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ARTURO CHIESA - RAFFAELE CHIESA

CELESTIAL NAVIGATION

ELEMENTARY ASTRONOMY

PILOTING

A text with 27 operative programs

Arturo and Raffaele Chiesa http://skyseasoftware.webs.com

Celestial Navigation, Elementary Astronomy, Piloting – Contents and Introduction

CONTENTS

INTRODUCTION 1 - A text and an operative software2 - Forms and running of the programs - 27 operative programs3 - Data stored in the software 3.1 - Celestial bodies How to deal with Sun and stars and with Moon and planets 3.2 - Places4 - LOG BOOK and LOG BOOK MANAGER5 - Format and check of the input and output quantities6 - Precision of the output data

First part - CELESTIAL NAVIGATION

I - WHAT IS CELESTIAL NAVIGATION Three programs to immediately obtain the true line of position and the fix and to select the celestial bodies to be sighted

II - SEXTANT DATA REDUCTION II.1 - From sextant altitude to true altitude and vice-versa. Correction for atmospheric refraction and dip. Stars and planets, Sun and Moon II.2 - Manual calculation of the true altitude from the altitude read with the sextant using data and tables available in the NAUTICAL ALMANAC II.3 - Running the program II.4 - Reverse procedure: from true altitude to observed altitude III - INTERPOLATION OF SEXTANT h - t READINGS III.1 - A program to obtain the most probable value of the true altitude ht of a celestial body at the time t by a series of h - t sextant altitude - time readings III.2 - Running the program III.3 - Example. True altitude of the centre of the Sun’s disk obtained by a series of sextant and chronometer readings

IV - THE TRUE LINE OF POSITION IV.1 - The “true line of position” as a part, that is an arc, of the circle of position IV.2 - Purpose and structure of the program IV.3 - Running the program IV.4 - Applications IV.4.1- The selected true lines of position IV.4.2 - Five selected true lines of position in a journey Gibraltar - Canary Islands IV.5 - Option: directly enter the true altitude IV.6 - Pointing out the shape of the true line of position: an arc IV.7 - Limits in the usual procedures of obtaining - the latitude and the longitude at the meridian passage of the Sun - the latitude by the sight of the PolarisAPPENDIX A SUB-PROGRAM TO IMMEDIATELY PLOT AN INTERCEPT

Arturo and Raffaele Chiesa http://skyseasoftware.webs.com2


V - CIRCLE OF AZIMUTHS V.1 - A circle of azimuths for the best choice of the celestial bodies V.2 - Running the program V.3 - Two examples in a passage from Cape Verde Islands to Canary Islands at the dawn and evening twilights of May 18, 2010

VI - VESSEL POSITION - FIX A method to immediately obtain a fix by sighting celestial bodies. Running fix

VI.1 - Principle of the method VI.2 - Mathematical and computer procedures VI.3 - Structure of the program and running VI.4 - Checking and warning of ERROR VI.5 - Procedure of verification VI.6 - A quick demonstration of the running of the program VI.7 - Two fixes by two sights. Observed bodies: the Sun and the Moon VI.7.1 - A journey from South Sardinia Island to Gibraltar - August 2010 VI.7.2 - Calculation and drawing of the two circles of position and their intersection points P1 and P2

VI.8 - Three examples of running fix procedure VI.8.1 - A preliminary statement - Problems in the running fix procedure VI.8.2 - A passage on the North Pacific Ocean Tokyo - Honolulu. Running fix with two planets and two stars when approaching the anti-meridian line September 2010 VI.8.3 - A passage in the South Pacific Ocean Tahiti - Melbourne. Running fix with the Moon, two planets and a star when approaching the anti-meridian line. October -November 2010. A case of ERROR VI.8.4 - A test of the program VESSEL POSITION - FIX with the greatest number of sights: seven. A passage in the South Atlantic Ocean from Cape Town to Cape Horn, November 2010 VI.9 - Planning sights in an Atlantic passage Gibraltar - New York. August 2010 VI.9.1 - Route. VI.9.2 - Programs used to plan the sextant observations VI.9.3 - Diurnal observations of Moon and Venus VI.9.4 - Checking the main compass of the vessel VI.9.5 - Availability of Moon, planets and stars at the twilights

APPENDIX A SUB-PROGRAM TO IMMEDIATELY PLOT THE INTERCEPT

VII - THE PRACTICE OF THE SEXTANT OBSERVATIONS VII.1 - Setting the sextant - Handling - Care in reading the altitude of the celestial bodies and the time VII.2 - Further cares to get a good sight Obtaining the index correction VII.3 - Daylight observations VII.4 - Observations at the twilights VII.5 - Preliminary use of the sextant in the reversed position VII.6 - Errors due to an inaccurate value of the time



Second part - ELEMENTARY ASTRONOMY

VIII - ARIES EPHEMERIDES VIII.1 - Sidereal time and equivalent Aries Local Hour Angle at a date, time and longitude. Direct and reverse procedure VIII.2 - Direct and reverse manual calculations by using the daily data and interpolation tables of the NAUTICAL ALMANAC VIII.3 - Direct and reverse procedure using the program ARIES EPHEMERIDES

IX - POSITION OF THE CELESTIAL BODIES IX.1 - RECALL OF SOME ELEMENTARY PHENOMENA

IX.2 - THE FIVE MAIN PROGRAMS IX.2.1 - First program: ALTITUDE AND AZIMUTH IX.2.2 - Second program: UT AND LT TIMES AT AN ALTITUDE IX.2.3 - Third program: UT AND LT TIMES AT AN AZIMUTH IX.2.4 - Fourth program : RISING, MERIDIAN PASSAGE, SETTING IX.2.5 - Fifth program : TWILIGHTS

IX.3 - Sixth program : REVERSE - TERRESTRIAL COORDINATES IX.3.1 - Purpose of the program IX.3.2 - Rule about the possible existence of terrestrial points IX.3.3 - Running the program and examples IX.3.4 - Obtaining four quadrangular terrestrial points APPENDIX

OBTAINING THE CELESTIAL COORDINATES (SHA* Dec ) OF INFINITELY FAR OBJECTS (STARS, GALAXIES, NEBULA, ETC) FROM THE TIME OF THE MERIDIAN PASSAGE AND THE ALTITUDE AT THE POLES

X - SOLAR SYSTEM X.1 - Recall of elementary notions X.2 - Visibility of the planets from the Earth. Elongation of the planets from the Sun X.3 - The constellations of the zodiacal band X.4 - Running the program X.5 - Applications X.5.1 - Configurations of the Solar System at the two equinoxes and solstices in the years 2010 and 2011 - Visibility of the planets in the night hours X.5.2 - Elongations of planets from the Sun Comparison with the data of THE ASTRONOMICAL ALMANAC X.5.3 - Dates of the conjunctions and oppositions of the planets in the years 2010 and 2011. Comparison with the data of THE ASTRONOMICAL PHENOMENA X.6 - Moving the planets on their orbits X.7 - Sub-program ZODIACAL DATES. Discrepancies with the dates of the “zodiacal signs” X.7.1 - The zodiacal signs of the Astrology X.7.2 - Sub-program ZODIACAL DATES - Years 2010 and 2011 X.7.3 - A curious anomaly: the names of the two tropic lines



XI – ALMANAC AND PLANETS THE TIME LINES XI.1 - Purpose of the two programs XI.2 - Structure of the program ALMANAC and running XI.3 - The shapes of the time lines of the fixed bodies and the ones of the wandering bodies Pointing up the periods of night visibility XI.4 - Applications - Program ALMANAC Sirius, Mars, Moon. A circumpolar star: Alioth A cultural notion. The phenomenon of the missing Moon XI.5 - Program PLANETS XI.5.1 - Purpose of the program and running XI.5.2 - Time lines of the meridian passage of the Sun and the seven planets for the years 2009 and 2010 XI.5.3 - Periods of night visibility of rising, meridian passage, setting of the seven planets seen in San Francisco in the year 2010 XII - DIURNAL ARCS XII.1 - Purpose of the program XII.2 - Structure of the program and running XII.3 - Applications XII.3.1 - Standard set of the three bodies Sirius, Mars, Moon observed in San Francisco on Mars 26, 2010 XII.3.2 - Circumpolar bodies Alioth in San Francisco, closed ring Alnair at Cape Horn, open arc

