On the physical nature of bursters. I

LITERATURE CITED

i. G. V. Abramyan, V. A. Lipovetskii, and Dzh. A. Stepanyan, Astrofizika, 32, 29 (1990). 2. G. V. Abramyan, V. A. Lipovetskii, A. M. Mikaelyan, and Dzh. A. Stepanyan, Astrofizika,

B-3, 213 (1990). 3. G. V. Abramyan, V. A. Lipovetskii, A. M. Mikaelyan, and Dzh. A. Stepanyan, Astrofizika,

33, 345 (1990). 4. P. G. Hayman, C. Hazard, and N. Sanitt, Mon. Not. R. Astron. Soc., 189, 853 (1979). 5. B. E. Markaryan, Astrofizika, ~, 55 (1967). 6. R. F. Green, M. Schmidt, and J. Fiebert, Astrophys. J. Suppl. Ser., 61, 305 (1986). 7. J. Berger and A.-M. Fringant, Astron. Astrophys. Suppl. Ser., 28, 123 (1977). 8. P. Pesch and N. Sanduleak, Astrophys. J. Suppl. Ser., 70, 163 (1989). 9. H. L. Giclas, R. Burnham (Jr.), and N. G. Thomas, Lowell Proper Motion Survey, Lowell

Obs. Bull., No. 166, 8, 6, 157 (1980). i0. J. L. Greenstein, Astrophys. J., 158, 281 (1969).

ON THE PHYSICAL NATURE OF BURSTERS. I

G. S. Saakyan, G. P. Alodzhants, and A. V. Sarkisyan

The phenomenon of burster flashes is examined. In a compact binary system consisting of a neutron star and an ordinary star, a characteristic quan- tity is an accretion rate h E z 1.3"i0 Is g/sec, which corresponds to the Eddington luminosity limit L E. If the accretion rate is M < ME, then there is a source of soft x rays that in the presence of a strong magnetic field is manifested as an x-ray pulsar. Bursters are objects with M ~ ME and apparently comparatively weak magnetic field. Finally, in objects with relativistic jets (for example, SS 433) M > ME" It is shown that in bursters the temperature of the neutron star is practically the same from the center to its surface, where the accretion flow is stopped; the temperature is T z 2.107 . The fluctuations of the temperature about this mean value are of order AT z 10-3T. It is shown that the bursts are due to the thermal state of the hadronic core, which is a huge reservoir of thermal energy of the neutron star. It is small fluctuations of the thermal energy of this reservoir that are the cause of the regular bursts. Before a burst, the thermal energy of the hadronic core reaches its greatest value, at which the luminosity L becomes slightly greater than L E. During a burst, the accretion to the neutron star is halted, and the losses increase. The hadronic core expends a small fraction of its thermal energy mechanically in ejecting mass and much less on radiation. The burst stops, and the accretion recommences once L becomes less than L E. After this, until a certain time t = ~i ([i is the time the burster is quiescent), some of the accretion energy released on the surface is transmitted into the star, heating its hadronic core, but with the passage of time the energy flux to the hadronic core decreases, and when L reaches the value L E there is a new burst, and the history is repeated. In bursters, neutrino energy losses do not play a significant part.

i. Introduction

Neutron stars are of exceptional scientific interest. This is not only because they are stellar configurations of nuclear matter and possess powerful magnetic fields but also, to no lesser degree, because of their unique astrophysical manifestations. Thus, isolated neutron stars that rotate sufficiently rapidly are manifested as pulsars, radiating a directed pulsating flux of radio radiation. A second important type of closely studied objects are binary systems in which one of the components is a neutron star. A characteristic feature of these objects is the continuous flow of mass from the diffuse component (an ordinary star), which has filled its Roche lobe, to the

Erevan State University. Translated from Astrofizika, Vol. 34, No. i, pp. 21-40, January-February, 1991. Original article submitted December 24, 1990.

0571-7256/91/3401-0015512.50 �9 1992 Plenum Publishing Corporation 15

superdense component, i.e., the accretion of matter to the neutron star. When the accretion flow is stopped, it heats a thin surface layer of the neutron star, and then the released thermal energy is radiated in the form of black body x rays. Depending on the accretion rate, the orientation of the axis of the magnetic dipole relative to the rota- tion axis of the neutron star, and the magnetic induction, a close binary system is manifested as an x-ray pulsar, a burster, an object with relativistic jets (for example, SS 433), or simply as a point source of x rays. During the last two decades, many papers devoted to the problem of accreting neutron stars have been published. It is not possible for us to give a complete list of them here. We merely mention that references to many of them can be found in [1--3].

We are interested here in the problem of bursters, which have been studied theo- retically in the papers [2--12]. These have proposed an interesting mechanism of genera- tion of the x-ray bursts in accreting neutron stars, and a corresponding theory has been developed. It is assumed that during the accretion of matter, in the interval between bursts, helium, which is present in the accretion flow and may in part also be formed by thermonuclear synthesis of hydrogen in the upper layer of the neutron star, accumulates at a depth at which p z 108 g/cm ~ (helium nuclei cannot exist at higher densities, since they rapidly disappear in pycnonuclear reactions). It is assumed, further, that when the number density of the helium nuclei reaches a certain critical value they undergo sudden instantaneous synthesis in so-called triple s-particle collisions [13] at a temperature T z 5"10 ~ (it is assumed that this high temperature can be achieved by thermonuclear synthesis of hydrogen in the same region). A corresponding amount of energy, as is observed in the bursts, is then released explosively. However, we believe it is most unlikely that the extremal conditions necessary for explosive burning of helium in triple collisions of its nuclei can arise on the surface of a neutron star. In our view, neither the high temperature nor the accumulation of helium is possible. It is more probable that the accretion flow, after it has been stopped on the surface, continues to penetrate slowly into the star. Of course, some of the hydrogen undergoes thermonuclear synthesis in the process, but we can expect the bulk of it to reach the layer with p(r 2) z 3.10 7 g / c m s , where the protons decay into neutrons, which are immediately absorbed by nuclei. With regard to the helium (including the part of it that is formed in the thermonuclear synthesis of hydrogen), it will disappear in pycnonuclear synthesis reactions when it penetrates below the surface at which p(r z) z 10 8 g/cm 3 [14]. It is our view that the origin of the bursts must be sought in the thermal state of the neutron star. The overwhelming bulk of the thermal energy of the star is accumulated in its hadronic core, which is a thermal reservoir that controls the state of the star. We shall show below that the bursts are responses to small changes of the temperature that occur in the hadronic core during accretion.

2. Basic Equations

The equations that determine the internal structure, parameters, and thermal state of a neutron star in a quasisteady accretion regime are [15, 16]

clP

.dr G (pc ~ 4- P ) / 4= pr3)

du - - = 4 ~pr 2, dr

dv 9 dP - - . y

dr pc~ + P dr

" d T \ .c}(Le') __ 4 = r e / �9

O ( Te','h

Or

We here use the Schwarzschild metric

3 Xpo Le ~1~ 64~a T3r 2 (1 -- 2 Guitar) '12

(i)

(2)

(3)

(4)

(5)

16

dr 2 d s ~ = e~ c g d t 2 r 2 (d 02 -6 sin a 0 dq~2),

1 - - 2 Gu/c2r

P is the pressure, pc 2 is the density of the total energy, including the rest energy

of the particles, the kinetic energy, and the nuclear and gravitational interaction en- ergies; p0(r) = Zmknk(r) is the rest-mass density, n k is the proper density of the particles

~n~ dVp = Nk, = dV(1 -- 9 Gu/c~r) - ,1~, dVp

u(r) is, with a certain reservation, the accumulated mass, c v is the proper specific heat of unit volume (~v = cn, c is the specific heat per particle), p0gs is the proper energy density of the source (the energy released in unit volume during unit proper time),

is the total energy released in unit time, p0gv is the proper power of the neutrino energy losses:

i s t h e t o t a l e n e r g y o f t h e n e u t r i n o l o s s e s d u r i n g 1 see ( t h e n e u t r i n o l u m i n o s i t y o f t h e s t a r ) , • i s t h e o p a c i t y c o e f f i c i e n t , and , f i n a l l y , �9 i s t h e p r o p e r t i m e .

The b o u n d a r y and i n i t i a l c o n d i t i o n s f o r Eqs. ( 1 ) - - ( 5 ) a r e

r=O, P ( O ) ~ P ~ , u ( 0 ) = O , ~ ( R ) = l n ( 1 - - r / R ) , T(O, 0), L(O,O)-=O, (6)

where R i s t h e r a d i u s o f t h e s t a r . The v a r i a b l e s P, u , u depends o n l y on t h e d i s t a n c e r f o r t h e c e n t e r , and T and L depend on t h e t ime ~ as w e l l as on r .

I n c e r t a i n e x t r e m a l c a s e s i t i s a l s o n e c e s s a r y t o t a k e i n t o a c c o u n t in Eq. (1 ) t h e contribution of radiation pressure on the very surface of the star. However, whatever the thermal state of the neutron star, the first three equations are essentially not entangled with the last two, and they can therefore be integrated separately. In other words, the mass distribution in the star and, therefore, the mass, radius, and moment of inertia of the neutron star are not significantly affected by the temperature. In contrast, Eqs. (4) and (5) are essentially related to (1)--(3), and they can be integrated only simultaneously with them.

3. Material Parameters

It is our view that on the transition from the Ae shell (where the plasma consists of nuclei and a degenerate gas of electrons) of the neutron star to its hadronic core the mass density changes from 5.8"i0 II g/cm 3 to 2.85-1014 g/cm 3, i.e., at the interface there is a jump by about 500 times at unchanged pressure P = 6.4"1029 erg/cm 3. The comparatively thin shell of Ae plasma is a kind of atmosphere for an hadronic ball, in which almost all the mass of the star is contained. For stable neutron configurations, there are no quarks at the center: At central pressures 6.5"1029 < P(0) ~ 6"1033 erg/cm 3, the hadronic core consists solely of nuclear matter (nucleons, ~- mesons, and a comparatively small admixture of electrons); for 6"1033 ~ P(0) ~ 4"103s erg/cm 3 in the central region, the core consists of hadronic matter (nucleons, hyperons, resonance particles, 7- mesons, and a small admixture of electrons) and in the outer layer it consists of nuclear matter. By nuclear matter we here understand a degenerate plasma at density p = 2.8"10 I~ g/cm 3. It is characterized by the possession of a high degree of incom- pressibility (liquid phase at P ~ 6"1033 g/cm 3) and consists of a mixture of neutrons, protons, and v- mesons with approximately equal concentrations with a small admixture of electrons:

