Jupsat Pro Astronomy Software

From time immemorial, humans have wondered about their place in the universe. Those in early ages believed that the earth was flat and at the center of the universe and that all celestial bodies which they could see above and around them revolved around the earth. This geocentric view prevailed for a long time in human history and was even supported by Aristotle in 320 B.C. and Ptolemy in the second century A.D. It was not until 1514 A.D. that a Polish priest by name Copernicus challenged the Aristotle-Ptolemic theory and shifted the earth from its proud central position to a position where it revolved around the sun like any other planet of the solar system. Copernicus knew that his heliocentric view would meet violent opposition from the then orthodox church which was wedded to the geocentric view of Ptolemy. So, he withheld publication of his book, 'De Revo-lutionibus Orbium Coelestium' concerning the revolutions of the Celestial Orbs, until the end of his life. His fears were well founded, for Copernicus's theory was condemned as heresy and his book remained locked up in papal custody until 1835.

Meanwhile, the heliocentric theory of Copernicus received strong support from the work of the astronomers, Johannes Kepler (1571-1630) in Germany and Galileo Galilei (1564-1642) in Italy. Kepler using the careful astronomical measurements of his predecessor, Tycho Brahe, enunciated in 1609 his celebrated three laws of planetary motion as follows:

(i) The planets revolve round the sun in elliptical orbits with the sun occupying one focus;

(ii) The orbital velocity of a planet sweeps out equal areas in equal times; and

(iii) The squares of the periods of revolution of the planets are proportional to the cubes of their orbital major axes.

But both Kepler and Galileo were afraid of coming out with the truth for fear of being persecuted by the then orthodox church. However, after Kepler enunciated his laws of planetary motion which were soon followed by Newton's law of universal gravitation in 1687, the truth ultimately triumphed and the scientific world accepted the heliocentric theory of our solar system.

According to the accepted view, the earth is one of the inner planets of the solar system (for detailed information on the sun's planetary system, the reader may consult a book on the solar system or astronomy) which revolve around the sun and are held in their orbits by the gravitational force of the sun. The earth orbits around the sun in an elliptical orbit under a centrally-directed gravitational force at an average distance of about 149.6 million km from the sun once in about 365 days. It also rotates about its own axis once in about 24 h. Its angular velocity is about 7.29 x 10"5 radians per second and is usually denoted by Q. The earth's equatorial plane is inclined to its orbital plane by an angle of 23.45°, which is called the earth's obliquity to the sun. While the rotation of the earth makes day and night, the obliquity gives us the seasons.

The shape of the earth departs slightly from being a sphere. Its polar radius happens to be about 21 km shorter than its equatorial radius, the average radius being about 6371 km. To understand the likely cause of the departure of the earth's surface from sphericity, we need to consider the earth's gravitational force and rotation from the time of its birth from the sun.

1.2 Earth's Gravitational Force - Gravity

According to Newton's law of universal gravitation, every body in the universe attracts every other body with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. This means that if the earth were truly spherical, its gravitational attraction F on a body of mass m placed at a point P on its surface (see Fig. 1.1) would be given by the relation where r is the radius vector of P, G is the Gravitational constant, M is the mass of the earth deemed to be concentrated at its center, and r is the mean radius of the earth.

Fig. 1.1 Gravitational force of the earth

[Vectors are indicated by bold letters; a summary of some commonly-used vector symbols and operations appears in Appendix-IA. Also, a list of some useful physical constants appears in Appendix-8]

The acceleration due to gravity, g that corresponds to the gravitational force in (1.2.1), is given by g =(GM/r2) (r/r) (1.2.2)

We assume that the earth's surface was once spherical in shape when it was born and that the effect of its rotation moulded its shape only gradually after its birth. Two forces must have acted on a body of unit mass at its surface then, viz., a gravitational force pulling it towards the center of the earth and centrifugal force acting away from the axis of rotation of the earth (see Fig. 1.2).

The resultant of the two forces which we may call the earth's effective gravity g, is given by the relation g = g + Q2 R (1.2.3)

where R is the radius vector of the body at a point P in a direction perpendicular to the axis of rotation and Q is the angular velocity of the earth. Because of the centrifugal force, the resultant acceleration due to gravity g no longer passes through the center of the earth except at the poles and the equator. The reason simply is this: If the earth's surface were truly spherical, the effective gravity would have a component parallel to the earth's surface and directed towards the equator. This force is denoted by the vector E in Fig. 1.2. The earth's surface has adjusted to this equatorward component by taking up a spheroidal shape with a bulge at the equator and contraction at the poles so that the local vertical at all points on the earth's surface would be parallel to the resultant gravity. It is because of the equatorial bulge and the polar contraction that the polar radius is shorter than the equatorial radius by about 21 km. The transformation envisaged here must have had occurred long ago in earth's history when its surface layers were cooling off from a hot molten plasma state to a solid crust.

The acceleration due to gravity, g, varies with latitude along the earth's surface, with a maximum at the poles and minimum at the equator. It also varies with altitude. The value of g at latitude 9 and height h meters above the earth's surface is empirically given by the approximate relation:

g(h)= 9.80616 (1 -0.0026373cos29 + 0.59 x 10-5cos29)(1 - 3.14 x 10-7h)

where 9.80616 ms-2 is the value of the acceleration due to gravity at mean sea level at latitude 45°.

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