Radiation of Angular Momentum by Neutrinos from Merged Binary Neutron Stars

12 Pages·0.23 MB·English

by  Thomas W. Baumgarte

Radiation of Angular Momentum by Neutrinos from Merged Binary
Neutron Stars
Thomas W. Baumgarte and Stuart L. Shapiro1
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Il 61801
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ABSTRACT
n
a We study neutrino emission from the remnant of an inspiraling binary neutron star
J
following coalescence. The mass of the merged remnant is likely to exceed the stability
8
2 limit of a cold, rotating neutron star. However, the angular momentum of the remnant
may also approach or even exceed the Kerr limit, J/M2 = 1, so that total collapse may
1
not be possible unless some angular momentum is dissipated. We find that neutrino
v
4 emission is very inefficient in decreasing the angular momentum of these merged objects
9 and may even lead to a small increase in J/M2. We illustrate these findings with a post-
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1 Newtonian, ellipsoidal model calculation. Simple arguments suggest that the remnant
0 may form a bar mode instability on a timescale similar to or shorter than the neutrino
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emission timescale, in which case the evolution of the remnant will be dominated by the
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/ emission of gravitational waves.
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1DepartmentofAstronomyandNationalCenterforSupercomputingApplications,UniversityofIllinoisatUrbana-Champaign,Urbana,
Il61801
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1. INTRODUCTION the ISCO exceeds unity for all binaries in which each
member has a rest mass < 0.94Mmax. Here Mmax is
∼ 0 0
Considerabletheoreticaleffortiscurrentlydirected
themaximumallowedrestmassofanonrotating,iso-
toward understanding the coalescence of binary neu-
lated,coldneutronstar,andM ,M andJ denotethe
0
tronstars. Theintereststemspartlyfromthepromise
totalrestmass,energyandangularmomentumofthe
such coalescence holds for generating gravitational
binary system. For more realistic, nonsynchronous
waves that can be detected by laser interferome- binaries, J/M2 is likely to be somewhat smaller, but
ters now under construction, like LIGO, VIRGO and
may still be appreciable. For many of these bina-
GEO.Additionalinterestistriggeredbytheideathat
riestherestmassofthemergedremnantalsoexceeds
neutrinos generated during the merger and coales- Mmax. Immediately following merger, these objects
0
cence mightbe the sourceofgammaraysbursts (see,
may be temporarily stabilized by thermal pressure
e.g., Paczyn´ski, 1986; Eichler et. al., 1989; Janka &
andangularmomentum. Onlyaftersomeofthether-
Ruffert, 1996). Most of the theoretical work has, to
mal energy and angular momentum have been radi-
date,beenperformedinaNewtonianframework(see,
ated away can the star later collapse to a black hole.
e.g.,Rasio&Shapiro,1994;Ruffert, Janka&Scha¨fer
The ultimate fate of such an object can only be es-
1996; Zhuge, Centrella & McMillan, 1996). Only a
tablished by a three-dimensional, relativistic hydro-
few hydrodynamical calculations have taken into ac-
dynamic numerical simulation which employs a hot
count post-Newtonian corrections (Oohara & Naka-
nuclear equation or state and full neutrino radiation
mura, 1995; Shibata, Oohara & Nakamura, 1997),
transport. While the development of such a code is
while results from calculations in full general relativ-
underway by several groups, it is still very far from
ity are very preliminary (Oohara & Nakamura, 1995;
complete.
Wilson & Mathews, 1995; Wilson, Mathews & Mar-
Our goal in this paper is to isolate the role of
ronetti, 1996).
the neutrino emission in the early evolution follow-
Thereareanumberofissues,however,thatcannot
ing merger and to estimate its effect on the angular
be addressed even qualitatively in Newtonian gravi-
momentum J and the angular momentum parameter
tation. Suchissuesincludethecollapseofthemerged J/M2. To doso,weconsiderasimplifiedscenariofor
remnant to a black hole. Such a collapse is quite
the post-merger neutron star remnant and imagine
likely because the mass of the remnant will proba-
that the remnant evolves to an axisymmetric, quasi-
bly exceed the maximum allowed mass for a single
equilibrium state on a dynamical (orbital) timescale
neutron star. However, the final fate of the collapse
immediately following merger. Since the merger fol-
is completely unknown if the remnant’s angular mo-
lows a rapid plunge from the ISCO, the merged rem-
mentumparameterJ/M2 exceedsunity. Inthis case,
nantformswiththesametotalmassM,therestmass
the remnant cannot form a Kerr black hole without
M and the angular momentum J of the progenitor
0
first losing some of its angular momentum. In prin-
binary at the ISCO (here we neglectthe small loss of
ciple, angular momentum could be radiated away by
massandangularmomentum viagravitationalwaves
theemissionofgravitationalwaves,neutrinos,and/or
and gas ejection during the dynamical plunge). Ac-
electromagnetic (e.g. magnetic dipole) radiation. cordingly, J/M2 will typically be near unity. We as-
During the coalescence of the two neutron stars sume that the merged configuration quickly settles
some fraction of the kinetic energy will be converted intouniformlyrotating,dynamicallystablequasiequi-
into thermal energy. Therefore the remnant is likely librium state and focus on its quasi-static evolution.