g = 0 . 5 9 l , gp = 0.409, y= = 0~4, y, = 0.0035,

where Yk = nk/n, n = n n +np = 1.7"10 s8 cm -3 is the number density of the nucleons [17, 18, 19]. Thus, in the degenerate plasma of neutron stars a situation in which the number of neutrons is predominant does not exist at any densities. In this connection,

17

it is appropriate to recall that the name of these objects is nominal and does not correspond to the real picture that exists in them. Below, we shall use an equation of state and values of the material parameters that correspond to the above description. We note, however, that in many studies on the physics of neutron stars the transition from the Ae phase to the nuclear-matter phase is assumed to occur smoothly, without any jump, and that the nuclear matter consists predominantly of neutrons. However, this difference between the descriptions of the superdense degenerate plasma does not have too strong an influence on the values of the integrated parameters of the neutron configurations.

For the power of the neutrino energy losses of the nuclear matter we use, taking into account the part played by the pion condensate, the formula

~ = I0-~ T 8 erg.g-l.sec -I,

which was obtained in [20]. For the well-known reason, the contribution of the Ae shell to the neutrino luminosity of the star is very small.

The power of the energy source due to the accretion of matter onto the neutrino star can be described by the convenient formula

4xpo (~) R' I f1 -- re/R

( 7 )

-- Mbt ],/1 - - re/r, XHMb. V1 rg/r, ~(r--r.)+ . . . . . . . . ~ ( r - - r , ) . (8) 4=m,P.(r,)d 4 ~ , Po(~,) q

H e r e , rg i s t h e g r a v i t a t i o n a l r a d i u s o f t h e s t a r , and M = dM/dm i s t h e a c c r e t i o n r a t e . This expression corresponds to the discussion in [14]. The first term in (8) is the kinetic energy of the accreting matter released when it is stopped on the surface of the star, where the density is of the order of a few grams per cubic centimeter [21]. The second term is the energy released by the decay of the protons with subsequent absorption of the neutrons by nuclei below the surface r = r 2 where the density p(r 2) = 3/2" 107 g/cm 3, b 2 = 7.13 MeV, and X H is the hydrogen mass concentration in the accretion flow. It is assumed below that the flow consists largely of hydrogen and helium with mass concentrations X H = 0.64 and XHe = 0.34, the fraction of the remaining elements being X z = 0.02. The final term is the energy released in the pycnonuclear reactions below the surface r = r I, where the density p(r I) z l0 s g/cm 3, b I = 1.63 MeV. In (8) the ~ func- tions are introduced for convenience.

The specific heat of unit volume of a degenerate gas of particles with spin 1/2 is

cm~ka x (1 ~ x:)l:2T, (9) C v 3~ ~

where m is the mass of the particles, and x is the limiting Fermi momentum in units of mc. Using this expression, we can estimate the specific heat of the hadronic core. We note first that in it the contribution of the electrons is negligibly small, since their number density is less than the baryon number density by several orders of magnitude. Also unimportant is the part played by the ~- meson condensate, since exceptionally high temperatures are required for its thermal excitation. Thus, the thermal energy of the hadronic core is basically determined by the baryons. The hadronic plasma has a very complicated chemical composition. Therefore, in practice we can only make estimates. To this end, we shall regard the hadronic plasma as a baryonic gas consisting of particles with mean mass m z 2.3.10 -24 g, which is a reasonable approximation for the problems considered here. We can now use the expression (9) to estimate the specific heat of such matter. Bearing in mind that here x ~ i, we find

Ir 1/3 k 2 m 2/3 1~3 c b ~ - - ~ 2 n T ~ --ta ~ p, T ~ 3 . 1 0 6 Tp'0/~, (10)

where n is the baryon number density.

18

Practically all the mass of the star and, therefore, the thermal energy are included in its hadronic core, and therefore the small contribution of the Ae shell to the thermal energy can be ignored.

We now consider the opacity of the hadronic core. For a strongly degenerate plasma, the dominant energy transport mechanism is heat conduction. The opacity corresponding to it is

X- 16= T~ , (Ii)

3 [,~),

where I is the thermal conductivity, which we estimate by means of the elementary formula of the kinetic theory of gases:

k~ c~ v ~ , (12) 3~

where c v is the specific heat (of unit volume) for the given particle species, ~ is the mean velocity of their random motion, and v is the number of collisions of a test particle in unit time. In the estimate, it must be borne in mind that due to the degeneracy the collisions with the test particle basically involve only baryons (the electron number is relatively small) with energy in a strip of width Agb ~ kT near the Fermi surface. The number of such particles is of order

3 Aeb 3 m k T . 1/3 ~ n - ~ - n ~ - , ( 1 3 )

2 % (3 =~)~/3 ~

where E b = (3~2)2/3N=n2/S/2m.

