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Classical Astronomy

Introduction

Astronomy is probably the oldest of sciences. Humans, with their innate curiosity

and intelligence have looked up and wondered about phenomena in the sky since

prehistoric times. People of different cultures scattered across the globe have

incorporated their observations of celestial objects and events into their creation

myths and religions. Civilizations such as the Babylonians and Egyptians made long-

term, systematic observations of the night sky and some of their records still

survive.

Five planets, Mercury, Venus, Mars, Jupiter and Saturn plus the Sun and Moon were

visible to the unaided eyes of the ancient astronomers. The planets could be

distinguished from stars in that through regular observation they were seen to

move relative to the stars. The very word planetderives from the classical Greek

term for wandering star. Unlike stars, planets also varied their brightness over

time. A final complication in the observed behavior of planets was that of

retrograde motion. This is where a planet seemed to back track on its path across

the sky through the constellations before reverting to its normal direction. The

diagram below clearly shows this for the planet Mars in late 2003.

Credit: Image was generated usingStarry Night Prosoftware.

Figure 1.1From mid-July to mid-September 2003 the planet Mars appeared tobackwards relative to the stars on the celestial sphere. This is an example of

retrograde motion.

This section does not seek to provide a detailed history of astronomy up till

Newton. If you wish to delve into this vast subject in more detail we encourage

you to follow some of the linksto other sites specialising in the topic. The depthpresented here nonetheless probably goes beyond that required by the NSW
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syllabus. In doing so it should provide some background to what is a fascinating

topic.

Classical Astronomers

The Ionians (6th - 4th Century BC)

The ancient Greeks, specifically the Ionian school of philosophers, are credited

with the move to a natural, mechanistic view of the Universe. Based on Miletus in

Asia Minor and founded by Thales, the Ionians are remembered not so much for

the specific models of the Universe that they suggested, but rather that they asked

questions that they could then attempt to answer through reason, observation and

the application of geometry. Anaximander refined Thales' ideas and proposed a

model which had a cylindrical Earth at rest in the centre of the Universe,

surrounded by air then one or more spherical shells with holes in them. These

appeared as stars due to the rim of fire that lay beyond the solid sphere.

Credit: R. HollowCSIRO

Figure 1.2Anaximander had a cylindrical Earth surrounded by air and a solidsphere. Holes in the solid sphere allowed the light from the outer rim of fire to

shine through, appearing as stars and the Sun.

Anaximander's model of the Universe was revolutionary for two main reasons.

Firstly it introduced a mechanistic view, moving beyond a mythological,

supernatural explanation for the Universe. It also proposed the concept of spheres

surrounding the Earth. This was to profoundly influence astronomy and cosmology

for the next two millennia.Anaximenesrefined Anaximander's model by suggesting


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that the stars were fixed on to a solid, transparent crystalline sphere that rotated

about the Earth.

Later Ionians contributed more ideas and discoveries. Anaxogoras (c. 450 BC)

realised that the Moon shone by reflected sunlight,had mountains and was

inhabited and that the Sun was not a god but a large fiery stone much larger than

Greece and a large distance from Earth. Empedoclessuggested that light traveled

fast but not at infinite speed. Democritusproposed not just at atomist model of

matter but also proposed that the Milky Way was composed of thousands of

unresolved stars.

The Pythagoreans

Pythagoras

Pythagoras(c. 580 - 500 BC) is credited with postulating a spherical Earth and withrealising that Phosphoros, the morning star and Hesperos, the evening star were in

fact the same object, the planet Venus. He and his followers believed in the

concept of cosmos, a well-ordered, harmonious Universe. They placed great

importance on the power and aesthetics of geometry and mathematics rather than

experiments. Regular geometrical solids, especially the sphere, were revered and

they sought to find harmonies and ratios in the natural world.

Herakleides, a student of Plato and Aristotle but heavily influenced by

Pythagorean ideas refined an earlier model by Philolausto develop one that had a

spherical Earth rotating on it axis. It also had Mercury and Venus revolving around

the Sun whilst the Sun and other planets revolved around the Earth. Stars again

were fixed on a revolving crystalline sphere. Models that had the Earth at the

centre of the Universe are termedgeocentricor earth-centered.

