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Announcements. HOMEWORK assignment 2 due Feb 16 See Assignment contains two tutorials; eclipses, phases of the moon

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Announcements l.jpg

HOMEWORK assignment 2 due Feb 16


Assignment contains two tutorials; eclipses, phases of the moon

Things like unopened hint bonus, allowable attempts per answer all changes as the semester progresses, these are written on the first page of the homework assignment

First Mid-Term Exam will be Feb 19 (Mon)

Second Mid-Term Exam will be Mar 14 (Weds)

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Part 1: Early Cosmological Ideas



The Birth of "Scientific" Cosmology

Aristotle & Ptolemy

The "State of the Universe" for the Greeks

The "Dark" Millennium

The Recovery of W. European Science

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The word Cosmology is from the Greek kosmos (world)

and logia (from legein: to speak).

All civilizations (probably) developed some form of "cosmology". Indeed all civilizations (we know about) seem to have "Creation myths" of some sorts.

In the earliest civilizations (& up to very recent times) cosmology was primarily (often totally) a branch of religion/myth.

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Certainly many (all ?) ancient civilizations performed astronomical observations of various levels of sophistication - from the neolithic observatories (eg.Stonehenge), to written reports (eg. The Chinese, Egyptians, etc).

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..a little humility required ?

Many earlier beliefs & models may seem "silly" or "absurd" now.

Our beliefs and our scientific methods are "clearly far superior" - aren't they ?

However it worth reminding ourselves from the very start that we have not

figured everything out yet !! (dark matter, cosmic acceln, WIMPs, MACHOS etc)

Many variations on Mythological/Cosmological Ideas, but many with themes

not so different from our own:

AND we should not underestimate the struggles/difficulties early thinkers had attempting to

explain the universe within their various cultural, religious & technological environments….

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..a little humility required ?

  • Imagine life before telescopes…

    • Earth seems flat and motionless

    • Sun, Moon, planets, stars move in sky (East to West)

    • Strange things appear (comets, meteors)

The Greeks are generally credited with promoting the idea that the

universe was understandable using logic, and could be described

by mathematics.

This is a huge leap forward, and the basis of all science today

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Birth of "Scientific" Cosmology

By ~400BCE, generally thought that the universe worked & evolved

through "natural" processes that can be observed on Earth.

...Divine intervention is not required (at least in the "running" of the universe).

Consensus - there are no limits to what can be observed & understood

(again, at least not concerning the "running" of the universe),

theories could be postulated,

predictions made,

theories revised as necessary

Cosmology was really a branch of philosophy at the time,

but the Empirical Scientific Method was developing.

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Birth of "Scientific" Cosmology

TheGreeks understood that the Earth is a sphere:-

Observation of ships sailing over the horizon

Observation of Earths shadow on the moon during lunar eclipses

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Geocentric Spheres

Only five planets were known to the Early Greeks

(Mercury, Venus, Mars, Saturn, Jupiter).

Thus early Greek cosmologists believed they had to account for

8 celestial entities -the 5 planets, the Sun, the Moon, and the "Stars".

Cosmologies were all "naturally"


(centered on the Earth)

Cosmologies generally included

perfect spheres

(Sphere count: Pythagoras of Samos,(c.530BCE)8)

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Aristotle (c.350 BCE)

The physical universe was finite - beyond the outer

sphere of the stars was the (non-material) spiritual realm.

Reasoned that that this "universe" must be unique

(& have a single center), and have existed for

eternity (& in a steady-state).

Believed celestial bodies move in

perfect circles

Forced to increase the number of spheres due

to refined observations of planetary motion

(Sphere count: Aristotle,(c.350BCE)55)

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Hipparchus (c.125 BCE)

Constructed a catalog of close to 1000 stars.

Discovered precession (1 degree/century)

- the change of the position of the stars with time

(now known to be due to the precession of the Earth's axis).

In 134 BCE he discovered a new star (a nova), direct contradiction to the paradigm that the "heavens" were unchanging.

Refined distances to (& hence size of) the Moon (via Parallax)

...made the first step determining the scale of the "cosmos"

Suggested that the Sun appeared to be much larger than the Earth.

...some aesthetic concerns for a geocentric universe

… but these were generally ignored…

Parallax l.jpg

...the apparent change of position

of a (closer) object as measured

against the positions of more

distant object(s) due to the

movement of the observer.

