Position of Sun on Celestial Sphere at Input Universal Time (Ut)
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A Study of Ancient Khmer Ephemerides
A study of ancient Khmer ephemerides François Vernotte∗ and Satyanad Kichenassamy** November 5, 2018 Abstract – We study ancient Khmer ephemerides described in 1910 by the French engineer Faraut, in order to determine whether they rely on observations carried out in Cambodia. These ephemerides were found to be of Indian origin and have been adapted for another longitude, most likely in Burma. A method for estimating the date and place where the ephemerides were developed or adapted is described and applied. 1 Introduction Our colleague Prof. Olivier de Bernon, from the École Française d’Extrême Orient in Paris, pointed out to us the need to understand astronomical systems in Cambo- dia, as he surmised that astronomical and mathematical ideas from India may have developed there in unexpected ways.1 A proper discussion of this problem requires an interdisciplinary approach where history, philology and archeology must be sup- plemented, as we shall see, by an understanding of the evolution of Astronomy and Mathematics up to modern times. This line of thought meets other recent lines of research, on the conceptual evolution of Mathematics, and on the definition and measurement of time, the latter being the main motivation of Indian Astronomy. In 1910 [1], the French engineer Félix Gaspard Faraut (1846–1911) described with great care the method of computing ephemerides in Cambodia used by the horas, i.e., the Khmer astronomers/astrologers.2 The names for the astronomical luminaries as well as the astronomical quantities [1] clearly show the Indian origin ∗F. Vernotte is with UTINAM, Observatory THETA of Franche Comté-Bourgogne, University of Franche Comté/UBFC/CNRS, 41 bis avenue de l’observatoire - B.P. -
Planet Positions: 1 Planet Positions
Planet Positions: 1 Planet Positions As the planets orbit the Sun, they move around the celestial sphere, staying close to the plane of the ecliptic. As seen from the Earth, the angle between the Sun and a planet -- called the elongation -- constantly changes. We can identify a few special configurations of the planets -- those positions where the elongation is particularly noteworthy. The inferior planets -- those which orbit closer INFERIOR PLANETS to the Sun than Earth does -- have configurations as shown: SC At both superior conjunction (SC) and inferior conjunction (IC), the planet is in line with the Earth and Sun and has an elongation of 0°. At greatest elongation, the planet reaches its IC maximum separation from the Sun, a value GEE GWE dependent on the size of the planet's orbit. At greatest eastern elongation (GEE), the planet lies east of the Sun and trails it across the sky, while at greatest western elongation (GWE), the planet lies west of the Sun, leading it across the sky. Best viewing for inferior planets is generally at greatest elongation, when the planet is as far from SUPERIOR PLANETS the Sun as it can get and thus in the darkest sky possible. C The superior planets -- those orbiting outside of Earth's orbit -- have configurations as shown: A planet at conjunction (C) is lined up with the Sun and has an elongation of 0°, while a planet at opposition (O) lies in the opposite direction from the Sun, at an elongation of 180°. EQ WQ Planets at quadrature have elongations of 90°. -
Sun Tool Options
Sun Study Tools Sophomore Architecture Studio: Lighting Lecture 1: • Introduction to Daylight (part 1) • Survey of the Color Spectrum • Making Light • Controlling Light Lecture 2: • Daylight (part 2) • Design Tools to study Solar Design • Architectural Applications Lecture 3: • Light in Architecture • Lighting Design Strategies Sun Tool Options 1. Paper and Pencil 2. Build a Model 3. Use a Computer The first step to any of these options is to define…. Where is the site? 1 2 Sun Study Tools North Latitude and Longitude South Longitude Latitude Longitude Sun Study Tools North America Latitude 3 United States e ud tit La Sun Study Tools New York 72w 44n 42n Site Location The site location is specified by a latitude l and a longitude L. Latitudes and longitudes may be found in any standard atlas or almanac. Chart shows the latitudes and longitudes of some North American cities. Conventions used in expressing latitudes are: Positive = northern hemisphere Negative = southern hemisphere Conventions used in expressing longitudes are: Positive = west of prime meridian (Greenwich, United Kingdom) Latitude and Longitude of Some North American Cities Negative = east of prime meridian 4 Sun Study Tools Solar Path Suns Position The position of the sun is specified by the solar altitude and solar azimuth and is a function of site latitude, solar time, and solar declination. 