Читать онлайн Jamolitdin Abdullayev, Шахло Иброхимова - Все науки. №12, 2024. Международный научный журнал

UDK: 530.1

Abstract. This study presents the process of theoretical modeling of the current value of the solar radiation energy by means of the Laplace equation, taking the Sun as a radiating object with the setting of appropriate boundary conditions in a spherical coordinate system. For the obtained solution of the equation, through the use of dynamic modeling and time dependence, taking into account thermonuclear processes in the Sun, the method of separation of variables is used. As a result of the study, the function and its graphs are presented, which allow us to observe the indicators of radiated power per square meter of the surface of the planet Earth.

Keywords: Laplace equation, method of separation of variables, thermonuclear fusion, solar radiation.

Аннотация. В данном исследовании представлен процесс теоретического моделирования текущего значения энергии солнечного излучения с помощью уравнения Лапласа, в котором Солнце рассматривается как излучающий объект с заданием соответствующих граничных условий в сферической системе координат. Для полученного решения уравнения, благодаря использованию динамического моделирования и временной зависимости, с учетом термоядерных процессов на Солнце, используется метод разделения переменных. В результате исследования представлены функция и ее графики, которые позволяют наблюдать показатели излучаемой мощности на квадратный метр поверхности планеты Земля.

Ключевые слова: уравнение Лапласа, метод разделения переменных, термоядерный синтез, солнечное излучение.

The development of energy and especially solar energy at the moment creates the need for a full-fledged mathematical model capable of determining, in the dynamic case, in the selected coordinate system, the amount of radiation power directed from the Sun to any of the specified coordinates. This simulation will allow you to determine the amount of power at a specified point on Earth at any given time period. However, in order to create a connection between this pattern and the dynamic characteristics of the described system, it is necessary to determine the rate of expenditure of hydrogen fuel from the Sun, taking into account all known characteristics.

At the moment, it is known that when extrapolating, based on data obtained from probes, including Voyagers, the Sun generates radiation equivalent to 4 million tons of matter at the time of 4.5 billion. The magnitude of the solar constant equal to 1,360.8 W/m2 up to the atmosphere has been established for the years of the star’s existence [1—3; 10—11]. Based on the current model of star formation and the model of mass expenditure, the value is clearly formed that at the time of 5.6 billion years after the formation of the star, the radiation power will increase by 11% and amount to 1,532.47 W/m2, respectively [4; 6—7]. The resulting value may be sufficient to transform the model into a dynamic form.

For further modeling, we will take an equation capable of describing the phenomenon of radiation in the presence of radiation, with the condition that in this case no energy is supplied to the source from the outside or, in comparison with the values of solar radiation, the power of cosmic radiation can be ignored, which is demonstrated in practice. Despite the fact that even taking into account the presence of not a small number of sources of cosmic radiation, the percentage of radiation – the flux of charged particles, even compared with the solar neutrino and the flux of solar charged particles, is extremely small in percentage [5; 7—10]. Based on this, the main radiation energy is transmitted by photonic radiation.

In many works, the processes of radiation formation in the Sun and the process of energy transfer from the Sun to other planets, including to Earth, are considered, but the proposed technology for mathematical analysis of available data and the development of further radiation processes are not considered. Therefore, this study is relevant.

To carry out the research, empirical materials that are well-known, relevant literature on the topic, as well as data obtained from international research papers were used [13—15; 18—19]. Numerous probes have been launched to measure the radiated energy, with the help of which radiation parameters are periodically studied. Within the framework of the ongoing research, methods of theoretical modelling, analysis, classification, the method of separation of variables (Fourier method), the method of using differential equations for modelling, with the resulting methods for determining their solutions, are applied.

At the moment, a dynamic problem has been formed with respect to the Laplace equation (1), with respect to the function (2), with known initial conditions (3—4), proceeding from the phenomenon of thermonuclear fusion.

To determine the boundary conditions, it is sufficient to adopt a spherical coordinate system, despite the fact that the coordinate at zero angles at a radius of 1 astronomical unit is the position of the planet Earth on New Year’s Day – the transition from the night of December 31 to January 1. The Sun is also accepted as an absolutely smooth body, spreading uniform radiation over the entire surface, due to which an error for the presence of black spots is initially stipulated, which can be eliminated later. Thus, based on the above, it is necessary to state the fact that, based on the conditions taken, the Earth is located at 0 degrees in the latitude angle of the Sun, also taking into account the deviation of 23.497 degrees of the Earth’s axis, while the maximum deviation to the poles of the planet in the form of the specified angle can be calculated.

Fig. 1. Spherical coordinate system

The conditions set lead to the fact that between the center of the Sun, the Earth and one of the poles of the Earth there are 3 imaginary straight lines forming at the time of the vernal equinox a right triangle with legs of 1 astronomical unit (1,496*108 km) and 1 radius of the Earth (for the polar case 6,356.8 km and for the equatorial 6,378.1 km), from where it is possible to calculate the angle according to the Pythagorean theorem (5).

Fig. 2. The modeling schemes

Also, from the same ratio, but in a transformed form, it is possible to obtain the angle of deviation at the time of June 21 and December 21 according to the cosine theorem (6—9).

Similar calculations are used to determine the boundary conditions with respect to the angle of longitude (10—13).

Based on certain data, it is possible to state the change of function (2) to (14) and the problem of the following boundary conditions (15—18), given that the boundary conditions and the dynamic phenomenon are known, the Fourier method of variable separation can be adopted as a solution for it.

Now that the initial and boundary conditions, as well as the corresponding equation, have been determined at the specified moment, it is necessary to pay attention to the effect of the Laplace equation in static form, and it, as a partial equation from the Helmholtz equation, can be interpreted as follows. Namely, for the reason that in this case the phenomenon of energy transfer is observed, and in this case the Laplace equation is used to display in a global sense a model of an emitter or an energy-emitting «charge» in the face of the Sun. Thus, on a more local scale, the harmonic function taken will satisfy, based on these conditions, the homogeneous equation of thermal conductivity or energy conductivity (19), including based on the transformation model for the connection of the Helmholtz equation and the wave equation.

Where, the coefficient of energy conductivity is determined in (20), along with all the determined parameters, including the coefficients of energy conductivity of the vacuum between the Sun and the Earth (21), the specific energy capacity (22) and the available energy density under the circumstances in the specified area (23).

Based on the calculated parameters according to (21—23), expression (20) obtains a numerical indicator (24).

Based on the conditions obtained, it is possible to determine that the problem can be solved by taking the form of an equation of the form (25), where, after substitution, a transformation can be obtained according to (26), with an equated coefficient (27).

From expression (27), 2 partial differential equations are formed – 1 ordinary with respect to time in the first degree and the second in the square of partial derivatives. The first equation is solved by adopting a general solution with an exponential form, where, after substitution, a characteristic form is presented, from which a general form of the function is formed – the solution of the resulting ordinary differential equation in time (28).