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

\setcounter{page}{266}

\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  suppl.\hfill X. Xu \& X. Wu: A Method of Estimation for
    Magnetic Inclination Angle of 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 \bc {\Large\bf
A New Method of Estimation for Magnetic Inclination\\[2mm] Angle of Pulsars   %%（文题）
%\\[1ex]
%$\!\!^*$
}

\vs{\normalsize Xuanbin Xu\hs\&\hs Xinji Wu}  %%（本文作者）

\vs ({\footnotesize {\it Department of Astronomy, Peking
University, Beijing}~ 100871})\\
({\footnotesize {\it CAS-PKU Joint Beijing Astrophysics Center,
Beijing}~ 100871}) \ec

\normalsize
\begin{quotation}

\no{\bf A{\scriptsize\bf BSTRACT }}~~~\small Several methods to
estimate magnetic inclination angle need good polarization
measurement. We propose a new method for different classes of
pulsars based on the classification of profile and the
distribution of $Q$ parameter. The method is applicable to pulsars
without good linear polarization measurement but with good
classification. We calculated the magnetic inclination angles for
14 triple and multiple pulsars.

\end{quotation}

%%%% 正文
 \para{\large\bf{1}~~ Introduction}

Magnetic inclination angle $\alpha$ is a very important parameter
for pulsar. Several methods have been proposed to calculate the
magnetic inclination angle in the frame of magnetic pole model,
but there is no perfect method for most pulsars now.

Based on the geometrical magnetic pole model, one obtains two
equations:
 \begin{equation}
\cos\triangle\phi=\frac{\cos\rho-\cos\alpha\cos\theta}{\sin\alpha\sin\theta}\,,
 \end{equation}
\begin{equation}
(\frac{d\psi}{d\phi})_{\max}=\frac{\sin\alpha}{\sin(\theta-\alpha)}\,,
 \end{equation}
where $\theta$ is the angle between the rotation axis and the
observer's line of sight, 2$\triangle\phi$ is the total pulse
width in longitude, $\rho$ is the angular radius of emission beam
and $\alpha$ is the angle between the rotation axis and magnetic
axis. The maximum rate of the position angle swing could be
measured from high quality polarization. Since $\triangle\phi$ and
$(\frac{d\psi}{d\phi})_{\max}$ could be measured from observation,
only 3 parameters need to be determined, so we need another
equation besides eqs. (1) and (2).

Kuzmin(1984) first proposed the equation of the relation between
the emission beam radius $\rho$ and the period P, which reads:
 \begin{equation}
\rho=5.0P^{-1/2},
 \end{equation}
Combing this equation with (1) and (2), he obtained the angle
$\alpha$. Lyne and Manchester (LM, 1988) revised the  $\rho-P$
relation:
\begin{equation}
\rho=6.5P^{-1/3}\,.
 \end{equation}
They calculated $\alpha$ for about 100 pulsars. Though many
authors did not agree with their relation, the results of LM were
still used widely.

The statistic relation of  $\rho-P$ has been debated for many
years. Some authors suggested that the index should be -0.5. After
analyzing a large body of profile data, Rankin (1993) proposed the
core-double-cone model, which suggests that the emission beam
consist of a central core and two outercones. The $\rho-P$
relation given by Rankin is:
\begin{equation}
\rho_{\rm outcone}=5^{o}.75P^{-1/2},
 \end{equation}
\no This method needs high quality linear polarization angle swing
curve to calculate the maximum rate, however, it is difficult to
be obtained. Till now not more than 1/4 of the pulsars have
reliable measurements.

For the triple($T$) and the five-component pulsars with core, the
line of sight must travel across the core component. So it is near
the center of the emission beam. Rankin presumed that the impact
angle $\beta$ equal 0, then she gave:
\begin{equation}
W_{50}=2^{o}.45P^{-1/2}/\sin\alpha,
 \end{equation}
She (1991) also used this equation to estimate the $\alpha$ for
those pulsars with core component. But in fact it is difficult to
measure the half width $W_{50}$ of the core because it is often
superposed by the nearby components. Here we develop a new method
which is based on the classification of integrated pulse profile
and the distribution of $Q$ for different class pulsars.

