DOCSLIB.ORG

Spherical Trigonometry of the Projected Baseline Angle

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spherical Trigonometry of the Projected Baseline Angle Spherical Trigonometry of the Projected Baseline Angle Richard J. Mathar∗ Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands† (Dated: October 29, 2018) The basic geometry of a stellar interferometer with two telescopes consists of a baseline vector and a direction to a star. Two derived vectors are the delay vector, and the projected baseline vector in the plane of the wavefronts of the stellar light. The manuscript deals with the trigonometry of projecting the baseline further outwards onto the celestial sphere. The position angle of the projected baseline is defined, measured in a plane tangential to the celestial sphere, tangent point at the position of the star. This angle represents two orthogonal directions on the sky, differential star positions which are aligned with or orthogonal to the gradient of the delay recorded in the u−v plane. The North Celestial Pole is chosen as the reference direction of the projected baseline angle, adapted to the common definition of the “parallactic” angle. PACS numbers: 95.10.Jk, 95.75.Kk Keywords: projected baseline; parallactic angle; position angle; celestial sphere; spherical astronomy; stellar interferometer I. SCOPE II. SITE GEOMETRY A. Telescope Positions The paper describes a standard to define the plane that contains the baseline of a stellar interferometer and the 1. Spherical Approximation star, and its intersection with the Celestial Sphere. This intersection is a great circle, and its section between the In a geocentric coordinate system, the Cartesian coor- star and the point where the prolongated baseline meets dinates of the two telescopes of a stellar interferometer the Celestial Sphere defines an oriented projected base- are related to the geographic longitudes λi, latitudes φi line. The incentive to use this projection onto the Ce- and effective Earth radius ρ (sum of the Earth radius and lestial Sphere is that this ties the baseline to sky coordi- altitude above sea level), nates, which defines directions in equatorial coordinates which are independent of the observatory’s location on cos λi cos φi Earth. This is (i) the introduction of a polar coordi- Ti ≡ ρ sin λi cos φi , i =1, 2. (1) sin φ nate system’s polar angle in the u − v coordinate system i g associated with the visibility tables of the Optical Inter- ferometry Exchange Format (OIFITS) [15, 16], detailing We add a “g” to the geocentric coordinates to set them the orthogonal directions which are well and poorly re- apart from other Cartesian coordinates that will be used solved by the interferometer, and (ii) a definition of ce- further down. A great circle of radius ρ ≈ 6380 km, cen- lestial longitudes in a coordinate system where the pivot tered at the Earth center, connects T1 and T2; a vector Jˆ 1 T T point has been relocated from the North Celestial Pole perpendicular to this circle is ≡ ρ2 sin Z 1 × 2, where arXiv:astro-ph/0608273v3 2 May 2009 (NCP) to the star. a baseline aperture angle Z has been defined under which the baseline vector In overview, Section II introduces the standard set b = T2 − T1 (2) of variables in the familiar polar coordinate systems. Section III defines and computes the baseline position is seen from the Earth center: angle—which can be done purely in equatorial or hori- zontal coordinates or in a hybrid way if the parallactic T1 · T2 = |T1| |T2| cos Z; (3) angle is used to bridge these. Some comments on relating cos Z = cos φ1 cos φ2 cos(λ2 − λ1) + sin φ1 sin φ2.(4) distances away from the optical axis (star) in the tangent plane to the optical path delay conclude the manuscript Jˆ is the axis of rotation when T1 is moved toward T2. in Section IV. The nautical direction from T1 to T2 is computed as follows: define a tangent plane to the Earth at T1. Two orthonormal vectors that span this plane are ∗ Electronic address: [email protected]; − cos λ1 sin φ1 URL: http://www.strw.leidenuniv.nl/ mathar ~ Nˆ 1 ≡ − sin λ1 sin φ1 ∼ ∂T1/∂φ1; |Nˆ 1| =1. †Supported by the NWO VICI grant 639.043.201 to A. Quirren- cos φ1 bach, “Optical Interferometry: A new Method for Studies of Ex- g trasolar Planets.” (5) 2 to the North and example − sin λ1 C x Eˆ 1 ≡ cos λ1 ∼ ∂T1/∂λ1; |Eˆ 1| =1. (6) C = Cy , (9) 0 C g z g to the East. The unit