Astroparticle Physics 33 (2010) 97–107

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Astroparticle Physics
journal homepage: www.elsevier.com/locate/astropart

The angular resolution of the GRAPES-3 array from the shadows of the Moonand the Sun
A. Oshima a, S.R. Dugad a, U.D. Goswami a, S.K. Gupta a,*, Y. Hayashi b, N. Ito b, A. Iyer a, P. Jagadeesan a,
A. Jain a, S. Kawakami b, M. Minamino b, P.K. Mohanty a, S.D. Morris a, P.K. Nayak a, T. Nonaka b, S. Ogio b,
B.S. Rao a, K.C. Ravindran a, H. Tanaka a, S.C. Tonwar a, The GRAPES-3 Collaborationab

Tata Institute of Fundamental Research, Mumbai 400 005, India
Graduate School of Science, Osaka City University, Osaka 558-8585, Japana r t i c l e

i n f oArticle history:
Received 3 May 2009
Received in revised form 27 October 2009
Accepted 17 December 2009
Available online 30 December 2009
Keywords:
Cosmic rays
Extensive air showerMoon

a b s t r a c t
The absence of a well established point source of very high energy ð J 10 TeVÞ c-rays in the sky, makes
the measurement of the angular resolution and the absolute pointing accuracy of an extensive air shower(EAS) array a challenging task. In the past, several groups have utilized the reduction in the isotropic ﬂux
of cosmic rays due to the shadows of the Moon and the Sun, to measure the angular resolution and theabsolute pointing accuracy of their arrays. The data collected from the GRAPES-3 EAS array, over the period of 4 years from 2000 to 2003, has been used to observe the shadow of the Moon at a level of  5r andthat of the Sun at a lower level of signiﬁcance. The high density of the detectors in GRAPES-3 enabled anangular resolution of 0.7° to be obtained at energies as low as 30 TeV. The angular resolution studies were
further extended by using two other techniques, namely, the even–odd and the left–right methods. All
three techniques have yielded nearly identical results on the energy dependent angular resolution.
Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction
The interstellar magnetic ﬁelds deﬂect the charged cosmic rays
to travel along complex trajectories within the galaxy resulting incomplete loss of information on the location of the cosmic ray
sources. This problem could be obviated if one could identify cosmic ray photons which would travel along straight lines, unaffected by the interstellar magnetic ﬁeld and thereby preserving
the directional information of their sources. Therefore, knowledgeof the high energy processes that lead to the production and acceleration of cosmic rays to ultra-high energies may be obtained
through the study of the cosmic c-rays.
It is generally agreed that the bulk of the cosmic rays of energies
below the ‘knee’ are produced within our galaxy and are mainly
accelerated in supernova explosions, through the diffusive shock
acceleration mechanism [1–3]. Therefore, the supernova remnants
within the galaxy have been an important target for the observations during the last several decades, over a broad energy region
from 100 MeV to beyond a PeV. The discovery of the pulsars in
the 1960s and their association with the supernova remnants
added to the interest in particle acceleration processes associatedwith the supernovae. The knowledge that the pulsars are capableof directly accelerating particles to TeV energies and beyond [4–* Corresponding author. Tel.: +91 9869439435; fax: +91 2222804610.
E-mail address: gupta@grapes.tifr.res.in (S.K. Gupta).
0927-6505/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.astropartphys.2009.12.004

8] have provided further impetus to these studies. Among the
known pulsars and the associated supernova remnants, the CRAB
pulsar and its associated nebula have been among the most-studied astronomical objects at all wavelength in the electromagnetic
spectrum. The observations of the CRAB pulsar and its associatednebula by the early satellite based instruments such as the SAS-2
and the COS-B, followed by the detailed studies by the EGRET
instrument on the Compton Gamma Ray Observatory, established
the CRAB as a standard candle over the MeV–GeV energy range.
These observations also imply the presence of highly relativistic
charged particles in the nebula that have been accelerated to energies beyond hundreds of GeV. The discovery of the synchronous
pulsed emission from the radio (105 eV) to GeV c-rays, emphasized the importance of the pulsar and its role in the accelerationof charged particles in this nebula.
The satellite borne detectors have also provided considerable
information on the c-ray sources and their galactic distribution
for energies below 10 GeV. The experimental studies on thesources of photons of energies above 100 GeV require large effective collection area and exposure time, which are presently possible only with the ground based detectors employing theatmospheric Čerenkov technique [9,10]. However, the cosmic
sources of ultra-high energy ð J 100 TeVÞ c-rays can only be studied using the extensive air shower technique, which permits a sufﬁciently large area and exposure time factors. Since the c-ray ﬂux
from a source has to be detected, against a huge background con-

