Celestial Mechanics III. Time and reference frames Orbital elements Calculation of ephemerides Orbit determination
|
|
- Kristin Pearson
- 5 years ago
- Views:
Transcription
1 Celestial Mechanics III Time and reference frames Orbital elements Calculation of ephemerides Orbit determination
2 Orbital position versus time: The choice of units Gravitational constant: SI units ([m],[kg],[s]) G = m 3 kg -1 s -2 Gaussian units ([AU],[M ],[days]) k = AU 3/2 M -1/2 days -1 Kepler III: k 2 replaces G m 1 = 1; m 2 = 1/ (Earth+Moon) P = (sidereal year); a = 1
3 Celestial mechanics units The usage of Gaussian units is typical of celestial mechanics applications In spaceflight applications, SI units are also necessary to use In Oort Cloud dynamics, one often uses the year as unit of time. The gravitational constant is then = 4π 2 to very good approximation
4 History of time keeping Before 1960: Earth s rotation provided the basic clock, measured by astronomical observations of the sidereal day = 23 h 56 m 4.1 s (Earth s spin period). One mean solar day = 24 h = s. Universal Time (UT) = Greenwich mean time (GMT). Locally observed UT = UT0 (the results differ due to polar motion). Corrected UT = UT1 (but the rate still varies because the Earth is not a perfect rotator). Lunisolar tides lead to Length Of Day (LOD) variations.
5 LOD variations From Time Service Dept., US Naval Obs. Note: the LOD differs from s by about 2 ms!
6 History of time keeping, ctd From 1960: Solar system orbital motions provided the basic clock, measured by astronomical observations of orbiting objects. Ephemeris time ET = uniform time, as required by the Newtonian equations of motion. The ephemeris second was defined as: 1/31,556, of the tropical year at epoch 1900 UT1 was used temporarily, until ET was available and the correction ΔT=ET-UT1 was published.
7 History of time keeping, ctd In 1967: the frequency of a hyperfine transition in the 133 Ce isotope defines the SI second to be the duration of 9,192,631,770 periods of the radiation corresponding to this transition. This is equivalent to one ephemeris second. International Atomic Time (TAI) is based on the SI second and maintained by a large number of clocks operating at standards laboratories. Zero point: UT1 TAI 0 on Jan. 1, 1958 Since 1984: Terrestrial Time (TT) is used in astrometry and ephemeris calculations, thus replacing ET. Zero point: TT TAI = s (ET UT1 in 1958).
8 History of time keeping, ctd Currently, on average, one mean solar day = UT seconds TT seconds. In about 500 days, the difference between TT and UT increases by 1 s. Coordinated Universal Time (UTC) increases at the TAI rate like TT, but by introducing leap seconds, it is kept close to UT1. It defines civil time and provides the connection between astronomical and atomic times. The first leap second was introduced in 1972 with a starting value of TAI UTC = 10 s. The latest one (nr. 24) was introduced on 31 Dec
9 Julian dates A continuous count of days and fractions since noon UT on Jan 1, 4713 BCE (Before Christian Era) on the Julian calendar; the day numbers are now approaching 2.5 million Very useful for ephemeris calculations Converters between Julian dates and calendar dates can be found at: A practical formula, valid for the 20 th and 21 st centuries, is:
10 Fundamental reference frames Ecliptic frame, basically heliocentric, couples to the orbits of objects Equatorial frame, basically geocentric, couples to the astrometric observations Each one is defined by a fundamental plane, which cuts the celestial sphere along a great circle. The points of intersection are the equinoxes the vernal equinox is used as reference direction.
11 Reference frames, ctd The angle between the two planes ( obliquity of the ecliptic ) was ε= on 1 Jan Due to lunisolar precession of the Earth s spin axis, the vernal equinox drifts by ~50 per year. Planetary precession of the ecliptic plane causes smaller effects in both ε and the equinox. Since we need fixed reference frames, we use a standard epoch (currently J ) to define the equator and equinox.
12 Angular coordinates Use (x eq,y eq,z eq ) and (x ec,y ec,z ec ) as Cartesian geocentric coordinates. Let the x eq and x ec axes point toward the vernal equinox, and let the z eq and z ec axes point toward the respective poles. In spherical geocentric coordinates we instead use: Δ = geocentric distance α = right ascension δ = declination λ = ecliptic longitude β = ecliptic latitude (index 2000 means they refer to the standard equator and equinox of 2000)
13 Coordinate transformation
14 Orientation of orbit w.r.t. ecliptic
15 Inclination i is the angle from the pole of the ecliptic to the pole of the orbital plane (0 < i < π) Prograde orbits: i < π/2 Retrograde orbits: i > π/2
16 Longitude of the ascending node Ascending node vector Ω is measured along the ecliptic, counterclockwise as seen from the North pole, from the vernal equinox to the ascending node
17 Argument of perihelion ω is counted along the orbital plane, counterclockwise as seen from its North pole, from the ascending node to the perihelion direction
18 Orbital elements We have identified six orbital elements, which can be grouped as follows: Semi-major axis a, or perihelion distance q = a(1 e) Eccentricity e Time of perihelion passage T, or mean anomaly at a given epoch M 0 Orbital position at given time Inclination i Longitude of the ascending node Ω Argument of perihelion ω, or longitude of perihelion ϖ = Ω+ω Conversion from orbital position to ecliptic frame
19 Conversion matrix This transforms vectors from the orbital frame to the ecliptic frame, e.g., the position and velocity vectors:
20 Ephemeris calculation Calculate mean motion n=2π/p, P=2πa 3/2 /k Calculate mean anomaly M=n(t-T) Solve Kepler s equation to obtain eccentric anomaly E Use E to calculate the position (and velocity) vector(s) X (and dx/dt) in the orbital frame Use {i,ω,ω} and Homer s transformation matrix to obtain position (and velocity) in the heliocentric ecliptic frame
21 Ephemeris calculation, ctd Find the heliocentric ecliptic coordinates of the Earth Calculate the geocentric position vector of the object Calculate the distance Δ of the object Correct for planetary aberration (light time correction) If the light time is Λ = Δ/c, repeat the calculation of the object for a time t Λ
22 Ephemeris calculation, ctd Convert to equatorial coordinates Find the right ascension and declination of the object Repeat for a set of regularly spaced dates
23 Orbit determination Essential in order to identify and keep track of moving objects like Near-Earth asteroids or comets Clearing house: IAU Minor Planet Center Search programs, discovery statistics for numbered minor planets as of March 10, 2009: LINEAR 104,780 Spacewatch 22,036 NEAT 19,670 LONEOS 13,000
24 Way of procedure Preliminary orbit determination according to one of several methods (calculate orbital elements from few observed positions) Orbit improvement (reduction of uncertainties in orbital elements by using many observed positions) Linkage of several oppositions of an asteroid or apparitions of a comet (including chance identifications)
25 The Method of Gauss Assume 3 observations available: (α j,δ j,t j ); j=1,3 Thus 3 geocentric equatorial unit vectors are known: We can also find the Sun s geocentric positions at the 3 times of observation We can write 3 heliocentric position vectors of the object using unknown geocentric distances: (at time t j )
26 The Method of Gauss, ctd Idea: find the three Δ values; then use the three r values to derive the heliocentric position and velocity at the middle observation Transform from position and velocity to orbital elements Since motion is planar: Coefficients given by:
27 The Method of Gauss, ctd We obtain a vector equation, equivalent to 3 scalar equations, from which Δ j can be solved, if c 1 and c 3 are known: The method will be to work with successive approximations and iterate until convergence Start with an initial guess for c 1 and c 3
28 Estimating the c parameters Half the vector product is the area of the triangle; A is the area overswept by r y j are the sector-to-triangle ratios (close to 1) by Kepler II: Similarly:
29 Transforming the vector equation Transform from the equatorial to a new coordinate system, which we denote C : Multiplying the vector equation for Δ j by the transformation matrix, in the new system we have:
30 Solving for the Δ s Due to our choice of axes in the C system, the scalar equations become separable in the three Δ s The scalar products ξ i, η i,ζ i are known quantities
31 Solving for the Δ s, ctd
32 Algorithm Use {α j } to calculate Δ j unit direction vectors Use these to calculate {ξ 2, η 2, ζ 2, ξ 3, η 3 } and R eq C Transform Δ j and R,j vectors to C system We need to know c 1 and c 3! Assume y 2 / y 1 = y 2 / y 3 =1 Calculate the geocentric distances! Calculate the approximate heliocentric position vectors of the object! But now we can estimate overswept areas, i.e., more realistic values of c 1 and c 3 can be calculated. Iterate!
33 Numerical example
34 Orbit Improvement Apply an algorithm to the orbital elements to obtain an ephemeris Π: Orbital elements Θ: Algorithm (e.g., two-body) Φ: Ephemerides Compute the Jacobian of the algorithm: Further observations show discrepancies with the ephemeris
35 Orbit Improvement, ctd Identify the Jacobian with the ratio of finite differences: With many observations, we get a system of many equations involving known residuals δφ. Use a statistical method like the least-squares method to find the δπ that minimizes the residuals.
Satellite 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 informationis a revolution relative to a fixed celestial position. is the instant of transit of mean equinox relative to a fixed meridian position.
PERIODICITY FORMULAS: Sidereal Orbit Tropical Year Eclipse Year Anomalistic Year Sidereal Lunar Orbit Lunar Mean Daily Sidereal Motion Lunar Synodical Period Centenial General Precession Longitude (365.25636042
More informationIAU 2006 NFA GLOSSARY
IAU 2006 NFA GLOSSARY Prepared by the IAU Division I Working Group Nomenclature for Fundamental Astronomy'' (latest revision: 20 November 2007) Those definitions corresponding to the IAU 2000 resolutions
More informationThe Position of the Sun. Berthold K. P. Horn. necessary to know the position of the sun in the sky. This is particularly
MASSACHUSETTS INSTITUTE OF TECHNOLOGY ARTIFICIAL INTELLIGENCE LABORATORY Working Paper No. 162 March 1978 The Position of the Sun Berthold K. P. Horn Abstract. The appearance of a surface depends dramatically
More informationGeometry of Earth Sun System
12S56 Geometry of Earth Sun System Figure below shows the basic geometry Northern Hemisphere Winter ω equator Earth s Orbit Ecliptic ω ω SUN equator Northern Hemisphere Spring Northern Hemisphere Fall
More informationFundamentals of Satellite technology
Fundamentals of Satellite technology Prepared by A.Kaviyarasu Assistant Professor Department of Aerospace Engineering Madras Institute Of Technology Chromepet, Chennai Orbital Plane All of the planets,
More informationMAHALAKSHMI ENGINEERING COLLEGE-TRICHY QUESTION BANK UNIT I PART A
MAHALAKSHMI ENGINEERING COLLEGE-TRICHY QUESTION BANK SATELLITE COMMUNICATION DEPT./SEM.:ECE/VIII UNIT I PART A 1.What are the different applications of satellite systems? *Largest International System(Intel
More informationPHYSICS 1030 Homework #9
PHYSICS 1030 Homework #9 (Due Dec. 5, 2018, 6:00 pm) Find the position of the planet Mars at time t D December 5, 2018, 7:50 pm EST. You will do this by following the steps shown below. (a) Convert the
More informationASTRONOMICAL COORDINATE SYSTEMS CELESTIAL SPHERE
ASTRONOMICAL COORDINATE SYSTEMS CELESTIAL SPHERE To the naked eye, stars appear fixed on the sky with respect to one another. These patterns are often grouped into constellations. Angular measurements
More informationEarth Science, 13e Tarbuck & Lutgens
Earth Science, 13e Tarbuck & Lutgens Origins of Modern Astronomy Earth Science, 13e Chapter 21 Stanley C. Hatfield Southwestern Illinois College Early history of astronomy Ancient Greeks Used philosophical
More informationModern Navigation. Thomas Herring
12.215 Modern Navigation Thomas Herring Review of Monday s Class Spherical Trigonometry Review plane trigonometry Concepts in Spherical Trigonometry Distance measures Azimuths and bearings Basic formulas:
More informationThese notes may contain copyrighted material! They are for your own use only during this course.
