Hill's Approximation in the Three-Body Problem
|
|
- Lucas Heath
- 5 years ago
- Views:
Transcription
1 Progress of Theoretical Physics Supplement No. 96, Chapter 15 Hill's Approximation in the Three-Body Problem Kiyoshi N AKAZA W A and Shigeru IDA* Department of Applied Physics, T.okyo Institute of Technology, Tokyo 152 *Geophysical Institute, University of Tokyo, Tokyo 113 (Received September 14, 1988) Hill's equation which describes approximately motions of interacting two bodies orbiting around the Sun is summarized, together with its properties. Hill's equation can be separated into equations of the relative and center of mass motions, and the center of mass motion can be integrated analytically. Further, it can be rewritten in a non-dimensional form scaled by the Hill radius and the heliocentric distance. These characteristics of Hill's equation reduce the degrees of freedom of particle motion and, hence, it is advantageous to use it in studies of gravitational scattering and collision between Keplerian particles, which will be described in later chapters. 1. Introduction Planetesimals which revolve around the protosun experience mutual gravitational scatterings and direct collisions through which they grow, finally, into planets. Mathematical descriptions of such processes (i.e., gravitational scatterings and mutual collisions), which are presented in the later chapters of this Part, have to be formulated, at least, as a three-body problem: the protosun and two interacting planetesimals. Unfortunately, solutions to the general three-body problem are rather complicated and, hence, it is very difficult to treat these problems analytically and, even numerically. However, noticing that masses of planetesimals (and also those of protoplanets) are much smaller than that of the protosun and that they revolve almost along circular orbits, we can make use of the so-called Hill's equation which was originally introduced for the study of the lunar orbit (Hill, 1878). As will be seen later in detail, Hill's equation has many advantages. Namely, (1) the relative and the center of mass motions can be separated and the equation for the motion of the center of mass can be integrated analytically, (2) the equation of relative motion can be scaled by the cube root of the sum of the masses; if we have a solution for the relative motion of planetesimals with certain masses, we can obtain orbits of planetesimals with arbitrary masses, and (3) there exists an energy integral (called the Jacobi integral) in a system described by Hill's equation. Because of the above properties of Hill's equation, we can considerably reduce the degrees of freedom of particle motion. Thus, it is very useful to study gravitational scatterings as well as mutual collisions
2 168 K. N akazawa and S. Ida in the framework of Hill's equation. Prior to the description of the physical processes in the solar gravitational field, we will summarize briefly Hill's equation and its properties for later use. Details have already been presented in the literature (e.g., Szebehely, 1967; Henon and Petit, 1986 ; Petit and Henon, 1987; N akazawa et al., 1988), which readers may refer to. 2. Hill's equation Suppose that a protoplanet with mass m1 and a planetesimal with mass mz (hereafter called particles 1 and 2) are revolving around the protosun. Let us first introduce rotating local Cartesian coordinates (x, y, z) called the Hill coordinates. They are l defined by x=iii-ao, y= ao(8-!jot), z=z, (15 2 1) where ao is a certain reference semimajor axis, (iii, 8, z) are cylindrical coordinates with the z-axis perpendicular to the ecliptic plane and with origin at the center of mass of the system. Further,!2o is the Keplerian angular velocity at iii=ao, given by (15 2 2) where Me is the solar mass. Now we will make the following assumptions, called Hill's approximation (Hill, 1878). The assumptions are and (j=1 and 2) (15 2 3) (15 2 4) where a prime denotes a derivative with respect to time t. In the above, it should be noticed that we need not require the condition IYil~ao. In terms of the heliocentric orbital elements introduced in a later section, the above conditions can also be written as ej, z i~1, lai- aol~ao, (15 2 5) where a;, ej, and ij are the instantaneous heliocentric semimajor axis, eccentricity, and inclination of the j-th particle, respectively. The above conditions (15 2 5) are usually satisfied in our problem. Expanding the equations of motion in a power series of the infinitesimals which appear in Eqs. (15 2 3) and (15 2 4) and retaining only the first order terms, we obtain Hill's equation which is given, for particle j, by
3 Chapter 15 Hill's Approximation in the Three-Body Problem 169 x - " 2.!2> y = ' 3.!2>2 x + -x -x 3J.I; ( ) j J J r3 z j, y/' + 2.!2Jx/ = 2 3J.I;( ) =-.!JJ Zi+r Z;-Zj (15 2 6) with (15 2 7) where r is the distance between two particles given by (15 2 8) In Eq. (15 2 6), the first and second terms on the right-hand side denote the solar tidal force (i.e., the sum of solar gravity and centrifugal force) and the mutual gravity force, respectively. The second terms on the left-hand side of Eq. (15 2 6) come from the Coirolis force. 3. Keplerian motion When the relative distance r is very large and the mutual interaction term can be neglected, Eq. (15 2 6) can be solved exactly (Hen on and Petit, 1986):! Xi(t)= bj-ejaocos(!&.jt- ri), Ylt)=~ ~ ~i(!&.jt- i)+2eiaosin(!&.jt-r;), zlt)=ziaosm(.!&jt-w;), (15 3 1) where the twelve parameters, bj, e;, ii, r.;, ;, and Wi are constants of integration. For the corresponding velocity components, we have! x/(t)=eiao!&.jsin(!&.jt-ri), y/(t)=- ~ bi+2ejaocos(!&.jt-r;)}.!&>, z/(t)= ijao.!&>cos(.!&jt- w;). (15 3 2) We can expect that the above solutions (15 3 1) and (15 3 2) coincide with those of Keplerian motion, because they are obtained under the assumption of J.lj=O. In fact, this can be shown as long as bi/ao, ei, and i; are sufficiently smaller than unity. Here, ao+ bi. ei, and ij are identified with the semimajor axis, eccentricity, and inclination of Keplerian orbit, respectively, and rj, tpi, and w; are the phase angles. It should be noted that the Keplerian motion given by Eq. (15 3 1) is composed of two motions: the motion of the guiding center (xg;, YG;, zgj and the epicyclic motion (xe;, YE;, zd. They are described, respectively, as
