Microscopic Interpretation of Neutron Star Structure
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ft A M i c r o s c o p i c Interpretation o f Ne u t r o n V': St a r St r u c t u r e
by
W. Da v i d Ar n e t t a n d R ichard L. Bo wers
- 3 .'A 2
I. Introduction published, and the authors have constructed neutron star
■odels based on their own equations of state. However, Since the pioneering work of Baade and Zwicky in IS34, there has been no systematic study using (1) a variety of and Oppenheiner and Volkoff in 1939, *any studies have equations of state and (2) consistent numerical techniques. been made of the equations of state and structure of neutron In several cases not all neutron star structure parameters stars. E a r ’v investigations of structural properties were of interest have been calculated. based on free gas equations of state, numerical evaluation Because of the uncertainties which plague our current of relativistic polytrope* (Thorne 1971; Tooper 1965}, or understanding of the interactions at supernuclear densities, have been based on the analysis of a small number of speci we have chosen fifteen equations of state and have calculated fic (and by now out-dated) models (Tsuruta and Cameron 1966;
Hartle and Thorne 1968, for example). An improvement in the structure of slowly rotating neutron stars for each. Thirteen of these (Models A-M) employ non-relativistic de this state of affairs is represented by the work of Borner scriptions of the interactions and a non-relativistic man;'- and Cohen (1973), who considered five equations of state at body formalism is used in constructing the equations of nuclear and supernuclear densities. They do not, however, state. Two (Models N and 0) are based on a relativistic discuss a number of parameters of interest in evolutionary description of the interactions and a consistent, relativis theory, nor did they obtain general conclusions about the tic many-body theory. With the exception of model G, all relationship between the equations of state (interactions) of the equations of state represent normal systems (Fermi and the resulting structure. In 1974, the duihors surveyed fluids). In this category, models L ana H represent pos sible upper and lower bounds on the stiffness expected in existing equations of state selecting those thought to be any reasonable equation of state. Six (Models B, D-G, and ■ost nearly representative of the known physics of superdense M) represent what might be called the most nearly realistic matter, and systematically investigated the structure of non-relativistic results calculated to date in the super cold neutron stars based on these. Since then a number of nuclear density range. With the exception of models A and N significant new results (two cf them relativistic) have been (which have bsen included to show the relative importance of hypcronization--compare models B and D) the remaining models cover the nucle«r and subnuclear density range. The equa reasonable alternative and has been included to test the sen tions of state used in this study are found in the papers by sitivity of models near mass peak on the low density region. Attention has been restricted to high densities. We are A. Pandharipande, V. R. 1971a, Nucl. Phys. A174, 641. primarily concerned with the mass peak beyond which gravita B. Pandharipande, V. R. 1971b, Nucl. Phys. A178, 125. tional collapse occurs, and the mass range around ' l-4Mg C. Be the, H. A., and Johnson, M. B. 1974, Nucl. Phys. which is made interesting by current evolutionary models for A230, I. pre-supernovae and observationally based mass estimates for D. Same as C. the Crab pulsar and nine compact x-ray sources. E. Moszkowski, S. 1974, Phys. Rev. D£, 1613. Many published studies of neutron stars do not give F. Arponen, J. 1972, Nucl. Phys. A191, 257. values for the total nucleon number or (equivalently) the 6. Canuto, V., and Chitre, S. M. 1974, Phys. Rev. D9, "mass” defined by the product of the total nucleon number 1587. and the a.m.u. This mass is determined by the number H. Oppenheimer, J. R . , and Volkoff, G. M. 1939, Phys. density, nor by the total mass energy density (the proper Rev. 55, 374. (Ideal neutron gas) mass or gravitational mass depending on whether integration I. Cohen, J. M., Langer, W. D., Rosen, L. C., and is over the proper or coordinate volume). It is M^ which Cameron, A.G.W. 1970, Astrophys. and Spacc can be related to pre-collapse evolutionary models. Sci. 6, 228. A simple analysis is given which connects the micro J. ftaym, G., Pethick, C. J., and Sutherland, P. 1971, scopic parameters (strength and range) of a non-relativistic Ap. J. 170, 299. interaction with macroscopic properties of the neutron star K. Bay*, G., Beth*. , H. A., and Pethick, C. J. 1971, models presented here. We have found this to be valuable Nucl. Phys. A17S. 223. in understanding the systematic trends in the numerical re L. Pandharipande, V. R., and Smith, R. A. 1975 sults. M. Pandharipande, V. R., and Smith, R. A. 197S, Nucl. Finally we have performed all of the calculations Phys. A237, 507. using a standard, "calibrated" numerical program. This N. Walecka, J. 0. 1974, Ann. Phys. 82, 491. eliminates differences such as those which exist between 0. Bowers, R. L., Cleeson, A. M., and Pedigo, R. D. some published results based on the same equation of state. 1975, Phys. Rev. D12. S043. We note that insufficiently accurate analytic approximations So attempt has been made to survey the density range to equations of state have been the source of considerable below 1015 g/cm3. In general wc use model K and J in com- error. Wc use a consistent numerical tabulation procedure cosite fora at lover densities. Model F also represent!- 3 for all equations of state. We believe we have eliminated 5
aatheaatical inconsistencies, and that such differences a* appear in tha neutron star aodels presented here reflect Tha growing interest in supernova reanants, pulsars, different input physics rather than different nuaerical coapact x-ray sources and gravitational collapse has lad to techniques. a corresponding need for accurate neutron star aodels. The results presented in Tables 2-S of this paper sua- These in turn demand a detailed knowledge of the interac narize and extend work reported in Publication Nuaber 9, tions between eieaentary particles (primarily the strong in Department of Astronony, The Univetsity of Texas at Austin, teractions) in the nuclear and supernudear density range. Austin, Texas, 7S712. In that work we pres?nt tables con The latter are reasonably well understood at and beiow nu taining nuaerical output for approximately SO out of ISO clear density - 2 » 10** g/c«J), but the situation above tones ranging from the center to the surface for neutron p ( j is less certain. Typically the density varies froa the stars based nn |k« ffinions of state A-K. surface (p^ * 10 g/cc) to the center by fourteen to fifteen orders of magnitude, but because of the relatively flat density profiles in neutron stars aost of the aatter aay be at densities comparable to the central density p£ . The aaxi- aua central density Pcrj, for stable cold neutron stars is expccted to lie in the range i0**-!0** g/ca^. The most •**.- sive have average densities greater than I0IS g/caJ and aay contain a significant nuaber of hyperons whose interactions should be treated relativisticalSy. Since little is known * with certainty about the equation of state in this density range, the resulting neutron star parameters vary consider ably depending on the equation of state u*ed in their con struction (published values of "he naiinun *us> vary by nearly a factor of five; G,S2Hg * In this study will be primarily concerned with cold 7 • non-aagnetic neutron stars. A* is well known, their struc consist of nucleons, hypcrons, electrons, auons and possibly ture is deterained (for a given equation of state) by the acsons. No fira justification exists for the tieataenl of central density, and by their observed angular velocity 0. the strong interactions between the hyperens via non-rtla- The relation between total gravitational aass N and *c »* tivistic potentials in this density regiae. Instead they shown schematically in Figure I (Zel’dovich and Novikov 1971) should be aescribed relativistically. Unfortunately rela for non-rotating stars. The region of stable neutton star tivistic aany-body theories foe strongly interacting natter aasses (solid line) nay be divided into two pari*. For are still in a state cf development {towers, Gleeson and P| < p < 10,S g/caJ the equation of state for noraal systeas Pedigo 197$; Kalasn 197#; ttalecka 19 / 0 , The possibility is reasonably welt understood. The principal constituents that real aesons aay fora gose-Einstein condensates at super- of these stats are nucleons. The leptons--electrons and nuclear densities « m also be faced. However, a Clear un auons--have only a saall effect on the static structure. derstanding of the quantitative effects that they wilt have The interactions of the nucleons are described adequately t*“ on neutron star structure has not yet developed. C»r,uto phenoacnological potentials, and neutron-rich nuclei or H9*<, 19‘Sl and 9*yn and rethick {Ifl’i) have publtthtd con- clusters described by seai-eapirical nass relations. The pletc review* of the profiles. aajor uncertainties in this region concern the possibility V*ur enphasis here wj|J be on the region 10** g/ca* * p * of neutron and proton superfluid states, pion condensates, **crit' **014v*1 >ttn *cr *kis choice results fron the fol and of neutron solidification above The foraer are expected lowing argument. St»r* aore aasstve than about apparently 12 P to he wore important for sagnetohydrodynaaics of neutron star for* ignite the *C • *C reaction in * nonviolent manner. tvo> nation and aay have no aore than *IOt effect on their static iufionary computation* suggest that ««»t cf these stars properties. Although the question of a solid core has sig t gow# ) proceed to neon, osygen and silicon burn' nificant consequences (priaarily for pulsar aodels), current in*; these stages seea to be non-oplosive. The nature of suggestions are that core solidification is uncertain neutrino cooling (see Arnett 19?)