Concerning an analytical solution of some families of Kepler’s transcendental equation Open Access

One of the classical laws of planetary motion due to Kepler says that a planet revolves around the sun in an elliptic orbit (Fig. 1(a)). The planet revolves around sun can be described by a well known Kepler’s equation. Furthermore, many problems in celestial mechanics require a solution to Kepler’s equation.^1,2 The Kepler transcendental equation of the form

which links the eccentric anomaly of elliptic motion E, mean anomaly M and eccentricity e, is of critical importance in celestial mechanics. The basic physical meaning of this equation is better explained by Fig. 1(b) in which is depicted an ellipse, with eccentricity e, that is the orbit of a body moving about the stationary gravitating center placed in focus of the ellipse S. Denote by C and A the center and pericenter of the ellipse, respectively. Construct also a circle with its center at point C and with a radius equal to the major semi axis of the ellipse. At some time let the position of the rotating body is determined by point P. From P we drop a perpendicular to the major axis of the ellipse and denote the foot of the perpendicular by letter R. Extend this perpendicular to intersect the circle at point Q. Then the angle ↑ ACQ is just the eccentric anomaly E. Suppose that the planet P, having passed thought perihelion A is at position P after elapsed time t, it is possible to express the polar coordinates of P (r, v), relative to the sun in terms of t.

Kepler’s transcendental equation relates E to time by means of a quantity

where T is time required for the planet complete one trip in its orbit around the sun. The quantity M represents the average angular speed of the radius vector SP. Classical formulae to find r and v is

and

Therefore if T and M are known it is possible to solve the Kepler’s equation for E and to determine position (r, v) at time t by using equations (3) and (4).

Many algorithms have been derived for solving the Kepler’s equation as a result of its importance in celestial mechanics.^1–7 Detailed characteristics of these methods will be discussed in the next section.

As it is noted in the Introduction, many algorithms have been derived for solving the Kepler’s equation. Thus, Colwell’s history of Kepler’s equation contains approximate half of thousand references.⁴ Our interest is oriented toward some analytical approaches^1–7 where is presented procedure for solving this equation which implies writing a E as a power series in e as follows

where the coefficients a_n are given by the Lagrange inversion theorem as

It should be noted that this series diverges for e > 0.6627434…

On the other hand, equation (1) can be solved using Bessel functions of the first kind,⁴ as follows

where J_k is the k_th-order Bessel function of the first kind, given by the following series:

All of the series solutions, at the first glance, appear to require an infinite number of terms, which are computationally impossible to generate.

In Ref. 5 polynomialization of Kepler’s equation through Chebyshev polynomial expansion of the sine is presented.

Also, Lambert W function can be used to truncate series solutions to Kepler’s equation.⁶ The above paper offers a simple solution for the truncation of previously known infinite series of solutions to the functions of Kepler’s transcendental equation. Presented method is based on many approximations, while its accuracy is not great, and expressions are complex and enormous. For example, in the above paper is presented analytical expansion for cos (v), within the 10⁻⁹ tolerance, correct to e = 0.1, (equation (24a)).

In Ref. 7 is presented the iterative solution to Kepler’s equation. Namely, in the above paper the solution for the relative motion is presented in the closed-form, in terms of the eccentric anomalies of the target and chaser orbits, while the eccentric anomalies themselves are expressed in terms of the orbits respective eccentricities, using an iterative method.

There would seem to the nothing further say.⁵ But, in this article we show that solving of the Kepler’s transcendental equation is possible within the Special Trans Functions Theory, by simple, and, also by advanced STFT iterative method. Advantage of this novel approach to the Kepler’s equation analytical solving, first of all, is conceptual simplicity, absent of initial values and unlimited of the theoretical accuracy of eccentric anomaly.

So, the subject of our theoretical analysis presented here is obtaining a new analytical solution of some families of Keller’s transcendental equation, by using some novel iterative methods within the Special trans functions theory. Consequently, a brief introductory observation concerning the STFT is presented.

Let us note that Perovich’s Special Trans Functions Theory has been proved to be a very powerful, consistent theory for solving a broad family of transcendental equations and obtaining exact analytical closed-form solutions in the real domain [Refs. 8–25 and 34–41 et al. ]. Examples of its application are shown in articles concerning the genesis of an exact analytical solutions in: theory of neutron slowing down,^8–10 nonlinear circuit theory,^8,11,39 linear transport theory,^8,12,13,19, Hopfield neuron analysis¹⁴ some families of transcendental equations,^{8,12,15,20,23–25,38} solar cell analysis,^16,21,35 Plutonium temperature estimation,^8,17,36 ambient temperature estimation,^18,11 Lambert transcendental equations analysis,^8,20 as well as in problem in engineering materials,^22,34,40,41 ect.

