Съдържание

Условие 12.1 12.2–12.4 12.5–12.6 12.a 12.b 12.c Protractor Решение

IAO 2021, Практически тур — Задача 12. Mars motion as seen from the Earth

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2021

IAO 2021 — Practical round (IRAO, α/β/γ) · 12 т.

Practical round. Problems to solve

International Remote Astronomy Olympiad. Milan, Italy, 5–13 XI 2021. Task 12: groups α, γ (β crossed out); task 13: α, β, γ; task 14: β, γ. The question pages carry the round label M&I. The text and matching solution/criteria identify the practical round.

Printed source details

The introduction calls the objects in task 13 open clusters, while question 13.1 says “globular cluster”; both wordings are retained. Table 14.a prints “rm” for Europa and has handwritten “km” for Ganymede and Callisto. The official solution retains “Sun-Eart”, the table-13.b heading refers to questions 12.1–12.3, and the final diagram prints “12th Nov, 021”. These are source defects, not transcription corrections.

Practical Round - Sketches for solutions. For jury job ONLY

These solutions are written for jury members, so they contain some explanations in the form of text. Students are instructed to use only formulas, calculations, drawings, graphs and filling tables. Writing texts is not allowed in the solutions.

Reference inventory

The separately archived constants sheet is labelled Theo (Russian/English). It was inspected as reference context; it supplies no additional practical task or requested answer form. The protractor sheet contains three identical copies; one complete copy is reproduced with task 12.

Условие

Assuming that the orbits of Earth and Mars are circular, picture 12.a shows such orbits to scale; positions of the Sun and of the Earth and Mars when Mars is at opposition are also represented.

In table 12.b you will report distance from Earth and angular data for Mars:

Column 1: day of observation (in the interval ±105 days from time of opposition $t_0$ ).
Column 2: distance to Mars from Earth, to be given in km.
Column 3: position angle $\alpha$ of Mars, to be given in degrees.
Column 4: day/days in which Mars is in quadrature with the Sun as seen from Earth (to be indicated by the symbol Q) or it appears “stationing” (to be indicated by the symbol S).

12.1

12.1 Taking into account the sidereal periods of the Earth and Mars with a good approximation and starting from the initial configuration where Mars is in opposition (i.e. along the up-down direction the orbit representation), compute the positions of the two planets every 15 days for a time interval of ±105 days and draw them on picture 12.a.

12.2–12.4

In picture 12.a draw the lines connecting the positions of Earth and Mars at the same epoch and, for each epoch, measure:

12.2 the Earth-Mars distance $D$ and report it, in km, in column 2 of table 12.b,

12.3 the position angle $\alpha$ of Mars as seen from the Earth with reference to the up-down direction (i.e. line connection Earth and Mars when the Mars is at opposition) and report it, in degrees, in column 3 of table 12.b.

12.4 Infer in which of the positions of Earth and Mars at the same epoch you have drawn Mars would be in quadrature with the Sun and mark them (one, two or more) with the symbol Q in column 4 of table 12.b.

12.5–12.6

In picture 12.c report the distance (in au, Y axis) Mars-Earth $D$ as a function of the position angle $\alpha$ (in degrees, X axis).

12.5 Infer the days (one, two or more with the precision allowed by the data) when Mars appears stationing in the sky during the considered time interval (±105 days) and mark them with the symbol S in column 4 of table 12.b.

12.6 Provide an estimation of the duration of the apparent retrograde motion of Mars $T(\mathrm{retrograde})$ and write it, in days, in the box at the bottom of picture 12.c.

12.a

Printed orbit drawing and complete plotting grid, rotated clockwise for reading

12.b

1 · Day	2 · $D$ (km)	3 · $\alpha$ (°)	4 · Quadrature / Stationing
$t_0 − 105^{\mathrm{d}}$
$t_0 − 90^{\mathrm{d}}$
$t_0 − 75^{\mathrm{d}}$
$t_0 − 60^{\mathrm{d}}$
$t_0 − 45^{\mathrm{d}}$
$t_0 − 30^{\mathrm{d}}$
$t_0 − 15^{\mathrm{d}}$
$t_0$ (opposition)
$t_0 + 15^{\mathrm{d}}$
$t_0 + 30^{\mathrm{d}}$
$t_0 + 45^{\mathrm{d}}$
$t_0 + 60^{\mathrm{d}}$
$t_0 + 75^{\mathrm{d}}$
$t_0 + 90^{\mathrm{d}}$
$t_0 + 105^{\mathrm{d}}$

