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III. The Ephemeral Model of a Spiral Nebula

The chance of proving the previous model right is much diminished by the fact that there is an explanation of the structure of a spiral nebula based on the classical wisdom. Indeed, we will describe now another candidate based on conservative forces with the magnitude depending only on the radial distance, allowing for an alternative model where the spiral arm is just an accident. As shown elsewhere, the forces in question are not necessarily central forces, but in case they are central the equation (10) holds. Notice that the same happens if the meridian and parallel components of the force are as in equation (17) even though the radial component of the force is now nonzero. Two cases are worth noticing as giving closed trajectories if the forces are conservative, namely

(25)

The first case is that of Newtonian gravitation describing the Kepler motion of celestial bodies, while the other is the case of elastic forces. Both of them have ellipses as closed trajectories, a fact classically known as the Bertrand theorem (Bertrand, 1873), the only difference being that in the case of gravitational force the center of force does not coincide with the center of the trajectory: it is located in one of the foci. We describe the case of Kepler motion in detail, not based on equation (10) but on the general observation that if the forces are conservative and their magnitude depends only on the radial distance, then they are necessarily central. Consequently the motion is plane and we refer it to the polar coordinates of the plane of motion (r, j). The Cartesian coordinates will be denoted (x, y) as usual. The equations of motion in that plane are

(26)

This system can be arranged in a complex equation as follows (Mittag, Stephen, 1992):

(27)

where z = x + i×y. This makes it particularly easy to integrate: in view of the fact that the motion is plane we use the area constant in order to eliminate r, to the effect that

(28)

Splitting again this last equation into its real components we get the components of speed in the plane of motion as

(29)

Now, considering the area constant in the form of determinant

(30)

and using equation (29) for the components of particle velocity we have

(31)

which is the equation of a centered conic in the plane of motion. The implicit equation of this conic can be written as

(32)

Its center has the coordinates

(33)

and it is thrown at infinity if

(34)

Consequently the parabolas are a kind of limit trajectories in the Newtonian framework and indeed they represent the “earthly” form of the Kepler motion. Indeed the ballistic trajectories, which are parabolas, are the first known Kepler motions, as Newton showed for the first time. To better understand this, notice that m_1,2 are the components of a velocity related to the initial conditions of the particle, and (K/h) is a kind of limit speed. When the initial velocity is smaller in magnitude than the limit speed characteristic to the trajectory the trajectory is an ellipse. When the initial velocity is greater in magnitude than the limit speed characteristic to the trajectory the trajectory is a hyperbola. Parabola is that trajectory for which the velocity equals the limit speed.

The spiral structure of nebulae can be explained in this framework by the fact that around a center of attraction there are many stars. Evidence shows millions – an ensemble! Then, as Kalnajs observed (Toomre, 1977) it is possible that some of these stars are, for a period of time, in points on their elliptic trajectories, which give the pattern of a spiral arm. One such example is shown in Figure 2. This fact can be accounted for by a certain relationship of the parameters of trajectories. The Figure 2 illustrates the Kalnajs’ exponential type correlation (Toomre, 1977). As mentioned before, this explanation of the spiral structure makes it ephemeral in contrast with the previous explanation, based on partially free motion, where it is a permanent structure.

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