# Calculating the time of New (and Full) Moon

## Author: Dr A.R. (Tom) Peters,

### ©2001, ©2022 Dr A.R. (Tom) Peters

## Note:

Derived from "Wikipedia: New Moon"

The calculation details on that Wikipedia page were largely added by me (Tom Peters) in 2001 and later. It was removed on 14 April 2021 because Wikipedia is not a manual. I have been hosting this more elaborate artice on my personal website since 6-Oct-2022.

# Introduction

In astronomy, the New Moon is the first lunar phase, when the Moon and Sun are in conjunction in ecliptic longitude, *i.e.* when they have the same ecliptic longitude (see Astron.Alg. (1998) Ch.49 p.349; Expl.Suppl.Astron.Alm. (2013) 12.2.1.3, p.507). At this phase the lunar disk is not visible to the unaided eye, except when silhouetted during a solar eclipse.

A **lunation** or synodic month is the time from one New Moon to the next. Around present, the average length of a lunation is about 29.53059 days (29 days, 12 hours, 44 minutes, 3 seconds). However, the length of any one synodic month can vary from 29.27 to 29.83 days due to the perturbing effects of the Sun's gravity on the Moon's eccentric orbit (see S&T Nov.1993 pp.76,77; More Math.Astron.Morsels (2002) Ch.4 pp.29,30). In a lunar calendar, each month corresponds to a lunation. Each lunar cycle can be assigned a unique lunation number to identify it.

# Formula for New Moons

An approximate formula to compute the mean moment of New Moons (mean conjunction in ecliptic longitude between Sun and Moon) for successive months is:

d = 5.597661 + 29.5305888610 × N + ( 102.026 × 10^{−12} ) × N^{2} [1a]

where **N** is an integer, starting with 0 for the first New Moon in the year 2000 C.E., and is incremented by 1 for each successive lunation; 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 the following approximate correction to the result **d** obtained above:

−ΔT = − 0.000739 − ( 235 × 10^{−12} ) × N^{2} days [1b]

Periodic perturbations change the time of true conjunction or opposition 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 (see the two references quoted in the Introduction above), *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 my article on the Full Moon Cycle for a fairly simple method to compute the moment of New (and Full) Moon more accurately.

The long-term error of the formula is approximately: 1 cy^{2} seconds in TT, and 11 cy^{2} seconds in UT (**cy** is centuries since 2000; see section Derivation of the formulae for details).

## Formula for Full Moon

For mean Full Moon (mean opposition of Moon to Sun in ecliptic longitude) the same formula applies but delayed by a half mean lunation: +14.765294 days: so the constant term for Full Moons is:

20.362955 days TT into January 2000.

# Derivation of the formulae

The moment of mean conjunction can easily be computed from an expression for the mean ecliptic longitude of the Moon minus the mean ecliptic longitude of the Sun (mean elongation: Delauney variable D), taken from a lunar theory. From the lunar and solar theory the various perturbation terms can be added and subtracted to obtain a series expansion to compute the elongation more precisely. P.A.Hansen published (1857) expressions to compute the time of conjunction directly, and apply them for eclipse calculations; he explained the derivation of the formulae in 1863.

Jean Meeus derived similar formulae for his "Astronomical Formulae for Calculators" (1979) based on the ephemerides of E.W. Brown and S. Newcomb (ca. 1900). In his first edition of "Astronomical Algorithms" (1991) he based them on the ELP2000-85 (Chapront *et alii* 1988, 1991). The value for the mean lunation length of 29.530588853 days comes from this work: it is quoted often in Wikipedia and elsewhere, but is somewhat outdated. In the second edition (1998) Meeus used ELP2000-82 with improved expressions from Chapront *et alii* (1997). Chapront *et alii* published more improved parameters in 2002, and I used those to derive the formula above and put it into Wikipedia in 2003.

## Derivation from D

In "More Morsels" Ch.5 pp.34..35, Meeus briefly explains how to compute the mean lunation length from the expression of variable **D**.

