DOCSLIB.ORG

General Relativity Fall 2019 Lecture 20: Geodesics of Schwarzschild

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
General Relativity Fall 2019 Lecture 20: Geodesics of Schwarzschild General Relativity Fall 2019 Lecture 20: Geodesics of Schwarzschild Yacine Ali-Ha¨ımoud November 7, 2019 In this lecture we study geodesics in the vacuum Schwarzschild metric, at r > 2M. Last lecture we derived the following equations for timelike geodesics in the equatorial plane (θ = π=2): d' L 1 dr 2 M L2 ML2 = ; + Veff (r) = ;Veff (r) + ; (1) dτ r2 2 dτ E ≡ − r 2r2 − r3 where (E2 1)=2 can be interpreted as a kinetic + potential energy per unit mass. The radial equation can also be rewrittenE ≡ as− d2r M 3 h i = V 0 (r) = r~2 L~2r~ + 3L~2 ; r~ r=M; L~ = L=M: (2) dτ 2 − eff − r4 − ≡ CIRCULAR ORBITS AND THE ISCO We show the effective potential in Fig. 1. In contrast to the Newtonian effective potential for orbits around a central 2 2 2 3 2 mass (i.e. Veff M=r + L =2r , without the last term ML =r ), which always has a minimum at rNewt = L =M, ≡ − − the relativistic effective potential has both a maximum and a minimun for L > p12 M, an inflection point for L = p12 M, and is strictly monotonic for L < p12 M. 0 Circular orbits (with r = constant) are such that Veff (r) = 0. Solving this equation, one finds that such orbits exist only for L > p12 M. When this condition is satisfied, the radii of circular orbits are L2 p r± = 1 1 12M 2=L2 : (3) c 2M ± − The Newtonian limit is obtained for L M, in which case r+ L2=M. c ! 2 2 From classical mechanics, we know that circular orbits where d Veff =dr > 0 are stable, whereas those with 2 2 d Veff =dr < 0 are unstable. This can simply be understood graphically: for a given energy, an orbit is confined to the interval in radii for which Veff (r) . At a circular orbit, this is an exact equality. Perturb the orbit away by ≤ E 2 2 2 2 giving it a bit of extra energy, and it will start oscillating between two radii if d Veff =dr > 0. If d Veff =dr 0, it will either plunge towards r 2M or diverge to infinity. ≤ ! The smallest radius of a stable circular orbit is obtained for L = p12 M, and has value rc = 6M { strictly speaking, only orbits with rc > 6M are actually stable. This orbit is called the innermost stable circular orbit (ISCO). The energy associated with a circular orbit at the ISCO is given by 1 p p isco = Veff (6M) = ; Eisco = 1 + 2 isco = 8=9 0:94: (4) E −18 ) E ≈ This number is often invoked to quantify the efficiency of accretion around Schwarzschild black holes: suppose material is orbiting around a central mass (typically a black hole) on a circular orbit, and slowly looses energy and angular momentum due to dissipative forces, while remaining on a quasi-circular orbit. If it starts at large distances with E 1, it will lose about 6% of its energy to dissipation before reaching the ISCO and plunging towards r 2M. Thus if≈ the rate of mass-energy accretion is M_ , the luminosity radiated by dissipation is at most 0:06 M_ .! This is usually quoted as a maximal efficiency of 6% for accretion onto Schwarzschild black holes. Spinning black holes (which we will explore in the last classes) can∼ be much more efficient. 2 ��� L/M = Newtonian effective potential 5 ��� circular orbits ) � 4.2 ( ��� ��� � 2p3 -��� 3 -��� � �� �� ��� �/� p FIG. 1. Effective potential for orbits in the Schwarzschild metric. The numbers label the value of L=M. For L=M < p12, the potential has no extremum, and there are no circular orbits { in fact, all orbits are plunging to r ! 2M. For L=M > 12, the effective potential has a local maximum, correspondingp to an unstable circular orbit, and a local minimum, corresponding to a stable circular orbit. The two merge together at L = 12M, for which the marginally stable circular orbit is at r = 6M. This is the innermost stable circular obit (ISCO). PRECESSION OF PERICENTER In this section we consider bound orbits, i.e. for which < 0, and the coordinate r is bound within a range E rmin r rmax, called the pericenter and apocenter, respectively. We adjust the polar angle ' such that, at some ≤ ≤ arbitrary pericenter passage, r(' = 0) = rmin, and dr/dτ(' = 0) = 0. Note that the next pericenter passage need not be at ' = 2π if the orbit does not close, as we will see now! From the angular equation, we have d/dτ = (L=r2)d=d', so the radial equation can be rewritten as 1 L2 dr 2 + Veff (r) = : (5) 2 r4 d' E Let us define the variable u L2=(Mr). It satisfies the following differential equation: ≡ 1 du 2 1 1 2L2 u + u2 (M=L)2u3 = (1 e2); 1 e2 (6) 2 d' − 2 − −2 − − ≡ − M 2 E Differentiating again, we get d2u + u 1 = 3(M=L)2u2: (7) d'2 − Recall that we have adjusted ' such that dr/dτ = 0 at ' = 0, hence du=d' = 0 at ' = 0. We will solve this equation perturbatively in (M=L)2 1, corresponding to the Newtonian limit: 2 u(') = u0(') + (M=L) u1(') + ::: (8) Note that since (L=M)2 (consider, for instance, circular orbits), the quantity 1 e2 is of order unity. The lowest-order termE satisfies ∼ − 2 d u0 + u0 = 1: (9) d'2 3 This has a simple solution: u0(') = 1 + A cos ', where we have adjusted the origin of ' such that du0=d' = 0 at ' = 0. To find the integration constant A, we plug into 2 1 du0 1 2 1 2 u0 + u = (1 e ); (10) 2 d' − 2 0 −2 − implying A2 = e2. For definiteness, we pick A = +e, corresponding to L > 0. The zero-the order solution is therefore the well-known ellipse, L2=M r0(') = ; (11) 1 + e cos ' where e is the eccentricity. This orbit happens to be periodic, which is a unique feature of the 1=r potential. Let us now find the next-order correction: 2 d u1 2 du1 + u1(') = 3u ('); (' = 0) = 0: (12) d'2 0 d' The solution is 2 2 2 2 u1(') = (3 + e ) + (C 3 e ) cos ' + e sin ' + 3e ' sin '; (13) − − where C u1(0) is an integration constant, which we won't need to calculate explicitly for the problem of pericenter precession,≡ but is easy to find from Eq. (6): C = (1 + e)3=e. Obviously this solution cannot be correct for arbitrary ', because of the growing term ' sin '. This is because deviations away from a Keplerian orbit accumulate over time. So we should only consider this perturbative solution to be valid over one orbital period 0 ' 2π { after that, one can reset the origin of ' and reiterate the perturbative expansion. ≤ ≤ We want to know at what angle the next pericenter passage will be. In the Kepler problem, it is at ' = 2π, but, because of the additional term in the effective potential, the next pericenter passage will be at ' = 2π + ∆', where we expect ∆' (M=L)2. The pericenter passage is defined by dr=d' = 0, hence u0(') = 0. Thus we must solve for ∼ 0 2 0 u0(2π + ∆') + (M=L) u1(2π + ∆') = 0: (14) We Taylor-expand this around ∆' = 0, and find, to lowest order, 0 2 u1(2π) 2 ∆' = (M=L) 00 = 6π(M=L) : (15) − u0 (2π) Thus we conclude that the pericenter (as well as apocenter) precesses by ∆' = 6π(M=L)2 per orbit. This is usually expressed in terms of the semi-major axis a and eccentricity. The semi-major axis is a = (rmax + 2 2 2 2 rmin)=2 = L =M=(1 e ), i.e. L = Ma(1 e ), thus − − 6πM ∆' = : (16) a(1 e2) − We now apply this to the orbit of Mercury around the Sun, which was one of the earliest successfull tests of 12 GR. Here we have M = M , aM 6 10 cm, e 0:2. This leads to a precession rate of ∆' 0:1 arcsec per orbital period, hence Ω = d∆'=dt ≈ 43× arcsec/century.≈ ≈ It is interesting to compare this to≈ the effect of Jupiter on the orbit. The tidal acceleration due to Jupiter is of order 3 MJaM=aJ. The amplitude of this perturbed acceleration relative to the relativistic correction is therefore of order 3 3 δfJupiter MJaM=aJ aM MJ aM −6 aM 2 3 10 10; (17) δfGR ∼ M =aM ∼ M M aJ ∼ M ∼ −3 13 where we used MJ 10 M and aJ 7 10 cm 10 aM. Therefore we see that the relativistic precession is a factor of 10 times∼ weaker than the∼ precession× due∼ to Jupiter's tidal force. To be aware of this tiny \anomalous precession"∼ in the early 20-th century, physicisists therefore had to model the effect of Jupiter (and Solar-sytem planets) to exquisite precision! 4 RADIAL ORBITS Consider a massive particle that is radially infalling, so that L = 0, and dr/dτ < 0. For L = 0 the radial geodesic equation reduces to dr p (dr/dτ)2 2M=r = E2 1 = E2 + 2M=r 1: (18) − − ) dτ − − The proper time elapsed between two radii ri > rf is Z ri dr Z ri=2M dx Z ri=2M px dx ∆τ = = 2M = 2M ; (19) p 2 p 2 2 r E + 2M=r 1 r =2M E + 1=x 1 r =2M p1 x + xE f − f − f − 2 2 For E > 0, this integral is clearly convergent, for any rf 2M and even for E , it converges and remains finite ≥ ! as rf 2M.
