Designing a cross‐correlation search for continuous‐wave gravitational radiation from a neutron star in the supernova remnant SNR 1987A★

Abstract

ABSTRACTA strategy is devised for a semicoherent cross-correlation search for a young neutron star in the supernova remnant SNR 1987A, using science data from the Initial Laser Interferometer Gravitational Wave Observatory (Initial LIGO) and/or Virgo detectors. An astrophysical model for the gravitational wave phase is introduced which describes the star’s spin-down in terms of its magnetic field strength B and ellipticity ε, instead of its frequency derivatives. The model accurately tracks the gravitational wave phase from a rapidly decelerating neutron star under the restrictive but computationally unavoidable assumption of constant braking index, an issue which has hindered previous searches for such young objects. The theoretical sensitivity is calculated and compared to the indirect, age-based wave strain upper limit. The age-based limit lies above the detection threshold in the frequency band 75 ≲ν≲ 450 Hz. The semicoherent phase metric is also calculated and used to estimate the optimal search template spacing for the search. The range of search parameters that can be covered given our computational resources (∼109 templates) is also estimated. For Initial LIGO sensitivity, in the frequency band between 50 and 500 Hz, in the absence of a detected signal, we should be able to set limits of B≳ 1011 G and ε≲ 10−4.

Full text

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arXiv:1102.4654v1 [gr-qc] 23 Feb 2011
Mon. Not. R. Astron. Soc. 000, 1–17 (2010) Printed 24 February 2011(MN LATEX style file v2.2)
Designing a cross-correlation search for continuous-wave
gravitational radiation from a neutron star in the
supernova remnant SNR 1987A
C. T. Y. Chung1, A. Melatos1, B. Krishnan2, J. T. Whelan3
1School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
2Max Planck Institut f¨ ur Gravitationsphysik, Am M¨ uhlenberg 1, D-14476 Golm, Germany
3Center for Computational Relativity and Gravitation and School of Mathematical Sciences,
Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USA
ligo-p1000089-v3
ABSTRACT
A strategy is devised for a semi-coherent cross-correlation search for a young neutron
star in the supernova remnant SNR 1987A, using science data from the Initial LIGO
and/or Virgo detectors. An astrophysical model for the gravitational wave phase is
introduced which describes the star’s spin down in terms of its magnetic field strength
B and ellipticity ǫ, instead of its frequency derivatives. The model accurately tracks the
gravitational wave phase from a rapidly decelerating neutron star under the restrictive
but computationally unavoidable assumption of constant braking index, an issue which
has hindered previous searches for such young objects. The theoretical sensitivity is
calculated and compared to the indirect, age-based wave strain upper limit. The age-
based limit lies above the detection threshold in the frequency band 75Hz ? ν ?
450Hz. The semi-coherent phase metric is also calculated and used to estimate the
optimal search template spacing for the search. The range of search parameters that
can be covered given our computational resources (∼ 109templates) is also estimated.
For Initial LIGO sensitivity, in the frequency band between 50Hz and 500Hz, in
the absence of a detected signal, we should be able to set limits of B ? 1011G and
ǫ ? 10−4.
Key words: gravitational waves — pulsars: general — stars: neutron — supernovae:
individual (SNR 1987A)
1 INTRODUCTION
The Laser Interferometer Gravitational Wave Observa-
tory (LIGO) achieved its design sensitivity during its
fifth science run [S5; (Abbott et al. 2009c)]. Analysis
of S5 data is progressing well, with new upper lim-
its being placed on the strength of various classes of
burst sources (Abbott et al. 2009d; Abadie et al. 2010a;
Abbott et al. 2010a), stochastic backgrounds (Giampanis
2008;Abbott et al. 2009a),
(Abbott et al. 2009e; Abadie et al. 2010c,d) and continuous-
wave sources (Abbott et al. 2009f,b, 2010b; Abadie et al.
