Astronomy & Astrophysics manuscript no. Popov2006˙6589 c© ESO 2018
October 23, 2018
Statistical properties of giant pulses from the Crab pulsar
M.V. Popov1 and B.Stappers2
1 Astro Space Center of the Lebedev Physical Institute, Profsoyuznaya 84/32, Moscow, 117997 Russia
2 Astronomical Institute ”Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam,The Netherlands;
Stichting ASTRON, Postbus 2, 7990 AA, Dwingeloo, The Netherlands
Received ; accepted
ABSTRACT
Aims. We have studied the statistics of giant pulses from the Crab pulsar for the first time with particular reference to their widths.
Methods. We have analyzed data collected during 3.5 hours of observations conducted with the Westerbork Synthesis Radio Telescope
operated in a tied-array mode at a frequency of 1200 MHz. The PuMa pulsar backend provided voltage recording of X and Y linear
polarization states in two conjugate 10 MHz bands. We restricted the time resolution to 4 µs to match the scattering on the interstellar
inhomogeneities.
Results. In total about 18000 giant pulses (GP) were detected in full intensity with a threshold level of 6σ. We analyzed cumulative
probability distribution (CPD) of giant pulse energies for groups of GPs with different effective widths in the range 4 to 65 µs. The
CPDs were found to manifest notable differences for the different GP width groups. The slope of a power-law fit to the high-energy
portion of the CPD evolves from −1.7 to −3.2 when going from the shortest to the longest GPs. There are breaks in the CPD power-
law fits indicating flattening at low energies with indices varying from −1.0 to −1.9 for the short and long GPs, respectively. The
GPs with a stronger peak flux density were found to be of shorter duration. We compare our results with previously published data
and discuss the importance of these peculiarities in the statistical properties of GPs for the theoretical understanding of the emission
mechanism responsible for GP generation.
Key words. pulsars: general – pulsars: individual: B0531+21 – Methods: statistical – Radiation mechanisms: non-termal
1. Introduction
Giant pulses are one of the most striking phenomena of
pulsar radio emission. Their flux density can exceed thou-
sands of times the average pulse-integrated flux. Although re-
cently there are reports of detecting excessively strong pulses
from a number of pulsars (Johnston et al. 2001; Kramer et al.
2002; Romani & Johnston 2001; Johnston & Romani 2002;
Kuzmin et al. 2004; Ershov & Kuzmin 2003; Knight et al.
2006), only the Crab pulsar and the millisecond pulsar
B1937+21 generate giant pulses numerous enough to study their
statistical properties. The very large fluxes of giant pulses are
coupled with an extremely short duration. Indeed, the over-
whelming majority of giant pulses from the millisecond pulsar
B1937+21 are shorter than 15 ns (Soglasnov et al. 2004), while
giant pulses from the Crab pulsar have a mean width of about
few microseconds (Hankins 2000) with occasional bursts shorter
than 2 ns (Hankins et al. 2003).
The longitude position of giant pulses is remarkable in
that it coincides with the position of high-energy emission
(Moffett & Hankins 1996; Cusumano et al. 2003; Johnston et al.
2004; Knight et al. 2006). Furthermore, giant pulses originate in
a very narrow phase window that in general does not correspond
to the phase window of regular radio emission. In the millisec-
ond pulsar B1937+21, giant pulses are observed at the very trail-
ing edge of the average profile in both the main pulse and the
interpulse (Cognard et al. 1996; Kinkhabwala & Thorsett 2000;
Soglasnov et al. 2004). Popov et al. (2006) has recently sug-
gested that giant pulses from the Crab pulsar are also seen at the
Send offprint requests to: M.V. Popov
Correspondence to: mpopov@asc.rssi.ru
trailing edge of the regular radio emission window, which they
consider to be the precursor. In other words, they consider that
radio emission in the main pulse and the interpulse of the Crab
pulsar consists completely of giant pulses. Lastly, an essential
property of giant pulses, by which they may be distinguished
from normal strong pulses, is the distribution of their flux den-
sities, which appears to follow a power law in contrast with
the Gaussian or exponential flux distribution typical of regular
(ordinary) individual pulses (Backer 1971; Hesse & Wielebinski
1974; Ritchings 1976). In this paper we present an analysis of
giant pulse-energy distribution based on the observations con-
ducted with the Westerbork Synthesis Radio Telescope (WSRT)
at 1200 MHz using the PuMa (Voute et al. 2002) pulsar backend.
2. Observations and data reduction
The observations were carried out in November 2003 as part
of a multi-frequency observing campaign that also included
Jodrell Bank at 1420 MHz, Effelsberg at 8350 MHz, Kalyazin
at 600 MHz, Pushchino at 111 MHz, and Kharkov (UTR-2) at
23 MHz. Simultaneous optical observations were also made
with the 6-m telescope of the Special Astrophysical Observatory
and 2.8-m telescope at La Palma. The MAGIC and HESS
gamma-ray telescopes in La Palma and Namibiya also partici-
pated. While some results obtained at the separate observatories
have already been published (Jessner et al. 2005; Popov et al.
2006), the joint analysis of multi-frequency observations will be
presented in future publications.
