[Journal of Astrophysics and Astronomy] Physics of Neutron Stars and Related Objects (Volume 38, Issue 3, September 2017)

252 Pages·2017·20.026 MB·English

by  Dipankar Bhattacharya, K. S. Dwarakanath and Sushan Konar (eds.)

J.Astrophys.Astr.(September2017)38:36 ©IndianAcademyofSciences
DOI10.1007/s12036-017-9477-2
Editorial
Wearedelightedtopresentthiscompilationofarticles cept of residual field was born. Over the years, closer
brought together on the special occasion of the 75th scrutiny has linked all significant field decay in neu-
birthyear of Prof. G. Srinivasan, a leading luminary of tron stars to the accretion process itself. Srini and
compactstarresearchinIndia.Afteranearlycareerin colleaguesproposedanelegantphysicalmechanismfor
condensed matter physics Prof. Srinivasan, ‘Srini’ to the reduction of interior field of a neutron star—via
hisfriendsandcolleagues,joinedtheRamanResearch an interaction between superfluid neutron vortices and
Institute in 1976 and was initiated into compact star quantized fluxoids threading superconducting protons.
research.Sincethenhehasmademanypioneeringand Thishasinspiredmanyfollow-upinvestigationsinthe
seminal contributions in this area as well as trained, literature.
mentored and inspired many students. His activities In addition to the binary evolution and interior
have, in a large measure, been responsible for shaping physics,hewasalsokeenlyinterestedinthehighenergy
thecourseofcompactstarresearchinIndia. emissionfrompulsarsandtheinteractionofpulsarswith
He was the first to realize and report in a publica- theirsurroundings.Basedonmagnetosphericemission
tion with Ed van den Heuvel the importance of the mechanisms,hepredictedthatmillisecondpulsarsmay
Eddingtonlimitonluminosityinlimitingtheaccretion- bestrongemittersofgammarays,eventhoughtillthen
driven spin-up of a neutron star. Today this is widely they had been seen only in radio bands. The launch
referredtoasthe‘spin-upline’.He,alongwithV.Rad- of the Fermi satellite dramatically confirmed his pre-
hakrishnan, predicted a whole population of spun-up diction; millisecond pulsars now constitute one of the
pulsarsandnamedthem‘recycled’pulsars—alabelthat mostprolificclassesofgammaraysources.
hasremainedpopularinthecommunity.Inastatistical Srini and colleaguesalso worked extensivelyon the
studyafewyearslater,Sriniandcolleagueswere able interactionofpulsarwindswiththesurroundingsuper-
toquantifythesizeofthispopulation. nova remnant, and modelled in detail the evolution of
One of the landmark discoveries in 1982 was that Crab-like supernova remnants. Their work led to the
ofthefirstmillisecondpulsar.Althoughthispulsardid surprising conclusion that the majority of pulsars start
nothaveabinarycompanion,RadhakrishnanandSrini- theiractivelifeasrelativelyslowrotators,notspinning
vasanquicklyrealizedthatitoweditsorigintorecycling nearbreak-upperiodashadusuallybeenthought.
and, based on the spin-up line, predicted its magnetic Apartfromhisscientificcontributionstoneutronstar
fieldstrengthtobefourordersofmagnitudebelowthat physics, many of which are referred to in several arti-
ofthegeneralpulsarpopulation,whichwasconfirmed clesinthisissueofJ.Astrophys.Astr.,hemadeseveral
by later observations. Two more millisecond pulsars, importantcontributionstothegrowthofAstronomyin
which did have binary companions, were then discov- India. In 1981, Srini started the biannual Neighbour-
eredinquicksuccession—initiatinganotherimportant hoodAstronomyMeetings(NAMs)oftheastronomers
threadofactivityinSrini’sresearchgroup. in Bangalore which comprised of people from the
All the millisecond pulsars had low magnetic field RamanResearchInstitute(RRI,Bangalore),theIndian
strengths, as required for spin-up to such short peri- InstituteofAstrophysics(IIA,Bangalore)andtheRadio
ods. It was thought at that time that old neutron stars Astronomy Centre (RAC, Ooty). It has been his belief
would lose their magnetic field strength due to Ohmic thatsuchinteractionsbetweenneighboursisanimpor-
decay. Srini and his colleagues realized that the exis- tantingredienttokeepthelevelofresearchactivityhigh.
tenceofthesemillisecondpulsarsimpliedthatthefield ManysuccessfulNAMswereheldatRRIandIIAinthe
decay had to stop for these objects. Thus the con- subsequentyears.