Interestingly whilst most classical models were variations on geocentric models,

one of the Pythagoreans, Aristarchus of Samos(c. 310 - 230 BC) proposed a model

that placed the Sun at the centre, that is a heliocentricUniverse. His model wouldbe familiar to us today as a reasonable description of the solar system. All the


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planets, including the earth, revolved around a fixed Sun in circular orbits. The

Earth rotated once a day on its axis and the Moon revolved about the Earth.

Credit: R. Hollow, CSIRO

Figure 1.3In Aristarchus' heliocentric model all the planets orbit the Sun alongcircular paths. The Moon orbits the Earth which in turn spins on its axis.

There are several reasons why Aristarchus' model did not gain wide acceptance and

was in effect lost for 18 centuries until Copernicus redeveloped it. Firstly his

original writings were lost in the destruction of the Great Library of Alexandria in

AD 415. Secondly his concept of a moving Earth defies common sense. We do not

feel the Earth spinning or moving through space. His idea contradicted the

prevailing view of motion as espoused by Aristotle. The final key objection to his

model was the failure of observers to detect any stellar parallax. Under

Aristarchus' model, the closer stars should show a periodic shift in position to and

fro against more distant stars over the course of a year as the Earth orbited the

Sun. In fact this was not detected until 1838 following careful telescopic

observations. Aristarchus had underestimated the distance of the earth from the

Sun thus the size of possible parallax was overestimated.
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Credit: R. Hollow, CSIRO

Figure 1.4In a heliocentric model, closer stars should show an apparent shift inposition relative to background stars due to parallax. This diagram greatly

exaggerates the effect and is not to scale.

Plato (428 - 348 BC)

Plato

An Athenian and a pupil of Socrates, Platohad a profound influence on philosophy

and he wrote widely on many different fields. Rather than being remembered for a

specific model of the Universe it was his views on its nature, put forward in his

dialogue Timaeus, that were to so strongly influence subsequent generations. To

Plato the Universe was perfect and unchanging. Stars were eternal and divine,

embedded in an outer sphere. All heavenly motions were circular or spherical as

the sphere was the perfect shape. Such was his influence that the concept of

circular paths was not challenged until Kepler, after many years of painstaking

calculations, discovered the elliptical orbits of planets nearly 2,000 years later.

Plato thought that the visible world was only a dim representation of the real

world. He was therefore not concerned with direct observations or how they


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correlated with his ideas but realised geometrical, arithmetical models could be

devised to fit observations and save appearances.

Aristotle (384 - 322 BC)

Aristotle

Aristotle'swork had a profound influence on western thought, eventually being

absorbed and molded into supporting Christian theology and dogma. He would

probably have been disturbed by this. A pupil of Plato, he in turn tutored

Alexander the Great. Whilst thought of as a theoretical philosopher he also

conducted experiments in several fields. His works on astronomy and the physics of

motion were written in On the heavensand Physics.

Like Empedoclesbefore him Aristotle saw all matter on Earth as being composed

of combinations of only four elements; earth, air, fire and water with theproperties of cool, moist, hot and dry. The stars were made of a separate fifth

element, quintessenceand were incorruptible and eternal. Motion in the heavens

was natural, unforced and circular so that the planets and Sun orbited a fixed,

unmoving spherical Earth in circular orbits. On Earth, however, matter was

corruptible and subject to decay. Motion was linear with objects requiring a force

acting on them to stay in motion. It is was not untilNewtonin the second half of

the seventeenth century that this concept of forced motion was overthrown.

Aristotle's own model of the Universe was a development of that of Eudoxuswho

had also studied under Plato. It had a series of 53 concentric, crystalline,

transparent spheres rotating on different axes. Each sphere was centered on a

stationary Earth so the model was both geocentric and homocentric. Stars were

fixed on the outer sphere. The Moon marked the boundary between the

unchanging, constant heavens and the corruptible Earth. According to Aristotelian

cosmology it was only within the sub-lunary sphere, that is between the Earth and

Moon, that changeable phenomena such as comets could exist.

Ptolemy (AD 120 - 180)
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Claudius Ptolemy

The last of the great classical astronomers, Claudius Ptolemylived in Alexandria.

He contributed to mathematics, optics, geography and music but is chiefly

remembered for his vast work on astronomy, known as the Almagest. In it he

detailed a model of the Universe that profoundly influenced Western and Arabic

thought for the next 1,500 years.

Ptolemy relied heavily on tools invented and observations made by earlier

astronomers. Apollonius (262 - 190 BC) had developed the concepts of the

eccentric and the epicycle to explain planetary motions (see Figure 1.5 below).

Hipparchus(161 - 126 BC) had organised earlier Babylonian records together with

his own observations to develop a catalogue of 850 stars. He plotted them on a

celestial sphere and introduced the concept of comparing brightnesses on a

magnitude scale that forms the basis of that still used today. Ptolemy synthesised

all this work and incorporated his own careful observations to produce a model

that was to become accepted as the standard model until the 1600s.

Credit: R. HollowCSIRO


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Medieval & Renaissance Astronomy

Medieval Astronomy

Contrary to common misconception the period between the end of the classical

era and the start of the Renaissance was not devoid of scientific progress. Islamic

scholars translated many of the surviving writings from Greek or Syriac into Arabic

from the late 700's onwards. These translations in turn were transported into

Islamic Spain where they eventually fell into Christian hands and were translated

into Latin. Islamic astronomers such as Muhammad al-Battani (c. 850 - 929)

refined Ptolemy's model and their published works and tables were later used byWestern astronomers. Even today the influence of Islamic astronomers is found in

the names of many of the bright stars such as Betelgeuse( Ori), Alnitak( Ori)

and Zubenelgenubi( Lib).

Nicole Oresme

As Western scholars studied the Latin translations of the classical philosophers they

incorporated many aspects of their work into the prevailing theology and world

view. Aristotle'sphysicsdescribed the motion of objects and the refined model of

Ptolemy was used to study the night sky. The Frenchman, Nicole Oresme(1320 -

82) applied an early concept of the centre of gravity, used mathematics to argueagainst astrology and even suggested the existence of other inhabited worlds in

space. Nicolas of Cusa(born c. 1401) supported this idea and rejected the concept

of a static Earth at the centre of all motions.

Georg Puerbach(1423 - 61) refined theAlmagestand wrote a popular textbook on

it. This prompted a renewal of interest in the need for accurate observations. His

pupil, Regiomontanus(1436 - 76) highlighted problems with Ptolemy's work based

on observations made at his purpose-built observatory. He published his own and

other writings on astronomy and the increasing availability of printed books did


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much to spread ideas among scholars. In 1482 he observed a bright comet that was

later identified as one of the visits of Comet Halley and was most likely working

towards a heliocentric model influenced by Aristarchus at the time of his death.

The Renaissance

Nicolaus Copernicus (1473 - 1543)

Nicolaus Copernicus

Copernicusstudied classics and mathematics at Krakow in his native Poland, canon

law in Bologna and Ferrara and medicine at Padua in Italy. His keen interest in

astronomy was fostered in Italy and developed back in Poland where he was canon

at the cathedral in Frauenberg (now Frombeck) where he spent most of his life.

A conjunction of Jupiter and Saturn in 1504 was observed to differ by 10 days from

the predictions of tables based on Ptolemy's work. This, combined with Copernicus'

abhorrence of the equant drove him to develop an improved model. Influenced by

the work of Regiomontanus (thus also Aristarchus) and neoplatonism (which viewed

the Sun as the Godhead and source of all knowledge) he produced his own model.

He withheld publication due to his conservative nature and fear of ridicule but was

eventually persuaded by Rheticus. Allegedly he received the first copy of his work

De revolutionibus orbium (On the revolution of the heavenly spheres) on his

deathbed in 1543.


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Manuscript example of Copernicus' model.

In Copernicus' model a spherical Earth rotates daily on it axis whilst it and the

other planets each orbit the Sun. The period of the planets' orbits increases with

increasing distance from the Sun. The Sun was not exactly at the centre of the

planetary orbits thus strictly speaking the model is heliostatic rather than

heliocentric.

There were several advantages of Copernicus' model over that of Ptolemy:

1. It could predict planetary positions to within 2, the same as that ofPtolemy.

2. Retrograde motion of planets was explained by the relative motion betweenthem and the Earth.

3. Distances between planets and the Sun could be accurately determined inunits of the Earth-Sun distance (ie Astronomical Units).

4. Orbital periods could be accurately determined.


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5. It explained the difference between the inferior planets (Mercury andVenus) that were always observed close to the Sun and the superior ones

(Mars, Jupiter and Saturn).

