# Excluding the case of a neutron star-black hole merger with gravitational wave GW 150914

The paper from LIGO's original gravitational wave detection (PRL 116, 061102 (2016)) defines the chirp mass as $$M=\frac{(m_1 m_2)^{3/5}}{(m_1+m_2)^{1/5}}=\frac{c^3}{G}\left[\frac{5}{96}\pi^{-8/3}f^{-11/3}\dot{f}\right]^{3/5}\approx 30M_{sun}$$

and states "a black hole neutron star binary with the deduced chirp mass would have a very large total mass..." I do not get this. I can get the deduced chirp mass with a neutron star mass of about 18 suns and a black hole mass of 70 suns, for a total mass of 98 suns. Why did they draw this conclusion?

The important thing to realize here is that neutron stars are limited to comparatively low masses. The classic mass of a "canonical" neutron star is $$\sim1.4M_{\odot}$$, and almost all observed examples fall into the $$1.4\mathrm{-}2.0M_{\odot}$$ range, with a few extremely tentative candidates approaching $$2.5M_{\odot}$$ (e.g. a possible $$2.50\mathrm{-}2.67M_{\odot}$$ compact object seen in GW190814, Abbott et al. 2020, which may or may not be a neutron star). The general relativistic bound is no higher than about $$3.0M_{\odot}$$, and an $$18M_{\odot}$$ neutron star is definitely out of the question.
With that in hand: Say GW 150914 was the result of a neutron star-black hole merger. If we take the neutron star to have the canonical mass of $$1.4M_{\odot}$$, the black hole would have to have a mass of $$\sim3000M_{\odot}$$ to produce the measured chirp mass. Even increasing the neutron star mass to $$2.0M_{\odot}$$ only reduces the black hole mass to a bit under $$2000M_{\odot}$$. It's certainly possible to have black holes in this mass range, but as the paper points out, this would lead to gravitational wave emission at a much lower frequency than was actually observed.
For comparison, the sources of the GW200105 and GW200115 detections, postulated to be neutron star-black hole mergers, had chirp masses of only $$\sim3.41M_{\odot}$$ and $$\sim2.42M_{\odot}$$, respectively, and total masses of $$\sim10.8M_{\odot}$$ and $$\sim7.3M_{\odot}$$. In both cases, the neutron stars appear to be comfortably within the observed range of neutron star masses.