DOCSLIB.ORG

Two-Body Dynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Two-Body Dynamics Two‐Body Dynamics: Orbits in 3D Chapter 4 Introductions • So far we have focus on the orbital mechanics of a spacecraft in 2D. • In this Chapter we will now move to 3D and express orbits using all 6 orbital elements Geocentric Equatorial Frame r X 2 Y 2 Z 2 sin1Z /r 1X /r cos (Y /r 0) cos X /r 2 cos1 (Y /r 0) cos Orbital Elements • Classical Orbital Elements are: a = semi‐major axis (or h or ε) e = eccentricity i = inclination Ω = longitude of ascending node ω = argument of periapsis θ = true anomaly Orbital Elements vr r v/r iˆ ˆj kˆ h r v XY Z Vx Vy Vz iˆ ˆj kˆ n kˆ h 001 hx hy hz Orbital Elements h i cos1 z h 1n e cos (ez 0) ne n e 2 cos1 (e 0) ne z 2 1 2 h 2 2 e v r rvrv and e e e or e 1 v r 2 r Orbital Elements 2 e r 1 1 h 1 cos 1 (v 0) cos (vr 0) r er e r e r 2 1 1 1 h 2 cos (vr 0) 2 cos 1 (v 0) er r e r Coordinate Transformation • Answers the question of “what are the parameters in another coordinate frame” y y’ x’ Q x Transformation z (or direction cosine) matrix z’ 100 T Q Q 1 010 Q is a orthogonal transformation matrix T 001 QQ 1 Coordinate Transformation x' Qx Where T x Q x' Q Q Q iˆ / iˆ iˆ / ˆj iˆ / kˆ 11 12 13 ˆ / ˆ ˆ / ˆ ˆ / ˆ And Q Q21 Q22 Q23 j i j j j k ˆ / ˆ ˆ / ˆ ˆ / ˆ Q31 Q32 Q33 k i k j k k Where is made up of rotations about the axis {a, b, or c} by the angle {θd, θe, and θf} Q R R R a d b e c f 3rd rotation 2nd rotation 1st rotation Coordinate Transformation For example the Euler angle sequence for rotation is the 3-1-3 rotation Q R3 R1R3 0 360 0 180 0 360 where you rotate by the angle α along the 3rd axis (usually z-axis), then by β along the 1st axis, and then by γ along the 3rd axis. sin cossin cos cos cos cossin sin cos sin sin Q sin coscos cos sin cos coscos sin sin sincos 313 sin sin cos sin cos Thus, the angles can be found from elements of Q Q31 Q13 tan cos Q33 tan Q32 Q23 Coordinate Transformation Classic Euler Sequence from xyz to x’y’z’ Coordinate Transformation For example the Yaw-Pitch-Roll sequence for rotation is the 1-2-3 rotation Q R1R2 R3 0 360 90 90 0 360 where you rotate by the angle α along the 3rd axis (usually z-axis), then by β along the 2nd axis, and then by γ along the 1st axis. cos cos sin cos sin Q cos sinsin sin cos sin sin sin cos cos cossin 123 cos sincos sin sin sin sin cos cos sin coscos Thus, the angles can be found from elements of Q Q12 Q23 tan sin Q13 tan Q11 Q33 Coordinate Transformation Yaw, Pitch, and Roll Sequence from xyz to x’y’z’ Transformation between Geocentric Equatorial and Perifocal Frame Transferring between pqw frame and xyz cos h 2 / r sin pqw 1 ecos 0 sin v e cos pqw h 0 Transformation from geocentric equatorial to perifocal frame Transformation between Geocentric Equatorial and Perifocal Frame Transformation from perifocal to geocentric equatorial frame is then Therefore Perturbation to Orbits Oblateness • Planets are not perfect spheres R R oblateness eq pole Req Perturbation to Orbits Oblateness ˙r˙ r p p p uˆ p uˆ p hˆ r3 r r t t h R2 p 1.5 J 1 3sin2 i sin2 r r2 2r 2 R 2 2 pt 1.5 J2 sin i sin 2 r2 r R2 p 1.5 J sin 2i sin2 h r2 2r Perturbation to Orbits Oblateness Perturbation to Orbits Oblateness Perturbation to Orbits Oblateness Sun‐Synchronous Orbits Orbits where the orbit plane is at a fix angle α from the Sun-planet line Thus the orbit plane must rotate 360° per year (365.25 days) or 0.9856°/day Finding State of S/C w/Oblateness • Given: Initial State Vector • Find: State after Δtassuming oblateness (J2) • Steps finding updated state at a future Δtassuming perturbation 1. Compute the orbital elements of the state 2. Find the orbit period, T, and mean motion, n 3. Find the eccentric anomaly 4. Calculate time since periapsis passage, t, using Kepler’s equation Me nt E esin E Finding State of S/C w/Oblateness 5. Calculate new time as tf = t + Δt 6. Find the number of orbit periods elapsed since original periapsis passage n p t f /T 7. Find the time since periapsis passage for the final orbit t n floor n T orbit _ n p p 8. Find the new mean anomaly for orbit n M nt e orbit _ n orbit _ n 9. Use Newton’s