Equations of motion
We can write expressions for both the angular momentum and the total energy. If
p
_{
θ
}
is the magnitude of the momentum in the tangential direction, then
since this perpendicular to
,
L = rp
_{
θ
}
. But
p
_{
θ
} = mv
_{
θ
} = m
= mr
= mr
.
Hence
L = r(mr
) = mr
^{2}
. Hence:
L = mr
^{2}


We can also write an expression for the total energy as a sum of the radial
kinetic energy term, the angular kinetic energy term and the potential term:
Rearranging and dividing through the left side by
m
^{2}
r
^{4}
and the right by
L
^{2}
, and canceling factors
of
dt
^{2}
we find:
To find the equations of motion we want to find
r
in terms of
θ
. In principle we could take the square root of both sides of
the above equation, separate the variables, integrate to find
θ(r)
, and then invert to find
r(θ)
. This involves a lot of
messy algebra which is not very enlightening, so we will just state the
result:
= (1 + εcosθ)


where
ε
is the eccentricity that we saw in
Ellipses
and Foci and is now given by:
ε =


This equation determines the motion of all orbital systems in the universe.
We can also find the maximum and minimum values of
r
. The minimum occurs
where the expression for
1/r
is a maximum. This is when
cosθ = 1
and
the maximum is therefore
. Thus:
r
_{min} =


The maximum occurs for the minimum of
1/r
. There are two cases: first, when
ε < 1
, the minimum of
1/r
is
.
Thus:
r
_{max} =


When
ε≥1
, the expression for
1/r
can take on the value zero when
cosθ =  1/ε
. Hence the maximum value
r
can take on is infinite
in this case.
We can also take the equation and using
r
^{2} = x
^{2} + y
^{2}
, and
cosθ = x/r
, we can write:
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 = 1/2mv
^{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}
.
Figure %: Potential well for a elliptical orbit.
We can also solve the equation for
ε
in this range. The algebra
turns out to be complicated, but we find that:
+ = 1


where
a =
and
b =
. This is an ellipse with its center at
( L
^{2}
ε/GMm
^{2}(1  ε
^{2}), 0)
, and with semimajor and semiminor axis length
a
and
b
respectively. It can also be shown that one focus of this ellipse is at the
origin.
Parabolic and Hyperbolic Orbits
Parabolic orbits occur when
ε = 1
, which means that
E = 0
.
Figure %: Potential graph for parabolic and hyperbolic orbits.
Hyperbolic orbits occur when
ε > 1
, which means that
E > 0
. In both
cases the particle will go off to infinity. In the parabolic case the particle
barely has enough energy to make it to infinity, but the hyperbolic orbit makes
it with energy to spare. In the parabolic orbit this equation simplifies
to
y
^{2} =
±2x
. This is an
equation for a parabola with its vertex at
(, 0)
.
Figure %: Shapes of various orbits.
Shows the shapes and locations of the circular,
elliptical and parabolic orbits.