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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 G. J. Qiao et al.: A Further Investigation on the Central Beam of PSR B1237+25 %%（作者：题目名称）
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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 \bc {\Large\bf
A Further Investigation on the Central Beam of PSR\\[2mm] B1237+25    %%（文题）
%\\[1ex]
%$\!\!^*$
}

\vs{\normalsize  G. J. Qiao,\hs\hs K. J. Li,\hs\hs H. G.
Wang,\hs\hs
R. X. Xu\hs\&\hs J. F. Liu\\}  %%（本文作者）
\vs
({\footnotesize {\it School of Physics, Peking University, Beijing}~ 100871}) %%（本文作者单位）
\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
By using Gaussian-components fitting, Qiao et al (2000) pointed
out that the central or core emission beam may be hollow. In this
paper, by developing a quantitative criterion for the fitting
procedure, we prove that the mean profile of the pulsar PSR
B1237+25 is composed of six Gaussian components, which confirms
that the core beam is hollow. From the fitted results, further
information of the radio emission region is obtained based on the
Invert Compton Scattering model (ICS model). The Lorentz factor of
secondary particles for this pulsar is estimated to be 1000
$\sim$3000.

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

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

\vs Various kinds of pulse shapes have been observed, which
results from a combination of certain configurations within the
radio emission beam and the viewing geometry (how the line of
sight sweeps across the emission beam). In the early days, the
limited handful of profiles are mostly single or double-peak ones,
leading to the hollow cone model (Radhakrishnan \& Cooke 1969,
Ruderman \& Sutherland 1975). As more and more complex profiles
were observed, Rankin (1983) firstly made an empirical
classification for pulsar average profiles and distinguished two
different kinds of emission components, the central 'core' and
outer 'cone' components, which was confirmed by Lyne \&
Mancherster (1988, hereafter LM88). However, whether the
mechanisms of the core and conal emission are the same (LM88) or
not (Rankin 83) is still an open issue.

Since the 90s' last century, the method of Gaussian decomposition
for the mean pulse profile has been developed (Wu et al 1992,
Kramer et al. 1994, Kuz'min \& Izvekova 1996, Wu et al 1998) to
study the emission structure of the pulsar. Qiao et al.(2000)
applied this method to PSR B1237+25, and found that six individual
components could be separated at five observing frequencies. They
suggested that the core beam may be hollow. This provides a severe
challenge to the previous concept that the core beam is solid
conceived by various models(e.g. Beskin et al 1988, Wang et
al.1989, Rowe 1992, Weaherfall \& Eilek 1997). The authors
suggested that the Inverse Compton Scattering (ICS) model (Qiao
1988 a,b, Qiao 1992, Qiao \& Lin 1998) could naturally explain the
very complex emission beam composed of a hollow core and two outer
cones.

Here we further investigate the profile decomposition of this
pulsar by improving the criterion of goodness-of-fit for the
fitting procedure. In section 2 the method of decomposition is
described and the results are presented. In section 3 we derive
the initial Lorentz factor of secondary particles and the impact
angle based on the ICS model. Conclusions and discussions are made
in section 4.


\para{\large\bf{2}~~The Gaussian Decomposition of PSR B1237+25 with New
Criteria}

\vs\vs In this section we introduce our new criteria and show the
numerical results of the decomposition of PSR B1237+25 at four
frequencies (0.1, 0.925, 1.4 \& 4.75\,GHz). The mean profiles of
high signal-to-noise ratio are chosen from the EPN database.

We follow the description of the decomposition approach given by
Wu et al. (1998). Suppose that N Gaussian functions are selected
to fit a mean profile, the reconstructed profile $f(x)$ is
\begin{equation}
f(x)= \sum_{i=1}^{N} \frac{h_{i}}{w_{i}
\sqrt[]{\pi/2}}e^{\frac{(x-p_{i})^{2}}{w_{i}^{2}}}\,,
\end{equation}

The difference between $f(x)$ and the observational data {$x_{\rm
i},y_{\rm i}$} is defined as

\begin{equation}
Q=\sum_{j=1}^{M}[f(x_{\rm i})-y_{\rm i}]^{2}\,,
\end{equation}

where $M$ is the number of the data. And the fitting residual is
\begin{equation}
Res=\sqrt\frac{Q}{M}=\sqrt\frac{\sum_{j=1}^M[f(x_i)-y_i]^2} {M}\,.
\end{equation}

It is easily to see that when $Q$ reaches its minimum so does the
$Res$.

For a given number of Gaussian components, two conditions must be
satisfied to obtain the best fitting. First, the residual should
be minimized, and second, the residual should have similar
amplitude to the off-pulse noise (Kuzmin et al 1996).

Obviously, the residual keeps decreasing when more and more
Gaussians are involved. But the best fitting is not to let the
residual as small as possible. Because the excessive ones are only
used to fit the noise, and these redundant components do not
reflect the reality. Here we present a new criterion to determine
the best-fitting number.

When we use less Gaussians than the real number of components to
fit the profile, each Gaussian fits a component. And the reduction
will become slower than before when excessive Gaussians are
involved, because the excessive Gaussians are just used to fit the
noise and the noise is much weaker than the emission component.
(That is why we chose the data with high signal to noise ratio.)
Therefore, the breaking point of the residual-to-number curve is a
good indicator of the real component number.

