On the evolution of low mas binary systems

O N T H E E V O L U T I O N O F L O W MASS B I N A R Y S Y S T E M S

P. R. A M N U E L and O. H. G U S E I N O V

Shemakha Astrophysical observatory, U.S.S.R.

(Received 8 February, 1982)

Abstract. The evolution of parameters of close binary systems containing a red dwarf (with mass loss) and a condensed star is investigated. The mass loss from the system and asynchronism of the red dwarf rotation are taken into consideration. The calculations show that if the initial mass ratio qo <~ 2 then during ~ 107 yr the instability of mass loss process arises, and a bright X-ray source forms with luminosity close to the Eddington one. If qo > 2 then during the mass loss phase the bright X-ray source arises twice. The models explain the existence of the 'forbidden' interval of orbital periods and the absence of systems with periods less than ~ 80 min.

1. Introduction

The physical nature of bright X-ray sources in the galactic bulge is now being investigated by many astrophysicists. In some cases the measured orbital periods are short (see Table I), and several optical stars identified with soft X-ray transient and steady sources possess spectral classes later than K0. This leads to an assumption that normal components in such systems are low-mass stars. An analogous problem exists in the theory of cataclysmic variables.

One may estimate the mass of a normal companion (i.e., the star losing mass) taking into account a hypothesis that this star is on the Main Sequence and fills their Roche lobe. Then the maximum value of mass may be estimated. The estimated masses for normal components of cataclysmic variables and low mass X-ray sources are listed in Table I. From Figure 1 one may see that ~ of all binaries listed in Table I have normal components with masses less than 0.8 Mo.* Such a star evolves on the Main Sequence during a time longer than the lifetime of the Galaxy. Then, the caus e of mass transfer in that system is not an increase of the radius of a normal star during red giant formation. On the other hand, in systems of separation less than ~ 2 x 1011cm (periods less than about 4 hr), the gravitation radiation becomes sufficiently signifi- cant. Therefore, it may be suggested that the contraction of a system during the gravitation radiation process leads to the filling of the Roche lobe by the Main- Sequence low-mass companion. The calculations concerning the evolution of such systems were developed by Faulkner (1971), Tutukov and Yungelson (1979), Kieboom and Verbunt (1981), Paczynski and Sienkiewicz (1981), Rappaport et al. (1982), and others. According to those calculations, a star losing mass is in a state of thermal equilibrium (i.e., on the Main Sequence) during the evolution, when its mass is larger than 0.1 M o. Then one may use for the calculations of orbital elements the dependence

* This also concerns the bursters. Walter et al. (1981) suggests eclipses in 4U 1916-05 (periods ~50min). van Paradijs et al. (1981) found an orbital period ~4hr in optical light of the burster MXB 1636-53.

Astrophysics and Space Science 86 (1982) 91-106. 0004-640X/82/0861-0091502.40. Copyright �9 1982 by D. Reidel Publishing Co., Dordreeht, Holland, and Boston, U.S.A.

92 P. R. AMNUEL AND O- H. GUSEINOV

TABLE I

System Period Spectrum Mx/M o M2/M o (hr) of primary

Cataclysmic variables T CrB 5462h24 m M3 2.6 1.8? G K Per 16 26 K2 IVp 0.6 1.3 BV Cen 14 38 G: 1.4 1.4 HZ 9 13 28 1.2 V 471 Tau 12 30 K0 V 0.8 1.2 V Sge 12 20 G 1.0 AE Aqr 9 53 K2 V 0.7 1.2 RU Peg 8 54 G8 IVn 0.8 1.0 EM Cyg 00 G - K 0.9 0.7 Z Cam 6 56 G1 1.0 1.2 SS Cyg 6 38 G5 0.8 1.0 RW Tri 5 34 0.5 1.3 RX And 5 05 0.6 1.0 T Aur 4 54 0.8 EY Cyg 4 53 K0 V 0.8 UX U M a 4 43 0.8 1.2 DQ Her 4 39 0.7 1.2 HR Del 4 35 Q 0.7 1.3 SS Aur 4 20 G - K 0.7 0.9 U Gem 4 10 0.6 0.4 W W Cet 3 50 0.6 RR Pic 3 29 0.5 VZ Scl 3 28 0.4 0.3 V 603 Aql 3 19 0.4 0.9 TT Ari 3 12 0.4 0.8 MV Lyr 1 55 0.2 AN U M a 1 55 0.2 VW Hyi 1 47 0.2 Z Cha 1 47 0.2 0.3 VV Pup 1 40 0.1 EX Hya 1 39 0.1 1.0 V 436 Cen 1 32 0.1 OY Car 1 29 0.1 0.7 WZ Sge 1 21 0.1 1.3 G 61-29 0 46 ? AM CVn 0 18 ?