vector from T1 to T2 along the plus local telescope coordinates 1 great circle is Jˆ × T1. The compass rose angle τ is ρ T ≡ C + t , i =1, 2. (10) Jˆ 1 T Nˆ i i computed by its decomposition × ρ 1 = cos τ 1 + sin τEˆ 1 within the tangent plane. It becomes zero if T2 is North of T1, and becomes π/2 if T2 is East of T1: Fig. B. Sky coordinates 1. Geodetic longitude λ, geodetic latitude φ and altitude North H above the geoid are defined with the array center [7, T2 21, 24, 25], Ab−A C (N + H)cos φ cos λ x Cy = (N + H)cos φ sin λ , (11) Ab τ>0 2 Cz N(1 − e )+ H sin φ g g East where e is the eccentricity of the Earth ellipsoid, and 2 2 N ≡ ρe/ 1 − e sin φ the distance from the array center A>0 p T to the Earth axis measured along the local vertical. 1 In a plane tangential to the geoid at C we define a star azimuth A, a zenith angle z, and a star elevation a = π/2 − z, star − cos A sin z − cos A cos a FIG. 1: In the horizontal coordinate system, the baseline di- s = sin A sin z = sin A cos a . (12) rection is characterized by the azimuth angle Ab. In the case cos z sin a shown, 0 <A<π<Ab and D < 0. t t We label this coordinate system t as “topocentric,” with (Note that the tables in [5] use a different definition.) the first component North, the second component West Straight forward algebra establishes τ from and the third component up. The value of A used in this script picks one of the (countably many) conventions: cos φ1 sin φ2 − sin φ1 cos φ2 cos(λ1 − λ2) cos τ = ; (7) South means A = 0 and West means A = +π/2. This sin Z conventional definition of the horizontal means the star cos φ2 sin(λ2 − λ1) coordinates can be transformed to the equatorial param- sin τ = . (8) sin Z eters of hour angle h = l − α, declination δ, and right ascension α [10, (5.45)][8, (2.13)], 2. Caveats cos a sin A = cos δ sin h; (13) cos a cos A = − sin δ cos φ + cos δ cos h sin φ; (14) The angle τ of the previous section does not transform sin a = sin δ sin φ + cos δ cos h cos φ; (15) exactly into τ ± π if the roles of the two telescopes are cos δ cos h = sin a cos φ + cos a cos A sin φ; (16) swapped, because the great circle, which has been used to define the direction, is not a loxodrome. This asym- sin δ = sin a sin φ − cos a cos A cos φ. (17) metry within the definition indicates that a more generic These convert (12) into framework to represent the geometry is useful. If we consider (i) geodetic rather than geocentric rep- cos φ sin δ − sin φ cos δ cos h resentations of geographic latitudes [14] or (ii) long base- s = cos δ sin h . (18) line interferometers build into rugged landscapes, the 5 sin φ sin δ + cos φ cos δ cos h t parameters above (ρ, λi, φi) are too constrained to han- dle the 6 degrees of freedom of two earth-fixed telescope A star at hour angle h = 0 is on the Meridian, positions. Nevertheless, a telescope array “platform” is helpful to define a zenith, a horizon, and associated co- sin(δ − φ) ordinates like the zenith distance or the star azimuth. s = 0 , (h = 0), (19) C cos(δ − φ) This leads to the OIFITS concept, an array center , for t 3 north of the zenith at A = π if δ − φ> 0, south at A =0 s if δ − φ < 0. A reference point on the Celestial Sphere D P is the North Celestial Pole (NCP), given by insertion of δ = π/2 into (18), θ T b T cos φ 1 2 s+ = 0 . (20) sin φ FIG. 2: Sign conventions of the co-planar vectors b, D and t P. The angular distance between s and s+ is where the angular distance θ between the baseline and s · s+ = sin δ. (21) star directions is introduced as C. Baseline: Generic Position cos θ = cos ab cos a cos(Ab − A) + sin ab sin a. (28) The star circles around the NCP in 24 hours, which The baseline vector coordinates of the geocentric changes the distance to the baseline in a centric periodic OIFITS system way (App. B). B x b = By = T2 − T1 (22) B III. PROJECTED BASELINE ANGLES z g A. Definition (defined to stretch from T1 to T2—the opposite sign is also in use [17]) could be converted with We define position angles of points on the Celestial bx t2x t1x Bx Sphere relative to a star’s position as the bearing angle b = by = t2y − t1y = Utg(φ, λ)· By by which the NCP must be rotated around the star direc- b t2 t1 B s z t z t z t z g tion (the axis) until the NCP and the particular point (23) of the object are aligned to the same direction (along a to the topocentric system via the rotation matrix celestial circle) from the star.
Load more

Recommended publications
  • A Basic Requirement for Studying the Heavens Is Determining Where In
    Abasic requirement for studying the heavens is determining where in the sky things are. To specify sky positions, astronomers have developed several coordinate systems. Each uses a coordinate grid projected on to the celestial sphere, in analogy to the geographic coordinate system used on the surface of the Earth. The coordinate systems differ only in their choice of the fundamental plane, which divides the sky into two equal hemispheres along a great circle (the fundamental plane of the geographic system is the Earth's equator) . Each coordinate system is named for its choice of fundamental plane. The equatorial coordinate system is probably the most widely used celestial coordinate system. It is also the one most closely related to the geographic coordinate system, because they use the same fun­ damental plane and the same poles. The projection of the Earth's equator onto the celestial sphere is called the celestial equator. Similarly, projecting the geographic poles on to the celest ial sphere defines the north and south celestial poles. However, there is an important difference between the equatorial and geographic coordinate systems: the geographic system is fixed to the Earth; it rotates as the Earth does . The equatorial system is fixed to the stars, so it appears to rotate across the sky with the stars, but of course it's really the Earth rotating under the fixed sky. The latitudinal (latitude-like) angle of the equatorial system is called declination (Dec for short) . It measures the angle of an object above or below the celestial equator. The longitud inal angle is called the right ascension (RA for short).
    [Show full text]
  • COORDINATES, TIME, and the SKY John Thorstensen
    COORDINATES, TIME, AND THE SKY John Thorstensen Department of Physics and Astronomy Dartmouth College, Hanover, NH 03755 This subject is fundamental to anyone who looks at the heavens; it is aesthetically and mathematically beautiful, and rich in history. Yet I'm not aware of any text which treats time and the sky at a level appropriate for the audience I meet in the more technical introductory astronomy course. The treatments I've seen either tend to be very lengthy and quite technical, as in the classic texts on `spherical astronomy', or overly simplified. The aim of this brief monograph is to explain these topics in a manner which takes advantage of the mathematics accessible to a college freshman with a good background in science and math. This math, with a few well-chosen extensions, makes it possible to discuss these topics with a good degree of precision and rigor. Students at this level who study this text carefully, work examples, and think about the issues involved can expect to master the subject at a useful level. While the mathematics used here are not particularly advanced, I caution that the geometry is not always trivial to visualize, and the definitions do require some careful thought even for more advanced students. Coordinate Systems for Direction Think for the moment of the problem of describing the direction of a star in the sky. Any star is so far away that, no matter where on earth you view it from, it appears to be in almost exactly the same direction. This is not necessarily the case for an object in the solar system; the moon, for instance, is only 60 earth radii away, so its direction can vary by more than a degree as seen from different points on earth.
