Astronomy

Is gravitational wave frequency always equal to double the orbital frequency?

Is gravitational wave frequency always equal to double the orbital frequency?


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If the binary does not evolve into merger stage (i.e.: it is still a steady binary), does the gravitational wave frequency have to be 2*orbital frequency? Could the frequency be, for instance, 2/3*orbital frequency and steady, rather than a chirp? And if not, could the frequency 2/3*orbital frequency be the result of interactions with other waves of this binary? The frequency range is about several milihertz, low frequency.


A binary will always emitted a spectrum of gravitational wave frequencies. As long as the binary is in the adiabatic regime (meaning that changes to the orbit due to the emission of gravitational waves happen on a timescale much longer than the orbital period) this spectrum is determined by the frequencies of the orbit.

When the binary is circular and non-precessing, this spectrum consists purely of integer multiples of the orbital frequency. Usually (although there are extreme counter examples) the mode at double the orbital frequency dominates the spectrum, often being strong then all other modes combined. The other modes tend be suppressed exponentially, but generically all integer multiples of the orbital frequency will be present.

General binaries however are not circular, but have some eccentricity and feature precessing spins. Such a system is characterized the by four fundamental frequencies, the orbital frequency $Omega_phi$, the frequency of the radial oscillation $Omega_r$, and the precession frequencies of the two spins $Omega_ heta$ and $Omega_psi$. The gravitational wave spectrum of such a binary consists of integer combinations $m Omega_phi+ nOmega_r + k_1Omega_ heta+k_2Omega_psi$, and generically all such combinations are present, but a few will dominate.

In the weak field regime all these frequencies are approximately equal, and the spectrum again consists (to first approximation) of only integer multiples of the orbital frequency. However, being in the weak field, is not the only way of being in the adiabatic regime. The evolution rate of a binary is proportional to the mass ratio. Consequently, binaries with small mass-ratios can be in the slowly evolving while being in the strong field regime, where the fundamental frequencies can be very different.

To achieve the milliHertz frequencies request while in the strong field regime the total mass of the binary needs to be of the order a million solar masses. So we could get an (extremely) small mass-ratio slowly evolving binary by considering a solar mass object, orbiting a million solar mass supermassive black hole (e.g. the one in the center of our own galaxy).

Now can we get a situation where there GW modes with 2/3s of the orbital frequency. The answer is yes. One situation is as follows. Consider a binary with eccentricity $e= 0.1$, and semi-latus rectum $p$ roughly 10.8 times $GM/c^2$. In this case, the radial frequency $Omega_r$ is roughly $2Omega_phi/3$. The spectrum of this binary contains a mode with frequency $omega = 2Omega_phi - 2 Omega_r = 2Omega_phi/3$. However such a mode will be fairly weak with a strain amplitude of only about 0.5% of the dominant mode at twice the orbital frequency.


The gravitational waves (GWs) from a perfectly stable, circular binary system are completely dominated by GWs at twice the orbital frequency.

If the binary orbit is eccentric then the GWs have a frequency spectrum; the waves are a combination of discrete frequencies at integer multiples of the orbital frequency (e.g. Wen 2003). At low eccentricity, most of the power is still at twice the orbital frequency, but for eccentricities greater than about 0.3, the peak frequency moves to higher and higher multiples.

Despite mmeents objections, I believe what I've written is (approximately) true for the circumstances posed in the question - i.e. a non-evolving binary, where the GW energy losses are small, the binary components are thus widely separated and can be treated as point-like, and the rate of periastron precession is small compared with the orbital frequency (note that if the eccentricity were really high, $e sim 1$, then the latter may not be true). Closer binaries may have significant higher order multipole emission and modifications to their frequency spectrum caused by the decay of the orbit, precession of their periastron, their spin etc.


Gravitational Wave Sources

Both bar and laser interferometers are high-frequency detectors, but there are a number of interesting gravitational wave sources which emit signals at lower frequencies. The seismic noise provides an insurmountable obstacle in any earth-based experiment and the only way to overcome this barrier is to fly a laser interferometer in space. LISA (Laser Interferometer Space Antenna) is such a system. It has been proposed by European and American scientists and has been adopted by the European Space Agency (ESA) as a cornerstone mission recently NASA joined the effort. The launch date is expected to be around 2008.

LISA will consist of three identical drag-free spacecraft forming an equilateral triangle with one spacecraft at each vertex ( Fig. 7 ). The distance between the two vertices (the arm length) is 5 × 10 6 km. The spacecraft will be placed into the same heliocentric orbit as earth, but about 20° behind earth. The equilateral triangle will be inclined at an angle of 60° with respect to earth's orbital plane. The three spacecraft will track each other optically by using laser beams. Because of the diffraction losses it is not feasible to reflect the beams back and forth as is done with LIGO. Instead, each spacecraft will have its own laser. The lasers will be phase locked to each other, achieving the same kind of phase coherence as LIGO does with mirrors. The configuration will function as three partially independent and partially redundant gravitational wave interferometers.

FIGURE 7 . Schematic design of the space interferometer LISA.

At frequencies f ≥ 10 −3 Hz, LISA's noise is mainly due to photon shot noise. The sensitivity curve steepens at f ∼ 3 × 10 −2 Hz because at larger frequencies the gravitational wave's period is shorter than the round-trip light travel time in each arm. For f ≤ 3 × 10 −2 Hz, the noise is due to buffeting-induced random motions of the spacecraft and cannot be removed by the drag-compensation system. LISA's sensitivity is roughly the same as that of LIGO, but at 10 5 times lower frequency. Since the gravitational wave energy flux scales as Ff 2 h 2 , this corresponds to 10 10 times better energy sensitivity than LIGO.


Effects of passing

The effect of a plus-polarized gravitational wave on a ring of particles.

The effect of a cross-polarized gravitational wave on a ring of particles.

Gravitational waves are constantly passing Earth however even the strongest is miniscule and sources are also generally very far away from us in space. For example, the waves given off by the cataclysmic final merger of GW150914 reached Earth after travelling over a billion lightyears , as a ripple in spacetime that changed the length of a 4-km LIGO arm by a ten thousandth of the width of a proton , proportionally equivalent to changing the distance to the nearest star outside the Solar System by one hair’s width.This tiny effect of even extreme gravitational waves makes them completely undetectable on Earth, by any means other than extremely sophisticated detectors.

The effects of a passing gravitational wave, in an extremely exaggerated form, can be visualized by imagining a perfectly flat region of spacetime with a group of motionless test particles lying in a plane (e.g., the surface of a computer screen). As a gravitational wave passes through the particles along a line perpendicular to the plane of the particles (i.e. following the observer’s line of vision into the screen), the particles will follow the distortion in spacetime, oscillating in a “ cruciform ” manner, as shown in the animations. The area enclosed by the test particles does not change and there is no motion along the direction of propagation.

The oscillations depicted in the animation are exaggerated for the purpose of discussion—in reality a gravitational wave has a very small amplitude (as formulated in linearized gravity ). However, they help illustrate the kind of oscillations associated with gravitational waves as produced, for example, by a pair of masses in a circular orbit . In this case the amplitude of the gravitational wave is constant, but its plane of polarization changes or rotates at twice the orbital rate and so the time-varying gravitational wave size (or ‘periodic spacetime strain’) exhibits a variation as shown in the animation. [18] If the orbit is elliptical then the gravitational wave’s amplitude also varies with time according to Einstein’s quadrupole formula .

As with other waves , there are a number of characteristics used to describe a gravitational wave:

  • Amplitude : Usually denoted h , this is the size of the wave — the fraction of stretching or squeezing in the animation. The amplitude shown here is roughly h = 0.5 (or 50%). Gravitational waves passing through the Earth are many sextillion times weaker than this — h ≈ 10 −20 .
  • Frequency : Usually denoted f , this is the frequency with which the wave oscillates (1 divided by the amount of time between two successive maximum stretches or squeezes)
  • Wavelength : Usually denoted λ , this is the distance along the wave between points of maximum stretch or squeeze.
  • Speed : This is the speed at which a point on the wave (for example, a point of maximum stretch or squeeze) travels. For gravitational waves with small amplitudes, this is equal to the speed of light ( c ).

