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OUR UNIVERSE
WEEK 2
Lectures 4 - 6
LEARNING GOALS
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.
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.
Astronomy seems to have
Astronomy
seems
to
have
been practised by
been practised by
most ancient civilisations.
most ancient civilisations.
Many
ideas,
myths
and
Many ideas, myths and misconceptions
misconceptions
haveand
occurred
have occurred over
over.
We follow over
a Western
and history
over. from the
Ancient
(400
BC to
thethe
present)
We
follow Greeks
a Western
history
from
Ancient
Greeks (400 BC to the present)
Gravitation & Planetary Motion
or
The Copernican Revolution
Geocentric
versus
Heliocentric
cosmogony
Explains
diurnal
motion of stars
Sun & Moon
rotate with
celestial
sphere,
Def. A theory of the
Earth's
but also drift
place in the Universe
slowly
with respect
to stars.
The Geocentric
cosmogony
Explains
diurnal
motion of stars
Merry-go-round
analogy
Sun & Moon
rotate with
celestial sphere,
but also drift
slowly
with respect
to stars.
Gravitation & Planetary Motion
The key difficulty is theMARS
retrograde motion
of the planets (wanderers)
In the geocentric view
this required epicycles
July 2005 to February 2006
Mars’
retrograde motion
MARS
Aristotle (384-322 BC)
• Earth does not feel
as if it’s moving
• Natural state for any body
is to be stationary
•The circle: the perfect form
• Cycles & epicycles required
Geocentric explanation
of retrograde motion
Ptolemy (140 AD)
in Alexandria’s Library
set up precise epicycles
to fit the observed
planetary motions.
Geocentric explanation
of retrograde motion
Ptolemy (140 AD)
in Alexandria’s Library
set up precise epicycles
to fit the observed
planetary motions.
Ptolemy (140 AD)
• Refined the geocentric model to a
high degree
•Very accurate, but also very
complicated - 80 circles!
•Refinements kept being added to
account for data.
•No coherent theory behind it.
Ptolemey’s
13 -Volume
Almagest
covered 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 Copernicus (1473 - 1543)
Polish Polymath: Lawyer, physician,
economist, canon of the church,
and artist.
Gifted in Mathematics and influenced by
the ideas of Aristarchus, he turned to
Astronomy in the early 1500’s.
Nicolaus Copernicus (1473 - 1543)
The heliocentric model explains
retrograde motion easily.
Nicolaus Copernicus (1473 - 1543)
Worked out many details:
Ordering of planetary orbits.
• Mercury & Venus, Inferior planets,
always seen near Sun.
• Mars, Jupiter, Saturn, Superior planets,
sometimes seen on opposite side of the
celestial sphere to Sun, high
above horizon - Earth between Sun and
these planets.
Nicolaus Copernicus (1473 - 1543)
Explained why planets appear in
different parts of the sky on different
dates
• Mercury & Venus, Inferior planets,
seen in west near Sunset, then in east
just before sunrise - elongation.
• Mars, Jupiter, Saturn, Superior planets,
best seen at night in 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
Venus45o – 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
Venus45o – 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 Period
Let
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
T
2

Hence
Elongation Period
T
2

But ω = ωp – ωe
Hence
Elongation Period
T
Hence
2

But ω = ωp – ωe
Hence
2
T
 p  e
Elongation Period
Since
Then
2

T
Hence
2
T  2 2 

Tp
Te
Tp/e are the
siderial periods
Elongation Period
Since
Then
2

T
Hence
Te
2
T  2 2  Te
T p  Te
Tp  1
Tp/e are the siderial periods
Elongation Period
Since
Then
2

T
Hence
Te
2
T  2 2  Te
T p  Te
Tp  1
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
S
S
360  360  360
P
E
Relationship between synodic and
siderial periods
S
S
360  360  360
P
E
SP
 360S 
 360 P
E
SP
S 
P
E
P P
1  
E S
1 1 1
  
