Kinematic Variables

(* Conversion between various kinematic variables. Definitions are given in Appendix 6-A of the thesis. *)

(* Simplification for gamma parameter *)
GSim[expr_]:= Simplify[expr,Assumptions->{m>0,ν>0,ν<1/4,γ>1}];

(* Conversion to gamma parameter *) 
repγ = {ET-> m*γ,σ->1+(γ^2-1)/(2*ν), ξ-> ((-1+(γ^4)+4 ν)/(4 (γ^4))),PS->(((m^2) (-1+(γ^2)) (-1+(γ^2)+4 ν))/(4 (γ^2))), ϵ->(γ-1)/ν, ζ->(γ^2-1)/(2ν)};
ToGamma[expr_]:= GSim[expr/.repγ];

PN Lagrangian and Canonical Velocity

https://arxiv.org/abs/gr-qc/0209089v2		Blanchet + Iyer, Actions

NG= 4;
NPN = 3;
Nη = 2*(NPN+1)

cL[0] = VS/2 + G*m/r;

cL[1] = (1/8)*VS^2*(1-3*ν) + (1/2)*(G*m/r)*(ν*VR^2 + (3+ν)*VS) - (1/2)*(G*m/r)^2;

cL[2] = (1/16)*VS^3- (7*ν/16)*VS^3 + (13*ν^2/16)*VS^3 + (G*m/r)*((3*ν^2/8)*VR^4 + (ν/2)*VS*VR^2 -(5*ν^2/4)*VS*VR^2 + (7/8)*VS^2 -(3*ν/2)*VS^2 - (9*ν^2/8)*VS^2) + (G*m/r)^2*((3*ν/2)*VR^2 + (3*ν^2/2)*VR^2 + 2VS -ν*VS + (ν^2/2)*VS) + (G*m/r)^3*(1/4 + 3*ν/4);

cL[3] = (((5/128)VS^4 -(59ν/128)VS^4 + (119ν^2/64)VS^4 - (323ν^3/128)VS^4)
		+ (G m/r) ((5ν^3/16)VR^6 + (9ν^2/16)VS VR^4 - (33ν^3/16)VS VR^4
				+ (ν/2)VS^2 VR^2 - (3ν^2)VS^2 VR^2 + (75ν^3/16)VS^2 VR^2
				+ (11/16)VS^3 - (7ν/2)VS^3 + (59ν^2/16)VS^3 + (65ν^3/16)VS^3)
		+ (G m/r)^2 (-(5ν/12)VR^4 + (17ν^2/12)VR^4 + (2ν^3)VR^4
				+ (39ν/16)VS VR^2 - (29ν^2/8)VS VR^2 - (27ν^3/4)VS VR^2
				+ (47/16)VS^2 - (15ν/4)VS^2 - (25ν^2/16)VS^2 - (9ν^3/4)VS^2)
		+ (G m/r)^3 ((77ν/16)VR^2 + (5ν^2/4)VR^2 + (7ν^3/2)VR^2
				+ (3ν π^2/64)VR^2 + (13/8)VS - (409ν/48)VS - (5ν^2/8)VS
				+ (ν^3/2)VS - (ν π^2/64)VS)
		+ (G m/r)^4 (-(1/8) - (1881ν/280) + (11λ ν/3) + (21 ν π^2/32)));

L =Sum[cL[i],{i,0,3}];

(* Canonical velocity in terms of canonical momentum (to 3PN order) *)
v[i] = p[i]+(1/2) PS (-1+3 ν) p[i]+(3/8) (PS^2) (1-5 ν+5 (ν^2)) p[i]+(5/16) (PS^3) (-1+7 ν-14 (ν^2)+7 (ν^3)) p[i]+(((G^2) (m^2) (5 r p[i]+8 r ν p[i]+3 PR ν q[i]))/(r^3))+G (-((m (3+ν) p[i])/r)-((m PR ν q[i])/(r^2)))+(G^3) ((((m^3) (-600-1340 ν+3 (π^2) ν-552 (ν^2)) p[i])/(96 (r^3)))+(((m^3) PR (-340 ν-3 (π^2) ν-112 (ν^2)) q[i])/(32 (r^4))))+(G^2) ((((m^2) (-54 PS+17 (PR^2) ν+272 PS ν+30 (PR^2) (ν^2)+218 PS (ν^2)) p[i])/(8 (r^2)))+(((m^2) (40 (PR^3) ν+51 PR PS ν+344 (PR^3) (ν^2)+90 PR PS (ν^2)) q[i])/(24 (r^3))))+G (-(((-5 m PS+20 m PS ν+m (PR^2) (ν^2)+3 m PS (ν^2)) p[i])/(2 r))-(((3 m (PR^3) (ν^2)+m PR PS (ν^2)) q[i])/(2 (r^2))))+G ((((m (PR^2) PS (2 (ν^2)-3 (ν^3)))/(4 r))-((3 m (PR^4) (-(ν^2)+(ν^3)))/(8 r))-((3 m (PS^2) (7-42 ν+53 (ν^2)+5 (ν^3)))/(8 r))) p[i]+(1/(8 (r^2)))(6 m (PR^3) PS (ν^2)+2 m PR (PS^2) (ν^2)-15 m (PR^5) (ν^3)-6 m (PR^3) PS (ν^3)-3 m PR (PS^2) (ν^3)) q[i]);

(* Replacement rules for scalars (v^2), (Subscript[v, r]), and (Subscript[v, ϕ]), respectively. *)
v2p={VS->PS+(PS^2) (-1+3 ν)-((2 G (3 m PS+m (PR^2) ν+m PS ν))/r)+(((G^2) (19 (m^2) PS+12 (m^2) (PR^2) ν+22 (m^2) PS ν+3 (m^2) (PR^2) (ν^2)+(m^2) PS (ν^2)))/(r^2))+(1/(48 (r^3)))(G^3) (-2040 (m^3) PS-2364 (m^3) (PR^2) ν-9 (m^3) (π^2) (PR^2) ν-4124 (m^3) PS ν+3 (m^3) (π^2) PS ν-1680 (m^3) (PR^2) (ν^2)-1320 (m^3) PS (ν^2))+(1/4) PS (4 (PS^2)-21 (PS^2) ν+24 (PS^2) (ν^2))+(1/r)G (8 m (PS^2)+m (PR^2) PS ν-28 m (PS^2) ν-3 m (PR^4) (ν^2)-5 m (PR^2) PS (ν^2)-6 m (PS^2) (ν^2))+(1/(6 (r^2)))(G^2) (-201 (m^2) (PS^2)+20 (m^2) (PR^4) ν+3 (m^2) (PR^2) PS ν+780 (m^2) (PS^2) ν+226 (m^2) (PR^4) (ν^2)+300 (m^2) (PR^2) PS (ν^2)+645 (m^2) (PS^2) (ν^2)+42 (m^2) (PR^4) (ν^3)+36 (m^2) (PR^2) PS (ν^3)+18 (m^2) (PS^2) (ν^3))+(1/8) PS (-8 (PS^3)+59 (PS^3) ν-130 (PS^3) (ν^2)+80 (PS^3) (ν^3))+(1/(4 r))G (-40 m (PS^3)-3 m (PR^2) (PS^2) ν+238 m (PS^3) ν+15 m (PR^4) PS (ν^2)+25 m (PR^2) (PS^2) (ν^2)-303 m (PS^3) (ν^2)-15 m (PR^6) (ν^3)-27 m (PR^4) PS (ν^3)-36 m (PR^2) (PS^2) (ν^3)-48 m (PS^3) (ν^3)),VR->PR-((G PR (3 m+2 m ν))/r)+(((G^2) (5 (m^2) PR+11 (m^2) PR ν))/(r^2))+(1/2) PR (-PS+3 PS ν)+(3/8) PR (PS^2) (1-5 ν+5 (ν^2))-(((G^3) (m^3) PR (300+1180 ν+3 (π^2) ν+444 (ν^2)))/(48 (r^3)))+(((G^2) (m^2) (-162 PR PS+91 (PR^3) ν+867 PR PS ν+434 (PR^3) (ν^2)+744 PR PS (ν^2)))/(24 (r^2)))+((G (5 m PR PS-20 m PR PS ν-4 m (PR^3) (ν^2)-4 m PR PS (ν^2)))/(2 r))+(5/16) PR (PS^3) (-1+7 ν-14 (ν^2)+7 (ν^3))+(1/(8 r))G (-21 m PR (PS^2)+126 m PR (PS^2) ν+3 m (PR^5) (ν^2)+10 m (PR^3) PS (ν^2)-157 m PR (PS^2) (ν^2)-18 m (PR^5) (ν^3)-12 m (PR^3) PS (ν^3)-18 m PR (PS^2) (ν^3)),VP->(((Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]))/r)-((G (Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]) (3 m+m ν))/(r^2))+(((G^2) (Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]) (5 (m^2)+8 (m^2) ν))/(r^3))+(((Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]) (-PS+3 PS ν))/(2 r))-(((G^3) (Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]) (600 (m^3)+1340 (m^3) ν-3 (m^3) (π^2) ν+552 (m^3) (ν^2)))/(96 (r^4)))-((G (Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]) (-5 m PS+20 m PS ν+m (PR^2) (ν^2)+3 m PS (ν^2)))/(2 (r^2)))+(1/(8 (r^3)))(G^2) (Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]) (-54 (m^2) PS+17 (m^2) (PR^2) ν+272 (m^2) PS ν+30 (m^2) (PR^2) (ν^2)+218 (m^2) PS (ν^2))+((3 (Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]) ((PS^2)-5 (PS^2) ν+5 (PS^2) (ν^2)))/(8 r))-(1/(8 (r^2)))G (Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]) (21 m (PS^2)-126 m (PS^2) ν-3 m (PR^4) (ν^2)-4 m (PR^2) PS (ν^2)+159 m (PS^2) (ν^2)+3 m (PR^4) (ν^3)+6 m (PR^2) PS (ν^3)+15 m (PS^2) (ν^3))+((5 (Sqrt[1-((PR^2)/PS)]) (Sqrt[PS]) (-(PS^3)+7 (PS^3) ν-14 (PS^3) (ν^2)+7 (PS^3) (ν^3)))/(16 r))};

