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9.4 Elliptical Orbits; Kepler’s Laws 283

MATH HELP ELLIPSES

An ellipse is defined geometrically by the condition that the If the semimajor axis of length a is along the x axis and the
sum of the distance from one focus of the ellipse and the dis- semiminor axis of length b is along the y axis, then the x and
tance from the other focus is the same for all points on the y coordinates of an ellipse centered on the origin satisfy
ellipse.This geometrical condition leads to a simple method
for the construction of an ellipse: Stick pins into the two foci x 2 y 2
1
and tie a length of string to these points. Stretch the string taut a 2 b 2
to the tip of a pencil, and move this pencil around the foci
while keeping the string taut (see Fig. 1a). The foci are on the major axis at a distance f from the origin
An ellipse can also be constructed by slicing a cone given by
obliquely (see Fig. 1b). Because of this, an ellipse is said to 2 2
f 2a b
be a conic section.
The largest diameter of the ellipse is called the major axis, The separation between a planet and the Sun is a f
and the smallest diameter is called the minor axis.The semimajor at perihelion and is a f at aphelion.
axis and the semiminor axis are one-half of these diameters,
respectively (see Fig. 1c).

(a) (b) (c)

semiminor
axis

b
Sun
focus focus f a

semimajor
axis

FIGURE 1 (a) Constructing an ellipse. (b) Ellipse as a conic section. (c) Focal distance f, semimajor axis a, and semiminor axis b of an ellipse.

Figure 9.10 illustrates this law. The two colored areas are equal, and the planet takes
Q
equal times to move from P to P and from Q to Q . According to Fig. 9.10, the speed
of the planet is larger when it is near the Sun (at Q) than when it is far from the Sun
(at P). Q' S
Kepler’s Second Law, also called the law of areas, is a direct consequence of the P'
central direction of the gravitational force. We can prove this law by a simple geo- P
metrical argument. Consider three successive positions P, P ,P on the orbit, sepa-
rated by a relatively small distance. Suppose that the time intervals between P, P and Radial line sweeps out
equal areas in equal times.
between P ,P are equal—say, each of the two intervals is one second. Figure 9.11
shows the positions P, P ,P . Between these positions the curved orbit can be approx- FIGURE 9.10 For equal time intervals, the
imated by straight line segments PP and P P . Since the time intervals are one unit areas SQQ and SPP are equal. The distance
of time (1 second), the lengths of the segments PP and P P are in proportion to the QQ is larger than the distance PP .