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 ( ) 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 ( ) 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 P 2 =a 3 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 P 2 =a 3 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….

The Geometry of Elliptical Motion Description in book (pp Done on blackboard)

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 ( ) 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/ ) 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 Influence of Spherical Body Derivation in book (pp 33-35) done on blackboard

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 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

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