For the number of collisions of an electron with the baryons, we have

_ A e 4 a e An A ce ~ /Xn �9 , ~ . ~ - . ~ - (14)

I n t h e h a d r o n i c p l a s m a , t h e e l e c t r o n s a r e u l t r a r e l a t i v i s t i c ; A i s t h e Coulomb l o g a r i t h m . T a k i n g i n t o a c c o u n t ( 1 2 ) - - ( 1 4 ) , we f i n d f r o m ( 1 1 ) f o r t h e e l e c t r o n o p a c i t y i n t h e h a d r o n i c core

96 A-~ ltrn";3 e 4 T 3 ~ 3 . t 0 _ 3 0 T3/p~,3 (15) X~ ~ (3 ~)2/3 kra4 c G x 4, ~ o'~"'3

Here, x e = Pe/meC z I00, and the Coulomb logarithm for the ultrarelativistic electron gas is A z 0.5 in(~cil/4e 2) z 2.5.

The number of collisions of a baryon with baryons is of order

~.~ ~ r ~ , ~ , (16)

where v b z (3~2)I/Shnl/S/m, and r 0 = 1.12"10 -13 is the range of the nuclear forces. Using (16), (13), and (12) we find from (ii) that the opacity determined by the baryons is

32 ~ r0: m */3 T ' T 3 lb ~ = 4.2 .10 -a9 (17)

3 ~k~ p~'~ p~/3 "

The ratio of the electron and baryon opacities is approximately

~/l b ~ 7.10-'np~2I%

For stable neutron configurations, the density within the hadrQnic core varies in the range 3"10 l~ ~ p ~ 2"1015 g/cm 3 (P0 ~ P); in this interval Xe << Xb, and therefore the opacity of the hadronic plasma is determined by the electron, i.e., by the expression (15).

We give the expressions for the opacities for the degenerate and nondegenerate regions of the neutron star. In the region of the hadronic core, the opacity is determined by (15). In the Ae shell

19

64=ae'hs . ( - - XiZ](ln26.3Zi)~ l + x J _ 10_.2o X,Z~ l n 2 6 . 3 Z t ) . _ _ T , 6 72, (18) X=n,arn;c~k.~\ ~ ~- ' X - - T T" ~-~- 4.45 - ( ~ Al

where X i is the mass concentration of the nucleus with parameters Zi, A i. If it is assumed that the Ae plasma consists on the average of iron nuclei, Z = 26, A = 56, then

2 X ~ 3 . 5 , 1 0 -Is T 2 ~ 1 4- x e

x 6 e

The expression (18) is taken from [23, 24]; it can be obtained directly in accordance with the simple scheme given above using (ii) and (12) and the corresponding expression for the number of collisions v. In the outer nondegenerate shell of the star [13]

P X1! ~ 3.68- 10 :~" (X. + Xne) (1 +Xn) -~--~s' 1-'-

( 1 9 )

Xb., -~ 4.34.10 ''~ X, (1 + XH) ;a.5 (20)

Finally, in the region in which p < i00 g/cm 3 the opacity is determined by Thomson scattering:

Xe = 0.19 (1 + X~. (21)

The approximate interface between the degenerate and nondegenerate regions is determined by the relation

P ( n ) ~ 2 2 " 1 0 s ~t~,

(we r e t a i n t h e n o t a t i o n adop t ed in [ 2 1 ] ) . The mass o f t h e r e g i o n r > r 4 i s of o r d e r

a,'d, ~ 10 '5 T~ILa~.

For a n e u t r o n s t a r i n a reg ime of c o n t i n u o u s a c c r e t i o n , t h e chemica l c o m p o s i t i o n o f t h e region r > r 4 will be similar to that of the matter that falls onto the surface of the star.

4. Thermal State of the Neutron Star

We choose a definite configuration of the neutron star [25] in order to make a more definite discussion of the problem:

P(0 )=S .96 .10a~e rg / cm3 , R 0=11 .19km, R = 1 1 . 7 3 km,

M=2.162"lOaag, AM=3.24 .10-SMo, rg=3 .206km,

where R 0 is the radius of the hadronic core, and AM is the mass of the Ae shell. To the given value of the central pressure there corresponds the mass density p(0) = 4.65 "~ 1014 g/cm 3 .

The next important parameter of the burster is its Eddington luminosity limit, which for the considered configuration is

(22)

4r~GMc Le = ~ 2 . 0 4 . 1 0 as e r g / s e c . (23) ze i / 1 - - ,.g/&

Here, Xe = 0.19(1 + X H) = 0.312 is the opacity for Thomson scattering under the assumption that on the surface of the star the mass fraction of hydrogen is 0.64. For a quasisteady regime and under the assumption that the neutrino luminosity is relatively small (this is indeed the case) the accretion rate corresponding to the Eddington limit for a star with the parameters (22) is

�9 4~GM ME

CXe V 1 -- rglR = 1.31-10 '8 g/sec.

Here, we have not taken into account the contribution of nuclear reactions taking place in the accreting matter at r z r 2 and r z rl; allowance for it leads to a small

20

correction to ME: ME = 1.~62-10 ~8 g / s e c . (24)

We note that for neutron stars the accretion rate corresponding to the Eddington luminosity limit is almost independent of the mass. We can see this by considering the non- relativistic approximation

4 r GM,(4e /.~ R

Hence

Me ~ 4 r.cR e~

- - -~- 1.2.10 TM Re, g / s e c .

Neutron stars have approximately the same radius, and therefore their limiting accretion rates differ little.

In the case of a constant accretion rate satisfying M < M0, the neutron star will undoubtedly be in a steady thermal state and, depending on the magnetic field, will be manifested either as an x-ray pulsar (in the case of a strong magnetic field) or as a compact x-ray source (when the magnetic field is relatively weak). We assume that bursters are objects with M ~ M0. Then in the neutron star there is established a quasisteady thermal state in which long quiescent periods (~ ~ i04--i0 ~ sec) of slow heating are interrupted by short bursts accompanied by the ejection of a certain amount of matter and radiative energy, when the luminosity, reaching the Eddington limit, just exceeds it.

For what follows, it is important to understand the obvious fact that during the short time �9 z I0 sec of the burst the star loses only a very small fraction of its thermal energy, and therefore its state and, accordingly, temperature profile do not change significantly. However, these small changes do lead to important consequences. This is because the accretion rate is close to its critical value ME, so that the star is in a metastable state. Indeed, as soon as the luminosity exceeds L E by a small amount, the radiation force on the surface exceeds the gravitational force, with the consequence that matter escapes from the star; but when the luminosity sinks just below LE, the gravitational force is again the stronger, and the accretion process immediately recommences. Since the internal energy of the star decreases when the outburst occurs, during this short interval of time T(r, ~) < 0. In the interval of time between two bursts, the star is in the accretion regime; it receives energy, but the bulk of this energy is radiated from the surface, and only a small fraction of it enters the star to heat the hadronic core. The star slowly accumulates the small amount of energy that is lost during the burst, and therefore during the quiescent period T(r, T) > 0. As the time passes, T decreases, since the star tends to a stationary state, but this process is interrupted by the next burst. The qualitative picture of the variation of T with time is shown in Fig. i.

i

C

A

'I0 t(sec)

10"

Fig. i. Curve of T as function of the time. The scales among the axes are arbitrary.

21

Thus, during the time between the bursts (between the points C and D on the curve of T, Fig. i), the temperature of the star slowly decreases and the additional thermal energy needed for the next burst is accumulated. In this part of the smooth increase in the temperature of the hadronic core of the star, a good approximation is

T(r, =)= r(r , %)] 1--~ 1-- , (25)

where �9 = t-exp(v/2) is the proper time measured from the start of the quiescent period of the burster, and ~ and ~0 are the parameters that determine the slow growth of the temperature in the interval between bursts. Essentially, ~0 is the time required by the star to reach a steady thermal state, and e = [T(r, ~0) -- T(r, 0)]/T(r, ~0) ~ i, as we shall soon show. For cooling isolated neutron stars, the temperature falls as we go from the center to the surface, but so slowly that it almost remains constant right up to the layer at which the degeneracy is lifted. A significant fall in the temperature occurs in the thin outer layer of the Ae shell where the plasma is nondegenerate. In the case of a neutron star subject to continuous accretion, this picture becomes more complicated. The star remains isothermal (we mean by this constancy of the product T'e v]2) right up to the surface at which the falling matter is stopped, so that in (25) the dependence of the temperature on r has a symbolic nature. The rate of change of the temperature of the star during the quiescent period is

Taking into account (7), equation (4) in the form

Y(r. ")=2~(1--~) T(~, %). % \

( 8 ) , ( 1 0 ) , and ( 2 6 ) , we r e p r e s e n t t h e e n e r g y t r a n s p o r t

--=d(l"e') Mc2( 1 ~ ------ . - -: . e + ~ ( , ' - - r ~ -~ e" g ( r - - r t l - - d r \ Ii/1 r~'/-R 1 / e ' ~ ( r - R) + XHMb+mp Mb,mp

~.'4~p~/~T2(, ", *.)e'r' ( ~ ) 10 -634~p~ ~)e+ 6.10 ~ �9 1- - �9

(26)

(27)

Here we have used the relations

We integrate Eq.

(v)l" T']'= 2--~ 1-- 1--~ i - - 7'~(r, Top~. 1-- T2(r,:). "% "~0

(27) from the center to the surface:

I +

~~ 6 .10~ -- a 10 -63 ~ 4 ~ P o r ~ 7 S ( r , z ) e ' d r __ , 4 ~ I~r~.e~ T~(r, ~) dr (28)

] --rg]R ~ V1 -- ~ Guitar ( 1 - - rg/R)r 1/1 -- 2 Guitar 0 o

where at the upper limit of the integral we have replaced the radius of the star by the radius R 0 of its hadronic core; this results in a totally negligible error, since the mass of the Ae shell is several orders of magnitude less than the mass of the hadronic

core.

