Transcript notes6b

Section 12.3. Orbital Motion of
Satellites and Kepler’s Laws

r

M
m

v
 Satellites move in circular
(or more generally, elliptical)
orbits
 Compute their period and
speed by applying Newton’s
2nd Law in the radial direction
mv 2
 Fr  r Orbital speed
GMm mv 2
GM

v
2
r
r
r
2r
2r
T

v
GM / r
3/ 2
2r
Orbital
T
GM period
Example
Venus rotates slowly about its axis, the period
being 243 days. The mass of Venus is 4.87 x 1024
kg. Determine the radius for a synchronous
satellite in orbit about Venus.
Solution:
Given: MV = 4.87 x1024 kg, TV = 243 days
Recognize: Synchronous means that the period of
the satellite equals the period of Venus, Ts=TV
Convert TV to seconds and find rs
 24 hr  60 min  60 sec 
7


TV  243 days 


  2.10x10 s

 1 day  1 hr  1 min 
Ts GM V
2rs
3/ 2
TS 
 rs 
2
GM V
3/ 2
7
rs3 / 2 
(2.10 x10 s) (6.6726 x10
-11 Nm 2
kg 2
)( 4.87 x10 24 kg)
2
 6.025 x 1013 m 3/2  rs  1.54 x 109 m
Compare this to the radius of Venus: 6.05x106 m
Kepler’s Laws of Orbital Motion
1st Law - planets follow elliptical orbits
with the Sun at one focus of the ellipse
2nd Law - the radius vector from the
Sun to the planet sweeps out equal
areas in equal time
3rd Law - the orbital period of a planet
is proportional to the radius to the 3/2
power (derived for circular orbit – just
replace r by a)