Quasi-Analytical Solution of Kepler’s Equation as an Explicit Function of Time

Abstract

Although Kepler’s laws can be empirically proven by applying Newton’s laws to the dynamics of two particles attracted by gravitational interaction, an explicit formula for the motion as a function of time remains undefined. This paper proposes a quasi-analytical solution to address this challenge. It approximates the real dynamics of celestial bodies with a satisfactory degree of accuracy and minimal computational cost. This problem is closely related to Kepler’s equation, as solving the equations of motion as a function of time also provides a solution to Kepler’s equation. The results are presented for each planet of the solar system, including Pluto, and the solution is compared against real orbits.

1. Introduction

Kepler’s laws offer an empirical mathematical model for describing the motion of orbiting bodies, later proven by Newton’s analytical results [1,2]. Therefore, they are of fundamental importance in astrophysics, celestial mechanics, and Earth science [3]. Nevertheless, Kepler’s laws do not describe the motion of celestial bodies as a function of time. It is the so-called Kepler’s equation (KE) that provides the time dependence of the position of orbiting celestial bodies. Since Kepler first established this equation in 1609, several solutions of different types have been proposed, continuing up to the present day [2,4,5,6,7,8,9]. One of the significant early attempts was made by Carlini in 1819, later refined by Jacobi [9]. Because of its transcendental nature, an exact explicit solution remains elusive. There exist a vast variety of numerical methods renowned for their remarkable accuracy. These methods typically involve calculating infinite series or employing high-order iterative approaches. It is hardly possible to refer here to all these works, but we cite the most relevant ones [3,10,11,12,13,14,15,16,17,18,19]. Recently, some methods that use artificial intelligence have been proposed [20].

In this work, we present a quasi-analytical solution that gives explicit equations of motion as a function of time. This solution is quasi-analytical, because even though it gives explicit formulas, it depends on six numerical coefficients, which in turn depend on the orbital eccentricity. Our results provide an explicit solution to KE.

The Orbit Equation

The classical problem begins then, with the description of the motion of the planets according to the following equations [21]:

$$t\left(\theta \right)=\frac{{\alpha }^2\mu }{l}{\int }_0^{\theta }\frac{1}{{(1+\epsilon cos\phi )}^2}d\phi ,$$

(1)

and

$$r\left(\theta \right)=\frac{\alpha }{1+\epsilon cos\theta }.$$

(2)

Here, $(r,\theta )$ are the coordinates of the planet in the polar coordinate system, with the origin at the Sun; t is the time; $\epsilon$ is the eccentricity; and $\alpha$, $\mu$, and l are the focal parameter, the reduced mass, and the angular momentum, respectively [2]. The angle $\theta$ is called the true anomaly, defined as $\theta =0$ at the pericenter [19].

The inverse of the solution of Equation (1), $\theta \left(t\right)$ is closely related to KE (the explicit relation will be shown in Section 4):

$$E-\epsilon sinE=M,$$

(3)

where E is the eccentric anomaly and M is the mean anomaly, expressed as

$$M=\frac{2\pi }{T}t,t\in [0,T],$$

(4)

T is the orbital period. As was pointed out before, usually, E needs to be estimated by iterative methods [8] or series expansions [22]. Thus, the accuracy of the position and/or velocity of a celestial object moving in the Keplerian orbit, which may be obtained from the solution of KE, depends on the approximation method used. Figure 1 illustrates the variables involved in the problem.

Equations (1) and (3) are both transcendental. The solution of Equation (1) gives the solution of Equation (3), and vice-versa, which has led to a vast literature concerning alternative approaches [3]. A detailed description of the different works presented in the literature on this problem can be found in [3,19].

The remainder of this paper is structured as follows. In Section 2, we present in detail, step by step, a quasi-analytical solution of Equation (1). In Section 3, the results are presented and compared with the real orbit of each planet of the solar system. In Section 4 we present the solution to KE based on the solution of Equation (1). Finally, in Section 5, we give a brief discussion and conclusions about the results obtained in this work.

2. Methodology

In the present section, a quasi-analytical method is established for solving Equation (1). For clarity, we divide the method into sections that we call steps. All calculations and graphics were performed with MATLAB 2020b (MathWorks Inc., Natick, MA USA). Now, we briefly describe each step.

Step 0: In this step, we integrate Equation (1) and introduce a normalized time $\tau$, which maps the real time t from the interval $[0,T]$ to the interval $[0,\pi ]$, which leads us to Equation (7) for $\tau \left(\theta \right)$.

Step 1: In this step, we separate $\tau \left(\theta \right)$ into two parts, ${\tau }_a\left(\theta \right)$ and ${\tau }_b\left(\theta \right)$. Based on the observation that ${\tau }_a\left(\theta \right)$ resembles the behavior of $\tau \left(\theta \right)$ (especially for small values of $\epsilon$), and, taking into account its analytical form, we find the inverse of ${\tau }_a\left(\theta \right)$, ${\theta }_0\left(\tau \right)$ as our zeroth approximation for $\theta \left(\tau \right)$.