The speed, wavelength, and frequency of a gravitational wave are related by the equation c = λ f , just like the equation for a light wave . For example, the animations shown here oscillate roughly once every two seconds. This would correspond to a frequency of 0.5 Hz, and a wavelength of about 600 000 km, or 47 times the diameter of the Earth.

In the above example, it is assumed that the wave is linearly polarized with a “plus” polarization, written h + . Polarization of a gravitational wave is just like polarization of a light wave except that the polarizations of a gravitational wave are at 45 degrees, as opposed to 90 degrees. In particular, in a “cross”-polarized gravitational wave, h × , the effect on the test particles would be basically the same, but rotated by 45 degrees, as shown in the second animation. Just as with light polarization, the polarizations of gravitational waves may also be expressed in terms of circularly polarized waves. Gravitational waves are polarized because of the nature of their sources.


The quietest place on Earth

The design of the detector used in LIGO (in Livingston, Louisiana and Hanford, Washington)—which is also used in the existing Virgo (near Pisa, Italy), GEO 600 (in Hanover, Germany) and KAGRA (in the Kamioka mines, Gifu Prefecture, Japan) detectors and will be used in the planned LIGO-India detector (in Hingoli, Maharashtra, India)—has an ‘L’ shape with equal-length arms connected to a corner station (see Fig. 1). When a typical gravitational wave passes by, at some phases of the wave one leg of the ‘L’ will be shortened and the other lengthened, and at other phases the reverse will happen. Thus, laser photons that bounce between the corner station and one end station return to the corner station later than laser photons that reflect off the other end station. As a result, the interference fringes produced when the light is combined at the corner station shift back and forth as the wave changes in phase. This shift can be compared with the expectations from different types of signals (for example, from binaries) to assess the probability that signal or noise is being observed.

a, Laser light is sent into the instrument to measure changes in the length of the two arms. b, A ‘beam splitter’ splits the light and sends out two identical beams along the arms. c, The light waves bounce off the mirror and return. d, A gravitational wave affects the interferometer’s arms differently: one extends and the other contracts as they pass from the peaks and troughs of the gravitational waves. e, Normally, the light returns unchanged to the beam splitter from both arms and the light waves cancel each other out. Image credit: ©Johan Jarnestad/The Royal Swedish Academy of Sciences.

This experimental setup raises an important question related to the smallness of the effect. To get a sense for the length changes that are measured, we note that the first directly detected gravitational waves had a maximum (dimensionless) fractional amplitude of 10 −21 , which means that the 4-km LIGO arms changed in length by 10 −21 × 4 × 10 5 cm = 4 × 10 −16 cm. Put differently, the effective force exerted by the gravitational waves is roughly 4 pN at 100 Hz, which is comparable to the weight of a eukaryotic cell at the frequency of a sonic toothbrush. Given that the proton radius is 10 −13 cm, we are trying to measure distance changes of the order of 1/200 of the proton radius, with light that has a wavelength of the order of 10 −4 cm. This seems impossible, even before we consider the many noise sources (for example, any shaking of the ground). The workaround is to have an enormous number of coherent photons that bounce around within the arms (in a Fabry–Perot configuration) many times before recombining. For N1 photons of wavelength λ, the location of the intensity peak can be measured with a precision of about (lambda /sqrt<_<1>>) . Similarly, for N2 bounces within the arms, the effective length of the interferometer, and thus the change in LIGO arm length, increases by a factor of N2. This means that for large enough N1 and N2, the necessary precision can be attained. A similar method is in fact used in astrometry observatories such as Gaia 26 , where the absolute angular localization of bright stars, about 10 −5 arcsec, is far better than the about 0.1-arcsec angular resolution of its telescope.

Other noise sources—although substantial—can be managed for frequencies that are not too low. As an example, one might think that seismic noise would be a serious problem, but the detectors can be strongly shielded from shaking by the use of pendulum suspensions for a pendulum of resonance frequency f0, the amplitude of oscillations with frequency f > f0 is reduced by a factor of about (f/f0) 2 . Thus the multi-stage pendulum suspension used by LIGO and Virgo can greatly reduce seismic noise. What cannot be shielded is so-called ‘Newtonian’ or ‘gravity gradient’ noise: pressure waves inside Earth travel because of a local (albeit small) change in the density of the rock a temporarily increased density means that the region in question has greater mass and therefore greater gravity than it did before the pressure wave. This gravity enhancement pulls on the detector mirrors and, because we cannot shield anything from gravity, the noise must be reduced by one of three methods: delicate subtraction using feed-forward cancellation 27 , building the detector underground (because seismic waves have much bigger amplitudes on Earth’s surface) or placing the detector in space.


Sources of Gravitational Waves

What are the main sources of GWs in the sky? Below you can find a list of astrophysical sources of GWs and even listen to their “gravitational sound”!

Indeed the GW frequencies are the same frequencies as the sound waves that are audible to humans. Therefore, any signal measured by Advanced Virgo can be sent to a loudspeaker (after some filtering), allowing us to hear the symphony of the Universe. However, let us specify that GWs are not sound waves: the analogy with sound is made just to allow us to have a better comprehension of changes in amplitude and frequency of a GW signal.

Coalescence of compact binary systems

A compact binary system is composed of two compact stellar objects, such as a Binary system of two Neutron Stars (BNS) or Binary system of two Black Holes (BBH) or a mixture of a Neutron Star (NS) and a Black Hole (BH) orbiting around each other this is a typical source of GWs. As the system evolves with time, the two compact objects get closer and closer to one another until they eventually merge, because the system loses energy through the emission of GWs. This phenomenon is known as coalescence. As the two bodies approach one another, the GWs that are generated increase in frequency and amplitude. This type of signal is called a “chirp”. You can hear an example of this kind of signal here.

During the final stage orbital decay and before the merging of the two bodies the GWs are strong enough to be observed in the frequency sensitive band of Advanced Virgo, the two bodies orbit around one another many times per second.

Artist’s impression of the merging phase of a binary neutron star system emitting gravitational waves (Credits:NASA/CXC/GSFC/T.Strohmayer). The two stars are orbiting each other and progressing (from left to right) to merger, while emitting gravitational waves.

The first announced GW to be detected jointly by Advanced LIGO and Virgo (called GW170814) is a signal of this kind, coming from a BBH merger. The GW signal revealed two BHs of 25 and 30 solar masses, coalescing at a distance of about 2.2 billion light-years from us. Just before their merger into a single BH of about 53 solar masses, the two objects were spinning around one another about 200 times per second. Advanced Virgo was crucial in pinpointing the source sky-location within a narrow sky area of 60 square degrees. A significant improvement with respect to the 1160 square-degree area obtained by analysing only the LIGO data.

On the 17th of August, 2017, another GW arrived at the Advanced Virgo and LIGO detectors: GW170817. This was a very long GW, lasting for about a minute in our detectors. Shortly after its arrival, the Fermi Gamma-ray Burst Monitor independently detected a gamma-ray burst coming from the same direction of the Sky, with a time delay of

1.7 seconds with respect to the merger time. An extensive observing campaign was launched across the electromagnetic spectrum, leading to the discovery of the corresponding optical and infrared emission. Combining these observations, we were able to understand that GW170817 was emitted during the merger of two NS of about 1.4 solar masses, merging into a single body of

The arrival of GW170817 opened a new era in the new-born GW astronomy: multi-messenger astronomy with Gravitational Waves, thanks to the observation of both GWs and light emitted by the same astrophysical source.

Rotating neutron stars

A neutron star is a very compact, rapidly rotating and highly magnetised star, which represents the remnant core of a massive star which has undergone a supernova explosion. They have roughly the same mass as our Sun, but concentrated into a sphere with a radius of 10 km, and they can spin up to 1000 times per second! If the star is not perfectly spherical, with only a tiny “mountain” on its surface, it will generate GWs. This type of GW is different from those emerging from compact binary coalescence, since they are emitted continuously by spinning neutron stars. We refer to this type of GW as continuous Gravitational Waves .

Advanced Virgo will search for gravitational waves coming from known pulsars (neutron stars from which we detect periodic light flashes). A pulsar is observed at the center of the image: it is a small, dense object, only 20km in diameter. It is responsible for this beautiful X-ray nebula that spans 150 light years. Image credit: NASA/CXC/SAO/P.Slane, et al.

Pulsars are special cases of magnetically asymmetric neutron stars, from which we receive periodic light flashes each time the star completes a rotation. Until now, Advanced Virgo has not detected any signals from known pulsars.

This allowed us to conclude that Neutron stars are almost perfect spherical stars. For instance, we know that the pulsar J1400-6325 should not have “mountains” higher than 4 centimetres, otherwise we would observe a GW from this source.

Observations of neutron stars and pulsars will provide important information about their internal structure. Advanced Virgo will search for GWs emitted by neutron stars in the neighbourhood of the Sun, and in the rest of our own galaxy, the Milky Way.

GW Bursts and Supernovae

“GW bursts” is a generic name given to short GW signals that last from a few milliseconds to a few seconds. This type of GWs has not yet been observed and are expected to be associated with short-lived and violent astrophysical events such as supernova explosions. Supernova explosions are one of the possible outcomes of a dying massive star. During this phase, the star collapses on itself due to the fact that the star’s gravitational attraction can no longer be sustained by the thermonuclear reactions within it.

Eventually, the collapsing material becomes so dense that the star’s core becomes a hot neutron star. The newly-born neutron star will cool by intense neutrino emission and the rest of the collapsing star will be disrupted in a supernova explosion. Advanced Virgo will be able to observe the GW emission from the collapse and explosion of such massive stars at the end of their life.

In a galaxy like the Milky Way there are only a few such supernovae per century. Their observation will allow us to better understand what is hidden behind the matter ejected during the explosion and how black holes are created.

Stochastic background

Most cosmological models predict that we are bathed in a random background of GWs generated by sources that individually can not be detected.

Eventually the combination of all of the GWs emitted by these sources will form a stochastic GW background that we will be able to observe.

For instance we expect that a background of cosmological GWs were emitted in the first moments after the Big Bang. Detection of such a background would give us insight into the evolution of the Universe, providing a unique probe of the very early Universe.

In addition, an astrophysical background of GWs must result from the superposition of all the faint and distant sources that have emitted GWs since the beginning of the Universe. Detection of this background would help elucidate the star formation history and the evolution of astrophysical sources.


Listening to the universe with gravitational-wave astronomy

The LIGO (Laser Interferometer Gravitational-Wave Observatory) detectors have just completed their first science run, following many years of planning, research, and development. LIGO is a member of what will be a worldwide network of gravitational-wave observatories, with other members in Europe, Japan, and—hopefully—Australia. Plans are rapidly maturing for a low frequency, space-based gravitational-wave observatory: LISA, the Laser Interferometer Space Antenna, to be launched around 2011. The goal of these instruments is to inaugurate the field of gravitational-wave astronomy: using gravitational waves as a means of listening to highly relativistic dynamical processes in astrophysics. This review discusses the promise of this field, outlining why gravitational waves are worth pursuing, and what they are uniquely suited to teach us about astrophysical phenomena. We review the current state of the field, both theoretical and experimental, and then highlight some aspects of gravitational-wave science that are particularly exciting (at least to this author).


Observing gravitational waves from white dwarf binaries

The typical orbital frequency of a binary system depends both on the component masses and on their compactness, and it directly determines the frequency of the gravitational waves emitted by that system. For White Dwarf binaries, the frequencies are such that the proper instrument to detect them is the Laser Interferometer Space Antenna, LISA, a space-borne gravitational wave detector that might be deployed around 2034.

Compared with all other gravitational wave sources, such as binary black holes, White Dwarfs have a great advantage. We know they are there and that, at the planned sensitivity, several of them must be detectable with LISA. Once LISA is launched, these “verification binaries” will provide an important “sanity check” for the detector – if all goes as planned, their signals should show up in the data.

Studies of evolution of stars predict that the number of White Dwarf binaries is enormous. In our galaxy alone, it is estimated that there are roughly 250 million detached binaries and 10 million interacting binaries. The sensitivity of LISA will allow the detection of thousands of these binaries as individual sources. The gravitational waves from the other millions of binaries, especially towards the lower end of LISA’s frequency range, will combine to form an unresolved background – similar to a lively party at which everyone is engaged in conversation: You are likely recognize a number of individual speakers close to you, while the other conversations will emerge to give a steady background noise.

How can we distinguish this from other kinds of noise – for instance from the noise produced by the detector itself? The sensitivity of a detector like LISA depends on the direction from which a gravitational wave reaches it. But the stars are not distributed evenly in the sky. You can see this for yourself if you look up into the night sky. If the sky is dark enough, you will be able to see a dim band where the density of stars is much higher than elsewhere – the Milky Way. The following image shows a schematic representation of the night sky above Warsaw on May 1 – shown are the brightest stars, the stellar constellations and a whitish band representing the Milky Way:

[Image uses data from XEphem]

In the same way, the White Dwarfs producing the background of galactic binaries are not distributed uniformly in the sky – they, too, are concentrated in the disk of our galaxy. As LISA orbits the sun, trailing behind the earth, its orientation will change continuously, and after one year and one full orbit, the cycle will repeat. But for LISA, like for all gravitational wave detectors, orientation is important. For any given orientation, LISA will be much more sensitive to gravitational waves coming from some directions than to waves coming from certain other directions. With the continuous changes in orientation, there will be times when LISA is most sensitive with regard to gravitational waves coming from those regions of the night sky that contain more White Dwarf binaries – namely the band of the Milky Way – and times when it will be less sensitive. Correspondingly, there will be times when the waves from the background binaries will lead to a stronger noise and times when their noise will be weaker. This characteristic modulation with a period of one year will allow the data analysts to unambiguously distinguish between the white dwarf background noise and other types of noise – notably from noise produced in the detector itself.

The following image shows how the background should look to LISA. The horizontal direction represents time (in years), the vertical direction represents the gravitational wave amplitude (multiplied by a factor of a hundred thousand billion billions, or 10 22 in exponential notation). The up-down-up-down of the waves is much too fast to show up distinctly in this plot – at this resolution, a sine wave would just look like a solid horizontal strip, as you wouldn’t be able to distinguish one crest (or trough) from the next. What you can see, however, is how the overall strength of the waves detected varies over time. The image covers a time period of three years each year exhibits two humps in which the total background amplitude is at a maximum, and two slender waists where it is at a minimum:


Acknowledgements

This summary is derived from a White Paper submitted 4 August 2019 to ESA’s Voyage 2050 planning cycle on behalf of the LISA Consortium 2050 Task Force [135]. Further space-based GW observatories considered by the LISA Consortium 2050 Task Force include a microhertz observatory μAres [136] a more sensitive millihertz observatory, the Advanced Millihertz Gravitational-wave Observatory (AMIGO) [137], and a high angular-resolution observatory consisting of multiple DOs [138].

The authors thanks Pete Bender for insightful comments, and Adam Burrows and David Vartanyan for further suggestions. MAS acknowledges financial support from the Alexander von Humboldt Foundation and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 138713538 – SFB 881 (“The Milky Way System”). CPLB is supported by the CIERA Board of Visitors Research Professorship. PAS acknowledges support from the Ramón y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain, as well as the COST Action GWverse CA16104. This work was supported by the National Key R&D Program of China (2016YFA0400702) and the National Science Foundation of China (11721303). TB is supported by The Royal Society (grant URF∖R1∖180009). EB is supported by National Science Foundation (NSF) Grants No. PHY-1912550 and AST-1841358, NASA ATP Grants No. 17-ATP17-0225 and 19-ATP19-0051, NSF-XSEDE Grant No. PHY-090003, and by the Amaldi Research Center, funded by the MIUR program “Dipartimento di Eccellenza” (CUP: B81I18001170001). This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 690904. DD acknowledges financial support via the Emmy Noether Research Group funded by the German Research Foundation (DFG) under grant no. DO 1771/1-1 and the Eliteprogramme for Postdocs funded by the Baden-Wurttemberg Stiftung. JME is supported by NASA through the NASA Hubble Fellowship grant HST-HF2-51435.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. MLK acknowledges support from the NSF under grant DGE-0948017 and the Chateaubriand Fellowship from the Office for Science & Technology of the Embassy of France in the United States. GN is partly supported by the ROMFORSK grant Project No. 302640 ‘‘Gravitational Wave Signals From Early Universe Phase Transitions'' . IP acknowledges funding by Society in Science, The Branco Weiss Fellowship, administered by the ETH Zurich. AS is supported by the European Union’s H2020 ERC Consolidator Grant “Binary massive black hole astrophysics” (grant agreement no. 818691 – B Massive). LS was supported by the National Natural Science Foundation of China (11975027, 11991053, 11721303), the Young Elite Scientists Sponsorship Program by the China Association for Science and Technology (2018QNRC001), and the Max Planck Partner Group Program funded by the Max Planck Society. NW is supported by a Royal Society–Science Foundation Ireland University Research Fellowship (grant UF160093).


