A. Review: 2body relative motion
1. Twobody problem: relative motion with reduced mass
μ
=
m
1
m
2
m
1
+
m
2
,
momentum
p
=
μ
˙
r
, angular momentum
l
=
r
×
p
. Conserved quantities
with conservative central forces:
1
and energy
E
=
p
2
2
μ
+ Φ
eff
=
μ
2
˙
r
2
+ Φ
eff
with an eﬀective potential
Φ
eff
= Φ(
r
) +
‘
2
2
μr
2
2. Motion remains in a plane perpendicular to
l
and a sketch of Φ
eff
(
r
)
shows the range of
r
for given
E
. The turning points
r
1
,r
2
of bounded
motion can be set “by inspection” and the period
τ
evaluated by
τ
= 2
Z
r
2
r
1
dr/
q
(2
/μ
)[
E

Φ
eff
]
3. The gravitational potential Φ =

Γ
/r
has the special feature that,
even for noncircular orbits, the periods of angular and radial motion
are equal.
4. The solution for
r
(
φ
) was obtained by direct integration of the equation
for
dr/dφ
obtained from
(2
/μ
)[
E

Φ
eff
] = ˙
r
2
= [(
‘/r
2
μ
)
dr/dφ
]
2
= [(
‘/μ
)
d
(1
/r
)
/dφ
]
2
and is
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 Fall '09
 BRUCH
 Angular Momentum, Energy, Force, Mass, Momentum, Kepler's laws of planetary motion, Celestial mechanics, Elliptic orbit, Kepler Problem, virial theorem

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