The abscissa interval of two asymptotic points gives the half period (P/2), and the ordinate interval of the same points gives the double amplitude (2×a). Again the arithmetic mean of abscissa components gives the initial epoch t₀, and the arithmetic mean of ordinate components gives the annual precession rate p₀ for the initial epoch. Error Dp₀ at the initial epoch for p₀ is computed by applying the derivative of the polynomials. Integral constant t_c is computed for J2000 by comparing to original polynomials. From these mean values, the prototype of the sinusoidal to define the general precession rate p_t for the epoch t can be written approximately as:

p_t = p₀ - Dp₀ cos g + a_p sin g    where    g = 2 p ( t - t₀ ) / P    and    t in Julian years.

It is obvious that the p_t curve fits exactly to the original one at the critical points, but deviates slightly elsewhere. Below is the centennial error check ( value = prototype - original ) for the interval J2000±500, in which the original curve is expected to be most reliable:

Table 1.3 - Deviation of the Prototype Curve for General Precession Rate (in seconds of arc)

As it is seen above, the error margins are quite satisfactory for a simple sinusoidal to substitute the polynomial for remote dates. Only F03r is an exception, yet it is not considered as a master for the prototype. The prototype with the mean parameters (Table 1.2) gives p = 50.2900027 for J2000, and when compared with the IAU proposal (50.2879695), the error is +0.0020332 arcseconds, which is acceptable with the present status of the prototype. The same prototype with the mean values of seven reliable models gives p = 50.2894799 for J2000 ( error = + 0.0015104).

It is possible to integrate p_t for ( t - t₀ ), to compare it to the original expressions. But, the significant period and the amplitude of the curve urge us to examine the polynomials for the obliquity of the ecliptic, before trying to improve the prototype. Examination of epsilon curves shows that model C03 must be also excluded next to F03 variants. In the end, we have only seven reliable models, and the above prototype can be integrated accordingly to compute the accumulated precession for remote dates as:

p_A = p₀ ( t - t_c ) - ( P / 2 p ) ( Dp₀ sin g + a_p cos g )    where    g = 2 p ( t - t₀ ) / P    and    t in Julian years.

The constants are mean values of the first seven models from Table 1.2, and the equation is to be considered as the preliminary version. Below is the centennial error check ( value = prototype - original ) for the interval J2000±500:

Table 1.4 - Deviation of the Prototype Curve for General Accumulated Precession (in seconds of arc)

As it is seen above, the error margins are quite satisfactory for a simple sinusoidal to substitute the polynomial for accumulated precession figures. F03r is an exception, yet it is not considered as a master for the prototype. S94, B97, and C03 are showing relatively rapid deviations, and need to be reconsidered. But, B03 is showing a perfect match, as well as La86. The rest; B88, W94, and B98 are not presenting huge deviations, yet supporting the prototype with less accuracy.   

Obliquity of the ecliptic was given as 23°27'08.26" for J1900.0, with the expressions of Newcomb:

e = 23.452294 - 0.0130125 T - 0.00000164 T² + 0.000000503 T³

But, after reconsidering ancient and medieval measurements, as well as those of the recent ones, Newcomb's expressions became questionable, particularly in the late fifties. After two decades, IAU adopted the expressions of Lieske in 1976. The table below is listing the major contributions since then, until the end of 2003, in conformity with Table 1.1:

Table 2.1 - Numerical Expressions for the Obliquity of the Ecliptic  e_A

Cells colored with pink are denoting coefficients, which deviates from the median of its kind.  

Graph 2.1 - Obliquity of the Ecliptic According to Thirteen Models

Graph for the obliquity of the ecliptic as proposed polynomials by thirteen sources is depicted above. The legends are similar to those of the previous graph (G1.1). Here the expression similarities are so close that it is almost impossible to discriminate the epsilon curves of B81, F03i, and F03r, as well as of S94 and B98. But, the similarity of IAU and L77 curves is not a surprise. The only problem, which cannot be understood easily, is the quasi-linearity of C03, in respect to others [N5]. A comparison of thirteen polynomials in figures for the interval J2000±500 is given as a separate table.

Applying the same method like before, critical points of the curves are computed by getting the first order derivatives, and tabulated here with truncated figures:

Table 2.2 - Critical Points of Twelve Epsilon Curves

Calculation method is the same of Table 1.2. From these mean values, the prototype of the sinusoidal to define the obliquity of the ecliptic e_t for the epoch t can be written approximately as:

e_t = e₀ - De₀ cos g - a_e sin g    where    g = 2 p ( t - t₀ ) / P    and    t in Julian years.

