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    {\small\rm\thepage\hfill
    \quad\quad ACTA ASTRONOMICA SINICA \hfill
    \hspace{2mm} Vol.44\vspace{2mm}}}} %%%%%%%
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    {\small\rm  suppl.\hfill X. Wu et al.:
Relation between Emission Altitude and Frequency of  Pulsars  %%�����ߣ���Ŀ���ƣ�
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\zihao{5}\songti

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    Vol.44~ Suppl.
    \hfill\hspace{8mm} %%% if ``October/January'' then \hspace{8/10 mm} July
    \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 The Study of the
Relation between Emission Altitude \\[2mm] and Frequency of  Pulsars   %%�����⣩
%\\[1ex]
%$\!\!^*$
}

\vs\vs{\normalsize Xinji Wu,\hs\hs Zhankui Huang\hs\&\hs Xuanbin Xu }  %%���������ߣ�

\vs
 (\footnotesize \pka)\\
\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
It is important to study the radial structure of pulsar radio
emission region; however, the altitude of emission region cannot
be derived directly from observation. Usually the altitude can be
expressed as a function of frequency, $r_{\nu}\propto\nu^{-\xi}$.
A method of K parameter analysis was applied to calculate the
power law index $\xi$ directly from the observational data. The
values of $\xi$  are obtained for 18 pulsars at two frequencies of
610\,MHz and 1408\,MHz. The mean value of the index $\xi$ is 0.27,
which indicates that the emission altitude increases with
decreasing frequency and that the radial structure is compact.

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

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

\vs\vs The radial structure of the emission region of pulsars has
been concerned since pulsar was discovered, but even the altitude
of emission can not be determined directly from observations.

The radius to frequency mapping model (RFM) was proposed by Cordes
(1978) to explain the systematic increase of the separation
between the profile components and the profile widths with
decreasing frequency. A few methods of calculating radial location
of radio emission at different frequencies have been put forward
by several authors (Cordes 1978, BCW91, Phi92, KG97, Wu99).

The interval of emission region for frequency 430\,MHz and
1400\,MHz was found to be less than $\sim10^2-10^3$\,km, or a few
percent of the light cylinder radius. Phi92 analyzed the timing
data of four pulsars. After subtracting the effect of retardation,
aberration and magnetic field sweep back, they found that the
radial separation of emission region at 47\,MHz and 4800\,MHz is
less than 200\,km. BCW91 found that the relativistic electron
emission and the rotation of pulsars could result in the delay of
the center of linear polarization position angle to the intensity
profile center by $\Delta=4r_{\rm delay}/c$. They obtained the
emission altitude $r_{\rm delay}$ at 430\,MHz and 1408\,MHz for 11
pulsars.

Based on the polar cap model, the altitude of emission can be
expressed as a function of period ($P$) and opening angle ($\rho$)
of the pulsar beam (KG97)
\begin{equation}
\rho=1.24sr_{6}^{1/2}P^{-1/2}
 \end{equation}
where $r_6=r/R$ is the emission altitude in unit of pulsar radius,
$s$ is the parameter that describes a locus of field lines
contributing to the emission. $0<s\leq1$, $s=0$ is the axis of the
magnetic field, $s=1$ the last opened field line. In KG's paper,
$s$ is assumed to be 1. In order to get the opening angle of
emission beams, KG97 used the geometric relation of polar cap
\begin{equation}
\rho=\cos^{-1}[\cos\beta-2\sin\alpha\sin\theta\sin^{2}(\triangle\phi/4)]\,,
 \end{equation}
in which $\theta=\alpha+\beta$, $\alpha$ is magnetic inclination
angle, $\beta$ is the angle between the line of sight and magnetic
axis, $\Delta\phi$ is the apparent width of mean profile in
degree. The emission altitudes can be calculated by equations (1)
and (2), the $\Delta\phi$ on different frequency, and $\theta$,
$\alpha$ published on journals. KG97 obtained the emission
altitude for 430MHz and 1420MHz. They argued that the altitude of
radio emission region depends on pulsar period approximately as
$P^{0.3}$(KG97). But KG's method is restricted by their adoption
of parameters $\beta$ and $\alpha$ from other literatures.
Obviously, the published data of $\theta$ and $\alpha$ given by
different authors are based on the different $\rho-P$ relations
(LM88, Kuzmin 1984, Xu \& Wu 1991, R93, G94). If one choose the
relationship $\rho\propto P^{-0.5}$, the emission altitude $r$
must be a constant. The value of parameter $s$ in equation (1)
also bothers us that only a part of Goldreich-Julian polar cap is
involved in emission processes. It means that $s$ is less than 1.

