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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 A. Zhou et al.:
    The Structure Function of PSR B0136+57 %%�����ߣ���Ŀ���ƣ�
    \vspace{1.5mm}
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\zihao{5}\songti

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    \zihao{6}\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 Structure Function of PSR B0136+57   %%�����⣩
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}

\vs\vs{\normalsize\zihao{4}\fangsong Aizhi Zhou$^{1,3}$,\hs Xinji Wu$^{1,2,3}$\hs\&\hs Ali Esamdin$^{2,1,3}$}  %%���������ߣ�

\vs
 (1.\pka)\\
 ({\footnotesize 2.{\it Urumqi Astronomical Observatory, NAO, CAS, Urumqi}~ 830011})\\
 (3.\bac)
 \vs \ec

\normalsize
\begin{quotation}

\no{\bf A{\scriptsize\bf BSTRACT }}~~~\small We calculate the
theoretical flux density structure function of PSR B0136+57 at
610MHz for both the Kolmogorov spectrum and the super-Kolmogorov
spectrum. The theoretical fitted curves for the case of
$\beta=11/3$ is in good agreement with the observation. The result
indicates that, the spectrum of the electron density fluctuations
is the Kolmogorov spectrum. To some extent, it supports the RISS
theory and the extended medium scattering model. It also provides
some helpful information of the modification of the model.

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

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

\vs Radio waves emitted by pulsars are dispersed or scattered by
the thermal plasma in the interstellar medium (ISM). This kind of
propagation effect can be classified into two different
scintillations: diffractive and refractive interstellar
scintillation (DISS and RISS). Small spacial scale
($10^6-10^8$\,m) irregularities in the interstellar medium can
cause DISS with characteristic time scale of $\sim$ minutes to
hours and frequency scale of $\sim$ 100\,KHz to 1\,MHz. The other
kind of scintillation, RISS, is often used to interpret the
long-term ($\sim$ days to months) flux variations (Huguenin 1973)
mainly at meter wavelengths.

As for the previous work, on the theoretical aspect, the nature
and magnitude of the fluctuations such as decorrelation bandwidth,
scintillation time scale, and so on have been predicted (Romani et
al. 1986). On the observational aspect, the flux modulations and
the variations in dynamic spectrum have been measured (Stinebring
\& Condon 1990; Labrecque et al. 1994; Gupta et al. 1994). Some
unusual multiple imaging and extreme scattering have also been
observed, owing to refractions by discrete structures in the ISM
(Fiedler et al. 1987).

However, the quantitative theoretical calculations of the
structure functions have rarely been carried out. In this paper,
we calculate the structure function for PSR B0136+57 with the RISS
theory and compare the result with the observation.

\para{\large\bf{2}~~ Theoretical Calculation}

\vs This pulsar was observed by Stinebring (2000, hereafter
S2000). Its structure function has the obvious saturation value
and the span of $D(\tau)$ is in a large range of magnitude.

Romani and Blandford developed the RISS theory (Romani et al.
1986; Blandford et al. 1986). For the case of $\beta=11/3$ and
$\beta=4$, the autocorrelation function is as follows:

 (1) $\beta$=11/3(Kolmogorov spectrum)
\begin{eqnarray}
F(\tau)&=&\frac{2Q_{\ast}(\frac{\lambda}{2\pi})^{4}\Gamma(7/6)}
{\pi^{3/2}2^{5/6}}H^{-1/3}\int^{d_{H}}_{0}dyy^{-1/3}
<\theta^{2}_{s}(y)>^{-7/6} \nonumber\\& &
(1-\frac{y}{d_{H}})^{2}\exp(-y^{2})M(7/6,1, -x),
 \end{eqnarray}
 where
 \begin{equation}
 Q_{\ast}=3.7\times10^{-18}D\ C_{-4}\,({\rm cm^{-11/3}})\,,
\end{equation}
 \begin{equation}
   \theta_{s}=2.2D^{0.6}C^{0.6}_{-4}\lambda^{2.2}\,({\rm mas})\,.
\end{equation}