5 Sun Study Tools Suns Position The rotation of the earth about its axis, as well as its revolution about the sun, produces an apparent motion of the sun with respect to any point on the altitude earth's surface. The position of the sun with respect to such a point is expressed in terms of two angles: azimuth The sun's position in terms of solar altitude (a ) and azimuth (a ) solar azimuth, which is the t s with respect to the cardinal points of the compass. -
Equatorial and Cartesian Coordinates • Consider the Unit Sphere (“Unit”: I.E
Coordinate Transforms Equatorial and Cartesian Coordinates • Consider the unit sphere (“unit”: i.e. declination the distance from the center of the (δ) sphere to its surface is r = 1) • Then the equatorial coordinates Equator can be transformed into Cartesian coordinates: right ascension (α) – x = cos(α) cos(δ) – y = sin(α) cos(δ) z x – z = sin(δ) y • It can be much easier to use Cartesian coordinates for some manipulations of geometry in the sky Equatorial and Cartesian Coordinates • Consider the unit sphere (“unit”: i.e. the distance y x = Rcosα from the center of the y = Rsinα α R sphere to its surface is r = 1) x Right • Then the equatorial Ascension (α) coordinates can be transformed into Cartesian coordinates: declination (δ) – x = cos(α)cos(δ) z r = 1 – y = sin(α)cos(δ) δ R = rcosδ R – z = sin(δ) z = rsinδ Precession • Because the Earth is not a perfect sphere, it wobbles as it spins around its axis • This effect is known as precession • The equatorial coordinate system relies on the idea that the Earth rotates such that only Right Ascension, and not declination, is a time-dependent coordinate The effects of Precession • Currently, the star Polaris is the North Star (it lies roughly above the Earth’s North Pole at δ = 90oN) • But, over the course of about 26,000 years a variety of different points in the sky will truly be at δ = 90oN • The declination coordinate is time-dependent albeit on very long timescales • A precise astronomical coordinate system must account for this effect Equatorial coordinates and equinoxes • To account -
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. -
2 Coordinate Systems
2 Coordinate systems In order to find something one needs a system of coordinates. For determining the positions of the stars and planets where the distance to the object often is unknown it usually suffices to use two coordinates. On the other hand, since the Earth rotates around it’s own axis as well as around the Sun the positions of stars and planets is continually changing, and the measurment of when an object is in a certain place is as important as deciding where it is. Our first task is to decide on a coordinate system and the position of 1. The origin. E.g. one’s own location, the center of the Earth, the, the center of the Solar System, the Galaxy, etc. 2. The fundamental plan (x−y plane). This is often a plane of some physical significance such as the horizon, the equator, or the ecliptic. 3. Decide on the direction of the positive x-axis, also known as the “reference direction”. 4. And, finally, on a convention of signs of the y− and z− axes, i.e whether to use a left-handed or right-handed coordinate system. For example Eratosthenes of Cyrene (c. 276 BC c. 195 BC) was a Greek mathematician, elegiac poet, athlete, geographer, astronomer, and music theo- rist who invented a system of latitude and longitude. (According to Wikipedia he was also the first person to use the word geography and invented the disci- pline of geography as we understand it.). The origin of this coordinate system was the center of the Earth and the fundamental plane was the equator, which location Eratosthenes calculated relative to the parts of the Earth known to him. -
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. -
Lecture 6: Where Is the Sun?
