On Surface Tension for Compact Stars

Transcription

1 J. Astophys. Ast. (2007) 28, On Suface Tension fo Compact Stas R. Shama & S. D. Mahaaj Astophysics and Cosmology Reseach Unit, School of Mathematical Sciences, Univesity of KwaZulu Natal, Pivate Bag X54001, Duban 4000, South Afica. Received 2007 Febuay 22; accepted 2007 May 15 Abstact. In an ealie analysis it was demonstated that geneal elativity gives highe values of suface tension in stange stas with quak matte than neuton stas.we geneate the modified Tolman Oppenheime Volkoff equation to incopoate anisotopic matte and use this to show that pessue anisotopy povides fo a wide ange of behaviou in the suface tension than is the case with isotopic pessues. In paticula, it is possible that anisotopy dastically deceases the value of the suface tension. Key wods. Relativity pulsas equation of state. 1. Intoduction Stas that ae moe compact than neuton stas, at pesent, have become a subject of consideable inteest as they povide us natual laboatoies fo testing QCD. Ove the last couple of decades, vaious models have been poposed to explain the compactness and popeties of some of the obseved compact objects. Pioneeing woks in this field have put fowad new concepts of compact matte, namely stange stas (Witten 1984; Fahi & Jaffe 1984) and boson stas (Kaup 1968; Ruffini & Bonazzola 1969; Colpi et al. 1986). Due to the high matte densities within such stas one expects pessue to be anisotopic in geneal, i.e., in the inteio of such stas the adial pessue and tangential pessue ae diffeent. An anisotopic enegy momentum is a topic which is often ignoed in the calculations of compact stas. Howeve, since the pioneeing wok of Bowes & Liang (1974) thee has been extensive eseach in the study of anisotopic elativistic matte in geneal elativity. The analysis of static spheically symmetic anisotopic fluid sphees is impotant in elativistic astophysics. Rudeman (1972) showed that nuclea matte may be anisotopic in the high density anges of ode gm cm 3 whee nuclea inteactions have to be teated elativistically. Anisotopy in compact objects may occu due to the existence of a solid coe o the pesence of type 3A supefluid (Kippenhahm & Weiget 1990), phase tansition (Sokolov 1980), pion condensation (Sawye 1972), slow otation (Heea & Santos 1997), mixtue of two gases (Letelie 1980) o stong magnetic fields (Webe 1999). Also objects made up of self-inteacting scala paticles known as boson stas ae natually anisotopic in thei configuations. Anisotopic models fo compact self gavitating objects have been studied by Heea & Santos (1997); Rao et al. (2000); Cocheo (2001); Mak & Hako 133

