Formula for New Moons
A graphic depicting the new moon phase
The length of a lunation is about 29.53 days. Its precise duration is linked to many phenomena in nature, such as the variation between spring and neap tides (the most and least profound tidal variances respectively).
An approximate formula to compute the mean moments of new moon (conjunction between Sun and Moon) for successive months is:
where N is an integer, starting with 0 for the first new moon in the year 2000, and that is incremented by 1 for each successive synodic month; and the result d is the number of days (and fractions) since 2000-01-01 00:00:00 reckoned in the time scale known as Terrestrial Time (TT) used in ephemerides.
To obtain this moment expressed in Universal Time (UT, world clock time), add the result of following approximate correction to the result d obtained above:
Periodic perturbations change the time of true conjunction from these mean values. For all new moons between 1601 and 2401, the maximum difference is 0.592 days = 14h13m in either direction. The duration of a lunation (i.e. the time from new moon to the next new moon) varies in this period between 29.272 and 29.833 days, i.e. −0.259d = 6h12m shorter, or +0.302d = 7h15m longer than average. This range is smaller than the difference between mean and true conjunction, because during one lunation the periodic terms cannot all change to their maximum opposite value.
See the article on the full moon cycle for a fairly simple method to compute the moment of new moon more accurately.
The long-term error of the formula is approximately: 1 cy2 seconds in TT, and 11 cy2 seconds in UT (cy is centuries since 2000; see section Explanation of the formulae for details.)
Explanation of the formula
Approximately a new Moon in the sky of Earth during sunset
The moment of mean conjunction can easily be computed from an expression for the mean ecliptical longitude of the Moon minus the mean ecliptical longitude of the Sun (Delauney parameter D). Jean Meeus gave formulae to compute this in his Astronomical Formulae for Calculators based on the ephemerides of Brown and Newcomb (ca. 1900); and in his 1st edition of Astronomical Algorithms based on the ELP2000-85 (the 2nd edition uses ELP2000-82 with improved expressions from Chapront et al. in 1998). These are now outdated: Chapront et al. (2002) published improved parameters. Also Meeus's formula uses a fractional variable to allow computation of the four main phases, and uses a second variable for the secular terms. For the convenience of the reader, the formula given above is based on Chapront's latest parameters and expressed with a single integer variable, and the following additional terms have been added:
- Sun: +20.496"
- Moon: −0.704"
- Correction in conjunction: −0.000451 days[note 3]
- For UT: at 1 January 2000, ΔT (= TT − UT ) was +63.83 s;[note 4] hence the correction for the clock time UT = TT − ΔT of the conjunction is:
- −0.000739 days.
- In ELP2000–85 (see Chapront et alii 1988), D has a quadratic term of −5.8681"T2; expressed in lunations N, this yields a correction of +87.403×10–12N2[note 5] days to the time of conjunction. The term includes a tidal contribution of 0.5×(−23.8946 "/cy2). The most current estimate from Lunar Laser Ranging for the acceleration is (see Chapront et alii 2002): (−25.858 ±0.003)"/cy2. Therefore, the new quadratic term of D is = -6.8498"T2.[note 6] Indeed, the polynomial provided by Chapront et alii (2002) provides the same value (their Table 4). This translates to a correction of +14.622×10−12N2 days to the time of conjunction; the quadratic term now is:
- +102.026×10−12N2 days.
- For UT: analysis of historical observations shows that ΔT has a long-term increase of +31 s/cy2. Converted to days and lunations,[note 7] the correction from ET to UT becomes:
- −235×10−12N2 days.
The theoretical tidal contribution to ΔT is about +42 s/cy2 the smaller observed value is thought to be mostly due to changes in the shape of the Earth. Because the discrepancy is not fully explained, uncertainty of our prediction of UT (rotation angle of the Earth) may be as large as the difference between these values: 11 s/cy2. The error in the position of the Moon itself is only maybe 0.5"/cy2,[note 8] or (because the apparent mean angular velocity of the Moon is about 0.5"/s), 1 s/cy2 in the time of conjunction with the Sun.