The sensitivity of past and near-future lunar radio experiments to ultra-high-energy cosmic rays and neutrinos

0.92 MB·

by  Justin Bray

The sensitivity of past and near-future lunar radio experiments to ultra-high-energy
cosmic rays and neutrinos
J.D. Braya,b,c,∗
aSchool of Chemistry & Physics, Univ. of Adelaide, SA 5005, Australia
bCSIRO Astronomy & Space Science, Marsfield, NSW 2122, Australia
cJBCA, School of Physics & Astronomy, Univ. of Manchester, Manchester M13 9PL, UK
6
1Abstract
0
2Various experiments have been conducted to search for the radio emission from ultra-high-energy particles interacting
in the lunar regolith. Although they have not yielded any detections, they have been successful in establishing upper
n
alimits on the flux of these particles. I present a review of these experiments in which I re-evaluate their sensitivity to
Jradio pulses, accounting for effects which were neglected in the original reports, and compare them with prospective

2near-future experiments. In several cases, I find that past experiments were substantially less sensitive than previously
1believed. I apply existing analytic models to determine the resulting limits on the fluxes of ultra-high-energy neutrinos

and cosmic rays. In the latter case, I amend the model to accurately reflect the fraction of the primary particle energy
]
Mwhich manifests in the resulting particle cascade, resulting in a substantial improvement in the estimated sensitivity to
cosmic rays. Although these models are in need of further refinement, in particular to incorporate the effects of small-
I
.scale lunar surface roughness, their application here indicates that a proposed experiment with the LOFAR telescope
h
would test predictions of the neutrino flux from exotic-physics models, and an experiment with a phased-array feed on
p
-a large single-dish telescope such as the Parkes radio telescope would allow the first detection of cosmic rays with this
otechnique, with an expected rate of one detection per 140 hours.
r
t
sKeywords: ultra-high-energy neutrinos, ultra-high-energy cosmic rays, radio detection, Moon
a
[

11. Introduction anindependentcalculationofitssensitivitytoradiopulses
v and, in most cases, an independent model for calculating
0 Observations of ultra-high-energy (UHE; >1018 eV) the resulting aperture for the detection of UHE particles.
8
cosmic rays (CRs), and attempts to detect their expected This situation calls for further work in two areas, both of
9
2counterpart neutrinos, are hampered by their extremely which are addressed here: the recalculation of the radio
0low flux. The detection of a significant number of UHE sensitivity of past experiments in a common framework,
.particles requires the use of extremely large detectors, or incorporating all known experimental effects, and the cal-
1
0the remote monitoring of a large volume of a naturally- culation of the resulting apertures for both UHECRs and
6occurring detection medium. One approach, suggested neutrinos using a common analytic model.
1by Dagkesamanskii and Zheleznykh [1], is to make use of An additionalbenefit of this workis to provide a com-
v:the lunar regolith as the detection medium by observing prehensive description of the relevant experimental con-
ithe Moon with ground-based radio telescopes, searching siderations,with past experiments as casestudies, to sup-
X
for the Askaryan radio pulse produced when the inter- port future work in this field. To that end, I also present
araction of a UHE particle initiates a particle cascade [2]. here a similar analysis of the radio sensitivity and par-
The high time resolution required to detect this coherent ticle aperture for several possible future lunar radio ex-
nanosecond-scale pulse puts these efforts in a quite differ- periments. The most sensitive telescope available for the
ent regime to conventional radio astronomy. application of this technique for the forseeable future will
Since the firstapplicationofthis lunarradiotechnique betheSquareKilometreArray(SKA),prospectsforwhich
with the Parkes radio telescope [3], many similar experi- have been discussed elsewhere [4], but phase 1 of this in-
ments have been conducted, none of which has positively strumentisnotscheduledforcompletionuntil2023;inthis
detectedaUHEparticle. Consequently,theseexperiments work, I instead evaluate three proposed experiments that
haveplacedlimitsonthefluxesofUHECRsandneutrinos. could be carried out in the near future (<5 yr) with ex-
To determine these limits, each experiment has developed istingradiotelescopes. Mostotherexperimentsthatcould
be conducted with existing radio telescopes will resemble
one of these.
∗Correspondingauthor.
This work is organised as follows. In Sec. 2 I address
Email address: [email protected] (J.D.Bray)
Preprint submittedto Astroparticle Physics January 13, 2016
the calculation of the sensitivity of radio telescopes to co- in both polarisations over the band, giving us
herentpulses,obtainingasimilarresulttoEq.2ofGorham
S
et al. [5], but incorporating a wider range of experimen- F =2h i (3)
tal effects. This provides the theoretical basis for the re- h i ∆ν
evaluation in Sec. 3 of past lunar radio experiments, in =2 Er2ms from Eq. 2 (4)
whichIcalculateacommonsetofparameterstorepresent Z0∆ν
their sensitivity to a lunar-origin radio pulse. Alongside
which is the SEFD again. Combining Eqs. 1 and 4 shows
these, I calculate the same parameters for proposed near-
that
future experiments. 1/2
kT Z ∆ν
In Sec. 4 I discuss the calculation of the sensitivity of E = sys 0 . (5)
rms
A
lunar radioexperiments to UHE particles. Foreachofthe (cid:18) eff (cid:19)
experiments evaluated inSec. 3,I calculatethe sensitivity It is also useful to define
to neutrinos based on the analytic model of Gayley et al.
E
rms
[6], and the sensitivity to UHECRs based on the analytic rms = (6)
E ∆ν
modelofJeongetal.[7]. Finally,inSec.5,Ibrieflydiscuss
1/2
kT Z
the implications for future work in this field. sys 0
= from Eq. 5, (7)
A ∆ν
(cid:18) eff (cid:19)
2. Sensitivity to coherent radio pulses the equivalent RMS spectral electric field for this band-
width, although for incoherent noise it should be borne
The sensitivity of a radio telescope is characterisedby in mind that, unlike the flux density, the spectral electric
thesystemequivalentfluxdensity(SEFD),conventionally field varies with the bandwidth. This is in contrastto the
measured in janskys (1 Jy = 10−26 W m−2 Hz−1), which behaviour of coherent pulses, for which the spectral elec-
is given by tric field is bandwidth-independent, and the flux density
kT
F =2 sys (1) scales with the bandwidth.
