IAO 2003, Теория, Група B — Задача 3
VIII International Astronomy Olympiad, Stockholm, Sweden, 2–8.10.2003 — Theoretical round. Problems to solve — Group B · 2 октомври 2003 г.
Внимание: Бележка към темата
Solutions for problems 1-3 are shared 'Group A,B' solutions as printed in the source; they were matched to the Group B problem statements which are identical in content for these three problems.
Бележка
Data from the "Table of planetary data" may be used for solving of every problem.
Условие
The White Bear from the previous International Astronomy Olympiad is still sitting at the North Pole. But this year a follower has appeared – a Penguin sitting at South Pole. Recently, after the ending of polar night, the Penguin observed the sunrise. What did the Bear observe at this time? Draw what the White Bear saw at the moment when the Penguin observed exactly half of the solar disk on the horizon. Assume that the Earth is spherical. The answer should be explained by drawing a figure with an image of the Bear on North Pole; necessary sizes or angular sizes should be in the picture. Recollect for yourself the necessary information about the animals.
Отговори
64 arcmin
The White Bear sees the Sun's centre about 64' (≈1°) above the horizon due to refraction and horizon depression, not sunset.
Решение
It seems that since the points are opposite, the situation should be absolutely symmetric: the White Bear should observe sunset at this time. But this is almost not correct. A few days more will pass till the sunset. So the answer "The White Bear observed the sunset" is not correct!
It is necessary to take into account two important circumstances.
First, for any observer, the physical horizon is slightly lowered. For example, if a person stands on a flat surface, the depression of physical horizon she sees is about 2.5'. The depression of physical horizon is easy to calculate: if R is the radius of the Earth, and h is the height of eye level above the surface of the Earth, the horizon dip is equal to
Penguins may be different. For an estimation let us take the sizes of an Imperial Penguin, whose eyes' height above the surface of the Earth is about 1 m. The horizon depression is then about 2'. The White Bear, as it is mentioned, is sitting, so height of his eyes above the surface of the Earth will be about 1 m too, and so the horizon depression is about 2' too (If it stands up, this height would be more than 2 m and about 3' accordingly).
This means that if, as seen by the Penguin, the centre of the solar disk is on horizon, this centre already in 2' below mathematical horizon. So at the opposite point of the Earth – at the North Pole – it is 2' above the mathematical horizon. Accordingly, the sitting Bear-observer sees it already in 4' above physical horizon, i.e. a quarter of radius of the Sun.
But horizon depression is actually not the main effect in this problem.
Secondly, there is refraction of light beams in the atmosphere. The value of a refraction depends slightly on weather conditions, but on average contributes about 35' at the horizon. So at the moment when the centre of the solar disk became visible on the horizon at the South Pole, at the opposite point of the Earth – at the North Pole – the Sun has still not begun to set at all. In fact the refraction at the South Pole yields 35', and at North Pole about 25' (not 35', since the Sun not at the horizon but higher). And this effect is the main one in the given problem.
Deduction: the White Bear sees centre of the Sun above the horizon, at approximately or about one degree.
A picture is necessary to describe this.
Оригинал в Архива: theo.doc · официални решения: theosol.doc