Low mass X-ray binaries Cyg X-2 216 00 > 1.0: A 0620~)0 187 12 K 4-5 V 0.7 Her X-1 40 48 A-F > 1.0: 4U 1908 + 00 31 48 K 0.7 Sco X-1 18 53 ~> 1.0: Cen X-4 8 11 K 3-7 V 0.6 4U 1822-37 5 36 0.9 4U 2129 + 47 5 12 0.8 2A 0526-328 5 12 0.8 Cyg X-3 4 48 0.8 H 22544)33 3 36 0.5 AM Her 3 06 0.4 2A 0311-227 1 21 M 7-8 0.1 4U 1626-67 0 42 ?

1.0:

1.0:

Note: Masses of primaries are estimated according to their spectral classes. If the spectrum is not known, the masses are estimated according to the values o f the orbital periods.

O N THE E V O L U T I O N OF L O W MASS BINARY SYSTEMS 93

Fig. 1.

N

10

4

P

al �9

0

0 0.2

e,

a.

b

N ! ! ! , I , , I

O.4 o.6 o.8 I

M/~ o

The dependence of numbers of binaries on the masses of primary components (see Table I).

of the radius of the star on their mass in the form R = kM". Rappaport et al. (1981) used n = 0.78. When M ~< 0.1 Mo, the value of n decreases to - �89 The orbital period of the system has a minimum value of about 80 min which does not contradict the observations. Note, however, that several systems (Table I) have periods of less than 80 min.

The models mentioned above do not explain the high mass transfer rates observed in many cataclysmic variables and low-mass X-ray sources. The average X-ray luminosity of bulge sources is about 2 x 1037 ergs s-1 in the 2-6 keV range (Amnuel and Guseinov, 1980), which corresponds to the accretion rate onto a neutron star of about 3 x 10 -9 Moyr -1. The brightest bulge sources have luminosities close to the Eddington limit (an accretion rate of more than 10-8 Mo yr-1). In several cataclysmic variables (SS Cyg, DQ Her, etc.) the mass transfer rate is also more than 10-s Mo yr-1. However, calculations by Rappaport et al. (1982) and others give the maximum of red dwarf mass loss to be about 10- 9 Mo y r - 1.

Verbunt and Zwaan (1981) suggested the hypothesis of the magnetic braking of a red dwarf star. The mass loss rate by the normal component in this case increases. This hypothesis still requires a more detailed investigation; however, one notes that the Alfv6n radius of a red dwarf is more than its geometric radius only if the magnetic field in the star surface is more than 100 G.

94 I ' . R. A M N U E L A N D O. H. G U S E I N O V

All investigations concluded that the high mass loss rates took place in systems where the mass-losing star is a white dwarf with mass ~<0.08 M o. Then the mass transfer rate is ~ 10 -8 Moyr -~ during several million years. The origin of such anomalous systems is not revealed.

All the calculations were made taking into account an assumption of the constant total mass of the system (conservative evolution). Rappaport et al. (1982) considered

two models where the mass accreted onto condensed star is 0.8 and 0.77 of the mass lost by the red dwarf. Exactly in those two models the high mass loss rate (--,2 x 10 -9 Moyr -~) was obtained. However, the nonconservative evolution models were not developed, although as has been shown by many investigators, the process of mass transfer in close binaries is not conservative.

