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A&A manuscript no. (will be inserted by hand later) ASTRONOMY AND Your thesaurus codes are: ASTROPHYSICS 01 (08.14.1; 08.09.2: Her X-1) 1.2.2008 On some features of free precession of a triaxial body: the case of Her X-1 1 2 1 N.I.Shakura , K.A.Postnov and M.E.Prokhorov 1 SternbergAstronomical Institute, Moscow University,119899 Moscow, Russia 2 Faculty of Physics, Moscow StateUniversity,119899 Moscow, Russia Received ... 1997, accepted ..., 1997 8 9 9 Abstract. We show that the free precession of a tri- ofthefreeprecessionmodel,wouldcorrespond,asthe au- 1 axial body can naturally explain the anomalously rapid thorsclaim,toaverylargechangeinthemomentofinertia n change of the X-ray pulse profile of Her X-1 observed by of the neutron star body corresponding to an oblateness a J the HEAO-1in September 1978without requiring a large of ∼ 8×10−6. Such a large moment of inertia changing change in the moment of inertia. wouldleadinturnto the pulse periodchangeby anorder 9 of magnitude higher than was actually observed during 2 Key words: Stars: neutron; stars: individual: Her X-1 this period (i.e. in August-September 1978) 8×10−7 s. v 3 The purpose of this Letter is to show that in fact the 3 free precession model cannot be so easily rejected if one 0 considersthe possibletriaxiality ofthe neutronstarbody. 1 Then the observed episode of an unusually rapid X-ray Hercules X-1 is the most famous and well studied X- 0 pulse shape changeinHer X-1cannaturallybe explained 8 ray pulsar containing an accreting neutron star with a by a sudden small deviation of the moment of inertia 9 spin period of P = 1.24 s in a circular orbit around ns / a 2M main-sequence star. The orbital period of the bi- along one axis without changing the characteristic period h ⊙ of the free precession (i.e. conservingthe gross oblateness p nary system is 1.7 days. Discovered in 1972 (Tananbaum - et al. 1972), it nevertheless has not been completely un- ofthebody).Themagneticpolesimplystartsmovingnon- o uniformlyalonganon-planartrajectorywhichapparently derstood until now. This mostly concerns the origin of r t its long-term35-dayX-ray periodicity, which has broadly manifests itself as the rapid change in the X-ray pulse s shape because the X-ray beam goes rapidly down to the a been discussed in the literature. The possible reason for v: this long-term period was suggested to be either (1) the rotational equator of the neutron star (then the observer sees two poles producing two equal X-ray pulses over one i neutronstarfreeprecession(Brecher1972)or(2)thepre- X spin period, as was observed by the EXOSAT during the cession of a tilted accretion disk controlled by the outer r low-onstateofHerX-1inMarch1984)andtravelstheway parts (i.e. by the precession of HZ Her (Roberts 1974) or a it usually takes a 17-day interval in a much shorter time by some intrinsic reasons (Boynton et al. 1980)). Notice of ∼ 1 day. Of course, the total moment of inertia of the that both the neutron star free precession and a complex neutron star remains practically unchanged and, subject precessing motion of the accretion disc may in fact si- to the angular momentum conservation, no appreciable multaneously operate in Her X-1/HZ Her binary system X-ray pulse period change should be observed. (Shakura et al. 1997, in preparation). A strong evidence favouring the free precession model Considerfirstthemorefamiliarcaseofanaxiallysym- wasfoundbyTru¨mperetal.(1986)intheEXOSATobser- metric body with I′ > I′ = I′ rotating around the an- 3 2 1 vations of the X-ray pulse profile phase and shape chang- gular momentum M (see Fig. 1). Then in the rotating ingoverthe35-dayperiod.Incontrast,Soongetal.(1987) frame the trajectories the magnetic pole of the neutron claimedthattheirobservationsofHerX-1byHEAO-1X- star moves along represent plane circles on the neutron raysatellitein1978donotsupportthismodel.InSeptem- star surface with the center at the largest moment of in- ber 1978, they observed an unusually short high-on state ertia (I′). This situation corresponds to a ”normal” free 3 of Her X-1 (the X-ray emission faded down very rapidly precessionin Her X-1 and the magnetic pole P uniformly during 7 days instead of 10, and the pulse profile shape goes alongsucha circle passinginone daya pathmarked changed over 20 hrs, as contrasted to about 17 days for by the short thick arrow.The precession period is simply theEXOSATobservations),which,ifinterpretedinterms P ≈ P I /(cosb(I −I )) >> P , where I and I pr ns || ⊥ || ns || ⊥ 2 N.Shakura,K.Postnov & M.Prokhorov. Triaxial free precession Fig.1.Aschematicviewoftheneutronstarbody.Mistheangularmomentumvector.Thecaseofaxisymmetricfreeprecession: ′ ′ ′ I3 >I2 =I1, the magnetic pole moves along a plane trajectory; the small thick arrow shows the way the pole passes in 1-day time interval.The case of thetriaxial free precession: I3∼>I2 >I1,two separatrices appear crossing at I2 and I˜2; anon-planar trajectory of M relative to the new axes of inertia is shown with the thin arrows indicating the direction of the angular momentummotion.Intheleft panel,thecasewhenMgoes towardI˜2 isshown,i.e.theneutronstarbodyturnsanti-clockwise aroundanaxisclosetoI1 .Intherightpanel,MmovestowardI2 andthestarturnsclockwisearoundI1.Thelongthickarrow indicates therapid motion of the magnetic pole P toward therotational equator. are the components of the moment of inertia parallel and which indeed corresponds to the observations of Soong et normalto the totalangularmomentumand b is the angle al. (1987)1. betweenthelargestmomentofinertia(I3′ inourcase)and Inthetriaxialcase,theangularmomentumvectorcan the angular momentum. move in two opposite directions depending on at which part of the trajectory the quake happened (left and right Letnow the body ofthe neutronstar experience some schemesinFig.1).Accordingly,intherotatingframewith quakeresultinginapracticallyinstantaneouschangeinall the z-axis along M, the magnetic pole will rapidly move moments of inertia with I3 > I2 > I1 (Fig. 1). As is well downward (left part of Fig. 1) or upward (right part of known (see Landau and Lifshits 1965), in this case the Fig.1)sincetheneutronstarbodyturnsaroundsomeaxis motion of the angular momentum vector relative to the (close to I1 in Fig. 1). Requiring that the magnetic pole axes of inertia becomes more complicated: two families of lies near the rotational equator shortly after the quake non-planar trajectories appear isolated by two separatri- (in order to make it possible to observe an X-ray pulse cespassingthroughI2,one aroundI3,anotheraroundI1. withtwoequalpeaks),itshouldbelocatednearthecircle The motion along a trajectory around the maximal mo- passing through I3 and I1 axes of inertia (as the angular ment of inertia (I3) becomes very nonuniform (see Fig. momentum vector ”freezes” near the axis I2). 2): the closer the trajectory to the separatrix, the more The long thick arrow in Fig. 1 illustrates the way the nonuniform the motion along it is. In Fig. 2 we show how magnetic pole now passes over one day. In Her X-1, the the angle between the angular momentum and the mag- transition from one trajectory to another occurs between netic pole θ changes with time over one precessionperiod September 22 and 23, 1978, which explains the appar- (see the Appendix for more detail). The angular momen- ent 10-fold increase in the free precessionrate. After that tum rapidly passes most part of the trajectory and slows the moments of inertia relaxes to their ”usual” values downitsmotionneartheturningpointclosetothepoints I2 andI˜2 (in the limiting case when the pole goes exactly andthe magnetic pole returns to anotherplanar”axially- symmetric” trajectory lying not far from the old one (be- along the separatrix, it would stay infinitely long at the separatrix crossing points I2 and I˜2, being in the state of cause the angular momentum spent most time near the separatrix ”crotch”). indifferent equilibrium). In the middle