IAO 2025, Теория, βγ — Задача 6. Model of the Universe

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.

6.1.

In this model, the Sun makes 1 revolution around the Earth (more precisely, around the Earth–Moon system) relative to the sphere of fixed stars in 1 year. According to Kepler's Third Law, comparing this motion with the orbital motion of the Earth–Moon system:

Thus, the distance to the Sun in this model is:

The angular size of the Sun is $32'$. The physical diameter of the Sun in this model is:

6.2.

In this model, the base for measuring angles could only be the diameter of the Earth. If the accuracy of measuring angles in the sky at that time was about $\delta=10''$, then the distances could only be measured to those objects from which the diameter of the Earth is visible at an angle greater than this value, in other words, to those objects whose doubled daily parallax is greater than $\delta=10''$. That is, the distance up to

This means that the sphere of fixed stars can be located at any distance greater than 260 million km.

In our real Universe, this is 1.8 au, that is, the characteristic distances of the inner part of the Solar System (the order of the distances to the Sun, Mercury, Venus and Mars).

Answer: greater than 260 million km.

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.