DOCSLIB.ORG

Use of Eiscat 3D for Observations of Space Debris

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Use of Eiscat 3D for Observations of Space Debris USE OF EISCAT 3D FOR OBSERVATIONS OF SPACE DEBRIS J. Vierinen(1), J. Markkanen(2), H. Krag(3), J. Siminski(3), and A. Mancas(3) (1)University of Tromsø, Hansine Hansens veg 18, 9019 Tromsø, Norway, Email: [email protected] (2)EISCAT Scientific Association, Sodankyla,¨ Finland (3)ESA/ESOC Space Debris Office, Darmstadt, Germany ABSTRACT large-aperture (HPLA) radar planned in Northern Fenno- Scandinavia. The primary mission for the radar is how- ever ionospheric science, which means that it the driver We investigate the capabilities of the next generation for the radar design was not space object observations. In ionospheric research radar EISCAT 3D (E3D) for ob- this study, we have set out to investigate in greater detail servations of space objects. The radar is multi-static, the utility of E3D for space object observations – both for and is therefore capable of observing instantaneous three- beam-park observations, and for surveillance purposes. dimensional vector velocity and position by observing We will use the current projected design of E3D as basis round-trip delay and Doppler shift between the transmit- of this study. ter and three receiver sites. The radar is to be located in Northern Scandinavia, which provides a high revisit- rate for high inclination objects. To model the perfor- mance of E3D for space object observations, we have in- To model the performance of E3D for space debris ob- cluded radar equation based analysis of object detectabil- servations, we have studied radar equation based de- ity as a function of range and size. To study the perfor- tectability and performed an error analysis for range and mance of the radar for orbital elements determination, we range-rate observables [MLHV02]. Because the radar have used a linearized error covariance analysis for ide- will be multi-static, it is capable of observing instanta- alized Keplerian elements. The analysis includes range neous three-dimensional vector velocities and positions and range-rate errors due to signal-to-noise and iono- by observing delay and Doppler shift between the trans- spheric radio propagation. To estimate the fraction of to- mitter and three or more receiver sites. We have included tal debris that can be observed with E3D, we have used linearized error estimates for Keplerian orbital elements the MASTER model [FGW+09]. E3D uses a relatively based on multi-static observations of targets. We have low VHF frequency (233 MHz), which experiences more intentionally chosen not to use a more complicated orbit radio wave propagation effects than more conventional model, in order to get a simple and intuitive first order higher frequency space surveillance radars. Our model- idea of orbit determination accuracy and repeat-rate of ing shows that ionospheric ray-bending and group delay observations. are severe enough that these effects need to be modeled in order to determine accurate orbital elements. As EIS- CAT 3D is an ionospheric research radar, there will be high quality ionospheric electron density measurements that can be utilized for radio propagation modeling. Our E3D uses a relatively low VHF frequency (233 MHz), simulations indicate that the radar can be used for ob- which experiences more radio wave propagation effects servations of orbital elements of objects down to 5 cm than more conventional higher frequency surveillance in diameter. It is therefore feasible that the radar could radars. Our modeling shows that ionospheric ray-bending provide to be a useful source of accurate information of and group delay are severe enough that these effects need orbital elements of space debris. to be addressed if one were to utilize E3D for accurate orbital elements determination. Without ionospheric cor- Keywords: Radar, Space object tracking. rections, measurements performed using E3D are unus- able for determining useful orbital elements. Because E3D is an ionospheric research radar, it can measure elec- tron density profiles at the same time with every measure- 1. INTRODUCTION ment of a space object, and this can be used to correct for propagation errors. In order to obtain a more realistic model of errors, we have estimated residual ionospheric Study of space debris is listed in the Science Case for propagation errors after ionospheric propagation correc- EISCAT 3D [MAA+15] as one of the application areas tions have been applied, and used these errors when esti- of the planned next generation multi-static high-power mating orbital elements determination performance. Proc. 7th European Conference on Space Debris, Darmstadt, Germany, 18–21 April 2017, published by the ESA Space Debris Office Ed. T. Flohrer & F. Schmitz, (http://spacedebris2017.sdo.esoc.esa.int, June 2017) 2. EISCAT 3D Site Gain Nbeams TX Trec Btx dBi K MHz Skibotn 43 100 yes 150 5 The EISCAT 3D radar (E3D) is a a multi-static HPLA Bergfors 43 100 no 150 ≤ radar, which is to be located in Northern Scandinavia. Karesuvanto 43 100 no 150 The radar uses phased array antennas for transmit and Jokkmokk 43 100 no 150 receive. The radar will initially have one transmit and Karesuvanto 43 100 no 150 receive site in Skibotn, Norway (69.340◦N, 20.313◦E). There will also be four receive-only sites: Karesuvanto (68.463◦N,22.458◦E), Bergfors (68.148◦N, 19.769◦E), Table 1. The performance parameters for E3D. Andoya Norway (69.251◦N, 16.096◦E) and Jokkmokk (66.60◦ N, 19.81◦ S). The technical description of the planned radar is described in detail in the technical note published by EISCAT [EIS14]. The planned locations are shown on a map in in Figure 1. The plan is to build E3D in two stages. In the first stage, a transmitter and receiver will be located in Skibotn. Re- ceivers will be located in Karesuvanto and Bergfors. In the next stage, two more receivers will be built: one in Andoya and one in Jokkmokk. Currently planned parameters for E3D are listed in the Table 1. In this table, Nbeams stands for the number of si- multaneous beams that can be formed, Trec stands for re- ceiver noise temperature, and Btx is transmit bandwidth. The transmit power Ptx is planned to be 10 MW. The radar will have a center frequency of 233 MHz. Each antenna array will consist of 9919 inverted-v dipole Figure 1. Planned E3D sites. The core site in Skibotn antenna elements similar to the PAVE PAWS design is colored with red. Receive sites are located in Karesu- [BPC+10], grouped under 91 antenna subgroups. The vanto, Bergfors, Andoya, and Jokkmokk are colored with array will be a planar array placed horizontally with the blue. on-axis direction pointing towards zenith. The antenna arrays have a full-width half maximum beam width of approximately 0.9◦. The antennas are steerable to 60◦ beams are then used by the second level beamformer to off zenith, with gain falling off approximately according form up to 100 dual polarization 30 MHz beams. On the to projected geometric area G(α) = G0 cos(α), where α core site, most of these beams will be utilized for inter- is the zenith angle. ferometry. On the receive sites, these beams will be used to form beams that intersect the transmitter beam across The Skibotn transmitter site will have interferometric out- all altitudes of interest. The number of beams is suffi- lier antennas, which are approximately 10 meters in di- cient to simultaneously cover all common volumes with ameter. The longest baseline will be approximately 1.5 the transmit beam. An example of a transmit beam inter- km. The main purpose of the spaced antenna interfer- secting with 100 receive beams is shown in Figure 2. ometer is to perform imaging of small-scale ionospheric structure in the E-region of the ionosphere during auroral energetic electron precipitation. The interferometry will The second level beamformer and the high level signal also have uses for measurements of trajectories of mi- processing will be implemented using a general purpose crometeoroids [KSN+12], and for planetary radar mea- high performance computing cluster. The e-infrastructure surements of the Moon [Tho87]. These interferometric for E3D is still in somewhat formative stages, and it is not antennas may also have utility when estimating the angle yet decided if the second level beamformed data will be of arrival of high signal-to-noise ratio space debris targets transmitted over a fast internet connection to a central su- with high precision. It is also possible to divide the core percomputer for analysis, or if the supercomputer will re- antenna into subarrays for interferometry. Figure 7 shows side at the location of the radar site. The most likely sce- an example of a mono-pulse type of an interferometer, nario is that the vast majority of computing will be per- achieved by dividing the core array into four quadrants. formed on-site and accomplished by having extra com- putational resources for analyzing and reducing the radar The antenna will have an all-digital beamformer, with a data before sending it to a central archive. The latter sce- direct sampling 105 MHz digitizer connected to two po- nario will allow low-latency analysis to be performed, al- larizations on each one of the 9919 antenna elements. A lowing that radar to quickly adapt to changing conditions. level one beamformer will reduce the bandwidth to 30 It is feasible that the radar will be able to allocate beams MHz and form up to 10 dual polarization beams on each in a sub-second timescale to follow transient targets, such one of the 91 element tiles. These first level wide angle as meteors, or space debris. Figure 2. Conceptual diagram of a 1◦ transmit beam in- Figure 3.
