DOCSLIB.ORG

The Transition from Adiabatic Inspiral to Geodesic Plunge for a Compact

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Transition from Adiabatic Inspiral to Geodesic Plunge for a Compact The 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 Ave., Cambridge, MA 02139 (Dated: November 4, 2018) 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 timescale 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. We generalize this technique to include inclination and eccentricity. PACS numbers: 04.25.-g, 04.80.Nn, 97.60.Lf I. INTRODUCTION AND MOTIVATION Teukolsky equation in the time-domain. The world line of the compact object serves as an input to this code. While the world line in the adiabatic phase can be calcu- Extreme mass ratio inspirals (EMRIs), in which stel- lated from a frequency-domain based Teukolsky equation lar mass compact objects radiate gravitational energy solver [5, 6], the trajectory in the transition regime for and fall into their massive black hole companions, are completely generic obits remains unknown. This calcu- promising sources of gravitational waves. LISA [1], the lation will provide the missing link needed to generate a proposed space based gravitational wave detector should complete inspiral trajectory. detect waves from the last stages of such inspirals. A clear theoretical understanding of the dynamics of EM- A number of researchers are working towards solving RIs is vital to the detection of these gravitational waves. the self-force problem exactly [7, 8]. Such an exact solu- tion can be separated (at least qualitatively) into time- The small mass ratios, which typically lie in the range reversal symmetric and asymmetric components. The µ/M = 10−5 10−8, allow EMRIs to be treated within − symmetric component (the “conservative self-force”) con- the framework of perturbation theory. The trajectory serves the integrals of motion. On the other hand, of the compact object can be roughly broken into three the asymmetric component (the “dissipative self-force”) regimes: (a) An adiabatic inspiral phase, during which leads to non-zero time derivatives of the integrals of mo- the dominant inspiral mechanism arises from the radia- tion. Recent advances demonstrate that we are mak- tion reaction force on the smaller object. In this stage, ing steady progress on this problem. For example, the the time scale over which the characteristic radial sep- self force is now essentially understood for circular orbits aration (between the compact object and its black hole around Schwarzschild black holes [8]. Although approxi- companion) changes is large compared to the orbital pe- mate, the results in this paper may serve as an indepen- riod. This allows us to approximate the trajectory as a dent check for these solutions. It is worth noting that if sequence of bound geodesics. (b) A plunge phase, during it becomes possible to include the conservative force in which stable geodesics do not exist. It has been shown a simple way, we should be able to build its impact into [2] that the effect of radiation reaction is negligible dur- the formalism developed here. This work may also be of arXiv:0803.4482v3 [gr-qc] 8 May 2008 ing the plunge and that this phase can be modeled as interest for numerical relativity — a perturbative inspiral a geodesic infall. (c) A regime where the spiraling com- constructed by the techniques discussed here may be an pact object transitions from adiabatic inspiral to geodesic accurate point of comparison for full numerical inspirals plunge. The course of motion at this juncture shows as- for small ratios (and may even be useful, if not so accu- pects of both, the self-force from radiation reaction and rate, for mass ratios that are not strictly perturbative). the effects of unstable geodesics. Ref. [9] discusses the transition when the compact ob- In [2], Ori and Thorne introduce a method to predict ject is in an eccentric, equatorial orbit. However, the the motion when the object is constrained to an approxi- focus of that paper is to calculate the transit time and mately circular, equatorial orbit. We generalize this pro- estimate the probability for LISA to observe such a tran- cedure to include inclined and eccentric trajectories. A sition. Our intent is to generate the world line during few modifications to the prescription in [2] are introduced the transition. We also choose our initial conditions dif- to handle such generic orbits. The results from