The mass-ratio distribution of spectroscopic binary stars

In order to determine the mass-ratio distribution of spectroscopic binary stars, the selection effects that govern the observations of this class of binary systems are investigated. The selection effects are modelled numerically and analytically. The results of the models are compared to the data inThe Eighth Catalogue of the Orbital Elements of Spectroscopic Binary Stars (DAO8) compiled by Battenet al. (1989). The investigations involve binary systems with Main-Sequence primary components only, in order to avoid confusion of evolutionary and selection effects.For single-lined spectroscopic binaries (SBI) it is found that the mass ratios (q=M _sec/M _prim) in general adhere to a distribution _q q ^-2 forq>q ₀, withq ₀=0.3. The observations are consistent with a distribution that is flat forq<q ₀. The turn-over value varies fromq ₀=0.3 for systems with B-type primaries, toq ₀=0.55 for systems with K-type primaries. The semi-major axesa ₁ are distributed according to _a (a ₁)a ₁ ^-a with an average value of _a =1.3. The power varies from _a =1.7 for systems with B-type primaries to _a =0 for systems with K-type primaries. The eccentricitiese of the orbits of SBI systems are distributed according to _e (e)e ^-1.For double-lined spectroscopic binary stars (SBII) it is found that the shape of theq-distribution, as derived from observations, is almost entirely determined by selection effects. It is shown that the distribution is compatible with theq-distribution found for SBI systems. A sub-sample, consisting of the SBII systems from DAO8 with magnitudesm _V 5 ^m , is less hampered by selection effects, and shows the same shape of theq-distribution as the SBI systems, at theq-interval (0.67, 1).It is estimated that 19–45% of the stars in the solar neighbourhood are spectroscopic binary systems.     

Some applications of Jacobi dynamics to the stellar evolution

The stars in the Main Sequence are seen as a hierarchy of objects with different massesM and effective dynamical radiiR _eff=R/ given by the stellar radii and the coefficients for the inner structure of the stars.As seen in a previous work (Paper I), during the lifetime in the Main SequenceR _eff(t) remains a near invariant when compared to the variation in the time ofR(t) and (t).With such an effectiveR _eff one obtains the amounts of actionA _c(M), the effective densities _eff(M)=(M)³(M), the densities of action and of energy (or mean presures in the stellar interior)a _c(M),e _c(M), and the potential energiesE _p(M).The amounts of action areA _cM ^k withk1.87 for the M stars,k5/3 for the KGF stars, andk1.83 for the A and earlier stars, representing very simples conditions for the other dynamical parameters. For instancek5/3 means a near invariant effective density _eff for the KGF stars, while for such stars the mean densities and coefficients present the strongest variations with masses (M)M ^–1.81, (M)M^0.6.The cases for the M stars (e _c(M)M ^–1) and for the A and earlier stars (betweena _c(M)=constant and _eff(M)M ^–1) and also discussed. These conditions for the earlier stars also represent reasonable mean values for the whole stellar hierarchy in the range of masses 0.2M M25M .With all this, one can build dynamical HR diagrams withA _c(M), E_p(M), _eff M ^–p , etc., whose characteristics are analogous to these in the photometrical HR diagram. A comparison is made betweenA _c(M) from the models here and the HR diagram with the best known stars of luminosity classes IV, V, and white dwarfs.The comparison of the potential energiesE _p(M)M ^–p according to the stellar models used here and the observed frequency function (MM _–q (number of stars in a given interval of masses) from different authors suggests the possibility that the productE _p(M)(M) is a constant, but this must be confirmed with further studies of the function (M) and its fine structure.There are analogies between the formulation used here for the stellar hierarchy and other physical processes, for instance, in modified forms of the Kolmogorov law of turbulence and in the formulation used for the hierarchy of molecular clouds in gravitational equilibrium. Besides, the function of actionA _c(M) for the stars has analogous properties to the relations of angular momenta and massesJ(M) for different types of objects. The cosmological implications of all this are discussed.     