h i A
eff The sensitivity of an experiment to detect a coherent
where k is Boltzmann’s constant, T the system temper- radio pulse can be expressed as , a threshold spectral
sys min
E
ature and A the effective aperture (i.e. the total col- electric field strength above which a pulse would be de-
eff
lecting area of the telescope multiplied by the aperture tected. This is typically measured with respect to ,
rms
E
efficiency). In the context of a lunar radio experiment, in terms of a significance threshold n . Note that the
σ
thesystemtemperatureistypicallydominatedbythermal additionofthermalnoisewillincreaseordecreasethe am-
radiation from the Moon — or, at lower frequencies, by plitude of a pulse, so that is actually the level at
min
E
Galactic backgroundemission — with a smaller contribu- which the detection probability is 50% rather than an ab-
tionfrominternalnoiseintheradioreceiver. However,the solute threshold, but this distinction becomes less impor-
strength of a coherent pulse, such as the Askaryan pulse tantwhenn islarge. furtherdependsontheposition
σ min
E
from a particle cascade, is expressed in terms of a spec- of the pulse origin within the telescope beam, as
tral electric field strength, in e.g. V/m/Hz. To describe
n η
the sensitivity of a radio telescope to a coherentpulse, we (θ)=f σ (8)
min C rms
must relate this quantity to the parameters in Eq. 1. E α (θ)E
rB
The factoroftwoinEq.1occursbecausethe fluxcon-
where (θ) is the beam power at an angle θ from its axis,
tains contributions from two polarisations, whether these B
normalised to (0)=1 and assumed here to be radially
are considered as orthogonallinear polarisations or as op- B
symmetric (e.g.anAiry disk). This same equationis used
posite circular polarisations (left and right circular polar-
to calculate as described in Sec. 3. The factor η is
isations; LCPand RCP).The bolometric flux density in a Emax
the ratio between the total pulse power and the power in
single polarisationis givenby the time-averagedPoynting
the chosen polarisation channel, typically found as
vector
E2
S = rms (2) 2 for circular polarisation
h i Z0 η = (9)
(1/cos2φ for linear polarisation
where E is the root mean square (RMS) electric field
rms
strength in that polarisation, and Z is the impedance of
0 with φ the angle between the receiver and a linearly po-
free space. If the received radiation has a flat spectrum
larised pulse such as that expected from the Askaryan ef-
over a bandwidth ∆ν, the total spectral flux density is
fect. The term α is the proportion of the original pulse
found by averaging the combined bolometric flux density
amplituderecoveredafterinefficienciesinpulsereconstruc-
tion, as described in Sec. 2.1. The remaining factor, f ,
C
accounts for the improvement in sensitivity from combin-
ingC independentchannelswithathresholdofn ineach,
σ
as described in Sec. 2.2.
2
The behaviour of coherent pulses as described above Whenasignalisconvertedtodigitalsampleswithafi-
is quite different to that of conventional radio astronomy nite samplingrate,the peak amplitude is further reduced,
signals. As a consequence of Eq. 7, sensitivity to coher- because the sampling times do not necessarily correspond
ent pulses scales as √A ∆ν in electric field and hence as to the peak in the original analog signal [10]. This ef-
eff
A ∆ν in power,whereas sensitivity to incoherent signals fect can be mitigated by oversampling the analog signal,
eff
scales as A √∆ν in power. Fundamentally, this is be- or by interpolating the digital data [11]. For a coher-
eff
cause the signal of a coherent pulse combines coherently ent sinc-function pulse with no oversamplingor interpola-
both acrossthe collecting area of the telescope and across tion,theworstcasecorrespondstosamplingtimesequally
its frequency range, while most radio astronomy signals spaced either side of the peak, giving a value for α of
combinecoherentlyacrossthe collectingareaandincoher- sinc(0.5)=0.64.
ently across frequency. Because of this difference it is not The interaction between these effects is complex, and
entirely appropriate to represent a detection threshold in not susceptible to a simple analytic treatment. I have in-
terms of an equivalent flux density, as the flux density of steaddevelopedasimulationtofindarepresentativevalue
a coherent pulse depends on its bandwidth, which defeats of α for a given experiment, described in App. A.
the purpose of using a spectral (rather than bolometric)
measure such as flux density in the first place. However, 2.2. Combining channels
this quantity is occasionally reported in the literature, so
Some coherent pulse detection experiments combine
I calculate it in several cases for comparative purposes;
thesignalsfrommultiplechannels,whichmaybedifferent
ensuring, to the best of my ability, that both values are
polarisations,frequencybands,antennas,oranycombina-
calculated for the same bandwidth, so that the compari-
tion of these. In this context, I take ∆ν to be the band-
son is valid. For a polarisedpulse at the detection thresh-
width ofa singlechannel,andEq.8withf =1givesthe
C
old, with spectralelectricfield andtotalelectric field
min thresholdforasinglechannelonitsown. Thesensitivityof
E
E = ∆ν,theequivalentfluxcanbefoundsimilarly
min min the combinedsignaldepends criticallyonwhetherthereis
E
to Eq. 4 — omitting the factor of 2, as the pulse appears
phase coherence between the channels, and whether they
in only a single polarisation — as
are combined coherently (i.e. direct summation of volt-
ages) or incoherently (summing the squared voltages, or
2 ∆ν
F = Emin . (10) power). The scaling of the sensitivity for C independent
min
Z
0 identical channels is as described below.
2.1. Amplitude recovery efficiency
Coherent channels, coherent combination
The spectral electric field of a pulse is, in general,
In this case, the pulses in each channel combine co-
E
a complex quantity. For a coherent pulse, its phase is
herently, and the combination acts as a single chan-
constant across all frequencies. If this phase is zero, then
nel with bandwidth C∆ν. The threshold in voltage
the time-domainfunction E(t) hasits powerconcentrated thus scales as f =C−1/2.
