Orbital Mechanics
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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. -
Astrodynamics
Politecnico di Torino SEEDS SpacE Exploration and Development Systems Astrodynamics II Edition 2006 - 07 - Ver. 2.0.1 Author: Guido Colasurdo Dipartimento di Energetica Teacher: Giulio Avanzini Dipartimento di Ingegneria Aeronautica e Spaziale e-mail: [email protected] Contents 1 Two–Body Orbital Mechanics 1 1.1 BirthofAstrodynamics: Kepler’sLaws. ......... 1 1.2 Newton’sLawsofMotion ............................ ... 2 1.3 Newton’s Law of Universal Gravitation . ......... 3 1.4 The n–BodyProblem ................................. 4 1.5 Equation of Motion in the Two-Body Problem . ....... 5 1.6 PotentialEnergy ................................. ... 6 1.7 ConstantsoftheMotion . .. .. .. .. .. .. .. .. .... 7 1.8 TrajectoryEquation .............................. .... 8 1.9 ConicSections ................................... 8 1.10 Relating Energy and Semi-major Axis . ........ 9 2 Two-Dimensional Analysis of Motion 11 2.1 ReferenceFrames................................. 11 2.2 Velocity and acceleration components . ......... 12 2.3 First-Order Scalar Equations of Motion . ......... 12 2.4 PerifocalReferenceFrame . ...... 13 2.5 FlightPathAngle ................................. 14 2.6 EllipticalOrbits................................ ..... 15 2.6.1 Geometry of an Elliptical Orbit . ..... 15 2.6.2 Period of an Elliptical Orbit . ..... 16 2.7 Time–of–Flight on the Elliptical Orbit . .......... 16 2.8 Extensiontohyperbolaandparabola. ........ 18 2.9 Circular and Escape Velocity, Hyperbolic Excess Speed . .............. 18 2.10 CosmicVelocities -
Stability of the Moons Orbits in Solar System in the Restricted Three-Body
Stability of the Moons orbits in Solar system in the restricted three-body problem Sergey V. Ershkov, Institute for Time Nature Explorations, M.V. Lomonosov's Moscow State University, Leninskie gory, 1-12, Moscow 119991, Russia e-mail: [email protected] Abstract: We consider the equations of motion of three-body problem in a Lagrange form (which means a consideration of relative motions of 3-bodies in regard to each other). Analyzing such a system of equations, we consider in details the case of moon‟s motion of negligible mass m₃ around the 2-nd of two giant-bodies m₁, m₂ (which are rotating around their common centre of masses on Kepler’s trajectories), the mass of which is assumed to be less than the mass of central body. Under assumptions of R3BP, we obtain the equations of motion which describe the relative mutual motion of the centre of mass of 2-nd giant-body m₂ (Planet) and the centre of mass of 3-rd body (Moon) with additional effective mass m₂ placed in that centre of mass (m₂ + m₃), where is the dimensionless dynamical parameter. They should be rotating around their common centre of masses on Kepler‟s elliptic orbits. For negligible effective mass (m₂ + m₃) it gives the equations of motion which should describe a quasi-elliptic orbit of 3-rd body (Moon) around the 2-nd body m₂ (Planet) for most of the moons of the Planets in Solar system. But the orbit of Earth‟s Moon should be considered as non-constant elliptic motion for the effective mass 0.0178m₂ placed in the centre of mass for the 3-rd body (Moon). -
Electric Propulsion System Scaling for Asteroid Capture-And-Return Missions
Electric propulsion system scaling for asteroid capture-and-return missions Justin M. Little⇤ and Edgar Y. Choueiri† Electric Propulsion and Plasma Dynamics Laboratory, Princeton University, Princeton, NJ, 08544 The requirements for an electric propulsion system needed to maximize the return mass of asteroid capture-and-return (ACR) missions are investigated in detail. An analytical model is presented for the mission time and mass balance of an ACR mission based on the propellant requirements of each mission phase. Edelbaum’s approximation is used for the Earth-escape phase. The asteroid rendezvous and return phases of the mission are modeled as a low-thrust optimal control problem with a lunar assist. The numerical solution to this problem is used to derive scaling laws for the propellant requirements based on the maneuver time, asteroid orbit, and propulsion system parameters. Constraining the rendezvous and return phases by the synodic period of the target asteroid, a semi- empirical equation is obtained for the optimum specific impulse and power supply. It was found analytically that the optimum power supply is one such that the mass of the propulsion system and power supply are approximately equal to the total mass of propellant used during the entire mission. Finally, it is shown that ACR missions, in general, are optimized using propulsion systems capable of processing 100 kW – 1 MW of power with specific impulses in the range 5,000 – 10,000 s, and have the potential to return asteroids on the order of 103 104 tons. − Nomenclature -
