Transcript Ancient Astronomy - University of North Texas

Chapter 2: The Copernican Revolution
The Birth of Modern Science
•Ancient Astronomy
•Models of the Solar System
•Laws of Planetary Motion
•Newton’s Laws
– Laws of Motion
– Law of Gravitation
Scientific Method
Gather data
Form theory
Test theory
Astronomy in Ancient Times
• Ancient people had a better, clearer chance
to study the sky and see the patterns of stars
(constellations) than we do today.
• Drew pictures of constellations;
created stories to account for the figures
being in the sky.
• Used stars and constellations for navigation.
• Noticed changes in Moon’s shape and
position against the stars.
• Created accurate calendars of seasons.
Ancient Astronomy
Stonehenge on the
summer solstice.
As seen from the center of
the stone circle,
the Sun rises directly over
the "heel stone" on the
longest day of the year.
The Big Horn Medicine
Wheel in Wyoming,
built by the Plains Indians.
Its spokes and rock piles are
aligned with the rising and
setting of the Sun and other
stars.
Astronomy in Early Americas
• Maya Indians developed
written language and
number system.
• Recorded motions of Sun,
Moon, and planets -especially Venus.
• Fragments of astronomical
observations recorded in
picture books made of tree
bark show that Mayans
had learned to predict solar
and lunar eclipses and the
path of Venus.
• One Mayan calendar more
accurate than those of
Spanish.
Ancient Contributions to Astronomy
• Egyptians
– recorded interval of floods on Nile
• every 365 days
– noted Sirius rose with Sun when floods due
– invented sundials to measure time of day from
movement of the Sun.
• Babylonians
– first people to make detailed records of movements
of Mercury, Venus, Mars, Jupiter, Saturn
• only planets visible until telescope
Greek Astronomy
• Probably based on knowledge from Babylonians.
• Thales predicted eclipse of Sun that occurred in
585 B.C.
• Around 550 B.C., Pythagoras noted that the
Evening Star and Morning Star were really the
same body (actually planet Venus).
• Some Greek astronomers thought the Earth
might be in the shape of a ball and that moonlight
was really reflected sunlight.
Time Line
• Ancient Greeks
– Pythagoras
– Aristotle
– Aristarchus
– Hipparchus
– Ptolemy
6th century B.C.
348-322 B.C.
310-230 B.C.
~130 B.C.
~A.D. 140
Pythagorean Paradigm
• The Pythagorean Paradigm had three key
points about the movements of celestial
objects:
– the planets, Sun, Moon and stars move in
perfectly circular orbits;
– the speed of the planets, Sun, Moon and
stars in the circular orbits is perfectly
uniform;
– the Earth is at the exact center of the
motion of the celestial bodies.
Aristotle’s Universe:
A Geocentric Model
• Aristotle proposed that
–the heavens were literally
composed of concentric,
crystalline spheres
–to which the celestial
objects were attached
–and which rotated at
different velocities,
–with the Earth at the
center (geocentric).
The figure illustrates the ordering of
the spheres to which the Sun, Moon,
and visible planets were attached.
Planetary Motion
• From Earth, planets
appear to move wrt
fixed stars and vary
greatly in brightness.
• Most of the time, planets
undergo direct motion moving W to E relative
to background stars.
• Occasionally, they
change direction and
temporarily undergo
retrograde motion motion from E to W before looping back.
(retrograde-move)
Planetary Motion: Epicycles
and Deferents
•Retrograde motion was first
explained as follows:
•the planets were attached,
not to the concentric spheres
themselves, but to circles
attached to the concentric
spheres, as illustrated in the
adjacent diagram.
•These circles were called
"Epicycles",and the
concentric spheres to which
they were attached were
termed the "Deferents".
•
(epicycle-move)
Epicycle/Deferent Modifications
In actual models, the
center of the epicycle
moved with uniform
circular motion, not
around the center of
the deferent, but
around a point that was
displaced by some
distance from the
center of the deferent.
This modification predicted planetary motions that
more closely matched the observed motions.
Further Modifiations
• In practice, even this was not
enough to account for the
detailed motion of the planets
on the celestial sphere!
• In more sophisticated epicycle
models further "refinements"
were introduced:
In some cases, epicycles were
themselves placed on epicycles,
as illustrated in the adjacent
figure.
The full Ptolemaic model
required 80 different circles!!
Ptolemy
• 127-151 A.D. in Alexandria
• Accomplishments
– completion of a “geocentric” model of solar system
that accurately predicts motions of planets by using
combinations of regular circular motions
– invented latitude and longitude
(gave coordinates for 8000 places)
– first to orient maps with
NORTH at top and EAST at right
– developed magnitude system to describe brightness of
stars that is still used today
Aristarchus
• 310-230 B.C.
• Applied geometry to find
– distance to Moon
• Directly measure angular diameter
• Calculate linear diameter using lunar eclipse
– relative distances and sizes of the Sun and Moon
• ratio of distances to Sun and Moon by observing angle
between the Sun and Moon at first or third quarter Moon.
• Proposed that the Sun is stationary and that the
Earth orbits the Sun and spins on its own axis
once a day.
Hipparchus
• ~190-125 B.C.
• Often called
“greatest astronomer of antiquity.”
• Contributions to astronomy
– improved on Aristarchus’ method for calculating the
distances to the Sun and Moon,
– improved determination of the length of the year,
– extensive observations and theories of motions of the
Sun and Moon,
– earliest systematic catalog of brighter stars ,
– first estimate of precession shift in the vernal equinox.
Time Line
• Ancient Greeks
– Pythagoras
6th century B.C.
– Aristotle
348-322 B.C.
– Aristarchus
310-230 B.C.
– Hipparchus
~130 B.C.
– Ptolemy
~A.D. 140
• Dark Ages
A.D. 5th - 10th century
– Arabs
– China
translated books, planets positions
1054 A.D. supernova; Crab Nebula
Heliocentric Model - Copernicus
• In 1543, Copernicus proposed
that:
the Sun, not the Earth, is the
center of the solar system.
• Such a model is called a
heliocentric system.
• Ordering of planets known to
Copernicus in this new system
is illustrated in the figure.
• Represents modern ordering
of planets.
• (copernican-move)
Stellar Parallax
• Stars should appear to
change their position with
the respect to the other
background stars as the
Earth moved about its orbit.
• In Copernicus’ day, no
stellar parallax was
observed, so the Copernican
model was considered to be
only a convenient calculation
tool for planetary motion.
• In 1838, Friedrich Wilhelm
Bessel succeeded in
measuring the parallax of
the nearby, faint star
61 Cygni. ( penny at 4 miles)
Time Line
• Ancient Greeks
Pythagoras
6th century B.C.
Aristotle
348-322 B.C.
Aristarchus
310-230 B.C.
Ptolemy
~A.D. 140
• Dark Ages
A.D. 5th - 10th century
• Renaissance
Copernicus
(1473-1543)
Tycho Brahe
Kepler
Galileo
(1546-1601)
(1571-1630)
(1564-1642)
Newton
(1642-1727)
Galileo Galilei
• Galileo used his telescope to
show that Venus went through a
complete set of phases, just like
the Moon.
• This observation was among the
most important in human
history, for it provided the first
conclusive observational proof
that was consistent with the
Copernican system but not the
Ptolemaic system.
Galileo and Jupiter
• Galileo observed 4 points of
light that changed their
positions with time around
the planet Jupiter.
• He concluded that these
were objects in orbit
around Jupiter.
• Galileo called them the
Medicea Siderea-the
“Medician Stars” in honor
of Cosimo II de'Medici,
who had become Grand
Duke of Tuscany in 1609.
