Transcript Kepler - ClassNet

History of Astronomy - Part II
• After the Copernican Revolution, astronomers
strived for more observations to help better
explain the universe around them
• During this time (1600-1750) many major
advances in science and astronomy occurred
– Kepler's Laws of Planetary Motion
– Newton's Laws of Motion and Gravity
• Warning! - Math and Equations Ahead!
Tycho Brahe - An Observer
• Tycho Brahe was a prominent
scholar and aristocrat in Denmark
in the mid-late 1500's
• He made a huge number of
observations of the stars and
planets, all with the naked eye
– Even without a telescope, he was
very accurate in his
measurements
• Also recorded the appearance of
comets and supernovae
– The Tycho supernova remnant is
still visible today
Tycho (1546-1601)
Johannes Kepler - A Theorist
• Shortly before his death,
Tycho began working with
another scientist named
Kepler
• Kepler was put to the task of
creating a model to fit all of
Tycho's planetary data
• Kepler spent the remainder
of his life formulating a set of
laws that explained the
motion of the planets
Kepler (1571 - 1630)
Kepler's First Law
• Kepler first noted that the orbital
path of a planet around the Sun is
an ellipse, not a perfect circle
• The Sun lies at one of the foci of
the ellipse
• The eccentricity of an ellipse is a
measure of how 'squished' from a
circle the shape is
• Most planets in the Solar System
are very close to a perfect circle
– Eccentricity, e ~ 0 for a circle
Focus
Focus
Kepler's 1st Law: The orbital
paths of the planets are elliptical
with the Sun at one focus.
Kepler's First Law
=closest to the Sun
=farthest from the Sun
Kepler's Second Law
• Kepler also noticed that the
planets sweep out equal
areas in their orbit over
equal times
• Notice that this means the
planet must speed up and
slow down at different points
• If it takes the same amount
of time to go through A as it
does C, at what point is it
moving faster?
– C, when it is closest to the
Sun
Kepler's 2nd Law: An imaginary line
connecting the Sun to any planet
sweeps out equal areas of the
ellipse over equal intervals of time.
Kepler's Third Law
• Finally, Kepler noticed that
the period of planet's orbit
squared is proportional to
the cube of its semi major
axis
Kepler's 3rd Law Simplified
P a
2
• This law allowed the orbits
of all the planets to be
calculated
• It also allowed for the
prediction of the location of
other possible planets
3
NOTE: In order to use the
equation as shown, you must be
talking about a planet in the Solar
System, P must be in years, and
a must be in A.U. !!!
Kepler's Third Law - Examples
• Suppose you found a new planet in the Solar
System with a semi major axis of 3.8 A.U.
P 2  a3
P 2  3.83  54.872
P  54.872
1
2
 54.872  7.41 years
• A planet with a semi major axis of 3.8 A.U.
would have an orbital period of 7.41 years
Kepler's Third Law - Examples
• Suppose you want to know the semi major
axis of a comet with a period of 25 years
a3  P2
a 3  252  625
a  625
1
3
 3 625  8.55 A.U.
• A planet with an orbital period of 25 years
would have a semi major axis of 8.55 A.U.
Isaac Newton
• Kepler's Laws were a revolution in
regards to understanding
planetary motion, but there was no
explanation why they worked
• That explanation would have to
wait until Isaac Newton formulated
his laws of motion and the concept
of gravity
• Newton's discoveries were
important because they applied to
actions on Earth and in space
• Besides motion and gravity,
Newton also developed calculus
Newton (1642-1727)
Some terms
• Force: the push or pull on an object that in some way
affects its motion
• Weight: the force which pulls you toward the center of the
Earth (or any other body)
• Inertia: the tendency of an object to keep moving at the
same speed and in the same direction
• Mass: basically, the amount of matter an object has
• The difference between speed and velocity
– These two words have become identical in common language, but in
physics, they mean two different things
– Speed is just magnitude of something moving (25 km/hr)
– Velocity is both the magnitude and direction of motion (35 km/hr to
the NE)
Newton's First Law
• Newton's first law states: An object at rest will remain at
rest, an object in uniform motion will stay in motion UNLESS acted upon by an outside force
Outside Force
• This is why you should always wear a seat belt!
Newton's Second Law
• Acceleration is created whenever there is a change in
velocity
– Remember, this can mean a change in magnitude AND/OR
direction
• Newton's Second Law states: When a force acts on a
body, the resulting acceleration is equal to the force
divided by the object's mass
F
a
m
or
F  ma
• Notice how this equation works:
– The bigger the force, the larger the acceleration
– The smaller the mass, the larger the acceleration
Newton's Third Law
• Newton's Third Law states:
For every action, there is an
equal and opposite reaction
• Simply put, if body A exerts
a force on body B, body B
will react with a force that is
equal in magnitude but
opposite direction
• This will be important in
astronomy in terms of
gravity
– The Sun pulls on the Earth
and the Earth pulls on the Sun
Newton and the Apple - Gravity
• After formulating his three
laws of motion, Newton
realized that there must be
some force governing the
motion of the planets around
the Sun
• Amazingly, Newton was able
to connect the motion of the
planets to motions here on
Earth through gravity
• Gravity is the attractive force
two objects place upon one
another
The Gravitational Force
Gm1m2
Fg 
r2
• G is the gravitational constant
– G = 6.67 x 10-11 N m2/kg2
• m1 and m2 are the masses of the two
bodies in question
• r is the distance between the two bodies
Gravity - Examples
• Weight is the force you feel due to the gravitational force
between your body and the Earth
– We can calculate this force since we know all the variables
Gm1m2
Fg 

2
r
(6.67 10
11
N m
24
)(
72
kg
)(
5
.
97

10
kg)
2
kg
6
2
(6.378 10 m)
2
Fg  705 N
1 Newton is approximately 0.22 pounds
0.22lbs
Fg  705 N 
 155lbs
1N
Gravity - Examples
• What if we do the same calculation for a person standing on
the Moon?
– All we have to do is replace the Earth's mass and radius with the
Moon's
Gm1m2
Fg 

2
r
(6.67 10
11
N m
22
)(
72
kg
)(
7
.
35

10
kg)
2
kg
6
2
(1.738 10 m)
2
Fg  117 N
1 Newton is approximately 0.22 pounds
0.22lbs
Fg  117 N 
 26lbs
1N
Gravity - Examples
• If gravity works on any two bodies in the universe, why don't
we all cling to each other?
– Replace the from previous examples with two people and the distance
with 5 meters
Gm1m2
Fg 

2
r
(6.67 10
11
N m
)(72kg)(65kg)
2
kg
2
(5m)
2
8
Fg  0.0000000125N  1.25 10 N
1 Newton is approximately 0.22 pounds
0.22lbs
Fg  1.25 10 N 
 2.75 10 9 lbs
1N
8
Revisions to Kepler's 1st Law
• Newton's law of gravity required
some slight modifications to
Kepler's laws
• Instead of a planet rotating around
the center of the Sun, it actually
rotates around the center of mass
of the two bodies
• Each body makes a small elliptical
orbit, but the Sun's orbit is much
much smaller than the Earth's
because it is so much more
massive
Revisions to Kepler's 3rd Law
• Gravity also requires a slight
modification to Kepler's 3rd
Law
3
a
P 
M1  M 2
2
• The sum of the masses of
the two bodies is now
included in the equation
• For this equation to work,
the masses must be in units
of solar mass (usually
written as M )

• Why did this equation work
before?
Remember - for this equation to work:
P must be in years!
a must be in A.U.
M1 and M2 must be in solar masses