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Appendix a Orbits
Appendix A Orbits As discussed in the Introduction, a good ¯rst approximation for satellite motion is obtained by assuming the spacecraft is a point mass or spherical body moving in the gravitational ¯eld of a spherical planet. This leads to the classical two-body problem. Since we use the term body to refer to a spacecraft of ¯nite size (as in rigid body), it may be more appropriate to call this the two-particle problem, but I will use the term two-body problem in its classical sense. The basic elements of orbital dynamics are captured in Kepler's three laws which he published in the 17th century. His laws were for the orbital motion of the planets about the Sun, but are also applicable to the motion of satellites about planets. The three laws are: 1. The orbit of each planet is an ellipse with the Sun at one focus. 2. The line joining the planet to the Sun sweeps out equal areas in equal times. 3. The square of the period of a planet is proportional to the cube of its mean distance to the sun. The ¯rst law applies to most spacecraft, but it is also possible for spacecraft to travel in parabolic and hyperbolic orbits, in which case the period is in¯nite and the 3rd law does not apply. However, the 2nd law applies to all two-body motion. Newton's 2nd law and his law of universal gravitation provide the tools for generalizing Kepler's laws to non-elliptical orbits, as well as for proving Kepler's laws. -
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 -
OCEANOGRAPHY an Additional 1.4 Tg of Carbon Per Year Atmospheric CO2 Concentrations from Ice Loss and Ocean Life Over This Period
research highlights OCEANOGRAPHY an additional 1.4 Tg of carbon per year atmospheric CO2 concentrations from Ice loss and ocean life over this period. A rise in phytoplankton Antarctic ice cores. They found that Glob. Biogeochem. Cycles http://doi.org/p27 (2013) productivity in the surface waters of the two younger periods of enhanced the Laptev, East Siberian, Chukchi and mixing coincided with a steep increase in Beaufort seas was responsible for the atmospheric CO2 levels, and the oldest with increase in carbon uptake. In contrast, net a small, temporary rise. carbon uptake declined in the Barents Sea, The team concluded that enhanced where a warming-induced outgassing of vertical mixing in these intervals brought surface-water CO2 countered the rise in CO2-rich waters from the deep ocean to the primary production. surface, allowing CO2 to escape from these The findings suggest that the continued upwelled waters to the atmosphere. AN decline of Arctic sea ice cover could be accompanied by a rise in the oceanic uptake PLANETARY SCIENCE of carbon dioxide, although uncertainties Weighing Phobos in the response of physical, chemical and Icarus http://doi.org/p26 (2013) biological processes to sea ice loss hinder reliable predictions at this stage. AA PALAEOCEANOGRAPHY Southern upwelling Nature Commun. 4, 2758 (2013). At the end of the last glacial period about 20,000 years ago, atmospheric CO2 concentrations rose in several steps. Radiocarbon measurements from a marine sediment core suggest that the upwelling of © FRANS LANTING STUDIO / ALAMY © FRANS LANTING STUDIO carbon-rich waters in the Southern Ocean contributed to the CO2 rise. -
Ionizing Feedback from Massive Stars in Massive Clusters III: Disruption of Partially Unbound Clouds 3
Mon. Not. R. Astron. Soc. 000, 1–14 (2006) Printed 30 August 2021 (MN LATEX style file v2.2) Ionizing feedback from massive stars in massive clusters III: Disruption of partially unbound clouds J. E. Dale1⋆, B. Ercolano1, I. A. Bonnell2 1Excellence Cluster ‘Universe’, Boltzmannstr. 2, 85748 Garching, Germany. 