# Spacecraft orbits and missions

Size: px
Start display at page:

Transcription

1 General Astrophysics and Space Research Course , Space Physics Module Spring 2009, Joachim Vogt Spacecraft orbits and missions Topics of this lecture Basics of celestial mechanics Geocentric orbits from LEO to GEO SSO, Lagrange points, gravity assists Appendix Review questions and further reading Additional problems and sample solutions Basics of celestial mechanics Celestial mechanics: motions and gravitational effects of celestial objects (stars, planets, moons,... ). The motion of planets around the Sun is described by Kepler s laws. Orbital mechanics or astrodynamics: motions of rockets, man-made satellites and spacecraft. Special attention is given to orbits around the Earth, but the field also deals with propulsion, transition between orbits, and interplanetary missions. Satellites: objects in orbit around a celestial body (star, planet, moon). Distinguish between natural satellites: planets around the Sun, moons around planets, moonlets around asteroids, and artificial satellites: spacecrafts in orbit around a celestial body. Orbital motion is also called revolution that has to be distinguished from planetary rotation: the Earth revolves around the Sun, and rotates around an axis that intersects the surface at the geographic poles. 1 2 Kepler I : The Law of Orbits Kepler II : The Law of Areas Planetary orbits are elliptical, with the Sun at one focus. An (imaginary) line connecting the planet to the Sun sweeps out equal areas in equal times. [Hyperphysics (1)] Important terms: foci of the ellipse, semi-major axis, semi-minor axis, eccentricity, perihelion, aphelion. The special case of zero eccentricity yields a circular orbit. [Hyperphysics (1)] The second law follows from the conservation of angular momentum L = m r v in central force fields such as the gravitational field. Planetary motion is fastest at the perihelion, and slowest at the aphelion. 3 4

2 Kepler III : The Law of Periods Exercise: The mass of the Sun [Hyperphysics (1)] For a planet revolving around the Sun, the square of its orbital period T is proportional to the cube of the semi-major axis a of its elliptical orbit: T 2 a 3. More precisely, we write T 2 = 4π2 a 3. GM Here GM is the product of the gravitational constant G and the mass M of the Sun. The planetary parameters T and a allow to deduce the solar mass M. Sample question: Using the Earth s orbital parameters, compute the mass of the Sun. Note that the orbit is almost circular (eccentricity e =0.0167), so you may write a =1AU= km. The Earth s orbital period is T/s = = , and the numerical value of the gravitational constant is G = m 3 kg 1 s 2. Hence we get for the mass of the Sun M = 4π 2 a 3 /(GT 2 ) in SI units, i.e., M kg = ( ) ( ) = Question A: Repeat this exercise with the orbital parameters of other planets to verify the result. 5 6 General two-body problem Kepler s laws apply also to objects that orbit celestial bodies other than the Sun (e.g., satellites around planets) as long as the central body is much more massive than the satellite. General case: two bodies of masses M 1 and M 2 orbit around the common barycenter (center-of-mass), and the distance parameters are measured with respect to it. The general form of the third law is given by Further terminology T 2 = 4π 2 G(M 1 + M 2 ) a3. Other terms for the points of closest approach (farthest excursion) on an elliptical orbit are as follows. General: periaspsis or pericenter (apoapsis or apocenter). Earth: perigee (apogee). Star: periastron (apoastron). Moon: periselene or perilune (aposelene or apolune). Special terms also for other planets, galaxy, black holes,... 7 Exercise: Barycenter of the Earth-Moon system Sample question: Compute the distance D bc of the barycenter of the Earth-Moon system from the Earth s center-of-mass. Note that for two masses M 1 and M 2 located on a line at distances D 1 and D 2, D bc =(M 1 D 1 + M 2 D 2 )/(M 1 + M 2 ). Let the subscripts 1 and 2 belong to Earth and Moon parameters, respectively. If we measure distances from the Earth s center, then D 1 = 0, and the formula can be divided by M 2 to yield D bc = D 2 /(μ + 1) where μ = M 1 /M 2 is the Earth-Moon mass ratio. With μ = 81, and D 2 = km, we obtain D bc = 4680 km. The barycenter of the Earth-Moon system is inside the Earth about 1700 km below the surface. Question B: Repeat this exercise for other planet-moon pairs in the solar system. 8

