3
VOLUME 30, NUMBER 21 PHYSICAL REVIEW LETTERS 21 MAY 1973 on Heavy Ion Physics, Dubna, 1971 (to be published). ^P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. 28, 409 (1966). ^F. Schmittroth, W. Tobocman, and A. A. Golestaneh, Phys. Rev. CI,, 377 (1970). •^A. J. Baltz and S. Kahana, Phys. Rev. Lett. 29, 1267 (1972). ^M. C. Bertin, Y. Eisen, G. Goldring, S. L. Tabor, and B. A. Watson, Nucl. Phys. A167, 216 (1971). ^R. M. DeVries and K. I, Kubo, Phys. Rev. Lett. 3^, 325 (1973). i^B. F. Bayman, Phys. Rev. Lett. 25, 1768 (1970). ^^L. S. Kisslinger and R. A. Sorensen, Kgl. Dan, Vi~ densk. Selsk., Mat.-Fys. Medd.^, No. 9 (1960). Rapidly Rotating Neutron Stars'^ James R. Wilson Laiorence Livermore Laboratory, University of California, Livermore, California 94550 (Received 20 February 1973) Models of differentially rotating neutron stars have been calculated. An upper limit to the rotational enhancement of the maximum gravitational mass is found to be about 50% when the Ostriker-Tassoul instability criterion is used. The search for black holes depends, to a large extent, on finding compact stars whose masses exceed the accepted maximum mass for a neu- tron star. Hartle and Thorne^ have calculated the increase in mass to be expected in a rigidly rotating neutron star. The object of the present work is to relax the condition of uniform rotation and see if the mass of a neutron star can be in- creased significantly. In the present work, sequences of increasingly faster-rotating, axially symmetric equilibrium configurations are found for a given equation of state, using the fully relativistic equations. In order to terminate the sequences, the Ostriker- Tassoul^ stability criterion is used. The notation used here and the equations for the metric and hydrostatic equilibrium are taken from Bardeen and Wagoner^ and Wilson.^ The method of solution is different and is as follows. An estimate of density versus radius of a neutron star is assumed, and the field equations are solved for the metric. Next, the Z component of Eq. (5) from Ref. 4 is solved for density by in- tegrating out from the equator. The boundary condition on the integration is that the mass per unit coordinate volume on the equator is fixed. Then the R component of Eq. (5) is solved for the angular velocity. Using the new densities and velocities, the field equations are solved again for the metric. Then the hydrostatic equation is again solved for the new densities and velocities. This alternation back and forth between field equations and equilibrium equations is repeated until a convergence is achieved. Thus, a nearly spherical configuration is first established. To find more rapidly rotating models, the density profile in the equator is simply stretched out in the R direction in proportion to the radius, and the above iteration procedure is carried through again. Since the equations are solved by finite-differ- ence methods by a rather small number of grid points (about 25x25), coarse (15x15) and fine- zoned (25x25) problems were run on two of the configurations. From this, the error in mass is estimated to be probably less than 5%. The Tolman-Oppenheimer-Volkoff equation was in- tegrated in a fine grid, using the same equation of state. The mass of<the spherical stars, as calculated with the two-dimensional program, agreed within better than 5% with the accurate spherical calculation. Wilson's^ equation of state III was used except where indicated. This equation of state yields a maximum gravitational mass of l.QOM®, and so it is probably as hard an equation of state as is now popular. Star configurations with three dif- ferent central densities were calculated with the above equation of state. Due to the calculational procedure used, the central density decreases as the rotation increases in a sequence with in- creasing rotation. Figure 1 gives the star mass- es and central densities for the three calculations. Ostriker and TassouP found that rotating, grav- itationally bound systems have nonaxially sym- metric modes of instability, independent of the degree of differential rotation. The density dis- tributions for the examples with high central den- sity {B and C of Fig. 1) are relatively flat. The isodensity contour ^ of the way in from the out- side of the star indicates a density ^ that of the central density. We infer from this that the lin- 1082

Rapidly Rotating Neutron Stars

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V O L U M E 30, N U M B E R 21 P H Y S I C A L R E V I E W L E T T E R S 21 M A Y 1973

on Heavy Ion Phys i c s , Dubna, 1971 (to be published). ^P. J . A. Buttle and L . J . B . Goldfarb, Nucl. Phys .

2 8 , 409 (1966). ^F. Schmit troth, W. Tobocman, and A. A. Golestaneh,

P h y s . Rev. CI , , 377 (1970). •̂ A. J . Baltz and S. Kahana, P h y s . Rev. Lett . 29, 1267

(1972).

^M. C. Ber t in , Y. Eisen, G. Goldring, S. L. Tabor , and B . A. Watson, Nucl. Phys . A167, 216 (1971).

^R. M. DeVries and K. I, Kubo, Phys . Rev. Let t . 3 ^ , 325 (1973).

i^B. F . Bayman, Phys . Rev. Let t . 25 , 1768 (1970). ^^L. S. Kiss l inger and R. A. Sorensen, Kgl. Dan, Vi~

densk. Selsk. , M a t . - F y s . M e d d . ^ , No. 9 (1960).

