# METHOD FOR DETERMINING THE LONGITUDE OF PLACES BY OBSERVING OCCULTATIONS OF FIXED STARS BY THE MOON *

Save this PDF as:

Size: px
Start display at page:

Download "METHOD FOR DETERMINING THE LONGITUDE OF PLACES BY OBSERVING OCCULTATIONS OF FIXED STARS BY THE MOON *"

## Transcription

1 Synopsis of Leonhard Euler's 1749 paper: METHOD FOR DETERMINING THE LONGITUDE OF PLACES BY OBSERVING OCCULTATIONS OF FIXED STARS BY THE MOON * By Jennifer Grabowski, Jeffrey Meyer, and Erik Tou Carthage College, June History of the problem Navigation has been important in traveling as well as safety. For example, in 1707, a British fleet made an error on their longitude findings that sent them in the wrong direction where four of the five ships were destroyed and nearly two thousand soldiers lost their lives. This problem may have been avoided if the sailors had an accurate method of finding longitude. In 1714, the British Parliament passed the Longitude Act, which offered 20,000 to anyone who could calculate longitude to within a half degree. Since one degree of longitude can be as much as 69 miles, even a fraction of a degree comes out to a significant distance. Therefore, the required method would need to be very accurate. In the late 1720s, a self-educated British watchmaker named John Harrison set out to construct a sea clock which could keep accurate enough time to calculate longitude. Harrison succeeded in constructing a sea clock in 1735 which was deserving of the 20,000 prize. However, he was very critical of his machine even though everyone else, including the royal board of longitude, was very impressed. The sextant was then invented by a couple of inventors during the same time period. A sextant had incorporated a longer measuring arc as well as a telescope. These additional pieces enabled the device to measure distances during the day from the moon to the sun and from the moon to the stars at night. With a star chart and a sextant a sailor now could measure all sorts of lunar distances. A sailor makes an observation at a set time and found the same observation through the charts in London at a different time; the sailor is now able to find how many degrees they are away from London. At this time, Leonhard Euler set out to predict the motions of the moon and from these particular motions he constructed lunar tables. [9] These tables were found through the studies of Isaac Newton who introduced the theory of gravity into astronomy. * Originally published as: "Methode de determiner la longitude des lieux par l'observation d'occultations des étoiles fixes par la lune," Mémoires de l'académie royale des sciences et belles-lettres de Berlin 3 (1747) 1749, pp A copy of the original text is available electronically at the Euler Archive: This paper is numbered E115 in the Eneström index.

2 2. The body of the paper I & II. These first two sections allow Euler to set up for the observations. He says, at some known location, in this case Paris; it will be α hours after noon and the moon s right ascension will be ζ when the occultation is observed. We can see this in Figure 1 where Paris is denoted by P 1. To determine the longitude of the unknown location, which he specifies should be west of Paris, the same occultation needs to be viewed with time and right ascension recorded. At that unknown location, which we say is X 2, he lets β be the hours past noon and η denotes the moon s right ascension when the occultation is observed (see Figure 2). III. Since these two observations would occur within a few hours of each other, it is necessary that α and ζ be predetermined. Mr. Euler explains that the hourly movement of the moon s right ascension can be deduced from "the Astronomical Tables." 1 These lunar tables give information about the occultation s occurrence in Paris and can be compared to what is observed in the unknown location. These tables were quite complex, including predictions of the moon s orbit and how it would be viewed from Paris at every hour on any day to come. Euler remarks that these tables are not completely accurate; however, he says that they are very close to the truth. He assigns γ to be the hourly movement of the moon s right ascension at this time. Figure 1 Figure 2 IV. Here, another variable is defined. Euler denotes the difference in longitude between Paris and the unknown location by the letter z (see Figure 3). He then states that it will be β + z hours after noon in Paris, when the angle of the moon s right ascension is η. V. In this section Euler first reminds the reader that at α hours the right ascension of the moon was said to be ζ. Then he makes the connection that during β + z α hours the right asc- Figure 3 ension of the moon has changed by η ζ. We will refer to β + z α hours as T, this value represents the total elapsed time between observations. P 1 denotes Paris s original location; after T hours Paris has moved to P 2. Similarly, X 1 and X 2 represent the positions of the unknown location (Figure 3). Euler goes on to say that T (time) multiplied by the earlier defined γ (speed) must equal the moon s total change in right ascension. 1 This presumably refers to Euler s "Tabulæ Astronomicæ Solis et Lunæ" [2]