For what follows, it is necessary to have a correct picture of the temperature profile of the neutron star. Bursters are objects with luminosities close to the Eddington limit (23), and therefore for them the temperature on the surface of the star is

Lr )'~= 2.14.107 . (29) T~= 4==R =

This number corresponds to the configuration with the parameters (22). The surface temperature is approximately the same for other burster neutron stars. Because of the high

22

thermal conductivity of the degenerate matter (under conditions of quasisteady accretion) the temperature will, apart from the relativistic factor exp(v/2), be the same in the

hadronic core. Thus, for accretion rates corresponding to the Eddington limit, the characteristic temperature of a neutron star with mass M~M O is approximately equal to 2"107 At such temperatures the neutrino luminosity of the star is many orders of magnitude less its photoluminosity:

L, ~ , 10 - ~ MT s , ~ 5-10 's M/M|

As we shall see, in the Ae shell of the neutron star the temperature is significantly below the value required for the reactions involving triple collisions of ~ particles.

(30)

5. Mechanism of the Bursts

We consider formula (28) for the luminosity for the model of a neutron star with the parameters (22). We have

( Z) L~s = 1.617M,8-- 1.785.10 ~ I - - ~ J , -7 .6 .10 J,, "%

( 3 1 )

where we have introduced the notation

10,o 4~p~/~r~e" T'(r ,~)dr , j _ 0.01 4~Por2e ~ T*(r,~)dr

J' = - - M - ~ ~ a . . M V I - 2 6ul~' % V ' l o C2r

I t i s e a s y t o show t h a t Jx ~ 2T~ z 10, J 2 z T~ z 3. N o t e t h a t Te ~/2 z c o n s t , and t h e r e f o r e the corresponding combinations of this expression can be taken in front of the integrals. Below, we shall omit in (31) the term representing the neutrino luminosity, this being an entirely justified approximation for the accretion rates considered here. Thus,

L3s ~ 1.617 t~18 - - 1.785-106 ~-- (1 - - - -~0 ) J~. ( 3 2 ) ~o

With the passage of time, the luminosity of the star increases to reach at z = z 0 a steady regime.

Essentially, the philosophy of the operation of a burster is contained in Eq. (32). Let us consider the situation in which the following conditions are satisfied:

1.617/14>10-~~ i . e . , M > M E , ( 3 3 )

1.617 M - 1.785.10 ~-' ~-- JL < 10-2~ �9 %

They are necessary and sufficient conditions for the accreting neutron star to operate as a burster. The second condition means that at the time T = 0 (end of a burst, be- ginning of the quiescent period) the luminosity of the neutron star must be below its Eddington limit :

L .~ 1.617- 1020 M - - 1.785.1044 ~ J1 < Le . '~o

This is the lowest luminosity of the neutron star at accretion rates compatible with the conditions (33) and (34).

In accordance with (26), from the time �9 = 0 the rate of change of the temperature is positive, T > 0, and this means that in the quiescent period there is an energy flow from the surface to the hadronic core, as a result of which the temperature of the star slowly rises. As can be seen from (32), the luminosity increases with the time, and if nothing happens, then at T = T O it reaches its largest value L38 z 1.1617~z8 , which is above the Eddington limit. But this does not occur, since as soon as L reaches L E and begins to exceed it the balance of the forces on the surface of the star is changed, and the burst takes place. During the burst, the hadronic core loses a comparatively small proportion of its energy, and in the quiescent period it acquires energy, creating the condition for the next burst. Thus, it is all very simple -- the huge reservoir of

(34)

23

thermal energy naturally regulates the chronology of the bursts.

The time of a burst is determined by the condition L = LE, i.e.,

1.617 M,s - 1.785-106 ! ( 1 - ~_t] j , ~ 2.04. (35) ~0 \ % /

This equation determines the time ~ between two successive bursts; of course, ~ < T 0. During the time A~ of the burst, an absorbing layer of matter is blown above the surface, and as a result the luminosity increases somewhat, this leading to a certain decrease in the internal energy of the star (the hadronic core). However, during this short period of time much more energy is lost in the form of work required to eject mass from the surface. The burst stops once the temperature of the star has been reduced sufficiently for the luminosity to again be below the Eddington limit. Then the quiescent period commences, during which the hadronic core replenishes the lost internal energy. From (35) we obtain

(1--J_.L~ ~ = 9 . 0 6 . 1 ~ - ~ (M- ME).- ( 3 6 ) ~0 k % /

As M + ME, ~I § ~0, i.e., the energy flux from the surface to the hadronic core stops and the star enters a stationary thermal regime; however, this does not happen, since before this the burst commences. Bearing in mind that Jt ~ i0, ~0 ~ 105, M -- ME ~ i018, and 1 -- Tz/T 0 z i, we see that a = AT/T ~ i0 -3, where AT is the maximal change in the temperature.

