A&A 404, 557-567 (2003)
DOI: 10.1051/0004-6361:20030480
V. I. Shishov^{1} - T. V. Smirnova^{1} - W. Sieber^{2} - V. M. Malofeev^{1,5} - V. A. Potapov^{1} - D. Stinebring^{3} - M. Kramer^{4} - A. Jessner^{5} - R. Wielebinski^{5}
1 - Pushchino Radioastronomy Observatory of Lebedev Physical
Institute, 142290 Pushchino, Russia, and Isaac Newton
Institute of Chile, Pushchino Branch
2 -
Hochschule Niederrhein, Reinarzstr. 49, 47805 Krefeld,
Germany
3 -
Oberlin College, OH 44074, Oberlin, USA
4 -
University of Manchester, Jodrell Bank Observatory,
Macclesfield SK11 9DL, UK
5 -
Max-Planck-Institut für Radioastronomie, Auf dem Hügel
69, 53121 Bonn, Germany
Received 2 July 2002 / Accepted 21 March 2003
Abstract
Interstellar scintillation multi-frequency observations
of PSR 0329+54 in the frequency range from 102 MHz to 5 GHz were
analysed to estimate the spectrum of interstellar plasma
inhomogeneities in the direction of this pulsar. Based on the
theory of diffractive scintillation, the composite structure
function of phase fluctuations covering a large range of
turbulence scales was constructed. We found that the spectrum is
well described by a power law with n = 3.5 for scales from
10^{6} to 10^{9} m, which differs from the value known for a
Kolmogorov spectrum. We can, however, within the accuracy of our
data not exclude a Kolmogorov spectrum. It became also clear that
angular refraction of emission must be taken into account to fit
the data points at all observing frequencies. The size of the
irregularities responsible for the angular refraction is estimated
to be about
m. They can be identified with clouds
of neutral hydrogen that can be considered as holes of electron
density.
Key words: stars: pulsars: general - turbulence - ISM: structure - stars: pulsars: individual: PSR B0329+54
A generally accepted point of view at the present time is that
scattering as well as diffractive and refractive interstellar
scintillation effects are caused by electron density fluctuations
in the interstellar medium (ISM) (Scheuer 1968;
Rickett 1969; Sieber 1982; Rickett et al. 1984; Armstrong et al. 1995;
Stinebring et al. 2000). It could be shown that
for the different kinds of pulsar observations connected with the
propagation through the interstellar plasma, a composite structure
function of phase fluctuations can be constructed and that this
structure function follows a power law over a very wide range of
scales (10^{6} to 10^{13} m). This function fits the
experimental data quite well for the near ISM (1 kpc)
(Armstrong et al. 1995) as well as for the distant
ISM (>1 kpc) (Shishov & Smirnova 2002). The
exponent of the structure function is about 1.7. This means that
the 3-dimensional spatial spectrum of electron density can be
described by a Kolmogorov spectrum
Although the Kolmogorov spectrum describes the data sufficiently
well in a statistical sense, the dispersion of points is large and
in particular directions the spectrum can differ from a Kolmogorov
one. Smirnova et al. (1998) showed that two types of
turbulent spectra can exist in the ISM as measurements of the flux
density variations of 21 pulsars at 610 MHz prove, which show two
different types of behaviour for the dependence of the modulation
index on the dispersion measure DM. The data can be explained if
one group follows a pure power law spectrum
(Eq. (1)) with
,
whereas the other
group corresponds to a medium with a power law spectrum pieced
together by an exponent of
and an inner scale of
length
cm and an exponent n_{1} > 4 at
higher space frequencies:
In general, there are two types of power law spectra known for turbulent plasmas: in addition to the aforementioned Kolmogorov spectrum with (Tu et al. 1984) the spectrum of weak plasma turbulence with (Iroshnikov 1963; Kraichnan 1965). The difference between and is unfortunately small, about 5%, whereas the accuracy of the measurements of the value of n is not better than 10%. New measurements with much higher accuracy are therefore urgently needed.
In this paper we propose a new method for the determination of the turbulent spectrum in the direction of a given pulsar using multi-frequency scintillation observations covering a wide frequency range from centimeter to meter wavelengths. We will then address the question whether the exponent of the power law spectrum equals a Kolmogorov spectrum or not.
In the theory of wave propagation through random media the fundamental
function is the gradient of the phase structure function computed in
the geometrical optics approximation (Prokhorov et al. 1975). This function is given for a plasma by the equation
(Shishov & Tokumaru 1996):
When the displacement of the line of sight is determined by a
movement of the source with the velocity ,
which is
assumed to be greater than internal motions within the medium or
the observer velocity, the structure phase function is given by
The phase structure function can be determined from observations
of intensity scintillation in the weak scintillation regime. In
this case the intensity correlation function of intensity
fluctuations is determined by the equation (Prokhorov et al. 1975; Malofeev et al. 1996)
may be measured at any frequency f; if a fixed
frequency f_{0} is given, the result may be converted by use of the
universal factor
An estimation of the value of the phase structure function can be
obtained from the scintillation index of refractive scintillation
(Smirnova et al. 1998). The
refractive scintillation index of strong scintillation is
described in case of isotropic turbulence by
Ostashov & Shishov (1977) showed that the
two-frequency field coherence function of a spherical wave
in the case of a power law turbulence spectrum and
for small values of the frequency difference
Another type of relation between the frequency correlation
function
of the intensity fluctuations and
the spatial phase structure function
is realised
in case of strong angular refraction. According to (Shishov 1973)
we may introduce the accumulated angle of refraction
at distance r, which is a random function of r and
depends only weakly on the coordinates in the plane perpendicular
to the line of sight. If the characteristic value of the
refraction angle
is much bigger than the
characteristic value of the diffractive or scattering angle
For the frequency correlation function of intensity fluctuations
one can show that for small values of
and in case of a
power law turbulence spectrum the following relations hold
The weak dependence of
on
in
comparison with that of
leads to dominant
diffraction effects for very small values of
It is important for practical observations that the diffractive and refractive model gives indeed for small values of different functional dependences of the intensity correlation function on . The power law indices of the temporal and frequency structure functions of intensity fluctuations are different: (n-2) and (n-2)/2 in case of the diffractive model. The indices are equal and have the value (n-2) for the refractive model.
The dynamic spectra of pulsars must show a strong frequency drift in case of a phase screen model, if refractive effects dominate over diffractive effects. The effect of the frequency drift is however weak as shown by Shishov (1973) in case of an extended random medium. It is possible that the frequency drift can in this case be the result of an interaction of refractive scintillation with strong angular refraction. This problem must be analyzed in more detail in future.
A comprehensive collection of observations of PSR B0329+54, covering the wide frequency range from 102 MHz to 5 GHz, was available for this paper to compare the theoretical results of Sect. 2 with measurements. The collection of data was used to construct the phase-structure function as outlined above.
The emission of PSR B0329+54 is fortunately strong at high frequencies - our observations were taken at 4.7 and 10.6 GHz - showing clearly visible scintillation (Malofeev et al. 1996). We completed and extended the available observational material with specific measurements in July 1997 at the 100-m radio telescope of the Max-Planck-Institut für Radioastronomie, when we used a filterbank receiver with four channels of 60 MHz bandwidth each. The system temperature was in average about 60 K and the channels were centered at 5060, 4940, 4760 and 4640 MHz. Individual pulses were sampled at 1/1024 of the observing period and integrated subsequently in a data logger over ten periods to improve the signal-to-noise ratio. These blocks of data were used in the following analysis separately for each channel. More details on the observing method, the receiver and the pulsar backend are presented by Kijak et al. (1997).
Figure 1: Flux density variations over 400 min at four frequencies. | |
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Examples of intensity variations in the four channels are shown in Fig. 1. The measurements were smoothed over 20 data points by a running mean to improve the signal-to-noise ratio. The intensity fluctuations contain fast noisy oscillations and variations with a temporal scale order of the order of 10 min due to scintillation. The sharp peaks in the first 110 min are due to interference. They were excluded before we started the correlation analysis. It is interesting to note that the scintillation patterns at 5060 and 4640 MHz are partly decorrelated. To establish the degree of decorrelation, normalized cross-correlation functions were calculated between frequency channels. We introduced also an artificial time shift of 20 data points (142.6 s) between the channels to minimize the influence of noise. The dependence of the cross-correlation coefficient on frequency separation is shown in Fig. 2. The effect of decorrelation is obvious for frequency lag of about 8% of the observing frequency f.
At these high observing frequencies we are in the weak scintillation regime and the flux density variations should be well correlated in all four channels, which is obviously not the case. The decorrelation indicates that we see the influence of strong refraction, changing with frequency. We will discuss this later.
Figure 2: Cross-correlation coefficients of the flux variations in dependence on frequency separation corrected for noise. | |
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The temporal structure function of intensity fluctuations for time shifts t>200 s (calculated for f=4640 MHz) is shown in Fig. 3. It was corrected for noise by subtracting the mean value of the structure function calculated over the first 10 points.
Figure 3: Structure function of intensity variations normalized by the mean intensity in the time domain at a frequency of 4640 MHz corrected for noise. | |
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At 610 MHz we used part of the data of a two year monitoring program of PSR B0329+54 at the NRAO 42-m telescope in 1994-1995. The observations were made using 1024 channels of the NRAO Spectral Processor covering a total bandwidth of 20 MHz in two orthogonal polarisations from which the total intensity was computed as the sum of the two signals. The spectra were accumulated during 59 s and written to magnetic tape for subsequent off-line analysis. The off-pulse spectra were subtracted from the on-pulse spectra. Each observation lasted for about 70 min.
We computed for all the 70-min dynamic spectra the normalized autocorrelation function (ACF) versus frequency and time to visualise the diffractive scintillation pattern, from which we calculated the characteristic frequency and time scales, and . and were defined as half of the ACF width at the level of 0.5 along the frequency axis and at the level of 1/e along the time axis after removing the spike at zero lag due to noise. Both, and , are changing with time up to a factor of 3, an effect which is in accordance with the observations of Bhat Ramesh et al. (1999) at 327 MHz and which we will consider in a forthcoming paper.
Figure 4: Mean auto- (top) and cross-correlation (bottom) function between successive spectra in dependence on frequency lag. One frequency lag corresponds to 19.5 kHz. The data points are averaged over 70 min. | |
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Figure 5: Mean auto- (top) and cross-correlation (bottom) function at f = 610 MHz in dependence on time lag. One time lag corresponds to 59 s. The data points are averaged over 1024 frequency channels. | |
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We computed in addition average autocorrelation functions for each day. The mean frequency ACF was calculated from the normalized ACFs averaged in time over all 59-s time bins whereas the mean time ACF was calculated by averaging over all normalised ACFs in each frequency channel. As an example the average time and frequency ACFs are shown for the observations of 12 March 1994 in the upper panels of Figs. 4 and 5. Figure 4 presents the frequency dependence, where one lag in frequency corresponds to 19.5 kHz, and Fig. 5 the time dependence with one lag representing 59 s in time (the spike at zero lag is due to noise). The corresponding crosscorrelation functions between intensity variations in neighbouring time bins () and frequency channels ( ) averaged over all pairs are shown at the bottom of these figures.
More informative for a comparison with theory are the structure functions (SF), since their slopes on a log-log scale contain information on the spectral distribution of the inhomogeneities in the ISM. We calculated therefore time and frequency structure functions separately for each channel and time bin and averaged them afterwards. The structure functions were normalized by the square of the mean intensity. The off-spectra, containing information on the noise, were treated in the same way. A special treatment is necessary for the first points of the SFs, , since these points include noise and correlated signal. We subtracted from the SFs a value of , where and for a correction of the time and frequency structure functions. The ACFs and CCFs in the time and frequency domain were calculated as described earlier.
