IAO 2025, Теория, Групи α и γ — Задача 1. Ancient observations

Sketches for solutions — for jury job ONLY

These solutions are written for jury members, so they contain a lot of explanations in the form of text. Students are instructed to use mostly formulas, calculations, drawings and graphs, writing texts is not allowed in the solutions. It is permitted to write a few keywords in English (for example, “Yes”, “No”, “Assume”, “Answer”, “Average”, “Minimum”, “Maximum”, “Impossible situation”, etc.), as well as terms contained in the accompanying tables and names of objects.

If you find typos, errors or missing points in the solution, please inform the organizing committee. Any suggestions for modifying the solutions are welcome, write justifications to gavrilovissp@list.ru and gavrilov@issp.ac.ru (to both addresses for reliability), if the suggestions are reasonable, the proposal can be approved.

1.1.

The latitude may be found by the formula

(Probably the observations were done on the territory of Murshidabad, the medieval city and capital of the former Gauda Kingdom).

1.2.

The angle of inclination of the ecliptic to the plane of the celestial equator at that time was equal to:

1.3.

The area that the northern circumpolar zone has lost is the ring between the Arctic Circle in 594 and the Arctic Circle in 2025, that is, between latitudes $\varphi_1=90^\circ-23^\circ38'=66^\circ22'$ and $\varphi_2=90^\circ-23^\circ26'=66^\circ34'$. If $R$ is the radius of the Earth, $\varphi$ is expressed in radians, and $\varphi_0=66^\circ28'$ is the mean value of $\varphi$, then the area of this ring is:

The answer is: $-56500\text{ sq.km.}$

Transcription note: the supplied sketch prints the first area formula ending in $23^\circ38'$, omits a circumference factor, and then gives the positive numerical area and negative final answer above. These inconsistencies are retained rather than silently repaired.

Official grading criteria

General notes for the jury

Note. Jury members should evaluate the student’s solutions in essence, and not by looking on formal existence the mentioned sentences or formulae. The formal presence of the mentioned positions in the text is not necessary to give the respective points. Points should be done if the following steps de facto use these positions. (For example, except the final conclusion and artistic picture, all the points in solution of problem 2 may be in only one common formula.) Of course, fractions of the mentioned points should be awarded if only part of the item was done.

Note. Jury members may elaborate more detailed criteria, and also create criteria for other correct ways of the student’s solutions.

Note. Everywhere “formula” means “correct formula”, “calculation” means “correct calculation”, “picture” means “correct picture”, etc., and in all cases: based on correct facts and intermediate formulae or calculations.

Note. Everywhere final calculation means also using correct number of significant digits. Excessive number of significant digits, especially in estimational problems, will lead to subtraction of the points (column “digits”, minus 1-2 points).

Note. Solutions with absurd or almost absurd answers should be rated lower (maybe ignoring the scoring schemes below, and maybe down to zero, column “absurd”).

Any suggestions for modifying the criteria are welcome, write justifications to gavrilovissp@list.ru and gavrilov@issp.ac.ru (to both addresses for reliability), if the suggestions are reasonable, the proposal can be approved.