UvA-DARE (Digital Academic Repository)
Advanced evolution of helium stars and massive close binaries
Habets, G.M.H.J.
Publication date 1985
Link to publication
Citation for published version (APA): Habets, G. M. H. J. (1985). Advanced evolution of helium stars and massive close binaries.
General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).
Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.
UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)
Download date:28 Sep 2021 IIIA.. THE EVOLUTION OF A SINGLE AND A BINARY HELIUM STAR OF 2.5 M^ UP TO NEONN IGNITION; CASE BB MASS TRANSFER.
Suanary. .
Thee evolution of a single helium star of 2.5 M0 is calculated from the heliumm Zero Age main sequence up to off-centre neon ignition. In the carbon- shelll burning phase the 2.5 M0 helium star becomes a red giant and possibly enterss the Cepheid pulsational instability strip. As a red giant the 2.5 M0 heliumm star may resemble a R Coronae Borealis (R CrB) star. The mass-exchange phasee and the subsequent evolution up to neon igntion of a 2.5 M0 helium star inn a binary is calculated, with the aim to explore the evolutionary history of thee B-emission X-ray (Be/X-ray) binaries. The orbital parameters and the mass off the companion star (a 17 M@ hydrogen star) were chosen such that Roche-lobe overfloww occurred during carbon-shell burning. [The progenitor of this system consistedd of 13.5 + 6 M0 hydrogen-rich stars which underwent case B mass transfer.. After the mass transfer the system is expected to have become a Be- starr binary]. The helium star loses about 0.3 M0 to its companion and (again) neonn is Ignited at "- 0.16 MQ in the 2.2 M0 (case BB) mass-exchange remnant. 122 16^ 20 2k i* Thee core region, mainly composed of C, 0, Ne, Mg, and some He at the boundaryy of the core (denoted in the following as the "core") of the single
2.55 M© helium star as well as that of the 2.2 M0 remnant, is larger than the Chandrasekharr limiting mass at neon ignition. Hence, these cores are expected too undergo core collapse and to produce neutron stars. NumericalNumerical instabilities are encountered when carbon burning is treated with a simplee two-point difference equation. Therefore, an implicit difference scheme basedd on a three-point difference equation is used to overcome most of these instabilities.. The characteristics of some other solutions for preventing the numericall instabilities are discussed. The effects of neutrino losses are investigatedd for the advanced evolutionary phases. The validity of the ideal gass law for the ionic equation of state is examined.
Thee orbital dimensions after the supernova (SN) explosion of the 2.2 M0 remnantt in the binary are calculated under the assumption of symmetric SN mass ejectionn and neglecting the effects of impact of the SN shell on the orbit and onn the companion (these effects are proven to be negligible). The orbital eccentricityy after the SN explosion is found to be small (•*• 0.043). The orbitall dimensions before and after the SN explosion are compared with those off some observed Be-star binaries, Be/X-ray binaries, and other X-ray binaries
-- 47 - withh known eccentricities. For the known Be/X-ray and other X-ray binaries the pre—SNN orbit and the neutron star progenitor masses are calculated (assuming circularr pre-SN orbits, a symmetric SN explosion, and neglecting the effects off impact of the SN shell). The assumption of a circular pre-SN orbit is found too be consistent with observations. Our calculcations thus confirm the model off the formation of Be/X-ray binaries by mass-transfer dominated evolution in ann intermediate-mass binary.
1.. Introduction.
PaczynskiPaczynski (1971a), Arnett (1972a, b), Savonije and Takens (1976), Woosley ett al. (1980, 1984), Nomoto (1980a, b, 1981, 1982a, b, 1984a, b, c, e), and Delgadoo and Thomas (1981) have calculated the advanced evolution of helium starss with masses of 2.0 MQ or larger. In the lower mass range (up to about 3 MQ)) the radii of these stars undergo considerable expansion, which in a binary cann cause the onset of a second phase of mass transfer (Delgado and Thomas, 1981).. An important question is: for which masses do these stars - even after aa second phase of mass transfer in a binary - still collapse to a neutron star.. In order to explore this problem we calculated in full detail the behaviourr with time of the radius, the core mass, and other parameters of a 2.55 MQ helium star during its advanced evolution. We investigate whether or nott the core of the 2.5 MQ helium star will collapse and form a neutron star [ass is suggested by the calculations of Woosley et al. (1980) and Hillebrandt (1982a),, cf. Habets (1984), but contradicted by the results of Delgado and Thomass (1981)]. Our results (paragraphs 2 to 4 below) confirm the evolution towardss core collapse. In paragraph 5 we compute the orbital dimensions after thee SN explosion of such a helium star and in paragraph 6 we compare these withh observed orbital dimensions of Be and other X-ray binaries, and Be-star binariess with known eccentricities in order to test the suggestion of Rappaportt and Van den Heuvel (1982) that Be/X-ray binaries are formed by a mass-transferr dominated evolution in an intermediate-mass binary.
Thee evolution from the helium Zero Age main sequence to the onset of neon burningg of a 2.5 MQ helium star is described In paragraphs 2 to 4. The mass
exchangee in a binary consisting of a 2.5 M0 helium star and a 17 M@ hydrogen
starr is calculated for the case in which the 2.5 M0 star fills its Roche lobe duringg the carbon-shell burning phase (paragraph 3). The orbital parameters of thiss binary were chosen such that the progenitor system could have been a 13.5
-- 48 - ++ 6 Mg system which underwent case B mass transfer. After the second phase of masss transfer (case BB) the evolution of the remnant of the 2.5 M0 helium star iss calculated until neon ignites off-centre. Inn paragraph 4 we investigate the effects of possible numerical instabilitiess - as found by Sugimoto (1970a) and Paczynski (1974a) - on the resultss for the carbon-burning phase. By using an improved difference scheme thee effects of these instabilities on the final results are expected to be negligible. .
2.. The evolution of a 2.5 M^ helium star - single and in a binary - up to neon ignition. .
Inn Figure 1 the evolutionary history of the central density, pc, and temperature,, T , (both in decimal logarithms) is shown. The evolutionary track off the 2.5 Mg helium star in the Hertzsprung-Russell (HR) diagram is depicted inn Figure 2. The fully drawn curves in Figures 1 and 2 are the tracks without masss exchange, the dotted curves are with mass transfer to the 17 M@ companion.. In Figure 3 the evolution of the radius versus core mass (both scaless are decimal logarithms) is given. The dotted track is with mass exchange,, the deviation from the solid curve is marked with an arrow. The obliquee lines in Figure 1 are the approximate lines of constant electron degeneracyy parameter £ as defined in Chapter II. The curve y - 1 is the approximatee demarcation line of the non-relativistic and relativistic regimes. Thee meaning of the letters in the Figures 1-3 is as follows: A-B, the helium main-sequencee phase and the exhaustion of helium in the core; B-C, the radiativee helium-shell burning phase and the gravitational contraction of the 122 16 C-- 0 core; C-D, the ignition of carbon, radiative and convective core- carbonn burning and radiative carbon-shell burning; D-E, the convective carbon- shelll burning phase and the gravitational contraction of the 0-Ne-Mg core; E 20 0 orr E' (in case of mass exchange), the radiative off-centre ignition of Ne, andd convective off-centre neon burning. Thee Figures 4 and 5 show the evolution of the inner regions with time. 10 0 Figuree 5 is for the case with mass exchange. Time is here defined as log r ((tf11 -. - t 0i)/y )> where t , is the evolutionary time in years elapsed
sincee the arrival of the 2.5 M0 helium star on the helium main sequence and
tf.. , is chosen to be (about) one year later than the t •, in the final XX 111 3. J. cvui model.. Thus, t ]_ is zero in the first model, which is derived with a
-- 49 - Schwarzschildd fitting technique (see Chapter II). The dotted lines In Figures 44 and 5 indicate the boundary of the core which is defined by the region interiorr to the helium shell with maximum energy generation. Convective regionss are shown hatched.
Inn Tables 2 and 3 the main data for the 2.5 M0 helium star without and withh mass exchange, respectively, are presented: number of the model in the 10 0 sequence,, teyol (in years), log Teff (Teff in K), Mbol, radius (in R0), logg T (T in K), log p (p in g cm" ), the central helium ( He), carbon 122 16 (( C), oxygen ( 0) abundances by mass or - in case of helium depletion in the 20 0 centree - the central carbon, oxygen, and neon ( Ne) abundances by mass, and * * inn case of mass exchange (Table 3): the mass-transfer rate M in MQ/yr, mass off the mass-losing component in M0, orbital separation (in RQ), and the orbitall period in days (see also the list of symbols in Chapter II, section a).. The standard computer exponential notation is used, e.g .2E+2 means .2 x 100 (= 20). Inn the following paragraphs a more detailed description of the phases A- B,, ..., to E is given.
(i)) core-helium burning (phase A-B in Figures 1-3)
Heliumm burns non-degenerately in the core (Figure 1). The convective core growss about 50% in mass and contains at maximum 1.35 M0 (Figure 4). This resultt was obtained by employing the diffusion approximation of convective mixing.. As an independent check on our results Th. J. Van der Linden computed partt of the core-helium burning phase of the 2.5 M0 helium star with his code. Hee used the same diffusion equation together with the implicit mesh point technique.. He found a maximum extent of the convective helium-burning core of 1.3566 Mg. But, although practically the same input physics was used, the differentt techniques for determining the convective boundary appear not to givee the same abundance variations with time. Figure 6 illustrates this. This Figuree depicts the changes in the central abundance (by mass) with time (in i++ 12 16 years)) of the elements He, C, and 0. The solid curves are our results; Vann der Linden's results are shown by the dashed curves. The change of the centrall helium abundance with time obtained by Savonije and Takens (1976) is drawnn by a dotted line, the variation with time of the central net energy _11 -1 generationn rate enet (in erg g s ) is indicated by the crosses. With differentt opacities, with another triple-alpha burning rate, and with some
-- 50 - 4.00 5.0 6.0 7.0 10logpp (gcrn3)
Fig.. 1.—The central p, T history of the 2.5 M0 helium star. The solid curvee is without mass exchange. The dotted curve is with mass exchange to the 177 MQ companion (see text). The oblique lines of constant electron degeneracy parameterr 5 demarcate (1) the non-degenerate/partially degenerate regime (£ = -- 2.6), (2) the partially degenerate/degenerate regime (5 = 4.A), and (3) the non-degenerate/degeneratee regime (£ = 0). The criteria defining these lines aree adopted from Cox and Giuli (1968). The curve y = 1 divides the non- relativisticc and relativistic regimes. The letters along the solid and dotted curvess are explained in the text.
minorr differences in other input physics Savonije and Takens obtained a maximumm convective helium-burning core mass of 1.23 M0 for a 2.5 M_ helium starr just after helium exhaustion in the core. The use of different opacity
-- 51 - tabless by Savonije and Takens is most probably the cause of the much smaller growthh of the convective helium-burning core in their calculations than in thosee of ours and of Van der Linden. For the C-0 core we find the following 122 16 finall abundances in the centre after helium exhaustion: C = .4905, 0 = 122 16 .4945,, i.e. quite close to the values of C = .463 and 0 = 0.507 found by Savonijee and Takens (see also section IIIB). If one regards only the behaviour withh time, the abundance variations found in our work agree better with those off the latter authors than with those of Van der Linden. We guess that the slightlyy different triple-alpha reaction rate adopted by Van der Linden and an errorr in the programming of the implicit mesh point technique caused this
5.00 4.8 U. 6 U-U U-1 4.0 38 10, , logg TefflK)
Fig.. 2.—The evolutionary track of the 2.5 M0 helium star in the HR diagramm (solid curve). The dotted curve represents the part of the track with masss exchange to the 17 M companion. For the meaning of the letters see the text. .
-- 52 - 2.5 5 [[ 1 1
2.0 0 -- 2.5 MQ
2.55 MQwith mass exchange
CHANDRASEKHAR R 'E E MASS S 1.5 5-- --
E'"1 1_^_ -- 10 logR/Ro o D D
1.0 0--
0.5 5-- r r
0.0 0 -- l l
A"-"" BI -0.5 5 _ _ 11 1 1 i i -0.11 0.0 0.1 0.2 10 logg Mc0re/Mo
Fig.. 3.—The change of the radius with core mass defined as the mass of thee convective helium-burning core between A and B, and from B to E as the 16 20 24 122 ** masss of the core (mainly constituted of C, 0, Ne, Mg, and some He at thee outer boundary of the core) which is limited by the helium-burning shell withh maximum energy generation. The dotted curve represents the part of the track,, with mass exchange to the 17 M0 companion. The dashed line indicates the positionn of the Chandrasekhar limiting mass. The arrow marks the onset of mass exchangee in the 2.5 -I- 17 Mg binary system. The letters have the same meaning ass in Figures 1 and 2 (see the text).