A Parsec is defined as the distance of an object that exhibits

parallax of 1 arcsec

(Easy to remember since the word parsec is a

construction from parallax and arcsec)

1 parsec = 3.085678 x 1016 m = 3.26 light years

Larger parallax = smaller distance

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Hipparchus, Parallax & the Moon

Hipparchus of Rhodes estimated the distance to the Moon from

measurements taken during a solar eclipse in189BCE.

eclipse was

"full" in Hellespont

(NW. Turkey),

partial in Alexandria


20% of the Sun's disk remained visible in Alexandria

Since 20% of the Sun's disk corresponds to 6 arcmins,

then by estimating the distance between the cities

one can derive the distance of the Moon.

Hipparchus estimated distance (4.5 to 5.2) x 108 m

(c.f. modern value of 3.8 x 108 m)

First attempt to scale the cosmos

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Ptolemy (c.150)

His (13 volume) master work

Megale Syntaxis("Great Compilation")

Is usually known as by the arab translation

Almagest("The Greatest")

Extended the system of Aristotle,

sticking "religiously" to the ideas of

a geocentric cosmology,

the perfection of spheres

a finite universe.

(There is an on-going debate whether he stole/plagiarized Hipparachus' data !)

[e.g. see Schaefer, Sky & Telescope 2002 Feb issue, p39]

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Ptolemy & Epicycles

The key elements are

the epicycle to account for the retrograde

motion of the planets

the deferent (main circle) to

account for the brightening &

speeding-up of the planets at

some times.

relative tilts between the various planes,

Large number of parameters

(for 8 celestial objects)

The “equant” is the point from which ang. vel

of epicycle ~const)

(Bothun, Fig1.2)

The model was able to make accurate predictions

Remained the "standard" cosmological model

for 1400 years

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Math as a Parameterization

Contrary to the Greek ideas that Nature is

simple, perfect, beautiful

In the middle ages some thinkers started suggesting that

the Ptolemaic system of eccentrics/epicycles

do not actually exist,

...but are merely convenient mathematical descriptions

of celestial motion & reality (not reality itself)

Note: The concept of Nature being simple, perfect, beautiful

can be argued to be back with some modern theories (!)

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The Ptolemy Monopoly

Alan W. Hirshfeld, in

"Parallax - The Race to Measure the Cosmos"

“In a sense, Ptolemy was the Bill Gates of his day.

His Ptolemaic "operating system", despite its known

deficiencies, grew to dominate - in fact, monopolize -

the astronomical market place."

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Rise of the Scientific Method

The modern Scientific Method

observations, theory, predictions, tests/revisions...

Roger Bacon (c.1250) helps popularize the scientific method

in W.Europe,

William of Ockham (or Occam; c.1300) suggests entities should not be

“multiplied unneccessarily”, leading to (interpreted as)

Ockham's Razor:

the simplest (most succinct) theory is more likely

to be correct, and certainly a better working model

(to attempt) to disprove first.

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Other Early Cosmologies (?)

It should be remembered that our knowledge of history

is solely dependent us on having written records

Also few (if any) of the original works survive,

so we must rely on later works (true & complete reporting ?)

Who knows what ideas have been lost...

One (radical) idea that was not developed (apparently ignored)

is due to Aristarchus (c280BCE - between Aristotle & Ptolemy)

a Heliocentric universe - the Earth orbiting the Sun (!)

Recap l.jpg

The following should be remembered:

Cosmology one of the oldest philosophies/sciences

Many ancient cosmologies grappled with some of the same deep

philosophical questions we still ponder with today.

The Greeks first (we think)

reasoned that Universe was formed by natural processes

which could be observed, understood/explained by mathematics

Developed the Empirical Scientific Method

Developed a geocentric system (Pythagoras of Samos, c.550BCE;

Aristotle,c.350BCE) culminating with that of Ptolemy,c.150

involving a complex arrangement of spheres & epicycles.

