# Attributing Gravitational waves to a source

At a given time, there would be n number of binary neutron stars or Black Holes or even Super Novae, all of which would be leaving a gravitational wave imprint ..So how do LIGO scientists know which wave is attributable to which formation ?

• Black holes are smaller so the final orbits have a shorter period. Super Novae would not have the inspiral signature. – Keith McClary Mar 1 '18 at 4:18
• Sources and Types of Gravitational Waves – Dhruv Saxena Mar 1 '18 at 7:24
• Are you interested in differentiating between different types of sources (e.g. neutron star vs. black hole), or between multiple sources of the same type (e.g. figuring out which specific black hole binary was a source)? – HDE 226868 Mar 16 '18 at 15:35

# The LIGO case

This mostly comes down to the range of frequencies that LIGO is sensitive to.

As a compact binary system inspirals, the orbital frequency increases. Based on the mass of the system and the current orbital frequency, you can calculate when the system will merge into a single object. LIGO is sensitive to GWs with frequencies from about 30 Hz to 2000 Hz.

By the time a binary neutron star (BNS) system inspirals to 30 Hz it will merge in less than 1 minute.

Compare this to the Hulse-Taylor Binary which has an orbital period of about 7.75 hours and emits GW with a frequency of about $7\times 10^{-5}$ Hz. It will merge in around 300 million years.

Most binary systems are nowhere close to merging, and emit GWs that LIGO has no hope of detecting. Because BNS spend such a tiny fraction of their lifetime observable to LIGO it is unlikely that two will merge at the same time and lead to confusion.

Even if two binaries did merge simultaneously, the frequencies of the GWs depends on the masses of the systems. The two mergers could be disentangled by isolating the different frequencies, in the same way our ears can tell the difference between a bass and violin playing together.

# Other GW experiments

You are correct that there are lots of gravitational waves overlapping all the time, just not in LIGO's sensitivity range.

LISA is a planned space based GW detector that is sensitive to GW frequencies around 0.1 to 10 mHz (millihertz). In our galaxy there are a ton of binary white dwarf systems that emit GWs in this range. This confusion foreground of GWs acts like a noise source for LISA. You can also learn about the average properties of galactic white dwarfs by studying this signal.

Pulsar timing arrays (PTA) try to detect nHz (nanohertz) frequency GWs by measuring signals from radio pulsars. The primary source for PTAs is the stochastic background of GWs from merging supermassive black holes. This is the sum of all the binary supermassive black hole GW signals from merging galaxies.

# the math

The first order approximation for the time till merger of a compact binary system is

$$t = \frac{5}{256} \frac{c^5}{G^3} \frac{r^4}{m_1 m_2 (m_1+m_2)}$$

From Kepler's third law we can determine the binary separation $r$ as a function of the orbital frequency $F$. (The Newtonian form is an approximation to full GR. It is closer to true the further apart the two bodies are.)

$$2\pi F = \sqrt{\frac{G(m_1 + m_2)}{r^3}}$$

The GW frequency $f$ is twice the orbital frequency.

$$f = 2F$$

Putting it all together:

$$t = \frac{5}{256} (\pi f)^{-8/3} \frac{c^5}{G^{5/3}} \frac{(m_1+m_2)^{1/3}}{m_1 m_2}$$

It looks a bit cleaner in terms of the chirp mass, $\mathcal{M}$.

$$t = \frac{5}{256} (\pi f)^{-8/3} \left( \frac{G\mathcal{M}}{c^3} \right)^{-5/3}, \quad \mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1+m_2)^{1/5}}$$

For an equal mass BNS with $m_1 = m_2 = 1.5 M_\odot$, $\mathcal{M}\sim 1.3 M_\odot$, where $M_\odot$ is the mass of the sun (about $2 \times 10^{30}$ kg). For $f=30$ Hz, $t \approx 47.6$ sec.

As we increase $\mathcal{M}$ the time to merge gets smaller. So black holes will spend even less time observable to LIGO.