Orbital Mechanics MARYLAND U N I V E R S I T Y O F. Orbital Mechanics. ENAE 483/788D - Principles of Space Systems Design
|
|
- Darlene Marshall
- 5 years ago
- Views:
Transcription
1 Lecture #05 September 15, 2015 Planetary launch and entry overview Energy and velocity in orbit Elliptical orbit parameters Orbital elements Coplanar orbital transfers Noncoplanar transfers Time in orbit Interplanetary trajectories Relative orbital motion ( proximity operations ) 2015 David L. Akin - All rights reserved
2 Space Launch - The Physics Minimum orbital altitude is ~200 km Circular orbital velocity there is 7784 m/sec P otential Energy kg in orbit = Kinetic Energy kg in orbit Total energy per kg in orbit µ + µ = r orbit r E = 1 2 µ r 2 orbit = J kg J kg T otal Energy kg in orbit = KE + P E = J kg
3 Theoretical Cost to Orbit Convert to usual energy units T otal Energy kg in orbit = J kg Domestic energy costs are ~$0.05/kWhr Theoretical cost to orbit $0.44/kg = kw hrs kg
4 Actual Cost to Orbit SpaceX Falcon 9 13,150 kg to LEO $65 M per flight Lowest cost system currently flying $4940/kg of payload Factor of 11,000x higher than theoretical energy costs
5 What About Airplanes? For an aircraft in level flight, Weight Thrust = Lift mg, or Drag T Energy = force x distance, so = L D Total Energy thrust distance = = Td kg mass m = gd L/D For an airliner (L/D=25) to equal orbital energy, d=81,000 km (2 roundtrips NY-Sydney)
6 Equivalent Airline Costs? Average economy ticket NY-Sydney round-roundtrip (Travelocity 9/3/09) ~$1300 Average passenger (+ luggage) ~100 kg Two round trips = $26/kg Factor of 60x more than electrical energy costs Factor of 190x less than current launch costs But you get to refuel at each stop
7 Equivalence to Air Transport 81,000 km ~ twice around the world Voyager - one of only two aircraft to ever circle the world non-stop, nonrefueled - once
8 Orbital Entry - The Physics 32 MJ/kg dissipated by friction with atmosphere over ~8 min = 66kW/kg Pure graphite (carbon) high-temperature material: c p =709 J/kg K Orbital energy would cause temperature gain of 45,000 K (If you re interesting in how this works out, you can take ENAE 791 Launch and Entry Vehicle Design next term...)
9 Newton s Law of Gravitation Inverse square law F = GMm r 2 Since it s easier to remember one number, µ = GM If you re looking for local gravitational acceleration, g = µ r 2
10 Some Useful Constants Gravitation constant µ = GM Earth: 398,604 km 3 /sec 2 Moon: km 3 /sec 2 Mars: 42,970 km 3 /sec 2 Sun: 1.327x10 11 km 3 /sec 2 Planetary radii r Earth = 6378 km r Moon = 1738 km r Mars = 3393 km
11 Energy in Orbit Kinetic Energy Potential Energy Total Energy K.E. = 1 2 mv2 = K.E. m P.E. = µm r = P.E. m = v2 2 = µ r Constant = v2 2 µ r = µ 2a v 2 = µ 2 1 r a <--Vis-Viva Equation
12 Classical Parameters of Elliptical Orbits
13 The Classical Orbital Elements Ref: J. E. Prussing and B. A. Conway, Oxford University Press, 1993
14 Implications of Vis-Viva Circular orbit (r=a) Parabolic escape orbit (a tends to infinity) v circular = v escape = µ r 2µ Relationship between circular and parabolic orbits v escape = 2v circular r
15 The Hohmann Transfer r 2 r 1 v 2 v apogee v 1 v perigee
16 First Maneuver Velocities Initial vehicle velocity Needed final velocity Delta-V v perigee = v 1 = µ v 1 = r 1 µ µ r 1 r 1 2r2 r 1 + r 2 ( ) 2r2 1 r 1 + r 2
17 Second Maneuver Velocities Initial vehicle velocity Needed final velocity Delta-V v apogee = v 2 = µ v 2 = r 2 µ r 2 ( 1 µ r 2 2r1 r 1 + r 2 2r1 r 1 + r 2 )
18 Implications of Hohmann Transfers Implicit assumption is made that velocity changes instantaneously - impulsive thrust Decent assumption if acceleration ~5 m/sec 2 (0.5 g Earth ) Lower accelerations result in altitude change during burn lower efficiencies and higher ΔVs Worst case is continuous infinitesimal thrusting (e.g., ion engines) ΔV between circular coplanar orbits r 1 and r 2 is V Low T hrust = V c1 V c2 = r µ r 1 r µ r 2
19 Limitations on Launch Inclinations Equator
20 Differences in Inclination Line of Nodes
21 Choosing the Wrong Line of Apsides
22 Simple Plane Change v 1 v perigee v apogee Δv 2 v 2
23 Optimal Plane Change v perigee v 1 v Δv apogee 1 v 2 Δv 2
24 First Maneuver with Plane Change Δi 1 Initial vehicle velocity Needed final velocity Delta-V v 1 = µ r 1 v p = µ r 1 2r 2 r 1 + r 2 v 1 = v v2 p 2v 1 v p cos i 1
25 Second Maneuver with Plane Change Δi 2 Initial vehicle velocity v a = µ 2r 1 r 2 r 1 + r 2 Needed final velocity Delta-V v 2 = µ r 2 v 2 = v v2 a 2v 2 v a cos i 2
26 Sample Plane Change Maneuver Optimum initial plane change = 2.20
27 Calculating Time in Orbit
28 Time in Orbit Period of an orbit Mean motion (average angular velocity) Time since pericenter passage M=mean anomaly P = 2 n = M = nt = E µ a 3 a 3 µ e sin E
29 Dealing with the Eccentric Anomaly Relationship to orbit r = a (1 e cos E) Relationship to true anomaly tan 2 = 1 + e 1 e tan E 2 Calculating M from time interval: iterate E i+1 = nt + e sin E i until it converges
30 Example: Time in Orbit Hohmann transfer from LEO to GEO h 1 =300 km --> r 1 = =6678 km r 2 =42240 km Time of transit (1/2 orbital period) a = 1 2 (r 1 + r 2 ) = 24, 459 km t transit = P 2 = a 3 µ = 19, 034 sec = 5h17m14s
31 Example: Time-based Position Find the spacecraft position 3 hours after perigee n = µ 4 rad = 1.650x10 a3 sec r p e = 1 a = E j+1 = nt + e sin E j = sin E j E=0; 1.783; 2.494; 2.222; 2.361; 2.294; 2.328; 2.311; 2.320; 2.316; 2.318; 2.317; 2.317; 2.317
32 Example: Time-based Position (cont.) Have to be sure to get the position in the proper quadrant - since the time is less than 1/2 the period, the spacecraft has yet to reach apogee --> 0 <θ<180 E =2.317 r = a(1 e cos E) = 12, 387 km tan 2 = 1+e 1 e tan E 2 = = 160 deg
33 Basic Orbital Parameters Semi-latus rectum (or parameter) Radial distance as function of orbital position Periapse and apoapse distances Angular momentum h = r v p = a(1 e 2 ) r = p 1+e cos θ r p = a(1 e) r a = a(1 + e) h = µp
34 Velocity Components in Orbit v r = dr dt = d dt v r = r = p 1+ecos p 1+e cos pe sin d (1 + e cos ) 2 dt = p( e sin d dt ) (1 + e cos ) 2 1+ecos = p v r = r2 d dt e sin r p h = r v
35 Velocity Components in Orbit (cont.) ~ h = ~r ~v h = rv cos = r r d dt = r 2 d dt
36 Patched Conics Simple approximation to multi-body motion (e.g., traveling from Earth orbit through solar orbit into Martian orbit) Treats multibody problem as hand-offs between gravitating bodies --> reduces analysis to sequential twobody problems Caveat Emptor: There are a number of formal methods to perform patched conic analysis. The approach presented here is a very simple, convenient, and not altogether accurate method for performing this calculation. Results will be accurate to a few percent, which is adequate at this level of design analysis.
37 Example: Lunar Orbit Insertion v 2 is velocity of moon around Earth Moon overtakes spacecraft with velocity of (v 2 -v apogee ) This is the velocity of the spacecraft relative to the moon while it is effectively infinitely far away (before lunar gravity accelerates it) = hyperbolic excess velocity
38 Planetary Approach Analysis Spacecraft has v h hyperbolic excess velocity, which fixes total energy of approach orbit Vis-viva provides velocity of approach v 2 2 v = v 2 h + 2µ r Choose transfer orbit such that approach is tangent to desired final orbit at periapse µ r = µ 2a = v2 h 2 v = vh 2 + 2µ µ r r
39 Patched Conic - Lunar Approach Lunar orbital velocity around the Earth Apogee velocity of Earth transfer orbit from initial 400 km low Earth orbit v a = v m v m = Velocity difference between spacecraft infinitely far away and moon (hyperbolic excess velocity) µ r m = 2r1 r 1 + r m = , , 400 =1.018 km sec , 400 =0.134 km sec v h = v m v a = v m = = km sec
40 Patched Conic - Lunar Orbit Insertion The spacecraft is now in a hyperbolic orbit of the moon. The velocity it will have at the perilune point tangent to the desired 100 km low lunar orbit is v pm = r vh 2 + 2µ m = r LLO r (4667.9) 1878 The required delta-v to slow down into low lunar orbit is v = v pm v cm =2.398 r Red text is a correction to original notes =2.398 km sec =0.822 km sec
41 ΔV Requirements for Lunar Missions From: To: Low Earth Orbit Lunar Transfer Orbit Low Lunar Orbit Lunar Descent Orbit Lunar Landing Low Earth Orbit km/sec Lunar Transfer Orbit km/sec km/sec km/sec Low Lunar Orbit km/sec km/sec Lunar Descent Orbit km/sec km/sec Lunar Landing km/sec km/sec
42 LOI ΔV Based on Landing Site
43 LOI ΔV Including Loiter Effects
44 Interplanetary Trajectory Types
45 Interplanetary Pork Chop Plots Summarize a number of critical parameters Date of departure Date of arrival Hyperbolic energy ( C3 ) Transfer geometry Launch vehicle determines available C3 based on window, payload mass Calculated using Lambert s Theorem
46 C3 for Earth-Mars Transfer
47 Earth-Mars Transfer 2033
48 Earth-Mars Transfer 2037
49 Interplanetary Delta-V Hyperbolic excess velocity V h C 3 = V 2 h V req = p V 2 esc + C 3 V = p V 2 esc + C 3 V c 2033 Window: V =3.55 km/sec 2037 Window: V =3.859 km/sec V in departure from 300 km LEO
50 Hill s Equations (Proximity Operations) x = 3n 2 x + 2n y + a dx y = 2n x + a dy z = n 2 z + a dz Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993
51 Clohessy-Wiltshire ( CW ) Equations x(t) = [ 4 3cos(nt) ]x o + sin(nt) n x o + 2 n y(t) = 6[ sin(nt) nt]x o + y o 2 n [ 1 cos(nt) ] x o + z(t) = z o cos(nt) + z o n sin(nt) z (t) = z o nsin(nt) + z o sin(nt) [ 1 cos(nt) ] y o 4sin(nt) 3nt n y o
52 V-Bar Approach Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On- Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001
53 R-Bar Approach Approach from along the radius vector ( R-bar ) Gravity gradients decelerate spacecraft approach velocity - low contamination approach Used for Mir, ISS docking approaches Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply, 15th USU Small Satellite Conference, 2001
