A&A 405, 795-801 (2003)
DOI: 10.1051/0004-6361:20030699
J. M. Smits ^{1} - B. W. Stappers ^{2,3} - J.-P. Macquart ^{4} - R. Ramachandran ^{2,3} - J. Kuijpers ^{1}
1 - Department of Astrophysics, University of Nijmegen, The Netherlands
2 - Stichting "Astron'', PO Box 2, 7990 AA Dwingeloo, The Netherlands
3 - Astronomical
Institute "Anton Pannekoek'', Amsterdam, The Netherlands
4 - Kapteyn Institute,
University of Groningen, Groningen, The Netherlands
Received 10 February 2003 / Accepted 6 May 2003
Abstract
We have examined data from pulsars B0950+08 and B0329+54 for evidence of
temporally coherent radiation using the modified coherence function (MCF) technique
of Jenet et al. (2001). We consider the influence of both instrumental bandpass and interstellar propagation effects.
Even after removal of the effects due to the instrumental bandpass, we detect a
signature in the MCF of our PSR B0329+54 data
which is consistent with the definition of a coherent signal.
However, we model the effects due to interstellar scintillation for this pulsar and show
that it reproduces the observed signature. In particular, the temporal coherence time is close to the
reciprocal of the decorrelation bandwidth due to diffractive scintillation.
Furthermore, comparison of the coherence times of three pulsars reported by
Jenet et al. (2001) with their expected diffractive decorrelation
bandwidths suggests that the detection of coherence in these pulsars is
also likely a result of interstellar scintillation, and is not intrinsic to
the pulsars.
Key words: radiation mechanisms: general - stars: neutron - pulsars: general - pulsars: individual: B0329+54, B0950+08
Jenet et al. (2001) (hereafter JAP) claim to have detected the existence of coherent non-Gaussian radiation on 100 ns time scales for pulsars B0823+26, B0950+08 and B1133+16 in observations made at the Arecibo observatory. They used their "modified coherence function'' (MCF), defined in Eq. (1) below, to find statistics inconsistent with amplitude-modulated Gaussian noise in the voltage time series from observations of these three pulsars. They show that a coherent model for pulsar radiation can account for the observed statistics. However, if the MCF of a time series of pulsar radiation shows statistics of a non-Gaussian nature, it does not yet prove that the pulsar radiation itself contains non-Gaussian statistics. We have to consider the effect of the ISM on pulsar radiation and the instrumental effects on the signal after detection. JAP assert that scintillation does not influence the statistics of the signal measured by the MCF. In particular, they state that if the statistics of the intensity fluctuations are well described by Gaussian noise, the MCF ought not to exceed zero as a result of propagation effects. Scintillation, however, can give rise to a quasi-periodic fluctuation in the frequency power spectrum (see Fig. 3b). Such fluctuations in the frequency domain can be expected to influence the autocorrelations from which the MCF is constructed.
We consider here an independent data set for pulsars B0950+08 and B0329+54 to try to confirm the detection of coherence. In Sect. 2 we describe our observations and the construction of the MCF. In Sect. 3 we present an analysis of our data and investigate whether scintillation and instrumental effects do influence the MCF. Here, we also present a numerical experiment that shows that scintillation quantitatively explains the coherent features observed by JAP and compare their values of the coherence time with the inverse of the diffractive decorrelation bandwidth for each pulsar. In Sect. 4 we discuss our results and present our conclusions.
These data were taken using the Westerbork Synthesis Radio Telescopes (WSRT) with its pulsar backend, PuMa (Voûte et al. 2002). In its tied array mode the WSRT is equivalent to a single dish with a diameter of 94 m and has a gain of 1.2 K/Jy. PSR B0950+08 was observed on 28 April 1999 at a centre frequency of 382 MHz, with a bandwidth of 10 MHz. PSR B0329+54 was observed on 13 August and 11 September 1999 at a centre frequency of 328 MHz, with a bandwidth of 5 MHz. In both observations a 10 MHz band was Nyquist-sampled in 2 linear polarisation channels. For PSR B0329+54 the 5 MHz band was formed by digitally filtering the data using a finite impulse response filter. After sampling, the data were 2 bit digitised. The high time resolution of 50 ns and 100 ns for pulsars B0950+08 and B0329+54, respectively, is necessary to study radiation which is expected to contain features with a coherence time of a few hundred nanoseconds (Jenet et al. 2001).