Time Line
• Ancient Greeks
Pythagoras
6th century B.C.
Aristotle
348-322 B.C.
Aristarchus
310-230 B.C.
Ptolemy
~A.D. 140
• Dark Ages
A.D. 5th - 10th century
• Renaissance
Copernicus
(1473-1543)
Tycho Brahe
Kepler
Galileo
(1546-1601)
(1571-1630)
(1564-1642)
Newton
(1642-1727)
Tycho Brahe
Tycho Brahe
• Danish astronomer
• Studied a bright new star in sky
that faded over time.
• In 1577, studied a comet
– in trying to determine its distance from Earth
by observing from different locations, noted
that there was no change in apparent position
– proposed comet must be farther from Earth
than the Moon.
• Built instrument to measure positions of
planets and stars to within one arc minute
(1’).
Quadrant
Sextant
Johannes Kepler:
Laws of Planetary Motion
Kepler: Elliptical orbits
The amount of "flattening" of the ellipse is the
eccentricity. In the following figure the ellipses become
more eccentric from left to right.
A circle may be viewed as a special case of an ellipse
with zero eccentricity, while as the ellipse becomes
more flattened the eccentricity approaches one.
(eccentricity-anim)
Elliptical Orbits and Kepler’s Laws
• Some orbits in the Solar System cannot be
approximated at all well by circles
- for example, Pluto’s separation from the Sun
varies by about 50% during its orbit!
According to Kepler’s First Law, closed orbits around
a central object under gravity are ellipses.
As a planet moves in an elliptical orbit, the Sun is at one
focus (F or F’) of the ellipse.
r
F’
C
F
The line that connects the planet’s point of closest approach
As a planet moves in an elliptical orbit, the Sun is at one
to the Sun, the perihelion ...
focus (F or F’) of the ellipse
perihelion
v
r
F’
C
F
… and its point of greatest separation from the Sun,
As a planet moves in an elliptical orbit, the Sun is at one
the aphelion
focus (F or F’) of the ellipse
perihelion
is called the major axis of the ellipse.
v
r
F’
aphelion
C
F
The only other thing we need to know about ellipses is how
to identify the length of the “semi-major axis”, because that
determines the period of the orbit.
“Semi” means half, and so the
semi-major axis a is half the
length of the major axis:
v
r
F’
a
C
F
a
Kepler’s 1st Law:
The orbits of the
planets are ellipses,
with the Sun at one
focus of the ellipse.
Kepler’s 2nd Law
• The line joining
the planet to the
Sun sweeps out
equal areas in
equal times as the
planet travels
around the ellipse.
Orbit-anim
An object in a highly elliptical orbit travels very slowly
when it is far out in the Solar System,
… but speeds up as it passes the Sun.
According to Kepler’s Second Law,
… the line joining the object and the Sun ...
… sweeps out equal areas in equal intervals of time.
equal areas
That is, Kepler’s Second Law states that
The line joining a planet and the Sun sweeps out
equal areas in equal intervals of time.
For circular orbits around one particular mass - e.g. the Sun we know that the period of the orbit (the time for one complete
revolution) depended only on the radius r
- this is Kepler’s 3rd Law:
M
For objects orbiting a common
central body (e.g. the Sun)
in approximately circular orbits,
r
m
v
the orbital period squared is proportional to the orbital
radius cubed.
Let’s see what determines the period for an elliptical orbit:
For elliptical orbits,
the period depends
not on r, but on the
semi-major axis
a instead.
v
r
F’
a
C
F
a
It turns out that Kepler’s 3rd Law applies to all elliptical
orbits, not just circles, if we replace “orbital radius”
by “semi major axis”:
For objects orbiting a common
central body (e.g. the Sun)
the
the orbital
orbital period
period squared
squared is
is proportional
proportional to
to
the
radius
cubed.
the orbital
semi major
axis
cubed.
