The Relativistic Inverse Stellar Structure Problem

0.29 MB·

by  Lee Lindblom

The Relativistic Inverse Stellar Structure Problem
Lee Lindblom
TheoreticalAstrophysics,CaliforniaInstituteofTechnology,Pasadena,CA91125
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Abstract. Theobservablemacroscopicpropertiesofrelativisticstars(whoseequationsofstateareknown)canbepredicted
0
bysolvingthestellarstructureequationsthatfollowfromEinstein’sequation.Forneutronstars,however,ourknowledgeof
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theequationofstateispoor,sothedirectstellarstructureproblemcannotbesolvedwithoutmodelingthehighestdensity

b partoftheequationofstateinsomeway.Thistalkwilldescriberecentworkondevelopingamodelindependentapproach
e todetermining thehigh-density neutron-star equationof statebysolvinganinversestellarstructureproblem. Thismethod
F usesthefactthatEinstein’sequationprovidesadeterministicrelationshipbetweentheequationofstateandthemacroscopic
observablesofthestarswhicharecomposedofthatmaterial.Thistalkillustrateshowthismethodwillbeabletodetermine
1
thehigh-densitypartoftheneutron-starequationofstatewithfewpercentaccuracywhenhighqualitymeasurementsofthe

massesandradiiofjusttwoorthreeneutronstarsbecomeavailable.Thistalkwillalsoshowthatthismethodcanbeused
]
E withmeasurementsofothermacroscopicobservables,likethemassesandtidaldeformabilities,whichcan(inprinciple)be
measuredbygravitationalwaveobservationsofbinaryneutron-starmergers.
H
Keywords: equationofstate,stars:neutron,relativity
.
h PACS: 04.40.Dg,97.60.Jd,26.60.Kp,26.60.Dd
p
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1. STANDARD STELLAR STRUCTURE PROBLEM
r
t
s
a Modelsof stars in generalrelativity theoryare constructedby solvingthe relativistic stellar structureequationsthat
[ followfromEinstein’sequation.Themostwidelyusedformoftheseequations,fornon-rotatingstellarmodels,was

1 firstderivedbyOppenheimerandVolkoff[1]:
v
5 dm = 4p r2e , (1)
3 dr
0 dp m+4p r3p
0 = −(e +p) . (2)
. dr r(r−2m)
2
0 Theseequationsdeterminem(r)and p(r),themassofthestarcontainedwithinasphereofradiusrandthepressure
4
respectively.Theseequationsareincompletehowever,andmustbesupplementedbygivinganequationofstatethat
1
characterizesthematerialcontainedwithinthestar. Inparticular,therelationshipbetweenthetotalenergydensitye
:
v andthepressure pmustbespecified.Theadditionoftherelationshipe =e (p)completesthestructureequations,(1)
i
X and(2),makingthemafirst-ordersetofordinarydifferentialequationsthatcanbesolvedusingstandardtechniques.
These equationsare usually integratedstarting at the center of the star, where r=0, using the boundaryconditions
r
a m(0)=0, to ensure thatthe stellar modelis non-singular,and p(0)= p to select the centralvalue of the pressure.
c
Thevalueofthecentralpressure p canbechosenarbitrarily,withlargerp generallycorrespondingtostellarmodels
c c
havinglargermasses.Thestellarstructureequations,(1)and(2),arethenintegratedtowardlargervaluesofruntilthe
surfaceofthestaratr=Risreached,where p(R)=0.ThetotalmassofthestarisgivenbyM=m(R).
Thecentralpressurep isafreelyspecifiableparameter,sothestellarstructureequationsdetermineaone-parameter
c
family of solutions, m(r,p ) and p(r,p ), for any given equation of state. In general, the macroscopic observables
c c
associatedwiththesestellarmodels,likethetotalmassMandradiusR,alsodependonthisparameterp .Thereforethe
c
macroscopicobservablesassociatedwithanyparticularequationofstatecanbethoughtofasacurve,parameterized
by p ,inthespaceofthoseobservables.ThistalkwillfocusontheparticularcaseofmassesM(p )andradiiR(p ).
c c c
Thestandardstellarstructureproblemcanbethoughtofasanabstractmapthatassociatesanequationofstate(which
isineffectacurvethoughthespaceofenergydensitiesandpressures)withacurvethroughthespaceofobservables.
Figure1illustrateswhatthisabstractmaplookslikefortheparticularcaseofthemass-radiuscurveofobservables.
FIGURE1. Standardrelativisticstellarstructureproblemcanbethoughtofasanabstractmapbetweenanequationofstate(a
curveinthespaceofdensitiesandpressures)andacurveofmacroscopicobservables,likethemass-radiuscurve.
