Detecting gravitational waves from mountains on neutron stars in the Advanced Detector Era

0.71 MB·English

by  Brynmor Haskell

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed29January2015 (MNLATEXstylefilev2.2)
Detecting gravitational waves from mountains on neutron
stars in the Advanced Detector Era
5
1 B. Haskell1, M. Priymak1, A. Patruno2,3, M. Oppenoorth4, A. Melatos1, P. D. Lasky1,5
0
1 School of Physics, The Universityof Melbourne, Parkville VIC3010, Australia
2
2 Leiden Observatory, LeidenUniversity,PO Box 9513, 2300 RA Leiden, the Netherlands

n 3 ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA Dwingeloo, the Netherlands
a 4 Copernicus Institute of Sustainable Development, Universityof Utrecht, P.O. Box 80.115, 3508 TA Utrecht, the Netherlands
J 5 Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, VIC3800, Australia.

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2
29January2015

]
R
S ABSTRACT
.
h
p Rapidly rotating Neutron Stars (NSs) in Low Mass X-ray Binaries (LMXBs) are
- thought to be interesting sources of GravitationalWaves (GWs) for current and next
o
generation ground based detectors, such as Advanced LIGO and the Einstein Tele-
r
t scope.ThemainreasonisthatmanyoftheNSinthesesystemsappeartobespinning
s
well below their Keplerian breakupfrequency, and it has been suggested that torques
a
[ associated with GW emission may be setting the observed spin period. This assump-
tion has been used extensively in the literature to assess the strength of the likely

1 gravitational wave signal. There is now, however, a significant amount of theoretical
v
andobservationworkthatsuggeststhatthismaynotbe thecase,andthatGWemis-
9
sion is unlikely to be setting the spin equilibrium period in many systems. In this
3
paper we take a different starting point and predict the GW signal strength for two
0
physical mechanisms that are likely to be at work in LMXBs: crustal mountains due
6
0 to thermal asymmetries and magnetically confined mountains. We find that thermal
. crustal mountains in transient LMXBs are unlikely to lead to detectable GW emis-
1
sion, while persistent systems are good candidates for detection by Advanced LIGO
0
and by the Einstein Telescope. Detection prospects are pessimistic for the magnetic
5
mountain case, unless the NS has a buried magnetic field of B ≈ 1012 G, well above
1
: the typically inferred exterior dipole fields of these objects. Nevertheless, if a system
v were to be detected by a GW observatory, cyclotron resonant scattering features in
i
X the X-ray emission could be used to distinguish between the two different scenarios.
r Key words: stars: neutron — X-rays: binaries — gravitationalwaves
a
1 INTRODUCTION crust (Bildsten 1998; Ushomirsky,Cutler & Bildsten
2000; Haskell, Jones & Andersson 2006;
Rapidlyrotating Neutron Stars(NSs)areconsidered an in-
Johnson-McDaniel & Owen 2013) or by a solid core of
teresting sourceof Gravitational Waves(GWs) and areone
exotic matter (Owen 1995; Haskell et al. 2007), unsta-
of the main targets for current searches with ground based
ble modes of oscillation of the star (Andersson 1998;
detectors, such as Virgo and LIGO (Riles 2013). The char-
Andersson,Kokkotas & Stergioulas 1999)andmagnetically
acteristicamplitudeoftheGWsignalscaleswiththesquare
supported mountains (Cutler 2002; Haskell et al. 2008;
of the rotation frequency, thus making the more rapidly
Melatos & Payne 2005; Priymak, Melatos & Payne 2011;
rotating NSs ideal candidates for detection. In particular
Vigelius & Melatos 2009a).
someofthemostpromisingtargetsarelikelytobeaccreting
NSs in Low Mass X-ray Binaries (LMXBs). Not only are LMXBs were originally invoked as a source of GWs
theseNSsrotatingwithmillisecondperiods,buttheprocess to solve an observational puzzle. In an LMXB the NS is
of accretion from the companion star can drive the growth spunupbymatter accreted from thecompanion viaadisc.
of a quadrupolar deformation. Plausible mechanisms that This is, in fact, how old NSs are thought to be recycled
may be at work are the creation of a “mountain” (i.e. any to millisecond periods and eventually produce a millisec-
kind of non-axisymmetric deformation that gives rise to ond radio pulsar after accretion stops (Alpar et al. 1982;
an l = m = 2 mass quadrupole) supported by the elastic Radhakrishnan & Srinivasan 1982; Papitto et al. 2013a).
2 B.Haskell et al.
One would therefore expect the NS to be spun up to its cuss the level at which GW emission may be expected. We
centrifugalbreak-upfrequency,whichisequationofstatede- also take the discussion one step further and ask, given a
pendent,butgenerally well above1KHz(Cook et al. 1994; GW detection, what constraints can be set on NS interior
Haensel et al.1999).Thisisnot,however,whatisobserved. physicsandhowonecoulddistinguishbetweenthedifferent
ThedistributionofspinsinbothLMXBsandmillisecondra- mechanisms giving rise to the mountain using electromag-
dio pulsars appears to have a statistically significant cutoff netic(e.g. X-ray) observations.
at around 730 Hz(Chakrabarty et al. 2003;Patruno 2010).
It is natural to ask what physical process removes an-
gular momentum from the NS and prevents it from spin- 2 THERMAL MOUNTAINS
ning up further. The first and most obvious candidate is
2.1 Crustal heating
the interaction between the stellar magnetic field and the
accretion disc. This possibility was examined in detail by Theouter,low densitylayersof aNSarethoughttoform a
White& Zhang(1997)whofoundthat,atleastforthedata crystalline crust of ions arranged in a Body Centred Cubic
available at the time, this scenario would involve an unex- (BCC)lattice[althoughrecentworkbyKobyakov& Pethick
pected correlation between theaccretion rateand magnetic (2014) suggests that much more inhomogeneous configura-
field strength (which would also need to be higher than ex- tions may be possible]. Above densities of ≈ 1011 g/cm3,
pected). This led to the alternative suggestion that GWs neutronsdripout ofnucleiandform asuperfluidinmature
maybeprovidingthetorqueneededtobalancetheaccretion NSs with internal temperatures T .109 K. At higher den-
torques,andsetthespinequilibriumperiodofthesesystems sities several phase transitions may occur, with nuclei no
(Papaloizou & Pringle1978;Wagoner1984;Bildsten 1998). longer being spherical but forming rods and plates, the so
A corollary of GW torque-balance is that the bright- called “pasta” phases (Lorenz, Ravenhall & Pethick 1970),
est X-ray sources should also be the loudest GW emitters until at ≈2×1014 g/cm3 there is a transition to a fluid of
(Bildsten 1998). This describes the nearby LMXB Scorpius neutrons,protonsandelectronswhich formsthecoreofthe
X-1, which has been the subject of a number of LIGO and NS.