So as all of these elliptical orbits have the same semi-major
axis a, so they have the same
period.
a
a
So if each of these orbits is around the same massive
object (e.g. the Sun),
So if each of these orbits is around the same massive
object (e.g. the Sun),
then as they all have the same
semi-major axis length a,
So if each of these orbits is around the same massive
object (e.g. the Sun),
then as they all have the same
semi-major axis length a,
then, by Kepler’s
Third Law,
they have the same
orbital period.
Ellipses and Orbits
Ellipse animation
Kepler’s 3rd Law
The ratio of the
squares of the
revolution periods (P)
for two planets is
equal to the ratio of
the cubes of their
semi-major axes (a).
P2 = a3 or P2/a3 = 1
where
P is the planet’s sidereal
orbital period
(in Earth years)
and
a is the length of the
semi-major axis
(in astronomical units)
Astronomical Unit
• One astronomical unit is the
semi-major axis of the Earth’s orbit
around the Sun, essentially the average
distance between Earth and the Sun.
• abbreviation: A.U.
• one A.U. ~ 150 x 106 km
Kepler’s 3rd Law for the Planets
P2 = a3 or P2/a3 = 1
Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
a (A.U.) P(Earth years) P*P/a*a*a eccentricity
0.206
0.98
0.241
0.387
0.007
0.99
0.615
0.723
0.017
1.00
1
1
0.093
0.99
1.881
1.524
0.048
0.99
11.86
5.203
0.054
0.99
29.42
9.537
0.047
0.99
83.75
19.19
0.009
0.99
163.7
30.07
0.249
0.99
248
39.48
Planetary Motions
• The planets’ orbits (except Mercury and
Pluto) are nearly circular.
• The further a planet is from the Sun, the
greater its orbital period.
• Although derived for the six innermost
planets known at the time, Kepler’s Laws
apply to all currently known planets.
• Do Kepler’s laws apply to comets
orbiting the Sun?
Do they apply to the moons of Jupiter?
Kepler’s Laws
• 1st Law: Each planet moves around the
Sun in an orbit that is an ellipse, with the
Sun at one focus of the ellipse.
• 2nd Law: The straight line joining a
planet and the Sun sweeps out equal
areas in equal intervals of time.
• 3rd Law: The squares of the periods of
revolution of the planets are in direct
proportion to the cubes of the semi-major
axes of their orbits.
What’s important so far?
• Through history, people have used the scientific method:
– observe and gather data,
– form theory to explain observations and predict behavior
– test theory’s predictions.
• Greeks produced first surviving, recorded models of universe:
– geocentric (Earth at center of universe),
– other celestial objects in circular orbits about Earth, and
– move with constant speed in those orbits.
• Geocentric models require complicated combinations of deferents
and epicycles to explain observed motion of planets.
Ptolemaic model required 80 such combinations.
• Copernicus revived heliocentric model of solar system, but kept
circular, constant speed orbits.
What’s important so far? continued
• Without use of a telescope, Tycho Brahe made very accurate
measurements of the positions of celestial objects.
• Johannes Kepler inherited Brahe’s data and determined three
empirical laws governing the motion of orbiting celestial objects.
– 1st Law: Each planet moves around the Sun in an orbit that is
an ellipse, with the Sun at one focus of the ellipse.
– 2nd Law: The straight line joining a planet and the Sun
sweeps out equal areas in equal intervals of time.
– 3rd Law: The squares of the periods of revolution of the
planets are in direct proportion to the cubes of the
semi-major axes of their orbits.
• Galileo used a telescope to observe the Moon and planets. The
observed phases of Venus validated the heliocentric model
proposed by Copernicus. Also discovered 4 moons orbiting
Jupiter, Saturn’s rings, named lunar surface features, studied
sunspots, noted visible disk of planets (stars - point sources).
Why do the planets move
according to Kepler’s laws?
Or, more generally,
why do objects move as they do?
