Finding the exact moment a tithi turns, the gist

Finding the exact moment a tithi turns

The Indian almanac's lunar day ends at an exact instant that no table lists, and you can find that instant to within a second or two using two lines of an ephemeris and one division.

This is the short way in. The essay has the working, the sources and the sky.

The moment a tithi ends is the same moment for the whole world. It is not a local sunrise or a local midnight. It is one instant, and the only thing that changes from country to country is what the clock on the wall says when it happens.

The Indian almanac counts a kind of day that has nothing to do with the sun coming up. It is called a tithi, and it measures how far ahead of the sun the moon has got.

Picture the sun and the moon both moving along the same track across the sky. The sun goes round that track in a year. The moon goes round in about a month, so the moon keeps overtaking. The gap between the two, measured along the track, is called the elongation. It is zero when they sit together, which is a new moon. It grows all month. It reaches 180 degrees when they are on opposite sides of us, which is a full moon. Then it shrinks back to zero.

Every time that gap grows by another twelve degrees, one tithi ends and the next begins. Twelve degrees, thirty times over, is 360, which is the whole month.

So finding the exact moment a tithi ends means finding the exact moment the gap passes a multiple of twelve. No table lists that moment. What you can get is a table of where the moon and the sun were on the hour.

The essay takes two such lines, an hour apart, for the morning of 28 August 2026. Subtract one longitude from the other and the gap at four o'clock is 179.8481 degrees. Do it again for five o'clock and it is 180.3401. The number 180 sits between them, so the full-moon tithi ends inside that hour.

Now the only real step. At four o'clock the gap had 0.1519 degrees left to grow, and across the hour it grew 0.4920 degrees. Divide the first by the second and you get 0.3087, so the crossing falls 0.3087 of the way through the hour: about eighteen and a half minutes past four.

Two published almanacs print the same minute. The essay then asks how far that answer can be trusted, and shows what goes wrong when the two lines you have are a day apart instead of an hour.

Words decoded

tithi (तिथि): the almanac's lunar day; the time the moon takes to gain another twelve degrees on the sun
purnima (पूर्णिमा): the full-moon tithi, the fifteenth one, which ends when the gap reaches 180 degrees
elongation: the gap between the moon and the sun, measured along the sky's main track
ecliptic: that track; the line the sun appears to follow through the year
longitude (in the sky): how far along that track something sits, counted in degrees from 0 to 360
ephemeris: a table of where the sun, moon and planets were, hour by hour or day by day
interpolation: working out a value that falls between two rows of a table
linear interpolation: doing that by assuming the change between the two rows is steady
arcminute: a sixtieth of a degree; the full moon is about thirty arcminutes wide
panchang (पञ्चाङ्ग): the Indian almanac, the printed or online sheet that lists these times
ayanamsha (अयनांश): the offset between two ways of numbering the sky; it cancels out here, because it is subtracted from both bodies
Universal Time: the world's reference clock, kept at Greenwich; Indian Standard Time runs five and a half hours ahead of it

Why anyone should careFast days, festival dates and family observances are decided from these boundary times, and almost nobody is shown where the times come from. They come from one subtraction and one division on numbers anybody can look up. Knowing that changes what a printed almanac is. It stops being an authority to be trusted or doubted and becomes a piece of work you could check yourself, including the parts where two honest sheets disagree by a minute.

Read it properly, 7 min Finding the exact moment a tithi turns