Positional Astronomy by Fiona Vincent Fiona Vincent Positional Astronomy A Collection of Lectures on Positional Astronomy for an undergraduate course at the. University of St. Andrews. Scotland in 1998 Positional Astronomy By Fiona Vincent Copyright : Fiona Vincent 1998. Revised and updated on 2003. Converted to .pdf by Alfonso Pastor on 2005 and revised on 2015 1 Copyright : Fiona Vincent 1998 Comments, complaints or queries? Please e-mail: Fiona Vincent at : [email protected] For a short Professional Curriculum of Fiona Vincent see the last page of this eBook Originally created by Fiona Vincent on March 1998. Revised and updated November 2003. Adapted to .pdf format by Alfonso Pastor on 2005. And revised on January 2015 Note: On the .pdf version, in the Chapter’s Index as well as in the references to other sources, the links to the original pages are active. Clicking on the links, in some .pdf readers with Internet connection, the original pages can be accessed whatever the University of St. Andrews keep them active and stored in their resources. If any flaw in the text of the pdf version is suspected, the original page can be accessed to compare it with the .pdf version. Some words in the original Fiona Vincent’s text are not recognized by the dictionaries of the modern Word-processors as the word “centre” , which was widely used in old English writings, is nowadays being replaced by the “Americanism” term “center” , but it has been conserved as in the original text in aims to be respectful with Fiona Vincent’s writing. Copyright : Fiona Vincent 1998. Revised and updated on 2003. Converted to .pdf by Alfonso Pastor on 2005 and revised on 2015 2 .Index of Chapters 00.- Introduction 01.- The Terrestrial Sphere 02.- Spherical Trigonometry 03.- Coordinate Systems: the horizontal or "alt-az" system 04.- Coordinate systems: the first equatorial or "HA-dec." system 05.- Coordinate systems: the second equatorial or "RA-dec." system 06.- Sidereal Time 07.- Conversion between horizontal and equatorial systems 08.- Galactic coordinates 09.- Ecliptic coordinates 10.- Equatorial and ecliptic coordinates 11.- The Sun's motion, and its effect on time-keeping 12.- The Moon 13.- Refraction 14.- Sunrise, sunset and twilight 15.- Geocentric or diurnal parallax 16.- Annual parallax 17.- Aberration 18.- Precession 19.- Calendars 20.- Final Exercise ….- Bibliography Copyright : Fiona Vincent 1998. Revised and updated on 2003. Converted to .pdf by Alfonso Pastor on 2005 and revised on 2015 3 Positional Astronomy: Chapter 00. Introduction. These pages were created by Fiona Vincent in 1998 to assist the students in the lectures on Positional Astronomy for second-year undergraduate at the University of St.Andrews, Scotland. Now are maintained as a help to the students of St.Andrews as well as a service to the world-wide community. This document was converted to . pdf format by Alfonso Pastor in 2005 as a tribute to Fiona Vincent in order to allow this extraordinary lectures to be loaded and read in computers and eBooks without the possibility of Internet connection. You are welcome to link to Fiona Vincent´s pages from your own pages, but please do not reproduce any of this material without Fiona Vincent’s acknowledgement. Notes about the pages: (1) Many of the equations of the original pages in. html format use Greek letters as symbols. (If you try to link to the original pages, if your browser does not distinguish between "a,b" and "α, β" (the Greek letters "alpha, beta") then you will not be able to make much sense of the equations. ) Here in the .pdf version this problem does not exist unless your reader is missing the Greek characters set. (2) These pages use "classical" definitions and terminology: that is, those used previous to the International Astronomical Union's 2000 resolutions at the site: http://syrte.obspm.fr/IAU_resolutions/Resol-UAI.htm (For more information about time scales and Earth Rotation Models, see USNO Circular No.179 ) at : http://www.usno.navy.mil/USNO/astronomical- applications/publications/Circular_179.pdf This course is intended to address the following problems: 1.- How to describe the position of an object in the sky. 2.- Which different coordinate systems are appropriate in different situations. 3.- How to transform between coordinate systems. 4.