IN THE SOUTHERN HEMISPHERE:
The longer days of sunlight between Sept. 21 and March. 21, centered around Dec. 21, causes summer weather.
The shorter days of sunlight between March 21 and Sept. 21, centered around June 21, causes winter weather.
The effect is of course reversed in the Northern Hemisphere.
An interesting and essential point, though, is the lag time involved. In the south, there is the same amount of daylight on Nov 21 as there is on Jan 21. But it is much warmer in January than November. This is because of the damping nature of the atmosphere and the ground, which retain heat or cold. Instead of looking just at the amount of daylight on any one day, we must look at the amount of daylight averaged over the last few weeks. In this case, one month before Nov 21 is Oct 21, which has much less daylight. One month before Jan 21 is Dec 21, which has the most daylight. Hence the warmer weather in January.
On the equinoxes (March 21 and Sept 21) the tilt, extended out from the poles, is tangent to the orbit around the sun. The effect is that the sun's path across the sky follows the equator, and day and night is of equal length in both northern and southern hemispheres.
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