We have analyzed the data obtained in observations with the
WSRT during about 3.5 hours in two conjugate bands of
10 MHz each at a central frequency of 1197 MHz. Baseband
http://arxiv.org/abs/0704.1197v2
2 M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses
voltages of X and Y linear polarization states were recorded
with two-bit sampling at Nyquist frequency. The coherent
predetection dedispersion technique originally developed by
Hankins (Hankins 1971; Hankins & Rickett 1975) was used to
remove dispersion smearing in the received pulsar signal. We
took the value of dispersion measure (56.757 pc cm−3) and the
timing model from the Jodrell Bank Monthly Ephemeris (Lyne
1982). The technique provides a formal time resolution of 100
ns, but the expected pulse-broadening time due to interstellar
scintillations is in the range 2 to 4 µs, based on the estimation
made by Kuzmin et al. (2002). Therefore, we averaged the
recorded signal after dispersion removal and square-low detec-
tion synchronously with a topocentric pulsar period into 8192
bins per period and with a resulting sampling interval close
to 4.1 µs. The total intensity time series was finally formed
and used for detection of giant pulses with the threshold level
of 6σ. To provide better sensitivity for wider giant pulses, we
developed a searching procedure which progressively tried to
increase the averaging time interval (τi) by 1, 2, 3, 4, 6, 8, 12,
16, 24, 32, and 48 samples, i.e.from 4.1 µs to 196 µs. The
averaging time with the best signal-to-noise ratio (SNR) was
selected as We for the GP, detected simultaneously at several
averaging times. The pulse energy E (or average flux density)
was calculated for the GP as a product of the SNR and the pulse
width, equal to the averaging time, which corresponds to the
best SNR value. In this approach we have different threshold
levels both in peak flux density Fp and in pulse energy E for
each separate averaging time. The value of the root-mean-square
deviation (RMS or σ) goes down with increasing time averaging
as the square root of time. Therefore, the threshold level in
peak flux density Fp decreases with increasing averaging time
as Fp(τ) ∝ 1/
τ, while the threshold level in pulse energy E
increases with τ as E(τ) ∝
For the Crab pulsar, the system temperature is notably influ-
enced by the impact of the Crab Nebula, whose flux density
can be approximated by the relation Fν = 955ν
−0.27 Jy (ν
in GHz) (Allen 1973; Bietenholz et al. 1997), which gives
F1200 = 909 Jy at 1200 MHz. In our observations we used all
14 dishes of the WSRT in tied-array mode, where all telescopes
are added coherently, with the width of synthesized beam being
equal to 34 arcsec, thus reducing the contribution from the Crab
Nebula to the system noise by a factor fν = ΩA/ΩCN = 0.14
to the value FCN = 127.6 Jy. Here ΩCN is a solid angle of the
Crab Nebula, and ΩA a solid angle of the intersection of the
synthesized beam with the Crab Nebula. However, there is still
the contribution from the individual dishes; the system tem-
perature of every 25-m dish was increased by the Crab Nebula
emission by ∆TCN = F1200G = 86.3 K, where G is the gain of
a single telescope, equal to 0.095 at 1200 MHz. The intrinsic
system temperature in the absence of the Crab Nebula is 30 K
at 1200 MHz, and the resulting total system temperature is thus
116.3 K, equivalent to Fsys = 87.4 Jy for the gain in the tied
array (1.33 K/Jy), which when combined with the nebula con-
tribution to the tied array becomes Ftot = Fsys + FCN = 215 Jy.
One can see that WSRT reduces the contribution of the Crab
Nebula to the system noise considerably when compared with
the single dish observations at this frequency. This improvement
will permit us to follow the statistics of giant pulses from the
Crab pulsar to the lower energies; namely, the limiting RMS (σ)
value in one polarization is Ftot/
Bτ, with B = 10 MHz, and
τ = 4.1 µs the RMS is equal to 33.5 Jy. For total intensity, the
RMS will be reduced by
2, and the threshold of 6σ in peak
flux density will be equal to 142 Jy. The threshold will be lower
 1000
 10000
 100000
 1e+06
 1  10  100  1000
Averaging time We in microseconds
Fig. 1. Peak flux density versus the effective puse width We for
all detected giant pulses.
for averaged records, with a limiting value of 20 Jy for giant
pulses, which have a maximum effective width of about 200 µs.
In total 17869 giant pulses were detected over 370000 pulse
periods with 14994 giant pulses located at the longitude of the
main pulse and 2875 giant pulses associated with the interpulse.
The background rate was determined by counting events at
quiescent phases of the pulsar period and the corresponding
values were taken into account in our statistical calculations. In
fact, at the lowest range, between 6σ and 7σ, only about 10%
of all pulses we potentially incorrectly identified as giant pulses.
3. Results
Figure 1 represents the complete set of data showing peak flux
density Fp versus the effective pulse width We. The striped na-
ture of the diagram is caused by the procedure of giant pulse de-
tection, which is based on the discrete values of averaging time
We. The diagram demonstrates the peculiarity of the statistics on
giant pulses: the strongest pulses clearly have a shorter duration;
namely, there are no pulses with peak flux density greater than
1000 Jy that are wider than 16 µs. As a consequence, giant pulses
with different durations have different distributions in pulse en-
ergy, and the distribution in giant pulse width depends on the
range of pulse energy used. Therefore, in our analysis, we sepa-
rated the giant pulses belonging to the main pulse longitudes into
five groups by their widths: (a) – GPs with the effective width We
of 4.1 µs, (b) – GPs with the We of 8.2 µs, (c) – combined group
of GPs with the We of 12.3 and 16.4 µs, (d) – combined group of
GPs with the We of 24.5 and 32.8 µs, and (e) – combined group
of GPs with the We of 49.2 and 65.6 µs. A separate group (f)
was formed for the GPs detected at the longitude of the inter-
pulse with the We 4.1, 8.2, and 12.3 µs. Then, pulse energy, or
integrated flux density, was calculated as E = Fp ×We.
In our analysis we use cumulative probability distribution
(CPD), which gives the number of pulses N(E), above pulse en-
M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses 3
Table 1. The list of parameters of the power-law fits to the CPDs
displayed in Fig.2.