The form of the radius-to-frequency mapping depends on the physics
of the emission mechanism. Usually a power-law relationship
between emission altitude and frequency is assumed as following

\begin{equation}
r_{\nu}\propto\nu^{-\xi}\,,
 \end{equation}
where the $\xi$ is a constant. The index $\xi$ is an important
parameter to understand the radial structure of emission region in
pulsar magnetosphere.

\para{\large\bf{2}~~The Method of $K$ Parameters Analysis }

\vs\vs In order to study the evolution of inclination angle and
emission angle, Xu et al (1991) and Wu et al. (1992) defined a
parameter $K$ under the polar cap model

\begin{equation}
K=\frac{\sin\triangle\psi}{\sin\triangle\phi}=\frac{\sin\alpha}{\sin\rho}\,,
 \end{equation}
where $\Delta\Psi$ is the half polarization position angle swing,
$\Delta\phi$ is the half apparent beam width. $K$ is an observable
parameter since both $\Delta\Psi$ and $\Delta\phi$ are observable
parameters. $\alpha$ is the inclination angle and $\rho$ is the
half angle of emission cone.

The altitude ratio of emission at two frequencies can be obtained
by using equation (1) and equation (4):

\begin{equation}
m=\frac{r(f_{1})}{r(f_{2})}=\frac{\rho^{2}(f_{1})}{\rho^{2}(f_{2})}
\approx\frac{K^{2}(f_{2})}{K^{2}(f_{1})}\,.
 \end{equation}

From this formula it is straightforward to get altitude ratio of
radio emission of two frequency. The power-law index  can be
derived :
 \begin{equation}
m=\frac{r(610)}{r(1408)}=(\frac{610}{1408})^{-\xi}\,,
 \end{equation}
 \begin{equation}
\xi=2.75\log(\frac{K(1408)}{K(610)})^{2}=2.75\log m\,.
 \end{equation}

The $\xi$ value can be obtained from the $K$ parameters. So the
$\xi$ is also determined by observational data.  The great
advantage of this method is that it does not require to calculate
the opening angle $\rho$ of beam cone from the published values of
$\theta$ and $\alpha$, and does not need to consider the parameter
s in equation (1).

To get the position angle swing $\Delta\Psi$ is much more
difficult even for high precision radio observations. For some
pulsars the swing of the position angle is an 'S' curve, which can
be well understood within the rotating vector model (Radhakrishnan
\& Cooke, 1969). In the case of  'S' shape, the parameter
$\Delta\Psi$ can be measured easily. But in real situations the
polarization position angle curves are very complex, such as the
position angle jumps. Some curves are asymmetric or lack of a
large part of the 'S' curve. An investigation of the relation
between the position angle jump and the depolarization due to
relative longitude shift of pulsar beams is presented by Xu et al
(1997).

For those pulsars with complex polarization position angle curves,
the $\Delta\Psi$ cannot be obtained. In this paper we need to know
$K$ parameters on two or more frequencies, which means that all of
the pulsars in our sample must have good polarization data on at
least two frequencies. This strict requirement prevents us from
collecting a large database.

The polarization study of mean pulse profile of pulsars is
fundamental. Since 1990s, several groups have carried out
systematic polarization observations (Wu et al. 1990, 1993, Qiao
et al. 1995, Manchester et al, 1998, G94). The largest sample of
mean pulse profile and the polarization are presented in the
thesis of G94. 18 pulsars with good data of the $\Delta\phi$ and
$\Delta\Psi$ at 610\,MHz and 1408MHz are selected from G94. The
profile boundary cannot be determined precisely, we use the
apparent beam width at 10\% intensity levels. Rankin (1983) found
the phenomena of profile absorption for some pulsars. All these
pulsars whose profile might be affected by profile absorption are
excluded in our samples.


