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Asteroseismology with ET Asteroseismology with ET Erich Gaertig & Kostas D. Kokkotas Theoretical Astrophysics - University of Tübingen nice, et-wp4 meeting,september 1st-2nd, 2010 Asteroseismology with gravitational waves as a tool to probe the inner structure of NS • neutron stars build from single-fluid, normal matter Andersson, Kokkotas (1996, 1998) Benhar, Berti, Ferrari(1999) Benhar, Ferrari, Gualtieri(2004) Asteroseismology with gravitational waves as a tool to probe the inner structure of NS • neutron stars build from single-fluid, normal matter • strange stars composed of deconfined quarks NONRADIAL OSCILLATIONS OF QUARK STARS PHYSICAL REVIEW D 68, 024019 ͑2003͒ NONRADIAL OSCILLATIONS OF QUARK STARS PHYSICAL REVIEW D 68, 024019 ͑2003͒ H. SOTANI AND T. HARADA PHYSICAL REVIEW D 68, 024019 ͑2003͒ FIG. 5. The horizontal axis is the bag constant B (MeV/fmϪ3) Sotani, Harada (2004) FIG. 4. Complex frequencies of the lowest wII mode for each and the vertical axis is the f mode frequency Re(␻). The squares, stellar model except for the plot for Bϭ471.3 MeV fmϪ3, for circles, and triangles correspond to stellar models whose radiation which the second wII mode is plotted. The labels in this figure radii are fixed at 3.8, 6.0, and 8.2 km, respectively. The dotted line Benhar, Ferrari, Gualtieri, Marassicorrespond (2007) to those of the stellar models in Table I. denotes the new empirical relationship ͑4.3͒ obtained in Sec. IV between Re(␻) of the f mode and B. simple extrapolation to quark stars is not very successful. We want to construct an alternative formula for quark star mod- fixed at Rϱϭ3.8, 6.0, and 8.2 km, and the adopted bag con- els by using our numerical results for f mode QNMs, but we stants are Bϭ42.0 and 75.0 MeV fmϪ3. The results of the cannot do this using the same method as Kokkotas et al., calculation are plotted in Fig. 5, and listed in Table III. In because a special relation between the f mode QNMs and the Fig. 5, the squares, triangles, and circles correspond to stellar stellar properties cannot be found. models whose radiation radii are fixed at 3.8, 6.0, and 8.2 In the previous section, we saw that the f mode frequency km, respectively. From our numerical results, we find the depends strongly not on the stellar radiation radius but on the following new empirical formula between the f mode fre- bag constant. So, in order to get the relationship between the quency Re(␻) and the bag constant B: FIG. 1. Relations between ͑a͒ mass M and central density ␳c , ͑b͒ M and radius R ͑solid line͒ and M and radiation radius Rϱ ͑broken line͒, 3 f mode frequency Re(␻) and the bag constant B, we calcu- and ͑c͒ ‘‘average density’’¯␳ϵ3M/4␲R and radiation radius Rϱ of quark stars for four values of the bag constant, Bϭ28.9, 56.0, 94.92, and TABLE III. Deviation from the new empirical formula ͑4.3͒ for Ϫ3 late f mode QNMs emitted by six more stellar models in 471.3 MeV fm . addition to the above nine stellar models (A1–A3, B1–B3, each stellar model in Fig. 5. In this table, Bemp is the value of the bag constant that is calculated by using the empirical formula ͑4.3͒ C1–C3). The radiation radii of these six added models are obtained in Sec. IV. from X-ray observations such as ͓5͔. We pay attention to out the possibility of adopting the value Bϭ471.3FIG. 3. Complex frequencies of both w and w modes for Bϭ(a) 28.9, ͑b͒ 56.0, ͑c͒ 94.92, and ͑d͒ for 471.3 MeV fmϪ3. For ͑a͒–͑c͒, the Ϫ3 TABLE II. Comparison between the f modeII QNMs of quark Rϱ B Re͑␻͒ Bemp Deviation stars whose radiation radii are in the range of 3.8–8.2 km. In MeV fm , which is rather greater than Bmax . We considerstarsupper, a and middle, the values and givenlower by sequences the empirical correspond formulas to͑ quark4.1͒ and stars͑4.2 of͒.Rϱϭ3.8 km, 6.0 km, and 8.2 km, respectively. For ͑d͒, the mass of the quark star mass of Mϭ0.7M for this value of the bag con- ͑km͒ (MeV/fm3) ͑kHz͒ (MeV/fm3) ͑%͒ this range of radiation radius, we pick up three values, Rϱ ᭪ Instellar this model table, is Re( 0.7␻M)emp᭪ .and Filled Im( and␻) nonfilledemp express marks the denote range ofw and fre- wII