Eccentric Waveforms

Eccentricity replaces the nearly single-frequency circular carrier by a comb of harmonics tied to radial and azimuthal motion. Relativistic periastron advance splits those frequencies, while large eccentricity concentrates acceleration and radiation near periastron into repeated bursts. Radiation reaction often drives an isolated adiabatic binary toward the circular limit, but the rate and the residual eccentricity depend on where and how eccentricity is defined.

Helpful background. Einstein Equations supplies the relativistic corrections; Waves, Green Functions, and Geometric Optics supplies harmonic and retarded-wave structure; Relativistic Two-Body Dynamics supplies the accepted eccentric orbit and its evolution; Gravitational Radiation and Waveforms supplies the multipolar strain and flux; and Computational Relativity supplies strong-field waveforms beyond an adiabatic expansion.

Two orbital frequencies and a harmonic comb

For bound planar relativistic motion, let $T_r$ be the interval between successive periastra and define

$$\Omega_r=\frac{2\pi}{T_r}, \qquad \Omega_\phi=K\Omega_r, \qquad K=1+k.$$

$2\pi k$ is the periastron advance per radial period. At leading post-Newtonian order for semimajor axis $a$ and a specified eccentricity $e$,

$$k=\frac{3GM}{a(1-e^2)}+\cdots.$$

This orbital result is used as input rather than rederived. Beyond Newtonian order, “the eccentricity” is not unique: quasi-Keplerian descriptions contain radial, time, and angular eccentricities. A waveform statement must name the choice, coordinate or invariant frequency variables, and reference epoch.

The rescaled strain

$$H=r(h_+-ih_\times) =\sum_{\ell,m}H_{\ell m}\,{}_{-2}Y_{\ell m}$$

uses unit-normalized spin-weight $-2$ harmonics. A doubly periodic eccentric mode has the schematic expansion

$$H_{\ell m}(t) =\sum_{n=-\infty}^{\infty} A_{\ell mn}\, e^{-i[m\psi_\phi(t)+n\psi_r(t)]}, \qquad \omega_{mn}=m\Omega_\phi+n\Omega_r,$$

consistent with the global $e^{-i\omega t}$ convention. Slow radiation reaction makes $A_{\ell mn}$ and both fundamental frequencies drift secularly. Blanchet 2024, §§3.5.1–3.5.2, pp. 220–227 develops this doubly periodic and quasi-Keplerian structure.

In the Newtonian limit, $\Omega_\phi=\Omega_r$, so different pairs $(m,n)$ can coincide at integer multiples of one mean motion. Periastron advance removes that degeneracy. The resulting sideband spacing is set by $\Omega_r$, not by spin-precession frequency.

Periastron bursts and harmonic redistribution

For a Kepler ellipse,

$$r=a(1-e\cos u), \qquad \ell_{\mathrm{mean}}=u-e\sin u,$$

where $u$ is the eccentric anomaly. Conservation of Newtonian angular momentum gives $r^2\dot\phi=\text{constant}$, so the angular speed and quadrupole acceleration are greatest at periastron. As $e$ grows, a smooth sinusoidal waveform becomes a train of sharper passages, and representing each passage requires more Fourier harmonics.

The leading orbit-averaged luminosity makes the enhancement explicit:

$$\left\langle\mathcal F\right\rangle =\frac{32}{5G}\nu^2 \left(\frac{GM}{a}\right)^5 \frac{1+\frac{73}{24}e^2+\frac{37}{96}e^4} {(1-e^2)^{7/2}}.$$

At $e=0$ this reduces to the circular quadrupole flux. The slow-motion approximation additionally requires weak periastron gravity,

$$\frac{GM}{a(1-e)}\ll1.$$

Thus $e\to1$ at fixed $a$ sends $r_p=a(1-e)\to0$ and is not a uniform limit of the Peters–Mathews formula: both slow-motion control and orbit-averaged adiabatic control can fail during a strong isolated passage. The distribution of power among integer harmonics within the controlled arbitrary-eccentricity slow-motion domain was derived in Peters and Mathews 1963, pp. 435–440.

The enhancement factor does not by itself give a waveform. Polarization, viewing direction, periastron phase, and the coefficients $A_{\ell mn}$ determine the coherent strain. Nor does a broad harmonic spectrum necessarily imply eccentricity: abrupt merger structure or precessional rotation can also create sidebands, but with different time-frequency organization.

