What a Two Solar Mass Neutron Star Really Means

26 Pages·2013·0.37 MB·English

July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
International JournalofModernPhysicsB
(cid:13)c WorldScientificPublishingCompany
0
1
0
2 What a Two Solar Mass Neutron Star Really Means

c
e
D
JAMESM.LATTIMER

5 Department of Physicsand Astronomy
1 Stae Universityof NewYork at Stony Brook
Stony Brook, NY 11794-3800, USA

] [email protected]
R
S
MADAPPAPRAKASH
.
h Department of Physics and Astronomy, Ohio University
p Athens, Ohio 45701, USA
- [email protected]
o
r
t ReceivedDayMonthYear
s
a RevisedDayMonthYear
[

Thedeterminationofneutronstarmassesisreviewedinlightofanewmeasurement of
1
v 1.97M⊙forPSRJ1614-2230andanestimateof2.4M⊙fortheblackwidowpulsar.Using
a simple analytic model related to the so-called maximally compact equation of state,
8
model-independent upper limitsto thermodynamic properties inneutron stars, suchas
0
energydensity,pressure,baryonnumberdensityandchemicalpotential,areestablished
2
which depend upon the neutron star maximum mass. Using the largest well-measured
3
. neutron star mass, 1.97 M⊙, it is possible to show that the energy density can never
2 exceed about 2 GeV, the pressure about 1.3 GeV, and the baryon chemical potential
1 about 2.1 GeV. Further, if quark matter comprises a significant component of neutron
0 star cores, these limits are reduced to 1.3 GeV, 0.9 GeV, and 1.5 GeV, respectively.
1 Wealsofindthemaximumbindingenergyofanyneutronstarisabout25% oftherest
: mass. Neutron matter properties and astrophysical constraints additionally imply an
v
i upper limitto the neutron star maximum mass of about 2.4 M⊙. A measured mass of
X 2.4M⊙wouldbeincompatiblewithhybridstarmodelscontainingsignificantproportions
ofexoticaintheformofhyperons,Bosecondensates orquarkmatter.
r
a
Keywords: neutronstarmasses;equationofstate.
1. Introduction
One of the most fascinating stories in astrophysics concerns the accumulation of
preciselymeasuredneutronstarmasses.Gerryhashadalong-timeinterestinthese
measurements,asformanyyearshehasmaintainedthattheneutronstarmaximum
mass is no greater than (1.5-1.6) M⊙. From a theoretical perspective, a maximum
mass of 1.5 M⊙ fits his fondness for the effects of kaon condensation proposed by
1
Kaplan and Nelson : the effective kaon mass falls with increasing density and the
eventual onset of kaon condensation at a few times the nuclear saturation density
softenstheequationofstate(hereafter,EOS)andleadstoarathersmallmaximum
1
July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
2 James M Lattimer and Madappa Prakash
mass.He takesasobservationalsupportofthis thesis the “missing”neutronstarin
the remnant of SN 1987A,which had a baryonmass not in excess of 1.7 M⊙ based
2
on estimates of the initial mass of the star and its ejected mass . He reasons that
the maximum stable mass of the proto-neutron star as it deleptonized must have
3
eventually fallen below its actual mass, prompting a collapse to a black hole . Of
course,thishadtohavehappenedmorethan12secondsafterthecorecollapsesince
neutrinos were observed during that time period and their emission is believed to
4
abruptly cease once the black hole’s event horizon forms.
Measurements of Neutron Star Masses
The most accurate measurements of neutron star masses are for pulsars in bound
binary systems. In these systems, five Keplerian parameters can be precisely mea-
5
sured by pulse-timing techniques , including the binary period P, the projection
of the pulsar’s semimajor axis on the line of sight a sini (where i is the binary in-
p
clinationangle),the eccentricitye,andthe time andlongitudeofperiastronT and
0
ω.Combiningtwooftheobservationalparameters,onecanformthemassfunction:
2π 2 apsini 3 M⊙ (Mcsini)3
fp = P c T⊙ = M2 M⊙, (1)
(cid:18) (cid:19) (cid:18) (cid:19)
where M = M + M is the total mass, M is the pulsar mass, and M is the
p c p c
companionmass(allmeasuredinM⊙ units).TheconstantT⊙ =GM⊙/c3 is4.9255
µ s. The mass function f is also equal to the minimum possible mass M for the
p c
companion.