Is gravitational wave frequency always equal to double the orbital frequency? - Astronomy

We present a study of the gravitational wave form from pulsars. Typically the observation times will be of the order of a few months. Due to the rotation and orbital motion of the Earth, a monochromatic signal becomes frequency and amplitude modulated. The effect of both these modulations is to smear out the monochromatic signal into a small bandwidth about the signal frequency of the wave. However, the effect on the Fourier transform of the frequency modulation is much more severe compared to the amplitude modulation in that the height of the peak is reduced drastically. The Fourier transform of the pulsar signal, taking into account the rotation of the Earth for one day observation period is studied. We have obtained an analytical closed form of the Fourier transform considering the rotational motion of the Earth only. With the inclusion of orbital corrections one obtains a double series of Bessel functions.


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3 Gravitational Wave Astronomy A major challenge and opportunity in relativistic gravitation is the direct detection of gravitational radiation from astrophysical sources. The expectation is that technology now under develop ment wiD within the next 10 years detect gravitational radiation or, if nature is insufficiently kind, that we will at least be able to set interesting astrophysical limits on the gravitational waves inci- dent on the Earth. These terrestrial observations will be made at gravitational wave frequencies above 10 Hz. The exploration of the potentially rich low-frequency spectrum of gravitational waves can only be carried out from space. The development of low-frequency gravitational wave detectors and ultimately a space gravitational wave observatory is a major new direction proposed in this study for the space program. The observation of gravitational radiation from astrophysical sources has several features. First, the direct detection of the waves will serve to test relativistic gravitation by measuring the propaga- tion speed and polarization states of the waves. The sources of the waves will most likely be regions in which the gravitational field strength is large. Thus, the detection of signals from these regions will serve to test gravitation in the strong-field, high-velocity limit. Second, gravitational radiation is very weakly coupled to matter and will not scatter even in the strongest sources. Observations 21

22 of phenomena deep In the interior of regions normally obscured in the electromagnetic astronomers will thus become accessible. The processes involved in stellar collapse and the pruneval cosmic kernel are two weD-known examples. Estimates of the gravitational wave flux incident on the Earth from astrophysical sources have been divided into three categories: sources of gravitational wave impulses or bursts, periodic sources that may produce continuous gravitational wave trains, and, fi- nally, sources of a gravitational wave background noise. The cat- egories are related to different techniques for detection. The es- tunates themselves are of varying quality. In some cases, such as the radiation from ordinary binary stellar systems, the estimate is reliable. Should no radiation be observed, it would indicate a failure of the theory. For other sources, such as supernova ex- plosions, the occurrence of the phenomena is wed established but our ignorance of the physical processes involves> leave the ampli- tude of the waves uncertain. Yet another class consists of posited sources whose number is unknown, but for which the amplitude of the waves is calculable black holes are the best example of this class. Prediction in astrophysics is always a hazardous exercise, especially when one opens a new field or when a profound change in sensitivity makes new observations possible. The theoretical predictions of what might be discovered at the birth of x-ray as- tronomy certainly clid not anticipate the enormous diversity of the phenomena that subsequently were uncovered. Supernova explosions are known to occur In our galaxy at a rate of between 1 and 10 per century. The collapse of the core of a supernova to a neutron star or black hole could be accom- panied by the release of gravitational radiation if the collapse Is not spherically symmetric. The time scales of the collapse dur- ing which gravitational radiation would be emitted lie between 1 and 10 ms, but the fraction of the explosion energy going into gravitational radiation is uncertain. A supernova at the center of our galaxy that releases

percent of its energy into gravitational radiation would produce a wave amplitude at the Earth having a strain of 10

. Present-day gravitational antennas would detect such a pube. In part, the goal of ground-based efforts to detect gravitational radiation has been set by the search for supernova events to achieve a strain sensitivity of 10-2i, which would observe these events to a distance of the Virgo cluster of galaxies at an event rate of 1 to 10 per year.

23 Pulsars are periodic sources whose number of occurrences is known but whose amplitude of gravitational radiation is uncertain. The amplitude depends on the mass eccentricity of the spinning star. Should the eccentricity be entirely due to distortion of the star by the magnetic fields trapped In the star during the col- lapse from an ordinary star, the wave amplitudes from pulsars in our own galaxy would give strains of 10-32 to 10-33, much too small to measure. However, the pulsar could have an intrinsic mass eccentricity, which, if it was as large as 10-5, would produce a measurable strain of 10-26. Earth-based detectors with wave frequencies above 10 Hz are now being planned with such sensitiv- ities. Many of the pulsars have frequencies around

Hz and thus would be candidates for space antennas. Ordinary binary stellar systems abound in the galaxy approx- imately

percent of all stars are members of binary systems. The closer and fast ordinary binaries are sources of gravitational radi- ation at strain levels of 10-2

with periods ranging between 1 to 10 h. These are clear candidates for detection by space antennas. In fact, there may be so many of them that they could constitute an unresolved stochastic background of gravitational waves. A subclass of binary stars includes the double neutron star systems such as PSR 1913+16. These are particularly interesting for both space- and ground-basec! gravitational antennas. PSR 1913+16 now radiates at submultiples of 8 h with strain amplitudes around 10-23, which would be detectable by a space antenna. After about one million years, as this system loses energy through gravitational radiation, it will produce a gravitational wave chirp that will be detectable by ground-based antennas. The system will then spend about 1 year near a period of 10 s and ultimately come to an abrupt end in 1 ms as the two stars collide. The gravitational wave strain multipliecl by the square root of the number of cycles lies in the vicinity of 1

. Three binary neutron star systems from a total population of several hundred pulsars are known to exist in our galaxy. Extrapolating to the rest of the universe, we could expect to detect an event of this type every few hours in an antenna with a strain sensitivity of 10-22 to 10-23. Binary systems composed of double white dwarf stars are fairly certain to exist. These systems radiate at periods of 1000 to 100 s with strain amplitudes in the region 10-2° to 10-22. Not observable by ground-based antennas, they fit well into the best performance region of a projected space antenna.

24 There are a host of more speculative sources. The gravita- tional radiation from the collision of a black hole vnth another black hole or other compact object as well as the radiation emutted in the formation of a black hole is well studied. The radiation originates both from the acceleration of the masses and from the excitation of ringing in the normal modes of the metric solution around the black hole. The gravitational wave bursts from such events have large strain amplitudes with frequency components that vary as the inverse of the black hole masses. The formation of a l

solar-mass black hole at a distance of 100 megaparsecs (Mpc) could produce a strain pulse of 10-2t lasting

ms. Should black holes form binary systems, the orbital decay of a l

solar- mass black hole binary system anywhere in the universe could be observed with an antenna having a strain sensitivity of 10-22. The space antennas are well suited to measure the radiation from massive black holes. The formation of a 107-solar-mass black hole anywhere in the universe would produce a strain at periods of several hours of 10-

6 or larger. One of the most interesting speculative sources of gravitational radiation under current theoretical consideration Is the radiation suffusing the universe that may have originated In the universe's earliest epoch. Should present thinking be correct, it is possible that quantum gravitational wave fluctuations during the Planck epoch were amplified in the subsequent universal expansion. The radiation would appear as a gravitational wave background noise. The spectrum of the radiation is not well understood however, it is believed to contain less than 10-4 of the energy density required to spatially close the universe. The search for cosmic background of gravitational radiation is a prime motivation for both space- and ground-based gravitational wave antennas. The sensitivity of gravitational wave observations on the ground is advancing rapidly. Acoustic bar detectors at cryogenic temperature using low noise position transducers are now able to search for gravitational wave bursts In the kilohertz band with a strain sensitivity of 10