3 So, one can say that (β + z α) γ = η ζ. This equation can be reordered so that one may find the change in longitude z = α β + (η ζ)/γ hours. VI. Now that Euler has set up the equation to find the difference in longitudes, a few precautionary statements are made. Although Euler had assumed that the unknown location was west of Paris, he says that if the value of z is found to be negative then it is clear that this location is actually east of Paris. In the equation used, if α and β were swapped and η and ζ were swapped, then the opposite sign would result because of the symmetry of the equation. Therefore, the connection between east and west and positive and negative values makes logical sense. VII. In this last statement, Euler says that if the unknown location is too far from the one that had been used to calculate the tables, in this case Paris, then the difference cannot be measured. Next, he apparently refers to paragraph six of his previous article [3], and says that it provides an alternate step that is needed when the two locations are too far from each other. Euler says that by repeating all the same steps and research, a new table could be developed that is appropriate for the unknown location. 3. Appraisal of method Although all of these observations and calculations can be quite accurate, there are some pitfalls that cannot be avoided. In the 1700s, celestial navigation was dependent on the human eye; there were no computers to help gain this knowledge, just the eye and a few simple tools. The sextant is a tool that was commonly used to find longitude and it is one thing that could cause error. It could be calibrated poorly, placement of the eye could be off, or an error could be made in calculation. Some of these problems could be corrected; however, human error is hard to escape. [1] Euler s theory is completely dependent on astronomical tables to provide at least half of the information needed, and he states that they are not completely accurate. These tables predicted the change in the sun and moon s orbits, in his lunar tables [2] Euler states, Although it is claimed of several lunar tables that they are based on this theory, I dare to assert that the calculations to which this theory leads are so intricate, that such tables must be considered to differ greatly from the theory. Nor do I claim that I have included in these tables all the inequalities of motion which the theory implies. [10] There are many gravitational factors that alter a planet s orbit derived from Newton s theory of gravity. Since the sun, moon, and earth are not always equidistant the amount of gravitational pull is also a variable. The table s predictions are difficult to calculate because of the irregular change that is caused by the gravity. Also, based on the complexity of these equations and the calculations that are done, the results should be expected to have some error. By setting up an example one can see how a small error in measurement can greatly affect the longitude that is found. One can calculate a correct z value from the equation Euler offers. Then alter γ by say 1, 0.5, and 0.1 and see how much a small change to γ will affect z. Below

4 is an example where α is set to be 10 hours past noon, β to be 11 hours, ζ is 75, and η is observed to be 88. Suppose the speed of the moon s right ascension is said to be 8.2 /hour, when this value is used the true z value comes out to hours. In order to receive the prizes discussed, the proposed technique needed to be within half a degree, 2 / 3 of a degree, or just within one degree of the actual longitude. If we make an error of ε in measurement of γ the calculated longitude would be, z = α β + η ζ γ + ε. Then, ε = z z = ( η ζ ) 1 γ 1, where ε' γ + ε represents the actual error in the results. In the table below this error is given in longitudinal hours and degrees. ε z z' ε' 1 /hour hrs hrs hrs /hour hrs hrs hrs /hour hrs hrs hrs In order to win the top prize, ε could be at most /hour. 4. Comparison with other methods The measurement of the moon has been successful in the past, but the surroundings have to be just right. If the sky is cloudy then the astronomical method of finding longitude fails because the moon and stars cannot be seen. At times this method is impractical because it requires one to know how to calculate longitude once a measurement has been taken. When one takes the measurement, the reference point cannot differ too much from the proposed location. Also, time must be kept at least on a daily basis, if not more frequently. The benefit to using the astronomical method is that no complicated machines are necessary, such as the chronometer. The chronometer method requires a tool that is very difficult to construct. It took Harrison a great deal of time to build this product and there still could have been problems with its operation. Even though the chronometer method does not need perfect weather, numerous chronometers are needed on board. This may have been a problem back in the 1700s because they were not mass-produced and the chronometer's size proved to be an issue. The process to mass-produce these chronometers was impractical and inconvenient. 5. Epilogue Euler's work on the longitude problem did not end with this paper. In Letters to a German Princess, published some 20 years later, Euler describes a total of six methods that could be used to calculate longitude. While he gives several different astronomical methods, he also mentions chronometers and compasses as providing plausible solutions to the longitude problem. [6] Near the end of his life, Euler wrote another paper on longitude, in which he describes a process now known as the "lunar distance method." [8] Additionally, Euler wrote a second book on lunar motion, namely his Theoria motuum lunae [7], which was published in 1772.

