Viewpoint the Weak Stability Boundary, a Gateway for Human
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Astrodynamics
Politecnico di Torino SEEDS SpacE Exploration and Development Systems Astrodynamics II Edition 2006 - 07 - Ver. 2.0.1 Author: Guido Colasurdo Dipartimento di Energetica Teacher: Giulio Avanzini Dipartimento di Ingegneria Aeronautica e Spaziale e-mail: [email protected] Contents 1 Two–Body Orbital Mechanics 1 1.1 BirthofAstrodynamics: Kepler’sLaws. ......... 1 1.2 Newton’sLawsofMotion ............................ ... 2 1.3 Newton’s Law of Universal Gravitation . ......... 3 1.4 The n–BodyProblem ................................. 4 1.5 Equation of Motion in the Two-Body Problem . ....... 5 1.6 PotentialEnergy ................................. ... 6 1.7 ConstantsoftheMotion . .. .. .. .. .. .. .. .. .... 7 1.8 TrajectoryEquation .............................. .... 8 1.9 ConicSections ................................... 8 1.10 Relating Energy and Semi-major Axis . ........ 9 2 Two-Dimensional Analysis of Motion 11 2.1 ReferenceFrames................................. 11 2.2 Velocity and acceleration components . ......... 12 2.3 First-Order Scalar Equations of Motion . ......... 12 2.4 PerifocalReferenceFrame . ...... 13 2.5 FlightPathAngle ................................. 14 2.6 EllipticalOrbits................................ ..... 15 2.6.1 Geometry of an Elliptical Orbit . ..... 15 2.6.2 Period of an Elliptical Orbit . ..... 16 2.7 Time–of–Flight on the Elliptical Orbit . .......... 16 2.8 Extensiontohyperbolaandparabola. ........ 18 2.9 Circular and Escape Velocity, Hyperbolic Excess Speed . .............. 18 2.10 CosmicVelocities -
Electric Propulsion System Scaling for Asteroid Capture-And-Return Missions
Electric propulsion system scaling for asteroid capture-and-return missions Justin M. Little⇤ and Edgar Y. Choueiri† Electric Propulsion and Plasma Dynamics Laboratory, Princeton University, Princeton, NJ, 08544 The requirements for an electric propulsion system needed to maximize the return mass of asteroid capture-and-return (ACR) missions are investigated in detail. An analytical model is presented for the mission time and mass balance of an ACR mission based on the propellant requirements of each mission phase. Edelbaum’s approximation is used for the Earth-escape phase. The asteroid rendezvous and return phases of the mission are modeled as a low-thrust optimal control problem with a lunar assist. The numerical solution to this problem is used to derive scaling laws for the propellant requirements based on the maneuver time, asteroid orbit, and propulsion system parameters. Constraining the rendezvous and return phases by the synodic period of the target asteroid, a semi- empirical equation is obtained for the optimum specific impulse and power supply. It was found analytically that the optimum power supply is one such that the mass of the propulsion system and power supply are approximately equal to the total mass of propellant used during the entire mission. Finally, it is shown that ACR missions, in general, are optimized using propulsion systems capable of processing 100 kW – 1 MW of power with specific impulses in the range 5,000 – 10,000 s, and have the potential to return asteroids on the order of 103 104 tons. − Nomenclature -
Aas 11-509 Artemis Lunar Orbit Insertion and Science Orbit Design Through 2013
AAS 11-509 ARTEMIS LUNAR ORBIT INSERTION AND SCIENCE ORBIT DESIGN THROUGH 2013 Stephen B. Broschart,∗ Theodore H. Sweetser,y Vassilis Angelopoulos,z David C. Folta,x and Mark A. Woodard{ As of late-July 2011, the ARTEMIS mission is transferring two spacecraft from Lissajous orbits around Earth-Moon Lagrange Point #1 into highly-eccentric lunar science orbits. This paper presents the trajectory design for the transfer from Lissajous orbit to lunar orbit in- sertion, the period reduction maneuvers, and the science orbits through 2013. The design accommodates large perturbations from Earth’s gravity and restrictive spacecraft capabili- ties to enable opportunities for a range of heliophysics and planetary science measurements. The process used to design the highly-eccentric ARTEMIS science orbits is outlined. The approach may inform the design of future eccentric orbiter missions at planetary moons. INTRODUCTION The Acceleration, Reconnection, Turbulence and Electrodynamics of the Moons Interaction with the Sun (ARTEMIS) mission is currently operating two spacecraft in lunar orbit under funding from the Heliophysics and Planetary Science Divisions with NASA’s Science Missions Directorate. ARTEMIS is an extension to the successful Time History of Events and Macroscale Interactions during Substorms (THEMIS) mission [1] that has relocated two of the THEMIS spacecraft from Earth orbit to the Moon [2, 3, 4]. ARTEMIS plans to conduct a variety of scientific studies at the Moon using the on-board particle and fields instrument package [5, 6]. The final portion of the ARTEMIS transfers involves moving the two ARTEMIS spacecraft, known as P1 and P2, from Lissajous orbits around the Earth-Moon Lagrange Point #1 (EML1) to long-lived, eccentric lunar science orbits with periods of roughly 28 hr. -
