1 General Astronomy

V. B. Bhatia

 

1. Learning Outcomes

 

After studying this module, you shall be able to

  • Understand the concept of a celestial sphere
  • Distinguish between great and small circles on a sphere
  • Pinpoint the poles of a great circle
  • Explain what a spherical triangle is and how it differs from a plane triangle
  • Recall relations between the sides and angles of a spherical triangle
  • Appreciate the terrestrial coordinate system

 

2. Introduction

 

Astronomy is traditionally defined as the science which studies the location and kinematics of celestial objects such as planets, stars, galaxies. Astrophysics, as the name suggests, studies the physical processes taking place inside the celestial objects. There is no real dividing line between these two subjects. The two together study every phenomenon taking place in objects beyond the atmosphere of the earth.  The internal structures of stars, nucleosynthesis in stars, white dwarf and neutron stars, supernova explosions, origin of black holes, activity in galaxies, emission of neutrinos by the sun, emission of gravitational waves by interacting black holes, origin of the universe are some of the phenomena in a very long list that forms the subject matter of astronomy and astrophysics.

 

3.  Celestial Sphere

 

When we look at the sky on a dark night, especially when there is little dust and city light, all the celestial objects appear to lie on the surface of a sphere. This imaginary sphere is called the celestial sphere.

Fig. 1.1. Celestial sphere, celestial equator, north and south celestial poles. Celestial equator is a plane parallel to the plane of the earth’s equator. North and south celestial poles are in the directions of the north and south poles.

 

All celestial objects are located on this sphere. Celestial equator is a plane parallel to the plane of the earth’s equator.  North and south celestial poles are in the directions of the north and south poles of the earth.

 

Celestial sphere is an important tool for astronomers.  Since our eyes do not give a good estimate of distances, we assume that the celestial sphere has a unit radius. With this provision one can locate any celestial object on this sphere by just two coordinates.   Considering the use and context, four coordinate systems have been devised.  To understand these systems, however, we need to understand elements of spherical trigonometry.

 

4. Elements of Spherical Trigonometry

 

4.1.  Great and Small Circles

 

Great Circle:  Any circle passing through the centre of the sphere is called a great circle.  Any other circle is called a small circle. On the earth, for example, equator is a great circle. A circle of latitude, say Tropic of Capricorn, is a small circle.  The points at which a line perpendicular to a great circle through the centre of the sphere meets the surface of the sphere are called the poles of the great circle.  In Fig. 1.2, P and P´ and Q and Q´ are the poles of the respective great circles. North pole and south pole are the poles of the terrestrial equator.

 

An angle between two great circle arcs is called a spherical angle.  For example, in Fig 1.3 angle KNL is a spherical angle.  Since the radius of the sphere is fixed, it is customary to measure angles in terms of arc lengths.  The measure of the angle KNL is the arc length KL, or the angle KOL submitted at the centre of the sphere. The arc length between a great circle and its pole on either side is 90 degrees.

Fig. 1.2.  Three examples of great circles and one example of a small circle on  a sphere. Points P and P´ are the poles of the great circles. Arc length from a great circle to its pole is 90°.

 

4.2.  Spherical Triangle

 

A spherical triangle is formed between three intersecting great circles. An example is the triangle KRL in Fig. 1.4 formed by three great circles, RLS, RKS and PKLQ (to avoid clutter,

Fig 1.3.  Angle KNL is spherical angle formed by the intersection of two great circles. Its measure is the arc length KL, or the angle KOL submitted by the arc length at the centre of the sphere.

 

instead of drawing full great circles, we draw only half circles, or circular arcs). An important thing to remember is that, unlike the plane triangles, the sum of the angles of a spherical triangle exceeds ?.

 

4.3.  Relations Governing Spherical Triangles

 

We state, without proof, the relations between the sides and angles of a spherical triangle. These relations will be useful for transforming astronomical coordinates from one system to another.

 

A, B and C are the angles of the spherical triangle and a, b, c are its sides (Fig. 1.5). The following relations hold:

Fig. 1.4. A spherical triangle NLK is formed by the intersection of three great circles NLS, NKS and PKLQ. To avoid clutter, we have drawn only half great circles.

Fig. 1.5. A spherical triangle formed by three great circles.