The biggest impact of this work is in the usage of a new STF theory approach in formulae genesis for the determining position (r, v) in time t. All investigations and analysis of the value E in this paper have been consistently accomplished with the usage of STF theory. STF theory ensures reaching extreme precisions in numerical results (arbitrary number of accurate digits in the numerical structure of the transcendental numbers), which is reflected in this paper as well, where we show the highest precision in defining the eccentric of elliptic motion E achieved so far. That, of course, implies that the relevant constants (π, a ect.) have been used with greater number of exact digits than in conventional approaches.

Driven by the thorough analysis of the results obtained, we believe that STF theory, supported by Mathematica software, represents a novel theoretical approach in analysis of the many problems in celestial mechanics which require a solution to Kepler’s transcendental equation.

Within this section our interest is oriented toward determining analytical solutions for the Kepler’s transcendental equation by using a simple iterative procedure based on the Special Trans Functions Theory -STFT.^8–25,34,35 The Kepler’s transcendental equation for determining the eccentric anomaly in celestial mechanics (1) after some symbolic modification of the type

takes the form

In addition, after an original structural modification, Kepler’s transcendental equation (10) takes the following form

and

where

and where

Structural modification of the transcendental equation (10) implies that solutions of equations (11) and (12) are unique in the real domain^8,15,17,22et al. Namely, within the Special trans functions theory, the transcendental equation (11) (or (12)), for fixed real values of B, has unique analytical closed form solution in the real domain, in the form of the special tran function, $tran + B$⁠. Consequently, we have $sinX= tran + B$ and $Y=αX−β= tran + B$⁠. or $X=M+e⋅ tran + B$⁠.

In some detail this special tran function is referred in [Refs. 8–11, 15, 22, and 35–41,] et al.

Also, from formulae (11) and (12) we have $sinXexp sin X =B$⁠, and, $Yexp Y =B$⁠, respectively. Consequently, since $abs sin X <1$⁠, and $abs Y <1$⁠, we have $B∈ − 1 exp 1 , exp 1$⁠.

The presented STFT concept implies application of the analytical closed form solution of Eqs. (11) and (12), as unique existing analytical closed-form expression for practical calculation of X, for all values of B.

Let us note that according to the usual traditional notations of quantities, which are used in the domain of Astronomy, the function that is an analytical solution of equations (11) and (12) is denoted Lambert W(x) function. This function is based and inspired by the works of famous scientist J. Lambert.^26–29 But, historically, in the literature, we have not analytical closed form solution to the implementation of the Lambert W function, for all values of real B! All known solutions are fragmental analytical or approximate solutions (See Section III A). Because of that, in this article, the Perovich $tra n + B$ function, previously referred in [Refs. 8–11, 25, 15, 22, and 35–41] et al., is implemented. Of course, the analytical closed form solution to the Lambert transcendental equations (11) and (12) is the Perovich $tran + B$ function. In some detal, explanations are given in the Section III A.

In conceptual sense, the Simple iterative method, within the Special trans function theory, implies the following form of iteration:

or, form

where

and,

where (k) is number of iteration. Note that equations (10) and (13) imply the statement $abs k Y < 1$⁠. Of course, $k X$ is value of X in k_th iteration? The Error Criterion is defined as

where ε is an arbitrary small real positive number, and, $k B$ is real value of B in k’th iteration.

Also, the initial iteration takes the form $1 B=exp 1$⁠. The outline of the simple STFT iterative process begins with the certain value of $B= 1 B=exp 1$⁠, and, for this value of $1 B$⁠, from equation (12) we have $1 Y= tran + 1 B$ and, from equation (17) follows $1 X=M+e⋅ tran + 1 B$⁠. After that, the second value of B is obtained from equation (18) in the form $2 B=sin 1 X exp sin 1 X$⁠. If $2 B$ does not satisfy the error criterion $1 B − 2 B ≤ ε$⁠, where ε is an arbitrary small real positive number, then using $2 B$⁠, from equation (12), we have $2 Y= tran + 2 B$ and $2 X=M+e⋅ tran + 2 B$⁠, or, more explicitly, $2 X=M+ etran + sin 1 X exp sin 1 X$⁠. After that, the next value of B is obtained from equation (18) in the form $3 B=sin 2 X exp sin 2 X$⁠. If $3 B$ does not satisfy the error criterion $2 B − 3 B ≤ ε$ the whole procedure is repeated. Let us note that general scheme of the Simple STFT iterative procedure for determining of values B takes the form (18), where (k) is number of iteration. Thus, for k=2, we have

Also,

ect. Let us note that within STF theory $tran + n S exp n S = n S$⁠, where $n S x =sin n X$⁠. Consequently, based on the Eqs. (16) and (14), we have:

Finally, for N iteration a simple STFT iterative formula takes the form

where

and

For practical analysis and numerical calculations the formula (20) takes the form

where $N X P$ denotes the value of the Nth iteration of X given with P accurate digits. P is defined as $P= 1 − abs log G$⁠, where the error function G is defined as G = sinX − αX + β.

