PHYSICAL REVIEW D VOLUME 49, NUMBER 8 15 APRIL 1994

BRIEF REPORTS

Brief Reports are accounts of completed research which do not warrant regular articles or the priority handling given to Rapid Communications; however, the same standards of scientific quality apply. (Addenda are included in Brief Reports.) A Brief Report may be no longer than four printed pages and must be accompanied by an abstract.

How large can the crust of a strange star be?

B. V. Martemyanov Institute of Theoretical and Ezperimental Physics, 11 7259, B. Cheremushkinskaya 25, Moscow, Russia

(Received 3 December 1992)

The problem of a crust of normal matter surrounding a strange star is considered. This crust is thinner by a factor 0.22-0.65 than the external crust of a neutron star.

PACS number(s): 97.60.Jd, 12.39.Ba, 21.65.+f

Strange quark matter (SQM) [I], whether or not it is absolutely stable, certainly becomes stable at high den- sities and pressures. These are the conditions that can be met inside compact astrophysical objects such as neu- tron stars (NS's). The conversion [2] of NS's to more stable strange stars (SS's) very well can start when the mass of NS1s becomes larger than some critical value [3]. The result of conversion can be different: the SS that is gravitationally unstable and converts further to a black hole [4], the SS where SQM forms the core surrounded by normal matter and is in chemical equilibrium with it [4], and the SS where the surrounding crust is not in chemical equilibrium with SQM but is supported by an electrostatic barrier of SQM [I]. What the result will be depends on the parameters used for the description of SQM: B, the bag constant; m,, the mass of the strange quark; and a,, the constant of quark interactions.

Here I shall explore the third possibility. In this case SQM is absolutely stable and energetically can absorb nuclei that are in contact with it. But the contact is prevented by the electrostatic barrier of SQM that acts on positively charged particles, nuclei. So, some part of the external crust (densities p < pdrip z 4 . 3 ~ 10" g/cm3) of the initial NS can survive in the process of conversion. How large this part can be is the subject of the following discussion.

Let us start with consideration of the bare surface of SQM [5]. SQM consists of u, d, s quarks and electrons that neutralize positive hadronic electric charge. Elec- trons do not feel confinement a t the boundary of SQM and contribute to the pressure of confined plasma due to electrostatic coupling to charged quarks. The distri- bution of electrons near the surface is governed by the relativistic Thomas-Fermi equation. If p is the electron chemical potential that changes from pi inside SQM to zero outside SQM, then it satisfies the equation

that is assumed to be positive inside SQM. Multiplying Eq. (1) by p' and integrating over x we can easily get

where PO is the electron chemical potential on the bound- ary (x = 0).

As can be seen from Eq. (2) PO = jpi. The left-hand side of Eq. (2) is the electric force per unit area acting on charged quarks near the surface of SQM. This force produces an additional pressure of quarks on the surface and the additional pressure is exactly equal to the pres- sure of relativistic electron gas inside SQM. In such a way electrons contribute to the pressure of the confined plasma.

Now let us consider the contact of SQM and nor- mal matter at a pressure below the neutron drip point (pdrip % 4.3 x 1011 g/cm3). Matter a t densities above pdrip cannot be in contact with SQM because of neutrons that are present in matter at these densities can be freely absorbed by SQM [6]. At p < pdrip matter consists of a crystal lattice of nuclei and degenerate electron gas [7]. The total pressure Pe (the index "e" means "external") is equal to

where pe is external electron chemical potential and PL is the pressure of crystal lattice (PL < 0). The data for PL were taken from the Baym-Pethick-Sutherland (BPS) equation of state (EOS) [7]. Equation (2) is now changed to

~i pe3 pi 4

s ( ~ -Po) + ( P o 3x - p e ) = ---- - -- Pe (4) 12x2 12x2 '

The additional pressure of quarks due to the electrostatic 37r2 37r2 coupling [the first term on the left-hand side (LHS) of

Eq. (4)] should be less or equal to ( ~ i ~ / l 2 7 r ' - Pe) in Here a is the fine structure constant, x is the coordinate order that the nuclei of crystal lattice have no contact (no

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FIG. 1. The difference of normal matter and SQM electron chemical potentials as a function of SQM electron chemical potential.

positive external pressure on SQM surface) with SQM surface. This requirement gives the inequality

So, the system of Eq. (4) and inequality (5) constrain the values of the external electron chemical potential LL, from above. I have solved this system for values of the electron chemical potential in SQM p, that lie in the range 5 MeV < p, < 15 MeV. This interval corresponds to the following range of MIT bag model parameters: O < a , < 0.6,70 MeV < m, < 170 MeV 151. The results are presented as a plot (ALL, p,) (see Fig. 1) where A p is defined by the formula

