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Some aspects of the evolutionary history of X-ray double stars

Sutantyo, W.

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Download date: 07 Dec 2020

SOME ASPEC'rS 01" 'PIlE F~VO"LU'l'IONJ\llY

HISTORY OF X - RAY nOUBLE STARS

AGADE~HSCH PROEFSCHHIFT

tar Verkrlj van de graBd van doctor in de Wiskunde en NBtuur­wetenschappen aan de Universiteit van Amsterdam, op gezng van de Rector MagnificuB, Dr A. de Froe, hoogleraar in de Faculteit dar Geneeekunde, in het openbaar to verdedigen in de aula dar Univer­6iteit (tijdelijk in de LutherSB Kerk, ingang Si 411, hoek Spui) op woensdag 1 oktober 1975

des namiddags te 13.30 uur

door

WINARD! SUTANTYO

geboren te Solo

PROMOTOR: PROF. DR. E:. P. J, VAN DEN HEUVEL,

CO-RE~ERE:NT: PROF. DR. C., DE JAGER

CON 'l'EN TS•

Samenvatt 5

Introduction and Summary 7

1. On the Detection

Component

of Binaries with B Collapsed

13

2. Are Binary Systems

always disrupted?

involved in SupernovB osiana

19

~. Supernovn Explosions in Close

Application to CentauruB X-3

Binary

?1

4. Supernova Explosions in Cloae Binary Systems.

II. Runaway Velocities of X-Ray Binaries

5. On the Tidal Evolution of Massive X-Ray Binaries

h. The EVolution of Massive Close Binaries.

III. The Possibility of a Tidal Instability during

Post Supernova Phase

the

7. The Formation of Globular Cluster

Neutron Star - Giant Collisions

X-Ray Burces through

67

Acknowledgements 83

3

Dit proefschrift beschrijft een onderzeek van anige aspecten van

de evolutie geschiedenis van R~ntgendubbelsterren. Hierbij werd

uitgecaan ven het door Van den Hauvel en Heise in 1972 Yoorgestelde

model veor de evolutie van R~ntgendubbelsterren van grote massa.

In artikel I worden enige begrenzingen afgeleid voar de physische

parameters van dubbelsterren met een compacta, in een supernova

geimplodeerde, component. De conclusie is dat dergelijke

dubbelsterren eerder gezocht moe ten worden onder dubbelntelaels

met een kleine waarde Yoar de messafuDctie, dan onder die met een

grote ""aarde.

Verschillende auteurs hebben twijfele geopperd met betrekking tot

de mogelijkheid dat aen dubbelstelsel een Bupernova exploBie van

een dar componenten zou overlaven. Dit probleem wordt in artikel

II en III nader bestudeerd. Het resultaat van dit onderzoek is dot

het dubbelstelsel in het algemeen nauw gebonden blijft ne de

explosle.

Het waarneminEsgegeven dat sterke R~ntgenbronnen een concentratie

vertonen naar het galactisch vlak ,. duidt ar op dat de IIIrunawayli

(terugstoot) snelheden van daze bronnen oiet groot zijn. In

artikel IV wordt aangetoond dat het systeem geen hoge enelheid

hoeft te krijgen, vooropgeateld dat de masse van de exploderende

ster niet te groat is. De verkregen bovengrens voor de massa is

in overeenstemming met het tegenwoor~ige beeid Van de evolutie

van nauwe dUbbelaterren.

Uit waarnemingen is gebleken dat sommige R~ntgendubbelsterren

cirkelvormige banen hebben. Dit is in atrijd met de verwachting

dat de banen excenttisch zouden moeten zijn tengevolge van de

supernoval explosiea. Geti j denkrachten kunnen aen mogelijke oorzDak

zijn voor het cirkelvormig worden Van de banen. Berekeningen van

5

de invloed Van getijdenkrachten cp de evolutie van R~ntgendubbel­

sterren met grote massa worden beschreven in artikel V. Het

cirkelvormig maken van de banen tijdens de hoofdreeksfase van de

niet gedegenereerde component vereist een enorme viecositeit 12 14 -1 -1)(ca. 10 - 10 g em 8. Voar deze grate viBcositjit is nog

geen verklaring gevonden.

In artikel VI wordt de invloed Van de getijdenwerking op de

evolutie van R~ntgendubbelsterren na het hoofdreeksstadium van de

sI1perreuBcomponentfie~chrBven.Voorsommige systemen kan een, door

getijdenkrachten veroorzaakte, instabiliteit optreden, voordat de

sterren hun Roche-oppervlakken vullen. Het lijkt waaTschijnlijk

dot de hanen van Cen X-3 en 3 U 1700-37 op het ogenblik reeds

instabiel zijn.

In artikel VII wordt de mogelijkheid geopperd dat R~ntgenbronnen

in bolvormige sterrenhopen dubbelsystemen zijn, gevormd als

eavolg Van directe botsingen van reuzen met compacte sterren. De

compacta sterren zouden afkomstig zijn van vroegera supernova

Bxplocies in de bolhoop. Het aantal R~ntgenbronnen, verwacht op

gx'ond van deze hypothese blijkt - binnen de onzekerheden ­

overeen te stemmen met de waargenomen aantallen R~ntgenbronnen in

bolhopen.

INTRODUCTION AND SUMMARY.

The launching of the first X-ray satellite Uhuru in late 1970

opened a new era in X-raJ' astronomy. More than "100 n~w X-r"t,Y

sources were discovered in addition to BomB previously known

sources. The accumulating X-ray data disclosed many interesting

facts concerning the nature of X-ray sources. Some sources exhibit

short time scale variabilities (less than one miliBecond to A few

seconds) in the X-ray emission. of them (Cen X~3 Bnd Her )(-1)

show regular pulsations with periods of 4.84 and 1.?4 seconds,

res~ectively. These facts suggest that these X-ray sources Bre

compact stars. For many reasons most investigators believe that

these X-ray Bources are neutron stars or black holes rather thBn

white dwarfs. Some sources have been identified with optical

objects which provides opportunities for further studies of these

sources in different frequency ranges.

One of the most important discoveries is that seven X-ray Bources

have firmly been proven to be members of close binAry systems. This

supports a hypothesis proposed in 1964 by HayAkawB Rnd MatSUOka,

ond in 1966 by Zeldovich and Gussynov, that R compact Ater in R

close binary system can be detected B8 an X-ray source if the

compact star is accreting matter from its I1normal" companion .. Part

of the kinetic energy of the infalling matter is transformed into

X-ray radiation. The_binary character provides a number of

observational advantages; the physical parameters of the sources

as well as those of the companions can readily be estimated from

the investigation of the binary motion. This has led to the most

fascinating (but still controversial) discovery in recent years,

that is the presumed discovery of a black hole in the system of

Cyg X-1.

There are at least two groups of X-ray binaries. In one group the

Gources are ~GRociated with extreme Population I stars, i.e., very

7

mD~sive And luminous OB stars, while in the other group, the

companions are stars of rather low mass, of spectral type A or

later. Among the known X-ray binaries, five (Cen X-3, eyg 1-1,

3 U 1700-37, 3 U 0900-40 and SHC X-1) belong to the first group

and two (Her 1-1 and eyg X-3) belong to the second group.

In this thesis, a study on some aspects of the evolutionary

history of X-ray binaries is presented. A particular emphasis is

given to systems which belong to the first group, although some

attempts were also made to consider systems in the second group.

We generally follow the scenario for the evolutionary history of

massive X-ray binaries proposed by van den Heuvel and Heise in

1972. This idea was further elaborated by De Loore, De Greve,

van den Heuvel and De Cuyper in 1975. In this evolutionary scenario

it is proposed that the compact star in an X-ray binary originated

by a supernova explosion that occurred after extensive mass

exchange.

In Paper I 90me evolutionary constraints on the physical parameters

of binaries with a collapsed component are considered. It is

concluded that the search for such binary systems should be

extended through the examination of binary systems with Bmall mass

functions rather than with large ones, as originally proposed by

Trimble ~nd Thorne.

Doubts have been raised by several Buthors (e.g. Cameron and Canute

at the Solvay Conference in 1973), whether a binary system can

survive a supernova explosion. This is mainly due to the. thou~hts

that (i) the more massive component will be the first one to

ode and , (ii) the momentum imparted the supernova shell onto

the unexploded star could be very large so that the star would

leave the system, i.e., the binary would he disrupted. Papers II

and III aTe devoted to the study of this problem. It is found that

in most Cases an evolved close binary system cannot be disrupted

a supernova explosion. The binary will in most Cases even remain

closely bound after the explosion. This is mainly due to the mass

8

transfer which preceded the explosion, causing the explod

component to have a smaller mass than its companion at the time

of the explosion.

There is a tendency for strong X-ray sources to be concentrBted

towards the galactic ane. Since a supernova oE'don rnig'h t gi ve

an additional space velocity to the system du~ to matter ejection

(which makes the system to I'i run aW!3Iyll from the place where it

originated\ i.e., the galactic plane), one might raise a question

whether the above fact is not an argument against the, view that

compact stars originated by supernova explosions. In Paper IV

this problem is studied. It is found that it is not necessary for

the system to become a high-velocity runaway system provided that

the mass of the exploding star is not large. The derived upper

limit for the mass is consistent with current views of close

binary evolution.

The existence of X-ray pulsars in the systems of Cen X-3 and

Her X-1 enables one to accurately determine the orbital parameters

of the systems. It has been shown that at least in these two

systems the orbits arB circular (the upper limits of e determined -3from Uhuru's observations are - 2 x 10 ). is is contrary to

the expectation that the orbits should be eccentric as a consequence

of the supernova explosions which have formed the compact objects.

It has been suggested that tidal forces are the mechanisms which

have circularized the orbits. Computations of the tidal evolution

of the orbital parameters of massive ~-ray binaries are given in

Paper V. In order to circularize the orbits within the main-sequence 12 14lifetime of the primaries, an enormous viscosity (_ 10 _ 10

-1 -1) , 'J" • t b ]'''' t fg em t, :18 requlree; WInerlt Canna' e exp","a].ne(~ l.n"erms 0

ordinary gas viscosity. This problem remains Bofar unsolved. The

large observed eccentricity in the orbit of the binary radio pulsar

PSR 1913+16 has given extra impetus to search for this unknown type

of viscosity.

9

The tidal evolution will in most cases lead to the circularization

3.nd synchronization of the orbi t. In this st,age the system reaches

the most stable configuration. However" the changing structure of

the primary during the post main-sequence evolution will

continuously disturb the stability of the orbit. If the tidal

forcBe are strong enough (this is the case which we expect if the

cirCUlarization of the orbit is indeed due to the tidal effects),

the synchronism will be restored on a time scale shorter than the

evolutionary time scale of the primary so that the orbit is nearly

synchronous all the time. During this process, orbital angular

momentum is transformed into rotational angular momentum of the

evolving star , leading to a reduction in the orbital separation.

Due to this effect, at a certain stage the system becomes unstable,

synchronism becomes impossible and the compact object will be

spiralling down onto and into the primary. In Paper VI, the tidal

evolution during the post main-sequence stage of the primaries of

ma~"l!:>ive X-ray binaries is consi.dered. For some systems. the tidal

instability might occur before the stars are filling their Roche

lobes. It is found that possibly Cen X-3 and 3 U 17°0-37 are

ready unstable at present.

The discovery of four X-ray BourceS in globular clusters creates

Borne specific problems concerning th'ir origin. The frequency of

occurrence of these sources is one to two orders of magnitude

higher than that of sourcer1 in the galactic plane. It has heen

argued that the formation of a compact object in a binary system

might give a velocity large enough for the system to leave the

cluster. In Paper VII it is suggested that such X-ray sources, if

they are indeed binaries, can be formed by direct collisions

between giant branch stars and compact remnants of ancient

supernova explosions in the cluster. The frequency of such

collisions and the lifetimes of the resulting X-ray binaries are

estimated. The number of X-ray sources expected on the basis of

this suggestion appears to agree -within the uncertainties- with

the observed frequency of globular cluster X-ray Bources.

10

j)eferences'IIl

Cameron, A.G.W., Canuto, V. 1973, Proe. Sixteenth Solvay

Conference on Physics, University of Brusaelst

De Loore, C., De Greve. J.P •• van den Heuvel, E.P.J., De Cuyper,

J.P. 1975. Proe. 2nd I.A.U. Re Meeting, Trieste (in

preparetion).

Ha nwa, S~, Matsuoka, M. 1964, Prog~ Theor. Phyn. Suppl~

.22.., 204.

vnJn den Heuvel, E.P.J., ]{eise, J. 1972, Nature Phyn. 1. 239. 67.

rrrimble, V.L., 'I'horne, K.S. 1969, Astrophy.s. J. 1:)6.11013.

Zeldovich, Ya. B., Gl.ls,eynov, O.H. 1966, Astrophys. ,J. 144, 840.

11 1

On the D'etection of Binaries with a Collapsed Con1ponent VIi. SUI<lnlyo* ASlrononlical In'liHHC. UinlversHY 01' An'Slad"ITl

R~:n;!Vt'cJ July 2. rtvised Juiy 30,,1971

SUIDumuy. Conditions are developed for !ht~ observable system parameters of evolved close binaries in which one of the components has collapsed through a ::;uper­nova explosion. It is adopted that the total mass of the system is conserved during the mass transfer preceding the explosion These conditions set an upper limit to t he mass of the possibly collapsed component in an observed single line spectroscopic binary or in an X-ray binary. Only one binary in the lists of non edipsing singile line spectroscopic binaries given by Trimble and Thome (1969) fulfils these conditions. The fact thai

mosl X-raybinarlcs aiso fulfil tbese condi'lions k::ads 10 the conclusion that il~ most cases Ihe amounl or mass loss during the nrst stage of mass tnmsfCf has 1101 bcen vcry large. From the very sinal! mass functions of !most rnassive X-Tay binaries II seems that mllssive hinaries with very sI11all mass functions could be mure likely candidates for having a 'collapsed componcili than, massive binaries with large mass funcl Ions.

Key ""fords: Close binaries X-ray SolHces collapsedi slars

-_._-~--_._-------.._-----_.._._._----­

I. mnlroductioll

1.1 has been suggested that in some non echpsing single line spectroscopic binaries wilh high mass functions, the invisible component may be it neutron star or a black hole (Zel'dovich and Guseynov, 1965; Trimble and Thorne. 1969; Gibbons and Hawking, 1971; 0011, 197 J). The facts that some X-ray sources are members or binary systems, and that strong X-ray radiation can easily be produced if one of the components of the binary system is a collapsed body such as a neulron star or a black hole (Shklovsky, 19(7), give evidence of the eXIslence of binaries with a collapsed component. Van den Heuvell1973 a) shows thai severa] thousands of binaries wilh a "sleeping" neutron star or black hole componenl (in a stage before the onset of X-ray radialion) are expected to exist in our Galaxy. Trimble and Thome (1969) give lists of binaries which possibly have a neutron star or a black hole as one of the componenls. They only consider binaries of which the rna.ss of the unseen component is larger than IA Mo (Chandrasckhar limit). It has not been possible to find observational criteria to select which or these binaries arc really systems with a collapsed component except in the case that such a system is an X-ray source, because we may not rule out the possibility that the unseen component is an ordinary star of which the light is concealed by the bright companion. Only one of the systems in the lists of Trimble and Thome (0 2 Orioni::;) has been suggested to be an X-ray source (Barbon et aI., 1972). It is well accepted that a neutron star or a black hole is the product of a supernova explosion. Van den Heuve1

" On leave from IlosschH Observalory, Lembang. Indonesia.

and Heise (J912) have [hilt iii most dmlc binaries the explosion occurs arter the first slage of mass transfer when the original primury has losl almost all of' iils hydrogen envelope, becoming a pure helium slar Furthermore. these authors suggcsl thai most probably the explosion willi happen only if th(~

mass of the helium star is larger limn 4- II will tiinish its life by exploding as a supernova, leaving behind a neutron sial' or a bla-.:k hole" II is most Hk,ely lhat the exploded star was lhe less massive component, because it has lost more Ihan 65 percenl or its mass during the mass transfer, while Ihe other component has coUecled much of this matter. The binary becomes an X-ray source after the original secondary, presently the more massive component, in lum has evolved to fill up .its Roche lobe a nd the second stage of mass trans­fer takes place. The fact that only one among the systems in the .lists of Trimble and Thome is suggesled to be an X-ray somce, suggestfilhal some other binaries, if they are really binaries with a collapsed component, arc in the stage after the explosiion of the original primary, but before the onset of the second stage of !liIaSS transfer (or the orlset of almospheric condirjons which produce a stellar wind; cJ. Ostriker and Davidson, j973). The aim of this paper is 10 find from lhe lists of Trhllblc and Thorne and the known X-ray binaries systems which may be the result. of such evolution ..

u. Mas.'" Transfer without Mass Loss (rom the System

In pedorming the calculation of Iheevolution of dose binaries most authors assume that the total mass and angular momentum of the system are conserved during

13

104 W. SUlanHyo

the mass lransl'er ( 1971 Inourflrsratt,empt we will this assumption. We denote the mass of the original primary and secondary a,,:; MIl and lVlf, respectively. If the first stage of mass transfer takes place ill. case B (i.e ..." during the hydrogen shell burning preoeding the helium ignition), the primary becomes a practically pure helium star with mass M , at the end of the mass transfer (Pac-zynski, t971 a). This occurs for some 80 percent of all OB stars (van den Hcuvd, 1969). We denote the mass of the other com­ponent as ,\;12 , If we do not give any limitation to Mf except, of course, Mf > 0, we have,

(t)

The mass of the helium star (the helium core of the original primary) can be approximated from the formula (d, Smith, (968)

M1 =, 0.4 /'1/[\) .- 2.0 (2)

for MlJ :> 10 the masses are expressed in solar masses, I-Ienee, the exchanged amount of matter is j M j = 0.6 +2.0. In case C mass transfer (after the helium core burning) the remaining mass of the primary is about the same as in case B (Paczynski, J971 f:.ILITIlllatlmg i\-'!.? from (l) and (2) we have

M1 < 0.67 M2 .-. 3.33 . P)

From the inequality (3) one can show that M1/l'v!z < 1 for all values of M I , so the exploding star will be the less massive component in all cases. Clearly this is due 10 the mass Iransfer. The necessary condition for a heliium star to finish its life as a neutron star or a black hole is that the 11l1aSS must be greater than 1.4 MCl

{Chcmdrasekhar limi.!). Therefore, we may from (3) that we cannot expect either a neutron stat or Ifl black hOle as the final! slate of the helium star if M] < 7. i Mo. This limit of Mz will be lower if we allow mass loss from the system, If the fraction of tile trans­ferred mass which leaves the system is j: then similarly the lower limit of lVll , above which the binary may have II coilapsed component, is 7.1 (I - f) Mo.

HI. Tht: ReilllltioJ'l;s betwct~11 nl(~' Physi.cal. Parameters before andancr the m<:xl'losion We assume that before the ex.plosion takes place the orbit is circular. The supernova explosion is supposed 10 be sphericaHy syrnrnetric <mel 10 take place in­s~antaneously. We: also assume that rem.aills con­stant during thcexploshon. The interaction bel\vCell the ejected mailer and the unexploded sial' will be neglected, so the change of the orbital clements can be considered only as the result of mass loss during the explosion, By those assum.ptions tbe eccentricity of the orbit after th.e explosion is (GoU, 1972) .

Ml~M{ e = -------- (4)

M2 + M{ ,

where M{ is the mass of the cutlapsed eomponent. From Eq. (4) it is clear that a sufficient conditnon to have a bound orbit after the explosion is

2 M{ > M1 - Ml . (5)

This condition is fulfilled if j\,11 ~ M]; so in our case, where the exploding star is always the less massive component, we must always have a bound orbit after the explosion. By eliminating M l from (3) and (4) we fi.nd an upper limit of the mass of the collapsed com­poneIlt

I {O.67- el ,~12 - 3.33M. = .._-_... . (6)1m.. t +e

For single line spectroscopic binaries one can in general estimate the mass of the visible component from its spectlt'Um. The mass function then will give the lower limit of the mass of the unseen component (we denote it as M{mi',). If we assume that since the el1d of the explosion e has not changed, one can rule a \.It the possibility that a binary is the result of evolution wi~h

conservation of mass during the first stage of mass transfer if M{m,," < M{ma<' We will apply this condition to the syslems in the lists of Trimble and Thorne (j 969) and to the known X-ray binaries. During the explosion the centre of gravity of ~he sys~em

will be accelerated and the system becomes a runaway system with space velocity (Gatt, t972)

MI K 1V = e -"·7'·--:-"--;- , {7)

1v11 Sllll

where K 1 is the radial-velocity semiamplitude of star 2 and i is the inclination angle. The relation between K l and the orbital parameters is given by the rela­tion

(8)

where a1 is the semi-major axis of the orbit of star 2 with respect to the centre of gravity and pI is the period a.fter the ,explosion. From Eqs. (7) and (8) and using Kepler third law one can obtain that the mnaway veloeily is given by

J , M1 e (9)

= 2nG plU _ e1]3ir {l+Ml7M;}2" '

where G is the graviul.tional constant By adopting the assumptions we discuss above, Boersma (1961) shows that the rela~ion between the semi-major axis before and after the explosion is

J a = aI ._- _c····_· · ..·•..·c ._._ (10)7

One can easily transform Eq. (10) into a relation of the periods, and using Eq. one can obtain

. (1 _e)3f2p=pJ.. " ( 11)

(1+

--------------------

lOS

the periods beforeiil.nd allter the e~~lplosl(Jln r'~·~"""'i'l'lw.i·hl pi as well 8S e C,1Ul be determined from observations. We do not compute the period the rnllliSS

Iransfer, as the value of is ve'I)' sensiilive to the exact values of the masses M I , 1\4;', Mi lvly are not accurately Known.

IV. Result and Di.scussiom

We made a survey through the lists of non eclipsing line spectroscopic binaries given by Trimble and

Thorne (1969) to find systems whkh might be the result of evolution without mass~oss from the system during Ihe first stage of m.ass transfer preceding the explosion. The result is given in Table j. By adopting M{ as the mean value between ]lltf{onl" and M{m..' one can cal­culate M I from Eq. (4) and M? from Eq. (2),. Then Mf can be cakulat,ed by using the condition of mass con­servation dming the mass transfer. In accordance with the conditions developed in Sections H and In we exclude lhe possibility that a binary is a result of such an evolution if M{max < M{"'ill; or M;' < M~, or both. The second condition is dear since the original primary must ha ve fi lied up its Roche lobe earlier than its k~S8

massive companion. Furthermore, we also ,exclude binaries with M I < 4 MG , as a helium star with such masses might nnishits life as a white dwarf instead of exploding as a supernova (Paczynski, 1971 b; van den Heuvel and Heise, j 972). We find that only one binary In the lists of Trimble and Thome (i.e., HD 199579) fulllls the conditions to be the result of such III type of evolution (Table n. In tbe table we also give the maximum value of runaway velocity for this system obtained from Eq. (9) by adopting M{ = M{mill; we lind that this system has a low flInaway velocity (V < 14.41 km s - 1). We also made a survey through the known X-ray binaries. Unforhmately the values of the eccentricities of most of them are very uncertain. Even if reliable values are available, one may argue that the eccentri­city after the explosion in the past was higher than the observed value now because the .interaction between the collapsed components of these binaries with the gas nows in th.e systems may have reduced it considerably.

Table l. The physical parameters of HD 199579, the only binary in Ihc lists of Trimble and Thorne (1969) which fulfils Ihe condilions for being the result of evolution wilhout mass loss from Ihc system during the firsl stage of mass transfer

e = 0_099 M, = 119Mo pi = 48.61 days Mf' = 34.8 ME)

M1 = 25.00 Me Mf'= 2.1 Me M{ min = 7.3 Me P = 39.65 days

M{ "''' = 9.9 Me V < 14.41 km s-- I

M{ = 8.6 Me

For this re,iil.son H'le has bt:l~n done for 0.1, 0.2. and 0.4 Onh' Ihe

Orionis the 'clllculatioll was don~ for the observed vaJue of e because the pet"ioel of Ihis is rather (2: I days) and the emission is weak, a alnOUIH of gas in the so the interaction may not ~ the ec(:entril)l1:y \le.r)! much. The resuHs are labul!lted in TaMe 2. In case the condition mass omillg the first

of mass transfer is ",M,.' < MO, \Vl~ estimate the fraction of the mass which has

the sysle.tn. (we denote it as f) by M~ "" I M(~ (and 114{ = jVt{"'i" OrI0l1is). 1'1 is re­markable thai mosl of the X-ray binaries fulfil the condition of mass conservati.ol1 during the first of mass transfer. Only two HZ H.erculis ilnd

Orionis, do not fulfil lhis condition. 'rhe \'1I1ue f is about 0.42 for OJ Orklilis and over 0.78 for HZ Hercll~is. We will discuss later thnl the results we oblaiJled for these sysh,:ms Me probably nol reailislic,. The hinaries Cyg X-I and Velin X- t !ilso do not fu~fil this condition if we take high values of e, bUI the requiredamouut of mass loss is small. Furthermore, ,if we requir.c M 1 > 4 MG , the eccentricity of 2 U~700-"37

must be great.er than 0,05, whH(\ for Vela X-I, Cen X-3 and SMC X·I the eccentriicity Inus! 00 largerlhin 0.1 (notice Ihal e ill. this case is the eccentricityl:lt the end of the explosion and not the presently observ,ed value). Vall den Heuvel and De Loore (1973) predict HUll the periods of X-ray binaries m,ly have been d".,·w,,,,,-.."''''''

'fbis predict]Ol1J soe:1113 to be cOlll1l'med by Scluc:ier et ai, (1973) from their observations of the decl'el:~se in binary period of Cen X-J So the orbital period we observe now may be smaller than the period in the past Uusl afler the explosion). Th·erefore the values of P' we give in Table 2 are certainly lower Il.mi.ts., We also give the values of runaway velocities in Tabl,e 2,

Some authors give further evidence of the existence of binaries in Trimble and Thorne's Hst.s which may have a collapsed component The binaries which are suggested to have a coJiapsed component are, HD 176318, HD 194495 (Gibbons and Hawking, 1971; Gatt, 1970, HD 59543 (Gott, 1972) and (J2 Orlonis (Barbon et al., 1972). The binaries HD 1.76318 and HD 194495 have relatively high eccentricities afld shol'! period3,lhe binary HD 59543 might be a moderate runaway syst,em and 8.2 Odonis is probably an X-ray source. None of these systems fulfils the requirements discussed above. It means that, if they are really the result of such a type of evolution, we must al.low mass loss from .those systems dming the first stage of mass transfer. By as­suming M~ = 1 Mo and adopting M{ = M{,,,;n' we find that, if these systems indeed have a collapsed component, f takes values up to 65 percent. The required amount of mass loss in this case is tabulated in Table 3, except for rP Orionis of which the result has already been given in Table 2.

15

W. Sul.anlyo

T:.lok 2. The physical parameters of the knowI1 X-ray binaries. For cases in which Eq. (6llridded anegailivc mas", AIL",. has becllP Ul,tO be eq ual lio Zero. In C,'S(,S OW con('l!ilkm of mass COrls.crvatiol1 during Ihe i1,rsl si.agc 0!1 mass transler lS v lola led IS adopted 10 be I ,1,.10 , Nolloc

(I m 226H6X, 114{ = pi "=

.lO Mr.) S<]607

.1

.2 4,7 31

d 6,6

111.0 215

2.8 .1

4.565 3.662

26.50 5381

,0

.0

.3 1.7 84 26.0 1,0 2,880 82.91 .20 l:>O

.4 .5 10.2 30.5 1,0 2.202 I 15.05 ,3 I l:> 0

Vela X-I! h)

'HD 775!i;IJ MJ. 1:5.0 Mo M{ "" 1.4 Mu I'J ~, 7dO10

.05

.1

.2

5.6 4.7

3.1

2.2 30 4.7

10.5 12.6 16.7

6~1·'.1

54 3,0

6.334 5.707 4.5N

l3,[j4 26,17 5.1.IS

.0

.0

.0

lvl n < 4 ~\{:D

M, <4Mo

.3 1,7 6.3 20,8 .5 3,601 81.89 .0

.4 5 8.0 24,9 10 2.753 13.64 .17 />0

10,(l Me, .05 8.6 i5 88 12.7 1.886 22.43 .0 iH I <4 Mo ' M? < Mi'

~;

05 it1c, 2~!OH7

I 2

7,3 5.0

25 4.6

11.4 165

I 1.2 8.1

1.699 1.363

45.03 9L46

,0

.0 M 1 <4 iVft'J

.J .1.11 6.6 21,.6 5.0 1,072 140.91 ,0

.4 1.4 8.7 26.7 2.0 820 19555 .0

srI/Ie X.I <lj M1 20.0 MI:, ,US 8.6 20 iO.l I 1.9 3518 17..93 .0 ,1,1, < 4 Mo. Mf < Aff (Sandulcak Nr. 1(0) M{ 1.0 Mo "I 7J 11 12.7 lOA ).169 36.00 .0 M" <4 iHc;,

pi 3~89] .2 5,0 5,2 18.0 7.2 2.543 73.1 1 ,0

.J 11 7,3 23.2 4.11 2000 I 12.64 ,0

A 14 9.4 28.S .9 1.529 156.32 .0

2 U 1.100 37') = 25.0 N() .05 I 1.5 2,.8 12.1 15.8 H183 20.06 .0

iHD'IS3919) ~.5 lW{) I 9,8 4.1 15.4 13.8 2.778 40.27 .0 = 3"412 2 6.9 ,6,,8 22J) 9,8 2229 81.78 .0

.3 4.5 94 28.6 :5,8 1.753 126.00 .0 4 2A 12.11 35,2 1.9 1.. 340 174.85 .0

ller X·I Ii !'V{l~" 2.5 Mc)' ,05 .n 1.5 8.7 1.0 ISHi 9.23 .79 M I <4 M(0'.!>0 (HZ Herculisl {Vf{ I,J Mr " ,I! .0 1.7 9.2 1.0 1.384 18.54 ,80 Mil <4 M"j' .!>O

pI 1~17 ,2 .0 2.11 IlU I.() I, II (J J7.65 .HI Mil <4 Me.I> 0 J .0 2.4 I 1.1 1.0 .873 58.01 .83 ,'HI; < 4 M(l' /:> () .4 .I) 2.8 12,0 1.0 ,668 80.51 .H4 Mil <4MfJ ./>0

2 U 0525 06"' M2 20.,0 M", I J I 6.5 18.5 5 ,I U) 16.1)21 19.56 .42 />0 (0" (hiolli'J NI[",· 14,OMo

pi = 21 '~;OJ2

Ihal Inosl of (lle,,-, hiTJiHic~ fumlthe ~(>I1dition of ma~s cOl1senraliol1

X-ray Ihmary

CYi~ X·I 'OJ M I = :ISO A{; .05 5,6 NI'",) 39 111,." 1,4,7 Ivl' 4.2 M(!) 5"067 0.20 )(111/' 0

"I Boh.ol1 (1972) '} .I ones "I al. (1972). van den Hcuvc.1 ( !971 b). 10, II illlfWf "I ,:11, (1972) ') Cramp'l (n] and H lltchmgs ( 1(72).

'I Schreier l'I Ill. i[ 119721. l'o1.n d"m !H,mvd and Heise (1'J12). 0) Trunble lind 'rhome ( J (69), Barbon 1'1 ai, (197'2), ,1 1 \1Ii'cbsl',>r ",/ <1.1. (19721,

'fahle ,1 Thrc'c bil;lUri,is i,n the !isls. of Trilnhl,~ rlndT'home (1%9) wl1ich !lI'C ;mgg'esl",d by sOIm: authors to be binaries with a colJapsed componel1t. r;'or al'l SjtstCITI" W(, '1I,lop't ,.~ I ,\"to and min' Neithcr of thc systems fulfils the condition of mass conservation dllring the first Mage (JI' nl,:,!;, tnlnS(l:r. Notice Ihat the sys,lcrm, e'in OIil!y have a collapsed componc'11 if ,1 considerable fraction f of [he Inlllslernxll11<Jss was lost from 'Ihe '~'D,lern, ,hlfll1!J, lhe IIl'sl slage of mass IransJer

-------,-------_._--~

e

HI) 59543 2.0 M 7,0 Me', 6.71\4", 21.7.MGl 520 17"'91 ! 4'!8~ I 8o.73km, I ,60

liD 176:liX 19 5.0 :I 1 '1'2.7 .169 2.912 2.040 :15.52 .65 HD 194495 2.H 6.0 4.0 1,4,9 .IIB 4.905 ~,720 2J.H4 .60

,--------_._-,-----,

Consid.ering the fact thaI Inos! X-ray binari'es fulfil the According to van den Heuvel (1969) the fraction of mass condition of mass conservation dur,ing the mass trans­ loss from close binaries with a tolal mass of 2,5 Mo is fer preceding Ihe explosial"lJ, fHld remembering that x.·ray probably not larger than 30 percent, while Yungel'son binaries are the most certain candidates far a (t 973) finds that for binaries with a total mass of about colla psed campol1.ent, we suggest that the amount of 3.01\10 the best agn~ement between theoretical models mass loss is probably not so great in most cas,e.s. and the observational data can be achieved .if f = 0.25,

101

if ~bis is 21150 'tfue for '?\lith masses, the amOI.ll1t of mass loss requir,erl for HZ

Orioni!; and all binaries in Table 3 is probably not realistic. the of most of those systems as binaries a collapsed compon.enl is not correct There has been no conclusive evidence that the binary Orion!s is identical with the source 2 1I 0525--06. The identifi:catiol1 of the eccentric

systems HD 116318 and HO 194495 as binaries with a collapsed component has been doubted by several authors (Hatten and Olowin, 1971; Evans and Bath, 1972). The evidence that the binary HD 59543 is the result of such evolution is also not co.nclusive. Further­more, considering the fact that the period of HZ H.er­culis is very sl10rt (1.7 days), it seems not unlikely that the mass transfer in tllis system occun',ed i.n case A (before the end of hydrogen burning in the core). In that case we cannot apply the above prescri.bed condi­lions to this system be,cause at the end of the rnas!> transfer the mass losing star is not a l1eliurn star, so

is not valid., Therdore, the large amount of mass loss we have obtained for this system is probably not reaL

from the fact that the coll.apsed component in some X-ray binaries has a very low mass (below the Chandrasekhar limit), and the mass I'tinction of Vela X- t and 2 U J 700--37 are only 0.'013 and O.OO5Mo respec­tively (Hiltner et .111'., 1972; van den Heuvel, 1973b), we suggest that the search of systems with a collapsed com.ponent should be extended beyond the lists of Trimble and Thorne through theexaminatiol1 of single line spectroscopic binaries wilh small mass functions..

.4cknowled"enl<.'tll.\:. The author H; paniclJlarly inJebwJ to Dr. E. P,l v,m den lIellvd for introdull.:ing him 10 the problem alld for his helpful ad>lce. llils Ilh"'nb also go to Dr I-I, R. 12, T.iin A Djle for discussing Ihis pap,:r He would I:ike 10 express hiB gratitude to the Dutch Min.ist.ry of Education and Science for granting him a scholal:­sl1lp.

13m!x).I1. R., BcrrHI,ccll, P. L., T,m:ughi. M" '1"I'('I'(1S,I\. 1912. l'hl'flU',,, 2.40" ~ ,~2

BiiII\wn, A. HL, Olowin" R. 11:', ill! L """'Hur,· 1.14. 341 Boersnm,J 1961, Mull. it~/./·OIL 1'11:.'1. i,'Ii'.,'Ii'l, 15.291 Bo.l.tO.lil,C T 19'12. ,,,II,m.,j·,' Phu·.,. Sdrll'''' 2:35,. 2"11 Cra'lnploll, D" Ihllchil1:l.gs"L B 1\ll2. i1SlrIlpll)'S. J. Lew?/'s 1"/'8, L 0$ EIIIIJ!s. \.V.D" BlHh .. O. T. '19721',,"1111"" PhI'S. Sc.fellor? .BS" ,} Oibbons.(t 'N, HiH.\!ltit'll!l.S. W. 191 it, ,"""1011'(> 1,31.46) GOIl,J.R. 1,'HII.l\h~Hm·1J". :~'U

Oon,J. R, 1972. AsII'O/lh,.·:t J. 17),.221 I'.an d~n I-hnlvel,E. P.J. 1969. ibll·on. J. i'4. 1095 111m den !'-lIeUI'd, E. P..!. 197'311, p"per .11 Summer Schoo!

on Physic, <JJlIt "ISH'OpJ'".l'sics of COllii'll!"! Ohjt"ds. !",l1g!nnd

Vllil dcn Ih:uvd,E. P.J. i9'Hb. '1A.U. "in:ulllR' Nr. 1526 (April 10, VIm den IHk:liI\'cl. E. P.L Heist:..!. 11l12. Nfllllr," 1'11)'.\ .. .)cie,I'J.c(, 2JIJ, 61 van dcn Hlcuv,el, E. P.L Dc Loorc,C. 197.'1. 'blnm. ,~ A"lrflplhys. 25,

JH7 Hil,'uncR', W.A" WenH:r,L Osmer, P·. 1972. ",Lslmphys, .1, L,'u",'s 175.

Ll9 Jones, C.. F"01'l11111:1, W.. 'l'<III,1Il1hmul1, 1''1., Sdu<l'i'Jr. E.. Chlrsky, 1'1., KeI·

logg, E.. Gill()Col1i.I~" 1(7). ASII'Opllys, .1. Leu"'l', 1111. L 4:\ OsHiker,l P., Davidson, K. 1973, x., ami Glllllmlll,Rliy Aslrolwmy.

I.A.U. syrnpOSilHl1 'Nr. :55. 143 Paczynski, B. 1971 a, .'11'111,,1111 1~,'I'. A.Hrllll'. illsrroIJII)'S.9, HB Paczynski,B. i97lb, Acla Asrroll, 21.1 Sdurchlr.l'., Levilrlsolrl, fl.... (husky, H.. KdloM;, E., 'flll1,mbllum., II.,

Glllccom, R. 1912., A.~rropiJJ·~. J. teNers 1'1':2, L 79 Schreier, E., Gillcconn.lt, HI.. E.. Lev iJ1S0il ,R., '1'111,11111'

baWil1,l-I. 1973. LA.U, N!' (April 17) Shkilo\l~ky,1. S. 196/, Ils/l'lJIJIIYs. .I. L,1un;II48, L I Smith, L p', 1968, Wo'llf Ral'ci Stlil'S, Eds.. R. N. ThOI11JIS lUll!!

K. B. Gebbie, Nat. BurellII' or StlThndllrds.. Spedal PlJ,blIcliliOll, ){)7, 2J

nimbli:,V,L.. TI'lol'nc.K.S. 1969. .1,1:56, WLI Websler,B.L, Martill.,W.L, FClIst,M .. W.., AIU!rC'wS,!l".J, '1912,

i'IIl.llllre240, 118J Yungcl'son,L.n. 1973, Soviel' ANII·O/'!. 1/,., 8M Zcl'doviich,Yll.R. Gliscynov,O.H. 1965, !J"'lI'l.ll,iJyS. .I. 144.840

W. Sulanlyo Slcrn.:nbllldig Ills\illiul UnivcI'si1eil "all A'insierdam Roclerss1raal 15 Amslerdam, Nederland

(Rej1rimed tram }\/al'ure, 1<"I,:t48, iVa'. 5445, pp. iVS-,Jo9, Mardll5, 1914)

Are, bin:ary sysfemsi,nv,olivedin supe:rnova explosions,cdways d:is,rup,tedl? No known is a mem.ber Does this mean, IlS Illls been fife tilt' l'l',mlt of supemol'a explosions O'l)ipc,th'~SIS 1 ,ar~d explOSions which occurred HI binary systems in most cases disrupted tbtl

systems (hYjJotliesl~ 2)? 'I'I'lf: 1\\'0 X-ray 'pulsars' (Cc[\X-3 1I11.d. Her X-I ('mup;H~t X-r'l:~"

s)'ste:l'tlsargumelll against hyuoth('sl8 I VII] id

sour'C(~s. Here I show Ihat :2 is f!'Om the point of view of present t heoth€'8 of dose e\'olution

A binary with period of s{wel'al y"iHS may he dose cnough the original primary to lose !L cm:lsil:lel:'able amount of its mass throu.gh the Illass transfer in t he late slage of its evolul.ion 2 , Van den :Hellvel~ shows that 80% of the known 013 spec:tl'Oseopic bin!uies will lUl.dergo clI,se 13 lnass j'r:msfer (dll ring hydrogen shell burning). Therefore it is reasonable to believe thlit, supernova explosions In close bimny systems moe preceded by mass 1I'ftillsfer.

It is intel'(,stmg that all known massive X-ray binal'les have a common property that consist of an OB super­giant with mass of abollt ao M0 ami a eomp1l.ct objl~ctwith

mass in the nHlge between-0.5 and -5 !lI0 (ref. 10, In fi ve of the six known X-ray binaries the compact object has M < 3M",; only in Cyg X-I [night it be more mlissive, The large mass ratio of these massive X-I'IIY binlliries sup­ports the hypothesis that the explosion is more likely to have occllrred after the mass transfer. In that elise the ex­ploded star is more likely co be tbe ],eSB massIve component~.

I assume that the mass of the original secondary (pres­ently the primary) is constant during the explosion. The explosion is SllppOS,ed to be spherically symmetric ami to take place instantaneously. I also aSSllme IIlli.t initially the orbit, is circular. Adopting these assumptions and taking the effects of the impact of the Sllpernova shell onto the un8xploded component into account. in the way formulated by Co!gateG, one can show that the condition f.or a binary system to remain bound after a supernova explosion is7

a + 2f3 - y _. 1 > 0, (1)

where £l! = M,/M" f3 = M,'/J1\ and y is the ratio (gain in kinetic energy of the unexploded star due to t.he impact)/ (initial orbital energy). M, is the mass of the unexploded component;I'II, ilnd M.' are the mass of the exploded star before and afler the explosion respectively.

I assume a typieal value for the velocity of matter ejected by :1 supernova expJosion is ]09 cm S-l (ref. 8). For a given combination of vlducs of M 2 , Ilf,'~L1JtJLjlli1iJll orbital period P, one can detennil1e fl"Om the mequality (I) the' ~riii(:;l-\;arue ofa (au) below or equal to which tlw condition (1) is unsatisfied (that is, the binary willi be dis­rupted). I take M, = ao Me and M,' = 1 and 5 M0 re­spectively. These values seem to be representatives for aU massive X-ray binaries. The values of «" as a function of P are given in ]l'ig. 1.

It is likely that in massive X-ray binaries only a small

fraction f of the transferred mass will escape from-

the sys­tem9. I assume that f :::;; 30% which is consistent whb the conclusions obtained from observations of evolved close binariesB,lo, If the explosion takes place after case B or

'fbi case C mass transfer and. 1. I :::;; 30%, the possi. Ie va ues of a are between -1.1 and -6 for M. = 30 M 0 (my un­published work). :from Fig._l one can see that all binaries

, I . d I h L 5 d '11 . b d fmtlpcflO Barger tanainJut' Wi. remam' oun a tel' t!ie explosion: Binaries' with such periods form about 50% of the O-B5 spectroscopic biuaries in Batten's Catalogue11 •

Printed! ii.B Gr:eat Britain b~ Henr}' liAng Ltd."

1 I 6 l-+__td'~ll.\a_.__. ~

5

I, I I \ \ \,2 ,,,

li i-- ~' .. ~ ~_{~~ '''''"::7~--==::===:::::::::::::::=== - -- - _ .... .... = "'" ....... "" ................ - ... - '- ..

p' (d)

}.'IO..... I .. '.'I'h.e.v,II!.. IlCS of '''or as Il. IlInct,io,n.ofP for s;vslenlswith Jlnlll 111,lllL'«'1'l 30 + 1M,", (~") and 00 :1- I:J.M~)\- "_ ..J: Fa I" 11 given vlll.ue of P the bin!lfY IilY!ltel~l\\'m he dliJl'up!,cd

<" a". Tbellorizolllu.1 line>! Il1oo11ed O:,n'" !l.lld ". resp<!ct,ively the lower and UPPO!' lim/.ct of

po~iblp vllllws orew (I1(JO text).

Since if the relTIrillnt, rIlass is large!' tJum 1 lllt!lj llil!!lO~Olnl~

binaties with i'I period of less thll.n 5 d rmxHlln bound ll.ft.llt

t.hoexplosion, the probabllity of mahlJ,alning, elOBe hlMry systems 1M leas,t i'or close binaries with an O-B5 ,component} through supernovaexplo~ionl!ilshouldbe eOllsider!JIbly hlrg,el' than 50%, Therefore the disl'uptioll of iJueh binllty SyBt'C'ffi1il

involved in supernova explosions is more liklJly to bfl m'l

exception rather than iii, rule. Even if J i.I!l 40%, sys:tems with P ~ 5 d in which the

remnant, mass is ~2 M0 will remain b?lUUd My. r~sllIts do not depend critically on the exact chOice of II. limit for j. So it Ileems necessary to seek another expl.anation for t.he l'iOD-exlsl,lluce of nldio pulsars in binary systems. The ab­sorption and smearing out of the radio pul~e8 by tJle gftS flows in the system might he such an alternative e)(p~lmation,

I thank Professor E. P. J. van de.n Reuve! for advice and Dr H. R. E. Tjin A Djie for rJisclission. A scholarship from the Dutch Ministry of Education and Science is acknowl­edged.

W. SU'I'AN'fyO*

Astronomical Imtit'Ute, University of Am8terdam, Roetl3rsstraat 15, Amosterdam-C

Recejved November 15, 11173; r,evifled January 24, 19'14.

'" On leave from BOBllcha Observatory, Institute o£ Technology,Bandllng, bdonesia.

1 Ruderman, M., A, Rev. Astr. Aslrllphys., 10, 4,27 (1912). 3 Paczynski, B, A. Rev. Astr. Aslropkys., 9, 183 (1911). 3 van den Rellvel, E. P. J .., Ast·r. J., 74, 1095 \1009). 4 Baheall, J. N., a.nd BaheRII, N. A., Proc. llixteentk ..r:;011)01/

Conjerence, 1973 (in the !Jl'eB8). 0 van den Reuvel, E. P. J., Bull. (UJtr. IMt. Neth., 19, 4132

(968), B Colgare, S, A., Na.ture, 225" 2417 (1970). 7 MeOhlBkey, G. Eo, and Kondo, Y., Alitrophys. Space Bci., HI,

4(j;ji 09'10. BShklovsky,1. S., Bupe'f"fUJvru (lnrerscience, New York, 1008), 9 Sutantyo, W., Asfr. Aefroph118., 2D, 103 (1973).

10 Yun.gel'son, I,. R, Boviet A8lr.,.16,864 (1973). 11 Batten, A. H., Publ. Dominion Ihtroph1/s. Ob8., xm, 119

(1008). at U:l,e l':)o,rSie£ P'~"e:lm. Do:r,!:·htl:ste"l Dorset 19

Supernova Explosions in Close Binary Systerns, with an Application to Centaurus X-3 VIi Sut<illly()* ASlrOlion'lKiiil In;;U!UiC. lhllversily of Arn,lerd,ull

Summary, Th~' effecl of a spherically symmetric super­nova ~~xplosion on !he orbital period of a close binary sy~tem is considered. Simple approximation of the effect of the impact is included. The upper and lower limit of the rna:~s ratio before lhe explosion can be determined from current theories of close binary evolution. It is shO\vn thaI lhe rmal period (after the explosion) as eSlimated from observalional data, yields an order of .--~--_..._------------ ­

1. introduction

Some discrete X-ray sources are members of binary systems. M osl investigators bel.ieve that onc component of Ihese binaries is a collapsed sIal' (a neutron star or a black hole) whi~e the other component is an ordinary star. The X-ray radiation is produced when the col:lapsed star is accreting matter from the "nor­mal" cornponcnt. The kinetic energy of the accreted mal tel' is transformated inlo X-ray radiation during this process {d. Pringle and Rees, 1972). It is gencrally accepted that a collapsed body like a neutmn star is the product of a catastrophic event such as a supernova explosion lee Ruderman, 1972). For this reason it is important to investigate the effeci of a supenwva explosion on the orbit of a close binary system in order to trace back lhe history of X-ray binaries. If we assume that the mass is ejected in a spherically symmelric sheil, the change of the orbital elements will due to two processes, namely the mass ~oss from the syslem andlhe impact of the ejected matter on lhe unexploded component. The change of the orbit due t() the mass loss in case lhat the exploded star is Ihe more massive component has been calculated numericadly by Boersma (l961j, while the nurncrical results in case thai the exploded slar is the less massive component were given by van den Heuvel (1968).

MOSI or the known X-ray binaries are very close systems. The periods before the exp~osions vvere pro­bably also short, since these systems are likely to have <!~~_en?_~.~lrom Wolf Rayet binaries (van den Heuvel, * On leave from Hosscha Observatory, Bandung, Instil.llle of T,edmology Indonesia.

magnitude estimate 11'01' lhe upper Ii.mil of the lion vcl.ocilLes or lhe supernova sheil!!. Application to t.he neulron sial' in the X-ray Cen X-3 ,indi­cates t.hal this st.ar is liikety 10 have originaled in a 'fype II supemovawith an cjccUon velocity of lile order or 10'1 cm/s.

Key words: supernovae close binH.ries X-my sources

1973), which i.ll llllinycases have periods !eHS Ihan to days (Sl1nilh, 1968). In such close systems lhe impact of the ejected InaHer becomes i'i11pOrlalll. Th~~ elTects of impact arc twofold (C'olgate, 1(70), viz .. : (0 Ihe supernova shell irrtpans mOlllcntuHI to I.he GOlnpanion, (2} an impact produced shock wave into the envelope of the companion and blows off Inaterial into the direction 01' lhe cxploding star. Canrwron and Canula (1973) have argued that, especially due to the second effect, ilwould b(~ unlikel,y f hat a neutron star could remain bound in a binary systenl. li"lowcvcr, we wiU show in this paper that in the case of Cen X"J the system may remain closely bound anerlhc explosion.

II. The Domain of Mass and Mass Ratio before the Exp.losion

Before discussing the supernova explosion It IS impor­tanl td know lhe physi.cal parameters of the syslem before the explosion occurs. Van den Heuvcl and I-Ieise (1972) suggest that the supernova explos.ion in a pre X-ray .binary luosl likely bappens aftcr lhe on'iginal primary has evolved 10 fill up ils H.oehe lobe and lost a large alUoulH of its mass to lhe secondary. Al the cnd of mass Iransfer the primary is pmcticaUy a pure helium star (originally the helium core of the primary) and is the less massive component. If Ihe mass of the original primary is M? then, from models computed by Kippenlhahn (cf. Smilh. 19(8), [he mass of lbe helium core (for A1? > 10 Mo ) can be approximated by

M, =O.4M? - 2.0, (I)

340 w. 'SUIIU1.lyo

where and M? arc cxpress,ed in solar masses. The hclwm star may finish its life by exploding as a super­nova. Ilowever; the cvo'lul!ionary Iracks of helium starS curnflutcLi hy l'ac/ynsk i ( 1971 a) show I.hal ilf M I < 4 M" lh,e hdiwil sial' wdl again expand to its Roche lobe and may again lose an amount of rnass 10 its cornpanion, SUL:h a star might finish ilS life as a white dwarf. There­fore a condition for a supernova explosion to occur is lhat lhe mass of the helium star is larger than about 4/111 . This is also in agreemcntwith the evolution of hehum sial'S computed by Arnett (1973). The corre­spomling mass of the original primary [ (I)j is,

(2'1:> 15 l\lJ" ' , , We will denote the mass of the original secondary as ,VIi and the origina.! mass ratio ,'HVMi( as rx o. Ciear!.y, 0< rxo < I, We d.SSUl11C Ihat a fraction I of the transJc~rred mass is 10tH from the syslem during the maBS transfer Imd (1 - I) is captllred. by the secondary. !,'rom the slatistics of vvolvcd close binaries (van den l-lellvel, 1969), the fraction of mass loss from close bi.naries with a total ma:,;s of 1,5 ,fl.1,. is probably not larger than 30'%, while Yungel'son (J97.l,l rinds that for binaries wilh a 3 totailihe hcst agreement between theoretical rnode! and lilc ohservational data can be achieved if I = 0.25. We conclude that the amount of mass loss during the mass transfer is genendly not very large, pl'Obably .~ 30%, This conclusion is supported by the fact that most of the known X·ray binaries fulfill the condition of mass conservat ion during t.he mass Iransler preceding the explosion ISutantyo, 1(73), Vv'c: dell0le Ihe mass of the or:iginal secondary after mass tnlllsfcr Ipn.:scnl.ly the more massive ,component) as All and Ihe maSS n:lt io f-d Ii as tI.. FrOID ( I), one can derive the following relat

,\i!2,,,,[o.,6(~-· + 1+2.0(1--/), (3)

and :x -'" 1V1 2/lOA-- 2,0) .

With lhe Giid of these equations one can transform the dornalll\n1 :> 15 i\{ [the inequ..ality (2)] and 0 < iXo < I into the dolO/Hll of AI, and ex. The results arc given in Fil.!, I for {,,,.O and / = OJ, respectively. The domains ar~' illustrakd by the shaded an:a, The points in these dornains represent aU pOSSible combinations of 1\11 2

and :1. for systems in the swge after the mass transfer. Since tile l",~ndHion that the core helium mass exceeds 4 Mo is indudcd, the domains represent ali for which a supernova expl,osion of the original primary 'lpresently the less massive component) is to be expected,

HI. l1w Change of tilt) Et~.ceIHricity and the period Due to the Supernovli KXldosion

'Nt: follow i'v1cCluskc'i and Kondo (1971) in considering tbe drecl of II supcr;lova explosion on the orbit of a

dose binary system. Assume thai the explosion lakes place In a circular orbit This seems permissible since ,ilrnos'~ all dos,e bmaries with P < 5 have practi­Lally circular orbits (Gibbons and Ihlwking, I 'ni I The 'l.:xplmiol1 is supposed sphericaHy symllletri.L: anu lakes place instantaneously. We also assume that line rHass of the unexploded cornponcmM 2 is constant dming the explosion, The ejected mailer wilt impart a velocity component in the radial directionl/, to the unexploded star which can be approximated from the formula (Colgate, 1970),

(5)

where R2 is the radius of the unexploded star, ilo is the initjal distance between the two components, M{ is the final mass of the exp~oded star, is the velocity of the ejected matter and 1/,,", is the escape velocity of the unexploded star. The factor In Ii',,) Ves in Eq, ~5) is only valid if Vej > V",. This factor is due to the el1ects of the propagation of the supernova shock wave in the interior of the unexploding staI' which causes stellar matter to he ejected into the direction of the exploding star (as dissipative effects are neglected and plane parallel layers arc assumed, this formula will overtsti· male the real momentum imparted to the companion; we will therefore lise itin the sense of an upper limit to the effecls of impact. H we find by using Eq, (5) that a system wil!. not be disrupted, we \vill be certain that it will not be disrupted in case of a more realistic treat­ment of the effecls of ill1lpact of a sphericaHy symmetric shell). If we assume that the unexploded componenl is an unevolved main sequence star (ef. van den Jleuvd and I-leise, 1972) then the radius can be approximated from the formula CPlavec, 1968),

(6)

where R. 2 and M2 are expressed in solar radius and solar masses respectively. This formula is good over an mterval I < .11/1 2 < 15 Afc,; as it is abo satJlsfied by 30 iHo

model computed by Stothers (1966} we ""ill assume that it is also valid for stars with larger masses. The semi major-axis and the eccentricity after the explosion are giv.en the formulae {McCluskey and Kondo, 1971V},

a"" ilo(C,i-/J)!(a + 2{'l --. y - I), (7)

, (lo{ I +:0:)I _. e- = .. -.--..... ----.; , (8)a(,:x + fJ)

where 0:, Ii and yare paramclcrs defined as follo'ws,

tJ.=M2/,vl l , p= I, =I} (9)

') Nollce lhal thl' factor He in the dCnOmin"lOr of F:q_ 16i oj' )I,~CChL'ikcl' .ano Kondo is .111'0 [() all error and should be ornlllcd.

1

1

u 7

~~·'·l

1

2

4JI

fOI" I~ compuH:d with \51 and G, the conslant Kepler's third law transforms Eq P) into a relation of lhe ~<"'"""k

v,rhere P and are the pernods bdon.~ and after the n respectively.

1\. All Applicatioll to Cen X-3 and Discussion

\iVe \,I.'ill now app~y the equations of the foregoing secl ion to the case of Cen X-3. keepi,ng in m:ind thill, in this wav, the effects of impact are likely to be over­estimat~d. We adopt the physical paramelers of the Cen X-3 system determined by Leach and Ruffini (1973). SlIlee the inclination i is indeLermined within wide limits we calcuLated several examples but will a"sume i~. 70" for compact discussion. Then the masses of the primary and the secondary are about 20 lHu and 0.2 Jl,1o , respec-

The secondary is it very Imv-mass neutron star, From Fig. 1 one can see that for 1'111 = 20 Mo tbe possible values of the mass ratio before the expllosion are 1.25 ~ 'J. ~ 5. or 422 1\1! ::5 16 11;10 , From these values one can easily sholv that the amount of maHer ejected by the supernova explos~on must be .18?: IM , - 1\.1{j;;2 15,g Al,J' Therefore the neutron Slar in this systen1 is morc likely to be a remnant of a Type II supernova than of arype I.

1-'---""-'-­50 L

F,g. l. 'I'll" Jomain of the possible values of the mass M1 (ill Mo ' and the mass ral io ?: before the explosion in cas,es I == 0 and J = 0.3. res peel ivdy

The orbilal period of Ct~l~ X·J is -:::: 2.nR? A recent report by Schrdt:r l'f ai, (197)) shows lhat the orbiad period of ('en )(-) is 11\ II nIle of about J s pCI' ye;H, but still 1I11,ore recenl obS{~rVilli()ns (G,iacconL 197J) show thaI lhe p(~I'iod is increasing llgll in. Tiler,,"

if the period is (which seems nOlal illl c,~rhlin 110w) 1m c:slilllat(~ of the: nlle of the decreast: will be II t less than J. Van dcnl lieUI'd and ,Hdse ( 972} sug,gesl lhal Cen X-JIUkS l)l~conlt~ an X"l'av soun:e after the prinlary, origmalty Il1l: s.c:co;\dary,IHis in 'tUrn e\!ohi'l~d 10 fill its Roch~; lobe: llnd the: second of mass lrallsk'r ,)ccurs, The rlldialior~ is prodnoc:d !"rom the tnmsfei'red mallcr accreted by the lleul.I"On SltH Ir we adopt III is suggestion thell Ihe age of Cen X~.3 as an X-ray source cannot bt~

much larger lhan 2)( 10" yellrs, SitlCC: lhe rapid [nass Iransfer can only happen Oil a IhermaI time scnl,e of tlh~

20 M) cmnrlOnenl, which is (1f Ibis order (Pacl':ynski, 1971 b, vall den Heuvel and Dc 19B}, asswlliing an upper lin1i! of 3 s/yenr ror 'lhe rale or tIlt: decrease of th,!:: period we Ilnd that the orbitlll period in the past. Jusl after lheexpl()sion, did nol exceed 2.x days. On lhe other hLlnd, it SC'l~nnS u111ikcly thaI il '\IllS

much less Ihan about I day since otherwis(: the system would have be,en iJI coni act. The observed velocities of matl.er eJected by arype II supernova are in the order of 109 (ShkJovsky, I%H}, Vv'e first assume lhal I~·,. j = I x IO'} cm/s, f·'or the phys ieal parameters of Cen X-J above we calculatt: Pi as II

function of P from Eq, (IOj (Note: PI is nOlll1e preset'llly observed value but the value just u.l't,er tJw explosion). The result. is shown ,in 2 for !1 cec :1, 4 lind 5 respec­lively, It is interesting to nole Ihal lhere is II minilm.ll11 value of Pj' for each value of .J: (we will denote il as

'~~~-·'~·_'····'-·l8 p~

(days) 7

6­

j 4 j 3

2 ~ r: ~02 ~: 1", Vr.~ ~'1)l.109c.m/!i

...L....._--"__-'I-__-'-- ..-----II__-1-_ J 2 3

P(dioys)

Fig, 2. The orb!la:1 period <Iller the explosion verSLlS I.he initial orbila.1 p~riod. The number Ofl each curvc indicatcs the value of !t,

No!ice thatlhere is il minimum final' period for each vatue of (j

1

4

23

342

l'Ok"" {·[joys)

6

'"

'2

5a 1·1iJ,. 3. Tbe mlmrm,m filla~ period verSII, the ma,s ralio before the cxpl"sl:oll. The III umber on cach curve mdicalCs (he velocity of tbe cJec!ed IiTlillicr m cmls. The horizonlal Iinc Jabekd ,hows tbe upper lim ir of Ihe period of Cell X·.l JUS:I afler rhe explosioll c,timaleu from the observational data (SCI: lexl)

The existence of this milllllltUn final period is due La Ihe effect of Iheimpaci of matler onto Ihe

unexploded component Til is effect becornes very strong if the initial period is very smaiL As one can see from Fig. 2 the value of P/ nlii ", is smaller if :x becomes larger. The dependence of PIm;n on :x is shown in Fig.:I for ex ~ S (as argued above these arc the only possible values of 0( for A12 =20 MDI. 111 this i1gme we also give the resul'ls for I/",) '''', 0.7 and 1.5 x 109 respectively.

For larger values V,'j' the values of the rinal period for a Ihed value of IX are also larger. We have ,drcadiy discussed thai the [inal periioc! of Cen X·J after the explosion probahly was 1101 than aboul 2.8 days (Wlllch we denote as PI I'm,,)' From Fig.3 one observes that in order 10 have PI"'''' ~ one must sct the condition Ii ::;; I X 109 Cl1ils. On other hand, ir we assume thal. the orbital period is constant since the end of (he IilHl 2,087 days), the upper limit of l:'i is IBore likely to be 0.8 x 109' em/so The above resul,tw,l,s lim.ited by the assumption Ihat the inclinalion of the orbital plane is about 70 , Values of A4;; and M{ were calculated by Leach and Ruffini (1973) for the cases i "" 90" gO, 70',60,50',40 and 30, respectively. For these i values, results in Table I are based on [he assumption that the upper limit of Ihe

·orbiltal just after the explosion is 2.8 days do not do the calculation for the case ,i =-:10 since il

= 120 Mo which seems 10 be unhkcly). Our results do 110t depend sensitively on the inclinatj:on: in all cases \ve lind nearly Ihe same reSUlt, i.e.. [hat ~ I x I em/s. Thus bOlh ejected mass and velocity are characteristic of Type \I supernovae. Therdorc, keeping in mind tha.1 in our equations the effects of impact are to be

24

Tabk L The upper IWllI of the velocity of maHC!' ey~cted by the ,uperrW1>ia explosion in Cen X-3 lor ViHiollS possible phySical parameters:

i ~)

'fO 16.0 0.275 OK x 10" kll 16.6 CL250 ll.9" 10·' 70 19.1 0.220 !.o 10" 60 24.2 CU'!4 1.1" 10" 50 lS 0.163 I J) ~ 10"

40 59 0.176 o.~." Hr'

overestimated (which means thaI the upper limn or Vc ) should be somewhat higher than given .in TabIc Ill, we can safely conclude that after a sllpernova event with a characteristic ejection velocity of the order of 109 cm/s, the Cen X"J system may have remained closely bOllnd.

Ackl1mvledYl'lIle1l1s. The author is indeb!edlo Professor E. P. J. I'MI

dell Ih~ovcl 1'01' his cont.inuous guidance dunng Ihc course of this inves:lig,Hion. ae also lhanks Dr I-I. R. E. Tim A Djie for diSCUSSing Ihis paper The scholarship from the Dutch Ministry of Education alld SClelKC is gratcfullyacknowledged.

Re~erellces

Ametl. \."1.0.11973, Lecwrcs al lhe Advanced Swdy lnslillilc on Physics and Astrophysics of Compact Objecls, Cambridge, England

Boersma,J, 1961, BuN. Aslmll, II1SL NeilL 15,291 Cameron, A. G. W., CamJ'lo, V. 1973, Report presented at the XVI'11>

Solvay Confcrence, University of Bmssels, Sepl. 24····28 Colgalc,S. A 1970, ,'Ii'U(Ufl' 225.247 Gi",:,;olli .. R, 197~, Repo'!'l presented at the XV'l'h Solvay Conf"rem:e.

Uliliversily of Bwssels, SCpL 24 28 GibbOlI'. (} W., H"wkinll,S. W. 1971. NI111UP 1:32. 465 Villi den Heuvd, E. PJ. 196H. 13,,1,(. /!s(I'O/l. II:1's," ,'Ilelli. 19',432 "an dcn I-Ieuvel, E, PJ. 1969, rh·lron. f 74. lO95 van den ~IeIH!cL E. PL l-Ieise,J, 11972. Nlllw'" r~i1)'s. SCI 2311,67 ,'an dell Hcuvci. E.I"J 197 J. "i/llw'" PllyS, Sci. 2,42,71 van den HClivel, E. P J., De Loore,C. 1973, ,-1>11'0/1. & "1.w·o/,lIy,,.

25.387 L~ach.IC Vi.. R11I11n.I.R. 1973, Asnopl.y". ./. Lmers 180" L 15 McCI.lIskey.G. E, Kondo. Y. 1971. ASlrnphys. SrliJce Sci. 10,4(,4 Paczynski, B. 1971 ,L A"1a ihrrOiL 21, 1 Paczynski. B. 1971 b, Am' ReI' ASlroll.'& ihirol'llrs. 9, 18.1 Plonec.!VI. 1968. Ad!". AstrmL Asrm/lI'ly's. 6,20 I Pringle,.!. E" ll.,ccs,M.J 1972. Allro". & ASII·ophys. 21, l Ruderman. M. 1972. !I'm. Rei'. 'h/fm' & ASII·0I,hr5. HI,427 Sducier, G'al:l:oni. R.. Gursky, 1"1., Kellog, E.., Lninson,R .. Tanan­

ballm.I-L 197.1, I.A.U. CirclIlar Nr. 2S24("pril t7)

Shklo\isky.~.S. 1968. SUPCII1MllC, 1l1Tcrscicnce, New York SmiTh. L. F. 1'l6/i, Woilf Rayci. Sial'S. Ed,. R. N. Thomas. K.Il Gcbbie,

Nair lillI', Sid, Special I'IIh1icnli(i'II 307 .. B Srolhcrs. R. 1966. AI ,1mI'll \'S J.143. l) I Slitanlyo. W. I'J7.~, ."1,',/(011, ,& ihfI'Ophl.'.I. 29. IWI YUlIgcl'son., L R. 19/.\ S"n"l ;blflill. 111.8,64

IN. SlIlllIHyo

SI<:rrenkundig Illslilulil Ul1i\'~rSllen van Amslerdam, ROcI<:rsslmau IS Amsterdam. Nederland

(Provisional page numbers)

Supernova Exp£osions in Close Binary Systems IL Runav.iay Veloel of X-ray Binaries

Sllmma.ry. The effects of a sphcl'icaHy symmetric ex­plosion on the runaway velocity of a close binary system with an initial circular orbit IS conslderea. It is shown thaI the runaway velocily is completely deter­mined by the final orbital parameters regardless of the Initi;d londilion. The galactic:: distribution of thc knowl] massive X-ray binaries indicates lhat, lhe I'Ll na W,~i\ vclocit ies of these systems arc very probably ,m!allcr than .~ 100 km/s with the most like,l!y values of .~ 25 50 krn/s. Such runaway vdocities call be obtained If the pilsl-cxplosion eccentricities are less than ~ 0.25 This [hell h;15 the consequence that the mass of the expl.oded Slar which produced the neutron stars in the massive X-ray binaries can in most cases not have been larger Ihan ~ 7--8 A10 with the most likely values of ...... 3A Ale if the supergiants in these systems have

L Introduction

The discovery of Iwo pulsars in supernova remnants (Crab alld Vela pulsars!, suppmts Ihe idea raised several decades before (Baade and Zwicky, 1934) that a neutron star call be formed by a supernovaexp,losion. Also the large space veloci.ties of the order of 100 km/s

and in som.c cases > ]00 km/s observed in some pulsars rrrimblc, 1970; Ewing ('/ 01., 1970; Man­chesler d .111., 1974) indicate thai the formation of these objects is associated wI:lh catastrophic events SllCh as supernova explosions. Two X-ray pulsars (Cen X-3 and Her X-I} and other compact X-ray sources have recently been discovered to be members of binary s)'siems. It is generally be­lieved that these compact objects, ~ike radio pulsars, have been formed in supernova explosions. It has been shown thai such systems may remain closely bound after an explosion with a characteristic ejection velocity of the order or H)9 em!s, even .if one takes the impact of tht~ supemova shell onto the unexploded star into ac­count (McCluskey and Kondo, 1971; Sutantyo, 197421; Wheeler r:1 ul.. 1974; Cheng, 1974). This is due to the

• On kaw from BlhSdlw OhSCrVi'!Ory, Balldullg IllslrlutC' of Tc·.:lmo!ogj. Lemo<l,jg_ Ind"llc,ia.

mass (A1~) of..... ,\<f(;). For Cyg X-I. the llpper ImlSS

limit of the expl!oded star is found to be " lfl M(). For == 30 !lit" these upper limil becomes ,,9\0

and 19 respectively. An argum.cnt is given suggl.'stinl;l that tbe explosion in Her X·I cannot have been lriggered by .llHlSs transfer ilnd lhal the SystCI11 most II:kdy origirlll.tcd from a PopUlation I objet:!. Assllmilng lhat the mass of the l:xploded sUtr was large.r :thulli ~ 4 td(}, the post-ex rlOSHlIl cccenHicity Ililust hall'~'

bCClt larger than ....,O.7, andl.lw runaway lIelocity is found to be "-' 150 km/s. Slich II vdoci.IY is approxi­malety the one required ror ex pl.nhling Hs :prcsent distance of about 3 kpc above the gllhu:!ic plane.

Key words: supernova explosions --- close binadcs X-ray sources

filC! thaI in a close binary system tbe c:~ploding star will always be the less massive component as a conse­quence of the mass transfer preceding the explosion, The tidal forces I1:HIY then circularize and synchroni:.zc Ihe orbit within the time scale 01'.5 x IOh years (Lea 'lI1d Margon, 1973: Sutanlyo, 19740: Wheeler 1:'/ al,. 1(74). If the compact objet:ts in the X-ray binaries did Indeed originate rrom explosions, we ,expect that these systems should have acquired additional sp~tce vclociti:cs (run­away velocities) as a result of l.hc explosions (Boersma, 1961; van den f-1cuvel, 19(8). In this paper we wilJ con" sider what runaway vcIoeilies one derives for I he X-ray binaries from the observed galactic z distri !:Jutian an4 we will then conSIder model explosions which may produce these val ues of the runaway vel!ocities.

u. The Runaway Velocity

We consi:der the orbilal motion of the components with respect to the original centre of gravity of the system. We denote the exploding slar as slar I and the uncx· pJoding star as sUn 2. Let !HI and M{ be the mass of star I before aml after the explosion, respectively. The

25

2 Vel SUlal1lyo

mas, of the uncxpJoding slar At2 is assumed 10 be conslanl dunng the ex Let the line connecting the two stars at the beginning the explosion be the x-axis and the orbital plane be the (x, yJ pl<ine. LeI the velocity components of both sIal' after the explosion be

, r·D and i1.l~, v~), respectively. The velocity of the centre or gravity of the system w.ith respect to the Miginal centre of gravity can then be calculated from a general formula,

OJ

where

!l4{d + /\12 ... - ,\1 -1- M(

2

We call 1<1 the runaway velocity of the system.

HI. Sl~hcli'il'any Symmetrk Explosion

Lei us a,sumc that the explosion is spherically sym­metric. 'Two kinds of explosion ,\lin be considered here. The fir'sl is an instantaneous explosion where the ci'recis of ,inlpal:t will be taken inlo account; and the second is slow ex plosi(jl1 where tile dreet, or iirnpacl will be neglected so thal the change of the orbital elements is only due 10 the mass loss from the system Since the explosion may take place on a tllne scale much ShOnell' Ihan the orbital period probably the assumptionl of instantaneous explosion is most reasonable. Inmost of t he discussions below we wid] assulTIe Ihis type of l:xploston, F'or beJth above of 'explosions we assume! hat the orbit. before explosion is circular.

II1SlmUi:II/('liIiS Explosion 11'1111 lmpacJ

In this case the sl'mi-major axis and the ecoentricily of the orbit ancr the explosion are given by (McCluskey and Kondo. 1971),

a'''' (/,,{:x +, /i}i('1 + I) • (2)

, lI,n{ I + cd t-· ('­ co:

(1IOI!- m where lin is the initial distance ibetween the two COI11­

pom~nls. 'fhe param.c'lers tl. f! ;ulid ',' arc dd'ined as ldlows.

'X "'" lI/lz 1\4 1 , fl '"" kli ;\11 • (4)

Ii.. is the velocity eOlnponcl1l in the radial directIOn of siar 2 imparted 'mailer ejected by the explosion and G is the lal:ional constant. Consider now the movemcm of the two c()mponenls with respect [0 the coordinate system defined in Sec­tion IL The velocity conlplments of star 1 and 2 before the exp~osion arc (0, - :\12 V!/M1) arid (0, 111) respec­

?6

livels, where V2 = [(;jl"f I2/{0oIM l + ,M2Hr i2 The veloc­

ity components after the explosion willi be (u{. r\) and (u~, 1'1) respectively, where,

=0, l'l = - AJ2 V1.!llyl, ' (5)

v{=v~.

Inserting (5) into OJ we have,

V;l = M2 [11/ + v} (l - M{j lll!, )2] li2 /iAI{ + ,\;t1 ) (6)

Wi,th the aid of Eqs.(2), (3) an.d (6) one finds that the runaway velocity of the system is given by the formuta (see also Gott, 1972),

, . M eJ 2= 211 G ---".,- ------~.---""-.---------"_._-.- (7.1

Pr(l- ( 2)311 (I + M{;fl,c12 )111 '

VI/here Pj is the final period, Therefore, the runaway velocity can be calculated entirely from the fin.al parameters.

Slow Explosion wit/lOW' f m[Joc.l

The solmion for this kind of explosion was computed numerically by Boersrna (196 nlor the case where the exploding star is the more massive component and by van den Hcuvel (196g) for the case where the exploding star is the less massiivc component From the anaJysi~

given by Boersma olle can show tbal,

where 112 is the initjal velocity of star 2 with respect te the original centre of gravity of tbe system. Further· more:. one can also derive that,

(9

Equations (8) and (9) are independ'ent from the HlOd( of mass loss. From Eqs. (8) and (9) one can then. ag3in find thatlhe runaway velocity of the system can be ex" pressed by Eq. (7), I·!ence, for both above considerec models of explosion Ihe rwwwaJi ue/oeily is c0I1'Ipletel.1 de/ermined hy the filial orhital parame.le/'s rl'flardless oJ Ihe illilj,ul condilion (exCl:,pt F)r fll£' nnuilion eo = 0 \vhid I'lwsl he ,wJI'isfied in I his i'IISe) ami i he del ailed lill!!

profile oj' the explosion.

IV. The Runaway Velocities of X-rny Binaries

From the galactic;; distribution of the known X-f3~

binaries we can make rough estimates of the runawa~

velocities of these systems .. In Table I we give th. galactic coordinates, I he esti:mated distances and the, Yalll{~S of the known galactic X-ray binaries. As we cal: see from the table the values oil' :: of all rnassjvc X-fa: binaries (we do Tlot consider Her X-I as a maSSIY' system) arc in the range ,,65·--135 pc. According t(

I·;-~hl:..· G].la,l~'l c(I()rdhn~U,l~5.. dtstances ani-I (ii~k~n\.'(~s (ron) th{~

g,aJai.:tlc p~~mc ()f ~ilc paL1cHC X-ray bUUifHL"S,

d A"'" I! In( Cell X-J .~ U 17(}O

~91 067 -,'ilyI5

n 161 2 19.1

HHpc, 1'1

Cy~X~i'J 71 :118 l084 2.5 135 Vda X-I "I ~6l065 I,J7 IkrX·1 58.258 J084 ----_..­.._-----------­"j Rjl'brd. 1974. !~) Hi~nsb~rge 1:'1 (,f" 1971 ~ I Bergman (',j I.A. 1971 ,j'~ IUlderWii!k d ~il' 1974

'~ SlnHrnall'l::T <'f ur. 11..)7.1

van HCllvel and Heise (1972; sec also Dc Loon: CI the agcs of massive X-ray binaries since the ex al'e of the order of 5 6 million years. nle valiucs of K: derived by 001'1 (1965) anecan show that III order to move aul of the galactic planc over a dislallCc '~. 130 pc in 5 x [0" years, the initial z velocity should be oilly -·25 kmjs Thi,s implies that the most iii run,llIiay velocities of these massive systems are ~. 25 50 is. The~-eforc, the runaw'ay velocities oflbese

sV';I.crns arc certainly lower than those of pulsars 12.: 1.00 " cr. Section I). The tendency of strong X-ray ",)urccs 10 Ix: concentraled in the galactic plane (Giac­coni eI al., 1974) also indicates that in. general the veloc­ities of Ihese objects are not very high. Since the estimated distances of the X-ray binaries listed in Table I may be uncertain by a factor 2, we will assuille for a general discussion below that the runaway veloci;lics of these massive systenls are ::S 100 km/s, The large z value~ of l-Ier X-I j,ndicates that the system has acquired velocity .~ 125 km/s if the system originated in the galac1ic piane (such a high velocity is necessal'y in order 10 overcome the galactic gravitational field),

V. Tbe Post-Explosion Orbital Parameters

Equation (7) can be applied Lo the X-rav binaries. I-Iow­ever, in applying .this formula one should not adopt thclrpreselll orbital parameters, since beLween the moment of explosion and the X-ray radiation stage tidal forces may have changed both I.he eccentricity and ~he period to the presently observed values (SuIanLyo, 1974b). Of COLII'Se, it is impossible to trace back the exact or bit al pa rameters of t he systems just after the explOSIOn. One may, however, make reasonable esti­mates of these values. In order to estimate these pantm­eters we assume that during the tida~ evolution the total angular momentum of the system .i.sconserved and the tidal evolution has lXeD completed before the primary (star 2) len the main-seq uence. The total angular momentum of the sys1em is given by,

H = A pi '1( 1_ ( 2 ) li2 + B/).P, (10)

J. is the ratio the rotHtlonal pCl'iod of slOlr:; and the orbi,t,l,l and Cz i,s tile mom.enl 01' inert.ia or slnr 2, Assurning thai attht: slate of the lidal evolution t' "'" n, ;, '" I and P = observed perwd). OI1C can detennin..:: H. heal1 be Ihal fOf a given value of H /lnd for A> 0, there an:: rca I values p' as rools of ( 0) If and on ly if:

(III

Assllming vallics (' and ,1 \ivl1icb fulfil till: concllt.lcm ( 1.1 j, we can calculate the orbillilperiod frorn (10), These val lies willi he lldioptcd 10 he the posH.Jxplosiol1 parameters of the systenl.

"I. Application l(l tilt, X·RI.lY Ilill~rIt'!li

We will apply Ihis above prcKcdure first to the Il.HISSIVe

X~ray binaries and then to the l·ler X-I system,

The l'lif [Is.~it'e X-fav Binafil's

We will Il,rsl assume 'I hat. mass 01' the unexploded star ,~42 20 Mo. We assume three Vl111iJl~s of the explosion eccentricity (0.5,0.25 and O. L for ~aeh s~st~IlL It is very likely thaI after. . explosion 11 <. I. rim IS due to the facts that the explosion wi:! I increase the orbi tal period and thal the rotation was probably synchronous before lhc expl.osion, 'fbe com­bination of e and): shou,ld fulfil several conditions. First, the values of e and 11 ShOilld be in the dornain where the eccentricity i.s during the tidal evolution (Alexander, 1973). Second, the condil:ion 111'1 should be satisfied. Fimdly, since I, mighl ahm represen; the ratio of the .orbital periods before and after the ex· plosion (assuming synchmnizHtiol1 be~~lf(: the ex­plosion), the condition thaI the pre-explosion mass of the exploded stal' should be <:: 3·-4 ll-1", (van den HClIvel and Heise, 1972; Arnell and Sehran~Jn, 1973) gives an upper limit to 1. For Cen X-3 it is impossible to find values of e and;' wh.ieh satisfy all th.ree conditions if tl)(; mass of l.he compact object is less tha.n -1.2 Mr.,. F'or this reas0 111 we assume that M{ ~-" 1.3 Mc) fo'~ this system. rt is very well possible Ihat the tidal evolution of Vela X-ll has not been c0!11Ipieted yet (there is stin eccentricity in the orbit; H utch.ings, 1974; Zuiderwijk et ai" 1974). In this case the assumption e=.O and ), ,= 1 at the present stage is certainly wrong. However, since we arc onlyi l1lerestcdi,n the global piet LIre of the ex­plosion and since the calculation of PI gives only a rough. estimate of lhe postcxplosion period, wc will retain the above assumption also for this svstem.,

27

-I \~,/, Suta nt:'lO

T,Jbie ~, Ad')PI~d nbri~rvcd a,nd pQJ~n~{;,~piO(;j(iill pa(drncten~ of the X~ray bl!11aric::i. C.i:t'!CUhilCd ru.na"""i~Y '\;'docitl'lb), and UPlJel rrM1S:\ IIm::il of the ,",pk,d"d "(,if I'll" Ir.. ,c.'" of the prirnafylsHlf 21 is assumed (0 btt 20 Mi ,:, for !hc mar,SIVC syslems and 2.5 ,~ ..if. for Her X·I

PrlcH ----- ..._~-------

( en X·J I. 1 2.nx"! () 50 0.25 fUn

0.49 I 0,61<6 (J,Xl2

2.469 1767 LXXI

D6,66

I OJ 4' J Ii

11/)5 6.63 1.4.1

.?cO .'AI2 050 (125 0.10

0524 0732 OX11

5.072 )(".1 J ..H5

11<2.14 9:1.'117

.\6.11

IH~1

151)

4.20

'-;\<1C X·I 15 UI'J1 0.50 (US O.JO

0.50i) 0.699 0.R32

5.71ll 4.i.mS 3.1<31<

17HS XX93 35 ..'4

1125 61\8

365

lJ.n 56(fJ 050 0.25 010

0.502 ()7UI

O.S27

~.5').)

6.149 5.675

127. L3 6J,57 1541

2350 Hi.lS 1190

20 X9W OjO 0.25 fJlG

0.524 0.7J2 OS.1 I

13679 9.788 9JH7

1.1090 65.45 2616

13.00 750 4.20

Ik! X·{ I 1 l ..7Un 0.85 (US

(J,050 (1)99

I 12J.1 5674

J57.44 j 38.92

4.53 4.15

1'/ !c1~ V;,.(km!-j MI rr:lH'I,~{.;))

CenX·J 16 2.01<7 050 U.561 2.529 270.74 1740 (1.25 071\4 1.80') 135.37 9'.50 0.1 () 0.851 1,6611 54,iS 4.76

J U PIHI n 1.0 .1.412 0.50 0,54.1 4()76 214.25 18.00 0.25 0.759 3.561 lOTI J IO.DO 0.10 0821 3.281 42.85 5.10

SMC X·J 1.5 11'9.1 O.SO 0,S26 5551 20875 li7 .25 0,25 0.735 3.973 10437 tUg (l.10 0.826 3.716 41.54 4.65

'i0 5607 OSo 05J4 g.54!l 15657 2K.50 1125 0747 6.142 71'.28 nus 0.10 01';24 5.66J J 1.31 12.90

Vdu X·I 20 8.9S'I 0.50 O.54J 1:'1629 IS3.1 J Is.no ()2~ {US9 9.752 76.56 1000 0.10 (l.S2J 8,')87 30..6.1 S.lO ----_._.__......... - ........_ .._...._----_._-_.._--_ .._ ...._-- ­

Since ;"in Eq. (2) rnllst bl: posillve, we can set an upp,er nclllron stars in these shoukJ in genera! have lirnit Ifl tile ITHISS of the exploded star been _. .3 4 . If we assume MI = 30 Mu rather than

20 ;'40 , these above results will increase by - 2 Mo:\1 1 mi" {' ( + MIl + ,M{. (12) (Table 3). FlIfthermme, I!\IC can also show that for small

The paralll.L:I.crs of 'l he syslen1s, the cakulalcd runaway fil1lai eccentricities it is very well possible that the posl­velocities, and lile values of ;\1 1"",' arc listed in Table 2. explosion period is less than the present period so that As we can sec from Lhe table, runaway velocities during the tidal evolution the period has increased :;; IIXI kn'lis can be obtained :ir (':;;:: O.. 25.rhis then has while the eccentricily decreased. the consequence lhal.. according 10 (12). the mass Using the combinations I! and ), given in Table 2 (as of the exphHJcd star in the sysh~lllS with :Iow mass cmn- is alrc,:ldy dis(;usscd ) is also likely to represent the

obJects (neunon strus with td' - 1·,2 i\4,~)) cannot ratIO of the~ orbital periodS before and after the ex­have been .inuch higher than...., 78 !l.4(), For Cyg X-I. plosion) we calculate 1\4, and V, from Eqs. (2) and (3). of which the mass of the compact object is assumed to The results arc given in Table 4.. As we can see from be 9 Me (Avny and Bahcall, 1975), the limit is ..... 16 fHe . the :In ble, the parameters hefore the ex plosion are con­II' one assumes thai the most likely runaw,-iy velocities sistent wito Wolf-Rayet binary progenitors of X-ray are 25- 50 klll!s (er Section IV) {hen (' ....... O. 10. This binaries (van den Heuvel, 197j). The resuhing values of implies that the rnasses of the progenitors of the II; can be used to calculate the ejection velocity of the

r-,!hi{~ 4 PIT~\,.C\pl"~~IOf~ raran'k't~;-TS (~r I !hI;.;: X·r~.) hilH,ric~ (.;:t'ktI1al~;d

hL),~n panHlX'h,;r", t!1't'::n fl1 Tahle 1: Ii 1:-1 dl(,~ l~CH·t~spondi,ng pn&t­

l.:'xpJosion l'\({'i,.'nl [].t.::1 tJi

--_._.._._"'~_._----~._-~~-~"_._~_._--_."._~.-

l 1..'j'l \-.1 112~

0. 10 I ) 12 I 564

../,,0 1 Ii"!

1!l073 21 5H

0 550 II 3"'('l

~ I.: I jUf~ ~7 Ik25 0 HI

~_659

2.XOS

~.so

~_95

R9.2~

lit50 I 124 051 :.

:­,\R' x- I 0.25 0 10

::8;;6 \ 191

·t50 I .40

7\U2 I 7.7B

I I 11 0.569

t yg X- I 025 () 10

4.1 10 4.691

I 300 I 1.65

·.'(I.S1

14.91 I 576 066.1

\' l,.~ 1~l X-I 02" O. 10

7 164

7.5 I I 4.50 J9S

64 i .1

1332 1.905 ~ .OM

HL'r X-I 0.85 0 -,,;

056 I 0.561!

4 00 4.00

I 3 .15 67.60

2.753 I .767

supernova shen '<'j' Applying the formu~a derived by Colgate {1970j \ve find, thai ~ 0.3 10 x 109 cmis which fits very well: with the observational data 011 ex­pansion veIocities of supernova shells (Shklovsky. 1961:\). \""'c: woul,d IhJle that if one takes IIllo account the blow­ulT (;!Inaterial from the oulerlayell" of sial' 2 due 10 the Impact (Wheeler et (/I., 1974, 1(75) all parameters given 111 Table 4, ex.cept V;'j (which will slighily increase due to the decrease of the cross section), and all above conclusions remain unaltered. This is true because only a negligible amount of mass is lost from the unexploded star (Wheeler et (/1., 1975).

Hen'liles X- J

The I-leI' X-I syslem consists ofa normal star of .... 2.5 Me and a compact object or ~ 1.3 llll10 (Crampton and Hutchings, 1972; Bahcall 1'1 aL, 1974). From the total mass Iiself it is unlikely that the system belongs to Population I r. "rhe high galactic latit ude of Her X-I [lwl! indICates thal the system must have Inoved out of the galactic plane where the system has originated. Since the latitude of this system is far higher t.han those of the massive X-ray bi naries one Illay conclude i hat the syst.em has a fairly large runaway velocity and the system should be much older than the massive systems. Based on 111(: later argument, it seems 10 be untikely that the explosion in Her X-I was triggered by mass transfer [the type of explosion considered by Whelan and Iben, (1973); in this type of explosion, the exploding star is a \vhitc dwarf which is pushed up above the Chandra­sckhar limit by the mass transfer]. If tbis lype of ex­plosion were responsi ble for the formation of the neutron star in Her X-I, it would not need a long time for the system to become an X-ray source. As one call show from Eqs. !9l and (10). the period before the cxp~osion

(I.e., Pu =. iPfl is uniquely de1erm.ined by the present parameters (char:H.:terized by A11• M{ and Ji) and the initial mass of the exploded star A1 1 (regardless of the

cllcels of irnpacH [This is due to the llte! that the sy:aem ~ll1er the (~xplosion muSl have had Ihe StUI'll:

ang\.llar moment urn as t he This cbno

In 111m, ~lxcs Po.] Assumi,llg M! = 2.5M l '" 14 r(~quin.:d inlhc CHst the \JVhelan-lhen and = 1.0 we I1nd

"". 136 d. '['Ilis pennet is so largt: 'lIUII mils:; t ransf'er which triggered lheexplosion should hav(~ happened during: the post main-sequence of the primary. Ho\\/cver, in that case the ell.vdol)C of the primtH)' would have ctYll'tlnuca 10 t:xpnnd on a tht:mlll1 time

followiinglhe explosion. As Ihis tilm: scale is only II levI" millIon ycnrs for a 25 shU, this would irnply t.hai·j List like in !be l1i\assive systems - the X-rlly singe in. li~h;r X-I starlcd! wil !lin II few million )Ielll'S the explosion. Sincelh\': n~sulting flJlHlway vdod!y W(lllid

be at 'most - 150 km/s. Ihe maxirl1lJ,m distance Oller which tile syslem could have moved out of the Ib,un\,.uu\,·

plane would in thaI case be only",· 750 pc, For I his reason il is vcry unbkd,y II\li! this type of eKplo:~ion has formed thl~ ncutrO'11 star in Her X-I. This them In(~Mns

that the ncullron star was probably formed in II type or explosion similar to that in the massive systems (Lc". not triggcTcd_by mass ! ransfcrJ, rcuch its pn~scnl

distance above d1l~ galactic plane with an inilial::-vclm.> ity component 01' ... 150 the needed al leaS! some 20 million years. imp.lic,s I hl! t certainly star 2 was still on lhe [nain-sequence at Ihe [nomenl of expliosion. In this case the time inter\lul between lhe momenl of cxp~osion and the X"ray radiation stagli: is about thc Inain-scqucncc lifctiJnc of the pll'imary which is some 3 x .tOS years for a 2.5 star, Therefore, the system is dearly much older Ihan the massIve systems. Assuming lhal also here the exploded star was a helium star, and afi::;uming a lower mass limit for this star of ,. 4lHcJ (van den I-kuvel and Heise. 1972; Arnell and Schramm, /9731, we find that the poSI. explosion ec­centric/I.y must have bl~en higher than "-' 0.7 (cf. Table 2). The runaway velocity of the system wilil then be ~ 150 km/s. With this velocity, the system may during its lifelune have undergone a few cycles of ()scilhllion with respect 1.0 the gahH.:tic plant: with ampliwdel)fa lew kpc (cr. Oort. 1965). In Table 4 we gi!ve the prccxplosiol1 parameters of the system..

VU. I;>i,scussion and Conclusions

From t.he ralher small runaway velocities ofthe massive X-ray binaries we conclude that in general, in the systems with compact components of ~ 1·-2 Me. the masses of t he exploded stars were not much liarger than ..... 7--8 l'vl0 with the most likdy values of JA Mo for M2 = 201,\;10 , In Ihe case of Cyg X-I, where the compact object is generally believed to be a black hole (which is assumed to have 9 M(O) here). thc upper mass limjlt of !he exploded star is ..... 16 .fvlC).These results will increase by ..... 2 1\10 if we assume Ml = 30 Mo.

29

I

W. SUI3111yo

The C.x which produced the neutron slar in I~er X-I seems to have been of a type similar to the ones ill the massive binaries. This then means, from the .rather low total mass of the system, that the system must have suffered heavy mass loss during the stage of mass transfer preceding the explosion ISutantyo, 1973). An idea proposed by 051ri1er and Pacl.ynskil (private communicalion to E.PJ, van dell Hem/eI) might ,ex· plain the origin ofa system hkc HcrX-I. According to this hypothesis, the original system consisled o'f two normal stMS with a very large rnass ratio, e.g. a star of ~ 15 together with one of .~. I in a binary system with a pt:riod of a ft:w days. 'When the primary leaves the ma.in-sequcllcc the syslcm may lose its sta­bility due to the tidal interaction (Sparks and Stecher, 1974; Dc Greve el' ai" J!(75) and lh,~ low mass star will be ,;piraJillg down onto and into the massive star. ()striker and Paczy,'lski suggest that the heating effects due to lhe large friction when the I approaches the IH:liulTI core or lhe massive star will cause the hydrogen­rich outer Jaycrs or IIK~ primary ito be lost. "rhe resulting syslt:fII wil~ probably be a short-period binary consisting of a helium star, which is originally (he helium core or ilk: (in the case of a 15 lvle' star this core has

~ 4 ), together with a normal low-mass star (the nwss or Ihis star will be somewhat larger than its original value, due to the llccretion).This configuration, e.g, consisling of a 4 Mo l1el.ium star and ~~ 2 Mo normal star, closely resemble 10 the ·..·0.5 d period pre­supernova system of Her X-I which we need in 'Table 4, "I'he explosion of the helium slar in such a system will Ihereforc result in a system like HerX-I. If the helium star Ivould be slighlly more massive (say> 5 M,J)' or if the was sOlnewhat Iargcr, a of Lhis type would be disrupted in the explosion Therefore, the of a I'fer X·I might be a rather rare cvenLfhis type of evolution a syswm wi 1h a vcry large mass ratio can be anticipated from. the classical "Roche-lobe" nmlss transfer picture leI' van den Heuvel 1'1 al,. jI975), In this picture, 1he

down of the small component is clue 10 the mass transfer and mass loss ralher Ihan dlle 10 1he 1lidal instability, .'k~:HOidi'''fW'''''/i1,', Tile 'luihor Illimb "rnles';"1 1:. P. .I. "1m den hlcu<d for h'is advlcc, 'rhe sdwlar,hlp frorn the Dutch Vli,ni,iry ,.1' FJllc;lIiro,', il.ild Scicnces i8 gralefully ad nowlcdgcd.

AlexillH:kr, M, E, 1:973. AsITophl'S. Space Sn 23.459 Amell, W. D., Schramm, D. N. 1973, As/rophy,. J 1.,1'11"'" 184. I AVll1Y,Y" BahcalU N'. 1975lprepriifllj Baade. W, ZWicky, F. 1934, f'IJ}·s. Rev. 45,. :138 lJahcalU N .. Joss, P.e., A''1:n,y., Y, 1974, A.,Ylrophys. J. 19'1, 21t Uo~rsma. .I.. 196 J. BIIll. As/ron. i nsL I'll el h. 15, 291 Bregman.L Butller.D" Kemper. L Koski,A. Kraft, I!; , P... SI,

R./'. 1971 A.,tmph)'". J. Lellef' 185. L 117 Cheng..". 1974. AS/lopl1y'. Space Sd. 31. 49 Colgalc. S. A. 1970. Nalllr" 215, 247 Cramplon,D" Hutchings. .! B. 1972. /h/rophl's. j 1."1.11'1'\ 1711. I Dc Greve.J. P,,, fk Loore.. C. SUlanlyo. 'IV 1975 lin ru.:par,m"nj Dc Loorc, C. Dc (ircvc.J. P vall ,Jen HcuwL L P.L De ellyp':LJ

1975, Proc:, 2nd 1.1\. U !'kglonal Meetmg, Tne';1le (111 prep~r:H'

Fw:ing, M. S, Batchelor. R.l" I'°rddd. R. D. Pnce. R. 1\1 .. SlClchn. n 1970, Aw./'OlliJy., 1. I.e/len 162, I. 16')

Giaccon", R.. I\hlrray, S.. (oursk y. H.. Kellogg, E.. ScllrCICr, L M~l il, r .. Ko~ch.D, Tananballm. H. 1'174. ASll'ophys. J. Suppl 2..17

(J(llt,J.R, 1972, ,4'/I'O'/,hl's. I 17J.227 Hensherge. G, V,1I1 den Hcuvd. E. P..I., Pae,;, de BarrelS, M. H. 1'(

A.'ltOl'I. & A.\fropiJys. 29, 69 van den Heuve!. E p,.!. 196M, HI/II Asmm. InSI. Nerh. 19,432 van defJ HellvcI.E. P.L Ikisc.J 1972, NllIure Phi". Sci. 239. 67 van den Hcuvd.I.. P.J 1973, ,IVa/ilre Phys. Sci. 242, 71 van dCIl HCUiv,,~1. E. 1'..1 .. Dc LOOlC.C .. lamers, l-I,J, 1975 (in

paralion} HUlchings,llt 1914. As/mphy". J. 188, 341 Lea,S.M .. Margon.B. 197:\, As.r rophys. Lelfcr~ IJ,.B Manchesler,R. N. Taylor../. H. VUfJ. Y. Y. 1974, AS/l'Ophys. J. 1.<'1

1119, L 1/9 J\.kCluskey, G. E, Kondo, Y. 19'71, /I.\;U·OpllyS. Spun' Sci. 10.464 Oort"J.H. 1965. Sta.I·8 and Stellar System, Vol. V, Eds. A. Bill.

:ll1d M. Schmidl, Uni~. Chicago Press, Chic<lgo. p. 445 Rickal'd.J..L 1974, As/roplill's. J. Leuersl89, L I Sllklo\!sky, IS t968, SllpCrflOVll~, IlltersCICflce, New York Sparks, W. M.. St,cl:hcr.T. P. 1974. Ai.\/rophys. J. 1!iS. 149 Slrillmaller.P.A .. Seoill,L Whclan.J., Wi.l:kramasinghe,D.T.. W,

N.J, 1973, Asl/'o/l. & A.'/rllp!lv.,. 25. 275 Sl.ilanlyo, W. 1973. A.'lrrm. & ASlrophl'-" 29. IOJ StJl,mlyo. W 1974a. A.\lrul1. ,~ As/wphy.\. ]1, J.W Swanl}'''. W. 1974b. As/nm. c~ A.\lruphys. ]5, 2S I Trimble, V. 970, PuM. Asrnlll. Sue. Pacific 82.375 Wheelcr,JC. McKee,C.F'" Lc.:ar,M. 19'74. A,.I mphI.'. J. Lei

192. I. 71 Whc'c!cr,.Lc.. Lee",r, M. M<:Kcc,CF. 1975. A:\frophy.~. J 1m pI' Whelan,.!.. lben. L 19B. As.rrophys. J. 186, t007 Zui,derwijk, E..I .. van den IkllVd, I:. P. L IIensbcrge, G. 1974. ..lSi

(\\ AS.l!rophy" 35, 35.1

W. SUW.l1lyo S!crn:nkulldig: 'Illsllhml Ulliversiteil vall II Ills[crdam RoelerS,;lnal 115 Amslerdam, NcdcTland

30

On the Tidal Evolution of Massive X-ray Binaries \\1 Sutantyo* Astronormcal Il15l1tute, UnIversIty of i\'rnslerdam

Summlill'J'. The tidal evolution of known massive X"ray binaries is considered in order 1.0 explain why in mosl cases their orbits are nearly circular. Only the tidal inrcrtlclion during the main-sequenoe life time of the primary (whICh is assumed to be a 20 Mo ) is considered starting after the supernova explosion which produced the compact object. AssUllling thai at the final stage of the tidal evolution the orbit is circular and the orbital motion is synchronized wit.1l the rotation of the primary, a lower limit of the mass of the compact object can be determined. The mean viscosity required to circularize the orbit of Vela X·I in a given lime interval is at least one order of magnitude higher than

At present si.x X-ray sources have been firmly proved to he members of bi.nary systems (cf. BahcaU and Bahcall, j 974). Most investigators believe that these X-ray sources are compact objects, i.e. neutron stars or black holes, orbiting "normal" stars. These compact objects, like pulsars, are prob.ably the result of supernova explosions. One therefore expects the orbits to be of considerable eccentricity as a consequence of the preceding explosions (van den Heuvel and Heise, 1972; van den Heuvel, 1973). However, contrary 10 this expectation, the orbits of most X-ray binaries are almost circular (Schreier el al., 1972; Bolton, 1972; Hensberge eral., 1973). Only in Vela X-I (3 U 0900-40) both the photometric study by Hutchings (1974) and the spectroscopic study by Zuiderwijk et al. (1974) indicate that the eccentricity seems to be as high as 0.22. Therefore in most systems some mechanisms seem to have reduced the eccentricity .. Lea and Margan 11973.1 suggested tidal forces to be the most likely cause, Other mechanisms, such as atmospheric drag exerted on the compact object grazing the atmosphere of the "normal" component or accretion of matter by the compact object, have been proved to be ineffective. Circular orbits and synchronization between orbital motion and rotation of the components are common among binary systems with periods less than about 5d (Pl~:!ec, J970; van den Heuvc/, 1970). This fact indicates ~. 011 leave from Bosscha Observatory; Bamjung Institute of Technology, Lembang, Indonesia.

svs(ml'lS due foille orbital period of ; this nlay explain why the orbil of this

has nol yel been circularized. Assum.ing the tidal evolution 10 take place in less t.hun 5 x [06 years

approximated age of Ihe thelowt:r limit of the mean viscosity n:quired to orbit of all systems, excepl Vela X-I, is about It)! 2 to 10 1 .. iii Cl11- I S-l. Such high can be produced by magnellte viscous foroes iJ magnetic with II

strength of a few gauss are present

Key words: tidal evolution 0­

sources

that tidal forces iuay plfty important role in Ihe evolution of normal dose btinary systems. Since the period of all X-ray binaries, except Vela X" I, arc shcu't (P;:;; 5.6d) it seems likely that tidal forces may ~IIso be important in lhe evolution of these systems.

U. The Basic As."llimpdons

The primaries of all massive X-my binaries arc OB supergiants:. We will for simplicity assume throughoul this paper that the mass of such stars is 20 Mo. The tidal evolution is assumed to start jWid aner theexplo" sian. According to van den Heuvel and I-leise (1972; see also van den HeuveJ, 1974) the 'unexploded 5tllris the more massive component as a COl[]isequence of the mass transfer preceding theexpIosiol1. This star was stiU close to the zero age main-sequence when lhe explosion occurred. The time span between the moment of expJO'!iion and the mornent at which the star begins to leave the main-sequence is of the order of 5 x 106 years. After this, the outer layers will rapidJy expand and within a few times 104 years the system becomes an X-tay source (van den HeuveI and Heise, j 972). The duration of the X-ray stag,e is also expected to be only a few times 104 years (i.e., the thermal time scale of the envelope of the primary). We assume that for all systems except Vela X-I. the tidal evolution had been completed (i.e. the orbit bad

31

W, SUlantyo

becornc circular) before the primary lefl the mam­sequ.ence. We also assume that at the final state we have a stable orbit with synchronization between the rotation of Il1e primary and the orbital motion. In such a slate there will be no further tidal dissipation of energy and for a gl ven lot'll angular momentum a state of minimuJrI energy is attained (Koral, 1972; Alexander, I Any reasonable set of values of the system parameters resulting from a supernova explosion wi.JI In general···Uuough tidal evolution·-lead to synchfO" I1IZalion in the final state Section V). Therefore this assumption does not seem unreasonable. There are also ra rer cases in which a stab I.e synchronolls !ina!. orbi'l, cannol be attained (cl: Section V). In such cases a secular change of the period will be expected. One may lhen expect that the orb.it will eventually decay in.ward tow;ml coalescence: of the two components; or decay outward toward escape, although at 11. continu­ollsly decreasing rate (Counselman, 1973). Such changes in p(;riod or X-ray binary have not been reported so far. (This does not mean that we may exclude Sllch a possib.ility since thes'e chang,es are probabl:y too small to be detected by means of the present technology.) Schreier el a" (1973) reported that the per'iod of Cc.n X..J was decreasing at a rate of about 3 s/year; however, a more recent report by Giacconii (1974) shows I his Irenid has reversed and the period is increasing. The reason of this apparent oscillation of the period is most probably due to the apsidal motion in an orbit whh eccel]lricity~O.0037(Thomas, 1974) or due to the orbi.!al motion of the pair with respect to a third body {cL Bauen, 1(73), i.e. not a real change in the orbital period). The <~ssumptlon that a final synchronous orbil can be auai.ned has an interesting consequence. In ordcl' 10 al.tain such a final stale, one has to seL the condition (Kopul, 1972; Alexander, 1(73),

(I)

where NOrl> iiS the orbital angular momentum and !:frill is the rotational rnornenlum of the system. The

r indicates Ihe value for the final state. This condition Can be .applied to set a lower limit to the mass t)j' the compact object provided that we know the mass and the moment of inertia of the primary. Application of this condi tion In Cen X-J yields a k)wer limit of about 0.5 for the mass of the compact object The upper lim it determined frmn the duration and the Roche klhc ge:omciry is only about 0.84/\-10 (van den lklIvel amI Heise. 1972; Davidson and Ostriker,l. 97]). Therefore we have only a narrow range of possible nU1SSCS for the compact object in Cen X-3. However, we have to lake Ihis lower limit cautiously since it is based 011 Ihe assumption that before the primary len the main-seq uence Ihe orbit had become slable and synchronous, which is notcomplelely certain. In the following discussion wev,,rill lor the sake of argument assume that the mass or the c:ol't1pact object in

Ceo X-3 is 07 roughly the mean of the above mentioned upper and Imiver limits). Application of the conditi on (! ) to the other systems is not of interest since it yields very low lower limits fOf the mass·es of the

objects (:5 0.26 A1o ),. while photometric and spectroscopic studies of these systems ha \Ie indicated that Ihe masses are <:. {.5 Mo (ef. Bahcall and Bahcall, iI974). We will. neglect Ihe tidal efj~ects in the compact object since its radius and thus its tidal. defonnation are negligibly small. The rotational angular momentLIm of the compact object is abo negligibly small in compari­son with the lotal angular momentum of the syslem. The tidal and rotational contribution to the rnoment of inertia of the primary (due to deformationj are neglected and the moment of inertia of ! his star is assumed to be constant during the tidal evolution. The rotational axis of this star is adopted to be parallel to the orbital axis. This seems a reasonable assumption since any obliquity will be settled to zero on a time scale much shorter than the lime scal,e of the t.ida! evolution (Alexander, 1973).

III. The Variation oftltle Orbita~

and the Rotationsl Parameters due to the Tidal Friction

To consider the effect of tidal friction on the orbital and rotatioflal paranl1eters we apply the fourth order harmonic tides theory with weak friction approximation devc:lioped by Alexander (1973). All variables arc written in dimensionless ~onn by defining the unit of length,

l = I~~\_+ 1142 Jj 2 (2)-0 GUvl 114 ' t 2

and t he unit of time,

All +1\42 .:l .....;.,-:._ _..,..:1' H ,CJ-(MJ IU2 , ­

where 114 1 and .\42 arc lhe masses of the primary and the compact object, respectively, 11 is the totaf angular momentum, G is the gravitational constant The dimen­sion~ess parameter x and T represent respectively the semi-major axis a and the time I through the following equations,

(4)

The mOl11elH of inertia of the primary C j and the rotational angutar WI are represented re­spectively by the parameters Xi and YI'

(6)WI =·Y.1

Under assumptions described in Section II. i.e. neg­lecting tbe tidal effects in the compacl star, clc.,. the formulae derived by Alexander (1973) for the tidal

evolution of the orbil.al and rnlaUOIl:frll ej~emellts can be 'l·abh.:~ fhe Zil~fO ngt~ ffll~~·~n"'$il;.~(lut')n{.)~~ ~1!i,~rHlii\'.l'1lc'r& (Jf :tht~ prtrn~,ry kuul the ~ll).tddal nl.nu~c"n c,nn::s.tn,H,ts

SIl1.ipllll.c'U 10,

dx = )'

dr

'[J - Ij + I" -- 'li -I- t I , I'll ,1'1

(1"1/)1," X J \ "'(i+ 21, -I)

+U+ I} 'I (; +- q. -- U+ I 2:

ill) ~ = II I:

ilT j~ 2

+3)F( --(j--l 2;11),)/\

- (1- 'I!' :ux' +2}(4j + I) f( -U + 0. -(j + 1/2),2;

4­

" )"L

i 2 (9),[F(-j. --(i-t/2), I;I/)Y,

---(1 --11r J12 x- J F(-'(j+ I), .... (j+ 1/2), i;ll)]. where,

(m0)

fV. 'fhe Mmnel1lt of Inertia or the Pdmary and the Apsidam Motion Constant..

We assume that the primary is still on the maln­sequence during the time that tidal ef~ects are opli;~rative

(cf. Section II). In order to solve Eqs. (7)--(9) it is necessary 10 calculate the moment of inertia of the primary and the apsidai motion conslants. -rhese parameters were derived from a 20 Mo zero age main­sequence model with composition X = 0,7 and Z = 0.03 computed with Paczynski's (1970) evolutionary code'), Neglecting the rotational and tidal contributions, the moment of inertia is given by

8n: R".4­C = ._-- I {u dr (If)

1 3 -0 ~ ,

where R I is the radius of tlu: star- The apsidal .motion constants are determined from the equations (Kopal, 1959)

k, = L±._~=_~1J(R 1) 2 3 4 (12)J 2U+ 1/ (R )) ' j= ,-, ,

j t

"J The author thanks Dr. 1-1. R. E_ Tjin.A. Dj;ie for his help 10 consl.ruct this modeL

M, 20M., J::, (JOt 70 N, 5JI113 R,. k.l 0.0'1),1'1 (\ 5,524'(lX liO~t' ~"''fnj' '" tUlOl4

is a I\HH.~liol1: which

+ I) + --;;;" (11) -I' I) ",·0, (Ill Q

and fulfils Il1t' bOUI1I;,1HIl'y cond,illon 'Ij=i :2 III i'ce,O. Q is the melln inside the radius r.

IJ= JQr'dr U4) o

1.~I.j;U",I"'''JlIl~ (I: I j and (13) were solwd , llH: results are lisled in L TIH~ accuracy of the integmtion is about III 'N,.

V. l'be Tldam Evol.uti~I'. with COllistrvad()11 or Angu]a., M.omentum

Implicit in the derivation of m-(!)} is tllat the total IIngular momentum of the :'lyslenl is "'t"t' """',,,"rl

during the tidal evolution. Let t, be the I'lltiiD of the rotational to the ol'bhal momibnl.um of the

~ 15)

where.

A=

and

B=2nC, .

As mentioned in Section 11 we assume lhal in the final state e ~= 0 and A. = I (Le., 2l circular and synchl'onmH, orbit). Fmm the value of the orbital In this slate (i.e., the presen tly observed period of the SYSLellnJ

we can determin(~ the total angular momentUl1t In Table 2 we list for each system th.e adopted obs,erved parameters M2 and PI (in VdaX-l, PI is the presently observe:cl period but not the final period) and the corresponding total angullar momentmn_ One can show that for a given H and for ). > 0, there ar,e two real values of P as roots of Eq. 115) if and Oldy if,

11 Z; 256BA 3 (l - e2 f1i2/27 H4 _ (16)

This means that, if at some stage of the tidal evolution values of e and Aoccur which violate the condition (16), synchronization of ~he revolution arid rotation at the present stage (when::: H iis derived) is impossible. In

.) 3

254 W Sutamyo

.8 'A 1.06.2.. 0

.8

UJ

Fig 1. The domain or the eccentricily e and Ille rallo of I.he rotational 10 Ihe orbilal period Awhere synchronization at lhe present slate is possibJe/Jmpo.ssible (10 the righl/lcf! hand side or Itach curve! for: Cen X·3 (aI, 3 U noo·)7 (Ii), SMC X·I (e) and Cyg X-j; id)

Fig. I, i he region at (hi: left hand side of each curve gives the domaine and A for each binary (except Vela X-I.) where: the condition (16) is nol satisfied .. From the theory of close binary evol.lJiion and the effects of tbe supernova explosion on the orbit one can estimate what values of the ,eccentricity can roughly be expected just aller the explosion (cf. van den Heuvel, 1973.; Smantyo, 1974); tbes": values are generally be­tween 0.3 and 0.6. For thiS reason we adopt eo ~ 0.5 as an initial condition for the tidal evolution of all systems. The value of At) for each system is given such that the condition (16) is satisfied; we adopt that ,1,0 -~ O.5~. 7. For a 20 Mo main-sequence star with radius of about 6 R(0 in a dose binary with period between 3d and lOcI. this corresponds toequ,HoriaJ velocities of rotatjon oetween 30 and 220 km/s, which seem very reasonable values for stars of this type (Plavec:. 1970; vall den Hicuvel, l.970). The initial period Po can then be calculated from (15). This obtained posslbl,e sets of initial values Po, eo and Ao for the systems are listed in Table 2. For the Vela x- 1 system. the initial condilion is a.djusted such that Ihe period is 8.96<1 when Ii' = 0.22 (the presently observed value; Zuiderwijk ift ai" f(74). Of course these sets of inilial vr:l!ues are not unique; the above given sets represent, however, entirely reasonable initial condi­tions and are therefore lIsed to show ho,w lhe tidal evolulion in realistic cases may have proceeded. (Due to ineversible viscous dissipation it is impossible 10 Irace back·~--from the present system parameter-·-what

X·ray bm.ary

Cell X,} 3 U 1700·37 SMC X-I CygX-1 Vela X, I ------,-_.­

0,7 2087 2.0 1412 1.5 ],893 5.0 5.607 1.5 8.959

--_._-----~--~.._._._-­

exactly the real initial conditions for each syst,em were.) 1.0 order to eliminate the time lag T from the right hand side of Egs. (7)--(9) we divide aU termS by a new dimensionless lime parameter T',

and define

dr' = ,ili dr ,

or,

dr' = .,)1)

_L__ dt, (17) To

where t is the real lime. Equations (7}-(9) are solved numenca]ly in terms of r' by using fourth order Runge­Kulta method, the accur.acy of the integration is better than 10- 5 %.. As a check we calculate the total angular momentum at each step al1d we find that it does nol vary at I. east up to the seventh decimal place. The results are given in F:igs. 2-6 which shows the variation of P, e and A as a function of the dimensionless time parameter r' for each sys1.em. From these figures one can see that 311 the final state the orbit is circularized and synchronized at the pres·ent observed period (except Vela X-I).. Synchronization between the revolution and the rota­tion in the final state always means that (Horb/Hrot)f ~ 3 [cf. the inequality (ll]' Therefore the most favourable conditions which may .lead to this state occur if

1.0 e, A

.8

.6

.~.

.2

.0

A _.-----.------1 33

P(d)

eEN X·]

p

e

.]005 ,01 D .015 :020 .025-(

Fig. 2 'ridal ,,"ohmon of lhe orbilal period p. lhe cccelllflcil.y e and the rMlo of tbc rot!lIional 10 the orbiJal period )_ with the dimension­less time parameter 't' for Cell X-3; inil.ial conditions Po, Co' J.o were taten frOln Tab!e 2

Tllb)c 2, f\dopled syslem parameters (eI'. BahcaH and Bahcall, I974} and reasonable sets of initialcondilloll'S for thc tidal evolution

-----_._----_.-._------_._-_...,--­j fJOOCj 0.5 0.7 3.6554 2.7811 OJ 05 5.0lnl 2.2065 05 05 5.6694 7.6011: 05 0.5 8.5554 27650 045 0-64 lUIOO

255

.8

1.0 P!dl'P. \

.8

.6

3 U l' 700-37

p .2

o

005 .1'0 .15 20 .25 l' Fig. 3. The same liS Fig. 1, for 3 U ! 7'00·37

,_--------15.51.0 e, '1\ P(d)

.8

6

X-1.~

.2

0

.0 05 .1'0 15 .2025 1;'

FIg 4. The same as Fig. 2, for SMC X·j

10 e. A \

.6

, CYG X·1.'"

.2

e .0

0 .1 1 3 ., .5 t" Fig. 5. The same as Fig. 2, for Cyg X·j

(Horb/Hro,)D is considerably larger than 3 (subscript 0

indicates the initial value). For the initial conditions listed in Table 2 one finds that (Horb/HroJo is - 5.37 fOl" Ceo X-3, .~ 15 for 3 U nOO-37 and SMC X-!, - 80 for Cyg X-I and ..... 140 for Vela X-I. These vallues will be

_'~-------'---1,100

P(d)

.6

VEl.A X·'I

p 2

.0

.02 3 .4 .5·C Fig 6. The Slime liS FilI. 2, for \1<11[1 X· 1

reduced to the values less than liIboul 3 if the ptimHry rotated very fnsl in the beginning [(Pro' I)";;; 2d for Cen X-3 and much less for other systems]. Allhough this is possible fot' Cen X-) (corresponding to an equatorial velocity of rotation of aboWl60 Km/s) it seems unlikely for other sysh:lms sincic the minimum observed rotational period of the eady type main­sequence stars in close binaluesis abollt 11,5d (C(H"

n~sponding to V;,,\ -- 210 km/s; Plavec, 1970; van den IIelJvel, i.970), 'Therefore it seems thai sylll.:hrollilz,H.!ion is possible from most initial conditions resulting rrom supernova explosions.

VI. The System Parameters; before the SupernovlI Expllosiol18

Adopting a value for the inliial rllass Mf (before t.he explosion) of the exploded star and using the approxi­mation of the effects of impact of the supernova shell onto the unexploded component as given by McCh.l$key and Kondo (1971) one can, from the values of M j ,. M2 •

M~, eo and Po, calculate the binary period P before the explosion and the ejection ve],odty v,,; of the supernova shell (d, Sutanl.yo, 1974) (in this computation it is assumed thai before tbe explosion the orbits were circular). For Cyg X-I we adopted Mg "" 8 Mo, for other systemsM~' = 5 Mo. Such values seem entirely reasonable for the masses of the helium star pro­genitors of the collapsed objects (d. van den Heuvel, 1973,1974; Sutantyo,1974). Table 3 lists thus cal.culat,ed

Table 3. The physical parameters of the syslelTls before the explosions, calculated for the selS of inilial conditions in Table 2

X-ray bi,nary !'(d)

Cen X·3 50 1.6277 0,99 3 U 1700-37' 5,0 2.5164 2.01 SMC X-I 5.0 2,7234 1.83 Cyg X·j 80 4.4279 3.54 Vela X·j 50 6,1678 3.21

35

256 W, SUI;Jilly,)

values of P and VeJ' The table shows that the reqUIred ejection velocities arc of the same order of magnitude as those observed in !I supernovae (Shkiovsky, 1(68) and that the injtia~ periods P are in the normal range for early type close binaries,

vn. The 'VillCosity

The n:maining question is now, are tidal forces slilficienily powerful to change the orbital elements from Po, eo and }'o values listed in 'rahle 2 10 compl.cte synchronism within about 5 x I years {eL Section I From Figs, 26 one can detennine ~he time scale l,lr' required to circularize the orbit We give this time scal:e in 'fable 4. Transformation of the dirnensionless time seale tl r' into the reaJ time A I involves knowledge of the

Or conversely, if a real time scale of 5x 106

years is adopted: as an upper limit, the lower limit of the required viscosity can he solved. By seuing the rate of the energy dissipation due to the Udal friction equal 10

the rale of work clone by the disturbing force one can express the lime lag T in lenns of viscosity as (Alexander, 1(73),

( 18)

Where </l) is the mean viscosity defined as follows,

9 II, . " ('1) = ---,. I ill' cit', (t 9), Hi lJ

Using ~.his equation one can derive the relation between y' and I from (17),

ill = . CI. .. dr' (20)(;.1)

where.

Thc valuc of a for each system ilS given in Table 4, Selling <If in (20) equal 10 5 x 106 ye.ars one finds from tbis equation (together with Figs, the lower Ilimit of the mean viscosity required to circularize the

'1"'I.ole 4, Dinl"IISIOl1lcs~ Ilmc scnles reqllll'ed locircllhrlz,c the orbits (starLing fwm the: e". I~(l. A" combimi'ilons of Table 2)" the COllVCrmll1 facll1r ?! and (he lowe!' I1mil of rhe meiln Viscosity <II) r,cquircd 10 clrculanze 'Ihe orbll ill ~css than 5 x 106 years

'~~----_._-'----'-'------.-

<"'>c",,(g erll IS-! i

Cen X-:J :JU 1700·)7 SMC X-I CygX-1 Vela X-I

OJ)15 0.275 0.275 0.500 0400

6,7022 .2Jj37 6,5i35 9.)130

3334060

fd8xIO" 4.07 10L'

1.14",10'· 102 x H)'" R45 x 1,0'"

-----~-- --._-~.~'-

orbit The results are in Tabie 4" These results do 110t sensilively on the choice oflne initial conditions.

v In. UiiscussiOill

From Table 4 one can sec that the minimum value of the mean viscosity required to circulanze t he orbit of VclaX-! in a time less than about 5x 10" years is at least one order of magnitude larger than for other systems, This is clearly due to the large period of the system (although the period of Cyg X- t IS also fairly

but the mass ratio Mz/lvf , is larger than in other systems; therefore Ihe tidal interaction is considerably stronger than in Vda X-Ii)" This may explain why the orbit of Vela X-I is stiB eccentric while the olhers have become circular. From Fig, 6 w,c can see thai the dimensionless time scale required to reach the present stage of Vela X-I (P=8.959d, e=0.223) is Ar'::::::O.05. If we assume that the present stage of Vely X-I is attained in 5 x 106 years after Ihe exptosion, we can calculate the mean viscosity from Eq" (20). We find that <It) ~ 10 15 g cm - I S - I, From results listed in Table 4 we mav conclude thai the tidal friction is ef~ective in red~cing the eccentricity only if (p) <: !O li 2 g COl I s- I, Such a large viscosity cannot be explained in terms of radiative or plasma viscosity which, even near the center of the star where it has largest value, is only of the order of 1 04 gem - I S - I

(K opal, 1968)" Turbulent viscosity seems also to be un lik.ely for main-sequence stars since we expect that turbulence only exists in the convective core whcr'e tidai. crfects are minor, We suggesl that probably magnetic plays important role in Ihis case. The magnetic viscosity can be estimaled from the formula (d. Jackson, 1962),

(21 )

where (J is the electric conductivity, B is the componem of the magnetic fields p<:rpendicular 10 the mass motion (for convenience we will call this the transverse magnetic fields}, i is the dimension of the system and c is Ihe ,,,'11,-,,("11 u of light The value of (J for stellar matter is of the order of lOU_lOIS S-1 (d" Alfven and Falthammar, 1963) If we adopt that the transverse magnetic fields is of the order of I Gauss and I is the dimension of the outer region of the star where tidal effects are effective (which is assumed to be ....... 0,1 RI]' ,U", will be abollt 10 13_10 15 gem 1 s·- I ,rllis value is of the same order of magnitude as the req uired Magnetic fields with a strength in excess 01' aile gauss. are almost certainly present in all young stars, as can be deduced from the presence of vcry strong magnetic fields in stellar remnants (pulsars). However, it seems unlikely that the transverse magnetic fields in the primary of

'le,:Jj X-i dunng 1hi: life ti,me (and also bter) WHS much higher than - 30 G,HISS since otherwise: the orbit must have been circularized. (This does not flll:an that Ihe IOlal 111agndK Helds. must also be 5maiL If the star possesses toroidal m;agnetlc fields with axis nearly coirtciding with rotal.ional axis. the largest component of the fields is in the direction or the mass motion Therefore only l~ small. cornponerH Iii/ill give a drag to the mass molion,) II seems therefore that by making very modest

assumptions about the fields present in these stars the rapid tidal circularization of the orbits or X-ray binaries 01 other close binaries as well) can be simply understood in terms of tidal forces. .,!,chmnl'icd"enilCnIS. The author is particularly iJldebted to ['rofess<li'

p, J. V"11 den !'!cuve! for suggeslio,1! him to rh" pl'Oh,lem and tor hi, UJIUlmIOl.iS gUldance during the course 01 \hi, Investigation.. He ilbo Ihanks Dr. H, R. E, Tjm .11, Djie and Mr. R. J. Takl~OS for miilny diStuss,I('ilsfhe ,cholarship frorn Ih,e DUlch M,nlsliry of [dim::ollon

,wei: SC1~i1Ce IS jI,nw:,ruily ill'k nowkd,~cd.

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Gi,ICCOI1J-, R 1972, ;l"t!'"{!/J!Il;\ J. L.e.II.-",\ In. L N Sd,re,cr,I'. U HICconi. R,. Cit;!',k y. ii, Kdlngg, F, I ,c'ViIWHl, R" Talllln'

b'"Jlll, II. 197.1, Lid! drcuhH NI. 2514IAIHU 171 SllkkJ\'sky, LS, 191\K SlipernOVi.\'), Inliel'sd011CC, N",w V'iI!'k Sulanlyo, W, 1974. /'1.\ InJl! , ,~ ASll'iJ{,liys, 31, 3.l9

Thomas.ll.l.'. 1'>74, AM,i'''I,III','.), IcdlU'S Ill? L 25 ZIJ,itkrwi'.!k,L.I,. van d0,n Iknvd,i' P.L, II(;ilSb(jl'~¢Jj. 1'J7'L

;I,I:/'I',-,n. & Ilsll'{il'hl'~' ~in Ih,' f!f0~,q

W SUlanlyo $Ierrenikundlig Insdtuul Univers.dci! \'3ll Am"ler(\fllTI

Roel,'r'SlrLllil 15' Amgt,~rd'lrlI·C. Nederland

After writinB this paper it was realized (through discussions with

Professor L. Mestel) that the reasoning given here is not correct. since ~n

cases with very large electric conductivity (as in stellar interiors) the

field is practically completely frozen in (the current density is practically

zero), resulting in small magnetic viscosity. In order to explain the

circularization of the orbit processes with high energy dissipation rates

will be required. This problem remains sofar unsolved.

3'7

THE EVOLUTION OF MASSIVE CLOSE BINARIES

III. T'HE POSSIBILITY OF' A "rIDAL INSTABILITY

DURING THE POS"l' SUPERNOVA PHASE i~

J.P. De Greve, C. De Loore,

Astrophysical Institute,

Vrije Universiteit BruBsel ,

and W. Sutantyo **,

Astronomical Institute,

University of Amsterdam.

* This research is supported by the National Foundation of

Collective Fundamental Research of Belgium (F.K.F.O.) under nr. 10303

** On leave from Bosscha Observatory, Bandung Institute of Technology,

Lembang, Ind6nesia

39

Abstract.

The influence of tidal interaction on the periods of massive X-ray

binaries during the post-supernova evolution is investigated. It is

assumed that after a certain time the orbit has become circular and

synchronous. The tidal effects of sUbsequent evolutionary changes

in the moment of inertia of the massive component are calculated.

It is shown that, as is already suggested by Sparks and Stecher

(1914), for small mass ratios and short binary periods a tidal

instability may occur resulting in an accelerating inward spiral

motion. Before the onset of the instability the tidal forces

maintain a nearly synchronous orbit. Possibly the orbits of Cen X-3

and 3 U 1700-37 are already unstable at present.

40

I. Introduction

In the first paper of this series (De Loorel'l.nd De Greve, 1 5),

hereafter called Paper I, detailed calculations were presented for

the pre X-ray evolution of massive binary X-ray sources. according

to the scenario given by VaD den Heuvel and Heise (1972). The

effects of the SN-explosion on the system were investigated by

several authors (van den Heuvel~ 19688; McCluskey and Kondo l 1970'

Sutantyo, 1974a; Wheeler at 1'1.1., 1974; Ch,eng~ 1974; De Loora &t aL,

1974). It follows from these computations that the system. remain

bound in most cases. The tidal evolution is then expected to

recircularize and synchronize the orbit (SutBDtyO, 1974b). Here we

investigate the further evolution of such a system.

In a preliminary investigation van den Heuvel and De Loore (1973)

found that during the second stage of mass exchange the binary

period probably will decrease very drastically. A subsequent

evolutionary scenario, leading to runaway or binary pulsars was

proposed in the foregoing paper (De Loore ,at a1., 1975; hereafter

called Paper II) and also by Flann,ary and van den Heu\I'el (1915).

A somewhat comparable scheme - though wi thou t a drastic redueti,on,

in the binary period - was given by Tutukov et al. (1914). A

particular end of the evolution of binary systems related to X-ray

sources was proposed by Sparks and Stecher (1914). Using tidal

instability arguments they suggested that collision between the

compact object and the primary might occur.

41

In this paper we will elaborate the idea of Sparks and Stecher

by including the effects of stellar evolution BS well as the

tidal interaction.

II. ThE! RadiuB of Gyration of uMas,sive sta.rs

One of the main parameters necessary to study the tidal instability

2of the orbit ie the radius of gyration r defined byg

;?, ') r= 1/(M. Fl."),

g

where I is the moment of inertia of the star, M is the mass and R

is the radius of the star.

SparkA and Stecher (1974) used in their computations the value 0.015

for ,derived by van den Heuvel (1968b), a value that is accurate g

for ZAMS stars with masses between 1 and 10 solar masses. In order

Lo complete this range, the radius of gyration was calculated for

stars \o,lith malSses between 10 and 30 M'0' The computation.::; were

performed from the zero age main-sequence to helium ignition stage.

The adopted initial chemical composition was X = 0.70 and Z = 0.03.

The results are shown in Figures 1 and 2. Figure 1a shows the

2evolution of r from the ZAMS to the end of core hy.drogen burning,e

for stars of 10, '15" 20, 251 30 and 35 M<=). During this stage the

radius of gyration decreases more or less linear with time until

towards the end of the core hydrogen burning it starts to decrease

rapidly. The final value is about 40 % of the initi!'i\l \falue. ]l"igure

1b g,)ive8 the evolution of r 2 startin~ from the end of core hydrogeng b a

burning (t = 0), through hydrogen shell burning up to helium ignition

in the core. The time scale in this diagram has been enlarged by e

factor 100 compared to Figure 181. For all models semiaonvection WBS

treated according to scheme R of Stothers ('970). Another treatment

of the semiconvective zone during this stage dOBS not alter the

2evolution of r seriously.g

The radius of gyration continues to decresse during hydrogen

burning and reaches a minimum soon after the onset of core helium

burning. When this nuclear process increases its contribution to

2the energy production, r increases sharply, climbing up to the

g

initial ZAMS-value. For stars of 1'0 "'\:;) the evolution was follo'wed

further during helium burning. It was found that after reaching a

2maximum value r remains nearly unchanged during this stage.g

III. Stable and Unstable Configurations

Sparks and Stecher (1974) have suggested that a binary system (in

their case a white dwarf binary) might become unstable if the

stellar radius R, exceeds a critical radius R*.

vJe '.-Jill now consider the physical meaning of the critical radius R*

1

43

in connection with the stability of the orbit of massive X-ray

binaries~

According to Sparks and Stecher the critical radius is given by

(2)

where A is the distance between the two components, M Bnd M BTB1 2

the masses of the massive star and the compact component respective­

1y, and N is a dimensionless orbital mean motion (N* = 0.76). The

quanti t~t N* is related to 1.* I the cri tical dimensionless total

angular momentum of the system, by N*2 =: 1.71902 (1,*)-2. The

condition R, < R* is eqUivalent to the condition that the ratio of

orbital angular momentum to rotational angular momentum is higher

than 3. This is the condition for the stability of a synchronous

and circular orbit used by other a.uthors (Alexander, 1973i Pringle,

1974; Wheeler et a1., 1974). During the evolution the rotational

period of the primary tends to increase or decrease due to the

changing structure of the star. On the other hand, the tidal forces

try to maintain the synchronism. Therefore angular momentUM will

be transferred from orbital to rotational motion or vice verSB.

If the tidal forces are strong enough to restore the synchronism on

~} time scale shQ&ter than the time scale of the stellar evolution,

the rotation will be nearly synchronized with the orbital motion

all the time as long as R < R*. The orbital period will decrease1

44

or increase with about the same time saale &s the evolutionary

time scale of the primary (quasi st.ble stage). However, if the

stellar radius exceeds R* l' no, synchronous orbit will b. p()saible 1

no matter how strong the tidal forces are. tlill period will

then decrease at an ever increasing rate DD a time Bcale much

shorter than the evolutionary time scale of the primary.

On the other hand I if the tidal forces are not su f'U,ciently strong I

the deviation from the state of synchronism b~comeB larger and

larger. In that case computations show that the v.riation of the

orbital period is small.

In order to determine whether the components of the sy~t.m will

collide or a new mass transfer stage will occur, we compare the

critical radius R* with the critical Roche lobe radius RR' which

can be approximated by the following formulae,

r R = RR/A = 0.37777' - 0.20247 x Q + 0.01838 x QC'}

+ 0.02275 x Q3

for q > 0.1 (;)

for q $: 0.1, (4)

where q = 1'1 /1'1 1 and Q = logW(q). If e.t a given time t the radiu,152

Ii of the massive star exceeds either R* or DR there are two1

possible ways for the final evolution:

45

Case I: RR < R{~. The star overflows its critical Roche lobe and a

second stage of masS lOBS will occur. The evolution may then proceed

BS outlined by van den Heuvel and De Loore (1973) and in Paper II.

It is still an open question whether or nat in this case the mass

lOBS will prevent the collision between the two components.

R* < RR" As already discussed, the compact abject will be

spiralling down to the companion. At a certain moment also in this

case the Roche lobe will be filled- But the accelerating spiral

motion takes place on a much shorter time scale than the mass

exchange. 'fherefore it is very well con'cei'\rable that the collision

cannot be prevented. The compact object enters the surfa£e of the

companion, but the occurrence of a collision with the core of the

primary, as haA been suggested by Sparks and Stecher (1974) is

still uncertain. The heating due to the large friction might blow

off the outer layers and a system consisting of a helium star and

a compact objeot with a very small period might be the result

(Ostriker and Paczynski; private communication to E.P.J. van den

lieu'ifel) •

Figures 2a and 2b show the relative Roche ra~ii and critical radii

of the primaries in systems with primary masses ranging between 10

and 30 M(.) and a neu trOD star secondary of 1 MG. 'rhe absiscae

represent the same time Sca~es as in Figures 1a and 1b: the main­

sequence stage (2a) and the hydrogen shell burning stage (2b). The

46

full curves represent (the cri ti radius expressed in terms

of A), the dashed 1 6 rR(the Roche rad s, also in terms of A).

The curves are labeled with the maSB of the primary. By setting

e to the radius of the star at the intersection of corresponding

curves, we can easily derive the minimum and maximum periods of the

range in which a second stage of mass 10sB is possible. p, .. andI1n.o

P ,as well as the corresponding times t and t. . are given in max 1 2c ••

no unstable situation occurs. It follows from this table that, aB

may be seen, the probability for a second stage of maSs 10SB

decreases with decreasing mass ratio. It is also a function of the

mass of the primary, since the condition of intersection,

( 5)

is, through the radius of gyration, related to M " 1

Condition (5) can be used to determine the mass ratio ranges where

only one of the Cases I and II describe in this section, can exist.

(This does not necessarily imply that they will occur). We therefore

transform equation (5) into the following mass ratio condition,

q

(q + 1 ) f(q) = K,

-.2 2vihere f( q) = l' and K ::::: r (N*)-4. The condition rfq/(q + 1~f(q) < KT R g

implies R* < RR" If we exclude the case that the stellar radius

does not increase sufficiently to exceed either R* or RR I thie

(·1AI '"1f

q leads to case II~ If on the other hand --:-Jf(q) > K then

the system unsvitablY evolves through case I.

Defining q1 as the smallest value of q that always leads to Case I,

and q2 as the largest value leading always to case II, q1 and q2

2':1re found by determining the maximum and minimum values of r during

g

the considered part of the evolution and solving equation (6)

2numerically. The results are listed in Table 2. For r the

g,max

ZAMS values have been taken. The table gives the limiting values

of K, q and M as a function of the mass of the primary "1 for2

both CaSBS. The vslues for q2 are much smaller than the values

2found by Sparks and Stecher due to differences in the values of r

g

and the use of N* instead of L* in equation (2).

IV. The Post Main-Sequence Evolution of Massive X-Ray Binaries

t that most X-ray binaries have (nearly) circular orbits

mi t indicate that the tidal forcBs have been operative since the

explosion which produced the compact object (Lea and Margon, 1913;

SutantYOl 1974b). At the end of tbe tidal evolution one expects a

circula.r and synchronous brbit. The system reaches the most stable

configuration at this stage. However, the gradually changing

structure of the primary during the post main-sequence evolution

might continuously disturb the stability of the orbit. As is

already mentioned in S~ation III, th~re will be transfer of angular

momentum from the orbital motion into the rotation or vice versa,

48

due to the tidal interaction. Consequently orbital and rotation period

will continuously vary. '['he orbit remains circub'lr during thi s

evolution.

In order to follow the tidal evolution we use the formulas derived

by Alexander (1973). We include the variations of the radius and

the moment of inertia of the primary and apply the formulas to the

known massive X-ray binaries. For each system the computations are

started from the present stage assuming that the orbit is synChronous

I'lt present. We assume that the mean dynAmic viscol9ity of th,e

1 1primary is - 10'3 g cm- s- , which is the order of magnitude

required to circularize and to synchronize the orbit within the

main-sequence lifetime of the primary (Sutantyo, 1974b).

In Table 3 we give the orbital parameters of the system, the starting

time t (t = 0 corresponds to the zero-age main-sequence) and the o

initial (i.e. present) radius of the primary R • It i5 very likelyo

that the primary has not yet filled the Roche lobe at present

since mass transfer through the first Lagrangian point will

extinguish the X-ray source (Pringle, 1973; Shakura and Sunyaev,

1973). The X-ray source is most probably powered by stellar wind

from the primary (Davidson and Ostriker, 1973). As within a certain

distance from the star a strong stellar wind might be opa~ue enough

to obscure the X-ray source, the radius should not necessarily be

the same ae the occulting radiu5 (Pringle, 1973). There i8 no

certain indication about the present stage of evolution of the

prlmarles; therefore, the stdrting parameters are in fact not

unique. 'I'he only criterium used to choose the starting evolutionary

parameters is t the primaries are in the stage of core hydrogen

exhaustion. In more advanced stages, the primary of Cen 1-3 should,

in view of its short orbital period, have filled the Roche lobe

which, as mentioned above, Beems to be unlikely. For other system

the stage of core hydrogen exhaustion may, however, precede the

present stage by several times 105 years and should certainly not

be considered as an exact representation of the present moment.

De te this uncertainty, we expect that our results will at least

give a qualitative picture of how the tidal interaction will

proceed. For each system we assume two values of the mass of the

primary Bnd of the compact star. As a general picture of the

evolution, the variation with time of A, R, R*, HH and the orbital

period of Cen X-3 (with an adopted primary of 20 He) are shown in

Figures 3a and 3b. The orbit is nearly synchronous all the time up

to the stage R1 '" R*. As one can Bee from the figures, in the case

M2 '" 1.5 Me the star is filling the Roche lobe before the condition

of instability (R > R4}) is reached. On the other hand, if 1'1 '= 1 M(:)1 .. 2... ,

the system has become unstable before the star fills the Roche lobe.

About 500 years later the radius exceeds RR' and within about 15000

~{ears it decayed virtually zero. Hence, here the inward spiral

cannot be prevented.

We summarize the post main-sequence tidal evolution of thE!! known

massive X-ray binaries in Table 3 (we do not include Vela X~1 since

the orbit is probably not yet circular; cf. Hutchings. 1974;

Zuiderwijk at al., 1974; in this Case our method is inapplicable).

In this table we give the time t* of the onset of the instability

and the time t R of the moment when the star starts filling its

Roche lobe. In case ttl < t* we leave t* a.s blank sinc8- tit,eo I1Ul,Se

loss might affect the further tidal evolution. The values of t*

and t R are given with respect to to. In Cen X-3 and 3 U 1700~37

it is possible that the systems are already unstable at the initial

(i.e. present) state Ct* < 0). We also list the orbital period of

the systems at the moment the primary starts filling the Roahe lobe

ePH) •

V. Discussion and Conclusion

It is observed from Table 1. 2 and 3 that a binary system might

become tidally unstable at the post main-sequence stages of the

primary if the system has a small masS ratio and a short period.

The X-ray binaries Cen X-3 and 3 U 1700-37 are good candidates for

being such systems. It is even possible that the systems are

already unstable at present. In case the systems are indeed

unstable the orbital periods are expected to decrease with a rate

of a rew seconds per year or more (depending on how advanced the

stage of instability is). The compact star will spiral down into

4the envelope of the companion on a time Beale of about,O years.

However, the collision with the core as suggested by Sparks and

Stecher ('974) is still uncertain; the large friction exerted on

the compact object when it is moving in the interior might prevent

Buch a collision. In case the systems are stable, our computations

indicate that, assuming the BamB viscosity as required for the

circularization of the orbit (~ 10'3 g cm-1s-1 ; Sutantyo, 1974b),

the tid forces are in general Bufficiently strong to maintain

the orbit to be nearly synchronous. ThBre~ore~ if future observations

show that the orbit is non-synchronous and, at the same time~ no

decrease in period ia found (i.e., there is no indication for

instability), then one might conclude that the tidal forces are

considerably wORker than what we expected, and the hypothesis that

the tidal forces Bre responsible for the circuLarization of the

orbit becomes doubtful. (As illustration we give the results for

3 U 1700-37, assuming that the system consists of components of 25

As one Can observe from Table 3, the system is still

stEOlble a.t t ::: t • Denote I.t as the mean dynamic viscosity and A. asR

the ratio of the rotational to the orbital period. In case

~ z 10 13 g cm-'s-1 we find that the maximum value of A before the

star fills the Roche lobe is ~ 1.1 Bnd the minimum is ~ 0.92, while

10 -1-1for ~ = 10 g em 5 find A ~ 1.28 and ~. ~ 0.8).maX m~n

Acknowledgements.

We are indebted to E.P.J. van den Beuvel for his critical reading

of the manuscript and some valuable comments. We also tha.nk

R.J. Takena for helpful discussions.

53

References.

Alextmder, M.E. 19731 Astrophys. Space Sci. £1, 459.

Cheng, A. 1974, Astrophyso Space Sci. .2.l, 49.

Davidson, K., Or~triker, J.P. 1973, Astrophys. J. 179, 585.

De Loore, C., De Greve, J.P. 1975, Astrophys. Space Sci. (in

De Loore, C., De Greve, J.P., van den Heuvel, E.P.J.,

De Cuyper J.P. 1974, Proc. 2nd lAD Regional Meeting,

Trieste (in preparation).

De Loore, C., De Greve, J.P., De Cuyper, J.P. 1975, Astrophys.

Space ScL (in the press).

Flannery, B.P., van den Heuvel, E.P.J. 1975, Astron. Astrophys.

Hutch 3, J.B. 1974, Astrophys. J. 188, 341'.

LI':lll, S.M., Margan, B. 1!973, Astrophys .• L,etters 2. 33.

McClu(':,l{e;y, G.E., Kondo, 'L 1971, Astrophys. Space Sci. 1'0, 464.

Pringle, J.E. 1973, Nature Phys. Sci. 243, 90.

Shakura, N.L, Sunyaev, R.A. 1!973. Astron. Astrophys •.24, 337.

Sparks, IN.H., Stecher, T.P. 1974, Astrophys. J. 188, 149,.

Stathers, R., 1970, Mon. Not. R. Astron. Soc. ill. 65.

Sutantyo • W.. 1974a" A,stron,. Astrophys. l!., 339.

Sutantyo, IN. 1974b Astron. Astrophys. 12., 251 . "

Tutukov, A.V., Yungel'son, L.R., Kraitcheva, Z.T. 1974, Proe.

2nd lAU Regional Meeting, Trieste (in preparation).

van den Heuvel, E.P.J. 19688, Bull. Astron. Inst. Netherlands

van den Heuvel, E.P.J. 1968b, Bull. Astron. last. Netherlsnds

van den Heuvel, E.P.J •. " Heil5€l,J. 19'"(2, Nature Phys •. Sci. .2'9,

6'7I •

van den Heuve1, E.P.J., De Loore, C. 197', AstroD. Astrophys.

L7 1 •

Zuiderwijk, E.J., van den Heuval, E.P.J., Hensberge, G. 1974,

Astren. Astrephys. 12, 353.

55

Figur~ Cap~ion6.

218. Evolution of the radius of gyration r from zero ageg

main-sequence to the end of core hydrogen burning for

starB of 10, 15, 20 1 25, 30 and 35 M0 •

2Fig. 1b. Evolution of the radius of gyration r from the end of g

Core hydrogen burning (t =0) to helium ignition in

the core for .stars of 10! l' 5, 20, 25, 30 and 35 M(l).

Fig. 2a. Comparison of the relative Roche radius r (dashed line)R

with the evolution of the relative critical radius r*

(full curve), for primary masses of 10,,1'5, 20 and 30 Me

and a RBcondary of 1 Me" The time scale is the same as

in Fig .. 1a.

Fig. 2b. Comparison of the relative Roche radius r (dashed line)R

with the evolution of the relative critical radius r*

(full curve·), for primary masse,s of 10, 15, 20 and 30 M(:i

and a secondary of 1 M ; t = 0 corresponds to the end ofm

core hydrogen burning.

li'ig. 3al. 'I'idl'll evolution of ' A, RR and Hi of the Ceo X-3Porb

system in Case the system has components of 20 M an~0

1.5 MG' The starting time to = 4.92897 x 106

yr is given

with respect to the zero age main-sequence_ The mean dyn

n -1 -1viscosity of the primary is assumed to be 10 ~ g em s

Fig. 3b. The same as Fig. 311 for components of 20 140 + 1 M ­m

T 1)}

IDlmum end maximum values of the orbital parameters r&quired for

the occurrence of a second stage of mass loss during core hydrogen

shell burning, for systems consist of

prlm2rles of 10, 15, 20 and 30 M(.)l accompanied b.)lll '1 n.eu.tron

c' tar.

(MASS r~ 1 10 15 20 30

I (, )t . \)(10 yr' 2.85 5.65 4.234mIn

n(R,~) 5.84 11 .34 ?3.41

A . (Rr:) 9.64 18. 11 35.84mIn ..

P . (days) 0.87 1.95 4.47 rl11. n

6t (x10yr) 8.44 6.18 4.2. 47 max

It (Hr.,) 400 671 649 max ,',

ft. (R) 660 107'1 994 mi-1X 17)

P (days) 491 887 652 mc3.X

57

Tnble ::

Limiting values for the mass ratio q and for the mRSS of the

econdary M~ for the unique occurrence of case I (second stage of t

mass loss) or case II (accelerating spiral motion) for various

inp~Q,~PS ()f t"h.p ))~-J.·mi~ry i'~"'.'0",. '.,' , • -, • _. ",,'" • '1'

CASE II CASE I

I ,I

~jl' 1 l<. min q ") (;"

(M 2

) maX

K max q, 01 ) .2 m1.D

I ~:)

I

I 30 I 0.033 69

0.04004

0.017

0.019

0.59

0.51

I 0.243°3

0.23592

0.090

0.088

3. 08

2.64 ,

25

20

I 15

,

0.04697

0.0'5419

0.06885

0.022

0.0?5

O.,0~\1

0.55

0.50

0.46

0.22825

0.21968

0.2°7°6

0.086

0.083

0.079

2.15

1.66

1• 18

10 0.07877 0.035 0.35 0.16998 0. 061 0.61

Table}

Adopted o~bital parameters of the four massive X-ray binaries with circular orbits Bnd correspondioE

results of the tidal evolution computations. The starting time t is (arbitrarilY) taken equal to the - ~ 0 ".

moment of core hydrogen eXhaustion (t= 0 corresponds to the zero age main-sequence), while t*

(corresponding to H, = R*) and t (R = R~) are evaluated with respect to t • The mean dynamic . .. . 13 1R 1":'1 _,.K 0

V1SCOS1ty 18 assumed to be 10 b em s for all systems (see text).

X-ray binary M,(N0 ) P (d) t. (106yr) H (Re:) 14 2 (Mr;-.) t* (104yr) t (104

yr) P (d)0 0 o R R

Cen X-3 2.087 3. 28332 12.536 1.5 :;.4 3.9 1.9153° 1.0 < ° 3.1,. 1.817

20 4.92897 10.259 1.5 - 25.0 1.933 1.0 12.0 12.05 1.766

3 U 1700 -37 35 3.412 3.3 11 32 15.532 2.5 - 48.1 3·533 1.0 < 0 9.6 2Aoo

25 3. 627 83 10.869 2.5 - 13 1 .2 3. 1 38

1 .0 58.8 60.2 1.943

St4C X-1 20 3. 893 4.92897 10.259 2.5 - 122.4 3.563

1.5 - 122.4 3.287

Cyg X-1 25 5.60 7 3.• 627 83 10.869 8.0 - 131.7 5.404

6.0 - 13 1 .7 5M324

\J1 '.D

--to

..........r-L....

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t ........

0 'l;:"­

m

00

LL

1 N

OJ CD to U:) ....::t ('I) N .., ­o o 0 0 0 ,0 ,0 to

•

60

. CI)

LL

c ------- ­f o

r

o N

aM LD

M -!---I-----jI----1f--::::=""--""'::l

---r---+--+-~'--_

N

'D) l:O LO o o o

61

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I L:.-.l. ~

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I L: I m0

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I,I

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I I I I I I I I I' I I I I I ill I I I I 0

I il NLDI 01

I C.9M

L, 'II

I LLiI I --SI ~ II

j I I I M Ii I I I

I N

I I I I I I! I' I' I I I I

1

L.. m o::J t- <..0 LD -...::t M

r

0' r­

01 00

l.D

o N

f I

U)I r- I

I I I I I I I I I I0:

~I

I I

01 (") I

I'

~

!I­>-..

"f o ,.-- ~

IN "'­......

o ,~ N!

