KEPLER'S LAWS

After a tedious and difficult research process, Kepler discovered three laws that could describe how the planets move with reference to the Sun with more precision as compared to Copernicus’ and Ptolemy’s planetary models. The following information is based on pages 98 – 99 and 111 – 114 of “Text-Book on Spherical Astronomy” written by W.M. Smart.

 Kepler’s First Law states that the path, or orbit, of a planet around the Sun is an ellipse, the position of the Sun being at a focus of the ellipse.

Figure 15

Figure 15 above shows an ellipse of which S and F are the two foci, C is the centre (midway between S and F) and ΠA is the major axis. The Sun will be supposed to be at S and the planet to move anticlockwise around the ellipse. At P, the planet is at perigee, while at A, it is at apogee. ΠC is the semimajor axis; its length is given by a. DC is the semiminor axis; its length is denoted by b. The eccentricity e is given by the ratio SC : SA. By properties of ellipses, b2 = a2 (1 - e2) . The perigee distance SΠ is a(1- e)  and apogee distance SA is a(1 + e).

Let ρ denote the distance of the planet Z from the Sun and the angle θ be the planet’s angular distance from perigee, or the true anomaly. The equation of its elliptical orbit is known to be

The time required for the planet to describe its orbit is called the period, denoted by T. In time T, the radius vector SZ sweeps out an angle of 2π and thus, the mean angular velocity of the planet, w, is .

  Kepler’s Second Law states that the radius vector SZ (in Figure 16) sweeps out equal areas in equal times.

Figure 16

Let Z correspond to the planet’s position at time t and Q its position at time Let denote the radius vector SQ and be Hence, If is sufficiently small, the arc ZQ may be regarded as a straight line and the area swept out in the infinitesimal interval is simply the area of triangle QSZ which is equal to or with sufficient accuracy, The area velocity or the rate of description of area is the previous expression for area divided by As this rate is constant according to Kepler’s second law, we can write,

where h is a constant.

Now, the whole area of the ellipse is πab and this is described in the period T. Hence,                               

 

or,                                     

because

By (1) and (2), we have,

Figure17

Theoretically, if the values of the semimajor axis a, the eccentricity e, the time at which the planet passed through perigee τ and the orbital period T  are known, Kepler’s second law would enable us to determine the position of the planet in its orbit at any instant.

Referring to Figure 17, Z is the position of the planet at time t. In the interval (t - τ) the radius vector moving from SΠ to SZ sweeps out the shaded area SZΠ. By Kepler’s Second Law,

                                  Area SZΠ : Area of ellipse = t - τ : T.

Hence,                         Area SZΠ =

Or,                               Area SZΠ =       

where w = and b2 = a2 (1 – e2) .

This method may seem easy but in practice, it is inconvenient; the alternative is explained in detail in the last section, "Determining Planetary Positions".