**by Dr Phill Edwards**

The celestial sphere ritates about the celestial axis which is an extension of the Earth's axis of rotation. The celestial equator is a plane perpendicular to the celestial axis which corresponds to the Earth's equatorial plane. The direction of the Vernal Equinox V in the celestial plane is the origin of celestial longitude. As the Vernal Equinox varies due to orbital factors, it is usual to define the Vernal Equinox of Date which is the position of the Vernal Equinox at a particular time, currently J2000 2000-01-01T12:00:00.

The declination δ is the angle subtended from the celestial equator to
the position of a body.

The right ascension α is the angle subtended from the Vernal Equinox to
the position of a body. Right ascension is often measured in hours, where 1
hour = 15°.

The Earth's axis of rotation is inclined to the plane of the ecliptic. The
angle of tilt precesses over time. It varies from 22.1° to 24.5° over a 42,000
year period. The P03
improvements to the International Astronomical Union (IAU) 2000 precession
model produced the series:

ε = 84381.406" - 0.024725"T + 0.0512621"T^{2} -
0.00772501"T^{3} - 0.467"×10^{-6}T^{4} +
3.337"×10^{-8}T^{5}

The precession in longitude, which is the precession of the Vernal Equinox,
is defined by the P03
improvements to the IAU 2000 precession model to be the series:

ψ = 5038.48209"T - 1.0789921"T^{2} - 0.0011404"T^{3} +
0.000132851T^{4} - 9.51x10^{-8}"T^{5}

The ecliptic is the plane of the Earth's orbit around the Sun. The Vernal Equinox V is defined as the direction of the line of intersection of the ecliptic plane with the celestial equatorial plane.

The ecliptic longitude λ is the angle subtended from the Vernal
Equinox to the position of a body.

The ecliptic latitude β is the angle subtended from the ecliptic to the
position of a body.

The axial tilt of the Earth's axis to the ecliptic ε = 23°26'21".
Then:

$$sin\beta =cos\epsilon sin\delta -sin\epsilon cos\delta sin\alpha $$ $$tan\lambda =\frac{sin\epsilon sin\delta +cos\epsilon cos\delta sin\alpha}{cos\delta cos\alpha}$$ The inverse transforms are:
$$sin\delta =cos\epsilon sin\beta +sin\epsilon cos\beta sin\lambda $$ $$tan\alpha =\frac{-sin\epsilon sin\beta +cos\epsilon cos\beta sin\lambda}{cos\beta cos\lambda}$$

Most astronomical calculations use the ecliptic coordinate system relative to the plane of the Earth's orbit. In order to calculate the position of a body for an observer on the Earth's surface, it is necessary to transform ecliptic coordinates into topocentric coordinates. As the Earth is not a sphere due to flattening it is necessary to determinate the position of the observer relative to the centre of the Earth - geocentric coordinates. For this purpose it is usually sifficiently accurate to model the Earth as an ellipsoid of revolution.

The Earth can be a modeled as an ellipsoid using the World Geodetic System
WGS 84 standard with:

- Semi major axis, the equatorial radius, a = 6378.1370 km
- Semi minor axis, the polar radius, b = 6356.7523142 km

Let ρ be the distance from the centre of the Earth to the observer as a fractor of the Earth's equatorial radius ( b/a ≤ ρ ≤ 1). Then calculate: $$tanu=\frac{b}{a}tan\phi $$ Then if H is the observer's height above sea level in kilometres we have the quantities: $$\rho sin\psi =\frac{b}{a}sinu+\frac{H}{a}sin\phi $$ $$\rho cos\psi =cosu+\frac{H}{a}sin\phi $$