OUR UNIVERSE WEEK 2 Lectures 4 - 6. 2. Know how Tycho Brahe revolutionized the practice of astronomy. Know Kepler's three laws and be able to explain

OUR UNIVERSEOUR UNIVERSE

WEEK 2 Lectures 4 - 6

2. Know how Tycho Brahe revolutionized the practice of astronomy. Know Kepler's three laws and be able to explain them. Understand how Galileo's telescopic observations supported a heliocentric cosmogony.

3. Know Newton's three laws of motion and be able to give examples of each. Know Newton's universal law of gravitation. Be able to explain how Newton used his laws of motion and gravity to obtain Kepler's laws.

1. Understand the difference between geocentric and heliocentric cosmogonies. Understand the Ptolemaic system and how the Copernican heliocentric system better explains our observations of the Moon and planets.

LEARNING GOALSLEARNING GOALS

Astronomy seems to havebeen practised by

most ancient civilisations. Many ideas, myths and

misconceptions have occurred over and over.

We follow a Western history from the Ancient Greeks (400 BC to the present)

Astronomy seems to havebeen practised by

most ancient civilisations. Many ideas, myths and misconceptions

have occurred over and over.We follow a Western history from the

Ancient Greeks (400 BC to the present)

Gravitation & Planetary MotionGravitation & Planetary Motion

oror

The Copernican Revolution The Copernican Revolution

GeocentricGeocentric

versusversus

HeliocentricHeliocentric

cosmogonycosmogony

The Geocentric The Geocentric

cosmogonycosmogonyDef. A theory of the Earth'sDef. A theory of the Earth's

place in the Universeplace in the Universe

Sun & Moonrotate with

celestial sphere,but also drift

slowly with respect

to stars.

Explains diurnal

motion of stars

Merry-go-roundMerry-go-round

analogyanalogy

Explains diurnal

motion of stars

Sun & Moonrotate with

celestial sphere,but also drift

slowly with respect

to stars.

Gravitation & Planetary MotionGravitation & Planetary Motion

The key difficulty is the The key difficulty is the

retrograde motionretrograde motion

of the planets (of the planets (wandererswanderers))

In the geocentric view In the geocentric view

this required this required epicyclesepicycles

MARSMARS

July 2005 to February 2006July 2005 to February 2006

Mars’ Mars’

retrograde motionretrograde motion

MARSMARS

Aristotle (384-322 BC)Aristotle (384-322 BC)• Earth does not Earth does not feel feel

as if it’s movingas if it’s moving• Natural state for any bodyNatural state for any body

is to be stationaryis to be stationary•The circle: the perfect formThe circle: the perfect form

• Cycles & epicycles requiredCycles & epicycles required

Geocentric explanation of retrograde motion

Ptolemy (140 AD)in Alexandria’s Libraryset up precise epicycles

to fit the observedplanetary motions.

Geocentric explanation of retrograde motion

Ptolemy (140 AD)in Alexandria’s Libraryset up precise epicycles

to fit the observedplanetary motions.

Ptolemy (140 AD)Ptolemy (140 AD)• Refined the geocentric model to a Refined the geocentric model to a

high degreehigh degree •Very accurate, but also very Very accurate, but also very

complicated - 80 circles!complicated - 80 circles!•Refinements kept being added to Refinements kept being added to

account for data.account for data.•No coherent theory behind it.No coherent theory behind it.

Ptolemey’s Ptolemey’s

13 -Volume13 -Volume

AlmagestAlmagestcovered elements of spherical astronomy,

solar, lunar, and planetary theory,

eclipses, and the fixed stars.

It remained the definitive authority

on its subject for nearly 1500 years.

Nicolaus CopernicusNicolaus Copernicus (1473 - 1543)(1473 - 1543)

Polish Polymath: Lawyer, physician, Polish Polymath: Lawyer, physician,

economist, canon of the church, economist, canon of the church,

and artist.and artist.

Gifted in Mathematics and influenced byGifted in Mathematics and influenced by

the ideas of Aristarchus, he turned tothe ideas of Aristarchus, he turned to

Astronomy in the early 1500’s.Astronomy in the early 1500’s.

Nicolaus CopernicusNicolaus Copernicus (1473 - (1473 -

1543)1543)

The heliocentric model explains The heliocentric model explains

retrograde motion easily.retrograde motion easily.

Nicolaus CopernicusNicolaus Copernicus (1473 - 1543)(1473 - 1543)Worked out many details:Worked out many details:

Ordering of planetary orbits.Ordering of planetary orbits.• Mercury & Venus,Mercury & Venus, Inferior planets,Inferior planets,

always seen near Sun.always seen near Sun.• Mars, Jupiter, Saturn, Mars, Jupiter, Saturn, Superior planets,Superior planets,

sometimes seen on opposite side of thesometimes seen on opposite side of the

celestial sphere to Sun, highcelestial sphere to Sun, high

above horizon - Earth between Sun andabove horizon - Earth between Sun and

these planets.these planets.

Nicolaus CopernicusNicolaus Copernicus (1473 - 1543)(1473 - 1543)Explained why planets appear in Explained why planets appear in

different parts of the sky on differentdifferent parts of the sky on different

datesdates• Mercury & Venus,Mercury & Venus, Inferior planets,Inferior planets,

seen in west near Sunset, then in east seen in west near Sunset, then in east

just before sunrise - just before sunrise - elongation.elongation.• Mars, Jupiter, Saturn, Mars, Jupiter, Saturn, Superior planets,Superior planets,

best seen at night in best seen at night in opposition.opposition.

• Conjunction:

The Earth, Sun and a Planet form a straight line in the direction of the Sun (as seen from the Earth)

• Opposition:

The Earth, Sun and a Planet form a straight line in the direction away from the Sun (as seen from the Earth,

• Inferior Planets:

Inferior planets can never be in opposition (they are cannot be away from the sun as seen from the earth).

• Two Types of Conjunction:

Inferior conjunction (same side as the earth)

Superior conjunction (opposite side)

• Elongation of a Planet

Elongation is the angular distance of an inferior planet from the Sun as seen from the earth.

• Elongation of Inferior Planets:

Greatest Elongation is the maximum angular distance of an inferior planet from the Sun.

Mercury 18o – 28o

Venus 45o – 47o (eliptical orbits)

If visible in the morning: (Eastern Elongation)

If visible in the evening: (Western Elongation)

Minimum Elongation occurs at …….?

• Elongation of Inferior Planets:

Greatest Elongation is the maximum angular distance of an inferior planet from the Sun.

Mercury 18o – 28o

Venus 45o – 47o (eliptical orbits)

If visible in the morning: (Eastern Elongation)

If visible in the evening: (Western Elongation)

Minimum Elongation occurs at conjunction (0o either inferior or superior)

• Elongation of Superior Planets:

The minimum elongation of a superior planet occurs at conjunction (= zero degrees)

The greatest elongation of a superior planet occurs at opposition ( = 180o)

Elongation Period• Greatest elongations of a planet happen

periodically, with a eastern followed by western, and vice versa.

• The period depends on the relative angular velocity of Earth and the planet, as seen from the Sun.

• The time it takes to complete this period is the synodic period of the planet.

Elongation PeriodLet

T be the period between successive greatest elongations,

ω be the relative angular velocity,

ωe Earth's angular velocity and

ωp the planet's angular velocity.

Then

Elongation Period

2

T

Hence

Elongation Period

But ω = ωp – ωe2

T

Hence

Elongation Period

But ω = ωp – ωe2

T

Hence

Hence

ep

T

2

Elongation Period

Since

T

2

Hence

ep TT

T

22

2

Then

Tp/e are the

siderial periods

Elongation Period

Since

T

2

Hence

1

222

p

e

ep TT

e

TT

TT

Then

Tp/e are the siderial periods

Elongation Period

Since

T

2

Hence

1

222

p

e

ep TT

e

TT

TT

Then

Tearth = 365 days: Tvenus = 225 days: T = 584 days

Relationship between synodic and siderial periods

• Copernicus devised a mathematical formula to calculate a planet's sidereal period from its synodic period.


