Thurs., Sep
. 16, 2010
Gravity, Tidal Force, & Light
Newton’s Law of Gravity
Tidal Force: a difference in gravity
Light as a wave and as a particle
Given:
(1) speed in a circular orbit is:
Circular Velocity
v
circ
=
2
"
R
P
=
K
R
P
#
$
%
&
'
(
where
K
is a constant, and
(2) Kepler’s Third Law,
P
2
= K
!
(R
3
)
where the value of
K
!
depends on the units (= 1 for
P
in years
and
R
in A.U.), and we replace
a
with
R
(it’s a circle).
Find the relationship between
v
circ
and
R
, by eliminating
P
.
That is, express
v
circ
as some power of
R
, the orbit’s radius (you
may combine or drop the constants).
Relating velocity to orbit size
v
circ
2
=
4
"
2
R
2
P
2
=
4
"
2
(
)
R
2
P
2
=
4
"
2
#
K
$
%
&
'
(
)
R
2
R
3
=
#
#
K
(
)
1
R
*
1
R
From Kepler’s Third Law we have:
P
2
= K
!
(R
3
)
"
R
3
Squaring the orbit equation and substituting for
P
2,
The instructions were not to worry about the constants. Then
v
circ
=
"
"
K
R
#
1
R
This is called “Keplerian” rotation (I wonder why!)
If we
use Newton’s form of Kepler’s 3rd law we can find the value
of
K
#
(we’ll see this later in the semester).
Kepler’s Third Law: Velocity
variation with orbital radius
!
We have just shown that the average speed of a planet in a
larger orbit is slower than a planet in a smaller orbit
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“escape velocity”

depending on its
initial velocity, the
cannonball will either
fall to Earth,
continually freefall
(stay in orbit)
, or
escape the force of
Earth’s gravity.
Fate of a Falling Object
Orbital Paths
•
Extending Kepler’s First
Law, Newton found that
ellipses were not the only
possible orbital paths:
–
ellipse
(bound)
–
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 Fall '10
 Dinerstein
 Astronomy, Celestial mechanics, Tidal force, Third Law, total orbital energy

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