2. INVERSE STELLAR STRUCTURE PROBLEM
Whentheequationofstate iswellunderstood,asinwhitedwarfstars,thestandardstellarstructureproblemisvery
useful. Detailed models of these stars can be computed, and their observable properties can be compared directly
with observations. For cases where the equation of state is poorly know however, like neutron stars, the standard
stellar structureproblemisnotsouseful.Figure2 illustratesthehigh-densityportionsof34theoreticalneutron-star
equationsof state [2]. The range of pressures correspondingto nuclear density (dashed red line) predictedby these
theoreticalequationsofstate havearangethatspansaboutanorderofmagnitude.Thismeansthatitisnotpossible
to make reliable quantitativepredictionsaboutthe structuresof neutron stars simply by solving the standard stellar
structureproblem.
15
10
e (p)
p
14
10
12 13 14
10 10 10
FIGURE2. Theneutron-starequationofstateispoorlyknown,asillustratedherebythehigh-densityportionsof34theoretical
neutron-starequationsofstate.Therangeofpressuresatnucleardensity(dashedredline)intheseequationsofstatespanaboutan
orderofmagnitude.
The range of densities and pressures inside typical neutron stars are well beyondthe range where any currentor
foreseeablefuturelaboratoryexperimentwillbeabletomeasuretheneutron-starequationofstatedirectly.Incontrast,
observationsof themacroscopicpropertiesof neutronstarsare becomingmorenumerousandmoreaccurateall the
time. What is needed then is an inverse stellar structure problem,where the equation of state is determinedfrom a
knowledgeofthemacroscopicobservablesofthosestars,e.g.theirmassesandradii.Thatis,weneedawayofsolving
thestellarstructureequationsfortheequationofstate,insteadofspecifyingitaprioriaswedidaspartofthedefinition
ofthestandardstellarstructureproblem.
Atanabstractlevelitisnothardtoimaginewhatsuchaninversestellarstructureproblemmightlooklike.InFig.1
we visualized the standard stellar structure problem as a map that takes a curve in density-pressure space (i.e. the
equation of state) and identifies it with a curve in the space of macroscopicobservables.From this perspective, the
inversestellar structureproblemcorrespondstothe inverseofthisabstractmap.Thatis, the inversestellar structure
problemistheabstractmap,illustratedinFig.3,thattakesacurveinthespaceofmacroscopicobservablesandfrom
itdeterminesthecorrespondingcurveindensity-pressurespace.
FIGURE 3. Inverse stellar structure problem can be thought of as an abstract map that takes a curve of observables, like the
mass-radiuscurve,andfromitdeterminestheequationofstateofthestellarmatter.
3. TRADITIONALSOLUTIONOFTHEINVERSESTELLAR STRUCTUREPROBLEM
Letmefirstdiscussaparticularsolutiontotheinversestellarstructureproblem,developedabout20yearsago[3],that
Icallthetraditionalsolution.LaterinthistalkIwillalsodiscussanewerandmoresophisticatedapproachtosolving
thisproblem.Thetraditionalapproach,however,providesaclearerinsightintothewaythestellarstructureequations
determinethesolutiontothisinverseproblem,soIthinkitisworthdiscussing.
Letusassumethattheentiremass-radiuscurve{R,M}isknown.Letusalsoassumethatthelowestdensitypartof
theequationofstateisknown,i.e.,e =e (p)isassumedtobeknownfor p≤ p ande (p)≤e (p)=e .Let{R,M}
i i i i i
denotethepointonthemass-radiuscurvewhosecentralpressureis p. We chooseanotherpointonthemass-radius
i
curve,{R ,M },thatis“close”tothepoint{R,M},buthasaslightlylargercentralpressure.Figure4illustrates
i+1 i+1 i i
these assumptions. The mass-radius curve is illustrated in the first panel of Fig. 4, including the particular points
{R,M} and {R ,M }. The red portion of the equation of state curve, shown in the middle panel of Fig. 4, is
i i i+1 i+1
alsoassumedtobeknown,includingtheparticularpoint{p,e }.Theredportionofthemass-radiuscurveinFig.4
i i
represents those stellar models whose central pressures are less than p. The panel on the right in Fig. 4 is a cross
i
section thatrepresentsthe structureofthe stellar model{R ,M }. Theouterportionsofthis model,shadedred,
i+1 i+1
arecomposedoflowdensitymaterialwhoseequationofstateisalreadyknown.Thesmallcentralcoreofthismodel,
shadedblue,iscomposedofmaterialwhosepressureanddensityexceed{p,e }.
i i
FIGURE4. Knowledgeofthecompletemass-radiuscurveplusaknowledgeofthelowdensityequationofstatedeterminesthe
structureoftheouter partsofthestarwithtotalmassM andradiusR .Alsodeterminedfromthisknowledgearethemass
i+1 i+1
m andradiusr ofthesmallinnercoreofmaterialhavingdensitiesandpressuresfromtheundeterminedhigh-densitypartof
i+1 i+1
theequationofstate.