Historical Views of Motion
• Aristotle:
two types of motion
– natural motion
– violent motion
• Galileo
– discredited Aristotelian
view of motion
Animations:
Air resistance
Free-fall
Galileo: Why do objects move
as they do?
Slope down,
Slope up,
speed increases. speed decreases.
No slope,
does speed
change?
Without friction,
NO, the speed is
constant!
What is a “natural” state of motion
for an object?
At rest?
Moving with
constant velocity?
Inertia and Mass
Inertia:
a body’s resistance to a change in its motion.
Mass:
a measure of an object’s inertia
or, loosely, a measure of the total amount of
matter contained within an object.
Newton’s First Law
• Called the law of inertia.
• Since time of Aristotle,
it was assumed that a body required
some continual action on it to remain
in motion, unless that motion were a
part of natural motion of object.
• Newton’s first law simplifies concept
of motion.
Animation: collision-1st-law
FORCES and MOTION
• An object will remain
• (a) at rest or
• (b) moving in a straight line at constant speed until
• (c) some net external force acts on it.
What if there is an
outside influence?
To answer this question,
Newton invoked the concept of a
FORCE acting on a body to cause a
change in the motion of the body.
Forces can act
instantaneously
through contact
(baseball bat making contact with the baseball),
or
or
continuously
at a distance.
(gravity keeping the baseball from flying into space).
Velocity and Acceleration
Velocity:
describes the change in position of a body divided by
the time interval over which that change occurs.
Velocity is a vector quantity,
requiring both the speed of the body and its direction.
Acceleration:
The rate of change of the velocity of a body,
any change in the body’s velocity:
speeding up, slowing down, changing direction.
Animation: circularmotion
Newton’s Second Law:
F = ma
• Relates
– net external force F applied to
object of mass m
– to resulting change in motion of object,
acceleration a.
If there is a NET FORCE on an object,
how much will the object accelerate?
For a given force,
F = ma
greater mass (greater inertia),
yields
ma
= ma
F =
smaller acceleration.
F
For a given mass,
F
greater force
yields
greater acceleration.
F
= ma
= ma
smaller mass,
yields
greater acceleration.
smaller force,
yields
smaller acceleration.
Newton and Gravitation
• Newton’s three laws of motion enable
calculation of the acceleration of a body and its
motion,
BUT must first calculate the forces.
• Celestial bodies do not touch -----do not exert forces on each other directly.
• Newton proposed that celestial bodies exert an
attractive force on each other at a distance,
across empty space.
• He called this force “gravitation.”
• Isaac Newton discovered that two bodies
share a gravitational attraction, where the
force of attraction depends on both their
masses:
• Both bodies feel the same force, but in
opposite directions.
This is worth thinking about - for example, drop a pen to
the floor. Newton’s laws say that the force with which
the pen is attracting the Earth is equal and opposite
to the force with which the Earth is attracting the pen,
even though the pen is much lighter than the Earth!
• Newton also worked out that if you keep the
masses of the two bodies constant, the force
of gravitational attraction depends on the
distance between their centers:
mutual force
of attraction
• For any two particular masses, the
gravitational force between them depends
on their separation as:
magnitude
of the
gravitational
force
between 2
fixed
masses
as the separation between the
masses is increased, the
gravitational force of attraction
between them decreases quickly.
distance between the masses increasing
Gravity and Weight
• The weight of an object is a measure of the
gravitational force the object feels in the
presence of another object.
• For example on Earth, two objects with
different masses will have different weights.
•
Fg = m(GmEarth/rEarth2) = mg
• What is the weight of the Earth on us?
• Mass
Mass and Weight
A measure of the total amount of matter
contained within an object;
a measure of an object’s inertia.
• Weight
The force due to gravity on an object.
• Weight and mass are proportional.
•
Fg = mg
where m = mass of the object
and g = acceleration of gravity
acting on the object
Free Fall
• If the only force acting on an object is force of
gravity (weight), object is said to be in a state of
free fall.