. Several detectors are now operating in coincidence, and results of the searches at this level of sensitivity should be available in 1986. The acoustic detectors will continue to improve but must be able to circumvent the Enliven quantum limit, which will set in at strain sensitivities between 10-2° and

25 10-2i depending on the frequency of observation. Acoustic detec- tors will be constructed at lower frequency and as the transducer technology unproves could have bandwidths of around 10 percent. The other ground-based technique for detecting gravitational radiation utilizes laser interferometer systems that measure the separation of a configuration of free masses. This technique is also a precursor for the most promising space antennas. At present the largest of these systems are 30 to 40 m long. Using light powers of several hundred milliwatts, they can attain strain sen- sitivities of 10-

7 at 1 kHz. The promise for greatly enhanced sensitivity lies in constructing these systems with 100 times larger length and increased position sensitivity by increasing the light power modulated by the interferometer. The systems are

nher- ently broadband, and as the techniques for reducing the stochastic forces on the masses unprove, they are expected to perform at a pube strain sensitivity of better than 10-22 between 10 Hz and 1 kHz. The sensitivity for periodic sources is expected to be less than 10-26 for integration times of a month. The present plan is to construct two

km-Iong antennas in the United States, and there are plans in Great Britain, Germany, and trance to construct an- tennas of comparable length. These antennas will be operated as a network to determine the position of gravitational wave sources in the sky. These systems will be limited by ground noise and gravity gradient noise to operate above 10 Hz. The exploration of the gravity wave flux at longer periods is clearly the domain of space research, where longer baselines are possible and smaller low-frequency stochastic forces will be encountered.


Is gravitational wave frequency always equal to double the orbital frequency? - Astronomy

The epoch of gravitational wave astronomy has now begun with the first detection [ 1 , 2 ] of the merger of binary black holes by Advanced LIGO [ 3 ]. Now that the first ground based gravitational wave detection has been achieved, observations of binary neutron star mergers should soon be forthcoming. This is particularly true as other second generation observatories such as Advanced VIRGO [ 4 ] and KAGRA [ 5 ] will soon be online. In addition to binary black holes, neutron star binaries are thought to be among the best candidate sources gravitational radiation [ 6 , 7 ]. The number of such systems detectable by Advanced LIGO is estimated [ 7 – 14 ] to be of order several events per year based upon observed close binary-pulsar systems [ 15 , 16 ]. There is a difference between neutron star mergers and black hole mergers however, in that neutron star mergers involve the complex evolution of the matter hydrodynamic equations in addition to the strong gravitational field equations. Hence, one must carefully consider both the hydrodynamic and field evolution of these systems.

To date there have been numerous attempts to calculate theoretical templates for gravitational waves from compact binaries based upon numerical and/or analytic approaches (see, e.g., [ 17 – 26 ]). However, most approaches utilize a combination of post-Newtonian (PN) techniques supplemented with quasicircular orbit calculations and then applying full GR for only the last few orbits before disruption. In this paper we analyze the hydrodynamic evolution in the spatially conformally flat metric approximation (CFA) as a means to compute stable initial conditions beyond the range of validity of the PN regime, that is, near the last stable orbits. We establish the numerical stability of this approach based upon many orbit simulations of quasicircular orbits. We show that one must follow the stars for several orbits before a stable quasicircular orbit can be achieved. We also illustrate the equation of state (EoS) dependence of the initial conditions and associated gravitational wave emission.

When binary neutron stars are well separated, the post-Newtonian (PN) approximation is sufficiently accurate [ 27 ]. In the PN scheme, the stars are often treated as point masses, either with or without spin. At third order, for example, it has been estimated [ 28 – 30 ] that the error due to assuming the stars are point masses is less than one orbital rotation [ 28 ] over the 縖,000 cycles that pass through the LIGO detector frequency band [ 7 ]. Nevertheless, it has been noted in many works [ 25 , 31 – 42 ] that relativistic hydrodynamic effects might be evident even at the separation (縐�   km) relevant to the LIGO window.

Indeed, the templates generated by PN approximations, unless carried out to fifth and sixth order [ 28 , 29 ], may not be accurate unless the finite size and proper fluid motion of the stars are taken into account. In essence, the signal emitted during the last phases of inspiral depends upon the finite size and the equation of state (EoS) through the tidal deformation of the neutron stars and the cut-off frequency when tidal disruption occurs.

Numeric and analytic simulations [ 43 – 51 ] of binary neutron stars have explored the approach to the innermost stable circular orbit (ISCO). While these simulations represent some of the most accurate to date, simulations generally follow the evolution for a handful of orbits and are based upon initial conditions of quasicircular orbits obtained in the conformally flat approximation. Accurate templates of gravitational radiation require the ability to stably and reliably calculate the orbit initial conditions. The CFA provides a means to obtain accurate initial conditions near the ISCO.

The spatially conformally flat approximation to GR was first developed in detail in [ 32 ]. That original formulation, however, contained a mathematical error first pointed out by Flanagan [ 52 ] and subsequently corrected in [ 34 ]. This error in the solution to the shift vector led to a spurious NS crushing prior to merger. The formalism discussed below is for the corrected equations. Here, we discuss the hydrodynamic solutions as developed in [ 31 – 34 , 53 , 54 ]. This CFA formalism includes much of the nonlinearity inherent in GR and leads set of coupled, nonlinear, elliptic field equations that can be evolved stably. We also note that an alternative spectral method solution to the CFA configurations was developed by [ 55 , 56 ], and approaches beyond the CFA have also been proposed [ 48 ]. However, our purpose here is to clarify the viability of the hydrodynamic solution without the imposition of a Killing vector or special symmetry. This approach is the most adaptable, for example, to general initial conditions such as that of arbitrarily elliptical orbits and/or arbitrarily spinning neutron stars.

Here, we summarize the original CFA approach and associated general relativistic hydrodynamics formalism developed in [ 32 , 34 , 53 , 54 ] and illustrate that it can produce stable initial conditions anywhere between the post-Newtonian to ISCO regimes. We quantify how long this method takes to converge to quasiequilibrium and demonstrate the stability by subsequently integrating up to 񾄀 orbits for a binary neutron star system. We also analyze the EoS dependence of these quasicircular initial orbits and show how these orbits can be used to make preliminary estimates [ 57 ] of the gravitational wave signal for the initial conditions.

This paper is organized as follows. In Section 2 the basic method is summarized and in Section 3 a number of code tests are performed in the quasiequilibrium circular orbit limit to demonstrate the stability of the technique. The EoS dependence of the initial conditions and associated gravitational wave frequency are analyzed in Section 4 . Conclusions are presented in Section 5 .

2. Method 2.1. Field Equations

The solution of the field equations and hydrodynamic equations of motion were first solved in three spatial dimensions and explained in detail in the 1990s in [ 31 , 32 ] and subsequently further reviewed in [ 53 , 58 ]. Here, we present a brief summary to introduce the variables relevant to the present discussion.

One starts with the slicing of space-time into the usual one-parameter family of hypersurfaces separated by differential displacements in a time-like coordinate as defined in the ( 3 + 1 ) ADM formalism [ 59 , 60 ].

In Cartesian x , y , z isotropic coordinates, proper distance is expressed as (1) d s 2 = - α 2 - β i β i d t 2 + 2 β i d x i d t + ϕ 4 δ i j d x i d x j , where the lapse function α describes the differential lapse of proper time between two hypersurfaces. The quantity β i is the shift vector denoting the shift in space-like coordinates between hypersurfaces. The curvature of the metric of the 3-geometry is described by a position-dependent conformal factor ϕ 4 times a flat-space Kronecker delta ( γ i j = ϕ 4 δ i j ). This conformally flat condition (together with the maximal slicing gauge, tr ⁡ K i j = 0 ) requires [ 60 ] (2) 2 α K i j = D i β j + D j β i - 2 3 δ i j D k β k , where K i j is the extrinsic curvature tensor and D i are 3-space covariant derivatives. This conformally flat condition on the metric provides a numerically valid initial solution to the Einstein equations. The vanishing of the Weyl tensor for a stationary system in three spatial dimensions guarantees that a conformally flat solution to the Einstein equations exists.