2Department of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS 30 August 2021 ABSTRACT We extend our previous SPH parameter study of the effects of photoionization from O–stars on star–forming clouds to include initially unbound clouds. We generate a set 4 6 of model clouds in the mass range 10 -10 M⊙ with initial virial ratios Ekin/Epot=2.3, allow them to form stars, and study the impact of the photoionizing radiation pro- duced by the massive stars. We find that, on the 3Myr timescale before supernovae are expected to begin detonating, the fractions of mass expelled by ionizing feedback is a very strong function of the cloud escape velocities. High–mass clouds are largely unaf- fected dynamically, while lower–mass clouds have large fractions of their gas reserves expelled on this timescale. However, the fractions of stellar mass unbound are modest and significant portions of the unbound stars are so only because the clouds themselves are initially partially unbound. We find that ionization is much more able to create well–cleared bubbles in the unbound clouds, owing to their intrinsic expansion, but that the presence of such bubbles does not necessarily indicate that a given cloud has been strongly influenced by feedback. -
ESO's VLT Sphere and DAMIT
ESO’s VLT Sphere and DAMIT ESO’s VLT SPHERE (using adaptive optics) and Joseph Durech (DAMIT) have a program to observe asteroids and collect light curve data to develop rotating 3D models with respect to time. Up till now, due to the limitations of modelling software, only convex profiles were produced. The aim is to reconstruct reliable nonconvex models of about 40 asteroids. Below is a list of targets that will be observed by SPHERE, for which detailed nonconvex shapes will be constructed. Special request by Joseph Durech: “If some of these asteroids have in next let's say two years some favourable occultations, it would be nice to combine the occultation chords with AO and light curves to improve the models.” 2 Pallas, 7 Iris, 8 Flora, 10 Hygiea, 11 Parthenope, 13 Egeria, 15 Eunomia, 16 Psyche, 18 Melpomene, 19 Fortuna, 20 Massalia, 22 Kalliope, 24 Themis, 29 Amphitrite, 31 Euphrosyne, 40 Harmonia, 41 Daphne, 51 Nemausa, 52 Europa, 59 Elpis, 65 Cybele, 87 Sylvia, 88 Thisbe, 89 Julia, 96 Aegle, 105 Artemis, 128 Nemesis, 145 Adeona, 187 Lamberta, 211 Isolda, 324 Bamberga, 354 Eleonora, 451 Patientia, 476 Hedwig, 511 Davida, 532 Herculina, 596 Scheila, 704 Interamnia Occultation Event: Asteroid 10 Hygiea – Sun 26th Feb 16h37m UT The magnitude 11 asteroid 10 Hygiea is expected to occult the magnitude 12.5 star 2UCAC 21608371 on Sunday 26th Feb 16h37m UT (= Mon 3:37am). Magnitude drop of 0.24 will require video. DAMIT asteroid model of 10 Hygiea - Astronomy Institute of the Charles University: Josef Ďurech, Vojtěch Sidorin Hygiea is the fourth-largest asteroid (largest is Ceres ~ 945kms) in the Solar System by volume and mass, and it is located in the asteroid belt about 400 million kms away. -
A Hot Subdwarf-White Dwarf Super-Chandrasekhar Candidate
A hot subdwarf–white dwarf super-Chandrasekhar candidate supernova Ia progenitor Ingrid Pelisoli1,2*, P. Neunteufel3, S. Geier1, T. Kupfer4,5, U. Heber6, A. Irrgang6, D. Schneider6, A. Bastian1, J. van Roestel7, V. Schaffenroth1, and B. N. Barlow8 1Institut fur¨ Physik und Astronomie, Universitat¨ Potsdam, Haus 28, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany 2Department of Physics, University of Warwick, Coventry, CV4 7AL, UK 3Max Planck Institut fur¨ Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching bei Munchen¨ 4Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA 5Texas Tech University, Department of Physics & Astronomy, Box 41051, 79409, Lubbock, TX, USA 6Dr. Karl Remeis-Observatory & ECAP, Astronomical Institute, Friedrich-Alexander University Erlangen-Nuremberg (FAU), Sternwartstr. 7, 96049 Bamberg, Germany 7Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA 8Department of Physics and Astronomy, High Point University, High Point, NC 27268, USA *[email protected] ABSTRACT Supernova Ia are bright explosive events that can be used to estimate cosmological distances, allowing us to study the expansion of the Universe. They are understood to result from a thermonuclear detonation in a white dwarf that formed from the exhausted core of a star more massive than the Sun. However, the possible progenitor channels leading to an explosion are a long-standing debate, limiting the precision and accuracy of supernova Ia as distance indicators. Here we present HD 265435, a binary system with an orbital period of less than a hundred minutes, consisting of a white dwarf and a hot subdwarf — a stripped core-helium burning star. -
Introduction to Astronomy from Darkness to Blazing Glory