Hamiltonians

3PM Hamiltonian

https://arxiv.org/pdf/1908.01493.pdf  (10.10)

(* PM/PN Order Parameters *)
NG = 3;
NPN = 3;
Nη = 2*(NPN+1);

(* PM Coefficients *)
(* See "Kinematic Variables" section of this notebook for defs. of ET, γ, ξ, etc.*)

cPM[0] = ET;

cPM[1] =(ν^2*m^2/(γ^2*ξ))*(1-2*σ^2);

cPM[2] = (ν^2*m^3/(γ^2*ξ))*((3/4)*(1-5*σ^2) - (4*ν*σ/(γ*ξ))*(1-2*σ^2) - (ν^2/(2*γ^3*ξ^2))*(1-ξ)*(1-2*σ^2)^2);

cPM[3] = ((ν^2*m^4)/(γ^2*ξ))*((1/12)*(3-6*ν+206*ν*σ - 54*σ^2 + 108*ν*σ^2 + 4*ν*σ^3) - (4*ν*(3+12*σ^2 - 4*σ^4)*ArcSinh[Sqrt[(σ-1)/2]])/Sqrt[σ^2-1] - (3*ν*γ*(1-2*σ^2)*(1-5*σ^2))/(2*(1+γ)*(1+σ)) -3*ν*σ*(7-20*σ^2)/(2*γ*ξ) + (2*ν^3*(3-4*ξ)*σ*(1-2*σ^2)^2)/(γ^4*ξ^3) - (ν^2*(3+8γ-3ξ-15*σ^2-80*γ*σ^2 +15*ξ*σ^2)*(1-2*σ^2))/(4*γ^3*ξ^2) + (ν^4*(1-2*ξ)*(1-2*σ^2)^3)/(2*γ^6*ξ^4));

HPM =Sum[cPM[i]*(G/r)^i,{i,0,NG}];

3PN,(G^3) Hamiltonian

https://arxiv.org/pdf/1401.4548.pdf (5.17)

(* PM order parameter *)
NG = 3;
(* PN order parameter *)
NPN = 3;
Nη = 2*(NPN+1);

(* PN Coefficients *)
cPN[0] = PS/2 -1/r;

cPN[1] = (1/8)*(3*ν-1)*PS^2 - (1/2)*((3+ν)*PS +ν*PR^2)*(1/r) + 1/(2*r^2);

cPN[2] = (1/16)*(1-5ν+5ν^2)*PS^3 + (1/8)*((5-20ν-3ν^2)*PS^2 - 2ν^2*PR^2*PS - 3ν^2*PR^4)*(1/r) + (1/2)*((5+8ν)*PS +3ν*PR^2)*(1/r^2) -(1/4)*(1+3ν)*(1/r^3);

cPN[3] = (1/128)*(-5+35ν-70ν^2+35ν^3)*PS^4 + (1/16)*((-7+42ν-53ν^2-5ν^3)*PS^3 + (2-3ν)*ν^2*PR^2*PS^2 + 3*(1-ν)*ν^2*PR^4*PS - 5ν^3*PR^6)*(1/r) + ((1/16)*(-27+136ν+109ν^2)*PS^2 + (1/16)*(17+30ν)*ν*PR^2*PS+(1/12)*(5+43ν)*ν*PR^4)*(1/r^2) + (((-25/8) + ((π^2/64) - (335/48))*ν -(23ν^2/8))*PS + ((-85/16)-(3*π^2/64) -(7ν/4))*ν*PR^2)*(1/r^3) + ((1/8)+((109/12)-(21π^2/32))*ν)*(1/r^4);

HPN =Expand[Sum[cPN[i],{i,0,NG}]]/.{r->r/(G*m)};

PM Radiated Energy and Angular Momentum

https://arxiv.org/pdf/2203.04283.pdf   	2/3PM ΔJ (matches our expression)

https://arxiv.org/pdf/2101.07255.pdf 	3PM ΔE

https://arxiv.org/pdf/2210.05541.pdf	(12, 13)	  4PM ΔE

Coefficients

(* Results first derived by H. Herrmann, J. Parra-Martinez, M. S. Ruf, and M. Zeng in 2101.07255*)
f1=(210*sigma^6-552*sigma^5+339*sigma^4-912*sigma^3+3148*sigma^2-3336*sigma+1151)/(48(sigma^2-1)^(3/2));
f2=(-((35*sigma^4+60*sigma^3-150*sigma^2+76*sigma-5)/(8(sigma^2-1)^(1/2))));
f3=(((2*sigma^2-3)(35*sigma^4-30*sigma^2+11))/(8(sigma^2-1)^(3/2)));
eSigma=Pi*(f1+f2 Log[(sigma+1)/2]+f3 (sigma ArcSinh[((sigma-1)/2)^(1/2)])/(sigma^2-1)^(1/2));

g1=1/(24(sigma^2-1)^2) (-237+386*sigma+111*sigma^2-683*sigma^3+537*sigma^4+240*sigma^5-411*sigma^6+105*sigma^7);
g2=1/(4(sigma^2-1)) (-62+155*sigma+16*sigma^2-70*sigma^3-90*sigma^4+35*sigma^5);
g3=-1/(4(sigma^2-1)^2) (2*sigma^2-3)*(-12+19*sigma+72*sigma^2-70*sigma^3-60*sigma^4+35*sigma^5);

iSigma=(-(16/3)+(2*sigma^2)/(sigma^2-1)+4(2*sigma^2-3)/(sigma^2-1)*(sigma ArcSinh[((sigma-1)/2)^(1/2)])/(sigma^2-1)^(1/2));

cSigma=Pi*(g1+g2*Log[(sigma+1)/2]+g3*(sigma ArcSinh[((sigma-1)/2)^(1/2)])/(sigma^2-1)^(1/2));
dSigma=3*Pi*(5 sigma^2-1)/8*iSigma;

h = Sqrt[1+2*ν*(σ-1)];

Radiated Charges

ΔE2LIT = 0;
ΔE2HYPLIT = 0;

ΔE3LIT = ((G*m/b)^3*m*ν^2/(h))*(eSigma/.{sigma->σ});
ΔE3HYPLIT=(ΔE3LIT* j^3)/.{j->b*Sqrt[PS]/(G*m^2*ν)};

ΔJ2LIT = (G*m/b)^2*(J0*(2*ν*(2*σ^2-1)*iSigma)/Sqrt[σ^2-1])/.{sigma->σ};
ΔJ2HYPLIT=(ΔJ2LIT* j^2)/.{j->b*Sqrt[PS]/(G*m^2*ν)};

ΔJ3LIT = (G*m/b)^3*(J0*(ν/Sqrt[σ^2-1])*(cSigma + 2*dSigma - (ν*Sqrt[σ^2-1]*eSigma/h^2)))/.{sigma->σ};
ΔJ3HYPLIT=(ΔJ3LIT* j^3)/.{j->b*Sqrt[PS]/(G*m^2*ν)};

(* From anc-4pm.m at https://arxiv.org/abs/2210.05541 *)
ΔE4LITCOEFF = -(1/((b^4) (Sqrt[1+2 ν (-1+σ)])))(m^5) (ν^2) ((1/(1440 (σ^7) (((-1+(σ^2)))^(5/2))))(1575-8250 (σ^2)+35710 (σ^4)-142640 (σ^6)-5560073 (σ^8)-417302 (σ^10)+4034092 (σ^12)-587336 (σ^14)+6144 (σ^16))+(1/(1440 (((-1+(σ^2)))^3)))(15 (π^2) (-1+(σ^2)) (27 (((1-5 (σ^2)))^2) (-1+(σ^2))+2 (1051+8535 (σ^2)+3249 (σ^4)-3747 (σ^6)))+64 (-13895-9916 (σ^2)+70104 (σ^4)-54616 (σ^6)+8464 (σ^8)-96 (σ^10)+45 (3-44 (σ^2)+152 (σ^4)-192 (σ^6)+80 (σ^8))))+(-((16 σ (-869-496 (σ^2)+204 (σ^4)+16 (σ^6)))/(5 (((-1+(σ^2)))^(3/2))))+(1/(480 (σ^8) (((-1+(σ^2)))^3)))(525-3100 (σ^2)+13690 (σ^4)-55260 (σ^6)+816595 (σ^8)+3752006 (σ^10)-1978290 (σ^12)-1029342 (σ^14)+213480 (σ^16)+24576 (σ^18))) ArcCosh[σ]+(-((16 (σ^2) (-9-6 (σ^2)+8 (σ^4)))/(((-1+(σ^2)))^2))-((32 σ (-3+2 (σ^2)) (-8-51 (σ^2)-6 (σ^4)+8 (σ^6)))/(((-1+(σ^2)))^(7/2)))) ((ArcCosh[σ])^2)-(((2057+15261 (σ^2)+3387 (σ^4)-4321 (σ^6)) ArcCosh[σ] Log[2])/(4 (((-1+(σ^2)))^2)))-(((2039+13155 (σ^2)+1221 (σ^4)-1823 (σ^6)) ArcCosh[σ] Log[σ])/(4 (((-1+(σ^2)))^2)))-(((9+1053 (σ^2)+1083 (σ^4)-1249 (σ^6)) ArcCosh[σ] Log[((1+σ)/2)])/(4 (((-1+(σ^2)))^2)))-(1/(8 (((-1+(σ^2)))^2)))(36-1209 σ+4212 (σ^2)-6422 (σ^3)+4332 (σ^4)+1755 (σ^5)-4996 (σ^6)+2100 (σ^7)) PolyLog[2,(Sqrt[((-1+σ)/(1+σ))])]+(1/(32 (((-1+(σ^2)))^2)))(36-1209 σ+4212 (σ^2)-6422 (σ^3)+4332 (σ^4)+1755 (σ^5)-4996 (σ^6)+2100 (σ^7)) PolyLog[2,((-1+σ)/(1+σ))]+(((2039+13155 (σ^2)+1221 (σ^4)-1823 (σ^6)) PolyLog[2,-(((σ-(Sqrt[-1+(σ^2)])))^2)])/(8 (((-1+(σ^2)))^2)))+(((9+1053 (σ^2)+1083 (σ^4)-1249 (σ^6)) PolyLog[2,-σ+(Sqrt[-1+(σ^2)])])/(2 (((-1+(σ^2)))^2)))+ν ((1/(96 (((-1+(σ^2)))^2)))(π^2) (108-1485 σ-1188 (σ^2)+4993 (σ^3)+3780 (σ^4)-3195 (σ^5)-2700 (σ^6)+1575 (σ^7)+54 (((1-5 (σ^2)))^2) (-1+(σ^2))-4 (1051+8535 (σ^2)+3249 (σ^4)-3747 (σ^6)))+(1/(45 (((-1+(σ^2)))^3)))4 (13895+9916 (σ^2)-70104 (σ^4)+54616 (σ^6)-8464 (σ^8)+96 (σ^10)+30 σ (-1+2 (σ^2)) (-109+258 (σ^2)-216 (σ^4)+64 (σ^6))-45 (3-44 (σ^2)+152 (σ^4)-192 (σ^6)+80 (σ^8))-30 σ (-1535+2292 (σ^2)+384 (σ^4)-1528 (σ^6)+384 (σ^8)))+(1/(352800 (σ^8) (((-1+(σ^2)))^(5/2))))(385875-1837500 (σ^2)+7188300 (σ^4)-21241500 (σ^6)+767410066 (σ^8)+3966858415 (σ^10)-3429240286 (σ^12)-791542442 (σ^14)+393897472 (σ^16)-490 σ (1575-8250 (σ^2)+35710 (σ^4)-142640 (σ^6)-5560073 (σ^8)-417302 (σ^10)+4034092 (σ^12)-587336 (σ^14)+6144 (σ^16)+3840 (σ^7) (((-1+2 (σ^2)))^3) (-8+5 (σ^2))))+(-(1/(15 (((-1+(σ^2)))^(3/2))))32 (15 (3+6 (σ^2)-28 (σ^4)+8 (σ^6))-3 σ (-869-496 (σ^2)+204 (σ^4)+16 (σ^6))-10 (-82-425 (σ^2)-32 (σ^4)+44 (σ^6)))+(1/(3360 (σ^9) (((-1+(σ^2)))^3)))(3675-19950 (σ^2)+79800 (σ^4)-246540 (σ^6)+222810 (σ^8)-25426269 (σ^10)-37185456 (σ^12)+46406238 (σ^14)+2662204 (σ^16)-3592192 (σ^18)+53760 (σ^10) (-3+2 (σ^2)) (((-1+2 (σ^2)))^3)-14 σ (525-3100 (σ^2)+13690 (σ^4)-55260 (σ^6)+816595 (σ^8)+3752006 (σ^10)-1978290 (σ^12)-1029342 (σ^14)+213480 (σ^16)+24576 (σ^18)))) ArcCosh[σ]+(-((16 (-2 (σ^2) (-9-6 (σ^2)+8 (σ^4))+σ (-151-226 (σ^2)+24 (σ^4)+16 (σ^6))))/(((-1+(σ^2)))^2))+(1/(((-1+(σ^2)))^(7/2)))64 (σ (-3+2 (σ^2)) (-8-51 (σ^2)-6 (σ^4)+8 (σ^6))+(σ^2) (60+35 (σ^2)-59 (σ^4)+4 (σ^8)))) ((ArcCosh[σ])^2)+(1/(4 (((-1+(σ^2)))^2)))(1263 σ-3883 (σ^3)+1065 (σ^5)-525 (σ^7)+2 (2057+15261 (σ^2)+3387 (σ^4)-4321 (σ^6))) ArcCosh[σ] Log[2]+(1/(4 (((-1+(σ^2)))^2)))(2 (2039+13155 (σ^2)+1221 (σ^4)-1823 (σ^6))+σ (1041-2773 (σ^2)-1065 (σ^4)+525 (σ^6))) ArcCosh[σ] Log[σ]+(1/(4 (((-1+(σ^2)))^2)))(2 (9+1053 (σ^2)+1083 (σ^4)-1249 (σ^6))+3 (37 σ-185 (σ^3)+355 (σ^5)-175 (σ^7))) ArcCosh[σ] Log[((1+σ)/2)]+(1/(4 (((-1+(σ^2)))^2)))(36-1209 σ+4212 (σ^2)-6422 (σ^3)+4332 (σ^4)+1755 (σ^5)-4996 (σ^6)+2100 (σ^7)-6 (6-37 σ-66 (σ^2)+185 (σ^3)+210 (σ^4)-355 (σ^5)-150 (σ^6)+175 (σ^7))) PolyLog[2,(Sqrt[((-1+σ)/(1+σ))])]-(1/(16 (((-1+(σ^2)))^2)))(36-1209 σ+4212 (σ^2)-6422 (σ^3)+4332 (σ^4)+1755 (σ^5)-4996 (σ^6)+2100 (σ^7)-6 (6-37 σ-66 (σ^2)+185 (σ^3)+210 (σ^4)-355 (σ^5)-150 (σ^6)+175 (σ^7))) PolyLog[2,((-1+σ)/(1+σ))]-(1/(8 (((-1+(σ^2)))^2)))(2 (2039+13155 (σ^2)+1221 (σ^4)-1823 (σ^6))+σ (1041-2773 (σ^2)-1065 (σ^4)+525 (σ^6))) PolyLog[2,-(((σ-(Sqrt[-1+(σ^2)])))^2)]-(1/(2 (((-1+(σ^2)))^2)))(2 (9+1053 (σ^2)+1083 (σ^4)-1249 (σ^6))+3 (37 σ-185 (σ^3)+355 (σ^5)-175 (σ^7))) PolyLog[2,-σ+(Sqrt[-1+(σ^2)])]));

ΔE4LIT = G^4*ΔE4LITCOEFF;

ΔE4HYPLIT=(G^4*ΔE4LITCOEFF* j^4)/.{j->Sqrt[PS]*b/(G*m^2*ν)};

PN Energy and Angular Momentum Fluxes

Instantaneous Energy Flux

https://doi.org/10.1103/PhysRevD.77.064035		Arun et. al E Flux:	(5.1),(5.2) [SH Coords]	(6.11) [ADM Transf.]