The second assumption (Rappaport et al., 1982, etc.) was in the hypothesis of the synchronism of red dwarf rotation and the orbital revolution of the system. In fact, according to Zahn (1977), a dwarf star with a convective envelope can synchronize its rotation with the orbital revolution during 104 yr. However, desynchronization effects caused by a loss of the total momentum of the system were not taken into consideration. Recent calculations by Papaloizou and Pringle (1981) showed that the orbit of a total convective red dwarf in a close binary system with an orbital period of 100min circularises during 3 x 10 7 yr. Gorbatsky (1974), and Amnuel and Guseinov (1977), listed binaries with asynchronous components. The problem of synchronism in a system containing a red dwarf is completely open - there are no observational data on rotational velocities of red dwarf stars in close systems. According to Gorbatsky (1974), one should raise the question of synchronism after the converse question has been answered: namely, what are all the consequences of asynchronism? The observations of such consequences should be evidence of asynchronism in the system. The problem of bright bulge X-ray sources and cataclysmic variables may be an indicator of such a reversal problem.

One also notes that the assumption of synchronism has been incorrectly intro- duced. Rappaport et al. (1982), Paczynski and Sienkiewicz (1981), etc., neglected the angular momentum of red dwarf rotation as compared with orbital angular momentum - i.e., co/g2 ~ 1, where co and ~2 are the angular velocities of red dwarf rotation and the orbital revolution of a system. However, if co/• ~ 1, the assumption of conservative mass transfer is not correct (see below). If synchronism takes place (co = s then the evolution is conservative when q ~ 0 (q is the mass ratio in system, q = M z / M 1 , where M~ and M 2 are the masses of normal and condensed components respectively). If q ~ 0.5-1 and co = f2, then the condensed star (M2) accretes only ~ 0.5 of the mass lost by the red dwarf. If q ~ 0.5-1, the mass transfer is conservative when co ~ f2 - i.e., synchronism does not take place. Therefore, the assumptions of synchronism and conservative mass transfer contradict each other, and, in general neither assumption is fulfilled.

Amnuel and Guseinov (1977) suggested the evolution of the orbital elements of a close binary taking into account the angular momentum of a star losing mass and the asynchronism of its rotation. The accretion disk near a compact star is formed, and mass

ON THE EVOLUTION OF LOW MASS BINARY SYSTEMS 95

transfer through the internal Lagrangian point takes place. The fraction fl of mass lost by the primary is accreted by the secondary; the rest of the mass (1 - t ) escapes the system through the external Lagrangian point. However, mass loss rates were not determined, and characteristic times of mass loss were not estimated. Below we shall determine these parameters for close systems with a low mass normal component, taking into account the effects of gravitation radiation.

2. The Evolution of the Parameters of a Binary System

Let us consider a binary system containing a main sequence red dwarf star with a mass of less than 0.8 M o and a condensed companion (white dwarf or neutron star). If the separation of the system is a, then this system goes to the semidetached phase during the time

t = 6.4 x 108 (a/1011 cm)4 q(1 + q)(M1/Mo) 3 yr (1)

due to gravitation radiation.

If a < 2 x 1011 cm, this time is less than the Main Sequence evolution time of a red dwarf. After the time (1) goes, a red dwarf flUs their Roche lobe and mass loss begins. Faulkner (1971), Tutukov and Yungelson (1979), Rappaport et al. (1982) used the

following approximations in their determination of the rate of mass loss: (i) At all times the radius of the red dwarf (R1) is equal to the effective radius of the

Roche lobe, or

R1 = RL, /~1 = /qL; (2)

(ii) the gravitation radiation is the only cause of the contraction of the system. Then the orbital momentum decreases as

J 32 GS/3qM~/3f2 s/3

Jorb 5 C5(1 + q)'/3 ' (3)

where Jorb is an orbital momentum of a system; G, the gravitation constant; and c, the velocity of light.

(iii) the radius of red dwarf R1 depends on their mass as

R~ = kM~. (4)

According to Rappaport et aL (1982), k = 0.7 and M = 0.78 (if the mass and radius

are determined in solar units). Besides, as we noted above, all previous investigators proceeded on the assumption that the rotation of the red dwarf is synchronous, and the mass transfer process conservative.

