A&A 408, 1057-1063 (2003)
DOI: 10.1051/0004-6361:20031028
S. A. Petrova
Institute of Radio Astronomy, 4, Chervonopraporna St., Kharkov 61002, Ukraine
Received 30 April 2003 / Accepted 3 June 2003
Abstract
Propagation of natural waves through the ultrarelativistic highly
magnetized plasma in the rotating magnetosphere of a pulsar is
considered. Based on the quantitative description of the
polarization-limiting effect, we develop a technique of density
diagnostics of pulsar plasma according to the observed
polarization profiles of radio pulses. For the first time, it
appears possible to obtain the profiles of the plasma density
distribution across the open field line tube. The density profiles
found for PSR B0355+54 and PSR B0628-28 show a perfect exponential
decrease towards the tube edge. The multiplicity factors derived
are compatible with those predicted by modern theories of pair
cascade in pulsars. The results of the plasma density diagnostics
may have numerous implications for the physics of the pulsar
magnetosphere. Further application of the suggested technique to
single-pulse polarization data seems to be promising.
Key words: plasmas - polarization - waves - stars: pulsars: general - stars: pulsars: individual: PSR B0355+54, PSR B0628-28
The diverse and complicated behaviour of the single pulse polarization is mainly caused by the presence of orthogonal polarization modes (OPMs), which are fundamental feature of pulsar radiation. The concept of completely polarized superposed OPMs has strong observational support (McKinnon & Stinebring 1998, 2000) and it is undoubtedly favoured by theoretical considerations, since direct generation and further propagation of the partially polarized disjoint modes are questionable. The superposed modes can be recognized as the independently propagating natural waves of pulsar plasma, which either are generated by different emission mechanisms (McKinnon 1997) or originate as a result of partial conversion of a single emission mode (Petrova 2001). As the magnetic field of a pulsar is extremely strong and ray propagation is typically quasi-transverse, the natural waves of the magnetospheric plasma should be linearly polarized, with the electric vector reflecting the orientation of the ambient magnetic field. However, the representation of the observed superposed OPMs as the natural waves of pulsar plasma faces two major difficulties. Firstly, the observed modes are elliptically polarized, with the degree of circular polarization varying stochastically from pulse to pulse (McKinnon 2002). Secondly, at a fixed pulse longitude, for each of the orthogonal states there is a substantial pulse-to-pulse spread in position angle (PA) of linear polarization (McKinnon & Stinebring 1998), and moreover, the average PAs of the two states may differ by not exactly (e.g. Gil et al. 1991, 1992).
From the theoretical point of view, if one consistently considers
propagation of natural waves in the open field line tube of a
pulsar, one cannot overlook the region where the waves decouple
from the magnetospheric plasma and become common vacuum
electromagnetic waves. This happens as soon as the plasma density
decreases sufficiently for the approximation of geometrical optics
to be broken, i.e. on condition that
(1) |
In the present paper, we develop a technique that allows us to separate the observational consequences of PLE from the polarization data and to obtain the plasma number density profiles. The technique is applied to the average polarization profiles of PSR B0355+54 (at 4.85 GHz) and PSR B0628-28 (at 0.408 and 0.61 GHz), where the contribution of pulse-to-pulse orthogonal transitions is believed to be negligible, so that the average polarization characteristics can be regarded as those of a "typical'' single pulse. Although our results are rather illustrative, they do encourage further application of the suggested technique to real single pulses of a variety of pulsars, in which case the orthogonal transitions and other pulse-to-pulse polarization fluctuations can be properly taken into account. Our present research has made use of the database of published pulse profiles maintained by the European Pulsar Network, available at http://www.mpifr-bonn.mpg.de/pulsar/data.
= | -2R(b_{x}+l_{y})(b_{y}-l_{x})v, | ||
= | R[(b_{x}+l_{y})^{2}-(b_{y}-l_{x})^{2}]v, | (2) | |
= | 2R(b_{x}+l_{y})(b_{y}-l_{x})q+R[(b_{y}-l_{x})^{2}-(b_{x}+l_{y})^{2}]u, |
The Stokes parameters evolve along the wave trajectory because of
the plasma density decrease,
,
and also because
of variation of the orientation of the ambient magnetic field as a
result of ray propagation and pulsar rotation. For the wave
propagation far enough from the emission region, to the first
order in
(where
is the light cylinder
radius) the geometrical terms can be presented as (for more
details see e.g. Petrova & Lyubarskii 2000):
b_{x}+l_{y} | = | ||
b_{y}-l_{x} | = | (3) |
Substituting Eq. (3) into Eq. (2), we find finally:
= | |||
= | (4) |
(5) |
A | = | ||
= |
The solution of the set of Eq. (4) at yields the final polarization of the natural waves. Below we are interested in the final degree of circular polarization, , and the final PA shift from the initial plane of magnetic lines, . Figure 1 shows the numerically calculated final polarization characteristics of the ordinary waves as functions of the parameters and . According to Fig. 1c, the pairs and exhibit a unique correspondence.
