At 40 degrees north the sunrise point shifts about 0.53 degrees a day at an equinox, roughly the width of the sun, but only about 0.22 degrees in the 7 days after a solstice, so one horizon mark fixes an equinox to about a day and a solstice only to about a week. Marking the two dates on either side of the solstice when the sun rises on the same mark, then taking the midpoint, fixes the solstice to about a day.
Solstice and equinox markers are fixed sighting lines and shadows that let a community date the sun’s turning points from observation, with a precision set by the geometry of the sun’s yearly path.
The Sun Stands Still At The Solstice, So The Date Is Hard To Pin
The carrier article calls the solstices and equinoxes “observable facts” that “require no cultural agreement”. They are observable, but not to the precision that statement suggests. Near a solstice the sun’s declination barely changes, so its rising point barely moves; near an equinox it changes fastest.
This page uses the corpus’s declination formula and altitude and azimuth relations from Seasonal Sun Angle Calculation, which cites Duffie and Beckman for Cooper’s equation. All numbers below were recomputed from those relations and are geometric: they ignore atmospheric refraction and the sun’s disc unless stated.
The Arithmetic At 40 N
Sunrise azimuth (measured from true north, on a flat horizon): cos A = sin δ ÷ cos φ.
- Summer solstice, δ = +23.44°, φ = 40°: cos A = 0.3977 ÷ 0.7660 = 0.5192, A = 58.7°.
- Winter solstice, δ = −23.44°: A = 121.3°. The sunrise point swings 62.6° of horizon between them.
- Equinox, δ = 0: cos A = 0, A = 90°, due east at every latitude.
Day length = 2 × arccos(−tan φ × tan δ) ÷ 15: summer solstice 14.84 h, equinox 12 h, winter solstice 9.16 h. At 35° south the pattern is mirrored: a June solstice (δ = +23.44°) gives sunrise at 60.95° and 9.64 h of daylight, a December solstice 119.05° and 14.36 h.
Rate of movement of the sunrise point.
- At an equinox, dδ/dn = 23.44° × 2π ÷ 365.25 = 0.403° a day, and dA = −dδ ÷ cos φ = 0.403 ÷ 0.766 = 0.53° a day. The sun’s disc is about 0.53° across (recalled, not re-checked), so the sunrise point moves about one sun-width a day. A mark placement error of 0.1° (an assumed figure) is then 0.1 ÷ 0.53 = 0.19 day, so the equinox is fixed to about a day.
- Near a solstice, δ ≈ 23.44° × cos(2πk ÷ 365.25) for k days from it. From the corpus formula the sunrise azimuth has moved 0.22° after 7 days, 0.46° after 10 days and 0.91° after 14 days.
What The Precision Limit Means
Half a sun-width is about 0.27°. A sun that has moved less than that is indistinguishable from the original mark, which happens for about 7 days either side of a solstice. A single mark therefore dates a solstice only to about a week.
Bisecting beats the single mark. Put a mark where the sun rises about 14 days before the solstice. Note the date it rises on that mark again after the solstice. The solstice lies at the midpoint, because the sun’s path is symmetric about it. At 14 days out the sunrise point moves about 0.13° a day (recomputed from the same formula), so a mark placement error of 0.1°, an assumed figure, shifts each date by 0.1 ÷ 0.13 = 0.8 day, and the midpoint by roughly a day. This is a calculated expectation and should be tested over two years.
Horizon Height Shifts Every Date
The flat-horizon formula holds only at sea level or a flat plain. For a horizon h degrees above level: cos A = (sin δ − sin φ × sin h) ÷ (cos φ × cos h).
| Horizon height | Summer solstice azimuth, 40 N | Winter solstice azimuth |
|---|---|---|
| 0° | 58.72° | 121.28° |
| 2° | 60.64° | 123.29° |
| 5° | 63.39° | 126.49° |
A 2° ridge moves the summer solstice sunrise by 60.64 − 58.72 = 1.92°, far more than the 0.1° precision target. Even the equinox sunrise is no longer due east: at 40 N and h = 2°, cos A = −tan 40° × tan 2° = −0.0293, so A = 91.7°. Mark the real horizon, not the ideal one, and calibrate each mark against a year of observation rather than the formula alone.
Noon Shadow As A Check
At solar noon, altitude = 90° − |φ − δ|. At 40 N this is 73.44° in summer and 26.56° in winter, so a vertical 1 m post casts a shadow of 1 ÷ tan 73.44° = 0.30 m in summer and 1 ÷ tan 26.56° = 2.00 m in winter. Around the winter solstice the noon shadow shortens by about 1.5 cm after 7 days and 3.0 cm after 10 days. The sun’s disc blurs the shadow tip: the tip is about √(1² + 2²) = 2.24 m from the post’s top, and 2.24 × 0.0093 rad (0.53° in radians) = about 2.1 cm. The blur is as large as the weekly change, so the shadow also dates the solstice only to about a week. The shadow’s real use is the north-south line (shortest shadow), from which true east is a right angle.
Tropics And The Southern Hemisphere
- South of the equator the solstices swap seasons. Name the events by the sun’s position, not by month or season names (Seasonal & Calendar Rituals).
- Near the equator, day length barely changes (12 h on the equator, geometric), and the sunrise point swings between 66.56° and 113.44°. Zenith passage, when the noon sun is overhead and a vertical post has no noon shadow, happens twice a year at latitudes within the tropics and is a better marker there.
- Prehistoric horizon markers for the sun are known in several places; I recall monuments in Europe and a ridge of towers in Peru but have not checked either.
What You Need
- A fixed observing place and a clear view of the eastern horizon
- A plumb rod or post for shadows, a flat level surface and a rope or stakes
- A record of dates and positions kept for at least two years
- A way to draw the north-south line from the shortest noon shadow
Method: Build A Horizon Calendar
Step 1: Find true north-south. Mark the tip of a post’s shadow every few minutes around midday; the shortest shadow points toward the nearer pole (north in the northern hemisphere, south in the southern; within the tropics it can point either way depending on the season).
Step 2: Find true east at right angles. Treat it as the equinox direction only on a flat horizon.
Step 3: Record the rising point daily for a year by the direction of a rod’s shadow at sunrise, not by looking at the sun, and mark each new position with a stake or stone.
Step 4: Locate the extremes and bisect with equal-position dates.
Step 5: Fix the equinox from the date the rising point passes the middle marker, noting the horizon height.
Step 6: Repeat in a second year and compare.
Common Mistakes
- Trusting a single mark for the solstice. It is good to a week, not to a day.
- Using the flat-horizon azimuth on a hilly horizon. The error exceeds the target precision.
- Calling the equinox due east on uneven ground.
- Copying another latitude’s markers. The swing and the day lengths change with latitude.
- Skipping the second year. One year of weather-affected sightings is not a calibration.
Never look at the sun
Do not look at the sun directly, even low on the horizon: it can damage the eye. Judge position by the shadow of a rod or by a pinhole image on a card. Solstice gatherings are often held around a fire at night, so apply the fire and alcohol precautions in Fire Management, keep the paths lit, and dress for the cold.
What’s Next
- Learn next: Planting & Harvest Ceremonies — tying events to crop state
- Combine with: Calendar Systems — building a year count from these markers
Unverified references
The sun’s angular width (about 0.53°) and the mention of prehistoric solar monuments are from recollection and not re-checked. The declination formula and sun-angle relations come from the corpus page cited and were recomputed in a scratch script; the 0.1° placement error is an assumption, not a measured figure. Refraction (which lifts the sun by roughly its own width at the horizon) is not included.