-- 53 - 2.55 M-
2.5 5
He e Helium-burning g He e 2.0 0 //convectivee shell _M_r r I \ agg agg M0 0 A A 1.5 5
1.0 0
0.5 5
ogdtf.nar^oiVy) )
Fig.. 4.—The evolution of the interior regions of the 2.5 M0 helium star withoutt mass exchange. Hatched regions are convective, the dotted line demar- catess the core and the helium-burning layer with maximum energy generation.
"II 1 T-ff
2.55 M0 with mass exchange
2.5 5
2.0 0
1.5 5
1.0 0
05 5 Neonn -burning convectivee shell—^4^
og((t,.. ,-t , )/yr) yy final evol. 3
Fig.. 5.—Same as Figure 4 with case BB mass transfer to the 17 MQ companionn (see paragraph 3).
-- 54 - 44 5 difference.. The kink in our He curve at tgvol = 10 years is due to a somewhatt coarse distribution of our mesh points. Savonije and Takens (1976) didd not use a diffusion approximation for mixing, and the only compositional 4 4 variablee which they implicitly calculated was the abundance of He in the i+ + implicitt mesh point. When Y ( = He) = 0.27 (Figure 6) one reaches the maximum
generationn rate emax in the centre. Up to that time the radius increasess (Figure 3). The star begins to shrink shortly afterwards. This maximumm rate is 2.7 x 10 erg g~ s~ (the same as the one found by Van der Lindenn when Y was - 0.36). When helium is becoming exhausted in the core the 44 1 _1 ratee drops quickly to 3 * 10 erg g s . The spread of enet (crosses) at Y - 0.022 is due to the mixing of fresh fuel into the core. A finer grid of mesh pointss on both sides of the convective boundary at that epoch dit not produce suchh a spread. This demonstrates that the evolution of the 2.5 Ms helium star
1.0 0 2.55 Mo o
-- '\ \ \ \ 0.8 8-- VHe -- 2. \\\ 12„ c c C 12 o o \\ r _ _ He^KK . / W- ££ 0.6 - v.\\ / / ++ -- 2.6 6 \'.. \ / / \\\ 16K xx '\ ' / \\ 0 a a \v\\ / v_ _ E E 'V'x\ \ 100 erg gV '' X \\ O.i. O.i. // / \ \ M M 2.4 4 12rr /' / x'u\ c c o o ~o o c c ,, / ^ \/ -J -J .. / \ V .* -DD 0.2 2.2 2 v A A ////// \ / . Ny\/ // 16o yy?<^.\^H e e ** * s* * r.... - —=e i i . - i 2.0 0 0.0 0 1.0 0 2.00 , „6 3.0 t/10yr r
Fig.. 6.—The variation with time of the central abundances by mass of 44 12 16 He,, C, 0 and the variations with time of the central net energy generationn rate e ,_ (crosses) during convective core-helium burning in the 2.55 MQ helium star. The solid curves are from the present investigation. Van derr Linden's test calculations are shown by the dashed lines. The He curve of Savonijee and Takens (1976) is given by the point-dash line.
-- 55 - iss very sensitive to the mesh point distribution and to the precise determinationn of the convective boundary.
(ii)) helium-shell burning (phase B-C in Figures 1-3)
55 _1 Ass soon as helium is burning radiatively in a shell with e = 10 erg g ss , the radius begins to increase again. The core (defined as before) grows inn mass due to the nuclear burning in this helium shell and becomes larger thann the Chandrasekhar limiting mass (Figures 3 and 4). The bolometric luminosityy increases by about a factor 3 (see also Figure 2). The maximum net energyy generation rate e_a„ achieved in the radiative helium-burning shell is 55 _1 _l *** 4.2 x 10 erg g s , i.e similar to the maximum in the previous phase. The 44 _3 C-00 core contracts, the central density changes from 1.0 x 10 g cm to 4.6 x 55 _3 100 g cm , but no temperature inversion occurs. Throughout phase B-C the neutrinoo losses are not large enough to outweigh the gravitational energy productionn by the contraction of the core. In phase A-B the neutrino losses aree negligible. In this phase the network for helium burning was used (with 3 compositionn variables). In the subsequent phases (B-E) 7 composition variables aree included. For the beginning of phase B-C we determined the equilibrium 11 23 abundancess of H and Na from the networks for thermonuclear burning of 20 0 heliumm and carbon as given in Table 1 and we assumed small abundances for Ne 24 4 andd Mg which are produced in small or negligible amounts by helium burning [cf.. Arnett and Truran (1969), Arnett (1972a), Arnett and Thielemann (1984a), andd section IIIB].
(iii)) core-carbon burning and radiative carbon-shell burning (phasee C-D in Figures 1-3)
8 8 Carbonn ignites radiatively in the centre at T = 5.2 x 10 K, p = 4.6 55 _3 c c xx 10 g cm , and Z, *• 0.7. From that moment nuclear burning of carbon by the 122 12 20 4 12 12 23 1 reactionss C + C —> Ne + He and C + C —> Na + H becomes the dominatingg energy source in the radiative core. The core-carbon burning takes placee in the partially degenerate regime and is therefore non-violent (Figure 1).. The temperature rises quickly after the ignition, the C-0 core becomes 88 5 3 convectivee at Tc = 6.13 x 10 K, pc = 9.28 x 10 g cm" , I *• 1.05, and central carbonn abundance Xc of 0.48, and contraction ceases. At first the density in thee core decreases due to the convective inwards mixing of fresh carbon. Since
-- 56 - thee nuclear energy production rate of the carbon-burning reactions is proportionall to p, also the temperature in the core gradually decreases. As soonn as the energy loss by gravitational expansion of the core can no longer diminishh the total energy production rate, i.e. if no more fuel is mixed inwards,, the central temperature of the core begins to increase again. Just as betweenn points A and B, there is a steady nuclear burning stage (emax - 1.2 x 77 11 100 erg g~ s~ ) during which both the central density and the central temperaturee increase while the bolometric luminosity becomes larger. The maximumm net energy generation in phase C-D occurs at the beginning of this phasee when the central temperature reaches its (first) maximum (see Figure 1, 100 7 _1 _1 att log pc - 5.8 and Xc - 0.47): Emax > 2.6 x 10 erg g s . The maximum
sizee of the convective carbon-burning core is 0.74 M0. When the carbon abundancee is low in the core, the central density still keeps increasing, but thee central temperature drops. The main energy source at that time is helium 66 _1 _1 burningg in a shell (E,,^ - 1.5 x 10 erg g s ). The central contraction finallyy results in a temperature increase. At this time, neutrino losses are noo longer non-negligible. We used the neutrino processes uncorrected for neutrall current effects as given by Beaudet et al. (1967). In the later evolutionaryy stages we corrected the rates of the pair-neutrino, the photoneutrino,, and the plasma-neutrino processes for the effects from the electron-- and muon-neutrinos according to the prescription of Dicus (1972).
(tv)) convective carbon-shell burning up to neon ignition (phasee D-E in Figures 1-3)
Att point D the first carbon-burning convective shell has developed. This 88 _1 _1 shelll produces at maximum 2.3 x 10 erg g s . The carbon shell provides the energyy necessary to maintain the luminosity of the star achieved in the foregoingg phase. The core expands slightly and the neutrino losses control the coolingg of the core during the development of the convective layer. As soon as thee energy production in the shell slows down, the core contracts and the centrall temperature rises again. The stellar centre passes the non-relativis- tic/relativisticc border line (the curve y = 1 in Figure 1) and enters the relativisticc degeneracy regime. Once the convective shell has disappeared, the temperaturee and density throughout the core increase. This process continues untill a second carbon-burning radiative layer at a larger mass coordinate becomess unstable to convection. This shell produces initially enough energy to
-- 57 - stopp the core contraction. The central regions cool again due to neutrino losses.. The above described behaviour repeats itself a number of times (Figuress 1, 4 and 5).
(v)) comparison with results of other calculations
Boozerr et al. (1973) presented details of a similar post-carbon-burning 122 16 stagee for stars (1 - 1.5 KQ) composed of 50% C and 50% 0 without a helium envelopee [Joss et al. (1973) reported a similar behaviour for such stars with aa helium envelope]. These authors detected a series of carbon shell flashes whichh each produced a convective zone. Each successive flash and its associa- tedd convective region occurred farther away from the centre. The evolutionary trackss calculated (but not published) were erratic. In our 2.5 M0 helium-star sequencee the maximum total luminosity in a mass zone of the first convective 6 6 carbonn shell was *• 4 x 10 L0, i.e. similar to the values of L (the rate of energyy generation by nuclear reactions) in the shell flashes of the 1.37 M0 modell of Boozer et al. The latter authors describe the shell-burning phase as aa sequence of thermal relaxation oscillations. [In fact, Ikeuchi et al. (1971, 122 16 1972)) for stars consisting of 50% C and 50% 0 and Kutter and Savedoff (1969)) for pure carbon stars were the first who found such a behaviour (with ann explicit and an implicit method, respectively), even though the last- mentionedd authors used an Incorrect equation of energy conservation from Kutterr and Savedoff (1967, cf. Strittmatter et al., 1970)].
(vi)) convective helium regions in the outer layers and the evolution of the outerr layers - pulsation instability
Duringg the first carbon-burning convective shell phase the outer radius off the star grows to its largest value of 200 R0 (Figure 3). A density 10 0 inversionn and convection occur in the envelope in the region log T = 4.15 - 10 0 4.55 and log p = -8.3 ± 0.3. The convective layer is at maximum about 2 x -22 _2
100 MQ thick. The density inversion extends over 1.5 x 10 M0 at maximum. In thesee low-temperature regions partial Ionization of helium dominates in the equationn of state [cf. Law (1982); our equation of state for a helium mix (Ibenn V) over the range of p and T in a helium red-giant envelope agrees perfectlyy with the tabulated equation of state of Fontaine et al. (1977)].
Whenn the radius is close to 200 R0 the envelope becomes also convectively
-- 58 - unstablee over a mass range of at most 1.5 x 10 MQ at log T *• 5.2 - 5.45 10 0 andd log p = -6.3 to -7.1, see Figure 4. This is the temperature-density regimee in which helium is nearly completely ionized, but in which no density inversionn occurs. The inversion at the lower temperatures and densities is probablyy due to an opacity peak in the helium ionization zone. Maeder (1981a) showedd that the effect of the density inversion in a hydrogen envelope will be aa Rayleigh-Taylor instability. Maeder's results for 60 M@ and 30 M0 hydrogen starss are that the internal structure is unaffected by his treatment of the 10 0 Rayleigh-Taylorr instability and that Teff is lowered by A log T = 0.09. Concludingg from Maeder's test calculations we may expect that the evolution throughh the "red-giant" phase is only slightly influenced by the instability resultingg from the density inversion in the envelope. In contrast to the resultt for our 2.5 MQ star, Law (1982) does not mention a density inversion in thee red-giant phase of a 1 MQ helium star. Before returning to the blue part off the HR diagram the luminosity of the star increases (Figure 2). If the resultss of Law (1982) can be extrapolated, the helium Hayashi track of fully convectivee models of 2.5 M^ will lie close to this turning point of our model inn the HR diagram. The helium Hayashi track is given by the relation:
a logg Teff d log L + constant, wheree d < 0.1, while d might be negative, cf. Law (1982).
