Gravitational Waves from Inspiralling Compact Binaries in ... - LUTH
Gravitational Waves from Inspiralling Compact Binaries in ... - LUTH Gravitational Waves from Inspiralling Compact Binaries in ... - LUTH
Gauge Invariant Variables Memmesheimer, Gopakumar and Schäfer (2004) stress use of gauge invariant variables in the elliptical orbit case Damour and Schäfer (1988) showed that the functional form of n and Φ as functions of gauge invariant variables like E and h is the same in different coordinate systems (gauges). From the explicit expressions for n and Φ in the ADM and modified harmonic coordinates the gauge invariance of these two parameters is explicit at 3PN. MGS04 suggest the use of variables xMGS = (Gmn/c 3 ) 2/3 and k ′ = (Φ − 2π)/6π as gauge invariant variables in the general orbit case. We propose a variant of the former: x = (Gmn Φ/2πc 3 ) 2/3 = (Gmn Kc 3 ) 2/3 = (Gmn (1 + k)c 3 ) 2/3 . Our choice is the obvious generalisation of gauge invariant variable x in the circular orbit case and thus facilitates the straightforward reading out of the circular orbit limit. BRI-IHP06-I – p.43/??
Orbital average - Energy flux -MHar To average the energy flux over an orbit requires use of 3PN GQKR → Modified Harmonic coords Involves evaluation of the the integral, 〈 ˙ E〉 = 1 P P 0 E(t) ˙ dt = 1 2 π 2 π 0 dl du ˙ E(u)du . Using GQKR, transform the expression for the energy flux ˙ E (r, ˙r 2 , v2 ) or more exactly (dl/du × ˙ E)(r, ˙r 2 , v2 ) to (dl/du × ˙ E)(x, et, u). Choices: GI97 uses Gm/ar and er. DGI04 employs Gmn/c 3 and et. ABIQ06 uses et and x = (GmnΦ/2πc 3 ) 2/3 Recall: 3PN flux contains log terms; convenient to rewrite the expression as dl du ˙ E = 11 N=3 αN (et) 1 (1 − et cos u) N + βN (et) Non-vanishing αN ’s, βN ’s and γN ’s are too long to be listed. βN ’s correspond to all the 2.5PN terms. γN represent the log terms at order 3PN. sin u (1 − et cos u) N + γN (et) ln(1 − et cos u) (1 − et cos u) N BRI-IHP06-I – p.44/??
- Page 1 and 2: Gravitational Waves from Inspiralli
- Page 3 and 4: Introduction Inspiralling Compact B
- Page 5 and 6: Kozai Mechanism One proposed astrop
- Page 7 and 8: Kozai Mechanism, Globular Clusters
- Page 9 and 10: Kicks, Eccentricity Compact binarie
- Page 11 and 12: Related Earlier Work ∗ Peters and
- Page 13 and 14: Earlier Work ∗ GW from an eccentr
- Page 15 and 16: FF as fn of initial eccentricity e0
- Page 17 and 18: Eccentric Signal Plots of s(t) (up
- Page 19 and 20: The Generation Modules Generation p
- Page 21 and 22: Present Work Represent GW from a bi
- Page 23 and 24: PN order of Multipoles For a given
- Page 25 and 26: Radiative moments - Source moments
- Page 27 and 28: 3PN EOM for ICB a i = dvi dt AE = 1
- Page 29 and 30: Instantaneous Terms dE dt inst d
- Page 31 and 32: 3PN Instantaneous Terms dE dt 2PN
- Page 33 and 34: Transfn of World lines Having obtai
- Page 35 and 36: Energy Flux - Modified Harmonic Coo
- Page 37 and 38: 3PN generalised Quasi-Keplerian rep
- Page 39 and 40: 3PN generalised Quasi-Keplerian rep
- Page 41 and 42: 3PN GQKR - Mhar n = (−2E) 3/2 +11
- Page 43: 3PN GQKR - Mhar + Φ = 2 π 70 16
- Page 47 and 48: Orbit Averaged Energy Flux - MHar <
- Page 49 and 50: Comments Note: No term at 2.5PN. 2.
- Page 51 and 52: Comments Useful internal consistenc
- Page 53 and 54: Gauge Invariant Variables < ˙ E >
- Page 55 and 56: Gauge Invariant Variables < ˙ E3PN
- Page 57 and 58: Hereditary Contributions F 3PN tail
- Page 59 and 60: Log terms in total energy flux Summ
- Page 61 and 62: Log terms in total energy flux FZ t
- Page 63 and 64: Complete 3PN energy flux - Mhar <
- Page 65 and 66: Complete 3PN energy flux - Mhar <
- Page 67 and 68: Present Work Extends the circular o
- Page 69 and 70: Angular Momentum Flux Hereditary co
- Page 71 and 72: Far Zone Angular Momentum Flux dJi
- Page 73 and 74: Far Zone Angular Momentum Flux dJi
- Page 75 and 76: 3PN AMFlux - Shar dJi dt dJi dt
- Page 77 and 78: Orbital Averaged AMF - ADM Using th
- Page 79 and 80: Orbital Averaged AMF - ADM 〈 dJ d
- Page 81 and 82: Orbital Averaged AMF - ADM 〈 dJ d
- Page 83 and 84: Evoln of orbital elements under GRR
- Page 85 and 86: Evoln of orbital element n under GR
- Page 87 and 88: Evoln of orbital element n under GR
- Page 89 and 90: Evoln of orbital element et under G
- Page 91 and 92: Evoln of orbital element ar under G
- Page 93 and 94: Evoln of orbital element ar under G
Orbital average - Energy flux -MHar<br />
To average the energy flux over an orbit requires use of 3PN GQKR →<br />
Modified Harmonic coords<br />
Involves evaluation of the the <strong>in</strong>tegral,<br />
〈 ˙<br />
E〉 = 1<br />
P<br />
P<br />
0<br />
E(t) ˙ dt = 1<br />
2 π<br />
2 π<br />
0<br />
dl<br />
du<br />
˙<br />
E(u)du .<br />
Us<strong>in</strong>g GQKR, transform the expression for the energy flux ˙ E (r, ˙r 2 , v2 ) or more<br />
exactly (dl/du × ˙ E)(r, ˙r 2 , v2 ) to (dl/du × ˙ E)(x, et, u).<br />
Choices: GI97 uses Gm/ar and er. DGI04 employs Gmn/c 3 and et.<br />
ABIQ06 uses et and x = (GmnΦ/2πc 3 ) 2/3<br />
Recall: 3PN flux conta<strong>in</strong>s log terms; convenient to rewrite the expression as<br />
dl<br />
du<br />
˙<br />
E =<br />
11<br />
N=3<br />
<br />
αN (et)<br />
1<br />
(1 − et cos u) N + βN (et)<br />
Non-vanish<strong>in</strong>g αN ’s, βN ’s and γN ’s are too long to be listed.<br />
βN ’s correspond to all the 2.5PN terms.<br />
γN represent the log terms at order 3PN.<br />
s<strong>in</strong> u<br />
(1 − et cos u) N + γN (et) ln(1 − et cos u)<br />
(1 − et cos u) N<br />
<br />
BRI-IHP06-I – p.44/??
Change language