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Motion of the Moon

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Computing the true longitude of the Moon:

True longitude of Moon = mean longitude + major inequality + evection + variation + annual inequality + reduction to ecliptic + parallactic inequality + more terms

 

		Period
mean longitude of the Moon: measured from the mean position of the perigee	L0 = 218.31617 + 481267.88088*T - 4.06*T*T/3600	~27.32 d
the Moon's mean anomaly:	M = 134.96292 + 477198.86753*T + 33.25*T*T/3600	~27.55 d
the Sun's mean anomaly:	MSun = 357.52543 + 35999.04944*T - 0.58*T*T/3600	365.25 d
the mean distance of the Moon from the ascending node	F = 93.27283 + 483202.01873*T - 11.56*T*T/3600	~27.21 d
the difference between the mean longitudes of the Sun and the Moon	D = 297.85027 + 445267.11135*T - 5.15*T*T/3600	~29.53 d
Time	T = (JD - 2451545)/36525

Source: Montenbruck, Pfleger: Astronomy on the Personal Computer

True longitude of Moon = mean longitude + major inequality + evection + variation + annual inequality + reduction to ecliptic + parallactic inequality + more terms

L = L0 + (1) + (2) + (3) + (4) + (5) + (6) + (7)

		Period
(1) Major Inequality (equation of center)	= 22640"*sin(M) + 769*sin(2M)	~27.5 d
(2) Evection	= -4586"*sin(M-2D)	31.8 d
(3) Variation	= 2370"*sin(2D)	~14.8 d
(4) Annual Inequality	= -668"*sin(MSun)	1 y
(5) Reduction to the Ecliptic	= -412"*sin(2F)	~13.6 d
(6) Parallactic Inequality	= -125"*sin(D)	~29.5 d
(7) More terms term1 term2 term3 term4 term4 term5 term6	= -212"*sin(2*M-2D) = -206"*sin(M+MSun-2D) = +192"*sin(M+2D) = -165"*sin(MSun-2D) = +148"*sin(L-MSun) = -110"*Math.sin(M+MSun) = - 55"*Math.sin(2F-2D)

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Example: 2000 Jan at 00:00 UT

Drawing the diagram the small terms (4) to (7) are multiplied by 15.

The error of the true longitude (L = L₀ + sum) is (0.52 ± 0.25)°,
reference HORIZONS Web-Interface (NASA JPL).


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The red circle is the position of the Moon on the first day of the month (2000 Jan 1 at 00:00 UT).

The position of the Sun is added. The motion is counterclockwise:





The relative positions of the Sun and the Moon are causing the phases of the Moon.


In 2000 Jan, New Moon occured on Jan 6 at 18 UT:




In 2000 Jan, Full Moon occurred on Jan 21 at 5 UT:





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Draconic month time between the Moon’s passages through the same node, or intersection of its orbit with the ecliptic.	27.21 d
Sidereal month time needed for the Moon to return to the same place against the background of the stars.	27.32 d
Anomalistic month time from perigee to perigee	27.55 d
Synodic month time of complete cycle of phases of the Moon	29.53 d

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The geocentric distance of the Moon is computed by Meeus (Astronomical Algorithms, Table 45.A, 46 trigonometric terms):




The mean absolute error (Jan 2000, 0 UT) is (33 ± 25) km,
reference: MICA


More accurate values are computed by my spreadsheet
moon_month

Download my LibreOffice spreadsheet: moon_mov_month.ods

Apache OpenOffice (4.1.6) and Microsoft Excel for Mac (16.16.10)
are showing an error computing the distance of the Moon
(formula overflow, Err:512)
a



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E	The eccentric anomaly (a parameterization of polar angle)
ν	The true anomaly specifies the position along the orbit.
M	The mean anomaly is the angle of the line joining the focus (Earth) to a hypothetical body that has the same orbital period but travels at a uniform angular speed: M = n·(t - T)

Kepler's Equation for a body orbiting on an ellipse with eccentricity e:

M = E - e·sin(E)

Books

More details:

Blue Moon Full Moon distance Lunar Perigee and Apogee Calculator

Web Links

Keplers Equation (MathWorld) Phases of the Moon (U.S. Naval Observatory)

Last update: 2019 May 31