method and Kepler’s equation to find the Eccentric anomaly (See slide 57) Finding State of S/C w/Oblateness 10. Find the new true anomaly 1 e E tan orbit _ n tan orbit _ n 2 1 e 2 11. Find position and velocity in the perifocal frame cos h 2 / r sin pqw 1 ecos 0 sin v e cos pqw h 0 Finding State of S/C w/Oblateness 12. Compute the rate of the ascending node 13. Compute the new ascending node for orbit n 14. Find the argument of periapsis rate 15. Find the new argument of periapsis Finding State of S/C w/Oblateness 16. Compute the transformation matrix [Q] using the inclination, the UPDATED argument of periapsis, and the UPDATED longitude of ascending node cos(i) 17. Find the r and v in the geocentric frame Ground Tracks Projection of a satellite’s orbit on the planet’s surface Ground Tracks Projection of a satellite’s orbit on the planet’s surface E 15.04 deg/hr Ground Tracks Projection of a satellite’s orbit on the planet’s surface Ground Tracks reveal the orbit period Ground Tracks reveal the orbit inclination i LATmax or min If the argument of perispais, ω, is zero, then the shape below and above the equator are the same. .
Load more

Recommended publications
  • Appendix a Orbits
    Appendix A Orbits As discussed in the Introduction, a good ¯rst approximation for satellite motion is obtained by assuming the spacecraft is a point mass or spherical body moving in the gravitational ¯eld of a spherical planet. This leads to the classical two-body problem. Since we use the term body to refer to a spacecraft of ¯nite size (as in rigid body), it may be more appropriate to call this the two-particle problem, but I will use the term two-body problem in its classical sense. The basic elements of orbital dynamics are captured in Kepler's three laws which he published in the 17th century. His laws were for the orbital motion of the planets about the Sun, but are also applicable to the motion of satellites about planets. The three laws are: 1. The orbit of each planet is an ellipse with the Sun at one focus. 2. The line joining the planet to the Sun sweeps out equal areas in equal times. 3. The square of the period of a planet is proportional to the cube of its mean distance to the sun. The ¯rst law applies to most spacecraft, but it is also possible for spacecraft to travel in parabolic and hyperbolic orbits, in which case the period is in¯nite and the 3rd law does not apply. However, the 2nd law applies to all two-body motion. Newton's 2nd law and his law of universal gravitation provide the tools for generalizing Kepler's laws to non-elliptical orbits, as well as for proving Kepler's laws.
    [Show full text]
  • Elliptical Orbits
    1 Ellipse-geometry 1.1 Parameterization • Functional characterization:(a: semi major axis, b ≤ a: semi minor axis) x2 y 2 b p + = 1 ⇐⇒ y(x) = · ± a2 − x2 (1) a b a • Parameterization in cartesian coordinates, which follows directly from Eq. (1): x a · cos t = with 0 ≤ t < 2π (2) y b · sin t – The origin (0, 0) is the center of the ellipse and the auxilliary circle with radius a. √ – The focal points are located at (±a · e, 0) with the eccentricity e = a2 − b2/a. • Parameterization in polar coordinates:(p: parameter, 0 ≤ < 1: eccentricity) p r(ϕ) = (3) 1 + e cos ϕ – The origin (0, 0) is the right focal point of the ellipse. – The major axis is given by 2a = r(0) − r(π), thus a = p/(1 − e2), the center is therefore at − pe/(1 − e2), 0. – ϕ = 0 corresponds to the periapsis (the point closest to the focal point; which is also called perigee/perihelion/periastron in case of an orbit around the Earth/sun/star). The relation between t and ϕ of the parameterizations in Eqs. (2) and (3) is the following: t r1 − e ϕ tan = · tan (4) 2 1 + e 2 1.2 Area of an elliptic sector As an ellipse is a circle with radius a scaled by a factor b/a in y-direction (Eq. 1), the area of an elliptic sector PFS (Fig. ??) is just this fraction of the area PFQ in the auxiliary circle. b t 2 1 APFS = · · πa − · ae · a sin t a 2π 2 (5) 1 = (t − e sin t) · a b 2 The area of the full ellipse (t = 2π) is then, of course, Aellipse = π a b (6) Figure 1: Ellipse and auxilliary circle.
