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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









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 Fourierlike 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 figureoftheEarth. FigureoftheEarth


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.