54 References for This Lecture Wernher von Braun, The Mars Project University of Illinois Press, 1962 William Tyrrell Thomson, Introduction to Space Dynamics Dover Publications, 1986 Francis J. Hale, Introduction to Space Flight Prentice- Hall, 1994 William E. Wiesel, Spaceflight Dynamics MacGraw- Hill, 1997 J. E. Prussing and B. A. Conway, Oxford University Press, 1993
Orbital Mechanics MARYLAND U N I V E R S I T Y O F. Orbital Mechanics. ENAE 483/788D - Principles of Space Systems Design
Planetary launch and entry overview Energy and velocity in orbit Elliptical orbit parameters Orbital elements Coplanar orbital transfers Noncoplanar transfers Time in orbit Interplanetary trajectories
More informationOrbital Mechanics MARYLAND
Orbital Mechanics Energy and velocity in orbit Elliptical orbit parameters Orbital elements Coplanar orbital transfers Noncoplanar transfers Time in orbit Interplanetary trajectories Planetary launch and
More informationOrbital Mechanics MARYLAND. Orbital Mechanics. ENAE 483/788D - Principles of Space Systems Design
Lecture #08 September 22, 2016 Planetary launch and entry overview Energy and velocity in orbit Elliptical orbit parameters Orbital elements Coplanar orbital transfers Noncoplanar transfers Time in orbit
More informationOrbital Mechanics MARYLAND. Orbital Mechanics. ENAE 483/788D - Principles of Space Systems Design
Lecture #08 September 20, 2018 Planetary launch and entry overview Energy and velocity in orbit Elliptical orbit parameters Orbital elements Coplanar orbital transfers Noncoplanar transfers Time in orbit
More informationOrbital Mechanics MARYLAND U N I V E R S I T Y O F. Orbital Mechanics. ENAE 483/788D - Principles of Space Systems Design
Discussion of 483/484 project organization Planetary launch and entry overview Energy and velocity in orbit Elliptical orbit parameters Orbital elements Coplanar orbital transfers Noncoplanar transfers
More informationMARYLAND U N I V E R S I T Y O F. Orbital Mechanics. Principles of Space Systems Design
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 informationMARYLAND U N I V E R S I T Y O F. Orbital Mechanics. Principles of Space Systems Design
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 informationOrbital Mechanics MARYLAND U N I V E R S I T Y O F. Orbital Mechanics. ENAE 483/788D - Principles of Space Systems Design
Lecture #03 September 8, 2011 ENAE 483/484 project organization Planetary launch and entry overview Energy and velocity in orbit Elliptical orbit parameters Orbital elements Coplanar orbital transfers
More informationCourse Overview/Orbital Mechanics
Course Overview/Orbital Mechanics Course Overview Challenges of launch and entry Course goals Web-based Content Syllabus Policies Project Content An overview of orbital mechanics at point five past lightspeed
More informationRobotic Mobility Above the Surface
Free Space Relative Orbital Motion Airless Major Bodies (moons) 1 2016 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu Propulsive Motion in Free Space Basic motion governed by Newton
More informationENAE 791 Course Overview
ENAE 791 Challenges of launch and entry Course goals Web-based Content Syllabus Policies Project Content 1 2016 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu Space Transportation System
More informationLecture D30 - Orbit Transfers
J. Peraire 16.07 Dynamics Fall 004 Version 1.1 Lecture D30 - Orbit Transfers In this lecture, we will consider how to transfer from one orbit, or trajectory, to another. One of the assumptions that we
More informationRobotic Mobility Above the Surface
Free Space Relative Orbital Motion Airless Major Bodies (moons) Gaseous Environments (Mars, Venus, Titan) Lighter-than- air (balloons, dirigibles) Heavier-than- air (aircraft, helicopters) 1 2012 David
More informationSatellite Orbital Maneuvers and Transfers. Dr Ugur GUVEN
Satellite Orbital Maneuvers and Transfers Dr Ugur GUVEN Orbit Maneuvers At some point during the lifetime of most space vehicles or satellites, we must change one or more of the orbital elements. For example,
More informationOrbit Characteristics
Orbit Characteristics 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 primary located at the focus of the conic
More informationExtending the Patched-Conic Approximation to the Restricted Four-Body Problem
Monografías de la Real Academia de Ciencias de Zaragoza 3, 133 146, (6). Extending the Patched-Conic Approximation to the Restricted Four-Body Problem Thomas R. Reppert Department of Aerospace and Ocean
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) 10. Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L) Motivation 2 6. Interplanetary Trajectories 6.1 Patched conic method 6.2 Lambert s problem
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) L06: Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L) Motivation 2 Problem Statement? Hint #1: design the Earth-Mars transfer using known concepts
More informationPatch Conics. Basic Approach
Patch Conics Basic Approach Inside the sphere of influence: Planet is the perturbing body Outside the sphere of influence: Sun is the perturbing body (no extra-solar system trajectories in this class...)
More informationRocket Science 102 : Energy Analysis, Available vs Required
Rocket Science 102 : Energy Analysis, Available vs Required ΔV Not in Taylor 1 Available Ignoring Aerodynamic Drag. The available Delta V for a Given rocket burn/propellant load is ( ) V = g I ln 1+ P
More informationFlight and Orbital Mechanics
Flight and Orbital Mechanics Lecture slides Challenge the future 1 Flight and Orbital Mechanics AE-104, lecture hours 1-4: Interplanetary flight Ron Noomen October 5, 01 AE104 Flight and Orbital Mechanics
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) 5B. Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L) Previous Lecture: Coplanar Maneuvers 5.1 INTRODUCTION 5.1.1 Why? 5.1.2 How? 5.1.3 How much? 5.1.4 When?
More informationOrbital Mechanics! Space System Design, MAE 342, Princeton University! Robert Stengel
Orbital Mechanics Space System Design, MAE 342, Princeton University Robert Stengel Conic section orbits Equations of motion Momentum and energy Kepler s Equation Position and velocity in orbit Copyright
More informationPrevious Lecture. Orbital maneuvers: general framework. Single-impulse maneuver: compatibility conditions
2 / 48 Previous Lecture Orbital maneuvers: general framework Single-impulse maneuver: compatibility conditions closed form expression for the impulsive velocity vector magnitude interpretation coplanar
More informationFundamentals of Astrodynamics and Applications
Fundamentals of Astrodynamics and Applications Third Edition David A. Vallado with technical contributions by Wayne D. McClain Space Technology Library Published Jointly by Microcosm Press Hawthorne, CA
More informationInterplanetary Mission Opportunities
Interplanetary Mission Opportunities Introduction The quest for unravelling the mysteries of the universe is as old as human history. With the advent of new space technologies, exploration of space became
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) 5B. Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L) Previous Lecture: Coplanar Maneuvers 5.1 INTRODUCTION 5.1.1 Why? 5.1.2 How? 5.1.3 How much? 5.1.4 When?
More information10 Orbit and Constellation Design Selecting the Right Orbit
Orbit and Constellation Design Selecting the Right Orbit.7 Design of Interplanetary Orbits Faster Trajectories Ron Noomen, Delft University of Technology Using the recipe given in Table -9, one can compute
More informationAstromechanics. 6. Changing Orbits
Astromechanics 6. Changing Orbits Once an orbit is established in the two body problem, it will remain the same size (semi major axis) and shape (eccentricity) in the original orbit plane. In order to
More informationPrevious Lecture. Approximate solutions for the motion about an oblate planet: The Brouwer model. The Cid- Lahulla model 2 / 39
2 / 39 Previous Lecture Approximate solutions for the motion about an oblate planet: The Brouwer model The Cid- Lahulla model 3 / 39 Definition of Orbital Maneuvering Orbital maneuver: the use of the propulsion
More informationThe B-Plane Interplanetary Mission Design
The B-Plane Interplanetary Mission Design Collin Bezrouk 2/11/2015 2/11/2015 1 Contents 1. Motivation for B-Plane Targeting 2. Deriving the B-Plane 3. Deriving Targetable B-Plane Elements 4. How to Target
More informationThe Astrodynamics and Mechanics of Orbital Spaceflight
The Astrodynamics and Mechanics of Orbital Spaceflight Vedant Chandra 11-S1, TSRS Moulsari 1 1 Introduction to Rocketry Before getting into the details of orbital mechanics, we must understand the fundamentals
More informationCelestial Mechanics Lecture 10
Celestial Mechanics Lecture 10 ˆ This is the first of two topics which I have added to the curriculum for this term. ˆ We have a surprizing amount of firepower at our disposal to analyze some basic problems
More informationMAE 180A: Spacecraft Guidance I, Summer 2009 Homework 4 Due Thursday, July 30.