The inclination angle i is often the most difficult aspect to infer, but even if it
was known a priori the above equation only specifies a relation between M and
p
M unless the mass function f of the companion is also measurable. This occurs
c c
in the rare case when the companion is itself a pulsar or a star with an observable
spectrum.
Fortunately, binary pulsars are compact systems and general relativistic effects
can often be observed. These include the advance of the periastron of the orbit
5/3
2π
ω˙ =3 (MT⊙)2/3(1 e2)−1, (2)
P −
(cid:18) (cid:19)
the combined effect of variations in the tranverse Doppler shift and gravitational
redshift around an elliptical orbit
P 1/3 M (M +M )
c c 2/3
γ =e T , (3)
2π M4/3 ⊙
(cid:18) (cid:19)
and the orbital period decay due to the emission of gravitationalradiation
5/3
P˙ = 192π 2πT⊙ 1+ 73e2+ 37e4 (1 e2)−7/2MpMc. (4)
− 5 P 24 96 − M1/3
(cid:18) (cid:19) (cid:18) (cid:19)
July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
What a Two Solar Mass Neutron Star Really Means 3
The inclination angle can be constrained by measurements of two or more of these
effects.However,only inextremely compactsystems is this preciselypossible.Oth-
erwise, additional effects, such as an eclipse or limits obtained from the lack of
an eclipse, or Shapiro time delay, which is caused by the propagation of the pul-
sar signal through the gravitational field of its companion, must be observed. The
Shapiro 6 delay in general relativity produces a delay in pulse arrival time 7,8
1+ecosφ
δS(φ)=2McT⊙ln , (5)
1 sin(ω+φ)sini
(cid:20) − (cid:21)
where φ is the true anomaly, the angular parameter defining the position of the
pulsar in its orbit relative to the periastron. δ is a periodic function of φ with an
S
approximate amplitude,
1+esinω 1+sini
∆S 2McT⊙ ln , (6)
≃ 1 esinω 1 sini
(cid:12) (cid:20)(cid:18) − (cid:19)(cid:18) − (cid:19)(cid:21)(cid:12)
(cid:12) (cid:12)
which is large only for edge-on(cid:12)binaries, sini 1 or those wh(cid:12)ich have both large
(cid:12) ∼ (cid:12)
eccentricities and large magnitudes of sinω. Only a fraction of pulsars in binaries
have two or more sufficiently well-measured relativistic effects to enable precise
measurements of the pulsar mass M .
p
In some cases, the companion star can be optically detected. In most cases, the
companion is a white dwarf. It is possible to estimate white dwarf masses from
observations of the optical flux, effective temperature and distance, with the latter
being especially problematic. Another possibility is to measure the surface gravity
fromspectralmeasurements,butcurrentlythisissubjecttolargesystematiceffects.
Less accurate measurements of neutron star masses are possible in X-ray bina-
ries, in which X-rays are emitted by matter accreting onto a neutron star from a
companionstar.BothX-rayandopticalobservationscanyieldbothmassfunctions
f and f , but f is subject to largeuncertainties due to the faintness ofopticalra-
p c p
diation.Inthecaseofthe blackwidowpulsar,PSRB1957+20,opticalobservations
of the companion yield both a mass function and an estimate of the inclination i
9
fromtheshapeofthelightcurve .Theinferredsemimajoraxisofthecompanion’s
orbit has to be corrected for the finite size of the companion, however (see below),
introducing a source of systematic error in the pulsar mass determination of this
49
system. Nevertheless, interesting lower limits to the pulsar mass are obtained .
Recent Measurements
The most recent information, as of November 2010,is summarized in Figure 1 and
Table 1 andcollects informationfromRefs. 10 to 50. This compilationrepresentsa
significant update to the figure of observed masses and references in Ref. 52.