4 170 K. N akazawa and S. Ida and l XE;=- eiaocos (!Jot- rj, YE;=~eiao_sin(!Jot- ri), ZE;=zjaosm(!Jot-wi). (15 3 4) The motion of the guiding center comes from the Keplerian shear due to the difference between ao and aj. On the other hand, the epicyclic motion draws an ellipse with axes ei and 2ei in a period of the Keplerian time at ao. Even in the general case where mutual interaction cannot be neglected, it is often convenient to express particle motion in the same form as Eqs. (15 3 1) and (15 3 2). In this case, however, the six elements bj, ei, ii, are no longer constant but are functions of time: They are the instantaneous Keplerian orbital elements. 4. Separation of the relative motion from the center of mass motion As is well known, if particles experience a potential force linear to their position vectors and if the mutual interaction term is a function of only the relative distance between two particles, then their motions can be separated into two parts: the center of mass motion and the relative motion. For the general three-body problem, the potential force GM., rj/r 3 is nonlinear in the position vector and, hence, the separation is impossible. On the other hand, since in Hill's equation (15 2 6) the solar gravity is expressed in a linear form, the motions are separable. Now, we will introduce the relative and center of mass coordinates, which are defined as x=x1-x2 and X= m1x1 + m2x2 m1+m2 ' (15 4 1) respectively. Using the above coordinates, we can rewrite Eq. (15 2 6): for the motion of the center of mass, we have l X"-2 JoY'=3 Jo 2 X, Y" +2 JoX'=O, Z" =-!Jo 2 Z (15 4 2) and, for the relative motion
5 Chapter 15 Hill's Approximation in the Three-Body Problem 171 y" + 2.!J>x' = z" (15 4 3) where f.1. is given by _ m1+mz 3nz f.j.- M. ao ~~. (15 4 4) As seen from Eq. (15 4 2), the center of mass motion is purely Keplerian so that the motion can be analytically expressed in the same form as that of Eq. (15 3 1). On the other hand, the equations of relative motion (15 4 3) are not solvable but have the same form as those of an individual particle (15 2 6). Hence, even in this case, solutions to Eq. (15 4 3) can be expressed in the form of Eqs. (15 3 1) and (15 3 2) with bi> ej, replaced by b, e, which are the instantaneous orbital elements of the relative motion. From Eqs. (15 3 1), (15 3 2) and (15 4 1), it follows that the orbital elements of the relative motion are described in terms of those of individual particles: (15 4 5) 5. Non-dimensional form of Hill's equation As seen in the previous section, Hill's equation can be separated into relative and center of mass motions. This enables us to reduce the degrees of freedom of particle motions because we have a general solution for the center of mass motion. It is also known that Hill's equation has another remarkable advantage (N akazawa et al., 1988 ; Hayashi et al., 1977). That is, Hill's equation can be rewritten in a non-dimensional form independent of mass m1 + mz and the heliocentric distance ao, in which time is normalized by.!jj- 1 and distance by hao: t =t.!jj and r=(x, y, z)=(x, y, z)/aoh' (15 5 1) where h is the reduced Hill radius defined by h=( m1 + m2 ) 1 ' 3 3M., (15 5 2) By the above scaling, Hill's equation for the relative motion (15 4 3) can be expressed as x" -2y'=3x -3x/r 3, y" +2x'= -3y/r 3, z" =- z-3z/r 3, (15 5 3)
6 172 K. N akazawa and S. Ida where a prime denotes a derivative with respect to t. Solutions to the above equations are also given by the same form as before. That is, x ( i) = b- e cos( i- r),.y( i)=- ~ b( i - )+2esin( i- r), z( i)= z sin( i -w)' (15 5 4) x'( i)= esin( i- r),.y'( t - )= b +2 ecos( t- r), z'( i)= z cos( i -w), (15 5 5) where b, e, and z are the instantaneous Keplerian orbital elements of relative motion. These elements are related to those used earlier by b =b/aoh, e=e/h and z =i/h. (15 5 6) By the above normalization, the results of an orbital calculation obtained for a particular mass and heliocentric distance ao are applicable for arbitrary mass and distance. This also reduces the degrees of kinematic freedom. Sometimes, it is useful to introduce the new variables ~. TJ, and (Henon and Petit, 1986), the convenience of which was first pointed out by Hayashi (1980). They are defined by (15 5 7) Using the above variables, solutions given by Eqs. (15 5 4) and (15 5 5) can be re-described as x(t)= b -.;1COS f -.;2sin f, y(f)=- ~ b f + +2.;1sin f -2.;2COS f, z(f)=7}1sin f +1]2COS f, (15 5 8) and '=.;1sin f -,;2COS f, y'(t)=-z-b +2.;1cos t +2.;2sin t, z'(t)= 111 cos i -172sin i. (15 5 9) Note that Eqs. (15 5 8) and (15 5 9) are expressed as linear combinations of b and the new variables. These will be referred to in the next chapter.
7 Chapter 15 Hill's Approximation in the Three-Body Problem The Jacobi integral As is well known, there is an energy integral in a system described by the set of equations (15 5 3), which is called the Jacobi integral: (15 6 1) where U< - - -) x,y,z--2 x 2 z-r 2. (15 6 2) The zero velocity curves U(x, y, z)=e with z=o are shown in Fig.1, together with the potential U(x, 0, 0). The minima of the potential barrier are at (x, y, z) =(±1, 0, 0). These are the liberation points which correspond to the so-called Lagrange points L1 and L2 in the general three-body problem. The constant 9/2 in Eq. (15 6 2) is added so that U = 0 at the liberation points. The Jacobi integral (15 6 1) can be also expressed in terms of the instantaneous Keplerian elements: with the help of Eqs. (15 5 4) and (15 5 5), we have (15 6 3) Note that Eq. (15 6 3) contains none of the phase angles L1 L2 y 0 u X X (a) (b) Fig. L Zero velocity curves in a plane with z=o (a). Dotted regions represent those of E>O and attached numbers show the values of E. The central unfilled region denotes the Hill sphere of a protoplanet P within which mutual gravity overcomes that of the Sun. Further, L1 and L2 denote liberation points which correspond to the Lagrange points. The potential U at y = z = 0 is also illustrated in (b).
8 174 K. Nakazawa and S. Ida As seen from the above mathematical arguments of Hill's equation, it is advantageous to use Hill's framework. In Hill's equation the relative and center of mass motions can be separated, and the center of mass motion can be described analytically by a Keplerian orbit. This reduces the degrees of freedom of orbital motions. Further, we can see that Eq. (15 5 3) does not contain m1 and mz and, hence, the solutions are applicable to any pair of particles, that is, for arbitrary mass ratio. The above-mentioned characteristics of Hill's equation enable us to use a numerical approach for obtaining the collisional probability as well as the scattering rate between Keplerian particles. References Hayashi, C., 1980 : private communication. Hayashi, C., Nakazawa, K. and Adachi, I., 1977: Pub!. Astron. Soc. Jpn. 29, 163. Henon, M. and Petit, J. M., 1986: Celestial Mech. 38, 67. Hill, G. W., 1878: Am. ]. Math. 1, 5; 129 ; 245. Nakazawa, K., Ida, S. and Nakagawa, Y., 1988: submitted to Astron. and Astrophys. Petit, ]. M. and Henon, M., 1987: Astron. Astrophys. 173, 389. Szebehely, V., 1967: Theory of Orbit (Academic Press, New York), p. 16.
Three objects; 2+1 problem
Three objects; 2+1 problem Having conquered the two-body problem, we now set our sights on more objects. In principle, we can treat the gravitational interactions of any number of objects by simply adding
More informationSatellite Communications
Satellite Communications Lecture (3) Chapter 2.1 1 Gravitational Force Newton s 2nd Law: r r F = m a Newton s Law Of Universal Gravitation (assuming point masses or spheres): Putting these together: r
More informationOrigin of high orbital eccentricity and inclination of asteroids
Earth Planets Space, 53, 85 9, 2 Origin of high orbital eccentricity and inclination of asteroids Makiko Nagasawa, Shigeru Ida, and Hidekazu Tanaka Department of Earth and Planetary Sciences, Tokyo Institute
More informationPeriodic Orbits in Rotating Second Degree and Order Gravity Fields
Chin. J. Astron. Astrophys. Vol. 8 (28), No. 1, 18 118 (http://www.chjaa.org) Chinese Journal of Astronomy and Astrophysics Periodic Orbits in Rotating Second Degree and Order Gravity Fields Wei-Duo Hu
More informationTP 3:Runge-Kutta Methods-Solar System-The Method of Least Squares
TP :Runge-Kutta Methods-Solar System-The Method of Least Squares December 8, 2009 1 Runge-Kutta Method The problem is still trying to solve the first order differential equation dy = f(y, x). (1) dx In
More informationStudy of the Restricted Three Body Problem When One Primary Is a Uniform Circular Disk
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Applications and Applied Mathematics: An International Journal (AAM) Vol. 3, Issue (June 08), pp. 60 7 Study of the Restricted Three Body
More informationDynamical properties of the Solar System. Second Kepler s Law. Dynamics of planetary orbits. ν: true anomaly
First Kepler s Law The secondary body moves in an elliptical orbit, with the primary body at the focus Valid for bound orbits with E < 0 The conservation of the total energy E yields a constant semi-major
More informationAstronomy 241: Review Questions #2 Distributed: November 7, 2013
Astronomy 241: Review Questions #2 Distributed: November 7, 2013 Review the questions below, and be prepared to discuss them in class. For each question, list (a) the general topic, and (b) the key laws
More informationUse conserved quantities to reduce number of variables and the equation of motion (EOM)
Physics 106a, Caltech 5 October, 018 Lecture 8: Central Forces Bound States Today we discuss the Kepler problem of the orbital motion of planets and other objects in the gravitational field of the sun.