Group We(µs) γhigh γlow Ebreak(Jy · µs)
(a) 4.1 1.7 1.1 4000
(b) 8.2 1.8 1.0 3500
(c) 12.3, 16.4 2.3 1.2 4000
(d) 24.5, 32.8 2.5 1.4 4000
(e) 49.2, 65.6 3.2 1.9 5000
(f) 4.1, 8.2, 12.3 1.6 1.6
ergy E. Some authors prefer to calculate the probability distri-
bution (PD) by giving the number of pulses n(E) per interval
of energy dE. If a power-law fit is used for the PD function as
n(E) ∝ E−β, then a power-law fit will be also valid for the CPD
function N(E) ∝ E−γ, since
N(E > Ey) =
n(E)dE ∝ E−β+1y
with γ = β − 1 in the absolute value.
Figure 2 displays the CPD of pulse energies for all the afore-
mentioned groups of GPs. The CPD for the GPs belonging
to the interpulse can be fitted with a power-law function with
γ = 1.6 ± 0.1. The CPDs for the GPs in the main pulse manifest
a break in the power-law index at certain values of pulse energy,
which is slightly different for the different groups. Table 1
contains the results of a least-square solution for the power-law
fit. The first column indicates the group of GPs, We is the
averaging time for the group in µs, γhigh and γlow are indices of
the power law function for the high energy tail and low energy
portion of the CPD, respectively, and Ebreak indicates the pulse
energy where the slope of the power law changes. The break
energy values were obtained as the crossing point between two
straight lines defined by the least-square solutions. Relative
inaccuracy is within 10% at the RMS level.
The absence of any break in the power-law fit for the CPD
of the GPs belonging to the interpulse longitudes is good proof
that the observed breaks in the CPDs of the main pulse GPs were
not caused by selection effects, since we used exactly the same
procedure for GPs detection irrespective of pulse longitude. A
possible explanation is that we observe the regions where the
GPs are generated in the the main pulse and the interpulse at
different impact angles relative to the GP beam, which has a half
width of about 5 degrees. Indeed, the comparison of the CPDs
(combined for the width groups (a), (b), and (c)) for the main
pulse and the interpulse shows that the CPDs would be identical
if one were to multiply the interpulse GP energies by a factor of
about 4 and increase their rate of occurrence by a factor of about
2. Therefore, the weakest detected interpulse GPs with a pulse
energy of about 1000 Jy · µs, when observed at the same impact
angle as the main pulse GPs (multiplied by 4), will have pulse
energies of 4000 Jy · µs, which is above the break point.
4. Discussion and conclusions
The CPD function of the GPs originating at the longitudes of
the main pulse manifests gradual changes in the power-law in-
dex from −1.7 to −3.2 at the high energy part of the CPD, and
from −1.0 to −1.9 at the low energy portion of the CPD. As
a rough guide, all GPs can be separated into two main groups:
short GPs belonging to groups (a) and (b), and long GPs with
an We greater than 8.1 µs. It is interesting to note that the CPD
power-law index of about −1.7 for short GPs is close to the ex-
ponent determined for the CPD of the GPs from the millisec-
ond pulsar B1937+21, which was found to be equal to −1.8
at 430 MHz (Cognard et al. 1996). The value was confirmed
by Kinkhabwala & Thorsett (2000), and recently estimated by
Soglasnov et al. (2004) as −1.4 at 1650 MHz. All GPs from the
millisecond pulsar are very short, lasting only 1-2 µs as mea-
sured by Kinkhabwala & Thorsett (2000), while Soglasnov et al.
(2004) have found that the majority of GPs from this pulsar are
shorter than 15 ns.
Argyle & Gower (1972) were the first to present the CPD for
GPs from the Crab pulsar. They combined the results of observa-
tions made at 146 MHz with the 26-m Penticton radio telescope
and with the 46-m dish at the Algonquin Radio Observatory and
found that the CPD was consistent with the power-law exponent
of −2.5 for the main pulse and with the exponent for the inter-
pulse events equal to −2.8.
Recently Cordes et al. (2004) in their multifrequency study of
the Crab pulsar’s giant pulses presented histograms of GP peak
amplitudes (S) at 0.43 and 8.8 GHz. The histograms represent
the PD functions, i.e. number of events per interval of SNR in
logarithmic binning d(lg S ). They found power-law segments
in the distributions with the slopes −2.3 and −2.9 at 0.43 and
8.8 GHz, respectively. To compare this result with our study of
the CPD functions one has to convert the number of events pre-
sented by Cordes et al. (2004) from logarithmic binning to linear
binning d(lg S ) = ln 10dS/S . Then, for a power-law fit with lin-
ear binning the slope will change from −β to −(β + 1):
N(S ) ∝ (S )−βd(lg S ) ∝ (S )−(β+1)dS
for the power-law fit of the CPD γ = β − 1, as was explained
in Sect. 3. Thus, the values of the power-law exponent β found
by Cordes et al. (2004) in their histograms may by immediately
compared with our values of the exponents γ for the power-law
fit of the CPDs. In fact the inspection of the histogram of
Cordes et al. (2004) for 0.43 GHz has enabled us to distinguish
a break at an S/N value of about 30 where the slope changes
from −2.3 to −0.7 in the peak amplitude distribution of the main
pulse GPs, while the distribution for the interpulse GPs does not
manifest such a break. With the given equivalent flux system
temperature of 1262 Jy, the receiver band of 12.5 MHz, and a
sample interval of 128 µs, the S/N = 30 break point at 0.43 GHz
will correspond to a pulse energy of about 120000 Jy · µs.
In his PhD thesis Moffett (1997) presentes the CPD functions
at 1.4 GHz based on observations conducted with the VLA
with a time resolution of 160 µs. In Fig. 4.3 (page 62) Moffett
distinguishes a break in power-law fit of the CPD for the main
pulse GPs at a level of 12 Jy corresponding to a pulse energy of
about 2000 Jy · µs, where the slope changes from −3.0 to −1.8
when going from the high flux densities to the lower. Again, the
CPD for the GPs originated at the longitudes of the interpulse
(Figure 4.4) does not show a break going straight with the slope
of −1.7 until it merges with noise at a level of about 6 Jy.