MAE 180A: Spacecraft Guidance I, Summer 2009 Homework 4 Due Thursday, July 30. Guidelines: Please turn in a neat and clean homework that gives all the formulae that you have used as well as details that
More informationASEN 5050 SPACEFLIGHT DYNAMICS Interplanetary
ASEN 5050 SPACEFLIGHT DYNAMICS Interplanetary Prof. Jeffrey S. Parker University of Colorado Boulder Lecture 29: Interplanetary 1 HW 8 is out Due Wednesday, Nov 12. J2 effect Using VOPs Announcements Reading:
More informationSUN INFLUENCE ON TWO-IMPULSIVE EARTH-TO-MOON TRANSFERS. Sandro da Silva Fernandes. Cleverson Maranhão Porto Marinho
SUN INFLUENCE ON TWO-IMPULSIVE EARTH-TO-MOON TRANSFERS Sandro da Silva Fernandes Instituto Tecnológico de Aeronáutica, São José dos Campos - 12228-900 - SP-Brazil, (+55) (12) 3947-5953 sandro@ita.br Cleverson
More informationChapter 13. Gravitation
Chapter 13 Gravitation e = c/a A note about eccentricity For a circle c = 0 à e = 0 a Orbit Examples Mercury has the highest eccentricity of any planet (a) e Mercury = 0.21 Halley s comet has an orbit
More informationUlrich Walter. Astronautics. The Physics of Space Flight. 2nd, Enlarged and Improved Edition
Ulrich Walter Astronautics The Physics of Space Flight 2nd, Enlarged and Improved Edition Preface to Second Edition Preface XVII Acknowledgments XIX List of Symbols XXI XV 1 Rocket Fundamentals 1 1.1 Rocket
More informationAP Physics 1 Chapter 7 Circular Motion and Gravitation
AP Physics 1 Chapter 7 Circular Motion and Gravitation Chapter 7: Circular Motion and Angular Measure Gravitation Angular Speed and Velocity Uniform Circular Motion and Centripetal Acceleration Angular
More informationSpacecraft Dynamics and Control
Spacecraft Dynamics and Control Matthew M. Peet Arizona State University Lecture 5: Hyperbolic Orbits Introduction In this Lecture, you will learn: Hyperbolic orbits Hyperbolic Anomaly Kepler s Equation,
More informationFlight and Orbital Mechanics. Exams
1 Flight and Orbital Mechanics Exams Exam AE2104-11: Flight and Orbital Mechanics (23 January 2013, 09.00 12.00) Please put your name, student number and ALL YOUR INITIALS on your work. Answer all questions
More informationASEN 6008: Interplanetary Mission Design Lab Spring, 2015
ASEN 6008: Interplanetary Mission Design Lab Spring, 2015 Lab 4: Targeting Mars using the B-Plane Name: I d like to give credit to Scott Mitchell who developed this lab exercise. He is the lead Astrodynamicist
More informationPowered Space Flight
Powered Space Flight KOIZUMI Hiroyuki ( 小泉宏之 ) Graduate School of Frontier Sciences, Department of Advanced Energy & Department of Aeronautics and Astronautics ( 基盤科学研究系先端エネルギー工学専攻, 工学系航空宇宙工学専攻兼担 ) Scope
More informationOptimal Generalized Hohmann Transfer with Plane Change Using Lagrange Multipliers
Mechanics and Mechanical Engineering Vol. 21, No. 4 (2017) 11 16 c Lodz University of Technology Optimal Generalized Hohmann Transfer with Plane Change Using Lagrange Multipliers Osman M. Kamel Astronomy
More informationSeminar 3! Precursors to Space Flight! Orbital Motion!
Seminar 3! Precursors to Space Flight! Orbital Motion! FRS 112, Princeton University! Robert Stengel" Prophets with Some Honor" The Human Seed and Social Soil: Rocketry and Revolution" Orbital Motion"
More informationParametric Design MARYLAND. The Design Process Level I Design Example: Low-Cost Lunar Exploration U N I V E R S I T Y O F
Parametric Design The Design Process Level I Design Example: Low-Cost Lunar Exploration U N I V E R S I T Y O F MARYLAND 2005 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu Parametric
More informationMARYLAND. The Design Process Regression Analysis Level I Design Example: UMd Exploration Initiative U N I V E R S I T Y O F.
Parametric Design The Design Process Regression Analysis Level I Design Example: UMd Exploration Initiative U N I V E R S I T Y O F MARYLAND 2004 David L. Akin - All rights reserved http://spacecraft.ssl.
More information5.12 The Aerodynamic Assist Trajectories of Vehicles Propelled by Solar Radiation Pressure References...
1 The Two-Body Problem... 1 1.1 Position of the Problem... 1 1.2 The Conic Sections and Their Geometrical Properties... 12 1.3 The Elliptic Orbits... 20 1.4 The Hyperbolic and Parabolic Trajectories...
