Unit 6 Celestial Sphere and Equinoctial System of Coordinates

Celestial Sphere and UNIT 6 CELESTIAL SPHERE AND EQUINOCTIAL Equinoctial System of SYSTEM OF COORDINATES Coordinates

Structure 6.1 Introduction Objectives 6.2 Celestial Sphere and References 6.3 Apparent Annual Motion of the Sun and the Concept of the Ecliptic and the ‘Obliquity of the Ecliptic’ 6.4 The Equinoctial System of Coordinates 6.4.1 Reference Great Circles – Equinoctial, Celestial Meridians of Greenwich and Celestial Meridian of the First Point of Aries 6.4.2 Coordinates – Declination, Greenwich Hour Angle (GHA), Local Hour Angle (LHA), Sidereal Hour Angle, Right Ascension 6.5 Relationship between GHA, LHA and Longitude 6.6 The Concept of the Earth’s Axial Rotation Causing Change in the Hour Angle of Bodies 6.7 Summary 6.8 Key Words 6.9 Answers to SAQs

6.1 INTRODUCTION

In order to use celestial bodies (the Sun, Planets, Moon, and stars) for position fixing it is required that we know their position in the sky. The position of a celestial body is defined on the celestial sphere by using different coordinate systems based on their purpose. The coordinate systems used are equinoctial system, Horizon system and ecliptic system. In this unit we will discuss the equinoctial coordinate system of defining position of the body in the sky, which is used in the Nautical Almanac. Objectives After studying this unit, you should be able to • define the celestial sphere, celestial poles, celestial meridians, equinoctial, declination circles, • explain the apparent annual motion of the sun and the concept of the ecliptic and the ‘obliquity of the ecliptic’, • explain the equinoctial as a fixed reference plane and the direction of the First Point of Aries as a reference direction (ignoring the effect of precession), • describe the equinoctial system of co-ordinates, • define ‘Greenwich Hour Angle (GHA)’, ‘Local Hour Angle (LHA)’ and longitude, and explain their relationship, • define sidereal hour angle, declination, and right ascension, and • describe the concept of the earth’s axial rotation causing change in the hour angle of bodies.

Celestial Navigation 6.2 CELESTIAL SPHERE AND REFERENCES

In celestial navigation, the earth is assumed to be a perfect sphere, located at the center of the universe. At night when we look at the sky we see all heavenly bodies located on a hemispherical dome called the celestial sphere. The earth is assumed to be stationary due to which all celestial bodies appear to move on the celestial sphere. The grid for pin pointing position of a celestial body may be imagined to be created by projecting the earth’s grid on the celestial sphere with light source at the center of the earth.

North Celestial Poles (NCP)

Declination Circles

Earth’s North Pole Earth’s Meridian

Equator

Equinoctial

Celestial Meridians

Celestial Sphere

South Celestial Poles (SCP)

Figure 6.1 6.2.1 Definitions Celestial Sphere It is a sphere of immense radius whose centre is same as centre of the earth. It is the sphere on which all celestial bodies appear to lie. As calculations in celestial navigation are based on angular measurements, the radius of the earth is irrelevant. It can be also defined as sphere of infinite radius concentric with the earth. Celestial Poles The earth’s poles extended to meet the celestial sphere are called celestial poles (north celestial pole and south celestial pole). They can also be defined as outward projection of the earth’s poles on the celestial sphere. Equinoctial or Celestial Equator Equinoctial is a great circle on the celestial sphere whose plane is same as plane of the earth’s equator. It can be also defined as projection of the equator on the celestial sphere. Celestial Meridians The semi-great circles on the celestial sphere joining the celestial poles or projection of the earth’s meridians on the celestial sphere. Celestial meridians cut the equinoctial and the declination circles at 90º. Declination Circles / Parallels of Declination These are small circles on the celestial sphere parallel to the plane of equinoctial. Declination circles may also be defined as projection of parallels of latitude on the celestial sphere.