Therelativisticstellarstructureequations,(1)and(2),canbeintegratedtodeterminequantitativelythestructureof
the stellar model{R ,M }. Generally these equationsare integratedfrom the center of the star, r=0, outward.
i+1 i+1
Butwecannotdothatinthiscase,becausewedonotknowtheequationofstateatdensitiescontainedinthecoreof
thismodel.Wecan,however,integratetheequationsstartingatthesurfaceofthisstar,r=R ,usingasinitialdata
i+1
m(R )=M and p(R )=0.Sincethelowdensityequationofstateisknown,thisintegrationcanbecontinued
i+1 i+1 i+1
up to the point where p= p. The mass and radius, {r ,m }, of this small high-density core can therefore be
i i+1 i+1
determinedquantitativelyfromthismodel: p(r )=p,andm(r )=m .
i+1 i i+1 i+1
Ifthepoint{R ,M }hasbeenchosentobesufficientlycloseto{R,M},thenthehigh-densitycorecomposedof
i+1 i+1 i i
materialfromtheunknownhigh-densitypartoftheequationofstatewillhaveverysmallmassandradius,{r ,m }.
i+1 i+1
Inthiscasethestructureequations,(1)and(2),canbesolvedaspowerseriesexpansions:
4p
m = e r3 +O(r5 ), (3)
i+1 3 i+1 i+1 i+1
2p
p = p − (e +p )(e +3p )r2 +O(r4 ). (4)
i i+1 3 i+1 i+1 i+1 i+1 i+1 i+1
Thecoefficientsintheseseriesdependon{p ,e },thecentraldensityandpressureofthelittlecore{r ,m }.
i+1 i+1 i+1 i+1
These series can therefore be inverted to determine {p ,e } in terms of the known quantities {p,e } and
i+1 i+1 i i
{r ,m }.Thisprovidesanadditionalpointontheequationofstatecurve,asillustratedinFig.5.Thepointsonthe
i+1 i+1
equationofstatecurvethatareintermediatebetween{p,e }and{p ,e }aredeterminedbyadoptingasuitable
i i i+1 i+1
interpolationformula.Thisprocedurecanbeiteratedtodeterminetheequationofstatefortheentirerangeofdensities
thatoccurwithinthecoresofneutronstars.Thismethodhasbeenusedsuccessfullytosolvetheinversestellarstruc-
ture problemusing mock mass radiusdata constructedfrom a known theoreticalequationof state in Lindblom[3].
FIGURE5. Traditionalsolutionoftheinversestellarstructureproblemiterativelydeterminesasequenceofpointsontheequation
of state curve by inverting the power series solutions for small inner cores of material whose masses and radii are determined
numericallyfromaknowledgeofthecompletemassradiuscurve.
Thistraditionalmethodforsolvingtheinversestellar structureproblemrepresentstheequationofstateasa table
{p,e } for i=1,...,N plus an interpolation formula to fill in intermediate points. The accuracy of a particular
i i EOS
implementationofthismethoddependsonhowmanytermsareretainedinthepowerseriesexpansions,(3)and(4),
aswellasthetypeofinterpolationformulaused.Theoverallaccuracyoftheequationofstatedeterminedinthisway
will also depend on the number,N , of mass-radius data available, typically scaling like N−n for some n. Since
stars stars
thismethodofrepresentingequationsofstateisratherinefficient,andsincethismethodofsolvingtheinversestellar
structureproblemconvergesratherslowly,manymass-radiusdatapointswouldbeneededtodeterminetheequationof
statewithreasonableaccuracyinthisway.Measurementsofthesedataareverydifficulttomake,soitseemsunlikely
thatthistraditionalmethodwilleverbeusefulforanalyzingrealastrophysicaldata.
4. SPECTRAL APPROACHTO THE INVERSESTELLAR STRUCTURE PROBLEM
Thekeyideaofthespectralapproachtotheinversestellarstructureproblemistofindamoreefficientwaytorepresent
equationsofstateusingspectralmethods.Fewerobservationsofthemacroscopicobservablesshouldthenbesufficient
todeterminetheequationofstatebysolvinganappropriatelymodifiedversionoftheinversestellarstructureproblem.
