IAO 2014, Теория, Група α — Задача 6. Brightest stars

Solution

Due to precession of the Earth’s axis, the North celestial pole moves in a circle around the North ecliptic pole, making one circle approximately every ~26000 years (the value can be clearly seen in the sky map of precession). The time till the CL century AD is approximately half of this period, and the North pole will change its position by approximately $2\varepsilon\approx47^{\circ}$ in the direction of the brightest star in Draco constellation, γ Dra or Eltamin (in Draco the order of α-β-γ… is not the traditional one that starts from the brightest star, this fact is also clearly seen from the supplement materials). That is, in the CL century the visible part of the celestial sphere will be shifted about 47° in the direction to about the current 18h RA.

Let us list the brightest stars (for example, twice more of them than the required four) and see whether they may be seen in Cholpon-Ata in the CL century. Now we can see stars with $\delta>-(90^{\circ}-\varphi)+5^{\circ}\approx-42^{\circ}$ for any RA. After 13000 years the limit will move, $\delta>-42^{\circ}+47^{\circ}=5^{\circ}$ for RA6h and $\delta>-42^{\circ}-47^{\circ}=-89^{\circ}$ for RA18h (according to current coordinates). The situation is more complicated for other right ascensions. Anyway, the center of invisible area of the sky will be at the point (6h,−43°) and its radius will be $\rho=\varphi+5^{\circ}=48^{\circ}$. By easy drawing we can find that

That is, we will lose from the visible stars Sirius, Procyon and Rigel, and get α Centauri. The four brightest stars in the night sky of Cholpon-Ata in the CL century AD are: α Centauri, Arcturus, Vega and Capella.

Official grading criteria

• Understanding nature, precession — 1.5 pt.
• Position of North celestial pole after 13000 years — 1 pt.
• Region of sky never rising above mountains — 1.5 pt.
• Speculation about Sirius, Canopus, Toliman, Arcturus, Vega, Capella — 3 pt.
• Final 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.