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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 X. H. Sun \& J. L. Han: Pulsar Motions in Our Galaxy  %%�����ߣ���Ŀ���ƣ�
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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
Pulsar Motions in Our Galaxy    %%�����⣩
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\vs\vs{\normalsize X.H. Sun\hs\&\hs  J.L. Han}  %%���������ߣ�

\vs
 (\footnotesize \nao)\\
 E-mail: xhsun@bao.ac.cn (XHS), hjl@bao.ac.cn (JLH)
\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
We simulate motions of $2\times10^5$ pulsars in our Galaxy with
Monte-Carlo method and try to understand the evolution of pulsar
 height distribution. Within about 8\,Myr, pulsar height distribution can be
fitted by a Gaussian function with a scale height linearly
increasing with time. An extra exponential function is necessary
to be added to the Gaussian function to fit the height
distribution afterwards. The Gaussian scale height increases
linearly until about 40\,Myr. After about 200\,Myr, the height
distribution is stabilized. Based on the simulation results, we
obtain the initial 1D velocity dispersion of
80\,km\,s$^{-1}$ for millisecond pulsars.

\end{quotation}

\vs\vs Pulsar motions have been involved in population synthesis (e.g.
Bhattacharya et al. 1992), but not explicitly clarified. Some
authors tried to deduce scale height evolution theoretically, but
they had to assume a simple Galactic potential (e.g. Arnaud \&
Rothenflug 1981) or a simplified height evolution (Narayan \&
Ostriker 1990). In this contribution, we explore the height
evolution with Monte-Carlo simulation.

The initial height distribution of pulsars is assumed to be a
Gaussian with a scale height of 60\,pc as that of the progenitors
i.e. OB stars (Ma\'iz-Apell\'aniz 2001). The radius
distribution is assumed to follow a Gamma function (Paczy\'{n}ski 1990). The
final results are insensitive to the form of initial height or
radius distribution. Initial velocity is the vectorial sum of
Galactic rotation velocity and kick velocity from asymmetric
supernovae explosions. We assumed the kick velocities follow a
Maxwellian distribution, with trial 1D velocity dispersion
$\sigma_{\rm k}=$100, 200, 300 and 400\,km\,s$^{-1}$,
respectively. After comparing all the potentials available, we
used the one given by Paczy\'{n}ski (1990) which could reproduce
the Galactic rotation curve and the local volume density near the Sun
better than others.

\begin{figure}[tbph]
\center {\includegraphics [scale=0.5,angle=-90]{Fig1.eps}}
\parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize
 Fig.\,1~~ The evolution of scale heights of Gaussian components $h_g$
(lower panel) and exponential components $h_e$ (upper panel).
Different lines correspond to different initial 1D kick velocity
dispersion ($\sigma_{\rm k}$).}

\end{figure}

Provided these initial conditions described above, we solved the
Newtonian kinetic equations using {\it 4}th order Runge-Kutta
method with an adaptive step size (Press et al. 1992), then
tracked all the pulsars from 0\,Myr to
 500\,Myr. The scale heights are shown in Figure 1. Within about 8\,Myr, the
distribution can be fitted very well by a Gaussian function with a
scale height $h_g$ increasing linearly with time as $h_g\approx
h_0+\sigma_{\rm k} t$. As time goes, a peak appears near the Galactic
plane more and more prominently, so the distribution has to be
fitted by a Gaussian function {\it plus an exponential function}
with a scale height $h_e$. The scale height $h_g$ keeps linear
increase until about 40\,Myr. After about 200\,Myr, both $h_g$ and
$h_e$ gets stabilized.



Now we try to understand the distribution of available pulsars
using the simulation results.

Because the height distribution is stabilized after about
200\,Myr, the birth rate, if does not vary much with time, has no
effects on the final distribution. We can compare height
distribution of our simulated pulsars with that of observed MSPs
directly. The K-S test was employed to check the consistency of
the distribution of known MSPs and the distribution from
simulation with initial 1D velocity dispersion  from
30\,km\,s$^{-1}$ to $180$\,km\,s$^{-1}$. Finally, we found that
$\sigma_{\rm k}=80$\,km\,s$^{-1}$ was the best (Figure 2). So the
most possible initial 1D velocity dispersion for MSPs is about
80\,km\,s$^{-1}$.

We binned all known pulsars with characteristic ages
1\,Myr$<\tau<$8\,Myr into 4 groups. Through Gaussian fitting, we
obtained four scale heights which should have linear relation with
ages. Surprisingly, the resultant 1D velocity dispersion is
31$\pm$13\,km\,s$^{-1}$, too small compared with 3D velocity
dispersion
 of 500\,km\,s$^{-1}$ by Lyne \& Lorimer (1994). We understand that the
distances of most pulsars are less that 8~kpc and they were
discovered in low latitude surveys (e.g. $|b|<5^\circ$ in Parkes
multibeam survey (Manchester et al. 2001)). The high velocity
pulsars with a velocity of 400~km~s$^{-1}$ will statistically move
to 800~pc in 2~Myrs, that is $5.7^\circ$ at 8~kpc. So the majority
of known pulsars are low speed old pulsars plus some high speed
young pulsars. The population synthesis would be difficult to
constrain the initial velocities using this sample of low speed
known pulsars available.

\begin{figure}[tbhp]
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[scale=0.25,angle=270]{Fig2.eps}
\par\vspace{0pt}
\parbox[tb]{6.5cm}{\footnotesize  Fig.\,2~~The K-S test for 1D velocity dispersion of MSPs between simulated
and known MSP populations. }
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[scale=0.25,angle=270]{Fig3.eps}
\par\vspace{0pt}
\parbox[tb]{6.5cm}{\footnotesize  Fig.\,3~~The scale heights versus characteristic ages of known normal
pulsars, indicating a kick velocity dispersion of
31$\pm$13\,km\,s$^{-1}$. The number of pulsars in each bin is
marked at the top.}
\end{minipage}
\end{figure}

\vspace*{1mm} \noindent{\bf ACKNOWLEDGEMENTS}~~ Many thanks to
Professor Andrew Lyne for helpful discussion. This work is
supported by the National Natural Science Foundation of China
(19903003 and 10025313) and the National Key Basic Research
Science Foundation of China (G19990754) as well as the partner
group of MPIfR at NAOC.


\bc {\bf References} \ec

\vspace{-4mm} \no \fns

 \nref Arnaud M., Rothenflug R.,  A\&A, 1981, 103: 263
 \nref
Bhattacharya D., Wijers R.A.M.J., Hartman J.W., et al,  A\&A,
1992, 254: 198
 \nref Lyne A.G., Lorimer D.R., Nat, 1994,
369: 127
 \nref Ma\'iz-Apell\'aniz J., AJ, 2001, 121: 2737
\nref Manchester R. N. et al., MNRAS, 2001, 328: 17 \nref Narayan
R., Ostriker J.P., ApJ, 1990, 352: 222
 \nref
Paczy\'{n}ski B., ApJ, 1990, 348: 485
 \nref Press W.H.,
Teukolsky S.A., Vetterling W.T., Flannery B.P., Numerical Recipes
in Fortran, second edition, 1992

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