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On the Observational Appearances of a Freely Precessing Neutron Star in Hercules X-1 K.A. POSTNOV, M.E. PROKHOROV, AND N.I. SHAKURA Sternberg Astronomical Institute ABSTRACT This paper is concerned with evidence for neutron star free precession, which is often presumed to be responsible for the observed 35-day cycle in Hercules X-1. The precise formula for the period derivative due to free precession is obtained under the assumption that the precession period is much longer than that of the neutron star rotation. The optical light curves to be seen from the binary, with freely processing accreting neutron star are simulated numencally. This simulation takes into account the redection effect on the surface of the secondary component and on the accretion disk itself for different diagrams of X-ray emission. INTRODUCTION The eclipsing, pulsating binary X-ray source Hercules X-1 is known to be one of the most interesting accreting binary stars. Its properties are deduced from a wealth of X-ray and optical observations made during the past 17 years (see Ogelman and lluemper 1988, and references therein for a review of X-ray and Boynton 1978, for a review of optical data). One of the most enigmatic X-ray features of Her X-1 is the Away intensity variation Cycle. A number of explanations of this period has been proposed thus far. The major ones among them are: (1) a freely processing neutron star (13recher 1972; Shl~lovsly 1973; Nov~ov 1973; Ih~emper et al. 1986~; (2) a processing accretion disk (Katz 1973; Roberts 1974; Petterson (1975, 1976~; (3) presence of a third body (Mazeh and Shaham 1977~; non-linear oscillations excited on the outer part of the disk 307

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308 AMERICAN AND SOVIET PERSPECTIVES (Meyer and Meyer-Ho~neister 1984). None of the explanations suggested is commonly accepted. Although the model of the processing disk which screens periodically the X-ray flux toward the Earth has managed to explain certain details in the optical and X-ray light curves of Her X-1 (Gerend and Boynton 1980; Howarth and Wilson 1983), its feasibilitiy remains unclear (Papaloizou and Pnngle 1982; Kondo et al. 1983~. The recent extensive X-ray observations from EXOSAT show evidence for the free precession model underlying the 35-day cycle in Her X-1 (Ogelman and lluemper 19883. This model has been discussed in more detail by Shakura (1988~. In this paper we investigate a precise formula for the timing of an X-ray pulsar due to the free precession of a neutron star. We show that the pulsar period variations with the precession phase has a special shape by measuring which one could check up the model In the framework of this model we also simulate numerically optical light cubes to be observed from such a binary, taking into account the reflection effect on the secondary star and on the disk itself, and investigate the impact of the X-ray emission diagram on the light curves. PERIOD DERIVATIVE OF THE PULSAR DUE TO FREE PRECESSION Consider a freely processing biaxial solid body rotating around awns OF (see Figure 1) with a frequency Q = 2,r/P. Let the main inertial ems pierce the body at the point I, and the magnetic pole direction constitute the angle X with the ems OI and intersect the body's surface at point M. The body processes slowly around ems OI with a frequent y ~ = 2,r/Ppr. Angle ~ between OF and OI keeps constant. As is well known, the precessional frequency in the case of a symmetrical rotator is ~ = Q costs (~-13~/13, where I' and Is denote moments of inertia, (I3 being directed along the precessional ems Ol). Let the X-ray emission originate at the magnetic pole M. Then the problem is to find the time interval fit between subsequent pulses registered by a distant observer. There are different ways to solve this problem (see e.g. Bisnovatyi- Kogan e! al. 1989, who independently have obtained the solution). Here we follow the way which seems to be the most simple and straightforward (Shakura 1988~. In the stationary spherical frame with the origin in the center of the body, He position of the emitting spot on the surface can be described in terms of polar distance b and longitude A, measured from the meridian plane composed by vector J and the line of sight: The change in b results in the periodical (with the precessional period Pro) disappearance 1 We are working under the condition Q/w ~ 1, when the rotational axis of the body practically coincides with the angular momentum J. In a more general case one ought to introduce the lati- tude and longitude of the emitted area with respect to the vector of instant rotational velocity Q which is not aligned with J and rotates with frequent Q arount it (we shank Dr. R Blandford for

