Gravitational Wave Strain

Gravitational Wave Strain

The gravitational wave strain $h$ is a dimensionless number that measures the fractional stretching of space. If a detector of length $L$ is stretched by $\delta L$, then $h = 2\delta L/L$. LIGO measures strains of order $10^{-21}$ — one part in a billion billion billion.

For a compact binary at luminosity distance $d$, with chirp mass $\mathcal{M}_c$ and instantaneous GW frequency $f$, the strain amplitude from the leading-order quadrupole formula is:

$h = \frac{4}{d} \cdot \frac{G\mathcal{M}_c}{c^2} \cdot \left(\frac{\pi G \mathcal{M}_c f}{c^3}\right)^{2/3}$

This grows during the inspiral as $f$ increases, reaching its peak at the ISCO frequency.

GW150914 Numbers

Distance Conversion

One megaparsec (Mpc): $1\text{ Mpc} = 3.0857 \times 10^{22}$ m.

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. The chirp mass and ISCO frequency formulae from previous lessons may be computed inline inside gw_strain_at_isco.