Slide 1

Kepler’s Laws of Planetary Motion

1.The orbits of the planets are ellipses with the sun at one focus.

Eccentricity e = c/a

c

Slide 2

e = Ra - Rp

Ra + Rp

Parameters: perihelion Rp, aphelion Ra, semimajor axis a = (Rp+Ra)/2, eccentricity e

Elliptical orbits

12

2

2

2

b

y

a

x

Slide 3

LAW 3: The squares of the periods of the planets are proportional to the cubes of their semimajor axes:

For the Earth P2 = 1 yr, a2 = 1 AU

)()( 31

21 AUayrP Note units!!

32

31

22

21

a

a

P

P

Slide 4

A New Era of Science

Mathematical foundation for physics

Slide 5

UNIQUE INFLUENCE

1. He was one of the most creative geniuses the world has ever seen and to many people the greatest scientist who ever lived.

2. While Kepler’s and Galileo's discoveries brought humankind to the brink of a new age, Newton took it the rest of the way.

3. He unified the work of Copernicus, Galileo, and Kepler into one scientific theory that has stood the test of time.

4. Principia Mathematica is still considered by many to be the greatest scientific book ever written. It is the fundamental work for all of modern science.

5. Newton was the integrator, the unifier, the organizer, of all the scientific knowledge available at the time. He established a solid platform on which all modern science could be built.

Slide 6

SOME ACCOMPLISHMENTS

1. Newton formulated the laws of classical mechanics.

2. He discovered the law of gravitation.

3. He discovered the origin of color.

4. He invented calculus.

4. He invented the first reflecting telescope.

5. He wrote and published the book Mathematica Principia, which provided a detailed explanation of the laws of gravity and motion, particularly as they applied to astronomy.

Slide 7

Newton’s Laws of Motion (1)

1. A body continues at rest or in uniform motion in a straight line unless acted upon by some net force.

An astronaut floating in space will continue to float forever in a straight line unless some external force is accelerating him/her.

Slide 8

Velocity and Acceleration

Acceleration (a) is the change of a body’s velocity (v) with time (t):

1. Acceleration in the conventional sense (i.e. increasing speed)

a = v/t

Different cases of acceleration:

Velocity and acceleration are directed quantities (vectors)!

3. Change of the direction of motion (e.g., in circular motion)

2. Deceleration (i.e. decreasing speed)

a

v

Slide 9

Newton’s Laws of Motion (2)

2. The acceleration a of a body is inversely proportional to its mass m, directly proportional to the net force F, and in the same direction as the net force.

a = F/m F = m a

Slide 10

Slide 11 p. 63

Curved path there must be some force!

Slide 12

m1m2

0221

21 rr

mGmFF

1F

2F

1|| 0 r

Gravity: by far the most important force in the Universe

kgs

m1067.6 2

311G

Slide 13

Newton’s Laws of Motion (3)

3. To every action, there is an equal and opposite reaction.

The same force that is accelerating the boy forward, is accelerating the skateboard backward.

Slide 14

Why Earth does not crash into the Sun?

No other force but gravity!

Why Moon does not crash into Earth?

Understanding orbital motion

Slide 15

Slide 16

Understanding Orbital MotionThe universal law of gravity allows us to understand orbital motion of planets and moons:

• Earth and moon attract each other through gravitation.

Example:

Earth

Moon

v v’

v

F

• Since Earth is much more massive than the moon, the moon’s effect on Earth is small.

• Earth’s gravitational force constantly accelerates the moon towards Earth.

• This acceleration is constantly changing the moon’s direction of motion, holding it on its almost circular orbit.

Slide 17

Center of Mass

(SLIDESHOW MODE ONLY)

Slide 18

m1m2

0221

21 rr

mGmFF

1F

2F

0r

1

2

2

1

d

d

m

m

If m1 << m2, then d2 << d1

Slide 19

Uniform circular motion

m

M

r

v

a

F

Fam

2

2

2

311

2 kg

mNor

kgs

m1067.6;|| G

r

GMmF

22;

r

GMa

r

GMmma

On the Earth’s surface

r = R = 6400 km; M = 6x1024 kg;

22

/8.9 smgR

GMa

Slide 20

m

M

r

v

a

F

rP

rvvPr

22

Uniform circular motion - continued

locityangular ve2

P

r

vrva

22

velocityorbital2

2

r

GMv

r

GM

r

v

GM

r

v

rP

v

rP

32

2

222 44

;2

III Kepler’s law:

Slide 21

Orbital Motion (2)In order to stay on a closed orbit, an object has to be within a certain range of velocities:

Too slow => Object falls back down to Earth

Too fast => Object escapes Earth’s gravity

Slide 22

Newton’s Cannon

(SLIDESHOW MODE ONLY)

Slide 23

Orbital Motion (3)

Geosynchronous Orbits

Slide 24

Geosynchronous Orbit

(SLIDESHOW MODE ONLY)

Slide 25

Escape condition: Kinetic Energy K Gravitational Potential Energy U

R

GMmUmvK ;2/2

At threshold:

orbesc vR

GMv

R

GMmU

mvK 2

2

2

2

Note: total energy E = K + U; E < 0 for bound orbitsE 0 for unbound trajectories

Slide 26

Object Mass Escape velocity

Ceres (largest asteroid)

1021 kg 0.64 km/s

The Moon 7x1022 kg 2.38 km/s

The Earth 6x1024 kg 11.2 km/s

Jupiter 2x1027 kg 60 km/s

The Sun 2x1030 kg 618 km/s

Slide 27

Celestial mechanics - summary

It all started with Galileo!

Slide 28

Clockwork universe