## Why doesn't the earliest sunset occur on the shortest day of the year? (Advanced)

I noticed this past December that the sun started setting later in the evening several days before the winter solstice. Why doesn't the earliest sunset occur on the shortest day of the year? Is the fact that the Earth is at perihelion in January have anything to do with the phenomenon?

Our clocks are synchronized with the "mean solar day", which is the motion of the Sun if the tilt of the Earth's axis (or obliquity) is zero and the orbit of Earth were perfectly circular (ie. zero ellipticity). So, first let us consider the case when the obliquity and ellipticity are zero. Then, the ecliptic (the plane of the solar system and the path of the Sun in the sky with respect to the stars) will coincide with the celestial equator. As the clock is perfectly synchronized with the Sun, it will always be on the meridian at 12:00 pm.

But actually the Earth's axis is tilted at 23.5 degrees. Hence, the ecliptic is tilted with respect to the celestial equator by 23.5 degrees. So what happens? Consider the fall equinox point. If the obliquity were zero, from day to day the Sun would move along the celestial equator (this is the motion due to Earth's revolution around the Sun not due to the Earth's rotation on its axis), but since the obliquity is actually 23.5 degrees, the Sun will move at an angle of 23.5 degrees to the celestial equator, or it will move both south and east.

Now, compare how much east the Sun has moved in the two cases (the case where obliquity is zero and the case where obliquity is 23.5 degrees). One can see that the Sun has moved more east in the case where obliquity is zero. Hence, the Sun will be on the meridian (meaning local noon) before the clock has indicated 12:00 pm since the Sun has moved less eastward as it would have if the Earth's obliquity is zero. Since the clock is always synchronized to the "mean Sun", this error (between local noon and the clock's 12:00 pm) will keep growing.

Now, as the Sun moves progressively south, note that the longitudinal lines of right ascension get more cramped together. Also, note that near the winter solstice point, the motion of the Sun is along the declination of -23.5 degrees, which means that the Sun is moving purely eastward. Combining the two facts, one finds that near solstice, the Sun moves eastward faster than what it would have if the obliquity were zero. Hence, at some point, the difference between local noon and the clock's noon will diminish and reduce to zero. This happens exactly at the solstice point. Beyond the solstice point, the Sun is still moving eastward faster than the "mean Sun". Hence, local noon will happen after the clock's noon.

As the Sun passes towards higher declination as it goes from winter solstice to spring equinox, the effect above reverses and the local noon and clock's noon coincide again at the equinox point. Thus, due to obliquity alone,

• local noon and the clock's noon will coincide at solstices and equinoxes.
• After equinox, the clock's noon will occur after local noon.
• After solstice, the clock's noon will occur before local noon.
• The maximum deviation between the clock's noon and local noon turns out to be about 9 minutes and 40 seconds.

Hence due to obliquity alone,

• Local noon will occur successively later between early November and early February.
• Local noon will occur successively earlier between early February and early May.
• Local noon will occur successively later between early May and early August.
• Local noon will occur successively earlier between early August and early November.

Hence, as one approaches winter solstice, sunrise wants to become earlier and sunset wants to become later. But as local noon is becoming later than the clock's noon, this shifts the sunrise and sunset time later. The combination of these two gives rise to the later sunset before actual winter solstice, even though the day is still becoming shorter. Note that we have considered only obliquity and have assumed that the orbit is circular.

Now, let us see what happens due to ellipticity of Earth's orbit alone. Hence, let us set obliquity to be zero. If the orbit of Earth were circular, then the time between two local noons will be the same throughout the year. But because the orbit of Earth is elliptical, it moves faster than average near perihelion (near January) and slower than average near aphelion (near July).

Draw a picture of the Sun in the center with the Earth going around it. A star is at infinite distance in comparison. Now, draw the Earth at one location and mark the location of noon. Now, as the Earth rotates, it also moves in its orbit; mark a location on the orbit where the point you marked on the Earth has noon again. This period is the mean solar day. (As a side fact, note that the mean solar day is different from the time taken for Earth to rotate once around itself). Now, if the Earth were moving faster than average, it would have moved more in its orbit in the same amount of time, and so the location you marked on Earth will not have noon. The Earth would have to rotate more for the location to have noon. Hence, local noon will occur later than the clock's noon near perihelion. The opposite effect happens near aphelion when Earth moves slower than average, and hence local noon happens earlier than the clock's noon. Here also, errors add progressively around perihelion and subtract progressively around aphelion.

Hence, due to ellipticity alone,

• Local noon occurs progressively later between October and April.
• Local noon occurs progressively earlier between April and October.
• The maximum deviation between local noon and clock's noon turns out to be around 8 minutes.

Now let us combine the two effects and see what happens near winter solstice: Obliquity and ellipticity both conspire to make local noon later than clock's noon near winter solstice. Hence, this shifts sunrise and sunset timings later (note that without this effect, sunrise will become later and sunset will become earlier). The overall effect is that sunset starts becoming later before actual solstice even though the day is becoming shorter. 