\documentstyle[twoside,epsfig,CJK]{carticle}
%\documentclass[twoside]{carticle}
%\usepackage{epsfig}
\pagestyle{empty}
\headsep=0.6cm %%%%%%%

\setcounter{page}{227}

%\input cchead.sty
%\input Vatola.sty

\def\thefootnoot{}

\textwidth=140mm \textheight=210mm \topmargin=0pt
\oddsidemargin=0pt \evensidemargin=0pt \footskip 0.2 truecm

%%...................
%\vspace*{1\baselineskip}
\doublerulesep=0pt
%%..................
% \renewcommand{\baselinestretch}{1.4} %%cc
\parindent=7.5mm

\setlength{\parskip}{0 true pt}
    \def\hang{\hangindent\parindent}
    \def\RE{\par\hang\textindent}
    \catcode`@=11
%%%双页
    \def\@evenhead{\underline{\parbox{\textwidth}
    {\small\rm\zihao{6}\songti\thepage\hfill
    \quad\quad ACTA ASTRONOMICA SINICA \hfill
    \hspace{2mm} Vol.44\vspace{2mm}}}} %%%%%%%
%%%单页
    \def\@oddhead{\underline{\parbox{\textwidth}
    {\small\rm \zihao{6}\songti suppl.\hfill Z. K. Huang \& X. J. Wu: Initial Periods of
    Millisecond Pulsars %%（作者：题目名称）
    \vspace{1.5mm}
    \hfill\thepage}}} %%%%%%%
    \normalsize\rm
\zihao{5}\songti

    \begin{CJK}{GB}{song}
    \begin{document}


%%篇首
    \thispagestyle{empty}
    \vspace*{-1.5 true cm}
    \small\rm
    \zihao{6}\songti
    \noindent
    \\%%%[-0.5 true mm]
    Vol.44~ Suppl.
    \hfill\hspace{8mm} %%% if ``October/January'' then \hspace{8/10 mm} July
    \small ACTA ASTRONOMICA SINICA
    \hfill
      Feb.,~2003 \\[-1.5 true mm] %%January
    \rule {\textwidth}{0.12 true mm}\\[-3.2 true mm]
    \rule {\textwidth}{0.12 true mm}
%%%%%%%

\input{../newcom.ctx}

\renewcommand{\baselinestretch}{1}

\setcounter{footnote}{1}

\vs\vs\vs \bc {\Large\bf
Initial Periods of Millisecond Pulsars   %%（文题）
%\\[1ex]
%$\!\!^*$
}

\vs\vs{\normalsize\zihao{4}\fangsong Z.K. Huang\hs\&\hs  X.J. Wu}  %%（本文作者）

\vs
(\pka)\\
(\bac)
\vs\vs \ec

\normalsize
%\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small Millisecond pulsars are accelerated to such short periods by
accretion of matter and angular momentum from their companion. It
is important to know the initial periods of millisecond pulsars,
since this quantity is closely related to the accretion process
and the equation of state of neutron star matters. For some
millisecond pulsars, the initial period can not be neglected in
the calculation of their characteristic ages, otherwise their
characteristic ages will be greater than the Galaxy age. Based on
off-centered dipole radiation theory, the initial periods of
millisecond pulsars can be obtained, and the initial periods are
found to be comparable to the current values.

\vs\para{\large\bf{1}~~ Introduction}

\vs\vs Millisecond pulsars are formed by accretion of matter from their
companions in the binaries (van den Heuvel 1987), which is
different from the process creating normal pulsars, so their
magnetic field strength, rotation periods and period derivative
differ remarkably from the corresponding quantities of normal
pulsars. As first noted by Shklovsky (1970), for near pulsars the
high transverse velocity may contribute to the period derivative.
Camilo et al. (1994) found after the correction of their high
transverse velocity, the corrected ages of several millisecond
pulsars are older than the age of the Galactic disk. They
reconcile this apparent paradox by suggesting some millisecond
pulsars were born with periods close to their current periods. In
order to find sub-millisecond pulsars, Edwards et al. (2001)
surveyed 19 globular clusters, but they discovered none. The new
X-ray observation of Crab (Weisskopf et al. 2000) and Vela pulsar
(Pavlov et al. 2001) indicate the proper motion direction, the
asymmetric axis of nebula and the jet direction are coincident.
The polarization observations of those two pulsars at radio
frequencies also indicate the spin axis and the proper motion is
parallel in projection on the plane of sky (Deshpande et al. 1999;
Moffett \& Hankins 1999). These facts are strong support of the
off-centered dipole radiation theory (Lai, D. et al. 2001;
Harrison \& Tademaru 1975). According to this theory, the velocity
at the end of acceleration is a strong function of the initial
period of pulsars. For these pulsars whose velocities have been
measured, we can got their initial periods. It is obvious the
period derived is the lower limit because we assume the last
velocities are the result of asymmetric electromagnetic radiation,
which dredge up the rotation energy.