Load more

Recommended publications
  • Effective Potential
    Murrary-Clay Group Notes By: John McCann Effective Potential Consider a three-body system with m3 << m2 ≤ m1, from this point narratored with m1 as a star, m2 as a planet and m3 as a small satellite. We shall use a rotating non-inertial coordinate system, which rotates about the barycenter but with the origin centered on the planet. Oriented such that the barycenter falls along the x{axis, in the x > 0 half, and the axis of rotation is parallel to the z{axis. Rewrite, m1 ≡ M∗ as the mass of the star, and m2 ≡ MP as the mass of the planet. Define ~a as the vector from center of the planet to the center of the star, ~` as the vector from the center of the planet to the barycenter, and ~r? ≡ ~ρ, as projection of the vector from the center of the planet to a given point into the plane normal to the axis of rotation (such given point denoted as ~r). ^ To be succinct, ~r? = j~rj sin(θ) sin(θ)^r + cos(θ)θ = ρρ^, where the angle is the usual spherical coordinate definition and ρ is the standard cylindrical coordinate, as used by physicist. We chose to define this last vector, since it is the relevant distance for determining the centrifugal potential, along with ` and Ω. The effective potential per unit mass, u, for a tertiary object in a planet-star system is GM GM 1 u (~r) = − P − ∗ − Ω2j~r − ~`j2: (1) eff j~rj j~a − ~rj 2 ? Respectively the terms are Newton's gravitational potential from the planet (thus defining G as Newton's gravitational constant), the gravitational potential from the star and the centrifugal potential from an object moving about the barycenter with angular frequency Ω.
    [Show full text]
  • Quantum Interferometric Visibility As a Witness of General Relativistic Proper Time
    ARTICLE Received 13 Jun 2011 | Accepted 5 Sep 2011 | Published 18 Oct 2011 DOI: 10.1038/ncomms1498 Quantum interferometric visibility as a witness of general relativistic proper time Magdalena Zych1, Fabio Costa1, Igor Pikovski1 & Cˇaslav Brukner1,2 Current attempts to probe general relativistic effects in quantum mechanics focus on precision measurements of phase shifts in matter–wave interferometry. Yet, phase shifts can always be explained as arising because of an Aharonov–Bohm effect, where a particle in a flat space–time is subject to an effective potential. Here we propose a quantum effect that cannot be explained without the general relativistic notion of proper time. We consider interference of a ‘clock’—a particle with evolving internal degrees of freedom—that will not only display a phase shift, but also reduce the visibility of the interference pattern. According to general relativity, proper time flows at different rates in different regions of space–time. Therefore, because of quantum complementarity, the visibility will drop to the extent to which the path information becomes available from reading out the proper time from the ‘clock’. Such a gravitationally induced decoherence would provide the first test of the genuine general relativistic notion of proper time in quantum mechanics. 1 Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria. 2 Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. Correspondence and requests for materials should be addressed to M.Z. (email: [email protected]). NATURE COMMUNICATIONS | 2:505 | DOI: 10.1038/ncomms1498 | www.nature.com/naturecommunications © 2011 Macmillan Publishers Limited.
