Model earth's atmosphere as a spherical shell \(100 \mathrm{~km}\) thick, with index of refraction \(n_{t}=1.00000\) at the top and \(n_{b}=1.00027\) at the bottom. Is a light ray's final angle \(\varphi_{\mathrm{f}}\) relative to the normal at the ground greater or less than its initial angle \(\varphi_{\mathrm{i}}\) relative to the normal at the top of the atmosphere? (Earth's radius is \(R=6400 \mathrm{~km}\).) Assume the ray strikes the upper atmosphere at a \(45^{\circ}\) angle.