Differentially-rotating neutron star models with a parametrized rotation profile

0.23 MB·English

by  Filippo Galeazzi

Astronomy&Astrophysicsmanuscriptno.rotlaw˙aa˙v6 c ESO2011
(cid:13)
January14,2011
Differentially-rotating neutron star models with a parametrized
rotation profile
FilippoGaleazzi1⋆,Shin’ichirou Yoshida2,Yoshiharu Eriguchi2
1 Max-Planck-Institutfu¨rGravitationsphysik,Albert-Einstein-Institut,AmMu¨hlenberg1,D-14476Potsdam,Deutschland
2 Department of EarthScienceandAstronomy, GraduateSchool of ArtsandSciences, Universityof Tokyo, Komaba, Meguro-ku
3-8-1,153-8902Tokyo,Japan
Recieved/Accepted
1
1 ABSTRACT
0
2 We analyze the impact of the choice rotation law on equilibrium sequences of relativistic differentially-rotating neutron stars in
axisymmetry.Themaximumallowedmassforeachmodelisstronglyaffectedbythedistributionofangularvelocityalongtheradial
n
directionandbytheconsequent degreeofdifferentialrotation.Inordertostudythewideparameterspaceimpliedbythechoiceof
a
J rotationlaw,weintroduceafunctionalformthatgeneralizesthesocalled“j-const.law”adoptedinallpreviouswork.Usingthisnew
rotationlawwereproducetheangularvelocityprofileofdifferentially-rotatingremnantsfromthecoalescenceofbinaryneutronstars
3
invarious3-dimensionaldynamicalsimulations.Wecomputeequilibriumsequencesofdifferentiallyrotatingstarswithapolytropic
1
equationofstatestartingfromthesphericallysymmetricstaticcase.Byanalyzingthesequencesatconstantratio,T/W,ofrotational
| |
kineticenergytogravitationalbindingenergy,wefindthattheparametersthatbestdescribethebinaryneutronstarremnantscannot
]
R produceequilibriumconfigurationswithvaluesofT/W thatexceed0.14,thecriterionfortheonsetofthesecularinstability.
| |
S Keywords.relativity–gravitation–stars:rotation–stars:interiors–stars:neutron
.
h
p
- 1. Introduction Ottetal.(2007)andTakiwaki&Kotake(2010)forrecentstud-
o
iesongravitationalwavemechanismfromstellarcorecollapse).
r A neutron star (NS), during most of its life, is considered to
t An exciting way in which a neutron star’s life can end
s be a stationary and rigidly rotating object, apart from a tiny
a is in the inspiral and consequent merger with a binary
lag between the rotation of the superfluid component and that
[ companion. Recent results of numerical 3-dimensional sim-
of the normal fluid and the crust (e.g. Baymetal. (1969) and
ulations of neutron star binary mergers (e.g., Ruffertetal.
1 Pines&Alpar(1985)).
(1996); Ruffert&Janka (2001); Shibata&Uryu¯ (2002);
v In fact a nascent neutron star which is born in a supernova
Shibata&Taniguchi (2006); Andersonetal. (2008);
4
event is likely to rotate differentially at first before its angular
6 Baiottietal. (2008); Giacomazzoetal. (2010)) have shown
velocity distribution evolves toward a uniform rotation. There
6 that there are cases in which the remnant of the merger has
are several mechanisms that account for the redistribution of
2 a significantly high degree of differential rotation such that it
the angular momentum.One such mechanism is the shear vis-
. can sustain a total mass considerably larger than that of a uni-
1 cosity of neutron matter (Sawyer 1989). An estimate of the
formly rotating star (Baiottietal. 2008). These hyper-massive
0
timescaleinwhichtheviscositydampsaneutronstar’sinternal
1 neutron stars (HMNSs) remain stable over many dynamical
shearmotionhasreceivedalotofattentionintheliterature(e.g.