• Copernicus devised a mathematical formula to calculate a planet's sidereal period from its synodic period.

• E = siderial period of the Earth

• P = siderial period of the Planet

• S = the synodic period.


• During the time S,

the Earth moves over an angle of (360°/E)S (assuming a circular orbit)

and the planet moves (360°/P)S.


• Let us consider an inferior planet.

which will complete one revolution before the earth by the time the two return to the same position relative to the sun.


360360360 E

S

P

S


SEP

S

P

E

P

PE

SPS

PE

SPS

E

S

P

S

111

1

360360

360360360


SESEP

S

P

E

P

PE

SPS

PE

SPS

E

S

P

S

11

P

1 PlanetsSuperior For :

111

1

360360

360360360

for for inferiorinferior planet planet

Box 4-1

SS is observed as time interval between successive is observed as time interval between successive

overtakings of one planet by the other.overtakings of one planet by the other.

for for superiorsuperior planet just swap: planet just swap: E ↔ P

for for superiorsuperior planet planet

1E

1P

1S

1P

1E

1S

Nicolaus CopernicusNicolaus Copernicus (1473 - (1473 -

1543)1543)Determined planetary distances from Determined planetary distances from

Sun by geometry in terms 1 AUSun by geometry in terms 1 AUPlanet------Copernicus---ModernPlanet------Copernicus---Modern

Mercury 0.38 AU 0.39 AUMercury 0.38 AU 0.39 AU

Venus 0.72 AU 0.72 AUVenus 0.72 AU 0.72 AU

Mars 1.52 AU 1.52.AUMars 1.52 AU 1.52.AU

Jupiter 5.22 AU 5.20 AUJupiter 5.22 AU 5.20 AU

Saturn 9.07 AU 9.54 AUSaturn 9.07 AU 9.54 AU

Nicolaus CopernicusNicolaus Copernicus (1473 - 1543)(1473 - 1543)• His results showed that the larger the His results showed that the larger the

orbit, the longer the period & the smaller orbit, the longer the period & the smaller

the speed.the speed.• Noticed variable speed on orbits and so Noticed variable speed on orbits and so

included epicycles to keep using circularincluded epicycles to keep using circular

motion!motion!• This made his model no better thanThis made his model no better than

Ptolemy’s geocentric one to astronomers Ptolemy’s geocentric one to astronomers

at the time. at the time. MORE EVIDENCE NEEDEDMORE EVIDENCE NEEDED

Copernicus’De Revolutionibus Orbium Coelestium(1543, year of his death)

On the Revolutionsof the Celestial

Spheres

Tycho BraheTycho Brahe (1546 - 1601)(1546 - 1601)Danish Astronomer: Danish Astronomer:

Observed Supernova Nov. 11, 1572Observed Supernova Nov. 11, 1572

Danish king financed observatory Danish king financed observatory

Uraniborg (sky castle) on Hven Island.Uraniborg (sky castle) on Hven Island.

Made measurements of stars and planetsMade measurements of stars and planets

with with unprecedented accuracyunprecedented accuracy..

Repeated measurements with differentRepeated measurements with different

instruments to assess errors - pioneerinstruments to assess errors - pioneer

of our modern practices.of our modern practices.

Tycho BraheTycho Brahe (1546 - 1601)(1546 - 1601)Danish Astronomer: Danish Astronomer:

Observed Supernova Nov. 11, 1572Observed Supernova Nov. 11, 1572

Danish king financed observatory Danish king financed observatory

Uraniborg (sky castle) on Hven Island.Uraniborg (sky castle) on Hven Island.

Made measurements of stars and planetsMade measurements of stars and planets

with with unprecedented accuracyunprecedented accuracy..

Repeated measurements with differentRepeated measurements with different

instruments to assess errors - pioneerinstruments to assess errors - pioneer

of our modern practices.of our modern practices.

