= Solution
{c}
The <geodetic precession> per orbit is
$$
\boxed{\Delta\psi=2\pi\left(1-\sqrt{1-\frac{3}{400}}\right)
\simeq0.0236063\,\mathrm{rad}=1.35254^\circ.}
$$
The orbital position and <four-velocity> repeat after a full orbit, so the direction to distant Earth in the instantaneous <rest frame> repeats. The freely transported <gyroscope> does not: its beam axis has turned through $\Delta\psi$ relative to the original aiming direction. This applies to any initial in-plane spin direction by linearity of <parallel transport>, not only the radial initial condition used in part (b). The beam's angular diameter is $1^\circ$, giving a half-angle of $0.5^\circ$. Thus \b[Earth lies outside the beam after one orbit]; comparing the offset with the diameter would already show a miss, but the correct acceptance angle is the half-angle.
The initial radial ray must be outward from the <black hole> for a direct path to Earth. A precisely inward radial <null geodesic> falls into the <black hole> and cannot provide the assumed initial communication. For <outgoing near-radial Schwarzschild light deflection>, the angular correction is especially small. If a static observer at $R$ measures a small emission angle $\psi$ from the outward radial direction, its <impact parameter> is $b=R\sin\psi/\sqrt{f(R)}$. The outgoing <null geodesic> gives
$$
\Delta\phi=\int_R^\infty\frac{b\,dr}{r^2\sqrt{1-f(r)b^2/r^2}}
=\frac{b}{R}+O\left(\frac{b^3}{R^3}\right)
=\frac{\psi}{\sqrt{f(R)}}+O(\psi^3).
$$
Relative to flat propagation, <curvature> changes the small pointing angle and beam radius by the fraction $f(R)^{-1/2}-1\simeq0.00251$. At an offset of $1.35^\circ$ this is only about $0.0034^\circ$; the beam's $0.5^\circ$ radius changes by about $0.0013^\circ$. The exactly radial central ray has zero angular deflection. This is not the geometry of the usual $4GM/(bc^2)$ scattering formula, which assumes a ray coming from infinity, passing the <black hole>, and returning to infinity.
<Relativistic aberration> also distinguishes a coordinate-radial light ray from a beam aimed radially in the moving crew's frame. Here $v/c=1/\sqrt{398}\simeq0.0501$. Successful initial aiming includes that fixed correction; the same correction applies at the repeated orbital position. More precisely, an outgoing static-radial ray initially has crew-frame angle $\psi'_0=-\arcsin(v/c)$ from the radial axis. The local small-angle aberration factor about that aim is $d\psi/d\psi'=\gamma=\sqrt{398/397}$; including the outgoing propagation factor gives $\gamma/\sqrt f=1/q\simeq1.00377$. Thus the asymptotic miss is about $1.358^\circ$ and the beam half-angle about $0.502^\circ$, with the same conclusion. The small distortion of the outward beam cannot correct the much larger accumulated <geodetic precession>.
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