Abstract
The earliest sunset of the year is easily taken to fall on the winter solstice, the shortest day. What this paper shows is that the gap between sundial and clock (the equation of time) is the sum of two causes, the eccentricity of the orbit and the tilt of the ecliptic, and that around the solstice it changes by about $0.5$ minutes a day, so the earliest sunset moves half a month before the solstice and the latest sunrise half a month after. No new theorem or law is claimed.
First, the equation of time has two peaks and two troughs of different sizes——with eccentricity $0.0167$ and obliquity $23.44$ degrees, the maximum is $\mathbf{16.4187}$ minutes and the minimum $\mathbf{-14.2455}$ minutes, the smaller peak and trough are $+3.6841$ and $-6.5099$ minutes, and it passes through zero four times a year (Section 2).
Second, the equation of time is the sum of two causes——the eccentricity part has amplitude $\mathbf{7.6550}$ minutes ($7.6547$ from the first-order $2e$), the obliquity part $\mathbf{9.8664}$ minutes ($9.8634$ from the first-order $\tan^2(\varepsilon/2)$), and their sum matches the whole to within $4.1\times10^{-13}$ minutes (Section 2).
Third, taking the causes one at a time changes the shape——with obliquity alone the peak and trough are equal at $\pm9.8664$ minutes and there are four zeros a year; with eccentricity alone they are $\pm7.6550$ minutes with two zeros. Large and small peaks alternate because a yearly and a half-yearly period overlap (Section 3).
Fourth, turning the perihelion by $90$ degrees changes the shape, but by $180$ degrees does not——at $90$ degrees the extremes become $\mathbf{14.5793}$ and $\mathbf{-16.6434}$ minutes; at $180$ degrees nothing changes, because the obliquity part has a half-year period (Section 3).
Fifth, and this is the core. At latitude $35.7$ degrees north, the earliest sunset comes $15.61$ days before the winter solstice and the latest sunrise $16.07$ days after——the equation of time at the solstice is $+1.6951$ minutes and falls by $\mathbf{0.4975}$ minutes a day. Solar noon arrives later every day, so the shortest day and the earliest sunset fall on different days (Section 4).
Sixth, the separator is whether one looks at the length of the day or also at the time of its middle (solar noon)——day length alone bottoms out at the solstice; when solar noon moves, the extremes of sunrise and sunset separate onto different days (Section 5).
Keywords: equation of time, sunset, winter solstice, Kepler’s equation, obliquity of the ecliptic.
Scope (what is not claimed)
No new theorem or law is claimed——the equation of time, Kepler’s equation and the equation of centre, conversion from ecliptic longitude to right ascension, and the sunrise equation are all standard. The orbital elements are representative——eccentricity $0.0167$, obliquity $23.44$ degrees and longitude of perihelion $282.94$ degrees, with no secular change. Sunrise and sunset are computed from geometry alone——as in Paper 414, in local mean time at latitude $35.7$ degrees north, without refraction or the sun’s apparent radius. No calendar dates are given——days are counted from perihelion and from the solstice. The history of calendars and clocks is not treated. Relation to earlier papers: Paper 414 showed that an hour of the old unequal-hour system varies by a factor of $1.5$ with the season, and stated in its scope note: “the equation of time is not included; the difference between apparent and mean solar time is not the subject”——this paper takes that seat and puts the equation of time into the same sunrise equation at latitude $35.7$ degrees. Paper 535 derived the moon’s libration in longitude from the eccentricity, $2e$——the eccentricity part of the equation of time here comes from the same $2e$ (the equation of centre in Kepler’s equation). Paper 300 showed that sharing a root is decidable——the moon’s libration in longitude and the sun’s equation of time are two appearances of $2e$ met in separate places, with one source, the equation of centre; in that sense they share a root. What is added is splitting the equation of time into its two causes and checking each amplitude against its first-order form, setting out the shapes with one cause at a time and with the perihelion turned, finding the days of earliest sunset and latest sunrise counted from the solstice, and placing the separator on day length versus the time of solar noon.
1. Notation and the Question
Setting: let $L$ be the sun’s mean longitude, $\lambda$ its true longitude and $\alpha$ its right ascension. The difference between sundial time (apparent solar time) and clock time (mean solar time), the equation of time $E$, is
$$E=(L-\lambda)+(\lambda-\alpha)\approx-2e\sin M+\tan^2\frac{\varepsilon}{2}\,\sin2\lambda,$$the first term arising because the sun does not move uniformly along the ecliptic in a non-circular orbit (eccentricity $e$, mean anomaly $M$), the second because the ecliptic is tilted to the equator (obliquity $\varepsilon$).