    [Show full text]
  • New Closed-Form Solutions for Optimal Impulsive Control of Spacecraft Relative Motion
    New Closed-Form Solutions for Optimal Impulsive Control of Spacecraft Relative Motion Michelle Chernick∗ and Simone D'Amicoy Aeronautics and Astronautics, Stanford University, Stanford, California, 94305, USA This paper addresses the fuel-optimal guidance and control of the relative motion for formation-flying and rendezvous using impulsive maneuvers. To meet the requirements of future multi-satellite missions, closed-form solutions of the inverse relative dynamics are sought in arbitrary orbits. Time constraints dictated by mission operations and relevant perturbations acting on the formation are taken into account by splitting the optimal recon- figuration in a guidance (long-term) and control (short-term) layer. Both problems are cast in relative orbit element space which allows the simple inclusion of secular and long-periodic perturbations through a state transition matrix and the translation of the fuel-optimal optimization into a minimum-length path-planning problem. Due to the proper choice of state variables, both guidance and control problems can be solved (semi-)analytically leading to optimal, predictable maneuvering schemes for simple on-board implementation. Besides generalizing previous work, this paper finds four new in-plane and out-of-plane (semi-)analytical solutions to the optimal control problem in the cases of unperturbed ec- centric and perturbed near-circular orbits. A general delta-v lower bound is formulated which provides insight into the optimality of the control solutions, and a strong analogy between elliptic Hohmann transfers and formation-flying control is established. Finally, the functionality, performance, and benefits of the new impulsive maneuvering schemes are rigorously assessed through numerical integration of the equations of motion and a systematic comparison with primer vector optimal control.
    [Show full text]
  • 2. Orbital Mechanics MAE 342 2016
    2/12/20 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 2016 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE342.html 1 1 Orbits 101 Satellites Escape and Capture (Comets, Meteorites) 2 2 1 2/12/20 Two-Body Orbits are Conic Sections 3 3 Classical Orbital Elements Dimension and Time a : Semi-major axis e : Eccentricity t p : Time of perigee passage Orientation Ω :Longitude of the Ascending/Descending Node i : Inclination of the Orbital Plane ω: Argument of Perigee 4 4 2 2/12/20 Orientation of an Elliptical Orbit First Point of Aries 5 5 Orbits 102 (2-Body Problem) • e.g., – Sun and Earth or – Earth and Moon or – Earth and Satellite • Circular orbit: radius and velocity are constant • Low Earth orbit: 17,000 mph = 24,000 ft/s = 7.3 km/s • Super-circular velocities – Earth to Moon: 24,550 mph = 36,000 ft/s = 11.1 km/s – Escape: 25,000 mph = 36,600 ft/s = 11.3 km/s • Near escape velocity, small changes have huge influence on apogee 6 6 3 2/12/20 Newton’s 2nd Law § Particle of fixed mass (also called a point mass) acted upon by a force changes velocity with § acceleration proportional to and in direction of force § Inertial reference frame § Ratio of force to acceleration is the mass of the particle: F = m a d dv(t) ⎣⎡mv(t)⎦⎤ = m = ma(t) = F ⎡ ⎤ dt dt vx (t) ⎡ f ⎤ ⎢ ⎥ x ⎡ ⎤ d ⎢ ⎥ fx f ⎢ ⎥ m ⎢ vy (t) ⎥ = ⎢ y ⎥ F = fy = force vector dt
    [Show full text]
  • Orbital Mechanics Course Notes
    Orbital Mechanics Course Notes David J. Westpfahl Professor of Astrophysics, New Mexico Institute of Mining and Technology March 31, 2011 2 These are notes for a course in orbital mechanics catalogued as Aerospace Engineering 313 at New Mexico Tech and Aerospace Engineering 362 at New Mexico State University. This course uses the text “Fundamentals of Astrodynamics” by R.R. Bate, D. D. Muller, and J. E. White, published by Dover Publications, New York, copyright 1971. The notes do not follow the book exclusively. Additional material is included when I believe that it is needed for clarity, understanding, historical perspective, or personal whim. We will cover the material recommended by the authors for a one-semester course: all of Chapter 1, sections 2.1 to 2.7 and 2.13 to 2.15 of Chapter 2, all of Chapter 3, sections 4.1 to 4.5 of Chapter 4, and as much of Chapters 6, 7, and 8 as time allows. Purpose The purpose of this course is to provide an introduction to orbital me- chanics. Students who complete the course successfully will be prepared to participate in basic space mission planning. By basic mission planning I mean the planning done with closed-form calculations and a calculator. Stu- dents will have to master additional material on numerical orbit calculation before they will be able to participate in detailed mission planning. There is a lot of unfamiliar material to be mastered in this course. This is one field of human endeavor where engineering meets astronomy and ce- lestial mechanics, two fields not usually included in an engineering curricu- lum.
    [Show full text]
  • A Delta-V Map of the Known Main Belt Asteroids
    Acta Astronautica 146 (2018) 73–82 Contents lists available at ScienceDirect Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro A Delta-V map of the known Main Belt Asteroids Anthony Taylor *, Jonathan C. McDowell, Martin Elvis Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA, USA ARTICLE INFO ABSTRACT Keywords: With the lowered costs of rocket technology and the commercialization of the space industry, asteroid mining is Asteroid mining becoming both feasible and potentially profitable. Although the first targets for mining will be the most accessible Main belt near Earth objects (NEOs), the Main Belt contains 106 times more material by mass. The large scale expansion of NEO this new asteroid mining industry is contingent on being able to rendezvous with Main Belt asteroids (MBAs), and Delta-v so on the velocity change required of mining spacecraft (delta-v). This paper develops two different flight burn Shoemaker-Helin schemes, both starting from Low Earth Orbit (LEO) and ending with a successful MBA rendezvous. These methods are then applied to the 700,000 asteroids in the Minor Planet Center (MPC) database with well-determined orbits to find low delta-v mining targets among the MBAs. There are 3986 potential MBA targets with a delta- v < 8kmsÀ1, but the distribution is steep and reduces to just 4 with delta-v < 7kms-1. The two burn methods are compared and the orbital parameters of low delta-v MBAs are explored. 1. Introduction [2]. A running tabulation of the accessibility of NEOs is maintained on- line by Lance Benner3 and identifies 65 NEOs with delta-v< 4:5kms-1.A For decades, asteroid mining and exploration has been largely dis- separate database based on intensive numerical trajectory calculations is missed as infeasible and unprofitable.
    [Show full text]
  • Orbital Mechanics
    Orbital Mechanics Part 1 Orbital Forces Why a Sat. remains in orbit ? Bcs the centrifugal force caused by the Sat. rotation around earth is counter- balanced by the Earth's Pull. Kepler’s Laws The Satellite (Spacecraft) which orbits the earth follows the same laws that govern the motion of the planets around the sun. J. Kepler (1571-1630) was able to derive empirically three laws describing planetary motion I. Newton was able to derive Keplers laws from his own laws of mechanics [gravitation theory] Kepler’s 1st Law (Law of Orbits) The path followed by a Sat. (secondary body) orbiting around the primary body will be an ellipse. The center of mass (barycenter) of a two-body system is always centered on one of the foci (earth center). Kepler’s 1st Law (Law of Orbits) The eccentricity (abnormality) e: a 2 b2 e a b- semiminor axis , a- semimajor axis VIN: e=0 circular orbit 0<e<1 ellip. orbit Orbit Calculations Ellipse is the curve traced by a point moving in a plane such that the sum of its distances from the foci is constant. Kepler’s 2nd Law (Law of Areas) For equal time intervals, a Sat. will sweep out equal areas in its orbital plane, focused at the barycenter VIN: S1>S2 at t1=t2 V1>V2 Max(V) at Perigee & Min(V) at Apogee Kepler’s 3rd Law (Harmonic Law) The square of the periodic time of orbit is proportional to the cube of the mean distance between the two bodies. a 3 n 2 n- mean motion of Sat.
