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By Frank Miller, Neil Gaiman et al.

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The phase is φ = ωdt where ω is the orbital frequency, from which one deduces φ = − (ω/L)dE. The result is simpler if we use instead of the local time t flowing in the observer’s frame (t = T − R/c with the notation of Section 3) the adimensional time variable [61] c3 ν (tc − t) , (66) 5GM where tc denotes the instant of coalescence (at which the frequency goes formally to infinity). In terms of this time variable the orbital phase is obtained as Θ= φ(t) = φ0 − 1 Θ5/8 + ν 3715 55 3π 1/4 + ν Θ3/8 − Θ 8064 96 4 + 284875 1855 2 9275495 + ν+ ν Θ1/8 14450688 258048 2048 − 38645 15 + ν π ln Θ + O(Θ−1/8 ) 172032 2048 , (67) where φ0 is some constant phase determined by initial conditions (for instance when the signal enters the frequency bandwidth of the detector).

197 (1975) 717. , Ann. Phys. ) 11 (1960) 169. [40] Bonnor W. , Rotenberg M. , Proc. R. Soc. London A 265 (1961) 109. [41] Bonnor W. , Rotenberg M. , Proc. R. Soc. London A 289 (1966) 247. [42] Hunter A. , Rotenberg M. , J. Phys. A 2 (1969) 34. [43] Bonnor W. , Ondes et radiations gravitationnelles (CNRS, Paris, 1974) p. 73. [44] Thorne K. , Kov´acs S. , Astrophys. J. 200 (1975) 245. [45] Crowley R. , Thorne K. , Astrophys. J. 215 (1977) 624. [46] Thorne K. , Rev. Mod. Phys. 52 (1980) 299. , Philos.

For convenience we denote by x = (GM ω/c3 )2/3 a small dimensionless post-Newtonian parameter of order ε2 , where ω is the instantaneous angular frequency of the orbit (ω = 2π/P , where P is the orbital period). 5PN–accurate luminosity L of the binary is obtained by application of the formalism above. Namely, one uses the decomposition of L in terms of observable moments UL and VL as given by (25). Then one relates the observable moments to the moments ML and SL by the equations (36)-(40). 5PN order the mass quadrupole moment comprises the dominant tail integral and the quadrupole-quadrupole interaction terms which include the nonlinear memory integral, but the tail of tail integral appearing in (39) is negligible.

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Batman Black and White Vol.2 by Frank Miller, Neil Gaiman et al.


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