Chapter 2 - Location and coordinates Updated 10 July 2006 -4 -3 -2 -1 0 +1 +2 +3 +4 Figure 2.1 A simple number line acts as a one-dimensional coordinate system. Each number describes the distance of a particular point-location from the origin, or zero. The unit of measure for the distance is arbitrary. 10 9 8 (4.5,7.5) 7 6 s i x a (3,5) - 5 y 4 3 (6,3) 2 1 0 0 1 2 3 4 5 6 7 8 9 10 x - axis Figure 2.2 Here is a simple x-y coordinate system with the coordinates of a few points shown as examples. Coordinates are given as an ordered pair of numbers, with the x-coordinate first. The origin is the lower left, at (0,0). The reference lines are the x and y axes. All grid lines are drawn parallel to the two reference lines. The purpose of the grid is to make it easier to make distance measurements between a location-point and the reference lines. 5 4 3 (2,3) 2 s 1 i x a (-2,0) - 0 y -1 -2 (1,-2) -3 -4 -5 -5 -4 -3 -2 -1 0 1 2 3 4 5 x - axis Figure 2.3 A more general coordinate grid places the origin (0,0) at the center of the grid. The coordinates may have either positive or negative values. The sign merely indicates whether the point is left or right (x), or above or below (y) the axis. z-axis (2, 3, 4) 4 origin y-axis 2 3 x-axis Figure 2.4 A three-dimensional space, such as a room, requires a three-dimensional coordinate system. Each point- location is specified by an ordered set of measurements of distances from the origin made along or parallel to a reference line. l 1055 l 427 a g a g d 243 d e 916 z e z 817 b 1342 k 721 k 773 b Figure 2.5 The stars of Orion are distributed through three-dimensional space but our view of those stars does not suggest any radial distance, except by the stars' apparent brightness. Apparent brightness however, is not a valid indicator of true distance. The Greek letters are the Bayer designations for each star (p. 45). The numbers are the actual distance to the stars, in light-years, as given by the Hipparcos Catalog. Arc Arc Arc 90º 25º 135º Figure 2.6 A few angles are shown with an arc line between the lines creating the angle. The 90° angle is also known as a “right angle.” Such angles are shown with a small square at the vertex. D C A B Figure 2.7 The angle between two points is independent of the distance of the points from the vertex of the angle. The angle between points A and C is the same as between B and D. Notice also, that this same angle occurs between points B and C. 90º 135º 45º (2, 30º) 180º 0º 1 2 3 4 (2.5, 225º) 225º 315º 270º Figure 2.8 Polar coordinate systems use the radial line distance from the origin to a location-point, and the angle between the location-point’s radial line and a reference line. The origin is sometimes called “the pole” because the angle coordinate has no meaning if the radial distance is zero. In this system, generally, there are no negative coordinate values. However, we could have the angle coordinate run from –180° to +180° rather than from \Angle{0} to \Angle{360}. This may have advantages in some situations. Small Circle Plane Sphere Figure 2.9 When a plane intersects a sphere the intersection creates a circle on the surface of the sphere and on the plane. Pole Line, perpendicular to plane. Great Circle (equator) Plane Sphere Pole Figure 2.10 A plane passing through the center of the sphere creates a great circle on the sphere. When a line matching the sphere’s rotational axis is drawn perpendicular to the plane, the line pierces the sphere at its poles. In this case the great circle is called the sphere’s equator. Pole Parallel M e r i d i a Parallel n s Equator Parallel Parallel Pole Figure 2.11 Planes, parallel to the plane creating the equator, create small circles that are parallel to the equator. On Earth, the lines are called parallels. Great circles created by planes containing the polar axis are called meridians. North Pole Greenwich England Meteor Crater Prime Center 32º 02' N Meridian 111º 01' W Equator Figure 2.12 This is an illustration of the terrestrial or latitude-longitude system. The location of the Barringer Meteor Crater, Arizona is, 111° 01' west longitude, 32° 02' north latitude. The angles are measured at the center of the Earth. North Pole H ori zon Equator Geocentric