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Kepler, Newton and Gravity 1 Intro to Stars

Kepler, Newton and

Gravitation


Using the unit of distance

1 AU = Earth-Sun distance

PLANETS COPERNICUS MODERN

Mercury 0.38 0.387

Venus 0.72 0.723

Earth 1.00 1.00

Mars 1.52 1.52

Jupiter 5.22 5.20

Saturn 9.17 9.54


Johannes Kepler - 1600 A.D.

• built on successes and failures

• demanded that his model’s predictions be

at least as accurate as the observations (1

arcmin error)

foundation for modern cosmological concepts

“I have discovered among the celestial movements

the full nature of harmony.”


Kepler - the Ellipse


Kepler - the Ellipse two focal points -

foci

focal distance



minor axis

semi-minor axis - half the minor axis



major axis

minor axis

semi-major axis - half the major axis (a)



Kepler - the Ellipse two focal points -

foci major axis

minor axis

focal distance

semi-major axis - half the major axis (a)


eccentricity = focal distance

major axis



major axis

focal distance

eccentricity = focal distance

major axis

e = 0 means ????

e = 1 means ????


Kepler’s Three Laws

I. Law of Ellipses

Each planet’s orbit is an ellipse with

the Sun at one of the foci.

implication: distance of the planet to

the Sun varies



II. Law of Equal Areas

A line drawn from a planet to the Sun

sweeps out equal areas in equal times.

implication: orbital speeds are non-uniform

yet vary in a regular way

Closer a planet is the the Sun, the faster

it moves in its orbit

(force ????)



III. Harmonic Law

Planet p (Earth years) a (AU)

============================

Mercury 0.24 0.39

Venus 0.62 0.72

Earth 1.0 1.0

Mars 1.9 1.5

Jupiter 12 5.2

Saturn 29 9.5



III. Harmonic Law

Square of the orbital period is proportional

to the cube of the average distance.

p2 = k a3

implication: planets with large orbits

move slowly

Proportion holds for all planets =>

A PHYSICAL CAUSE !!


Kepler’s Third Law

p2 = k a3

k = proportionality constant

p (years)

a (AU’s)

=> k = 1

True for any body orbiting the Sun, even spacecraft!

“I contemplate its beauty with incredible

and ravishing delight.


Kepler

made perhaps the greatest leap in scientific thinking

predictions were 10 times more accurate than either

Ptolemaic model (geocentric)

Copernican model (heliocentric)

gave birth to astronomy as a physical science


Orbital Eccentricities

Planet Orbital Eccentricity

Mercury 0.206

Venus 0.007

Earth 0.017

Mars 0.094

Jupiter 0.048

Saturn 0.054

Uranus 0.048

Neptune 0.007

Pluto 0.253


GALILEO, NEWTON,

and GRAVITY


Mechanics: the study of falling bodies

speed: how fast you are going

velocity: how fast you are going

in a specific direction


Albany to Portland:

speed = 100 km/hr

velocity =

Portland to Albany:

speed = 80 km/hr

velocity =

?????? 100 km/hr north

?????? 80 km/hr south


• change in speed

• change in direction

• change in both speed and direction

velocity: speed and direction

acceleration: CHANGE in velocity


• You are stopped at a red light.

You step on the gas.

Your speed changes from 0 to 15 km/hr.

Are you accelerating?

speed: measured in km/hr

velocity: measured in km/hr in a direction

acceleration: measured in km/hr

unit of time

Three Cases

Yes, it is your speed that is changing.


• You are driving in a circle around a racetrack

at a constant 20 km/hr.


Yes. Your direction is changing.

• You are braking from 25 km/hr to 15 km/hr.


Yes. Your speed is changing.


Galileo and Motion

downward motion:

gravity pulling downward

forced motion

attractive force

horizontal motion:

would continue forever if no forces acted

natural motion


Galileo’s Experiment

inclined plane, perfectly smooth

perfectly smooth ball


Galileo

inertia: natural tendency for a body in motion

to remain in motion or natural tendency

for a body at rest to remain at rest


acceleration due to gravity

• constant

• velocity changes at a constant rate

• at earth’s surface g = 9.8 m/s

Galileo also experimented with falling bodies

falling is NOT a natural motion

motion due to the force of gravity

s

We can use 10 m/s/s


DROPPED ROCK

How fast is it going after

5 seconds of falling?

10 m/s

30 m/s

Starts at

0 m/s


1 second of falling?



50 m/s


PARACHUTIST

How fast is she going after


10 m/s

30 m/s

Starts at

0 m/s


1 second of falling?