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tributed by the isotropic ﬂux of the cosmic rays, it is necessary toachieve as good a rejection of this background as possible for thedetection system. For a point source, the signal to background ratio
is inversely proportional to the square of the angular resolution.
Therefore, the most important requirement of an EAS detector system is, its angular resolution for maximizing the signal to background ratio. The technique of relative arrival timing has beenused extensively in the EAS arrays, for the determination of the arrival direction of the showers [9,11,12].
The studies of the EAS at energies below 100 TeV involve thedetection of the shower electrons that suffer considerable scattering in the atmosphere prior to their detection at the ground level.
Therefore, it is expected that the angular resolution attainable with
an EAS array would be relatively poorer as compared to an atmospheric Čerenkov detector which involves detection of the Čerenkov photons. Furthermore, the angular resolution of an EAS arrayis also expected to depend on the shower size ‘N e ’ at the detectionlevel and it improves with the increase in the shower size. In thepast, attempts have been made to improve the angular resolution
for small size showers, by artiﬁcially increasing the detected particle density. This is normally done by covering the scintillator
detectors with a thin lead sheet, for the conversion of the low energy c-rays in the EAS that are far more numerous, into electron–
positron pairs [13].
An ideal approach for improving the angular resolution of an EAS
array would be to deploy a carpet of detectors to ensure the detectionof a large fraction of the shower particles in the EAS. This is also
important to achieve a larger triggering efﬁciency for lower energy
showers. The MILAGRO and the ARGO-YBJ experiments are two of
the examples of this approach. The MILAGRO [14] experimentachieved full particle detection efﬁciency by instrumenting a 8 m
deep water pond of 80  60 m2 area, to detect Čerenkov emission
produced by the shower particles inside the water pool, by thephotomultipliers immersed in it. The ARGO-YBJ collaboration [15]employs resistive plate chambers to detect and time the shower particles. They have installed a carpet of resistive plate chambers, covering an area of 74  78 m2, which is surrounded by 50% coverageover an area of 100  100 m2. However, the total carpet area in bothof these experiments is [6000 m2, thereby maximizing the sensitivity of these experiments to energies below 5 TeV.
An alternative approach for improving the angular resolution of
an EAS array is to decrease the inter-detector separation of thescintillator detectors. This approach provides an excellent compromise between a smaller area carpet array and a very large area
sparsely instrumented array. However, the practical constraints
limit the number of detectors employed in an EAS array, thereby
providing only a sampling of the particle density qe ðrÞ at a limited
number of points. In the recent times, for example, in the Tibet ASc[16], the CASA [17], the EAS-TOP [18], the KASCADE [19] and the
GAMMA [20] arrays, the area covered by the detectors is [1% of
the physical area over which the array is spread out. The physicalarea covered by the detectors in a given array, which is determinedby the density of the detectors, is a key factor. This factor contributes signiﬁcantly to the accuracy of the measurement of theparameters, such as the shower size ‘N e ’ and consequently the primary energy, as well as the arrival direction. These parameters areimportant in the studies in the c-ray astronomy, at multi-TeV
energies. With these deﬁnitive considerations in mind, the
GRAPES-3 air shower array [21] has been designed to have oneof the most compact conﬁguration of the conventional type of
EAS arrays, with a separation of only 8 m between the adjacentdetectors. The GRAPES-3 detectors are arranged in a symmetrichexagonal geometry, resulting in a detector coverage of 2% of
the total area of the array.
In 1957, Clark had proposed, the use of the decrease in the isotropic ﬂux of the cosmic ray showers, caused by the shadows of theMoon and the Sun, for the estimation of the angular resolution of
an EAS array [22]. In the past, different groups have utilized thistechnique of the reduction in the cosmic ray ﬂux, due to the shadow of the Moon or the Sun, to measure not only the angular resolution, but also to obtain the absolute pointing accuracy of their
EAS arrays. However, this technique works well only for those arrays that possess an angular resolution comparable to the apparent
angular size (0.5°) of these two objects in the solar system. However, the presence of the magnetic ﬁeld on the surface of the Sun
results in some deﬂection of the charged cosmic rays leading to a
distortion and displacement of its shadow. The CASA [17] and HEGRA [23] collaborations had reported  6r deﬁcit due to the shadow of the Moon.
The measurement of the shadow of the Moon by the Tibet ASc
experiment yielded an angular resolution of 0.5° at 10 TeV. This
experiment also reported the displacement of the cosmic ray shadow of the Sun, by 0.9°W and 0.4°S relative to its optical position[24]. However, at lower energies even the magnetic ﬁeld of the
Earth would deﬂect the primary cosmic ray particles, resulting ina measurable displacement of the Moon shadow relative to its
optical position. The Tibet ASc collaboration reported a shift of
0.23°W in the shadow of the Moon, relative to its actual position.
This shift was attributed to the deﬂection of the positively charged
cosmic rays in the geomagnetic ﬁeld [24].
A marginal detection of the Moon shadow in the muon ﬂux
underground was reported by the BUST group [25]. A 6.5r deﬁcit
due to the Moon shadow in the muon ﬂux, was detected by the
MACRO collaboration [26]. The L3 collaboration had also studied
the shadow of the Moon on the ﬂux of high energy muons. A displacement of 0.8° in the Moon shadow was measured for themuons in the energy range 65–100 GeV [27,28]. Similarly the
ARGO-YBJ collaboration have also reported, a displacement of
0.7°W and 0.5°N of the Moon shadow in the cosmic ray ﬂux detected by them [29].
The present work describes the results obtained on the angularresolution and the absolute pointing accuracy of the GRAPES-3 array using the data collected over the period of 4 years from 2000 to
2003. The relevant details of the experimental system are summarized in Section 2 and the issues pertaining to the angular resolution are presented in Section 3. An extensive study of the angularresolution was carried out by using three different techniques.
Two of these include, the even–odd and the left–right methods
that are described in Sections 4 and 5, respectively. In the third
method we have used these observations to detect the shadow of
the Moon at a level of  5r and that of the Sun at a lower levelof signiﬁcance that are described in Section 6. All three techniques
have yielded nearly identical results on the energy dependent
angular resolution. The angular resolution of the GRAPES-3 arrayis found to be 1.4° at 10 TeV which improves to 0.7° at 30 TeV.
These results are discussed in Section 7 and a summary is presented in the ﬁnal section.

2. Experimental system
The experimental system of the GRAPES-3 (Gamma Ray Astronomy at PeV EnergyS Phase-3) experiment consists of a densely
packed array of scintillator detectors and a large area tracking
muon detector as shown in Fig. 1. The EAS array consists of 257
plastic scintillator detectors, each of 1 m2 in area. These detectors
are deployed with an inter-detector separation of only 8 m. The array is being operated at Ooty in south India (11.4°N, 76.7°E, 2200 m
altitude) [21].
In order to achieve the lowest possible energy threshold, a simple 3-line coincidence of detectors has been used to generate theLevel-0 trigger, which acts as the fast GATE and START for the ana-

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80

200
60

a

40

150

b

[m]

20

100

0
−20

50

−40

0

−60
−80

−80

−60

−40

−20

0
[m]

20

40

60

Fig. 1. The GRAPES-3 experimental system with 257 scintillator detectors ðÞ and
16 muon detector modules ðÞ. The distances along the X- and Y-axes are in meters.

log to digital and time to digital converters (ADCs and TDCs),
respectively. As expected, this trigger selects a large number ofvery small and local showers and also larger showers whose cores
land very far from the physical area of the array. Therefore, it is also
required that at least 10 out of the inner 127 detectors should havetriggered their discriminators within 1 ls of the Level-0 trigger.
This Level-1 trigger with an observed EAS rate of 13 Hz is used torecord the charge (ADC) and the arrival time (TDC) of the pulses
from each detector. The pulse charge is later converted into theequivalent number of minimum-ionizing particles (MIPS) using
the most probable charge for a single MIP measured using the trigger from a small area (20  20 cm2) scintillation counter telescope[21].
Fig. 1 also shows 16 squares in the lower left side. Each of thesesquares represents a 4-layer muon tracking detector with an energy threshold of 1 GeV for the vertical muons. Each layer consistsof 58 proportional counters, each 6 m long with 10  10 cm2 crosssectional area. The 560 m2 GRAPES-3 muon detector consists offour super-modules, each in turn having four modules. The detailsof the muon detectors and their performance in relation to these
physics objectives have been described in detail elsewhere [30].

