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 README.planetmath:  Understanding Planetary Positions in KStars.
 copyright 2002 by Jason Harris and the KStars team.
 This document is licensed under the terms of the GNU Free Documentation License
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 0. Introduction: Why are the calculations so complicated?
 
 We all learned in school that planets orbit the Sun on simple, beautiful 
 elliptical orbits.  It turns out this is only true to first order.  It 
 would be precisely true only if there was only one planet in the Solar System, 
 and if both the Planet and the Sun were perfect point masses.  In reality, 
 each planet's orbit is constantly perturbed by the gravity of the other planets
 and moons.  Since the distances to these other bodies change in a complex way,
 the orbital perturbations are also complex.  In fact, any time you have more 
 than two masses interacting through mutual gravitational attraction, it is 
 *not possible* to find a general analytic solution to their orbital motion.  
 The best you can do is come up with a numerical model that predicts the orbits 
 pretty well, but imperfectly.  
 
 
 1. The Theory, Briefly
 
 We use the VSOP ("Variations Seculaires des Orbites Planetaires") theory of 
 planet positions, as outlined in "Astronomical Algorithms", by Jean Meeus.  
 The theory is essentially a Fourier-like expansion of the coordinates for 
 a planet as a function of time.  That is, for each planet, the Ecliptic 
 Longitude, Ecliptic Latitude, and Distance can each be approximated as a sum:
 
   Long/Lat/Dist = s(0) + s(1)*T + s(2)*T^2 + s(3)*T^3 + s(4)*T^4 + s(5)*T^5
 
 where T is the number of Julian Centuries since J2000.  The s(N) parameters
 are each themselves a sum:
 
   s(N) = SUM_i[ A(N)_i * Cos( B(N)_i + C(N)_i*T ) ]
 
 Again, T is the Julian Centuries since J2000.  The A(N)_i, B(N)_i and C(N)_i
 values are constants, and are unique for each planet.  An s(N) sum can
 have hundreds of terms, but typically, higher N sums have fewer terms.
 The A/B/C values are stored for each planet in the files
 <planetname>.<L/B/R><N>.vsop.  For example, the terms for the s(3) sum
 that describes the T^3 term for the Longitude of Mars are stored in
 "mars.L3.vsop".
 
 Pluto is a bit different.  In this case, the positional sums describe the
 Cartesian X, Y, Z coordinates of Pluto (where the Sun is at X,Y,Z=0,0,0).
 The structure of the sums is a bit different as well.  See KSPluto.cpp
 (or Astronomical Algorithms) for details.
 
 The Moon is also unique, but the general structure, where the coordinates
 are described by a sum of several sinusoidal series expansions, remains
 the same.
 
 
 2. The Implementation.
 
 The KSplanet class contains a static OrbitDataManager member.  The
 OrbitDataManager provides for loading and storing the A/B/C constants
 for each planet.  In KstarsData::slotInitialize(), we simply call
 loadData() for each planet.  KSPlanet::loadData() calls
 OrbitDataManager::loadData(QString n), where n is the name of the planet.
 
 The A/B/C constants are stored hierarchically:
  + The A,B,C values for a single term in an s(N) sum are stored in an
    OrbitData object.
  + The list of OrbitData objects that compose a single s(N) sum is
    stored in a QVector (recall, this can have up to hundreds of elements).
  + The six s(N) sums (s(0) through s(5)) are collected as an array of
    these QVectors ( typedef QVector<OrbitData> OBArray[6] ).
  + The OBArrays for the Longitude, Latitude, and Distance are collected
    in a class called OrbitDataColl.  Thus, OrbitDataColl stores all the
    numbers needed to describe the position of any planet, given the
    Julian Day.
  + The OrbitDataColl objects for each planet are stored in a QDict object
    called OrbitDataManager.  Since OrbitDataManager is static, each planet can
    access this single storage location for their positional information.
    (A QDict is basically a QArray indexed by a string instead of an integer.
    In this case, the OrbitDataColl elements are indexed by the name of the
    planets.)
 
 Tree view of this hierarchy:
 
 OrbitDataManager[QDict]: Stores 9 OrbitDataColl objects, one per planet.
 |
 +--OrbitDataColl: Contains all three OBArrays (for 
    Longitude/Latitude/Distance) for a single planet.
    |
    +--OBArray[array of 6 QVectors]: the collection of s(N) sums for 
       the Latitude, Longitude, or Distance.
       |
       +--QVector: Each s(N) sum is a QVector of OrbitData objects
          |
          +--OrbitData: a single triplet of the constants A/B/C for 
             one term in an s(N) sum.
 
 To determine the instantaneous position of a planet, the planet calls its
 findPosition() function.  This first calls calcEcliptic(double T), which
 does the calculation outlined above: it computes the s(N) sums to determine
 the Heliocentric Ecliptic Longitude, Ecliptic Latitude, and Distance to the
 planet.  findPosition() then transforms from heliocentric to geocentric
 coordinates, using a KSPlanet object passed as an argument representing the
 Earth.  Then the ecliptic coordinates are transformed to equatorial
 coordinates (RA,Dec).  Finally, the coordinates are corrected for the
 effects of nutation, aberration, and figure-of-the-Earth.  Figure-of-the-Earth 
 just means correcting for the fact that the observer is not at the center of 
 the Earth, rather they are on some point on the Earth's surface, some 6000 km
 from the center.  This results in a small parallactic displacement of the 
 planet's coordinates compared to its geocentric position.  In most cases, 
 the planets are far enough away that this correction is negligible; however, 
 it is particularly important for the Moon, which is only 385 Mm (i.e., 
 385,000 km) away.