Licensed for Personal Use Only DO NOT DISTRIBUTE These notes may contain copyrighted material! They are for your own use only during this course. Distributing them in anyway will be considered a breach
More informationPHYSICS 1030 Homework #9
PHYSICS 1030 Homework #9 (Due Dec. 6, 2017) Find the position of the planet Mars at time t D December 6, 2017, 5:00 am EST. You will do this by following the steps shown below. (a) Convert the time t to
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) 3. The Orbit in Space Gaëtan Kerschen Space Structures & Systems Lab (S3L) Motivation: Space We need means of describing orbits in three-dimensional space. Example: Earth s oblateness
More informationKnowing the Heavens. Goals: Constellations in the Sky
Goals: Knowing the Heavens To see how the sky changes during a night and from night to night. To measure the positions of stars in celestial coordinates. To understand the cause of the seasons. Constellations
More informationKnowing the Heavens. Goals: Constellations in the Sky
Goals: Knowing the Heavens To see how the sky changes during a night and from night to night. To measure the positions of stars in celestial coordinates. To understand the cause of the seasons. Constellations
More informationTime frames, leap seconds and GPS
Time frames, leap seconds and GPS Andy Shearer Centre for Astronomy NUI, Galway How important is absolute timing accuracy... I How important is absolute timing accuracy... II important enough for particle
More informationPHSC 1053: Astronomy Time and Coordinates
PHSC 1053: Astronomy Time and Coordinates Astronomical Clocks Earth s Rotation on its Axis Time between two successive meridian transits of the sun 1 solar day (our adopted clock time) 24 hours (86,400
More informationDid you Fall Back Today?
Did you Fall Back Today? Yes this morning at 2AM Daylight Savings Time (DST) we switched to Standard Time (ST) and turned our clocks back 1 hour. Time is basically calculated based on Earth s rotation,
More informationSatellite meteorology
GPHS 422 Satellite meteorology GPHS 422 Satellite meteorology Lecture 1 6 July 2012 Course outline 2012 2 Course outline 2012 - continued 10:00 to 12:00 3 Course outline 2012 - continued 4 Some reading
More informationMAE 180A: Spacecraft Guidance I, Summer 2009 Homework 2 Due Tuesday, July 14, in class.
MAE 180A: Spacecraft Guidance I, Summer 2009 Homework 2 Due Tuesday, July 14, in class. Guidelines: Please turn in a neat and clean homework that gives all the formulae that you have used as well as details
More informationEarth Science, 11e. Origin of Modern Astronomy Chapter 21. Early history of astronomy. Early history of astronomy. Early history of astronomy
2006 Pearson Prentice Hall Lecture Outlines PowerPoint Chapter 21 Earth Science 11e Tarbuck/Lutgens This work is protected by United States copyright laws and is provided solely for the use of instructors
More informationOn Sun-Synchronous Orbits and Associated Constellations
On Sun-Synchronous Orbits and Associated Constellations Daniele Mortari, Matthew P. Wilkins, and Christian Bruccoleri Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843,
More informationCoordinate Systems. Basis for any 3D Coordinate System. 2. Locate the x-y plane (the fundamental plane ) Usual approach to define angles:
Coordinate Systems Basis for any 3D Coordinate System Basic steps for the definition of a 3D coordinate system:. Locate the origin. Locate the -y plane (the fundamental plane ) 3. Decide on direction of
More informationEssential Astrophysics
ASTR 530 Essential Astrophysics Course Notes Paul Hickson The University of British Columbia, Department of Physics and Astronomy January 2015 1 1 Introduction and review Several text books present an
More informationOrbits in Geographic Context. Instantaneous Time Solutions Orbit Fixing in Geographic Frame Classical Orbital Elements
Orbits in Geographic Context Instantaneous Time Solutions Orbit Fixing in Geographic Frame Classical Orbital Elements Instantaneous Time Solutions Solution of central force motion, described through two
More informationIntroduction To Modern Astronomy I: Solar System
ASTR 111 003 Fall 2007 Lecture 02 Sep. 10, 2007 Introduction To Modern Astronomy I: Solar System Introducing Astronomy (chap. 1-6) Planets and Moons (chap. 7-15) Chap. 16: Our Sun Chap. 28: Search for
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 informationThe Measurement of Time
CHAPTER TWO The Measurement of Time Solar Time In antiquity the time of day was measured by the direction of a shadow cast in sunlight. This resulted in the development of a wide variety of sophisticated
More informationOn the definition and use of the ecliptic in modern astronomy
On the definition and use of the ecliptic in modern astronomy Nicole Capitaine (1), Michael Soffel (2) (1) : Observatoire de Paris / SYRTE (2) : Lohrmann Observatory, Dresden Technical University Introduction
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) 3B. The Orbit in Space and Time Gaëtan Kerschen Space Structures & Systems Lab (S3L) Previous Lecture: The Orbit in Time 3.1 ORBITAL POSITION AS A FUNCTION OF TIME 3.1.1 Kepler
More informationAST 443 / PHY 517. Astronomical Observa<onal Techniques. Prof. F.M. Walter
AST 443 / PHY 517 Astronomical Observa
More informationRECOMMENDATION ITU-R S Impact of interference from the Sun into a geostationary-satellite orbit fixed-satellite service link
Rec. ITU-R S.1525-1 1 RECOMMENDATION ITU-R S.1525-1 Impact of interference from the Sun into a geostationary-satellite orbit fixed-satellite service link (Question ITU-R 236/4) (21-22) The ITU Radiocommunication
More informationChapter 1: Discovering the Night Sky. The sky is divided into 88 unequal areas that we call constellations.
Chapter 1: Discovering the Night Sky Constellations: Recognizable patterns of the brighter stars that have been derived from ancient legends. Different cultures have associated the patterns with their
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 8 PLANETARY MOTIONS
1 CHAPTER 8 PLANETARY MOTIONS 8.1 Introduction The word planet means wanderer (πλάνητες αστέρες wandering stars); in contrast to the fixed stars, the planets wander around on the celestial sphere, sometimes
More informationNGA GNSS Division Precise Ephemeris Parameters
NGA GNSS Division Precise Ephemeris Parameters Precise Ephemeris Units. Earth-centered, Earth-fixed Coordinate system Position Velocity GPS time Trajectory interval Standard Trajectory Optional Trajectory
More informationM2 GLOSSARY aphelion: the point in an orbit that is the most distant from the Sun. apocenter: the point in an orbit that is farthest from the origin o
M1 T: the difference between Terrestrial Time (TT) and Universal Time (UT): T = TT UT1. UT1 (or UT): the value of the difference between Universal Time (UT) and Coordinated Universal Time (UTC): UT1 =
More informationaberration (of light): aberration, annual: aberration, diurnal: aberration, E-terms of: aberration, elliptic: aberration, planetary:
M1 aberration (of light): the relativistic apparent angular displacement of the observed position of a celestial object from its geometric position, caused by the motion of the observer in the reference
More informationUnit 2: Celestial Mechanics
Unit 2: Celestial Mechanics The position of the Earth Ptolemy (90 168 AD) Made tables that allowed a user to locate the position of a planet at any past, present, or future date. In order to maintain circular
More informationTime, coordinates and how the Sun and Moon move in the sky
Time, coordinates and how the Sun and Moon move in the sky Using the colors and magnitudes of quasars drawn from the SDSS Catalog Archive Server to distinguish quasars from stars using the light they emit
More informationThird Body Perturbation
Third Body Perturbation p. 1/30 Third Body Perturbation Modeling the Space Environment Manuel Ruiz Delgado European Masters in Aeronautics and Space E.T.S.I. Aeronáuticos Universidad Politécnica de Madrid
More informationLecture Module 2: Spherical Geometry, Various Axes Systems
1 Lecture Module 2: Spherical Geometry, Various Axes Systems Satellites in space need inertial frame of reference for attitude determination. In a true sense, all bodies in universe are in motion and inertial
More informationCELESTIAL MECHANICS. Part I. Mathematical Preambles
Chapter 1. Numerical Methods CELESTIAL MECHANICS Part I. Mathematical Preambles 1.1 Introduction 1.2 Numerical Integration 1.3 Quadratic Equations 1.4 The Solution of f(x) = 0 1.5 The Solution of Polynomial
More informationChapter S1 Lecture. The Cosmic Perspective Seventh Edition. Celestial Timekeeping and Navigation Pearson Education, Inc.
Chapter S1 Lecture The Cosmic Perspective Seventh Edition Celestial Timekeeping and Navigation 2014 Pearson Education, Inc. Celestial Timekeeping and Navigation 2014 Pearson Education, Inc. S1.1 Astronomical
More informationAST111, Lecture 1b. Measurements of bodies in the solar system (overview continued) Orbital elements
AST111, Lecture 1b Measurements of bodies in the solar system (overview continued) Orbital elements Planetary properties (continued): Measuring Mass The orbital period of a moon about a planet depends
More informationorbits Moon, Planets Spacecrafts Calculating the and by Dr. Shiu-Sing TONG
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources the and Spacecrafts orbits Moon, Planets Calculating the 171 of by Dr. Shiu-Sing TONG 172 Calculating the orbits
More informationINTRODUCTION TO ORBITAL MECHANICS - MODEL & SIMULATION SOFTWARE (OM-MSS) Earth, Sun, Moon & Satellites Motion in Orbit - Model & Simulation Software
Return to Website INTRODUCTION TO ORBITAL MECHANICS - MODEL & SIMULATION SOFTWARE (OM-MSS) Earth, Sun, Moon & Satellites Motion in Orbit - Model & Simulation Software RC Chakraborty (Retd), Former Director,
More informationIn all cases assume the observer is located at the latitude of Charlottesville (38 degrees north).
1. Recalling that azimuth is measured around the sky from North (North is 0 degrees, East is 90 degrees, South is 180 degrees, and West is 270 degrees) estimate (do not calculate precisely) the azimuth
More informationAstronomy. The Seasons
Astronomy The Seasons The seasons are caused by the inclination of the Earth s axis: when a hemisphere is tipped toward the Sun, the Sun is more directly above it. At the Summer Solstice the tilt is most
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 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 informationWeek 02. Assist. Prof. Dr. Himmet KARAMAN
Week 02 Assist. Prof. Dr. Himmet KARAMAN Contents Satellite Orbits Ephemerides GPS Review Accuracy & Usage Limitation Reference Systems GPS Services GPS Segments Satellite Positioning 2 Satellite Orbits
More informationEESC 9945 Geodesy with the Global Posi6oning System. Class 2: Satellite orbits
EESC 9945 Geodesy with the Global Posi6oning System Class 2: Satellite orbits Background The model for the pseudorange was Today, we ll develop how to calculate the vector posi6on of the satellite The
More informationOrbit Representation
7.1 Fundamentals 223 For this purpose, code-pseudorange and carrier observations are made of all visible satellites at all monitor stations. The data are corrected for ionospheric and tropospheric delays,
More informationAy 1 Lecture 2. Starting the Exploration
Ay 1 Lecture 2 Starting the Exploration 2.1 Distances and Scales Some Commonly Used Units Distance: Astronomical unit: the distance from the Earth to the Sun, 1 au = 1.496 10 13 cm ~ 1.5 10 13 cm Light
More informationCelestial mechanics in a nutshell
Celestial mechanics in a nutshell Marc van der Sluys CMiaNS.sf.net October 13, 2018 Copyright c 2013 2018 by Marc van der Sluys All rights reserved. No part of this publication may be reproduced, distributed,
More information1 Determination of the Orbit of a Minor Planet Using 3 Observations
1 Determination of the Orbit of a Minor Planet Using 3 Observations If the ecliptic latitudes of the three observations being used are greater than 2 it is possible to use the method of three observations
More informationThe sky and the celestial sphere
Chapter 1 The sky and the celestial sphere The Sun, and sometimes the Moon are, by and large, the only astronomical objects visible in the day sky. Traditionally, astronomy has been a nocturnal activity.
More informationLecture 2c: Satellite Orbits
Lecture 2c: Satellite Orbits Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Universal Gravita3on 3. Kepler s Laws 4. Pu>ng Newton and Kepler s Laws together and applying them to the Earth-satellite
More informationChapter S1 Celestial Timekeeping and Navigation. How do we define the day, month, year, and planetary time periods?
Chapter S1 Celestial Timekeeping and Navigation S1.1 Astronomical Time Periods Our goals for learning:! How do we define the day, month, year, and planetary time periods?! How do we tell the time of day?!
More informationEquatorial Telescope Mounting
Equatorial Telescope Mounting Star Catalogs simbad IRSA The Meridian Every line of celestial longitude is a meridian of longitude, but we recognize the line of longitude, or simply the great circle line,
More informationCoordinates on the Sphere
Survey Observations Coordinates on the Sphere Any position on the surface of a sphere (such as the Earth or the night sky) can be expressed in terms of the angular coordinates latitude and longitude Latitude
More informationt S 18. Determining Planetary Co-ordinates
8. Determining Planetary Co-ordinates θ θ 0 ω R In the heliocentric reference frame which rotates with the Earth s orbital motion, suppose that initially a planet s unplanet line makes an angle θ 0 with
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 informationASTRONOMICAL REFERENCE SYSTEMS AND FRAMES, ASTROMETRIC TECHNIQUES AND CATALOGS
1 ASTRONOMICAL REFERENCE SYSTEMS AND FRAMES, ASTROMETRIC TECHNIQUES AND CATALOGS Jan Vondrák, Astronomical Institute Prague P PART 1: Reference systems and frames used in astronomy:! Historical outline,
More informationFUNDAMENTAL ASTRONOMY
FUNDAMENTAL ASTRONOMY Magda Stavinschi Astronomical Institute of the Romanian Academy No indication of the distance to the objects The astrometric information is generally NOT the direction from which
More informationLOCATING CELESTIAL OBJECTS: COORDINATES AND TIME. a. understand the basic concepts needed for any astronomical coordinate system.
