On the universal stellar law for extrasolar systems

Author's Accepted Manuscript

On the universal stellar law for extrasolar systems Alexander M. Krot

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S0032-0633(14)00129-9 http://dx.doi.org/10.1016/j.pss.2014.05.002 PSS3743

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Planetary and Space Science

Received date: 20 August 2013 Revised date: 28 March 2014 Accepted date: 1 May 2014 Cite this article as: Alexander M. Krot, On the universal stellar law for extrasolar systems, Planetary and Space Science, http://dx.doi.org/10.1016/j. pss.2014.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On the universal stellar law for extrasolar systems    Alexander M. Krot Laboratory of Self-Organization System Modeling, United Institute of Informatics Problems, National Academy of Sciences of Belarus, 6, Surganov Str., 220012 Minsk, Belarus

[email protected] ∗

Fax.: +375 17 231 84 03.

Abstract In this work, we consider a statistical theory of gravitating spheroidal bodies to derive and develop an universal stellar law for extrasolar systems. Previously, it has been proposed the statistical theory for a cosmogonic body forming (so-called spheroidal body). The proposed theory starts from the conception for forming a spheroidal body inside a gas-dust protoplanetary nebula; it permits us to derive the form of distribution functions, mass density, gravitational potentials and strengths both for immovable and rotating spheroidal bodies as well as to find the distribution function of specific angular momentum. If we start from the conception for forming a spheroidal body as a protostar (in particular, proto-Sun) inside a prestellar (presolar) nebula then the derived distribution functions of particle as well as the mass density of an immovable spheroidal body characterizes the first stage of evolution: from a prestellar molecular cloud (the presolar nebula) to a forming core or a protostar (the proto-Sun) together with its shell as a stellar nebula (the solar nebula). This paper derives the equation of state of an ideal stellar substance based on conception of gravitating spheroidal body. Using this equation we obtain the universal stellar law (USL) for the planetary systems connecting temperature, size and mass of each of stars. This work also considers the Solar corona in the connection with USL. Then it is accounting under calculation of the ratio of temperature of the solar corona to effective temperature of the Sun’ surface and modification of USL. To test justice of the modified USL for different types of stars, temperature of the stellar corona is estimated. The prediction of parameters of stars is carrying out by means of the modified USL as well as the known Hertzsprung–Russell’s dependence is derived from USL directly. This paper also shows that knowledge of some characteristics for multi-planet extrasolar systems refines own parameters of stars. In this connection, comparison with estimations of temperatures using of the regression dependences for multi-planet extrasolar systems testifies the obtained results entirely.

1. Introduction In 1911–24 the astronomers Russell, Hertzsprung and Eddington established that for stars of the Main sequence there is a dependence of luminosity of a star on temperature of its stellar surface (the diagram of Hertzsprung–Russell), and also there is a connection between luminosity L and mass M of star (the diagram of mass–luminosity) [1, 2]. According to this diagram for stars of the Main sequence, the mass– luminosity dependence looks like L ∝ M s , where s = 2.6 for stars of small masses ( − 1.1 < lg M / M Sun < −0.2 ), s = 4.2 for stars of medium masses ( − 0.2 < lg M / M Sun < 0.4 ) and s = 3.3 for stars of big masses ( 0.6 < lg M / M Sun < 1.7 ), M Sun is the mass of the Sun. In the monograph [3 p.416], the different versions of invariant relations between temperature T , concentration n and parameter of gravitational compression α of Sun-like stars have been derived on the basis of the theory of rotating and gravitating spheroidal bodies. Recently Pintr et al. have found heuristic regression dependences, i.e. they have studied the regression dependence of the distance of planets an from the central stars on the parameter of specific angular momentum a n vn ( an is a planetary distance and vn is a planetary velocity) and then they have applied the regression analysis to other physical parameters of stars, namely, anTeff , an L , and an J , where Teff is an effective temperature of stellar surface, L is a luminosity of a star, J is a stellar irradiance for the multi-planet extrasolar systems [4]. Thereupon there is a question: whether there exist like the Kepler’s laws a universal law for the planetary systems connecting temperature, size and mass of each of stars? According to the statistical theory of gravitating spheroidal bodies [3, 5, 6] under the usage of laws of celestial mechanics in conformity to cosmogonic bodies (especially, to stars) it is necessary to take into account an extended substance called a stellar corona. In this connection the stellar corona can be described by means of model of rotating and gravitating spheroidal body. Moreover, the parameter of

gravitational compression α of a spheroidal body (describing the Sun, in particular) has been estimated on the basis of the linear size of its kernel, i.e. the thickness of a visible part of the solar corona. Really, NASA’ astronomer Dr. S. Odenwald in his notice «How thick is the solar corona?» wrote: “The corona actually extends throughout the entire solar system as a “wind” of particles, however, the densist parts of the corona is usually seen not more than about 1–2 solar radii from the surface or about 690,000 to 1.5 million kilometers at the equator. Near the poles, it seems to be a bit flatter...” [7]. In the fact, a recession of plots of dependences of relative brightness of components of spectrum of the solar corona occurs on distance of 3–3.5 radii from the center, i.e. on 2–2.5 radii from the edge of the solar disk. Thus, accepting thickness of a visible part of the solar corona equal to Δ = 2 R (here R is radius of the solar disk) we find that r* = R + Δ = 3R , where r* = 1 / α . In other words, the parameter of gravitational compression α = 1 / r*2 of a spheroidal body in case of the Sun with its corona (for which the equatorial radius of disk R = 6.955 ⋅108 m) can be estimated by the value [3, 6]: α=

1 (3R ) 2

-2 ≈ 2.29701177718 ⋅10 −19 (m ).

(1)

So, the procedure of a finding α is based on the known 3σ -rule in the statistical theory, where σ = 1 / α is a root-mean-square deviation of a random variable. Really, the solar corona accounting under calculation of perturbed orbit of the planet of Mercury allows to find the estimation of a displacement of perihelion of Mercury’orbit for the one period within the framework of the statistical theory of gravitating spheroidal bodies. As it is known, on a way of specification of the law of Newton using the general relativity theory the Mercury problem solving was found [8]. Nevertheless, from a common position of the statistical theory of gravitating spheroidal bodies the points of view as Leverrier (about existence of an unknown matter) and Einstein (about insufficiency of the theory of Newton) practically differ nothing. Really, there exist plasma as well as gas-dust substance around of kernel of cosmogonic body (in particular, the solar corona in case of the Sun), i.e. the account of circumstance that forming cosmogonic bodies have not precise outlines and are represented by means of spheroidal forms demands some specification of the Newton’ law in connection with a gravitating spheroidal body. Using the Binet’ formula the equation of disturbed orbit of a planet (the Mercury) in a vicinity of a kernel of a rotating and gravitating spheroidal body has been derived. The obtained relation expresses the equation of the so-called “disturbed” ellipse in polar coordinates with the origin of coordinates in focus, i.e. the planet Mercury is moving on a precessing elliptic orbit in view of the fact that there is a modulating multiplier of a phase (or azimuth angle). So, within the framework of the statistical theory of gravitating spheroidal bodies the required angular moving of Newtonian ellipse during one turn of Mercury on the disturbed orbit (or displacement of perihelion of its orbit for the period) has been estimated [3, 6]: 2π (3 + e) ⋅ ε 0 2 , (2)  δε = α ⋅ a 2 (1 − e 2 ) 2 where through a and e a major semi-axis and an eccentricity of Mercury’s orbit are designated respectively, α is a parameter of gravitational compression and ε 0 is a geometrical eccentricity of kernel of a rotating and gravitating spheroidal body (the Sun). Thus, according to the proposed formula (2) the turn of perihelion of Mercury’ orbit is equal to 43.93'' in century that well is consistent with conclusions of the general relativity theory of Einstein (whose analogous estimation is equal to 43.03'') and astronomical observation data (43.11 ± 0.45'') [3, 6]. This work also considers the solar corona in the connection with so-called universal stellar law (USL) introduced in the Section 3. Then it is accounting under calculation of the ratio of temperature of the solar corona to effective temperature of the Sun’ surface and modification of USL in the next Section 4. To test justice of USL for different types of stars, temperature of the stellar corona is estimated in the Section 5. The Section 6 shows that knowledge of some characteristics for multi-planet extrasolar systems permits us to refine own parameters of stars. Really, the numerous papers are devoted to investigations of

exoplanetary systems in the last time (for example, [9-43]). In this connection, comparison with estimations of temperatures using of the mentioned above regression dependences for multi-planet extrasolar systems testifies the obtained results entirely. 2. The strength, potential and potential energy of the gravitational field of spheroidal body formed by a collection of particles

According to the statistical theory of gravitating spheroidal bodies [3, 5, 6] the probability density function of a particle having distance r being confined between r and r + dr from the centre of a spheroidal body can be expressed by the following formula: ⎛α ⎞ f (r ) = 4π ⎜ ⎟ ⎝ 2π ⎠

