Effect of the obliquity of the ecliptic / tilt of the Earth on the equation of time

Question

In attempting to answer the question “why aren’t changes in sunset and sunrise times symmetrical around the solstices ?” I’ve got stuck on being able to explain the effect of the obliquity of the ecliptic.

I understand the general idea of the equation of time and how the difference in clock time and solar time answer my original question, but I’m struggling to grasp how the tilt of the Earth contributes to the equation of time.

I’m trying to visualise the effect, and assuming it is to do with the point at which the sun is highest in the sky not being in the middle of sunset and sunrise (except at solsitces and equinoxes?). So, is this correct? Is there any good way of trying to visualise it ?

I’m a bit confused by the wikipedia entry https://en.wikipedia.org/wiki/Equation_of_time#Obliquity_of_the_ecliptic saying

1. “If this effect operated alone, then days would be up to 24 hours and 20.3 seconds long (measured solar noon to solar noon) near the solstices, and as much as 20.3 seconds shorter than 24 hours near the equinoxes”
2. “the inclination of the ecliptic results in the contribution of a sine wave variation with an amplitude of 9.87 minutes and a period of a half year to the equation of time. The zero points of this sine wave are reached at the equinoxes and solstices”. Is point 1 correct ?

Also found these links helpful to some extent, but haven’t fully explained it for me. https://web.archive.org/web/20150910174438/http://www.rmg.co.uk/explore/astronomy-and-time/time-facts/the-equation-of-time http://totaleclipse.eu/Astronomy/EOT.html

Answer 1

As the solar time Wikipedia page illustrates, we can split a solar day into a constant 23h56m to complete a rotation relative to the stars (a sidereal day) and a variable ~4m to compensate for one day's orbital motion around the Sun. The Sun's apparent motion around the equator determines how much more than 360$^\circ$ the Earth must rotate from one solar noon to the next.

But the Earth orbits the Sun in the ecliptic plane, which is oblique to the equator. If that orbit were circular, the Sun's ecliptic longitude would change at a constant rate (see Note below). One degree along the equator spans exactly 4m00s of right ascension; one degree along the ecliptic, measured around the equator, may span ~8% more or less than that.

At an equinox, the ecliptic crosses the equator at a 23.4$^\circ$ angle, and one degree of ecliptic longitude spans 3m40s of right ascension. The Sun appears to move slower than average around the equator, so the apparent solar day is slightly shorter.

At a solstice, the ecliptic is tangent to a $\pm$23.4$^\circ$ declination parallel, and one degree of ecliptic longitude spans 4m22s of right ascension. The Sun appears to move faster than average around the equator, so the apparent solar day is slightly longer.

Stellarium images around the March equinox and the June solstice, showing 30 degrees of ecliptic longitude and 2 hours of right ascension for comparison. Equatorial grid is blue, ecliptic grid is orange, ecliptic is yellow.

The equation of time is the cumulative sum of differences between mean and apparent solar day length. Like a mathematical function and its derivative, zero crossings of one should roughly correspond to maxima or minima of the other. Differences of a few seconds per day, several weeks in a row, add up to a few minutes.

The earliest/latest sunrise and latest/earliest sunset of the year do occur several days on either side of the summer/winter solstice. However, on any given day, sunrise, noon, and sunset are all behind or ahead of mean solar time by approximately the same amount.

Note: the Earth's orbit is not quite circular. Following Kepler's second law, the Sun's ecliptic longitude changes ~3% slower at aphelion and faster at perihelion, making the apparent solar day slightly shorter or longer respectively. This Wikipedia entry discusses it in more detail.