From Chapront *et alii* 2002 table 4:

D = 297°51'00.6902" + 1602961601.0312"×t − 6.8498"×t^{2} + 0.006595"×t^{3} − 0.00003184"×t^{4}

Converted to revolutions (*i.e.* divide by 60×60×360):

D = 0.8273616437 + 1236.8530872154×t − 5.2853E-6×t^{2} + 5.089E-9×t^{3} -24.57E-12×t^{4}

Derivative:

v = 1236.8530872154 − 10.5707E-6×t + 15.266E-9×t^{2} − 98.27E-12×t^{3} revolutions per Julian century (of 36525 days)

= 1236.8530872154 × (1 − 8.54643E-9×t + 12.343E-12×t^{2} − 79.45E-15×t^{3})

So the polynomial for the length of the synodic month around epoch J2000.0 is:

(36525 d/cy) / [ 1236.8530872154 rev/cy × (1 − 8.54643E-9×t + 12.343E-12×t^{2} − 79.45E-15×t^{3}) ] = 29.530588860986 (d/rev) / (1 − 8.54643E-9×t + 12.343E-12×t^{2} − 79.45E-15×t^{3})

This has the form p/(1−f), and to first order of approximation this can be re-written as p×(1+f) . This is accurate for our purpose because f is very small (8 orders of magnitude smaller than the main term):

P = 29.530588860986 + 252.381E-9×t − 364.49E-12×t^{2} + 2.346E-12×t^{3}

Change unit from t (Julian centuries of 36525 days) to N (lunation of 29.53059... days) by multiplying by (29.53059.../36525)^{n} :

P = 29.530588860986 + 204.051E-12×N - 238.26E-18×N^{2} + 1.2400E-21×N^{3}

D is 0 modulo 1 after (1−0.8273616437)×29.530588860986 = 5.098112 days after the epoch. Integrating now with this offset as the integration constant yields:

d = 5.098112 + 29.530588860986×N + 102.026E-12×N^{2} - 79.42E-18×N^{3} + 310.0E-24×N^{4}

N=0 for the first mean New Moon in 2000, and that is actually the epoch of the formula. At that time (0.17262308 revolutions after J2000) the period has already increased to 29.530588861021 days, so round to 10 decimals.

### The quadratic term

In ELP2000–85 (Chapront *et alii* 1988), D has a quadratic term of −5.8681"×T^{2}. To express this in lunations N, divide this number by the factor −(60×60×360 "/circle) × (36525/29.53059... lunations per Julian century)^{2} / (29.53059... days/lunation) = −67.138105E+9 . This yields a correction of +87.403×10^{–12}×N^{2} days to the mean time of conjunction. The term includes a contribution from the tidal acceleration of the Moon of 0.5×(−23.8946 "/cy^{2}), which was the value obtained after about a decade of Lunar Laser Ranging (LLR) measurements. At the time of the article by Chapront *et alii* in 2002, after 2 more decades of measurements, an improved value had been obtained: (−25.858 ±0.003)"/cy^{2}. Therefore, the correction to the quadratic term of D is +0.5×(−25.858 − −23.8946)" = −0,9817"/cy^{2}. This translates to a correction of +14.622×10^{−12}×N^{2} days to the time of conjunction. The new quadratic term of D becomes −5.8681" − 0,9817" = −6.8498"×T^{2}. Indeed, the polynomial provided by Chapront *et alii* (2002) has the same value (their Table 4). The new quadratic term for the conjunction then becomes:

+102.026×10^{−12}×N^{2} days, as given in the polynomial of formula [1a] above.

The latest published value for the tidal acceleration appears to be: (−25.97 ±0.05)"/cy^{2} (Williams and Boggs 2016).

## Corrections

To compute the actual apparent phase as seen by an observer on the surface of the Earth, many periodic corrections must be added to the time of mean conjunction computed above. This is beyond the scope of this article (again, see my article on the Full Moon Cycle for the most important periodic corrections). However constant and polynomial offsets can be easily added to the polynomial derived above.

### start of day

Add 0.5 days to the constant to shift the zero point to midnight instead of noon (epoch 2000.0 is JD 2451545 falls at noon): 5.598112 days.

