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Mon.Not.R.Astron.Soc.000,1–6(2009) Printed13January2009 (MNLATEXstylefilev2.2) A Roche Model for Uniformly Rotating Rings Stefan Horatschek⋆ and David Petroff⋆ Theoretisch-PhysikalischesInstitut,UniversityofJena,Max-Wien-Platz1,07743Jena,Germany 9 0 13January2009 0 2 n ABSTRACT a ARochemodelfordescribinguniformlyrotatingringsispresentedandtheresultsarecom- J paredwithnumericalsolutionstothefullproblemforpolytropicrings.Inthethinringlimit, 3 thesurfacesofconstantpressureincludingthesurfaceoftheringitselfaregiveninanalytic 1 terms,eveninthemass-sheddingcase. ] Keywords: gravitation– methods:analytical– hydrodynamics– equationof state – stars: h rotation. p - o r t s a 1 INTRODUCTION 2 THEROCHEMODELWITHARINGSOURCE [ Auniformlyrotatingstarwithasufficientlysoftequationofstate 2.1 BasicEquations 2 can be described approximately using the Roche model (Roche If we use cylindrical coordinates (̺,z,ϕ), then the gravitational v 1873). In this model the matter is treated as a test fluid in a potentialofacircularlinecentredontheaxisofmassMandradius 9 1/r-potential, i.e. one considers the whole mass of the star to 6 be concentrated at the centre. By doing so, self-gravitating ef- breads 6 π fects of the outer mass shells are neglected. Such Roche mod- GM dϕ 3 U = (1) 8. eblysZhaevl’edobveiecnh&conNsoidveikreodv,(e1s9p7e1c)ia;lSlyhainpirtohe&mSahsisb-asthaed(2d0in0g2)l.imFoitr, − π Z0 b2+̺2+z2−2b̺cosϕ 0 polytropeswithindicesn & 2.5thisapproximationyieldsresults 2GpM 2√b̺ 8 that differ by less than about a percent from their correct values = K , (2) 0 −π (b+̺)2+z2 (b+̺)2+z2! (Meineletal.2008).Whereasmost analyticalsolutions tofigures v: ofequilibriumdescribebodieswithconstantmassdensity(e.g.the whereGisthegpravitationalconstantapndK denotesthecomplete i Maclaurin spheroids, Jacobi ellipsoids, etc.), the Roche model is ellipticintegralofthefirstkind, X veryusefulbecauseitisapplicabletonon-homogeneousmatter. π 2 r Aside from spheroidal figures of equilibrium, there exist dθ a K(k):= . (3) configurations with toroidal topology. Such rings have been 1 k2sin2θ studiedbothanalytically(Kowalewsky 1885;Poincare´ 1885a,b,c; Z0 − Dyson 1892, 1893; Ostriker 1964b; Petroff&Horatschek 2008) Sometimesitwillbeconvenientptousethepolar-likecoordinatesr and numerically (Wong 1974; Eriguchi&Sugimoto 1981; andχdefinedby Eriguchi&Hachisu 1985; Hachisu 1986; Ansorgetal. 2003b; ̺=b rcosχ and z=rsinχ. (4) Fischeretal.2005). − ForatestfluidrotatinguniformlywiththeangularvelocityΩ,Eu- Here we apply the basic idea of the Roche model to rings. ler’sequationcanbewrittenas Thus,wedonotchoosetohavethetestfluidrotateinthefieldofa pointmass,butinthatofacircularlineofmasswithconstantlinear 1 U+h Ω2̺2 =0, (5) massdensity.Inadditiontothemass,wethusalsohavetospecify ∇ − 2 „ « theradiusofthecircle. wherehisdefinedby ForthecomparisonofthesolutionsoftheRochemodelwith p those of the full problem for polytropes, we use a multi-domain dp′ spectral program, much like the one described in Ansorgetal. h:= µ(p′), (6) (2003a) and a similar one tailored to Newtonian bodies with Z0 toroidaltopologies(seeAnsorg&Petroff2005formoreinforma- andwherepisthepressure,µthemassdensityandU thepotential tion). givenabove1.Weintegrate(5)andget 1 U+h Ω2̺2=V , (7) − 2 0 ⋆ E-mail:[email protected](SH); [email protected](DP) 1 Forisentropicmatter,hissimplythespecificenthalpy. 