OWAO 2022, Теория — Задача 2. Mysterious Mercury

• August 27, 2022; Mercury was in Virgo (ecliptic longitude ≈ 183°).

The synodic period of Mercury is

Using Kepler’s third law, we obtain

The occultation of Mercury by the Moon will occur the day before the eclipse. Consequently, Mercury at this time is west of the Sun and is observed in the morning.

Let us calculate the time interval between the eastern and western elongations of Mercury. Mercury has to orbit the angle $2\arccos\frac{a_M}{a_\oplus} = 2\arccos 0.387 = 134^{\circ}$ relative to the Earth.

It will pass this angle in $\frac{134^{\circ}}{360^{\circ}}\cdot S = 43^{\mathrm{d}}$, as $S$ is period of relative motion of Mercury and Earth, and Mercury orbits around the Sun faster than the Earth.

Thus, the last greatest eastern (evening) elongation of Mercury occurred 43 days before October 9, on August 27, 2022. We are lucky, this is an absolutely accurate answer, despite the actual ellipticity of Mercury’s orbit!

At that time Mercury was $\arcsin 0.387 \approx 23^{\circ}$ east of the Sun, so it has the ecliptic longitude of $\approx 160^{\circ} + 23^{\circ} = 183^{\circ}$. It corresponds to Virgo.

Marking Scheme:

• Concept of synodic period — 1 pt.
• Calculation of synodic period — 2 pt.
• Description or geometric scheme of the situation — 2 pt.
• The moment of the greatest evening elongation — 2 pt.
• Identification of the constellation — 3 pt.