P E S
Relationship between synodic and
siderial periods
S
S
360  360  360
P
E
SP
 360S 
 360 P
E
SP
S 
P
E
P P
1  
E S
1 1 1
1 1 1
   :
For Superior Planets  
P E S
P E S
for interval
inferiorbetween
planetsuccessive
S is observed as time
overtakings of one planet by the other.
for superior planet just swap:
1
E
1
P
1
P
1
E
1
S
1
S
E ↔P
for superior planet
Box 4-1
Nicolaus Copernicus (1473 - 1543)
Determined planetary distances from
Sun by geometry in terms 1 AU
Planet------Copernicus---Modern
Mercury
0.38 AU
0.39 AU
Venus
0.72 AU
0.72 AU
Mars
1.52 AU
1.52.AU
Jupiter
5.22 AU
5.20 AU
Saturn
9.07 AU
9.54 AU
Nicolaus Copernicus (1473 - 1543)
• His results showed that the larger the
orbit, the longer the period & the smaller
the speed.
• Noticed variable speed on orbits and so
included epicycles to keep using circular
motion!
• This made his model no better than
Ptolemy’s geocentric one to astronomers
at the time. MORE EVIDENCE NEEDED
Copernicus’
De Revolutionibus
Orbium Coelestium
(1543, year of his death)
On the Revolutions
of the Celestial
Spheres
Tycho Brahe (1546 - 1601)
Danish Astronomer:
Observed Supernova Nov. 11, 1572
Danish king financed observatory
Uraniborg (sky castle) on Hven Island.
Made measurements of stars and planets
with unprecedented accuracy.
Repeated measurements with different
instruments to assess errors - pioneer
of our modern practices.
Tycho Brahe (1546 - 1601)
Danish Astronomer:
Observed Supernova Nov. 11, 1572
Danish king financed observatory
Uraniborg (sky castle) on Hven Island.
Made measurements of stars and planets
with unprecedented accuracy.
Repeated measurements with different
instruments to assess errors - pioneer
of our modern practices.
Tycho Brahe (1546 - 1601)
• Attempted to test Copernicus’s ideas
about the planets orbiting the Sun.
• Failed to measure any stellar parallax;
concluded Earth was stationary and
Copernicus wrong. (We now know the stars
were too far away to measure parallax without a
telescope)
• Compiled a massive data base with
1 = 1 arcmin accuracy
(best one can do without a telescope)
Tycho Brahe (1546 - 1601)
• Attempted to test Copernicus’s ideas
about the planets orbiting the Sun.
• Failed to measure any stellar parallax;
concluded Earth was stationary and
Copernicus wrong. (We now know the stars
were too far away to measure parallax without a
telescope)
• Compiled a massive data base with
1 = 1 arcmin accuracy
(best one can do without a telescope)
Johannes Kepler (1571 - 1630)
Employed by Tycho in 1600 in Prague.
After Tycho’s death Kepler inherited his
data and his position as
Imperial Mathematician
of the
Holy Roman Empire.
Johannes Kepler (1571 - 1630)
Employed by Tycho in 1600 in Prague.
After Tycho’s death Kepler inherited his
data and his position as
Imperial Mathematician
of the
Holy Roman Empire.
Johannes Kepler (1571 - 1630)
Kepler could be said to be the first astrophysicist
He could also be said to be the last scientific
astrologer.
(except maybe me)
Johannes Kepler (1571 - 1630)
Astrology was once kind of scientific
Johannes Kepler (1571 - 1630)
Astrology was once kind of scientific
What happened last time Venus rose in the
constellation of the goat? Maybe something like it
will happen again.
Johannes Kepler (1571 - 1630)
Astrology
Disaster:
Johannes Kepler (1571 - 1630)
Astrology
Disaster: from the Greek for bad star
Johannes Kepler (1571 - 1630)
Astrology
Disaster: from the Greek for bad star
Influenza:
Johannes Kepler (1571 - 1630)
Astrology
Disaster: from the Greek for bad star
Influenza: the influence of the stars
Johannes Kepler (1571 - 1630)
Astrology
Even today, how many papers have a regular
astrology column?
Johannes Kepler (1571 - 1630)
Astrology
Even today, how many papers have a regular
astrology column?
But how many have a regular astronomy column?
Johannes Kepler (1571 - 1630)
Astrology
Based on the idea that the position of the planets in
the sky fundamentally affect our lifes.
But there are greater influences.
Johannes Kepler (1571 - 1630)
Kepler believed in the heliocentric model.
29 years of struggle with the data led him to try
elliptical orbits with dramatic success.
He confirmed this by mapping out the shape of
orbits by observations with Earth’s orbit (1 AU) as
baseline.
Johannes Kepler (1571 - 1630)
In Kepler’s time there were only 6 known planets:
Mercury, Venus, Earth, Mars, Jupiter and Saturn.
Johannes Kepler (1571 - 1630)
In Kepler’s time there were only 6 known planets:
Mercury, Venus, Earth, Mars, Jupiter and Saturn.
Why not 20, or 100?
Why these particular spacings?
Before Kepler no one had asked such questions.
Johannes Kepler (1571 - 1630)
Consider an equilateral triangle,
Draw a circle outside and one inside
Johannes Kepler (1571 - 1630)
Consider an equilateral triangle,
Draw one circle outside, one inside and remove the
triangle.
Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the
orbit of Jupiter to the orbit of Saturn.
Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the
orbit of Jupiter to the orbit of Saturn.
Spooky eh!
Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the
orbit of Jupiter to the orbit of Saturn.
Spooky eh! But Kepler was intrigue and expanded
on it.
Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the
orbit of Jupiter to the orbit of Saturn.
Spooky eh! But Kepler was intrigue and expanded
on it. A triangular prism is a tetrahedron
Johannes Kepler (1571 - 1630)
These two circles have the same ratio as did the
orbit of Jupiter to the orbit of Saturn.
Spooky eh! But Kepler was intrigue and expanded
on it. A triangular prism is a tetrahedron
Johannes Kepler (1571 - 1630)
Could a similar geometry relate the orbits of the
other planets?
Johannes Kepler (1571 - 1630)
Could a similar geometry relate the orbits of the
other planets?
Kepler recalled the regular solids of Pythagoras.
There were five.
Johannes Kepler (1571 - 1630)
Could a similar geometry relate the orbits of the
other planets?
Kepler recalled the regular solids of Pythagoras.
There were five.
Johannes Kepler (1571 - 1630)
He believed they nested one within another.
Hence the invisible supports of the 5 solids was the
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)
But no matter how he tried, he could not make it
work very well.
Johannes Kepler (1571 - 1630)
But no matter how he tried, he could not make it
work very well.
Why not?
Johannes Kepler (1571 - 1630)
But no matter how he tried, he could not make it
work very well.
Why not?
Because it was wrong.
Johannes Kepler (1571 - 1630)
But no matter how he tried, he could not make it
work very well.
Why not?
Because it was wrong.
The later discovery of Uranus, Neptune, Pluto, and
the others prove that
Johannes Kepler (1571 - 1630)
He spent 29 years trying to make it work, but in
the end decided that it was the observations that
were right, not his ideas.
Hence, he finally abandoned them.
Astronomy wins over astrology
Johannes Kepler (1571 - 1630)
In abandoning his regular solids, he was also able
to free his mind of the perfect sphere/circle for
orbital motion.
Hence he considered that they may be elliptical.
Drawing
an Ellipse
Johannes Kepler (1571 - 1630)
Kepler’s 3 Laws of planetary motion:
1) Orbital paths of planets are ellipses,
with the Sun at one focus.(1609)
2) Line joining the planet to the Sun
sweeps out equal areas in equal times.
3) The square of a planet’s orbital period
is 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.
Planet
P
Sun at a focus
Empty focus
Kepler’s 1st Law

r
1  e cos 
r and  are polar coordinates
e is the eccentricity of the ellipse
 is the semi-latus rectum
Planet
P
Sun at a focus
Empty focus
Kepler’s 1st Law
Planet

rrP
r and  are polar coordinates
Major axis
Kepler’s 1st Law
Eccentricity e
a b
a
e
 1  
2
a
b
2
Planet

rrP
Semi Major axis a
Semi Minor Axis b
2
2
Kepler’s 1st Law
Eccentricity e
a b
a
e
 1  
2
a
b
2
2
2
Kepler’s 1st Law
Semi Latus Rectum
Planet
=b2/a

rrP

r
1  e cos 
Note that a circle is a special type is ellipse (one with e = 0)
Kepler’s 2nd Law
The line between the sun and a planet sweeps out equal
areas in equal time.
Kepler’s 2nd Law
The 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 Law
The orbit is an ellipse.
Thus, the planet must move faster when near perihelion
than it does near aphelion.
Kepler’s 2nd Law
The 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
Written symbolically,
d
dt

1
2

r   0
2
1
2
r  is the " areal velocity"
2
Kepler’s 3rd Law
The square of the orbital period of a planet is
proportional to the cube of its semi-major axis.
P2  a3
Kepler’s 3rd Law
The 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 Law
The 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 was
the first astronomer to predict
a transit of Venus (for the year 1631)
Galileo Galilei (1564 - 1642)
One of the first to use a telescope
From 1610 onwards he saw:
mountains on the Moon, sunspots on
the Sun, the rings of Saturn,
Jupiter’s moons ( providing a counter
example to the view that Earth is the
centre of the universe)
Galileo Galilei (1564 - 1642)
One of the first to use a telescope,
His observations constitute the
beginnings of modern astronomy. His
defence of the Copernican heliocentric
solar system was published in
The Starry Messenger.
(Siderius Nuncius)
Galileo Galilei (1564 - 1642)
One of the first to use a telescope,
His observations constitute the
beginnings of modern astronomy. His
defence of the Copernican heliocentric
solar system was published in
The Starry Messenger.
(Siderius Nuncius)
Galileo Galilei (1564 - 1642)
He noted that as
the phases of Venus changed,
so did its apparent size.
This provided
decisive evidence against
Ptolemaic geocentric system.
Phases of Venus
as it orbits
a = angular
diameter
(arcsec)
Venus in the Heliocentric system
Venus in the Geocentric system
Galileo Galilei (1564 - 1642)
1610: Using his telescope he
discovered 4 moons orbiting
Jupiter
(the Galilean satellites)
This provided a counterexample to
the view that Earth is the centre of
the universe
Jupiter’s moons
Jupiter’s moons
1610
Galileo
observed
Jupiter’s
moons.
Isaac Newton (1642 - 1727)
One of the greatest scientists
who ever lived:
was a great experimentalist,
mathematician,
&
philosopher of the
scientific method
.
Isaac Newton (1642 - 1727)
One of the greatest scientists
who ever lived:
was a great experimentalist,
mathematician,
&
philosopher of the
scientific method
.
Isaac Newton (1642 - 1727)
Principia Mathematica 1667
Newton’s Laws of Motion:
1) A particle will continue moving in a
straight line unless acted on by a force.
2) Application of a force, F causes an
acceleration, a, given by ma = F
3) Action & reaction are equal and
opposite.
Isaac Newton (1642 - 1727)
Principia 1667
Newton’s derivation of Centripetal
Acceleration for motion in a circle
using:
1) A particle will continue moving in a
straight line unless acted on by a force.
2) Application of a force, F causes an
acceleration, a, given by ma = F
Centripental
Position
Velocity
Centripental
Position
Draw a position vector
Velocity
Centripental
Position
Velocity
Draw a position vector
r
Centripental
Position
Velocity
Draw a position vector
v
r
Centripental
Position
Velocity
Draw a position vector
Draw that velocity vector
v
r
Centripental
Position
Velocity
Draw a position vector
Draw that velocity vector
v
r
Centripental
Position
Velocity
Draw that velocity vector
v
r
Draw a position vector
some time dt later
Centripental
Position
Velocity
Draw that velocity vector
r
v
r
Draw a radius vector
some time dt later
Centripental
Position
Velocity
v
r
v
r
Draw a position vector
some time dt later
Centripental
Position
Velocity
v
r
v
r
Draw a position vector
some time dt later
Draw that new velocity vector
Centripental
Position
Velocity
v
r
v
r
Draw a position vector
some time dt later
Draw that new velocity vector
Centripental
Position
Velocity
v
r
v
r
Now draw an acceleration vector
Centripental
Position
Velocity
v
r
v
r
Now draw an acceleration vector
Centripental
Position
Velocity
v
r
v
r
And here
Now draw an acceleration vector
Centripental
Position
Velocity
v
r
v
r
Centripental
Position
Velocity
v
r
v
r
The time taken for both the position vector and the velocity vector to
complete one cycle must be the same.
How long does it take the position to complete one cycle?
v
r
v
r
How long does it take the position to complete one cycle?
Circumference divided by the velocity.
v
r
v
r
How long does it take the position to complete one cycle?
Circumference divided by the thing that is changing: v.
2r
P
v
v
r
v
r
How long does it take the velocity to complete one cycle?
2r
P
v
v
r
v
r
How long does it take the velocity to complete one cycle?
The circumference divided by the thing that is changing: a
2r
P
v
v
r
v
r
How long does it take the velocity to complete one cycle?
The circumference divided by the thing that is changing: a
2r
P
v
2v
P
a
v
r
v
r
But the periods P are the same for both.
2r
P
v
2v
P
a
v
r
v
r
But the periods P are the same for both. Hence,
2r 2v

v
a
v
r
v
r
But the periods P are the same for both. Hence,
2r 2v
v

a
v
a
r
2
v
r
v
r
Centripetal Acceleration
v
v
dx
dt
 dx  vdt
d 
dx
r
vdt
d 
r
dx
r
d
r
v’
Centripetal Acceleration
vdt
d 
r
d 
dv
dv
v
vdt


v
r
dv v 2

dt r
v
v at A
d
v’
dv
v at B
Apply Newton’s 2nd Law
2
v
F  ma  m
r
Apply Newton’s 2nd Law
2
v
F  ma  m
r
v

