, we finally obtain:

                                                                                           1.14

This is the exact relationship, as compared to equation 1.10, and one of the Kepler’s Laws for circular orbits which must be used when dealing with bodies with comparable masses.

As an additional example, let us consider that as first and rather crude approximation the Moon moves in a circular orbit around the Earth.

The lunar period is approximately 27.32 days, from which we can calculate:

Mean movement:

Distance between the centers of the Earth an d Moon (equation 1.14) :

  Lunar circular speed:

  Masses of the planets

  The masses of the planets with at least one satellite can be determined using the basic concepts discussed above, as a fraction of the mass of the Sun, as follows:

  Let us define as the gravitational parameter as:

                                                                                                  1.15

Where M is the Sun’s mass and m that of a given planet.

For the Earth, we have:

Equation 1.7;  

Equation 1.14 ;            

Then:

                                                                                      1.16

  For this case, we have that the planet is the central body and the satellite is the secondary body. If we denote their masses as m_P y ms, respectively, we have:

                                                                                     1.17

  where rs is the distance between the satellite and the planet and Ps is its orbital period.

  By dividing equation 1.17 by equation 1.16, we obtain:

                                                                                      1.18

If we assume that we can discard the mass of the satellite with respect to that of the planet and the mass of the Earth with respect to that of the Sun, the above equation is :

                                                                                           1.19

  For the practical application of equation 1.19, we must know the mean distance and orbital period of the satellite under study. As a matter of example, let us try to determine the mass of the planet Mars in units of the mass of the Sun, by considering the orbital data of its natural satellite Deimos, as follows:

  Mean distance Deimos-Mars:  0.00015695 Astronomical Unit  (23,480 km. aprox.)

Orbital period : 1.26244 days = 0.003456 years

  According to equation 1.19, the mass of Mars is then found to be:

  That is, the mass of Mars is less than 3 millionths of the Sun's mass.

2. The of "two body" problem

  The cause and explanation of the planetary movements is the mutual gravitational interaction that, as mentioned earlier, is directly proportional to the masses of the involved bodies and inversely proportional to the square of their distances. The movement of a system of masses is regulated by the law of universal gravitation; however, in most cases it is not possible to calculate directly its effects and it is necessary to use approximate or numerical methods. Nevertheless, if we restrict our analysis to a system of two bodies, it is possible to obtain analytical expressions to study their movement without major complications.

  The study of the movement of two bodies interacting only under their mutual gravitational actions is traditionally known as the two body problem and constitutes a fundamental chapter in celestial mechanics, being perhaps the basis for any attempt to mathematically understand the movement of planets in space.

  During the development of the equations for the two body problem, we shall see that we will be able to conclude the following points:

  -         The two bodies move in a fixed and common plane for which there exists a center of mass with constant and uniform motion through space.

  -         Each of the two bodies moves around the center of mass of the system in a conic orbit.

  -         The ratio of the distances to the center of mass depends only on the masses of the bodies and thus constant, therefore both orbits are similar.

  Normally, the problem of the two bodies refers to a system formed by the Sun and another secondary body and can be used as a first approximation for determining and explain the movement in a system of several bodies like that of the solar system, but bearing in mind that it is only an approximation to the physical reality.

  Additionally, for the purpose of deriving a general, though simplified, expression for the relative movement of our two body system we must assume the following simplifications:

  -         Both bodies are to be considered as spherically symmetrical with their masses concentrated in their geometric centers.

  -         We will neglect other external or internal additional forces and thus only consider gravitational effects between the centers of mass of the system.

  Figure 2.1 depict a system of masses M and m, with their respective position vectors rM and rm which are supposed to be connected to a rectangular and non-accelerated inertial system X’, Y’, Z’ and an similar parallel rectangular X, Y, Z system which conveniently has its origin with the center of mass of M.. The connecting vector of the masses is r = rm- rM.

Figure 2.1

  Applying Newton’s law, as seen above, to the system X', Y', Z', we have:

  The first of these forces is negative as it has direction opposite to r.

  Now, both equations can also be written as:

                                                                                                  2.1

                                                                                                     2.2

  If we subtract equation 2.2 from equation 2.1, we obtain:

                                                                              2.3

  Equation 2.3 is the differential equation of relative movement of our two body system, expressed in vectorial terms.

  If we define, as we did in last chapter, the gravitational parameter as = G (M+m), we can express equation 2.3 as:

                                                                                                    2.4

As also mentioned before, we must say that if the two bodies under study represent the movement of an artificial satellite around the Earth, then we can assume that the mass of M is much greater than that of m and thus we can express = GM as an additional simplification but normally adequate.

As the gravitational fields are of conservative nature we can intuitively state, as will be proven rigorously ahead, that a moving body under the sole gravitational action of another does not gain nor lose mechanical energy but rather only experiences an exchange of kinetic energy by potential energy and that additionally its angular momentum is kept constant by virtue of the fact that the gravitational force is directed along the centers of mass and no other tangential forces exist that could alter the angular momentum of the system.

Mechanical energy conservation:

The energetic constant of the movement of the two bodies can be derived as follows:

1. Multiply equation 2.4 by  :

2. As in general we have that:  , then: