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    {\small\rm\zihao{6}\songti\thepage\hfill
    \quad\quad ACTA ASTRONOMICA SINICA \hfill
    \hspace{2mm} Vol.44\vspace{2mm}}}} %%%%%%%
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    {\small\rm \zihao{6}\songti suppl.\hfill Y. Q. Xue et al.: An Accretion Disk Model
    for Periodic Variations of PSR B1828-11  %%（作者：题目名称）
    \vspace{1.5mm}
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\zihao{5}\songti

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    Vol.44~ Suppl.
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    \small ACTA ASTRONOMICA SINICA
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      Feb.,~2003 \\[-1.5 true mm] %%January
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\vs\vs\vs \bc {\Large\bf An Accretion Disk Model
    for Periodic Variations of\\[2mm]
 PSR B1828-11 %%（文题）
%\\[1ex]
%$\!\!^*$
}

\vs\vs{\normalsize Y. Q. Xue,\hs\hs G. J. Qiao,\hs\hs R. X. Xu,\hs\hs H. G. Wang\hs\&\hs B. W. Xiao}  %%（本文作者）

\vs ({\footnotesize {\it Department of Astronomy, School of
Physics, Peking
University, Beijing }~ 100871}) \\%%（本文作者单位）
\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
At the present time, the debate about the nature of Soft Gamma-ray
Repeaters (SGRs) and Anomalous X-ray Pulsars (AXPs) has aroused
researchers' great interests in the pulsar (PSR) astronomy. There
coexist two prevalent lines of thoughts, the magnetar models
(neutron stars with super-strong magnetic fields) and the
accretion models (neutron stars or strange stars with fossil disks
and ordinary pulsar magnetic fields), which try to explain the
SGRs/AXPs. In this paper, we present an accretion disk model for
periodic variations of PSR B1828-11. Fitted with reasonable
parameters, the observed phenomenon of PSR B1828-11 (Stairs et al.
2000) can be explained. This may link radio pulsars and AXPs and
give a support for the fossil disk model of SGRs \& AXPs.

 %\vn
 %\no{\bf Key words}~~  pulsar-mode  change-spectrum
\end{quotation}

%%%% 正文
 \para{\large\bf{1}~~ Debate about SGRs/AXPs}

\vs\vs SGRs/AXPs distinguish themselves from ordinary radio
pulsars by many distinct observational properties, among which the
fact, that their X-ray luminosities are about 1 - 4 magnitude
greater than their rates of rotational energy loss while the
reverse for radio pulsars, is the most prominent one. Hence, any
model attempting to interpret SGRs/AXPs should have a ``new''
energy source other than rotational energy loss, which powers the
radiation of radio pulsars. Among the debates about SGRs/AXPs,
magnetism-powered or accretion-powered, has been the focus. In the
magnetar models, SGRs/AXPs are thought to be neutron stars with
super-strong magnetic fields (10$^{14}-10^{15}$\,G) and their
luminosities are produced by the magnetic field decay. On the
other hand, the accretion models suppose that SGRs/AXPs are
neutron stars or strange stars with ordinary pulsar magnetic
fields (10$^{11}-10^{12}$\,G) and their radiations are powered by
the accretion from fossil disks. The direct criteria to test these
two kinds of models lies in whether the super-strong magnetic
field or the accretion disk exists or not. Our accretion disk
model for periodic variations of PSR B1828-11 may cast some light
on the debates.


\para{\large\bf{2}~~ Free or Disk-Induced Precession}

\vs\vs Recently, Stairs, Lyne and Shemar (2000) discovered
long-term, highly periodic and correlated variations in both the
pulse shape and the rate of spin-down of the pulsar PSR B1828-11
(see Figure 1). They emphasized this is evidence for free
precession which is caused by asymmetry in the shape of the
pulsar. Some authors (Jones \& Andersson 2001, Link \& Epstein
2001, Rezania 2002) proposed different models to explain this
phenomenon within the framework of free precession. Unfortunately,
this scenario seriously challenges our understanding of the liquid
interior of the neutron star. Torque-free precession of a solitary
pulsar should be damped out due to the dissipation caused by the
vortices pinning in its superfluid interior within the timescale
of several hundred precession periods. Thus we present an
accretion disk model instead. In our model, the total torque
exerted on the pulsar by the fossil disk is nonzero due to the
asymmetry in pulsar shape and nonalignment between the pulsar
rotation axis and the axis of pulsar moment of inertia. The long
time scale precession of the pulsar can be well elucidated with
our model. Hence, we suggest a forced precession model, other than
the free precession model, might be more reasonable.
\begin{figure}[h]
\center {\includegraphics [bb=0 0 750 720,scale=0.3]{f2.eps}}