6.3 ANNUAL APPARENT PATH OF THE SUN

The solar system consists of the Sun, the planets (Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto), the planetary satellites (moons), asteroids, comets and meteors. The most important member of the Solar system is the Sun. All the planets 30

revolve eastward in elliptical orbits around the sun. The earth also rotates eastward on its Celestial Sphere and Equinoctial System of axis. Coordinates For the purposes of celestial navigation the earth is assumed to be stationary. Therefore due to eastward revolution of the Earth, the Sun appears to move eastwards on the celestial sphere, in the plane of the Earth’s orbit.

CelestialError! Sphere

Ecliptic

Earths Orbit

Figure 6.2 Figure 6.2 shows the apparent motion of the Sun along the great circle on the celestial sphere, due to the eastward motion of the earth in its orbit. The great circle on the celestial sphere in the plane of the earth’s orbit is called Ecliptic. It is so called because the Sun, Moon and Earth must be on this plane for a solar or lunar eclipse to occur. The projection of the Sun on the ecliptic from successive positions of the earth in its orbit, appears to constantly move eastwards. The earth’s axis is inclined to its orbiting plane by about 23° 26.7′. Therefore the equinoctial also makes the same angle with the plane of ecliptic. The angle at which the plane of ecliptic cuts the equinoctial plane is called Obliquity of Ecliptic. First Point of Aries γ The ecliptic intersects the equinoctial at two points called the equinoctial points. The equinoctial point when the Sun appears to cross the Equinoctial from South to North is called first point of Aries and is denoted by symbol γ. This occurrence takes place on 21st March, at Vernal Equinox. The first point of Aries is considered as fixed point on the celestial sphere (ignoring precession of equinoxes will be discussed in later units). First Point of Libra Ώ The equinoctial point when the Sun appears to cross the Equinoctial from North to South is called first point of Libra, and is denoted by the symbol K. The First point of Aries and the First point of Libra were named after the constellations in which they once lay. These points are however moving westward slowly, along the Ecliptic. Due to this, the 1st point of Aries is no longer in the constellation of Aries. It is now in the constellation of Pisces.

Celestial Navigation 6.4 THE EQUINOCTIAL SYSTEM OF CO-ORDINATES

The coordinates used to define the position of a celestial body on the celestial sphere in the equinoctial system are Declination and Greenwich Hour Angle. This system of coordinates is used in Nautical Almanac to define the position of celestial body at any instant of the year. For determining these coordinates reference used are as follows. 6.4.1 The Reference Great Circles Used in the Equinoctial System Equinoctial Equinoctial is the great circle on the celestial sphere which is equidistant from the celestial poles or whose plane is same as the plane of equator. This is the reference plane from which declination of astronomical bodies is measured north or south. Celestial Meridian Passing of Greenwich It is the celestial meridian that passes through Greenwich. Greenwich Hour Angle (GHA) of all celestial bodies is measured westward from this reference great circle. Celestial Meridian of First Point of Aries It is the celestial meridian that passes through first point of Aries. Sidereal Hour Angle of all celestial bodies is measured with reference to this great circle. 6.4.2 The Coordinates in the Equinoctial System Declination It is the arc of celestial meridian passing through the body contained between the Equinoctial and the body. It may be also defined as arc of celestial meridian or the angle at the centre of the earth contained between the Equinoctial and the declination circle passing through the body. If the body is North of the equinoctial the declination is North and if the body is South of the equinoctial declination is named South. As it is angular measure it is expressed in degrees and minutes and its value is in between 0º to 90º N or S. Hour Angle (HA) Hour Angle is arc of the Equinoctial or the angle at the celestial poles contained between two celestial meridians. Greenwich Hour Angle (GHA) GHA of a celestial body is the arc of the Equinoctial or the angle at the celestial poles contained between the celestial meridian of Greenwich and celestial meridian passing through the body, measured westward from celestial meridian of Greenwich. It is expressed in degrees and minutes and its value is in between 0º to 360º. Sidereal Hour Angle (SHA) SHA of a celestial body is an hour angle the arc of the Equinoctial or the angle at the celestial pole contained between the celestial meridian of the First point of Aries and that through the body, measured westward from Aries. Right Ascension (RA) RA of a celestial body is an hour angle contained between the celestial meridian of the First point of Aries and the celestial meridian passing through the body, measured eastward from Aries. RA is generally expressed in hours, minutes and seconds, instead of, in arc. Since SHA is measured westward and RA eastwards from the same point, the SHA and RA of any body will together always add up to 360°. SHA* + RA* = 360º 32

