**by Phill Edwards**

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 |

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}

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

Then: $$S=\rho sin\psi $$ $$C=\rho cos\psi $$ $$N=cos\lambda cos\beta -Csin\pi sin\theta $$

$$tan\mathrm{\lambda \text{'}}=\frac{sin\lambda cos\beta -sin\pi (Ssin\epsilon +Ccos\epsilon sin\theta )}{N}$$

$$tan\mathrm{\beta \text{'}}=\frac{cos\mathrm{\lambda \text{'}}(sin\beta -sin\pi (Scos\epsilon -Csin\epsilon sin\theta \mathrm{))}}{N}$$

$$sin\mathrm{s\text{'}}=\frac{cos\mathrm{\lambda \text{'}}cos\mathrm{\beta \text{'}}sins}{N}$$

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_{MS}^{2} = r_{S}^{2} +
r_{M}^{2} - 2r_{S}r_{M}cosβ

$${r}_{\mathrm{UM}}=\frac{{R}_{M}{r}_{\mathrm{MS}}}{{R}_{S}-{R}_{M}}$$

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:

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

$$I=\frac{Kxk}{|Kxk|}=(-sina,cosa,0)$$ $$J=\frac{IxK}{|IxK|}=(-cosasind,-sinasind,cosd)$$The Besselian Elements x, y and d plus the z coordinate are:

$$x=r.I$$ $$y=r.J$$ $$z=r.K$$ $$sind=K.k$$The hour of the shadow axis μ in terms of the right ascension a and
Grenwich Sidereal Time T_{s} is:

The penumbral and umbral cone angles f1 and f2 are:

$$sin\mathrm{f1}=\frac{{R}_{S}+{R}_{M}}{|R-r|}$$ $$sin\mathrm{f2}=\frac{{R}_{S}-{R}_{M}}{|R-r|}$$The penumbral and umbral cone vertices c1 and c2 are:

$$\mathrm{c1}=z+\frac{{R}_{M}}{{R}_{E}}cosec\mathrm{f1}$$ $$\mathrm{c2}=z-\frac{{R}_{M}}{{R}_{E}}cosec\mathrm{f2}$$The penumbral and umbral cone radii in the fundamental plane L1 and L2 are:

$$\mathrm{L1}=\mathrm{c1}tan\mathrm{f1}$$$$\mathrm{L2}=\mathrm{c2}tan\mathrm{f2}$$

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:

The observer's location in the fundamental plane is:

$${\rho}_{F}=(\xi ,\eta ,\zeta )$$The x, y and z axis rotation matrices are R_{1}, R_{2} and
R_{3}.

Then the relationship between observer coordinates is:

$${\rho}_{F}={R}_{1}(\frac{\pi}{2}-d){R}_{3}(\frac{\pi}{2}-\mu ){\rho}_{G}$$