Shape of the Earth, Motion of the Planets and the Method of Least Squares. Probal Chaudhuri
|
|
- Suzanna Moore
- 6 years ago
- Views:
Transcription
1 Shape of the Earth, Motion of the Planets and the Method of Least Squares Probal Chaudhuri Indian Statistical Institute, Kolkata IMS Public Lecture March 20, 2014
2 Shape of the Earth French astronomer Jean Reicher in South America in 1672 Reicher s pendulum clock was losing 2 minutes 28 seconds every day in French Guyana.
3 Shape of the Earth Isaac Newton and his Principia in 1687 Newton claimed that the Earth s rotation around its axis made it an Oblate Sphere. Rotation caused flattening at the Poles and bulging at the Equator. Many French Scientists, however, believed that Newton was wrong. There was yet another conflict between the English and the French scientists.
4 Shape of the Earth The Earth viewed from a Satellite
5 Shape of the Earth (Contd.)
6 Relationship between the Arc Length and the Latitude Let Y be the length of a 1 0 meridian arc centered at latitude θ. Then, assuming the Earth to be an ellipsoid, Y m sin 2 θ + c = mx + c, where X = sin 2 θ. Ellipticity of the Earth can be measured by m/3c. Newton estimated the ellipticity of the Earth to be 1/230.
7 Shape of the Earth French arc measurements around 1720 over a range of 9 0 latitude. The project was led by Dominico Cassini and Jacques Cassini of the Royal Observatory in Paris. Their measurements and analysis contradicted Newton s model for the Earth. French Geodesic Mission during Expeditions to Quito in Ecuador and Lapland in Finland.
8 Meridian Arc Length Data
9 The Problem of Inconsistent Equations m 12 =(y 2 y 1 )/(x 2 x 1 ) (x 2,y 2 ) (x 5,y 5 ) m 34 =(y 4 y 3 )/(x 4 x 3 ) (x 1,y 1 ) (x 4,y 4 ) (x 3,y 3 ) m 45 =(y 5 y 4 )/(x 5 x 4 )
10 Roger Boscovich Roger Joseph Boscovich ( )
11 Boscovich s Least Absolute Deviations Line Fitting We have data points (X 1, Y 1 ), (X 2, Y 2 ),, (X 5, Y 5 ). Y = arc length, X = sin 2 (latitude) To find the line : Y = mx + c that fits the data best. Minimize m,c 5 Y i mx i c. i=1 This is the least absolute deviations problem formulated by Roger Boscovich in Boscovich worked on the problem during
12 Least Absolute Deviations Line Fitting (Contd.) Minimize m Y 1 mx 1 c + + Y n mx n c = Minimize m Y i mx i c. i=1 Y axis X axis
13 The Pairwise Solution m 12 =(y 2 y 1 )/(x 2 x 1 ) (x 2,y 2 ) (x 5,y 5 ) m 34 =(y 4 y 3 )/(x 4 x 3 ) (x 1,y 1 ) (x 4,y 4 ) (x 3,y 3 ) m 45 =(y 5 y 4 )/(x 5 x 4 )
14 The Pairwise Solution (Contd.)
15 The Solar System The Solar System
16 The Problem of the Saturn and the Jupiter : In 1676, Edmund Halley was analyzing the data on the positions of the Jupiter and the Saturn that were tabulated over many centuries. Halley s analysis implied that the Jupiter was accelerating while the Saturn was retarding. The Academy of Sciences in Paris announced a prize in 1748 for a scientific resolution of this crisis in the solar system. Leonhard Euler was the winner of the prize but only for making some interesting attempts to solve the problem.
17 Leonhard Euler Leonhard Euler ( )
18 Pierre-Simon Laplace Pierre-Simon Laplace ( )
19 The Problem of the Saturn and the Jupiter (Contd.) : The solar system was eventually rescued by Pierre Simon Laplace in Laplace showed that the inequalities in the motions of the Jupiter and the Saturn are NOT Secular but Periodic. Laplace was dealing with 24 linear equations in 4 unknowns. He created a system of 4 equations by forming linear combinations of original 24 equations.
20 Laplace s First Attempt to Least Absolute Deviations Problem: Laplace changed Boscovich s formulation of the problem and worked on it during He wanted to minimize max 1 i n Y i mx i c w.r.t. m and c. In 1799 Laplace decided to go back to Boscovich s original least absolute deviations problem and concentrated on that.
21 Laplace s Second Attempt to Boscovich s Problem: Question 1 : How to minimize n i=1 Y i m w.r.t. m? Answer 1 : The minimizing m will be the middle most Y -value after the Y i s are ordered. Question 2 : How to minimize n i=1 Y i mx i w.r.t. m? Answer 2 : Note that Y i mx i = (Y i /X i ) m X i = i=1 i=1 Z i m W i, where Z i = Y i /X i and W i = X i. Order the Z i s, where Z i has weight W i. The minimizing m is the middle most Z -value in the weight distribution. i=1
22 Laplace s Solution (Contd.) : Question 3 : How to minimize n i=1 Y i mx i c w.r.t. m and c? Partial Answer 3 : Assume that the minimizing line passes through ( X, Ȳ ). Then we need to choose m only by minimizing (Y i Ȳ ) m(x i X). i=1
23 The Metric System in Europe : In 1792, after the French Revolution, France and many other European counties decided switch to a new system of measurement - the Metric System. One Meter was defined to be 1/10, 000, 000 of a Meridian Quadrant, which is the distance from the Equator to the North Pole. In 1792, Adrien Marie Legendre, was associated with the French Commission charged with the measurement of a Meridian Quadrant. Legendre invented and published the Principle of Least Squares in 1805.
24 Adrien-Marie Legendre Adrien-Marie Legendre ( )
25 French Meridian Arc Measurements
26 French Meridian Arc Measurements (Contd.)
27 Legendre s Least Squares Line Fitting We have data points (X 1, Y 1 ), (X 2, Y 2 ),, (X n, Y n ). To find the line : y = mx + c that fits the data best. Minimize m,c (Y i mx i c) 2. i=1 (Y i mx i c) 2 = {(Y i Ȳ ) m(x i X) + (Ȳ m X c)} 2. i=1 i=1 = (Y i Ȳ )2 +m 2 (X i X) 2 2m (Y i Ȳ )(X i X)+n(Ȳ m X c) 2. i=1 i=1 i=1 (Ȳ m X c)} 2 is minimized if c = Ȳ m X. So, the optimal line (Y Ȳ ) = m(x X) passes through the point ( X, Ȳ ).
28 Least Squares Line Fitting (Contd.) (Y i Ȳ )2 + m 2 (X i X) 2 2m (Y i Ȳ )(X i X) i=1 i=1 i=1 The optimal m that minimizes the above expression is m = (Y i Ȳ )(X i X) i=1 = (X i X) 2 i=1 (Y i Y j )(X i X j ) i=1 j=1 i=1 j=1. (X i X j ) 2
29 Least Squares Line Fitting (Contd.) m = 1 i<j n ( ) Yi Y j (X i X j ) 2 X i X j. (X i X j ) 2 i=1 j=1 So, optimal m is the weighted average of the slopes of the lines joining pairs of points like (X i, Y i ) and (X j, Y j ).
30 French Meridian Arc Measurements
31 Carl Friedrich Gauss Carl Friedrich Gauss ( )
32 Gauss and the Ceres : In January-February 1801, Italian astronomer Joseph Piazzi observed and recorded data on the dwarf planet Ceres for 41 days before it vanished in the glare of the Sun. Astronomers wanted to predict the position and the time of re-appearance of the Ceres. Many scientists including Laplace thought that it is an impossible problem due to inadequate data. Carl Friedrich Gauss made a very accurate prediction of the time and the position of re-appearance. The Ceres was observed again in November-December 1801.
33 Gauss and the Method of Least Squares : Gauss finished writing his work related to the Method of Least Squares in 1807, and the book was published in Gauss claimed Least Squares as his method and said that he was using it since This is one of the most famous priority disputes in the history of Statistical Science.
34 A Major Breakthrough due to Gauss : Recall Question 3 : How to minimize n i=1 Y i mx i c w.r.t. m and c? Solution due to Gauss : One of the elemental lines is an optimal line.
35 The Linear Programming Formulation : Problem : To minimize n i=1 Y i mx i c w.r.t. m and c Equivalent Linear Programming Problem : To minimize n i=1 R i subject to the constraints R i Y i mx i c R i + mx i + c Y i R i Y i + mx i + c R i mx i c Y i for all i = 1, 2,..., n
SECOND, METRE AND SPEED OF LIGHT. HISTORICAL AND EXPERIMENTAL APPROACH
SECOND, METRE AND SPEED OF LIGHT. HISTORICAL AND EXPERIMENTAL APPROACH Francis Berthomieu EAAE Summer School Working Group (France) Abstract For millenniums, units of time were defined in astronomical
More informationBOUGUER FAILS TO WEIGH THE EARTH, 1739
BOUGUER FAILS TO WEIGH THE EARTH, 1739 Newton s theory of the motion of the planets was very successful. The law of universal gravitation that he used said that any two bodies attract each other: not just
More informationLecture 3 - Least Squares
Lecture 3 - Least Squares In January 1, 1801, an Italian monk pages 1,2 are from Giuseppe Piazzi, discovered a faint, http://www.keplersdiscovery. nomadic object through his telescope in com/asteroid.html
More informationLecture 3 - Least Squares
Lecture 3 - Least Squares In January 1, 1801, an Italian monk pages 1,2 are from Giuseppe Piazzi, discovered a faint, http://www.keplersdiscovery. nomadic object through his telescope in com/asteroid.html
More informationJohannes Kepler ( ) German Mathematician and Astronomer Passionately convinced of the rightness of the Copernican view. Set out to prove it!
Johannes Kepler (1571-1630) German Mathematician and Astronomer Passionately convinced of the rightness of the Copernican view. Set out to prove it! Kepler s Life Work Kepler sought a unifying principle
More information11 Newton s Law of Universal Gravitation
Physics 1A, Fall 2003 E. Abers 11 Newton s Law of Universal Gravitation 11.1 The Inverse Square Law 11.1.1 The Moon and Kepler s Third Law Things fall down, not in some other direction, because that s
More informationRevolution and Enlightenment. The scientific revolution
Revolution and Enlightenment The scientific revolution Background in Revolution In the middle ages, educated europeans relied on ancient authorities like Aristotle for scientific knowledge. By the 15th
More informationGETTING STARTED. Chapter PROBABILITY CURVES
Chapter 2 GETTING STARTED The goals in Part I of this book are to describe and explain basic methods typically covered in an applied statistics course, but from a perspective that helps the reader appreciate,
More informationWhat is a Revolution? A Revolution is a complete change, or an overthrow of a government, a social system, etc.
CW10 p374 Vocab What is a Revolution? A Revolution is a complete change, or an overthrow of a government, a social system, etc. The Scientific Revolution In the 1500s and 1600s the Scientific Revolution
More informationThe Scientific Revolution
The Scientific Revolution What is a Revolution? A Revolution is a complete change, or an overthrow of a government, a social system, etc. The Scientific Revolution In the 1500s and 1600s the Scientific
More informationNEWTON-RAPHSON ITERATION
NEWTON-RAPHSON ITERATION Newton-Raphson iteration is a numerical technique used for finding approximations to the real roots of the equation where n denotes the f ( φ ) = 0 φ φ = n+ 1 n given in the form
More informationSSWH13 The student will examine the intellectual, political, social, and economic factors that changed the world view of Europeans.
SSWH13 The student will examine the intellectual, political, social, and economic factors that changed the world view of Europeans. a. Explain the scientific contributions of Copernicus, Galileo, Kepler,
More informationAPPROXIMATING THE PATH OF A CELESTIAL BODY WITH A CIRCULAR ORBIT FROM TWO CLOSE OBSERVATIONS
1 PPROXIMTING TH PTH OF CLSTIL BODY WITH CIRCULR ORBIT FROM TWO CLOS OBSRVTIONS Thomas J. Osler, Joseph Palma Mathematics Department Rowan University Glassboro, NJ 08028 Osler@rowan.edu bstract Data from
More informationGravity and the Orbits of Planets
Gravity and the Orbits of Planets 1. Gravity Galileo Newton Earth s Gravity Mass v. Weight Einstein and General Relativity Round and irregular shaped objects 2. Orbits and Kepler s Laws ESO Galileo, Gravity,
More informationVenus Project Book, the Galileo Project, GEAR
1 Venus Project Book, the Galileo Project, GEAR Jeffrey La Favre November, 2013 Updated March 31, 2016 You have already learned about Galileo and his telescope. Recall that he built his first telescopes
More informationPlanetary Orbits: Kepler s Laws 1/18/07
Planetary Orbits: Kepler s Laws Announcements The correct link for the course webpage http://www.lpl.arizona.edu/undergrad/classes/spring2007/giacalone_206-2 The first homework due Jan 25 (available for
More informationDEPARTMENT OF ASTRONOMY UNIVERSITY OF FLORIDA
DEPARTMENT OF ASTRONOMY UNIVERSITY OF FLORIDA AST-3043 Exam # 3 - Test Form A Section 0486 Fall, 2011 Periods: T5-6/R6 Dec 6 Special Code 05 Dr. Haywood Smith NAME (PRINT): Last, First, I The scores for
More informationGalileo Galilei. Trial of Galileo before the papal court
Rene Descartes Rene Descartes was a French philosopher who was initially preoccupied with doubt and uncertainty. The one thing he found beyond doubt was his own experience. Emphasizing the importance of
More informationScientific Revolution
Age of Revolutions Scientific Revolution Scientific Revolution Period of time in which a new way of thinking came about. The beliefs held by many for so long were now being questioned. Use logic and reason
More informationKentucky SkyTalk. A History of Gravity: 300 Years of an Attractive Theory. Tim Knauer University of Kentucky 8 January 2015
Kentucky SkyTalk A History of Gravity: 300 Years of an Attractive Theory Tim Knauer University of Kentucky 8 January 2015 Aristotle (384-322 BC) Knew that there were five, and only five, regular solids.