*} is such .that at each (Caauto and Chitr- 1974; laya and Fethick |»?S; Pandharipande. burning stage only * fraction of the available fuel--that Pirns and Saith 197S) near the center--»s converted to nuclear ashe*. The ashes Neutron stars in the range 10** g/caS < ~e « of one stage are fuel for the neat. Consequently even very 9 10
Mssiw* start develop m i l cores by the tl*e they reach would be the expected result. Svperaovae are observed ?o silicon burning. These cores cannot get too snail However. sxplode. che Crab and Vela supernova rewuats contain pulsars, Tha advanced burning stages require high temperatures (as and pulsars are thought to be slowly rotating neutron stars. high as T > 3.4 * I09*S for silicon burning). Saall cores Middled itch. Mast and Nelson (1974) have recently determined cannot reach these temperatures because their contraction the mass of the compact x-ray source (neutron star) to be will be halted by pressure due to electron degeneracy. 1.3.1 : 0.1JM# . and Kappaport and McCIintack (197$) have found Only for core masses near or greater than the Chandrasekhar the aass of the compact x-ray source Vela X-l (newly identi ■ass (Hchan;J = 1.4M0 ) can such teaperaturcs he attained. If fied as pulsating) to he M » l.?H# . The pulsar (PS» 1913 • 16) a core is initially too snail, shell burning of the previous in a binary systea scens to have characteristics consistent theraonuclear process will add its ashes to the core, causing with a mass around 1.4Mfl (Taylor and Itulse 1975; Masters, it to increase in aas*. Contraction and an increase in cen Roberts and Arnett 19?6). Therefore as an interesting work tral tenperature result. Eventually the ignition point for ing hypothesis we propose that ihc pulsar* have a aas* * the new fuel is reached. l.4Me . although it m m be reeenbered vhat Bonification* These two «:fects "funnel'* the core s j j j iiito u narrow during the dynamic collapse and explosion cannot be reliably range near M^ * the **■* silicon burning has been estimated as yet. This possibly 1 tailed aass range for rca- reached. After silicon burning little additional nuclear nants would have significant implications as is shown by the energy can be released. The core mist collapse to beyond three tcheaatic aass curves in Figure 2. The horizontal the white dwarf Mate because of electron capture. For ex- cross-hashed band corresponds to s The three aass anplc a ZBSi white dwarf can have a aass up to about !.*Mg curves represent the result of three possible equations of while a white dwarf coaiposed of the products of silicon stase for supcrdsnse aatter. The hardness ("degree of re burning (NiS6 which can electron capture to forn 56Fe for pulsion”) of the underlying equation of state increases froa exacple) has a aaxiaua r i s s of only 1.26 M^. If the core a so c. The ordinate gives the AMU aass (to be discussed collapse causes (or is accompanied by) an explosion, a rea- in aore detail in section tit). The curves have maxima M # , nant of 1 l.4Mg sceas likely. If the collapse continues, Mh and Mc> where Ma « • t-^M^ « M£ , and suggest three or if sufficient ailditional miis is accreted, a black hole possible predictions tiependlng on which equation of state is >1 12
the aore nearly representative of neutron stars: collapse is as yet insufficiently explored; this adds to Case a: < I.4H0 and supernovae will not be expected our uncertainty. to produce stable neutron stars (unless some as yet unknown A survey of the literature rapidly deaonstrates that mechanisa exists which can reduce the Mass of the reanant the aass range shown schematically in Figure 2 is represen during its collapse). The result is expected to be a black tative of published values of Mn|X. Many values aay be dis* hole. carded as the result of unre.ilist 1c or incomplete physics at Case b: » 1 5 0 that stable neutron stars will the equation of state level. However, there still remains result in Many eases. The accretion of supernova debris, nearly the sane spread based on models none of which can or mass transfer if the reanant remains bound in a close claim undebatable precedence over the others. This situa binary system (Paczynski 1971; Bekenstein and Bowers 1974) tion is further complicated by the abscnce of analytic solu can lead *o black holes in a reasonable fraction of cases. tions to the relativistic structure equations for any equa Some black hoies nay be formed directly. tion of state except the unphysical example of an incompres Case c : Mc > 1. U(r) ♦ P(r)/c2][m(r) ♦ «.r^(r)/c2l 3 ? ■ «T*p(r), (3.2) is appropriate since precollapse models contain large amounts of *2C or nuclei of similar binding energy. The following where ii(r) is the mass contained within a sphere of coordi observations may be made: in general nate radius r (surface area 4«r2). The boundary conditions on (3.1*3.2) are that the center be free of singularities ■'V * * °» (J*6) and that the pressure or the density at the surface be speci fied : and the difference (Mp - Mc)/Mc represents the fraition of the star's energy which has gone into binding energy. The difference (MA - Mc)/Mc may be positive or negative. As -(O) - 0 P(R> * Surface ( 3 J ) have defined it with R the radius of the configuration. Two conventional choices of the pressure boundary condition are that P(R) • 0 ■2 ■ represents an equilibrium system. The locus of points M( pc ) is a curve similar to that shown in Figure 1. Stable equilib rium obtains only if dM(oc)/doc > 0 (this is necessary but from r • 0 to r • R; and the AMU mass defined by the in not sufficient for stability). tegral In »he limit ZMG/Rc2 << 1, (2.1-2.2) reduces to the ML- m. f ---f-r2n(r) dr (J>5) usual Newtonian structure equations. Me see that the ratio, A AJ (1 • 2»(r)fi/rc | ^ 17 II 2MG/Rc* is a measure of the extent to which general rela ♦(») • 0 and w(R) • 0 - j(dw/dr)R. An arbitrary value of tivity is significant in determining the star's structure. uc • u(0) is chosen which determines 0; a new value of «c * Indeed when 2MG/Rc2 - 1 the surface approaches the event corresponding to the observed Q' may then be trivially ob horizon; for 2HG/RcZ < 1 a black hole results. For most tained from the relation neutron star models considered here we find 2Mm ](C/Rc 2 ’ 2/3. Slow rigid rotation with angular velocity 0 produces (Zc/c:) - (ucvn*). (3.12) structural changes which are proportional to powers of n/Qc , where dc' • (MG/R3)1^2 is the angular velocity above which Finally the angular momentum is given by J » It). Hartle equatorial mass loss occurs. To lowest order in (n/flc) rota (1973) and Thorne (1971) show that second order corrections tion produces the Lense-Thirring effect (dragging of inertial ~(fl/Oc)* result in mass shifts (centrifugal force due to ro frames) and alters the moment of inertia I given by tation helps support the star) and deformations away from sphericity. I.«I f IpW * P(W cV.-*:2 “ dr (3.8) The utility of the approximations leading to (3.8-3.10) 3 J 0 [I - 2m(r)G/rc ) * where depends on the smallness of the ratio (fi/»l ) which may be _ di . ------J --- — dP (3.9) written as 3F o(r)c ♦ P(r) Ur The angular velocity u is obtained from (n/n.) ■ a * 1.4 « io's(R/r )3/2 (m/m0) (3.13) where rg • 2MG/c2. For pulsars more than a few months old, the ratio is expected.to be small. For the Crab pulsar j(r) - [I - 2m(r)G/rcZ]e'*(rl (5.11) (P 9 0.033 sec) with reasonable estimates for its mass and radius (M - Mg and R = 10km), we ^n/®c 'crat, * 10 *• The in The non-rotating gravitational mass distribution m(r) and crease in maximum mass due to rotations has been studied by the unperturbed equations of state are used to solve (3.8- llartlc and Thorne. For the relatively stiff V equation of 3.11). The boundary conditions are du/dr » 0 at r * 0, state of Tsuruta and Cameron, and the Harrison-Wheeler equation 19 20 of state (nearly free gas at supernuclear densities), their IV. Equations of State results show that for rotation rates applicable ta all but The macroscopic properties of neutron stars are fixed the very youngest pulsars the increase in maximum stable by the structure equations of the last section and the two ■ass is much less than two-tenths of a percent, which is far equations of state P ■ P(n) and o * p(n). Elimination of n saaller than the uncertainties resulting froa our inadequate gives P » P(p), which is sufficient to determine the mass knowledge of the equation of state at supernuclear densities. .M ■ M( pc). We now consider general properties of P ■ P(p), A nuaber of additional parameters relating to the run and discuss the basic features of the equations of state of density from the center to the surface of neutron stars used to calculate neutron star properties rep.