C
at a single point in time with peak amplitude ∆ν, as
|E| Coherent channels, incoherent combination
implicitly assumed in the above discussion. However, an
Squaring the voltages in this case converts them to
Askaryan pulse has a phase close to the worst-case value
the power domain, in which the sensitivity scales as
of π/2 [8], for which it takes on a bipolar profile with the
C1/2. Thesensitivityinthevoltagedomainscalesas
powersplitbetweenthepoles,causingthepeakamplitude
thesquarerootofthis,orC1/4,andhencef =C−1/4.
tobereducedbyafactor √2. Ifthispulseisrecordeddi- C
∼
rectlywithoutcorrectingthephase,thisgivesα 0.71. If Incoherent channels, coherent combination
∼
the signalundergoesfrequencydownconversion,the phase Sincethereisnophasecoherencebetweenthepulses
israndomised,givingαsomewherebetweenthisvalueand in different channels, they sum incoherently, in the
unity [9]. same way as the noise. The signal-to-noise ratio
A pulse originating from the Moon is smeared out in therefore does not scale with the number of chan-
time, also reducing its peak amplitude, by dispersion as nels, so fC =1.
it passes through the Earth’s ionosphere. The frequency- Incoherent channels, incoherent combination
dependent delay is
Squaring the voltages converts them to the power
domain, in which the sensitivity scales as C1/2, re-
STEC ν −2
∆t=1.34 109 s (11) gardlessoftheoriginalphases. Thesensitivityinthe
× TECU Hz
(cid:18) (cid:19)(cid:16) (cid:17) voltage domain therefore scales as C1/4, and hence
where STEC is the electron column density or slant total fC =C−1/4.
electron content measured in total electron content units
Conventionalradio astronomyoperates in the first regime
(1 TECU=1016 electronsm−2). Typicalvaluesareinthe
for the combination of multiple antennas, as the signal is
range5–100TECU,depending onthe time ofday,season,
coherent across the collecting area; and in the last regime
solar magnetic activity cycle, and slant angle through the
for the combination of multiple frequency channels, as
ionosphere.
3
most astronomical radio signals are not coherent across workissimilarinconcepttopreviousworkbyJaegeretal.
a range of frequencies. [13], but contains a more detailed analysis of previous ex-
Caremustbe takenindefining the significancethresh- periments, including all the effects described in Sec. 2. I
old n when the signal is in the power domain. For a determine the following parameters.
σ
voltage-domain signal s, which has a Gaussian distribu-
Observing frequency: ν
tion,thesignificanceisdefinedsimplyintermsofthepeak
I take this to be the central frequency of the trig-
and RMS signal values as n =s /s . If this signal
σ peak rms gering band. Generally speaking, a lower frequency
is squaredto produce the power-domainsignal S, it has a
results in a larger effective aperture for UHE par-
χ2 distributionwithonedegreeoffreedom,andthesignifi-
ticles, while a higher frequency reduces the thresh-
canceisinsteadfoundasn =(S /S)1/2intermsofthe
σ peak olddetectable particle energy. As the analytic mod-
mean value S, since S =s2 and S =s2 . The ra-
peak peak rms els used in this work all assume a small fractional
tioS /S isthesameastheratiobetweentheequivalent
peak bandwidth, I also report the width ∆ν of the trig-
fluxdensityofthepulse(fromEq.10)andthemeanback-
gering band as an indication of the accuracy of this
ground flux in a single polarisation (i.e. half the SEFD).
assumption. However,this does not include the sec-
WhenC identicalindependentpower-domainchannelsare
ondary 1.4 GHz band of the Kalyazin experiment
summed, the resulting signalhas a χ2 distribution withC
(see Sec. 3.3).
degrees of freedom, but the scaling factor f corrects for
C Minimum spectral electric field:
this,withnσ remainingthesignificanceinasinglechannel. This is the spectral electric fielEdmsitnrength of a co-
Some experiments operate with multiple channels,but
herent pulse for which the detection probability is
do not combine them either coherently or incoherently as
50%, as described in Sec. 2; its interpretation as
described above. Instead, they combine them in coinci-
anabsolutethresholdwillslightlyunderestimatethe
dence, requiring a pulse to be simultaneously detected in
sensitivity for weaker pulses and overestimate it for
allchannelssimultaneously. Thisincreasestheeffectivede-
strongerones. AnAskaryanpulsefromalunarUHE
tection threshold: taking f =1 gives the threshold
C Emin particle interaction is expected to have linear polar-
at which the detection probability is 50%, due to Gaus-
isation oriented radially to the Moon, and to origi-
sian thermal noise increasing or decreasing the pulse am-
nate from the lunar limb [12]. For telescope beams
plitude, but the probability of simultaneous detection in
pointed at the limb of the Moon I use the minimum
C channels is only 2−C. To scale so that the detec-
Emin value = (0) at the centre of the beam; oth-
tion probability remains 50%,for C identical independent Emin Emin
erwise, I take (θ ) at the closest point on the
channels, we require fC such that limb. I represeEnmtitnheLpulse reconstruction efficiency
C ∞ dsi e−s2i/2 =0.5 (12) wcuitlahtethdewmitehanthvealsuimeuαlafotiroanfldaets-csrpibecetdruinmAppupls.e,Ac.al-
iY=1 Znσ(1−fC)√2π ! Limb coverage: ζ
wheretheintegralisovertheGaussian-distributedvoltage- Asingletelescopebeamtypicallycoversonlypartof
domain signal s in each channel. Solving for f gives us the Moon, which reduces the probability of detect-
i C
ing a UHE particle. As the probability of detection
√2 is dominated by radio pulses originating from the
f =1 erf−1 1 2(C−1)/C (13)
C − n − outermost fraction of the lunar radius, at least at
σ
(cid:16) (cid:17) higher frequencies [14], I take the effective coverage
where erf−1 is the inverse of the standard error function. to be the fraction of the circumference of the lunar
The value of fC approaches unity for large nσ, for which limb within the beam, multiplied by the number of
the effects of thermal noise become insignificant, and for beamsn whentherearemultiplesimilarbeams
beams
small C. pointed at different parts of the limb. For this pur-
pose,I considera point onthe limbto be within the
3. Past and near-future lunar radio experiments beamif the effective threshold min(θ) in that direc-
E
tion is no more than √2 times the minimum thresh-
Lunar radio experiments have been carried out with old as defined above. For a beam pointed at
min
E
a diverse range of telescopes, with a variety of different thelimb,thiscorrespondstothecommonly-usedfull
receivers and trigger schemes to balance their sensitivity width at half maximum (FWHM) beam size. The
with their ability to exclude radio-frequency interference analytic models used in this work assume full sensi-
(RFI). Here I attempt to represent them with a unified tivity within this beam andzero outside of it, which
set of parameters, so their sensitivity to UHE particles will slightly overestimate the sensitivity to weaker
can be calculatedwith the analytic models used in Sec. 4. pulses near the detection threshold, which cannot
Althoughthisrepresentationisinevitablyonlyanapprox- be detected throughout the beam, and underesti-
imation to the inputs to numerical simulations (e.g. [12]), mate the sensitivity to stronger pulses, which can
it lends itself more easily to use in future models. This be detected even when they are slightly outside of
4
it. Where available, I have used the dates of obser- terrestrial RFI. However, they calculated the relative dis-
vationstodeterminethemedianapparentsizeofthe persive delay across a band ∆ν as
Moon when calculating the limb coverage, although
∆ν STEC ν −3
thishasonlyaminoreffectontheresult: theappar-
∆t=0.012 s (14)
ent size of the Moon varies across the range 29–34′, Hz electrons cm−2 Hz
(cid:18) (cid:19)(cid:18) (cid:19)(cid:16) (cid:17)
butmostexperimentsprovideafairlyevensampling
whereas, to be equivalent (for small ∆ν) to Eq. 11, the
of this range,so their median values are within 1′ of
leadingconstantshouldbe0.00268[17]. Consequently,the
one another.
