Supernova Seismology: Gravitational Wave Signatures of Rapidly Rotating Core Collapse

1.1 MB·English

by  Jim Fuller

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed30April2015 (MNLATEXstylefilev2.2)
Supernova Seismology: Gravitational Wave Signatures of
Rapidly Rotating Core Collapse
Jim Fuller1,2,(cid:63) Hannah Klion1, Ernazar Abdikamalov3, and Christian D. Ott1
1TAPIR, Walter Burke Institute for Theoretical Physics, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA
2Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106, USA
3Physics Department, School of Science and Technology, Nazarbayev University, 53 Kabanbay Batyr Ave., Astana, 010000, Kazakhstan
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30April2015
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r ABSTRACT
p
A
Gravitational waves (GW) generated during a core-collapse supernova open a
window into the heart of the explosion. At core bounce, progenitors with rapid core
9
2 rotation rates exhibit a characteristic GW signal which can be used to constrain the
properties of the core of the progenitor star. We investigate the dynamics of rapidly

] rotating core collapse, focusing on hydrodynamic waves generated by the core bounce
E
andtheGWspectrumtheyproduce.Thecentrifugaldistortionoftherapidlyrotating
H proto-neutron star (PNS) leads to the generation of axisymmetric quadrupolar oscil-
. lations within the PNS and surrounding envelope. Using linear perturbation theory,
h
we estimate the frequencies, amplitudes, damping times, and GW spectra of the os-
p
cillations. Our analysis provides a qualitative explanation for several features of the
-
o GW spectrum and shows reasonable agreement with nonlinear hydrodynamic simu-
r lations, although a few discrepancies due to non-linear/rotational effects are evident.
t
s ThedominantearlypostbounceGWsignalisproducedbythefundamentalquadrupo-
a laroscillationmodeofthePNS,atafrequency0.70kHz(cid:46)f (cid:46)0.80kHz,whoseenergy
[
is largely trapped within the PNS and leaks out on a 10 ms timescale. Quasi-radial
∼
2 oscillations are not trapped within the PNS and quickly propagate outwards until
v theysteepenintoshocks.BoththePNSstructureandCoriolis/centrifugalforceshave
1 astrongimpactontheGWspectrum,andadetectionoftheGWsignalcantherefore
5 be used to constrain progenitor properties.
9
6 Key words: supernovae, gravitational waves, waves, oscillations
0
.
1
0
5
1 INTRODUCTION down, shedding its remaining kinetic energy in a few pul-
1
sations, then settles to its new postbounce equilibrium and
: Rotating iron core collapse in a massive star (M (cid:38) 8M ,
v (cid:12) becomes the core of the newly formed proto-neutron star
resulting in a core-collapse supernova [CC SN]) was one of
i (PNS). The entire rotating bounce–ring down process in-
X thefirstpotentialsourcesofgravitationalwaves(GWs)con-
volves rapid changes of the inner core’s quadrupole mo-
r sidered in the literature (Weber 1966; Ruffini & Wheeler mentandtremendousaccelerations.TheresultingGWburst
a 1971; see Ott 2009 for a historial overview). GWs are of
signal has been investigated extensively both with ellip-
lowest-orderquadrupolewavesandrotationnaturallydrives
soidalmodels(e.g.,Saenz&Shapiro1978)andwithdetailed
quadrupole deformation (oblateness) of the homologously
multi-dimensional numerical simulations (e.g., Mu¨ller 1982;
(v ∝ r) collapsing inner core of a rotating massive star.
Mo¨nchmeyer et al. 1991; Yamada & Sato 1995; Zwerger &
When the inner core reaches nuclear densities, the nuclear
Mu¨ller 1997; Dimmelmeier et al. 2002; Kotake et al. 2003;
equation of state stiffens, stopping the collapse of the in-
Ott et al. 2004; Dimmelmeier et al. 2008; Obergaulinger
nercore.Thelatterovershootsitsnewequilibrium,bounces
et al. 2006; Ott et al. 2007, 2012).
back (a process called “core bounce”) and launches the hy-
drodynamic supernova shock at its interface with the still Based on this extensive volume of work, it is now clear
collapsing outer core. Subsequently, the inner core rings that rotating core collapse proceeds mostly axisymmetri-
callyandthatnonaxisymmetricdynamicssetsinonlywithin
tens of milliseconds after bounce (Ott et al. 2007; Schei-
(cid:63) Email:[email protected] degger et al. 2008, 2010; Kuroda et al. 2014). Only rapidly
(cid:13)c 0000RAS
2 J. Fuller et al.
rotating iron cores (producing PNSs with central spin pe- 4.5
riods (cid:46) 5ms) generate sufficiently strong GW signals from
4.0
core bounce to be detected throughout the Milky Way by Bounce
Advanced-LIGO-classGWobservatories(Aasietal.(LIGO 3.5
Scientific Collaboration) 2014; Ott et al. 2012, hereafter 3) 3.0
−
O12). Since the cores of most massive stars are believed to m
c 2.5
beslowlyrotatingatcorecollapse((cid:38)90%,e.g.,Hegeretal. g
2005;Ottetal.2006;Langer2012),thedetectionofGWsof 140 2.0
1
a rotating core collapse event may be exceedingly rare and ( 1.5
GW emission in CC SNe may be dominated by neutrino- ρc
1.0
drivenconvectioninstead(e.g.,Mu¨ller&Janka1997;Mu¨ller
et al. 2004; Ott 2009; Kotake 2013; Ott et al. 2013; Mu¨ller 0.5 ρc(1014gcm−3)
etal.2012,2013;Murphyetal.2009).However,ifarapidly
rotating core collapse event were to be detected, it could 200
possibly be linked to an energetic CC SN driven by mag- RingDown
netorotationalcoupling(e.g.,Burrowsetal.2007;Takiwaki 100
et al. 2012; Mo¨sta et al. 2014). Abdikamalov et al. (2014)
(A14 hereafter) have shown that the angular momentum of )m 0
c
the inner core can be measured from the observed GW sig- (
D
nal. + 100
h −
The morphology of the GW signal from rotating core
collapse, bounce, and ring down is uniform across the en- 200 BounceSignal
−
tire parameter space of plausible initial conditions (Dim-
melmeieretal.2007,2008,O12)andessentiallyindependent 300 h+D(cm)
−
ofprogenitorstarmass.Itconsistsofafirstprominentpeak
5 0 5 10 15 20 25
associated with core bounce (the “bounce signal”, cf. Fig- − t t (ms)
bounce
ure1)andalower-amplitudebutlongerdurationoscillatory −
ring down signal, persisting for ∼ 20ms after bounce. The
ringdownsignalispeakedataGWfrequencyof0.7−0.8kHz Figure1. Top:CentraldensityρcoftherapidlyrotatingA3O05
(somewhatdependentonequationofstateandrotationrate, simulationofA14.Timeismeasuredfromthemomentofbounce,
Dimmelmeieretal.2008;O12;?,inprep,hereafterK15)and definedasthemomentwhenρcpeaks.Bottom:GWstrainfrom
the same simulation, computed at a distance D = 10kpc. The
may be correlated with variations in the early postbounce
large amplitude signal near bounce is the “bounce signal” cre-
neutrino luminosity (O12), suggesting that the ring-down
atedprimarilybythequasi-radialbounceofthecentrifugallydis-
oscillations of the PNS (here defined as the inner 20km of
tortedinnercore(O12,A14).Thesmalleramplitude“ringdown”
the postbounce star) are connected to the excitation of a oscillationsafterbouncearecreatedprimarilybyquadrupolaros-
global PNS oscillation mode at core bounce (O12). cillationsofthePNS.