    [Show full text]
  • Assignment Week 5 Introduction
    MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics 8.224 Exploring Black Holes General Relativity and Astrophysics Spring 2003 ASSIGNMENT WEEK 5 NOTE: Exercises 6 through 8 are to be carried out using the GRorbits program, programmed in JAVA, available as a compressed file on the Assignments page. Download the zip file, decompress it and click on the icon GRorbits.html. If that does not work, click on the icon GRorbitsConverted.html. INTRODUCTION Now we get to the heavy lifting in general relativity! By this time we are accustomed to the surprising predictions of the Schwarzschild metric, have learned to change coordinate systems the way we change clothes, and easily use total energy as a constant of the motion to describe a stone plunging into a black hole. The topic for this week: Orbiting satellites. For orbits, there are two constants of the motion: (1) energy-measured-at-infinity, which is given by the same expression as for radial plunge, and (2) angular momentum, which is the same expression as in Newtonian mechanics, except the Newtonian universal time increment dt is replaced by the proper time increment dτ, that is, the time read on the wristwatch of the orbiting stone. What is difficult about this chapter? For many people the greatest difficulty is manipulating angular momentum, whether Newtonian or relativistic. For both theories, the angular momentum is just the radius to the particle multiplied by the particle’s component of linear momentum perpendicular to this radius. There is a different expression for linear momentum in the two cases: mds/dt for Newton, mds/dτ for Einstein.
    [Show full text]
  • 4. Central Forces
    4. Central Forces In this section we will study the three-dimensional motion of a particle in a central force potential. Such a system obeys the equation of motion mx¨ = V (r)(4.1) r where the potential depends only on r = x .Sincebothgravitationalandelectrostatic | | forces are of this form, solutions to this equation contain some of the most important results in classical physics. Our first line of attack in solving (4.1)istouseangularmomentum.Recallthatthis is defined as L = mx x˙ ⇥ We already saw in Section 2.2.2 that angular momentum is conserved in a central potential. The proof is straightforward: dL = mx x¨ = x V =0 dt ⇥ − ⇥r where the final equality follows because V is parallel to x. r The conservation of angular momentum has an important consequence: all motion takes place in a plane. This follows because L is a fixed, unchanging vector which, by construction, obeys L x =0 · So the position of the particle always lies in a plane perpendicular to L.Bythesame argument, L x˙ =0sothevelocityoftheparticlealsoliesinthesameplane.Inthis · way the three-dimensional dynamics is reduced to dynamics on a plane. 4.1 Polar Coordinates in the Plane We’ve learned that the motion lies in a plane. It will turn out to be much easier if we work with polar coordinates on the plane rather than Cartesian coordinates. For this reason, we take a brief detour to explain some relevant aspects of polar coordinates. To start, we rotate our coordinate system so that the angular momentum points in the z-direction and all motion takes place in the (x, y)plane.Wethendefinetheusual polar coordinates x = r cos ✓, y= r sin ✓ –48– Our goal is to express both the velocity and acceleration y ^ ^ θ r in polar coordinates.