r
F  m r
2
Isaac Newton (1642 - 1727)
Principia Mathematica 1667
Newton’s Laws of Motion:
1) A particle will continue moving in a
straight line unless acted on by a force.
2) Application of a force, F causes an
acceleration, a, given by ma=F
3) Action & reaction are equal and opposite.
Isaac Newton (1642 - 1727)
Principia 1667
Newton’s Law of
Universal Gravitation
Newton’s Law of Universal
Gravitation
mP
r
MSun
Newton’s Law of Universal
Gravitation
mP
r
MSun
F= G
M S un m P
r
2
Newton’s Law of Universal
Gravitation
mP
r
MSun
F= G
M S un m P
r
2
Newton’s Law of Universal
Gravitation
m
P
How did Newton derive this law?
r
MSun
F= G
M S un m P
r
2
Newton’s Law of Universal
Gravitation
m
P
He made it up
r
MSun
F= G
M S un m P
r
2
Newton’s Law of Universal
Gravitation
m
P
Its an educated guess
r
MSun
F= G
M S un m P
r
2
Newton’s Law of Universal
Gravitation
m
P
He made a few educated guesses
r
Until he found one that worked.
MSun
F= G
M S un m P
r
2
Isaac Newton (1642 - 1727)
To keep the planet in an
orbit of radius r, requires a
centripetal force F(centripetal).
This is provided by the Sun’s
gravitational force F(grav).
F(centripetal) = F(grav)
Using the astronomer’s notation,
r = a = semi-major axis
Notice that this law
applies to all planets, asteroids etc
orbiting the sun.
 4π 2 
2
3


P =
a
 G M sun 


P = period
a = semi-major axis
MSun= Solar mass ( M⊙)
Notice that this law applies to all objects
orbiting the sun.
P
2
4 2
3
a
GM Sun
Earth has
P = 1 yr, a = 1 AU
2
3
P (yrs) = a (AU)
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 skater
Conserves
Angular
Momentum
Angular Momentum is
L = Momentum  lever arm
Illustrate for circular motion:
L = mvr = mr
Conservation is
L = constant
r
2
v
m
v
r
A
r
Area swept out on one second is:
A
r
2
P
v
r
A
r
Area swept out on one second is: but P = 2p/w
A
r
2
P
v
r
A
r
Area swept out on one second is: but P = 2/
r 2
r 2
A

P
2
v
r
A
r
Area swept out on one second is: but P = 2/ and v = r
r 2
r 2
A

P
2
v
r
A
r
Area swept out on one second is: but P = 2/ and v = r
r 2
r 2 vr
A


P
2
2
v
r
A
r
Conservation of Momentum
L  mvr  constant
Conservation of Momentum
L  mvr  constant
vr
A
2
Conservation of Momentum
L  mvr  constant
vr
L
A

2 2m
Conservation of Momentum
L  mvr  constant
vr
L
A

2 2m
L, 2, and m are all constant, hence A must be a constant.
Real Planetary Orbits
Both bodies orbit
about a
common centre of mass.
MASSES
Real Planetary Orbits
1:1
Both bodies orbit
1:2
about a
1000:1
common centre of mass.
SUN:Jupiter
reflex motion
of SUN 12.4 m/s
Real Planetary Orbits
rd
Kepler's 3 Law
(Newton's Form)

 3
4π
a
P = 
 GM 1 + M 2  
2
2
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.
4a
m J  mE 
2
GP
3
4a
m J  mE 
2
GP
3
We know 4, , a, G, and P; but neither of the two masses,
giving one equation with two unknowns.
4a
m J  mE 
2
GP
3
We know 4, , a, G, and P; but neither of the two masses,
giving one equation with two unknowns.
4a
m J  mE 
2
GP
3
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.
4a
m J  mE 
2
GP
3
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).
4a
mJ 
2
GP
3


4 6.7110
27
mJ 
 1.9 10 kg
2
11
6.67 10 3.55  86400
8 3
In Solar Units
a in AU P in years
M in solar masses
3
a
M≈
P2
a E u r op a = 6 7 1× 1 0 6 /1 .4 9 6× 1 0 1 1 = 4 .4 9× 1 0 -3 A U
P E u rop a = 3 .5 5 /3 6 5 .2 5 = 9 .7 × 1 0 -3 y e ars
-3
M Ju p it e r = 0.9 6 2 × 1 0 M S u n
THE END
OF LECTURES 46