Kshaya and vriddhi tithis across a decade

Background

In the panchang a civil day is named after the Tithi that is current at sunrise. A tithi lasts while the Moon gains twelve degrees on the Sun. Because the Moon moves faster near perigee and slower near apogee, a tithi can last from a little under twenty hours to a little over twenty-six. A civil day, sunrise to sunrise, is close to twenty-four hours.

Two consequences follow. When a short tithi begins after one sunrise and ends before the next, no day is named after it. The tradition calls it kshaya, a lost or diminished tithi. When a long tithi spans two sunrises, two consecutive days carry the same tithi name. This is vriddhi, an increased tithi.

Neither is an anomaly. Over a lunar month of thirty tithis and about twenty-nine and a half days, the count must lose roughly one tithi a month on average, so kshaya tithis are expected to outnumber vriddhi tithis.

Method

1. Fix a city and record its coordinates and time zone.
2. Fix the sunrise convention: upper limb of the Sun on the geometric horizon with standard refraction, or the centre of the disc. State it, because a change of convention moves sunrise by a minute or two and can flip borderline cases.
3. For each day in a ten-year span, compute the Sun–Moon elongation at local sunrise and hence the tithi number.
4. Compare consecutive sunrises. A jump of two in tithi number indicates a kshaya; no change indicates a vriddhi.
5. For each event, record the Gregorian date, the amanta lunar month, the tithi involved and the Moon's angular speed at that time.
6. Count events per year and per month and compare with a printed panchang for spot checks.

What we expect†

We expect kshaya tithis to be somewhat more frequent than vriddhi tithis across the decade, in keeping with the arithmetic above. We expect both to cluster according to the Moon's speed: kshaya near perigee, when tithis are short, and vriddhi near apogee, when they are long. Because the perigee itself moves around the lunar month over a cycle of roughly 8.85 years, the months in which these events cluster should drift across the decade.

We also expect location to matter at the margins. Two cities a few degrees of longitude apart have different sunrise times, so a tithi that ends a few minutes after sunrise in one may end before sunrise in the other. That is why two almanacs printed for different places can list a kshaya on different days.

Why it matters

Kshaya and vriddhi tithis decide which day a vrata or festival is observed, and the classical rule books, including the dharmashastra digests, give detailed rules for these cases. A clear count also helps readers of our article on tithis understand why a fast may be kept on a different day from the one a calendar app suggests.

Limits

The method studies one location with modern computation. It does not reproduce the counts of almanacs that follow the Sūrya Siddhānta, which would need its own run. Rules for festival observance involve more than sunrise and are outside the scope of this note.

† Marked dates or readings are debated among scholars or still being confirmed by our faculty.