5 In the end, however, it was Euler's first lunar theory that achieved the most recognition. The astronomer Tobias Mayer used Euler's Theoria motus lunae [5] to develop his method for calculating longitude, which earned him (posthumously) a 3,000 prize from the Board of Longitude. Euler was awarded 300 for his contributions to Mayer's success. From a historical perspective, Harrison's chronometers were more successful than the astronomical methods. Eventually, the problems of design and production were overcome, thus permitting their widespread use. Since the astronomical methods required difficult calculations, the mass-production of chronometers (which required no such calculations) was sufficient to make these methods obsolete. Bibliography [1] Bowditch, N. The New American Practical Navigator. Newburyport, Mass.: Cushing & Appleton, Reprinted by the National Imagery and Mapping Agency, [2] Euler, L. "Tabulae astronomicae solis et lunae," Opuscula varii argumenti , pp [E87 in Eneström's index] [3] "Methode de trouver le vrai lieu geocentrique de la lune par l'observation de l'occultation d'une étoile fixe," Mémoires de l'académie royale des sciences et belles-lettres de Berlin 3 (1747) 1749, pp [E114] [4] "Methode de determiner la longitude des lieux par l'observation d'occultations des étoiles fixes par la lune," Mémoires de l'académie royale des sciences et belles-lettres de Berlin 3 (1747) 1749, pp [E115] [5] Theoria motus lunae exhibens omnes eius inaequalitates, [E187] [6] Lettres à une princesse d'allemagne sur divers sujets de physique & de philosophie [E343/344] [7] Theoria motuum lunae, [E418] [8] "De inventione longitudinis locorum ex observata lunae distantia a quadam stella fixa cognita," Acta Academiae Scientiarum Imperialis Petropolitinae (1780) [E570] [9] Sobel, D. Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of his Time. New York: Walker & Company, [10] Wilson, C. "Euler and Applications of Analytical Mathematics to Astronomy," Leonhard Euler: Life, Work and Legacy, Bradley, R. and Sandifer, C. E., eds. Elsevier Science, 2007.

### Local Sidereal Time is the hour angle of the First Point of Aries, and is equal to the hour angle plus right ascension of any star.

1 CHAPTER 7 TIME In this chapter we briefly discuss the several time scales that are in use in astronomy, such as Universal Time, Mean Solar Time, Ephemeris Time, Terrestrial Dynamical Time, and the several

### Name: Earth 110 Exploration of the Solar System Assignment 1: Celestial Motions and Forces Due in class Tuesday, Jan. 20, 2015

Name: Earth 110 Exploration of the Solar System Assignment 1: Celestial Motions and Forces Due in class Tuesday, Jan. 20, 2015 Why are celestial motions and forces important? They explain the world around

### Section 1 Gravity: A Force of Attraction

Section 1 Gravity: A Force of Attraction Key Concept Gravity is a force of attraction between objects that is due to their masses. What You Will Learn Gravity affects all matter, including the parts of

### Earth In Space Chapter 3

Earth In Space Chapter 3 Shape of the Earth Ancient Greeks Earth casts a circular shadow on the moon during a lunar eclipse Shape of the Earth Ancient Greeks Ships were observed to disappear below the

### Lecture 13. Gravity in the Solar System

Lecture 13 Gravity in the Solar System Guiding Questions 1. How was the heliocentric model established? What are monumental steps in the history of the heliocentric model? 2. How do Kepler s three laws

### CELESTIAL MOTIONS. In Charlottesville we see Polaris 38 0 above the Northern horizon. Earth. Starry Vault

CELESTIAL MOTIONS Stars appear to move counterclockwise on the surface of a huge sphere the Starry Vault, in their daily motions about Earth Polaris remains stationary. In Charlottesville we see Polaris

### Solar System Formation

Solar System Formation Background Information System: Many pieces that make up a whole Solar System: Anything that orbits the Sun Just like in the formation of of stars.. Gravity plays a major role. Gravitational

### 1.2 Investigations and Experiments

Science is about figuring out cause and effect relationships. If we do something, what happens? If we make a ramp steeper, how much faster will a car roll down? This is an easy question. However, the process

### 2. Orbits. FER-Zagreb, Satellite communication systems 2011/12

2. Orbits Topics Orbit types Kepler and Newton laws Coverage area Influence of Earth 1 Orbit types According to inclination angle Equatorial Polar Inclinational orbit According to shape Circular orbit

### Teaching astronomical navigation at the university: an historical overview

Highlights of Spanish Astrophysics VI, Proceedings of the IX Scientific Meeting of the Spanish Astronomical Society held on September 13-17, 2010, in Madrid, Spain. M. R. Zapatero Osorio et al. (eds.)

