Binary Inspiral Time

Binary Inspiral Time

How long does it take two compact objects to spiral together from an initial separation $r_0$? Integrating the orbital decay rate $|dr/dt| = (64/5) G^3 m_1 m_2 (m_1+m_2)/(c^5 r^3)$ from $r_0$ to $0$ gives the Peters formula for a circular orbit:

$T_{\rm merge} = \frac{5\,c^5}{256\,G^3} \cdot \frac{r_0^4}{m_1 m_2 (m_1+m_2)}$

This $r_0^4$ dependence is dramatic: halving the initial separation cuts the merger time by 16.

Separation as a Function of Time

Running the integral in reverse, the separation remaining at time $t$ before merger is:

$r(t) = \left(r_0^4 - \frac{256}{5} \frac{G^3 m_1 m_2 (m_1+m_2)}{c^5}\,t\right)^{1/4}$

At $t=0$ this gives $r_0$; at $t = T_{\rm merge}$ the bracket vanishes and $r = 0$.

Physical 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.