Spinning Compact Binary Dynamics and Chameleon Orbits

Home , Orbital state vectors

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Relativity large evolv General at parameters that hyperbolic orbital appl towards We local, pericenter, evolution. near whose pe the orbits, we by chameleon preserved systems of are dynamical constraints constrained l the of conservation that theory by generic encom constrained variables the are dimensionless syst by of evolutions closed set These a a derive for angles. we form, result compact main a a in spin- As spin-orbit, order included. leading tributions with accuracy, order (2PN) eaayetecnevtv vlto fsinn opc b compact spinning of evolution conservative the analyse We eatet fTertcladEprmna hsc,Univ Physics, Experimental and Theoretical of Departments .INTRODUCTION I. pnigcmatbnr yaisadcaeenorbits chameleon and dynamics binary compact Spinning v ≪ c u otevra theorem, virial the to Due . G . ) fe hc merger a which after 1), h rvttoa con- gravitational the m László ε and = Gm/c r r´ dGrey ota Keresztes Zoltán Gergely, Arpád ´ r h to- the are Dtd ue2,2021) 24, June (Dated: 2 r ≪ ε r to is 1 vligi iesols ie nafr utbeto suitable form a in time, dimensionless variables monitor dimensionless a of in set in a derived of evolving full first terms in the equations, [25]-[26], rewrite evolution Refs. We conservative system. of binary set compact spinning a galactic 35]. on [30, implications Refs. and in 27–33], ra- jets [23, gravitational Refs. correc- of in backreaction These in diation the extensively while discussed spin. [20–26], been the have Refs. of dynamics the terms dynamics to to in tions contributions expressed QM rota- be the to moment which could of quadrupole the case back in trace tion, to (QM). bi- usual couplings the is quadrupole-monopole modifies It mass also with The orbits objects nary compact effects. the (SS) leading of spin-spin quadrupole themselves, and or- (SO) with the spin-orbit with and to couple momentum also angular components the bital of orders, spins gen- 2PN the and dynamics 1PN but conservative at large contribute the at effects In relativistic arise eral features, [19]. these parameters of numerical PN contribu- some capture and Corresponding that 16] 18]. tions, [15, [17, simulations calculations orbits, relativity zoom-whirl geodesic mention we in Keple- example from arising beyond an different As and extremely ones. be orders rian the could 2.5PN orbits including In the at without [14], arising effect. even 1PN effects regimes an dissipative gravity is the strong orbits relativity, planetary general the of of shift tests perihelion classical A conservative. 2032). date (launch [13] LISA with or [12] eaiiyi h omo h P n P contributions 2PN and 1PN the of form Ke- the the general in of from relativity force corrections perturbing encompassing in evolution, the Then plerian introduce and angles. we spin motion III c) orbital Section finally the momentum, by angular selected total defined systems variables conserva- reference angular 2PN in b) the constants, to are evolution up include tive which These quantities, dynamics. dynamical order a) leading the to tied closely Newtonian local, a in sense). elements orbital osculating their by nti ae esuytecnevtv yaisof dynamics conservative the study we paper this In is dynamics the orders 2PN to up inspiral, the During nScinI eitouetedmninesvariables dimensionless the introduce we II Section In pnadms uduoemnpl con- quadrupole-monopole mass and spin itne hnNwointer does. theory Newtonian than distances sbhvo scnitn ihtepicture the with consistent is behavior is rmelpi i h etna sense) Newtonian the (in elliptic from e w odn t2Nodr srequired As order. 2PN at holding aws asn ohobtleeet n spin and elements orbital both passing h omls oso h existence the show to formalism the y equations differential order first of em ohbuddadubuddorbits unbounded and bounded both fr ossec hc n prove and check consistency a rform riyo zgd Hungary Szeged, of ersity nre oscn post-Newtonian second to inaries a defined (as 2 and spin related corrections, namely the leading order quadrupole coefficient wi. SO, SS and QM couplings. The contributions to the pre- The mass of neutron stars is typically of 1.4 solar cessional evolutions are also enlisted. masses. Black holes on the other hand could have The full 2PN conservative dynamics is presented in masses extending from a few solar masses (for stellar Section IV in the form of a generalisation of the Lagrange mass black holes), up to 1010 solar masses (for the largest planetary equations. First the evolution of 5 dimension- mass supermassive black holes). We will frequently em- less orbital elements and of 4 spin angles is given in terms ploy the total and reduced masses m = m1 + m2 and of the true anomaly. This is complemented by the evo- µ = m1m2/m, the mass ratio ν = m2/m1 ∈ [0, 1] and its lution of the true anomaly in terms of the dimensionless symmetrical counterpart η = µ/m ∈ [0, 0.25]. time. As a result we obtain a closed system of first order Kerr black holes in extreme rotation provide the up- differential equations. They are involved, nevertheless per bound of the dimensionless spin parameter, which they exhibit a simple structure. Suitable notations made for general relativistic black holes is constrained as χi ∈ this structure transparent. [0, 1]. Faster rotation would destroy the horizon, ren- The dimensionless variables however do not evolve dering them into naked singularities. In order to es- unconstrained. At the 2PN accuracy of the conserva- timate the range of χi for neutron stars, we proceed tive motion there are constants of motion, expressible as follows. From the expression of the spin magni- 2 in terms of these variables. We give these constraints in tudes Si = (G/c) mi χi we rewrite the dimensionless terms of the dimensionless dynamical variables in Section spin as a ratio of two dimensionless parameters: χi = 2 V. (Si/miRic)/(Gmi/c Ri). For a neutron star of 1.4 so- As for any constrained dynamical system, consistency lar masses and radius of 10 km the denominator is 2 checks need to be performed. This implies to take the (Gmi/c Ri) ≈ 0.2. Approximating the neutron star to time derivative of the constraints and to investigate their leading order by a rigid sphere, the numerator becomes role from a dynamical point of view. In principle such (Si/miRic) ≈ (2/5)(RiΩi/c), hence χi ≈ 2(RiΩi/c). Un- constraints could lead to A) new equations of motion, less the surface rotational velocity RiΩi of the neutron B) new constraints, or C) identities. Section VI is de- star is higher than half of the speed of light (typical ob- voted to this involved analysis, with some of the compu- served rotational velocities are much smaller), for neu- 1 tational details shifted to Appendix A. We prove that the tron stars χi ∈ [0, 1] also holds. dynamical equations given in Section IV are exhaustive When the quadrupole moment arises from rotation in describing the binary and spin evolution, as the time rather than asymmetric mass distribution, wi = 1 holds derivatives of the constraints lead to identities. We fulfil for general relativistic black holes [36], while for neutron the task by performing a series of consistency checks of stars wi ∈ [4, 8], depending on their equation of state the system of differential equations at each PN and spin [31], [37]. order. As an application of the derived formalism in Section VII we analyse the possibility of having orbits which B. Keplerian dynamical constants change from hyperbolic to elliptic and vice-versa, in terms of the eccentricity of the osculating ellipse, thus in The Newtonian expressions of the energy, orbital an- a Newtonian sense. The existence of such evolutions are gular momentum and Laplace-Runge-Lenz vector of a to be expected, as General Relativity predicts stronger binary system are gravity at short distances than the Newtonian theory, as well known from the study of the stellar equilibrium. µv2 Gmµ E ≡ − , (1) Indeed, we find orbits dubbed chameleon, which appear N 2 r elliptic (locally, in a Newtonian sense) at short range, but LN ≡ µr × v , (2) transform into hyperbolic (in the same sense) at larger AN ≡ v × LN − Gmµˆr . (3) distances. Finally Section VIII contains the concluding remarks. They obey the constraints Throughout the paper an overhat denotes the direction of the respective vector. LN · AN = 0 , 2E L2 (Gmµ)2 + N N = A2 , (4) µ N II. DIMENSIONLESS VARIABLES

A. Dynamical characteristics of axially symmetric compact objects 1 We note that this is not the case for ordinary stars, where 2 (Gmi/c Ri) has a much smaller value due to their non- compactness. Hence in the dynamics spin-orbit coupling terms Compact binary components with axial symmetry are containing χi could dominate over general relativistic correc- characterised by their mass mi, their proper rotation tions, while spin-spin terms with χ1χ2 or quadrupole-monopole 2 encompassed in their dimensionless spin χi and their terms with χi could become even larger. 3 and are constants to Keplerian order. As it is well-known, tron rmin = pN / (1 + er) as the Keplerian orbit is a conic section characterised by these constants. c2r 1/2 l = min (1 + e )1/2 . (8) r Gm r

