#### Transcript Inferior planets.

```Astronomy 2
Overview of the Universe
Winter 2006
2. Lectures on Copernicus to Modern Times.
Joe Miller
Nicholas Copernicus (1473-1543)
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Proposed heliocentric system.
Inferior planets, superior planets, planetary configurations.
Seasons.
Sidereal, synodic periods.
Distances of planets from the sun, assuming circular orbits.
Parallax problem still there.
Siderial period: one revolution around the sun with respect to
the stars.
Synodic period: the time between two successive identical
geometrical configurations.
Geometrical configurations:
Opposition: a planet is in opposition when it is opposite the
sun in the sky.
Conjunction: a planet is lined up with the sun.
A superior planet has its orbit outside the earth’s orbit, an
inferior planet’s orbit is inside.
Planetary Configurations
The time between two oppositions is one synodic period- easy to observe.
But how do you determine the siderial period of another planet? We know
what the earth’s is- one siderial year.
Inferior planet case:
Let S = synodic period, E =period of the earth,
and P = siderial period of planet.
In one day earth goes 360o/E degrees around the sun.
In one day planet goes 360o/P degrees around the sun.
In one synodic period, earth goes S(360o/E), and
planet goes S(360o/P) . But an inferior planet makes
one extra trip around the sun (it “laps” the earth),
so it goes an extra 360o. Therefore
S(360o/P) - 360o = S(360o/E),
or dividing through by S and 360o, we have
1/P - 1/S = 1/E. Since E = 1 year, we can write 1/P = 1 + 1/S Inferior planets.
In like manner, for superior planets it is the earth that travels the extra 360 o, so 360o has to
be subtracted from the distance the earth travels and we get 1/P = 1 - 1/S. Superior
planets.
The general formula:
1/S = 1/P(faster) - 1/P(slower)
The earth is the faster for superior planets, while it is
the slower for exterior planets.
Copernicus determined the relative distances of the
planets from the sun. He made direct use of the
heliocentric system, the known sidereal and synodic
periods of the planets, and observations of the times
of specific planetary configurations: greatest
elongation for inferior planets, and quadrature and
opposition for superior planets.
1) Orbits of planets are perfect circles.
2) All planets travel at constant speed in their orbits.
He defined the distance of the earth to the sun to be
1.0, so that all other distance would be in terms of
the earth-sun distance.
Inferior planets (Mercury and Venus)
Superior planets (Mars, Jupiter, and Saturn)
Results
The seasons: primarily the result of the tilt of the earth’s axis.
Tycho Brahe (1546-1601)
• A Danish nobleman a disagreeable snob.
• A clever instrument builder and excellent
observer.
• Supernova of 1572 was beyond the moon’s orbit.
• Comet of 1577 was at least three times the
distance to the moon- no detectable parallax- and
probably in orbit around the sun.
• 30 years of very accurate data on planetary
positions.
• Hired Johannes Kepler.
• Invented the “Tychonic System.”
The “Tychonic” System
Johannes Kepler (1571-1630)
Kepler was a staunch believer in the Copernican
System
• His assignment by Tycho was to go over the
observations of Mars’ positions. He spent 25 years
doing this.
• He was driven to find order and harmony in the
heavens. His reasoning could flip rapidly
between the mystical and the rigorously scientific.
• He fit the five regular solids of Euclid between the
orbits of the planets, which he thought was
demonstration of the harmony of the universe.
Kepler first investigated the shape of Mars’ orbit.
The approach was to use pairs of observations of
Mars separated by one sidereal period of Mars.
These pair would be made when Mars was in the
same position in its orbit, but the earth was at a
different position in its orbit.
By this approach Kepler finally worked out that the
orbit of Mars was not a circle, as had been believed
for nearly 20 centuries, but a mathematical curve
called an ellipse.
The ellipse is a conic section:
Another view of conic sections:
C
a
Mathematics of ellipses :
a  semi - major axis
b  semi - minor axis
c  distance of focus from center
c
 e the ellipticity or eccentricity of the ellipse
a
c
 0 is a circle.
a
Kepler’s First Law:
A planet travels around the sun in an orbit of
elliptical shape with the sun at one focus.
Kepler’s Second Law:
The line from a planet to the sun sweeps over equal
areas in equal times.
In modern terminology, this is a result of the
conservation of angular momentum.
Kepler' s Third Law : The Harmonic Law
The squares of the periods of the planets
are proportional to the cubes of their mean
distances from the sun.
p 2  a 3 or p 2  Ka3 , where K is a constant
that depends on the choice of units for time and distance.
If units of earth years and astronomical units are
used, then K  1.0 and
2
3
p
a
1
1
p2  a3
or you could write

when you are
2
3
p2
a2
comparing two objects in orbit around the same central object.
P. 70
Galileo Galilei (1564-1642)
• Contemporary of Kepler
• Modern scientist- rejected Aristotelian approach.
• Excellent observer and experimenter.
– Explained earthshine.
– Discovered the law of the pendulum
– Noticed speed of hailstones.
• Founded the science of mechanics.
• Heard about a telescope and immediately built
one. His observations of the sky profoundly
confronted modern beliefs.
Galileo founded modern mechanics: the study
of the motions of objects.
• The Law of Falling Bodies:
– In a vacuum, all bodies fall with the same
uniform acceleration, regardless of their size
or mass.
– Acceleration is the rate of change of velocity. It
has funny-looking units: km/sec/sec or
ft/sec/sec, etc.
• This contrary to Aristotle, who held that the
heavier a body was, the faster it would fall
compared to a lighter body.
Uniformly accelerated motion
Velocity changes the same amount over each
identical period of time. Consider dropping an
object:
Acceleration of gravity is 9.8m/sec/sec=32 ft/sec/sec
After 1 sec
2 sec
3 sec
falling 9.8 m/sec
19.6 m/sec
29.4 m/sec
At any time t, the velocity v is given by
v  at, where a is the acceleration.
Distance fallen:
The distance d traveled by an object is given by
d  vt. If the velocity is changing during the
time period, then one must use the average velocity.
For uniform acceleration starting from rest,
velocity is the final velocity divided by 2
The appropriate average velocity
v
v(final)
at

2
2
d
:
v is
Therefore d  vt 
1 2
at
2
the average
at
1
t  at 2
2
2
Galileo’s Law of Inertia
A body set into uniform motion will remain in
uniform motion until interfered with.
By “uniform motion” we “mean moving with a
constant speed in a straight line.” So the object
neither changes speed nor direction.
This also means that an object at rest will remain at
rest until it is interfered with. This idea is
fundamental to our understanding of physics. Why
did it take us 2000 years to get to this? Aristotle
taught it is “natural” for all objects to remain at rest.
Motion is unnatural and would be resisted.
Some examples:
1. A weight dropped from the mast of a ship.
2. Playing catch in an airplane.
3. A billiard ball.
Galileo’s discoveries with a telescope
• Craters and mountains on the moons- a
tremendous challenge to prevailing.thinking.
• Sunspots.
• The Milky Way consists of many unresolved stars.
• Saturn was not always round, but sometimes had
two blobs near it.
• Venus went through phases: a real challenge to
Aristotelian views.
• Discovered four moons of Jupiter! Caused
tremendous excitement.
The Beginnings of Modern Physics
C. Huygens.
Explaining it all--the laws of mechanics.
First, a brief digression:
There are two kinds of quantities:
Scalar quantities have just a size or
magnitude
Vector quantities have both a size and
direction
Examples:
Temperature is a scalar. It is just an amount.
Speed is a scalar. It measure how fast
something is going.
Velocity is a vector. It specifies both speed
and direction.
Acceleration is a vector quantity. It is a measure of
the rate of change of velocity, a vector quantity,
with respect to time. A change in velocity can be the
result of any of three things:
•
•
•
•
A change in speed, but not direction.