12.c

Protractor

Решение

12.1

The sidereal period of Mars, with a good approximation, can be obtained from the III Kepler law considering that orbits of Earth and Mars are in scale. Starting from the configuration where Mars is in opposition, the positions of the two planets every 15 days are as follows:

Dated positions of Mars and Earth with lines and angles — 12.a - positions of Mars and Earth

12.2–12.4

12.2 By measuring in the graph the Earth-Mars distance $D$ , it then can be converted to km considering that the Sun-Eart distance is equal to 1 AU

12.3 Once the position of the two planets every 15 days are known, the angle $\alpha$ of Mars as seen from the Earth with reference to the up-down direction can be measured (for clarity only the value for $t_0-105^{\mathrm d}$ is shown in picture 12.a)

12.4 Looking at the positions of Earth and Mars at the same epoch, Mars would be in quadrature (or nearby) for $t_0-105^{\mathrm d}$ and for $t_0+105^{\mathrm d}$

12.b - solution for questions 12.2, 12.3, 12.4 and 12.5

1 · Day	2 · $D$ (km)	3 · $\alpha$ (°)	4 · Quadrature / Stationing
$t_0 − 105^{\mathrm{d}}$	$169 \cdot 10^6$	-15	Q
$t_0 − 90^{\mathrm{d}}$	$153 \cdot 10^6$	-7.5
$t_0 − 75^{\mathrm{d}}$	$133 \cdot 10^6$	-1
$t_0 − 60^{\mathrm{d}}$	$118 \cdot 10^6$	4.5
$t_0 − 45^{\mathrm{d}}$	$102 \cdot 10^6$	8	S
$t_0 − 30^{\mathrm{d}}$	$89.7 \cdot 10^6$	8	S
$t_0 − 15^{\mathrm{d}}$	$80.8 \cdot 10^6$	4
$t_0$ (opposition)	$77.8 \cdot 10^6$	0
$t_0 + 15^{\mathrm{d}}$	$80.8 \cdot 10^6$	-4
$t_0 + 30^{\mathrm{d}}$	$89.7 \cdot 10^6$	-8	S
$t_0 + 45^{\mathrm{d}}$	$102 \cdot 10^6$	-8	S
$t_0 + 60^{\mathrm{d}}$	$118 \cdot 10^6$	-4.5
$t_0 + 75^{\mathrm{d}}$	$133 \cdot 10^6$	1
$t_0 + 90^{\mathrm{d}}$	$153 \cdot 10^6$	7.5
$t_0 + 105^{\mathrm{d}}$	$169 \cdot 10^6$	15	Q

12.5–12.6

12.5 Once the distance Mars-Earth as a function of the position angle is plotted in Figure 12.c, the days, with the precision allowed by the data, when Mars appears stationing in the sky are inferred

12.6 The motion in the sky of Mars appears retrograde among the points of stationing

12.c - Position angle of Mars as function of Earth-Mars distance

Mars distance in AU versus elongation angle, with retrograde loop

$T(\mathrm{retrograde})=75$ days

Practical Round - Basic criteria. For work of Jury

Note. Everywhere terms as “positioning” means “correct positioning”, “values” means “correct values”, “identification” means “correct identification”, etc., ...

N	Description	Points (up to)
1	Positioning of Earth and Mars on the graph	4
2	Values of the distance	2
3	Values of the α angle	2
4	Identification of the days of quadrature	1
5	Identification of the days of stationing	1
6	Plot of position angle of Mars as function of Earth-Mars distance	1
7	Time interval of retrograde motion of Mars (the point will be given only if the reported value, 75 days ± 15, is inferred from correct assumptions; it will not be given if it results by wrong assumptions)	1

Теми и трудност

Трудност: неоценена.

Слънчева система · Небесна механика · Планетни конфигурации · Закони на Кеплер

Предпоставки: Разширена олимпиадна подготовка

Оригинал в Архива: prac-EN.pdf · официални решения: Practical_Round_IRAO_2021_Solutions.pdf · допълнителни данни: Practical_Round_IRAO_2021_Criteria.pdf · допълнителни данни: protractor.pdf · допълнителни данни: Ia21_TABLE_Constants.pdf