expression (6).

The surface of the primary is defined in terms of energy," if we identify it with a Lagrangian surface. This is demonstrated in figure 2: the stellar material fills the potential well around the center of gravity of the primary up to some energy level U =-fi . Due to its internal evo- s s lution the'primary may expand, which results in a decrease of Ü ; when ._ . S J2_ becomes equal to Q the star fills its critical lobe. In the case of S ' C further expansion Qsa becomes smaller than Üc and that part of the stellar material below Q can freely reach the secondary: matter is transfered from the primary to the secondary by Roche-lobe overflow through the inner Lagrangian point. Therefore, the critical lobe is the maximum possible volume that can be occupied by the primary while the gas is still energetically bound to the star.

2.2 The Locally Emergent Flux - von Zeipel's "Theorem"

The distribution of the emergent flux over the stellar surface is estimated in our model by using von Zeipel's (1924) gravity-darkening law, which states that the radiation flux in any point of the stellar surface is proportional to the local acceleration of gravity. In short, this can be demonstrated in the following way (cf. Mestel, 1965; Clayton, 1968): For a star in hydrostatic equilibrium the pressure and density at any point in the stellar interior are related through the equation:

VP = (8)

"1 40

where VP is the (local), gradient of the total pressure (Mihalas, 1978, p. 170), p is the density of the stellar material, and the potential gradient V ¥ represents all external forces acting on the fluid. (It is assumed that these forces are conservative.) In the case of a star in a binary system corotating with the orbital angular velocity, ¥ is given by the expression (2). Since the density p is a scalar quantity, the local gradients of P and ¥ are everywhere parallel to each other which means that the pressure P is constant on any given equipotential surface. Therefore, P is a function of ¥ only, so P= P(¥) and, because of equation (8) one will also have: p = p(¥). Given the equation of state this leads to the conclusion that also the gas temperature is a function of ¥ only, thus:

T = T (P,p) = T (¥) (9)

In radiative equilibrium the (radiative) heat flux is given by:

where K = K (p,T) is the mass absorption coefficient and a is the Stefan - dT Boltzman constant. Sxnce the quantities T, p, K and —— are all functions dT of ¥ only, we see that the flux propagating through the stellar interior should be proportional to the magnitude of the local potential gradient. (Starting from equation (10) it can be demonstrated that V.H^O, which is a violation of the condition of radiative equilibrium, and will' cause the development of slow meridional circulation currents. This inconsis- tency in the derivation of the gravity-darkening law is usually ignored since the velocities of the currents are very small and do not influence the hydrostatic equilibrium (Schwarzschild, 1958, p.179; Nestel, 1965).)

In section 4 we discuss the limitations of the assumptions used in our model calculations. 41

3 The Computer Program and Results

3.1 Representation of the Model Atmospheres

A set of 37 LTE model atmospheres given by Kurucz, Peytremann and Avrett (KPA, 1974) was used to estimate the contribution of each surface element of the primary to the total flux emerged into the direction of the observer and also to compute the local contribution to several spectral absorption line profiles. Table 1 lists parameters of the available model atmospheres. The contribution OF per unit area of a surface element on the primary to the total flux in a specified photometric passband is given by:

OF = /r^I^O.U) ydX (ID A.

where y is the cosine of the angle between the surface normal and the line of sight, X is the wavelength, r. is the spectral response function of the passband filter and 1,(0,U) is the specific intensity. The latter quantity is dependent on the local values of the effective temperature T and the acceleration of gravity g. In the case of a narrow passband filter (with a FHMW of 200 8 to 300 A ), such as those of "the Strömgren uvby system, the limb-darkening coefficient L, (y) is effectively constant over the passband; therefore we replace expression (11) by:

OF = ƒ yd A = L^ (U) y ƒ dX (12) X o X

where X is the wavelength at the maximum transmission of the filter, and L°(y) is defined by: V°'y) (13)

This approximation allows us to represent 6 F by using two functions which give:

i) The perpendicularly emerging intensity integrated over the pass-, band (a function of T and g): 6

-| 42

Tabl~ 1; parcaneters of the LTE model atmospheres (KPA) used in the computations

No T log g No T log g eff eff 1 12000 K 2.0 21 25000K 4.5 2 12000 3.0 22 25000 5.0 3 12000 4.0 23 30000 3.5 4 14000 2.0 24 30000 4.0 5 14000 3.0 25 30000 4.3 6 14000 4.0 26 30000 5.0 7 16000 2.0 27 35000 3.5 8 16000 3.0 28 35000 4.0 9 16000 3.5 29 35000 4.5 10 16000 4.0 30 35000 5.0 11 18000 3.0 31 40000 4.0 12 18000 3.5 32 40000 4.5 13 18000 4.0 33 40000 5.0 14 18000 4.5 34 45000 4.5 15 20000 3.0 35 45000 5.0 16 20000 3.5 36 50000 4.5 17 20000 4.0 37 50000 5.0 18 20000 4.5 19 25000 3.5 20 25000 4.0

Table 2: wavelengths and p values for which the' limb darkening ratio I (0,y) / I. (0,1) was computed. A A

A 3000 X X 4000 & ]i = 0.00 0.15 3200 4500 0.01 0.20 3400 5000 0.02 0.40 3500 5500 0.04 0.60 3600 6000 0.05 0.80 - 3700 6500 0.10 1.00 3800 . 7000 43

ƒ r. 1,(0,1) dX X X A

ii) The limb-darkening ratio L. (]i), which is a function of T , g and X.

