Transcript Lecture 4

ASTR 1200
Announcements
..
First Problem Set Assigned. ..Due next Tuesday in class.
Observatory Sessions all now at 8:30pm
Lecture Notes going up on the website
Schedule has been updated. Exam dates set.
Text reading still needed
Website
http://casa.colorado.edu/~wcash/APS1200/APS1200.html
Orbits
Falls Freely
Hits Surface
Misses Surface
From Sideways Motion
Physics of Orbits
sideways velocity
central acceleration
If you balance things just right, you have an orbit.
Too Little Sideways V = Crash
Too Much = Escape to Infinity
Earth
Circular Orbit
circle
implies
v r

a v
F  ma
v2
a
r
F
GMm
ma 
r2
v
GM
r
Gm1m2
r2
mv2 GMm
 2
r
r
2r
r3
P
 2
v
GM
Kepler’s Law
Example
Calculate the period of a planet that is the same distance from the star
Sirius that the Earth is from the Sun.
r=1.5x1011m
G=6.7x10-11
M=4x1030kg
(1.5 x1011 )3
4 x1033
P  6.28
 6.3
 6.3 1.5 x1013  6.3x 15 x 1012  25 x106  2.5 x107 s
11
30
20
6.7 x10 x 4 x10
2.5 x10
About 9 Months
Non-Circular Formula
2r
Focus
Same Formula for P
Use half orbit length for r
Fast
Slow
Escape Velocity
R
Fall from Large Height
1 2 GMm
mv 
2
R
Same Energy Needed to Reverse and Fly Away
2GM
ve 
R
Escape velocity is the speed
at which object must be thrown
upward to escape and never come
down.
Example
Earth
R=6x106m
M=5x1024kg
2 x6.7 x10 11 x6 x10 24
ve 
6.4 x106
8 x1014
4
ve 

1
.
1
x
10
m / s  11km / s
6
6 x10
Surface Gravity
Force of Gravity at Surface of Body is
So
GMm
ma 
R2
or
GM
a 2
R
F
GMm
R2
The acceleration of a body
near the surface.
Independent of mass.
Remember Galileo and the Leaning Tower of Pisa
Gravity on Earth
GM 6.7 x1011 x6 x1024
a 2 
 9.80m / s / s
2
6
R
6.4 x10


1 gee is 9.8m/s per second
Spin Up
When a body is spinning “Angular Momentum” is conserved
For spherical body:
2
2  2 
A  MR 

5
 P 
2
As R decreases P decreases
Spin up of ice skater.
Same for Collapsing Stars
Escape Velocity
R
Fall from Large Height
1 2 GMm
mv 
2
R
Same Energy Needed to Reverse and Fly Away
2GM
ve 
R
G=6.7x10-11 in mks units
Escape velocity is the speed
at which object must be thrown
upward to escape and never come
down.
Question
What is the escape velocity of the Moon?
M=7.35x1022kg, R=1738km
a)2000km/s
b)200km/s
c)20km/s
d)2km/s
e)0.2km/s
Answer: D
Moon
R=1700km
M=7x1022kg
2 x6.7 x10 11 x7 x10 22
ve 
1.7 x106
9 x1012
4
ve 

2
x
10
m / s  2km / s
6
1.7 x10
The Sun
The Sun
Falls into “Disk Stability”
99.9% Ended in Sun (0.1% in Jupiter)
Probably the Same Around All Stars -- Planets are Common
Shoots Planet-Size
Bullets into Space
Most of Mass Forms Ball
in Center
A Star Is a Hot
Ball of Hydrogen
(plus 11% Helium)
One Million Miles
Question
• Why is the Sun spherical while the planets
orbit in a disk?
• A) They actually orbit in a sphere
• B) The planets hit each other if they don’t
all lie in the same plane
• C) The Sun has lost spin over time, while
planet orbits cannot
• D) The Sun will eventually evolve to a disk
Answer: C
• The surface of the Sun is orbiting at far
less than orbital speed. (just like the
Earth) Back when it was first forming it
spun much faster and was closer to a disk.
What Stops the Fall?
Gravity Gets Stronger As Material Gets More Dense
GMm
Fg 
R2
R smaller implies F greater
The smaller it gets, the faster it falls in!
Why doesn’t it just become a black hole?
Or worse yet, a point-like singularity of mass?