Sunrise….Sunset on GG’s poor astronomy

(Minor update, GG added his own graphics to the end)

Another solstice, another reminder of the screwy ways that sunrise and sunset don’t align nicely with the solstice.  GG has written on this a few times before, but is guilty of being lazy.  A post over at Category Six (Weather Underground) examines the same misalignment but points to another web page where an effect GG overlooked–the variable changes in the rate of change of the solar noon due simply to the Earth’s tilt–is brought in as significant. GG’s posts have focused on how the solar day varies because of the ellipticity of the Earth’s orbit, but these posts argue that the length of a solar day (high noon to high noon) is longer at both solstices, not just the December one.

So a quick check is to look at the changes in solar noon near each solstice; if it is just ellipticity of orbit, we’d expect long solar days near one solstice and shorter solar days near the other. Here in Boulder (using timeanddate.com) solar noon on June 1 is 12:58 pm (20 days before the solstice), on June 21st (solstice) it is 1:02 pm and on July 11 is 1:06 pm–so over those 40 days, the solar day was 8/40’ths of a minute (or 12 seconds) longer than 24 hours, and it is quite symmetrical about the solstice. On Dec 1 it will be 11:50 am, on Dec 21 (solstice) it is 11:59 am and on January 12 it will be 12:08 pm, so over the forty days at the winter solstice the length of the solar day is longer by 18 minutes, so each solar day was 18/40 minute longer than 24 hours, or 27 seconds longer each day.

We can do the same calculation for equinoxes.  Around the spring equinox it is 10 minutes shorter than 24 hours for the 40 days centered on the equinox, and in the fall it is 14 minutes shorter, so on average the solar day is short by 12 minutes/40 days or 18 seconds/day.

So we see the effect of ellipticity accounts for the 15 seconds/day difference between the solstices (it is a bit larger if you compare between perihelion and aphelion), but the 38 seconds/day difference in solar day between solstices and equinoxes is simply because the rate of change of solar day varies through the year even in a circular orbit, as discussed in the posts above.

Earliest sunrise is 7 days before the summer solstice here in Boulder, but the earliest sunset comes 14 days before the winter solstice–that difference is due to ellipticity, but the average offset of about 10 days is due to the tilt.

So GG’s explanation was mostly touting the wrong feature. Oops. Sorry. Shows how attractive a nice (but wrong) story can be.

It took GG a bit to really get his head around this, so here is an alternate to the postings linked above. Imagine that the Earth is fixed with respect to the stars.  If it went around the Sun in a circular orbit, then the sun would move 1/365th of the way around the Earth.  For simplicity, let’s pretend the year is 360 days long.  So in one day, the Sun moves 1° along the circle describing the points directly under the Sun. In ten days, 10°.  If the Earth’s pole pointed normal to its orbit, all would be well, because the circle of points below the Sun would be the equator: 10° along the circle is 10° of longitude everywhere.

But the Earth is tilted, so we might look at something like this if we look down from directly above the ecliptic (i.e., normal to the plane of Earth’s orbit):

The outer ring of spokes is where the Sun is every 10 days of our slightly fake 360 day year (assuming a circular orbit). The outer circle of the globe is again where the Sun is overhead–but now it is well south of the equator at the December solstice and well above it at the June solstice. But where the lines of longitude hit they no longer match up with the spokes. If you look closely at the 10 days from Dec 21 to Dec 31, you see the Sun moved over more than 10° of longitude:

This is saying that 10 solar days were longer than 10 “rotation” days.  And if we look at the September equinox, we see the opposite trend: 10 solar days cover a smaller number of degrees of longitude (it is a bit harder as the lines of longitude are angled here):

Put another way, each solar day the Sun moves 111 km along the circle (which is the circumference of Earth).  When the circle is near the tropics, though, one degree of longitude is about 111 km  * cos (23°) = 102 km. The Sun, moving due west, overshoots its target and is shining down on a point about 9 km west of where it should. Now the Earth rotates through 15° of longitude or about 1665 km an hour at the equator, so 9 km is 9/1665 x 60 minutes x 60 seconds or about 19 seconds too far–which is exactly what we saw above from the sun tables.*

At the equinoxes, the path of the Sun overhead is now angled about 23° from east-west as it crosses the equator.  So the Sun moves to the west only 111 km x cos (23°) = 102 km, but at the equator one degree of longitude is 111 km, so it now is 9 km east of where it belongs, and the solar day should be about 19 seconds shorter than the rotation day.

It is really simple once you get it, and GG is embarrassed that he overlooked this before and didn’t check the numbers the way he has done it above.  Well, better late than never.

* OK, that was kind of an accident where approximations canceled.  On the Earth with 365.25 days in a year, the Sun would go through 40,070 / 365.25 =109.7 km each day.  At 23° N or S, 1/365.25th of the way around a line of latitude would be 101.0 km, so the Sun overshoots by 8.7 km. At 23 N or S, the Earth moves at a rate of 1515 km/hr, so 8.7 km would be 20.7 s, which would be the maximum movement of the Sun’s sub solar point relative to where it should be. There’s probably some care that should be taken with sidereal rotation rates and what not, but we’re pretty much there given the accuracy of the tables we used up above–they only give solar noon to within a minute.