3 Geocentric orbits from LEO to GEO Orbital elements To completely characterize a Keplerian (unperturbed) orbit, six parameters must be specified. Keplerian elements: semi-major axis a, eccentricity e, inclination i, argument of the periapsis ω, longitude of ascending node Ω, mean anomaly M 0 (at epoch). Important categories of satellite orbits around Earth LEO Low Earth Orbits MEO Medium Earth Orbits GSO/GEO Geosynchronous/Geostationary Earth Orbits Distinguish also between polar orbits and equatorial orbits, circular orbits (small eccentricity) and highly elliptical orbits (large eccentricity). [Wikipedia (2)] Instead of M 0, other parameters are in use also Low Earth Orbit (LEO) Exercise: Orbital periods of LEO satellites [NASA (3)] Altitude range: altitudes below about 2000 km, practical upper limit of about 1000 km due to the increased radiation exposure (Van Allen belts), practical lower limit of about 160 km due to atmospheric drag (significant at altitudes below 500 km). LEO satellites: space stations (ISS), astronomy (Hubble ST), weather monitoring, communication (Iridium), reconnaissance (spy) missions (Keyhole, SAR-Lupe). 11 Sample question: How large is the orbital period T of an Earth satellite on a circular orbit at an altitude of h = 500 km? We apply Kepler s third law T 2 = 4π2 GM a3 (SI units) with central body mass M = M E = kg and semi-major axis a = R E + h = mto get T s = 2π ( ) 3/2 = The orbital period of the satellite is about 94 minutes. Question C: Repeat this exercise for an Earth satellite on a circular orbit at altitude h = 800 km. 12

4 Equatorial and polar LEOs Equatorial LEO: low inclination, least energy requirements. Polar LEO: high inclination, used for Earth monitoring and surveillance. Geosynchronous Orbit and Geostationary Orbit Geosynchronous Orbit (GSO) Orbital periods are exactly one sidereal Earth day: T =23 h 56 m. Ground paths are repeated once per day. At the farthest point of a GSO, the distance from the Earth s center (semi-major axis) is a =6.6 R E. GSO satellites are often used for telecommunication purposes. Geostationary Orbit (GEO) or Clarke Orbit: = circular equatorial GSO, i.e., a GSO with zero eccentricity and zero inclination. From the ground, a satellite at a GEO always occupies the same point in the sky. [GFZ/CHAMP (4)] Supersynchronous orbits: above GSO/GEO, westward drift, are used for disposal (graveyard orbits) or storage of satellites. Subsynchronous orbits: below GSO/GEO, eastward drift, can be used for station changes in an eastern direction Special GSOs: Tundra orbits Tundra orbits are examples of GSOs with high values of inclination and eccentricity. Geostationary Transfer Orbits (GTO) Direct insertion into GEO only by Heavy Lift Launch Vehicles (e.g., Delta IV, Space Shuttle, Proton, Ariane 5). [Chris Peat (5)] [Wikipedia (6)] Launch vehicles of smaller capacity use a geostationary transfer orbit (GTO): launch into a (circular) LEO, upper stage of the launch vehicle fires a rocket (tangent to the orbit), increase of velocity (delta-v) lifts the apogee, once in GTO, the satellite itself fires at the agogee (apogee motor) to lift the perigee. Such a procedure is also called a Hohmann transfer

5 Medium Earth Orbit (MEO) The term Medium Earth Orbit (MEO) refers to the region in space between LEO and GEO. Special MEOs: Molniya Orbits A Semi-Synchronous Orbit has an orbital period of half a sidereal day (i.e., T = 11 h 58 m ). Examples are the GPS constellation, and also the Russian Molniya satellites. [Wikipedia (6)] The most common use for satellites in MEO is navigation (e.g., GPS, Glonass, Galileo). Typical example: GPS circular orbits, orbital radius km (altitude km), each satellite completes two orbits each day. [Wikipedia (6)] SSO, Lagrange points, gravity assist Orbits and missions: SSO, Lagrange points, gravity assist Kepler s laws describe the motion of an object in the gravitational force field of a single idealized (spherically symmetric) celestial body. Not considered in the case of geocentric satellites are, e.g., (a) atmospheric drag, (b) non-spherically symmetric components of the Earth s gravitational field, (c) gravitational attraction of other celestial bodies (Sun, Moon,... ). Such perturbations give rise to variations of the orbital elements which are monitored and corrected in the process of station-keeping. A Sun-Synchronous Orbit (SSO) is a special polar LEO that takes advantage of an orbit perturbation imposed by (b). Lagrange points are special positions in a configuration of two bodies in circular revolution around each other, e.g., the Earth-Moon system or the Sun-Earth system. Gravity assists at the Moon or other planets are important to gain sufficient energy for interplanetary missions. Sun-Synchronous Orbits Earth s deviation from spherical symmetry is mainly due to its equatorial bulge caused by centrifugal action (which makes the planet an oblate ellipsoid rather than a perfect spheroid). For inclined (and not exactly polar) satellite orbits, this leads to a precession (i.e., a slow rotation) of the orbital plane. The precession rate matches the effect of Earth s revolution around the Sun for special choices of the altitude: typically km, and the inclination: about 98 (i.e., an almost polar orbit but slightly retrograde revolution with respect to the sense of the Earth s rotation). Then the orbital plane is fixed with respect to the Sun which explains the term Sun-Synchronous Orbit. Caution: Do not confuse with the term Heliosynchronous Orbit : that refers to a orbital motion around the Sun (!) with a period that matches the solar rotation period