Introduction to Astronomy From Darkness to Blazing Glory Published by JAS Educational Publications Copyright Pending 2010 JAS Educational Publications All rights reserved. Including the right of reproduction in whole or in part in any form. Second Edition Author: Jeffrey Wright Scott Photographs and Diagrams: Credit NASA, Jet Propulsion Laboratory, USGS, NOAA, Aames Research Center JAS Educational Publications 2601 Oakdale Road, H2 P.O. Box 197 Modesto California 95355 1-888-586-6252 Website: http://.Introastro.com Printing by Minuteman Press, Berkley, California ISBN 978-0-9827200-0-4 1 Introduction to Astronomy From Darkness to Blazing Glory The moon Titan is in the forefront with the moon Tethys behind it. These are two of many of Saturn’s moons Credit: Cassini Imaging Team, ISS, JPL, ESA, NASA 2 Introduction to Astronomy Contents in Brief Chapter 1: Astronomy Basics: Pages 1 – 6 Workbook Pages 1 - 2 Chapter 2: Time: Pages 7 - 10 Workbook Pages 3 - 4 Chapter 3: Solar System Overview: Pages 11 - 14 Workbook Pages 5 - 8 Chapter 4: Our Sun: Pages 15 - 20 Workbook Pages 9 - 16 Chapter 5: The Terrestrial Planets: Page 21 - 39 Workbook Pages 17 - 36 Mercury: Pages 22 - 23 Venus: Pages 24 - 25 Earth: Pages 25 - 34 Mars: Pages 34 - 39 Chapter 6: Outer, Dwarf and Exoplanets Pages: 41-54 Workbook Pages 37 - 48 Jupiter: Pages 41 - 42 Saturn: Pages 42 - 44 Uranus: Pages 44 - 45 Neptune: Pages 45 - 46 Dwarf Planets, Plutoids and Exoplanets: Pages 47 -54 3 Chapter 7: The Moons: Pages: 55 - 66 Workbook Pages 49 - 56 Chapter 8: Rocks and Ice: -
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 -
An Overview of New Worlds, New Horizons in Astronomy and Astrophysics About the National Academies
2020 VISION An Overview of New Worlds, New Horizons in Astronomy and Astrophysics About the National Academies The National Academies—comprising the National Academy of Sciences, the National Academy of Engineering, the Institute of Medicine, and the National Research Council—work together to enlist the nation’s top scientists, engineers, health professionals, and other experts to study specific issues in science, technology, and medicine that underlie many questions of national importance. The results of their deliberations have inspired some of the nation’s most significant and lasting efforts to improve the health, education, and welfare of the United States and have provided independent advice on issues that affect people’s lives worldwide. To learn more about the Academies’ activities, check the website at www.nationalacademies.org. Copyright 2011 by the National Academy of Sciences. All rights reserved. Printed in the United States of America This study was supported by Contract NNX08AN97G between the National Academy of Sciences and the National Aeronautics and Space Administration, Contract AST-0743899 between the National Academy of Sciences and the National Science Foundation, and Contract DE-FG02-08ER41542 between the National Academy of Sciences and the U.S. Department of Energy. Support for this study was also provided by the Vesto Slipher Fund. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the agencies that provided support for the project. 2020 VISION An Overview of New Worlds, New Horizons in Astronomy and Astrophysics Committee for a Decadal Survey of Astronomy and Astrophysics ROGER D. -
THE EARTH's GRAVITY OUTLINE the Earth's Gravitational Field
GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY OUTLINE The Earth's gravitational field 2 Newton's law of gravitation: Fgrav = GMm=r ; Gravitational field = gravitational acceleration g; gravitational potential, equipotential surfaces. g for a non–rotating spherically symmetric Earth; Effects of rotation and ellipticity – variation with latitude, the reference ellipsoid and International Gravity Formula; Effects of elevation and topography, intervening rock, density inhomogeneities, tides. The geoid: equipotential mean–sea–level surface on which g = IGF value. Gravity surveys Measurement: gravity units, gravimeters, survey procedures; the geoid; satellite altimetry. Gravity corrections – latitude, elevation, Bouguer, terrain, drift; Interpretation of gravity anomalies: regional–residual separation; regional variations and deep (crust, mantle) structure; local variations and shallow density anomalies; Examples of Bouguer gravity anomalies. Isostasy Mechanism: level of compensation; Pratt and Airy models; mountain roots; Isostasy