IAO 2014, Теория, Група β — Задача 4. Hydroplanet

Solution

When passing the border between the airless space and water, rays are reflected:

As a result the rays go steeper after refraction. Due to the total reflection effect, only those rays reach observers at the outer surface of the ocean, which are directed, in the water medium, so that their angle with respect to the normal to the water surface is below the critical value that occurs for $\sin\alpha=1$, i.e. for $\alpha=90^{\circ}$.

For low depth of the ocean, this is the largest zenith distance from which rays reach observers. When increasing the ocean’s depth, this critical zenith angle increases because of the surface curvature. Drawing a plot, it is easy to find that light from the horizon line begins to come in the case of the minimal ocean depth that satisfies the condition:

4.1. Answer: $H=0.334R$.

4.2. It appears from the same plot that the central star on the sky of the Hydroplanet passes, from its rise to its set, the angle

In hour units, this is $17^{\mathrm h}29^{\mathrm m}$.

4.3. It appears from the same plot that it is the angle $90^{\circ}-\beta$ that is analogous to the refraction angle at the horizon for such a planet.

Source note: the key prints 262.12° in 4.2; its printed β=48°33′ gives 262°54′ for 360°−2β. The printed angle and time are retained.

Official grading criteria

Common part

• Understanding nature of effect, total reflection effect — 2 pt.
• Correct drawing (also showing total reflection effect) — 1 pt.
• Using refractive index of water n=1.334, calculation sinβ and β — 1.5 pt.

4.1. Answer H=0.334R — 1.5 pt.
4.2. Calculation and correct answer — 1 pt.
4.3. Picture, calculation and correct answer — 1 pt.

Jury note: evaluate the solution in essence rather than formal presence of sentences/formulae; award points when subsequent steps de facto use the required positions.