6 Orbits and missions: SSO, Lagrange points, gravity assist Orbits and missions: SSO, Lagrange points, gravity assist Sun-Synchronous Orbit (cont d) Lagrange points Advantages: near-constant solar illumination conditions of the satellite s surface footprint, satellite orientation with respect to the Sun can be fixed (e.g., for a constant energy supply by solar panels, solar and astronomical observations), If two celestial bodies are in circular orbits around each other, there are five positions where another (much less massive) object can co-rotate with the configuration. [NASA/Landsat (7)] SSOs are typically used for remote sensing of the Earth surface (Landsat, ERS), meteorological satellites, spy satellites, observation of the Sun (TRACE, Yohkoh, ACRIMSat, Hinode), astronomical telescopes (IAS). [NASA/WMAP (8)] Spacecraft missions at Lagrange points L1: SOHO, ACE. L2: WMAP, and future space observatories like Herschel, Planck, and Gaia. L4 and L5: STEREO Orbits and missions: SSO, Lagrange points, gravity assist Orbits and missions: SSO, Lagrange points, gravity assist Gravity assists or Swing-bys: = close flybys at a planet to alter the velocity of a spacecraft. Gravity assists are also used to slow down space vehicles. Useful for missions to the inner planets, e.g., MESSENGER to Mercury. Missions to the outer planets, e.g., Cassini to Saturn, make use of gravity assists to gain speed. [NASA (9)] [JHU/APL (10)] 23 24

7 Orbits and missions: SSO, Lagrange points, gravity assist Figures and references Gravity assist analogy: elastic collision Light objects can change their speed substantially when they bounce off a heavy object in motion. [JPL/NASA (11)] In head-on elastic collision between a light object (mass m and velocity v) and a heavy object (mass M and velocity V ), the post-collision velocity of the light object is given by v (m M)v +2MV = m + M which simplifies to v = v +2V in the limit m/m (1) The illustrations of Kepler s laws are taken from the HyperPhysics web page hosted by the Department of Physics and Astronomy at Georgia State University: (URL checked on 22 February 2008). (2) Image credit: Wikipedia, the free encyclopedia ( URL checked on 22 February 2008). (3) Image credit: NASA ( URL checked on 22 February 2008). (4) Image credit: Geoforschungszentrum (GFZ) Potsdam, CHAMP mission web pages ( SYSTEMS.html, URL checked on 22 February 2008). (5) Image credit: Heavens Above web pages are developed and maintained by Chris Peat at (URL checked on 23 February 2008). The diagram shows the orbit of Radionet-3 operated by Sirius Satellite Radio. 26 Orbits and missions: Figures and references Figures and references (cont d) (6) Image credit: Wikipedia Commons, File names of images: Orbits around earth scale diagram.svg, Molniya.jpg, Hohmann transfer orbit.svg (URLs checked on 23 February 2008). (7) Image credit: NASA Landsat Handbook web pages ( toc.html, URL checked on 23 February 2008). (8) Image credit: NASA, public outreach pages of the WMAP mission ( mm/ob techorbit1.html, URL checked on 23 February 2008). (9) Image credit: JPL/NASA, public outreach web page of the Cassini-Huygens mission to Saturn and Titan ( URL checked on 24 February 2008). (10) Image taken from the MESSENGER web site at JHU/APL ( mission/trajectory.html, URL checked on 24 February 2008). (11) Cartoon taken from the the JPL web page A Gravity Assist Primer at (URL checked on 24 February 2008). 27 Review questions and further reading Review questions Explain the following key terms of celestial and orbital mechanics: astrodynamics, natural satellites, artificial satellites, revolution, planetary rotation. Explain and discuss Kepler s three law of planetary motion. Draw an ellipse and explain the following terms: foci, semi-major and semi-minor axis, eccentricity. Which law determines the velocity along the orbit? How is the orbital period related with the semi-major axis? Which law allows you to work out the mass of the central body, and how? What is the barycenter of a two-body system? Which form does Kepler s third law assume if the two bodies are equal in mass? How is the pericenter (apocenter) of a Kepler orbit called if the central body is (a) the Sun, (b) the Earth, and (c) the Moon? Name and explain three of the six Keplerian orbital elements. What is the altitude range of Low Earth Orbits (LEOs), and which kind of satellites populate LEOs? What are the advantages and disadvantages of equatorial and polar LEOs? 28