Reason and beauty/perfection were a strong influence of their


The universe was reasoned to be finite but eternal/unchanging

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Greeks thought Earth was stationary, if it were moving, wouldn’t we

feel a sense of motion (great winds, loose objects whizzing by us etc)

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Recap (cont)

Mathematics (ie. the Ptolemaic system) seen as a parameterization,

By c.1400, the Ptolemaic (geocentric) system had remained

essentially unchallenged as the cosmology for 1300 years

You should be familiar with

the concept of Parallax

the basics of how the Ptolemaic system works

(how epicycles, deferent etc account for retrograde


the concept of Ockham's Razor

Again, a detailed knowledge of names, dates and places is not required

However, you should be familiar with at least the names & approximate

dates of Aristotle (c.350BCE) and Ptolemy (c.150).

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Foundatn of Modern Cosmo


The Earth moves from Center Stage

And Then the Apple Dropped…

Summary at the beginning of the C20th

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State of the Universe, 1400

By 1400, the geocentric cosmology of Aristotle & Ptolemy

(based on concentric spheres, epicycles etc) had been essentially unchallenged for well over a thousand years.

However, in the 15th & 16th centuries, following the years of the

"Black Death" & centuries of strife, the start of the Renaissance in W.Europe finally allowed scientific & technological progress.

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Rumblings of Discontent

In c.1430, Nicholas de Cusa published

On Learned Ignorance

In which he suggested

the universe is infinite

(the universe does not have a center,

the pattern of stars would look the

same at all locations.

all motion is relative,

& that the Earth might not be stationary

Homogeneity & Relativity

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Earth moves from Center Stage...

The suggestion by Nicholas de Cusa (c.1430) that

the Earth might not be stationary,

was supported by Leonardo da Vinci (c1490),

who amongst many (!) other things also suggested

the Earth moves (rather than the Sun).

However it was not until 1543 when

Nicholas Copernicus publishes his

Revolutionibus Orbium Coelestium

(The Revolution of the Celestial Spheres)

that this idea was put of a more rigorous footing.

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Heliocentric Cosmology

Copernicus suggested

the planets rotate (on circles)

around a central Sun

….with "slower" planets being

further from the Sun.

Heliocentric Cosmology

Copernicus also acknowledged

the Earth rotates on its axis

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A “Good” (Simpler) Model

The heliocentric model of Copernicus obviously could be used

to make predictions, that could be compared to observations.

It was simpler than the model of Ptolemy that it replaced.


it’s predictions were not any better than those of Ptolemy’s model

unless (much smaller) epicycles were added

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The Cosmological Principles

One important aspect of Copernicus’ work - he took his heliocentric model, went further and made a model for the cosmos by saying, lets assume several things, then use observations to test whether this is a good model

Cosmological principles are the assumptions which allow us to deduce the whole of nature on the basis of the observable to the unobservable. Not surprisingly, any study of cosmological principles must combine elements of astronomy, physics and philosophy.

The Copernican Cosmological Principle

This is sometimes simply referred to as simply

“The Cosmological Principle”

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The Copernican Cosmo Principle

The Copernican Cosmological Principle is a logical

extension of the the Copernican theory that the Earth is not the center of the universe.

Thus the Earth is not "special", thus the "laws of nature" on (or around) Earth are not special.

It is essentially a philosophical requirement/simplification necessary/assumed for all modern cosmologies:

- our laws of physics are otherwise "irrelevant"

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The CCP itself

The Copernican Cosmological Principle is that

On a large scale, the universe is both

homogeneous and isotropic(in 3-D space),

and has/will always be so.

Note that the statement "has/will always be so" refers to the universe continuing to display the properties of homogeneity & isotropy.

The CCP does not imply that any actual observable parameter (e.g. the density of matter in the universe) will remain constant with time. Indeed, the CCP allows the properties of the universe to evolve with time, but states that at any given time the universe will be both homogeneous and isotropic (in 3-D space).

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The CCP again

Another way of expressing

the Copernican Cosmological Principle is that ...


will see identical properties & laws everywhere- homogeneity

will NOT see any preferred direction - isotropy

i.e. this was the suggestion that we do not occupy a special place in the universe

Homogeneity isotropy l.jpg

homogeneous - same properties everywhere

isotropic - no special direction, uniform in all directions

homogeneous but not isotropic

isotropic but not homogeneous

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The CCP - an analogy

A (small) sentient being living in the center of a "perfect" loaf of bread…!

There may be obvious structure on small scales (air bubbles etc), but on the large scale the loaf can be considered

uniform and isotropic

The laws of physics (e.g. which caused the dough to rise)

are the same throughout the loaf.