More informationOptimal Gravity Assisted Orbit Insertion for Europa Orbiter Mission
Optimal Gravity Assisted Orbit Insertion for Europa Orbiter Mission Deepak Gaur 1, M. S. Prasad 2 1 M. Tech. (Avionics), Amity Institute of Space Science and Technology, Amity University, Noida, U.P.,
More informationASTRIUM. Interplanetary Path Early Design Tools at ASTRIUM Space Transportation. Nathalie DELATTRE ASTRIUM Space Transportation.
Interplanetary Path Early Design Tools at Space Transportation Nathalie DELATTRE Space Transportation Page 1 Interplanetary missions Prime approach: -ST has developed tools for all phases Launch from Earth
More informationASEN 5050 SPACEFLIGHT DYNAMICS Prox Ops, Lambert
ASEN 5050 SPACEFLIGHT DYNAMICS Prox Ops, Lambert Prof. Jeffrey S. Parker University of Colorado Boulder Lecture 15: ProxOps, Lambert 1 Announcements Homework #5 is due next Friday 10/10 CAETE by Friday
More informationIntroduction to Astronomy
Introduction to Astronomy AST0111-3 (Astronomía) Semester 2014B Prof. Thomas H. Puzia Newton s Laws Big Ball Fail Universal Law of Gravitation Every mass attracts every other mass through a force called
More informationChapter 8. Precise Lunar Gravity Assist Trajectories. to Geo-stationary Orbits
Chapter 8 Precise Lunar Gravity Assist Trajectories to Geo-stationary Orbits Abstract A numerical search technique for designing a trajectory that transfers a spacecraft from a high inclination Earth orbit
More information14.1 Earth Satellites. The path of an Earth satellite follows the curvature of the Earth.
The path of an Earth satellite follows the curvature of the Earth. A stone thrown fast enough to go a horizontal distance of 8 kilometers during the time (1 second) it takes to fall 5 meters, will orbit
More informationwhere s is the horizontal range, in this case 50 yards. Note that as long as the initial speed of the bullet is great enough to let it hit a target at
1 PHYS 31 Homework Assignment Due Friday, 13 September 00 1. A monkey is hanging from a tree limbat height h above the ground. A hunter 50 yards away from the base of the tree sees him, raises his gun
More informationLesson 11: Orbital Transfers II. 10/6/2016 Robin Wordsworth ES 160: Space Science and Engineering: Theory and ApplicaCons
Lesson 11: Orbital Transfers II 10/6/2016 Robin Wordsworth ES 160: Space Science and Engineering: Theory and ApplicaCons ObjecCves Introduce concept of sphere of influence Study the patched conics approach
More informationDesign of Orbits and Spacecraft Systems Engineering. Scott Schoneman 13 November 03
Design of Orbits and Spacecraft Systems Engineering Scott Schoneman 13 November 03 Introduction Why did satellites or spacecraft in the space run in this orbit, not in that orbit? How do we design the
More informationLecture 22: Gravitational Orbits
Lecture : Gravitational Orbits Astronomers were observing the motion of planets long before Newton s time Some even developed heliocentric models, in which the planets moved around the sun Analysis of
More informationInterplanetary Spacecraft. Team 12. Alliance: Foxtrot
Interplanetary Spacecraft Team 12 Alliance: Foxtrot February 26 th 2010 Team Name : Impala Cover Art : Parthsarathi Team Members: Parthsarathi Trivedi (L) Michael Thompson Seth Trey Mohd Alhafidz Yahya
More informationAPPENDIX B SUMMARY OF ORBITAL MECHANICS RELEVANT TO REMOTE SENSING
APPENDIX B SUMMARY OF ORBITAL MECHANICS RELEVANT TO REMOTE SENSING Orbit selection and sensor characteristics are closely related to the strategy required to achieve the desired results. Different types
More informationA SIMULATION OF THE MOTION OF AN EARTH BOUND SATELLITE
DOING PHYSICS WITH MATLAB A SIMULATION OF THE MOTION OF AN EARTH BOUND SATELLITE Download Directory: Matlab mscripts mec_satellite_gui.m The [2D] motion of a satellite around the Earth is computed from
More informationRocket Performance MARYLAND U N I V E R S I T Y O F. Ballistic Entry ENAE Launch and Entry Vehicle Design
Rocket Performance Parallel staging Modular staging Standard atmospheres Orbital decay due to drag Straight-line (no gravity) entry based on atmospheric density 1 2014 David L. Akin - All rights reserved
More informationLecture Outline. Chapter 13 Gravity Pearson Education, Inc. Slide 13-1
Lecture Outline Chapter 13 Gravity Slide 13-1 The plan Lab this week: exam problems will put problems on mastering for chapters without HW; will also go over exam 2 Final coverage: now posted; some sections/chapters
More informationCHAPTER 3 PERFORMANCE
PERFORMANCE 3.1 Introduction The LM-3A performance figures given in this chapter are based on the following assumptions: Launching from XSLC (Xichang Satellite Launch Center, Sichuan Province, China),
More informationLecture 15 - Orbit Problems
Lecture 15 - Orbit Problems A Puzzle... The ellipse shown below has one focus at the origin and its major axis lies along the x-axis. The ellipse has a semimajor axis of length a and a semi-minor axis
More informationConcurrent Trajectory and Vehicle Optimization for an Orbit Transfer. Christine Taylor May 5, 2004
Concurrent Trajectory and Vehicle Optimization for an Orbit Transfer Christine Taylor May 5, 2004 Presentation Overview Motivation Single Objective Optimization Problem Description Mathematical Formulation
More informationUniversal Gravitation
Universal Gravitation Newton s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely
More informationSession 6: Analytical Approximations for Low Thrust Maneuvers
Session 6: Analytical Approximations for Low Thrust Maneuvers As mentioned in the previous lecture, solving non-keplerian problems in general requires the use of perturbation methods and many are only
More informationCelestial Mechanics II. Orbital energy and angular momentum Elliptic, parabolic and hyperbolic orbits Position in the orbit versus time
Celestial Mechanics II Orbital energy and angular momentum Elliptic, parabolic and hyperbolic orbits Position in the orbit versus time Orbital Energy KINETIC per unit mass POTENTIAL The orbital energy
More informationYear 12 Physics. 9.2 Space
Year 12 Physics 9.2 Space Contextual Outline Scientists have drawn on advances in areas such as aeronautics, material science, robotics, electronics, medicine and energy production to develop viable spacecraft.
