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The Swinging Spring: Regular and Chaotic Motion
References The Swinging Spring: Regular and Chaotic Motion Leah Ganis May 30th, 2013 Leah Ganis The Swinging Spring: Regular and Chaotic Motion References Outline of Talk I Introduction to Problem I The Basics: Hamiltonian, Equations of Motion, Fixed Points, Stability I Linear Modes I The Progressing Ellipse and Other Regular Motions I Chaotic Motion I References Leah Ganis The Swinging Spring: Regular and Chaotic Motion References Introduction The swinging spring, or elastic pendulum, is a simple mechanical system in which many different types of motion can occur. The system is comprised of a heavy mass, attached to an essentially massless spring which does not deform. The system moves under the force of gravity and in accordance with Hooke's Law. z y r φ x k m Leah Ganis The Swinging Spring: Regular and Chaotic Motion References The Basics We can write down the equations of motion by finding the Lagrangian of the system and using the Euler-Lagrange equations. The Lagrangian, L is given by L = T − V where T is the kinetic energy of the system and V is the potential energy. Leah Ganis The Swinging Spring: Regular and Chaotic Motion References The Basics In Cartesian coordinates, the kinetic energy is given by the following: 1 T = m(_x2 +y _ 2 +z _2) 2 and the potential is given by the sum of gravitational potential and the spring potential: 1 V = mgz + k(r − l )2 2 0 where m is the mass, g is the gravitational constant, k the spring constant, r the stretched length of the spring (px2 + y 2 + z2), and l0 the unstretched length of the spring. -
Exomars Schiaparelli Direct-To-Earth Observation Using GMRT
TECHNICAL ExoMars Schiaparelli Direct-to-Earth Observation REPORTS: METHODS 10.1029/2018RS006707 using GMRT S. Esterhuizen1, S. W. Asmar1 ,K.De2, Y. Gupta3, S. N. Katore3, and B. Ajithkumar3 Key Point: • During ExoMars Landing, GMRT 1Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA, 2Cahill Center for Astrophysics, observed UHF transmissions and California Institute of Technology, Pasadena, CA, USA, 3National Centre for Radio Astrophysics, Pune, India Doppler shift used to identify key events as only real-time aliveness indicator Abstract During the ExoMars Schiaparelli separation event on 16 October 2016 and Entry, Descent, and Landing (EDL) events 3 days later, the Giant Metrewave Radio Telescope (GMRT) near Pune, India, Correspondence to: S. W. Asmar, was used to directly observe UHF transmissions from the Schiaparelli lander as they arrive at Earth. The [email protected] Doppler shift of the carrier frequency was measured and used as a diagnostic to identify key events during EDL. This signal detection at GMRT was the only real-time aliveness indicator to European Space Agency Citation: mission operations during the critical EDL stage of the mission. Esterhuizen, S., Asmar, S. W., De, K., Gupta, Y., Katore, S. N., & Plain Language Summary When planetary missions, such as landers on the surface of Mars, Ajithkumar, B. (2019). ExoMars undergo critical and risky events, communications to ground controllers is very important as close to real Schiaparelli Direct-to-Earth observation using GMRT. time as possible. The Schiaparelli spacecraft attempted landing in 2016 was supported in an innovative way. Radio Science, 54, 314–325. A large radio telescope on Earth was able to eavesdrop on information being sent from the lander to other https://doi.org/10.1029/2018RS006707 spacecraft in orbit around Mars. -
Aerothermodynamic Analysis of a Mars Sample Return Earth-Entry Vehicle" (2018)