I t is seen that A/I << pt . This is the result of the inequality PL I < < pe"127r2: the pressure P, is de- fined mainly by the pressure of degenerate electron gas. The number of wiggles on Fig. 1 is due to cha~lges in "preferred nucleus" as a function of density for BPS EOS [ 7 ] . At the point of neutron drip pedrip z 26.1 MeV. The maximum of the electron chemical potential a t the bas? of the SS criist lies correspondingly in t,he range 5.1 MeV < p, < 15.3 MeV (densities 1.7 x 10' g/crn3 < p < 6.6 x lo1' g/cn13).

Now we can calculate the maximal thickness of SS crust. For this purpose. let us integrate the equation of stellar structure :8]

dP - Gmp (1 + 47rr"/m) (1 + Pip) - -- - --

dr r 2 (1 - 2Gmlr ) (7)

where P is the pressure, r the current radius. m the mass within the sphere of this radius, p the mass density, and G the gravitational constant.

FIG. 2. The ratio of SS and NS crust heights as a function of the SQM electron chemical potential (NS crust means here outer part of NS crust).

Because the crust does not contribute significantly to the total mass (M) and radius (R) of SS, we can approx- imate the calculations of SS crust thickness (hss) by the formula

where r, = 2MG/R, the terms 47rr3P/m and P l p are neglected.

I11 the case of NS p, = pedri, and the RHS of Eq. (8) is equal to lo-'. For a Pr'S with a mass 1.4Mo and radius 10 km this means that the thickness of its outer crust is about hNS = 300 m. In the case of a SS of the same (for example) mass and radius hss is less than hNS and the ratio h s s l h ~ s lies in the range 0.22-0.65 (see also Fig. 2). The ratio of crusts' masses (moments of inertia) is in t,his case approximately equal to the ratio of pressures a t t,he crusts' bases Pe(p , ) lP , (pedr ip) and lies in the range 0.9 x 10-"1. x 10-'.'

So, the main conclusion is that the crust of a SS is less thick than the outer crust of a NS. In this sense the

'TO imagine the crust of SS's we can draw here some analogy. The density of the crust and the density of SQM differs by 4-5 orders of magnitude. The height of the crust and the radius of SS's differs approximately by 2 orders of magnitude. Analogously, the density of the air and the density of the ground differs approximately by 4 orders of magnitude, the height of the atmosphere and the radius of the Earth differs approximately by 2 orders of magnitude. The analogy is no full: the crust of SS's is solid and SQM is gaseous (or liquid), reversely. the atmosphere of the Earth is gaseous and the ground of the Earth is solid.

49 - BRIEF REPORTS 4295

argument [9] against the SS nature for pulsars exhibiting thickness of its crust: a thinner crust is less insulating glitches is even stronger. The uncertainty in the exact and gives a faster cooling of the SS surface during the value of SS crust characteristics is due t o the uncertainty first few years. This fact can be important for future of electron chemical potential of SQM. I n Refs. [6,10] this observations of young compact objects. question was not discussed and the crust of the SS goes u p t o the pressure Pe(pedrip). AS was noted in Ref. [ll] the I highly appreciate discussion of the problem consid- early cooling history of SS's essentially depends on the ered with M. Krivoruchenko and 0. Benvenuto.

[l] E. Witten, Phys. Rev. D 30, 272 (1984). [2] A. Olinto, Phys. Lett. B 192, 71 (1987). [3] B. Martemyanov, ITEP report, September 1992 (unpub-

lished). [4] J . Madsen and M.L. Olesen, in Strange Quark Matter

in Physics and Astrophysics, Proceedings of the Inter- national Workshop, Aarhus, Denmark, 1991, edited by J . Madsen and P. Haensel [Nucl. Phys. B (Proc. Suppl.) 24B, 170 (1991)l.

[5] E. Farhi and R.L. Jaffe, Phys. Rev. D 32, 2452 (1985).

[6] C. Alcock, E. Farhi, and A. Olinto, Astrophys. J . 310, 261 (1986).

[7] G. Baym, C. Pethick, and P. Sutherland, Astrophys. J . 170, 299 (1971).

[8] J.R. Oppenheimer and G.M. Volkoff, Phys. Rev. 55, 374 (1939).

[9] M.A. Alpar, Phys. Rev. Lett. 58, 2152 (1987). [lo] N.K. Glendenning and F. Weber, Astrophys. J. 400, 647

(1992). [ll] P.M. Pizzochero, Phys. Rev. Lett. 66, 2425 (1991).