Two Unified Algorithms for Fundamental Planetary Ephemeris

In the present paper, we established two unified algorithms. The first algorithm is for the transformations between J2000.0 Keplerian orbital elements and B1950.0 elements, while the second is for the transformations between the equatorial orbital elements and the ecliptic orbital elements. Mathematica Modules of the algorithms are given together with some numerical applications.


Introduction
Precession is a change in the orientation of the rotational axis of a rotating body.In astronomy, precession refers to any of several slow changes in an astronomical body's rotational or orbital parameters, and especially to the Earth's precession of the equinoxes.The shift in the position of the Earth's axis and of the ecliptic caused by forces exerted by the Sun, Moon and planets not only causes a slight change in the angle between the equator and the ecliptic, but also a shift of the vernal equinox by about 1.5˚ per century (1' per year).This effect is negligibly small for casual observing, but is an important correction for precise observations.To get accurate observations, the International Astronomical Union (IAU) 2000 recommended significant improvements in the definition of the International Celestial Reference System (ICRS) [1].It follows that changes in Earth's orbital pa-rameters (e.g.orbital inclination, the angle between Earth's rotation axis and its plane of orbit) is important to the study of Earth's climate, in particular to the study of past ice ages.
For precise calculations therefore, the equinox of the coordinate system used must be stated.The equinoxes most frequently used are the equinox of date, equinox J2000 and equinox B1950 [2] and [3]."Equinox of the date" means that the values used are those for the equator, ecliptic, and vernal equinox for actual date under consideration.Such daily alteration of the coordinate system is sensible if one requires the coordinates of a planet, for example, for use in conjunction with the setting circles on an equatorial mounted telescope, or on a transit circle.Since the shift in the Earth's axis, the orientation of the polar axis of a telescope will also alter.On the other hand, if one wants to study the actual spatial motion of a planet then it is better to use a fixed equinox, such as that for Julian Epoch J2000 (2000 January 1.5 = JD = 2451545.0),which was generally introduced in 1984.Before that, the older equinox B1950 has been used for a long time, and was employed for many stellar catalogues and atlases (such as the SAO Star Catalog and Atlas Coell).The beginning of the Basselian year 1950 (1950 Jan.0.9232 = JD 2433282.423).
On the other hand, there are two standard reference planes to specify the orbits of celestial objects, the equatorial plane of the Earth and the ecliptic plane (the plane of Earth's orbit around the Sun).However, due to precession, the equatorial and the ecliptic planes are slowly changing their positions relative to the background stars [4].
In this paper, we established two new unified computational algorithms each one is applicable to get ephemeris in both directions simultaneously.That is to mean it can be used as a switch between: 1) J2000.0Keplerian orbital elements and B1950.0 elements and 2) between the equatorial orbital elements and the ecliptic orbital elements.Such artifices do not exist in any other numerical ephemeris methods.Furthermore, in our algorithms the number of the utilized equations was reduced via applying some mathematical operations, a matter, which facilitates the computations.Mathematica Modules for the two algorithms are also included.
Finally, it should mentioned that, although all numerical ephemeris methods utilize the same equations, but their accuracy may be differ greatly depending on a) the computational package adopted for their evaluation; b) the form of the equations, such that, the more they are explicit, the more their satiability and accuracy will be.
Due to the quite simple explicit forms of the reduced equations of our algorithms, and the usages of the most power full computational packages of Mathematica, consequently as regarding to that two points a & b, our algorithms may be more accurate than that given by JPL system or other numerical ephemeris methods.

Duality of Theorems Relating to the Spherical Triangle
The duality of theorems relating to the spherical triangle [5] was stated as: Any theorem relating the sides and angles of any spherical triangle will remain true when the angles are changed into the supplements of the corresponding sides and the sides into the supplements of the corresponding angles.

The Basic Equations for the Transformation
Applying the duality property (see Section 2.1) to the spherical triangle ABC of Figure 1, we get ( ) From the spherical triangle ABC of Figure 1 we get ( ) ( ) Clearly the right hand side of each of the Equations (2) contained mixture of the unknown quantities (dented by primes) and known quantities (without primes) e.g. and ω ω ′ , while their left hand sides are known quantities.To overcome this difficulty, we have to apply the transformation rules of Equations (1) to Equations (2) and we get for the transformation from , , to , , 3. Unified Transformation Formula for J2000.0 and B1950.0Keplerian Elements For practical applications, we can unified the two sets of Equations ( 1) and ( 2) as: ( )

Transformation Formulae between Equatorial and Ecliptic Orbital Elements
Let the equatorial orbital elements of celestial body, be denoted by , , i ω Ω and its corresponding ecliptic elements be , , i ω Ω (Figure 2 and Figure 3).The rest of the elements ( ) , , e a τ , which determine the orbit of the body, do not change by changing the coordinates systems.
From the spherical triangle γNL we get: sin sin sin sin , sin cos cos sin sin cos cos , i i i ( )

Unified Transformation Formulae for Equatorial and Ecliptic Orbital Elements
For practical applications, we can unify the two sets of Equations ( 7) and (8) as: where , the given elements.
, the required elements.
• η, an integer takes the values 1 + or 1 − such that: • 1 η = + , for the transformation from the ecliptic orbital elements ( ) , , i ω Ω .• 1 η = − , for the transformation from the equatorial orbital elements ( ) , , i ω Ω .Since the right hand side of each of the above equations is known (note ε is known angle), consequently we can get ( ) 6. Computational Developments of J2000.0 and B1950.0Keplerian Elements Transformations

Mathematica Module: KeplerB1950TJ2000
• Purpose Transfer the Keplerian orbital elements with respect to reference system B1950.0 to the Keplerian orbital elements with respect to reference system J2000.0 and vice versa.

Numerical Examples
Table 1 gives the transformations of the orbital Kepler elements 1950.0 to the corresponding elements J2000.0 and vice versa as computed from the above Module.

Numerical Examples
Table 2 gives the transformations of the ecliptic orbital elements to the equatorial orbital elements and vice versa.In concluding, the present paper introduced two unified and simple algorithms that are capable of executing calculations in both directions and in one program run to the 1) transformations between J2000.0 Keplerian orbital elements and B1950.0 elements and 2) transformations between the equatorial orbital elements and the ecliptic orbital elements.The algorithms are elaborated using Mathematica package, which is qualified for accurate computations.The proposed algorithms are checked by numerical examples given in Table 1 and Table 2.
, for the transformation from the Keplerian orbital elements ( ) , for the transformation from the Keplerian orbital elements ( ) previous numerical values.•The unified formulae for the required elements 1

Figure 2 .Figure 3 .
Figure 2. The orbit in space with respect to the fundamental planes.L By the same way as above, we get for the transformation from , , to , , 2.2.2.The Basic Equations for the Transformation ′ ′ ′, , I Ω ω to , , I Ω ω

Table 1 .
The transformations of the orbital Kepler elements 1950.0 to the corresponding elements J2000.0 and vice versa.

Table 2 .
Transformations of the ecliptic orbital elements to the equatorial orbital elements and vice versa.