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Astrowidget. Position Calculations 1 P AstroWidget. Position Calculations 1 P. de Vicente Informe Tecnico´ IT-CAY 1997/8 CONTENTS 1 Contents 1 Introduction 2 2 Coordinate systems 2 3 Time 3 3.1 Civil Calendar to Julian Day . 4 3.2 Julian Day to Civil Calendar . 4 4 Precession 4 5 Nutation 5 6 Anual Aberration 6 7 Local Standard of Rest 10 8 Parallax 12 9 Sidereal time 13 10 Apparent Coordinates 13 11 Topocentric Coordinates 14 12 Planets 14 13 Equatorial to Galactic Coordinate Conversion 14 1 INTRODUCTION 2 1 Introduction Astrowidget is a package created to aid in the astronomical observations of celestial objects. It calculates the position of sources and makes transformations of coordinates. This package is a partial result of the work done to write a control program for the 14m radiotelescope at CAY. Astrowidget is a collection of C programs and functions mostly written from scratch following some excellent books on astronomy (see bibliography). I chose C as the programming language because the control of the 14m radiotelescope will be performed from a VME machine with a real time OS in Unix, with a C compiler. In order to keep the maximum compatibilty I have used Ansi C. All functions have been grouped in libraries and the different programs have been glued in a single package using Tcl/Tk (see http://sunscript.sun.com) Tcl/Tk, is a very powerful script language which provides a very comfortable and nice Graphical User Interface. This feature makes Astrowigdet easy to use and to port to other Unix OSs. This report only concentrates on the lower layer of AstroWidget. That is, the basic functions used for computing positions and making transformations of coordinates. 2 Coordinate systems In astronomy there are 4 basic coordinate systems which may be distinguished by the origin of the coordinate system and the XY plane. In all cases except in the topocentric system, the X axis is given by the intersection of the ecliptic with the true equator. • heliocentric equatorial. The origin is the center of the sun. The XY plane is the true equator. This is seldom used. • geocentric equatorial. The origin is the center of the earth. The XY plane is the true equator. The cartesian coordinates are written in the following way: x = r cos δ cos α (1) y = r cos δ sin α (2) z = r sin δ (3) α and δ are usually known as rigth ascention and declination. This is the most widely system because it is used for galactic and extragalactic sources. • heliocentric ecliptic. The origin is the center of the sun. The XY plane is the ecliptic. The cartesian coordinates are written as follows, x = r cos b cos l (4) y = r cos b sin l (5) z = r sin b (6) where l and b are the ecliptic longitude and ecliptic latitude respectively. This is the usual system for planets and other objects in the solar system. • geocentric ecliptic. The origin is the center of the earth, and the XY plane is the ecliptic. The cartesian coordinates are written as, x = r cos β cos λ (7) y = r cos β sin λ (8) z = r sin β (9) 3 TIME 3 where λ and β are the ecliptic longitude and ecliptic latitude respectively. This system is also used for objects in the solar system. • topocentric. The origin is the local place of observation on the surface of the Earth, the XY plane is the horizon and the X axis the local meridian. The cartesian coordinates are written as, x = r cos Az cos El (10) y = r cos Az sin El (11) z = r sin El (12) where Az and El are the azimuth and elevation of the source latitude respectively. The true equator is affected by the nutation. We remind here that there is no true ecliptic, since the ecliptic is defined by the regular motion of the earth around the sun. Since these systems are linked to the Earth they depend on time. Therefore it is the common practice to refer the coordinates of celestial objects to standard epochs, like B1950 or J2000, and then transform to the desired epoch. In order to go from the equatorial to the ecliptic coordinate system one has to make a rotation. If one desires to go from heliocentric to geocentric it is necessary to make a translation from the baricenter of the solar system to the center of the earth. This implies to calculate the position of the earth relative to the sun. All transformations can be easily described within a vectorial space. Let (xg; yg; zg) be the coordinates of an object in the equatorial geocentric system for a certain epoch. The coordinates in the ecliptic heliocentric system for the same epoch should