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.
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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.
Relationship between synodic and siderial periods
• 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.
Relationship between synodic and siderial periods
• 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.
Relationship between synodic and siderial periods
• 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.
Relationship between synodic and siderial periods
360360360 E
S
P
S
Relationship between synodic and siderial periods
SEP
S
P
E
P
PE
SPS
PE
SPS
E
S
P
S
111
1
360360
360360360
Relationship between synodic and siderial periods
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)
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
Astrology was once kind of scientific Astrology was once kind of scientific
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
Astrology Astrology
Disaster:Disaster:
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
Astrology Astrology
Disaster: from the Greek for bad star Disaster: from the Greek for bad star
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
Astrology Astrology
Disaster: from the Greek for bad star Disaster: from the Greek for bad star
Influenza:Influenza:
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
Astrology Astrology
Disaster: from the Greek for bad star Disaster: from the Greek for bad star
Influenza: the influence of the starsInfluenza: the influence of the stars
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
Astrology Astrology
Even today, how many papers have a regular Even today, how many papers have a regular
astrology column?astrology column?
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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?
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
Consider an equilateral triangle,Consider an equilateral triangle,
Draw a circle outside and one insideDraw a circle outside and one inside
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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!
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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! But Kepler was intrigue and expanded Spooky eh! But Kepler was intrigue and expanded
on it.on it.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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! But Kepler was intrigue and expanded Spooky eh! But Kepler was intrigue and expanded
on it. A triangular prism is a tetrahedronon it. A triangular prism is a tetrahedron
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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! But Kepler was intrigue and expanded Spooky eh! But Kepler was intrigue and expanded
on it. A triangular prism is a tetrahedronon it. A triangular prism is a tetrahedron
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
Could a similar geometry relate the orbits of the Could a similar geometry relate the orbits of the
other planets?other planets?
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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
Spheres enclosing solids
Spheres enclosing solids
All this, is an attempt to fit the orbits of the planets with harmonics in music.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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?
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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?
Because it was wrong.Because it was wrong.
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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?
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
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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
Johannes Kepler (1571 - 1630)Johannes Kepler (1571 - 1630)
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
Kepler’s 3rd LawThe square of the orbital period of a planet is proportional to the cube of its semi-major axis.
P2 a3
Example
Uranus was found to have a period of 84 years.
What is its distance from the Sun?
Kepler’s 3rd LawThe square of the orbital period of a planet is proportional to the cube of its semi-major axis.
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)
Galileo Galilei (1564 - Galileo Galilei (1564 -
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)
Galileo Galilei (1564 - Galileo Galilei (1564 -
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)
Galileo Galilei (1564 - Galileo Galilei (1564 -
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
Galileo Galilei (1564 - Galileo Galilei (1564 -
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
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
Centripental Position Velocity
Centripental Position Velocity
Draw a position vector
Centripental Position Velocity
Draw a position vector
r
Centripental Position Velocity
Draw a position vector
r
v
Centripental Position Velocity
Draw a position vector
r
v
Draw that velocity vector
Centripental Position Velocity
Draw a position vector
r
v
Draw that velocity vector
Centripental Position Velocity
Draw a position vector some time t later
r
v
Draw that velocity vector
Centripental Position Velocity
Draw a radius vector some time t later
r
v
Draw that velocity vector
r
Centripental Position Velocity
Draw a position vector some time t later
r
vr
v
Centripental Position Velocity
Draw a position vector some time t later
r
v
Draw that new velocity vector
r
v
Centripental Position Velocity
Draw a position vector some time t later
r
v
Draw that new velocity vector
r
v
Centripental Position Velocity
r
v
Now draw an acceleration vector
r
v
Centripental Position Velocity
r
v
Now draw an acceleration vector
r
v
Centripental Position Velocity
And here
r
v
Now draw an acceleration vector
r
v
Centripental Position Velocity
r
vr
v
Centripental Position Velocity
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
How long does it take the position to complete one cycle?
Circumference divided by the velocity.
r
vr
v
How long does it take the position to complete one cycle?
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
How long does it take the velocity to complete one cycle?
The circumference divided by the thing that is changing: a
v
rP
2
r
vr
v
How long does it take the velocity to complete one cycle?
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
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, 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
Newton’s Law of Universal Newton’s Law of Universal
GravitationGravitation
MSun
mPr
F=GMSunmP
r 2
Newton’s Law of Universal Newton’s Law of Universal
GravitationGravitation
MSun
mPr
F=GMSunmP
r 2
Newton’s Law of Universal Newton’s Law of Universal
GravitationGravitation
MSun
mPr
F=GMSunmP
r 2
How did Newton derive this law?
Newton’s Law of Universal Newton’s Law of Universal
GravitationGravitation
MSun
mPr
F=GMSunmP
r 2
He made it up
Newton’s Law of Universal Newton’s Law of Universal
GravitationGravitation
MSun
mPr
F=GMSunmP
r 2
Its an educated guess
Newton’s Law of Universal Newton’s Law of Universal
GravitationGravitation
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
Conservation of Momentum
constant mvrL
2
vrA
Conservation of Momentum
constant mvrL
m
LvrA
22
Conservation of Momentum
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.
Real Planetary OrbitsReal Planetary Orbits
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
Real Planetary OrbitsReal Planetary Orbits
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
We know 4, , a, G, and P; but neither of the two masses, giving one equation with two unknowns.
2
34
GP
amm EJ
We know 4, , a, G, and P; but neither of the two masses, giving one equation with two unknowns.
Make the reasonable assumption that the mass of Europa is zero.
2
34
GP
amm EJ
We know 4, , a, G, and P; but neither of the two masses, giving one equation with two unknowns.
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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