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: