NIST's relativistic framework for lunar clocks

In plain language

Every popular article about lunar time quotes a number of microseconds per day. This paper is where the careful version of that number comes from ^[1]. Neil Ashby and Bijunath Patla, both at NIST, published it in The Astronomical Journal on 12 August 2024, and NIST announced it the same day ^[3]Official.

The paper does one job: given a position in the Earth-Moon system, it estimates how fast a clock there runs compared with a clock on Earth ^[1]. It covers the lunar surface and the Lagrange points, and it states an accuracy of a few nanoseconds per day ^[1].

It is not a standard and does not pretend to be one. A rate is a physical quantity; a time scale is an institution. The article on why clocks run faster on the Moon works through the two physical effects behind the rate ^[4].

What it decides

• A method, with a stated accuracy. The framework estimates clock rates on the Moon and at the Lagrange points to within a few nanoseconds per day ^[1].
• The surface rate. A clock at rest on the selenoid gains 56.0199(12) µs per day on an identical clock at rest on Earth’s geoid, with a small periodic term ^[1]. The selenoid is the lunar counterpart of the geoid.
• The coordinate rate. With the Moon’s own potential removed, a clock at the Moon’s distance runs 58.721 µs per day faster than an Earth clock ^[1]. That is the figure the 2024 OSTP memorandum rounds to 58.7 ^[5]Official.
• The lunar constants. The paper introduces the lunar potential constant and the selenoid by direct analogy with the terrestrial constant and the geoid ^[1], which is what lets later work reuse the definitions of IAU Resolution II ^[6]Official.

What it leaves open

• Everything normative. The paper computes and does not establish. No scale, no ensemble, no publication schedule follows from it ^[1].
• The realization. Which clocks would carry a lunar scale, where they would stand and who would weight them are questions for agencies, not for a journal ^[5]Official.
• The tie to UTC. Traceability to Coordinated Universal Time is a policy requirement stated elsewhere, and the paper does not address it ^[5]Official.

Key passages

The result the rest of the field quotes is a rate in microseconds per day, for a clock at rest on the selenoid against a clock at rest on Earth’s geoid. The constant term is the secular rate; the term in cos(f) is a small periodic variation over the lunar orbit ^[1].

56.0199(12) − 0.10843417·cos(f)

Written out with units, that is 56.0199(12) µs/day minus 0.10843417 µs/day multiplied by cos(f) ^[1]. Rounded to the two decimals this site uses, it is the 56.02 µs/day that appears on our clock methodology page ^[2].

Pair of clocks	Rate	Source
Clock at rest on the selenoid against a clock at rest on Earth’s geoid	56.0199(12) µs/day, less a periodic term of 0.10843417 µs/day in cos(f)	[1]
Clock at the Moon’s distance with the Moon’s own potential excluded, against an Earth clock	58.721 µs/day	[1]
Difference between the two, which is the Moon’s own potential at the selenoid	2.71 µs/day, from a monopole term of 3.14 × 10⁻¹¹	[7]
The same surface result in round numbers, as NIST states it	about 56 µs/day	[3]Official
A lunar coordinate time against Terrestrial Time, as the OSTP memorandum quotes it	58.7 µs per Earth-day	[5]Official

The paper also tabulates rates for the Earth-Moon Lagrange points L1 to L5 ^[8]Official. Those values are not reprinted here: the two rates above are the ones this site uses, and every number we display is recomputed from them as a model value ^[2]. Which pair of clocks each figure belongs to is the whole subject of 56 vs 58.7 microseconds, and the OSTP memorandum is where the rounded figure entered policy.