Astronomical latitude latitude South Pole Figure 2.13 Two definitions of latitude are caused by the Earth’s polar flattening, which in turn is caused by its rotational motion. The flattening shown here is drastically exaggerated to make the difference clearer. Figure 2.14 This is the “transit house” through which the terrestrial coordinate system’s prime meridian passes, at the Old Royal Observatory in Greenwich, England. The line on the ground in front of the house is the prime meridian. NCP Earth SCP Celestial Equator Figure 2.15 The celestial sphere is an imaginary sphere, centered on the Earth. The celestial equator is directly above the Earth’s equator and the celestial poles are directly above the Earth’s poles. They share the axis of rotation. (Thanks to Kirk Korista for providing the sphere shown in this picture.) Big Dipper NCP Polaris Figure 2.16 Use the “pointer stars” at the end of the Big Dipper bowl to find the North Star (Polaris). The north celestial pole (NCP) is next to Polaris. Dorado Volans Mensa g Crux Chamaeleon g a Hydrus Musca SCP s b Apus Octans b Triangulum Australe Figure 2.17 To find the south celestial pole, draw a line through g and a Crucis (Crux) toward b Hydri (Hydrus), then draw a line from b Trianguli Australis to g Hydri. The intersection of these two lines is near the SCP. Also, the dim star s Octantis is near the SCP. (The Greek letters used here are the Bayer designations for these stars. Bayer designations are introduced on p. 45.) Altitude = +90º (Zenith) ian rid e Almucantar l M a c o L West 270º H orizon e d u t i t South l North 180º A A 0º ltitu de=0 º Azimuth n East ia d 90º ri e m ti n A Nadir (-90º) Figure 2.18 This is the basic structure of the horizon coordinate system. Azimuth is measured from 0° to 360° along the horizon, starting from due north. Altitude is measured from 0° to 90° starting from the horizon. Negative altitudes are positions below the horizon. Star Ruler Protractor 0 0 9 String Small Weight Eyeball Figure 2.19 Here is a simple, inexpensive quadrant. Measure a star’s altitude by sighting the star along the top edge of the ruler. Pinch the string against the protractor to read the altitude angle. 20º 1º 10º Figure 2.20 Your hand can be used to measure altitudes and other angles when a quadrant is not available. Hold your hand at arm’s length and the equivalent angles are shown above. NCP n ia id er M c e D NP e r n u Prime o l i t o Meridian a Sun C n i l l a c i t e c D o n i Equator u q E R igh tor t As qua cens tial E ion Celes Vernal Ecliptic Equinox SCP Figure 2.21 The equatorial system works similar to the terrestrial system. Declination measures angles above (+) and below (–) the celestial equator in units of degrees. The north celestial pole is declination +90°. The south celestial pole is declination –90°. Right Ascension measures along the celestial equator in units of time. The origin is the vernal equinox. Figure 2.22 A Tycho Globe has a clear plastic sphere representing the celestial sphere. Inside is a plastic half-sphere which floats on a small amount of water on the bottom of the celestial sphere. The position of the celestial sphere can be adjusted for observer latitude with the metal ring which acts like the local meridian. The celestial sphere can rotate, simulating the daily motion of the sky. Zenith ian erid l M ca Lo Q 90º NCP (Polaris) C e West A = 50º le CE s 40º t ia l E q u a to r l South North East Figure 2.23 At latitude 40° north the altitude of the north celestial pole is 40° and the transit altitude of the celestial equator is 90° – 40° = 50° off the southern horizon. Zenith Q n ia d ri 90° e M l a c o L NCP C (Polaris) e A = 64º l CE e s West t i a l E q u a 26º t o r l South North East Figure 2.24 At latitude 26° north (Miami, Florida) the altitude of the north celestial pole is 26° and the transit altitude of the celestial equator is 90° – 26° = 64° off the southern horizon. Zenith Loc al M er idi an Q SCP 90º West r to a A = 50º u CE q E l a ti 40º s le e South l C North East Figure 2.25 At latitude 40° south, the altitude of the south celestial pole is 40° from the southern horizon and the altitude of the celestial equator is 90° – 40° = 50° off the northern horizon.