A change of direction, but not speed.
A change of both speed and direction.
Examples:
– Car speeding up or slowing down on a straight
– Car traveling at constant speed on a circular
track.
Huygens’ formula for acceleration in circular motion
a  acceleration, a vector quantity
r  radius of circular path
v  velocity, a vector quantity
The two triangles are similar, so
v vt

, where  means change.
v
r
Therefore
v
v2
a
t
r
Isaac Newton (1643-1727)
• In his early 20’s, most of it in two years, he
– Invented calculus
– Developed the laws of motion
– Developed the Law of Gravity
– Developed optics theory
– Developed a theory of colors
• He performed a great synthesis of the results of
Copernicus, Galileo, Kepler, and Huygens,
ultimately published in
Philosophae Naturalus Principia Mathematica
Among many other things, the “Principia” vastly
simplified a great range of behavior associated with
moving objects into three apparently “simple” laws.
But first Newton introduced a new concept, mass.
Mass is the quantity of matter, of stuff, that an object
has. The mass of an object is the same on the moon
as it is on earth.
Second, he made clear that another important
physical quantity is force, which is defined as a
push or pull. Weight is a force, the amount of force
an object exerts on a scale. An object’s weight is
different on the moon than on the earth.
Newton’s Three Laws of Motion.
First Law:
In the absence of forces, the motion of an object does
not change.
By “motion” we mean the “velocity.” Therefore,
without forces there can be no accelerations. An
object at rest will remain at rest unless it is acted
upon by a force.
An alternative look at the First Law:
Newton introduced another new quantity,
momentum.
Momentum = mass times velocity= mv, a vector
quantity.
The First Law can be stated as
The momentum of an object does not change when
no forces act on it.
It takes the action of a force to change the
momentum of an object.
Newton’s Second Law:
A force acts so as to accelerate an object. The
amount of acceleration is directly proportional to the
applied force, but is inversely proportional to the
object’s mass.
In terms of simple formulae:
F
a
m
F  ma
acceleration equals force divided by mass
force = mass times acceleration.
An alternative way of writing Newton’ Second Law
Force = rate of change of the momentum, or
(mv)
F
t
"" means " change in"
Newton’s Third Law
For every action there is equal and opposite
reaction.
or
Forces come in opposing pairs.
or
In any system of objects, the total momentum is a
constant.
P. 80
Newton’s three laws provide a complete prescription
for motion. They are very powerful, though they
appear very simple.
Using these three laws and Kepler’s Laws, Newton
was able to explain the motion of the planets in
terms of forces operating and masses involved. This
is called dynamics, as opposed to what Kepler did,
kinematics, a description of motion.
Newton’s starting assumption: there is some kind
of force of attraction between all things in the
universe. Let’s call it gravity. It is responsible for
holding the planets in their orbits. What kind of
force do we need to explain what we observe?
The force required to maintain an object in circular
motion:
F  ma Newton' s Second Law
v2
a
Huygens' Formula
r
mv 2
F 
r

"" means " therefore"
A simple derivation of the Law of Gravity using
Newton' s Laws and Kepler' s Third Law :
Let r  the radius of the planet' s orbit
mp  the planet' s mass
ms  the sun' s mass
v  the orbital speed of the planet
v2
The acceleration a is given by Huygen' s Formula : a =
r
Therefore the force needed to keep the planet in its orbit is
mp v
F =
r
2
Now we have to figure out how to calculate
d
Speed is distance divided by time, or v 
t
Let us use for d the distance around the orbit,
v.
2 r, and the
time t to get around the orbit P, the orbital period of the planet. Then
d 2r
4 2 r2
2
v 
and v 
t
P
P2
4 2 r2
mp ( 2 ) 4 2 m r 2
p
P
Therefore F 

r
rP2
4 2 m pr
2
3
3
F
.
