# An Implementation of Sgp4 in Non-Singular Variables Using a Functional Paradigm

Total Page：16

File Type：pdf, Size：1020Kb

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]
• Kepler's Equation—C.E. Mungan, Fall 2004 a Satellite Is Orbiting a Body
Kepler’s Equation—C.E. Mungan, Fall 2004 A satellite is orbiting a body on an ellipse. Denote the position of the satellite relative to the body by plane polar coordinates (r,) . Find the time t that it requires the satellite to move from periapsis (position of closest approach) to angular position . Express your answer in terms of the eccentricity e of the orbit and the period T for a full revolution. According to Kepler’s second law, the ratio of the shaded area A to the total area ab of the ellipse (with semi-major axis a and semi-minor axis b) is the respective ratio of transit times, A t = . (1) ab T But we can divide the shaded area into differential triangles, of base r and height rd , so that A = 1 r2d (2) 2 0 where the polar equation of a Keplerian orbit is c r = (3) 1+ ecos with c the semilatus rectum (distance vertically from the origin on the diagram above to the ellipse). It is not hard to show from the geometrical properties of an ellipse that b = a 1 e2 and c = a(1 e2 ) . (4) Substituting (3) into (2), and then (2) and (4) into (1) gives T (1 e2 )3/2 t = M where M d . (5) 2 2 0 (1 + ecos) In the literature, M is called the mean anomaly. This is an integral solution to the problem. It turns out however that this integral can be evaluated analytically using the following clever change of variables, e + cos cos E = .
[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]
• Synopsis of Euler's Paper E105
1 Synopsis of Euler’s paper E105 -- Memoire sur la plus grande equation des planetes (Memoir on the Maximum value of an Equation of the Planets) Compiled by Thomas J Osler and Jasen Andrew Scaramazza Mathematics Department Rowan University Glassboro, NJ 08028 [email protected] Preface The following summary of E 105 was constructed by abbreviating the collection of Notes. Thus, there is considerable repetition in these two items. We hope that the reader can profit by reading this synopsis before tackling Euler’s paper itself. I. Planetary Motion as viewed from the earth vs the sun ` Euler discusses the fact that planets observed from the earth exhibit a very irregular motion. In general, they move from west to east along the ecliptic. At times however, the motion slows to a stop and the planet even appears to reverse direction and move from east to west. We call this retrograde motion. After some time the planet stops again and resumes its west to east journey. However, if we observe the planet from the stand point of an observer on the sun, this retrograde motion will not occur, and only a west to east path of the planet is seen. II. The aphelion and the perihelion From the sun, (point O in figure 1) the planet (point P ) is seen to move on an elliptical orbit with the sun at one focus. When the planet is farthest from the sun, we say it is at the “aphelion” (point A ), and at the perihelion when it is closest. The time for the planet to move from aphelion to perihelion and back is called the period.
[Show full text]
• Analytical Low-Thrust Trajectory Design Using the Simpliﬁed General Perturbations Model J
Analytical Low-Thrust Trajectory Design using the Simpliﬁed General Perturbations model J. G. P. de Jong November 2018 - Technische Universiteit Delft - Master Thesis Analytical Low-Thrust Trajectory Design using the Simpliﬁed General Perturbations model by J. G. P. de Jong to obtain the degree of Master of Science at the Delft University of Technology. to be defended publicly on Thursday December 20, 2018 at 13:00. Student number: 4001532 Project duration: November 29, 2017 - November 26, 2018 Supervisor: Ir. R. Noomen Thesis committee: Dr. Ir. E.J.O. Schrama TU Delft Ir. R. Noomen TU Delft Dr. S. Speretta TU Delft November 26, 2018 An electronic version of this thesis is available at http://repository.tudelft.nl/. Frontpage picture: NASA. Preface Ever since I was a little girl, I knew I was going to study in Delft. Which study exactly remained unknown until the day before my high school graduation. There I was, reading a ﬂyer about aerospace engineering and suddenly I realized: I was going to study aerospace engineering. During the bachelor it soon became clear that space is the best part of the word aerospace and thus the space ﬂight master was chosen. Looking back this should have been clear already years ago: all those books about space I have read when growing up... After quite some time I have come to the end of my studies. Especially the last years were not an easy journey, but I pulled through and made it to the end. This was not possible without a lot of people and I would like this opportunity to thank them here.
[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]