    [Show full text]
  • Assignment 5 Solutions
    Physics 106a: Assignment 5 (Solutions prepared by Alvin Li) 8 November, 2019; due 5pm Friday, 15 November in the \Ph106 In Box" in East Bridge mailbox. One Dimensional Systems: Central Forces and the Kepler Problem Reading: Hand and Finch Chapter 4. Problems 1. Binary-sun solar system: Consider a binary pair of identical suns of mass M orbiting in the x − y plane in an orbit centered at the origin. The gravitational constant is G. Now add a planet of small mass m with an initial condition on the z axis above the center of mass of the two suns and with a velocity in the z direction; by the symmetry of the system the small mass will remain on the z axis and suns will have equal z coordinates and their center of mass will also remain on the z axis. We can choose as coordinates describing the dynamics: the z coordinate of the planet z; the z coordinate of the two suns Z; and the polar coordinates (R; θ) giving the x; y coordinates of the suns (±R cos θ; ±R sin θ) | see figure. m z M Z R M R e (a) What is the Lagrangian of the system in terms of these coordinates and their time derivatives? [Solution] For each of the large suns i, the kinetic energy is 1 1 1 T = MR_ 2 + MZ_ 2 + MR2θ_2; (1) Mi 2 2 2 and for the small planet, the kinetic energy is 1 T = mz_2: (2) m 2 The potential energy comes from three contributions: The gravitational potential between M1 and M2 (separated by a distance of 2R), that between M1 and m (separated by a distance of p 2 2 p 2 2 R + (z − Z) ), and lastly that between M2 and m (separated by a distance of R + (z − Z) ).
    [Show full text]
  • Relativity in the Global Positioning System
    Relativity in the Global Positioning System Neil Ashby Dept. of Physics, University of Colorado Boulder, CO 80309–0390 U.S.A. [email protected] http://physics.colorado.edu/faculty/ashby_n.html Published on 28 January 2003 www.livingreviews.org/Articles/Volume6/2003-1ashby Living Reviews in Relativity Published by the Max Planck Institute for Gravitational Physics Albert Einstein Institute, Germany Abstract The Global Positioning System (GPS) uses accurate, stable atomic clocks in satellites and on the ground to provide world-wide position and time determination. These clocks have gravitational and motional fre- quency shifts which are so large that, without carefully accounting for numerous relativistic effects, the system would not work. This paper dis- cusses the conceptual basis, founded on special and general relativity, for navigation using GPS. Relativistic principles and effects which must be considered include the constancy of the speed of light, the equivalence principle, the Sagnac effect, time dilation, gravitational frequency shifts, and relativity of synchronization. Experimental tests of relativity ob- tained with a GPS receiver aboard the TOPEX/POSEIDON satellite will be discussed. Recently frequency jumps arising from satellite orbit adjust- ments have been identified as relativistic effects. These will be explained and some interesting applications of GPS will be discussed. c 2003 Max-Planck-Gesellschaft and the authors. Further information on copyright is given at http://www.livingreviews.org/Info/Copyright/. For permission to reproduce the article please contact [email protected]. Article Amendments On author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date informa- tion possible is provided.
    [Show full text]
  • VADEMECUM ‘Go with Me’
    VADEMECUM ‘Go With Me’ 2006 Submission to the Student Space Settlement Contest NASA Ames Research Center ALEXANDER BRIDI 56 Rue de Tenbosch, 1050 Brussels, Belgium [email protected] +011 32 473 898 170 Table of Contents 1. Introduction ..................................................................................................................7 2. Why do we need a space station?................................................................................7 3 .What should Vademecum be like? ..............................................................................8 3.1 Design Options.........................................................................................................8 3.2 Vademecum’s Life Support System........................................................................20 3.2.1 Air System........................................................................................................20 3.2.2 Vademecum’s Biomass System ......................................................................22 3.2.3 Food Production ..............................................................................................24 3.2.4 Thermal System...............................................................................................25 3.2.5 Waste Management System ............................................................................26 3.2.6 Water Management System ..........................................................................27 3.2.7 Energy Generation, Storage and Distribution System: .............................27
    [Show full text]
  • The Kerr Black Hole Hypothesis: a Review of Methods and Results
    The Kerr black hole hypothesis: a review of methods and results João Miguel Arsénio Rico Thesis to obtain the Master of Science Degree in Engineering Physics Examination Committee Chairperson: Doctor José Pizarro de Sande e Lemos Supervisor: Doctor Vitor Manuel dos Santos Cardoso Members of the Committee: Doctor Jorge Miguel Cruz Pereira Varelas da Rocha Doctor Paolo Pani September 2013 ii Acknowledgements Life and physics are hard. It is often easy to forget how much there is beyond the formulas we understand and that some of our basic assumptions are wrong. Experiment rules regardless of pet theories, and we are fortunate for the opportunity to at least refine our intuitions. It is essential to keep questioning, keep working, keep humble. Life and physics are a pleasure, and for the most part people are the reason why. I want to start by thanking Vitor Cardoso, my thesis supervisor, for all the help, patience and encouragement throughout this project. An excellent researcher and an excellent supervisor are not the same thing, and I have been extremely fortunate because Vitor really is both. His sheer energy, constant good spirits and relentless enthusiasm for Physics are an inspiring example for me, and I have yet to measure all that I have learned from him. I also want to thank Paolo Pani for being such a great co- supervisor! Thank you for all the guidance and fun through this initiation to research and black hole physics. It has been a real pleasure and privilege to work with both of you. I thank Jorge Rocha, my thesis examiner, for a very careful reading of the manuscript and valuable comments and suggestions.
