# LIGO Detection Question: Why Lighter Masses Mean More Time

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• verve825
In summary, the second LIGO detection, from Symmetry, involved lighter masses which resulted in a longer time spent in the sensitive band of the detectors. This is because the chirp mass, which controls the binary's evolution, is inversely proportional to the time it takes for the signal to rise from "too weak to detect" to "too high frequency to detect". This was shown through the formula for the chirp mass and its role in determining the masses from the observed data.

#### verve825

Re: the second LIGO detection, from Symmetry: “Because of their lighter masses compared to the first detection, they spent more time—about one second—in the sensitive band of the detectors.”

As an absolute (albeit deeply fascinated) novice here, I'm unclear as to why lighter masses would allow for the waves' greater time in the sensitive band of the detectors.

These were lower mass objects, so everything happens more slowly. Orbital speeds are lower and orbital decay happens slower. Since the frequency of the gravitational waves depends on the time it takes the black holes to go round each other, the result is that the frequency change is slower. So it takes longer for the signal to rise from "too weak to detect" to "too high frequency to detect".

verve825 said:
Re: the second LIGO detection, from Symmetry: “Because of their lighter masses compared to the first detection, they spent more time—about one second—in the sensitive band of the detectors.”

As an absolute (albeit deeply fascinated) novice here, I'm unclear as to why lighter masses would allow for the waves' greater time in the sensitive band of the detectors.

If you look at http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.241103 you'll see the following hint:

The chirp mass [26,45], which controls the binary’s evolution during the early inspiral, is determined very precisely. The individual masses, which rely on information from the late inspiral and merger, are measured far less precisely.

Now, if you look at the first Ligo paper, http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102, you can see how the chirp mass is defined and how it controls the evolution of the inspiral. The formula is:

$$\mathcal{M}^{\frac{5}{3}}= k \, f^{\,-\frac{11}{3}} \, \dot{f}$$

Here f is the observed instantaneous frequency of the gravitational wave (i.e the chirp), ##\dot{f}## is it's time derivative, and k is some constant given in the paper. By letting ##\dot{f} = df/dt## you can find an integral for the time t to pass through the "sensitivity band"

$$dt = \int_{f_{low}}^{f_{hi}} \, \frac{k \, f^{\,-\frac{11}{3}} \, df}{\mathcal{M}^{\frac{5}{3}}}$$

So the time to pass through the "sensitivity band" from ##f_{low}## to ##f_{hi}## is given by the above integral, which is inversely proportional to ##\mathcal{M}##. Thus a lower chirp mass means a greater time to pass through the band.

There is a reference for where the formula for the "chirp mass" was derived in the paper, but I don't actually know the details of the derivation. But given the existence of the formula, you can see how you can compute ##\dot{f}## given the value of ##f## and ##\mathcal{M}##, and thus controls the evolution of the inspiral (at least during the early phase). You can also see the importance of this parameter in determining how the masses were computed from the observed data.

Last edited:
PeterDonis, vanhees71 and m4r35n357
Thank you for your great and detailed response, Pervect- I really appreciate it.

jb

## 1. What is LIGO and why is it important?

LIGO stands for Laser Interferometer Gravitational-Wave Observatory. It is a scientific facility that uses laser interferometry to detect gravitational waves, which are ripples in the fabric of space-time. LIGO is important because it allows scientists to study and understand some of the most violent and energetic events in the universe, such as black hole mergers and supernovas.

## 2. How does LIGO detect gravitational waves?

LIGO uses a technique called laser interferometry, where a laser beam is split and sent down two perpendicular arms. When a gravitational wave passes through the detector, it causes the arms to change length by an incredibly small amount. This change is measured by comparing the two beams of light, allowing scientists to detect the presence of a gravitational wave.

## 3. What does the "LIGO Detection Question: Why Lighter Masses Mean More Time" mean?

The question refers to the fact that when two objects with different masses are in orbit around each other, they emit gravitational waves. These waves cause the objects to gradually lose energy and spiral towards each other, eventually colliding. The time it takes for this to happen is directly related to the masses of the objects, with lighter masses taking longer to collide.

## 4. What are some potential applications of LIGO's gravitational wave detection?

LIGO's gravitational wave detection has the potential to revolutionize our understanding of the universe. It can help us study the behavior of black holes, test Einstein's theory of general relativity, and even provide insight into the early moments of the universe. Additionally, LIGO technology can also be used for practical applications, such as improving precision in GPS systems and developing new forms of communication.

## 5. Has LIGO made any important discoveries so far?

Yes, LIGO has made several significant discoveries since its first detection of gravitational waves in 2015. Most notably, it detected the collision of two massive black holes, providing strong evidence for the existence of these objects. LIGO has also detected several other black hole mergers and a neutron star merger, which was also observed by other telescopes and provided valuable insights into the nature of these events.