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

<p>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. <a href="https://arxiv.org/abs/astro-ph/0211492" rel="nofollow noreferrer">Wen 2003</a>).
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.</p>

<p>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, <span class="math-container">$e \sim 1$</span>, 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.</p>

CC BY-SA 4.0
2020-05-23T23:18:51.087Z
1) Even perfectly circular binaries produce other frequencies than the dominant 22 mode. 2) the frequencies of eccentric binaries are not integer multiples of the orbital frequency. They are integer combinations of the orbital frequency and the pericenter precession frequency.
TimRias
CC BY-SA 4.0
2020-05-24T08:20:23.247Z
Thank you Rob and mmeent for your reply. I just don't quite understand the comment of mmeent saying that "Even perfectly circular binaries produce other frequencies than the dominant 22 mode". what kind of other frequencies for low eccentricity orbit binary can produce?
Chen
CC BY-SA 4.0
2020-05-24T08:27:44.630Z
@mmeent (1) Did I say otherwise? Is what I said untrue? (2) I'll get back to you on that. The statement I made is in lots of peer-reviewed papers along with calculations and plots of the nth multiple of the orbital frequency.
ProfRob
CC BY-SA 4.0
2020-05-24T14:12:30.177Z
@RobJeffries  Does the "Closer binaries" you mention include those ultra-compact x-ray binaries? Those binaries have only serval tens of minutes orbital period and are very closed, but they still in steady state and don't step into merge stage. Does the frequency spectrum of these binaries can produce frequency lower than orbital frequency? They should be circle, at least are low eccentricity.
Chen
CC BY-SA 4.0
2020-05-24T17:48:51.573Z
@Chem Even the effects that mmeent talks about produce higher harmonics of the orbital frequency, not frequencies below the rbital frequency.
ProfRob
CC BY-SA 4.0
2020-05-24T22:00:05.103Z
@RobJeffries Actually, pretty much any integer combination of the fundamental frequencies will show up in the spectrum. This includes differences between the frequencies, and allows you to get modes with frequencies lower than the orbital frequency. (But these tend to be quite weak.)
TimRias
CC BY-SA 4.0
2020-05-25T02:33:40.780Z
@mmeent Could you show me any example or theory articles that discribe in producing such low frequencies? Even if it is weak.
Chen
CC BY-SA 4.0
2020-05-23T15:57:22.520Z
2020-05-24T10:10:49.617Z
2020-05-24T10:10:49.617Z
ProfRob
<p>A binary will always emit 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.</p>
<p>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 stronger than all other modes combined. The other modes tend to be suppressed exponentially, but generically all integer multiples of the orbital frequency will be present.</p>
<p>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 <span class="math-container">$Ω_ϕ$</span>, the frequency of the radial oscillation <span class="math-container">$Ω_r$</span>, and the precession frequencies of the two spins <span class="math-container">$Ω_θ$</span> and <span class="math-container">$Ω_ψ$</span>. The gravitational wave spectrum of such a binary consists of integer combinations <span class="math-container">$mΩ_ϕ + nΩ_r + k_1Ω_θ + k_2Ω_ψ$</span>, and generically all such combinations are present, but a few will dominate.</p>
<p>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 slowly evolving while in the strong field regime, where the fundamental frequencies can be very different.</p>
<p>To achieve millihertz frequencies while in the strong field regime the total mass of the binary needs to be of the order of 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).</p>
<p>Now can we get a situation where there are GW modes at ⅔ of the orbital frequency? The answer is yes. One situation is as follows. Consider a binary with eccentricity <span class="math-container">$e= 0.1$</span>, and semi-latus rectum <span class="math-container">$p$</span> roughly 10.8 times <span class="math-container">$GM/c²$</span>. In this case, the radial frequency <span class="math-container">$Ω_r$</span> is roughly <span class="math-container">$⅔Ω_ϕ$</span>. The spectrum of this binary contains a mode with frequency <span class="math-container">$ω = 2Ω_ϕ - 2 Ω_r = ⅔Ω_ϕ$</span>. 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.</p>

CC BY-SA 4.0
2020-05-26T10:24:47.647Z
@mmeent  Is this method suitable to the closed binaries, i.e.: those binaries that have matter exchange, one accrete matter from another one?
Chen
CC BY-SA 4.0
2020-05-26T11:20:44.830Z
I had not considerd such extreme mass ratios. This is slowly evolving in terms of the ratio of orbital period to inspiral time, but still merges in a few decades. So not at the chirp phase but still very short-lived.
ProfRob
CC BY-SA 4.0
2020-05-26T12:31:17.850Z
@Chen I don't think you would be able to get down to the require mass-ratios for two stellar objects.
TimRias
CC BY-SA 4.0
2020-05-26T13:47:50.607Z
@mmeent  Well, to my understand, this is a situation that is basically impossible to observe for a close binary of mass-ratio at about 1:serval tens. Anyway, thank you and Rob.
Chen
CC BY-SA 4.0
2020-05-26T09:15:52.670Z
2024-08-30T18:22:09.070Z
2024-08-30T18:22:09.070Z
TimRias
<p>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.</p>