8 Orbits and missions: Review questions and further reading Review questions (cont d) Define and discuss: Geosynchronous Orbit (GSO), Geostationary Orbit (GEO), Geostationary Transfer Orbit (GTO). What is the altitude range of Medium Earth Orbits (MEOs), and which kind of satellites populate MEOs? What is a Semi-Synchronous Orbit? What causes the precession of the orbital plane of satellites at Sun- Synchronous Orbits (SSOs)? Which possibilities do SSOs offer, and what kind of satellites are placed at SSOs? What are Lagrange points, and which spacecraft missions take advantage of them? What is meant by a gravity assist? Which spacecraft missions make use of this maneuvers and how? Textbooks Celestial mechanics is covered in textbooks on classical mechanics. Orbital mechanics is also addressed in printed and electronic material on spaceflight, rockets, and propulsion systems. 29 Orbits and missions: Review questions and further reading Web resources The Jet Propulsion Laboratary maintains a suite of web pages that provides a comprehensive introduction to the subjects discussed in this lecture. Visit Basics of Space Flight at David Stern s educational web pages at provide information on a variety of space-related topics. In particular, have a look at the suite of pages entitled From Stargazers to Starships at HyperPhysics web page hosted by the Department of Physics and Astronomy at Georgia State University: A public outreach programme on space physics is coordinated at the Rice University (Project Manager: Patricia Reiff), see A very comprehensive list of links is A nice glossary can be found on the web site of the IMAGE spacecraft, see intro.html. The Open House web site of the Space Physics & Aeronomy section of the American Geophysical Union also provides useful information, see house.html. To reach the Oulu Space Physics Textbook, see 30 Additional questions and problems Sample solutions of the problems Problem 1 Compute the first cosmic velocity. It is defined as the (hypothetical) orbital velocity of an Earth satellite on a circular orbit at zero altitude, i.e., with semi-major axis a =1R E (one Earth radius). Problem 2 How does the first cosmic velocity relate to the escape velocity? The latter is also called the second cosmic velocity. Problem 3 Verify that the semi-major axis of a geosynchronous orbit (GSO) is a =6.6 R E. Problem 4 How long does it take to go from Earth to Mars by means of a Hohmann tranfer orbit? Where should Mars be located when the maneuver starts? (This is a somewhat longer exercise.) Sample solution of problem 1 On a circular orbit, the velocity V 1 is given by the ratio of the circumference of the circle and the orbital period T, i.e., V 1 =2πa/T, and therefore, V1 2 = 4π2 a 2. T 2 Here a denotes the radius of the circle. We insert Kepler s third law T 2 = 4π 2 a 3 /(GM) to obtain V1 2 = 4π2 a 2 4π 2 a GM = GM 3 a and, finally, V 1 = GM/a. Inserting the values M = M E = kg, G = m 3 kg 1 s 2, and a = R E = m yields V 1 = 7.9km/s for the first cosmic velocity

9 Orbits and missions: Sample solutions Sample solution of problem 2 The escape velocity is given by V 2 = 2GM a = 2 V 1 and thus differs from the first cosmic velocity by a factor of 2= Sample solution of problem 3 We apply Kepler s third law T 2 =4π 2 a 3 /(GM) with central body mass M = M E = kg. The orbital period is given (T =23 h 56 m = s), and we solve for the semi-major axis a and to yield a m = and a =6.6 R E. ( ) (86160) 2 1/3 = Sample solution of problem 4 4π 2 The solution of the problem can be found on the web page From Stargazers to Starships #21b Flight to Mars: How Long? Along what Path? by David Stern: 33

### Orbital Mechanics and Space Geometry

Orbital Mechanics and Space Geometry AERO4701 Space Engineering 3 Week 2 Overview First Hour Co-ordinate Systems and Frames of Reference (Review) Kepler s equations, Orbital Elements Second Hour Orbit

### 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,

### 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

### 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

### Penn State University Physics 211 ORBITAL MECHANICS 1

ORBITAL MECHANICS 1 PURPOSE The purpose of this laboratory project is to calculate, verify and then simulate various satellite orbit scenarios for an artificial satellite orbiting the earth. First, there

### 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,

### 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

### Lecture L17 - Orbit Transfers and Interplanetary Trajectories

S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L17 - Orbit Transfers and Interplanetary Trajectories In this lecture, we will consider how to transfer from one orbit, to another or to

### 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

### 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

### Astrodynamics (AERO0024)

Astrodynamics (AERO0024) 6. Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L) Course Outline THEMATIC UNIT 1: ORBITAL DYNAMICS Lecture 02: The Two-Body Problem Lecture 03:

### Astromechanics Two-Body Problem (Cont)

5. Orbit Characteristics Astromechanics Two-Body Problem (Cont) We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the

### Understanding Orbital Mechanics Through a Step-by-Step Examination of the Space-Based Infrared System (SBIRS)

Understanding Orbital Mechanics Through a Step-by-Step Examination of the Space-Based Infrared System (SBIRS) Denny Sissom Elmco, Inc. May 2003 Pg 1 of 27 SSMD-1102-366 [1] The Ground-Based Midcourse Defense

