Penn State University Physics 211 ORBITAL MECHANICS 1
|
|
- Ralph Anthony
- 7 years ago
- Views:
Transcription
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 is the analysis of the two-body problem with respect to Newton s Law of Universal Gravitation, and then the examination of the specific scenario known as Kepler s problem in relation to his Three Laws of Planetary Motion. THEORY Newton s Law of Universal Gravitation provides an equivalent expression of his second law of motion,, with respect to the force of attraction between two bodies. The general form given in Eq. 1 (1) where G is the gravitational constant, G = x10-11 m 3 /kg s 2, m 1 and m 2 are the masses of the respective bodies and r 2 is the square of the distance separating the bodies. This force is a central force 2 and is therefore a conservative force, meaning that it is independent of path and equals the first derivative of the potential energy between the masses. When there exists a large difference between the masses of the objects, such that the mass of the smaller object can be neglected a specific form of the two body problem known as Kepler s problem is achieved. The actual problem to be solved though is not one of finding forces, but is instead one of finding the position or speed of the bodies as a function of time given their masses, initial positions and respective velocities. The solution to this problem is a Keplerian orbit and can be expressed with what are called the Classical Orbital Elements (COE) 3. In a simplified model of the Keplerian problem, disregarding outside forces from other bodies, atmospheric drag and relativistic effects, it is found that the only force acting on the satellite causes an acceleration directed towards the center of the inertial body, Earth in our case. Since the gravitational 1 Portions of this lab are derived from U.S. Air Force Academy s ASTRO 310 class A central force: a force being exerted by one object onto another object that is directed along the line connecting both objects. 3 Classical Orbital Element (COE) Six parameters that describe all the aspects of a specific orbit, also known as the Keplerian elements. They are: semi-major axis a, eccentricity e, inclination i, ascending node Ω, argument of perigee ω, and true anomaly ν. - 1
2 force is conservative, we conclude that the total mechanical energy of the system is a constant and the sum of the potential and kinetic energies equal that constant. The total mechanical energy is given by the expression in Eq. 2. (2) where is the semi-major axis, is the mass of the Earth, and is the mass of the orbiting body or satellite. Additionally, we can derive an expression for the tangential velocity of the satellite as given in Eq. 3. (3a) (3b) Tangential or linear velocity is the velocity vector tangent to the circular path as show in Figure 1. FIGURE I As the satellite orbits the Earth, it also has constant angular momentum resulting from the conservative gravitational force. Though today we can derive all of the constants and equations of motion for an orbiting body from Newton s Law of Gravitation, Kepler did not have such luxury and thus derived his laws of planetary motion from observation. Kepler s first law tells us that the shape of the orbit will be an ellipse, and thus we can derive a complete description of the orbit using the definition of a conic section. Specifically, given the position and velocity of a satellite at apogee 4 or perigee 5 or given the eccentricity and semi-major axis, we can determine the angular momentum and period of the orbit. Table 1 provides the related formulas for a conic section with Figure 1 showing an ellipse for reference. 4 Apogee point at which the distance between the orbiting body and Earth is at a maximum. 5 Perigee point at which the distance between the orbiting body and Earth is at a minimum. - 2
3 Table I Elliptical Parameters radius of perigee or periapsis radius of apogee or apoapsis reduced mass - ( ) Figure I Eccentricity: Semi-major axis: ( ) Parameter the Ellipse: ( ) Specific Angular Momentum: Period of the Orbit: Polar Equation of an Ellipse: ( ) ( ) Area of an Ellipse: : Orbit is elliptical, the greater the value, the more eccentric. : Orbit is circular. : Orbit is a parabola. Using the information above, you will be analyzing different orbital scenario problems first on paper, then using Satellite Tool Kit 9 (STK) 6 to verify results and simulate the scenario. STK is a software package designed specifically for simulating air and space missions. The STK engine allows a user to create a virtual world to configure and test potential or current real world situations. The allowable level detail within the software makes it the preferred application used by aerospace organizations worldwide including NASA, Boeing and Lockheed Martin to name a few. 6 Analytical Graphics, Satellite Tool Kit 9 (STK) - 3
4 PART I Newton s Law of Universal Gravitation and Ballistic Trajectories In this part of the experiment you will use STK to place a missile at the North Pole approximately km above the Earth s surface. Using Newton s Law of Universal Gravitation and the related derived formula for tangential velocity (Eq. 3b) where r is the radius of the earth added to the given altitude of the missile ( ( ) ( ) ) determine what minimum velocity that is required for the missile to complete one revolution around the Earth. Calculations: (Remember to check the units of any constants and make sure they are consistent.) Tangential Velocity Required for a Complete Revolution: Period of One Revolution: PROCEDURE 1. Open STK. 2. On the startup screen, click Create New Scenario (Figure 1). Click Create a New Scenario FIGURE 1 3. In the New Scenario Window, name the scenario and set the analysis time. (Figure 2) a) Do not use spaces in the scenario name. You can use the underscore _ in place thereof. b) The analysis start and stop time determine how long of a period you need to generate and collect information for. In this activity, 6 hours will be more than sufficient. Change the end time so it is exactly 6 hours later than the start time. (Reference Figure 2). - 4
5 Type a name for the scenario. The location to save the scenario and related files. Leave as it. Description is optional. Start time. End time. Change this to be 6 hours after the start time. FIGURE 2 4. Add and configure a FACILITY OBJECT to the scenario a) Left click on the arrow next to the Satellite Object ( ) on the default toolbar to open the Object Catalog list (as shown to the right). b) Double click on the Facility icon it will close the window and a Facility will appear in the Object Browser under and attached to the Scenario icon. c) Right click on the icon and select Rename. d) Rename the Facility North_Pole the underscore between North and Pole is important STK does not allow spaces in its object names. e) Double click on the icon in the Object Browser this will open the Properties Browser showing the properties for the object. f) Type in the Latitude, Longitude (pick any value -180 to 180 deg), and Altitude (use 0 km) for the North Pole (See Figure 3 and note the values in the figure are NOT for the North Pole. Change them!). g) Click Apply at the bottom of the Properties Browser, click OK deg 0.0 deg FIGURE 3-5
6 5. Verify in the 3-D window that your facility is now at the North Pole a) Place the mouse cursor in the 3-D window. Holding down the left mouse button, rotate the Earth to verify your facility is at the North Pole. (If it is not there, repeat the preceding step.) b) Click on the View From/To icon ( ). c) Select the North_Pole in the View From frame and click OK. (See Figure 2) d) Move the view around using the left mouse button. Zoom in and out by clicking and holding the right mouse button. e) After verifying, click the Home View ( ) button on the vertical tool bar. 6. Add and configure a Missile to the Scenario Figure 4: Select North_Pole Select the North_Pole a) Pull down the Object Catalog ( ) from the default toolbar. Double click on the Missile to add a missile to the scenario. b) Double click on the Missile to open the Properties Browser to view the missile s Property Pages. c) Select 3D Graphics/Trajectory Property Page. Change the Ground Track Lead Type to None, change the Trajectory Lead Type to None and set the Trajectory Trail Type to All. Then click Apply. (See Figure 5) Change to None Change to None Change to All Select 3D Graphics / Trajectory Figure 5: Ground Track/Trajectory Type - 6
7 d) Select the Basic/Trajectory Property Page. e) Type in the Launch Latitude Geodetic and Launch Longitude of the North Pole. f) Type in a Launch Altitude of km. g) Change the Impact Latitude Geodetic to Launch Elevation. h) Set the Elevation to 0 0 a horizontal launch. i) Set the Launch Azimuth to the angle from true north measured clockwise to the launch direction. j) Set the Fixed Delta V to 5.0 km/sec. k) Click Apply at the bottom of the Properties Browser. Do Not click inside the 2-D window. This will cause the missile launch location to change. If you accidentally do, reset the Latitude/Longitude/Azimuth/Elevation in the Properties Browser. Instead of selecting the 2-D window by clicking on its banner, select the 2-D tab at the bottom of the Workspace. 7. Evaluate the effects of launch velocities a) Arrange your Workspace windows so you can see the 3-D and 2-D window and the Basic Trajectory Property Page (see Figure 6). Do Not click inside the 2D window. Time Step Figure 6: Suggested Workspace Window Arrangement b) You can change the time step or speed of your scenario and also step forward or backward frame by frame: You may want to slow down the simulation using the Time Step Controls ( ) to get a more accurate reading. You can also use the scenario Step Forward ( ) or Step Reverse ( ) buttons move the scenario forward or backward one time step for a more accurate reading. The Reset button ( )will restart the scenario, and the Pause button ( ) will pause the scenario. c) Press Start ( ) to run the animation until the 3-D trajectory disappears. Record the data in Table 2 below. d) Change the Fixed Delta V value from 5.0 km/sec to 8.5 km/sec in 0.5 km/sec intervals. e) Remember to click Apply after each change and to Reset the animation before replaying the scenario. - 7