In what follows we shall determine the values of mass loss and accretion rates using assumptions (i)-(iii), but we shall take into consideration a possibility that the synchronism of red dwarf rotation is not conserved after the onset of mass loss, and only

96 P.R. AMNUEL AND O. H. GUSEINOV

the fraction/~ of mass lost is accreted by the compact star. According to Amnuel and Guseinov (1977), the value of/? is to be determined as

[ I q, /~ = 1 - (0.5 + 0.2251ogq) 1 + q-]~/2 1 -- ~-(0.5-0.2251og (5) q J

We conclude that the radius of the Roche lobe of a red dwarf is given by

R L = Ra = a(0.5-0.225 log q). (6)

The error of this determination is no more than ~ 10~. The difference of real critical lobe geometry from the Roche lobe has not been taken into account. According to Pratt and Strittmatter (1976), if co =p Q the value of critical radius becomes

RL = a(1 -- 0.45 log q) [0.53-0.3(09/0) 2]

which does not differ from that given by Equation (6) by more than a few per cent. If co = O, then /~ < 1 for all values of q, and evolution is not conservative. The dependence of/~ on q and co/f2 is shown in Figure 2. The shaded region corresponds to /~=0.

The total angular momentum of the system is the sum of the orbital momentum and the rotational momentum of the red dwarf (the momentum of the condensed star is neglected); and is given by

G2/3qM~/3 z J = Jorb q- J1 = ~,-21/3(1 + q)1/3 "q- MlcoRef"

For a Main-Sequence star of polytropic index 1.5, the value of Ref is 0.48tR~ (Guseinov and Kasumov, 1972); and for a red giant star, Ref ~ 0.3R1 (Pringle, 1976). The total momentum of the system decreases because of gravitation radiation and mass loss. According to Amnuel and Guseinov (1977) and taking into account (3), one finds that

J 35 GS/3qM~/3f28/3 +

Jorb = -- 5 C5(1 + q)1/3 ~/1 1 + q ( 1 /~)( 1 ) - + 0.5 + 0.225 logq x

-~ M I q l + q

i 1 ql/2 ] x ~ + (1 + q)1/2(0.5 + 0.2251ogq)~/2 �9 (7)

On the other hand, from (2), (4), (6) combined with Kepler's third law we find that

g) _ 3 5)/1 ~ 0.0977 q + f l 1 q + / / (n--l)-] . (8) (2 2 M1 [_0.5-0.2251ogq q 3 1 + q A

From (7) and (8) one finds the value of )9/~ to be given by

64 GS/3q2(f2M1) s/3 (9) i f / l = - 5 cSF(1 +q)t/3 0


4

0.25

0.5

0.75

0.75

0.5

Fig. 2.

0 9 0 0.2 0.4 0.6 0.8 1

The dependence of J3 on the values of mass ratio q and synchronization degree o~/~. The solid lines correspond to j~ = const. In the dashed regions the value of j~ is zero.

98 1'. R. AMNUEL AND O. H. GUSE1NOV

where the value of F is determined from

CO F = 2(2 + q)(0.5-0.225 log q)~- +

( 1 0 . 0 9 7 7 ) +/~ ~ + 0 . 4 5 1 o g q - +

+ q 0.5-0.225 log q

/ q - 1 0.45 l o g - 0.0977 + ~ q ~ - i - + n - q J 0.5-0.225 log q

(3 - \ ~ + 0 . 9 1 o g

When F ~< 0, the stationary mass loss does not take place. The radius of the Roche lobe changes more rapidly than the radius of the red dwarf; conclusion (i) then becomes incorrect, for the red dwarf loses mass on a dynamical time-scale (Tutukov and Yungelson, 1979).

For the determination of parameters of a binary system and mass loss rate as functions of time, one must specify the system of equations which relate ~I 1 with fl, co,

and O. Using assumption (i), one obtains from the change of angular rotational momen-

tum of a red dwarf that

~ / i O} il~ef )~/i a2 - - + - - + 2 - (0.5-0.225 log q)2 Mi co Ref Ma R~

and

_ ( _ - 1). (10) co M1 \Re~ J

The solution of Equations (5), (8)-(10) determines the mass loss and accretion rates, angular velocities co and/2 as functions of time. Note, that, in past calculations, the red dwarf was assumed to be in an equilibrium state (n -- 0.78) when M1 ~> 0.1-0.15 Mo (Rappaport et aL, 1982).

