Winter Solstice Sunset and Sunrise Times

 The winter solstice arrives every year on December 21 or 22.  However, for mid-northern latitudes the date of earliest sunset is almost two weeks earlier, on December 7.  Likewise, the date of latest sunrise is later, about January 5.  Why?

Short Answer:  For most of the year the rapidly changing number of Daylight Hours determines whether sunset is earlier or later than the day before.  But as we approach the Solstice on December 21 the Solar Day 24 (amount of time between successive zenith crossings of the Sun, approximately 24 hours) becomes more important and pushes the time of sunset later.  Larger than average Solar Day 24 wins the battle with shrinking Daylight Hours on December 7.  A similar interaction combines to push the date of latest sunrise to January 5.  

Details:

If the Earth’s orbit around the Sun was a perfect circle, and the Earth’s axis was not tilted with respect to its orbit, then latest sunset, earliest sunrise, and the solstice would all happen on the same date.  Likewise if every day we set our clocks to noon when the sun crossed the zenith, then sunrise, sunset, and solstice would all be simultaneous.

But the Earth’s axis is tilted 23.5 degrees with respect to the plane of the earth’s orbit.  This results in a large variation in the amount of Daylight Hours we get during the year in mid northern latitudes.  The Earth’s orbit is slightly elliptical, and that produces a small variation in the “Solar Day 24”, the amount of time between successive crossings of the zenith by the Sun.  For mid northern latitudes the Daylight Hours can fluctuate during the year by 5 hours (300 minutes), whereas the Solar Day 24 varies by only 1 minute, 30 seconds on either side of 24 hours. Therefore for most of the year the rapidly changing Daylight Hours determines whether today’s sunset is earlier or later than yesterday’s.

Approaching the Solstice, the Numbers

In the Fall the Daylight Hours are getting smaller rapidly.  The difference between Daylight Hours on two successive days can be as much as 3 minutes.  This means that sunsets are getting earlier each day by as much as 1.5 minutes, since the 3 minute difference is shared between sunrise and sunset times.  As we approach the Solstice, the daily difference in Daylight Hours begins to shrink, and in early December it is only .4 minutes.  The Daylight Hours curve is flattening, preparing for its turnaround on the Solstice.

Shortening Daylight Hours implies earlier sunset times.  However, there is another factor that affects sunset times and that is the Solar Day 24, the amount of time between successive zenith crossings (“solar noon”) by the Sun.   As mentioned before, the Solar Day 24 varies throughout the year by up to 30 seconds.  In early December, the Solar Day 24 is about 24 hours plus .4 minutes.  The Earth is approaching its closest point to the sun on January 3.  As a result it is speeding up, and that means that the Earth has to rotate a little more each day to put the Sun at zenith, since the Earth rotates towards the east and the Earth’s revolution around the sun is to the east. If Solar Day 24 was the only thing that mattered, then sunset would be later each day by .4 minutes in early December.

Daylight Hours Changes vs the Length of the Solar Day

So we have two opposing forces:  Daylight Hours are decreasing, which would cause earlier sunsets, but the Solar Day 24 is more than 24 hours, which would cause later sunsets. 

As we approach winter solstice Daylight Hours is shortening but not so dramatically.  On November 1 the day to day difference of Daylight Hours/2 is -1.07 minutes.  But on December 1 the day to day difference is only -.59 minutes.

Meanwhile the solar zenith time is later each day by about .4 mins due to the longer Solar Day.  On December 1 the zenith time is .37 minutes later, but Daylight Hours/2 has shortened by .59, so sunset is .22 mins earlier.  But as we step along through the next few days the day to day differences converge.  On December 6 the Solar Day 24 is .42 mins and the Daylight Hours/2 difference is .46, so sunset is still earlier. 

On December 8 the Solar Day 24 is .44 mins longer than 24 hours, and the Daylight Hours/2 difference is .41, so now the Solar Day 24 dominates and sunset is later than the previous day by about .03 minutes.  As long as the Solar Day 24 is longer then 24 hours, which implies later zenith times day to day, it will cause later sunsets *before* the solstice since the day to day difference for Daylight Hours/2 goes to zero as we approach solstice.  



In figure 1 above, the blue line is the Solar Day 24, the time between successive zenith crossings of the Sun, shown as the amount of minutes relative to 24 hours.

