Astronomy and Cosmologies week 3 – Thursday 15
April 2004:
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Gravity
Minilecture by Brian Ott and Tristen Bristow
Star Date
Logistics: only one weekly seminar paper
Gravity lecture and applications
break
• Research projects
• Friday: take Quiz 3
• Next Thus. gravity workshop on moons of
Jupiter and dark matter
Guiding Questions
1. How did ancient astronomers explain the motions of the
planets?
2. Why did Copernicus think that the Earth and the other planets
revolved around the Sun?
3. What did Galileo see in his telescope that confirmed that
planets orbit the Sun?
4. How did Tycho Brahe attempt to test the ideas of Copernicus?
5. What paths do the planets follow as they move around the
Sun?
6. What fundamental laws of nature explain the motions of
objects on Earth as well as the motions of the planets?
7. Why don’t the planets fall into the Sun?
8. What keeps the same face of the Moon always pointed toward
the Earth?
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.
Use Kepler’s 3d law to find the mass
of the Sun:
2
4
p
Algebraically (no numbers yet), solve T 2 =r 3
GM
for M=____________________
G = universal gravitational constant
r = radius of Earth’s orbit about Sun = ________ km = ________ m
T = period of Earth’s orbit = ____yrs
=________sec
Knowing Earth’s orbit radius and period, you can solve for the
Sun’s mass M =
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 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 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
GmM
Egrav 
* 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) =_____________________