Panchanga Formulations & Root-Finding Mathematics

1. Limb 1: Tithi (Lunar Phase Elongation)

A Tithi is the temporal interval required for the Moon’s celestial longitude to advance by exactly 12 degrees ($12^\circ$) relative to the Sun. The synodic lunar month comprises $360^\circ / 12^\circ = 30$ Tithis, divided into two lunar fortnights:Śukla Pakṣa (Tithis 1–15, culminating in Pūrṇimā) andKṛṣṇa Pakṣa (Tithis 16–30, culminating in Amāvāsyā).

2. Limb 2: Nakṣatra & Pāda

The sidereal zodiac of $360^\circ$ is divided into 27 equal lunar mansions (Nakṣatras), each spanning an arc of $360^\circ / 27 = 13^\circ 20' = 13.3333^\circ$. Each Nakshatra is further subdivided into 4 quarters (Pādas) of $13^\circ 20' / 4 = 3^\circ 20' = 3.3333^\circ$, creating 108 padas matching the Navāṁśa D9 harmonic division.

3. Limb 3: Solilunar Yoga

Unlike Tithi (which measures the difference between lunar and solar longitudes),Yoga measures the sum of the Sun and Moon’s sidereal longitudes, divided into 27 equal segments of $13^\circ 20'$.

4. Limb 4: Karaṇa (Half-Tithi Intervals)

A Karaṇa is equal to half a Tithi, spanning 6 degrees ($6^\circ$) of lunar elongation. In a 30-Tithi lunar month, there are exactly 60 Karaṇas composed of:

• 4 Fixed (Sthira) Karaṇas: Occur only once per month: Śakuni (Kṛṣṇa 14-2), Catuṣpada (Amāvāsyā-1), Nāga (Amāvāsyā-2), and Kiṃstughna (Śukla 1-1).
• 7 Movable (Cara) Karaṇas: Repeat cyclically 8 times across the remaining 56 half-tithis: Bava, Bālava, Kaulava, Taitila, Gara, Vaṇija, and Viṣṭi (Bhadrā).

5. Continuous Boundary Root-Finding via Brent’s Algorithm

Finding the exact second a Tithi, Nakshatra, or Yoga concludes requires locating the root $t^*$ of the continuous nonlinear angular function:

def find_boundary_transition(t_start, t_end, target_boundary, arc_func):
    """Brent's method root-finder with continuous circular phase unwrap."""
    def objective(t):
        val = arc_func(t) % 360.0
        diff = (val - target_boundary) % 360.0
        if diff > 180.0:
            diff -= 360.0
        return diff
    
    return scipy.optimize.brentq(objective, t_start, t_end, xtol=1e-5)