[dickermoshe]

On some days the sun only just reaches an angle like 16.1° or 18° below the horizon, right around midnight. NOAACalculator and MeeusCalculator return no zman (NaN) on those days, even though the zman exists.

Why

Both calculators start from the sun's declination at noon. If the angle is out of reach at that declination, they stop and return NaN. But the declination changes during the day, by up to about 0.2° in 12 hours. So near the start and end of the season when a zman stops existing, the sun can reach the angle around midnight even though noon's figures say it can't.

Examples (UTC):

Location	Date	Zman	Before	After
London	2024-06-03	16.1° alos	NaN	00:10
London	2024-07-20	18° tzais	NaN	23:52
75.5°N	2093-04-22	sunrise	NaN	00:13

This isn't a polar edge case. For London, Manchester, Edinburgh, Copenhagen, Stockholm, Toronto and Montreal, every day from 2024 to 2030, at alos/tzais angles from 16.1° to 26°, each calculator was missing 234 zmanim. It's a few days every year, on the same dates.

Change

When the first pass finds the angle out of reach, it now tries once more using the sun's declination at the solar midnight next to the event. That's the night before for sunrise and the night after for sunset, when the sun is lowest. If the angle is reachable there, the existing passes continue as usual. Nothing else changes.

Testing

I checked results against an independent search that steps through the half day minute by minute, using each calculator's own sun-position model:

• Seven cities, 2024–2030: all 468 missing zmanim now come back, and no other result changed.
• 300,000 random cases up to 66.5° latitude: 148 missing zmanim recovered per calculator, and no other result changed.
• Accuracy: the recovered times are as accurate as each calculator's normal results, within 16 seconds for NOAA and 3 seconds for Meeus.
• New tests: NOAACalculatorTest and MeeusCalculatorTest each get a test covering the three examples above.

Not covered

• The reverse case. On a night when the sun misses the angle by a tiny amount, the calculators can still return a time. In the city data above this happened about once in 24,000 results for NOAA and once in 100,000 for Meeus. Each time, the sun came within 0.004° of the angle, which is smaller than the calculators' own position error. So I left it alone.
• Above about 80° latitude, a recovered time can be off by several minutes. The fixed number of passes already produces errors like that at those latitudes.
• SPA tests on master. Master currently fails 10 SPACalculatorTest tests since 7cab372. This PR doesn't touch SPA.