More informationA study upon Eris. I. Describing and characterizing the orbit of Eris around the Sun. I. Breda 1
Astronomy & Astrophysics manuscript no. Eris c ESO 2013 March 27, 2013 A study upon Eris I. Describing and characterizing the orbit of Eris around the Sun I. Breda 1 Faculty of Sciences (FCUP), University
More informationorbits Moon, Planets Spacecrafts Calculating the and by Dr. Shiu-Sing TONG
A Science Enrichment Programme for Secondary 3-4 Students : Teaching and Learning Resources the and Spacecrafts orbits Moon, Planets Calculating the 171 of by Dr. Shiu-Sing TONG 172 Calculating the orbits
More informationPADEU. Pulsating zero velocity surfaces and capture in the elliptic restricted three-body problem. 1 Introduction
PADEU PADEU 15, 221 (2005) ISBN 963 463 557 c Published by the Astron. Dept. of the Eötvös Univ. Pulsating zero velocity surfaces and capture in the elliptic restricted three-body problem F. Szenkovits
More informationCelestial Mechanics I. Introduction Kepler s Laws
Celestial Mechanics I Introduction Kepler s Laws Goals of the Course The student will be able to provide a detailed account of fundamental celestial mechanics The student will learn to perform detailed
More informationSymplectic Correctors for Canonical Heliocentric N-Body Maps
Symplectic Correctors for Canonical Heliocentric N-Body Maps J. Wisdom Massachusetts Institute of Technology, Cambridge, MA 02139 wisdom@poincare.mit.edu Received ; accepted 2 ABSTRACT Symplectic correctors
More informationChapter 9 Circular Motion Dynamics
Chapter 9 Circular Motion Dynamics Chapter 9 Circular Motion Dynamics... 9. Introduction Newton s Second Law and Circular Motion... 9. Universal Law of Gravitation and the Circular Orbit of the Moon...
More informationStress, Strain, Mohr s Circle
Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected
More informationLecture 13. Gravity in the Solar System
Lecture 13 Gravity in the Solar System Guiding Questions 1. How was the heliocentric model established? What are monumental steps in the history of the heliocentric model? 2. How do Kepler s three laws
More informationGravitation and the Waltz of the Planets
Gravitation and the Waltz of the Planets Chapter Four Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets
More informationGravitation and the Waltz of the Planets. Chapter Four
Gravitation and the Waltz of the Planets Chapter Four Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets
More informationORBITS WRITTEN Q.E. (June 2012) Each of the five problems is valued at 20 points. (Total for exam: 100 points)
ORBITS WRITTEN Q.E. (June 2012) Each of the five problems is valued at 20 points. (Total for exam: 100 points) PROBLEM 1 A) Summarize the content of the three Kepler s Laws. B) Derive any two of the Kepler
More informationarxiv: v1 [astro-ph.ep] 23 Mar 2010
Formation of Terrestrial Planets from Protoplanets under a Realistic Accretion Condition arxiv:1003.4384v1 [astro-ph.ep] 23 Mar 2010 Eiichiro Kokubo Division of Theoretical Astronomy, National Astronomical
More informationOrbit Characteristics
Orbit Characteristics We have shown that the in the two body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the primary located at the focus of the conic
More informationKepler, Newton, and laws of motion
Kepler, Newton, and laws of motion First: A Little History Geocentric vs. heliocentric model for solar system (sec. 2.2-2.4)! The only history in this course is this progression: Aristotle (~350 BC) Ptolemy
More informationGravitation and the Motion of the Planets
Gravitation and the Motion of the Planets 1 Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets go around
More informationRegular n-gon as a model of discrete gravitational system. Rosaev A.E. OAO NPC NEDRA, Jaroslavl Russia,
Regular n-gon as a model of discrete gravitational system Rosaev A.E. OAO NPC NEDRA, Jaroslavl Russia, E-mail: hegem@mail.ru Introduction A system of N points, each having mass m, forming a planar regular
More informationGravitation. chapter 9
chapter 9 Gravitation Circular orbits (Section 9.3) 1, 2, and 3 are simple exercises to supplement the quantitative calculations of Examples 4, 5, and 6 in Section 9.3. 1. Satellite near Earth s surface
More informationNewton s Laws of Motion and Gravity ASTR 2110 Sarazin. Space Shuttle
Newton s Laws of Motion and Gravity ASTR 2110 Sarazin Space Shuttle Discussion Session This Week Friday, September 8, 3-4 pm Shorter Discussion Session (end 3:40), followed by: Intro to Astronomy Department
More information2. FLUID-FLOW EQUATIONS SPRING 2019
2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid
More informationAST111, Lecture 1b. Measurements of bodies in the solar system (overview continued) Orbital elements