The loaf might still be rising - but (in this perfect loaf) this happens

uniformly & following then same lawsthroughout the loaf

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The CCP Evidence for & against

The best support for the Copernican Cosmological Principle is the Cosmic Microwave Background (CMB), which is isotropic to 1 part in 105

The obvious observational evidence against the Copernican Cosmological Principle seems to be the structure seen in the universe on a variety of scales

(stars, galaxies, clusters, super-clusters..the cosmic web)

This is why the qualifier "On a large scale.." is required to be added to the principle.

The question them becomes a question of scale (now large is "large" ?), and whether the observed structures on large scales are indeed representative of the universe on these scales (or are "perturbations" which "happen” to be visible to us).

Recap36 l.jpg

We are up to 1400

We briefly mentioned the ideas of Nicholas de Cusa

Homogeneity & Relativity

We discussed the Heliocentric system of Nicholas Copernicus

We then discussed the Copernican Cosmological Principle

On a large scale, the universe is both

homogeneous and isotropic (in 3-D space)

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What have we learned?

  • How can we distinguish science from non-science?

    • It’s not always easy, but science generally exhibits at least three hallmarks.

    • (1) Modern science seeks explanations for observed phenomena that rely solely on natural causes.

    • (2) Science progresses through the creation and testing of models of nature that explain the observations as simply as possible (Ockhams Razor)

    • (3) A scientific model must make testable predictions about natural phenomena that would force us to revise or abandon the model if the predictions do not agree with observations.

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What have we learned?

  • What is a theory in science?

    • A model that explains a wide variety of observations in terms of just a few general principles, which has survived numerous tests to verify its predictions and explanations.

  • How were astronomy and astrology related in the past, and are they still related today?

    • Astronomy and astrology both grew out of ancient observations of the sky. Astronomy grew into a modern science. Astrology has never passed scientific tests and does not qualify as science (ALTHOUGH EARLY ON IT HELPED US DEVELOP ASTRONOMY).

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What else have we learned?

  • Copernicus created a Sun-centered model of the solar system designed to replace the Ptolemaic model, but was more fundamentally correct yet it was no more accurate because he still used perfect circles.

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Tycho Brahe (c.1570))

Also famous for having lost his nose in a swordfight

Tycho Brahe (c.1570)

did accept that the (other) Planets

move around the Sun

but did not accept that the Earth & Stars

move around the Sun

Why ?

… Falling bodies fall towards the Earth

... The lack of Stellar Parallax

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What was his problem ?)

Falling Bodies fall towards the Earth

Indeed if you throw something vertically upwards,

it falls vertically downwards (to the same spot)

Tycho Brahe reasoned this surely meant

the Earth was the center of the universe

Tycho Brahe was unable to detect (by naked-eye) Stellar Parallax

and reasoned that in a Copernican system this would require

the Stars to be so far away they would have to be

"unreasonably" large/bright.

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Tycho Brahe the observer

Tycho Brahe was primarily an observer

a Supernova

position did not change (so it was not a comet or meteor),

- lack of Parallax must be in one of the “outer spheres”

-therefore the outer sphere of stars does change!

a Comet

position did not change significantly throughout the night.

- lack of Parallax, must lay beyond the orbit of the Venus

Observed positions of Mars twice-daily

which implied its orbit intersects that of the Sun.

- apparent crossing/smashing of the “crystalline spheres”

“there are no solid spheres "holding" the celestial bodies”

- observations that strengthened the rejection of the cosmology of Aristotle & Ptolemy

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Tycho Brahe - his contribution

So, even though Tycho Brahe never found the “right” model…

… his observations did play a major role in the final


of the notions of Aristotle/Ptolemy that

the celestial bodies are carried by crystalline spheres,

with everything beyond the Moon eternal & unchanging.

In addition Tycho Brahe also actually published his data !

In particular his twice-daily measurements of the position of Mars

provided Johannes Kepler with a crucial database a few years later.

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Johannes Kepler (1571-1630)

  • Greatest theorist of his day

  • a mystic

  • there were no heavenly spheres

  • forces made the planets move

  • Modifications to Copernicus idea showed heliocentric models were correct

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Kepler’s Laws - #1

  • Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus.

