Do Autumn Days Really "Drop Like a Well Bucket"?
– Around the Autumn Equinox, Daytime Shrinks by 2 Minutes a Day and Dusk Is Shortest of the Year
In autumn, evenings seem to arrive suddenly early. People in the old days had a saying for it: "In autumn, the sun drops like a well bucket" (tsurube-otoshi). The idea is that the sun sinks straight down, the way a bucket plunges down a well. This is not your imagination. Around the autumn equinox, the length of the day shrinks faster than at any other time of year. On top of that, the dusk between sunset and darkness is also the shortest of the year.
At the end of August, the park at 6 p.m. is still bright, and children are playing.
A month later, at the end of September, you walk the same path at the same 6 p.m. The sky is already deep blue, and the street lamps are on.
And once you think "ah, the sun is setting," it seems to get dark much faster than in summer. What changed in just one month?
Just two reasons
Around the summer and winter solstices, the length of the day hardly changes. It changes fastest around the spring and autumn equinoxes. In Tokyo around the autumn equinox, daytime gets shorter by about 2 minutes each day.
Around the autumn equinox, the sun sinks at the steepest angle to the horizon. As a result, the dusk between sunset and the sky going dark is the shortest of the year.
The first is the change from day to day. The second is the change within a single evening. Both peak around the autumn equinox, so autumn evenings close in twice as fast.
Around the autumn equinox, the day's "downhill slope" is steepest
Look at the left side of Figure 1. It is a line joining the length of daytime in Tokyo across a whole year. Around the summer solstice it is about 14.5 hours, flat like the top of a hill. Around the winter solstice it is just under 10 hours, flat like the bottom of a valley.
But the slope that joins them is steepest at the spring and autumn equinoxes. In Tokyo around the autumn equinox, daytime shrinks by about 2 minutes a day. Over a month, that is more than an hour. Sunset alone comes about 40 minutes earlier between early September and early October. Around the summer solstice, sunset moves by less than 10 minutes in a whole month.
The slope exists because the sun's path swings north and south with the seasons. That swing is like a pendulum. A pendulum stops for a moment at each end and passes the middle at top speed. The sun's path likewise nearly stops at the "ends," the summer and winter solstices, and moves south fastest through the "middle," the spring and autumn equinoxes.
Around the autumn equinox, dusk itself is short too
Even after the sun sets, it doesn't go pitch dark at once. Air high in the sky catches light from the sun below the horizon and glows. This lingering brightness is called "twilight." Once the sun reaches 6 degrees below the horizon, reading print outdoors becomes difficult.
What sets the length of dusk is how fast the sun sinks through those 6 degrees. Look at the right side of Figure 1. At the equator, the sun drops straight onto the horizon. Dusk ends in about 20 minutes. In Tokyo the sun sets at a slant, so it takes a little longer to descend the same 6 degrees.
That slant changes with the seasons. Around the autumn and spring equinoxes, the path is steepest. Around the summer solstice, the sun slides sideways toward the north as it sets, so the path lies flatter. Calculated for Tokyo, the time from sunset to 6 degrees below is about 25 minutes around the autumn equinox and about 30 minutes around the summer solstice. Measured to near-total darkness, the gap widens to more than 10 minutes.
Around the autumn equinox, sunset in Tokyo is a little after 5:30 p.m. Only a month earlier, you were used to evenings that stayed light past 6. The gap between the "usual evening" your body remembers and the actual sky widens by 2 minutes every day. This habit is thought to add to the "bucket-drop" feeling.
From autumn into winter, as darkness comes earlier, pedestrians become harder for drivers to see in the dim evening hours. The National Police Agency and local police forces urge people at this time of year to turn on headlights early and to wear bright clothing or reflective gear. The season of the bucket-drop is also a season when the way home looks different.
Summary
It is not your imagination that autumn evenings close in suddenly. Daytime shrinks faster than at any other time of year, and the sun sets at a steep angle so that dusk is short. These two things coincide right around the autumn equinox.
Autumn days darken fastest,
both from one day to the next and within each evening.
