IAO 2003, Теория — Задача 3

• Центърът на Слънцето е приблизително $64^\prime$, около $1^\circ$, над хоризонта.

Official English sketches for solutions — theosol.doc.

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 $4'+35'+25' = 64'$ or about one degree.

A picture is necessary to describe this.

Бележка за източника: решението изисква рисунка, но самият официален файл не съдържа такава; не е добавена измислена фигура. Във формулата е запазено отпечатаното H вместо h.