Equation of State Effects on Gravitational Waves from Rotating Core Collapse

3.4 MB·

by  S. Richers

Equation of State Effects on Gravitational Waves from Rotating Core Collapse
Sherwood Richers,1,2,3,4,∗ Christian D. Ott,1,5 Ernazar Abdikamalov,6 Evan O’Connor,7,8 and Chris Sullivan9,10,11
1TAPIR, Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA, USA
2DOE Computational Science Graduate Fellow
3NSF Blue Waters Graduate Fellow
4Los Alamos National Lab, Los Alamos, NM, USA
5Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan
6Department of Physics, School of Science and Technology,
Nazarbayev University, Astana 010000, Kazakhstan
7Department of Physics, North Carolina State University, Raleigh, NC, USA
8Hubble Fellow
7 9National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, USA
1 10Department of Physics and Astronomy, Michigan State University, East Lansing, MI, USA
0 11Joint Institute for Nuclear Astrophysics: Center for the Evolution of the Elements,
2
Michigan State University, East Lansing, MI, USA

r (Dated: January 10, 2017)
a
M Gravitationalwaves(GWs)generatedbyaxisymmetricrotatingcollapse,bounce,andearlypost-
bounce phases of a galactic core-collapse supernova will be detectable by current-generation gravi-

3 tational wave observatories. Since these GWs are emitted from the quadrupole-deformed nuclear-
density core, they may encode information on the uncertain nuclear equation of state (EOS). We

] examine the effects of the nuclear EOS on GWs from rotating core collapse and carry out 1824 ax-
E
isymmetricgeneral-relativistichydrodynamicsimulationsthatcoveraparameterspaceof98differ-
H entrotationprofilesand18differentEOS.WeshowthatthebounceGWsignalislargelyindependent
. oftheEOSandsensitiveprimarilytotheratioofrotationaltogravitationalenergy,T/|W|,andat
h
highrotationrates,tothedegreeofdifferentialrotation. TheGWfrequency(f ∼600−1000Hz)
peak
p
of postbounce core oscillations shows stronger EOS dependence that can be parameterized by the
- √
o core’s EOS-dependent dynamical f√requency Gρ¯c. We find that the ratio of the peak frequency
r to the dynamical frequency fpeak/ Gρ¯c follows a universal trend that is obeyed by all EOS and
t rotation profiles and that indicates that the nature of the core oscillations changes when the rota-
s
a tionrateexceedsthedynamicalfrequency. Wefindthatdifferencesinthetreatmentsoflow-density
[ nonuniform nuclear matter, of the transition from nonuniform to uniform nuclear matter, and in

the description of nuclear matter up to around twice saturation density can mildly affect the GW
2
signal. More exotic, higher-density physics is not probed by GWs from rotating core collapse. We
v
furthermore test the sensitivity of the GW signal to variations in the treatment of nuclear electron
2
capture during collapse. We find that approximations and uncertainties in electron capture rates
5
canleadtovariationsintheGWsignalthatareofcomparablemagnitudetothoseduetodifferent
7
nuclearEOS.Thisemphasizestheneedforreliableexperimentaland/ortheoreticalnuclearelectron
2
captureratesandforself-consistentmulti-dimensionalneutrinoradiation-hydrodynamicsimulations
0
. of rotating core collapse.
1
0
7
I. INTRODUCTION coresurpassesnucleardensities. Thecollapseisabruptly
1
stoppedasthenuclearequationofstate(EOS)israpidly
:
v Massivestars(M (cid:38)10M )burntheirthermonu- stiffened by the strong nuclear force, causing the inner
ZAMS (cid:12)
Xi clear fuel all the way up to iron-group nuclei at the top core to bounce back and send a shock wave through the
of the nuclear binding energy curve. The resulting iron supersonically infalling outer core.
r
a core is inert and supported primarily by the pressure of The prompt shock is not strong enough to blow
relativistic degenerate electrons. Once the core exceeds through the entire star; it rapidly loses energy dissoci-
itseffectiveChandrasekharmass(e.g.,[1]),collapsecom- ating accreting iron-group nuclei and to neutrino cool-
mences. ing. The shock stalls. Determining what revives the
Asthecoreiscollapsing,thedensityquicklyrises,elec- shock and sends it through the rest of the star has been
tron degeneracy increases, and electrons are captured the bane of core-collapse supernova (CCSN) theory for
onto protons and nuclei, causing the electron fraction half a century. In the neutrino mechanism [2], a small
to decrease. Within a few tenths of a second after the fraction((cid:46)5 10%)oftheoutgoingneutrinoluminosity
−
onset of collapse, the density of the homologous inner fromtheprotoneutronstar(PNS)isdepositedbehindthe
stalled shock. This drives turbulence and increases ther-
mal pressure. The combined effects of these may revive
theshock[3]andtheneutrinomechanismcanpotentially
∗ [email protected] explain the vast majority of CCSNe (e.g., [4]). In the
2
magnetorotational mechanism [5–10], rapid rotation and strain the nuclear EOS [44? –46]. In this paper, we ad-
strong magnetic fields conspire to generate bipolar jet- dress the question of how the nuclear EOS affects GWs
like outflows that explode the star and could drive very emitted at core bounce and in the very early post-core-
energetic CCSN explosions. Such magnetorotational ex- bounce phase (t t (cid:46) 10ms) of rotating core col-
bounce
−
plosionscouldbeessentialtoexplainingaclassofmassive lapse. Stellar core collapse and the subsequent CCSN
star explosions that are about ten times more energetic evolutionareextremelyrichinmulti-dimensionaldynam-
than regular CCSNe and that have been associated with ics that emit GWs with a variety of characteristics (see
long gamma-ray bursts (GRBs) [11–13]. These hyper- [47, 48] for reviews). Rotating core collapse, bounce,
novae make up (cid:38)1% of all CCSNe [11]. and early postbounce evolution are particularly appeal-
Akeyissueforthemagnetorotationalmechanismisits ing for studying EOS effects because they are essentially
needforrapidcorespinthatresultsinaPNSwithaspin- axisymmetric (2D) [49, 50] and result in deterministic
period of around a millisecond. Little is known obser- GWemissionthatdependsonthenuclearEOS,neutrino
vationally about core rotation in evolved massive stars, radiation-hydrodynamics, and gravity alone. Complicat-
even with recent advances in asteroseismology [14]. On ing processes, such as prompt convection and neutrino-
theoretical grounds and on the basis of pulsar birth spin driven convection set in only later and are damped by
estimates (e.g., [15–17]), most massive stars are believed rotation (e.g., [44, 47, 51]). While rapid rotation will
to have slowly spinning cores. Yet, certain astrophysical amplify magnetic field, amplification to dynamically rel-
conditions and processes, e.g., chemically homogeneous evantfieldstrengthsisexpectedonlytensofmilliseconds
evolutionatlowmetallicityorbinaryinteractions,might after bounce [7, 10, 52, 53]. Hence, magnetohydrody-
still provide the necessary core rotation in a fraction of namiceffectsareunlikelytohaveasignificantimpacton
massive stars sufficient to explain extreme hypernovae the early rotating core collapse GW signal [54].
and long GRBs [18–21]. GWs from axisymmetric rotating core collapse,
Irrespective of the detailed CCSN explosion mecha- bounce, and the first ten or so milliseconds of the post-
nism, it is the repulsive nature of the nuclear force at bounce phase can, in principle, be templated to be used
short distances that causes core bounce in the first place inmatched-filteringapproachestoGWdetectionandpa-
andthatensuresthatneutronstarscanbeleftbehindin rameterestimation[44,55–57]. Thatis,withoutstochas-
CCSNe. ThenuclearforceunderlyingthenuclearEOSis tic (e.g., turbulent) processes, the GW signal is deter-
an effective quantum many body interaction and a piece ministic and predictable for a given progenitor, EOS,
of poorly understood fundamental physics. While essen- and set of electron capture rates. Furthermore, GWs
tial for much of astrophysics involving compact objects, fromrotatingcorecollapseareexpectedtobedetectable
we have only incomplete knowledge of the nuclear EOS. by Advanced-LIGO class observatories throughout the
Uncertainties are particularly large at densities above a Milky Way and out to the Magellanic Clouds [58].
few times nuclear and in the transition regime between Rotating core collapse is the most extensively studied
uniform and nonuniform nuclear matter at around nu- GW emission process in CCSNe. Detailed GW predic-
clear saturation density [22, 23]. tions on the basis of (then 2D) numerical simulations go
The nuclear EOS can be constrained by experiment back to Mu¨ller (1982) [59]. Early work showed a wide
(see [22, 23] for recent reviews), through fundamental variety of types of signals [59–65]. However, more re-
theoretical considerations (e.g., [24–26]), or via astro- cent2D/3Dgeneral-relativistic(GR)simulationsthatin-
nomical observations of neutron star masses and radii cluded nuclear-physics based EOS and electron capture
(e.g., [22, 27, 28]). Gravitational wave (GW) obser- during collapse demonstrated that all GW signals from
vations [29] with advanced-generation detectors such rapidlyrotatingcorecollapseexhibitasinglecorebounce
as Advanced LIGO [30], KAGRA [31], and Advanced followed by PNS oscillations over a wide range of rota-
Virgo[32]openupanotherobservationalwindowforcon- tion profiles and progenitor stars [44, 49, 50, 55, 57, 66].
straining the nuclear EOS. In the inspiral phase of neu- Ott et al. [55] showed that given the same specific angu-
tron star mergers (including double neutron stars and lar momentum per enclosed mass, cores of different pro-
neutron star – black hole binaries), tidal forces distort genitorstarsproceedtogiveessentiallythesamerotating
the neutron star shape. These distortions depend on the core collapse GW signal. Abdikamalov et al. [57] went
nuclear EOS. They measurably affect the late inspiral a step further and demonstrated that the GW signal is
GW signal (e.g., [33–36]). At merger, tidal disruption of determinedprimarilybythemassandratioofrotational
a neutron star by a black hole leads to a sudden cut off kinetic energy to gravitational energy (T/W ) of the in-
| |
of the GW signal, which can be used to constrain EOS ner core at bounce.
properties [36–38]. In the double neutron star case, a The EOS dependence of the rotating core collapse
hypermassive metastable or permanently stable neutron GW signal has thus far received little attention. Dim-
starremnantmaybeformed. Itistriaxialandextremely melmeieret al.[44]carriedout2DGRhydrodynamicro-
efficiently emits GWs with characteristics (amplitudes, tating core collapse simulations using two different EOS
frequencies,time-frequencyevolution)thatcanbelinked (LS180[67,68]andHShen[69–72]),fourdifferentprogen-
to the nuclear EOS (e.g, [39–43]). itors(11M 40M ), and16differentrotationprofiles.
(cid:12) (cid:12)
−
CCSNe may also provide GW signals that could con- They found that the rotating core collapse GW signal
3
changes little between the LS180 and the HShen EOS, II. EQUATIONS OF STATE
but that there may be a slight ( 5%) trend of the GW
∼
spectrum toward higher frequencies for the softer LS180 Thereissubstantialuncertaintyinthebehaviorofmat-
EOS.Abdikamalovetal.[57]carriedoutsimulationswith teratandabovenucleardensity,andassuch,therearea
theLS220[67,68]andtheHShen[69–72]EOS.However, large number of proposed nuclear EOS that describe the
theycomparedonlytheeffectsofdifferentialrotationbe- relationship between matter density, temperature, com-
tween EOS and did not carry out an overall analysis of position (i.e. electron fraction Y in nuclear statistical
e
EOS effects. equilibrium [NSE]), and energy density and its deriva-
tives. Properties of the EOS for uniform nuclear matter
areoftendiscussedintermsofapower-seriesexpansionof
In this study, we build upon and substantially extend
the binding energy per baryon E at temperature T = 0
previous work on rotating core collapse. We perform 2D
around the nuclear saturation density n of symmetric
s
GR hydrodynamic simulations using one 12-M progen-
(cid:12) matter (Y =0.5) (e.g., [22, 23, 78, 81]):
e
itor star model, 18 different nuclear EOS, and 98 dif-
ferent initial rotational setups. We carry out a total
of 1824 simulations and analyze in detail the influence K K(cid:48)
E(x,β)= E + x2+ x3+...+ (x,β) , (1)
of the nuclear EOS on the rotating core collapse GW − 0 18 162 S
signal. The resulting waveform catalog is an order of
magnitude larger than previous GW catalogs for rotat- where x = (n ns)/ns for a nucleon number density n
−
ing core collapse and is publicly available at at https: andβ =2(0.5 Ye). Thesaturationdensityisdefinedas
−
//stellarcollapse.org/Richers_2017_RRCCSN_EOS. where dE(x,β)/dx = 0. The saturation number density
n 0.16fm−3 and the bulk binding energy of symmet-
s
≈
ric nuclear matter E 16MeV are well constrained
0
≈
The results of our study show that the nuclear EOS from experiments [22, 23] and all EOS in this work have
affects rotating core collapse GW emission through its a reasonable value for both. K is the nuclear incom-
effect on the mass of inner core at bounce and the cen- pressibility,anditsdensityderivativeK(cid:48) isreferredtoas
traldensityofthepostbouncePNS.Wefurthermorefind the skewness parameter. All nuclear effects of changing
that the GW emission is sensitive to the treatment of Ye away from 0.5 are contained in the symmetry term
the transition of nonuniform to uniform nuclear matter, (x,β), which is also expanded around symmetric mat-
S
to the treatment of nuclei at subnuclear densities, and ter as
to the EOS parameterization at around nuclear satura-
tion density. The interplay of all of these elements make (x,β)= 2(x)β2+ 4(x)β4+... 2(x)β2 . (2)
S S S ≈S
it challenging for Advanced-LIGO-class observatories to
There are only even orders in the expansion due to the
discern between theoretical models of nuclear matter in
chargeinvarianceofthenuclearinteraction. Coulombef-
theseregimes. Sincerotatingcorecollapsedoesnotprobe
fects do not come into play at densities above n , where
densitiesinexcessofaroundtwicenuclear,verylittleex- s
protons and electrons are both uniformly distributed.
otic physics (e.g., hyperons, deconfined quarks) can be
The term is dominant and we do not discuss the
probed by its GW emission. We also test the sensitivity S2
higher-order symmetry terms here (see [22, 23, 81]).
ofourresultstovariationsinelectroncaptureduringcol-
(x) is itself expanded around saturation density as
lapse. Since the inner core mass at bounce is highly sen- S2
sitive to the details of electron capture and deleptoniza-
1
tionduringcollapse,ourresultssuggestthatfullGRneu- (x)= J + Lx+... . (3)
2
S 3
trinoradiation-hydrodynamicsimulationswithadetailed (cid:18) (cid:19)
treatment of nuclear electron capture (e.g., [73, 74]) will
J corresponds to the symmetry term in the Bethe-
be essential for generating truly reliable GW templates
Weizs¨acker mass formula [82, 83], so J is what the lit-
for rotating core collapse.
eraturereferstoas“thesymmetryenergy“atsaturation
density and L is the density derivative of the symmetry
term.
Theremainderofthispaperisorganizedasfollows. In
Itisimportanttonotethatnoneoftheaboveparame-
SectionII,weintroducethe18differentnuclearEOSused
ters can alone describe the effects an EOS will have on a
inoursimulations. Wethenpresentoursimulationmeth-
core collapse simulation. This can be seen, for example,
odsinSectionIII.InSectionIV,wepresenttheresultsof
from the definition of the pressure,
our2Dcorecollapsesimulations,investigatingtheeffects
of the EOS and electron capture rates on the rotating
∂E(n,Y )
core collapse GW signal. We conclude in Section V. In P(n,Ye)=n2 e , (4)
∂n
Appendix A, we provide fits to electron fraction profiles
obtained from 1D GR radiation-hydrodynamic simula- which depends directly on K and the first derivative of
tions and, in Appendix B, we describe results from sup- (n). Since the matter in core-collapse supernovae and
S
plemental simulations that test various approximations. neutron stars is very asymmetric (Y =0.5), large values
e
(cid:54)
4
TABLE I. Summary of the employed EOS. Names of EOS in best agreement with the experimental and astrophysical
constraintsinFigure1areinboldfont. ForeachEOS,welisttheunderlyingmodelandinteraction/parameterset,thehandling
of nuclei in nonuniform nuclear matter, and give the principal reference(s). We use CLD for “compressible liquid drop”, RMF
for“relativisticmeanfield”,andSNAfor“singlenucleusapproximation”. Wereferthereadertotheindividualreferencesand
to reviews (e.g., [22, 23]) for more details. Note that we use versions of the EOS provided in tabular form that also include
contributions from electrons, positrons, and photons at https://stellarcollapse.org/equationofstate.
Name Model Nuclei Reference
LS180 CLD, Skyrme SNA, CLD [67]
LS220 CLD, Skyrme SNA, CLD [67]
LS375 CLD, Skyrme SNA, CLD [67]
HShen RMF, TM1 SNA, Thomas-Fermi Approx. [69–71]
HShenH RMF, TM1, hyperons SNA, Thomas-Fermi Approx. [71]
GShenNL3 RMF, NL3 Hartree Approx., Virial Expansion NSE [75]
GShenFSU1.7 RMF, FSUGold Hartree Approx., Virial Expansion NSE [76]
GShenFSU2.1 RMF, FSUGold, stiffened Hartree Approx., Virial Expansion NSE [76]
HSTMA RMF, TMA NSE [77, 78]
HSTM1 RMF, TM1 NSE [77, 78]
HSFSG RMF, FSUGold NSE [77, 78]
HSNL3 RMF, NL3 NSE [77, 78]
HSDD2 RMF, DD2 NSE [77, 78]
HSIUF RMF, IUF NSE [77, 78]
SFHo RMF, SFHo NSE [79]
SFHx RMF, SFHx NSE [79]
BHBΛ RMF, DD2-BHBΛ, hyperons NSE [80]
BHBΛΦ RMF, DD2-BHBΛΦ, hyperons NSE [80]
for J and L can imply a very stiff EOS even if K is not electrons, positrons, and photons. Of the 18 EOS we
particularly large. use, only SFHo [79, 89] appears to reasonably satisfy all
current constraints (including the recent constraint pro-
The incompressibility K has been experimentally con-
posed by [26]).
