A&A 455, 195-201 (2006)
T. V. Smirnova1 - V. I. Shishov1 - W. Sieber2 - D. R. Stinebring3 - V. M. Malofeev1 - V. A. Potapov1 - S. A. Tyul'bashev1,4 - A. Jessner4 - R. Wielebinski4
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 - Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
Received 28 May 2004 / Accepted 9 March 2006
Multi-frequency observations of interstellar scintillation toward the pulsar PSR B1642-03 were analyzed to estimate the spectrum of interstellar plasma inhomogeneities in the direction of this pulsar. Using data over the frequency range from 103 MHz to 5 GHz, we constructed the composite structure function (SF) of phase fluctuations, which covers a corresponding wide range of turbulence scales. The structure function shows that the interstellar plasma spectrum in the direction to this pulsar follows a piecewise power law. The power law is well described by two exponents: n = 3. 7 for scales from 109 to 1015 cm (Kolmogorov spectrum) and n = 3.35 for scales less than 109 cm. We interpret the unusual behaviour of the spectrum to be caused by the line of sight passing through the North Polar Spur, which may have plasma properties similar to the anisotropic plasma of the solar wind, although at a very different density.
Key words: stars: pulsars: general - turbulence - ISM: structure - stars: pulsars: individual: PSR B1642-03
It is well known that the analysis of diffractive and refractive scintillation of pulsar signals bears information about the shape of the interstellar plasma spectrum. Such measurements allow the construction of the composite structure function (SF) of phase fluctuations. The SF follows a power law as shown by Armstrong et al. (1995) over a very wide range of scales (106 to 1013 m) in the nearer interstellar medium (ISM) ( kpc) as well as for greater distances (R > 1 kpc) (Shishov & Smirnova 2001). The data are in a statistical sense well described by a Kolmogorov spectrum. However, the dispersion of the points is large, and the spectrum may differ from a Kolmogorov law in specific directions to given sources. We showed previously (Shishov et al. 2003) that the analysis of multi-frequency observations gives useful information about the ISM in a chosen direction with good accuracy. We found that the spectrum in the direction to PSR B0329+54 is well described by a power law with n = 3.5 for scales from 106 to 109 m, differing from the Kolmogorov value n = 3.67. It is notable that strong angular refraction was also observed for this line of sight.
In this paper we analyze multi-frequency observations of PSR B1642-03 in the range from 103 MHz to 5 GHz. The method of analysis and the basic equations are described in Shishov et al. (2003). We construct the composite structure function of phase fluctuations in the time and frequency domain and compute the spectrum of interstellar plasma inhomogeneities in the direction to this pulsar.
The observations of PSR B1642-03 at 102.6 MHz were carried out in June 1994 with the Large Phased Array (BSA) at Pushchino using a 128-channel receiver with a bandwidth, , of 1.25 kHz per channel. The 128 channels were sampled every 5.12 ms in a pulse window of 200 ms duration synchronized with the pulsar period and were written to disk after detection and summation over eight pulsar periods (3.1 s), which we call a "block'' of data. The window included on-pulse and off-pulse contributions. A typical observation lasted for about 70 blocks. The methods of observation are described in more detail by Malofeev et al. (1995).
The signals in adjacent channels were shifted to remove the dispersion delay and scaled so that the variance due to noise had the same amplitude in all channels. We then averaged three bins at the maximum of the mean pulse profile to represent the on-pulse intensity of each channel and three bins off the pulse to represent the noise (off-pulse) intensity. On-pulse and off-pulse correlation functions could then be evaluated from these data. In keeping with convention (Cordes et al. 1985), the characteristic frequency and time scales and were defined as half of the autocorrelation function (ACF) width at a level of 0.5 along the frequency axis and at a level of 1/e along the time axis after removing the spike at zero lag due to noise. We also calculated cross-correlation functions (CCF) between spectra of pairs of data blocks (minimum separation 3.1 s) separated by 8, 16, and 24 pulses (corresponding to one, two and three blocks).
