Phase transitions in rotating neutron stars cores: back bending, stability, corequakes and pulsar timing

0.4 MB·English

by  J. L. Zdunik

Astronomy & Astrophysics manuscript no. 4260 (cid:13)c ESO 2008
February 5, 2008
Phase transitions in rotating neutron stars cores: back bending,
stability, corequakes, and pulsar timing
1 1 1 2
J. L. Zdunik , M. Bejger , P. Haensel , and E. Gourgoulhon
1 N.Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka18, PL-00-716 Warszawa, Poland
2 LUTH,UMR 8102 du CNRS,Observatoire deParis, F-92195 Meudon Cedex, France
6 [email protected], [email protected], [email protected], [email protected]
0
0 Received:28.09.2005, Accepted:06.01.2006
2

n ABSTRACT
a
J Aims. We analyze potentially observable phenomena during spin evolution of isolated pulsars, such as back bending and

5 corequakes resulting from instabilities, which could result from phase transitions in neutron star cores.
2 Methods.Westudytheseaspectsofspinevolutionofisolatedcompactstarsbymeansofanalyticalmodelsofequationsofstate,
for both constant-pressure phase transitions and the transitions through the mixed-phase region. We use high-precision 2-D

2 multi-domainspectralcodeLORENEforthecalculation oftheevolutionarysequencesofrotatingneutronstars.Thisallowsus
v
to search the parameter space for possible instability regions, and possible changes in the stability character of rotating stars
6
with phase transitions in their cores.
0
Results.Wedeterminetheconditionsonthedensityjumpinconstant-pressurephasetransitionswhichleadstothebackbending
8
phenomenaortotheexistenceoftheunstablesegmentsintheevolutionarysequencesofspinningdownisolatednormalneutron
9
0 stars. We formulate the conjectures concerning the existence of two disjoint families of non-rotating and rotating stationary
5 configurations of neutron stars. To clarify the effect of rotation on the stability of neutron star we present the particular
0 case of EOSs leading to marginal instability of static and rotating configurations: marginal instability point in non-rotating
/ configurationscontinuestoexistinallevolutionaryspin-downtracks.Wediscussthefateofrotatingstarsenteringtheregionof
h
instability calculating thechangein radius,energy release, and spin-upassociated with thecorequakein rotating neutronstar,
p
triggeredbytheinstability.Theenergyreleaseisfoundtobeveryweaklydependentontheangularmomentumofcollapsingstar.
-
o
r
t
s Key words.densematter – equation of state – pulsars: general – stars: neutron – stars: rotation
a
:
v
i
X 1. Introduction ply information on the properties of matter at density
r >∼ 1015 g cm−3 expected at neutron star center. We can
Thepropertiesofthehigh-densitymatterincompactstars
a only hope that observations of neutron stars will provide
arenow studied by means bothof nuclearphysics andas-
uswithconstraintswhichwillenableustoselectacorrect
trophysics. Many different equations of state (EOS) were
dense matter model, or at least to limit the number of
proposed in the literature, based on different theoretical
acceptable dense matter theories.
models of dense matter. In particular, it has been postu-
Particularly interesting method of searching for the
lated that at high enough density, matter contains hy-
phase transition in neutron star cores via pulsar timing
perons, and that phase transitions to exotic phases of
was proposed by Glendenning et al. (1997). As a pul-
dense matter, such as meson (pion, kaon) condensates, or
sar spins down, its central density increases, and for a
quark matter can occur in the dense compact star cores
certain density a new phase of matter can appear. In
(for a recent review of possible states of matter in com-
the case considered by Glendenning et al., the new phase
pactstarcoresseee.g.Glendenning2000).Unfortunately,
consisted of quark matter. The authors suggested, that
the lack of precise knowledge of strong interactions be-
the softening of the EOS, induced by the formation of
tweenhadronsindense matter,as wellasdeficiencies and
the new dense phase, leads to a temporary spin-up era,
approximations plaguing the many-body calculations of
thephenomenoncalledback-bending.Originally,thename
the EOS prevent us from knowing the actual structure of
comes from nuclear physics, where the phenomenon of
neutron star cores. Terrestrial experiments cannot sup-
“back-bending” was observed in the systematics of the
Send offprint requests to: J.L. Zdunik moment of inertia of excited states of rapidly rotating
2 J. L. Zduniket al.: Rotating NSsand phase transitions
nuclei, see e.g., Ring & Schuck (1980). The calculations Zdunik et al.(2004) pointed out that many casesclaimed
of Glendenning et al. were performed within the slow- beforetocorrespondtotheback-bending,actuallycannot
rotation approximation (Hartle 1967; Hartle & Thorne berealizedinnaturebecauseoftheinstabilitywithrespect
1968), supplemented with additional relations resulting to the axi-symmetric perturbations.