and free–air gravity, examples of isostatic balance and isostatic anomalies. Background reading: Fowler §5.1–5.6; Lowrie §2.2–2.6; Kearey & Vine §2.11. GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY FIELD Newton's law of gravitation is: ¯ GMm F = r2 11 2 2 1 3 2 where the Gravitational Constant G = 6:673 10− Nm kg− (kg− m s− ). ¢ The field strength of the Earth's gravitational field is defined as the gravitational force acting on unit mass. From Newton's third¯ law of mechanics, F = ma, it follows that gravitational force per unit mass = gravitational acceleration g. g is approximately 9:8m/s2 at the surface of the Earth. A related concept is gravitational potential: the gravitational potential V at a point P is the work done against gravity in ¯ P bringing unit mass from infinity to P. -
Year XIX, Supplement Ethnographic Study And/Or a Theoretical Survey of a Their Position in the Article Should Be Clearly Indicated
III. TITLES OF ARTICLES DRU[TVO ANTROPOLOGOV SLOVENIJE The journal of the Slovene Anthropological Society Titles (in English and Slovene) must be short, informa- SLOVENE ANTHROPOLOGICAL SOCIETY Anthropological Notebooks welcomes the submis- tive, and understandable. The title should be followed sion of papers from the field of anthropology and by the name of the author(s), their position, institutional related disciplines. Submissions are considered for affiliation, and if possible, by e-mail address. publication on the understanding that the paper is not currently under consideration for publication IV. ABSTRACT AND KEYWORDS elsewhere. It is the responsibility of the author to The abstract must give concise information about the obtain permission for using any previously published objective, the method used, the results obtained, and material. Please submit your manuscript as an e-mail the conclusions. Authors are asked to enclose in English attachment on [email protected] and enclose your contact information: name, position, and Slovene an abstract of 100 – 200 words followed institutional affiliation, address, phone number, and by three to five keywords. They must reflect the field of e-mail address. research covered in the article. English abstract should be placed at the beginning of an article and the Slovene one after the references at the end. V. NOTES A N T H R O P O L O G I C A L INSTRUCTIONS Notes should also be double-spaced and used sparingly. They must be numbered consecutively throughout the text and assembled at the end of the article just before references. VI. QUOTATIONS Short quotations (less than 30 words) should be placed in single quotation marks with double marks for quotations within quotations. -
1. Two Objects Attract Each Other Gravitationally. If the Distance Between Their Centers Is Cut in Half, the Gravitational Force A) Is Cut to One Fourth
MULTIPLE CHOICE REVIEW – GRAVITATION (Dynamics Part 3) 1. Two objects attract each other gravitationally. If the distance between their centers is cut in half, the gravitational force A) is cut to one fourth. B) is cut in half. C) doubles. D) quadruples 2. Two objects, with masses m1 and m2, are originally a distance r apart. The magnitude of the gravitational force between them is F. The masses are changed to 2m1 and 2m2, and the distance is changed to 4r. What is the magnitude of the new gravitational force? A) F/16 B) F/4 C) 16F D) 4F 3. As a rocket moves away from the Earth's surface, the rocket's weight A) increases. B) decreases. C) remains the same. D) depends on how fast it is moving. 4. A hypothetical planet has a mass of half that of the Earth and a radius of twice that of the Earth. What is the acceleration due to gravity on the planet in terms of g, the acceleration due to gravity at the Earth? A) g B) g/2 C) g/4 D) g/8 5. Two planets have the same surface gravity, but planet B has twice the mass of planet A. If planet A has radius r, what is the radius of planet B? A) 0.707r B) r C) 1.41r D) 4r 6. A planet is discovered to orbit around a star in the galaxy Andromeda, with the same orbital diameter as the Earth around our Sun. If that star has 4 times the mass of our Sun, what will the period of revolution of that new planet be, compared to the Earth's orbital period? A) one-fourth as much B) one-half as much C) twice as much D) four times as much 7.