Significant correlation exists for adjacent blocks (time separation 3.1 s or eight pulse periods) falling down to the noise level at a separation of three blocks (24 pulse periods). The correlation levels are , and at time separations of 3.1 s, 6.2 s and 9.3 s, respectively. We conclude therefore that s. Numerical simulations of Lee (1975) show that smoothing of the frequency structure by the receiver bandwidth (which is the case for our data; see below) causes a decrease of the modulation index but changes the time scale of intensity fluctuations only weakly.
Table 1: Frequency and time resolution and in comparison to the characteristic frequency and time scales and .
Our frequency resolution (
insufficient to evaluate the decorrelation
directly; it is clear only that
(see Table 1). We may, however, estimate
the time scale from the scintillation index m and the value of
the cross-correlation coefficient at zero frequency lag CCF(0):
At 335 MHz, we analyzed the observations of Roberts & Ables (1982) made with the Parkes 64-m telescope. These observations were made in September 1974 with a 63-channel receiver with a bandwidth of 10 kHz for each channel. Roberts and Ables added 26 pulse periods (10 s) to build one spectrum. The total observing time was 147 min corresponding to 880 successive spectra.
Figures 12a and b of their paper show the derived auto-covariance functions in frequency and time. They compute - after removing the spike at zero lag due to noise - a scintillation time of s and a frequency scale of kHz. Assuming a frequency scaling law as for a Kolmogorov spectrum of inhomogeneities - that is f4.4 and f1.2 for and - one would get kHz and s at 102.6 MHz.
The data at 408 MHz were taken from the paper of Gupta et al. (1994). They quote a characteristic frequency scale of kHz and time scale of s. Their frequency and time resolutions were insufficient, however, to compute the frequency and time structure functions in the region of small lags.
The observations at 610 MHz and 800 MHz were made with the NRAO 42-m telescope in 1995 and 1992, respectively. We used the 1024 channel NRAO Spectral Processor. At 610 MHz, a total bandwidth of 20 MHz in two orthogonal polarizations was used from which the total intensity was computed as the sum of the two signals. The off-pulse spectra were subtracted from the on-pulse spectra. The spectra were accumulated for 59 s and written to magnetic tape for subsequent off-line analysis. Each observation lasted for about 105 min. At 800 MHz, a total bandwidth of 79.8 MHz in one linear polarization was recorded. The spectra were accumulated for 29 s and the total observing time was 48 min.
The signal-to-noise ratio of the frequency ACFs can be enhanced by averaging over time. We have done this by averaging over 59 s at 610 MHz (the accumulation time, one block) and over 29 s at 800 MHz. We increased the signal-to-noise ratio of the time ACFs by averaging over the frequency channels, which was done to compute mean time ACFs. In addition, we computed mean cross-correlation functions by cross-correlating the intensity variations in neighbouring time blocks (a separation of either 59 s or 29 s) or frequency channels and then by averaging over all available pairs. The mean frequency and time ACFs and CCFs are presented in Fig. 1 for f = 610 MHz and Fig. 2 for f = 800 MHz. All calculated correlation functions were normalized by for the ACF or for the CCF where is the standard deviation of intensity variations. We did a linear extrapolation of the ACFs near zero time and frequency lags to correct for noise. The frequency resolution at 610 MHz is satisfactory (see Fig. 1), about 20 times narrower than (see Table 1), but the time resolution is only about twice that of . The frequency and time resolution at 800 MHz is quite good.
