Fast and accurate computational tools for gravitational waveforms from binary stars with any orbital eccentricity

Pierro, V.; Pinto, I. M.; Spallicci, A.; Laserra, Ettore; Recano, F.

doi:10.1046/j.1365-8711.2001.04442.x

Cited by 45 publications

(78 citation statements)

References 14 publications

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“…One then finds that [41] sin u 1 − e cos u = 2 11) where primes stand for derivatives with respect to the argument, and 12) with Γ the Gamma function (see e.g. [52] …”

Section: The Basics Of the Kepler Problemmentioning

confidence: 99%

“…Let us re-express the cosine and sine of the orbital phase in terms of the mean anomaly, through the wellknown Keplerian relations [41] r a cos φ = cos u − e, (2.6)…”

Section: The Basics Of the Kepler Problemmentioning

confidence: 99%

“…(2.2) for the sine of the eccentric anomaly both numerically and with Bessel functions. The latter solution is simply given by [41] sin u = 2 e In Fig. 4 we plot the absolute value of the fractional relative difference between the numerical solution and the Bessel-expanded solution, keeping 9 terms in the sum.…”

Section: The Basics Of the Kepler Problemmentioning

confidence: 99%

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Post-circular expansion of eccentric binary inspirals: Fourier-domain waveforms in the stationary phase approximation

et al. 2009

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We lay the foundations for the construction of analytic expressions for Fourier-domain gravitational waveforms produced by eccentric, inspiraling compact binaries in a post-circular or smalleccentricity approximation. The time-dependent, "plus" and "cross" polarizations are expanded in Bessel functions, which are then self-consistently re-expanded in a power series about zero initial eccentricity to eighth order. The stationary phase approximation is then employed to obtain explicit analytic expressions for the Fourier transform of the post-circular expanded, time-domain signal. We exemplify this framework by considering Newtonian-accurate waveforms, which in the post-circular scheme give rise to higher harmonics of the orbital phase and amplitude corrections both to the amplitude and the phase of the Fourier domain waveform. Such higher harmonics lead to an effective increase in the inspiral mass reach of a detector as a function of the binary's eccentricity e0 at the time when the binary enters the detector sensitivity band. Using the largest initial eccentricity allowed by our approximations (e0 < 0.4), the mass reach is found to be enhanced up to factors of approximately 5 relative to that of circular binaries for Advanced LIGO, LISA, and the proposed Einstein Telescope at a signal-to-noise ratio of ten. A post-Newtonian generalization of the post-circular scheme is also discussed, which holds the promise to provide "ready-to-use" Fourier-domain waveforms for data analysis of eccentric inspirals.

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“…One then finds that [41] sin u 1 − e cos u = 2 11) where primes stand for derivatives with respect to the argument, and 12) with Γ the Gamma function (see e.g. [52] …”

Section: The Basics Of the Kepler Problemmentioning

confidence: 99%

“…Let us re-express the cosine and sine of the orbital phase in terms of the mean anomaly, through the wellknown Keplerian relations [41] r a cos φ = cos u − e, (2.6)…”

Section: The Basics Of the Kepler Problemmentioning

confidence: 99%

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Post-circular expansion of eccentric binary inspirals: Fourier-domain waveforms in the stationary phase approximation

et al. 2009

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“…The interaction of stars and stellar remnants amongst each other as well as with the large mass at the center of the Milky Way may provide tools to explore the nature of SgrA* using gravitational waves (e.g. Amaro-Seoane et al, 2012;Freitag, 2003;Pierro et al, 2001). …”

Section: Future Progress In Observationsmentioning

confidence: 99%

The Milky Way’s Supermassive Black Hole: How Good a Case Is It?

