On the Periodic Change of the Luminosity of the Cosmic Sources with an Active Medium

The presence of an internal layer with an active medium in a hot radiation source is considered, which can be described by a quantum two-level system located near equilibrium. The population of the upper and lower levels is approximately equal. It is shown that during convection from deeper hot layers, which supports the inversion of the populations of the active system, generation of induced radiation pulses is possible, the intensity of which is comparable to or greater than the intensity of the background spontaneous radiation of the source. With a sufficient thickness of the surface layers due to the effects of radiation scattering in them, the emission spectrum of a completely black body may well form there. Pulse generation near a previously detected new threshold of induced radiation can lead to a periodic change in the radiation intensity of the source as a whole. This threshold is determined by the equality of the squared population inversion to the total number of states. The generation of pulses of induced radiation is considered both in Einstein's representation, on the basis of balanced equations, and using a semiclassical description for small values of population inversion and for low levels of electric field intensity, when the Rabi frequency is less than the line width. The description of the induced radiation process is reduced to a one-parameter system of equations. Periodic solutions are represented by closed trajectories on the phase plane (relative density of quanta, relative density of population inversion). A similar layer with an active medium, which can be described by a quantum two-level system located near equilibrium, can exist in stars and is most likely localized in the photosphere. If there is significant convection in the star’s atmosphere, conditions can be realized for generating pulses of induced radiation. It turns out that one can see the similarity of the obtained solutions with known observations of changes in the luminosity of Cepheid stars (Cepheus delta and the North Star).


Spontaneous and induced radiation.
In the most general case, particle radiation can be either spontaneous, independent of external influence, or stimulated, imposed by an intense external field [1,2]. Consider a particle that has its own field. If this field or part of it is not able to propagate independently of the particle, accompany the particle, then there is no radiation. If, however, during particle motion, translational or vibrational, the particle field is considered, then under certain conditions there will be a part in it that will be able to propagate independently in the medium. The work of this part of the field on the particle will not be zero and will lead to its deceleration or to a decrease in the amplitude of the oscillations, respectively, which is a sign of the presence of radiation of field energy by the particle [3].
It is also important to note that the value of the work of the particle's own field on its own current is always signdefined and describes only the radiation process, which is one of the important characteristic features of spontaneous processes. Another characteristic feature of spontaneous emission is the fact that its sources are independent and the radiation process is not imposed by a wave at this frequency existing in a given medium or system [4].
The induced radiation is due to the fact that an external field in the entire interaction space modulates the movement of particles of the medium. In this case, radiation (or absorption) of many particles located at different points in space may well occur in phase with this field. Ch. H. Townes drew attention to this in his Nobel lecture: "... the energy emitted by molecular systems has the same field distribution and the same frequency as the inducing radiation, and, consequently, a constant (possibly zero) difference phases" [5]. Such radiation synchronized by an external field, as well as particle absorption, leads to a sharp increase in the efficiency of interaction between particles and the field. The cause of the induced (stimulated) radiation, as it turned out, is the presence of population inversion (a positive definite difference of particles at higher and lower energy levels) [1].
Formulation of the problem. Consider the radiation source and select in it the layer where the medium is located, capable of generating periodic radiation pulses.
For example, such a layer in stars can be localized in the region of the photosphere. In some cases, there may be several such layers. It is possible to simulate the active zone of this layer by a two-level quantum system, and the populations of these two levels will be considered approximately equal. With a very insignificant excess of the upper level population (the population inversion is positive in this case), in addition to spontaneous emission, the generation of induced radiation is possible, the threshold of which was found in [6,7]. With a sufficient thickness of the overlying layers due to the effects of scattering of radiation emanating from the active zone of the two-level system in them, the emission spectrum of a completely black body may well form there [8].
In addition, we will assume the existence of strong convective flows emanating from layers closer to the hot region of the source of the source, which are capable of introducing a certain fraction of excited atoms of this active substance into the active zone. That is, such convective flows are able to increase the population inversion in the core.

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On The Periodic Change of the Luminosity of the Cosmic Sources...

EEJP. 2 (2020)
One can verify that the intensity of the induced radiation of a given spectral line, described by a two-level system, can significantly exceed not only the spontaneous intensity of atoms of a given substance, but also the spontaneous emission of the entire spherical source as a whole. The purpose of this work is to consider the generation of periodic pulses of induced radiation in a heated gas with a fraction of active particles in two states (two-level system). It is important that in a certain active zone of the radiation source, the number of active particles at two energy levels is approximately equal, the population inversion is much less than the total number of particles. Under these conditions, the intensity of the induced radiation is comparable to the intensity of spontaneous emission. It is assumed that in this active zone, which is transparent to radiation at the frequency of the energy transition, due to convection, active particles from the underlying dense layers enter the excited state. That is, in the core there is a source of inverted (excited) active particles.