It should be remarked that the value of the CCF at (t,
Fig. 1 is greater than that of the ACF
because the intensity variations in neighbouring frequency channels are more highly correlated than
the variations at the same frequency with a time shift of one lag (
so that the correlation along the time axis is smaller.
|Figure 1: Left: normalized mean frequency auto- ( top) and cross-correlation ( bottom) function at 610 MHz. One frequency lag corresponds to 19.5 kHz. The integration time is 105 min and the time shift for cross-correlated spectra amounts to 59 s. Right: mean time auto- ( top) and cross-correlation ( bottom) function at 610 MHz. One time lag corresponds to 59 s. The integration covers 1024 frequency channels and the frequency shift for the cross-correlation is 19.5 kHz.|
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|Figure 2: Bottom: mean frequency auto-correlation function ( right) combined with the cross-correlation function ( left) between successive spectra at 800 MHz. One frequency lag corresponds to 77.9 kHz. The integration time is 48 min. Top: mean time auto-correlation function ( right) combined with the cross-correlation function ( left). One time lag corresponds to 29 s. The integration covers 1024 frequency channels and the frequency shift for the cross-correlation is 77.9 kHz.|
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|Figure 3: Top: flux variations (in arbitrary units) at 4.85 GHz covering 200 min. Each data point corresponds to 39 pulsar periods (about 15 s). Middle: time auto-correlation function of these variations. Bottom: time structure function. Intensity variations were normalized by the mean value.|
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We have about 90 independent measurements of and at 610 MHz so that their standard deviation with time, , can be evaluated. The standard deviation amounts to 25% as shown in Table 1. These variations are, however, mainly caused by refractive scintillation. The relative variations of the diffractive frequency and time scales are proportional to the modulation index of refractive scintillation (Shishov 2001) so that one may evaluate these variances at other frequencies considering the modulation index for refractive scintillation, (Smirnova et al. 1998). In Table 1 error bars are computed for the frequencies 335 MHz, 308 MHz, and 800 MHz according to . The error at 4850 MHz was estimated by where T is the total observing time.
The observations at 4.85 GHz were carried out in July 2001 with the 100-m radio telescope of the Max-Planck-Institut für Radioastronomie. We used a receiver with a bandwidth of 500 MHz. The mean system temperature was about 60 K. Individual pulses were sampled 1024 times per pulsar period and integrated subsequently in a data logger over 39 pulsar periods (about 15 s) to improve the signal-to-noise ratio. Intensity variations for the 200 min observing time are shown in Fig. 3 (top). The corresponding ACF is shown in the middle panel of Fig. 3. The modulation over our still limited observing time shows a quasiperiodic appearance, which might smooth out for longer observations. We used three days of observations for the following analysis and computed a modulation index in the range .
The diffractive parameters as derived from our correlation analysis, as
well as the frequency and time resolutions, are presented for all
frequencies in Table 1. Figure 4 presents the dependence of the
on the observing
frequency. Linear fits to the log-log scale result in
For a power law spectrum of inhomogeneities with slope n one would expect
which means nshould be near to 3.3 based on the derived values of
will show later that this value of n is in good agreement with the results of
the structure function analysis.
|Figure 4: Frequency dependence of the diffractive scintillation frequency scale ( top) and time scale ( bottom). The data are taken from Table 1.|
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|Figure 5: Time ( right) and frequency ( left) intensity structure function at 610 MHz calculated for different days of observation. Structure functions were normalized by the intensity variance squared.|
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|Figure 6: Time structure function of phase fluctuations (in radians2) reduced to f0 = 1 GHz as compiled from data based on different frequency ranges: MHz ( bottom), 4.85 GHz ( middle) and 340 MHz 4.85 GHz. Data points are marked with the following symbols: 340 MHz - open circles; the mean SF based on data at 610 MHz and 800 MHz - filled squares; 4.85 GHz - triangles. The solid lines correspond to the best fits to the corresponding data points.|
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|Figure 7: Time structure function of phase fluctuations (in radians2) reduced to f0 = 1 GHz as compiled from the following data points: 1) 102 MHz observations - stars; 2) 340 MHz - open circles; 3) 800 MHz - filled squares; 4) 4.85 GHz - triangles; 5) refractive scintillation at 610 MHz - cross; 6) timing at 103 MHz - filled circle. The two solid lines correspond to fits to the structure function with a slope of 1.35 in the range s and with a slope of 1.7 for time lags larger than 200 s.|
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|Figure 8: Frequency structure function of phase fluctuations (in radians2) reduced to f0 = 1 GHz based on data at 102 MHz, 340 MHz, 610 MHz and 4.85 GHz (symbols as in Fig. 7) for two models: refractive model ( top) and diffractive model ( bottom). The solid line corresponds to the best fit to the first data points of the structure function at 610 MHz (diffractive model).|
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Using the equations of Appendix A we have constructed time and frequency structure functions (SFs) based on the derived ACFs which were corrected for noise and reduced to the frequency f0 = 1000 MHz. The structure functions can be modified by two factors in addition to noise. The first is a modulation of the diffractive pattern by refractive scintillation. This is demonstrated in Fig. 5, where examples of frequency structure functions are shown at 610 MHz calculated for different days of observation. The shape of the structure functions varies clearly for small frequency lags. The variation is mainly caused by different values of the flux variance used for normalization. The SFs are very similar for small frequency lags without this normalization.
The second factor that may modify a SF is given by the smoothing of the frequency structure by integration over time. Integration along the time axis leads to a suppression of small-scale structure along the frequency axis and to a distortion of the frequency structure function at small values of frequency lags. Similarly, integration along the frequency axis (or insufficient frequency resolution) can lead to a distortion of the time structure function. Our measured frequency structure functions show, as a consequence of this mutual interdependence, a steepening at small frequency lags (Fig. 5). This steepening is caused by poor time resolution. The lack of short-time measurements causes the time structure function to stop at a level of 0.2 (Fig. 5 right), which influences the frequency structure function at these low levels. The minimum levels of the time and frequency structure functions should therefore be consistent with each other.
Figure 6 shows the time structure functions in a log-log diagram calculated for different frequency ranges converted to one reference frequency, 1000 MHz (Eq. (A.9)). The frequency ranges are: MHz, 4.85 GHz, and 340 MHz 4.85 GHz. We used the exact relation (Eq. (A.4)) to compute the SF in the range from 340 MHz to 800 MHz and Eq. (A.3) for 4.85 GHz. The main effect that causes scatter in the calculated SF at different observation epochs is the modulation of the diffractive pattern by refractive scintillation. It is well known that the characteristic time and frequency scales of diffractive scintillation can change in time by up to a factor of 2 to 3. We averaged the structure functions of three days at 610 MHz and one day at 800 MHz to improve the statistics. To use the same time resolution (29 s) at 610 MHz as for 800 MHz we interpolated the corresponding SFs before averaging. The estimation errors for structure functions were defined using equation (B12) from the paper of Rickett et al. (2000) taking into account that all SFs were reduced to f0 = 1 GHz. We mention that the error of the SF( ) at 340 MHz is about a factor of two greater than the statistical error at this scale.
The structure function at 4.85 GHz was calculated by averaging the SFs from 3 days of observation and using time lags from 120 s to (to exclude the steepening caused by insufficient frequency resolution). The data points at different frequencies are marked by different symbols. The fits, corresponding to different frequency ranges, are shown in Fig. 6 by solid lines. The structure functions based on the 340 MHz to 800 MHz and 4.85 GHz data points have different slopes: and , respectively. The fit over the whole range from 340 MHz to 4.85 GHz gives . which differs significantly from the Kolmogorov value of .
We believe that a piece-wise power law exists for different time scales of scintillation. Figure 7 shows the resulting composite time structure function including data from 102.6 MHz up to 5 GHz and also the evaluation of the SF obtained from refractive scintillation and timing data of PSR B1642-03. We used the refractive scintillation index ( ) and the refractive time scale ( days) at 610 MHz from Stinebring et al. (2000) to compute a value of the SF at a time lag of about 105 s according to Eqs. (A.18) and (A.19). For this computation one has to assume a power law index, which we took to be n = 3.67(a Kolmogorov spectrum). The error of this estimation of the SF is about 30, which is based on errors of and . We supplemented the compilation of the SF by the addition of a point at s - by far the largest time lag - which is based on timing residuals from 8.5 years of timing observations of PSR B1642-03 at 103 MHz (Shabanova et al. 2001) (Fig. 7, filled circle). We used for this compilation the relation with and ms at 103 MHz. was taken from the amplitude of timing residuals (Fig. 5 of Shabanova et al. 2001). We estimated the error of to be 50 over the 8.5 years interval. The fitted power laws (solid lines) to the composite time SF have a value of up to a scale of about 200 s and for time lags above 200 s. They match our data well.
The compiled frequency structure function reduced to f0 = 1 GHz is shown in Fig. 8. We used two different models for the evaluation: pure diffractive scintillation (shown at the bottom) and scintillation with strong angular refraction (at the top of the figure). At 610 MHz we used the mean structure function (from 3 days of observation) with errors corresponding to deviations. The errors of the SFs at 102.6 MHz, 340 MHz and 800 MHz presented in Fig. 8 correspond to variations due to refractive scintillation as mentioned earlier. Added to this figure is a point (triangle) which is based on an analysis of the structure function of the 5 GHz data.
The structure function at 5 GHz becomes steeper at the level (Fig. 3, bottom panel). This steepening is caused by insufficient frequency resolution (our frequency band is B=500 MHz) and corresponds to the steepening of the spatial structure function due to averaging of intensity fluctuations in the frequency band (see Lee 1976). We modeled this effect, using our data at 800 MHz, since at this frequency we have good frequency resolution. We averaged these data over 20 frequency channels and detected the level where we have a steepening of the calculated . Then we used the same level of the frequency structure function to obtain the frequency lag corresponding to this level. This lag was 1000 kHz so the ratio of the real frequency resolution kHz) to this lag is: a = 1.56. This coefficient, a, will be the same for small time lags of based on data obtained at different frequencies of observation. We conclude therefore that the frequency lag, MHz corresponds to the level 0.03 at 5 GHz. We converted this level and frequency lag to f0 = 1 GHz and show this point in Fig. 8.
It is obvious that the diffractive model describes our data better: nearly all data points can be fit by one power law with a slope . It might be that the SF steepens at frequency lags higher than 2000 kHz, but to decide this 5 GHz observations with better frequency resolution would be needed. The refractive model, by comparison, fails with respect to the 103 MHz and 5 GHz observations, and there are difficulties with the 340 MHz measurements. We are therefore confident that diffractive effects dominate the refractive effects at all frequencies that we used. The slopes of the temporal and frequency structure functions (see Figs. 6 and 8) differ by about a factor of two ( and ) as they should for a pure diffractive model. They should be about equal in the refractive model (Shishov et al. 2003), which is ruled out by our data.
We were able to construct the composite time and frequency structure
function in the direction to PSR B1642-03 over a wide range of
frequencies and decorrelation times based on a comprehensive collection
of observations. The measurements show that the interstellar plasma
along the line of sight to PSRB1642+03 consists of turbulent
irregularities that can be described by a piecewise power law
spectrum. The slope of the spectrum is given by
is the slope of the structure function in a log-log
diagram, which we found to be either
for small scales (time structure function,
Fig. 7). This means that the spectrum is close to a Kolmogorov
spectrum (n = 3.67) for large temporal or spatial scales, similar to
the spectrum in the direction to PSR B0329+54 (Shishov et al. 2003).
For small scales the spectrum is clearly flatter, with a power law
(Fig. 6). The flattening is obvious, both in
the time and the frequency structure functions. We did not see such a
feature in the direction to PSR B0329+54, and we do not see any
influence of angular refraction on the shape of the frequency structure
function for PSR B1642-03.
For this line of sight we then have
The decorrelation time of weak scintillation at 5 GHz,
s, can be explained if the thickness z of the
scattering layer is
If the turbulent medium is uniformly distributed between source and observer one finds from Eq. (4) a distance of kpc to the pulsar, given the measured proper motion of mas/year (Brisken et al. 2003). The pulsar velocity would be about 930 km s-1 at such a distance, which is at the extreme end of the pulsar velocity range. The computed value of R is also much larger than the value estimated by Prentice & ter Haar (1969) which gives 160 pc and the value 2900 pc given by Taylor et al. (1993), so that the model of a uniformly distributed medium is unlikely.
It appears realistic, therefore, to assume that the layer of turbulent medium is located near to the source. If we use in Eq. (4) and use the known value of we find z/R = 0.2R/R0, where R0 = 1 kpc and R is the distance to the pulsar. The same relation was obtained previously (Smirnova et al. 1998). This means that z is either 5 pc for a distance of 160 pc or 1.7 kpc for a distance of 2.9 kpc. The corresponding transverse pulsar velocities are 22 km s-1 and 400 km s-1 for the assumed distances. PSR B1642-03 is a high latitude pulsar ( ). It is therefore unlikely that there is strong scattering at z = 1.7 kpc. We believe that it is a nearby pulsar, and we will use R = 160 pc and V = 20 km s-1 for the conversion of the temporal scale to the spatial scale. The corresponding spatial scale of the structure function is shown in Fig. 7 (top axis). We see that it consists of two components. The SF corresponds to a Kolmogorov spectrum for large scales (from 109 to 1015 cm), which is in accordance with the results obtained for other pulsars (Armstrong et al. 1995; Shishov & Smirnova 2002). The level of this component is higher than the level of the SF in the direction to PSR B0329+54. This enhancement may be due to a higher level of the electron density.
The small-scale component (scales less than 109 cm) corresponds to an unusually flat spectrum of turbulence (n about 3.3). We note that PSR B1642-03 is located in the region of an old supernova remnant - the North Polar Spur (Spoelstra 1972) - which means that the flattening of the turbulence spectrum might be caused by the special characteristics of the turbulent medium in this supernova remnant. Such flat spectra were also detected for the strongly anisotropic plasma near to the Sun at distances less than 0.1 astronomical units (Woo & Armstrong 1979; Armstrong et al. 1990). An enhancement of the solar wind fluctuations at the proton thermal gyroradius was described by Neugebauer (1975) for the more distant regions of the interplanetary plasma. The enhancement of the interstellar plasma turbulence spectrum is located in our case in a region of spatial scales near to cm, which is close to the value of the thermal proton gyroradius as estimated by Chashei & Shishov (1980).
The measurements of turbulence spectra by use of multi-frequency observations in the direction to a given pulsar show new features which were undetectable by the usual methods based on ensembles of sources. We evaluated the position and thickness of the turbulent medium causing the scintillation of the pulsar emission. We found unusual features like the enhancement of the ISM turbulence spectrum in a narrow range of scales as detected in the direction to PSR B1642-03 and strong angular refraction as detected in the direction to PSR B0329+54. We hope that the use of this new method will allow us to investigate turbulence spectra for a larger number of pulsars in fine detail, enabling us to obtain information on the physical conditions of the interstellar turbulent plasma.
This work was supported by INTAS grant No. 2000-849, NSF grant No. AST 0098685, the Russian Foundation for Basic Research, project codes 03-02-509, 03-02-16522, and the Russian Federal Science and Technology Program in Astronomy. We thank the NRAO, operated by Associated Universities under arrangement with the NSF, for support with the 610 MHz and 800 MHz observations and L. B. Potapova for technical assistance.
The frequency and time structure functions were evaluated from the
corresponding correlation functions in the same way as explained in the
paper of Shishov et al. (2003). The relationship between the phase
and the correlation function of intensity
variations BI (t) is given in the saturated scintillation regime by
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 is determined by the value of the turbulence spectrum in the
region of spatial frequencies near
is the characteristic spatial scale of refractive scintillation.
can be expressed as (Shishov & Smirnova 2002)