WhiletheGWsignalofrapidlyrotatingCCSNecanbe
straightforwardly computed from complex nonlinear multi-
dimensional hydrodynamic simulations, its features are not
understood at a fundamental level. The goal of the present
investigationistoprovidesuchabasicunderstandingofthe
signal features. To do this, we employ semi-analytical, lin-
ear calculations of the wave-like fluid response produced by
the bounce of the inner core. These calculations shed light ofatrainofaxisymmetricquadrupolarwaves.Someofthese
on the physical mechanisms responsible for the GW signals wavesarereflectedattheedgeofthePNS,causingthemto
discussedabove,andtheirsimplicitycomplementsthecom- interfere to create standing waves, whose energy is peaked
plexityofthesimulationresults.However,ourmethodsonly neartheoscillation“modes”ofthePNS.ThedominantGW
provide a qualitative explanation for GW signals, the sim- signalisproducedbythefundamentalPNSoscillationmode
ulations are needed for precise quantitative predictions. Al- at a frequency f ∼0.75kHz, accounting for the most
f−mode
though oscillations of PNSs have previously been examined prominentpeakintheGWringdownsignal.TheGWsignal
in several papers (see, e.g., Ferrari et al. 2003, 2004), these damps on 10ms timescales as the wave energy leaks out of
worksfocusonPNSoscillationmodeswellafter((cid:38)100ms) the PNS into the surrounding envelope.
bounce, and they do not investigate the physics of waves Our paper is organized as follows. In Section 2 we in-
excited by the bounce itself. troduce our semi-analytical framework for calculating the
We find that the postbounce fluid response has a few spectrum of waves excited at core bounce, and we discuss
distinguishing characteristics. First, the bounce excites a the properties of the resulting waves. Section 3 investigates
train of radial, outwardly propagating acoustic waves. Be- the subsequent wave damping, and discusses the complica-
causethebackgroundstructureiscentrifugallydistortedby tions introduced rotational and relativistic effects. In Sec-
rotation, these waves are only quasi-radial and contain a tion 4 we present the GW spectra produced by the waves
quadrupolemoment,allowingthemtoemittheGWrespon- and compare with the GW spectra seen in simulations. We
sibleforthebouncesignaldiscussedabove.Second,thecen- conclude in Section 5 with a discussion of our results and
trifugal distortion of the progenitor leads to the excitation future avenues for theoretical development.
(cid:13)c 0000RAS,MNRAS000,000–000
Supernova Seismology 3
2 OSCILLATIONS EXCITED AT BOUNCE 102
As described above, the GW spectrum of a rapidly rotat- ρ(1013gcm−3) Ni(kHz)
ing supernova near core bounce consists of a bounce signal L2(kHz) Ω(kHz)
andaringdownsignal.Thebouncesignalhasashortdura- Nr(kHz) ωf(kHz)
tion in time and is thus characterized by a broad frequency
spectrum,whiletheringdownsignallastslongerandhasa 101
spectrum peaked at discrete frequencies. Our main goal is
tounderstandthephysicsoftheringdownsignal,although Ω
our methods also shed some light on the spectrum of the ,
N
bounce signal.
,
2
During CC, the inner core of a massive star progenitor L
collapsesintoaPNS,whiletheoutercoreformstheshocked ,ρ 100
region surrounding the PNS core during the on-going su-
pernova. In this paper, we refer to the inner r (cid:46) 30km
(ρ (cid:38) 1012gcm−3) as the PNS, while the low density sur-
roundingregionsaretheenvelope.Wechoosethisdefinition
becausethebounceexcitedwavescanbecometrappedinthe
inner∼30km.However,notethatatafewmsafterbounce, 10−1
the PNS only has a mass of M ∼0.6M while the mass
PNS (cid:12)
within the inner 300km is M∼1M(cid:12), and we therefore ex- 0 20 40 60 80 100
pectallofthematerialwithinourcomputationaldomainto r(km)
eventually accrete onto the central compact object. In the
brief(lessthanasecond)periodfollowingthePNSbounce,
Figure 2. Density(ρ),Brunt-V¨aisa¨l¨afrequency(N),Lambfre-
but preceding the supernova, the inner ∼100 km (i.e., re-
gions below the shock radius) of the supernova core is in quency(L2),andangularspinfrequency(Ω)profilesintheinner
100 km of the early postbounce evolution of model A3O04 of
approximate hydrostatic equilibrium (Janka 2001). A14. Positive values of N2 are denoted by the real part of the
Brunt-V¨ais¨al¨afrequency(Nr)whilenegativevaluesofN2 corre-
spond to imaginary values of the Brunt-V¨ais¨ala¨ frequency (Ni).
ThePNSoccupiestheinner∼30kmoftheremnant.Wavesnear
thef-modefrequencyhavef ∼0.8kHz,correspondingtoangular
2.1 Models frequenciesω=2πf ∼5kHz,markedbythehorizontalline.The
model is made by averaging the simulation output over 10 ms,
To generate background models for wave excitation and starting 3 ms after bounce. At these times, the shock is located
propagationwithinthepostbouncesupernovastructure,we nearr∼95km.
usesimulationoutputsgeneratedbyA14.Thesesimulations
arerunwiththe2-dimensionalversionoftheCoCoNuTcode
2.2 Wave Excitation and Computation
(Dimmelmeier et al. 2002, 2005) in axisymmetry and con-
formallyflatgeneralrelativity.Theseapproximationsareall Thesuddendecelerationatcorebounceexciteswaveswhich
appropriateforunderstandingtheaxisymmetricwavesofin- propagate within the PNS and surrounding material. Here,
terest.Inthepostbouncephase,aneutrinoleakage/heating we semi-analytically calculate the spectrum of waves ex-
scheme approximates the effects of neutrinos. We choose cited by the bounce. We use linear and adiabatic approxi-
snapshots of the supernova structure beginning 3 ms after mationsfordisplacementsfromthebackgroundstate,which
bounceandaveragethenext10msofevolution(sampledby weassumetobeinhydrostaticequilibrium.Wediscussnon-
snapshots every 1 ms) to determine the background struc- adiabatic and non-linear effects in Section 3. We also tem-
ture. This procedure smooths out most of the waves, tur- porarily ignore special/general relativistic effects and the
bulence, and other short-lived features present within the impact of Coriolis and centrifugal forces, which we address
supernova core without allowing for significant evolution of in Section 3.2
the background structure. Our fiducial model is the A3O04 Applying the approximations listed above, the lin-
model of A14. The rapid rotation of this model is suffi- earized momentum equation is
cient to generate a rotationally dominated GW signal, but ∂2 1 δρ
slow enough to be reasonably approximated by our semi- ξ=− ∇δP −∇δΦ−g ˆr+f(r,t) . (1)
∂t2 ρ ρ
analytical techniques described below.
Here, ξ is the Lagrangian displacement, ρ is the density, g
Figure 2 shows a density profile and propagation di-
isthegravitationalacceleration,δP andδρaretheEulerian
agram for the central 100 km of our fiducial model, aver-
pressure and density perturbations, and f is the force per
aged over the time period 3-12 ms after bounce. The inner
unitmassprovidedbythebounce.Wealsousethecontinuity
∼30kmofthestarcomprisethehighdensityPNS,whichis
equation
surroundedbyamuchlowerdensityenvelope.Theshockra-
diusisnearr∼95kmduringthistime,andtheshockdoes δρ+∇·(cid:0)ρξ(cid:1)=0 , (2)
notstronglyaffectthenatureofwavespropagatingnearthe
PNS.Inourmodels,thePNSisalwaysstablystratified;con- the adiabatic equation of state
vection driven by a lepton gradient does not develop until 1 ρN2
δρ= δP + ξ , (3)
later times, which we do not study here. c2 g r
s
(cid:13)c 0000RAS,MNRAS000,000–000
4 J. Fuller et al.
and Poisson’s equation do this, we decompose all perturbation variables into their
components per unit frequency, e.g.,
∇2δΦ=4πGδρ . (4)
(cid:90)
ξ(t)= dω(cid:48)ξ (ω(cid:48))e−iω(cid:48)t . (10)
Here, δΦ is the gravitational potential perturbation, c is ω(cid:48)
s
the sound speed, N is the Brunt-Va¨isa¨la¨ frequency, and G
Inserting this expression into Equation 1, multiplying by
is Newton’s gravitational constant.
eiωt, and integrating over time, we obtain
Waves are excited by the force applied during the
bounce of the inner core. For a spherically symmetric col- −ω2ξ + 1∇δP +∇δΦ +gδρωˆr=δf , (11)
lapse, the strength of the force will be comparable to the ω ρ ω ω ρ ω
amount of force required to halt the collapse of a shell of
with
material at radius r infalling at the escape velocity. Its di-
(cid:20) (cid:21)
rection will be radially outwards. A rough estimate of the δf =δf 2Y ˆr+r∇ Y , (12)
ω ω 20 ⊥ 20
magnitude of the force per unit mass is
and
f(r)∼gˆr , (5)
δf ≡ √A (cid:15)gt e−(ωtdyn/2)2. (13)
where g is the local gravitational acceleration in the hydro- ω dyn
2π
static postbounce material, and ˆr is the radial unit vector.
In Equation 11, each perturbed quantity is the perturba-
The force peaks at the moment of the bounce, which we
tion per unit frequency. Similar equations can easily be de-
define as t = 0. The duration of the bounce is approxi-
rived from Equations 2-4, and are given in Appendix A.
mately equal to the local dynamical time, t ∼ t =
bounce dyn
(cid:112) The Gaussian frequency dependence of the forcing term in
r3/GM(r). We approximate the time dependence of the
Equation 11 entails that only waves with angular frequen-
bounceasaGaussianofwidthequaltothedynamicaltime
cies ω (cid:46) 1/t will be strongly excited at a particular lo-
such that dyn
cation. The dynamical time is smallest at the center of the
f(r,t)∼ge−(t/tdyn)2ˆr . (6) PNS where the density is highest, we therefore do not ex-
pectsignificantexcitationofwaveswithangularfrequencies
√
Notethatbothgandt arefunctionsofradius,soboththe
dyn ω(cid:29) Gρ ∼10kHz.
c
magnitudeanddurationoftheforcearestronglydependent
Tosolveforthefrequencycomponentresponse,wemust
on radial position. Because the gravitational acceleration g
also implement boundary conditions. At the center of the
peaksinthe coreofthe PNS, theforcingis concentratedin
PNSweadoptthestandardcentralreflectiveboundarycon-
this region (see Figure 3).
ditions (see Appendix A). However, in the outer regions of
In a non-rotating progenitor, the collapse would be
ourcomputationaldomain(r∼250km),thereisnosurface
spherically symmetric and only radial oscillations would be atwhichthewaveswillreflect.1Insteadweexpectthewaves
generated. In rapidly rotating progenitors, the background
to propagate outwards until they dissipate (see Section 3).
structure is centrifugally distorted, leading to the genera-
Intheouterregions,thewavesgenerallybehavelikeacoustic
tion of axisymmetric non-radial waves. We may expect the
(pressure) waves because their frequencies are greater than
degree of the non-spherical component of the force to be
the local dynamical frequency. We therefore adopt a radia-
proportional to the centrifugal distortion of the collapsing
tive outer boundary condition (described in Appendix A)
star, (cid:15), defined such that the background density structure
thatensuresonlyoutgoingacousticwavesexistattheouter
has the form
boundary.
(cid:2) (cid:3)
ρ(r,θ)=ρ(r) 1+(cid:15)(r)Y (θ) . (7)
20
2.3 Wave Propagation and Characteristics
Here, Y is the l = 2, m= 0 spherical harmonic, and ρ(r)
20
is the spherically averaged density profile. The centrifugal The behavior of waves can be understood from their WKB
acceleration has magnitude ∼ rΩ2, where Ω is the local dispersion relation
spin frequency. Thus, the centrifugal distortion scales (in
the limit (cid:15)(cid:28)1) as k2 = (L2l −ω2)(N2−ω2) , (14)
r ω2c2
s
(cid:15)∼(Ω/Ω )2, (8)
dyn where k is the radial wavenumber,
r
where Ω =t−1 is the local dynamical frequency. In Ap-
dyn dyn l(l+1)c2
pendixA,weshowthattheperturbationinthebounceforce L2 = s (15)
l r2
per unit mass due to the centrifugal distortion is
istheLambfrequencysquared,andN2istheBrunt-V¨aisa¨la¨
(cid:114) (cid:20) (cid:21)
δf (cid:39)A 2 (cid:15)ge−(t/tdyn)2 2Y ˆr+r∇ Y , (9) frequency squared. In regions where ω > Ll and ω > N,
π 20 ⊥ 20 waves behave like acoustic waves, while they behave like
buoyancy (gravity) waves (not to be confused with GWs)
where∇ isthenon-radialcomponentofthegradient,and
⊥ where ω < L and ω < N. Radial profiles of N and the
A∼1 parameterizes the magnitude of the force. l
Withanestimateoftheforcingfunctioninhand,wecan
solve Equations 1-4 for the forced response of the fluid due 1 Apossibleexceptionistheshockfront,however,weexpectthe
tothebounce.Itismostconvenienttosolvetheseequations waves to be largely dissipated before they reach the shock (see
in the frequency domain rather than the time domain. To Section3).