More informationChapter 13. Universal Gravitation
Chapter 13 Universal Gravitation Planetary Motion A large amount of data had been collected by 1687. There was no clear understanding of the forces related to these motions. Isaac Newton provided the answer.
More informationGravitational Potential Energy and Total Energy *
OpenStax-CNX module: m58347 Gravitational Potential Energy and Total Energy * OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 By the end of
More informationASE 366K Spacecraft Dynamics
ASE 366K Spacecraft Dynamics Homework 2 Solutions 50 Points Total: 10 points each for 1.16, 1.19, 2.6, 2.7, and 10 points for completing the rest. 1.13 Show that the position vector is a min or max at
More informationSpacecraft Dynamics and Control
Spacecraft Dynamics and Control Matthew M. Peet Arizona State University Lecture 4: Position and Velocity Introduction In this Lecture, you will learn: Motion of a satellite in time How to predict position
More informationSpace Travel on a Shoestring: CubeSat Beyond LEO
Space Travel on a Shoestring: CubeSat Beyond LEO Massimiliano Vasile, Willem van der Weg, Marilena Di Carlo Department of Mechanical and Aerospace Engineering University of Strathclyde, Glasgow 5th Interplanetary
More informationHW Chapter 5 Q 7,8,18,21 P 4,6,8. Chapter 5. The Law of Universal Gravitation Gravity
HW Chapter 5 Q 7,8,18,21 P 4,6,8 Chapter 5 The Law of Universal Gravitation Gravity Newton s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that
More informationInterplanetary Mission Analysis
Interplanetary Mission Analysis Stephen Kemble Senior Expert EADS Astrium stephen.kemble@astrium.eads.net Page 1 Contents 1. Conventional mission design. Advanced mission design options Page 1. Conventional
More informationChapter 13 Gravity Pearson Education, Inc. Slide 13-1
Chapter 13 Gravity Slide 13-1 12.12 The system shown below consists of two balls A and B connected by a thin rod of negligible mass. Ball A has five times the inertia of ball B and the distance between
More informationCelestial Mechanics and Satellite Orbits
Celestial Mechanics and Satellite Orbits Introduction to Space 2017 Slides: Jaan Praks, Hannu Koskinen, Zainab Saleem Lecture: Jaan Praks Assignment Draw Earth, and a satellite orbiting the Earth. Draw
More informationGRAVITATION. F = GmM R 2
GRAVITATION Name: Partner: Section: Date: PURPOSE: To explore the gravitational force and Kepler s Laws of Planetary motion. INTRODUCTION: Newton s law of Universal Gravitation tells us that the gravitational
More informationMINIMUM IMPULSE TRANSFERS TO ROTATE THE LINE OF APSIDES
AAS 05-373 MINIMUM IMPULSE TRANSFERS TO ROTATE THE LINE OF APSIDES Connie Phong and Theodore H. Sweetser While an optimal scenario for the general two-impulse transfer between coplanar orbits is not known,
More informationList of Tables. Table 3.1 Determination efficiency for circular orbits - Sample problem 1 41
List of Tables Table 3.1 Determination efficiency for circular orbits - Sample problem 1 41 Table 3.2 Determination efficiency for elliptical orbits Sample problem 2 42 Table 3.3 Determination efficiency
More informationChapter 4. Integrated Algorithm for Impact Lunar Transfer Trajectories. using Pseudo state Technique
Chapter 4 Integrated Algorithm for Impact Lunar Transfer Trajectories using Pseudo state Technique Abstract A new integrated algorithm to generate the design of one-way direct transfer trajectory for moon
More informationChapter 13 Gravity Pearson Education, Inc. Slide 13-1
Chapter 13 Gravity Slide 13-1 The plan Lab this week: there will be time for exam problems Final exam: sections posted today; some left out Final format: all multiple choice, almost all short problems,
More informationAstrodynamics (AERO0024)
Astrodynamics (AERO0024) 5A. Orbital Maneuvers Gaëtan Kerschen Space Structures & Systems Lab (S3L) Course Outline THEMATIC UNIT 1: ORBITAL DYNAMICS Lecture 02: The Two-Body Problem Lecture 03: The Orbit
More informationThe Design Process Level I Design Example: Low-Cost Lunar Exploration Amplification on Initial Concept Review
Parametric Design The Design Process Level I Design Example: Low-Cost Lunar Exploration Amplification on Initial Concept Review U N I V E R S I T Y O F MARYLAND 2008 David L. Akin - All rights reserved