C. Osculating orbit Note that rmin = 0 and lr = 0 are equivalent, thus a collision course is possible only for vanishing orbital When general relativistic corrections in the weak field angular momentum. For bounded orbits we can also 2 and slow motion approximation are taken into account introduce the semimajor axis ar = pN / 1 − er = as PN and 2PN corrections; also by including the modi- Gm/c2 l2/ 1 − e2 .2 r r fications to the dynamics due to leading SO, SS and QM For the three Euler angles, defining the orientation couplings, the orbit ceases to be a conic section in the of the plane of motion and the orientation of the or- strict sense, nevertheless it can be well approximated by bit in this plane we chose the following: the inclination a conic section locally. This local approximant is the os- α = arccos Jˆ · LˆN of the plane of orbit with respect culating orbit, defined as the Keplerian orbit with the same orbital state vectors (position and velocity) as for to the reference plane (which is chosen perpendicular to the orbit realised in the presence of the perturbations. the total angular momentum J), the longitude of the as- ˆx It is easy to see then that the dynamical constants of cending node −φn (measured from an arbitrary axis lying in the reference plane to the ascending node ˆl, de- the osculating orbit are EN , LN, AN, restricted by the constraints (4). fined by the intersection of the reference plane with the The perturbed orbit can be envisaged as a sequence plane of motion) and the argument of the periastron ψp of conic sections, the orbital elements of which slowly (measured from the ascending node to the periastron in evolve. One can therefore characterise the osculating or- the plane of motion, see Fig 2. of Ref. [25]). These three bit (the instantaneous Keplerian orbit, the orbital param- Euler angles together with (lr,er) are equivalent with the eters of which evolve in time) by the above introduced 5 set of dynamical variables (EN , LN, AN), as only five of independent and time-evolving variables. The additional the latter are independent due to the constraints (4). information encoded in the orbital state vectors isr ˙. Note that the definition of the above angles is mean- ingful only when the ascending node and the periastron can be defined, thus alternative definitions of the angles are required in the cases a) of evolutions which are ei- D. Dimensionless orbital elements and spin variables ther nonspinning or the spins are aligned to the orbital angular momentum, when the ascending node cannot be defined in the above sense, and b) for circular orbits, The 5 independent dynamical variables are equivalent when there is no periastron. We regard these however to a similar number of orbital elements. To show this, as configurations of measure zero in the generic param- first we define two independent variables characterising eter space, which need special attention. The formalism the shape of the osculating orbit, which are both dimen- developed in this paper is well suited for precessing con- sionless and equally apply for bounded or unbounded or- figurations and noncircular orbits. bits. These are a dimensionless version of the Newtonian The spin polar and azimuthal angles are κi = orbital angular momentum and the orbital eccentricity, Sî LˆN defined as: arccos · and ψi (when measured from the ascending node), or ζi = ψi − ψp (when measured from the cLN periastron). In this paper we employ the latter, as this lr = , (5) Gmµ will simplify the notations. A Finally we mention the last angular variable necessary e = N . (6) r Gmµ for the description of the compact binary dynamics. The position of the reduced mass particle is characterised by In these variables the Newtonian expression of the energy its azimuthal angle, this is χp when measured from the reads (see the second constraint (4)): periastron, or ψp +χp when measured from the ascending node. The other quantity, which defines its instantaneous 2 2 er − 1 position is the distance r measured from the focal point, EN = µc 2 , (7) where the potential generated by the (fixed) total mass 2lr which is manifestly negative for circular (er = 0) or ellip- tical orbits (0 < er < 1), zero for parabolic orbits (er = 2 Note that it is possible to define in a similar way a semi-latus 1) and positive for hyperbolic orbits (er > 1). Note that l 2 2 rectum pKerr and radial eccentricity eKerr in terms of the con- r is related to the semi-latus rectum pN = LN /Gmµ of served energy and z-component of the orbital angular momentum 1/2 the conic orbit as lr = c (pN /Gm) and to the perias- of Kerr orbits, as introduced for bounded orbits in Ref. [15]. 4 m is centered. Its relation with the already introduced e.g. to the osculating orbit), the total energy and angu- quantities will be discussed next. lar momentum (also their dimensionless versions) characterise the real orbit. J It is also possible to define the periastron distance rmin E. Parametrization of the radial evolution J and eccentricity er of the fictious Keplerian motion with energy E and orbital angular momentum J through the The true anomaly parametrization r (χp) of the oscu- relations lating orbit is the same as for the Keplerian motion c2rJ J2 min J L2 = 1+ er , (17) r = N , (9) Gm ! µ (Gmµ + AN cos χp) J A = Gmµer , (18) with the important difference that the dynamical quantities LN and AN evolve with the osculating orbit. The where A is the Laplace-Runge-Lenz vector of the Keple- parametrization obeys rian motion with energy E and orbital angular momentum J, defined in the usual way as AN r˙ = sin χp , (10) 1/2 LN A = Gmµ 1+2EJ2 . (19) (Gmµ)2 + A2 +2GmµA cos χ v2 = N N p . (11) L2 These relations lead to the expressions of the orbital ele- N ments of the fictious Keplerian motion in terms of E and In terms of osculating orbital elements introduced above, J as these relations read J 2 1/2 2 er = 1+2EJ , Gm lr 2 J 2 r = 2 , (12) c r J c 1+ er cos χp min = 1/2 . (20) er Gm 1+ 1+2EJ2 r˙ = c sin χp , (13) lr J J 1+ e2 +2e cos χ To 2PN accuracy both r and e are constants.3 v2 = c2 r r p . (14) min r 2 With this we have all ingredients to obtain the dynam- lr ics of a spinning, precessing compact binary on noncircular orbit at 2PN order accuracy. F. Dimensionless constants of motion

At the level of accuracy discussed in this paper (with III. RELATIVISTIC AND SPIN INDUCED PERTURBATIONS PN, SO, 2PN, SS, QM contributions to the dynamics included) there are several constants of the motion: 2i−3 (a) the magnitudes Si = (G/c) mµν χi of the spins We will characterise the deviation from Keplerian evo- Si (as the spins undergo a purely precessional evolution lution in terms of a generic perturbing force, which [21]), receives contributions from 1PN relativistic corrections (b) the total energy E = EN + EPN + ESO + E2PN + (given in terms of relative coordinates in Ref. [20]), the ESS +EQM of the system, with the various contributions 2PN relativistic, the leading order SO (in the Newton- given in Refs. [22], [23], [32], Wigner-Pryce SSC) and SS corrections (all given in Ref. (c) the total angular momentum [23]) and quadrupolar contributions (given in Ref. [32]). J = LN+LPN+LSO+L2PN + S1+S2, with contributions enlisted in Ref. [23]; however for the contribution LSO we adopt the expression given in Ref. [26], 3 which holds true in the Newton-Wigner-Price [38] spin Note that similar deﬁnitions can be also introduced as L 2 1/2 supplementary condition (SSC), employed in this paper. er = 1 + 2EL , We introduce the dimensionless versions for the total 2 L 2 c rmin L = , (21) energy and angular momentum magnitude as 2 1/2 Gm 1+ 1 + 2EL E E = , (15) where L = cL/(Gmµ), with L the magnitude of the total orbital µc2 angular momentum. As L is not a constant when SS and QM L L cJ contributions to the dynamics are present, rmin and er vary on J = . (16) the orbit, therefore they are not particularly useful. Alterna- Gmµ tively, an orbital average L¯ of the magnitude of orbital angular momentum was introduced and employed in Refs. [30], [32] and Note that unlike other quantities employed in this section [39] together with the corresponding orbital elements in Ref. [24], (characteristic to the local approximant of the real orbit, but only for closed orbits. 5

PN 2PN The spins, with the exception of the aligned case also where ζ(±) = ζ2 ± ζ1 and the coeﬃcients c1(k) and c1(k) induce precessions of the spins and of the orbital plane. are given as We deﬁne the dimensionless versions of the perturbing force ∆a acting on unit mass and of the spin angular cPN = −2 (2 − η) e , frequencies Ωi as follows 1(0) r

PN 3η 2 Gm c1(1) = C1 − 1+ er , a = ∆a , 2 c4 cPN = 6(1 − η) e , Gm 1(2) r ωi = Ωi . (22) 3η c3 cPN = − e2 , (24) 1(3) 2 r The components of ∆a and Ωi expressed in the system ˆ ˆ ˆ ˆ ˆ f(i) = (AN, QN ≡ LN × AN, LN) were given in Ap- and pendix B of Ref. [26] in terms of the variables (r, r,˙ v). Starting from those, by employing the parametrization 2PN 2 2 (12)-(14) and by rewriting all quantities in terms of the c1(0) = 2+13η +2η − η (3 − η) er er , dimensionless variables introduced in the previous sec- 71η η tion, also by switching to a description in terms of the c2PN = C + 4+ +2η2 e2 − (3 − 29η) e4 , 1(1) 2 2 r 8 r spin azimuthal angles ζi (rather than ψi), ﬁnally organ- 2PN 2 ising the expressions such that they can be written in c1(2) = C3 +(2+27η) er er , a compact form, which emphasizes their true anomaly 3η dependence, we explicitly give the projections of the per- c2PN = C − (1+7η)e2 e2 , 1(3) 4 4 r r turbing accelerations and spin angular frequencies below. 59η c2PN = −2 − + 13η2 e3 , 1(4) 2 r 15η A. Perturbing force c2PN = − (1 − 3η) e4 , (25) 1(5) 8 r The dimensionless version of the component of the perturbing force along the periastron line is: with the shorthand notations

C1 = 3 − η , PN 2PN SO SS QM a · Aˆ N = a + a + a + a + a , (23) 1 1 1 1 1 73η 2 2 3 C2 = −9 − +2η , PN (1 + er cos χp) PN k 4 a = c cos χ , 2 1 6 1(k) p C = −20 − 49η +8η , lr 3 Xk=0 223η 5 2 (1 + e cos χ )2 C4 = −13 − + 16η . (26) a2PN r p 2PN k 4 1 = 8 c1(k) cos χp , lr Xk=0 η (1 + e cos χ )3 (e + cos χ ) aSO r p r p The dimensionless version of the perturbing force com- 1 = 7 2lr ponent in the plane of motion, but perpendicular to the 2 periastron line is 2k−3 × 4ν +3 χk cos κk , k=1 X 4 3η (1 + er cos χp) PN 2PN SO SS QM aSS = χ χ − cos κ cos κ a · Qˆ N = a + a + a + a + a , (27) 1 l8 1 2 1 2 2 2 2 2 2 r 2 n (1 + e cos χ )2 sin χ 1 aPN = r p p cPN cosk χ , × cos χp + sin κ1 sin κ2 2cos ζ(−) cos χp 2 6 2(k) p 4 lr Xk=0 2 4 +cos χp − ζ(+) +5cos 3χp − ζ(+) , (1 + e cos χ ) sin χ a2PN r p p 2PN k 2 = 8 c2(k) cos χp , 4 2 o lr QM 3η (1 + er cos χp) 2k−3 2 k=0 a = − wkν χ cos χp X 1 2l8 k η (1 + e cos χ )3 sin χ r k=1 aSO r p p X n 2 = 7 1 2lr − sin2 κ [cos ζ +5cos(2χ − ζ )] 2 k k p k 2 2k−3 × 4ν +3 χk cos κk , × cos(χp − ζk) , k=1 o X 6

4 3η (1 + e cos χ ) B. The precessional angular velocities aSS r p 2 = 8 χ1χ2 − cos κ1 cos κ2 lr 1 n The precessions arise due to the SO, SS and QM con- × sin χ + sin κ sin κ 2cos ζ − sin χ p 4 1 2 ( ) p tributions to the dynamics. The components of the dimensionless angular velocity are: − sin χp − ζ(+) + 5 sin 3χp − ζ(+) , 3 4 2 o η (1 + er cos χp) 2j−3 3η (1 + e cos χ ) − ωi · Aˆ N = ν χ sin κ aQM = − r p w ν2k 3χ2 sin χ 2l6 j j 2 2l8 k k p r r k=1 X n × [3 cos (2χp − ζj )+cos ζj ]+3wiχi 1 2 − sin κk [sin ζk + 5 sin (2χp − ζk)] × sin κi [cos(2χp − ζi)+cos ζi]} , (31) 2 3 η (1 + er cos χp) 2j−3 i Qˆ N × cos(χp − ζk) , ω · = 6 ν χj sin κj 2lr o PN 2PN × [3 sin (2χp − ζj ) + sin ζj ]+3wiχi where the coefficients c2(k) and c2(k) are given as × sin κi [sin (2χp − ζi) + sin ζi]} , (32) 7η 3 PN 2 η (1 + er cos χp) 3−2i c2(0) = C1 + 3 − er , ωi · LˆN = 4+3ν 2 l5 2 r PN PN 3 c2(1) = c1(2) , η (1 + e cos χ ) − − r p ν2j 3χ cos κ , (33) PN PN l6 j j c2(2) = c1(3) , (28) 2 r and with j 6= i. The first term in ωi · LˆN is due to the SO interaction. The terms containing wi are due to the QM 2PN 2 21η 4 c2(0) = C2 + 17ηer + (1 + η) er , interaction, while the other terms containining χi due to 8 the SS interaction. 2PN 2 c2(1) = C3 +(26+3η) ηer er , Note that all terms in the equations above carry the −2 3η same power of the PN parameter lr . Whether any of c2PN = C + (5 − 3η) e2 e2 , 2(2) 4 4 r r the terms dominate, depends on the mass ratio. 2PN 2PN c2(3) = c1(4) , 2PN 2PN IV. 2PN CONSERVATIVE DYNAMICS c2(4) = c1(5) . (29) The dimensionless version of the perturbing force com- In Ref. [26] the evolutions of the independent variables ponent perpendicular to the plane of motion has only spin were derived as a system of first order coupled ordinary induced contributions differential equations. We rewrite below these evolutions explicitly in terms of the dimensionless variables of this SO SS QM a LˆN a a a · = 3 + 3 + 3 , (30) paper. We will also switch from ψi to ζi, and switch to 3 2 a dimensionless time variable, defined as SO η (1 + er cos χp) 2k−3 a3 = 4ν +3 χk sin κk 4l7 3 r k=1 c X t = t . (34) × [er cos ζk +4cos(χp − ζk) Gm

+3er cos(2χp − ζk)] , We will denote by an overdot the derivative with respect 3η (1 + e cos χ )4 to t (as opposed to Ref. [26], where an overdot was the aSS r p 3 = − 8 χ1χ2 [cos κ1 sin κ2 derivative with respect to t): lr × cos(χp − ζ2)+cos κ2 sin κ1 cos(χp − ζ1)] , d Gm d = 3 . (35) 3η (1 + e cos χ )4 dt c dt aQM r p 3 = − 8 2lr With all these changes in the notation and in the choice 2 of independent variables the equations simplify consider- 2k−3 2 × wkν χk sin 2κk cos(χp − ζk) . ably. k=1 For the osculating orbital elements we obtain the cou- X pled system of the evolutions: An important remark we make here is that the PN l2 order of various terms can be evaluated from the relative ˙ r ˆ l l−1 lr = − a · AN sin χp powers of r in the respective terms, r counting for 1+ er cos χp 0.5PN orders. As lr is much larger than unity, it is also a Qˆ N much larger than the dimensionless spins χi. + · cos χp , (36) 7

lr ˆ and of the auxiliary angle γ span by the spin vectors: e˙r = − a · AN (er + cos χp) sin χp 1+ er cos χp sin γ γ˙ = ω − · Aˆ N [− cos κ sin κ sin ζ ˆ 2 ( ) 1 2 2 + a · QN 1+2er cos χp + cos χp , (37) + sin κ cos κ sin ζ ]+ ω − · Qˆ N 1 2 1 ( ) lr ˆ sin (ψp + χp) × [cos κ sin κ cos ζ − sin κ cos κ cos ζ ] ψ˙p = − a · LN 1 2 2 1 2 1 (1 + er cos χp) tan α ˆN h 2 + ω(−) · L sin κ1 sin κ2 sin ζ(−) . (45) 1+ er cos χp + sin χp + a · Aˆ N Here we denoted ω − = ω2 − ω1. er ( ) These evolution equations in terms of dimensionless ˆ sin χp cos χp − a · QN , (38) variables stand as the main result of the paper. er i ˆ cos(ψp + χp) α˙ = lr a · LN , (39) V. CONSTRAINTS ON THE VARIABLES 1+ er cos χp At 2PN order accuracy, with the leading order SO, ˆ sin (ψp + χp) SS and QM contributions included, the total energy and φ˙n = −lr a · LN . (40) (1+ er cos χp) sin α total angular momentum are conserved. These primary The spin angles evolve as: constraints can be expressed in terms of the dimensionless dynamical variables for which we derived evolution ˆ ˆ κ˙ i = − ωi · AN sin ζi + ωi · QN cos ζi equations. Therefore in this section we derive these constraints. ˆ sin (χp − ζi) −lr a · LN , (41) 1+ er cos χp A. Total energy

ζ˙ = − ωi · Aˆ N cos ζ + ωi · Qˆ N sin ζ cot κ i i i i Starting from the expression of the total energy, with h lr i the PN and 2PN contributions explicitly given in [23], + ωi · LˆN + (1 + er cos χp) SO (in the Newton-Wigner-Price SSC) in [24], SS in [30] and QM in [32], rewritten in the notations of the present cos(χp − ζi) × a · LˆN paper as4 tan κi h 2 3 v4 1 Gm v2 ˆ 1+ er cos χp + sin χp E = µc2 (1 − 3η) + (3 + η) + a · AN PN 8 c4 2 c2r c2 er n 2 2 sin χp cos χp η Gm r˙ 1 Gm a Qˆ N + + , − · , (42) 2 c2r c2 2 c2r er i o while the true anomaly 6 2 5 2 v 2 E2PN = µc 1 − 7η + 13η 6 (1 + er cos χp) lr 16 c χ˙ p = 3 + n 4 lr er (1 + er cos χp) 3 Gm r˙ − η (1 − 3η) 2 4 2 8 c r c × a · Aˆ N 1+ e cos χ + sin χ r p p 1 Gm v4 + 21 − 23η − 27η2 h ˆ 2 4 − a · QN sin χp cos χp . (43) 8 c r c 1 Gm 2 v2 i + 14 − 55η +4η2 This latter equation allows to replace (dimensionless) 8 c2r c2 time-derivatives with derivatives with respect to χp in η Gm v2r˙2 1 Gm 3 all previous evolution equations. + (1 − 15η) − (2+15η) Although not independent from the previous ones, for 4 c2r c2 c2 4 c2r completeness we also give the evolutions of the auxiliary 1 Gm 2 r˙2 spin azimuthal angles ψ : + 4+69η + 12η2 , i 8 c2r c2 ˆ ˆ o ψ˙i = − ωi · AN cos ζi + ωi · QN sin ζi cot κi h i ˆ lr ˆ + ωi · LN − a · LN 4 1+ er cos χp G and c were reintroduced in all 1PN and 2PN terms on dimen- sional grounds. In the SS and QM terms χ of Refs. [30] and [32] sin (χp + ψp) cos(χp − ζi) was replaced with χp, as to leading order they agree. Also ψ0 is × − , (44) − − tan α tan κ denoted in this paper as ψp and δi = 2 (ψ0 ψi) as 2ζi. i 8

ESO =0 , and 3 2 3 4 2 η (1 + e cos χ ) G m η E r p 2 2i−3 QM = − 6 wiχi ν ESS = − 4 3 χ1χ2 3cos κ1 cos κ2 − cos γ 2l 2c r r i=1 X 2 2 −3 sin κ1 sin κ2 cos 2χp − ζ(+) , × 1 −3sin κi cos (χp − ζi) . (52) 3 4 2 2 G m η − E = − w χ2ν2i 3 B. Total angular momentum QM 2c4r3 i i i=1 X2 2 × 1 −3sin κi cos (χp − ζi) , (46) The projections along the basis vectors ˆ ˆ ˆ ˆ with γ related to the other variables by the spherical l, ˆm ≡ LN × l, LN of the expression of the to- cosine identity tal angular momentum give constraint relations. In the Newton-Wigner-Price SSC these were given as Eqs. cos γ = cos κ cos κ + sin κ sin κ cos ζ − , 1 2 1 2 ( ) (B26)-(B28) of [26]. We rewrite these relations in terms we ﬁnd that the osculating orbital elements, spin vari- of the dimensionless variables employed in this paper, ables and true anomaly obey the constraint also employ trigonometric identities to give them in the most simple form containing the spin azimuthal angles E = EN + EPN + E2PN + ESS + EQM , (47) ζi (rather than ψi). We obtain with the contributions 2 e2 − 1 2i−3 E = r , (48) 0= χi sin κi ν cos(ζi + ψp) N 2l2 r i=1 h X 2i−3 η 4ν +3 (1 + er cos χp) 3 − 1 l2 E k k 2 r PN = 4 qker cos χp , (49) 8lr k=0 × sin (χp + ψp) sin (χp − ζi) , (53) X 2 4 q0 = 19 − 5η +2(9 − 5η) er +3(1 − 3η) er , i 2 q1 = 4 14 − 6η + (6 − 7η) er , 2 J sin α = χ sin κ ν2i−3 sin (ζ + ψ ) q2 = 8(5 − 4η) , i i i p i=1 q3 = −4η , h X 2i−3 η 4ν +3 (1 + er cos χp) 5 + 2 1 2lr E = s ek cosk χ , (50) 2PN 16l6 k r p r k=0 × cos(χp + ψp) sin (χp − ζi) , (54) X 2 2 2 s0 = 67 − 251η + 19η + 135 − 165η + 59η er i 2 4 2 +3 19 − 51η + 33η er J cos α = lr (1 + ǫPN + ǫ2PN )+ χi cos κi 2 6 +5 1 − 7η + 13η e , i=1 r X 2 2i−3 s = 2 2 82 − 265η + 38η − η 4ν +3 (1 + er cos χp) 1 × ν2i 3 − , 2 2 2 +2 96 − 157η + 85η e 2lr r h i 2 4 (55) +312 − 43η + 49η er , 2 with s2 = 4 126 − 415η + 106η 2 2 2 7 − η + (1 − 3η) er +4(2 − η) er cos χp + 66 − 140η + 125η er , ǫPN = , (56) l2 2 2 2 r s3 = 4 60 − 258η + 113η +2 η(1+3η) er , 2 and s4 = −2 4+76η − 57η , 3 16 2PN 1 k k s = c . ǫ2PN = pke cos χp , (57) 5 5e4 2(4) 8l4 r r r k=0 X 2 2 2 3 p0 = 59 − 143η + 11η +2 17 − 45η + 11η er η (1 + er cos χp) ESS = − χ1χ2 2 4 6 +3 1 − 7η + 13η er , 2lr 2 2 2 p1 = 4 38 − 92η + 16η + 10 − 33η + 25η er , × 2cos κ1 cos κ2 − sin κ1 sin κ2 p = 2 48 − 119η + 56η2 , n 2 × cos ζ − +3cos 2χ − ζ , (51) ( ) p (+) p3 = 4η(2+5η) . o 9

For aligned configurations the constraints (53)-(54) be- angles given in Section IV. We will do this order by or- come identities. der, starting with the Keplerian order, then proceeding We note that all three total angular momentum con- with the relativistic 1PN and 2PN contributions, finally −2 straints have a leading order and an lr contribution. It discussing the leading order consistency for the SO, SS is instructive to discuss the leading order contributions and QM contributions. The calculations will be some- to Eqs. (53)-(55): what simplified by taking into account that only χp has a Newtonian order evolution; the orbital elements and 2 2i−3 −2 spin angles being conserved at this order. 0 = χi sin κiν cos ψi + O lr , (58) i=1 X 2 A. Time derivative of the total angular momentum 2i−3 −2 J sin α = χi sin κiν sin ψi + O lr , (59) constraint i=1 X 2 For calculating the time derivative of the total angular J l 2i−3 l−2 cos α = r + χi cos κiν + O r , (60) momentum constraint one has to remember that the basis i=1 ˆ ˆ ˆ ˆ X l, ˆm ≡ LN × l, LN employed for the decomposition while the ratio of the last two becomes (53)-(55) itself changes, being a precessing basis. Hence for the consistency condition we need to prove 2 χi 2i−3 sin κiν sin ψi tan α = i=1 lr + O l−2 . (61) 2 χi 2i−3 r d 1+ cos κ ν Jˆˆ J ˆm Jˆ ˆN P i=1 lr i 0= ll + ˆm + LN L . (63) dt P1/2 As χi/lr = O ε , for comparable masses tan α = For the evolutions of the basis vectors we start from the 1/2 O ε sin κi, hence α is of 0.5PN order. By contrast, precession relations (12)-(13) given in Ref. [26] for the − for small mass ratios tan α = tan κ / 1+ O ε 1/2 ν basis f(i), and rewrite them in terms of the dimensionless 1 which is approximated as α ≈ κ1 (not necessarily small) variables as: when O ε−1/2 ν ≪ 1. Another useful formula holding to leading order, which f˙ i Ω f i will be explored later on in the paper is: ( ) = A × ( ) , (64)