1 timescales before collapsing to a BH (e.g. Shibata&Uryu¯
Cutler&Lindblom(1987)).Foratypicalneutronstarweexpect
: (2002), Baiottietal. (2008)). In order to study the effect of
v atimescaleof10 100yrs.Inthepresenceofmagneticfields,
the centrifugal force in supporting the HMNS, most of these
i themagneticbrak−ing(Spruit1999)orthemagnetorotationalin-
X papers show the angular velocity profile of the merged object
stability(MRI)(Balbus&Hawley1991)maydrasticallyaccel-
beforetheeventualcollapsetoablackhole.Thetypicalprofile
r
a eratetheredistributionofangularmomentumtotheorderof1s extracted from simulations characteristically shows a plateau
(Duezetal.2006).
neartherotationaxisandacertaindistancefromtheaxisshows
Differentialrotationplaysanimportantroleinthebeginning a nearly power law behavior. The power index seems to differ
andattheendofthelifeofaNS.Inotsearlylife,strongdifferen- from simulation to simulation, but generally does not agree
tialrotationofamassivecoreinasupernovamayaffectthecol- with the so-called “j-const. law” (Eriguchi&Mu¨ller 1985) of
lapseandbouncedynamics(Dimmelmeieretal.2002;Ottetal. rotation, which has been so far the only choice available to
2004). A newly-bornneutron star with strong differential rota- constructequilibriummodelsofdifferentiallyrotatingstars.
tion may lose its stability in the course of the rotational (and
Models of rapidly rotating stars in general relativity have
thermal) evolution, and may collapse to a black hole or other
been studied since the 70s, when large numerical comput-
type of compact star (quark star, hybrid star). This may give
ing facilities became available. The centrifugal deformation
a unique neutrino signal and a strong gravitational wave emis-
and general relativistic gravity make these investigations fully
sion from the supernova explosion of massive stars (see, e.g.,
reliant on numerical methods. Since the pioneering work of
Butterworth&Ipser (1976), these studies have included pro-
⋆ fi[email protected] gressively more sophisticated aspects, such as nuclear matter
1
FilippoGaleazzi,Shin’ichirouYoshida,YoshiharuEriguchi:Differentiallyrotatingneutronstars
EOS, degreesof differentialrotation, magnetic fields and quite transformedinto convenientintegralequationsby using appro-
recently meridional flows (Birkletal. 2010). We here name priateGreen’sfunctions.
the following references of such studies: Butterworth&Ipser The spatial linear velocity V of the flow with respect to an
(1976) ; Komatsuetal. (1989) ; Bonazzolaetal. (1993) ; observerwithzeroangularmomentumisgivenby
Stergioulas&Friedman (1995) ; Baumgarteetal. (2000) ;
V =eβ νrsinθ(Ω ω) (3)
Ansorgetal. (2002). For morecomprehensivecollectionoflit- −
−
eratures,seeStergioulas(2003).
whereΩ = uϕ/ut istheangularfrequencyofthefluidmeasured
However,itissurprisingtonotethatallofthesestudieshave
in theasymptoticinertialframe.We nextintroducethe specific
usedjustasinglekindofansatzontherotationalangularveloc-
angularmomentum2 j:
ityprofile(namelythe“j-const.law”,whichincludeuniformly-
rotatingstarsasalimitingcase(Section2.2)). uϕ
j= . (4)
In this work we introduce a new parametrized functional −u
t
form of the rotation profile that enables us to investigate a
broader class of differentially rotating stars. In particular, our Using these quantities, we can write the equilibrium equations
newrotationlawreproducesdifferentpowerlawsintheouteren- forastationaryconfigurationas
velopeoftheangularvelocitydistribution(seeSection2.2)with
∂ p 1 j
whichitispossibletomatchtheprofileoftheHMNS.Toana- A +∂ ν V∂ V + ∂ Ω=0 (A=r,θ). (5)
ǫ+p A − 1 V2 A 1 jΩ A
lyzetheimpactofthelawofrotationonthemaximumallowed − −
mass for the differentially-rotating neutron star we construct Assumingthefluidtobebarotropic,theconditionofintegrabil-
equilibriumsequencesbyimposingafixedvalueofT/W close ityforEq.(5),canbewrittenas
| |
to what is the classical limit for the onset of the secular insta-
bility;rotatingneutronstarsareknowntobedestabilizedbythe j
=g(Ω), (6)
effectofdissipativemechanismslikeviscosityandgravitational 1 jΩ
−
radiation reaction (Chandrasekhar-Friedman-Schutz (CFS) in-
byintroducinganarbitraryfunctionalg,whichisdetailedinthe
stability; Chandrasekhar (1970); Friedman&Schutz (1978)).
nextsubsection.Thefirstintegralofmotionforastationaryso-
We show that the classical secular instability criterion is satis-
lutioncanbewrittenas
fiedonlyforalimitedclassoftherotationprofiles.