II

...D ' N

M

N

L

0"\

120..p­

r~ --...!:O:-b20

r (RaJ i p(d) A

2oM0 + 1.5M - "

-, 1.515

R'

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10 -. 1.0

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I I 4 0F

2 3 5 )t. 1 t (10 years

,/

............ In '­t,

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LO,.. ---.J__..JL.._--''--_......J'-_....o.­ ---I__..J-_-J.__...1-_-J.__...I-_-J ...:

o N

65

The Formation of Globular Cluster X.. Hay SourcelS through

Neutron Star - Giant Collisions

N. Sutantyo ill)

Astronomical Institute, University of Amsterda.m, the Netherlandl:'3

[{unning title The Formation of Globular Cluster X-Ray Sour\ces

Address Roetersstraat 15, Amsterdam, the Netherlandts

Subd.ivision Stars and Stellar Evolution

Publication in . Research Note

*) On leave from Bosscha Observatory, Bandung Institute of

Technology, Lembang, Indonesia

('7

:?1Jmm8I~:L. 'rha }loGsibili ty is examined that the globular cluster

X-ray sources are binaries with B compact (neutron star or black

hole) component which were formed by direct collisions between

compact stars and giant branch stars in the clueter. It is argued

that such a collision will lead to capture of the compact star

and to the formation of an X-ray binary that may live for some

109 yr. The anticipated number of - 2 x 104

compact stars in the

globular cluster H 3 produces a few collisions with giant branch

12stars in 10 yr. In the densest globular clusters this frequency

10 can be a few collisions per 10 yr. Such a frequency is sufficient

to explain the observed frequency of occurrenCe of globular

oluster X-ray sources. X-ray sources produced by such collisions

are not necessarily to be confined to the central region of the

Clusters.

Key words: globular clusters - X-ray sources - stellar collisions

68

I. Introduction

The presence of X-ray sources in at least four globular clusters

indicates that X-ray sources occur some one to two orders of

magnitude more frequently among stars in globular clusters than

Bmong stars in the galactic disk (Gursky, 1973; Katz, 1975;

Clark, 1975). The short-time variability and the luminosity of

the globular cluster Bources closely resemble those of the known

binary X-ray sources, and suggest that also here accretion of

mGtter towards a neutron star or a black hole in a binary system

is likely to be the cause of the X-ray emission (Katz, 1975;

Clark et a1., 1915; Clark, 1975).

The companions of the compact stars must be stars with mass

- 0.8 M since more massive stars must have completed their8

evolution. From pulsar statistic6 Bnd from the presence of white

dwarfs in galactic cluaters it seems likely that e compact etar

(8 neutron star or a black hole) Can only be formed from a star

that was originally more massive than 3 - 4 M0 (Gunn and Ostriker,

1970; van den Reuvel, 1975). If not more than the core mass is

left after the explosion, the formation of compact star in

non-massive binary systems will most likely disrupt the system

or give it such a large recoil velocity that it escapes from the

cluster (Katz, 1975).

Clark (1975) has .suggested that the globular cluster X-ray

binaries were formed by capture. In his hypothesis a solar-type

69

main-Acquence star was captured by a massive black hole cluster

member (the remnant of a massive star that underwent a supernova

in the early days of the cluster's life). When the solar-type

component finally evolved into a red giant and overflowed its

Roche lobe, Buch B system would have become an X-ray source.

The arguments advanced by Clark are that many-body gravitational

encounters Bre known to caUBB massive cluster members to sink to

the cluster center (Aarseth , 1972). Here , due to the high

stellar density and multiple encounters, capture of a normal

star would becomB possible. Thus several binaries consisting of

a massive black hole Bnd a normal star might exist near the

cluster center. However, it is difficult to quantitatively

estimate the formation rate of such binary systems by this

proceSl5.

In the present paper we examine the alternative possibility that

these BourceS were formed by direct collisions between collapsed

ch.wter members (neutrDn stars or black holes) and one of the

many thousands of red giant cluster members. In order to do so

we first estimate the number of collapsed stars expected in a

globular cluster, and from this number, we estimate the collision

f:r('!quency.

II. The Number of Collapsed Cluster Members

A. EquRtions and Assumptions

To estiMate the number of dead stars (white dwarfs, neutron

stars and black holes) in the cluster, one should know (cf.

Sandage, 1957; Tinsley, 1974)

a. the number N of main-sequence stars within a certain mass12

interval H, and

b. the .shape of the initial mass distribution fUDction ~(M) of

the main-sequence stars in the cluster.

Then, if all stars that initially had maSses in excess of a

certain value M terminated their life as supernoVae (leavingG

behind a neutron star or a black hal.) and all stars less mas ve

than M left white dwarf remnants, the number of collapsed stars c

in the cluster will be equal to (cf. Tinsley, 1914):

IJj ") c

N ;:; N r (V ( M) dM / J (HM)dM,c 12 oJ

M c M,!

nnd the number of white dwarf members is ,

14 Mc 2

N ;:; N (Jr t)J ( M) dM - N h·. }/ J ~(M)dM,wd 12 rg+· b, . M 1"1

t 1

\\lhere N denotes the number of red giants and horizontalrg+hb

branch stars, and M is the mass at the turn-off of the rnain­t

sequence.

Well studied open clusters of variety of tyP~s, richness and age

all show a fairly similar masa function ~(M) with a shape closely

approximated by (Balpeter, 1955; Sandage, 1957; Limber, 1960;

Taft', 1974),

in which the constant C is proportional to the total number of

cluster stars in a given mass interval. Since the mass

distribution on the observed part of the main-sequence of the

globular cluster M 3 ie, within the observational uncertainties,

similar to this function (Sandage, 1957) we will also adopt the

mass spectrum of Equation (3) to hold for globular clusters (the

outcome will not be very different if exponents between -2 and

-3 Bre used, as the reader Can verify).

B. '1:'he Number of Compact. Stars and Vlhi te dwarfs in a Globular

Cluster

We estimate the constant C in Equation (3) from the number of

stt'ars in the post main-sequence expansion phase. Due to their

brightness, this number is well determined. E.g. for M 3 it is

nbout 6000 (Sandage, 1957). The theory of stellar evolution

allows one to determine, with good accuracy, the mass-interval

on the main-sequence from which these stars have originated,

as follows. The main-sequence lifetime of stars with masses of

around 0.8 M0 (the turn-off mass of the main-sequence in a

obulsr cluster i Cf. Si'mdage, 1970) depends on 1'M.sa in good

approximation as (cf. Iben, 1967)j

J,,4t -= const;n , (4)

Thus, if stars in this mass range spend a time 6t between

leaving the main-sequence Bnd reaching the top of th~ ant

branch, the red giants in B globular cluster

originated from a mass interval 6M which accord to Equation

(4) is related to 6t by,

ot!t = - 4 6M/M .

Evolutionary computations for M-., 0.8 M show that for these0

eta.rs L!.t/t 0.10 (Iben, 1964) which yield!':,; 6M""" 0.02

and N.~ - 6000 (the number of stars along the giant j i~.

branch) one obtains C ::.: 1.8 x 105• With Me= 3 M0 Equations (1)

4and (2) yield Nc ::.: 2 x 10 and N

Wd ::: 1.3 x 105 • With 1'\ ::.: 4 M(.'l

4 one obtains N ::.: 1.3 x 10 and N d - 1.5 x 105 • c w,

III. Collision and Capture

The velocity of the compact star when it enters the atmosphere

of the red giant is given bYl

(6)

where M is the total mass, V is the velocity of the compacttot oo

7 7,)

star at infinity and R is the radius of the red giant. The

compact star will be captured in a cloee orbit around the red

giant if it leaves the atmosphere with a velocity les6 than the

escape velocity '/(2 (J Mtot/H). Neglecting the amount of mass

'whi.ch is possibly blown off from the atmosphere, one can show

that in order to be captured, the compact star must lose at

least all of ite kinetic energy at infinity which is about

1045 _ 1'046 sr' g f·or t s t . th".. M M ( correspon :lng t 0compo,c arsw~ =0 d'

relative velocities of cluster stars of - 10 -30 km/e).

Novikov and Thorne (1973) indicate that if a compact star moves

supersonically through medium of finite density, it will develope

a shock front with a characteristic size comparable to the

2capture rr!dius 2 G m/v , whe're m is the mass of the compact

star. In a strong shock, the gas will lose most of its velocity

perpendicular to the shock front, but will retain the component

parallel to it. The amount of kinetic energy dissipated in the

shock can then be apprOXimated by,

d'~/!('.t_l _ ~.. ~2 TIl 2 -1"_rtu pv

It should be noted that since the strong shock approximation is

used, Equation (7) will 1n fact give an upper limit to the rate

of the energy dissipation. However, we expect that the real

amount of the dissipated energy will not differ by more than

one to two orders a.:f magnitude from that calculated from

Equation (7).

Consider now a compact star of 1 Mm moving through the

ntmosphere of B red giant of abnut the same mass th radius of

10 • From Equations (6) Bnd (7) and adopting p ...... 10-4 IS cm-} I

one can c~lculate that dE/dt - 1042

erg/so Since the compact

star spends about 105 s inside the red gi&nt1the total amount

of energy dissipation is - 1047 ergs which is some cne to two

orders of magnitude larger than the initial kinetic energy of

the compact star. Therefore, even if the dissipation rate is Doe

to two orders of magnitude lower, the capture is atill easily

POt3Bi ble.

The above numbers indicate that the rate of energy release

d,Ur:lfle: ' t1-,Ie,d'lSSl, pa t"1.on ).8. ~ 1040 -,104.2 Since this :1.13

considerably larger than the Eddington critical luminosity

("'-' 10 3(?; erglE> for R star with M :::: M(-'.) one expects that part of

the m~tter in the outer layers of the red giant attains

sufficient energy to escape from the system during the passage

of the neutron star through the envelope. The diBS tion is

expected to terminate when most of the matter outside the

(instantaneous) critical Lagrangian surface defined by the two

DtiJrCi, h:H3 been expelled. Therefore, we ~'Xpect that the drrJ;g

will ce~se after one or two periastron passBBec.

After thiG moment, further mass transfer from 'the str~ipped red

Bient towards the neutron star is expected to continue, since

75

the stripped giant thermally unstable. The thermal time scale

8of a 0.8 1'1<7\ red ant is of the order of Borne 10 yr. It is

quite likely in this case that the compact star is more massive

thrin i tEl fitripped companion I such that the mass trnnBfer takes

place from the less massive component towards the more massive

component. The binary separation becomes wider during such a

type of mnss transfer. Therefore, after mass transfer on the

thermal time scale, the mass transfer may still continue on a

nuclear time Bcole Ccf. Kippenhahn at al., 1967), which is of

the order of 10!) Y1' for a 0.8 M0

star with ahell hydrogen burning

around a degenerate helium core. Therefore, the system may

become an X-ray source for a time interval of the order of 109 yr.

IV. The Number of Compact star - Giant Branch Star Collisions

vie fH,n;;ume that only collisions between compact stars and giant

branch CGB) stars ere effective in forming binaries with B

collapsed component since only GB stars have large collision

cross-sections snd exist in B con8ider~ble number. The number of

collisions per unit time in a cluster is given by,

\ :cluster

Fcall - 2 L gb

! o.i 0

PC

P&b. b'

v 1re en R2

)gb 4 n r 2

dr, (8)

where v .1· re is the relative velocity of two cluster stars, Rgb is

the radius of the GB star, P and P .b are the space densities of c g

compact stars and GB stars respectively. (The factor 2 accounts

for tOe fact that the orginally straight path of the Rpproachin~

two stars is deflecteri by their grBvitaicnal attraction which

lTICrpaS€s the collision probability). We assume that v ;;,:reI

and r _ ..t = 10 pc. The number and the radius of GB stars BSclus·er .

a function of their position on the giant branch Bre estimated

from the data given by Sandage (19 ) for M 3. The integr is

computed for each magnitude interv/:'ll of M1 =: 0.2alon[t, the v

ant branch. the summation of these integrals then gives the

expected frequency of collisions.

UsinE the space density distribution of BtBrB in M 3 erived

from star count by Sandage and van Zeipel (cf. Gort ~nd VAn Herk,

1959) and assuming that all types of stars ere distributed in a

similar way, we obtain, for a cluster like M 3:

11 -1F _, - 0.19 x 10- yr

CO.l.,­

The expected number of collisions is a quadratic function of

the compactness (the mean star density) of the cluster. For

clusters with the same density distribution it is simply a

quadratic function 0 f the total number oi;' staTs in the clusterEl 1

i.e., if a cluster contains 10 times more stars than M 3. one

1:0 expects a few collisions in 10 yr.

77I ,

v. Discussion and Conclusion

There are about 130 known globular clusters in our Galaxy, rOUT

of which contain X-ray sources (which means that the probability

of obeerving an X-ray source in a cluster ie - 0.03). If we

assume that the lifeti~e of such X-ray sources ie - 109 yr (cf.

Section III), it is required that in the average 3 binary systems

11with a collapsed component must be formed in 10 yr in each

cluster. This is about one order of magnitUde higher than the

result we obtain for M 3.

In order to estimate the mean frequency of collisions per cluster

from Equation (9), and assuming that all clusters have the same

density distribution, we have to know the mass (i.e., the total

number of stars) of M 3 as compared to the average mass of a

e:lOtJl11ar cllLster. 'J'he IfH1SS afM '3 derived from its luminofdty

function is 2.45 x 10 7C'

(Sandage, 1957). However, this mass

determination is rather uncertain since it depends 00 the

BBBumed shape of the luminosity function. The viri~l masses of

the obular clusters M 92 and 47 Tucanae are also a few times

·510 M~ (Schwarzschild and Bernstein, 1955; Feast and Thackeray,

). Using high dispersion spectroscopy of the integrated light,

Illingworth and Freeman (1974) have determined a total mass of

6·1.3 x 10 M for the globular cluster NGC 6388. Poveda and Allenm

(1975) determined the vial ma.sses of m Cen to be ~. 3.2 x 10 Mm.

'?s

1

6

t r :J t i 0 () r abo II t t. W 0, 0 n () r i n d E> t h L til f'

r chest clusters hAve mRsses up to - 5.2 x t,

10 t·llm'0..."., ('I'l~emilinf~

t [11 co ,

The above derived frequency of collisions for M 3 w 1 fit with

the observed frequency of globular cluster X-ray sources if in

the averaGe, the number of 8tarE5 in a obular cluster 1s 3 Dr

4 times larger than in M 3 or, alternatively, if some ten

ohular clusters in the Galaxy have ten times more stArs thnn

In the Ii t of the above discussed maSS determinations

both theRe alternatives seem very well p05sj.ble~

'l'l-le above dincussionr: on,l)' y if we assume that ~ll clueterH

h~ve the same denRlty distributions.

et ~l. (1968) indicate that the degree of compactness muy vary

from cluster to cluster. For B fixed total number of stars, the

fre'·luenc;y of collisions is proportional to the square of

compactness (the mean density), thuB inversely proportional to

the sixth power of the linear cluster dimension. Therefore, we

may also conclude that the existence of some ten clusters with

a mass similar to that of M 3 but with roughly two times smaller

lineAr riimension is sufficient to give the required frequency of

collisions. Since some clusters in King's investigation are

indeed more compact than M 3, this seems very well possible.

It is 31so interesting to notice that the frequency of ant ­

'79

white dwarf collisions is higher by a factor 10 than that of

Eiant - compact star collisions. Such collisions will form close

binaries with a white dwarf component. This might possibly account

for the existence of (8 part of the) novae and U Gem variables

in globular cluaters (Kukarkin and Mironov, 1971).

Stellar collisions have been proposed to be important in the

galactic nuclei (Van den Bergh, 1965; Shara and Shaviv, 1974).

Such collision::; might lead to runaway nuclear reactions which

might produce a peculiar kind of novae. It is also interesting

to notice that the above discussed energy dissipation through the

formation of shock waves <cf. Section III) is also large enough

to considerably contribute to the energy generation in the

galactic nuclei, The frequency of collisions is considerably

hi in galactic nuclei than in globular clusters (cf. Van

den Bergh, 1965). However, due to the large random velocities

of the otars, the condition in the galactic nuclei is not

favourable for such collisions to form binary systems.

Ack nowl e dgements.

The author is indebted to Professor E.P.J. van den Heuvel for

suggesting him the idea and for his helpfUl advice. He also

thanks J.A. Petterson for discussions.

80

1972, Gravitational N-Body Problem. Ed. M. Lecar

(J: .. A~U. Colloquium No .. 10, D. Reidel Publ. Go., Dor-drecht), 8S

G .. w. 1975, Astrophys .. J~ Letters (in the press)Clark,

G .. w. ~ 1975, Astrophya. J. Letters

(in the preas)

",p-.c-t ,_ "'" ,,'"

Ml .. '1/'1., Thackeray, A.. D.. ',. d.,.::) ....

Gunn, ,) .. goO" Ostriker. J .. P .. 1970. AstrophYB~ Jo 160, 979

(,ura k.:{, H .. 1973, Lectures at the Advanced Study Institute on

phya~CS and Astrophysics of Compact Objects, CambridRe, Rn and

van den Heuvel, E. P .. J. , Astrophys. J. Letters 196, L121

Iben~ To

Iben, I ..

1974-. As tro

K;ippenhahn, R .. , Kohl, K•• Weigert, A.

Limber, D .. N e

Novikov, I .. D... Thorne, K.S.

B. De Witt, Gordon and Breach, Ne~ York

Oort,J .. II .. .van Herk, G.. 1959 t Bull. Astron. lost. Neth .. 14, 299

1975. Astrophys. J .. 197, 155

Sal peter - I' EeE. 1955, Astrophy.s .. J .. 121. 161

Sand.age " 1957, Astrophys .. J .. 125 9 422

1970, Astrophys .. J .. 162, 841

Bernstein, S .. 1 5, Astrophys. J. 122, 200

81'

Shara, M.M., Shaviv, G. 1974. Nature 248, 398

Taff, L.G. 197L~, Astron. J. 79" 1:280

Tinsley, 8.M. 1974, Publ. Astron. Soc. Pasific 86, 554

Tremaine, S.D., Ostrikerf J.R., Spitzer, L. 1975, Astrophys. J.

196 t 4,07

Van den Bergh, S. 1965, Astron. J. 70, 124

82

I would like to express my gratitude to all who have helped me

to make this work possible, particularly:

Professor E.P.J. van den Heuvel for introducing me to problems

in X-ray astronomy and for his guidance dUT the COUTee of

this work,

the late Professor G.B. VBn AlbadB for hlR help which has ennbled

me to study in the Netherlands and for his guidance during my

earliest period of study at the Astronom cal Institute in Amsterdam,

Dr. P.S. The for his help in various problems, not only during my

stay here, but also during my study in Indonesia,

Dr. H.R.E. Tjin A Djie and Mr. R.J. Takena for Many discussions

which helped to clarify many problems,

all my colleagues in the Astronomical Institute or the Univereity

of Amsterdam for their hospitality,

the Dutch Ministry of Education and Sciences for a scholarship,

the Rector of Bandung Institute of Technology and the Director of

BDsGcha Observatory for the leave of absence,

the Indonesian Department of Education and Culture in

the Indonesian Embassy in The Hague for their help to mske offieinl

3rrangements and renewals of my stay in the Netherlands,

my dear wife Asih Trisnawati and my daughter Nani, who did not see

~e for a few years since I started my study here, for their

understanding and their patience.

83

STELLINGEN.

1. The observed orbi tal period charwes of Ceh X... 3 cannot set

any constraint on the initial eccentricity, contrary to a

suggestion by Chevalier.

Chev3lier, R.A. 1975, Astrophys. J. 199, ,89.

2. The observed space velocities of OB runaway stars suggest that

these stars originated from rather wide binary systems (p ~ 10 d).

Bekenstein, J.D., Bowers, R.L. 197/;, Astrophys. J. "120, 653.

3. The existence of a third body with mass of few hundreths of a

solar mass at a distance of a few astronomical unlta from the

Gen X-3 system might account for the apparent orbital period

changes in this system.

Lt. I'he discovery of X~ray pulsations with a period of 282 s in the

X-ray binary Vela X-1 suggests that the previously discovered

pulsating transient X-ray sources do not form a new class of

X-ray sources. The existence of such sources Can be understood

from current theories of the evolution of X-ray binaries.

Rappaport, S.. lv1cClintock, J. 1975, LA.U. Circular Nr. 2794.

5. A number of X-ray binaries with low-mass non-degenerate oomponents

should exist in low galactic latitudes.

van den Heuvel, E.P.J., De Loore, c. 1973, Astron. Astrophys.

25, 387.

Paper IV of this thesis.

6. Low-mass X-ray binaries with short periods (e.g. Her X-"l and Sea X-1)

cannot have originated from Type r supernovae of the kind

hypothesized by Whelan and Iben.

'Ihelan, J., Iben I. 1973, Astrophys. J. 186, 1007.

7. The behaviour of the limi ting maSE] of a neutron star with an

anisotropic equation of state as derived by Heintzmann and

Hillebrandt is a consequence of the imposed spherical Eiymmt~try of

13.11 variables.

Heintzmann, H., Hillebrandt, W. 1975, Aatron. Astrophye. ~, 51.

8. The 13ahcall-Ostriker black hole model for the X... ray sources 1.n

Globular clusters is unlikely.

Bahcall, J., Ostriker, J. 1975, Nature, 256, 23.

9. Rapid publication of preliminary results in letter journals is

necessary in rapidly developing fields and should not be suppressed

because of the rapid growth of the astronomical journals.

Allen, D.A. 1974 i The Observatory 94, 320.

10. The custom of sending reprints as observatory's or institute's

pUbli.cations has no obvious advantages.

Amsterdam, October 1, 1975. Winardi Sutantyo.