3A total of 1.4  109 showers have been collected over a total
live-time of 9.4  107 s, spread over a 4-year period, from 2000
to 2003. For each EAS, the core location, the shower age ‘s’ representing the steepness of the Nishimura–Kamata–Greisen (NKG)
lateral distribution function and the shower size N e have been
determined [31]. This was done by using the observed particle densities, following the minimization procedure described below. For
each shower, the zenith ðhÞ and the azimuthal ð/Þ angles have been
calculated using the time information from the TDCs, also described below.
Although the number of detectors triggered by an EAS is a quantity that is experimentally well-deﬁned, but it is not very usefulwhen comparing the results from different experiments. For thisreason we have studied the relationship between the shower size
Ne and the number of detectors triggered by it. In Fig. 2, we display
the mean number of detectors triggered as a function of the

4
4.5
Log 10 (N e )5

5.5

Fig. 2. Mean number of detectors triggered as a function of the calculated shower
size, N e .

shower size, N e . Clearly the number of triggered detectors is a lesssensitive function of shower size for N e 6 104 , and thereafter theincrease is more rapid. The reason for this behavior is that for low
shower sizes, the lateral extent of the showers is relatively small
and therefore the number of detectors triggered increases rather
slowly with N e . We have estimated the uncertainty in the shower
size determination due to the reconstruction errors to be 10%.
However, the errors due to the uncertainty in the hadronic interaction models are expected to be signiﬁcantly larger. The error in theshower core location has been found to be 1 m [32].
The initial values of the parameters for the shower ﬁt were derived as described below. The timing information from the triggered detectors was used to determine the initial arrival
direction ðh; /Þ using the least square ﬁt method assuming a plane
shower front. A crude estimate of the shower size Ne was obtained
from the total number of detected particles nd in all detectors by,

Ne ¼ 102  n0:97dð1ÞThis relationship was derived from the Monte Carlo simulationsof the array response using the CORSIKA code. The initial core location vector Rcore was estimated from the weighted mean of thelocation vectors ri of the top seven detectors that had recorded
the highest particle densities ni ,P

top ni ri

Rcore ¼ P 7
3. GRAPES-3 angular resolution

3.5

top7 nið2Þ

It was assumed that the initial shower age s = 1. Next, the NKG
function was used to ﬁt the observed density in the detectors having ADC counts greater than equivalent of 0.5 particles, and the initial values of the shower direction, size, core location, age as
described above. The minimization of the NKG function was carried out iteratively, by using the maximum likelihood algorithm
MINUIT [33], to obtain more accurate values of various shower
parameters, namely, the core location, size and age, etc. Now, byusing the distance of each detector from the shower core, correction was applied at the rate of 215 ps m1 to the recorded arrival
time at each triggered detector, for the cone-shaped shower front
as explained subsequently. This value of the correction was obtained from the shower data. This correction effectively transforms
the shape of the shower front from cone into a plane. The arrival
times were then ﬁtted to a plane-shape using the least square
method to obtain the ﬁnal values of the arrival direction, namely,
h and /.

A. Oshima et al. / Astroparticle Physics 33 (2010) 97–107

As mentioned in the Section 1, one of the most critical parameter in the search for the point sources of cosmic c-rays, using a particle detector array is its angular resolution. An improvement in
the angular resolution allows the rejection of a larger fraction of
the background EAS, initiated by the charged cosmic rays. This inturn would increase the signal to background ratio and may enable
the discovery of the new cosmic c-ray sources.
An improvement in the angular resolution in turn requires anaccurate determination of the relative arrival time of the shower
front at various detectors and a precise knowledge of the shapeof the shower front. It may be noted that the assumption of a plane
shower front is not adequate even for relatively small shower sizes.
This is primarily because the particles spread out to beyond 50 mfrom the shower core. At distances 50 m from the core, theshower particles are delayed relative to the core because of; (i)multiple Coulomb scattering resulting in a lateral spread and consequently an added path length, (ii) transverse momenta imparted
during secondary particle production also results in movement at
an angle relative to the shower axis, leading to additional pathlengths. It has been experimentally shown, that a cone-shaped
shower front provides an excellent description of the EAS front[24,34].
Since the detectors in the EAS array are distributed over a large
area and exposed to the outside weather, the timing accuracy is affected by the temperature dependence of the response of various
components of the scintillation detector system. Observations haveshown that the temperature coefﬁcient for the scintillator, thephotomultiplier, the signal cables, and the electronic modules(ampliﬁers, discriminators and the TDCs) are signiﬁcantly different
from each other and undergo measurable variations.
The observed arrival time of the shower front at a detector has
two components, namely tsi , which depends on the arrival direction(h and /) of the EAS and tdi , which is the transit time from the scintillator to the TDC, which includes the time response of the scintillator, the photomultiplier, the signal cables and the electronics, etc.
Therefore, the time difference ti;j between two detectors i and j for
an EAS may be written as,

Temperature (C) Nanosecond Nanosecond Nanosecond

100

1

(a)

0
−1
−2

(b)
−1
−2
−3

(c)

0
−1
−2

(d)

25
20
15
12

13

14

15

16

17

18

19

Fig. 3. Variation of the residual time for detectors 1, 3 and 5, labeled (a), (b), (c),respectively, obtained from a ﬁt of the observed times to a cone-shaped shower
front, for a period of 7 days. The minimum of the residual time occurs in theafternoon when the temperature is maximum.

are slightly different, varying within a range of 2 ns. Since themagnitude of this variation is relatively small and changes nearly
simultaneously for all detectors, it gets nearly eliminated whencalculating the relative arrival time of shower particles in the
EAS for direction reconstruction. At present, attempts are being
made to determine the transit time in our detector system over a
period as short as an hour, for making the necessary corrections
for the observed temperature dependence. However, no correction
was made in the present analysis of the data discussed here. The
accuracy of the angular resolution of the array discussed below is
only marginally affected by the temperature dependent systematics discussed above.

 
t i;j ¼ t si  t sj þ tdi  t dj

4. The even–odd method

In a perfectly stable system, the second component should largely remain unchanged with time which can, in principle, be determined from the EAS incident vertically. However, in practice, thesecond component varies with time, for example, with the ambient
temperature as mentioned above, or due to the changes in; (a) therise and the transit times in the photomultiplier, (b) the transit
time in the cable from the detector in the ﬁeld to the signal processing electronics in the control room, and (c) the effect of thetemperature coefﬁcient of various electronic modules such as theampliﬁers, discriminators and TDCs, etc. on their propagation delay. It is easy to observe, the temperature dependence by plotting
the residual time between the observed and the expected time, obtained from the arrival direction (h and /) of the EAS, from a ﬁt of
the relative arrival times, to a cone-shaped shower front throughthe v2 minimization technique.
In Fig. 3, the variation of the residual time, for three GRAPES-3
detectors, numbered 1, 3, and 5 are shown for a period of 7 days
from 12 to 19 February 2001. These three detectors are locatedwithin 8 m from each other and have been selected to study thetemperature dependent effects. Clearly the residual time is inversely correlated with the temperature, reaching its maximum value
during the early morning hours. We understand that this effect isprimarily caused by the change in the length of the signal cablefrom the detector to the control room, due to the change in thetemperature. Due to different levels of exposure of the signal
cables to the sunlight, the residual time for different detectors

A method extensively used for estimating the angular resolution for a speciﬁc EAS array relies on the Monte Carlo simulationsof the showers in the atmosphere and the response of the array tothose simulated showers. This method suffers from the limitation,
due to the inadequacies in the Monte Carlo simulations caused by
the approximations used to reduce the computation time.
The EAS detectors in the GRAPES-3 array are sequentially numbered from the centre outwards, with the detector number increasing clockwise over each successive hexagonal ring. Therefore, it
was decided to divide the array into two sub-arrays, namely, theﬁrst with the even-numbered and the second with the odd-numbered detectors. Next, a comparison of the arrival directions determined independently with the two sub-arrays, labeled ‘even’ and‘odd’, respectively, is carried out. Since these two sub-arrays havea very substantial spatial overlap, they provide similar estimateof the EAS direction and also an important insight into the angulardetermination capability of the experiment [35]. An important limitation of this approach arises from the systematic errors that arecommon to both of the sub-arrays and therefore can not be measured. For example, there can be large systematic errors in the angles determined for a major fraction of the showers that have theiraxes striking the periphery of the GRAPES-3 array. But, due to thespatial overlap of the ‘even’ and ‘odd’ sub-arrays, the direction
determined by them would contain the same systematic error.
Therefore, the angle between these two directions would be freefrom this systematic error.

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o

100

1.6

80

1.4

o

o

1.2
60

o

1.0

b

40

o

0.8

a

o

0.6

20

0.4o
o

0

1

2

o

3

o

4

o

5

o

6

o

Fig. 4. Distribution of the angle between shower directions determined from ‘even’and ‘odd’ sub-arrays, for P100 detectors triggered, (a) all events, (b) shower core
within dashed hexagon labeled ‘a’ in Fig. 1. Peak value on y-axis normalized to 100
for easy comparison.

40000

a

30000

b
20000

10000

0

o

1

2

o

3

o

4

o

5

o

6

o

Fig. 5. Distribution of the angle between shower directions determined from, (a)even and odd sub-arrays for P100 detectors triggered, (b) 2-D Gaussian ﬁt yields ar ¼ 0:94 .

Since the shielding coverage of the muon detector extends only
up to 45°, the showers for further analysis have been selected up toa maximum zenith angle of 40°. The distribution of the angle between the EAS arrival directions, determined by the ‘even’ and‘odd’ numbered sub-arrays are shown in Fig. 4, by the dashed histogram labeled ‘a’. The selected showers were required to trigger at
least 100 detectors, which corresponds to a shower size N e of 104.5
as shown earlier in Fig. 2. Thus on an average each sub-array would
have information from P50 detectors. A further cut was imposedon the location of the shower core to lie within the hexagon labeled‘a’, as shown in Fig. 1. The resulting angular distribution for theeven and odd sub-arrays is shown in Fig. 4, by the histogram labeled ‘b’. The peak value of the two distributions in Fig. 4 has beennormalized to 100 for an easy comparison. This ﬁgure clearlyshows that the error in the determination of the direction improves
when a cut on the core location is imposed. The error in arrival
direction for the shower size considered here, is relatively small.
To estimate the angular resolution, we have ﬁtted the distribution
labeled ‘b’ in Fig. 4 to a 2-dimensional Gaussian [36]. The even–odd
angular distribution with the core cut is shown in Fig. 5 by the histogram labeled ‘a’, and the 2-D Gaussian ﬁt with the label ‘b’. A value of the r ¼ 0:94 is obtained for the standard deviation from this2-D Gaussian ﬁt.
Since the data from these two sub-arrays are independent ofeach other, the actual error in the determination of the angle is

o

0.2

0

50

100

150

200

Fig. 6. Variation of the median angle aM as a function of the number of detectors
triggered.

smaller by a factor of p1ﬃﬃ2, than the value quoted above. Since, only
half of the total number of detectors triggered, are used in eachof the two sub-arrays, the angular resolution for the full array
would be smaller by a further factor of p1ﬃﬃ2. Therefore, the true angular resolution would be 0.94°/2 = 0.47°, for showers triggering
P100 detectors, which corresponds to a shower size of P104.5 asis shown earlier in Fig. 2. However, it needs to be emphasized here,that the even–odd method provides an estimate of only the statistical error in the angle determination, and it does not permit theevaluation of the systematic error in the measurement of the arrival direction. Therefore, the value of 0.5° represents the best possible angular resolution attainable for the GRAPES-3 array, for ashower size N e of P104.5.
In the earlier discussion, we had used only those showers that
had triggered P100 detectors. But the angular resolution of an
EAS array is a function of the number of detectors triggered. Therefore, next we examine the distribution of the angle aeo determinedfrom even and odd sub-arrays as a function of the number of detectors triggered and then calculate the median angle aM of the aeo
distribution. In Fig. 6, the median angle aM , which is also an alternative measure of the angular resolution of our array, is shown as afunction of the number of detectors triggered. The magnitude of
the median angle aM initially decreases rapidly with increasing
number of triggered detectors up to 50 detectors, indicating rapid
improvement in the angular resolution. However, thereafter the
improvement is relatively slower.
As described in the beginning of Section 3, the size of eachshower is determined by ﬁtting the NKG function [31] to the lateral
distribution of the observed shower densities. Thereafter, theshowers in a narrow size range of 10x  10xþ0:2 are selected, where
3:2 6 x 6 5:4, and the distribution of the number of detectors
triggered is obtained. In Fig. 7, the distribution of the number ofdetector triggered for a shower size range of 104:4 6 N e 6 104:6
is shown. A Gaussian ﬁt to this distribution yields a mean of 111
with a rather narrow rms spread of 15, implying a fractional widthof only 13%. Therefore, the number of detectors triggered appears
to be a fairly good measure of the shower size, provided a sufﬁciently narrow range of the shower size is selected.
The dependence of the angular resolution on the zenith angle
has been studied by dividing the zenith angle range from 0 to
40° into ﬁve bins, namely, 0–13°, 13–19°, 19–25°, 25–32°, and
32–40°. This particular choice of angular segmentation was implemented to ensure that the number of showers present in each of
the ﬁve bins are rather similar to each other. The angular resolu-

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A. Oshima et al. / Astroparticle Physics 33 (2010) 97–107

3500

3000

b

3000

2500

2500
2000

2000

1500

1500

a

1000
1000

500

500

0

0

50

100

150

200

2

o

o

4

o

6

o

8

o

10

o

12

o

14

Fig. 9. Distribution of angle ap between EAS directions determined from (a) leftand the right-half arrays for 104:4 6 N e 6 104:6 , (b) 2-D Gaussian ﬁt with ar ¼ 1:64 .

Fig. 7. The distribution of the number of detectors triggered for a narrow size rangeof 104:4 6 N e 6 104:6 . Gaussian ﬁt yields a mean of 111 and r ¼ 13%.

Ne

4

4

4

5

1.0x102.7x106.1x101.6x10

50

100

150

200

o

1.6

o

1.4

o

1.2

o

1.0

o

0.8

o

0.6

o

0.4

o

0.2

0

Fig. 8. Variation of angular resolution with number of triggered detectors for ﬁve
zenith angle bins, (a) 0–13°, circle , (b) 13–19°, square j, (c) 19–25°, inverted
triangle ., (d) 25–32°, triangle N, (e) 32–40°, star H, respectively. Shower size N e isshown at top along secondary x-axis.

tion as a function of the number of detectors triggered for, (a) 0–
13° by circle , (b) 13–19° by square j, (c) 19–25°, by inverted triangle ., (d) 25–32°, triangle N, and (e) 32–40°, star H are shown in
Fig. 8. The corresponding shower size N e is displayed at the top
along the secondary x-axis in Fig. 8 for an easy comparison, atthe location of the markers. The angular resolution shows a rather
weak dependence on the zenith angle for the range of 0–40° studied here. The magnitude of the angular resolution increases gradually with the increasing zenith angle as seen from Fig. 8. The
increase in the angular resolution relative to the ﬁrst bin shownas  is; 0.8% for the second bin j, 1.6% for the third bin ð.Þ,
2.9% for the fourth bin ðNÞ, and 5.6% for the ﬁfth bin ðHÞ, respectively. In the subsequent discussion we examine the data by twoother alternative methods to evaluate the actual angular resolution
achieved.
5. The left–right method
In the beginning of Section 3, we had mentioned that a coneshaped shower front has been used for determining the arrival
direction of an EAS. Here, we present the results of a study that
demonstrates the presence of the curvature in the shower front.
A considerable improvement is seen in the angular resolution

when a cone-shape is used to describe the curvature of the shower
front. The curvature of the shower front is readily studied by dividing the array into two independent halves, namely, the left- andthe right-half arrays. For this purpose the following procedure isadopted. First, the location of the shower core is determined andthen the line joining the shower core with the centre of the arrayis used to divide the array into the left- and the right-half arrays.
Although this array division is implemented on an event-to-event
basis, yet this procedure ensures that the number of detectors triggered in each of the two half-arrays are approximately equal.
In the even–odd case, the density of the detectors is reduced bya factor of 2, while the area of the each sub-array remains nearly
the same as the area ‘A’ of the full array. However, in the left–rightcase the density of the detectors remains unchanged, but the areaof each half-array gets reduced to A2. Therefore, we expect the ratioof the areas in the two cases to be 2. Since the angular resolution isinversely proportional to the linear extent of the array, we expect
the angular resolution to worsen by a factor of p1ﬃﬃ2 in the left–rightcase, as compared to the even–odd case. Therefore, correction by
an additional factor of p1ﬃﬃ2 is required when comparing the resultsfrom the above methods.
A stringent cut is imposed, on the location of the shower core tolie within the hexagon labeled ‘b’ in Fig. 1. This is done to ensure
the selection of the showers landing in the central region of the array. The direction of a given shower is determined independently,
by each half-array (hL ; /L and hR ; /R ), by assuming a plane shower
front. Next, the angle ap between these two shower directions given by hL ; /L and hR ; /R is calculated. In Fig. 9, the distribution of
ap is shown labeled ‘a’, for a shower size range of
104:4 6 N e 6 104:6 . Clearly the large width of ap distribution witha mean value in excess of 6° indicates that the plane-shape does
not provide an appropriate description of the shower front. Several
studies done in the past had demonstrated that the shower front isadequately represented by a cone-shape [37–40].
We have also found that the GRAPES-3 data can be very well ﬁtted to a cone-shaped EAS front. The cone-shaped front also results
in a large reduction in the angle ac between the left and the righthalf-arrays. The best ﬁt yielded a value of 215 ps m1 for the slopeof the cone, relative to the plane perpendicular to the shower axis.
The distribution of ac is shown labeled ‘b’ in Fig. 9. The large
improvement in the angular resolution due to the use of thecone-shaped front may be judged from the difference in the widthof the distributions of ap and ac . Also shown on the same plot is a
2-D Gaussian ﬁt to the ac distribution. A value of r ¼ 1:64 was obtained for the standard deviation from this ﬁt. However, the valueof 1.64° has to be corrected for the systematically smaller area

A. Oshima et al. / Astroparticle Physics 33 (2010) 97–107

7

o

6

o

5

o

4

o

3

o

2

o

103

a

b
o

1

0

50

100

150

200

Fig. 10. Variation of the median angle, (a) amp for plane ﬁt shown by ﬁlled circles ,(b) amc for cone ﬁt shown by ﬁlled squares j, as a function of number of detectors
triggered.

ðAL ; AR Þ of the left–right arrays, as compared to the area ðAE ; AO Þ ofthe even–odd arrays as explained above. But instead of applying
the correction factor due to area calculated above, we have investigated this problem by utilizing the experimental data.
We have classiﬁed the showers according to the number ofdetectors triggered. The values of the areas AE ; AO ; AL and AR are calculated using the area covered by the triggered detectors for eachshower. The distribution of the ratio of AE and AO is extremely narOas
row and centred at 1. Therefore, we decided to use AEO ¼ AE þA2
the area for the even–odd case, for a given number of detectors
triggered. Nearly identical results have also been obtained in theRas theleft–right cases as well. Therefore, we have used ALR ¼ AL þA
2
area for the left–right case. The values of AEO and ALR are found to
be different and both increase with increasing number of detectors
is found to be constant andtriggered. However, the ratio R ¼ AAEO
LR
independent of the number of detectors triggered. The value of R
is found to be 2.002, almost same as the expected value of 2 as discussed above. When this area dependent correction is applied, we
get an effective angular resolution of 1.15° which is only marginally larger than the value of 0.94° obtained by the even–odd method as shown in Fig. 5. This is not surprising, because as explained
earlier in Section 4, the even–odd method represents the best possible angular resolution attainable by a given array.
As described in Section 4, the angular resolution of an EAS arrayis a function of the number of detectors triggered. In the earlier discussion, we had used the EAS in the shower size range of
104:4 6 N e 6 104:6 that triggered a mean of 111 detectors. Next
we examined, for a given number of triggered detectors the distribution of the angle ap determined from the left- and right-half arrays using a plane EAS front. We then calculated the median angle
amp from the ap distribution. In Fig. 10, the median angle amp isshown as ﬁlled circles, as a function of the number of detectors
triggered. The magnitude of amp shows a very small change withincreasing number of triggered detectors, right up to 200 detectors.
Using an identical procedure the value of the median angle amc is
calculated for the cone-shaped EAS front. In Fig. 10, the median angle amc is displayed as ﬁlled squares, as a function of the number ofdetectors triggered. The magnitude of amp initially decreases rapidly with increasing number of triggered detectors up to 50
detectors, indicating a rapidly improving angular resolution, however, thereafter the improvement becomes relatively slower. A
study of the dependence of the angular resolution on the zenith angle was carried out using the method described in Section 4. The

Fig. 11. Concentric circles drawn around Moon or Sun directions of equal solid
angle. First ring has a diameter of 1.26°.

results of this study also showed a weak dependence of the angularresolution on the zenith angle, similar to the one shown in Fig. 8.

6. Shadows of Moon and Sun
It is well-known that the observations of the shadow of theMoon or the Sun cast on the ﬂux of the cosmic rays observed at
the Earth provides an ideal method to determine the angular resolution of an EAS array. It allows both, the statistical error in the angle determination, as well as the systematic error in ‘pointing’towards a given direction to be measured. The shadows due tothe Moon and the Sun on the cosmic ray ﬂux has been observedfor various primary energies by several groups. The shadow of
the Moon in the ﬂux of the secondary muons has been observed;
in a deep mine by the Soudan group [41], in the underground
MACRO detector [26] and inside the L3 detector at CERN [42].
The MILAGRO collaboration have also studied the shadow of Moonat energies 1 TeV using a water pond as an EAS array [43].
The following procedure has been adopted for the analysis of the
GRAPES-3 shadow data; divide the ﬁeld of view of radius 4° aroundthe centre of the Moon or the Sun, into 40 concentric circles of equal
solid angle area. The inner 10 rings are shown schematically in
Fig. 11. Next, the number of showers incident along directions lying
within each concentric bin are counted. The inner most bin (disk) has
a diameter of 1.26°, which is 2.5 times the angular diameter of theMoon. The solid angle of each bin is 3.8  104 sr, which is 6.4
times the solid angle projected by the Moon or the Sun. This choiceof binning was dictated by the anticipated angular resolution of
the GRAPES-3 array which was 1° and a solid angle segmentation
smaller than this value would have resulted in excessive amountof over-sampling. For a comparison, similar concentric circles weredrawn centred on two different directions, on either side of the Moon(or the Sun) that are offset by 8° in the azimuthal angle, but at thesame zenith angle. This selection of the background region ensuresthat the shower attenuation in both the Moon/Sun and backgroundregions is identical and concurrent. This analysis showed a uniform
distribution indicating the absence of any shadowing effects along
these two directions away from the Moon and the Sun. However, thisdata was not used for any further studies.

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A. Oshima et al. / Astroparticle Physics 33 (2010) 97–107

24000

23800

23600

a

16400
16200
16000

b

8900
8800
8700

c

8600

4200
4100
4000

d
3900
0

o

1

o

2

o

3

o

4

Fig. 12. Number of EAS in 3.8  10-4 sr concentric bins around Moon for shower
size N e , (a) P 103:2 , angular resolution r ¼ 1:39  0:17 , (b) P 103:5 ; r ¼ 1:07 
0:13 , (c) P 103:75 ; r ¼ 0:89  0:10 , (d) P 104:0 ; r ¼ 0:73  0:09 . Dashed lines
represent 2-D Gaussian ﬁts.

The data collected over a 4-year period from 2000 to 2003 areused in the study described below. The histogram in Fig. 12ashows, the variation in the number of showers of size
N e P 103:2 , for the incident EAS directions in the annular regionsof the concentric rings of equal solid angle as displayed inFig. 11. The total exposure time for the Moon is 2.1  107 s. The
ﬁrst bin of the histogram in Fig. 12a is centred on the Moon. A
reduction in the ﬂux of showers in the direction of the Moon isclearly seen. The dashed line in Fig. 12a is a 2-D Gaussian ﬁt to
the deﬁcit in the data and the bar represents the 1r error. The
deﬁcit in counts described by the 2-D Gaussian is given by the following equation:NðhÞ ¼ No