(* Energy Prefactor *)
(* Not reduced!!! *)
FEPRESH = (32/5)*ν^2*(G*m/r)^3*(m/r);

(* 0PN Rel.   3PN Abs. *)
cFEPNSHLIT[0] = (VS - (11/12)VR^2);

(* 1PN Rel.   4PN Abs. *)
cFEPNSHLIT[1] =((785/336 -71ν/28)VS^2
		+(-1487/168 +58ν/7)VS VR^2
		+(687/112 -155ν/28)VR^4
		+(G m/r)(-170/21 + 10ν/21)VS
		+(G m/r)(367/42 - 5ν/14)VR^2);

(* 2PN Rel.   5PN Abs. *)
cFEPNSHLIT[2] = ((47/14 - 5497ν/504 + 2215ν^2/252)VS^3 
			+ (-573/56 + 1713ν/28 - 1573ν^2/42)VS^2 VR^2
			+ (1009/84 - 5069ν/56 + 631ν^2/14)VS VR^4
			+ (-2501/504 + 10117ν/252 -2101ν^2/126)VR^6
			+ (G m/r)(-247/14 + 5237ν/252 - 199ν^2/36)VS^2
			+ (G m/r)(4987/84 - 8513ν/84 + 2165ν^2/84)VS VR^2 
			+ (G m/r)(-5585/126 + 60971ν/756 - 7145ν^2/378)VR^4);

(* 2.5PN Rel.   5.5PN Abs. *)
cFEPNSHLIT[3] = (G m/r)((-12349ν/210)VS^2 VR +(4524ν/35)VS VR^3 +(-985ν/14) VR^5);
(*cFEPNSHLIT[3] = 0;*)

(* 3PN Rel.   6PN Abs. *)
cFEPNSHLIT[4] = ((80315/14784 - 694427ν/22176 + 604085ν^2/11088 - 16985ν^3/462)VS^4
			+ (-31499/1008 + 1119913ν/5544 - 44701ν^2/132 + 38725ν^3/231)VS^3 VR^2
			+ (204349/2464 - 3522149ν/7392 + 2354753ν^2/3696 - 109447ν^3/462)VS^2 VR^4
			+ (-1005979/11088 + 2589599ν/5544 -1322141ν^2/2772 +90455ν^3/693)VS VR^6
			+ (1507925/44352 -20365ν/126 + 687305ν^2/5544 -32755ν^3/1386)VR^8
			+ (G m/r)(-61669/3696 + 95321ν/1008 -955013ν^2/11088 + 47255ν^3/1386)VS^3
			+ (G m/r)(136695/1232 -202693ν/336 + 744377ν^2/1232 -931099ν^3/5544)VS^2 VR^2
			+ (G m/r)(-715157/3696 + 35158037ν/33264 -3672143ν^2/3696 + 871025ν^3/4158)VS VR^4
			+ (G m/r)(5476951/55440 - 671765ν/1232 + 5205019ν^2/11088 -860477ν^3/11088)VR^6);

(* Combine terms. *)
FEPNSHLIT = FEPRESH*Sum[cFEPNSHLIT[i],{i,0,4}];

(* Conversion to ADM coordinates: Begins at G^4, 5PN Abs. *)
δFEPRE = (-ν^2)(G m/r)^4 (m/r);

(* 2PN Rel.   5PN Abs. *)
cδFEPNLIT[0] =  ((184ν/5)VS^2 - (736ν/5)VS VR^2 + (320ν/3)VR^4);

(* 3PN Rel.   6PN Abs. *)
cδFEPNLIT[1] =   ((5886ν/35 - 1616ν^2/7)VS^3 
			+ (-129866ν/105 +21598ν^2/15)VS^2 VR^2
			+ (689434ν/315 - 714608ν^2/315)VS VR^4 
			+ (-116138ν/105 + 110986ν^2/105)VR^6);

(* Combine terms. *)
δFEPNLIT = δFEPRE*Sum[cδFEPNLIT[i],{i,0,1}];

(* Compute ADM Flux *)
FEPNLIT = FEPNSHLIT + δFEPNLIT;

Instantaneous Angular Momentum Flux

https://doi.org/10.1103/PhysRevD.80.124018		Arun et. al. J Flux	(3.1)-(3.4) [SH Coords]

(* Angular Momentum Prefactor *)
FJPRE =  ν^2 (G*m/r)^2 m r VP;

(* 0PN Rel.   2PN Abs. *)
cFJPNLIT[0] = ((16/5)VS - (24/5)VR^2 + (16/5)*(G m/r));

(* 1PN Rel.   3PN Abs. *)
cFJPNLIT[1] = ((614/105 - 1096ν/105)VS^2
			+ (-296/35 + 1108ν/35)VS VR^2
			+ (38/7 - 144ν/7)VR^4
			+ (G m/r)(-464/105 - 152ν/21)VS
			+ (G m/r)(496/35 + 788ν/105)VR^2);

(* 2PN Rel.   4PN Abs. *)
(* ADM GAUGE *)
cFJPNLIT[2] =( (533/63 - 353ν/9 + 614ν^2/15)VS^3 
		+ (-2246/105 + 12653ν/105 - 15637ν^2/105)VS^2 VR^2
		+ (715/21 - 3361ν/21 + 448ν^2/3)VS VR^4
		+ (-52/3 + 652ν/9 - 388ν^2/9) VR^6
		+ (G m/r) (11/21-1333ν/63 +4022ν^2/315)VS^2
		+ (G m/r)(21853/315 +2942ν/105 +2551ν^2/315)VS VR^2
		+ (G m/r)(-22312/315 + 1999ν/45 -277ν^2/9)VR^4);

(* 2.5PN Rel.   4.5PN Abs. *)
cFJPNLIT[3] = (1/5)(G m/r)((-27744ν/35)VS^2 VR +(19144ν/7)VS VR^3 +(-1960ν) VR^5)+ (1/5)(G m/r)^2((-100816ν/105)VS VR  + (39056ν/35)VR^3);

(* 3PN Rel.   5PN Abs. *)
(* ADM GAUGE *)
cFJPNLIT[4] =( (145919/13860 - 110423ν/1260 + 1079083ν^2/4620 -30229ν^3/165)VS^4
		+ (-2473/70 + 763409ν/2310 - 2155249ν^2/2310 + 543171ν^3/770)VS^3 VR^2
		+ (18695/231 -632111ν/924 + 1552525ν^2/924 -61970ν^3/77) VS^2 VR^4
		+ (-451/6 + 59870ν/99 -14841ν^2/11 +32342ν^3/99) VS VR^6
		+ (93/4 - 4035ν/22 +4294ν^2/11 -410ν^3/11) VR^8
		+ (G m/r) (483097/13860 - 17429ν/154 +693331ν^2/4620 +91ν^3/165)VS^3
		+ (G m/r) (205817/13860 - 22549ν/105 -6112567ν^2/13860 -9979ν^3/66)VS^2 VR^2
		+ (G m/r) (-100999/1980 + 6446971ν/6930 -512285ν^2/2772 + 738973ν^3/3465)VS VR^4
		+ (G m/r) (1932907/69300 - 40997ν/70 + 955543ν^2/2772 -50525ν^3/1386)VR^6);

(* Combine terms. *)
FJPNLIT = FJPRE*Sum[cFJPNLIT[i],{i,0,4}];

