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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Precession and free core nutation of neutron stars
A. Gusev, I. Kitiashvili
Kazan University, Russia
In 1992 Wolszczan and Frail discovered a planetary system around the pulsar
PSR 1257+12 [1]. The system is the most interesting and #rich by number of
planets (A, B, C and D). It is located in the Virgo constellation at the distance of
300 pc. Two of them, B and C, move in the resonance 3:2. At the present time the
question is under discussion whether there exist other planetary systems around
three pulsars: PSR 0329+54 (1 planet), PSR B1620--26 (1 planet) and PSR 1828--
11 (3 planets --- A, B, C). The orbital periods PA = 0:68 yr, PB = 1:35 yr,
PC = 2:71 yr show that the extrasolar planets around PSR 1828­11 follow the
same 1:2:4 harmonic relationship as those of Jovian satellites. In 2000 Stairs,
Lyne and Shemar [2] reported the discovery of the long--term, highly periodic
and correlated variations pulse shape and the rate of slow--down of pulsar PSR
B1828--11 with the period variations of approximately 1000, 500 and 250 days.
There is also strong indication of the presence of a further harmonically--related
periodicity of approximately 167 days in both shape and rotation. There are three
potential explanations of the arrival time from pulsar related with the interior
of the neutron star, planetary bodies and free precession. The radial velocity
of a star is obtained by measuring the magnitude of the Doppler effect in its
spectrum. The stars showing small amplitudinal variation of radial velocity have
been interpreted as systems having planetary companions. Assuming that the
pulsar has the mass of 1:35M fi , the Keplerian orbital radii identified with the
three harmonically related sinusoids are 0.9, 1.4 and 2.1 AU while the masses are
3:1M \Phi sin i, 10:2M \Phi sin i, 4:6M \Phi sin i, where i is the orbital inclination. Second
explanation: the periods of Tkachenko oscillations of the neutron superfluid vortex
array, which carries much of the angular momentum of the neutron star, depends
on the size of the star and the square root of the pulse period. For Crab pulsar
(PSR 0531+21) Tkachenko oscillations are expected to amount to 4 months; for
the PSR B1828--11 the period of the Tkachenko oscillations are 13 months [2].
Unfortunately, the theory of the Tkachenko oscillations does not easily explain
the existence of multiple harmonics in the PSR B1828--11 timing residuals. Third
case: the most likely possibility is the free precession of the pulsar with a rigid
81

neutron star. It will occur if the star is deformed so that the spin axis is not
aligned to its angular momentum vector. The time scale for precession depends
on the degree of deformation of the pulsar. To produce a periodicity of 1000 days,
the deformation of a rigid­body neutron star would amount to approximately
5 \Theta 10 \Gamma9 . This value is comparable with or smaller than that expected for a
spining neutron star [2]. Another family of neutron stars, the anomalous X--ray
pulsars, are known to display ``wobbles'' in their slowdown rates.
It is known that rotation of the terrestrial planets having rigid mantle and
elliptical liquid core is characterized by Free Core Nutation (FCN). Any celestial
body whose rotation axis does not coincide with the main inertia axis is char­
acterized by Chandler Wobble (CW). These phenomena of FCN and CW are
manifested as periodical oscillations of the rotation axis of the pulsar in iner­
tial reference system. For rotating pulsar we deal with the case of modulation
of pulses emitted around the direction of the magnetic axis of a pulsar whose
symmetry axis is misaligned with the angular velocity vector. Two modes in a
polar oscillation are obtained when a free rotation of the two--layer pulsar was
studied:
PCW = P pulsar
2 p
fffi
A core
A
P FCN = P pulsar
p
fffi
A core
A
`
A core
C core \Gamma A core
'
(1)
Correctly extending the theory of core--mantle differential rotation of the planets
to neutron star [3] we have obtained the period of CW and free nutation of
liquid part of the N--star. It was made in the frame of the Hamiltonian approach
for description of rotation of two--layer deformable pulsar having rigid crust and
liquid mantle. Two modes of free pulsar libration are obtained in the case of polar
motion from the Hamilton's equations: a Chandler--like wobble with a period PCW
and a FCN with a period P FCN :
PSR B1828--11: P FCN ъ 250 days for a \Gamma c = 5m, a core \Gamma c core = 5:6m;
PSR 1257+12: P FCN ъ 67 days for a \Gamma c = 1m, a core \Gamma c core = 1:1m;
PSR 0531+21: P FCN ъ 120 days for a \Gamma c = 1m, a core \Gamma c core = 3:2m.
References
1. Wolszczan A., Frail D. A. Nature, 1992, 355, 145.
2. Stairs I. H., Lyne A. G., Shemar S. L. Nature, 2000, 406, 484--486.
3. Petrova N., Gusev A. Cel. Mech. & Dyn. Astron., 2001, 80, 215.
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