-fesented by are contained in Arnett and Bowers (1974). Tables 2-12. An excellent review of many aspects of questions touched on here nay be found in the review article by Canuto (1974), by Baym and Pethick (I97S) and by Tandharipande, Pines and Smith (1975). The exact nature of the ground state for neutron stars is still a matter of debate. It is usually assumed to be a normal Fermi liquid. Other possibilities include: non-normal ground states (superfluids and meson condensates); solid cores of baryons; Ferromagnetic ground states based on coupled baryon spins through the itrong interactions (Bowers, Pedigo, Cleeson and Zimmerman 1975; Baym and Pcthick >97S; Pandharipande, Pines and Smith 1975). A representative equation of state applicable to neu tron stars is shown in Figure 3, along with chat of a free neutron gas (dashed). At densities below neutron drip (Op = S ■ 10** g/cm1) repulsion dominates. Above .-p the equation of state softens, first due to removal of 12 1 f « pressure-producing electrons; then as nuclear densities are densities > 10 g/caT*. In fact for this case models with approached, because of the long range attractive portion of masses as low as IMg consist of matter more than 981 of which the strong interactions. In the vicinity of I01* g/cm3 the is at supernuclear densities. short range repulsive strong interactions dominate, and the Tables 2-12 contain basic structural properties of slowly equation of state becomes significantly harder than that of rotating neutron stars based on thirteen equations of state a free gas. Two possibilities are shown in this region. for densities > 1015 g/cm3. The latter are summarized in The boxed portion of the P - p plane corresponding to Table 1. In models A-E, and 1 the high density equation is 14.S < log p < 16.0 is shown in Figures 4 and 5, where w* joined to the composite BBP (p > 4.3 « 1011 g/cm*) - BPS have included all of the equations of state considered below. (10* g/cm5 < p * 4.3 ■ 10U g/cm1); model c uses PC above Other equations of state have been derived in the sub-nuclear 2.4 - 101S g/cm3 , and BPS (10* g/cm3 « f. < 3.1 * 1011 g/cm3 ). density range, but they are nearly indistinguishable Fro* the Model C uses !>C (7 « 10H g/cm3 < p * 2.4 ■ 10iS g/cm3) and BPS-BBP results shown in Figure 3. BBP-BPS sequence of models A-C at lower densities. Models As has been previously stressed, we focus attention on t-M join to BPS at neutron drip. Models N-0 contain broad the equation of state in the density range p > 1015 g/cm1. phase transitions which include the density range 3.4 « 10*^ - The astrophysical motivation for this has been reviewed in 1.7 * 1014 g/cm3. !n both cases we have interpolated between section II. Since the determination of ntutron star param the pnase transition region and neutron drip. The exact eters near Pcrjt involves the equation of state down to the method has no significant effect on the structure of our density of iron (oFe • 7.86 g/cm3), one might consider in models. Finally model H is an ideal neutron gas at all den corporating a variety of equations of state in the nuclear sities. All models with the exception of model II use the and sub-nuclear density range. Figure 6 gives the density Feynman-Metropolis-Teller equation of state (FMT) at densi profile (left scale) and mass fraction m(r)/M (right scale) ties below 10* g/cm3 . based on Pandharipande’s hyperon equation of state (model B) The interactions assumed, as well as the many-body for several values of oc- This figure demonstrates a feature theory used for models except N and G, are non-relativistic. typical of neutron stars near the critical point: most of Most arc based on a Reid type potential of the form the mass (in the cases shown 8 5 - ft 6 %) consists of natter at (4.1) 23 24 where u * me/"ft, n is the pion mass, I designates the angular of phenomenological meson-exchange in relativistic dense momentum state and A, B stand for the interacting baryons. matter. Model N is restricted to a system of neutrons. In most cases n takes the values 1, 2, 4, 6 and 7 (Bethe and Model H has been included to set what we consider to • Johnson consider non-integer values). Generally speaking the be reasonable jounds on the softness of realistic normal first terra corresponds to the long range pion-exchange type equations of state. In a somewhat broader sense it may be attraction, the next two terms represent intermediate range argued that a lower bound on softness is represented by a attraction, and the last two terms the short range repulsion relativistic pion-exchange model (Bowers, Pedigo, Gleeson at interparticle separations near 0.2 fm (Bethe and Johnson and Zimmerman 1974). 1974; Canuto 1974). In all models containing hyperons the It is not impossible that a substantial reduction in nucleon-nucleon (N-N) and baryon-hyperons (B-Y) interactions the lower limit could be achieved through pion condensation. are assumed to be different. Since current data is insuffi Assessments of this possibility are quite difficult since cient to establish the relative strengths of (B-Y) interac they require a consistent treatment not only of the conden tions, various authors have employed a variety of models to sate but of the strong interactions as well. Broadly speak describe them. These range from assumed equality of N-N, ing the problem involves two processes. The first, which is N-B and Y-Y couplings with an arbitrary reduction by 10* in relatively simple, leads to a reduction in pressure at a the intermediate range attraction between, hyperons given density as the baryons are redistributed within multi- (Pandharipande 1971b), to more sophisticated assumptions plets (n * p for example) to produce pions. This process based on the quark model (Moszkowski 1974). The correct does not alter the total number of fermions in the star situation is as yet not known. which are responsible for the bulk of the pressure. Conse: Models N-0 represent equations of state based on fully quently it can lead only to a slight softening of the equa: relativistic descriptions of the interactions and the many- tion of state (Hartle, Sawyer and Scalr^ino 197S). Further body theory. Both are fit to simple models of nuclear more since the process competes with hype/oniiation, it is , matter and treat the interactions self-consistently. Model expected to be most significant near threshold, and in al-, 0, which includes the low jtiass spin 1/2 fermions, is based tering equations of state which are relatively soft at den;, on an approximation scheme motivated by a detailed analysis sities > 101S g/cmS. The second aspect of pion considerations « 25 26 is much ■ore subtle, and •* yet has not been fully explored V. Numerical Analysis even at the equation of state level, let alone as it regards In order to place the overall accuracy of our computa neutron star structure. It involves the dynamical effects tion in perspective, we briefly review the numerical analysis of the baryons and pions on one another and can only be used in constructing Tables 2-12. answered when more detailed studies have been completed. - 26 The physical constants used were: ft » 1.05459 * 10 It is, in fact, not clear whether the dominant effects will erg sec; c - 2.9979 x io10 cm/sec; G - 6.6732 x io*8 ca3/g/sec2; lead to a reduction or increase in pressure. • '*4 the "baryon mass” > 1.659 * 10 g; the mass unit based on C12, l/»^ 13 6.022529 * 1023. In converting masses in grams to solar units we have used » 1.987 * 1933 g. The equations of state were all used in tabular form even when analytic approximations were available, thus giving 2 Uformity to our treatment. Entries for P (dynes/cm ), (i/cm3) and n (cm'3) were read in to four significant t 'es. Values between two tabulated points such as (Pj.P^) a were obtained by logarithmic interpolation. log P - log Pj , log P.tl - log Pj log p - log Pj log pj+1 - log oj CS.l) The join between different equations of state (as for ex ample between PC and BBP) was affected by (5.1). As a gen eral rule the first and last point? were taken to be sepa rated by an interval somewhat largur than the