The chase, the wobble and the brake

Two numbers have been sitting quietly in every table so far: the Moon orbits Earth once every 27.32 days (the sidereal month), yet the cycle of phases — new moon to new moon — takes 29.53 days (the synodic month). They differ by more than two days, and that gap is not a rounding error or an approximation. It is a real, calculable consequence of the fact that Earth itself is moving.

This layer is about the why behind everything Understand told you to accept: why the two months differ by exactly the amount they do, why a tithi is never quite 24 hours, why an extra month has to be inserted into the Hindu lunar calendar every few years, why eclipses only happen a handful of times a year instead of every month, and why the Moon is quietly leaving us at a rate you could almost measure with a ruler and a stopwatch, if the stopwatch ran for a very long time.

Expect harder arithmetic than before, some genuine derivations, and a few numbers that will surprise you.

Picture two runners on a circular track, both running the same direction. The Moon covers the 360° track in 27.32 days (it laps Earth once every sidereal month). The Sun — really Earth's own motion around it, seen from Earth — covers the same 360° in 365.256 days (one year).

The Moon is the faster runner. "New moon" is the moment the Moon catches up to the Sun's position in the sky and laps it exactly once. So the question "how long is a synodic month?" is really "how long until the faster runner gains one full lap on the slower one?" — a classic overtaking problem, solvable with nothing but ratios.

Here is a clean consequence of the same chase, easy to miss and satisfying once seen.

In one year, how many sidereal months fit (laps of the Moon around Earth against the stars)? And how many synodic months fit (new moon to new moon)?

Understand told you a tithi is one thirtieth of a lunar month — 23.62 hours on average. "On average" is doing real work in that sentence. A tithi is not defined as a fixed slice of time at all; it is defined as the time the Moon takes to gain exactly 12° of elongation on the Sun, and that time genuinely stretches and shrinks through the month.

Twelve synodic months make a lunar year of 354.37 days. A solar (tropical) year, the one the seasons actually follow, is 365.24 days. The gap is 10.88 days every single year — over ten days, gone missing, unless something is done about it.

Left unpatched, festivals tied to a purely lunar calendar would drift steadily earlier every year, cycling all the way through the seasons over a few decades. (You will meet a calendar that actually does this in the next chapter.) The Hindu calendar avoids that drift by being lunisolar: it keeps lunar months, but reinserts a whole extra month, an adhik maas, whenever the shortfall has built up to about one lunar month's worth.

If the Moon's orbit lay exactly in the same plane as Earth's orbit around the Sun, every single new moon would produce a solar eclipse and every full moon a lunar eclipse — twelve or thirteen of each, every year. That obviously is not what happens. The reason is a single number: the Moon's orbit is tilted 5.145° to the plane of Earth's orbit (the ecliptic).

Twice a year, on average about 173.31 days apart, the Sun's apparent path carries it close to one of these nodes. Each such eclipse season lasts roughly a month, during which the new moon and/or full moon that falls inside it has a real chance of producing an eclipse. Outside the two eclipse seasons, the geometry simply cannot work, no matter how new or full the Moon is.

Understand told you the Moon always shows the same face because it is tidally locked: its rotation period exactly equals its orbital period. This chapter explains the actual mechanism, which is a beautiful piece of physics involving nothing but gravity and a slight lag.

Billions of years ago, the young Moon almost certainly spun much faster than it orbited Earth, the way most moons and planets do when they form. But Earth's gravity does not pull equally on every part of the Moon: it pulls harder on the near side than the far side, because the near side is closer. That difference in pull stretches the Moon very slightly into an elongated shape, with a bulge pointing roughly towards Earth and another pointing roughly away — a tidal bulge, the same effect that raises ocean tides on Earth (that connection is explored fully in the tides topic).

The same tidal bulge that locked the Moon's rotation is still doing work today, on a much larger and slower scale: it is very gradually pushing the Moon away from Earth, at a measured rate of about 3.8 cm per year.

The mechanism is the mirror image of Chapter 6's story, but now the bulge in question is Earth's ocean tidal bulge, and it is Earth's much faster 24-hour spin (compared with the Moon's monthly orbit) that drags that bulge slightly ahead of the Earth-Moon line. Earth's gravity, acting through that misaligned bulge, now speeds the Moon up very slightly along its orbit. A slightly faster-moving satellite settles into a slightly higher, wider orbit — so the Moon spirals gently outward, while by Newton's third law, Earth's spin correspondingly slows down (this is the same mechanism from Chapter 6, running today, with Earth and Moon swapping roles).

The Moon's elliptical orbit means its distance from Earth genuinely varies, between roughly 363,300 km at its closest (perigee) and 405,500 km at its farthest (apogee) in an extreme month. A popularly reported "supermoon" is a full moon that happens to fall close to perigee. How much bigger does it actually look?

A last set of mix-ups, sharper than Understand's, because they trip up people who already know the basics.