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Microscopic Interpretation of Neutron Star Structure

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Microscopic Interpretation of Neutron Star Structure ORD 257 Iftwi' *aiP' (^4x6 m s NOTICE March 1976 PORTIONS OF THIS REPORT A?.Z JLLZG1HLE. It has boen reproduced from th b best csvailcbl© copy to pormit tho broadest possible avail­ ability, ... T hu tepoJt wit piepwed a* an account o iponh'-ed by the United State* Government »he United S tafe i not I he United Stale* rn ftewajch and Development Adminutration, r their employee*, not *ny of iheu n icbcontractort. of iheu employee*, warranty, expteu of implied, o» ittumet i la h ^ jly ot >e*pormbd»y tor «he accuracy, coir or LKfulne* of any information. apparatua, p proceM ducloted, ot tepreienu th*t iti w * * infringe ptm tely owned ttfht*. 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.
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