inter-point spacing near the join. The stellar structure equations for slow rotation were then simultaneously integrated from the center to the surf&ce 27 28 using a four-point Runge-Kutta method with variable step ways; first by recalculating the parameters for the PC-BBP-BPS- size A given by (Baym, Pethick and Sutherland 1971) FMT equations of state (Baym, Pethick and Sutherland 1971). Our results agree with their published values except for a *-*(11? - i S?)'1 <»•» small range of models with central density in the range where the 'C- and BBP equations of state join, and there the differ with 6 • O.SO. Near neutron drip we used 6 * 0.23, since too ence is only a few percent. By replacing the pressure gradient large a step size in this region leads to errors in radius (3.1) with the Newtonian expression (change of a single card) by as much as a factor of two. Initial steps may be chosen we were able to calculate the mass, radius and density pro in two ways: as 10** * (average neutron star radius taken file of a non-rotating Newtonian polytrope. Using a poly to be -10 km) * a; or from the scale distance set by the trope with n * 1, which is exactly soluble, we found that our • _ 2 _ | Newtonian pressure gradient 10 x p x (dP/dr)* . The latter program reproduced the correct results to better than five leads to a decreasing step size with decreasing central den significant figures. sity due to the fact that lower mass stars have slightly less An overall change in step size by setting 4 * 0.23 flat density and pressure profiles than those of higher mass. throughout ied to no changes in five significant figures for However, the range of variation was found to be small, so sample cases. Approximately 150 integration steps were re the first method was employed throughout. . The numerical quired for each model, which took an average of two seconds 1 integration for each model was terminated with the last step on the CDC-6600. before p s. 7.86 g/cm5. When treating the equations for rotation we reduced (3.10) for £> to two first order differential equations by setting n = dui/dr and solving for n(r) and w(r). As initial conditions we took at r ■ 0, u(0) » 1.82342 sec"* and n(0) ■ 0. For fl » SJC the renormalized u(0) was obtained by requiring that u(r) » n - n(R) R/3 (see Thorne 1971). The overall accuracy of the program was checked in two 29 30 VI. Discussion all -of its range. Since the stiffar models of Figure S con tribute to stable neutron stars only for p i 3 * 1015 g/cm3 The masses and moments of inertia given in Tables 2-12 we see that their relative stiffness roughly .follows the are summarized in Figures 7-10 (Mg and MA vs. pc). Figure 11 order M<0 The maximum stable value of Mg is given in parenthesis for The mass-radius curves Mg vs. R for several models are each model. The spread in Mg at * l-4Mg is less than 101. shown in Figure 11. It is observed that the radius for all In all cases Mg Is below the critical mass. This suggests but the lowest mass models fall within a restricted region, that case a discussed in section II may not be applicable, the maximum variation in radius at critical nass and for which is comforting in view of the difficulties with pulsar M ~ lMg being about 2.5 km. Generally speaking the smallest theory that would result if were greater than Mmix. How stable radius is 8-9 km, which is about twice the gravita ever, the remnant mass is quite close to the critical mass tional radius r^ « 2 MG/c2. for models B, G and F (the difference amounts to 141, 81 and Moments of Inertia (Figures 12-13): The range of Im#x 131 of the critical mass, respectively). In fact, for models for the models A-G considered is 6.OS * 10 44 g/cm 2 < I „ < max A-G, the fractional difference (Mcrj| ‘ MR ^ Mcrit * 30*‘ 1.65 * 10^5 g/cm^ (excluding CCLR and free neutrons). When This means that if MR car, be increased by about 201 for the stifier equations of state of Figure 5 are included, models B, G and F (or about 50t for the others) by accretion the maximum can reach 4.9 x 1 0 g/c«* in model L, and of supernovae ejecta, for example, or as a result of mass 3.1 * 104^ g/cm2 in models M and 0. Two notable features transfer in close binaries, a black hole would be expected are seen: the maximum I always occurs before that in (or to fora. This suggests that if the assumptions of a narrow M a ) i and increases with maximum mass. The latter is physically range of MR discussed in section III are correct, many super reasonable. The former will be shown below to follow fro* novae remnants which remain bound in close binaries night an analysis of Newtonian polytropes. undergo gravitational collapse. Recent observations which Observational data on pulsars may be used to infer lower suggest that some discrete x-ray sources may be black holes limits for the neutron star's moment of inertia (Gold 1E69; (particularly Cyg X-l, SMC X-l, 2U 1700-37 and possibly Arnett 1974; Borner and Cohen 1973). These may then the used Cen X-3 and Vel X-l) are consistent with this suggestion. to rule out some equations of state. For example, the If, on the other hand, superdense matter is nore accurately luminosity estimates of Trimble and Rees (1970; L * 2 - 4 * 103® described by equations of state such as N or 0, collapse erg/sec) require that I > (Z to I) » 10*4 g cm* for the. Crab would require that the remnant mess be enhanced by one or pulsar. The upper limit would rule out models B and G, and nore solar nasses. would suggest that the minimum nass be greater than O..7SM0 .. I 33 34 Unfortunately th*it estimates are still quit* uncertain. The onset of instability occurs when dM/dpc * 0, which yields Although a trend toward a greater h u < h i i i m n t of the familiar result Ycrjt ~ 4/3. Similarly tha aoaent of inertia with increasing stiffness of the equation of state inertia is given by I * MR* - M5^3 p"*^3 which, when coabined is seen in Figures 12-13, it is not as strong as was the with (6.2) becomes variation in critical mss. This is to be expected, in that the aoaent of inertia 1 depends not just or. total aass, but , % B V2(r - 8/S) (6 J) on its distribution as well. Thus the equation of state in model D is generally less repulsive than Models A and E, The aaxiaua occurs for - 8/S. It then follows Chat contains aore aatter at lower densities (larger radii) and thus a larger This is a consequence of the rather TI * W (»•«) sudden dip in pressure in the range IS.5 < log p < 1S.7S (see Figure 4). This is observed in all of the models considered here. It The qualitative behavior of the I vs. M curves near may further be argued that Chs inclusion of general rela the aaiiaum aay be shown to be a general feature of systeas tivistic cffects, at least in the post Newtonian approxima described by a single polytropic equation of state tion, will not alter (6.4;. If 2MG/fcc* < 1, the leading order effect of general relativity is to increase the value 9 • to' (*•»> of Ycr|t at which instability occurs by an additive tera proportional to M/R (Chandrasekhar 196S): in hydrostatic equilibrium with » ■ 1 * 1/n. In particular we wf.il show that l ||a|( occurs at a lower central density rcrit * Tcrit “ 4/3 * 1C’M/R • ( 6 'S) than does Consider the non-relativistic case first. Diaensionally the equation of hydrostatic equilibriua and Diaensional arguaents allow us to cast the last tera in the the density are P * p M / R - M*/R* and p - M/R*. Combining fora kpy'^. Rather than thinking of the curvature correction these with (6.1), it follows that ■rp*'1 as increasing the critical v*lue of y for the onset of instability, we may take it to the left hand side of (6.S). M , ccV 2 ( * ' 4/3) («-2> 39 It then reduces the actual adiabatic exponent, and instability Here VQ > 0 (VQ < 0) corresponds to repulsion (attraction), occurs at y • 4/3 as before. Thus we formally consider and the range of the interaction is determined by a. curvature as equivalent to softening the equation of state. Standard no.i-rclat ivistic many-body theory may be used to Y -1 This means that we replace y in (6.2) and (6.3) by y - rp' calculate the thermodynamic properties of the system at If the effect of the curvature term is not too large, we may zero temperature (Fetter and Walecka 1971). We are justified approximate this by y - in the region near Pcrjt» where in using non-relativistic expressions (with possible inclu y » 4/3. Then (6.2-6.3) become sion of leading order relativistic corrections to the equa tion of hydrostatic equilibriun as in 6.5) as long as we are M - p0V2Cy ' 4/3 ' « p 1/3> (6 .6 ) only interested in the qualitative dependence of masses and moments of inertia on VQ and a. This should be evident from r ,, D S/2(y - 8/5 - * p 1/3) 1 Pq (6.7) the fact that the qualitative features of neutron star struc ture are already contained within the framework of Newtonian Treating p as constant in the exponents, we see that (6.4) gravitation (Zel’dovich and Novikov 1971). Starting with is still valid. (6.9) it is easily shown that the single fermion Green's A simple analysis in the spirit of our discussion above function is given by for I-ax and Mcrit may be used to study the effects of inter actions on the structure of neutron stars. Suppose that we G(P,-) - [« - p2/2m - r(p, U)]'X (6.10) have N fermions of number density where, to order V0 in the self-energy, and for dilute systems n - qj|/3ii2 (6 .8 ) (qpa « 1), ZV„a5 ( 7 v whose interactions are described by a Yukawa potential Z(p, u.) - — “— qjj/p2 ♦ 3/s qj; j*.... (6.11) Terms of order (qFa)6 and higher have been neglected. The poles u(p) of G(pu) give the excitation energies of the . 