Gravitational Wave Power

Gravitational Wave Power

Einstein's quadrupole formula gives the power (luminosity) radiated as gravitational waves by a compact binary in a circular orbit of separation $r$:

$P_{\rm GW} = \frac{32}{5} \frac{G^4}{c^5} \frac{(m_1 m_2)^2 (m_1+m_2)}{r^5}$

The steep $r^{-5}$ dependence means power increases enormously as the binary shrinks. At merger, a stellar-mass binary briefly outshines the entire visible universe in gravitational-wave luminosity.

Orbital Decay Rate

This radiated energy must come from the orbital energy $E_{\rm orb} = -Gm_1 m_2/(2r)$. Differentiating with respect to $r$ and setting $P_{\rm GW} = -dE_{\rm orb}/dt$:

$\left|\frac{dr}{dt}\right| = \frac{64}{5} \frac{G^3}{c^5} \frac{m_1 m_2 (m_1+m_2)}{r^3}$

The orbit shrinks slowly at large $r$ and catastrophically fast near merger.

Earth–Sun as a Sanity Check

For the Earth–Sun system ($m_1 = 5.972 \times 10^{24}$ kg, $m_2 = 1.989 \times 10^{30}$ kg, $r = 1.496 \times 10^{11}$ m), the GW power is only about 200 W — negligible compared to the Sun's $3.8 \times 10^{26}$ W electromagnetic luminosity. The orbit decays by ~$10^{-20}$ m/s, imperceptible on human timescales.

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. Return the positive magnitude for power and decay rate (energy is being lost, separation is decreasing). Return the negative value for orbital energy (bound orbits have $E < 0$).