Volume 147, number 1 PHYSICS LETTERS A 25 June 1990

The problem of the maximum mass of neutron stars

Alfredo B. Henriques CFMC/INIC, Av. Prof. Gama Pinto 2, 1699 Lisbon-Codex, Portugal

Received 7 March 1990; accepted for publication 25 April 1990 Communicated by J.P. Vigier

Using an equation of state for the nuclear matter parameterized by the nucleon number density n and applying a method developed by Rhoades and Ruffini, we calculate the maximum mass of a neutron star to be no larger than four solar masses. The meaning and reliability of this result are discussed.

It is known that the best method to find whether wc are in the presence of a black hole or of a neutron star is still based on the mass of these objects, as it seems to bc extremely difficult to distinguish be- tween these two possibilities on the sole basis of the properties of the emitted electromagnetic radiation [ 1 ]. Moreover, the existence of compact objects in binary systems affords a method of determining their masses with reasonable accuracy [2].

The problem wc wish to address in this paper is the following one: above which mass are we sure that wc are in the presence of a black hole and not in the presence of a very massive neutron star? In other words, is there an absolute limit to the maximum mass of a neutron star? Unfortunately the range of values for the maximum mass of neutron stars varies considerably, spanning an interval from 0.7Mo up to 2.5Mo, essentially duc to the uncertainty in our knowledge of the equation of state of nuclear matter at nuclear and above nuclear densities. How can wc then bc sure that an equation of state is not found giving a maximum mass considerably higher than the previously quoted values? Such questions and an at- tempt to find a solution to them wcrc addressed for the first time, to the best of our knowledge, by Rhoades and Ruffini [ 3 ] who tried to establish an absolute upper limit, independent of the detailed characteristics assumed for the neutron matter, us- ing only the following general assumptions: (a) gen- eral relativity and the general relativistic equation for hydrostatic equilibrium, (b) causality (speed of

sound smaUcr or equal to the speed of light), which implies dpldp~< 1 and, finally, (c) no local sponta- neous collapse of matter, dp/dp>O (sec also ref. [4]) .

Their method is based on an cxtrcmization tech- nique, allowing a path in the (p, p) plane to be found which maximizes the mass of the star, the path sat- isfying dp /dp= 1. In a practical application we have to assume knowledge of the equation of state up to a fiducial density Pl which is usually taken to be of the order of the nuclear matter density; Rhoades and Ruffini used the Harrison-Wheeler equation of state [ 5] up to Pl = 4.6 × 1014 g/cm 3, and found an upper limit of 3.2Mo to the maximum mass of a neutron star.

What wc here propose to do is much more modest; it is to investigate the influence that the equation of state, to bc used up to the density Pl, may have on the value obtained for the maximum mass. More- over, wc still have some freedom in the choice of the fiducial density pl, consequently we also study its in- fluence on the final result which, as we shall see, is not negligible. Instead of using a specific equation of state, we shall construct a general equation of state parameterized in terms of n, the nucleon number density, with the coefficients appearing in the equa- tion fixed once we give the values of (¢ /n )o= ( p i n ) o - r n c 2, the binding energy per nucleon at sat- uration density no, and the compression modulus Ko. The nucleon number density at saturation is taken to be no ~ 0.16 fm-3. Wc first describe our equation

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Volume 147, number I PHYSICS LETTERS A 25 June 1990

of state and then comment on our choices for the in- put quantities, (e/n)o and Ko.

Let us assume that we have n particles (nucleons) interacting among themselves; then, the total energy density will get contributions, not only from the rest mass density nmc 2 and the Fermi energy 8n 5/3, but also from the two-body, three-body, etc. interactions among these particles. Such contributions will be proportional to the combinations nC2, nC3, etc.; these correlation terms will give rise to terms proportional to r/2, /7 3, ..., lower order powers being absorbed, in each case, in the lower order terms of the series. This means that the rest mass term itself will have to be redefined, with the mass m replaced by an effective mass m'. Adding the terms, we find the following expression for the energy density,

p=m' c2n+~nS/3+an2+bn3+ .... (1)

where a, b .... are constants. This expansion in fact has some theoretical justification and can be found in models based on the phenomenological Skyrme interaction [6-9 ].

Using as experimental input the values of (e/n)o and Ko at saturation, we shall have to limit our series to the terms quoted in ( 1 ). Knowing that no ~ 0.16 fm -3 and p o ~ 2 × 1014 g / c m 3, o n c e (~/n)o is given we only have a limited freedom in the choice of m' , values in the interval between 0.8 and 1.1 GeV being acceptable. To be definite we shall keep m' equal to the nucleon mass m = 0.94 GeV, although other pos- sibilities could be explored. Now a few words on (~/n)o and Ko.

The binding energy per nucleon (¢/n)o is known to have the value - 16 MeV at saturation density no and for symmetric nuclear matter. For the case of neutron-rich matter a symmetry coefficient of ~ 30 MeV seems to be favored in studies of nuclear sys- tematics [10]; we take our values for (E/n)o from Baym, Bethe and Pethick's paper (ref. [11 ], eq. (3.19) and fig. 1 ). As a typical sample we quote 18 MeV for a negligible proton concentration, 13 MeV corresponding to a proton concentration of the order of 5% and 7 MeV for a concentration of ~ 10%.

With pressure defined by

p=n2d(p /n ) /dn , (2)

the compressibility and the compression modulus are given by

x=ndp/dn, K=9dp/dn=9/n,z. (3)

We use ordinary derivatives because we are studying equilibrium configurations of matter at zero tem- perature and entropy S= 0. We are now dealing with nonsymmetric nuclear matter and, therefore, we cannot take po=0 or (~/n)o to be a minimum; we shall nevertheless assume that X and K are indepen- dent of x=Z/A and vary the value of the compres- sion modulus at saturation density, Ko, between 120 MeV, appropriate to soft nuclear matter, and 420 MeV, for very stiff matter. We have thus a suffi- ciently large range of values for (~/n)o and Ko and once these are given the constants a and b can be found (a comes out negative, as befits an attractive two-body interaction term).

Eq. ( 1 ) becomes supefluminal at a value of n = n~ for which cs=c, with the speed of sound given by

(cjc)2 - dp/dn (4) dp/ dn "

To keep it subluminal, a modification in the form of an extra cut-off factor could be used, as seen in applications of the Skyrme interaction [6,9 ]; this cut- off factor also helps in bringing down the large Ko that results from ( 1 ) when we take Po = 0 at satu- ration [6]. Being interested in the application of Rhoades and Ruffini results, we keep the form of eq. ( 1 ) and match it to the maximized equation obeying dp/dp = 1, which can be parameterized by

p=AnZ+B. (5)

The constants A and B are obtained by matching p and p, as given by ( 1 ) and (5), at the point where cs=c or at any lower point (cs<c). We shall gener- ically denote the matching point by n~ or by t~, where t=n/no, and use different possibilities for tl like tl = 1, corresponding to nl = no and an energy density of the order of the nuclear matter density, and t~ = to, when we take the matching point to be cs= c. A value t~ = 1.5 corresponds roughly to p~ .~4× 1014 g/cm 3, close to the fiducial point used by Rhoades and Ruffini.

To solve our problem we use the Schwarzschild metric

cls2=-B(r) dt2 +A(r) dr2+dt02, (6)

and the energy-momentum tensor for a perfect fluid,

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Volume 147, number 1 PHYSICS LETTERS A 25 June 1990

Tu~ = (P + P )UuUv + Pgu~ , (7 )

where uu is the fluid velocity four-vector defined so that gUVuuuv= - 1.

The problem can now be formulated in terms of two first-order differential equations [ 12,13 ], the general-relativity equation of hydrostatic equilibri- um,

GM(r ) p' = r2 p ( r )

P + 4~pr 3 . - 1 ×(, + (, (, and the equation for the mass of the star,

M' (r) =4gr2M(r ) , (9)

with the initial condition M ( 0 ) = 0 . The total mass of the star is then M = M ( r = R ), R the radius of the star, defined as the point where p ( r ) drops to zero (here an accent denotes d /dr ) .

The remaining equation is the equation of state given by expressions ( 1 ) and (5), allowing us to ex- press the pressure p and the energy density p in terms of n. It is more convenient to use the variable t= n/no and reexpress eq. (8) in terms of r , with the help of dp /dr= ( d p / d t ) ( d t / d r ) . The other initial condition is then the value of the central nucleon density t (r = 0) = n (r = 0) / no. Our input parameters will thus be (~/n)o, Ko (determining the coefficients a and b, as well as A and B, of the equation of state) and t ( r = 0 ) . The integration starts at the center of the star, with eq. (5); when t becomes smaller than a previously specified value h, the integration switches to eq. ( 1 ) and then proceeds till we reach the surface, where p = 0.

To test the equation of state, we begin with a cal- culation of the maximum mass of a neutron star for the case where the matching point of eqs. ( l ) and (5) is taken to be the point where c= c,, correspond- ing approximately to a density of the order of 10 times Po, a high density. This means that throughout most of the star the neutron matter will be described by eq. ( 1 ). We take (¢ /n )o= 18 MeV, K0=220 MeV and find that c~=c at a density p = 2 . 9 × 1015 g /cm 3. The maximum mass is obtained for a central nu- cleon number density n = 10no, p ( r = 0 ) = 3 . 9 X 1015 g /cm 3, and is equal to 1.82Mo ( M o = 2 X l033 g),

with a radius R=8 .59 km; the part for which t> q, described by eq. (5), has a mass of ~0 .25Mo and is comprised within a radius of 3.3 kin. Approxi- mately 70% of the mass of the star lies at densities n>4.5n0. These results seem very reasonable and compare well with those of ref. [ 9 ], although in our case the star is slightly more compact, R = 8.59 km instead of R>9 .3 km (table II of ref. [9], case F ( u ) =u~/2). Varying the input variables (¢/n)o and Ko we find masses for the neutron star between 1.46Mo (for (~ /n)o= 18 MeV, K0= 120 MeV) and 2.45Mo (for (~ /n )o=7 MeV, Ko=420 MeV).