The paper is organized as follows. In Section 2 we briefly dp 1
+ν+ ln(1 V2)+ g(Ω)dΩ=0. (7)
reviewtheformalismoftheequationsandthenumericalmethod Z ǫ+p 2 − Z
used to solve them. Then we describe the new functional
form of the rotation profile. In Section 3 we construct equilib- ThecomponentsofEinstein’sequationandthefirstintegral
rium sequencesof rotating stars using the new rotation profile. ofhydrostaticequationcanbeiterativelysolvedasdescribedin
Discussionsaregiveninthefinalsection. Komatsuetal.(1989)
2. Formulation 2.2.Rotationprofile
To exploit the first integral of the hydrostatic equation,
2.1.Equationsforfluidandspacetime
we need to impose the integrability condition for Eq.(6).
Weconstructconfigurationsforanaxisymmetricandstationary Differentchoicesoffunctionalformg(Ω)wouldleadtovarious
rotating perfect fluid in general relativity. The spacetime is as- classes of differential-rotation profiles. All previous studies of
sumed to be asymptotically flat and the flow is assumed to be differentially-rotatingrelativisticstellarmodelsassumethesim-
circular(thevelocityisonlyintheazimuthaldirection).Inthis plestlinearfunctionalform,
casewehavetwoKillingvectors(Bardeen1970)andcanchoose
acoordinatesysteminsuchawaythatthemetricofthespace- g(Ω)=A2(Ω Ω), (8)
c
−
timeiswrittenas(e.g.,Komatsuetal.(1989)),
whereAisaconstantwiththedimensionsofalengthandΩ is
c
ds2 = e2νdt2+e2α dr2+r2dθ2 +e2βr2sin2θ(dϕ ωdt)2, (1) theangularvelocityontheaxisofrotation.
− −
(cid:16) (cid:17) Thischoiceistermedas“j-const.law”sincetheNewtonian
wherethesphericalpolarcoordinates(r,θ,φ)areused.Themet- limitofthespecificangularmomentum jisthatof“j-const.law”
ric potentialsα,β,ν and ω are functionsof r and θ only.1 The inNewtonianrotatingstars(Eriguchi&Mu¨ller1985).
energymomentumtensorTabofaperfectfluidis Wehereintroduceamoreflexibleformoftherotationprofile
as
Tab =(ǫ+p)uaub+pgab (2)
R20Ω(Ωα Ωα)
whereǫ, pandua arethetotalenergydensity,thepressureand g(Ω)= Ωαc − c , (9)
thefourvelocity,respectively. 1 R20Ω2(Ωα Ωα)
Thebasicequationsare:1)Einstein’sequationforthemetric − Ωαc − c
potentialsG = 8πT ,2)restmassconservation (ρua) = 0, where α, R and Ω are constants. The corresponding specific
ab ab ∇a 0 c
whichistriviallysatisfiedunderthepresentassumptionsand3) angularmomentumis
stress-energyconservation Tab =0.
b
As described in Komat∇suetal. (1989), the components of R2
j= 0Ω(Ωα Ωα). (10)
Einstein’s equation are cast into 4 equations for the potentials, Ωα − c
c
threeofwhichareellipticpartialdifferentialequations.Theyare
2 This quantity is one definition of specific angular momentum in
1 Weusegeometrizedunits,c=G=1. generalrelativity.Fordifferentdefinitions,seeKozlowskietal.(1978).
2
FilippoGaleazzi,Shin’ichirouYoshida,YoshiharuEriguchi:Differentiallyrotatingneutronstars
thestarweadoptthe“compactified”coordinateasinCooketal.