8 f) Record your observations in Table 2 Remember, if you click in the 2-D map window without first clicking the Zoom In button, the launch location will be changed to the cursor location. You may want to zoom into a particular area of the 2-D map using the Zoom Control ( ) buttons. Click on the Zoom In button and then draw a box around the area you want to enlarge. Click the Zoom Out button to reset the 2-D map. To get the Time of Flight, watch the Current Scenario Time window in the default tool bar at the top of the screen or use the time readout at the bottom of the screen. To get the Lat/Long of a location on the 2-D map, place your cursor over the location and look to the bottom of the screen for the Lat/Long readout. Latitude, Longitude Table 2: Simulation Data Record Launch Delta V (Km/Sec) Time of Flight (Approximate to impact or 1 revolution) Impact Location (Approximate Latitude and Longitude) 4.5 km/sec 16 min 5 sec N Latitude, 4 0 W Longitude 5.0 km/sec Shape of trajectory/orbit (Circular, Elliptical, Parabolic, Hyperbolic) Earth intersecting elliptical or ballistic trajectory 5.5 km/sec 6.0 km/sec 6.5 km/sec 7.0 km/sec 7.5 km/sec 8.0 km/sec 8.5 km/sec Questions: 1. At what Fixed Delta V does the missile make one complete circular orbit around the Earth? 2. How does this Delta V compare to your theoretical calculation? 3. What is the gravitational force and centripetal acceleration acting on the missile that makes the first complete revolution? (Neglecting the mass of the missile and using in place of GM E m s.) 4. How does the time for one revolution compare to your period calculation? - 8
9 Part II Kepler s First Law Johannes Kepler was a German astronomer who is best known for his laws of planetary motion. Kepler derived his three laws of planetary motion over a 16 year period from observation data collected by Tycho Brahe, a Danish astronomer, whom Kepler was an assistant to. Newton, who was born 12 years after Kepler s death, demonstrated that Kepler s laws of planetary motion are the consequences of the gravitational force between the planets and the sun. Kepler s laws of planetary motion are: 1. Each planet in the Solar System moves in an elliptical orbit with the Sun at one focus. 2. The radius vector drawn from the Sun to any planet sweeps out equal areas in equal time intervals. 3. The square of the orbital period of any planet is proportional to the cube of the semi-major axis of the elliptical orbit. Using Kepler s first law and the definition of an ellipse along with our knowledge of central and conservative forces, we can determine the shape and attributes of a satellites orbit about Earth. Recall that our Earth-satellite system operates under the conservative force of gravity, therefore the sum of the kinetic and potential energy is constant as give in Equation 4. (4) Where E is the total energy of the system, is the kinetic energy and is the potential energy. Gravitational potential energy is dependent on position and therefore where is the distance from the center of the Earth to the satellite we can express the gravitational potential energy of the system as a function of position, Eq. 5. (5) 1. Given a satellite s perigee and apogee radius 7, determine the constants of motion from Newton s laws and the geometry of an ellipse. Calculate the semi-major axis, eccentricity, specific angular momentum and period of this satellite. (Using the information provided in Table 1.) a. Radius at Perigee: (km) b. Radius at Apogee: (km) Constant a Semi-major Axis e Eccentricity h Specific Angular Momentum T - period Table 3: Constants of Motion 2. What is gravitational potential energy of the system? 3. What shape is the orbit? Highly elliptical, moderately elliptical, circular? Values 7 As measured from the center of the Earth. - 9
10 Procedure 1. Open STK and open the Scenario named COE_DEMO1 2. Configure the Satellite object in your Scenario a) Double click on the Satellite icon to open the Properties Browser. b) On the Basic Orbit property page, change the Coord Type to Cartesian c) You will see six boxes for the three components of the R R iˆ R ˆj R kˆ vectors. Enter the values from step 1 above in Table 4. x y z and V V iˆ V x y ˆj V kˆ z OPTION VALUE Table 4: Basic Orbit Properties X R x = 0 Y R y = (radius at apogee) Z R z = 0 X Velocity V x = Y Velocity V y = 0 Z Velocity V z = 0 Click Apply. d) Now change the Coord Type to Classical Do the semi-major axis and eccentricity agree with those you calculated in mission planning? Record STK s values in Table 5. Click OK to close the Properties Browser for the Satellite. Table 5: STK Calculated COEs CLASSICAL ORBITAL ELEMENT a Semi-major Axis e Eccentricity VALUES (4 decimal places) Select the 3-D window you should see part of the satellite s orbit, the three principal vectors, and the fundamental plane. If all of the vectors do not show up immediately, run the scenario for a short time and they should appear. 3. Run the scenario a) Note the current start time in the toolbar at the top of your window. Start time: b) Run the scenario and pause it after one complete orbit. You will need to use the Pause button ( ) on the toolbar as well as the Decrease and Increase Time Step buttons ( button ( ). ), and/or Reverse - 10