to be hot and emit thermal neutrinos. During the We show that the emission of neutrinos is very inef-
rapid plunge of the binary once it reaches the inner- ficient in carrying off J. Depending on the stiffness
most stable circular orbit (ISCO), the angular mo- of the equation of state, this emission may lead to
mentum of the system will be conserved approxi- either an increase or a decrease of J/M2. Also, we
mately, so the remnant will be rotating very rapidly. argue that these objects typically develop a bar in-
In their fully relativistic treatment of corotating, stability on a timescale comparable or shorter than
polytropic binaries in quasiequilibrium circular or- the neutrino emission timescale. We therefore con-
bit, Baumgarte et. al. (1997a, 1997b) found that, clude that while neutrino emissionmay be important
for a polytropic equation of state with polytropic in- for the cooling of the remnant, it is negligible for the
dex n = 1, the total J/M2 of the binary system at emission of angular momentum.
2
The paper is organized as follows. In Section 2 where the dot denotes a derivative with respect to τ.
we estimate the rate of angular momentum loss by Equation (6) is fully relativistic. The relation be-
neutrino emission. In Section 3 we incorporate our tween L and J˙ depends on the characteristics of the
estimate in a post-Newtonian, ellipsoidal model cal- emission of the neutrinos from the star’s surface, as
culation, that follows the post-merger evolution of well as the relativistic structure of the star. Estab-
the remnant. The equations and formalism for this lishing this relationship therefore requires detailed
approximation are presented in Appendix A. In Sec- numerical models of radiating, relativistic, rotating
tion4wecomparethe relevanttimescalesforangular stars. Instead, we adopt a simple, first-order Newto-
momentumlossbycompetingprocesses,andwesum- nian description to find the approximate relationship
marize our conclusions in Section 5. given by equation (14) below. We assume that the
neutrinos are emitted isotropically in the rest frame
2. ANGULAR MOMENTUM LOSS BY of a local stationary observer comoving with the sur-
NEUTRINO EMISSION face. An observer in a nonrotating static frame will
therefore find that, on average, each neutrino carries
Consider a neutrino ν with four-momentum pα
off some angular momentum. This angular momen-
emitted from the surface of a uniformly rotating ax-
tum can be found by Lorentz transforming from the
isymmetric equilibrium star into an arbitrary direc-
stationary frame into the static frame. To lowest or-
tion. Alocalobserverwithfourvelocityuα comoving
derinvwefindthat,onaverage,eachneutrinocarries
withthesurfacewillmeasuretheneutrino’senergyto
off a linear momentum p =vE =Ω̟E , where Ω
be ϕˆ ν ν
is the angular velocity of the star, ̟ the distance of
δW(e) =−uαp =−utp −uϕp . (1)
α t ϕ the surface elementfrom the axisof rotation,andE
ν
Hereandthroughoutthepaperweadoptgeometrized is the energy of the neutrino in the comoving frame
units, c ≡ 1 ≡ G. Since the spacetime is stationary (equation (1)). The corresponding angular momen-
andaxisymmetric,thecomponentsp andp arecon- tum is
t ϕ
servedalongtheneutrino’spath. Therefore,adistant p =̟2ΩE . (7)
ϕ ν
observer at rest with respect to the star’s center of
The total rate of loss of angular momentum is there-
mass will measure the neutrino’s energy to be
fore (see also Kazanas, 1977)
δW(r) =−p =−dM, (2)
t J˙= −<̟2 >ΩL, (8)
Adistantobserverwillidentifytheneutrino’sangular
where L is the neutrino luminosity and < ̟2 > de-
momentum to be
notes the average of ̟2 over the surface of the star.
We can get a reasonable estimate for this average by
dJ =−p . (3)
ϕ
adopting a rotating, compressible ellipsoid model for
Combiningthelastthreeequationsyieldsthefamiliar the star (see Lai, Rasio & Shapiro, 1993, hereafter
result (e.g., Thorne, 1971) LRS). If we denote the principal axes by a1, a2 and
a (wherea =a ismeasuredintheequatorialplane
3 1 2
1 and a is along the rotation axis), we find
dM =ΩdJ − δW(e), (4) 3
ut
2
where Ω = uϕ/ut is the angular velocity of the rota- <̟2 >= 3a21f(e), (9)
tion. The luminosity as measured by a local, comov-
where
ing observer can be defined as
δW(e) f(e) ≡ 6π a1̟3 a21−e2̟2 1/2d̟
L= dτ , (5) a21A Z0 (cid:18) a21−̟2 (cid:19)
1
where τ is the observer’s proper time. Equation (4) = 1− e2+O(e4). (10)
15
can then be rewritten
Here
1
M˙ =ΩJ˙− L, (6) e2 ≡1−(a /a )2 (11)
ut 3 1
3
is the eccentricity, and stars U /M ∼ 0.1, we have U /M ∼ few per-
cold hot
centfollowingmerger. Hence, fromequation(15),we
A=4π a1̟ a21−e2̟2 1/2d̟ (12) expect that the relative change of the angular mo-
Z (cid:18) a2−̟2 (cid:19) mentum due to neutrino emission can be at most a
0 1
few percent. We conclude that neutrinos are very
isthesurfaceareaoftheellipsoid. Notethatf(e)∼1
inefficient in reducing the merged remnant’s angular
evenforlargeeccentricities. Itisconvenienttorewrite
momentum.