\para{\large\bf{2}~~ The Acceleration of Pulsars by Off-centered Dipole Radiation}

\vs\vs Harrison \& Tademaru (1975) showed that the electromagnetic
radiation from an off-centered magnetic dipole imparts a kick to
the pulsars along its spin axis. This acceleration is attained on
the initial spindown time scale of the pulsar. The spindown
luminosity is
\begin{equation}
L=\frac{2\Omega^{4}}{3c^{3}}(\mu^{2}_{\rho}+\mu^{2}_{\phi}+
\frac{2\Omega^{2}s^{2}\mu^{2}_{z}}{5c^{2}})\,,
 \end{equation}
where $\Omega$ is the angular rotation frequency of pulsar, the
$\mu_{\rho}, \mu_{\phi}$ and $\mu_{z}$, are components of the
magnetic momentum, $s$ is the distance of dipole center to the
pulsar center. Another effect of off-centered radiation is the
asymmetry of the emission intensity from the two poles. This
asymmetric intensity can give a force to the pulsar
 \begin{equation}
F=\frac{8}{15}(\frac{\Omega
s}{c})\frac{\Omega^{4}\mu^{2}_{z}\mu^{2}_{\phi}}{c^{4}},
 \end{equation}
For a typical situation,
$\mu^{2}_{\rho}\sim\mu^{2}_{\phi}\sim\mu^{2}_{z}$, the asymmetry
parameter $\varepsilon\equiv F/(L/c)$ is $0.4(\Omega s/c)$. For a
given $\Omega$ the maximum $\varepsilon=0.63$ is achieved when
$\mu^{2}_{\rho}/\mu^{2}_{z}=0$ and
$\mu^{2}_{\phi}/\mu^{2}_{z}=0.4(\Omega s/c)$.
 From
\begin{equation}
M\dot{V}=\varepsilon(L/c)=-\left(I\Omega \dot{\Omega}\right)/c\,,
\end{equation}
we can get the kick velocity
\begin{equation}
V=140R^{2}_{10}(\frac{s}{10\,{\rm km}})(\frac{\nu_{i}}{1\,{\rm kHz}})^{3}
[1-(\frac{\nu}{\nu_{i}})^{3}] \,\rm{km/s}\,,
 \end{equation}
where the $R_{10}$ is radius of pulsar, $\nu_{i}$ is the rotation
frequency of pulsar at birth, $\nu$ is the rotation frequency at
present. If we assume $s=10$\,km, equation (4) can be expressed in
another way
\begin{equation}
\nu_{i}=(\nu^{3}+\frac{V}{140(\displaystyle\frac{s}{10})R^{2}_{10}})^{3}\
\rm{kHz}\,,
 \end{equation}

Usually the diameter of a typical neutron star is $20-30$\,km, so
we assume the radius $R_{10}$ to be 1. If $s=10$\,km, we can get
the initial period and rotation frequency from equation (5).

\para{\large\bf{3}~~ The Initial Periods of  Millisecond Pulsars}

It is believed millisecond pulsars are formed through different
process from that of normal pulsars, most of millisecond pulsars
lie in the binary systems, the accreted material from the
companion form a disk around the pulsar, and the accretion of this
matter accelerated the pulsar to milliseconds. The velocity of
millisecond pulsars are about one third of normal pulsars (Toscano
et al. 1998). The parameters of millisecond pulsar are given in
Table 1, these four columns are pulsar name, velocity in km/s,
period in ms, and initial period in ms respectively. It is obvious
the initial periods of millisecond pulsar are close to their
current present observed period, for most of millisecond pulsars,
it is about 30 percent of the current period. The shortest initial
period is 0.8 ms and the longest is 2.1 ms. For the fastest
rotational pulsar PSR B1937+21, the initial period is 0.777 times
of its present period.