MODEL OF PULSES GENERATION IN EINSTEIN REPRESENTATIONS
According to the ideas of A. Einstein, the description of the simplest one-dimensional two-level system in the presence of radiation at the transition frequency using the so-called balance equations is as follows [1]: moreover, the total number of particles of the system in the first and second levels is constant 2 1 N n n Const    , 21 u -is the rate of change in the number of quanta of the second excited level due to spontaneous emission processes, 2 1 n n    -is the population inversion. The rate of change in the number of quanta (particles) at these levels due to the induced processes of radiation 21 2 k w N n   and absorption 1 12 Here k N is the number of radiation quanta (spectral density) at the transition frequency, for which the equation 21 2 21 where D  -is the decrement of absorption (or radiation loss) of quantum energy. Generally speaking, the relationship between the coefficients 21 u and 21 w in the general case can be represented as follows where 21 A and 21 B are the corresponding Einstein coefficients. The dimension of this ratio in the three-dimensional case is time per unit volume, since we are talking about the spectral density of quanta. For yellow light the numerical value 0.25 g  , for violet at the edge of the visible spectrum 0.6 g  . The traditional idea of the instability threshold in neglecting spontaneous processes meets the requirement of positivity of the right-hand side of (3) or, what is the same One can detect the threshold for the appearance of such radiation due to competition with spontaneous processes at a certain critical value of population inversion 2 TH  [6,7]. In the absence of absorption of quantum energy ( 0 D   ), the new threshold for the generation of induced radiation is determined by the relation Near this threshold 2 1 / 2 n n N   , the inversion value 2 1 , n n   is very small. It is important to note that under these conditions the induced radiation is comparable to spontaneous. Collisions with fast electrons of the medium can provide a transition from the lower energy level of the quantum system to the upper energy level: where  is the effective frequency of collisions with fast electrons of the medium and 2 1 / 2 n n N   . In this case 21 u   , we can introduce into consideration the quantity where 0 ( 0) t     is some initial population inversion value that exceeds the threshold (6).
The condition 0 0 I  describes the equilibrium state of collisional excitation by free electrons of the main gas of active atoms and their radiative relaxation (due to spontaneous emission). A transition from a higher nonradiative level n u n     ) can take an active part in the excitation of the upper working level, then a dark line (absorption line at the frequency of this transition) appears in the spectrum, which allows one to determine the sort of particles and the working levels of the quantum system. Under the discussed conditions, at various levels of initial inversion both near the threshold (6) and far from it, equations (3), (7) are similar to the Statz-DeMars equations [9], which describe relaxation (that is, damped) oscillations with the establishment of a stationary state. The situation changes if transfer of population inversion from other, for example, inner layers of the radiation source, can occur is responsible for convective transfer of inversion with speed V from denser underlying source layers [10].
In these works, it was shown that the inversion flux into the region of the active medium can lead to a change in the relaxation nature of oscillations to periodic. In this case, one can observe the appearance of characteristic sawtooth periodic pulses of induced radiation against the background of the average radiation flux. It is important to note that such pulses occur only under equilibrium conditions 0 0 I  and also in the case of proximity of the generation thresholds (5) and (6), and their repetition rate depends on the quantum absorption decrement D  and the rate of change of the inversion due to its transfer. In the discussed models, for convenience of description, the field quanta were divided by origin: ( ) The change in the relative number of quanta of spontaneous emission When 0 0 I  , when the collisional excitation is large, the induced radiation is monotonic in nature, in the case of 0 0 I  you can see that there is no induced radiation. That is, the generation of periodic pulses reveals itself only under conditions of equilibrium of the processes of collisional excitation by free electrons of the main gas of active atoms and their radiative relaxation mainly due to spontaneous emission 0 0 I  [10]. In this case, the integrated radiation intensity during the appearance of a pulse can increase several times. It should be noted that the authors of [10] noticed the considered phenomenon only under the condition of equilibrium 0 0 I  and also in the case of proximity of the generation thresholds (5) and (6).

DESCRIPTION OF THE PULSE GENERATION PROCESS USING THE QUASICLASSIC THEORY
With a higher field intensity or with a significant inversion, the induced radiation in a bounded waveguide system should be described by a semiclassical theory [11], where the medium is described quantum mechanically and the electric field in the classical representation. The nature of the process depends on the ratio of the line width of the wave packet to the Rabi frequency, which determines the probabilities of induced radiation or absorption of field quanta [12,13] in each region of the waveguide along its length. With a significant population inversion or in strong fields, the line width can be neglected. For small population inversions and for low electric field intensities, when the Rabi frequency is less than the line width of the wave packet, the behavior of a two-level system is similar to the description of the process presented in [1].
Let us consider a one-dimensional model for perturbations of the electric field, polarization, and population inversion slowly varying with time, which describe the excitation of electromagnetic waves in a two-level active medium, the equations of which can be represented as [11]