### Orientation to the Sky: Apparent Motions

Chapter 2 Orientation to the Sky: Apparent Motions 2.1 Purpose The main goal of this lab is for you to gain an understanding of how the sky changes during the night and over the course of a year. We will

### Exam # 1 Thu 10/06/2010 Astronomy 100/190Y Exploring the Universe Fall 11 Instructor: Daniela Calzetti

Exam # 1 Thu 10/06/2010 Astronomy 100/190Y Exploring the Universe Fall 11 Instructor: Daniela Calzetti INSTRUCTIONS: Please, use the `bubble sheet and a pencil # 2 to answer the exam questions, by marking

### Douglas Adams The Hitchhikers Guide to the Galaxy

There is a theory which states that if ever anybody discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.

### Chapter 25.1: Models of our Solar System

Chapter 25.1: Models of our Solar System Objectives: Compare & Contrast geocentric and heliocentric models of the solar sytem. Describe the orbits of planets explain how gravity and inertia keep the planets

### The Solar System. Unit 4 covers the following framework standards: ES 10 and PS 11. Content was adapted the following:

Unit 4 The Solar System Chapter 7 ~ The History of the Solar System o Section 1 ~ The Formation of the Solar System o Section 2 ~ Observing the Solar System Chapter 8 ~ The Parts the Solar System o Section

### Grade 6 Standard 3 Unit Test A Astronomy. 1. The four inner planets are rocky and small. Which description best fits the next four outer planets?

Grade 6 Standard 3 Unit Test A Astronomy Multiple Choice 1. The four inner planets are rocky and small. Which description best fits the next four outer planets? A. They are also rocky and small. B. They

### Solar System. 1. The diagram below represents a simple geocentric model. Which object is represented by the letter X?

Solar System 1. The diagram below represents a simple geocentric model. Which object is represented by the letter X? A) Earth B) Sun C) Moon D) Polaris 2. Which object orbits Earth in both the Earth-centered

### COASTLINING THE ZODIAC

COASTLINING THE ZODIAC Astronomy books and skywatching guides offer a wide variety of charts for naked-eye observation of the skies. What works best for each person will depend on various factors such

### Pre and post-visit activities - Navigating by the stars

Pre and post-visit activities - Navigating by the stars Vocabulary List Adult Education at Scienceworks Pre-visit Activity 1: What is longitude and latitude? Activity 2: Using the Southern Cross to find

### Stellarium a valuable resource for teaching astronomy in the classroom and beyond

Stellarium 1 Stellarium a valuable resource for teaching astronomy in the classroom and beyond Stephen Hughes Department of Physical and Chemical Sciences, Queensland University of Technology, Gardens

### The Observed Calendar of the Second Temple Dates For 2014

www.120jubilees.com The Observed Calendar of the Second Temple Dates For 2014 The Observed Calendar of the Second Temple Era was used by the official priests of the Second Temple from at least 520 BC to

### Educator Guide to S LAR SYSTEM. 1875 El Prado, San Diego CA 92101 (619) 238-1233 www.rhfleet.org

Educator Guide to S LAR SYSTEM 1875 El Prado, San Diego CA 92101 (619) 238-1233 www.rhfleet.org Pre-Visit Activity: Orbital Paths Materials: Plastic Plate Marble Scissors To Do: 1. Put the plate on a flat

### Lecture 17 Newton on Gravity

Lecture 17 Newton on Gravity Patrick Maher Philosophy 270 Spring 2010 Introduction Outline of Newton s Principia Definitions Axioms, or the Laws of Motion Book 1: The Motion of Bodies Book 2: The Motion

### Stellar Aspirations A ticking reminder of one s place in the universe, the astronomical timepiece makes a lofty addition to your watch collection.

Left to right: Chopard L.U.C Lunar Big Date (\$32,180); Arnold & Son HM Perpetual Moon (\$27,845 in red gold). STORY BY BY JONATHON KEATS PHOTOGRAPHY BY JEFF HARRIS Stellar Aspirations A ticking reminder

### ASTRONOMY 161. Introduction to Solar System Astronomy

ASTRONOMY 161 Introduction to Solar System Astronomy Seasons & Calendars Monday, January 8 Season & Calendars: Key Concepts (1) The cause of the seasons is the tilt of the Earth s rotation axis relative

### How Do Paper Airplanes Fly?

West Ashley Intermediate School Charleston, South Carolina Summer 2004 Research Host: Charles Hossler Dr. Carolyn Jenkins Medical University of South Carolina Lesson # 10 Appropriate citation: Herron,

### Name Period 4 th Six Weeks Notes 2015 Weather

Name Period 4 th Six Weeks Notes 2015 Weather Radiation Convection Currents Winds Jet Streams Energy from the Sun reaches Earth as electromagnetic waves This energy fuels all life on Earth including the

### Artificial Satellites Earth & Sky

Artificial Satellites Earth & Sky Name: Introduction In this lab, you will have the opportunity to find out when satellites may be visible from the RPI campus, and if any are visible during the activity,

### 4 Gravity: A Force of Attraction

CHAPTER 1 SECTION Matter in Motion 4 Gravity: A Force of Attraction BEFORE YOU READ After you read this section, you should be able to answer these questions: What is gravity? How are weight and mass different?

### The Moon. Nicola Loaring, SAAO

The Moon Nicola Loaring, SAAO Vital Statistics Mean distance from Earth Orbital Period Rotational Period Diameter 384,400 km 27.322 days 27.322 days 3476 km (0.272 x Earth) Mass 7.3477 10 22 kg (0.0123

### How Rockets Work Newton s Laws of Motion

How Rockets Work Whether flying a small model rocket or launching a giant cargo rocket to Mars, the principles of how rockets work are exactly the same. Understanding and applying these principles means

### STUDY GUIDE: Earth Sun Moon

The Universe is thought to consist of trillions of galaxies. Our galaxy, the Milky Way, has billions of stars. One of those stars is our Sun. Our solar system consists of the Sun at the center, and all

### Scale of the Solar System. Sizes and Distances: How Big is Big? Sizes and Distances: How Big is Big? (Cont.)

Scale of the Solar System Scale of the Universe How big is Earth compared to our solar system? How far away are the stars? How big is the Milky Way Galaxy? How big is the Universe? How do our lifetimes

### 1-2. What is the name given to the path of the Sun as seen from Earth? a.) Equinox b.) Celestial equator c.) Solstice d.

Chapter 1 1-1. How long does it take the Earth to orbit the Sun? a.) one sidereal day b.) one month c.) one year X d.) one hour 1-2. What is the name given to the path of the Sun as seen from Earth? a.)

### Science Standard 4 Earth in Space Grade Level Expectations

Science Standard 4 Earth in Space Grade Level Expectations Science Standard 4 Earth in Space Our Solar System is a collection of gravitationally interacting bodies that include Earth and the Moon. Universal

### Cycles in the Sky. Teacher Guide: Cycles in the Sky Page 1 of 8 2008 Discovery Communications, LLC

Cycles in the Sky What is a Fun damental? Each Fun damental is designed to introduce your younger students to some of the basic ideas about one particular area of science. The activities in the Fun damental

### A long time ago, people looked

Supercool Space Tools! By Linda Hermans-Killam A long time ago, people looked into the dark night sky and wondered about the stars, meteors, comets and planets they saw. The only tools they had to study

### Dynamics of Iain M. Banks Orbitals. Richard Kennaway. 12 October 2005

Dynamics of Iain M. Banks Orbitals Richard Kennaway 12 October 2005 Note This is a draft in progress, and as such may contain errors. Please do not cite this without permission. 1 The problem An Orbital

### Size and Scale of the Universe

Size and Scale of the Universe (Teacher Guide) Overview: The Universe is very, very big. But just how big it is and how we fit into the grand scheme can be quite difficult for a person to grasp. The distances

### Name Class Date. true

Exercises 131 The Falling Apple (page 233) 1 Describe the legend of Newton s discovery that gravity extends throughout the universe According to legend, Newton saw an apple fall from a tree and realized

### Your guide to Using a solicitor

www.lawsociety.org.uk 1 Most of us need expert legal help at some time in our lives. Some of the most common issues are to do with buying a house, getting a divorce or making a will. But you might also

### DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION

1 DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION Daniel S. Orton email: dsorton1@gmail.com Abstract: There are many longstanding

### Essential Question. Enduring Understanding

Earth In Space Unit Diagnostic Assessment: Students complete a questionnaire answering questions about their ideas concerning a day, year, the seasons and moon phases: My Ideas About A Day, Year, Seasons

### Misconceptions in Astronomy in WA High School students (in preparation)

Misconceptions in Astronomy in WA High School students (in preparation) Michael Todd Department of Imaging and Applied Physics, Curtin University of Technology The purpose of this study was to examine

### Periods of Western Astronomy. Chapter 1. Prehistoric Astronomy. Prehistoric Astronomy. The Celestial Sphere. Stonehenge. History of Astronomy

Periods of Western Astronomy Chapter 1 History of Astronomy Western astronomy divides into 4 periods Prehistoric (before 500 B.C.) Cyclical motions of Sun, Moon and stars observed Keeping time and determining

### INTRUSION PREVENTION AND EXPERT SYSTEMS

INTRUSION PREVENTION AND EXPERT SYSTEMS By Avi Chesla avic@v-secure.com Introduction Over the past few years, the market has developed new expectations from the security industry, especially from the intrusion