2 134 R. Shama & S. D. Mahaaj (2003); Ivanov (2002); Dev & Gleise (2003); Henández & Núñez (2004); Chaisi & Mahaaj (2005), and many othes. Anisotopic models fo compact objects have been shown to achieve high ed-shift values (Bowes & Liang 1974; Heea & Santos 1997; Ivanov 2002; Mak & Hako 2003), and they ae stable (Heea & Santos 1997; Dev & Gleise 2003). In this aticle, we show that pessue anisotopy may also affect the suface tension of compact stas. We believe that this aspect has not been consideed yet in the context of anisotopic stella models. 2. Suface tension of stange stas In a ecent pape by Bagchi et al. (2005), it has been shown that objects composed of u, d and s quaks populaly known as stange stas give highe values of suface tension than neuton stas, a necessay citeion fo the existence of stable stange stas in the Univese. This calculation is based on equations of state (EOS) fo stange matte fomulated by Dey et al. (1998). In an appoximated lineaized fom, the EOS may be witten as (Zdunik 2000; Gondek-Rosińska et al. 2000) p = a(ρ ρ b ), (1) whee ρ is the enegy density, ρ b is the density at the suface, p is the isotopic pessue, and a is a paamete elated to the velocity of sound (a = dp/dρ). To calculate the suface tension, one assumes that the sta is a huge spheical ball composed of stange matte which is self-bound and non-otating. The excess pessue on the suface of the sta can be expessed as p =R = 2S R, (2) whee S is the suface tension of the sta and R is the adius of cuvatue. At the suface dp p =R = n d =R, (3) whee n is the adius of the quak paticle given by n = (1/πn) 1/3 whee n is the bayon numbe density. As stange stas ae vey compact, a elativistic teatment is necessay to find thei configuations and othe physical paametes. Thus fo a given EOS, one uses the Tolman Oppenheime Volkoff (TOV) equation (Oppenheime & Volkoff 1939) dp G(ρ + p) d = ( c 2 [ m() c 2 1 2Gm() ] + 4π2 p c 4 ) (4) to find the suface tension of the sta, making use of equations (2) and (3). This method helps to yield highe values of suface tension as compaed to neuton stas including the possible explanation fo the existence of stange stas in the Univese and othe elated phenomena like delayed γ -ay busts (Bagchi et al. 2005). Howeve, at vey high densities, anisotopy may be significant in such stas which may contibute to the suface tension. If we assume that pessue within such a sta is anisotopic in geneal then the TOV equation (4) gets modified yielding diffeent esults as obtained by Bagchi et al. (2005). In the following sections, we deive the modified TOV equation with anisotopic pessue and pefom some numeical calculations to show the effects of pessue anisotopy on the suface tension of compact stas.

3 On Suface Tension fo Compact Stas Anisotopic TOV equation We fist fomulate the modified TOV equation with anisotopic pessue. We assume the line element fo a static spheical object in the standad fom ds 2 = e γ() c 2 dt 2 + e µ() d (dθ 2 + sin 2 θdφ 2 ), (5) whee γ() and µ() ae the two unknown metic functions. Without any loss of geneality, the enegy momentum tenso fo an anisotopic sta may be witten as T ij = (ρc 2 + p )u i u j + p g ij + (p p )n i n j, (6) whee u i is the fluid fou-velocity, n i is a adially diected unit space-like vecto. We assume that p p and p p = gives the measue of pessue anisotopy in this model. The Einstein s field equations ae then given by ( 1 e µ ) 8πG ρ = + µ e µ c 4 2 8πG p c 4 = γ ( e ) µ 1 e µ 8πG p c 4 = (2γ e µ + γ 2 γ µ + 2γ 4, (7), (8) 2 ), (9) 2µ whee pimes denote diffeentiation with espect to the adial coodinate. Equations (7) (9) may be combined togethe to yield (ρ + p )γ + 2p + 4 (p p ) = 0 (10) which is a consevation equation. If we wite the metic function µ in tems of mass function m() as e µ = 1 Gm() c 2 then equation (10) becomes dp d = (ρ + p ) ( ) Gm() + 4πG2 p c 2 c 4 ( 1 2Gm() c 2 (11) ) + 2 (p p ). (12) Equation (12) is the the modified TOV equation in the pesence of pessue anisotopy. Fo a given cental density ρ c o cental pessue p c and anisotopic paamete, equation (12) may be integated to find the mass M = m(r) and adius R of the sta povided the EOS p = p (ρ) is known. Local anisotopy thus effects the geomety of the sta. At the suface of the sta = R, the adial pessue p vanishes. Howeve, the tangential pessue p is not necessaily zeo at the suface. The two pessue pofiles within the sta should satisfy the following conditions: p > 0 and p > 0. The

4 136 R. Shama & S. D. Mahaaj maximum value of the anisotopic paamete vis-a-vis the tangential pessue p is constained by the physical equiement that the adial pessue gadient dp /d should be negative in the stella inteio; othe physical equiements may, howeve, put a moe stingent estiction on the values of. Thus fo finite values of p at the bounday ( = R) = p b, equation (12) becomes dp d = =R ρ b GM c 2 R R ( 1 2GM c 2 R ) + 2pb R. (13) If p b is not negligible at the bounday, equation (13) shows that it is possible to get diffeent sets of values of suface tension as obtained by Bagchi et al. (2005) fo isotopic matte. Thus it is possible to geneate a wide ange of behaviou in the suface tension fo anisotopic matte than is the case fo isotopic pessues. 4. Numeical esults To get an estimate of the effects of pessue anisotopy on the suface tension, we conside the stange matte EOS given by equation (1). We conside two paticula cases as discussed by Gondek-Rosińska et al. (2000): EOS SS1: whee, a = 0.463, ρ b = gm cm 3, ρ c = gm cm 3, n( = R) = fm 3, n( = 0) = 2.35 fm 3, M = M, R = 7.07 km. EOS SS2: whee, a = 0.455, ρ b = gm cm 3, ρ c = gm cm 3, n( = R) = fm 3, n( = 0) = fm 3, M = M, and R = 6.55 km. Numeical calculations show that fo a given mass and adius, if we gadually intoduce anisotopy, the absolute value of the suface tension deceases as can be seen in Fig. 1. Fo example, it is obseved that even if we conside a tangential pessue of 100 MeV fm 3 at the suface, the suface tension deceases dastically. It is to be noted hee that the anisotopy paamete should be so chosen that all the egulaity conditions (Delgaty & Lake 1998) ae satisfied. Thus, although in Fig. 1, the suface tension inceases beyond a cetain value of the anisotopic paamete, we ignoe this egion as the adial pessue gadient becomes positive in this egion. The esults ae given in Table 1. Figue 1. Suface tension S plotted against. The solid line is fo EOS SS1 and the dotted line is fo EOS SS2.

5 On Suface Tension fo Compact Stas 137 Table 1. Anisotopic effect on the suface tension of stange stas. dp d =R (MeV fm 3 km 1 ) S (MeV fm 2 ) EOS n (fm) p = 0 p = 100 (MeV fm 3 ) p = 0 p = 100 (MeV fm 3 ) SS SS Discussions We have shown that anisotopy plays an impotant ole in the calculation of suface tension of compact stas. The oigin of such anisotopies within compact objects may be diffeent fo diffeent objects. We may, howeve, ask whethe it is necessay at all to conside anisotopic effects on the suface tension of stange stas. The answe is in the affimative since one possibility fo the oigin of anisotopies within stange stas could be the pesence of chaged paticles at the suface. It has ecently been epoted that in stange stas, the electic field could be as high as ev/cm (Usov 2004), which indicates the possibility of a lage chage distibution within such objects. Theefoe we need to conside the effect of chage while deiving the the goss featues of such stas. It can be shown that in the pesence of chage, the TOV equation is modified to dp d = (ρ + p) ( ) Gm() + 4πG2 p c 2 c 4 ( 1 2Gm() c 2 ) + Q() dq(), (14) 4π 4 d whee, Q() is the total chage confined within a sphee of adius. Note that the Einstein Maxwell system is always anisotopic which is often teated as an isotopic system of field equations fo mathematical simplicity (see fo example, Ray et al. 2004). Also ecent woks (Schmitt 2005) suggest that a natual mechanism to explain the stong pulsa kicks in neuton stas could be the existence of asymmetic phases in quak matte. It is to be noted hee that, fo simplicity, we ignoed the effect of otation in the pesent wok although pulsas ae magnetized otatos and a stong magnetic field ( G) is obseved at the suface of such stas. Pulsas known as magnetas may even have a magnetic field as stong as G. Though we do not have an established theoy fo the micoscopic oigin of such a stong magnetic field, it is ageed that Feo-magnetization may occu in the high density quak matte which, in tun, may modify the EOS fo stange matte. The deivation and the fom of the modified EOS in the pesence of stong magnetic field o supefluidity (esponsible fo anisotopy) is a complex issue and a moe detailed analysis is equied to see the effect of the modified EOS on the oveall configuation vis-a-vis suface tension of compact stas. To conclude, without going into the micoscopic details of a sta, it can be shown that suface tension is affected in the pesence of anisotopy. Fo the vey existence of stange stas in ou Univese a cucial condition put fowad was a lage value of S by Alcock & Olinto (1989) which accoding to Bagchi et al. (2005) can be achieved by a geneal elativistic teatment of stange stas. Howeve, in this aticle we have shown that a wide ange of values of S ae possible if we conside anisotopy in the enegy momentum tenso; an issue ignoed in the pevious calculation (Bagchi et al. 2005).

6 138 R. Shama & S. D. Mahaaj Theefoe, on the basis of suface tension fo compact stas, no conclusive emaks at this moment can pehaps be made on the possible existence of stange stas. Thee could be, howeve, some othe means to justify the existence of such stas which will be taken up elsewhee. Acknowledgements RS acknowledges the financial suppot (gant no. SFP ) fom the National Reseach Foundation (NRF), South Afica. SDM acknowledges that this wok is based upon eseach suppoted by the South Afican Reseach Chai Initiative of the Depatment of Science and Technology and the National Reseach Foundation. Refeences Alcock, C., Olinto, A. 1989, Phys. Rev. D, 39, Bagchi, M., Sinha, M., Dey, M., Dey, J., Bhowmick, S. 2005, Aston. & Astophys., 440, L33. Bowes, R. L., Liang, E. P. T. 1974, J. Astophys., 188, 657. Chaisi, M., Mahaaj, S. D. 2005, Gen. Relat. Gav., 37, Colpi, M., Shapio, S. L., Wasseman, I. 1986, Phys. Rev. Lett., 57, Cocheo, E. S. 2001, Astophysics and Space Science, 275, 259. Delgaty, M. S. R., Lake, K. 1998, Comput. Phys. Commun., 115, 395. Dev, K., Gleise, M. 2003, Gen. Relat. Gav., 35, Dey, M., Bombaci, I., Dey, J., Ray, S., Samanta, B. C. 1998, Phys. Lett. B, 438, 123. Fahi, E., Jaffe, R. L. 1984, Phys. Rev. D, 30, Gondek-Rosińska, D., Bulik, T., Zdunik, J. L., Gougoulhon, E., Ray, S., Dey, J., Dey, M. 2000, Aston. & Astophys., 363, Henández, H., Núñez, L. A. 2004, Can. J. Phys., 82, 29. Heea, L., Santos, N. O. 1997, Phys. Rep., 286, 53. Ivanov, B. V. 2002, Phys. Rev. D, 65, Kaup, D. J. 1968, Phys. Rev., 172, Kippenhahm, R., Weiget, A. 1990, Stella Stuctue and Evolution, Spinge, Belin. Letelie, P. 1980, Phys. Rev. D, 22, 807. Mak, M. K., Hako, T. 2003, Poc. Roy. Soc. London A, 459, 393. Oppenheime, J. R., Volkoff, G. M. 1939, Phys. Rev., 55, 374. Rao, J. K., Annapuna, M., Tivedi, M. M. 2000, Pamana J. Phys., 54, 215. Ray, S., Malheio, M., Lemos, J. P. S., Zanchin, V. T. 2004, Baz. J. Phys., 34, 310. Rudeman, R. 1972, Ann. Rev. Aston. Astophys., 10, 427. Ruffini, R., Bonazzola, S. 1969, Phys. Rev., 187, Sawye, R. F. 1972, Phys. Rev. Lett., 29, 382. Schmitt, A., Shovkovy, I. A., Wang, Q. 2005, Phys. Rev. Lett., 94, Sokolov, A. I. 1980, JETP, 79, Usov, V. V. 2004, Phys. Rev. D, 70, Webe, F. 1999, Pulsas as astophysical obsevatoies fo nuclea and paticle physics, Institute of Physics, Bistol. Witten, E. 1984, Phys. Rev. D, 30, 272. Zdunik, J. L. 2000, Aston. & Astophys., 359, 311.