More informationMain Themes: 7/12/2009
What were some of the major achievements of scientists during this period? Why has this period been labeled a revolution? Why was the Scientific Revolution seen as threatening by the Catholic Church? How
More informationDevelopment of Thought continued. The dispute between rationalism and empiricism concerns the extent to which we
Development of Thought continued The dispute between rationalism and empiricism concerns the extent to which we are dependent upon sense experience in our effort to gain knowledge. Rationalists claim that
More informationAnnouncements. Topics To Be Covered in this Lecture
Announcements! Tonight s observing session is cancelled (due to clouds)! the next one will be one week from now, weather permitting! The 2 nd LearningCurve activity was due earlier today! Assignment 2
More informationContents: -Information/Research Packet. - Jumbled Image packet. - Comic book cover page. -Comic book pages. -Example finished comic
Contents: -Information/Research Packet - Jumbled Image packet - Comic book cover page -Comic book pages -Example finished comic Nicolaus Copernicus Nicholas Copernicus was a Polish astronomer who lived
More informationAstronomy- The Original Science
Astronomy- The Original Science Imagine that it is 5,000 years ago. Clocks and modern calendars have not been invented. How would you tell time or know what day it is? One way to tell the time is to study
More informationAstronomy 291. Professor Bradley M. Peterson
Astronomy 291 Professor Bradley M. Peterson The Sky As a first step, we need to understand the appearance of the sky. Important points (to be explained): The relative positions of stars remain the same
More informationChapter 16 The Solar System
Chapter 16 The Solar System Finding the Standard Time and Date at Another Location Example When it is 12 noon in London, what is the standard time in Denver, Colorado (40 N, 105 W)? Section 15.3 Finding
More informationSERIES ARTICLE Dawn of Science
Dawn of Science 19. Measuring the Heavens T Padmanabhan His improved telescope gave Huygens a better vision of the sky, but his ideas about light lay buried for a century. T Padmanabhan works at IUCAA,
More informationTHE SCIENTIFIC REVOLUTION
THE SCIENTIFIC REVOLUTION REVOLUTION: a sudden, extreme, or complete change in the way people live, work, etc. (Merriam-Webster) THE SCIENTIFIC REVOLUTION Time of advancements in math and science during
More informationAP Physics C Textbook Problems
AP Physics C Textbook Problems Chapter 13 Pages 412 416 HW-16: 03. A 200-kg object and a 500-kg object are separated by 0.400 m. Find the net gravitational force exerted by these objects on a 50.0-kg object
More informationDirected Reading B. Section: The Outer Planets
Skills Worksheet Directed Reading B Section: The Outer Planets 1. What is one way that gas giants differ from the terrestrial planets? a. They are much smaller. b. They are rocky and icy. c. They are made
More informationAn Introduction to Celestial Mechanics
An Introduction to Celestial Mechanics This accessible text on classical celestial mechanics the principles governing the motions of bodies in the solar system provides a clear and concise treatment of
More informationTransit of Mercury 9 May 2016
Silversea Discoverer - 12-March-2016 Kris Delcourte MrBlacksun@gmail.com www.flickr.com/photos/krisdelcourte/albums What? 2 Inner planets Mercury and Venus SomeMmes they pass in front of the Sun: the move
More informationLecture 10: General Relativity I
Lecture 10: General Relativity I! Einstein Tower Experiment! Gravitational redshifting! Strong Equivalence Principal! Read Chapter 8! Due to snow and confusion the mid-term is delayed to Thursday March
More informationTycho Brahe ( )
Tycho Brahe (1546-1601) Foremost astronomer after the death of Copernicus. King Frederick II of Denmark set him up at Uraniborg, an observatory on the island of Hveen. With new instruments (quadrant),
More informationNewton s Three Law of Motion
Born in England on Christmas day 1643. Overview Chapter 2b Copernican Revolution Bubonic Plague 1665 While home for 2 years with nothing to do he made his most profound discoveries and proposed his most
More informationThe comic strip of transits
http://mercurio2016.ub.edu The comic strip of transits Josep Manel Carrasco & Jordi Del Río (from a text by Salvador Ribas) Dear reader: On 9th of May of 2016 a Mercury transit is going to happen. A transit
More informationThe Solar System LEARNING TARGETS. Scientific Language. Name Test Date Hour
Name Test Date Hour Astronomy#3 - Notebook The Solar System LEARNING TARGETS I can describe the objects that make up our solar system. I can identify the inner and outer planets. I can explain the difference
More informationThe Metric System. The International System of Units
The Metric System The International System of Units Origin of the Metric System Idea of standardized system of measurement based on the decimal was first proposed as early as ~1670. The first practical
More informationAS3010: Introduction to Space Technology
AS3010: Introduction to Space Technology L E C T U R E S 8-9 Part B, Lectures 8-9 23 March, 2017 C O N T E N T S In this lecture, we will look at factors that cause an orbit to change over time orbital
More informationPhysical Science 1 Chapter 16 INTRODUCTION. Astronomy is the study of the universe, which includes all matter, energy, space and time.
INTRODUCTION Astronomy is the study of the universe, which includes all matter, energy, space and time. Although the universe is vast and almost beyond imagination, much is known about its make-up and
More informationSolar System Perambulations
Solar System Perambulations Bob Albrecht & George Firedrake This work is licensed using a Creative Commons Attribution-Noncommercial 3.0 license. http://creativecommons.org/licenses/by-nc/3.0/ Version
More informationTP 3:Runge-Kutta Methods-Solar System-The Method of Least Squares
TP :Runge-Kutta Methods-Solar System-The Method of Least Squares December 8, 2009 1 Runge-Kutta Method The problem is still trying to solve the first order differential equation dy = f(y, x). (1) dx In
More informationResonance In the Solar System
Resonance In the Solar System Steve Bache UNC Wilmington Dept. of Physics and Physical Oceanography Advisor : Dr. Russ Herman Spring 2012 Goal numerically investigate the dynamics of the asteroid belt
More informationPerihelion Precession in the Solar System
International Journal of Multidisciplinary Current Research ISSN: 2321-3124 Research Article Available at: http://ijmcr.com Sara Kanzi * Hamed Ghasemian! * Currently pursuing Ph.D. degree program in Physics
More informationThe Solar System. Name Test Date Hour
Name Test Date Hour Astronomy#3 - Notebook The Solar System LEARNING TARGETS I can describe the objects that make up our solar system. I can identify the inner and outer planets. I can explain the difference
More informationThe Earth, Moon, and Sky. Lecture 5 1/31/2017
The Earth, Moon, and Sky Lecture 5 1/31/2017 From Last Time: Stable Orbits The type of orbit depends on the initial speed of the object Stable orbits are either circular or elliptical. Too slow and gravity
More informationFluxions and Fluents. by Jenia Tevelev
Fluxions and Fluents by Jenia Tevelev 1 2 Mathematics in the late 16th - early 17th century Simon Stevin (1548 1620) wrote a short pamphlet La Disme, where he introduced decimal fractions to a wide audience.