Load more

Recommended publications
  • Turbulent Heat Transfer and Pressure Drop
    Proceedings of the XI DINAME, 28th February-4th March, 2005 - Ouro Preto - MG - Brazil Edited by D.A. Rade and V. Steffen Jr.© 2005 - ABCM. All rights reserved. ANALYTICAL AND NUMERICAL APPROACHES TO STUDY THE GRAVITATIONAL CAPTURE IN THE FOUR-BODY PROBLEM Antonio Fernando Bertachini de Almeida Prado Instituto Nacional de Pesquisas Espaciais – INPE, São José dos Campos - SP - 12227-010 - Brazil [email protected] Abstract. The objective of this paper is to show a survey of results in the problem of gravitational capture in the bi- circular restricted four-body problem using numerical and analytical techniques. A gravitational capture occurs when a massless particle changes its two-body energy around one celestial body from positive to negative without the use of non-gravitational forces. In this paper it is mainly studied the effect of a fourth-body included in the dynamics. The Earth-Moon system with the inclusion of the Sun is used for the simulations. The results show the savings obtained in this more realistic model when compared with the more traditional restricted three-body problem model, as well as several examples of trajectories. Keywords: astrodynamics, gravitational capture, celestial mechanics 1. Introduction The application of the phenomenon of gravitational capture in spacecraft trajectories has been studied in details in the literature in the last few years. Among the first studies are the ones performed by Belbruno (1987, 1990, 1992); Krish (1991); Krish, Belbruno and Hollister (1992); Miller and Belbruno (1991); Belbruno and Miller (1990). They all studied missions in the Earth-Moon system that use this technique to save fuel during the insertion of the spacecraft in its final orbit around the Moon.
    [Show full text]
  • General Relativity 2020–2021 1 Overview
    N I V E R U S E I T H Y T PHYS11010: General Relativity 2020–2021 O H F G E R D John Peacock I N B U Room C20, Royal Observatory; [email protected] http://www.roe.ac.uk/japwww/teaching/gr.html Textbooks These notes are intended to be self-contained, but there are many excellent textbooks on the subject. The following are especially recommended for background reading: Hobson, Efstathiou & Lasenby (Cambridge): General Relativity: An introduction for Physi- • cists. This is fairly close in level and approach to this course. Ohanian & Ruffini (Cambridge): Gravitation and Spacetime (3rd edition). A similar level • to Hobson et al. with some interesting insights on the electromagnetic analogy. Cheng (Oxford): Relativity, Gravitation and Cosmology: A Basic Introduction. Not that • ‘basic’, but another good match to this course. D’Inverno (Oxford): Introducing Einstein’s Relativity. A more mathematical approach, • without being intimidating. Weinberg (Wiley): Gravitation and Cosmology. A classic advanced textbook with some • unique insights. Downplays the geometrical aspect of GR. Misner, Thorne & Wheeler (Princeton): Gravitation. The classic antiparticle to Weinberg: • heavily geometrical and full of deep insights. Rather overwhelming until you have a reason- able grasp of the material. It may also be useful to consult background reading on some mathematical aspects, especially tensors and the variational principle. Two good references for mathematical methods are: Riley, Hobson and Bence (Cambridge; RHB): Mathematical Methods for Physics and Engi- • neering Arfken (Academic Press): Mathematical Methods for Physicists • 1 Overview General Relativity (GR) has an unfortunate reputation as a difficult subject, going back to the early days when the media liked to claim that only three people in the world understood Einstein’s theory.