"
#!
h2M
h2
1  2 exp  2
2r
2r

ð3Þ

where hM is the radius of the Moon or the Sun, N o is a constantthat represents the unshadowed background shower rate and r is
the standard deviation of the 2-D Gaussian ﬁt and represents
the angular resolution of the array. The values of No and r are
obtained from the 2-D Gaussian ﬁt. Based on this ﬁt, the angular resolution of the GRAPES-3 array was estimated to be
r ¼ ð1:39  0:17 Þ; No ¼ 23886 for the showers of size larger than
103.2 particles. For calculating the deﬁcit in the number of showers,
the optimum angular radius for a Gaussian distribution is given by
1:58  r ¼ 1:58  1:39 ¼ 2:2 , which corresponds to the 12 innermost solid angle bins. The deﬁcit N d in the number of showers
around the direction of the Moon, relative to the mean shower rateof No per bin may be calculated as follows. Following the proceduredescribed above we ﬁnd the total number of showers in the innerP
most 12 bins, Nmoon ¼ 12
i¼1 N i ¼ 284; 029 as against an expected
mean N expt ¼ 12  N o ¼ 12  23; 886 ¼ 286; 632. The observed deﬁTherefore, thecit Nd ¼ Nexpt  N moon ¼ 286; 632—284;
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ029 ¼ 2603.
signiﬁcance of this deﬁcit is, N d = Nexpt ¼ 2603= 286; 632 ¼ 5r.
The histogram in Fig. 12b represents the data for the shower
size N e P 103:5 . Here also a clear decrease in the shower rate isvisible. The 2-D Gaussian ﬁt to this data yields an angular resolution r ¼ ð1:07  0:13 Þ for the shower size threshold listed above.
Similarly the data for the shower size N e P 103:75 is shown in
Fig. 12c and the 2-D Gaussian ﬁt to the data yields an angular resolution r ¼ ð0:89  0:10 Þ. Finally, the data for the shower size
N e P 104:0 are shown in Fig. 12d and the 2-D Gaussian ﬁt yields
an angular resolution r ¼ ð0:73  0:09 Þ.
Since the EAS recorded by the GRAPES-3 experiment are of relatively high energy (P15 TeV), the deﬂection in the geomagnetic
ﬁeld is expected to be very small. Therefore, the location of theMoon shadow may be used to obtain the absolute pointing accuracy of the GRAPES-3 array. The minima of the Moon shadow appears to lie within ±0.2° in the right ascension and declination of
the centre of the Moon. Therefore, we conclude that the absolute
pointing accuracy of the GRAPES-3 array is 0.2°.
As compared to the 1-dimensional histogram shown in Fig. 12
a 3-dimensional color coded map of the decrease in the numberof showers due to the shadow of the Moon is more informative.
In Fig. 13, the 3-dimensional map of the number of showers arriving from the directions around the Moon are shown. The map iscentred on the Moon and covers ±5° both in the right ascension
and the declination. The magnitude of the shadow is parametrized as percentage change in the ﬂux of isotropic cosmic rays.
The shape of the shadow of the Moon on the primary cosmic
ray ﬂux is clearly visible in Fig. 13. In the earlier discussion in
Section 4, we had employed a 2-D Gaussian in the study of theangular resolution of our array. Similarly in Section 6, to measure
the effect of the shadows of the Moon and the Sun, we have again
employed the 2-D Gaussian ﬁt. Although there are sizable ﬂuctuations near the outer periphery of the shadow, a sharp and reasonably symmetric shadow, centred on the direction of theMoon is clearly visible.
It may be mentioned here that the shadow of the Moon on thecosmic ray ﬂux is a straightforward case of the physical blocking by
the Moon. However, in the case of the Sun, the magnetic ﬁeld of the
Sun and the solar activity in its vicinity also plays an important rolein causing a deﬂection in the path of cosmic rays, in addition to thephysical blocking. The structure of the magnetic ﬁeld around the
Sun is also very complex with frequent temporal changes. As reported by the Tibet ASc [44] and ARGO-YBJ [45] collaborations,the shadow pattern due to the Sun evolves with time, resulting
in a rather blurred image when integrated over a period of several
years. We observed the shadowing effects from the direction of theSun, although with signiﬁcantly larger ﬂuctuations. This is not
unexpected, since our observations during 2000–2003 coincidedwith the period of maximum solar activity.

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A. Oshima et al. / Astroparticle Physics 33 (2010) 97–107

0.6%
0.4%
0.2%
0
-0.2%
-0.4%
-0.6%
-0.8%
-1.0%
-1.2%
-1.4%
-1.6%

0.6%
0.4%
0.2%
0
-0.2%
-0.4%
-0.6%
-0.8%

17

50

180

#D

4

10

5

Ne

TeV
500

100

Moon

-1.0%
-1.2%
-1.4%
2

-1.6%
-4 -3

-2 -1

0
RA (deg.)

1

2

3

4

-4

-3

-2

-1

0

3

4

20

1
DEC (deg.)

Fig. 13. A 3-D proﬁle of the number of showers arriving from directions around theMoon. The map is centred on the Moon and covers ±5° in R.A. and declination. A
clear and reasonably symmetric shadow, centred on the Moon is visible.

0.6%
0.4%
0.2%
0
-0.2%
-0.4%
-0.6%
-0.8%
-1.0%
-1.2%

0.6%
0.4%
0.2%
0
-0.2%
-0.4%

Sun

-0.6%
-0.8%
-1.0%
2

-1.2%
-4 -3

-2 -1

0
RA (deg.)

1

2

3

4

-4

-3

-2

-1

0

3

4

1
DEC (deg.)

Fig. 14. A 3-D proﬁle of the number of showers from directions around the Sun. The
map is centred on Sun and covers ±5° in R.A. and declination. shadow centred on
Sun is visible.

The total GRAPES-3 exposure time for the Sun is 2.2  107 s.
Now, we display the shadow of the Sun in a 3-dimensional mapof the number of showers arriving from the directions aroundthe Sun in Fig. 14. This map is centred on the Sun and covers ±5°
both in the right ascension and declination. The magnitude of the
shadow is parametrized as percentage change in the ﬂux of isotropic cosmic rays. The effect of the shadow of the Sun is visible inFig. 14, although it appears to be a lot more fuzzy than the shadowof the Moon which was shown in Fig. 13. Although the Sun data
show signiﬁcantly larger ﬂuctuations specially in the outer regionof the shadow, yet a clear shadow is still discernible.
The smaller deﬁcit with lesser signiﬁcance for the shadow of the
Sun with a more fuzzy shape as seen in Fig. 14 may be due to theeffect of the turbulent magnetic ﬁeld around the Sun. In addition,
the structure of the magnetic ﬁeld around the Sun is rather complex with frequently occurring time dependent changes. Thesechanges in the solar magnetic ﬁeld are expected to inﬂuence theshadow pattern, resulting in a blurred images when integrated
over a long period of time. A signiﬁcant contribution to the observed blurring of the shadow may also be due to the turbulent

10

Fig. 15. The x-axis shows the shower size N e and corresponding primary protonenergy in TeV, is shown on the y-axis. Number of detectors triggered #D are shownat top along secondary x-axis.

interplanetary magnetic ﬁeld associated with the higher solar
activity due to the solar maximum, during a major part of the 4
year observation period of the GRAPES-3 data. It may also be notedthat in Fig. 14, the minima of the shadow appears to be offset, by
0.4°W and 0.2°S relative to the centre of the Sun. This displacement(0.4°W, 0.2°S) of the Sun shadow relative to its optical position, isalong the same direction (0.9° W, 0.4°S) as was reported by the Tibet ASc collaboration [24]. The magnitude of the displacement observed by us is about a factor of 2 smaller as compared to Tibet
ASc. This is not unreasonable, in view of the fact that the energy
threshold for the GRAPES-3 is larger than Tibet ASc, which would
result in a smaller deﬂection of the charged particles due to a given
magnetic ﬁeld in the solar vicinity. However, given the relatively
large ﬂuctuations in the shape of the shadow, the magnitude of
the observed offset (0.4°W, 0.2°S) is also not inconsistent with anull value.
7. Discussion
For a direct comparison of the angular resolution of the
GRAPES-3 array by the use of three different methods, we have
converted the shower size N e into the energy of the primary protons, using the CORSIKA Monte Carlo simulation program, version
6.500 with the QGSJet generator [46]. Although the primary cosmic
rays contain heavier nuclei, their contribution at any given shower
size Ne is relatively small for energies below 100 TeV consideredhere. In Fig. 15, the derived median primary proton energy isshown as a function of the observed shower size N e . The shower
size N e is plotted on the x-axis and the primary energy in TeV isdisplayed on the y-axis. This plot is used to convert the shower size
into the primary proton energy for further discussions. In the earlier discussion, the dependence of the angular resolution was studied, as a function of either the number of detectors triggered or theshower size N e . However, in the following, the relation between
shower size and number of detectors triggered as shown in
Fig. 2, and the relation between shower size and primary energy
as shown in Fig. 15 has been used for conversion into equivalent
primary energy. At the top of Fig. 15, the corresponding numberof detectors triggered for a given Ne are also displayed along thesecondary x-axis for an easy comparison.
A comparison of the angular resolution of the GRAPES-3 array,as determined by the three completely independent methods

106

A. Oshima et al. / Astroparticle Physics 33 (2010) 97–107

3

6.0x104

1.3x104

2.1x104

3.2x10

Ne

o

1.4

1.2

o

1.0

o

0.8

o

0.6

o

o

0.4

0

20

40

60

80

100

TeV
Fig. 16. Variation of angular resolution with primary energy (TeV) by threemethods, (a) even–odd, ﬁlled squares ðjÞ, (b) left–right, ﬁlled triangles ðNÞ, (c)Moon shadow, ﬁlled circles ðÞ. Shower size N e is shown at top along secondary xaxis.

namely; the even–odd, the left–right and the Moon shadow, as
outlined in Sections 4–6, respectively, are shown in Fig. 16. For
the even–odd case the angular resolution is extracted from the2-D Gaussian ﬁt to the angular distribution of events for a given
number of detectors triggered. This value is divided by 2, as explained in Section 4, to obtain the true value of the angular resolution. The number of triggered detector is converted into shower
size by using the relationship shown in Fig. 2 and then into the primary energy by using the dependence shown in Fig. 15. Also
shown at the top of Fig. 16 is the corresponding shower size N ealong the secondary x-axis for an easy reference. Finally, the angular resolution of the GRAPES-3 measured by the even–odd method,
as a function of the primary energy is shown by ﬁlled squares inFig. 16. The resolution of the array improves with the increasing
primary energy from 1.3° at 10 TeV to 0.4° at 80 TeV.
Using the same technique used for the even–odd case, the angular resolution as a function of primary proton energy is calculated
for the left–right method. However,the value of the resolution ispﬃﬃﬃ
divided by a further factor of 2 for the reasons outlined in Section 5. The resolution of the array is 1.3° at 10 TeV, which is nearly
the same as measured by the even–odd method. However, theimprovement is marginally smaller at higher energies with a valueof 0.5° being observed at 80 TeV. The angular resolution measured
by the left–right method, as a function of the primary energy is displayed by ﬁlled triangles in Fig. 16.
The Moon shadow data also show a very similar behavior withan angular resolution of 1.4° at 10 TeV, which improves to 0.5° at
80 TeV. The resolution measured by the Moon shadow, as a function of the primary energy is shown by ﬁlled circles in Fig. 16.
However, due to the limited statistics of the data on the Moon shadow, the statistical errors are not very small. It is to be noted that
the Moon shadow results are consistent with those obtained fromthe other two methods. Clearly, the values of the angular resolution derived from the three completely independent methods arenearly identical. This fact provides the conﬁdence in the coneshaped shower front being an adequate description of the observed
EAS in the GRAPES-3 experiment. It also implies that the determination of the angular resolution of the GRAPES-3 is reliable and largely free from systematic errors. Using the Moon shadow weestimate the absolute pointing accuracy to be 0.2°.

8. Summary
We have analyzed the EAS data collected by the GRAPES-3
experiment over the 4-year period from 2000 to 2003. The divisionof the array into even and odd sub-arrays is used to estimate theshower energy dependent angular resolution of the GRAPES-3 array. An alternative method of estimating the angular resolution
by using the concept of left and right half-arrays with cone-shaped
shower front has yielded a value of 0.5°, which is slightly larger
than the value of 0.4° obtained by the even–odd method for showers of energy P80 TeV. The even–odd method provides the best
estimate of the angular resolution attainable, in an array, free fromsystematic errors. The agreement between the angular resolution
obtained from the even–odd and the left–right methods provides
the evidence, that the cone-shaped shower front used in determining the shower direction is an adequate representation of theshower disk. Finally, the reduction in the isotropic ﬂux of the cosmic rays due to the shadow of the Moon yields an angular resolution of 0.5° for showers of energy P80 TeV. The absolute pointing
accuracy of the GRAPES-3 arrays is 0.2° and the small offset observed, based on the present data set, is consistent with a null value. The Sun data also yields a similar value of the angularresolution, although with signiﬁcantly larger uncertainty for thereasons outlined in Section 6. We conclude that the high densityof the detectors in the GRAPES-3 experiment has been instrumental in obtaining the observed angular resolution.
Acknowledgments
We thank D.B. Arjunan, S. Kingston, K. Manjunath, S. Murugapandian, B. Rajesh, C. Ravindran, V. Santosh Kumar and R. Suresh
Kumar for their help in the testing, installation and operation of
the detectors and the electronic modules. The administrative services of V. Viswanathan during the experiment are acknowledged.
We thank G.P. Francis, I.M. Haroon, V. Jeyakumar, K. Ramadass, fortheir help in the fabrication, assembly and the installation of various mechanical components and detectors. The Japanese membersof the GRAPES-3 collaboration acknowledge the partial ﬁnancial
support from the Ministry of Education and Science of the Government of Japan for the experiment. We thank the two anonymousreferees whose precise and critical comments have led to a significant improvement in the presentation and discussion of our results. This paper is dedicated to the memory of our brilliant
colleague S. Karthikeyan who unfortunately died in a tragic roadaccident on 20 June 2008.
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