Rapidly Rotating Neutron Stars'^

James R. Wilson Laiorence Livermore Laboratory, University of California, Livermore, California 94550

(Received 20 Feb rua ry 1973)

Models of differentially rotat ing neutron s t a r s have been calcula ted. An upper l imit to the rotat ional enhancement of the maximum gravi ta t ional m a s s is found to be about 50% when the O s t r i k e r - T a s s o u l instabi l i ty c r i t e r i on i s used .

The search for black holes depends, to a large extent, on finding compact s tars whose masses exceed the accepted maximum mass for a neu­tron star. Hartle and Thorne^ have calculated the increase in mass to be expected in a rigidly rotating neutron star. The object of the present work is to relax the condition of uniform rotation and see if the mass of a neutron star can be in­creased significantly.

In the present work, sequences of increasingly faster-rotating, axially symmetric equilibrium configurations are found for a given equation of state, using the fully relativistic equations. In order to terminate the sequences, the Ostriker-Tassoul^ stability criterion is used.

The notation used here and the equations for the metric and hydrostatic equilibrium are taken from Bardeen and Wagoner^ and Wilson.^ The method of solution is different and is as follows. An estimate of density versus radius of a neutron star is assumed, and the field equations are solved for the metric. Next, the Z component of Eq. (5) from Ref. 4 is solved for density by in­tegrating out from the equator. The boundary condition on the integration is that the mass per unit coordinate volume on the equator is fixed. Then the R component of Eq. (5) is solved for the angular velocity. Using the new densities and velocities, the field equations are solved again for the metric. Then the hydrostatic equation is again solved for the new densities and velocities. This alternation back and forth between field equations and equilibrium equations is repeated until a convergence is achieved. Thus, a nearly spherical configuration is first established. To find more rapidly rotating models, the density

profile in the equator is simply stretched out in the R direction in proportion to the radius, and the above iteration procedure is carried through again.

Since the equations are solved by finite-differ­ence methods by a rather small number of grid points (about 25x25), coarse (15x15) and fine-zoned (25x25) problems were run on two of the configurations. From this, the error in mass is estimated to be probably less than 5%. The Tolman-Oppenheimer-Volkoff equation was in­tegrated in a fine grid, using the same equation of state. The mass of<the spherical s tars , as calculated with the two-dimensional program, agreed within better than 5% with the accurate spherical calculation.

Wilson's^ equation of state III was used except where indicated. This equation of state yields a maximum gravitational mass of l.QOM®, and so it is probably as hard an equation of state as is now popular. Star configurations with three dif­ferent central densities were calculated with the above equation of state. Due to the calculational procedure used, the central density decreases as the rotation increases in a sequence with in­creasing rotation. Figure 1 gives the star mass­es and central densities for the three calculations.

Ostriker and TassouP found that rotating, grav-itationally bound systems have nonaxially sym­metric modes of instability, independent of the degree of differential rotation. The density dis­tributions for the examples with high central den­sity {B and C of Fig. 1) are relatively flat. The isodensity contour ^ of the way in from the out­side of the star indicates a density ^ that of the central density. We infer from this that the lin-

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VOLUME 30, NUMBER 21 PHYSICAL R E V I E W LETTERS 21 MAY 1973

:5

Q .

J / M (lO^^cmVsec)

FIG. 1. The solid curves are the masses of rotating stars in units of the solar mass as a function of the specific angular momentum for three different calcula­tions {A,B, C). The dashed curves give the central den­sities of the corresponding labeled mass curves.

ear trial function of Ostriker and Tassoul should be adequate. Instability arises when the ratio of kinetic to potential energy exceeds 0.14, This criterion is difficult to apply directly to the p res ­ent situation, since neither kinetic nor potential energies are well defined in a relativistic star. By examining several Newtonian, differentially rotating stars of a shape similar to the neutron stars considered in this paper, the average ratio of the equatorial to polar distances of equal-den­sity contours was found to lie between 1.7 and 1.8 whenever the kinetic- to potential-energy ratio was 0.14. (For the onset of Jacobi ellipsoids in the classical star of uniform angular velocity and density, the equatorial-to-polar radius ratio is 1.72.)

In our neutron star models, the ratio of proper distance to coordinate distances varies from pole to equator by less than 5% for the highest-density model. In the higher-density models, the central red shift is about 2, and this leads to different formulas for the kinetic- to potential-energy ratios that give results that are different by as much as a factor of two. Hence, we will use the shape of the star as the best guess for

FIG. 2. The mass of the rotating stars relative to the mass of spherical stars of the same central density versus the average ratio of equatorial to polar dis­tances of equal density contours. Note that instability is estimated to set in at i ^ / Z s 1.75. The dashed curve is for stars with the very hard equation of state. See Fig. 1 for the meaning of labels A, B, and C.