We remove the effects of
interstellar dispersion by means of coherent
dedispersion (Hankins 1971). Finally, for both pulsars, we take two
sets of small successive time windows for each individual pulse.
One set contains small windows centered on the peak of the
average pulse profile,
the other set contains a region outside the pulse, which is used for
system plus sky noise corrections.
Pulsar | B0950+08 | B0329+54 |
Period | 0.2531 s | 0.7145 s |
Period derivative | ||
Dispersion measure | 2.9702 pc cm^{-3} | 26.7765 pc cm^{-3} |
Centre frequency | 382 MHz | 328 MHz |
Bandwidth | 10 MHz | 5 MHz |
Number of pulses | 2 320 | 5 200 |
Size of window | 102.4 s | 204.8 s |
Number of windows | 16 | 16 |
Time-resolution | 50 ns | 100 ns |
From these windows we calculate the MCF for real voltages. The MCF
tests the fourth moment of the signal against the square of the
second moment and is given by
The MCF is sensitive to phase relationships between measurement points separated in time, called temporal coherence. The two polarisations are also separated and treated as different pulses. We thus obtain 32 time series from each pulse.
In order to calculate the second term of the MCF we note that the recorded on-pulse voltage time series V(t) contains both the pulsar signal and
system plus sky noise. It can therefore be expressed as
To obtain the first term, we calculate the AAC's of the intensity of the voltage
time series of on-pulse signal and off-pulse signal. The AAC of the
intensity of the noise-subtracted pulsar signal is given by
The peak in PSR 0329+54 is at a that is the reciprocal of the bandwidth. This makes us suspect that the frequency-dependent gain of the system, the bandpass, can influence the MCF. We can understand the presence of a peak in the MCF due to the bandpass as follows^{}. Define as the Fourier transform of the voltage signal S(t)and define the bandpass as . The bandpass contains modulation with a width equal to the width of the bandpass, and also modulation with a width of about one third the width of the bandpass (see Figs. 3c, d). This implies that S(t) and S(t)^{2} contain a peak with a width on the order of . The autocorrelation of the signal and the autocorrelation of the intensity will then contain a peak at and decorrelate on a timescale on the order of . The MCF contains the peak from the autocorrelation of the intensity minus 2/3 the square of the peak of the autocorrelation of the signal itself. This last contribution works as a partial bandpass correction, but not as a full bandpass correction, as the two contributions are not necessarily equal.
To correct for the bandpass, we calculate and apply a bandpass
correction as follows. First, we estimate the average
bandpass as the power of the off-pulse signal as a function of
frequency averaged over all pulses.
We then normalise by dividing each frequency value by the average of the bandpass.
Before the calculation of the MCF, we Fourier transform the signal
inside each window into the frequency domain, divide the value at
each frequency by the square root of the corresponding value of
the bandpass and Fourier transform back into the time domain. This
process does not correct for a possible frequency dependent phase
shift introduced by the system.
We carry out the procedure described above separately for both
polarisations. High-level interference peaks are replaced in the frequency domain by the average of their
left and right neighbours. Adjacent peaks are considered as one. Remaining interference peaks should only have a small influence on
the MCF and only on long timescales.
Figure 2: Our results of the MCF calculated for PSR B0329+54 after bandpass correction. See Fig. 1 for explanation. |
In Fig. 2 we see the corrected MCF for PSR B0329+54. The peak at low has gone down slightly, but remains present. Looking at Fig. 3b, we see that there is strong modulation in the on-pulse bandpass of PSR B0329+54 which is absent in the off-pulse bandpass. This reduces the effectiveness of the bandpass correction, as this modulation makes the on-pulse bandpass different from the off-pulse bandpass. We did not do a bandpass correction for PSR B0950+08, as there is no sign of coherence present in our data from this pulsar.
Having established that variations in the bandpass shape can theoretically influence the MCF, we now consider the additional influence of interstellar scintillation (ISS). Diffractive ISS introduces structure in the instantaneous spectrum of a pulsar and can therefore, in principle, influence the MCF. In particular, we argue that ISS induces a signal in the MCF that is likely to masquerade as a false detection of temporally coherent pulsar radiation.
Diffractive interstellar scintillation is observed in pulsars, including B0950+08 and B0329+54, at centimetre wavelengths and lower. It imposes large (>100%) temporal and spectral modulations in the intensity of the radiation. The decorrelation timescale and bandwidth of the fluctuations are chiefly determined by two parameters: the diffractive scale , and the Fresnel scale . It is often convenient to consider the phase fluctuations imposed by the scattering medium confined to a thin phase screen of distance L from the observer. Then the diffractive scale is the distance on the phase screen over which the root mean square phase difference is one radian, and the Fresnel scale is set by the distance to the screen and the wavelength .
Diffractive scintillation imposes random spectral variations of
characteristic bandwidth
(7) |
Let us consider the effect of diffractive spectral variations on the quantity . The instantaneous power spectrum of the pulsar signal contains large variations with a characteristic bandwidth . Thus, since the amplitudes of the Fourier transformed voltages contain ripples, the observed voltages are broadened. In the case of scintillation, the broadening can be described by the pulse broadening function (PBF) (Williamson 1972). The instantanous PBF is expected to have wiggles on a time scale set by the reciprocal of the decorrelation bandwidth, . This implies that the intensity also contains variations on the same characteristic time scale. Neglecting any intrinsic temporal coherence due to the pulsar, the intensity autocorrelation function is expected to peak at and decorrelate on a timescale set by the inverse of the scintillation decorrelation bandwidth. The average intensity autocorrelation function, obtained by combining the autocorrelations from many individual pulses and even over many diffractive timescales, is expected to exhibit the same decorrelation timescale. This is because every set of data is expected to exhibit spectral structure with a similar decorrelation bandwidth.
The second contribution to the MCF comes from the autocorrelation of the
signal voltages,
.
The autocorrelation function of the
pulsar voltages, S(t), and the on-pulse bandpass,
are related by a Fourier transform:
The simulations are conducted by constructing a set of phase
fluctuations as a function of position and wavelength,
,
frozen onto a thin phase-changing screen located a
distance L from the observer's plane. We take a plane wave of unit amplitude
incident on the phase screen, so that the phase of the wave upon exiting the
screen is
.
The phase fluctuations are generated
according to a von Karman power spectrum (Goodman & Narayan 1989)
(10) |
Temporal fluctuations in the wavefield are obtained by moving the phase screen
relative to the observer, however, due to the limited number of grid points
short time scale variations could not be simulated. Spectral variations are obtained by scaling
the phase fluctuations according to
(11) |
Figure 4: The MCF of simulated scintillation at a frequency of 328 MHz and a bandwidth of 5 MHz. The input parameters are set to generate scintillation similar to that in our data from B0329+54. |
Input | Description |
parameters | |
Minimum wavenumber | |
Maximum wavenumber | |
( and are the edges of | |
the observed frequency range) | |
D = 0.7 kpc | Distance between scintillation |
screen and observer | |
Q_{0} | Amplitude of the power spectrum |
of the phase fluctuations | |
N = 128 | Number of discrete points across |
the wavefield (total number of | |
grid points = N^{2}) | |
Measured | |
parameters | |
m | Fresnel scale ( ) |
m | Diffraction length scale |
m | Refraction length scale |
(= ) | |
MHz | Diffractive decorrelation |
bandwidth |
We find that frequency modulation due to scintillation and possibly the shape of the WSRT bandpass have an effect on the MCF at small delay values. This may be problematic in determining a coherence time. For PSR B0950+08 we find no signature of coherence. For PSR B0329+54 we find two features at small delay values: a peak in the first two bins and a broad excess reaching up to 35 s. We have shown that the shape of the bandpass can theoretically cause a peak in the first bins of the MCF. The bandpass correction, which uses the off-pulse, might not be effective in the case of PSR B0329+54 due to the frequency modulation present in the on-pulse. By comparing Fig. 4 with Fig. 1b, we see that scintillation reproduces the broad rise observed in the MCF of PSR B0329+54. Furthermore, a simulation of an increasing scintillation pattern on top of a flat bandpass, where the modulation due to the scintillation was overlapping, has shown that small changes in the scintillation pattern can cause a peak in the first bins of the MCF. This is similar to a time dependent variation in the bandpass itself. Moreover, when smaller timescales are used to calculate the MCF the peak is seen to decrease also indicating that scintillation may play a role here.