Thee turning point of the track of the 2.5 M@ helium star lies just before the theoreticall Cepheid pulsational instability strip for hydrogen-rich stars deducedd by Iben and Tuggle (1975). The position of this strip was derived by thesee authors for low values of Y (= He), i.e. 0.25 to 0.38; higher values of YY move the blue edge of the Instability strip further towards the blue. Recentlyy Cox et al. (1980) and Saio et al. (1984) made linear non-adiabatic stabilityy analyses for helium-rich red-giant stars of 1 - 3 M0 (with Y = 0.98 andd Z = 0.02 and with Y = 0.90 and Xj, = 0.10) with luminosities larger than a feww times 10 L0, such that we are now able to verify our expectation that the trackk might cross a pulsational instability strip. [Non-linear pulsation ana- lysess are available for 1 to 2 MQ model helium stars with luminosities less 't t thann 1 to 2 times 10 L@, but these models are dynamically unstable (Saio and Wheeler,, 1983), see also section IIIB]. The blue edge of the pulsational instabilityy strip for the fundamental mode for 1 - 3 M0 helium stars has a bluewardd extension and a critical luminosity which increases with increasing
-- 59 - masss of the red giant (see Saio et al., 1984); this blueward extension makes ann obtuse angle with the blue edge (see Figure 16a in section IIIB); the blue edgee itself is nearly vertical in the HR diagram and may be analytically expressedd by a relation of the form similar to the above-mentioned equation forr the helium Hayashi track. From Figure 16a in section IIIB (showing the bluewardd extensions for different helium-star masses and different combina- tionss of Y and Z values in the envelope) it can be seen that the 2.5 M0 helium starr may enter the pulsation instability region for the fundamental mode at thee time when its radius is at maximum (<- 200 RQ). This region is thought to bee occupied by R Coronae Borealis (R CrB) variables of which the bolometric luminositiess are poorly known (Saio and Wheeler (1983) and Cox (1984)). In the casee of the 2.5 Mg helium star the period of the R CrB type pulsations has to bee > 40 days (assuming that the R CrB stars pulsate in the fundamental mode, cf.. Saio and Wheeler). However, during part of the "red-giant" phase the 2.5 MQQ helium star will resemble a non-pulsating R CrB star like XX Cam (cf. Saio andd Wheeler). There is only a small probability that the 2.5 M0 helium star cann be observed at the effective temperatures attached to R CrB variables: the 33 10 2.55 MQ helium spends 3.3 x 10 years at values of log T « between 3.7 and 3 3 4,, and 2.6 * 10 years at values between 3.7 and 3.9; see also section HIE.
3.. Roche-lobe overflow from a 2*5 Me helium star in the carbon-shell burning phase* * (i)) evolution before neon ignition (phase D-E' In Figures 1-3)
Inn the following the evolution of a 2.5 M@ helium star in a binary up to neonn ignition is described. The mass transfer is assumed to be "conservative" (Paczynski,, 1971c) and Is treated in a way analogous to a procedure developed byy Van der Linden (1982). As a companion we took a 17 M0 hydrogen star at an d initiall orbital separation of 84.2 R0 ^ = 20 25). Such a binary can be a remnantt of case B mass transfer (P = 2.58) in a system with initial component massess of 13.5 + 6 MQ which evolved conservatively. The 2.5 M0 star appears to filll its Roche lobe several times in phase D-E. Once the radius of the 2.5 MQ heliumm star is > 18 R0 a major amount of mass (0.3 M0) is exchanged. Figure 7 •• _1 showss the mass-loss rate M In MQ yr as a function of time (in years) duringg the first convective carbon-burning shell phase. In view of the new boundaryy conditions during the mass-transfer phase one obtains a smaller
convectivee carbon-burning shell (cf. Figures 4 and 5). In 2860 years 0.3 M0 is
-- 60 - lostt to the companion. The Kelvin-Helmholtz time scale of the star in this phasee (i.e. neglecting neutrino losses) is *• 240 years. At the onset of mass transferr the convective part of the stellar envelope is thermally unstable and iss expanding on roughly this time scale. On the other hand mass transfer causess also the critical radius to increase, because mass is exchanged from thee less massive to the more massive star. Initially, the change of the criticall radius cannot follow the growth of the stellar radius and a sharp _*•• _1 risee in M occurs (Figure 7). A maximum M of 2.3 x 10 M0 yr is reached.
Thee helium star detaches from the Roche lobe at Pf = 28.09; the orbital separationn at that time is 104.6 RQ. In the fourth (and last calculated) convectivee carbon-burning shell phase the helium star again fills its Roche _5 5 lobe.. In about 4 years of Roche-lobe overflow a maximum M of 2 x 10 MQ/yr
-3.6F F
-4.0 0
111 1-UM -UM 5 5 8-44 8
I/) ) Iff = 20.29 3-5.2 2 2.5+17M© © d o o fjj = 28 .08 o o -5.6 6
-6.0 0 0.00 0.5 10 1.5 2.0 2.5 5 TIMEE (x1000 YRS)
Fig.. 7.—The variations with time of the mass-loss rate M in M0/yr duringg the first mass-transfer phase in the 2.5 + 17 Ma binary system. The initiall period (P.) and final period after the mass exchange (Pf) are indicated. .
-- 61 - iss reached. (This exchange phase lasts much shorter than the foregoing one, becausee the evolution of the interior regions proceeds much faster just before neonn ignites.) A third mass-transfer phase lasting about 10 years may occur afterr the disappearance of this convective shell. This phase was not calculated.. (The total amount of mass loss in the last-mentioned two phases is negligiblee (< 10 M@)). Thee expansion of the outer helium layers was not large enough to cause 10 0 thee formation of a thin convective layer at log T ^ 5.2 - 5.5 or to cause a densityy inversion in the stellar envelope. Several helium-burning convective andd semi-convective shells were formed in a region at the base of the helium shelll with maximum energy generation (the dotted curve in Figure 5). At the endd of the calculations the base of this shell is convectively burning helium ass well as carbon (see Figure 5: a convective helium-burning layer of the helium-burningg shell penetrates the core region and merges together with a helium-- and carbon-burning convective layer in the core. This causes a slight reductionn of the core mass (Figures 3 and 5). In case of no mass loss such a penetrationn is avoided and the convective shell becomes radiative by burning 122 16 off helium and carbon due to the triple-alpha and the C(a, y) 0 reactions (Figuree 4). Inn the binary as well as in the single star computations a temperature Inversionn develops in the centre. A second small temperature inversion occurs
inn a carbon-burning layer with a mass coordinate of 1.38 MQ.
(ii)) neon ignition (phase E, E* in Figures 1-3)
Neonn ignites off-centre at a mass coordinate of "• 0.16 Ma in the 2.5 M0
heliumm star and In the 2.2 M0 mass-exchange remnant. The computations were terminatedd a few models after the moment at which the neon-burning shell becamee convectively unstable. Figure 8 depicts the thermodynamic structure of
thee interior of the 2.5 and the 2.2 M0 stars in the last models (at the time
off neon burning in a convective shell). At the edge of the core at 1.54 M0 the densityy and temperature fall off extremely sharply (the solid and dash-dotted ,, 8 liness of T/10 K are of the 2.5 M0 and 2.2 M@ helium stars, respectively; the 10 0 dashedd and dotted lines of log p are of the 2.5 M@ and 2.2 M0 helium stars, respectively).. The pressure gradient is less extreme (the pressure gradient of thee 2.2 MQ mass-exchange remnant is depicted). Notice also (Figure 8) that
moree than 1.4 M^ is contained within 0.01 R0> while at the helium main
-- 62 - sequencee only 0.01A M0 was concentrated within that radius (the solid and 10 0 dash-dottedd lines of log T/RQ are of the 2.5 Ma and 2.2 M0 helium stars, respectively).. Figures 9 and 10 show the composition structure of the last modelss in the 2.5 MQ and 2.2 M0 helium stars. The compositional gradients were
16 6 11 1— 1 11 1 relatrvisticc [ non-relativistic 10 degenerate e non-degenerate e logg P tt jolly H H 10 |degenera e e logpp --™-Ii:L_=i,
= 8 ^^ *^SSV-II I ^v -- 12 2 T/10 K K I I —-—___J___ _ ii ^^vA 21 1 10 0 \\ u ulogP P £? ? ulogp p 10K K J11 1 19 9 8 8 ,() 2 2 ^""~~~\\ | U logr/R0 6 --6 1 -- i. 0 0 \%\% YX IS-- i. i... ,, log r/R0 10 logg r/R0 r-V-^!jj pdP IS-' ' 2 2 11 \ ' v 1 -- 2 ''uu 10logP JJ ii 0 I0|„nn ,-"- \\ ' «V logp p T/10K K 10K -2 2 i i 1!! ', II 1 10, % %
^—-^—-——"—'——"—' ! \ 10log'JOx. x -U U 10logx x :\\ .; logp -2 2 11 I \ ' -6 6 !! ' \ * _ _ ^partiallyy ' e\ i idegeneratee ' \ ; . -8 8 !! « 1 11 1 1 ,, relativtstic , J non-relatiyistic -10 0-2. 0 0 -1.5 5 -1.00 -0.5 i°iogMr/M0 0
Fig.. 8.—The thermodynamic structure in the 2.5 and 2.2 M0 stars in the
lastt calculated models with convective neon-shell burning. The run of the 88 10 _3 10 temperaturee (T/10 K), density ( log p; p in g cm ), and radius ( log r/R0) aree depicted for both stars, see the text. Also the variation of the total 100 _2 pressuree ( log P; P in dyn cm ), of the electron degeneracy parameter E, ( = 100 2 _1 u/kT),, and of the opacity ( log K; K in cm s ) through the 2.2 M0 star are shown. .
-- 63 - Abundancee by mass(fraction) O O o o O O O O o o 9 9 o o o o o o o o O O ID D o o o o o o o o o o o o o o o o M M en n NJ J ai i NJ J L/1 1
£ £
O O ai i
ai i era a
C C 3 3
r-o o ö ö
a. a.