More informationPHYS 160 Astronomy Test #1 Fall 2017 Version B
PHYS 160 Astronomy Test #1 Fall 2017 Version B 1 I. True/False (1 point each) Circle the T if the statement is true, or F if the statement is false on your answer sheet. 1. An object has the same weight,
More informationThe Earth and its representation
GEOGRAPHY UNIT 1 The Earth and its representation THE SOLAR SYSTEM AND THE EARTH THE SOLAR SYSTEM The solar system is a planetary system. Is a group of astronomical objects who surround a star, in this
More informationEarly Theories. Early astronomers believed that the sun, planets and stars orbited Earth (geocentric model) Developed by Aristotle
Planetary Motion Early Theories Early astronomers believed that the sun, planets and stars orbited Earth (geocentric model) Developed by Aristotle Stars appear to move around Earth Observations showed
More informationhttp://radicalart.info/physics/vacuum/index.html The Scientific Revolution In the 1500s and 1600s the Scientific Revolution changed the way Europeans looked at the world. People began to make conclusions
More informationThe Scientific Revolution & The Age of Enlightenment. Unit 8
The Scientific Revolution & The Age of Enlightenment Unit 8 Unit 8 Standards 7.59 Describe the roots of the Scientific Revolution based upon Christian and Muslim influences. 7.60 Gather relevant information
More informationBoardworks Ltd Asteroids and Comets
1 of 20 Boardworks Ltd 2011 Asteroids and Comets 2 of 20 Boardworks Ltd 2011 What are asteroids? 3 of 20 Boardworks Ltd 2011 Asteroids are large rocks which normally orbit the Sun. Scientists believe that
More informationAstrophysics/Astronomy Problems
Astrophysics/Astronomy Problems Every question counts as a single point, except starred problems (*) which count as 2 points. Double starred problems (**) also count as 2 points, and will be used to decide
More informationLecture 10: General Relativity I
Lecture 10: General Relativity I! Recap: Special Relativity and the need for a more general theory! The strong equivalence principle! Gravitational time dilation! Curved space-time & Einstein s theory
More informationHow Astronomers Learnt that The Heavens Are Not Perfect
1 How Astronomers Learnt that The Heavens Are Not Perfect Introduction In this packet, you will read about the discoveries and theories which changed the way astronomers understood the Universe. I have
More informationThe Scientific Method
The Scientific Method Objectives: 1. Outline the scientific method. 2. Explain why the scientific method has been more successful than other approaches to understanding the universe. 3. Distinguish between
More informationUnit 5 Gravitation. Newton s Law of Universal Gravitation Kepler s Laws of Planetary Motion
Unit 5 Gravitation Newton s Law of Universal Gravitation Kepler s Laws of Planetary Motion Into to Gravity Phet Simulation Today: Make sure to collect all data. Finished lab due tomorrow!! Universal Law
More informationCelestial Mechanics and Orbital Motions. Kepler s Laws Newton s Laws Tidal Forces
Celestial Mechanics and Orbital Motions Kepler s Laws Newton s Laws Tidal Forces Tycho Brahe (1546-1601) Foremost astronomer after the death of Copernicus. King Frederick II of Denmark set him up at Uraniborg,
More informationThe Earth is a Rotating Sphere
The Earth is a Rotating Sphere The Shape of the Earth Earth s Rotation ( and relative movement of the Sun and Moon) The Geographic Grid Map Projections Global Time The Earth s Revolution around the Sun
More informationQuantile Regression: Past and Prospects
Quantile Regression: Past and Prospects Roger Koenker University of Illinois, Urbana-Champaign Northern Illinois University: 31 October 2014 Roger Koenker (UIUC) Quantile Regression NIU: 31.10.2014 1 /
More informationUse of reason, mathematics, and technology to understand the physical universe. SCIENTIFIC REVOLUTION
Use of reason, mathematics, and technology to understand the physical universe. SCIENTIFIC REVOLUTION Background Info Scientific rev gradually overturned centuries of scientific ideas Medieval scientists
More informationSTATION #1: NICOLAUS COPERNICUS
STATION #1: NICOLAUS COPERNICUS Nicolaus Copernicus was a Polish astronomer who is best known for the astronomical theory that the Sun was near the center of the universe and that the Earth and other planets
More informationLecture 5. The Digital Fourier Transform. (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith)
Lecture 5 The Digital Fourier Transform (Based, in part, on The Scientist and Engineer's Guide to Digital Signal Processing by Steven Smith) 1 -. 8 -. 6 -. 4 -. 2-1 -. 8 -. 6 -. 4 -. 2 -. 2. 4. 6. 8 1
More informationCritical Thinking: Sir Isaac Newton
Critical Thinking: Sir Isaac Name: Date: Watch this NOVA program on while finding the answers for the following questions: https://www.youtube.com/watch?v=yprv1h3cgqk 1.In 19 a British Economist named
More informationPhysics Unit 7: Circular Motion, Universal Gravitation, and Satellite Orbits. Planetary Motion
Physics Unit 7: Circular Motion, Universal Gravitation, and Satellite Orbits Planetary Motion Geocentric Models --Many people prior to the 1500 s viewed the! Earth and the solar system using a! geocentric