The red line is the day to day difference in the amount of Daylight Hours divided by 2.  It is positive but decreasing as we approach early December.  When Solar Day 24 becomes larger than successive differences in Daylight Hours/2 the wall clock time of sunset will start to get later.  The crossover happens on about December 7. 

After the solstice the Daylight Hours will start to increase again which will push the sunset to earlier times *after* the Daylight Hours difference is greater than the Solar Day 24, which happens on about January 5.

The Equation of Time

Most discussions on this topic focus on the Equation of Time (EOT).  As we’ve discussed, the Solar Day 24 varies throughout the year by about 30 seconds above and below 24 hours.  Imagine you had a watch that was 30 seconds slow every day.  After 10 days your watch would be 300 seconds or 5 minutes slow.  The cumulative error is 5 minutes.  The EOT is just a table of this cumulative error for every day in the year.  If your watch then started running fast by 30 seconds per day, after 10 days the EOT would be back down to zero.  

If the solar day is 24 hours and .3 minutes, the .3 mins will be subtracted from the EOT for that day.  It is the daily difference of the EOT vs the daily difference of Daylight Hours that will determine the latest sunset.  But the daily difference of the EOT is just the Solar Day 24 (minus 24 hours) for that day.  So it seems simpler to concentrate on Solar Day 24 rather than EOT.  Here is Figure 1 recast with EOT.

 


EOT is influenced by both the elliptical nature of the Earth’s orbit and the tilt of the Earth’s axis.  Figure 3 shows the contribution of both:

 


 

Axis Tilt Also Affects EOT

Previously we talked only about the contribution of the elliptical orbit to EOT.  We also see that axis tilt comes into the equation, but this time in a more subtle way.  As the Earth approaches the Solstice the Sun’s path gets lower and lower in the sky.  As the US Naval Observatory puts it:  “The geometry of the sky is such that the Sun's east-west rate is slower when the north-south rate is large and faster when the north-south rate is small (in such a way that the combined rate from the Earth's orbital motion is constant, considering this effect alone). For timekeeping purposes, it is only the east-west rate that matters. The cumulative effect of the east-west rate variations is that the true Sun falls behind then moves ahead of the fictitious mean sun in several cycles over the course of a year. The part of the Equation of Time due just to this effect is zero at the equinoxes and solstices, and can reach ±10 minutes at other times of the year. If the Earth's axis were perpendicular to its orbit, this effect would disappear (as would our seasons).”

 

Asymmetric Change of Sunrise vs Sunset

A reader noticed that sunrise times did not change as fast as sunset times during January.

Here is a table for the first two months for Tulsa, OK.  Sunrise times decrease more slowly in January than sunset times increase, but that effect largely disappears in Feb.  

       Sunrise Diff  Sunset  Diff

Jan 1   7:35             5:20

Feb 1   7:24     11   5:51    31

Mar 1   6:53    31    6:19   28

The solar day (SD24) is the length of time it takes the Sun to go from zenith to zenith.  It varies a little over the year, 24 hours plus or minus .5 minutes.  On January 1 the SD24 is about 24 hours and .4 minutes.  The SD24 is longer then its average of 24 hours because the earth sped up a little as it approached perihelion (closest to the Sun) on January 3.  The earth has to rotate a little bit further to the east to put the Sun on the zenith, day to day.

If the SD24 were the only thing that mattered, then both sunrise and sunset would be later each day by .4 minutes in early January.  However, the amount of daylight hours is lengthening each day by about a minute in early January and 2 minutes on Feb 1.  So with regard to sunrise times, the two effects are "fighting" each other -- SD24 wants later sunrises, and daylight hours wants earlier sunrises.  But with regard to sunset times, SD24 and daylight hours both imply later sunsets.  The effect is additive for sunsets, so later sunset times happen faster than earlier sunrise times, where the two effects offset each other.  

By February 10 the SD24 has reduced to exactly 24 hours, and then it goes slightly under 24 hours for the rest of February, rendering it neutral with respect to sunrise/set times in February.  Now daylight hours dominates, and you can see from the table above that in Feb sunrise goes earlier by 31 minutes, and sunset later by 28 minutes, so both times are changing at about the same rate.  The difference in daylight hours during Feb is about 60 minutes.  So it makes sense that sunrise and sunset both change by about 30 minutes in February with SD24 temporarily out of the equation.

References:

https://aa.usno.navy.mil/faq/rs_solstices

https://aa.usno.navy.mil/faq/eqtime

https://gml.noaa.gov/grad/solcalc/calcdetails.html


Questions or Comments;

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