UNIT 2 UNIT 2 LOCATING CELESTIAL OBJECTS: COORDINATES AND TIME Goals After mastery of this unit, you should: a. understand the basic concepts needed for any astronomical coordinate system. b. understand
More informationOrbit Definition. Reference Vector. Vernal (March) Equinox Vector. Sun Vector
Simulation: TMG Thermal Analysis User's Guide Orbit Definition TMG can model three types of orbits: Beta Angle, Geostationary and Classical. For Earth, three special classical orbits are already partially
More informationFundamentals of Astrodynamics and Applications
Fundamentals of Astrodynamics and Applications Third Edition David A. Vallado with technical contributions by Wayne D. McClain Space Technology Library Published Jointly by Microcosm Press Hawthorne, CA
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) The Orbit in Space and Time Gaëtan Kerschen Space Structures & Systems Lab (S3L) Newton s laws F = ma F g mm = G r 1 2 uˆ 2 r Relative motion μ r = r 3 r Energy conserv. 2 v μ
More informationDRAFT OF NOMENCLATURE & TERMINOLOGY FOR IAU WG 1
s TIO terrestrial intermediate origin DRAFT OF NOMENCLATURE & TERMINOLOGY FOR IAU WG 1 This is not a complete list. Symbols are not, and need not necessarily be unique, eg φ is used for latitude, both
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 informationDiscovering the Night Sky
Discovering the Night Sky Guiding Questions 1. What role did astronomy play in ancient civilizations? 2. Are the stars that make up a constellation actually close to one another? 3. Are the same stars
More informationDiscovering the Night Sky
Guiding Questions Discovering the Night Sky 1. What role did astronomy play in ancient civilizations? 2. Are the stars that make up a constellation actually close to one another? 3. Are the same stars
More informationEarth-Centered, Earth-Fixed Coordinate System
Fundamentals of Global Positioning System Receivers: A Software Approach James Bao-Yen Tsui Copyright 2000 John Wiley & Sons, Inc. Print ISBN 0-471-38154-3 Electronic ISBN 0-471-20054-9 CHAPTER FOUR Earth-Centered,
More informationKnowing the Heavens. Chapter Two. Guiding Questions. Naked-eye (unaided-eye) astronomy had an important place in ancient civilizations
Knowing the Heavens Chapter Two Guiding Questions 1. What role did astronomy play in ancient civilizations? 2. Are the stars that make up a constellation actually close to one another? 3. Are the same
More informationNumerical Model for the Orbit of the Earth
Universal Journal of Geoscience 5(2): 33-39, 2017 DOI: 10.13189/ujg.2017.050203 http://www.hrpub.org Numerical Model for the Orbit of the Earth S. Karna 1,*, A. K. Mallik 2 1 Physics Department, Tri-Chandra
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 informationlightyears observable universe astronomical unit po- laris perihelion Milky Way
1 Chapter 1 Astronomical distances are so large we typically measure distances in lightyears: the distance light can travel in one year, or 9.46 10 12 km or 9, 600, 000, 000, 000 km. Looking into the sky
More informationCoordinate Systems for Astronomy or: How to get your telescope to observe the right object
Coordinate Systems for Astronomy or: How to get your telescope to observe the right object Figure 1: Basic definitions for the Earth Definitions - Poles, Equator, Meridians, Parallels The rotation of the
More informationCelestial Mechanics and Satellite Orbits
Celestial Mechanics and Satellite Orbits Introduction to Space 2017 Slides: Jaan Praks, Hannu Koskinen, Zainab Saleem Lecture: Jaan Praks Assignment Draw Earth, and a satellite orbiting the Earth. Draw
More informationThe Celestial Sphere. GEK1506 Heavenly Mathematics: Cultural Astronomy
The Celestial Sphere GEK1506 Heavenly Mathematics: Cultural Astronomy Helmer Aslaksen Department of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/ The
More informationObservational Astronomy - Lecture 4 Orbits, Motions, Kepler s and Newton s Laws
Observational Astronomy - Lecture 4 Orbits, Motions, Kepler s and Newton s Laws Craig Lage New York University - Department of Physics craig.lage@nyu.edu February 24, 2014 1 / 21 Tycho Brahe s Equatorial
More informationObservational Astronomy - Lecture 5 The Motion of the Earth and Moon Time, Precession, Eclipses, Tides
Observational Astronomy - Lecture 5 The Motion of the Earth and Moon Time, Precession, Eclipses, Tides Craig Lage New York University - Department of Physics craig.lage@nyu.edu March 2, 2014 1 / 29 Geosynchronous
More information3) During retrograde motion a planet appears to be A) dimmer than usual. B) the same brightness as usual C) brighter than usual.