Figure 1: The final ellipticity a) and PA shift b) of the ordinary waves as functions of parameters and ; c) versus at different ( ), each of the closed lines corresponds to . | |
Open with DEXTER |
Both polarization characteristics of the natural waves, and
,
can be derived from observations. Given
that the observed radiation is an incoherent mixture of the
orthogonal elliptical modes with intensities I_{1} and I_{2}, the
observed Stokes parameters (I, Q, U, V) are written as
I | = | ||
V | = | v_{m}(I_{1}-I_{2}), |
(6) |
(7) |
Figure 2: Polarization profile of PSR B0355+54 at 4.85 GHz (von Hoensbroech & Xilouris 1997): solid line - total intensity, dashed line - linear polarization, dotted line - circular polarization, asterisks - reliable values of mode ellipticity. | |
Open with DEXTER |
The profile of PSR B0355+54 is plotted in Fig. 2. The reliable values of mode ellipticity, v_{m}, calculated in accordance with Eq. (6) are shown by asterisks. The errors in v_{m} are assumed to be normally distributed; hence, the standard error reads: where are the rms-deviations of the Stokes parameters in the off-pulse region. Our consideration involves the points with the relative errors . The standard errors in PA, , and the corresponding relative errors are typically much less.
Figure 3: Parameter ( ) versus pulse longitude (asterisks) according to the polarization data on PSR B0355+54 at 4.85 GHz; the observed values, , are shown by triangles. | |
Open with DEXTER |
For the pulse longitudes considered, the tangent of the observed
PA is presented in Fig. 3 (triangles). According to the sign of the PA
derivative,
.
Then, as can be seen from Eq. (4)
and Fig. 1, the negative sign of v_{m} testifies to the dominance
of the ordinary waves. Using the numerical solution of
PLE-equations (4) at various
and ,
for each pair of
the observed values
one can find a unique pair
which provides such values of
and
that satisfy both equalities (7)
simultaneously.
Figure 4: Parameter as function of pulse longitude (asterisks) for PSR B0355+54 at 4.85 GHz. The normalized plasma density at estimated following Eq. (8) is shown by diamonds, cm^{-3}. The emission altitude, z_{0}, found from the relation is shown by circles. For more details see text. | |
Open with DEXTER |
It should be mentioned that fitting the RVM-curve directly to the observed PA usually faces difficulties, even in the absence of prominent orthogonal transitions (e.g. Everett & Wiesberg 2001). Moreover, in those rare cases when the fitting procedure yields satisfactory results, the geometrical parameters of a pulsar derived from the fitted curves at different frequencies appear to differ drastically. Probably, these discrepancies can be attributed to PLE. Further on it would be reasonable to allow for this effect and fit the RVM-curve to rather than to . Although in the case considered, and differ slightly, in general one can expect that taking into account PLE can improve the technique of RVM-fits, especially in application to the single-pulse data.
According to the theory of PLE, the linearly polarized natural wave originating with the RVM position angle, , acquires both the observed and v_{m}at a certain value of , which characterizes the physical properties of the ambient plasma. The quantity is related to the polarization-limiting radius, , with . In our case C=2.3, corresponding to the slope of a straight line in Fig. 3. Then (cf. Fig. 4).
Proceeding from Eq. (5), one can find the plasma number density at
:
(8) |
In view of the continuity of the plasma flow within the open field
line tube, it is convenient to introduce the multiplicity factor
of the plasma along a fixed field line,
(9) |
The polar angle of the ray trajectory in the open field line tube,
,
can be expressed in terms of pulse phase of the ray along
with parameters of observational geometry. One can notice that far
from the emission region the radius-vector of the ray trajectory
makes approximately the same angle with the magnetic axis as does
the wavevector,
,
where
.
Hence, at the polarization-limiting radius, ,
the
polar angle reads:
(10) |
(11) |
The dependence presents the plasma density profile in the open field line tube. Although far from the emission region, at , the tube is much wider than the radio beam, it is still possible to probe the plasma density over a substantial part of the tube. The point is that the polarization-limiting radius can vary considerably across the pulse profile (e.g., as in case of PSR B0355+54, see Fig. 4); then, because of the magnetosphere rotation, for different rays can differ markedly (cf. Eq. (10)) and, in addition, for distinct values of even close values of may correspond to distant field lines (cf. Eq. (11)).
Strictly speaking, the rays observed at different pulse longitudes
pass through the polarization-limiting region at different
azimuths, so that the plasma is probed only along the line
.
However, if the azimuthal density
gradient is insignificant, one can suppose that
completely describes the plasma density distribution,
i.e. yields the density in any cross-section of the tube.
Figure 5: Plasma density profiles of PSR B0355+54 (squares) and PSR B0628-28 (triangles and circles for the polarization data at 0.408 and 0.61 GHz, respectively). | |
Open with DEXTER |
Substituting the calculated values of and into Eqs. (8)-(11), we find the plasma density profile of PSR B0355+54 (squares in Fig. 5). Here it is taken that . The dependence exhibits a perfect exponential behaviour, with the exponent being well fitted by polynomial of the second order. The values of the multiplicity factor appear to be compatible with those given by modern theories of pair cascade, -100 (e.g. Hibschman & Arons 2001a,b; Arendt & Eilek 2002). At the same time, strong nonuniformity of the plasma distribution across the tube is an essentially new result awaiting theoretical explanation.