Step 2: Assuming the exact solution has the form of Equation (14), we introduce the true argument of the arctangent function, $\phi \left(\tau \right)$, which gives the exact solution. Then, we introduce the function ${\psi }_0\left(\tau \right)$ (Equation (17)) as the quotient of the true argument $\phi \left(\tau \right)$ and the argument of the zeroth approximation, ${\phi }_0\left(\tau \right)$. Finally, we define the first approximation ${\theta }_1\left(\tau \right)$ by adjusting the argument ${\phi }_0\left(\tau \right)$ of the arctangent function, which appears in the zeroth approximation ${\theta }_0\left(\tau \right)$. The main idea is to make a series of adjustments in such a way that the quotient mentioned above approaches 1 as closely as possible.

Step 3: Following the logic of Step 2, we introduce the function ${\psi }_1\left(\tau \right)$ (Equation (22)) and the argument ${\phi }_2\left(\tau \right)$ (Equation (25)) in order to obtain the second approximation ${\theta }_2\left(\tau \right)$ (Equation (26)). First, we propose two kinds of approximations for ${\psi }_1\left(\tau \right)$: linear and harmonic. This gives the approximations ${\theta }_{2.1}\left(\tau \right)$ and ${\theta }_{2.2}\left(\tau \right)$ (Equations (27) and (28)), respectively. These approximations give corresponding precisions of 3400 km and 470 km, respectively, for the position of the Earth and are completely analytical. They may be used in certain applications that do not require high precision.

Next, we proceed to improving the function ${\psi }_1\left(\tau \right)$ as an analytical expression (Equation (29)), involving an arctangent function, with an argument which, in turn, is an expression that depends on six numerical coefficients (Equation (30)). In order to calculate these coefficients, we propose two methods, namely Method A and Method B.

Method A consists in calculating the coefficients for each given value of the orbital eccentricity, while Method B allows us to express these coefficients as another analytical function with corresponding numeric coefficients that are calculated for ranges of the eccentricity.

2.1. Step 0

Our first step is integrating Equation (1) to obtain $t\left(\theta \right)$. The solution is (see Appendix A).

$$t\left(\theta \right)=\frac{{\alpha }^2\mu }{l}\left[\frac{2}{{(1-{\epsilon }^2)}^{\frac{3}{2}}}arctan\left(\sqrt{\frac{1-\epsilon }{1+\epsilon }}tan\frac{\theta }{2}\right)-\frac{2\epsilon }{1-{\epsilon }^2}\frac{tan\frac{\theta }{2}}{1+\epsilon +(1-\epsilon ){tan}^2\frac{\theta }{2}}\right].$$

(5)

This solution is periodic, with a period equal to $2\pi$, and coincides with the actual time behavior in the interval $\theta \in [-\pi ,\pi ]$. Since the time evolution should be monotonic, we illustrate a way to extend Equation (5) to depict the time evolution by applying the following mapping:

$$t_1\left(\theta \right)=2t\left(\pi \right)⌊1+\frac{\theta -\pi }{2\pi }⌋+t\left(\theta \right),$$

(5a)

where $2t\left(\pi \right)=max\left(t\right(\theta \left)\right)-min\left(t\right(\theta \left)\right)$. In order to avoid complications related to the periodicity of the functions of $\theta$ involved, and the monotony of time, in what follows, we focus our work only on the half-period of the orbit, namely $\theta \in [0,\pi ]$. The second half of the orbit may be obtained from the first part using the symmetry of the orbit.

We now propose the following change of variable:

$$\tau =\frac{l}{{\alpha }^2\mu }\frac{{(1-{\epsilon }^2)}^{\frac{3}{2}}}{2}t.$$

(6)

Thus, Equation (5), in terms of $\tau$ and $\theta$ has the form:

$$\begin{array}{c}\hfill \tau \left(\theta \right)=\left\{\begin{array}{c}arctan\left(\sqrt{\frac{1-\epsilon }{1+\epsilon }}tan\frac{\theta }{2}\right)-\epsilon \sqrt{1-{\epsilon }^2}\frac{tan\frac{\theta }{2}}{1+\epsilon +(1-\epsilon ){tan}^2\frac{\theta }{2}}\mathrm{for}\theta \in [0,\pi ),\hfill \\ \frac{\pi }{2}\mathrm{for}\theta =\pi .\hfill \end{array}\right.\end{array}$$

(7)

Let us seek for the solution to Equation (7) in the form $\theta \left(\tau \right)$ on the interval $\theta \in [0,\pi ]$, and, as deduced from Equation (7), in $\tau \in [0,\frac{\pi }{2}]$.

