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Suppose we have neutron stars of 8 and 9 solar masses, they both merge and produce a GW. How does one find the frequency of the GW? Is there any specific formulae to compute it?

What I want to know is the relation between the mass and frequency of GW? How is the frequency calculated numerically?

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    $\begingroup$ The frequncy is twice the orbital period frequency, and so depends on the distance between the masses as well. $\endgroup$
    – James K
    Jul 29, 2023 at 12:22
  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Jul 31, 2023 at 8:36
  • $\begingroup$ @James K thanks for addressing my problem. My query is suppose we have neutrons of 8solar and 9 solar masses, they both merged and produced a GW how to find the frequency of the GW? is there any specific formulae to compute it? $\endgroup$ Aug 1, 2023 at 6:48

2 Answers 2

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If you have a system containing two black holes with masses $m_1$ and $m_2$, you can calculate a quantity called their chirp mass,

$$ \mathcal{M}_c = (m_1 m_2)^{3/5} / (m_1 + m_2)^{1/5}, $$

which is very useful for calculating various characteristics of the gravitational waves that they emit.

If you want to work out (a good approximation to) the gravitational wave frequency for the system at some time before the components merge, you can rearrange equation 5 from this paper (note that it uses the geometric units system, so excludes some factors of $G$ and $c$ that I'll add back in) to give:

$$ f = \frac{1}{\pi}\left(\frac{5}{256 \tau}\right)^{3/8}\left(\frac{c^3}{G\mathcal{M}_c}\right)^{5/8}\,\text{Hz}, $$

where $\tau$ is the time (in seconds) until merger at which you want to calculate the frequency, $G$ is the Gravitational constant, $c$ is the speed of light and $\mathcal{M}_c$ is the chirp mass defined above.

So, for your 8 and 9 solar mass black holes (as noted in the other answer these cannot be neutron stars at that mass), we would have a chirp mass of $7.4\,{\rm M}_{\odot}$ (or $1.47\!\times\!10^{31}\,{\rm kg}$). Calculating the frequency at 100 s, 50 s, 10 s, 1 s and 0.1 s before merger we have 7.7 Hz, 10.0 Hz, 18.3 Hz, 43.3 Hz and 102.7 Hz, respectively.

To calculate this in Python, you could use:

from numpy import pi

# get speed of light and G from scipy
from scipy.constants import c, G

# 1 solar mass in kg
Msun = 1.98854695e30

def chirp_mass(m1, m2):
    # chirp mass function
    return (m1 * m2)**(3 / 5) / (m1 + m2)**(1 / 5)

def freq_at_time(m1, m2, tc):
    # calculate frequency at a given time to coalesence
    Mc = chirp_mass(m1, m2)

    return (
        (1 / pi) *
        (5 / (256 * tc))**(3 / 8) *
        (c**3 / (G * Mc))**(5 / 8)
    )

# masses
m1 = 8 * Msun
m2 = 9 * Msun

# time to coalesence
tcs = [100, 50, 10, 1, 0.1]

for tc in tcs:
    f = freq_at_time(m1, m2, tc)

    print(f"The frequency at {tc} s before coalesence is {f:.2f} Hz")
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Neutron stars collapse to black holes at about 2.2 solar masses, so I will assume you mean "black holes".

If you have black holes of those masses and they merge, the frequency will not be constant, it will be a chirp, so the frequency will be a function of time. To find this function would require relativity, and to do it right would need numerical relativity (and hence no formula). But the frequency range would be from very low frequency increasing to two or three hundred hertz as the black holes merge.

Among real GW detections, the one on the 8th of June, 2017 matches your example. This has been determined to be two black holes with masses of about 12 and 7 solar masses. You can see on the frequency spectrum below that the signal first becomes visible at about 30Hz, and rapidly (over about one second) increases to 200Hz, at which point it abruptly ends.

(Image from Ligo Collaboration, via Wikipedia)

enter image description here

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  • $\begingroup$ thanks a bunch @James K. It's a great annotation. $\endgroup$ Aug 4, 2023 at 9:52

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