AST111, Lecture 1b Measurements of bodies in the solar system (overview continued) Orbital elements Planetary properties (continued): Measuring Mass The orbital period of a moon about a planet depends
More informationAST1100 Lecture Notes
AST1100 Lecture Notes 5 The virial theorem 1 The virial theorem We have seen that we can solve the equation of motion for the two-body problem analytically and thus obtain expressions describing the future
More informationLecture 15 - Orbit Problems
Lecture 15 - Orbit Problems A Puzzle... The ellipse shown below has one focus at the origin and its major axis lies along the x-axis. The ellipse has a semimajor axis of length a and a semi-minor axis
More informationOptimal Generalized Hohmann Transfer with Plane Change Using Lagrange Multipliers
Mechanics and Mechanical Engineering Vol. 21, No. 4 (2017) 11 16 c Lodz University of Technology Optimal Generalized Hohmann Transfer with Plane Change Using Lagrange Multipliers Osman M. Kamel Astronomy
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More informationGravity: Motivation An initial theory describing the nature of the gravitational force by Newton is a product of the resolution of the
Gravity: Motivation An initial theory describing the nature of the gravitational force by Newton is a product of the resolution of the Geocentric-Heliocentric debate (Brahe s data and Kepler s analysis)
More informationPhysics for Scientists and Engineers 4th Edition, 2017
A Correlation of Physics for Scientists and Engineers 4th Edition, 2017 To the AP Physics C: Mechanics Course Descriptions AP is a trademark registered and/or owned by the College Board, which was not
More informationI ve Got a Three-Body Problem
I ve Got a Three-Body Problem Gareth E. Roberts Department of Mathematics and Computer Science College of the Holy Cross Mathematics Colloquium Fitchburg State College November 13, 2008 Roberts (Holy Cross)
More informationAccretion of Planets. Bill Hartmann. Star & Planet Formation Minicourse, U of T Astronomy Dept. Lecture 5 - Ed Thommes
Accretion of Planets Bill Hartmann Star & Planet Formation Minicourse, U of T Astronomy Dept. Lecture 5 - Ed Thommes Overview Start with planetesimals: km-size bodies, interactions are gravitational (formation
More informationEXTENDING NACOZY S APPROACH TO CORRECT ALL ORBITAL ELEMENTS FOR EACH OF MULTIPLE BODIES
The Astrophysical Journal, 687:1294Y1302, 2008 November 10 # 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A. EXTENDING NACOZY S APPROACH TO CORRECT ALL ORBITAL ELEMENTS
More informationANNEX 1. DEFINITION OF ORBITAL PARAMETERS AND IMPORTANT CONCEPTS OF CELESTIAL MECHANICS
ANNEX 1. DEFINITION OF ORBITAL PARAMETERS AND IMPORTANT CONCEPTS OF CELESTIAL MECHANICS A1.1. Kepler s laws Johannes Kepler (1571-1630) discovered the laws of orbital motion, now called Kepler's laws.
More informationOn the Tides' paradox.
On the Tides' paradox. T. De ees - thierrydemees @ pandora.be Abstract In this paper I analyse the paradox of tides, that claims that the oon and not the Sun is responsible for them, although the Sun's
More informationEFFICIENT INTEGRATION OF HIGHLY ECCENTRIC ORBITS BY QUADRUPLE SCALING FOR KUSTAANHEIMO-STIEFEL REGULARIZATION
The Astronomical Journal, 128:3108 3113, 2004 December # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A. EFFICIENT INTEGRATION OF HIGHLY ECCENTRIC ORBITS BY QUADRUPLE SCALING
More informationEngg. Math. I. Unit-I. Differential Calculus
Dr. Satish Shukla 1 of 50 Engg. Math. I Unit-I Differential Calculus Syllabus: Limits of functions, continuous functions, uniform continuity, monotone and inverse functions. Differentiable functions, Rolle
More informationPhysics Test 7: Circular Motion page 1
Name Physics Test 7: Circular Motion page 1 hmultiple Choice Read each question and choose the best answer by putting the corresponding letter in the blank to the left. 1. The SI unit of angular speed
More informationLecture Tutorial: Angular Momentum and Kepler s Second Law
2017 Eclipse: Research-Based Teaching Resources Lecture Tutorial: Angular Momentum and Kepler s Second Law Description: This guided inquiry paper-and-pencil activity helps students to describe angular
More informationConstrained motion and generalized coordinates
Constrained motion and generalized coordinates based on FW-13 Often, the motion of particles is restricted by constraints, and we want to: work only with independent degrees of freedom (coordinates) k
More informationThe 3D restricted three-body problem under angular velocity variation. K. E. Papadakis
A&A 425, 11 1142 (2004) DOI: 10.1051/0004-661:20041216 c ESO 2004 Astronomy & Astrophysics The D restricted three-body problem under angular velocity variation K. E. Papadakis Department of Engineering
More informationAST111 PROBLEM SET 2 SOLUTIONS. RA=02h23m35.65s, DEC=+25d18m42.3s (Epoch J2000).