strainedto240 10MeV[84],thoughthereissomemodel
±
dependenceininferringthisvalue,makinganerrorbarof Historically, the EOS of Lattimer & Swesty [67, 68]
20MeV more reasonable [79]. A combination of exper- (hereafter LS; based on the compressible liquid drop
±
iments,theory,andobservationsofneutronstarssuggest model with a Skyrme interaction) and of H. Shen et
that 28MeV (cid:46) J (cid:46) 34MeV (e.g., [85]). Several exper- al.[69–72](hereafterHShen;basedonarelativisticmean
iments place varying inconsistent constraints on L, but field [RMF] model) have been the most extensively used
they all lie in the range of 20MeV(cid:46)L(cid:46)120MeV (e.g., in CCSN simulations. The LS EOS is available with in-
[86]). K(cid:48)andhigherorderparametershaveyettobecon- compressibilities K of 180, 220, and 375 MeV. There is
strainedbyexperiment,thoughastudyofcorrelationsof also a version of the EOS of H. Shen et al. (HShenH)
these higher-order parameters to the low-order parame- that includes effects of Λ hyperons, which tend to soften
ters (K, J, L) in theoretical EOS models provides some the EOS at high densities [71]. Both the LS EOS and
estimates [87]. Additional constraints on the combina- the HShen EOS treat nonuniform nuclear matter in the
tion of J and L have been proposed that rule out many single-nucleus approximation (SNA). This means that
of these EOS (most recently, [26]). Finally, the mass of they include neutrons, protons, alpha particles, and a
neutronstarPSRJ0348+0432hasbeendeterminedtobe single representative heavy nucleus with average mass A¯
2.01 0.04M [88],whichisthehighestwell-constrained and charge Z¯ number in NSE.
(cid:12)
±
neutron star mass observed to date. Any realistic EOS
Recently, the number of nuclear EOS available for
model must be able to support a cold neutron star of
CCSN simulations has increased greatly. Hempel et
at least this mass. Indirect measurements of neutron
al. [77, 78, 89] developed an EOS that relies on an RMF
star radii further constrain the allowable mass-radius re-
modelforuniformnuclearmatterandnucleonsinnonuni-
gion [27].
form matter and consistently transitions to NSE with
In this study, we use the 18 different EOS de- thousandsofnuclei(withexperimentallyortheoretically
scribed in Table I. We use tabulated versions that determined properties) at low densities. Six RMF EOS
are available from https://stellarcollapse.org/ by Hempel et al. [77, 78, 89] (hereafter HS) are avail-
equationofstate that also include contributions from able with different RMF parameter sets (TMA, TM1,
5
alue M22m0aMx>eV1.<97KM<(cid:12)260MeV 2.5
V
J0348+0432
er 28MeV<J<34MeV 2.0
et 20MeV<L<120MeV
m ]
(cid:12)
ra x M 1.5
a a
P m [
d M
ne 1.0 LS220 GShenFSU2.1
straiM97(cid:12)min SSFFHHox BHHSDBΛDΦ2
on.1 0.5
C Parametersbetweenthelinessatisfyconstraints.
0.0
0 0 5 o x Λ Φ 2 F G 7 1 3 3 1 n H A 7 8 9 10 11 12 13 14 15
LS18 LS22 LS37 SFH SFH BHB BHBΛ HSDD HSIU HSFS SFSU1. SFSU2. GSNL HSNL HSTM HShe HShen HSTM R[km]
G G
FIG. 2. Neutron star mass-radius relations. The rela-
FIG. 1. EOS Constraints from experiment and NS tionship between the gravitational mass and radius of a cold
mass measurements. The maximum cold neutron star neutron star is plotted for each EOS. The EOS employed in
gravitational mass M , the incompressibility K, symme- this study cover a wide swath of parameter space. EOS that
max
try energy J, and the derivative of the symmetry energy L lie within the constraints depicted in Figure 1 are colored,
are plotted. For M , the bottom of the plot is 0, the min and the color code is consistent throughout the paper. We
max
line is at 1.97M , and the max line is not used. The other show the 2σ mass-radius constraints from “model A” of [27]
(cid:12)
constraints are normalized so the listed minima and maxima as a shaded region between two dashed lines. These con-
lie on the min and max lines. EOS that are within all of straints were obtained from a Bayesian analysis of observa-
these simple constraints are colored, and the color code is tions of type-I X-ray bursts in combination with theoretical
consistent throughout the paper. Note that there are addi- constraintsonnuclearmatter. TheEOSthatagreebestwith
tional constraints on the NS mass-radius relationship, which these constraints are SFHo, SFHx, and LS220.
we show in Figure 2, and joint constraints on J and L [26]
that we do not show.
has no influence on the EOS.
Few of these EOS obey all available experimental and
FSU Gold, NL3, DD2, and IUF). Based on the Hempel observational constraints. In Figure 1 we show where
model, the EOS by Steiner et al. [79, 89] require that each EOS lies within the uncertainties for experimental
experimental and observational constraints are satisfied. constraints on nuclear EOS parameters and the obser-
They fit the free parameters to the maximum likelihood vational constraint on the maximum neutron star mass.
neutron star mass-radius curve (SFHo) or minimize the We color the EOS that satisfy the constraints, and use
radius of low-mass neutron stars while still satisfying all the same colors consistently throughout the paper.
constraints known at the time (SFHx). SFH o,x differ The mass-radius curves of zero-temperature neutron
{ }
from the other Hempel EOS only in the choice of RMF stars in neutrino-less β-equilibrium predicted by each
parameters. EOS are shown in Figure 2. We mark the mass range
The EOS by Banik et al. [80, 89] are based on the for PSR J0348+0432 with a horizontal bar. We also in-
Hempel model and the RMF DD2 parameterization, but clude the 2σ semi-empirical mass-radius constraints of
also include Λ hyperons with (BHBΛφ) and without “model A” of N¨atill¨a et al. [27]. They were obtained via
(BHBΛ) repulsive hyperon-hyperon interactions. a Bayesian analysis of type-I X-ray burst observations.