Tycho BraheTycho Brahe (1546 - 1601)(1546 - 1601)• Attempted to test Copernicus’s ideasAttempted to test Copernicus’s ideas

about the planets orbiting the Sun.about the planets orbiting the Sun.• Failed to measure any stellar parallax;Failed to measure any stellar parallax;

concluded Earth was stationary andconcluded Earth was stationary and

Copernicus wrong. (We now know the starsCopernicus wrong. (We now know the stars

were too far away to measure parallax without a were too far away to measure parallax without a

telescope)telescope)• Compiled a massive data base with Compiled a massive data base with

11 = 1 arcmin accuracy = 1 arcmin accuracy

(best one can do without a telescope)(best one can do without a telescope)

Tycho BraheTycho Brahe (1546 - 1601)(1546 - 1601)• Attempted to test Copernicus’s ideasAttempted to test Copernicus’s ideas

about the planets orbiting the Sun.about the planets orbiting the Sun.• Failed to measure any stellar parallax;Failed to measure any stellar parallax;

concluded Earth was stationary andconcluded Earth was stationary and

Copernicus wrong. (We now know the starsCopernicus wrong. (We now know the stars

were too far away to measure parallax without a were too far away to measure parallax without a

telescope)telescope)• Compiled a massive data base with Compiled a massive data base with

11 = 1 arcmin accuracy = 1 arcmin accuracy

(best one can do without a telescope)(best one can do without a telescope)

Johannes KeplerJohannes Kepler (1571 - 1630)(1571 - 1630)

Employed by Tycho in 1600 in Prague.Employed by Tycho in 1600 in Prague.

After Tycho’s death Kepler inherited his After Tycho’s death Kepler inherited his

data and his position as data and his position as

Imperial Mathematician Imperial Mathematician

of the of the

Holy Roman Empire.Holy Roman Empire.

Johannes KeplerJohannes Kepler (1571 - 1630)(1571 - 1630)

Employed by Tycho in 1600 in Prague.Employed by Tycho in 1600 in Prague.

After Tycho’s death Kepler inherited his After Tycho’s death Kepler inherited his

data and his position as data and his position as

Imperial Mathematician Imperial Mathematician

of the of the

Holy Roman Empire.Holy Roman Empire.

Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)

Kepler could be said to be the first astrophysicistKepler could be said to be the first astrophysicist

He could also be said to be the last scientific He could also be said to be the last scientific

astrologer. astrologer.

(except maybe me)(except maybe me)


Astrology was once kind of scientific Astrology was once kind of scientific


Astrology was once kind of scientificAstrology was once kind of scientific

What happened last time Venus rose in the What happened last time Venus rose in the

constellation of the goat? Maybe something like it constellation of the goat? Maybe something like it

will happen again.will happen again.


Astrology Astrology

Disaster:Disaster:


Astrology Astrology

Disaster: from the Greek for bad star Disaster: from the Greek for bad star


Astrology Astrology


Influenza:Influenza:


Astrology Astrology


Influenza: the influence of the starsInfluenza: the influence of the stars


Astrology Astrology

Even today, how many papers have a regular Even today, how many papers have a regular

astrology column?astrology column?


Astrology Astrology

Even today, how many papers have a regular Even today, how many papers have a regular

astrology column?astrology column?

But how many have a regular astronomy column?But how many have a regular astronomy column?


Astrology Astrology

Based on the idea that the position of the planets in Based on the idea that the position of the planets in

the sky fundamentally affect our lifes.the sky fundamentally affect our lifes.

But there are greater influences.But there are greater influences.


Kepler believed in the Kepler believed in the heliocentric heliocentric model. model.

29 years of struggle with the data led him to try 29 years of struggle with the data led him to try

elliptical orbits with dramatic success. elliptical orbits with dramatic success.

He confirmed this by mapping out the shape of He confirmed this by mapping out the shape of

orbits by observations with Earth’s orbit (1 AU) as orbits by observations with Earth’s orbit (1 AU) as

baseline.baseline.


In Kepler’s time there were only 6 known planets:In Kepler’s time there were only 6 known planets:

Mercury, Venus, Earth, Mars, Jupiter and Saturn.Mercury, Venus, Earth, Mars, Jupiter and Saturn.


In Kepler’s time there were only 6 known planets:In Kepler’s time there were only 6 known planets:

Mercury, Venus, Earth, Mars, Jupiter and Saturn.Mercury, Venus, Earth, Mars, Jupiter and Saturn.

Why not 20, or 100?Why not 20, or 100?

Why these particular spacings?Why these particular spacings?

Before Kepler no one had asked such questions.Before Kepler no one had asked such questions.


Consider an equilateral triangle,Consider an equilateral triangle,

Draw a circle outside and one insideDraw a circle outside and one inside


Consider an equilateral triangle,Consider an equilateral triangle,

Draw one circle outside, one inside and remove the Draw one circle outside, one inside and remove the

triangle.triangle.


These two circles have the same ratio as did the These two circles have the same ratio as did the

orbit of Jupiter to the orbit of Saturn.orbit of Jupiter to the orbit of Saturn.




Spooky eh!Spooky eh!




Spooky eh! But Kepler was intrigue and expanded Spooky eh! But Kepler was intrigue and expanded

on it.on it.





on it. A triangular prism is a tetrahedronon it. A triangular prism is a tetrahedron





on it. A triangular prism is a tetrahedronon it. A triangular prism is a tetrahedron


Could a similar geometry relate the orbits of the Could a similar geometry relate the orbits of the

other planets?other planets?




Kepler recalled the regular solids of Pythagoras.Kepler recalled the regular solids of Pythagoras.

There were five. There were five.




Kepler recalled the regular solids of Pythagoras.Kepler recalled the regular solids of Pythagoras.

There were five. There were five.


He believed they nested one within another.He believed they nested one within another.

Hence the invisible supports of the 5 solids was the Hence the invisible supports of the 5 solids was the

spheres of the 6 planets. spheres of the 6 planets.

Spheres enclosing solids



All this, is an attempt to fit the orbits of the planets with harmonics in music.


But no matter how he tried, he could not make it But no matter how he tried, he could not make it

work very well.work very well.




Why not?Why not?




Why not?Why not?

Because it was wrong.Because it was wrong.




Why not?Why not?

Because it was wrong.Because it was wrong.

The later discovery of Uranus, Neptune, Pluto, and The later discovery of Uranus, Neptune, Pluto, and

the others prove thatthe others prove that


He spent 29 years trying to make it work, but in He spent 29 years trying to make it work, but in

the end decided that it was the observations that the end decided that it was the observations that

were right, not his ideas.were right, not his ideas.

Hence, he finally abandoned them. Hence, he finally abandoned them.

Astronomy wins over astrologyAstronomy wins over astrology


In abandoning his regular solids, he was also able In abandoning his regular solids, he was also able

to free his mind of the perfect sphere/circle for to free his mind of the perfect sphere/circle for

orbital motion.orbital motion.

Hence he considered that they may be elliptical.Hence he considered that they may be elliptical.

Drawing Drawing

an Ellipsean Ellipse

Johannes KeplerJohannes Kepler (1571 - 1630)(1571 - 1630) Kepler’s 3 Laws of planetary motion:Kepler’s 3 Laws of planetary motion:

1) Orbital paths of planets are ellipses, 1) Orbital paths of planets are ellipses,

with the Sun at one focus.(1609)with the Sun at one focus.(1609)

2) Line joining the planet to the Sun 2) Line joining the planet to the Sun

sweeps out equal areas in equal times.sweeps out equal areas in equal times.

3) The square of a planet’s orbital period 3) The square of a planet’s orbital period

is proportional to the cube of its semimajor axisis proportional to the cube of its semimajor axis

Kepler’s 1st Law• The orbit of every planet is an ellipse with

the Sun at one focus.

P

Planet

Sun at a focus

Empty focus

Kepler’s 1st Law

P

Planet

Sun at a focus

Empty focus

cos1 er

r and are polar coordinates

e is the eccentricity of the ellipse

is the semi-latus rectum

Kepler’s 1st Law

P

Planetr and are polar coordinates

Major axisrr

Kepler’s 1st Law

P

Planet

Eccentricity e

Semi Major axis a

Semi Minor Axis b

rr

2

2

22

1

b

a

a

bae

Kepler’s 1st LawEccentricity e

2

2

22

1

b

a

a

bae

Kepler’s 1st Law

P

PlanetSemi Latus Rectum

=b2/a

rr

cos1 er

Note that a circle is a special type is ellipse (one with e = 0)

Kepler’s 2nd LawThe line between the sun and a planet sweeps out equal areas in equal time.

Kepler’s 2nd LawThe line between the sun and a planet sweeps out equal areas in equal time.

If the planet moves from A to B in one day.

Then the Sun A and B roughly form a triangle.

The area of that triangle is the same no matter where the planet is on its orbit.

Kepler’s 2nd LawThe orbit is an ellipse.

Thus, the planet must move faster when near perihelion than it does near aphelion.

Kepler’s 2nd LawThe orbit is an ellipse.

Thus, the planet must move faster when near perihelion than it does near aphelion.

This is because the net tangential force involved in an elliptical orbit is zero.

As the areal velocity is proportional to angular momentum, Kepler's second law is a statement of the law of conservation of angular momentum..

Kepler’s 2nd Law

0221 r

dt

d

Written symbolically,

velocity"areal" theis 221 r

Kepler’s 3rd LawThe square of the orbital period of a planet is proportional to the cube of its semi-major axis.

P2 a3


P2 a3

Example

Uranus was found to have a period of 84 years.

What is its distance from the Sun?


P2 a3

Example

Uranus was found to have a period of 84 years.

What is its distance from the Sun?

a = P2/3 = 842/3 = 19 AU

Using his laws Kepler wasUsing his laws Kepler was

the first astronomer to predictthe first astronomer to predict

a transit of Venus (for the year 1631)a transit of Venus (for the year 1631)

Galileo Galilei (1564 - Galileo Galilei (1564 -

1642)1642)One of the first to use a telescopeOne of the first to use a telescope

From 1610 onwards he saw: From 1610 onwards he saw:

mountains on the Moon, sunspots on mountains on the Moon, sunspots on

the Sun, the rings of Saturn,the Sun, the rings of Saturn,

Jupiter’s moons ( providing a counterJupiter’s moons ( providing a counter

example to the view that Earth is the example to the view that Earth is the

centre of the universe)centre of the universe)


1642)1642)One of the first to use a telescope,One of the first to use a telescope,

His observations constitute the His observations constitute the

beginnings of modern astronomy. His beginnings of modern astronomy. His

defence of the Copernican heliocentric defence of the Copernican heliocentric

solar system was published in solar system was published in

The Starry Messenger. The Starry Messenger.

(Siderius Nuncius)(Siderius Nuncius)


1642)1642)One of the first to use a telescope,One of the first to use a telescope,

His observations constitute the His observations constitute the

beginnings of modern astronomy. His beginnings of modern astronomy. His

defence of the Copernican heliocentric defence of the Copernican heliocentric

solar system was published in solar system was published in

The Starry Messenger. The Starry Messenger.

(Siderius Nuncius)(Siderius Nuncius)


1642)1642)

He noted that as He noted that as

the phases of Venus changed,the phases of Venus changed,

so did its apparent size.so did its apparent size.

This providedThis provided

decisive evidence againstdecisive evidence against

Ptolemaic geocentric system. Ptolemaic geocentric system.

Phases of Venusas it orbits

= angulardiameter(arcsec)

Venus in the Heliocentric system

Venus in the Geocentric system


1642)1642)1610: Using his telescope he 1610: Using his telescope he

discovered 4 moons discovered 4 moons orbitingorbitingJupiterJupiter

(the Galilean satellites)(the Galilean satellites)

This provided a counterexample toThis provided a counterexample to

the view that Earth is the centre of the view that Earth is the centre of

the universethe universe

Jupiter’s moonsJupiter’s moons

Jupiter’s moonsJupiter’s moons

16101610

GalileoGalileo

observedobserved

Jupiter’s Jupiter’s

moons.moons.