### Section 2. Satellite Orbits

Section 2. Satellite Orbits References Kidder and Vonder Haar: chapter 2 Stephens: chapter 1, pp. 25-30 Rees: chapter 9, pp. 174-192 In order to understand satellites and the remote sounding data obtained

### 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

### RS platforms. Fabio Dell Acqua - Gruppo di Telerilevamento

RS platforms Platform vs. instrument Sensor Platform Instrument The remote sensor can be ideally represented as an instrument carried by a platform Platforms Remote Sensing: Ground-based air-borne space-borne

### Flight and Orbital Mechanics

Flight and Orbital Mechanics Lecture slides Challenge the future 1 Material for exam: this presentation (i.e., no material from text book). Sun-synchronous orbit: used for a variety of earth-observing

### EDMONDS COMMUNITY COLLEGE ASTRONOMY 100 Winter Quarter 2007 Sample Test # 1

Instructor: L. M. Khandro EDMONDS COMMUNITY COLLEGE ASTRONOMY 100 Winter Quarter 2007 Sample Test # 1 1. An arc second is a measure of a. time interval between oscillations of a standard clock b. time

### Astronomy 1140 Quiz 1 Review

Astronomy 1140 Quiz 1 Review Prof. Pradhan September 15, 2015 What is Science? 1. Explain the difference between astronomy and astrology. (a) Astrology: nonscience using zodiac sign to predict the future/personality

### Learn From The Proven Best!

Applied Technology Institute (ATIcourses.com) Stay Current In Your Field Broaden Your Knowledge Increase Productivity 349 Berkshire Drive Riva, Maryland 1140 888-501-100 410-956-8805 Website: www.aticourses.com

### 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

### 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

### USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION

USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION Ian Cooper School of Physics The University of Sydney i.cooper@physics.usyd.edu.au Introduction The numerical calculations performed by scientists and engineers

### Newton s Law of Gravity

Gravitational Potential Energy On Earth, depends on: object s mass (m) strength of gravity (g) distance object could potentially fall Gravitational Potential Energy In space, an object or gas cloud has

### Solar System Fundamentals. What is a Planet? Planetary orbits Planetary temperatures Planetary Atmospheres Origin of the Solar System

Solar System Fundamentals What is a Planet? Planetary orbits Planetary temperatures Planetary Atmospheres Origin of the Solar System Properties of Planets What is a planet? Defined finally in August 2006!

### Orbital Dynamics: Formulary

Orbital Dynamics: Formulary 1 Introduction Prof. Dr. D. Stoffer Department of Mathematics, ETH Zurich Newton s law of motion: The net force on an object is equal to the mass of the object multiplied by

### Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13.

Chapter 5. Gravitation Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13. 5.1 Newton s Law of Gravitation We have already studied the effects of gravity through the

### Can Hubble be Moved to the International Space Station? 1

Can Hubble be Moved to the International Space Station? 1 On January 16, NASA Administrator Sean O Keefe informed scientists and engineers at the Goddard Space Flight Center (GSFC) that plans to service

### 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

### astronomy 2008 1. A planet was viewed from Earth for several hours. The diagrams below represent the appearance of the planet at four different times.

1. A planet was viewed from Earth for several hours. The diagrams below represent the appearance of the planet at four different times. 5. If the distance between the Earth and the Sun were increased,

### Satellite Posi+oning. Lecture 5: Satellite Orbits. Jan Johansson jan.johansson@chalmers.se Chalmers University of Technology, 2013

Lecture 5: Satellite Orbits Jan Johansson jan.johansson@chalmers.se Chalmers University of Technology, 2013 Geometry Satellite Plasma Posi+oning physics Antenna theory Geophysics Time and Frequency GNSS

### 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

### Does currently available technology have the capacity to facilitate a manned mission to Mars?

Furze Platt Senior School Does currently available technology have the capacity to facilitate a manned mission to Mars? Daniel Messias Date: 8/03/2015 Candidate Number: 7158 Centre Number: 51519 Contents

### Introduction to satellite constellations orbital types, uses and related facts

Introduction to satellite constellations orbital types, uses and related facts Dr Lloyd Wood space team, Cisco Systems http://www.cisco.com/go/space Guest lecture, ISU summer session July 2006 created

### A. 81 2 = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great.

Q12.1 The mass of the Moon is 1/81 of the mass of the Earth. Compared to the gravitational force that the Earth exerts on the Moon, the gravitational force that the Moon exerts on the Earth is A. 81 2

### Examination Space Missions and Applications I (AE2103) Faculty of Aerospace Engineering Delft University of Technology SAMPLE EXAM

Examination Space Missions and Applications I AE2103 Faculty of Aerospace Engineering Delft University of Technology SAMPLE EXAM Please read these instructions first: This are a series of multiple-choice

### Planetary Orbit Simulator Student Guide

Name: Planetary Orbit Simulator Student Guide Background Material Answer the following questions after reviewing the Kepler's Laws and Planetary Motion and Newton and Planetary Motion background pages.