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On The Periodic Change of the Luminosity of the Cosmic Sources... EEJP. 2 (2020) 2 , where the transition frequency  between the levels corresponds to the field frequency, we neglect the relaxation of the inversion due to external reasons, D  is the decrement of field absorption in the medium, and 2 1 d is the matrix element of the dipole moment (more precisely, its projection on the direction of the electric field), 2 1 ( ) n n    -the population difference per unit volume, 2 n and 1 n -levels populations in the absence of a field, 12  -the width of the spectral line, 1 2 ( ) N n n   is the density of the dipoles of the active medium. Fields are represented as . Spontaneous emission is absent in this representation, but as shown above, it is not difficult to take into account, for example, using equation (12). For slowly varying amplitudes, the equations That is, it is easy to see that the previous description (10 -(11) corresponds to the equality, whence it follows. You can make the system of equations one-parameter by dividing the right and left parts by and passing to the values, write the equations in the form That is, it is easy to see that the previous description (10 -11) (12). A significant part of the radiation of a quantum source is induced radiation, which is almost monochromatic. However, if the quantum source is surrounded by a fairly extended atmosphere, due to multiple scattering, as noted in [8], the radiation characteristics will approach the radiation characteristics of an absolute black body. The total number of radiation quanta per unit volume is equal to . In the case of small deviations 2 2 , M N from the singular point, the system can be linearized and the equation for the phase trajectory obtained: The phase trajectory turns out to be a closed line, which indicates the presence of stable periodic solutions.
In Figure 1

EEJP. 2 (2020)
takes the form of a saw: a sharp increase and a slow decrease. Such behavior is characteristic of the Cepheid luminosity, where fusion has reached its ultimate element -iron: significant sawtooth fluctuations in the luminosity of the Cephei delta and small sinusoidal oscillations of the North Star.  , here m 1 is the apparent magnitude at the maximum luminosity of the star, m 2 -at the minimum luminosity. Below we will focus on the values of the apparent magnitude obtained from observations. Using the known apparent magnitude m , knowing the distance to the star d in parsec, we can calculate the luminosity [14]: To compare the luminosities of stars with each other, the absolute magnitude (apparent magnitude from a distance of 10 parsecs) (25) Table 2 shows the calculation results for the Cepheus delta and the North Star. For this, the values of Table 1 are taken from the first and fifth rows.  The lower curve on each fragment represents the number of quanta of induced radiation. The upper curve takes into account the constant component of spontaneous emission due to other elements of the radiation source, in addition to the quantum system. Fig. 5 and Fig. 6 below show the change in the stellar magnitude of the star of the Cepheus delta [15] and the North Star with time [16]. The ordinate shows the values of the apparent magnitude, while the abscissa shows time in fractions of the period of change in the brightness of the star.

CONCLUSIONS
The state of a gas radiation source close to equilibrium under collisional excitation by free electrons of the main gas of active atoms and their radiative relaxation due mainly to spontaneous emission was considered [10]. In such a medium, for a two-level system, when the population inversion is much less than the total number of states, there are conditions for the generation of pulses of induced radiation.
Moreover, the intensity of such pulses is comparable with the intensity of spontaneous emission of the system. It was shown earlier that during convection from deeper hot layers, which supports (is the source) the population inversion of the active system, it is possible to generate periodic pulses of induced radiation, the intensity of which is comparable to or greater than the intensity of the background spontaneous radiation of the source [10]. With a sufficient thickness of the surface layers due to the effects of scattering of radiation in them, the emission spectrum of a completely black body may well form there [8].
Pulse generation near a previously detected new threshold of induced radiation [10] under the conditions of such convection can lead to a periodic change in the radiation intensity of the source as a whole. This threshold is due to competition between spontaneous and induced radiation, which weaken the population inversion and is determined by the equality of the square of the population inversion to the total number of states. A consideration of the generation of pulses of induced radiation in the present work is carried out both using Einstein's balanced equations and on the basis of a semiclassical description for small population inversions and at low levels of electric field intensity, when the Rabi frequency is less than the width of the radiation line.
It is shown in the paper that the description of the induced radiation process reduces to a one-parameter system of two equations (21) -(22). Periodic solutions are represented by closed trajectories on the phase plane (relative density of quanta, relative density of population inversion). A similar layer with an active medium, which can be described by a quantum two-level system located near equilibrium, can exist in stars and is most likely localized in the photosphere. If there is significant convection in the star's atmosphere, conditions can be realized for generating pulses of induced radiation. It turns out that one can see the similarity of the obtained solutions with the well-known observations of changes in the luminosity of Cepheid stars (Cepheus delta and the North Star). Moreover, the ratio of periods of change in luminosity is (according to table 1.) also similar (accurate to 2.5%). It is important to note that the solutions of equations (21) -(22) for these two cases are on the same diagram in Fig. 3, which corresponds to one type of object. As well as small amplitude Cepheids (DCEPS) of the North Star type, they belong to the class of classical Cepheids, the brightest representative of which is the Cepheus delta.