Virgo searches (Abbottet al. 2007a,b; Abadie et al. 2011; InLMXBsaccretedmatter,composedoflightelements,
Aasi et al. 2014a) that have led to a 90% confidence up- is buried by accretion and compressed to higher densities,
per limit for the gravitational wave strain of hrms ≈ 10−25 whereitundergoesaseriesofnuclearreactionssuchaselec-
around150Hz.Withadvanceddetectorsnowcomingonline troncaptures,neutronemission andpycnonuclearreactions
thereisastrongcasetodevelopdirecteddataanalysisalgo- (Haensel & Zdunik1990).Theobservedcoolingoftransient
rithms (Aasi et al. 2014b) and all-sky pipelines that search LMXBs,astheyenterquiescence,isconsistent with acrust
for unknown binary systems (Goetz & Riles 2011). that has previously been heated by such reactions [see e.g.
Although GW searches with initial LIGO are still Wijnands, Degenaar & Page (2013) and references therein,
not sensitive enough to probe the predictions of the GW although not all details of the cooling processes are fully
torque balance scenario, the problem has been recently re- understood (Degenaar et al. 2013; Schatzet al. 2014)].
assessed by several authors. Patruno, Haskell & D’Angelo Accretion asymmetries can produce asymmetries in
(2011) found that with current data the strong correla- composition and in heating, which in turn deform the star
tion between magnetic field and accretion rate found by and lead to a quadrupole (Ushomirsky,Cutler & Bildsten
White& Zhang(1997)isnolongerneededandthemeasured 2000).OncethequadrupleQ22 isknowntheGWamplitude
spin period of most systems can be understood in terms can becalculated as:
of the disc/magnetosphere interaction (Andersson et al. 16 π 1/2 GQ22Ω2
2te0m05s).sFhuowrtshertmhaotremaandyetaoilfedthaenmalysdios,ofinindfaicvtid,uaaplpseyasr- h0 = 5 (cid:16)3(cid:17) dc4 , (1)
whereGandcarethegravitational constantandthespeed
to be close to a propeller phase in which the spin-up
of light respectively, d is the distance to the source and Ω
torque is much weaker than in standard accretion models
is the angular frequency of the star. Note that we are con-
(Haskell & Patruno 2011; Ferrigno et al. 2013). Finally the
measurements of spins and surface temperatures for most sidering a quadrupolar Y22 deformation, as this harmonic
dominates GW emission. In this case GWs are emitted at
NSs in LMXBs are not consistent with theoretical predic-
twice the rotation frequency of the star. An approximate
tions for GW emission due to an unstable r-mode (or at
expression for the quadrupole due to asymmetric crustal
least not at a level that would allow for spin-equilibrium
heating from nuclear reactions in the crust is given by
due to torque balance) (Ho, Andersson & Haskell 2011;
(Ushomirsky,Cutler & Bildsten 2000):
Haskell, Degenaar & Ho 2012; Mahmoodifar & Strohmayer
2013G) Wtorquebalancesuppliesausefulupperlimittocal- Q22 ≈1.3×1035R64(cid:18)1δ0T5Kq (cid:19)(cid:18)30MQeV(cid:19)3g cm2, (2)
ibratesearches.Howeverifitisnotthedrivingforcebehind
pulsar spin evolution, it is natural to ask at what level the where R6 is the stellar radius in units of 106 cm, δTq is the
physical mechanisms mentioned above will give rise to GW quadrupolar componentofthetemperaturevariation dueto
emission, andwhetherit islikelytobedetected.Thisques- nuclear reactions and Q is the reaction threshold energy.
Higher threshold energies correspond to higher densities.
tion is crucial, given that Wattset al. (2008) showed that
evenatthetorquebalancelevelthesesystemswouldbechal- In general the reactions will heat the region by an amount
lenging to detect. In this paper we explore the non-torque- (Ushomirsky & Rutledge 2000):
bsuaplapnocretesdceenitahreiorbinymeloarseticdietytaoilr. mWaegfnoectuicssotnre“ssmeso,uanntadindsi”s-, (10δ6TK) ≈Ck−1p−d1Qn∆M22, (3)
Detecting gravitational waves from mountains on neutron stars in the Advanced Detector Era 3
Source ν d hM˙i ∆t Ref.
(Hz) (kpc) (10−10M⊙yr−1) (d)
SAXJ1808.4–3658 401 3.5 4 30 Patrunoetal.(2009)
XTEJ1751–305 435 7.5 10 10 Milleretal.(2003)
XTEJ1814–338 314 8 2 60 thiswork
IGR J00291+5934 599 5 6 14 Falangaetal.(2005)
HETEJ1900.1–2455 377 5 8 3000 Papittoetal.(2013b)
AqlX-1 550 5 10 30 Gu¨ng¨or,Gu¨ver&Eksi(2011)
SwiftJ1756.9–2508 182.1 8 5 10 Krimmetal.(2007)
NGC6440X-2 204.8 8.5 1 4 thiswork
IGR J17511–3057 244.9 6.9 6 24 Falangaetal.(2011)
IGR J17498–2921 400.9 7.6 6 40 Falangaetal.(2012)
SwiftJ1749.4-2807 518 6.7 2 20 Ferrignoetal.(2011)
EXO0748–676 552 5.9 3 8760 Degenaaretal.(2011)
4U1608–52 620 3.6 20 700 Gierlinski&Done(2002)
KS1731–260 526 7 11 4563 Narita,Grindlay&Barret(2001)
SAXJ1750.8–2900 601 6.8 4 100 thiswork
4U1636–536 581 5 30 pers. thiswork
4U1728–34 363 5 5 pers. Egronetal.(2011)
4U1702–429 329 5.5 23 pers. thiswork
4U0614+091 415 3.2 6 pers. Pirainoetal.(1999)
Table 1.LMXBs forwhich wehave obtained an estimate of the outburst duration ∆t andaverage accretion rate
hM˙i. Where the reference column indicates ‘this work’, we have used a fiducial power law index of Γ = 2 and
the galactic absorption column from Kalberlaetal. (2005). We also listthe distance d of the system and the spin
frequency ν. Sources in the top half of the table are AMXPs, while those in the bottom half are nuclear powered
pulsarsandtheirfrequencyisinferredfromthefrequencyofburstoscillations,asexplainedinthetext. Wedonot
attempt to explicitly estimate the errors associated with these measurements. The most uncertain quantity is the
distance, but our mainconclusions on the detectability of the GW signals are unlikely to change unless there is a
substantialerrorinthevaluesabove.
where C is the heat capacity per baryon in units of the system, the GW strain would be comparable to that of a
k
Boltzman constant k , p is the pressure, in units of 1030 persistent system.