2 S. Horatschek&D. Petroff whereV istheconstantofintegration.Evaluatingthisequationat where a comma indicates partial differentiation. The equatorial 0 thesurfaceofthering,alongwhichthefunctionhvanishes, then symmetryimpliesthatfirstderivativeswithrespecttoz vanishat leadsto z=0,thusleadingto 1 Us− 2Ω2̺2˛˛s =V0. (8) ̺=̺o, z=0: dzds(̺̺) =s−VV,,z̺z̺ =sΩ2msU−,zzU,̺̺. (16) Viathisequation,thesurfaceofthe˛ring,describedbyr = r(χ) ˛ s s Hencethefullangleis orz =z(̺),andthustheratioofinnerandouterradius̺ and̺ s s i o A:= ̺i, (9) δ= 2arctan Ω2ms−U,̺̺ ̺o s U,zz ˛ ˛̺=̺o,z=0 (17) aregivenimplicitlyforprescribedM,b,Ω2 and̺o.Analogously, 2 1 ¯b2 E ¯b˛˛ equation(7)canbeusedtofindsurfacesofconstanth. =1 − ˛ , − (1 ¯b2)K ¯b 2E ¯b Inwhatfollows,wesimplifytheequationsbyintroducingthe −` ´−` ´ dimensionlessquantities cf.Fig.4. ` ´ ` ´ r¯ ¯b ̺¯ z¯ 1 = = = = , (10a) r b ̺ z ̺ 2.3 TheShapeoftheSurface o Ω¯2 ̺3 U¯ V¯ h¯ ̺ = o , = 0 = = o , (10b) To treat both mass-shedding configurations and the general case, Ω2 GM U V0 h GM weparametrizetheangularvelocityby whichimplies̺¯ = Aand̺¯ = 1.Inthedimensionlessversions i o Ω2 =αΩ2 , α [0,1]. (18) of the above equations, GM cancels out. Except for two scaling ms ∈ constants(e.g.Mand̺ ),twoparametersarenecessarytodescribe Thischoiceofαarisesfromthefactthatthemass-sheddinglimit o aringintheRochemodel(e.g.¯bandΩ¯2). (α=1)posesanupperboundfortheangularvelocityandasolu- tionto(8)canbefoundforarbitrarilysmallΩ2. Evaluating(8)atthepoint(̺¯=̺¯ =1,z¯=0)yields o 2.2 Mass-SheddingConfigurations 1 αE ¯b Ofparticularinterestareconfigurationsatthemass-sheddinglimit. V¯ = +2K ¯b (19) Inthislimit,afluidparticleattheouterrimrotateswiththeKepler 0 −π "1−`¯b2´ # ` ´ frequency,whichmeansthatthegravitationalforceisbalancedby andatthepoint(̺¯=̺¯ =A,z¯=0)gives i thecentrifugalforcealone–thepressuregradientvanishes.Forthe squaredangularvelocityatthemass-sheddinglimitwefind 1 αA2E ¯b 2 A V¯ = + K . (20) Ω¯2 = ∂U¯ . (11) 0 −π " 1−¯b`2´ ¯b „¯b«# ms ∂̺¯ RequiringthatbothexpressionsforV agreeleadsto ˛̺¯=1,z¯=0 0 ˛ Withthepotential(2)thisgives ˛ 1 A α 1 A2 E ¯b ˛ K ¯b = K − . (21) Ω¯2ms = π21E−`¯b¯b´2 , (12) Atanarbitrary`su´rface¯bpoi„nt¯b(̺¯«,z¯−s(̺¯)`)2wè1g−et´¯b2´` ´ whereEdenotesthecompleteellìpticintégralofthesecondkind, 1 α̺¯2E ¯b V¯ = + π2 0 −π" 1−¯b`2´ E(k):= 1 k2sin2θdθ. (13) (22) − 2 2 ¯b̺¯ Z K . 