2010b). In some cases, the LIGO limits on astrophysical
parameters beat those inferred from electromagnetic astron-
omy, e.g. the maximum ellipticity and internal magnetic field
strength of the Crab pulsar (Abbott et al. 2008, 2010b). Re-
cently, an S5 search was completed which placed upper lim-
its on the amplitude of r-mode oscillations of the neutron
compact binarysources
star in the supernova remnant Cassiopeia A (Abadie et al.
2010b).
Aspherical,
promising
(Ostriker & Gunn 1969). The origin of the semi-permanent
quadrupole in these objects can be thermoelastic (Melatos
2000; Ushomirsky et al.2000;
Haskell 2008) or hydromagnetic (Bonazzola & Gourgoulhon
1996; Cutler 2002; Haskell et al.
Akg¨ un & Wasserman 2008; Mastrano 2010). Thermoelastic
deformations arise due to uneven electron capture rates in
the neutron star crust. A persistent 5% temperature gra-
dient at the base of the crust produces a mass quadrupole
moment of ∼ 1038g cm2(ǫ ∼ 10−7; Ushomirsky et al.
2000). Hydromagnetic deformations, on the other hand, are
produced by large internal magnetic fields, and misaligned
magnetic and spin axes. For example, a neutron star
with spin frequency 300Hz and internal toroidal field
Bt ? 3.4 × 1012G has an ellipticity ǫ ∼ 10−6??Bt?/1015G?
isolated
of
neutronstars constitute
candidates
one
classcontinuous-wave source
Nayyar & Owen2006;
2008; Haskell2008;

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2C. T. Y. Chung, A. Melatos, B. Krishnan, J. T. Whelan
(Cutler 2002). The deformation of an ideal fluid star with
an arbitrary magnetic field distribution and a barotropic
equation of state can be computed, with ellipticities as high
as 10−5predicted for some configurations (Haskell et al.
2008). Additionally, some exotic neutron star models (e.g.
solid strange quark stars) allow for ellipticities as large
as 10−4(Owen 2005). Accreting neutron stars in binary
systems can form magnetic mountains, with ǫ ? 10−5
(Melatos & Payne 2005; Vigelius & Melatos 2009a). Defor-
mations of all kinds relax viscoelastically and resistively
over time, so that young neutron stars are expected to
be generally stronger gravitational wave emitters. For
example, thermoelastic deformations relax on the thermal
conduction time-scale (∼ 104yr), after the temperature
gradient in the crust has switched off (Brown & Bildsten
1998; Vigelius & Melatos 2010). Magnetic mountains relax
as the accreted matter diffuses through the magnetic field
on the Ohmic time-scale 105–108yr (Vigelius & Melatos
2009b).
A coherent search for 78 known radio pulsars was per-
formed on S3 and S4 LIGO and GEO 600 data. Upper limits
on the ellipticities of these pulsars were obtained, the small-
est being ǫ ? 7.1×10−7for PSR J2124–3358 (Abbott et al.
2007c). More recently, a coherent search for 116 known pul-
sars was carried out using data from both the LIGO and
Virgo detectors, placing an upper limit of ǫ < 7.0×10−8for
PSR J2124−3358.
The youngest isolated neutron star accessible to LIGO
probably resides in the supernova remnant SNR 1987A.
The coincident detection of neutrino bursts from the su-
pernova by detectors all over the world confirmed the core-
collapse event, strongly indicating the formation of a neutron
star (Aglietta et al. 1987; Hirata et al. 1987; Bionta et al.
1987; Bahcall et al. 1987).1Constraints have been placed
on the magnetic field strength, spin period, and other
birth properties of the putative neutron star (Michel 1994;
¨ Ogelman & Alpar 2004); see Section 2 for details. However,
searches for a pulsar in SNR 1987A have yielded no con-
firmed sightings; upper limits on its luminosity have been
placed in the radio, optical and X-ray bands (Percival et al.
