pray-calc-ml/academic_formula.md

Academic Best Formula: Solar Depression Angle at True Fajr and Isha

Date: 2026-03-23 Dataset: pray-calc-ml (48,668 Fajr + 34,529 Isha raw records) Method: Filtered to Tier 1+2 sources only, source-weighted harmonic regression

---

1. Source Tier Classification

Each source in the dataset was classified by how closely its methodology matches the Islamic astronomical definitions:

• Fajr Sadiq: horizontal white light band appears on the eastern horizon, visible to a trained naked eye at a dark site. Not zodiacal light. Not instrument-detected pre-dawn glow.
• Shafaq al-Abyad: white twilight glow disappears from the western horizon. Not red glow (ahmar, which ends earlier at ~14 degrees).

Tier 1: Directly measures the definition

Naked-eye observers at dark sites watching the horizon, or CCD cameras pointed at the horizon.

Source	Records (Fajr)	Mean Angle	Latitude
OpenFajr Birmingham (CCD horizon camera)	3,723	13.36	52.5N
Shaukat/Miftahi 2007 (naked eye, Blackburn UK)	29	13.55	53.8N
Abed 2015 (naked eye group, Jordan desert)	18	16.94	31.5N
Taha 2025 (naked eye, Mauritania + Riyadh desert)	23	14.60	20-26N
Omar Afzal (naked eye, Ithaca NY)	2	11.48	42.4N
Ouared (camera, Pampigny Switzerland)	1	13.25	46.6N
Asim Yusuf 2017 (naked eye, Exmoor UK)	4	14.65	51.1N
Ludhianvi 1975 (naked eye, 11 ulama, Sindh Pakistan)	2	15.38	25.8N
TOTAL	3,804	13.38

Tier 2: Good proxy

SQM at dark sites (MPSAS >= 21.3), DSLR photometry pointed at the horizon, per-night observations.

Source	Records (Fajr)	Mean Angle	Latitude
Basthoni 2022 PhD, dark sites only (MPSAS >= 21.3)	590	15.77	0-10S
Kassim Bahali 2018-2019 (DSLR, Malaysia/Indonesia)	90	17.30	1-7N
BRIN multistation SQM 2018	34	17.61	1-7S
Khalifa 2018 (deep desert, Saudi)	11	14.59	22N
Marzouk 2025 (DSLR, Egypt desert)	11	14.41	24-30N
Various Indonesian SQM/DSLR studies	~30	16-19	0-10S
Hidayatulloh 2023 (SQM+SOOF, Bortle 1)	10	18.17	5S
TOTAL	1,109	15.77

Tier 3: Loose proxy (excluded from formula fitting)

SQM at light-polluted sites (MPSAS < 21.3), citizen science with approximate timestamps. Records: 3,702 (Fajr). Mean: 12.40 degrees. These systematically read shallower angles because light pollution masks the true dawn signal.

Tier 4: Wrong definition (excluded from formula fitting)

TESS-W zenith SQM network, Globe at Night citizen science, Madrid UCM zenith SQM. Records: 40,054 (Fajr). These measure zenith sky brightness, not horizon light. The inflection point in zenith brightness occurs at a different depression angle than the onset of visible light on the horizon.

---

2. Fajr Formula: Tier 1+2 Source-Weighted Medium-9

Model specification

angle = base
      + a1 * |lat|
      + a2 * sin(D) + a3 * cos(D) + a4 * sin(2D) + a5 * cos(2D)
      + a6 * |lat| * sin(D) + a7 * |lat| * cos(D)
      + a8 * |lat| * sin(2D) + a9 * |lat| * cos(2D)

where D = 2 * pi * day_of_year / 365.25
      |lat| = absolute latitude in degrees

Weighting scheme

Tier 1 sources receive 3x weight. Per-source weight is scaled by 1/sqrt(n) to prevent OpenFajr's 3,723 records from overwhelming the 80 non-OpenFajr Tier 1 records and the 1,109 Tier 2 records.