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HIGH-ENERGY ASTROPHYSICS ~ . ~/ / \ \ 309 Q / / 1` \ \ / / / ' ~ b \ I Act / : M \ - / FIGURE 1 The schematic representation of the processing neutron star geometry (see the text). Ibe star rotates around vector OP with frequency Q and slowly processes around axis OI. the X-ray emission escapes from the magnetic pole M, whose coordinates (>,b3 are measured with respect to the pole P and meridian of the observer. Of the spot from the line of sight and wee versa. The change in ~ can lead to variations in the observing pulsar period (Shefier 1987~: Robe = Q + dA/dt = const. If dA/dt = const, Robs is constant as well. But this is not the case for a processing neutron star. Making use of familiar spherical trigonometric formulae one easily finds dA/dt sin X cos ,o-sin ~ cos X cos at (1) 1-(cosxcos~ ~ s~xsm~coswt)2 Obviously, d>/dt is related to the change in time of pulse arrival through the relation POP Q = dA/dt. The same expression can be obtained from differentiating directly moments of amval for individual pulses, which is expressed by the equation drawing our attention to this note). However, the approximation used is well suited for Hercules X-1.

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310 AMERICAN AND SOVIET PERSPECTIVES At = AN-arctan [ sinxsin~t ], N = 1, 2, (2) sin X cos y cos At - cos X sin ~ Note that (1) Is valid to the first order of (~/Q), which is <<1 in our case (P = 1.24 s and Ppr = 35 days). The function ~P/P(t) has a specific shape, by measunog which one could in principle check the predictions of the modes The processing NS model implies that dP/dt should change its sign two times or four times per 35-day period depending on the geometry. ~ check this one needs simply to differentiate dA/dt. The standard investigations show that this derivative vanishes at the following phases: 1. if X < So (i.e. the spot never crosses the pole) there are only two obvious roots defined by sinwt = 0, which correspond to moments of crossing the meridian by the spot. In this case dA/dt = 0 at the points count = tgxJtg~e 2. if X > ~ (the spot can be behind the pole) dA/dt never vanishes; nevertheless two additional minima can arise as the solutions to the follow- ing equation: cos At >/sin2 X - sins tog cos% sin X sing (3) The necessary condition I cos At I <1 put angles X and y within the boundaries shown by the hatched area in Figure 2: OX end co if Vx+w>,r/2; X < 1/2 ~ arcsin(3 sin ~) - So] X > ~r/2 - 1/2 [arcsin(3 since - ~] if X + ~ ~ ~/2 (4) Cling geometrical parameters of Her X-1 from Ogelman and lluem- per (1988) to be ~ = 71, X = 12, we find the POP curve of Her X-1 should pass through two additional extreme on the phases of the 35~ay c ycle to be found from (4~. We conclude this section by stressing the neces- sity to make the precise X-ray pulsar timing in Her X-1 to check directly by this way the model of the free precession of the neutron star. OPTICAL LIGHT CURVES HZ HER IN THE MODEL OF A FREELY PROCESSING NEUTRON STAR Synthesis of the light curies to be seen from binaries is a well-developed field of modern astrophysics (see e.g., Antothina and Cherepashchok 1987

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HIGH-ENERGY ASTROPHYSICS Icl2 X~35o 311 ~ 1 1 ~7 O arcsin1/3 up ~/2 FIGURE 2 The ~ vs X plane for magnetic pole location on the surface of the neutron star relative to pole P. The hatched area represents locus of ~P/P(t) curve baring four extreme. Asterisk marks the spot for Her X-1 geometry (~ = 12, X = 71) according to pulse deconvolution analysis (Ogelman and Temper 1988~. for a review). Recently we have developed a computer code to calculate the light curie from any binary system (the description of the method used will be published elsewhere). Here we present some preliminary results showing synthetical optical light curves from Her X-1/HZ Her system In the framework of the processing neutron star model. The parameters of the binary and of the neutron star have been chosen as follows. The binary parameters: Masses: Men = 2.2M~, ME = 1.3M~; semimajor ens a = 9.06R~; the binary inclination i = 85. The aeon disk parameters: The accretion disk is supposed to be coplanar with the orbital plane. Its outer radius is ran' = 3.2R~ (=0.8 of the neutron star's Roche lobe).

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312 AMERICAN AND SOVIET PERSPECTIVES The inner disk radius is rin = 2108cm. The disk is deserted in the terms of c'-theo~y and has thickness/radius dependence H/r oc ri/8 with H(rOu~/rou' = 0.1. The neutron star parameters: The X-ray to optical luminosity kx = LXILopt = 300. The X-ray beam Is taken to have a sine-like shape of width W and is inclined towards OF (spin axis) at the angle b. We assume that the spin axis of the neutron star can be directed arbitrarily relative to the orbital momentum L (at angle 8) and to the line of sight at the phase zero of the binary period (at angle Id. The optical star parameters: The optical star is assumed to fill its Roche lobe and has a polar temperature of 8000 K The limb-darkening coefficient u = 0.4. The effectiveness of the X-ray conversion for reflection effect is arc = I. The reflection effect from the optical star and the accretion disk itself have been accounted according to Khruzina et al. (1988~. The synthesized light curves are presented in Figures 3~. In the case when the spin awns of the NS is normal to the orbital plane (8 = 0) the light curies are symmetrical with respect to phase 0.5 for any precession phase. The curves, which look like those observed from HZ Her can be obtained only for an inclined neutron star. The figures also show the dependence of the curie shapes upon the X-ray beam width W = 45 and 90 for the neutron star orientations ~ = 50, ~ = 50. DISCUSSION There are two general objections to the freely processing neutron star model First is that the difference of moments of inertia HI required to give rise to the precession frequency observed is LEVI ~ w/Q ~ 4~10-7 which cannot be sustained by stresses in the neutron star crust (G.S. Bisnovatyi- Kogan). The second objection is that the crust of the processing neutron star may break when relaxing tO equilibrium shape caused by rotation (R. Blandford). Let's make use of simple estimations to show the feasibility of free precession in the Bade of Her X-1. The limiting crust deformation can be deduced from pulsar glitch observations to be ^e ~ ~Q/Q ~ 210-6 (for the Vela pulsar). Supposing the same crust properties for Her X-1, we can get Al/I < ~e which is enough to explain the observations. On the other hand, Her X-1 has a rotational oblateness =e 10-7, which is less than the limiting deformation Ac. Then the precessional changing of the rotational axis in the body of the neutron star could not lead to the crust cracldng. In our opinion, the attractiveness of the model is dictated further by a possibility for an Secreting magnetic neutron star to obtain a slight difference in its moments of inertia (AI), which is enough to gme rise to

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HIGH-ENERGY ASTROPHYSICS AmB 1.2 A amB a+ o Ems 2- . o 313 ~ ~ 50O, Q ~ 50, W ~ 45 _r \J 0.25 0.5 0.75 it/ 0.25 0.5 0.75 \ / 0.25 0.5 0.75 b . 60 ~_.1 0.25 0.5 0.75 ~1 2- , ~ ~1 . b ~ 45 1 b = 75 b . 90. 0.25 0.5 0.75 , , 1 0.25 0.5 0.75 FIGURE 3 1-ne synthesized B-light curares of Her X-1 in the model of a freely processing neutron star ninth the parametem as described in the Section 3 for X-ray beam of width W = 45 shown at different latitudes b of the emitted area. the free precession with the observed period in Her X-1 during the mass infall onto the star's surface throughout the magnetic poles (Novikov 1973~. This idea nicely fits with the observational absence of free precession in radiopulsars. The clock mechanism for the X-ray emission in Her X-1 then can be provided by change in the emitted area location on the stellar surface. The minimum of X-ray flux (so-called "main-on" state) is expected when the beam is in the upper position over the accretion disk plane. The "off"-state and "low-on"-state of Her X-1 might correspond to the situation when the magnetic poles are close to the plane of the disk and differ from each other by some azimuthal inhomogeneities on the disk caused by the X-ray heating. This does not seem to contradict the X-ray observations of low- massive X-ray binaries, which show modulations resulting from obscuration by materials located in Hick azimuthally structured accretion disks (Parmar and White 1988~.