AST111 PROBLEM SET 2 SOLUTIONS Home work problems 1. Angles on the sky and asteroid motion An asteroid is observed at two different times. The asteroid is located at RA=02h23m35.65s, DEC=+25d18m42.3s (Epoch
More informationA STUDY OF CLOSE ENCOUNTERS BETWEEN MARS AND ASTEROIDS FROM THE 3:1 RESONANCE. Érica C. Nogueira, Othon C. Winter
A STUDY OF CLOSE ENCOUNTERS BETWEEN MARS AND ASTEROIDS FROM THE 3: RESONANCE Érica C. Nogueira, Othon C. Winter Grupo de Dinâmica Orbital e Planetologia UNESP -- Guaratinguetá -- Brazil Antonio F.B. de
More informationProperties of the stress tensor
Appendix C Properties of the stress tensor Some of the basic properties of the stress tensor and traction vector are reviewed in the following. C.1 The traction vector Let us assume that the state of stress
More informationOn the Estimated Precession of Mercury s Orbit
1 On the Estimated Precession of Mercury s Orbit R. Wayte. 9 Audley Way, Ascot, Berkshire, SL5 8EE, England, UK e-mail: rwayte@googlemail.com Research Article, Submitted to PMC Physics A 4 Nov 009 Abstract.
More informationThe Heliocentric Model of Copernicus
Celestial Mechanics The Heliocentric Model of Copernicus Sun at the center and planets (including Earth) orbiting along circles. inferior planets - planets closer to Sun than Earth - Mercury, Venus superior
More informationDynamics of Distant Moons of Asteroids
ICARUS 128, 241 249 (1997) ARTICLE NO. IS975738 Dynamics of Distant Moons of Asteroids Douglas P. Hamilton Astronomy Department, University of Maryland, College Park, Maryland 20742 2421 E-mail: hamilton@astro.umd.edu
More informationEVOLUTIONS OF SMALL BODIES IN OUR SOLAR SYSTEM
EVOLUTIONS OF SMALL BODIES IN OUR SOLAR SYSTEM Dynamics and collisional processes Dr. Patrick MICHEL Côte d'azur Observatory Cassiopée Laboratory, CNRS Nice, France 1 Plan Chapter I: A few concepts on
More informationChapter 3. Algorithm for Lambert's Problem
Chapter 3 Algorithm for Lambert's Problem Abstract The solution process of Lambert problem, which is used in all analytical techniques that generate lunar transfer trajectories, is described. Algorithms
More informationHamilton-Jacobi Modelling of Stellar Dynamics
Hamilton-Jacobi Modelling of Stellar Dynamics Pini Gurfil Faculty of Aerospace Engineering, Technion - Israel Institute of Technology Haifa 32000, Israel N. Jeremy Kasdin and Egemen Kolemen Mechanical
More informationLecture Notes for PHY 405 Classical Mechanics
Lecture Notes for PHY 405 Classical Mechanics From Thorton & Marion s Classical Mechanics Prepared by Dr. Joseph M. Hahn Saint Mary s University Department of Astronomy & Physics September 1, 2005 Chapter
More information8 Rotational motion of solid objects
8 Rotational motion of solid objects Kinematics of rotations PHY166 Fall 005 In this Lecture we call solid objects such extended objects that are rigid (nondeformable) and thus retain their shape. In contrast
More informationA REGION VOID OF IRREGULAR SATELLITES AROUND JUPITER
The Astronomical Journal, 136:909 918, 2008 September c 2008. The American Astronomical Society. All rights reserved. Printed in the U.S.A. doi:10.1088/0004-6256/136/3/909 A REGION VOID OF IRREGULAR SATELLITES
More informationRestricted three body problems in the Solar System: simulations
Author:. Facultat de Física, Universitat de Barcelona, Diagonal 645, 0808 Barcelona, Spain. Advisor: Antoni Benseny i Ardiaca. Facultat de Matemàtiques, Universitat de Barcelona, Gran Via de les Corts
More informationPHYS 160 Astronomy Test #1 Fall 2017 Version B
PHYS 160 Astronomy Test #1 Fall 2017 Version B 1 I. True/False (1 point each) Circle the T if the statement is true, or F if the statement is false on your answer sheet. 1. An object has the same weight,
More informationLecture 8. Stress Strain in Multi-dimension
Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
More informationOrbital Mechanics! Space System Design, MAE 342, Princeton University! Robert Stengel
Orbital Mechanics Space System Design, MAE 342, Princeton University Robert Stengel Conic section orbits Equations of motion Momentum and energy Kepler s Equation Position and velocity in orbit Copyright
More informationPhysics 103 Homework 5 Summer 2015
Physics 103 Homework 5 Summer 2015 Instructor: Keith Fratus TAs: Michael Rosenthal and Milind Shyani Due Date: Friday, July 31 st, IN LECTURE 1. A particle moving in three dimensions is constrained to
More informationMotion under the Influence of a Central Force
Copyright 004 5 Motion under the Influence of a Central Force The fundamental forces of nature depend only on the distance from the source. All the complex interactions that occur in the real world arise
More informationBasics of Kepler and Newton. Orbits of the planets, moons,
Basics of Kepler and Newton Orbits of the planets, moons, Kepler s Laws, as derived by Newton. Kepler s Laws Universal Law of Gravity Three Laws of Motion Deriving Kepler s Laws Recall: The Copernican
More informationChapter 5 Circular Motion; Gravitation
Chapter 5 Circular Motion; Gravitation Units of Chapter 5 Kinematics of Uniform Circular Motion Dynamics of Uniform Circular Motion Highway Curves, Banked and Unbanked Newton s Law of Universal Gravitation
More informationLearning Objectives. one night? Over the course of several nights? How do true motion and retrograde motion differ?
