According to the Reciprocal System the main energy generation process in stars is by way of thermal destruction of the atoms of the elements present in the stellar core.
The Destructive Limit
The destructive limit Td, that is, the temperature at which the neutralization of one of the two-dimensional displacements of the atom takes place can be worked out as follows. A temperature T in Kelvin, expressed in the natural units, is given by
Since speed displacement is the deviation of the speed from the natural datumthe natural datum in the universe of motion being unitythe space displacement. corresponding to a temperature T is
This, therefore, is the space displacement available per each rotational unit of the atom when it is at a temperature of T Kelvin. If the net number of rotational units (the atomic number) of the atom is Z, the total space displacement available due to the thermal energy is
The temperature of the atom is a linear (vibratory) motion in the time-space region while the rotational motion that constitutes the atom is in the time region (inside unit space). The total number of equipossible orientations for a unit of linear motion in the time-space region is shown to be 8 . As such, the portion of the space displacement of the temperature that becomes effective in the time region is
Thermal destruction of the atom implies the neutralization of one of its two-dimensional time displacement units since the basic rotation constituting an atom is two-dimensional. The one-dimensional equivalent of a two-dimensional displacement of magnitude n being 2 * n² , where the atomic displacements are a - b - c , the time displacement units needing to be neutralized would be either 2 * (a - 1)² or 2 * b². Thus, we have, at the destructive limit Td
In fig. 1 are shown plotted the values of the thermal destructive limit of the elements against Z. As can be seen, this temperature increases as the atomic number decreases. But the most conspicuous feature of the curve is that, instead of being monotonous, it dips at several locations where there is a change in the displacement of the atom in one of the magnetic dimensions. These dips, occurring respectively at Z = 70, 27 and 6 are of paramount significance in determining the course of stellar evolution as we will presently see.
The Intrinsic Variables
Under normal stellar conditions, where there is no severe large-scale turbulence, gravitational segregation of the elements according to their masses would take place, the heavier ones migrating toward the core. Taking this gravitational segregation into consideration, if we plot the thermal limit Td of the material of the star at different radii we obtain a curve of the general nature shown in fig. 2.
The distribution of the actual temperature Ts in the star at various radii is also shown plotted in the figure. We see at the center, P, that the temperature is the same as the destructive limit of the heaviest element present. As such, this element gets thermally neutralized to yield the energy output of the star. We shall refer to this process as the regular burning in order to distinguish it from the secondary burning which we will presently explain.
As the element burning (that is, the thermal neutralization) continues, elements of lower Z (and with higher Td) keep arriving at the center. At the same time the stellar temperature gradually keeps on rising so that each of these lower Z elements reaches the thermal limit successively at the center. Thus, the Td vs. radius curve goes on shifting horizontally to the left in the diagram, while the Ts vs. radius curve gradually keeps on rising, signifying higher stellar temperatures, as evolution progresses. Eventually the group of elements with Z = 74 to 70 arrives near the center. The state of affairs is now as shown in fig. 3.
It can be seen that the stellar temperature curve now begins to intersect the Td curve at two points P and Q. Therefore we find that while principally elements 74 and 73 are burning at the center P, element 70 also arrives at its destructive limit at location Q, slightly farther out. The consequent ignition of element 70, however, upsets the previous equilibrium between the thermal and the gravitational forces, for three reasons.
Firstly, the element 70 is relatively more plentiful compared to the elements of higher Z and thus a potentially larger energy source is switched on, and switched on suddenly, in addition to the existing source. This new source we shall refer to as the secondary source. Secondly, this happens not quite at the center (where the regular source has been operating) but at a slightly larger radius rs (see fig.3), which we shall call the secondary burning radius, where there was no energy generation previously. Thirdly, the extent of the spherical area at radius rs, where s the thermal limit of the secondary fuel is reached, is comparatively larger than available to the regular fuel at the center, and in consequence the proportion of the secondary fuel that is ignited is much greater.
The additional energy thus released causes an expansion of the star. This drops the stellar temperature and acts as a negative feedback, shutting off the new energy source. Subsequent contraction repeats the cycle and we have the phenomenon of the intrinsic variable. In fact, the Cepheids could be identified as the stars burning elements around the ytterbium-dip at Z = 70 (fig.l).
Larson proposes , in the context of the Reciprocal System, that in the regular course of the energy generation process, with elements of lower atomic numbers successively arriving at the center to be neutralized, the appearance of an element such as lead (Pb), with comparatively higher-than-normal abundance, initiates the variability cycle. This might well be one of the causes of the variability and the long period variables seem to fall into this category. However, as the stellar mass (and consequently its temperature) increases, the ratio of the additional energy produced to the total stored energy decreases. As such, the variations become damped out and unobservable. In the case of the higher temperature stars, therefore, the principal cause for the variability could be attributed to the Yb-dip as explained in the foregoing.
The phenomenon of the intrinsic variable occurs whenever there is a cyclic upsurge in the energy production. This also happens when the central temperature of a gravitationally contracting aggregate first reaches the destructive limit of the heaviest element present there. In fact, Larson identifies this category of stars as the long period variables .
The Main Sequence Stars
The phenomenon of the variable luminosity manifests only in case the accretion rate is high and the star follows the path AC shown in the Color-Magnitude diagram (see fig.6 of Reference [SD] The story of a main sequence star, with about the same surface temperature as that of the variable is somewhat different. This is primarily because when- the stellar temperature archives at the Yb-dip, the temperature gradients in the variable and the main sequence star are markedly different. In the variable the Yb-dip occurs much farther from the core as compared with the main sequence star. As shown in fig.4 (a) and (b) the temperature gradient in the core is much steeper in the main sequence star.
Figure 4: Effect of Temperature Gradient on the Dip
If dZ is the difference between the atomic number of the element currently burning at the center and that of the new arrival (the secondary source), the effect of the steep temperature gradient is to keep dZ to a small value (1 or 2). Therefore no marked difference in the magnitude of energy generation will result by the initiation of the secondary burning. Whatever little difference there may be is successfully damped out by the larger heat capacity and the larger mass of the overlying material (in view of the smaller secondary burning radius rs). Consequently the main sequence star of comparable surface temperature passes less conspicuously than the variable through this Td dip. In the case of the variable the dZ is larger (4 or 5) (see fig.4 b)) which results in considerable amount of secondary energy production.
The Type I Supernova
As the element-burning continues and the cobalt-dip (at Z = 27) arrives at the core while element 31 or 30 is burning at the center, a spectacularly different result ensues. Firstly, the secondary source triggered suddenly is proportionately very largenot just three or four times the regular, as in the case of the Yb-dip, but nearly a hundredfold biggerowing to the much greater relative abundance of the Co-group of elements. Secondly, because of the large size of the dip in Td at Z = 27 the secondary burning radius is appreciably large. As such, a large number of elements (with Z between 32 and 27), and in quantities more plentiful than the regular fuel now burning at the center, are present within the sphere of radius rs, waiting to be ignited but for their higher destructive temperatures.
The initial spate of the secondary energy released by the onset of the thermal destruction of Co at the radius rs does two things: on the one hand, it causes the expansion of the overlying material and results in a drop in the temperature, which thereby acts as a negative feedback switching off the Co ignition. On the other hand, it compresses the material inside the radius rs . This sudden implosion raises the temperature in the region sharply and brings all the high-Td material within the radius rs catastrophically to its destructive limit. Consequently, a pilot explosion takes place in the core, liberating considerably large quantities of energy greater nearly by a magnitude or two than was being released hithertoand liberating this energy in a short interval. This acts as a positive feedback and retriggers the burning of the Co-Fe group of materials at the radius rs at a substantially high rate. This high rate of temperature rise raises the temperature of a large portion of. the Co-group well above its destructive limit, culminating in the supernova explosion before the negative feedback of the drop in temperature owing to the expansion of the outer layers has time to operate. In fact, if the star is quite large, a few outer luminosity pulsations may be apparent before the core explodes, activating the final cataclysm of the cobalt explosion.
As Larson points out , the supernova explosion disperses the major portion of the Co-group out into space before it had a chance to get destroyed in the event. Hence their cosmic abundance keeps on building up unlike that of the other elements of higher Z. The elements of Z = 28 through 30, which are inside the secondary burning radius, and are involved in the pilot explosion, also seem to share this good fortune to a limited degree by virtue of their higher destructive limits.
Highlighting the effect of the Td dip on stellar evolution we summarize: