2 Coordinate Systems, Measurement of Time

V. B. Bhatia

1. Learning Outcomes

After studying this module, you should be able to

transform coordinates from Alt – Azimuth system to Right – Ascension system and vice versa
understand the essential requirements of the ecliptic system of coordinates
describe the ecliptic system
appreciate the galactic plane as a great circle of the celestial equator
discuss the galactic system
grasp the concept of time
comprehend the diurnal motion of celestial objects
explain how solar time is derived from the diurnal motion of the Sun

2. Introduction

We have already noted that locating celestial objects in space is important from the point of view of studying their kinematics, their motion in space for example. Since our eye does not give a reliable estimate of distance, an objects is located in terms of two directions, or angles, on the celestial sphere. We have so far studied two of such coordinate systems. There is an obvious need to be able to transform from one system of coordinates to another. For this purpose, we need to solve spherical triangles, formulae for which were given in Module 01. We shall also describe two other coordinate systems.

3. Inter conversion from horizon system to equatorial system and vice versa

3.1. Conversion from Alt – Azimuth to Right Ascension – Declination

The inter-conversion can be done with the help of Equations (1.1) to (1.5) in Module 01 pertaining to a spherical triangle. In Fig. 3.1 are shown the two sets of coordinates. Suppose

Fig. 3.1. Two sets of coordinates of the same object X. Solution of triangle PZX will give the relation between the two sets.

zenith distance ?(= 90° – a) and the azimuth ?(measured from N) are given. To find the hour angle ? and declination ?, we use the relations (1.2) and (1.3) applied to the spherical triangle PZX. In writing these relations we shall use the fact that the altitude of the pole star is equal to the latitude of the observer, which we have already proved. Accordingly, arc length NP = ? and arc length PZ = 90° – ?.

1.1. Conversion from Right Ascension – Declination to Alt – Azimuth

2. The Ecliptic System of Coordinates

This system has the ecliptic – the intersection of the apparent annual orbit of the Sun round the earth with the celestial sphere – as the fundamental great circle. We know that most planets and other objects in the solar system have orbits which are close to the ecliptic and make only small angles with its plane. Therefore, the choice of ecliptic as the reference great circle makes this system very useful in representing the positions and orbits of these objects. Objects outside the solar system are far off and move slowly; for these objects this system is not very useful.

The ecliptic is inclined to the celestial equator at an angle of about ? = 23° 27´. Points M and M´ are the poles of the ecliptic. The point of intersection of the ecliptic and the celestial equator, the First point of Aries (vernal equinox, or spring equinox), denoted by ϒ, is taken as the reference point for this system of coordinates (Fig. 3.2). The coordinates of the object X are found in the usual way by drawing a half great circle joining M and M´ and passing through X.

Fig. 3.2. Ecliptic system of coordinates gives ecliptic longitude and ecliptic latitude.

The point of intersection of this half great circle with the ecliptic is L. Then the arc XL is called celestial latitude (�), measured positive towards north and negative towards south of the ecliptic. The arc ϒL is the celestial longitude (?) of the object X. It is measured eastwards (in the direction of earth’s motion) from vernal equinox from 0° to 360°.

The gravitational forces due to the Sun, moon and (to a lesser extent) other planets make the axis of the earth precess slowly. This results in the vernal equinox advancing towards west at the annual rate of about 50 arcseconds. Thus, the ecliptic longitude of celestial objects increases continuously at this rate.

2.1. Transformation to Ecliptic System from the Equatorial System

The ecliptic coordinates cannot be measured directly; they must be obtained by transforming coordinates from other systems. Fig. 3.3 shows equatorial and ecliptic coordinates together. Solution of the spherical triangle MPX would give the transformational equations from one set to another set of coordinates. Fig. 3.3 also shows, incidentally, that the maximum declination of the Sun is ∓23° 27´, and at equinoxes its declination is zero.

Conversion from equatorial coordinates to the ecliptic coordinates can be done by applying Equations (1.1), (1.2) and (1.3) discussed in Module 01 to triangle MPX. The equation we get for conversion from (?, ?) to (?, ?) are:

Fig. 3.3. The equatorial set of coordinates and the ecliptic set for the same object X are shown here. Solution of the spherical triangle gives the transformational equations.

3. The Galactic System of Coordinates

The great circle formed by the intersection of the plane of our galaxy, the Milky Way Galaxy or just the Galaxy, is called the galactic equator (Fig. 3.4).

Fig. 3.4. Side on view of the Milky Way Galaxy. (Credit: NASA)

Fig. 3.5. A sketch of the side on view of the Milky Way Galaxy. The central plane, which appears a line seen side on. is called the galactic equator.