    [Show full text]
  • In-Depth Analysis of the Clustering of Dark Matter Particles Around Primordial Black Holes I: Density Profiles
    Prepared for submission to JCAP LUPM:21-008, LAPTH-019/21 In-depth analysis of the clustering of dark matter particles around primordial black holes I: density profiles a; a b Mathieu Boudaud, y Thomas Lacroix, Martin Stref, Julien Lavalle,c and Pierre Salatib aInstituto de F´ısicaTe´oricaUAM/CSIC, Universidad Aut´onomade Madrid, Ciudad Uni- versitaria de Cantoblanco, 28049 Madrid, Spain bUniversit´eGrenoble Alpes, USMB, CNRS, LAPTh, 9 chemin de Bellevue, Annecy-le-Vieux, F-74941 Annecy, France cLaboratoire Univers et Particules de Montpellier (LUPM), Universit´ede Montpellier & CNRS, Place Eug`eneBataillon, 34095 Montpellier Cedex 05, France yDeceased E-mail: [email protected], [email protected], [email protected], [email protected] Abstract. Primordial black holes may have been produced in the early stages of the thermal history of the Universe after cosmic inflation. If so, dark matter in the form of elementary particles can be subsequently accreted around these objects, in particular when it gets non- relativistic and further streams freely in the primordial plasma. A dark matter mini-spike builds up gradually around each black hole, with density orders of magnitude larger than the cosmological one. We improve upon previous work by carefully inspecting the computation of the mini-spike radial profile as a function of black hole mass, dark matter particle mass and temperature of kinetic decoupling. We identify a phase-space contribution that has been overlooked and that leads to changes in the final results. We also derive complementary an- alytical formulae using convenient asymptotic regimes, which allows us to bring out peculiar power-law behaviors for which we provide tentative physical explanations.
    [Show full text]
  • Arxiv:1702.00030V1 [Physics.Space-Ph] 28 Jan 2017 O Ceeainrceswudb Icl.Hg Ceeain,O the on Accelerations, High Difficult
    The optimisation of low-acceleration interstellar relativistic rocket trajectories using genetic algorithms Kenneth K H Fung ✩,a,b, Geraint F Lewis a, Xiaofeng Wu b a Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney, NSW 2006, Australia b School of Aerospace, Mechanical and Mechatronic Engineering, J07, The University of Sydney, NSW 2006, Australia Abstract A vast wealth of literature exists on the topic of rocket trajectory optimisation, particularly in the area of interplanetary trajectories due to its relevance to- day. Studies on optimising interstellar and intergalactic trajectories are usually performed in flat spacetime using an analytical approach, with very little focus on optimising interstellar trajectories in a general relativistic framework. This paper examines the use of low-acceleration rockets to reach galactic destinations in the least possible time, with a genetic algorithm being employed for the opti- misation process. The fuel required for each journey was calculated for various types of propulsion systems to determine the viability of low-acceleration rock- ets to colonise the Milky Way. The results showed that to limit the amount of fuel carried on board, an antimatter propulsion system would likely be the minimum technological requirement to reach star systems tens of thousands of light years away. However, using a low-acceleration rocket would require sev- eral hundreds of thousands of years to reach these star systems, with minimal time dilation effects since maximum velocities only reached about 0.2c. Such arXiv:1702.00030v1 [physics.space-ph] 28 Jan 2017 transit times are clearly impractical, and thus, any kind of colonisation using low acceleration rockets would be difficult.
    [Show full text]
  • M. Sc. Dissertation
    Gda´nsk University of Technology Faculty of Electronics, Telecommunications and Informatics Department: Katedra System´owGeoinformatycznych Author: Tomasz Mrugalski Studies type: M. Sc. Studies M. Sc. Dissertation Dissertation topic: Trajectory determination and orbital maneuvers in Earth vicinity Polish title: Wyznaczanie trajektorii i manewry orbitalne w otoczeniu Ziemi Supervisor: prof. dr hab. in_z. Cezary Specht Technical Supervisor: dr in_z. Mariusz Specht Gda´nsk, 2020-09-27 Contents 1. Introduction 2 1.1. Astrodynamics.....................................................2 1.2. Thesis overview and goals...............................................2 1.3. History of orbital mechanics..............................................3 1.3.1. Antiquity....................................................3 1.3.2. Middle Ages..................................................4 1.3.3. Age of Experiments..............................................5 1.4. Orbital elements....................................................6 1.5. Orbit classification by shape.............................................. 12 1.6. Orbit classification by altitude............................................ 12 1.7. Orbit classification by inclination........................................... 13 1.8. Special purpose orbits................................................. 14 1.9. Lagrangian points and exotic orbits.......................................... 15 1.10. Reference systems................................................... 16 1.11. Popular orbital notations..............................................