4.430 Daylighting Massachusetts Institute of Technology Christoph Reinhart Department of Architecture 4.430 Where is the sun? Building Technology Program Goals for This Week Where is the sun? Designing Static Shading Systems MIT 4.430 Daylighting, Instructor C Reinhart 1 1 MISC Meeting on group projects Reduce HDR image size via pfilt –x 800 –y 550 filne_name_large.pic > filename_small>.pic Note: pfilt is a Radiance program. You can find further info on pfilt by googeling: “pfilt Radiance” MIT 4.430 Daylighting, Instructor C Reinhart 2 2 Daylight Factor Hand Calculation Mean Daylight Factor according to Lynes Reinhart & LoVerso, Lighting Research & Technology (2010) Move into the building, design the facade openings, room dimensions and depth of the daylit area. Determine the required glazing area using the Lynes formula. A glazing = required glazing area A total = overall interior surface area (not floor area!) R mean = area-weighted mean surface reflectance vis = visual transmittance of glazing units = sun angle 3 ‘Validation’ of Daylight Factor Formula Reinhart & LoVerso, Lighting Research & Technology (2010) Graph of mean daylighting factor according to Lynes formula v. Radiance removed due to copyright restrictions. Source: Figure 5 in Reinhart, C. F., and V. R. M. LoVerso. "A Rules of Thumb Based Design Sequence for Diffuse Daylight." Lighting Research and Technology 42, no. 1 (2010): 7-32. Comparison to Radiance simulations for 2304 spaces. Quality control for simulations. LEED 2.2 Glazing Factor Formula Graph of mean daylighting factor according to LEED 2.2 v. Radiance removed due to copyright restrictions. Source: Figure 11 in Reinhart, C. F., and V. -
Impulsive Maneuvers for Formation Reconfiguration Using Relative Orbital Elements
JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 38, No. 6, June 2015 Impulsive Maneuvers for Formation Reconfiguration Using Relative Orbital Elements G. Gaias∗ and S. D’Amico† DLR, German Aerospace Center, 82234 Wessling, Germany DOI: 10.2514/1.G000189 Advanced multisatellite missions based on formation-flying and on-orbit servicing concepts require the capability to arbitrarily reconfigure the relative motion in an autonomous, fuel efficient, and flexible manner. Realistic flight scenarios impose maneuvering time constraints driven by the satellite bus, by the payload, or by collision avoidance needs. In addition, mission control center planning and operations tasks demand determinism and predictability of the propulsion system activities. Based on these considerations and on the experience gained from the most recent autonomous formation-flying demonstrations in near-circular orbit, this paper addresses and reviews multi- impulsive solution schemes for formation reconfiguration in the relative orbit elements space. In contrast to the available literature, which focuses on case-by-case or problem-specific solutions, this work seeks the systematic search and characterization of impulsive maneuvers of operational relevance. The inversion of the equations of relative motion parameterized using relative orbital elements enables the straightforward computation of analytical or numerical solutions and provides direct insight into the delta-v cost and the most convenient maneuver locations. The resulting general methodology is not only able to refind and requalify all particular solutions known in literature or flown in space, but enables the identification of novel fuel-efficient maneuvering schemes for future onboard implementation. Nomenclature missions. Realistic operational scenarios ask for accomplishing such a = semimajor axis actions in a safe way, within certain levels of accuracy, and in a fuel- B = control input matrix of the relative dynamics efficient manner. -
Using the SFA Star Charts and Understanding the Equatorial Coordinate System
Using the SFA Star Charts and Understanding the Equatorial Coordinate System SFA Star Charts created by Dan Bruton of Stephen F. Austin State University Notes written by Don Carona of Texas A&M University Last Updated: August 17, 2020 The SFA Star Charts are four separate charts. Chart 1 is for the north celestial region and chart 4 is for the south celestial region. These notes refer to the equatorial charts, which are charts 2 & 3 combined to form one long chart. The star charts are based on the Equatorial Coordinate System, which consists of right ascension (RA), declination (DEC) and hour angle (HA). From the northern hemisphere, the equatorial charts can be used when facing south, east or west. At the bottom of the chart, you’ll notice a series of twenty-four numbers followed by the letter “h”, representing “hours”. These hour marks are right ascension (RA), which is the equivalent of celestial longitude. The same point on the 360 degree celestial sphere passes overhead every 24 hours, making each hour of right ascension equal to 1/24th of a circle, or 15 degrees. Each degree of sky, therefore, moves past a stationary point in four minutes. Each hour of right ascension moves past a stationary point in one hour. Every tick mark between the hour marks on the equatorial charts is equal to 5 minutes. Right ascension is noted in ( h ) hours, ( m ) minutes, and ( s ) seconds. The bright star, Antares, in the constellation Scorpius. is located at RA 16h 29m 30s. At the left and right edges of the chart, you will find numbers marked in degrees (°) and being either positive (+) or negative(-). -
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 .