More informationNewton s Gravitational Law
1 Newton s Gravitational Law Gravity exists because bodies have masses. Newton s Gravitational Law states that the force of attraction between two point masses is directly proportional to the product of
More informationMission Trajectory Design to a Nearby Asteroid
Mission Trajectory Design to a Nearby Asteroid A project present to The Faculty of the Department of Aerospace Engineering San Jose State University in partial fulfillment of the requirements for the degree
More informationProjectile Motion. Conceptual Physics 11 th Edition. Projectile Motion. Projectile Motion. Projectile Motion. This lecture will help you understand:
Conceptual Physics 11 th Edition Projectile motion is a combination of a horizontal component, and Chapter 10: PROJECTILE AND SATELLITE MOTION a vertical component. This lecture will help you understand:
More informationHow Small Can a Launch Vehicle Be?
UCRL-CONF-213232 LAWRENCE LIVERMORE NATIONAL LABORATORY How Small Can a Launch Vehicle Be? John C. Whitehead July 10, 2005 41 st AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit Tucson, AZ Paper
More informationWiley Plus. Final Assignment (5) Is Due Today: Before 11 pm!
Wiley Plus Final Assignment (5) Is Due Today: Before 11 pm! Final Exam Review December 9, 009 3 What about vector subtraction? Suppose you are given the vector relation A B C RULE: The resultant vector
More informationOverview of Astronautics and Space Missions
Overview of Astronautics and Space Missions Prof. Richard Wirz Slide 1 Astronautics Definition: The science and technology of space flight Includes: Orbital Mechanics Often considered a subset of Celestial
More informationBravoSat: Optimizing the Delta-V Capability of a CubeSat Mission. with Novel Plasma Propulsion Technology ISSC 2013
BravoSat: Optimizing the Delta-V Capability of a CubeSat Mission with Novel Plasma Propulsion Technology Sara Spangelo, NASA JPL, Caltech Benjamin Longmier, University of Michigan Interplanetary Small
More informationConceptual Physics 11 th Edition
Conceptual Physics 11 th Edition Chapter 10: PROJECTILE AND SATELLITE MOTION This lecture will help you understand: Projectile Motion Fast-Moving Projectiles Satellites Circular Satellite Orbits Elliptical
More informationNAVIGATION & MISSION DESIGN BRANCH
c o d e 5 9 5 National Aeronautics and Space Administration Michael Mesarch Michael.A.Mesarch@nasa.gov NAVIGATION & MISSION DESIGN BRANCH www.nasa.gov Outline Orbital Elements Orbital Precession Differential
More informationAAE 251 Formulas. Standard Atmosphere. Compiled Fall 2016 by Nicholas D. Turo-Shields, student at Purdue University. Gradient Layer.
AAE 51 Formulas Compiled Fall 016 by Nicholas D. Turo-Shields, student at Purdue University Standard Atmosphere p 0 = 1.0135 10 5 Pascals ρ 0 = 1.5 kg m 3 R = 87 J kg K γ = 1.4 for air p = ρrt ; Equation
More informationAS3010: Introduction to Space Technology
AS3010: Introduction to Space Technology L E C T U R E S 8-9 Part B, Lectures 8-9 23 March, 2017 C O N T E N T S In this lecture, we will look at factors that cause an orbit to change over time orbital
More information(b) The period T and the angular frequency ω of uniform rotation are related to the cyclic frequency f as. , ω = 2πf =
PHY 302 K. Solutions for problem set #9. Non-textbook problem #1: (a) Rotation frequency of 1 Hz means one revolution per second, or 60 revolutions per minute (RPM). The pre-lp vinyl disks rotated at 78
More informationEffect of Coordinate Switching on Translunar Trajectory Simulation Accuracy
Effect of Coordinate Switching on Translunar Trajectory Simulation Accuracy Mana P. Vautier Auburn University, Auburn, AL, 36849, USA This paper focuses on the affect of round-off error in the accurate
More informationParametric Design MARYLAND. The Design Process Regression Analysis Level I Design Example: Project Diana U N I V E R S I T Y O F.
Parametric Design The Design Process Regression Analysis Level I Design Example: U N I V E R S I T Y O F MARYLAND 2003 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu Parametric Design
More information