 Solution

Using the source's rounded solar mass, each hole has $m\simeq 3\times 10^{31}\mathrm{kg}$. Its Schwarzschild radius isThus $\boxed{r_s\sim 10^5\mathrm{m}}$ at order-of-magnitude accuracy. Using the more accurate solar mass would give about $9\times 10^4\mathrm{m}$.

The stated orbit is not a consistent separated-hole weak-field binary. Its nominal center-to-center separation is only $20\mathrm{km}$, below even the above single-hole radius. Applying two isolated Schwarzschild spacetime descriptions and a circular Newtonian orbit at that separation is not controlled. Likewise an estimate ${\bar{h}}_{11}^{\prime }\sim r_s/r$ at $r=10\mathrm{km}$ is of order one or larger, contradicting the small-perturbation assumption. The intended near-source statement is therefore a strong-field scale,not a reliable prediction from linearized gravity at the specified orbit. The exact radiative component is also gauge dependent near the source.

One can still estimate the intended distant gravitational wave strain scale. A radiative amplitude decreases as $1/D$ in the wave zone. A compact strong-field source has a characteristic length of order $r_s$, giving a dimensional strain estimate $h_{\mathrm{far}}\sim r_s/D$, up to mass, orientation and efficiency factors of order unity. This is also the strong-field dimensional scale suggested by far-zone quadrupole strain; it is not an application of the slow-motion formula to the inconsistent orbit.

With the supplied conversion,For a ring radius $a=4\times 10^3\mathrm{m}$, part (c) gives a single principal semiaxis displacement amplitudeThe difference of the two principal semiaxes has twice this amplitude. This is a conditional dimensional estimate of a physically plausible merger signal at that distance; the supplied close circular orbit does not define a self-consistent waveform or a precise amplitude.