- What corrections have to be applied. Objects in the sky appear to be positioned on the celestial sphere, an indefinite distance away. A sphere is a three-dimensional object, but its surface is two-dimensional. Spherical geometry is carried out on the surface of a sphere: it resembles ordinary (plane) geometry, but it involves new rules and relationships. Copyright : Fiona Vincent 1998. Revised and updated on 2003. Converted to .pdf by Alfonso Pastor on 2005 and revised on 2015 4 Positional Astronomy: Chapter 01 The terrestrial sphere We will start with a familiar sphere: the Earth (assume for the moment that it is spherical), spinning around an axis. The North & South Poles are where this axis meets the Earth's surface. The equator lies midway between them. The equator is an example of a great circle: one whose plane passes through the centre of the sphere. Every great circle has two poles. We can define these: (a) as the points which are 90° away from the circle, on the surface of the sphere. (b) as the points where the perpendicular to the plane of the great circle cuts the surface of the sphere. These two definitions are equivalent. The length of a great-circle arc on the surface of a sphere is the angle between its end-points, as seen at the centre of the sphere, and is expressed in degrees (not miles, kilometres etc.). A great circle is a geodesic (the shortest distance between two points) on the surface of a sphere, analogous to a straight line on a plane surface. To describe a location X on the surface of the Earth, we use latitude and longitude (two coordinates, because the surface is two-dimensional). Copyright : Fiona Vincent 1998. Revised and updated on 2003. Converted to .pdf by Alfonso Pastor on 2005 and revised on 2015 5 Draw a great circle from pole to pole, passing through location X: this is a meridian of longitude. The latitude of X is the angular distance along this meridian from the equator to X, measured from -90° at the South Pole to +90° at the North Pole. The co-latitude of X is the angular distance from the North Pole to X co-latitude = 90° - latitude. There is no obvious point of origin for measuring longitude; for historical reasons, the zero-point is the meridian which passes through Greenwich (also called the Prime Meridian). The longitude of X is the angular distance along the equator from the Prime Meridian to the meridian through X. It may be measured east or west 0° to 360°, or both ways 0° to 180°. Small circles parallel to equator are parallels of latitude. The circumference of a small circle at any given latitude is 360 * cos(latitude) in degrees. The length of arc of a small circle between two meridians of longitude is (difference in longitude) * cos(latitude). The length of arc of a great-circle distance is always shorter than the length of arc of a small circle, as we shall see in the next section. The length of arc of a great- circle distance is called Orthodromic distance. Note that a position on the surface of the Earth is fixed using one fundamental circle (the equator) and one fixed point on it (the intersection with a meridian referred to Greenwich Meridian). Celestial navigation used at sea (and in the air) involves spherical trigonometry, so the results are in angular measure (degrees). These must be converted to linear measure for practical use. We define the nautical mile as 1 arc-minute along a great circle on Earth's surface. This comes out about 15% greater than the normal "statute" mile (6080 feet instead of 5280 feet). Note: terrestrial coordinates are actually more complicated than this, because the Earth is not really a sphere. One source where you can find out more about this is the Ordnance Survey's "Guide to coordinate systems in Great Britain". Copyright : Fiona Vincent 1998. Revised and updated on 2003. Converted to .pdf by Alfonso Pastor on 2005 and revised on 2015 6 Exercise: 1 Alderney, in the Channel Islands, has longitude 2°W, latitude 50°N. Winnipeg, in Canada, has longitude 97°W, latitude 50°N. How far apart are they, in nautical miles, along a parallel of latitude? Distance along a parallel of latitude is (difference in longitude) x cos(latitude) = (97° - 2°) cos(50°) = 61.06° But 1° = 60 nautical miles. So the distance is 61.06 x 60 = 3663 nautical miles. (Space left intentionally blank for notes) Copyright : Fiona Vincent 1998. Revised and updated on 2003. Converted to .pdf by Alfonso Pastor on 2005 and revised on 2015 7 Positional Astronomy: Chapter 02 Spherical trigonometry Spherical trigonometry note : The diagrams on this pages are only correct if the observer is imagined to be at a finite distance from the outside of the sphere, as in this diagram. If the observer is imagined to be infinitely far away, the diagram should be drawn with the z-axis emerging from the sphere a little way "in front of" the edge, thus: In my opinion, this version, although equally correct, is slightly more difficult to understand. So throughout the rest of these pages, I am using diagrams of the type shown in the next page. My thanks to Dr Friedrich Firneis of the Austrian Academy of Sciences, to Ryu Izawa of the University of Colorado, USA, and to engineering consultant David Bosher, who provided the diagram above for helpful discussions on this point.