The sky over Mainz on the night of Hoshana Rabbah in every year of the Maharil's rabbinate, 1387 to 1426, plus this year for comparison. Pick a year and every panel follows.
Height of the moon and the sun above the horizon from sunset to sunrise. Shaded bands are the stages of darkness; the dashed line marks alos (sun 16° below the horizon), and the hatched hour before it is when the mistake would have happened.
See where the moon actually was on Hoshana Rabbah night, and that it was high and bright, not rising, in the hour before dawn.
High in the south-southeast, about 50° to 67° above the horizon depending on the year. Not low in the east.
No. On every one of these mornings the moon set between about 11:30 AM and 2:30 PM, Mainz solar time. Something else made the sky go dark.
Because it had been up since the evening. It was not a new light; it was the same moon everyone had seen all night.
Students often assume the moon rose just before dawn and was mistaken for the sun. The chart shows it rose in the evening. The page uses 16° for alos; other shitos exist (see the Zmanim app on this site), and the point of the lesson does not depend on which one.
Each dot is one year. Across the bottom is the date of Hoshana Rabbah, which slides almost a month because the Hebrew calendar is lunar. Switch what the dots measure. Click a dot to select that year.
| Year | Hoshana Rabbah | Sunset | Moonrise | After sunset | Alos | Sunrise | Moonset | Lit | Distance km |
|---|
See that the date of Hoshana Rabbah moves almost a month on the solar calendar, and that the moonrise moves with it.
The Hebrew month follows the moon, so 21 Tishrei drifts against the sun's calendar, between about 23 September and 21 October in these years (Gregorian reckoning).
The late-September years, about 1.5 to 2 hours after sunset. Late-October years run 3.5 to 4.5 hours.
No. The latest was about 4.5 hours after sunset.
The common rule 'the moon rises 50 minutes later each day, so on the 21st it rises around midnight' does not hold at Sukkos. Sections 3 and 3b show why.
The moon's orbit is an ellipse with Earth at one focus. At its true proportions it is almost a circle: the nearest point (perigee) is about 13% closer than the farthest (apogee). The dots trace where the moon was each day of Tishrei in the selected year; the large dot is the pre-dawn hour of Hoshana Rabbah.
The shape does not change the rising time much. What moves moonrise from year to year is mostly the moon's north-south swing; section 3b splits the two apart. Distance mostly changes how big and bright the moon looks.
Show that the moon's path is an ellipse, so its distance and speed change, and that this changes how big and bright it looks more than when it rises.
About 13%, roughly 363,000 km against 405,000 km.
The moon goes from nearest point to nearest point in about 27.5 days, but the month is about 29.5 days, so the nearest point falls on a different day of the Hebrew month each time.
No. It moves the daily delay by only a few minutes. Section 3b shows the bigger cause.
The perigee is drawn on the left for clarity. In reality the ellipse itself slowly turns, about once every 8.85 years.
Each night the moon rises later than the night before. On average about 49 minutes later, if it moved at its average speed straight along the equator. Two things push each night away from that average: the moon swinging north or south, and its speed along the ellipse. The charts split each night of Sukkos in the selected year into those two parts.
Below zero means the moon rose sooner than the average delay; above zero, later. The labels under the bars give the moon's north-south position (declination) and which way it was heading. Heading north makes the delay shrink, because each night it comes up farther north along the eastern horizon, and at 50° north that point clears the horizon earlier.
Method: consecutive moonrises computed for Mainz, then split with the rising-time formula: the moon's eastward motion at average speed, the change in its rising point from its change in declination, and the difference made by its actual speed. What is left over is under half a minute on every night.
Separate the two causes of the moon's changing rise time, and show which one matters at Sukkos.
The moon is moving north each night, so it comes up farther north along the eastern horizon, and at Mainz's latitude that point clears the horizon sooner.
The north-south swing. Across all forty years it changes a night's delay by up to about 36 minutes; the speed along the ellipse by at most about 8.