from accounting for the rotational stretching and frame- One of the aims of the present work is to determine
dragging effects (Weber & Glendenning, 1991, 1992). reliably and precisely the stability regions on the back
Several other authors (Heiselberg & Hjorth-Hensen, bending segments of the spin evolution tracks. In order
1998; Chubarian et al., 2000) carried out their calcula- to avoid any precision problem, and to investigate large
tions of the back-bending phenomenon using the slow ro- andpossiblycompleteparameterspace,wewillworkwith
tation approximation of Hartle. However, as shown by analytical EOSs of dense matter exhibiting a softening
Salgado et al.(1994),the Hartlemethod, whencompared at supra-nuclear density. Two examples of softening by a
withresultsofexact2-Dnumericalcodes,breaksdownfor phase transition will be considered. We will study EOSs
angular velocity close to the Keplerian one. withconstant-pressurephasetransitions,characterizedby
First calculations concerning back-bending based a density jump obtained using the Maxwell construction.
on exact 2-D code were performed by Cheng et al. Second,wewilluseEOSswithphasetransitionextending
(2002). These authors used the version of KEH overafinitepressurerangeinwhichtwopurephasescoex-
code (Komatsu, Eriguchi & Hachisu, 1989a,b), improved ist forming a mixed-phase state. Such EOSs are obtained
by Stergioulas & Friedman (1995) (see also references forthefirst-orderphasetransitionbetweentwopurephase
therein). In their work, Cheng et al. focused on the role byrelaxingtheconditionoflocalelectricchargeneutrality
of the crust for the very existence of the back-bending. and replacing it by less stringent condition of the global
Indeed, as they show,evena slightchange in the physical neutrality (Glendenning 1992). Mixed-phase state can be
state ofthe crust(for example,a changein the crust-core realized provided the surface tension at the interface be-
transition pressure) may significantly affect the results. tween the two pure phase is not too large.
This shows that high precision is mandatory for reliable The plan of the article is as follows. In Sect. 2 we
calculation of the back-bending phenomenon. present various types of analytical EOSs used in the cal-
Another important article containing results based culation of the spin evolution tracks. Numerical methods
on 2-D computations with the Stergioulas & Friedman used in exact 2-D calculations are briefly presented in
(1995) code, was published by Spyrou & Stergioulas Sect. 3. Our numerical results are described in Sect. 4.
(2002). They showed that the results obtained by We firstdescribe thegeneralcriteriaforthe back-bending
Glendenning et al. (1997) are plagued by large numeri- andthestabilityforspinning-downstars.Then,theresults
cal uncertainties. For example, the very same EOS as for the EOSs with a mixed-phase segment are studied in
that used by Glendenning et al. (EOS from Table 9.2 of Sect. 4.1, and those with constant pressure phase transi-
Glendenning 2000) did not yield the back-bending phe- tion with density jump are reviewed in Sect. 4.2. In Sect.
nomenonatallinSpyrou & Stergioulas(2002)!Itbecame 5 we describe a link between the existence of unstable
evident that the back-bending problem is much subtler segments in the families of static and rotating configura-
than previously considered, and that it requires careful tionsofneutronstars.Changeinneutronstarparameters,
handlingaswellashigh-precision2-Dcomputations.Also, accompanyingtransitionsbetweentworotatingconfigura-
Spyrou & Stergioulas pointed out some errors in previ- tions,triggeredbyinstabilitiesofisolatedrotatingneutron
ous papers on back-bending. For example, the formula stars, are studied in Sect. 6. Modifications in the pulsar
for braking index must be corrected by taking into ac- timing and pulsar age evaluations, due to phase transi-
count the rotational flattening of the star (Sect. 6 in tions in spinning-down isolated neutron stars are studied
Spyrou & Stergioulas 2002). in Sect. 7. A summary of our results and their discussion
Most recently, Zdunik et al. (2004) showed that the is presented in final Sect. 8. Formulae referring to analyt-
back-bending phenomenon can also occur for the EOSs ical models of EOSs with phase transitions are collected
different from the mixed-phase one. They also pointed in the Appendix.
out importance of the stability with respect to the axi-
symmetric perturbations. The appearance of hyperons
2. Analytical EOS
in the dense matter (Balberg & Gal, 1997) softens some
EOSs so much, that pulsars losing angular momentum Studies of stability of stationary rotating configurations
actually spin up during a period of time. Paradoxically, requirehighaccuracyofthe 2-Dcalculations.This is par-
during this spin-up phase pulsars could lose a significant ticularly true for the determination of a critical configu-
amount of their angular momentum. ration which separates the stable configurations from the
Itshouldbestressedthatallpreviousworks(exceptof unstableones.Ontheotherhand,inoursearchforcriteria
Zdunik et al. 2004) considered back-bending as a feature which link the properties of the phase transition in dense
of the the dependence of the moment of inertia, I, on the matter with its effect on the stability of stellar configura-
rotation frequency, f ≡ 1/rotationperiod: I = I(f). As tions with new-phase cores we wanted to explorepossibly
shownrecentlybyZdunik et al.(2004),thiscaneasilylead large space of the phase transition parameters. In view
toincorrectdeterminationofthestabilityofrotatingstars. of this, we decided to use analytic forms of the EOSs. In
J. L. Zduniket al.: Rotating NSsand phase transitions 3
density phase B is a polytrope of index Γ . Details of
B
the construction of the analytical EOSs are given in the
Appendix A.1.