Fitted coefficients (source-weighted)

Coefficient	Value	Feature
base	16.982	Intercept (equatorial equinox angle)
a1	-0.066561	abs_lat
a2	-0.080311	sin(D)
a3	-0.100276	cos(D)
a4	+0.150842	sin(2D)
a5	-0.662638	cos(2D)
a6	-0.004398	abs_lat * sin(D)
a7	+0.000249	abs_lat * cos(D)
a8	+0.003904	abs_lat * sin(2D)
a9	+0.002038	abs_lat * cos(2D)

Fit statistics

Metric	Value
Records	4,912 (Tier 1 + Tier 2)
Weighted R-squared	0.426
Weighted MAE	1.14 degrees
Baseline 95% CI	[16.733, 17.222] degrees
Night-to-night noise floor	~0.35 degrees std (irreducible atmospheric variation, measured from OpenFajr)

Predictions vs literature

Location	Day of Year	Formula	Literature / Observed	Source
Equator, equinox	80	17.55	~17-18	MWL/Karachi convention
Equator, Jun solstice	172	16.39	~17-18	MWL/Karachi convention
Mecca (21.4N), equinox	80	16.03	14.7	Taha et al. 2025
Riyadh (25N), equinox	80	15.77	14.7	Taha et al. 2025
Cairo (30N), equinox	80	15.42	14.4	Marzouk et al. 2025
Jordan (31.5N), equinox	80	15.31	17.0	Abed 2015
Madrid (40.4N), equinox	80	14.68	15-16 (est)
Birmingham (52.5N), equinox	80	13.81	13.73	OpenFajr observed monthly
Birmingham, June solstice	172	12.87	12.26	OpenFajr observed monthly
Birmingham, Sep equinox	265	14.45	14.42	OpenFajr observed monthly
Birmingham, Dec solstice	355	12.81	12.53	OpenFajr observed monthly

---

3. OpenFajr Birmingham: Purest Single-Source Analysis

OpenFajr is the single most controlled dataset: one CCD camera, one location (52.483N, -1.894W), community-voted true dawn timestamps, running since 2018.

Single-location seasonal fit (4-parameter)

angle = 13.361
      - 0.308 * sin(D)
      - 0.085 * cos(D)
      + 0.350 * sin(2D)
      - 0.761 * cos(2D)

Metric	Value
Records	3,723
R-squared	0.767
MAE	0.240 degrees
Baseline	13.361 degrees
Seasonal range	12.26 (June) to 14.42 (September)

Monthly observed means

Month	Observed Mean	Model Prediction	N
Jan	12.96	13.02	341
Feb	13.12	13.57	311
Mar	13.73	13.82	327
Apr	14.05	13.54	300
May	12.75	13.00	310
Jun	12.26	12.82	300
Jul	13.18	13.30	310
Aug	14.14	14.10	310
Sep	14.42	14.47	300
Oct	13.92	14.10	304
Nov	13.29	13.30	300
Dec	12.53	12.83	310

Key finding: the seasonal pattern at 52.5N shows a DOUBLE peak (April and August/September), not the single summer-winter cycle many conventions assume. This is captured by the sin(2D) and cos(2D) terms. The deepest angles occur at equinoxes (~14 degrees) and the shallowest near solstices (~12.3-12.5 degrees).

---

4. Comparison with pray-calc's Current DPC

The current pray-calc DPC (v2.0) uses a three-layer model:

1. MSC piecewise seasonal base (Khalid Shaukat): 75 minutes at equator, latitude-scaled
2. Physics corrections: Earth-Sun distance, Fourier smoothing, atmospheric refraction
3. Environmental: elevation dip, pressure/temperature

The DPC is documented as producing ~18 degrees at the equator and ~12-14 degrees at 50-55N in summer.

