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The present study deals with a traditional physical problem: the solution of the Kepler’s equation for all conics (ellipse, hyperbola or parabola). Solution of the universal Kepler’s equation in closed form is obtained with the help of the two-dimensional Laplace technique, expressing the universal functions as a function of the universal anomaly and the time. Combining these new expressions of the universal functions and their identities, we establish one biquadratic equation for universal anomaly ( χ) for all conics; solving this new equation, we have a new exact solution of the present problem for the universal anomaly as a function of the time. The verifying of the universal Kepler’s equation and the traditional forms of Kepler’s equation from this new solution are discussed. The plots of the elliptic, hyperbolic or parabolic Keplerian orbits are also given, using this new solution.

In the Keplerian problem, a body of mass

where

The traditional form of Kepler’s equation, which can be obtained directly from Equation (1) (see [

For elliptic orbits

for hyperbolic orbits

where

(see [

Johannes Kepler announced the relevant laws of above equation early in 1609 and 1619 [

Early analytical solution of Kepler’s equation was considered in a comprehensive study of Tisserand [

In virtually every decade from 1650 to the present, there have appeared papers devoted to the solution of this Kepler’s equation. Its exact analytical solution is unknown, and therefore, efficient procedures to solve it numerically have been well discussed in many standard text books of Celestial Mechanics and Astrodynamics as well as in a large number of papers. Colwell [

In the current study, an analytical investigation of the Kepler’s equation real roots in closed form is presented. In Section 2, we will establish the general form of Kepler’s equation and will clear up the useful identities of the universal functions. In Section 3, using the two-dimensional Laplace transform technique, we will present an analytical solution for the universal Kepler’s equation, obtaining the universal functions

Finally, discussion of the results, thus obtained, is presented in Section 5; the new solution of the problem will prove that verifies the traditional form of Kepler’s equations for elliptic, hyperbolic or parabolic orbits. The elliptic, hyperbolic or parabolic Keplerian motion is easily plotted, using this new solution.

In order to solve the Kepler’s Equation (2), we use here the generalized form of this equation with the universal functions and the universal anomaly instead of the eccentric anomaly (see [

Working for the Kepler’s Equation (2), we consider an object following a path of same eccentricity

(see [

where E is the eccentric anomaly angle of elliptic or hyperbolic orbit and D the parabolic eccentric anomaly of parabolic orbit with dimension

Depending on the sign of

From the initial condition

where

Remark that the

Now, using the universal functions defined by

with their following useful properties:

(see [

which is a standard form of the traditional Kepler’s Equations (2) with the epoch at pericenter passage (see [

To find out the expression of many orbital quantities, e.g. the magnitude of the position vector

In order to obtain the analytical solution of the present problem, we shall solve first the universal Kepler’s equation Equation (10), obtaining the universal functions

The universal functions

(see Equations (9)) and the universal Kepler’s Equation (10) becomes

From the initial condition

The application of double Laplace transform (with respect to

where

Now, the universal function

where we have abbreviated

with dimensions:

The universal functions

(see Equations (9c)) and the universal Kepler’s Equation (10) becomes

For

Similarly as in the case of

(cf., Appendix). Inverting

where we have defined the non-dimensional function

Then, substituting the results of Equation (14) and (19) into the relations

where

In order to obtain a solution for the universal anomaly

where

Substituting Equation (24) (with

To find out two more relations between

Further, we can find one more relation using the basic identity

The three Equations (25), (26) and (27) are a system of the three unknowns:

Finally, substituting Equations (28) into Equation (27), we obtain the following biquadratic equation for universal anomaly

where we have abbreviated

with dimensions:

The solution of the biquadratic Equation (29) gives the relation between the universal anomaly

Solving the new biquadratic Equation (29), we get the solution of the present problem for the universal anomaly

where we have abbreviated

particularly, for elliptic orbits

for hyperbolic orbits

and for parabolic orbits

Remark that the discriminant of the biquadratic Equation (29) is

where

In the case of parabolic orbits where the limiting case

The Equation (31) is the solution of the present problem for all conics (ellipse, hyperbola or parabola) and expresses the relation between the universal anomaly