B d
erg cm−3, at which the reaction occurs, Q is the heat
n
per unit baryon (in MeV) deposited by the reactions and
∆M22 isthedepositedmassinunitsof1022 g.NotethatδT 2.2 Maximum quadrupole
in (3) is the total increase in temperature; only a fraction
δT /δT ≪1islikelytobeasymmetricingeneralandspecif- Large stresses can break the crust, so one should
q
ically quadrupolar. Ushomirsky,Cutler & Bildsten (2000) also ask how large a mountain the star can sustain.
estimatethatδT /δT 60.1,butinrealitytheratioispoorly This problem has been studied by different authors in
q
known. Newtonian physics (Ushomirsky,Cutler & Bildsten 2000;
After an accretion outburst, as the system returns to Haskell, Jones & Andersson 2006) and, more recently, in
quiescence, thedeformations are erased on thecrust’s ther- general relativity (Johnson-McDaniel & Owen 2013). The
mal timescale (Brown, Bildsten & Rutledge1998): results depend critically on the breaking strain σ¯ of the
c
τ ≈0.2p3/4 yr (4) crust, i.e. the average strain σ¯ = T¯/µ that can be built up
th d beforethecrustcracks,whereµistheshearmodulusofthe
If the system is in quiescence for longer than the thermal crustand T¯ theaveragestress. Thebreakingstrain ofaNS
timescale in (4), Q22 is likely to be washed out and a new crust is not well known, but is known to be σ¯c ≈ 10−2 for
mountain is rebuilt during the next outburst. A shorter re- perfect crystals in a laboratory setting, and recent molec-
currencetime,ontheotherhand,couldleadtoanincremen- ular dynamics simulations have shown that it may reach
talaccumulationofmaterial.However,compositionalasym- σ¯ ≈10−1inNScrusts(Horowitz & Kadau2009).Themax-
c
metries may be frozen into the crust, and not be erased imumquadrupolesthatcanbesustainedarethusofthethe
on a thermal timescale, allowing for the mountain to be order of Q22 ≈ 1038 −1039 g cm2 for more massive stars
built incrementally (Ushomirsky,Cutler & Bildsten 2000). (M ≈ 2M⊙) and Q22 ≈ 1039−1040 g cm2 for less massive
Thisscenariopredictstheformation oflargequadrupolesin stars (M ≈ 1.2M⊙), depending on the exact equation of
alltransientsystems(1038 .Q.1040 gcm2),aswediscuss state.
insection2.4.Theimpliedspin-downrate,inthecaseoffour
transientsystems(SAXJ1808.4–3658,XTEJ1751–305,IGR
J00291+5934andSWIFTJ1756.9–2508)isalreadyexcluded
2.3 Gravitational radiation
by measurements of the spin-down rate between outbursts
(Patruno & Watts 2012). We do not consider this scenario It is natural to ask, for the currently known LMXBs, how
further, but note that if it were to occur in any transient large a thermal mountain can grow and if it is detectable
4 B.Haskell et al.
bycurrentandnextgenerationinterferometers,suchasAd- 2.4 Transient sources
vanced LIGO or the Einstein Telescope (ET). To answer
In the left panel of figure 1 we show the GW strain corre-
these questions let us examine the LMXBs whose spins are
sponding to the maximum mountain that could be created
known. They can be divided in two classes: the Accreting
during an outburst [equations (2) and (1)], assuming that
Millisecond X-ray Pulsars (AMXPs), which are detected as
δT /δT = 0.1. We consider this to be a reasonable upper
pulsars and can thus be timed, and the Nuclear Powered q
limit,asasignificantlylargerfractionδT /δT wouldleadto
(NP) pulsars which do not pulsate, but exhibit quasi peri- q
detectable pulsations in quiescence for some of the sources,
odicoscillationsinthetailsoftypeIInuclearbursts.Thefre-
as we shall see in section 2.6. Note, however, that there is
quencyof these oscillations is ameasure of thespin period,
currentlynophysically motivated estimate for δT /δT, and
asconfirmedbyobservationsofburstoscillations insources q
the true value may be much smaller. We consider two cap-
that are also detected as X-ray pulsars (Patruno& Watts
turelayers,ashallowoneclosetoneutrondrip(wheremost
2012).ThedetailsoftheLMXBsweusearepresentedinta-
of the heat is predicted to be deposited (Haensel & Zdunik
ble1.Theotherquantitieslistedin table1arethedistance
1990)) with athreshold of Q=30 MeV,and a deeperlayer
tothesource and additionally theaverage duration ∆t and
average mass accretion rate < M˙ > during outbursts. The at a pressure of p = 1032 dyne/cm2, with a threshold en-
ergy of Q = 90 MeV. All values lie below the maximum
amount of mass that is accreted during an outburst can
then be obtained as ∆M =< M˙ > ∆t, and inserted into quadrupole that the crust can sustain. Note also that the
increase in the quadrupole Q22 due to the higher thresh-
equations (3) to calculate the temperature increase due to
old energy Q is more than offset by thedecrease in heating
nuclear reactions in the crust.
at higher pressures, as obtained from eq. (3). The results
To calculate the average mass accretion rate for all for the deep and shallow capture layers are thus very sim-
sources we followed two different approaches. For AMXPs ilar. The thermal timescale for the deeper capture layers
we used the data collected by the Rossi X-ray Timing Ex- is, however, τ ≈ 6 years. Hence a ‘deep’ mountain may
th
plorer (RXTE)andrecordedwiththeProportionalCounter never relax entirely in systems such as Aql X-1 that have
Array(PCA;seeJahoda et al.2006).WeusedtheStandard- frequent outbursts, with recurrence times shorter than τ .
th
2datamodeandextractedthe2–16keVX-rayfluxesforall These systems may effectively behave as persistent sources
outbursts, following the procedure by van Straaten et al. for our purposes, and harbour larger mountains.
(2003). The fluxes for each outburst were averaged assum- Intheleft paneloffigure(1)wecompareourresultsto
ingafiducialspectral index(i.e.assuming thatthespectral thesensitivityachievedbyAdvancedLIGO(assumingboth
index remains constant during the course of the outburst detectorshavethesamesensitivity)andET,firstbyassum-
and between different outbursts) taken from the literature. inganintegration timeof1month(anaveragedurationfor
Wethen extrapolated to the 0.1-100 keV (bolometric) flux. anoutburst)andthenof2years.Itisquiteclearfromthefig-
Weusedtheunabsorbed luminosity (wherewetakethehy- urethat,evenfora2yearintegration,mostsystemsfallwell
drogen absorption column NH reported in the literature, belowthesensitivitycurve.StrainsensitivitycurvesforAd-
seetable1).Thebolometricluminositywasthencalculated vancedLIGOandETarerespectivelytakenfromthepublic
from the distance in table 1 and the mass accretion rate LIGO document1 and Hild, Chelkowski & Freise (2008). A
as given by Lacc = GMM˙ R = M˙ηc2. Here, Lacc is the fully coherent search overtime, Tobs, issensitive toa strain
bolometric accretion luminosity, and we assumed a mass of of
M = 1.4M⊙, and a radius R = 10 km and η is the con-
version efficiency for therest-mass into energy. Aftercalcu- h≈11.4 Sn(ν)/Tobs, (5)
lating theaverage mass accretion rate for each outburst we p
whereS (ν)isthedetectornoisepowerspectraldensity,and
selected(andreportedintable1)thehighestvalueobtained n
thefactor 11.4 accountsfor asingle trial false alarm rateof
(i.e., we consider the biggest possible mountain).