1995; Burrows et al. 2000; Manchester 2007). An uncon-
firmed detection of a transitory 467.5 Hz optical/infra-red
pulsation in SNR 1987A was reported by Middleditch et al.
(2000).
The likely existence of a young neutron star in SNR
1987A makes it a good target for gravitational wave searches
(Piran & Nakamura 1988; Nakamura 1989). A coherent
matched filtering search was carried out in 2003 with the
TAMA 300 detector, searching 1.2×103hours of data from
its first science run over a 1-Hz band centered on 934.9Hz,
assuming a spin-down range of (2–3)×10−10Hz s−1. The
search yielded an upper limit on the wave strain of 5×10−23
1An unconfirmed correlation was also reported between data
taken by the Mont Blanc and Kamioka neutrino detectors
and gravitational wave detectors
(Amaldi et al. 1989). Taken at face value, these observations are
consistent with a weak neutrino pulsar operating briefly during
the core-collapse event. However, a serious flaw in the original
analysis was found by Dickson & Schutz (1995), whose reanalysis
led them to conclude that the correlations were not physically
significant.
in Maryland and Rome
(Soida et al. 2003). An earlier matched filtering search was
conducted using 102hours of data taken in 1989 by the
Garching prototype laser interferometer. The latter search
was carried out over 4-Hz bands near 2kHz and 4kHz, did
not include any spin-down parameters, and yielded an upper
limit of 9×10−21on the wave strain (Niebauer et al. 1993).
There are two main types of continuous-wave LIGO
searches: coherent and semi-coherent. The former demand
phase coherence between the signal and search template over
the entire time series. Although sensitive, they are restricted
to small observation times and parameter ranges as they
are computationally intensive. Semi-coherent searches break
the full time series into many small chunks, analyse each
chunk coherently, then sum the results incoherently, trad-
ing off sensitivity for computational load. Santostasi et al.
(2003) discussed the detectability of gravitational waves
from SNR 1987A, estimating that a coherent search based
on the Middleditch et al. (2000) spin parameters requires 30
days of integration time and at least 1019search templates
covering just the frequency and its first derivative. In reality,
the task is even more daunting, because such a young object
spins down so rapidly, that five or six higher-order frequency
derivatives must be searched in order to accurately track
the gravitational wave phase. A Bayesian Markov Chain
Monte Carlo method was proposed as an alternative to
cover the parameter space efficiently (Umst¨ atter et al. 2004;
Umst¨ aetter et al. 2008). As yet, though, SNR 1987A has not
been considered a feasible search target, because even Monte
Carlo methods are too arduous. In this paper, we show how
to reduce the search space dramatically by assuming an as-
trophysically motivated phase model.
In this paper, we discuss how to use a cross-correlation
algorithm to search for periodic gravitational waves from a
neutron star in SNR 1987A. The search is semi-coherent
(Dhurandhar et al. 2008). The signal-to-noise ratio is en-
hanced by cross-correlating two data sets separated by an
adjustable time lag, or two simultaneous data sets from dif-
ferent interferometers, thereby nullifying short-term timing
noise (e.g. from rotational glitches). This is a modification
of the method used in searches for a cosmological stochastic
background (Abbott et al. 2007a, 2009a) and for the low-
mass X-ray binary Sco X-1 (Abbott et al. 2007b). In Section
2, we review the properties of SNR 1987A and its putative
neutron star. Section 3 briefly describes the cross-correlation
algorithm and the data format. We estimate the theoretical
sensitivity of the search in Section 4. Section 5 describes
an astrophysical model, which expresses the gravitational
wave phase in terms of the initial spin, ellipticity, magnetic
field, and electromagnetic braking index of the neutron star.
We calculate the semi-coherent phase metric and the num-
ber of templates required for the search in the context of
the astrophysical phase model. Given the computational re-
sources available to us, we derive upper limits on the grav-
itational wave strain, ellipticity and magnetic field which
can be placed on a neutron star in SNR 1987A with a cross-
correlation search. Finally, Section 7 summarises the results.