Local Hour Angle (LHA) Celestial Sphere and Equinoctial System of LHA of a celestial body is an hour angle contained between the observer’s Coordinates celestial meridian and the celestial meridian through that body, measured westward from the observer. If the angle or arc is measured eastward from the observer, it is known as the Easterly Hour Angle (EHA) and not LHA. LHA* + EHA* = 360º

NCP

GHA*

X X′ LHA*

L E W G Declination

Equinoctial

Celestial Sphere

SCP

Figure 6.3 : Equinoctial Coordinate System

6.5 RELATIONSHIP BETWEEN GHA, LHA, AND LONGITUDE

As we know the longitude is measured East or West with reference to Greenwich meridian hence the GHA and LHA of a celestial body are also related. The relationship can be easily understood and shown on the plane of Equinoctial that is projecting the celestial meridians on the plane of Equinoctial from the celestial pole. In this projection the celestial meridians will appear as radial lines and the centre of the equinoctial circle is celestial pole. Looking from North celestial pole West-ward angles and arcs are measured clockwise. Eastward angles and arcs are measured counter-clockwise. The angle at the Pole, between any two meridians is equal to the corresponding arc on the Equinoctial.

W X X

P P

G G Figure 6.4 Figure 6.5

In Figure 6.4 In Figure 6.5 33

Celestial Navigation LHA* = ∠ WPX = ∠ GPX − ∠ GPW LHA* = ∠ EPX = ∠ GPX + ∠ GPE = GHA* − Long. (West) = GHA* + Long. (East) LHA* = GHA* − Long. (West) LHA* = GHA* + Long. (East) GHA* = LHA* + Long. (West) GHA* = LHA* − Long. (East) (Longitude West GHA BEST) (Longitude East GHA LEAST)

Example 6.1 Calculate the LHA of a star whose GHA is 70°, for an observer in longitude 147°E. Solution Longitude East GHA Least LHA* = GHA* + Long (E) = 70° + 147° = 217° LHA* = 217° Example 6.2 Calculate the GHA of Sun which is on the meridian for an observer in longitude 97°W. Solution When meridian of the observer is also the celestial meridian of the body then : LHA = 000° Longitude is West ∴ GHA is Best LHA = GHA − Long (W) 0° = GHA − 97° GHA = 0° + 97° GHA = 97° The above questions can also be solved with the help of drawing the diagram on the plane of the equinoctial. SAQ 1 (a) What is equivalent coordinate for declination in geographical system of defining position on the surface of the Earth? (b) What is the obliquity of the ecliptic? (c) To an observer the Sun’s LHA was 342°, when its GHA was 35°. Find the longitude of the observer. (d) For an observer in DR 20º 12′S 164º 44′E, the EHA of Moon was 71º. Find GHA of the Moon. (e) Calculate LHA γ, when GHA was 49° and the longitude of the observer is 119°20′W.