6. Energy Balance Between Bursts

The thermal energy of the hadronic core is

r Ro L

Q= dvp e 'c . dT-: 1.5.10 6J ?1__iGu/c..,r ' 0 0

where we have used (i0) for the specific heat, and for the well-known reason the contribution of the Ae shell is not taken into account. The energy Q is approximately equal to 3"i043j ~ 3"1044 erg. During the interval of time between two neighboring bursts i

the thermal energy of the neutron star increases by the amount

R. ~ 4.~p~3[r'(r, ~,)-- T2 (r, o)]e * r*dr

A Q = 1.5 .10 6 o V 1 - - 2 ' d u / c 2 r

Taking into account (25), the smallness of the parameter a, and the relation (36), we find ~o~R:] ) 4~':~ T2 (r, % ) e ' r ~ dr 1--~.d2% ~1[ ~1 ro

AQ = 6. I0 ~ T o t \1 - = 1.2-10as(Ma8 - 1.262) ca �9 ( 3 8 ) , V~l __ 2Gu /c i r 1 - - ~1/~o O

Here, we have noted that for the considered configuration ME = 1.262. The time ~i ~ 104--105 sec and the last factor is of order unity since ~0 > TI (we recall that M * ME as T ! § To), and therefore AQ ~ 2-10saT1, this exceeding by approximately three orders of magnitude the energy (E ~ i039 erg) which is released in the burst. At the same time, in accordance with the proposed burster mechanism, all the energy accumulated during the time T I must be expended during the burst. We noted above that it is mainly expended on the work required to eject a certain amount of stellar matter.

We estimate the mass m b that is ejected during the burst from the positions on the star at which L > L E. In order of magnitude

G M m e ---g--~ ~O, (39)

where T is the Lorentz factor for the ejected masses. Taking into account (38), we obtain

24

m B ~ 1 0 ' s f ( Mts-I'2621 1--='/2"~ -1--~t/~o " ~10'8 7z-i~ (Mt~--1"262)" (40)

This is a mass of order m B ~ 1022-102s g.

Hitherto, we have not taken into account the role of the magnetic field in the phenomenon. It is generally assumed that in bursters the magnetic field of the neutron star is relatively weak, since otherwise their radiation would undoubtedly have, in addi- tion to the bursts, a periodically interrupted nature, as occurs in x-ray pulsars. We attempt to take into account the possible influence of a magnetic field on the considered questions under the assumption that it is significantly weaker than the field of pulsars.

The magnetosphere of a neutron star consists of the region of the Alfv~n sphere, which is formed by closed lines of force of the magnetic field and a narrow channel of open lines of force. The last line of force that is "unperturbed" by the accretion flow and closes within the Alfvgn sphere leaves the surfaces of the magnetic polar caps at polar angle

/ R \':,

, i

where r A is the radius of the Alfv~n sphere,

1.35. lo7 ,4, M,~M} '

and ~ is the magnetic moment of the star. The open lines of force emanating from the magnetic polar caps form a solid angle

-, '2-_ 0.23 } ~'6. (41)

The accretion flow occurs mainly within this solid angle.

Part of the mass m B ejected during the burst of the neutron star can be expected to reach the region of closed lines of force surrounding the star. Between the star and the plasma in the region of closed lines of force surrounding it there must obviously exist a dynamical equilibrium, since the particles move freely along the force tubes resting on the surface of the star (under the influence of the radiation, gravitational, and centrifugal forces). The other part, equal to

is ejected along the open lines of force in the form of diametrically opposite jets. Here, measuring the magnetic moment in units I02s erg/G, we assume that the magnetic field of a bursting neutron star is approximately two orders of magnitude less than for a pulsar.

We now estimate the jet parameters. In accordance with (29), the temperature at the base of a column at which a jet commences is approximately equal to 2.14"107 . The initial velocity of the outward flow must obviously be of the order of the speed of sound,

vR ~ ( F P ~'/~= ( F k T )'/2 ~.-67 !07 cm/sec, (43) \ P / ~A mp

where F = 5/3 is the adiabatic exponent, and ~A = 0.65 is the mean molecular weight. The initial density is determined by the condition of conservation of the flux,

4~ R2pRvR A~--~-mB, (44)

where At is the duration of the burst. Hence, taking into account (40) and (43), we find

25

TABLE i. Temperature Profile of Accreting Neutron Star with the Parameters Given in ( 2 2 ) .

p (r) 7"7 (r) e "/~ T7 e "/2 R (g/cm 3 )

0.000 1.17 10 -5

0.102

0.275

0. 550

0.805

.0.922

0.973

0.991

0.999

l. 000

4.65.10 TM

4.65.1014

4.62-10 TM

4.49.1014

4.06.101'

3.52-10 t4

3.19.1014

3.91- lOre

1.49.109 2.77.10s

3.13

2.410

2.410

2.407

2.387

2.320

2.227

2.175

2.154

2.146

2.142

2.142 I

0.757

0.757

0.758

0.765

0.787

0.819

0.839

0.847

0.851

0.852

0.852

1.826

1.826

1.826

1.826

1.826

1.826

1.826

1.826

1.826

1.826

1.826

PR ~-~ 1 . 2 . 1 0 _ 3 ~, M~8 - 1 .262 mB t~, R~ o 4~R~ VR •

Assuming ~I/~AT z 500, we obtain DR z 1 g/cm s. From (45), we can conclude that

(45)

(46)

The burst duration At is approximately the same for all objects, and therefore it follows from (46) that the time ~ between bursts is approximately inversely proportional to the background luminosity of the burster.