XIII - CHOOSING PLANETS FOR SEXTANT OBSERVATIONS A SYNOPTIC TABLE FOR THE YEARS 2010 2011

XIV - IMAGE OF THE SKY XIV.1 - Structure of the program. Images of the sky with or without the names of the celestial bodies and constellations. Lines of the ecliptic and celestial equator. Adding other bodies. Sequence of images (function STEP) XIV.2 - Different images of the sky provided by the options of the program XIV.3 - Adding celestial bodies XIV.4 - Occultations of planets by the Moon XIV.5 - Images of the sky during four eclipses of the Sun occurring in the current century XV - MOTION OF THE CONSTELLATIONS XV.1 - Purpose of the program XV.2 - Choice of the groups of constellations and their main star

XVI - IDENTIFICATION OF STARS AND CONSTELLATIONS XVI.1 - Purpose of the program XVI.2 - Running the program XVI.3 - Applications Identification of stars in the sky of Auckland Outlining the position of the estimated point in the sky



Third part - PILOTING

XVII - ROUTES XVII.1 -The four programs - First program: GREAT CIRCLE ROUTE - Course, distance, duration Plotting a great circle line point by point - Second program: GREAT CIRCLE ROUTE - Coordinates of the destination point, duration - Third program: RHUMB LINE ROUTE- Course, distance, duration - Fourth program: RHUMB LINE ROUTE - Coordinates of the destination point, duration. XVII.2 - Correlation between the first and the second program (great circle route) and between the third and fourth program (rhumb line route) A South Pacific passage Tahiti - Melbourne across the anti-meridian line

XVIII - LEGS AND TACKS ROUTES LEGS XVIII.1 - A zig-zag route along a series of WayPoints TACKS XVIII.2 - Dead reckoning in a zig-zag route Sailing tacks in a passage Cape Verde Islands - Canary Islands XIX - GREAT DISTANCES ROUTES XIX.1 - A great circle route travelled with rhumb line legs XIX.2 - Two examples - A North Atlantic passage Gibraltar - New York. Eleven legs - A South Pacific passage Cape Horn - Auckland. Thirteen legs XX - DRAWING THE GRID OF A MERCATOR CHART XX.1 - Features of a Mercator chart and purpose of the program XX.2 - Two examples 1 - Drawing the grid of a Mercator chart on a graph paper sheet 2 - Drawing the grid of a Mercator chart on a squared paper sheet

______________________

FUNCTIONS OF THE PROGRAMS



I N T R O D U C T I O N



1 - A text and an operative software

The book embodies three parts- Celestial Navigation- Elementary Astronomy- Pilotingsupported by an interactive software with 27 operative programs.

The outstanding feature in Celestial Navigation is a new method of immediately obtaining a fix (vessel position) by entering the sequences h - t of the sextant altitudes and chronometer time readings of at least two celestial bodies in a programmed computer. No need of tables like HO tables or similar, no graphical construction on a nautical chart. The computer immediately provides the fix and optionally provides course, distance and time to reach a destination point. The procedure applies to any kind of celestial body: Sun, Moon, planets, stars.

A set of astronomical programs offers wide fields of application to astronomy hobbyists. Position of celestial bodies for any date, time and place. Times for an altitude or an azimuth. Solar System. Image of the sky. Diagrams of time lines of rising, meridian passage and setting of any celestial body in a point of the Earth for periods chosen from three days up to one year. Validity of the data: two centuries from January 1st 1900, to December 31, 2100.

A set of piloting programs. Great circle routes and rhumb line routes. Plotting point by point a great circle route. Crossing point of the route for any longitude. Great Distances: route run along rhumb line legs. Plotting the grid of a Mercator nautical chart.

The programs of the three parts are tightly connected in many examples worked out in the book. But each part also covers kinds of application specific of that single part. The text of the book has twenty chapters, structured as follows:- relevant notions- the program and its running- examples, aimed to highlight possible fields of application.

The software includes 27 operative programs. See farther the table of the programs. Each program includes a LOG BOOK and a HELP.

The LOG BOOK is a list of all input and output data of the last application carried out. It can be reviewed by the user, printed and saved in a file with the function LOG BOOK MANAGER.

The HELP has three choices:− a synthetic description of the matter dealt with in the called program and its running.

To easy an overall view of the matters dealt with in the book, the texts of all 27 operative programs are summarized in the section FUNCTIONS OF THE PROGRAMS at the end of the book.

− information about the formats of the input and output quantities and automatic check of validity of the input quantities

− visualization of the CHAPTER OF THE BOOK related to the program.



The use of the programs is greatly facilitated by two DATA BASE:- DATA BASE of the celestial bodies: 1241 celestial bodies, each with its name: - celestial coordinates of Sun, Moon, eight planets, 63 main stars - coordinates of 1044 minor stars and 24 non-stellar objects: galaxies, nebulas of the Messier Catalogue

- DATA BASE of the places: 60 terrestrial places of the five continents (coordinates and Standard Zone). Other 40 places at choice can be stored, substituted and cancelled.

2 - Forms and running of the programs - 27 operative programs

The software includes 27 operative programs:- 5 main CELESTIAL NAVIGATION PROGRAMS - 14 ASTRONOMICAL PROGRAMS, the first of them, POSITION OF CELESTIAL

BODIES, is a main program based on 6 programs - 8 PILOTING PROGRAMS, the first of them, ROUTES, is a main program based on 4

programs

SEXTANT DATA REDUCTION INTERPOLATION OF SEXTANT READINGS

- CELESTIAL NAV. PROGRAMS TRUE LINE OF POSITION CIRCLE OF AZIMUTHS VESSEL POSITION - FIX

ARIES EPHEMERIDES POSITION OF CELESTIAL BODIES

6 PROGRAMS → SOLAR SYSTEM ALMANAC

- ASTRONOMICAL PROGRAMS PLANETS DIURNAL ARCS IMAGE OF THE SKY MOTION OF THE CONSTELLATIONS STARS IDENTIFICATION

- PILOTING PROGRAMS ROUTES PROGRAMS 4 PROGRAMS →

LEGS TACKS GREAT DISTANCES MERCATOR CHART

- AUXILIARY FUNCTIONS CALENDAR PLACES DATA BASE LOG BOOK MANAGER

- EXIT



from the main program POSITION OF CELESTIAL BODIES:

ALTITUDE AND AZIMUTH AT A TIME TIMES AND AZIMUTHS AT AN ALTITUDE

-POS. CELESTIAL BODIES TIMES AND ALTITUDES AT AN AZIMUTH RISING MERIDIAN PASSAGE SETTING TWILIGHTS REVERSE - TERRESTRIAL COORDINATES

from the main program ROUTES:

- ROUTES

GREAT CIRCLE ROUTE - COURSE DISTANCE TIMEGREAT CIRCLE ROUTE - DESTINATION POINT

RHUMB LINE ROUTE - COURSE DISTANCE TIMERHUMB LINE ROUTE - DESTINATION POINT

3 - Data stored in the software

3.1 - Celestial bodies How to deal with Sun and stars and with Moon and planets

The astronomical coordinates of 1240 celestial bodies are permanently stored with their names.

Sun and starsDirectly enter their name - 1207 stars of the 88 constellations defined by the Astronomical Association, see the list in the

table of the second following page: - 63 stars of 1st level (fig. 0.1) - 163 stars of 2nd level - 981 stars of 3rd level - 24 non stellar objects of the Messier’s catalogue (galaxies, nebulas), visible and not visible at the naked eye (fig. 0.2).The astronomical data of the Sun and the 63 stars of 1st level are automatically corrected for the date within two centuries from January 1st, 1900 to December 31, 2100 .Any other celestial body can be dealt with by entering its pair of celestial coordinates (Sidereal Hour Angle SHA« or the equivalent Right Ascension R.A. and Declination δ 1).

1 THE NAUTICAL ALMANAC, United States Naval Observatory, The United Kingdom, Nautical Almanac Office, uses the coordinate Sidereal Hour Angle SHA« for the stars and non stellar objects. THE ASTRONOMICAL ALMANAC uses the equivalent coordinate Right Ascension R.A. = (360° - SHA«) hh.mm.ss



Fig. 0.1 - The 63 stars of 1st level permanently stored in the software with their coordinates (R.A. or SHA and Decl) and magnitude. To operate with a star, select it from the DATA BASE. Any other celestial body can be entered: press with the mouse on the box SHA/R.A. and Decl

Fig. 0.2 - The 24 non stellar objects permanently stored in the software



The 88 constellations defined by the Astronomical Association

Latin name main star Latin name main star Latin name main star

Ara

And

Aqr

Aql

Ari

Cet

Boo

Lib

Cae

Pix

Cha

Cnc

CMa

CMi

CVn

Cap

Car

Cas

Equ

Cep

Cen

Com

Cyg

Aur

Col

Cir

CrA

CrB

Crv

Cru

Ara

Andromeda

Aquarius

Aquila

Aries

Cetus

Bootes

Libra

Caelum

Pyxsis

Chamaleon

Cancer

Canis Major

Canis Minor

Canes Venatici

Capricornus

Carina

Cassiopeia

Equuleus

Cepheus

Centaurus

Coma Berenic.