We now go over to the other extreme and fix tl = 1, nl = no. In this case only a small proportion of the total matter in the star ( ~ 6%) is described by eq. ( l ) , making the results much less sensitive to the values of (¢ /n)o and Ko. Indeed we find a maximum mass varying only between 4.06M o and 4.09Mo, for (E/n)o=18 MeV, Ko=120 MeV and (¢ /n )o=7 MeV, Ko=420 MeV, respectively. The central den- sity corresponding to these configurations is typi- cally of the order of 8.2 × 1014 g /cm 3 and the radius ~17 km.

We also considered an intermediate situation with t1=1.5 for which we obtained a density p (tl) ~ 4.1 X 1014 g / c m 3, close to the fiducial density assumed by Rhoades and Ruffini [ 3 ]. We got a max- imum mass of ,~3.3Mo ( R ~ 14.0 kin), in agree- ment with their results.

The results for the maximum mass of the neutron star are thus seen to be quite sensitive to the match- ing point assumed in the calculation, the uncertainty in the maximum mass of the neutron star being of the order of 1.6 solar masses. From our calculations we conclude that if a compact body is found with a mass < (2-2 .5)Mo it can well be a neutron star, such masses being accounted for by known equations of state; if the mass lies between 2.5Mo and 4.1Mo, it is probably a black hole, although it is in this range that the uncertainties are specially annoying. On the other hand, if the mass of the object is bigger than 4.1 solar masses, then we are almost certainly in the presence of a black hole. Our confidence in this re- sult comes not only from the fact that the matching point was already taken at a value of the density as low as the normal nuclear matter density, but also from the approximate independence of the maxi-

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Volume 147, number I PHYSICS LETTERS A 25 June 1990

m u m mass wi th respect to va r i a t i ons o f (~/n)o a n d

Ko. In ou r ca lcula t ions we d id no t take in to accoun t

the in f luence o f ro ta t ion on the m a x i m u m mass o f the star; this will lead to an increase o f the m a x i m u m mass, d e p e n d i n g on the per iod. Fo r a rapid ly rota t - ing n e u t r o n star ( P < 0. I m s ) the mass could change by a factor as large as 1.5, bu t m u c h less so for smal ler frequencies, as 8 M / M is p ropor t iona l to o9 2 [3 ]. Also

no t d iscussed is the compa t ib i l i t y o f ou r e q u a t i o n o f state wi th very large f requencies , a l though a na ive app l i ca t ion o f the a p p r o x i m a t e fo rmu l a for the ter- m i n a t i o n po in t g iven by F r i e d m a n , Ipser a n d Parker [14] ,

• -s[ M / M o ,,/2 o 9 = 2 . 3 × l o ~,(R/1 km)3,] s - ' , ( l O )

indica tes the m a x i m u m mass n e u t r o n star o f 4 . 1 M o to be compa t ib l e wi th a pe r iod o f ~ 1 ms #2

I t hank Lid ia Fer re i ra for helpful discussions.

~ Large pulsar frequencies are also discussed in ref. [ 15 ] in terms of a quark-matter equation of state with some similarities to eq. ( 5 ) - see ref. [15],p. 2631.

R e f e r e n c e s

[ 1 ] D. Christodoulou and R. Ruffini, in: Black holes, eds. B. De Witt and C. De Witt (Gordon and Breach, New York, 1973).

[2 ] S.L. Shapiro and S.A. Teukolsky, Black holes, white dwarfs and neutron stars, the physics of compact objects (Wiley, New York, 1983 ).

[3] C.E. Rhoades and R. Ruffini, Phys. Rev. Lett. 32 (1974) 324.

[4] M. Nauenberg and G. Chapline, Astrophys. J. 179 (1973) 277.

[5 ] B.K. Harrison, K.S. Thorne, M. Wakano and J.A. Wheeler, Gravitation theory and gravitational collapse (Univ. Chicago Press, Chicago, 1965 ).

[6] S,A. Bludman and C.B. Dover, Phys. Rev. D 22 (1980) 1333.

[7] H.A. Bethe, G.E. Brown, J. Applegate and J.M. Lattimer, Nucl. Phys. A 324 (1979) 487.

[8] D. Vautherin and D.M. Brink, Phys. Rev. C 5 (1972) 626. [9] M. Prakash, T.L. Ainsworth and J.M. Lattimer, Phys. Rev.

Lett. 61 (1988) 2518. [ 10] H.A. Bethe, Annu. Rev. Nucl. Part. Sci. 38 (1988) 1. [ 11 ] G. Baym, H.A. Bethe and C.J. Pethick, Nucl. Phys. A 175

( 1971 ) 225. [ 12 ] S. Weinberg Gravitation and cosmology (Wiley, New York,

1972) ch. 11. [ 13 ] J.R. Oppenheimer and G.M. Volkoff, Phys. Rev. 55 (1939)

374. [ 14] J.L. Friedman, J.A. Ipser and L. Parker, Phys. Rev. Lett. 62

(1989) 3015. [ 15 ] N.K. Glendenning, Phys. Rev. Lett. 63 ( 1989 ) 2629.

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