Old Dominion University ODU Digital Commons Mechanical & Aerospace Engineering Theses & Dissertations Mechanical & Aerospace Engineering Summer 2018 Aerothermodynamic Analysis of a Mars Sample Return Earth- Entry Vehicle Daniel A. Boyd Old Dominion University, [email protected] Follow this and additional works at: https://digitalcommons.odu.edu/mae_etds Part of the Aerodynamics and Fluid Mechanics Commons, Space Vehicles Commons, and the Thermodynamics Commons Recommended Citation Boyd, Daniel A.. "Aerothermodynamic Analysis of a Mars Sample Return Earth-Entry Vehicle" (2018). Master of Science (MS), Thesis, Mechanical & Aerospace Engineering, Old Dominion University, DOI: 10.25777/xhmz-ax21 https://digitalcommons.odu.edu/mae_etds/43 This Thesis is brought to you for free and open access by the Mechanical & Aerospace Engineering at ODU Digital Commons. It has been accepted for inclusion in Mechanical & Aerospace Engineering Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. AEROTHERMODYNAMIC ANALYSIS OF A MARS SAMPLE RETURN EARTH-ENTRY VEHICLE by Daniel A. Boyd B.S. May 2008, Virginia Military Institute M.A. August 2015, Webster University A Thesis Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE AEROSPACE ENGINEERING OLD DOMINION UNIVERSITY August 2018 Approved by: __________________________ Robert L. Ash (Director) __________________________ Oktay Baysal (Member) __________________________ Jamshid A. Samareh (Member) __________________________ Shizhi Qian (Member) ABSTRACT AEROTHERMODYNAMIC ANALYSIS OF A MARS SAMPLE RETURN EARTH-ENTRY VEHICLE Daniel A. Boyd Old Dominion University, 2018 Director: Dr. Robert L. Ash Because of the severe quarantine constraints that must be imposed on any returned extraterrestrial samples, the Mars sample return Earth-entry vehicle must remain intact through sample recovery. -
Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda
Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda • Introduction to the elastic pendulum problem • Derivations of the equations of motion • Real-life examples of an elastic pendulum • Trivial cases & equilibrium states • MATLAB models The Elastic Problem (Simple Harmonic Motion) 푑2푥 푑2푥 푘 • 퐹 = 푚 = −푘푥 = − 푥 푛푒푡 푑푡2 푑푡2 푚 • Solve this differential equation to find 푥 푡 = 푐1 cos 휔푡 + 푐2 sin 휔푡 = 퐴푐표푠(휔푡 − 휑) • With velocity and acceleration 푣 푡 = −퐴휔 sin 휔푡 + 휑 푎 푡 = −퐴휔2cos(휔푡 + 휑) • Total energy of the system 퐸 = 퐾 푡 + 푈 푡 1 1 1 = 푚푣푡2 + 푘푥2 = 푘퐴2 2 2 2 The Pendulum Problem (with some assumptions) • With position vector of point mass 푥 = 푙 푠푖푛휃푖 − 푐표푠휃푗 , define 푟 such that 푥 = 푙푟 and 휃 = 푐표푠휃푖 + 푠푖푛휃푗 • Find the first and second derivatives of the position vector: 푑푥 푑휃 = 푙 휃 푑푡 푑푡 2 푑2푥 푑2휃 푑휃 = 푙 휃 − 푙 푟 푑푡2 푑푡2 푑푡 • From Newton’s Law, (neglecting frictional force) 푑2푥 푚 = 퐹 + 퐹 푑푡2 푔 푡 The Pendulum Problem (with some assumptions) Defining force of gravity as 퐹푔 = −푚푔푗 = 푚푔푐표푠휃푟 − 푚푔푠푖푛휃휃 and tension of the string as 퐹푡 = −푇푟 : 2 푑휃 −푚푙 = 푚푔푐표푠휃 − 푇 푑푡 푑2휃 푚푙 = −푚푔푠푖푛휃 푑푡2 Define 휔0 = 푔/푙 to find the solution: 푑2휃 푔 = − 푠푖푛휃 = −휔2푠푖푛휃 푑푡2 푙 0 Derivation of Equations of Motion • m = pendulum mass • mspring = spring mass • l = unstreatched spring length • k = spring constant • g = acceleration due to gravity • Ft = pre-tension of spring 푚푔−퐹 • r = static spring stretch, 푟 = 푡 s 푠 푘 • rd = dynamic spring stretch • r = total spring stretch 푟푠 + 푟푑 Derivation of Equations of Motion -
Huygens Probe Set to Detach from Cassini Orbiter Tonight 24 December 2004
Update: Huygens Probe Set to Detach From Cassini Orbiter Tonight 24 December 2004 mission is approximately $3 billion. Many of these sophisticated instruments are capable of multiple functions, and the data that they gather will be studied by scientists worldwide. Aerosol Collector and Pyrolyser (ACP) will collect aerosols for chemical-composition analysis. After extension of the sampling device, a pump will draw the atmosphere through filters which capture aerosols. Each sampling device can collect about 30 micrograms of material. Descent Imager/Spectral Radiometer (DISR) can take images and make spectral measurements using sensors covering a wide spectral range. A few hundred metres before impact, the instrument will switch on its lamp in order to acquire spectra of the surface material. The highlights of the first year of the Cassini- Doppler Wind Experiment (DWE) uses radio Huygens mission to Saturn can be broken into two signals to deduce atmospheric properties. The chapters: first, the arrival of the Cassini orbiter at probe drift caused by winds in Titan's atmosphere Saturn in June, and second, the release of the will induce a measurable Doppler shift in the carrier Huygens probe on Dec. 24, 2004, on a path signal. The swinging motion of the probe beneath toward Titan. (read PhysOrg story) its parachute and other radio-signal-perturbing effects, such as atmospheric attenuation, may also The Huygens probe, built and managed by the be detectable from the signal. European Space Agency (ESA), is bolted to Cassini and fed electrical power through an Gas Chromatograph and Mass Spectrometer umbilical cable. It has been riding along during the (GCMS) is a versatile gas chemical analyser nearly seven-year journey to Saturn largely in a designed to identify and quantify various "sleep" mode, awakened every six months for atmospheric constituents. -