There is now ample observational support from pulsars for neutron stars with
massessignificantlygreaterthan1.5M⊙.TheseincludePSRJ1903+0327whichhas
a main-sequence companion; the pulsars I and J in the globular cluster Ter 5, the
July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
4 James M Lattimer and Madappa Prakash
Fig. 1. Measured neutron star masses. References in parenthesis following source numbers are
identifiedinTable1.
July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
What a Two Solar Mass Neutron Star Really Means 5
Table1. Neutronstarmassmeasurements with1σ uncertainties. Referenceletters correspondtoFig.
1.Foreachgroup,meanandweightedmassesareindicated.
Object Mass(M⊙) Reference Object Mass(M⊙) Reference
X-Ray/Optical Binaries (mean =1.609 M⊙, weighted mean =1.393 M⊙)
4U1700-377 2.44+−00..2277 a(10) VelaX-1 1.86+−00..1166 b,c(11,12)
CygX-2 1.71+−00..2211 d(13) 4U1538-52 0.96+−00..1196 k(14)
SMCX-1 1.06+−00..1110 E(15) LMCX-4 1.25+−00..1110 E(15)
CenX-3 1.34+−00..1164 E(15) HerX-1 1.47+−00..1128 k(14)
XTEJ2123-058 1.53+−00..3402 l(16,17) 2S0921-630 1.4+−00..11 F(18)
4U1822-371 1.96+−00..3365 n(19) EXO1722-363 1.545+−00..446655 f(47)
B1957+20 2.39+−00..3269 Q(49)
Neutron Star – Neutron Star Binaries (mean =1.325 M⊙, weighted mean =1.403 M⊙)
J1829+2456 1.338+−00..030328 z(20) J1829+2456 (c) 1.256+−00..304063 z(20)
J1811-1736 1.608+−00..066068 A(21) J1811-1736 (c) 0.941+−00..708271 A(21)
J1906+07 1.694+−00..061924 B(22) J1906+07(c) 0.912+−00..701004 B(22)
J1518+4904 0.72+−00..5518 C(23) J1518+4904 (c) 2.00+−00..5581 C(23)
1534+12 1.3332+−00..00001100 K(24) 1534+12(c) 1.3452+−00..00001100 K(24)
1913+16 1.4398+−00..00000022 q(25) 1913+16(c) 1.3886+−00..00000022 q(25)
2127+11C 1.358+−00..001100 x(26) 2127+11C (c) 1.354+−00..001100 x(26)
J0737-3039A 1.3381+−00..00000077 i(27) J0737-3039B 1.2489+−00..00000077 i(27)
J1756-2251 1.312+−00..001177 J(28) J1756-2251 (c) 1.258+−00..001177 J(28)
Neutron Star – White Dwarf Binaries (mean =1.599 M⊙, weighted mean =1.362 M⊙)
B2303+46 1.38+−00..0160 e(29) J1012+5307 1.68+−00..2222 m(30)
J1713+0747 1.53+−00..0084 r(31,51) B1802-07 1.26+−00..0187 e(29)
B1855+09 1.57+−00..1121 g(32,51) J0621+1002 1.70+−00..1107 y(33)
J0751+1807 1.26+−00..1144 y(33) J0437-4715 1.76+−00..2200 p(34)
J1141-6545 1.27+−00..0011 j(35) Ter5I 1.874+−00..30268 t(36)
Ter5J 1.728+−00..016366 t(36) J1909-3744 1.47+−00..0032 u(37)
J0024-7204H 1.41+−00..0048 w(38) B1802-2124 1.24+−00..1111 R(50)
J0514-4002A 1.497+−00..040987 D(39) B1516+02B 2.08+−00..1199 I(40)
J1748-2021B 2.74+−00..2211 H(41) J1750-37A 1.26+−00..3399 H(41)
J1738+0333 1.55+−00..5555 L(42) J1911-5958A 1.34+−00..0088 O(43)
J1614-2230 1.97+−00..0044 P(48) J1045-4509 <1.48 e(29)
J1804-2718 <1.70 e(29) J2019+2425 <1.51 h(44)
Neutron Star– Main Sequence Binaries
J0045-7319 1.58+−00..3344 M(45) J1903+0327 1.67+−00..0011 N(46)
globular cluster binaries PSR J1748-2021Band PSR B1516+02B,and PSR J1614-
2230,allwithwhite-dwarfcompanions;andtheblackwidowpulsarB1957+20.The
inclinationanglesofthetwobinariescontainingpulsarsIandJintheglobularclus-
terTer5areunconstrainedbyobservation,butiftheirinclinationsareindividually
randomlydirected,thereisa95%chancethatatleastoneofthesepulsarsisgreater
36
than1.68M⊙ . Assumptions relatedto randomorientationalso haverolesin the
mass determinations for PSR J1748-2021B and PSR B1516+02B. One has to be
careful about the assumption of random orientation, however, as the selection of