Eckart

Hüttemann²,

Kiefer³

et al. 2017

Found Phys

122

105

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The compact and, with 4.3±0.3×106 M ⊙ , very massive object located at the center of the Milky Way is currently the very best candidate for a supermassive black hole (SMBH) in our immediate vicinity. The strongest evidence for this is provided by measurements of stellar orbits, variable X-ray emission, and strongly variable polarized near-infrared emission from the location of the radio source Sagittarius A* (SgrA*) in the middle of the central stellar cluster. Simultaneous near-infrared and X-ray observations of SgrA* have revealed insights into the emission mechanisms responsible for the powerful near-infrared and X-ray flares from within a few tens to one hundred Schwarzschild radii of such a putative SMBH at the center of the Milky Way. If SgrA* is indeed a SMBH it will, in projection onto the sky, have the largest event horizon and will certainly be the first and most important target of the Event Horizon Telescope (EHT) Very Long Baseline Interferometry (VLBI) observations currently being prepared. These observations in combination with the infrared interferometry experiment GRAVITY at the Very Large Telescope Interferometer (VLTI) and other experiments across the electromagnetic spectrum might yield proof for the presence of a black hole at the center of the Milky Way. The large body of evidence continues to discriminate the identification of SgrA* as a SMBH from alternative possibilities. It is, however, unclear when the ever mounting evidence for SgrA* being associated with a SMBH will suffice as a convincing proof. Additional compelling evidence may come from future gravitational wave observatories. This manuscript reviews the observational facts, theoretical grounds and conceptual aspects for the case of SgrA* being a black hole. We treat theory and observations in the framework of the philosophical discussions about "(Anti)Realism and Underdetermination", as this line of arguments allows us to describe the situation in observational astrophysics with respect to supermassive black holes. Questions concerning the existence of supermassive black holes and in particular SgrA* are discussed using causation as an indispensable element. We show that the results of our investigation are convincingly mapped out by this combination of concepts.

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“…Equations (7) and (8) can be immediately used to construct these waveforms. However, by setting a required tolerance on the accuracy of sin Φ and F cos for a given eccentricity, we can truncate the infinite summations (Pierro et al 2001;Yunes et al 2009) to accelerate calculations. We investigate the minimum number of terms required for the Fourier series expansion of F cos in Equation (7) to maintain accuracy with the exact numerical solution of Equations (1)-(3), by demanding that the error in the two solutions (determined by the normalized scalar product between the two solution vectors) is less than 0.001% over 2π of meananomaly.…”

Section: Eccentric Binary Orbitsmentioning

confidence: 99%

Detecting Eccentric Supermassive Black Hole Binaries With Pulsar Timing Arrays: Resolvable Source Strategies

Taylor

Huerta

Gair

et al. 2016

ApJ

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The couplings between supermassive black hole binaries (SMBHBs) and their environments within galactic nuclei have been well studied as part of the search for solutions to the final parsec problem. The scattering of stars by the binary or the interaction with a circumbinary disk may efficiently drive the system to sub-parsec separations, allowing the binary to enter a regime where the emission of gravitational waves can drive it to merger within a Hubble time. However, these interactions can also affect the orbital parameters of the binary. In particular, they may drive an increase in binary eccentricity which survives until the system's gravitational-wave (GW) signal enters the pulsar-timing array (PTA) band. Therefore, if we can measure the eccentricity from observed signals, we can potentially deduce some of the properties of the binary environment. To this end, we build on previous techniques to present a general Bayesian pipeline with which we can detect and estimate the parameters of an eccentric SMBHB system with PTAs. Additionally, we generalize the PTA  e -statistic to eccentric systems, and show that both this statistic and the Bayesian pipeline are robust when studying circular or arbitrarily eccentric systems. We explore how eccentricity influences the detection prospects of single GW sources, as well as the detection penalty incurred by employing a circular waveform template to search for eccentric signals, and conclude by identifying important avenues for future study.

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Fast and accurate computational tools for gravitational waveforms from binary stars with any orbital eccentricity

Cited by 45 publications

References 14 publications

Post-circular expansion of eccentric binary inspirals: Fourier-domain waveforms in the stationary phase approximation

Post-circular expansion of eccentric binary inspirals: Fourier-domain waveforms in the stationary phase approximation

The Milky Way’s Supermassive Black Hole: How Good a Case Is It?

Detecting Eccentric Supermassive Black Hole Binaries With Pulsar Timing Arrays: Resolvable Source Strategies

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