The Celestial Sphere: Coordinate Systems
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Basic Principles of Celestial Navigation James A
Basic principles of celestial navigation James A. Van Allena) Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242 ͑Received 16 January 2004; accepted 10 June 2004͒ Celestial navigation is a technique for determining one’s geographic position by the observation of identified stars, identified planets, the Sun, and the Moon. This subject has a multitude of refinements which, although valuable to a professional navigator, tend to obscure the basic principles. I describe these principles, give an analytical solution of the classical two-star-sight problem without any dependence on prior knowledge of position, and include several examples. Some approximations and simplifications are made in the interest of clarity. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1778391͔ I. INTRODUCTION longitude ⌳ is between 0° and 360°, although often it is convenient to take the longitude westward of the prime me- Celestial navigation is a technique for determining one’s ridian to be between 0° and Ϫ180°. The longitude of P also geographic position by the observation of identified stars, can be specified by the plane angle in the equatorial plane identified planets, the Sun, and the Moon. Its basic principles whose vertex is at O with one radial line through the point at are a combination of rudimentary astronomical knowledge 1–3 which the meridian through P intersects the equatorial plane and spherical trigonometry. and the other radial line through the point G at which the Anyone who has been on a ship that is remote from any prime meridian intersects the equatorial plane ͑see Fig. -
Equatorial and Cartesian Coordinates • Consider the Unit Sphere (“Unit”: I.E
Coordinate Transforms Equatorial and Cartesian Coordinates • Consider the unit sphere (“unit”: i.e. declination the distance from the center of the (δ) sphere to its surface is r = 1) • Then the equatorial coordinates Equator can be transformed into Cartesian coordinates: right ascension (α) – x = cos(α) cos(δ) – y = sin(α) cos(δ) z x – z = sin(δ) y • It can be much easier to use Cartesian coordinates for some manipulations of geometry in the sky Equatorial and Cartesian Coordinates • Consider the unit sphere (“unit”: i.e. the distance y x = Rcosα from the center of the y = Rsinα α R sphere to its surface is r = 1) x Right • Then the equatorial Ascension (α) coordinates can be transformed into Cartesian coordinates: declination (δ) – x = cos(α)cos(δ) z r = 1 – y = sin(α)cos(δ) δ R = rcosδ R – z = sin(δ) z = rsinδ Precession • Because the Earth is not a perfect sphere, it wobbles as it spins around its axis • This effect is known as precession • The equatorial coordinate system relies on the idea that the Earth rotates such that only Right Ascension, and not declination, is a time-dependent coordinate The effects of Precession • Currently, the star Polaris is the North Star (it lies roughly above the Earth’s North Pole at δ = 90oN) • But, over the course of about 26,000 years a variety of different points in the sky will truly be at δ = 90oN • The declination coordinate is time-dependent albeit on very long timescales • A precise astronomical coordinate system must account for this effect Equatorial coordinates and equinoxes • To account -
Capricious Suntime
[Physics in daily life] I L.J.F. (Jo) Hermans - Leiden University, e Netherlands - [email protected] - DOI: 10.1051/epn/2011202 Capricious suntime t what time of the day does the sun reach its is that the solar time will gradually deviate from the time highest point, or culmination point, when on our watch. We expect this‘eccentricity effect’ to show a its position is exactly in the South? e ans - sine-like behaviour with a period of a year. A wer to this question is not so trivial. For ere is a second, even more important complication. It is one thing, it depends on our location within our time due to the fact that the rotational axis of the earth is not zone. For Berlin, which is near the Eastern end of the perpendicular to the ecliptic, but is tilted by about 23.5 Central European time zone, it may happen around degrees. is is, aer all, the cause of our seasons. To noon, whereas in Paris it may be close to 1 p.m. (we understand this ‘tilt effect’ we must realise that what mat - ignore the daylight saving ters for the deviation in time time which adds an extra is the variation of the sun’s hour in the summer). horizontal motion against But even for a fixed loca - the stellar background tion, the time at which the during the year. In mid- sun reaches its culmination summer and mid-winter, point varies throughout the when the sun reaches its year in a surprising way. -
Earth-Centred Universe