panel of Fig. 2 we also reproduce the phase change of cosθ in the axisym- metrical case with the angles taken from Tru¨mper et al. 1 WeremindthatconventionallytheprecessionphaseΨ35 = (1986)(the thick sinusoid). Clearly, the quake must have 0 corresponds to the maximum X-ray flux so that the main taken place somewhere between Ψ35 = 0 and Ψ35 = 0.1, X-ray turn-onstarts at Ψ35 =−0.15 N.Shakura,K.Postnov & M.Prokhorov. Triaxial free precession 3 Fig. 2. The dependence of the angle θ between the magnetic pole P and the vector of angular momentum M on the precession phase Ψ35 in the case of the triaxial precession. The relative differences in moments of inertia are both 10−6. The magnetic pole position is close to I1 (left panel), I2 (middle panel), and I3 (right panel). The five curves in each figure are shown for the trajectories around I3 (see Fig. 1) characterized by different maximal angles χmax between M and the axis I3: cotχmax = 1,1/3,1/10,1/100,1/1000. The closer the trajectory to the separatrices, the more nonuniform the motion of M along it is. The thick sinusoid in the middle panel depicts the phase behaviour of cosθ for axisymmetric precessional motion with the angles taken from Tru¨mper et al. (1996): cosθ = cos(25o)cos(75o)+sin(25o)sin(75o)cosΨ35. The quake must have taken place close to Ψ35 =0.05 Duringthetriaxialmotiondescribedabovethepreces- thankDr.A.N.Rakhmanovforhelpindrawingthefigures sion period should not change appreciably since the gross and the anonymous referee for valuable notes. differenceintheparallelandperpendicularmomentsofin- ertia remains practically the same. After the body of the Appendix neutron star has returned into its axisymmetric form, it shouldberecognizedthattheX-raypulseshouldgenerally For I3 >I2 >I1, given the total energy be phase-shifted. This effect can in principle be detected 2 2 2 by accurate timing of X-ray pulses in different 35-daycy- 2E =I1Ω1+I2Ω2+I3Ω3, cles. In the free precession model for Her X-1 the vector of and angular momentum theneutronstarangularmomentumshouldbeinclinedto the line of sightby an angle of ∼−40 degrees tilted away M2 =I12Ω21+I22Ω22+I32Ω23, from the observer (see Tru¨mper et al. 1986 for a detailed discussion of all relevant angles in the case of the axially themotionoftheangularmomentumvectorinthecoordi- symmetric free precession). That the angular momentum natesystemrelatedtoaxesofinertiaintherotatingframe of the neutron star proves to be tilted with respect to isgivenbythefollowingsystemofequations(Landauand the orbital angular momentum is naturally explained in Lifshits 1965): the framework of the free precession model because the torquesappliedtoastronglymagnetizedrotatingneutron 2EI3−M2 starbytheaccretiondiskchangethesignforsomecritical Ω1 =sI1(I3−I1)cnτ inclination (∼ 55 degrees) of the magnetic dipole axis to the neutron star spin axis (Lipunov 1992). 2EI3−M2 Note to conclude that free precession of neutron stars Ω2 =sI2(I3−I2)snτ makes them an interesting potential source of gravita- tionalradiation(Jones1998),andthe confirmationof the M2−2EI1 free precession model for Her X-1 should stimulate such Ω3 =sI3(I3−I1)dnτ, studies. where cnτ, snτ, and dnτ are elliptic Jacobi functions TheworkwaspartiallysupportedbyRussianFundfor and the dimensionless time τ is Basic Research through Grant No 95-02-06053-a, by the INTAS GrantNo 93-3364and by the RussianMinistry of (I3−I2)(M2−2EI1) τ =t . Science NTP “Astronomija”,project1.4.4.1.The authors s I1I2I3 4 N.Shakura,K.Postnov & M.Prokhorov. Triaxial free precession Specifying the relative differences ∆I12/I3, ∆I23/I3 and expressing the precession phase in units of the dimensionless precession period π/2 du Π=4 0 1−k2sin2u Z with the parameter k definepd as 2 (I2−I1)(2EI3−M2) k = , (I3−I2)(M2−2EI1) we calculate the curves shown in Fig. 2 for the position of the magnetic pole on the neutron star surface and the angular momentum vector as explained in the figure cap- tion. 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