More informationBe on time Switch off mobile phones. Put away laptops. Being present = Participating actively
A couple of house rules Be on time Switch off mobile phones Put away laptops Being present = Participating actively Het basisvak Toegepaste Natuurwetenschappen http://www.phys.tue.nl/nfcmr/natuur/collegenatuur.html
More informationStatistical Methods in Particle Physics
Statistical Methods in Particle Physics Lecture 10 December 17, 01 Silvia Masciocchi, GSI Darmstadt Winter Semester 01 / 13 Method of least squares The method of least squares is a standard approach to
More informationStudent Guide. 2. Estimate: One possible estimate 2,900,000. Copyright Kendall Hunt Publishing Company
Self-Check: Questions Planet Problems Before you solve the following problems, decide if you need to find an exact answer or an estimated answer. Then choose a strategy to solve each problem. Be ready
More informationSCIENTIFIC REVOLUTION
SCIENTIFIC REVOLUTION VOCABULARY: SCIENTIFIC REVOLUTION Revolution a sweeping change Geocentric earth-centered universe Astronomer scientist who studies the motion of stars and planets Heliocentric sun-centered
More informationAstronomy 311 Professor Menningen January 2, Syllabus overview books & supplies course goals assignments & grading About the professor
1 Astronomy 311 Professor Menningen January 2, 2014 Syllabus overview books & supplies course goals assignments & grading About the professor 2 How to Learn Astronomy Stay curious Interact with the same
More information9.2 Taylor Polynomials
9.2 Taylor Polynomials Taylor Polynomials and Approimations Polynomial functions can be used to approimate other elementary functions such as sin, Eample 1: f at 0 Find the equation of the tangent line
More informationSCIENTIFIC REVOLUTION
SCIENTIFIC REVOLUTION What IS Science? What IS Science? a branch of knowledge or study dealing with a body of facts or truths systematically arranged and showing the operation of general laws: the mathematical
More informationThe N-body Problem. Luis A. Aguilar Instituto de Astronomía/UNAM México
The N-body Problem Luis A. Aguilar Instituto de Astronomía/UNAM México The N-body problem is, perhaps, the oldest and most fruitful unsolved problem in the history of science. The N-body problem is, perhaps,
More informationNewton s Law of Universal Gravitation (attraction between. Gravitation field strength surrounding the Earth g E
VISUAL PHYSICS ONLINE MODULE 5 ADVANCED MECHANICS GRAVITATIONAL FIELD: FORCES Newton s Law of Universal Gravitation (attraction between Gm1m two masses) F F r Gravitational field strength g F m GM Gravitation
More informationGravity and the Laws of Motion
Gravity and the Laws of Motion Mass Mass is the amount of stuff (matter) in an object. Measured in grams (kg, mg, cg, etc.) Mass will not change unless matter is added or taken away. Weight Weight is the
More informationChapter 3 - Gravity and Motion. Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Chapter 3 - Gravity and Motion Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In 1687 Isaac Newton published the Principia in which he set out his concept
More informationResearch Article Geodesic Effect Near an Elliptical Orbit
Applied Mathematics Volume 2012, Article ID 240459, 8 pages doi:10.1155/2012/240459 Research Article Geodesic Effect Near an Elliptical Orbit Alina-Daniela Vîlcu Department of Information Technology, Mathematics
More informationDay 4: Scientific Ideas Change the World
Day 4: Scientific Ideas Change the World Learning Goal 4: Describe how the ideas of Copernicus, Galileo, Newton and Boyle and the invention of the printing press contributed to the Scientific Revolution
More informationPHYSICS. Chapter 13 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.
PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 13 Lecture RANDALL D. KNIGHT Chapter 13 Newton s Theory of Gravity IN THIS CHAPTER, you will learn to understand the motion of satellites
More informationThe Sphere of the Earth Activities
The Sphere of the Earth Activities December 20, 2012 Daniel Ramos MMACA (Museu de Matemàtiques de Catalunya) These activities are designed to be done with the materials, posters and software of the exhibit
More informationHOMEWORK AND EXAMS. Homework Set 13 due Wednesday November 29. Exam 3 Monday December 4. Homework Set 14 due Friday December 8
HOMEWORK AND EXAMS Homework Set 13 due Wednesday November 29 Exam 3 Monday December 4 Homework Set 14 due Friday December 8 Final Exam Tuesday December 12 1 Section 8.6. The Kepler orbits Read Section
More informationTo Frame the World 1 Oct
To Frame the World 1 Oct Test 1 will be returned on Wed. Homework 4 is on el. Due Mon, Oct 11 Suggest way to improve Ast207 Angel > Lessons > Survey after 4 weeks Anonymous Newton amends Kepler s 3 rd
More informationIsaac Newton and the Laws of Motion and Gravitation 2
Isaac Newton and the Laws of Motion and Gravitation 2 ASTR 101 3/21/2018 Center of Mass motion Oblate shape of planets due to rotation Tidal forces and tidal locking Discovery of Neptune 1 Center of Mass
More informationEarth s Motions. Rotation -!! Period of Rotation - amount of time to make one complete rotation Example: Earth rotates in hours.