1% and a false dismissal rate of 10% (Abbott et al. 2007a;
For the nuclear-powered accreting pulsars we used in- Wattset al. 2008).
steaddatafromtheAll-SkyMonitor(ASM)onboardRXTE, It is unlikely that transient systems will be strong
whichoperatedinthe1.3–12.1keVband.WeusedtheASM enoughsourcesforAdvancedLIGO,buttheyarepromising
rather than the PCA because all the 8 sources selected are sources for ET. This is essentially the same conclusion of
either persistent sources or have long outbursts. The ASM, Wattset al. (2008), who considered emission at the torque
beingamonitoringinstrument,hasamuchbetterdatacov- balance level, which is higher than the strain we calcu-
erage(althoughwithlowersensitivityandanarrowerenergy late[note thatboth ourestimates and thoseof Watts et al.
band). In this case we selected the absorption column NH (2008) assume mountains that are smaller than the max-
and spectral index Γ from the literature (whenever avail- imum that the crust can sustain before breaking]. A few
able) or, when no spectral analysis was available, used the systems appear to be close to the threshold for detection.
galactic NH and a simple power law model with spectral However these systems are unlikely to be good targets for
index Γ=2. upcoming GW searches, as they have all just entered qui-
escenceafterlongoutbursts,duringwhichlarge amountsof
Wecautionthattheresultsmaysufferfrom systematic
errors in both distance d and spectral index Γ. However mass were accreted and the crust was heated considerably.
The mountain is currently relaxing on a timescale τ and
the estimates are likely to be sufficiently accurate for our th
therecurrencetime between accretion outburstsis likely to
purposes.Themainconclusionsofthispaperwillnotchange
unless there is a substantial error in our assumptions that
can change the mass accretion rate by orders of magnitude
(e.g., a substantial error in thedistance). 1 https://dcc.ligo.org/LIGO-T0900288/public
Detecting gravitational waves from mountains on neutron stars in the Advanced Detector Era 5
1e-24 1e-23
Advanced LIGO (2 yr integration)
1e-25 ET (2 yr integration)
1e-24 Persistent - Maximum Mountain
KS 1731 Persistent - Spin Equilibrium
1e-26
EXO 0748
4U 1608 1e-25
h 1e-27 HETE J1900.1 h

n n
i1e-28 i1e-26
a ET - 2 year integration a
r r
t Adv Ligo- 2 year integration t
s1e-29 Adv Ligo- 1 month integration s
Q=30 (shallow layer) 1e-27
1e-30 Q=90 (deep layer)
ET - 1 month integration
1e-28
1e-31
1e-32 1e-29
10 100 1000 10000 10 100 1000 10000
GW frequency (Hz) GW frequency (Hz)
Figure1.GravitationalwavestrainvsfrequencyformountainsinAMXPsandNPpulsars.Intheleftpanelweshowtransientsources,
forwhichthemountainisthelargestthatcanbecreatedduringanoutburst,bothinthecaseofashallow(Q=30MeV)andofadeep
(Q=90 MeV)capture layer. Intherightpanel weshow thepersistent sources,forwhichweassumeboth themaximum mountainthe
crustcansustain(crosses),andamountainthatwouldgivespinequilibriumfromtorquebalance(solidtriangles).Thebarsindicatethe
rangegivenbyuncertainties onthebreakingstrain,asdescribedinthetext.
belong.Itisthusprobablethattheywillnotbe‘on’ascon- require δT /T > 1, even for deeper capture layers. Such
q
tinuousGW sources duringAdvancedLIGO observations. high values of δT /T > 1 would lead to pulsations in qui-
q
escence at a level that is not observed. However for lower
values of Q22, deeper capture layers and higher activation
2.5 Persistent sources
energy Q, the temperature perturbation is δT /T 6 0.1.
q
For the persistently accreting sources the situation is dif- During accretion outbursts the resulting perturbations to
ferent.Weassumethat ongoing accretion buildsthelargest the luminosity are δLbol . 1032ergs−1, and are not visible
mountainthatcanbesustained.Wetakethequadrupoleto (Ushomirsky,Cutler & Bildsten 2000),astheemission isat
be in the range 1038g cm2 . Q22 . 1040g cm2, to account muchhigherlevels(Lbol =Lacc ≃1035−1037ergs−1).How-
for the uncertainty in mass and equation of state, as es- eversuchlevelsofheatingcanmakethequiescentfluxvary,
timated by Johnson-McDaniel & Owen (2013). The results as we argue below.
are shown in the right hand panel of figure 1. The error Wecansummarisethediscussionabovebyaskingwhat
barsaccountfortherangediscussedabove.Wealsopresent a GW detection implies for deep crustal heating. We use
the torque balance upper limits on Q22, as in Watts et al. equations (1) and (2) to represent the sensitivity curve of
(2008).Theresultsforthemaximummountaincomfortably AdvancedLigoandETintermsofanequivalentquadrupo-
exceedthetorquebalancelimits.Ifaccretionisongoing,the lar temperature deformation δT , as shown in figure 2. We
q
quadrupole can thus become larger than the value needed consider two fiducial systems at a distance d = 5 kpc:
for torquebalance. In thisscenario thereis thusanet spin- a system that undergoes shorter outbursts and is colder
downtorqueduetoGWemission,andthepredictionforthe (T =5×107 K),forwhichweintegratetheGWsignalover
spin-down rate is: thefiducialdurationoftheoutburst(1month);andahotter
(T = 5×108 K), persistent system for which we integrate
ν˙ ≈6×10−13(cid:16)500νHz(cid:17)5(cid:18)1038Qg22cm2(cid:19) Hz/s, (6) tvhaencGedWLsIiGgnOalaonvderEaT2yweiallrppreorbioedt.hFeigδuTre/2Tsh≈ow0s.1thraegtiAmde-,
q
(where we have assumed a moment of inertia I = 1045 g with ET probing the possibly more realistic δTq/T 6 0.01
cm2 for the star). Such spin-down is sufficiently strong to regime.Thisisalsotheorderofmagnitudeoftheperturba-
be detectable with current instrumentation. However, none tions expected in thequiescent flux(see section 2.6),which
of the persistent sources considered here have ever shown may be detectable with future X-ray satellites such as the
accretion powered pulsations that would allow us to test LargeObservatoryforX-rayTiming(LOFT)ortheNeutron
thisprediction.Continueddeepsearchesforpulsationsfrom star Interior Composition Explorer (NICER).