    [Show full text]
  • Arxiv:Astro-Ph/0608273 V1 13 Aug 2006 Ln Oteotclpt Ea Ocuetemanuscript the Conclude IV
    CORE Metadata, citation and similar papers at core.ac.uk Provided by CERN Document Server Spherical Trigonometry of the Projected Baseline Angle Richard J. Mathar∗ Leiden Observatory, P.O. Box 9513, 2300 RA Leiden, The Netherlands (Dated: August 15, 2006) The basic geometry of a stellar interferometer with two telescopes consists of a baseline vector and a direction to a star. Two derived vectors are the delay vector, and the projected baseline vector in the plane of the wavefronts of the stellar light. The manuscript deals with the trigonometry of projecting the baseline further outwards onto the celestial sphere. The position angle of the projected baseline is defined, measured in a plane tangential to the celestial sphere, tangent point at the position of the star. This angle represents two orthogonal directions on the sky, differential star positions which are aligned with or orthogonal to the gradient of the delay recorded in the u−v plane. The North Celestial Pole is chosen as the reference direction of the projected baseline angle, adapted to the common definition of the “parallactic” angle. PACS numbers: 95.10.Jk, 95.75.Kk Keywords: projected baseline; parallactic angle; position angle; celestial sphere; spherical astronomy; stellar interferometer I. SCOPE II. SITE GEOMETRY A. Telescope Positions The paper describes a standard to define the plane that 1. Spherical Approximation contains the baseline of a stellar interferometer and the star, and its intersection with the Celestial Sphere. This In a geocentric coordinate system, the Cartesian coor- intersection is a great circle, and its section between the dinates of the two telescopes of a stellar interferometer star and the point where the stretched baseline meets the are related to the geographic longitude λi, latitude Φi Celestial Sphere defines an oriented projected baseline.
    [Show full text]
  • Astronomical Coordinate Systems
    Appendix 1 Astronomical Coordinate Systems A basic requirement for studying the heavens is being able to determine where in the sky things are located. To specify sky positions, astronomers have developed several coordinate systems. Each sys- tem uses a coordinate grid projected on the celestial sphere, which is similar to the geographic coor- dinate system used on the surface of the Earth. The coordinate systems differ only in their choice of the fundamental plane, which divides the sky into two equal hemispheres along a great circle (the fundamental plane of the geographic system is the Earth’s equator). Each coordinate system is named for its choice of fundamental plane. The Equatorial Coordinate System The equatorial coordinate system is probably the most widely used celestial coordinate system. It is also the most closely related to the geographic coordinate system because they use the same funda- mental plane and poles. The projection of the Earth’s equator onto the celestial sphere is called the celestial equator. Similarly, projecting the geographic poles onto the celestial sphere defines the north and south celestial poles. However, there is an important difference between the equatorial and geographic coordinate sys- tems: the geographic system is fixed to the Earth and rotates as the Earth does. The Equatorial system is fixed to the stars, so it appears to rotate across the sky with the stars, but it’s really the Earth rotating under the fixed sky. The latitudinal (latitude-like) angle of the equatorial system is called declination (Dec. for short). It measures the angle of an object above or below the celestial equator.
    [Show full text]
  • Instrument Manual
    MODS Instrument Manual Document Number: OSU-MODS-2011-003 Version: 1.2.4 Date: 2012 February 7 Prepared by: R.W. Pogge, The Ohio State University MODS Instrument Manual Distribution List Recipient Institution/Company Number of Copies Richard Pogge The Ohio State University 1 (file) Chris Kochanek The Ohio State University 1 (PDF) Mark Wagner LBTO 1 (PDF) Dave Thompson LBTO 1 (PDF) Olga Kuhn LBTO 1 (PDF) Document Change Record Version Date Changes Remarks 0.1 2011-09-01 Outline and block draft 1.0 2011-12 Too many to count... First release for comments 1.1 2012-01 Numerous comments… First-round comments 1.2 2012-02-07 LBTO comments First Partner Release 2 MODS Instrument Manual OSU-MODS-2011-003 Version 1.2 Contents 1 Introduction .................................................................................................................. 6 1.1 Scope .......................................................................................................................... 6 1.2 Citing and Acknowledging MODS ............................................................................ 6 1.3 Online Materials ......................................................................................................... 6 1.4 Acronyms and Abbreviations ..................................................................................... 7 2 Instrument Characteristics .......................................................................................... 8 2.1 Instrument Configurations .......................................................................................
