Answer
0.2 m.
Work Step by Step
Use the definition of the radian. Calculate the angle subtended by the star-planet system by dividing the orbital radius by the distance to the star from the Earth.
$$\theta=\frac{r}{d}$$
Use equation 25–7 to relate the minimum aperture diameter D for an Earth-based telescope to resolve the objects, and the angle of separation, $\theta$. The observations are made at a wavelength $\lambda$.
$$\theta=\frac{1.22\lambda}{D}$$
Combine the two equations and solve for D.
$$D=\frac{1.22\lambda d}{r}$$
$$=\frac{1.22(550\times 10^{-9}m)(4\;ly)(9.461\times10^{15}m/ly)}{(1AU)(1.496\times10^{11}m)}=0.17m\approx0.2m$$
