IAO 2014, Теория, Група α — Задача 5. Argali

Solution

Argali means the constellation Aries. The Sun is in this constellation in late April – early May, now it is almost in the opposite position on the ecliptic, so Aries is perfectly visible at night now. The brightest star in the constellation is Hamal, α Ari, its right ascension is $02^{\mathrm h}07^{\mathrm m}$. The upper culmination of a celestial body occurs when the sidereal time is equal to its right ascension. Sidereal time is equal to mean solar time at the date of autumnal equinox, and then begins to outstrip it with the speed of $1^{\mathrm d}/365.2422$ per day (about $3^{\mathrm m}56^{\mathrm s}$ per day). 23 days have passed between the night of September 22/23 (autumnal equinox in 2014) and October 15–16 (the next 24 hours). The difference between mean solar time and sidereal time is now:

So now sidereal time $02^{\mathrm h}07^{\mathrm m}$ corresponds to

of mean solar time. Cholpon-Ata is located in 13° to the West of the meridian 90° (the standard meridian for UT+6 Timezone), so the mean solar time here is behind the local time UT+6 used in Kyrgyzstan by $13^{\circ}\times4^{\mathrm m/\circ}=52^{\mathrm m}$. So, to get the answer according to time UT+6, we have to add these 52 minutes, and we have:

of Kyrgyz time. So, tonight it will be a favorable period for the mountain sheep to observe the constellation of Aries.

The altitude of α Ari in upper culmination may be found using the formula:

Also we should calculate the altitude of α Ari in lower culmination:

Source note: the lower-culmination line prints the same symbolic expression as the upper line, but substitutes −90°+φ+δ. The original mismatch is retained.

Official grading criteria

Solution: 7 pt in total, including:

• Argali means constellation Aries — 1 pt.
• Position of constellation in October, Aries now perfectly visible at night — 1 pt.
• Understanding brightest star is Hamal, α Ari — 0.5 pt.
• Correct calculation of time of upper culmination — 3 pt, including: difference with September 23 (or other reference point) — 1 pt; Cholpon-Ata 13° west of 90° meridian — 1 pt; correct use of RA and final result — 1 pt.
• Correct altitude of upper culmination — 1 pt.
• Correct altitude of lower culmination — 0.5 pt.

Artistic picture of Argali — 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.