Figure 61: Paths traced by the shadow of the nodus of the times of the equinoxes and solstices in latitude 34°N. The gnomon is a vertical pin of the length shown in the insert at the top.
As the Sun moves across the sky from east to west each day, the shadow which is cast on a horizontal surface by the upper tip of the gnomon traces a path from west to east. At the times of the equinoxes, this path is a straight line. At other times of the year, the path can be a circle, ellipse, parabola or hyperbola. When the Sun is north of the equator during the summer months, this path is concave toward the gnomon which casts the shadow, but during the remainder of the year, it is concave away from the gnomon. The three situations are shown in Figure 61. The upper curve is formed during the winter solstice and the lower curve is formed during the summer solstice. The line in between the two curves is formed during both the equinoxes.
While the lines are shown for only four days of the year (the two solstices and two equinoxes), there would be a different but distinctive path on any other dates, depending solely on the declination of the Sun. These other lines would all lie within the limits of those shown in Figure 61. This is because the Sun's maximum and minimum declinations are at the summer and winter solstices respectively. Thus, the shadows formed during the solstices will serve as the two extremes of the shadows cast over a year. If we know the Sun's declination on any day, we can draw the line of declination for that day, so that our dial will serve as a calendar to tell the day of the year as well as a timepiece to tell the time of day.
With the help of a few animations, we shall explain how we get the curves in Figure 61.
Click here for the animations.
At the equator on both the equinoxes, the Sun's declination is 0°. The sun rays over the course of a day form a plane with the tip of the gnomon, which intersects the ground to give a line.
When the Sun goes below the horizon, no shadow is formed. At the equinoxes for all latitudes, the Sun is above the horizon for half of the day and below the horizon for the other half of the day. Hence, we get 12 hours of day and night each.
At the equator during the summer solstice, the sun rays over the course of the day form a shadow cone with the tip of the gnomon, which intersects the ground to give a hyperbola. Similarly, during the winter solstice, the sun rays over the course of the day form a shadow cone with the tip of the gnomon, which intersects the ground to give another hyperbola on the opposite side of the gnomon.
Again, we note that we have 12 hours of day and night each since the Sun is above the horizon for half of the day and below the horizon for the other half of the day on both the solstices.
At latitude 45°N, on both the equinoxes, the Sun's declination is 0°. The sun rays over the course of a day form a plane with the tip of the gnomon, which intersects the ground to give a line.
At latitude 45°N, during the summer solstice, the Sun is at a declination of +23.5°. The sun rays over the course of the day form a shadow cone with the tip of the gnomon, which intersects the ground to give a hyperbola. During the winter solstice, the Sun is at a declination of -23.5°. The sun rays over the course of the day forms a shadow cone with the tip of the gnomon, which intersects the ground to give another hyperbola on the opposite side of the gnomon.
At latitude 66.5°N on both the equinoxes, the Sun's declination is 0°. The sun rays over the course of a day form a plane with the tip of the gnomon, which intersects the ground to give a line.
When the Sun is at declination +15°, the sun rays over the course of the day form a shadow cone with the tip of the gnomon, which intersects the ground to give a hyperbola. When the Sun's declination is -15°, the sun rays over the course of the day forms a shadow cone with the tip of the gnomon, which intersects the ground to give another hyperbola on the opposite side of the gnomon.
During the summer solstice, the Sun is at declination +23.5°. The sun rays over the course of the day form a shadow cone with the tip of the gnomon, which intersects the ground to give a parabola. However, the Sun is below the horizon for the whole of the winter solstice. Hence, we do not get any shadow on the other side of the gnomon.
Since the Sun is above the horizon for the whole of the summer solstice, we get 24 hours of daylight. On the other hand, for the whole of the winter solstice, the Sun is below the horizon and we do not get any sunlight.
At latitude 80°N on both the equinoxes, the Sun's declination is 0°. The sun rays over the course of a day form a plane with the tip of the gnomon, which intersects the ground to give a line.
When the Sun is at declination +5°, the sun rays over the course of the day form a shadow cone with the tip of the gnomon, which intersects the ground to give a hyperbola. When the sun's declination is -5°, the sun rays over the course of the day forms a shadow cone with the tip of the gnomon, which intersects the ground to give another hyperbola on the opposite side of the gnomon.
When the Sun is at declination +10°, the sun rays over the course of the day form a shadow cone with the tip of the gnomon, which intersects the ground to give a parabola. However, at declination -10°, the sun is below the horizon for the whole day. Hence, we do not get any shadow on the other side of the gnomon.
During the summer solstice, the sun rays over the course of the day form a shadow cone with the tip of the gnomon, which intersects the ground to give an ellipse. However, the Sun is below the horizon for the whole of the winter solstice. Hence, we do not get any shadow on the other side of the gnomon.
Again, the Sun is above the horizon for the whole of the summer solstice. Hence, we get 24 hours of daylight.
At the North Pole, on both the equinoxes, the Sun is moving along the horizon. Hence, we do not get any shadows on both days.
During the summer solstice, the sun rays over the course of the day form a shadow cone with the tip of the gnomon, which intersects the ground to give a circle. However, the Sun is below the horizon for the whole of the winter solstice. Hence, we do not get any shadow on the other side of the gnomon.
Since the Sun is above the horizon for the whole of the summer solstice, we have 24 hours of daylight.