Descriptive Astronomy (ASTR 108) Exam 1 B February 17, 2010 Name: In each of the following multiple choice questions, select the best possible answer. In the line on the scan sheet corresponding to the
More information1) Kepler's third law allows us to find the average distance to a planet from observing its period of rotation on its axis.
Descriptive Astronomy (ASTR 108) Exam 1 A February 17, 2010 Name: In each of the following multiple choice questions, select the best possible answer. In the line on the scan sheet corresponding to the
More informationDynamics of the Earth
Time Dynamics of the Earth Historically, a day is a time interval between successive upper transits of a given celestial reference point. upper transit the passage of a body across the celestial meridian
More informationASTRO 6570 Lecture 1
ASTRO 6570 Lecture 1 Historical Survey EARLY GREEK ASTRONOMY: Earth-centered universe - Some radical suggestions for a sun-centered model Shape of the Earth - Aristotle (4 th century BCE) made the first
More informationThe following terms are some of the vocabulary that students should be familiar with in order to fully master this lesson.
Lesson 211: EARTH'S SEASONS Students learn the complex geometry and planetary motions that cause Earth to have four distinct seasons. Fundamental Questions Attempting to give thorough and reasonable answers
More informationThe asteroids. Example for the usage of the Virtual Observatory
Example for the usage of the Virtual Observatory The asteroids Florian Freistetter, ZAH, Heidelberg florian@ari.uni-heidelberg.de Asteroids in the solar system There are not only planets in our solar system.
More informationGravitation and the Waltz of the Planets
Gravitation and the Waltz of the Planets Chapter Four Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets
More informationGravitation and the Waltz of the Planets. Chapter Four
Gravitation and the Waltz of the Planets Chapter Four Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets
More informationTIME & FREQUENCY. Overview from artefact to current definition & Realisation UTC. Frank Coutereel.
TIME & FREQUENCY Overview from artefact to current definition & Realisation UTC Frank Coutereel Legal Time in Belgium Past: based on GMT or UT (observations in sky) Today: based on UTC (working atomic
More informationAST 1002 Section 1 (Dobrosavljevic) PLANETS, STARS, GALAXIES
Your name (print) Your FSUID AST 1002 Section 1 (Dobrosavljevic) PLANETS, STARS, GALAXIES Midterm Exam 1, Fall 2018 Instructions: 1. Use a pencil for marking the machine scoring sheet. 2. Enter and encode
More informationGravitation and the Motion of the Planets
Gravitation and the Motion of the Planets 1 Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets go around
More informationA study upon Eris. I. Describing and characterizing the orbit of Eris around the Sun. I. Breda 1
Astronomy & Astrophysics manuscript no. Eris c ESO 2013 March 27, 2013 A study upon Eris I. Describing and characterizing the orbit of Eris around the Sun I. Breda 1 Faculty of Sciences (FCUP), University
More informationEquator SOME NOTES ON THE EQUATION OF TIME. λ α. Ecliptic. by Carlos Herrero
SOME NOTES ON THE EQUATION OF TIME by Carlos Herrero Version v2.1, February 214 I. INTRODUCTION Since ancient times humans have taken the Sun as a reference for measuring time. This seems to be a natural
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) 3A. The Orbit in Space and Time Gaëtan Kerschen Space Structures & Systems Lab (S3L) Previous Lecture: The Two-Body Problem 2.1 JUSTIFICATION OF THE 2-BODY MODEL 2.2 GRAVITATIONAL
More informationExperimental Analysis of Low Earth Orbit Satellites due to Atmospheric Perturbations
Experimental Analysis of Low Earth Orbit Satellites due to Atmospheric Perturbations Aman Saluja #1, Manish Bansal #2, M Raja #3, Mohd Maaz #4 #Aerospace Department, University of Petroleum and Energy
More information