Download Document

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Kepler (spacecraft) wikipedia, lookup

Nebular hypothesis wikipedia, lookup

Aquarius (constellation) wikipedia, lookup

Astrobiology wikipedia, lookup

Rare Earth hypothesis wikipedia, lookup

Tropical year wikipedia, lookup

History of astronomy wikipedia, lookup

Planets beyond Neptune wikipedia, lookup

Astronomical unit wikipedia, lookup

Copernican heliocentrism wikipedia, lookup

Lunar theory wikipedia, lookup

Modified Newtonian dynamics wikipedia, lookup

Planets in astrology wikipedia, lookup

Extraterrestrial life wikipedia, lookup

Geocentric model wikipedia, lookup

Definition of planet wikipedia, lookup

Exoplanetology wikipedia, lookup

IAU definition of planet wikipedia, lookup

Philosophiæ Naturalis Principia Mathematica wikipedia, lookup

Planetary habitability wikipedia, lookup

Solar System wikipedia, lookup

Planetary system wikipedia, lookup

Satellite system (astronomy) wikipedia, lookup

Dialogue Concerning the Two Chief World Systems wikipedia, lookup

History of Solar System formation and evolution hypotheses wikipedia, lookup

Formation and evolution of the Solar System wikipedia, lookup

Orrery wikipedia, lookup

Timeline of astronomy wikipedia, lookup

Transcript
The Motion of the Planets
7/17/06
ISP 209 -- 3B
1
The Motion of the Planets
7/17/06
ISP 209 -- 3B
2
The Solar System
Precisely how do the planets (including
Earth) move around the sun?
What are the fundamental laws of
nature that govern this motion?
7/17/06
ISP 209 -- 3B
3
Kepler’s Laws of Planetary Motion
Johannes Kepler – a contemporary of Galileo –
studied the data on astronomical observations of
the planets, seeking to describe accurately their
motion around the sun. After 20 years of work, he
deduced three empirical laws of planetary motion.
1. A planet moves on an elliptical orbit with the
sun at one focal point of the ellipse.
2. The radial line – the line from the sun to the
planet – sweeps out equal areas in equal times.
3. The square of the period is proportional to the
cube of the semi-major axis.
7/17/06
ISP 209 -- 3B
4
Ellipse geometry
An ellipse with semi-major
axis a and eccentricity e.
This ellipse has a = 1
and e = 0.5 .
Hint:
perihelion+aphelion = 2a
7/17/06
ISP 209 -- 3B
5
The motion of the planets
Diagram of a planet revolving
around the sun.
The eccentricity e is grossly
exaggerated ― real orbits are
very close to circular.
In fact there are nine planets. The center of mass
of the solar system is fixed (). To a first
approximation the center of mass is at the Sun.
() actually it moves around the
7/17/06
ISP 209 -- 3B
center of the galaxy
6
Centripetal
acceleration
For an object in circular
motion, the centripetal
acceleration is a = v 2/r .
(Christian Huygens)
Example. Determine the string tension if a mass of 5 kg is
whirled around your head on the end of a string of length 1
m with period of revolution 0.5 s.
Answer : 790 N
7/17/06
ISP 209 -- 3B
7
Three concepts --–
• Centripetal acceleration
• Centripetal force
• Centrifugal force
7/17/06
ISP 209 -- 3B
v
ar 
r
2
8
Centripetal force and “centrifugal force”
View from above
View from rear
7/17/06
ISP 209 -- 3B
9
Isaac Newton (1642 – 1727)
Newton solved the premier scientific problem of his
day – to explain why the planets move as they do.
To solve this problem he developed …
• the three laws of motion,
• the theory of universal gravitation,
• calculus, a branch of mathematics.
Newton quote:
“If I have been able to see farther than
others it is because I stood on the shoulders
of giants.”
-- in a letter to Robert Hooke
He could be referring to Galileo and Kepler.
7/17/06
ISP 209 -- 3B
10
Circular Orbits
(a pretty good approximation for all the
planets because the eccentricities are
small.)
(velocity)
Centripetal acceleration
v2
ar  
r
(acceleration)
Newton’s second law
mv
GMm

r
r2
2
There is a subtle approximation here: we are approximating the center of
mass position by the position of the sun. This is a good approximation.
7/17/06
ISP 209 -- 3B
11
mv
GMm

