where p=h^2 /GM and =C/GM. This result is known as Kepler’s first law and implies
that all the planets of the solar system describe elliptical paths with the sun at one
focus.
Kepler’s second law states that the position vector r sweeps out equal areas in
equal time intervals. Consider the area swept out by the position vector of a planet
during a time interval ∆t. This element of area, in polar coordinates, is written as
dA =
1
2
r^2 dθ
and therefore the rate of change of this area with respect to time is
dA
dt
=^1
2
r^2 dθ
dt
.
It has been demonstrated that the angular momentum per unit mass h=r ×v
is a constant. For r =rcos θˆe 1 +rsin θˆe 2 ,the angular momentum has components
which can be calculated from the determinant
h=r ×dr
dt
=
∣∣
∣∣
∣∣
ˆe 1 ˆe 2 ˆe 3
rcos θ r sin θ 0
−rsin θθ ̇+ ̇rcos θ r cos θθ ̇+ ̇rsin θ 0
∣∣
∣∣
∣∣
By expanding the above determinant and simplifying one can verify that
h=r^2 dθ
dt
ˆe 3 =hˆe 3 =Constant (9 .116)
which in turn implies
dA
dt =
1
2 r
2 dθ
dt =is a constant. (9 .117)
This result is known as Kepler’s second law. Analysis of this second law informs us
that the position vector sweeps out equal areas during equal time intervals.
The time it takes for mass mto complete one orbit about mass Mis called the
period of the motion. Denote this period by the Greek letter τ. Note that equation
(9.117) tells us that when r^2 is small dθdt becomes large and, conversely, when dθdt is
small r^2 becomes large. The resulting motion is for planets to move faster when
they are closer to the Sun and slower when they are farther away. Express equation
(9.117) in the form dA =^12 h dt and integrate the result from t= 0 to t=τ, to show
A=
h
2 τ, (9 .118)
