Environmental Engineering Reference
In-Depth Information
This equation is familiar to mathematicians, for it is none other than the polar
equation for what are called the 'conic sections'. Although we shall not prove it,
these are so named because they are the curves that are generated upon slicing
through a right circular cone in the various possible ways. The parameter
ε
determines which type of curve we are dealing with. For
ε>
1thecurveis
a hyperbola, for
ε<
1 it is an ellipse, for
ε
=
0 it is a circle and for
ε
=
1a
parabola. These are the curves which were plotted in Figure 9.9.
We can connect the results we have just derived to the qualitative statements
we made above. In particular, notice that
ε>
1 corresponds to
E>
0 and so the
unbound orbit we described earlier can now be seen to correspond to motion along
a hyperbola. Similarly,
ε<
1 corresponds to
E<
0 and we have an ellipse whilst
ε
=
=
0andacircle.Noticethatwehavetradedofftheenergy
and angular momentum of the orbit for the parameters
ε
and
α
respectively.
To conclude this section we shall spend a little time exploring the content of
Eq. (9.59) in the case of elliptical orbits, i.e.
ε<
1. Figure 9.10 shows an elliptical
orbit with the semi-major axis
a
and semi-minor axis
b
marked. One of the masses
is located at
r
1 corresponds to
E
0 whilst the other follows the elliptical orbit, sweeping out area
at a constant rate in accord with Kepler's Second Law. Note how the mass at the
origin is not at the centre of the ellipse. In fact, the point
r
=
0 is known as the
focus of the ellipse and it is displaced by a distance
from the centre of the ellipse.
The fact that all of the planets in the solar system orbit around the Sun in ellipses
with the Sun at one focus was first established by Johannes Kepler (1571-1630)
and it is known as Kepler's First Law.
Let us compute the semi-major axis
a
. It is defined such that
=
α
α
2
a
=
r(
0
)
+
r(π)
=
ε
+
1
+
1
−
ε
α
=
∴
a
ε
2
.
(9.60)
−
1
b
r
q
a
∆
Figure 9.10
The various parameters that define an elliptical orbit.