Observation 1.1 the earliest sunset of the year is easily taken to fall on the winter solstice, the shortest day. If the day is shortest, sunset should be earliest——whether day length and sunset time bottom out on the same day is another question.
Question: does the earliest sunset fall on the winter solstice?
2. Two Causes
Computation 2.1: with eccentricity $0.0167$, obliquity $23.44$ degrees and longitude of perihelion $282.94$ degrees, Kepler’s equation was solved and the equation of time found through the year in steps of $0.01$ days.
| peak or trough (days from perihelion) | equation of time |
|---|---|
| $39.01$ days | $\mathbf{-14.2455}$ minutes |
| $130.84$ days | $+3.6841$ minutes |
| $203.91$ days | $-6.5099$ minutes |
| $303.96$ days | $\mathbf{+16.4187}$ minutes |
The range is $30.6642$ minutes, and the zeros fall at $102.28$, $161.00$, $241.12$ and $355.95$ days from perihelion, four times a year. The eccentricity part has amplitude $\mathbf{7.6550}$ minutes ($7.6547$ from the first-order $2e$) and the obliquity part $\mathbf{9.8664}$ minutes ($9.8634$ from the first-order $\tan^2(\varepsilon/2)$); the sum of the two differs from the whole by at most $4.1\times10^{-13}$ minutes.
Observation 2.2 the equation of time is the sum of two causes. The eccentricity part comes from the sun moving faster near perihelion; the obliquity part from uniform motion along the ecliptic becoming uneven when projected on the equator——each agrees well with its first-order form, and together they make the whole.
3. One Cause at a Time
Computation 3.1: the same computation with one cause removed, and with the perihelion turned.
| case | maximum | minimum | zeros |
|---|---|---|---|
| both | $+16.4187$ min | $-14.2455$ min | $4$ |
| obliquity only (eccentricity $0$) | $+9.8664$ min | $-9.8664$ min | $4$ |
| eccentricity only (obliquity $0$) | $+7.6550$ min | $-7.6550$ min | $2$ |
| perihelion turned $90$ degrees | $\mathbf{+14.5793}$ min | $\mathbf{-16.6434}$ min | $4$ |
| perihelion turned $180$ degrees | $+16.4187$ min | $-14.2455$ min | $4$ |
Observation 3.2 large and small peaks alternate because two periods overlap. The obliquity part has a half-year period with equal peaks and troughs; the eccentricity part has one of each per year——added together, one peak reinforces and the other cancels; the difference between $+16.4187$ and $+3.6841$ minutes comes from that.
Observation 3.3 turning the perihelion by half a turn leaves the shape unchanged. The obliquity part has a half-year period, so on days counted from perihelion a $180$-degree turn lands on the same shape——a $90$-degree turn changes the overlap, giving a maximum of $14.5793$ and a minimum of $-16.6434$ minutes.
4. Around the Winter Solstice
Computation 4.1: at latitude $35.7$ degrees north, as in Paper 414, sunrise and sunset (local mean time, geometry only) were found with the equation of time included. The winter solstice (longitude $270$ degrees) falls $352.54$ days from perihelion (on the side of the previous year).
| days from solstice | time | |
|---|---|---|
| shortest day | $+0.00$ days | |
| earliest sunset | $-15.61$ days | $16.6952$ h (solstice $16.7616$ h) |
| latest sunrise | $+16.07$ days | $7.2505$ h (solstice $7.1819$ h) |
The earliest sunset is $3.9798$ minutes earlier than at the solstice, and the latest sunrise $4.1116$ minutes later. The equation of time at the solstice is $+1.6951$ minutes and falls by $\mathbf{0.4975}$ minutes a day.
Observation 4.2 this is the core. Around the solstice the day length barely changes, but the equation of time falls by about $0.5$ minutes a day, so solar noon by the clock moves about $0.5$ minutes later each day——before the minimum of day length (the solstice), the later noon pushes sunset later, so the earliest sunset comes before the solstice; for sunrise it is the reverse, and the latest sunrise comes after.
5. The Separator
Observation 5.1 the separator is whether one looks at day length or also at the time of solar noon. Day length alone bottoms out at the solstice. When solar noon moves, the extremes of sunrise and sunset separate onto different days on either side of the solstice——“the solstice has the earliest sunset” is right only if solar noon does not move.
Observation 5.2 taking the seat left by Paper 414. It showed with the sunrise equation at latitude $35.7$ degrees that an unequal hour varies by a factor of $1.5$ with the season, and set aside the equation of time in its scope note——this paper puts the equation of time into the same equation and shows that the extremes of sunrise and sunset move half a month from the solstice.