I I ai i
-- 64 - Abundancee by mass(fpaction) o o o o o o o o o o ö ö o o o o O O p p o o o o o o o o o o O O o o o o -» » ro o ai i M M en n —* * ro o Ti Ti '' ' I -i—i—111111 1 III I I I M era a
I I I I ro o en n
o o en n 3 3 O O o o 3 3 (£3 3 i/> i/> to o f» » o o TT TT O O 3" " \ \ O O e e (Jl l Troo <ü IPP _, O. . \ \ ro o ö ö X X 1) ) ^ ^ ro o ,, ,1 1 1 ai i -- 65 10 0 ^.convectivee neon-burning layer e 3log|c| | itj j _.^_-.^-iT-- v.'-/A-^- - n n P-net1 1 44 h l^netl l 10loglLlI l I 2 2 0 0 -radiativee neon-burning layer r >> i -2 2 ^?net t 1II I Legend: : 1 1 ' ' ""-;-—ICnetll /,.-* /"'\; -/. . tn n A A c / l neri,' ' 11 . \ II lie I £ ne£ tt * I I -- ./v '' + 1 .. V -6 6 -Cnet t "4 4 -L L ** * ' ' V.. \ L L j j ~~-~~~~-~~ __convectrve He- :: I burningg layers i i ]] L .111 .13 .15 .17 .19 .21 .23 .25 .27 -10 0 11 1 1 1_ -2.0 0 -1.5 5 -1.00 -0.5 0.0 0 10|, , logMr/M0 0 Fig.. 11.—The variation with fractional mass of the absolute values of _11 _1 thee energy generation rates (in erg g s ) and of the heat flow L (in Ls) inn the last calculated model of the 2.2 M0 mass exchange remnant. The convec- tivee and radiative regions are indicated. Since e and L can be negative we takee e and |L in the case these quantities are negative. The inset gives thee variation of £ and L at the edge of the core. The parts of the curves off e and L in which e and L have different sign are denoted by differently drawnn curves (see the legend). The nuclear energy generation rate e peaks in thee convective neon- and radiative carbon-burning shells. The net energy generationn rate Enet is not given for the outermost layers (see the text). In thesee regions enet- is equal to the gravitational energy generation rate e . -- 66 - producedd by nuclear burning as well as convective mixing. The abundance of N iss kept constant in all computations. Notice the large extent of the helium- burningg layer in the 2.2 M@ star just before neon ignites; the inner 1.4 M0 is nott much affected by the mass exchange, cf. Figures 4, 5, 9, and 10. Figuree 11 gives for the last model of the 2.2 MQ remnant the variation of thee energy generation rate by nuclear burning (£„)» of the energy loss rate by neutrinoo emission (~ev), and of energy production by gravitational contraction +e or OI ( e)) energy loss by gravitational expansion (-e ). The net energy e s snown generationn rate ( net^ * also, together with the total luminosity (whichh may be negative due to either neutrino or gravitational energy losses orr both). The region with the steep density gradient at 1.5 M0 has been enlargedd in the inset. Decimal logarithms are taken of the absolute values of thee energy generation rate and of the heat flow Lr. The alternating positive andd negative enet-values in adjacent mesh points in radiative central regions requiree an explanation. The reason for this is outlined in the following paragraph. . 4.. The influence of the numerical procedures and neutrino losses on the modelss in the very advanced evolutionary stages. a.. Possible effects of our numerical procedure. Thee use of the three-point difference equations (see Appendix I and Chapterr II) stabilizes the solution for the luminosity. It may, however, for smalll absolute values of the net energy generation rate enet» cause £net to changee sign in a series of adjacent mesh points. These changes need not always bee real, but may be the result of the numerical procedure in which enet is obtainedd by the subtraction of two large and almost equal quantities. These cancellationn effects are introduced by the gravitational term In enet» That termm is expressed as: AQ Q ee = - - (QT • AlnT + QR • Alnp + Z QX • AlnX )/At gg At ill wheree AQ = T AS and AS is the change in the specific entropy of the gas per gramm S in a time step At; QT, QR and QX^^ are T times the derivatives of S with respectt to InT, lnp, and lnX^ at constant p and X., , at constant T and X., or att constant p, T and X. (j*i). respectively. The summation is over all -- 67 - chemicall elements i. The cancellation is primarily caused by the terms QTT • AlnT and QR • Alnp. A slight change in the evolutionary corrections AlnT andd Alnp has great consequences for the final value of e . These corrections are,, however, more strongly determined by the solutions of the other structure equationss (2)-(4) (Chapter II) than by the equation of energy conservation (equationn (5) in Chapter II). Therefore, for small jenet|-values, enet can changee sign in adjacent points although the corrections to InT and lnp do only changee gradually in these points. The nature of our difference equation gua- ranteess a smooth variation of the total luminosity, but the main contribution too the luminosity is not from the parts in the interior of the star in which ee „r changes sign in a series of adjacent points (see Figure 11). In other words,, for small |enet|-values, the spurious sign variations may occur. The reall and often large sign variations, however, are caused by nuclear burning orr by convective mixing which requires energy. E.g. the sharp rise of log |e|| and log |Lr| at log mr/M0 = 0.188 in the inset of Figure 11 is entirely duee to the occurrence of a convective helium-burning shell and to nuclear i++ 12 burningg of the He and C in the radiative layer beneath this shell. The relativelyy weak dependence of the final model on the sign variations of enet att small IenetI-values therefore justifies the use of our numerical procedure. b.. Alternative numerical procedures, advantages and disadvantages* Althoughh our procedure might sometimes be unstable In radiative regions wee do not believe that other difference methods (i.e. as suggested by Kutter andd Sparks (1972) and references therein, or by Sugimoto (1970a) and by Christyy (1964)) used in stellar evolution calculations are the best available forr these regions. Sugimoto (1970a) proposed a hybrid difference scheme: if in aa part of the interior the temperature distribution is determined by the heat floww L , this region is treated with the ordinary difference equations like thee set (10) of Chapter II, cf. Sugimoto (1981); and if in a region of the starr L does not determine the stellar structure and if the temperature distributionn depends only weakly on heat diffusion (e.g. during rapid evolu- tion)) a modified implicit scheme (which tends to an explicit one, see below) iss used. In the latter case the ordinary difference scheme gives an unstable solution,, cf. Sugimoto (1970a, 1981) and see Chapter II. Sugimoto (1970a) deducedd that the time step to be used in the ordinary implicit difference schemee should at least be larger than the time scale of heat diffusion t^(r) -- 68 - inn order to obtain stable solutions. In our last models indeed t, (r) is always lesss than the time step At in regions of the star beneath the envelope. In thosee models the ordinary difference (or simple-minded) scheme ought to be stablee according to Sugimoto. In our last models with non-equidistant mesh it Is,, however, not stable. Sugimoto et al. (1981) argue that the hybrid scheme iss stable even in the intermediate region where the stability conditions for thee explicit scheme (for equidistant mesh points) as well as the simple-minded implicitt scheme are weakly violated. In the limit of a small time step the hybridd scheme tends to the explicit scheme (Sugimoto, 1970a, Sugimoto et al., 1981)) and by experience this scheme always gives stable solutions. However, onee of its undesirable aspects Is that the physical picture Is completely determinedd by the choice of the mesh points for which the modified implicit differencee scheme is used. In such scheme the heat flow Lr is not evaluated at thee same mesh points as the temperature, but for example "shifted by half a masss zone with respect to the temperature" (Paczynski (1974a), loc. cit.). We concludee that a shift by a full mass zone implies that the differences Ln - Lnn and Tn - Tn for the mesh points n+1 and n are approximated by (cf., Sugimoto,, 1970a): n+11 n . n+1 n. , n n n. LL - L = (m - m ) • enet(T , p , X±) (1)) and n+1 n+11 n n+1 n+ n+1 +1 Gm 2 n+1 T - T = (m - m") • Vrad(T \ p , x£ ) • n+1/^(r P) (rr )" orr by: n+11 n . n+1 n. .n+1 n+1 v n+1, LL - L = (m - m ) • £net(T , p , X ) (2)) and +11 n n+1 n n n 2 n T" - T = (m - m ) • Vrad(T , p , jj) • -^/4*(r P) (rr ) Clearly,, only part of the physical information at each mesh point goes into thesee equations. Therefore it is expected that the schemes (1) and (2) do not givee the same results. Since the hybrid implicit scheme tends to the explicit scheme,, also, quite different results are expected for the intermediate regime inn which the stability conditions for the explicit and the simple-minded implicitt schemes are violated and in which the differential equations cannot bee solved exactly with the hybrid scheme (cf., Sugimoto, 1970a). In that regimee Sugimoto (1970a) proposes to obtain the stable, approximate solutions byy using double precision in the calculations (which is then also needed in -- 69 - thee matrix solving routine). It is worthwhile to compare the evolutionary sequencess of models which are calculated with either the difference scheme (1) orr (2) and to explore if differences found in the first models would accumulatee in subsequent models, due to either the explicit nature of the schemee or the approximations made. Nevertheless, Sugimoto (private communi- cation)) reported that similar results have been obtained for problems in which thee hybrid scheme and a difference method based on a staggered distribution of meshh points were used. However, there are stellar evolution codes which