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The Celestial Sphere: Coordinate Systems

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The Celestial Sphere: Coordinate Systems The astrophysics for 4th class Department of Physics/ College of Science/ University of Kirkuk 2019-2020 Prof. Dr. Wafaa H. A. Zaki Chapter one/ 4 weeks The Celestial Sphere: Coordinate Systems 1. Spherical Astronomy Spherical Astronomy is a science studying astronomical coordinate frames, directions and apparent motions of celestial objects, determination of position from astronomical observations, observational errors, etc. For simplicity we will assume that the observer is always on the northern hemisphere, we will also use degrees to express all angles unless otherwise mentioned. The Celestial Sphere is an imaginary sphere of arbitrarily large radius, concentric with Earth. The Zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. The opposite direction is toward the nadir. The zenith is the "highest" point on the celestial sphere. The Meridian is the great circle passing through the celestial poles, the zenith, and the nadir of an observer's location. Consequently, it contains also the horizon's north and south points, and it is perpendicular to the celestial equator and horizon. The Celestial Meridian is coplanar with the analogous terrestrial meridian projected onto the celestial sphere. Hence, the number of astronomical meridians is infinite. The celestial sphere. Earth is depicted in the center of the celestial sphere. 2. Celestial Coordinate System The Celestial Coordinate System is a system for specifying positions of celestial objects: satellites, planets, stars, galaxies, and so on. Coordinate systems can specify a position in 3-dimensional space, or merely the direction of the object on the celestial sphere, Since the distances of the stars are ignored, we need only two coordinates to specify their directions. Each coordinate frame has some fixed reference plane passing through the center of the celestial sphere and dividing the sphere into two hemispheres along a great circle. One of the coordinates indicates the angular distance from this reference plane. There is exactly one great circle going through the object and intersecting this plane perpendicularly; the second coordinate gives the angle between that point of intersection and some fixed direction. Each coordinate system is named for its choice of fundamental plane. The meridian on the celestial sphere. A. Horizontal coordinate system The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane. It is expressed in terms of altitude (or elevation) angle and azimuth. This coordinate system divides the sky into the upper hemisphere where objects are visible, and the lower hemisphere where objects cannot be seen since the Earth obstructs vision. The great circle separating the hemispheres is called the celestial horizon. The Celestial Horizon is a great circle on the celestial sphere having a plane that passes through the center of the Earth and is parallel to an observer’s horizon. The pole of the upper hemisphere is called the zenith. The pole of the lower hemisphere is called the nadir. There are two independent horizontal angular coordinates: Altitude (E), or elevation, is the angle between the object and the observer's local horizon. The altitude lies in the range [−90◦,+90◦]; it is positive for objects above the horizon and negative for the objects below the horizon. The zenith distance, or the angle between the object and the zenith, is obviously z = 90◦- E…………….(1) The horizontal coordinate system Azimuth (A), is the angle of the object around the horizon, usually measured from the north or south. Azimuth is measured from the north point (sometimes from the south point) of the horizon around to the east (0 -360 ). The main disadvantage of this system: The horizontal coordinate system is fixed to the Earth, not the stars. Therefore, the altitude and azimuth of an object in the sky changes with time, as the object appears to drift across the sky with the rotation of the Earth. Because the horizontal system is defined by the observer's local horizon, the same object viewed from different locations on Earth at the same time will have different values of altitude and azimuth. Horizontal coordinates are very useful for determining the rise and set times of an object in the sky. When an object's altitude is 0°, it is on the horizon. If at that moment its altitude is increasing, it is rising, but if its altitude is decreasing, it is setting. One can determine whether altitude is increasing or decreasing by instead considering the azimuth of the celestial object: if the azimuth is between 0° and 180° (north–east–south), it is rising. if the azimuth is between 180° and 360° (south–west–north), it is setting. B. Equatorial Coordinate System The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be applied in spherical or rectangular coordinates, both defined by an origin at the center 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. The