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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. Jessner: Some Observational
    Constraints on Theoretical Models %%（作者：题目名称）
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\zihao{5}\songti

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    \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
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\vs\vs\vs \bc {\Large\bf
How Pulars Provide Some Observational Constraints \\[2mm] on Theoretical Models    %%（文题）
%\\[1ex]
%$\!\!^*$
}

\vs\vs{\normalsize A. Jessner}  %%（本文作者）

\vs
({\footnotesize {\it Max-Planck-Institut f\"ur Radioastronomie,
Auf dem H\"ugel 69,  D-53121Bonn, Germany}}) %%（本文作者单位）

\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
Pulsar observations at Effelsberg have a long history and cover a
wide frequency range
 (800\,MHz --- 43\,GHz). Our equipment enables us to make high quality radio %astronomical
 observations in all important aspects of pulsar study.

High precision pulsar timing observations enabled us to find an
upper limit to the radius of the companion star close to the
theoretical minimum from the influence of
 spin-orbit coupling on the orbital parameters of  PSR J2051--0827.
Studies of interstellar scattering for nine pulsars with high
dispersion measure show a much flatter dependence of the
scattering time with frequency than predicted by simple single
screen Gaussian or Kolmogorov ISM models. A determination of
rotation measure for 46 pulsars enabled us to conclude,
 that no large scale reversals of the magnetic field occur within
 $85\deg <l<245\deg$ and the
 magnetic field follows the local spiral closely between $150\deg<l< 245\deg$,
 but the turbulent
structure of the intervening field defies modelling with a single
scale height and  typical size of the random magnetic field
component.
 Investigating
single pulses of  PSR B0329+54 simultaneously received in Jodrell
Bank and Effelsberg,
 it became evident, that  between the two frequencies, the centre component correlates
 strongly for circular polarisation, but not for linear polarisation, whereas both outer
 components exhibit  the converse behaviour.  The emission process has a wide bandwidth,
but the outer (conal) profile components  cannot originate at a
lower height  compared to the central component, if the observed
fluctuations are to be a magnetospheric propagation effect.
Pulsars also show unexpected behaviour at the very high radio
frequencies ($> 10\rm GHz$)
 where they are still visible,- if only for the Effelsberg instrument. In a number
 of cases
 the received fluxes are higher than expected, suggesting an up-turn in their spectrum.
 Furthermore, their variability is higher than predicted and it is likely that
 we observe intrinsic intensity variations at these frequencies.
Observations indicate that radioemission is generated within the
light-cylinder, but that fact in itself places a tight upper limit
on the product of particle density and Lorentz factor. In the case
of the crab pulsar, this density is much less than the assumptions
of standard pulsar models. Coherent curvature radiation  is also
shown to be too inefficient to account for radio emission below
10-20\,GHz.

\end{quotation}

\para{\large\bf{1}~~ High Precision Pulsar Timing: \\[2mm]Spin-Orbit Coupling in
the J2051--0827 System}

\vs\vs
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
The eclipsing binary pulsar J2051--0827 was discovered at Parkes
 (Stappers et. al. 1996).
It has a very short orbital period of $P_b > 2.4$~hrs and an
almost circular orbit, eclipsing about 10\% of the orbital period.
The companion is a white dwarf with $0.027 M_{sun} < m_{comp} <
0.055 M_{sun}$. The semi-major axis has a radius of  $1.03
R_{sun}$ and the orbital period decreases at a rate of  $\dot P_b
= -15.8(8)\cdot 10^{-12}$ which implies a decay time of about
$\sim 25\cdot 10^6$ years. Observations were made in Effelsberg (
0.86, 1.4, 1.7, 2.7\,GHz) and Jodrell Bank (0.325, 0.41, 0.61,
1.4\,GHz) for over 6.5 yrs yielding 584 TOA's (Doroshenko et al.,
2001). A complete orbit was observed each time, but orbital phases
from 0.2 to .35 (near the eclipse) were excluded from the fit. The
TOA's were fitted using the TIMAPR package (Dorsohenko \& Kopeikin
1995) with average post-fit residuals of $\sigma=21.2 \mu\rm s$. A
number of effects, such as the emission of gravitational waves,
acceleration of the binary system, mass loss from the system,
tidal dissipation and gravitational quadrupole coupling can in
principle contribute to the observed orbital decay. We found
however, that except for the case of gravitaional quadrupole
coupling, the mentioned effects are too small by at least  two
orders of magnitude if compared to the observed value
 (Doroshenko et al., 2001).   We also measured the second derivative of
$P_b$ as $\ddot P_b=+2.1(3)\cdot 10^{-20} \rm s^{-1}$.  Its magnitude implies that the presently
decreasing orbital period will begin to increase after $-2\dot P_b / \ddot P_b \sim 33 \rm yrs$ and the
that the system is undergoing quasi-periodic oscillations with a cyclic spin-up and spin-down
similar to PSR B1957+20. Like in some normal stellar binaries (Algol, RS CV) the outer
layers
of the companion of J2051-0807 are thought to spin up and down in
cycles providing a variable inertial quadrupole moment and an exchange
between  orbital angular momentum {\bf L} and spin of the companion {\bf S}.
This will result in a precession of both {\bf L} and {\bf S} which can be detected
as a variation of the
 projected semi-major axis a. Our measurements
yielded
\footnote[1]{distances are given as light travel times}
 $\dot x = -23(3)\cdot 10^{-14} \rm s\cdot s^{-1}$.
 In this case the semi-major axis $a_p$ will remain constant, but its projection
$x=a_p \sin i$ will undergo secular variations resulting in a change of the inclination
of
${di \over dt} ={\dot x \over x}\tan i$. Depending on the actual inclination
$30\deg < i < 89\deg$ we get a range of change for $i$ as
$0.007\deg {\rm yr^{-1}}< di/dt < 0.67\deg {\rm yr^{-1}}$.
Again, an investigation of other possible causes of the inferred $\dot x$ showed
them to be too small by
four to five orders of magnitude. For spin-orbit coupling, the secular change of
$i$ depends on the fifth power
of the radius $R_c$ of the companion:
\begin{equation}
{di \over dt}={\Omega_b\over 2}{k_{\rm apsid} R_c^2 \over a_p
(1-e^2)}{\Omega_c^2 R_c^3 \over \Gamma m_c}\,,
\end{equation}
here $k_{\rm apsid}$ is the apsidal motion constant and  for
synchronous rotation of the companion the orbital $\Omega_b $ and
companion spin $ \Omega_c$  frequencies are the same. One can thus
find an upper limit for the radius of  companion of $R_c < 0.06
R_{\rm sun}$ and a most probable value of $R_c \le 0.03 R_c$ for
$i=40\deg$. The upper limit is still only half the minimum size of
the Roche lobe $R_L = 0.13 R_{\rm sun}$ for a $0.027 M_{\rm sun}$
companion, but the most probable value is alarmingly close to the
minimum size of the smallest  $0.027 M_{\rm sun}$ star, a pure He
white dwarf. In fact, B.P. Gong (these proceedings) has pointed
out that the spin-orbit coupling of the neutron star itself will
also contribute to ${di \over dt}$ which would make that estimate
even tighter.

\para{\large\bf{2}~~Scatter Broadening:\\[2mm] An Example of a Multifrequency Profile
Study}

 \vs\vs On its way from the source the pulsar signal is
scattered by inhomogeneities in the interstellar medium,
propagates on multiple paths and thus aquires random phases. This
results in an exponential tail in the observed pulsar profile
$P_{obs}(t)$
 which can be described as a convolution
of the unscattered profile $P_0(t)$ with an exponential scattering kernel
\begin{equation}
P_{obs}=P_0(t)\otimes e^{-t \over \tau_{sc}}\,,
\end{equation}
where $\tau_{sc}$ is the scattering timescale.
Profiles of nine highly dispersed pulsars were obtained in Effelsberg
 (1.4, 2.7 and 4.9\,GHz) and in Jodrell Bank (0.6, 0.9 and 1.4\,GHz).
They were carefully selected for negligible profile evolution (L\"ohmer et al, 2001).
 Their outer pulse components have flatter spectra and pulse widths decrease with
increasing frequency. These effects are however quite small above
1\,GHz. Their decorrelation bandwidth $\Delta\nu \sim 50 \rm Hz$
at 1\,GHz is hard to measure, hence these were the first
measurements of  the scattering spectral index for high-DM
pulsars.