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Properties of an Ellipse

a + b = constant

The amount of "flattening" of the ellipse is termed the eccentricity

All ellipses have eccentricities lying between zero and one.

e=ratio; dist. between foci compared to major axis

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Kepler’s Laws - #1

  • Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus.

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Kepler's Laws:

  • Kepler's First Law:

  • Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus.

Kepler's Second Law: Line joining planet and the Sun sweeps out equal areas in equal times

Kepler's Third Law:The squares of the periods of the planets are proportional to the cubes of their semi-major axes:

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Kepler’s Laws

  • A convenient unit of measurement for periods is in Earth years, and a convenient unit of measurement for distances is the average separation of the Earth from the Sun, which is termed an astronomical unit and is abbreviated as AU. If these units are used in Kepler's 3rd Law, the denominators in the preceding equation are numerically equal to unity and it may be written in the simple form

Kepler’s 3rd Law: version 2

This equation may then be solved for the period P of the planet, given the length of the semi-major axis axis,

or for length of the semi-major axis, given the period of the planet

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Kepler’s Laws

  • As an example of using Kepler's 3rd Law:

  • let's calculate the "radius" of the orbit of Mars (that is, the length of the semi-major axis of the orbit) from the orbital period. The time for Mars to orbit the Sun is observed to be 1.88 Earth years. Thus, by Kepler's 3rd Law the length of the semi-major axis for the Martian orbit is

Second example, let us calculate the orbital period for Pluto, given that its observed average separation from the Sun is 39.44 astronomical units. From Kepler's 3rd Law

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Galileo Galilei (1564-1642)

  • First man to point a telescope at the sky

  • wanted to connect physics on earth with the heavens

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Galileo saw shadows cast by the mountains on the Moon.

He observed craters.

The Moon had a landscape; it was a “place”, not a perfect heavenly body.

Galileo’s Observations

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Galileo discovered that Jupiter had four moons of its own.

Jupiter was the center of its own system.

Heavenly bodies existed which did not orbit the earth.

Galileo’s Observations

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Ptolemy: -recap

  • What was the Ptolemaic model?

    • Ptolemy’s synthesis of earlier Greek ideas about the geocentric universe, which was a sophisticated (or some may say incorrect and overly complex) model which, most importantly, allowed prediction of planetary positions. Epicycles ‘explained’ retrograde motion…but of course this was not physically a correct model (by far!!!).

      • In use for ~1500 years!!)

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Galileo’s observation of the phases of Venus was the final evidence which buried the geocentric model.



No gibbous or full phases!

All phases are seen!

His astronomical work got him in trouble with the Church!

Galileo observed all phases!

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KEPLER: -recap evidence which buried the geocentric model.

  • What are Kepler’s three laws of planetary motion?

  • (1) The orbit of each planet is an ellipse with the

  • Sun at one focus.

  • (2) As a planet moves around its orbit, it sweeps out equal areas in equal times.

  • (3) More distant planets orbit the Sun at slower average speeds, following a precise mathematical relationship (P2 = R3).

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KEPLER: 3rd law -recap evidence which buried the geocentric model.

  • (3) More distant planets orbit the Sun at slower average speeds, following a precise mathematical relationship (P2 = R3).

This version of 3rd Law can use *any* units for P and R (hours, years, weeks, km, feet etc etc

This version of 3rd Law is simpler, because we are constrained to use years and AU

Rearranged versions ‘isolate’ P or R

All you need is to be able to use the the squared and cubed buttonson your calculator, plus the square-root  or cube-root 3 buttons

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KEPLER: 3rd law -recap evidence which buried the geocentric model.

Rearranged versions ‘isolate’ P or R

All you need is to be able to use the the “squared” and “cubed” buttonson your calculator, plus the square-root  or cube-root 3 buttons

Calculate R cubed (RxRxR) then take the square-root of that number

Calculate R squared (RxR) then take the cube-root of that number

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What else have we learned? evidence which buried the geocentric model.

  • Copernicus created a Sun-centered model of the solar system designed to replace the Ptolemaic model, but was more fundamentally correct yet it was no more accurate because he still used perfect circles.

  • Tycho provided observations used by Kepler to refine the model by introducing orbits with the correct characteristics.

  • Galileo’s experiments and telescopic observations (sunspots,; craters on moon; moons of Jupiter, phases of Venus…)

  • --overcame remaining objections to the Copernican idea of the Earth is a planet orbiting the Sun.