Gravitational Waves from Inspiralling Compact Binaries in ... - LUTH
Gravitational Waves from Inspiralling Compact Binaries in ... - LUTH Gravitational Waves from Inspiralling Compact Binaries in ... - LUTH
Method of variation of constants (l) are periodic in l with a period of 2π. n denotes the unperturbed ‘mean motion’, given by n = 2π P , P being the radial (periastron to periastron) period, while k = ∆Φ/2π, ∆Φ being the advance of the periastron in the time interval P . Functions S(l) and W (l) and therefore ∂S ∂l (l) and ∂W ∂l Angles l and λ satisfy, for the unperturbed system, ˙ l = n and ˙λ = (1 + k)n, which integrate to l = n(t − t0) + cl , λ = (1 + k)n(t − t0) + cλ , t0 some initial instant and the constants cl and cλ, corresponding values for l and λ. Unperturbed solution depends on four integration constants: c1, c2, cl and cλ. BRI-IHP06-I – p.104/??
Method of variation of constants At 2PN, one can write down explicit expressions for the functions S(l) and W (l). GQKR yields: S(l; c1, c2) = ar(1 − er cos u) , W (l; c1, c2) = (1 + k)(v − l) + fφ c 4 sin 2v + gφ sin 3v , c4 where v and u are some 2PN accurate true and eccentric anomalies, which must be expressed as functions of l, c1, and c2, say v = V(l; c1, c2) = V (U(l; c1, c2)) and u = U(l; c1, c2). ar and er are some 2PN accurate semi-major axis and radial eccentricity, while fφ and gφ are certain functions, given in terms of c1 and c2. v = V (u) ≡ 2 arctan 1 + eφ 1 − eφ 1/2 tan u . 2 Fn u = U(l) defined by inverting the ‘Kepler equation’ l = l(u) l = u − et sin u + ft gt sin V (u) + c4 c4 (V (u) − u) . BRI-IHP06-I – p.105/??
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- Page 95 and 96: PART II Based on Phasing of Gravita
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Method of variation of constants<br />
(l) are<br />
periodic <strong>in</strong> l with a period of 2π. n denotes the unperturbed<br />
‘mean motion’, given by n = 2π<br />
P , P be<strong>in</strong>g the radial<br />
(periastron to periastron) period, while k = ∆Φ/2π, ∆Φ be<strong>in</strong>g<br />
the advance of the periastron <strong>in</strong> the time <strong>in</strong>terval P .<br />
Functions S(l) and W (l) and therefore ∂S<br />
∂l<br />
(l) and ∂W<br />
∂l<br />
Angles l and λ satisfy, for the unperturbed system, ˙ l = n and<br />
˙λ = (1 + k)n, which <strong>in</strong>tegrate to<br />
l = n(t − t0) + cl ,<br />
λ = (1 + k)n(t − t0) + cλ ,<br />
t0 some <strong>in</strong>itial <strong>in</strong>stant and the constants cl and cλ,<br />
correspond<strong>in</strong>g values for l and λ. Unperturbed solution<br />
depends on four <strong>in</strong>tegration constants: c1, c2, cl and cλ.<br />
BRI-IHP06-I – p.104/??
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