## Measuring the Motion of Celestial Bodies

The earth rotates around its own axis in approximately 24 hours, and moves on an ellipse, with the sun at one focus, completing a revolution in approximately 365 days. When they are seen by a fixed observer on the surface of the moving earth (the only point of view that concerns us here), then, all heavenly bodies move.

If we extend the plane on which the earth moves around the sun, imagining we are cutting the celestial sphere, we can define a circle known as the ecliptic. The earth's axis is not perpendicular to the plane of the ecliptic, but is inclined about 23^ degrees with respect to it. This is thus the angle that the ecliptic forms with the celestial equator, that is, with the projection into the sky of the terrestrial equator. The inclination, or obliquity, of the ecliptic gives us the alternating seasons and is thus essential for life itself. (Over thousands of years it undergoes only slight variations). For us earthlings the ecliptic also represents the path of the sun in the course of the year in relation to the stars dotted around in the background.

If we imagine prolonging the terrestrial axis onto the celestial sphere, a point in the sky is identified that we call the celestial pole (north or south depending on the hemisphere of the observer; but to simplify matters, I shall refer only to the north pole). If we lower the perpendicular to the horizon from this point, geographical north can be determined. Once geographical north is known, two coordinatesâ€”that is, two numbersâ€”are needed to determine the position of a star in the celestial sphere at any moment, just as longitude and latitude are needed to identify a point on the earth's surface. There are many systems, that is, pairs, of possible coordinates, but two pairs are fundamental. The most intuitive system, hence the one preferred in this book, is based on azimuth and altitude. Given a point P whose coordinates

Figure A1.1: The path of the sun during the year, viewed by an observer at northern tropical latitude.
Figure A1.2: The north celestial pole

we wish to find, we imagine tracing the vertical plane that passes through this point. This plane intersects the horizon of the observer at point A; the azimuth is the angle between north and the point A on the horizon, counting positively from north to east, and the altitude is the angle measured on the vertical circle from A to P (in particular, the altitude reached by a star when it passes the celestial meridian, that is, the ideal projection of the observer's meridian into the sky, is called the culmination).

The second system of coordinates is easily understood if one imagines measuring the latitude and longitude of a point on the celestial sphere. The angular distance of the point from the celestial equator toward the pole gives the analogue of the latitude and is called the declination-, the analogue of the longitude is called the right ascension, and it is measured from a point (the vernal equinox) along the celestial equator to the "hour circle," the maximal circle passing through the pole and the point.

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