α -1 -4/3 -2 -4
(1992),totakeintoaccounttheexactboundaryconditionatthe
Ω R 2 R 3/2 R 1 R 1/2
outer − − − −
radialinfinity.
powerlawinenvelope j-const. Keplerian v-const. HMNS
ForagivenEOS,afunctionaloftherotationprofileandmax-
Table1.Choosingdifferentα,itispossibletointroducediffer- imum density, the code requires information on ”how rapidly”
ent powerlaw distributionsof the angularvelocityin the outer thestarisrotating.Tothisend,itismoreconvenienttospecifya
region of a star. We here tabulate the Newtonian limits of par- parameterthatmeasuresthedeformationinducedbytherotation
ticularchoicesofα.“j-const.”referstoconstantspecificangular rather than the angular momentum or the rotational frequency.
momentum.“v-const.” is for constant linear velocity. “HMNS” We followKomatsuetal.(1989)infixingtheratioofthepolar
is the approximate power law of some of numerical relativis- coordinateradiusofthestartotheequatorialradius.Thephysi-
tic simulations with HMNS formation (e.g., Shibata&Uryu¯ calparametersofrotationaswellasotherphysicalquantitiesare
(2002);Baiottietal.(2008)). thencomputed.
3.2.Equationofstate
Thephysicalsignificanceoftheparametersα,R andΩ in
0 c
theprofileiseasilyseenifweconsideritsNewtonianlimit.Then Inthe currentstudywe use a polytropicequationof state, with
theangularfrequencyiswrittenas theindexN =1tomodelneutronstars,
R 2 α1 p=κρ1+1/N,ǫ =ρ+Np. (13)
Ω=Ω 1+ (11)
c R !  Typicalneutronstarmassesandradiarerecoveredwhenweset
where R = rsi0nθ.For R R0, we have Ω Ωc, while for sκeq=ue1n0c0eisntogehoamveeatriczreuddeunmitosd.eWloeffithxethniuscpleaarraemqeutaetriofonroaflslttahtee
≪ ∼
R R0, forneutronmatter.Inasubsequentpaperwewillinvestigatethe
≫
dependence of the space of configurations on the EOS, using
2
Ω Ω R α . (12) realisticcoldandfinite-temperatureEOSs.
c
∼ R !
0
This means that the rotationprofile has an inner plateau inside 3.3.Parameters
R R and a power law envelope for R R . The introduc-
0 0
∼ ≫ To obtain an equilibrium model, we fix the parameters of the
tionoftheindexαisanimportantfeatureofthenewprofile:as
rotation profile, α and R , then choose the maximum density
shown in Table 1, it is possible to reproduce different rotation 0
ρ andthecoordinateaxisratio.Inourcodethissetofparam-
lawsbychoosingdifferentvaluesforα. max
eters uniquely determines an equilibrium configuration, whose
Forsomeinterestingcases(α 1, 2, 4),itispossible
∈ {− − − } physical characteristics such as mass, compactness and angu-
tointegrateanalyticallytheexpressionforg(Ω)(seeAppendix).
lar momentum, angular frequency at the centre are computed
It is important to note that for α = 1 the j-constant type of
− oncethesolutionisobtained.Restmassandgravitationalmass
law is recovered – the two functional forms must agree in the
are computed by using the standard formula (see e.g. Bardeen
limit of weak gravity. This particular case is marginally stable
(1970)).Animportantmeasureofrotation,theT/W parameter,
underRayleigh’slocalstabilitycriterionofaxisymmetricinsta-
| |
is introduced as in Komatsuetal. (1989). This is used both in
bility,whichstatesthatthespecificangularmomentummustnot
Newtonian and generalrelativistic studies of rotating stars and
decrease outward in a stable star. Values of α smaller than 1
− defined as the ratio of rotationalkinetic energyto gravitational
satisfy this condition (and hence are stable). α > 0 also satis-
bindingenergy(seeKomatsuetal.(1989)fordetails).Itcharac-
fiesthiscondition,buta star cannotbespunuprapidlyenough
terisestheoverallstrengthofrotationofthestar.Classicalstud-
toallowtheappearanceoftheCFSinstability,becauseiteasily
iessuggestthatwhenT/W 0.14,bar-shapedequilibriabifur-
reachesitsmass-sheddinglimit.
| |∼
cate (into ”Jacobi” or ”Dedekind”type equilibriumsequences)
Wenowapplythenewrotationlawtoconstructequilibrium
andT/W 0.27markstheonsetofdynamicalinstabilityofax-
configurationsof differentiallyrotating neutron stars and study
| |∼
isymmetric configurations(Tassoul 1978). In general relativity
theirdependenceonα.Eachconfigurationisdefinedbyfivepa-
numericalsimulationspointtoaratherlowerlimit,T/W 0.25
rameters:theaxisratior /r ,thepolytropicindexN,themaxi-
p e | |∼
(Mancaetal.2007).
mumdensityρ andthetwoparametersoftherotationlaw,R
max 0
andα.
3.4.Numericalequilibriumsequences
3. Results For a given rotation profile and a maximum density, we start
from a non-rotatingmodel(the polar-to-equatorialaxis ratio is
3.1.Numericalcode
equal to unity), and decrease the ratio to obtain stars with in-
We construct sequences of axisymmetric and differentially ro- creasingly rapid rotation. We terminate the sequence if one of
tatingobjectsusingthecodediscussedinKomatsuetal.(1989). thefollowingconditionsissatisfied:1)T/W =0.14.Thisisthe
| |
ThecodeiterativelysolvesEinstein’sequationandthefirstinte- classical criterion at which CFS instability or viscous instabil-
gralofmotionforastationaryfluidconfiguration. itysetsinforabar-shapeddeformation(theactualcriticalpoint
Forallcomputationweusedagridwith600pointsinther dependsweaklyonboththeequationofstateanddegreeofdif-
−
directionand300intheθ direction.Theradialgridissetupin ferentialrotation(Karino&Eriguchi2002)).2)Mass-shedding
−
sucha way thatthe inneruniformly-distributed300gridpoints limit.Thelastconditionisreachedwhentheangularfrequency
just cover the whole star (on the equator the 300th point from ofmatterattheequatorialsurfacecomesclosetolocalKeplerian
theoriginalwayscorrespondstothesurfaceofthestar).Outside frequency and a further spin-up of a star leads to shedding of
3
FilippoGaleazzi,Shin’ichirouYoshida,YoshiharuEriguchi:Differentiallyrotatingneutronstars
mass. Beyond this point it is not possible to construct equilib- Asinthecaseofuniformrotation,forstarswithweakdifferen-
rium configurations. 3) Topology change. When the degree of tial rotation the centrifugal force at the equator on the surface
differential rotation is large the maximum density may move reaches a value at which the star sheds mass before its T/W
| |
away from the rotation axis. When the value of the axis ratio value reaches 0.14. In this case all the sequences terminate at
reacheszero, the star developsa hole on the rotation axis. The mass-sheddinglimitbeforereachingT/W = 0.14orthe point
| |
equilibrium sequence may be continued beyond this point, but oftopologychange.Howeverastarwithsufficientlystrongdif-
we are not interested in this toroidal shape configuration as a ferentialrotationcanstorelargerotationalenergydeepinsidethe
modelofaneutronstar. star andallowthe surfaceangularfrequencytobe smallerthan
Concerning the last point above, we should note that the that of the mass-shedding limit. Therefore a star with α = 1
−
structureoftheparameterspaceisquitecomplex,ashasbeenan- or 2 reaches the critical point T/W = 0.14 before encoun-
alyzedbyAnsorgetal.(2009). Thetoroidalandthespheroidal teri−ng mass-shedding or topology c|ha|nge, provided that Rˆ is
0
configurationsformdisjointfamiliesofsolutionsseparatedbya small enough (that is a smaller core region of nearly uniform
masssheddingregion.Increasingthedegreeofdifferentialrota- rotation). For these choices of α and sufficiently small Rˆ , we
0
tionbyreducingthevalueoftheR parameter,thetwofamilies alsoseetheappearanceoftopologicalchangebeforethecritical
0
joinagainintheparameterspace.Theareaoftheregionseparat- T/W = 0.14isreached(representedasthe trianglesinFig.1).
| |
ingthetwotopologicallydifferentfamiliesdependsontheρ Indeed,thesequenceswhichweterminateatT/W = 0.14will
max
| |
oftheconfiguration,becomingsmallerasthelatterisincreasing. eventuallyseethetopologychangeifextendedtofasterrotation.