One consequence of this conformally flat approximation to the three-metric is that the emission of gravitational radiation is not explicitly evolved. Nevertheless, one can extract the gravitational radiation signal and the back reaction via a multipole expansion [ 32 , 61 ]. An application to the determination of the gravitational wave emission from the quasicircular orbits computed here is given in [ 57 ]. The advantage of this approximation is that conformal flatness stabilizes and simplifies the solution to the field equations.

As a third gauge condition, one can choose separate coordinate transformations for the shift vector and the hydrodynamic grid velocity to separately minimize the field and matter motion with respect to the coordinates. This set of gauge conditions is key to the present application. It allows one to stably evolve up to hundreds and even thousands of binary orbits without the numerical error associated with the frequent advocating of fluid through the grid.

Given a distribution of mass and momentum on some manifold, then one first solves the constraint equations of general relativity at each time for a fixed distribution of matter. One then evolves the hydrodynamic equations to the next time step. Thus, at each time slice a solution to the relativistic field equations and information on the hydrodynamic evolution is obtained.

The solutions for the field variables ϕ , α , and β i reduce to simple Poisson-like equations in flat space. The Hamiltonian constraint [ 60 ] is used to solve for the conformal factor ϕ [ 32 , 62 ] (3) ∇ 2 ϕ = - 2 π ϕ 5 W 2 ρ 1 + ϵ + P - P + 1 16 π K i j K i j . In the Newtonian limit, the RHS is dominated [ 32 ] by the proper matter density ρ , but in strong fields and compact neutron stars there are also contributions from the internal energy density ϵ , pressure P , and extrinsic curvature. The source is also significantly enhanced by the generalized curved-space Lorentz factor W (4) W = α U t = 1 + ∑ U i 2 ϕ 4 1 / 2 , where U t is the time component of the relativistic four velocity and U i are the covariant spatial components. This factor, W , becomes important near the last stable orbit as the specific kinetic energy of the stars rapidly increases.

In a similar manner [ 32 , 62 ], the Hamiltonian constraint, together with the maximal slicing condition, provides an equation for the lapse function, (5) ∇ 2 α ϕ = 2 π α ϕ 5 3 W 2 ρ 1 + ϵ + P - 2 ρ 1 + ϵ + 3 P + 7 16 π K i j K i j .

Finally, the momentum constraints yields [ 60 ] an elliptic equation for the shift vector [ 34 , 52 ], (6) ∇ 2 β i = ∂ ∂ x i 1 3 ∇ · β + 4 π ρ 3 i , where (7) ρ 3 i = 4 α ϕ 4 S i + 1 4 π ∂ ln ⁡ α / ϕ 6 ∂ x j ∂ β i ∂ x j + ∂ β j ∂ x i - 2 3 δ i j ∂ β k ∂ x k . Here S i are the spatial components of the momentum density one-form as defined below.

We note that, in early applications of this approach, the source for the shift vector contained a spurious term due to an incorrect transformation between contravariant and covariant forms of the momentum density as was pointed out in [ 34 , 52 ]. As illustrated in those papers, this was the main reason why early hydrodynamic calculations induced a controversial additional compression on stars causing them to collapse to black holes prior to inspiral [ 31 ]. This problem no longer exists in the formulation summarized here.

2.2. Relativistic Hydrodynamics

To solve for the fluid motion of the system in curved-space time it is convenient to use an Eulerian fluid description [ 63 ]. One begins with the perfect fluid stress-energy tensor in the Eulerian observer rest frame, (8) T μ ν = P g μ ν + ρ 1 + ϵ + P U μ U ν , where U μ is the relativistic four velocity one-form.

By introducing the usual set of Lorentz contracted state variables it is possible to write the relativistic hydrodynamic equations in a form which is reminiscent of their Newtonian counterparts [ 63 ]. The hydrodynamic state variables are the coordinate baryon mass density (9) D = W ρ the coordinate internal energy density (10) E = W ρ ϵ the spatial three velocity (11) V i = α U i ϕ 4 W - β i and the covariant momentum density (12) S i = D + E + P W U i .

In terms of these state variables, the hydrodynamic equations in the CFA are as follows: the equation for the conservation of baryon number takes the form (13) ∂ D ∂ t = - 6 D ∂ log ⁡ ϕ ∂ t - 1 ϕ 6 ∂ ∂ x j ϕ 6 D V j . The equation for internal energy evolution becomes (14) ∂ E ∂ t = - 6 E + P W ∂ log ⁡ ϕ ∂ t - 1 ϕ 6 ∂ ∂ x j ϕ 6 E V j - P ∂ W ∂ t + 1 ϕ 6 ∂ ∂ x j ϕ 6 W V j . Momentum conservation takes the form (15) ∂ S i ∂ t = - 6 S i ∂ log ⁡ ϕ ∂ t - 1 ϕ 6 ∂ ∂ x j ϕ 6 S i V j - α ∂ P ∂ x i + 2 α D + E + P W W - 1 W ∂ log ⁡ ϕ ∂ x i + S j ∂ β j ∂ x i - W D + E + P W ∂ α ∂ x i - α W D + Γ E ∂ χ ∂ x i , where the last term in ( 15 ) is the contribution from the radiation reaction potential χ as defined in [ 32 , 57 ]. In the construction of quasistable orbit initial conditions, this term is set to zero. Including this term would allow for a calculation of the orbital evolution via gravitational wave emission in the CFA. However, there is no guarantee that this is a sufficiently accurate solution to the exact Einstein equations. Hence, the CFA is useful for the construction of initial conditions.

2.3. Angular Momentum and Orbital Frequency

In the quasicircular orbit approximation (neglecting angular momentum in the radiation field), this system has a Killing vector corresponding to rotation in the orbital plane. Hence, for these calculations the angular momentum is well defined and given by an integral over the space-time components of the stress-energy tensor [ 64 ] that is, (16) J i j = ∫ T i 0 x j - T j 0 x i d V . Aligning the z -axis with the angular momentum vector then gives (17) J = ∫ x S y - y S x d V .

To find the orbital frequency detected by a distant observer corresponding to a fixed angular momentum we employ a slightly modified derivation of the orbital frequency compared to that of [ 53 ]. In asymptotically flat coordinates the angular frequency detected by a distant observer is simply the coordinate angular velocity that is, one evaluates (18) ω ≡ d ϕ d t = U ϕ U 0 .

In the ADM conformally flat ( 3 + 1 ) curved space, our only task is then to deduce U ϕ from code coordinates. For this we make a simple polar coordinate transformation keeping our conformally flat coordinates, so (19) U ϕ = Λ ν ϕ U ν = x U y - y U x x 2 + y 2 . Now, the code uses covariant four velocities, U i = g i ν U ν = β i U 0 + ϕ 4 U i . This gives U i = U i β i ( W / α ) / ϕ 4 . Finally, one must density weight and volume average ω over the fluid differential volume elements. This gives (20) ω = ∫ d 3 x ϕ 2 D + Γ E α / W x U y - y U x - x β y - y β x / x 2 + y 2 ∫ d 3 x ϕ 6 D + Γ E . This form differs slightly from that of [ 53 ] but leads to the similar results.

A key additional ingredient, however, is the implementation of a grid three velocity V G i that minimizes the matter motion with respect to U i and β i . Hence, the total angular frequency to a distant observer ω t o t = ω + ω G , and in the limit of rigid corotation, ω t o t → ω G , where ω G = x V y + y V x .

For the orbit calculations illustrated here we model corotating stars, that is, no spin in the corotating frame. This minimizes matter motion on the grid. However, we note that there is need at the present time of initial conditions for arbitrarily spinning neutron stars and the method described here is equally capable of supplying those initial conditions.

