Transcript Tycho: The most accurate pre

Models of the Solar System
• Positions of planets change, whereas stars
appear relatively ‘fixed’
• Greeks held on to the Geocentric model
because they could not observe stars to
change their positions, and therefore
thought that the earth must be stationary
• Ptolemy, Aristotle and others refined the
geocentric model
• But there were problems…….such as the
path reversal by Mars  Retrograde motion
Retrograde motion of Mars
(path reversal seen in the Sky)
Epicycles – Ptolemic Geocentric Model
How do we know the Earth is
spherical ?
• Stars differ from place to place
• Northern and southern hemispheres
• What kind of an object always has a round
shadow ?
Earth Shadow during Lunar
Eclipse
Multiple Exposure
Photograph
Cyrene
Alexandria
Syene
Tropic of Cancer
The Spherical Earth
• The ancient Greeks had deduced not only
that the Earth is spherical but also
measured its circumference !
Eratosthenes’s method to measure the circumference of the earth
At noon on summer solstice day the
Sun is directly overhead at Syene,
but at an angle of 7o at Alexandria
7º
7 = Distance (Alexandria - Syene)
---------------------------------------360
Circumference of the Earth
Sunlight
Alexandria
Answer: 40,000 stadia
= 25,000 mi !
Syene
Earth
Earth-Moon-Sun Geometry
Aristarchus’s determination of distances
(Closer the S-E-M angle to 90, the farther the Sun)
If we replace the moon with a planet, then can determine relative distances,
as done by Copernicus
Copernicus
Copernican Model:
Inferior and Superior Planets
(orbits inside or outside the Earth’s orbit)
Configurations of Inferior
Planets, Earth, and the Sun
Earth
Inferior
Conjunction
Superior
Conjunction
Configurations of Superior
Opposition
Earth
Planets, Earth, and the Sun
Conjunction
Synodic (apparent) period – one conjunction to next (or one opposition to next)
Synodic and Sidereal Orbital Periods
• Inferior planets are never at opposition;
superior planets can not be at inferior
conjunction
• Copernican model of orbital periods
• Synodic period is the apparent orbital
period of a planet, viewed from the earth,
when the earth-planet-sun are in
successive conjunction or opposition
• Sidereal (with respect to stars) period is
the real orbital period around the Sun
• Synodic periods of outer planets (except
Mars) are just over one year
Apparent (Synodic) and true (Sidereal –
with respect to stars) orbital periods of
planets differ due to Earth’s relative motion
Synodic periods of all outer planets (except mars) are just over 1 year because
their Sidereal periods are very long and they are in opposition again soon after
an earth-year
Earth-Venus-Sun
Inferior planets appear farthest away from the Sun at ‘greatest elongation
Measurements of Distances to Planets
Maximum
Eastern Elongation
P
Angle of max
elongation
= P-E-S
90 deg
Earth
E
P-E-S
S
Maximum Western
Elongation
Sin (P-E-S) = PS / ES
ES = 1 AU
Copernicus first determined the
relative distances of planets
Copernican Heliocentric Model:
(Retrograde motion of Mars seen when
Earth overtakes Mars periodically)
Earth is closer to the Sun, therefore moves faster than Mars
Tycho: The most accurate
pre-telescopic observer
Tycho charted very accurately the movement of Mars in the Sky, but still believed
In the Geocentric Universe
Kepler – Tycho’s assistant
(used Tycho’s data to derive Kepler’s
Laws)
Planetary Orbits
• The Copernican heliocentric model is
essentially correct
• But it consisted of circular orbits which did
not exactly fit observations of planetary
positions
• Kepler realized, based on Tycho’s data of
the orbit of Mars, that orbits are elliptical
 Kepler’s First Law
• However, the difference for Mars is tiny, to within the
accuracy of drawing a circle with a thick pen !
Kepler’s First Law:
All planetary orbits are elliptical,
with the Sun at one focus
Eccentricity ‘e’:
e = distance between foci/major axis
= AB / ab
a
A
B
b
A circle has e = 0, and a straight line has e = 1.0
Kepler’s Second Law:
Planetary radius sweeps equal area
triangles in equal time
It follows that the velocity of the planet must vary according to distance from
the Sun -- fastest at Perihelion and slowest at Aphelion
Kepler’s Third Law: P2 = a3
P – Orbital Period, a – semi-major axis
What is the size ‘a’ of the orbit of a comet with the period ‘P’ of 8 years?
Kepler’s Laws
• Empirically derived from observational
data largely from Tycho (e.g. observations
of the positions of Mars in its orbit around
the Sun)
• Theoretical explanation had to await
Newton’s discovery of the Law of
Gravitation
• Universally valid for all gravitationally
orbiting objects (e.g. stars around black
holes before falling in)
Galileo
Galileo’s Discoveries With Telescope
• Phases of Venus
- Venus displays phases like the Moon as it revolves around the Sun
• Mountains and “seas” on the Moon
- Other objects in the sky are like the Earth (not therefore special)
• Milky Way is made of stars like the Sun
• Sunspots
- “Imperfections” or “blemishes” in otherwise perfect “heavenly” objects
• 4 Galilean satellites of Jupiter
- Objects in the sky revolve around other objects, not the Earth (i.e. other moons)
All of these supported the Copernican System
Galileo also conducted experiments on gravity :
Regardless of mass or weight objects fall at the same rate
Phases of Venus
Venus is never too far from the
Sun, therefore can not be in
opposition like the Moon.
Changing phases of Venus
demonstrate that it orbits the
Sun like the Earth.