0 p π ¯b+̺¯ 2+z¯2 0 ¯b+p̺¯ 2+z¯21# Contrarytothegeneralcase,onlyoneparameterisfree,theother s s oneisfixedbyequation(11). Togetherwith(1q9)`thisisánimplicit@eqqua`tionfo´rthesurAfacefunc- Mass-shedding configurations haveacusp at theouter rim2, tionz¯(̺¯)orr¯(χ).Unfortunatelythisfunctioncannotbefoundin s s andwewillnowcalculatetheassociatedangle.Atthesurface,the analyticterms,however itisnotdifficulttohandleitnumerically. quantity Furthermore,itwillbetreatedanalyticallyforthinringsinthenext 1 subsection. V :=U− 2Ω2̺2 (14) For aprescribed α, the requirement A > 0 means that (21) canonlybesatisfiedfor¯blargerthansomeminimalvalue.Itturns isconstant,see(8),whichmeansthatalongthesurfacewehave outthatsmallervaluesof¯bnolongerdescriberings,butspheroidal d2V dz dz 2 d2z figures.Inthiscase,(20)and(21)mustbereplacedby 0= =V +2V s +V s +V s, (15) d̺2 ,̺̺ ,̺z d̺ ,zz d̺ ,z d̺2 1 „ « ̺¯=0, z¯=z¯ : V¯ = (23) p 0 − ¯b2+z¯2 p 2 Onecaneasilyshowthataconfigurationwithacuspmustbeatthemass- q and sheddinglimit.Theconversestatement,thatmass-sheddingconfigurations haveacusp,canbeprovedifonemakestheveryreasonable assumption π αE ¯b hthaalft-tshpeacpere(sPsaührtezd2o0e0s7n).otincreaseforincreasingvaluesofzintheupper K ¯b = 2 ¯b2+z¯p2 − 2 1−`¯b´2 , (24) ` ´ q ` ´ ARocheModelforUniformlyRotatingRings 3 wherez¯ =z /̺ denotesthedimensionlesspolarradius,inother p p o words,theratioofthepolartotheequatorialradius.Thetransition fromspheroidaltotoroidaltopologiesisdescribedwhenz¯ =0or p equivalentlyA = 0.Inthelimit¯b 0,thepotential(2)becomes → that of a point mass M at the star’s centre and one recovers the ‘standard’Rochemodel,cf.AppendixA. 2.4 ThinRings In the thinring limit in which A 1, clearly¯b also tends to1. → Thislimitisofparticularinterest,especiallysinceaself-gravitating thin ring is also tractable to analytical methods (Kowalewsky 1885; Poincare´ 1885b,c,d; Dyson 1892, 1893; Ostriker 1964b; Petroff&Horatschek2008).Byexpandingthesurfacefunction Figure 2. Lines of constant pressure (corresponding to constant h¯) are ∞ shownforthemass-sheddingringinthethinringlimit.Thefunctionth¯(χ) r¯s(χ)= si(χ) 1 ¯b i (25) withα=1isdepictedforvariousvaluesofh¯inpolarcoordinates,w1here − i=1 theaxisofrotationistotheleftofthecross-sectionandinfinitelyfaraway. aboutthethinringlimit¯b X1,equatio`n(8)yíelds Thesurface,whichcorrespondstoh¯ =0,ispreciselys1(χ)andcanalso → befoundinFig.1.Thevaluesofh¯ startingfromthesurfaceandmoving 0=α[1+s (χ)cosχ]+lns (χ), (26) inwardare0,0.001,0.01,0.05,0.1,0.2,0.5and1.Thevalueofh¯tendsto 1 1 infinityatthepoint,wherethesourceislocated. whichcanbesolvedbyusingtheLambertWfunction,whichfulfils theequationW(x)eW(x) =x: s (χ)=e−W(αe−αcosχ)−α= W(αe−αcosχ). (27) 3 COMPARISONWITHTHEFULLPROBLEM 1 αcosχ ThecomparisonoftheRochemodelwiththefullproblemrequires For α 0, we find s (χ) = 1 meaning that the cross-section the identification of various physical quantities. Whereas M, Ω, 1 tends to→ward a circle for ¯b 1. For α = 1 (and only for this V0,Aandzs(̺)areclearlydefinedinboth,thevalueofb,which value),s (χ)isnotdifferenti→ableatthepointχ=πalthoughboth describesthe locationof thesingular source intheRoche model, 1 one-sidedderivativesexist.Onefinds is not defined in the full problem. There, we choose b to be the positionofthecentreofmassofameridionalcross-sectionofthe