2 A YOUNG NEUTRON STAR IN SNR 1987A
SNR 1987A is the remnant of a Type II core-collapse su-
pernova which occurred in February 1987, 51.4kpc away

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Cross-correlation search for continuous-wave gravitational radiation from a neutron star in SNR 1987A3
in the Large Magellanic Cloud (α = 5h 35m 28.03s, δ
= −69◦16′11.79′′) (see reviews by Panagia 2008 and in
Immler et al. 2007). Its progenitor was the blue supergiant
Sk 1 (Panagia 1987; Gilmozzi et al. 1987; Barkat & Wheeler
1988; Woosley et al. 2002). The color of the progeni-
tor, as well as the origin of the complex three-ring
nebula in the remnant, are still unexplained. Detailed
simulations of the evolutionary history of Sk 1, per-
formed by Podsiadlowski et al. (2007), support the the-
ory that two massive stars merged to form an oversized
20M⊙ red supergiant 2 × 105years before the supernova,
which eventually shrank as its envelope evaporated (e.g.
Podsiadlowski & Joss 1989; Podsiadlowski et al. 1990). An
alternative theory suggests that Sk 1 was instead a single 18–
20M⊙ red supergiant which evolved into a blue supergiant
via wind-driven mass loss (e.g. Woosley 1988; Saio et al.
1988; Sugerman et al. 2005).
There is strong evidence for the existence of a neu-
tron star in SNR 1987A. The progenitor mass range re-
quired to produce Type II supernovae, 10–25M⊙, which in-
cludes the above evolutionary scenarios, is the same range
required to produce neutron star remnants (Woosley et al.
2002; Heger et al. 2003). The secure neutrino detections
mentioned in Section 1 support this conclusion. Although
there have been no confirmed pulsar detections, numerous
searches have placed upper limits on the flux and luminos-
ity at radio (< 115 µJy at 1390 MHz, Manchester 2007),
optical/near-UV (< 8×1033ergs s−1, Graves & et al. 2005),
and soft X-ray (< 2.3 × 1034erg s−1, Burrows et al. 2000)
wavelengths. Middleditch et al. (2000) reported finding an
optical pulsar in SNR 1987A with a frequency of 467.5Hz,
modulated sinusoidally with a ∼ 1-ks period, consistent
with precession for an ellipticity of ǫ ∼ 10−6. However,
the pulsations were reported to have disappeared after 1996
(Middleditch et al. 2000) and were never confirmed indepen-
dently.
There are several possible reasons why a pulsar in SNR
1987A has not yet been detected. If its spin period is greater
than 0.1s, it would not be bright enough to be detectable in
the optical band (Pacini & Salvati 1987; Manchester 2007).
If the radio emission is incoherent or the emission region is
patchy, the pulses may have been missed, even if the beam
width is as wide as is typical for young pulsars (Manchester
2007). Shternin & Yakovlev (2008) argued that, although
the neutron star’s theoretical X-ray luminosity exceeds the
observational upper limits by a factor of 20–100, the current
upper limits still allow for concealment behind an opaque
shell formed by fallback (Woosley & Weaver 1995). How-
ever, simulations by Fryer et al. (1999) suggest that, once
fallback ceases, the accreted material cools, leaving no ob-
scuring atmosphere.
Another possible reason why a pulsar has not yet been
detected is that its magnetic field is too weak. The weak-
field theory is supported by theoretical models, in which the
field grows only after the neutron star is formed and can
take up to 103years to develop (e.g. Blandford & Romani
1988; Reisenegger 2003). A growth model for SNR 1987A
was proposed by Michel (1994), in which the magnetic
field of a millisecond pulsar intensifies from 1010G at
birth to ∼ 1012G after several hundred years (exponen-
tial and linear growth were considered, yielding growth
times of ∼ 0.3–0.7 kyr), before the pulsar has time to spin
down significantly. In an alternative model, the neutron
star is born with a strong magnetic field, which is ampli-
fied during the first few seconds of its life by dynamo ac-
tion (e.g. Duncan & Thompson 1992; Bonanno et al. 2005).