Name: Date: Period: Earth In the Solar System The Physical Setting: Earth Science CLASS NOTES! Rotation -! Period of Rotation - amount of time to make one complete rotation Example: Earth rotates in hours
More informationRadial Acceleration. recall, the direction of the instantaneous velocity vector is tangential to the trajectory
Radial Acceleration recall, the direction of the instantaneous velocity vector is tangential to the trajectory 1 Radial Acceleration recall, the direction of the instantaneous velocity vector is tangential
More informationEvidence that the Earth does not move: Greek Astronomy. Aristotelian Cosmology: Motions of the Planets. Ptolemy s Geocentric Model 2-1
Greek Astronomy Aristotelian Cosmology: Evidence that the Earth does not move: 1. Stars do not exhibit parallax: 2-1 At the center of the universe is the Earth: Changeable and imperfect. Above the Earth
More informationPHAS 1511: Foundations of Astronomy
PHAS 1511: Foundations of Astronomy Dr Roger Wesson Research interests: deaths of stars. Planetary nebulae, novae and supernovae. Astronomy: some maths You can see that distances in astronomy are huge.
More informationGat ew ay T o S pace AS EN / AS TR Class # 19. Colorado S pace Grant Consortium
Gat ew ay T o S pace AS EN / AS TR 2500 Class # 19 Colorado S pace Grant Consortium Announcements: - Launch Readiness Review Cards - 11 days to launch Announcements: - Launch Readiness Review Cards - 11
More informationANNEX 1. DEFINITION OF ORBITAL PARAMETERS AND IMPORTANT CONCEPTS OF CELESTIAL MECHANICS
ANNEX 1. DEFINITION OF ORBITAL PARAMETERS AND IMPORTANT CONCEPTS OF CELESTIAL MECHANICS A1.1. Kepler s laws Johannes Kepler (1571-1630) discovered the laws of orbital motion, now called Kepler's laws.
More informationEnergy Efficiency, Acoustics & Daylighting in building Prof. B. Bhattacharjee Department of Civil Engineering Indian Institute of Technology, Delhi
Energy Efficiency, Acoustics & Daylighting in building Prof. B. Bhattacharjee Department of Civil Engineering Indian Institute of Technology, Delhi Lecture - 05 Introduction & Environmental Factors (contd.)
More informationModern Navigation. Thomas Herring
12.215 Modern Navigation Thomas Herring Today s Class Latitude and Longitude Simple spherical definitions Geodetic definition: For an ellipsoid Astronomical definition: Based on direction of gravity Relationships
More informationUNIT 6 CELESTIAL SPHERE AND EQUINOCTIAL SYSTEM OF COORDINATES
UNIT 6 CELESTIAL SPHERE AND EQUINOCTIAL SYSTEM OF COORDINATES Structure 6.1 Introduction Objectives 6.2 References 6.3 Apparent Annual Motion of the Sun and the Concept of the Ecliptic and the Obliquity
More informationPaper Reference. Paper Reference(s) 1627/01 Edexcel GCSE Astronomy Paper 01. Friday 15 May 2009 Morning Time: 2 hours
Centre No. Candidate No. Paper Reference(s) 1627/01 Edexcel GCSE Astronomy Paper 01 Friday 15 May 2009 Morning Time: 2 hours Materials required for examination Calculator Items included with question papers
More informationSatellite Communications
Satellite Communications Lecture (3) Chapter 2.1 1 Gravitational Force Newton s 2nd Law: r r F = m a Newton s Law Of Universal Gravitation (assuming point masses or spheres): Putting these together: r
More informationLecture 10: General Relativity I
Lecture 10: General Relativity I Einstein Tower Experiment Gravitational redshifting Strong Equivalence Principal Sidney Harris 10/2/13 1 O: RECAP OF SPECIAL RELATIVITY Einstein s postulates Laws of physics
More informationSOLAR SYSTEM 2019 SAMPLE EXAM
SOLAR SYSTEM 2019 SAMPLE EXAM Team Name: Team #: No calculators are allowed. All questions are of equal weight unless otherwise noted. Turn in all materials when you have completed the test! Make sure
More informationThe Birth of Astronomy. Lecture 3 1/24/2018
The Birth of Astronomy Lecture 3 1/24/2018 Fundamental Questions of Astronomy (life?) What is the shape of the Earth? How big is the planet we live on? Why do the stars move across the sky? Where is Earth
More information