equation(8)intermsoftheangularmomentumJ and
We now focus on the change in J/M2 associated
themomentofinertiaI. ThelatterisI =2/5κ Ma2,
n 1 with the neutrino luminosity L:
where κ is a dimensionless structure constant of or-
n
der unity which depends onthe star’sdensity profile. M2 d J J˙ M˙
For a polytropic equation of state = −2
J dτ M2 J M
P =Kρ1+1/n, (13) J˙ T 6 κ
rot n
= 1−4 − .(16)
J (cid:18) M 5 utf(e)(cid:19)
whereP isthepressure,ρtherestmassdensity,nthe
polytropic index and K the polytropic constant, κ
n Here we have used equations (6) and (14) as well as
can be derived from the Lane-Emden function (see, T = ΩJ/2. The magnitude of J/M2 therefore in-
rot
e.g, LRS). Equation (8) can then be written
creases if
J˙=−f(e) 5 J L. (14) 1−4Trot − 6 κn <0. (17)
3κn M M 5 utf(e)
From equation (14), we find that the angular mo- For a dynamically stable rotating ellipsoid with rota-
mentum J changes by at most (Huet, 1996) tional velocity v ≪ 1 and T /|W| < 0.27 (see be-
rot ∼
low), we have T /M ≪ 1, ut ∼ 1 and f(e) ∼ 1, so
∆J 5 U rot
hot
∼− , (15) that this criterion reduces to
J 3κ M
n
5
assuming that the merged remnant radiates away all κn > . (18)
6
of its thermal energy U ∼ L∆τ prior to collapse.
hot
The ratio U /M following merger can be estimated The critical value of κ =5/6 corresponds to a poly-
hot n
bytakingtheprogenitorstarstobecold,withU = tropic index of n = 0.45 (see, e.g., LRS, Table 1).
hot crit
0,priorto reachingthe ISCO.Duringthe subsequent Neutrinoradiationthusleadstoanincreaseofthean-
plunge and merger, a part of the kinetic energy will gular momentum parameter J/M2 for all equations
be converted into thermal energy by contact shocks of state with n < n . Interestingly, the value of
crit
(Rasio & Shapiro, 1994; Ruffert, Janka & Scha¨fer, n = 0.45 is very close to typical values that are
crit
1996). In the extreme case of a head-on collision, a expected in realistic neutron star equations of state.
verylargepartofthekineticenergywillbeconverted Notethatwehavederivedthisnumericalvalueinthe
into heat. The ISCO, however, usually occurs at a limit of slow rotation. For rapid rotation, as in the
very small separation of the stars, and therefore we numericalexampleofSection3,wefindthatthevalue
expect that only a small part of the kinetic energy for n increases in the ellipsoidal approximation.
crit
will be converted into heat. For values of n close to n , the terms on the
crit
To estimate U , we construct post-Newtonian, left hand side of equation (17) will always be close
hot
compressibleellipsoidmodelsofhot,uniformlyrotat- to zero. According to equation (16), the fractional
ing stars, generalizing the cold post-Newtonian mod- change of J/M2 will therefore be much smaller than
elsofLombardi,RasioandShapiro(1997). Matching thefractionalchangeofJ,whichwehaveestimatedto
these configurations to binary models at the ISCO beintheorderofafewpercentatmost. Nevertheless,
calculated by Baumgarte et. al. (1997b), we typi- itisinterestingtonotethatthe emissionofneutrinos
cally find U /U ∼ 0.2−0.3, where U is the can lead to an increase in J/M2, and that for nearly
hot cold cold
total internal energy, excluding the thermal energy Newtonian configurations the sign of the change is
(i. e. kinetic energy of degenerate nucleons; see Sec- determinedessentiallybythestiffnessoftheequation
tion 3 and Appendix A). Since for typical neutron of state.
4
Fig. 1.— Time evolution of a rapidly rotating neu- Fig. 2.— Evolution of Model 2 (see text). The plots
tron star in a post-Newtonian ellipsoidal approxima- are labeled as in Figure 1. Although Model 2 is ini-
tion (Model 1, see text). We show the luminosity tially stabilized against collapse by a combination of
L, the angular momentum J/M2, the principle axes thermal pressure and angular momentum, a delayed
0
a (solid line in the lower left panel) and a (dashed collapseisinducedassoonasitradiatesawayasmall
1 3
line), and the angular momentum parameter J/M2. part of its thermal energy. Note that the neutrino
AlthoughthemergedremnantinModel1exceedsthe luminosity L increases during the collapse.
maximumallowedmassofacold,nonrotatingconfig-
urations, it is stabilized against collapse by angular
ergy M and angular momentum J match those of
momentum.
binary configurations at the ISCO as calculated by
Baumgarte et. al (1997a,b). Here, we summarize re-
3. A NUMERICAL EXAMPLE sults for two different progenitor binary models with
n =1 and J/M2 ≥1 (see lines 5 and 6 of Table 2 in
To illustrate these effects, we dynamically evolve
Baumgarteet. al., 1997b,forthe binaryparameters).
axisymmetric post-Newtonian, compressible ellipsoid
The twomodels differ byhowclosethe stellarmasses
modelsofhotneutronstarmergerremnants. Weem-
are to the maximum allowed mass of an isolated star
ploy the formalism of Lai, Rasio & Shapiro (1994),
in spherical equilibrium. We pick these two particu-
supplemented by post-Newtonian corrections (Lom-
lar cases since their evolution turns out to be quali-
bardi, Rasio & Shapiro, 1997) and thermal contribu-
tatively different. For both models, we fix K such
cold
tions to the internal energy. The thermal corrections
that the rest mass of the remant is M =2×1.5M ,
are taken into account by naively decomposing the 0 ⊙
since observed binary neutron stars all have gravita-
polytropic constant K in (13) into a linear sum of a
tional masses close to 1.4 M , corresponding to rest
⊙
cold and a hot contribution, K =K +K . The
cold hot masses ∼ 1.5M . K is determined by the rela-
complete formalism and equations can be found in tion K =(M⊙/M¯ )c2o/ldn, where the nondimensional
Appendix A. quantitcyoldM¯ is g0iven0in Table 2.
0
For initial data we construct equilibrium configu-
For Model 1 (line 5 of Table 2 in Baumgarte
rations as described in Appendix A.3. These models
et. al., 1997b), we have K = 208 km2. In our
cold
of hot, rotating neutron stars form a three parame-
post-Newtonian approximation, this yields a maxi-
ter family, which can be uniquely determined by the
mum allowed rest mass of Mmax = 1.92M (corre-
centraldensity, the eccentricity andthe thermalheat 0 ⊙
sponding to a maximum allowed gravitational mass
content (K /K ). We can choose these three pa-
hot cold of Mmax = 1.71M ) for isolated, nonrotating cold
⊙
rameters such that the stars’ rest mass M , total en-
0 stars. Thesevaluesareconsistentwiththoseobtained
5
from recent, realistic nuclear equations of state (see, In the high temperature limit, ǫ ∼ ǫ , so that
hot rad
e.g., Pandharipande, 1997). The initial gravitational T4 ∼ ρ1+1/n ∼ (a2a )−(1+1/n). Taking the surface
1 3
energy of Model 1 is M = 2.75M and the initial areaof the ellipsoid to be ∼4πa2 we find from(A33)
⊙ 1
angular momentum parameter is J/M2 = 1.05. The that in the diffusion approximation the luminosity
remnant has an eccentricity of e = 0.935, a thermal scales as
heat content K /K = 0.25 and a = 28.0 km. L∼a3−1/na−(1+1/n)/2 (19)
hot cold 1 1 3
The thermal energy in this model corresponds to a
For spherically symmetric collapse, a = a and
1 3
maximum temperature of ∼14 MeV. L ∼ a(5−3/n)/2. For all n > 3/5 (which accomodates
For Model 2 (line 6), we have K = 173 km2, 3
cold mostrealisticequationsofstate),sphericallysymmet-
corresponding to Mmax = 1.75M and Mmax =
0 ⊙ ric collapse therefore leads to a decrease of the lumi-
1.56M , so that this model’s rest mass exceeds the
⊙ nosity: the neutrino optical thickness increases faster
maximumallowedrestmass by a largeramountthan
than the thermal energy increases. If, however, the
Model 1. It has an initial gravitational mass of
star collapses to a pancake, as in Figure 2, a will
1
M = 2.72M⊙ and J/M2 = 1.0, corresponding to remain finite, and L ∼ a−(1+1/n)/2. In this case the
a remnant of e = 0.94, K /K = 0.33, and 3
hot cold increaseofthethermalenergyovercomestheincrease
a = 23.3 km. Here the maximum temperature is
1 ofthe opticalthickness,andtheluminosity increases.