### for aberration

We can easily add – as Meeus did as well – one correction that is a constant offset, namely the constant terms of the aberration.- Annual aberration is the ratio of Earth's orbital velocity (around 30 km/s) to the speed of light (about 300,000 km/s), which shifts the Sun's apparent position relative to the celestial sphere toward the west by about 1/10,000 radian.
- Light-time correction for the Moon is the distance it moves during the time it takes its light to reach Earth divided by the Earth-Moon distance, yielding an angle in radians by which its apparent position lags behind its computed geometric position.
- Light-time correction for the Sun is negligible because it is almost motionless relative to the barycenter (center-of-mass) of the solar system during the 8.3 minutes that light travels between Sun and Earth.
- The aberration of light for the Moon is also negligible: the center of the Earth moves too slowly around the Earth-Moon barycenter (0.002 km/s).
- Finally the so-called diurnal aberration, caused by the motion of an observer on the surface of the rotating Earth (0.5 km/s at the equator), is variable and therefore cannot be added to the polynomial, but it is small enough to be ignored.

Although aberration and light-time are often combined as "planetary aberration" (Astron.Alg. (1998) Ch.33 p.224), Meeus separated them for the computation of the apparent conjunction of Sun and Moon.

The constant term of the annual aberration is Derived Constant No. 14 from the IAU (1976) System of Astronomical Constants (p.58): 20,49552"; also see Seidelmann (1977).

Expressions for the aberration in the Moon's apparent position were derived by Clemence *et alii* (1952). Using the later conventional value for the Moon's mean distance of 384400 km, I compute a value that gives a different rounding in the last digit: -0.704".

The apparent mean solar longitude is (rounded) −20.496" from mean geometric longitude; the apparent mean lunar longitude −0.704" from mean geometric longitude. The correction to D = Moon − Sun is −0.704" + 20.496" = +19.792" that the apparent Moon is ahead of the apparent Sun; divided by 360×3600"/circle is 1.527×10^{−5} part of a circle; multiplied by 29.530... days for the Moon to travel a full circle with respect to the Sun is 0.000451 days that the apparent Moon reaches the apparent Sun ahead of time. So subtract 0.000451 days from the constant term 5.598112 of the polynomial derived from D, to correct for aberration: this yields the constant value of 5.597661 days after midnight of 1 Jan. 2000 (TT) in the formula [1a] at the beginning of this article.

### for clock time

- Ephemerides use their own time scale (ephemeris time), which these days is defined as Terrestrial Time:
**TT**. The expressions derived above are valid in TT. - On Earth, we count time with atomic clocks that provide the International Atomic Time scale:
**TAI**. - World time, known as Universal Time (
**UT**), is based on the somewhat irregular rotation of the Earth. Because it determines what is above the horizon, including the Sun, it remains important for astronomical and civil use. - World clock time, known as Coordinated Universal Time (
**UTC**) is counted with atomic clocks, so based on**TAI**, but kept within 0.9s from**UT**by adding a leap second whenever necessary: so it increasingly differs from TAI.

TT was equal to UT around 1900 C.E., and TAI was set equal to UT at 1 January 1958 C.E. In the intermediate half century, UT had slowed down, so the difference between TT and TAI is around 32.184 seconds.

The difference between ephemeris time (TT) and world time (UT) is known as ΔT: it was 0 around 1900 C.E. and is slowly increasing.

The International Earth Rotation and Reference Systems Service (IERS) keeps track of these time scales; they provide TAI−UTC here and UT1−UTC here; ΔT = 32.184s + (TAI−UTC) − (UT1−UTC).

At 1-Jan-2000 TAI-UTC was 32s and UT1-UTC was 0.3553880s. So ΔT(2000) = 32.184 + 32 - 0.355 = 63.829s = TT - UT; dividing by (24*60*60 = 86400) yields 0.000739 days. UT = TT - ΔT so subtract the number from the constant term to convert from TT to UT time scale, as specified in formula [1b] above.

#### quadratic term for clock time:

Analysis of historical observations shows that ΔT has a long-term increase of +31 s/cy^{2} (Stephenson 1997, p.507, eq.14.3). Converting to days and lunations, the correction from TT to UT becomes:

−31 s / (86400 s/d) / [(36525 d/cy) / (29.53059... d/lunation)]^{2} = −235E−12×N^{2} days in formula [1b].

Although tidal deceleration of the Earth's rotation should be the dominant factor over long time scales (millennia), in the shorter time scale covered by historical observations ΔT shows erratic evolution. Many attempts have been made to fit polynomials to the ΔT curve for different time periods. Here I compute ΔT at the time of conjunction only with the measured offset at the epoch (2000) and the observed long-time quadratic factor.

