Transcript Ch.2: Celestial Mechanics

Physics of Astronomy
week 2 – Thursday 15 Jan 2004
Celestial Mechanics
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Star Date
Boas: Spherical Coordinates
Gravity lecture and applications
Workshop moons of Jupiter
Learning plans and assignments
Boas: Spherical coordinates
Guiding Questions: Celestial Mechanics
1. How did ancient astronomers explain the motions of the
planets?
2. Why did Copernicus propose that the Earth and the other
planets revolved around the Sun?
3. What did Galileo see in his telescope that supported the
geocentric model?
4. How did Tycho Brahe attempt to test the ideas of
Copernicus?
5. What phenomenological laws did Kepler induce from
Tycho’s data?
6. How do Newton’s laws explain Kepler’s conclusions?
7. Why don’t the planets fall into the Sun?
Derive Kepler’s 3d law from Newton’s
second law:
F=ma
Gravitational force
F=GmM/r2
acceleration in circular orbit
a = v2/r
Solve for v2:
Speed v = distance/time = 2pr/T. Plug this into v2 and solve for T2:
This is Kepler’s third law: T = period and r = orbit radius.
Apply Kepler’s 3d law:
For objects orbiting the Sun,
a=radius in AU and p=period in years
A satellite is placed in a circular orbit around the Sun, orbiting
the Sun once every 10 months. How far is the satellite from
the Sun?
2
 10 
a = p =    _______
 12 
3
2
a  ______
NB: This simple form of K3 only works for our solar system. Why?
Sidereal and Synodic periods
A satellite is placed in a circular orbit around the Sun, orbiting
the Sun once every 10 months. How often does the satellite
pass between the Earth and the Sun?
1
1
1


sidereal period Earth ' s sidereal year synodic period
1
1 1


P P S
1
1 1
 
10
1 S
12
1
 ________________
S
S  ________________
We can use Newton’s gravity to approximate
the size of a black hole!
Gravitational energy  kinetic energy
GmM 1
 m v2
r
2
Solve for r  ____________
Not even light can escape (v=c) if it is closer than r to a black
hole. This is the Schwarzschild radius:
R=_____________________
Keplerian orbits: close = 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= M=
Saturn data:
period ~ 30 year = __________________ sec
orbit radius = 10 AU ~ ________________ m
M= M  =
Use Kepler’s 3d law to weigh Jupiter.
Then, use Kepler’s 3d law to weigh
galaxies and discover dark matter
Learning Plan for weeks 3-4:
Mon.19.Jan: Holiday; Tuesday Boas HW due.
Tues.20.Jan: ML on Universe Ch.3: Moon & Eclipses
Thus.22.Jan: ML on Universe Ch.4: Gravity & Orbits
Mon.26.Jan: Workshop on Jupiter’s moons & Dark matter
Tues.27.Jan: HW due on Universe 3+4; Quiz
Review Physics Ch.3+4; ML Physics Ch.6
Thus. 28.Jan: Lecture and ML on Astrophysics Ch.2
Tues.3.Feb: HW due on Physics Ch.6; CO Ch.2; Quiz
Assignments for week 3:
Tues.20.Jan: Universe Ch.3: Moon & Eclipses (Boas due)
Team 1: Ch.3.1, Motion of Moon, # 23
Team 2: Ch.3.2, Motion of Moon, # 43
Team 3: Ch.3.3-4, Lunar Eclipses, # 30
Team 4: Ch.3.5, Solar Eclipses, #33
Team 5: Ch.3.5, Distances: demonstrate and diagram
Eratosthenes’ calculation
Thus.22.Jan: Universe Ch.4: Gravity & Orbits
Team 1: Ch.4.1-2, Retrograde motion, #48
Team 2: Ch.4.3, Galileo’s observations, #50 or 52
Team 3: Ch.4.4-5, Tycho + Kepler, #35
Team 4: Ch.4.6-7, Newton + orbits, #39
Team 5: Ch.4.8, Tides #44 (or Physics Ch.6 #54)
Assignments for week 4:
Mon.26.Jan: Workshop on Jupiter’s moons & Dark matter
Tues.27.Jan: HW due: Physics Ch.6 # 54
Universe Ch.3 # 23, 43, 30, 43, Ch.4 # 48, 52, 35, 39, 44
Review Physics Ch.3+4; ML Physics Ch.6, Gravitation:
Team 1: Ch.6.1-3, #13, 14
Team 2: Ch.6.4-5, #28
Team 3: Ch.6.6-8, #47
Thus. 28.Jan: Lecture on Astrophysics Ch.2
Team 4: Ch.2.1, #2.1 & 2.2
Team 5: Ch.2.3, #2.7
Tues.3.Feb: HW due: Physics Ch.6 # 13, 14, 28, 47, 57, 60
CO (Astrophysics) Ch.2 # 1, 2, 7, 8, 11