Earth-centred Universe The fixed stars appear on the celestial sphere Earth rotates in one sidereal day The solar day is longer by about 4 minutes → scattered sunlight obscures the stars by day The constellations are historical → learn to recognise: Ursa Major, Ursa Minor, Cassiopeia, Pegasus, Auriga, Gemini, Orion, Taurus Sun’s Motion in the Sky The Sun moves West to East against the background of Stars Stars Stars stars Us Us Us Sun Sun Sun z z z Start 1 sidereal day later 1 solar day later Compared to the stars, the Sun takes on average 3 min 56.5 sec extra to go round once The Sun does not travel quite at a constant speed, making the actual length of a solar day vary throughout the year Pleiades Stars near the Sun Sun Above the atmosphere: stars seen near the Sun by the SOHO probe Shield Sun in Taurus Image: Hyades http://sohowww.nascom.nasa.g ov//data/realtime/javagif/gifs/20 070525_0042_c3.gif Constellations Figures courtesy: K & K From The Beauty of the Heavens by C. F. Blunt (1842) The Celestial Sphere The celestial sphere rotates anti-clockwise looking north → Its fixed points are the north celestial pole and the south celestial pole All the stars on the celestial equator are above the Earth’s equator How high in the sky is the pole star? It is as high as your latitude on the Earth Motion of the Sky (animated ) Courtesy: K & K Pole Star above the Horizon To north celestial pole Zenith The latitude of Northern horizon Aberdeen is the angle at 57º the centre of the Earth A Earth shown in the diagram as 57° 57º Equator Centre The pole star is the same angle above the northern horizon as your latitude. -
Chapter 7 Mapping The
BASICS OF RADIO ASTRONOMY Chapter 7 Mapping the Sky Objectives: When you have completed this chapter, you will be able to describe the terrestrial coordinate system; define and describe the relationship among the terms com- monly used in the “horizon” coordinate system, the “equatorial” coordinate system, the “ecliptic” coordinate system, and the “galactic” coordinate system; and describe the difference between an azimuth-elevation antenna and hour angle-declination antenna. In order to explore the universe, coordinates must be developed to consistently identify the locations of the observer and of the objects being observed in the sky. Because space is observed from Earth, Earth’s coordinate system must be established before space can be mapped. Earth rotates on its axis daily and revolves around the sun annually. These two facts have greatly complicated the history of observing space. However, once known, accu- rate maps of Earth could be made using stars as reference points, since most of the stars’ angular movements in relationship to each other are not readily noticeable during a human lifetime. Although the stars do move with respect to each other, this movement is observable for only a few close stars, using instruments and techniques of great precision and sensitivity. Earth’s Coordinate System A great circle is an imaginary circle on the surface of a sphere whose center is at the center of the sphere. The equator is a great circle. Great circles that pass through both the north and south poles are called meridians, or lines of longitude. For any point on the surface of Earth a meridian can be defined. -
Calibrating the Fundamental Plane with SDSS DR8 Data⋆
A&A 557, A21 (2013) Astronomy DOI: 10.1051/0004-6361/201321466 & c ESO 2013 Astrophysics Calibrating the fundamental plane with SDSS DR8 data Christoph Saulder1,2,Steffen Mieske1, Werner W. Zeilinger2, and Igor Chilingarian3,4 1 European Southern Observatory, Alonso de Córdova 3107, Vitacura, Casilla 19001, Santiago, Chile e-mail: [csaulder,smieske]@eso.org 2 Department of Astrophysics, University of Vienna, Türkenschanzstraße 17, 1180 Vienna, Austria e-mail: [email protected] 3 Smithsonian Astrophysical Observatory, Harvard-Smithsonian Center for Astrophysics, 60 Garden St. MS09 Cambridge, MA 02138, USA e-mail: [email protected] 4 Sternberg Astronomical Institute, Moscow State University, 13 Universitetski prospect, 119992 Moscow, Russia Received 13 March 2013 / Accepted 27 May 2013 ABSTRACT We present a calibration of the fundamental plane using SDSS Data Release 8. We analysed about 93 000 elliptical galaxies up to z < 0.2, the largest sample used for the calibration of the fundamental plane so far. We incorporated up-to-date K-corrections and used GalaxyZoo data to classify the galaxies in our sample. We derived independent fundamental plane fits in all five Sloan filters u, g, r, i,andz. A direct fit using a volume-weighted least-squares method was applied to obtain the coefficients of the fundamental plane, which implicitly corrects for the Malmquist bias. We achieved an accuracy of 15% for the fundamental plane as a distance indicator. We provide a detailed discussion on the calibrations and their influence on the resulting fits. These re-calibrated fundamental plane relations form a well-suited anchor for large-scale peculiar-velocity studies in the nearby universe. -
And Are Lines on Sphere B That Contain Point Q
