Sunrise Delayed after the Solstice
The winter solstice has passed, but the sunrise continues to delay every morning. According to the webpage of National Astronomical Observatory of Japan, on the solstice of December 22, the sunrise in Tokyo was at 6:47am. Since then, it continues to delay to 6:51am on January 2 - 13 in 2019. It begins to get earlier starting on January 14. Why does this happen ? Utsusemi on December 26, 2014 explained why. But use of 3D geometrical illustrations made it difficult, and I myself take a long time to understand it again after some time. Well, I'll try again this time without using illustrations to explain easily only with texts.
[Assumptions of explanation]
1. The earth is assumed to revolve around the sun in the perfect circle. Actually it is elliptical and its effect is discussed at the end of this memo.
2. One year is assumed to be exactly 365 days. Let's represent 360 degrees/365 by .
3. How many times does the earth rotate in a year ? It is not 365 times but 365 + 1 times. We have 365 times of days and nights, but even if we assume no rotation and the sun would be always above Tokyo, the earth would rotate one time while it revolve around the sun.
4. The rotation axis of the earth is about 23 degrees tilted from the line perpendicular to the plane of revolution.
[Explanation]
1. In 24 hours, the earth rotates 360 + degrees around its rotation axis. Thus in a year it rotates 365 + 1 times. Namely it rotates degrees excessively every day, or 360 degrees excessively in a year.
2. The angle to see the sun from the center of the earth changes degrees every day on the plane of revolution. Those changes sum up to 360 degrees in a year to return to the same original angle.
3. The angle in 1. above and in 2. above are the same angle on different planes. When the angle around the rotation axis is mapped onto the plane of revolution, how many degrees will it be on the plane of revolution ? Let's represent this angle by '. [Namely, when a sheet of paper is folded to make degrees and place the crease onto the rotation axis, ' is the angle of two intersection lines between the paper and the plane of revolution.] The angle ' may be bigger or smaller than depending on the season, namely the relationships between tilting direction of the rotation axis and the direction of the sun.
4. If we assume that the rotation axis were perpendicular to the plane of revolution, = ' would result. If we tilt this perpendicular axis by 23 degrees to various directions, ' will become bigger or smaller than .
5. In the spring or fall equinox, the rotation axis is tilted 23 degrees perpendicularly to the direction of the sun. In these cases, ' > results. [If the axis were tilted 90 degrees onto the plane of renvolution, ' would become 90 degrees.n Namely, in comparison with the movement of the sun, the earth rotates excessively and today's sunrise and sunset would be earlier than yesterday's.
6. In the summer or winter solstice, the rotation axis is tilted 23 degrees towards or away from the sun. In these cases, ' < results. [If the axis were tilted 90 degrees onto the plane of renvolution, ' would become 0 degrees.n Namely, in comparison with the movement of the sun, the earth rotates insufficiently and today's sunrise and sunset would be later than yesterday's.
7. Thus, if we look at movement of the sun geocentrically, following things should occur;
(1) In the spring equinox as the peak and on days near the equinox, the sun moves earlier day by day.
(2) In early May, the sun turns around and the sun moves later day by day.
(3) In the summer solstice as the peak and on days near the solstice, the sun moves later day by day.
(4) In early August, the sun turns around and the sun moves earlier day by day.
(5) In the fall equinox as the peak and on days near the equinox, the sun moves earlier day by day.
(6) In early November, the sun turns around and the sun moves later day by day.
(7) In the winter solstice as the peak and on days near the solstice, the sun moves later day by day.
(8) In early February, the sun turns around and the sun moves earlier day by day.
(Note) Under the assumption of the perfect circle revotution, (2)(4)(6)(8) are correct. But because it is actually elliptical revolution, (4) is actually near the end of July, and (6) is early December.
8. Let's call the differences described above "Change A", which has two cycles in a year. In addition, we must consider "Change B", in which we have longer or shorter day-time in one cycle in a year. Right after the winter solstice, Change A makes the sunrise later and Change B makes the sunrise earlier. In this competition, Change A prevails in the beginning to make the sunrise later before Change B takes over to make it earlier.
Then, the effect of elliptical revolution should be considered. The perihelion, in which the earth comes nearest to the sun, occurs at the end of the year. As the distance becomes smaller, the revolution speed becomes higher according to the Kepler's second law. Therefore becomes larger and delay of the sun movement in and around the winter solstice becomes larger. In the aphelion, in which the earth comes farthest from the sun, at the end of June, becomes smaller and delay of the sun movement is smaller. END