    [Show full text]
  • Lunar Dynamics and a Search for Objects Near the Earth-Moon Lagrangian Points
    Lunar Dynamics and A Search for Objects Near the Earth-Moon Lagrangian Points Student Researcher : Jillian Eymann Mentor and Professor : Dr. Ronald P. Olowin September 13, 2009 Table of Contents Pages I. Introduction ……………………………………. 2 II. Background - What are Lagrangian points? ……… 2 II. Background - Lagrangian points: A History ……... 3 II. Background - L4 and L5 ………………………... 4 III. Equipment / Tools ……………………………… 6 IV. Procedure - Ideal ………………………………... 7 IV. Procedure - Problems …………………………….. 8 V. Results …………………………………………… 10 VI. Conclusion / Continuation …………………….. 11 Appendix A ……………………………………... 12 Appendix B ……………………………………... 13 Appendix C ……………………………………... 14 Works Cited …………………………………….. 16 I. Introduction The goal at the outset of this project was to begin a CCD (charge-coupled device) imaging search of the L4 and L5 Lagrangian points in the Earth-Moon system in an attempt to discover natural satellites in those regions. Using the 0.5m SC telescope at the Saint Mary’s College Norma Geissberger Observatory, the observing plan spanned 60° along the lunar orbital plane x 5° around Earth-Moon L5, 45° x 5° around L4. The limiting magnitude for the detection of libration objects in this search was assumed to be 17-19th magnitude; however it is understood that in order to detect such faint objects, perfect seeing conditions would be necessary. An automated search of selected priority was to be attempted using the Faint Object Classification and Analysis System (FOCAS) software package from the National Optical Astronomy Observatories. II. Background - What are Lagrangian points? A precise but technical definition of the Lagrangian points is that they are the stationary solutions of the circular restricted three-body problem in gravitational physics. When looking at the Earth-Moon system, for example, if we assume that the two massive bodies (the Earth and Moon, see Figure 1) are in circular orbits around their common center of mass, there are five positions in space (referred to as L1-L5) where a third body of comparatively negligible mass (i.e.
    [Show full text]
  • Deep Space Communication and Exploration of Solar System Through Inter-Lagrangian Data Relay Satellite Constellation
    Deep Space Communication and Exploration of Solar System through Inter-Lagrangian Data Relay Satellite Constellation Miftahur Rahman1), Monirul Islam2) and Rashedul Huq2) 1) Department of Mathematics and Physics, North South University, Dhaka, Bangladesh 2) Department of ECE, North South University, Dhaka, Bangladesh Solar system interplanetary communication and deep space exploration research have been undertaken by most of the leading global space agencies. It is required to have an uninterrupted communication links among the planets of the solar system to support future space missions and human colonization in and around the solar system. At present, we don’t have an integrated interplanetary network links and infrastructure that can be used to meet the vision and mission of the proposed future deep space exploration. To support these types of communication, our proposal is to develop a ‘Cluster Based Data Relay Satellite Network Architecture’ across the whole solar system. For developing this architecture, we need to split the current Direct to Earth (DTE) communication system and transform it to communicate through multiple relay satellites. The proposed system is based on satellite constellations in the Earth-Moon Lagrangian orbits, alongside with Earth-Sun Lagrangian orbits. It creates a direct link to the satellites in Lagrangian orbit of other planets and hence it will extend the network to support uninterrupted interplanetary data links and continuous communications in near future. Thus through the support of multiple data relay satellites, transmission of signals Direct to the Earth (DTE) from any deep space spacecraft will not be required. Therefore the spacecrafts or probes will be much smaller in size allowing the CubeSat missions to operate throughout the solar system.