11 Look at the Current Scenario Time window ( ) on the default toolbar at the top of the screen. Is the time one orbital period after the initial time? If not, check your calculations for the period. 4. Display the COEs and Earth Coordinate Inertial (ECI) data for the satellite a) Open the Satellite s property page and go to 3-D Graphics/Data Display. b) Check the Show box of the Classical Orbital Elements (Figure ). Select check-box next to Classical Orbit Elements Figure 7: Configure the 3-D COE Display c) Click Apply. d) Switch to the 3-D window you should see a list of the six COEs. e) Back in the 3-D Graphics/Data Display property page, check the Show box next to J ECI Position Velocity. f) In the position box, set the X Origin: to 700. g) Click Apply. h) Switch to the 3-D window. You should see the COEs displayed on the left and the R and V vectors displayed on the right. (Figure 8) Figure 8: Position the 3-D J2000 Position/Velocity Vector Display 8 J2000 J2, J4 and TwoBody are the propagation engines that perform the calculations to determine and animate the satellite in the scenario. This propagation engine performs a simplified analytical solution. Other propagation engines perform numeric solutions and account for the various other forces that may be acting on a satellite. - 11
12 5. Run the scenario a. Click the Start ( ) button and observe the orbit in three dimensions from various angles. Which COEs change and which remain constant? Which R and V value(s) is/are changing? 6. Reporting. a) Click the Pause( ) button. Right click on the Satellite icon to open the Properties Drop Down Menu (Figure 9). Scroll down to Report & Graph Manager and move the mouse pointer over the arrow ( ) to the right. Select Apogee and Perigee Report. Print this report. Figure 9: Report & Graph Manager b) Right click on the Satellite icon to open the Properties Drop Down Menu (Figure 9). Scroll down to Report & Graph Manager and move the mouse pointer over the arrow ( ) to the right. Select Angular Momentum Report. You only need the data for a short period of time, therefore locate the STOP time directly above the report data in the toolbar (Figure 10). Refresh Button Highlight the value you want to change. 15 to 30 minutes of data is sufficient. Figure 10: Adjust Report Period - 12
13 Change the hour by highlighting it and advance it two hours from the start time. Now select the day by highlighting it and correct the value so it matches the day of the start time. Click the refresh button to generate the new report data. Part III - Explore Real World Orbits 1. Close the current STK scenario. Click on File, Close. Click NO when prompted to save changes. 2. Open a scenario. Click File, Open. Select the ORBITS.SC scenario in the Open Dialogue and click OK. 3. This scenario contains satellites that are currently in orbit and operational today. Run the scenario and observe. a. You can change the viewing perspective in the 3D window by clicking on the View From/To button. (Reference Figure 11) Figure 11: Change the 3D perspective. What different artificial satellites and space vehicles are orbiting? What are their orbits? Circular, elliptical, LEO 9, MEO 10, GEO 11, HEO 12? Can you determine which one of these satellites is the research satellite operated by PSU from our Mission Operations Center 13 at University Park? 9 LEO Low Earth Orbit km altitude (from the surface of the Earth) 10 MEO Mid Earth Orbit 20,100 km altitude (from the surface of the Earth) 11 GEO Geosynchronous Earth Orbit 35,786 km altitude (from the surface of the Earth) with a period equal to 23 hrs. 56 min 12 HEO High Earth Orbit 39,500 km altitude (from the surface of the Earth) 13 Mission Operations Center at PSU:
14 Questions: 1. How do your calculated values from Table 2 compare with those as reported by STK? 2. What do you notice about the specific angular momentum value with respect to position and velocity? We show below the derivation of the area swept out by the radius vector r of the satellite using the vector equation for angular momentum where L is the vector form of angular momentum, r is the position vector, v is the velocity vector and p is the linear momentum. (6) (6a) (6b) Note, in the simplified model we assume m sat=1kg. (6c) ( ) (7) 3. How is Eq. 7 related to Kepler s Second Law? - 14
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
More informationOrbital 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
More informationAstromechanics 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
More information2. 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
More informationHalliday, 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
More informationSection 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,
More informationHalliday, 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
More informationUSING 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
More informationPlanetary 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.