our gen- ferently than they are chosen in Ref. [9]; we discuss these eralized prescription are in excellent agreement with [2]. differences in more detail in Sec. III. The simple calculation described in this paper is meant The rest of the paper is organized as follows: Section II to serve as a stopgap for many other open and important discusses circular orbits with arbitrary inclination. Sec. problems. There has been recent progress ([3, 4] and ref- III generalizes the formalism developed in Sec. II to in- erences therein) in the development of a code to solve the clude eccentricity. Finally, we summarize our results in 2 Sec. IV. The potentials can be expressed as: 1 2 R = E a2 + r2 aL µ2 − z − ∆ + (L aE)2 + r2µ2 + Q , (2.5) µ2 z − II. THE TRANSITION TRAJECTORY FOR 1 CIRCULAR ORBITS V = [Q θ µ2 − cos2 θ a2 µ2 E2 + L2/ sin2 θ , (2.6) − z Up to initial conditions, a set of three constants, the en- 1 ergy, E, the component of the angular momentum along V = L /sin2 θ aE φ µ z − the spin axis, Lz, and the Carter constant, Q define a a 2 2 geodesic. The Carter constant has an approximate in- + E r + a Lza , (2.7) terpretation of being the square of the component of µ∆ − angular momentum perpendicular to the spin axis. As 1 2 Vt = a Lz aE sin θ + the compact object radiates, the “constants” that de- µ − fine its geodesic will gradually evolve. (We will refer to r2+a2 E r2 + a2 L a . (2.8) [E(t),Lz(t),Q(t)] as the “constants”, although they are µ∆ − z slowly evolving.) A common approach to model the adi- abatic regime consists of treating the motion as the se- The parameters (r,θ,φ,t) are the Boyer-Lindquist coor- quence of geodesics [5, 6] defined by these evolving con- dinates, M is the black hole mass, µ is the perturbing stants. As pointed out in [10], this limit amounts to a “ra- mass, Σ = r2 + a2 cos2 θ,∆= r2 2Mr + a2 and a diative” or “dissipative” approximation. A true adiabatic is the spin parameter of the black hole.− The constants approximation would be a sequence of orbits in which (E,Lz,Q) represent the actual energy, momentum and each orbit included conservative self corrections. Since Carter constant (in units of M, M 2 and M 4 respectively), we currently use purely geodesic orbits as our background not the dimensionless versions of them. By introduc- motion (in lieu of a self-force enhanced description), we ing the perturbing mass explicitly, our notation deviates will refer to a sequence of geodesics as an “adiabatic in- from previous literature. We do this in order to show the spiral” throughout this paper. Thus, within the adiabatic dependence of the transition phase on the mass of the approximation, the world line of a particle is computed perturbing object. We also set G = c = 1 everywhere. by mapping [E(t),Lz(t),Q(t)] to [r(t),θ(t), φ(t)]. The symbols r, θ and φ are the usual Boyer-Lindquist coor- dinates. B. The last stable orbit In contrast, the plunge can be treated as a single unsta- A standard but not unique definition of the “inclina- ble geodesic with almost constant E, Lz and Q. Thus, the passage from adiabatic inspiral to geodesic plunge tion” of a Kerr geodesic is given by must contain both these features — slowly evolving “con- L cos ι = z , (2.9) stants” and marginal stability. 2 Lz + Q L2 Q = p z . (2.10) ⇒ cos2 ι L2 − z It is possible to use ι to eliminate the Carter constant. A. Kerr Geodesics Thus, any circular orbit can be parametrized by its radius (r) and inclination (ι). The last stable orbit (LSO) serves as an important ref- The following system of first order equations describes erence point — the inspiral is adiabatic well before the geodesics in a Kerr [11, 12] geometry: compact object crosses the LSO and is approximately a plunge well after the crossing. Since the transition occurs in the vicinity of the LSO, a preliminary step in our com- dr Σ = √R, (2.1) putation is to determine r and (E,Lz,Q) at the LSO for dτ ± a given inclination at the LSO, ιLSO. Note that ι changes dθ with time because it is a function of [E(t),L (t),Q(t)]. Σ = Vθ , (2.2) z dτ ± Circular orbits satisfy dφ p Σ = V , (2.3) dτ φ R =0and (2.11) dt ′ dR Σ = V . (2.4) R = =0 . (2.12) dτ t dr 3 ′ We must have R = 0 because the LSO lies at an ex- ing points, θmax and θmin, where tremum of R. We also require that 2 ′′ d R 0 θmin θmax π .
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]
  • 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].
    [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]