OWAO 2022, Теория — Задача 1. Slow down and shine

• 0.75 mag

  Δm = 2.5 lg 2 = 0.75 m; the star is now farther away (r₂ = √2 r₁) and fainter.

Proper motion $\mu$ is related to the tangential velocity $V_\tau$ of the star and the distance $r$ to it:

Let $M_1$ be the initial position of the star, $M_2$ be the position of the star after 10 000 years. The tangential velocity $V_{\tau,i}$ is related to the total velocity $V$:

The ratio of the proper motions is equal to 0.5,

Using the law of sines, we express the ratio of the sine angles in terms of the ratio of the sides of the $\triangle M_1M_2S$:

Therefore,

So we conclude that $r_2 = \sqrt{2}r_1$. The illuminance is inversely proportional to the square of the distance, so the illuminance also halved. The change in magnitude is

Marking Scheme:

• Relation between proper motion and distance — 3 pt.
• Relation between angles and distances — 3 pt.
• Correct use of Pogson’s equation — 3 pt.
• Correct answer — 1 pt.