Thebasicideaistoconstructefficientparametricrepresentationsoftheequationofstate,e.g.e =e (p,g ),where
k
the g are a set of parameters that fix the functionalform of a particular equation of state. One example of such a
k
representationisasimplespectralexpansionoftheform,e (p,g )=(cid:229) g F (p),wheretheF (p)aresuitablychosen
k k k k k
basis functions.Examplesof such basis functionsincludethe Fourier basis functionsF (p)=eikp, and the various
k
collectionsoforthonormalpolynomials,liketheChebyshevpolynomialsF (p)=T (p).Spectralexpansionsofthis
k k
typeareexactwhenaninfinitenumberofbasisfunctionsareincludedinthesum.Truncatingtheseexpansionsatsome
finitenumberofterms,k≤Ng ,givesanapproximationtotheequationofstate,whoseerrorsdecreasefasterthanany
k
power of Ng (e.g. typically exponentially) for smooth functions. I will discuss the particular choice of parametric
k
representationthathasbeenusefulforequationsofstatelaterinthistalk.
For now, let us simply assume that we have a parametric representation, e =e (p,g ), that provides reasonably
k
accurateapproximateequationsofstatewithjustafewparametersg .Weusethisfamilyofequationsofstatetosolve
k
thestandardstellarstructureequations,(1)and(2).Theresultsincludethevariousmacroscopicobservablesassociated
withthesestellarmodels,liketheirtotalmassesM(p ,g )andradiiR(p ,g ).Thesemodelobservablesnowdepend
c k c k
explicitlyintheparametersg thatdeterminetheparticularequationofstate,aswellastheparameterp thatspecifies
k c
thecentralpressureofthe model.Thespectralapproachtothe inversestellarstructureproblemfixestheseequation
ofstate parameters,g ,bymatchingthe modelobservables{R(p ,g ),M(p ,g )} toa collectionofpoints{R,M},
k c k c k i i
fori=1,...,N ,fromtheexactcurveofmacroscopicobservablesforthesestars.Theparametersg ,alongwiththe
stars k
parameters pi that representthe centralpressures of the models whose observablesare {R,M}, are determinedby
c i i
minimizingthequantityc (g ,pi),definedby,
k c
c 2(g ,pi) = 1 N(cid:229)stars log R(pci,g k) 2+ log M(pci,g k) 2 , (5)
k c N R M
stars i=1 ((cid:20) (cid:18) i (cid:19)(cid:21) (cid:20) (cid:18) i (cid:19)(cid:21) )
withrespecttoeachoftheparametersg and pi Thisnon-linearleastsquaresproblemcanbesolvedusingstandard
k c
numerical methods. Once fixed in this way, the parameters g now determine an equation of state, e = e (p,g ),
k k
thatprovidesanapproximatesolutiontotheinversestellarstructureproblem.Theaccuraciesofspectralexpansions
typicallydecreaseexponentiallywiththenumberofparametersNg usedintheexpansion.Theexpectationisthatthe
k
approximateequationsofstateconstructedwiththisspectralapproachtotheinversestellarstructureproblemshould
thereforeconvergerapidlytotherealequationofstateasthenumberofparametersg isincreased.
k
5. SPECTRAL REPRESENTATIONS OFNEUTRON-STAR EQUATIONSOFSTATE
The standard Oppenheimer-Volkoff representation of the stellar structure equations, (1) and (2), which use the
pressure as the fundamental thermodynamic variable, is not actually the best form to use for numerical solutions
(cf.Lindblom[3]).Foravarietyofreasonsitisbettertore-writethoseequationsintermsoftherelativisticenthalpy
h,definedby
p dp′
h(p) = , (6)
0 e (p′)+p′
Z
insteadofthepressurep.Iwilldiscusssomeofthebenefitsoftheenthalpy-basedstellarstructureequationslaterinthis
talk.Butletusfocusfirstontheproblemoffindingsuitableparametricenthalpy-basedrepresentationsoftheequation
of state. The enthalpy-basedequation of state consists of the pair of functions, e =e (h) and p= p(h). The simple
spectralexpansionsoftheseequationswouldhavetheformse (h,a )=(cid:229) a F (h)andp(h,b )=(cid:229) b F (h).These
k k k k k k k k
simpleexpansionshavetwoseriousdrawbacks.First,thespectralbasisfunctionsF (h)aretypicallyoscillatory,while
k
physicalequationsofstatemustbemonotonicallyincreasingfunctions.Whileitispossibletoexpresseverymonotonic
functionusinganexpansionofthissort,doingsorequiresthatthespectralcoefficientssatisfyaverycomplicatedset
ofinequalities.Theseconddrawbackisthattheseexpansionsofe (h)and p(h)requiretwosetsofparameters,while
the original e (p) expansion considered previously required only one. These simple enthalpy based expansions are
thereforeveryinefficient,whichmakestheminappropriateforourpurposes.
Weneedfaithfulspectralrepresentationsoftheequationofstate.Faithfulrepresentationsinthiscontextmeansones
whereeverychoiceofspectralparameterscorrespondsto a possiblephysicalequationof state, andconverselyones
whereeveryphysicalequationofstatecanberepresentedbysomechoiceofspectralparameters(cf.Lindblom[4]).
