5.3 Waveforms in the case of circular orbit

UpdateJump To The Next Update Information In the previous two subsections, we only considered the luminosity formulas for the energy and the angular momentum. Here, focusing on circular orbits, we review the previous calculation of the gravitational waveforms.

On the other hand, the gravitational waveforms have also been calculated. In the case of circular orbit around a Schwarzschild black hole, Poisson [83Jump To The Next Citation Point] derived the 1.5PN waveform and Tagoshi and Sasaki [100Jump To The Next Citation Point] derived the 4PN waveform. These were done by using the post-Newtonian expansion of the Regge–Wheeler equation discussed in Section 3. Recently, Fujita and Iyer [39Jump To The Next Citation Point] derived the 5.5PN waveform by using the MST formalism. They also discussed factorized re-summed waveforms which is useful to obtain better agreement with accurate numerical data.

In the case of circular orbit around a Kerr black hole, Poisson [83Jump To The Next Citation Point] derived the 1.5PN waveform under the assumption of slow rotation of the black hole. In [94Jump To The Next Citation Point] and [101], although the luminosity was derived up to 2.5PN and 4PN order respectively, the waveform was not derived up to the same order. Recently, Tagoshi and Fujita [97] computed the all multipolar modes &tidle;Z lmω necessary to derive the waveform up to 4PN order, and the results were used to derive the factorized, re-summed, multipolar waveform in [78].

From Equations (46View Equation) and (47View Equation), we have

2 ∑ −2Saℓωmn iωn(r∗−t)+im φ h+ − ih × = − -- Zℓmωn -√----e r ℓmn 2π ≡ ∑ (h+ − ih × ) (182 ) ℓm ℓm ℓm

Since the formulas for the waveform are very complicated, we only show the mode for ℓ = m = 2 up to 4PN order in the Schwarzschild case. We define +,× ζℓ,−m as [83]

μ h+,ℓm× + h+ℓ,,−×m = − -(M Ωφ)2∕3ζ+ℓ,,m×. (183 ) r

ζ+2,,×2 are given as

( + 107v2 cos(2 ψ) ζ2,2 = (3 + cos(2 𝜃)) cos(2ψ ) − --------------- ( ( ) 42 ) 4 +v3 2π cos(2 ψ) + − 17-+ 4 ln 2 sin(2 ψ) − 2173-v-cos(2-ψ)- ( 3( ) ) 1512 5 − 107 π cos(2 ψ ) 1819 214 ln2 +v ----------------+ ----- − -------- sin(2ψ ) ( 21( 126 212 ) +v6 cos(2ψ ) 49928027- − 856-γ-+ 2π--+ 668-ln-2-− 8 (ln 2)2 − 856-ln-v- ( 1940400) 105 ) 3 105 105 − 254 π + --35--- + 8π ln 2 sin(2ψ ) ( ( ) ) +v7 −-2173-π-cos(2ψ-)+ 36941- − 2173-ln-2 sin(2 ψ) ( 75(6 4536 378 8 326531600453 45796 γ 107 π2 +v cos(2ψ ) − -12713500800--+ -2205---− --63-- 2 ) 35738-ln-2 428-(ln-2)- 45796-lnv- − 2205 + 21 + 2205 ( ) ) ) + 13589-π-− 428-π-ln-2- sin(2 ψ) , (184 ) 735 21 ( 107 v2 sin(2 ψ) ( ( 17 ) ) ζ×2,2 = 4 cos(𝜃) sin(2 ψ) − ---------------+ v3 cos(2ψ ) ---− 4 log(2) + 2 π sin(2ψ ) ( 42( ) 3 ) 2173v4-sin(2ψ-)- 5 1819- 214-log(2) 107-π-sin-(2-ψ)- − 1512 + v cos(2 ψ) − 126 + 21 − 21 ( ( 254 π ) +v6 cos(2ψ ) ------− 8 π log(2) ( 35 ) ) 49928027- 856-γ- 2-π2 668-log(2) 2 856-log(v) + 1940400 − 105 + 3 + 105 − 8 log(2) − 105 sin (2 ψ) ( ( 36941 2173 log(2)) 2173 π sin(2 ψ)) +v7 cos(2ψ ) − ------+ ----------- − --------------- ( ( 4536 378 ) ( 756 8 −-13589-π- 428π-log(2)- 326531600453-- 45796-γ- +v cos(2ψ ) 735 + 21 + − 12713500800 + 2205 2 2 ) ) ) − 107-π- − 35738--log-(2)+ 428-log(2)--+ 45796--log-(v-) sin(2ψ ) , (185 ) 63 2205 21 2205
where
∗ 3 ψ = Ω (t − r ) − φ − 2v (γ + 2 ln 2 + 3ln v). (186 )

Other modes are given in [100] up to 4PN order. (Note however the following errors which were pointed out in [61, 17Jump To The Next Citation Point, 4Jump To The Next Citation Point, 5Jump To The Next Citation Point, 39Jump To The Next Citation Point]. The signs of + ζℓm are opposite. The sign of × ζ7,3 is also opposite. × ζ8,7 and ζ+10,6 have errors and the corrected formulas are given in Equations (6.6) and (6.7) in [39Jump To The Next Citation Point].) In the literature on the post-Newtonian approximation [17, 4, 5], the post-Newtonian waveforms are express in terms of (n∕2) H +,× defined as

2μx ∑ n∕2 (n∕2) h+,× = ---- x H +,× , (187 ) r n
where x ≡ (M Ω )2∕3 φ. The expression of H (n∕2) +,× up to 5.5PN order are given in [39].
  Go to previous page Go up Go to next page