36 Page 2 of 3 J.Astrophys.Astr.(September2017)38:36
A year later, in 1982, the highly acclaimed Joint Hehasalwayshadaverycloseassociationwiththe
Astronomy Program (JAP) started. The idea for this Indian Academy of Sciences. He was elected to be a
uniquegraduateprogram,combiningthestrengthsofa Fellow in 1984 and has served in the Council during
numberofresearchinstitutessituatedatdifferentparts 1986–2003. He has been the Secretary during 1986–
ofthecountrywasconceivedbySriniincollaboration 1988,theEditorofPublicationsduring1989–94andthe
withcolleaguesfromRRI,IIAandtheIndianInstitute Treasurer during 1995–2003. He has also been instru-
of Science (IISc, Bangalore). This initiative was very mental in bringing out a decadal vision for astronomy
important coming at a time when none of these places onbehalfoftheAcademy.
could have been able to sustain an astrophysics grad- HewasawardedtheJawaharlalNehruFellowshipfor
uate program due to the small number of astrophysics 2007–08,whichhehasutilizedtoauthortwointroduc-
faculty available. JAP draws its faculty from the par- toryastrophysicstextbookstitled“WhataretheStars?”
ticipating institutes and the students selected for this and “Can Stars find peace?”. In these, he brings in his
program,aftercompletingthefirstyearofcoursework, insightsandrichexperienceinteachingtoexplainastro-
have the choice to carry out their research in any of physicalconceptsinasimpleandelegantfashion.Not
the participating institutes. Apart from RRI, IIA and surprisingly, these books are on their way to become
IISc,TataInstituteofFundamentalResearchandIndian astrophysics‘mustreads’attheundergraduatelevel.
Space Research Organisation also joined this program Neutronstars,thecompactstellarremnantsofcore-
inthesubsequentyears.Thishasbeenaverysuccessful collapsesupernovae,aresomeofthemostexoticobjects
program and continues to run even today. Not surpris- tobeencounteredintheuniverse.Theyprovideunique
ingly, he is extremely passionate about teaching and it cosmic laboratories for exploring both novel phases
has been an integral part of his life. Naturally JAP has of matter as well as extreme physics, impossible to
benefitedfrommanyyearsofSrini’sexcellentteaching. access in terrestrial conditions. This particular area of
Inthe80’s,thereusedtobe(andcontinuestobe)very astrophysics, therefore, challenges our understanding
little astronomy and astrophysics at the undergraduate of almost every branch of known physics. It has been
levels.Srinistartedorganizingannualsummerschools ouraimtobringoutthisflavourofneutronstarphysics
inastronomyandastrophysics,asameanstointroduce in this volume honouring Srini, as well as provide an
therecentexcitementsinthefieldtoyoungstudentswho updateontheproblemsthathehimselfworkedon.We
would come from all over the country to attend these werepleasantlysurprisedwhenspecialists,fromSrini’s
schools.Manyofthosewhoattendedtheseschoolslater long-termassociatestobrightyoungexperts,agreedto
took up astronomy as a career and went on to become devote their valuable time to be a part of this project.
world-famous astronomers. Some of them remember, Here is a brief summary of the areas covered by these
eventothisdate,theimpactthesesummerschoolshad articles.
onthem. Thecompositionandthenatureofthematerialinside
MostintheIndianastronomicalcommunityarealso a neutron star continue to motivate a lot of current
awareofhisenormouscontributionstotheAstronomi- research. Recent discovery of a number of massive
calSocietyofIndia(ASI).HewasthePresidentofASI (M ≥ 2M(cid:3)) neutron stars has given rise to an intense
during2001–2003.Hehasalsobeenveryactivelyasso- debateonthecomposition(inparticular,thequestionof
ciatedwiththeInternationalAstronomicalUnion(IAU) thepresenceofexoticphases)andconsequentequation
andwasthePresidentofCommission44onhighenergy of state of the interior material, a discussion of which
astrophysicsduring1997–2000,aswellasof Division can be found in the first article of this issue. On the
XIonspaceandhighenergyastrophysicsduring2000– other hand, the occurrence of superfluidity in neutron
2003. stars and its effect on the observable properties have
He has always believed that the astronomical com- been long-standing problems of the field. A couple of
munityofIndiashouldhaveitsownjournalandisone articles review our current understanding of this area
ofthepersonsresponsibleforstartingJ.Astrophys.Astr. ofresearch.Intheearly90s,Sriniandhiscollaborators
whichispublishedbytheIndianAcademyofSciences proposedasimpleyetelegantexplanationforthereduc-
(nowjointlywithSpringer).Hehaspublishedmanyof tionofthemagneticfieldofaneutronstarresidinginan
hispapersinthisjournalandhasalwaysencouragedthe X-ray binary via the coupling of the quantized vortex
youngsters to do so. Till date, he has been the longest lines of the neutron superfluid with the quantized flux
serving editor of this journal and ran it quite success- lines of the proton superconductor. The current status
fullyforadecadeduringtheperiod1992–2002. ofthisideahasalsobeendiscussed.