### Newton s Law of Universal Gravitation

Newton s Law of Universal Gravitation The greatest moments in science are when two phenomena that were considered completely separate suddenly are seen as just two different versions of the same thing.

### Satellites and Space Stations

Satellites and Space Stations A satellite is an object or a body that revolves around another object, which is usually much larger in mass. Natural satellites include the planets, which revolve around

### From Aristotle to Newton

From Aristotle to Newton The history of the Solar System (and the universe to some extent) from ancient Greek times through to the beginnings of modern physics. The Geocentric Model Ancient Greek astronomers

### Celestial Sphere. Celestial Coordinates. Lecture 3: Motions of the Sun and Moon. ecliptic (path of Sun) ecliptic (path of Sun)

Lecture 3: Motions of the and Moon ecliptic (path of ) ecliptic (path of ) The 23.5 degree tilt of Earth s spin axis relative to its orbital axis around the causes the seasons Celestial Sphere Celestial

### Orbital Dynamics with Maple (sll --- v1.0, February 2012)

Orbital Dynamics with Maple (sll --- v1.0, February 2012) Kepler s Laws of Orbital Motion Orbital theory is one of the great triumphs mathematical astronomy. The first understanding of orbits was published

### Satellite Communications

Satellite Communications Department of Electrical Engineering Faculty of Engineering Chiangmai University Origin of Satellite Communications Arthur C. Clark (1945) British Science fiction writer propose

### EN4 Dynamics and Vibrations. Design Project. Orbital Design for a Lunar Impact Mission. Synopsis

EN4 Dynamics and Vibrations Design Project Orbital Design for a Lunar Impact Mission Synopsis NASA has identified a need for a low-cost mission to launch a satellite that will impact the moon. You will

### SUN-SYNCHRONOUS ORBIT SLOT ARCHITECTURE ANALYSIS AND DEVELOPMENT. A Thesis. Presented to. the Faculty of California Polytechnic State University

SUN-SYNCHRONOUS ORBIT SLOT ARCHITECTURE ANALYSIS AND DEVELOPMENT A Thesis Presented to the Faculty of California Polytechnic State University San Luis Obispo In Partial Fulfillment of the Requirements

### Chapter 6. Orbital Mechanics. Maj Edward P. Chatters IV, USAF; Maj Bryan Eberhardt, USAF; and Maj Michael S. Warner, USAF

Chapter 6 Orbital Mechanics Maj Edward P. Chatters IV, USAF; Maj Bryan Eberhardt, USAF; and Maj Michael S. Warner, USAF Knowledge of orbital motion is essential for a full understanding of space operations.

### Astronomy 110 Homework #04 Assigned: 02/06/2007 Due: 02/13/2007. Name:

Astronomy 110 Homework #04 Assigned: 02/06/2007 Due: 02/13/2007 Name: Directions: Listed below are twenty (20) multiple-choice questions based on the material covered by the lectures this past week. Choose

### Binary Stars. Kepler s Laws of Orbital Motion

Binary Stars Kepler s Laws of Orbital Motion Kepler s Three Laws of orbital motion result from the solution to the equation of motion for bodies moving under the influence of a central 1/r 2 force gravity.

### Chapter 2. Mission Analysis. 2.1 Mission Geometry

Chapter 2 Mission Analysis As noted in Chapter 1, orbital and attitude dynamics must be considered as coupled. That is to say, the orbital motion of a spacecraft affects the attitude motion, and the attitude

### Trajectory design for the Solar Orbiter mission

Monografías de la Real Academia de Ciencias de Zaragoza. 25: 177 218, (2004). Trajectory design for the Solar Orbiter mission G. Janin European Space Operations Centre. European Space Agency. 64293 Darmstadt,

### DEVELOPMENT OF AN ARCHITECTURE OF SUN-SYNCHRONOUS ORBITAL SLOTS TO MINIMIZE CONJUNCTIONS. Brian Weeden Secure World Foundation

DEVELOPMENT OF AN ARCHITECTURE OF SUN-SYNCHRONOUS ORBITAL SLOTS TO MINIMIZE CONJUNCTIONS Brian Weeden Secure World Foundation Sun-synchronous orbit (SSO) satellites serve many important functions, primarily

### 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

### Development of a Sun Synchronous. Conjunctions

Development of a Sun Synchronous Zoning Architecture to Minimize Conjunctions Kevin Shortt Brian Weeden Secure World Foundation www.secureworldfoundation.org Overview Current Situation in Sun synchronous

### Lab 7: Gravity and Jupiter's Moons

Lab 7: Gravity and Jupiter's Moons Image of Galileo Spacecraft Gravity is the force that binds all astronomical structures. Clusters of galaxies are gravitationally bound into the largest structures in

### 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

### 1. The orbit of each planet is an ellipse with the Sun at one focus. 2. The line joining the planet to the Sun sweeps out equal areas in equal times.

Appendix A Orbits As discussed in the Introduction, a good first approximation for satellite motion is obtained by assuming the spacecraft is a point mass or spherical body moving in the gravitational

### G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M

G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD

### 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.

### Physics Competitions Vol 13 No 2 2011 & Vol.14 No 1 2012. A few good orbits. 400 088. # Corresponding author: anikets@hbcse.tifr.res.

A few good orbits Chiraag Juvekar 1, Mehul Jain 1 and Aniket Sule 2,# 1 Indian Institute of Technology (Bombay), Mumbai, Maharashtra, India - 400 076. 2 Homi Bhabha Centre for Science Education (HBCSE).

### Lab 6: Kepler's Laws. Introduction. Section 1: First Law

Lab 6: Kepler's Laws Purpose: to learn that orbit shapes are ellipses, gravity and orbital velocity are related, and force of gravity and orbital period are related. Materials: 2 thumbtacks, 1 pencil,

### 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

### 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

### Development of an automated satellite network management system

Development of an automated satellite network management system Iasonas Kytros Christos Porios Nikitas Terzoudis Varvara Chatzipavlou Coach: Sitsanlis Ilias February 2013 Abstract In this paper we present

### Use the following information to deduce that the gravitational field strength at the surface of the Earth is approximately 10 N kg 1.

IB PHYSICS: Gravitational Forces Review 1. This question is about gravitation and ocean tides. (b) State Newton s law of universal gravitation. Use the following information to deduce that the gravitational

### Orbital Mechanics. Orbital Mechanics. Principles of Space Systems Design. 2001 David L. Akin - All rights reserved

Energy and velocity in orbit Elliptical orbit parameters Orbital elements Coplanar orbital transfers Noncoplanar transfers Time and flight path angle as a function of orbital position Relative orbital

### An Introduction to Astronomy and Cosmology. 1) Astronomy - an Observational Science

An Introduction to Astronomy and Cosmology 1) Astronomy - an Observational Science Why study Astronomy 1 A fascinating subject in its own right. The origin and Evolution of the universe The Big Bang formation

### The Four Seasons. A Warm Up Exercise. A Warm Up Exercise. A Warm Up Exercise. The Moon s Phases

The Four Seasons A Warm Up Exercise What fraction of the Moon s surface is illuminated by the Sun (except during a lunar eclipse)? a) Between zero and one-half b) The whole surface c) Always half d) Depends

### 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.)

### Coverage Characteristics of Earth Satellites

Coverage Characteristics of Earth Satellites This document describes two MATLAB scripts that can be used to determine coverage characteristics of single satellites, and Walker and user-defined satellite

### 4.1.6. Interplanetary Travel. Outline. In This Section You ll Learn to...

Interplanetary Travel 4.1.6 In This Section You ll Learn to... Describe the basic steps involved in getting from one planet in the solar system to another Explain how we can use the gravitational pull

### The Gravitational Field

The Gravitational Field The use of multimedia in teaching physics Texts to multimedia presentation Jan Hrnčíř jan.hrncir@gfxs.cz Martin Klejch martin.klejch@gfxs.cz F. X. Šalda Grammar School, Liberec

### Quasi-Synchronous Orbits

Quasi-Synchronous Orbits and Preliminary Mission Analysis for Phobos Observation and Access Orbits Paulo J. S. Gil Instituto Superior Técnico Simpósio Espaço 50 anos do 1º Voo Espacial Tripulado 12 de

### Mobile Computing. Chapter 5: Satellite Systems

Mobile Computing Chapter 5: Satellite Systems Prof. Sang-Jo Yoo History of satellite communication 1945 Arthur C. Clarke publishes an essay about Extra Terrestrial Relays 1957 First satellite SPUTNIK by

### Lecture 19: Planet Formation I. Clues from the Solar System

Lecture 19: Planet Formation I. Clues from the Solar System 1 Outline The Solar System:! Terrestrial planets! Jovian planets! Asteroid belt, Kuiper belt, Oort cloud Condensation and growth of solid bodies

### Chapter 5: Circular Motion, the Planets, and Gravity

Chapter 5: Circular Motion, the Planets, and Gravity 1. Earth s gravity attracts a person with a force of 120 lbs. The force with which the Earth is attracted towards the person is A. Zero. B. Small but

### Mobile Communications: Satellite Systems

Mobile Communications: Satellite Systems Mobile Communication: Satellite Systems - Jochen Schiller http://www.jochenschiller.de 1 History of satellite communication 1945 Arthur C. Clarke publishes an essay

### Niraj Sir GRAVITATION CONCEPTS. Kepler's law of planetry motion

GRAVITATION CONCEPTS Kepler's law of planetry motion (a) Kepler's first law (law of orbit): Every planet revolves around the sun in an elliptical orbit with the sun is situated at one focus of the ellipse.

### The Sun. Solar radiation (Sun Earth-Relationships) The Sun. The Sun. Our Sun

The Sun Solar Factoids (I) The sun, a medium-size star in the milky way galaxy, consisting of about 300 billion stars. (Sun Earth-Relationships) A gaseous sphere of radius about 695 500 km (about 109 times

### Correct Modeling of the Indirect Term for Third-Body Perturbations

AAS 07-47 Correct Modeling of the Indirect Term for Third-Body Perturbations Matthew M. Berry * Vincent T. Coppola The indirect term in the formula for third body perturbations models the acceleration

### Satellite Mission Analysis

CARLETON UNIVERSITY SPACECRAFT DESIGN PROJECT 2004 FINAL DESIGN REPORT Satellite Mission Analysis FDR Reference Code: FDR-SAT-2004-3.2.A Team/Group: Satellite Mission Analysis Date of Submission: April

### Earth Is Not the Center of the Universe

Earth Is Not the Center of the Universe Source: Utah State Office of Education Introduction Have you ever looked up at the night sky and wondered about all the pinpoint lights? People through the ages

### Study Guide: Solar System

Study Guide: Solar System 1. How many planets are there in the solar system? 2. What is the correct order of all the planets in the solar system? 3. Where can a comet be located in the solar system? 4.

### Problem Set #13 Solutions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.0L: Physics I January 3, 06 Prof. Alan Guth Problem Set #3 Solutions Due by :00 am on Friday, January in the bins at the intersection of Buildings

### Gravitation and Newton s Synthesis

Gravitation and Newton s Synthesis Vocabulary law of unviversal Kepler s laws of planetary perturbations casual laws gravitation motion casuality field graviational field inertial mass gravitational mass

### 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

### Satellites and Space Probes Space System Design, MAE 342, Princeton University Robert Stengel

Satellites and Space Probes Space System Design, MAE 342, Princeton University Robert Stengel Atmospheric science and meteorology satellites Earth resources satellites Navigation satellites Communications

### 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

### CELESTIAL CLOCK - THE SUN, THE MOON, AND THE STARS

INTRODUCTION CELESTIAL CLOCK - THE SUN, THE MOON, AND THE STARS This is a scientific presentation to provide you with knowledge you can use to understand the sky above in relation to the earth. Before

### 5. Satellite Systems. History of Satellite Communications

5. Satellite Systems History and Orbits Routing, Localization, and Hand-over Systems 2005 Burkhard Stiller and Jochen Schiller FU Berlin M5 1 History of Satellite Communications 1945 Arthur C. Clarke about

### 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

### Pluto Data: Numbers. 14b. Pluto, Kuiper Belt & Oort Cloud. Pluto Data (Table 14-5)

14b. Pluto, Kuiper Belt & Oort Cloud Pluto Pluto s moons The Kuiper Belt Resonant Kuiper Belt objects Classical Kuiper Belt objects Pluto Data: Numbers Diameter: 2,290.km 0.18. Earth Mass: 1.0. 10 22 kg

### Explain the Big Bang Theory and give two pieces of evidence which support it.

Name: Key OBJECTIVES Correctly define: asteroid, celestial object, comet, constellation, Doppler effect, eccentricity, eclipse, ellipse, focus, Foucault Pendulum, galaxy, geocentric model, heliocentric

### SATELLITE ORBIT DETERMINATION AND ANALYSIS (S.O.D.A) A VISUAL TOOL OF SATELLITE ORBIT FOR SPACE ENGINEERING EDUCATION & RESEARCH

SATELLITE ORBIT DETERMINATION AND ANALYSIS (S.O.D.A) A VISUAL TOOL OF SATELLITE ORBIT FOR SPACE ENGINEERING EDUCATION & RESEARCH 1 Muhammad Shamsul Kamal Adnan, 2 Md. Azlin Md. Said, 3 M. Helmi Othman,

### The Solar Wobble or Gravity, Rosettes and Inertia

The Solar Wobble or Gravity, Rosettes and Inertia john.erich.ebner@gmail.com http:blackholeformulas.com February 10, 2015 Abstract Our objective is to show that the sun moves. At least it wobbles. Any

### Mobile Communications Chapter 5: Satellite Systems

Mobile Communications Chapter 5: Satellite Systems History Basics Localization Handover Routing Systems History of satellite communication 1945 Arthur C. Clarke publishes an essay about Extra Terrestrial

### Tidal forces in the Solar System

Tidal forces in the Solar System Introduction As anywhere else in the Universe, gravity is the basic and fundamental principle that rules the shape and permanent motion of all the celestial bodies inside

### Class 2 Solar System Characteristics Formation Exosolar Planets

Class 1 Introduction, Background History of Modern Astronomy The Night Sky, Eclipses and the Seasons Kepler's Laws Newtonian Gravity General Relativity Matter and Light Telescopes Class 2 Solar System