3/ 2

−

α

r 2e 2

r2

,

(3)

so that a mass density function of a spheroidal body is equal to ρ ( r ) = ρ 0 ⋅ e −αr

2 /2

,

(4)

where ρ0 = M (α / 2π ) is a density in the center of a spheroidal body, M is a mass of a spheroidal body. Let us calculate the characteristics of the gravitational field produced by a collection of isolated particles in the form of a spheroidal body. We shall use the gravitational field equation in nonrelativistic mechanics written down in the form of the Poisson equation [44]: Δϕg = 4πγρ , (5) 3/ 2

where Δ is the Laplacian operator, γ = 6.673 ⋅10−11 N⋅ m 2 /kg 2 is the Newtonian constant of gravitation, ϕg is a gravitational field potential, ρ is a body mass density. We shall seek a spherically symmetric solution ϕg depending on r only, therefore (5) becomes: α

− r2 1 ⎡ d ⎛ 2 dϕ g (r ) ⎞⎤ ⎜ ⎟ ⎢ ⎜r ⎥ = 4πγρ0 e 2 . dr ⎟⎠⎥⎦ r 2 ⎢⎣ dr ⎝

(6)

Since ϕg is a function of r alone, one obtains: r

dϕg (r ) dr

On the other hand, the derivative

∫

α

− x2

x2e 2

= 4πγρ0 0

dϕg (r ) dr

dx

.

r2

(7)

determines the gradient value a (r ) = − grad ϕg (r ) ,

the gradient being also termed the strength of the gravitational field [44]. Indeed, the gradient expression in spherical coordinates: dϕg (r ) dϕg (r ) r a (r ) = − gradϕg (r ) = − (8) ⋅ er = − ⋅ . dr

dr

r

Resulting from (7), (8), the strength of the gravitational field produced by a collection of particles is expressed in the following relation [44]: r

∫

α

− x2

x2e 2

a (r ) = − 4πγρ0 0

r

2

dx r ⋅ . r

(9)

At infinitely large distances r → ∞ , the gravitational interaction forces tend to zero just as 1 / r 2 . Similarly, in the case of large r the value of gravitational field strength (9) takes the form: a(r ) = γ

M r2

.

(10)

α

In the case of small r , the function

− x2 e 2

≈1−

α 2

x 2 is in the subintegral expression in (9). Because of

this formula (9) transforms into r

∫x

α 2⎞ ⎜1 − x ⎟dx 2 ⎠ ⎝

2⎛

1 3 α 5 r − r 3 10 ≈ 4πγρ0 r . a(r ) = 4πγρ0 0 πγρ 4 = (11) 0 2 2 3 r r In (11), the values greater than the second order of smallness of r were ignored. Thus, the formula

(11) coincides with the value of strength inside of homogeneous sphere of constant density ρ0 [44]. As follows from (11), a(r ) → 0 with r → 0 , i.e. there is no field in the centre of the body. In such a way, the obtained relation (9) for the strength of gravitational field of a spheroidal body, the body having been formed from a large number of particles, includes the known results of (10) and (11) as particular cases with r values large and small, respectively. It should be noted that classical formulas (10) and (11) contradict each other if there is not a preliminary information about the value r . Thus, according to (11) a(r ) → 0 when r → 0 , whereas according to (10) a(r ) → ∞ when r → 0 , which is absurd; and on the contrary: according to (38) a(r ) → 0 with r → ∞ , at the same time, according to (11) a(r ) → ∞ with r → ∞ . It is obvious that (11) is valid for small r only, while (10) is only for large ones. But in the case of medium-size r we should use the obtained formula (9) owing to it the relations (10), (11) are conformed just as solutions are “sewn together” at domain boundaries in mathematical physics problems. Now let’s return to formula (7) and calculate the gravitational potential ϕg (r ) = 4πγρ0 ∫

1

α

− x2

r

∫

x2e 2 2 r 0

dxdr + C ,

(12)

where C is the integration constant defined from the condition that the potential is equal to zero on the infinity: ϕg (∞ ) = 0 . With the aim of simplifying (12) we shall transform the indefinite integral on the basis of the integration formula by parts: α 2⎤ α 2 r −α x 2 ⎡ r α 2 1 1 −2x 1 − r 1 − r 1 dxdr = − ⎢ e dx − r e 2 ⎥ − e 2 = − e 2 dx . ⎥ α α αr r ⎢α 0 ⎣ 0 ⎦ From the condition ϕg (∞ ) = 0 and formulas (12) and (13) we obtain [3, 5, 6, 45]:

1

α

− x2

r

∫ ∫

∫

x2e 2 2 r 0

∫

ϕg (r ) = −

α

r

2

4πγρ0 − 2 x e dx . αr

∫

(13)

(14)

0

Using the error function erf (x ) =

2

x

e π ∫

−s2

ds [46], we shall transform (14) into:

0

ϕg (r ) = −

(

)

4πγρ0 αr

2

α

r α /2

∫

2

e − s ds = −

0

γM r

(

)

erf r α / 2 .

(15)

Since lim erf r α / 2 = 1 , then for large r the last expression turns into r →∞

ϕg (r ) = −

γM r

.

(16)

Relation (16), as is known, describes the gravitational potential of a field produced by one particle (or a spherical body) of mass M . α

In the case of small r the function e

− r2 2

≈ 1−

α 2

r 2 , which leads to the transformation of formula (14):

ϕg (r ) = −

r

4πγρ0 ⎛ α 2 ⎞ 4πγρ0 ⎛ α 3 ⎞ 4πγρ0 . ⎜1 − x ⎟dx = − ⎜r − r ⎟ ≈ − r 2 6 αr α α ⎝ ⎠ ⎝ ⎠ 0

∫

(17)

In expression (17), the values higher than the second order of smallness of r were ignored. As is seen from (17), at small distances from the centre the field potential is proportional to the density near the 4π centre ρ0 and to the sphere area of radius r* = 1 / α : S* = = 4πr*2 . Thus, expression (17) describes the α gravitational potential in the near zone of the field, while (16) describes that in the remote one. The potential energy of a particle in a gravitational field is equal to its mass multiplied by the potential of the field. The potential energy of any distribution of masses is described by expression [44]: Eg =

1 ρϕ g dV . 2

∫

(18)

V

Since ρ = ρ (r ) and ϕg = ϕg (r ) are functions independent of angle variables θ and ε then, having done the integration over θ and ε in (18), we obtain: ∞

Eg = 2π ρ (r )ϕg (r ) r 2 dr .

∫

(19)

0

Substituting expressions (4) and (14) for ρ (r ) and ϕg (r ) in (19) and then using the formula of integrating by parts, one obtains [3, 45]: ⎛π ⎞ Eg = −4γρ0 2 ⎜ ⎟ ⎝α ⎠

5/ 2

=−

γM 2 α . 2 π

(20)

From (20) it is not difficult to see that α=

2 4π Eg . ⋅ γ2 M4

(21)

According to (20) one can deduce a distance called the effective radius of the body: π . r+ = α On account of (22) one obtains [3, 45]: γM 2 Eg = − . 2r+

(22)

(23)

Let us consider a single small body (a test particle) of mass m in the gravitational field of its own collective body of mass M , situated at distance r from the body centre. Now evaluate the potential energy of interaction of the particle and the spheroidal body: Eg (r ) = mϕg (r ) . (24) int Using relations (24) and (14) one can easily calculate the energy of interaction of a spheroidal body and a test particle placed at different distances from the centre of masses. Since energy depends on distance at which a test particle is, and particles themselves are distributed over space, one can determine the average gravitational potential energy of interaction of a test particle with a spheroidal body formed by a collection of such particles: ∞

∫

∞

Eg = Eg 0

(r ) f (r )dr = m ∫ ϕg (r ) f (r )dr . int

(25)

0

Let us use relation (14) for calculating ϕg (r ) , on account of it and (3) relation (25) will take the form [3, 45]: Eg = −

4πγρ0 m

=−

γmM

. r+ α 2 Let m = dM in (26). Taking this into account formula (26) is transformed into:

(26)

dEg = −

γM dM

By integrating both parts of (27) one obtains the formula (23), i.e. γM 2

∫ dEg = −

.

r+

2r+

= Eg .