have a differencee scheme based on an staggered mesh (cf. Lax, 1954), e.g. as describedd by Christy (1964) and used by Arnett (1972a) and as given by Kutter andd Sparks (1972). An explicit scheme with a shift by half a mass zone was developedd by Christy (1964) and by Edwards (1969) who gives a good introductionn to the method. An implicit scheme with such a shift was published byy Kutter and Sparks (1972) with references to other similar schemes. In all thesee half-mass-zone schemes the energy generation rates of one "half" mesh pointt enters into the equation of energy conservation, i.e. quite similar to thee prescription of Sugimoto (see scheme (1)). Moreover, Kutter and Sparks use ee and e from the previous model to derive the new model. Thus, their code is nn v r inn fact partly explicit. With this code Sparks and Endal (1980) were able to evolvee a 15 Mg hydrogen star from neon burning to core collapse. Christyy (1964) introduced a rather cumbersome explicit difference equa- tionn for energy conservation with which Arnett (e.g. 1972a, 1978b) obtained stablee solutions for Lr• With some modifications during the computations this differencee equation proved to be succesful in stellar evolution - from helium burningg to core collapse. Christy's difference equation for the energy flow containss the information of 4 mesh points. Therefore It needs a special treatmentt (like our diffusion equation) in order to be built into the computer code.. The difference expression which we use here as derived by Takens (see Appendixx I of Chapter II) has the favourable feature that the physical informationn of three mesh points, weighted according to the mesh point division,, goes into the difference equation. Such a feature is not built into Christy'ss expression. Hence, each of the available difference schemes has Its ownn undesirable properties, and as concluded in the foregoing paragraph, our resultss are not expected to have been much affected by the slight numerical instabilitiess in the very interior regions of the models of the final stages. c.. The effects of neutrino losses* -- 70 - Comingg from the interior the neutrino loss rates increase towards lower 99 5 _3 density,, and a maximum is obtained at 1.38 M0 (T * 10 K, p - 10 g cm ). In thiss density-temperature regime the pair-and photoneutrino loss rates are closee to their maximum possible values (cf. figure 3 in Beaudet et al., 1967). Thee pair-neutrino process dominates over all other processes. The plasma- neutrinoo emission contributes most to the losses in the higher temperature regime.. From the work of Itoh and Kohyama (1983) it follows that if the correctionss for neutral current effects (see e.g. Itoh et al., 1984a, b, c), forr electron screening (Itoh et al., 1977, 1979, Ichimaru, 1982, but see also Ichimaruu and Utsumi, 1983, 1984) and for ionic correlations are included that thee neutrino-Bremsstrahlung emission rate is about 0.25 of the total of the 99 7 8 _3 otherr neutrino loss rates at T ^ 10 K, p »• 10 - 10 g cm . The other neutrinoo loss rates are approximated by an analytic formula which is precise 88 10 too 5-15% for 10 K < T < 10 K. The uncorrected neutrino-Bremsstrahlung rate iss uncertain by 5% for extremely relativistic regions according to Festa en Ruderraann (1969), but uncertain by less than about 17% in that regime according too Dicus et al. (1976). In the calculations presented here the neutrino- Bremsstrahlungg losses were not included. Moreover the other neutrino loss ratess which we actually used differ somewhat from the rates of Beaudet et al. correctedd for neutral current effects, but always by less than 17%, which is aboutt the accuracy of the "corrected" rates. We included neutral current effectss due to the electron- and muon-neutrinos only, and not due to the tau- neutrino.. Test calculations show that neon ignites convectively at the same zonee in the star if all neutrino loss rates are Included properly. Inn these very outermost layers the neutrino losses fall off somewhat less rapidlyy in comparison with the energy generation rate by nuclear reactions (Figuree 11). This may be due to the fact that in our program the neutrino loss ratess in the very outermost layers are evaluated slightly beyond the range of validityy of their analytical expressions. This less steep decline may not be realistic. . ,10 0 Thee irregular behaviour of |L| and |Enet| at log mr/M0 = 0.19 and at logg m /MQ = 0.25 is due to convection and semi-convection which occur in narroww shells (Figure 11). The curve of |enet| is not shown for log mr/MQ > 0.27.. In that mass range |£net| is rather similar to E , because in this range ee and e go to zero. [As the contribution of these layers to the overall energyy generation rate is negligible, this slight discrepancy does not affect thee overall structure of the star]. -- 71 - Thee central temperatures and densities of the 2.5 M@ helium star evolve duringg core-neon burning towards the region where the electrons become stronglyy degenerate and in which the Coulomb interaction energies become of thee same order as the thermal energies of the ions. Hence, the equation of statee of the ions is inadequately described by that of an ideal gas and should bee improved with formulae for one-component-plasmas as given by Slattery et al.. (1980) for the liquid state of the ions. Care must be taken to generalize thesee formulae to the more-components plasma, see Mochkovitch (1983); for a revieww see Ichimaru (1982). These adjustments for the ionic equation of state weree not made in the present calculations. Test calculations show that the use off the proper equation of state gives higher temperatures and densities (by severall percents in the very advanced evolutionary stages), but due to this onlyy minor quantative changes to the evolutionary paths are to be expected. Inn Figure 8 the run of the electron degeneracy parameter £ throughout the 2.22 MQ mass-transfer remnant is displayed. The inner 0.87 Ms is degenerate (£ >> 4.4), the inner 1.10 MQ is relativistic, and the outer 0.75 MQ is non- degenerate.. In the whole convective neon-burning shell the degeneracy is liftedd to E, *• 8. Remarkably, the part of the star with the steep temperature andd density gradients is non-degenerate. The central regions of the star are quitee strongly degenerate (£ = 11.22), but the electron-capture regime is not yett reached. Threee temperature inversions occur in the last models (Figure 8). At 10 0 logg mT/'HQ < -0.8 the inversion is due to neutrino losses and it is removed byy the energy produced in the convective neon-burning shell. The radiative helium-- and carbon-burning shell causes the spike in the temperature track at 10 0 logg m /M0 = 0.24. The outer helium-burning shell gives rise to a small temperaturee Inversion at log tnr/M0 = 0.2 7. The opacity in the interior regionss of the 2.2 M0 mass-exchange remnant (Figure 8) increases with fractionall mass, except at £ = 5 where the effect of electron degeneracy on thee opacity gives a discontinuity. 5.. The orbital dimensions after the supernova explosion of the 2»2 Me remnant. Inn order to derive these orbital dimensions we first neglect the impact off the SN shell on the Be-star companion. The effects of impact will be estimatedd under the assumption of a symmetric explosion. -- 72 - (i)) orbital parameters after an explosion with symmetric mass ejection, withoutt impact Inn this case, if we assume a circular pre-supernova orbit, equations (1)- (8)) of Flannery and Van den Heuvel (1975) yield the final period, Pf, the finall eccentricity ef, and the final orbital separation, af, as a function of uf,, the final total mass of the system in units of the initial total mass. Thesee parameters are given by: 3/2 (1)) Pf = (uf/(2uf- 1) ) P0 ; ef - (1 - uf)/uf ; af = aQuf/(2^f- 1), wheree uc = —h £— , M°, M? are the pre-SN masses of the exploding and the ff M° + M2 non-explodingg component, and PQ and aQ are the initial period and initial orbitall separation, respectively. d Withh P0 =* 28 09, Mf = 2.2 M0, M° = 17.3 M0, M| = M°, aQ = 104.6 R0 and assumingg that the SN explosion of the 2.2 M0 star leaves a 1.4 M0 compact d remnantt we readily obtain: Pf = 30 63, ef - 0.043, and af = 109.3 R0. (Ü)) the effects of impact on the orbital dimensions Dee Cuyper (1982, 1984) derived expressions for Pf, ef, and af if the SN shelll impacts on the companion: 1/2 n££ s (2)) Pf = -3^ P ; e - {1 pi ; af - nf • -g- , SS 2 ^ wheree S = 2uf -1 - (-5^) , vim and v° are the velocity of impact of the SN shelll onto the companion and the relative velocity in the initially circular orbitt of the system, respectively. These velocities are given by: 00 G(M° + M£) vimm = If' FIN * vSNshell and v a^ » wheree If is the ratio of the momentum effectively imparted to the companion to thee momentum of the impacting part of the SN shell. If is adopted to be 0.3 [seee Fryxell (1979) and Fryxell and Arnett (1981), and De Cuyper (1982)]. Note thatt equations (2) reduce to equations (1) in the case vlm = 0. The ratio of thee mass of the supernova shell which interacts with the companion to the totall mass of the companion, FIN, is: -- 73 - (M°° - Mf) R° * FlNN = ~T4 (^ * Substitutingg the values of M^, M^, a , M^ and assuming R? = 5.5 R0 for the Be starr (the radius of a main-sequence star of 17.3 M0 with Z = 0.03 according to -55 ' _h Vann der Linden, 1982) we obtain FIN = 3.2 x 10 . Hence, 5.5 x 10 M0 of the SNN shell of 0.8 MQ interacts with the Be star. De Cuyper gives also an expressionn for M2 [which is a fit to analytic impact calculations for a star expressionn for M2 [which is a fit to analytic in withh polytropic index 3 by Wheeler et al. (1975)] xo T J Mee = M2 [1 -{ * *4 }°], snel1 wheree <\> = FIN [--y- - 1] with vSNshell being the ejection velocity of the supernovaa shell and v is the escape velocity from the surface of the companion: : 2GM°° !