Kepler s Laws Learning Objectives! Do the planets move east or west over the course of one night? Over the course of several nights? How do true motion and retrograde motion differ?! What are geocentric
More informationOn a time-symmetric Hermite integrator for planetary N-body simulation
Mon. Not. R. Astron. Soc. 297, 1067 1072 (1998) On a time-symmetric Hermite integrator for planetary N-body simulation Eiichiro Kokubo,* Keiko Yoshinaga and Junichiro Makino Department of Systems Science,
More informationPOLYNOMIAL REPRESENTATION OF THE ZERO VELOCITY SURFACES IN THE SPATIAL ELLIPTIC RESTRICTED THREE-BODY PROBLEM
POLYNOMIAL REPRESENTATION OF THE ZERO VELOCITY SURFACES IN THE SPATIAL ELLIPTIC RESTRICTED THREE-BODY PROBLEM Ferenc Szenkovits 1,2 Zoltán Makó 1,3 Iharka Csillik 4 1 Department of Applied Mathematics,
More informationPHYS-2010: General Physics I Course Lecture Notes Section VIII
PHYS-2010: General Physics I Course Lecture Notes Section VIII Dr. Donald G. Luttermoser East Tennessee State University Edition 2.4 Abstract These class notes are designed for use of the instructor and
More informationRotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep.
Rotational Motion Chapter 4 P. J. Grandinetti Chem. 4300 Sep. 1, 2017 P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. 1, 2017 1 / 76 Angular Momentum The angular momentum of a particle with respect
More informationThe History of Astronomy. Please pick up your assigned transmitter.
The History of Astronomy Please pick up your assigned transmitter. When did mankind first become interested in the science of astronomy? 1. With the advent of modern computer technology (mid-20 th century)
More informationLecture 1a: Satellite Orbits
Lecture 1a: Satellite Orbits Meteorological Satellite Orbits LEO view GEO view Two main orbits of Met Satellites: 1) Geostationary Orbit (GEO) 1) Low Earth Orbit (LEO) or polar orbits Orbits of meteorological
More informationarxiv:physics/ v2 [physics.gen-ph] 2 Dec 2003
The effective inertial acceleration due to oscillations of the gravitational potential: footprints in the solar system arxiv:physics/0309099v2 [physics.gen-ph] 2 Dec 2003 D.L. Khokhlov Sumy State University,
More informationKeywords : Restricted three-body problem, triaxial rigid body, periodic orbits, Liapunov stability. 1. Introduction
Bull. Astr. Soc. India (006) 34, 11 3 Periodic orbits around the collinear liberation points in the restricted three body problem when the smaller primary is a triaxial rigid body : Sun-Earth case Sanjay
More informationMost of the time during full and new phases, the Moon lies above or below the Sun in the sky.
6/16 Eclipses: We don t have eclipses every month because the plane of the Moon s orbit about the Earth is different from the plane the ecliptic, the Earth s orbital plane about the Sun. The planes of
More information5.1. Accelerated Coordinate Systems:
5.1. Accelerated Coordinate Systems: Recall: Uniformly moving reference frames (e.g. those considered at 'rest' or moving with constant velocity in a straight line) are called inertial reference frames.
More informationCelestial Orbits. Adrienne Carter Ottopaskal Rice May 18, 2001
Celestial Orbits Adrienne Carter sillyajc@yahoo.com Ottopaskal Rice ottomanbuski@hotmail.com May 18, 2001 1. Tycho Brache, a Danish astronomer of the late 1500s, had collected large amounts of raw data
More informationPerturbations to the Lunar Orbit
Perturbations to the Lunar Orbit th January 006 Abstract In this paper, a general approach to performing perturbation analysis on a two-dimensional orbit is presented. The specific examples of the solar
More informationThe changes in normalized second degree geopotential coefficients. - f P 20 (sin4) J.), (la) /5 'GM*ft r] (lb) Ac 21 -ia* 21 = i n ^ S i f RI J^ GM.
CHAPTER 7 SOLID EARTH TIDES The solid Earth tide model is based on an abbreviated form of the Wahr model (Wahr, 98) using the Earth model 66A of Gilbert and Dziewonski (975). The Love numbers for the induced
More informationParticles in Motion; Kepler s Laws
CHAPTER 4 Particles in Motion; Kepler s Laws 4.. Vector Functions Vector notation is well suited to the representation of the motion of a particle. Fix a coordinate system with center O, and let the position
More informationHomework 1. Due whatever day you decide to have the homework session.