10 ns dedispersive delay they introduced between the two
Effective observing time: t
obs subbands exceeded the required value by a factor of 4.
This is the effective time spent observing the Moon ∼
Since the delay error is comparable to the 10 ns length
afterallowingforinefficiencyinthetriggeralgorithm,
of a band-limited pulse in a 100 MHz subband, a lunar-
instrumental downtime while data is being stored,
origin Askaryan pulse would have no significant overlap
and the false positive rates of anti-RFI cuts.
betweenthetwosubbands,andwouldnotmeetthetrigger
criteria. Even if such a pulse were recorded, it would be
Some experiments have used an anticoincidence filter in
excludedbylatertestsonthestoredfull-banddata,which
which they exclude any event which is detected in multi-
requiredthatapulsedisplayanincreasedamplitudewhen
ple receivers pointed at different parts of the sky, as these
‘correctly’dedispersed. Thisexperimentwasthereforenot
are typically caused by local RFI detected through the
appreciably sensitive to UHE particles.
antenna sidelobes. These filters are critical for exclud-
Thetelescopebeamforthisexperimentwasdirectedat
ing pulsed RFI which might otherwise be misidentified as
thecentreoftheMoon,reflectingthecontemporaryexpec-
a lunar-origin pulse, but they also have the potential to
tationthatthiswasthemostlikelypointatwhichtodetect
misidentifyasufficientlyintenselunar-originpulseasRFI,
the Askaryanpulse from an interacting UHE neutrino [1].
which may substantially decrease the sensitivity of an ex-
Because of this, the beam had only minimal sensitivity
periment to UHE particles [15]. To reflect this, for these
at the lunar limb, where detectable Askaryan pulses are
experiments I calculate another quantity.
now known to be most likely to originate,which limits its
Maximum spectral electric field: sensitivity to UHE particles [18], even if the dedispersion
max
E
This is the spectralelectricfield strengthofa coher- problem described above is ignored. This experiment did,
entpulsewhich,ifdetectedinonebeam,wouldhave however,serveanimportantroleintriggeringfurtherwork
a 50% chance of also being detected through a side- in this field.
lobe of another beam and hence being misidentified
asRFI.Itis otherwisedefinedsimilarlyto , and 3.2. GLUE
min
E
calculatedwithEq.8withnσ asthesignificancelevel The Goldstone Lunar Ultra-high-energy Neutrino Ex-
for exclusion and (θ) as the sidelobe power of one periment (GLUE) made use of the 34 m DSS13 and 70 m
B
beam at the centre of another. A lunar-origin pulse DSS14 antennas at the Goldstone Deep Space Communi-
is considered to be detected and identified as such cationsComplexinaseriesofobservationsover2000–2003,
only if its spectral electric field strength is between withatotalof124hoursofeffectiveobservingtime [5,19,
min and max. 20]. They observed around 2.2 GHz on both antennas,
E E
forming two non-overlapping 75 MHz RCP channels on
I derive these values for past experiments in Secs.3.1–3.8,
DSS13, and a 40 MHz LCP channel and a 150 MHz RCP
calculatingthem separatelyfor eachpointing ifthe exper-
channel(latertwo75MHzRCPchannels)onDSS14. Each
iment used multiple pointing strategies. I also consider
channel was triggered by a peak in the signal power as
possible near-future experiments in Secs. 3.9–3.11. The
measuredbyasquare-lawdetector. Aglobaltrigger,caus-
results are presented in Table 1, and are used in the rest
ing an event to be stored, required a coincidence between
of this work.
all four (or five) channels within a 300 µs time window.
Subsequent cuts eliminated RFI by tightening the coin-
3.1. Parkes
cidence timing criteria, aided considerably by the 22 km
The first lunar radio experiment was conducted with
baseline between the two antennas, as well as by exclud-
the 64 m Parkes radio telescope in January 1995 [3, 16].
ing extended pulses, pulses clustered in time, and pulses
They observed for 10 hours with a receiver that Nyquist-
detected by an off-axis 1.8 GHz receiver on DSS14. A
sampled the frequency range 1175–1675 MHz in dual cir-
range of beam pointings were used, ranging from the cen-
cularpolarisations. The storageofthis data wastriggered
tre to the limb ofthe Moon, reflectingthe realisationthat
whenathresholdwasexceededbythepowerinbothoftwo
Askaryan pulses were most likely to be observed from the
subbands, eachof width 100MHz in a single polarisation,
limb.
centredon1325MHzand1525MHz,atadelayoffsetcor-
Williams [20] excluded thermal noise by applying sig-
responding to that expected from ionospheric dispersion.
nificance cuts at n =4 (DSS13 RCP), n =6 (DSS14
σ σ
This last criterion was effective in discriminating against
RCP) and n =3 (DSS14 LCP), with these thresholds
σ
5
Table1: Observationparametersforpastandnear-futurelunarradioexperiments.
Pointing ν ∆ν ζ t
Experiment Emin Emax obs
( n ) (MHz) (MHz) (µV/m/MHz) (%) (hr)
beams
×
limb 2200 150 0.0221 0.3695 11 73.5
GLUE half-limb 2200 150 0.0500 0.2527 20 39.9
centre 2200 150 0.4737 0.2527 100 10.3
Kalyazin limb 2250 120 0.0235 — 7 31.3
LUNASKA limb 1500 600 0.0153 — 36 13.6
ATCA centre 1500 600 0.0207 — 100 12.6
NuMoon limb ( 2) 141 55 0.1453 — 14 46.7
×
RESUN limb ( 3) 1425 100 0.0549 — 100 200.0
×
LUNASKA limb ( 2) 1350 300 0.0053 0.0241 16 127.2
×
Parkes half-limb 1350 300 0.0142 0.0489 15 99.4
Future experiments
LOFAR face ( 50) 166 48 0.0313 0.0768 100 183.3
×
Parkes PAF limb ( 12) 1250 1100 0.0043 0.0303 100 170.0
×
AuScope centre 2300 200 0.0830 — 100 2900.0
chosenbyscalingbasedonbandwidth(but notoncollect- mined by the least sensitive channel or channels. Most of
ingarea)toequalisetheirsensitivity,andconsideredthese, theobservingtimeforthisexperimentwasspentwithboth
ratherthanthetriggerthresholds,todefinethesensitivity antennas pointed on the limb of the Moon, in which con-
oftheexperiment. Thetriggerthresholdsarenotstraight- figuration the least sensitive channels are those of DSS13
forward to determine, as they depend on the characteris- RCP: given the reported values of 105 K for the system
tics ofthe signaloutput of the square-lawdetectors, but I temperature and 75% for the aperture efficiency, I find
assumethatthe 10nsintegrationtimeofthesquare-law them by Eq. 7 to have =0.0033 µV/m/MHz. Under
rms
∼ E
detectorseffectivelyremovesanydependenceonthephase the assumption that any event which exceeds the trigger
of the original signal while not further smearing out any threshold on both DSS13 RCP channels will almost cer-
peaks, and take the output to be the square of the sig- tainly also trigger the more sensitive channels, Eq. 13 can
nal envelope. This analog output was searched for peaks then be applied to find that the coincidence requirement
by SR400 discriminators which act on a continuous sig- between the two DSS13 RCP channels gives f =1.13.
C
nal[21], andsoare notsubjectto the amplitude lossfrom From Eq. 8, taking the above values and η =2 for cir-
a finite sampling rate described in Sec. 2.1. Given these cular polarisation, I find = 0.022 µV/m/MHz at the
min
E
assumptions, the 30 kHz single-channel trigger rates for centreofthebeam. Notethatthisishigher(lesssensitive)
DSS13 RCP and DSS14 RCP imply thresholds equivalent than the value 0.00914 µV/m/MHz found by Williams
to n =4.2 and 4.4 respectively in the originalunsquared [20], which was based on the cut threshold (rather than
σ
voltages, and the 45 kHz trigger rate for DSS14 LCP im- the trigger threshold) and the more sensitive 150 MHz
plies n =4.0(fromRef.[9],Eq.46). I thereforefindthat DSS14 RCP channel. Fig. 1 shows the relationship be-
σ
the trigger thresholds are higher than the cut thresholds, tween the cut and trigger thresholds, calculating (θ)
min
E
and thus limit the sensitivity, for the DSS13 RCP and for all channels through the same procedure as above and
DSS14 LCP channels. Note that my assumptions, and assuming an Airy disk beam shape. Although the DSS14
the insignificance of dispersion at this experiment’s high LCP channelis moresensitive thanDSS13 RCP,its beam
observing frequency, imply α=1. If my assumptions are is narrower, so it limits the effective beam width to 11′,
invalid then the true trigger thresholds will be lower than giving a limb coverageof 11%.
found here, but the amplitude reconstruction efficiency α The GLUE experiment spent a shorter period of time
will be decreased,leading to a net increasein the effective (see Table 1) pointing either directly at the lunar centre,
threshold and a decrease in the sensitivity of this experi- or in a half-limb position offset 0.125◦ from this. In these
ment. cases,theDSS14antennawasdeliberatelydefocused,which
Duetotherangeofdifferentchannelsusedinthecoin- reduced its aperture efficiency but improved its sensitiv-
cidencetriggerrequirement,thescalingrelationinSec.2.2 ity on the limb of the Moon. The degree of defocusing
is not directly applicable: instead, the threshold is deter- waschosento matchthe DSS13 beam size,so under these
6
averaged over 1 µs, which is 80 the Nyquist sampling
×
0.04 interval for the 40 MHz bandwidth of the receiver; hence,
a band-limited pulse wouldneedanamplitude of√80σ to
increase the averagedpowerby a factor of two, whichwas
beam width: 10.8′
the threshold for the cut. I assume a system temperature
0.03
z) DSS13 RCP two-channel forthereceiverofonly30K,asitwasoffsetfromthemain
H 75 MHz coincidence
M beamby0.5◦ andhencenotdirectedatthe Moon. Dueto
m/
V/0.02 thisoffset,itwasonlyminimallysensitivetoalunar-origin
(θ)µ (minE DS4S01 M4 HLzCP ibpsiunolisnnelg:ytt0hh.ee1s6be%epaafmorrapmDoewStSeer1rs4B,w(oθirt)h1o.Ef4aq3.%18.8,wtGhheHenzthdAreeifrsoyhcoudlsideskd.atC0o.fm5o◦r-
max
0.01 E
DSS14 RCP exclusion of a pulse by this effect is 0.370 µV/m/MHz, or
150 MHz
0.253 µV/m/MHz when DSS14 was defocused. Since this
trigger threshold
per Williams (2004)
cut threshold latter value is below the detection threshold for the
min
0.00 E
8 6 4 2 0 2 4 6 8 centre-pointing configuration, I conclude that this config-
θ (arcmin) uration was not sensitive to UHE particles, as any pulse
from the limb of the Moon which was detected in the pri-
Figure1: ThresholdelectricfieldstrengthEmin(θ)overangleθfrom mary DSS14 beam would also be detected in the off-axis
the beam axis for different channels of the GLUE experiment, for
receiver and thus be excluded as RFI.
a limb pointing. Solid lines show the trigger thresholds I calculate
for each channel, with the dashed line showing the threshold for a Therearesubstantialuncertaintiesassociatedwiththis
coincidence on both DSS13 RCP channels, while dotted lines show analysisofthe effects ofthe anti-RFI cut with the off-axis
thresholds based on the cuts of Williams [20]. The cut threshold
receiver. The exclusionthresholdis highlysensitivetothe
calculated by Williams for DSS14 RCP at the centre of the beam
(starred) corresponds closely to my curve. The sensitivity is deter- assumed system temperature and beam shape, and real-
minedbythehighestthreshold,whichisatriggerthreshold(rather istically it will vary with the power of the off-axis beam
than a cut threshold) across the entire beam. I take min at the at different points on the limb, rather than taking a sin-
E
centre of the beam to be given by the two-channel coincidence re-
gle value (for the centre of the on-axis beam) as assumed
quirement for DSS13 RCP, as described in the text, and the beam
widthtobethat atwhichthe triggerthresholdfortheDSS14 LCP here. There is a less serious approximation involved in
channel reaches √2timesthisvalue,asshown. conflating the 2.2 GHz primary observing frequency with
the 1.8 GHz frequency of the off-axis receiver, effectively
assumingthatanAskaryanpulsewillhaveaflatspectrum
circumstances I treat DSS14 as a 34 m antenna, and find
acrossthisfrequencyrange. Finally,thisanti-RFIcutwas
the sensitivity to be limited by the 40 MHz DSS14 LCP
not applied to all of the data, so some fraction of the ob-
channel. Asthereisonlyonesuchchannel,f =1. Given
C serving time will be free of this effect. However, this is
thereportedsystemtemperaturesof170K(half-limb)and
the best representation of this effect that can be achieved
185K(centre),Ifind inthischanneltobe0.0057and
Erms with the chosen set of parameters, and I expect it to be
0.0059 µV/m/MHz respectively.