Generators and Transformations

E Flux Schott Terms and Generators

ESchottGenG3PN6 = (1/(2661120 (m^3) (r^3) (ν^5)))(G^3) (192 (PR^7) (-301585+1433696 ν-1099688 (ν^2)+209632 (ν^3))-3 PR (-1300992 (m^6) (ν^6)+6336 (m^4) PS (ν^4) (-1193+928 ν)+176 (m^2) (PS^2) (ν^2) (20609-33236 ν+24064 (ν^2))+(PS^3) (-9670325+38024912 ν-2340480 (ν^2)+189696 (ν^3)))+24 (PR^5) (176 (m^2) (ν^2) (2501-20234 ν+8404 (ν^2))+PS (4836297-19193432 ν+4204056 (ν^2)+982464 (ν^3)))-4 (PR^3) (-6336 (m^4) (ν^4) (-687+620 ν)+176 (m^2) PS (ν^2) (-1171-119024 ν+39268 (ν^2))+(PS^2) (21522507-69081008 ν-36985248 (ν^2)+18245088 (ν^3))));

ESchottG3PN6 = -(1/(887040 (m^4) (r^4) (ν^6)))(G^3) (640 (PR^8) (-301585+1433696 ν-1099688 (ν^2)+209632 (ν^3))+256 (PR^2) (20328 (m^6) (ν^6)-4092 (m^4) PS (ν^4) (-43+30 ν)-66 (m^2) (PS^2) (ν^2) (2202-1239 ν+2272 (ν^2))+(PS^3) (580778-2007391 ν-368718 (ν^2)+101952 (ν^3)))+512 (PR^6) (22 (m^2) (ν^2) (2501-20234 ν+8404 (ν^2))+PS (840913-3348556 ν+727568 (ν^2)+216712 (ν^3)))-768 (PR^4) (66 (m^4) (687-620 ν) (ν^4)+44 (m^2) PS (ν^2) (-43-7164 ν+1992 (ν^2))+(PS^2) (482697-1586600 ν-684608 (ν^2)+388040 (ν^3)))+PS (-1300992 (m^6) (ν^6)+2112 (m^4) PS (ν^4) (-3271+1860 ν)+176 (m^2) (PS^2) (ν^2) (39311-100502 ν+60316 (ν^2))+(PS^3) (-13911925+59922304 ν-32738056 (ν^2)+14721312 (ν^3))));

ESchottGenG4PN4 = (1/(2100 (r^4) ν))(G^3) m PR (20 (PR^2) r (-687+620 ν)-15 PS r (-1193+928 ν)-2 (m^2) (ν^2) (-1540 r+G m (14897+4040 ν)));

ESchottG4PN4 = (1/(2100 (r^5) (ν^2)))(G^3) (5 (PS^2) r (3271-1860 ν)-3080 G (m^5) (ν^4)-120 (PR^4) r (-687+620 ν)+(m^2) PS (ν^2) (3080 r-G m (61549+3400 ν))+80 (PR^2) (31 PS r (-43+30 ν)+2 (m^2) (ν^2) (-77 r+4 G m (299+87 ν))));

(* ESchottGenDiff = ESchottGenG4PN4 - ESchottGenG3PN4 *)
ESchottGenDiff = -(((G^4) (m^4) PR ν (14897+4040 ν))/(1050 (r^4)));

J Flux Schott Terms and Generators

JSchottGenG3PN5 = (1/(6652800 (m^4) (r^3) (ν^5)))(G^2) (372556800 G (m^4) (PR^4) (ν^4)-429732864 G (m^4) (PR^2) PS (ν^4)+65083392 G (m^4) (PS^2) (ν^4)+16800 (PR^7) r (-1023+8070 ν-17176 (ν^2)+1640 (ν^3))-20 (PR^3) (24 (PS^2) r (-23721+40429 ν+156575 (ν^2)+118344 (ν^3))+352 (m^4) (ν^4) (54 r (19-72 ν)+G m (-20147+36994 ν+21636 (ν^2)))+(m^2) PS (ν^2) (-1056 r (-681+2179 ν+308 (ν^2))+G m (-7848865+43677456 ν+40412984 (ν^2)-6894048 (ν^3))))-15 PR (32 (PS^3) r (14493-72457 ν+47373 (ν^2)+34734 (ν^3))+2112 (m^6) (ν^6) (-336 r+G m (2099+1836 ν))-352 (m^4) PS (ν^4) (24 r (7-86 ν)+G m (-14359+4230 ν+17840 (ν^2)))+(m^2) (PS^2) (ν^2) (1408 r (-293+1144 ν+975 (ν^2))-G m (-1002767+18120288 ν+34681992 (ν^2)+4122720 (ν^3))))+24 (PR^5) (100 PS r (2013-35150 ν+104636 (ν^2)+35760 (ν^3))+(m^2) (ν^2) (17600 r (39-163 ν+97 (ν^2))-G m (11200991-56213840 ν+9962040 (ν^2)+11956960 (ν^3)))));

JSchottG3PN5= -(1/(2217600 (m^5) (r^4) (ν^6)))(G^2) (869299200 G (m^4) (PR^5) (ν^4)-1212963840 G (m^4) (PR^3) PS (ν^4)+351571968 G (m^4) PR (PS^2) (ν^4)+50400 (PR^8) r (-1023+8070 ν-17176 (ν^2)+1640 (ν^3))-160 (PR^4) (15 (PS^2) r (-13385+10388 ν+140004 (ν^2)+99312 (ν^3))+88 (m^4) (ν^4) (45 r (19-72 ν)+G m (-23981+50975 ν+22958 (ν^2)))+(m^2) PS (ν^2) (-1320 r (-215+713 ν+52 (ν^2))+G m (-4964873+23884350 ν+13265232 (ν^2)-2308928 (ν^3))))-160 (PR^2) (12 (PS^3) r (15059-59302 ν-565 (ν^2)+25512 (ν^3))+1056 (m^6) (ν^6) (-63 r+G m (753+466 ν))-44 (m^4) PS (ν^4) (396 r (1-8 ν)+G m (-30169+32028 ν+27656 (ν^2)))-3 (m^2) (PS^2) (ν^2) (-44 r (-971+2786 ν+3056 (ν^2))+G m (-345373+2207260 ν+4298180 (ν^2)+559248 (ν^3))))+5 (709632 G (m^9) (ν^8)+64 (PS^4) r (13588-76747 ν+76206 (ν^2)+46572 (ν^3))+2112 (m^6) PS (ν^6) (-336 r+G m (3639+2500 ν))-352 (m^4) (PS^2) (ν^4) (120 r (-7+8 ν)+G m (-3690-20993 ν+1328 (ν^2)))+(m^2) (PS^3) (ν^2) (1408 r (-461+1768 ν+804 (ν^2))-G m (-3519111+30374200 ν+39143552 (ν^2)+10166784 (ν^3))))+32 (PR^6) (350 PS r (2871-33500 ν+86650 (ν^2)+24944 (ν^3))+(m^2) (ν^2) (30800 r (39-163 ν+97 (ν^2))-G m (27158107-133214730 ν+34654880 (ν^2)+26092120 (ν^3)))));

Canonical Transformations and Generators

(* Only need CanonGenG1PN2.5 to do G3PN6,G4PN4 E flux matching and G3PN5 J flux matching. The G3PN2.5 expression below is required to match the ADM and Isotropic gauge Hamiltonians at G3PN3 order. *)
CanonGenG3PN3 = G (((PR^3)/(8 m))-((PR PS (-2+ν))/(8 m ν)))+(1/2) G m PR ν+(((G^2) (m^2) PR (-ν+(ν^2)))/(4 r))+(((G^3) (m^3) PR (304 ν+3 (π^2) ν+36 (ν^2)+12 (ν^3)))/(192 (r^2)))+(((G^2) (-(((PR^3) (5+34 ν-5 (ν^2)))/(48 ν))-((PR PS (21+2 ν-(ν^2)))/(16 ν))))/r)+G (((PR^5)/(16 (m^3) ν))-(((PR^3) PS)/(48 (m^3) ν))-((PR (PS^2) (1-3 ν+(ν^2)))/(16 (m^3) (ν^3))));

(* transformed q[i] *)
CanonQG3PN3 = q[i]+((G m ν q[i])/(2 r))-((G (-4 PR r p[i]+2 PR r ν p[i]-2 PS q[i]-3 (PR^2) ν q[i]+PS ν q[i]))/(8 m r ν))+(((G^2) (-(m^2) ν q[i]+(m^2) (ν^2) q[i]))/(4 (r^2)))+(((G^3) (304 (m^3) ν q[i]+3 (m^3) (π^2) ν q[i]+48 (m^3) (ν^2) q[i]))/(192 (r^3)))+(1/(16 (r^2) ν))(G^2) (-42 PR r p[i]-4 PR r ν p[i]+2 PR r (ν^2) p[i]-5 (PR^2) q[i]-21 PS q[i]-32 (PR^2) ν q[i]-4 PS ν q[i]+4 (PR^2) (ν^2) q[i]+2 PS (ν^2) q[i])+(1/(48 (m^3) r (ν^3)))G (-12 PR PS r p[i]+36 PR PS r ν p[i]-2 (PR^3) r (ν^2) p[i]-12 PR PS r (ν^2) p[i]-3 (PS^2) q[i]+9 (PS^2) ν q[i]+15 (PR^4) (ν^2) q[i]-3 (PR^2) PS (ν^2) q[i]-3 (PS^2) (ν^2) q[i]);

(* transformed p[i] *)
CanonPG3PN3= p[i]+((G m ν (-r p[i]+PR q[i]))/(2 (r^2)))+(1/(8 m (r^2) ν))G (-2 PS r p[i]-3 (PR^2) r ν p[i]+PS r ν p[i]+2 PR PS q[i]+3 (PR^3) ν q[i]-PR PS ν q[i])+(((G^2) ((m^2) r ν p[i]-2 (m^2) PR ν q[i]+(m^2) PR (ν^2) q[i]))/(4 (r^3)))+(1/(48 (r^3) ν))(G^2) (15 (PR^2) r p[i]+63 PS r p[i]+108 (PR^2) r ν p[i]+24 PS r ν p[i]-12 PS r (ν^2) p[i]-20 (PR^3) q[i]-126 PR PS q[i]-148 (PR^3) ν q[i]-24 PR PS ν q[i]+8 (PR^3) (ν^2) q[i]+12 PR PS (ν^2) q[i])+(1/(16 (m^3) (r^2) (ν^3)))G ((PS^2) r p[i]-3 (PS^2) r ν p[i]-5 (PR^4) r (ν^2) p[i]+(PR^2) PS r (ν^2) p[i]+(PS^2) r (ν^2) p[i]-PR (PS^2) q[i]+3 PR (PS^2) ν q[i]+5 (PR^5) (ν^2) q[i]-(PR^3) PS (ν^2) q[i]-PR (PS^2) (ν^2) q[i])+(1/(192 (r^4)))(G^3) (-304 (m^3) r ν p[i]-3 (m^3) (π^2) r ν p[i]-96 (m^3) r (ν^2) p[i]+24 (m^3) r (ν^3) p[i]+912 (m^3) PR ν q[i]+9 (m^3) (π^2) PR ν q[i]+192 (m^3) PR (ν^2) q[i]-24 (m^3) PR (ν^3) q[i]);

(* The additional scaling parameters are also fixed by matching the Hamiltonians. *)

μ = 1;
ν = 1/(m ν); (* Take care not to confuse the momentum-scaling parameter on the LHS of this equation (ν) with the symmetric mass ratio appearing on the RHS (also called ν). *)

3PM Fluxes

E Flux

AEG3 =(((ν^3) (1151-3336 σ+3148 (σ^2)-912 (σ^3)+339 (σ^4)-552 (σ^5)+210 (σ^6)))/(24 (γ^3) ξ (-1+(σ^2))));
BEG3 =-(((ν^3) (-5+76 σ-150 (σ^2)+60 (σ^3)+35 (σ^4)))/(4 (γ^3) ξ));
CEG3 =(((ν^3) σ (-33+112 (σ^2)-165 (σ^4)+70 (σ^6)))/(4 (γ^3) ξ (-1+(σ^2))));

FLUXEG2 = 0;

FLUXEG3 = (G*m/r)^3*(m/r)*( AEG3 + BEG3* Log[((1+σ)/2)] + CEG3*ArcSinh[((Sqrt[-1+σ])/(Sqrt[2]))]/(Sqrt[-1+(σ^2)])) ;

FLUXE = FLUXEG2 + FLUXEG3;

J Flux

AJG2 =-((2 (ν^2) (-1+2 (σ^2)) (-8+5 (σ^2)))/(3 (γ^2) ξ (-1+σ) (1+σ)));
BJG2 =0;
CJG2 =((4 (ν^2) σ (-1+(σ^2)) (-3+2 (σ^2)) (-1+2 (σ^2)))/((γ^2) ξ (((-1+σ))^2) (((1+σ))^2)));
AJG3 =(1/(24 (γ^9) (ξ^3) (((-1+(σ^2)))^2)))(ν^2) (16 (ν^4) (((-1+σ))^3) (8-40 σ-37 (σ^2)+121 (σ^3)+52 (σ^4)-92 (σ^5)-20 (σ^6)+20 (σ^7))+64 (ν^3) (8-8 σ-21 (σ^2)+13 (σ^3)+10 (σ^4)+19 (σ^5)-31 (σ^7)+10 (σ^9))+2 (γ^5) (ξ^2) (-141+386 σ-525 (σ^2)-683 (σ^3)+1377 (σ^4)+240 (σ^5)-711 (σ^6)+105 (σ^7)+16 (γ^3) (((1-2 (σ^2)))^2) (-8+5 (σ^2)))+16 (ν^2) (-1+2 (σ^2)) (8-5 (σ^2)+2 (γ^4) ξ (8-25 σ-4 (σ^2)+37 (σ^3)-6 (σ^4)-18 (σ^5)+8 (σ^6)))+(γ^4) ν ξ (16 σ (-25+87 (σ^2)-92 (σ^4)+36 (σ^6))+γ ξ (1715-5444 σ+5641 (σ^2)+3056 (σ^3)-11049 (σ^4)+4908 (σ^5)+3675 (σ^6)-2712 (σ^7)+210 (σ^8))));
BJG3 =((ν^2) (2 (-62+155 σ+16 (σ^2)-70 (σ^3)-90 (σ^4)+35 (σ^5))+ν (253-944 σ+701 (σ^2)+360 (σ^3)-105 (σ^4)-440 (σ^5)+175 (σ^6))))/(4 (γ^4) ξ (-1+σ) (1+σ));
CJG3 =-(1/(4 (γ^9) (ξ^3) (((-1+σ))^3) (((1+σ))^3)))(ν^2) (-1+(σ^2)) (32 (γ^8) (ξ^2) σ (((1-2 (σ^2)))^2) (-3+2 (σ^2))+16 (ν^2) σ (3-8 (σ^2)+4 (σ^4)) (-1+(ν^2) (((-1+σ))^3) (1-5 σ-2 (σ^2)+2 (σ^3))+4 ν (1-σ-(σ^3)+(σ^5)))+(γ^5) (ξ^2) σ (-3+2 (σ^2)) (-48+38 σ+288 (σ^2)-140 (σ^3)-240 (σ^4)+70 (σ^5)+ν (85-172 σ-459 (σ^2)+856 (σ^3)+135 (σ^4)-620 (σ^5)+175 (σ^6)))+16 (γ^4) ν ξ (-1+2 (σ^2)) (-3+21 (σ^2)-28 (σ^4)+12 (σ^6)+2 ν (3-21 (σ^2)+10 (σ^3)+28 (σ^4)-16 (σ^5)-12 (σ^6)+8 (σ^7))));

FLUXJG2 = (G*m/r)^2*(J0/r)*( AJG2 + BJG2* Log[((1+σ)/2)] + CJG2*ArcSinh[((Sqrt[-1+σ])/(Sqrt[2]))]/(Sqrt[-1+(σ^2)]));

FLUXJG3 = (G*m/r)^3*(J0/r)*( AJG3 + BJG3* Log[((1+σ)/2)] + CJG3*ArcSinh[((Sqrt[-1+σ])/(Sqrt[2]))]/(Sqrt[-1+(σ^2)])) ;

FLUXJ= FLUXJG2 + FLUXJG3;

4PM E Flux

(* Only the G^4 term in the 4PM flux. *)