Dynamical Analysis of Rendezvous and Docking with Very Large Space Infrastructures in Non-Keplerian Orbits
DYNAMICAL ANALYSIS OF RENDEZVOUS AND DOCKING WITH VERY LARGE SPACE INFRASTRUCTURES IN NON-KEPLERIAN ORBITS Andrea Colagrossi ∗, Michele` Lavagna y, Simone Flavio Rafano Carna` z Politecnico di Milano Department of Aerospace Science and Technology Via La Masa 34 – 20156 Milan (MI) – Italy ABSTRACT the whole mission scenario is the so called Evolvable Deep Space Habitat: a modular space station in lunar vicinity. A space station in the vicinity of the Moon can be exploited At the current level of study, the optimum location for as a gateway for future human and robotic exploration of the space infrastructure of this kind has not yet been determined, Solar System. The natural location for a space system of this but a favorable solution can be about one of the Earth-Moon kind is about one of the Earth-Moon libration points. libration points, such as in a EML2 (Earth-Moon Lagrangian The study addresses the dynamics during rendezvous and Point no 2) Halo orbit. Moreover, the final configuration of docking operations with a very large space infrastructure in a the entire system is still to be defined, but it is already clear EML2 Halo orbit. The model takes into account the coupling that in order to assemble the structure several rendezvous and effects between the orbital and the attitude motion in a Circular docking activities will be carried out, many of which to be Restricted Three-Body Problem environment. The flexibility completely automated. of the system is included, and the interaction between the Unfortunately, the current knowledge about rendezvous in modes of the structure and those related with the orbital motion cis-lunar orbits is minimal and it is usually limited to point- is investigated. -
What Is the Interplanetary Superhighway?
What is the InterPlanetary Suppgyerhighway? Kathleen Howell Purdue University Lo and Ross Trajectory Key Space Technology Mission-Enabling Technology Not All Technology is hardware! The InterPlanetary Superhighway (IPS) • LELow Energy ObitfSOrbits for Space Mi Miissions • InterPlanetary Superhighway—“a vast network of winding tunnels in space” that connects the Sun , the planets, their moons, AND many other destinations • Systematic mapping properly known as InterPlanetary Transport Network Simó, Gómez, Masdemont / Lo, Howell, Barden / Howell, Folta / Lo, Ross / Koon, Lo, Marsden, Ross / Marchand, Howell, Lo / Scheeres, Villac/ ……. Originates with Poincaré (1892) Applications to wide range of fields Different View of Problems in N-Bodies • Much more than Kepler and Newton imagined • Computationally challenging Poincaré (1854-1912) “Mathematics is the art of giving the same name to different things” Jules Henri Poincaré NBP Play 3BP Wang 2BP New Era in Celestial Mechanics Pioneering Work: Numerical Exploration by Hand (Breakwell, Farquhar and Dunham) Current Libration Point Missions Goddard Space Flight Center • z WIND SOHO ACE MAPGENESIS NGST Courtesy of D. Folta, GSFC Multi-Body Problem Orbit propagated for 4 conic periods: • Change our perspective 4*19 days = 75.7 days Earth Earth Sun To Sun Inertial View RotatingRotating View View Inertial View (Ro ta tes w ith two b odi es) Aeronautics and Astronautics Multi-Body Problem • Change our perspective • Effects of added gravity fields Earth Earth TSTo Sun Rotating View Inertial View Equilibrium -
Orbit and Spin
Orbit and Spin Overview: A whole-body activity that explores the relative sizes, distances, orbit, and spin of the Sun, Earth, and Moon. Target Grade Level: 3-5 Estimated Duration: 2 40-minute sessions Learning Goals: Students will be able to… • compare the relative sizes of the Earth, Moon, and Sun. • contrast the distance between the Earth and Moon to the distance between the Earth and Sun. • differentiate between the motions of orbit and spin. • demonstrate the spins of the Earth and the Moon, as well as the orbits of the Earth around the Sun, and the Moon around the Earth. Standards Addressed: Benchmarks (AAAS, 1993) The Physical Setting, 4A: The Universe, 4B: The Earth National Science Education Standards (NRC, 1996) Physical Science, Standard B: Position and motion of objects Earth and Space Science, Standard D: Objects in the sky, Changes in Earth and sky Table of Contents: Background Page 1 Materials and Procedure 5 What I Learned… Science Journal Page 14 Earth Picture 15 Sun Picture 16 Moon Picture 17 Earth Spin Demonstration 18 Moon Orbit Demonstration 19 Extensions and Adaptations 21 Standards Addressed, detailed 22 Background: Sun The Sun is the center of our Solar System, both literally—as all of the planets orbit around it, and figuratively—as its rays warm our planet and sustain life as we know it. The Sun is very hot compared to temperatures we usually encounter. Its mean surface temperature is about 9980° Fahrenheit (5800 Kelvin) and its interior temperature is as high as about 28 million° F (15,500,000 Kelvin). -
