A · Canonical textTutorial 1 • 15 min read

Deriving a Panchanga by Hand from Raw Coordinates

A pedagogical masterclass on the manual mathematics of Vedic calendrics. Follow this step-by-step calculation to compute the Five Limbs (Pañcāṅga) using paper, pencil, and standard planetary ephemeris longitudes.

1. The Astronomical Input Coordinates

Suppose an observer consults an ephemeris for a specific date and time and records the following apparent sidereal longitudes (Nirayana, Lahiri Ayanāṁśa):

Sidereal Sun Longitude (λ_Sun):

45.5000° (45° 30′ 00″)

Location: 15° 30′ into Vṛṣabha (Taurus)

Sidereal Moon Longitude (λ_Moon):

112.3000° (112° 18′ 00″)

Location: 22° 18′ into Karka (Cancer)

2. Step 1: Calculating the Tithi (Lunar Phase)

The Tithi measures the angular separation between the Moon and Sun along the ecliptic:

1. Solilunar Elongation: Δλ = (λ_Moon - λ_Sun) mod 360°

Δλ = (112.3000° - 45.5000°) = 66.8000°

2. Tithi Index = ⌊ Δλ / 12.0° ⌋ + 1

Tithi Index = ⌊ 66.8000 / 12.0 ⌋ + 1 = ⌊ 5.5667 ⌋ + 1 = 6

3. Fortnight Check: Δλ < 180° → Śukla Pakṣa (Waxing)

✓ Result: Śukla Pakṣa Ṣaṣṭhī (6th Tithi)

Elapsed: (66.8° mod 12°) / 12° = 6.8° / 12° = 56.67% of the Tithi has elapsed.

3. Step 2: Calculating Nakṣatra & Pāda

Nakshatras span exactly $13^\circ 20' = 13.3333^\circ$ based strictly on the Moon’s longitude:

1. Nakṣatra Index = ⌊ λ_Moon / 13.333333° ⌋ + 1

Nakṣatra Index = ⌊ 112.3000 / 13.333333 ⌋ + 1 = ⌊ 8.4225 ⌋ + 1 = 9

✓ Asterism #9: Āśleṣā (आश्लेषा) Nakṣatra

2. Residual Arc in Āśleṣā:

Arc = 112.3000° - (8 × 13.333333°) = 112.3000° - 106.6667° = 5.6333°

3. Pāda Calculation (3°20′ = 3.3333° per quarter):

Pāda = ⌊ 5.6333 / 3.3333 ⌋ + 1 = ⌊ 1.690 ⌋ + 1 = 2

✓ Result: Āśleṣā Nakṣatra, Pāda 2

4. Step 3: Calculating Solilunar Yoga

Yoga evaluates the longitudinal sum of Sun and Moon:

1. Longitude Sum: Σλ = (λ_Sun + λ_Moon) mod 360°

Σλ = (45.5000° + 112.3000°) = 157.8000°

2. Yoga Index = ⌊ Σλ / 13.333333° ⌋ + 1

Yoga Index = ⌊ 157.8000 / 13.333333 ⌋ + 1 = ⌊ 11.835 ⌋ + 1 = 12

✓ Result: Dhruva (ध्रुव) Yoga (Yoga #12)

5. Step 4: Calculating Karaṇa

A Karana represents a 6-degree half-tithi interval:

1. Karaṇa Half-Tithi ID = ⌊ Δλ / 6.0° ⌋ + 1

ID = ⌊ 66.8000 / 6.0 ⌋ + 1 = ⌊ 11.133 ⌋ + 1 = 12

2. Identifying the Karana:

Half-Tithi 1 is Kiṃstughna (Fixed). Half-Tithis 2 through 57 cycle through the 7 movable Karanas:

[Bava, Bālava, Kaulava, Taitila, Gara, Vaṇija, Viṣṭi]

3. Modular Cycle Mapping:

Movable Index = (ID - 2) mod 7 = (12 - 2) mod 7 = 10 mod 7 = 3

✓ Result: Kaulava (कौलव) Karaṇa

Final Verified Panchanga Output

From the raw celestial coordinates λ_Sun = 45.5° and λ_Moon = 112.3°, the mathematical limbs of time are decisively established:

Tithi: Śukla Pakṣa Ṣaṣṭhī (56.7% elapsed)
Nakṣatra: Āśleṣā, Pāda 2
Yoga: Dhruva Yoga (#12)
Karaṇa: Kaulava Karaṇa