∼28MeV,whichissimilartotypicalvaluesfoundby
Whenever the surface of a collapsing star approaches
Ruffert, Janka & Scha¨fer (1996).
a newly formed event horizon, the luminosity will al-
We show the dynamical evolution of these two
ways be suppressed by the increasing redshift and
models in Figures 1 and 2. Even though the rest black hole capture. These effects are absent in our
mass of Model 1 greatly exceeds Mmax, it is dynam-
0 model calculations. Nevertheless, our results suggest
ically stabilized, within our approximations, by its
that the collapse of a rapidly rotating, hot neutron
high angular momentum, even after all the thermal starfollowingmergermayleaveaverydistinctsigna-
energyhasbeenradiatedaway. By contrast,Model2
ture in the neutrino signal.
collapses after it has emitted part of its thermal en-
ergy. InModel1,theangularmomentumdecreasesby 4. ESTIMATE OF TIMESCALES
∼1.5%,whileJ/M2 increasesby∼0.3%. Thisshows
that post-Newtonian corrections in an ellipsoid ap- So far we have focussed on the emission of neu-
proximation, as well as rotation and deviations from trinos and their role in the evolution of the merged
sphericity,causethethecriticalpolytropicindexn remnant. To estimate if this evolution is indeed gov-
crit
to increase to a value greater than one. In Model 2, erned by neutrinos, we now compare the timescale
thechangesinJ andJ/M2 showthesametrendsbut for the neutrino emission with those associated with
are smaller, because the star emits only a small part eletromagnetic and gravitationalradiation.
of the thermal energy before it collapses. In a fully The timescale for the emission of the neutrinos is
relativistictreatment,the collapsewouldnotproceed givenbythediffusiontime,τ . TakingκtobeRosse-
ν
directly to a black hole, because the Kerr limit on landmeanopacityduetoscatteringoffnondegenerate
J/M2wouldbeexceeded. Binarieswithmassescloser neutrons and protons (see, e.g., Baumgarte, Shapiro
to the maximum nonrotating mass have J/M2 < 1 & Teukolsky, 1996), we find
(see Table 2 in in Baumgarte et. al., 1997b). Their
M
remnants could collapse directly to Kerr black holes. τ ∼ κρR2 ∼κ (20)
ν
R
Notealsothattheneutrinoluminosityincreasesas
Model 2 undergoes ’delayed’ collapse. This is in con- kT 2 M R −1
∼ 4 s.
trastto sphericallysymmetric results for delayedcol- (cid:18)20MeV(cid:19) (cid:18)2M (cid:19)(cid:18)15km(cid:19)
⊙
lapse(Baumgarte,Shapiro&Teukolsky,1996;Baum-
garte et. al., 1996), in which the luminosity always Assuming that electromagnetic emission is domi-
decreases during collapse. This difference can be un- nated by a magnetic dipole radiation, its timescale
derstoodfromasimplegeometricalscalingargument. can be estimated from (see, e.g., Shapiro & Teukol-
As explained in detail in Appendix A, we assume for sky, 1983)
simplicitythatwecanwritethethermalenergyinthe
T M
rot
polytropic form ǫhot =nKhotρ1+1/n (equation (A4)). τEM ∼ E˙ ∼ 3B2sin2αR4Ω2
6
−2 −2 −4
M Bsinα M M/R RΩ
∼ 7×107 ∼4×10−3 s.(24)
(cid:18)2M (cid:19)(cid:18) 1012G (cid:19) (cid:18)2M (cid:19)(cid:18) 0.2 (cid:19) (cid:18)0.5(cid:19)
⊙ ⊙
−4 −2
R Ω For t < t , the total gravitational radiation time-
s. (21) dyn
(cid:18)15km(cid:19) (cid:18)104s−1(cid:19) scaleτGW ∼τbar+τgrav isthereforedominatedbythe
initial growth time τ . Since τ strongly depends
bar bar
Clearly, τ is by many orders of magnitudes larger
EM on t, this timescale is quite uncertain. However, the
thanτ . Electromagneticradiationis therefore negli-
ν estimate (22) suggests that for compact stars with
gibleintheearlyevolutionofhotneutronstars,unless
J/M2 > 1, we have τ < 3 s. We then see that
the magnetic fields are extreme (B >1015 G). ∼ GW ∼
∼ the evolution of these merged stars is characterized
An axisymmetric, stationary star does not emit by the formation of a bar mode (see Lai & Shapiro,
gravitational waves. However, if the ratio of the 1995)andemissionofgravitationalwaves,whichmay
kinetic and potential energy t ≡ Trot/|W| is large then carry off angular momentum very efficiently. In
enough, the star will develop a bar instability and some cases, t > t , and a bar may develop on a
dyn
will then emit gravitational radiation. More specifi- dynamical timescale (∼ Ω−1 ∼ few ms; see Rasio &
cally,thestarissecularlyunstabletotheformationof Shapiro (1994) for a numerical demonstration).
