_{1}

^{*}

A simple methodology was shown in order to determine when a lonely star of a certain type might have a planetary family like the one of the Sun’s.

The Solar system is unique in the Universe in the sense that we do not know any other star which has planets revolving about it, and resembling those of the Sun’s family. It is important to detail the very special structure that the Solar system has.

The Solar system consists of the Sun and a large number of smaller objects gravitationally associated with it. These other objects are the planets, their satellites, the asteroids or minor planets, the comets, the meteoroids, and an interplanetary medium of very sparse gas and microscopic solid particles. Differently from the Sun, the planets are small, relatively cool, and five of them are solid. They give off no light of their own but shine only by reflected sunlight [

The four innermost planets are Mercury, Venus, Earth, and Mars, and also they are called Inner planets. The four largest planets are gaseous and often are called the Jovian planets; and the others, including Pluto, are the Terrestrial planets [

Between 1911 and 1913, the astronomers E. Hertzsprung and H. N. Russell, led independently to an extremely important discovery concerning the relation between the luminosities and surface temperatures of stars. The discovery is exhibited graphically on a diagram named in honor of the two Scientists, the Hertzsprung-Russell or H-R diagram [

The most significant feature of the H-R diagram is that the stars are not distributed over it at random, exhibiting all combinations of absolute magnitude and temperature, but rather cluster into certain parts of the diagram. The majority of stars are aligned along a narrow sequence running from the upper left of hot, highly luminous part of the diagram to the lower right of cool, less luminous part. This band of points is called the Main Sequence [

In order to give a solution to the problem of identifying stars which possess planetary systems similar to the Solar System all the binary and multiple systems, must be discarded from the set of Main-Sequence stars, only focusing the attention on lonely stars; in such a way that the problem is reduced to localize lonely yellow dwarfs, of slow rotation, and of dimensions and basic characteristics similar to those of the Sun’s (see Appendix).

In the specialized literature [

As will be seen next, the values of the main parameters of any gaseous star can be obtained by direct calculation with only two observational data, such as the effective temperature and the absolute magnitude [

The radius of any gaseous star can be calculated from those observational data, and using the magnitude of the self-generated magnetic field at the surface of the star [

where T_{e} is the effective temperature, the subscript s indicates the surface, and

is a universal constant [

On the other hand, from the well known relationships [

and

where L is the luminosity, c the velocity of light in the empty space, M_{bol} the bolometric or absolute magnitude, and the symbol ¤ indicates the Sun, considering it as the unit of measure [

Then the luminosity can be easily calculated from the relationship (3) using the value of the effective temperature.

In order to determine the mass of any gaseous star, it is first necessary to calculate the value of the b parameter for such star. This parameter is defined as follows [

where p_{r}(x, t) is the pressure from the radiation, p_{g}(x, t) is the pressure from the hot gases and p(x, t) is the total pressure. From the modified mass-luminosity relationship [

where G is the gravitational constant, M is the stellar mass, k_{c} is the opacity coefficient which for numerical calculations is customary to assume that k_{c} = k , and a = 2.5 is a constant indicating an intense and uniform concentration of energy generating sources at the center of the star [

which is the commonly accepted law for radiation absorption [

Which is a formula that can be used to calculate the temperature at the center of the star [

it is obtained the following [

In these last equations, r(x, t) is the mass density, and T the temperature both that the center of the star, R is the gas constant, m is the average molecular weight whose numerical value is almost always taken equal to 2.11 [_{1} is a constant, n is a positive integer number, and M´ = 2.015, and R’ = 6.901 [

is obtained [

was used to obtain the following formula [

in order to eliminate from 85,299.20-(11) the mass M.

With the numerical values of the constants (see Appendix) substituted in (12) the following result, which is valid for any gaseous star [

According to the previous statement, any two stars can be compared if one of them is considered as the appropriate unit of measure. It is common practice to use the brightest component of Capella binary system [

where the asterisk refers to the particular star which is being compared to Capella. With the numerical data for this last star (see Appendix) one gets [

such that in (16) the following is obtained

Since the effective temperature is a conventional measure used to specify the flow of radiant heat emitted per unit area by any gaseous star [_{1} and k_{1} appearing in equation(17), keep the same proportion among themselves as that of their respective effective temperatures [_{1}/k_{1} and T_{e}/T_{e} are numerically equal, such that it is always true that [

For any couple of gaseous stars which can be compared between themselves [

where

is a constant for each particular star under study. For the solution of numerical equations, Newton’s method is used [

Once the parameter b has been obtained, the following fourth degree equation is used to determine the mass of the star [

In this way, one gets all the necessary results to calculate the volume, and the average mass density. The central mass density is obtained from the following formula from the theory of the polytropic gas sphere [

where the subscripts indicates the central and the average mass density, respectively.

With the former method, the opacity coefficient, the temperature at the center of the star, and the magnitude of the self-generated magnetic field at any point inside the star can also be estimated [

where r_{*} is the distance in parsecs.

Galileo first demonstrated that the Sun rotates on its axis when he observed the apparent motions of the Suns pots across the solardish [

The exact formula for the Doppler shift is

wherel is the wavelength emitted by the star, Dl is the difference between l and the wavelength measured by the observer, c is again the velocity of light in the empty space, and v is the relative line of sight velocity of the observer and the star, which is counted as positive if the velocity is one of recession and negative if it is one of approach [

On the other hand, for a gaseous star, it is always true that [

where E is the energy density radiated by the star considered as a black body [_{V} is the total energy E_{T} radiated by the star as thermal radiation. On the other hand, it is

easy to see that E_{T} =_{l} the energy of each photon. Thus,

where n/V is the density of thermal radiation, and the Equation (26) was used. Moreover, and according to Planck’s equation

with h Planck’s constant [

Substituting (29) in (26) the following result valid for any gaseous star is obtained

In this case, any gaseous star can be compared with the Sun to obtain the following relationship

Nevertheless, it is easy to see that in (31) the term corresponding to the one of the effective temperatures is the dominant; so that it can be assumed that Dl_{*} » Dl

where the numerical values of

With the help from the theoretical methodology presented, it is proposed the configuration of an observational program over the Main-Sequence stars, which consists of a systematical search of lonely yellow dwarfs, whose basic parameter is similar to those of the Sun’s; in order to obtain a group of stars which have high probability to possess a planetary family of the solar type, and where in some of those systems may be likely the natural conditions for forms of living organisms exist.

Angel Fierros Palacios, (2016) The Problem of the Existence of Planetary Systems Similar to the Solar System. Journal of High Energy Physics, Gravitation and Cosmology,02,161-167. doi: 10.4236/jhepgc.2016.22015