\para{\large\bf{2}~~ The Relation Curve of Magnetic Inclination Angle $\alpha$ and $Q$}

\vs Wu et al.\,(1986) defined the parameter $Q$:
\begin{equation}
Q=\frac{|\theta-\alpha|}{\rho},
\end{equation}
The parameter is used to describe the trajectory of the line of
sight across the emission cone. It is different from $\beta$, the
impact angle, which can not represent the position of the sight
line traveling across the emission beam. However, the $Q$ is the
ratio of $\beta$ to $\rho$. A bigger $Q$ represents the sight line
sweeps across the outer emission beam; and a smaller one the sight
line nearer to the magnetic axis. When $Q=0$, the sight line
travels across the center of the beam. The range of $Q$ is: $0\leq
Q \leq1$.

Rankin\,(1993) proposed the core-double-cone model and she
classified the pulse profile into different types as
$M$(multiple), $T$(triple), $D$(double), $Sd$(conal single), $St$
(core single), $cQ$(conal quartre) and $cT$(conal triple).
According to the structure of the emission beam, the different
types will have different $Q$ values. Apparently for the $M$ and
$T$ pulsars, since the sight line across the core component, $Q$
should be small. For $cQ$, $cT$ and $D$ pulsars, the value of $Q$
should be medium, and for the $Sd$, $Q$ should be the largest. So
it is possible to give the $Q$ for different class.

  On the basis of the classification of Rankin we pick out the
pulsars with clear profile classification and good polarization
data, then calculate their values of $Q$ using the maximum rate of
position angle swing $(\frac{d\psi}{d\phi})_{\max}$ and the
relation between $\alpha$ and $\rho$. That is, we can obtain the
$Q$ from equation (1) and (2), and then using these values we can
calculate the average $Q$ for different classes. The average value
of $M$ and $T$ pulsars is $Q_{M}=0.29\pm0.12$.

\para{\large\bf{3}~~ The Relation Between $\alpha$ and $Q$}

\vs From equation (1) and (7), we can get:
 \begin{equation}
cos(2\alpha\pm\rho Q)=\cos\rho
Q-2\frac{\sin\rho(1+Q)/2\sin\rho(1-Q)/2}{\sin^{2}\triangle\phi/2},
\left\{\begin{array}{r@{\quad\quad}} \theta>\alpha
,+\\\theta<\alpha,-
\end{array}\right.,
 \end{equation}
where, $\theta<\alpha$ means that the sight-line's trajectory is
between the magnetic and rotation axes, and $\theta>\alpha$ means
the outer. The sign in the equation (8) is decided by the sign of
$(\frac{d\psi}{d\phi})_{\max}$:
\be (\frac{d\psi}{d\phi})_{\max}>0
\quad\quad \theta>\alpha,
 \ee
  \be
(\frac{d\psi}{d\phi})_{\max}<0 \quad\quad \theta<\alpha\,.
 \ee
The relation between $\alpha$ and $Q$ can be obtained from
equation (8). For the same value of $Q$, the magnetic inclination
($\alpha$) for $\theta<\alpha$ is larger than that for
$\theta>\alpha$. When $Q=0$, the $\alpha$ has the same value of
$\arcsin(\frac{\sin(\rho/2)}{\sin\triangle\phi/2})$ for the two
cases and then decreases with $Q$. When $\theta>\alpha$, $\alpha$
limits to $0^{o}$; and when $\theta<\alpha$, $\alpha$ limits to
$\rho$. According to the increase of $\triangle\phi$, the $\alpha$
falls quickly at first, but when $\triangle\phi/\rho$ becomes
larger, the magnetic inclination angle will not be sensitive with
$Q$. That is, the $\alpha$ will vary slowly with $Q$.

\para{\large\bf{4}~~ New Method to Measure the Magnetic Inclination Angle}

\vs After we get the average $Q$ value of different classes, we
can regard the mean value as the real value of this type pulsar.
Then using equation (8), $Q$ value, the $\rho-p$ relation and the
direction of the polarization position swing, we can obtain the
magnetic inclination angle. Even if the direction of the swing is
not available, we can also get two values for $\theta<\alpha$ and
$\theta>\alpha$ separately. Since the distinction of these two
values is so little, we take the average of them as the
inclination angle of this pulsar. The error of $\alpha$ results
from the dispersion of $Q$. We know that the less the $Q$, the
less the error; the larger $\triangle\phi/\rho$, the less the
error. The $Q$ of the triple and multiple is the least, so we
choose 14 pulsars of triple and multiple. In table 1 we list their
magnetic inclination angles estimated by our new method. Their $Q$
we used is 0.29 and the $\triangle\phi$ of mostly pulsars are much
larger than their $\rho$. So the estimation error is very small.

\bc{\small\bf

 Table 1~~The result of the inclination angle $\alpha$
}\\[2mm]

\fns \tabcolsep 4mm
\begin{tabular}{ccccccc}
\hline\\[-3.6mm]\hline
 PSR & Class  &$P(s)$ &$\triangle\phi (^o)$&$\rho(^o)$&
 $\frac{\triangle\psi}{\triangle\phi}$&$\alpha(^o)$\\
 B   &        &       & & & &\\
\hline
$0011+47$& T & 1.24  & 15.8   & 5.2 & $-$ & 17.3\\
$0059+65$& T & 1.68  & 10.9   & 4.4 & $-$ & 21.8\\
$0727-18$& T & 0.51  & 10.5   & 8.1 & $+$ & 50.3\\
$1607-13$& T & 1.02  & 17.7   & 5.7 & $+$ & 19.7\\
$1657-13$& T & 0.64  & 19.0   & 7.2 & $-$ & 19.7\\
$1738-08$& M & 2.04  & 10.1   & 4   & $+$ & 23.6\\
$1746-30$& T & 0.61  & 65.4   & 7.4 & $-$ & 4.8\\
$1756-22$& T & 0.46  & 7.6    & 8.5 & $-$ & 42\\
$1851-14$& T & 1.15  & 20.4   & 5.4 & $-$ & 18.7\\
$1857-28$& M & 0.61  & 18.9   & 7.4 & $-$ & 28.8\\
$1944+17$& T & 0.44  & 10.3   & 8.7 & $+$ & 35.9\\
$2000+40$& M & 0.91  & 13.7   & 6   & $+$ & 20.2\\
$2154+40$& T & 1.53  & 13.0   & 4.6 & $+$ & 15.5\\
$2319+60$& T & 2.26  & 26.1   & 3.8 & $-$ & 7.2\\
 \hline\\[-3.6mm]\hline%\\[-4.5mm]\hline
 \end{tabular}
 \ec

\vn{\bf ACKNOWLEDGEMENT}~~ This work was supported by the National
Natural Science Foundation of China under Grant 10073001.

%\newpage
\bc {\bf References} \ec

\vspace{-4mm} \no \fns

 \no Lyne A G, Manchester R N.  MNRAS, 1988, 234: 477\\
 \no Rankin J M, ApJ, 1983, 274: 333\\
 \no Rankin J M, ApJ, 1990, 352: 247\\
 \no Rankin J M, ApJ, 1993, 405: 285\\
 \no Rankin J M, ApJ, 1993, 85: 145\\
 \no Wu Xinji, Qiao Guojun, Xia Xiaoyang, Li Fang,  Astrophys. Space Science, 1986,
 119: 101\\

\end{CJK}
\end{document}