If one invokes the synchronization effects, then the function

M1 I f 2 _ 1~ �9 1.727 x 108 f(1 + q)1/3

q2( )s,3(10_3s -lf2 is to be added to the right-hand side of (10); where t~ is the approximated syn-

chronization time (in years). We calculated the evolution of a system's parameters for 8 models whose initial and

final parameters are listed in Table II. Dependences of the accretion rate (and X-ray luminosities for accretion onto a neutron star),/~ and co/f2 on time are shown in Figures 3-5. In the models 1-4 the mass loss process becomes unstable after ~ 107 yr. During

ON THE EVOLUTION OF LOW MASS BINARY SYSTEMS

TABLE IIa Initial parameters of models

99

Model Mlo/Mo M2o/Mc9 qo (2o(10-3s - t ) Notes

1 0.65 1.100 1.692 0.619 2 0.65 1.170 1.8 0.619 3 0.65 1.235 1.9 0.619 4 0.65 1.300 2.0 0.618 5 0.65 1.365 2.1 0.618 6 0.65 1.625 2.5 0.618 7 0.65 1A00 1.692 0.619

8 0.65 1.100 1.692 0.619

coo = f2o, synchronism is not conserved coo = f2o, synchronism is not conserved coo = f2o, synchronism is not conserved coo = f2o, synchronism is not conserved co o = f2o, synchronism is not conserved coo = f2o, synchronism is not conserved rotation of red dwarf is synchronous during all the time of mass loss process synchronism is lost when mass loss begins, but is restored during 107 yr.

TABLE IIb Final parameters of models

Model Oct a f2max b Per M1 er M2er ter tbr e (10-3s -1) hr M o M o yr yr

1 0.66 - 2.64 0.60 1.12 1.05 107 1.05 x 107 2 0.67 - 2.60 0.58 1.20 1.92 107 1.92 x 107 3 0.70 - 2.50 0.53 1.28 2.85 107 2.30 x 107 4 0.76 - 2.30 0.47 1.37 4.36 107 2.20 x 107 5 1.59 1..59 1.10 0.14 1.49 1.43 10 s (1) 3 X 107

(2) 3 x 106 6 1.55 - 1.13 0.12 1.75 3.38 108 3.0 X 106 7 1.42 1.69 1.23 0.10 1.34 6.76 108 (1) 1.5 X 107

(2) 5 X 106 8 1.73 1.76 1.01 0.10 1.33 5.87 108 (1) 2.5 x 107

(2) 4 x 106

" At this value of f2 the stability of mass loss breaks (F = 0). b The frequency corresponding to minimum of orbital period. c in models 5, 7, and 8 bright X.ray sources arise twice, tcr corresponds to the time ofstability break; tbr is the lifetime of the bright source.

(1 -2) x 107 yr a b r i g h t X - r a y s o u r c e a r i ses (Lx is m o r e t h a n 1037 e rgs s -1 ) . T h e n t he

m a s s loss is in t he d y n a m i c a l t ime-sca le , a n d t h e r e d d w a r f w o u l d be c o m p l e t e l y

d i s r u p t e d . I n t h a t case t he g a s e o u s d isc n e a r t he n e u t r o n s t a r (wh i t e dwar f ) f o r m s w i t h a

m a s s o f a b o u t 0.5 Mo , a n d i n s t a b i l i t i e s in t he a c c r e t i o n p r o c e s s l ead to X - r a y b u r s t

p h e n o m e n a . Af te r t h e f o r m a t i o n o f t he d isc t h e g r a d u a l s e t t l i ng of t h e i r m a t t e r o n t o t he

n e u t r o n s t a r t a k e s p l a c e (a f r a c t i o n o f t he m a s s is e j ec t ed i n t o e x t e r n a l space) . T h e X - r a y

l u m i n o s i t y is c lose to t h e E d d i n g t o n one ; t h e l i fe t ime of s u c h a b r i g h t s o u r c e is a b o u t

5 x 106 to 107 yr. T h e r e d d w a r f is d i s r u p t e d w h e n t he o r b i t a l p e r i o d of t h e s y s t e m in

m o d e l s 1 - 4 is 2 .3 -2 .6 h r ( T a b l e II).

I f t h e i n i t i a l m a s s - r a t i o is m o r e t h a n a b o u t 2, t h e n t he m a s s loss ( a n d a c c r e t i o n ) r a t e

100 P. R, AMNUEL AND O. H. GUSEINOV

M2 (Me/yr) L• (erg/s)

10 - 8 . . . . . . . . r ' ~ ' '

4

10 - 9 ,

1 0 - 1 0 I , , ! II . . . . . . f ,

0 1 2 3 4 5

t { l O 7 y r )

Fig. 3a.

