Download Lecture2 - University of Waterloo

Celestial Mechanics
Planetary motions
• The planets move relative to
the background stars.
• Sometimes they show
complex retrograde motions
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.
Sidereal and synodic periods
It is easy to observe the synodic period: this is the time between
successive oppostions (when the Earth, Sun and planet are
aligned).
But how do we know when a planet has completed one sidereal
period (i.e. is in the same position relative to the background
stars?
The angular velocity of a circular
orbit is 360/P. The synodic rate
is the rate of the planet relative
to the Earth. So:
360
360
360


PSynodic PEarth P sidereal
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.
Break
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
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
Example: the dwarf planet Eris has a
small moon, Dysnomia. This moon
orbits at a distance of about 30,000
km, with a period of about 14 days.
What is the combined mass of the
Eris/Dysnomia system?