The earliest sunset of the year comes not at the autumn equinox but in early December. You can read how that works in Is it true that the earliest sunset isn't on the winter solstice? and why the date of the autumn equinox shifts from year to year in Why is Autumn Equinox Day sometimes September 23 and sometimes the 22nd?
- Every day at the same time (for example, 5:30 p.m.), take a photo of the western sky from the same spot. Over a week, you will see the brightness change clearly.
- On the National Astronomical Observatory of Japan's Ephemeris Computation Office page, look up sunset times for where you live and write down a week's worth. Work out how many minutes earlier it gets each day.
- Use a clock to time how long it takes, from sunset, until you can no longer read a newspaper outdoors. Do the same around next year's summer solstice and compare.
Weather, shadows from buildings, and nearby lights will change the results. Comparing clear days with clear days is best.
For those who want to know more – terms, formulas, and links to textbooksFrom middle-school science to university courses, with the level of each topic marked
- Middle schoolCovered in middle-school science
- High schoolCovered in high-school Earth science and mathematics (trigonometry)
- High school+Advanced high-school material, or textbook sidebar content
- UniversityUniversity courses not taught in high school (positional and spherical astronomy)
- ResearchNot taught even at university as settled fact; being studied by researchers now
Middle schoolTerms: this phenomenon has names
- Day length (period of illumination): The time from sunrise to sunset. This is the day length by the calendar, and does not consider clouds.
- Declination: The angle showing how far the sun is north or south of the celestial equator. It is about 23.4 degrees north at the summer solstice, about 23.4 degrees south at the winter solstice, and 0 degrees at the spring and autumn equinoxes.
- Twilight (civil twilight): The time after sunset, until the center of the sun reaches 6 degrees below the horizon, in which some sky brightness remains. Down to 12 degrees below is nautical twilight, and down to 18 degrees is astronomical twilight.
Middle schoolHigh schoolChecking with a formula: how many minutes does daytime shrink per day around the autumn equinox?
We find the main quantity, the change in day length per day, from the sunrise and sunset formula. In the second half, we also estimate the length of dusk (civil twilight).
| In symbols | ΔL ＝ 2 × tan φ × Δδ ÷ 15 |
| In words | Daytime lost per day (hours) = 2 × tangent of latitude × the angle the sun's declination moves per day (degrees) ÷ 15 |
| Where the formula comes from | It is the sunset-time formula cos H ＝ −tan φ × tan δ, rewritten for small changes around the autumn equinox (declination 0 degrees). The Earth turns 15 degrees per hour, and daytime stretches or shrinks at two ends, sunrise and sunset, which is where the 2 and the 15 come from. |
| Dusk formula | Twilight length ＝ angle the sun must descend ÷ (15 × cos φ). It comes from the fact that when the sun crosses the horizon on the autumn equinox, it drops 15 × cos φ degrees per hour. |
| Symbol | Meaning and unit |
|---|---|
| ΔL | Change in day length per day (hours) |
| φ | Latitude of the observing place (degrees) |
| Δδ | Angle the sun's declination changes per day (degrees) |
| H | Hour angle at sunset (how many degrees the sun has turned from due south) |
| Latitude of Tokyo | about 35.7 degrees |
| Tangent of 35.7 degrees | about 0.72 |
| Angle the sun's declination moves per day around the autumn equinox | about 0.40 degrees (the fastest of the year) |
| Cosine of 35.7 degrees | about 0.81 |
| Center of the sun at sunset (the moment its upper edge vanishes) | said to be about 0.8 degrees below the horizon. About 5.2 degrees remain to reach 6 degrees below |
| Effect of latitude × movement of declination | 0.72 × 0.40 ＝ 0.288 |
| Two ends, sunrise and sunset | 0.288 × 2 ＝ 0.576 |
| Convert the angle to time (15 degrees per hour) | 0.576 ÷ 15 ≒ 0.0384 hours |
| Convert to minutes | 0.0384 × 60 ≒ 2.3 minutes (per day) |
| If it continues for a month | 2.3 × 30 ＝ 69 minutes |
| Angle the sun sinks per hour on the autumn equinox | 15 × 0.81 ＝ 12.15 degrees |
| Time to sink the remaining 5.2 degrees | 5.2 ÷ 12.15 ≒ 0.43 hours |
| Convert to minutes | 0.43 × 60 ＝ 25.8 minutes |
In Tokyo around the autumn equinox, daytime shrinks by about 2.3 minutes a day, which is more than an hour in a month. Dusk, from sunset to darkness, is about 25 minutes. By the same method, it comes to about 21 minutes at the equator and about 30 minutes in Tokyo at the summer solstice (for the summer solstice, the sun's path is curved, so this value was calculated with the original formula rather than this simple one).
