The equation of centre, the gist

The equation of centre

Every calculation of where the moon is starts with a pretend moon that moves at one steady speed, and a single line of arithmetic, the equation of centre, pulls that pretend moon most of the way back to the real one.

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

The moon a computer starts with and the moon you can see can be eight degrees apart, which is about fifteen full moons laid side by side. You can watch that gap open and close: it is why the moon appears to rock slowly from side to side, showing a little more of one edge and then a little more of the other.

Every program that works out where the moon is begins by pretending. It pretends the moon goes round the sky at one unchanging speed, like the hand of a clock. That pretend moon has a name, the mean longitude, and it is easy to work out, because it is nothing but a starting angle plus a fixed amount added for every day that has gone by since.

The real moon does not behave. Its path round us is a slightly squashed circle, so it is sometimes nearer to us and sometimes further, and it moves faster when it is nearer. Across a month it pulls ahead of the pretend moon and then falls behind it again.

The gap between the two can reach eight degrees. That is nearly fifteen hours of the moon's own travel. So the pretence has to be paid for, and the payment is one line of arithmetic with an old name: the equation of centre.

The line says: take how far the moon is past its closest point, take the sine of that angle, and multiply by 6.29 degrees. The 6.29 is not measured off the sky. It comes straight out of how squashed the orbit is.

The essay works it for four o'clock in the morning on 28 August 2026. The pretend moon says 339.33 degrees. The correction says take off 5.96, which gives 333.37. The real answer, from NASA's published tables, is 334.74. So the one correction was a little too big, and 1.37 degrees were left for smaller corrections to mop up.

It still does most of the job. Across thirty-five years of data the same line cuts the typical error from 4.57 degrees to 1.04. Put in practical terms: a lunar day that really ended at 4:18 in the morning would have been placed nine hours early with no correction, and under three hours late with this one.

Words decoded

longitude (in the sky): how far along the sky's main track something sits, counted in degrees from 0 to 360
ecliptic: that main track; the line the sun appears to follow through the year
mean longitude: where the moon would be if it moved at one steady speed and never varied; a useful fiction, not a real position
true longitude: where the moon actually is
perigee: the point in the moon's path where it is closest to us
apogee: the point where it is furthest
eccentricity: a number saying how squashed a circle is; the moon's orbit scores 0.0549, which is very slightly squashed
mean anomaly: how far the moon has gone since perigee, again counted at a steady pretend speed
equation of centre: the correction that turns the pretend position into very nearly the real one; here the word equation means a correction, not an equals sign
mandaphala (मन्दफल): the Sanskrit name for the same correction, literally the slow correction, worked in the classical texts from a circle-on-a-circle diagram and a table of sines
sine: a standard mathematical function of an angle, found on any calculator
sidereal month (27.32 days): the time the moon takes to go once round the sky
anomalistic month (27.55 days): the time from one perigee to the next; slightly longer, because perigee itself slowly moves
evection, variation, annual equation: the names of the next three corrections, all smaller, all caused by the sun's pull
tithi (तिथि): the Indian almanac's lunar day, which ends whenever the moon has gained another twelve degrees on the sun
purnima (पूर्णिमा): the full-moon tithi
ephemeris: a published table of where the sun, moon and planets were, hour by hour
Universal Time: the world's reference clock, kept at Greenwich; Indian Standard Time runs five and a half hours ahead
Terrestrial Time: a steadier scientific clock used in these formulas, currently 69.184 seconds ahead of the civil one
libration: the slow rocking of the moon's face, which lets us see a little way round each edge in turn

Why anyone should careFasting days and festival dates are decided from the times a tithi begins and ends, and those times are computed from the moon's position. Skip this one correction and a boundary can move by nine hours, which is enough to land on the other side of a sunrise and change which day carries the name. The correction itself is one multiplication. Anyone who can find a sine on a calculator can check the number their almanac was built on, which is a better relationship to have with a printed sheet than trust or suspicion.

Read it properly, 7 min The equation of centre