Note the relationship between $\tau$ and the mean anomaly M:

$$\tau =\frac{M}{2}.$$

(8)

2.2. Step 1

We rewrite Equation (7) as follows:

$$\begin{array}{c}\hfill \tau \left(\theta \right)=\left\{\begin{array}{c}{\tau }_a\left(\theta \right)-{\tau }_b\left(\theta \right)\mathrm{for}\theta \in [0,\pi ),\hfill \\ \frac{\pi }{2}\mathrm{for}\theta =\pi ,\hfill \end{array}\right.\end{array}$$

(9)

where

$${\tau }_a\left(\theta \right)=arctan\left(\sqrt{\frac{1-\epsilon }{1+\epsilon }}tan\frac{\theta }{2}\right),$$

(10)

$${\tau }_b\left(\theta \right)=\epsilon \sqrt{1-{\epsilon }^2}\frac{tan\frac{\theta }{2}}{1+\epsilon +(1-\epsilon ){tan}^2\frac{\theta }{2}}.$$

(11)

Figure 2 shows the behavior of $\tau \left(\theta \right)$, ${\tau }_a\left(\theta \right)$, and ${\tau }_b\left(\theta \right)$ with $\epsilon$ as parameter.

Figure 2 shows how ${\tau }_a\left(\theta \right)$ is similar to $\tau \left(\theta \right)$; therefore, as our zeroth approximation ${\theta }_0\left(\tau \right)$, we will solve the following equation:

$$\tau \left(\theta \right)={\tau }_a\left(\theta \right)=arctan\left(\sqrt{\frac{1-\epsilon }{1+\epsilon }}tan\frac{\theta }{2}\right).$$

(12)

To solve the Equation (12), we apply the tangent function to both parts. Solving for $\theta$, we obtain

$$\begin{array}{c}\hfill {\theta }_0\left(\tau \right)=\left\{\begin{array}{c}2arctan\left(\sqrt{\frac{1+\epsilon }{1-\epsilon }}tan\tau \right)\mathrm{for}\tau \in [0,\frac{\pi }{2}),\hfill \\ \pi \mathrm{for}\tau =\frac{\pi }{2}.\hfill \end{array}\right.\end{array}$$

(13)

Figure 3 shows plots of $\theta \left(\tau \right)$ and ${\theta }_0\left(\tau \right)$ with $\epsilon$ as parameter. $\theta \left(\tau \right)$ was obtained using Equation (7) and interchanging axes.

2.3. Step 2

We introduce the functions $\phi \left(\tau \right)$ and ${\phi }_0\left(\tau \right)$ as follows:

$$\theta \left(\tau \right)=2arctan\left[\phi \right(\tau \left)\right],$$

(14)

$${\phi }_0\left(\tau \right)=\sqrt{\frac{1+\epsilon }{1-\epsilon }}tan\tau ,$$

(15)

so that

$${\theta }_0\left(\tau \right)=2arctan\left[{\phi }_0\left(\tau \right)\right].$$

(16)

Now, we introduce another function ${\psi }_0\left(\tau \right)$, defined on the open interval $\tau \in (0,\frac{\pi }{2})$:

$${\psi }_0\left(\tau \right)=\frac{\phi \left(\tau \right)}{{\phi }_0\left(\tau \right)}.$$

(17)

Appendix B proves the following limits (the former is used immediately to introduce the function ${\phi }_1\left(\tau \right)$, and the latter is used in step 3):

$$\underset{\tau \to 0}{lim}{\psi }_0\left(\tau \right)=\frac{1}{1-\epsilon },$$

(18)

and

$$\underset{\tau \to \frac{\pi }{2}}{lim}{\psi }_0\left(\tau \right)=1+\epsilon .$$

(19)

Now, we introduce, using Equations (15) and (18), the function ${\phi }_1\left(\tau \right)$ as follows:

$${\phi }_1\left(\tau \right)=\underset{\tau \to 0}{lim}{\psi }_0\left(\tau \right){\phi }_0\left(\tau \right)=\frac{1}{1-\epsilon }{\phi }_0\left(\tau \right)=\frac{\sqrt{1+\epsilon }}{{(1-\epsilon )}^{\frac{3}{2}}}tan\tau .$$

(20)

We define approximation one, ${\theta }_1\left(\tau \right)$, as follows:

$$\begin{array}{c}\hfill {\theta }_1\left(\tau \right)=\left\{\begin{array}{c}2arctan\left[{\phi }_1\left(\tau \right)\right]=2arctan\left(\frac{\sqrt{1+\epsilon }}{{(1-\epsilon )}^{\frac{3}{2}}}tan\tau \right)\mathrm{for}\tau \in (0,\frac{\pi }{2}),\hfill \\ 0\mathrm{for}\tau =0,\hfill \\ \pi \mathrm{for}\tau =\frac{\pi }{2}.\hfill \end{array}\right.\end{array}$$

(21)

Figure 4 shows plots of $\theta \left(\tau \right)$ and ${\theta }_1\left(\tau \right)$ with $\epsilon$ as parameter.

In Figure 4, we observe an evident improvement in the ${\theta }_1\left(\tau \right)$ approximation compared to ${\theta }_0\left(\tau \right)$ shown in Figure 3.

2.4. Step 3

We introduce a new function, ${\psi }_1\left(\tau \right)$, defined on the open interval $\tau \in (0,\frac{\pi }{2})$:

$${\psi }_1\left(\tau \right)=\frac{\phi \left(\tau \right)}{{\phi }_1\left(\tau \right)}.$$

(22)

As follows from Equations (18)–(20),

$$\underset{\tau \to 0}{lim}{\psi }_1\left(\tau \right)=1,$$

(23)

$$\underset{\tau \to \frac{\pi }{2}}{lim}{\psi }_1\left(\tau \right)=1-{\epsilon }^2.$$

(24)

Figure 5 shows the plot of ${\psi }_1\left(\tau \right)$ for different values of the parameter $\epsilon$.