Secular evolution toward circularity

At leading quadrupole order for an isolated adiabatic binary, the accepted orbit-averaged eccentricity evolution is

$$\left\langle\frac{de}{dt}\right\rangle =-\frac{304}{15} \frac{G^3M^3\nu}{a^4} \frac{e}{(1-e^2)^{5/2}} \left(1+\frac{121}{304}e^2\right).$$

This is the secular decay law of Peters 1964, eq. (5.7). It is negative for $0\lt e\lt1$ and vanishes at $e=0$, making circular motion a fixed point within this approximation; the semimajor-axis decay is given alongside it in eq. (5.6). Because $e$ changes while the binary chirps, quoting “the eccentricity of a waveform” without a reference frequency or time is incomplete.

This circularization statement has boundaries. A recently captured system may retain substantial eccentricity close to merger; a third body or environment can continually excite eccentricity; and different relativistic eccentricity parameters agree only in the weak-field limit. This page organizes the resulting radiation but does not derive the binary equations of motion, environmental dynamics, production numerical algorithms, or observational parameter estimation.

For a cosmological source, every source-frame harmonic frequency is observed as

$$f_{\mathrm{det}}=\frac{f_{\mathrm{src}}}{1+z}.$$

The ratios of harmonic frequencies and the dimensionless eccentricity at a correspondingly mapped reference epoch are unchanged by homogeneous cosmological redshift.

Circular and high-eccentricity checks

The circular limit requires more than setting one coefficient to zero. As $e\to0$:

• radial amplitude oscillations vanish;
• the Newtonian harmonic power collapses to the circular quadrupole structure;
• periastron orientation becomes physically meaningless;
• the leading eccentricity decay is linear in $e$ and approaches zero smoothly.

At high eccentricity, the opposite checks apply. A calculation retaining only one or two harmonics cannot resolve a narrow periastron passage. Orbit averaging must be compared with the periastron timescale, and the approximation should be abandoned if radiation reaction changes the orbit substantially during one burst.

For small eccentricity the circularization rate has a simple quantitative interpretation. Combining the displayed $\dot e$ with $\dot a=-64G^3M^3\nu/(5a^3)+O(e^2)$ gives

$$\frac{de}{da}=\frac{19}{12}\frac{e}{a}+O(e^3), \qquad e\propto a^{19/12}\propto f_{22}^{-19/18}.$$

Here $f_{22}\propto a^{-3/2}$ is the nearly circular quadrupole frequency, and both values of $e$ must use the same definition. Doubling that frequency therefore reduces a sufficiently small eccentricity by $2^{-19/18}\simeq0.481$ at leading order. This local scaling explains why a reference frequency is part of an eccentricity measurement; it should not be extrapolated to a capture or a strong-field separatrix.

Common pitfalls

Using one undefined eccentricity at high PN order. Radial, time, and angular eccentricities differ. A model must name its parameter and reference frequency.

Confusing periastron and spin precession. Eccentric sidebands follow the radial–azimuthal frequency split and recur at periastron. Spin precession rotates the radiation frame on a separate timescale.

Assuming every eccentric binary circularizes before merger. The sign of the adiabatic quadrupole formula establishes a tendency in its regime, not a universal statement about captures, perturbations, or strong-field mergers.

Exercise

For a Newtonian ellipse, $r_p=a(1-e)$ and $r_a=a(1+e)$. Use angular-momentum conservation to find the ratio of angular speeds at periastron and apastron. Interpret its circular and high-eccentricity limits.

Solution

Conservation gives $r_p^2\dot\phi_p=r_a^2\dot\phi_a$. Hence $\dot\phi_p/\dot\phi_a=(r_a/r_p)^2=[(1+e)/(1-e)]^2$. At $e=0$ the ratio is one, consistent with uniform circular motion and the collapse of the eccentric harmonic comb. As $e\to1$, the ratio diverges: most angular motion and the strongest quadrupole acceleration occur in a short periastron interval. A Fourier representation must then retain many harmonics. At fixed $a$, however, $GM/r_p=GM/[a(1-e)]$ also grows without bound, so the Newtonian slow-motion premise fails before the formal endpoint; orbit averaging must independently be checked against the shrinking burst timescale.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024) 4. DOI. Open PDF.
• Peters, P. C. “Gravitational Radiation and the Motion of Two Point Masses.” Physical Review 136 (1964): B1224–B1232. DOI.
• Peters, P. C., and J. Mathews. “Gravitational Radiation from Point Masses in a Keplerian Orbit.” Physical Review 131 (1963) 435–440. DOI.