modes, respectively. These stellar models are listed in Table I. ϭ3.8, 6.0, and 8.2 km. stant to compare with other models. For the casequencyB and damping rate, respectively, given by using the empirical 3.8 28.90 6.822 28.67 Ϫ0.813 ϭ471.3 MeV fmϪ3 also, we plot the relation between M and On the other hand, Nakamura argued that the mass of the formulas. IV. EMPIRICAL FORMULA 3.8 42.00 8.2423 41.024 Ϫ2.344 ␳ in Fig. 1͑a͒, the relations between M and R and between Im͑␻͒ M 10 km compact star reported in ͓5͔ should be roughly 0.7M ᭪ in c 3.8 56.00Х2␲ 9.540 54.10͑22.85Ϯ1.51Ϫ3.385͒ ¯ Kokkotas,Re͑␻͒ Apostolatos,Im͑␻͒ andRe( Andersson␻) constructedIm(␻) em- 1 Hz ͩ 1.4M ᭪ ͪ ͩ R ͪ ͫ order to account for its observed X-ray luminosity ͓6͔. If we M and Rϱ in Fig. 1͑b͒, and the relation between ␳ and Rϱ in emp emp 3.8 75.00 11.076 71.79 Ϫ4.285 piricalModel16 formulas͑kHz͒ for͑Hz the͒ f modes͑kHz of͒ neutron stars͑Hz͓14͒ ͔: use a value smaller than Bmax , however, the mass of a quark Fig. 1͑c͒. In this case, each plot is for 4BϽ␳cр5.0ϫ10 3.8 94.92 12.503 90.30 Ϫ4.868 Ϫ3 M 10 km star whose radiation radius is in the range of 3.8–8.2 km is g/cm . The properties of our quark star models are tabu-A1 6.822 2.98ϫ10Ϫ3 10.52–10.72 (2.97Ϫ3.42)ϫ10Ϫ2 6.0 28.90Ϫ͑14.65Ϯ 6.8721.32͒ 29.07 0.581. ͑4.2͒ ͩ 1.4M ᭪ ͪͩ R ͪͬ far below 0.7M , as seen in Fig. 1͑b͒. Therefore he pointed lated in Table I. Ϫ2 Ϫ1 ᭪ A2 6.872Re͑␻͒ 2.67ϫ10 10.55–10.74 (2.66Ϫ3.10)ϫ10 6.0 42.00 8.331 41.85 Ϫ0.355 A3 6.945 1.12Х2ϫ␲10͑Ϫ0.781 10.58–10.78Ϯ0.01͒ϩ͑1.63Ϯ0.01 1.11–1.32͒ 6.0 56.00 9.676 55.57 Ϫ0.768 1 kHz ͫ We apply these empirical formulas to quark star models and 024019-3 Ϫ2 Ϫ1 B1 9.540 2.05ϫ10 12.77–12.99 (2.04–2.37)ϫ10 show6.0 the results 75.00 in Table 11.286 II. As these 74.39 results show,Ϫ0.819 although Ϫ1 1/2 3/2 B2 9.676 1.69ϫ10 M 12.84–13.0610 km 1.67–2.00 these6.0 formulas 94.92 were deduced 12.803 by employing 94.44 variousϪ0.503 EOSs B3 9.873 6.40ϫϫ10Ϫ1 12.95–13.17, 6.17–7.74 ͑4.1͒ 8.2 28.90 6.945 29.65 2.608 ͩ 1.4M ᭪ ͪ ͩ R ͪ ͬ including the realistic one for neutron stars, it is found that a C1 12.50 9.18ϫ10Ϫ2 15.20–15.46 0.913–1.07 8.2 42.00 8.458 43.07 2.554 C2 12.80 6.83ϫ10Ϫ1 15.36–15.62 6.63–8.24 8.2 56.00 9.873 57.74 3.101 024019-7 C3 13.26 2.28 15.64–15.90 20.4–28.1 8.2 75.00 11.597 78.31 4.415 N1 38.10 20.6 34.90–35.39 22.5–251.3 8.2 94.92 13.257 100.89 6.294 024019-8 Asteroseismology with gravitational waves as a tool to probe the inner structure of NS H. SOTANI AND T. HARADA PHYSICAL REVIEW D 68, 024019 NONRADIAL OSCILLATIONS OF QUARK STARS PHYSICAL REVIEW D 68, 024019 • neutron stars build from single-fluid, normal matter ͑2003͒ NONRADIAL OSCILLATIONS OF QUARK STARS PHYSICAL REVIEW D • strange stars composed of deconfined quarks mode for each FIG. 5. The horizontal axis is the bag constant and the vertical axis is the , for circles, and triangles correspond to stellar models whose radiation f mode frequency Re( B (MeV/fm • superfluid neutron stars radii are fixed at 3.8, 6.0, and 8.2 km, respectively. The dotted line 3) Andersson, Comer (2001) Andersson, Comer, Langlois (2003) and to test GR Asteroseismology with gravitational waves as a tool to probe the inner structure of NS H. SOTANI AND T. HARADA PHYSICAL REVIEW D 68, 024019 NONRADIAL OSCILLATIONS OF QUARK STARS PHYSICAL REVIEW D 68, 024019 • neutron stars build from single-fluid, normal matter ͑2003͒ NONRADIAL OSCILLATIONS OF QUARK STARS PHYSICAL REVIEW D • strange stars composed of deconfined quarks mode for each FIG. 5. The horizontal axis is the bag constant and the vertical axis is the , for circles, and triangles correspond to stellar models whose radiation f mode frequency Re( B (MeV/fm • superfluid neutron stars radii are fixed at 3.8, 6.0, and 8.2 km, respectively. The dotted line 3) • NS oscillations in TeVeS Sotani, Kokkotas (2004, 2005) Asteroseismology with gravitational waves as a tool to probe the inner structure of NS and to test GR • neutron stars are the most extreme laboratories in the universe ‣ rapid (differential) rotation ‣ relativistic effects ‣ superfluidity, superconductivity ‣ strong magnetic fields ‣ solid crust ‣ etc..
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