We introduce the function ${\phi }_2\left(\tau \right)$ as follows:

$${\phi }_2\left(\tau \right)={\psi }_1\left(\tau \right){\phi }_1\left(\tau \right)={\psi }_1\left(\tau \right)\frac{\sqrt{1+\epsilon }}{{(1-\epsilon )}^{\frac{3}{2}}}tan\tau .$$

(25)

The definition of ${\phi }_2\left(\tau \right)$ as the product of ${\psi }_1\left(\tau \right)$ and ${\phi }_1\left(\tau \right)$ leads us to an apparent logical paradox, since, as follows from Equation (22), ${\phi }_2\left(\tau \right)=\phi \left(\tau \right)$. The solution to this paradox is the fact that ${\psi }_1\left(\tau \right)$ is an unknown function, which we are trying to approximate with the best precision possible. The existence of such a function is obvious, for it can be theoretically constructed point-wise. However, in this work, we are looking for a quasi-analytical expression for ${\psi }_1\left(\tau \right)$, since the exact analytical expression for it, in principle, may not even exist.

Now, we define approximation two, ${\theta }_2\left(\tau \right)$, as follows:

$$\begin{array}{c}\hfill {\theta }_2\left(\tau \right)=\left\{\begin{array}{c}2arctan\left[{\phi }_2\left(\tau \right)\right]=2arctan\left({\psi }_1\left(\tau \right)\frac{\sqrt{1+\epsilon }}{{(1-\epsilon )}^{\frac{3}{2}}}tan\tau \right)\mathrm{for}\tau \in (0,\frac{\pi }{2}),\hfill \\ 0\mathrm{for}\tau =0,\hfill \\ \pi \mathrm{for}\tau =\frac{\pi }{2}.\hfill \end{array}\right.\end{array}$$

(26)

It should be noted that the approximation ${\theta }_1\left(\tau \right)$ (Equation (21)) can be considered as the approximation ${\theta }_2\left(\tau \right)$ with ${\psi }_1\left(\tau \right)=1$. For small values of $\epsilon$, such as Earth’s ($\epsilon =0.0167$), ${\psi }_1\left(\tau \right)$ has a small range (for the Earth, it will be of [0.9997, 1]) so that the approximation ${\psi }_1\left(\tau \right)=1$ already gives us an acceptable result for certain applications. Thus, for the Earth, the maximum error of ${\theta }_1\left(\tau \right)$ is of $1.8\times {10}^{-4}$ rad, which, translated to the error in the position and taking into account the average distance from the Earth to the Sun as $1.5\times {10}^8$ km, is equivalent to 27,000 km, which is a little more than two diameters of the Earth. Now, approximating ${\psi }_1\left(\tau \right)$ in a linear form, and as a cosine with period $\pi$ and amplitude adjusted to the range [$1-{\epsilon }^2$, 1], the following corresponding approximations are obtained:

$$\begin{array}{c}\hfill {\theta }_{2.1}\left(\tau \right)=\left\{\begin{array}{c}2arctan\left[\left(1-\frac{2{\epsilon }^2}{\pi }\tau \right)\frac{\sqrt{1+\epsilon }}{{(1-\epsilon )}^{\frac{3}{2}}}tan\tau \right]\mathrm{for}\tau \in [0,\frac{\pi }{2}),\hfill \\ \pi \mathrm{for}\tau =\frac{\pi }{2},\hfill \end{array}\right.\end{array}$$

(27)

$$\begin{array}{c}\hfill {\theta }_{2.2}\left(\tau \right)=\left\{\begin{array}{c}2arctan\left[\left(1+\frac{{\epsilon }^2}{2}(cos2\tau -1)\right)\frac{\sqrt{1+\epsilon }}{{(1-\epsilon )}^{\frac{3}{2}}}tan\tau \right]\mathrm{for}\tau \in [0,\frac{\pi }{2}),\hfill \\ \pi \mathrm{for}\tau =\frac{\pi }{2}.\hfill \end{array}\right.\end{array}$$

(28)

The approximation ${\theta }_{2.1}\left(\tau \right)$ gives a maximum error of $2.24\times {10}^{-5}$ rad, which translates to 3400 km in this position (slightly more than half the Earth’s radius), and the approximation ${\theta }_{2.2}\left(\tau \right)$ gives a maximum error of $3.11\times {10}^{-6}$ rad, which corresponds to approximately 470 km in this position.