6. It preserved the concept of uniform circular motion without the need forequants.

7. It preserved Aristotle's concept of real spheres nestled inside one another.8. Unlike Ptolemy's model it did not require the Moon to change in size.

Copernicus' model also had several problems which contributed to its failure to

immediately supplant Ptolemy's model:

1. No annual stellar parallax could be detected. Copernicus explained this asdue to the fact that the stars were a vast distance hence any parallax would

be very small and difficult to detect.

2. It required a moving Earth, This would contradict Aristotelian physics andCopernicus presented no new laws of motion to replace Aristotle.

3. By removing the Earth from its natural place it was philosophically andtheologically unacceptable to many scholars.

4. It was no more accurate than Ptolemy's in predicting planetary positions.5. It was actually more complicated then Ptolemy's model. In his efforts to

avoid the equant but retain uniform circular motion he had to introduce

more devices to fit his observations.

Tycho Brahe (1546 - 1601)

Tycho Brahe and his great mural quadrant

Tycho Brahe, of Danish noble stock, was probably the greatest astronomical

observer of the pre-telescope era. Early observations in the 560's revealed

inaccuracies with existing tables and spurred him onto making systematic, long-


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term observations and records. This task would occupy the rest of his life. With

generous funding from the King of Denmark he established a dedicated

observatory, Uraniborg, on the island of Hven (now Ven). He built large

instruments such as quadrants from wood and brass that improved on earlier

designs. The measurements he made were up to ten times more accurate than any

preceding ones and were at the limit of that obtainable by the unaided eye. The

investment by the Danish King amounted to 5% of his total income, still a record

for investment on scientific research. Brahe eventually fell out with the Danish

court and moved to Prague for his final years.

In November 1572 a new star appeared in the constellation Cassiopeia. Brahe's

observations showed that it was motionless relative to nearby stars suggesting to

him that it was in fact a star and not a tail-less comet. Five years later heobserved a bright comet and discerned no parallax and placed it at least six times

further from Earth than the Moon. Both of these observations challenged the

Aristotelian orthodoxy. The stars were supposed to be changeless and perfect

whilst comets were supposed to be confined to the sub-lunary sphere, that is

between the Earth and Moon. Further observations revealed that the comet would

move through the solid crystalline spheres of an Aristotelian Universe.

To reconcile his observations with his philosophy Brahe developed his own model,

incorporating some aspects of Copernicus' but rejecting the idea of a moving Earth.

Although his hybrid model enjoyed a brief period of popularity it was soon

replaced by the work of his assistant, Johannes Kepler.


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Tycho Brahe's Model:Brahe's model was somewhat of a hybrid and drew upon Herakleide's earlier

concepts. It had a static Earth at the centre of the Universe with the Moon orbitingit. A rotating sphere of fixed stars also revolved around the Earth once every 24

hours. The planets however orbited about the Sun which itself orbited the Earth. Itutilised epicycles, deferents and equants. In his model there is no need for stellar

parallax. The diagram above shows a simplified representation.

Brahe's lasting legacy was his long-term and meticulous observations of planetary

motions, especially those of Mars. This data was used after his death by Kepler,

who worked as his assistant during Brahe's last year.

Johannes Kepler (1571 - 1630)

Johannes Kepler


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Best known for his key works on astronomy, Johannes Kepler made valuable

contributions in other fields. In his works on optics he examined the refraction of

light, correctly explained the working of the eye for the first time and provided a

theoretical basis for telescopes with suggested means of improving them. His

explanation on the new Napierian logarithms did much to encourage their wide

acceptance. Given a challenge to calculate volumes of wine casks he ended up

developing an approach to infinitesimal calculus well ahead of the ideas of Liebniz

and Newton. Kepler had studied under the renowned astronomer Michael Maestlin,

one of the first proponents of Copernicus' work.

In his first astronomical work, Mysterium cosmographicum(The cosmic mystery) in

1596, Kepler upheld his belief in the Copernican system. He also discovered a

geometrical relationship for the orbits of the planets around the Sun. Between thesphere of each planet's orbit he found he could place one of the five regular solids,

for example a cube between Jupiter and Saturn, so that the six planets were

separated by five regular solids. This system reflects the influence on Kepler of the

Platonic-Pythagorean tradition of matching order in nature with the regularities of

mathematics. Of greater long-term importance however was his suggestion that

the Sun somehow affected the orbits of the planets, perhaps by magnetism.