at least approximately correct. Note that the complete
The sensitivityinthese cases,however,is dramatically
exclusion of the centre-pointing configuration makes little
affected by the large angle between the beam centre and
differencetothetotalsensitivityoftheGLUEexperiment,
thelunarlimb. AssuminganAirydiskbeamshapeandan
asonlyasmallfractionofthe observingtime wasspentin
apparent lunar size of 31′, the beam power at the closest
this configuration,and previous work which neglected the
point on the lunar limb is 40.7% for a half-limb pointing,
anti-RFIcut[12]hasalreadyshownthatthisconfiguration
and only 0.5% for a centre pointing. Including these fac-
had only minimal sensitivity to UHE neutrinos.
tors as (θ ) in Eq. 8, I obtain values for of 0.050
L min
B E
and 0.474 µV/m/MHz respectively, greatly increasing the
3.3. Kalyazin
threshold relative to that for a limb pointing. The advan-
Beresnyak et al. [22] conducted a series of lunar ra-
tage of these configurations is that the limb coverage is
dio observations with the 64 m Kalyazin radio telescope,
increased: 20% for a half-limb pointing, and 100% for a
with an effective duration of 31 hours, using 120 MHz of
centre pointing since the beam is equally sensitive to the
bandwidth (RCP only) at 2.25 GHz. Pulses in this band
entire limb.
triggeredthestorageofbuffereddatabothforthischannel
The off-axis 1.8 GHz receiver on DSS14 used to iden-
andfor a 50 MHz band with dualcircularpolarisationsat
tify RFIwasoperatedthroughouttheexperimentand,for
1.4 GHz. RFI was excluded by requiring a corresponding
most of the data, a cut was applied to exclude events
pulsetobevisibleinbothpolarisationsat1.4GHzatade-
in which this receiver detected a significant increase in
lay corresponding to the expected ionospheric dispersion,
noise power. Since a lunar-origin pulse could be detected
along with further cuts on the pulse shape and the clus-
through a sidelobe of its beam, this cut places an upper
teringoftheir timesofarrival. Of15,000eventsexceeding
limit on the intensity of a pulse that could be identified
the2.25GHztriggerthresholdof13.5kJy,nonemetthese
by this experiment. The cut was applied to the power
criteria.
7
Interpreting this trigger threshold as an equivalent to- centre by 14′, effectively on the limb. The resulting limb
tal flux density in both polarisations, it is equivalent by coverage for the 2.25 GHz beam, with an FWHM of 7′,
Eq. 10 to a threshold of 0.0206 µV/m/MHz in a radially- is 7%. The 1.4 GHz beam is larger than this, and is thus
aligned linear polarisation. (If it is instead interpreted as able to confirmadetection anywherewithin the 2.25GHz
thefluxdensityintheRCPchannelalone,theelectricfield beam, so it does not further constrain the limb coverage.
threshold will be increased by a factor of √2.) This value Dagkesamanskiietal.[24]reportfurtherobservationswith
for neglects several of the scaling factors in Eq. 8, a new recordingsystem anda lowertriggerthreshold, but
min
E
which I will now apply. For a single channel in a beam do not provide enoughdetail to evaluate the sensitivity of
directedatthelimb, f = (θ)=1,soonlyαneedstobe these observations, so they are not included here.
C
B
calculatedtocompensateforinefficiencyinreconstruction
of the peak pulse amplitude. 3.4. LUNASKA ATCA
Dispersionis negligible at 2.25GHz overthe relatively TheLunarUltra-high-energyNeutrinoAstrophysicswith
narrowbandofthis experiment. Thetriggersystemis de- theSquareKilometreArray(LUNASKA)projectconducted
scribed as having a time resolution of 2 ns, which I take lunarradioobservationswiththreeofthe22mantennasof
to be the sampling interval, giving a sampling rate of 500 the Australia Telescope Compact Array (ATCA), requir-
Msample/s, compared with a Nyquist rate of 240 Msam- ing a three-way coincidence for a successful detection, in
ple/s. Thisoversamplingsubstantiallymitigatesthesignal Februaryand May2008[10, 25]. The pointing ofthe tele-
loss from a finite sampling rate. (Note that this sampling scope in the two observation runs was at the centre and
rateislowerthanthemaximum2.5Gsample/srateofthe the limb of the Moon respectively, with a total effective
TDS3034digitaloscillographusedinthisexperiment[23]; duration of 26 hours. The radio frequency range was 1.2–
possibly itwassettolessthanthe maximumvalue, orthe 1.8GHz, with ananalogdedispersionfilter to compensate
triggeralgorithmonlyprocessedeveryfifthsample. Inany for ionospheric dispersion over this wide band, and sam-
case,theimprovementinsensitivityfromfurtheroversam- plingat2.048Gsample/swhichaliasedthesignalfromthe
pling is minimal.) Due to the frequency downconversion, 1.024–2.048GHz range to 0–1.024 GHz.
the final phase of the pulse is essentially random, as de- Theyreportamedianthresholdovertheirobservations
scribed in Sec. 2.1. I simulate these effects as described of 0.0153 µV/m/MHz, not significantly different between
in App. A, assuming the downconverted signal to be at thetwoobservingruns,possiblybecausethereducedther-
baseband (0–120 MHz), and find a mean signal loss of mal emission from the Moon in the limb pointing of May
13% (i.e. α=0.87), almost entirely from this last effect. 2008 was counteractedby the introduction of an anti-RFI
Applying this correction, I find an effective threshold of filter that removed part of the band. Their figure already
=0.0235µV/m/MHz,equivalenttoF =17.6kJy. includes most of the effects considered here: it is aver-
min min
E
For a pulse to be detected by this experiment it must agedovera rangeof linear polarisationalignments, scaled
also have sufficient amplitude to be visible in the 1.4 GHz for a 50% detection probability given the requirement of
band, todistinguishitfromRFI.Assumingasystemtem- a three-way coincidence, and increased to compensate for
perature of120 K andan aperture efficiency of 60%,both the signalloss from the finite sampling rate, and from the
polarisationsatthis frequencyhaveanoiselevelof = mismatch between the fixed dedispersion characteristic of
rms
E
0.0025 µV/m/MHz. Given η =2 for circular polarisa- their filter and the varying ionospheric STEC. These last
tion and α=0.90 for this band calculated as above, a two effects are treated with greater sophistication than
pulse with an amplitude matching the threshold at in this work, because they simulate them for pulses with