1037

1036

increases to (2-3) x 10 -9M e yr-1 and decreases (models 5, 6). During (1-3) x 108 yr the accretion rate is low and an X-ray weak source arises. When the mass of the red dwarf decreases to 0.1~).15 M e, the value of n in (4) decreases from 0.78 to - �89 The mass-loss rate rapidly increases, the process begins to occur on the dynamical time- scale, which leads to complete disruption of the dwarf. The bright X-ray source with luminosity close to the Eddington one also arises and the lifetime is (3-6) x 106 yr. In this

case the orbital periods are concentrated near 1.2 hr. Therefore, the evolution of a low-mass close binary depends sensitively on the initial

mass-ratio q. Ifq < 2 (i.e., when the mass of the red dwarf has not sufficiently decreased) the stability is quickly lost. Ifq > 2 the dwarf becomes unstable only in the last stage of mass transfer, when the mass decreases to 0.1-0.15 M e. In the first case, the bright X-ray source arises only once, in the second case the bright source arises twice (the duration of the first stage is several times longer than the second one). It is sufficient that, in both cases (qo < 2 and qo > 2), the X-ray sources with luminosities close to the Eddington one can arise when the system has a normal component on the verge of disruption (or after their disruption). In many cases such X-ray sources (brightest bulge sources) are single neutron stars surrounded by the gaseous clouds which arise after the red dwarf disruption. That is why the search for eclipses and orbital periods in X-ray bulge sources may be successful when their luminosities are sufficiently smaller than the


M2 (Mo/yr) 10-8

4

10-9 i

10-1a

10-11 o 1

'I !

5 6 1 8 I 7

I t, ,,! I l I, 2 3 4 5 6 7

t (108 yr)

Fig. 3b.

L x (erg/s)

Fig. 3a-b. The accretion rate (and X-ray luminosity in the case of neutron star) as a function of time. (a) models 1-4; (b) models 4-8.

1037

1036

1035

Eddington limit - this signifies that the red dwarf in a binary does not lose its stability, but the accretion rate already begins to increase (see Figure 3).

In our models two observational facts connected with cataclysmic variables and bulge X-ray sources receive the following natural explanation.

(i) The absence of systems in the period interval from 2 to 3 hr (Table I). In our models (Table II) the bright sources in intervals of periods from 1.23 to 2.3 hr are also absent. This does not signify that systems with those periods are really absent; however, the mass loss (and accretion) rate in such cases is low, and the possibilities of discovery

102 P. R. A M N U E L A N D O. I-I. G U S E I N O V

Y 0.5 w �9 ' , �9 . . . . �9 �9 ~ r ' I

. . . . 7

0 , 4 1

o

0 . 2

0.1 1 f I f If I I . . . . . . . l I , , ~ - ~

0 1 2 3 4 5 6 7 8 s 10 t ( 10 7 yrs)

Fig. 4. The fraction of mass fl accreted by a second component as a function of time.

1

0 . 8

0 .6

0 .4

0 . 2

i ,, ' 8 1 1 |

1 6

2 3

, , I , ~ ! I _ _ . . 1[ i 1 , | , I _

0 1 2 3 4 5 6 7 8 9 10 t (10 7 yrs)

Fig. 5. The synchronization degree ~o/f2 as a function of time.


of the systems are limited. The presence of the 'forbidden' interval of orbital periods in models by Rappapor t et al. (1982), etc. remains unexplained.

(ii) The absence of cataclysmic variables with orbital periods of less than 80 rain. In our models this corresponds to instability which arises when the mass of the red dwarf decreases to 0.1-0.15 M o.

In the models by Rappaport et al. (1982) the instability in mass loss is also possible in principle. However, such instability arises only when qo < 1.2 - i.e., in rare cases when the neutron star in the system is of low mass. In our models (without synchronization) the possibility of instability depends on the initial correlations between q and co/O. If

co o = O o, the stationary mass loss takes place to M 1 ~ 0.1q).l 5 M o only in the case qo > 2.