Kepler's geometrical relationship in the Solar System as shown in his Mysteriumcosmographicumof 1596.

Kepler tried to fit Brahe's data to the Copernican model but consistently arrived at

errors of at least eight seconds of arc, small but not insignificant. He was finally


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forced to abandon the concept of uniform circular orbital paths but it was to take

him several years of painstaking, methodical calculations before he arrived at an

alternate model that fitted Brahe's 20 years of data on Mars. The results were

published in 1609 in his work Astronomica nova (New Astronomy). In it he

explained what are now known as his first two laws of planetary motion.

Kepler's 1st Law: The Law of Ellipses.All planets orbit the Sun in elliptical orbits with the Sun as one common focus.Note the eccentricity of the ellipse has been greatly exaggerated in the above

diagram. For most planets their orbits are almost circular.

Kepler's 2nd Law: The Law of Equal Areas.The line between a planet and the Sun (the radius vector) sweeps out equal areas

in equal periods of time.In the diagram, the time interval t2-t1= t4-t3so the areas swept through in equal

times are equal,that isA1=A2.

This effect is very noticeable in comets such as Comet Halley that have highlyelliptical orbits. When in the inner Solar System, close to the Sun at perihelion,

they move much faster than when far from the Sun at aphelion.


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Kepler actually formulated the law of equal areas first and it then led him to the

law of ellipses. His third law was not published until 1618 in Harmonice mundi(The

Harmony of the World). This resulted from his attempts to find a relationship

between the distance of a planet from the Sun and its orbital period.

Kepler's Third Law: The Law of Periods or the Harmonic Law*.

The square of a planet's period, T, is directly proportional to the cube of its

average distance from the Sun, r.

Mathematically this can be expressed as:

T2r3

or T2/r3= k(1.1)

wherekis a constant and the same for all planets or orbitalbodies (such as comets) in a given system.

If Tis measured in Earth years and rin astronomical units (AU) then for the Earth,

T= 1 and r= 1 so:

T2/r3= k

1/1 = k

ie. k=1

(* Note this equation is not explicitly required for 8.5 The Cosmic Engineunit inthe NSW Preliminary Course. It is, however, explicitly required in Unit 9.2 Spacein

the HSC course.)

The implication of Kepler's Third Law is that planets more distant from the Sun

take longer to orbit the Sun. Let us see how this can be used to determine the

mean distance of Mars from the Sun if its orbital period is 1.88 Earth years.

If T2/r3= k(1.1)

Then rewriting for r

r3= T2/k

r= ((1.88)2/1)1/3

so r= 1.524 AU

So Mars is 1.524 astronomical units from the Sun.

Kepler's laws of planetary motion were empirical, they could predict what would

occur but could not account for why planets behaved in such a manner. His

Rudolphine tables of planetary motion published in 1627 were more accurate than

nay previous ones. He came close to uncovering the concept of gravitation and

corresponded withGalileoand was aware of his telescopic discoveries.
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Galileo & Newton

Galileo Newton

Galileo Galilei (1564 - 1642)

Credit: LeoniGalileo

Galileo was born in the same year as Shakespeare and on the day of Michelangelo's

death. Appointed to the Chair of Mathematics at the University of Pisa when he

was 25 his studies of motion there and later at Padua provided the foundation of

the study of dynamics. His contributions to the the development of gravitational

theory and motion were to terminally undermine the tenets of Aristotelian motionand physics.

In 1604 a bright new star appeared in the constellation Serpentarius. Galileo's

observations detected no parallax, suggesting it was a star and not some

atmospheric phenomenon. This result confirmed Brahe's findings from the nova of

1572 that stars could change and again challenged the Aristotelian orthodoxy.

When Galileo heard about a new optical device, the telescope, in 1609 he quickly

built his own version. He then used it and more refined telescopes tosystematically observe the night sky. Details on Galileo's use of the telescopecan

be found in the HSC Astrophysics section. His findings, published in 1610 in

Sidereus nuncius(The starry messenger) had important implications.