   \parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,1~~ Variations in rotation
and pulse shape in PSR B1828-11. The displayed time series are the
residuals in arrival time $\bigtriangleup t$, the period
$\bigtriangleup P$, the period derivative $\bigtriangleup
\dot{P}$, and the mean pulse shape parameter $<S>$ for the most
recent 2000 days. These periodic patterns continue back through
the entire 13-year data set. The solid curves indicate a fit of
three harmonically-related sinusoids with the fundamental period
1009 $\pm$ 8 days (Stairs, Lyne \& Shemar 2000).}

    \end{figure}

\para{\large\bf{3}~~ Possible Evolutions in the Accretion Model}

\vs\vs Although the exact evolutionary tracks of neutron stars in
the accretion scenario are unclear, there exist some hypotheses
trying to elucidate them. Here a possible series of evolutionary
tracks for neutron stars are presented (cf. Chatterjee et al.
2000).

After the supernova explosion, there are 3 possible outcomes of
the newly born neutron star: (a) The neutron star may be born
without any accretion disk and present itself as a radio pulsar.
(b) An accretion disk may be formed by the matter falling back
onto the neutron star. The accretion disk will exist for radii
only beyond the magnetospheric radius
$R_{m}\cong0.5(B^{4}R_{*}^{12}/8GM\dot{M}^{2})^{1/7}$ (Frank et
al. 1992), where $M$, $R_{*}$ and $B$ are the mass, radius and
magnetic field of the neutron star, respectively, $\dot{M}$ is the
rate of accretion, $G$ is the gravitational constant. Once
$R_{m}>R_{LC}$ ($R_{LC}=c/\Omega$, the light cylinder radius,
where $c$ is the speed of light and $\Omega$ is the angular
velocity of the neutron star), the coupling between the disk and
the neutron star is unlikely to happen. Thus most of the neutron
stars may live as ordinary radio pulsars even if they acquire
debris disks through fallback. (c) In other cases, accretion will
be permitted and the neutron star will behave as an accreting
X-ray source. If $R_{LC}>R_{m}\gg R_{co}$
($R_{co}=(GM/\omega^{2})^{1/3}$, the corotation radius defined by
$\Omega=\Omega_{K}(R_{co})$, where $\Omega_{K}(R)$ is the
Keplerian rotation rate at radius $R$), which corresponds to
$\Omega\gg\Omega_{K}(R_{m})$, the neutron star may be in the
phase, in which accretion onto the neutron star is inefficient due
to centrifugal forces acting on the matter. The neutron star will
spin down and be X-ray faint (e.g. Compact Central Objects (CCOs)
may be the kind of X-ray faint sources which have just begun to
spin down). As $\Omega$ approaches $\Omega_{K}(R_{m})$, the system
enters a quasi-equilibrium phase (happening at the time of
$t_{trans}$), in which the spin of the star roughly matches the
rotation of the disk at $R_{m}$. In this phase, the neutron star
will be X-ray bright due to accretion. It is possible that the
advection-dominated accretion flow (ADAF) will occur at the time
of $t_{ADAF}$, when much of the mass will be ejected prior to
reaching the neutron star surface. The X-ray luminosity of the
neutron star will decline rapidly with time due to $\dot{M}_{X}$
(the rate of accretion onto the star surface) diminishing.
Therefore, the bright AXPs may exist only between $t_{trans}$ and
2$t_{ADAF}$ (about 5000 - 40000 years). Finally, the weak X-ray
luminosity will be powered by rotational energy loss since
accretion ceases (the matter of the disk is exhausted).
Consequently, the neutron star will be observed as a dim isolated
neutron star near our galaxy (e.g. Dim Thermal Neutron Stars
(DTNs) may be the kind of dim X-ray isolated neutron stars with
the distances $\leq$ 0.4\,kpc).

