Physics of Astronomyweek 2 – Thursday 15 Jan 2004

Celestial Mechanics

• 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 acceleration in circular orbit

F=GmM/r2 a = v2/r

Solve for v2:

Speed v = distance/time = 2r/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

3 2 10 a = p = _______

12

a ______

NB: This simple form of K3 only works for our solar system. Why?

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

'

1 1 1

1 1 110 1

121

________________

________________

sidereal period Earth s sidereal year synodic period

P P S

S

SS

Sidereal and Synodic periods

We can use Newton’s gravity to approximate the size of a black hole!

21

2____________

Gravitational energy kinetic energy

GmMmv

rSolve 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=_____________________

42r3 = GM T2 a3=p2 for planets around the SunOrbit radius: r(m) or a(AU); Period T(sec) or p(years)

http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html

Keplerian orbits: close = faster

Solve 42r3 = 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 the Sun

Use Kepler’s 3d law to weigh Jupiter.

Then, use Kepler’s 3d law to weigh galaxies and discover dark matter

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

Learning Plan for weeks 3-4:

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 3:

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

Assignments for week 4: