Motion in the Earths Gravitational Field The Law of Central Forces

Kepler's discoveries of the laws of planetary motion around the sun, followed by Newton's law of universal gravitation, paved the way for a better understanding of several phenomena in the planetary atmospheres under the centrally-directed force of the planet's gravitation in the same way as for the motion of planets around the sun. Two phenomena immediately come to mind. The first is the present composition of the earth's atmosphere. Scientists believe that the primordial atmosphere billions of years ago had an abundance of many lighter gases, such as hydrogen and helium which exist in traces only in to-day's atmosphere (see chap. 2).

What happened to all those lighter elements? Why and how did they escape or disappear? We shall have an occasion to look into this question in the next chapter where we discuss the composition of the present-day atmosphere. Secondly, there is the problem of space travel which we are by now all familiar with. Why must we use the booster rockets to go into space? A review of the general problem of motion under centrally-directed forces may throw light on some of these issues.

The discoveries of Kepler and Newton prescribed two conditions to be satisfied by a moving object in or above the earth's atmosphere. These are that: (a) the sum of its kinetic energy and potential energy has to remain invariant; and (b) the areal velocity of the object has to remain constant (Kepler's second law). These conditions in the case of a body of mass m moving with a velocity v in the earth's atmosphere, assuming there is no friction, are expressed by the following relations

where r is the radial distance of the object from the earth's center at time t, dG/dt is the angular velocity of the object, and the variables with suffix 0 denote values at time t = 0 (see Fig. 1.3).

Using simple mathematics and eliminating t from the Eqs. (1.4.1) and (1.4.2), it can be shown (see Appendix-1B) that the path of the moving body will be a conic the polar equation of which is given by r = k/{1+ e cos(G + a)} (1.4.3)

where k = 4A2/GM, e= 2AB/GM, with B = [v2 - 2GM/r0 + (GM/2A)2]1/2, which is called the eccentricity, and a = phase angle.

If 6 be measured from the maximum value of r, a = n. (1.4.3) then reduces to the familiar polar equation of a conic with the origin at one focus (Kepler's First law). The orbit is an ellipse, parabola or hyperbola, according as the numerical value of the eccentricity e is less than, equal to, or greater than, unity. This condition requires that in the expression for e above,

< 2GM/r0, for an ellipse vo2 = 2GM/r0, for a parabola > 2GM/r0, for a hyperbola

The condition (1.4.4) implies that if the initial kinetic energy of the moving object is greater than its potential energy with sign changed, the object will move in an hyperbolic path and escape from the earth's gravitational pull. The minimum velocity required for such an escape, as calculated from the relation, is about 11.3 km s^1. This explains why a booster rocket is required to give an earth-orbiting satellite a velocity greater than the velocity of escape in order to propel it to space.

The relation (1.4.3) also enables us to derive an expression for Kepler's third law, in the case of an elliptical orbit.

The parameter of an ellipse is vo = (2GM/r)1/2

where a is semi-major axis, and b is semi-minor axis.

From (1.4.5), (1- g2)=4 A2/GMa If T be the orbital period,

AT = nab = n a2(1- G2)1/2 = 2n a2A/(GMa)1/2 (1.4.6)

Thus,

This is Kepler's third law.

Chapter 2

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