In Figure 5, we notice that the shape of the curves resembles an arctangent function, with a mapping of the argument from ($-\infty$, ∞) to ($\frac{\pi }{2}$, 0), with the range mapping from ($-\frac{\pi }{2}$, $\frac{\pi }{2}$) to ($1-{\epsilon }^2,1$), and with the behavior of the argument being asymmetric and nonlinear. As before, we search for the approximation of the function ${\psi }_1\left(\tau \right)$ in the following form:

$${\psi }_1\left(\tau \right)=1+\frac{{\epsilon }^2}{2}\left(\frac{2}{\pi }arctan\left[\xi (\epsilon ,\tau )\right]-1\right),$$

(29)

where

$$\xi (\epsilon ,\tau )=a_1{\tau }^{-2}+a_2{\tau }^{-1}+a_3\tau +b_1{\left(\tau -\frac{\pi }{2}\right)}^{-2}+b_2{\left(\tau -\frac{\pi }{2}\right)}^{-1}+b_3\left(\tau -\frac{\pi }{2}\right),$$

(30)

with $a_i$ y $b_i$ ($i=1,2,3$) being functions of $\epsilon$.

In order to evaluate the coefficients $a_i$ and $b_i$ (the six numerical coefficients mentioned in the introduction), two methods were developed, Method A and Method B. In Method A, the coefficients are calculated for each value of $\epsilon$, while in Method B, they are expressed as analytical functions that, in turn, depend on other numeric coefficients, which are calculated for ranges of $\epsilon$. As will be seen subsequently, Method A is more accurate, but less general, while Method B is slightly less accurate (especially after a certain value of $\epsilon$), but more general and easier to use in applications.

2.4.1. Method A

The coefficients $a_i\left(\epsilon \right)$ and $b_i\left(\epsilon \right)$ were calculated numerically for each eccentricity value, corresponding to every planet of the solar system, by means of RMSE (root mean square error) minimization, using MATLAB 2020b.

Figure 6 shows the behavior of $a_i\left(\epsilon \right)$ and $b_i\left(\epsilon \right)$ in the range of $\epsilon$∈ (0, 0.5) (this range was selected because some absolute values of $a_i\left(\epsilon \right)$ and $b_i\left(\epsilon \right)$ grow drastically after $\epsilon =0.5$, which does not allow us to appreciate the behavior before $\epsilon =0.5$).

Table 1. Values of $a_i$ and $b_i$ and error in $\theta \left(\tau \right)$ for the Earth and Pluto.

Planet	$\mathit{\epsilon }$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$	${\mathit{b}}_1$	${\mathit{b}}_2$	${\mathit{b}}_3$	ME	MAE	RMSE
Earth	0.0167	0.310	−0.073	−0.363	−0.340	−0.087	−0.357	$6.1\times {10}^{-9}$	$2.9\times {10}^{-9}$	$3.4\times {10}^{-9}$
Pluto	0.2488	0.146	−0.001	−0.472	−0.647	−0.274	−0.331	$6.6\times {10}^{-6}$	$2.8\times {10}^{-6}$	$3.4\times {10}^{-6}$

shows the values of $a_i\left(\epsilon \right)$ and $b_i\left(\epsilon \right)$, as well as the error of $\theta \left(\tau \right)$ for the Earth and Pluto, in radians, where ME is the maximum absolute error, MAE is the mean absolute error, and RMSE is the root mean square error.

Since Table 1 serves just for illustrative purposes, the values of $a_i\left(\epsilon \right)$ and $b_i\left(\epsilon \right)$ were rounded to three decimal places. A detailed table including more accurate values for all planets will be shown in Section 3. For now, let us mention that the absolute maximum error in the position of the Earth is 915 meters and approximately 39,000 km (close to the value of the circumference of the Earth, or 16 Pluto diameters) for the position of Pluto (the average distance of Pluto to the Sun is approximately 5.9 billion kilometers).

2.4.2. Method B

Analyzing Figure 6, it can be seen that the shapes of the curves have the behavior of a cubic polynomial, so we search for the coefficients $a_i\left(\epsilon \right)$ and $b_i\left(\epsilon \right)$ in the following form:

$$\begin{array}{c}\hfill a_i\left(\epsilon \right)=a_{ij}{\epsilon }^j,\\ \hfill b_i\left(\epsilon \right)=b_{ij}{\epsilon }^j,\\ \hfill i=1,2,3,\\ \hfill j=0,1,2,3,\end{array}$$

(31)

where repeated indices imply summation.

We introduced the approximation ${\theta }_3\left(\tau \right)$ with the same form of ${\theta }_2\left(\tau \right)$ given by Equations (26), (29) and (30), where Equation (31) was now used to approximate the $a_i$ and $b_i$ in Equation (30).

The attempt to calculate the coefficients $a_{ij}$ and $b_{ij}$ in the complete range of $\epsilon$∈ (0,1) was not successful. Thus, it was decided to divide the range into five intervals: $\epsilon$∈ (0,0.1], $\epsilon$∈ (0.1,0.25], $\epsilon$∈ (0.25,0.5], $\epsilon$∈ (0.5,0.7], and $\epsilon$∈ (0.7,1). In the next section, we will present the values of $a_{ij}$ and $b_{ij}$ by ranges, and compare the precision of the approximation ${\theta }_2\left(\tau \right)$, which uses values of ${\psi }_1\left(\tau \right)$ calculated using Method A, with that of ${\theta }_3\left(\tau \right)$.