Figure 6: Mean time structure function of intensity variations at f = 610 MHz corrected for noise as described in the text and normalized by 2 m^{2}, where m is the modulation index. The line is a fit to the first points. | |
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Figure 7: Mean frequency structure function of intensity variations at f = 610 MHz corrected for noise as described in the text and normalized by 2 m^{2}, where m is the modulation index. The line is a fit to the first points. | |
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Figures 6 and 7 present the mean time and frequency structure functions at 610 MHz. The lines delineate fits to the first points of the SFs. To present the data of different frequencies in the same scale, the mean SFs were normalized by , where mis the modulation index computed on the basis of a two-dimensional frequency time array (70 min 20 MHz).
At 102 MHz we used observational material published by Popov & Soglasnov (1984); the observations were made with the Large Phased Array (BSA) at Pushchino in 1978. The analysis followed a different normalization procedure since the noise contribution in their correlation functions is large with the result that the correlation for small lags in time and frequency is significantly less then 1. We decided therefore to renormalize the time and frequency correlation functions and so that the correlation at zero lag becomes indeed 1: and . These modified functions were used for a comparison with our data at cm and dm wavelengths as shown in Figs. 8 to 10.
Popov & Soglasnov (1984) mention two frequency scales for the intensity variations: 100 Hz and 750 Hz, the latter feature amounting to about 20% of the 100 Hz component in the correlation function. The 100 Hz structure corresponds quite well to observations at other frequencies and the estimated scintillation time scale, whereas the weaker 750 Hz feature might very well be due to statistical fluctuations. We consider therefore the 100 Hz fluctuations to be real and caused by ISS, whereas the 750 Hz feature is doubtful.
Figure 8: Time structure function of phase fluctuations at f_{0} = 1 GHz as compiled from the following data points 1.) 102 MHz observations: filled circles; 2.) 610 MHz: open circles; 3.) 5 GHz: stars. The solid line corresponds to the best fit to the first points of the structure functions. The slope is 1.50. The dashed line corresponds to n = 1.67 (Kolmogorov model). | |
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Using the theory developed in Sect. 2 one can reduce the
measurements obtained at many different frequencies to one given
frequency f_{0}. Figure 8 shows the resulting time
structure function
at the frequency f_{0} = 1 GHz
as compiled from Eqs. (10) and (13). Data
points at different frequencies are marked by different symbols.
The first points obtained at 5 GHz (stars) correspond very well to
those at 610 MHz proving that we used the right correction
procedure for measurements at different frequencies. The
uncertainty increases, however, at large time lags (>2000 s)
for the 5 GHz observations because the ratio between the time
duration of the observations and the characteristic scintillation
time scale becomes smaller and smaller. For a fit only the first
three points were used. The measurements follow a power law:
For the computation of the frequency dependence, i.e. for the computation of at f_{0} = 1 GHz, one may use two different approaches. One approach is to base the analysis on the diffractive scintillation model, described by Eqs. (20), (22), and (23). The result of such a compilation is shown at the top of Fig. 9. One recognises immediately a strong discrepancy between the data sets at different frequencies.
Figure 9: Frequency structure function of phase fluctuations at f_{0} = 1 GHz based on data at 102 MHz, 610 MHz and 5 GHz (the different frequencies are characterized by different sizes of the circles) for two models: 1.) diffractive scintillation model; 2.) model with angular refraction. The solid line corresponds to the best fit to the first data points of the structure function. The slope is 1.47. | |
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Alternatively one may assume the strong refraction model (model 2)
using Eqs. (23), (31), and (33).
It should be remembered that
is reduced to f_{0} in
this case according to
It is interesing to compare the angular refractive effect discussed above with the slanting features which are sometimes observed in the dynamical spectra of pulsars (see, for example, Rickett 1969; Gupta et al. 1994; Stinebring et al. 1996; Bhat et al. 1999). The angular refractive effect, which is evident in the frequency correlation function of intensity fluctuations, is stable over more than 15 years, because the time lag between the observations at 102 and 610 MHz as well as 5 GHz is about 15 years. The observations do, however, not show corresponding stable slanting features. This is an additional argument in favour of an extended model of the interstellar turbulent plasma near to the Sun because the frequency drift of diffraction patterns is strong for a phase screen and weak for an extended medium (Shishov 1973).
In case of a power law turbulence spectrum, the theory predicts small angle refraction of the order of more or less , where is the scintillation index of refactive scintillation and the scattering angle. The observations show however sometimes strong frequency drifts, that correspond to very large values of the refraction angle . It is difficult to explain such values of the refraction angle in terms of a power law turbulence spectrum only. It may be necessary to introduce into the model additional large scale inhomogeneities that give strong angular refraction, a modification which may open up new possibilities. The problem of refractive scintillation in presence of strong angle refraction is difficult and needs obviously special consideration.
One may reduce the temporal structure function to a spatial
structure function
using the pulsar velocity
and the equation
The whole compilation for is shown in Fig. 10, where the data points for are taken from Fig. 8 and converted to the spatial scale using Eq. (39).
Figure 10: Structure function of phase fluctuations at f_{0} = 1 GHzversus spatial scale of the inhomogeneities. The signs are the same as in Fig. 8; the cross designates the value computed from an analysis of refractive scintillation; the point at 10^{14} m is an upper limit from timing observations at 102 MHz. The points at the highest spatial scales are from DM variations of pulsar pairs in globular clusters. The line 1 corresponds to the best fit to the first points of the structure function, the line 2 to the Kolmogorov spectrum, and the line 3 is computed from Eqs. (41) and (44) as explained in the text. | |
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Further data points for the determination of can be obtained from measurements of refractive scintillation at 610 MHz as given by Smirnova et al. (1998) and from timing data at 102 MHz as published by Shabanova (1995). These points have been added to Fig. 10. The 610 MHz measurements allow one to calculate a value of at m by use of Eqs. (16) and (17) with n=3.5 and assuming a scintillation index m = 0.37 and a characteristic time scale of days. In addition, the variations of the pulse arrival times at 102 MHz with ms provide an upper limit for at m if we use Eq. (6).
Data of dispersion measure variations for pulsars in globular clusters (Shishov & Smirnova 2002) were used to estimate for large values of in the range of one parsec (see Fig. 10). These data were reduced to our reference value of the dispersion measure, i.e. . The errors for these values are about 50% of . A detailed discussion of this data material can be found at Shishov & Smirnova (2002).
All the experimental data points were fitted by two lines where line 1 has a slope of which corresponds to an exponent of n = 3.5 in the spectrum and where the line 2 has a slope of corresponding to a Kolmogorov spectrum. The point characterising the refractive scintillation falls above line 1 with but we know that this point depends on the used theoretical model for the spectrum and the distribution of inhomogeneities along the line of sight to PSR 0329+54 and can be reduced.
We show in addition in Fig. 10 by line 3 the relation
The observations show that the interstellar plasma near to the Sun along the line of sight to PSR 0329+54 consists of two types of inhomogeneities: turbulent irregularities, which produce the scattering and scintillation effects, and much larger scale irregularities, which are responsible for angular refraction effects only.
The turbulence spectrum can be described by the Karman model (Tatarskii 1971)
Using this model one obtains for small values
For large values of
is given by
The type of irregularities which are responsible for the angular
refraction are characterised by much smaller values of L_{0}
Using the value (50) in the Eq. (44) one obtains
This scenario is not universal. There is evidence that different types of turbulent spectra exist in different regions of the Galaxy (see, for example, Smirnova et al. 1998; Lambert and Rickett 2000). Multifrequency scintillation observations of different pulsars should allow one to investigate the real change of interstellar plasma turbulence parameters in different regions of the Galaxy.
Acknowledgements
The authors thank the referee B. J. Rickett for useful comments, which improved the paper considerably. This work was supported by INTAS grant No. 00-00849, NSF grant No. AST 0098685, the Russian Foundation for Basic Research, project code 00-02-17850, and the Russian Federal Science and Technology Program in Astronomy. We thank the NRAO operated by Associated Universities for support with the 610 MHz observations and L. B. Potapova and G. Breuer for technical assistance.