July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
6 James M Lattimer and Madappa Prakash
potential candidates for follow-up observations necessary for accurate mass deter-
minations are biased. The case of PSR J0751+1807 illustrates the point: initially
estimated to have a mass 2.2 0.2 M⊙, the mass was recently established to be
±
1.26 M⊙ upon measurement of another relativistic parameter. Nevertheless, about
46
2yearsago,ameasurementofthe inclinationofPSRJ1903+0327 usingShapiro
time delayleadtoanaccuratelydeterminedmassof1.67 0.01M⊙,replacingPSR
±
B1913+16as establishing the value for the minimum maximum mass.
Butevenmoreoutstandingistherecentdeterminationof1.97 0.04M⊙ forthe
48 ±
massofPSRJ1614-2230 ,alsousingShapirotimedelaytomeasuretheinclination.
This3.15mspulsarisinan8.7dnearlycircularorbitaboutan0.5M⊙ companion,
with a sini=11.3 light-second and sini=0.99989, i.e., it is almost edge on. The
p
Shapiro time delay amplitude, from Eq. (6), becomes
∆s 4McT⊙ln(2/cosi) 48.8 µs. (7)
≃ ≃
By virtue of its accuracy,this mass measurement has become the new standardfor
the minimum value of the neutron star maximum mass.
In addition, a number of X-ray binaries seem to contain high-mass neutron
stars:about1.9M⊙ inthe caseofVela X-1and2.4 M⊙ inthe caseof4U1700-377.
Nonetheless, the large systematic errors inherent in X-ray binary mass measure-
ments warrantcaution.
The black widow pulsar represents an intriguing case, with a best estimate of
about 2.4 M⊙. This system has both pulsar timing and optical light curve infor-
mation. It consists of a 1.6 ms pulsar in a nearly circular 9.17 h orbit around an
extremely low mass companion, Mc 0.03 M⊙. The pulsar is eclipsed for about
≃
50–60 minutes of each orbit, but considering that a sini = 0.089 light-second =
p
0.038 R⊙, and acsini 3 R⊙ is Mp/Mc 80 times larger,the eclipsing object has
∼ ≃
to be much larger ( 3 R⊙/10) than the size of the companion star. It is believed
∼
that irradiation of the companion by the pulsar strongly heats its nearside to the
point of ablation, leading to a comet-like tail and a large cloud of plasma which
is responsible for the eclipsing. The pulsar is literally consuming its companion,
hence the name “black widow”, and has reduced its mass to a small fraction of its
original mass. The irradiation also produces an enormous (factor of 100) variation
in the brightness of the companion depending on how much of its nearside is visi-
ble during its orbit. The companion is bloated and nearly fills its Roche lobe. The
companion’s optical light curve allows the mass ratio M /M and the inclination
p c
angle i to be estimated. However, the large size of the companion means that the
“center of light” of the system is not equivalent to its “center of mass”:the optical
light curve depends on the projected semi-major axis of the irradiated nearside of
the companion rather than the projected semi-major axis of the center of mass of
the companion.Theextremecasesareeitherthatthe companionhaszeroradiusor
that it completely fills its Roche lobe, but estimates based on modeling consider-
ably reduce the allowed range. The extremes lead to a range 1.7 < Mp/M⊙ < 3.2,
but the likely value is 2.4 0.4 M⊙. It will clearly be valuable to extend observa-
±
July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
What a Two Solar Mass Neutron Star Really Means 7
tionsandmodelingofthissystemsincea2.4M⊙ neutronstarwouldhaveprofound
implications.
What Gerry Would Say
We would love to be able to argue with Gerry about these observations and their
implications, and so we felt this was an ideal topic to explore in this contribution.
We can, however, guess what Gerry would have said: “We can’t tell nature how
to behave; we have to explain it” and, as he often quoted Bismarck, “Was schert
mich mein Geschwaetz von gestern?” (“Why do I care about my twaddle from
yesterday?”)
What We Say
One of the most surprising consequences of a large pulsar mass measurement is
that it severely restricts values of the maximum central density and pressure in
thatstar,and,byextension,thatofanyotherneutronstaraswell.Thisrealization
waselaboratedinRef.52,butitcanbe extendedtorestrictthe possibilities forthe
presenceofexoticmatter,suchashyperons,Bosecondensatesand/orquarkmatter,
in neutronstar interiors.Moreover,we show here that upper limits to the chemical
potentials of quark or baryonic matter can be established using the available mass
data,andthatthesearesmallerthanoftenemployedincalculationsoftheproperties
of quark matter. The consequences of well-determined large neutron star masses is
the topic to which we now turn our attention.
2. Neutron Star Structure
The global properties of neutron stars are determined, assuming general relativity,
53
by the Tolman-Oppenheimer-Volkov (TOV) equations
dp G(ε+p)(m+4πr3p/c2)
= , (8)
dr − r(rc2 2Gm)
−
dm
=4πr2ε/c2, (9)
dr
d 2Gm −1/2
N =4πr2n 1 , (10)
dr − rc2
(cid:18) (cid:19)
where m is the enclosedgravitationalmass, is the enclosedbaryonnumber, p is
N
the pressure, ε is the total energy density, n is the baryonnumber density, and r is
the radialvariable.Choosing a value of the pressureor baryonor energy density at
the star’scenter,andintegratingtothe surfacewherethe pressurevanishes,results
inaspecificmassandradius,asshowninFig.2.Aconsequenceofgeneralrelativity
istheexistenceofamaximummass:configurationswithlargercentralpressuresare
dynamically unstable and collapse into black holes.
July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
8 James M Lattimer and Madappa Prakash
Fig.2. SolutionoftheTOVequations.Integrationsbeginningatthestar’scenterwithpressures
and baryon number densities (in units of the nuclear saturation density ns) shown by the dots
inthe leftpanel resultinspecificmasses andradii,shownbythecorrespondingdots intheright
panel.
It is convenientto recastthe TOVequations in terms of the dimensionless vari-
able h defined by dh = dp/(ε +p) = dµ/µ where µ = (ε + p)/n is the baryon
chemical potential, which in beta equilibrium is the same as the neutron chemical
potential. Therefore,
µ=µ eh, (11)
o
where µ =ε /n , ε andn arethe chemicalpotential, energydensity andbaryon
o o o o o
number density, respectively, where p = h = 0. For a star with a normal crust,
µ 930 MeV, the approximate mass-energy of iron nuclei per baryon. Note that
o
≃
the quantity h is related to the metric function ν, where g =eν and dh= dν/2.
tt
−
One finds
dr2 2r2c2 rc2 2Gm
= − , (12)
dh − G mc2+4πr3p
(cid:20) (cid:21)
dm 4πr3ε rc2 2Gm
= − , (13)
dh − G mc2+4πr3p
(cid:20) (cid:21)
d 4πr7/2nc3 √rc2 2Gm
N = − . (14)
dh − G "mc2+4πr3p#
Writing the TOV equations this way ensures that near both the center and the
surface the right-hand sides are finite. The equations are integrated from a central
value h = h to h = 0. In order to obtain a specific total gravitational mass M =
c
m(h=0),oneiteratestofindthe appropriatevalueofh .ForagivenEOS,thatis,
c
the functions ε(h) and p(h), a specific value or h , h , results in the maximum
c max
mass configuration where M =M ,N =N and R=R where R=r(h=
max max max
0) is the stellar radius and N = (h=0) is the total baryon number. The baryon
N
July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
What a Two Solar Mass Neutron Star Really Means 9
Fig.3. ThesolidlineisatypicalneutronstarEOS.Thedashedlineshowsthemaximallycompact
EOS formed by minimizing the pressure p at low energy density ε (i.e., p= 0) and maximizing
p,subjecttocausality, athighdensity(i.e.,p=ε−εo).pandεareinthe samearbitraryunits,
withεo inthisexamplesetto2.