The dependence of the critical values co/~ on the q0 is shown in Figure 6. In the dashed region the stationary mass loss is absent.

1 .5

1 . 0

0 .5

Fig. 6.

0 I 2 3 4 q

The critical value of ~/Q as a function of the mass ratio q. The dashed region corresponds to the instability of mass loss process.

104 1'. R. A M N U E L A N D O. H. GUSE1NOV

3. The Evolution of Low Mass X-ray Binaries

Amnuel and Guseinov (1977) estimated the expected number of bright low mass X-ray binaries in the Galaxy and suggested that such systems arise after the evolution of Main Sequence binaries with the initial mass of the primary more than 12 M o and the initial mass ratio less than 0.1. If, as is shown above, the lifetime of a bright low mass X-ray source is (2-3) x 107 yr, the probability of a binary remaining after the Supernova

explosion is 2-5%. If the hypothesis of the origin of low mass X-ray binaries from systems with qo < 0.1

is correct, then one needs to explain that during the evolution of stars on the Main Sequence (i.e., before the Supernova explosion) all the mass lost by the primary is ejected from the system. Let us consider a system consisting of the Main Sequence stars of masses t2 and 0.6 M| If only 10% of mass lost by the primary is accreted by the secondary, then their mass increases to 1.5 Mo. Amnuel and Guseinov (1980) suggested observational data on the total ejection of mass lost by the primary. Analogous data were suggested by the Gr6ve (1980). One can see from formula (5), that in the case of

q < 0.1 the value of/~ is close to zero when e) < 0.6[2. In the majority of cases, the type B evolution is supposed to take place - a primary

begins to lose its mass only after the Main Sequence evolution is finished. During the evolution on the Main Sequence, the synchronism of rotation of the primary is established. However, when the star begins to expand, the synchronism of rotation and revolution breaks down. One concludes from the conservation of the angular momentum of the primary that, when the star expands and fills its Roche lobe, the angular velocity of rotation becomes smaller than the orbital one. It is, therefore, possible that at the moment when mass loss begins, the value of(~ is less than 0.60, and/~ ~ 0. If during the mass transfer process (before the helium core has been formed) the synchronism is not restored, then all the time the value of/~ is close to zero.

Let us propose the following qualitative picture of the possible evolution of a system initially consisting of Main Sequence stars with masses 12 and 0.6 Mo and separation

10 ~ cm. When the more massive primary fills its Roche lobe, the mass loss begins (fl ~ 0) and the system contracts. When the mass loss ends and the helium core forms, the separation of the system may be < 2 x 10 ~ ~ cm. Then the primary explodes (the red dwarf is still on the Main Sequence). In 95-98% of supernova explosions in such systems, the binaries disrupt; in the rest of the cases, the system containing a red dwarf and a neutron star is formed. The eccentricity of that binary is large. During some 107 yr the value of eccentricity decreases to zero. Then, during ,-~ 109 yr, the separation of the system decreases due to the gravitation radiation, and the red-dwarf component fills its own Roche lobe. Two ways of foli0wing the evolution are possible, depending on the

mass ratio ~q at the moment when the mass loss began. (i) ~/o < 2. During (1-5) x 107 yr the mass loss increases, and the process leads to

instability. The mass loss will take place on a dynamical time-scale; and the red dwarf will be completely disrupted. Around the neutron star the gaseous cloud forms. The process of the accretion of this cloud by the neutron star lasts about 10 vyr; the


luminosity of the X-ray source is close to the Eddington one; and X-ray bursts are likely

t o o c c u r .

(ii) qo > 2. During (1-3) x 107 yr the accretion rate is greater than 10- 9 Mo y r - 1; the luminosity of the X-ray source is greater than 1037 ergs s-1. Then the mass loss (and accretion) rate decreases; during the next (1-7) x 108 yr the luminosity of the X-ray source is less than some 1036 ergs s- 1. When the mass of the red dwarf decreases to O. 1-0.15 M o, the value ofn in the radius-mass dependence begins to decrease; the mass loss rate rapidly increases; and the process turns into a dynamical one. The dwarf may be disrupted, with the formation of a gaseous cloud around the neutron star. At this stage an X-ray bright source arises again with a lifetime of (3-6) • 106 yr.

The models calculated above concern the cataclysmic variables also (systems with a red and a white dwarf).

