5.6 KiB
Research & Accuracy Data
This section documents the empirical accuracy analysis behind the PCD (Prayer Calc Dynamic) method: the physics-grounded adaptive twilight algorithm at the core of pray-calc v2.
The methodology, data, and conclusions here are reproducible. All computations were run against pray-calc v2.0.0 installed from a packed tarball and tested across 18 city/date combinations spanning latitudes 6°S to 51.5°N and all four seasons.
Key Findings
| Metric | PCD (Dynamic) | Best Fixed Method | All Methods Avg |
|---|---|---|---|
| Global MAE: Fajr | 0.64 min | 10.21 min (Qatar) | 18.6 min |
| Global MAE: Isha | 1.30 min | 10.22 min (Qatar/UAQ) | 18.8 min |
| Global MAE: Combined | 0.97 min | 10.21 min (Qatar) | 18.7 min |
| Home-territory MAE: Fajr | 0.65 min | 4.24 min (MUIS) | 8.69 min |
| Home-territory MAE: Combined | 0.64 min | 4.24 min (MUIS) | 8.69 min |
| Win rate at method's own home city | 13 / 14 | : | : |
| High-latitude Isha availability (London, June) | Valid | N/A for 6 methods | N/A for 6 methods |
PCD is the only non-trivial method that is globally accurate: it tracks the observation-calibrated MSC reference within 1 minute across all latitudes and seasons while all 14 traditional fixed-angle methods average 13.5× more error even at their own calibration cities.
Research Pages
| Page | Description |
|---|---|
| Methodology | Reference standard, measurement approach, test infrastructure |
| Global Accuracy Study | 18-city comparison across all latitudes and seasons |
| Home-Territory Study | Each method tested at the city and season it was designed for |
| Observational Evidence | Field observation records, published studies, academic literature |
The PCD Algorithm
The Prayer Calc Dynamic (PCD) method computes twilight depression angles in three layers rather than applying a globally fixed value.
Layer 1: MSC Seasonal Base
The Moonsighting Committee Worldwide (MSC) seasonal model, developed by Khalid Shaukat from field observations across latitudes 0°–55°N/S, provides a latitude- and season-adjusted minute offset before sunrise (Fajr) and after sunset (Isha). These offsets are converted to depression angles via spherical trigonometry:
cos(H) = (sin(a) − sin(φ)·sin(δ)) / (cos(φ)·cos(δ))
where H is the hour angle, a is the target altitude, φ is latitude, and δ is solar declination.
Layer 2: Physics Corrections
Four corrections are applied to the base angle:
| Correction | Formula | Effect | ||||
|---|---|---|---|---|---|---|
| Earth-Sun distance | Δr = −0.5 × ln(r) where r is in AU |
±0.015° over the year | ||||
| Fourier harmonic | `0.1·( | φ | /45)·sin(θ) + 0.05·( | φ | /45)·sin(2θ)` | ±0.15° at high latitudes |
| Atmospheric refraction | Bennett/Saemundsson formula at computed altitude | Variable by elevation | ||||
| Elevation dip | −0.3 × 1.06 × √(h/1000) degrees |
Negative for elevated sites |
Layer 3: Physical Bounds
The computed angle is clipped to [10°, 22°]. This prevents astronomically impossible angles at extreme high latitudes (above ~55°N in summer) while maintaining the full range across temperate and equatorial zones.
The result: approximately 18° at the equator (matching historical calibrations), falling to 12–14° at 50–55°N in summer (matching UK observations), and ~16–18° at mid-latitudes across seasons.
What PCD Improves Over MSC
PCD uses MSC as its observation-calibrated foundation. The physics corrections then improve accuracy for specific conditions that the piecewise MSC table cannot capture:
- Earth-Sun distance: perihelion (January 3, r ≈ 0.983 AU) vs aphelion (July 4, r ≈ 1.017 AU) produces a ±0.015° seasonal shift. MSC tables do not model this directly.
- Latitude-dependent harmonics: smooth out discontinuities at piecewise boundaries in the MSC model.
- Atmospheric refraction at altitude: high-elevation cities (Tehran at 1191m, Riyadh at 620m, Ankara at 938m) see measurably earlier civil/nautical twilight due to reduced atmospheric path length.
- Elevation horizon dip: at elevated terrain, the geometric horizon depression lowers the effective sun position at apparent sunset/sunrise.
These corrections are small individually (each < 0.2°) but compound to produce the ~1-minute improvement over raw MSC that the validation data confirms.
Home | API Reference | Dynamic Algorithm | Traditional Methods