Comparing the values of break-point pulse energy (BPPE)
120000 Jy · µs at a frequency of 0.43 GHz (Cordes et al.(2004)),
5000 Jy · µs at a frequency of 1200 MHz (this paper), and
2000 Jy · µs at a frequency of 1400 MHz (Moffett (1997)),
we have found that the BPPE follows a simple power-law
frequency dependence BPPE(ν) ≈ 7(ν)−3.4kJy · µs, with the
exponent −3.4 close to the mean spectral index for the main
pulse component of the average profile −3.0 (Moffett 1997).
The conclusion can be considered as support for the suggestion
4 M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
Fig. 2. The CPD of giant pulses with pulse energy above the displayed value for different groups of giant pulses classified by their
effective width We as described in the Table 1. Straight lines represent the fit by power-law functions with the parameters indicated
in the Table 1. The last plot (f) represents giant pulses belonging to the interpulse.
that all emission in the main component consists entirely of
giant pulses (Popov et al. 2006).
Lundgren et al. (1995) has collected about 30000 GPs from
the Crab pulsar at 812 MHz with the Green Bank 43-m radio
telescope in 10 days of observations simultaneous with the
Compton Gamma Ray Observatory (CGRO) in May 1991.
They did not distinguish GPs belonging to the main pulse and
the interpulse. Their flux-density distribution was fitted with a
power-law function for S > 200 Jy, and the exponent was found
to be equal to −3.46, the exponent being equal to −2.46 for the
CPD used in our analysis. The BPPE value is expected to be
14 kJy · µs at 812 MHz according to the frequency relation we
derive above. The value corresponds to 50 Jy flux density with
307.5 µs averaging time used by (Lundgren et al. 1995), and it is
well below their threshold of 120 Jy. The dramatic roll-off at the
rather high pulse energy of about 200 Jy × 300µs = 60kJy · µs
found by (Lundgren et al. 1995) was not observed in the CPDs
at 430, 1200, and 1400 MHz discussed above. It was not
observed in the CPD at 600 MHz either, which goes straight
with the slope of −2.2 at least down to a pulse energy of about
M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses 5
30 kJy · µs (Popov et al. 2006). Therefore, it is difficult to
reconcile their results with those from the many independent
data sets mentioned above.
The power-law form of the observed CPDs can be compared
with the field statistics of possible emission mechanisms
responsible for the generation of GPs, as discussed by Cairns
(2004) who used the values of the exponents of PD functions
for comparison. The very short-time flux density variations are
of particular interest, since they are closely tied to the physics
of the emission process. The power-law indices γ of the CPDs
at high energies for the shortest GPs are similar enough for both
the Crab pulsar and the PSR B1937+21, and the values were
found to be in the range 1.4 to 1.8. Converting to indices of the
probability distribution function (PD) β = γ + 1 gives the range
2.4 – 2.8 for the values of β. According to the normalization
conditions used by Cairns, the exponent α of the PD of the
field (P(E) ∝ E−α) is connected with the exponent β of the
observed PD power-law fit by the relation α = 2β − 1, giving
us the range of α from 3.8 to 4.6 to be compared with the
theoretical predictions of the field statistics determined by the
emission mechanism and propagation effects. The observed
breaks in the slope of the CPD functions at certain energies
have to be included in theoretical explanations, giving us an
extra parameter to constrain the source physics and emission
mechanism.
The break in the slope of the CPD functions has important con-
sequences for the estimation of the total rate of GP generation.
Popov et al. (2006) made such an estimation for the Crab pulsar
under the suggestion that radio emission in the main pulse and
in the interpulse consists entirely of giant pulses. They found
that about 10 giant pulses are generated during one rotation
period of the neutron star. In the estimation they considered that
pulse energies of the GPs follow a power-law function with
the exponent of −2.2 down to the threshold of about 100 Jy
in the peak flux density at a frequency of 600 MHz, and with
the threshold considered as a real lower limit equivalent to
the minimum pulse energy of about 5000 Jy · µs. With their
threshold of about 20000 Jy · µs in GP detection, they did not
notice the break in the CPD slope. The break from −2.2 at high
energies to −1.2 at low energies will notably change the estimate
of the lower limit for GP energies to about 1000 Jy · µs, and the
rate of GP generation will increase at least by a factor of 2. The
lower energy limit for GP generation, if it exists, will serve as
a crucial constraint on the physics of the emission mechanism.
To solve the problem, it is necessary to test the low-intensity
portion of the CPD function with better sensitivity. Such a study
would be possible using VLA or GMRT observations in phased
array mode, thereby, significantly reducing the impact from the
Crab Nebula. A notable increase in the recording band from
10 to 160 MHz recently achieved for the PuMa II recording
systems also makes new observations with the WSRT very
promising.
Finally, we summarize the main results of our analysis:
1) The CPDs were found to be notably different for the GPs
detected at the longitudes of the main pulse and the interpulse.
We suppose that the difference can be explained by the simple
attenuation caused by a beaming factor.
2) For the main pulse longitudes, the CPD are different for
the GPs of different effective widths with breaks in the CPD
power-law indices indicating steepening at high energies (see
Table 1).
3) The CPD power-law indices (γhigh ≈ 1.7) for the group of
short GPs for the Crab pulsar (We < 10µs) are close to the value
observed for the millisecond pulsar B1937+21, which seems to
generate only the shortest GPs;
4) GPs with a stronger peak flux density were found to be of
shorter duration.
The last of these properties testifies in favor of nonlinear
temporal models that suggest that the higher the intensity,
the narrower the pulse width. Such an emission model was
considered, for example, by Mikhailovskii et al. (1985), who
treated the micropulses as solitons of the radio-wave envelope
propagating through the magnetospheric plasma of the pulsar.
Acknowledgements. This investigation was supported in part by the Russian
Foundation for Fundamental Research (project number 04-02-16384).
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Voute, J. L. L., Kouwenhoven, M. L. A., Langerak, J. J., et al. 2002, A&A, 385,
	Introduction
	Observations and data reduction
	Results
	Discussion and conclusions

  We have studied the statistics of giant pulses from the Crab pulsar for the
first time with particular reference to their widths. We have analyzed data
collected during 3.5 hours of observations conducted with the Westerbork
Synthesis Radio Telescope operated in a tied-array mode at a frequency of 1200
MHz. The PuMa pulsar backend provided voltage recording of X and Y linear
polarization states in two conjugate 10 MHz bands. We restricted the time
resolution to 4 microseconds to match the scattering on the interstellar
inhomogeneities. In total about 18000 giant pulses (GP) were detected in full
intensity with a threshold level of 6 sigma. Cumulative probability
distributions (CPD) of giant pulse energies were analyzed for groups of GPs
with different effective widths in the range 4 to 65 microseconds. The CPDs
were found to manifest notable differences for the different GP width groups.
The slope of a power-law fit to the high-energy portion of the CPDs evolves
from -1.7 to -3.2 when going from the shortest to the longest GPs. There are
breaks in the CPD power-law fits indicating flattening at low energies with
indices varying from -1.0 to -1.9 for the short and long GPs respectively. The
GPs with a stronger peak flux density were found to be of shorter duration. We
compare our results with previously published data and discuss the importance
of these peculiarities in the statistical properties of GPs for the heoretical
understanding of the emission mechanism responsible for GP generation.

Introduction
Giant pulses are one of the most striking phenomena of
pulsar radio emission. Their flux density can exceed thou-
sands of times the average pulse-integrated flux. Although re-
cently there are reports of detecting excessively strong pulses
from a number of pulsars (Johnston et al. 2001; Kramer et al.
2002; Romani & Johnston 2001; Johnston & Romani 2002;
Kuzmin et al. 2004; Ershov & Kuzmin 2003; Knight et al.
2006), only the Crab pulsar and the millisecond pulsar
B1937+21 generate giant pulses numerous enough to study their
statistical properties. The very large fluxes of giant pulses are
coupled with an extremely short duration. Indeed, the over-
whelming majority of giant pulses from the millisecond pulsar
B1937+21 are shorter than 15 ns (Soglasnov et al. 2004), while
giant pulses from the Crab pulsar have a mean width of about
few microseconds (Hankins 2000) with occasional bursts shorter
than 2 ns (Hankins et al. 2003).
The longitude position of giant pulses is remarkable in
that it coincides with the position of high-energy emission
(Moffett & Hankins 1996; Cusumano et al. 2003; Johnston et al.
2004; Knight et al. 2006). Furthermore, giant pulses originate in
a very narrow phase window that in general does not correspond
to the phase window of regular radio emission. In the millisec-
ond pulsar B1937+21, giant pulses are observed at the very trail-
ing edge of the average profile in both the main pulse and the
interpulse (Cognard et al. 1996; Kinkhabwala & Thorsett 2000;
Soglasnov et al. 2004). Popov et al. (2006) has recently sug-
gested that giant pulses from the Crab pulsar are also seen at the
Send offprint requests to: M.V. Popov
Correspondence to: mpopov@asc.rssi.ru
trailing edge of the regular radio emission window, which they
consider to be the precursor. In other words, they consider that
radio emission in the main pulse and the interpulse of the Crab
pulsar consists completely of giant pulses. Lastly, an essential
property of giant pulses, by which they may be distinguished
from normal strong pulses, is the distribution of their flux den-
sities, which appears to follow a power law in contrast with
the Gaussian or exponential flux distribution typical of regular
(ordinary) individual pulses (Backer 1971; Hesse & Wielebinski
1974; Ritchings 1976). In this paper we present an analysis of
giant pulse-energy distribution based on the observations con-
ducted with the Westerbork Synthesis Radio Telescope (WSRT)
at 1200 MHz using the PuMa (Voute et al. 2002) pulsar backend.
2. Observations and data reduction
The observations were carried out in November 2003 as part
of a multi-frequency observing campaign that also included
Jodrell Bank at 1420 MHz, Effelsberg at 8350 MHz, Kalyazin
at 600 MHz, Pushchino at 111 MHz, and Kharkov (UTR-2) at
23 MHz. Simultaneous optical observations were also made
with the 6-m telescope of the Special Astrophysical Observatory
and 2.8-m telescope at La Palma. The MAGIC and HESS
gamma-ray telescopes in La Palma and Namibiya also partici-
pated. While some results obtained at the separate observatories
have already been published (Jessner et al. 2005; Popov et al.
2006), the joint analysis of multi-frequency observations will be
presented in future publications.
We have analyzed the data obtained in observations with the
WSRT during about 3.5 hours in two conjugate bands of
10 MHz each at a central frequency of 1197 MHz. Baseband
http://arxiv.org/abs/0704.1197v2
2 M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses
voltages of X and Y linear polarization states were recorded
with two-bit sampling at Nyquist frequency. The coherent
predetection dedispersion technique originally developed by
Hankins (Hankins 1971; Hankins & Rickett 1975) was used to
remove dispersion smearing in the received pulsar signal. We
took the value of dispersion measure (56.757 pc cm−3) and the
timing model from the Jodrell Bank Monthly Ephemeris (Lyne
1982). The technique provides a formal time resolution of 100
ns, but the expected pulse-broadening time due to interstellar
scintillations is in the range 2 to 4 µs, based on the estimation
made by Kuzmin et al. (2002). Therefore, we averaged the
recorded signal after dispersion removal and square-low detec-
tion synchronously with a topocentric pulsar period into 8192
bins per period and with a resulting sampling interval close
to 4.1 µs. The total intensity time series was finally formed
and used for detection of giant pulses with the threshold level
of 6σ. To provide better sensitivity for wider giant pulses, we
developed a searching procedure which progressively tried to
increase the averaging time interval (τi) by 1, 2, 3, 4, 6, 8, 12,
16, 24, 32, and 48 samples, i.e.from 4.1 µs to 196 µs. The
averaging time with the best signal-to-noise ratio (SNR) was
selected as We for the GP, detected simultaneously at several
averaging times. The pulse energy E (or average flux density)
was calculated for the GP as a product of the SNR and the pulse
width, equal to the averaging time, which corresponds to the
best SNR value. In this approach we have different threshold
levels both in peak flux density Fp and in pulse energy E for
each separate averaging time. The value of the root-mean-square
deviation (RMS or σ) goes down with increasing time averaging
as the square root of time. Therefore, the threshold level in
peak flux density Fp decreases with increasing averaging time
as Fp(τ) ∝ 1/
τ, while the threshold level in pulse energy E
increases with τ as E(τ) ∝
For the Crab pulsar, the system temperature is notably influ-
enced by the impact of the Crab Nebula, whose flux density
can be approximated by the relation Fν = 955ν
−0.27 Jy (ν
in GHz) (Allen 1973; Bietenholz et al. 1997), which gives
F1200 = 909 Jy at 1200 MHz. In our observations we used all
14 dishes of the WSRT in tied-array mode, where all telescopes
are added coherently, with the width of synthesized beam being
equal to 34 arcsec, thus reducing the contribution from the Crab
Nebula to the system noise by a factor fν = ΩA/ΩCN = 0.14
to the value FCN = 127.6 Jy. Here ΩCN is a solid angle of the
Crab Nebula, and ΩA a solid angle of the intersection of the
synthesized beam with the Crab Nebula. However, there is still
the contribution from the individual dishes; the system tem-
perature of every 25-m dish was increased by the Crab Nebula
emission by ∆TCN = F1200G = 86.3 K, where G is the gain of
a single telescope, equal to 0.095 at 1200 MHz. The intrinsic
system temperature in the absence of the Crab Nebula is 30 K
at 1200 MHz, and the resulting total system temperature is thus
116.3 K, equivalent to Fsys = 87.4 Jy for the gain in the tied
array (1.33 K/Jy), which when combined with the nebula con-
tribution to the tied array becomes Ftot = Fsys + FCN = 215 Jy.
One can see that WSRT reduces the contribution of the Crab
Nebula to the system noise considerably when compared with
the single dish observations at this frequency. This improvement
will permit us to follow the statistics of giant pulses from the
Crab pulsar to the lower energies; namely, the limiting RMS (σ)
value in one polarization is Ftot/
Bτ, with B = 10 MHz, and
τ = 4.1 µs the RMS is equal to 33.5 Jy. For total intensity, the
RMS will be reduced by
2, and the threshold of 6σ in peak
flux density will be equal to 142 Jy. The threshold will be lower
 1000
 10000
 100000
 1e+06
 1  10  100  1000
Averaging time We in microseconds
Fig. 1. Peak flux density versus the effective puse width We for
all detected giant pulses.
for averaged records, with a limiting value of 20 Jy for giant
pulses, which have a maximum effective width of about 200 µs.
In total 17869 giant pulses were detected over 370000 pulse
periods with 14994 giant pulses located at the longitude of the
main pulse and 2875 giant pulses associated with the interpulse.
The background rate was determined by counting events at
quiescent phases of the pulsar period and the corresponding
values were taken into account in our statistical calculations. In
fact, at the lowest range, between 6σ and 7σ, only about 10%
of all pulses we potentially incorrectly identified as giant pulses.
3. Results
Figure 1 represents the complete set of data showing peak flux
density Fp versus the effective pulse width We. The striped na-
ture of the diagram is caused by the procedure of giant pulse de-
tection, which is based on the discrete values of averaging time
We. The diagram demonstrates the peculiarity of the statistics on
giant pulses: the strongest pulses clearly have a shorter duration;
namely, there are no pulses with peak flux density greater than
1000 Jy that are wider than 16 µs. As a consequence, giant pulses
with different durations have different distributions in pulse en-
ergy, and the distribution in giant pulse width depends on the
range of pulse energy used. Therefore, in our analysis, we sepa-
rated the giant pulses belonging to the main pulse longitudes into
five groups by their widths: (a) – GPs with the effective width We
of 4.1 µs, (b) – GPs with the We of 8.2 µs, (c) – combined group
of GPs with the We of 12.3 and 16.4 µs, (d) – combined group of
GPs with the We of 24.5 and 32.8 µs, and (e) – combined group
of GPs with the We of 49.2 and 65.6 µs. A separate group (f)
was formed for the GPs detected at the longitude of the inter-
pulse with the We 4.1, 8.2, and 12.3 µs. Then, pulse energy, or
integrated flux density, was calculated as E = Fp ×We.
In our analysis we use cumulative probability distribution
(CPD), which gives the number of pulses N(E), above pulse en-
M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses 3
Table 1. The list of parameters of the power-law fits to the CPDs
displayed in Fig.2.
Group We(µs) γhigh γlow Ebreak(Jy · µs)
(a) 4.1 1.7 1.1 4000
(b) 8.2 1.8 1.0 3500
(c) 12.3, 16.4 2.3 1.2 4000
(d) 24.5, 32.8 2.5 1.4 4000
(e) 49.2, 65.6 3.2 1.9 5000
(f) 4.1, 8.2, 12.3 1.6 1.6
ergy E. Some authors prefer to calculate the probability distri-
bution (PD) by giving the number of pulses n(E) per interval
of energy dE. If a power-law fit is used for the PD function as
n(E) ∝ E−β, then a power-law fit will be also valid for the CPD
function N(E) ∝ E−γ, since
N(E > Ey) =
n(E)dE ∝ E−β+1y
with γ = β − 1 in the absolute value.
