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2.13 The two angles

 

Frank L. Preuss

 

The following image is to demonstrate the difference, which is discussed here:

Die2Winkel

With the example of Cairo it is to be shown here, how two angles, which play a role in the examination of the position of the sun, are to be differentiated.

It is the difference of the inclination of the three coloured lines compared to the inclination of the two dotted black lines.

The circle shows the apparent celestial sphere.

The centre of the circle is an axis, which goes through the city of Cairo. Before Cairo is the west and behind Cairo is the east. It is therefore a sight from the west towards the east and in the centre is Cairo.

The horizontal line is the horizontal surface of the earth. To the left is north and to the right is south, or the south point of the firmament. The vertical line is a vertical line in Cairo. On top is the zenith and below it the nadir.

Since the circle presents the celestial sphere, the horizontal line is also the equator of the celestial sphere.

The red line presents the apparent orbit of the sun at the time of the high point of summer, therefore on the 21. June, at solstice. The full line is that part of orbit, which is visible during the day. The dotted part the nocturnal.

The green line is the sun orbit at the time of equinox, therefore at 20. March and 23. September.

The blue line is the orbit of the sun at the high point of winter, therefore at 22. December, at solstice.

The points where the full coloured lines touch the circle, present the sun at midday, and the points where the dotted coloured lines touch the circle present the sun at midnight.

The transitions of the full coloured lines to the dotted colours lines present the points of sunrise and sunset.

The diagonal continues black line is the axis about which the apparent sun orbit revolves. Where the diagonal continues black line cuts the celestial sphere, to the left at the top, is the North Pole, and, to the right below, the South Pole.

The angle between the full coloured lines and the horizontal always remains the same for Cairo. It is angle δ, Delta, which results from the degree of latitude on which Cairo is. Delta is the fourth letter of the Greek alphabet.

The declination δ gives the angle difference of the celestial body from the equator of the celestial sphere.

The dotted black lines are those lines, which go from the centre, therefore from the city of Cairo, to the sun.

The angle between the dotted lines and the horizontal is the angle φ, Phi. The angle does not remain the same as the angle δ, but changes. Phi is the twenty first letter in the Greek alphabet.

The angle φ is the angle between the horizontal and the sun at midday, therefore when the sun is in the highest position during the course of one day.

The angle δ is the same for all places on a parallel. It is the inclination of the orbit of the sun to the horizontal. The angle results from the vertical minus the number of the parallel. The parallel for Cairo is 30o. Therefore the angle is 90o - 30o = 60o

The angle φ is the midday elevation of the sun above the south point. For places in the southern hemisphere it would be the north point.

In Cairo the following values result for φ:

On the 20.03. the midday elevation is 90o - 30o         = 60o.

On the 21.06. the midday elevation is 90o - 30o + 23,5o = 83,5o.

On the 23.09. the midday elevation is 90o - 30o         = 60o.

On the 22.12. the midday elevation is 90o - 30o - 23,5o = 36,5o.

The difference between the two angles therefore is the starting point of the respective angle.

The first starting point, the starting point of the angle δ, is the intersection of the horizontal, therefore of the celestial sphere, with the sun orbit.

The second starting point, the starting point of the angle φ, is the intersection of the horizontal, therefore of the celestial sphere, with the vertical, therefore with the line between the zenith and the nadir, the city of Cairo.

The difference between the two angles one can also describe so that with both angles the horizontal is the one side of the angle and the line to the sun the other side of the angle, only that the line to the sun in the first case starts from the point, which results from the intersection of the line from the north point to the south point with the line from the sun at midday to the sun at midnight, but in the second case from the centre of the celestial sphere.

At the time of the equinox the two lines to the sun coincide with one line.

The inclination of the sun orbit remains the same the whole year, therefore does not change with the seasons. The centre of the sun orbit just shifts; at the equinox it is in the centre of celestial sphere and towards summer it shifts away from the equator; therefore the sun comes closer, and towards winter it shifts towards the equator, the sun therefore moves away.

The angle of the sun orbit towards the horizontal is therefore simply 90 degrees minus the degree of the parallel.

When for example the angle is measured, by which the sun rotates in one hour, then this should be done at equinox. At equinox the position of the observer is in the plane of the orbit of the sun. When the sun is away from equinox, the centre of its orbit is also away from the observer. The centre of the plane is moved away from the plane, and the plane becomes the mantle of a cone. The greater the distance, the smaller the angle for one hour.

The angle of the mantle of the cone can get as great as 23.5o.

 

^

 

This is the end of "2.13 The two angles"
To the German version of this chapter: 2.13 Die Jahreszeiten in den Tropen

 

 

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