The Motion of Planets

Birth of Modern AstronomyOR

How Nerds Changed the World!!!

• explain qualitatively Kepler’s first and �second laws and apply quantitatively Kepler’s third law

• explain and apply the law of universal gravitation to orbital notations by using appropriate numeric and graphic analysis �

• distinguish between scientific questions and technological problems as applied to orbital situations

Learning Outcomes (Students will be able to…):

Assumptions of Early Models of the Solar System (from the time of Aristotle…)

• Geocentric - Earth in the middle• Everything orbits the Earth• Stars are located on the Celestial Sphere• Everything moves in uniform circular

motions

Earth

Deferent

Epicycle

Mars

Equant

Claudius Ptolemy (87-165)

Nicolaus Copernicus (1473-1543)

•Errors building up

•Must be a better way!

•Let’s try a Heliocentric (or Sun-centered) system!

•Not any better though

Tycho Brahe (1546-1601)

•Comet – beyond the Moon

•Supernova – far away

•Naked eye observations of planets

•Accuracy through repetition

•Best observations of planetary positions

•Hired “nerd” to help calculate model

•Died….

Johannes Kepler (1571-1630)

•Worked for Brahe

•Took data after his death

•Spent years figuring out the motions of the planets

•Came up with…

Three Laws of Planetary Motion

1st Law: Planets move in elliptical orbits with the Sun at one foci

Sun

Foci (sing. Focus)

Perihelion Aphelion

Average distance from the Sun = 1 Astronomical Unit (1 A.U.) = approx. 150 000 000 km

2nd Law: Planets move faster at perihelion than at aphelion OR a planet sweeps out equal areas in equal time periods.

1 Month1 Month

3rd Law: Period is related to average distance

T = period of the orbit

r = average distance

T2 = k r3

•Longer orbits - greater average distance

•Need the value of k to use the formula

•k depends upon the situation

•Can be used for anything orbiting anything else

Special version of Kepler’s third Law –

If the object is orbiting the Sun

T – measured in years,

r – measured in A. U., then….

T2 = r3

For planets A and B, Kepler’s 3rd Law can look like this…

2

3

2

3

B

B

A

A

Tr

Tr

Galileo Galilei (1564-1642)

•Knew of Copernicus’s & Kepler’s work

•Used a telescope to look at the sky

•What did he see?

The Moon was an imperfect object

Venus has phases

Jupiter has objects around it

Saturn is imperfect

The Sun is imperfect

Isaac Newton (1642-1727)

•The ultimate “nerd”

•Able to explain Kepler’s laws

•Had to start with the basics -

The Three Laws of Motion

1. Law of Inertia - Objects do whatever they are currently doing unless something messes around with them.

2. Force defined

F = maF=forcem=massa=acceleration (change in motion)

3. For every action there is an equal and opposite reaction.

The three laws of motion form the basis for the most important law of all (astronomically speaking)

Newton’s Universal Law of Gravitation

221

RMGM

F

F=force of gravity

G=constant (6.67 x 10-11 Nm2/kg2)

M1, M2 = masses

R=distance from “centers”

Gravity is the most important force in the Universe

Newton’s Revisions to Kepler’s Laws of Planetary Motion:•Kepler’s 1st and 2nd Laws apply to all objects (not just planets)•3rd Law rewritten:

3

21

22

32

4 rMMG

T

krT

)(

became... was...

•4π2 and G are just constant #s (they don’t change)

•M1 and M2 are any two celestial bodies (could be a planet and Sun)

•Importance: if you know period and average distance of a planet, you can find mass of Sun (2 x 1030 kg) or any planet!

Mass of Sun is 2 000 000 000 000 000 000 000 000 000 000 kg

Mass of Earth is 6 000 000 000 000 000 000 000 000 kg

Mass of Mr. J is 100 kg! WOW!

An Inverse Square Law…

Another way to look at “g”…

rGMmr

rGMmEp

hrGMmmghEp

2

2

Another way to look at gravitational potential energy of an object… (h is height but since it is arbitrary, it can be chosen as the distance from the center of the Earth to the position of the object…or r)

Some important orbital applications…

Geosynchronous means having an orbit around the Earth with a period of 24 hours

Einstein viewed gravity and the motion of celestial objects, like planets, VERY differently…

(1875 – 1955)

Curved space-time effects both mass and light!