Isaac Newton (1642 - 1727)Isaac Newton (1642 - 1727)One of the greatest scientistsOne of the greatest scientists

who ever lived: who ever lived:

was a great experimentalist,was a great experimentalist,

mathematician,mathematician,

&&

philosopher of the philosopher of the

scientific method scientific method

..

Isaac Newton (1642 - 1727)Isaac Newton (1642 - 1727)One of the greatest scientistsOne of the greatest scientists

who ever lived: who ever lived:

was a great experimentalist,was a great experimentalist,

mathematician,mathematician,

&&

philosopher of the philosopher of the

scientific method scientific method

..

Isaac Newton (1642 - 1727)Isaac Newton (1642 - 1727)Principia Mathematica 1667Principia Mathematica 1667

Newton’s Laws of Motion:Newton’s Laws of Motion:1) A particle will continue moving in a1) A particle will continue moving in a

straight line straight line unless unless acted on by a force.acted on by a force.

2) Application of a force, 2) Application of a force, F causes an causes an

acceleration, acceleration, a, given by, given by ma = F

3) Action & reaction are equal and 3) Action & reaction are equal and

opposite.opposite.

Isaac Newton (1642 - 1727)Isaac Newton (1642 - 1727)Principia 1667Principia 1667

Newton’s derivation of CentripetalNewton’s derivation of Centripetal

Acceleration for motion in a circle Acceleration for motion in a circle

using:using:

1) A particle will continue moving in a 1) A particle will continue moving in a


2) Application of a force, 2) Application of a force, F causes an causes an

acceleration, acceleration, a, given by, given by ma = F

Centripental Position Velocity


Draw a position vector



r



r

v



r

v

Draw that velocity vector



r

v



Draw a position vector some time t later

r

v



Draw a radius vector some time t later

r

v


r



r

vr

v



r

v

Draw that new velocity vector

r

v



r

v

Draw that new velocity vector

r

v


r

v

Now draw an acceleration vector

r

v


r

v


r

v


And here

r

v


r

v


r

vr

v


r

vr

v

The time taken for both the position vector and the velocity vector to complete one cycle must be the same.

r

vr

v

How long does it take the position to complete one cycle?

r

vr

v


Circumference divided by the velocity.

r

vr

v


Circumference divided by the thing that is changing: v.

v

rP

2

r

vr

v

How long does it take the velocity to complete one cycle?

v

rP

2

r

vr

v


The circumference divided by the thing that is changing: a

v

rP

2

r

vr

v


The circumference divided by the thing that is changing: a

v

rP

2

a

vP

2

r

vr

v

But the periods P are the same for both.

v

rP

2

a

vP

2

r

vr

v

But the periods P are the same for both. Hence,

a

v

v

r 22

r

vr

v

But the periods P are the same for both. Hence,

r

va

a

v

v

r 222

Centripetal Accelerationv

v’

r

r

x

r

tvr

x

tvxdt

xv

Centripetal Acceleration

v at Av

v’

r

v

t

v

r

tv

v

vv

v

r

tv

2

v at B

v

Apply Newton’s 2nd Law

r

vmmaF

2

Apply Newton’s 2nd Law

r

vmmaF

2

r

v

rmF 2

Isaac Newton (1642 - 1727)Isaac Newton (1642 - 1727)Principia Mathematica 1667Principia Mathematica 1667

Newton’s Laws of Motion:Newton’s Laws of Motion:1) A particle will continue moving in a 1) A particle will continue moving in a


2) Application of a force, 2) Application of a force, FF causes an causes an

acceleration, acceleration, aa, given by, given by ma=Fma=F

3) Action & reaction are equal and opposite.3) Action & reaction are equal and opposite.

Isaac Newton (1642 - 1727)Isaac Newton (1642 - 1727)Principia 1667Principia 1667

Newton’s Law of Newton’s Law of

Universal GravitationUniversal Gravitation

Newton’s Law of Universal Newton’s Law of Universal

GravitationGravitation

MSun

mPr



MSun

mPr

F=GMSunmP

r 2



MSun

mPr

F=GMSunmP

r 2



MSun

mPr

F=GMSunmP

r 2

How did Newton derive this law?