The use of galactic equator as the basis for a system of coordinates gives us another system of celestial coordinates. This system, called the galactic system, is used extensively in the study of galactic structure and galactic rotation.

The galactic equator is inclined to the celestial equator at an angle of 62.6° (Fig. 3.8). The reference point, A, is the direction of the centre of the Galaxy whose coordinates are (? = 17 h 42 m and ? = – 28.9°). The North galactic pole has coordinates: ? = 12 h 49 m, ? = +27° 24´.

The declination of the north galactic pole can be easily inferred from Fig. 3.8. The centre of the Galaxy is in the direction of the constellation Sagittarius. It is to be noted that the centre of the Galaxy is not visible through optical wavelengths. It is visible in radio, infrared and x-ray region of the spectrum.

Fig. 3.6. Milky Way as it would appear if seen from the top. (Credit: NASA). The Milky Way Galaxy is a spiral galaxy. We cannot actually photograph our own galaxy. This photographs and the one in Fig. 3.4 are the impressions we have formed by observing similar other galaxies. The nearest spiral galaxy, believed to be quite like our own, is the Andromeda Galaxy.

Fig. 3.7. Teapot of the Sagittarius is a part of the constellation Sagittarius. The centre of the Milky Way Galaxy lies in the direction of the constellation Sagittarius.

Points G and H in Fig. 3.8 are the poles of the galactic equator. A half great circle joining G and H and passing through X intersects the galactic equator at L. Then the arc length AL is called the galactic longitude of X, measured from 0° – 360° eastward along the galactic equator.

Fig. 3.8. Galactic system of coordinates. Galactic equator is the central plane of the Milky Way Galaxy. It makes an angle of 62.6° with the celestial equator. The reference point is A, the direction of the centre of the Galaxy.

Galactic longitude is denoted by ???. The arc length LX is called the galactic latitude of X. It is denoted by ???. It is measure from 0° – 90° positive (+) towards north or 0° – 90° negative (-) towards south of the galactic equator. Notice that galactic longitude and galactic latitude are defined in a similar way as longitude and latitude of a place on the earth. Earlier the galactic longitude and latitude (??, ??), used to be reckoned from the point of intersection of the galactic equator with the celestial equator. Unlike A, this point had no physical significance.

Fig. 3.9. Galactic coordinates. Direction to the galactic centre defines the reference point of the system. Angle ??? is in the plane of the Galactic equator and angle ??? is perpendicular to this plane.

4. Summary

Coordinates can be transformed from one system to another by solving relevant spherical triangle.
Ecliptic system of coordinates is used extensively in studying the solar system.
Objects of the solar system have orbits quite close to the ecliptic. Therefore, the reference great circle for the ecliptic system is the ecliptic.
The reference point is the First Point of Aries, or Vernal equinox.
A half great circle joining the two poles of the ecliptic is drawn through the object whose coordinates are required.
The celestial latitude is the arc length from the ecliptic to the object along this great half circle.
The celestial longitude is the arc length from vernal equinox along the ecliptic to the point of intersection of this great half circle with the ecliptic.
Another system of coordinates, the galactic system, is used in the study of galactic structure and galactic rotation.
The reference circle for the galactic system is the great circle parallel to the plane of our Milky Way galaxy.
This plane is inclined to the celestial equator at an angle of 62.6°.
The reference point is the direction of the galactic centre in the direction of the constellation Sagittarius.
The two coordinates on this system are called galactic longitude and galactic latitude.
Our Galaxy is a spiral galaxy. Seen side-on it shows the characteristic bulge at the centre. The solar system is near the edge of the Galaxy.
Seen from the top the spiral structure of the Galaxy is clearly seen.

KNOW – MORE

Material for further reading on the topics discussed in this module is available on the following websites:

https://en.wikipedia.org/wiki/Spherical_trigonometry
http://www.krysstal.com/sphertrig.html
http://jwilson.coe.uga.edu/EMAT6680Fa2013/Lively/Spherical%20Triangles/Solving_Spherical_Triangles.pdf
http://astrowww.phys.uvic.ca/~tatum/celmechs/celm3.pdf
http://faculty.trinityvalleyschool.org/hoseltom/labs/Lab-01 %28Spherical%20Trigonometry%20Intro%29.pdf
http://csep10.phys.utk.edu/astr161/lect/celestial/celestial.html
http://stars.astro.illinois.edu/celsph.html
http://astro.wsu.edu/worthey/astro/html/lec-celestial-sph.html