these objects is thus of significant importance for GW sci- In the analysis above we make many approxima-
ence. tions. First and foremost we only consider two capture
Anotherissuetoconsideristheamountofinternalheat- layers in the stars. In reality all layers contribute to
ing required to sustain a large quadrupole. Rearranging (3) Q22, leading to larger quadrupoles than those discussed
weseethat,forafiducialstarofradiusR=12km,onehas: above (Ushomirsky,Cutler & Bildsten 2000). For shorter
3 outburststhereducedheatingathigherdensitiesoffsetsthe
7 Q22 30MeV
δTq ≈3×10 (cid:18)1038g cm2(cid:19)(cid:18) Q (cid:19) K, (7) higher quadrupole Q22 in those regions. The reactions that
depositthemostheatthusdominate,independentlyofden-
ForhotsourceswithinternaltemperaturesT =108 K,high sity.Weareaccountingforwhatisconsideredtobethemost
values of the quadrupole (around Q22 ≈1039 g cm2) would important layer at neutron drip (Haensel & Zdunik 1990),
6 B.Haskell et al.
10 whereT isthetemperature,C theheatcapacity,Kthecon-
ALIGO (1 month) v
ductivity,ρthedensityandǫ=ǫ −ǫ ,withǫ theneutrino
ν h ν
emissivityandǫ theenergydepositionrate.Tosimplifyour
h
1
treatment and make a first assessment of detectability, we
ET (1 month) will use an n = 1 polytrope for the equation of state, and
analytic expressions for the contribution of electrons in the
0.1
Τ
Τ / q chreuastt(tMoatxhweecloln1d9u7c9t)i,viftryom(Fwlohwiechrso&neItoobhta1i9n8s1:)andspecific
δ 0.01
ALIGO (2 year) 16 ρ 1/3 T
K = 10 erg/(cm s K), (9)
(cid:18)106g/cm3(cid:19) (cid:18)108K(cid:19)
0.001 AAEETTLL::II GG21 OOyme::o a21nr ,tym hQe,o a=Qnr9,t= h0Q3,, =0QT,9 ==T053,= e0T58, =eT75=e58e7 ET (2 year) Cv = 3.72×1017(cid:18)ρρ0(cid:19)4/3(cid:18)10T8K(cid:19) erg/(cm3 K), (10)
0.000110 100 1000 10000 where ρ0 is the nuclear saturation density. We set
GW frequency (Hz)
ǫ = 0 (note this is true for the background, but
h
for the perturbations we will have δǫ 6= 0), and
Figure 2.SensitivityofcurrentandnextgenerationGWdetec- h
for the neutrino emissivity we approximate the results
tors to gravitational waves sourced by quadrupolar temperature
of Haensel, Kaminker& Yakovlev (1996) for νν¯ electron
deformations δTq in deep (Q = 90 MeV) and shallow (Q = 30
MeV) layers of the NS crust. The gravitational wave strain is Brehmstrahlung as:
expressedintermsofthetemperature perturbations δTq/T that 6
give rise to the mountain, as described in the text. Both 2nd ǫ =6.46×1018 ρ T cm2/s3 (11)
and 3rd generation detectors will probe regimes of physical in- ν (cid:18)1012g/cm3(cid:19)(cid:18)109K(cid:19)
terest, with Advanced LIGO probing the δTq/T ≈ 0.1 regime,
and ET the δTq/T ≈ 0.01 regime. We show the sensitivity of Attheboundarywiththecoreweassumeaconstanttemper-
Advanced LIGO both for a 1 month integration, Q = 30 MeV, atureandfor theouterboundarywe assumethat theemis-
and background temperatures T = 5×107 K (dotted curve), sionfromthesurfaceisthermal,i.e.−K∇T =(R2/R2)σT4,
∗ s
corresponding to the case of a short outburst, and for a 2 year with σ the Stefan-Boltzman constant. The stellar radius is
integration, Q = 90 MeV and a background temperature of
T =5×108 K(dashedcurve),moreappropriateforapersistent R, and R∗ is the radius at which we fix the outer bound-
aryof ournumericalgrid. Thesurface temperatureT at R
system.SimilarlyweshowthesensitivityofETforT =5×107K, s
Q=30MeV,anda1monthintegration(dot-dashed curve)and is then obtained from the temperature T at R∗ using the
T =5×108 K,Q=90MeV,anda2yearintegration(solidline prescription of Gudmundsson,Pethick & Epstein (1983):
curve). Theregionenclosedbytheredboxisthatmostrelevant
forLMXBs. (cid:18)10T6sK(cid:19)=g14(cid:18)18.110T9K(cid:19)2.42, (12)
so unless there is significantly more heating in deeper lay- with g14 the gravitational acceleration in units of 1014 cm
ers that previous calculations have not accounted for, it is s−2. Note that one can model the composition of the outer
unlikelythatourresultsseverelyunderestimateQ22.Ingen- layersin moredetail (see e.g. Haskell, Degenaar & Ho 2012
eral the result of our analysis is that thermal mountainson and Mahmoodifar & Strohmayer2013).However, given the
NSsintransientLMXBsarelikelytobeverychallengingto manysimplifying assumptions andtheuncertainties associ-
detect,evenwiththirdgeneration detectors.Persistent sys- atedwiththemeasurementsintable1,weusetheexpression
tems,however,offerapromisingtargetandelectromagnetic in(12),asitisunlikelytobethemainsourceoferrorinour
observationsmayallowfurtherconstraintsonthephysicsof analysis.
thesystem, as we shall see in section 3.4. We obtain our background model by specifying a core
temperatureattheinnerboundary(thecrust/coreinterface)
and evolving equation (8) until we obtain an equilibrium.
2.6 X-ray flux variations We are now ready to evolve the quadrupolar temperature
perturbationsduetothemountainonthisbackground.The
We now focus on the observable X-ray flux variations in-
evolution equation for an l=m=2 perturbation takes the
duced by a thermal perturbation due to a mountain in the
form (Ushomirsky,Cutler & Bildsten 2000):
crust.Asalready discussedwerestrictourattentiontoper-
turbations of the thermal quiescent emission. We thus as- C ∂δTq = − 1 ∂ r2K∂δTq −l(l+1)KδTq +
sumethatamountainhasbeencreatedduringanaccretion v ∂t r2∂r (cid:18) ∂r (cid:19) r2
outburst, and study how the associated thermal perturba-
δK δǫ ∂ δK
tions evolve as the system returns to quiescence. To un- ρǫ − +F , (13)
(cid:18) K ǫ (cid:19) Q∂r(cid:18) K (cid:19)
derstand how the surface flux is affected we consider the
quadrupolar flux variations as linear perturbations on a with F = −K∇T the background flux obtained
Q
spherically symmetric background. We start by obtaining from the equilibrium solution of (8). We obtain δK
thespherical background model for thetemperature profile from equation (9) with the condition δρ = 0, as in
from the Newtonian heat transport equations in thecrust: (Ushomirsky,Cutler & Bildsten 2000). At the boundary
∂T with the core we assume that δTq = 0, while at the outer
Cv ∂t =∇(K∇T)−ρǫ, (8) boundaryweperturbthethermalfluxcondition,sothatone
Detecting gravitational waves from mountains on neutron stars in the Advanced Detector Era 7
has δF =4(R2/r2)σT3δTs, with
s q
(cid:18)1δ0T6qKs (cid:19)=2.42g14(cid:18)18.110T9K(cid:19)2.42 δTTq. (14) 10 ALIGO (1 month)
Wetakeδǫ=δǫ and assumeδǫ tobedueto quadrupolar 1
h h
energy deposition in the capture layers. For the deep cap- ET (1 month)
ture layer, we specify an energy deposition term δǫ with
h
a Gaussian radial profile, located at a pressure of P =1032 FQ 0.1
ldayyneer,/camnd2,aatnPdw=it1h03a0hdaylfnwe/idcmth2oafn5dmwiftohrtahheadlfeewpidctahptoufr1e δ F / 0.01
mfortheshallowlayer.Evolvingequation(13)wefindthat, ALIGO (2 year)
astheproblemislinearintheperturbations,toaverygood
approximation thefollowing relations hold: 0.001 ALIGO: 2 year, Q=90, T=5e8
ALIGO: 1 month, Q=30, T=5e7
δF δT ET: 2 year, Q=90, T=5e8 ET (2 year)
≈1.29 q, (deep layer) (15) ET: 1 month, Q=30, T=5e7
F T
Q 0.0001
10 100 1000 10000
δF ≈1.48 δTq, (shallow layer) (16) GW frequency (Hz)
F T
Q
with very little dependenceon the chosen background tem- Figure 3. The pulsed fraction δF/FQ corresponding to a GW
detection at threshold [as obtained from equation (5)] for Ad-
perature T. We remind the reader that we are normalising
vanced LIGO and ET. For deep capture layers (Q = 90 MeV)
to the quiescent (thermal) flux FQ obtained from the equi- and a background temperature of T = 5×108 K, we show the
librium solution of (8). results for an integration time of 2 years (dashed curve for Ad-
In quiescence the quadrupolar temperature perturba- vanced LIGO, solid curve for ET). Physically this is due to the
tions associated with a mountain and gravitational wave factthat inthe deep crust τth >2 years,and the mountain will
emission [equations(2)and(3)]thusperturbtheX-rayflux thus not relax significantly during the observation. For shallow
from the surface [equations (15) and (16)], and as the star capture layers (Q = 30 MeV) and T = 5×107 K, we show the
resultsforanintegrationtimeof1months(dottedcurveforAd-
rotates this leads to pulsations at twice the rotation fre-
vancedLIGO,dot-dashedcurveforET).Theregionenclosedby
quency(i.e.thesamefrequencyasthegravitational waves).
