Lecture 4: Kepler orbits (11 Sep 09)
A. Review
1. Relative motion with reduced mass
μ
=
m
1
m
2
/
(
m
1
+
m
2
), potential
Φ = Γ
/r
, relative angular momentum
‘
.
2. Integration of equation of motion gives
1
r
=
C
[1

cos(
φ
)]
after choosing an origin for
φ
,
C
≡
μ
Γ
/‘
2
, eccentricity
=
q
1 + (2
E‘
2
/μ
Γ
2
)
3. Virial theorem relates average potential and kinetic energies:
2
h
K
i
=
h
r
· ∇
Φ
i ⇒
2
h
K
i
=
h
Φ
i
E
=
h
Φ
i
/
2
h
...
i
is a time average, so we will need to evaluate some trigonometric
integrals. For the period and time average over a period, use
μr
2
˙
φ
=
‘
:
τ
=
Z
2
π
0
dφ/
˙
φ
=
μ
‘
Z
2
π
0
r
2
dφ
h
1
r
i
=
1
τ
Z
τ
0
dt
r
=
1
τ
Z
2
π
0
dφ
r
˙
φ
=
μ
‘τ
Z
2
π
0
rdφ
B. Analytical geometry of an ellipse
1. Ellipse has foci at
f
=
±
f
ˆ
x
and is traced out as the points for which
sum of the distances from the foci is a constant:
d
1
+
d
2
= 2
a
a
is the semimajor axis of the ellipse. Define the focal position
f
=
a
;
is found to be the eccentricity already defined.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2. Vector relation
d
2
=
d
1
+ 2
f
ˆ
x
. Then calculate
d
2
1
from this equation
and from the B.1 equation using scalar product
d
2
·
ˆ
x
=
r
cos
φ
, with
plane polar coordinates
r, φ
for
d
2
. The result of the algebra is:
r
(1

cos
φ
) =
a
(1

2
)
3. The correspondence to the Kepler orbit is 1
/C
=
a
(1

2
). The center
of force is at one of the foci;
r
is measured from that focus.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 BRUCH
 Angular Momentum, Mass, Momentum, Orbits, Kepler's laws of planetary motion, dφ µ

Click to edit the document details