Knowing the solution of the universal anomaly

Then, the universal functions (19), (14), (21) and (22) are expressed as functions of the time t as show below

The magnitude of the position vector

(see [

Furthermore, if we work in the orbital reference system with the origin at the attracting center (or focus), we chose the

where

where _{ }and

Using the standard form of the universal Kepler’s equation (10) with the epoch at pericenter passage, we have derived a new biquadratic equation (29) for universal anomaly

The new solution (31) of the present problem was obtained solving Equation (29) with initial-value conditions for the orbiting body at time

The solution (31) is a solution of the present problem. Indeed, the new expressions of the universal functions

The solution (31) verifies also the traditional forms of Kepler’s Equation (2). Particularly, in the case of elliptic orbits

where

Note that

Similarly, in the case of hyperbolic orbits

where

Finally, in the case of parabolic orbits

From the other hand, the standard and hyperbolic trigonometric functions of Equations (2) are expressed as

where we have use Equation (35) and the relationship between the function U_{1} and the standard and hyperbolic

trigonometric functions of Equations (2):

and hyperbola, respectively (see [

Now, we will prove that the Equations (40) and (42) represent the solutions of the traditional forms of Kepler’s Equations (2). Indeed, it can be shown that the left-hand sides of Equations (2) are reduced to the right- hand sides, namely

in accordance with the Equations (40), (42) and (45).

It should be pointed out that our solutions for eccentric anomaly (cf., Equations (40) and (42)) are ready for physical applications in the corresponding Keplerian orbits.

In addition to above Keplerian orbits, the new solution (33b) of the present problem for the case of parabola (

Remark that the parabolic Keplerian equation is called Barker’s equation (see [

where

In order to study the Keplerian orbits with the help of the new solutions, we use also the cartesian coordinates

where we have defined the non-dimensional relation

in accordance to the Equations (39). Then, we introduce the non-dimensional coordinates

The new expressions (52) verify the following equations of ellipse

where

In the other hand, for the case of parabola

Then, we have, from Equations (54),

The last Equation (55) is the equation of parabola, which passes through its pericenter with coordinates

(see [

with the non-dimensional coordinates

In addition to above results for the non-dimensional coordinates we have (for physical application) the corresponding expressions:

For elliptic orbits

for hyperbolic orbits

for parabolic orbits

where

In order to get a physical insight into the new solution of the Kepler’s problem, we apply the above results for the system Earth-Moon. For this system the eccentricity of the Moon is

Note that the use of the upper limit of the mean anomaly, given from the relation

eccentricity

Now, varying the mean anomaly

with the eccentricity

Finally, varying the mean anomaly

This work presents a solution to the well known Keplerian two body physical problem. From the investigation

for this new solution, the main conclusions have been drawn as following:

1) An analytical solution for the universal Kepler’s equation has been determined, obtaining the universal functions

2) Using an explicit expression for the universal anomaly

3) The solution

4) This new analytical solution for the universal anomaly has been discussed and proved that verifies the universal Kepler’s equation (cf., Equation (10)), since the time depended universal functions U_{3} and U_{1} verify this equation. Then, the solutions for the eccentric anomaly (cf., Equations (40) and (42)) were also proved that verify the traditional form of Kepler’s equations for elliptic or hyperbolic orbits. This new solution for the universal anomaly has also proved that verifies the traditional Barker’s equation for parabolic orbits [

5) To our knowledge, this work gives in closed form the actual analytical solution of the Kepler’s problem. The advantage of the new solution is simple and ready for physical applications in the elliptic, hyperbolic or parabolic Keplerian orbits.

Solution of partial differential equations using two-dimensional Laplace transforms

The general form of second-order linear partial differential equation in two variables is given as following

where

and their one-dimensional Laplace transformations

in accordance with the two-dimensional analysis formula, which can be written as one-dimensional analysis in the

Now, applying double Laplace transformation to both sides of Equation (A1) and using Equations (A3), we obtain the solution of Equation (A1) in the transform domain as

with the abbreviation

In order to invert this two-dimensional Laplace transform

where we keep the second transform variable