Quasi-stellar objects as rotating magnetic superstars

In this paper the magnetic superstar model is used to discuss QSO luminosity and density evolution. Our main hypotheses are that (i) mass loss from old stars in massive galaxies cools and then falls into the centre to form a nuclear disc (Bailey, 1980); and (ii) magnetic superstars in galactic nuclei condense out of gaseous material at the centre of a supermassive-magnetised disc (Kundt, 1979). On this generalised model we find that the non-thermal (synchrotron) optical luminosity scales asL _opt L ³ t ^–7/3, whereL is the total blue luminosity of old stars in the galaxy and t is cosmic time. In addition we show that QSO co-moving density follows the lawD(t)exp-(t/t _Evol)^16/15 with an evolution timescalet _Evol = 1.95 × 10⁹ yr. The model as a whole is in good agreement with observations.     

Late stages of the evolution of close binaries

Close binaries can evolve through various ways of interaction into compact objects (white dwarfs, neutron stars, black holes). Massive binary systems (mass of the primaryM ₁ larger than 14 to 15M ₀) are expected to leave, after the first stage of mass transfer a compact component orbiting a massive star. These systems evolve during subsequent stages into massive X-ray binaries. Systems with initial large periode evolve into Be X-ray binaries.Low mass X-ray sources are probably descendants of lower mass stars, and various channels for their production are indicated. The evolution of massive close binaries is examined in detail and different X-ray stages are discussed. It is argued that a first X-ray stage is followed by a reverse extensive mass transfer, leading to systems like SS 433, Cir X1. During further evolution these systems would become Wolf-Rayet runaways. Due to spiral in these system would then further evolve into ultra short X-ray binaries like Cyg X-3.Finally the explosion of the secondary will in most cases disrupt the system. In an exceptional case the system remains bound, leading to binary pulsars like PSR 1913+16. In such systems the orbit will shrink due to gravitational radiation and finally the two neutron stars will coalesce. It is argued that the millisecond pulsar PSR 1937+214 could be formed in this way.A complete scheme starting from two massive ZAMS stars, ending with a millisecond pulsar is presented.Paper presented at the Lembang-Bamberg IAU Colloquium No. 80 on Double Stars: Physical Properties and Generic Relations, held at Bandung, Indonesia 3–7 June, 1983.     

Spin parameter and scale-free density perturbations in hierarchical clustering

The present attempt aims to predict the dependence of the spin parameter, , the angular momentum,J, and the typical radius,a _vir, on the mass,M, which have been found inN-body simulations of expanding density perturbations in hierarchical clustering, when virialization is attained. We show that M ⁰ for systems with same adimensional density distribution and velocity distribution, and in particular for scale-free density perturbations in hierarchical clustering. In the special case of ellipsoidal perturbations, it is also found:J M ^7/4,a _vir M ^1/2. All these results turn out to be in close agreement withN-body simulations, despite the simple model adopted. Expanding and virialized perturbations are modelled, respectively, by homogeneous and heterogeneous, similar ellipsoids which allow flat rotation curves far from the centre. Both energy and angular momentum maintain from a given time on, lying between the beginning of strong decoupling from the Hubble flow and the occurrence of maximum volume. Then the whole set of virialized ellipsoidal configurations with same energy and angular momentum are derived, and the dependence of the spin parameter on the anisotropy parameter, _pec is investigated. Turning our attention to the formation of galaxies, we derive the total mass as a function of the collapse factor, using the empirical anticorrelation between dark to visible mass ratio within the optical radius of disk galaxies and their luminous masses. Observational data related to a sample of elliptical galaxies provide evidence that the contraction in these bodies occurred in proportion to the square root of the ratio of total to luminous mass. On the contrary, it is deduced that dissipation of angular momentum in elliptical galaxies occurred more or less at the same rate. If both shape and anisotropy parameter are preserved during the collapse, typical axis rations ₂₁ = 0.98, ₃₁ = 0.69, are found to correspond to a moderate anisotropy, _pec 0.27, with a small dependence on the spin parameter in the range allowed.     