/2 vess = (—7o> ' R2 2 100 10 Thiss equation only holds for log 4" > - 3; for log Substitutingg the above-mentioned values of M° and R°. we derive for v : _11 _5 l l es 1095.33 km s .As F-j-^ < 5 x 10 we have to invoke extremely large ejection velocitiess in order to obtain deviations from a circular orbit due to impact, 44 _1 i.e.. vSNshell snould be larger than 3.5 x 10 km s . However, the observed *tt _1 valuess of vSNghell are in the range 0.5 to 3 x 10 km s , cf. Schatzman (1965),, Zwicky (1965), Shklovsky (1968), and Minkowski (1969). Hence, we can safelyy neglect the effects of impact on the companion and on the orbit of the systemm in the case of a symmetric SN explosion. Preliminary results of De Cuyperr (1984, private communication) for an asymmetric explosion indicate that v loww kick velocites ( wcir ^ 100 km/s) are sufficient to yield large eccen- tricitiess (> 0.4) for the final orbit (neglecting impact of the SN shell). 6.. Comparison with observations. -- 74 - Kelleyy et al. (1983) obtained orbital dimensions for the transient X-ray binaryy 2S1553-542. These authors suggest that the system 2S1553-542 contains a Bee star with a mass of 5 - 10 M^ according to the mass function. The period of thee system is 30d6 ± 2d2. The orbital eccentricity is at most 0.09. Kelley et al.. conclude from this small eccentricity that either the system is circula- rizedd or that the supernova explosion left the orbit nearly circular. The firstt conclusion is very unlikely, because the tidal circularization time 8 8 scalee is very large in a binary system with P > 5 - 10 days, i.e. > 10 years,, according to Savonije and Papaloizou (1983). The second conclusion, if true,, would require, according to Kelley et al., that the mass of the progenitorr of the neutron star was less than 4 MQ (under the assumptions of an upperr limit of 20 MQ for the mass of the companion, a mass of the neutron starr < 2 Mg, and a supernova mass loss in the system < 2 M0). The final orbitall period and eccentricity after the SN explosion which we obtained for thee 2.2 + 17.3 MQ system happen to be quite similar to those of 2S1553-542. Hence,, a case BB mass transfer as proposed in this paper could very well have takenn place in the progenitor system of 2S1553-542. Inn Table 5 the periods and orbital eccentricities of 2S1553-542, some otherr Be/X-ray binaries, Be-star binaries, and GX301-2 are listed. The massive oness among the Be-star binaries are likely candidates for an evolution towards Be/X-rayy binaries (cf. Rappaport and Van den Heuvel, 1982). Since circula- rizationn occurs on a large time scale, the large observed orbital eccentri- citiess in the wide Be/X-ray binaries listed in Table 5 are due either to an asymmetricc supernova explosion or to a symmetric supernova explosion with a largee supernova mass loss from the system. This conclusion holds especially forr a circular pre-supemova orbit, but is weakened for non-circular initial orbits.. If we assume a symmetric supernova explosion to occur in an initially circularr orbit with no effects of impact on the companion and on the orbit, we cann obtain the masses of the progenitor systems and the initial pre-supernova periodd (PQ) of 4U0115+63, V0332+53, A0535+26, A0538-66, and GX301-2. To this endd we assume the mass of the neutron star to be 1.4 M0 and we estimate the finall mass of the secondary M2 from the spectral type; in the case of GX301-2, Moo was taken from Rappaport and Joss (1983). In this way we find for the d systemm 4U0115+63: PQ = ll 24, MJ = 2.9 - 8.68 M0 if M° ranges from 3 to 20 M0 [thee mass of an 0 or Be star, cf. Hutchings and Crampton (1981)]. Taking M° = 99 - 15 Mg, for the mass of the B-type optical component in A0538-66 and e = 0.4 wee derive: P = 6?5 and M? = 5.56 - 7.96 M0. However, for these latter two -- 75 - systemss the neglect of impact is questionable. If for the system V0332+53 we adoptt the lower limit of ef (= 0.25) and the final period of 34.2 and if we takee the mass of the optical component to be 3 - 20 M@ we derive: PQ = 19.87, M?? = 2.5 - 6.75 MQ. These resulting values for the mass of the progenitor of thee neutron star in V0332+53 suggest that the neutron star originated from a moree massive progenitor than in the system 2S1553-42 which has a nearly circularr orbit and a comparable period [under the assumption of a symmetric SN explosionn without impact and a circular pre-supernova orbit, after a (case B) mass-exchangee phase]. Taking M£ = 15 - 20 M@ (the mass of the 09.7 Hie companion)) and ef = 0.3 we obtain for the system A0535+26: M° = 6.32 - 7.82 MQQ , P = 57.0. With M? - 40 M0 (the minimum mass in absence of eclipses in the X-rays),, ef = 0.47 and Pf = 41^4 we obtain for the system GX301-2: PQ = 13^21 andd M? = 20.9 MQ. From the above derived initial orbital periods and masses it iss likely that the systems 4U0115+63, V0332+53, A0535+26, and GX301-2 all ori- ginatedd from progenitor systems which underwent case B mass transfer. In such aa system after the mass exchange a Wolf-Rayet (WR) system is formed in which lateronn the WR star collapsed to a neutron star. The WR star was the origi- nallyy more massive star in an unevolved early-type binary. Such unevolved early-typee binaries often have a mass ratio close to unity [Garmany et al. (1980)) and Lucy and Ricco (1979)]. Hence, in general, due to the mass transfer thee period of the pre-SN orbit (i.e. after the mass-transfer phase) was larger thann the initial (Zero age) period. Therefore also A0535+26 may have had an initiall (Zero age) orbital period < 30 . As a consequence of the mass exchange thee pre-SN orbit Is expected to have been nearly circular. Massey (1982) pointedd out that all observed WR + 0 systems with P < 30 days have nearly circularr orbits. Therefore, the above-made assumption of an initially circular orbitt is plausible for the pre-SN orbit of 4U0115+63, V0332+53, A0535+26, andd GX301-2, such that the conclusions about the pre-SN masses and orbits seem warranted. . -- 76 - .;M<*ME+I>JJ 4.9 7»»5 -^.!3S* .15U4F.HHJ n.4i:4 1,8001 JtUI5 .4950 .4*85 1 22 .620597E+04 4.6B5fc -4.4157 .lH25F*O0 «. 19J1 l.ÜMTi ,8h73 .n|m .CKI75 ,3»>2ÏÏ JHE-HI. 4.979^ -5.14*1 .1472E+OÜ 8.Ü78 l.SW* .0011 .4938 -4*01 .2*2228F«-UU -.9813 -5.1529 .1454F+00 *.4JL0 1.8280 .0009 .4911 .4908 111 .425708E+05 4.FB55 -4.4222 .lAltSE+Ofl «.mei ! .9507 V0">M ,i>,7S 177 .874873E+05 4.8555 -4.431» .3652EHK. 8.194. 1.2 70B .41irit .047^ .mi:,, .2*2i;iFnJÏÏ 4.9(134 -5.1*09 .3*34E«O0 8.4248 3.6404 .0O07 .492 7 .49L6 222 .D1326E+06 4.BB57 -4.4461 .1675E+O0 ft.14™ 1.77:4 ."KV*; ..>7r\4 .171171» .2h241 MM1U.07 4.17HH 4.M72 .1790140 0 1.4**5 5 1.14*1 1 .tM5 5 .«9*5 5 .0000 0 ,, .271444*4' *.6*41 1 -«.594* *.29*0*4 1 1 1.851** 5.74)9 .46300 .479* .03*9 .i«*JH«7 7 «.HUU -4.JM7 .. 101HW1 9 9 1.217* * .4909 9 .*9*1 1 .0000 0 .271*64*47 7 4.6IM M -6.9941 1 .2191*41 1 8.15100 5.7140 .46900 .6797 .0170 .>*»HI*07 7 *.aM«« -i.yut .10*4141 1 2 2 1.2701 1 .4*09 9 .«9*9 9 .0000 0 II .271467147 4.61)2 2 -«.9951 1 .2600*41 1 8.89066 5.7455 .4*433 .479] .03*0 .IIOMW-KII I 4.A4.77 -é.407* .1144*41 1 7 7 5.1217 7 .4*09 9 .*»*J J .0000 0 >> .271449*47 4.4*2* * -6.9992 2 .2606*41 1 1.15011 5.7)71 .46466 .47*1 .0]» .270I4HW W 4.HO00 -*.*1H .11*4*41 1 t.472* * 9.15*1 1 .4905 5 .*9*1 1 .0000 0 II .271471*47 4.4124 4 -6.991* * 26I1E4I a.*4977 5.7)01 .4*411 .4794 .0171 .I10»«-K17 7 4.amm -6..3H .1292141 1 a.«*oo o5.411 9 9 .4105 5 .0000 0 .271472147 7 4.6(20 0 -6.9996 6 .2616*41 1 8.84922 1.7214 ,1*111 ,179* .036* .ÏKHOM-MT T 4.(1100 -*.»»*) .itmw w 9.4*71 1 .4*09 9 .41*9 9 ll .271476*47 4.6*11 1 -6.9161 1 .2616*41 1 44 5.710* .66MM .«1*9 .0)92 .11DMIS407 7 4.«1177 -i.4t04 .1171*41 1 8.4*17* 7* 1.4*77 7 .*909 9 .4*44 4 .2714*0*47 7 4.6805 5 -6.59«7 7 .16)5*41 1 1.8*755 1.6*91 .461** .47(0 .061* .tiottimn .tiottimnt.Ml èè -«.4714 .1449241 1 *.«**4 4 9.9*7* * .*905 5 .49*4 4 .0000 0 .271461*47 7 4.6796 6 -6.597* * .2647*41 1 S.**677 5.66*1 .4*077 .477) .0411 .I7«77MM7 7 4.7W11 -4.HH .1515*41 1 1.7090 0 9.9171 1 .*«05 5 .0001 1 .271488*47 7 4.67(6 6 -6.9983 3 .2699*41 1 I.4U344 1.6777 .46000 .6769 .0*62 .1J«70I*<17 7 4.7.UU -t.4N« .1*07*41 1 1.7114 4 1.4*57 7 -*90* * .«944 4 .0002 2 .2714*9*47 7 4.4769 9 -6.1600 0 .26*2*41 1 «.**««« 5.6624 .49*22 .47» .0*42 .*«** * .17141*147 7 .2691*41 1 .27MHI«0> > 4.77122 -t.MU .lt*9t41 1 1.71*1 1 9.4115 5 ..901 1 .000) ) 4.676) ) -6.1406 6 8.8*422 9.6171 .41799 .4797 .0670 .171107*47 7 .2719*41 1 .17L0UKO7 7 4.74011 4.9147 .17*9*41 1 1.71*7 7 9.7*07 7 .**01 1 .*9*2 2 .0005 5 6.674) ) -6.16)1 1 8.84)00 9.6433 .6M11 .67)1 .0*86 .171319*47 7 4.6717 7 .2716*41 1 «.1*111 5.4290 .41444 .471* .0119 .271MXM7 7 4.71199 -4.9202 .1*91*41 1 1.7111 1 5.7714 4 .*900 0 .*9*2 2 .0001 1 -6.36*6 6 .171923*47 7 t.6707 7 .1771*41 1 8.64199 9.6242 .11)77 .47H .0530 .jm.«*07 7 4.7**55 -t.9211 .1*14*41 1 1.7)35 5 s.aoi. . .*B98 8 .*9*0 0 .0010 0 -6.9680 0 .2715*0*47 7 .2*31*41 1 8.84099 9.6097 .41200 .4724 .019* .I7117««7 7 4.74199 -*.HU .1*4*141 1 1.71*9 9 5.au* * .4899 9 .*939 9 .001* * -6.1736 6 .171373*47 7 .2158*41 1 8.81922 5.3*»* .44*44 .4704 .060) .J71HW-M» » 4.71777 -4.9111 .2001*41 1 1.7115 5 5.1)15 5 ) ) .*9H H .0014 4 6.65I) ) -6.186) ) .171614*47 7 .)12*C4I I 8.81*11 3.1727 .44422 .4U1 .0661 .1711191407 7 4.7M** -4.1112 .201*141 1 t.7*15 5 5.8514 4 .4*92 2 ..937 7 .0011 1 4.6479 9 -6.6097 7 .271657*47 7 .)312*41 1 1.13422 9.1602 .44111 .4666 .0700 .1711JOI407 7 4,71244 -4.5)44 2091*41 1.7433 3 s.Mia a.1*9 0 0 .411» » .0021 1 4.6)7* * -6.6229 9 .271717*47 7 .1569*41 1 a.S»)) 1.1666 .43166 .