Assuming this model, measurements of the known spin
periods of isolated radio pulsars imply a distribution of
birth magnetic field strengths between 1012G and 1013G
(Arzoumanian et al. 2002; Faucher-Gigu` ere & Kaspi 2006).
Several birth scenarios for the pulsar in SNR 1987A were
considered by¨ Ogelman & Alpar (2004) in this context, who
concluded that the maximum magnetic dipole moment is
< 1.1×1026G cm3, 2.5×1028G cm3, and 2.5×1030G cm3
for birth periods of 2ms, 30ms, and 0.3s respectively. How-
ever, the dynamo model also accommodates a magnetar in
SNR 1987A, with magnetic dipole moment > 2.4 × 1034G
cm3, regardless of the initial spin period (¨ Ogelman & Alpar
2004).
Estimates of the birth spin of the pulsar in SNR
1987A are more uncertain. Simulations of the bounce and
post-bounce phases of core collapse were performed by
Ott et al. (2006) to determine the correlation between pro-
genitor properties and birth spin. These authors found
proto-neutron star spin periods of between 4.7–140 ms,
proportional to the progenitor’s spin period. A Monte
Carlo population synthesis study using known velocity
distributions (Arzoumanian et al. 2002) favoured shorter
millisecond periods, but a similar population study by
Faucher-Gigu` ere & Kaspi (2006) argued that the birth spin
periods could be as high as several hundred milliseconds.
Faint, non-pulsed X-ray emission from SNR 1987A was first
observed four months after the supernova and decreased
steadily in 1989 (Dotani et al. 1987; Inoue et al. 1991), lead-
ing to the suggestion that a neutron star could be powering
a plerion that is partially obscured by a fragmented super-
nova envelope. Bandiera et al. (1988) modelled the X-ray
spectrum from a nebula containing a central pulsar, with a
magnetic field of 1012G and an expansion rate of 5×108cm
s−1. The authors found a fit to the SNR 1987A data for a
pulsar spin period of 18ms.
3 THE CROSS-CORRELATION ALGORITHM
In this section, we briefly summarise the cross-correlation
method described in Dhurandhar et al. (2008), a semi-
coherent search algorithm designed specifically to search
for continuous-wave gravitational radiation. It operates on
Short Fourier Transforms (SFTs) of data segments of length
∆T = 30 min, whose duration is chosen to minimise the
Doppler effects due to Earth’s rotation. In each SFT, the
kth frequency bin corresponds to the frequency νk= k/∆T
for 0 ? k ? N/2 and νk= (k−N)/∆T for N/2 ? k ? N−1,
where N is the total number of frequency bins in the SFT.
The output x(t) of a detector is the sum of the instan-
taneous noise, n(t), and the gravitational wave signal, h(t).
The noise is assumed to be zero mean, stationary, and Gaus-
sian. Its power is characterised by Sn(ν), the single-sided
power spectral density (i.e. the frequency-dependent noise
floor) in the following way:
?˜ n(ν)∗˜ n(ν′)? =1
2Sn(ν)δ(ν − ν′), (1)
where∗denotes complex conjugation. Therefore, in the low

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4 C. T. Y. Chung, A. Melatos, B. Krishnan, J. T. Whelan
signal limit (|h(t)| ≪ |n(t)|), the power in the k-th frequency
bin of SFT I can be approximated by
?|˜ xk,I|2? ≈∆T
2
Sn(νk), (2)
where we apply the finite time approximation to the delta
function in (1), i.e. δ∆T(ν) = sin(πν∆T)/(πν) ≈ ∆T.
