Gravitational Waves from Inspiralling Compact Binaries in ... - LUTH

Phasing of GWF
TT radn field is given by wave generation formalisms, as a PN
expansion of the form
h TT
ij = 1
c 4
h 0 ij + 1
c h1 ij + 1
c 2 h2 ij + 1
c 3 h3 ij + 1
c 4 h4 ij + 1
c 5 h5 ij + 1
c 6 h6 ij + · · ·
Leading (‘quadrupolar’) approximation is given in terms of the
relative separation vector x and relative velocity vector v as
1
c 4 (h0 km) =
4 G µ
c4 Pijkm(N)
R ′
vij −
G m
r nij
Pijkm(N) TT projection operator projecting normal to N, N = R ′ /R ′ ,
R ′ radial distance to the binary.
When inserting the explicit expression of h 0 ij, and its higher-PN
analogues h 1 ij, h 2 ij · · · which are currently known up to h 4 ij one ends
up with a corresponding expression for the two independent
polarization amplitudes, as functions of the relative separation r and
the ‘true anomaly’ φ, i.e. the polar angle of x, and their time
derivatives,
,
BRI-IHP06-I – p.97/??