1. The Moon:According to Aristotelian principles the Moon was above the sub-lunary

sphere and in the heavens, hence should be perfect. Galileo found the

"surface of the moon to be not smooth, even and perfectly spherical,...,but

on the contrary, to be uneven, rough, and crowded with depressions and
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bulges. And it is like the face of the earth itself, which is marked here and

there with chains of mountains and depths of valleys."He calculated the

heights of the mountains by measuring the lengths of their shadows and

applying geometry. He also detected earthshine on the lunar surface, that is

the Moon was lit up by reflected light from the Earth just like we receive

reflected light from the Moon.

2. Stars in the Milky Way:

Galileo's drawing of the Pleiades shows many more stars than visible to the

unaided eye.

Even through a telescope the stars still appeared as points of light. Galileo

suggested that this was due to their immense distance from Earth. This then

eased the problem posed by the failure of astronomers to detect stellarparallax that was a consequence of Copernicus' model. On turning his

telescope to the band of the Milky Way Galileo saw it resolved into

thousands of hitherto unseen stars. This posed the question as to why there

were invisible objects in the night sky?

3. The Moons of Jupiter:

The moons of Jupiter as drawn by Galileo over successive nights.

Observations of the planet Jupiter over successive night revealed four star-

like objects in a line with it. The objects moved from night to night,

sometimes disappearing behind or in front of the planet. Galileo correctly

inferred that these objects were moons of Jupiter and orbited it just as ourMoon orbits Earth. Today these four moons are known as the Galilean


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satellites; Io, Europa, Ganymede and Callisto.

For the first time, objects had been observed orbiting another planet, thus

weakening the hold of the Ptolemaic model. The Earth was clearly seen to

not be at the centre of all motions.

Two subsequent observations also undermined the Arisotelian-Ptolemaic Universe.

Galileo found that Venus exhibits phases, just like the Moon. This of course could

be accounted for in a Copernican system but not in a Ptolemaic one. He published

a letter in 1613 announcing his discovery of sunspots in which he also proclaimed

his belief in the Copernican model. Monitoring sunspots showed that the Sun

rotated once every 27 days and that the spots themselves changed. The concept of

a perfect, unchanging Sun thus also became untenable.

In presenting his views in his Dialogue concerning the two chief systems of the

world, the Ptolemaic and the Copernican in 1632 Galileo reignited his earlier

conflict with the authorities of the Catholic Church. Eventually forced into publicly

recanting his belief in the Copernican system and being placed under comfortable

house arrest his Dialogue, along with the works of Copernicus and Kepler was

placed on the Index of Forbidden Books.

Galileo spent the last years of his life working once again on trying to understand

motion. The resultant final book Dialogues concerning two new scienceshad to besmuggled out of Italy before being published in Holland in 1638. It primarily dealt

with describing motion, kinematics, but also revealed that acceleration resulted

from the application of a force and that he was aware of the concept of inertia. He

rejected Aristotle's ideas of forced and natural motions after studying falling or

rolling objects and projectiles and realised that gravity was some type of force

acting in terrestrial situations though he does not seem to have extended this to

heavenly motions.

Whilst Galileo did not propose his own model of the Universe, his observational,

experimental and theoretical work provided the conclusive evidence need to

overthrow the Aristotelian-Ptolemaic system. His work on forces was to help

Newton develop his dynamics. Galileo died in 1642, the year that Newton was

born.

Sir Isaac Newton (1642 - 1727)


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Sir Isaac Newton

Isaac Newton is the pivotal figure in the scientific revolution of the 16th and 17th

centuries. He discovered the composition of white light, and laid the foundations

of modern optics. In mathematics he invented infinitesimal calculus and the

binomial theorem. His work on the laws of motion and of universal gravitation

became the basis of modern physics. Whilst today remembered for his immense

contributions to science the bulk of his writings were actually in the fields of

theology and alchemy though as his views on both of these was contrary to the

establishment he kept many of them secret.

During 1665-6 Newton returned to his home at Woolsthorpe from Cambridge when

the University closed due to the Great Plague. This period allowed him time to

develop his ideas on optics and light, planetary motions and the concept ofgravitation. By 1670 he was Lucasian Chair of Mathematics at Cambridge, had

developed his corpuscular theory of light and built the first successful reflecting

telescope, thus avoiding the chromatic aberration problems inherent in the lenses

of refracting telescopes. For this he was elected a Fellow of the Royal Society. He

withheld publication of chief work on light, Optiks, until 1704, the year after his

adversary Robert Hookedied.