In the cross-correlation algorithm, SFTs are paired ac-
cording to some criterion (e.g. time lag or interferome-
ter combination) and multiplied to form the raw cross-
correlation variable
Yk,IJ =˜ x∗
kI,I˜ xkJ,J
(∆T)2
, (3)
where I and J index the SFTs in the pair. The gravitational
wave signal is assumed to be concentrated in a single fre-
quency bin in each SFT (because ∆T ≪ ν/˙ ν due to sidereal
or intrinsic effects), whose index is denoted by kI or kJ.
The frequency bins in the two SFTs are not necessarily the
same; they are related by the time lag between the pair and
between interferometers, as well as spin-down and Doppler
effects. For an isolated source, the instantaneous frequency
at time t is given by
ν(t) = ˆ ν(t) + ˆ ν(t)v · n
c
, (4)
where ˆ ν(t) is the instantaneous frequency in the rest frame of
the source, v is the detector velocity relative to the source,
n is the position vector pointing from the detector to the
source, and c is the speed of light. The instantaneous signal
frequencies in SFTs I and J, νI and νJ, are calculated at
the times corresponding to the midpoints of the SFTs, TI
and TJ. The frequency bin kJ is therefore shifted from kI by
an amount ∆TδνIJ, with δνIJ = νJ−νI (Dhurandhar et al.
2008). For convenience, we now drop the subscripts kI and
kJ.
In the low signal limit, YIJ is a random, complex vari-
able. The cross-correlation statistic comprises a weighted
sum of YIJ over all pairs IJ. YIJ has variance σ2
S(I)
tral density of SFT I at frequency νI, and S(J)
power spectral density of SFT J at frequency νJ.
The parameters describing the amplitude and the phase
of the signal are contained within the signal cross-correlation
function˜GIJ, defined as
˜GIJ
+FI×FJ×A2
with ∆ΦIJ = ΦI(TI) − ΦJ(TJ). ΦI(TI) and νI(TI) are the
phase and frequency at time TI, whereas ΦJ(TJ) and νJ(TJ)
are evaluated at time TJ. Note that there is an error in equa-
tion (3.10) of Dhurandhar et al. (2008), which omits the fac-
tor of e−iπ∆T[νI(TI)−νJ(TJ)]arising from the choice of time
origin of the Fourier transforms. The phase factors are deter-
mined by the astrophysical phase model described in Section
5.
The terms in square brackets in (5) depend on the po-
larization angle ψ, and the inclination angle ι between n and
the rotation axis of the pulsar, in the following way:
IJ =
n (νI)S(J)
n (νJ)/(4∆T2), where S(I)
n (νI) is the power spec-
n (νJ) is the
=
1
4e−i∆ΦIJe−iπ∆T[νI(TI)−νJ(TJ)]?FI+FJ+A2
×− i(FI+FJ×− FI×FJ+)A+A×
+
?,(5)
A+
A×
=
1 + cos2ι
2
cosι,
, (6)
= (7)
F+(t;n,ψ)=a(t;n)cos2ψ + b(t;n)sin2ψ, (8)
F×(t;n,ψ)=b(t;n)cos2ψ − a(t;n)sin2ψ,(9)
where a(t;n) and b(t;n) are the detector response functions
for a given sky position, and are defined in equations (12)
and (13) of Jaranowski et al. (1998). A geometrical defini-
tion is also given in Prix & Whelan (2007). The gravitational
wave strain tensor is
← →
h (t) = h0A+cosΦ(t)← →
where h0 is the gravitational wave strain, and← →
basis tensors for the + and × polarizations in the transverse-
traceless gauge.
In principle, one should search over the unknowns cosι
and ψ, but this adds to the already sizeable computational
burden. Accordingly, it is customary to average over cosι
and ψ when computing˜GIJ, with
?˜GIJ?cos ι,ψ=
e ++ h0A×sinΦ(t)← →
e ×
(10)
e +,× are the
1
10exp−i∆ΦIJe−iπ∆T[νI(TI)−νJ(TJ)](aIaJ+bIbJ),
(11)
where aI,J = a(TI,J;n) and bI,J = b(TI,J;n). Once the first-
pass search is complete, a follow-up search on any promising
candidates can then be performed, which searches explicitly
over cosι and ψ. Preliminary Monte Carlo tests indicate that
the detection statistic resulting from (11) is approximately
10−15% smaller than if the exact cosι and ψ values are used.
