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Kepler’s Laws
Kepler and thePhysics of Planetary Motion
Kepler and thePhysics of Planetary Motion Laws of Planetary Motion
Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic Law (r3/T2 = C)
Kepler’s laws provide a concise and simple description of the motions of the planets
Kepler’s First Law
axis major of length
points focus between distancetyeccentrici
The Law of Ellipses: The planets move in elliptical orbits with the Sun at one focus.
Major
Axis
Minor
Axis
aphelion
perihelion
Focus PointsCente
r
90°
axismajor of length
points focus between distancetyeccentrici
e=0 perfect circle
e=1 flat line
Semi-major Axis = ½ Major Axis
The Ellipse
perihelion
aphelion
Center
Verifying Kepler’s 1st
P1L1
L2
P2
L3
L4
L1 + L2= L3 + L4 ??
Kepler’s Second Law
As a planet orbits the Sun, it moves in such a way that a line drawn from the Sun to the planet sweeps out equal areas in equal time intervals.
Lunar Orbit of Explorer
35
Moon
Points represent satellite positions separated by equal time intervals.
periluna
apoluna
Verifying Kepler’s 2nd
Equal area in equal time.
Area = ½ base X height
A2
A1 = A2 ??
A1
base
height
Kepler’s Third Law
The ratio of the average distance* from the Sun cubed to the period squared is the same constant value for all planets.
* Semimajor axisr3 = CT2
Summarizing Kepler’s Laws
Kepler's Second Law: Line joining planet and the Sun sweeps out equal areas in equal times
Kepler's First Law:Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus.
Kepler's Third Law: The squares of the periods of the planets are proportional to the cubes of their semi-major axes or:
r3 = CT2
5. Universal Laws of Motion
“If I have seen farther than others, it is because I have stood on the shoulders of giants.”
Sir Isaac Newton (1642 – 1727)Physicist
Newton’s Universal Law of Gravitation
Isaac Newton discovered that it is gravity that plays the vital role of determining the motion of the planets - concept of action at a distance.
Gravity is the force that results in centripetal acceleration of the planets.
Orbital Paths Extending Kepler’s
Law #1, Newton found that ellipses were not the only orbital paths.
possible orbital paths ellipse (bound) parabola (unbound) hyperbola (unbound)
Newton’s Universal Law of Gravitation
Between every two objects there is an attractive force, the magnitude of which is directly proportional to the mass of each object and inversely proportional to the square of the distance between the centers of the objects.
Newton’s Universal Law of Gravitation
G=6.67 x 10-11 m3/(kg s2)
Newton’s Version of Kepler’s Third Law
Using calculus, Newton was able to derive Kepler’s Third Law from his own Law of Gravity.
In its most general form:
T2 = 42 r3 / G M If you can measure the orbital period of two
objects (T) and the distance between them (r), then you can calculate the mass of the central object, M.
What have we learned?
• What is the universal law of gravitation?• The force of gravity is directly proportional to the
product of the objects’ masses and declines with the square of the distance between their centers (Inverse Square Law).
• What types of orbits are possible according to the law of gravitation?
• Objects may follow bound orbits in the shape of ellipses (or circles) and unbound orbits in the shape of parabolas or hyperbolas.
What have we learned?
• How can we determine the mass of distant objects?• Newton’s version of Kepler’s third law
allows us to calculate the mass of a distant object if it is orbited by another object, and we can measure the orbital distance and period.
Combining Newton’s and Kepler’s Laws, we can . . . . Determine the mass of an unknown
planet. Determine the escape and orbiting
velocities for a satelite. Determine the acceleration due to
gravity on a planet. You should be able to derive equations
for the above determinations.
Derivations
Escape velocityOrbiting velocity
“g”Kepler’s “C”
G and g Geosynchronous satellites orbit the
Earth at an altitude of about 3.58 x 107 meters. Given that the Earth’s radius is 6.38 x 106 meters and its mass is 5.97 x 1024 kg, what is the magnitude of the gravitational acceleration at the altitude of one of these satellites?
Orbiting velocity The International Space Station
orbits the Earth at an average altitude of 362 kilometers. Assume that its orbit is circular, and calculate its orbital speed. The Earth’s mass is 5.97 x 1024 kg and its radius is 6.38 x 106 meters.
Gravity is a source of energy
Because gravity is a force, it can be associated with potential energy:Recall:
Solving, the formula for gravitational PE is:
The minus sign indicates that PE decreases as the masses get closer together.
dx
dUF
r
GmMU g
Gravitational Potential Energy
•Gravitational PE is negative.•PE increases as r decreases.
Potential energy vs. separation distance
Sample problem What is the minimum escape
speed from Earth? KEat Earth’s surface = PEouter space
½ mvesc2 = GmM/r
vesc =
r
GM2
Sample problem. Calculate the total energy of a
satellite in circular orbit about Earth with a separation distance of r?
The change in a system’s energy equals work. How much work is required to
move the satellite to an orbit with a separation distance of 2r?