Moon Phases and Eclipses

by Phill Edwards

Moon Phases

To calculate the Moon phase, first calculate the longitude of the Earth λ_E and the longitude of the Moon λ_M for the date. Then calculate the lunar solar elongation D = λ_E - λ_M. The Moon phase can then be determined from the phase angle as shown in the table.

Phase Angle	Phase
D = 0°	New Moon
0° < D < 90°	Waxing Crescent
D = 90°	First Quarter
90° < D < 180°	Waxing Gibbous
D = 180°	Full Moon
180° < D < 270°	Waning Gibbous
D = 270°	Last Quarter
00° < D < 180°	Waning Crescent

Eclipses

For a Solar eclipse to occur it needs to be close to a New Moon. New and Full Moons where the Sun, Earth and Moon are aligned are called syzygy. There is not an eclipse at every New Moon because the Moon's orbit is inclined at about 5° to the ecliptic. So, for a Solar eclipse to occur the lunar latitude at the syzygy β_syz needs to be within a certain range.
The angular radius of the Sun ρ_S = R_S/r_E
The angular radius of the Moon ρ_M = R_S/r_M
The maximum parallax of the Moon δ_M = R_E/r_M
An eclipse, either total, annular or partial, occurs if |β_syz| < δ_M + ρ_M + ρ_S
A total eclipse occurs if ρ_M > ρ_S and |β_syz| < δ_M + ρ_M - ρ_S
An annular eclipse occurs if ρ_S > ρ_M and |β_syz| < δ_M + ρ_S - ρ_M

Lunar Parallax

The Moon is sufficiently large that the Moon as seen from a hypothetical viewer at the centre of the Earth (geocentric) will appear in a different postion to an observer on the Earth's surface (topocentric). In fact observers at different latitudes on the Earth's surface will see the Moon at a different position. This effect is due to parallax. There is also a similar, but much smaller parallax offset for the Sun. The Solar parallax is considered a derived astronomical constant with a value of 8".794. Only the sine of this angle is required for calculations.

Given the following parameters:

λ is the geocentric ecliptical longitude of the Moon
β is the geocentric ecliptic latitude of the Moon
s is the semi diameter of the Moon
φ is the observer's geographic latitude
ρ and ψ define the observer's geocentric latitude as described in Coordinates
ε is the obliquity of the ecliptic
θ is the local sidereal time
π is the equatorial horizontal parallax of the Sun

We need to calculate the topocentric values λ', β' and s'.

Then: $S = ρ sin ψ$ $C = ρ cos ψ$ $N = cos λ cos β - C sin π sin θ$

$tan λ' = \frac{sin λ cos β - sin π (S sin ε + C cos ε sin θ)}{N}$

$tan β' = \frac{cos λ' (sin β - sin π (S cos ε - C sin ε sin θ))}{N}$

$sin s' = \frac{cos λ' cos β' sin s}{N}$

Eclipses Revisited

Terms:

λ_S the solar longitude
λ_M the lunar longitude
D = λ_S - λ_M the lunar solar elongation
β the lunar latitude
R_S the solar radius
R_M the lunar radius
r_S the Earth Sun distance
r_M the Earth Moon distance
r_MS the Moon Sun distance
r_UM the Moon umbral point distance

At the new moon syzygy D = 0
By the cosine rule r_MS² = r_S² + r_M² - 2r_Sr_Mcosβ
$r_{UM} = \frac{R_{M} r_{MS}}{R_{S} - R_{M}}$

Besselian Elements

Besselian Elements were introduced in 1824 by the Prussian astronomer and meathematician Frederick Bessel. They are a good method of calculating the characteristics and locations of an eclipse which are still valid. The principle is to define a fundamental plane, which is a plane which passes through the Earth's centre and is normal to the Sun-Moon centre line. The x axis point East, the y axis point North and the z axis point towards the Sun. There are six Besselian Elements which are:

x, y the x and y positions of the centre of the shadow cone in units of the Earth's equatiorial radius
d the declination of the shadow axis
μ the hour angle of the shadow axis
L1 the radius of the penumbral shadow on the fundamental plane
L2 the radius of the umbral shadow on the fundamental plane
f1 the angle which the penumbral shadow cone makes with the shadow axis
f2 the angle which the umbral shadow cone makes with the shadow axis

The Besselian Elements are normally calculated accurately at five hourly intervals and then a least squares interpolation is performed. This may not be necessary using modern computers. The elements are calculated for the nearest hour to the maximum eclipse and for the two hours before and after the greatest eclipse.

Let r be the Earth-Moon vector and R be the Sun-Earth vector and k = (0, 0, 1) is the unit vector pointing North.
The unit vector K which is normal to the fundamental plane, pointing in the direction of right ascension a and declination b is:

K = \frac{R - r}{| R - r |} = (cos d cos a, cos d sin a, sin d)

The other two unit vectors in the fundamental plane are the unit vectors created from the cross products:

I = \frac{K x k}{| K x k |} = (- sin a, cos a, 0)

J = \frac{I x K}{| I x K |} = (- cos a sin d, - sin a sin d, cos d)

The Besselian Elements x, y and d plus the z coordinate are:

x = r . I

y = r . J

z = r . K

sin d = K . k

The hour of the shadow axis μ in terms of the right ascension a and Grenwich Sidereal Time T_s is:

μ = T_{s} - a

The penumbral and umbral cone angles f1 and f2 are:

sin f1 = \frac{R_{S} + R_{M}}{| R - r |}

sin f2 = \frac{R_{S} - R_{M}}{| R - r |}

The penumbral and umbral cone vertices c1 and c2 are:

c1 = z + \frac{R_{M}}{R_{E}} cosec f1

c2 = z - \frac{R_{M}}{R_{E}} cosec f2

The penumbral and umbral cone radii in the fundamental plane L1 and L2 are:

L1 = c1 tan f1

L2 = c2 tan f2

Besselian to Geocentric Coordinates

Given the Earth's equatorial radius a = 6378.1km and the Earth's polar radius b = 6356.8. Then the flattening f and the ellipticity e are defined as:

f = \frac{a - b}{b}

e = \sqrt{1 - \frac{b^{2}}{a^{2}}} Given an observer's longitude λ, latitude φ and distance from the
centre of the Earth ρ then the observer's Geocentric position ρ

_G is:

ρ_{G} = ρ (cos φ cos λ, cos φ sin λ, sin φ)

The observer's location in the fundamental plane is:

ρ_{F} = (ξ, η, ζ)

The x, y and z axis rotation matrices are R₁, R₂ and R₃.

Then the relationship between observer coordinates is:

ρ_{F} = R_{1} (\frac{π}{2} - d) R_{3} (\frac{π}{2} - μ) ρ_{G}