50 m/s

Ignore air resistance


ALL falling bodies

fall with the

SAME acceleration !!


Newton 1600 A.D.

Principia - Physics of motion

and Concept of Gravitation

Concept of Force

force produces an acceleration

acceleration is in the same direction

as the force


Increase the force

=> greater acceleration

=> object reaches a greater velocity

What kind of relationship is this?

ACCELERATION

is directly proportional to

FORCE


Increase the MASS

apply the original force

=> less acceleration

What kind of relationship is this?

ACCELERATION

is inversely proportional to

MASS


Can you come up with a Pot O’ Gold type

relationship for acceleration, force & mass?

a = F

m

Forces have direction (same direction

as the acceleration)


Newton clarified these definitions:

• mass : the measure of an object’s resistance

to a change in motion

• velocity : how fast an object moves in a

particular direction

• acceleration : how much the velocity (or

direction) changes with time


Newton’s Three Laws of Motion

• INERTIAL LAW

A body at rest remains at rest unless

acted on by an outside force.

A body in motion at a constant velocity

along a straight line remains in motion

unless acted on by an outside force.



INERTIAL LAW

Implication:

If we see an acceleration, we know

there’s a net force acting on the

body in question (that is, change

in speed direction, or both)



• FORCE LAW

rate of change in a body’s velocity,

due to an applied force (in other words,

a body’s acceleration) is

– in the same direction as the force

– proportional to the force

– inversely proportional to its mass



FORCE LAW

Implication: can apply a force, measure

acceleration and infer the mass

of an object



• REACTION LAW

For every applied force, a force of

equal size by opposite direction arises.

Implication: forces act in pairs


AN EXAMPLE

You are an astronaut out in space.

You push on your space capsule.

What is the equal and opposite force?

Since the forces are the SAME,

exactly what is different?

Spaceship pushes back on you.


You Spaceship

F F


You Spaceship

F F

m M


You Spaceship

F F

m M

A a


You Spaceship

F F

m M

A a

Which has the greater velocity?


NEWTON’S GREAT INSIGHT

Gravitation is an interaction

between two (or more)

bodies, such as the Sun and

the planets.


Newton’s Law of Gravitation

• What direction does the force

of gravity act? (nature of the force)

• What is the amount of force?

(physical properties that determine

the strength of the force)


LAWS OF MOTION

+

Kepler’s Planetary Laws

Law of Universal Gravitation


GRAVITATION

• central force : type of force that causes

elliptical orbits; force directed towards

the center of motion

• planets moving in orbits under the influence

of a central force followed Kepler’s

Second Law (Law of Areas)

• from the geometric properties of ellipses,

force is described by a specific type of

force law => re-derived Kepler’s Third

Law using this force law

Newton’s Laws and Kepler’s Laws were in

total agreement


• Direction - changing? or not?

Consider the moon in orbit:

curved path => changing

Newton’s First Law:

direction is changing

there is an acceleration

Therefore, there is a force acting

on the moon


Force is directed towards the center

of the Earth. We call a center-directed

force a centripetal force.

At every point in its orbit, a centripetal

force acts on the moon to keep it

bound to Earth.


Billiard Ball Analogy

• Ball makes a

collision with

side

• Ball changes

direction

• Ball accelerates

from applied

force of the

wall

Force is

directed towards

the center of the

table


Pentagonal

Table

Force still

points to

the center Angle of strike and rebound

is smaller


Hexagonal

Table

Force still

points to

the center Angle of strike and rebound

is even smaller


Circular

Table

Infinite number

of sides, angle

of strike and

rebound is

zero

Force points to

center at every

point


Moon in its ‘circular’ orbit: at every

point in its orbit, a centripetal force

acts on the moon to keep it bound to

Earth.

GRAVITY !!


Law of Gravitation

Every body in the Universe attracts

every other body with a gravitational

force.


Consider two objects only

• amount of gravitational force depends DIRECTLY

on the amount of material each object has (mass)

What would happen if you doubled the mass

of one of the objects?


The force between them was F.

After the pink object doubles, the gravitational

force is 2 times F or 2F

TWICE AS MUCH


The force between them was 2F

After the purple object doubles, the gravitational

force is 2 times 2F or 4F

4 TIMES the original

amount of force

Now what happens

if we double the mass

of the other object?


To express that in algebra:

Fg is directly proportional to

mpurple x masspink

or

Fg m1 x m2


• objects at greater separations have less gravitational

force between them than those closer together

Decrease of force with

distance happens in a

special way

Force is inversely

proportional to the

square of the distance.