The cross-correlation detection statistic is a weighted
sum of YIJ over SFT pairs. The number of pairs which can
be summed over are limited by the available computational
power. We discuss the computational costs of running the
search in Section 6.2. The cross-correlation detection statis-
tic is given by
ρ = ΣIJ(uIJYIJ + u∗
IJY∗
IJ),(12)
where the weights are defined by
uIJ =
˜G∗
σ2
IJ
IJ
.(13)
For each frequency and sky position that is searched, we
obtain one real value of ρ, which is a sum of the Fourier
power from all the pairs. Ignoring self-correlations (i.e. no
SFT is paired with itself), the mean of ρ is given by µρ =
h2
0
?
and if self-correlations are included, µρ and σ2
(Dhurandhar et al. 2008).
IJ|˜GIJ|2/σ2
ρ= 2ΣIJ|˜GIJ|2/σ2
IJ. In the low signal limit, the variance of ρ
IJ. In the presence of a strong signal, is σ2
ρscale as h2
0
4 SENSITIVITY
4.1 Detection threshold
Detection candidates are selected if they exceed a threshold
value, ρth. For a given false alarm rate Fa, this threshold is
given by (Dhurandhar et al. 2008)
ρth= 21/2σρerfc−1(2Fa/N),(14)
where erfc is the complementary error function, and N is
the number of search templates used. In the presence of a
signal, the detection rate for events with ρ > ρthis given by
γ =1
2erfc
?ρth− µρ
√2σρ
?
.(15)

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Cross-correlation search for continuous-wave gravitational radiation from a neutron star in SNR 1987A5
As µρ
tional wave strain that is detectable by the search to be
(Dhurandhar et al. 2008)
∝h2
0, one can calculate the lowest gravita-
hth(ν) =
S1/2
√2?|˜GIJ|2?1/4N1/4
pairs
?Sn(ν)
∆T
?1/2
. (16)
In (16), we define S = erfc−1(2Fa)+erfc−1(2Fd), Fd is the
false dismissal rate, ?|˜GIJ|2? is the mean-square of the signal
cross-correlation function defined in (5), Npairsis the number
of SFT pairs, and Sn(ν) is the single-sided power spectral
density of the interferometers (assumed to be identical).
One can estimate ?|GIJ|2?1/4theoretically for the spe-
cial case where TI = TJ and˜GIJ is averaged over cosι, ψ,
and sidereal time. In this case, the primary contribution to
∆ΦIJ is the term [r(TI) − r(TJ)] · n/c, where r(t) is the
position of the detector at time t in the frame of the solar
system barycentre. Under these assumptions, equation (11)
can be expressed in terms of the overlap reduction func-
tion (Whelan 2006), which depends only on ν,α, and δ. For
SNR 1987A, we have (α,δ) = (1.46375 rad, −1.20899 rad),
and hence ?|˜GIJ|2?−1/4= 4.6882. Assuming Fa = Fd = 0.1,
∆T = 1800s, and Npairs = 105(approximately 1 year of
SFTs), equation (16) gives
hth(ν) = 5.92 × 10−3
?Sn(ν)
Hz−1
?1/2
. (17)
Figure 1 is a graph of hth as a function of ν. The val-
ues of Sn(ν) are based on LIGO’s S5 noise characteristics.2
The S5 run began in November 2005 and accumulated a
year’s worth of triple coincidence data. For a signal from
SNR 1987A to be detectable, we must have hth? h0.