(cid:13)c 0000RAS,MNRAS000,000–000
Supernova Seismology 5
0.20
0.6 Ur, l=2, f =0.77kHz δQω , l=2, f =0.77kHz
| |
U, l=2, f =0.77kHz
i
0.4
0.15
)
0.2 d
) e
d l
e a
l c
a 0.0 0.10s
c (
s |
( ω
U Q
0.2
δ
−
|
0.05
0.4
−
0.6
−
0.00
1.0 dE/rd(rk,ml=) 2, f =0.77kHz rf(ωkml=) 2 f =0.77kHz 1.0
dE/dr, l=2, f =0.50kHz fω l=2, f =0.50kHz
f l=0, f =1.2kHz
dE/dr, l=0, f =1.2kHz ω
0.8 0.8
)
d
e )
l d
ca0.6 0.6le
s a
( c
r s
d (
/ ω
E0.4 0.4f
d
0.2 0.2
0.0 0.0
0 20 40 60 80 0 20 40 60 80
r (km) r (km)
Figure 3. Top Left: Real (Ur) and imaginary (Ui) parts of the radial wave displacement per unit frequency (Equation A5) for
bounce-exited waves with f =0.77 kHz. Top Right: Magnitude of the quadrupole moment |δQω| (Equation 18) for the same waves.
BottomLeft:Timeintegratedwaveenergyperunitradius(EquationB5)forwavesofdifferentfrequencies.Thethreeshownfrequencies
approximately correspond to the PNS quadrupolar (l = 2) f-mode (f = 0.77 kHz), the PNS l = 2 g1-mode (f = 0.50 kHz), and an
outgoingquasi-radialwave(f =1.2kHz).BottomRight:Strengthoftheexcitationforcefω(Equation13).Allquantitiesarenormalized
viatheirmaximumvalueswithinthecomputationaldomain(seeFigure4fordimensionalvaluesofthewaveamplitude).
quadrupolar Lamb frequency, L , are shown in Figure 2. wave amplitude scales as ξ ∝ (ρc )−1/2, and so the waves
2 r s
Recallthattypicalbounce-excitedwaveshavel=0orl=2 grow in amplitude (although their energy flux remains con-
√
andangularfrequenciesnearω∼ Gρ ∼5kHz,correspond- stant) as they propagate outwards. These factors cause the
c
ing to f∼0.8kHz. waves to become increasingly non-linear as they propagate
Near the outer boundary, L and N are much smaller outwards, i.e., they steepen into shocks. The radial group
2
than typical wave frequencies ω, and in this limit the dis- velocity of the waves in the outer regions is simply v (cid:39)c ,
g s
persion relation reduces to that of acoustic waves, so that wave energy travels outwards at the sound speed.
At the center of the PNS, L → ∞ and N → 0, so
ω2 2
k2 (cid:39) . (16) quadrupolar waves are evanescent for r (cid:46) 5km. In this re-
r c2
s gion, the response is not wave-like, but is characterized by
The wavelength shortens as the waves propagate outwards the coherent fundamental mode-like oscillation of the PNS.
intoregionswithsmallersoundspeeds.Moreover,theWKB The value of N2 is large where the density gradient is large
(cid:13)c 0000RAS,MNRAS000,000–000
6 J. Fuller et al.
atradii5km(cid:46)r(cid:46)20km.Thisregionofthestarcanharbor 4.0
buoyancy waves for frequencies ω(cid:46)5 kHz. ξr(km)(f 0.77kHz)
Figure 3 shows a plot of the radial component of the | | ≈
wavedisplacementperunitfrequency,U,foraquadrupolar 3.5 |krξr|(f ≈0.77kHz)
wave with angular frequency ω (cid:39) 5kHz (f (cid:39) 0.8 kHz) as
a function of radius. The wave contains both a real part 3.0
(U ) and an imaginary part (U ), which are perpendicular
r i
inphaseforapropagatingwaveandinphaseforastanding ξr|2.5
wave.ThesewavesreflectattheedgeofthePNS(atradiiof kr
r∼20km), so the response within the PNS is composed of |
bothingoingandoutgoingwaves,whichinterferetoproduce ),m 2.0
a standing wave, or oscillation mode. There are no nodes k
(
in the radial wave function within the PNS (r (cid:46) 15km), ξr|1.5
|
therefore we refer to this mode as the fundamental mode
(f-mode) of the PNS.2 The standing waves are not totally
1.0
reflected, and gradually leak into the surrounding material.
For r (cid:38) 50km, the real and imaginary component of U(r)
are perpendicular in phase, characteristic of an outwardly 0.5
propagating acoustic wave.
Figure 3 also plots the strength of the forcing, fω (in- 0.0
0 20 40 60 80 100
tegrand of Equation 13), and the time-integrated wave en-
r(km)
ergy per unit radius, given by Equation B5, for waves of
different frequencies. The forcing is localized to near the
PNS,especiallyforhigherfrequencywaves.Forquadrupolar Figure 4. Approximatemaximumamplitudesoftheradialdis-
waves, the displacements are largest outside the inner core placements, ξr, associated with waves with frequencies near the
(r (cid:38)30km), although the wave energy density is primarily l=2f-mode.Wealsoplottheapproximatenon-linearityparame-
localizedtotheinner∼20km.Thisindicatesthatquadrupo- ter,|krξr|,associatedwiththewaves.Wavesarehighlynon-linear
lar waves are trapped within the PNS, and only gradually and are expected to generate shocks where |krξr|(cid:38)1. Since the
leak out into the outer regions. For quasi-radial waves, the waves are somewhat non-linear, our linear results are only ap-
proximate, and may differ from simulation results by a factor of
time-integratedwaveenergyissmallerinthePNSandlarger
orderunity.Wavedampingisnotincludedinthisplot,andthese
intheenvelope,indicatingthesewavesarenotwell-trapped
amplitudeswilllikelybediminishedatlargeradiiandlatetimes.