««2* .0*06 .171UW.07 7 4.71022 -4.51*0 .2074*41 1 1.7*** * 9.*7U U ..9)3 3 .002) ) 4.62» » -6.6*84 4 ,27ia9)C47 7 .1756*41 1 8.8)911 3.129) .«20)) .411) .09*1 .2712tMM>7 7 4.729** -4.5M* .212X41 1 1.7*1) ) i.aaa* * .4*8* * .493) ) .0029 9 *.61«6 6 -6.6677 7 * * .272107*47 7 .1911*41 1 8.8*177 5.5083 .3**44 .4444 .1217 ,77m»«7 7 4.72111 -4.9411 .214K41 1 1.791* * 5.W94 4 **79 .41)0 0 .00)6 6 4.4061 1 -6.6723 3 .2723*0*47 7 .4124*41 1 .17711 .4)12 .1117 .Jïl)l«*07 7 4.71711 -«.9411 .2212*41 1 B.. 755* 5.9204 4 .487) ) .«127 7 «.5158 8 -6.6«95 5 .0045 5 .2227*7*47 7 .9122*41 1 a.***ii 5.«4)1 .14177 .4171 .1941 .27UJJ««7 7 4.11122 -«.9491 .2299*41 1 *.75M M 5.9342 2 4.5516 6 -4.7113 3 .484* * .4122 2 .0057 7 .2710** «147 .9671E41 1 1.8*822 5.«796 .12044 .4062 .2216 ,!71)i||«7 7 4.7»77 -4.94*» 1.7**1 1 4.5310 0 -6.7264 4 .22*1*41 1 3.1441 1 .4*9* * .1114 4 .0071 1 .27)225*47 7 .6074E41 1 «.89100 1.4717 .»4** .3990 .11*1 .171MUM7 7 4.70*44 -«.9**4 *.5171 1 -6.7347 7 .2110*41 1 a.749) ) 1.9597 7 ..8*1 1 .4909 9 .«Hl l .271)00*47 7 .6266E4I I 8.81111 9.474) h.5111 1 .2*611 .1931 .2692 .171H««7 7 4.702** -4.550] .2179*41 1 4 4 9.W5I I .*81* * .4194 4 .0127 7 .27)941*47 7 -6.740* * 6*62*41 1 (.85*11 5.4746 4.19M M .27166 .1*11 ,2740 .17H.)t«T T *.70011 -4.9111 .21*4*41 1 a,7*71 1 1.9*7* * .4802 2 .41*7 7 .01*5 5 .27)767*47 7 -4.75377 .7*06X41 1 1 8.85722 5.4754 «.«7«( ( .24611 .17)1 .3060 .27I40MM7 7 *.«*711 -«.9921 .2424*41 1 1.7173 3 5.H94 4 .4839 9 .«901 1 .0095 5 .27)907*47 7 -4.741) ) 7*02*41 1 8.86099 5.480* «.«662 2 .22777 .Mil .1266 .nitomtm .nitomtm4.4,7 .. -4.551* .2412*41 1 «.71*1 1 5.94M M .*8)3 3 .«905 5 .0101 1 .77«0*1E47 7 -6.7709 9 8230(41 1 8.86366 5.4861 4.«552 2 .20*11 .1170 .)** .ITI.10CM7 7 *.»» 4.W17 .24*0141 1 a.toi. . 9.9t*e e .4116 6 .0112 2 .77*2)0*47 7 -6.7770 0 8962*41 1 8.86700 5.4934 4.4)75 5 .17*88 .1447 ,3714 .171.11147 7 *.«H22 -«.1921 .2*47*41 1 1.S047 7 5.9695 5 ,*»19 9 .0121 1 .77*40**47 7 -4.785) ) 91)3*41 1 *.*l*l l 8.67233 5.5076 .14744 .311) .406* .171.10t.O7 7 t.«M77 4.9519 .2*4**41 1 4.1110 0 1.9468 8 .*M00 0 .«!«.4*9«6 6« .014 1 1 .22 74906E47 -6.7911 1 1051(42 2 8.871)) 5,5279 .126** .12» .6276 .I7141WM1 1 4.499** -«.lil* .2*79*41 1 ..116* * 5.16*0 0 .4)95 5 .27140IE47 7 -6.7959 9 1I30E42 2 8.68499 5.149) .106** .3154 .4419 ,27141U407 7 t.«M22 -4.99*1 .24*2*41 1 M.II9* * 5.9617 7 ,*7M M 1 1 .0(44 4 .171675E47 7 79*5 5 207E42 2 8.69166 5.5687 .0*399 .1074 .4690 .171UKV>7 7 4.«M99 -i.9541 .2**0*41 1 1.42)7 7 5.9571 1 *7«1 1 4.7* * 0177 7 .17«721E47 7 268*42 2 8.89955 5.5*82 .06333 .3004 ,4*64 .271*121^)7 7 4.t*iaa -4.55*4 ,2*HI41 1 1.1171 1 5.9513 3 .77* * .0184 4 .174791E47 7 II 319*42 8.90788 5.6122 .05099 .2910 .5000 ,27141««7 7 4.490.. -4.5541 .2901*41 1 «,»J37 7 5,9*00 0 44 768 «647 7 1145 5 .15611 -6.7912 II. 3)8*42 166 7* .0)611 ,289) .513* .I71.1UM17 7 4.44044 -4.554* .2515*41 1 *.*)47 7 5.9)25 5 *762 2 «843 3 0105 5 27.770E47 .15188 -6.7986 11*6*47 7 8.923** 5.71)8 .0)044 .7874 .3189 .271.41C47 7 *.***77 -4.5541 .2521*41 1 1.4*11 1 5.918* * 0215 5 .271775(47 7 .15099 -6.74** 1151*42 2 8.92411 5.7)56 .02711 .2862 .5219 .17l«lt«7 7 *.«*«« -4,5590 .212*141 1 «.«4» » 5.9070 0 «7*4 4 «851 1 0112 2 .27177SE47 7 ,)*977 -4.7981 1116*42 2 8.427** 5.7497 .02466 .269) .9242 27U4 ««7 *.4IR77 -4.55*1 2515*41 1 4.fl*h) ) 5.81)1 1 »7*) ) ««57 7 021) ) 27«7*UC47 .1*922 -6,7*112 I357E42 2 8.92822 9.7614 .02177 .2*41 .9266 1)59*47 7 .02011 .2(37 .528) .271*471+07 7 *.«»822 -4.55*. .15*1*41 1 1.N*S* * 5.1796 6 *'I9 9 «8)8 8 n248 8 .27«7«lt47 7 .3*877 -6.79MI 8.92866 5.7750 136IE42 2 .018** .2811 .3289 .27I44WM7 7 4.41777 -4.55U 22 544E41 5.1**) ) *7I) ) njr: : ,27178)t«i7 7 8.92877 5.7825 1162E47 7 .014)) .261) .5)17 .271451t«7 7 029* * 8.92855 5.7907 ««)* * .)«*** -«.;««! 11 966E42 .01*88 .1817 .5)11 ,22 714)2147 4.44700 -4.55*7 2559*41 1 1.1515 5 5.»*07 7 *70* * 4*29 9 1291 1 8.927»» 1.7995 ,)«6II -6.79*0 11 36 7E42 ,01199 .2807 .1)57 .271*11(47 7 4.4M77 -4.55*7 255IF.41 1 5.152* * 4101 1 Ii21« « .Ü788 -6.7980 II69C42 2 6.92699 5.«(186 .01099 .780) .5166 .171414(47 7 u)nl l .)*766 -6.7971 I571E42 2 8.92)22 5.«791 .01011 .1800 .5)7* .1714UC47 7 k.»*»c)) -4.59*4 2544*41 1 1 1 7 7 i.77 .2 7*789FH i.asii i 13211 1 .)«700 -6.797* 1171*41 1 «.92111 1.819) .009)) .2798 .51*0 .27141W47 7 b.4.944 -4.55*4 2571*41 1 3 3 .«nil l «Wi7 7 .)**** -4.7977 I176EXI1 1 6.91900 5.«1(12 ,00866 .1791 .118 7 .J7141W47 7 .«4511 -4.59*4 ).. 1521 2574F4I I .Til* * «)*nl*7« 7« .)«r,SS -4.7974 8.91*11 1,86!» 27I462E47 7 .4M44 -4.55*7 ?5A1FH«I I .«51. . .775. . *h*l l J..M. . ) ) 6.91)99 5.6776 -- 77 - ,ir.,ir. 9.. SF*, 4.]2n9 9 -6.7561 1 .1511 1F+0 .7547 7 6.1113 3 04114 4 .2772 2 .5444 5 2Ï4747E+07 7 .5*45 5 '968 8 .. HSJE-«'2 .9045 5 4155 5 'W71 1 779' ' siqa a ..V14I9F+0 0 4.115' ' -- T«bL« Z CMctaiMd - Tobl* 2 cent li ^fff "wi tl*» » .27524»t*07 7 5.8142 2 -4.87») ) .16)41*0) ) (.7477 7 6 6 3)09 9 .0019 9 .2772 2 .5445 5 .275)641*07 7 3.9425 5 -6.8462 2 .90101*02 2 8.(402 2 6.7479 9 0016 6 .27700 . .1751761907 7 ).820* * -6.«7*7 7 .14051*0) ) 8.7309 9 )62t t .0014 4 .2771 1 .5445 5 .27)3441*07 7 3.94(0 0 -4.. Wil .(774K01 1 (.8171 1 4.7474 4 0016 6 .17700 . .17)2(21+07 7 1.U11 1 -6.8777 7 .11471*0) ) (.7))) ) ))52 2 .0019 9 .2772 2 .5445 5 .275)681*07 7 3.453) ) -6.8*13 3 .(54(t*02 2 (.8349 9 4.7497 7 0015 5 .17700 . .2732421*07 7 LD» » -*.»71t t .15291*05 5 8.757* * 1 1 37)7 7 .0014 4 .2772 2 .5445 5 .275)701*07 7 3.9)84 4 -6.8404 4 .8)421*02 2 8.8540 0 4.714) ) 0015 5 .17700 . .27)1021+07 7 3.1)74 4 -4.8741 1 .14771*0) ) (.7423 3 4 4 34*5 5 .0014 4 .1772 2 .5445 5 .27))72t*07 7 3.94*2 2 -4.()«1 1 .81241*02 2 8.(5)5 5 4.7601 1 0015 5 .17700 . .1715)121*02 2 1.W76 6 -6.«72» » .14121*0) ) 1.7846 6 4150 0 .0014 4 .27)2 2 .5*43 3 .2'))74t+07 7 ).944) ) -6.1)77 7 .74)11*01 1 1.(5» » 6.7661 1 0015 5 .17700 . .1211121907 7 3.857* * -4.1701 1 .1344C*0) ) 8.7787 7 4331 1 .0019 9 .27)2 2 .5443 3 .2733761+07 7 ).9743 3 -6.8)44 4 .77)71*01 1 8.(3)0 0 6.77)1 1 0013 3 .17700 . .1751191907 7 3.. HM -4.8710 0 .132**03 3 8.78)1 1 <6 6465 <7 7 .0014 4 .2772 2 .5445 5 .2733781*07 7 3.47*6 6 -6.(1)2 2 .. 7)531*01 8.8531 1 6.7(0) ) OOII 3 .17700 . .1711171*07 7 )-84*l l -4.8720 0 .130)1*03 3 8.7815 5 6 6 4(28 8 .0019 9 .2772 2 .5*45 5 .2733(01*07 7 1.96*2 2 -6.(1*0 0 .. 714)1*01 (.8310 0 6.7871 1 0015 5 .17700 . .1711181901 1 1.M71 1 -4.8727 7 .118(1*0) ) 8.7928 8 4 4 4916 6 .0019 9 .2772 2 .5443 3 .2753811+07 7 3.987* * -6.81)4 4 .71(21*01 1 8.85)2 2 1.7(11 1 0015 5 .17700 . .17)111 «1907 ).Htt t -4.lil! ! .12711*0) ) 1.795) ) 6 6498 5 5 .0014 4 .2772 2 .5445 5 .275)821+07 7 7.9894 4 -4.028 8 .720(1+02 2 8.8519 9 1.794) ) 0015 5 .17700 . .1711101907 7 1.171) ) -4.(7)9 9 .12601*0) ) 8.797) ) 4 4 5048 8 .0014 4 .2772 2 .5445 5 .275)8)1*07 7 3.9927 7 -6.8)22 2 .710*1+02 2 8.B547 7 6.102) ) 0015 5 .17700 . -17))30B*O7 7 3.1717 7 -4.8724 4 .12511*0) ) 8.7982 2 50*9 9 .0019 9 .2712 2 .5445 5 .275)841*07 7 ).9984 4 -4.(107 7 .48981*01 1 8.8514 4 6.1113 3 0015 5 .17700 . .171)111907 7 1.8746 6 -4.8721 1 .12441+0) ) 8.79») ) 50S2 2 ,0019 9 .2772 2 .5415 5 .275)1(1*07 7 4.0049 9 -6.624) ) .47051*02 2 (.85)4 4 6.(217 7 0015 5 .27700 . .17)1)11*07 7 1.87)2 2 -4.8719 9 .12431*0) ) 8.79(3 3 «6 6 508«8 8 .0014 4 .2772 2 .5445 5 .27))90E*07 7 4.0107 7 -4.627* * .4524E+02 2 8.8357 7 6.8)07 7 0011 1 .27700 . .1711)11*07 7 1.I7H H -4.1717 7 .12401*0] ] 8.7H) ) 4 4 50*7 7 .0014 4 .2772 2 .5445 5 .173)951+07 7 4.0165 5 -4.8257 7 .11451*02 2 ) ) 6.(4)2 2 O0I1 1 .27700 . .1713)11*07 7 3.8760 0 -4.8713 3 .121(1*03 3 8.78(2 2 5087 7 .0019 9 .2772 2 .5445 5 .275)951+07 7 4.0219 9 -4.82)8 8 .11841+02 2 8.(1(0 0 6.(3» » 0011 1 .27700 . .2711111*07 7 ).S7tl l -4.6713 3 .12)71*0) ) 8.7978 8 6 6 908I I .0019 9 .2772 2 .5445 5 .279)96(+07 7 4.0247 7 -4.6229 9 .61021+02 2 B.8591 1 6.(408 8 001* * .27700 . .Z7Ï)32R*07 7 1.876) ) 4.171* * .l2Wt*0) ) 8.7976 6 4 4 5074 4 .0019 9 .277! ! .5445 5 .2753991*07 7 4.0)0) ) -4.8210 0 .59431*02 2 6.6404 4 6.87)4 4 0014 4 .27)00 . .271)121*07 7 1.1764 4 -4.8712 2 .12)W«0) ) 6.7940 0 6 6 5051 1 .0019 9 .2)71 1 .5445 5 .273401E+O7 7 *.0)54 4 -6.8192 2 .38001*02 2 B.8420 0 5.(151 1 0014 4 .27700 . .211)121*07 7 1.6777 7 -4.1708 8 .12281*03 3 8.7924 4 4994 4 .0019 9 .2171 1 .5445 5 .2754021*07 7 t.0)7* * -4.6179 9 .57441*02 2 1.(612 2 6.(877 7 0014 4 .27700 . .27113)1*07 7 1.6716 6 -4.1704 4 .12111*0) ) 8.7411 1 6 6 4945 5 .0019 9 .7"! ! .5*46 6 .1754041*07 7 4.042) ) -4.8158 8 2 2 1.(427 7 6.4000 0 0014 4 .27700 . -273)))E*07 7 1.8755 5 -4.170) ) .12181*01 1 8.7901 1 6 6 4459 9 .0019 9 .1771 1 .1445 5 .2754051+07 7 4.0446 6 -4.81*9 9 .55471*02 2 1.(654 4 5.4104 4 0011 1 .27700 . .22 711)41*07 ).1800 0 -4.1702 2 .12151*0) ) 8.7807 7 4968 8 .0014 4 .2"! ! .5445 5 .2754061*07 7 4.0470 0 -6.81*2 2 .54851*02 2 1.(6(7 7 6.9224 4 0013 3 .27700 . .2733)41+07 7 i.ttu u-4.M M M .120*1*0) ) 8.7975 5 5018 8 .0014 4 .22 772 .5445 5 .2756081+07 7 4.0494 4 -4.813* * .54231*02 2 8.6721 1 6.9)47 7 OOI] ] .27700 . .271)1)1*07 7 1.8821 1 -6.8494 4 .. U9(t*0) 8.794) ) 6 6 5116 6 .0019 9 .2772 2 .5444 5 .27540911 +07 4.0504 4 -6.8123 3 .33841*02 2 1.8758 8 b.4481 1 001) ) .27700 . .27)))IC*07 7 ).*VI35 5 -4.1489 9 .I18»*0) ) B.êOll l 530) ) 0019 .2772 2 .5445 5 .275*101*07 7 4.0528 8 -6.8117 7 .33351*02 2 8.(796 6 6.9617 7 OOI) ) .27699 . .273)181*07 7 ).M67 7 -6.868' ' .11771+0) ) 1.80" " 6 6 5)91 1 .2"! ! .2754I1E*07 7 4.054) ) -6.8108 8 52841*02 9.68)5 5 6.9756 6 0012 2 .27699 . .27)))*E* -- 78 - lo» » 811 1 77S11 «< . 275428E+077 4 :7^i.2flF.^ll 4 22 754?9f.*07 i. dftM M :75i29E*t>77 1. -ft.' ' -6.. 7 IS SI;;:™S JJ :0B7 4 4 «50 0 0B7i i 275431E*U77 t -6.7 7 814 4 Z75431E*077 4 50*5 5 .OrtOO O.27* 5 5 .541-1 1 836 6 27543IE+077 * (J«7(, , ii .4478E+02 5165 5 .27(,5 5 ."j^bl l «39 9 275431E+077 4 -6.7 7 till .4477F.+OZ 52B5 5 flOOO flOOO-Z7ft l l 5*M «40 0 275431E»077 4 0875 5 777 ,4*76£+02 27S431E*077 4 Z'.l Z'.l666 .44 75E+A2 »52r»» .dOon .27*S 842 2 27543|E*077 4 0175 5 bLL .4472E*.J2 27S431E*077 4 0877 7 -6.7 7 622 .4d6ftE^2 275432E+077 4 11887 7 400 .4459E+C2 847 7 275432EW77 4 0887 7 622 .4448E+02 2754371*077 4 08* * 849 9 275432E+077 4 -6.7 7(. 55 .4436Ë+C2 b«0 0 .0000 0 .2765 5 .54*1 1 851) ) 2754}2E»077 4 um+oz z «29 9 .0000 0 27*5 ,5461 1 851 1 J7S43!E»077 4 0M7 7 -t:J J*55 .442SE+02 6AR1 1 .22 766 SI 27543IE+077 4 -t.7 76 33 .4425E+02 bB?b b .0000 0 .2*66 6 iS 2754321*077 4 *33 .4423E+02 7018 8 .0000 0 .2766 6 ,5**n n 1ST T 275»3JE«077 4 7168 8 .0000 0 .2767 7 59 2754338*077 4 0801 1 -6.7 7 5** .44L9E.+42 7J17 7 .276» » .544 54 BK> > 275433E-K>77 4 0802 2 599 .441R402 7391 1 .0000 0 .2769 9 .54)2 2 HI I 1754331*077 4 0802 2 5** .441BE-K92 741* * .0000 0 ,2769 9 ,5*51 1 2 2 275433E*077 4 -6.7 7 5** .4417E+02 .0000 0 .2770 0 .5450 0 8*3 3 275433E+D77 4 -6.7 7 588 .4M7E+02 7470 0 .2771 1 864 4 2754331*077 4 577 .4417E402 7477 7 .0000 0 .2771 1 .5440 0 US S 27M33E*D77 4 0902 2 577 .44L6C+02 7480 0 .5*47 7 IM M 275*3)E*077 4 0802 2 -6.7 7 577 .4416E*«2 7481 1 .2771 1 W7 7 2754331*077 4 0802 2 577 .*416E*02 7481 1 .2771 1 .5447 7 2754)31*077 4 09Q2 2 577 .4416E402 74R1 1 .2771 1 .5**7 7 8 8 27543K4077 0802 2 -6.7 75 77 .441W+02 7479 9 .2771 1 70 275433E-H377 4 0902 2 577 .4*16E -- 79 - l* * ett * . . . . 79*4 4 001 1 713** 2 3091 1 04.3 3 t\t t\t"ÏS S S !Ï1" "-1 : : ":S M"S J J;2 5 5 .7772 2 144 3 2017 7 04.1 1 -'IJlti i .7772 2 9*0)) 2 30*7 7 .7521) ) -JU* * 1.74U U ,44*4 4 44244 2 2074 4 .25)5 5 7*1) ) 21.01 1 4,79*7 7 .2771 1 )) 2 107* * 04.4 4 * Hl Hl!7321 7 7 4.7*1* * .4114 4 ooi i 00 1 04.4 4 ».M M 4.7*U U .4*44 4 ,2771 1 2*.01 1 «2 2 viï J[;JviïJ J .4441 1 oo! ! 422)) 7 7070 0 :!"!! !.11);ïï ü1 1ü *'. »»* ? *'.? 21,03 3 M*)l l ,44*7 7 91444 2 2070 0 04.4 4 ,11)1 1 21,02 2 1.7*0* * .4170 0 99700 2 304* * 0*.* * 94944 2 3044 4 04.4 4 42» » .1771 1 2047 7 .7522) ) -11)2 2 7*ï* * .4441 1 99*77 2 2047 7 2.74*43 3 .27» » -4.7241 1 11.1! ! .nu u .1464 4 .001* * .ir7i i .1*41 1 .11)2 2 7»)) ) 21.02 2 99744 2 2044 4 04.4 4 7.-7**94 4 -4.721* * 11.20 0 .ui* * .14** * .ooi* * 'X: 'X: 1,4000 0 ,5049 9 204) ) 04.3 3 1.7**92 2 .27*4 4 11.3* * .1727 7 .001* * ,27'2 2 .1**1 1 001 1 .1773 3 00044 2 «255 2 1041 1 2.7**94 4 .37» » .2772 2 -)T|**22 J.***l A4.1* 3L.( 4)1 1 4 4 2.73003 3 11.M M .730* * .)**) )-1.900 22 2.*S4* êfc-I* IL.; 1-7122» » .1111 1 '931 1 001 1 00)77 2 20*0 0 04.3 3 3.7340* * 171* -4.707» » 11.«I I .77 SO* .1*47 7 .001* * .2772 2 ^*! ! 11.0) ) 1.. 4211 :ü;i i 7773 3 2019 9 04.3 3 2.73017 7 1701 -4.70» » 14.4* * .74*7 7 .143* * ,001* * .2772 2 .1*41 1 M377 7 .5401 1 001 1 .7773 3 7.73024 4 .2440 0 .74H H 27J2 .5**1 1 -1,47»» J.*10* 4*.t>0 Ï2.: Ï-71ÏÏ4 4 4.1140 0 .1773 3 El122** i2 i -4.7004 4 Ü!u u.74* 4 4 Jon n.«! I I -).**?!! I.W** *0.i[ 22,- 1-717)4 4 .HU U .1771 1 1.7W31 1 .2444 4 ia.«4 4 .2117 7 4 4 749* * 31.0* * 173** 2 203) ) .1773 3 20)2 2 04.4 4 -4.7012 2 i*.oi i .74*0 0 .JOT* * .'OOI* * -l.éi*]] 2.1*12 *n.Aï 22.: iü ü 744* * 21,0* * ».*12t t .411) ) Hl l t-TWI I .2**ï ï -4.7011 1 14.14 4 .7*77 7 .212) ) .177\\\\2 2 :ï*» \\\\ï ï-J.44 MM 3.177* *n.Tfc 21.1 .4721 1 -2773 3 m\ m\ 2.7)0*7 7 .34» » .2772 2 .1**1 1 -3.4*1** M4*ü *2.hl n.« 44) ) ï.71J41 1 7*44 4 9.44)4 4 .4*1* * .2772 2 «1400 3 -4.7074 4 -Ï.**MM j.iifj «.n n,< .2122 2 7*55 5 4.4490 0 1)200 2 1030 0 20)0 0 04.4 4 1.730)7 7 -4.7121 1 n.« « .744) ) .001* * 74*3 3 11,0* * 1-4704 4 :HÜ Ü 2030 0 .25*] ] -*.72JJ J 19.7* * .74H H 3341 J772 -3.497** ï.1117 *4.0* 24.: 't 't 1.75077 7 .75*3 3 -4.7114 4 11.« « .7»** * .2*0* * .001* * .377] ] .54*4 4 -1.724** 2.300) *7.t4 11.1 3»» I.7WI1 4.237* -1.7*7** l,ï*)| *7.»9 71.' 473 3 -712** * .7177 7 3*11 2.730*9 I .71211 1 21J7 7 10.10 0 .4740 0 04.4 4 750*77 4.2344 -4.7314 3 ,7151 1 4 4 7712 2 30-4* * I-9110 0 2030 0 1.23422 -«.7404 1 1.744** 4.2413 441 1 74*7 7 10.31 1 4.93*2 2 2090 0 04.4 4 3*44 1.7)10* I 2'73214 4 "J J ooi i 2771 1 04.4 4 1.73117 7 .2317 7 -4-7704 4 20.34 4 -74711 4.1*1 0 0 .001* * I77J -3"! ! -4..01UU 2.241* 101. . 34 4 T| | 447 7 30M M t.73122 2 .2333 3 -4.77» » 10.41 1 .74*00 I.2*3 * * .17014 4 .2772 2 .3*43 3 -4.0)011 1,2*U 14 4 4U U 20VO O >.73127 7 .1334 4 -4.7771 1 10.44 4 .74133 1 -4.0*4** 1.237* 101, , 17 70 3 3 1,71244 4 ,2190 0 4.9H0 0 .4944 4 2930 0 04.4 4 1.73131 1 .233] ] -4.7102 2 30.70 0 .74177 1 .2*44 4 .3772 2 .34*3 3 -4.1*011 1.2111 27 7 In In !ÏÏ1 110.0 * * ! ! Ml l lm! ! 20)0 0 0*.4 4 1.73137 7 -4.7U1 1 10.74 4 .74*44 \ .30Ï3 3 .240* * ^ 719^0 0 19.97 7 1.1342 2 .7*43 3 1D30 0 1.731*1 1 .2350 0 -4.7134 4 20.77 7 .749** 1.301 1 1 .1772 2 .3443 3 -4.11WW 3.3377 2.71247 7 75*7 7 1*.H H 4.4*10 0 -771* * »:t t 1.731*7 7 134* -4.7173 3 70,40 0 .73044 I .31*1 1 .0019 9 -2771 1 .3*41 1 702 2 1.4*17 7 .7917 7 :££ £ 20» » -4.7**} } .730** I.319 * * .3771 1 -4.31411 3.2124 103. . 7' ' 30 0 ^ ^ 30.01 1 t.alo) ) OOI I .2771 1 04.4 4 1.73137 7 -4.7*01 1 1134 -4.31111 2.2107 103, , 17 7 14 4 20.17 7 1.470) ) .1771 1 ».*) ) 1030 0 .2344 4 -4.7111 1 20.47 7 -73344 4 001* .3773 3 .W*) ) -4.4)111 2.Ill* 1,712*7 7 .!«-ÏÏ7»1 1» -4 4 7114 4 .9012 2 .2771 1 04.4 4 t.73147 7 2343 -4.7*34 4 20,4* * .73322 i DOL* .3*41 1 -4.«Orr 7.3173 704 4 .71100 0 7114 4 20-34 4 1.4*3* * 001 1 .2770 0 1030 0 04.4 4 2.73172 2 .2344 4 -ft.7943 3 20,, tl .73422 i .1434 4 17 7 73 3 74*2 2 10.44 4 1.9100 0 .9**4 4 OO) ) 04.4 4 1.73171 1 1343 -4.. f Hl .73344 1-)».333 *3 3 *.001 * * .1771 1 .3**3 3 -4.374»» 3.2144 101. . 17 7 .0171 1 OOMM .274* 04.4 4 1.73111 1 .2341 1 -4.7fS* * 20.MM «.7371 < -4.41100 2.2132 103, , 27 74 1 1 10.3* * 1.9174 4 .0404 4 000 0 .1747 7 1030 0 0*.* * Ï.TIM7 7 -4.7*40 0 20.. *3 .73144 1.34* 9 9 .001* * .3443 3 Z.73101 1 .2912 2 -4 4 74*3 3 30.57 7 .0344 4 .2747 7 »H H 0*.4 4 1,713*1 1 .1341 1 -4.7**) ) 10*97 7 ,74099 ' .17*9 9 .1772 2 -4 4 7417 7 20.3* * 1.9390 0 0734 000 0 .2747 7 04.4 4 712 2 2030 0 04.4 4 «-- «II "l-l »» %fl S.1 4*00 0 .3744 4 .3*40 0 1.791044 4.1911 4.7447 20.41 I.WI ; 11 ÏO90 in* * 11 73131 11 4.044* .7103 3 1.7)10** *.1SU -t.nw ».** 0.91*1 .14133 -4.7044 II. 99 9.0447 .7101 1 .141** -4.70» 11. 44 9.0444 .72*7 7 1,791077 4. 55 9.0730 1.711077 ». 1*1?? 4.7*14 20.** ft.*10* 7. .341)) -4.4941 31- II *.0731 73*9 1.7UI77 *. .24144 -4.4494 11. 44 9.07H l.TWW *131. 77 -«.7*0* 10.4* I.MM 7. .1*133 -4.4133 21. 11 9.041* .7*02 2 1.711077 4291. 77 4.71*7 10.» l.tM* 7. .1*177 -4.4414 31. 33 4.04*1 .7*97 7 1.71)000 *tl. »» 4.717* 10.11 1.»S11 7. 11.0* * 1.7)122 2 34.04 4 443 3 t.. 7)111 t.34144 -4.47*9 21.HH 9.0*41 i.riMOO *IJl. tt -».7»7 M.M ».**71 7. I.ltHO O l.JJJOll *. 3.HM M104 . . 14.01 1 1.7UMM *. 1.1030 0 104. . 34.0* * 1.711100 *.11 »» 4.71*1 M.»7 I.M ! 2.2OS0 0 1*4. . 14,04 4 2.2030 0 24.0* * 1.711100 4. 34.04 4 1.711111 4. 2.2030 0 1.711111 *. 2.2030 0 104. . 34.0» » 1.711111 2*2034 4 .04. . 29.04 4 1.711111 4.111 11 4.7144 10.4* 0.*1*9 7. 2.10» » 10*. . 39-0* * 1.711144 4,991 ** 4.7149 ».ll *.**» 7. 1.111144 4.111 11 4.719* 10.17 ».*M1 7. 1.711111 4,IM ** 4.71*0 10-41 *.*799 7 1.711111 4.1.9 00 4.1141 20.71 0.W0* 7, 1.7111** 4.14* 00 4.7174 10.79 t.WOt 7 1.1111»» 4,14* 44 4.717* ».*« *.**!* 7 1.20300 10*.4 : 1.711177 4.1*7 77 4.7101 20.01 t.0011 7 1.71117 7 .14711 4.7107 10.M 0.O110 7 1.71117 7 .14*00 4.71*0 11.01 «.01*1 7 t.. 71117 .14*77 4.710* 11.01 «.0177 7 2.203OO I 2.71911 1 .!**** 4.7171 11.» *.0171 7 1.7111» » ,|4*ll 4.72.5 11.04 ».*l»9 7 -).*B11 2.1044 1 >> 21-Oft 2.71111 1 .14*11 4.71*1 21.04 9.01*0 7 -*,9**|| 2.204* 104.4 24.00 1.7U10 0 „24*11 4.7117 21.04 9.01*1 7 -4.O300 1-1044 104.4 24.04 1.7911* * „24*22 4.7211 21.04 9.01*7 7 -4.Kill 1.20*9 10*.4 24.04 1.7991* * I.14*11 4.71*1 ll.O» ».Ol*t 7 -4.4)100 1*20*9 104.4 24.04 2.7911* * I.14HH 4.729* ll.O. 9.01*4 7 -1.1*177 2.20*9 10*.4 24.04 1.79111 1 1.14*00 4.71M ll.O. *.0111 7 II iOt.4 2*.< 1.7»10 0 ,.24477 4.714* 11.11 «.01*3 r 24,04 1.7U1I I 1.14411 4.714* 21.17 9.0141 7 2.7391* * 1.14499 4.7240 II.IS t.0101 7 ,.14300 4.7213 21.1' 9.0104 7 ,.24499 4.711» 21.1. 9.0201 7 1.7311* * !!l4lll 4.71»* 11.09 4.02» 7 1.7U1* * .24400 4.7171 21-10 9.0270 7.341. .0000 .17*1 .3*62 .75)200 4,14*4 - 1.7)1200 4.2**7 -4.7141 11.11 1 .0)4]] 7.39*1 .0000 . (.73)311 *.3**1 -*.7l«) 31.1* ' 1.711211 *.2443 -4.720* 21.19 ' !,7)J311 4.1414 -4.7144 21,2* i -- 80 - Tablee 5. The orbital periods and eccentricities of so«e Be/X-ray binaries, Be-star binaries,, and the X-ray binaries V0332+53, GX301-2, and 2S1553-542. System m Periodd (days) Type e References s 4U0115+633 24.31 0.34 4 Be/X X Rappaportt et al. (1978) V0332+533 34.2 1.5 0.355 (+0.25,-0.10) Be/X X Whitee et al. (1984) Nagasee et al. (1982) A0535+266 111.0 0.4 0.33 - 0.4 (?) Be/X X Prledhorskyy and Terrell (1983) Hutchingss (1984) Charless et al. (1983) A0538-66 6 16.6 6 -- 0.7, > 0.4 Be/X X Skinnerr et al. (1982) Rappaportt and Joss (1983) GX301-2 2 41.4 4 0.47 7 B2Ia/X X Kelleyy et al. (1983) 2S1553-542 2 30.66 2.2 << 0.09 Be/XX ? CXX Dra 6.697 7 0.0 0 Koubskyy (1978) Tii Ori 7.98 8 0.0 0 Battenn et al. (1978) PP Lyr 12.93 3 0.0 0 HDD 187399 27.97 7 0.399 (?) RXX Cas 32.3 3 0.00 - 0.16 Krizz et al. (1980) HDD 698 55.93 3 0.03 3 Battenn et al. ** Per 126.7 7 0.15 5 CC Tau 132.97 7 0.188 (+0.11,-0.05) Harmanecc (1984) AXX Mon 232.5 5 0.02 2 Battenn et al. 177 Up 260 0 0.13 3 CC Aur 972.2 2 0.41 1 322 Cyg 1147.8 8 0.30 0 311 Cyg 3784.3 3 0.22 2 VVV Cep 7430.5 5 0.35 5 KQQ Pup 9752 2 0.46 6 Hackk (1984) and Batten et al. ee Aur 9890 0 0.20 0 -- 81 -