How a tithi is worked out

The Moon's orbit is an ellipse. At perigee, its closest point, the Moon covers more than fifteen degrees in a day. At apogee it manages under twelve. The gap it is opening against the Sun therefore widens fast in some weeks and slowly in others, and a twelve degree arc takes as little as about nineteen hours or as much as about twenty six.

The Earth's orbit is an ellipse too, so the Sun's own apparent speed changes through the year, from about 61 arcminutes a day in January to about 57 in July. The Sun is the thing the Moon is pulling away from, so that moves the gap as well.

Which is why a tithi is not a day, and why a panchangam has to say which tithi was running at some particular moment. A tithi shorter than the stretch between two sunrises can begin and end without ever touching a sunrise, and that is a skipped tithi, a kshaya. A tithi longer than that stretch can cover two sunrises and be counted twice. Both happen several times a year.

No ayanamsa appears anywhere on this page, and that is not an oversight. The ayanamsa is subtracted from the Sun and from the Moon alike, so it cancels out of the gap between them. Whichever ayanamsa you prefer, the tithi boundaries are the same to the millisecond. The solver checks that across five of them.

Honest note. Two paths, because there are two. The Moon goes round the Earth. The circle the Sun travels is our own orbit seen from here, which is what makes it look like the Sun is doing the going. Neither of them is the ring with the ticks on it: that ring is longitude, the measuring circle, and it is why each line runs from the Earth past its body and out to the rim. What a tithi counts is the angle between those two directions. Distance does not enter it anywhere.

Which is lucky, because nothing here is to scale and nothing could be. The Sun is about 390 times further off than the Moon, so with its path drawn where it is, the whole of the Moon's orbit would fit inside one pixel. Both paths are ellipses, and both are drawn as ellipses now, fitted to this month's own distances rather than assumed, with the Earth at a focus. But they are drawn four times further out of round than they are. An orbit this round does not look like a squashed circle at any honest scale: all you would see is the Earth sitting a few percent off the middle, which at this size is a couple of units. The real figures are printed under the picture, and they are one of the reasons no two tithis are the same length: the bars above are that unevenness, measured. The Moon's path is also tilted about five degrees out of the Sun's, which is why there is not an eclipse every month, and that tilt is not drawn either, because a tithi is defined on longitude alone and latitude never enters it. The positions come from astronomy-engine, pinned, which agrees with the worked examples in Meeus to 2.3 arcseconds for the Moon and 10.6 for the Sun.