5
$\begingroup$

The first gravitational wave observations were noted for their "chirp-like" behavior; the objects would speed up as they got closer, radiating more strongly and therefore loosing energy faster. An example of a "chirp" is shown below.

The new PRL GW190521: A Binary Black Hole Merger with a Total Mass of 150 M⊙ seems to show the first observation of the formation of an intermediate mass black hole (IMBH). The image below doesn't seem to exhibit much chirpiness compared to lower-mass mergers. For example from P. B. Abbott et al. (2016)

On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz...

Question: Why do some gravitational wave events chirp more than others? Which ones produce the best, longest, and most obvious chirp in frequency?


enter image description here above: not very chirpy GW190521 from Abbott et al. (linked above)

below: "chirpy behavior" from this answer and P. B. Abbott et al. PRL 116, 061102 (2016) Observation of Gravitational Waves from a Binary Black Hole Merger.   

enter image description here

enter image description here

$\endgroup$
2
  • $\begingroup$ I dont think i fully follow as this is the first time I have heard of "chirps" but would the different masses effect the wave that can be received? when a larger black holes merges with a smaller the waves are one sided and distorted. $\endgroup$
    – user34615
    Sep 2, 2020 at 16:13
  • $\begingroup$ @Orochi luckily Rob Jeffries has posted an excellent answer $\endgroup$
    – uhoh
    Sep 3, 2020 at 7:31

1 Answer 1

11
$\begingroup$

The chirpiness of the chirp is determined to first order by how massive the merging binary sytem is and whether the gravitational wave (GW) signal is detectable when it reaches the sensitive frequency window (roughly 30-2000 Hz) of LIGO. Thus the most chirpy signals will be from nearby merging neutron stars and the least chirpy will be distant, massive black hole mergers.

All the binary mergers chirp - the GW emission extracts energy from the system causing the components to get closer together and orbit with higher frequency, leading to stronger and higher frequency GWs... But the overall timescale of the event depends on the total system mass (or more accurately, the chirp mass - see below). The more massive the system, the more rapid the evolution of the amplitude and frequency and the lower the orbital frequency when it finally merges. What you observe is also governed by the response of the detector - that is, its strain sensitivity as a function of frequency.

High-mass stellar black hole mergers have rapidly evolving signals and end their chirps at relatively low frequencies and spend less than a second inside LIGO's sensitive frequency band. Lower mass black hole mergers or neutron star mergers are much more slowly evolving, end their chirps at much higher frequencies and can spend a minute or more at detectable frequencies.

The key equations (assuming circular orbits and to first order) are: $$ \frac{df}{dt} \simeq \left(\frac{96}{5}\right)\left(\frac{G\mathcal{M}_c}{c^3}\right)^{5/3}\pi^{8/3} f^{11/3}\, ,$$ where $\mathcal{M}_c$ is the "chirp mass" and is given by $(m_1m_2)^{3/5}/(m_1+m_2)^{1/5}$, where $m_1$ and $m_2$ are the component masses. The chirp mass basically gets larger with the total mass of the system for a given mass ratio, but strictly speaking, what I said in the first paragraph about high- and low-masses means high and low chirp masses.

You can see from this formula that at a given frequency $f$ (in this case in the LIGO sensitivity band), that the rate of change of frequency is higher for higher chirp masses.

The chirp ends when the objects "merge". Exactly what this means is a bit loose, but is when the separation of the components is a small multiple of their radii in the case of neutron stars or a small multiple of their Schwarzschild radii for black holes. A simple use of Kepler's third law (not strictly applicable in General Relativistic conditions, but it gives the right scaling), and noting that the dominant GW frequency is twice the orbital frequency, yields $$ f_{\rm max} \sim \left(\frac{GM}{\pi^2 a^3_{\rm merge}}\right)^{1/2}\ ,$$ where $ a_{\rm merge}$ is the separation at merger and $M$ is the total system mass.

If we let $a_{\rm merge} \sim 4GM/c^2$ (for black holes), then we see that $f_{\rm max} \propto M^{-1}$. $f_{\rm max}$ is around 130 Hz for a $30+30$ solar mass ($M_\odot$) black hole merger (like GW150914, which had $\mathcal{M}_c \simeq 28M_{\odot}$), and would be just $\sim 50$ Hz for a merger resulting in a $150M_\odot$ black hole like GW190521 (which had $\mathcal{M}_c \simeq 64M_{\odot}$). On the other hand, a pair of $1.5 M_\odot$ neutron stars of radius 10 km would chirp at $a_{\rm merge}\sim 20$ km and $f_{\rm max}\sim 2250$ Hz.

So that I think is your answer. The sensitivity of LIGO is quite poor below 30 Hz due to seismic noise. Thus in GW190521, the merger and chirp is barely seen at all in the sensitive LIGO frequency window of 30-2000 Hz - just the final few orbits (in fact only two for GW190521) before the merger and the ring-down phase begins at a peak frequency of $\sim 50$ Hz.

To get a more chirpy signal that is detectable by LIGO you need (a) a lower mass system with a higher $f_{\rm max}$ and (b) it needs to be close enough so that LIGO can detect the lower amplitude GWs with $f<f_{\rm max}$ that are emitted prior to the merger.

$\endgroup$
1
  • 1
    $\begingroup$ Adding to the point about LIGO's low frequency sensitivity, for this object they only saw 2 orbits (4 cycles) of the binary black hole which is really going to limit how much "chirp" you can see. $\endgroup$
    – Rob
    Sep 3, 2020 at 8:38

This site is temporarily in read-only mode and not accepting new answers.

Not the answer you're looking for? Browse other questions tagged .