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Kepler’s Laws

Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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Page 1: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

Kepler’s Laws

Page 2: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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

Page 3: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

Kepler’s First Law

Page 4: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

axis major of length

points focus between distancetyeccentrici

The Law of Ellipses: The planets move in elliptical orbits with the Sun at one focus.

Page 5: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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

Page 6: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

perihelion

aphelion

Center

Verifying Kepler’s 1st

P1L1

L2

P2

L3

L4

L1 + L2= L3 + L4 ??

Page 7: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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.

Page 8: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic
Page 9: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

Lunar Orbit of Explorer

35

Moon

Points represent satellite positions separated by equal time intervals.

periluna

apoluna

Page 10: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

Verifying Kepler’s 2nd

Equal area in equal time.

Area = ½ base X height

A2

A1 = A2 ??

A1

base

height

Page 11: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic
Page 12: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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

Page 13: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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

Page 14: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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

Page 15: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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.

Page 16: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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)

Page 17: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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.

Page 18: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

Newton’s Universal Law of Gravitation

G=6.67 x 10-11 m3/(kg s2)

Page 19: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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.

Page 20: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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.

Page 21: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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.

Page 22: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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.

Page 23: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

Derivations

Escape velocityOrbiting velocity

“g”Kepler’s “C”

Page 24: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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?

Page 25: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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.

Page 26: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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

Page 27: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

Gravitational Potential Energy

•Gravitational PE is negative.•PE increases as r decreases.

Potential energy vs. separation distance

Page 28: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

Sample problem What is the minimum escape

speed from Earth? KEat Earth’s surface = PEouter space

½ mvesc2 = GmM/r

vesc =

r

GM2

Page 29: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

Sample problem. Calculate the total energy of a

satellite in circular orbit about Earth with a separation distance of r?

Page 30: Kepler’s Laws. Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic

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?