From the above, we conclude that scintillation can be responsible for both the broad excess as well as the peak in the first two bins of the MCF of PSR B0329+54.
We now discuss whether the excess found in the MCF of three
pulsars by JAP could also be due to scintillation.
In Table 3 we show the similarity between the
diffractive decorrelation bandwidth of the different pulsars and their
coherence time, defined
as the point where the MCF becomes zero. The values for
were scaled from a frequency of 51 MHz for PSR B0950+08 and from
1 GHz for the other pulsars, assuming
(Cordes et al. 1985). No errors were quoted
in Cordes (1986), however it is known that
can vary significantly with time, in
some cases as much as a factor of 2-3 (Bhat et al. 1999).
Assuming that the rise in the MCF is due to
scintillation, we estimate the MCF to become zero when
is on
the order of
.
Looking at
Table 3, we see that the values of the
reciprocal of the diffractive decorrelation bandwidth and the measured
coherence time (fourth and fifth column, respectively) are indeed
similar. For our result of PSR B0950+08 there is no measured
coherence time, while JAP find a coherence time for
this pulsar of 0.4 s. Assuming the diffractive decorrelation
bandwidth from Phillips & Clegg (1992) the time resolution of our data and
that of JAP are too large to see the scintillation. The observed
feature in the MCF for PSR B0950+08 of JAP (see their Fig. 1) might then
result from a time variation in the bandpass of their data. There is,
however, some controversy as to what
the diffractive decorrelation bandwidth for PSR B0950+08 is. According
to Cordes (1986)
is 4.0 MHz at an observing
frequency of 430 MHz. This gives a value for
of
0.25 s, which is in the order of size of the coherence time of
0.4 s, measured by JAP. Assuming the value given by Cordes (1986),
it would appear that the signal-to-noise ratio in our data is
insufficient to see the scintillation, as we see no excess in the MCF.
For PSR B1937+21, JAP also found no coherence time. Although this
pulsar is known to scintillate, the value for
is so large that we can expect
the excess in the MCF due to scintillation to be smeared out over such
a large range of delays as to make it unmeasurable with the sensitivity of their observation.
Pulsar | Frequency | Time resolution | Measured coherence | ||
(MHz) | (ns) | (s) | time (s) | ||
B0329+54 | 328 | 100 | 29 kHz [1] | 34 | 35 [4] |
B0823+26 | 430 | 100 | 0.81 MHz [1] | 1.2 | 1.5 [5] |
B0950+08 | 382 | 50 | GHz [3] | 4.5 | none [4] |
B0950+08 | 430 | 100 | GHz [3] | 2.9 | 0.4 [5] |
B1133+16 | 430 | 100 | 1.47 MHz [1] | 0.67 | 1.1 [5] |
B1937+21 | 430 | 100 | 4.2 0.9 kHz [2] | 238 | none [5] |
[1] | Cordes (1986). |
[2] | Cordes et al. (1990). |
[3] | Phillips & Clegg (1992). |
[4] | Our data. |
[5] | Data from JAP. |
We conclude that scintillation and possibly the shape of the bandpass causes the excess in the MCF of our data from PSR B0329+54. Furthermore, we conclude that the coherent features, found by JAP, also appear to be the result of scintillation. We therefore cannot confirm that the MCF is clearly showing us the presence of coherence in these pulsars.
Acknowledgements
The authors would like to thank J. Cordes for his extensive comments and F. A. Jenet for his helpful discussions which have both greatly contributed to the accuracy and clarity of this paper.