\para{\large\bf{3}~~The Power Index $\xi$ of Altitude-Frequency Relation }

\vs\vs Table 1 shows the observation parameters at 610\,MHz and
1408\,MHz and the calculated results of the altitude ratio $m$ and
the parameter $\xi$ for 18 pulsars. Obviously the $K$ analyses at
more than two frequencies are valuable to understand the property
of structure of emission region. However, there are only a few
pulsars with good polarization data at three or more than three
frequencies.

\bc{\small\bf

 Table 1~~  The observed data and derived parameters
}\\[2mm]

\fns \tabcolsep 2mm
\begin{tabular}{ccccccccc}
\hline\\[-3.6mm]\hline
  PSR B &$\triangle\phi_{610}$ &$\triangle\phi_{1408}$ & $\triangle\psi_{610}$ &
$\triangle\psi_{1408}$ & $K_{610}$ & $K_{1408}$ & $m$ & $\xi$ \\
\hline
0301+19& 18.2 & 16.3  & 117.5   & 122.5  & 5.41 & 6.18 &1.31  &0.32\\
0525+21& 20.4 & 18.8  & 153.2   & 153.2  & 5.49 & 5.96 &1.18  &0.19\\
0628-28& 38.0 & 35.6  & 120.3   & 120.3  & 2.66 & 2.84 &1.13  &0.15\\
1039-19& 19.6 & 17.7  & 131.5   & 142.7  & 5.36 & 6.16 &1.32  &0.33\\
1133+16& 12.1 & 11.4  & 88.6    & 88.6   & 6.63 & 7.03 &1.13  &0.14\\
1702-19& 18.0 & 17.3  & 101.3   & 101.3  & 4.94 & 5.14 &1.08  &0.09\\
1737+13& 23.8 & 22.0  & 154.4   & 177.2  & 4.73 & 5.24 &1.23  &0.24\\
1811+40& 15.7 & 15.1  & 139.2   & 145.6  & 6.86 & 7.27 &1.12  &0.14\\
1839-04& 81.9 & 70.7  & 182.3   & 183.5  & 1.53 & 1.73 &1.28  &0.30\\
1916+14& 9.4  & 8.0   & 139.2   & 139.2  & 11.4 & 13.4 &1.38  &0.38\\
2011+38& 51.1 & 43.7  & 31.6    & 32.9   & 0.63 & 0.76 &1.45  &0.45\\
2021+51& 19.7 & 16.9  & 62.0    & 62.0   & 3.01 & 3.50 &1.36  &0.36\\
2045-16& 17.4 & 16.1  & 143.0   & 158.2  & 6.27 & 7.01 &1.25  &0.27\\
2154+40& 27.5 & 25.6  & 126.6   & 125.3  & 3.76 & 4.01 &1.14  &0.15\\
2306+55& 26.6 & 24.1  & 153.2   & 154.4  & 4.23 & 4.67 &1.22  &0.24\\
2319+60& 26.1 & 22.9  & 101.3   & 101.3  & 3.42 & 3.90 &1.29  &0.31\\
2323+63& 37.0 & 34.2  & 183.5   & 183.5  & 3.15 & 3.40 &1.16  &0.18\\
2324+60& 31.3 & 26.4  & 145.6   & 177.2  & 3.54 & 4.38 &1.53  &0.51\\
 \hline\\[-3.6mm]\hline%\\[-4.5mm]\hline
 \end{tabular}
 \ec

All of the 18 pulsars have $m>1$ and $\xi>0$, which indicate that
the emission region of 610\,MHz is higher than the emission region
of 1408\,MHz. This conclusion is in agreement with RFM model. The
mean value of $m$ is 1.25$\pm$0.12, from 1.08 to 1.53; the power
law index ($\xi$) of altitude-frequency
($r_{\nu}\propto\nu^{-\xi}$) is 0.27$\pm$0.12, from 0.09 to 0.51.
These estimates show that the pulsar emission region is very
compact.

