Problem: Orbits with a 64 central potential
(a)
The equation of motion for the onedimensional radial problem is
μ
··
r
=
f
ef
(
r
) =
−
dV
dr
+
l
2
μr
3
,
(1)
with the 64 potential is de
fi
ned as
V
(
r
) =
k
∙
³
a
r
´
6
−
³
a
r
´
4
¸
,
(2)
where
k
is a constant that determines the strength of the potential,
a
is a
constant with dimensions of length,
l
is the constant angular momentum,
μr
2
·
θ
,
and
μ
is the reduced mass of the two particle system.
This equation can be
written explicitly as
m
··
r
=
f
ef
(
r
) =
k
a
∙
6
³
a
r
´
7
−
4
³
a
r
´
5
¸
+
l
2
μr
3
=
−
dV
0
dr
,
(3)
where
V
0
is the e
ff
ective potential for this problem,
V
0
=
V
(
r
) +
l
2
2
μr
2
.
(4)
To discuss the qualitative nature of the motion, we need to make a plot of
V
0
versus
r
.
It will be convenient to introduce a scaled radial coordinate
x
=
r/a
.
This allows us to write
e
V
(
x
) =
V
0
k
=
c
2
x
2
+
x
−
6
−
x
−
4
,
(5)
where
c
is the positive constant
c
=
l
2
μka
2
.
(6)
We can make convenient choices for the value of
c
by investigating when circular
orbits are possible. The condition to be satis
fi
ed is that the e
ff
ective force should
vanish,
f
ef
(
r
c
) =
k
a
"
6
μ
a
r
c
¶
7
−
4
μ
a
r
c
¶
5
#
+
l
2
μr
3
c
= 0
,
(7)
where
r
c
is the radius of the circular orbit. Using Eq.(6) it follows that
6
x
−
7
c
−
4
x
−
5
c
+
cx
−
3
c
= 0
,
(8)
where
x
c
=
r
c
/a
. After multiplying by
x
7
c
, Eq.(8) becomes a quadratic equation
for the variable
x
2
c
,
cx
4
c
−
4
x
2
c
+ 6 = 0
,
(9)
1
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 Fall '10
 Wilemski
 mechanics, Orbits, circular orbit, Celestial mechanics, Elliptic orbit, unstable circular orbit

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