Съдържание

IAO 2005, Теория, практически тур и наблюдения — Задача 1. The Bear. · Groups α β

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IAO 2005

X Международная астрономическая олимпиада · X International Astronomy Olympiad · 25.X. – 2.XI.2005 · Beijing, China · Китай, Пекин · 25 октомври 2005 г. · 8 т.

Внимание: Бележка към темата

This photographed English/Russian mixed-round booklet includes nine distinct theory tasks with native solution sketches, practical αβ7 and β8 without keys, and three bilingual observational tasks without keys. The α practical task 8, planetary data table, observational sky map and refraction drawing grid referenced in the source are absent. The Sun sketch provides only methods and no calculated distance or magnitude. Jury-result companions print 8 points per theory task, 10 per practical task and 20 for the observational round together, with 88 per cohort; those two cohort totals are not a total for this combined 14-task edition. The 12-minute allowance applies only to all observational jobs. Printed source errors are preserved: the Andromeda sketch uses exp(17.1/5)=2630; the α5 magnitude-difference sign is positive; β4 switches to Δα in one sentence; β5 refers to α2; the refraction formula prints 273°; the planetary-nebula question describes an explosion. Native diagram lines and spectrum peaks are faint in the original photograph.

Theoretical round. Problems to solve · General note

Groups α β

General note. Maybe not all problems have correct questions. Some questions (maybe the main question of the problem, maybe one of the subquestions) may have no real sense. In this case you have to write in your answer (in English or Russian): «situation is impossible - ситуация невозможна». Of course, this answer has to be explained numerically or logically.

При решении каждой задачи можно использовать данные прилагаемой “Таблицы данных о планетах”

Data from the “Table of planetary data” may be used for solving every problem.

Theoretical round. Sketches for solutions · Тезисы решений задач теоретического тура

Русский · English

Условие

The White Bear (whom was already met in the texts of the VII, VIII and IX International Astronomy Olympiads) has returned from Crimea to the North Pole and decided to observe a sunset. Calculate how long this sunset lasts. The solution has to include a picture with an image of the Bear on the North Pole; necessary sizes or angular sizes should be in the picture. Assume that the Earth is spherical. Recollect for yourself the necessary information about the White Bear.

But can the Bear prolong the observation of sunset, not moving from the Pole? For what time?

Отговори

Решение

For an observer on the North Pole the sunset happens once a year, in time close to the day of the autumnal equinox. For the observer on the North Pole the angle between the ecliptic and the horizon is equal to $\varepsilon=23.5^{\circ}$ . During the sunset the Sun lowers to the angle equal to its apparent angular diameter, that is $\alpha=32'$ . During this time the Sun transits angular distance on ecliptic is equal to $\varphi=\alpha/\sin\varepsilon$ . Thus, the duration of sunset is $\tau=\varphi/v=\alpha/(v\cdot\sin\varepsilon)$ , where $v$ is the speed of the Sun on the ecliptic, equal to $v=360^{\circ}/365.25^{\mathrm d}=0.986^{\circ}$ per day. Thus,

\tau=(32/60)^{\circ}/(360^{\circ}/365.25^{\mathrm d})/\sin23.5^{\circ}=1.36\ \text{days or}\ 32.5\ \text{hours}.

Yes, during the sunset the Bear will come to twist the head 490 degrees!

As for the prolongation of the observation of the sunset, the Bear can use the effect of depression of the physical horizon, if it rises on the back feet. That is, he can begin to observe sunset lying on ground (pardon, on ice), where the horizon dip can be disregarded, and to finish, costing on back feet. For the last case the depression of the physical horizon makes

\Delta\varphi=\arccos(R/(R+h))\approx((R+h)^2-R^2)^{1/2}/R\approx(2h/R)^{1/2}\quad\text{(in radians)}.

Where $R$ is the radius of the Earth, and $h$ is the height of eye level above the surface of the Earth (ice). For the height of an eye of a White Bear about 2 m above the surface, the horizon depression is about $3'$ . So, the Bear can prolong observation of sunset almost by 10 %, namely,

\Delta\tau=(3/60)^{\circ}/(360^{\circ}/365.25^{\mathrm d})/\sin23.5^{\circ}\approx3\ \text{hours}.

Observational round. Problems · Задачи наблюдательного тура · Groups α, β

English

Maximum total time for all jobs is 12 minutes.

Русский

Максимальное время выполнения задания – 12 минут.

Теми и трудност

Оценена трудност: 4/5.

Небесна сфера и сферична астрономия · Светлина и астрономически измервания · Сезони и път на Слънцето · Небесна сфера · Ъглов размер

Предпоставки: Гимназиална подготовка, Разширена олимпиадна подготовка

Оригинал в Архива: iao2005.pdf · официални решения: iao2005.pdf