FLUXEG4 = (G*m/r)^4*(m/r)*(-(1/(24 (γ^10) (ξ^3) (((-1+(σ^2)))^2)))(ν^5) (1151-3336 σ+3148 (σ^2)-912 (σ^3)+339 (σ^4)-552 (σ^5)+210 (σ^6)) (-1-2 (σ^2)+4 (σ^4)+(ν^2) (((-1+σ))^3) (3-7 σ-6 (σ^2)+6 (σ^3))+4 ν (1-2 σ+2 (σ^2)-4 (σ^4)+3 (σ^5)))-(1/(24 (γ^6) (ξ^2) (((-1+(σ^2)))^3)))(ν^4) (1-3 (σ^2)+2 (σ^4)) (3366-6881 σ+726 (σ^2)+4502 (σ^3)-1182 (σ^4)+1071 (σ^5)-2910 (σ^6)+1260 (σ^7)+2 ν (-1064+3575 σ-3613 (σ^2)+1072 (σ^3)+66 (σ^4)-1533 (σ^5)+3723 (σ^6)-3066 (σ^7)+840 (σ^8)))+(1/(470400 (γ^3) ξ (σ^8) (((-1+(σ^2)))^2)))(ν^3) (-245 σ (1575-8250 (σ^2)+35710 (σ^4)-142640 (σ^6)-5560073 (σ^8)-417302 (σ^10)+4034092 (σ^12)-587336 (σ^14)+6144 (σ^16))+ν (-385875+771750 σ+1837500 (σ^2)-4042500 (σ^3)-7188300 (σ^4)+17497900 (σ^5)+21241500 (σ^6)-69893600 (σ^7)-752357266 (σ^8)-2724435770 (σ^9)-4066583215 (σ^10)-204477980 (σ^11)+3666321886 (σ^12)+1976705080 (σ^13)+558224042 (σ^14)-287794640 (σ^15)-318633472 (σ^16)+3010560 (σ^17)))-(1/(1920 (γ^3) ξ (((-1+(σ^2)))^(3/2))))(ν^3) (512 (-1+(σ^2)) (-1720-4927 (σ^2)+1484 (σ^4)-12 (σ^6)+2 ν (1720+6165 σ+4927 (σ^2)+1950 (σ^3)-1484 (σ^4)-960 (σ^5)+12 (σ^6)))+15 (π^2) (2075+17367 (σ^2)+5553 (σ^4)-6819 (σ^6)+ν (-4150-1485 σ-34734 (σ^2)+4993 (σ^3)-11106 (σ^4)-3195 (σ^5)+13638 (σ^6)+1575 (σ^7))))+((1/((γ^3) ξ (((-1+(σ^2)))^(3/2))))12 (ν^3) σ (σ (-9-6 (σ^2)+8 (σ^4))+ν (-151+18 σ-226 (σ^2)+12 (σ^3)+24 (σ^4)-16 (σ^5)+16 (σ^6)))-(1/((γ^3) ξ (((-1+(σ^2)))^3)))24 (ν^3) σ (-3+2 (σ^2)) (8+51 (σ^2)+6 (σ^4)-8 (σ^6)+2 ν (-8-20 σ-51 (σ^2)-25 (σ^3)-6 (σ^4)+3 (σ^5)+8 (σ^6)+2 (σ^7)))) ((ArcCosh[σ])^2)-(1/(4 (γ^10) (ξ^3) (((-1+(σ^2)))^(3/2))))(ν^5) (4 (σ^3) (-89+389 (σ^2)-600 (σ^4)+280 (σ^6))+(ν^2) (((-1+σ))^2) (-66-165 σ+1068 (σ^2)+178 (σ^3)-3890 (σ^4)+1167 (σ^5)+5600 (σ^6)-2800 (σ^7)-2520 (σ^8)+1540 (σ^9))+ν (33-666 (σ^2)+1424 (σ^3)+1589 (σ^4)-6224 (σ^5)+312 (σ^6)+9600 (σ^7)-4340 (σ^8)-4480 (σ^9)+2800 (σ^10))) ArcSinh[((Sqrt[-1+σ])/(Sqrt[2]))]+((1/(4 (γ^10) (ξ^3) (((-1+(σ^2)))^2)))(ν^5) (5-76 σ+145 (σ^2)+16 (σ^3)-185 (σ^4)+60 (σ^5)+35 (σ^6)) (-1-2 (σ^2)+4 (σ^4)+(ν^2) (((-1+σ))^3) (3-7 σ-6 (σ^2)+6 (σ^3))+4 ν (1-2 σ+2 (σ^2)-4 (σ^4)+3 (σ^5)))+(1/(2 (γ^6) (ξ^2) (-1+(σ^2))))(ν^4) (-1+2 (σ^2)) (-38+145 σ+24 (σ^2)-370 (σ^3)+150 (σ^4)+105 (σ^5)+2 ν (33-107 σ-24 (σ^2)+378 (σ^3)-335 (σ^4)-15 (σ^5)+70 (σ^6)))) Log[((1+σ)/2)]+ArcCosh[σ] (-(1/(5 (γ^3) ξ (-1+(σ^2))))4 (ν^3) (3 σ (869+496 (σ^2)-204 (σ^4)-16 (σ^6))+2 ν (-865-2607 σ-4340 (σ^2)-1488 (σ^3)+100 (σ^4)+612 (σ^5)+320 (σ^6)+48 (σ^7)))+(1/(4480 (γ^3) ξ (σ^9) (((-1+(σ^2)))^(5/2))))(ν^3) (-7 σ (525-3100 (σ^2)+13690 (σ^4)-55260 (σ^6)+816595 (σ^8)+3752006 (σ^10)-1978290 (σ^12)-1029342 (σ^14)+213480 (σ^16)+24576 (σ^18))+ν (-3675+7350 σ+19950 (σ^2)-43400 (σ^3)-79800 (σ^4)+191660 (σ^5)+246540 (σ^6)-773640 (σ^7)-222810 (σ^8)+11432330 (σ^9)+25264989 (σ^10)+52528084 (σ^11)+38260656 (σ^12)-27696060 (σ^13)-48986718 (σ^14)-14410788 (σ^15)-81724 (σ^16)+2988720 (σ^17)+2732032 (σ^18)+344064 (σ^19)))+(1/(16 (γ^3) ξ (((-1+(σ^2)))^(3/2))))3 (ν^3) (2057+15261 (σ^2)+3387 (σ^4)-4321 (σ^6)+ν (-4114-1263 σ-30522 (σ^2)+3883 (σ^3)-6774 (σ^4)-1065 (σ^5)+8642 (σ^6)+525 (σ^7))) Log[2]-(1/(16 (γ^3) ξ (((-1+(σ^2)))^(3/2))))3 (ν^3) (-2039-13155 (σ^2)-1221 (σ^4)+1823 (σ^6)+ν (4078+1041 σ+26310 (σ^2)-2773 (σ^3)+2442 (σ^4)-1065 (σ^5)-3646 (σ^6)+525 (σ^7))) Log[σ]+(1/(16 (γ^3) ξ (((-1+(σ^2)))^(3/2))))3 (ν^3) (9+1053 (σ^2)+1083 (σ^4)-1249 (σ^6)+ν (-18-111 σ-2106 (σ^2)+555 (σ^3)-2166 (σ^4)-1065 (σ^5)+2498 (σ^6)+525 (σ^7))) Log[((1+σ)/2)])-(1/(32 (γ^3) ξ (((-1+(σ^2)))^(3/2))))3 (ν^3) (-36-3 (-403+658 ν) σ+36 (-117+256 ν) (σ^2)+(6422-15064 ν) (σ^3)+12 (-361+512 ν) (σ^4)+15 (-117+518 ν) (σ^5)+(4996-8192 ν) (σ^6)+2100 (-1+ν) (σ^7)) PolyLog[2,(Sqrt[((-1+σ)/(1+σ))])]+(1/(128 (γ^3) ξ (((-1+(σ^2)))^(3/2))))3 (ν^3) (-36-3 (-403+658 ν) σ+36 (-117+256 ν) (σ^2)+(6422-15064 ν) (σ^3)+12 (-361+512 ν) (σ^4)+15 (-117+518 ν) (σ^5)+(4996-8192 ν) (σ^6)+2100 (-1+ν) (σ^7)) PolyLog[2,((-1+σ)/(1+σ))]+(1/(32 (γ^3) ξ (((-1+(σ^2)))^(3/2))))3 (ν^3) (-2039-13155 (σ^2)-1221 (σ^4)+1823 (σ^6)+ν (4078+1041 σ+26310 (σ^2)-2773 (σ^3)+2442 (σ^4)-1065 (σ^5)-3646 (σ^6)+525 (σ^7))) PolyLog[2,-(((σ-(Sqrt[-1+(σ^2)])))^2)]-(1/(8 (γ^3) ξ (((-1+(σ^2)))^(3/2))))3 (ν^3) (9+1053 (σ^2)+1083 (σ^4)-1249 (σ^6)+ν (-18-111 σ-2106 (σ^2)+555 (σ^3)-2166 (σ^4)-1065 (σ^5)+2498 (σ^6)+525 (σ^7))) PolyLog[2,-σ+(Sqrt[-1+(σ^2)])]);

3PM Radiation Reaction Force

(Subscript[F, r])