Homework 1. Due whatever day you decide to have the homework session. Problem 1. Rising Snake A snake of length L and linear mass density ρ rises from the table. It s head is moving straight up with the
More informationISIMA lectures on celestial mechanics. 3
ISIMA lectures on celestial mechanics. 3 Scott Tremaine, Institute for Advanced Study July 2014 1. The stability of planetary systems To understand the formation and evolution of exoplanet systems, we
More informationChapter 13. Gravitation
Chapter 13 Gravitation e = c/a A note about eccentricity For a circle c = 0 à e = 0 a Orbit Examples Mercury has the highest eccentricity of any planet (a) e Mercury = 0.21 Halley s comet has an orbit
More informationA = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great Pearson Education, Inc.
Q13.1 The mass of the Moon is 1/81 of the mass of the Earth. Compared to the gravitational force that the Earth exerts on the Moon, the gravitational force that the Moon exerts on the Earth is A. 81 2
More informationEscape Trajectories from the L 2 Point of the Earth-Moon System
Trans. Japan Soc. Aero. Space Sci. Vol. 57, No. 4, pp. 238 244, 24 Escape Trajectories from the L 2 Point of the Earth-Moon System By Keita TANAKA Þ and Jun ichiro KAWAGUCHI 2Þ Þ Department of Aeronautics
More informationPhysics Mechanics Lecture 30 Gravitational Energy
Physics 170 - Mechanics Lecture 30 Gravitational Energy Gravitational Potential Energy Gravitational potential energy of an object of mass m a distance r from the Earth s center: Gravitational Potential
More informationLecture Outline. Chapter 13 Gravity Pearson Education, Inc. Slide 13-1
Lecture Outline Chapter 13 Gravity Slide 13-1 The plan Lab this week: exam problems will put problems on mastering for chapters without HW; will also go over exam 2 Final coverage: now posted; some sections/chapters
More informationDynamics of the Earth
Time Dynamics of the Earth Historically, a day is a time interval between successive upper transits of a given celestial reference point. upper transit the passage of a body across the celestial meridian
More informationTides The Largest Waves in the Ocean
Tides Tides The Largest Waves in the Ocean Understanding Tides Understanding Tides You will study several topics: Why Earth has tides Why tides vary daily Why tides vary monthly Tide Generation Tide Generation
More informationIn this section, mathematical description of the motion of fluid elements moving in a flow field is
Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small
More informationThe Law of Ellipses (Kepler s First Law): all planets orbit the sun in a
Team Number Team Members Present Learning Objectives 1. Practice the Engineering Process a series of steps to follow to design a solution to a problem. 2. Practice the Five Dimensions of Being a Good Team
More informationAdvection Dominated Accretion Flows. A Toy Disk Model. Bohdan P a c z y ń s k i
ACTA ASTRONOMICA Vol. 48 (1998) pp. 667 676 Advection Dominated Accretion Flows. A Toy Disk Model by Bohdan P a c z y ń s k i Princeton University Observatory, Princeton, NJ 8544-11, USA e-mail: bp@astro.princeton.edu
More informationCALCULATION OF POSITION AND VELOCITY OF GLONASS SATELLITE BASED ON ANALYTICAL THEORY OF MOTION
ARTIFICIAL SATELLITES, Vol. 50, No. 3 2015 DOI: 10.1515/arsa-2015-0008 CALCULATION OF POSITION AND VELOCITY OF GLONASS SATELLITE BASED ON ANALYTICAL THEORY OF MOTION W. Góral, B. Skorupa AGH University
More informationDynamics of the solar system
Dynamics of the solar system Planets: Wanderer Through the Sky Planets: Wanderer Through the Sky Planets: Wanderer Through the Sky Planets: Wanderer Through the Sky Ecliptic The zodiac Geometry of the
More informationINTRODUCTION TO CENTRAL FORCE FIELDS
INTRODUCTION TO CENTRAL FORCE FIELDS AND CONIC SECTIONS Puneet Singla Celestial Mechanics AERO-624 Department of Aerospace Engineering Texas A&M University http://people.tamu.edu/ puneet/aero624 20th January
More informationClassical Mechanics. Luis Anchordoqui
1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body
More informationName Period Date Earth and Space Science. Solar System Review
Name Period Date Earth and Space Science Solar System Review 1. is the spinning a planetary object on its axis. 2. is the backward motion of planets. 3. The is a unit less number between 0 and 1 that describes
More informationTwo types of co-accretion scenarios for the origin of the Moon
Earth Planets Space, 53, 213 231, 2001 Two types of co-accretion scenarios for the origin of the Moon Ryuji Morishima and Sei-ichiro Watanabe Department of Earth and Planetary Sciences, Nagoya University,
More information9 Kinetics of 3D rigid bodies - rotating frames
9 Kinetics of 3D rigid bodies - rotating frames 9. Consider the two gears depicted in the figure. The gear B of radius R B is fixed to the ground, while the gear A of mass m A and radius R A turns freely
More information