2 with the angular velocity vector (redefined by a factor of 2i−3 −2 dt/dt as compared to Ref. [26]) lr = ν χi [sin κi sin (ζi + ψp)cot α − cos κi]+O lr . i=1 X l (62) r a ˆ ˆ ˆ l ΩA = · LN cos χpAN + sin χpQN For comparable masses sin κi cot α = O ( r), hence it is 1+ er cos χp ( large, while cos κi is of order unity. By contrast, for small 1 ˆ 2 mass ratios, where α ≈ κ1, the two terms are of compara- − a · AN 2+ er cos χp − cos χp ble unit order, nevertheless the prefactor ν−1χ is large, er 1 h of order l . r ˆ ˆ The equations (47), (53)-(55) are primary constraints − a · QN sin χp cos χp LN . (65) ) for the 2PN accurate binary dynamics, the consistency of i which with the dynamical equations has to be analysed. ˆ This will be done in the next section. Next we take into account that the basis l, ˆm is trans- ˆ ˆ formed into AN, QN by a rotation with angle ψp: VI. CONSISTENCY ˆ ˆ ˆ l = cos ψpAN − sin ψpQN , (66) ˆ ˆ According to the general theory of dynamical systems ˆm = sin ψpAN + cos ψpQN , (67) with constraints, the derivatives of the constraints could lead to either new dynamical equations, new constraints, which leads to a precession of the basis vectors or be identically satisfied. In case of new constraints ˆl, ˆm, LˆN with the angular frequency vector arising by this procedure, the check of the consistency ˆ conditions should be repeated. Therefore in the present ΩL = ΩA − ψ˙pLN . (68) section we discuss these consistency conditions. We will verify the consistency of the above lengthy sys- (Note, that in contrast with the expression (24) given tem of evolution and constraint equations by taking the in Ref. [26] here the dot refers to the derivative with time derivatives of the four dynamical constraints (47), respect to the dimensionless time t.) The detailed form (53)-(55) derived in the previous section and inserting of ΩL was also given as Eq. (30) in Ref. [26], which, after in them the evolution of the orbital elements and spin a proper rescaling to account for the evolution in terms 10 of t and rewritten in terms of the dimensionless variables B. Keplerian evolution reads With only the leading order terms due to the vanishing ˆ lr a · LN a ˆ ˆ of and ωi, Eq. (43) reduces to ΩL = cos χpAN + sin χpQN 1+er cos χp h 2 sin (ψ + χ ) (1 + er cos χp) p p ˆ χ˙ p = . (76) + LN , (69) l3 tan α r i Combining this with the definition of the true anomaly, or rewritten in the basis ˆl, ˆm, LˆN as Eq. (12) we obtain Kepler’s second law for the area: ˆ 2 2 lr a · LN 2 G m ˆ r χ˙ p = lr . (77) ΩL = cos(ψp + χp) l + sin (ψp + χp) ˆm c4 1+er cos χp h sin (ψp + χp) The constraint equations reduce to + LˆN . (70) tan α i e2 − 1 E r Hence the consistency condition to be proven reads = 2 , 2lr α = 0 , J˙ˆˆ J˙ ˆm J˙ ˆ ˆN 0 = ll + ˆm + LN L J = lr . (78) Jˆ ˆ J ˆm Jˆ ˆN + lΩL × l + ΩL × ˆm + LN ΩL × L .(71) Then Eq. (39) becomes an identity, Eqs. (36)-(37), (41)- We rewrite this by inserting the components of the (42) imply constant lr, er, κi and ζi (although at this Jˆ J ˆm Jˆ normalized total angular momentum l, , LN = accuracy there are no spins, thus κi and ζi have no in- (0, J sin α, J cos α) and by exploring Eqs. (67) and (70): terpretation). With α = 0, Eqs. (38) and (40) become ill-defined, ˆ lr a · LN unless we multiply them with sin α, when they give iden- 0 = J˙ˆˆl + J˙ ˆm ˆm + J˙ ˆ LˆN + J tities, but no information on ψ and φ . This is related l LN 1+e cos χ p n r p to the ill-definedness of the node line ˆl when the two ˆ ˆ × cos(ψp + χp) − cos α ˆm+ sin α LN . (72) planes coincide. Therefore some l has to be chosen in an arbitrary way to define the argument of the periastron. Hence the desired consistency conditions are This last remark also holds when only the 1PN or 1PN+2PN contributions are included, or when the spins are perpendicular to the orbit (±S1 k ±S2 k LN) thus 0 = J˙ˆl , (73) they do not precess. In these cases by definition α = 0, ˆ lr a · LN consistent with a · LˆN = 0 (when spins are present then 0 = J˙ ˆm − J cos α cos(ψ + χ ) , (74) 1+e cos χ p p due to κi = 0) in Eq. (39). For all these cases the ref- r p erence plane and node line should be defined by another ˆ lr a · LN vector, not aligned to J. J˙ ˆ J 0 = LN + sin α cos(ψp + χp) .(75) 1+er cos χp

In order to prove them, for the derivatives of the normal- C. 2PN level consistency, non-spinning case ized total angular momentum we take the derivatives of the rhs of the constraints (53)-(55). We discuss the 1PN and 2PN consistency conditions ˙ Note that the component Jˆl of the normalized total an- together below, by switching oﬀ the spin. gular momentum (which vanishes for nonprecessing evolutions) is a conserved scalar for precessing evolutions. We have seen at the end of Section V that this constraint 1. The energy condition decouples into two independent conditions, each obeyed by one of the spin directions. Another remark is that the second terms of the rhs in The time derivative of the total energy, the constraint Eqs. (74)-(75) are the sign ﬂipped versions of the deriva- equation (47), without the spin and quadrupole contri- tives of the lhs expressions of the constraints (54)-(55), butions, to 2PN accuracy gives withα ˙ taken from Eq. (39). Hence the same consistency conditions could be obtained by simply taking the dimen- e l˙ e2 − 1 d d r r r E E 0 =e ˙r 2 − 2 + PN + 2PN , (79) sionless time derivative of the constraints (54)-(55). lr lr lr dt dt 11

PN PN with Inserting a1 and a2 we obtain for the coefficients of the powers 0, 1, 2 and 3 of the arbitrary cos χp the rela- d 1 l˙ 3 E r k k tions PN = 4 −4 qker cos χp dt 8lr lr ( k=0 PN PN X q1er = 8 −c +c er , 3 k 1(0) 2(0) d qker k +e ˙r cos χp 2 PN PN PN der q2er = 4 −c1(1) +c2(1)er +c2(0) , k=0 X 3 3 8 PN PN PN k k−1 q3er = −c1(2) +c2(2)er +c2(1) , −χ˙ p sin χp kqker cos χp , (80) 3 k=0 ) PN PN X 0 = c1(3) − c2(2) , (85) and which can easily be verified to hold with the definitions d 1 l˙ 5 (24), (28) and (49) of this paper. E = −6 r s ek cosk χ t 2PN l6 l k r p As expected, the 2PN part of the consistency condition d 16 r ( r Xk=0 (79), Eq. (83) gives a much more cumbersome equation: 5 k d sker k 2PN 2PN +e ˙r cos χp 0 = (1+ er cos χp) a1 sin χp − a2 (er + cos χp) der k=0 X aPN 3 5 1 sin χp k k + −4 qke cos χp k k−1 8l2 r −χ˙ p sin χp ksker cos χp , (81) r k=0 X k=0 ) 3 k X d qker k ˙ + (er + cos χp) cos χp The 1PN and 2PN contributions toe ˙r and lr/lr carry der −5 −7 k=0 factors of l and l , respectively; whileχ ˙ p has New- X r r 3 tonian, 1PN and 2PN contributions, carrying factors of 2 k−1 −3 −5 −7 −2 + 1+ er cos χp + sin χp kqk (er cos χp) lr , lr and lr , respectively. Remembering that lr rep- k=0 resents one relative PN order it is easy to separate the X PN 3 1PN and 2PN contributions to the consistency condition a2 k k −7 −9 + 4cos χ q e cos χ (79). These are the terms scaling with l and l , re- 2 p k r p r r 8lr spectively (while higher order terms should be dropped, Xk=0 3 being beyond the accuracy of the present calculations). 2 k−1 We find at 1PN − sin χp cos χp kqk (er cos χp) k=0 ˙PN 2 X PN er lr er − 1 3 k 0 =e ˙r 2 − 2 2 d qker k lr lr lr − 1+2er cos χp + cos χp cos χp der N 3 k=0 χ˙ p sin χp k k−1 X − kqke cos χp , (82) 3 5 4 r (1 + er cos χp) sin χp − 8lr k k 1 k=0 + 8 ksker cos χp . (86) X 16lr and at 2PN kX=0 3 Inserting aPN , aPN , a2PN and a2PN , we can simplify e 1 d q ek 1 2 1 2 2PN r PN k r k with sin χ , then after a long, but straightforward calcu- 0 =e ˙r 2 +e ˙r 4 cos χp p l 8l der r r k=0 lation we obtain a rank 6 polynomial in cos χp, the coef- X 3 ficients of which have to vanish one by one, as discussed l˙2PN e2 − 1 l˙PN 1 r r r k k in Appendix A. − 2 − 4 qker cos χp lr lr lr 2lr Xk=0 sin χ 3 − p χ˙ PN kq ek cosk−1 χ 2. The angular momentum conditions l4 p k r p 8 r Xk=0 χ˙ N 5 With the method for verifying the consistency shown + p ks ek cosk−1 χ . (83) in detail above, we can proceed to verify the consistency l2 k r p 2 r ! of the other constraints. Xk=0 For the nonspinning 2PN evolution α =0= a · LˆN, Inserting the evolutions of er, lr and χp, the 1PN accurate hence Jˆ,J ˆm,Jˆ = (0, 0, J) and the consistency con- consistency condition (82) becomes l LN ditions (73)-(75) simply state that all components of PN PN 0 = a1 sin χp − a2 (er + cos χp) the dimensionless total angular momentum vector should 2 3 be conserved independently (there is no precession in- (1 + er cos χp) sin χp k k−1 volved). The time derivative of the nontrivial compo- + 6 kqker cos χp (84). 8lr nent gives (the same equation emerges by taking the time Xk=0 12 derivative of Eq. (55) with κi = α = 0): D. Consistency of spin and SO contributions l˙ 0= r (1 + ǫ + ǫ ) +ǫ ˙ +ǫ ˙ . (87) l PN 2PN PN 2PN In the Newton-Wigner-Price SSC the total energy does r not contain SO contributions, therefore the time deriva- Following the steps of the proof of consistency of the tive of Eq. (47) will not led to any constraints on the energy constraint we obtain, to 1PN order accuracy leading SO part of the dynamics. 3 In order to proceed with the consistency of the total 2 (2 − η) er (1 + er cos χp) sin χp 0 = 6 angular momentum constraints, by including the contri- lr −2 butions linear in the spin, we need to remember that lr PN PN +a1 sin χp − a2 cos χp , (88) represents one relative PN order. The rhs of the constraints (53)-(54) contain projections of the spin and of then NW P LSO , which are linear in the spins. We will consider 3 2 PN k PN k+1 only contributions linear in the spins and to leading order 0 = c1(k) cos χp − c2(k) cos χp −2 in lr . Xk=0 Xk=0 The consistency condition of the constraint (53), given +2(2 − η) er (1 + er cos χp) . (89) by Eq. (73) is

This again holds true in each polynomial rank of cos χp, 2 confirming 1PN level consistency of the total angular mo- d 0= χ sin κ ν2i−3 cos ψ mentum constraint. i dt i i i=1 At 2PN order accuracy Eq. (87) gives X n h η − − 4ν2i 3 +3 (1 + e cos χ ) aPN aPN l2 r p a2PN 1 b a2PN 2 b 2 r 0 = − 1 + 2 1 sin χp + 2 cos χp + 2 2 2lr 2lr × sin (χp + ψp) sin (χp + ψp − ψi) . (94) 3 (1 + e cos χ )3 sin χ − r p p kp ek cosk−1 χ , (90) io 8l8 k r p r k=0 From among the time derivatives we explore thatχ ˙ p has X a Newtonian partχ ˙ N = O l−3 ; then l˙ = O l−4 ,e ˙ = with the notations p r r r r O l−5 and ψ˙ PN = O l−5 , respectively. We also need 1 r p r l to keep in mind that the spin terms appearing in Eq. b1 = b1(l) cos χp , (62) combine to lr. Hence in Eq. (94) we will take into l=0 l−5 X 2 account the leading order O r terms, but also those of b = 9 − 7η + (1 − 3η) e , − − 1(0) r O l 6 which could combine to O l 5 terms by virtue r r b1(1) = 10(1 − η) er , (91) of Eq. (62). 2 2 2 l SO 2i−3 b2 = b2(l) cos χp , 0= κ˙ i ν χi cos κi cos ψi − χi sin κi l=0 i=1 i=1 X X X − η − b2(0) = 2(1 − 3η) er , × ψ˙ PN + ψ˙ SO ν2i 3 sin ψ +χ ˙ N 4ν2i 3 +3 i i i p 2l2 b2(1) = b1(0) + erb2(0) , r n d b2(2) = b1(1) . (92) × [(1 + er cos χp) sin (χp + ψp) dχp By inserting the coefficients, simplifying with sin χp we obtain a fifth order polynomial in cos χp: × sin (χp + ψp − ψi)] . (95) 5 4 o 2PN k 2PN k+1 To the required order the derivatives are 0 = −2 c1(k) cos χp +2 c2(k) cos χp k=0 k=0 X X 2 2 2 1 3 N (1 + er cos χp) PN PN k+l χ˙ p = 3 , (96) + b2(l)c2(k) − b1(l)c1(k) cos χp lr ! Xl=0 kX=0 Xl=0 Xk=0 3 (1 + er cos χp) − k k 1 SO η 2 − kpker cos χp , (93) κ˙ = − (1+ e cos χ ) sin (χ + ψ − ψ ) 4 i 6 r p p p i k=0 4lr X 2 the coefficients of which can be verified to vanish one by 2k−3 one, as indicated in Appendix A. × 4ν +3 χk sin κk [er cos(ψk − ψp) k=1 Therefore we fulfilled the task to prove the consistency X of the nonspinning evolution and constraint equations up +4cos(χp + ψp − ψk) to 2PN accuracy. +3er cos(2χp + ψp − ψk)] , (97) 13

η ˙ SO 3 3−2i or by inserting the respective time evolutions ψi = 5 (1 + er cos χp) 4+3ν 2lr η 2 − 6 (1 + er cos χp) 4lr 2 2 × [sin (χp + ψp)cot α − cos(χp + ψp − ψi)cot κi] η (1 + er cos χp) 2i−3 J˙ ˆm = 4ν +3 χ sin κ 2 2l5 i i 2k−3 r i=1 × 4ν +3 χk sin κk [er cos(ψk − ψp) X k=1 d X × [(1 + er cos χp)cos(χp + ψp) sin (χp − ζi)] +4cos(χ + ψ − ψ ) dχp p p k h +3er cos(2χp + ψp − ψk)] . (98) +(1+ er cos χp)cos(ψp + ζi) , (102) i We get

2 aSO l−7 J l 2i−3 Then as 3 = O r and to leading order cos α = r, 0= χi sin κi 4ν +3 (1 + er cos χp) sin ψi −5 the second term in Eq. (74) is also O lr . As the spin i=1 X n magnitudes are arbitrary constants, to leading order the d + [(1 + er cos χp) sin (χp + ψp) sin (χp + ψp − ψi)] consistency condition (74) splits into two equations, one dχp for each spin direction: 2 o 1 − ν2i−3χ [sin κ sin ψ cot α − cos κ sin (χ + ψ )] 2l i i i i p p r i=1 X 2 d 2k−3 0=2 [(1 + er cos χp)cos(χp + ψp) sin (χp − ζi)] × sin (χp + ψp) 4ν +3 χk sin κk dχp k=1 X +2(1 + er cos χp)cos(ψp + ζi) − cos(ψp + χp) × [er cos(ψk − ψp)+4cos(χp + ψp − ψk) × [er cos ζi +4cos(χp − ζi)+3er cos(2χp − ζi)] . +3er cos(2χp + ψp − ψk)] , (99) (103) which, after exploring the constraint (62) in the third line, becomes This can be shown to identically hold, hence to leading 2 order in the SO contributions the consistency condition 2i−3 0= χi sin κi 4ν +3 (1 + er cos χp) sin ψi (74) is obeyed. i=1 X n d Finally for the consistency condition (75) we calculate + [(1 + er cos χp) sin (χp + ψp) sin (χp + ψp − ψi)] J˙ ˆ dχp the derivative LN as the rhs of Eq. (55), with only the 2 o Newtonian and SO terms included. We again explore 1 − sin (χ + ψ ) 4ν2k−3 +3 χ sin κ the orders at which the dimensionless variables change, 2 p p k k k=1 obtaining X × [er cos(ψk − ψp)+4cos(χp + ψp − ψk)

+3er cos(2χp + ψp − ψk)] . (100) 2 SO N ηer sin χp 2i−3 This can be shown to identically hold, hence to leading J˙ ˆ = l˙ +χ ˙ 4ν +3 χ cos κ =0 . LN r p 2l2 i i order in the SO contributions the consistency condition r i=1 (73) is obeyed. X (104) For the consistency condition (74), we calculate the The second identity emerges by inserting the explicit expressions of l˙SO, aSO, aSO andχ ˙ N and holds at O l−5 . derivative J˙ ˆm as the rhs of Eq. (54). We count the orders r 1 2 p r SO −7 at which the dimensionless variables change, obtaining to Then as a3 = O lr and J sin α = O (1), the second − leading order term in Eq. (75) is O l 6 , to be dropped. Hence to r leading order in the SO contributions this last consis- 2 2i−3 tency condition is also obeyed. PN 2i−3 N η 4ν +3 J˙ ˆm = χ sin κ ψ˙ ν cos ψ +χ ˙ i i i i p 2l2 i=1 r Remarkably by Eq. (104) we have proven that in the X h d ˆ ˆ basis l, ˆm, LN not only Jˆl, but also JLˆ is conserved × [(1 + er cos χp)cos(χp + ψp) sin (χp − ζi)] , N dχp to leading order. This indicates how well this precessing i (101) basis is adapted to the dynamics of the binary. 14

E. Consistency of SS contributions Our aim here is to apply the equations we derived for the study of conservative dynamics in order to test The time derivative of the Keplerian + SS part of the general relativistic features of gravity. Indeed, it is energy constraint (47) gives well known (from example from the general relativistic Oppenheimer-Volkoff equation) that general relativity SS SS 0 = a1 sin χp − a2 (er + cos χp) predicts stronger gravity at short distances, than aris- 4 ing in the Newtonian description. Therefore we expect 3η (1 + e cos χ ) − r p χ χ that for sufficiently large values of the PN parameter the l8 1 2 2 r highly eccentric orbits could produce the following feature. Orbits which (in terms of the eccentricity e of the × 2e cos κ cos κ sin χ − sin κ sin κ r r 1 2 p 1 2 osculating orbit) are hyperbolic, could become elliptic close to the periastron. Another way to see that is due × e sin χ cos ζ − +3cos 2χ − ζ r p ( ) p (+) the potential well deepening faster in general relativity than in Newtonian gravity. Such orbits locally look hy- +2(1 + er cos χp) sin 2χp − ζ(+) . (105) perbolic at large distances and elliptic at short distances. Hence we call them chameleon orbits. After inserting the dimensionless perturbing force components, it is not difficult to verify that the coefficients It has been known earlier [18] that due to gravitational k radiation hyperbolic orbits can turn into elliptic. Our of cos κ1 cos κ2 and er sin κ1 sin κ2 (with k = 0, 1, 2) all vanish, therefore the above equation is an identity. analysis shows a similar effect already at the conservative The total angular momentum does not contain SS con- level. We were able to illustrate this behavior already by tributions, therefore the time derivatives of Eqs. (53)- including the 1PN corrections to the Keplerian dynamics, (55) do not impose any constraints at the leading SS by evolving numerically the system of equations (36)-(37) order of the dynamics. and (43). For this case of zero spins (χ1 = χ2 = 0) the system of differential equations is closed. The chameleon behavior is presented on Fig. 1, both for equal mass bi- F. Consistency of QM contributions naries ν = 1 (left panel) and for a highly asymmetric system, with mass ratio ν =1/30 (right panel). The initial values were chosen at the periastron as e (χ =0)=0.96 The time derivative of the Keplerian + QM part of the r p and ε (χ =0)= Gm/c2r =0.01 in both cases. Then energy constraint (47) gives the QM order equation: p min lr (χp = 0) is derived from (8). The function er (χp) is QM QM symmetric to the periastron and its asymptotic values 0 = a1 sin χp − a2 (er + cos χp) are larger for decreasing mass ratios. The orbits R cos χp 4 2 2 3η (1 + er cos χp) 2 2i−3 vs. R sin χp with R = c r/Gm are represented by the + wiχ ν 2l8 i green curve on Fig. 1. The domains with er < 1 and r i=1 er > 1 are also indicated. X2 × 2(1+ er cos χp) sin κi cos(χp − ζi) sin (χp − ζi) We then proceeded to study the modifications induced 2 2 −er sin χp 1 −3sin κi cos (χp − ζi) . (106) by the spins on these orbits. For this we supplemented the 1PN corrections with the leading order SO contri- After inserting the dimensionless perturbing force com- bution. For simplicity we have chosen non-precessing ponents, it is not difficult to verify that the coefficients configurations, with the spins of the components either 0 k 2 of sin κi and er sin κi (with i =1, 2 and k =0, 1, 2) all aligned or anti-aligned with the orbital momentum. In vanish, therefore the above equation is an identity. this case the angles κ1 and κ2 remain constants during The total angular momentum does not contain QM the motion and the system of equations (36)-(37) and contributions, therefore the time derivatives of Eqs. (53)- (43) is again closed. The orientations of the orbital angu- (55) do not impose any constraints at the leading QM lar momentum and spin vectors are indicated by arrows order of the dynamics. on the panels. Both dimensionless spin parameters are taken as χi = 0.9982, which is the canonical spin limit, achieved by black holes with radiating accretion disks VII. CHAMELEON ORBITS leading to photon capture [44]. The initial conditions were the same as for the chameleon orbits represented In this section we investigate highly eccentric orbits, on Fig. 1. For anti-parallel spins the two SO contribu- with er ≈ 1. Such orbits could be induced by three- tions cancel in the equations, therefore the orbit is iden- body interactions, also could arise in the central regions tical to the one represented on the left panel of Fig. 1. of galaxies. Stellar orbits in these regions were already When the spins are parallel, the orbits become asymmet- investigated in order to test the spin of the central su- ric with respect to the periastron, as shown on Fig. 2. permassive black hole [42]. Gravitational radiation from Then a further distinction comes from the alignment or such highly eccentric orbits was recently discussed in anti/alignment of the spins with the orbital angular mo- Refs. [15–19, 43]. mentum. In the anti-aligned case (left panel), the asymp- 15 totic value of er (χp) is larger before the periastron than metric evolutions occur than in the case of equal masses after it. For spins aligned with LN (right panel) the evo- but the difference between the asymptotic values of er is lution of er shows an opposite trend, also the difference enhanced by the small mass ratio. The asymmetric char- between the asymptotic values becomes slightly smaller. acter (e.g. which asymptotic value of er is bigger) of the On Fig. 3 we show various chameleon orbits due the orbits is determined by the orientation of the spin of the 1PN and SO contributions in the equations of motion larger mass with respect to the orbital angular momen- for unequal mass (ν =1/30) spinning binaries, again for tum, as shown on the upper right and lower left panels. χi = 0.9982. Each of the spins could be either aligned The orientation of the second spin has but little influence or anti-aligned with the orbital angular momentum. The on the precise asymptotic values of er, while the generic various possibilities are represented by arrows on the pan- shape of the chameleon evolutions is unaffected, as can be els of the figure. When the spins are parallel with each seen by comparing the upper panels or the lower panels. other (upper left and lower right panels), similar asym-

orbit orbit

1.08 100 1.08 100 ) )

p 50 p 50

χ 0 χ 0 1.0610 1.0610 1.06 -50 1.06 -50 -100 -100

1.0444 R cos( -150 1.0444 R cos( -150

-200 >1 <1 >1 -200 >1 <1 >1 1.04 r r r 1.04 r r r -250 e e e -250 e e e e -300 e -300 r -400 -300 -200 -100 0 100 200 300 400 r -400 -300 -200 -100 0 100 200 300 400 1.02 χ 1.02 χ R sin( p) R sin( p)

1 1 ν=1 ν=1/30 0.98 χ χ 0.98 χ χ 1= 2=0 1= 2=0 0.96 0.96 -2 -1 0 1 χ 2 -2 -1 0 1 χ 2 p p

FIG. 1: Chameleon orbits due to the 1PN order eﬀects are shown for equal (ν = 1, left panel) and asymmetric (ν = 1/30, right panel) mass binaries. The chameleon behaviour is characterized by the trespassing of the function er (χp) across the value 1 2 (indicated in blue). Initial conditions are ﬁxed at the periastron as er (χp = 0) = 0.96 and ε (χp =0) = Gm/c rmin = 0.01. The asymptotic values of er are given (in blue) on the left and right sides on each panel. The orbits R cos χp vs. R sin χp with 2 R = c r/Gm are shown by the (green) curve in the smaller boxes. The domains with er < 1 and er > 1, respectively are also indicated.

orbit orbit

1.08 100 1.08 100 ) )

p 50 p 50

χ 0 χ 0 1.06 -50 1.06 -50 1.0514 -100 -100 1.0476 R cos( -150 R cos( -150 1.0410 -200 >1 <1 >1 -200 >1 <1 >1 1.04 r r r 1.04 1.0374 r r r -250 e e e -250 e e e e -300 e -300 r -400 -300 -200 -100 0 100 200 300 400 r -400 -300 -200 -100 0 100 200 300 400 1.02 χ 1.02 χ R sin( p) R sin( p) ^ ^ ^ ^ LN S1 LN S2 1 1 ν=1 ν=1 0.98 χ χ 0.98 χ χ 1= 2=0.9982 1= 2=0.9982 S^ S^ 0.96 1 2 0.96 -2 -1 0 1 χ 2 -2 -1 0 1 χ 2 p p

FIG. 2: Chameleon orbits due to 1PN and SO eﬀects for binaries with equal masses and spins (χ1 = χ2 = 0.9982). The curves and initial conditions are as on Fig. 1. On the left (right) panel the spins are anti-aligned (aligned) with the orbital angular momentum.

VIII. CONCLUDING REMARKS Newtonian order accuracy, by including the leading order

In this paper we have considered the conservative evolution of spinning compact binaries up to the second post- 16

orbit orbit

1.08 100 1.08 100 ) )

1.0690 p 50 1.0686 p 50 χ 0 χ 0 1.06 -50 1.0573 1.06 -50 1.0574 -100 -100

R cos( -150 R cos( -150

-200 >1 <1 >1 -200 >1 <1 >1 1.04 r r r 1.04 r r r -250 e e e -250 e e e e -300 e -300 r -400 -300 -200 -100 0 100 200 300 400 r -400 -300 -200 -100 0 100 200 300 400 1.02 χ 1.02 χ R sin( p) R sin( p) ^ ^ ^ LN LN S2 1 1 ν=1/30 ν=1/30 0.98 χ χ 0.98 χ χ 1= 2=0.9982 1= 2=0.9982 S^ S^ S^ 0.96 1 2 0.96 1 -2 -1 0 1 χ 2 -2 -1 0 1 χ 2 p p

orbit orbit

1.08 100 1.08 100 ) )

p 50 p 50

χ 0 1.0645 χ 0 1.0647 1.06 -50 1.06 -50 1.0534 -100 1.0530 -100

R cos( -150 R cos( -150

-200 >1 <1 >1 -200 >1 <1 >1 1.04 r r r 1.04 r r r -250 e e e -250 e e e e -300 e -300 r -400 -300 -200 -100 0 100 200 300 400 r -400 -300 -200 -100 0 100 200 300 400 1.02 χ 1.02 χ R sin( p) R sin( p) ^ ^ ^ ^ ^ S1 LN S1 LN S2 1 1 ν=1/30 ν=1/30 0.98 χ χ 0.98 χ χ 1= 2=0.9982 1= 2=0.9982 S^ 0.96 2 0.96 -2 -1 0 1 χ 2 -2 -1 0 1 χ 2 p p