High schoolHigh school+Why change is fastest at the "middle of the pendulum"
High schoolThe sun's declination follows roughly a sine wave that completes one cycle a year. A sine wave has zero slope at its peaks and troughs, and its slope is greatest where it crosses zero. The greatest slope is the amplitude of 23.4 degrees times the angular speed of one cycle a year. This works out to about 0.40 degrees per day.
High school+The higher the latitude, the larger the tangent, so day length changes faster. It is faster in Sapporo than in Tokyo, and at the equator day length stays at about 12 hours nearly all year. Near the Arctic Circle, there are places where daytime shrinks by more than 10 minutes a day around the autumn equinox.
UniversitySunset and twilight through spherical trigonometry
From the spherical law of cosines, the sun's altitude h can be written sin h ＝ sin φ × sin δ ＋ cos φ × cos δ × cos H. Sunset takes h as about −0.83 degrees (allowing for atmospheric refraction and the sun's apparent radius), and the end of civil twilight takes −6 degrees; solving for the hour angle H gives the values above. It is known that twilight is shortest not exactly at the autumn equinox but a little after it, when the declination has shifted slightly south (and in spring, a little before the spring equinox).
📖 Derivations and further reading: Twilight (Wikipedia) ／ Declination (Wikipedia)
ResearchWhat is still not well understood
- How strong the psychology of "darkening early" is It is still unclear how much the speed at which day length changes affects people's sense of time or mood.
- Link to seasonal low mood It is known that some people feel low from autumn into winter, but research continues into whether the "amount" of daylight or the "speed of its decline" matters more.
- Changes in sky brightness Fine dust in the air and volcanic plumes change the brightness and color of twilight from year to year. Research continues to estimate the size of that effect accurately.
In other words, the content of this article is also "an explanation within what is known today." The calculations of day length and twilight are solid, but how people feel about them is still being investigated.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Middle school | Science "Earth and Space" (seasonal change and the sun's noon altitude) | Day length changes with the seasons |
| High school | Earth science "The celestial sphere and the sun's motion," mathematics "trigonometric functions" | Declination change follows a sine wave |
| High school+ | Mathematics "differentiation" | Change is fastest around the autumn equinox; differences by latitude |
| University | Spherical astronomy, positional astronomy | The formula for the sun's altitude; twilight calculation |
| Research | Time perception, seasonal mood research, atmospheric optics | Link to perception; changes in twilight brightness |
| ― | Links to daily life | Outdoor play and getting home on autumn evenings; poor visibility at dusk |
- National Astronomical Observatory of Japan, Ephemeris Computation Office (国立天文台 暦計算室) (sunrise, sunset, and twilight times for each place)
- National Astronomical Observatory of Japan (国立天文台), ed., Rika Nenpyo (理科年表, Chronological Scientific Tables), Maruzen Publishing (calendar section: sunrise/sunset, twilight)
- Wikipedia, "Twilight" (薄明)
- Takumi Nagasawa (長沢工), Hinode-Hinoiri no Keisan: Tentai no Shutsubotsu Jikoku no Motomekata (日の出・日の入りの計算 ― 天体の出没時刻の求め方, Calculating Sunrise and Sunset: How to Find Rising and Setting Times of Celestial Bodies), Chijin Shokan
*This article is a general-audience science explainer. The figures given are rough guides to help you understand how things work. Sunrise, sunset, and twilight times vary with location, terrain, and weather. For exact times, please check the National Astronomical Observatory of Japan's Ephemeris Computation Office or similar sources.