11-5 Spherical Geometry Name each of the following on sphere B. 3. a triangle SOLUTION: are examples of triangles on sphere B. 1. two lines containing point Q SOLUTION: and are lines on sphere B that contain point Q. ANSWER: 4. two segments on the same great circle SOLUTION: are segments on the same great circle. ANSWER: and 2. a segment containing point L SOLUTION: is a segment on sphere B that contains point L. ANSWER: SPORTS Determine whether figure X on each of the spheres shown is a line in spherical geometry. 5. Refer to the image on Page 829. SOLUTION: Notice that figure X does not go through the pole of ANSWER: the sphere. Therefore, figure X is not a great circle and so not a line in spherical geometry. ANSWER: no eSolutions Manual - Powered by Cognero Page 1 11-5 Spherical Geometry 6. Refer to the image on Page 829. 8. Perpendicular lines intersect at one point. SOLUTION: SOLUTION: Notice that the figure X passes through the center of Perpendicular great circles intersect at two points. the ball and is a great circle, so it is a line in spherical geometry. ANSWER: yes ANSWER: PERSEVERANC Determine whether the Perpendicular great circles intersect at two points. following postulate or property of plane Euclidean geometry has a corresponding Name two lines containing point M, a segment statement in spherical geometry. If so, write the containing point S, and a triangle in each of the corresponding statement. If not, explain your following spheres. reasoning. 7. The points on any line or line segment can be put into one-to-one correspondence with real numbers. -
Solar Engineering Basics
Solar Energy Fundamentals Course No: M04-018 Credit: 4 PDH Harlan H. Bengtson, PhD, P.E. Continuing Education and Development, Inc. 22 Stonewall Court Woodcliff Lake, NJ 07677 P: (877) 322-5800 [email protected] Solar Energy Fundamentals Harlan H. Bengtson, PhD, P.E. COURSE CONTENT 1. Introduction Solar energy travels from the sun to the earth in the form of electromagnetic radiation. In this course properties of electromagnetic radiation will be discussed and basic calculations for electromagnetic radiation will be described. Several solar position parameters will be discussed along with means of calculating values for them. The major methods by which solar radiation is converted into other useable forms of energy will be discussed briefly. Extraterrestrial solar radiation (that striking the earth’s outer atmosphere) will be discussed and means of estimating its value at a given location and time will be presented. Finally there will be a presentation of how to obtain values for the average monthly rate of solar radiation striking the surface of a typical solar collector, at a specified location in the United States for a given month. Numerous examples are included to illustrate the calculations and data retrieval methods presented. Image Credit: NOAA, Earth System Research Laboratory 1 • Be able to calculate wavelength if given frequency for specified electromagnetic radiation. • Be able to calculate frequency if given wavelength for specified electromagnetic radiation. • Know the meaning of absorbance, reflectance and transmittance as applied to a surface receiving electromagnetic radiation and be able to make calculations with those parameters. • Be able to obtain or calculate values for solar declination, solar hour angle, solar altitude angle, sunrise angle, and sunset angle. -
Michael Perryman
Michael Perryman Cavendish Laboratory, Cambridge (1977−79) European Space Agency, NL (1980−2009) (Hipparcos 1981−1997; Gaia 1995−2009) [Leiden University, NL,1993−2009] Max-Planck Institute for Astronomy & Heidelberg University (2010) Visiting Professor: University of Bristol (2011−12) University College Dublin (2012−13) Lecture program 1. Space Astrometry 1/3: History, rationale, and Hipparcos 2. Space Astrometry 2/3: Hipparcos science results (Tue 5 Nov) 3. Space Astrometry 3/3: Gaia (Thu 7 Nov) 4. Exoplanets: prospects for Gaia (Thu 14 Nov) 5. Some aspects of optical photon detection (Tue 19 Nov) M83 (David Malin) Hipparcos Text Our Sun Gaia Parallax measurement principle… Problematic from Earth: Sun (1) obtaining absolute parallaxes from relative measurements Earth (2) complicated by atmosphere [+ thermal/gravitational flexure] (3) no all-sky visibility Some history: the first 2000 years • 200 BC (ancient Greeks): • size and distance of Sun and Moon; motion of the planets • 900–1200: developing Islamic culture • 1500–1700: resurgence of scientific enquiry: • Earth moves around the Sun (Copernicus), better observations (Tycho) • motion of the planets (Kepler); laws of gravity and motion (Newton) • navigation at sea; understanding the Earth’s motion through space • 1718: Edmond Halley • first to measure the movement of the stars through space • 1725: James Bradley measured stellar aberration • Earth’s motion; finite speed of light; immensity of stellar distances • 1783: Herschel inferred Sun’s motion through space • 1838–39: Bessell/Henderson/Struve -
