Kepler's first law

Kepler's first law of planetary motion states that a planetary orbit is an ellipse, with the Sun located at one of the focii. Now, an ellipse is defined as the locus of a moving point whose distance from a fixed point, called the focus, has a constant ratio to its perpendicular distance from an infinite fixed straight-line, known as the directrix. This definition is illustrated in Fig. 2. Here, $F$ is the focus, and $AB$ the directrix. If

$$\frac{FP}{PT} = e,$$ (1)

where $0<e<1$ is a constant called the eccentricity, then point $P$ traces out an ellipse. The two symmetry axes of the ellipse, $RS$ and $R'S'$, which are known as the major and minor axes, respectively, meet one another at the figure's geometric center, $C$. The ellipse actually possesses a second focus $F'$, and a second directrix $A'B'$, which are the reflections of the original focus and directrix, respectively, in the minor axis. Thus, the ellipse is also the locus of the point whose distance from $F'$ is $e$ times its perpendicular distance from $A'B'$. Note, finally, that both focii lie on the major axis, and are equidistant from $C$.

Figure 2: The ellipse.

Figure 3 illustrates a planetary orbit viewed from the direction of the northern ecliptic pole (all subsequent figures are also drawn from this viewpoint). The planet is at $P$, and the Sun at $S$. Note that all planets in the solar system orbit the Sun in an anti-clockwise direction (when viewed from the northern ecliptic pole). The major and minor axes of the orbit are $MM'$ and $OO'$, respectively, whereas $C$ is its geometric center. Incidentally, the major axis is sometimes referred to as the line of apsides. The two focii lie at $S$ and $Q$, where the empty focus, $Q$, is called the equant. The distances $a\equiv CM$ and $b\equiv CO$ are known as the semi-major and semi-minor radii, respectively. The point of closest approach of the planet to the Sun, $M$, is called the perihelion. Likewise, the point of furthest distance from the Sun, $M'$, is known as the aphelion.

Figure 3: A planetary orbit with polar coordinates centered on the Sun. Here, $S$ is the Sun, $P$ the planet, $C$ the geometric center of the orbit, $Q$ the equant, and $MM'$ the line of apsides.

Consider plane polar coordinates centered on the Sun. If $r\equiv SP$ is the distance of the planet from the Sun, and $\theta$ the angle $MSP$, then the planetary orbit satisfies the well-known equation

$$r = \frac{a\,(1-e^2)}{1+e\,\cos\theta}.$$ (2)

It follows from this equation that the distances $QC$ and $CS$ are both $e\,a$.

Figure 4: A planetary orbit with polar coordinates centered on the geometric center. See previous caption.

Figure 4 illustrates a more convenient set of plane polar coordinates whose origin lies at the orbit's geometric center. Thus, $s\equiv CP$, and $\psi$ is the angle $MCP$. When expressed in these coordinates, the orbit satisfies

$$s = a\left(\frac{1-e^2}{1-e^2\,\cos^2\psi}\right)^{1/2}.$$ (3)

It follows that

$$b = (1-e^2)^{1/2}\,a.$$ (4)

Incidentally, it is clear, from the previous two equations, that the eccentricity, $e$, measures the deviation of the shape of the orbit from a circle (with $e=0$ corresponding to a circle, and $e=1$ to an infinitely elongated ellipse).