The post-RLOF structure of the secondary components in close binary systems,with an application to masses of Wolf-Rayet stars

The post-RLOF structure of the secondary after relaxation towards thermal equilibrium is calculated for a large grid of massive close binaries evolving through an early caseB of mass transfer. The initial primary masses range between 15 and 30M _o, the initial mass ratio between 0.3 and 0.9. The possibility that matter leaves the system during RLOF is included using an additional free parameter . The calculations are based on the accretion and relaxation properties of massive accretion stars. Conclusions on the post-RLOF secondaries are presented in function of , M1i, andq _i , in the form of tables and figures on the post-RLOF positions in the HR diagram, the final masses, mass ratios, chemical profiles and the remaining core-hydrogen burning lifetime. It is found that all systems starting from initial conditions in the grid specified above evolve sequentially, i.e. the primary evolves into a supernova before the end of core H burning of the secondary. No WR+WR systems are encountered. The results are used to determine the masses of ten double lined spectroscopic WR+OB binaries. Most of the WR masses are in the range 8–14M _o, although the sample is subject to some important selection effects. One WR+OB binary has a WR mass between 4 and 5M _o. It is argued that mass determinations based only on the spectral type of the secondary yield WR masses that are too high up to a factor two.     

The distinctive feature of relativistic stellar systems

In this paper we adopt the method of relativistic fluid dynamics to examine the number density distribution of stars around a massive black hole in the core of stellar clusters. We obtain extensive results,n(r) r ^–a, 3/2a9/2, which include, respectively, then(r) r ^–7/4 power law obtained by Bahcall and Wolf and then(r) r ^–9/4 power law by Peebles. Sincen(r) is not an observable quantity for star clusters, we also consider general relativity effects, i.e., the consequence of the bending of light, in calculating the projected density of stars in such a system. As an example we employ a massive black hole 10³ M inlaid in the center of a globular cluster and calculate various projected densities of stars. The results show that cusp construction occurs in all cases unless the central black hole massM=0, and the polytropic index does not affect at all the position of the capture radiusr _a. The obvious differences in the surface density is only embodied in the interior of the capture radius. At the outer regions of the core, the surface density of stars declines rapidly with ar ^–5 power law in all cases. These results can be applied to cases of unequal-mass and non-steady state.     

The mass-ratio distribution of Visual Binary Stars

The selection effects that govern the observations of Visual Binary Stars are investigated, in order to obtain a realistic statistical distribution of the mass-ratioq=M _sec/M _prim. To this end a numerical simulation programme has been developed, which generates binary stars and looks at them to determine whether an observer on Earth would be able to detect them. The simulations show that for mass-ratiosq>0.35, observations are expected to reveal the realq-distributions, while for mass-ratiosq<0.35 selection effects begin to play a major part. It is found that the observed mass-ratio distribution for Main-Sequence systems, derived from theIndex Catalogue of Visual Binary Stars (IDS), can be explained by a distribution of secondary masses according to the Initial Mass Function (IMF), i.e., (M)M ^–2.7.From theFourth Catalogue of Orbits of Visual Binary Stars (OVB) authors find aq-distribution that peaks strongly forq-values close toq=1. It is shown that this mass-ratio distribution may be the result of a sampling selection effect. Due to this sampling selection effect, the OVB is a considerably morebiased sample of the binary population in our Galaxy than the IDS. Numerical simulations of biased sampling show that theq-distribution, found from the OVB, is not incompatible with the distribution of secondary masses according to the IMF (forq>0.35), found from the IDS.Because of the selection effects, it is difficult to establish the realq-distribution forq<0.35. If the realq-distribution departs from (q)q ^–2.7 forq0.35, about 36% of all stars are in visual binaries (i.e., if theq-distribution is assumed to be flat for 0<q<0.35); if the distribution flattens forq0.25, about 60% of the stars must be primaries of visual binaries.     

Bifurcation points in close binary evolution: Medium-mass binaries

The remaining core hydrogen burning lifetime after case B of mass transfer of the secondary (mass gaining) component in a medium mass close binary star is estimated, for mediummass binaries with primaries in the mass range 5M to 9M . From the comparison of this quantity with the helium burning time-scale of the remnant primary a critical mass ratioq _c is derived such that for larger values ofq, mass transfer from the secondary towards the primary starts before the latter has evolved into a white dwarf. Consequences for the advanced stages of medium mass binaries are discussed.     

Massive X-ray binaries: Their physics and evolution

We review various aspects of the evolutionary history of massive X-ray binaries. It is expected that moderately massive close binaries evolve to Be X-ray binaries, while very massive systems evolve to standard X-ray binaries.The compact objects are formed through supernova explosions. The fairly low galactic latitudes of those systems indicate that the explosion should, in general, not have accelerated the system to a velocity larger than 50kms^–1. This implies that the mass of the exploding stars is in general less than 5 to 6M .After the explosion, tidal forces will circularize the orbit of short period systems. Even if the tidal evolution has been completed, the expansion of the optical star during the course of its evolution will continously disturb the stability of the orbit. Short period systems with large mass ratio may eventually become tidally unstable. Cen X-3 may be an example of such a system. The predicted rate of the orbital period decrease of Cen X-3 is in agreement with the observed rate.A way to represent the rotational and magnetic evolution of neutron stars in close binary systems is presented. The observed distribution of the pulsation periods of X-ray pulsars with Be companions is consistent with initial magnetic fields of 10¹²–10¹³ G of the neutron stars. We suggest that the fast X-ray pulsars 4U 0115+63 and A 0538-66 are young neutron stars, while Cen X-3 and SMC X-1 are recycled pulsars.The evolutionary relationship between massive X-ray binaries, binary pulsars, and millisecond pulsars is also discussed.Invited paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Dynamical conditions in planets and stars

With the available data in planets, stars and galaxies, it is studied the functions of angular momentaJ(M) and amounts of actionA _c(M) (associated to the non rotational terms in the kinetic energy). The results indicate that independently of how are these functionsJ(M),A _c(M) their ratioA _c/J remains a near invariant. It is independent also from the type of angular momenta: intrinsic spins of the bodies or the total angular (orbital) momenta of the bodies forming a system; for instance, the Solar System and the planets.The relationA _c(M) for the Solar System are analogous to these in the FGK stars of the main sequence, and the relationJ(M) (also for the Solar System) is analogous to the lower possible limit for binary stars.The different types of binary stars from the short period, detached systems to contactary systems, gives a range of functionsJ(M),A _c(M) that are the same that one can expect in stars with planetary systems. According to the detection limits given for planetary companions by Campbell, Walker and Yang (1988) (masses of less than 9 Jupiter masses and orbital periods of less than 50 years) we calculate the limits forJ(M) andA _c(M) This gives a lower limitA _c/J 1 associated to stars with planetary systems as 61 Cygni and to short period detached binaries. The upper limitA _c/J 16 correspond to planetary systems as the ours and probably to cataclysmic binaries. There are reasons to suspect that systems as the ours and in range 4 A _c/J 16 (with a lower limit analogous to contactary binaries as Algols and W Ursa Majoris) must be the most common type of planetary systems. The analogies with the functionsJ(M)A _c(M) for galaxies suggest cosmogonical conditions in the stellar formation.Independently of this, one can have boundary conditions for the Jacobi problem when applied to a collapsing cloud. Namely, from the initial stage (a molecular cloud) to the final stage (a formed stellar system: binary or planetary) the angular momenta and amounts of action decayed to 10^~4 the initial values, but in such a form thatA _c(t)/J(t) remains a near invariant.     

Evolution of close binaries and origin of Algol-type systems

The evolution of close binary systems was followed for ten systems with the initial mass of the primary in the range 1–4M and with different initial mass ratios and initial separations. A brief discussion of the evolution of the contact component is presented for two separate cases: when the primary reaches its Roche lobe during central hydrogen burning (case A) and after the exhaustion of hydrogen in the center (case B).The models obtained are compared with observed semi-detached systems separately for massive (with total mass greater than 5M ) and low mass (with total mass below 4M ) binaries. It is shown that the contact components of the observed massive binaries are probably burning hydrogen in the core. On the contrary, the majority of contact components of the observed low-mass binaries are burning hydrogen in the shell. The observed distribution of such binaries as a function of different luminosity excesses of contact components seems to indicate that their origin is connected with case A rather than with case B.     