2.2. Phase transition via the mixed-phase state
Normal phase is described by a polytrope of index Γ .
A
Mixed phase starts at baryon density n , and ends at n ;
1 2
it has a polytropic EOSwith index Γ <Γ . This approx-
A
imation is rather good for a sizable lower-density part of
the mixed-phase segment (see Bejger et al. 2005). To be
specific, pure high-density phase, which exists at n > n ,
2
was assumed to be quark matter, with a bag-model EOS.
Details of construction of the analytical EOSs are given
in the Appendix A.2. Depending on the values of param-
eters, we obtained three possible types of the EOS with
mixed phase region. First, we obtained EOSs which give
exclusivelystablemodelsbetweentheminimumandmaxi-
mumallowablemass;theyarecalledMSt(MixedStable).
Second, we obtained the EOSs which produced two dis-
tinctstablefamiliesofneutronstars,i.e.,lowdensityfam-
ily and high density family. They are represented by two
segments in the mass-central density plane and are sep-
arated by a segment of unstable configurations. Such an
EOS is denoted by MUn (Mixed Unstable). There are
also EOSs which yield a vanishingly short unstable seg-
ment of static configurations, so that the configurations
are “marginally stable there” (an inflection point). This
classofEOSswillbe denotedby MM(MixedMarginally
Stable).ParametersofexamplesofanalyticalEOSofeach
of these three types are given in Table 1, and their P(ρ)
Fig.1. Two examples of EOSs employed in our calcu- plots are shown in Fig. 2. These examples of the EOSs
lations: constant-pressure phase transition (dashed line, have n similar to the models tabulated in Tables 9.2-9.3
1
EOS with ΓA = ΓB = 2.25, the density jump λ = 1.3, by Glendenning (2000). Our values of n2 ∼ 3n1 are sys-
can be found in Table 2) and transitionthrougha mixed- tematicallylowerthanthoseofGlendenning(2000)which
phase state (solid line, thin dotted lines mark n1 and n2; are characterized by n2 ∼ 4n1. However, this difference
EOS MM of Table 1). Top panel: p(ρ) diagrams. Bottom is not relevant for the general results of our analysis of
panel: density profiles in stellar core, for a M =1.33 M⊙ stability of rotating stars.
neutron star models.
3. Numerical methods
this way, the EOSs were easy to handle, and we could Hydrostationary stellar models have been computed
easilychangethe phase-transitionparametersinacontin- by solving the Einstein equations for stationary axi-
uousway,detecting withhighprecisionthe appearanceof symmetric space-time. The numerical calculations were
instability.Moreover,withournumericalmethodsofsolv- performed by means of the rotstar code, a part of the
ing the hydrostationary equilibrium problem, we had no LORENE public-domain object-oriented C++ scientific li-
unnecessary preoccupation with control of precision (see brary based on spectral methods. For the complete set
Sect. 3). of equations as well as the tests and the description of
the numerical code we refer the reader to the articles
by Bonazzola et al. (1993), Bonazzola et al. (1998) and
2.1. Constant pressure phase transition
Gourgoulhon et al. (1999).Forthepurposeofthepresent
Normal phase is described by a polytrope of index Γ . work, we have employed two (in the case of a constant-
A
The phase transition to a dense phase takes place at a pressure phase transition)and three (phase transition via
specific pressure, P , at which normal phase of baryon amixed-phasestate)spectraldomainstodescribethestel-
AB
density n coexists with dense phase of baryon density lar interior and surface. The division into domains makes
A
n , with baryon density jump n −n >0. Coexistence, use of the adaptive coordinates while setting the bound-
B B A
equivalent to phase equilibrium, corresponds to equality ariesbetweendifferentphases.Therefore,thedensityfield
of baryon chemical potential of both phases. The high- is smooth in each domain and the resulting accuracy is
4 J. L. Zduniket al.: Rotating NSsand phase transitions
Table 1.MainparametersoftheEOSswithmixed-phase
segment. Below the mixed-phase transition point n a
1
polytropic EOS with Γ is used. Mixed phase extends
A
withinn <n <n ,andisdescribedbyapolytropewith
1 b 2
adiabaticindexΓ .Abovethedensityn weassumepure
m 2
quarkmatterwithMITbagmodelEOSp= 1(ρ−ρ )c2.In
3 0
all cases the dimensionless polytropic pressure coefficient
K was equal 0.025 (see Appendix A for details). Mstat
b,max
and Mstat denote the maximum allowable baryon and
max
gravitationalmassofthenon-rotatingstar.TheEOSsare
labeled as follows: MSt produces a stable back bending,
MUn - an unstable one, and MM produces a marginally
stable case (for more details see the text).