    [Show full text]
  • Observer's Handbook 1967
    THE OBSERVER’S HANDBOOK 1967 Fifty-ninth Year of Publication THE ROYAL ASTRONOMICAL SOCIETY OF CANADA Price One Dollar THE ROYAL ASTRONOMICAL SOCIETY OF CANADA Incorporated 1890 — Royal Charter 1903 The National Office of the Royal Astronomical Society of Canada is located at 252 College Street, Toronto 2B, Ontario. The business office of the Society, reading rooms and astronomical library, are housed here, as well as a large room for the accommodation of telescope making groups. Membership in the Society is open to anyone interested in astronomy. Applicants may affiliate with one of the Society’s seventeen centres across Canada, or may join the National Society direct. Centres of the Society are established in St. John’s, Halifax, Quebec, Montreal, Ottawa, Kingston, Hamilton, Niagara Falls, London, Windsor, Winnipeg, Edmonton, Calgary, Vancouver, Victoria, and Toronto. Addresses of the Centres’ secretaries may be obtained from the National Office. Publications of the Society are free to members, and include the J o u r n a l (6 issues per year) and the O b s e r v e r ’s H a n d b o o k (published annually in November). Annual fees of $7.50 are payable October 1 and include the publi­ cations for the following year. Requests for additional information regarding the Society or its publications may be sent to 252 College Street, Toronto 2B, Ontario. VISITING HOURS AT SOME CANADIAN OBSERVATORIES David Dunlap Observatory, Richmond Hill, Ont. Wednesday afternoons, 2:00-3:00 p.m. Saturday evenings, April through October (by reservation). Dominion Astrophysical Observatory, Victoria, B.C.
    [Show full text]
  • Revised Geometric Estimates of the North Galactic Pole and the Sun's
    Mon. Not. R. Astron. Soc. 000, 1{13 (2016) Printed 27 October 2016 (MN LATEX style file v2.2) Revised Geometric Estimates of the North Galactic Pole and the Sun's Height Above the Galactic Midplane M. T. Karim 1? and Eric E. Mamajek1;2 1Department of Physics & Astronomy, University of Rochester, Rochester, NY 14627, USA 2Current Address: Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA Accepted 2016 October 24. Received 2016 October 22; in original form 2016 September 5 ABSTRACT Astronomers are entering an era of µas-level astrometry utilizing the 5-decade-old IAU Galactic coordinate system that was only originally defined to ∼0◦.1 accuracy, and where the dynamical centre of the Galaxy (Sgr A*) is located ∼0◦.07 from the origin. We calculate new independent estimates of the North Galactic Pole (NGP) using recent catalogues of Galactic disc tracer objects such as embedded and open clusters, infrared bubbles, dark clouds, and young massive stars. Using these cata- logues, we provide two new estimates of the NGP. Solution 1 is an \unconstrained" NGP determined by the galactic tracer sources, which does not take into account the location of Sgr A*, and which lies 90◦:120 ± 0◦:029 from Sgr A*, and Solution 2 is a \constrained" NGP which lies exactly 90◦ from Sgr A*. The \unconstrained" NGP ◦ ◦ ◦ ◦ has ICRS position: αNGP = 192 :729 ± 0 :035, δNGP = 27 :084 ± 0 :023 and θ = 122◦:928 ± 0◦:016. The \constrained" NGP which lies exactly 90◦ away from Sgr A* ◦ ◦ ◦ ◦ has ICRS position: αNGP = 192 :728 ± 0 :010, δNGP = 26 :863 ± 0 :019 and θ = 122◦:928 ± 0◦:016.
    [Show full text]
  • Spherical Trigonometry of the Projected Baseline Angle
    Serb. Astron. J. } 177 (2008), 115 - 124 UDC 520{872 : 521{181 DOI: 10.2298/SAJ0877115M Professional paper SPHERICAL TRIGONOMETRY OF THE PROJECTED BASELINE ANGLE R. J. Mathar Leiden Observatory, Leiden University, P.O. Box 9513, 2300RA Leiden, The Netherlands E{mail: [email protected] (Received: March 11, 2008; Accepted: May 16, 2008) SUMMARY: The basic vector geometry of a stellar interferometer with two telescopes is de¯ned by the right triangle of (i) the baseline vector between the telescopes, of (ii) the delay vector which points to the star, and of (iii) the projected baseline vector in the plane of the wavefront of the stellar light. The plane of this triangle intersects the celestial sphere at the position of the star; the intersection is a circular line segment. The interferometric angular resolution is high (di®raction limited to the ratio of the wavelength over the projected baseline length) in the two directions along this line segment, and low (di®raction limited to the ratio of the wavelength over the telescope diameter) perpendicular to these. The position angle of these characteristic directions in the sky is calculated here, given either local horizontal coordinates, or celestial equatorial coordinates. Key words. Methods: analytical { Techniques: interferometric { Reference systems 1. SCOPE with the visibility tables of the Optical Interferome- try Exchange Format (OIFITS) (Pauls et al. 2004, 2005), detailing the orthogonal directions which are The paper describes a standard to de¯ne the well and poorly resolved by the interferometer, and plane that contains the baseline of a stellar interfer- (ii) a de¯nition of celestial longitudes in a coordinate ometer and the star, and its intersection with the Ce- system where the pivot point has been relocated from lestial Sphere.
    [Show full text]
  • Bowditch on Cel
    These are Chapters from Bowditch’s American Practical Navigator click a link to go to that chapter CELESTIAL NAVIGATION CHAPTER 15. NAVIGATIONAL ASTRONOMY 225 CHAPTER 16. INSTRUMENTS FOR CELESTIAL NAVIGATION 273 CHAPTER 17. AZIMUTHS AND AMPLITUDES 283 CHAPTER 18. TIME 287 CHAPTER 19. THE ALMANACS 299 CHAPTER 20. SIGHT REDUCTION 307 School of Navigation www.starpath.com Starpath Electronic Bowditch CHAPTER 15 NAVIGATIONAL ASTRONOMY PRELIMINARY CONSIDERATIONS 1500. Definition ing principally with celestial coordinates, time, and the apparent motions of celestial bodies, is the branch of as- Astronomy predicts the future positions and motions tronomy most important to the navigator. The symbols of celestial bodies and seeks to understand and explain commonly recognized in navigational astronomy are their physical properties. Navigational astronomy, deal- given in Table 1500. Table 1500. Astronomical symbols. 225 226 NAVIGATIONAL ASTRONOMY 1501. The Celestial Sphere server at some distant point in space. When discussing the rising or setting of a body on a local horizon, we must locate Looking at the sky on a dark night, imagine that celes- the observer at a particular point on the earth because the tial bodies are located on the inner surface of a vast, earth- setting sun for one observer may be the rising sun for centered sphere. This model is useful since we are only in- another. terested in the relative positions and motions of celestial Motion on the celestial sphere results from the motions bodies on this imaginary surface. Understanding the con- in space of both the celestial body and the earth. Without cept of the celestial sphere is most important when special instruments, motions toward and away from the discussing sight reduction in Chapter 20.
    [Show full text]
  • Brightest Stars : Discovering the Universe Through the Sky's Most Brilliant Stars / Fred Schaaf
    ffirs.qxd 3/5/08 6:26 AM Page i THE BRIGHTEST STARS DISCOVERING THE UNIVERSE THROUGH THE SKY’S MOST BRILLIANT STARS Fred Schaaf John Wiley & Sons, Inc. flast.qxd 3/5/08 6:28 AM Page vi ffirs.qxd 3/5/08 6:26 AM Page i THE BRIGHTEST STARS DISCOVERING THE UNIVERSE THROUGH THE SKY’S MOST BRILLIANT STARS Fred Schaaf John Wiley & Sons, Inc. ffirs.qxd 3/5/08 6:26 AM Page ii This book is dedicated to my wife, Mamie, who has been the Sirius of my life. This book is printed on acid-free paper. Copyright © 2008 by Fred Schaaf. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada Illustration credits appear on page 272. Design and composition by Navta Associates, Inc. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copy- right.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions.
