• Illuminated Fraction of the Moon

    The illuminated Fraction of the Moon, k, depends of the Phase Angle i, so, we have:

k = (1 + cosi)/2

    This value represents not only the ratio between the illuminated area and the total area, but also the ratio between the illuminated length of the diameter perpendicular to the line of cusps and the complete diameter.

    The Moon's phase angle, for a geocentric observer can be determinate by the calculation of the Moon's elongation y regarding to the Sun:

cosy = sind₀sind + cosd₀cosdcos(a₀-a)

cosy = cosb cos(l-l₀)

    where:    a₀, d₀, l₀: geocentric right ascension, declination and longitude of the Sol;

                 a, d, l: geocentric right ascension, declination and longitude of the Moon;

                 b: Moon's geocentric latitude

    So we have:

tani = Rsiny/(D-Rcosy)

    where R it's the distance from the Earth to the Sun and D the distance from the Earth to the Moon, both in the same units.

    The angles y and i are always between 0 and 180 degrees. So, when we have i, the illuminated fraction k can be easily obtained by the previous formulas.

    It's important to salient that for the calculation of k it's not really necessary to determinate with much precision the geocentric positions of the Sun and the Moon, it will be sufficient to make cosi = -cosy, since that the final error will never exceed 0.0014. With a lower precision, we can even ignore the Moon's latitude.

    So we can obtain a value for i by the following expression:

i=180⁰-D-6⁰.289sinM'+2⁰.100sinM-1⁰.274sin(2D-M')

-0⁰.658sin2D-0⁰.214sen2M'-0⁰.110sinD

    In this case, the geocentric position of the Sun and the Moon will not be necessary.

    • Position angle of the Moon's bright Limb:

    This angle it's the position angle c of the Moon's bright limb midpoint (C point from the previously image). It can be obtained by (this expression it's also valid for the planets):

tanc = cosd₀sin(a₀ - a)/(sind₀cosd - cosd₀sindcos(a₀ -a))

    with the same meaning as before.

    The angle c during the first quarter will be near the vicinity of 270º and near 90º after the Full Moon.

    If we consider c as the position angle of the bright limb midpoint, then the position angle of the poles will be (c-90⁰) and (c+90⁰).

    The big advantage that we obtain it's to define easily and without any doubt the bright limb of the Moon.

    • New Moon, First Quarter, Full Moon and Last Quarter:

    By definition, the times for New Moon, First Quarter, Full Moon and Last Quarter are the times when the Moon's apparent geocentric longitude exceed the Sun's apparent geocentric longitude at 0°, 90°, 180° e 270°.

    In this section we will describe the procedure that can be followed to determinate the time of New Moon, since the other times will be similar.

    To occur New Moon the Sun and Moon's ecliptic longitudes must be equals. The calculation mentioned before needs the determination of the difference between the mean longitude (D) and the difference between the periodical perturbations. The difference between the ecliptics longitudes is the sum of these two values.

l_L - l_S = D + (Dl_L - Dl_S)

    So the value for D will be:

D = D₀ + D₁T

= 297º.85027 + 445267º.11135T

    where:    T=(JD-2451545)/36525;

                  D₁: Modification in D by a Julian century.

    One result that simplified the New Moon's calculation during the year is that the mean interval between two successive New Moons is approximately 29.53 days.

    The periodically perturbations of the lunar and solar orbit are minimal in a short period of time Dt, which means that over that interval (Dl_L - Dl_S) essentially varies D₁´Dt/36525. An initial approximation t₀, for the time of New Moon can be now improved by the next formula:

t₁ = t₀ - [D(t₀)+(Dl_L(t₀) - Dl_S(t₀))]/D₁ ´ 36525

    If we continue to repeat this process it's possible to determinate the time with New Moons with a sufficiently accuracy. To optimize the calculation we can also determinate (l_L - l_S) regarding to the Sun and Moon's longitude and anomalies. As good approximation we can use the following results:

l_L = 22640''sin(l) - 4586''sin(l-2D) + 2370''sin(2D) +

+ 769''sin(2l) - 668''sin(l') - 412''sin(2F) - 212sin(2l-2D) -

- 206''sin(l+l'-2D) + 192''sin(l+2D) - 165''sin(l'-2D) +

+ 148''sin(l-l') - 125''sin(D) - 110''sin(l+l') - 55''sin(2F-2D)

l_S = 6893''sin(l') + 72''sin(2l')

    where:    l = 134º.96292 + 477198º.86753T + 33''25T² (Moon's Mean Anomaly);

                 l' = 357º.52543 + 35999º.04944T - 0''.58T² (Sun's Mean Anomaly);

                 F = 93º.27283 + 483202º.01873T - 11''.56T² (Mean distance from the ascend node to the Moon).

    It's also possible, using the same parameters, determinate the Moon's Ecliptic Latitude as follows:

b_L » 18520''sin(F+Dl) - 526''sin(F-2D)