Spinning and Eccentric PN Orbits

Spin and eccentricity turn a relativistic binary from a slowly precessing ellipse into a system with several coupled phases. Eccentricity separates radial and azimuthal frequencies; spin causes the orbital plane and the individual angular momenta to precess; spin-induced multipoles modify the radial dynamics. A useful PN solution therefore identifies which elements are conserved in the time-symmetric problem, which oscillate on the orbital timescale, and which evolve only after radiation reaction is restored.

Required background. Weak Metric Perturbations and Gauge Freedom supplies perturbative gauge transformations; Presymplectic Potentials and Currents supplies the constrained and canonical mechanics of spin variables. Helpful background. The Einstein Equation supplies the parent dynamics; Gravitational Radiation and Waveforms supplies the fluxes that drive secular circularization and spin-dependent phasing.

Spin variables and their leading couplings

For body $A$, define the spin angular momentum $\mathbf S_A$ and dimensionless spin

$$\boldsymbol{\chi}_A =\frac{c\,\mathbf S_A}{Gm_A^2}.$$

The spin tensor has more components than a spatial rotation, so a spin supplementary condition (SSC) is required to choose the representative center of mass. A covariant choice is $S_A^{\mu\nu}p_{A\nu}=0$; canonical formulations use a Newton–Wigner-type condition or an equivalent reduced set of variables. Changing the SSC shifts the worldline and spin together while preserving total angular momentum. It is a variable transformation, not a physical displacement of the body. The relation between the SSC and the chosen position is explained in Schäfer and Jaranowski 2024, §§ 3.3 and 7.1.

Canonical spins satisfy

$$\{S_A^i,S_B^j\} =\delta_{AB}\epsilon^{ijk}S_A^k,$$

so any Hamiltonian generates precession:

$$\dot{\mathbf S}_A =\frac{\partial H}{\partial\mathbf S_A}\times\mathbf S_A \equiv\boldsymbol{\Omega}_A\times\mathbf S_A.$$

This immediately preserves each spin magnitude. Let $\mathbf L=\mathbf r\times\mathbf P$, $\mathbf S=\mathbf S_1+\mathbf S_2$, and

$$\mathbf S_*=\frac{m_2}{m_1}\mathbf S_1+\frac{m_1}{m_2}\mathbf S_2.$$

The leading spin-orbit Hamiltonian in center-of-mass canonical variables is

$$H_{\rm SO}^{\rm LO} =\frac{G}{c^2r^3}\, \mathbf L\cdot \left(2\mathbf S+\frac32\mathbf S_*\right).$$

For compact objects with $\chi_A=O(1)$, this is a relative 1.5PN correction. The leading mutual spin-spin term is

$$H_{S_1S_2}^{\rm LO} =\frac{G}{c^2r^3} \left[ 3(\mathbf n\cdot\mathbf S_1)(\mathbf n\cdot\mathbf S_2) -\mathbf S_1\cdot\mathbf S_2 \right],$$

which enters at relative 2PN. Spin-squared terms also contain the object’s spin-induced quadrupole coefficient; the black-hole value is not automatically the neutron-star value. The canonical leading interactions and their normalizations appear in Schäfer and Jaranowski 2024, § 7.3, eqs. (7.72)–(7.75).

In the conservative problem the energy and total angular momentum

$$\mathbf J=\mathbf L+\mathbf S_1+\mathbf S_2$$

are constant, while the individual vectors generally precess. At leading spin-orbit order, special configurations can exhibit “simple precession,” but generic two-spin motion has nutation as well as precession. Aligned or antialigned spins form a symmetry-preserving sector: the orbital plane does not precess, although the radial potential and periastron advance still receive spin corrections.

Eccentric motion and three eccentricities

For a nonspinning conservative binary, radial motion remains periodic while azimuthal motion advances by more than $2\pi$ per radial period. Let $P_r$ be the radial period, $n_{\rm orb}=2\pi/P_r$ the mean motion, and

$$K=\frac{\Omega_\phi}{\Omega_r}=1+k$$

the periastron-advance factor. The 1PN quasi-Keplerian representation introduces an eccentric anomaly $u$:

$$r=a_r(1-e_r\cos u),$$ $$\ell\equiv n_{\rm orb}(t-t_p) =u-e_t\sin u,$$

and an angular anomaly $v$ defined by

$$\tan\frac{v}{2} =\sqrt{\frac{1+e_\phi}{1-e_\phi}}\tan\frac{u}{2}, \qquad \phi-\phi_p=K\,v.$$

The radial, time, and angular eccentricities $e_r$, $e_t$, and $e_\phi$ agree in the Newtonian limit but differ at PN order. This is not redundant notation: each is defined by a different part of the parametric solution. The construction, including the 1PN orbital elements as functions of conserved energy and angular momentum, is given in Blanchet 2024, § 3.5.2, eqs. (532)–(537).

At 2PN and 3PN the generalized Kepler equation and angular motion acquire additional harmonics. Schematically,

$$\ell =u-e_t\sin u +f_t\sin v +g_t(v-u)+\cdots ,$$ $$\frac{\phi-\phi_p}{K} =v+f_\phi\sin 2v+g_\phi\sin 3v+\cdots .$$

The coefficients are functions of the conserved orbital energy and angular momentum. Memmesheimer, Gopakumar, and Schäfer provide the complete 3PN representation in ADM-type and modified harmonic coordinates in Memmesheimer, Gopakumar, and Schäfer 2004, §§ III–IV, eqs. (19)–(25) and (39)–(41).