\begin{figure}[h]
\center {\includegraphics [bb=0.0 0.0 845.7600
576.9600,scale=0.3]{scatsp0.eps}}

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

      Fig.\,1~~ The graph shows
the average scatter broadening time for the nine objects with
their $3\sigma$ errors and a power-law fit. Dashed lines are
examples of the expected dependence from a Kolomogorov spectrum
with $\nu^{-4.4}$. The lower rightmost panel shows the observed
average dependency of $\tau_{sc} \propto \nu^{-3.4 \pm 0.13}$.}

    \end{figure}
A fit of  several Gaussian components to the high-frequency
(4.85\,GHz) profile was used as a template for convolution with
the exponential scattering kernel. The optimal scattering time was
determined for each frequency, by fitting the convolved templates
to the observations. In contrast to previous measurements of low
DM objects which were compatible with a $\tau_{sc} \propto
\nu^{-4.4}$ Kolmogorov spectrum
 (Cordes et al., 1985;  Johnston et al., 1998),
 highly dispersed pulsars show a much flatter decay of $\tau_{sc}$ than even
 the simple Gaussian model (Lambert \& Rickett 1999) with it $\tau_{sc} \propto \nu^{-4}$  dependence.
While individual measurements may indeed be affected by the geometry and spectral model
represented by the $3\sigma$ error bars , their trend isn't and the derived spectral
index
cannot be an artifact of the template fitting procedure.
An attempt was also made to use an alternative scattering kernel $t^{-1/2} e^{-t\over
\tau_{sc}}$
 (Cordes, priv. comm.) but this could not fit the scattering tail as accurately
 as the simple exponential.

A simple model with a single thin screen cannot describe the observed frequency
dependence for high DM pulsars. But multiple screens of smaller transverse extent
could in principle flatten the frequency dependence by reducing the amount of
scattering at lower frequencies

\para{\large\bf{3}~~Rotation Measures And The Large Scale Structure of The
Galactic\\[2mm]Magnetic Field}

 \vs\vs
Pulsars are excellent probes of the ISM and the magnetic field.
From a determination of the rotation measure $RM$ and the
dispersion measure $DM$ one can calculate the effective magnetic
field projected  onto the line of sight $B_{||}=1.23 {RM\over DM}
\rm \mu G$. For the observations we used the coherent
Effelsberg-Berkeley Pulsar Processor (EBPP) in its bandspread
mode, providing 4 x 8 polarisation channels across the receiver
band width of 98\,MHz. The bandwidth of each of the individual 32
channels depends on the DM of the observed pulsar, but is limited
to 0.875\,MHz for the polarization mode at around 1.4\,GHz. In
total, a bandwidth of 28\,MHz is avaliable, spread around in four
sub-bands of 7\,MHz, placed at the very edges of the receiver
bandwidth and in two symmetrically placed central regions.  All
signals are detected and folded in phase with the topocentric
pulsar period.  Each sub-integration, which typically lasts for
180--300 s,  is transferred to a computer for offline processing.
The four Stokes parameters are computed for each of the 32
channels and a RM search is made so that their superposition
maximises the linear polarisation. Altogether 45 PSR's were
observed in that mode, but only 35, unaffected by electron density
anomalies like HII regions and SNRs, were selected to estimate the
effective magnetic field strength of the Perseus arm within
$85\deg < l < 245\deg$ and $|b| < 10\deg$. We consider a model
where the regular uniform magnetic field of magnitude $B_{\rm o}$
traces the spiral arm as depicted by Georgelin \& Georgelin (1976)
and use a cubic spline fit as adopted by Taylor \& Cordes (1993)
to construct the spiral arm. For each respective Galactic
longitude of the pulsar we obtain the direction of $\vec{B_{\rm
o}}$, we then project onto the radial vector to obtain $B_{\rm p}$
as a function of $l$.  The resulting dependence of $B_{\rm p}$ on
$l$ is then used to fit the data points.  In addition to the field
strength, $B_{\rm o}$, we allow for another free parameter to be
determined, i.e. $\theta$ which describes the possibility that the
so called `magnetic arm' might actually be at an angle with
respect to the spiral arm, a circumstance which is also commonly
observed in external galaxies (Beck 2000). A positive value of
$\theta$ would mean that the spiral arm needs to be rotated around
the Galactic Centre in the counter clockwise direction by that
amount to match the `magnetic arm' to the regular component of the
field. A least-square fit (Mitra et al. 2002) to the data gives
$B_{\rm o}$ as 1.5$\pm$1 $\mu$G and $\theta = 12^{\circ} \pm
8^{\circ}$.
 The estimate of $B_{\rm o}$ is consistent with that
of Lyne \& Smith (1989).
 But even with the carefully selected sources we find a great variability of  $B_{||}$
 in our data
and evidently a model with a uniform magnetic field along the spiral arm, or even along
the line-of-
sight is too simple! In fact we should expect random variations of the electron density
$n_e$,
 as was indicated i.e. by scattering (see above)!
 Similarly,  the fit residuals should reflect the distance $D$ dependence along
 the ``random walk" through
 the variable field: $\sigma_{\rm rm}(D)=0.81 \times \delta n_e B_r\sqrt{L_r D}
 \rm{rad\,m^{-2}}$.
Here $B_r \sim 5\rm\mu G$ is the typical amplitude of the random magnetic field
component, and $L_r\sim 30 pc$
the scale of the the field fluctuations. The density fluctuations and their correlation
with $B_{||}$ are unknown,
hence their size $\delta n_e$ was assumed to be of order unity and only $B_r$ remained
as a free parameter.
However, if $B_{\rm o}$ and the product $ \delta n_e B_r\sqrt{L_r} $ were independent
of the distance, we would
find the expected square root dependence of $\sigma \propto \sqrt{D} $.
From our data, and from previous work by Rand \& Kulkarni (1989), we cannot confirm
such a simple view
of the turbulent structure of the ISM and its magnetic field.

\para{\large\bf{4}~~Multifrequency Single Pulse Polarisation Study of  PSR 0329+54}

 \vs\vs
Simultaneous multifrequency pulsar observations are an obvious way
of investigating the radio emission region more closely. First
studies were made by Bartel and Sieber (1978) at 0.3\,GHz and
2.7\,GHz using both Effelsberg and Jodrell Bank in total power
mode. The data processing of simultaneous observations is much
easier if a common data format is supported by the participating
observatories. In 1995, the EPN, the European Pulsar Network was
launched to further the cooperation of European pulsar astronomers
and one of its successes was the definition of such a common data
format (Lorimer, Jessner, Seiradakis et. al. 1998) and a public
on-line database (http://www.mpifr-bonn.mpg.de/div/pulsar/data) of
profiles and spectra. The EPN format is now supported by a large
number of radio-observatories, amongst them
 Effelsberg, Jodrell Bank, Bologna, Westerbork, Pushchino, Torun, Ooty and GMRT.
The existence of the new standard format  has made it easy and
 nearly a routine matter to participate in simultaneous multifrequency-multitelescope
 single pulse pulsar observations with our partners in Europe and in India.
In the case at hand  1912 single pulses of PSR B0329+54 were
recorded simultaneously with a $S/N>100$ in
 Effelsberg    (2.695\,GHz,   BW 80\,MHz) and Jodrell Bank. Calibrated and aligned
 EPN data was then
rebinned (Karastergiou et al. 2001) to a common time resolution.
\begin{figure}[h]
\hspace{0.5cm}\parbox[b]{7cm} {\includegraphics [bb=0.0 0.0
751.6800 585.6000,scale=0.24]{stokeslb.eps}}

\vspace{-50mm}
 \ \hfill \
\parbox[b]{6.6cm} {\includegraphics [bb=0.0 0.0
747.8400 584.6400,scale=0.24]{stokesvb.eps}}

   \parbox[tb]{14cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

    {\vspace{2mm}  Fig.\,2~~ Cross-correlation contours for linear (left) and circular
      polarisation (right) of single pulses from
PSR B0329+54, received at 1.41\,GHz (vertical direction) and
2.69\,GHz (horizontal direction). For  circular polarisation, the
sign of the correlation is marked near the contour boundaries.}}

    \end{figure}
We find that intensities correlate well over the full width of the
profile, but as seen in Fig. 2 the linear polarisation L and
circular polarisation V shows a more complex behaviour. In L only
the outer components correlate well, although the centre shows the
strongest linear polarisation. In V, the wings appear uncorrelated
because of their random signs but here the centre is strongly
correlated. Anti-correlation regions (offset from the centre)
indicate that RHC-LHC transition direction is maintained over the
band. As the magnitude of circular polarisation $\rm |V|$
correlates well over the full profile we have evidence that
uncorrelated flips between LHC \& RHC are observable in the outer
components of the profile but not in the centre. Our observations
lead us to the conclusion that the emission process has a wide
bandwidth ($> 2 \nu_{\rm min}$) and that either the assumption of
a lower origin of outer components (Rankin, 1990) is wrong or the
observable  fluctuations cannot be a propagation effect!

\para{\large\bf{5}~~Unexpected Properties of Pulsars at High Radio Frequencies}

 \vs\vs
The large and precisely formed collecting area of the 100\,m
telescope has been used for high frequency ($\nu \rm > 10\,GHz$)
pulsar observations since the beginning in the early seventies
(Wielebinski et al., 1972).  We extended the frequency range for a
number of pulsars up to 43\,GHz and even detected PSR B0355+54 at
90\,GHz on Pico Veleta (Morris et al., 1997).
\begin{figure}[h]
\hspace{0.5cm}\parbox[b]{7cm} {\includegraphics [bb=0.0 0.0
1417.9200 1195.2000,scale=0.12]{7mmspec.eps}}

\vspace{-47mm}
 \ \hfill \
\parbox[b]{6.6cm} {\includegraphics [bb=0.0 0.0 681.6000
536.6400,scale=0.27]{9mmod.eps}}

   \parbox[tb]{14cm}
   %{\hspace{3.5cm}
   {%\center
   \footnotesize

      Fig.\,3~~ {\em left:} Spectra of four pulsars observed up to 43\,GHz,
      PSR 0355+54 has at least 0.5\,mJy at 87\,GHz,
{\em right:} modulation index for five pulsars at 33\,GHz, black
triangles: predictions assuming weak interstellar scintillation.}

    \end{figure}
Using the Effelsberg telescope, equipped with a primary focus
receiver at $\nu =34\,{\rm GHz}$ with a bandwidth of 2\,GHz and a
system temperature of $T_{\rm sys}= 100\,\rm K$ we detected 8
pulsars (Wielebinski et. al. 1997). Even at $\nu=43\,\rm GHz$
(secondary focus, BW= 2\,GHz,  $T_{\rm sys}= 250\,\rm K$) we
succeeded in detecting four of the sources (Kramer et at., 1997).
An unexpected flattening or turn-over of their Spectra (Fig.3,
left), or their particular strength (i.e. PSR B0355+54 with
0.5\,mJy at 87\,GHz) made the Pulsars visible at high radio
frequencies (Kramer et al., 1996). This flattening of the spectrum
has also been detected at a lower frequency limit ($> 23\,\rm
GHz$) for a further pulsar: PSR B0144+59. In all cases we observed
simple, Gaussian profile shapes,- the profile development
saturates above 10\,GHz, a high and increasing depolarisation, but
the  position angle is  the same as at low $\nu$. From the profile
widths we estimated the high frequency emission heights to be
around 100\,km, - except for B1929+10 with 600\,km. There was no
indication of a deviation from dipolar geometry (Xilouris et al.,
1996). Another surprising property of the high frequency radio
emission from pulsars is its strong variability (Kramer et al.,
1997). Fig. 3 (right) shows the observed modulation index $m$ for
five pulsars at 33\,GHz and the theoretical predicitions from the
$m \propto (\nu/ \nu_{crit})^{-1.42}$ law pertaining to the weak
insterstellar scattering regime. The critical frequencies were
below 7\,GHz in all the discussed cases. Such a variability cannot
be explained by ordinary propagation effects
 in the ISM,-  hence it is an intrinsic property of the propagation or
 the emission within the
 PSR magnetosphere. Furthermore,- we do not expect the spectrum to become
 flatter because of the propagation in
the magnetosphere,- so there could be an indication of another emission mechanism
  appearing  above  20-30\,GHz and then taking over into the optical band.



\para{\large\bf{6}~~Observational Limits on Plasma Densities and Energies}

 \vs\vs
Electric charges in the inner parts of the PSR magnetosphere move
only along the field lines,- their plasma frequency determines the
lower frequency limits of effective plasma waves and
instabilities. Depending on the actual emission mechanism, the
minimum observable frequency at a given distance $r$
 from the neutron star is given by
\begin{equation}
\nu ={\gamma^{\alpha-1/2}\over 2\pi} \sqrt{{ n_{GJ}(r) \xi q_e^2\over{\epsilon_0 m_e}}}
\end{equation}
The index is given as $\alpha=1$ for a curvature emission process and
$\alpha=2$ for emission by
coherent inverse Compton scattering and
$ n_{GJ}(r) $ is the distance-dependent co-rotation charge
density together with the particle multiplicity  $\xi \sim 10^4$.
Solving for density and Lorentz factor we get a universal constraint linking
the emission distance $r$, the Sturrock factor $\xi$ and the Lorentz factor $\gamma$:
\begin{eqnarray}
\xi \gamma^{2\alpha-1}= {{2\pi^2 m_e \nu^2} \over {q_e \Omega B_0}} {r^3 \over r_{ns}^3}
  \qquad \hskip5.0cm \nonumber \\
= 1.8\cdot 10^{-3} \left({\nu\over 100\,\rm MHz} \right)^2
\left({P \over  \rm s} \right)  \left({10^{12} {\rm G}\over B_0}
\right)  \left({r \over r_{ns}} \right)^3 \quad \end{eqnarray} For
a ``standard pulsar'' with $B_0 = 10^{12}\,{\rm G}, P=0.5\,{\rm
s}, r=50 r_{ns}$ we obtain $\xi\gamma < 1786$. This already shows,
that the usual assumptions of $\xi=10^4$ and $\gamma=100$ are not
compatible with the escape of low frequency radioemission from the
observed emission regions. The situation is even more dramatic in
the well researched case of PSR 0531+21 (Crab) where the maximum
of the radio emission is at $\nu= 160\,\rm MHz$, and the other
parameters are $B_0=3.0\cdot 10^{12}\,{\rm G}, P=0.033\,{\rm s}$.
It is reasonable to assume, that the radio emission originates
{\em within} the light cylinder ($r < 157 r_{ns}$), but in that
case the 160\,MHz radiation can only be produced (Kunzl et al.,
1998) in a region where $\xi\gamma < 163$ which effectively rules
out any high particle density inner gap pair production model for
the crab pulsar.