According to the rough evolutionary scenario above, PSR B1828-11
may be presently in the phase of just beginning to spin down
rapidly and being X-ray faint. Its X-ray radiation (if have) may
be too faint to be observed, thus it is regarded as a radio
pulsar. If our interpretation for the precession phenomenon is
right, it may show some hints for the future studies on the
accretion disk models.
\begin{figure}[h]
\center {\includegraphics [bb=0 0 611
581,scale=0.4]{Figure1-1.eps}}

   \parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {\center
   \footnotesize
Fig.\,2~~ The geometry and parameters for the calculations}

    \end{figure}

%\newpage\vspace*{1mm}
\para{\large\bf{4}~~ Modelling the Precession}

\vs\vs Consider the pulsar as a rotation ellipsoid with the
principal moment of inertia $I_{1}=I_{2} \neq I_{3}$ and the
corresponding radii $a=a \neq b$. The geometry of the system is
shown in Figure 2: the axis $OI$ is the principal axis of moment
of inertia of the neutron star, $O$-$\xi\eta\zeta$ is a coordinate
system fixed in the space, $O$-$xyz$ is the coordinate system
fixed with the neutron star, $\theta$, $\varphi$ and $\phi$ are
Euler angles, $\omega$ is the angular velocity of neutron star
rotation. The parameters about PSR B1828-11 are as follows: the
rotational period of this pulsar $P=0.405$s ($\omega=2\pi/P$), the
period derivative $\dot{P}=6.0\times10^{-14}$ ss$^{-1}$, the
precession period $P_{pre} \approx 1000$ days, the magnetic field
$B=5.0\times10^{12}$G (Stairs et al. 2000). Thus we obtain the
gravitational potential from the pulsar to the disk:
\begin{equation}
V=-\frac{2GMM_{0}}{c+d}[1-\frac{b^{2}+2a^{2}}{10cd}+\frac{3b^{2}}{20cd}sin^{2}
\theta+\frac{3a^{2}}{20cd}(1+cos^{2}\theta)]\,.
\end{equation}
where $M$, $a$ and $b$ are the mass, the equatorial and
longitudinal radii of the pulsar, respectively, $M_{0}$ is the
mass of the disk, $c$ and $d$ are the inner and outer radii of the
disk, respectively.
The precession angular velocity of the pulsar
is derived according to the Lagrange's equation (Qiao et al.
1989):
\begin{equation}
\dot{\phi}=-\frac{3GM_{0}cos\theta}{2cd(d+c)\omega}(1-\frac{b^{2}}{a^{2}})\,.
\end{equation}
For simplification, $c\approx d\approx R$ is assumed, so that
\begin{equation}
\dot{\phi}=-\frac{3GM_{0}cos\theta}{4R^{3}\omega}(1-(1-\epsilon)^{2})\,,
\end{equation}
where $\epsilon=(a-b)/a$ is the oblateness of the pulsar. Through
the numerical simulations, we gain the relations between $M_{0}$
and $R$ derived from the 1000-day precession period for a group of
$\theta$. We find, in principle, $M_{0}$ can vary from less than
$0.001 M_\odot$ to even $0.1 M_\odot$, and $R$ from about $1
R_{co}$ (or smaller) to $2 \sim 3 R_{co}$ when $\theta$ varies
within $0 \sim 90^{o}$.
% where
%$R_{co}=(\frac{GM}{\omega^{2}})^{1/3}$ is the co-rotation radius
%of the disk with the pulsar.


\para{\large\bf{5}~~ Discussion and conclusion}

\vs\vs Besides the accretion disk model, we think a pulsar with a
small-mass companion star may also produce the precession
phenomenon. In that case, two limits should be considered: the
Roche Limit $d_{R}$ (if the companion approached the neutron star
closer than this limit, it would be torn asunder by tidal forces)
and the Instability Limit $d_{I}$ (if the company strayed farther
than this limit, it would escape due to the differential
perturbations from other celestial bodies). The details of that
case will be discussed in another article.



Our accretion disk model for periodic variations of PSR B1828-11
can explain the observed $\sim$ 1000-day precession period
avoiding the puzzle of the damping-out effect due to the vortices
in the superfluid interior of neutron star. If our model is proved
to be true, the link between radio pulsars and AXPs can be
established and the fossil disk model for AXPs can be
strengthened. It will also suggest the existence of the neutron
star-fossil disk systems or neutron star-planet like systems.



\bc {\bf References} \ec

\vspace{-4mm} \no \fns

 \no Chatterjee P., Hernquist L., Narayan R.,  ApJ, 2000, 534: 373\\
\no Frank J., King A., Raine, D., apa.book., Accretion Power in
Astrophysics, 1992\\
\no Jones D.I., Andersson N.,  MNRAS, 2001, 324:
 811\\
  \no Link B., Epstein R.I., ApJ, 2001, 556: 392\\
\no Qiao G.J., Cheng J.H., ApJ, 1989, 340: 503\\
\no Rezania V.,  preprint(astro-ph/0205180), 2002\\
\no Stairs H., Lyne A.G., Shemar S.L., Nature, 2000, 406: 484\\

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