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Ever Wonder What’s in Molniya? We Do. John T. McGraw J. T. McGraw and Associates, LLC and University of New Mexico Peter C. Zimmer J. T. McGraw and Associates, LLC Mark R. Ackermann J. T. McGraw and Associates, LLC ABSTRACT Molniya orbits are high inclination, high eccentricity orbits which provide the utility of long apogee dwell time over northern continents, with the additional benefit of obviating the largest orbital perturbation introduced by the Earth’s nonspherical (oblate) gravitational potential. We review the few earlier surveys of the Molniya domain and evaluate results from a new, large area unbiased survey of the northern Molniya domain. We detect 120 Molniya objects in a three hour survey of ~ 1300 square degrees of the sky to a limiting magnitude of about 16.5. Future Molniya surveys will discover a significant number of objects, including debris, and monitoring these objects might provide useful data with respect to orbital perturbations including solar radiation and Earth atmosphere drag effects. 1. SPECIALIZED ORBITS Earth Orbital Space (EOS) supports many versions of specialized satellite orbits defined by a combination of satellite mission and orbital dynamics. Surely the most well-known family of specialized orbits is the geostationary orbits proposed by science fiction author Arthur C. Clarke in 1945 [1] that lie sensibly in the plane of Earth’s equator, with orbital period that matches the Earth’s rotation period. Satellites in these orbits, and the closely related geosynchronous orbits, appear from Earth to remain constantly overhead, allowing continuous communication with the majority of the hemisphere below. Constellations of three geostationary satellites equally spaced in orbit (~ 120° separation) can maintain near-global communication and terrestrial surveillance. -
Stellar Dynamics and Stellar Phenomena Near a Massive Black Hole
Stellar Dynamics and Stellar Phenomena Near A Massive Black Hole Tal Alexander Department of Particle Physics and Astrophysics, Weizmann Institute of Science, 234 Herzl St, Rehovot, Israel 76100; email: [email protected] | Author's original version. To appear in Annual Review of Astronomy and Astrophysics. See final published version in ARA&A website: www.annualreviews.org/doi/10.1146/annurev-astro-091916-055306 Annu. Rev. Astron. Astrophys. 2017. Keywords 55:1{41 massive black holes, stellar kinematics, stellar dynamics, Galactic This article's doi: Center 10.1146/((please add article doi)) Copyright c 2017 by Annual Reviews. Abstract All rights reserved Most galactic nuclei harbor a massive black hole (MBH), whose birth and evolution are closely linked to those of its host galaxy. The unique conditions near the MBH: high velocity and density in the steep po- tential of a massive singular relativistic object, lead to unusual modes of stellar birth, evolution, dynamics and death. A complex network of dynamical mechanisms, operating on multiple timescales, deflect stars arXiv:1701.04762v1 [astro-ph.GA] 17 Jan 2017 to orbits that intercept the MBH. Such close encounters lead to ener- getic interactions with observable signatures and consequences for the evolution of the MBH and its stellar environment. Galactic nuclei are astrophysical laboratories that test and challenge our understanding of MBH formation, strong gravity, stellar dynamics, and stellar physics. I review from a theoretical perspective the wide range of stellar phe- nomena that occur near MBHs, focusing on the role of stellar dynamics near an isolated MBH in a relaxed stellar cusp. -
Pointing Analysis and Design Drivers for Low Earth Orbit Satellite Quantum Key Distribution Jeremiah A
Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-24-2016 Pointing Analysis and Design Drivers for Low Earth Orbit Satellite Quantum Key Distribution Jeremiah A. Specht Follow this and additional works at: https://scholar.afit.edu/etd Part of the Information Security Commons, and the Space Vehicles Commons Recommended Citation Specht, Jeremiah A., "Pointing Analysis and Design Drivers for Low Earth Orbit Satellite Quantum Key Distribution" (2016). Theses and Dissertations. 451. https://scholar.afit.edu/etd/451 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. POINTING ANALYSIS AND DESIGN DRIVERS FOR LOW EARTH ORBIT SATELLITE QUANTUM KEY DISTRIBUTION THESIS Jeremiah A. Specht, 1st Lt, USAF AFIT-ENY-MS-16-M-241 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio DISTRIBUTION STATEMENT A. APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. AFIT-ENY-MS-16-M-241 POINTING ANALYSIS AND DESIGN DRIVERS FOR LOW EARTH ORBIT SATELLITE QUANTUM KEY DISTRIBUTION THESIS Presented to the Faculty Department of Aeronautics and Astronautics Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command In Partial Fulfillment of the Requirements for the Degree of Master of Science in Space Systems Jeremiah A. -
Positioning: Drift Orbit and Station Acquisition
Orbits Supplement GEOSTATIONARY ORBIT PERTURBATIONS INFLUENCE OF ASPHERICITY OF THE EARTH: The gravitational potential of the Earth is no longer µ/r, but varies with longitude. A tangential acceleration is created, depending on the longitudinal location of the satellite, with four points of stable equilibrium: two stable equilibrium points (L 75° E, 105° W) two unstable equilibrium points ( 15° W, 162° E) This tangential acceleration causes a drift of the satellite longitude. Longitudinal drift d'/dt in terms of the longitude about a point of stable equilibrium expresses as: (d/dt)2 - k cos 2 = constant Orbits Supplement GEO PERTURBATIONS (CONT'D) INFLUENCE OF EARTH ASPHERICITY VARIATION IN THE LONGITUDINAL ACCELERATION OF A GEOSTATIONARY SATELLITE: Orbits Supplement GEO PERTURBATIONS (CONT'D) INFLUENCE OF SUN & MOON ATTRACTION Gravitational attraction by the sun and moon causes the satellite orbital inclination to change with time. The evolution of the inclination vector is mainly a combination of variations: period 13.66 days with 0.0035° amplitude period 182.65 days with 0.023° amplitude long term drift The long term drift is given by: -4 dix/dt = H = (-3.6 sin M) 10 ° /day -4 diy/dt = K = (23.4 +.2.7 cos M) 10 °/day where M is the moon ascending node longitude: M = 12.111 -0.052954 T (T: days from 1/1/1950) 2 2 2 2 cos d = H / (H + K ); i/t = (H + K ) Depending on time within the 18 year period of M d varies from 81.1° to 98.9° i/t varies from 0.75°/year to 0.95°/year Orbits Supplement GEO PERTURBATIONS (CONT'D) INFLUENCE OF SUN RADIATION PRESSURE Due to sun radiation pressure, eccentricity arises: EFFECT OF NON-ZERO ECCENTRICITY L = difference between longitude of geostationary satellite and geosynchronous satellite (24 hour period orbit with e0) With non-zero eccentricity the satellite track undergoes a periodic motion about the subsatellite point at perigee. -
Molniya" Type Orbits
EXAMINATION OF THE LIFETIME, EVOLUTION AND RE-ENTRY FEATURES FOR THE "MOLNIYA" TYPE ORBITS Yu.F. Kolyuka, N.M. Ivanov, T.I. Afanasieva, T.A. Gridchina Mission Control Center, 4, Pionerskaya str., Korolev, Moscow Region, 141070, Russia, [email protected] ABSTRACT Space vehicles of the "Molniya" series, launched into the special highly elliptical orbits with a period of ~ 12 hours, are intended for solution of telecommunication problems in stretched territories of the former USSR and nowadays − Russia, and also for providing the connectivity between Russia and other countries. The inclination of standard orbits of such a kind of satellites is near to the critical value i ≈ 63.4 deg and their initial minimal altitude normally has a value Hmin ~ 500 km. As a rule, the “Molniya" satellites of a concrete series form the groups with determined disposition of orbital planes on an ascending node longitude that allows to ensure requirements of the continuous link between any points in northern hemisphere. The first satellite of the "Molniya" series has been inserted into designed orbit in 