• A heavier body is attracted to the Earth with
more force than a light body.
• Does the heavier object free fall faster?
• NO, the acceleration of the body depends on both
– the force applied to it and
– the mass of the object, resisting the motion.
•
g=
F/m =
F/m
Newton’s Law of Gravitation
• We call the force which keeps the Moon in
its orbit around the Earth gravity.
Sir Isaac Newton’s conceptual leap in understanding
of the effects of gravity largely involved his realization
that the same force governs the motion of a falling object
on Earth - for example, an apple - and the motion of the
Moon in its orbit around the Earth.
• Your pen dropping to the floor and a
satellite in orbit around the Earth have
something in common - they are both in
freefall.
Planets, Apples, and the Moon
• Some type of force must act on planet;
otherwise it would move in a straight line.
• Newton analyzed Kepler’s 2nd Law and saw that
the Sun was the source of this force.
• From Kepler’s 3rd Law, Newton deduced that
the force varied as 1/r2.
• The force must act through a distance,
and Newton knew of such a force the one that makes an apple accelerate
downward from the tree to the Earth
as the apple falls.
• Could this force extend all the way to the Moon?
To see this, let’s review Newton’s thought experiment
Is it possible to throw an object into
orbit around the Earth?
On all these trajectories,
the projectile is in free fall
under gravity.
(If it were not, it would travel
in a straight line that’s Newton’s
First Law of Motion.)
If the ball is not given enough “sideways” velocity, its
trajectory intercepts the Earth ...
that is, it falls to Earth eventually.
trajectories
which
complete
orbits,
OnOn
allthe
these
trajectories,
themake
projectile
is in free
fall.the
projectile is travelling “sideways” fast enough ...
… that as it falls, the Earth curves away underneath
On all these trajectories, the projectile is in free fall.
it, and the projectile completes entire orbits
without ever hitting the Earth.
Gravity and Orbits
The Sun’s inward pull of gravity on the planet
competes with the planet’s tendency to continue
moving in a straight line.
Apparent Weightlessness in Orbit
This astronaut on a
space walk is also
in free fall.
The astronaut’s
“sideways” velocity
is sufficient to keep
him or her in orbit
around the Earth.
Let’s take a little time to answer the following question:
• Why do astronauts
in the Space Shuttle
in Earth orbit feel
weightless?
• Some common misconceptions which
become apparent in answers to this
question are:
(a) there is no gravity in space,
(b) there is no gravity outside the Earth’s atmosphere, or
(c) at the Shuttle’s altitude, the force of gravity is very small.
In spacecraft (like the Shuttle) in Earth orbit, astronauts
are in free fall, at the same rate as their spaceships.
On allisthese
That
why they
trajectories,
experience
the weightlessness:
projectile is in free
just
fall.
as
a platform diver feels while diving down towards a pool,
or a sky diver feels while in free fall.
Newton’s Form of Kepler’s 3rd Law
• Newton generalized Kepler’s 3rd Law to include
sum of masses of the two objects in orbit about
each other (in terms of the mass of the Sun).
–
(M1 + M2) P2 = a3
– Observe orbital period and separation of a planet’s
satellite, can compute the mass of the planet.
– Observe size of a double stars orbit and its orbital
period, deduce the masses of stars in binary system.
• Planet and Sun orbit the common center of mass
of the two bodies.
– The Sun is not in precise center of orbit.
Mass of Planets, Stars, and Galaxies
• By combining Newton’s Laws of Motion and Gravitation
Law,
the masses of
astronomical objects can be calculated.
•
•
•
•
•
•
a = v2/r , for circular orbit of radius r
F = ma = mv2/r
mv2/r = Fg = GMm/ r2
v = (GM/r)1/2
P = 2r/v = 2 (r3/GM)1/2
M = rv2/G
• If the distance to an object and the orbital period of the
object are known, the mass can be calculated.