The EOS by G. Shen et al. [75, 76, 90] are also based This analysis assumed a particular three-body quantum
on RMF theory with the NL3 and FSU Gold parame- Monte Carlo EOS model near saturation density by [93]
terizations. The GShenFSU2.1 EOS is stiffened at cur- and a parameterization of the super-nuclear EOS with
rently unconstrained super-nuclear densities to allow a a three-piece piecewise polytrope [94, 95]. Similar con-
maximum neutron star mass that agrees with observa- straintsareavailablefromothergroups(see,e.g.,[28,96–
tions. G. Shen et al. paid particular attention to the 98]).
transition region between uniform and nonuniform nu- Throughout this paper, we use the SFHo EOS as a
clearmatterwheretheycarriedoutdetailedHartreecal- fiducial standard for comparison, since it represents the
culations [91]. At lower densities they employed an EOS most likely fit to known experimental and observational
based on a virial expansion that self-consistently treats constraints. While many of the considered EOS do not
nuclear force contributions to the thermodynamics and satisfy multiple constraints, we still include them in this
composition and includes nucleons and nuclei [92]. It re- study for two reasons: (1) a larger range of EOS will al-
duces to NSE at densities where the strong nuclear force low us to better understand and possibly isolate causes
6
oftrendsintheGWsignalwithEOSpropertiesand,(2),
0.50
many constraint-violating EOS likely give perfectly rea-
sonable thermodynamics for matter under collapse and
PNS conditions even if they may be unrealistic at higher 0.45
densities or lower temperatures.
0.40
Ye
III. METHODS
0.35
LS220 GShenFSU2.1
SFHo HSDD2
Asthecoreofamassivestariscollapsing,electroncap-
0.30 SFHx BHBΛΦ
ture and the release of neutrinos drives the matter to be
increasinglyneutron-rich. TheelectronfractionY ofthe
e
inner core in the final stage of core collapse has an im- 0.25
portant role in setting the mass of the inner core, which, 107 108 109 1010 1011 1012 1013 1014
in turn, influences characteristics of the emitted GWs. ρ[gcm 3]
−
Multidimensional neutrino radiation hydrodynamics to
account for these neutrino losses during collapse is still
FIG. 3. Y (ρ) Deleptonization Profiles. For each EOS,
too computationally expensive to allow a large param- e
radial profiles of the electron fraction Y as a function of
eter study of axisymmetric (2D) simulations. Instead, e
density ρ are taken from spherically-symmetric GR1D radia-
we follow the proposal by Liebend¨orfer [99] and approx-
tion hydrodynamics simulations using two-moment neutrino
imate this prebounce deleptonization of the matter by
transportatthepointintimewhenthecentralY issmallest
e
parameterizing the electron fraction Ye as a function (roughly at core bounce) and are plotted here. We manually
of only density (see Appendix B1 for tests of this ap- extendthecurvesouttohighdensitieswithaconstantY to
e
proximation). Sincethecollapse-phasedeleptonizationis ensurethatsimulationsneverencounteradensityoutsidethe
EOS dependent, we extract the Ye(ρ) parameterizations range provided in these curves. In the 2D simulations, Ye is
from detailed spherically symmetric (1D) nonrotating determined by the density and one of these curves until core
GRradiation-hydrodynamicsimulationsandapplythem bounce.
to rotating 2D GR hydrodynamic simulations. We mo-
tivate using the Y (ρ) approximation also for the rotat-
e
ing case by the fact that electron capture and neutrino-
matterinteractionsarelocalandprimarilydependenton
density in the collapse phase [99]. Hence, geometry ef-
from 0 to 287MeV. This allows us to treat the effects
fectsduetotherotationalflatteningofthecollapsingcore
of neutrino absorption and emission explicitly and self-
can be assumed to be relatively small. This, however,
consistently. The neutrino interaction rates are calcu-
has yet to be demonstrated with full multi-dimensional
lated by NuLib [102] and include absorption onto and
radiation-hydrodynamic simulations. Furthermore, the
emission from nucleons and nuclei including neutrino
Y (ρ) approach has been used in many previous stud-
e
blocking factors, elastic scattering off nucleons and nu-
ies of rotating core collapse (e.g., [44, 57, 66, 100]) and
clei, and inelastic scattering off electrons. We neglect
using it lets us compare with these past results. We ig-
bremsstrahlung and neutrino pair creation and annihi-
nore the magnetic field throughout this work, since are
lation, since they are unimportant during collapse and
expected to grow to dynamical strengths on timescales
shortlyaftercorebounce(e.g.,[103]). Toensureaconsis-
longer than the first 10ms after core bounce that we
∼ tent treatment of electron capture for all EOS, the rates
investigate [7, 10, 52, 53].
for absorption, emission, and scattering from nuclei are
calculated using the SNA. To test this approximation,
in Section IVE, we run additional simulations with ex-
A. 1D Simulations of Collapse-Phase
perimentalandtheoreticalnuclearelectroncapturerates
Deleptonization with GR1D
instead included individually for each of the heavy nu-
clei in an NSE distribution. In Appendix B1, we test
We run spherically symmetric GR radiation hydrody-
the neutrino energy resolution and the resolution of the
namic core collapse simulations of a nonrotating 12M
(cid:12) interaction rate table.
progenitor(Woosleyet al.[101], models12WH07)inour
open-sourcecodeGR1D[102],onceforeachofour18EOS.
The fiducial radial grid consists of 1000 zones extending To generate the Y (ρ) parameterizations, we take a
e
outto2.64 104km,withauniformgridspacingof200m fluid snapshot at the time when the central Y is at a
e
×
out to 20km and logarithmic spacing beyond that. We minimum ( 0.5ms prior to core bounce) and create a
∼
test the resolution in Appendix B1. list of the Y and ρ at each radius. We then manually
e
The neutrino transport is handled with a two-moment enforce that Y decreases monotonically with increasing
e
scheme with 24 logarithmically-spaced energy groups ρ. The resulting profiles are shown in Figure 3.
7
whereAisameasureofthedegreeofdifferentialrotation,
TABLE II. Rotation Profiles. A list of the differential ro-
Ω isthemaximuminitialrotationrate,and(cid:36)isthedis-
tation A and maximum rotation rate Ω parameters used in 0
0 tance from the axis of rotation. Following Abdikamalov
generating rotation profiles. The Ω ranges imply a rotation
0
profileateach0.5rads−1interval. Intotal,weuse98rotation et al. [57], we generate a total of 98 rotation profiles us-
profiles. ing the parameter set listed in Table II, chosen to span
the full range of rotation rates slow enough to allow the
Name A[km] Ω [rads−1] # of Profiles
0 star to collapse. All 98 rotation profiles are simulated
A1 300 0.5 - 15.5 31 using each of the 18 EOS for a total of 1764 2D core col-
A2 467 0.5 - 11.5 23 lapse simulations. However, the 60 simulations listed in
A3 634 0.0 - 9.5 20 Table III do not result in core collapse within 1s of sim-
A4 1268 0.5 - 6.5 13 ulation time due to centrifugal support and are excluded
from the analysis.
