NJIT Physics 320: Astronomy and Astrophysics – Lecture II Carsten Denker Physics Department Center for Solar–Terrestrial Research

September 10, 2003NJIT Center for Solar-Terrestrial Research Celestial Mechanics  Elliptical Orbits  Newtonian Mechanics  Kepler’s Laws Derived  The Virial Theorem

September 10, 2003NJIT Center for Solar-Terrestrial Research Elliptical Orbits  Kepler’s 1 st Law: A planet orbits the Sun in an ellipse, with the Sun at on focus of the ellipse.  Kepler’s 2 nd Law: A line connecting a planet to the Sun sweeps out equal areas in equal time intervals.  Kepler’s 3 rd Law: The average orbital distance a of a planet from the Sun is related to the planets sidereal period P by:

September 10, 2003NJIT Center for Solar-Terrestrial Research Ellipses  Focal points F 1 and F 2 (sun in principal focus)  Distance from focal points r 1 and r 2  Semimajor axis a  Semiminor axis b  Eccentricity 0  e  1  Ellipse defined:

September 10, 2003NJIT Center for Solar-Terrestrial Research Conic Sections

September 10, 2003NJIT Center for Solar-Terrestrial Research Distances in the Planetary System  Astronomical unit [AU], average distance between Earth and Sun: 1 AU =  10 8 km  Light year: 1 ly =  km  Light minute:  10 7 km (1 AU = 8.3 light minutes)  Parsec: 1 pc =  km = ly

September 10, 2003NJIT Center for Solar-Terrestrial Research Newtonian Physics  Galileo Galilei (1564–1642)  Heliocentric planetary model  Milky Way consists of a multitude of stars  Moon contains craters  not a perfect sphere  Venus is illuminated by the Sun and shows phases  Sun is blemished possessing sunspots  Isaac Newton (1642–1727)  1687 Philosophiae Naturalis Principia Mathematica  mechanics, gravitation, calculus  1704 Optiks  nature of light and optical experiments

September 10, 2003NJIT Center for Solar-Terrestrial Research Laws of Motion  Newton’s 1 st 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 unbalanced force.  Newton’s 2 nd Law: The net force (the sum of all forces) acting on an object is proportional to the object’s mass and it’s resultant acceleration.  Newton’s 3 rd Law: For every action there is an equal and opposite reaction.

September 10, 2003NJIT Center for Solar-Terrestrial Research Gravitational Force (Kepler’s 3 rd law, circular orbital motion, M >> m) (constant velocity) (centripetal force)(law of universal gravitation) Universal gravitational constant: 6.67  10 –11 Nm 2 / kg 2

September 10, 2003NJIT Center for Solar-Terrestrial Research Gravity Near Earth’s Surface

September 10, 2003NJIT Center for Solar-Terrestrial Research Potential Energy

September 10, 2003NJIT Center for Solar-Terrestrial Research Work–Kinetic Energy Theorem

September 10, 2003NJIT Center for Solar-Terrestrial Research Escape Velocity Total mechanical energy: Conservation of mechanical energy: Minimal launch speed:

September 10, 2003NJIT Center for Solar-Terrestrial Research Group Problem  What is the minimum launch speed required to put a satellite into a circular orbit?  How many times higher is the energy required to to launch a satellite into a polar orbit than that necessary to put it into an equatorial orbit?  What initial speed must a space probe have if it is to leave the gravitational field of the Earth?  Which requires a a higher initial energy for the space probe – leaving the solar system or hitting the Sun?

September 10, 2003NJIT Center for Solar-Terrestrial Research Center of Mass

September 10, 2003NJIT Center for Solar-Terrestrial Research Binary Star System in COM Reference Frame Reduced mass

September 10, 2003NJIT Center for Solar-Terrestrial Research Energy and Angular Momentum In general, the two–body problem may be treated as and equivalent one–body problem with the reduce mass moving about a fixed mass M at a distance r.

September 10, 2003NJIT Center for Solar-Terrestrial Research Kepler’s 2 nd Law The time rate of change of the area swept out by a line connecting a planet to the focus of an ellipse is a constant.

September 10, 2003NJIT Center for Solar-Terrestrial Research Kepler’s 3 rd Law Virial Theorem

September 10, 2003NJIT Center for Solar-Terrestrial Research Kepler’s 3 rd Law (cont.) Virial Theorem: For gravitationally bound systems in equilibrium, it can be shown that the total energy is always one–half of the time averaged potential energy.

September 10, 2003NJIT Center for Solar-Terrestrial Research Class Project Exhibition Science Audience

September 10, 2003NJIT Center for Solar-Terrestrial Research Homework Class Project  Read the Storyline hand–out  Prepare a one–page document with suggestions on how to improve the storyline  Choose one of the five topics that you would like to prepare in more detail during the course of the class  Homework is due Wednesday September 23 rd, 2003 at the beginning of the lecture!

September 10, 2003NJIT Center for Solar-Terrestrial Research Homework Solutions

September 10, 2003NJIT Center for Solar-Terrestrial Research Homework  Homework is due Wednesday September 16 th, 2003 at the beginning of the lecture!  Homework assignment: Problems 2.3, 2.9, and 2.11  Late homework receives only half the credit!  The homework is group homework!  Homework should be handed in as a text document!