Question #43739

A binary star system consists of two stars, $S_1$ and $S_2$, emitting light predominantly at wavelengths $\lambda_1 = 500 \text{ nm}$ and $\lambda_2 = 505 \text{ nm}$, respectively. The true angular separation between the stars is $\phi_0 = 3.22 \times 10^{-5} \text{ rad}$. An astronomer observes this system using a telescope of diameter $D$. To correct for certain atmospheric effects, a thin dispersive prism of refracting angle $A = 0.2 \text{ rad}$ is placed directly in front of the telescope objective. The refractive index of the prism material follows the Cauchy relation $n(\lambda) = n_0 + \frac{b}{\lambda^2}$, where $b = 1.25 \times 10^{-15} \text{ m}^2$. The prism is oriented such that its dispersion reduces the apparent angular separation between the two stars. Assuming the stars are just resolved when their apparent angular separation equals the Airy disk radius ($\theta = 1.22 \lambda / D$) of the longer wavelength component, what is the minimum required diameter $D$ of the telescope objective? (Use binomial approximation for small changes in wavelength).