abarmode ift>t ∼0.14,anddynamicallyunsta-
sec
ble if t > tdyn ∼ 0.27 (Chandrasekhar, 1969; Shapiro 5. CONCLUSIONS
&Teukolsky,1983;seeStergioulas&Friedman,1997,
andFriedman&Morsink,1997,forarecentrelativis- We conclude that neutrinos are very inefficient
tic treatment). Writing t in terms of the ellipsoidal in carrying off angular momentum from hot, mas-
expressions (A10) and (A18), we find sive remnants of neutron star binary mergers, even
if these are rapidly rotating and have an angular
5(5−n) e(1−e)1/6 J 2 M momentum parameter J/M2 > 1. Curiously, neu-
t = ∼
12κn sin−1e (cid:18)M2(cid:19) R trino emission may even increase J/M2 by a small
amount. Wefindthatthetimescalefortheformation
2
J M
∼ 2 . (22) ofa bar mode may,for suchobjects, be muchshorter
(cid:18)M2(cid:19) R
than the neutrino emission timescale. For pure grav-
SincerelativisticconfigurationstypicallyhaveM/R> itational radiation the third term in equations (16)
∼
0.1,thissuggeststhatallneutronstars,rotatingwith and (17) vanishes, so that J/M2 always decreases.
J/M2 > 1, have t > 0.2 and are at least secularly This suggests that gravitational waves will be faster
∼ ∼
unstable to the formation of a bar mode. This is and more efficient in reducing J/M2 than neutrinos.
in agreement with the general relativistic, numerical Together, these arguments may have important con-
models of Cook, Shapiro & Teukolsky (1992). For sequencesforgravitationalwavedetectors,since they
t < t < t , the timescale for the formation of suggest that the evolution of mergered binary rem-
sec dyn
a bar mode driven by gravitational radiation can be nants will be dominated by the emission of gravita-
approximatedby(see,e.g.,Friedman&Schutz,1975; tional waves. In particular, should J/M2 be larger
Lai & Shapiro, 1995) than unity upon merger, gravitationalwaves seem to
be the only means by which J/M2 can be efficiently
M M/R −4 t−t −5 reduced to a value smaller than unity, so that the
sec
τ ∼3 s. (23)
bar (cid:18)2M (cid:19)(cid:18) 0.2 (cid:19) (cid:18) 0.1 (cid:19) entire merged configuration can collapse to a black
⊙
hole. Otherwise, hydrodynamic stresses will have to
Note that shear viscosity can also drive a bar mode
support or expel some of the matter to allow the in-
instability, but it is inefficient in hot neutron stars
terior regions to collapse.
with T ∼ 10 MeV as is the case here (see Bonaz-
These arguments imply that, except for cooling
zola, Frieben & Gourgoulhon, 1996, and references
anddeleptonizing,neutrinoemissionplaysaverymi-
therein). Once a bar has fully developed, the grav-
norroleindetermining the massandspinofthe final
itational radiation dissipation timescale can be esti-
configuration (black hole or rotating neutron star).
mated from the quadrupole emission formula
However,because of the role in inducing delayed col-
T IΩ M lapse in some cases, the neutrinos will have to be in-
rot
τ ∼ ∼ ∼
grav E˙ (M/R)2(RΩ)6 (M/R)2(RΩ)4 cludedincalculationsofthelatestagesofcoalescence.
7
Gravitationalradiationwillplayaveryimportantrole where U is the internal energy, W the potential en-
in determining the final mass and spin of the black ergy, T the kinetic energy, and U and W are
PN PN
holeorneutronstar. Thisresultstrengthensthe con- post-Newtonian corrections to the internal and po-
clusion that a fully general relativistic description of tential energy.
binary neutron star mergers is essential for full un- Including the post-Newtonian terms not only im-
derstanding. proves the solution for relativistic stars quantita-
Notethatourargumentsapplytomassive,hot,ro- tively, but also, more importantly, it changes the
tating neutron stars in general, and not only to the solution qualitatively. Without the post-Newtonian
remnants of binary neutron stars. For example, they corrections, a graph of the rest mass M as a func-
0
equally apply to newly formed, hot neutron stars in tion of the central density ρ exhibits no turning
c
supernovae. Ithasrecentlybeensuggestedthatthese point. By including the post-Newtonian terms, we
stars may be stable initially, but may later delep- find turning points reasonably close to the turning
tonize, undergo a phase transition and collapse to a pointsofthecorrespondingfullyrelativistic(Tolman-
blackhole (see, e.g.,Brown& Bethe, 1994). Alterna- Oppenheimer-Volkov) solution (see Lombardi, Rasio
tively, such a delayed collapse might result from the & Shapiro, 1997). Since these turning points give
loss of angular momentum from a nascent, rapidly the maximum allowed mass configuration, the post-
rotating neutron star which is stabilized initially by Newtonian corrections allow us to approximate the
its high spin. Neutrino emission could, in principle, onset of radial instability and to mimic the delayed
reducethisangularmomentumandhence inducecol- collapse of hot, rotating neutron stars to black holes.
