• Homework 1 due Tuesday Jan 15

Celestial MechanicsFun with Kepler and Newton

•Elliptical Orbits

•Newtonian Mechanics

•Kepler’s Laws Derived

•Virial Theorem

Elliptical Orbits ITyco 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 2Kepler’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 3Kepler’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 LawKepler’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 LawKepler’s Third Law:

The Harmonic Law P2=a3

• Semimajor axis vs Orbital Period on a log-log 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….

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.

• 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…

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 • Total Mechanical Energy• Conservation of Energy• Gravitational Potential

energy

• Escape velocity

Derivations on pp37-39

Kepler’s Laws DerivedCenter 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

• Central Force

Angular Momentum Conserved

• Consideration of quantity 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

• 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

• • Results in

• 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

• The Virial Theorem can be proven by considering the quantity and its time derivative along with Newton’s laws

and vector identities

• Many applications in Astrophysics…stellar equilibrium, galaxy clusters,….

• Derivation on pp 50-53