3. Results

We now compare the results of our quasi-analytical solution with the real orbits of the planets of the solar system. First, we review the results obtained above in the following equations. By a change of variable,

$$\tau =\frac{1}{{\alpha }^2\mu }\frac{{(1-{ϵ}^2)}^{\frac{3}{2}}}{2}t,$$

(32)

the equations of motion in the polar coordinate system with the origin at the Sun, and in the interval of time $\tau$∈$[0,\frac{\pi }{2}]$$(t\in [0,\frac{T}{2}])$, $\theta$∈$[0,\pi ]$ are the following:

$$\begin{array}{c}\hfill \theta \left(\tau \right)=\left\{\begin{array}{c}2arctan\left[\left(1+\frac{{\epsilon }^2}{2}(\frac{2}{\pi }arctan\left[\xi (\epsilon ,\tau )\right]-1)\right)\frac{\sqrt{1+\epsilon }}{{(1-\epsilon )}^{\frac{3}{2}}}tan\tau \right]\mathrm{for}\tau \in (0,\frac{\pi }{2}),\hfill \\ 0\mathrm{for}\tau =0,\hfill \\ \pi \mathrm{for}\tau =\frac{\pi }{2},\hfill \end{array}\right.\end{array}$$

(33)

$$r\left(\tau \right)=\frac{\alpha }{1+\epsilon cos\theta \left(\tau \right)},$$

(34)

where $\xi (\epsilon ,\tau )$ is given by the following expression:

$$\xi (\epsilon ,\tau )=a_1{\tau }^{-2}+a_2{\tau }^{-1}+a_3\tau +b_1{\left(\tau -\frac{\pi }{2}\right)}^{-2}+b_2{\left(\tau -\frac{\pi }{2}\right)}^{-1}+b_3\left(\tau -\frac{\pi }{2}\right),$$

(35)

$a_i$ and $b_i$$(i=1,2,3)$ are functions of $\epsilon$. In what follows, we present the two methods developed for the calculation of the coefficients $a_i$ and $b_i$.

3.1. Method A

As mentioned above, in this method, the values of $a_i$ and $b_i$ were calculated for each eccentricity by means of RMSE minimization in MATLAB 2020b.

Table 2. Values of $a_i$, $b_i$, corresponding errors in $\theta \left(\tau \right)$, and absolute position for celestial bodies.

	Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune	Pluto
$\mathbf{\epsilon }$	0.2056	0.0067	0.0167	0.0935	0.0489	0.0565	0.0457	0.0113	0.2488
$a_1$	0.17053517	0.31862395	0.30977503	0.24694331	0.28234256	0.27609996	0.28499728	0.31453377	0.14561949
$a_2$	−0.01166110	−0.07705003	−0.0728542	−0.04432541	−0.06014655	−0.05731287	−0.06135776	−0.07510434	−0.0012376
$a_3$	−0.43861374	−0.36086494	−0.36274460	−0.38289299	−0.36985666	−0.37179675	−0.36907139	−0.36171170	−0.47217706
$b_1$	−0.57282655	−0.33074668	−0.34005421	−0.42023391	−0.37170483	−0.37957121	−0.36843946	−0.33499871	−0.64694223
$b_2$	−0.22240872	−0.08288098	−0.08742593	−0.12920751	−0.10334339	−0.10741403	−0.10166725	−0.08494978	−0.27396065
$b_3$	−0.33681377	−0.35855961	−0.35697508	−0.34764211	−0.35253989	−0.35161930	−0.35294044	−0.35781706	−0.33108759
ME	$3.6\times {10}^{-6}$	$9.5\times {10}^{-10}$	$6.1\times {10}^{-9}$	$3.5\times {10}^{-7}$	$6.6\times {10}^{-8}$	$9.4\times {10}^{-8}$	$5.6\times {10}^{-8}$	$2.7\times {10}^{-9}$	$6.6\times {10}^{-6}$
MAE	$1.5\times {10}^{-6}$	$4.5\times {10}^{-10}$	$2.9\times {10}^{-9}$	$1.5\times {10}^{-7}$	$2.9\times {10}^{-8}$	$4.1\times {10}^{-8}$	$2.5\times {10}^{-8}$	$1.3\times {10}^{-9}$	$2.8\times {10}^{-6}$
RMSE	$1.9\times {10}^{-6}$	$5.3\times {10}^{-10}$	$3.4\times {10}^{-9}$	$1.8\times {10}^{-7}$	$3.5\times {10}^{-8}$	$5.0\times {10}^{-8}$	$3.0\times {10}^{-8}$	$1.5\times {10}^{-9}$	$3.4\times {10}^{-6}$
$D_s$	$5.8\times {10}^7$	$1.1\times {10}^8$	$1.5\times {10}^8$	$2.3\times {10}^8$	$7.9\times {10}^8$	$1.4\times {10}^9$	$2.9\times {10}^9$	$4.5\times {10}^9$	$5.9\times {10}^9$
$E_p$	208	0.11	0.92	80	52	131	163	12	39,000

shows the values of $a_i$ and $b_i$, ME, MAE, and RMSE errors in $\theta \left(\tau \right)$, in radians, with $D_s$ being the average distance of the planet to the Sun in kilometers, and $E_p$ being the maximum error of the position in kilometers with respect to the real orbit.