Celestial Sphere and 6.6 CHANGE IN HOUR ANGLE OF BODIES DUE TO Equinoctial System of ROTATION OF THE EARTH Coordinates

The Earth rotates on its axis from West to East i.e. counter clockwise as viewed from above the North Pole, at the rate of 15º 2.46′ per hour that is completing one rotation in 23 hours 56 minutes 04.1 seconds. Due to the rotation of the Earth the entire celestial sphere appears to rotate in the opposite direction, i.e. from East to West completing an apparent rotation of 360° in about 24 hours. Due to rotation of the Earth the meridian of Greenwich also rotates and so does the celestial meridian of Greenwich. Since GHAs of all celestial bodies are measured westward from the celestial meridian of Greenwich therefore the GHAs of all celestial bodies increase by 15° 2.46′ per hour due to rotation of the Earth. The GHA of celestial bodies change due to the following reasons : • Rotation of the Earth • Revolution of the Earth • Movement of the Celestial bodies, e.g. Revolution of planets and the Moon The GHA of stars changes only due to rotation of the Earth, the stars being at immense distance revolution of the Earth doesn’t register any angle at the stars and stars motion with respect to Sun or other stars is very slow, e.g. abt 3.7 seconds per year of Arcturus. The GHA of stars and first point of Aries γ increases by 15° 2.46′ per hour.

G Figure 6.6 : GHA* at Initial Stage

G’ G Figure 6.7 : GHA* after 1 hour increases due to Eastward Motion of the Celestial Meridian of Greenwich due to Rotation of the Earth The GHA of Sun changes due to rotation of the Earth and revolution of the Earth and it increases by about 15° per hour. The GHA of planets and the Moon changes due to rotation of the Earth, revolution of the Earth and their own motion. Change in GHA of the Sun, planets and the Moon will be further discussed in Unit 7 of this block.

Celestial Navigation 6.7 SUMMARY

If the geographical coordinates and reference circles are expanded to meet the celestial sphere we have the grid, references and coordinates of the equinoctial system. The similarity can be drawn from the table below :

Geographical Equinoctial Coordinate Coordinate System System Sphere Earth Celestial Sphere Grid Meridians Celestial Meridians Parallels of Latitude Parallels of Declination References Equator Equinoctial Meridian of Greenwich Celestial Meridian of Greenwich Coordinates Latitude Declination Longitude Greenwich Hour Angle

The equinoctial system is used in Nautical Almanacs for defining the position of heavenly bodies used for navigation purpose at any given Greenwich Mean Time. The relationship between GHA and Longitude is given by : LHA* = GHA* + Long (E) and LHA* = GHA* − Long (W) The above can be easily remembered by : “Longitude East GHA Least” and “Longitude West GHA Best” The annual apparent path of the Sun on the celestial Sphere is called Ecliptic. The Equinoctial cuts the Ecliptic at an angle of about 23º 30′ at two points called equinoctial points namely First Point of Aries and First Point of Libra. The point when the Sun appears to cross the Equinoctial from North to South is called First Point of Aries and is reference point for measuring SHA. The GHAs of celestial bodies’ increases by about 15º due to rotation of the Earth.

6.8 KEY WORDS

Celestial Sphere : A sphere of infinite radius whose centre is the Earth’s Centre. Equinoctial : A great circle on every point on which is 90º from the celestial poles. Greenwich Hour Angle : An hour angle contained between celestial meridian of Greenwich and celestial meridian of the body measured westward. Sidereal Hour Angle : An hour angle contained between celestial meridian of γ and celestial meridian of the body measured westward. Local Hour Angle : An hour angle contained between celestial meridian of the observer and celestial meridian of the body measured westward. Right Ascension : An hour angle contained between celestial meridian of γ and celestial meridian of the body measured eastward. 36

Easterly Hour Angle : An hour angle contained between celestial Celestial Sphere and Equinoctial System of meridian of the observer and celestial meridian of Coordinates the body measured eastward. Declination : It is arc of celestial meridian passing through the body contained between equinoctial and the body.

6.9 ANSWERS TO SAQs

SAQ 1 (a) Latitude (b) It is the angle between the plane of Equinoctial and Ecliptic and is about 23º 26.7′. (c) 04 h 35 m 00 s (d) 53º W (e) 124º 16′E