Analytical images of Kepler’s equation solutions and its analogues

. Approximate, but highly accurate analytical solutions of Kepler’s equation were found by reducing it to an algebraic equation. With the help of this approach and usage of the iterative algorithm there were obtained solutions of a similar equation for the hyperbolic motion near the orbit pericenter


Introduction
Kepler's equation is one of the main relations in celestial mechanics, as Barker's equation and a similar equation for the case of hyperbolic motion.It is well known from celestial mechanics that the relative motion of two point-like gravitating bodies with masses m 1 and m 2 occurs on Keplerian orbits (Barger & Olsson, 1995;Vallado & McClain, 2001) where r and v are polar coordinates, the focal parameter p and eccentricity e are determined by the masses of bodies and integrals of motion -angular momentum l and energy Here K = G(m 1 +m 2 ) is the so-called gravitational parameter, G is the gravitational constant and µ = m 1 m 2 (m 1 + m 2 ) −1 .Using an expression for the angular momentum l = µr 2 dv dt (3) and relation (2), we obtain a well known equation for the time dependence of true anomaly v(t) in the form Here, taking into account that v(0) = 0, equation (4) determines the time of motion from the orbit pericenter to the point with a fixed value of v(t).The integral in equation ( 4) is expressed by elementary functions, but it has different images depending on the eccentricity value.The trivial case e = 0 corresponds to the uniform motion on a circular orbit of radius p with the angular velocity and v(t) = ω p t.At non-zero eccentricities, the integral in equation ( 4) is calculated with the help of a universal substitution x = tan(v/2).In the case e = 1 and E = 0 the motion occurs on a parabolic orbit, and equation ( 4) takes the form (Barger & Olsson, 1995) This equation is known as Barker's equation.When the time varies in the region −∞ < t < ∞, the true anomaly changes in the interval −π < v < π.
For 0 < e < 1, equation ( 4) takes the form where t * = 2πt/T = tK 1/2 /p 3/2 (1−e 2 ) 3/2 (T is the orbital period of an elliptical motion).For e > 1, equation (4) has the following form in elementary functions where Solving equation ( 6) relative to tan(v/2) by Cardano's formulae (Abramowitz & Stegun, 1972), we find an exact solution for the true anomaly, where and v(−t) = −v(t).Function (10) has the asymptotics For the point which follows a parabolic path, the function v(t) is proportional to the time if the point is near the pericenter; if it is far away -then v(t) → ±π.This behavior is illustrated in Fig. 1, which depicts v(t) as a function of the dimensionless variable ω p t.
. Dependence of the true anomaly v(t) on the dimensionless variable ωpt.