Figure 2 displays the CPD of pulse energies for all the afore-
mentioned groups of GPs. The CPD for the GPs belonging
to the interpulse can be fitted with a power-law function with
γ = 1.6 ± 0.1. The CPDs for the GPs in the main pulse manifest
a break in the power-law index at certain values of pulse energy,
which is slightly different for the different groups. Table 1
contains the results of a least-square solution for the power-law
fit. The first column indicates the group of GPs, We is the
averaging time for the group in µs, γhigh and γlow are indices of
the power law function for the high energy tail and low energy
portion of the CPD, respectively, and Ebreak indicates the pulse
energy where the slope of the power law changes. The break
energy values were obtained as the crossing point between two
straight lines defined by the least-square solutions. Relative
inaccuracy is within 10% at the RMS level.
The absence of any break in the power-law fit for the CPD
of the GPs belonging to the interpulse longitudes is good proof
that the observed breaks in the CPDs of the main pulse GPs were
not caused by selection effects, since we used exactly the same
procedure for GPs detection irrespective of pulse longitude. A
possible explanation is that we observe the regions where the
GPs are generated in the the main pulse and the interpulse at
different impact angles relative to the GP beam, which has a half
width of about 5 degrees. Indeed, the comparison of the CPDs
(combined for the width groups (a), (b), and (c)) for the main
pulse and the interpulse shows that the CPDs would be identical
if one were to multiply the interpulse GP energies by a factor of
about 4 and increase their rate of occurrence by a factor of about
2. Therefore, the weakest detected interpulse GPs with a pulse
energy of about 1000 Jy · µs, when observed at the same impact
angle as the main pulse GPs (multiplied by 4), will have pulse
energies of 4000 Jy · µs, which is above the break point.
4. Discussion and conclusions
The CPD function of the GPs originating at the longitudes of
the main pulse manifests gradual changes in the power-law in-
dex from −1.7 to −3.2 at the high energy part of the CPD, and
from −1.0 to −1.9 at the low energy portion of the CPD. As
a rough guide, all GPs can be separated into two main groups:
short GPs belonging to groups (a) and (b), and long GPs with
an We greater than 8.1 µs. It is interesting to note that the CPD
power-law index of about −1.7 for short GPs is close to the ex-
ponent determined for the CPD of the GPs from the millisec-
ond pulsar B1937+21, which was found to be equal to −1.8
at 430 MHz (Cognard et al. 1996). The value was confirmed
by Kinkhabwala & Thorsett (2000), and recently estimated by
Soglasnov et al. (2004) as −1.4 at 1650 MHz. All GPs from the
millisecond pulsar are very short, lasting only 1-2 µs as mea-
sured by Kinkhabwala & Thorsett (2000), while Soglasnov et al.
(2004) have found that the majority of GPs from this pulsar are
shorter than 15 ns.
Argyle & Gower (1972) were the first to present the CPD for
GPs from the Crab pulsar. They combined the results of observa-
tions made at 146 MHz with the 26-m Penticton radio telescope
and with the 46-m dish at the Algonquin Radio Observatory and
found that the CPD was consistent with the power-law exponent
of −2.5 for the main pulse and with the exponent for the inter-
pulse events equal to −2.8.
Recently Cordes et al. (2004) in their multifrequency study of
the Crab pulsar’s giant pulses presented histograms of GP peak
amplitudes (S) at 0.43 and 8.8 GHz. The histograms represent
the PD functions, i.e. number of events per interval of SNR in
logarithmic binning d(lg S ). They found power-law segments
in the distributions with the slopes −2.3 and −2.9 at 0.43 and
8.8 GHz, respectively. To compare this result with our study of
the CPD functions one has to convert the number of events pre-
sented by Cordes et al. (2004) from logarithmic binning to linear
binning d(lg S ) = ln 10dS/S . Then, for a power-law fit with lin-
ear binning the slope will change from −β to −(β + 1):
N(S ) ∝ (S )−βd(lg S ) ∝ (S )−(β+1)dS
for the power-law fit of the CPD γ = β − 1, as was explained
in Sect. 3. Thus, the values of the power-law exponent β found
by Cordes et al. (2004) in their histograms may by immediately
compared with our values of the exponents γ for the power-law
fit of the CPDs. In fact the inspection of the histogram of
Cordes et al. (2004) for 0.43 GHz has enabled us to distinguish
a break at an S/N value of about 30 where the slope changes
from −2.3 to −0.7 in the peak amplitude distribution of the main
pulse GPs, while the distribution for the interpulse GPs does not
manifest such a break. With the given equivalent flux system
temperature of 1262 Jy, the receiver band of 12.5 MHz, and a
sample interval of 128 µs, the S/N = 30 break point at 0.43 GHz
will correspond to a pulse energy of about 120000 Jy · µs.
In his PhD thesis Moffett (1997) presentes the CPD functions
at 1.4 GHz based on observations conducted with the VLA
with a time resolution of 160 µs. In Fig. 4.3 (page 62) Moffett
distinguishes a break in power-law fit of the CPD for the main
pulse GPs at a level of 12 Jy corresponding to a pulse energy of
about 2000 Jy · µs, where the slope changes from −3.0 to −1.8
when going from the high flux densities to the lower. Again, the
CPD for the GPs originated at the longitudes of the interpulse
(Figure 4.4) does not show a break going straight with the slope
of −1.7 until it merges with noise at a level of about 6 Jy.
Comparing the values of break-point pulse energy (BPPE)
120000 Jy · µs at a frequency of 0.43 GHz (Cordes et al.(2004)),
5000 Jy · µs at a frequency of 1200 MHz (this paper), and
2000 Jy · µs at a frequency of 1400 MHz (Moffett (1997)),
we have found that the BPPE follows a simple power-law
frequency dependence BPPE(ν) ≈ 7(ν)−3.4kJy · µs, with the
exponent −3.4 close to the mean spectral index for the main
pulse component of the average profile −3.0 (Moffett 1997).
The conclusion can be considered as support for the suggestion
4 M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
 1000
 10000
 1000  10000
Pulse energy (Jy*µs)
Fig. 2. The CPD of giant pulses with pulse energy above the displayed value for different groups of giant pulses classified by their
effective width We as described in the Table 1. Straight lines represent the fit by power-law functions with the parameters indicated
in the Table 1. The last plot (f) represents giant pulses belonging to the interpulse.
that all emission in the main component consists entirely of
giant pulses (Popov et al. 2006).
Lundgren et al. (1995) has collected about 30000 GPs from
the Crab pulsar at 812 MHz with the Green Bank 43-m radio
telescope in 10 days of observations simultaneous with the
Compton Gamma Ray Observatory (CGRO) in May 1991.
They did not distinguish GPs belonging to the main pulse and
the interpulse. Their flux-density distribution was fitted with a
power-law function for S > 200 Jy, and the exponent was found
to be equal to −3.46, the exponent being equal to −2.46 for the
CPD used in our analysis. The BPPE value is expected to be
14 kJy · µs at 812 MHz according to the frequency relation we
derive above. The value corresponds to 50 Jy flux density with
307.5 µs averaging time used by (Lundgren et al. 1995), and it is
well below their threshold of 120 Jy. The dramatic roll-off at the
rather high pulse energy of about 200 Jy × 300µs = 60kJy · µs
found by (Lundgren et al. 1995) was not observed in the CPDs
at 430, 1200, and 1400 MHz discussed above. It was not
observed in the CPD at 600 MHz either, which goes straight
with the slope of −2.2 at least down to a pulse energy of about
M.V. Popov & B. Stappers: Statistical Properties of Giant Pulses 5
30 kJy · µs (Popov et al. 2006). Therefore, it is difficult to
reconcile their results with those from the many independent
data sets mentioned above.
The power-law form of the observed CPDs can be compared
with the field statistics of possible emission mechanisms
responsible for the generation of GPs, as discussed by Cairns
(2004) who used the values of the exponents of PD functions
for comparison. The very short-time flux density variations are
of particular interest, since they are closely tied to the physics
of the emission process. The power-law indices γ of the CPDs
at high energies for the shortest GPs are similar enough for both
the Crab pulsar and the PSR B1937+21, and the values were
found to be in the range 1.4 to 1.8. Converting to indices of the
probability distribution function (PD) β = γ + 1 gives the range
2.4 – 2.8 for the values of β. According to the normalization
conditions used by Cairns, the exponent α of the PD of the
field (P(E) ∝ E−α) is connected with the exponent β of the
observed PD power-law fit by the relation α = 2β − 1, giving
us the range of α from 3.8 to 4.6 to be compared with the
theoretical predictions of the field statistics determined by the
emission mechanism and propagation effects. The observed
breaks in the slope of the CPD functions at certain energies
have to be included in theoretical explanations, giving us an
extra parameter to constrain the source physics and emission
mechanism.
The break in the slope of the CPD functions has important con-
sequences for the estimation of the total rate of GP generation.
Popov et al. (2006) made such an estimation for the Crab pulsar
under the suggestion that radio emission in the main pulse and
in the interpulse consists entirely of giant pulses. They found
that about 10 giant pulses are generated during one rotation
period of the neutron star. In the estimation they considered that
pulse energies of the GPs follow a power-law function with
the exponent of −2.2 down to the threshold of about 100 Jy
in the peak flux density at a frequency of 600 MHz, and with
the threshold considered as a real lower limit equivalent to
the minimum pulse energy of about 5000 Jy · µs. With their
threshold of about 20000 Jy · µs in GP detection, they did not
notice the break in the CPD slope. The break from −2.2 at high
energies to −1.2 at low energies will notably change the estimate
of the lower limit for GP energies to about 1000 Jy · µs, and the
rate of GP generation will increase at least by a factor of 2. The
lower energy limit for GP generation, if it exists, will serve as
a crucial constraint on the physics of the emission mechanism.
To solve the problem, it is necessary to test the low-intensity
portion of the CPD function with better sensitivity. Such a study
would be possible using VLA or GMRT observations in phased
array mode, thereby, significantly reducing the impact from the
Crab Nebula. A notable increase in the recording band from
10 to 160 MHz recently achieved for the PuMa II recording
systems also makes new observations with the WSRT very
promising.
Finally, we summarize the main results of our analysis:
1) The CPDs were found to be notably different for the GPs
detected at the longitudes of the main pulse and the interpulse.
We suppose that the difference can be explained by the simple
attenuation caused by a beaming factor.
2) For the main pulse longitudes, the CPD are different for
the GPs of different effective widths with breaks in the CPD
power-law indices indicating steepening at high energies (see
Table 1).
3) The CPD power-law indices (γhigh ≈ 1.7) for the group of
short GPs for the Crab pulsar (We < 10µs) are close to the value
observed for the millisecond pulsar B1937+21, which seems to
generate only the shortest GPs;
4) GPs with a stronger peak flux density were found to be of
shorter duration.
The last of these properties testifies in favor of nonlinear
temporal models that suggest that the higher the intensity,
the narrower the pulse width. Such an emission model was
considered, for example, by Mikhailovskii et al. (1985), who
treated the micropulses as solitons of the radio-wave envelope
propagating through the magnetospheric plasma of the pulsar.
Acknowledgements. This investigation was supported in part by the Russian
Foundation for Fundamental Research (project number 04-02-16384).
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	Introduction
	Observations and data reduction
	Results
	Discussion and conclusions