EOS ΓA n1 Γm n2 Mbst,matax (Mmstaaxt)
[fm−3] [fm−3] [M⊙]
MSt 2 0.35 1.5 0.8 1.508 (1.393)
MUn 2.5 0.2 1.3 0.65 1.586 (1.453)
MM 2.25 0.25 1.25 0.57 1.685 (1.534)
Fig.3. Total angular momentum versus rotation fre-
quencyf (leftpanels),andmomentofinertiaI ≡J/Ωver-
susf (rightpanels),forEOSsMStandMUn.Thestability
criterioniseasilyappliedto leftpanels.Itis clearthatfor
the MSt EOSback bending feature is notassociatedwith
aninstability,withallconfigurationsbeingstable.Onthe
contrary, the MUn EOS produces back bending with a
large segment of unstable configurations. Simultaneously,
theI(f)curvesforbothEOSsareverysimilarandappar-
ently show very similar back-bending shapes.
fixed f. More generally, the softening of the EOS leads
to the flattening of the M(R ) and M(ρ ) curves repre-
eq c
senting these sequences. Here, R is the circumferential
eq
equatorial radius. The effect is larger for larger values of
f,anditappearsabovesomecriticalvalue off.Thesoft-
ening of the EOSresults then in a local maximumof M .
b
In such a case, there exists a region, close to the local
maximum mass, in which the decrease of J leads to the
Fig.2.Examplesofthethree EOSswithphasetransition
increase of f, and this is equivalent to back bending. The
through the mixed-phase state, considered in the present
fragment of the curve, for which M decreases as a func-
paper. The parameters of the EOSs are given in Table 1.
tion of ρ , does not necessarily correspond to the insta-
c
bility region - the decrease at fixed f does not imply the
veryhigh – in terms ofGRV2 andGRV3 virialerrorindi- decreaseatfixedJ.Onlythelatterconditionisequivalent
cators (see Nozawa et al. 1998) the relative error was on to the instability ofrotatingconfigurationwithrespectto
average ∼10−7. the axi-symmetric perturbations (Friedman et al., 1988).
Summarizing, the particular shape of the M(R ) curve
eq
for a fixed f indicates the presence of the back bending
4. Back-bending and stability of rotating
phenomenon, whereas the shape of M(R ) at a fixed to-
eq
configurations
tal angular momentum J tells us if the configurationsare
Asithashasbeenalreadymentionedbyusintheprevious stable.
paper treating specifically the back bending in hyperon InthepresentpaperwewillshowexamplesoftheEOS
stars(Zdunik et al.,2004),thebackbendingphenomenon with constant-pressure (density-jump) phase transitions
itselfisstrictlyconnectedwiththeexistenceofaminimum and mixed-phase transitions for which one of the two sit-
of M along the sequence of rotating configurations with uations occurs:
b
J. L. Zduniket al.: Rotating NSsand phase transitions 5
– 1. all configurations are stable but the back bending
exists.
– 2. the phase transition results in the instability region
for rotating stars.
To distinguish between these two cases we can look at
the behavioreither of the curves M (ρ ) or J(Ω) ; the
b c J Mb
quantity fixed along a sequence is indicated by the lower
index.Forsuchcurvestheinstabilitycriteriondirectlyap-
plies.However,isnotsoeasytodetecttheinstabilityusing
theI(f) plot,whichmaybesimilartothatcorrespond-
Mb
ing to a fully stable sequence. An example is presentedin
Fig. 3, where we plot two functions: J(f) and I(f)
Mb Mb
for the two models of EOS (MSt and MUn) which are
described in detail in Sect. 4.1. Upper panels correspond
to the stable model MSt and lower to the MUn model
for which a region of unstable configurations exists. The
difference between the MSt and MUn cases is clearly vis-
ible in left panels (J(f) curves), where we can easily
Mb
recognizethe instability regionfor MUn model,by apply-
ing the condition (dJ/dρ ) > 0, and keeping in mind
c Mb
thatρ ismonotonicalongthiscurve.However,the upper
c
andlowerrightpanels(I(f) )arequitesimilar,without
Mb
anyqualitativedifference.Simultaneously,thebackbend-
ingphenomenoninthe MUncaseis verylarge(almostby
200 Hz), and obviously the phase transition results in an
instability(increaseofJ forincreasingρ ).Andyet,I(ρ )
c c
is a monotonic decreasing function all the time.
4.1. Transition to a mixed phase
In Figs. 4-6 we present the results obtained for three dif-
ferent choices of the EOS’s parameters describing mixed Fig.4.Upperpanel:stellarbaryonmassvsradiusforfixed
phase phase transitions (Table 1).
frequency (dashed lines) and fixed total angular momen-
Model MM has been chosen to be a marginal one -
tum(solidlines),forthe modelMMofmixed-phaseEOS.
thereexistsaregionwherethecurveM (R )(orM (ρ ))
b eq b c Bottompanel:stellar angularmomentumas a function of
is locally horizontal, which means the marginal stability.
the rotational frequency for fixed baryon mass (indicated
The function M (R ) for non-rotating star (f = 0) has
b eq as a label, in solar masses) for the same MM EOS. This
an inflection point slightly above the configuration where
EOS corresponds to the marginal case from the point of
phase transition to the mixed phase occurs. At a fixed f,
view of stability – the curves M(R ) and J(f) have
eq J Mb
the condition for an inflection point reads,
flat horizontal regions. The regions of back-bending are
∂M ∂2M drawn by thick lines.