Observation 5.3 one root with Paper 535, in the sense of Paper 300. Paper 535 derived the moon’s libration in longitude from $2e$; the eccentricity part of the equation of time comes from the same $2e$——two appearances of $2e$, met in separate places, the visible area of the moon and the gap between sundial and clock, have one source, the equation of centre in Kepler’s equation.
Observation 5.4 what is not said. No calendar dates are given and the history of clocks is not treated. Without refraction and the sun’s apparent radius, real sunrise and sunset times differ from these——what is shown is the form, the extremes of sunrise and sunset moving half a month from the solstice, and the reason for it.
6. Roll Call and Closing
- the equation of time reaches $\mathbf{16.4187}$ and $\mathbf{-14.2455}$ minutes (Computation 2.1): four zeros a year.
- eccentricity part, amplitude $\mathbf{7.6550}$ minutes (Computation 2.1): first-order $2e$ gives $7.6547$.
- obliquity part, amplitude $\mathbf{9.8664}$ minutes (Computation 2.1): first-order gives $9.8634$.
- sum and whole agree to $4.1\times10^{-13}$ minutes (Computation 2.1): the sum of two causes.
- obliquity alone gives four zeros, eccentricity alone two (Computation 3.1): half-year and year periods.
- perihelion turned $90$ degrees: $\mathbf{14.5793}$ and $\mathbf{-16.6434}$ minutes (Observation 3.3): unchanged at $180$.
- earliest sunset $15.61$ days before the solstice (Observation 4.2): the core.
- latest sunrise $16.07$ days after the solstice (Computation 4.1): the extremes separate.
- the equation of time falls $\mathbf{0.4975}$ minutes a day at the solstice (Computation 4.1): solar noon moves later daily.
- the separator is day length versus solar noon (Observation 5.1): sharing $2e$ with Paper 535.
Closing: The earliest sunset of the year is easily taken to fall on the winter solstice, the shortest day. The gap between sundial and clock, the equation of time, is the sum of two causes——with eccentricity $0.0167$ and obliquity $23.44$ degrees it reaches $\mathbf{16.4187}$ and $\mathbf{-14.2455}$ minutes, the eccentricity part (amplitude $\mathbf{7.6550}$ minutes) and the obliquity part (amplitude $\mathbf{9.8664}$ minutes) adding to the whole. A yearly and a half-yearly period overlap, so large and small peaks alternate; turning the perihelion by $90$ degrees changes the extremes to $\mathbf{14.5793}$ and $\mathbf{-16.6434}$ minutes, and by $180$ degrees changes nothing. Around the solstice the equation of time falls by $\mathbf{0.4975}$ minutes a day——solar noon by the clock moves later every day, so at latitude $35.7$ degrees north the earliest sunset comes $15.61$ days before the solstice and the latest sunrise $16.07$ days after. The shortest day and the earliest sunset are not the same day. The separator is whether one looks at day length or also at solar noon——day length alone bottoms out at the solstice; when noon moves, the extremes of sunrise and sunset separate on either side of it. Placed among the earlier papers——this takes the seat for the equation of time that Paper 414 left in its scope note. The eccentricity part comes from the same equation of centre as the $2e$ with which Paper 535 derived the moon’s libration in longitude; as two appearances of $2e$ met in separate places, they share a root in the sense of Paper 300. To be honest——the orbital elements are representative with no secular change, and sunrise and sunset come from geometry alone, without refraction or the sun’s apparent radius. No calendar dates are given, and the history of clocks is not treated.
References
- Hughes, D. W., Yallop, B. D., & Hohenkerk, C. Y. (1989). The Equation of Time. Monthly Notices of the Royal Astronomical Society.
- Meeus, J. (1998). Astronomical Algorithms (2nd ed.). Willmann-Bell.
- Smart, W. M. (1977). Textbook on Spherical Astronomy (6th ed.). Cambridge University Press.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Same “Hour” Is Half as Long Again in Summer [Paper 414]. Zenodo. 10.5281/zenodo.22699766
- Yamagishi, Y. (2026). What Fixes How Much of the Moon Is Visible Is Which Side Two Normals Lie On [Paper 535]. Zenodo. 10.5281/zenodo.23014119
Reference
Yamagishi, Y. (2026). The Earliest Sunset Comes Half a Month Before the Winter Solstice: the Gap Between Sundial and Clock Is the Sum of Two Causes [Paper 596]. YOI: https://www.yamagishiyuuki.com/yoi/596
DOI: 10.5281/zenodo.23014410
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 596. This paper is a preprint and has not been peer reviewed.