The Panchang
System
The Panchang encodes five independent astronomical measurements at every moment — using planetary positions computed with the same precision as modern ephemeris software, 1,500 years before computers. Each limb is a different mathematical projection of celestial mechanics.
01
Tithi
02
Nakshatra
03
Yoga
04
Karana
05
Vara
The Astronomical Foundation
All five Panchang limbs are derived from just two inputs: the ecliptic longitude of the Sun (θ_Sun) and the ecliptic longitude of the Moon (θ_Moon). These are the same inputs used in modern ephemeris computation (JPL Horizons, Stellarium).
θ_Sun = Sun ecliptic longitude (sidereal)
θ_Moon = Moon ecliptic longitude (sidereal)
JD = Julian Day Number
Ayanamsha = ~24° 7' (Lahiri, 2025) — correction from tropical to sidereal
The Surya Siddhanta (c. 400 CE) provides algorithms to compute planetary positions from scratch — using integer arithmetic and table lookups — achieving accuracy within a few arcminutes over centuries. The Aryabhatiya (499 CE) refined these to match modern values within 0.1%.
How Each Limb Is Computed
Tithi (Lunar Day)
तिथि
Formula
T = ⌊(θ_Moon − θ_Sun) / 12°⌋ + 1
Variables
θ_Moon = Moon's ecliptic longitude, θ_Sun = Sun's ecliptic longitude
Period: ~23h 37m to 26h 6m (varies due to elliptical orbits)
Worked Example
If Moon is at 150° and Sun at 30°: T = ⌊(150−30)/12⌋ + 1 = ⌊10⌋ + 1 = 11th Tithi (Ekadashi)
Extended Explanation
There are 30 Tithis in a lunar month (15 Shukla + 15 Krishna). Because the Moon's orbit is elliptical, a Tithi can range from under 20 hours to over 26 hours. A Tithi can be skipped (Kshaya) when a solar sunrise occurs within the same Tithi twice, or doubled (Adhika) when the Tithi spans two sunrises. The entire festival calendar is keyed to specific Tithis.
Nakshatra (Lunar Mansion)
नक्षत्र
Formula
N = ⌊θ_Moon / (360°/27)⌋ + 1 = ⌊θ_Moon / 13.333°⌋ + 1
Variables
θ_Moon = Moon's sidereal longitude. Each nakshatra spans 13° 20'.
Period: ~1 sidereal day (27.32 days / 27 = ~24.5 hours per nakshatra)
Worked Example
Moon at 46°: N = ⌊46/13.33⌋ + 1 = ⌊3.45⌋ + 1 = 4th Nakshatra = Rohini
Extended Explanation
The 27 Nakshatras divide the ecliptic into equal arcs of 13° 20'. Each is associated with a ruling planet, a deity, a symbol, and specific qualities. The Moon transits one nakshatra approximately every 24.5 hours — providing a daily astronomical coordinate system far more fine-grained than the 12 zodiac signs. Vedic months are named after the nakshatra where the full moon falls.
Yoga (Sun+Moon Sum)
योग
Formula
Y = ⌊(θ_Moon + θ_Sun) / 13.333°⌋ + 1 (mod 27)
Variables
Sum of both longitudes divided into 27 equal segments of 13°20'.
Period: Averages ~1 day but ranges 21.5h–26h
Worked Example
Moon at 150°, Sun at 30°: Y = ⌊180/13.33⌋ + 1 = 14th Yoga = Vajra
Extended Explanation
Yoga is unique to Vedic astronomy — it uses the combined longitudinal sum of Sun and Moon, providing a third independent coordinate. The 27 Yogas range from auspicious (Siddha, Amrita, Brahma) to inauspicious (Vishkambha, Vajra, Vyaghata). It measures the rate at which the combined Sun-Moon "energy" moves through the zodiac — an abstract but mathematically precise quantity.
Karana (Half-Tithi)
करण
Formula
K = ⌊(θ_Moon − θ_Sun) / 6°⌋ (mod 60) → map to 11 Karana types
Variables
4 fixed Karanas (Shakuni, Chatushpada, Naga, Kimstughna) + 7 repeating (Bava to Vishti)
Period: ~11h 45m to 13h 3m (half a Tithi)
Worked Example
Each Tithi contains exactly 2 Karanas. Vishti Karana = inauspicious for new work.
Extended Explanation
Karana divides each Tithi into two halves, providing an intra-day astronomical coordinate. The 11 Karana types repeat in a fixed pattern through the lunar month except for 4 fixed Karanas that appear only at specific positions. Vishti (also called Bhadra) Karana is specifically noted as inauspicious for new ventures — its timing correlates with a specific angular Moon-Sun relationship.
Vara (Weekday)
वार
Formula
R = (JD + 1.5) mod 7 (Julian Day Number)
Variables
0=Sunday, 1=Monday, 2=Tuesday, 3=Wednesday, 4=Thursday, 5=Friday, 6=Saturday
Period: Exactly 24 hours (solar day)
Worked Example
JD 2451545.0 (Jan 1.5, 2000) → (2451545 + 1.5) mod 7 = 6 → Saturday ✓
Extended Explanation
The 7-day week in Vedic astronomy is planetary: Ravi (Sun), Soma (Moon), Mangala (Mars), Budha (Mercury), Guru (Jupiter), Shukra (Venus), Shani (Saturn). The Hora system assigns each hour of the day to a planet in a specific sequence — the first hora of each day gives the day its name. This is identical to the system that produced Sunday-Monday-Tuesday... in Western calendars.