ds (χ) ds (χ) lim 1 = lim 1 =1, (28) ring. χ↑π dχ −χ↓π dχ TheRochemodelcontainstwoscalingparameters,whichcan whichimpliesthattheangleat thispoint isδ = π/2, whichcan be‘eliminated’byusingthedimensionlessquantitiesintroducedin alsobederivedfrom(17)inthelimit¯b 1.Aseriesofpictures (10),andtwophysicalparameters.Ontheotherhand,foragiven → showingtheshapesofs1(χ)forvariousvaluesofαcanbefound equation of state, the full problem is determined by specifying a in Fig. 1. Higher terms si(χ) from the series (25) can be found scalingparameterandonlyonephysicalparameter.Whencompar- iterativelyandcontainpowersofln 1 ¯b . ingaspecificRochemodelwithasolutiontothefullproblem,one − Tocalculatesurfacesofconstan`th,r¯h¯´(χ),wecaneasilygen- thushasfreedomastohowtomakesuchacomparison.Onecan, eralizetheabovecalculation.Like(25),weexpandthesesurfaces forexample,choosetohave¯bandΩ¯ agreeandthencompareV¯ ,A 0 aboutthethinringlimit andtheshapeofthesurface.Onecouldalsochoosetohave¯band r¯h¯(χ)= ∞ thi¯(χ) 1−¯b i. (29) AagrTeheearnedmthuestnbceomsopmareewΩ¯e,lVl¯-0deafinndetdhewsahyapoefocfhothoesisnugrftahcee.addi- i=1 tionalparameterintheRochemodelifthefullproblemistotend X ` ´ As mentioned above, the function h vanishes at the surface, i.e. toitinsomelimit.Letusconsiderringsmadeupofmatterobeying r¯(χ)=r¯ (χ)ands =t0.Equation(5)thengives theequationofstate s 0 i i 0=α 1+th¯(χ)cosχ +lnth¯(χ)+π¯h, (30) p=Kµ1+1/n, (34) 1 1 andthus h i i.e.polytropes,whereKisthepolytropicconstantandnthepoly- tropic index. If such rings are expanded about the limit A 1, th¯(χ)=e−W(αe−α−πh¯cosχ)−α−πh¯ = W(αe−α−πh¯cosχ). thenasolutionresults,thefirstfewtermsofwhichprovidea→good 1 αcosχ approximation to the full problem over some range of values for (31) A(Ostriker1964b;Petroff&Horatschek2008).Thisrangeofval- ThisfunctionisdepictedinFig.2forα=1andvariousvaluesof uesshrinkstothesinglepointA = 1inthelimitn ,i.e.in h¯.Forlargevaluesofh¯,wehave the‘isothermallimit’.Thismeansthatthe‘isotherma→l’ri∞ngsmust be infinitelythin. Because we expect that only such rings can be lim th¯(χ)eα+πh¯ =1 (32) 1 treatedarbitrarilywellintheRochemodel,wedohaveawellde- h¯→∞ finedwayofchoosingthe‘additionalparameter’:namelyA = 1. meaningthatthecurvesofconstanthbecomecircularasthesource Theoneremainingfreephysicalparameter,e.g.α,correspondsto isapproached.Inthelimitα 0wefindthat → the single physical parameter one has in the full solution and in- GM 1 h= lnt, t (0,1] (33) terpolatesbetweenmass-shedringsandthosewithcircularcross- − b π ∈ sections.Wethusexpectthatthereexistisothermal(andnecessarily onlydependsonthecoordinatet:=r¯/(1 ¯b),butnotonχ. infinitelythin)ringswithnon-circularcross-sections,cf.Fig.1. − 4 S. Horatschek&D. Petroff α 0 0.3 0.6 0.8 0.95 1 Figure1.Cross-sectionsofringsinthethinringlimitscaledsuchthatthehorizontalextentofeachisequal.Thefunctions1(χ)isdepictedforvariousvalues ofαinpolarcoordinates,wheretheaxisofrotationistotheleftofeachcross-sectionandinfinitelyfaraway.Thedotsindicatetheorigin,wherethesourceof thepotentialislocated. Thosewithcircularcross-sectionswerestudiedinthepapers Table1.ComparisonbetweenpolytropicringswithA=0.7andthecor- mentionedinthelastparagraphandtheresultspresentedheremust respondingRocheconfigurationwiththesame¯b.∆denotesthedifference coincide with those for α = 0. Indeed expression (148) for the between thevalues ofthe Rocheconfiguration andthe values ofthefull potential and (35) with (151) for the angular velocity in Ostriker solution. (1964b)together withβ−1 ξ 1show thatΩ2b2 isnegligi- blecompared toU, asimpl≫iedby≫α = 0and(18).Furthermore, n κ ¯b ∆Ω/Ωnum ∆V0/V0num thelogarithmicbehaviour(33)canberecoveredfrom(120),(122), 1.0 6.3×10−1 8.48×10−1 2.6×10−2 7.4×10−3 (125) and (127) in Ostriker (1964b)3 (see also Ostriker 1964a; 1.5 4.9×10−1 8.46×10−1 1.9×10−2 6.0×10−3 3.0 2.7×10−1 8.42×10−1 8.2×10−3 3.1×10−3 Petroff&Horatschek2008) 5.0 1.3×10−1 8.34×10−1 2.6×10−3 1.2×10−3 ξ2 7.0 7.2×10−2 8.26×10−1 8.6×10−4 4.7×10−4 h =h 2Kln 1+ , ξ [0, ). (35) iso c− 8 ∈ ∞ „ « Inexpression(33)thefunctionhdivergesatthecentret = 0and the surface is located at the finite radius t = 1, whereas in (35) thenκisdefinedtobetheratioofthedistancebetweenthesetwo thecorrespondingfunctionremainsfiniteatthecentreξ = 0,but pointstothetotalwidthofthering’scross-section diverges at the surface ξ . To compare the expressions for ̺ ̺ h, we thus take a derivati→ve i∞n order to eliminate the physically κ:= b− a. (40) ̺ ̺ o i irrelevant constant andconsider onlytheregularportion ofthet- − interval, t > 0, which corresponds to infinite values of ξ. This Onlyforκ 1istheRochemodelexpectedtogivegoodresults. ≪ leadsto Wefindthatκ . 0.2forpolytropicringswithn & 3.Suchpoly- dh GM tropes always have values for the radius ratio A & 0.45, which t = (36) impliesthatRochemodelswithasmallerradiusratiodonotpro- dt − πb videaparticularlycloseapproximationtoanytoroidalpolytrope. and Theshapesofcross-sectionsofmass-sheddingringsforvar- dh lim ξ iso = 4K, (37) iouspolytropicindicesareplottedinFig.3.Here,thedifferences ξ→∞ dξ − thatariseduetodifferent prescriptionsof theparametersareevi- whichcanindeedbeseentobeinagreementupontakingintoac- dentforsmallvaluesofthepolytropicindex.WhenAischosento count coincidebetweentheRochemodelandthefullsolution,thenthe GM =4πKb, (38) two surfaces do not differ appreciably, even for n = 1. The ra- diusratioitself,andconsequentlytheshape,isquitedifferentfora see(146)andthedefinitions(97)inPetroff&Horatschek(2008). Rochemodelwith¯basprescribedfromthesolutiontothefullprob- Thesituationisanalogoustothatinthesphericallysymmetriccase lemwithn=1.Thesedifferencesvanishasnisincreasedandthe asdiscussedinAppendixA. twoprescriptivechoicesconvergetothefullsolutionastheymust. Ifwedepartfromthethinringlimit,thenthecomparisonwith Fig. 4 provides an interesting comparison between mass- numerical solutions to the full problem of a uniformly rotating, sheddingconfigurationsforpolytropesandtheRochemodel.The self-gravitatingringallowsustoaddresstheparticularlyimportant angleδ ofthecuspasgivenby(17)isplottedover thewholein- questionofhowgoodtheRochemodelis.Tables1and2showhow terval ¯b [0,1] as a solid line. This curve is compared to two thismodel convergestowardthefullsolutionforpolytropicrings ∈ distinctpolytropicmass-sheddingsequencesgeneratedbyvarying asthepolytropicindexnisincreased.Thefirstoftheseprovidesa thepolytropicindexn.Thedottedlinedescribesringsandmerges comparisonofringswiththeradiusratioA = 0.7andthesecond togetherwiththatoftheRochemodelinthethinringlimit,i.e.for formass-sheddingconfigurations(α=1fortheRochemodel).In ¯b 1.Onetheotherhand,thedashedlinedescribes(spheroidal) bothcases,thevalueof¯bfortheRochemodelwaschosentoagree sta→rsandmergestogetherwiththatoftheRochemodelfor¯b 0. withthenumericalone.Thesecondcolumn,κ,givesanindication → The change in topologies in the Roche model from toroidal to ofhow concentrated themassis.Ifweconsider thecross-section spheroidaltakesplacefor¯b 0.56ascanbecalculatedbysolving ofaring,anddefinethetwopointsintheequatorialplaneatwhich for¯bin(21)withA=0and≈α=1. thedensityfallsofftohalfofitsmaximalvalue Insummary, wecansaythataswithstars,theRochemodel µ µ(̺ ,z=0)=µ(̺ ,z=0)= max, ̺ >̺ , (39) presentedhereforringsisseentoyieldaverygoodapproximation a b b a 2 to the full problem in many cases. Moreover, it presumably pro- videsexactresultsintheappropriatelimitandoffersnewsolutions 3 Equation(122)inOstriker(1964b)shouldreadρ=λe−Ψ=ρ0e−Ψ. thathadnotbeenfoundusingperturbativeapproaches. ARocheModelforUniformlyRotatingRings 5 Table2.Comparisonbetweenpolytropicringsinmass-sheddingandthecorrespondingRocheconfigurationwiththesame¯b.∆denotesthedifferencebetween thevaluesoftheRocheconfigurationandthevaluesofthefullsolution. n κ ¯b ∆A/Anum ∆Ω/Ωnum ∆δ/δnum ∆V0/V0num 1.0 5.3×10−1 5.77×10−1 −3.7×10−1 −3.8×10−2 2.7×10−2 −4.1×10−2 1.5 4.0×10−1 6.04×10−1 −1.0×10−1 −2.0×10−2 1.3×10−2 −2.1×10−2 3.0 2.1×10−1 6.53×10−1 −1.2×10−2 −3.9×10−3 2.1×10−3 −4.4×10−3 5.0 1.1×10−1 6.92×10−1 −2.0×10−3 −7.7×10−4 3.4×10−4 −8.8×10−4 7.0 5.7×10−2 7.20×10−1 −4.7×10−4 −2.0×10−4 7.8×10−5 −2.4×10−4 n 120◦ 1.0 110◦ 1.5 δ 100◦ 3.0 5.0 90◦ 0.0 0.2 0.4 0.6 0.8 1.0 ¯ b 7.0 Figure4.Theangleδofthecuspforthemass-sheddingconfigurationsin 0 ̺ ̺o 0 ̺ ̺o theRochemodel(solidline)iscomparedwiththenumericalresultforpoly- tropicrings(dottedline)andpolytropicspheroidalconfigurations(dashed Figure3.Comparisonbetweenthecross-sectionsofpolytropicringsatthe line).Thelargerdotsdenotetheringconfigurationswithn=1/2,1,3/2, mass-shedding limit(dotted lines)andthoseofthecorresponding Roche 3,5andn=7,wheresmallervaluesofncorrespondtosmallervaluesof¯b, configurations,i.e.withα = 1(solidlines).IntheleftpanelstheRoche andthewhitesquaresdenotethespheroidalconfigurationswithn=1/2, modelwaschosensuchthat¯bagreeswiththenumericalvalue,whereasin 1,2,3andn = 4,wherelargervaluesofncorrespondtosmallervalues therightpanels,thevaluesofAwerechosentocoincide.Theticksinthe of¯b.Inthethinringlimit,whereweexpectthattheRochemodelprovides ‘centre’ indicate the values of¯b. WhenA is prescribed (right panels), it arbitrarilygoodresultstothefullproblemfor‘isothermal’rings(n→∞), turnsoutthat¯bnum<¯bRoche. wefindδ→90◦. ACKNOWLEDGMENTS We are grateful to Reinhard Meinel for a careful reading of this MeinelR.,AnsorgM.,Kleinwa¨chterA.,Neugebauer G.,Petroff paper.Wealsowanttothankourreferee,YoshiharuEriguchi,for D., 2008, Relativistic Figures of Equilibrium. Cambridge Uni- his helpful comments. This research was funded in part by the versityPress,Cambridge DeutscheForschungsgemeinschaft(SFB/TR7–B1). OstrikerJ.,1964a,Astrophys.J.,140,1056 OstrikerJ.,1964b,Astrophys.J.,140,1067 Pa¨htzT.,2007,UntersuchungendesMass-Shedding-Limitsrotie- REFERENCES renderFlu¨ssigkeiteninNewtonscherundEinsteinscherGravita- tionstheorie,Diplomarbeit,Friedrich-Schiller-Universita¨tJena Ansorg M., Kleinwa¨chter A., Meinel R., 2003a, Astron. 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Soc.,364,943 Vol.1,TheUniversityofChicagoPress,Chicago HachisuI.,1986,Astrophys.J.Suppl.Ser.,61,479 KowalewskyS.,1885,AstronomischeNachrichten,111,37 6 S. Horatschek&D. Petroff APPENDIXA: THESTANDARDROCHEMODEL where x 3K As mentioned in the introduction, the Roche model with a 1/r- ξ= , l2 = (A13) potential was discussed by Roche (1873); Zel’dovich&Novikov l 2πGµ4c/5 (1971); Shapiro&Shibata (2002) and provides a unique surface isthedimensionlessradiusandxthetrueradius.Thesurfaceex- functiondescribing amass-shedding star.Toderivethisfunction, tendsouttoinfinityandsincetheregularportionoftheR-interval, wefollowMeineletal.(2008)andbeginbywritingdownequation R>0,correspondstoinfinitevaluesoftheradiusx,weconsider (8)withthepotentialU ofapointmass s dh V0+ ̺2G+Mz(̺)2 + 12Ω2̺2=0. (A1) xl→im∞x2 dx5 =−6√3Klµ1c/5. (A14) s Calculatingthemassforthefullproblem, Byconsideringtheppoint̺ = 0,wecanrelatetheconstantV0 to ∞ thepolarradiusz p GM =4πG µ x2dx=4π√3l3µ =6√3Klµ1/5, (A15) 5 c c GM Z V = . (A2) 0 0 − zp showstheequivalenceof(A10)and(A14). Formass-sheddingstars,therelation ∂U =̺ Ω2, (A3) o ∂̺ ˛̺=̺o,z=0 ˛ where̺ istheequatorial˛radius,leadsto o ˛ GM =̺3Ω2, (A4) o andthesurfaceequation(A1)becomes 1 ̺2 z + =1. (A5) p ̺2+z2 2̺3o! Evaluating thisexpresspion at theequator, z = 0, tellsus thatfor mass-sheddingfluidsintheRochemodel,theradiusratio zp = 2 (A6) ̺ 3 o follows.Withthisrelationship,thecurvedescribingthefluid’ssur- facecanberewrittenas 4̺2 ̺2 ̺2 ̺2 z = o − o − . (A7) s 3̺2 ̺2 p o −` ´ Itthenfollowsthat dz lim s = √3, (A8) z↓0 d̺ − whichmeanstheinteriorangleofthemass-sheddingcuspis120◦ (seeFig.4for¯b 0). → Ingeneral,theRochemodelisexpectedtoapproachthefull solutionwhenthematterisarbitrarilyconcentrated.Forstaticpoly- tropes,thefullproblemleadstotheLane-Emdenequation,which turnsouttodescribearbitrarilyconcentratedmatterifn = 5.We shallnowshowtheagreementbetweenthesestarsandtheresults givenbytheRochemodel. Inthestaticcase,theRochemodelgives 1 1 h=GM , R (0,R ] (A9) „R − R0« ∈ 0 andthus dh R2 = GM. (A10) dR − Thesolutiontothefullproblemis ξ2 −5/2 µ =µ 1+ , ξ [0, ) (A11) 5 c 3 ∈ ∞ „ « andthus h =6Kµ1/5, (A12) 5 5