1 m

When the distance

is 1 meter, the force

between them is F.


1 m

Start with the objects

1 meter apart.

What happens if we move

them to 2 meters apart? (that is,

we double the distance)

2 m


2 m

The force, when they were 1 meter apart, was

F - now at 2 meters, the force is

Less is it 1/2 as much?

is it 1/4 as much?

Force is proportional to 1

(distance) 2 Force is as much as it was. 1 4


1 m 3 m

What happens to the force if we move

them 3 meters apart?

Force is then 1/9 as much as it was


F is proportional to m1

F is proportional to m2

Force is proportional to 1

(distance) 2

Force m1 m2

(distance) 2 or

m1 m2

R 2


Constant of proportionality is

G - Universal Gravitational Constant

G = 6.67 x 10-11

Force (newtons)

masses (kgs)

R (meters)

m1 m2

R 2

Force = G


Law of Gravitation

Fgravity = G m1m2

R2

Interpretation:

Force due to gravity is directly

proportional to the masses of

the objects involved.


Law of Gravitation

Fgravity = G m1m2

R2

Interpretation:

Force due to gravity is inversely

proportional to the distance

between them squared.

1 over

R2 Law


Newton worked this out mathematically

(it’s a model) - how did he test this?

1 Re

60 Re Moon is 60 times

farther from the center

of the Earth than the

apple is.

How much less gravitational

force is felt at the location of

the Moon ?


Force at Moon is 1

(60)2

= 1

3600

Newton’s mathematical model predicted

the force (acceleration) should be

3600 times less.

Comparing to measured observations of

the Moon in its orbit, this was

‘pretty nearly’ the same.


Newton’s Concept Extension

• Earth’s gravity keeps the moon

swinging around the Earth

• Sun’s gravity keeps the planets

swinging around the Sun


Newton Revised


p2 = k a3


Newton Revised


p2 = k a3

p2 = 4 2

G(Msun + mplanet)

a3


Kepler’s Third Law Revised

p2 = 4 2

G(Msun + mplanet)

a3

TREMENDOUSLY

IMPORTANT !!

Can use this to find the

mass of the Sun !!


p2 = 4 2

G(Msun + mplanet)

a3

• measure the period of the Earth’s orbit

• measure the average distance from the Sun

• approximate

Msun + mearth » Msun

• look up the value of G

• calculate the mass of the Sun !!


We can find the mass of the

Sun using any body orbiting

the Sun.

We can find the mass of the

Earth using any body orbiting

the Earth.


What would you do to find

the mass of a planet that

has no orbiting body?


Center of Mass

center of mass : balance point of a group of objects

Where’s the balance point?


1 AU (150 million km)

What’s your best

guess as to the location

of the center of mass

between the Earth

and the Sun ?

Earth

A mere 500 km

from the center of

the Sun !!!


5 AU (750 million km)

What’s your best

guess as to the location

of the center of mass

between Jupiter and

the Sun ?

Jupiter

Just barely

outside the

surface of the Sun


Newton’s Successful Predictions

• return of Halley’s comet

• discovery of planet Neptune

• binary star systems follow Kepler’s

Laws


Newton’s Accomplishments

• found the physical interaction between

the Sun and the planets

• revised Kepler’s Third Law so that it

became a tool for calculating masses of

distant objects

• answered the question of how planets move

• made predictions far more accurate than any

before

• physical support for the heliocentric model


Newton’s Law of Gravitation

Finally, a universal law, that is,

one that is the same for the

heavens and the Earth.


Escape Speed:

minimum speed an object

must have to break free of

gravity

11 km/sec escape speed from

Earth


Mass and radius of planet

determines the escape speed.

vescape = 2 Gm

R


Newton’s Cosmology

cosmology : study of the origin, the nature and

the evolution of the Universe

Universe is infinite in extent

If it were not infinite, but finite, gravitation would

eventually pull all the matter in the Universe back

together => one large mass !


In an infinite universe, there would be

an infinite number of small blobs of matter.

=> exactly the universe we see !


One Small Teeny, Tiny Problem

Orbit of Mercury had an unexplained ‘wobble’

41 arcsec per century

Could not be explained with Newton’s physics

Hypothesis of a planet, hidden behind the Sun,

never proved to be true (Vulcan)

kepler, newton and gravitationcf.linnbenton.edu/.../physci/rajabza/upload/6_gravity.pdfintro to...

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