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Sun-Earth Kinematics, the Equation of Time, Insolation and the Solar

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Sun-Earth Kinematics, the Equation of Time, Insolation and the Solar Sun-Earth Kinematics, the Equation of Time, Insolation and the Solar Analemma Phil Lucht Rimrock Digital Technology, Salt Lake City, Utah 84103 last update: January 8, 2013 Maple code is available upon request. Comments and errata are welcome. The material in this document is copyrighted by the author. The graphics look ratty in Windows Adobe PDF viewers when not scaled up, but look just fine in this excellent freeware viewer: http://www.tracker-software.com/pdf-xchange-products-comparison-chart . Overview ..................................................................................................................................................4 Summary..................................................................................................................................................5 1. Geometry of the Earth's Orbit.........................................................................................................10 (a) The Heliocentric Picture................................................................................................................10 (b) Development of the Geocentric Picture ........................................................................................13 (c) Relation between the three angular velocities ...............................................................................15 (d) Equation of the ellipse...................................................................................................................16 (e) The Anomalies...............................................................................................................................17 2. The Sun's Apparent Path in the Sky ...............................................................................................18 3. The Celestial Sphere and the Sun's Daily and Annual Paths Thereon ........................................20 (a) The Celestial Sphere and the Tropics of Cancer and Capricorn....................................................20 (b) Why is R >> REarth for the celestial sphere? ................................................................................22 (c) Annual apparent motion of the sun around the earth: the ecliptic.................................................23 (d) Precession effects ..........................................................................................................................27 (e) Daily sun paths for various observation points..............................................................................29 (f) The Arctic and Antarctic Circles ...................................................................................................33 4. How the Earth's Orbital Azimuthal Position ψ Varies with Time due to Eccentricity ..............35 (a) A simple solution for ψ(t)..............................................................................................................35 (b) A systematic solution for ψ(t) and estimate of error in using (4.17).............................................40 5. How the Earth's Tilt Angle θt Varies with Time ...........................................................................44 6. Coordinates of the Sun on the Celestial Sphere .............................................................................47 (a) Cartesian coordinates x,y,z and velocity of the sun in Frame S ....................................................47 (b) Spherical coordinates θ, φ of the sun in Frame S..........................................................................48 (c) The Equation of Time....................................................................................................................51 (d) Plots of φ (right ascension of the sun) and Δφ (equation of time) ................................................52 (e) Plots of θt (declination of the sun) ................................................................................................59 7. Coordinates of the Sun in an Arbitrary Frame S' Glued to the Earth.........................................61 (a) Setup: θ1, φ1, φb, φb0, τ, tMe...........................................................................................................61 (b) Obtaining a value for φb0 in terms of φG0 ......................................................................................63 (c) Cartesian coordinates x',y',z' and velocity of the sun in Frame S' .................................................64 (d) Spherical coordinates θ',φ' of the sun in Frame S'.........................................................................67 1 8. The Equation of Time Revisited .....................................................................................................70 (a) Definition of new azimuths Φ' and Φ's..........................................................................................70 (b) Local Time, Sundial Time and the Equation of Time...................................................................72 9. Noon and Midnight, Sunrise and Sunset.........................................................................................75 (a) Conditions for Noon and Midnight ...............................................................................................75 (b) Local time of Noon and Midnight : approximations.....................................................................77 (c) Another approach to noon .............................................................................................................79 (d) Sunrise and Sunset, Length of Day, Noon Elevation of the Sun...................................................80 10. Insolation ........................................................................................................................................86 (a) The tss function and hours of daylight ..........................................................................................86 (b) Daily Insolation versus latitude and day of year ...........................................................................90 (c) Annual Insolation plotted versus latitude .....................................................................................95 11. The Solar Analemma ......................................................................................................................97 (a) The Sun Sphere and the Analemma ..............................................................................................97 (b) Qualitative appearance of the inscribed analemma in photos .......................................................99 (c) Calculation of the analemma in angle space................................................................................101 1. Location of the actual sun and the mean fictitious sun .............................................................102 2. Relationship between local time LMT and master time τ : shooting times τN .........................103 3. Location of the actual and fictitious strobed suns in terms of LMT .........................................104 4. View of the continuous θ',Φ' data before strobing ....................................................................105 (d) Calculation of the analemma in film-space.................................................................................106 1. Overview...................................................................................................................................106 2. The transformation from Frame S to Frame S'..........................................................................107 3. The transformation from Frame S' to Frame Sc of the camera (eye transformation)................107 4. The transformation from Frame Sc to the camera film (viewport transformation)...................109 5. Putting the pieces together ........................................................................................................110 (e) A tour of Maple code which computes angle-space and screen-space analemmas.....................112 (f) An Analemma Gallery .................................................................................................................118 1. Analemmas for an observer at the North Pole ..........................................................................118 2. Analemmas for observers at other latitudes in the northern hemisphere. .................................121 (g) Analemma case study from the Crimea (Ukraine)......................................................................128 Appendix A: A Note on Various Arctangent Functions ...............................................................132 (a) arctan2Pi(y,x) and comparison with arctan(y/x) and arctan(y,x) ................................................132 (b) arctan2Pis(y,x).............................................................................................................................137 Appendix B: The Active and Passive Views of Vector Rotation...................................................141 (a) Active view..................................................................................................................................141 (b) Passive view ................................................................................................................................141 (c) Summary......................................................................................................................................142 Appendix C: Shape of the north pole analemma when e = 0 .........................................................144
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