  (2) $\beta$=4
   \begin{equation}
F(\tau)=\frac{Q_{\ast}(\frac{\lambda}{2\pi})^{4}}
{\pi^{3/2}}\int^{d_{H}}_{0}dy<\Theta^{2}_{s}(y)>^{-1}
(1-\frac{y}{d_{H}})^{2}\exp(-y^{2})
 M(1, 1, -x),
 \end{equation}
where
 \begin{equation}
 Q_{\ast}=1.6\times10^{-21}DC_{-4}\,({\rm cm^{-4}}),
\end{equation}
\begin{equation}
\Theta_{s}=2.2D^{0.5}C^{0.6}_{-4}\lambda^{2}\,({\rm mas})\,.
\end{equation}
The structure function $D$($\tau$) can be written as:
\begin{equation}
D(\tau)=2\left[\times F(0)-F(\tau)\right]\,.
\end{equation}
We calculated the structure function under formulas (1)-(7) using
the software Methematical 4.0.

Note that, there are five parameters in the calculation: $\beta,
H, D, V, C_{-4}$. $\beta$ is the power law index. $D$ and $H$ are
pulsar distance and the distance from the observer to the
scattering screen, respectively. $V$ is the transverse velocity of
the pulsar relative to the earth and the scattering screen, namely
${\bm V}_f={\bm V}_{\rm pm}+{\bm V}_s+{\bm V}_{e}$ , where ${\bm
V}_{\rm pm}, {\bm V}_{s}, {\bm V}_{e}$ represent the proper motion
velocity, the velocity of the scattering screen and the earth,
respectively. $C_{-4}$ is written as
$C_{-4}=\overline{C^{2}_{N}}/10^{-4}$, where $C_{N}^{2}$
characterizes the scattering strength and is a function of
location($z$) in the Galaxy. The selection of the parameters is
very important. We calculate the structure function for both
$\beta=11/3$ and $\beta=4$ by taking the same parameters. The
distance 2.9\,kpc is derived from DM, based on Taylor \& cordes'
(1993) electron density model. The fitted distance from the
observer to the scattering screen, $H$, is 1.45\,kpc, which means
that the scattering screen locates midway between the observer and
the pulsar.

The velocity is also a very important parameter. The proper motion
velocity of this pulsar is 300\,km/s (S2000). We take this value
and get a relatively good result for $\beta=11/3$. As for the
scattering strength, the fitted value of $10^{1.5}$ ${ m^{-20/3}}$
is equal to the observed value from S2000.

Finally, the fitted results for the two cases are quite different.
For $\beta=11/3$, the calculated result is consistent with the
observation very well. However, for $\beta=4$, the theoretical
curve departs very far from the observation. So there is a strong
evidence that the spectrum of the electron density irregularities
along the line of sight of this pulsar is the Kolmogorov spectrum,
not the super-Kolmogorov spectrum, which is in agreement with
Stinebring' opinion (S2000).

\para{\large\bf{3}~~ Analysis of the Parameters}

\vs\vs In figure 2, we show the effects of the parameters on the
structure function. In each figure, the solid line is the best
fitted curve with the parameters: $\beta=11/3;\ V=300\,{\rm
km/s};\ D=2.9\,{\rm kpc};\ H=1.45\,{\rm kpc};\
C_{-4}=10^{1.5}\,{m^{-20/3}}$. Keeping the other four parameters
invariable, we change one parameter during calculation. Figure
2a-2d show the variations of $H, V, C_{-4}$ and $D$, respectively.
\begin{figure}[h] \center {\includegraphics [bb=0 0 283 237,scale=1]{figure1.eps}}

   \parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,1~~ Observed (the solid dots) and theoretical flux density structure functions of pulsar
      PSRB0136+57. Solid and dotted lines represent the
     structure functions for the case of $\beta=11/3$ and $\beta=4$, respectively.}

    \end{figure}
    \begin{figure}[h]
\center {\includegraphics [bb=0 0 316 256,scale=1]{figure2.eps}}

