Revised as web pages, June 1997
My wife, after studying Weather Channel data on the sun, moon, and tides, one day early in a New Year asked me why the latest sunrise occurred later in the year than the shortest day. The original paper form of this note was the result. A similar enquiry to the "Last Word" column of the New Scientist prompted me to reformat the note as this WWW document.
The main effect in shifting latest sunrise several days after the shortest day of the year is an oscillation of the time of day when the sun reaches its highest elevation in the sky. This time oscillates about noon throughout the year in a sine wave with an amplitude of 8.8 minutes (at latitude 45 degrees) and with a period of 6 months. At the solstices and equinoxes, highest elevation is at noon. A graph of this sine wave is shown as the dotted blue line in the figure.
Sunrises and sunsets each day occur equally before and after the time of sun's highest elevation. If, for the moment, we redefine morning and afternoon to be those intervals, then at the winter solstice, the length of morning is minimum and so changes very little for a few days. But although the time of sun's highest elevation is noon at the solstice, that time gets later quite quickly and shifts sunrise to be later for a few days. After that few days, morning begins to get longer and overwhelms the 8.8 minute sine wave so we get earlier sunrises again. These effects can be seen in the same figure, where the solid red line is the variation in time of sunrise through a year and incorporates both the effect of change of length of morning and shift of the time of sun's highest elevation.
Why does the time of sun's highest elevation oscillate around noon?
Mainly because the Earth's axis is tilted with respect to its orbit
around the sun. Fix a frame of reference at
the center of the Earth,
aligned with the axis of the Earth, but rotating uniformly around that
axis only once per year, not once per day. In that frame,
the Earth rotates uniformly once per 24 hours,
and the sun
mainly goes up and down by 23 degrees once per year. But the sun
side to side
a little, and this gives the oscillation in time of sun's highest
elevation each day around noon.
The equation is complicated so simplifying it by leaving the expressions for the Sun's vector components in unevaluated form, gives us an easy check of sunrise and sunset times at the winter solstice.
The complicated equation can be given a simple exact form which shows that sunrise and sunset occur equally before and after the time when the sun is highest in the sky at point P. This time is obviously near noon and is calculated from our vectors P and S. Thus, as P rotates through the day, it gets nearest to S when the sun is highest in the sky. At this nearest place, the angle between P and S is smallest, and this in turn occurs when the azimuthal angles of P and S are the same in our frame of reference, i.e. and are equal.
The rearranged equation can then be approximated to get a simple form for the time of sunrise and sunset which is the sum of a sine term and a cosine term. A graph of the time of sun's highest elevation with respect to noon, and the time of sunrise is given.