\para{\large\bf{7}~~Shortcomings of the  Curvature Radiation Model}

 \vs\vs
Curvature radiation can be described in terms of emission by a
relativistic particle moving around the arc of a circle chosen
such that the actual acceleration corresponds to centripetal
acceleration. The critical frequency where most of the radiation
is emitted depends only on the Lorentz factor $\gamma$ of the the
particles and the local radius of curvature $R_c$ of the field
line which is a simple function of distance from the star. It is
approximately given by (e.g. Zheleznyakov 1996, p. 231)
$$\nu_{ c}\simeq {3c\over{4\pi R_{ c}}}\gamma^3\,,\eqno(5a)$$
Or if we observe a particular frequency we find the corresponding Lorentz factor via

$$\gamma\simeq \left[{\nu_{\rm c} 4\pi R_{\rm c}\over{3c}}\right]^{1/3}\,.\eqno(5b)$$

Together with the expression for the plasma frequency (3)
 we have two equations for the emitted frequency $\nu(\gamma, r)$ which enable
us to  determine the minimum emission height from
$\nu\ge \omega_{\rm pe}/2\pi$  by using eq.(5b) to eliminate $\gamma$
%\setcounter{equation}{6}
%\begin{equation}
 $$\nu = {3^{2/5}\over {4 \pi c^{1/5} } } \Bigl({{8 q_e \Omega B_0 r_{ns}^3}
 \over{3 m_e}} \Bigr)^{3/5}r^{-9/5}R_c^{1/5}\,.\eqno(6)$$
%\end{equation}

Coherent emission is required to explain the observed brightness
temperatures and coherence exists in volumes of the order of $V_c
= c^3\gamma^2/\pi\nu^3$ (Melrose 1991). The power output of such a
coherence volume is then given by $P_{N}={2\over 3}\gamma^4
{(N_c\cdot q_e)^2 c / {4\pi \epsilon_0 R_{c}^2}}$ with
$N_c=V_c\cdot n_{\rm GJ}$ being the number of particles within a
coherence volume. To account for the observed luminosity L one
requires $N_v ={L\over P_N} = 7\cdot 10^{11}$ of these volumes,
which implies a minimal size of the emission region $\Lambda_c =
\left(N_v\cdot V_c\right)^{1/3}$ which is about 16\,km for a
typical pulsar. The energy loss of any single charge within the
emission region is now $ P_\Lambda={\Lambda_c\over c}{P_N\over
N_c}  $, which must never exeed the energy of the particle $\gamma
m_e c^2$ itself. We can express the efficiency, the ratio of
emitted energy and  total kinetic energy as
%\begin{equation}
 $$\eta_1 = {P_\Lambda \over \gamma m_e c^2 }\eqno(7)$$
%\end{equation}
 and evalute it along any field line where it cannot exceed unity
 because of {\em energy conservation}. Fig. 4 shows the results for
 five frequencies in the case
of the well known Pulsar B0355+54 from which radio emission has
been observed up to 87\,GHz.
\begin{figure}[h]
\center {\includegraphics [bb=0.0 0.0 720.0000
446.4000,scale=0.5]{Curv3a.eps}}

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

    {\vspace{2mm}  Fig.\,4~~ PSR 0355+54:
Required efficiency for conversion of kinetic beam energy into
curvature radiation at five different frequencies.
 The upper half of the panel corresponds to a regime where the observed
 luminosity exceeds the energy budget of the
streaming particles.}}

    \end{figure}
Particles have only sufficient energies to provide the observed
radio luminosities for frequencies above 10\,GHz. Increases in
density or particle energy would aggravate the problem. An
increase in density corresponds to an increase in frequency
$\propto n^{1\over 3}$ at a given radius and similarly an increase
in the Lorentz factor of the particles increases the frequency
again $\propto  \gamma^3$. Hence coherent curvature radiation
cannot be the mechanism for the low frequency radio emission,
where most of the radio power is emitted.

\vs\vn{\bf ACKNOWLEDGEMENTS}~~ A. Karastergiou, M. Kramer, O.
L\"ohmer, D. Mitra and R. Wielebinski provided material and I am
grateful for their many helpful discussions and their support for
this article.

\bc {\bf References} \ec

\vspace{-4mm} \no \fns

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%\end{CJK}
\end{document}
