Makar-Sankranti
- Round
- Theory
- Points
- 10
- Level
- Silver
- Taught in
- Section 2.7
Adapted from IOAA 2025 (Mumbai), theory problem 2. We have restated it in our own words and kept the original data, adding the length of the sidereal year. The solution was last checked on 17 October 2026.
The Indian festival of Makar-Sankranti is celebrated when the Sun appears to enter the zodiacal region of Capricorn, measured against the stars. It now falls around 14 January. Long ago it fell on the winter solstice, which we take to be 21 December. Take the precession of the equinoxes as 1° every 71.6 years and the sidereal year as 365.2564 days.
(a) In what year did the festival last fall on the winter solstice? [3 points]
(b) In 2006 the Sun entered Capricorn at 11:50:13 local time in Mumbai on 14 January. On what date and at what time did it enter Capricorn in 2013? [3 points]
(c) The festival is celebrated on the date of the first sunset after the Sun enters Capricorn. Taking sunset in Mumbai in January as 18:30, give the date of the festival in each year from 2006 to 2013. [4 points]
- (a) From 21 December to 14 January is 24 days, in which the Sun moves
- (b) The entry into Capricorn comes round every sidereal year, 365 d 6 h 9 min 13 s. Seven sidereal years are
- (c) Year by year the entry moves 6 h 9 min 13 s later, and a day earlier after each leap day. It falls at 11:50 on 14 January 2006, 17:59 on 14 January 2007, 00:08 on 15 January 2008, 06:17 on 14 January 2009, 12:27 on 14 January 2010, 18:36 on 14 January 2011, 00:45 on 15 January 2012 and 06:54 on 14 January 2013. The festival is on the date of the first sunset at 18:30 after the entry. In 2011 the entry comes after sunset, so the festival moves to the next day.
Answer(a) Around AD 331. (b) At 06:54:44 on 14 January 2013. (c) The festival falls on 14 January in 2006, 2007, 2009, 2010 and 2013, and on 15 January in 2008, 2011 and 2012.
This problem also appears among the olympiad problems of Section 2.7, Precession and nutation, which teaches the ideas it uses. Equation and figure numbers refer to the course.