in the PNS and quickly propagate outward. We shall see in
Section4thatwaveenergyissharplypeakedatcharacteris-
ticfrequenciesthatcorrespondtothePNSoscillationmode
shown in Figure 4. Precise wave amplitudes require an in-
frequencies. The waves shown in Figure 3 have frequencies
tegral over the response per unit frequency at a given time
approximatelycorrespondingtothePNSl=2fundamental
t. However, as discussed in Section 4, the wave response
oscillationmode(f-mode,f (cid:39)0.8kHz),thefirstl=2grav-
is sharply peaked near discrete frequencies approximately
ity mode (g -mode, f (cid:39) 0.5kHz), and an outgoing l = 0
1 corresponding to oscillation mode eigenfrequencies. We can
pressure wave (p-wave, f (cid:39)1.2kHz).3
therefore integrate the response over each of these narrow
Finally, Figure 3 shows the m = 0 component of the
peaks to estimate wave amplitudes. We justify this proce-
wave quadrupole moment per unit frequency,
dureinAppendixB3.Theresultingamplitudesimplyradial
(cid:90) wave displacements of ∼kilometers and velocities of a few
δQω = dV r2δρω(r)Y2∗0 . (17) percentthespeedoflight.Theseamplitudesaremoderately
non-linear, which we discuss in more detail below.
For the l=2, m=0 waves, Equation 17 reduces to
(cid:90)
δQ = dr r4δρ (r) . (18)
ω ω
3 WAVE DAMPING AND ROTATION
The magnitude of the quadrupole moment is somewhat os-
3.1 Wave Damping
cillatory,butgenerallyincreaseswithradius.Intheabsence
ofwavedamping,GWsaremoreefficientlygeneratedasthe The analysis presented above did not include any sources
wavespropagateoutward,however,wefindbelowthatwaves of wave damping. In reality the waves will damp out on
are generally dissipated before reaching large radii. a relatively short (∼10ms) timescale. Our goal here is to
The approximate physical amplitude of the waves is identify the primary damping mechanism and estimate the
wave lifetime.
Wavedampingduetophotondiffusionisordersofmag-
nitude longer than any relevant timescales and can be ig-
2 In our models the f-mode has ω < N in much of the PNS,
nored. The radiative diffusion of neutrinos, however, could
thereforeithasgravitymodecharacteristics,whichwediscussin
potentially provide a significant damping mechanism. In-
moredepthinSection4.2.1.
3 We label the modes by the number of nodes in the radial dis- deed, the simulations of O12 show correlated neutrino and
placement Ur within the PNS. The f-mode has no nodes, while GWstrainoscillations,implyingthatsomewaveenergymay
theg1-modehasonenode,andsoon. be carried away by neutrinos. However, these simulations
(cid:13)c 0000RAS,MNRAS000,000–000
Supernova Seismology 7
also showed very little difference between the gravitational 100
waveforms with neutrino leakage turned on or off. Ferrari
t (r=50km), l=2
cross
etal.(2003)findthatneutrinosdampPNSoscillationmodes
t (r=50km), l=2
onaneutrinodiffusiontimescaleof∼seconds.Wetherefore leak
consider it unlikely that neutrinos can significantly damp tcross(r=50km), l=0
the waves considered here on timescales as short as tens of t (r=50km), l=0
leak
milliseconds.
GWs generated by the waves carry away wave energy
andcouldpotentiallybeanimportantsourceofwavedamp- 10 l=2f mode
−
ing.Asthewavespropagateoutward,theirquadrupolemo- )
s
mentincreases(seeFigure3)andsotheirGWenergyemis- m
(
sionrateincreases.However,thewavesalsobecomeincreas- t
ingly non-linear (see Figure 4). We find that waves nearly
always become non-linear before they radiate a significant
fraction of their energy into GWs. This is consistent with
simulations (Ott 2009; Kotake 2013), which find that the
energy radiated in GWs is a small fraction of the energy 1
containedinfluidwave-likemotions.Thereforeitisunlikely
thatGWemissionisasignificantsourceofdampingformost
waves. 4
If non-linear wave breaking occurs, the waves generate 0.5 1.0 1.5
shocksatwhichpointtheirenergyisrapidlyconvertedinto f (kHz)
increased entropy of the fluid where the shock forms. We
expectthewavestonon-linearlydissipatewhentheirampli-
Figure 5. Wave crossing timescale tcross and wave leakage
tude is comparable to their wavelength, i.e., when
timescale t across the inner 50 km of the postbounce super-
leak
novastructureasafunctionofwavefrequencyforl=2andl=0
k ξ ∼1 . (19)
r r
waves.Eachpeakint correspondstoaPNSoscillationmode;
leak
Figure 4 shows an estimate of the physical displacements thepeakatf ∼0.77kHzisthel=2f-mode.Quasi-radial(l=0)
of the waves, and their degree of non-linearity. The largest waveshavetcross≈tleak becausethesewavespropagateoutward
withoutsignificantreflection.
amplitude waves (with frequencies f ∼0.8kHz) are moder-
ately non-linear, reaching amplitudes k ξ ∼0.5 within the
r r
PNS.Indeed,thesimulationsofA14andK15showevidence
for the first harmonic of these waves in the GW spectra,5 Thelastequalityarisesbecausetheregionsofwavepropaga-
which is one possible outcome of non-linear wave coupling. tionareapproximatelyvirialized,sothatthesoundcrossing
In the absence of other sources of damping, the waves be- time is comparable to the dynamical time. Low frequency
comeverynon-linearwhenr(cid:38)90km,andwillnon-linearly wavesarebuoyancywavesinthePNS,andthesewaveshave
break if they are able to propagate that far. We also find muchlargerwavecrossingtimesbecausetheradialgroupve-
thatg-modesareverynon-linearwithinthePNS,andlikely locityofbuoyancywavesismuchsmallerthanthatofsound
break and dissipate within the PNS on short timescales. waves. Figure 5 shows tcross evaluated at r = 50km as a
The arguments above suggest the waves are likely dis- function of wave frequency. Wave crossing times over this
sipated via non-linear effects and/or turbulent damping in region range are ∼1ms for high frequency waves.
regionsabovethePNS(r(cid:38)40km).Animportanttimescale However, tcross(r) is not always a good estimate for a
is the wave crossing timescale wave damping timescale because waves can reflect off the
PNSsurfaceandbecometrappedwithinthePNS.Lowfre-
(cid:90) r dr
t (r)= , (20) quency waves only gradually tunnel through the overlying
cross v
0 g evanescent region (cf. Figure 3), and therefore a more rel-
where vg is the wave radial group velocity, vg = ∂ω/∂kr. evant timescale is the wave leakage timescale tleak(r). This
Outside of the inner core (r (cid:38) 30 km), v ≈ c and so the timescale reflects the rate at which wave energy leaks out
g s
wave crossing time for acoustic waves is approximately of regions below r, and is calculated via Equation B13 in
Appendix B2.
(cid:90) r dr
tcross(r)≈ c (r) ∼tdyn(r) . (21) Figure 5 plots tleak, evaluated at r = 50km, as
0 s a function of wave frequency. For high frequency waves
(f (cid:38) 1kHz), t (cid:39) t because these waves are not
cross leak
trapped within the PNS. They should therefore damp out
4 High frequency waves (f (cid:38)2 kHz) may radiate much of their ontimescalesofmilliseconds.Lowerfrequencywavesexhibit
energy in GWs due to the f6 dependence of the GW energy a series of peaks in the value of t . These peaks corre-
leak
emission rate. However, since these waves are weakly excited at
spond approximately to PNS oscillation modes, for which
bounce, they contain little energy and cannot generate a strong
the waves are mostly reflected at the PNS surface. Waves
GWsignature.
5 These peaks are more prominent in postbounce spectra, i.e., with frequencies f (cid:39) 0.77kHz, corresponding to the PNS
spectrawherethebounceiswindowedout.Thisallowsthenon- f-mode, leak outwards on timescales of tleak ∼ 10ms. This
linear harmonic peak to be separated from the broad spectrum timescaleisquitesimilartothedampingtimesfoundinsim-
ofGWscontributingtothebouncesignal. ulations (e.g., O12, A14). We therefore conclude that the
(cid:13)c 0000RAS,MNRAS000,000–000
8 J. Fuller et al.
lifetimeofthesewavesiswellapproximatedbyt .Waves and the modes instead undergo an “avoided” crossing in
leak
atthisfrequencyarelikelydampedviaturbulent/non-linear which they exchange character. During the avoided cross-
effects only after they are able to leak into the outer enve- ing, the mode frequency separation is approximately con-
lope(r(cid:38)30km).Wavesatlowerfrequenciescorresponding stant,andthemodesaresuperpositionsoftheunperturbed
togravitymodeswithinthePNSmayhavemuchlongerlife modes,givingthemahybridmodecharacter.Werevisitthe
times, although we shall see in Section 4 that their contri- observational consequence of rotational mode coupling in
bution to the GW spectrum is quite small. Section 4.2.1.
Finally, rotation introduces inertial waves/modes,
which are restored by the Coriolis force. In uniformly ro-
3.2 Rotation
tating bodies, inertial modes generally exist within the an-
Themostobviousimpactofrotationistoprovidecentrifugal gularfrequencyrangeω(cid:46)2Ω.Inourmostrapidlyrotating
support for the PNS and surrounding material. As the an- models, the f-modes and g-modes lie in the range ω (cid:46) 2Ωc
gularspinfrequencyΩincreases,centrifugalsupportcauses (where Ωc is the peak angular velocity in the postbounce
thepostbouncestructuretobelesscompact(i.e.,lowercen- model).Thereforeinertialmodescouldpotentiallyinfluence
tral densities and larger PNS radii) and more oblate. The the wave spectrum, either directly (by producing a peak in
changeinbackgroundstructureiswell-capturedbythesim- theGWspectrum)orindirectly(byaffectingthefrequency
ulations A14 used to generate our background structures. of the f-mode).
We have also attempted to approximately account for the
effect of the oblateness in the strength of the forcing that
excites the waves (see Section 2). Because the centrifugally
supported PNSs are less compact, their corresponding dy- 3.3 Special and General Relativity
namicalfrequenciesΩ andmodefrequenciesω aregen-
dyn α In this work, we largely ignore the effects of special rel-
erally lower. Including only the effect of rotation on the
ativity (SR) and general relativity (GR). Although rele-
backgroundstructure,wewouldexpectthefrequenciesofall
vant for the PNS, they greatly complicate the analysis.
axisymmetric modes to decrease with increasing rotation.
The effects of GR on NS oscillation modes have been stud-
However, in the wave calculations presented in Ap-
ied extensively (see, e.g., Thorne 1969a,b, Detweiler 1975,
pendix A, we have ignored the Coriolis and centrifugal
Cutler & Lindblom 1992, Andersson 1998, Lockitch et al.
terms in the momentum equations. These terms become
2001, 2003, Boutloukos & Nollert 2007, Gaertig & Kokko-
very important when the oscillation mode frequency f be-
tas2009,Burgioetal.2011).Theseresultsindicatethatwe
comes comparable to 2f . We are primarily interested
spin can anticipate GR to affect mode frequencies by at most
in waves with frequencies f (cid:46) 1kHz, whereas the inner O(cid:2)2GM(r)/(rc2)(cid:3) < 10% at all radii within our model.
corespinfrequenciesofthebackgroundmodelsareoforder
However, Dimmelmeier et al. (2002) compared Newtonian
2f ∼ 0.6kHz. Rotation therefore has a strong effect on
spin andconformallyflatGRCCsimulations,findingsignificant
thewavedynamicsandmaysignificantlychangeourresults.
differences in postbounce structure and GW spectra. Since
Details on the influence of rapid rotation on the oscillation
our background structures are generated from GR simula-
modes of NSs can be found in Bildsten et al. (1996), Dim-
tions, we expect that GR effects on wave dynamics will be
melmeier et al. (2006), and Passamonti et al. (2009).
smaller than the effects of rapid rotation.
We can attempt to predict the influence of rotation
SR effects also become important when fluid motions
based on perturbation theory. For the axisymmetric oscil-
approach the speed of light. The velocity associated with a
lation modes of interest, the first order change in mode fre-
perturbation with ξ = 2km at a frequency f = 1kHz is
quency (proportional to Ω) vanishes. Second order correc- r
δv≈1.2×109cm/s≈4×10−2c.Thereforethewaveampli-
tions include the Coriolis and centrifugal forces, the non-
tudes predicted by our calculations (see Figure 4) generate
spherical background structure, and spin-induced coupling
fluidvelocitiesfarbelowspeedoflight,andsoSRcorrections
betweenoscillationmodes.Foraxisymmetricl=2f-modes,
are small.
rotationhasonlyasmalleffectonthemodefrequency(Dim-
melmeier et al. 2006). For low frequency axisymmetric g-
modes, however, rotation typically increases axisymmetric
modefrequencies(seeBildstenetal.1996,Lee&Saio1997).
