We derive an analytic expression for the energy spectrum of gravitational waves from a parabolic Keplerian binary by taking the limit of the Peters and Matthews spectrum for eccentric orbits. This demonstrates that the location of the peak of the energy spectrum depends primarily on the orbital periapse rather than the eccentricity. We compare this weak-field result to strong-field calculations and find it is reasonably accurate (~10%) provided that the azimuthal and radial orbital frequencies do not differ by more than ~10%. For equatorial orbits in the Kerr spacetime, this corresponds to periapse radii of rp > 20M. These results can be used to model radiation bursts from compact objects on highly eccentric orbits about massive black holes in the local Universe, which could be detected by LISA.
Deep Dive into Gravitational wave energy spectrum of a parabolic encounter.
We derive an analytic expression for the energy spectrum of gravitational waves from a parabolic Keplerian binary by taking the limit of the Peters and Matthews spectrum for eccentric orbits. This demonstrates that the location of the peak of the energy spectrum depends primarily on the orbital periapse rather than the eccentricity. We compare this weak-field result to strong-field calculations and find it is reasonably accurate (~10%) provided that the azimuthal and radial orbital frequencies do not differ by more than ~10%. For equatorial orbits in the Kerr spacetime, this corresponds to periapse radii of rp > 20M. These results can be used to model radiation bursts from compact objects on highly eccentric orbits about massive black holes in the local Universe, which could be detected by LISA.
An important source of gravitational waves for the proposed space-based gravitational wave detector, the Laser Interferometer Space Antenna (LISA) [1,2], are the inspirals of stellar-mass compact objects into massive black holes in the centres of galaxies. During the last few years of inspiral these systems generate continuous gravitational waves in the LISA band, which will allow the detection of as many as several hundred systems out to redshift z ∼ 1 [3]. However, prior to this phase, the inspiraling object spends many years on a highly eccentric orbit, generating bursts of gravitational radiation at each periapse passage. LISA could resolve individual bursts from sources in the nearby Universe. Initial estimates [4] suggested a LISA event rate of ∼ 18 yr -1 , including ∼ 15 yr -1 from the centre of the Milky Way, although this was subsequently revised downwards to ∼ 1 yr -1 with negligible contribution from extragalactic sources [5]. If even a single burst from the Galactic centre is detected during the LISA mission, this will provide an unparalleled probe of the structure of spacetime there.
The spectrum of radiation from these bursts will be well approximated by the spectrum of a parabolic orbit. 1 In this note we derive an analytic approximation to this spectrum by taking the limit of the Peters and Mathews [6,7] (PM) energy spectrum for eccentric Keplerian binaries. We show that the peak of the spectrum depends primarily on the orbital periapse and only weakly on the eccentricity. We also estimate the range of validity of the approximation (in Sec. III) by comparing to numerical Teukolsky data, finding that it is a good approximation for equatorial orbits in Kerr with peri-apse r p 20M . The parabolic spectrum takes a neat analytic form; deriving it from the bound spectrum will allow corrections for high-eccentricity bound orbits to be found in the future. We hope this note will be a useful resource for future work on gravitational radiation from high-eccentricity orbits.
For an orbit of eccentricity e with periapse radius r p , Peters and Mathews [6] give the power radiated into the nth harmonic of the orbital angular frequency as
where the function g(n, e) is defined in terms of Bessel functions of the first kind
The Keplerian orbital frequency is
where ω c is defined as the angular frequency of a circular orbit of radius r p . The energy radiated per orbit into the nth harmonic, that is, at frequency ω n = nω 1 , is
as e → 1 for a parabolic orbit, ω 1 → 0 as the orbital period becomes infinite. The energy radiated per orbit is then the total energy radiated. The spacing of harmonics is ∆ω = ω 1 , giving energy spectrum
Changing to linear frequency 2πf = ω,
= 4π 2 5
where the function ℓ(n, e) is defined in the last line. For a parabolic orbit, we must take the limit of ℓ(n, e) as e → 1.
We simplify ℓ(n, e) using the recurrence formulae (Watson [8] 2.12)
and eliminate n using
where f = ω n /ω c = f n /f c is a dimensionless frequency.
To find the limit we define two new functions
(11) To give a well-defined energy spectrum, both of these must be finite.
The Bessel function has an integral representation
we want the limit of this for ν → ∞, z → ∞, with z ≤ ν.
Using the stationary phase approximation, the dominant contribution to the integral comes from the regime in which the argument of the cosine is approximately zero (Watson [8] 8.2, 8.43), for small ϑ:
this last expression is an Airy integral and has a standard form (Watson [8] 6.4)
where K ν (z) is a modified Bessel function of the second kind. Using this to evaluate the limit gives
For our case,
and the first limiting function is well defined,
To find the derivative we combine ( 9) and ( 16), and expand to lowest order yielding
We may re express the derivative using the recurrence formula (Watson [8] 3.71)
to give
And so finally we obtain the well-defined
Having obtained expressions for A( f ) and B( f ) in terms of standard functions, we can now calculate the energy spectrum for a parabolic orbit. From ( 7)
where we have used the limit This agrees with the e = 1 form of Turner’s result, which was computed by direct integration along unbound orbits [9]. Figure 1 shows how ℓ(n, e) changes with eccentricity including our result for a parabolic encounter (cf. Figure 3 of Peters and Mathews [6]). Although more power is radiated into higher harmonics, the peak of the spectrum does not move much: it is always between f = f c and f = 2f c , with f = 2f c for e = 0 and f ≃ 1.637f c for e = 1.
To check the validity of this limit we can calculate the total energy radiated by integrating (23) over all frequencies, or by summing the energy radiated into each harmonic. These must yield the same result. Summing:
where we have used equations ( 1), ( 3) and ( 4). Peters and Mathews [6]
Integrating the energy spectrum ( 23) gives
The integral can be evaluated numerically as
The two total energies are consistent, E int = E sum .
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