37 3« system-, in particular the Fermi energy <*>( u ‘ 4 * - - ( 6 -12) n r S/i . 7/3 ^ _ 8/3 ,, P«CjP - Cj j ♦ Cj p (6.17) Thermodynamic identities may now be used to find the ground state pressure Here Cj and are positive, and Cj a VQ. From (-6.1) we find ** ndii (6.13) , - (p/P)(dP/dp) (6.18) I '0 where.n is given by (6.8), and the ground state energy density Using (6.17) we find after some straightforward algebra the following expression for y: . m c ! . »-2n ( H l r W , (6.14) J. y - | - o2 p2/3 ♦ (6.19) Eq. (6.13) is the constant volume integral, of the Gibbs-Duhem relation, and (6.14) follows on integration of the definition The constant a2 is proportional to m"8^3. Eq. (6.17) or of the pressure P ■ n >(p/n)/Jn. The equation of state ob (6.19) show that attractive (repulsive) interactions soften tained from (6.12-6.14) is (harden) the equation of state as expected. The term »a2 is the contribution of the kinetic energy which, in this approxi / 10 V 5" J \ mation, has a larger effect on the energy density than on the pressure (see (6.1S-6.16)). Finally let us include the lead ing order effects of general relativity by incorporating the ^ / 4 V0aSn J \ *(qF> * I X ♦ ^ ( 1 * 5 - ^ r q?* »s ) <6,6> curvature corrcction as in the discussion following (6 .S). We find Nhere Oj represents terms of order (qF/ m ) 5 or higher. 39 40 t • j - a2 o2'5 ♦ o1/J . (6,20) by parts and the identity i(52d9/d?) ■ -C2fln obtained from d€ the equation of hydrostatic equilibrium yields the above ex The corresponding polytropic index becomes approxiaately pression for kn- Ne may now consider the effect of variations in y or n " " I * T (“2 1,2/3 * • V * * ‘ pl/3) (6 .2 1 ) on the polytrope's mass and nonent of inertia at constant central density. Examination of (6.22) for y - 4/3 and at In general the kinetic energy and curvature soften the equa densities applicable to neutron stars shows that tion of state as expected. %n approximate expression for the stellar mass is ob ’ 0 tained if we treat the densities p in y and n as suitable averages or constants. Ne then use standard expressions for Ne see that an increase in y results in an upward shift of the mass of a polytrope with index n given by (6.21): the maximum value of M( p c ) for fixed values of p£. The qualitative behavior of the M( p c ) curve near Pc r ^ t may be „ . ■ .IV'2" • ,/!) [ (7el'dovich and Novikov.1971). He aay also express the poly from (6.20) for pQ satisfying trope's moaent of inertia in the convenient fora 4 5 2 2/3 S 1/3 f(l ! • J - a »0 B V °0 0 (6-26) 1 • knMR2 (6.23) Ne have taken the average value of p in (6.20-6.21) to be Pq. An approximate expression for C/q may be obtained froa (6.26), kn • I * « £ 4 X 1 *20dt (6-2<) and the stellar mass obtained from (6.22). Three cases are Me arrive at (6.23-6.24) by expressing the aoaent of inertia of particular interest: # ■* I • .T^da ?n teras of the diaensionless variables £ - r/rn Case 1: No interactions (Vo • 0). Then only the and ? • (p/oc )1/n, with r2 ■ (n*l)K/4«Gpc . An integration kinetic energy and curvature terms enter into the adiabatic i 41 42 index which is, at t ■ 4/3, is seen to result. The priaary effect of general relativity is to reduce the aass of a star, and to reduce the central 4 5 2 2/3 1/3 j • J ' • »5 "**>0 (••«’) density at which collapse begins. Case II: Newtonian gravity with interactions (k ■ 0 ) . Solving this iteratively for KPj*^ < »ZPq ^ we find The adiabatic index at the critical point is equivalent to the expression 4 . 5 2 2/3 r, 5 I I ' “ p0 * 6V p0 (6-31J . for the density at which collapse begins. Notice that in the absence of general relativistic effects k • 0 and we Assuming that the kinetic energy doainates the interaction conclude that energy we obtain the approximite value (Pcrit)GKT ‘ (,>crit)Newt C‘-» > Furthermore, k increases the polytropic index (6.21), de at which instability results. It is evident that repulsion creases y end thus by (6.2S) decreases the. star’s aass: (attraction) increases (decreases) the critical density. Since the polytropic ind'x given by (6.21) decreases (in • o w , * , creases) in the presence of repulsion (attraction), we see that Mmax and p__i. crit aove in the same direction as shown Figure 14 shows the M(pc ) curve obtained by integrating schematically in Figure IS. first the fully relativistic equations of hydrostatic equi Case III: General relativity plus interactions. The librium, and than using the Newtonian equations (Ruffini and adiabatic exponent is given by (6.20). For attractive in Wheeler 1969). Both calculations use the Harrison-Wheeler teractions (Vq < 0) the situation is similar to case I, equation of state (basically a free gas for densities near where the third term on the right side of (6.20) combines aass peak). The general behavior indicated by (6.29-6.30) with the curvature term to give the sane qualitative effects. 43 44 For repulsive interactions near the onset of collapse we schematically in Figure 16 where curves a, b, c correspond write (6.20) as to increasing repulsion. Comparison with Figure 7 and the discussion at the beginning of this section shows that our y - § - 3 ♦ «2lv„Use0 - « »o1/3 <6-33) aodel is consistent with detailed calculations. The results for cases I*III are summarized in Table 14, Ne have seen above that the first two teras tend to cancel where we list the effect of kinetic energy, interaction en near collapse. Since the last two teras coapete (the inter ergy and curvature (general relativity) on the steilar struc action teras tending to stability, while the curvature tera ture (a (♦) entry means that the quantity is increased; a tends to destabilize the system) we expect (-) entry means a decrease). The moment of inertia for the equations of state shown « p01/3 > 62|V0|a5p0 (6.34) in Figures 4-5 is given in Figure 17 as a function of cen tral density. We see a general trend toward increased IM X in the liait y * 4/3. This gives, for the critical density and decreased Oj (the central density of the model with the approximate expression maximum moment of inertia) with increased stiffness of the 1 equation of state. pcrit ’ («/«2|V0ia5)3/2 . (6.35) Examination of (6.23-6.24) in the neighborhood of Y - ' 8/5 shows that Increasing the strength Vq or the range a of the repulsive interactions decreases the critical density and the poly ( d4 . . « >0 tropic index. The latter results in an increased mar.i'jtim aass (discussion following (6.22)). However, there is alto (see Appendix A). We may now use (C.?*) ?nd the earlier ex a weak tendency for k to increase with increasing n and vice- pressions for n and y to investigate the effects of interac versa (Chandrasekhar 196S). Consequently the density ir. Pcrjt tions and curvature on the maximum value of >, and on the will not be as rapid as if k were strictly constant. The be density pj for which it occurs. We note that although our havior of the aaxiaua mass and critical density are shown discussion leading to (6.28-6.35) assumed r - 4/3, the 4S 46 ttMlitttiw u p w t tit I k t i f aquations applies for t * 8/S VII. Conclusion as wall. T t i u l «• find in case 1 (VQ - 0) that k decreases y The nuarrical results discussed above aay he suaaarized and thus decreases tha value of IMax: by enuaerating what eaerges as the general properties of neutron stars based on a large saapling of equations of (,aa«)CRT * (laax}Newt. t6‘37) state. The saapling represented by A-G and M of Figures 4-S is believed to cover a broad range of possibilities, and in Tha shift ia density Pj at which JM X occurs saows the saae clude soae of the aost sophisticated non-relativistic work in qualitative behavior as does Pcrjt in (6.29). In case II this line to date. Although variation in M(pQ) and I(pQ) (« • 0) we see that repulsion increased the aaxiaua value of exists, it does not seen large considering the the aoaent of inertia and increases the value of as well. differences between the interactions used in con Finally, for case III (c 4 0) we see that increased repul structing the equations of state. We note that the range in sion increases I # while decreasing Pj. Reference to aaxiaua gravitational mass is 1.2-2.0Mg, and the correspond Figure 17, and the discussion of relative stiffness of the ing range in the AMU mass is 1.3-2.2Mg. If, as argued in equations of state in Figure 4 (beginning of section VI), section II, we tentatively assume that supernova remnant shows that our siaple aodel