Pioneers in Optics: Christiaan Huygens
Downloaded from Microscopy Pioneers https://www.cambridge.org/core Pioneers in Optics: Christiaan Huygens Eric Clark From the website Molecular Expressions created by the late Michael Davidson and now maintained by Eric Clark, National Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306 . IP address: [email protected] 170.106.33.22 Christiaan Huygens reliability and accuracy. The first watch using this principle (1629–1695) was finished in 1675, whereupon it was promptly presented , on Christiaan Huygens was a to his sponsor, King Louis XIV. 29 Sep 2021 at 16:11:10 brilliant Dutch mathematician, In 1681, Huygens returned to Holland where he began physicist, and astronomer who lived to construct optical lenses with extremely large focal lengths, during the seventeenth century, a which were eventually presented to the Royal Society of period sometimes referred to as the London, where they remain today. Continuing along this line Scientific Revolution. Huygens, a of work, Huygens perfected his skills in lens grinding and highly gifted theoretical and experi- subsequently invented the achromatic eyepiece that bears his , subject to the Cambridge Core terms of use, available at mental scientist, is best known name and is still in widespread use today. for his work on the theories of Huygens left Holland in 1689, and ventured to London centrifugal force, the wave theory of where he became acquainted with Sir Isaac Newton and began light, and the pendulum clock. to study Newton’s theories on classical physics. Although it At an early age, Huygens began seems Huygens was duly impressed with Newton’s work, he work in advanced mathematics was still very skeptical about any theory that did not explain by attempting to disprove several theories established by gravitation by mechanical means. -
A Phenomenology of Galileo's Experiments with Pendulums
BJHS, Page 1 of 35. f British Society for the History of Science 2009 doi:10.1017/S0007087409990033 A phenomenology of Galileo’s experiments with pendulums PAOLO PALMIERI* Abstract. The paper reports new findings about Galileo’s experiments with pendulums and discusses their significance in the context of Galileo’s writings. The methodology is based on a phenomenological approach to Galileo’s experiments, supported by computer modelling and close analysis of extant textual evidence. This methodology has allowed the author to shed light on some puzzles that Galileo’s experiments have created for scholars. The pendulum was crucial throughout Galileo’s career. Its properties, with which he was fascinated from very early in his career, especially concern time. A 1602 letter is the earliest surviving document in which Galileo discusses the hypothesis of pendulum isochronism.1 In this letter Galileo claims that all pendulums are isochronous, and that he has long been trying to demonstrate isochronism mechanically, but that so far he has been unable to succeed. From 1602 onwards Galileo referred to pendulum isochronism as an admirable property but failed to demonstrate it. The pendulum is the most open-ended of Galileo’s artefacts. After working on my reconstructed pendulums for some time, I became convinced that the pendulum had the potential to allow Galileo to break new ground. But I also realized that its elusive nature sometimes threatened to undermine the progress Galileo was making on other fronts. It is this ambivalent nature that, I thought, might prove invaluable in trying to understand crucial aspects of Galileo’s innovative methodology. -
Exomars Atmospheric Mars Entry and Landing Investigations and Analysis (AMELIA)
ExoMars Atmospheric Mars Entry and Landing Investigations and Analysis (AMELIA) F. Ferri1, F. Forget2, S.R. Lewis3, O. Karatekin4 and the International AMELIA team 1CISAS “G. Colombo”, University of Padova, Italy 2LMD, Paris, France 3The Open University, Milton Keynes, U.K. 4Royal Observatory of Belgium, Belgium [email protected] IPPW9 Toulouse, F 16-22 June 2012 ExoMars Entry, Descent and Landing Science F. Ferri & AMELIA team ESA ExoMars programme 2016-2018 The ExoMars programme is aimed at demonstrate a number of flight and in-situ enabling technologies necessary for future exploration missions, such as an international Mars Sample Return mission. Technological objectives: • Entry, descent and landing (EDL) of a payload on the surface of Mars; • Surface mobility with a Rover; • Access to the subsurface to acquire samples; and • Sample acquisition, preparation, distribution and analysis. Scientific investigations: • Search for signs of past and present life on Mars; • Investigate how the water and geochemical environment varies • Investigate Martian atmospheric trace gases and their sources. ESA ExoMars 2016 mission: Mars Orbiter and an Entry, Descent and Landing Demonstrator Module (EDM). ESA ExoMars 2018 mission: the PASTEUR rover carrying a drill and a suite of instruments dedicated to exobiology and geochemistry research IPPW9 Toulouse, F 16-22 June 2012 ExoMars Entry, Descent and Landing Science F. Ferri & AMELIA team EDLS measurements • Entry, Descent, Landing System (EDLS) of an atmospheric probe or lander requires mesurements in order to trigger and control autonomously the events of the descent sequence; to guarantee a safe landing. • These measurements could provide • the engineering assessment of the EDLS and • essential data for an accurate trajectory and attitude reconstruction • and atmospheric scientific investigations • EDLS phases are critical wrt mission achievement and imply development and validation of technologies linked to the environmental and aerodynamical conditions the vehicle will face. -
Sarlette Et Al
COMPARISON OF THE HUYGENS MISSION AND THE SM2 TEST FLIGHT FOR HUYGENS ATTITUDE RECONSTRUCTION(*) A. Sarlette(1), M. Pérez-Ayúcar, O. Witasse, J.-P. Lebreton Planetary Missions Division, Research and Scientific Support Department, ESTEC-ESA, Noordwijk, The Netherlands. Email: [email protected], [email protected], [email protected] (1) Stagiaire from February 1 to April 29; student at Liège University, Belgium. Email: [email protected] ABSTRACT 1. The SM2 probe characteristics The Huygens probe is the ESA’s main contribution to In agreement with its main purpose – performing a the Cassini/Huygens mission, carried out jointly by full system check of the Huygens descent sequence – NASA, ESA and ASI. It was designed to descend into the SM2 probe was a full scale model of the Huygens the atmosphere of Titan on January 14, 2005, probe, having the same inner and outer structure providing surface images and scientific data to study (except that the deploying booms of the HASI the ground and the atmosphere of Saturn’s largest instrument were not mounted on SM2), the same mass moon. and a similar balance. A complete description of the Huygens flight model system can be found in [1]. In the framework of the reconstruction of the probe’s motions during the descent based on the engineering All Descent Control SubSystem items (parachute data, additional information was needed to investigate system, mechanisms, pyro and command devices) were the attitude and an anomaly in the spin direction. provided according to expected flight standard; as the test flight was successful, only few differences actually Two years before the launch of the Cassini/Huygens exist at this level with respect to the Huygens probe. -
Computer-Aided Design and Kinematic Simulation of Huygens's