The results in Table 2 show that for small values of $\epsilon$, the accuracy was high. However, for values of $\epsilon \sim 0.2$ the error increased significantly (for Mercury and Pluto the error increased by 2–3 orders in comparison to the errors for the other planets).

3.2. Method B

As was pointed out before, in this method, we calculate the coefficients of a third-degree polynomial (Equation (31)) by minimizing the RMSE for the ranges of $\epsilon$. Therefore, in this case, it is not necessary to calculate the $a_i$ and $b_i$ for each value of the eccentricity.

Table 3. Values of $a_{ij}$ and $b_{ij}$ calculated in ranges of $\epsilon$.

Range of $\mathit{\epsilon }$	(0,0.1]	(0.1,0.25]	(0.25,0.5]	(0.5,0.7]	(0.7,1.0)
$a_{10}$	0.32464090	0.32455984	0.32493519	0.33117795	0.34799892
$a_{11}$	−0.90342437	−0.90136299	−0.90443788	−0.94434644	−1.02378174
$a_{12}$	0.79798292	0.77956682	0.78688870	0.87179816	0.99672027
$a_{13}$	−0.24897861	−0.19074605	−0.19470806	−0.25486105	−0.32027157
$a_{20}$	−0.07992819	−0.07920359	−0.07197522	−0.06646582	−0.11098658
$a_{21}$	0.43404410	0.41648507	0.33665965	0.29615322	0.50088661
$a_{22}$	−0.64017363	−0.49188569	−0.19208276	−0.09396227	−0.40748236
$a_{23}$	0.75341116	0.31132787	−0.07256170	−0.15093252	0.00894242
$a_{30}$	−0.35968044	−0.35742442	−0.15675162	6.29837377	154.50791377
$a_{31}$	−0.17220655	−0.22278632	−2.20357719	−41.92862343	−715.24316797
$a_{32}$	−0.64666864	−0.26055305	6.28024901	87.40520959	1105.82612363
$a_{33}$	−1.78408496	−2.80408480	−10.06884121	−65.08749455	−577.96129585
$b_{10}$	−0.32463507	−0.32142203	−0.09294215	6.21933680	138.39317241
$b_{11}$	−0.90434048	−0.97722350	−3.24616805	−42.18802414	−643.15766068
$b_{12}$	−1.11071449	−0.54525397	7.00356134	86.74796714	996.53861794
$b_{13}$	−1.63153199	−3.15712303	−11.61829996	−65.86504267	−524.44755555
$b_{20}$	−0.07992299	−0.07527391	0.29080466	11.97863173	287.23614950
$b_{21}$	−0.43571269	−0.54094614	−4.16198706	−76.07319110	−1326.12648115
$b_{22}$	−0.78063318	0.03353877	12.02291626	158.84151291	2048.91985911
$b_{23}$	−2.10660888	−4.29567159	−17.65734989	−117.20310761	−1068.68369232
$b_{30}$	−0.35968350	−0.36025835	−0.44346730	−3.54853992	−81.17535211
$b_{31}$	0.17171182	0.18402819	0.99888552	20.06864682	372.42195877
$b_{32}$	−0.59524043	−0.68312683	−3.34688052	−42.20159066	−574.67488149
$b_{33}$	1.45594036	1.66666199	4.58818969	30.87266047	298.77577747

shows the values of the coefficients $a_{ij}$ and $b_{ij}$ for the different ranges of $\epsilon$. Obviously, using this method results in errors greater than those of Method A.

Figure 7 shows a comparison of the errors between methods A and B. As can be seen, the error in Method B was slightly higher (especially for ME) up to values of $\epsilon =0.8$. For values of $\epsilon >0.8$, the errors of Method B increased significantly with respect to those of Method A.

3.3. Equations of Motion for the Planets

Finally, let us express the equations of motion of both position and velocity in the Cartesian coordinate system, with the origin at the Sun.

The equations for the position are as follows:

$$\begin{array}{c}\hfill x\left(\tau \right)=r\left(\tau \right)cos\theta \left(\tau \right),\\ \hfill y\left(\tau \right)=r\left(\tau \right)sin\theta \left(\tau \right),\end{array}$$

(36)

where $\theta \left(\tau \right)$ and $r\left(\tau \right)$ are given by Equations (33) and (34), respectively.

The equations for the velocity are as follows:

$$\begin{array}{c}\hfill v_x=\dot{r}\left(\tau \right)cos\theta \left(\tau \right)-r\dot{\theta }\left(\tau \right)sin\theta \left(\tau \right),\\ \hfill v_y=\dot{r}\left(\tau \right)sin\theta \left(\tau \right)+r\dot{\theta }\left(\tau \right)cos\theta \left(\tau \right),\end{array}$$

(37)

where

$$\begin{array}{c}\hfill \dot{\theta }\left(\tau \right)=2\frac{\sqrt{1+\epsilon }}{{(1-\epsilon )}^{\frac{3}{2}}}\frac{\dot{\psi }\left(\tau \right)tan\tau +\psi \left(\tau \right){sec}^2\tau }{1+\frac{1+\epsilon }{{(1-\epsilon )}^3}{\psi }^2\left(\tau \right){tan}^2\tau },\\ \hfill \dot{r}\left(\tau \right)=\frac{\alpha \epsilon \dot{\theta }\left(\tau \right)sin\theta \left(\tau \right)}{{(1+\epsilon cos\theta \left(\tau \right))}^2},\end{array}$$