theredboxistheregionofinterestforLMXBs.
Infigure3weshowthesensitivitycurveforAdvancedLIGO
and ET in terms of an equivalent pulsed fraction of the X-
ray flux. We can see that if it is possible to integrate the timescale τ , but the slower Ohmic dissipation timescale
th
signal for 2 years (physically this corresponds to a capture τ >108 years (Vigelius & Melatos 2009b). Hence a moun-
o
layer deep enough that τth ≫ 2 years, and the mountain tainformsgraduallyoverseveraloutbursts.Grad-Shafranov
is not dissipated significantly during theobservation), both calculations indicate that the hydromagnetic structure of
AdvancedLIGO and ET can probe an interesting region of a mountain conforms to a single-parameter family of so-
parameter space, with δF/FQ .0.01. lutions which, once the size of the accreting polar cap is
fixed, are function only of the mass accreted over the sys-
tems lifetime, M . This suggests that magnetic mountains
a
can be treated as the persistent sources of the previous
3 MAGNETIC MOUNTAINS
section. The main difference is that the quadrupole does
3.1 Hydromagnetic evolution not depend on crustal rigidity, but on the magnetic field
strength when accretion begins, which we denote B∗ (note
Accretion not only perturbs the structure of the star by
that this is different from, and generally lower than, the
affecting nuclear reactions in the crust, but it also de-
expected NS magnetic field at birth, as obtained form pop-
forms the stellar magnetic field. As matter is accreted
ulation synthesis models (Faucher-Gigu`ere & Kaspi 2006)),
and spreads towards the equator it drags the field with
the initial field structure, and M (Melatos & Payne 2005;
a
it, and compresses it. This leads to a locally strong field
Payne& Melatos 2004; Priymak, Melatos & Payne2011).
that can sustain a “magnetic” mountain (Payne& Melatos
As more mass is accreted the external dipolar com-
2004; Melatos & Payne 2005; Vigelius & Melatos 2009b).
ponent of the field, B , is quenched according to
ext
Thiscanleadtomuchlargerdeformationsthanthosedueto
Shibazakiet al. (1989):
the overall background magnetic field, even if the internal
−1
tinofreorirdeadl ecxotmerpnoanlenmtaogfnetthiec fideipldoleis(mCiuoclhfi &strRonegzezrolltaha2n01t3h)e. Bext =B∗(cid:18)1+ MMac(cid:19) (17)
Recent calculations have shown that for realistic equations and themass quadrupoleis given by
of state the mountain could lead to a detectable GW sig-
−1
nal(Priymak, Melatos & Payne2011).Notealso thatmag- Q22≈1045 A Ma 1+ Ma g cm2 (18)
netic mountains are not sustained by crustal rigidity and (cid:18)M⊙(cid:19)(cid:18) Mc(cid:19)
the resulting quadrupole can thus be larger than the value where A ≈ 1 is a geometric factor that depends on the
required to crack thecrust. equationofstateandaccretiongeometry(Melatos & Payne
One of the main differences with respect to thermal 2005), while M is the critical amount of accreted mat-
c
mountains is that the time-scale on which a magnetic de- ter at which the mechanism saturates, which also de-
formation relaxes, after an outburst, is not the thermal pendson the equation of state (Priymak, Melatos & Payne
8 B.Haskell et al.
1e-24 1e-24
1e-25
1e-25
1e-26
1e-26
1e-27
h h

n n
i1e-27 i1e-28
a a
r r
t t
s s1e-29
1e-28 ET - 1 month integration ALigo - 2 year integration
ET - 2 year integration ET - 2 year integration
1e-30
ALigo - 1 month integration ET - 1 month integration
1e-29 ALigo - 2 year integration ALigo - 1 month integration
B birth = 10e10 G 1e-31 B birth=10e10 G
B birth = 10e12 G B birth=10e12 G
1e-30 1e-32
10 100 1000 10000 10 100 1000 10000
GW frequency (Hz) GW frequency (Hz)
Figure4.GWstrainvsfrequencyforthesystemsintable1,fortwomagneticmountainscenarios.Intheleftpanelweshowthestrain
that can be achieved assuming that the magnetic mountain does not decay between outbursts, for two values of the magnetic field,
B∗=1010 GandB∗=1012 G.Intherightpanelweconsiderthescenarioinwhichthemountaindecays betweenoutbursts.Detection
willbeverychallengingforbothAdvanced LIGOandET,unlessB∗≈1012 Gbeforetheonsetofaccretion.
2011). The estimates above are valid to leading order tion of the field (Konar& Bhattacharya 1997, 1999)]. This
in M /M ; for M ≈ M they are no longer accurate would lead to larger spin-down rates than those observed.
a c a c
and numerical solutions are necessary. General relations Ontheotherhand,ifthefieldremainsburiedandthemag-
for the critical mass were derived by Melatos & Payne netic mountain is stable on long timescales [as simulations
(2005) and Payne& Melatos (2004) for isothermal moun- by Vigelius & Melatos (2009b) indicate], then the results
tains, while for more realistic equations of state [models of Priymak, Melatos & Payne (2011) imply a quadrupole
C and E of Priymak, Melatos & Payne (2011)], one has Q22 >1036(B∗/1011G)4/3 g cm2 for Ma =Mc. From equa-
Mc ≈ 10−7(B∗/1012G)4/3M⊙. In the regime Ma ≫ Mc, tion (6), this gives a spin down rate ν˙ > 10−14 Hz/s for a
bothrelationsareexpectedtodeviatesignificantlyfromthe 500 Hzpulsar, close to themaximum spin down rates mea-
simple estimates above in (17) and (18). Numerical simula- suredformillisecondpulsars.HighinitialfieldsofB∗ &1011
tions cannot probe this regime; instead one finds that, for G would thuschallenge current observations.