number is useful in the definition of the binding energy
BE =(Nm M)c2, (15)
b
−
where m is the baryon mass.
b
It is an inescapable consequence of general relativity that all EOSs result in a
maximum mass configuration, in contrast to Newtonian gravity for which no such
limit exists. It is intuitive that the largest maximum masses will be a consequence
of the stiffest EOSs, i.e., those with the largest pressures for a given density. It is
commonlyassumedthatthe maximumrealisticlimit to the pressureisobtainedby
assuming causality, in other words, that the sound speed is limited everywhere by
54
lightspeed.TheassumptionofcausalityledRhoadesandRuffini totheirestimate
forthemaximumneutronstarmass.Itislogical,therefore,thatthemaximummass
configuration so obtained must also have the largest possible values for the central
pressure,energydensity andchemicalpotential.The mainmotivationofthis paper
is to explore the consequences of this in light of new observations of large neutron
star masses.
55
ItwasarguedbyHaenselandZdunik andbyKoranda,StergioulasandFried-
56
man that the mostcompactconfigurationsresultfromthe combinationofa soft
EOSat small densities and a stiff EOS at large densities. The extreme limit of this
situation is shown in Fig. 3, where the pressure vanishes below the density ε and
o
is assumed causal at all higher densities (the so-called maximally compact EOS):
p=0 ε ε , p=ε ε ε ε . (16)
o o o
≤ − ≥
Since ε is the only EOS parameter, the TOV equations scale with it according
o
July 24, 2013 8:15 WSPC/INSTRUCTION FILE gebfest˙jl
10 James M Lattimerand Madappa Prakash
57
to
ε ε , p ε , n ε , m ε−1/2, r ε−1/2. (17)
∝ o ∝ o ∝ o ∝ o ∝ o
58
Following Lattimer, Prakash, Masak and Yahil (hereafter LPMY), we wish to
study a somewhat more general EOS, which encompasses the maximally compact
EOS as a special case, namely,
p=s(ε ε ), (18)
o
−
where s is a constantrepresentingthe square of the adiabatic soundspeed. We can
then express ε, p and the baryon number density n in terms of h as
ε=t−1ε (s−1eht+1), p=t−1ε (eht 1), n=ε eh/s/µ (19)
o o o o
−
where t=1+1/s. Using the dimensionless variables
x=r2ǫ G/c4, y =mǫ1/2G3/2/c4, z = ǫ1/2µ G3/2/c6 (20)
o o N o o
the scale-free TOV equations become
dx 2x(x1/2 2y)
= − , (21)
dh −y+4πx3/2p/ε
o
dy ε dx
=2πx1/2 , (22)
dh ε dh
o
dz nµ x dx
o
=2π . (23)
dh ε (x1/2 2y)1/2dh
o
−
For a given value of s, the value of h is the value of h that maximizes the
max c
total gravitational mass y(h = 0). By virtue of Eq. (11), h then determines
max
the maximum possible chemical potential for that EOS. Table 2 summarizes the
propertiesofthemaximummasssolutionsforthecasess=1,4/5,2/3ands=1/3.
The case s=1 represents the maximally compact star; as discussed in 4, the case
§
s=1/3 is potentially applicable to quark stars or stars with quark cores. For each
s are listed the central values of h, ε and p, and the surface values of x,y and z,
together with other dimensionless quantities of interest.
We will demonstrate that the maximally compact and the s=1/3 EOSs result
in useful approximate limits for the maximum values of thermodynamic quantities
in the interior of neutron and quark stars as well as their maximum masses.
3. Maximally Compact EOS
We first examine the maximally compact EOS, i.e., the case s = 1. Expressing
the results for s = 1 shown in Table 2 in astrophysical units (note that 1 MeV