Study of physical properties of spectroscopic binary stars

The main results of a study of a catalogue of physical parameters of 1041 spectroscopic binaries are presented. The distribution of spectroscopic binaries over all main parametersM ₁, a, e, M₁/M₂, P, and certain dependencies between some of them have been found.

1. It appears that among bright (m _v?3 ^m –5 ^m ) stars withM?1M _⊙, about 40% are apparently spectroscopic binaries with comparable masses of components.
2. The majority of spectroscopic binaries with the ratio of the large semiaxis of the orbit to the radius of the primarya/R ₁?20, have eccentricities close to zero. This is probably a consequence of the tidal circularization of orbits of close binaries by viscous friction.
3. The discovery of duplicity of double-line spectroscopic binaries is possible only if the semiamplitude of radial velocityK ₁ is almost 10 times higher than the semiamplitude of the radial velocity of a single-line spectroscopic binary of the same mass.
4. Double-line spectroscopic binaries witha/R _⊙?6(M ₁/M _⊙)^1/3,M ₁≈M ₂?1.5M _⊙ are almost almost absent, and the number of stars witha/R _⊙?6(M ₁/M _⊙)^1/3,M ₁≈1.5M _⊙ is relatively low.
5. The distribution of unevolved SB stars over the large semiaxis may be described by the expression d(N _d/N_t)≈0.2 d loga for 6(M ₁/M _⊙)^1/3?a/R _⊙?100.
6. The intial mass-function for primaries of spectroscopic binaries is the same Salpeter function dN _d≈M ₁ ^?2.35 dM ₁ for 1?M ₁/M _⊙?30.
7. It is possible to explain the observed ratio of the number of single-line spectroscopic binaries to the number of double-line binaries if one assumes that the average initial mass ratio is close to 1 and that the mass of the postmass-exchange remnant of the primary exceeds the theoretical one and/or that half of the angular momentum of the system is lost during mass-exchange.
8. The above-mentioned distributions ofM ₁ anda and assumptions on the mass of remnant and/or momentum loss also allow us to explain the observed shapes of dN/dM, dN/dq, and dN/da distributions after some selection effects are taken into account.

     

Close binary systems before and after mass transfer: A comparison of observations and theory

From a search through the literature 174 close binaries with known absolute dimensions have been sampled. Distinction is made between systems before and after mass exchange. Mass, period and mass ratio distributions and relations of the group of unevolved binaries (i.e., prior to mass exchange) are transformed into corresponding distributions and relations of evolved binaries. The transformations are based upon theM _1f=g(M _1f) relation derived from an extended set of published theoretical computations of the evolution of close binaries. From this relation the following characteristics of the system after mass exchange are computed:M _1f,M _2f (andq _f),P _f. Five different modes of mass transfer were applied for the computation of the values ofP _f andM _2f. The variation of the period was calculated using the formalism given by Vanbeverenet al. (1979). The results are compared to the observations of binary systems after mass exchange, and are discussed together with an analysis of the effect of several selection effects present in the distributions. The main conclusion is that, during mass exchange in close binaries, more than 50% of the mass is lost to the system in the process of transfer, removing a large amount of angular momentum.This research is supported by the National Foundation of Collective Fundamental Research of Belgium (F.K.F.O.) under No. 2.9009.79.     

Liminosity distribution function for visual binary stars

The cosmic scatter of 147 of the best known visual binary stars on the Main Sequence is discussed and a new estimation of the luminosity distribution function for multiple star systems is presented. As long as the mass ratioq of a close binary is not smaller than 0.5, the distribution of close binary components is identical to the van Rhijn liminosity-function. For smaller mass ratios (q<0.5) the number of close companions decreases rapidly. It appears that less than 13% of visual binaries in our sample are simple binary systems.Communication presented at the International Conference on Astrometric Binaries, held on 13–15 June, 1984, at the Remeis-Sternwarte Bamberg, Germany, to commemorate the 200th anniversary of the birth of Friedrich Wilhelm Bessel (1784–1846).     