2
r
r
2
Circular Orbits
The planetary mass m cancels out.
The speed is then
GM
v
r
Period of revolution
Time = distance / speed
i.e., Period = circumference / speed
2 r
r
4 r
2
T 
 2 r
, or T 
v
GM
GM
2
3
 Kepler’s third law: T 2  r 3
7/17/06
ISP 209 -- 3B
12
Generalization to elliptical orbits
(and the true center of mass!)
2 3
4 a
T 
G( M  m)
2
2 3
4 a

GM
where a is the semi-major axis of the ellipse
The calculation of elliptical orbits is difficult
mathematics.
The story of Newton and Halley
Many applications ...
7/17/06
ISP 209 -- 3B
13
Kepler’s second law -- The Law of
Equal Areas
Perihelion : fastest
Aphelion : slowest
Newton: Angular momentum is conserved—in fact
that’s true for any central force—so the areal rate
is constant.
7/17/06
ISP 209 -- 3B
14
Westminster Abbey
7/17/06
ISP 209 -- 3B
15
Here is buried Isaac Newton, Knight, who by a
strength of mind almost divine, and
mathematical principles peculiarly his own,
explored the course and figures of the planets,
the paths of comets, the tides of the sea, the
dissimilarities in rays of light, and, what no
other scholar has previously imagined, the
properties of the colors thus produced.
Diligent, sagacious and faithful, in his
expositions of nature, antiquity and the holy
Scriptures, he vindicated by his philosophy the
majesty of God mighty and good, and
expressed the simplicity of the Gospel in his
manners. Mortals rejoice that there has existed
such and so great an ornament of the human
race! He was born on 25th December, 1642,
and died on 20th March 1727.
Newton’s monument in
Westminster Abbey.
7/17/06
ISP 209 -- 3B
16
Comets
7/17/06
ISP 209 -- 3B
17
Ocean Tides -- an effect due to the
gravity of the moon and the sun
Bay of Fundy
7/17/06
ISP 209 -- 3B
18
If you have ever lived near the ocean, you have
observed the tides at the beach.
7/17/06
ISP 209 -- 3B
19
Newton: Ocean tides are due to the gradient of the
gravitational force
7/17/06
ISP 209 -- 3B
20
Newton said of himself…
“ I know not what I appear to the world,
but to myself I seem to have been only like
a boy playing on the sea-shore, and
diverting myself in now and then finding a
smoother pebble or a prettier sea-shell,
whilst the great ocean of truth lay all
undiscovered before me.”
7/17/06
ISP 209 -- 3B
21
Quiz question
7/17/06
ISP 209 -- 3B
Consider the star (S),
which has two planets (P1
and P2).
P2 is twice as far from S as
P1. The period of
revolution P1 is 1 year.
What is the period of P2?
22
Newton’s figure to explain
planetary orbits
Newton’s theory has stood the
test of time. We use the same
theory today for planets,
moons, satellites, etc.
7/17/06
ISP 209 -- 3B
23
Here is the general formula. We
can always neglect m (mass of
the satellite) compared to M
(mass of the center).
4 a
4 a
T 

G (M  m )
GM
2
2
3
2
3
Applications
• Earth and Sun
• Other planets
• Moon and Earth
• Artificial satellites
• Exploration of the solar system
Some of these are covered in the CAPA problems.
7/17/06
ISP 209 -- 3B
24
Comets
Comet Hale-Bopp
7/17/06
ISP 209 -- 3B
25