Kepler also investigated the speeds of the planets and found that the closer
in its orbit a planet is to the Sun, the faster it moves. Drawing a straight
line connecting the Sun and the planet (the radius vector), he
discovered that he could express this fact in his second law  the law of
areas:
For any planet the radius vector sweeps out equal areas in equal
times.
figure 32:

Kepler's second law.

Figure 32 shows the positions of a planet,
P_{1}, P_{2}, P_{3},
P_{4}, at times
t_{1}, t_{2}, t_{3},
t_{4}. Between times t_{1} and
t_{2} the planet's radius vector sweeps out the area
bounded by the radius vectors SP_{1},
SP_{2} and the arc
P_{1}P_{2}. Similarly, the area
swept out by the radius vector in the time interval
(t_{4}t_{3}) is the area
SP_{3}P_{4}. Then, Kepler's second
law states that:
area SP_{1}P_{2} /
(t_{2}t_{1}) =
area SP_{3}P_{4} /
(t_{4}t_{3}) = constant.
If t_{2}t_{1} =
t_{4}t_{3}, then
area SP_{1}P_{2} =
area SP_{3}P_{4}.
If the area is the area of the ellipse itself (which is given by
ab), the
radius vector will be back to its original position and Kepler's
second law therefore implies that the planet's orbital period is
constant.
It is possible to derive a mathematical expression for Kepler's
second law by supposing that the time interval
(t_{2}t_{1}) is very small
and equal to interval (t_{4}t_{3}).
Position P_{2} will be very close to
P_{1}, just as P_{4} will be close
to P_{3}. The area
SP_{1}P_{2} is then approximately
the area of triangle SP_{1}P_{2},
or
½SP_{1} x SP_{2} x
sin P_{1}SP_{2}.
If angle P_{1}SP_{2} is expressed
in radians we may write
sin P_{1}SP_{2} = angle
P_{1}SP_{2} =
_{1},
since angle P_{1}SP_{2} is very small.
Also,
SP_{1} = SP_{2} =
r_{1}, say,
so that the area SP_{1}P_{2} is given
by
½r^{2}_{1}_{1}.
Similarly, area SP_{3}P_{4} is given
by
½r^{2}_{2}_{2},
where SP_{3} = r_{2} and angle
P_{3}SP_{4} =
_{2}.
Let t_{4}t_{3} =
t_{2}t_{1} = t. Then, from
Kepler's second law,
½r^{2}_{1}(_{1} / t) =
½r^{2}_{2}(_{2} / t) = constant.
But /t is the
angular velocity, ,
in the limit when t tends to zero. Hence
½r^{2}_{1}_{1} =
½r^{2}_{2}_{2}
= constant,
is the mathematical expression of Kepler's second law. In order for this
law to be obeyed, the planet has to move fastest when its radius vector
is shortest, at perihelion, and slowest when it is at aphelion, as
shown in this java applet.
©Vik Dhillon, 30th September 2009