Download Celestial Mechanics Fun with Kepler and Newton

• Homework 1 due Tuesday Jan 15
Celestial Mechanics
Fun with Kepler and Newton
•Elliptical Orbits
•Newtonian Mechanics
•Kepler’s Laws Derived
•Virial Theorem
Elliptical Orbits I
Tyco Brahe’s (1546-1601) Observations
• Uraniborg Observatory - Island of Hveen, King
Frederick II of Denmark
• Large Measuring Instruments (Quadrant)
High Accuracy (better than 4’)
• Demonstrated that comets farther than the moon
• Supernova of 1572
Universe Changes
• No clear evidence of the motion of Earth through
heavens
concluded that Copernican model
was false
Elliptical Orbits 2
Kepler’s (1571-1630) Analysis
• Painstaking analysis of Brahe’s Data
• Heliocentrist but still liked spheres/circles…until he
could not get agreement with observations. Two
points off by 8’
considered the possibility that
orbits were elliptical in shape.
• Minor mathematical but major philosophical change
• Assuming elliptical orbits enabled Kepler to fit all of
Tycho Brahe’s data
Elliptical Orbits 3
Kepler’s Laws of Planetary Motion
Kepler’s First Law: A planet orbits the Sun in an ellipse,
with the Sun at one focus of the ellipse.
Kepler’s Second Law: A line connecting a planet to the
Sun sweeps out equal areas in equal time intervals
Kepler’s Third Law: The Harmonic Law
P2=a3
Where P is the orbital period of the planet measured in
years, and a is the average distance of the planet
from the Sun, in astronomical units (1AU = average
distance from Earth to Sun)
Kepler’s First Law
Kepler’s First Law: A
planet orbits the Sun in
an ellipse, with the Sun at
one focus of the ellipse.
• a=semi-major axis
• e=eccentricity
• r+r’=2a - points on
ellipse satisfy this
relation between sum
of distance from foci
and semimajor axis
Kepler’s Second Law
Kepler’s Second Law:
A line connecting a
planet to the Sun
sweeps out equal
areas in equal time
intervals
Kepler’s Third Law
Kepler’s Third Law:
The Harmonic Law
P2=a3
• Semimajor axis vs
Orbital Period on a loglog plot shows harmonic
law relationship
The Geometry of Elliptical Motion
Can determine distance from
focal point to any point along
elliptical path by using
Pythagoras’s help….
b 2 = a 2 (1- e 2 )
r¢2 = r 2 sin 2 q + (2ae + r cos q ) 2
r¢2 = r 2 = 4ae(ae + r cos q )
r + r¢ = 2a
a(1- e 2 )
r=
1+ ecosq
Conic Sections
• By passing a plane
though a cone with
different orientations
one obtains the
conic sections
–
–
–
–
Circle (e=0.0)
Ellipse (0<e<1.0)
Parabola (e=1.0)
Hyperbola (e>1.0)
Galileo Galilei
(1564-1642)
• Experimental Physicist
• Studied Motion of Objects
– formulated concept of inertia
– understood acceleration (realized that objects of
different weights experienced same acceleration
when falling toward earth)
• Father of Modern Astronomy
–
–
–
–
Resolved stars in Milky Way
Moons of Jupiter
Craters on Moon
Sunspots
• Censored/Arrested by Church…Apology
1992
Sir Isaac Newtonian
(12/25/1642- 1727)
• Significant Discoveries and theoretical
advances in understanding
–
–
–
–
motion
Astronomy
Optics
Mathematics
• …published in Principia and Optiks
Newton’s Laws of Motion
• Newton’s First Law: The Law of Inertia. An object at rest will
remain at rest and an object in motion will remain in motion in a
straight line at a constant speed unless acted upon by an
external force.
• Newton’s First Law: The net force (thesum of all forces) acting
on an object is proportional to the object’s mass and its
resultant acceleration.
n
Fnet = å Fi = ma
i=1
• Newton’s Third Law: For every action there is an equal and
opposite reaction
Newton’s Law of Universal
Gravitation
• Using his three laws of motion along
with Kepler’s third law, Newton
obtained an expression describing the
force that holds planets in their orbits…
Mm
F =G 2
r
Gravitational Acceleration
• Does the Moon’s
acceleration due to
the earth “match”
the acceleration of
objects such as
apples?
Work and Energy
•
•
•
•
•
•
Energetics of systems
Potential Energy
Kinetic Energy K = 1 mv 2
2
Total Mechanical Energy
Conservation of Energy
Gravitational Potential
Mm
energy
U = -G
r
• Escape velocity
v esc = 2GM /r
Derivations on pp37-39
Kepler’s Laws Derived
Center of Mass Reference Frame
and Total Orbital Angular Momentum
• Displacement vector
• Center of mass position
• Reduced Mass
• Total Orbital Angular
Momentum
• Definitions on pp 39-43.
Derivation of Kepler’s First Law
• Consider Effect of Gravitation on the
Orbital Angular Momentum
L = mr ´ v = r ´ p
• Central Force
Angular Momentum Conserved
• Consideration of quantity a ´ L leads
to equation of ellipse describing orbit!!!
• Derivation on pp43-45
Derivation of Kepler’s Second Law
• Consider area element
swept out by line from
principal focus to planet.
• Express in terms of
angular momentum
dA 1 L
=
dt 2 m
• Since Angular
Momentum is conserved
we obtain the second law
• Derived on pp 45-48
Derivation of Kepler’s Third Law
• Integration of the
expression of the 2nd
law over one full period
dA 1 L
=
dt 2 m
•
• Results in
A=
1L
P
2m
2
4
p
P2 =
a3
G(m1 + m2 )
• Derived on pp 48-49
Virial Theorem
• Virial Theorem: For gravitationally bound systems in
equilibrium the Total energy is always one half the time
averaged potential energy
U
E=
2
• The Virial Theorem can be proven by considering the quantity
Q º å pi · ri
and its time derivative along with Newton’s laws
i
and vector identities
• Many applications in Astrophysics…stellar equilibrium, galaxy
clusters,….
• Derivation on pp 50-53