be: 0 1 0 1 0 1 0 1 xh 1 0 0 xg x⊕ B C B C B C B C @ yh A = @ 0 cos " sin " A @ yg A + @ y⊕ A zh 0 − sin " cos " zg z⊕ where " is the obliquity of the ecliptic and (x⊕; y⊕; z⊕) are the coordinates of the Earth referred to an ecliptic heliocentric coordinate system. The obliquity of the ecliptic depends on time and does not have an exact expression. However we use the following formula, adopted by the IAU [?], " = "0 + δ" (13) ◦ 0 00 00 00 2 00 3 "0 = 23 26 21 :448 − 46 :8150 · T − 0 :00059T + 0 :001813T (14) where T is the time in julian centuries from the epoch J2000, and where δ" is obtained when the nutation is computed. This expression has an error of 1 arcsec over a period of 2000 years, and 10 arcsec over a period of 4000 years. Figure ?? shows the variation of the obliquity in an interval of 20000 years around J2000, and the approximation that we use. 3 Time In Physics the second is the fundamental time unit. It is provided by a Cesium clock. The scale associated with it is called TAI (International Atomic Time). However in astronomy, the earth rotation and its rotation around the sun, are fundamental for the observation of celestial bodies. The scale associated with the earth rotation is called UT1 (Universal Time). The 1 suffix accounts for many corrections to the earth rotation. See [?] for an explanation. There is a third scale which is the one used for legal purposes, called UTC, which links UT1 and TAI. At any time the UTC differs a integer number of seconds from 4 PRECESSION 4 TAI and less than a second from UT1. Therefore the UTC jumps 1 second each 1 or 2 years. When we compute an astronomical event we will use the UTC scale, since it is the legal one. That means that we will have to supply DUT1=UT1−UTC. The origin of the time scale as accorded by the IAU is at 12 AM of January 1, on year −4712 (juliano calendar), at the Greenwich meridian. Days counted from this origin are called Julian days (JD). One days has 86400 seconds. One of the fundamental epochs is J2000, which corresponds to January 1, at 12 AM at the Greenwich meridian on year 2000. Another fundamental epoch is B1950 which corresponds to Julian Day 2433282.423. Sometimes a Modified Julian Day is used. The advantage is that the number is smaller. MJD = JD − 2400000:5 (15) Through all the calculations we will use an algorithm taken from [?] to convert from the civil calendar to Julian day and back. We have used day October 12, 1582, as the time when the gregorian calendar began to be used. This date differs from that used by the english colonies which adhered to the calendar much later. 3.1 Civil Calendar to Julian Day (16) 3.2 Julian Day to Civil Calendar (17) 4 Precession The gravitational pull of the Sun and the Moon on a non spherical body produces a torque which causes a movement of the body’s rotation axis. The Earth is an oblate body and the movement of its axis can be splited artificially in a long-term periodic (secular) movement, and a short-term periodic one. The first one is called luni-solar precession and makes the axis of the Earth rotate slowly around another axis. This effect causes the ecliptic longitude of a celestial object to change, while its ecliptic latitude stays fixed. The gravitational influence of the planets also affects the movement of the Earth. They cause the earth to slightly depart from the elliptic orbit it had if there were no planets, and therefore the pole of the ecliptic changes slightly. This effect causes the equatorial declination of a celestial object to be fixed while the right ascention changes with time. Since these two terms are long term they have been grouped in a single expression called general precession. The general precession formula describes the change of the Earth axis from an initial to a final epoch given by Julian Days JDi and JDf respectively. We give here the general precession expression in cartesian equatorial coordinates. Let T be, JD − JD2000 T = i = JD2000 = 2451545:0 (18) 36525 JD − JD t = f i (19) 36525 5 NUTATION 5 Let the variables χ, z and θ be χ = (2306:2181 + 1:39656T − 0:000139T 2)t + (0:30188 − 0:000344T )t2 + 0:017998t3 z = (2306:2181 + 1:39656T − 0:000139T 2)t + (1:09468 + 0:000066T )t2 + 0:018203t3 θ = (2004:3109 − 0:85330T − 0:000217T 2)t − (0:42665 + 0:000217T )t2 − 0:041833t3 where the units are sexagesimal arcseconds Let xi, yi and zi be the initial cartesian equatorial coordinates of the object and xf , yf and zf the final cartesian equatorial coordinates.
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