(38)

and

$$\begin{array}{c}\hfill \psi \left(\tau \right)=\left\{\begin{array}{c}1+\frac{{\epsilon }^2}{2}(\frac{2}{\pi }arctan\left[\xi (\epsilon ,\tau )\right]-1)\mathrm{for}\tau \in (0,\frac{\pi }{2}),\hfill \\ 1\mathrm{for}\tau =0,\hfill \\ 1-{\epsilon }^2\mathrm{for}\tau =\frac{\pi }{2},\hfill \end{array}\right.\end{array}$$

(39)

$$\begin{array}{c}\hfill \dot{\psi }\left(\tau \right)=\left\{\begin{array}{c}\frac{{\epsilon }^2}{\pi }\frac{\dot{\xi }\left(\tau \right)}{1+{\xi }^2\left(\tau \right)}\mathrm{for}\tau \in (0,\frac{\pi }{2}),\hfill \\ 0\mathrm{for}\tau =0,\hfill \\ 0\mathrm{for}\tau =\frac{\pi }{2}.\hfill \end{array}\right.\end{array}$$

(40)

The function $\xi \left(\tau \right)$ defined in the interval $\tau$∈$(0,\frac{\pi }{2})$ is given by Equation (35) and its derivative is the following:

$$\dot{\xi }\left(\tau \right)=-2a_1{\tau }^{-3}-a_2{\tau }^{-2}+a_3-2b_1{\left(\tau -\frac{\pi }{2}\right)}^{-3}-b_2{\left(\tau -\frac{\pi }{2}\right)}^{-2}+b_3.$$

(41)

Figure 8 shows a comparison plot between the real orbit, calculated numerically using Equation (7), and the orbit obtained by numerical integration of Equation (37) (where Method A was used). Additionally, the velocity vectors are shown at an arbitrary scale at certain points of the trajectory.

As can be seen in Figure 8, the integrated orbit closely resembles the behavior of the real orbit. This shows that our Method A is sufficiently accurate, even for eccentricities of the order of up to $\epsilon =0.5$.

4. Solution to Kepler’s Equation

As can be seen from Figure 1b, it is trivial to obtain the following expression, which gives the formula for E (the eccentric anomaly), appearing in Equation (3) as a function of the polar coordinates $(\rho ,\theta )$ with the center in the focus F. In what follows, we set the major semi-axis equal to one ($a=1$):

$$E=arccos(F+\rho cos\theta ),$$

(42)

where $F=\left|OF\right|$ (Figure 1b). As is well known from the theory of conic sections, $F=\epsilon a=\epsilon$, since we set $a=1$. Thus, Equation (42) acquires the following form:

$$E=arccos(\epsilon +\rho cos\theta ).$$

(43)

Due to the normalization $a=1$, the expression for $\rho$ is as follows:

$$\rho =\frac{r}{a}.$$

(44)

The functions $\theta \left(\tau \right)$ and $r\left(\tau \right)$ are given by our final quasi-analytical solution (Equations (33) and (34)).

Now, in order to obtain the final explicit expression for the solution of KE, $E\left(M\right)$, we need to express $\theta$ and $\rho$ as functions of M (the mean anomaly given by Equation (4)). So far, $\theta$ and r are functions of $\tau$ (where $\tau$ is given by Equation (6)). Thus, by expressing $\tau$ as a function of M, $\theta$ and r automatically become functions of M. The expression for $\tau \left(M\right)$ is the following:

$$\tau \left(M\right)=\frac{M}{2}.$$

(45)

In order to obtain $\rho$ as a function of M, we use Kepler’s third law to express a as follows:

$$a={\left(\frac{l}{2\pi \mu \sqrt{1-{\epsilon }^2}}T\right)}^{\frac{1}{2}}.$$

(46)

The final solution to KE is given by Equation (43), using the quasi-analytical solution for the motion given by Equations (33) and (34) and taking into account the relations (44), (45) and (46).

5. Remarks and Conclusions

In this work, we obtained a quasi-analytical solution for the motion of the celestial bodies as an explicit function of time in four steps. We called this solution quasi-analytical, due to the dependency on certain numerical coefficients, which in turn themselves depend on the orbital eccentricity. We proposed two methods for evaluating these coefficients: Method A and Method B. The former gives a higher degree of accuracy, but involves calculations for each value of the eccentricity, while the latter is less accurate (especially for values of $\epsilon >0.8$) but more practical, since it works for ranges of the eccentricity. Although there exist methods for solving Kepler’s equation up to machine precision, e.g., [11], these methods are completely numerical and require an initial guess. The aim of our work was to find a relatively simple explicit analytical solution with acceptable precision. In future work, we plan to use our results as an initial guess for the iterative procedures mentioned previously. We hypothesize that this approach will lead to highly precise results with fewer iterations.