(27)

(28)

Thus, according to (23), (26) and (28), the potential energy of the gravitating spheroidal body is only then equal to the total average potentional energy of the gravitational interaction of particles when these particles have infinitely small masses [3, 45]. Indeed, in this case particles do not possess the own gravitational energy, because, according to (23), it is a value of the second order of smallness with respect to dM . In fact, with putting the question in this way, we deal with massless particles whose gravitational energy is the potentional energy of interaction of particles between one another. Further, supposing m = m0 for each particle of one-component gas it follows from (22) and (26) that ⎛ Eg α = π ⎜⎜ ⎝ γm0 M

2

⎞ ⎟ . ⎟ ⎠

(29)

where Eg is an average gravitational potential energy of interaction of a particle m0 with a gravitational field of a spheroidal body. Thus, a spheroidal body has a “strict” (distinct) outline if the potential energy of the gravitational interaction of the body particles is sufficiently great, and the body mass itself and its particle masses are relatively small. Ordinary macroscopic bodies possess distinct outlines due to their relatively small masses and to sufficiently great energies of interaction of particles the bodies consist of. On the contrary, giant cosmic objects (star formations, nebulae, etc.) have fuzzy contours because of their huge masses and enormous numbers of particles forming them. For instance, the Earth’s atmosphere has an indistinct outline, the temperature being different at various altitudes, so does the Sun photosphere which also locks temperature balance. 3. Derivation of the universal stellar law

As shown in the monograph [3 p.398], under a condition of virial balance of a rotating and gravitating spheroidal body its parameter of gravitational compression α is expressed by the formula: 2πγm0 ρ 0 , (30) α= 3k BT

where γ and k B are constants of Newton and Boltzmann respectively, m0 is a mass of particle. Taking into account (4), i.e. that according to the statistical theory of gravitating spheroidal bodies [3, 5, 6, 45] the density in the center of a spheroidal body is equal to 3/ 2 ⎛α ⎞ (31) ρ0 = M ⎜ ⎟ , ⎝ 2π ⎠ let’s rewrite the formula (30) for the parameter of gravitational compression: ⎛ 3k T α = 2π ⋅ ⎜⎜ B ⎝ γm0 M

2

⎞ ⎟⎟ . ⎠

(32)

It is not difficult to see that invariant relations follow from Eqs.(30) and (32) directly (see also [3 p.416]): 2πγ αT = const = ; (33a) m0 ρ 0 3kB α⋅

m0 M k = const = 3 2π ⋅ B . γ T

(33b)

Let’s try to justify them. Really, within the framework of the statistical theory of gravitating spheroidal bodies (under supposition that geometrical eccentricity of kernel of a rotating and gravitating spheroidal

body ε 0 → 0 , in particular, for the solar disk ε 02 ≈ 1.7999919 ⋅10 −5 [3, 6]) it has just been derived the similar formula (29) for one-component gas in the previous Section 2. The following result as well as others of a more general kind may also be obtained from a virial theorem first given by Poincaré [47]: 2 E k + Eg = 0 , (34) where Ek is the total kinetic energy of translation and Eg is the total gravitational potential energy of a steady state system in the form of a collection of detached masses moving under no forces except their own mutual gravitational attraction. Let’s apply the Poincaré’s virial theorem to a cloud-like configuration of ideal gas as gravitating spheroidal body in the steady state. In this connection we can write the kinetic energy Ek in the form [2, 44]: 1 (35) Ek = ∑ m0l vl2 = ∑ Ek l , 2 l

l

where vl is the velocity of translation of a l -th particle of mass m0l , Ek l is the average kinetic energy of a moving l -th particle. The gravitational potential energy Eg may similarly be written in the form [2, 44]: Eg =

1 m0lϕ g l = Eg l , 2 l l

∑

∑

(36)

where ϕg l is the gravitational potential at the l -th point occupied by the mass m0 , Eg l is the

average potential energy of interaction of a particle with a cloud. Taking into account (35) and (36) the Poincaré’s theorem (34) takes the form that

∑ m0l ⎜⎝ vl2 + 2 ϕg l ⎟⎠ = ∑ (2Ek l + Eg l ) = 0, ⎛

l

1

⎞

(37)

l

so that, in the steady state, the average value of vl2 , averaged over all the separate masses, is equal 1 2

to the average value of − ϕg l [2, 44], or the absolute value of average potential energy of interaction Eg l of a particle is equal to the double average kinetic energy Ek l of a moving particle (in view of arbitrariness of l -th particle, further we will omit the index l in (37)).

As Sir J. Jeans marked, this virial theorem of the kind (37) “provides a convenient rough measure of the average velocity of agitation of a system of gravitating masses in a steady state: it is equally applicable to systems of stars, star-clusters, nebulae, and masses of gravitating gas” [2 p.68]. Taking into account the condition of mechanical balance and using the virial theorem (37) as well as the theorem of uniform distribution of energy on freedom degrees for each particle [48] (under a condition k BT >> hν , where k B and h are constants of Boltzmann and Planck respectively, T is a temperature, ν is a frequency) we obtain: i − Eg = 2 Ek = 2 ⋅ ⋅ k BT = ik BT , (38a) 2 where Ek is a kinetic energy of heat movement of a particle, i is a number of freedom degree of a moving particle; in particular i = 3 for the translational movement of a particle (for example, atom of hydrogen H with mass m0 = mH ), so 3 Eg = 2 ⋅ kBT = 3k BT , (38b) 2

If the particles which constitute the system are taken to be the molecules of a gas, or other independently moving units such as atoms, free electrons, ions etc., then according to (38b) v 2 is equal to 3kB / mH μ r times the temperature of the gas, where μ r is its mean relative molecular weight. Thus, according to the virial theorem (37) the mean temperature of the gas is of the order of magnitude of 1 ⎛ m μ ⎞ γM ⎛ mH μ r ⎞ ⎜⎜ ⎟⎟ , (39) − ϕg ⎜⎜ H r ⎟⎟ ∝ 2

⎝ 3k B ⎠

r ⎝ 3k B ⎠

so that the mean internal temperatures of different stars are approximately proportional to the values of μ r M / r for these stars [2 p.68], i.e. stellar temperatures are very high as consequence their enormous masses M . From (29) and (38a) it follows directly that the parameter of gravitational compression α of a spheroidal body, being formed by a collection of particles of one-component ideal gas, is equal: ⎛ ik T α = π ⋅ ⎜⎜ B ⎝ γm0 M

2

2

⎛k ⎞ i T ⎞ ⎟⎟ = π ⎜⎜ B ⋅ ⋅ ⎟⎟ . ⎝ γ m0 M ⎠ ⎠

(40)

Nevertheless, even if the spheroidal body was initially formed exclusively on the basis of a cloud of molecular hydrogen (with μ r = 2 ) then as a result of gravitational heating (39), ionization and later nuclear reactions other atoms appear (for example, atoms of helium He, carbon C, oxygen O, etc.). Thus, in this case of formation of a spherical body on the basis of multi-component ionized gas the formula (40) becomes: 2

⎛k i T ⎞ ⋅ ⎟ , α =π⎜ B ⋅ ⎜ γ m0 M ⎟⎠ ⎝

(41)

where < > is an operation of statistical averaging. Really, in process of rise in temperature, the composition of existing particles becomes simpler in stellar atmosphere. The spectral analysis of stars belonging to the spectral classes O, B, A (with temperatures from 52 500 K till 7 550 K) shows lines of the ionized hydrogen and helium as well as ions of metals in their atmospheres, whereas in the spectral class K (4050–5250 K) radicals are already found out, and there exist even molecules oxides in the spectral class of M (2500–3850 K). For stars belonging to the first four classes, hydrogen and helium lines prevail, but in process of temperature fall lines of other elements also become. Besides, there appear even the lines pointing to existence of chemical compounds though these compounds are still very simple (CH, OH, NH, CH2, C2, C3, СаН, etc.). External layers of stars consist of hydrogen mainly, so on the average, there are only 1000 atoms of helium, 5 atoms of oxygen and less than one atom of other elements per 10000 atoms of hydrogen. As follows directly from (41), the equation is true: α ⋅M k i . (42) = π⋅ B⋅ T γ m0 The operation of statistical averaging in (42) means that i i , = m0 m0

(43а)

where i is an average number of degrees of freedom for a particle with averaged mass m0 . Following S. Сhandrasekhar [49], the averaged mass of a particle m0 it is convenient to express through a mean relative molecular weight μ r of a highly ionized stellar substance and the mass of proton mp : m0 = μ r ⋅ mp ,

(43b)

where mp = 1.67248 ⋅ 10−27 kg (instead of mH though mH ≈ mp ). Similarly (43b), the average number of degrees of freedom of a particle with averaged mass m0 is equal:

i = i ⋅i p ,

(43c)

where i p = 3 is the number of translation degrees of freedom of a proton with mass mp , i is an average number of all degrees of freedom for a particle with a mean relative molecular weight μr . Substitution (43b), (43c) in (42) allows us to write down the following equation connecting macroscopic and microscopic physical values: k 3i α ⋅M . (44) = π⋅ B⋅ T γ mp μ r Now let’s introduce a new constant called the universal stellar constant: k 1.38049 ⋅ 10 −23 J K κ = 3 π ⋅ B ≈ 3 3.14159265 ⋅ = 1.10003963 ⋅ 10 −12 (kg 2 K ⋅ m) . (45) γ 6.673 ⋅ 10−11 m3 kg⋅ s 2 Taking into account (44), (45) we obtain the equation of state of an ideal stellar substance: κi ⋅T , α ⋅M = (46) mp μ r named so by analogy to the known Clapeyron–Mendeleev’ equation of state of a ideal gas (or the usual Boyle–Charles law [2]). The stars obeying the equation of state of an ideal stellar substance (46) we can call as ideal ones. Having rewritten the equation of state of an ideal stellar substance (46) in the form: μ mp M α⋅ r⋅ =κ , (47) i

T

we obtain the universal stellar law (USL) [3]: α⋅

μ r mp M ⋅

i

T

= const .