Ma & Mc one has 0.01 6 Bext/B∗ 6 0.1 and quadrupoles
in the range 1037 . Q22 . 1038 g cm2, for an initial field
of B∗ = 1012.5 G. Note, however, that the main difficulty 3.3 Gravitational radiation
in pushing the simulations to M > 10M is numerical.
a c
The only firm upper limit on the suppression of the ex- Intheleft paneloffigure4weplot GWstrain vsfrequency
ternal dipole field come from Ohmic diffusion, which lim- for magnetic mountains that do not decay between out-
its the burial of the field at a level of Bext/B∗ ≈ 10−4 bursts. We consider first a scenario in which B∗ ≈ 1010
(Vigelius & Melatos 2009b). G and the critical mass M has been accreted over a sys-
c
tem’s lifetime (i.e. M = M ). The GW emission is pre-
a c
dictablyweak.Giventheuncertaintiesassociatedwithmod-
elling field burial, in figure 4 we also consider the case in
3.2 Pre-accretion magnetic field
which the birth field is B∗ ≈ 1012 G. In this case some of
WhatlimitscanwesetonB∗,thestrengthofthemagnetic the systems could be emitting detectable gravitational ra-
fieldattheonsetofaccretion?Observationalconstraintscan diation, and a detection would provide evidence for a high
be obtained from measurements of the spin down between degreeoffieldburial.Thelatterscenariocanbeexcludedin
outbursts for 4 systems [see Patruno & Watts (2012) and thethreesystems[SAXJ1808.4–3658,XTEJ1751–305,IGR
references therein], which are consistent with B ≈ 108 J00291+5934 (Patruno & Watts2012)]forwhichwehavea
ext
G.Themagneticfieldsinferredformillisecondradiopulsars measuredspindownbetweenoutbursts.Inallcasesthespin
are also in the range B ≈ 108 G. Furthermore observa- downrateisν˙ ≈−10−15 Hz/sanditimpliesanupperlimit
ext
tionsofaslow(11Hz)pulsarinTerzan5,IGRJ17480-2446, of Q22 ≈1036 gcm2,if weassume that GW emission is the
indicatethatthissystem,whichisthoughttohavebeenac- dominantspin-downmechanism.GWemission atthislevel,
cretingforashorterperiodoftimethanmostoftheLMXB due to a magnetic mountain, implies B∗ ≈ 5×1010 G and
population, may have a stronger field 109G . B . 1010 wouldbeunlikelytobedetected,ascanbeseenfromfigure
ext
G(Cavecchi et al.2011).Itisthusplausiblethatonestarts 4. For our models with Ma = Mc, B∗ ≈ 5×1010 G leads
with B∗ & 109 G, and that the external field is reduced to to Bext ≈ 2.5×1010 G (Priymak, Melatos & Payne 2011).
B ≈108 by accretion. Sucha strong dipolefield would, however,lead to agreater
ext
If B∗ & 1011 G, and polar magnetic burial is very than observed spin down due to magnetic dipole radiation.
short lived, we would expect Bext ≈ B∗ in the millisecond Infact,ifthespindownisattributedtodipoleradiation,the
radio pulsars [unless accretion leads to significant dissipa- implied magneticfield isB ≈108 Gforall thesesystems
ext
Detecting gravitational waves from mountains on neutron stars in the Advanced Detector Era 9
(Patruno & Watts2012).Anupperlimit of |ν˙|<2×10−15 1e+15
Hz/salsoexistsonthespin-downrateofSwiftJ1756.9-2508.
In this case the limit on the dipole field from electromag- 1e+14
netic spin down is of B .5×108 G butthe field needed
ext
to explain the spin down in terms of GWs from magnetic 1e+13
mountainsisB∗ ≈1012 GcorrespondingtoBext ≈5×1011 G)
GthofuogrhMthaat=whMilce sinimouulartimonosdienlsd.icIatteisthimatptohretaqnutadtorunpootlee B (ext1e+12
saturates for M & M (Wette,Vigelius & Melatos 2010),
a c 1e+11
no such effect is observed for the decay of the exter-
ET - 1 month
nal field, and the limits on evolving the field further are ET - 2 years
mainly numerical. One cannot thus exclude high degrees 1e+10 ALIGO - 1 month
ALIGO - 2 years
of field burial. In fact the harmonic content of thermo-
nuclear bursts suggests that in some systems burning oc- 1e+09
10 100 1000 10000
curs in patches and is confined by locally strong and com- GW frequency (Hz)
pressedmagneticfields(Bhattacharyya& Strohmayer2006;
Figure 5.ThesensitivityofAdvancedLIGOandETtoamag-
Misanovic, Galloway & Cooper 2010; Cavecchi et al. 2011;
neticmountain.TheGWstrainisexpressedintermsofthemag-
Chakraborty & Bhattacharyya2012).
netic field Bext of the star, for a fiducial system at 5 kpc and
We also analyse the scenario in which the magnetic
modelEofPriymak,Melatos&Payne(2011).WetakeMa=Mc
mountaindecaysonshorttimescalesbetweenaccretionout- and,asdescribedinthetext,onehasBext=B∗/2forthesemod-
bursts. Time-dependent MHD simulations show that mag- els. We plot both the case of a 1 month integration (dot-dashed
neticlinetyingatthestellarsurfacestabilisesthemountain curveforETanddottedcurveforAdvancedLIGO)anda2year
againstinterchangeinstabilities.Current-drivenParker-type integration (solid curve for ET and dashed curve for Advanced
instabilities do occur, but they do not disrupt the moun- LIGO).WecanseeAdvancedLIGOwillprobehighfieldscenar-
tain, saturating in a state where the quadrupole is reduced ios,with1011 G.Bext.1012G,whileETwillbeabletoprobe
by .60% (Vigelius & Melatos 2009b). Simulations confirm fieldsofBext<1011 G.
stability up tothetearing-modetimescales butthey donot
resolve slower instabilities and modes below the grid-scale. tering features should appear in the X-ray emission and
Differentchoicesofboundaryconditionscanalsodestabilise Priymak, Melatos & Lasky(2014)havestudiedtheproblem
thesystem(Mukherjee, Bhattacharya & Mignone2013a,b). in detail for thecase of an accretion buried field.
