How we calculate lunar time

Everything the lunar clock does, written out: one formula, two rate figures, three epochs, and an explicit list of what the model leaves out.

The formula

The clock on the lunar clock page evaluates one expression and shows the result on four lines:

LTC_model(t) = UTC(t) + r · (t − t0)

• t — the current instant, taken from the browser clock and used only in UTC.
• t0 — the epoch you choose: the TCL epoch of 1977 January 1, J2000.0, or a date of your own. ^[4]Official
• r — the rate figure you choose, in microseconds per day, always carrying the pair of scales it compares. ^[2]
• LTC_model(t) — UTC plus the accumulated offset. It is a model value: no lunar time scale has been realized, so there is no clock anywhere that reads it.^[5]Official

The expression is deliberately thin. Everything that would make it thicker — periodic terms, tides, the position of a particular station — is listed below under what we do not model, rather than folded in silently.

The same expression, applied to two dates you name rather than to the current instant, isthe drift calculator; the offset it returns becomes a distance on Earth-Moon light time, which is why any of this matters to navigation at all.

Which two scales each rate compares

A rate in microseconds per day is meaningless without the pair of clocks it compares. The two figures in circulation are not rivals; they answer different questions.

Epochs we offer

• 1977 January 1 — the TCL epoch. The IAU resolution fixes TCL to read 1977 January 1, 0h 0m 32.184s when TCB reads the same at the centre of the Moon; our model counts from 1977-01-01T00:00:00Z in UTC. ^[4]Official
• J2000.0 — defined as 2000 January 1, 12:00 TT, which in UTC is 2000-01-01T11:58:55.816Z, not noon UTC. The two differ by 64.184 s.^[1]
• A custom date — any UTC date, including dates before the epoch. Then the model runs behind rather than ahead, and the counter shows the sign explicitly.^[1]

That J2000.0 detail is worth 64.184 s of pedantry: taking noon UTC instead gives 9 730.500000 days rather than 9 730.500743 days, and so 545 102.61 µs of accumulated drift rather than 545 102.65 µs — a difference of 0.042 µs at 56.02 µs/day, lunar surface against Earth geoid. Nobody will ever notice it. We take it correctly anyway, because it costs nothing.^[1]

What we round

• The published secular rate is 56.0199(12) µs/day for a selenoid clock against a geoid clock, and we use 56.02 µs/day. ^[2]
• The same published rate carries a periodic term of about 0.108 µs/day, varying with the phase of the Moon's orbit. Our model drops it and keeps the secular part only.^[6]Official
• For TCL against TT the published figure is 58.721 µs/day, and we use 58.7 µs/day, because that is the figure the OSTP memorandum states.^[3]Official

Rounding 58.721 down to 58.7 costs 380.75 µs of accumulated TCL-against-TT drift over the 18 131 days since the epoch. We prefer the official figure to the more precise one here, and say so, rather than quietly publishing a number no document contains. ^[1]

Leap seconds and the JavaScript clock

JavaScript's Date treats UTC as uniform: it does not know leap seconds. Elapsed time computed from epoch milliseconds therefore misses every leap second inserted between the epoch and now, and our model inherits that.

The error has a hard ceiling. Even if every second of the current TAI − UTC difference, 37 s, were missing from the interval, at 56.02 µs/day for a lunar surface clock against an Earth geoid clock that is 0.0240 µs of drift — against 1 015 698.62 µs accumulated since the 1977 epoch. The true figure is smaller still, because most of that offset was inserted before 1977. We do not correct for it, and we would rather write the paragraph than hide the approximation. ^[1]

Why the microsecond field is a model, not a measurement

A browser clock reports milliseconds. High-resolution timers are deliberately coarsened by browsers, so no page can honestly display the current instant to the microsecond — ours included.

That is why the clock is built the way it is: the EARTH · UTC line has no microsecond field at all, and the lunar lines carry their microseconds in a separate, smaller field that shows the modelled offset — arithmetic on a rate figure, not a reading. The caption under the clock says it in one sentence: the microsecond field shows a modelled offset, not a measurement, and browser clocks are accurate to milliseconds at best.^[1]

What we do not model

• Periodic terms in the rate, including the term of about 0.108 µs/day. ^[2]
• Tidal effects and the deformation of both bodies.
• Orbital eccentricity and the varying Earth-Moon distance.
• Station-dependent terms: altitude, latitude, and velocity of a particular site on the lunar surface. ^[6]Official
• Any realization: there is no ensemble of clocks on the Moon, no steering, and no published offset series to compare against. ^[5]Official

Every constant above is quoted from a document in our registry, and every derived number is our own calculation, marked as such. How we pick those documents, and what happens when one of these numbers turns out to be wrong, is set out in about lunartime.org.

Reference values you can check

Every number below is produced by the same functions that drive the clock, evaluated for 2026-08-23. The working is printed next to the result so you can redo it on paper.

F5 and F6 are there for one reason: a leap year must add exactly one more day of drift, 56.02 µs at 56.02 µs/day for a lunar surface clock against an Earth geoid clock, and not "about the same". ^[1]

You can redo any row of that table without a calculator: the JSON API runs these same functions and returns the interval, the rate and the result as data, with the sources attached to every number. The same functions drive the widget you can put on your own page — embed the lunar clock — so a number shown there is this arithmetic and nothing else.

How to store a lunar timestamp so that it survives a revision of the rate, and what the API and the lunar-time package each give you, is set out in lunar time for developers.

Last verified 23 August 2026