Orbits and Motions
• Orbits can not be circular since objects do
NOT revolve around each other, but
around their common center-of-mass
• The Earth and the Moon both revolve
around each other
• This motion is in addition to Earth’s
Rotation, Revolution, Precession
The Earth-Moon Barycenter
• The earth and the moon both revolve
around a common center of mass called
the Barycenter
• The barycenter of Sun-planet systems lies
inside the Sun
• As the earth is much more massive, the
barycenter lies 1700 Km inside the earth
• Calculate its position ‘O’ from
M(E) x EO = M (M) x MO
M
E
O
Gravity
• Galileo’s observations on gravity led to
Newton’s Law of Gravitation and the three
Laws of Motion
• Objects fall at the same rate regardless of
mass because more massive objects have
more inertia or resistance to motion
•
Fgrav = G (m1 x m2) / r2
• Force of gravity between two masses is
proportional to the product of masses divided
by distance squared  ‘inverse square law’
Newton – Three Laws of Motion
1. Inertia
2. F = ma
3. Action = Reaction
Newton’s Laws of Motion
• Law of Inertia: A body continues in state of
rest or motion unless acted on by an external
force; Mass is a measure of inertia
• Law of Acceleration: For a given mass m, the
acceleration is proportional to the force applied
F=ma
• Law of Action equals Reaction: For every
action there is an equal and opposite reaction;
momemtum (mass x velocity) is conserved
Velocity, Speed, Acceleration
• Velocity implies both speed and direction;
speed may be constant but direction could
be changing, and hence accelerating
• Acceleration implies change in speed or
direction or both
• For example, stone on a string being whirled
around at constant speed; direction is
constantly changing therefore requires force
Ball Swung around on a String:
Same Speed,
(in uniform circular motion)
Changing Direction
(swinging around the circle)
Donut Swung around on a String
Acceleration
Force
Conservation of momemtum:
action equal reaction
• The momemtum (mv) is conserved before and
after an event
• Rocket and ignited gases:
M(rocket) x V(rocket) = m(gases) x v(gases)
• Two billiard balls:
m1 v1 + m2 v2 = m1 v1’ + m2 v2’
v1,v2 – velocities before collision
v1’,v2’ – velocities after collision
• Example – you and your friend (twice as
heavy) on ice!
Action = Reaction
Equal and Opposite
Force from the Table
Net Force is Zero,
No Net Motion
Force = (apple’s mass)  (acceleration due to gravity)
Acceleration due to gravity
• Acceleration is rate of change of velocity, speed or
direction of motion, with time  a = v/t
• Acceleration due to Earth’s gravity : a  g
g = 9.8 m per second per second, or 32 ft/sec2
•
Speed in free-fall
T (sec)
v (m/sec)
v (ft/sec)
0
0
0
1
9.8
32
2
19.6
64
3
29.4
96
60 mi/hr = 88 ft/sec (between 2 and 3 seconds)
Galileo’s experiment revisited
• What is your weight and mass ?
• Weight W is the force of gravity acting
on a mass m causing acceleration g
• Using F = m a, and the Law of Gravitation
W = m g = G (m MEarth) /R2
(R – Radius of the Earth)
The mass m of the falling object cancels
out and does not matter; therefore all
objects fall at the same rate or acceleration
g = GM / R2
i.e. constant acceleration due to gravity 9.8
m/sec2
Galileo’s experiment on gravity
• Galileo surmised that time differences
between freely falling objects may be too
small for human eye to discern
• Therefore he used inclined planes to slow
down the acceleration due to gravity and
monitor the time more accurately
v
Changing the angle of the incline changes the velocity v
‘g’ on the Moon
g(Moon) = G M(Moon) / R(Moon)2
G = 6.67 x 10-11 newton-meter2/kg2
M(Moon) = 7.349 x 1022 Kg
R(Moon) = 1738 Km
g (Moon) = 1.62 m/sec/sec
About 1/6 of g(Earth); objects on the
Moon fall at a rate six times slower than on
the Earth
Escape Velocity and Energy
• To escape earth’s gravity an object must have
(kinetic) energy equal to the gravitational
(potential) energy of the earth
• Kinetic energy due to motion
K.E. = ½ m v2
• Potential energy due to position and force
P.E. = G m M(Earth) / R
(note the similarity with the Law of Gravitation)
• Minimum energy needed for escape: K.E. = P.E.
½ m v2 = G m M / R
Note that the mass m cancels out, and
• v (esc) = 11 km/sec = 7 mi/sec = 25000 mi/hr
The escape velocity is the same for all objects of
mass m
Object in orbit  Continuous fall !
Object falls towards the earth at the same rate as the earth curves away from it
Angular Momentum
Conservation of angular momentum says that
product of radius r and momentum mv must be
constant  radius times rotation rate (number of
rotations per second) is constant
Angular Momentum
• All rotating objects have angular momentum
• L = mvr ; acts perpendicular to the plane of
rotation
• Examples: helicopter rotor, ice skater, spinning
top or wheel (experiment)
• Gyroscope (to stabilize spacecrafts) is basically
a spinning wheel whose axis maintains its
direction; slow precession like the Earth’s axis
along the Circle of Precession
Conservation of Angular Momentum
• Very important in physical phenomena
observed in daily life as well as throughout
the Universe. For example,
• Varying speeds of planets in elliptical
orbits around a star
• Jets of extremely high velocity particles,
as matter spirals into an accretion disc and
falls into a black hole
1
Relativistic
Jet “From” Black
Hole
1. “Relativistic velocities are close to the speed of light