Observational manifestations of early mixing in B- and O-type stars

The helium and nitrogen enrichment of the atmospheres of early B-type stars during the main sequence (MS) evolutionary phase is re-analysed. It is confirmed that the effect depends on both the aget and the stellar massM. For example, the helium abundanceHe/H increases by 0.04 (60–70% of initial value) for stars withM=8–13M and by 0.025 (about 30%) for stars withM=6M . The nitrogen abundance rises by three times forM=14M and by, two times forM=10M . According to the latest theoretical computations, the observed appearance of CNO-cycled material in surface layers of the stars can be a result of the rotationally induced mixing, in particular, of the turbulent diffusion. Carbon is in deficiency in B stars, but unexpectedly does not show any correlation with the stellar age. However it is shown that the total C+N abundance derived for early B stars conflicts with the theory.Basing on modern data the helium enrichment is first examined in O-type MS stars, as well as in components of binaries. As compared with early B stars, the He abundance for more massive O stars and for components of binaries show a different relation with the relative aget/t _MS . Namely during short time betweent/t _MS 0.5 and 0.7 a sharp jump is observed up toHe/H=0.2 and more. In particular, such a jump is typical for fast rotating O stars (v sini200 km s^–1),. Therefore the effect of mixing depends on massM, relative aget/t _MS , rotational velocityv and duplicity.The mass problem (the discrepancy betweenM _ev andM _sp ) is also analysed, because some authors consider it as a possible evidence of early mixing, too. It is shown that the accurate data for components of binaries lead to the conclusion that the discrepancy is less than 30%. Such a difference can be removed at the expense of theM _ev lowering, if the displacement of evolutionary tracks, owing to the rotationally induced mixing is taken into consideration.     

The product of the globular cluster collapse

In this paper we calculate the number of close binaries formed during the evolution process of a globular cluster core. The globular cluster core is assumed to contain a massive black hole at its center. We show that the central black hole can drive binaries formation in the core and the rate of binaries formation depends on the mass of the black hole at its center. When the massM of the black hole is between 10² M and 3×10³ M , there will be a few binaries formed. When the mass of the black hole is 4×10³ M M6×10³ M , the number of binary star formation will suddenly increase with a jump to the maximum value 58. When the mass of the black hole is 7×10³ M M9×10³ M , the number of binary star will immediately decrease. Whether cluster X-ray is produced mainly by the central black hole or by binaries in the core depends on the mass of the central black hole. Therefore, two cases arise: namely, black hole accretion domination and binaries radiation domination. We do think that we cannot exclude the possibility of the existence of a central black hole even when binary radiation characteristics have been observed in globular cluster X-ray sources.     

Angular momenta in binary systems

The correlations angular momentaL to massesM are studied for different types of spectroscopic binaries. The functionsL=AM ^b have the coefficientb with the values expected from a Keplerian mechanics, but the valuesA(q, T), A(q, a), A(q, v), associated tob=5/3, 3/2, and 2, respectively, are given (statistically speaking) by multiples or submultiples of discrete values of: the mass ratiosq, the semi-major axesa, periodsT, and velocitiesv of the reduced mass. This indicates the existence of a discrete unit of actionL=(1/2)×potential energy xperiod. Postulates about equivalent states of angular momenta for different orbital parameters are introduced, being this coherent with the analysis of the up-to-date data. Among other examples of the application of such equivalence postulates, we haveL(M) (W-type of the WUMa systems)L(M) (main group of the Algol binaries). The quantum units of action seen here are equivalent to those seen in the solar system in one of our previous works. From comparisons with galaxies and single stars, it is evidence that there is not an unique universal functionL=AM ^b, when the fine structure of the relation is analysed: each type of object has its own coefficients,A, b. It sems to be that there are an upper and a lower limit for all the possible functions. The upper limit isL=A _gM^5/3, withA _g1 associated to periodsT Hubble time, and the lower limit isL=GM ^2/c, with 1. The existence of the upper limit can be investigated with studies of pairs of galaxies, and the lower limit can be tested with analysis of single G, K, M stars. The quantical hypothesis introduced here can be checked definitely, when available larger samples of data with low errors, with similar quality as the selected list of almost 80 eclipsing binaries (mainly detached systems) analysed here.     