Integrated perpendicularly emerging intensities were computed for each KPA model atmosphere for the passbands of the Strömgren uvby system and the Walraven VBLUW system. The spectral response functions for the Ström- gren system were adopted from Crawford (1975) and for the Walraven system from Lub and Pel (1977). The value of the integrated intensity at a particular effective temperature and gravity is then obtained by means of a two dimensional logarithmic 6-point interpolation in this table (Abramo- witz, 1970). It should be noticed that the particular choice of the basic points used in the interpolation procedure slightly influences the final magnitude estimates. We also have, in the case of HD77581 (log g-2.8), to extrapolate to gravities somewhat lower than those covered by the KPA models. Therefore, several test calculations were performed for a range of realistic binary system parameters using different sets of basic points for the interpolation between the KPA data. These tests showed that the thus produced variations of the individual magnitudes were of the order of 0.005, but that the amplitude of the light curves was hardly affected. The deviations in this amplitude were always smaller than 0.002. The approximation given by equation (12) might give some trouble for the Walraven V and B passbands, which are rather wide (800ft and 500ft , respectively). Several test calculations using a Planckian continuum intensity distribution and the KPA limb-darkening law (which will be discussed later) showed that the approximation is very good for the small passbands, and that the amplitude of the light curve is not affected. The differential discrepancies for the V and B passbands (due to the filter width) are of the order of 0i001 over a range of effective temperatures of 5000 K ( which is about the temperature variation over the surface of the primary in an X-ray binary ) centered on central temperatures ranging from 15000 K to 35000 K. Therefore, only a minor increase of the errors in the light-curve amplitudes is introduced.

The limb-darkening function L.(y) was derived from the models in the A following way: For each KPA model at 14 wavelength values in the spectral

1 1.0

0.5

1(0.1)

0.0 1.0 0.5 0.0

Figure 3: The definition of the generalized limb darkening function G(\i) (of. eq.(14)). a) The limb darkening law is written as : 1^(0,y) /1^(0,1) =l-f(0.1) G(y). b) The generalized limb darkening function G(]i) for T' = 20000 K. 6 ' 45

range 3000% - 7000 % , and for each wavelength at 12 values of ]i (see table 2), the specific intensity I°(0,y) of the emergent radiation was computed. This grid - 6216 specific intensities - was used to construct a simple computer subroutine to represent the limb-darkening law. We first de-

fined a generalized limb-darkening function GT ,(y) by writing:

(14)

1,(0,1) -I, (0,U) So: (y) A A (15) lA(0,1)-Ix(0,0.1)

The function G - normalized to y = 0.1-turned out to be practically independent of both the gravity g and the wavelength X, to a high degree of

as a stan<3ard accuracy. Therefore, we used the function G _2oooo^ " " generalized limb-darkening function (see figures 3a and 3b). For a given value of y the differences between the "true" limb darkening ratio and the one derived from equation (14) using this standard function, correla- te nicely with the effective temperature and could be represented by simple linear functions of T . -Starting from the standard function, the (temperature dependent) generalized limb darkening ratio is evaluated by applying this (small) correction. The scaling factor f , (0.1) also turned out to be virtually inde- Te,g,X pendent of g, and could be represented easily in terms of a standard function for T = 20000 K (f ,) together with some linear correc- e Te=20000,m nnnnn A tion functions of T .

The limb-darkening function constructed in this way was checked against the KPA data in order to estimate the accuracy. The discrepancies are the

Table 3: Accuracy of the limb darkening representation

y Al^(y)

y i; o.01

largest for values y<0.05 (table 3). In the computation of the total flux, however, only the product yi^(O,y).is used. Therefore, the uncer- tainty in the final result due to this limb darkening representation is always smaller than 0.1 percent, corresponding to one thousandth of a magnitude. The limb-darkening laws at the central wavelength of the four passbands of the Strömgren system are shown in figure 4.

1.0

A: X5500 B: A4680 C:\4100 D:\3450

0.5

G.5 1.0 Figure 4: KPA limb-darkening taws for the Strömgren uvby passbands; T =-25000 K.

In a similar way subroutines were constructed for the calculation of the local contribution of a surface element to the profiles of the absorption lines Hel X4387, Oil A4367 and SilV A4089. The profiles of these lines were computed for each KPA model at 12 \i values (table 3). It is possible to reproduce the line profiles with reasonable accuracy, by using - for each ion individually - several simple semi-empirical interpolation subroutines which give:

i) The central line depth relative to continuum, for perpendicularly emerging intensity R (1), which is a function of T and g, and is c e obtained by two-dimensional six point interpolation between the KPA data stored in the memory.

ii) The perpendicularly emerging continuum intensity I. (0,1) at the computer caicuiat-ions or tms type nave neen periormea oy several au-

central wavelength of the spectral line X . This quantity is also obtained by the two-dimensional interpolation between the KPA data.

iii) The limb-darkening law for the product of continuum intensity and central line depth Lcr ,(y), defined by:

Lcr(y) (16)

This quantity turned out to be function of only y and T The limb-darkening law for the continuum intensity itself L.c (y, ) A c was already available. iiii) A line shape function S(X - X ) depending only on the central line

cr depth Rc(y) = Rc(l) L (y) / L?(y).

The local contribution per unit area of a surface element to the flux at wavelength X at a distance AX = X - X from the line center in the frame c of the observer, is then given by:

OF, (17) where I. .. is given by:

V R (1) (y) S(AX--^ c ~cC

In this expression V is the radial velocity of the surface element with respect to the observer; C denotes the velocity of light. (V is measured negative for a surface element which approaches the observer.) 48

3.2 Logical Organisation of the Program a) Introductory Remarks

The light curve synthesis program is written in Fortran IV; it was developed for use on the CDC 7300 Cyber computer of the SARA (Amsterdam). For obvious reasons the program was designed to use a minimum of central core memory at an optimum computational speed. We adopted the policy to store all those position-dependent quantities needed for each surface element which are independent of the foreshortening factor ]i. This resulted in the use of (5+NC) NP machine words storage in the case of light curve synthesis for NC photometric passbands and NP grid points. The synthesis of absorption line profiles and the computation of radial velocity orbits requires 10 NP words storage. It would have been possible to compute several parameters for each grid point at the moment they are needed; although this approach saves a lot of central memory, it is much more time consuming. The grid parameters are stored in "blank common". In order to ensure program flexibility, all essential operations are performed using a sequence of subroutines, which create, use or modify these data in "blank common". As input for the program one can use binary system parameters such as the mass ratio q, the dimensionless potential Q, the corotation factor (cf. section 4.2), the inclination angle :L, the optical luminosity of the primary. X-ray luminosity of the secondary and some combination of the

following observational quantities: asini, a t sin i, a sini, the period opt X P and/or the mass of the primary M . The proper choice of the input data determines the system completely.