lapse. WhilefortypeIIsupernovaeUhot/M isslightly We assume a polytropic equation of state
larger than in the examples presented in Section 3,
ourargumentsshowthattheneutrinoemissionisstill P =Kρ1+1/n, (A2)
very inefficient in carrying off angular momentum.
and crudely account for thermal pressure by taking
This fact makes this scenario very unlikely (Huet,
the constant K to have a cold contribution due to
1996). Similarly, our analysis suggests that neutrino
degenerate nucleons, K , and a hot contribution
cold
emission plays a negligible role in determining the
due to thermal heating, K :
hot
quasi-steady spin rate of accreting neutron stars at
the center of Thorne-Zytkow objects, which have re- K =K +K . (A3)
cold hot
cently been suggested to be likely sources of gravita-
The internal thermal energy density ǫ is therefore
tional waves for detection by GEO (Schutz, 1997). hot
taken to be of the form
It is a pleasure to thank Patrick Huet and Chris ǫhot =nKhotρ1+1/n. (A4)
Pethick for stimulating our interest in some of the
While K is assumed to be constant in both space
issues discussed in this paper, and Fred Lamb and cold
and time, we allow K to depend on time, keeping
James Lombardi for useful discussions. This paper hot
it constant throughout the star. Integrating the en-
was supported in part by NSF Grant AST 96-18524
ergy density overthe star allowsus to write the total
and NASA Grant NAG 5-3420 to the University of
internalenergyinterms ofthe centraldensityρ asa
Illinois at Urbana-Champaign. c
sum of a cold and a hot contribution
U =U +U =k (K +K )ρ1/nM (A5)
cold hot 1 cold hot c 0
A. POST-NEWTONIAN, ELLIPSOIDAL
MODELS OF HOT, ROTATING NEU- where k1 is a dimensionless structure constant that
TRON STARS depends on the density profile, i.e. the polytropic in-
dex n. Numerical values for k can be found, for
1
A.1. The Euler-Langrange Equations
example, in LRS (see Table 1). In terms of the mean
radius
Thetotalenergyofanaxisymmetric,rotating,hot
R≡(a2a )1/3, (A6)
neutronstarcan,inapost-Newtonian,ellipsoidalap- 1 3
proximation, be written as the sum the central density can be written
M
M =U +UPN +W +WPN +T, (A1) ρc =βR30, (A7)
8
where we have defined and a contribution due to spin
β ≡ ξ1 , (A8) T = 1IΩ2 = 5 β−2/3J2M−5/3ρ2/3(1−e2)1/3.
4π|θ′(ξ1)| rot 2 4κn c
(A18)
andwhereθ andξ arethe usualLane-Emdenparam- Here the moment of inertia is given by
etersforapolytrope(see,e.g.,Chandrasekhar,1939).
The internal energy can therefore be rewritten I = 2κ M a2, (A19)
5 n 0 1
M1+1/n
U =k1Kβ1/n R03/n . (A9) andκn isastructureconstantsoforderunitythatde-
pends only on the polytropic index n (see, e.g., LRS)
The potential energy W is given by Following Lombardi, Rasio & Shapiro (1997), the
post-Newtonian corrections can be written
3 M2sin−1e
W = −5−n R0 e (1−e2)1/6 (A10) UPN = −l1Kρ1c/n+1/3M05/3
= − 3 β−1/3M5/3ρ1/3sin−1e(1−e2)1/6. = −l Kβ1/n+1/3M02+1/n (A20)
5−n 0 c e 1 R1+3/n
Introducing the dimensionless coefficients A (Chan- and
i
drasekhar, 1969,§17)
M3
W =−l ρ2/3M7/3 =−l β2/3 0. (A21)
sin−1e 1−e2 PN 2 c 0 2 R2
A =(1−e2)1/2 − (A11)
1 e3 e2 Numericalvaluesforthestructureconstantsl andl
1 2
are calculated in Lombardi, Rasio & Shapiro (1997).
and
Both post-Newtonian terms are correct in this form
A =2−2A (A12)
3 1
only for spherical configurations. However, we shall
(for a1 >a3), the potential energy can be rewritten assume the slow rotation limit (Trot/|W| ≪ 1), for
whichcorrectionsarisingfromdeviationsfromspheric-
3 M2 J
W =− 0 , (A13) ity are higher order and will be neglected.