A5 10000 0.5 - 5.5 11
CoCoNuTsolvestheequationsofGRhydrodynamicson
a spherical-polar mesh in the Valencia formulation [105],
usingafinitevolumemethodwithpiecewiseparabolicre-
TABLEIII.No Collapse List. Welistthesimulationsthat
construction [106] and an approximate HLLE Riemann
do not undergo core collapse within 1s of simulation time
solver [107]. Our fiducial fluid mesh has 250 logarithmi-
due to sufficiently large centrifugal support already at the
callyspacedradialzonesouttoR=3000kmwithacen-
onsetofcollapse. Thesesimulationsareexcludedfromfurther
tralresolutionof250m,and40equallyspacedmeridional
analysis.
angular zones between the equator and the pole. We as-
A[km] Ω0[rads−1] EOS sume reflection symmetry at the equator. The GR CFC
300 15.5 GShenNL3 equations are solved spectrally using 20 radial patches,
467 10.0 GShenNL3 each containing 33 radial collocation points and 5 angu-
10.5 GShenNL3 lar collocation points (see Dimmelmeier et al. [104]). We
perform resolution tests in Appendix B2.
11.0 GShen{NL3,FSU2.1,FSU1.7}
The effects of neutrinos during the collapse phase are
11.5 GShen{NL3,FSU2.1,FSU1.7}
treatedwithaY (ρ)parameterizationasdescribedabove
634 8.0 GShenNL3 e
and in [44, 99]. After core bounce, we employ the neu-
8.5 GShen{NL3,FSU2.1,FSU1.7}
trino leakage scheme described in [55] to approximately
9.0 GShen{NL3,FSU2.1,FSU1.7} account for neutrino heating, cooling, and deleptoniza-
9.5 GShen{NL3,FSU2.1,FSU1.7} tion, though Ott et al. [55] have shown that neutrino
LS{180,220,375} leakage has a very small effect on the bounce and early
1268 5.5 GShenNL3 postbounce GW signal.
6.0 GShen{NL3,FSU2.1,FSU1.7} We allow the simulations to run for 50ms after core
bounce, though in order to isolate the bounce and post-
6.5 GShen{NL3,FSU2.1,FSU1.7}
bounce oscillations from prompt convection, we use only
LS{180,220,375}
about 10ms after core bounce. Gravitational waveforms
10000 4.0 GShen{NL3,FSU2.1,FSU1.7}
are calculated using the quadrupole formula as given in
4.5 GShen{NL3,FSU2.1,FSU1.7}
Equation A4 of [65]. All of the waveforms and reduced
5.0 GShen{NL3,FSU2.1,FSU1.7} datausedinthisstudyalongwiththeanalysisscriptsare
LS{180,220,375} available at https://stellarcollapse.org/Richers_
5.5 all but HShen,HShenH 2017_RRCCSN_EOS.
IV. RESULTS
B. 2D Core Collapse Simulations with CoCoNuT
Webeginbybrieflyreviewingthegeneralpropertiesof
We perform axisymmetric (2D) core collapse simula- the GW signal from rapidly rotating axisymmetric core
tions using the CoCoNuT code [65, 104] with conformally collapse, bounce, and the early postbounce phase. The
flatGR.WeuseasetupidenticaltothatinAbdikamalov GWstraincanbeapproximatelycomputedas(e.g.,[110,
et al. [57], but we review the key details here for com- 111])
pleteness. We generate rotating initial conditions for the
2D simulations from the same 12M(cid:12) progenitor by im- h 2G I¨, (6)
posingarotationprofileontheprecollapsestaraccording + ≈ c4D
to (e.g., [60])
where G is the gravitational constant, c is the speed of
light, D is the distance to the source, and I is the mass
(cid:36) 2 −1 quadrupole moment. In the left panel of Figure 4 we
Ω((cid:36))=Ω 1+ , (5)
0 A show a superposition of 18 gravitational waveforms for
(cid:20) (cid:16) (cid:17) (cid:21)
8
10
tb be ms A=634km
t 6 Ω0=5.0s−1
5 +
e 10 22
tb 2] −
/
1
+ 0 −
h + Hz
210 h∆ [f
1 5 √ o
g
− LSFSH22o0 HGSSDheDn2FSU2.1 h+ 1e0−23 Vir← K A G R A
10 b
− SFHx BHBΛΦ
aLIGO
15
− 5 0 5 10 15 20 102 103
−
t tb[ms] f[Hz]
−
√
FIG. 4. EOS Variability in Waveforms. The time-domain waveforms (left panel) and Fourier transforms scaled by f
(right panel) of signals from all 18 EOS for the A = 634km, Ω = 5.0rads−1 rotation profile (moderately rapidly rotating,
T/|W|=0.069−0.074atcorebounce,dependingontheEOS)areplottedassumingadistanceof10kpcandoptimalorientation,
alongwiththeAdvancedLIGO[30,108],VIRGO[32],andKAGRAinthezerodetuningVRSEconfiguration[31,109]design
sensitivity curves. t is the time of core bounce, t is the end of the bounce signal and the beginning of the post-bounce
b be
signal. Weusedataonlyuntilt +6mstoexcludetheGWsignalfrompromptconvectionfromouranalysis. Thedifferences
be
in post-bounce oscillation rates can be seen both in phase decoherence of the waveform and the peak location of the Fourier
transform. The colored curves correspond to EOS that satisfy the constraints depicted in Figure 1.
the A3=634km, Ω =5.0rads−1 rotation profile using signal, generating a stochastic GW strain whose time
0
each of the 18 EOS and assuming a distance of 10kpc domain evolution is sensitive to the perturbations from
and optimal source-detector orientation. which prompt convection grows (e.g., [46, 47, 57, 113]).
As the inner core enters the final phase of collapse, We exclude the convective part of the signal from our
its collapse velocity greatly accelerates, reaching values analysis. For our analysis, we delineate the end of the
of 0.3c. At bounce, the inner core suddenly (within bounce signal and the start of the postbounce signal at
1∼ms) decelerates to near zero velocity and then re- tbe, defined as the time of the third zero crossing of the
∼bounds into the outer core. This causes the large spike GW strain. We also isolate the postbounce PNS oscilla-
in h seen around the time of core bounce t . We de- tionsignalfromtheconvectivesignalbyconsideringonly
+ b
termine tb as the time when the entropy along the equa- the first 6ms after tbe.
torexceeds3k baryon−1,indicatingtheformationofthe In the right panel of Figure 4, we show the Fourier
b
bounce shock. The rotation causes the shock to form in transforms of each of the time-domain waveforms shown
theequatorialdirectionafewtenthsofamillisecondafter in the left panel, multiplied by √f for comparison with
the shock forms in the polar direction. GWdetectorsensitivitycurves. Thebouncesignalisvis-
The bounce of the rotationally-deformed core excites ibleinthebroadbulgeintherangeof200 1500Hz. The
−
postbounce “ring-down” oscillations of the PNS that are postbounce oscillations produce a peak in the spectrum
a complicated mixture of multiple modes. They last for of around 700 800Hz, the center of which we call the
−
afewcyclesafterbounce, aredampedhydrodynamically peak frequency fpeak. Both the peak frequency and the
[112], and cause the postbounce oscillations in the GW amplitudeofthebouncesignalingeneraldependonboth
signalthatareapparentintheleftpanelofFigure4. The the rotation profile and the EOS.
dominantoscillationhasbeenidentifiedasthe(cid:96)=2,m=
0 (i.e., quadrupole) fundamental mode (i.e., no radial
nodes)[55,112]. Thequadrupoleoscillationscanbeseen A. The Bounce Signal
in the postbounce velocity field that we plot in the left
panelofFigure5. Withincreasingrotationrate,changes ThebouncespikeistheloudestcomponentoftheGW
in the mode structure and nonlinear coupling with other signal. In Figure 6, we plot ∆h , the difference be-
+
modesresultinthecomplexflowgeometriesshowninthe tween the highest and lowest points in the bounce sig-
rightpanelofFigure5. ThedensitycontoursinFigure5 nal strain, as a function of the ratio of rotational kinetic
alsovisualizehowthePNSbecomesmoreoblateandless energytogravitationalpotentialenergyT/W ofthein-
dense with increasing rotation rate. nercoreatcorebounce(seethebeginningS|ect|ionIVfor
After the PNS has rung down, other fluid dynamics, details of our definition of core bounce). We assume a
notably prompt convection, begin to dominate the GW distanceof10kpcandoptimaldetectororientation. Just
9
t t =4.5 ms
b
−
30 8(GM)2 T A1
Rc4D W →
| |
+
h 20
1 Ω ∆ A2
−
s 0 21
d = 10
a A5 A4
r 8 10
0 0. LS220 GShenFSU2.1 A3
.
4 r
a SFHo HSDD2
= d
SFHx BHBΛΦ
Ω0 −s 0
1 0.00 0.05 0.10 0.15 0.20
T/ W
| |
FIG.6. Bounce Signal Amplitude. Weplotthedifference
between the maximum and minimum strain ∆h before t
+ be
assumingD=10kpcandoptimalsource-detectororientation
r = 15km
asafunctionoftheratioofrotationaltogravitationalenergy
T/|W| of the inner core at bounce. Each 2D simulation is a
single point and the SFHo simulations with the same differ-
1 2 3 4 5
ential rotation parameter A are connected to guide the eye.
Entropy[k baryon 1] A1−A5correspondstoA=300,467,634,1268,10000km,re-
B −
spectively. Simulations with all EOS and values of A behave
similarly for T/|W| (cid:46) 0.06, but branch out when rotation
FIG. 5. Velocity Field. We plot the entropy, density, and becomes dynamically important. We plot a dashed line rep-
velocity for the Ω0 = 4.0rads−1 (left) and Ω0 = 8.0rads−1 resentingtheexpectedperturbativebehaviorwithT/|W|,us-
(right) simulations with A = 634km at 4.5ms after core ing representative values of M =0.6M and R=65km. All
(cid:12)
bounce. Thecolormapshowsentropy. Blueregionsbelongto 1704 collapsing simulations are included in this figure.
the unshocked inner core. The density contours show densi-
ties of 10{13.5,13.75,14.0,14.25}gcm−3 from outer to inner. The
vectorsrepresentonlythepoloidalvelocity(i.e. therotational mass quadrupole moment I M(x2 z2), where M is
velocityisignored)andarecoloredforvisibility. Atlowrota- the mass of the oscillating i∼nner core−and x and z are
tionrates(left)theflowintheinnercoreislargelyquadrupo-
the equatorial and polar equilibrium radii, respectively.
lar. At high rotation rates (right), rotation significantly de-
Ifwetreattheinnercoreasanoblatesphere,wecancall
forms the inner core and couples (cid:96) = 2,m = 0 quadrupole
the radius of the inner core in the polar direction z =R
oscillations to other modes.
and the larger radius of the inner core in the equatorial
direction (due to centrifugal support) x = R+δR. To
first order in δR, the mass quadrupole moment becomes
as in Abdikamalov et al. [57], we see that at low rota-
tion rates, the amplitude increases linearly with rotation I M((R+δR)2 R2) MR(δR) . (7)
rate, with a similar slope for all EOS. At higher rota- ∼ − ∼
tion rates, the curves diverge from this linear relation- The difference between polar and equatorial radii in our
ship due to centrifugal support as the angular velocity simplified scenario can be determined by noting that the
Ω at bounce approaches the Keplerian angular velocity. surfaceofarotatingsphereinequilibriumisanisopoten-
Rotation slows the collapse, softening the violent EOS- tialsurfacewithapotentialof (cid:36)2Ω2/2 GM/r,where
− −
driven bounce and resulting in a smaller acceleration of (cid:36) is the distance to the rotation axis, r is the distance
the mass quadrupole moment. However, the value of to the origin, Ω is the angular rotation rate, and G is
T/W = 0.06 0.09 at which simulations diverge from the gravitational constant. Setting the potential at the
| | −
the linear relationship depends on the value of the dif- equator and poles equal to each other yields
ferential rotation parameter A. Stronger differential ro-
GM GM
tation affords less centrifugal support at higher rotation (R+δR)2Ω2+ = . (8)
energies,allowingthelinearbehaviortosurvivetohigher (R+δR) R
rotation rates.
Assuming differences between equatorial and polar radii
The linear relationship between the bounce amplitude
are small, we can take only the O(δR/R) terms to get
and T/W of the inner core at bounce can be derived δ((cid:36)2Ω2) R2Ω2 GM(δR)/R2. Solving for δR,
| |
in a perturbative, order-of-magnitude sense. The GW ∼ ∼
amplitude depends on the second time derivative of the δR Ω2R4/GM . (9)
∼
10
The timescale of core bounce is the dynamical time
t−2 Gρ GM/R3. In this order-of-magnitude esti- TABLE IV. Bounce Amplitude Linear Fits. We calcu-
dyn ∼ ∼ late a linear least squares fit for the bounce amplitudes in
mate we can replace time derivatives in Equation 6 with
Figure 6 to the function ∆h = m(T/|W|)+b. We only
+
divisionbythedynamicaltime. Wecanalsoapproximate
include data with T/|W| ≤ 0.04. All fitted lines have a y-
T/W R3Ω2/GM. This results in intercept b of approximately 0 and slopes m in the range of
| |∼
237−315×10−21. The three LS EOS have the shallowest
GMΩ2R2 T (GM)2
slopes and the ten Hempel-based EOS (HS, SFH, and BHB)
h . (10)
+ ∼ c4D ∼ W Rc4D havethesteepest. Them columnshowsthepredicted
predicted
| | slope of m =T/|W|×8(GM)2/Rc4D using the mass
predicted
ThoughthemassandpolarradiusofthePNSdependon andradiusofthenonrotatinginnercoreatbounce. Wechoose
rotation as well, the dependence is much weaker (in the the arbitrary factor of 8 to make the predicted and actual
slow rotating limit) [57], and T/W contains all of the SFHo slopes match. We list the mass of the nonrotating in-
| |
first-order rotation effects used in the derivation. Hence, nercoreatbounce(MIC,b,0)foreachEOSinthelastcolumn.
in the linear regime, the bounce signal amplitude should The SFHo ecap{0.1,1.0,10.0} rows use detailed electron cap-
depend approximately linearly on T/W , which is re- ture rates in the GR1D simulations for the Ye(ρ) profile (see
| | Section IVE).
flected by Figure 6.
Differences between EOS in the bounce signal ∆h+ EOS m b mpredicted MIC,b,0
enter through the mass and radius of the inner core at [10−21] [10−21] [10−21] [M ]
(cid:12)
bounce (cf. Equation 10). Neither M nor R of the in-
BHBL 318 -0.03 321 0.598
ner core are particularly well defined quantities since
BHBLP 317 0.02 322 0.599
they vary rapidly around bounce – all quantitative re-
HSDD2 316 0.00 322 0.599
sults we state depend on our definition of the bounce
time and Equation 10 is expected to be accurate only SFHo 306 0.03 304 0.582
to an order of magnitude. With that in mind, in or- HSFSG 306 -0.00 325 0.602
der to test how well Equation 10 matches our numeri- SFHx 305 0.09 303 0.581
cal results, we generate fits to functionals of the form HSIUF 304 0.06 316 0.593
h = m(T/W )+b. b is simply the y-intercept of the
+ HSNL3 298 0.07 324 0.600
| |
line, which should be approximately 0 based on Equa-
HSTMA 295 0.15 315 0.593
tion 10. m is the slope of the line, which we expect to
HSTM1 295 0.18 314 0.591
bem =8(GM )2/R c4DbasedonEqua-
predicted IC,b,0 IC,b,0
HShenH 281 0.28 311 0.604
tion 10, using the mass and radius of the nonrotating
PNS at bounce. We include the arbitrary factor of 8 HShen 280 0.29 310 0.604
to make the order-of-magnitude predicted slopes similar SFHo ecap0.1 274 0.22 262 0.562
to the fitted slopes. In Table IV we show the results GShenNL3 267 0.32 298 0.592
of the linear least-squares fits to results of slowly rotat- GShenFSU1.7 264 0.24 294 0.587
ing collapse below T/W 0.04 for each EOS. Though GShenFSU2.1 263 0.24 293 0.587
| | ≤
m is of the same order of magnitude as m, sig-
predicted LS180 242 0.16 245 0.536
nificant differences exist. This is not unexpected, con-
LS375 237 0.15 284 0.562
sidering that our model does not account for nonuni-
LS220 237 0.20 258 0.543
form density distribution and the increase of the inner
SFHo ecap1.0 210 0.08 207 0.506
core mass with rotation, which can significantly affect
the quadrupole moment. SFHo ecap10.0 174 0.03 198 0.482
At a given inner core mass, the structure (i.e. radius)
oftheinnercoreisdeterminedbytheEOS.Furthermore,
the mass of the inner core is highly sensitive to the elec- ner core mass at bounce for the GShenFSU2.1 EOS is
tron fraction Ye in the final stages of collapse. In the ∼0.59M(cid:12) while that for the LS220 EOS is ∼0.54M(cid:12).
simplest approximation, it scales with M Y2 [114], We further investigate the EOS-dependence of the
IC ∼ e
which is due to the electron EOS that dominates until bounce GW signal by considering a representative quan-
densities near nuclear density are reached. The inner- titative example of models with precollapse differential
core Y in the final phase of collapse is set by the delep- rotation parameter A3 = 634km, computed with the
e
tonizationhistory,whichvariesbetweenEOS(Figure3). six EOS identified in Section II as most compliant with
Inaddition,contributionsofthenonuniformnuclearmat- constraints. In Table V, we summarize the results for
terEOSplayanadditionalYe-independentroleinsetting these models for three precollapse rotation rates, Ω0 =
M . For example, we see from Figure 3 that the LS220 2.5,5.0,7.5 rads−1, probing different regions in Fig-
IC
{ }
EOS yields a bounce Y of 0.278, while the GShen- ure 6.
e
FSU2.1 EOS results in 0.2∼67. Naively, relying just At Ω =2.5rads−1, all models reach T/W of 0.02,
0
∼ | | ∼
on the Y dependence of M , we would expect LS220 hence are in the linear regime where Equation 10 holds.
e IC
to yield a larger inner core mass. Yet, the opposite is The LS220 EOS model has the smallest inner core mass
the case: our simulations show that the nonrotating in- and results in the smallest bounce GW amplitude of all