b b
=0 , =0 . (1)
(cid:18) ∂ρ (cid:19) (cid:18) ∂ρ2 (cid:19)
c f c f
In a special static case (f = 0) this condition is fulfilled are monotonously increasing, independently of the rota-
forMb ≃1.42M⊙ andR≃14km.ForthismarginalEOS tionrate(solidcurvesinFig.4).Strictlyspeaking,foreach
model, the back bending phenomenon exists at all rota- J there exists an inflection point of the function Mb(ρc)J
tion rates, and for baryon masses larger than 1.42 M⊙. (within the accuracy of our calculation). The size of back
The region for which back-bending exists is bounded by bending,definedasafrequencyrangeforwhichf increases
thepointscorrespondingtothelocalmaximaandminima fordecreasingJ (andincreasingρc),depends onthe mass
ofthefunctionM (R )atfixedrotationalfrequencyf.In ofthestar(seebottompanelofFig.4).InthemodelMM,
b eq
the marginal model for which the inflection-point condi- the back bending range varies from 0 (for Mb ≤1.42M⊙)
tion,Eq.(1),isfulfilledfornon-rotatingstars,evenanin- to ∼180Hz for the starwith mass Mb =1.685M⊙,equal
finitesimallyslowrotationleadstotheminimumandmax- to the maximum mass of non-rotating configurations (i.e.
imum of the function M (R) (albeit infinitesimally close not supra-massive ones)
b f
to each other), which is equivalent to the back-bending Model MSt produces a set of stable configurations in
(see dashed curvesin Fig. 4). However,imposing rotation the region of phase transition (Fig. 5). Here, M is an
b
does not result in the instability - the functions M (ρ ) increasing function of ρ for non-rotating configurations,
b c J c
6 J. L. Zduniket al.: Rotating NSsand phase transitions
Fig.5.ThesameasinFig.4butfortheMStEOS.Phase Fig.6. The same as in Fig. 4 but for MUn EOS. Upper
transition does not result in the stability loss - all config- panel: Phase transition results in the stability loss, i.e.,
urations are stable. there exists a region where M decreases with increas-
b
ing ρ at fixed J (marked by dotted lines). Lower panel:
c
plots in the J − f plane, where on the dotted seg-
ments J increases as ρ increases (unstable configura-
c
as well as along all rotating sequences of configurations
tions). Both features indicate instability with respect to
at fixed J, terminating at the globalmaximum mass con-
the axi-symmetric perturbations.
figurations.The back-bending phenomenon is in this case
limited to the baryon masses larger than that for which
thecurveM (R )(orequivalentlyM (ρ ))atfixedf has
b eq b c 4.2. Constant pressure phase transition
a flat, horizontal region (strictly speaking, where the Eq.
(1) is fulfilled). The numerical values for the MSt model For non-rotating configurations, the reaction of the star
are Mb > 1.45M⊙ and f > 400 Hz (see for example the to a constant pressure (first order) phase transition has
bottompanelofFig.5,wherethecurveforMb =1.45M⊙ been studied in detail in the second half of 1980s (see
has almost vertical part). Zdunik et al. 1987 and references therein). The appear-
For the MUn model there exists a region for which ance of a new, dense phase in the center of the star re-
configurations are unstable, i.e. the baryon mass is de- sults in the change of the derivatives of the global stel-
creasing function of ρ at fixed J (Fig. 6). In some sense lar parameters with respect to ρ (see the formula B6 in
c c
this instability is not very strong - the difference between the appendix of an article by Zdunik et al. 1987). Two
themaximum(local)andminimummassisoftheorderof important dimensionless parameters are: fractional den-
0.3%.Howeverthisfeature(existenceofinstabilityregion) sity jump λ = ρ /ρ and the relativistic parameter
B A
is characteristic to all rotational frequencies - for all val- x ≡ P /(ρ c2). There exists a critical value of λ,
A AB A
ues of angular momentum (J) fixed along the curve, the λ = 3(1+x ), such that for λ > λ configurations
crit 2 A crit
baryonmass has local maximumand localminimum con- with an infinitesimally small B-phase core are unstable
nected by an unstable sequence of stellar configurations. with respect to collapse into a new configuration with
J. L. Zduniket al.: Rotating NSsand phase transitions 7
a large core of the dense phase. Putting it differently, a at maximum mass point) and if λ > λ we have two
max
phase transition with λ > λ destabilizes the star at branchesofstableconfigurationforrotatingstars(forany
crit
central pressure P = P at which the phase transition J).
c AB
occurs. It should be stressed that while λ < λ guar- Picking up the onset parameters is visualized in Fig.
crit
antees stability of small-core configurations, it does not 7 where we display M (R ) for one of the EOSs from
b eq
assure the stability of configurations with a finite, or – Table 2. The curves are plotted for three frequencies,
in an extreme case – a large core. In such a case the in- with middle one corresponding to the back bending on-
stability would result from the softness of the B phase set, f = f . Last column of Table 2 gives the maximum
on
somewhataboveρ and not directly froman over-critical allowable baryon mass for static configurations, Mstat .