Cygnus

Auriga

Columba

Circinus

Corona Austr.

Corona Boreal.

Corvus

Crux Australis

Alpheratz

Altair

Hamal

Menkar

Arcturus

Zubenelge.

Sirius

Procyon

Canopus

Schedar

Rigilk

Deneb

Capella

Alphecca

Gienah

Acrux

Del

Dor

Dra

Her

Eri

Phe

For

Gem

Cam

Gru

Hya

Hyi

Ind

LMi

Leo

Lep

Lyn

Lyr

Lac

Lup

Ant

Men

Mic

Mus

Oph

Ori

Hor

UMa

UMi

Oct

Delphinus

Doradus

Draco

Hercules

Eridanus

Phoenix

Fornax

Gemini

Chamaleon

Grus

Hydra

Hydrus

Indus

Leo Minor

Leo

Lepus

Lynx

Lyra

Lacerta

Lupus

Antlia.

Mensa

Microscopiu.

Musca

Ophiucus

Orion

Horologium

Ursa Maior

Ursa Minor

Octans

Achernar

Pollux

Alphard

Regulus

Vega

Rasalhagu

Betelgeuse

Dubhe

Polaris

Pav

Peg

Per

PsA

Vol

Psc

Pic

Pup

Ret

Sge

Sgr

Sco

Sct

Scl

Ser

Sex

Nor

Crt

Tel

Tau

TrA

Tri

Tuc

Aps

Mon

Vel

Vir

Vul

Pavo

Pegasus

Perseus

Piscis Austr .

Volans

Pisces

Pictor

Puppis

Reticulum

Sagitta

Sagittarius

Scorpius

Scutum

Sculptor

Serpens

Sextans

Norma

Craters

Telescopium

Taurus

Triang. Austr.

Triang. Bor.

Tucana

Apus

Monoceros

Vela

Virgo

Vulpecula

Markab

Mirfak

Fomalhaut

Kaus Aust.

Antares

Aldebaran

Atria

Suhail

Spica

.



Moon and planets (Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune)Two procedures of calculation are available according to the required degree of precision. - Direct procedure. Simply enter the name of the body, as for the Sun and the 1st level stars. The

apparent semi-diameters of Sun and Moon, the parallax and the age of the Moon are directly provided by the program or can be read in the NAUTICAL ALMANAC.

Results of good approximation are obtained, sufficient for many kinds of application, as the ones of the GRAPHICAL PROGRAMS: medium level of precision.

Estimated precision of the data provided by the programs: altitude ±1’ azimuth ±0.1°- Ephemerides procedure. When a high precision is required, as in the calculations of Celestial

Navigation: line of position and fix, the pair of the integer GHA and Dec values of the previous and following integer hour of the relevant fractional time must be taken from the daily pages of the NAUTICAL ALMANAC.

Enter: Dec Hour - Dec Hour + GHA Hour - GHA Hour + The programs lead the operator step by step.

3.2 - Places The latitude, longitude and Standard Zone of 60 places are permanently stored. 40 other places or Waypoints (WP) can be stored, cancelled and substituted (fig. 0.3). When running a program, it is not necessary to use the PLACES DATA BASE. The place can be directly chosen in the input form of the program by shifting the mouse along the box of the places.Any other terrestrial point can be dealt with by entering its latitude, longitude and Standard Zone.

Fig. 0.3 - The 60 places permanently stored in the software with their latitude, longitude and Standard Zone. Other 40 places can be entered, named and cancelled. Any other terrestrial point can be entered with its coordinates and Standard Zone. The chosen place in the figure is San Francisco.



4 - LOG BOOK and LOG BOOK MANAGER

The LOG BOOK is a list of all the input and output data of the last application carried out. It can be saved and printed.The LOG BOOK MANAGER function allows to list, view and delete each saved LOG BOOK.