CC BY-SA 4.0
2020-05-23T14:21:03.390Z
Why 2/3 of the orbital frequency?
ProfRob
CC BY-SA 4.0
2020-05-23T14:41:02.690Z
That is just an example. I want to express whether the frequency lower than the orbital frequency may be the gravitational wave or the changed gravitational wave frequency? And whether gravitational wave can cause the continue x-ray energy spectrum to drift, since it can cause changes in the density of the material.
Chen
<gravitational-waves>
Is gravitational wave frequency always equal to double the orbital frequency?
Chen
# Is gravitational wave frequency always equal to double the orbital frequency?

(Asked by: username_0 on 2020-05-23)

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.

**Tags:** gravitational-waves
> **Comments:**
> **username_1**: Why 2/3 of the orbital frequency?  
> **username_0**: That is just an example. I want to express whether the frequency lower than the orbital frequency may be the gravitational wave or the changed gravitational wave frequency? And whether gravitational wave can cause the continue x-ray energy spectrum to drift, since it can cause changes in the density of the material.  

---

## 2 answers

### Answer by username_2 on 2020-05-26. Score: 6 | Accepted answer

A binary will always emit 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 stronger than all other modes combined. The other modes tend to 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 $Ω\_ϕ$, the frequency of the radial oscillation $Ω\_r$, and the precession frequencies of the two spins $Ω\_θ$ and $Ω\_ψ$. The gravitational wave spectrum of such a binary consists of integer combinations $mΩ\_ϕ + nΩ\_r + k\_1Ω\_θ + k\_2Ω\_ψ$, 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 slowly evolving while in the strong field regime, where the fundamental frequencies can be very different.

To achieve millihertz frequencies while in the strong field regime the total mass of the binary needs to be of the order of 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 are GW modes at ⅔ 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 $GM/c²$. In this case, the radial frequency $Ω\_r$ is roughly $⅔Ω\_ϕ$. The spectrum of this binary contains a mode with frequency $ω = 2Ω\_ϕ - 2 Ω\_r = ⅔Ω\_ϕ$. 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.

> **Comments:**
> **username_0**: @mmeent  Is this method suitable to the closed binaries, i.e.: those binaries that have matter exchange, one accrete matter from another one?  
> **username_1**: I had not considerd such extreme mass ratios. This is slowly evolving in terms of the ratio of orbital period to inspiral time, but still merges in a few decades. So not at the chirp phase but still very short-lived.  
> **username_2**: @Chen I don't think you would be able to get down to the require mass-ratios for two stellar objects.  
> **username_0**: @mmeent  Well, to my understand, this is a situation that is basically impossible to observe for a close binary of mass-ratio at about 1:serval tens. Anyway, thank you and Rob.  

---

### Answer by username_1 on 2020-05-23. Score: 5

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](https://arxiv.org/abs/astro-ph/0211492)).
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.

> **Comments:**
> **username_2**: 1) Even perfectly circular binaries produce other frequencies than the dominant 22 mode. 2) the frequencies of eccentric binaries are not integer multiples of the orbital frequency. They are integer combinations of the orbital frequency and the pericenter precession frequency.  
> **username_0**: Thank you Rob and mmeent for your reply. I just don't quite understand the comment of mmeent saying that "Even perfectly circular binaries produce other frequencies than the dominant 22 mode". what kind of other frequencies for low eccentricity orbit binary can produce?  
> **username_1**: @mmeent (1) Did I say otherwise? Is what I said untrue? (2) I'll get back to you on that. The statement I made is in lots of peer-reviewed papers along with calculations and plots of the nth multiple of the orbital frequency.  
> **username_0**: @RobJeffries  Does the "Closer binaries" you mention include those ultra-compact x-ray binaries? Those binaries have only serval tens of minutes orbital period and are very closed, but they still in steady state and don't step into merge stage. Does the frequency spectrum of these binaries can produce frequency lower than orbital frequency? They should be circle, at least are low eccentricity.  
> **username_1**: @Chem Even the effects that mmeent talks about produce higher harmonics of the orbital frequency, not frequencies below the rbital frequency.  
> **username_2**: @RobJeffries Actually, pretty much any integer combination of the fundamental frequencies will show up in the spectrum. This includes differences between the frequencies, and allows you to get modes with frequencies lower than the orbital frequency. (But these tend to be quite weak.)  
> **username_0**: @mmeent Could you show me any example or theory articles that discribe in producing such low frequencies? Even if it is weak.  

---