The suggested technique of plasma diagnostics has also been applied to the polarization profiles of PSR B0628-28 at 0.408 and 0.61 GHz first published by Gould & Lyne (1998). The absence of orthogonal transitions over a broad frequency range (Suleymanova & Pugachev 2002) allows one to suppose that the average profiles of PSR B0628-28 are appropriate for our treatment. The resultant density profile is shown in Fig. 5 by triangles (0.408 GHz) and circles (0.61 GHz) (it is taken that (Lyne & Manchester 1988), ; P=1.244 s, ). One can see that the polarization data at different frequencies lead to such multiplicity factors which lie on the same curve, testifying to the validity of the suggested technique. At the same time, this demonstrates that the multifrequency polarization studies can extend the region suitable for plasma probing, since the polarization-limiting radius is frequency-dependent. Indeed, as the rays reach , the magnetosphere turns at an angle , so that the rays of different frequencies observed at the same pulse longitudes probe the plasma at different field lines.
In the case of PSR B0628-28, the plasma multiplicity factor also shows an exponential decrease, however, in contrast to the density profile of PSR B0355+54, the second derivative of the exponent is 0, hinting at a flat maximum at smaller polar angles. Note that in both density profiles, the main uncertainty comes from the assumed values of . As has already been mentioned above, the customary RVM-fits usually yield questionable results. Therefore the density profiles obtained perhaps appear considerably shifted as a whole along both axes, though the characteristic exponential form of should remain unaltered. The multiplicity factors derived are also dependent on the assumed value of the characteristic Lorentz-factor of the plasma particles (cf. Eqs. (8) and (9)). Different models of the pair cascade certainly yield distinct distribution functions of the secondary plasma and therefore the characteristic Lorentz-factor should somewhat vary. However, is believed to be merely a scaling factor constant throughout the pulse profile for a pulsar, so that the shape of the density profile and its pulse-to-pulse variations are not affected.
The polarization profile of a pulsar at a fixed frequency typically does not allow one to reproduce the whole plasma density profile. Firstly, the rays propagating close enough to the magnetic axis remain invisible because of observational geometry. Secondly, far from the pulse profile peaks both V and L are too small to give reliable values of v_{m}. Thirdly, the asymmetry of the observed total-intensity profiles may hint at a marked azimuthal dependence of the plasma density, in which case the polarization profile yields the multiplicity factor only along a line in the transverse cross-section of the tube. Note that joint observations at different frequencies can enlarge the region suitable for the plasma diagnostics, since the polarization-limiting radius is frequency-dependent.
The technique of plasma diagnostics has been applied to the average polarization profiles of PSR B0355+54 at 4.85 GHz and PSR B0628-28 at 0.408 and 0.61 GHz. The resultant multiplicity factors are compatible with those predicted by the modern theories of electron-positron cascade. In addition, we have found that the density distribution across the tube is essentially nonuniform, exhibiting a perfect exponential decrease towards the tube edge.
Since we have used the average polarization profiles in place of the singe-pulse ones, the results are expected to be rough. At the same time, they demonstrate the possibilities of the technique suggested and urge its further application to the single-pulse data. With the present level of observational facilities, it seems realistic to obtain the instanteneous plasma density profiles for a number of pulsars.
The plasma multiplicities and their variability, which is believed to underlie polarization fluctuations in single pulses, can impose constraints on the models of pair cascade and stimulate more detailed theoretical studies of the physics of the polar gap. Furthermore, explicit knowledge of the structure of the secondary plasma flow may provide new insights into emission physics of pulsars. At first, it would be interesting to compare the profiles of total intensity and plasma density as well as their temporal variations. In particular, the plasma diagnostics can help to find out whether the drifting and nulling phenomena are indeed associated with the spatial and temporal changes in the plasma density. A rigorous description of the density distribution of pulsar plasma can also provide a basis for quantitative studies of the propagation effects. For example, observational evidence for a non-uniformity of the plasma distribution across the tube confirms that refraction should be mainly determined by a transverse rather than radial density gradient, since the transverse scale length of the tube is much less. This strongly supports our model of refraction (see e.g. Petrova & Lyubarskii 2000), where the transverse density gradient was merely postulated. At the same time, the results of the present paper testify against the alternative model of refraction (e.g. McKinnon 1997) based on 'ducted' propagation of subluminal waves along the magnetic lines, since in the presence of a significant transverse density gradient this regime should be broken (Barnard & Arons 1986).
On the whole, the technique of plasma diagnostics suggested in the present paper can become a useful tool for studying the physics of the pulsar magnetosphere.
Acknowledgements
This research has made use of the polarization profiles from the database of the European Pulsar Network, which is operated by Max-Plank Institut fur Radioastronomie. I am grateful to the referee, A. Jessner, for useful comments and suggestions.