J.Astrophys.Astr.(September2017)38:36 Page 3 of 3 36
Srini made fundamental contributions to our under- presented. An important issue for the study of neutron
standing of the formation and evolution of neutron starsisthe precise measurement of theirdistancesand
starsinbinarysystems,muchofwhichhavebeencon- velocities.Anewlookatthissuggestsadescriptionof
firmed by the discovery of the double radio pulsar the velocity distribution by two Gaussians instead of
system PSR J0737-3039AB in 2003. In particular, the one.
modelfortheformationofmillisecondpulsars,putfor- Last but not the least, it is important for a volume
ward independently by Radhakrishnan and Srinivasan like this to mention two related objects, namely the
(1982), Alpar et al. (1982) and Fabian et al. (1983), fastradiobursts(FRBs)andgamma-raybursts(GRBs).
has since become the ‘standard model’ for the for- Thestudyofthesetransients(bothinradioandinhigh
mation of these objects, confirmed by the observation energybands)isofgreatcurrentattention.Tworeviews
of a large number of accreting millisecond X-ray pul- consider these two subjects discussing their general
sarsinrecentyears,firstdiscoveredin1998.Naturally, properties,progenitormodelsandtheirpotentialascos-
the evolution of magnetic fields in neutron stars has mologicaltools.
beenanimportantingredientindefiningsuchevolution- Amoderndaycompendiumonneutronsstarphysics
arypathways.Discussionsonthetheoreticalmodelsof isincompletewithoutamentionofgravitationalwaves.
fieldevolutioninaccreting,aswellasisolatedneutron Attention has been focused on neutron stars both as
stars bring us up to date with the current status of this emitters of gravitational waves, as well as detectors
area. The evolution of the magnetic field, in its turn, (viamillisecondpulsartimingarrays)ofthem.Thefinal
depends upon the thermal evolution of a neutron star. twoarticlesconcentrateontheseissuesanddiscussthe
A review sheds light on the cooling behaviour of the importance of neutron stars as tools for probing this
coreandthecrustalregionofaccretion-heatedneutron newestfrontiersofphysics.
stars. Neutron star astrophysics started with the 1967 dis-
In recent years, with the advent of advanced high- coveryofaradiopulsarbyHewishetal.Aswecelebrate
energy instruments, X-ray studies of binary neutron the 50th year of that discovery and the 75th birth year
stars have made spectacular progress. Studies of the of Srini, it is only appropriate that Indian Academy of
process of accretion onto high magnetic field neutron Sciences is bringing out a special issue of J. Astro-
stars provide excellent opportunities to understand the phys. Astr. on a topic close to his heart as a salute
evolution of such binary systems. The relativistic iron to his many contributions to Indian and International
Kα spectral emission line, first detected in 2007, has science.
opened up exciting new ways for probing the strong Finally,itisourpleasanttasktoacknowledgethehelp
gravity and dense matter providing another means of wehavereceivedfromalargenumberofpeople,with-
constraining the equation of state of the neutron star out which this task could never be completed. We are
matter. On the other hand, cyclotron lines from being extremelythankfultotheChiefEditorofJ.Astrophys.
theonlydirectestimatorofthemagneticfieldstrength, Astr., Prof. Ram Sagar, who has been very enthusias-
have now become a tracer of the accretion geometry. tic about our idea of bringing out a special issue to
It is only appropriate that this issue contains an article honour Srini; and to the Editorial Board for readily
describing India’s AstroSat mission which is a power- endorsingtheproposal.Wewouldalsoliketotakethis
ful space based observatory for compact star research opportunity to express our heartfelt gratitude to all the
intheX-rayandUVband. contributors for enthusiastically agreeing to write for
A study of the pulsar magnetosphere is of crucial thisissuedespitetheirothercommitments.Lastbutnot
importance since it is the key to understanding how theleast,wegreatlyappreciatetheworkofShubhankar
energy outflow of a neutron star is produced. Yet, Biswas for providing the cover art within a very short
the emission mechanism in the magnetosphere is as time; and the editorial team at the Indian Academy of
much a topic of debate now as it had been in the Sciences for their invaluable help in bringing out this
early days. A possible model for the emission mech- issue.
anismusingchargedsolitonshasbeendiscussedinthis
context. Reviews of the magneto-rotational and Tay- DipankarBhattacharya,IUCAA,Pune
lor instabilities, as well as the well-documented but K.S.Dwarakanath,RRI,Bangalore
not-yet-completely understood phenomena of nulling, SushanKonar,NCRA-TIFR,Pune
mode-changinganddriftingsub-pulseshavealsobeen GuestEditors
J.Astrophys.Astr.(September2017)38:37 ©IndianAcademyofSciences
DOI10.1007/s12036-017-9456-7
Review
Neutron Stars: Laboratories for Fundamental Physics Under Extreme
Astrophysical Conditions
DEBADESBANDYOPADHYAY1,2
1AstroparticlePhysicsandCosmologyDivision,SahaInstituteofNuclearPhysics,1/AFBidhannagar,
Kolkata700064,India.
2HomiBhabhaNationalInstitute,TrainingSchoolComplex,Anushaktinagar,Mumbai400094,India.