The simple spectral expansions proposed above are not faithful. It is not possible to represent the space of all
monotonically increasing functions as all possible linear combinations of any choice of basis functions, because
monotonic functions do not form a vector space. Faithful spectral expansions can, however, be constructed for the
adiabaticindex,G (h),definedas
G (h)= e +pdp =exp (cid:229) g F (h) . (7)
p de k k
" k #
Monotonicityoftheequationofstateimpliesthattheadiabaticindexispositive,G (h)>0,butthereisnorequirement
thatitmustalsobemonotonic.Anyspectralexpansionoftheformgiveninequation(7)isthereforefaithful.
Unfortunately, the adiabatic index G (h) does not appear in the stellar structure equations, while e (h) and p(h)
do. Fortunately, however, the adiabatic index G (h) determines both e (h) and p(h). The definitions of the enthalpy
in equation(6), and the adiabatic indexin equation(7), can be written as the followingpair of ordinarydifferential
equationsfore (h)and p(h):
dp
= e +p, (8)
dh
de (e +p)2
= . (9)
dh pG (h)
Given an adiabatic index, G (h), this first-order system of equations can be integrated exactly (cf. Lindblom [4]) to
determineboth p(h)ande (h):
heh′dh′
p(h) = p exp , (10)
0 "Zh0 m (h′)#
eh−m (h)
e (h) = p(h) , (11)
m (h)
m (h) = p0eh0 + hG (h′)−1eh′dh′, (12)
e0+p0 Zh0 G (h′)
whereh isthe lowerboundontherangeofenthalpiesoverwhichthe spectralexpansionisperformed,e =e (h ),
0 0 0
and p = p(h ). While theintegralsthatappearin equations(10)–(12)cannotgenerallybe doneanalytically,these
0 0
integralscan neverthelessbe donenumericallyveryefficiently.I havefoundthatthey can be evaluatednumerically
withdoubleprecisionaccuracyusingGaussianquadratureusingonlytenortwelvecollocationpoints.
Alltheequationsofstate,e (h,g )andp(h,g ),definedbyequations(10)–(12),usingtheadiabaticindexexpansion
k k
giveninequation(7)arefaithfulforanychoiceofbasisfunctions.Todeterminewhetherexpansionsofthistypeare
efficientandpractical,Ihaveusedthemtoconstructspectralrepresentationsof34theoreticalneutron-starequations
of state. These theoretical equation of state models are cataloged in Read, et al. [2], and referencesare given there
to the original papers on which they are based. These theoretical equations of state are given as tables {p,e } for
i i
i=1,...,N ,fromwhichasetofcorrespondingh canbeevaluatedusingequation(6),onceasuitableinterpolation
EOS i
formula is adopted. These data were fit to these parametric equations of state, e = e (h,g ) and p = p(h,g ), by
k k
minimizingthequantityD EOS,definedby
Ng
k
D EOS 2 = 1 N(cid:229)EOS log e (hi,g k) 2,
(cid:16) Ngk (cid:17) NEOS i=1 (cid:26) (cid:20) ei (cid:21)(cid:27)
withrespecttoeachofthespectralparametersg .Suchfitswereperformedusingseveralchoicesforthespectralbasis
k
functionsF (h). The following simple choice was found to be quite effective for a wide range of these theoretical
k
equationsofstate,
G (h)=exp Ng(cid:229)k−1g log h k . (13)
k
h
( k=0 (cid:20) (cid:18) 0(cid:19)(cid:21) )
Thissimplepower-lawexpansionhasthereforebeenadoptedasthestandardtouseforenthalpy-basedrepresentations
oftheequationofstate(cf.Lindblom[4]).TheresultsofthesefitsaresummarizedinTable1.Theseresultsshowthat
only two or three spectral coefficients g are needed to produce representationsof these 34 theoretical neutron-star
k
TABLE1. Summariesof theaverageaccuraciesof
the direct spectral fits to 34 theoretical equations of
state.Theexamplesincludedinthistablearethebest
case,PAL6,atypicalaveragecase,MS1,andtheworst
case,BGN1H1.Alsolistedaretheaverageerrorsfor
thespectralfitstoall34theoreticalequationsofstate.
EOS D EOS D EOS D EOS D EOS
2 3 4 5
PAL6 0.0032 0.0016 0.0005 0.0002
MS1 0.0277 0.0055 0.0035 0.0003
BGN1H1 0.0868 0.0495 0.0439 0.0403
Average 0.0323 0.0166 0.0124 0.0089
equations of state with average errors of only a few percent errors over the entire range of densities that occur in
the cores of neutron stars. Figure 6 illustrates in more detail the errors in these spectral equationsof state for three
particular cases: PAL6, MS1 and BGN1H1. These three represent the equations of state having the best (PAL6), a
typical average (MS1), and the worst (BGN1H1) spectral representations. These results show that even the worst
case,theBGN1H1equationofstatewhichhasastrongphasetransitionwithintheneutron-stardensityrange,canbe
representedfairlyaccuratelywithaspectralexpansionhavingonlyafewspectralcoefficients.
