<p>Dimensionally speaking, the luminosity of a gravitationally radiating binary system, consisting of two objects of mass <span class="math-container">$M$</span>, separated by <span class="math-container">$R$</span>, goes as <span class="math-container">$(M/R)^5$</span>.  The timescale of the chirp for such a system goes as <span class="math-container">$M^{-3} R^4$</span>. (<a href="https://cds.cern.ch/record/407258/files/9911034.pdf" rel="nofollow noreferrer">Schutz 1999</a>).</p>
<p>Thus the total energy released goes as <span class="math-container">$M^2/R$</span>, i.e. it is proportional to the gravitational potential energy of the system.</p>
<p>Because of the <span class="math-container">$R^{-1}$</span> dependence, it is basically the mass and radius of the &quot;final&quot; state that determines the energy lost. For the black hole case, the final mass is just less than <span class="math-container">$2M$</span> and the final configuration event horizon is <span class="math-container">$4M$</span> (with <span class="math-container">$G=c=1$</span>). So <span class="math-container">$M^2/R = M$</span>. Thus I would expect the mass-energy released in gravitational waves to be a fixed fraction of the combined mass of the black holes (note that unequal mass black holes will lead to complications).</p>
<p>Looking at the <a href="https://en.m.wikipedia.org/wiki/List_of_gravitational_wave_observations" rel="nofollow noreferrer">data for the list of mergers</a> this model seems reasonable, with the fixed fraction being about 5%.</p>
<p>Extending this to neutron stars, well the &quot;final&quot; radius is going to depend on the physics of the neutron star material and so will be model dependent. However, that radius will be <span class="math-container">$&gt;4M$</span> (i.e. probably several times the Schwarzschild radius). Another way of saying this, is that neutron stars can't get as close together before the merger takes place.
So from that point of view I would expect <span class="math-container">$&lt;$</span>5% of the combined mass-energy is radiated as gravitational waves.</p>
<p>Observationally, there is no accurate estimate of the final remnant mass for GW170817.</p>

CC BY-SA 4.0
2020-09-05T08:11:26.677Z
2020-09-05T08:11:26.677Z
ProfRob
<p><a href="https://astronomy.stackexchange.com/q/38726/7982">GW190521 black hole merger total mass calculation and missing mass, how does this happen?</a> notes that there are about 9 solar masses missing from the final black hole.</p>
<p><a href="https://en.wikipedia.org/wiki/GW170817" rel="nofollow noreferrer">GW170817</a> is the first observed merger of two neutron stars, detected in several ways including a weak gravitational wave.</p>
<p>Do neutron star mergers also radiate several percent of their mass as gravitational waves, or is the fraction much smaller. They comprise ordinary matter rather than being singularities in spacetime, so my guess is that the fraction is much smaller, but I have no idea.</p>
<p>My question is motivated by <a href="https://astronomy.stackexchange.com/a/38732/7982">this answer</a>.</p>
<hr />
<p>Related:</p>
<ul>
<li><a href="https://astronomy.stackexchange.com/q/23187/7982">&quot;Who saw&quot; the binary neutron star merger first? What was the sequence of events? (GRB/GW170817)</a></li>
<li><a href="https://astronomy.stackexchange.com/q/28404/7982">What does &quot;GPU-accelerated butterfly matched filtering over dense bank of time-symmetric chirp-like templates&quot; mean? (GW170817)</a></li>
</ul>

<astrophysics><gravitational-waves><neutron-star>
Fraction of initial mass lost (radiated) by neutron star mergers compared to black hole mergers?
uhoh
# Fraction of initial mass lost (radiated) by neutron star mergers compared to black hole mergers?

(Asked by: username_0 on 2020-09-05)

[GW190521 black hole merger total mass calculation and missing mass, how does this happen?](https://astronomy.stackexchange.com/q/38726/7982) notes that there are about 9 solar masses missing from the final black hole.

[GW170817](https://en.wikipedia.org/wiki/GW170817) is the first observed merger of two neutron stars, detected in several ways including a weak gravitational wave.

Do neutron star mergers also radiate several percent of their mass as gravitational waves, or is the fraction much smaller. They comprise ordinary matter rather than being singularities in spacetime, so my guess is that the fraction is much smaller, but I have no idea.

My question is motivated by [this answer](https://astronomy.stackexchange.com/a/38732/7982).

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Related:

* ["Who saw" the binary neutron star merger first? What was the sequence of events? (GRB/GW170817)](https://astronomy.stackexchange.com/q/23187/7982)
* [What does "GPU-accelerated butterfly matched filtering over dense bank of time-symmetric chirp-like templates" mean? (GW170817)](https://astronomy.stackexchange.com/q/28404/7982)

**Tags:** astrophysics, gravitational-waves, neutron-star


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## 1 answers

### Answer by username_1 on 2020-09-05. Score: 1 | Accepted answer

Dimensionally speaking, the luminosity of a gravitationally radiating binary system, consisting of two objects of mass $M$, separated by $R$, goes as $(M/R)^5$. The timescale of the chirp for such a system goes as $M^{-3} R^4$. ([Schutz 1999](https://cds.cern.ch/record/407258/files/9911034.pdf)).

Thus the total energy released goes as $M^2/R$, i.e. it is proportional to the gravitational potential energy of the system.

Because of the $R^{-1}$ dependence, it is basically the mass and radius of the "final" state that determines the energy lost. For the black hole case, the final mass is just less than $2M$ and the final configuration event horizon is $4M$ (with $G=c=1$). So $M^2/R = M$. Thus I would expect the mass-energy released in gravitational waves to be a fixed fraction of the combined mass of the black holes (note that unequal mass black holes will lead to complications).

Looking at the [data for the list of mergers](https://en.m.wikipedia.org/wiki/List_of_gravitational_wave_observations) this model seems reasonable, with the fixed fraction being about 5%.

Extending this to neutron stars, well the "final" radius is going to depend on the physics of the neutron star material and so will be model dependent. However, that radius will be $>4M$ (i.e. probably several times the Schwarzschild radius). Another way of saying this, is that neutron stars can't get as close together before the merger takes place.
So from that point of view I would expect $<$5% of the combined mass-energy is radiated as gravitational waves.

Observationally, there is no accurate estimate of the final remnant mass for GW170817.


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