In this paper, we synthesize the above three points, viz., (a) the
breaking point, (b) the residual should have similar amplitude as
the off-pulse noise, and (c) the residual should be minimized, to
decide the best-fitting number.

The mean profiles of PSR B1237+25 are fitted by 4, 5, 6 and 7
Gaussians at four frequencies; an additional 8-component fitting
is made at 1.4\,GHz. The residual-to-number curve is presented by
Fig.1. We can see that when N=5 the residual is larger than the
off-pulse noise at each frequency. So the N should be greater than
5. The residual of 6 Gaussians is similar to that of 7. But the
braking point apparently indicates that the number should be less
than 7. So 6 is the best fitting.

\begin{figure}[h]
\center {\includegraphics [bb=60   187   545
589,scale=0.5]{1.eps}}

   \parbox[tb]{10cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,1~~ This figure give the relation between the
Q defined by (2) and the number of fitting components}

    \end{figure}



\para{\large\bf{3}~~Discussion}

\vs\vs The improved criterion make the result a further and
solider support to the previous suggestion that the core beam is
hollow (Qiao et al. 2000). In is section we reanalyze the physical
implications of the new data in the frame of ICS model.

The main idea of this model is: the low frequency ($\nu_{0}\sim
10^{6}$\,Hz) photons produced by the inner gap sparking are
inverse Compton scatted up to high frequency (observed radio
bands) photons by the relativistic particles moving along the
magnetic fields, when they are propagating through the magnetic
field line, (Qiao 1988a,b, Qiao \& Lin 1998). The frequency of the
up-scattered radio photons is
\begin{equation}
\nu\simeq2\gamma^{2}\nu_{0}\left(1-\beta\cos\theta_{i}\right)\,,
\end{equation}

where $\gamma$ is the Lorentz factor of the relativistic
particles, $\beta=(1-1/\gamma^{2})^{1/2}$, and $\theta_{i}$ is the
incidence angle between a particle and a low frequency photon.

From emission geometry we have the relation between the
$\theta_{\mu}$ and $\theta_{i}$, where $\theta_{\mu}$ is the
emission angle (cf. Qiao \& Lin 1998). $\theta_{\mu}$ is a
function of the peak separation $\Delta\phi$ between the two
counterparts of the same emission component (Gil 1984).
\begin{equation}
\sin^{2}\left(\theta_{\mu}\right/2)=\sin^{2}\left(\Delta\phi/4\right)
\sin\left(\alpha\right)\sin\left(\alpha+\beta\right)+\sin^{2}\left(\beta/2\right)\,,
\end{equation}

wherein $\alpha$ is the inclination angle between the rotation and
magnetic axes, and $\beta$ is the impact angle. we obtain
$\nu(\Delta\phi)$, which is dependent on a series of parameters
($\nu_{0}$, $\gamma_{0}$, $P$, $R$, $\alpha$, $\beta$) ($P$ is the
pulsar rotation period.).

We are now concerning the core emission. For PSR B1237+25,
$P=1.38$ s, $\nu_{0}$ and $R$ are assumed to be $10^{6}$ Hz and 10
km, respectively, $\alpha=48.2^{o}$ given by LM88 is accepted.
Then the free parameters reduce to two, $\gamma_{0} $ and $\beta$,
which are expected to be constrained by fitting the peak
separation of the core components. As demonstrated by Fig.2,
$\beta$ and $\gamma_{0}$ are confined from 0.9 to 1.1 degrees and
from 1000 to 3000, respectively. Our result of $\beta$ is
coincident with that of LM88 ($0.9^{o}$)
\begin{figure}[h]
\center {\includegraphics [bb=60   201   545
600,scale=0.5]{2.eps}}

   \parbox[tb]{10cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,2~~ The theoretical separations and the
separation given by decomposition. By this fitting procedure we
can constrain the $\beta$ and the initial Lorentz factor from 0.9
to 1.1 and from 1000 to 3000. The delta symbol is the results by
decomposition and the line is the theoretical results}

    \end{figure}

\para{\large\bf{4}~~Conclusions}

\vs\vs Basing on the Gaussian decomposition method, we develop a
new criterion for the decomposition procedure to determine the
exact number of Gaussians. The mean pulse profiles of PSR B1237+25
are thus decomposed to six Gaussians at four frequencies, which
are consistent with the results of Qiao et al.(2000) and provide a
further and solider support to their viewpoint that the core beam
is actually hollow.

The Inverse Compton Scattering model (ICS model) can naturally
explain the hollow core. By comparing the model results and the
phase separation between the two counterparts of the core, the
initial Loretz factor of secondary particles and the impact angle
are constrained to be from 1000 to 3000 and the impact angle from
0.9 to 1.1 degrees. The impact angle is consistent to LM88's
results.

\vs\vn{\bf ACKNOWLEDGEMENTS}~~ We are grateful to Mr.Wang X.D for
illuminative discussions. This work is partly supported by NSF of
China, the Climbing Project-the National Key Project for
Fundamental Research of China, and the Doctoral Programm
Foundation of Institution of Higher Education in China.

\bc {\bf References} \ec

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