Some elements are coordinate dependent. The frequencies $\Omega_r$ and $\Omega_\phi$, their ratio $K$, and relations among conserved invariants are the safer comparison quantities. A numerical “eccentricity” inferred from coordinate separation, orbital frequency, or waveform oscillations need not equal any one of $e_r,e_t,e_\phi$. Its estimator and regime must be stated.

Coupled orbital and spin precession

With spin present, $\mathbf L$ is no longer fixed even in the conservative problem. Hamilton’s equations and the spin brackets give

$$\dot{\mathbf L} =-\dot{\mathbf S}_1-\dot{\mathbf S}_2,$$

which follows from $\dot{\mathbf J}=0$. The orbital triad can be defined by $\mathbf n$, $\boldsymbol{\lambda}=\hat{\mathbf L}\times\mathbf n$, and $\hat{\mathbf L}$. Its angular velocity contains rapid orbital rotation and slower precession. A practical multiscale solution separates the orbital phase, periastron phase, spin-precession phases, and—when dissipation is included—the radiation-reaction time.

For eccentric orbits the precession vector varies strongly around the orbit because the leading coupling scales as $r^{-3}$. Orbit averaging is justified when precession is slow compared with $P_r$ and no relevant orbital–precession commensurability invalidates that separation. It removes oscillations on the orbital timescale, but it retains the slower two-spin nutation and can describe spin-orbit resonant configurations. A separate average over a precession cycle removes that slower nutation. Neither average is a pointwise solution. Kidder’s leading spin equations and conserved structures are given in Kidder 1995, §§ II B–II C, eqs. (2.4)–(2.15); the distinction between the two averages is developed on Precession-Averaged Dynamics and Morphologies.

One can combine the quasi-Keplerian radial solution with osculating elements: at each instant the orbit is represented by a member of a conservative family, while weak perturbations evolve its elements. This is useful for spin couplings or radiation reaction, but the element equations inherit the chosen coordinates and SSC. Frequency-based diagnostics are needed before comparing two implementations.

Conserved versus secular variables

In a strictly conservative, autonomous Hamiltonian, $E$, $\mathbf J$, and the spin magnitudes are conserved. For nonspinning planar motion, the magnitude $L$ is also conserved and the radial action labels the orbit. Periastron advance is secular in the angle but does not imply loss of energy. Spin precession likewise changes directions without changing spin magnitudes.

Radiation reaction changes the hierarchy. Energy and angular momentum acquire slow flux-driven evolution; eccentricity usually decreases during an isolated inspiral, but “circularization” is not a theorem for every strong encounter, resonance, or environmentally perturbed binary. Spin directions can evolve both conservatively and dissipatively. This page does not derive fluxes: source-independent radiation theory belongs to Gravitational Radiation and Waveforms, and systematic dissipative orbital evolution belongs to the radiation-reaction chapter.

Checks and regime limits

A spin-orbit calculation should preserve $S_A^2$ and total $\mathbf J$ in the conservative sector. Its test-mass limit must agree with spin precession or geodesic frequencies in Kerr under a stated SSC and variable map. Exchange $1\leftrightarrow2$ must leave the binary Hamiltonian invariant. Spin-induced quadrupole terms must carry the correct object-dependent coefficient and reduce to the Kerr multipole relation for black holes.

An eccentric solution should recover Kepler motion as $c\to\infty$, the circular limit smoothly as the chosen eccentricity tends to zero, and the invariant 1PN periastron advance. The quasi-Keplerian parameterization is for bound motion and becomes delicate near the separatrix, where the radial period grows and zoom-whirl behavior appears. High eccentricity can produce a relativistic pericenter even when the semimajor-axis PN parameter is small. Near that boundary, orbit averaging and a single global PN error estimate both fail.

Exercise

Use the 1PN result

$$K=1+\frac{3GM}{c^2p}+O(c^{-4}), \qquad p=a(1-e^2),$$

to find the periastron advance per radial period. Then explain why $K$ is a better formulation-comparison quantity than $e_r$.

Solution

During one radial period the anomaly $v$ increases by $2\pi$, so the azimuth increases by $2\pi K$. Relative to a closed Newtonian ellipse,

$$\Delta\Phi =2\pi(K-1) =\frac{6\pi GM}{c^2a(1-e^2)} +O(c^{-4}).$$

The result depends only on the invariant ratio of azimuthal and radial frequencies once $K$ is defined from asymptotic time and angle. By contrast, $e_r$ is the coefficient of $\cos u$ in a chosen radial coordinate; a contact transformation changes that coordinate and therefore changes $e_r$. Two formulations can assign different $e_r$ to the same physical orbit while agreeing on $K(E,J)$.

References

• Blanchet, Luc. 2024. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27, 4. DOI.
• Kidder, Lawrence E. 1995. “Coalescing Binary Systems of Compact Objects to (Post)$^{5/2}$-Newtonian Order. V. Spin Effects.” Physical Review D 52, 821–847. DOI.
• Memmesheimer, Raoul-Martin, Achamveedu Gopakumar, and Gerhard Schäfer. 2004. “Third Post-Newtonian Accurate Generalized Quasi-Keplerian Parametrization for Compact Binaries in Eccentric Orbits.” Physical Review D 70, 104011. DOI.
• Schäfer, Gerhard, and Piotr Jaranowski. 2024. “Hamiltonian Formulation of General Relativity and Post-Newtonian Dynamics of Compact Binaries.” Living Reviews in Relativity 27, 2. DOI.