More informationLecture 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
More informationG 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
More informationNewton 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
More informationChapter 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.
More informationThe Two-Body Problem
The Two-Body Problem Abstract In my short essay on Kepler s laws of planetary motion and Newton s law of universal gravitation, the trajectory of one massive object near another was shown to be a conic
More informationUnderstanding 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
More informationLecture 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
More informationChapter 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
More informationThe 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
More informationNotes: 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
More informationOrbital 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
More informationPhysics Midterm Review Packet January 2010
Physics Midterm Review Packet January 2010 This Packet is a Study Guide, not a replacement for studying from your notes, tests, quizzes, and textbook. Midterm Date: Thursday, January 28 th 8:15-10:15 Room:
More informationOrbital 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
More informationArtificial 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,
More informationCoverage 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
More informationNiraj 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.
More informationSection 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
More informationOrientation 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
More informationDIRECT 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
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More informationWhere On Earth Will Three Different Satellites Provide Simultaneous Coverage?
Where On Earth Will Three Different Satellites Provide Simultaneous Coverage? In this exercise you will use STK/Coverage to model and analyze the quality and quantity of coverage provided by the Earth
More informationBinary 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.
More informationSolar 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
More informationDevelopment 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
More informationNewton 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.
More informationTrajectory Design with STK/Astrogator. New Horizons Mission Tutorial
Trajectory Design with STK/Astrogator New Horizons Mission Tutorial STK/Astrogator New Horizons Mission Tutorial Page 2 Mission Overview In this tutorial, we will model a Mission to Pluto. Starting from
More informationOrbital 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
More information1. 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
More informationVocabulary - 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
More informationFlight 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
More informationRS 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
More informationcircular motion & gravitation physics 111N
circular motion & gravitation physics 111N uniform circular motion an object moving around a circle at a constant rate must have an acceleration always perpendicular to the velocity (else the speed would
More informationFrom 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
More informationMath 1302, Week 3 Polar coordinates and orbital motion
Math 130, Week 3 Polar coordinates and orbital motion 1 Motion under a central force We start by considering the motion of the earth E around the (fixed) sun (figure 1). The key point here is that the
More informationSatellite 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
More informationEN4 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
More informationCentripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.
Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.
More informationUse 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
More informationEDMONDS 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
More informationVELOCITY, ACCELERATION, FORCE
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
More informationPhysics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam
Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry
More informationChapter 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
More informationA. 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
More informationAcceleration of Gravity Lab Basic Version
Acceleration of Gravity Lab Basic Version In this lab you will explore the motion of falling objects. As an object begins to fall, it moves faster and faster (its velocity increases) due to the acceleration
More informationHyperspectral Satellite Imaging Planning a Mission
Hyperspectral Satellite Imaging Planning a Mission Victor Gardner University of Maryland 2007 AIAA Region 1 Mid-Atlantic Student Conference National Institute of Aerospace, Langley, VA Outline Objective
More informationCATIA V5 Tutorials. Mechanism Design & Animation. Release 18. Nader G. Zamani. University of Windsor. Jonathan M. Weaver. University of Detroit Mercy
CATIA V5 Tutorials Mechanism Design & Animation Release 18 Nader G. Zamani University of Windsor Jonathan M. Weaver University of Detroit Mercy SDC PUBLICATIONS Schroff Development Corporation www.schroff.com
More informationIntroduction to Aerospace Engineering
Introduction to Aerospace Engineering Lecture slides Challenge the future 1 Introduction to Aerospace Engineering AE1-10 Dept. Space Engineering Astrodynamics & Space Missions (AS) Prof. ir. B.A.C. Ambrosius
More informationName: 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
More informationPresentation of problem T1 (9 points): The Maribo Meteorite
Presentation of problem T1 (9 points): The Maribo Meteorite Definitions Meteoroid. A small particle (typically smaller than 1 m) from a comet or an asteroid. Meteorite: A meteoroid that impacts the ground
More informationAstronomy 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
More informationAstrodynamics (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:
More informationFree Fall: Observing and Analyzing the Free Fall Motion of a Bouncing Ping-Pong Ball and Calculating the Free Fall Acceleration (Teacher s Guide)
Free Fall: Observing and Analyzing the Free Fall Motion of a Bouncing Ping-Pong Ball and Calculating the Free Fall Acceleration (Teacher s Guide) 2012 WARD S Science v.11/12 OVERVIEW Students will measure
More informationExercise 5.0 LUNAR MOTION, ELONGATION, AND PHASES
Exercise 5.0 LUNAR MOTION, ELONGATION, AND PHASES I. Introduction The Moon's revolution in orbit around the center of gravity (barycenter) of the Earth- Moon System results in an apparent motion of the
More informationSUN-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
More informationLab 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
More informationGravitation 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
More informationUnit 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
More informationSatellite 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
More informationAngular Velocity vs. Linear Velocity
MATH 7 Angular Velocity vs. Linear Velocity Dr. Neal, WKU Given an object with a fixed speed that is moving in a circle with a fixed ius, we can define the angular velocity of the object. That is, we can
More information4.1.6. 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
More informationSATELLITE 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,
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationPractical Work DELMIA V5 R20 Lecture 1. D. Chablat / S. Caro Damien.Chablat@irccyn.ec-nantes.fr Stephane.Caro@irccyn.ec-nantes.fr
Practical Work DELMIA V5 R20 Lecture 1 D. Chablat / S. Caro Damien.Chablat@irccyn.ec-nantes.fr Stephane.Caro@irccyn.ec-nantes.fr Native languages Definition of the language for the user interface English,
More informationKINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
More informationApplying a circular load. Immediate and consolidation settlement. Deformed contours. Query points and query lines. Graph query.
Quick Start Tutorial 1-1 Quick Start Tutorial This quick start tutorial will cover some of the basic features of Settle3D. A circular load is applied to a single soil layer and settlements are examined.
More informationUnderstand the Sketcher workbench of CATIA V5.
Chapter 1 Drawing Sketches in Learning Objectives the Sketcher Workbench-I After completing this chapter you will be able to: Understand the Sketcher workbench of CATIA V5. Start a new file in the Part
More informationLecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014
Lecture 07: Work and Kinetic Energy Physics 2210 Fall Semester 2014 Announcements Schedule next few weeks: 9/08 Unit 3 9/10 Unit 4 9/15 Unit 5 (guest lecturer) 9/17 Unit 6 (guest lecturer) 9/22 Unit 7,
More informationExercise: Estimating the Mass of Jupiter Difficulty: Medium
Exercise: Estimating the Mass of Jupiter Difficulty: Medium OBJECTIVE The July / August observing notes for 010 state that Jupiter rises at dusk. The great planet is now starting its grand showing for
More informationHow To Understand The Theory Of Gravity
Newton s Law of Gravity and Kepler s Laws Michael Fowler Phys 142E Lec 9 2/6/09. These notes are partly adapted from my Physics 152 lectures, where more mathematical details can be found. The Universal
More informationThe 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
More informationMET 306. Activity 8a. Mechanism Design Creo 2.0 Level 7 POINT A GROUND LINK LINK 1 LINK 2 LINK 3 POINT B 10/15/2010 1
Mechanism Design Creo 2.0 Level 7 POINT A LINK 1 GROUND LINK LINK 2 LINK 3 POINT B 10/15/2010 1 Download parts ground, key, link_1, link_2, link_3 and pulley from the V:/MET_306/Activity_8_Creo drive.
More informationBasic Coordinates & Seasons Student Guide
Name: Basic Coordinates & Seasons Student Guide There are three main sections to this module: terrestrial coordinates, celestial equatorial coordinates, and understanding how the ecliptic is related to
More informationLab 7: Rotational Motion
Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125
More informationChapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.
Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular
More informationBackground Information
Background Information The Second Law of Motion and The Law of Gravitation Student Activities 1. Round and Round They Go! 2. onic Sections - Movement in Newton s Gravitational orce Notes to Teachers Teacher
More informationHybridSail. Hybrid Solar Sails for Active Debris Removal Final Report
HybridSail Hybrid Solar Sails for Active Debris Removal Final Report Authors: Lourens Visagie (1), Theodoros Theodorou (1) Affiliation: 1. Surrey Space Centre - University of Surrey ACT Researchers: Leopold
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationSpacecraft orbits and missions
General Astrophysics and Space Research Course 210142, Space Physics Module Spring 2009, Joachim Vogt Spacecraft orbits and missions Topics of this lecture Basics of celestial mechanics Geocentric orbits
More informationSatellite 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
More informationCuriosity's Fight Path to Mars. A Project for Differential Equations (Math 256)
Curiosity's Fight Path to Mars A Project for Differential Equations (Math 56) On November 5 th, 011, NASA launched a rocket that will carry a rover called Curiosity to Mars. The rover is scheduled to land
More informationSatellites 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
More informationEVOLUTION OF THE DEBRIS CLOUD GENERATED BY THE FENGYUN-1C FRAGMENTATION EVENT
EVOLUTION OF THE DEBRIS CLOUD GENERATED BY THE FENGYUN-1C FRAGMENTATION EVENT Carmen Pardini and Luciano Anselmo Space Flight Dynamics Laboratory Istituto di Scienza e Tecnologie dell Informazione Alessandro
More informationWorkspaces Creating and Opening Pages Creating Ticker Lists Looking up Ticker Symbols Ticker Sync Groups Market Summary Snap Quote Key Statistics
Getting Started Workspaces Creating and Opening Pages Creating Ticker Lists Looking up Ticker Symbols Ticker Sync Groups Market Summary Snap Quote Key Statistics Snap Report Price Charts Comparing Price
More informationOrbital Mechanics Course Notes. David J. Westpfahl Professor of Astrophysics, New Mexico Institute of Mining and Technology
Orbital Mechanics Course Notes David J. Westpfahl Professor of Astrophysics, New Mexico Institute of Mining and Technology March 31, 2011 2 These are notes for a course in orbital mechanics catalogued
More informationMobile 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
More informationastronomy 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,
More informationTABLE OF CONTENTS. INTRODUCTION... 5 Advance Concrete... 5 Where to find information?... 6 INSTALLATION... 7 STARTING ADVANCE CONCRETE...
Starting Guide TABLE OF CONTENTS INTRODUCTION... 5 Advance Concrete... 5 Where to find information?... 6 INSTALLATION... 7 STARTING ADVANCE CONCRETE... 7 ADVANCE CONCRETE USER INTERFACE... 7 Other important
More informationMicrosoft Excel Tutorial
Microsoft Excel Tutorial by Dr. James E. Parks Department of Physics and Astronomy 401 Nielsen Physics Building The University of Tennessee Knoxville, Tennessee 37996-1200 Copyright August, 2000 by James
More informationCompleting Baseline s Site Survey Request Form
Completing Baseline s Site Survey Request Form The first step in successfully implementing a radio network for your irrigation controllers is to identify the proposed locations for each radio. These radios
More informationDoes 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
More informationBHS Freshman Physics Review. Chapter 2 Linear Motion Physics is the oldest science (astronomy) and the foundation for every other science.
BHS Freshman Physics Review Chapter 2 Linear Motion Physics is the oldest science (astronomy) and the foundation for every other science. Galileo (1564-1642): 1 st true scientist and 1 st person to use
More information