Remember
Kepler'
s
Third
Law
:
P

Ka

Kr
P2
F 
4 2 m p r
Kr3

Cmp
r2
Thus we see that there is a constant part and a variable part to this
force law :
mp
Fp  C 2
r
The force on the planet depends only on the mass of the planet divided by
the square of its distance from the sun. This is an example of an
inverse square law.
Now Newton could apply his Third Law. There must be an equal and
opposite force on the sun to accelerate the sun and conserve momentum.
Newton showed this will only happen if this equal force on the sun obeys
a similar law
m
Fs  C2 2s , where is C2 some constant.
r
Thus the force depends on both the mass of the sun and the mass of the planet
F
msmp
r2
or
FG
msmp
, where G is called the constant of gravity
2
r
:
Newton then made one of the most extreme
generalizations in science. He postulated that this
gravitational force exists between any two masses:
m1 m2
FG 2
d
This applies to any two masses at any separation!
Newton derived Kepler’s three laws using his three laws of
motion and his law of gravity.
• First Law: all orbits of one object traveling around another
are conic sections: circles, ellipses, parabolas, or
hyperbolas.
• Second Law: Equal Areas rule is just an example of the
conservation of angular momentum.
• Third Law. Since how gravity works is directly involved,
masses have to be introduced, and Newton modified
Kepler’s Third Law:
m
2
3

m
P

Ka
where K is a constant. If we use earth years and the

s
p
astronomical unit for distances,
m
then
2
3

m
P

a
. If we let the mass of the sun

s
p
= 1.0,
then the masses of most planets are essentially negligable and we get back to
Kepler' s version.
How is G measured?
The Cavendish Balance:
The Law of Falling Bodies Revisited
Remember that Newton' s Second Law states
a
F
m1
m1 .
But the Law of Gravity gives
F G
m1m2
. Therefore
2
d
m1m2
G 2
m
d
a
 G 22 . Acceleration does not depend on m1!
m1
d
But what is the acceleration near the surface of the earth caused by the earth?
The effect of the earth’s gravity at its surface
Newton showed that the combined effect of all the
mass of the earth acted as if all the mass of the earth
was in one point at its center.
Inside the earth, only the matter closer to the center
mattered.
Tides were explained by Newton as a result of
gravitational forces:
Thus the tidal force is a differential gravitational
force. The effect of the moon is about three times
that of the sun.
Newton’s greatest triumph- the discovery of Neptune
• Herschel- discovered Uranus by accident on
March 13, 1781. By 1790 it was realized that its
orbit was not as expected. Conclusion: another
unknown planet was “perturbing” its motion.
• 1845- Adams calculated a position for the
unknown planet and sent it to the Astronomer
Royal Airy. Airy gave him a test problem to check
him out, but Adams refused to do it.
• 1845- Simultaneously Leverrier did the same
calculation and sent the position to Airy. He did
the problem, and Airy started looking for the
planet. He botched the job!
Neptune (cont.)
• September 23, 1845- Leverrier got tired of waiting
and sent a note to Galle in Berlin with the
position. The note arrived on this date. That very
night Galle checked it out and found what we now
call Neptune only 52 minutes of arc from the
predicted position!
• Thus Neptune was discovered because of its
gravitational effects, not its light.
Are Newton’s Laws true?
Until the 19th century everything looked good,
except for one “minor problem”: the rate that
Mercury’s orbit was precessing:
Newton’s physics predicted this would change at
531 seconds of arc per century. The observation was
574, a discrepancy of 43. Why. Another planet?
Newton’s physics not accurate. Einstein to the
rescue!
Finally, the first parallax of a star was observed by
Bessel in 1838. The parallax of the nearest star is
only 0.74 seconds of arc!
Summary of historical developments in astronomy
• Without telescope:
– Eclipses of the sun and moon predictable
– The earth’s axis precesses.
– The relative distances of the planets from the sun
accurately worked out.
– The orbits of the planets are shown to be ellipses with
the sun at one focus.
– The speed of the planets in their orbits varies according
to a simple law
– The periods and distances of the planets from the sun