ArG2 = -((2 (ν^2) (-1+2 (σ^2)) (-8+5 (σ^2)))/(3 (γ^2) ξ (-1+σ) (1+σ)));
BrG2 = 0;
CrG2 = ((4 (ν^2) σ (-1+(σ^2)) (-3+2 (σ^2)) (-1+2 (σ^2)))/((γ^2) ξ (((-1+σ))^2) (((1+σ))^2)));
Ar1G3 = -(((ν^3) (1151-3336 σ+3148 (σ^2)-912 (σ^3)+339 (σ^4)-552 (σ^5)+210 (σ^6)))/(24 (γ^2) (-1+(σ^2))));
Br1G3 = (((ν^3) (-5+76 σ-150 (σ^2)+60 (σ^3)+35 (σ^4)))/(4 (γ^2)));
Cr1G3 = -(((ν^3) σ (-1+(σ^2)) (-33+112 (σ^2)-165 (σ^4)+70 (σ^6)))/(4 (γ^2) (((-1+σ))^2) (((1+σ))^2)));
Ar2G3 = (1/(24 (γ^9) (ξ^3) (((-1+(σ^2)))^2)))(ν^2) (16 (ν^4) (((-1+σ))^3) (8-40 σ-37 (σ^2)+121 (σ^3)+52 (σ^4)-92 (σ^5)-20 (σ^6)+20 (σ^7))+64 (ν^3) (8-8 σ-21 (σ^2)+13 (σ^3)+10 (σ^4)+19 (σ^5)-31 (σ^7)+10 (σ^9))+2 (γ^5) (ξ^2) (-141+386 σ-525 (σ^2)-683 (σ^3)+1377 (σ^4)+240 (σ^5)-711 (σ^6)+105 (σ^7)+16 (γ^3) (((1-2 (σ^2)))^2) (-8+5 (σ^2)))+16 (ν^2) (-1+2 (σ^2)) (8-5 (σ^2)+2 (γ^4) ξ (8-25 σ-4 (σ^2)+37 (σ^3)-6 (σ^4)-18 (σ^5)+8 (σ^6)))+(γ^4) ν ξ (16 σ (-25+87 (σ^2)-92 (σ^4)+36 (σ^6))+γ ξ (1715-5444 σ+5641 (σ^2)+3056 (σ^3)-11049 (σ^4)+4908 (σ^5)+3675 (σ^6)-2712 (σ^7)+210 (σ^8))));
Br2G3 = ((ν^2) (2 (-62+155 σ+16 (σ^2)-70 (σ^3)-90 (σ^4)+35 (σ^5))+ν (253-944 σ+701 (σ^2)+360 (σ^3)-105 (σ^4)-440 (σ^5)+175 (σ^6))))/(4 (γ^4) ξ (-1+σ) (1+σ));
Cr2G3 = -(1/(4 (γ^9) (ξ^3) (((-1+σ))^3) (((1+σ))^3)))(ν^2) (-1+(σ^2)) (32 (γ^8) (ξ^2) σ (((1-2 (σ^2)))^2) (-3+2 (σ^2))+16 (ν^2) σ (3-8 (σ^2)+4 (σ^4)) (-1+(ν^2) (((-1+σ))^3) (1-5 σ-2 (σ^2)+2 (σ^3))+4 ν (1-σ-(σ^3)+(σ^5)))+(γ^5) (ξ^2) σ (-3+2 (σ^2)) (-48+38 σ+288 (σ^2)-140 (σ^3)-240 (σ^4)+70 (σ^5)+ν (85-172 σ-459 (σ^2)+856 (σ^3)+135 (σ^4)-620 (σ^5)+175 (σ^6)))+16 (γ^4) ν ξ (-1+2 (σ^2)) (-3+21 (σ^2)-28 (σ^4)+12 (σ^6)+2 ν (3-21 (σ^2)+10 (σ^3)+28 (σ^4)-16 (σ^5)-12 (σ^6)+8 (σ^7))));

FORCErG2 = (G*m/r)^2*(1/(r*PR))*(J0/r)^2( ArG2 + BrG2* Log[((1+σ)/2)] + CrG2*ArcSinh[((Sqrt[-1+σ])/(Sqrt[2]))]/(Sqrt[-1+(σ^2)]));

FORCErG3 = (G*m/r)^3*(1/(r*PR))*(m^2*( Ar1G3 + Br1G3* Log[((1+σ)/2)] + Cr1G3*ArcSinh[((Sqrt[-1+σ])/(Sqrt[2]))]/(Sqrt[-1+(σ^2)])) + (J0/r)^2*( Ar2G3 + Br2G3* Log[((1+σ)/2)] + Cr2G3*ArcSinh[((Sqrt[-1+σ])/(Sqrt[2]))]/(Sqrt[-1+(σ^2)])));

FORCEr = FORCErG2 + FORCErG3;

(Subscript[F, ϕ])

AϕG2 =((2 (ν^2) (-1+2 (σ^2)) (-8+5 (σ^2)))/(3 (γ^2) ξ (-1+σ) (1+σ)));
BϕG2 =0;
CϕG2 =-((4 (ν^2) σ (-1+(σ^2)) (-3+2 (σ^2)) (-1+2 (σ^2)))/((γ^2) ξ (((-1+σ))^2) (((1+σ))^2)));
AϕG3 =-(1/(24 (γ^9) (ξ^3) (((-1+(σ^2)))^2)))(ν^2) (16 (ν^4) (((-1+σ))^3) (8-40 σ-37 (σ^2)+121 (σ^3)+52 (σ^4)-92 (σ^5)-20 (σ^6)+20 (σ^7))+64 (ν^3) (8-8 σ-21 (σ^2)+13 (σ^3)+10 (σ^4)+19 (σ^5)-31 (σ^7)+10 (σ^9))+2 (γ^5) (ξ^2) (-141+386 σ-525 (σ^2)-683 (σ^3)+1377 (σ^4)+240 (σ^5)-711 (σ^6)+105 (σ^7)+16 (γ^3) (((1-2 (σ^2)))^2) (-8+5 (σ^2)))+16 (ν^2) (-1+2 (σ^2)) (8-5 (σ^2)+2 (γ^4) ξ (8-25 σ-4 (σ^2)+37 (σ^3)-6 (σ^4)-18 (σ^5)+8 (σ^6)))+(γ^4) ν ξ (16 σ (-25+87 (σ^2)-92 (σ^4)+36 (σ^6))+γ ξ (1715-5444 σ+5641 (σ^2)+3056 (σ^3)-11049 (σ^4)+4908 (σ^5)+3675 (σ^6)-2712 (σ^7)+210 (σ^8))));
BϕG3 =-(((ν^2) (2 (-62+155 σ+16 (σ^2)-70 (σ^3)-90 (σ^4)+35 (σ^5))+ν (253-944 σ+701 (σ^2)+360 (σ^3)-105 (σ^4)-440 (σ^5)+175 (σ^6))))/(4 (γ^4) ξ (-1+σ) (1+σ)));
CϕG3 =(1/(4 (γ^9) (ξ^3) (((-1+σ))^3) (((1+σ))^3)))(ν^2) (-1+(σ^2)) (32 (γ^8) (ξ^2) σ (((1-2 (σ^2)))^2) (-3+2 (σ^2))+16 (ν^2) σ (3-8 (σ^2)+4 (σ^4)) (-1+(ν^2) (((-1+σ))^3) (1-5 σ-2 (σ^2)+2 (σ^3))+4 ν (1-σ-(σ^3)+(σ^5)))+(γ^5) (ξ^2) σ (-3+2 (σ^2)) (-48+38 σ+288 (σ^2)-140 (σ^3)-240 (σ^4)+70 (σ^5)+ν (85-172 σ-459 (σ^2)+856 (σ^3)+135 (σ^4)-620 (σ^5)+175 (σ^6)))+16 (γ^4) ν ξ (-1+2 (σ^2)) (-3+21 (σ^2)-28 (σ^4)+12 (σ^6)+2 ν (3-21 (σ^2)+10 (σ^3)+28 (σ^4)-16 (σ^5)-12 (σ^6)+8 (σ^7))));

FORCEϕG2 = (G*m/r)^2*(J0/r)*( AϕG2 + BϕG2* Log[((1+σ)/2)] + CϕG2*ArcSinh[((Sqrt[-1+σ])/(Sqrt[2]))]/(Sqrt[-1+(σ^2)]));

FORCEϕG3 = (G*m/r)^3*(J0/r)*( AϕG3 + BϕG3* Log[((1+σ)/2)] + CϕG3*ArcSinh[((Sqrt[-1+σ])/(Sqrt[2]))]/(Sqrt[-1+(σ^2)])) ;

FORCEϕ = FORCEϕG2 + FORCEϕG3;