As a practical approach the simulation [ 32 ] of initial conditions is best run first with viscous damping in the hydrodynamics for sufficiently long time (a few thousand cycles) to relax the stars to a steady state. Subsequently, one can run with no damping. In the present illustration we examine stars at large separation which are in quasiequilibrium circular orbits and stable hydrodynamic configurations. In the initial relaxation phase the orbits are circularized by damping any radial velocity components. During the evolution, the rigorous conservation of angular momentum is imposed by adjusting the orbital angular velocity in ( 20 ) such that ( 17 ) remains constant. The simulated orbits described in this work span the time from the last several minutes up to orbit inspiral. Here, we illustrate the stability of the multiple orbit hydrodynamic simulation and examine where the initial conditions for the strong field orbit dynamics computed here deviates from the post-Newtonian regime.

3. Code Validation 3.1. Code Tests

To evolve stars at large separation distance it is best [ 53 ] to decompose the grid into a high resolution domain with a fine matter grid around the stars and a coarser domain with an extended grid for the fields. Figure 1 shows a schematic of this decomposition from [ 54 ].

Schematic representation of the field and hydrodynamics grid used in the simulations. The inner blue grid represents the higher resolution matter grid and the outer white grid represents the field grid.

As noted in [ 53 ] it is best to keep the number of zones across each star between 25 and 40 [ 54 ]. This keeps the error in the numerics below 0.5%. It has also been pointed out [ 53 ] that an artificial viscosity (AV) shock capturing scheme has an advantage over Riemann solvers because only about half as many zones are required to accurately resolve the stars when an AV scheme is employed compared to a Riemann solver.

The time steps d t n are taken as the minimum of the time step as determined by several conditions. Each condition is also multiplied by a number less than one to accommodate the nonlinear nature of the equations.

The first condition is known as the Courant condition, that is, a search over all zones i for the zone with the minimum sound crossing time: (21) d t 1 = min ⁡ d x b i C s i , where C s i is the sound speed in the i th zone.

The Newtonian sound speed is given by the variation of pressure with density. In relativity the wave speed is given instead by the adiabatic derivative of the pressure with respect to the relativistic inertial density. In terms of relativistic variables the local sound speed in zone i then becomes [ 53 ] (22) C s = Γ i Γ i - 1 E i D i + Γ i E i .

The second condition is a search for the zone with minimum time for material to flow across a zone (23) d t 2 = min ⁡ d x a i V i . This constraint is introduced to ensure stability and accuracy in the numerical advection calculation.

The third condition is introduced to maintain stability of the artificial viscosity algorithm. The viscous equations are analogous to a diffusion equation in four velocity with a diffusion coefficient D ≈ k 1 d x i | δ U i | , where δ U i ≡ U i + 1 - U i . We then can define a minimum viscous diffusion time across a zone derived from the stability condition for explicit diffusion equations (24) d t 3 = 1 4 min ⁡ W i d x b i δ U i .

The time step d t is then assigned to be some fraction (referred to as the Courant parameter) of the minimum of these three conditions (25) d t = k min ⁡ d t 1 , d t 2 , d t 3 , Obviously, smaller values for k increase the accuracy of the calculation but also increase the computation time.

Figure 2 shows a plot of orbital velocity versus time for various values of the Courant parameter. This figure establishes that the routines for the hydrodynamics are stable (e.g., changing the Courant condition has little to effect) as long as k ≤ 0.5 .

Comparison of the orbital angular velocity ω versus time for different values of the Courant parameter k . As can be seen, the simulations with k = 0.25𠄰.5 result in stable runs that converge to the same value, implying that a smaller k or equivalently a smaller δ t is not necessary and would only use extra CPU time. For comparison, we plot a simulation with k = 0.6 to show that the stability is lost for k > 0.5 .

Figure 3 illustrates the difference between the central density ρ c and the central density ( ρ 52 ) at the highest resolution of 52 zones for single isolated stars. This is expressed as the fractional error as a function of the number of zones across a star. This plot was calculated using the relatively soft MW EoS, that is, the zero temperature and zero neutrino chemical potential limit of the EoS that has previously been used to model core-collapse supernovae [ 32 , 53 , 65 ].

Plot of the error in the central density versus the number of zones across the star. It is clear that there is only a 1% error with � zones across the star. Increasing the number of zones across the star so that there are 㸵 zones across the star produces less than a 0.1% error.

This figure illustrates that here is only a 1% error in central density with � zones across the star, while increasing the number of zones across the star to 㸵 produces less than a 0.1%. In the illustrations below we maintain k = 0.5 and � zones across each star as the best choice for both speed and accuracy needed to compute stable orbital initial conditions.

As an illustration of the orbit stability Figure 4 shows results from a simulation [ 54 ] in which the angular momentum was fixed at J = 2.7 × 1 0 11 𠂜m 2 and the Courant parameter set to k = 0.5 . For this orbital calculation the MW EoS was employed and each star was fixed at a baryon mass of M B = 1.54   M ⊙ and a gravitational mass in isolation of M G = 1.40   M ⊙ .

Plot of the orbital angular velocity, ω in geometrized units (cm 𢄡 ) versus cycle number. When ω stops changing the simulation has reached a circular binary orbit solution. This run was extended to over 30,000 cycles, corresponding to � orbits.

Figure 4 shows the evolution of the orbital angular velocity ω versus computational cycle for the first 30,000 code cycles corresponding to � orbits. The stars were initially placed on the grid using a solution of the TOV equation in isotropic coordinates for an isolated star. The stars were initially set to be corotating but were allowed to settle into their binary equilibrium. Notice that it takes ߥ,000 cycles, corresponding to ߣ orbits, just to approach the quasiequilibrium binary solution. Indeed, the stars continued to gradually compact and slightly increase in orbital frequency until 縐 orbits afterward, the stars were completely stable.

Figure 5 shows the contours of the lapse function α (roughly corresponding to the gravitational potential) and corresponding density profiles at cycle numbers, 0, 5200, and 25800 (𢒀, 5, and 19 orbits). Figure 6 shows the contours of central density and the orientation of the binary orbit corresponding to these cycle numbers. One can visibly see from these figures the relaxation of the stars after the first few orbits and the stability of the density profiles after multiple orbits.

Contours of the lapse function (left) and central density (right) at cycle numbers 0 (a), 5,200 (b), and 25,800 (c) corresponding to roughly 0, 5, and 19 orbits.

Contours of the central density for the binary system at the approximate number of orbits as labelled.

We note, however, that this orbit is on the edge of the ISCO. As such it could be unstable to inspiral even after many orbits. Figures 7 and 8 further illustrate this point. In these simulations various angular momenta were computed with a slightly higher neutron star mass ( M b = 1.61   M ⊙ and M g = 1.44   M ⊙ ), but the same MW EoS. In this case the binary system was followed for nearly 100 orbits.

Plot of the orbital angular velocity, ω , versus cycle. When ω stops changing with time the simulation has reached a circular binary orbit solution. The run ( a for J = 2.7 × 1 0 11 𠂜m 2 ) goes over 縐 obits and then becomes unstable to inspiral and merger after 縐 4 cycles. The stable two runs ( b for J = 2.8 × 1 0 11 𠂜m 𢄢 and c for J = 2.9 × 1 0 11 𠂜m 𢄢 ) were run for 100,000 cycles and � orbits.

Plot of the central density, ρ c , versus the number of orbit. The solid lines from top to bottom are for J = 3.0,3.2,3.4,3.6,3.8,4.0 × 1 0 11 𠂜m 2 .

Figure 7 illustrates orbital angular frequency versus cycle number for three representative angular momenta bracketing the ISCO. The orbital separation for the lowest angular momentum ( J = 2.7 × 1 0 11 𠂜m 𢄢 ) shown on Figure 7 is just inside the ISCO. Hence, even though it requires about 10 orbits before inspiral, the orbit is eventually unstable. Similarly, Figure 8 shows the central density versus number of orbits for 6 different angular momenta. Here one can see that orbits with J ≥ 3.0 × 1 0 11 𠂜m 𢄢 are stable. Indeed, for these cases, after about the first 3 orbits the orbits continue with almost no discernible change in orbit frequency or central density.

As mentioned previously, the numerical relativistic neutron binary simulations of [ 43 ] all start with initial data that are subsequently evolved in a different manner compared to those with which they were created. One conclusion that may be drawn from the above set of simulations, however, is that the initial data must be evolved for ample time (ϣ orbit) for the stars to reach a true quasiequilibrium binary configuration. That has not always been done in the literature.