7. Temperature Profile

As was noted above, from the very surface of the neutron star to its center, the temperature hardly changes when allowance is made for the red-shift factor. This con- clusion was confirmed by numerical integration of Eqs. (1)--(5) using (25) and (26). The result was found to be entirely insensitive to the values of the parameters ~, t0, and ~. As can be seen from the final column of Table i, the expression T exp(~/2) remains constant along the radius of the star to four significant figures. The result also does not depend strongly on the accretion rate. All this indicates that an accreting neutron star is in a quasisteady thermal state almost the entire time.

The changes that take place during the bursts and in the intervals between them are only small fluctuations on the general constant temperature background -- provided, of course, the accretion rate does not change significantly.

Let us determine the fluctuations of the temperature around the mean background. In accordance with (25), the change in the temperature of the star during the quiescent period is

a T = T( 'h) - - T(o)=--2a'~% ( 1 - - "~"o) T{'~

U s i n g ( 3 6 ) a n d b e a r i n g i n m i n d t h a t J 1 z 2T72, we f i n d

aT .~ 2.3.10_a ( , , ) 2 `2 - -

1 --,i/~0 '

For the assumed accretion rates M ~ 1.3, the ratio AT/T ~ 0.001. It is obvious that

(47)

26

the temperature also undergoes a similar relative change during a burst.

We thank the participants of the seminar of the Department of Theoretical Physics at the Erevan State University for discussions.

LITERATURE CITED

i. W. H. G. Lewin and P. C. Joss, Space Sci. Rev., 28, 3 (1981). 2. S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars (1983). 3. P. C. Joss and C. A. Rappaport, Ann. Rev. Astron. Astrophys., 22, 537 (1984). 4. S. E. Woosley and R. E. Tamm, Nature, 263, i01 (1975). 5. E. V. Ergma and A. V. Tutukov, Astron. Astrophys., 84, 123 (1980). 6. L. Maraschi and A. Cavaliere, Highlights Astron., 4, 127 (1978). 7. P. C Joss, Astrophys. Lett., 225, L 123 (1978). 8. P. C Joss, Nature, 270, 310 (1977). 9. R. E Tamm and R. E. Picklum, Astrophys. J., 233, 327 (1979).

i0. R. E Tamm, Astrophys. J., 241, 358 (1980). ii. R. E Tamm, Astrophys. Space Sci., 77, 257 (1981). 12. R. E Tamm, Astrophys. J., 247, 257 (1981). 13. M. Schwarzschild, Structure and Evolution of the Stars, Princeton (1958). 14. G. P. Alodzhants, L. Sh. Grigoryan, G. S. Saakyan, and A. V. Sarkisyan, Astrofizika,

29, 573 (1988). 15. G. S. Saakyan, Equilibrium Configurations of Degenerate Gas Masses [in Russian],

Nauka (1972). 16. K. S. Thorne, Astrophys. J., 21___22, 825 (1977). 17. L. Sh. Grigoryan and G. S. Saakyan, Fiz. Elem. Chastits At. Yadra, i_O0, 1075 (1979). 18. L. S. Grigorian and G. S. Sahakian, Astrophys. Space Sci., 95, 305 (1983). 19. L. Sh. Grigoryan and G. S. Saakyan, Astrofizika, 13, 669 (1977). 20. L. Sh. Grigoryan, Astrofizika, 17, 395 (1981). 21~ G. P. Alodzhants, L. Sh. Grigoryan, G. S. Saakyan, and A. V. Sarkisyan, Astrofizika,

30, 558 (1989). 22. G. P. Alodzhants and A. A. Saaryan, Astrofizika, 20, 571 (1984). 23. R. E. Marshak, Astrophys. J., 92, 321 (1940). 24. E. Schatzman, White Dwarfs, North-Holland, Amsterdam (1958). 25. L. Sh. Grigorian and G. S. Sahakian, Astrophys. Space Sci., 95, 305 (1983).

QUASARS, TYPE 1SEYFERT GALAXIES, AND OBSERVATIONAL SELECTION

R. A. Vardanyan

A study is made of the influence of the absolute magnitudes on the red- shift dependence of the colors of quasars and type 1Seyfert galaxies. The red-shift dependence of these colors is shown to be a consequence of observational selection. New evidence is given for a connection between type 1Seyferts and quasars, and it can be regarded as support for a cosmological nature of the red shift of these objects.

i. Introduction

Quasars and type 1Seyfert galaxies are among the most interesting and unusual objects in the universe.

To a large degree, they have similar optical properties. In particular, this is true of the variability of the brightness and the spectrum, in which, besides broad emis- sion lines, one sometimes finds absorption lines, the ultraviolet power, etc.

These objects have been comprehensively studied by many authors. This is already indicated by the book Quasi-Stellar Objects by G. Burbidge and M. Burbidge, which was

Byurakan Astrophysical Observatory. Translated from Astrofizika, Vol. 34, No. I, pp. 41-50, January-February, 1991. Original article submitted December 12, 1990; accepted for publication December 25, 1990.

0571-7256/91/3401-0027512.50 �9 1992 Plenum Publishing Corporation 27

Documents

On the physical nature of bursters. I