   \parbox[tb]{12cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize Fig.\,2~~ Structure function vs time lag on the parameters of
      $H, V, C_{-4}$ and
      $D$ for PSR B0136+57. Solid dots and lines are the
      observational data (S2000) and the best fitted curves, respectively.
(a) The dashed, solid and dotted lines represent $H=2\,{\rm kpc},
1.45\,{\rm kpc}$ and $1\,{\rm kpc}$, respectively. (b)The dashed,
solid and dotted lines represent $V=400\ \rm{km/s},\ 300\
\rm{km/s}$ and $200\,{\rm km/s}$, respectively. (c)The dashed,
solid and dotted lines represent $C_{-4}=10^{1.6}$, $10^{1.5}$ and
$10^{1.4}$, respectively. (d)The dashed, solid and dotted lines
represent $D=3.6\,{\rm kpc},\,2.9\, {\rm kpc}$ and $2.2\,{\rm
kpc}$, respectively.}

    \end{figure}

    As for the velocity, it does not affect the
saturation value, but it changes the structure function of the
structure region. The structure function of the structure region
increases with the velocity. Seen from figure 2b, the proper
motion velocity value of 300\,km/s may be larger. The smaller
velocity is more realistic and better. For example, the dotted
curve adopting the velocity value of 200\,km/s is better
consistent with the observation. One reason is that, the
scintillation velocity is obviously different from the proper
motion velocity. Maybe the velocity of the scattering screen is
too high, thus leading to the lower transverse velocity. The other
likely interpretation is the measurement error of the proper
motion velocity.

Figure 2c shows that, the theoretical curve descends with the
scattering strength. Talor \& Cordes (1993) thought that, for most
known pulsars, the new electron density model provides distance
estimates accurate to $\sim$25\% or better. So we modulated the
distance to calculate the structure function, and found that the
curve descends with the distance.

The effect of $H$ on structure function is shown in figure 2a.
When it increases, the structure function of the structure region
decreases, while the saturation value increases. But when the
scattering screen is near to the pulsar, it has little effect on
the structure function.

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

\vs\vs The fitted curves and comparison with the observation are
shown in figure 1. There are three important quantities to
characterize the structure functions of the pulsars: $m, \tau_{r},
p$. $m$, the modulation index, denotes the flux variation
intensity. We calculate it by this formula:
$m=\sqrt{D_{\infty}/2}$, where $D\infty_{}$ is the saturation
value. $\tau_{r}$ expresses the value of refractive time scale and
is defined as the time lag at which the structure function reaches
one-half its saturation value. The theoretical time scale is
calculated by $D(\tau_{r})=D_{\infty}/2$. The relationship
$D(\tau)\propto\tau^{p}$ is used to calculate $p$.

The theoretical modulation index value of 0.12 is almost equal to
the observed value of $0.15\pm0.02$. The calculated time scale 21
days is shorter than the observed value of 26 days, and the
theoretical slope 0.74 is different from the observed value of
1.4. However, the fitting curve is still in the range of the error
bars, so the calculated result is consistent with the observation.
It proves that, to some extent, the RISS theory can account for
the long-term flux variation of this pulsar. Maybe we can adopt
the ``two component'' model (Qian et al. 1995) to get a better
result.

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

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

\vspace{-5mm} \no \fns

\no Blandford R, Narayan R, Romani R W.  ApJ, 1986, 301: L53\\
 \no   Cordes J M, Weisberg J M, Boriakoff V. ApJ, 1985, 288:
 221\\
\no  Cordes J M.  ApJ, 1986, 311: 183\\
\no  Fiedler R L, Dennison B, Johnston K J, Hewish A. Nature,
1987, 326: 675\\
\no  Gupta Y, Rickett B J, Lyne A G,  MNRAS, 1994, 269: 1035\\
\no  Huguenin G R, Taylor J H,  Helfand D J,  ApJ, 1973, 181L:
L139\\
\no LaBrecque D R, Rankin J M, Cordes J M,  AJ, 1994, 108: 1854\\
\no Qian S J, et al.  A\&A, 1995, 295: 47\\
\no Romani R W, Narayan R, Blandford R.  MNRAS, 1986, 220: 19\\
\no Stinebring D R, et al.  ApJ, 2000, 539: 300\\
\no Stinebring D R, Condon J J.  ApJ, 1990, 352: 207\\

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