4. Sensitivity of Initial Condition Orbital Parameters to the Equation of State 4.1. Equations of State

One hope in the forthcoming detection of gravitational waves is that a sensitivity exists to the neutron star equation of state. For illustration we utilize several representative equations of state often employed in the literature. These span a range from relatively soft to stiff nuclear matter. These are used to illustrate the EoS dependence of the initial conditions. One EoS often employed is that of a polytrope, that is, p = K ρ Γ , with Γ = 2 , where in cgs units, K = 0.0445 ( c 2 / ρ n ) , and ρ n = 2.3 × 1 0 14  g𠂜m 𢄣 . These parameters, with ρ c = 4.74 × 1 0 14  g𠂜m 𢄣 , produce an isolated star having a radius = 17.12 km and baryon mass = 1.5   M ⊙ . As noted in previous sections we utilize the zero temperature and zero neutrino chemical potential MW EoS [ 32 , 53 , 65 ]. The third is the equation of state developed by Lattimer and Swesty [ 66 ] with two different choices of compressibility, one having compressibility K = 220  MeV and the other having K = 375  MeV. We denote these as LS 220 and LS 375. The fourth EoS has been developed by Glendenning [ 67 ]. This EoS has K = 240  MeV, which is close to the experimental value [ 68 ]. We denote this EoS as GLN. Table 1 illustrates [ 54 ] the properties of isolated neutron stars generated with each EoS. For each case the baryon mass was chosen to obtain a gravitational mass for each star of 1.4   M ⊙ .

Table presenting central density, baryon mass, and gravitational mass for the five adopted equations of state.

In Figure 9 we plot the equation of state index Γ versus density, ρ , for the various EoSs considered here. These are compared to the simple polytropic Γ = 2 EoS often employed in the literature.

EoS index Γ versus central density for various equations of state. Large Γ implies a stiff EoS.

4.2. EoS Dependence of the Initial Condition Orbit Parameters

Table 2 summarizes the initial condition orbit parameters [ 54 ] at various fixed angular momenta for the various equations of state. In the case of orbits unstable to merger, this table lists the orbit parameters just before inspiral. These orbits span a range in specific angular momenta J / M 0 2 of ߥ to 10. The equations of state listed in Tables 1 and 2 are in approximate order of increasing stiffness from the top to the bottom.

Orbital parameters for each EoS.

As expected, the central densities are much higher for the relatively soft MW and GLN equations of state. Also, the orbit angular frequencies are considerably lower for the extended mass distributions of the stiff equations of state than for the more compact soft equations of state. These extended mass distributions induce a sensitivity of the emergent gravitational wave frequencies and amplitude due to the strong dependence of the gravitational wave frequency to the quadrupole moment of the mass distribution.

4.3. Gravitational Wave Frequency

The physical processes occurring during the last orbits of a neutron star binary are currently a subject of intense interest. As the stars approach their final orbits it is expected that the coupling of the orbital motion to the hydrodynamic evolution of the stars in the strong relativistic fields could provide insight into various physical properties of the coalescing system [ 58 , 69 ]. In this regard, careful modeling of the initial conditions is needed which includes both the nonlinear general relativistic and hydrodynamic effects as well as a realistic neutron star equation of state.

Figure 10 shows the EoS sensitivity of the gravitational wave frequency f = ω / π as a function of proper separation d p between the stars for the various orbits and equations of state summarized in Table 2 . These are compared with the circular orbit condition in the (post) 5 / 2 -Newtonian approximation, hereafter PN, analysis of reference [ 70 ]. In that paper a search was made for the inner most stable circular orbit in the absence of radiation reaction terms in the equations of motion. This is analogous to the calculations performed here which also analyzes orbit stability in the absence of radiation reaction.

Computed gravitational wave frequency, f , versus proper separation for each EoS as labelled. The black line corresponds to the (post) 5 / 2 -Newtonian estimate. Frequencies obtained from the stiff and polytropic equations of state do not deviate by more than 縐% from the PN prediction until a frequency greater than 񾌀 Hz. The grey line is an extrapolation of the frequencies obtained using the soft MW and GLN EoSs. These begin to deviate by more than 10% from the PN prediction at a frequency of 񾄀 Hz.

In the (post) 5 / 2 -Newtonian equations of motion, a circular orbit is derived by setting time derivatives of the separation, angular frequency, and the radial acceleration to zero. This leads to the circular orbit condition [ 70 ] (26) ω 0 2 = m A 0 d h 3 , where ω 0 is the circular orbit frequency and m = 2 M G 0 , d h is the separation in harmonic coordinates, and A 0 is a relative acceleration parameter which for equal mass stars becomes (27) A 0 = 1 - 3 2 m d h 3 - 77 8 m d h + ω 0 d h 2 + 7 4 ω 0 d h 2 . Equations ( 26 ) and ( 27 ) can be solved to find the orbit angular frequency as a function of harmonic separation d h . The gravitational wave frequency is then twice the orbit frequency, f = ω 0 / π .

Although this is a gauge-dependent comparison, for illustration we show in Figure 10 the calculated gravitational wave frequency compared to the PN expectation as a function of proper binary separation distance up to 200   km. One should keep in mind, however, that there is some uncertainty in associating proper distance with the PN parameter ( m / r ). Hence, a comparison with the PN results is not particularly meaningful. It is nevertheless instructive to consider the difference in the numerical results as the stiffness of the EoS is varied. The polytropic and stiff EoSs begin to deviate (by 㸐%) from the softer equations of state (MW and GLN) for a gravitational wave frequency as low as 񾄀 Hz and more or less continue to deviate as the stars approach the ISCO at higher frequencies.

Indeed, a striking feature of Figure 10 is that as the stars approach the ISCO, the frequency varies more slowly with diminishing separation distance for the softer equations of state. A gradual change in frequency can mean more orbits in the LIGO window and hence a stronger signal to noise (cf. [ 57 ]).

Also, for a soft EoS the orbit becomes unstable to inspiral at a larger separation. At least part of the difference between the soft and stiff EoSs can be attributed to the effects of the finite size of the stars which are more compact for the soft equations of state [ 37 ].

We note that, for comparable angular momenta, our results are consistent with the EoS sensitivity study of [ 37 ] based upon a set of equations of state parameterized by a segmented polytropic indices and an overall pressure scale. Their calculations, however, were based upon two independent numerical relativity codes. The similarity of their simulations to the results in Table 2 further confirms the broad validity of the CFA approach when applied to initial conditions. For example, their orbit parameters are summarized in Table II of [ 37 ]. Their softest EoS is the Bss221 which corresponds to an adiabatic index of Γ = 2.4 for the core and a baryon mass of 1.501   M ⊙ and an ADM mass of 1.338   M ⊙ per star for a specific angular momentum of 1.61 × 1 0 11 𠂜m 2 (in our units) with a corresponding gravitational wave frequency of 530 Hz at a proper separation of 46 km. This EoS is comparable to the polytropic, MW, and GLN EoSs shown on Figure 10 . For example, our closest orbit with the Γ = 2 polytropic EoS corresponds to a specific angular momentum of 1.8 × 1 0 11 𠂜m 2 and an ADM mass of 1.39   M ⊙ compared to their ADM mass of 1.34   M ⊙ at J = 1.6    ×    1 0 11 𠂜m 2 for the same baryon mass of 1.5   M ⊙ . Although, for the softer EoSs, their results are for a closer orbit than the numerical points given on Figure 10 , an extension of the grey line fit to the numerical simulations of the soft EoSs would predict a frequency of 540 Hz at the same proper separation of 46 km compared to 530 Hz in the Bss221 simulation of [ 37 ].

The main parameter characterizing the last stable orbit in the post-Newtonian calculation is the ratio of coordinate separation to total mass (in isolation) d h / m . The analogous quantity in our nonperturbative simulation is proper separation to gravitational mass, d P / m . The separation corresponding to the last stable orbit in the post-Newtonian analysis does not occur until the stars have approached 6.03 m . For M G 0 = 1.4   M ⊙ stars, this would correspond to a separation distance of about 25 km. In the results reported here the last stable orbit occurs somewhere just below 7.7 m G 0 at a proper separation distance of d P ≈ 30  km for both the polytropic and the MW stars.