is consistent with the results ■asses have ■ 1.4Hg, then the nodels considered here i t detailed calculations. yield a gravitational mass for the reanant in the range 1.2S < MR/Mg < 1 .28 . Nith the exception of aodel M, the ■aximua moment of inertia is S - 20 * 1044 g/cm2 and always occurs at a lower central density than does the aaxinum aass. Roughly speaking, the radius of the maximum miss aodel is 2 -4MaaxC/c (twice gravitational radius). Models H, L and N-0 lead to more massive neutron stars (2.39 < Mma x /M0 s 2.70) having moments of inertia in excess 4S 2 of 10 g/cm for all but the least massive models. The radii arc somewhat larger than in the previous group, and 47 48 are seen to be insensitive to the mass (except for M i 0.5Mg). in an increased maximum stellar mass, and higher central Both nodels L and M yield R = 15-16 k« while model 0 gives densities. Repulsion also increases the maximum moment of R : 12 kn. In both nodels L and 0 there is a slight tendency inertia and the central density of the model for which it for the radius to decrease with decreasing Mass in the inter* occurs. When the effects of general relativity are included, mediate mass range. We find that the more massive models even the qualitative nature of the situation is changed. have radii (at maximum mass) of about 3 to & times their Increased repulsion again leads to an increased maximum gravitational radius 2MG/RcZ. mass and moment of inertia (interactions dominate curvature); The maximum mass range above argues against the likeli but the central density for which each occurs is reduced by hood that supernova cores will in general be too Massive to increased repulsion when curvature effects are important evolve to the neutron star state. Direct formation of a (Figure 13). Reference to (6.32) and (6.35) shows that the black hole is not impossible, but it might not be expected interaction strength (VQ) and range [a) enter as a product. to occur in the majority of cases. Note however that if This feature, well known in simple models of nuclear matter mixing of the core and mantle is induced, massive cores theory, shows that quite different interactions may lead to might be formed which could collapse directly to b?ack holes. relatively small changes in the maximum neutron star mass, On the other hand it is unlikely that all supernova a result which is not inconsistent with our findings (Figures remnants bypass the neutron star state during their final 4 and 7). All of the equations of state (with the exception of evolutionary stages (case a discussed in section II). models N, 0, and the free neutron gas) used in the discussion The simple model discussed in section VI illustrates above are based on nen-relativistic interactions and many- the way in which the effects of general relativity, and the body theory. Despite the differences in detail between nature (attractive or repulsive) and range of the interac models A-G, we observe a general trend in the neutron star tions influence the maximum nass and moment of inertia of parameters. Based on these results it is tempting to con neutron stars, as well as the central density at which each clude that the maximum mass and moment of inertia of actual occurs. In the absence of general relativistic effects, in neutron stars should lie in the range spanned by Models A-G creased repulsion in the microscopic interactions results (Figures 7 and 17). Unfortunately this conclusion May be 49 SO premature, as illustrated by models 1-0, which lead to aaxi earlier. Since at least 751 of the aatter in the aassive aua masses greater than those of aodels A-G by a factor of stars of aodel M is handled in a self-consistent aanner, it as much as two. In two cases (aodels L-M) the increased is tempting to consider it the least objectionable of the m.'^s and rcoaents of inertia are due to an increase in the non-relativistic examples considered here. st.'ffness of the equation of state just above nuclear density Finally we note that the only two aodels built on (Figure 5). The situation is further coaplicated by the relativistic interactions and aany-body theories also yield characteristic feature, illustrated in Figure 6, that most increased maximum masses and aomcnts of inertia. natter in maximun mass neutron stars is at sufficiently high Despite the spirited defense of nuclcar matter theory densities that the use of non-relativistic descriptions of based on potentials at high density (for example see Bethe interactions and many-body theory nay be invalid. and Johnson 1974), it is doubtful that a fira consensus will A possible exception to this state of affairs is offered emerge before the results of realistic relativistic calcula by the following observation. The central density oi tions at the equation of state level are known. maximum mass neutron stars decreases with increased stiffness of the equation of state. It is possible, therefore, that increased repulsion at, or slightly above, nuclear density could reduce pc sufficiently such that lit.tle or none of the aatter in aassive neutron star cores would be at densities where relativistic effects would be important. Model M represents a possible step in this direction. If density profile and mass fraction curves are examined for this case it is found that 241 of the matter lies at densities greater than I01* g/cm3, and only about 40t is at densities greater than 8 * 10** g/cn3. Furthermore, the increased stiffness at lower densities gives increased radii and thus larger moments of inertia, which appear to be favored as mentioned S2 51 Appendix A Acknowledgements The analysis leading to (6.25} is straightforward. It follows directly upon differentiation of (6.22) with respect We are indebted to J. Arponen, H. A. Bethe, M, Johnson, to y at fixed pg. The resulting expression contains terms V. Pandharipande, and R. Smith for numerical tables of their which, by reference to tables of Lane-Eraden functions (see equations of state prior to publication, and for additional Table A1 below) are shown to b'e positive for densities ap information concerning their work. We express our gratitude plicable to neutron star interiors. The proof of dl/dy > 0 to V. Canuto for discussions concerning recent equation of for y - 8/5 is similar. However in this case we encounter state calculations and for a copy of his review article terms not given in standard tabulations of the Lane-Emden 'prior to publication. We also wish to thank A. Gleeson, functions. We therefore give in Table A1 values for kn de D. Pedigo and members of the Center for Particle Theory for fined by (6.23-6.24). In obtaining these values we have many helpful discussions. One of us (RLB) acknowledges par numerically integrated the equation tial support by the R. W. and Jane E. Olson Foundation the Department of Astronomy and the Center fbr Relativity GP'32051p MP-574-20755 and the Alfred P. Sloan Foundation* 8(6) are then used numerically to obtaine tj (the value of £ for which 9(?) * 0) and kn as given by (6.24). As a check on our results we include in parenthesis values of pre sented in the literature, or the values of Cj, '-?|[d0/dt]^ for k for n * 1 and 0 which may be obtained fro* exact' n ' solutions to (A.l) and (6.24). The proof of dl/dY > 0 (y = 8/5) is as follows. Fro« (6.22-6.24) and the relation r - rflC we *ay express I in the form 53 S4 Table Al ■ ■ V . « > n -CidB/dCj where wn (() s -tJde/dCj which is positive (see Table Al). *1 kn Differentiation with respect to r at constant pg and the re 0 2.4S (2.45) 4.892 (4.899) 0.399 (.400) lationship y • 1 ♦ 1/n gives 1.0 3.14 (3.14) 3.142 (3.142) 0.261 (0.261) 1.5 3.6S (3.65) 2.714 (2.714) 0.204 g - « » r 3/2 «c/ G ) s / 2 p05/2^ -%>x 2.0 4.35 (4.35) 2.411 (2.412) 0.1S4 2.S 5.3S 2.187 0.111 *(f(n*l)3/Z knwnl(n*l)lnn0 - n2J (A.3) 2.6 6.60 2. ISO 0.102 2.7 5.88 2.114 0.09S1 - n V l J 5 ' 2 & kawn J 2.6 6.19 2.080 0.0889 2.9 6.52 2.048 0.0810 As in the evaluation of dn/dv it y • 4/3 we see that the sun 3.0 6.895 2.018 (2.018) 0.0749 of the first two terms in () is positive. Examination of Table Al shows that k w_ is nearly constant in n for l.S < n n ' n < 3 and therefore that the last tern in.(A.3) is snail. Caption: Numerically constructed values of &n for the The first two terns dominate, are positive and thus (dl/d-y)y.8/j moment of inertia (see (6.24)) for polytropes of 0. Ne see a general trend in Table Al toward increased index n between 0 and 3.0. Models for which n ■ moment of inertia with increased n and thus increased raass, 0 or 1.0 are exactly soluble. as expected. Finally we note that dl/dy 5 ' for y ' 4/3 as well. Table Captions Table 7: Neutron Stars Based on Pandharipande (Hyperon), Table 1: Equations of State. Summary of equations of state Arponen and BPS Equation of State. used to obtain neutron star models of Tables 2-8. The den Columns same as in Table 2. sity range in g/cc in which the equation of state is used is given in column 4; the notation 6.97E14 means 6.97 x 10**. Table 8 : Neutron Stars Based on Canuto-Chitre, BBP and BPS Equation of State. Table 2 : Neutron Stars Baseil or. Pandharipande (Neutron}, Columns same as in Tzble 2. BBP, and BPS Equation of State. The columns give: (a) central density in g/cc; fb) gravita Table 9; Neutron Stars Based on Pandharipande and Snith tional mass in solar units; (c) proper mass in solar units; (Mean Field Model) Equation State. (d) AMU mass in solar units; (e) radius in km; (f) moment of * 2 IS Table 10: Neutron Stars Based on Pandharipande and Smith inertia in g/cm . The notation 7.943E+1S means 7.943x10 (Tensor Interaction) Equation of State. Table 3: Neutron Stars Bas^d on Pandharipande (Hyperon), Table 11: Neutron Stars Based on Walecka (Neutrons) Equation BBP, and BPS Equation of State. of State. Columns the same as in Table 2. Table 12: Neutron Stars Based on Bowers, Gleeson and Pedigo Table 4 : Neutron Stars Based on Bethe-Johnson I, BBP, and Equation of State. BPS Equation of State. Columns same as in Table 2. Table 13: Summary of Remnant Masses and Moments of Inertia. The colums give: (a) model; (b) central density for remnant Table 5: Neutron Stars Based on Bethe-Johnson V, and BPS mass in g/cc; (c) gravitational mass of supernova remnant Equation of State. (solar units) assuming MA/Mg * 1.41; values in parenthesis Columns same as in Table 2. are maximum mass of model; (d) moment of inertia of the rem Table 6 : Neutron Stars Based on Moszkowski and BPS Equation nant in g/cm2 . of State. Table 14: Effect of Interactions and General Relativity on Columns same as in Table 2. Neutron Star Structure. A ♦ (-) in the table implies that an increase in the quantity in the column heading results in an increase (decrease) in Table 1 the quantity in the row heading. Model Equation of State Density Range Gcafnsition A Pandharipande PN p > 6.97E14 neutrons B Pandharipande PC p > 7.000E14 n. p, A. I1'0, i**° C Bethe-Johnson BJ1 1.71E14 - 3.23E16 n, p, A r*° . Ai>0 and■ a♦♦ 0 Bethe-Johnson BJ5 1.7E14 - 2.26E16 Same E Maszkowski M 2.2E14 - 3.08E15 n, p, I , A0,, b F Arponen A 3.1E11 - 2.0E1S Nucleons, e and muons G Canuto-Qiitre CC 2.37E15 - 7.23E1S neutrons (solid) H Ideal neutron gas’ IN entire range neutrons I Cameron-Cohen- CCLR 1.0E14 - S.35E15 nucleons, e~ and Langer-Rosen muons L Pandharipande p > 4.386E11 neutrons and Snith (Mean field) M Pandharipande p > 8.428E13 neutrons and Smith (Ten sor Interaction) N Wa lecka P > 1.723E14 neutrons . .0 -,0 0 Bowers, Gleeson p > 1.732E14 P 1 * 2- * - and Pedigo Table 1 - Continued Model Interactions Many-body Theory .0E1 265-1 .0E0 262-1 128 ?.724E«43 11.248 2.632E-01 2.703E-01 2.605E-01 6.000E*14 .8E1 463-1 .3E0 477-1 017 Z.170E»44 3.026EM4 10.117 10.C30 4.767E-01 6.024E-01 4.934E-01 6.2S9E-01 4.643E-01 5.813E-017.080E»14 8.000E+14 .1E1 694-1 .2E0 728-1 005 3.S91E*44 10.015 7.218E-01 7.S28E-01 6.9C4E-01 8.913E*14 I-+ *-• N w o* i/i in o 9> -j ON C/1 U o o © Ni (4 o o A Re id soft core - adapted to Variational principle applied m V*I-* CD O'VO lo Jk o 5V*j CO O o <5 o Ul 00%o o CO N o o v» o nuclear natter to correlation function m m m tflCO tnin A n: m n m m m rs t-1 m ♦ m ♦ n♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ M ♦ ❖*-* ♦ ♦ ** ♦ >-• M B Same as A; arbitrary reduction Same as A U1 I/I tn %n tn in Kn m VI Vi in in tn V* VI for hvpcron-hvperon attraction C Modified Reid soft core; non Constrained variational integer n in (3.1) principle M O (A NW N> Lrt O'0 4 *» <#•in 1-40\ O N N N O 0 * 4 m m n m pi « CO U tA♦ ♦n m n ♦ D Same as C; nore nearly Sane o o o o realistic adaption to hyperon o o o o o o o o o witter E .Reid soft core; modified Reaction matrix MMNNIKtMNK»NN t/t in 9 to a: hypcron interactions based on O A U •9 quark theory. ml> mH Om OBpi Um X o o o o o o o o o © 2 Table F Thomas Fermi model Brueckner G-matrix o o o o G Modified Reidsoft core. T-matrix; includes spin Localization via non-rela dependence tivistic harmonic oscillators. Ui M O* x s|O' M|/i VI|* O v* "1 o M o Ch U> «5 M > H None Ferrai statistics tnnnniwniMWWtiriMriwnni I Levinger-Siimons velocity de Hartree'Fock approximation 8 S 8 § pendant V . with two-body potential L Nuclear attraction due to Vian field approxination VO VO <0 J© u> OB c m 0* Pi •a v| vo VO •**4 si * K> OD 4k o •>4 * scalar exchange or scalar; variational VO s is VO O' o» s hi O' ui tj 1* method. -J O' **4 o <0 1/4 VI v% o cn OB SO 2 N Nuclear attraction due to Constrained variational pion exchange tensor inter method a» O (• a 0 Non-perturtibative, pheno Relativistic menological approximation to relativistic meson exchange TabU I V* N « ® o •»••<> O C '» , v , »v w, ^ ' C' h » O > < • • » • o ® « N ^ ^ ^ m ^ f 3 ) f 9 ^ f » « » ^ f ^ * N • 9 N 9» •« N *f ^ f* ^ »v «• »4 ft <9| ^ «f « V) 13 t n o wt «n o I,•••••••* I I I I # I • • S i l § BiSijaawS&SsH&WSiKSlifKfcfWWa * * * v Q O O » * l V « r ^ O A < 8§8§S88SS8oOOOOOOO o o O O ? e O O O O Q C O O O O O O O O O O O O O Q % s ^ f,* r^ fs i% « 00000000<«M4M*«MM#«MrM««MMM M M « « M r M « # M M « * M 4 M « < 0 0 0 0 0 0 0 0 C% ^ •I w» W g • «■* »« w S s • » •r« *• a o o ■ m* K mm 8 • %* « ii O O mm 2 « O •* « N £ lA U 3 o is «* ■» ** v* i u O •« / • * * t^ (Jl ^ *t »* #* *• N */ 4 « o 4 «i « «4 «o •« Ui «p © *4 v* r># v* w •-• o mm i #’ O mm ••••• «* o «t */» £ o «« ** N N e» •» ► «9 ta O o 1 o s Ukt l.00C£*l» l.MTH.OC 2.3ME*00 1.7i2£*00 7.972 ».S6J!><4 7.*43£*1S l.W4E*00 l.i^EoOO l.«T2E»00 S.25$ 9.94*E.« •.200EMS l.7$B>00 2.4»4!> fSI +4 •n *A W r* M i 3 o O A r> St mm s tA mm »* Ui o O N 0 § § »« r» £ «« ** •4 r. m & « «» « M s 2 O u u u • 2 «* us O £ a «A i u s I *-« ^ § o O ut *> W”l h fr Ki O M h - N ri N - f4 fj« Mm. # 4 * 0 <6 n s t r t r I s I «r o «o U UJ U £ • » • • n n 3.200E*14 3.S92E*01 3.724E-C1 J.6J0K-01 14.246 2.S9<[>44 2.600014 2.664E-01 2.730E-01 2.67OE-01 IS.S60 I.7«4I>44 2.000014 1.I0SE-01 1.840F.-01 l.mME-OI 1‘J.ZZS t.0%l*-U Table S °c V Me M p/M„ V M6 R(km) I(g c » 2) 1.000E*16 1.4SSK*00 2.107E*00 1.632E*00 7.S41 6.918E*44 7.94 JEMS l.S3SE«00 2.148E*00 1.709E»00 7.8S0 7.818E*44 6.?Q0E»15 1.S8SE*00 2.180E+00 1.786E«00 8.237 8.912E*44 S.012E»1S 1.621E*00 2.191E»00 1.84QE*00 8.?i06 9.923E*44 3.981E*1S 1.646E*00 2.178E*00 1.877E»00 9.040 1.101E*4S 3.S48EMS 1.6S2E*00 2.162E*00 1.886E+00 9.262 1.153E+45 3.000E»15 1.649E»00 2.122E*00 1.883E*00 9.S97 1.222E+45 2.S12E*1S 1.632E*00 2.061E*00 1.860E*00 9.945 1.279E*45 2.2J9E«1S '1.611E+00 2.010E»00 1.831E*00 10.150 1.3Q1E*4S 1.99SE*1S 1.581E*00 1.950E+00 1.791E+00 10.331 1.306EMS 1.778E*1S 1.S49E*00 1.893E»00 1.747E+00 10.448 1.291E+45 l.S8SE«)5 1.S08E«00 1.827E*00 1.694E*00 10.S41 1.259E*45 1.413E*15 1.42SE»00 1.700E-00 l.S86E*00 10.678 1.183E+45 1.259E-* 15 1.313E*00 1.537E*00 1.444E+00 10.829 l.073E*45 1.322E+1S 1.187E«00 1.364E«00 1.290E*00 10.968 9.491E+44 1.000E*1S l.061E*00 1.198E+00 1.140I>00 11.089 8.247E*44 8.000E+14 8.211E-01 9.003E-01 8.637E-0J 11.306 5.933E*44 6.000E«14 S.493E-01 5.842E-0I S.6S4E-01 11.703 3.S22E*44 S.000E*14 4.130E-01 4.528E-01 4.207E-01 12.203 2.447E*44 4.000E*14 2.903E-01 3.001E-01 2.931E-01 13.3S7 1.590E+44 3.200E-14 M12E-01 2.16SE-01 2.120E-Ot IS.506 1.106E+44 2.600E*14 1.613E-01 1.643E-01 1.612E-01 19.240 8.3S3E*43 ,2.0!WE*14 1.211E-01 1.228E-01 1.207E-01 30.103 6.614E*43 Table 6 pc H c/Me Mp/Me MA/Me R(k«) 1(S C«2) 3.070E+1S 1.7!ZE*0Q 2.304E*00 2.056E+00 9.021 1.246E*4S 2.818E+15 1.713E*00 2.252E*00 2.008E+00 9.171 1.254E*45 2.512E+15 1.676E+00 2.J66E+00 1.954E+00 9.370 1.250E+45 2.239E+15 1.626E*00 2.064E+00 1.881E+00 9.562 1.227EM5 1.778E»1S 1.476E*00 1.083E+00 1.675E+00 9.91S 1.119E-45 1.585E+15 1.378E*00 1.650E*00 1.546E-00 10.066 1.03SE+45 1.413E*15 1.266E+00 1.488E+00 1.404E+00 10.200 9.347E+44 1.259E«1S 1.145E*00 1.321E+00 1.254E+00 10.316 8.239E+44 1.000E*1S 8.920E-01 9.930E-01 9.531E-01 10.495 S.947E*44 7.943E«14 6.516E-01 7.043E-01 6.813E-01 10.695 3.918E«44 6.310E*14 4.S23E-01 4.778E-01 4.648E-01 11.071 2.414E+44 S.012E*14 3.0S0E-01 3.168E-01 3.094E-01 11.976 1.150E+44 3.981E*14 Z.062E-01 2.117E-01 2.074E-01 ■ 14.0S9 9.046E»43 3.162E*14 1.461E-01 1.488E-01 1.461E-01 18.926 6.406E*43 2.512E*i4 1.138E-01 1.1S3E-01 1.135E-01 30.802 5.628E+43 2.20QE*14 1.037E-01 1.050E-01 1.033E-01 44.174 S.84SE*43 • •••#••••••••••• * * H H M W w y ? r4 .M ;t^8KSSS££SSS2SS8S M !o o w *o © 9 0( H H » O SiflSlfi^KSSSKRSSSBSfM is 8 ' H N O W O (A in hhhhn^mmmh^ mhhhnhn m m wwwwwi/iwu*(/»t/»w«ini/iwwwtf»V' H H N N * U1 IO H H HNMUWbtU**** 2 ©OCDOO>^ONHWP0*©lOK>tO«‘«*CA©W a*Itf» al 7 continued 7, Table hhmmhhhhmmmh•— »— ^ m m n n n U li «• V» ® V ^ » » # « « JO « O O O X • > 4 0 ^ 0 0 V» C* ‘o TJ S i/l H H O (/I O' mN ^ n 9> in O a s to 8 s i/i o ^ o vo m m m m X m m m X to...... pi ro ro ro S 00 h-» H K) oi to O 00 M 9k O O o ot ►-* H» O tttoOAOBN^SlJSfll'O'Pfg'iS o O k M o -■J NJ N ►-» H*OM*9CI^HaMvJN22SNS *-» *■* N o to OO lymflmijftniTimmniTiiTiijiftijiijiij rp tfl m rr. n m m m m * ? + ♦ ♦ ® o O o o o o © o o sssisssssissssssss M ►— ►-» H ►- H* H* •-• o to H ►* t/1 to rsi H p—» M »—• H H o o a to •o to o 00 00 m m oe ■M -a •J o O to K) MMM H M HI— o O O o to Is) M sj to to K) i/i 14 to IM W o O’ CO **4 to cn M o 00 ** o VO N o o s . VO *o N N N 8 o* tsl N i/i co o to (O H o VO 3 N to to K K» N) U1 VO to O O' O VI to fsi VI 00 o 00 o 00 ~v» trf W tn -J to «o to to oo oo oo ao vj ■*4 v| O' hO N rs> o oo tn VI o oo •«4 o to ■*4 o o w ^ m to o o 1-4 Hp/Me ft (km) °c V M0 V * 9 K l cm2) 6.700F.