Equation of elliptical motion
Using the substitution equation ( 7) is reduced to Kepler's equation (Vallado & McClain, 2001) and the additional function E is called an eccentric anomaly.As it is shown from equation ( 14), E(−t * ) = −E(t * ).Many works are devoted to finding an approximate solutions of this equation.The most famous iterative Lagrange method (Alexandrov, 2003), in which for a zero approximation there is chosen the function E (0) (t * ) = t * that corresponds to an uniform motion on a circular orbit, and the term e • sin E is considered as a perturbation.In this way, the solution is represented as an infinite series by powers of eccentricity As it was first shown by Laplace, this series coincides absolutely only in the region 0 < e ≤ ē = 0.66274 . . .(Alexandrov, 2003).Usage of the Fourier series leads to the solution in the form (Alexandrov, 2003) where are Bessel functions of the first kind with an integer index value (Abramowitz & Stegun, 1972).For the eccentricity values that are close to unity and small values of t * , series ( 16) has a weak convergence and requires taking into account dozens of terms, which makes this method cumbersome and irrational.Usage of the theory of a complex variable allowed the second authors to obtain solutions of Kepler's equation in a finite analytical form (Siewert & Burniston, 1972;Philcox et al., 2021).However, the obtained solutions are too complicated for usage and require the additional calculations.Because of that, they can only be considered as proving the solution existence.
From equation ( 14) it follows that the function E(t * ) is a periodic function with period 2π.Therefore, variables E and t * change in the range (0 ÷ 2π) and E = 0 at t * = 0, E = π at t * = π and E = 2π at t * = 2π.It also follows from equation ( 14) that in the interval π ≤ t * ≤ 2π therefore, it is sufficient to find solutions in the region 0 ≤ t * ≤ π.
There can be distinguished two regions of variables for a sufficiently great value of eccentricity, in which the curve behavior E(t * ) has a different character.Namely, in the plane (E, t * ) 0 ≤ E ≤ π, 0 ≤ t * ≤ π it can be selected the region of a rapid change of E(t * ) and the region of a slow change of As it follows from equation ( 14), for A different behavior of the function E(t) is caused by the different motion velocity of the material point along the orbit -rapid motion in the pericenter region and slow in the apocenter region.
To find analytical solutions of equation ( 14), we rewrite it in an equivalent form.Using the substitution we obtain an equation in the region I The substitution allows us to rewrite equation ( 14) in the region II in the form To obtain approximate analytical solutions of equations ( 23) and ( 25), we use the approximation cos F in the interval (0 ÷ π/2) in the form of a polynomial of the fourth order with coefficients We will consider equation ( 25) in detail and represent it in the form Function ( 26) is sufficiently accurately approximated by cos F in the interval (0 ÷ π/2), and deviation does not exceed 3 • 10 −4 even in the vicinity of F 2 = π/2.Because of that, for equation ( 28) is applied the method of successive approximations, and the zero approximation is determined by the solution of an algebraic equation The first iteration yields a specified solution where t (1) and etc.The solution of the equation of zero approximation is found by Cardano's formulae (Abramowitz & Stegun, 1972).The solution that corresponds to the condition 0 ≤ F (0) 2 (t * ) ≤ π/2 is determined by the expression Here, we used the following notations In the region I the solution of the equation of zero approximation is if in notations (33) we replace α 1 → −α 1 and α 3 → −α 3 .In the first iteration, we obtain the expression where t (1) The time dependence of functions F The maximal deviation E (0) (t * ) from the solution of equation ( 14) found by the numerical method does not exceed 5 • 10 −4 , and iterative corrections E (1) (t * ) − E (0) (t * ) are negligible.

Equation of hyperbolic motion
Using the substitution equation ( 8) is reduced to the form which is an analogue of Kepler's equation.Herewith −∞ < t H < ∞, and −∞ < H < ∞.Since the function H(−t H ) = −H(t H ), then it is sufficient to find the solution of equation ( 38) in the region 0 ≤ t H < ∞.
Unfortunately, the function sinh H cannot be approximated by a polynomial of the fourth order in a sufficiently wide region of change H. Thereby, we will consider the calculation of the eccentric anomaly asymptotics near the pericenter, which is precisely of practical interest.We rewrite equation ( 38) in the form choosing the approximation function in the form for a = 0.188479.The solution of equation ( 39) is found by the iterations method, using in the role of zero approximation the root of the equation namely Specified of this solution we perform by the iterating of equation ( 39 first iteration H ), t (1) (43) in the second iteration, we have H )}, and etc.
(44) Fig. 3 illustrates the function H (0) (t H ) (curve 1), as well as the numerically found solution of equation (38) (curve 2) for e = 1.4.In the vicinity of t H = 2.0 the relative deviation of these curves equals 0.59%.The deviation of the function H (1) (t H ) from the numerical solution does not exceed 0.06% for t H = 2.0, which indicates a rapid convergence of the iterative process.
* ) is shown in Fig.2in the case e = 0.6 and eccentric anomaly in the same approximation E (0) (t * ).

Figure 3 .
Figure 3.The solution of Kepler's equation in the case of hyperbolic motion for e = 1.4 in different approximations.Curve 1 corresponds to approximation (42), curve 2 -a numerical solution of equation (38).