The theoretical tidal contribution to ΔT is about +42 s/cy^{2} (Stephenson 1997 *op.cit.* p.38 eq.2.8). The smaller observed value of +31 s/cy^{2} is thought to be mostly due to changes in the shape of the Earth (Stephenson 1997 *op.cit.* par.14.8). 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/cy^{2}. The error in the position of the Moon itself is only maybe 0.5"/cy^{2} (from differences of various earlier determinations of the tidal acceleration, see *e.g.* Stephenson 1997 *op.cit.* par.2.2.3), or (because the apparent mean angular velocity of the Moon is about 0.5"/s), 1 s/cy^{2} in the time of conjunction with the Sun.

# References

- Chapront-Touzé, M.; Chapront, J. (1988): "ELP2000-85: a semi-analytical lunar ephemeris adequate for historical times".
*Astronomy & Astrophysics***190**, 342..352 .

Article at ADSabs.harvard.edu - Chapront-Touzé, M.; Chapront, J. (1991):
*Lunar tables and programs from 4000 B.C. to A.D. 8000*. Willmann-Bell. ISBN 0-943396-33-6 . - Chapront, J.; Chapront-Touzé, M.; Francou, G. (1997): "New expressions of the secular terms in
*Lunar tables and programs from 4000 B.C. to A.D. 8000*".*Notes Scientifiques et Techniques du Bureau des Longitudes S055*. ISSN 1243-4272; ISBN 2-910015-15-7.

PDF at IMCCE - Chapront, J.; Chapront-Touzé, M.; Francou, G. (2002): "A new determination of lunar orbital parameters, precession constant and tidal acceleration from LLR measurements".
*Astronomy & Astrophysics***387**, 700..709 .

Article at AandA.org - Clemence, G.M.; Porter, J.G.; Sadler, D.H. (1952): "Aberration in the lunar ephemeris",
*Astronomical Journal***57**(5) (#1198) pp.46..47 . *Explanatory Supplement to the Astronomical Almanac*, third edition (2013). Urban, Sean E.; Seidelmann, P.Kenneth (eds). University Science Books. ISBN 978-1-891389-85-6 .- Hansen, Peter Andreas (1857): "Ecliptische Tafeln für die Conjunctionen des Mondes und der Sonne."
*Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse*, 1857 Heft**II,III**, pp.75..112 . - Hansen, Peter Andreas (1863): "Analyse der ecliptischen Tafeln."
*Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse*, 1857 Heft**II**pp.125..174 . - IAU (1977): Proceedings of IAU Sixteenth General Assembly (1976):
*Transactions of the IAU***XVIB**. West, R.M. (ed.), D.Reidel, Dordrecht 1977 .

See PDF at IAU site. - Jawad, Ala'a H. (1993): "How Long Is a Lunar Month?". Roger W. Sinnott (ed.).
*Sky & Telescope*November 1993, pp.76,77 . - Meeus, Jean (1979, 1980):
*Astronomical Formulae for Calculators*. Monografieën over Astronomie en Astrofysica vol.4 . Volssterrenwacht Urania en Vereniging voor Sterrenkunde, Belgium. - Meeus, Jean (1982, 1985, 1988):
*Astronomical Formulae for Calculators, Enlarged and revised*. Willmann-Bell. ISBN 0-943396-22-0 . - Meeus, Jean (1991, 1998):
*Astronomical Algorithms*. Willmann-Bell. ISBN 978-0-943396-35-4 . - Meeus, Jean (2002):
*More Mathematical Astronomy Morsels*. Willmann-Bell. - Seidelmann, P.K. (1977): "Numerical values of the constants of the Joint Report of the Working Groups of IAU Commission 4".
*Celestial Mechanics***16**, pp.165..177; DOI: 10.1007/BF01228598.

Article at ADSabs.harvard.edu - Stephenson, F.R (1997):
*Historical Eclipses and Earth's Rotation*. Cambridge University Press. ISBN 0-521-46194-4 . - Williams, J.G.; Boggs, D.H. (2016): "Secular tidal changes in lunar orbit and Earth rotation".
*Celestial Mechanics and Dynamical Astronomy***126**, 89..129 . DOI: 10.1007/s10569-016-9702-3. ISSN 0923-2958. S2CID 124256137.

Abstract at Springer