The full moon of Tishrei sits opposite the sun near the autumn equinox, which is where the moon's path rises most steeply northward. This is the 'harvest moon' effect.
The split is computed, not estimated: the two parts add up to the real delay within half a minute on every night.
No. Three things change it, and the phase matters far more than the distance. Values below are for the hour before alos in the selected year, compared with a full moon overhead at average distance.
Brightness by phase: Krisciunas and Schaefer (1991). Distance follows the inverse square law. Air absorption uses 0.2 magnitudes per airmass. Twilight levels are typical published values for a clear sky and are approximate.
Show that the moon's light is not constant, and compare it with the light of real dawn.
Its phase. A moon two-thirds to three-quarters lit gives only about a fifth of a full moon's light. Distance changes it by about 15% either way.
Yes, many times brighter. In a clear sky you can still see it is the moon. Section 5 shows what happens under cloud.
Twilight values on the chart are typical published figures and approximate.
Slide the cloud thickness. Thickness is measured as optical depth: the number of times, on average, a ray of light is scattered on its way through. The picture is the sky looking east to south-southwest in the selected year, with the moon where it actually stood. Move through the night with the time slider, or click the chart below it.
Drag the height slider to see which years had the moon at that height at the same time relative to alos. "Set by hand" moves only the moon's height, keeping everything else about the chosen moment, so you can see the angle's effect on its own. The night-long chart always uses the real heights.
Why the moon's height matters: τ is fixed by the cloud, but light coming in at a slant crosses more of it. The path is τ ÷ sin(height). A moon 65° up crosses 1.1 times the layer; at 30° it crosses 2 times; at 10° almost 6 times. The same cloud that only dims a high moon can hide a low one completely.
Model: direct beam transmitted as e−τ/μ (μ is the sine of the moon's height). Cloud reflectance R = [(1 − g)τ + (2/3 − μ)(1 − e−τ/μ)] ÷ [4/3 + (1 − g)τ] (two-stream, no absorption), transmission 1 − R, asymmetry g = 0.85. The forward glow around the moon widens as τ grows, because each scattering spreads the light further, until the layer glows evenly. The rendering is a sketch of that physics, not a photograph.
Recreate the false dawn: show how a cloud layer turns a high moon into an even grey sky that looks like morning, and how real dawn differs.
With the moon behind a cloud layer, its light spreads across the whole sky and the eastern horizon is as bright as anywhere. With no moon disc to see, a grey sky reads as morning.
A thicker cloud bank, or the thin layer breaking up. Either fits the word החשיך. This is a reading of the physics; the source does not say why.
It starts in the east and only gets brighter. A moonlit sky changes with every cloud.
Light coming in at a slant crosses more cloud. A moon at 30° crosses twice the cloud of one overhead.
The picture is a model, not a photograph. Keep the distinction for students: the text gives what they saw; the slider shows one way it could have happened.
Every hour and minute on this page is clock time. Gaps like "moonrise 2h 56m after sunset" or "an hour before alos" are given so a reader today can follow the astronomy. They are not how the people in Mainz experienced that night.
They reckoned in sha'os zmaniyos, seasonal hours. By day those can be read from the sun, with a sundial. At night there is no sun to read, so the hour has to be judged by other means: the stars, the moon, candles, the length of the learning. All of these are far harder and less exact than a sundial.
So when this page says the moon rose 1h 35m after sunset in one year and 4h 31m in another, that is a precision nobody that night had. For them it was a judgment made by eye in the dark, and the text shows the final call was made by looking at the sky: היו סבורים שעלה עמוד השחר, "they thought dawn had risen."
Close the lesson: the precise times on the page are ours, not theirs.
By day the sun's position shows the hour, for example with a sundial. At night there is no sun to read.
On Erev Rosh Chodesh Cheshvan, the Maharil and the whole community fasted, read Vayechal and said selichos, because they had put on tallis and tefillin and said Pesukei D'Zimra while it was still night.
The source gives the date but not the reason. Students can be asked to suggest reasons, marked as their own.