min
E
2.25 GHz would be visible at 1.4 GHz with a significance a range of spectra, rather than only for a flat spectrum.
of n =5.9 in each polarisation. This exceeds the 4σ They implicitly assume the pulse to have a base phase of
σ
∼
maximumlevelexpectedfromthermalnoiseforthe15,000 zero, whereas the inherent phase of an Askaryan pulse is
storedevents,makingitsufficienttoconfirmthedetection closetotheworst-casevalueofπ/2[11],whichwillbepre-
of a pulse. The coincidence requirement is thus not the servedwhenthesignalisdownconvertedbyaliasingrather
limiting factoronthe sensitivityofthisexperiment,which than by mixing with a localoscillatorsignal,but the orig-
is instead determined entirely by the trigger threshold at inal phase will most likely be near-completelyrandomised
2.25GHz. Note,however,thatIhaveassumedaflatpulse by the remnant dispersion, which is included in their cal-
spectrum between 1.4 GHz and 2.25 GHz: a pulse could culation.
stillfailthecoincidencerequirementifitsspectrumpeaked Ithereforeadopttheirthresholdof0.0153µV/m/MHz
towardthelatterfrequency. Ihavealsoneglectedthescal- without modification as (0), the threshold at the cen-
min
E
ing factor f for the coincidence requirementbetween the tre of the beam. For the limb pointing, I take this value
C
2.25 GHz band and both 1.4 GHz channels, and my as- directly as , and use the apparent lunar size of 30′
min
E
sumptions for the system temperature and aperture effi- and an FWHM beam size of 32′ when averaged over the
ciency may be inaccurate,but these effects areunlikely to band from the empirical model of Wieringa and Kesteven
reducethesignificanceofapulsesomuchthatitsdetection [26], which should provide a more precise result than an
cannot be confirmed. Airy disk in this case, to find the limb coverage to be
Thisexperimentobservedapointoffsetfromthelunar 36%. For the centre pointing, the same model gives a
8
beam power at the limb of (θ )=55.1% and hence a Buitink etal.[27]givearangeforthe systemtemperature
L
B
threshold of (θ ) = 0.0207 µV/m/MHz, with equal of 400–700 K, with the range being due to the varying
min L
E
sensitivity around the entire limb. contribution from Galactic background noise; I take the
centralvalueof550K.Giventheseparameters,Icalculate
3.5. NuMoon from Eq. 7 the value of for a single 20 MHz band in
rms
E
The NuMoon project [27] conducted a series of lunar a single polarisation as 0.020 µV/m/MHz.
radioobservationsfromJune2007toNovember2008with All C =8 channels (two polarisations and four fre-
the Westerbork Synthesis Radio Telescope (WSRT), us- quency bands) for a single beam were separately down-
ing the PuMa-II backend[28] to combine the signals from convertedtobasebandsignals,introducingarbitraryphase
eleven of its fourteen 25 m antennas to form two tied- factors which were not calibrated, so there is no phase
array beams pointing at opposite sides of the Moon, in coherence between them. This is irrelevant, however, be-
four overlapping 20 MHz bands covering the effective fre- causetheywerecombinedinthepowerdomain,whichputs
quencyrange113–168MHz. Theyrecordedbasebanddata this experimentinthe fourthregimedescribedinSec.2.2,
continuously during their observations, and retroactively so that the sensitivity scales as f =C−1/4 regardless of
C
applied dedispersion and a series of cuts to remove RFI phase coherence. I modify this slightly because the bands
based on pulse width, regular timing, and coincidence be- were overlapping and thus not completely independent,
tween the two beams. The effective observing time was and instead take f based on the ratio between a single
C
46.7 hours, spread out over 14 observing runs. 20 MHz band and the 55 MHz total bandwidth, with an
They represented their sensitivity in terms of a pa- additionalfactorof2 forthe combinationofpolarisations,
rameter S which is a measure of the power in a single as (2 55/20)−1/4 = 0.65. This is slightly optimistic, as
×
beam summed across all four bands, both polarisations, the combination of the bands applies a suboptimal un-
and five samples (125 ns) in time, such that S =8 cor- even weighting between overlapping and non-overlapping
responds to the mean power or SEFD. The summation frequency ranges, but this discrepancy should be minor.
over time compensates for uncertainty in the STEC dur- The threshold in S already incorporates the effects of
ing the observations,whichleads to some remnantdisper- dispersion, and the averaging of power over five consec-
sion or excess dedispersion extending a pulse. The events utive samples will minimise the loss of pulse amplitude
remaining after cuts show a large excess over the distri- through finite sampling and randomisation of the pulse
bution expected from thermal noise, the most significant phase,soIdonotcalculateαasinSec.2.1. Theamplitude
event having S =76 compared to an expected maximum of a pulse will, however, be decreased when it is averaged
of S 30, with hundreds of other events falling between intime,andItakeα=1/√5toreflectthis. Thesumming
∼
thesetwovalues. Duetothelargenumberoftheseevents, of power between polarisations ensures that η =2 regard-
theyareunlikelytooriginatefromUHEparticlesinteract- less of the alignment between the linear polarisations of
ing in the Moon, but they are not positively identified as the receiversandofthe pulse,the latterofwhichisinthis
RFI, and so they limit the sensitivity of this experiment: case strongly frequency-dependent due to Faraday rota-
the detection threshold must be raised to exclude them. tion. Given these parameters, and with n as calculated
σ
Duetothelowobservingfrequencyofthisexperiment, earlier, I calculate from Eq. 8 the threshold electric field
dispersion is a large effect, and even small errors in the forthisexperimenttobe0.136µV/m/MHz,equivalentby
STEC used for dedispersion can lead to pulses being ex- Eq. 10 to a flux density over the 55 MHz bandwidth of
tended in time beyond a five-sample window, prevent- 272 kJy. The originally-reported value was 240 kJy, but
ing the parameter S from recording their entire power. this was for a detection efficiency of 87.5% (rather than
Buitink et al. [27] simulated this effect and found that a 50%) and assumed perfect aperture efficiency, which will
pulse with an original power equivalent to S >90 would respectively increase and decrease the threshold.