origin at the center of the Earth means the coordinates are geocentric. The fundamental plane and the primary direction mean that the coordinate system, while aligned with the Earth's equator and pole, does not rotate with the Earth, but remains relatively fixed against the background stars. A right-handed convention means that coordinates are positive toward the north and toward the east in the fundamental plane. A star's spherical coordinates are often expressed as a pair, rightascension and declination, without a distance coordinate. The declination symbol δ, measures the angular distance of an object perpendicular to the celestial equator, positive to the north, negative to the south. For example, the north celestial pole has a declination of +90°. Declination is analogous to terrestrial latitude,(+90°, -90°). The right ascension symbol ,measures the angular distance of an object eastward along the celestial equator from the vernal equinox to the hour circle passing through the object (the vernal equinoxⱱ point is one of the two where the ecliptic intersects the celestial equator). Analogous to terrestrial longitude, right ascension is usually measured in sidereal hours, minutes and seconds instead of degrees, a result of the method of measuring right ascensions by timing the passage of objects across the meridian as the Earth rotates. There are (360° / 24h) = 15° in one hour of right ascension, 24h of right ascension around the entire celestial equator. The hour angleh, measures the angular distance of an object westward along the celestial equator from the observer's meridian to the hour circle passing through the object. Unlike right ascension, hour angle is always increasing with the rotation of the Earth. Hour angle may be considered a means of measuring the time since an object crossed the meridian. A star on the observer's celestial meridian is said to have a zero-hour angle. Hour circle Observers celestial meridian Hour angle (h) Equatorial Coordinates system. The sidereal time The sidereal time Θ (the hour angle of the vernal equinox) equals the hour angle plus right ascension of any object. Θ =h+ ……..(2) Equatorial Coordinates system with ecliptic plane C. Ecliptic Coordinate System The ecliptic coordinate system is a celestial coordinate system commonly used for representing the positions and orbits of Solar System objects. Because most planets (except Mercury), and many small Solar System bodies have orbits with small inclinations to the ecliptic, it is convenient to use it as the fundamental plane (The fundamental plane is the plane of the Earth's orbit, called the ecliptic plane). The system's origin can be either the center of the Sun or the center of the Earth, its primary direction is towards the vernal equinox. The angle between equator plane and ecliptic plane is 23.5o, which is known as the obliquity of the ecliptic (ε ). Ecliptic longitude or celestial longitude (symbols: λ) measures the angular distance of an object along the ecliptic (counterclockwise) from the primary direction (vernal equinox) (0° ecliptic longitude)(0 -360 ). Ecliptic latitude or celestial latitude (symbols: β), measures the angular distance of an object from the ecliptic towards the north (positive) or south (negative) ecliptic pole. For example, the north ecliptic pole has a celestial latitude of +90° (+90 ,-90 ). D. Galactic Coordinate System The galactic coordinate system is a celestial coordinate system in spherical coordinates, with the Sun as its center, the primary direction aligned with the approximate center of the Milky Way galaxy, and the fundamental plane approximately in the galactic plane. It uses the right-handed convention, meaning that coordinates are positive toward the north and toward the east in the fundamental plane. Galactic longitude Longitude (symbol l) measures the angular distance of an object eastward along the galactic equator from the galactic center. Analogous to terrestrial longitude, galactic longitude is usually measured in degrees (°)(0 -360 ). Galactic latitude Latitude (symbol b) measures the angular distance of an object perpendicular to the galactic equator, positive to the north, negative to the south. For example, the north galactic pole has a latitude of +90°. Analogous to terrestrial latitude, galactic latitude is usually measured in degrees (°)(+90 ,-90 ). galactic coordinate system Fundamental Coordinates Coordinate Center point Primary direction plane Poles system (origin) (0° longitude) (0° latitude) Latitude Longitude Zenith, Altitude (E) or North or south point Horizontal Observer Horizon Azimuth (A) nadir elevation of horizon Right Celestial Celestial ascension (α) Equatorial Declination (δ) Center of equator poles or hour the Earth (geocentric), angle (h) Vernal equinox or Sun (heliocentric) Ecliptic Ecliptic Ecliptic Ecliptic Ecliptic poles latitude (β) longitude (λ) Galactic Galactic Galactic Galactic Galactic Center of the Sun Galactic center plane poles latitude (b) longitude (l) 3.
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