4.2Minimum ellipticity and indirect, age-based
limit
The deformation of a neutron star is parameterised by its
ellipticity ǫ. The gravitational wave strain at Earth emitted
by a biaxial neutron star is
h0 =4π2G
c4
Iǫν2
D
(18)
where G is Newton’s gravitational constant, c is the speed
of light, I is the moment of inertia, D is the distance to the
source, and ν is the gravitational wave frequency, assumed
to be twice the spin frequency (Jaranowski et al. 1998).
An upper limit on h0 can be derived from existing elec-
tromagnetic data by assuming all the observed spin down
comes from the gravitational wave torque, i.e. the observed
frequency derivative ˙ ν satisfies ˙ ν = −(32π4Gǫ2Iν5)/(5c5)
(Wette et al. 2008). Combining this with (18) to eliminate ǫ
gives
h0 ?1
D
?5GI|˙ ν|
2c3ν
?1/2
.(19)
Hence, for SNR 1987A to be detectable (i.e. hth ? h0), we
require
hth(ν)
?
1.66 × 10−20
2Available at http://www.ligo.caltech.edu/˜jzweizig/distribution/LSC Data
×
?
I
1038kgm2/s
?1/2?|˙ ν|
ν
?1/2?
D
51.4kpc
?−1
(20)
Unfortunately, without having observed any pulsations
from SNR 1987A, it is impossible to determine ν or |˙ ν| a
priori. Instead, we note that ˙ ν can be re-expressed in terms of
the characteristic age of the source, τc = ν/(4|˙ ν|), assuming
that ν today is much less than ν at birth. The factor 4
arises if one assumes the gravitational radiation dominates
electromagnetic spin down, in order to remain consistent
with (19); in reality, electromagnetic spin down is expected
to dominate, with τc = ν/(2|˙ ν|). Equation (20) then reduces
to
hth(ν) ? 3.39 × 10−25
?
τc
19yr
?−1/2?
D
51.4kpc
?−1
. (21)
The right-hand side of (21) is graphed as a horizontal red
line in Figure 1. The detectability condition (21) is then
satisfied for spins in the range 75Hz ? ν ? 450Hz. Note
that we have chosen τc = 19yr, the age of SNR 1987A in
2006 when the S5 search began.
It is important to note here that the assumption that ν
is currently much less than at birth is likely untrue for the
object in SNR 1987A, as it is so young. Hence, the indirect,
age-based limit in equation (21) and the horizontal line in
Figure 1 are only indicative of the expected gravitational-
wave emission strength (in fact, they are upper limits). Ex-
act calculations of ν and ˙ ν are performed in Section 5.1.
5 AN ASTROPHYSICAL MODEL FOR THE
GRAVITATIONAL WAVE PHASE
All continuous wave searches to date have used the stan-
dard model for the gravitational wave phase, described in
terms of a Taylor expansion involving spin frequency deriva-
tives (Jaranowski et al. 1998). For a young object like SNR
1987A, which spins down rapidly, it is not computationally
feasible to search over the six or more frequency derivatives
typically needed to track the phase accurately. In this sec-
tion, we present an alternative model for the gravitational
wave phase, stated in terms of astrophysical parameters (i.e.
the magnetic field strength and the neutron star ellipticity)
instead of spin frequency derivatives. It tracks the phase ex-
actly using four parameters, under the restrictive assump-
tion (justified further below) that the braking index is con-
stant.
The phase of a slowly evolving gravitational wave signal,
Φ(t) = Φ(t0) + 2π
?t
the
t0
dtν(t),(22)
can
(Jaranowski et al. 1998)
beapproximated by Taylorexpansion
Φ(t) = Φ(t0) + 2π
s
?
k=0
ν(k)
tk+1
(k + 1)!+2πn · r(t0)
c
s
?
k=0
ν(k)tk
k!
(23)
where ν(k)is the k-th derivative of the gravitational wave
frequency at time t0, and s is the number of spin-down pa-
rameters required to achieve a given accuracy. The compu-
tational cost of using (23) is substantial for rapidly deceler-
ating objects. For a maximum allowable phase error of one
cycle, the maximum bin size in the k-th derivative is ν(k)is