The Hilal Dashboard employs the rigorous Yallop (1997) criterion to predict lunar crescent visibility. Dr. B.D. Yallop of the HM Nautical Almanac Office developed this statistical model based on 295 historical observations.
The q-value elegantly predicts visibility across specific zones ranging from naked-eye certain (Zone A) to wholly invisible (Zone F).
The Odeh (2004) criterion was developed by Mohammad Sh. Odeh of the Islamic Crescents' Observation Project (ICOP), refining Yallop's model with 737 observations including modern CCD and telescopic sightings.
In 1932, French astronomer André Danjon discovered that sighting a crescent is physically impossible if the lunar elongation (angular distance between the sun and moon) is less than 7 degrees.
Due to the roughness of the lunar surface and shadowing from lunar mountains, the crescent simply does not extend enough to be visible when the moon is too close to the sun.
Our geometry computations account for standard atmospheric refraction — which "lifts" objects above the true horizon.
When the moon appears to just touch the horizon, its true geometric center is actually roughly 34 arcminutes below the horizon plane. The atmosphere bends the light, creating a mirage of the moon higher than it physically lies.
Calculations employ the Bennett (1982) formula for astronomical refraction, using default standard temperature and pressure unless overridden in Pro settings.