### Note S1: Eclipses & Predictions

The Moon's Orbit The first part of this note gives reference information and definitions about eclipses [14], much of which would have been familiar to ancient Greek astronomers, though not necessarily

### Adult cognition of large-scale geometric facts

Roberto Casati, David Mark, Ira Noveck Adult cognition of large-scale geometric facts Draft 4, April 1999 Project description Objectives 1. Adult cognition of large-scale geometric facts The primary objective

### Vocabulary - Understanding Revolution in. our Solar System

Vocabulary - Understanding Revolution in Universe Galaxy Solar system Planet Moon Comet Asteroid Meteor(ite) Heliocentric Geocentric Satellite Terrestrial planets Jovian (gas) planets Gravity our Solar

### Plato gives another argument for this claiming, relating to the nature of knowledge, which we will return to in the next section.

Michael Lacewing Plato s theor y of Forms FROM SENSE EXPERIENCE TO THE FORMS In Book V (476f.) of The Republic, Plato argues that all objects we experience through our senses are particular things. We

### Astrock, t he A stronomical Clock

Astrock, t he A stronomical Clock The astronomical clock is unlike any other clock. At first glance you ll find it has similar functions of a standard clock, however the astronomical clock can offer much

### Today. Solstices & Equinoxes Precession Phases of the Moon Eclipses. Ancient Astronomy. Lunar, Solar FIRST HOMEWORK DUE NEXT TIME

Today Solstices & Equinoxes Precession Phases of the Moon Eclipses Lunar, Solar Ancient Astronomy FIRST HOMEWORK DUE NEXT TIME The Reason for Seasons Hypothesis check: How would seasons in the northern

### Announcements. Due Monday, 11:59pm, Sept 9th. Can submit answers (and see score) more than once: persistence pays off!

Announcements Homework 1 posted on Compass Due Monday, 11:59pm, Sept 9th. Can submit answers (and see score) more than once: persistence pays off! Register those iclickers! link on course site Venus/Saturn

### Orbital Mechanics. Angular Momentum

Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely

### Motions of the Earth. Stuff everyone should know

Motions of the Earth Stuff everyone should know Earth Motions E W N W Noon E Why is there day and night? OR Why do the Sun and stars appear to move through the sky? Because the Earth rotates around its

### Solar Angles and Latitude

Solar Angles and Latitude Objectives The student will understand that the sun is not directly overhead at noon in most latitudes. The student will research and discover the latitude ir classroom and calculate

### Halliday, Resnick & Walker Chapter 13. Gravitation. Physics 1A PHYS1121 Professor Michael Burton

Halliday, Resnick & Walker Chapter 13 Gravitation Physics 1A PHYS1121 Professor Michael Burton II_A2: Planetary Orbits in the Solar System + Galaxy Interactions (You Tube) 21 seconds 13-1 Newton's Law

### Crafting an Argument. Students need to know how to state facts and express their opinions. Organized, Well-Supported, Convincing

Crafting an Argument Organized, Well-Supported, Convincing Students need to know how to state facts and express their opinions effectively so that they can positively contribute to classroom conversations

### PROCLAMATION OF 2009 AS THE UNITED NATIONS INTERNATIONAL YEAR OF ASTRONOMY OUTLINE

U General Conference 33rd session, Paris 2005 33 C 33 C/67 11 October 2005 Original: English Item 5.30 of the agenda PROCLAMATION OF 2009 AS THE UNITED NATIONS INTERNATIONAL YEAR OF ASTRONOMY Source: 172

### Today FIRST HOMEWORK DUE NEXT TIME. Seasons/Precession Recap. Phases of the Moon. Eclipses. Lunar, Solar. Ancient Astronomy

Today FIRST HOMEWORK DUE NEXT TIME Seasons/Precession Recap Phases of the Moon Eclipses Lunar, Solar Ancient Astronomy How do we mark the progression of the seasons? We define four special points: summer

### Introduction to the Solar System

Introduction to the Solar System Lesson Objectives Describe some early ideas about our solar system. Name the planets, and describe their motion around the Sun. Explain how the solar system formed. Introduction

### Earth, Moon, and Sun Inquiry Template Eclipses

One Stop Shop For Educators The following instructional plan is part of a GaDOE collection of Unit Frameworks, Performance Tasks, examples of Student Work, and Teacher Commentary. Many more GaDOE approved

### Science in the Elementary and Middle School

15-0 Science in the Elementary and Middle School Naturally Occurring Inquiry Process, Which Can Be Made More Effective With Experience Uses Observable Data Science Search for Regularity Involves Information