Newton's scientific legacy rests on his other work, the Philosophiae Naturalis

Principia Mathematica (Mathematical Principles of Natural Philosophy), generally

known as Principiapublished due to Edmond Halley'surging and funding, in 1687.

His detailed exposition of the concepts of force and inertia is summarised

eloquently in his three axioms or Laws of Motion(from the translation in On the

Shoulders of Giants, ed. by Stephen Hawking, Running Press, 2002).

Newton's Laws of Motion:

1. Law I:Every body preserves in its state of rest, or of uniform motionin a right line, unless it is compelled to change that state by forces


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impressed thereon.

This is more commonly stated as: An object remains at rest or in a

state of uniform motion unless acted on by an unbalanced force.

2. Law II: The alteration of motion is ever proportional to the motiveforce impressed; and is made in the direction of the right line in

which that force is impressed.

This is now commonly referred to as F= maand emphasises the

vector nature of force.

3. Law III: To every action there is always opposed an equal reaction: orthe mutual actions of two bodies upon each other are always equal,

and directed to contrary parts.

The true genius of his work is that he then went on and applied them not just to

motion on Earth but realised that they applied equally to the motions of other

bodies such as planets in space. He applied his mathematical techniques to

investigate the nature of the force between the Earth and the Moon, and the Earth

and the Sun. His solution revealed the force to obey an inverse-square relationship

and result in elliptical orbits as calculated by Kepler.

Newton's Law of Universal Gravitation

(The formulae used in this section are not required for the NSW Stage 6

Preliminary Course. They are explicitly required for unit 9.2 Space in the HSC

course)

As he showed in Book 3, System of the Worldof his Principia, Newton could apply

his law of universal gravitation to accurately predict the motions of planets, the

orbits of comets and even account for tides on Earth. His law can be

mathematically expressed as follows:

Fm1m2 / r2

where Fis the force between any two objects of masses m1and

m2respectively and separated by a distance r.

As there are no other variables involved the equation becomes:

F= Gm1m2 / r2(1.2)

where G is a constant known as the Universal Gravitational

Constant.

(G = 6.673 10-11

Nm2

kg-2

)


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This can also be expressed in words as:

The force of attraction between any two bodies in the Universe is proportional to

the product of their masses and inversely proportional to the square of their

distance apart.

Having shown that gravitationally all the mass of an object can be assumed to be

at its centre of mass, the gravitational force therefore acts along a line joining the

two bodies. It is always an attractive force. The gravitational massof an object

was shown to be identical to its inertial mass (that which hinders its change in

motion).

For a two-body system such as the Sun-Earth an equilibrium exists such that the

gravitational force = centripetal force. Using this relationship Newton was able to

deriveKepler's Third Law.

Since FG= FC

then: F= GmSmE/ r2= mEr

2(1.3)

where mSand mEare the masses of the Sun and Earth and is

the angular velocity of the Earth around the Sun.

Simplifying (1.3) gives

GmS /2= r3(1.4)

Now the time taken for one complete revolution of the Eartharound the Sun,is its orbital period, Tsuch that:

T= 2 / (1.5)

so 2= 42 / T2(1.6)

substituting this into (1.4) gives:

GmST2/ 42= r3

which can be rewritten as:

T2

= 42r3

/ GmSNote this is of the form:

T2= kr3

which is Kepler's Third Law, and the value of kis:

k= GmS / 42(1.7)

This value of kis a constant for all bodies orbiting the Sun as it only depends upon

the mass of the Sun and the constant, G.
http://outreach.atnf.csiro.au/education/senior/cosmicengine/renaissanceastro.html#keplerthirdhttp://outreach.atnf.csiro.au/education/senior/cosmicengine/renaissanceastro.html#keplerthirdhttp://outreach.atnf.csiro.au/education/senior/cosmicengine/renaissanceastro.html#keplerthirdhttp://outreach.atnf.csiro.au/education/senior/cosmicengine/renaissanceastro.html#keplerthird


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Newton's contributions profoundly influenced subsequent generations. His view of

the Universe was a mechanistic one that ran like clockwork and had a designer.

The success of his law of gravitation was confirmed in 1758 when a bright comet

returned as predicted earlier by Edmond Halley. He realised that it would be the

same comet that had previously been seen in 1531, 1608 and 1682. This comet was

subsequently named in his honour and we now know it was the same comet shown

on the Bayeaux Tapestry commerating the Norman invasion of England in 1066.

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