5−n R 2R2
From these energy contributions we can now con-
where struct the Lagrangian
J =2A a2+A a2. (A14)
1 1 3 3
L(q ;q˙ )=T −U −U −W −W , (A22)
i i PN PN
ThelasttermJ/(2R2)takesintoaccountcorrections
tothepotentialenergyduetodeviationsfromspheric- where the q are the independent variables a and φ
i i
ity. Note that partial derivatives of J take the form (with φ˙ = Ω). The dynamics of the system is then
governedby the Euler-Lagrangeequations
∂J 1
= (J −a2A ), (A15)
∂a a i i d ∂L ∂L
i i − =0. (A23)
dτ ∂q˙ ∂q
i i
so that
∂W = 3 M02a A (A16) The angular momentum can be defined as
∂a 5−n2R3 i i
i
∂L
(see Chandrasekhar,1969). J ≡ . (A24)
∂Ω
Assumingthestartobeuniformlyrotating,theki-
neticenergyT canwrittenasasumofacontribution Since φ is an ignorable coordinate of the Lagrangi-
due to expansion an(A22),the Euler-Lagrangeequationsimply thatJ
is conserved on a dynamical timescale. However, we
T = 1 κ M (2a˙2+a˙2) (A17) wanttoallowforthe radiationofangularmomentum
exp 10 n 0 1 3 on a secular timescale by the emission of neutrinos,
so we will therefore allow for an external dissipation
9
term on the right hand side of equation (A23) when neutrinos
q =φ: ǫ =(3+7N /8)aT4. (A30)
i rad ν
dJ
=−(J˙) . (A25) The total thermal energy density is then given by
ν
dτ
Note also that when we allow Khot to decrease with ǫhot =ǫrad+ǫnucl. (A31)
timeasthestarcools(seeequation(A39)below),the
Thisexpressioncanbeusedtoestimatethe tempera-
system will no longer conserve energy.
tureT. AdoptingtheRosselandmeanopacityarising
We now find that the Euler-Langrange equations
from scattering off nondegenerate neutrons and pro-
yield
tons, the neutrino opacity takes the form
a¨1 = Ω2a1+Kβ1/nn5κk1nRM3/01n/an1 − (A26) κ= 7(22π0)24σm0B (cid:18)mkTe(cid:19)2, (A32)
K 1 + 1 β1/n+1/35l1 M01/n+1 − where mB is the baryon mass, me the electron mass,
(cid:18)n 3(cid:19) κn R3/n+1a1 and σ0 = 1.76×10−44cm2. The neutrino luminosity
can then be estimated via the diffusion formula
10 l M2 5 3 M
3 β2/3κ2nR2a01 − 2κn5−n R30a1A1, L= A |∇ǫrad|∼ A ǫrad, (A33)
3ρ κ 3ρ κ a
a a 3
5k M1/n
a¨3 = Kβ1/nnκ1 R30/n − (A27) whereρa =3M0/(4πR3)istheaveragedensity. Since
n a < a , we expect that the “preferred” direction of
3 1
1 1 5l M1/n+1 diffusionisalongtherotationaxis,andthereforeesti-
K + β1/n+1/3 1 0 −
(cid:18)n 3(cid:19) κn R3/n+1a3 matethegradientofǫradbydividingǫradbya3. Using
equation(8), the angularmomentumlosscannowbe
10 l M2 5 3 M
β2/3 2 0 − 0a A , written
3 κnR2a3 2κn5−n R3 3 3 (J˙)ν =<̟2 >ΩL. (A34)
and
Next we have to relate the luminosity L to K .
1 a a˙ hot
Ω˙ =−I(J˙)ν −2 a121Ω. (A28) From equation (5) we have
1
In the absence of neutrino emission ((J˙)ν =0), these L= δW(e) =−<TdS >, (A35)
ordinary differential equations completely describe dτ dτ
the adiabatic evolution of a uniformly rotating, ax- where the brackets denote an average over the star
isymmetric neutron star in a post-Newtonian ellip- and S is the total entropy of the star. From the first
soid approximation. They represent in this limit the law of thermodynamics we have
ellipsoidal analogue of the post-Newtonian equations
ǫ 1
of hydrodynamics for an adiabadic gas. In order to Tds=d +Pd . (A36)
(cid:18)ρ(cid:19) (cid:18)ρ(cid:19)
allow for cooling and loss of angular momentum, we
have to supplement this system ofequations with ex- Here ǫ = nKρ1+1/n is the total internal energy den-
pressions for K˙ and (J˙) .
hot ν sity,ands isthe specificentropy(perrestmass). For
adiabatic changes, K is constant and Tds=0. Here,
A.2. Neutrino Cooling and Angular Momen-
K = K +K is not constant, however, and we
tum Loss cold hot
find
We approximate the thermal energy contribution Tds=nρ1/ndK. (A37)
from hot nucleons by
InaNewtoniantreatment,integrationoverthewhole
3 star now yields
ǫ = nkT (A29)
nucl
2
<TdS > = (Tds)dm=dK nρ1/ndm
(strictly true in the limit of high temperatures), and Z Z
add to it the thermal radiation of photons, electron-
= dlnK ǫdm=UdlnK. (A38)
positron pairs and Nν =3 flavors of (nondegenerate) Z
10