Let us note that convergence condition within the simple STFT iterative method takes the form

The STF theory genetic implies the above condition of convergence. Of course, the Kepler’s transcendental equation parameters determine the dynamic of the convergence process.

In this section, as above mentioned, the problem of finding an exact analytical solution of transcendental equations (11) and (12), will be discussed. First of all, note that equations (11) and (12) can be presented in the unique form

For convenience, we restrict ourselves to the one dimensional transcendental equations. Let us note that equation (24) has one real solution: $ψ ξ >0$⁠, for $B ξ ∈ 0 , exp 1$⁠, and $ψ ξ <0$ for $B ξ ∈ − 1 exp 1 , 0$⁠. Since equation (10) implies the statement of the form $abs k Y <1$⁠, consequently, we have that parameter B (ξ) is within $B ξ ∈ − 1 exp 1 , exp 1$⁠.

Let us note, that historically, an equation proposed by Lambert (1758) and studied by Euler in 1779

when α → β, becomes

and, for β < 0 has the solution

where W is Lambert’s W function.

The Lambert W-function has the series expansion

The Lagrange inversion theorem gives the equivalent series expansion

However, the series oscillates between ever larger positive and negative values for real z ≥ 0.4, and so cannot be used for practical numerical computation.

An asymptotic formula which yields reasonably accurate results for z ≥ 3 is

where

Another expansion due to Gosper is the double series

where S1 is a nonnegative Stirling number of the first kind, and a is a first approximation which can be used to select between branches. The Lambert W-function is two-valued for $−1/exp 1 ≤x<0$⁠. For $W x ≥−1$ the function is denoted $W 0 x$ or simply $W x$⁠, and this is called the principal branch. For $W x ≤−1$ the function is denoted as $W − 1 x$⁠.

Numerical methods for solving modified Lambert transcendental equation is presented in many different domain.³⁰ Let us briief review some of thet: Recursion, Halley’s iteration, Fritsch’s iteration, et al. Presented methods are complicated and complexed. For an instance below is given a formulae (24a) as an example of the complexity formulae which appears in Ref. 6 

Of course, within Maple, Matlab, and Mathematica different solver are developed. While the Lambert W function is simply called W in the mathematics software tool Maple³¹ and lambertw in the Matlab programming environment,³² in the Mathematica computer algebra framework this function is implemented under the name ProductLog.³³ 

Fortunately, within the Perovich’s Special Trans Functions Theory [Refs. 8–11, 15, 22, and 35–41] et al., we have analytical closed form solution, to the Lambert Transcendental Equation, in the form

where the Perovich trans₊(B) function is defined as:

where

with [t] denotes the greatest integer less than or equal to t.

For practical analysis and numerical calculation, an expression (25) (including Eq. (26)) takes the form:

where 〈ψ〉_P denotes numerical values of the transcendental number ψ given with

accurate digits, where the error function G is defined as:

More explicitly, for fixed values of sumlimits, [t], 〈ψ〉_P takes the form:

Expression (31) gives the number of accurate digits in the numerical structure of the transcendental number ψ, which depend of [t].

In addition, solution of equation (12) directly follows by formula (26) in the form

For practical numerical calculations, we have

Let us note, that we have already had experience with some forms similar to equation (31) by considering the problem of number of accurate digits in the numerical structure of transcendental numbers, as certain physical parameters, in different scientific areas (Refs. 8, 11, 15–17, 22, and 25 et al.).

The concept of the STFT exact analytical solutions is fundamentally different from that of conventional numerical techniques and analytical methods. Thus, the STFT solutions for Kepler’s transcendental equation should yield significant information that could be successfully used for the Kepler parameters recognition and classification. It is possible, due to the fact that the theoretical accuracy of the STFT numerical results is unlimited, and extreme precision is attainable with this approach. Consequently, we believe that novel STFT techniques represent an original theoretical and numerical analysis of many problems in celestial mechanics requires a solution to Kepler equation for different structures. Note, they should be extended in several directions in order to be both accurate enough and accept table regarding the calculation complexity.

In the other words, from a theoretical point of view STFT analytical solutions can be found with an arbitrary order of accuracy. Namely, the number of accurate digits in the numerical structure of the analytical STFT result depends upon the sumlimit in the STFT formulae ([t], in this case). Of course, this may be a disadvantage in some cases (large sumlimit for small number of accurate digits, for example). Within Tables I-IX, given below, the above statement concerning the sumlimit significance will be represented by obtained numerical results for error criterion (19).