Transcript Kepler`s Laws

Planetary
Motion
Kepler’s Laws apply to any celestial
body orbiting any other celestial body.
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•
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Any planet around a sun
The moon around the Earth
Any satellite around the Earth
The international space station
Any rings around any planet
KEPLER’S BREAKTHROUGH
• Kepler used Brahe’s data to develop three laws that
could be used to describe planetary motion.
• All of the laws are based upon an understanding of
the ellipse.
After Tycho Brahe’s death,
Johannes Kepler (pictured here
with Tycho in the background)
used Tycho’s observations to
deduce the three laws of
planetary motion.
KEPLER’S THREE LAWS OF PLANETARY MOTION
LAW #1. The orbit of a planet around the Sun is an ellipse with the
Sun at one focus.
The amount of elongation in a planet’s orbit is defined as its
orbital eccentricity. An orbital eccentricity of 0 is a perfect
circle while an eccentricity close to 1.0 is nearly a straight line.
In an elliptical orbit, the distance from a planet to the Sun
varies. The point in a planet’s orbit closest to the Sun is
called perihelion, and the point farthest from the Sun is
called aphelion.
Perihelion – location of planet at its closest
position to the star it is orbiting
Aphelion– location of planet at its furthest
position to the star it is orbiting
KEPLER’S THREE LAWS OF PLANETARY MOTION
LAW #2: A line joining the planet and the Sun sweeps out equal
areas in equal intervals of time.
Planet moves slower
in its orbit when
farther away from
the Sun.
Planet moves faster
in its orbit when
closer to the Sun.
KEPLER’S THREE LAWS OF PLANETARY MOTION
LAW #3: The square of a planet’s sidereal period around the Sun is
directly proportional to the cube of its semi-major axis.
This law relates the amount of time for the planet to complete one
orbit around the Sun to the planet’s average distance from the Sun.
If we measure the orbital periods (P) in years and distances (a) in
astronomical units, then the law mathematically can be written as P2
= a3 .
ORBITAL PERIODS FOR PLANETS
• Sidereal Period
• The true orbital period of a planet with respect to the
background stars.
• Synodic Period
• The period of time that elapses between two successive
identical configurations as seen from Earth (example: for
Venus, greatest eastern elongation to greatest eastern
elongation)
NEWTON’S PHYSICS—MOTION AND GRAVITY
• Newton’s Three Laws of Motion
• A body remains at rest or moves in a straight line at a
constant speed unless acted upon by an net outside
force. (Inertia)
• The acceleration of an object is proportional to the
force acting on it and dependent upon its mass. (F=ma)
• Whenever one body exerts a force on a second body,
the second body exerts and equal and opposite force
on the first body. (for every action there is a opposite
and equal reaction)
• Newton’s Universal Law of Gravitation:
Fgravity = G x m1m2
r2
F = gravitational force between two objects
m1 = mass of first object
m2 = mass of second object
r = distance between objects or d
G = universal constant of gravitation
• If the masses are measured in kilograms and the distance
between them in meters, then the force is measured in
Newtons
• Laboratory experiments have yielded a value for G of
G = 6.67 × 10–11 Newton • m2/kg2
JOHANNES KEPLER
THE SOLAR SYSTEM
LAWS OF PLANETARY MOTION
Austrian mathematician
Johannes Kepler (1571-1630),
interested in how the planets
move around the sun, went to
Tyco’s island to get these
accurate measurements.
At that time, many astronomers
believed that planets orbited around
the sun in perfect circles, but Tyco’s
accurate measurements for Mars
didn’t fit a circle.
Instead, the mathematician Johannes
Kepler found that the orbit of Mars fit
an ellipse the best…
What is an ellipse?
2 foci
An ellipse is a geometric shape
with 2 foci instead of 1 central
focus, as in a circle. The sun is
at one focus with nothing at
the other focus.
FIRST LAW OF PLANETARY MOTION
(Law of Ellipses)
An ellipse also has…
…a major axis
Perihelion
…and a minor axis
Aphelion
Semi-major axis
Perihelion: When Mars or any another planet
is closest to the sun.
Aphelion: When Mars or any other planet is
farthest from the sun.
Kepler also found that Mars changed
speed as it orbited around the sun:
faster when closer to the sun, slower
when farther from the sun…
But, areas A and B,
swept out by a line
A
B
from the sun to
Mars, were equal
over the same
amount of time.
SECOND LAW OF PLANETARY
MOTION
Kepler found a
relationship between the
time it took a planet to
go completely around
the sun (T, sidereal
year), and the average
distance from the sun
(R, semi-major axis)…
T1
R1
T2
R2
T1 2
T2 2
=
R1 3
R2 3
T2=TxT
( R3 = R x R x R )
THIRD LAW OF PLANETARY MOTION
R2
T2 Earth’s sidereal year (T)
and distance (R) both
equal 1. The average
distance from the Earth
to the sun (R) is called 1
astronomical unit (AU).
Kepler’s Third Law, then, changes to
T1 2
R1 3
T1 2
R1 3
2 = R 3
or
T
or
=
=
1
1
2
3
T2
R2
1
1
When we compare the orbits
of the planets…
Planet T(yrs) R(au)
T2
R3
Venus
0.62
0.72
0.38 0.37
Earth
1.00
1.00
1.00 1.00
Mars
1.88
1.52
3.53 3.51
Jupiter 11.86
5.20
141
141
We find that T2 and R3 are essentially equal.
Determining the Distances to Astronomical
Objects
Parallax
• Parallax view: the variation in angle that occurs when viewing a
nearby object from different places.
• Importance of parallax: Danish astronomer Tycho Brahe
reasoned that the distance of the object may be determined
by measuring the amount of parallax. A smaller parallax
angle meant the object was further away.
The apparent change in
the location of an object
due to the difference in
location of the observer
is called parallax.
Their views
differ because
of a change in
position
relative to the
mountain
Parallax – apparent difference in position of object
viewed from two different locations
Because the parallax of the “star” was too small to
measure, Tycho knew that it had to be among the other
stars, thus disproving the ancient belief that the “heavens”
were fixed and unchangeable.
http://www.astronomy.ohiostate.edu/~pogge/Ast162/Movies/parallax.gif
http://instruct1.cit.cornell.edu/courses/astro101/
java/parallax/parallax.html
Limitation to using parallax
• Eventually, the parallax shift will no longer be measurable.
• This is because the distance is too great for the effect to be observed.