(48)

Obviously, a verification of USL (48) for different stars requires estimating their parameters α , M , T and also μ r , i.e. chemical composition of stars.

4. Estimation of mean relative molecular weight of a highly ionized stellar substance and verification of universal stellar law As already noted, the spectral analysis of different stars reveals lines of the ionized hydrogen and helium mainly. There are stars having the heightened content of certain chemical element (carbon stars, silicon stars, iron stars etc.). Stars with abnormal composition of chemical elements are various enough. At the same time, the stellar chemical composition depends on a site of the star in Galaxy. Old stars in the spherical part of Galaxy contain a few atoms of heavy elements; on the contrary, there are many heavy elements in stars belonging to the peripheral spiral branches of Galaxy as well as its flat part where new stars are arising. Therefore, it is possible to connect presence of heavy elements with features of the chemical evolution characterizing life of a star. It is well-known [49] that a mean relative molecular weight μr of a highly ionized stellar substance can be found by the formula: μr =

1

∑ xZ ⋅ν Z

,

(49)

Z

where xZ is a relative content of an element with atomic serial number Z in a mass unit of a stellar matter, ν Z is a number of the free particles per unit of atomic weight AZ produced by each atom of an element as a result of its ionization. As shown by Сhandrasekhar [49], as a first approximation under condition of full ionization for an element with atomic serial number Z and relative atomic weight AZ the value ν Z is equal:

νZ =

Z +1 . AZ

(50)

As follows from the periodic system of chemical elements, under condition of excluding the easiest elements (namely, hydrogen H and helium He) the ratio ν Z is equal approximately ½ in accord with (50) since a serial number Z of elements defines a total number of electrons in atom. Hence, if we suppose that there are xH grams of hydrogen, xHe grams of helium and xZ = 1 − xH − xHe grams of «heavy elements» in 1 gram of a stellar matter then we can find that ν H = 2 ; ν He = 3 / 4 ; ν Z = 1 / 2 , that allows us to define the mean relative molecular weight μr of a highly ionized stellar substance as the first approximation according to the formula (49): μr =

1 2 . = 3 1 2 xH + xHe + (1 − xH − xHe ) 1 + 3 xH + 0.5 xHe 4 2

(51)

The second approximation takes into account the ionization state of stellar substance, so that the value of ν Z can be more precisely calculated by the formula of Strömgren [50]: ⎡ ⎢ 1 ⎢⎢ 2n 2 νZ = ⋅ 1+ χ n( Z ) AZ ⎢ n Ne ⎢ 1+ ⋅ e k BT ⎢ G (T ) ⎣

∑

⎤ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

(52)

where χ n(Z ) is an average ionized potential of various electron layers defined according to the theory of Bohr as 2π 2 e 4 me Z 2 ; (53a) χ n(Z ) = 2 2 n h

n is a main quantum number ( n = 1 for the K-electron, n = 2 for the L-electron, n = 3 for the Melectron, etc.); N e is a number of free electrons in a volume unit; G (T ) is calculated by the formula of statistical physics [51]:

G (T ) = 2 ⋅

(2πme kBT )3 2 ,

(53b) h3 besides me is a mass of electron, h is the Planck’s constant, and other designations have the usual sense. Comparing (52) with (50) we can see that the value in square brackets in the formula (52) is the corrected (by Strömgren) number of free particles per one nucleus with charge Z e [50]. Then according to (49) a mean relative molecular weight μ r of a mixture of chemical elements with the values ν Z (calculated by means of (52)) and the determined values of abundance xZ as previous it is equal: μr =

1

∑ xZ ⋅ν Z

=

1

ν

,

(54a)

Z

where ν = ∑ xZ ⋅ν Z . Z

(54b)

Having used (52)-(53b), (54a,b), B. Strömgren calculated μr for the so-called «mixture of Russel» in which elements O, (Na+Mg), Si, (K+Ca), Fe meet in the weight proportion 8:4:1:1:2 at the preassigned

values T and G (T ) N e . In particular, at T = 107 K and ln[G (T ) N e ] = 5 he found that ν R = 0.52 and μ R = ν R−1 = 1.92 [50]. Reasoning in accord with the mentioned above, i.e. that 1 gram of a stellar substance contains xH grams of Н, xHe grams of Не and (1 − xH − xHe ) grams of the «mixture of Russel», we obtain that the required mean relative molecular weight μ r is estimated by the formula of Strömgren: μr =

1 . 3 2 xH + xHe + ν R (1 − xH − xHe ) 4

(55)

which is more exact in comparison with the formula (51). Using his method [49] Strömgren calculated the values μr and deduced the conclusions relatively the hydrogen content of those stars for which there were reliable data concerning their luminosity L , mass M and radius R . Thus, estimating the hydrogen content of the Sun he obtained that ⎡ G (T ) ⎤ ln ⎢ ⎥ = 3; ⎣ Ne ⎦ xH = 0.36 ; μ r = 1.00 .

(56a)

(56b) (56c) Concerning the hydrogen content of the Capella A (or α Aurigae, HD340029=HIP24608) he found that ⎡ G (T ) ⎤ ln ⎢ ⎥ = 7 , xH = 0.30 and accordingly μ r = 1.01 . ⎣ Ne ⎦

Really, according to the modern data on relative quantity content of atoms in the stars the photospheric composition of the Sun includes hydrogen H (73.46 %), helium He (24.85 %), oxygen O (0.77 %), carbon C (0.29 %), iron Fe (0.16 %), neon Ne (0.12 %), nitrogen N (0.09 %), silicon Si (0.07 %), magnesium Mg (0.05 %), sulfur S (0.04 %) [52]. Then the relative content of hydrogen xH , helium xHe and «heavy elements» xZ = 1 − xH − xHe in a mass unit of stellar substance of the Sun is estimated by the values: xH = 0.3527 ; xHe = 0.4772 ; xZ = 0.17 . As noticed by Сhandrasekhar [49], in the assumption that the helium content can be neglected, formulas (51) and (55) give us approximately identical estimation of mean relative molecular weight of a stellar matter: μr =

2 1 . ≈ 1 + 3 xH 2 xH + ν R ⋅ (1 − xH )

(57)

Then according to (57) in case of the Sun we can obtain that μr =

2 = 0.97 , 1 + 3 ⋅ 0.3527

that practically coincides with the estimation of Strömgren (56c). So, being guided by the Strömgren’s estimation μ r = 1.00 and the proposition i = 1 , let’s verify the equation of state of an ideal stellar substance (46) as well as USL (47) in case of the Sun: α Sun ⋅

mp ⋅ M Sun TcorSun

=κ .

(58)

Relatively TcorSun , the analysis of references [53] - [57] shows “…that the temperature of the corona was mostly recorded in a range such as 1,000,000 to 2,000,000 K, whereas the temperatures of the other layers were more exact numbers. This is because the temperature drops slowly as you move from the corona into space. The variance in temperature is also due to the fact that the sun's corona has no defined boundary…” [58]. This means that average temperature of the Sun’s corona can be chosen as 1,500,000 K [52], [58]. According to the right part (58) and taking into account that mass of the Sun M Sun = 1.9891 ⋅1030 kg ,

average temperature of the Solar corona TcorSun = 1.5 ⋅106 K [52], [58], parameter of gravitational compression of the Sun α Sun= 2.29701⋅10−19 m −2 we calculate: α Sun ⋅

mp M Sun TcorSun

= 2.29701 ⋅10 −19 ⋅

1.67248 ⋅10 −27 ⋅1.9891 ⋅1030 1.5 ⋅106

= 1.062937513 ⋅10 −12 (kg 2 K ⋅ m) .

(59)

Comparison (59) with (45) shows a coincidence up to the relative error equal mp M Sun κ − α Sun ⋅ TcorSun δ= ⋅ 100% = 3.37% ,

κ

that testifies the validity of USL for the Sun which is a star of type G2V. Starting from the formula (59) it is possible to verify USL for other Sun-like stars since only in our Milky Way galaxy there are from above 100 billion stars of type G2. Really, the average value of effective temperature of surface of the Sun is Teff Sun = 5.778 ⋅ 103 K [52] whereas the temperature of its corona is accordingly equal to Tcor Sun = 1.5 ⋅ 106 K , i.e. after finding the ratio Tcor Sun Teff Sun

=

1.5 ⋅106 5.778 ⋅10

3

≈ 2.596 ⋅10 2 ,

(60)

we can use the effective temperature Teff Sun instead of the temperature of its corona Tcor Sun in USL (47). Then expressing Tcor through Teff for a star belonging to the spectral class G (in particular, the Sun) we obtain the modified USL (relatively Teff ): α⋅

mp M Teff

=Ξ ,

(61)

where Ξ = κ ⋅ 2.596 ⋅102 = 2.856296878 ⋅10−10 (kg 2 /K⋅ m) . (62) Besides the Solar system, now we will focus in the study of the multi-planet extrasolar systems Kepler - 20 [42], HD10180 [39], HIP14810 [43], 61Virginis [43], 55Cnc [14,27], Alpha Centauri [43], Upsilon Andromedae [10] and Gliese 876 [13] whose stars are belonging to the different spectral classes G, F, K and M. We would like to verify correctness of the modified USL (61) for these multi-planet systems. First of all, we note that satisfiability of the equality (61) for the Sun belonging to the spectral type G2V is confirmed by the calculations in (59), i.e. the modified USL (61) is carried out for the Sun with the relative accuracy δ = 3.37% . Example 4.1. For the second representative of this spectral class of stars, namely for the star Kepler-20 of type G8, it is known [4, 43] that M Kepler - 20 = 0.912 M Sun = 0.912 ⋅1.9891 ⋅1030 kg = 1.8140592 ⋅1030 kg ; RKepler - 20 = 0.944 RSun = 0.944 ⋅ 6.955 ⋅108 m = 6.56552 ⋅108 m ;

Teff Kepler- 20 = 5466 K .

i.e. the square root of the parameter of gravitational compression for the star Kepler-20 is equal α Kepler - 20 ≈

1 1 = = 5.07702868 ⋅10 −10 (m − 2 ) . 3RKepler - 20 3 ⋅ 6.56552 ⋅108

Then, according to (61) we obtain: α Kepler-20 ⋅

mp M Kepler-20 Teff Kepler-20

= 5.07702868⋅10−10 ⋅

1.67248⋅10−27 ⋅1.8140592⋅1030 3

5.466 ⋅10

= 2.8180739⋅10−10 (kg2 K⋅ m) . (63)

Comparison (63) with (62) shows that the modified USL (61) is carried out with a relative accuracy δ=

2.8562969 − 2.8180739 ⋅100% = 1.34% for the star Kepler-20 of type G8. 2.8562969

Example 4.2. Let’s test realizability of the modified USL (61) for a representative of spectral class G, namely for the star HD10180 of type G1V [39, 43]: M HD10180 = 1.06 ⋅ M Sun = 1.06 ⋅1.9891 ⋅1030 kg = 2.108446 ⋅1030 kg ; RHD10180 = RSun = 6.955 ⋅ 108 m ;

Teff HD10180 = 5911K , whence α HD10180 ≈

1 1 ≈ ⋅ 10 −8 m −1 = 4.792715 ⋅ 10 −10 m −1 . 3RHD10180 3 ⋅ 6.955

Substituting the estimated parameter for HD10180 in the left part of eq.(61) we have: α HD10180 ⋅

mp M HD10180 Teff HD10180

=

4.792715⋅10−10 ⋅1.67248⋅10−27 ⋅ 2.108446⋅1030 5.911⋅103

= 2.8591969⋅10−10 (kg 2 K⋅ m) . (64)

Thus, according to (64) the modified USL (61) is carried out with the relative error 2.8562969 − 2.8591969 δ= = −0.1% for the star HD10180 of type G1V. 2.8562969 Example 4.3. Now let’s verify the modified USL (61) for a representative of spectral class G which is the star HIP14810 of type G5 [43]: M HIP14810 = 0.99 ⋅ M Sun = 0.99 ⋅1.9891 ⋅1030 kg = 1.969209 ⋅1030 kg ; RHIP14810 = RSun = 6.955 ⋅ 108 m ; Teff HIP14810 = 5485 K ,

whence α HIP14810 = α Sun ≈ 2.29701 ⋅10 −19 = 4.7927132 ⋅10 −10 m −1 . Then, according to (61) we obtain: α HIP14810 ⋅

mp M HIP14810 Teff HIP14810

= 4.7927132⋅10−10 ⋅

1.67248⋅10−27 ⋅1.969209⋅1030 3

5.485 ⋅10

= 2.8777798⋅10−10 (kg2 K⋅ m) . (65)

Comparison (65) with (62) shows that the modified USL (61) is carried out with a relative accuracy 2.8562969 − 2.8777798 ⋅ 100% = −0.75% for the star HIP14810 of class G5. δ= 2.8562969 Example 4.4. For one more representative of spectral class G, namely for the star 61Virginis of type G5V, it is known [43] that M 61Vir = 0.95M Sun = 0.95 ⋅1.9891 ⋅1030 kg = 1.889645 ⋅1030 kg ; R61Vir = 0.94 RSun = 0.94 ⋅ 6.955 ⋅108 m = 6.5377 ⋅108 m ; Teff 61Vir = 5531 K ,

i.e. the square root of the parameter of gravitational compression for the star 61Virginis is equal α 61Vir ≈

1 3R61Vir

=

1 8

3 ⋅ 6.5377 ⋅10

= 5.098633056 ⋅10−10 (m − 2 ) .

Then, according to (61) we obtain: α61Vir ⋅

mp M 61Vir Teff 61Vir

= 5.098633056⋅10−10 ⋅

1.67248⋅10−27 ⋅1.889645⋅1030 5.531⋅103

= 2.9133406⋅10−10 (kg2 K⋅ m) . (66)

Comparison (66) with (62) shows that the modified USL (61) is carried out with a relative accuracy δ=

2.8562969 − 2.9133406 ⋅100% = −1.997% for the star 61Virginis of type G5V. 2.8562969

A quite high accuracy of this law for the given stellar class of G, most likely, can be explained by good approximation Tcor for these stars on the basis of the formula (60).

Example 4.5. Let’s investigate the law (61) for a representative of spectral class K which is the star 55 Cancri (55Cnc=HD75732=HIP43580) of type KOIV-V. According to the paper [27] as well as the catalog [43] the mass and the effective temperature of stellar surface of the star 55 Cancri are equal respectively M 55Cnc = 0.905M Sun = 1.800136 ⋅1030 kg ; Teff 55Cnc = 5196 K , and the radius is R55Cnc = 0.943RSun = 0.943 ⋅ 6.955 ⋅108 m = 6.558565 ⋅108 m .

Let’s find an estimation of the value α 55Cnc ≈

1 3R55Cnc

=

1 8

3 ⋅ 6.558565 ⋅10

= 5.082412591 ⋅10 −10 (m −1 )

and then calculate the left part of the equation (61): mp M 55Cnc 1.67248 ⋅ 10 −27 ⋅ 1.8001355 ⋅ 10 30 α 55Cnc ⋅ = 5.0824126 ⋅ 10 −10 ⋅ = 2.9448753 ⋅ 10 −10 (kg 2 K⋅ m) .(67) Teff 55Cnc 5.196 ⋅ 10 3 The modified USL (61) for the star 55Cancri of type KOIV-V is carried out with a relative error δ=

2.8562969 − 2.9448753 ⋅ 100% = −3.1% which confirms precise enough estimations of the effective 2.8562969

temperature, mass and radius of this star. Example 4.6. For the second representative of this spectral class K, namely for the star α Centauri of type K1V, it is known [43] that M αCent = 0.934 M Sun = 0.934 ⋅ 1.9891 ⋅ 10 30 kg = 1.8578194 ⋅ 10 30 kg ; RαCent = 0.863RSun = 0.863 ⋅ 6.955 ⋅ 10 8 m = 6.002165 ⋅ 10 8 m ; Teff αCent = 5214 K . i.e. the square root of the parameter of gravitational compression for the star α Centauri is equal 1 1 α αCent ≈ = = 5.5535516 ⋅ 10 −10 (m − 2 ) . 3RαСent 3 ⋅ 6.002165 ⋅ 10 8 Then, according to (61) we obtain: mp M αСent 1.67248⋅ 10−27 ⋅ 1.8578194⋅ 1030 α αCent ⋅ = 5.5535516⋅ 10−10 ⋅ = 3.3095139⋅ 10−10 (kg 2 K⋅ m) . 3 Teff αСent 5.214 ⋅ 10

The modified USL (61) is carried out with a relative accuracy δ =

(68)

2.8562969 − 3.3095139 ⋅100% = −15.87% 2.8562969

in accord with (68) for the star α Centauri of type K1V which is caused, probably, by an inexact estimation of temperature of its corona T55Cnc for this class K, i.e. by means of a directly proportional dependence (62) as in the special case of G2V. Example 4.7. Now we shall consider the modified USL (61) for a star of new class F using example of the Ups Andromedae of type F8V [4, 43]: MυAnd = 1.27 M Sun = 2.526157 ⋅1030 kg ; RυAnd = 1.631RSun = 1.1343605 ⋅109 m ; Teff υAnd = 6212 K .

To this end we shall find the following value: αυAnd ≈

1 3RυAnd

=

1 3.4030815 ⋅109

= 2.9385132 ⋅10 −10 (m −1 ) ,

after that we shall be able to estimate the left part of the equation (61): m ⋅M 1.67248 ⋅ 10−27 ⋅ 2.526157 ⋅ 1030 αυAnd ⋅ p υAnd = 2.9385132 ⋅ 10−10 ⋅ = 1.9985613 ⋅ 10−10 (kg 2 K⋅ m) . (69) 6.212 ⋅ 103 Teff υAnd

According to (69) for the star υ Andromedae of type F8V, the law (61) is carried out approximately with a relative error δ = 30% because the estimation of corona temperature of the star υ Andromedae on the basis of (60) or (62) is rough for stars of class F. Example 4.8. At last, we shall consider characteristics of the star Gliese 876 of type M4V: M Gl 876 = 0.334 M Sun = 0.334 ⋅1.9891 ⋅1030 kg = 6.643594 ⋅10 29 kg ; RGl 876 = 0.36 RSun = 0.36 ⋅ 6.955 ⋅108 m = 2.5038 ⋅108 m ; Teff Gl 876 = 3350 K ,

then also check up them on conformity to the law (61) for what we shall calculate preliminary: α Gl 876 ≈

1 3RGl 876

=

1 ⋅10 −8 = 1.3313097 ⋅10 −9 (m −1 ) 7.5114

and after that we shall find: α Gl 876 ⋅

mp ⋅ M Gl 876 Teff Gl 876

= 1.3313097 ⋅10 −9 ⋅

1.67248 ⋅10 −27 3.35 ⋅103

⋅ 6.643594 ⋅10 29 =

(70)

= 4.4156873 ⋅10 −10 (kg 2 K⋅ m)

For the star Gliese 876, the modified USL (61) is carried out very approximately with the high relative error δ = −54.59% as this star belongs to the class M (type M4V) for which the relation between temperature of its corona Tcor Gl 876 and effective temperature of its surface Teff Gl 876 is not known exactly (because the expression (60) is true for the class G mainly). Low accuracy of this law for the stars belonging to the spectral classes F or M, most likely, can be explained by too rough approximation Tcor for the more bright or dim stars on the basis of the formulas (60) or (62). Stellar parameters being estimated in the mentioned above examples are placed in Table 1. This table also contains data relatively other stars belonging to the high order spectral classes O, B and A (for simplification we suppose that μ r = 1 and i = 1 for all types of stars). Really, it is well-known that μ r ≈ 1 for numerous stars (in particular, μr = 1.01 for Capella A [49]). As shown in Table 1, there exists an abnormal group of intermediate-mass red giants (belonging to the spectral class K mainly) as 24 Sextanis, 18 Delphini, Capella A, 14 Andromedae, γ Cephei, β Ceti, ξ Aquilae and 11 Сomae whose chemical composition includes even radicals except ions and electrons. In this connection our preliminary assumption μ r ≈ 1 gives us a very high relative error δ ≈ 75% here, therefore following A.Eddington [2, 49], it is reasonable to suppose μ r ≈ 2 for this group of red giants (see Table 2). In other word, the ratio μ r / i can be an increasing function in the case of stars belonging to the spectral class K in accord with the modified USL(61), as shown in Table 2. Nevertheless, the investigation of character of function of relative error δ (see Fig. 1) depending on different types of stars reveals determinate regularity for very bright and dim stars: low accuracy of the modified USL (61) for stars belonging to the high order spectral classes O, B and A ( δ ≈ +50% ) as well as for stars belonging to the last spectral class M ( δ ≈ −60% ). This fact can be explained only by simple linear dependence Tcor (Teff ) according to (60) in the case of Sun. Really, the Sun is belonging to the spectral class G therefore we can reach the good approximation Tcor for the stars of this spectral class (see Example 4.1-4.4 and Table 1) using the formula (60). Thus, the dependence Tcor (Teff ) finding is an important task. Table 3 contains the data from Tables 1 and 2 together with the ± errors in measurements of Mass, Radius and Effective temperature as well as estimations for the modified USL with relative errors. To recalculate the obtained estimations in Table 1 and 2 for modified USL relatively different classes of stars, the following simple operations are used: (a ± x) + (b ± y ) = (a + b) ± ( x + y ); (a ± x) ⋅ (b ± y ) = ab ± (ay + bx);

(a ± x) −1 = 1 / a ∓ (1 / a ) 2 x, where x and y are the instrumental errors in measurements of the physical values a and b , so that x << a and y << b . As seen in Table 3, the instrumental errors in measurements of Mass, Radius and Effective temperature define lower limit of Relative error δ , % testifying the validity of the modified USL at the level of 3% for ideal stars, i.e. for the stars obeying the equation of state of an ideal stellar substance (46).

Table 1. Verification of the modified USL for stars belonging to the different spectral classes and types Stars

Spectral class and type

Mass

M , kg

Radius R, m

Effective tempera -ture

Teff , K

ξ Persei τ Scorpii γ Pegasi α Andromedae Sirius A WASP-12 υ Andromedae KOI-94 HD 74156 Kepler-60 HD 10180 Kepler-33 HD 155358 47UrsaeMajoris Sun HD 1461 μ Andromedae Kepler-11 HAT-P-13 HD 37124 KOI-730 61 Virginis HIP 14810 Kepler-20 α Centauri 55 Cancri 24 Sextanis 18 Delphini Capella A 14 Andromedae

O7.5III B0.2V B2IV A3V A1V G0 F8V None G0 None G1V None G0 G0V G2V G0V G3IV-V G G4 G4V None G5V G5 G8 K1V K0IV-V G5 G6III K0III K0III

31

7.16076⋅10 2.98365⋅1031 1.770299⋅1031 7.16076⋅1030 4.017982⋅1030 2.685285⋅1030 2.526157⋅1030 2.486375⋅1030 2.466484⋅1030 2.18801⋅1030 2.108446⋅1030 2.567928⋅1030 1.829972⋅1030 2.048773⋅1030 1.9891⋅1030 2.148228⋅1030 2.148228⋅1030 1.889645⋅1030 2.426702⋅1030 1.810081⋅1030 2.128337⋅1030 1.889645⋅1030 1.969209⋅1030 1.814059⋅1030 1.857819⋅1030 1.800136⋅1030 3.063214⋅1030 4.57493⋅1030 5.350679⋅1030 4.37602⋅1030

9

9.737 ⋅10 4.52075 ⋅109 3.3384 ⋅109 1.87785⋅109 1.190001⋅109 1.091935⋅109 1.134361⋅109 1.151748⋅109 1.09889⋅109 1.04325⋅109 6.955⋅108 1.26581⋅109 6.955⋅108 8.6242⋅108 6.955⋅108 7.615725⋅108 8.658975⋅108 7.6505⋅108 1.08498⋅109 5.7031⋅108 7.6505⋅108 6.5377⋅108 6.955⋅108 6.56552⋅108 6.002165⋅108 6.558565⋅108 3.40795⋅109 5.91175⋅109 8.4851⋅109 7.6505⋅109

35000 29850 21179 13800 9940 6300 6212 6116 6039 5915 5911 5904 5900 5892 5778 5765 5700 5680 5638 5610 5590 5531 5485 5466 5214 5196 5098 4979 4940 4813

Estimation of constant Ξ in the modified USL,

Relative error

δ,%

kg K⋅ m 2

1.1714009⋅10-10 1.2326299⋅10-10 1.3958619⋅10-10 1.5404897⋅10-10 1.8937129⋅10-10 2.1761695⋅10-10 1.9985613⋅10-10 1.9678019⋅10-10 2.0720429⋅10-10 1.9767231⋅10-10 2.8591969⋅10-10 1.9156125⋅10-10 2.4861938⋅10-10 2.2477705⋅10-10 2.760955⋅10-10 2.727781⋅10-10 2.4264909⋅10-10 2.4242742⋅10-10 2.2116139⋅10-10 3.1540156⋅10-10 2.7744599⋅10-10 2.9133406⋅10-10 2.8777798⋅10-10 2.8180739⋅10-10 3.3095139⋅10-10 2.9448753⋅10-10 9.8293312⋅10-11 8.6649477⋅10-11 7.1164706⋅10-11 6.6254182⋅10-11

58.9 56.8 51.1 46.1 33.7 23.8 30 31.1 27 30.8 -0.1 32.9 12.9 21 3.37 4.5 15 15.1 22.6 -10.4 2.87 -1.99 -0.75 1.34 -15.87 -3.1 65.6 69.7 75.1 76.8

γ Cephei β Ceti ξ Aquilae 11 Сomae HIP 57274 Groombridge34 Gliese 581 Gliese 876

2.78474⋅1030 5.56948⋅1030 4.37602⋅1030 5.37057⋅1030 1.452043⋅1030 8.035964⋅1029 6.16621⋅1029 6.643594⋅1029

K1III-V K0III G9III G8 III K5V M1.5V M2.5V M4V

3.40795⋅109 1.167049⋅1010 8.346⋅109 1.32145⋅1010 4.7294⋅108 2.635945⋅108 2.0865⋅108 2.5038⋅108

9.4905172⋅10-11 5.5461999⋅10-11 6.1152286⋅10-11 4.7780179⋅10-11 3.6888851⋅10-10 4.5565116⋅10-10 4.7099829⋅10-10 4.4156873⋅10-10

4800 4797 4780 4742 4640 3730 3498 3350

66.8 80.6 78.6 83.3 -29.15 -59.5 -64.9 -54.59

100 80 60

relative error, %

40 20 0 −20 −40 −60 −80 M

K

G

F

A

B

O

spectral classes

Figure1. The plot of relative error δ , % of constant estimating Ξ in the modified USL for stars belonging to the different spectral classes

Table 2. Refinement of the modified USL for the group of intermediate-mass red giants

Red giant stars

24 Sextanis 18 Delphini Capella A

Effective tempera -ture

Teff , K

5098 4979 4940

Mean relative molecular weight μ

2 2 2

Mass

Radius

M , kg

R, m

3.063214⋅1030 4.57493⋅1030 5.350679⋅1030

3.40795⋅109 5.91175⋅109 8.4851⋅109

Estimation of constant Ξ in the modified USL,

Relative error

δ, %

kg 2 K⋅ m 1.9658662⋅10-10 1.7329895⋅10-10 1.4232941⋅10-10

31.17 39.33 50.17

14 Andromedae γ Cephei β Ceti ξ Aquilae 11 Сomae 42 Draconis

4813 4800 4797 4780 4742 4200

2 2 2 2 2 2

2.148228⋅1030 2.78474⋅1030 5.56948⋅1030 4.37602⋅1030 5.37057⋅1030 1.949318⋅1030

8.658975⋅108 3.40795⋅109 1.167049⋅1010 8.346⋅109 1.32145⋅1010 1.53219⋅1010

1.3250836⋅10-10 1.8981034⋅10-10 1.1092399⋅10-10 1.2230457⋅10-10 9.5560358⋅10-11 3.3774696⋅10-11

53.61 33.55 61.17 57.18 66.54 88.18

Table 3. Estimation of the modified USL for the different classes of stars with regard to the errors in measurements

Stars

Mass

M , kg

Radius R, m

Effective temperature

Teff , K

υ Andromedae HD 10180 61 Virginis HIP 14810 Kepler-20

α Centauri 55 Cancri 24 Sextanis 18 Delphini

γ Cephei 11 Сomae Gliese 876

(2.526157 ± ± 0.119346)⋅1030 (2.108446 ± ± 0.099455)⋅1030 (1.889645 ± ± 0.0597)⋅1030 (1.969209 ± ± 0.079564)⋅1030 (1.8140592 ± ± 0.069619)⋅1030 (1.857819 ± ± 0.011935)⋅1030 (1.8001355 ± ± 0.029836)⋅1030 (3.063214 ± ± 0.159128)⋅1030 4.57493⋅1030 (2.78474 ± ± 0.238692)⋅1030 (5.37057 ± ± 0.59673)⋅1030 (6.643594 ±

(1.134361 ± ± 0.009737)⋅109 6.955⋅108 (6.5377 ± ± 0.202)⋅108 (6.955 ± ± 0.4173)⋅108 (6.56552 ± ± 0.660725)⋅108 6.002165⋅108 (6.558565 ± ± 0.06955)⋅108 (3.40795 ± ± 0.05564)⋅109 5.91175⋅109 3.40795⋅109 (1.32145 ± ± 0.1391)⋅1010 2.5038⋅108

(6.212 ± ± 0.08) ⋅103 (5.911 ± 0.019) ⋅103 5.531⋅103 (5.485 ± 0.044)⋅103 (5.466 ± 0.093) ⋅103 (5.214 ± 0.033) ⋅103 (5.196 ± 0.024) ) ⋅103 (5.098 ± 0.044) ⋅103 (4.979 ± 0.018) ⋅103 (4.8 ± ± 0.1) ⋅103 (4.742 ± ± 0.1) ⋅103 (3.35 ±

Estimation of constant Ξ in the modified USL,

kg 2 K⋅ m (1.9985613 ± ± 0.051527)⋅10-10 (2.8591969 ± ± 0.125677)⋅10-10 (2.913305 ± ± 0.002027)⋅10-10 (2.87778089 ∓ ∓ 0.079478)⋅10-10 (2.81807391 ∓ ∓ 0.223396)⋅10-10 (3.3095139 ± ± 0.000314)⋅10-10 (2.9448753 ± ± 0.003979)⋅10-10 (1.96586625 ± ± 0.05306)⋅10-10 (1.7329895387 ∓ ∓ 0.006265)⋅10-10 (1.8981034 ± ± 0.123151)⋅10-10 (9.5560358 ± ± 0.884368)⋅10-11 (4.415687396 ±

Relative error

δ,%

30.0 ∓ 1.8 -0.1 ∓ 4.4 -1.99 ∓ 0.07 -0.75 ± 2.78 1.34 ± 7.8 -15.9 ∓ 0.01 -3.1 ∓ 0.14 31.17 ∓ 1.86 39.33 ± 0.22 33.55 ∓ 4.31 66.5 ∓ 3.1 -54.6 ∓ 0.04

± 0.59673)⋅1029

± 0.3) ⋅103

± 0.001184)⋅10-10

5. Estimation of temperature of the stellar corona

To test justice of USL for other stars, it is necessary to estimate temperature of their stellar coronas Tcor using the value of effective radiative temperature Teff of star’s surface. To measure the real (thermodynamic) temperature T of body by means of its effective radiative temperature Teff the following

relation can be used: T T = eff , 4 A T

(71)

where factor AT is an integral absorptivity of a body, besides for a real body AT < 1 [59], i.e. radiative temperature of body is always less than real temperature. Really, AT is the ratio of the powers of integral radiant emittance of the given body ( ET ) and the perfect black body ( ε T ) at temperature T in accord with the Kirchhoff’ law: AT = ET / ε T ,

(72) therefore AT has meaning of the power of blackness of a body: 0 ≤ AT ≤ 1 , i.e. for the black body AT = 1 and for the mirror body AT = 0 . According to the Stefan-Boltzmann law, the power of integral radiant emittance of perfect black body is equal: 4 ε T = σ ⋅ Teff , (73) where σ = 5.6686 ⋅ 10 −8 (W/m 2 ⋅ K 4 ) is the Stefan-Boltzmann constant. Taking into account (71) we write USL (46) for an arbitrary case of a remote star:

α⋅

μ r mp ⋅ M ⋅

=

κ

. (74) AT Let’s note that except temperature of body the value AT in (74) depends on its chemical composition as well as the form and a surface state condition [59]. As a first approximation, the value 4 AT for each considered star depends on its temperature T , i.e. on the mentioned above value i . Nevertheless, in the general case AT is also a function of the star's chemical composition defining the mentioned parameter μ r , the star's form determining its radius R and star's surface state condition depending on its stellar belonging to the different spectral classes {...,F,G,K,...}, i.e. AT = AT ( μ r , R, i , {..., F, G, K,...}) . In this connection it should be reasonable to consider the dependences on the spectral classes directly as ATF = ATF (i , μ r , R ) , ATG = ATG (i , μ r , R ) ,

i

Teff

4

ATK = ATK (i , μ r , R ) . In particular (for the Sun), the effective radiative temperature of its solar surface is equal: TeffSun = 5.778 ⋅103 K ,

whereas the temperature of its corona is TcorSun = 1.5 ⋅ 10 6 K . Using (71) we can estimate a 4-th degree root of an integral absorptivity of the solar corona:

5.778 ⋅ 10 3 (75) = 3.852 ⋅ 10 −3 . 6 Tcor Sun 1.5 ⋅ 10 Since for any other star the temperature of its stellar corona is unknown then it is not possible to estimate 4A T directly as for the Sun. Thereupon, let’s consider the relation (72) for the Sun separately and for any other star belonging to the spectral class G: 4

AT cor Sun =

Teff Sun

=

AT corSun = ET corSun / ε T corSun ;

(76а)

AT cor G = ET cor G / ε T cor G .

(76b)

Dividing (76b) on (76a) we obtain: AT cor G E ε = T cor G ⋅ T corSun . AT corSun ET corSun ε T cor G Taking into account the Stefan-Boltzmann law (73) the relation (77) passes in the following:

(77)

4

AT cor G ET corSun ⎛ Teff Sun ⎞ ⎟ . (78) ⋅ =⎜ AT corSun ET cor G ⎜⎝ Teff G ⎟⎠ 4+ s As follows from the equation (78) choosing ET corSun / ET cor G = (Teff Sun / Teff G ) the ratio (77) becomes:

AT cor G ⎛ Teff Sun =⎜ AT corSun ⎜⎝ Teff G

⎞ ⎟⎟ ⎠

−s

⎛ T = ⎜ eff G ⎜T ⎝ eff Sun

s

⎞ ⎟ , ⎟ ⎠

(79)

.