    [Show full text]
  • INPE-11281-PRE/6718 SOME ANALYTICAL RESULTS in the PROBLEM of GRAVITATIONAL CAPTURE Antonio Fernando Bertachini De Almeida Prado
    INPE-11281-PRE/6718 SOME ANALYTICAL RESULTS IN THE PROBLEM OF GRAVITATIONAL CAPTURE Antonio Fernando Bertachini de Almeida Prado ADVANCES IN SPACE DYNAMICS 4: CELESTIAL MECHANICS AND ASTRONAUTICS, H. K. Kuga, Editor, 27-36 (2004). Instituto Nacional de Pesquisas Espaciais – INPE, São José dos Campos, SP, Brazil. ISBN 85-17-00012-9 INPE São José dos Campos 2004 ADVANCES IN SPACE DYNAMICS 4: CELESTIAL MECHANICS AND ASTRONAUTICS, H. K. Kuga, Editor, 27-36 (2004). Instituto Nacional de Pesquisas Espaciais – INPE, São José dos Campos, SP, Brazil. ISBN 85-17-00012-9 SOME ANALYTICAL RESULTS IN THE PROBLEM OF GRAVITATIONAL CAPTURE Antonio Fernando Bertachini de Almeida Prado Instituto Nacional de Pesquisas Espaciais - São José dos Campos-SP-12227-010-Brazil Phone (55-12)345-6201 - Fax (55-12)345-6226 - E-mail: [email protected] ABSTRACT The objective of the present paper is to study the possibility of using ballistic gravitational capture to maneuver a spacecraft to an orbit close to the largest primary of a binary system. The most practical application is to make a transfer from the Moon to the Earth using this technique. An analytical study is performed to evaluate the magnitude of the forces involved in the ballistic gravitational capture in a trajectory going to the largest primary as a function of time. Then, this phenomenon is explained in terms of the integration of the perturbing forces with respect to time. The relation between those integrals and the reduction of the two-body energy with respect to the Earth is derived. Analytical equations for those forces are derived to estimate their magnitude and to show the best directions of approach for the ballistic gravitational capture.
    [Show full text]
  • Transition from Adiabatic Inspiral to Geodesic Plunge for a Compact Object Around a Massive Kerr Black Hole: Generic Orbits
    Transition from adiabatic inspiral to geodesic plunge for a compact object around a massive Kerr black hole: Generic orbits The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Sundararajan, Pranesh A. “Transition from adiabatic inspiral to geodesic plunge for a compact object around a massive Kerr black hole: Generic orbits.” Physical Review D 77.12 (2008): 124050. (C) 2010 The American Physical Society. As Published http://dx.doi.org/10.1103/PhysRevD.77.124050 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/51333 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 77, 124050 (2008) Transition from adiabatic inspiral to geodesic plunge for a compact object around a massive Kerr black hole: Generic orbits Pranesh A. Sundararajan Department of Physics and MIT Kavli Institute, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA (Received 31 March 2008; published 27 June 2008) The inspiral of a stellar mass compact object falling into a massive Kerr black hole can be broken into three different regimes: An adiabatic inspiral phase, where the inspiral time scale is much larger than the orbital period; a late-time radial infall, which can be approximated as a plunging geodesic; and a regime where the body transitions from the inspiral to plunge. In earlier work, Ori and Thorne have outlined a method to compute the trajectory during this transition for a compact object in a circular, equatorial orbit.
    [Show full text]
  • Tidal Disruption of Inclined Or Eccentric Binaries by Massive Black Holes
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by LJMU Research Online MNRAS 477, 5682–5691 (2018) doi:10.1093/mnras/sty1069 Advance Access publication 2018 April 27 Tidal disruption of inclined or eccentric binaries by massive black holes Harriet Brown,1‹ Shiho Kobayashi,1 Elena M. Rossi2 and Re’em Sari3 1Astrophysics Research Institute, LJMU, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK 2Leiden University, Oort Building, Niels Bohrweg 2, NL-2333 CA Leiden, the Netherlands 3Racah Institute of Physics, Hebrew University, Jerusalem 91904, Israel Accepted 2018 April 24. Received 2018 April 2; in original form 2017 August 25 ABSTRACT Binary stars that are on close orbits around massive black holes (MBHs) such as Sgr A∗ in the centre of the Milky Way are liable to undergo tidal disruption and eject a hypervelocity star. We study the interaction between such an MBH and circular binaries for general binary orientations and penetration depths (i.e. binaries penetrate into the tidal radius around the BH). We show that for very deep penetrators, almost all binaries are disrupted when the binary rotation axis is roughly oriented towards the BH or it is in the opposite direction. The surviving chance becomes significant when the angle between the binary rotation axis and the BH direction is between 0.15π and 0.85π. The surviving chance is as high as ∼20 per cent when the binary rotation axis is perpendicular to the BH direction. However, for shallow penetrators, the highest disruption chance is found in such a perpendicular case, especially in the prograde case.
    [Show full text]
  • Phys 326 Discussion 7 – Kepler Orbits
    Phys 326 Discussion 7 – Kepler Orbits ! ! m2 ! ! ! m1 ! m1m2 • Coordinates & Reduced Mass : r1 = R + r , r2 = R − r , µ = M M M ! L2 L2 ˆ • Centrifugal force & PE : Fcf = 3 r , Ucf = 2 , effective U* = U +Ucf µr 2µr L 1 • Angular EOM : φ! = • Radial EOMs : µ!r! = F(r) + F (r) , E = T + U(r) = µr!2 + U (r) + U(r) 2 cf cf µr 2 µ F(1/ u) µr! • Path Equation : u(φ) ≡ 1/ r(φ) → u′′ + u = − 2 2 and u′ = − L u L 1 a c a2 ∓ b2 • Ellipse / Hyperbola with (r,φ) centered on a focal point : = (1+ ecosφ) , e = = r b2 a a 2 2 2 2 ! γ r0 L b γ γ (e − 1) 2 4π µ 3 • Kepler Orbits for F = − 2 rˆ : r(φ) = with r0 = = , E = ∓ = , τ = a r 1+ ecosφ µγ a 2a 2r0 γ 2r r rmax − rmin Also, the r(φ) formula in the last line immediately gives r = m in m a x and e = . 0 r r rmin + rmax max + min Problem 1 : Satellites Checkpoints for 1 & 2 To keep our mess of formulae under control, realize that it takes exactly two parameters to define the shape of an ellipse ... there are just many choices for those shape parameters: a, b, c, e, rmin, rmax, r0. When the ellipse is a Kepler orbit, i.e. the solution of a 2-body system with a 1/r2 central force, more choices appear: energy E, angular momentum L, period τ, the velocities at the apses, … but once the reduced mass μ and force constant γ are specified (they define the Kepler problem), there are still only two independent parameters for the solution.
    [Show full text]
  • Some New Aspects of First Integrals and Symmetries for Central Force
    SOME NEW ASPECTS OF FIRST INTEGRALS AND SYMMETRIES FOR CENTRAL FORCE DYNAMICS STEPHEN C. ANCO1, TYLER MEADOWS2,1 VINCENT PASCUZZI3,1 1 department of mathematics and statistics, brock university st. catharines, on l2s3a1, canada 2 department of mathematics and statistics, mcmaster university hamilton, on l8s 4k1, canada 3 department of physics, university of toronto toronto, on m5s 1a7, canada Abstract. For the general central force equations of motion in n> 1 dimensions, a com- plete set of 2n first integrals is derived in an explicit algorithmic way without the use of dynamical symmetries or Noether’s theorem. The derivation uses the polar formulation of the equations of motion and yields energy, angular momentum, a generalized Laplace- Runge-Lenz vector, and a temporal quantity involving the time variable explicitly. A variant of the general Laplace-Runge-Lenz vector, which generalizes Hamilton’s eccentricity vector, is also obtained. The physical meaning of the general Laplace-Runge-Lenz vector, its vari- ant, and the temporal quantity are discussed for general central forces. Their properties are compared for precessing bounded trajectories versus non-precessing bounded trajectories, as well as unbounded trajectories, by considering an inverse-square force (Kepler problem) and a cubically perturbed inverse-square force (Newtonian revolving orbit problem). 1. Introduction Classical central force dynamics have been long studied from both physical and mathemat- ical viewpoints. On one hand, many basic physical models are described by central forces, e.g. planetary motion and Coulomb scattering, which have inverse-square radial forces; vi- brations of atoms in a crystal, which have linear radial forces; interactions between a pair of neutral atoms or molecules, which have complicated nonlinear radial forces.