    [Show full text]
  • NOAA Technical Memorandum ERL ARL-94
    NOAA Technical Memorandum ERL ARL-94 THE NOAA SOLAR EPHEMERIS PROGRAM Albion D. Taylor Air Resources Laboratories Silver Spring, Maryland January 1981 NOAA 'Technical Memorandum ERL ARL-94 THE NOAA SOLAR EPHEMERlS PROGRAM Albion D. Taylor Air Resources Laboratories Silver Spring, Maryland January 1981 NOTICE The Environmental Research Laboratories do not approve, recommend, or endorse any proprietary product or proprietary material mentioned in this publication. No reference shall be made to the Environmental Research Laboratories or to this publication furnished by the Environmental Research Laboratories in any advertising or sales promotion which would indicate or imply that the Environmental Research Laboratories approve, recommend, or endorse any proprietary product or proprietary material mentioned herein, or which has as its purpose an intent to cause directly or indirectly the advertised product to be used or purchased because of this Environmental Research Laboratories publication. Abstract A system of FORTRAN language computer programs is presented which have the ability to locate the sun at arbitrary times. On demand, the programs will return the distance and direction to the sun, either as seen by an observer at an arbitrary location on the Earth, or in a stan- dard astronomic coordinate system. For one century before or after the year 1960, the program is expected to have an accuracy of 30 seconds 5 of arc (2 seconds of time) in angular position, and 7 10 A.U. in distance. A non-standard algorithm is used which minimizes the number of trigonometric evaluations involved in the computations. 1 The NOAA Solar Ephemeris Program Albion D. Taylor National Oceanic and Atmospheric Administration Air Resources Laboratories Silver Spring, MD January 1981 Contents 1 Introduction 3 2 Use of the Solar Ephemeris Subroutines 3 3 Astronomical Terminology and Coordinate Systems 5 4 Computation Methods for the NOAA Solar Ephemeris 11 5 References 16 A Program Listings 17 A.1 SOLEFM .
    [Show full text]
  • N-Body Problem: Analytical and Numerical Approaches
    N-body Problem: Analytical and Numerical Approaches Ammar Sakaji Regional Center for Space Science and Technology Education for Western Asia. Jordan/ United Nations Center of Theoretical Physics and Astrophysics CTPA, Amman-Jordan Jordanian Astronomical Society The Arab Union For Astronomy and Space Sciences Abstract • We present an over view of the Hamiltonian of the N-Body problem with some special cases (two- and three-body problems) in view of classical mechanics and General Theory of Relativity. Several applicational models of orbital motions are discussed extensively up to High eccentric problems (Barker’s equation, parabolic eccentricity…etc.) by using several mathematical techniques with high accurate computations and perturbation such as: Lagrange’s problem, Bessel functions, Gauss method…..etc. • As an example; we will give an overview of the satellite the position and velocity components of the Jordan first CubeSat: JY-SAT which was launched by SpaceX Falcon 9 in December 3, 2018. N-Body Problem N-Body Problem with weak gravitational interactions without Dark Matter (galactic dynamics) • Solving N-Body Problem is an important to understand the motions of the solar system ( Sun, Moon, planets), and visible stars, as well as understanding the dynamics of globular cluster star systems Weak Gravitational Interactions • The Lagrangain for the N-Body system for the weak gravitational interactions can be written as: (Newtonian Dynamics Limit) 푁 1 2 퐺푚푖푚푗 ℒ = ෍ 푚푖 ∥ 푞ሶ푖 ∥ − ෍ 2 ∥ 푞푖 − 푞푗 ∥ 푖=1 1≤푖<푗≤푁 • Where is 푞푖 is the generalized coordinates, 푞ሶ푖 is related to generalized momenta, and ℒ is the Lagrangian of the N-Body system. • Using the principle of least action, the above equation can be written in terms of Euler-Lagrange equations: 3N nonlinear differential equations as: 푑 휕ℒ 휕ℒ − 푑푡 휕푞ሶ푖 휕푞푖 General Relativity orbits (strong gravitational interactions) The two-body problem in general relativity is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity EFEs.
    [Show full text]
  • Elliptical Orbits
    CHAPTER ELLIPTICAL e ORBITS (0< <1) CHAPTER CONTENT: Page 120 / 338 8- ELLIPTICAL ORBITS (0<e<1) Page 121 / 338 8- ELLIPTICAL ORBITS (0<e<1) (NOTE13,P55,{1}) If f ( ) r 0 2 2 h 1 h 1 (1) rp r ra 1 e 1 e 0 180 The curve defined by orbit equation is an ellipse: Page 122 / 338 8- ELLIPTICAL ORBITS (0<e<1) Let 2a be the distance measured along the apse line from periapsis P to apoapsis A, as illustrated in figure, then (2) Substituting and values into (2), we get: (3) a is the semimajor axis of the ellipse. Page 123 / 338 8- ELLIPTICAL ORBITS (0<e<1) Solving equation (3) for and putting the result into orbit equation yields an alternative form of the orbit equation: (4) Let F denote the location of the body , which is the origin of the polar coordinate system. The center C of the ellipse is the point lying midway between the apoapsis and periapsis. Page 124 / 338 8- ELLIPTICAL ORBITS (0<e<1) From equation (4) we have: (5) So as indicated in the previous figure. If the true anomaly of point B is , then according to equation (4), the radial coordinate of B is: (6) Page 125 / 338 8- ELLIPTICAL ORBITS (0<e<1) The projection of onto the apse line is ae: Solving this expression for e, we obtain (7) Substituting this result into equation (6) we get According to the Pythagorean theorem, (8) Page 126 / 338 8- ELLIPTICAL ORBITS (0<e<1) Let an xy cartesian coordinate system be centered at C, In terms of , we see that: From, this we have: (9) For the y coordinate we have (by using equation (8)): Therefore: (10) Page
    [Show full text]
  • Orbital Mechanics Joe Spier, K6WAO – AMSAT Director for Education ARRL 100Th Centennial Educational Forum 1 History
    Orbital Mechanics Joe Spier, K6WAO – AMSAT Director for Education ARRL 100th Centennial Educational Forum 1 History Astrology » Pseudoscience based on several systems of divination based on the premise that there is a relationship between astronomical phenomena and events in the human world. » Many cultures have attached importance to astronomical events, and the Indians, Chinese, and Mayans developed elaborate systems for predicting terrestrial events from celestial observations. » In the West, astrology most often consists of a system of horoscopes purporting to explain aspects of a person's personality and predict future events in their life based on the positions of the sun, moon, and other celestial objects at the time of their birth. » The majority of professional astrologers rely on such systems. 2 History Astronomy » Astronomy is a natural science which is the study of celestial objects (such as stars, galaxies, planets, moons, and nebulae), the physics, chemistry, and evolution of such objects, and phenomena that originate outside the atmosphere of Earth, including supernovae explosions, gamma ray bursts, and cosmic microwave background radiation. » Astronomy is one of the oldest sciences. » Prehistoric cultures have left astronomical artifacts such as the Egyptian monuments and Nubian monuments, and early civilizations such as the Babylonians, Greeks, Chinese, Indians, Iranians and Maya performed methodical observations of the night sky. » The invention of the telescope was required before astronomy was able to develop into a modern science. » Historically, astronomy has included disciplines as diverse as astrometry, celestial navigation, observational astronomy and the making of calendars, but professional astronomy is nowadays often considered to be synonymous with astrophysics.
    [Show full text]
  • The Effects of Barycentric and Asymmetric Transverse Velocities on Eclipse and Transit Times
    The Astrophysical Journal, 854:163 (13pp), 2018 February 20 https://doi.org/10.3847/1538-4357/aaa3ea © 2018. The American Astronomical Society. All rights reserved. The Effects of Barycentric and Asymmetric Transverse Velocities on Eclipse and Transit Times Kyle E. Conroy1,2 , Andrej Prša2 , Martin Horvat2,3 , and Keivan G. Stassun1,4 1 Vanderbilt University, Departmentof Physics and Astronomy, 6301 Stevenson Center Lane, Nashville TN, 37235, USA 2 Villanova University, Department of Astrophysics and Planetary Sciences, 800 E. Lancaster Avenue, Villanova, PA 19085, USA 3 University of Ljubljana, Departmentof Physics, Jadranska 19, SI-1000 Ljubljana, Slovenia 4 Fisk University, Department of Physics, 1000 17th Avenue N., Nashville, TN 37208, USA Received 2017 October 20; revised 2017 December 19; accepted 2017 December 19; published 2018 February 21 Abstract It has long been recognized that the finite speed of light can affect the observed time of an event. For example, as a source moves radially toward or away from an observer, the path length and therefore the light travel time to the observer decreases or increases, causing the event to appear earlier or later than otherwise expected, respectively. This light travel time effect has been applied to transits and eclipses for a variety of purposes, including studies of eclipse timing variations and transit timing variations that reveal the presence of additional bodies in the system. Here we highlight another non-relativistic effect on eclipse or transit times arising from the finite speed of light— caused by an asymmetry in the transverse velocity of the two eclipsing objects, relative to the observer. This asymmetry can be due to a non-unity mass ratio or to the presence of external barycentric motion.
    [Show full text]