Spacecraft Trajectories in a Sun, Earth, and Moon Ephemeris Model
SPACECRAFT TRAJECTORIES IN A SUN, EARTH, AND MOON EPHEMERIS MODEL A Project Presented to The Faculty of the Department of Aerospace Engineering San José State University In Partial Fulfillment of the Requirements for the Degree Master of Science in Aerospace Engineering by Romalyn Mirador i ABSTRACT SPACECRAFT TRAJECTORIES IN A SUN, EARTH, AND MOON EPHEMERIS MODEL by Romalyn Mirador This project details the process of building, testing, and comparing a tool to simulate spacecraft trajectories using an ephemeris N-Body model. Different trajectory models and methods of solving are reviewed. Using the Ephemeris positions of the Earth, Moon and Sun, a code for higher-fidelity numerical modeling is built and tested using MATLAB. Resulting trajectories are compared to NASA’s GMAT for accuracy. Results reveal that the N-Body model can be used to find complex trajectories but would need to include other perturbations like gravity harmonics to model more accurate trajectories. i ACKNOWLEDGEMENTS I would like to thank my family and friends for their continuous encouragement and support throughout all these years. A special thank you to my advisor, Dr. Capdevila, and my friend, Dhathri, for mentoring me as I work on this project. The knowledge and guidance from the both of you has helped me tremendously and I appreciate everything you both have done to help me get here. ii Table of Contents List of Symbols ............................................................................................................................... v 1.0 INTRODUCTION -
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Cis-Lunar Autonomous Navigation via Implementation of Optical Asteroid Angle-Only Measurements Mark B. Hinga, Ph.D., P.E. ∗ Autonomous cis- or trans-Lunar spacecraft navigation is critical to mission success as communication to ground stations and access to GPS signals could be lost. However, if the satellite has a camera of sufficient qual- ity, line of sight (unit vector) measurements can be made to known solar system bodies to provide observations which enable autonomous estimation of position and velocity of the spacecraft, that can be telemetered to those interested space based or ground based consumers. An improved Gaussian-Initial Orbit Determination (IOD) algorithm, based on the exact values of the f and g series (free of the 8th order polynomial and range guessing), for spacecraft state estimation, is presented here and exercised in the inertial coordinate frame (2-Body Prob- lem) to provide an initial guess for the Batch IOD that is performed in the Circular Restricted Three Body Problem (CRTBP) reference frame, which ultimately serves to initialize a CRTBP Extended Kalman Filter (EKF) navigator that collects angle only measurements to a known Asteroid 2014 EC (flying by the Earth) to sequentially estimate position and velocity of an observer spacecraft flying on an APOLLO-like trajectory to the Moon. With the addition of simulating/expressing the accelerations that would be sensed in the IMU plat- form frame due to delta velocities caused by either perturbations or corrective guidance maneuvers, this three phase algorithm is able to autonomously track the spacecraft state on its journey to the Moon while observing the motion of the Asteroid. -
GCSE Astronomy Course Sample N Section 3 Topic 2 N the Moon’S Orbit
Sample of the GCSE Astronomy Course from Section 3 Topic 2 The Moon’s orbit Introduction We can see that the Moon changes its appearance by the day. In this topic you will study the orbit of the Moon and the changes that this brings about. You will learn about the phases of the Moon, why you can only see one side of the Moon from Earth, and what gives rise to our ability to see a small fraction of the far side. You will probably need 2 hours to complete this topic. Objectives When you have completed this topic you will be able to: n explain the rotation and revolution (orbit) of the Moon n describe the phases of the lunar cycle n explain the synchronous nature of the Moon’s orbit and rotation n explain the causes of lunar libration and its effect on the visibility of the lunar disc. Rotation and orbit of the Moon The Moon rotates about its axis in 27.3 days. The Moon orbits round the Earth in a period which is also 27.3 days. This means that we only see one side of the Moon (around 59 per cent of the lunar disc). The axis of the Moon has a slight tilt. The Moon’s equator is tilted by 1.5° to the plane of its orbit around the Earth. You may recall from Section 2 that the plane in which the Earth orbits the Sun is called the ecliptic. The plane of the Moon’s orbit is 5.1° to the ecliptic and the orbit is elliptical. -