 

By permuting a, b, c and A, B, C, we can get four more relations of this kind. These relations will help us in converting astronomical coordinates from one system to another.

 

5.  Terrestrial Coordinate System as an Example of Astronomical Coordinate Systems

 

Before we explain the coordinate systems used in astronomy, we explain their basis taking the longitude- latitude coordinate system (Fig. 1.6.) used for pints on the earth as an example. Let NQSP represent the globe, N and S being the north and the south poles respectively. The great circle PKLQ, which is perpendicular to the line joining N and S, is the equator. Any half great circle through N and S is called a meridian. Meridian through Greenwich, by international agreement, is called the prime meridian.

 

The point of intersection of the prime meridian with the equator, K, is used as the reference point for measuring the arc lengths along the equator. The meridian through X, whose position is to be fixed, cuts the equator at L. The longitude (?) of X is then defined as the spherical angle GNX, or the angle KOL in

Fig. 1.6. Coordinate system used for points on the surface of the earth is the longitude- latitude system.  NXS is the meridian through X and NGS is the meridian through Greenwich.  The great circle PKLQ is the equator of the earth. The coordinates of X are longitude (?) and latitude (?).

 

the plane of the equator, or the arc length KL (since the radius of the globe is fixed). The longitude of Greenwich is assigned a value zero. To the east of Greenwich longitudes are measured from 0 to 180°, and to the west of Greenwich, too, longitudes are measured from 0 to 180. Longitude of Shillong, for example, is 91° 52´ East. Longitude of Toronto is 79.4° West, or 79° 24´ West.

 

The other coordinate of point X on the earth, called latitude (?), is the angle LOX, or the arc length LX (Fig. 1.6). Latitude is measured from 0 to 90° North or South.   Latitude of North Pole is 90° North. Latitude of Kanyakumari is 8° 05´ North and that of Sydney is 33° 52´ South. All locations having the same latitude lie on a small circle. Tropic of cancer, for example, is the circle of latitude 23.5° N (Fig. 1.7).

 

It is obvious that 360 degrees of longitude equal a complete rotation of the earth on its axis, which takes 24 hours. Thus, 1 degree of longitude equals 4 minutes of time. Thus based on longitude 82.5 degrees East, the Indian Standard Time is ahead of Greenwich time by 5 hours and 30 minutes.

Fig. 1.7.  Circles of constant latitude.

 

The longitude–latitude system illustrates the essential requirements of an astronomical coordinate system. These are:

 

(I)  a fundamental great circle (such as the equator in the case of the earth, and

(II)  a reference point, or origin (such as point K in the above case).

 

Depending upon the choice of the fundamental circle and the reference point, the following astronomical coordinates systems have been devised:

 

(1)   The Horizon System

(2)   The Equatorial System

(3)   The Ecliptic system

(4)   The Galactic System

 

We shall study these systems in subsequent Modules.

  1. Summary
  • Astronomy and astrophysics study together kinematics of celestial objects and the physical processes taking place inside these objects.
  • Celestial sphere is an imaginary sphere on which all celestial objects appear to lie.
  • A great circle on a sphere is the one passing through the centre of the sphere.
  • A line through the centre perpendicular to a great circle intersects the sphere at two points, called the poles of the great circle.
  • A spherical angle is formed when two great circles intersect.
  • Since the radius of the sphere is fixed, spherical angles can be measured in terms of arc lengths.
  • A spherical triangle is formed when three great circles intersect.
  • The sum of the angles of a spherical triangle exceeds 2?.
  • Since the radius of the celestial sphere is taken as unity, any object on it can be represented by two angle coordinates.
  • The terrestrial coordinate system is an example of an astronomical coordinate system.
  • A half great circle joining the two poles is called a meridian.
  • By international agreement, the meridian through Greenwich is called the prime meridian.
  • A point on the terrestrial sphere is represented by two coordinates, longitude and latitude.
  • All points on the earth with the same latitude lie on a small circle.
  • Latitudes range between 0° – 90° North and 0° – 90° South.
  • Longitudes range between 0° – 180° East and 0° – 180° West from Greenwich.
  • Depending upon the choice of the great circle and the reference point, there are four systems of astronomical coordinates.

 

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