It is obvious that the e_t curve fits exactly to the original one at the critical points, but deviates slightly elsewhere. Below is the centennial error check ( value = prototype - original ) for the interval J2000±500, in which the original curve is expected to be most reliable:

Table 2.3 - Deviation of the Prototype Curve for Obliquity of the Ecliptic (in seconds of arc)

The discord between the two prototypes is caused by the inconsistency of the master models. The maximal and minimal points of two curves are expected to fit respectively, on the horizontal axis. Namely, when the obliquity reaches its maximal/minimal value, the precession rate at the corresponding epoch is expected to reach its minimal/maximal value. Since the annual amount of the precession is inversely related to the obliqueness of the ecliptic plane, the derivatives of the two polynomials are expected to join at least at two abscissa points, when the ordinates are reduced to zero. But, this special case is not possible with any of these polynomial couples. Yet, we may consider the corresponding roots of the differential equation:

p_A" = e_A'

which is expected to give the approximated T₁ and T₂ values of mutual asymptotic points of both curves. In a similar way, a further differentiation:

p_A"' = e_A''

may lead us to the origin, where the root is expected to give the approximated mutual point of both curves, with the maximal slant. The availability of such an approximation is only possible with the models  La86,  B88,  S94,  W94,  B97, B98, and  B03, with acceptable errors ( Dp_A⁽ⁿ⁺¹⁾ = De_A⁽ⁿ⁾ < 1" ; 0 < n < 3 ). The rest will be inappropriate for the purpose.

From these two equations, we can easily obtain the two approximations as:

T_1,2 = T₀ - ( P / 2p ) arctg_1,2 [ ( a_e + a_p ) / ( De₀ - Dp₀ ) ]

T₀ = T_m - ( P / 2p ) arctg [ ( De₀ - Dp₀ ) / ( a_e + a_p ) ]

which justify:

T_1,2 = T_m - ( P / 2p ) { arctg [ ( De₀ - Dp₀ ) / ( a_e + a_p ) ] + arctg_1,2 [ ( a_e + a_p ) / ( De₀ - Dp₀ ) ] }

Computed values are:

Table 3.1 - Approximated Mutual Critical Points of Precession & Epsilon Curves

Theoretically, a further similar examination of  p_A" = - e_A'  and  p_A"' = - e_A''  must give identical figures. Computed values for the case, and the differences between two points are:

Table 3.2 - Approximated Mutual Critical Counter Points of Precession & Epsilon Curves with Differences

Table 3.1 and 3.2 show that the prototype is supported mostly by B03, and partially by W94 and La86. Thus, we may focus on these models to develop our model for further applications. But, a final general comparison between the precession and epsilon curves (p_A' and e_A) is also necessary, regarding the critical points, and the value of an auxiliary s parameter, which is expected to be zero for the ideal sinusoidal match:

Since if    p_t = p₀ - Dp₀ cos g + a_p sin g    and    e_t = e₀ - De₀ cos g - a_e sin g    where    g = 2 p ( t - t₀ ) / P    then

p₀ + ( e₀ - p₀ ) + ( Dp₀ cos g - a_p sin g ) ( a_e / a_p )  = e₀ - De₀ cos g - a_e sin g

adjustment returns a parameter    s = ( a_e / a_p ) + ( De₀ / Dp₀ ) = 0

Comparison of table 1.2 and 2.2, and the value of the adjustment parameter is:

Table 3.3 - Comparison of Critical Points of Precession & Epsilon Curves

Table 3.3 shows that the prototype is supported by W94 and B03, and partially by La86, which confirms the previous result. Thus, the two models, B03 as the essential and La86 as the secondary, will be taken into consideration while developing the prototype, and W94 will be used as a theoretical model only, due to huge s deviation. At the bottom of the table, just to compare as a sample, F03r results are attached. It is clear that this model needs further development, in order to claim consistency.

It is interesting that S94 is presenting such a great deviation, relative to the other six models. Also, the predecessors of B03, (B97 and B98, and to some extent B88) do not seem to be well developed. But, the success achieved with B03 prevails the others. Although it is the earliest model, La86 is presenting more accuracy than B88, S94, B97, and B98, amazingly. Next to La86, W94 can also be regarded as a good theoretical work, despite the misestimated period for the obliquity of the ecliptic.[N6]

Thirteen models for numerical expressions for General Precession and Obliquity of the Ecliptic are examined for long-term validity, and only seven of them found worthy to consider in this sense. Among these seven models, only two (La86 and B03) seem to be more accurate and meticulously framed than the others, and one (W94) is theoretically remarkable.

Among the four models, which appeared after the IAU 2000 resolutions, B03 seems to be the only referable one for long-term applications. Yet, it seems quite possible that a more precise model can be developed in the future, by merging the theories of Laskar and Williams, and applying the new VLBI and LLR results, to build accurate numerical expressions for precession and the obliquity, which can be valid for a few ten thousand years.

The next part of this article will be focused on this subject.