Physics of Astronomy
Winter Week 6 - Thus.16.Feb.2006
Astro-A: Universe 4 – Gravity and Orbits
Pre-lab – brief discussion
Astro-B: Finish C&O Ch.2
2:30 Seminar
4:00 Lab in CAL 1234
Let’s look at the sky together after lab!
Looking ahead (remember to take Universe Ch.4 online quiz)
Astro-A: Universe 4 – Gravity & Orbits
Ptolemy: Circular orbits about Earth
Copernicus: Circular orbits about Sun
Kepler: Elliptical orbits about Sun
We derived Kepler’s 3d law from
Newton’s 2d 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.
For objects orbiting the Sun, Kepler’s law simplifies
to a3 = p2,
where 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  ______
Sidereal (real: P) and
Synodic (apparent: S) 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 E 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!
Knowing the gravitational force between two bodies m and M,
we can find their gravitational energy:
Energy  Force * distance
Egrav
GmM

* r  ________
2
r
In order for an object (say, m) to escape M’s gravity,
It needs sufficient kinetic energy K=1/2 mv2 …
Use energy conservation to find the size
of R of a black hole:
Gravitational energy  kinetic energy
GmM 1
2
 mv
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(for v=c) =_____________________
Lab
height and speed vs time
25.00
20.00
15.00
height(m)
Hand in before lab:
• Excel practice results
• Pre-lab
speed(m/s)
10.00
5.00
0.00
0
0.5
1
1.5
2
2.5
tim e(s)
Kinetic and Potential energy vs time
70
60
50
40
Kinetic(J)
30
Potential(m/s)
20
10
0
0
0.5
1
1.5
tim e(s)
2
2.5
Lab this afternoon
• Read lab guidelines
• Take notes in a bound
notebook
• Write up report this
weekend
Astro-B: C&O Ch.2: Gravity & orbits
Kepler’s laws (for M>>m)
Generalization (for M~m)
K1: orbits are elliptical
- about center of mass
K2: equal areas in equal times - conservation of L
K3:
GMT  4p R
2
Center of mass
Virial Theorem
2
3
G  M  m P2  4p 2 a3
Center of Mass reference frame
Total mass = M = m1+ m2
Reduced mass = m
m1m2
m
m1  m2
Total angular momentum
L=m r v = m rp vp
(Pick any point, e.g. perihelion:
rp and vp) p.50
L  m GMa 1  e2 
Virial Theorem <E> = <U>/2
where <f> = average value of f over one period
For gravitationally bound systems in equilibrium, the
total energy is always one-half of the potential energy.
Ex: Use K3 in Eorbit = T + U for circular orbit:
New HW: 2.8, 2.9
Looking ahead
Monday Tuesday
Wed
Thus
Fri