This document is part of https : // dcc.ligo.org/LIGO - T1500554/public

ENS[x_]:=Expand[Normal[Simplify[x]]];

1212.5520, 4.7

Δ[η_]:=Sqrt[1-4η];
Ss=S1+S2;
Σs=M(S2/m2-S1/m1);
δm=m1-m2;

Lv=(3/2+η/6);
Lv3=(27/8-19/8η+1/24η^2);
Lv5=(135/16+(-6889/144+41/24Pi^2)η+31/24η^2+7/1296η^3);

Lv2L=(-35/6)Ss+(-5/2)δm/M Σs;
Lv2a=(-3)Ss-δm/M Σs;
Lv2n=(1/2)Ss+(1/2)δm/M Σs;

Lv4L=(-77/8+427/72η)Ss+(-21/8+35/12η)δm/M Σs;
Lv4a=(-7/2+3η)Ss+(-1/2+4/3η)δm/M Σs;
Lv4n=(11/8-19/24η)Ss+(11/8-5/12η)δm/M Σs;

Lv6L=(-405/16+1101/16η-29/16η^2)Ss+(-81/16+117/4η-15/16η^2)δm/M Σs;
Lv6a=(-29/4+1/12η-4/3η^2)Ss+(-1/2-79/24η-2/3η^2)δm/M Σs;
Lv6n=(61/16-1331/48η+11/48η^2)Ss+(61/16-367/24η+5/48η^2)δm/M Σs;

Print["**1.5PN**"]
Print["n"];
ENS[SeriesCoefficient[Lv2n,{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv2n,{S2,0,1}]/.m1->M-m2]
Print["λ"]
ENS[SeriesCoefficient[Lv2a,{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv2a,{S2,0,1}]/.m1->M-m2]
Print["L"]
ENS[SeriesCoefficient[Lv2L,{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv2L,{S2,0,1}]/.m1->M-m2]
Print["----Time averaged"]
Print["     * Si"]
ENS[SeriesCoefficient[1/2(Lv2n+Lv2a),{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[1/2(Lv2n+Lv2a),{S2,0,1}]/.m1->M-m2]
Print["     * Li(L.S)"]
ENS[SeriesCoefficient[Lv2L-1/2(Lv2n+Lv2a),{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv2L-1/2(Lv2n+Lv2a),{S2,0,1}]/.m1->M-m2]

Lv4L=(-77/8+427/72η)Ss+δm/M(-21/8+35/12η)Σs;
Lv4n=(11/8-19/24η)Ss+δm/M(11/8-5/12η)Σs;
Lv4a=(-7/2+3η)Ss+δm/M(-1/2+4/3η)Σs;

Print["n"];
ENS[SeriesCoefficient[Lv4n,{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv4n,{S2,0,1}]/.m1->M-m2]
Print["λ"]
ENS[SeriesCoefficient[Lv4a,{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv4a,{S2,0,1}]/.m1->M-m2]
Print["L"]
ENS[SeriesCoefficient[Lv4L,{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv4L,{S2,0,1}]/.m1->M-m2]

Print["n"];
ENS[SeriesCoefficient[Lv6n,{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv6n,{S2,0,1}]/.m1->M-m2]
Print["λ"]
ENS[SeriesCoefficient[Lv6a,{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv6a,{S2,0,1}]/.m1->M-m2]
Print["L"]
ENS[SeriesCoefficient[Lv6L,{S1,0,1}]/.m2->M-m1]
ENS[SeriesCoefficient[Lv6L,{S2,0,1}]/.m1->M-m2]

Absorption: 2.11--2.17 di 1705.03869

F0=-32/5η^2 v^(10)(1-v^2(1247/336+35/12η));

Cv1=-16/5(m1/M)^4η^2v^(12)(1+Sqrt[1-χ^2])(1+3χ^2+v^2/4(3(2+χ^2)+2(m1/M)(2+3m1/M)(1+3χ^2))+pippo v^3);
δF=sgn v^3(1-η v^2+pippo v^3)(χ/(2m1/M/ (1+Sqrt[1-χ^2]))-sgn v^3(1-η v^2+pippo v^3))Cvi;

Fluxabs=ENS[Series[SeriesCoefficient[δF/F0/.χ->χLAL M^2/m1^2,{χLAL,0,1}],{v,0,7}]/.sgn^2->1/.η->m1(M-m1)/M^2]

T4

ENS[SeriesCoefficient[(1-v^2(1247/336+35/12η)-Fluxabs)/(1+4.5v^2(-3/4-η/12))/.η->m1/M(1-m1/M),{v,0,7}]]

F2

ENS[SeriesCoefficient[(1+9/2v^2(-3/4-η/12))/(1-v^2(1247/336+35/12η)-Fluxabs)/.η->m1/M(1-m1/M),{v,0,7}]]

ENS[SeriesCoefficient[(1-v^2(1247/336+35/12η)-Fluxabs)/(1+2v^2(-3/4-η/12))/.η->m1/M(1-m1/M),{v,0,7}]]

ENS[Series[δF/F0/.χ->0,{v,0,10}]/.sgn^2->1/.η->m1(M-m1)/M^2]

Series[1/(1+α v^2+β v^3+v^5x+v^7 y),{v,0,7}]
Series[(1+α v^2+β v^3+v^5x+v^7 y),{v,0,7}]

Sdot da absorption

Ωt=v^3/M(1-η v^2);
ΩH=χ/(2m1(1+Sqrt[1-χ^2]));

ENS[Series[(ΩH-Ωt)Cv1,{v,0,12}]]

Sdot = 1/M Ω x S

1 PN

ENS[3/4+η/2-3/4δm/M/.η->m1 m2/M^2/.m2->(M-m1)]

ENS[η/2 (4+3m2/m1)/.η->m1 m2/M^2/.m2->M-m1]

2 PN

ENS[9/16+5η/4-1/24η^2+δm/M(-9/16+5η/8)/.η->m1 m2/M^2/.m2->(M-m1)]

3 PN

ENS[piripippo9/16+5η/4-1/24η^2+δm/M(-9/16+5η/8)/.η->m1 m2/M^2/.m2->(M-m1)]

3.5 PN

ENS[Series[δF sgn/v^3/.χ->0,{v,0,15}]/.sgn^2->1/.sgn^3->sgn]

ENS[Series[SeriesCoefficient[δF sgn/v^3,{χ,0,1}],{v,0,12}]/.sgn^2->1]

ENS[Series[SeriesCoefficient[δF sgn/v^3,{χ,0,3}],{v,0,14}]/.sgn^2->1]

Spin cube

κp=κ1+κ2;
κm=κ1-κ2;
λp=λ1+λ2;
λm=λ1-λ2;

EScube=Ss^3 (2κp+4λp−20)+Ss^2 Σs(2δm/M κp+6δm/M λp−32δm/M+4κm−6λm)+Ss Σs^2(5δm/M κm−6δm/M λm−5κp+6λp−12+η(−2κp−12λp+68))+Σs^3(−3δm/M κp+2δm/M λp+3κm−2λm+η(−2δm/M λp+12δm/M−6κm+6λm));
FScube=Ss^3 (-16/3κp-4λp+40/3)+Ss^2 Σs(-35/6δm/M κp-6δm/M λp+73/3δm/M-3/4κm+6λm)+Ss Σs^2(-35/12δm/M κm+6δm/M λm+35/12κp-6λp+32/3+η(22/3κp+12λp-172/3))+Σs^3(67/24δm/M κp-2δm/M λp-δm/M/8-67/24κm+2λm+η(δm/M λp/2+2δm/M λp-11δm/M+61/12κm-6λm));

EScubeBH=-8Ss^3-16δm/M Σs Ss^2-10(1-4η)Σs^2 Ss-2(1-4η)δm/M Σs^3;
ENS[(EScube/.κ1->1/.λ1->1/.κ2->1/.λ2->1)-EScubeBH]

FScubeBH=-16/3Ss^3+2/3δm/M Σs Ss^2+(9/2-56/3η)Σs^2 Ss +(35/24-6η)δm/M Σs^3;
ENS[(FScube/.κ1->1/.λ1->1/.κ2->1/.λ2->1)-FScubeBH]

ES3S1cube=ENS[SeriesCoefficient[EScube,{S1,0,3}]];
ES3κ1=ENS[SeriesCoefficient[ES3S1cube,{κ1,0,1}]];
ES3λ1=ENS[SeriesCoefficient[ES3S1cube,{λ1,0,1}]];
ES3S2cube=ENS[SeriesCoefficient[EScube,{S2,0,3}]];
ES3κ2=ENS[SeriesCoefficient[ES3S2cube,{κ2,0,1}]];
ES3λ2=ENS[SeriesCoefficient[ES3S2cube,{λ2,0,1}]];
ENS[ES3S1cube-κ1 ES3κ1-λ1 ES3λ1/.m2->M-m1]
ENS[ES3S2cube-κ2 ES3κ2-λ2 ES3λ2/.m1->M-m2]

ES3S1sq=ENS[SeriesCoefficient[EScube,{S1,0,2}]];
ES1sqκ1=ENS[SeriesCoefficient[ES3S1sq,{κ1,0,1}]];
ES1sqλ1=ENS[SeriesCoefficient[ES3S1sq,{λ1,0,1}]];
ES3S2sq=ENS[SeriesCoefficient[EScube,{S2,0,2}]];
ES2sqκ2=ENS[SeriesCoefficient[ES3S2sq,{κ2,0,1}]];
ES2sqλ2=ENS[SeriesCoefficient[ES3S2sq,{λ2,0,1}]];
ENS[ES3S1sq-κ1 ES1sqκ1-λ1 ES1sqλ1/.M->m1+m2]
ENS[ES3S2sq-κ2 ES2sqκ2-λ2 ES2sqλ2/.M->m1+m2]

Example SO3

EnergySO3=2/3+2 M/m1

FluxSO3=-3/2-5/4M/m1

PhasingSO3=ENS[m1/M(25+38/3 m1/M)]

dtdvSO3=SeriesCoefficient[(1+5/2v^3EnergySO3)/(1+v^3FluxSO3),{v,0,3}]

CPhasingSO3=ENS[dtdvSO3*40/(3-5)/(3-8)]