5 - Format and check of the input and output quantities General ruleWhen an input quantity contains fractional parts, any arbitrary number of fractional digits can be entered: decimal fraction of the last group of digits. For instance: 21.0273 for an altitude 21 degrees, 2 minutes and 73 hundredths of a minute

345.593 for a distance 345 nautical miles and 593 thousandths of a mile simply 23 for 23h00m00sThe whole entered figure is taken into account in the calculations, even if the figure displayed on the screen of the computer is rounded off according to a standard format of limited number of digits, as shown in the following examples of input/output formats. The LOG BOOK provided by the program shows the values of the entered quantities, last decimal parts included. DATE The current date set in the computer is taken as default by the softwareInput format The day and month are chosen by clicking on their lists with the mouse. The year is to be entered with four digits.Output format dd, name of the month, yyyy.Input check Only years between 1900 - 2100, months between 1-12, dates of days between 1 and 28, 29, 30 or 31 according to the month are accepted.

TIME The current time set in the computer, expressed in hours, minutes and seconds, is taken as default. Input time is expressed in UT Output time is generally expressed both in UT and in ST (Standard or Zone Time) 2.Input format hh.mmss (e.g. 21.0735 for 21h07m35s). Integer hours can be expressed without fractional digits (e.g. 21). Output format xxhxxmxxs (e.g. 21h07m35s). Input check Times less than 0 and equal to or more than 24 hours are not accepted (instead of entering 24, enter 0 of the following day). Values whose first fractional digit is higher than 5 are not accepted, being not compatible with the

sexagesimal format.

2 Daylight-saving time is not adopted, owing to the different dates in the years.



DURATION (of time)Input format hh.mmss if greater than 24 hours, the days are to be converted in hours and added e.g.: 3d08h45m34s → 80.4534 Output format (xxd)xxhxxmxxs

DISTANCEnautical miles (nm) and decimal fractions. In some cases kilometres (km)

SPEED knots (kn) and decimal fractions.

LATITUDE and LONGITUDE These are expressed in degrees, minutes and decimal fractions of a minute. South latitudes and West longitudes are conventionally expressed with the minus sign.Input format ±xxx.xxx...(e.g. -12.023 for 12 degrees, 2 minutes and 3 tenths of a minute, South or West). Alternatively, the name of a place can be chosen from the list of the PLACES DATE BASE (which already contains the latitude, longitude and Standard Zone correction).Output format ±xx°xx.x' Input check Latitude less than -90° or greater than 90° and longitude less than -180° or greater than 180° are not accepted. Values where the first digit of the fractional part is over 5 are not accepted.

ALTITUDE (of a celestial body) This is expressed in degrees, minutes and decimal fractions of a minute . The minus sign indicates altitudes below the horizon.Input format ±xx.xxx…(e.g. 71.0432 for 71 degrees, 4 minutes and 32 hundredths of a minute).Output format ±xx°xx.x' . In some programs and for some celestial bodies only the integer degrees are displayed.Input check Altitudes greater than 90° or less than -20° (for the Sun) or less than -5° (for every other celestial body) are not accepted. Values whose first fractional digit is higher than 5 are not accepted.

AZIMUTH This is expressed in degrees and decimal fractions of a degree.Input format xxx.xx.... (e.g. 321.75 for 321 degrees and 75 hundredths of a degree).Output format xxx.x° . In some programs and for some celestial bodies, only integer degrees are displayed.Input check Values less than 0° or greater than 360° are not accepted.



DECLINATION OF A CELESTIAL BODY This is expressed in degrees, minutes and decimal parts of a minute. The minus sign conventionally indicates South declinations.Input format ±xx.xx...(e.g.-7.0583 for 7 degrees, 5 minutes and 83 hundredths of a minute South).Output format ±xx°xx.x' Input check Values equal to or lower than -90° and greater than or equal to 90° are not accepted. Values whose first fractional digit is greater than 5 are not accepted.HOUR ANGLES GHA, SHA, LHA, are expressed in degrees, minutes and decimal parts of a minute.

RIGHT ASCENSION (R.A.), LOCAL SIDEREAL TIME (L.S.T.) are expressed in hours, minutes and seconds.

6 - Precision of the output data

The calculations of all programs of the software work at medium level of precision.

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