For qo < 2, binaries with orbital periods of less than ~2.3 hr cease to form. When qo > 2, such a limit is ~ 1.0 hr. One predicts the absence of bright X-ray sources (and cataclysmic variables) within the interval of periods 1.2-2.3 hr due to low-mass loss rate (and accretion). Nova flares are also possible at this stage, but the discovery of such systems between the flares is difficult because of their low luminosities. In nature the 'forbidden' interval lies between 2 and 3 hr (Table I). The qualitative difference with our calculated interval one explains by the inexactitude of our models.

Note that Tutukov et al. (1982) calculated the models of red dwarfs losing mass, to determine the dependence of the value of n in the radius-mass relation on the mass loss rate. The calculations showed that the value of n rapidly decreases when mass loss rate increases. We re-calculated our models 6 and 7 taking into consideration the n - A~/ relation in the form suggested by Tutukov et al. (1982). In both cases the red dwarf reaches instability during ~ 107 yr; and the existence of a 'forbidden' interval of periods and minimdm value of periods --~ 80 min remains unexplained. Possibly the value of n used by Rappaport et al. (1982) is more correct.

4. Conclusions

We calculated 8 models of the evolution of close binary systems containing a neutron star (white dwarf) and red dwarf of spectral class later than K0. The mass loss by the red dwarf is induced by the system's contraction due to gravitation radiation. In our calculation we take into consideration that

(i) mass loss from the system exists;

(ii) asynchronism of red dwarf rotation with the system's revolution exists. The calculated models explain the existence of bright X-ray sources in the galactic

bulge, X-ray bursters and their observed peculiarities: high X-ray luminosities and the lack of eclipses. The existence of the 'forbidden' interval in orbital periods of cataclysmic variables and low mass X-ray binaries is also explained. The sufficient factor, which determines the character of the evolution of such systems, is the mass ratio qo in the moment when the red dwarf fills its Roche lobe and mass loss begins. Ifqo < 2, the stability of the mass loss process disappears when red dwarf loses only ~ 20% of its

106 P.R. AMNIJEL AND O. H. GUSEINOV

mass. If qo > 2, there are two stages of bright X-ray source radiation (stages of high mass loss) and one prolonged stage of low X-ray luminosity (low mass loss).

Future observations of bulge X-ray sources, bursters and cataclysmic variables (for example, observations of the rotation of red dwarfs in such systems) are needed for the examination of our assumptions and models.

References

Amnuel, P. R. and Guseinov, O. H.: 1977, Astron. Astrophys.,54, 23. Amnuel, P. R. and Guseinov, O. H.: 1980, Astrophys. Space Sci. 68, 315. Faulkner,.J.: 1971, Astrophys. J. 170, L99. Gorbatsky, V. G.: 1974, Novalike and Nova Stars, Moscow. Greeve, de J.: 1980, Astrophys. Space Sci. 72, 411. Guseinov, O. H. and Kasumov, F. K.: 1972, Astrofizika 8, 425. Kieboom, F. and Verbunt, F.: 1981, Astron. Astrophys. 95, Ll l . Paczynski, B. and Sienkiewicz, R.: 1981, Astrophys. J. 248, L27. Papaloizou, D. and Pringle, J.: 1981, Monthly Notices Roy. Astron. Soc. 195, 743. Paradijs, van D., Pedersen, H., and Lewin, W.: 1981, IAU Circ., No. 3628. Pratt, G. and Strittmatter, P.: 1976, Astrophys. J. 204, L29. Pringle, J.: 1974, Monthly Notices Roy. Astron. Soe. 168, 13 p. Rappaport, S., Joss, P., and Webbink, R.: 1982, Astrophys. J. (in press). Tutukov, A. V. and Yungelson, U R.: 1979, Acta Astron. 29, 665. Tutukov, A. V., Fyodorova, A. L., and Yungelson, L. R.: 1982, Pis'ma v Astron. Zhurn. (in press). Verbunt, F. and Zwaan, S.: 1981, Astron. Astrophys. 100, L7. Walter, F., White, N., and Swank, J.: i981, IAU Circ., No. 3611. Zahn, J.-P.: 1977, Astron. Astrophys. 57, 383.

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On the evolution of low mas binary systems