With reference to figure 9-8, a conic section is defined in terms of the ratio OPP D =
,
where OP =r, and P D = 2q−rcos θ. From this ratio, solve for the radius rand
obtain the representation
r=p
1 + cos θ, (9 .104)where p= 2 q
. Equation (9.104) is the equation of a conic section. The following
terminology is applied to the variables and parameters in this equation:
1. The angle θis called the true anomaly associated with the orbit.
2. The symbol ais introduced to denote the semi-major axis of an elliptical orbit.
The symbol acan be shown to be related to r, p and
.
3. The quantity pis called the semiparameter of the conic section and is illustrated
in figure 9-8. Note that when θhas the value π/ 2 ,then r=p.
An important relation connecting the parameters p, a and is obtained from
equation (9.104) by setting θequal to zero. This gives
r=p1 + =q=a(1 −^ ) which implies p=a(1 −^
(^2) ). (9 .105)
In order to demonstrate that the motion of mass m with respect to mass M
is a conic section, show that the magnitude r of the position vector r satisfies an
equation having the exact same form as equation (9.104).
Kepler’s Laws
Johannes Kepler^7 , an astronomer and mathematician, discovered three laws con-
cerning the motion of the planets. He discovered these laws from experimental data
without the aid of calculus or vector analysis. Newton, using calculus, verified these
laws with the model for the inverse square law of attraction. These three laws are
now derived.
To derive Kepler’s three laws one can make use of the following vector identities:
r ׈er=rˆer×eˆr= 0 (9 .106)
d
dt(r ×dr
dt) = r ×d(^2) r
dt^2
(9 .107)
ˆer·dˆer
dt
= 0 (9 .108)
ˆer×(ˆer×dˆer
dt
) = −dˆer
dt
(9 .109)
(^7) Johannes Kepler (1571-1630), German astronomer and mathematician.