have a >50% probability of being detected with power in The limb coverage is dependent on the shape of the
excessofthemostsignificanteventactuallyrecordedinthe tied-arraybeams,whichistheFouriertransformofthein-
experiment. ThisvalueofSdefinesthesignificancethresh- stantaneousu-v coverageofthetelescope. TheWSRTisa
old, equivalent in the voltage domain to n = 90/8 = linear array, which results in an elongated beam oriented
σ
3.4. The detection efficiency declines again for stronger perpendicular to the array axis. The tied-array beam is
p
pulses, as they may have sufficient power dispersed over further tapered by the primary beam of a single antenna,
a sufficient interval to be excluded by the cut on pulse but this is extremely wide (FWHM of 5◦) and so does
width, but the threshold width for this cut was chosen to not significantly affect the tied-array beam power around
minimise this effect, and I neglect it here. the Moon. The scale of the beam pattern is determined
Since the tied-array beams were formed coherently, I by the angle between the Moon and the east-west array
treat all antennas, for a single polarisation and 20 MHz axis, which determines the projected array length; I take
band,asasinglechannel. Forelevenantennaseachwitha this angle to be 65◦, which is its median value during the
diameter of 25 m, and with an aperture efficiency of 33% scheduled time listed for this experiment in the WSRT
for the Low Frequency Front End (LFFE) receivers used
in this experiment[29],the totaleffective areais 1782m2.
9
region only once, the fraction of the limb covered by the
two beams is 14%. Given the low observing frequency of
this experiment, at which the Askaryan pulse from a par-
ticle cascade is very broadly beamed and hence may be
detected away from the limb of the Moon, it is arguable
thatthe metric shouldinsteadbe the fractionofthe near-
side lunar surface area within the FWHM beams, which
is 21% in this pointing configuration. By either of these
metrics,thecoverageissubstantiallylowerthanthefigure
of 67% given in the original report.
The original report of this experiment also neglected
thepossibilityofalunar-originpulsebeingsimultaneously
detected in both beams, leading to it being excluded by
the anticoincidence cut. A pulse was considered to be de-
tected, and hence eligible for the anticoincidence cut, if it
exceeded a threshold of S =20 or n = 20/8 = 1.58 in
σ
the combined power in both polarisations,simultaneously
Figure2: WSRTbeamsasusedintheNuMoonexperiment,averaged p
acrossthefourbands,fortheMoonatthemedianangleof65◦ from in all four bands. The scaling factor fC must therefore
be calculated as the product of factors corresponding to
the WSRT array axis. Solid lines show the two tied-array beams,
pointedatoppositesidesoftheMoon;thestrongsidelobesat50–60′ bothmethodsofcombiningchannelsdescribedinSec.2.2:
are due to the regular spacing of the majorityof the WSRT anten- oneforthe incoherentcombinationofthe twopolarisation
nas, withthe sidelobewidth due to the largefractional bandwidth.
channels, and one for the required coincidence between
The upper dashed line shows the primary beam of a single WSRT
antenna, assumed to be an Airydisk. The lower dashed lineshows the four bands. The first of these is 2−1/4 for the two
the meansidelobe levelcorresponding to1/11 of theprimarybeam polarisations, as in the earlier calculation of for this
min
power, expected for random incoherent combination of the signals experiment. For the second factor Eq. 13 canEnot be used
from eleven antennas. Starred points show the power of each beam
directly, as the channels being combined in coincidence
at the centre of the other (the cross-beam power), which is 27.5%.
Theoverlappingpositionsofthe FWHMbeamswithrespecttothe do not have a Gaussian distribution: they have a χ2 dis-
Moonareshownabovetheplot;inthetransversedirection(vertical tribution with ten degrees of freedom (for the incoherent
in this figure) they will extend out to the 5◦ scale of the primary sum of two polarisations and five consecutive samples in
beam.
time), and are in the power domain. Instead, I approxi-
mate this distribution with a Gaussian distribution with
schedule archive1. The eleven WSRT antennas used in equal variance, and apply Eq. 13 with C =4 bands and a
this experimentconsistedofnineofthetenfixedantennas significanceof√2 10n2 (withthefactorof2forthevari-
withregular144mspacing(RT0–RT4andRT6–RT9),and ance of a χ2 distri×butionσ, the factor of 10 for the number
twoofthefourmoveableantennas(RTAandRTB),which of degrees of freedom, and the square of n to convert to
σ
are respectively36 mand 90m distantfromthe lastfixed the powerdomain), takingthe squarerootofthe resultto
antenna when the array is in the “Maxi-Short” configura- returnit to the voltagedomain. This gives a value of 1.04
tion used in this experiment. I calculate the beam shape for the factor of f describing the four-band coincidence
C
based on the u-v coverage of these antennas, neglecting requirement, which I multiply by the factor of 2−1/4 for
the minor effect of any phase errors between antennas in the combinationof the two polarisationchannels to find a
forming the tied-array beams, with the results shown in combined value of f =0.88. Finally, a lunar-originpulse
C
Fig. 2: each beam has an FWHM size of 4.2′ in the direc- detected at the centre of one beam will be detected in the
tion parallel to the array, and is highly elongated in the other beam with its intensity scaled by the power (θ) of
transverse direction. the secondbeam atthis point, whichis showninFBig. 2to
Fromtheoriginalpointingdataforthisexperiment[30], be 27.5%.
I find that the separation between the beams was scaled Applying these values for n , f and (θ) in Eq. 8,
σ C
during each observation to match the changing resolution with η =2 and α=1/√5 as in the calculaBtion of , I
min
of the array. The 2.8′ separation between the centres of find the maximum detectable pulse strengthto be E =
max
the beams shown in Fig. 2 is for the resolution when the 0.165 µV/m/MHz. As this exceeds by a faEctor of
min
Moon is at 65◦ to the array axis, as assumed for the cal- E
only 1.2, a lunar-originpulse must have a strength within
culation of the beam pattern. Since this is less than the a quite narrow range for it to be detected without being
FWHM beam size, the FWHM beams overlap as shown; excluded as RFI, which severely limits the sensitivity of
andsince the scalingofthe beamseparationmatchesthat this experiment. As for the GLUE centre-Moon pointing
of the beampattern, the proportionaloverlapwill be con- discussed in Sec. 3.2, I note that the exclusion threshold
stant throughout the observations. Counting the overlap will vary across the beam, so it may be less restrictive at
some points. The contribution from thermal noise may
1http://www.astron.nl/wsrt-schedule also assist in some cases by chance, elevating the power
10