Why the wave from an eccentric orbit is a comb of lines

Turning points and the third root

For an equatorial Kerr geodesic, $r^4(dr/d\tau)^2=R(r)$ with $R=P^2-\Delta\,(r^2+x^2)$, where $P=E(r^2+a^2)-aL$, $\Delta=r^2-2r+a^2$ and $x=L-aE$. $R$ is a quartic in $r$ with roots $0$, $r_3$, $r_p$ and $r_a$. Comparing the constant terms of the cubic factor fixes the third root.

Glampedakis and Kennefick give $E$ and $x^2$ as functions of $p$, $e$ and $a$. For the Lab’s default orbit, $a=0.7,\ p=7,\ e=0.4$, they give $E=0.94461$, $L=3.10553$ and $r_3=1.902$, safely inside the periapsis at $5$.

Reading: Glampedakis & Kennefick (2002)

A parameter that removes the turning points

Integrating in $r$ runs into square-root zeros at both turning points. The substitution $r=p/(1+e\cos\chi)$ removes them. One finds $(r_a-r)(r-r_p)=r^2e^2\sin^2\chi/(1-e^2)$ and $dr/d\chi=r^2e\sin\chi/p$, so the factor $\sin\chi$ cancels and the equations below are smooth for every $\chi$.

We derived these ourselves from the radial potential and checked them against a direct integration of Hamilton’s equations in Boyer–Lindquist coordinates. For the default orbit the two radial periods agree to two parts in ten billion.

Reading: Glampedakis & Kennefick (2002)

The two fundamental frequencies

One radial period is one full turn of $\chi$. Integrate the time and azimuth rates from $0$ to $2\pi$ to get the period $T_r$ and the azimuth $\Delta\phi$ swept in it. The integrands are smooth and periodic, so the trapezoid rule converges exponentially: 32 points already give fifteen digits.

Default orbit. $T_r=226.95\,M$ and $\Delta\phi=9.8259$ rad, which is $563.0^\circ$, so the periapsis advances by $203^\circ$. For $M=10^6\,M_\odot$ the frequencies are $f_r=0.8946\,\mathrm{mHz}$ and $f_\phi=1.3990\,\mathrm{mHz}$.

Reading: Glampedakis & Kennefick (2002)

Why the wave is a comb

Place the orbit in flat space and read $r$ and $\phi$ as polar coordinates. Its mass quadrupole has two independent pieces: $A=r^2e^{2i\phi}$ and $B=r^2$. A short calculation gives $\sum_{jk}\dddot Q_{jk}^2=\mu^2\bigl(|\dddot A|^2/2+\dddot B^2/6\bigr)$ for the trace-free quadrupole.

The azimuth is a steady rotation plus a part that repeats every radial period: $\phi=\Omega_\phi t+\psi(t)$. So $A=e^{2i\Omega_\phi t}\,G(t)$ with $G$ periodic in $T_r$. A Fourier series of $G$ puts the lines of $A$ at $2\Omega_\phi+n\Omega_r$. $B$ depends on $r$ only, so its lines sit at $n\Omega_r$.

If the orbit did not precess, $\Omega_\phi$ would equal $\Omega_r$ and every line would land on a whole multiple of $f_r$. The Lab draws those multiples as dashed grey ticks; the offset of the gold lines from them is the precession, seen in the spectrum.

Reading: Babak et al. (2017)

Power in each line and two checks

The quadrupole formula $\mathcal F=\tfrac15\langle\dddot Q_{jk}\dddot Q_{jk}\rangle$ then splits into a sum over lines. Each $A$ line contributes $\tfrac1{10}|A_n|^2\omega^6$. Because $B$ is real, its lines at $+n$ and $-n$ carry equal power and combine into $\tfrac1{15}|B_n|^2\omega^6$.

Circular check: only $A_0=r^2$ survives, at $\omega=2\Omega_\phi$, and the sum gives $\tfrac{32}{5}r^4\Omega_\phi^6$ exactly. Weak-field check: at $p=1000$ and $e=0.4$ the line sum is 1.0008 times the Peters–Mathews orbit-averaged power of a Newtonian ellipse. The 0.08% is the relativistic change in the orbit at that distance.

For the default orbit the four strongest lines are rotating ones. The breathing lines carry about 0.01% of the power. Eccentric orbits push power to higher $n$: at $e=0.7$ fourteen lines miss up to a fifth of the total, so the Lab sums forty.

Where this model sits among EMRI waveforms

Babak and collaborators describe two cheap families. The numerical kludge follows a Kerr geodesic and applies a flat-space emission formula, which is what this chapter does. The analytic kludge they use for their forecasts starts from a Keplerian orbit and adds precession and inspiral with post-Newtonian rules. They run it in two versions, stopped at the Schwarzschild or the Kerr ISCO, to bracket the modelling error.

Our line frequencies are exact for the geodesic. The line powers are only as good as the quadrupole formula, which the previous chapter showed is off by up to about 50% close to the hole when compared with a Teukolsky calculation. For the default orbit, the line sum is 0.834 times the Peters–Mathews value at the same $p$ and $e$. That ratio compares two approximations and says nothing about which is closer to the truth.

Reading: Babak et al. (2017) · Fujita & Tagoshi (2004)