    [Show full text]
  • Using Perturbation Theory to Understand the Two Body Problem in General Relativity Pranesh Adhyam Sundararajan AUG 0 4 2009
    Using perturbation theory to understand the two body problem in general relativity by Pranesh Adhyam Sundararajan M. Sc. (Hons) Physics, B. E. (Hons) Electrical & Electronics Engineering Birla Institute of Technology & Science, Pilani, India (2005) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2009 @ Massachusetts Institute of Technology 2009. All rights reserved. A uthor ........................................ ...................... Department of Physics -e 30, 2009 Certified by ............................... .S... Hu h ScttScottA. Hughes Associate Professor of Physics Thesis Supervisor Accepted by............. ............ omas J. Greytak MASSACHUSETTS INSTIJTE rofessor of Physics OF TECHNOLOGY Associate Departmen Head for Education AUG 0 4 2009 ARCHIVES LIBRARIES Using perturbation theory to understand the two body problem in general relativity by Pranesh Adhyam Sundararajan Submitted to the Department of Physics on June 30, 2009, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract Binary systems composed of compact objects (neutron stars and black holes) radi- ate gravitational waves (GWs). The prospect of detecting these GWs using ground and space based experiments has made it imperative to understand the dynamics of such compact binaries. This work describes several advances in our ability to model compact binaries and extract the rich science they encode. A major part of this dissertation focuses on the subset of binaries composed of a massive, central black hole (105 - 10SM®) and a much smaller compact object (1 - 100M®). The emission of gravitational energy from such extreme mass ratio inspirals (EMRIs) forces the separation between the two components to shrink, leading to their merger.
    [Show full text]
  • Particles of Lunar Origin. Part 1-Determination of Elements Of
    General Disclaimer One or more of the Following Statements may affect this Document This document has been reproduced from the best copy furnished by the organizational source. It is being released in the interest of making available as much information as possible. This document may contain data, which exceeds the sheet parameters. It was furnished in this condition by the organizational source and is the best copy available. This document may contain tone-on-tone or color graphs, charts and/or pictures, which have been reproduced in black and white. This document is paginated as submitted by the original source. Portions of this document are not fully legible due to the historical nature of some of the material. However, it is the best reproduction available from the original submission. Produced by the NASA Center for Aerospace Information (CASI) National Aeronautics and Space Administration Goddard Space Flight Center Contract No.NAS-5-12487 ST —CM —LS —IM —10857 PARTICLES OF LUNAR ORIGIN 1. DETERMINATION OF ELEMENTS CF GEOCENTRIC ORBITS OF LUNAR PARITICLES IN THE SPATIAL CASE BY V. P. Orlov (USSR) FN'I 1 6 9 - (THRUI -(ACCES9)0_N NUMBER) 0 jC009) CR OR ?MX OR AS) NUMBER) -7aX8A 4 6 AUGUST *1969`- 0 • ST-- CM—LS—IM— 10857 PARTICLES OF LUNAR ORIGIN I. DETERMINATION OF ELEMENTS OF GEOCENTRIC ORBITS OF LUNAR PARTICLES IN THE SPATIAL CASE Astronomicheskiy Vestnik, by V. P. Orlov .: Tom"3, No.2,-te r, 76-81, Astronomical Observatory Jzdatel'stvo "NAUKA", 1969 of the Odessa University 3 Al ABSTRACT Elements are derived of geocentric orbits of par- ;, ticles ejected from the Moon by shock explosions of meteo- rites as a function of primary conditions of ejection: selenocentric longitude as and latitude ^o', initial velo- city Vo, azimuth Ao and the zenithal distance 6 0 of the di- rection of ejection.
    [Show full text]