**2.5 Shape of cometary’s orbits**

Concerning the comets, the Newtonian equation of forces is:

(2.21) where: (2.22) and

h_{N} – energy constant remaining constant during the motion.

For:

a), h_{N}<0, E<0 that is (2.23) the comet describes an ellipse;

b) h_{N}=0, E=0 that is (2.24) the comet describes a parabola;

c) h_{N}>0, E>0 that is (2.25) the comet describes an hyperbola;

From the approximately 830 cometary’s orbits we know till now, according to the Newtonian theory, 73% have a quasi-parabolic eccentricity 0.99<e<1.01 and only 27% have e<0.99, that is they are moving on clearly elliptical orbits. From the nearly parabolic orbits there are 14% hyperbolic, 16% very lengthened ellipses, and 43% parabolic. We infer from this distribution of cometary’s orbits in the Newtonian theory that these comets are not permanent members of our solar system.

But there are many arguments in favor of the thesis that the comets are effectively permanent members of the solar system, and most of astronomers adhere to this view. In 1932, E. Opik enunciated for the first time the hypothesis of the existence of a "reservoir of comets" situated at the end of our solar system, which the comets leave under the disturbing action of the stars. J. van Woerkom (1948) and J. Oort (1950-1951) demonstrated, statistically and on celestial mechanics basic, that the comets are permanent members of our solar system.

These comets may start from the Sun on an elliptical initial (primary) orbit, but under the influence of the planets their trajectories may get a parabolic or hyperbolic shape.

Till now only about 25 such primary orbits were determined, being proved that they were really elliptical. It is nevertheless insufficient to draw any final conclusion. A.O. Leuschner published a statistical study on cometary’s orbits. The result is: The longer is the comet observation, the more probable is that its orbit is not a parabola (and the less is it a hyperbola).

This observation is illustrated by the following table:

Visibility period of the comet |
Registered parabolic orbits |

1-99 days |
68% |

100-239 days |
55% |

240-511 days |
13% |

* <For a comet visible during 240 days or more, it is extremely dubious that a parabolic orbit be definitively established...>*

The theory, according to which the comets generally are permanent members of the solar system, seems to be confirmed by the above mentioned statistic data.

In the theory of vortex, the equation of the forces becomes by introducing the corrective term:

(2.26)

where: (2.27) and

As (2.28), the speed of the comet (V_{g}) may be greater than that in the Newtonian theory and, notwithstanding this, its motion may be elliptical.

Suppose that at a comet, according to the Newtonian theory, there is , the orbit is a parabola, while by the new theory the same comet moves on an ellipse, because the comet does not reach the necessary parabolic speed. All parabolic comets of the Newtonian theory become elliptically orbited comets by introducing the corrective term.

Adding the corrective factor, there is the following distribution of the shapes of cometary’s orbits:

- clearly elliptic orbits: 27%+16%+43%=86%

- possible hyperbolic and parabolic orbits: 14%.

Based on this kind of distribution, we may assert that the comets are permanent members of the system, Of course, there are also comets with parabolic or hyperbolic orbits, but they are a result of the elliptical orbit disturbing by the planets (Jupiter, Saturn). When a comet gets near to a big planet, the gravitational field of the comet may either accelerate or decelerate it depending on the relative motion between planet and comet before approaching.

If the comet speed is increased by the gravitational field of the planet, the comet orbit may get either parabolic or hyperbolic; if it still remains elliptical, it modifies its orbital parameters.

If the comet speed is reduced by the gravitational field of the planet, the cometary’s orbit gets an ellipse which is less than before the "capture".