Cislunar Tether Transport System
FINAL REPORT on NIAC Phase I Contract 07600-011 with NASA Institute for Advanced Concepts, Universities Space Research Association CISLUNAR TETHER TRANSPORT SYSTEM Report submitted by: TETHERS UNLIMITED, INC. 8114 Pebble Ct., Clinton WA 98236-9240 Phone: (206) 306-0400 Fax: -0537 email: [email protected] www.tethers.com Report dated: May 30, 1999 Period of Performance: November 1, 1998 to April 30, 1999 PROJECT SUMMARY PHASE I CONTRACT NUMBER NIAC-07600-011 TITLE OF PROJECT CISLUNAR TETHER TRANSPORT SYSTEM NAME AND ADDRESS OF PERFORMING ORGANIZATION (Firm Name, Mail Address, City/State/Zip Tethers Unlimited, Inc. 8114 Pebble Ct., Clinton WA 98236-9240 [email protected] PRINCIPAL INVESTIGATOR Robert P. Hoyt, Ph.D. ABSTRACT The Phase I effort developed a design for a space systems architecture for repeatedly transporting payloads between low Earth orbit and the surface of the moon without significant use of propellant. This architecture consists of one rotating tether in elliptical, equatorial Earth orbit and a second rotating tether in a circular low lunar orbit. The Earth-orbit tether picks up a payload from a circular low Earth orbit and tosses it into a minimal-energy lunar transfer orbit. When the payload arrives at the Moon, the lunar tether catches it and deposits it on the surface of the Moon. Simultaneously, the lunar tether picks up a lunar payload to be sent down to the Earth orbit tether. By transporting equal masses to and from the Moon, the orbital energy and momentum of the system can be conserved, eliminating the need for transfer propellant. Using currently available high-strength tether materials, this system could be built with a total mass of less than 28 times the mass of the payloads it can transport. -
Pdf [35] Ghoddousi-Fard, R
International Journal of Astronomy and Astrophysics, 2021, 11, 343-369 https://www.scirp.org/journal/ijaa ISSN Online: 2161-4725 ISSN Print: 2161-4717 Updating the Historical Perspective of the Interaction of Gravitational Field and Orbit in Sun-Planet-Moon System Yin Zhu Agriculture Department of Hubei Province, Wuhan, China How to cite this paper: Zhu, Y. (2021) Abstract Updating the Historical Perspective of the Interaction of Gravitational Field and Orbit Studying the two famous old problems that why the moon can move around in Sun-Planet-Moon System. International the Sun and why the orbit of the Moon around the Earth cannot be broken Journal of Astronomy and Astrophysics, = 2 11, 343-369. off by the Sun under the condition calculating with F GMm R , the at- https://doi.org/10.4236/ijaa.2021.113016 tractive force of the Sun on the Moon is almost 2.2 times that of the Earth, we found that the planet and moon are unified as one single gravitational unit Received: May 17, 2021 2 Accepted: July 20, 2021 which results in that the Sun cannot have the force of F= GMm R on the Published: July 23, 2021 moon. The moon is moved by the gravitational unit orbiting around the Sun. It could indicate that the gravitational field of the moon is limited inside the Copyright © 2021 by author(s) and Scientific Research Publishing Inc. unit and the gravitational fields of both the planet and moon are unified as This work is licensed under the Creative one single field interacting with the Sun. -
Moon-Earth-Sun: the Oldest Three-Body Problem
Moon-Earth-Sun: The oldest three-body problem Martin C. Gutzwiller IBM Research Center, Yorktown Heights, New York 10598 The daily motion of the Moon through the sky has many unusual features that a careful observer can discover without the help of instruments. The three different frequencies for the three degrees of freedom have been known very accurately for 3000 years, and the geometric explanation of the Greek astronomers was basically correct. Whereas Kepler’s laws are sufficient for describing the motion of the planets around the Sun, even the most obvious facts about the lunar motion cannot be understood without the gravitational attraction of both the Earth and the Sun. Newton discussed this problem at great length, and with mixed success; it was the only testing ground for his Universal Gravitation. This background for today’s many-body theory is discussed in some detail because all the guiding principles for our understanding can be traced to the earliest developments of astronomy. They are the oldest results of scientific inquiry, and they were the first ones to be confirmed by the great physicist-mathematicians of the 18th century. By a variety of methods, Laplace was able to claim complete agreement of celestial mechanics with the astronomical observations. Lagrange initiated a new trend wherein the mathematical problems of mechanics could all be solved by the same uniform process; canonical transformations eventually won the field. They were used for the first time on a large scale by Delaunay to find the ultimate solution of the lunar problem by perturbing the solution of the two-body Earth-Moon problem.