Implements three criteria for predicting the visibility of the new crescent moon: Yallop (1997), Odeh (2006), and Schaefer (1988/2000).

All three functions accept a normalized location and a moment (midnight UTC for the date of interest) and return {:ok, visibility} where visibility is one of :A, :B, :C, :D, or :E.

Yallop (1997)

An empirical single-parameter criterion based on the geocentric arc of vision (ARCV) and crescent width (W). The q-value is the scaled residual between the observed ARCV and a cubic polynomial fit to Bruin's (1977) visibility curves:

ARCV' = 11.8371 − 6.3226·W + 0.7319·W² − 0.1018·W³
q = (ARCV − ARCV') / 10

Odeh (2006)

An updated empirical criterion using the same polynomial family as Yallop but fitted to 737 observations (vs Yallop's 295). Uses topocentric ARCV and a different intercept:

ARCV' = 7.1651 − 6.3226·W + 0.7319·W² − 0.1018·W³
V = ARCV − ARCV'

Odeh also enforces a Danjon limit of 6.4° (ARCL < 6.4° → not visible).

Schaefer (1988/2000)

A physics-based model that computes the contrast between the crescent moon's brightness and the twilight sky brightness, then compares against the human contrast detection threshold (Blackwell 1946). The best observation time is found by scanning from sunset to moonset.

The visibility parameter is:

Rs = log₁₀(contrast / threshold)

where Rs > 0 means the crescent is detectable. This implementation uses the V-band sky brightness model from Schaefer (1993) with default atmospheric parameters suitable for a sea-level site with clean air.

Visibility categories

References

• Yallop, B. D. (1997). A Method for Predicting the First Sighting of the New Crescent Moon. NAO Technical Note No. 69.

• Odeh, M. Sh. (2004). New Criterion for Lunar Crescent Visibility. Experimental Astronomy, 18, 39–64.

• Schaefer, B. E. (1993). Astronomy and the Limits of Vision. Vistas in Astronomy, 36, 311–361.

• Schaefer, B. E. (2000). New Methods and Techniques for Historical Astronomy and Archaeoastronomy. Archaeoastronomy, XV, 121–136.