1964. Up to now it is launched over 160 space vehicles of such a kind. As well as all artificial satellites, space vehicles "Molniya" undergo the action of various perturbing forces influencing a change of their orbital parameters. However in case of space vehicles "Molniya" these changes of parameters have a specific character and in many things they differ from the perturbations which are taking place for the majority of standard orbits of the artificial satellites with rather small eccentricity. In particular, to strong enough variations (first of all, at the expense of the luni-solar attraction) it is subjected the orbit perigee distance rπ that can lead to such reduction Hmin when further flight of the satellite on its orbit becomes impossible, it re-entries and falls to the Earth. -
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. -
Interplanetary Trajectories in STK in a Few Hundred Easy Steps*
Interplanetary Trajectories in STK in a Few Hundred Easy Steps* (*and to think that last year’s students thought some guidance would be helpful!) Satellite ToolKit Interplanetary Tutorial STK Version 9 INITIAL SETUP 1) Open STK. Choose the “Create a New Scenario” button. 2) Name your scenario and, if you would like, enter a description for it. The scenario time is not too critical – it will be updated automatically as we add segments to our mission. 3) STK will give you the opportunity to insert a satellite. (If it does not, or you would like to add another satellite later, you can click on the Insert menu at the top and choose New…) The Orbit Wizard is an easy way to add satellites, but we will choose Define Properties instead. We choose Define Properties directly because we want to use a maneuver-based tool called the Astrogator, which will undo any initial orbit set using the Orbit Wizard. Make sure Satellite is selected in the left pane of the Insert window, then choose Define Properties in the right-hand pane and click the Insert…button. 4) The Properties window for the Satellite appears. You can access this window later by right-clicking or double-clicking on the satellite’s name in the Object Browser (the left side of the STK window). When you open the Properties window, it will default to the Basic Orbit screen, which happens to be where we want to be. The Basic Orbit screen allows you to choose what kind of numerical propagator STK should use to move the satellite. -
Open Rosen Thesis.Pdf
THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE DEPARTMENT OF AEROSPACE ENGINEERING END OF LIFE DISPOSAL OF SATELLITES IN HIGHLY ELLIPTICAL ORBITS MITCHELL ROSEN SPRING 2019 A thesis submitted in partial fulfillment of the requirements for a baccalaureate degree in Aerospace Engineering with honors in Aerospace Engineering Reviewed and approved* by the following: Dr. David Spencer Professor of Aerospace Engineering Thesis Supervisor Dr. Mark Maughmer Professor of Aerospace Engineering Honors Adviser * Signatures are on file in the Schreyer Honors College. i ABSTRACT Highly elliptical orbits allow for coverage of large parts of the Earth through a single satellite, simplifying communications in the globe’s northern reaches. These orbits are able to avoid drastic changes to the argument of periapse by using a critical inclination (63.4°) that cancels out the first level of the geopotential forces. However, this allows the next level of geopotential forces to take over, quickly de-orbiting satellites. Thus, a balance between the rate of change of the argument of periapse and the lifetime of the orbit is necessitated. This thesis sets out to find that balance. It is determined that an orbit with an inclination of 62.5° strikes that balance best. While this orbit is optimal off of the critical inclination, it is still near enough that to allow for potential use of inclination changes as a deorbiting method. Satellites are deorbited when the propellant remaining is enough to perform such a maneuver, and nothing more; therefore, the less change in velocity necessary for to deorbit, the better. Following the determination of an ideal highly elliptical orbit, the different methods of inclination change is tested against the usual method for deorbiting a satellite, an apoapse burn to lower the periapse, to find the most propellant- efficient method.