FIGURE6. Accuraciesofthespectralrepresentationsofthreeofthetheoreticalneutron-starequationsofstate.Thesethreecases
representtheequationsofstatewithspectralrepresentationshavingthebest(PAL6),theaverage(MS1),andtheworst(BGN1H1)
accuracies.
6. SPECTRAL SOLUTION OFTHE INVERSE STELLAR STRUCTURE PROBLEM
The standard Oppenheimer-Volkoff version of the relativistic stellar structure equations, (1) and (2), are not ideal
fornumericalwork.The problemis thatthe rightside ofthe pressureequationvanishesas thesurfaceof thestar is
approached.Theleadingorderterminthesolutionforthepressurep(r)nearthesurfacegoeslikep(cid:181) (R−r)G 0/(G 0−1),
whereG isthesurfacevalueoftheadiabaticindex.Consequentlyitisverydifficulttosolve p(R)=0numericallyto
0
determinethelocationofthesurfaceofthestar.Thisproblemcanbeavoidedbyswitchingtotherelativisticenthalpy,
hdefinedinequation(6),asthefundamentalthermodynamicvariable.Inthiscasethestructureequation(2)becomes
dh 1 dp m+4p r3p(h)
= =− . (14)
dr e +p dr r(r−2m)
Thisequationimpliesthattheleadingorderterminthesolutionfortheenthalpyh(r)goeslikeh(cid:181) (R−r)nearthe
surface.Consequentlyitismucheasiernumericallytosolveh(R)=0todeterminethelocationofthesurfaceofthe
starinthiscase.
Equation (14) implies that the enthalpy h(r) is a monotonically decreasing function in any star. It is possible,
therefore to transform the relativistic structure equations one more time by using h instead of r as the independent
variable:
dr r(r−2m)
= − , (15)
dh m+4p r3p(h)
dm 4pe (h)r3(r−2m)
= − . (16)
dh m+4p r3p(h)
Theboundaryconditionsfortheseequationsatthecenterofthestarwhereh=h arer(h )=m(h )=0.Thesolution
c c c
to these equations is a relativistic stellar model with the mass m(h) and radius r(h) expressed as functions of the
enthalpy.Thisversionoftheequationshasseveraladvantagesfornumericalwork.First,therangeoftheindependent
variableh ≥h≥0isfixedbeforetheintegrationbegins.Therefore,itisnotnecessarytohuntforthelocationofthe
c
surfacebytryingto solveh(R)=0onthe flyaspartof thesolutionprocess.Second,the surfacevaluesofthetotal
massMandradiusRofthemodelaredeterminedsimplybyevaluatingthesolutionattheendpointoftheintegration
where h=0: M =m(0) and R=r(0). The disadvantage to this approach is that the standard representation of the
equationofstate,e =e (p)mustbetransformedintoapairoffunctionse =e (h)and p=p(h).Thiscanbedoneina
straightforwardwaynumerically,usingthedefinitionoftheenthalpyinequation(6).Inpracticethisdisadvantageis
notarealproblem.
These enthalpy based representations of the stellar structure equations, (15) and (16), along with the enthalpy
basedspectralrepresentationsoftheequationofstate,equations(10)–(13),canbesolvedtoevaluatethemacroscopic
observablesM(h ,g )and R(h ,g ) accuratelyandefficiently.These observablesdependon the centralvalueof the
c k c k
enthalpyh inthesemodels(inmuchthesamewayasthestandardrepresentationsdependonthecentralpressurep ),
c c
alongwiththespectralparametersg thatspecifyaparticularequationofstate.
k
My collaboratorNathanielIndikandI haveusedthese techniquesto constructsolutionsto the relativistic inverse
stellar structure problem [5, 6]. We compute the spectral coefficients g that determine the approximate solution to
k
the inversestellar structureproblembyminimizingthe quantity c (hi,g )thatmeasuresthe differencesbetweenthe
c k
model values of the observablesR(hi,g ) and M(hi,g ) and the data points {M,R} taken from the exact curve of
c k c k i i
observables:
c 2(hi,g ) = 1 N(cid:229)stars log M(hci,g k) 2+ log R(hci,g k) 2 . (17)
c k N M R
stars i=1 ((cid:20) (cid:18) i (cid:19)(cid:21) (cid:20) (cid:18) i (cid:19)(cid:21) )
The quantity c (hi,g ) is minimized with respect to variations in the g , as well as the parameters hi that specify
c k k c
the centralvaluesof the enthalpiesin the stars with observables{R,M}. This minimization is accomplishedusing
i i
standardnumericalmethods.
Wehavetestedthismethodbyconstructingsetsofmockobservables{R,M}basedonthe34theoreticalneutron
i i
star equations of state described earlier. For each equation of state, we select 2≤N ≤5 pairs of observables.
stars
Themassesofthesemockdataarechosentobeuniformlyspacedbetween1.2M (atypicalminimumastrophysical
⊙
neutron-starmass),andthemaximum-massmodelassociatedwiththatparticularequationofstate.Figure7illustrates
thesemockdatapointsfortheN =5casesofthePAL6,theMS1andtheBGN1H1equationsofstate.
stars
Wethensolvetheinversestellarstructureproblembysolvingfortheparametersg andhi thatminimizec (hi,g )
k c c k
forthedatasetsconsistingofN observabledatapairsfromeachofthe34theoreticalequationsofstate.Wetested
stars
theaccuracyoftheequationsof state determinedin thisway bycomparingtheresultingspectralmodelequationof
statee (h,g )withthetabulatedexacttheoreticalequationofstate{p,e ,h}thatwasusedtoconstructthemockdata.
k i i i
WeevaluatethedifferencesbetweentheseusingtheaverageerrormeasureD MR definedby
Ng
k
D MR 2 = 1 N(cid:229)EOS log e (hi,g k) 2. (18)
(cid:16) Ngk(cid:17) NEOS i=1 (cid:26) (cid:20) ei (cid:21)(cid:27)
TheseresultsaresummarizedinTable2.Theaverageerrorsinthespectralequationsofstatedeterminedbysolving
the inverse stellar structure problem D MR are given in the left columns of Table 2. These error values, D MR, can
Ng Ng
k k
be compared to those constructed earlier using direct spectral fits to the equation of state tables, D EOS, which are
Ng
k
reproducedintherightcolumnsofTable2.Weseethattheaccuraciesoftheequationsofstatedeterminedbysolving
the inverse stellar structure problem using spectral methods are somewhat less accurate than the optimal spectral
fits to those equationsof state. This is not surprising. For the smoothest equationsof state however,like PAL6, the
FIGURE7. Mockdataforthemassesandradii{Ri,Mi}wereselectedfromthemass-radiuscurvesofeachofthe34theoretical
neutron-starequationsofstate.Shownherearethemass-radiuscurvesforthePAL6,MS1andtheBGN1H1equationsofstatewith
theNstars=5datapointsselectedfromeachcurve.Mockdatawerechosentobeequallyspacedinmassbetween1.2M⊙(atypical
minimumastrophysicalneutron-starmass)andthemaximummassmodelforeachparticularequationofstate.
differencebetweentheinversestellarstructureequationofstateandtheonesdeterminedbydoingdirectspectralfits
arenegligible.Evenforequationsofstatehavingsharpphasetransitions,likeBGN1H1,theerrorsintheinversestellar
structureequationsofstateareonlyafewtimesworsethanthebestfits.
TABLE 2. Accuracies of the equations of state determined by the spectral approach to the
inversestellarstructureproblembasedonmass-radiusdata.Theexamplesincludedinthistable
arethebestcase,PAL6,atypicalmediancase,MS1,andtheworstcase,BGN1H1.Alsolisted
aretheaverageerrorsforall34theoreticalequationsofstate.TheaccuraciesfromTable1of
thedirectspectralfitstotheseequationsofstatearealsolistedintherightcolumnsofthistable
forcomparison.
EOS D MR D MR D MR D MR D EOS D EOS D EOS D EOS
2 3 4 5 2 3 4 5
PAL6 0.0034 0.0018 0.0007 0.0003 0.0032 0.0016 0.0005 0.0002
MS1 0.0474 0.0157 0.0132 0.0009 0.0277 0.0055 0.0035 0.0003
BGN1H1 0.1352 0.1702 0.1356 0.1382 0.0868 0.0495 0.0439 0.0403
Average 0.0396 0.0289 0.0276 0.0239 0.0323 0.0166 0.0124 0.0089
The overall accuracies of the equations of state determined by solving the spectral version of the inverse stellar
structureproblemarequiteimpressive.Theseequationsofstatecanbedeterminedusingspectralmethodswithaverage
errorsofjust a fewpercentoverthe entire rangeof neutronstar densities,using (highquality)measurementsof the
massesandradiiofjusttwoorthreeneutronstars.Eventheworstcase,anequationofstatewithasharpphasetransition
intheneutron-stardensityrange,isdeterminedwithaccuracylevelsaround15%usinghighqualitymeasurementsof
justafewmassesandradii.Figure8illustratesinmoredetailtheerrorsintheequationsofstateobtainedwithmock
mass-radius data from the three equations of state PAL6, MS1 and BGN1H1. These error graphs are qualitatively
similartothosedisplayedinFig.6forthedirectspectralfits.
7. INVERSESTELLAR STRUCTURE PROBLEMUSING TIDALDEFORMABILITIES
Thediscussionoftheinversestellarstructureprobleminthistalkhasfocuseduptothispointondevelopinganuclear-
physicsmodelindependentmethodforusingmass-radiusdatatodeterminetheequationofstateofneutronstarmatter.
Theadvancesinobservationalmethodsoverthepastfewyearshasdramaticallyincreasedthenumberandqualityof
theobservationsofneutronstarmassesandradii.Itseemslikelythattheseimprovementswillcontinue,andthatthese
observationswilleventuallyprovidehighenoughqualitydatatodeterminetheequationofstatewithgoodaccuracy.
Other types of observations can provide measurements of other neutron-star observables, and in many cases these
observablescan also be used to solve the inversestellar structureproblem.InparticularFlanniganand Hinderer[7]
havepointedout(andthishasbeenconfirmedbyanumberofothers,e.g.,Lackey,etal.,[8,9])thatthemassesand
tidal deformabilitiesof neutronstars could be measured by observing the gravitationalwaveforms producedby the
FIGURE8. Accuraciesofthespectralrepresentationsdeterminedbysolvingtheinversestellarstructureproblemusingmock
mass-radius data fromthree of the theoretical neutron-star equations of state.These three casesrepresent theequations of state
whosespectralrepresentationshavethebest(PAL6),theaverage(MS1),andtheworst(BGN1H1)accuracies.
inspiralandmergersofneutron-starbinarysystems.InthislastpartofmytalkIwilldescribesomerecentwork(done
incollaborationwithNathanielIndik)thattestshowwellthespectralapproachtotheinversestellarstructureproblem
workswhenappliedto(ideal)massandtidaldeformabilitydata.
Thetidaldeformability,L ,measurestheamountbywhichastardistortsitshapeinresponsetotheapplicationof
a tidalforcefroma nearbystar (orblackhole).Thetidaldeformabilitydependsonthe equationof state of the star,
and therefore its measurement can in principle be used to determine the equation of state of the stellar matter. For
a givenequationofstate, the tidaldeformabilitycan becalculatedformodelstars usingmethodsfirst developedby
Hinderer [10, 11], and then transformedinto the somewhat more efficient method described here by Lindblomand
Indik[6].Thestellar structureequations,(15)and(16),aresolvedform(h)andr(h), togetherwitha thirdequation
thatdeterminesafunctiony(h)whichdescribesthetimeindependentquadrapoledeformationsofarelativisticstar,
dy (r−2m)(y+1)y (m−4p r3e )y 4p r3(5e +9p)−6r 4p r3(e +p)2 4(m+4p r3p)
= +y+ + + − . (19)
dh m+4p r3p m+4p r3p m+4p r3p pG (m+4p r3p) r−2m
Theappropriateinitialconditionfory(h)atthecenterofthestar,h=h ,isy(h )=2.Afterthisintegration,thevalues
c c
oftheresultingfunctionsareevaluatedatthesurfaceofthestar,h=0,todeterminethequantitiesM=m(0),R=r(0),
andY =y(0).The(dimensionless)tidaldeformabilityL isthendeterminedbytheexpression,
16
L (C,Y)= (1−2C)2[2+2C(Y−1)−Y], (20)
15X
whereC=M/R,andX isgivenby
X (C,Y) = 4C3[13−11Y+C(3Y−2)+2C2(1+Y)]+3(1−2C)2[2−Y+2C(Y−1)]log(1−2C)
+2C[6−3Y+3C(5Y−8)]. (21)
For a givenequationof state, the mass M(h ,g ) and tidaldeformabilityL (h ,g ) of a stellar modelwill depend
c k c k
on the central value of the enthalpy h as well as the parameters g that characterized the equation of state. Using
c k
thesamespectralapproachtotheinversestellarstructureproblemdescribedearlier,theseparameterscanbefixedby
matchingthemodelobservables,{L (hi,g ),M(hi,g )},withdata{L ,M}fromtheexactcurveofmassesandtidal
c k c k i i
deformabilities.Asbefore,thismatchingisdonebyminimizingthequantity,c (hi,g ),definedby
c k
c 2(hi,g ) = 1 N(cid:229)stars log M(hci,g k) 2+ log L (hci,g k) 2 , (22)
c k N M L
stars i=1 ((cid:20) (cid:18) i (cid:19)(cid:21) (cid:20) (cid:18) i (cid:19)(cid:21) )
withrespectto variationsintheparametershi andg . Theresultingvaluesofg determineanequationofstatethat
c k k
servesasanapproximatesolutiontotheinversestellarstructureproblem.Theresultsofoursolutionstothisproblem