Theoretically the value of the spectral index $\xi$ is a constant
from $\xi=0$ (Barnad \& Arons 1986) to $\xi=2/3$ (Ruderman \&
Sutherland 1975). The results of 18 pulsars in this paper are
clearly desirable to discriminate the competing theory, and our
results consistent with the assumption of dipolar field in the
radio emission zone.

The average value $\xi$ we derived in this paper is comparable to
those obtained by using other methods (Table 2). All those values
of $\xi$ given by this paper and other papers listed in Table 2
are less than 0.66, which is an upper limit of $\xi$ given by
Phi92.

\bc{\small\bf

 Table 2~~   The comparison of the $\xi$ given by this paper and other
papers
}\\[2mm]

\fns \tabcolsep 4mm
\begin{tabular}{llll}
\hline\\[-3.6mm]\hline
0.21$\pm$0.1 &   0.43-1.42 &  11 & BCW91\\
 $\leq$0.66  &   0.05-4.80 &  4  & Phi92\\
 0.12$\pm$0.08 &  0.43-1.42 &  6  & GK93\\
  0.11$\pm$0.02  & 1.41-10.6 &  5 & K94\\
  0.21$\pm$0.07  & 0.43-1.42 &  10 & KG97\\
   0.26$\pm$0.09 &  0.43-1.42  & 16 & KG98\\
   0.21$\pm$0.1  &  0.43-1.42  & 8  & Wu99\\
   0.27$\pm$0.12  & 0.61-1.41   &18 & This paper\\
 \hline\\[-3.6mm]\hline%\\[-4.5mm]\hline
 \end{tabular}
 \ec

For some individual pulsars, the relations between emission
altitude and frequencies were obtained. For PSR1451-68 at
0.17-1.62\,GHz: $r({\rm km})=225f_{\rm GHz}^{-0.37}$ (Wu et al.
1998); for PSR 1857-26 at 0.17---2.65\,GHz: $r({\rm km})=249f_{\rm
GHz}^{-0.30}$ (R93); for PSR 2111+40 at 0.408---4.85\,GHz.


\para{\large\bf{4}~~Summary and Discussion}

\vs\vs The method of the $K$ parameter analyses was used to
calculate the power law index $\xi$ of altitude-frequency relation
($r_{\nu}\propto\nu^{-\xi}$) directly from observational data at
different frequencies. The values of $\xi$ are obtained for 18
pulsars at two frequencies of 610\,MHz and 1408\,MHz and for 3
pulsars at more than three frequencies. The average of the index
$\xi$ is 0.27, which is consistent with those obtained using
different estimation methods (Table 2). The emission altitude
increases with decreasing frequency, and the radial structure is
compact.

The altitudes of emission region at 610\,MHz and 1408\,MHz are
calculated using the method of R93 and GK93. The values of
$\alpha$ and $\beta$ were calculated using the new data of
apparent beam width ($\Delta\phi_{10}$) according the method of
LM88. The averaged value of emission altitude is 290.1$\pm$100.6
km above the surface of neutron star at 610\,MHz, and
236.4$\pm$86.4\,km at frequency of 1408\,MHz. The total size of
radio emission regions in the range between 0.61 and 1.408\,GHz is
only 54.2$\pm$29.2\,km.

There is close relation between the emission altitude and pulsar
period, which is in agreement with the results of GK93 and KG97.
But the method of calculating $\alpha$, $\beta$ may affect these
results of the relation between the altitude and period, due to
the $\rho-P$ relations introduced into their simultaneous
equations, such as $\rho\propto P^{-0.33}$ at 400\,MHz of LM88. If
such relation of $\rho\propto P^{-0.33}$ is applicable to
different frequencies, then we can get the relation $r\propto
P^{0.33}$ from equation (1), which is similar with the results of
$r\propto P^{-0.38}$ at 610\,MHz and $r\propto P^{-0.36}$ at
1408\,MHz given by this paper.


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

\bc {\bf References} \ec

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