On some features of free precession of a triaxial body: the case of Her X-1

4 Pages·0.13 MB·English

by  N. I. Shakura

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 pre-
cession 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 di-
mensionless 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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