*1S 1 .JS7E»00 l.S62E«0a l.S18E*00 6. ISO S.S3M>44 6.310E*15 1 • !SSt'*00 1.850E*00 l.S21E*00 6.944 S.649t><4 6.04ZE»IS 1.3S8E»00 1.84DE«00 l.$2U>00 7.013 S.72JE«44 S.2S2E*1S 1.349I>00 1.79SE*00 1.511F.*0C 7.237 5.916T*44 4. SO JEMS 1 .328E*Q0 1.7435*00 1.48SE*00 ,471 6.0281*44 4 .I6IE»I5 1.3UE*00 1.69IE*00 1.463E*00 .601 6.046E*44 3.829EM5 1 •287E»00 1.639E*00 1.432E*00 7.742 6.027E*44 3,498E*IS 1.2S4E*Q0 l.S7SE«00 l.392E*00 7.899 S.964E»44 3.198E*15 1.215E*00 l.S03E*00 1.343E*00 8.0SR S. 8541X4 2.912t*lS 1.169E*00 1.424E*00 1.287l>00 8.227 S.699l>44 2.631EMS l.USE*00 1.337E*00 1.222E*00 S. 401 5.49JE»44 2.376E*1S 1.0S7E*00 1.249E*QO 1 .1S3E*00 I.S56 5.232E*44 Table 9 Hc/H a Hp/Ma R(k«) K g M 2) °c V* 7.943E*1S 2.169E*00 3.12?E*00 2 .37SE*00 10.923 2 .IS3E*45 6,310E*1S 2 . 270E*00 3.247E*00 2.i3JE*00 11.269 2 .472E*45 S.012E*15 2 . 373E*00 3.3656*00 2.69SE*00 11.644 2 .83SE*4S 4.467E*15 2.422E*00 3.420E*00 2.773E*00 11.838 3.026E*45 3.9S1E+1S 2 . 470E*00 3.473E+00 2.850E*00 12.036 3.224E*45 3.S48E*15 2.514E*00 3.S18E*00 2.919E*00 12.225 3.414E«45 2.162E*15 2.553E*00 3.557E*00 2.982E*00 12.412 3 .601E*45 2.818E*15 2.589E*00 3. 591E*00 3.040E*00 12.596 3 .783E*45 2.S12E+15 2.621E*00 3.616E*00 3.090E*00 12.781 J.958E*45 2.239E-15 2.649E*00 3.635E*00 3.135E*00 12.96 4 .130E*45 1 .99SE+15 2.67SE*00 3.646E-00 3.172E*00 13.16 4.?95E«45 1.778E+15 2.690E-00 3.644E+00 3.200E*00 13.3S9 4 .451E*45 1.585E*15 • 2.700E+00 3.630E*00 3 .216E*00 13.S61 4 . 591E+4S 1.413E*15 2 . 702E+00 3.600E*00 3 . 218E*00 13.773 4.712E*45 1.2S9E+15 2.619E*00 3.549E+00 3.202E*00 13.991 4.803E*4S 1.122E*15 2.664E*00 3.472E*00 3.161E*00 14.211 4.848E*45 1.000E+15 2.618E-00 3.368E*00 3 . 093E*00 14.424 4.833E*4S 8.913E*14 2.S44 3.222 2.986 14.638 4 . 739E*45 7.943E*14 2.448 3.050 2.850 14.815 4.561E+45 Tabic 9, continued pc h c / mb *(k «) »(« c « Z) 6.310E*14 2.131 2.S60 2.432 I S .001 3.I63E*4S S.012E*14 1.722 1.911 1.904 IS.US 2.862E*4S 4.467E*14 1.473 1.651 1.602 • IS. 026 2.2I6E*4S 3.1911*14 1.223 1.3S0 1.311 14.191 1 .746E*4S 3.5«IE*14 0 .9ISS 1.061 1.041 14.741 1.274E*4S 2.I1IE*14 0.S931 0.6244 0.61 IS 14.601 6.074E*44 2.239E*14 0.3296 0.3401 0.3343 IS.437 2.603E*14 T a b U 10 Mp/M9 ec V Me V H« R(k*) K « « * ) 3 . 162E*1S 1.199 2.S12 2.101 11.211 1.72SE*4S 2.239E*1S 1.941 2.477 2.16S 12.343 2 . 100E*4S 1.778E*1S 1.9S4 2.43S 2.181 13.172 2.3I9E*45 1.SISE*1S 1.9S6 2.410 2.183 13.S83 2.542E*4S 1.2S9E*1S 1.9S2 2.3SS 2.17S 14.371 2.84SE*4S 1.000E*1S 1.934 2.298 2 . ISO 14.9S8 3 . 061E*4S 7.943E*14 1.902 2.246 2.109 I S .099 3 . 046E*4S 7.0I0E«14 1.868 2.196 2.066 I S . 181 2.99SE*4S 6.310E*14 1.734 2.00S 1.899 1S.4S8 2.791E*4S S.623E*14 1.60S 1.829 1.743 15.688 2. S80E*4S S.012E*14 1.478 1.663 1.S93 15.877 2 . 360E*4S 3.9I1E*14 1.211 1.329 1.283 16.204 1 . 87SE*4S 3.162E*14 0.9482 1.019 0.9904 1 6 .SOS 1 .395E*4S 2.S12E*14 0.7148 0.7544 0.7373 16.841 9.804E*44 1.995E*14 0.5200 0.S412 0.S310 17.421 6.S39E*44 l.S85E«14 0.3677 0.378S 0.3723 18.566 4.175E*44 Table Mc/Md Bc _ V ^ e _ R(k«) l(g cm1) S.162E»1S 2,485 3.526 2.919 11.271 3 . 176E>4S 2 . 512E*15 2.S43 3 .560 3.016 11.644 ).479E*45 2.000E*I5 2.S79 3.551 3.147 12.027 3 . 752E*4S 1. 58SE*15 2.583 3.483 3.1SS 12.433 3.970E*45 I.SOOEMS 2.577 3.456 3.14S 12.£21 4.000E.4S 1.259E*15 2. $30 .3.324 3.072 12.798 4.029E*45 I.OOOE-1S 2.31? 3.036 2 .(5 7 13.081 3 . 832E*45 8.913E«14 2.261 2.130 2.685 13.167 3.600E»45 7.943E*14 ‘ 2.119 2.590 2.476 13.118 3 . 2SQE*45 7.080E*14 1.919 2.292 2.206 13.126 2 . 841E»45 6 .3 1 0 E M 4 1.670 1.944 1.683 12.949 2 . 305E*45 5.623E*14 1.400 1.S90 1.549 12.651 1 . 761E*4S 5.3 Table XI R(kn} ' °c V M0 V " 9 V Me 1(8 « 2) 2.365E+15 2.380 3.288 2.889 11.138 2 . 929E*45 2.000E»1S 2.38a 3.245 2.899 11.422 3.0SOE«45 1 . 780E*1S 2.373 3.186 2.879 11.614 3.110E«45 1 . 500E*15 2.330 3.065 2.812 11.886 3 . 134E*4S 1 . 2S9E»15 2.239 2.873 2.675 12.135 3.048E*4S 1.000E+1S 2.017 2.486 2.358 12.368 2 . 701E*4S 8.000E*14 1.681 1.982 1.910 12.397 2.117E*4S 7.499E*14 1.566 1.822 1.763 12.358 1 . 918E*45 6.683E+14 1.3S8 1.546 1.50S 12.226 1.570E*45 6.390E*14 1.279 1.444 1.408 12.158 1.440E*4S 6.310E*14 1.257 1.417 1.382 12.137 1 . 406E*45 5.623E*14 1.061 1.174 1.150 11.898 1.099E«45 5.000E*14 0.8739 '0 .9 5 0 6 0.9342 11.587 8 . 238E*44 4.000E>14 0.5404 0.5715 0.5636 10.777 3 . 926E*44 3.000C+14 0.1640 0.1677 0.16S9 9.773 6.019E*43 Table 13 T«bl« 14 Kinetic Interaction Enemr General Energy Attraction Repulsion Relativity Model I(g cm2) »c (g c* 3) W Adiabatic index • - r -• Polytropic index ♦ « - ♦ A 1.6 • 101S 1.2S (1.66) 8.7 « 1044 n Critical B 3.0- 101S 1.22 (1.41) 6.1 « 1044 nst -- ♦ - *crit C 9.6 « 10M 1.28 (1.85) 1.3 « 1045 Critical density ♦ • D 1.2 * 101S 1.25 (1.65) 1.0 » 1045 - • ecrit E 1.4 « 101S 1.27 (1.73) 9.3 « 1044 F l.B * 101S 1.27 (1.46) 9.3 « 1044 G 3.6 « 1015 1.23 (1.36) 6.0 ■ 1044 L 1.4 « 101S 1.31 (2.70) 3 .9 * 1045 M 1.6 * 10lS 1.35 (1.96) 2 .1 x 1045 M 1.6 * 101S 1.29 (2.S8) 1.6 * 104S 0 2.0 * 1015 1.28 (2.39) 1.5 * 104S Figure Captions Figure 5. Equations of state corresponding to aodels L, M, N and 0. The region of the p,P plan* is the Figure 1. Schematic plot of the gravitational or AMU aass same as in the previous figure. vs. central density. Stable neutron stars exist I , for Pj < p < Pcrit* Stable configurations are Figure 6 . Density profile (left hand scale) and mass frac- 1 ‘' i denoted by (-) ; unstable models by {-----). tion (right hand scale) vs. radius froa the center of a cold neutron star for equatisn of state B. Figure 2. The AMU aass given by (3.5} vs. central density The tick marks on the right hand scale for 0.8 < for densities applicable to neutron stars. The a(r)/M < 0.9 gives for'the three aodels the ftac- cross hatched region corresponds to reanants with tion of mass consisting of Batter at densities AMU mass * 1.4Mg . The degree of repulsion in the p > 10** g/cc. Nuclear density is denoted by underlying equation of state increases in the the arrow on the left hand ordinate. sense a < b < c. Figure 7. Gravitational mass vs. central density for equa- Figure 3. Representative equations of state for cold neutron , » • ^ I * tions of state A-I. stars based on non-relativistic calculations. » 1 •# For comparison the equation of state for a free Figure, 8. .Gravitational mass vs. central density for aodels gas of neutrons is shown (-----). The region con * * , L, M,,N and 0. Also shown■ ■ i for• ■ ■ coaparisoni is tained in the rectangular region (upper right . model B of previous figure. Note reduction in corner) is shown enlarged in Figure 4. . , , vertical scale. 'Mi-.. I I | Figure < . Equations of state used in obtaining the results Figure 9. AMU mass vs. central density for equations of in Tables 2-8. For comparison we include a free state A-I. neutron gas (H) and the early work of Cameron, Figure 10. AMU mass vs. central density for aodels L, M, N Cohen, Langer and Rosen (I), letters denote and 0. Model B i4 included 'foncoaparfcsorij i Note equation of state referenced in the introduction. reduction in vertical•' ■ i scale. Figure 11. Gravitational aass-radius relation for selected Figure 16. Schematic shift in M(Pq ) near critical m s s show equations of state. ing affects of general relativity and repulsion in the equation of state. The degree of hardness Figure 12. Moment of inertia vs. gravitational mass. Note of the latter Increases in the sense a < b < c. the occurrence of lmMX at densities below which Maax occuts in all cases. Models I through 0 not Figure 17. Moment of inertia vs. central density for all included. ■(Mels calculated. Figure 13. Moment of inertia vs. AMU aass. All curves ex cept the topnost portions of I refer to the lower abscissa. The uppermost portion of I re fers to the scale at the top of the figure. Models L through 0 not included. Figure 14. Newtonian gravity vs. general relativity. The gravitational mass vs. central density is shown for Harrison-Wheeler equation of state. The upper curve results from the Tolnan-Oppenheiaer- Volkoff equation (3.1-3.2) which includes gen eral relativity; the lower curve results from the use of the Newtonian equations of hydro static equilibfium. Figure 15. Schematic shift in M(Pg} near critical mass re sulting from variations in the amount of repul sion or attraction in the underlying equation of state. Bibliography Bowers, R. I., Pedigo, D., Gieeson, A. M., and Zianeraan, R. L. 1974, to appear in the Astrophysical Journal. Arnett, W. P. 1973a, invited paper in late Stages of Stellar Brownell, D. H., and Calloway, J. 1969, II Nuovo Ciaento 60. Evolution, Nuclear Reactions and Neutrinos in Stellar 169. Evolution: Invited paper, IAU Syaposiua No. 66, Warsaw, Caaeron, A.G.W., and Canuto, V. 1973, Neutron Stars: General Poland, 5eptenber 10*12. Properties. 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