### Moon. & eclipses. Acting out celestial events. (oh my)

phasestides & eclipses Moon (oh my) Acting out celestial events Developed by: Betsy Mills, UCLA NSF GK-12 Fellow Title of Lesson: Moon Phases, Tides, & Eclipses (oh my)! Grade Level: 8 th grade Subject(s):

### Page. ASTRONOMICAL OBJECTS (Page 4).

Star: ASTRONOMICAL OBJECTS ( 4). Ball of gas that generates energy by nuclear fusion in its includes white dwarfs, protostars, neutron stars. Planet: Object (solid or gaseous) that orbits a star. Radius

### CHAPTER 18 TIME TIME IN NAVIGATION

CHAPTER 18 TIME TIME IN NAVIGATION 1800. Solar Time The Earth s rotation on its axis causes the Sun and other celestial bodies to appear to move across the sky from east to west each day. If a person located

### Moon Phases and Tides in the Planning the D-Day Invasion Part I: The Phases of the Moon

The Science and Technology of WWII Moon Phases and Tides in the Planning the D-Day Invasion Part I: The Phases of the Moon Objectives: 1. Students will determine what causes the moon to go through a cycle

RETURN TO THE MOON Lesson Plan INSTRUCTIONS FOR TEACHERS Grade Level: 9-12 Curriculum Links: Earth and Space (SNC 1D: D2.1, D2.2, D2.3, D2.4) Group Size: Groups of 2-4 students Preparation time: 1 hour

### Tropical Horticulture: Lecture 2

Lecture 2 Theory of the Tropics Earth & Solar Geometry, Celestial Mechanics The geometrical relationship between the earth and sun is responsible for the earth s climates. The two principal movements of

### Discover the Universe AST-1002 Section 0427, Spring 2016

Discover the Universe AST-1002 Section 0427, Spring 2016 Instructor: Dr. Francisco Reyes Office: Room 12 Bryant Space Science Center Telephone: 352-294-1885 Email: freyes@astro.ufl.edu Office hours: Monday

### The Mathematics of the Longitude

The Mathematics of the Longitude Wong Lee Nah An academic exercise presented in partial fulfilment for the degree of Bachelor of Science with Honours in Mathematics. Supervisor : Associate Professor Helmer

### Unit 8 Lesson 2 Gravity and the Solar System

Unit 8 Lesson 2 Gravity and the Solar System Gravity What is gravity? Gravity is a force of attraction between objects that is due to their masses and the distances between them. Every object in the universe

### 1. In the diagram below, the direct rays of the Sun are striking the Earth's surface at 23 º N. What is the date shown in the diagram?

1. In the diagram below, the direct rays of the Sun are striking the Earth's surface at 23 º N. What is the date shown in the diagram? 5. During how many days of a calendar year is the Sun directly overhead

### Gravity? Depends on Where You Are!

Gravity? Depends on Where You Are! Overview Gravity is one of the fundamental concepts of Physics. It is an abstract concept that benefits from activities that help illustrate it. This lesson plan involves

### Your years of toil Said Ryle to Hoyle Are wasted years, believe me. The Steady State Is out of date Unless my eyes deceive me.

Your years of toil Said Ryle to Hoyle Are wasted years, believe me. The Steady State Is out of date Unless my eyes deceive me. My telescope Has dashed your hope; Your tenets are refuted. Let me be terse:

### Section 4: The Basics of Satellite Orbits

Section 4: The Basics of Satellite Orbits MOTION IN SPACE VS. MOTION IN THE ATMOSPHERE The motion of objects in the atmosphere differs in three important ways from the motion of objects in space. First,

### Gravity. in the Solar System. Beyond the Book. FOCUS Book

FOCUS Book Design a test to find out whether Earth s gravity always pulls straight down. A pendulum is a weight that hangs from a string or rod that can swing back and forth. Use string and metal washers

### SIERRA COLLEGE OBSERVATIONAL ASTRONOMY LABORATORY EXERCISE NUMBER III.F.a. TITLE: ASTEROID ASTROMETRY: BLINK IDENTIFICATION

SIERRA COLLEGE OBSERVATIONAL ASTRONOMY LABORATORY EXERCISE NUMBER III.F.a. TITLE: ASTEROID ASTROMETRY: BLINK IDENTIFICATION DATE- PRINT NAME/S AND INITIAL BELOW: GROUP DAY- LOCATION OBJECTIVE: Use CCD

### Produced by Billy Hix and Terry Sue Fanning. As part of the TeachSpace Program. For more ideas and an image of the current phase of the moon, visit:

The Moon Phase Book Produced by Billy Hix and Terry Sue Fanning As part of the TeachSpace Program For more ideas and an image of the current phase of the moon, visit: www.teachspace.us Printing Date: 10/29/2010

### Content Area: Earth Science Grade(s) 4. Essential Question(s) and Enduring Understandings

Content Area: Earth Science Grade(s) 4 Unit Plan Title: Objects in the Universe Overview of Unit Students will make observations and record data to understand patterns of movement and relationships among

### AE554 Applied Orbital Mechanics. Hafta 1 Egemen Đmre

AE554 Applied Orbital Mechanics Hafta 1 Egemen Đmre A bit of history the beginning Astronomy: Science of heavens. (Ancient Greeks). Astronomy existed several thousand years BC Perfect universe (like circles

### galaxy solar system supernova (noun) (noun) (noun)

WORDS IN CONTEXT DAY 1 (Page 1 of 4) galaxy A galaxy is a collection of stars, gas, and dust. We live in the Milky Way galaxy. One galaxy may contain billions of stars. solar system A solar system revolves

### Full credit for this chapter to Prof. Leonard Bachman of the University of Houston

Chapter 6: SOLAR GEOMETRY Full credit for this chapter to Prof. Leonard Bachman of the University of Houston SOLAR GEOMETRY AS A DETERMINING FACTOR OF HEAT GAIN, SHADING AND THE POTENTIAL OF DAYLIGHT PENETRATION...

### Video Poker in South Carolina: A Mathematical Study

Video Poker in South Carolina: A Mathematical Study by Joel V. Brawley and Todd D. Mateer Since its debut in South Carolina in 1986, video poker has become a game of great popularity as well as a game

### What we will cover. Types of orders How to place an order Instruc8ons from the DPR So>ware and what they mean

What we will cover Types of orders How to place an order Instruc8ons from the DPR So>ware and what they mean What is a STOP Order? This type of order is what we use at How Do I Trade Stock.com regularly

### Gravity Field and Dynamics of the Earth

Milan Bursa Karel Pec Gravity Field and Dynamics of the Earth With 89 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Preface v Introduction 1 1 Fundamentals

### Why Is the Moon Upside Down?

CONNECTED, LEVEL 3 2013, Food for Thought Why Is the Moon Upside Down? by Trish Puharich Overview This article explore how three friends made observations of the phases of the Moon from different locations.

### Fractions Packet. Contents

Fractions Packet Contents Intro to Fractions.. page Reducing Fractions.. page Ordering Fractions page Multiplication and Division of Fractions page Addition and Subtraction of Fractions.. page Answer Keys..

### The orbit of Halley s Comet

The orbit of Halley s Comet Given this information Orbital period = 76 yrs Aphelion distance = 35.3 AU Observed comet in 1682 and predicted return 1758 Questions: How close does HC approach the Sun? What

### Stellar, solar, and lunar demonstrators

Stellar, solar, and lunar demonstrators Rosa M. Ros, Francis Berthomieu International Astronomical Union, Technical University of Catalonia (Barcelona, España), CLEA (Nice, France) Summary This worksheet

### Tips on Reading This Book with Children:

Level: N Word Count: 331 100th Word: planets (page 8) Teaching Focus: Phonics: Word Study Look at the words float and plants. How are the words alike? How are they different? Tips on Reading This Book

APPENDIX D: SOLAR RADIATION The sun is the source of most energy on the earth and is a primary factor in determining the thermal environment of a locality. It is important for engineers to have a working

### Phases of the Moon. Objective. Materials. Procedure. Name Date Score /20

Name Date Score /20 Phases of the Moon Objective Working with models for the Earth-Moon-Sun system, the student will simulate the phases the Moon passes through each month. Upon completion of this exercise,

### Georgia Performance Standards Framework for Science Grade 6. Unit Organizer: UNIVERSE AND SOLAR SYSTEM (Approximate Time 3 Weeks)

The following instructional plan is part of a GaDOE collection of Unit Frameworks, Performance Tasks, examples of Student Work, and Teacher Commentary. Many more GaDOE approved instructional plans are

### Users Guide to the Ecliptic Calendar by Damon Scott, Author of the Ecliptic System

Users Guide to the Ecliptic Calendar by Damon Scott, Author of the Ecliptic System Introduction The currently prevalent method of marking time is called the Gregorian Calendar and is filled with arbitrary

### Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

### God is Eternal Lesson 1

Preschool Curriculum (4K / 5K) August 15 th & 16 th, 2009 God is Eternal Lesson 1 Divine Attribute: Heart Response: God is Eternal Be Amazed Memory Verse: Jesus Christ is the same yesterday, today and