FIG. 3: Chameleon orbits created by 1PN and SO effects are represented for unequal mass (ν = 1/30) spinning binaries. The same functions as on Fig. 1 are shown for the same initial conditions. The relative direction of the spins and the orbital momentum are indicated by arrows. The dimensionless spin values are the same χ1 = χ2 = 0.9982 in all panels. spin-orbit, spin-spin and mass quadrupole-monopole con- evolution equations, and proved that they are preserved tributions. The novel feature of the discussion is that it by the evolution. had been presented in terms of suitably chosen dimen- We applied the formalism to show the existence of or- sionless variables. These are i) the variables replacing bits with unusual features. Close to the periastron, the the traditional orbital elements of celestial mechanics: a osculating orbits of these trajectories with eccentricity dimensionless version of the orbital angular momentum, close to one change from hyperbolic to elliptic, then back the eccentricity and three Euler angles characterising the to hyperbolic. Hence these orbits (as characterised by orientation of the orbit and the orbital plane with re- the eccentricity of their osculating orbit) look open, then spect to the total angular momentum vector, also ii) di- closed, then open again during the passage through the mensionless spin magnitudes (smaller than one for both periastron. These chameleon orbits evolve from elliptic black holes and neutron stars) together with the spin az- (locally, in a Newtonian sense) close to the periastron into imuthal and polar angles. The preferred reference system hyperbolic (in the same sense) at large distances. Such a of this analysis is tied to the orbital angular momentum property emerges due to the fact that General Relativity and periastron. predicts stronger gravity (deeper potential wells) at short As a main result we derived a system of first order distances than Newtonian theory does, as also illustrated differential equations in a compact form, for a set of 9 by the hydrostatic equilibrium in relativistic stars. dimensionless variables encompassing both the orbital el- We analysed the chameleon orbits as function of ements and the spin angles (the spin magnitudes being mass ratios and spin orientations, for aligned and anti- conserved). These are supplemented by the evolution aligned spin and orbital angular momentum configura- equation of the true anomaly, which closes the differen- tions. Without spin, these orbits are symmetric with tial system. respect to the periastron. The farther the mass ratio These evolutions are constrained by the conservation is from unity, the larger is the change in the eccentric- laws of energy and total angular momentum vector hold- ity of the osculating orbit, hence the easier to find such ing at 2PN order. As required by the generic theory of chameleon orbits. constrained dynamical systems we analyzed the consis- The presence of spins can not be detected when the tency of the constraints, e.g. their compatibility with the masses are equal and the spins anti-aligned with each 17 other. In all other cases with spin, they induce an asym- The coefficient of cos χp gives: metry with respect to the periastron. One aspect of this asymmetry is that the minimum of the eccentricity is not 2 2 2PN 2PN in the periastron, as can be seen on Figs. 2-3. As a rule 0 = 2s2er +s1er + 16 c1(1) − c2(0) we found that the alignment of the total spin S1 + S2 2PN 2PN 2 2PN +16erc − 16erc − 16erc with the orbital angular momentum shifts the minimum 1(0) 2(1) 2(0) dq dq dq eccentricity point of the trajectory before the periastron, −2cPN 1 + 0 e − 2cPN 0 2(0) de de r 2(1) de while the anti-alignment shifts it after the periastron. r r r These results hold both in the equal mass and in the PN PN dq0 asymmetric mass cases. +2 c1(1) − c2(0) 2 (q1 − 2q0)+ er der This feature of relativistic orbits is complementary to PN dq1 2 dq0 +2c1(0) 2(2q2 − q1) er + er + ,(A2) how the rotation or counterrotation of a particle in circu- der der lar orbit about a rotating black hole affects the location of the innermost stable orbit. In our case co-rotation adding up to zero after inserting the definitions of the apparently speeds up the (reduced mass) particle, while coefficients cPN , c2PN , q , s . counter-rotation slows it down, after leaving the perias- i(k) i(k) i i 2 tron. The coefficient of cos χp gives:

3 3 2PN 2PN 0 = 3s3er + 2s2er + 16 c1(2) − c2(1)

2PN 2 2PN 2PN 2PN −16erc2(2) − 16erc2(1) + 16er c1(1) − c2(0) IX. ACKNOWLEDGEMENTS dq dq +2cPN 6q e + 1 e − 2cPN 0 1(0) 3 r de r 2(2) de r r LAG´ was supported by the European Union and the PN dq1 dq0 PN dq1 2 −2c2(1) + er − 2c2(0) er State of Hungary, cofinanced by the European Social der der der Fund in the framework of TAMOP´ 4.2.4. A/2-11-/1- PN PN dq1 2 dq0 2012-0001 ‘National Excellence Program’. ZK has been +2 c1(1) − c2(0) 2(2q2 − q1) er + er + der der supported by OTKA Grant No. 100216. PN PN dq0 +2 c1(2) − c2(1) 2 (q1 − 2q0)+ er , (A3) der where the terms in the last line cancel by virtue of the relations between the coefficients cPN , while the first five Appendix A: Computational details for verifying the i(k) 2PN accurate, nonspinning consistency conditions lines add up to zero after inserting the definitions of the PN 2PN coefficients ci(k), ci(k) , qi, si. 3 The coefficient of cos χp gives: In this Appendix we give computational details for the proof of the 2PN accurate consistency conditions in the 4 4 2PN 2PN absence of spins. 0 = 4s4er + 3s3er + 16 c1(3) − c2(2)

First we discuss the energy consistency condition (86). 2PN 2 2PN 2PN 2PN PN PN −16erc2(3) − 16erc2(2) + 16er c1(2) − c2(1) After inserting the acceleration components a1 , a2 , 2PN 2PN a1 and a2 , then simplifying with sin χp the coef- PN dq1 dq0 PN dq1 2 −2c2(2) + er − 2c2(1) er ficients of the 6th order polynomial in cos χp, enlisted der der der below, have to vanish. PN PN dq1 PN 3 +2 c1(1) − c2(0) 6q3er + er + 4c1(0)q3er The terms without cos χp give: der PN PN dq1 2 dq0 +2 c1(2) − c2(1) 2(2q2 − q1) er + er + dq der der 0 = s e + 16c2PN − 16e c2PN − 2cPN 0 1 r 1(0) r 2(0) 2(0) dq0 der PN PN +2 c1(3) − c2(2) 2 (q1 − 2q0)+ er , (A4) der PN dq0 +2c1(0) 2q1 + er − 4q0 , (A1) der and the terms in the last two lines cancel by virtue of PN the relations between the coefficients ci(k), while the first which add up to zero after inserting the definitions of the four lines add up to zero after inserting the definitions of PN 2PN PN 2PN coefficients ci(k), ci(k) , qi, si. the coefficients ci(k), ci(k) , qi, si. 18

4 The coefficient of cos χp gives: where all terms cancel by virtue of the relations between PN 2PN the coefficients ci(k), ci(k) . 5 2 2PN 2PN 2PN 0 = 4s4er − 16erc2(3) + 16er c1(3) − c2(2) In summary all these 6 equations reduce to identities, dq confirming the consistency condition arising from energy −2cPN 1 e2 +4 cPN − cPN q e3 conservation. 2(2) de r 1(1) 2(0) 3 r r The consistency condition (93), arising from the total 4 2PN 2PN 2PN +er 5s5er − 16c2(4) + 16 c1(4) − c2(3) angular momentum conservation takes the explicit form 5 0= k=0 h(k) cos χp, with the coefficients PN PN dq1 2 dq0 +2 c1(3) − c2(2) 2(2q2 − q1) er + er + der der P dq1 1 +2 cPN − cPN 6q e + e , (A5) PN PN 2PN 1(2) 2(1) 3 r r h(0) = b2(0)c2(0) − b1(0)c1(0) − 2c1(0) − p1er der 4 h = b cPN − b cPN + b cPN − b cPN and the terms in the last three lines cancel by virtue (1) 2(0) 2(1) 1(0) 1(1) 2(1) 2(0) 1(1) 1(0) PN 2PN 1 1 of the relations between the coefficients ci(k), ci(k) , si, 2PN 2PN 2 +2 c − c − p2 + p1 e while the first two lines add up to zero after inserting the 2(0) 1(1) 2 4 r PN 2PN definitions of the coefficients c , c , qi, si. PN PN PN PN i(k) i(k) h(2) = b2(0)c2(2) − b1(0)c1(2) + b2(1)c2(1) − b1(1)c1(1) 5 The coefficient of cos χp gives: PN 2PN 2PN 3 1 3 +b2(2)c2(0) +2 c2(1) − c1(2) − p3 + p2 er 2 4 2PN 2PN 2PN 4 2 0 = e 5s5e − 16c + 16 c − c r r 2(4) 1(5) 2(4) PN PN PN PN h(3) = b2(1)c2(2) − b1(1)c1(2) − b1(0)c1(3) + b2(2)c2(1) 2PN 2PN PN PN 3 +16er c1(4) − c2(3) +4 c1(2) − c2(1) q3er 3 +2 c2PN − c2PN − p e4 2(2) 1(3) 4 3 r PN PN dq1 +2 c1(3) − c2(2) 6q3er + er , (A6) PN PN 2PN 2PN der h(4) = b2(2)c2(2) − b1(1)c1(3) +2 c2(3) − c1(4) 2PN 2PN where all terms cancel by virtue of the relations between h(5) = 2 c2(4) − c1(5) , (A8) the coefficients cPN , c2PN , s . i(k) i(k) i all of which vanishing by virtue of the relations between 6 PN 2PN The coefficient of cos χp gives: the coefficients ci(k), ci(k) , bi(k) and pi. Therefore the consistency condition arising from total angular momen- 2PN 2PN PN PN 2 0=4 c1(5) − c2(4) + c1(3) − c2(2) q3er , (A7) tum conservation also holds.

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