B b,max
λ. In other words the compressibility of a matter leads to We restrictourselvestobackbending forthe normal(non
the larger mean density in the core than the value ρ at supra-massive)stars,whichappearsduringthespin-down
B
thephaseboundary. Theresponseofthewholestartothe evolution which terminates eventually by a non-rotating
appearance of the dense core built of the S-phase of the stable configuration. The dependence between the back-
matter is determined by the mass and radius of this core bendingonsetparameters-f =f ,correspondingbaryon
on
(strictly speaking, this statement is true for non-rotating mass, and the intrinsic parameters of the EOS - the den-
configurations, see Zdunik et al. (1987); for rotating ones sity jump λ, as well as the ”departure” from the critical
also rotation rate and resulting oblateness play role). As configuration(∆λ=λ −λ)is presentedinFig.8.The
max
a result even if λ < λ the first order phase transition three families of curves visualize the data from Table 2
crit
can lead to the unstable configurations for finite size of (solid lines for Γ = 2, dotted for Γ =2.25 and dashed
A A
the core. As a result for the given model of the matter for Γ =2.5). The value λ defines the onset of back-
A max
in the phases A and B there exist the maximum value bending at the limit f =0; in this case the back-bending
of density jump λ for which all configurations below phenomenon is present for any rotational frequency. As
max
maximummassarestable.Ofcourseλ ≤λ andthe it can be seen on the left panel, the onset frequency f
max crit on
difference between λ and λ is larger for softer EOS depends very weakly on the EOS in the dense core - the
max crit
in the phase B. main parameter describing the reaction of the star to the
Numerical results for a collection of sets of EOSs with appearance of this phase transition is the density jump.
constantpressurephasetransitionarecollectedinTable2. The right panel presents the same data not normalized
The parameters presented in this table correspond to the with respectto the maximumdensity jumpλ -the re-
max
onset of back bending, i.e., the rotational frequency and sults can be very well approximated by the dependence
baryon mass for which the curve M (R ) or f(J) f2 =a∆λ+b(∆λ)3/2.Thesetwoplotscanbe treatedas
b eq f Mb on
starts to have a flat region. More precisely, at these val- a slice through the parameter space to search for regions
ues of frequency and mass, an inflection point appears in of the back-bending appearance - in the right panel, the
the curves under consideration. We also included param- back-bending is present, for a particular model, above its
eters of those EOSs for which all non-rotating stars with curve.
M < Mstat are stable. This means that for such EOSs
b b,max
theM (ρ )curveforstaticconfigurationsincreasesmono-
b c 5. Rotation and stability/instability of normal
tonically up to Mstat . The parameter λ gives then
b,max max configurations
the maximum value of the density jump for a fixed set of
otherEOSparameters(adiabaticindicesΓ andΓ ,num- Rotationcaninfluencestabilityofastarofagivenbaryon
A B
ber density threshold n ) for which this property of neu- mass.In particular,stars with M >Mstat cannotexist
A b b,max
tronstarsisvalid;inotherwordsλ correspondstothe without rotation and collapse into black hole as soon as
max
”marginally stable” case. Increasing λ implies increasing their rotation frequency falls below a certain minimum
softeningoftheEOSbythephasetransition.Ifλ>λ , value.Here,however,wewillrestricttoadifferentproblem
max
the phase transition leads to the existence of an unsta- of stability, which will concern the normal configurations
ble branchofthe non-rotatingstellar configurations.This only.
unstable branch separates stable family of neutron stars Inwhatfollowswewillusetheterm“stability”inare-
with A-phase cores from a second family of superdense strictedsense.Namely,bystability(instability)ofanequi-
neutron stars with B-phase cores: these are two distinct librium configuration we will mean stability (instability)
neutron-star families. It should be mentioned that this with respect to radial perturbations in the non-rotating
feature(existenceoftheunstableregion)doesnotdepend case,andwith respectto axi-symmetricperturbations for
on rotation- the unstable branches exist also for rotating rotating configurations.
configurations (strictly speaking for any value of a total We studied a very large set of EOSs with phase tran-
angularmomentumofthe starJ thereexistaregionwith sitions at constant pressure, as well as those with tran-
(∂M /∂ρ ) < 0). We have tested this feature (existence sition through a mixed phase state. We then produced
b c J
of or the lack of unstable regions) for very small depar- static sequences and normal rotating sequences for these
tures from marginally stable case (|λ −λ| < 0.005). EOSs. Our calculations were very precise, because we
max
From numerical results it follows that if λ < λ all ro- used analytic forms of the EOSs. The results for both
max
tating configurations are stable (before loosing stability constant pressure phase transitions, and those proceed-
8 J. L. Zduniket al.: Rotating NSsand phase transitions
Table 2. Selected sets {K ,Γ ,Γ ,n ,λ} of EOSs with
A A B A
a constant pressure phase transition. For all cases, K =
A
0.025 and n =0.25 (see the text and Appendix A.1).
A
EOS λ fon [Hz] Mbon [M⊙] Mbst,matax [M⊙]
1.344 247 1.033 1.256
1.318 368 1.071 1.283
ΓA=2,
1.265 482 1.131 1.342
ΓB=2,
1.212 555 1.193 1.407
λmax=1.364
1.159 601 1.248 1.477
1.106 624 1.286 1.554
1.371 189 1.021 1.475
ΓA=2, 1.318 407 1.087 1.534
ΓB=2.25, 1.265 507 1.149 1.597
λmax=1.382 1.212 570 1.208 1.667
1.159 620 1.272 1.743
1.371 257 1.034 1.712
ΓA=2, 1.318 453 1.110 1.776
ΓB=2.5, 1.265 522 1.160 1.844
λmax=1.393 1.218 576 1.215 1.918
1.159 608 1.258 1.999
1.376 412 1.307 1.642
Fig.7. The definition of the frequency of the onset of ΓA=2.25, 1.322 537 1.387 1.709
1.268 611 1.463 1.782
back-bending phenomenon, fon, and corresponding mass, ΓB=2.25,
1.215 662 1.536 1.861
Mbon. In the presented example fon =449 Hz and Mbon = λmax=1.432 1.161 694 1.597 1.947
1.325M⊙ 1.107 709 1.636 2.041
1.430 222 1.242 1.803
1.376 449 1.325 1.859
ΓA=2.25,
1.322 556 1.403 1.940
ΓB=2.5,
1.268 623 1.477 2.016
λmax=1.444
1.215 668 1.545 2.099
1.161 696 1.602 2.190
1.437 361 1.559 1.967
1.382 532 1.661 2.040
ΓA=2.5, 1.329 624 1.754 2.119
ΓB=2.5, 1.273 686 1.844 2.205
λmax=1.472 1.219 731 1.933 2.297
1.164 757 2.001 2.398
1.109 780 2.076 2.509
Fig.8. The onset frequency of the back-bending phe- 1.437 428 1.590 2.348
nomenon as a function of the departure of λ from the 1.382 566 1.689 2.427
ΓA=2.5,
1.329 646 1.871 2.512
maximum density jump λmax (λmax −λ, left panel), and ΓB=3,
1.273 701 1.870 2.603
densityjump λasafunction offon (rightpanel)formod- λmax=1.487
1.219 738 1.945 2.703
els presentedin Table 2 (solidlines for Γ =2, dottedfor
A 1.164 761 2.011 2.811
Γ =2.25 and dashed for Γ =2.5).
A A
ing through mixed phase, turned out to be qualitatively persistedwithintherotatingconfigurations,ateachvalue
the same. In all cases, if non-rotating configuration were of J. These two cases are illustrated in the Figs. 5 and 6.
stable(monotonicallyincreasingM (ρ )andM(ρ )),then WestudiedalsothecaseofmarginallystableEOS.An
b c c
for any value of the total angular momentum J the func- inflection point, witnessing marginal stability, present in
tions Mb(ρc)J and M(ρc)J were monotonically increas- the Mb(R) or M(ρc) curvesfor non-rotatingstars contin-
ing, too. Thus, when all non-rotating configuration with ued to exist in the M (R ) or M(ρ ) curves for normal
b eq c J
M < Mstat were stable (with respect to radial pertur- rotating stars (Figs 4, 9).
b b,max
bations), all normal rotating configurations were stable The analysis of numerical results leads us to an in-
too(withrespecttoaxi-symmetricperturbations).Onthe teresting conclusion. Namely, for an EOS with a phase
otherhand,iffornon-rotatingstarsthereexistedaregion transition(constant pressure one or through mixed-phase
with decreasing M (ρ ) and M(ρ ), even extremely small state), rotation neither stabilizes nor destabilizes normal
b c c
onewithaveryshallowminimum,thenanunstableregion sequencesofstationaryconfigurationsbasedonthis EOS.
J. L. Zduniket al.: Rotating NSsand phase transitions 9
Fig.9. The mass M of the star as a function of central Fig.10. Evolution of an isolated pulsar loosing angular
baryonnumberdensityncfortheMMmodel.Dashedlines momentum,afterit reachesthe instabilityregioninJ−f
-fixedrotationalfrequency.Solidlines-fixedtotalangular plane and then collapses. Arrows lead from unstable con-
momentum J. The vertical lines correspond to the densi- figurationtoacollapsedstableone,withthe samebaryon
ties of transitions from the nuclear to mixed phase and massandangularmomentum.Dottedlines-unstablecon-
from the mixed phase to the pure denser phase (here - figurations.
quark matter) - see Table 1.
We define a family of configurations as a compact set
(in mathematical sense) of configurations. Similarly, an
EOSleadingto amarginallystablepointfornon-rotating
stars, produces also spin-down evolution tracks with a
marginally stable point. Our result can be formulated as
three conjectures:
I If an EOS with a phase transition gives a single family
of static stable neutron stars then it produces also a single
family of stable rigidly rotating normal stars.
II If an EOS with a phase transition gives two disjoint
families of stablestaticstars thenit gives also twodisjoint
families of stable rigidly rotating normal stars.
III If an EOS with a phase transition gives two fami-
lies of stable static stars separated by a marginally sta-
ble configuration, then it gives also two families of stable
rigidly rotatingnormal stars separated bya lineconsisting
of marginally stable configurations.
6. Corequake resulting from instability
Aswasshownonmanyoccasionsintheprecedingsections,
an isolated neutron star, loosing its angular momentum,
Fig.11. Changes of stellar parameters of a rotating soli-
moves down along the line of fixed M in the J(f) plane,
b
tary neutron star, due to a collapse which occurs after
and can at some moment reach the instability point (i) (
a pulsar loosing angular momentum reaches an unstable
the minimum of J(f) at fixed M in Fig. 10). The sub-
b
configuration.
sequent behavior of the star cannot be described by our
10 J. L. Zduniket al.: Rotating NSsand phase transitions
In order to discuss in more detail the energy release
during collapsesi−→f, we plottedinFig. 12the gravita-
tional mass of the star, M, as a function of angular mo-
mentum,J,atfixedbaryonmass:M =M(J) .Consider
Mb
aninitialconfigurationC .Asthestarloosesangularmo-
1
mentum, it moves down along line a, and reaches eventu-
ally the cusp C (corresponding to the value of J = J).
i i
To continue movingonthe dotted segmentCC ,the star
i 2
would have to gain angular momentum and energy!
As we already mentioned, the evolution of the star
beyond the instability point cannot be described by our
hydrostationary model. The star can only collapse to the
final configurationC , with the same values of M and J,
f b
i.e. along vertical arrow in Fig. 12. Then, it evolves down
the line b.
We notice a very special role played by the point C
x
Fig.12. Total gravitational mass of the star as a func-
at which line a and b cross. This is a degeneracy point,
tion of its angular momentum, for fixed baryon number which corresponds to two very different configurations of
of the star for our EOS model MUn. The central density
the same M , M, and J. However, transitions between
b
is increasing along this curve as marked by the arrows.
these two configuration are prevented by the huge energy
The upper segment (dotted) corresponds to the unstable
barrier.
configurations. Two cusps reflect strict property that the
The existence of the sharp cusps at C and C on
i 2
mass and angular momentum have simultaneous extrema
the M(J) track is a very stringenttest of the precision
Mb
along the path with fixed baryon number.
of the numerical code: it means that mass and angular
momentum extrema (for fixed baryon mass) are reached
exactly at the same point. This property follows from the
stationary rigidly rotating model. In real world, the star
generalrelativisticrelationdM =ΩdJ+γdM (Bardeen
b
has to collapse, rearranging the angular momentum dis-
1972). Here, Ω ≡ 2πf, and γ is the “injection energy per
tribution in its interior. What we can do, is determining
unit mass”. This relation has to be strictly fulfilled by
the final stationary state (f), which by construction will
the stationary configurations. A graph, analogous to Fig.
be astablerigidlyrotatingneutronstar.We assume,that
12, can be plotted in the M − M plane for configura-
b
the transition conserves the baryon number of the star tions with fixed angular momentum J. Also in this case,
and is sufficiently rapid so that the angular momentum
the existence of sharp cusps proves the correctness of the
loss can be neglected. Therefore, the final stable configu-
numerical code.
rationwillhavethesameM andJ astheunstableinitial
b
one, M = M , J = J. The difference between the
b,f b,i f i
parameters of these configurations (mass, equatorial ra- 7. Phase transition and pulsar timing and age
dius, moment of inertia) gives us the energy release, and
It has been already pointed out by Spyrou & Stergioulas
changes in equatorial radius and rotation frequency, due
(2002),thatthebackbendingphenomenon,resultingfrom
to the collapse implied by the instability.
the growth of a dense-phase core, can lead to significant
Examplesareshownin Figs10-12forourEOSmodel
difference between the actual pulsar age, τ , and that in-
MUn. As the star becomes more compact, collapse is ac- 1
ferred from the measurements of the period P and pe-
companied by the decrease of the equatorial radius (by a
riod derivative, P˙, and denoted τ . The calculation of τ
few kilometers) and by a significant spin up. For a given 2 2
is based on quite strong assumptions. Firstly, pulsar ki-
EOS, the changes in radius, energy release, and spin-up
neticenergylossduetoradiationisgivenbythemagnetic
are function of the angular momentum at the instability
dipole formula. Secondly, non-relativistic approximation
point: ∆R (J), ∆E(J), ∆f(J). As we see in Fig. 11,
eq i i i is used, with pulsar kinetic energy given by 1IΩ2, where
the energy release depends rather weakly on the rotation 2
Ω ≡ 2π/P and pulsar moment of inertia is constant, in-
of the unstable configuration (i.e., on the value of J). It
i
dependent of Ω.
should be mentioned that our MUn model is only an ex-
Following Spyrou & Stergioulas (2002), we will use
ampleoftheEOSresultingintheinstabilityregionwithin
general relativistic notion of total pulsar energy, Mc2.
the hydrostatic equilibria. For EOSs with a weaker phase
Then the pulsar energy balance is
transition this instability region would be narrower and
the changesofstellarparametersinthe collapsewouldbe
dM κ
smaller.Howeveranapproximateconstancyofthe energy =− Ωα (2)
dt c2
release (i.e., its very weak dependence on J = J = J )
i f
seems to be a generic property of rotating neutron stars wheretheright-hand-sideisthepulsarenergylossratevia
undergoingacollapseduetoafirstorderphasetransition. radiation of electromagnetic waves and particles.