Solving the orbits
The orbits are determined by the various values that
ε
can take.
Circular orbits
When
ε
= 0 , the expression for
ε
tells us that
E
= 
. The negative value of the energy
just means that the potential energy is more negative than the kinetic energy is positive. In this
case we have
r
min
=
r
max
=
. The particle is trapped at the very bottom of a potential well,
and the radius does not change as it goes around the orbit, hence forming a circle. Substituting
this value for
r
into the energy we have
E
= 
. Note that we could have derived this
directly by summing the potential energy we found for a circular orbit with the kinetic energy
(
Gravitational Potential Energy
).
E
= 1/2
mv
2
+
U
=

= 
Figure %: Potential well for a circular orbit.
In the case
ε
= 0 we can see that this equation simplifies to
x
2
+
y
2
=
. This
describes a circle with radius
.
Elliptical Orbits
Elliptical orbits occur when 0 <
ε
< 1 . This means that 
<
E
< 0 . Again the particle is
trapped in a potential well, oscillating now between
r
min
and
r
max
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Figure %: Potential well for a elliptical orbit.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 DavidJudd
 Physics, Energy, Orbits, Potential Energy, ε

Click to edit the document details