How our empirical formula compares

Latitude Band	DPC Claims	Our Formula (equinox)	Agreement
Equator (0)	~18	17.6	Close. DPC may be ~0.5 degrees high
Low (21-30N)	~16-18	15.4-16.0	Our formula 1-2 degrees lower. Field data (Taha, Marzouk) confirms lower values around 14.5 degrees
Mid (40N)	~14-17	14.7	Reasonable agreement
High (52.5N)	~12-14	13.8 (equinox), 12.9 (solstice)	Good agreement
Very high (55N)	~11-15	12.7 (solstice)	Good agreement

Key discrepancies

1. At 20-30N (Saudi, Egypt, Mauritania): Our empirical data (Taha 2025, Marzouk 2025, Khalifa 2018) consistently shows 14.4-14.7 degrees. The formula predicts 15.4-16.0 because it is pulled upward by Indonesian Tier 2 SQM data (mean 15.8 at similar latitudes). The pray-calc DPC likely overestimates at these latitudes by 1-3 degrees relative to human-perceived Fajr Sadiq.

2. At 31.5N (Jordan): Abed 2015 reports 17.0 degrees from naked-eye group observation, but the formula predicts 15.3. This is a conflict. Abed's observers may be reporting a slightly different threshold than OpenFajr's CCD, or atmospheric conditions in Jordan differ systematically from Mauritania/Egypt.

3. At the equator: The formula predicts 17.6 degrees, consistent with the traditional 18-degree convention. However, this is extrapolated, as we have no Tier 1 equatorial data. The Tier 2 Indonesian data ranges widely (13-20 degrees), suggesting SQM inflection at tropical sites has high variance.

---

5. Isha: Insufficient Data

Available Tier 1+2 Isha records: 45 total

Source	Records	Mean Angle	Latitude
Shaukat/Miftahi 2007 (naked eye, Blackburn UK)	32	13.33	53.8N
Asim Yusuf 2017 (naked eye, Exmoor UK)	6	16.78	51.1N
Khalifa 2018 (desert, Hail Saudi)	3	14.14	27.5N
Niri & Zainuddin (Sabah Malaysia)	3	17.84	6.0N
Tayebi (naked eye, Michigan USA)	1	11.63	42.1N

With only 45 records across 4 latitudes, no reliable multi-latitude formula is possible. The bootstrapped 95% CI on the intercept spans from -161 to +322 degrees (meaningless).

Tentative findings for Isha (NOT suitable for production use):

• Shafaq al-Abyad at 53-54N: ~13.3 degrees (Shaukat/Miftahi, n=32)
• Shafaq al-Abyad at equator: ~17.8 degrees (Niri, n=3, single site)
• These are consistent with Fajr findings (deeper angle near equator, shallower at high lat)
• Isha needs its own dedicated data collection campaign before a formula can be produced

---

6. Recommended Coefficients for pray-calc

Fajr: ready for adoption (with caveats)

// Solar depression angle at true Fajr (Subh Sadiq)
// Source: pray-calc-ml Tier 1+2 source-weighted harmonic regression
// 4,912 records from CCD, naked-eye, SQM dark-site, and DSLR horizon observations
//
// Caveats:
//   - Equatorial prediction (17.6 deg) is extrapolated; no Tier 1 tropical data
//   - 15-30N prediction (15-16 deg) may be ~1 deg high vs field data (Taha, Marzouk ~14.5)
//   - 45-55N predictions are well-calibrated by OpenFajr (R^2 = 0.77)
//   - Night-to-night atmospheric noise: ~0.35 deg std (irreducible)

const FAJR_BASE = 16.982;
const FAJR_COEF = {
  abs_lat:     -0.066561,
  sinD:        -0.080311,
  cosD:        -0.100276,
  sin2D:       +0.150842,
  cos2D:       -0.662638,
  lat_sinD:    -0.004398,
  lat_cosD:    +0.000249,
  lat_sin2D:   +0.003904,
  lat_cos2D:   +0.002038,
};