    [Show full text]
  • Ariadna Final Report 09-1301
    Mapping the Spacetime Metric with a Global Navigation Satellite System Final Report Authors: Andrej Čadež1, Uroš Kostić1 and Pacôme Delva2 Affiliation: 1Faculty of Mathematics and Physics, University of Ljubljana, 2Advanced Concepts Team, European Space Agency Date: February 28th, 2010 Contacts: Andrej Čadež Tel: +38(0)614766533 Fax: +38(0)612517281 e-mail: [email protected] Advanced Concepts Team Tel: +31(0)715654192 Fax: +31(0)715658018 e-mail: [email protected] Ariadna ID: 09/1301 Study Type: Standard Available on the ACT w ebsite http://www.esa.int/act Contract Number: 22709/09/NL/CBI Contents Introduction 1 1 Global navigation satellite systems and Relativity3 1.1 The newtonian conception.............................3 1.2 The relativistic positioning system........................7 2 Schwarzschild space-time 11 2.1 Time-like geodesics................................. 13 2.2 Light-like geodesics................................. 14 2.3 Ray-tracing..................................... 15 3 Numerical algorithms 17 3.1 Calculating null-coordinates from Schwarzschild coordinates.......... 17 3.2 Calculating Schwarzschild coordinates from null-coordinates.......... 18 3.3 Accuracy and speed of the algorithms...................... 21 4 Non-gravitational perturbations 25 Conclusion 29 A Equations in Schwarzschild space-time 33 A.1 Time-like geodesics................................. 35 A.2 Light-like geodesics................................. 38 A.3 Ray-tracing..................................... 40 B Calculating Schwarzschild coordinates from null-coordinates 43 C Elliptic integrals and functions 57 iii List of Figures 1.1 Dening null-coordinates with the null past cone of a space-time event.....8 1.2 Dening null-coordinates with four one-parameter families of null future cone.8 2.1 The orbital plane in equatorial coordinates...................
    [Show full text]
  • Fowles & Cassidy
    are in direct impact with each other and have equal ani Defore impact, rebound with velocities that are, apart from the sign, the same." "The sum of the products of the magnitudes of each hard body, multiplied by the square of the velocities, is always the same, before and after the collision." — Christiaan Huygens, memoir, De Motu Corporum ex mutuo impulsu Hypothesis, composed in Paris, 5-Jan-1669, to Oldenburg, Secretary of the Royal Society 7.11 Introduction: Center of Mass and Linear Momentum of a System We now expand our study of mechanics of systems of many particles (two or more). These particles may or may not move independently of one another. Special systems, called rigid bodies, in which the relative positions of all the particles are fixed are taken up in the next two chapters. For the present, we develop some general theorems that apply to all sys- tems. Then we apply them to some simple systems of free particles. Our general system consists of n particles of masses m1, m2,. , whose position vectors are, respectively, r1, r2, . , We define the center of mass of the system as the point whose position vector (Figure 7.1.1) is given by = m1r1 + m2r2 +• + r = (7.1.1) m1 + m2 + m m = E the total mass of the system. The definition in Equation 7.1.1 is equiv- alent to the three equations — — — (7.1.2) m m m 275 276 CHAPTER 7 Dynamics of Systems of Particles z S S S S Center of mass y Figure 7.1.1 Center of mass of a system S of particles.
    [Show full text]