Gravitational Wave Frequency

Gravitational Wave Frequency

Gravitational waves from a compact binary are emitted at twice the orbital frequency:

$f_{\rm GW} = 2 f_{\rm orb}$

The orbital frequency follows from Kepler's third law for a circular orbit of separation $r$:

$f_{\rm orb} = \frac{1}{2\pi} \sqrt{\frac{G M_{\rm tot}}{r^3}}$

Combining these:

$f_{\rm GW} = \frac{1}{\pi} \sqrt{\frac{G(m_1+m_2)}{r^3}}$

ISCO: Peak Frequency at Merger

The inspiral ends at the Innermost Stable Circular Orbit (ISCO), located at $r_{\rm ISCO} = 6GM_{\rm tot}/c^2$ for a Schwarzschild (non-spinning) black hole. The GW frequency at ISCO is the highest frequency emitted before merger:

$f_{\rm ISCO} = \frac{c^3}{6^{3/2}\,\pi\,G\,M_{\rm tot}}$

For two 30 $M_\odot$ black holes (like GW150914), $f_{\rm ISCO} \approx 73$ Hz — squarely in LIGO's sensitive band.

Inverting for Orbital Separation

Given $f_{\rm GW}$, the corresponding orbital separation is:

$r = \left(\frac{G(m_1+m_2)}{(\pi f_{\rm GW})^2}\right)^{1/3}$

Constants (define inside each function)

• $G = 6.674 \times 10^{-11}$ m³ kg⁻¹ s⁻²
• $c = 299792458$ m/s

Your Task

Implement the three functions below. Use $M_\odot = 1.989 \times 10^{30}$ kg for tests.