Astrometry and Optics During the Past 2000 Years
1 Astrometry and optics during the past 2000 years Erik Høg Niels Bohr Institute, Copenhagen, Denmark 2011.05.03: Collection of reports from November 2008 ABSTRACT: The satellite missions Hipparcos and Gaia by the European Space Agency will together bring a decrease of astrometric errors by a factor 10000, four orders of magnitude, more than was achieved during the preceding 500 years. This modern development of astrometry was at first obtained by photoelectric astrometry. An experiment with this technique in 1925 led to the Hipparcos satellite mission in the years 1989-93 as described in the following reports Nos. 1 and 10. The report No. 11 is about the subsequent period of space astrometry with CCDs in a scanning satellite. This period began in 1992 with my proposal of a mission called Roemer, which led to the Gaia mission due for launch in 2013. My contributions to the history of astrometry and optics are based on 50 years of work in the field of astrometry but the reports cover spans of time within the past 2000 years, e.g., 400 years of astrometry, 650 years of optics, and the “miraculous” approval of the Hipparcos satellite mission during a few months of 1980. 2011.05.03: Collection of reports from November 2008. The following contains overview with summary and link to the reports Nos. 1-9 from 2008 and Nos. 10-13 from 2011. The reports are collected in two big file, see details on p.8. CONTENTS of Nos. 1-9 from 2008 No. Title Overview with links to all reports 2 1 Bengt Strömgren and modern astrometry: 5 Development of photoelectric astrometry including the Hipparcos mission 1A Bengt Strömgren and modern astrometry .. -
Southwest Florida Astronomical Society SWFAS
Southwest Florida Astronomical Society SWFAS The Eyepiece September 2013 A MESSAGE FROM THE PRESIDENT I hope everyone had a good Labor Day weekend! Well, as I said last month, Comet ISON is on its way. How it will perform is still uncertain. I will be talking about it and other upcoming events as our program this month. This summer the rain has really hurt observing. Did anyone get to see Nova Delphini at its brightest? We have a very busy fall/winter season with a lot of events and requests. Please let me know if you can help at any of the events. I will be sending out updated calendars regularly. Our October meeting program is to be determined. If you have a presentation or idea for a program, please let me know! Carole Holmberg has a public event planned for Friday Sept 13th at 8pm at the CNCP. The November meeting is our annual Telescope Renaissance night (with observing) starting at 7pm on the 7th at the CNCP. There will be no formal meeting that night. Moon: Sep New 5th, 1st Quarter 12th, Full 19th, Last Quarter 26th Oct New 4th, 1st Quarter 11th, Full 18th, Last Quarter 26th Star Parties: September 14, October 5, November 2, November 30, December 28. The planets: Venus is dominating the evening sky after sunset as Saturn slips further towards the sun. Mercury will make a brief appearance midmonth after sunset. Mars is reappearing in the morning sky (key to finding ISON) and Jupiter rises a few hours after midnight. Club Positions President: Program Coordinator: (239-940-2935) Brian Risley Vacant Club Historian: swfasbrisley@embarqmail. -
The Sundial Cities
The Sundial Cities Joel Van Cranenbroeck, Belgium Keywords: Engineering survey;Implementation of plans;Positioning;Spatial planning;Urban renewal; SUMMARY When observing in our modern cities the sun shade gliding along the large surfaces of buildings and towers, an observer can notice that after all the local time could be deduced from the position of the sun. The highest building in the world - the Burj Dubai - is de facto the largest sundial ever designed. The principles of sundials can be understood most easily from an ancient model of the Sun's motion. Science has established that the Earth rotates on its axis, and revolves in an elliptic orbit about the Sun; however, meticulous astronomical observations and physics experiments were required to establish this. For navigational and sundial purposes, it is an excellent approximation to assume that the Sun revolves around a stationary Earth on the celestial sphere, which rotates every 23 hours and 56 minutes about its celestial axis, the line connecting the celestial poles. Since the celestial axis is aligned with the axis about which the Earth rotates, its angle with the local horizontal equals the local geographical latitude. Unlike the fixed stars, the Sun changes its position on the celestial sphere, being at positive declination in summer, at negative declination in winter, and having exactly zero declination (i.e., being on the celestial equator) at the equinoxes. The path of the Sun on the celestial sphere is known as the ecliptic, which passes through the twelve constellations of the zodiac in the course of a year. This model of the Sun's motion helps to understand the principles of sundials.