MSun

mPr

F=GMSunmP

r 2

He made it up



MSun

mPr

F=GMSunmP

r 2

Its an educated guess



MSun

mPr

F=GMSunmP

r 2

He made a few educated guesses

Until he found one that worked.

Isaac Newton (1642 - 1727)Isaac Newton (1642 - 1727)To keep the planet in anTo keep the planet in an

orbit of radius orbit of radius rr, requires a , requires a

centripetal force centripetal force FF(centripetal)(centripetal). .

This is provided by the Sun’sThis is provided by the Sun’s

gravitational force gravitational force FF(grav)(grav)..

FF(centripetal)(centripetal) = F = F(grav)(grav)

Using the astronomer’s notation,r = a = semi-major axis

Notice that this law applies to all planets, asteroids etcall planets, asteroids etc

orbiting the sun.

3242 a

M sunG

π=P

P = period a = semi-major axis

MSun= Solar mass ( M⊙)

Notice that this law applies to all objects all objects orbiting the sun.

Earth has P = 1 yr, a = 1 AUP2 4 2

GMSuna3

(AU)a=(yrs)P 32

Kepler’s 2nd Law

Kepler’s 2nd Law

The line joining a planet to the sun sweeps out equal areas in equal time.

A consequence of the law of conservation of momentum

The ice skaterThe ice skater

Conserves Conserves

AngularAngular

MomentumMomentum

Angular Momentum isAngular Momentum isL = = Momentum lever arm

Illustrate for circular motion:Illustrate for circular motion:

r m

vConservation isConservation isL = = constant

2mr=mvr=L

A

r

r

v

A

r

r

v

Area swept out on one second is:

P

rA

2

A

r

r

v

Area swept out on one second is: but P = 2p/w

P

rA

2

A

r

r

v

Area swept out on one second is: but P =

2

22 r

P

rA

A

r

r

v

Area swept out on one second is: but P = and v = r

2

22 r

P

rA

A

r

r

v

Area swept out on one second is: but P = and v = r

22

22 vrr

P

rA

Conservation of Momentum

constant mvrL


constant mvrL

2

vrA


constant mvrL

m

LvrA

22


constant mvrL

m

LvrA

22

L, 2, and m are all constant, hence A must be a constant.

Real Planetary OrbitsReal Planetary Orbits

BothBoth bodies orbit bodies orbit

about a about a

common centre of mass.common centre of mass.


Both bodies orbitBoth bodies orbit

about a about a

common centre of mass.common centre of mass.

MASSESMASSES

1:1

1:2

1000:1SUN:Jupiter

reflex motionof SUN 12.4 m/s


Kepler's 3Kepler's 3rdrd Law Law

(Newton's Form)(Newton's Form)

3

21

22 4

aM+MG

π=P

Earth’s Moon 27.32 days 0.055 5.14

Example

• Jupiter’s moon Europa has a period of 3.55 days and its average distance from the planet is 671,000 km. Determine the mass of Jupiter.

2

34

GP

amm EJ

2

34

GP

amm EJ

We know 4, , a, G, and P; but neither of the two masses, giving one equation with two unknowns.

2

34

GP

amm EJ


2

34

GP

amm EJ


Make the reasonable assumption that the mass of Europa is zero.

2

34

GP

amm EJ


Make the reasonable assumption that the mass of Europa

is zero (i.e., that mJ + mE = mJ).

2

34

GP

amJ

kg 109.18640055.31067.6

1071.64 27211

38

Jm

In Solar Units In Solar Units

a in AU P in years a in AU P in years

M in solar massesM in solar massesM≈ a3

P2

aEu ropa=671×106 /1.496×1011=4.49×10-3 AU

PEuropa=3.55/365.25=9.7×10-3 years

M Jupiter=0.962×10-3MSun

THETHE END END OF LECTURES 4-OF LECTURES 4-

66

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OUR UNIVERSE WEEK 2 Lectures 4 - 6. 2. Know how Tycho Brahe revolutionized the practice of astronomy. Know Kepler's three laws and be able to explain