the onset of the nonaxial symmetric instability. In Fig. 2 we have plotted the ratio of the mass

of the rotating s tars to the mass of a spherical star of the same central density versus the ratio of equatorial radius to polar radius. Also in­cluded in this figure is a calculation done with a very hard equation of state that had a maximum spherical gravitational mass of S.SM^. As est i ­mated by Rhodes and Ruffini (to be published), S.SMQ is the maximum neutron star mass. This very hard equation of state deviates from equa­tion of state n i in Ref. 5 in that the "hard core' ' is multiplied by a factor of 10. It deviates from standard equations only for densities above 10"̂ ^̂ g/cm^ and has sound speed always less than light speed. It can be seen from this figure that the maximum rotational enhancement of the neutron star is around 1.5 to 1.7 and is only mildly de­pendent on the central density and equation of state. The isodensity contours of these models are elliptical in the interior and become oval toward the outside. The angular velocity is near­ly constant on lines of equal cylindrical-coordi-

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VOLUME 30, NUMBER 21 PHYSICAL REVIEW LETTERS 21 MAY 1973

nate radius. Hartle and Thorne^ found, for rigidly rotating

neutron stars , a maximum mass enhancement of 1.31. Their rotation was limited by equatorial shedding, rather than instability. Hence, the dif­ferential-rotation mass enhancement of 1.5 to 1.7 is only a slight increase over the rigid rota­tor, It may be noted, however, that the rotation­al enhancement found by Hartle and Thorne is strongly dependent on the equation of state and central density, while with the differential rota­tion the mass enhancement is a weak function of density and equation of state.

I would like to thank R. V. Wagoner, J. LeBlanc, M. Alme, and R. Ruffini for helpful discussions.

*Work performed under the auspices of the U. S. Atomic Energy Commission.

^J. B. Hartle and K. S. Thorne, Astrophys. J. 153, 807 (1968).

^J. P. Ostriker and J. L. Tassoul, Astrophys. J. 155, 987 (1969).

^J. M. Bardeen and R. V. Wagoner, Astrophys. J. 167, 359 (1971).

^J. R. Wilson, Astrophys. J. m , 195 (1972). ^J. R. Wilson, Astrophys. J. 1 ^ , 209 (1971).

Early Observation of Neutrino and Antineutrino Events at High Energies*

A. Benvenuti, D. Cheng, D. Cline, W. T. Ford, R. Imlay, T. Y. Ling, A. K. Mann, F . Messing, J . P i l c h e r , t D. D. Reeder , C. Rubbia, and L. Sulak

Department of Physics, Harvard University,% Cambridge, Massachusetts 02138, and Department of Physics, University of Pennsylvania, % Philadelphia, Pennsylvania 19104, and Department of Physics, University of

Wisconsin,% Madison, Wisconsin 53706 (Received 23 April 1973)

Presented here are preliminary results of two short runs with a broad-band neutrino-antineutrino beam at National Accelerator Laboratory incident on about 120 tons of tar­get which is part of a detector consisting of an ionization calorimeter and a muon mag­netic spectrometer. These results include (i) the observed distribution in transverse muon momentum, dN/dp^, (ii) the average neutrino cross section at a mean neutrino en­ergy of roughly 30 GeV, and (iii) the ratio of the antineutrino to the neutrino total cross section at a mean neutrino energy of 40 GeV.

The energy (300 GeV) and intensity (-10^^ pro­tons per pulse) of the external proton beam now available at the National Accelerator Laboratory (NAL) are sufficiently large to permit the obser­vation of a substantial number of high-energy neutrino interactions, even without any focusing of the secondary pions and kaons which, through their decays, produce the neutrino beam. In order to provide an early description, albeit crude, of neutrino interactions in this new ener­gy region, we report here the preliminary r e ­sults of two short runs (approximately 10^^ inter­acting protons on target) with such a broad-band neutrino-antineutrino beam incident on about 120 tons of target-detector.

The experimental arrangement is shown sche­matically in Fig. 1(a). The hadron-producing target (a collision length of iron) on which the extracted proton beam impinges is about 1400 m from the accelerator. The secondary hadrons travel unvexed through a drift region 350 m in length and 1 m in diameter. The drift region is in turn followed by a muon shield, primarily of

earth, about 1000 m long, at the end of which our neutrino detection apparatus is located.

The main outlines of the target-detector are sketched in Fig. 1(b). The neutrinos are incident on an ionization calorimeter (IC) consisting of four main sections in series along the beam axis, each main section of cross-sectional area 3x3 m^ and length along the beam of 1.8 m, and con­taining about 15 metric tons of mineral-oil-based liquid scintillator. Each main section is divided into four optically separated subsections viewed from two sides, as indicated in Fig. 1(b), by twelve photomultipliers (5-in. diam). There are wide-gap optical spark chambers of area 3x3 m^ after each main section of the ionization calori­meter as shown in Fig. 1(b). Immediately down­stream of the IC is a magnetic spectrometer made up of four units of toroidal iron magnets, one behind the other along the beam line, with narrow-gap optical spark chambers following each magnet unit. The toroids have an inside diameter of 0.3 m, an outside diameter of 3.6 m, and are 1.2 m long; they are driven into satura-

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