Transcript Lecture2 - University of Waterloo

The size of the solar system
• The size of Earth, Moon and the Sun
• Distances to SS objects
• Kepler’s laws and elliptical orbits
The Solar System
• Remarkably, with a few careful
observations it is possible to
measure the scale of the solar
system
Size and shape of Earth
•
The Earth has been known to be spherical since the time of the
early Greeks. Some of the evidence in favour of this was:
Size and shape of Earth
•
The Earth has been known to be spherical since the time of the
early Greeks. Some of the evidence in favour of this was:
1. at sea, land at sea level disappears before hills; hulls of ships at sea
vanish before their masts
Size and shape of Earth
•
The Earth has been known to be spherical since the time of the
early Greeks. Some of the evidence in favour of this was:
1. at sea, land at sea level disappears before hills; hulls of ships at sea
vanish before their masts
2. the altitude of stars in the sky depends on how far north or south
the observer is
Size and shape of Earth
•
The Earth has been known to be spherical since the time of the
early Greeks. Some of the evidence in favour of this was:
1. at sea, land at sea level disappears before hills; hulls of ships at sea
vanish before their masts
2. the altitude of stars in the sky depends on how far north or south
the observer is
3. in lunar eclipses (Earth passing between Sun and Moon) the shadow
is always circular
Size and Shape of Earth
• Eratosthenes used the assumption of a spherical Earth and his
observation of the difference of altitude of the Sun at Syene
(directly overhead on a known date) and at Alexandria, 5000
stadia farther north.
• At Alexandria the Sun was 7.2° north of
overhead
• Using basic geometry he related the
distance on the spherical Earth (5000
stadia) to the angular distance around the
circumference (7.2°) and found a radius of
~39000 stadia.
• We don’t know exactly what this unit is in
our terms, but best estimates suggest 1
stadium = 157 m; with this conversion
Eratosthenes’ method gives a radius for
Earth of ~6250km. This is very close to
the modern value of 6378km.
The Moon: Eclipses
Eclipses occur when the Moon comes between the Earth and Sun.
 Provides clear evidence that Moon is closer than Sun
Solar Eclipses
• Eclipses are so spectacular because of
the purely coincidental fact that the
moon and Sun have similar angular sizes
Lunar Eclipses
Lunar eclipses occur when Earth blocks sunlight to the Moon
Lunar eclipses always
have rounded edge:
further evidence that
Earth is spherical.
Distance to the Moon
Lunar eclipses can be used to determine distance to the Moon
• Angular diameter of the Sun is 0.53 degrees
• Knowing Earth’s diameter (13,000 km) you can find the extent of
Earth’s shadow: 1.4 million km.
• From observing the radius of curvature of the shadow we see the
angular size of Earth’s shadow at the distance of the Moon is
about 1.5 degrees.
• Can use geometry to show distance to Moon is about 350,0000 km
Given the angular size of the moon (0.5 deg) and its distance of
350,000 km we can find its size.
Distance to the Sun
• Aristarchos observed the angle between the Moon and Sun at
quarter phase; this told him the relative distances of Sun and
Moon.
 He measured this angle to be 87 degrees. The modern value is
89.75 degrees. (Why is this measurement hard?)
 Sun is about 400 times farther away than Moon
 Since Sun and Moon have the same apparent diameter when viewed
from Earth, the Sun must also be 400 times larger than the Moon
Planetary motions
• The planets move relative to
the background stars.
• Sometimes they show
complex retrograde motions
• Skygazer demonstration
Epicycles
• Epicycles were introduced
to explain the non-uniform
velocities of planets, in a
geocentric, circular-orbit
theory
Retrograde motion
• Retrograde motion is a
natural outcome of the
heliocentric model
• Inner planets orbit more
quickly than outer planets,
and so “overtake” them
Distances to Interior planets
• Venus and Mercury follow the
Sun around the ecliptic: means
their orbits are smaller than
Earth’s
• At greatest elongation a line
between the Sun and planet is
perpendicular to a line
between Earth and planet.
rPlanet Sun
sin  
rEarth Sun
• E.g. for Venus, =46 degrees, so
the distance from Venus to the
Sun is 0.72 times the EarthSun distance
Distances to exterior planets
• Exterior planets can be found anywhere in the zodiacal belt
• The true orbital period of the planet (sidereal period) tells how
long it takes the planet to return to point P.
• Observe the angles PES(initially) and PES (one superior planet
period later).
• The angle ESE’ is known from the Earth’s orbital period vs. the
planets. And the triangles can be solved.
Break
Tycho Brahe
• Brahe (1546-1601) believed in a geocentric
Universe: the Sun and moon go around the
Earth (but the other planets go around the
Sun)
• However, he also believed that this theory
could be tested by making sufficiently
accurate observations
 At time this was a revolutionary approach:
different from the idea that phenomena
could be understood through philosophical
discourse alone
 Arguably the first application of the
scientific method
Tycho Brahe’s observations
• Made very accurate, naked eye
observations of planetary motion
 Used devices for measuring angles
and positions
 To measure time, he used the
planetary motions themselves.
Clocks were rare and the pendulum
clock had not been invented
sextant
wall quadrant
Kepler’s Laws
Johannes Kepler derived the following 3 empirical
laws, based on Tycho Brahe’s careful
observations of planetary positions
(astrometry).
1.
A planet orbits the Sun in an ellipse, with the
Sun at one focus (supporting the Copernican
heliocentric model and disproving Brahe’s
hypothesis)
2. A line connecting a planet to the Sun sweeps out
equal areas in equal time intervals
3. PP2 2
=aa33, where P is the period and a is the average
distance from the Sun.
What is an ellipse?
Definition: An ellipse is a closed curve defined by the locus of all points such that the sum of the distances
from the two foci is a constant:
r  r  2a
Ellipticity: Relates the semi-major (a) and
semi-minor (b) axes:
a 2  a 2e 2  b 2
b
 1  e2
a
Equation of an ellipse:
r 2  r 2 sin 2   2ae  r cos  
2
Substituting r  r   2a
and rearranging we get:


a 1  e2
r
1  e cos 
Ellipses
Calculate the aphelion and perihelion distances for Halley’s
comet, which has a semi-major axis of 17.9 AU and an
eccentricity of 0.967.
Kepler’s Second Law
2. A line connecting a planet to the Sun sweeps out equal areas in equal time
intervals
  
This is just a consequence of angular momentum conservation. L  r  p
 mrv zˆ
Angular momentum conservation
Since L is constant,
La  L p
(aphelion=perihilion)
mra va  mrp v p
va rp 1  e 

 
v p ra 1  e 
Angular momentum conservation
How much faster does Earth move at perihelion compared
with aphelion? The eccentricity is e=0.0167
vp
va

1  e

1  e 
1.0167
0.9833
 1.034

i.e. 3.4% faster
Orbital angular momentum
We know the angular momentum is constant; but what is its value?
  
Lrp
 rv zˆ
Since L is constant, we
can take A and t at any
time, or over any time
interval.
dA
L  2m
dt
L  2m
 2m
Aellipse
P
a 2 1  e 2
P
Kepler’s Third Law
The general form of Kepler’s third law can be derived from
Newton’s laws.
4 2 a 3
P 
G ( M  m)
2
Circular Velocity
• A body in circular motion will have a constant velocity determined
by the force it must “balance” to stay in orbit.
• By equating the circular acceleration and the acceleration of a
mass due to gravity:
vcirc
GM

r
• where M is the mass of the central body and r is the separation between
the orbiting body and the central mass.
Escape velocity
• Escape velocity is the velocity a mass must have to escape the
gravitational pull of the mass to which it is “attracted”.
• We define a mass as being able to escape if it can move to an
infinite distance just when its velocity reaches zero. At this point
its net energy is zero and so we have:
GMm 1
2
 mvesc
r
2
vesc
2GM

r
Escape velocity
What is the escape velocity at
a) the surface of the Earth?
b) the surface of the asteroid Ceres?
Orbital Energy
GMm m 2 GMm
E
 v 
2a
2
r
• In the solar system we observe bodies of all orbital types:
 planets etc. = elliptical, some nearly circular;
 comets = elliptical, parabolic, hyperbolic;
 some like comets or miscellaneous debris have low energy orbits and we see them
plunging into the Sun or other bodies
orbit type
v
Etot
e
circular
v=vcirc
E<0
e=0
elliptical
vcirc<v<vesc
E<0
0<e<1
parabolic
v=vesc
E=
0
e=1
hyperbolic
v>vesc
E>0
e>1
Vis-Viva Equation
• Since we know the relation between orbital energy, distance, and
velocity we can find a general formula which relates them all –
the Vis Viva equation
1 1 
v (r )  2GM  

 r 2a 
2
• It is derived by integrating the equation for total energy as a
function of distance and incorporating the assumption that total
orbital energy is constant no matter what the distance from the
centre of mass.
• This powerful equation does not depend on orbital eccentricity.
• For instance, if we observe a new object in the SS and know its
current velocity and distance, we can determine its orbital
semimajor axis and thus have some idea where it came from.
Vis-viva equation
A meteor is observed to be traveling at a velocity of 42 km/s as it
hits the Earth’s atmosphere. Where did it come from?
Next Lecture
Physical processes in the SS
 Tidal forces
 Resonances
 Solar Wind