function fajrAngle(lat: number, dayOfYear: number): number {
  const absLat = Math.abs(lat);
  const D = (2 * Math.PI * dayOfYear) / 365.25;
  const sinD = Math.sin(D);
  const cosD = Math.cos(D);
  const sin2D = Math.sin(2 * D);
  const cos2D = Math.cos(2 * D);

  return FAJR_BASE
    + FAJR_COEF.abs_lat * absLat
    + FAJR_COEF.sinD * sinD
    + FAJR_COEF.cosD * cosD
    + FAJR_COEF.sin2D * sin2D
    + FAJR_COEF.cos2D * cos2D
    + FAJR_COEF.lat_sinD * absLat * sinD
    + FAJR_COEF.lat_cosD * absLat * cosD
    + FAJR_COEF.lat_sin2D * absLat * sin2D
    + FAJR_COEF.lat_cos2D * absLat * cos2D;
}

Isha: NOT ready

Do not replace the current pray-calc Isha algorithm. The dataset has only 45 Tier 1+2 records. The current MSC-based approach (shafaq abyad mode) remains the best available estimate until a dedicated Isha observation campaign is completed.

---

7. Per-Latitude-Band Validation Summary

Band	Lat Range	N (T1+2)	Observed Mean	Predicted Mean	MAE	R-squared
Equatorial	0-15	1,083	15.78	15.81	2.14	0.073
Subtropical	15-30	51	14.97	14.38	0.91	-0.342
Mid-latitude	30-45	20	16.39	14.63	2.46	-1.546
High-latitude	45-60	3,758	13.36	13.37	0.25	0.737

The formula performs well at high latitudes (R-squared = 0.74, MAE = 0.25 degrees) where we have dense, high-quality data. Performance degrades significantly at equatorial and mid-latitudes where the Tier 2 Indonesian SQM data has high variance (std > 2 degrees) and the mid-latitude band has only 20 records.

---

8. Data Gaps and Next Steps

Critical data needed

1. Equatorial Tier 1 data. Need naked-eye or CCD horizon observations at 0-15 degree latitudes. No such data currently exists in the dataset.

2. 15-30N data density. Only 51 Tier 1+2 records. Need at least 200-500 per-night observations from Saudi, Egypt, Pakistan, or North Africa to constrain the subtropical Fajr angle.

3. Mid-latitude Tier 1 data. Only 20 records from Jordan. Need observations from Turkey (39-41N), Iran (32-36N), or southern Europe.

4. Isha dedicated campaign. The 45-record Isha dataset is useless for modeling. Need the same source types as Fajr: CCD horizon cameras, naked-eye expert panels, and dark-site SQM series.

What would change the formula

• If equatorial Tier 1 data shows 15-16 degrees instead of 17-18, the baseline would drop by 1-2 degrees, bringing it closer to the Taha/Marzouk mid-latitude values. This would imply the traditional 18-degree convention is too deep.
• If equatorial Tier 1 data confirms 17-18 degrees, the current formula is approximately correct and the low-latitude field studies (Taha, Marzouk) may reflect atmospheric or site-specific factors.
• Either outcome is scientifically significant. The equatorial Fajr angle is the single most important measurement still missing.

---

Appendix: Tier 1 Only Formula (for reference)

This formula is fitted to Tier 1 data alone (3,804 records, 97.9% from OpenFajr). It should be viewed as the "OpenFajr-anchored" formula, extrapolated to other latitudes using only 80 non-OpenFajr records.

base    = 17.657
abs_lat = -0.081784
sinD    = -0.760861
cosD    = -1.468562
sin2D   = +0.317475
cos2D   = +0.677708
lat_sinD  = +0.008716
lat_cosD  = +0.026438
lat_sin2D = +0.000662
lat_cos2D = -0.027300

R-squared = 0.723, MAE = 0.259 degrees. The higher R-squared (vs 0.43 for T1+2 weighted) reflects that 98% of the data is at one location, so the seasonal model captures the seasonal signal very well. It should not be interpreted as indicating a better formula for global use.