#### Transcript Gravity

```Astronomy and Cosmology
week 4 – Tuesday 22 April 2003
Weighing the Sun and Planets
•
•
•
•
Star Date
Keplerian orbits
Weigh the Sun
Workshop: use moons to weigh Jupiter
break
• Minilecture by Derek and Andy
• Thursday: Ch.5 Light
We derived Kepler’s 3d law from
Newton’s second law, F=ma:
Gravitational force
acceleration in circular orbit
F=GmM/r2
a = v2/r
GmM/r2= mv2/r
v2= GM/r
Speed v = distance/time = 2pr/T
v2 = 4p2r2/T2 = GM/r
4p2r3 = GM T2
This is Kepler’s third law: T = period and r = orbit radius.
Algebra review at http://regentsprep.org/Regents/physics/phys06/keplers/default.htm
Keplerian orbits: closer = faster
http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html
4p2r3 = GM T2  a3=p2 for planets around the Sun
Orbit radius: r(m) or a(AU); Period T(sec) or p(years)
Use Kepler’s 3d law to weigh the Sun.
Solve 4p2r3 = GMT2 for central mass M=_______
Earth data:
period = 1 year ~ 3 x 107 sec
orbit radius = 1 AU ~ 150 x 109 m
M= Msun=
Saturn data:
period ~ 30 year = __________________ sec
orbit radius = 10 AU ~ ________________ m
M= Msun=
Use Kepler’s 3d law to weigh Jupiter.
Week 8: use Kepler’s 3d law to weigh
galaxies and discover dark matter
```