Question #43560

A transparent medium occupies the region $y \ge 0$. The refractive index of the medium varies with temperature $T$ as $n(T) = \sqrt{\frac{T_0}{T}}$, where $T_0$ is a constant. The temperature profile in the medium is given by $T(y) = \frac{T_0}{1 + ky}$, where $k = \frac{1}{\sqrt{3}} \text{ m}^{-1}$. A narrow light ray traveling in the $xy$-plane from vacuum ($y < 0$, $n=1$) enters the medium at the origin $(0,0)$. The incident ray makes an angle $\theta_0 = 60^\circ$ with the positive $y$-axis and has a positive $x$-component of velocity. An observer is located in the medium at a height $H = 2\sqrt{3} \text{ m}$. When the observer looks back along the tangent to the ray at $y=H$, the ray appears to originate from a virtual point source on the $x$-axis at $x = X_{\text{app}}$. Find the value of $X_{\text{app}}$.