E-mail:[email protected]
MSreceived4May2017;accepted6June2017;publishedonline7September2017
Abstract. Wediscussdifferentexoticphasesandcomponentsofmatterfromthecrusttothecoreofneutron
stars based on theoretical models for equations of state relevant to core collapse supernova simulations and
neutronstarmerger.Parametersofthemodelsareconstrainedfromlaboratoryexperiments.Itisobservedthat
equationsofstateinvolvingstrangenessdegreesoffreedomsuchashyperonsandBose–Einsteincondensates
arecompatiblewith2M neutronstars.Theroleofhyperonsisexploredontheevolutionandstabilityofthe
solar
protoneutronstarinthecontextofSN1987A.Momentofinertia,massandradiuswhicharedirectprobesof
neutronstarinteriorarecomputedandtheirobservationalconsequencesarediscussed.Wecontinueourstudy
onthedensematterunderstrongmagneticfieldsanditsapplicationtomagnetoelasticoscillationsofneutron
stars.
Keywords. Neutronstars—equationsofstate—magneticfields.
1. Introduction advent of X-ray, gamma-ray and radio telescopes, the
observational study of neutron stars has entered into
James Chadwick wrote to Niels Bohr about the a new era. Space-based Indian Observatory AstroSat
discovery of the neutron in a letter dated 24 February is the newest addition in this pool. Observations using
1932(Yakovlevetal.2013).Thepaperonthediscovery thesefacilitiesaswellasothertelescopesarepouringin
oftheneutronwaspublishedinNatureon27February very exciting data on neutron stars. From those obser-
1932.ItisamazingtonotethatLevLandauthoughtof vations, it might be possible to estimate masses, radii,
a highly dense astrophysical object as a giant nucleus moment of inertia, surface temperatures and magnetic
in 1931 well before this discovery and wrote an arti- fieldsofneutronstars(Konaretal.2016).Thenextgen-
cle on this subject which was published almost at the erationradiotelescopeknownastheSquareKilometer
sametimeofthediscoveryoftheneutronon29Febru- Array (SKA) is to be co-located in South Africa and
ary1932(Landau1932).IntheStanfordmeetingofthe Australia.Withthedetectionofgravitationalwavesig-
American Physical Society in 1933, Baade & Zwicky nalfromtheeventinGW150914byLIGOObservatory,
(1934)declared:“Withallreserveweadvancetheview gravitationalwaveastrophysicsopensanewwindowto
that supernovae represent the transition from ordinary probe the neutron star interior. It would be possible to
starstoneutronstarswhichintheirfinalstagesconsist studyfundamentalphysicsinstronggravitationalfields
ofextremelycloselypackedneutrons.”Thesedevelop- of pulsars and black holes using the SKA and LIGO-
mentsmarkedthebeginningofresearchinphysicsand Indiaalongwithothertelescopes.
astrophysicsofneutronstars(Yakovlevetal.2013). Neutronstarsharbourthedensestformofmatterinits
Shortly after the discovery of a pulsar in 1967 interior.Thesecompactastrophysicalobjectsareunique
(Hewish et al. 1968), the study of dense matter in the laboratoriesforcoldanddensematterasthesecannot
core of neutron stars gained momentum. With the beproducedinterrestriallaboratories.Awiderangeof
37 Page 2 of 9 J.Astrophys.Astr.(September2017)38:37
density,fromthedensityofironnucleusatthesurface simulations and neutron stars and discuss how com-
of the star to several times normal nuclear matter den- positions and EoS of matter can be constrained from
sity (2.7×1014 g/cm3) in the core are expected to be observations. In section 2, theoretical models of EoS
presentinneutronstars.Thecompositionandstructure in the crust and core are introduced. In connection to
ofaneutronstararedeterminedbythenatureofstrong SN1987A,theapplicationofthisEoSinsupernovasim-
interaction.Severalnovelphaseswithlargestrangeness ulationsiselaboratedinsection3.Calculationsofmass,
fraction such as hyperon matter (Glendenning 1992, radiusandmomentofinertiaandtheirobservablecon-
1996; Chatterjee & Vidana 2016), Bose–Einstein con- sequences are presented in section 4. Matter in strong
densates of strange mesons (Kaplan & Nelson 1986; magneticfieldsandoscillatorymodesofmagnetarsare
Pal et al. 2000; Banik & Bandyopadhyay 2001a; discussedinsection5.Finallyconclusionsaredrawnin
Knorren et al. 1995) and quark matter (Farhi & Jaffe section6.
1984)mayappearinthehighdensityregimeinneutron
starsduetoPauliexclusionprinciple.Furthermore,the
2. TheoreticalmodelingofEoS
recent accurately measured 2.01±0.04M neutron
solar
star puts stringent condition on the composition and
2.1 Matterinneutronstarcrust
equationofstate(EoS)(Antoniadisetal.2013).
On the other hand, there is a growing interplay
Neutron star interior is broadly separated into two
between the physics of dense matter found in labora-
regions – crust and core. Again the crust is divided
toriesandneutronstars.Thoughthequantumchromo-
into the outer crust and inner crust; so is the core.
dynamics (QCD) predicts a very rich phase structure
There is a huge variation of matter density starting
of dense matter, we can only probe a small region of from 104 g/cm3 in the outer crust to ∼1015 g/cm3 in
it in laboratories. Relativistic heavy ion experiments
thecore.Consequently,thisleadstointerestingphases
produce a hot (a few hundreds MeV) and dense mat-
and compositions of matter in different layers of neu-
ter (a few times normal nuclear matter density). The
tron stars. The outer crust is composed of nuclei in
study of dense matter in heavy ion collisions reveals
the background of a uniformly distributed relativistic
many new and interesting results such as the modi- electrongas.Ataround4×1011 g/cm3,neutronsstart
fications of hadron properties in dense medium, the
drippingoutofnucleiwhentheneutronchemicalpoten-
properties of strange matter including hyperons and
tialisequaltobareneutronmass.Thisistheendofthe
(anti)kaons and the formation of quark-gluon plasma
outercrustandthebeginningoftheinnercrust.Inthis
(Wattsetal.2016;Oerteletal.2017).Theseempirical
layer of matter, the components of matter are neutron-
information from heavy ion collisions may be useful
richnuclearcluster,freeneutronsandauniformgasof
in understanding dense matter in neutron star interior.
relativistic electron gas. As the density increases, the
The properties of finite nuclei obtained in laboratories
matter passes through an interesting phase called the
such as incompressibility of matter, symmetry energy,
pasta phase where various geometrical shapes such as
etc. also contribute to the understanding of matter in
rod,slab,bubble,etc.mightappearduetocompetition
neutronstars.
betweenthesurfacetensionandCoulombinteractionin
Extremely high magnetic fields might be produced
nuclearclusters.Itshowsthatthematterishighlynon-
in heavy ion collisions when moving charges of two
uniform in neutron star crusts. Neutron-rich nuclear
heavynucleisaygoldorleadcollidewitheachotherat
clusters dissolve into neutrons and protons which, in
thespeedoflight.Itwasestimatedthatthisfieldcould
turn,produceauniformnuclearmatter,atthecrust–core
beashighas1019 G(Kharzeevetal.2008).However, interfacearoundthematterdensity2.7×1014 g/cm3.
suchastrongmagneticfieldisproducedforashorttime
Weintroduceherethenuclearstatisticalequilibrium
∼afewfm/c.Ontheotherhand,itwasobservedthata
(NSE) model for the description of matter of light
newclassofneutronstarsknownasmagnetarshadvery
and heavy nuclei together with unbound but interact-
strongsurfacemagneticfields∼1015G.Itwasinferred
ingnucleonsatlowtemperatureandmassdensitybelow
fromthevirialtheorem(Chandrasekhar&Fermi1953) ∼2.7×1014 g/cm3(Hempel&Schaffner-Bielich2010).
that the interior magnetic field could be several times
Inthismodel,thenuclearchemicalequilibriumisregu-
higherthanthesurfacefieldsofmagnetars.
latedbythemodifiedSahaequation.Thetotalcanonical
Thisshowsthatneutronstarsareuniquelaboratories
partitionfunctioninthismodelisgivenby
forfundamentalphysicsunderextremedensities,mag- (cid:2)
neticfieldsandstronggravitationalfields.Inthisarticle, Z(T,V,{Ni}) = Znuc ZA,Z ZCoul , (1)
we describe different phases of matter in supernova A,Z
J.Astrophys.Astr.(September2017)38:37 Page 3 of 9 37
(cid:3)
with V denoting the volume of the system. The LB = ψ¯B(iγμ∂μ−mB+gσBσ −gωBγμωμ
Helmholtzfreeenergyinvolvingfreeenergiesofnucle-
B
ons (Fnuc), nuclei (FA,Z) and Coulomb (FCoul) is −gφBγμφμ−gρBγμτB·ρμ)ψB
computedas
1 1
F(T,V,{N })=−T lnZ (2) + (∂μσ∂μσ −m2σσ2)− ωμνωμν
i (cid:3) 2 4
= Fnuc+ FA,Z + FCoul . (3) +1m2ωωμωμ− 1φμνφμν + 1m2φφμφμ
A,Z 2 4 2
1 1
The number density of each nuclear species (A,Z) is − ρμν ·ρμν + m2ρρμ·ρμ. (5)
4 2
obtained from modified Saha equation (Banik et al.
2014) Here ψ denotes the baryon octets, τ is the isospin
B B
(cid:4) (cid:5) (cid:3)
nA,Z = κ gA,Z(T) M2Aπ,ZT 3/2 coopuerpaltionrgsa.nIdtigstsoabreedneontesditythdaetpφenmdeesnotnmsaerseonm–ebdairaytoend
(cid:6) (cid:7) betweenparticleshavingstrangenessquantumnumber.
exp (A−Z)μ0n+Zμ0p−MA,Z−ECAo,Zul−Pn0ucVA,Z , Next we can calculate the grand-canonical thermo-
T dynamicpotentialperunitvolume
(4)
(cid:12) 1 1 1
= m2σ2− m2ω2− m2ρ2
V 2 σ 2 ω 0 2 ρ 03
(cid:3)
1
− m2φ2−(cid:13)r n −2T
2 φ 0 B
B
(cid:8)
(cid:3)
× d3k [ln(1+e−β(E∗−νi))+ln(1+e−β(E∗+νi))], (6)
(2π)3
i=n,p,(cid:14),(cid:13)−,(cid:13)0,(cid:13)+,(cid:15)−,(cid:15)0
where gA,Z is the nuclear spin degeneracy; κ is the where t(cid:9)he temperature is defined as β = 1/T and
volumefractionavailablefornucleiandapproachesto E∗ = (k2+m∗2). This involves a term called the
zeroatthecrust-coreboundary.Finallyoneobtainsthe i
rearrangement term (cid:13)r (Banik et al. 2014; Hofmann
energydensityandpressureinthismodel.
et al. 2001) due to many-body correlations which is
givenby
2.2 Densematterinneutronstarcore
(cid:3)
(cid:13)r = [−g(cid:3) σns +g(cid:3) ω n +g(cid:3) φ n
σB B ωB 0 B φB 0 B
Neutrons and protons in neutron star core become
B
relativistic as baryon density increases. Furthermore, +g(cid:3) τ ρ n +g(cid:3) φ n ], (7)
dense matter in neutron star interior is a highly many ρB 3B 03 B φB 0 B
bodysystem.TheQCDmightbethefundamentalthe- where (cid:3) denotes the derivative with respect to baryon
ory to describe such a dense matter. Here we focus on densityofspeciesB.
a relativistic field theoretical model involving baryons We also study the Bose–Einstein condensation of
and mesons. In this Lorentz covariant theory, baryon– antikaons (K− mesons) in neutron star matter. In this
baryon interaction is mediated by the exchanges of case, baryons are embedded in the condensate. We
mesons. Meson–baryon couplings are made density treat the kaon–baryon interaction in the same foot-
dependent. Being a relativistic model, this ensures ing as the baryon–baryon interaction described by the
causalityintheEoS. Lagrangian density (5). The Lagrangian density for
Thestartingpointinthedensitydependentrelativistic (anti)kaons in the minimal coupling scheme is (Glen-
hadron (DDRH) field theory is the Lagrangian den- denning & Schaffner-Bielich 1999; Banik & Bandy-
sitywhichdescribesbaryon–baryoninteractionthrough opadhyay2001b)
exchangesofscalarσ,vectorω,φandρmesons(Banik
etal.2014;Typeletal.2010), LK = Dμ∗K¯DμK −m∗K2K¯K , (8)
37 Page 4 of 9 J.Astrophys.Astr.(September2017)38:37
¯
where K and K denote kaon and (anti)kaon doublets; after the explosion over 11 s. It was evident from the
the covariant derivative is Dμ = ∂μ + igωKωμ + detection of neutrinos that a hot and neutrino-trapped
igφKφμ + igρKtK · ρμ and the effective mass of protoneutron star was born and existed for about 11 s.
(anti)kaonsism∗K = mK −gσKσ.Thethermodynamic There is no detectionof a neutronstar in SN1987A so
potentialforantikaonsisgivenby far.Itisbelievedthataneventhorizonwasformedafter
(cid:8)
11sandthePNScollapseintoablackhole.Theques-
(cid:12) d3p
K = T [ln(1−e−β(ωK−−μ)) tioniswhatmadethePNSmetastableanddroveitinto
V (2π)3
ablackhole.
+ln(1−e−β(ωK++μ))]. (9) Different groups investigated the problem of
− stabilityofaPNSforshorttimes.WhenaPNSismade
Thein-mediumenergyof K mesonisgivenby
(cid:9) upofnucleonsandleptons,ithasaslightlysmallermax-
ωK− = ((cid:4)p2+m∗K2) (cid:5) imummassthanthatoftheneutronstar.However,this
situationchangeswiththeappearanceofexoticmatter
− gωKω0+gφKφ0+ 1gρKρ03 , (10) such as hyperons or K− condensation in dense matter
2
during the evolution of the PNS (Banik 2014; Brown
and μ is the chemical potential of K− mesons and is & Bethe 1994). The PNS including hyperon and/or
givenbyμ = μ −μ = μ .Thethresholdcondition Bose–Einsteincondensatehasahighermaximummass
n p e
for s-wave (p = 0) K− condensation is given by μ = thanthatofacoldneutronstar(Brown&Bethe1994;
ωK− = m∗K−gωKω0−gφKφ0− 21gρKρ03 .Meanfield Prakashet al.1995; Banik& Bandyopadhyay2001b).
valuesofmesonsareσ,ω ,φ andρ . Neutrino and thermal pressure could stabilize much
0 0 03
Thermodynamicquantitieslikeenergydensity,pres- larger maximum mass for a protoneutron star during
sure, etc. in the hadronic and kaon condensed phases theevolution.However,thePNSmightbeunstableafter
arecomputedfromthegrand-thermodynamicpotentials de-leptonizationandcooling.
(Baniketal.2008, 2014;Char&Banik2014).Charge The role of (cid:14) hyperons on supernova explosion
neutralityandβ-equilibriumconstraintsareimposedon mechanism and the evolution of PNS has been stud-
neutronstarmatter. ied using a general relativistic one-dimensional core
Finally,meson-nucleondensitydependentcouplings collapsesupernovamodel(O’Connor&Ott2011).Ear-
areobtainedbyfittingpropertiesoffinitenuclei(Banik liersimulationsweredonewiththehyperonEoSwhich
etal.2014;Typeletal.2010).Vectormesoncouplings was not compatible with the two solar mass neutron
for hyperons and kaons are estimated theoretically star(Banik2014).Furthermore,thelongdurationevo-
usingthesymmetryrelations(Weissenbornetal.2012; lution of the PNS with enhanced neutrino heating in
Schaffner&Mishustin1996)whereastheirscalarcou- thesupernovasimulationwith23solarmassprogenitor
plings are obtained from hyeprnuclei and kaonic atom is denoted as s23WH07 and is investigated to test the
data(Char&Banik2014). hypothesisofmetastabilityinthePNS.The(cid:14)hyperon
Recently,Banik,HempelandBandyopadhyay(BHB) EoSofBanik,HempelandBandyopadhyay,BHB(cid:14)φis
constructed a hyperon EoS for supernova and neutron usedasmicrophysicalinputinthissimulation.(cid:14)hyper-
starmatterinvolving(cid:14)hyperonsandtherepulsive(cid:14)–(cid:14) ons appear just after core bounce and its population
interaction mediated by φ mesons (Banik et al. 2014). becamesignificantasthePNSevolves.Thissimulation
This hyperon EoS is compatible with 2M neutron leadstoasuccessfulsupernovaexplosionandthePNS
solar
starsandisdenotedbyBHB(cid:14)φ (Baniketal.2014). evolvestoastableneutronstarof2.0Msolar over3sas
In the following sections, we describe the role of isevidentfromFig.1.Thisiscomparedwiththeresult
compositions and EoS on the evolution of the PNS in ofourearlierCCSNsimulationof20Msolar progenitor
corecollapsesupernovasimulations,masses,radiiand denotedass20WH07thatledtoastableneutronstarof
momentsofinertiaofneutronstarsandmagnetoelastic 1.6Msolar (Charetal.2015).Thesefindingsareatodds
oscillationsofstronglymagnetizedneutronstars. with the prediction about the collapse of the PNS into
ablackholeafterde-leptonizationandcooling.
3. MysteryofthemissingcompactstarinSN1987A 4. Probingneutronstarinterior:Mass,radius
andmomentofinertia
Overthepastthirtyyears,SN1987Ahasbeenthemost
studied core-collapse supernova event. It is the only Neutron star masses have been estimated to very
supernova event from which neutrinos were detected high degree of accuracy due to the measurement of
J.Astrophys.Astr.(September2017)38:37 Page 5 of 9 37
2.5 observations. Observations indicate that neutron stars
areslowlyrotatingandthefastestrotatingneutronstar
s23WH07
s20WH07 among them has a frequency of 716 Hz. Structures
of non-rotating neutron stars are computed from the
2
Tolman–Oppenheimer–Volkoff(TOV)equation
M)solar dp Gε(r)m(r) (cid:4) p(r)(cid:5)(cid:4) 4πr3p(r)(cid:5)
ss ( 1.5 dr =− c2r2 1+ ε(r) 1+ m(r)c2
a
M (cid:10) (cid:11)
al 2Gm(r) −1
on × 1− . (11)
ati 1 c2r
vit
a
Gr WeneedanEoStoclosetheTOVequation.
Slowly rotating neutron stars are investigated by
0.5
perturbing the spherical space-time metric (Hartle &
Thorne 1968). Moment of inertia is calculated from
I = J/(cid:12),where
0
0 1 2 3
(cid:8)
Post Bounce Time (sec) I = 8π Rr4e(λ−ν)(p(r)+ε(r)) ((cid:12)−ω(r))dr ,
Figure 1. Longdurationevolutionoftheprotoneutronstar 3 0 (cid:12)
using20and23M progenitorsandBHB(cid:14)φEoS. (12)
solar
andtheframe-draggingangularvelocity(ω)isobtained
post-Keplerian parameters in relativistic binary by solving the Hartle equation; (cid:12) is the spin of the
systems.Theaccuratelymeasuredhighestneutronstar neutronstarandλ,ν aremetricfunctions.
