kepler's first law back to teaching back to the course first page of this section previous page next page last page of this section help on navigating these pages

Using the planetary distance determination method that he derived in conjunction with the positional observations of the planets that Tycho Brahe had painstakingly accumulated over 20 years, Kepler set about tracing the orbit of Mars. He soon found that the age-old idea of circular orbits had to be discarded, as the distance of Mars from the Sun appeared to vary. Kepler tested many models of orbital shapes and finally showed that the orbital planes of the planets pass through the Sun and that the orbital shape was an ellipse. These findings were announced in 1609 as Kepler's first law - the law of ellipses:

The orbit of each planet is an ellipse with the Sun at one focus.

An ellipse is defined mathematically as the locus of all points such that the sum of the distances from two foci to any point on the ellipse is a constant. Figure 31 shows the elliptical orbit of a planet, P, with the Sun, S, at one of the foci. The other focus, F, is often called the empty focus. From the definition of an ellipse, we have

r + r´ = 2a = constant.

figure 31: An elliptical planetary orbit.

We define the following properties of the ellipse in Figure 31:
It is important to know the equation of an ellipse, as this can tell us the distance from one focus to a point on the ellipse (e.g. the Earth-Sun distance) as a function of the position of the point on the ellipse. If we centre a polar coordinate system (r, ) at S and let the line SA correspond to =0, then r measures the distance SP, and , the true anomaly, measures the counter-clockwise angle ASP. Using

cos (180° - ) = - cos

and the cosine law of plane trigonometry, we have

r´2 = r2 + (2ae)2 + 2r(2ae)cos.

From the definition of an ellipse, we have

r´ = 2a - r

and so
r = a (1 - e2) / (1 + e cos ).

This is the equation of an ellipse in polar coordinates. In Cartesian coordinates (x, y) the equation of the ellipse can be derived using Figure 31 and the theorem of Pythagoras, as follows:

r´2 = (x + ae)2 + y2

r2 = (x - ae)2 + y2

Subtracting these two equations and using the relation r´ = 2a - r derived earlier, we find r´ = a + ex. Substituting back into the first of the above two equations and employing the relation b2 = a2(1 - e2), we obtain

(x/a)2 + (y/b)2 = 1

which is the equation for an ellipse in Cartesian coordinates. It can be seen that this equation reduces to the equation of a circle when a=b.

©Vik Dhillon, 30th September 2009