Thef-modewithf (cid:39)0.8kHzinFigure5issomewhatmixed
in character. It behaves like an f-mode at r (cid:46)10 km where 4 GRAVITATIONAL WAVE SIGNATURES
it is evanescent, but behaves like a g-mode in the range
GWandandneutrinoemissionaretheonlywayofdirectly
10km(cid:46)r(cid:46)20km.Itsmixedcharactermakesisdifficultto
observing the core dynamics of CC SNe (Ott 2009). Here
easily predict the effect of rotation on its frequency.
weattempttoquantifytheGWsignaturesproducedbythe
An additional complication is that rotation couples
bounce-excited oscillations and compare them with simula-
spherical harmonics of Y and Y . Axisymmetric l=2,
l,m l±2,m tion results.
m=0 waves of interest couple to both l=0 (radial) waves
and l=4 waves. Consequently both l=0 and l=4 waves will
obtainquadrupolecomponentsthatallowthemtogenerate
GWs.Rotationalsoinducesmodemixingbetweenmodesof
4.1 Gravitational Wave Spectrum
thesamedegreel,e.g.,betweenthel=2PNSf-modeandg-
modes.Rotationalmixingbetweenthemodespreventstheir We now turn our attention to GW wave emission induced
frequenciesfromcrossing(seeSection4ofFulleretal.2014), bythefluidwaves.Thetime-integratedGWenergyemitted
(cid:13)c 0000RAS,MNRAS000,000–000
Supernova Seismology 9
per unit frequency is
10−6 dEGW/df dEGW/df,l=2
2π(cid:90) dt E˙ = dEGW dEGW/df,l=0 dEGW/df,l=4
GW,ω df )Hz 10−7
2G k
= 5c5ω6|δQω|2 , (22) 2/Mc(cid:12)10−8
where δQ is the quadrupole moment per unit frequency (
ω f
(calculated from Equation 17) and the second line is from /d 10−9
W
O12.TheGWenergycorrespondstoacharacteristicdimen- G
E
sionless wave strain (Flanagan & Hughes 1998) d 10−10
(cid:115)
2 G dE
h = GW . (23)
char π2c3D2 df 10−19 0.5 1l.=02ff(mkoH1dez.5) 2.0hchar(f)2.5
− hchar(f),l=0
We use the fiducial distance D = 10kpc in our presented
results. l=2g1−mode hchar(f),l=2
Caution must be used when evaluating Equation 22. 10−20 hchar(f),l=4
)
Although the wave frequency is constant in radius, the (f
quadrupolemoment|δQ |generallyincreasesatlargerradii har
ω hc
(see Figure 3). The GW energy flux is therefore dependent
onwhichradiuswechoosetoevaluate|δQ |.Moreover,the 10−21
ω
total energy emitted by the GWs could be larger than the l=4
f mode
wave energy, especially for high frequency waves. This un- − D=10kpc
psihoynsiicnatolitaycrceofluencttsinthtehefaflcutitdhaotscwilelahtaiovneneqoutattaikoenns;GinWreeamliitsy- 10−22 0.5 1.0 1.5 2.0 2.5
f(kHz)
the waves are attenuated as they emit GWs.
In what follows, we calculate GW energies and ampli-
tudeswithδQω evaluatedatrGW =30km.Thisradius isa Figure6. Top:GWenergyspectrum,dEGW/df,duetobounce-
goodchoiceaslongasthefluidwavesdampoutatradiijust excited oscillations, calculated using our A3O04 model (from
abover .Iftheyareabletopropagatetolargerradii,they A14).Wehaveplottedcontributionsfroml=2,l=4,andl=0
GW
will obtain larger quadrupole moments (see Figure 3) and waves.Thel=0andl=4wavesemitGWsbecausetheygaina
mayemitmoreenergyinGWs.Therefore,theenergyfluxes quadrupolemomentduetotheasphericalbackgroundstructure,
we calculate should be viewed as order of magnitude es- andtheirenergyspectrumisonlyanorderofmagnitudeestimate
(see text). The broad spectrum of GWs at higher frequencies is
timates, and only full non-linear hydrodynamic simulations
createdbyoutgoingpressurewavesthatcreatethebouncesignal.
canyieldquantitativelyreliablepredictions.Thefrequencies
This signal is likely to be overestimated, as these waves quickly
of the peaks in the GW spectrum are not strongly affected
dissipateviashockformationbeforetheycanradiateGWs.Bot-
by our choice of rGW because these peaks are primarily de- tom:Dimensionlesscharacteristicwavestrain(Equation23)for
termined by the values of the PNS mode frequencies. thesamemodel,computedatD=10kpc.Wehavelabeledpeaks
Figure 6 shows a plot of the total energy radiated in correspondingtosomeofthePNSoscillationmodes.
GWs per unit frequency and the associated characteristic
wave strains. For frequencies f (cid:46) 1 kHz, the GW energy
is sharply peaked around characteristic frequencies. These quencies, especially in the low frequency part of the spec-
frequencies essentially correspond to the oscillation mode trum.
frequencies of the PNS. The widths of the peaks are the We have also attempted to calculate the GW spectra
inverse of the mode lifetimes, i.e., the timescale on which due to quasi-radial (l=0) and l=4 waves, which obtain
the mode energy is able to leak out of the PNS, t (see quadrupole moments due to rotational mixing with l=2
leak
EquationB13).Therearenosharppeaksathighfrequencies modes (see Section 3.2) and due to the centrifugally dis-
f (cid:38)1kHzbecausethesefrequenciescorrespondtopressure torted background structure. To estimate GW spectra for
waves(whicharenotreflectedattheedgeofthePNS)that l=0 waves, we remove the (cid:15) = (Ω/Ω )2 dependence of
dyn
quickly propagate outwards on a wave crossing time t . the forcing term (see Equation 13) because centrifugal dis-
cross
Thepeakatf (cid:39)0.8kHzcorrespondstothel=2axisym- tortion is not needed to excite radial waves. To calculate
metric PNS oscillation mode (see Section 2.3). This mode their quadrupole moment, we multiply the right hand side
containsmoreenergythananyotherbecauseitcoupleswell of Equation 18 by (cid:15), which approximately accounts for the
(in both physical and frequency space) to the forcing pro- quadrupolemomentofthebackgroundstructureandallows
ducedatbounce.WethereforeconfirmthehypothesisofO12 thel=0wavestoemitGW.Forl=4waves,wereplace(cid:15)with
that the peak centered at f ∼0.8 kHz in their GW spectra (cid:15)2 in Equation 13 to estimate the reduced strength of the
isgeneratedbytheaxisymmetricquadrupolarPNSf-mode. waveexcitation,andusethesameprocedureasl=0waves
The f-mode is expected to dominate the early postbounce to calculate an approximate quadrupole moment. This pro-
GW signature of rapidly rotating CC SNe. The peaks at cedure is rudimentary and should not be expected to yield
lower frequencies correspond to l=2 PNS g-modes; the first accuratequantitativepredictionsfortheenergyradiatedby
is the g -mode at f (cid:39)0.5 kHz. We reiterate that rotational l =0 and l =4 waves, although it can be used for a quali-
1
effects are important and may substantially alter mode fre- tative understanding.
(cid:13)c 0000RAS,MNRAS000,000–000
10 J. Fuller et al.
The GW spectrum produced by the l=0 waves is a Our calculations predict total GW energy outputs and
smooth continuum rather than being peaked at mode fre- wave strains roughly consistent with the results of O12 and
quencies.Thereasonisthatl=0wavesarenotwellreflected A14whenweuseaforcingstrengthofA∼2(Equation13),
fromthePNSedge,andsoenergyinl=0wavesleaksoutof although there are significant uncertainties in calculating
the PNS on a wave crossing (dynamical) timescale. More- the GW spectrum. Here, we claim only that our method
over, l=0 g-modes do not exist, instead low frequency l=0 produces a sensible order of magnitude estimate for GW
waves are evanescent in the PNS when ω < N. This is in energies,andthatitprovidesaphysicalexplanationforsome
stark contrast to the l=2 waves, which can be trapped in ofthefeaturesintheGWspectrafromrotatingcorecollapse
regions with ω <N to form oscillation modes. Instead, the simulations.
force exherted by the bounce is transferred to l=0 waves of Finally, we comment on the widths of the GW spec-
a broad range in frequencies, which quickly travel outward tralpeaks.AsdiscussedinSection3,thedampingtimescale
and steepen into shocks; it is this process which generates for the oscillation modes is determined by the wave leak-
the outgoing shock created by bounce. Although the l=0 age timescale into the envelope. For the l=2 f-mode, this
waves are important for the GW spectrum of Figure 6 for leakage time is ∼ 10 ms (in good agreement with the GW
f (cid:38) 1kHz, their GW strain peaks near bounce and con- decaytimescaleseeninO12andA14),correspondingtothe
tributes primarily to the bounce signal (see Figure 1 and width of ∆f ∼ 0.1kHz for the peak at f . The g-mode
max
Section1).Thesameistrueforhigherfrequency(f (cid:38)1kHz) peaks in Figure 6 are narrower on account of the long leak-
quadrupolarwaves.Afterbounce,thehighfrequencywaves age timescale for the g-modes, but their widths are under-
are quickly dissipated by shocks, and the l=2 modes dom- estimatedsincetheg-modesmaybedampedvianon-linear
inate the GW spectrum. This idea is consistent with the processes or modified by the background structural evolu-
resultsofK15,whofindtheGWspectrumismorestrongly tion.
peaked around mode frequencies when the bounce is win-
dowed out.
4.2.1 Rotation
Thel=4wavesarelessefficientlyexcitedbythebounce
thanl=2waves,buttheresponseissimilarlypeakedaround A14 computed GW energy spectra of rapidly rotating pro-
mode frequencies. The largest peak is centered around the genitorsasafunctionoftherotationrate.Tocomparewith
l=4 f-mode at f ≈1kHz, although we expect this mode to their results, we generate background structures and pre-
radiate considerably less GW energy than the l=2 f-mode. dictedGWspectrafromthesimulationsofA14fordifferent
Nonetheless, given our rudimentary methods, we speculate rotation rates, using the same techniques described in the
that the l=4 f-mode may be detectable, especially for very preceding sections.
rapidly rotating progenitors. Figure7displaysourcomputedGWspectra(toppanel)
for the rapidly rotating models of A14, in addition to the
GWspectraobtainedfromthesimulationsthemselves(bot-
4.2 Comparison with Non-linear Simulations
tom panel). As the rotation rate increases, the PNS and
We now compare our semi-analytical results with the sim- surroundingmaterialhavemorecentrifugalsupportandbe-
ulation results of O12 and A14. The GW energy spectra of come less compact, decreasing their dynamical frequency,
thesesimulationsgenerallycontainafewdistinguishingfea- LambfrequenciesL ,andBrunt-Va¨isa¨la¨frequenciesN.For
2
tures: this reason, the prominent peaks computed from our semi-
1. A prominent peak of maximum GW energy in the range analytical analysis shift to lower frequencies with increas-
0.7kHz(cid:46)f (cid:46)0.8kHz. ing rotation rate. They also decrease in power due to the
max
2.AbroadspectrumofGWenergyatfrequenciesf (cid:46)2kHz. strongdependenceofE˙ onfrequency(seeEquation22).
GW
The main peak at f ∼ 0.75 kHz is due to the l=2, Rapidly rotating models also contain significant power in a
max
m=0 f-mode of the PNS, as speculated by O12. However, broad peak centered near f ∼ 1.2kHz. This power is cre-
boththefrequencyandwavefunctionofthismodearelikely atedbypressurewaveswhichquicklyleakintotheenvelope,
influenced by rotational interaction with other modes (see although this power may be overestimated (see discussion
Section 3.2 and discussion below). The g -mode may be re- above).
1
sponsible for peaks near f ∼ 0.5 kHz, and the l=4 f-mode A monotonic trend of decreasing f with spin fre-
max
may produce a peak at f ∼ 0.95 kHz. Our linear calcula- quency is not observed in A14. Instead, their results indi-
tions do not easily account for any peaks at f (cid:38)1 kHz. We cateamorecomplexdependenceinwhichf increasesin
max
speculate that peaks near f ∼ 1.5kHz are due to the first frequency at moderate rotation rates and decreases at very
harmonic of f , generated due to the non-linearity of the high rotation rates, although over most of the parameter
max
f-mode responsible for f (see Appendix B3). space 0.7kHz<f <0.8kHz.
max max
The broad spectrum of GW energy (visible as broad We attribute the difference to our neglect of the ef-
peaks near f ∼ 1.2kHz in Figure 6 and the top panel of fects of the Coriolis and centrifugal forces on the wave dy-
Figure7)isproducedbybothquasi-radialandquadrupolar namics, although GR effects could have some influence as
waves which are not efficiently reflected and quickly propa- well. As discussed in Section 3.2, the f-mode is really a hy-
gateoutofthePNS.However,wecautionthatourmethods bridf-mode/g-modethatpropagatesnearthesharpdensity
may overpredict the GW signal from these waves, as they gradient at the interface between the PNS and surround-
quickly steepen into shocks before generating GWs. This ing material. Due to its g-mode component, the Coriolis
mayaccountforthelackofaverybroadpeakatf ∼1.2kHz force will tend to increase its frequency (Bildsten et al.
in the rapidly rotating simulations of O12 and A14 (lower 1996) in rapidly rotating models. We speculate that the in-
panel of Figure 7). creasing impact of the Coriolis force with rotation largely
(cid:13)c 0000RAS,MNRAS000,000–000