mass (M) is 2.01±0.04M(cid:5) so far. However, the esti- We consider different compositions for the compu-
mationofradiusfromobservationsisstillproblematic tationofEoS,mass–radiusrelationshipandmomentof
(Bhattacharyyaetal.2017).Thediscoveryofhighlyrel- inertia. Neutron star matter made of neutrons and pro-
ativisticbinarysystemssuchasthedoublepulsarsystem tons is denoted by np. In this calculation, (cid:14) hyperons
PSR J0737-3039 for which masses of both pulsars are appear first at baryon density n = 2.2n where the
b 0
knownaccurately,opensupthepossibilityforthedeter- saturation density is n = 0.149fm−3. The repulsive
0
minationofmomentofinertia(I)ofpulsar Awhich,in (cid:14)–(cid:14) interaction is mediated by φ mesons. This com-
turn,mightovercometheuncertaintiesinthedetermina- position of matter involving neutrons, protons and (cid:14)
tionofradius(R).Itisexpectedthatthehighprecision hyperons is represented by np(cid:14)φ. Being heavier, (cid:13)
timingtechniqueintheupcomingSKAwouldfacilitate and (cid:15) hyperons are populated at much higher densi-
theextractionofthemomentofinertiaofapulsarear- tiesandexcludedfromthiscalculation.Anotherexotic
lier than that in the present day scenario. Higher order phase of matter considered here is the Bose–Einstein
−
post-Newtonian(PN)effectsinrelativisticneutronstar condensed matter of K mesons in which neutrons,
binariescouldbeprobedintheSKAera.Furthermore, protons and (cid:14) hyperons are embedded in the conden-
therelativisticspin-orbit(SO)couplingmightresultin sateandisdenotedbynp(cid:14)K−φ.Thethresholddensity
−
an extra advancement of periastron above the PN con- for K condensation is obtained from the equality of
tributions.ThemeasurementoftheSOcouplingeffect in-mediumenergy(ωK−)of K− andelectronchemical
overandabovethecontributionofthesecondPNterm potential (μ ). This is exhibited in Fig. 2. In this case,
e
couldleadtothedeterminationofmomentofinertiaof theonsetofthecondensateoccursatn = 3.69n .
b 0
a pulsar in relativistic neutron star binaries in general Figure 3 shows the relation between pressure (P)
(Damour & Schaefer 1988) andthe double pulsarsys- and energy density (ε) which is known as the EoS,
tem in particular (Lattimer & Schutz 2005). Observed for the above mentioned compositions of matter. It is
masses,radiiandmomentsofneutronsaredirectprobes evidentfromthefigurethatadditionaldegreesoffree-
−
of compositions and EoS in neutron star interior. The domintheformofhyperonsand K condensatemake
theoretical mass-radius, moment of inertia – compact- an EoS softer. This is also reflected in the structures
nessparameter(ratioofmassandradius)relationships of neutron stars. Mass-radius relationships for differ-
of neutronstars couldbe directlycomparedwithmea- ent compositions and EoS are shown in Fig. 4. Being
suredmasses,radiiandmomentsofinertiafromvarious the stiffest among all other cases considered here,
37 Page 6 of 9 J.Astrophys.Astr.(September2017)38:37
500 3
ωΚ− np
μ npΛφ
e 2.5 npΛΚ−φ
400
V) 2
e
M
ergy ( 300 M) solar1.5
n
E M (
200
1
0.5
100
0 2 4 6 0
8 12 16 20
Normalised Baryon Density (n/n) R (km)
b 0
Figure 2. In-medium energy of K− mesons (ωK−) and Figure 4. Mass–radius relationship for neutron star com-
electronchemicalpotential(μe)asafunctionofnormalized positionsnp,np(cid:14)φandnp(cid:14)K−φ.
baryondensity.
800
200
np
np
nnppΛΛφΚ−φ npΛΚ−φ
600
150
)
3
m
/f 2m)
V400 k
Me Msolar
P ( I ( 100
200
50
0
200 400 600 800 1000 1200 1400
ε(MeV/fm3) 0.1 0.15 0.2 0.25
M/R
Figure 3. Pressureasafunctionofenergydensityforcom- Figure 5. Moment of inertia versus compactness for neu-
positionsnp,np(cid:14)φandnp(cid:14)K−φ. tronstarcompositionsnpandnp(cid:14)K−φ.
nuclearmatterEoSresultsinthehighestmaximummass 2.01±0.04M .Itdemonstratesthatthereisroomfor
solar
neutronstarof2.42M .Ontheotherhand,(cid:14)hyper- exoticmatterinneutronstarinterior.Momentofinertia
solar
onsand K−condensatemaketheEoSsofterleadingto isplottedagainstthecompactnessparameter(M/R)in
smaller maximum mass neutron stars. The maximum Fig.5.Itisevidentfromthefigurethatthemomentof
masscorrespondingtonp(cid:14)φcaseis2.1M ,whereas inertia corresponding to nuclear matter EoS is signifi-
solar
it is 2.09M for the np(cid:14)K−φ case. It is impor- cantlyhigherthanthatoftheBose–Einsteincondensed
solar
tant to note that for exotic phases of matter maximum matter for compactness above 0.2. If the moment of
massesarewellabovetheobservationalbenchmarkof inertia of pulsar A in the double pulsar is estimated in