Question #43232

An astronomical telescope consists of an objective lens and an eyepiece, both treated as thin lenses with vacuum focal lengths $f_0$ and $f_e$ respectively. The telescope is placed in a vacuum, but its cylindrical tube of length $L$ (the distance between the objective and eyepiece) is filled with a transparent gas of non-uniform density. The gas has a radially varying refractive index $n(r) = n_0 (1 - k r^2)$, where $r$ is the radial distance from the optical axis, $n_0 > 1$, and $k > 0$ is a small constant ($k r^2 \ll 1$). Assuming paraxial rays and that the ray trajectory $r(x)$ inside the tube satisfies the differential equation $\frac{d}{dx} \left( n \frac{dr}{dx} \right) = \frac{\partial n}{\partial r}$, determine the required tube length $L$ for the telescope to be in normal adjustment.