    [Show full text]
  • II: "Representations of Celestial Coordinates in FITS"
    A&A 395, 1077–1122 (2002) Astronomy DOI: 10.1051/0004-6361:20021327 & c ESO 2002 Astrophysics Representations of celestial coordinates in FITS M. R. Calabretta1 and E. W. Greisen2 1 Australia Telescope National Facility, PO Box 76, Epping, NSW 1710, Australia 2 National Radio Astronomy Observatory, PO Box O, Socorro, NM 87801-0387, USA Received 24 July 2002 / Accepted 9 September 2002 Abstract. In Paper I, Greisen & Calabretta (2002) describe a generalized method for assigning physical coordinates to FITS image pixels. This paper implements this method for all spherical map projections likely to be of interest in astronomy. The new methods encompass existing informal FITS spherical coordinate conventions and translations from them are described. Detailed examples of header interpretation and construction are given. Key words. methods: data analysis – techniques: image processing – astronomical data bases: miscellaneous – astrometry 1. Introduction PIXEL p COORDINATES j This paper is the second in a series which establishes conven- linear transformation: CRPIXja r j tions by which world coordinates may be associated with FITS translation, rotation, PCi_ja mij (Hanisch et al. 2001) image, random groups, and table data. skewness, scale CDELTia si Paper I (Greisen & Calabretta 2002) lays the groundwork by developing general constructs and related FITS header key- PROJECTION PLANE x words and the rules for their usage in recording coordinate in- COORDINATES ( ,y) formation. In Paper III, Greisen et al. (2002) apply these meth- spherical CTYPEia (φ0,θ0) ods to spectral coordinates. Paper IV (Calabretta et al. 2002) projection PVi_ma Table 13 extends the formalism to deal with general distortions of the co- ordinate grid.
    [Show full text]
  • 19690018939.Pdf
    General Disclaimer One or more of the Following Statements may affect this Document This document has been reproduced from the best copy furnished by the organizational source. It is being released in the interest of making available as much information as possible. This document may contain data, which exceeds the sheet parameters. It was furnished in this condition by the organizational source and is the best copy available. This document may contain tone-on-tone or color graphs, charts and/or pictures, which have been reproduced in black and white. This document is paginated as submitted by the original source. Portions of this document are not fully legible due to the historical nature of some of the material. However, it is the best reproduction available from the original submission. Produced by the NASA Center for Aerospace Information (CASI) SELENOCENTRIC AND LUNAR TOPOCENTRIC SPHERICAL COORDINATES B. KOLACZEK 6 LACCESS, M NUMD)RI 0 WApas) 'COD[" 'A ," (NA • CRORTMXORADNUMMER) 3 ICATWOOFty) 4 4N, Ai V14. 0 Smithsonian Astrophysical Observatory SPECIAL REPORT 286 1 Research in Space Sciei.ce SAO Special Report No. 286 r SELENOCENTRIC AND LUNAR TOPOCENTRIC COORDINATES OF DIFFERENT SPHERICAL SYSTEMS B. Kolaczek September 20, 1968 Smithsonian Institution Astrophysical Observatory Cambridge, Massachusetts 02138 PRECEDING PAGE BLANK NOT FIL WO, TABLE OF CONTENTS Section Pale ABSTRACT ............................... vii 1 INTRODUCTION ............................ 1 2 SELENOEQUATORIA& COORDINATE SYSTEM ....... 5 2.1 Definition; Transformation of Mean Geoequatorial into Mean Selenoequatorial Coordinates ....... 5 2.2 Transformation of Geo-apparent Geoequatorial into Seleno-apparent Selenoequatorial Coordinates ... 10 2.3 Calculation of the Mean Selenoequatorial Coordinates .... .. ... .... .. .. .... .. 12 2.4 Transformation of Mean into Seleno-apparent Selenoequatorial Coordinates .
    [Show full text]