\bc{\small\bf

 Table 1~~   The initial period of millisecond pulsars
}\\[2mm]

\fns \tabcolsep 6mm
\begin{tabular}{llll}
\hline\\[-3.6mm]\hline
 PSR J-     & velocity(km/s) & period(\rm ms ) & $p_{0}\rm (ms) $\\
 \hline
 0437-4715  & 120.5 & 5.757  &  1.0513\\
 0613-0200  & 77    & 3.062  &  1.2205\\
 0711-6831  & 78    & 5.491  &  1.2153\\
 1024-0719  & 62    & 5.162  &  1.3119\\
 1045-4509  & 52    & 7.474  &  1.3912\\
 1300+1240  & 284   & 6.219  &  0.79\\
 1455-3330  & 100   & 7.987  &  1.1187\\
1603-7202   & 27    & 14.842 &  1.7308\\
1643-1224   & 159   & 4.622  &  0.9585\\
1713+0747   & 28    & 4.57   &  1.71\\
1730-2304   & 51    & 8.123  &  1.4002\\
1744-1134   & 33    & 4.075  &  1.6188\\
1857+0943   & 17    & 5.362  &  2.0194\\
1911-1114   & 183   & 3.626  &  0.9146\\
1939+2134   & 79    & 1.558  &  1.2101\\
1959+2048   & 190   & 1.607  &  0.9032\\
2019+2425   & 83    & 3.935  &  1.1904\\
 2051-0827  & 14    & 4.509  &  2.1544\\
 2124-3358  & 53    & 4.931  &  1.3823\\
  2129-5721 & 56    & 3.726  &  1.3572\\
  2145-0750 & 38    & 16.052 &  1.5445\\
  2317+1439 & 94    & 3.445  &  1.142\\
  2322+2057 & 80    & 4.808  &  1.2051\\
 \hline\\[-3.6mm]\hline%\\[-4.5mm]\hline
 \end{tabular}
 \ec

The closeness of initial period and the current period have great
importance for millisecond pulsars. We get the initial period from
the velocity considerations, but our results are in agreement with
Camilo et al. (1994) who arrived at the same conclusion by
considering the oldest pulsar must be younger than the Galaxy.

A few factors may contribute to the velocity of pulsars, including
orbital velocity of its progenitor, asymmetric neutrino and
hydrodynamic kick when the neutron star was born and asymmetric
dipole radiation after the neutron star was born. Because
millisecond pulsars are still in the binary the contribution by
these three effects are not as big as that imparts to normal
pulsars. Assuming the off-centered dipole radiation is the only
factor, we should get the lower limit of the initial period.

\bc
{\bf References}
\ec

\vspace{-4mm}
\no \fns

 \nref Camilo F, Lorime D R, Kulkarni S R. ApJ, 1994, 421: L5
   \nref Deshpande A A, Ramachandran R, Radhakrishnan V. A\&A, 1999, 351: 195
   \nref Edwards R T, Straten W van, Bailey M. ApJ, 2001, 560: 365
   \nref Lyne A G, Manchester R N.  MNRAS,  1988, 234: 477
   \nref Rankin J M, Rathnasree N. AJ, 1995, 452: 814
   \nref Rankin J M. AJ, 1983, 274: 333
   \nref Rankin J M. AJ, 1986, 301: 901
   \nref Rickett B J. ARA\&A, 1977, 15: 479
  \nref Romani R W, Narayan R, Blandford R. MNRAS, 1986, 220: 19
   \nref Suleymanva S, Izvekova V, Rankin J M.  In: Johnston S, Walker M A eds.
   IAU Colloquium 160, Pulsar: Problem and Progress. San Francisco: Astronomical
   Society of the Pacific, 1996. 223
   \nref Zhang B, Qiao G J, Lin W P, et al. AJ,  1997, 478: 313
   \nref Harrison E R, Tademaru E. ApJ, 1975,  216: 842
   \nref Lai D, Chernoff D F, Cordes J M. ApJ,2001, 549: 1111
   \nref Moffett D A, Hankins T H. ApJ, 1999, 522: 1046
   \nref Pavlov G G et al, ApJ, 2001, 554L, 189
   \nref Shklovsky I S. Sov  Astron, 1970, 13, 562
   \nref Toscano M et al,  MNRAS, 1999, 307: 925
   \nref van den Heuvel E P J. 1987, Millisecond pulsar formation
and evolution, in D. J. Helfand and J. Huang (eds.), The origin
and evolution of neutron stars, IAU Symposium No. 125, pp393-406,
Reidel, Dordrecht
 \nref Weisskopf M C et al,  ApJ, 2000, 536L: 81


\end{CJK}
\end{document}