Statistical properties of spectroscopic binary stars

As part of a study of the mass-ratio distribution of spectroscopic binary stars, the statistical properties of the systems in theEighth Catalogue of the Orbital Elements of Spectroscopic Binary Stars, compiled by Battenet al. (1989), are investigated.Histograms are presented of the distributions of various parameters of the systems in the catalogue as a whole, and compared to those of the previous edition.Histograms of binaries of various spectral types are presented. It is noted that the early-type binaries in the catalogue have on average higher radial-velocity amplitudes, shorter periods, and smaller semi-major axes than late-type binaries. Late-type binaries have relatively more eccentric orbits. Whether the differences noticed between the early- and late-type binaries have any significance with respect to the population of binary stars in the Galaxy is not clear, because it is very hard to assess the extent to which the catalogue is a statistically representative and complete sample.The distribution of semi-major axesa ₁ sini varies considerably among different subsamples.The mass-ratio distribution of single-lined spectroscopic binaries is, for all (sub-)samples, characterized by a decrease in the number of systems according to a power law asq1, forq=M _sec/M _prim>0.25.The mass-ratio distribution of double-lined spectroscopic binaries (SBII) is, for all (sub-)samples but one, characterized by an increase in numbers according to a power law, asq1. The exception to this general behaviour is the sample of SBII systems with magnitudesm5 ^m , which has its maximum atq0.65.The distributions are presented as they are, without corrections for selection effects.     

Period distributions in pulsating and binary stars

We analyze the hypothesis of quantization in bands for the angular momenta of binary systems and for the maount of actionA _c in stable and pulsating stars. This parameter isA _c=Mv _eff R _eff, where the effective velocity corresponds to the kinetic energy in the stellar interior and the effective radius corresponds to the potential energyGM ²/R _eff. Analogous parameters can be defined for a pulsating star withm=M where is the rate of the massm participating in the oscillation to the total massM andv _osc,R _osc the effective velocity and oscillation radius.From an elementary dimensional analysis one has thetA _c (energy x time) (period)^1/3 independently ifA _c corresponds to the angular momentum in a binary system, or to the oscillation in a pulsating star or the inner energy and its time-scaleP _eff in a stable star.From evolving stellar models one has that P _effP _eff(solar)1.22 hr a near-invariant for the Main Sequence and for the range of masses 0.6M <M<1.6M .With this one can give scalesn _k=kn ₁ withk integers andn ₁=(P/P ₁)^1/3 withP ₁=P _eff1.22 hr. In these scales proportional toA _c, one sees that the periods in binary and pulsating stars are clustered in discrete unitsn ₁,n ₂,n ₃, etc.This can be seen in pulsating Scuti, Cephei, RR Lyrae, W Virginis, Cephei, semi-regular variables, and Miras and in binary stars as cataclysmic binaries, W Ursa Majoris, Algols, and Lyrae with the corresponding subgroups in all these materials. Phase functions (n _k) in RR Lyrae and Cephei are also associated with discrete levelsn _k.the suggested scenario is that the potential energies and the amounts of actionE _p(t), A_c(t) are indeed time-dependent, but the stars remain more time in determinated most proble states. The Main Sequence itself is an example of this. These most probable states in binary systems, or pulsating or stable stars, must be associated with velocities sub-multiplesc/ _F , given by the velocity of light and the fine structure constant.Additional tests for such a hypothesis are suggested when the sufficient amount of observational data are available. They can made with oscillation velocities in pulsating stars and velocity differences of pairs of galaxies.