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314 AmB AmB ems 2- ~ - . , 0.25 0.5 0.75 2 o 2a o 0.25 0.5 0.75 - . I ~ ~1 2- , Y ~ l b . 45 j ,45~ t . , . o b . 30 0.25 0.5 0.75 0.25 0.5 0.75 FIGURE 4 The same as Figure 3 for W = 90. AMERICAN AND SOVIET PERSPECTIVES ~ ~ 50. O ~ 50, W ~ 90 . b . 0 2 ~ i . O \ b - 60 _ b A 75 / / 0.25 0.5 0.75 / - \ 0.25 0.5 0.75 In a stationary situation, the electrodynamical interaction of a rotating neutron star having a dipole magnetic field with a diamagnetic accretion disk, tends to make the dipole ens coplanar to the disk plane (Lipunov and Shakura 1980~. We speculate that the precession of the neutron star might force the rotational axis to be non-parallel to the disk plane. In general, the body of a neutron star could have a biaxial shape. As is well known, this would lead to a more complicated picture of the processing motions (Landau and T`i~hitz 1971~. A situation is possible when the magnetic poles achieve their upper location above the disk twice during the precession period. Then the "low~n"-state can be interpreted in terms of the second upper position of the poles. Due to a large width of the beam Bosch the poles could be visible as in fact is likely to be observed (Ogelman and lluemper 1988~. The similar situation can take place in case of symmetrical rotator when the one magnetic pole crosses the disk plane and the second pole becomes visible as well

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HIGH-ENERGY ASTROPHYSICS 315 CONCLUSIONS The model of freely the processing neutron star underlying the famous 35-day cycle in Her X-1 has been considered. We have deduced a strict formula for the amval times of X-ray pulses in the frame of this model. This expression can be used to check up the predictions of the model. The preliminary results of the optical light curve synthesis from Her X-1/HZ Her show that the model of the processing neutron star seems to be capable of describing the asymmetries and some features of the observed light curves. The neutron star, however, has to be inclined with respect to the orbital plane to make possible an anisotropic heating of the secondary. Obsenationally, this should manifest itself by a more pronounced mass outflow in the corresponding precessional phases. The more detailed calculation aiming to explain HZ Her optical light curves are now in progress and will be published later. ACKNOWLEDGEMENTS We would like to thank Drs. G.S. BisnovaWi-Kogan, R. Blandford, and E.K Sheffer for discussions. REFERENCES Antokhina, EM, and AM. Cherepashchu~ 1987. Astron. Zh. (in Russian) 64~3~: 562. Bisnovatyi-Kogan, G.S., G.A Mersov, and E.K Shelter. 1989. Preprint IKI. 1533 Boynton, P.E. 1978. In: Giacconi, R., and R Ruffini (eds.~. Physics and Astrophysics of Neutron Stars and Black Holes. Bologna. 121. Brecher, K 1972. Nature. 239: 325. Gerend, D., and P.E. Boynton. 1980. Astrophys. J. 209. 56~ Howarth,??, and ??W~lson. 1983. Mon. Notice Roy. Astron. Son 213. Katz, J.I. 1973. Nature Phys. Scie. 246: &7. Kondo, Y., 1:~ van Flandren, and AL Wolf. 1983. 273: 716. Khruzina, I: S., ~M. Cherepashchuk, N. I. Shakura, and Red Sunyaev. 1988. Adv. Space Res. 8:237. Landau, AD., and E.M. Lif~hitz. 1971. Mechanics Lipunov, V.M., and N.I. Shakura. 1980. Pistma Astron. Zh. 6: 28. Mazeh, T., and J. Shahs. 1977. Astrophys. J. 213: 117. Meyer, F., and E. Meyer-Ho~meister. 1984. Astron. Astrophys 140: L35. Novikov, I.D. 1973. Astron. Zb. 50: 459. Ogelman, H., and J. Temper. 1988. Mem. Son Astron. Ital. 59: 169. Papaloizou, J. and J.E. Pnagle. 1982. Mon. Notic Roy. Astron. Soc 200: 49. Parmar, AN., and N.E. White. 1988. Mem. Soc Astron. Ital. 59. 147. Petterson, Joy 1975. Astrophys. J. 201: Lot. Petterson, Joy 1977. Astrophys. J. 218: 783. Roberts, J.M. 1974. Astrophys J. 187: 575. Shakura, N.I. 1988. In: Physics of Neutron Stam. Pulsars and Bursters. PTI. Leningrad. 34 . Shelter, E.K 1987. Pistma Astron. Zll. 13: 204. Shklovsk~r, I.S. 1973. Astron. Zh. 50: 233. Ihemper, J., P. Kahabica, H. Ogelman, et al. 1985. Preprint MPI 41.