Stated differently, starting from the Newtonian limit and going Animportantobservationhereisthatforα= 4noneofthe
−
tothemoregeneralrelativisticcase,itbecomesmoreandmore models reach T/W = 0.14 before mass shedding occurs, and
complicatedtofindavalueofR forwhichthesolutioncanreach for α = 1, and| 2| we need to set Rˆ sufficiently small (i.e.,
0 0
− −
thetoroidalfamily,infacttheseparationincreasesbetweenthe enablinghighdegreeofdifferentialrotation)tohavethecritical
twotypesofsolutions.We havestudiedtheparameterspaceof T/W beforetopologychangeormasssheddingoccurs.
| |
thesolutionsforthethreedifferentrotationlawsanalyzedhere,
andwillcollecttheresultsalongwiththedependencyhiddenin
4. SummaryandDiscussion
theEOSinthefollowuppaper.
We haveintroduceda newrotationprofileto studyequilibrium
sequences of differentially rotating relativistic stars. Compared
3.5.Stellarmassinparameterspace
with the previousstudies,which assume onlyonetype of rota-
To visualize our results, we plot gravitational mass of stellar tion profile, we are now able to investigate a broader class of
models in the parameter space. Figure 1 shows the surfaces of rotatingstars.
equilibriummodelsembeddedintheR ρ M/M space, As a first step towards systematic studies, we focus on the
0 max sun
− −
where M is the gravitational mass in units of solar masses, simplest neutron star model with a polytropic EOS with in-
for three different values of the parameter α: α = 1 (top dex N = 1. For the rotation profiles that allow analytic ex-
−
panel), α = 2 (middle panel) and α = 4 (bottom panel). pressions in the Eq. (5), we computedsequencesof spheroidal
− −
In order to have the same degree of differentialrotation for all equilibria that start from non-rotating stars. Special attention
the models with a constant R , following the prescription of is paid to the appearance of the critical point to the secular
0
Baumgarteetal.(2000),wedefine: instability (CFS or viscous instability) to bar-shaped deforma-
tionof thestar. We see thatthe appearanceofthe criticalpoint
Rˆ =R /R , (14)
0 0 c may occur for rather strong differential rotation in the case of
whereR isthecircumferentialradius. α = 1and 2inEq.(9),whichcorrespondsrespectivelytoso-
c
− −
Onthebasesoftheseplots,wemarkedeachparameterpair called “j-const.” and “v-const.” rotation profiles in Newtonian
(ρ ,Rˆ )withdifferentsymbolstoindicatehowtheequilibrium stars(Eriguchi&Mu¨ller1985).Eveninthesecases,thecritical
max 0
sequence ends. The triangles correspond to the parameters for point does not appear before the configuration changes from a
which T/W of the model reaches 0.14 (and the computations spheroidaltopologytoatoroidalone,unlesstheparameterR in
0
| |
are stopped there). The dots correspond to the parameter pairs Eq.(9)issufficientlysmall.
for which the sequence terminates at the mass-shedding limit. When α = 4 instead there seemsno critical pointof CFS
−
Thesolidlinesarefortheparametersforwhichthesequenceof instability before the equilibrium sequences reach their mass-
spheroidsendswithatopologicalchange. shedding limit. This last case is relevant since it appears to
AsmentionedinSection3.4,oursequencesterminateatthe mimic the approximate rotation profile of some of the quasi-
threedifferentcriteria.Thereforeoneneedstobecarefulinread- stationaryHMNSsseeninthenumericalsimulationsofneutron
ingthesefigures.Itisimportanttostressthatforthesequences starmergers.Howeverthesestarsmaybesusceptibletotherel-
terminatingatT/W = 0.14and atthe topologicalchange,the ativelynewclassofdynamicalinstability(so-called”lowT/W
| | | |
valueofmassplottedhereis not themaximummass ofthe se- instability”, e.g. Centrellaetal. (2001); Shibataetal. (2002);
quence.Thesolutionsofthetoroidalclassdonotpossessavalue Wattsetal. (2003); Saijo&Yoshida (2006); Ou&Tohline
for the maximummass, instead this quantity can arbitrarilyin- (2006); Baiottietal. (2008)), whose existence relies on the
crease as the torus becomes thinner and thinner along the se- strong shear flow (Wattsetal. (2005); Corvinoetal. (2010)).
quenceofequilibriumfigures(see.Ansorgetal.(2003)). Indeed,numericalsimulationsofbinaryNSsdoindicatethatthe
It is seen from Eq.(9) that the angular velocity profile be- HMNSdevelopsabardeformation(e.g.Shibata&Uryu¯ (2002);
comes uniform as Rˆ . In this rigid rotation limit, the se- Baiottietal.(2008))ontopofthebackgroundstarwithapprox-
0
→ ∞
quence of N = 1 polytropesis known to terminate with mass- imateaxisymmetry.Itwouldbeinterestingtostudytheappear-
shedding state before reaching T/W = 0.14. The degree of anceoflowT/W instabilitybyusingourequilibriumstarswith
| | | |
differentialrotationdependsbothonthepower-lawexponentα different rotation profiles since the survival time of the merger
and on the radius R . We may regardit as weak when the pro- remnant could be an important diagnostic. The observation of
0
fileisclosetouniformrotation(i.e.R and/orα ). the delay between detection of the gravitational wave signal
0
→ ∞ → −∞
4
FilippoGaleazzi,Shin’ichirouYoshida,YoshiharuEriguchi:Differentiallyrotatingneutronstars
formabinaryneutronstarsmergerandthepossibleobservation
oftheshortgammarayburstcounterpartwouldthenhelptoput
constraintsontheinternalstructureofaNS.
Acknowledgements. WethankMarcus AnsorgandCarlos Palenzuela foruse-
22..22 fuldiscussions.ThisworkissupportedinpartbyaGrant-in-AidforScientific
Research (C) ofJapan Society forthe Promotion ofScience (20540225). FG
22..00
MM wantstothankAarynTonitafortheessentialsupportinthiswork,SamLander
//MM 11..88 forthefruitfuldiscussionsandfinallyLucianoRezzollaforthecontinuoussup-
ssuunn portthathedevotedinthisproject.
11..66
11..44
AppendixA: Integrationofg(Ω)
11..22
11..00 For some interesting cases, we have an analytic expression for
theintegralofg(Ω)whichappearsinEq.(7).
2255 2200 33..00 33..55 We define a parameter k := (R0Ωc)2 and normalize Ω as
RRˆˆ00 1155 1100 11..55 22..00((1100115522gg..55//ccmm33 )) ξas:=Ω/Ωc.Thentheintegrationofthefunctionalg(Ω)iswritten
55 11..00 (cid:0)(cid:0)
00..55 kξ(ξα 1)
g(Ω)dΩ= − dξ. (A.1)
Z Z 1 kξ2(ξα 1)
− −
Forα= 1,therighthandsidereducesto
−
k k 1
22..22 Arctan (2ξ 1) ln(1+kξ(ξ 1)). (A.2)
r4 k r4 k − − 2 −
MM//MM 2211....0088 Forα−= 2,itreduces−to 
ssuunn −
11..66 ln(1+k(ξ2 1)) 2klnξ
11..44 − − . (A.3)
2(k 1)
11..22 −
Finally,forα= 4wehave
11..00 −
2255 33..55 1 1+ √1+4k2 2kξ2
2200 33..00 lnξ+ ln − − . (A.4)
RRˆˆ00 1155 1100 55 11..00 11..55 22..00(cid:1)(cid:1)((1100115522gg..55//ccmm33 )) − 2√1+4k2 · +1+ √1+4k2+2kξ2
00..55
References
Anderson, M., Hirschmann, E. W., Lehner, L., et al. 2008, Physical Review
Letters,100,191101
Ansorg,M.,Gondek-Rosin´ska,D.,&Villain,L.2009,MNRAS,396,2359
Ansorg,M.,Kleinwa¨chter,A.,&Meinel,R.2002,Astron.Astrophys.,381,L49
Ansorg,M.,Kleinwa¨chter,A.,&Meinel,R.2003,ApJ,582,L87
Baiotti,L.,Giacomazzo,B.,&Rezzolla,L.2008,Phys.Rev.D,78,084033
22..00
Balbus,S.A.&Hawley,J.F.1991,ApJ,376,214
ssuunnMM//MM 1111....8866 BBBaaaruydmmeeg,naG,r.tJ,e.,PMTet..hW1ic9.k,7,S0Ch,.aA,p&pirJo,P,1inS6e.2sL,,7.D,1&.1S9h6i9b,aNtaa,tMure.,2202040,,8A7p2J,528,L29
Birkl,R.,Stergioulas,N.,&Mu¨ller,E.2010,ArXive-prints
11..44
Bonazzola,S.,Gourgoulhon,E.,Salgado,M.,&Marck,J.A.1993,A&A,278,
11..22 421
Butterworth,E.M.&Ipser,J.R.1976,ApJ,204,200
11..00 Centrella, J.M.,New,K.C.B.,Lowe,L.L.,&Brown,J.D.2001,ApJ,550,
L193
2255 33..55 Chandrasekhar,S.1970,PhysicalReviewLetters,24,611
2200 33..00
RRˆˆ00 1155 1100 11..55 22..00((1100115522gg..55//ccmm33 )) CCooorvki,nGo,.GB..,,SRheazpziorlola,,SL.L.,.B,&ernTuezuzkio,lSsk.,yD,Se.PAie.t1ri9,9R2.,,A&pGJ,i3a9co8m,2a0z3zo,B.2010,
55 11..00 (cid:2)(cid:2) ClassicalandQuantumGravity,27,114104
00..55 Cutler,C.&Lindblom,L.1987,ApJ,314,234
Dimmelmeier,H.,Font,J.A.,&Mu¨ller,E.2002,A&A,393,523
Duez,M.D.,Liu,Y.T.,Shapiro,S.L.,Shibata,M.,&Stephens,B.C.2006,
Fig.1.Two-dimensionalsurfacesof equilibriummodelsofdif- Phys.Rev.D,73,104015
ferentially rotating neutron stars embedded in the space Rˆ Eriguchi,Y.&Mu¨ller,E.1985,A&A,146,260
0 − Friedman,J.L.&Schutz,B.F.1978,ApJ,222,281
ρ M/M . The first panel is for α = 1, the second one
max sun Giacomazzo,B.,Rezzolla,L.,&Baiotti,L.2010,ArXive-prints
− −
is for α = 2 and the last one is for α = 4. The lines at the Karino,S.&Eriguchi,Y.2002,ApJ,578,413
− −
bottomofeachpanelaredrawntoclarifywhichofthethreeter- Komatsu,H.,Eriguchi,Y.,&Hachisu,I.1989,MNRAS,237,355
minalconditions(seeSection3.4)ofsequenceapplies.Triangles Kozlowski,M.,Jaroszynski,M.,&Abramowicz,M.A.1978,A&A,63,209
Manca, G. M., Baiotti, L., DePietri, R., & Rezzolla, L. 2007, Classical and
areforsequencesreachingT/W =0.14,dotsareforsequences
| | QuantumGravity,24,171
terminatingwithmasssheddingandthicksolidlinesareforthe
Ott,C.D.,Burrows,A.,Livne,E.,&Walder,R.2004,ApJ,600,834
casewhentopologychangeoccurs.
5
FilippoGaleazzi,Shin’ichirouYoshida,YoshiharuEriguchi:Differentiallyrotatingneutronstars
Ott,C.D.,Dimmelmeier,H.,Marek,A.,etal.2007,PhysicalReviewLetters,
98,261101
Ou,S.&Tohline,J.E.2006,ApJ,651,1068
Pines,D.&Alpar,M.A.1985,Nature,316,27
Ruffert,M.&Janka,H.2001,A&A,380,544
Ruffert,M.,Janka,H.,&Schaefer,G.1996,A&A,311,532
Saijo,M.&Yoshida,S.2006,MNRAS,368,1429
Sawyer,R.F.1989,Phys.Rev.D,39,3804
Shibata,M.,Karino,S.,&Eriguchi,Y.2002,MNRAS,334,L27
Shibata,M.&Taniguchi,K.2006,Phys.Rev.D,73,064027
Shibata,M.&Uryu¯,K.2002,ProgressofTheoreticalPhysics,107,265
Spruit,H.C.1999,A&A,349,189
Stergioulas,N.2003,LivingReviewsinRelativity,6,3
Stergioulas,N.&Friedman,J.1995,Astrophys.J.,444,306
Takiwaki,T.&Kotake,K.2010,ArXive-prints
Tassoul, J. 1978, Theory of rotating stars, Princeton Series in Astrophysics,
Princeton:UniversityPress
Watts,A.L.,Andersson,N.,Beyer,H.,&Schutz, B.F.2003,MNRAS,342,
1156
Watts,A.L.,Andersson,N.,&Jones,D.I.2005,ApJ,618,L37
6