applied sciences Article Computer-Aided Design and Kinematic Simulation of Huygens’s Pendulum Clock Gloria Del Río-Cidoncha 1, José Ignacio Rojas-Sola 2,* and Francisco Javier González-Cabanes 3 1 Department of Engineering Graphics, University of Seville, 41092 Seville, Spain; [email protected] 2 Department of Engineering Graphics, Design, and Projects, University of Jaen, 23071 Jaen, Spain 3 University of Seville, 41092 Seville, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-953-212452 Received: 25 November 2019; Accepted: 9 January 2020; Published: 10 January 2020 Abstract: This article presents both the three-dimensional modelling of the isochronous pendulum clock and the simulation of its movement, as designed by the Dutch physicist, mathematician, and astronomer Christiaan Huygens, and published in 1673. This invention was chosen for this research not only due to the major technological advance that it represented as the first reliable meter of time, but also for its historical interest, since this timepiece embodied the theory of pendular movement enunciated by Huygens, which remains in force today. This 3D modelling is based on the information provided in the only plan of assembly found as an illustration in the book Horologium Oscillatorium, whereby each of its pieces has been sized and modelled, its final assembly has been carried out, and its operation has been correctly verified by means of CATIA V5 software. Likewise, the kinematic simulation of the pendulum has been carried out, following the approximation of the string by a simple chain of seven links as a composite pendulum. The results have demonstrated the exactitude of the clock. -
SETTING up and MOVING a PENDULUM CLOCK by Brian Loomes, UK
SETTING UP AND MOVING A PENDULUM CLOCK by Brian Loomes, UK oving a pendulum This problem may face clock with anchor the novice in two different Mescapement can ways. Firstly as a clock be difficult unless you have that runs well in its present a little guidance. Of all position but that you need these the longcase clock to move. Or as a clock is trickiest because the that is new to you and that long pendulum calls for you need to assemble greater care at setting it in and set going for the very balance, usually known as first time—such as one you have just inherited or bought at auction. If it is the first of these then you can attempt to ignore my notes about levelling. But floors in different rooms or different houses seldom agree Figure 1. When moving an on levels, and you may eight-day longcase clock you eventually have to follow need to hold the weight lines through the whole process in place by taping round the of setting the clock level accessible part of the barrel. In and in beat. a complicated musical clock, Sometimes you can such as this by Thomas Lister of persuade a clock to run by Halifax, it is vital. having it at a silly angle, or by pushing old pennies or wooden wedges under the seatboard. But this is hardly ideal and next time you move the clock you start with the same performance all over again. setting it ‘in beat’. These My suggestion is that you notes deal principally with bite the bullet right away longcase clocks. -
Based Observations of Titan During the Huygens Mission Olivier Witasse,1 Jean-Pierre Lebreton,1 Michael K
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111, E07S01, doi:10.1029/2005JE002640, 2006 Overview of the coordinated ground-based observations of Titan during the Huygens mission Olivier Witasse,1 Jean-Pierre Lebreton,1 Michael K. Bird,2 Robindro Dutta-Roy,2 William M. Folkner,3 Robert A. Preston,3 Sami W. Asmar,3 Leonid I. Gurvits,4 Sergei V. Pogrebenko,4 Ian M. Avruch,4 Robert M. Campbell,4 Hayley E. Bignall,4 Michael A. Garrett,4 Huib Jan van Langevelde,4 Stephen M. Parsley,4 Cormac Reynolds,4 Arpad Szomoru,4 John E. Reynolds,5 Chris J. Phillips,5 Robert J. Sault,5 Anastasios K. Tzioumis,5 Frank Ghigo,6 Glen Langston,6 Walter Brisken,7 Jonathan D. Romney,7 Ari Mujunen,8 Jouko Ritakari,8 Steven J. Tingay,9 Richard G. Dodson,10 C. G. M. van’t Klooster,11 Thierry Blancquaert,11 Athena Coustenis,12 Eric Gendron,12 Bruno Sicardy,12 Mathieu Hirtzig,12,13 David Luz,12,14 Alberto Negrao,12,14 Theodor Kostiuk,15 Timothy A. Livengood,16,15 Markus Hartung,17 Imke de Pater,18 Mate A´ da´mkovics,18 Ralph D. Lorenz,19 Henry Roe,20 Emily Schaller,20 Michael Brown,20 Antonin H. Bouchez,21 Chad A. Trujillo,22 Bonnie J. Buratti,3 Lise Caillault,23 Thierry Magin,23 Anne Bourdon,23 and Christophe Laux23 Received 17 November 2005; revised 29 March 2006; accepted 24 April 2006; published 27 July 2006. [1] Coordinated ground-based observations of Titan were performed around or during the Huygens atmospheric probe mission at Titan on 14 January 2005, connecting the momentary in situ observations by the probe with the synoptic coverage provided by continuing ground-based programs.