Abstract

In this paper, we obtain the solution of the two-body problem with variable masses by posing some assumptions on the classical equation and thereby reducing the solution of the problem to investigating the solutions of some Ricatti type differential equations. Along this process, we also give a simple proof of the well-known Mestscherskii Theorem and establish some related formal relations under these assumptions.

1. Introduction

The attempts to solve the two-body problem with variable masses go back to the middle of 19th century. The most meticulous and comprehensive of these endeavors appeared in the works of H. Glyden ¹, J. Mestscherskii ², G. Armellini ³, Sir James Jeans ⁴, William D. MacMillan ⁵, G. N. Doubochine ^{6, 7}, and K. Sawtchenko ⁸. These were further expanded by the seven articles published by E. L. Martin in 1934 ^{9, 10, 11, 12, 13, 14, 15}. Most of these investigations relied upon the classical equations of motion in a gravitational field modified suitably to accommodate for the variability of the masses.

Then on, the problem kept on enjoying a distinguished existence at the confluence of physics, astrophysics, and applied mathematics, and was analyzed in many different ways and under many different assumptions.

Our goal in this paper is to investigate the solution of this problem under some reasonable restrictions. Throughout, we assume some basic familiarity with some physical and mathematical techniques, most of which can be found in Goldstein ¹⁶ and Betounes ¹⁷.

Since it is going to directly affect our system of equations, let us briefly talk about Kepler’s Second Law. As is well known, the law states that a line joining a planet and the Sun sweeps out equal areas during equal intervals of time, implying a planet travels faster when closer to the Sun, and slower when farther from the Sun.

We note that in a small time interval the planet sweeps out a small triangle having base and height

and consequently, an area of

Thus, the rate of change of area would be

Note that the area enclosed by the elliptical orbit is Thus, the period satisfies the equation

Now let us establish the system of differential equations that we will use to solve the problem.

Suppose the variable masses are given by two holomorphic functions and We will assume that

where is a positive analytic real function of

Let denote the constant of gravitation, which in SI units is approximately It can be shown that ¹⁸ the solution of the problem is equivalent to finding the solution of the system of differential equations

The first equation is, of course, Kepler’s Second Law with where are axes of the ellipse and the period of revolution. Clearly, its integrability depends on that of the second one. Thus, from now on, we assume that the solution of our problem is reduced to the integration of the second equation.

2. Solution of the Differential Equation in the Astronomical Case

In the astronomical case, we assume that the total mass varies very slowly with respect to time and thus and are negligible compared to

Let

Let denote the eccentricity and denote the true anomaly, that is, the parameter that defines the position of a body moving along a Keplerian orbit.

Thus,

Let us put

Hence,

Let us now substitute this in the equation

We obtain

Taking the constants as unities, and putting

the equation becomes

We refer to this equation as the equation of the two-body problem in the astronomical case.

Assuming the equation further simplifies to

A Special Case

If the total mass function is linear, say the equation

with can easily be solved by The Frobenius series method.

Putting

One easily obtains that for and

Giving us the solution

where is an arbitrary constant. It is easy to see that this series is convergent for all finite values of

Remark.

In the equation

Let us now put

Thus, we can rewrite the equation as

Recall that a differential equation of the form

with and is called a Ricatti equation, after the Italian mathematician Jacopo Ricatti (1676 – 1754). Thus, in this case, we can think of the solution of the two-body problem as a solution of a Ricatti equation. See Biernacki ¹⁹ and Milloux ²⁰.

3. Solution of the Differential Equation in Case of Small Eccentricity

Recall we had

with

and

Let us now introduce a new variable by

Thus,

Let us now substitute this in the equation

to obtain

Since

we have

We thus have

which is, of course, the well-known Armellini equation

²¹

Note that the homogenous form of this equation is a Ricatti equation, implying in these two cases the two-body problem with variable masses is reduced to finding the solution of a Ricatti equation, a theorem first proved by Armellini in 1935 ²¹.

Since in this case, the Armellini equation becomes

that is,

We get,

or

which, of course, is an alternate way of obtaining Mestscherkii’s theorem.

Special Cases

1. If a constant, then

2. If a constant, then

Since

and we still have

3. If a constant, then we get the equation

which can be solved easily. To this end, we write

This implies

and the equation becomes

4. The case was also analyzed by Armellini. In this case, if we put

in the equation

since

and consequently,

Thus, we now get a Sturm-Liouville type equation

We will now show that this equation can be transformed into a Bessel equation. To this end, let us put

Since

we get

implying

Differentiating both sides with respect to one obtains

Substituting this in the equation

we get

Since

this equation can be rewritten as

which is of course the Bessel equation

We can again apply the Frobenius series solution method to this equation and get

where

Here, is the gamma function defined as

over all complex with

References