• b) Construction of the Grid of Surface Elements

We constructed the grid of surface elements by dividing the surface of the primary into a number of rings N , using the polar coordinates introduced in section 2.1 . The first and the last rings are small circles centered on the X-axis. Starting from the number of rings the construction proceeds as follows. We define A 0 by:

A6 = ir/ (N - 1 ) (18) 49

Then the area of ring i (with i = 2f ,1*^- 1) is determined by:

where: 0° = (i - 1) A 6

The solid angle subtended by this ring is:

. Qo . Ae (19) iii. = 4 IT sin 6. sin —r-

The solid angle subtended by each polar element is given by:

Ü) = 2 TT ( 1 - COS —T- ) (20) P 2

Each ring is divided into Nj surface elements in such a way that the area of each element on the stellar surface, after correcting for the variation of the radius as a function of 6 and ij) , is as nearly constant as possible. In order to achieve this we used the equation:

b). f 0) N* = int ( -i- x { } + 0.5) (21) i ü) r ( it , 0)

which means that we used the radius of the primary r ( TT , 0) 180 degrees away from the direction to the neutron star to define a unit of surface area. The variation in the size of the individual surface elements obtained in tills way is less than 3 percent. The solid angles to. . of the NT elements are stored, together with the values of the polar coordinates 0. and d>.; these quantities are defined by: 0' ij i

2ir< j- (22) 1=1 •!

(Because of the symmetry of the geometry of the standard Roche model one would expect that only one-fourth of the total number of grid elements equiiiorium:

) We adopted the sign convention of Kopal (1959) and Kruszewski (1966)

have to be stored. However, if one does this, the number of elements in •i/t each ring should be an integer multiple of 4. This would cause the differences in area between the grid elements to increase and would make a somewhat larger number of rings necessary to achieve the same intrinsic accuracy of the program (cf. section 3.3) . One also looses the possibility to introduce asymmetries - such as realistic effects of non-corotation (cf. section 4.2) -in future models.) An impression of the distribution of the grid elements over the stellar surface is shown in figure 5; typical numbers of rings and surface elements are given in table 4.

Table 4: The total number of surface elements N for several numbers N . of rvngs r

Nr N N = E N N r P 1 P

10 135 25 950 15 320 30 1370 20 590 35 1900

c) Computation of the Grid Parameters

For each grid element the surface parameters are computed by numerous calls to a subroutine RADIUS, which solves equation (6) for r. (8. ,((>. ,ïï,q) with the Newton-Raphson method? each computed radius is used as the ini- tial approximation to the next radius. The convergence is such that in -6 one or two iterations the accuracy of 10 is reached (in r). The radius r. of the lobe is returned as the function value of RADIUS; other output of RADIUS is produced in a "named common block" and consists of:

- The surface normal components at the grid point (xn , yn , zn.). •

- The coordinates of the grid point (x., y , z.).

- The length of the potential gradient (g ), which is the local acceleration of gravity apart from a scaling factor due to the use of the dimensionless coordinates. ourselves to the primary, the (limiting) convex surface enclosing the

Figure 5: The grid of surface elements, computed fov N = 25, q= 0.076 and ft = 1.89. (The secondary is indicated with the small dot.)

In order to set the weight w. of each surface element correctly, we com- pensate for the varying radius of the primary and for the small tilt of the element with respect to the. radius vector. Therefore, we replace the values U. (stored in "blank common") by:

w± = to r * / co,s 3. (23)

where B. is the angle between the surface normal and the radius, so we have:

cos 3± = yn ± + z± zn± ) / r L (24)

The distribution of the locally emergent flux, which defines the lo- call.lyy "effective temperature" T , is computed fojr each surface element according to von Zeipel's gravity darkening, law: 52

(25) ei

The constant o£ proportionality a is fixed by prescribing the total luminosity F of the primary:

NP NP (26) where a is the distance between the two components (section 2.1) Therefore, we have:

NP a = F / < a2 E g w ) (27) il from which it follows:

, NP 2 1 ?±/or = {FTg±/(oa S SiV (28)

For use in the light curve synthesis the following quantities are stored: w. (replacing u.), T (replacing 9.), xn. (replacing (j).), yn. and 1 1 © • . 1 1 * 11 zn ; the acceleration of gravity is stored as: log (Gm g. /a2 )(cf. section 2.1). If one wishes to compute line profiles, or to incorporate X- ray heating, also x., y. and z. have to be stored. X-ray heating is treated in our model in a rough way: we use the "deep heating" approach, which implies that the local effective temperature is increased by absorption of X-rays below the photosphere of the primary. * Using an albedo r| we compute the new, increased effective temperature T according to the equation:

CTT* =(JT'+ (1-n ) L q/Ca^J) (29) °i ei . X where Te. is the undisturbed local effective temperature computed from the gravity darkening, L is the luminosity of the x-ray source and r. A 1 and X. are the distance of the X-ray source from the surface element and the angle between the surface normal and the direction to the X-ray source, respectively. Therefore, r. and X- are defined by:

(30) VP

and: cos x^^ = - { ( x± - 1 ) xn± + y± yn± + z± zn±} / r± (31)

Of course, the temperature is replaced only for those surface elements for which cosx. > 0.

d) Light Curve Synthesis

As was pointed out in section 3.1 the local contributions to the total luminosity in a photometric passband is computed from a table of perpendicularly emerging intensities and from the limb-darkening law. We therefore store these perpendicularly emerging intensities for each grid point, as they are only dependent of the local effective temperature (the flux) and the local gravity.. The NC x NP quantities are stored in "blank common", where NC is the number of photometric passbands used: 4 in the case of uvby, and 5 in the case of VBLUW photometry. (Since the limb-darkening function - cf. section 3.1-is independent of the gravity, the value g. for each surface element can be replaced by one of the intensities.) Por a given orbital phase $ (with respect to X-ray eclipse) and an orbital inclination i, the line of sight is defined by the three coordinates:

sx = - sin i cos 2ir$

sy=-sini sin 2TT$ (32)

sz = cos i,

The totally emerging flux in each passband in this direction is given by: NP F = a2 I I (0,y )y w (33) il

where y. is the cosine of the angle between the line of sight and the surface normal of the grid element:

U± = sx± xn± + sy± yn± + szi z^ (34)

and the integrated specific intensity I (0,y.) for each grid element is given by:

I (0,y ) =L°(y ).l (0,1) (35) 1 1 A X X J

For every phase $, NP values of y are computed while the limb-darkening function L°(y) has to be evaluated roughly NC*NP/2 times. A.

e) Line Profile Synthesis

As for the light curve synthesis we store all those quantities that are independent of the orbital phase $ and the inclination i. The velocity components in terms of wavelength shifts Xx., Xy. and Xz. are defined by: X , o. (36)

Xz.

where X is the central wavelength of the line, P is the orbital period in seconds and C is the velocity of light. (The third equation implies that it is not necessary to store z..) The quantities 1,(1,0) and R (1), which are the perpendicularly emer- \> c gent intensity at wavelength X and the relative central depth of the ab— o sorption line, respectively (cf. section 3.1) are computed from the KPA tables in the memory by means of the six-point interpolation, and are stored. (Because both limb-darkening functions L° and LCr are independent " i of the gravity, the values g. are replaced in the memory by R (1).) For a given line of sight (equations (32)) and a prescribed wavelength X = X + AX in the frame of the observer, the contribution to the totally observed flux at wavelength X is given by: S(A X ) (37) ""X +AX l R (1) ok c V" S

where vi is given by equation (34) and the wavelength shift due to the velocity of the surface element X is given by:

X = Xx. sx. + Xy. sy. + Xz. sz. (38) s 1 i i i i l

The profile of an absorption line is computed by evaluating the quantity F , defined by: NP . Wi (39) x=l xkJ

P±>0 55

for a number of wavelength shifts AX^. In practise, we calculated P for 60 equally spaced wavelength values (AX = 0.2A). For each line pro- c file NP values of u have to be computed; the limb-darkening functions L^ and L°r have to be evaluated roughly -r- times each and the line shape function S has to be evaluated roughly —-— times, where K is the number of wavelength points used. The theoretically computed line profiles can be used to estimate the "radial velocity" of the line, by computing the apparent wavelength of the line. We used for this apparent wavelength the first moment of the profile, defined by:

X = ƒ X D, d X / ƒ D.. d X (40) a X X where D. is the depth of the line as a function of wavelength. It is not fully certain whether this is indeed the quantity that we observe when measuring line positions with a comparator, and it would be interesting to repeat the computations using other definitions of the apparent wavelength of the line, for example, a realistic simulation of the "Grant comparator" method.

3.3 Stability and Accuracy of the Results

In the figures 6 and 7 are presented a theoretically computed light curve and a theoretical radial velocity curve for the system Vela X-l. Apart from the model dependent intrinsic approximations in the computational procedure, the precision of the computations is mainly determined by the accuracy with which the surface of the primary can be represented by the grid of surface elements. We expect that with an increasing number of elements, the computational results derived with the program converge towards an asymptotically "true" value. In order to check this expectation and to have an estimate of the errors introduced by the f ini- teness of the number of surface elements, the following tests were performed:

We considered N values VVr computed with the program using N rings for the construction of the grid. In the case of light curve synthesis we used a series of computed y magnitudes at N phase angles, equally spaced v 56

.f 0 v y A \ / \ . 1 q=0.076 \ / 0.05 \ / iUi.89 \ / - - \ / \ / M=20.3M« \_y \ / L= 1.2 1039 erg/s \^y P= 8.966 o i = 90* mo - d i l *^"*-—. ' ml

- 1 1 1 1 1 ill 0.5 1.0 PHASE

Figure 6: Synthetic Light Curve for the y passband of the StrOmgren photometrie system and theoretical Color Indices cl are. ml. The parameters indicated are those of the Vela X-l system. Corotation of the primary and a circular orbit of the secondary are assumed. The variations of the theoretical el index with orbital phase are due to the variation of the gravity over the surface of the primary. This predicted variation is observed

in the Vela X-l system (cf. Zuideruyigk et al.3 1977-, paper IC of this thesis) 1 1 1 1 1 1 | i . i i 1 i

10 - , f.

o -

/ Hel 4387 / q=0.076 -10 / 1=90* \ /

-20 il i i i i lii i i 1 0.5 1.0 PHASE

Figure 7: Synthetic Line Profiles for Eel 438? 8. (left)3 and the corresponding "First Moment" Radial Velocity Orbit (right). Notice the asymmetry in the profiles around phase 0.25 and phase 0.75. System parameters and internal assumptions are the same as for figure 6. J

• • . J - 58

over the binary period? in the case of line profile synthesis we used the relative depth (compared with the' continuum level) of the absorption line profiles at a large number of wavelengths (typically 60). Also the final- ly derived radial velocity was considered at a number of phase angles equally spaced over the binary period. We defined the quantity a(M,N) as follows: N

CT(M,N) ={• (41) N V

The quantity a(N ,N -1) gives an estimate of the differences between the results using N -1 rings and those using N rings; c(N ,36) indicates the differences between the results computed with N rings and those computed with 36 (a very large number) rings. These two quantities were computed for values of N between 10 and 35, using several sets of realistic X-ray binary system parameters as input of the program. It turned out that with increasing number of rings N__ the quantity a(N ,36) converges more systematically (but slower) than a(N ,N -1). Therefore o(N ,36) gives a more realistic estimate of the accuracy of the results. From table 5 it can be seen that the computational errors in the results for a reasonable number of rings used - 25 for light curve synthesis and 36 • for line profile synthesis - are smaller than, or comparable with, the errors introduced by the representation of the KPA model atmospheres (cf. section 3.1). Therefore, we conclude that the internal accuracy of the computed amplitudes of X-ray binary light curves is of the order of 0m002, which corresponds to 2 percent in the case of HD77581 (Am=0.10).

This implies that any discrepancies that are.found between the observed and theoretically computed light curves will not be due to computational inaccuracies, but must be intrinsic. Therefore, comparison of observed X-ray binary light curves with synthetic light curves provides us with information about the validity of, and deviations from the standard model (section 2). In the next section we will discuss the limitations of various assumptions underlying the standard model., and qualitatively discuss the deviations that might be expected between our computed curves and real,-life X-ray binary light curves.

| U.i>U <• ]i S l.UU <: 0.1 %

Table 5; Accuracy of the computational results

measured quantity number of rings

V^=y( -y N 1 < 0.01 magnitude V (V r Ü l

With: N.r > 25 < 0.0010 magnitude

. _i2i ... N > 30 < 0.0005 magnitude V r

P F vN. 60- i V± ~V P60 where: (N =60) N > 20 < 0.001 r X. =X + (i-30) 0.2 N > 32 < 0.0005 1 O with: i=l,...,60

v r-v C4 } a o i ~ rad 'i ~ X C N > 10 < 1.0 km/s o with: N > 15 < 0.5 km/s r N > 31 < 0.1 km/s V r

Note to table 5: . indicates the orbital phase.

4. Limitations of the Assumptions used in the Computational Model

4.1 The Definition of the Stellar Surface

In this section our computational mciel, as presented in section 2, is discussed, still under the assumption that the primary rotates synchronously with the orbital angular velocity and that the orbit of the secondary is circular.

a) Effects of Extended Atmospheres

For stars with extended atmospheres some difficulty may arise regar- ding the- definition of the surface level in the primary that we observe photometrically (i.e. the level with continuum optical depth T =1). "1 Ac

For example, let us consider the case of a primary which is slightly overfilling its critical lobe and transfers matter towards the secondary. It is in principle possible that while the total mass fraction above the energy level -Q (cf. section 2.1) is sufficient to maintain the Roche- lobe overflow, the level surface with continuum optical depth x =1 may at the same time be located considerably inside the critical lobe. This situation may occur in massive X-ray binaries, where the primaries are in most cases hot luminous supergiants with extended and expanding atmospheres. In such stars the distance between the levels x =0 and x =1 may be a considerable fraction of the stellar radius. For instance, Kunasz et al. (1975.) present for a 30 M low gravity model (log g =3.2; M. .= -9) the following parameters computed with a non-LTE extended atmosphere mo-

del: Te=29800K, = 22.5R@ and R /R =1.28. Although this particular example may be an extreme case, their results indicate that a distance between the levels X =0 and x =1 of about 10 percent of the stellar radius might very well be possible in the atmospheres of hot luminous supergiants. When the primary is a normal main- sequence star, the distance between these two levels is a much smaller fraction of the stellar radius, i.e. smaller than 1 percent, which can for example be seen from the data given by Allen (1973, p.213). Such an extended atmosphere will have the following effect on the apparent light curve of a massive X-ray binary. If we assume that the star is filling the potential well up to ti , then the photometrically obser- s ved surface level can be written as:

(42)

where 6(0,) can be of the order 0.05 to 0.10, corresponding roughly to 5 to 10 percent of the stellar radius. If 6 has a constant value over the surface of the primary, the photometrically observed level is again a Lagrangian equipotential surface, but with a higher value of Ü. In general S depends on the position at the surface, as the opacity of the stellar material is dependent of the gravity en the temperature. If, however, the variation of 6 is small compared with the value of 5 itself, the photometrically observed level is close to a Lagrangian surface and the resulting deviations in the light curve can be considered as higher order 61

effects. This is certainly a good assumption for main-sequence primaries, and probably still a reasonable one for B-type supergiants.

b) Applicability of von Zeipel's Gravity Darkening Law-, the Thermalisation Condition

The derivation of von Zeipel's gravity-darkening law, as presented in section 2.2, is in fact only allowed under a more stringent condition than those of hydrostatic and radiative equilibrium alone, for the following reasons: The use of the equation of hydrostatic equilibrium (8) implies that the total pressure P is treated as an isotropic hydrostatic pressure, which is a scalar quantity. Part of the pressure, however, is due to the radiation pressure, which.has a tensor character (Mihalas, 1978, p.12). Therefore, the application of equation (8) is only valid if the radiation pressure tensor degenerates into a scalar (i.e. if the elements outside the diagonal vanish). This condition is satisfied when we are dealing with a (nearly) isotropic radiation field. The same condition applies for the validity of.equation (10). This is basically a diffusion equation which states that the radiation flux is driven by a small anisotropy in the local radiation field, described by the temperature T. Furthermore, the application of the equation of state (9) to show that T is a function of the potential ¥ only, followed by the use of this temperature in equation (10), is only allowed when the temperatures of the gas and the radiation field are identical. We conclude that the gravity darkening law is only valid if we restrict ourselves to that part of the stellar interior where the radiation is thermalized, which is a stronger restriction than the condition of radiative equilibrium alone. The above derived result can easily be understood intuitively, as follows. The physical mechanism of gravity darkening requires some interaction between the external force field, described by the potential Y, and the radiation; the radiation must "know" about the potential field, and the only way it can do so is through absorption and emission processes in the stellar material. If we now follow the radiation flux from the central parts of the star towards the stellar "surface", we will reach a level above which an increasing fraction e of the photons can escape di-

"1 1 (18)

rectly into interstellar space, without further interaction with the gas. From this point on the information link between the radiation field and the force field becomes less and less efficient and, as e approaches uni- ty, it. disappears completely. If we are dealing with a stellar atmosphere where scattering processes give a negligible contribution to the total continuum opacity, which is the case for spectral types later than BO, <"hen the radiation field can be considered as being thermalized when the continuum optical depth is larger than about 1. In this case the surface level that we observe photometrically is also the limiting level were we can apply von Zeipel's gravity-darkening law. In very hot stars, however, (T >3.010l>K) where electron scattering contributes considerably to the total continuum opacity, the thermalization level can be situated far below the T ^1 level. R In this case the distribution of the surface brightness is determined at a level in star that is situated considerably deeper than the photometrically observed surface. a) Canelusions

If we restrict ourselves to binary systems with a corotating primary and a circular orbit, we can summarize the above given discussion as follows:

- In the primary, three different levels are of importance: a) The level that defines the boundary of the star energetically; this level is given by a Lagrangian equipotential; b) The "surface" level that we observe photometrically; c) The thermalization level, below which von Zeipel's "theorem" can be applied to compute the distribution of the' emergent flux. In the case of primaries of spectral type later than BO the photometric and thermalization levels coincide.

- In main-sequence primaries the photometrically observed surface level coincides to a high degree of accuracy with a Lagrangian equipotential} the distance between this surface and the boundary of the star is negligible compared with the radius of the star, due to the small scale • height (Allen, 1973). The Roche model gives an accurate description of the primary in this case. 63

- For supergiants with extended atmospheres the approximation of the photometrically observed surface by an equipotential surface is considered to be permissible if the factor 6, defined in equation (42), is less than a few times 0.01. The latter is most likely the case in the primaries of Cyg X-l (HDE 226868) , Vela X-l (HD 77581) and 401538-52, which are all early B-type supergiants. For the primaries of very early spectral type, like in 401700-37 (HD 153919) and Cen X-3, larger deviations from an equipotential surface can be expected. In these stars the thermalisation level will be located deeper then the photometrically observed surface.

The effects of extended atmospheres and NLTE effects on the light curves of X-ray binaries with a supergiant primary will be such that the level Q , to which the primary fills its potential well, can be underestimated if one derives it from the apparent amplitude of the brightness variations. The discrepancy will be the largest for O-type primaries. Even if such a primary fills its critical lobe completely, the amplitude of the light curve will suggest serious underfilling.

4.2 Further Effects that may Produce Light Curve Deviations a) The Point-Mass Approximation

The density distribution in the distorted primary and the gravitational potential mutually infuence each other, which might result in deviations of the equipotential surfaces from those computed with the centrally condensed model. Calculations based on the analysis of a tidally and rotationally distorted polytropic stellar model by Martin (1970), however, show that the shape of the equipotential surfaces is virtually independent of the run of the internal density. The deviations from the Roche model are of the order of 10"3 for a model with poly tropic index n=3, and become smaller with increasing value of n (which is as expected,- because then the central concentration is higher). As the polytropic index of a supergiant primary is somewhat larger than 3 (see for example the model given by Schwarzschild, 1958, p.258) these deviations can be considered negligible (see also Kruszewski, 1966) the dimensionless coordinates.

b) Deviations from Covotaiion

When the primary is not corotating with the orbital angular velocity, or if the orbital and rotational axes are rot aligned, the determination of its shape becomes a dynamical problem because the star is moving with respect to the corotating frame, and the position of the secondary with respect to any fluid element in the primary is time dependent. This produces a tidal bulge on the primary, which will probably be stationary in the corotating frame. A possible way to treat this situation mathemati- cally is to construct a rotating stellar model for the primary in which the motion of every fluid element satisfies equation (1), and search for a time-independent density distribution (if such exists) in the frame corotating with the orbital angular velocity. This type of computations is very complicated due to the presence of the Coriolis force, which cannot be derived from a potential; it is not possible to apply a transformation to some reference frame, such that this force vanishes. An approximate treatment is given by Kruszewski (1966): he considers a £» H» ? frame fixed to the primary and rotating with angular velocity til. The equation óf motion in this frame is, in analogy with (1) :

f + 2pu x -£= - n, ?, t) (43)

dt2 • dt

where the time dependent potential is given by:

Gm Gm m C, t) = (44) S

If the rotation axis of the primary is perpendicular to the orbital plane, we have the following relations between the £, n, C frame and the X, y, 2 frame: ~ x2+ y2

(45)

while:

where f is the corotation factor. The potential ¥ is time dependent in the frame fixed with the primary, but time independent in the frame rotating with angular velocity ID . K according to von Zeipel's gravity darkening, law;

• In this frame ¥ can be written in dimensionless form:

a- i (46)

This equation was used by Wickramasinghe (1975) and Avni and Bahcall (1975) to study the effects of non-synchronous rotation of the primary on the light curves of X-ray binaries, assuming that the star adjusts it-

i self instantaneously to the shape of the equipotential surfaces of ¥ . This assumption is commonly referred to as "instantaneous hydrostatic equilibrium" (IHE) and is basically identical with the condition that in the frame of the primary the equation of motion (43) reduces to its hydrostatic form: VP = pVf' (47)

This assumption cannot be satisfied in the case of non-synchronous rotation (Kruszewski, 1966) • In that case the equipotential surfaces of ¥ are time dependent in the frame fixed inside the primary, which means that if IHE is satisfied, large-scale velocity fields must exist in the star: therefore equation (47) cannot be satisfied. In this approach the Coriolis force is ignored in the frame corotating with the primary. This is also reflected in the fact that ¥ is symmetrical with respect to the X2 plane, which means that a trailing tidal bulge is a priori impossible. Also the hydrostatic equilibrium condition cannot be satisfied in an a- synchronously rotating primary (Avni and Bahcall, 1975). It is not known how this affects the gravity darkening and thus the emergent flux. Although the assumption of IHE can only be considered as a rough first approximation, equation (46) produces some qualitative results that could also be inferred intuitively, such as the increase of the equatorial diameter of the primary (compared with the polar diameter) with increasing value of the corotation factor f . Therefore, these results can be consi- c dered as an indication of the effects of deviations from corotation on the light curves. It can be seen from the tables given by Wickramasinghe (1975) that for HD77581 (f ^ 0.7; Mikkelson and Wallerstein, 1974) the c amplitude of the light curve decreases by about 0.005 magnitude. (30)

c) Eccentric Orbits

In two X-ray binaries, Vela X-l and Cyg X-l, the eccentricity of the orbit is well established (van Paradijs et al., 1977; Bolton, 1975). The effects of an eccentric orbit on the shape of the primary are hard to estimate; no theoretical calculations are available in the literature. The assumption of IHE is certainly not valid for the same reason as in the case of non-synchronous rotation. Computations using IHE would there-1 fore presumably be as inaccurate as those based on the standard model. It is reasonable to assume that the primary will exhibit some type of forced pulsation (not necessarily radial), due to the periodically varying gravitational potential to which the star is subjected. The correlated deviations from the mean light curve of HD77581 may perhaps be interpreted in this wayj the observations suggest that-a quasi-periodic variation with a period of about 1.6 days is present in the uvby light curve of this star (cf. Zuiderwijk et al., 1977).

d) Conclusions

In table 6 the known deviations from the standard model are indicated for the best studied massive X-ray binaries. The table shows that the Roche model should be-considered as a rather rough first approximation, and that the analysis of observed light curves by means of comparison with theoretical ones can be subject to considerable model-dependent errors, although it is possible to estimate at least qualitatively the influence of deviations from corotation and of the effects of extended atmospheres. The system HD77581/4U0900-40 is illustrative in this respect: because both deviations tend to decrease the amplitude of the light curve we can conclude that the lower boundary value for the inclination angle of the o • orbital plane (70 ), obtained from light curve analysis, is not affected. The fact that the variation of the cl index is reproduced at least qualitatively indicates that the standard model seems to be a reasonable approximation for this system. Table 6; Deviations from the Standard Model in Massive X-ray Binaries

Binary System Spectral Effects of Extended Corotation Circular Other Complications Type Atmospheres Orbit

2 HD77561 (Vela X-l) B0.5lb minor no no HD153919 06-5f important ? no3 (?) Strong Stellar Wind 4 O Skl60 (SMC X-l) B0.5la minor yes yes x-ray Heating / Disk

Wray 977 B1I minor ? ? II 6 7 Cen X-3 06III important no yes 5 8 HDE226868 (Cyg X-l) BOI minor yes no o 9 LMC X-4 08V-III moderate ? ? Disk H

4U1538-52 BOIa minor ? ?

Notes to table 6: 1) Wallerstein (1974) 6) Conti (1978) 2) van Paradijs et al. (1977) 7) Pabbiano and Schreier (1977) 3) Hammerschlag Hensberge (1978) 8) Bolton (1975) 4) Hutchings et al. (1977) 9) Chevalier and Ilovaiski (1977) 5) Primini et al. (1976)

ui UI 68

References

Abramowitz, M.,- Stegun, I.A. (1970):"Handbook of Mathematical Functions", "Handbook of Mathematical Functions", Dover Publ. (New York)

Allen, C.W. (1973): "Astro-physical Quantities", The Athlone Press (London)

Avni, Y., Bahcall, J.N. (1975): Astrophys.J. 197_, 675

Bahcall, J.N. (1978): Ann. Rev. Astron.Astrophys. 16, 241

Bolton, C.T. (1975): Astrophys.J. 200, 269

Chevalier, C., Ilovaisky, S.A. (1977): Astron.Astrophys. 59_, L9

Clayton, D.D. (1968): "Principles of Stellar Evolution and Nucleo- synthesis", Mac Graw Hill (New York)

Conti, P.S. (1978): Astron.Astrophys. 63_, 225

Crawford, D.L. (1975): Astron.J. 75_, 977

Fabbiano, G., Schreier, E. (1977): Astrophys.J. 214, 235

Hammerschlag-Heiisberge, G. (1978): Astron.Astrophys. 64_, 399

Hutchings, J.B., Crampton, D., Cowley, A.P., Osmer, P. (1977): Astrophys.J. 217_, 186.

Kitamura, M. (1970): Astrophys.Space Sci., 7_, 272

Kopal, Z. (1959): "Close Binary Systems", Ch.III, Wiley (New York)

Kopal, Z. (1972): Adv.Astron.Astrophys. 9, 1

Kruszewski, A. (1966): Adv.Astron.Astrophys., 3_, 89

Kunacz, P.B., Hummer, D.G., Mihalas, D. (1975): Astrophys*J. 202, 92

Kurucz, R., Peytremann, E., Avrett, E.H. (1974): "Blanketed Model Atmos- pheres for Early-Type Stars", Smithonian Astrophysical Institute, Washington, D.C.

Lub, J., Pel, J.w. (1977): Astron.Astrophys. 5£, 137

Martin, P.G. (1970): Astrophys.Space Sci., 7_, 119

Mestel, L. (1965): in "Stellar Structure", ed. L.H. Aller and D.B. McLaughlin (Chicago: University of Chicago Press) I. 69

Mihalas, D. (1978): "stellar Atmospheres" (Freeman, San Francisco)

Mikkelson, D.R., Wallerstein, G. (1974): Astrophys.J. 194, 459 van Paradijs, J., Zuiderwijk, E.J., Takens, R.J., Hammerschlag-Hensberge, G., van den Heuvel, E.P.J., de Loore, C. (1977): Astron.Astrophys.Suppl. 3£, 195

Primini, F., Joss, P.C., Rappaport, S. (1977): Astrophys.J. 217, 543

Schwarzschild, M. (1958): "Structure and Evolution of the Stars" Dover Pub. (New York)

Strittmatter, P.A., Scott, J., Whelan, J., Wickramasinghe, D.T., Woolf, N.J. (1973): Astron.Astrophys. 25_, 275

Wallerstein, G. (1974): Astrophys.J. 194, 451

Wickramasinghe, D.T. (1975): M.N.R.A.S. 173_, 21

Wickramasinghe, D.T... Whelan, J. (1975): M.N.R.A.S. 172^, 175

Wilson, R.E. (1972): Astrophys.J. (Letters) 174_, L27

von Zeipel, H- (1924): M.N.R.A.S. 8£, 665

Zuiderwijk, E.J., Hammerschlag-Hensberge, G., van Paradijs, J., Sterken, C, Hensberge, H. (1977): Astron.Astrophys. 54, 167 70

1977, Astron. Astrophys. Suppl. 27,433-434.

