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# 6: The Celestial Sphere

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If you look up in the sky, it appears as if you are at the centre of a vast crystal sphere with the stars fixed on its surface. This sphere is the celestial sphere. It has no particular radius; we record positions of the stars merely by specifying angles. We see only half of the sphere; the remaining half is hidden below the horizon. In this section we describe the several coordinate systems that are used to describe the positions of stars and other bodies on the celestial sphere, and how to convert between one system and another. In particular, we describe altazimuthequatorial and ecliptic coordinates and the relations between them. The relation between ecliptic and equatorial coordinates varies with time owing to the precession of the equinoxes and nutation, which are also described in this chapter.

• 6.1: Introduction to the Celestial Sphere
If you look up in the sky, it appears as if you are at the centre of a vast crystal sphere with the stars fixed on its surface. This sphere is the celestial sphere. It has no particular radius; we record positions of the stars merely by specifying angles. We see only half of the sphere; the remaining half is hidden below the horizon. In this section we describe the several coordinate systems that are used to describe the positions of stars and other bodies on the celestial sphere.
• 6.2: Altazimuth Coordinates
In the altazimuth system of coordinates, the position of a star is uniquely specified by its azimuth and either its altitude or its zenith distance. Of course the altitude and azimuth of a star are changing continuously all the time, and they are also different for all observers at different geographical locations.
• 6.3: Equatorial Coordinates
The equatorial coordinate system is  used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere (forming the celestial equator), a primary direction towards the vernal equinox, and a right-handed convention.
• 6.4: Conversion Between Equatorial and Altazimuth Coordinates
Whereabouts in the sky will a given star be at a certain time? This as a typical problem involving conversion between equatorial and altazimuth coordinates. We have to solve a spherical triangle. That is no problem – we already know how to do that. The problem is: which triangle?
• 6.5: Ecliptic Coordinates
Because most planets (except Mercury) and many small Solar System bodies have orbits with slight inclinations to the ecliptic, using the ecliptic coordinate system as the fundamental plane is convenient. The system's origin can be the center of either the Sun or Earth, its primary direction is towards the vernal (northward) equinox, and it has a right-hand convention. It may be implemented in spherical or rectangular coordinates.
• 6.6: The Mean Sun
The bright yellow (or white) ball of fire that appears in the sky and which you could see with your eyes if ever you were foolish enough to look directly at it is the Apparent Sun. It is moving eastward along the ecliptic, and its right ascension is increasing all the time. The hour angle of the Apparent Sun might have been called the local apparent solar time, except that we like to start our days at midnight rather than at midday.
• 6.7: Precession
From the point of view of classical mechanics, Earth is an oblate symmetric top. That is to say, it has an axis of symmetry and two of its principal moments of inertia are equal and are less than the moment of inertia about the axis of symmetry. The phenomena of precession of such a body are well understood and are studied in courses of classical mechanics. It is necessary, however, to be clear in one’s mind about the distinction between torque-free precession and torque-induced precession.
• 6.8: Nutation
Earth’s axis of rotation nutates because it is subject to varying torques from Sun and Moon – the former varying because of the eccentricity of Earth’s orbit, and the latter because of both the eccentricity and inclination of the Moon’s orbit. This means that the equinox  does not move at uniform speed along the ecliptic, and the obliquity of the ecliptic varies quasi-periodically. These two effects are known as the nutation in longitude and the nutation in the obliquity.
• 6.9: The Length of the Year
The calendar that we use in everyday life is the Gregorian Calendar, in which there are 365 days in most years, but 366 days in years that are divisible by 4 unless they are also divisible by 100 other than those that are also divisible by 400.  The Anomalistic Year is the interval between consecutive passages of the Earth through perihelion and is a little longer than the sidereal year.
• 6.10: Problems
• 6.11: Solutions

Thumbnail: Earth rotating within a relatively small-radius, geocentric celestial sphere. Depicted here are stars (white), the ecliptic (red), and circles of right ascension and declination (cyan) of the equatorial coordinate system. Image used with permission (CC SA-BY 3.0; Tfr000).