In this scenario we take Ma =∆thM˙i for each system, and We repeat the analysis here for a 1.4 M⊙ NS with an
calculate the quadrupole from equation (18). The results accreted outer envelope described by equation of state E
forthepredicted gravitational wavestrain are shownin the of Priymak, Melatos & Lasky (2014). We vary B∗ between
right panel of figure 4. This scenario leads to small moun- 1011Gand1012 GandstudytheemissionfeaturesforM =
a
tains and weak GW emission, that would be undetectable M . In figure 6 we show an example of the kind of spectra
c
formostsystems,evenforET.Theonlysystemsthatwould that such a setup produces. The solid line represents the
leadtodetectableGWsarethepersistentones,ifB∗ ≈1012 phase averaged spectrum, while the dotted lines represent
G. phase resolved spectra for two extreme rotational phases,
Infigure5weshowtheGWstrainexpressedintermsof ω = π/2 and ω = 3π/2. We can see that in all cases the
an equivalent Bext obtained from equation 18, using model energyofthefirstlineisfairlystable,butthedepthcanvary
E of Priymak, Melatos & Payne (2011) and Ma = Mc, for strongly with phase, as can the shape of the higher energy
whichBext =B∗/2.WecanseethatAdvancedLIGOisex- features.Astrongphasedependenceofthefundamentalline
pected to probe high field scenarios, with 1011 G . B∗ . fordifferentsizesofpolarmountainshasalsobeenfoundby
1012 G, while ET will probe a physically more realistic sec- Mukherjee, Bhattacharya & Mignone (2012).
tion of parameter space, i.e. B∗ <1011 G. Letusfocusonthephaseaveragedspectrum.Oursim-
ulations show that for B∗ .1012 G no cyclotron resonance
scattering features are present. The results for higher field
3.4 Distinguishing magnetic from thermal
strengths are shown in figure (7), where we plot the dif-
mountains
ference in depth between the first and second line and the
Aninterestingquestionisif,givenaGWdetection,itwould ratiobetweentheenergiesat whichthelinesappear,versus
bepossible tounderstand whetherwe are observinga ther- the pre-accretion magnetic field B∗. The effects are small,
mal or magnetic mountain. We have already discussed the but may be measurable by future X-ray observatories such
electromagnetic counterpart of a thermal mountain in sec- as NICER and LOFT, which will both be capable of re-
tion 2.6, and showed that a quadrupolar deformation could solving modulations of less than 1% at energies of ≈1 keV
lead to flux modulations and pulsations in quiescence at (Gendrau et al. 2012; Feroci et al. 2012). Furthermore the
twice the spin frequency. The results of the previous sec- cyclotron features appear to bemore pronounced in there-
tion suggest that if a magnetic mountain were to be de- gion of interest, i.e. the field strengths that would lead to
tected in a hypothetical system, such a NS would have GW emission at the Advanced LIGO and ET threshold.
a strong ’birth’ (i.e. at the onset of the LMCB phase) Thismethodthushasthepotentialtobeagood diagnostic
magnetic field B∗ ≈ 1012 G, although the external dipo- for distinguishing different kinds of continuous GW emis-
lar field may be lower, due to accretion induced magnetic sion. Additionally instruments such as NICER and LOFT
burial. In such a circumstance cyclotron resonance scat- will also be able to carry out phase resolved spectroscopy,
10 B.Haskell et al.
7 4.5
6 4
5
3 3.5
-0 4
1
)/1 3 E1 3
D /2
- 2 2 E 2.5
D 2
( 1
0 1.5
-1 1
1 12 10 1 12 10
B /(10 G) B /(10 G)
* *
Figure 7. Difference in depth D in normalised flux units (left panel) and ratio between the energies (right panel) of the second and
first cyclotron line for a 1.4M⊙ NS obtained with model E of Priymak,Melatos&Lasky (2014), for varying field strengths B∗. The
differentcoloursrepresentdifferentinclinationsoftheobserver(ι)andofthemagneticfiledaxis(α)withrespecttotherotationalaxis:
ι = 0,α = 0 (blue squares), ι = 0,α = 0.5π (green diamonds), ι = 0.5π,α = 0.5π (red triangles), and ι = 0.25π,α = 0.25π (orange
circles).ForB∗<1012 thefeaturescannotbedistinguished.
currently in quiescence) were to become visible and emit
1
detectable GWs, the presence of a cyclotron line would
point to a magnetic mountain. Its absence, on the other
0.999 hand,combined with theestimates in section 5,would sug-
x
u gest that the quadrupole is more likely to be due to ther-
Fl mally generated crustal mountain [although mountains in
d 0.998
e the core of the star are also a possibility (Haskell et al.
z 2007)]. Especially for weaker fields, however, several com-
mali 0.997 binations of orientation and inclination could lead to cy-
r clotron resonance scattering features not being detectable
o
N (see Priymak, Melatos & Lasky (2014) for an in depth dis-
0.996
cussion), so theirabsence is inconclusive.
0.995
0 10 20 30 40 50 60
4 CONCLUSIONS
Energy (keV)
In this paper we asses the likely gravitational wave sig-
Figure 6. Example of a cyclotron spectrum, obtained with the
nal strength and detection prospects for deformations, or
code of Priymak,Melatos&Lasky (2014) for a M = 1.4M⊙ ‘mountains’onneutronstarsinLMXBs.Unlikemostprevi-
NS described by EOS E, with the following parameters: ι =
ous work on this topic we do not assume that the grav-
π/4 (observer inclination relative to the rotation axis), α =
itational wave spin-down torque has to balance the ac-
π/4 (inclination of the magnetic axis relative to the rotation
axis), B∗ = 1012.8 G, Ma = Mc = 3.01426×10−7M⊙ (see cretion induced spin-up torque, as several studies have
Priymak,Melatos&Lasky(2014)forafulldescriptionofthepa- indicated that this is unlikely to be the case for many
rameters).Thesolidlinerepresentsthephaseaveragedspectrum, systems (Andersson et al. 2005; Haskell & Patruno 2011;
while the dashed and dotted lines represent the phase resolved Patruno, Haskell & D’Angelo 2011). Rather, we calculate
spectrafortwoextremerotationalphases,ω=π/2(dashedline) the gravitational wave signal strength due to the two main
andω=3π/2(dottedline).Whiletheenergyofthelinesremains mechanisms that havebeen suggested for building a moun-
fairlyconstantthedepthvariessignificantlywithphase.Theflux
tain: asymmetric thermal deposition in the crust (thermal
isnormalisedtogiveunitpeakflux.
mountains)andmagneticallyconfinedmountains(magnetic
mountains). We calculate the gravitational wave strain for
both mechanisms in known LMXBs for which we can mea-
allowingforamuchmoredetailedcharacterisationofthecy- sure the spin frequency, average accretion rate during out-
clotronresonancescatteringfeaturesinthesesystems,which burstsand outburst duration.
can vary significantly with phase, as illustrated in figure 6. Oneofthemainuncertaintiesisthetimescaleonwhich
We stress here that no cyclotron lines have been de- themountainisstableonceaccretion ceasesandthesystem
tectedinLMXBscontainingneutronstarsrotatingwithmil- enters quiescence. For thermal mountains it is likely that
lisecondperiods,andthefieldsofthesesystemsaregenerally thequadrupolewilldissipateonathermaltimescaleτ .6
th
thought to be reasonably weak (B ≈ 108 G). Neverthe- years,leadingtolargemountainsonlyinpersistentlyaccret-
ext
less if an as yet unobserved system (e.g. a system that is ingsystems.InthisscenariotheGWsignalstrengthformost