IB FOUR-COLOUR PHOTOMETRIC OBSERVATIONS OF THE JT-RAY BINARY STAR HD 77581 (VELA X-l) I. OBSERVATIONS

E.J. ZUIDERWUK, G. HAMMERSCHLAG-HENSBERGE, J. VAN PARADIJS Astronomical Institute, University of Amsterdam, The Netherlands and European Southern Observatory, La Silla, Chile C. STERKEN and H. HENSBERGE Astrophysical Institute, Vrije Universiteit, Brussels, Belgium European Southern Observatory, La Silla, Chile

Received June 1, 1976

Results or the uvby observations of HD 77S81, used in the analysis of the light curves (Zuiderwijk et al. 1976) are presented.

Key ironic.- Jf-ray binaries- uvby photometry

We have observed the BO.S Ib supergiant HD 77S81, the optical counterpart of the X-ray binary system 3U 0900-40 (Vela X-l). A discussion of the data presented here is given in a companion paper in the Main Journal (Zuiderwijk el al. 1976, paper II). The observations were made with the 50 cm Danish telescope on La Silla, Chile, with a four-channel photometer designed for simultaneous measurements in the Strömgren uvby system. The photometer is used in combination with a photon counting system, and is described in detail by Granbech et al. (1976). HD 77581 was observed on 46 nights during the periods February 16 to March 14 (by E.J.Z. and G.H.-H.), May 27 to June 8 (by E.J.Z.) and December 3 to December 21, 1975 (by C.S.). The most extensive observations were made during the first period. Table 1 lists the 429 magnitude differences which were obtained for HD 77581 with respect to the comparison star HR 3656 (B8). The table is ordered according to JD. Each tabulated observation is the result of three integrations of 20 sec. for HD 77581 (P) and two integrations of 20 sec. for the comparison star (S). The typical standard deviation of such a sequence (one observation in table I) is 0.005 mag. All observations of magnitude differences are given in the instrumental system, as an accurate transformation of reddened supergiants to the standard uvby system is very difficult (E(B-V) =0.13 for HD 77581, see puper II). Especially the indices c\ and ml are highly affected by the reddening. Acknowledgements are given in paper II.

REFERENCES

Cirunbcch. B.. Olsen. E.H. and Slriimgrcn. B.: 1976. Aslron. Astrophys. Suppl. 26, 15S. Zuiderwijk. E.J.. Hammcrschlag-Hcnshcrgc. G.. Paradijs, J. van. Sterken. C. and Hensberge, H.: 1976, Astron. Astrophys. (paper II).

E.J. Zuiderwijk Astronomical Institute G. Hammcrschkig-Hcnshcrgc University of Amsterdam J. van Paradijs Roetersstraat 15 N - 1004 Amsterdam, The Netherlands

C. Sterken - Astrophysical Institute H. Hensbergc Vrije Universiteit Brussel Pleinlaan 2 B - 10S0 Brussel. Belgium

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ASTRONOMY Astron. Aslrophys. 54,167-173 (1977) AND ASTROPHYSICS 1C Four Colour Photometric Observations of the X-ray Binary Star HD 77581 (Vela X-l), II: Analysis of the Light Curve

E. J. Zuiderwijk1, G. Hammerschlag-Hensberge1, J- van Paradijs1, C, Sterken2 and H. Hensberge2 1 Astronomical Institute, University of Amsterdam. Roetersstraat 15,1004 Amsterdam, The Netherlands and European Southern Observatory, La Silla, Chile 2 Astrophysical Institute, Vrije Universiteit, Pleinlaan 2, 1050 Brussels, Belgium and European Southern Observatory, La Silla, Chile

Received June 1,1976

Summary. Extended photometric observations of HD tions of HD 77581 in the Strömgren uvby system. 77581, the optical counterpart of the X-ray source Our observations in this intermediate band system Vela X-l, in the Strömgren four colour system are reveal light and colour changes which are in phase discussed. These observations show changes in the with the orbital period. Colour-index changes were not light curve from one orbital period to another. A regular noticed in the UBV data obtained by Jones and Liller double-wave variation of the colour index cl (and (1973) and Vidal (1974). possibly b—y) is observed. These observations are In the following sections we discuss these observa- consistent with light and colour variations predicted tions and give a preliminary analysis using a theoretical from a model of a tidally distorted, rotating primary, model of a tidally distorted rotating star. These light using orbital parameters derived from spectroscopie and colour variations can be understood from the observations. temperature and gravity variations on the surface of Key words: X-ray binaries — uvby photometry the distorted primary.

2. Observations I. Introduction The observations of HD 77581 weie made on 46 nights during several observing runs in 1975 with the 50 cm The B0.5Ib supergiant HD 77581 is the optical counter- Danish telescope at the ESO, Chile. The results are part or the X-ray binary source 3 U0 900-40 (Vela X-l). presented in tabular form in a separate paper (Zuider- X-ray observations show an eclipsing light curve with a period of 8.95 days (Forman et al., 1973). The optical wijk et al., 1976, paper I). In Figure 1 we show the uvby identification was first achieved by photometric observa- magnitudes of HD 77581 during the first observing tions, and confirmed by spectroscopie observations run, as a function of JD. A double wave in phase with (Hiltner et al, 1972: Jones and Liller, 1973; Vidal et al., the binary period of 8.966 days is clearly visible. [This 1973: Zuiderwijk et al.. 1974). These observations orbital period has been derived from the radial velocity showed that HD 77581 is a member of a binary system variations of HD 77581 during a time interval of 17 with a period of 8.96 days. Recently a regular X-ray years, cf. van Paradijs et al. (1976).] It is apparent that pulse with a period of 283 s was discovered with the the minima occurring at X-ray eclipse time (=JD SAS-3 satellite (Rappaport and McClintock, 1975: 2442468.7+nx 8.966, « integer) are less pronounced McCliniock et al.. 1976). The modulation of the pulse than those occurring half a period later. We also arrival times by the Doppler effect, together with the notice the disappearance of a maximum twice during optical radial velocity curve of the supergiant enabled our observations. A similar phenomenon has also been for the first time an accurate mass determination for found by Jones and Liiler (1973) in their UBV light both the compact object and the supergiant in the curves, although it should be noted, that their missing system (van Paradijs et al.. 1976: Rappaport et al., maximum occurs after X-ray eclipse (

168 E. J. Zuidetwijk et al.: Photometry of HD 77581 (Vela X-l) I \ l