(80)

whence 4

AT cor G =

4

AT corSun

⎛ Teff G ⋅⎜ ⎜T ⎝ eff Sun

⎞ ⎟ ⎟ ⎠

s/4

Substitution (80) in (74) leads to the equation:

α⋅

μ r mp M i

⋅

Teff G

=

κ 4

AT corSun

⎛ Teff Sun ⋅ ⎜⎜ ⎝ Teff G

⎞ ⎟⎟ ⎠

s/4

,

(81)

which allows to present USL in the form: 1/ 4

⎛ Teffs Sun ⎞ ⎟ . α ⋅ ⋅ 1− s / 4 = κ ⋅ ⎜ (82) ⎜ AT corSun ⎟ i Teff G ⎝ ⎠ Choosing some value s and also α ≈ 1 / 3R ( R is a star’s radius) according to the mentioned 3σ -rule we

μ r mp M

can derive the empirical variant of USL: ⎛ Teffs Sun ⎞ ⎟ ⋅ = κ ⋅⎜ ⎜ AT corSun ⎟ 3Ri Teff1 −Gs / 4 ⎝ ⎠ If parameters of a star are given in units relatively the Sun’ parameters:

μr

mp M

1/ 4

.

(83)

R = k R ⋅ RSun ;

(84a)

M = k M ⋅ M Sun , then having divided eq.(81) on the similar equation concerning the Sun we obtain:

κ

μ mp ⋅ k M ⋅ M Sun 1 ⋅ r⋅ 3 ⋅ k R ⋅ RSun i Teff G = 1 1 mp ⋅ M Sun ⋅ ⋅ 3 ⋅ RSun 1 Teff Sun

AT corSun

4

⎛ Teff Sun ⋅ ⎜⎜ ⎝ Teff G

(84b)

⎞ ⎟ ⎟ ⎠

s/4

,

κ 4

AT corSun

whence Teff Sun ⎛ Teff Sun 1 μr ⋅ ⋅ kM ⋅ = ⎜⎜ kR i Teff G ⎝ Teff G

⎞ ⎟ ⎟ ⎠

s

−1

s/4

.

In other words, the relation is valid:

μr kM

⋅ i kR besides it allows to find as a first approximation

⎛ Teff Sun = ⎜⎜ ⎝ Teff G

⎞4 ⎟⎟ ⎠

,

(85)

s ≈4. More exact value s can be found by taking logarithm of the equation (85):

ln

whence directly follows that

μr kM i

⋅

kR

⎛ s ⎞ ⎛ Teff Sun = ⎜ − 1⎟ ln⎜⎜ ⎝ 4 ⎠ ⎝ Teff G

(86)

⎞ ⎟, ⎟ ⎠

μr kM

⋅ ⎡ ln μ r − ln i + ln k M − ln k R ⎤ i kR s = 4 + 4⋅ (87) = 4 ⎢1 + ⎥ . ln T ln T − ⎛ Teff Sun ⎞ ⎢ ⎥ eff Sun eff G ⎣ ⎦ ⎟⎟ ln⎜⎜ T ⎝ eff G ⎠ Coming from formulas (71), (80) and (87), it is easy to calculate temperature of a stellar G-corona through temperature of a solar corona: ln

Teff G

Tcor G = 4

⎛ T AT Sun ⋅ ⎜ eff G ⎜ Teff Sun ⎝

⎞ ⎟ ⎟ ⎠

⎛μ k ln ⎜⎜ r ⋅ M i kR 1+ ⎝ ⎛ Teff Sun ⎜ ln ⎜ ⎝ Teff G

⎞ ⎟⎟ ⎠ ⎞ ⎟ ⎟ ⎠

= Tcor Sun

⎛ Teff Sun ⋅ ⎜⎜ ⎝ Teff G

⎞ ⎟⎟ ⎠

ln( μ r / i ) + ln( k M / k R ) ln(Teff Sun / Teff G )

.

(88а)

Using from formula (88а) we can calculate temperature of stellar corona Tcor G enough accurate for stars belonging to the spectral class G; approximately this formula can be applied to the different spectral classes (see Table 4, the values μr for the Xi Persei and the Tau Scorpio were calculated in accord with [49]). This Table 4 shows that functional dependence Tcor G on stellar spectral classes M, K, G,.., O is described by monotonically increasing function entirely excepting area of red giants (K0III- K1III-V). Probably, it can be explained by origin of radicals and other chemical elements in stellar substance of red giants, i.e. by chemical composition changing from μr = 1 to μ r = 2 approximately. In other words, if i = 1 as usual then μ r / i = 2 . This means that it takes place a new virial equilibrium with new temperature state of stellar corona Tcor G in the case of red giant stars. Obviously, there are also radicals in stellar substance of the spectral class M but in this case μ r = 2 and i = 2 simultaneously due to temperature of stellar corona of class M becomes higher than for red giants (K0III- K1III-V). Obviously, if for each spectral class we can find a representive (as the Sun for the spectral class G) then the relation (88а) determines a sufficiently exact dependence between Tcor of the given spectral class and Teff of its representative (for example, between Tcor F and Teff υAnd if the last value is supposed to be known). Evidently, the formula (88a) does not have high relative accuracy of estimation of Tcor for all distant spectral classes from G. Let us note that the formula (88a) can be easily transformed to the more simple form. Really, according to the Eqs. (87), (88a) and taking into account (85) we obtain Tcor G = Tcor Sun

⎛ Teff Sun ⋅ ⎜⎜ ⎝ Teff G

s

−1

⎞4 μ k ⎟⎟ = Tcor Sun ⋅ r ⋅ M , i kR ⎠

i.e. the relation in the form: Tcor G = Tcor Sun ⋅

μr kM i

⋅

kR

.

(88b)

This form does not comprise Tcor G or Teff Sun at all. Analogously the dependences, say, Tcor F (Teff υAnd ) , Tcor K (Teff 55Cnc ) and Tcor М (Teff Gl 876 ) can be written. In this connection the usage (88a, b) for estimation of Tcor F , Tcor K or Tcor М though Tcor Sun leads to more big errors.

6. Comparison with estimations of temperatures on the basis of regression dependences for multiplanet extrasolar systems

In this Section, following Pintr et al. [4] we shall pay our attention to the spectral classes of stars F, G, K and M mainly. This is justified since the life times of spectral classes of stars O, B and A are so small that the complex life will never form on the planets associated to them [4]. According to the Schneider’ catalogue [43], spectral classes of our interest can be characterized as: – spectral class F with Teff between 6000 − 7500 K; – spectral class G with Teff between 5200 − 6000 K; – spectral class K with Teff between 3700 − 5200 K; – spectral class M with Teff less than 3700 K. Recently P. Pintr, V. Peřinová, A. Lukš and A. Pathak have obtained the regression dependences of effective temperature of stellar surface Teff from specific angular momentum an vn ( an is the planetary distance and vn is the planetary velocity) for the different spectral classes of stars [4]: a nTeff F ≈ 4 ⋅ 10 −17 (a n v n )1.9963 (89a)

with the coefficient of determination R2=0.997 for stars of the spectral class F; a nTeff G ≈ 1 ⋅ 10 −16 (a n v n )1.976

(89b)

2

with the coefficient of determination of regression R =0.995 for stars belonging to the spectral class G; (89c) a nTeff K ≈ 4 ⋅ 10 −14 ( a n v n )1.807 with the coefficient of determination R2=0.776 for the stellar spectral class K; a nTeff М ≈ 5 ⋅ 10 −14 (a n v n )1.814

(89d)

2

with the coefficient of determination R =0.974 for stars of the spectral class M.

Table 4. Theoretical estimations of temperatures of stellar corona Tcor for stars belonging to the different spectral classes

Stars Xi Persei Tau Scorpio Sirius A UpsAndromedae HD10180 Sun Kepler-20 55 Cancri Capella A 14 Andromedae Gamma Cephei Groombridge 34 Gliese 876

Spectral class and type O7.5III B0.2V A1V F8V G1V G2V G8 K0IV-V K0III K0III K1III-V M1.5V M4V

kM

kR

26 15 2.02 1.27 1.06 1 0.912 0.905 2.69 2.2 1.4 0.404 0.334

14 6.5 1.711 1.631 1 1 0.944 0.943 12.2 11 4.9 0.379 0.36

μr / i

Teff , K

Tcor , K

0.764 0.7296 1 1 1 1 1 1 2 2 2 1 1

35000 31440 9940 6212 5911 5778 5466 5196 4940 4813 4800 3730 3350

2.1288⋅106 2.5255⋅106 1.7886⋅106 1.1680⋅106 1.5899⋅106 1.5 ⋅106 1.4492⋅106 1.4396⋅106 6.6148⋅105 5.9999⋅105 8.5714⋅105 1.5989⋅106 1.3917⋅106

Table 5. Orbital and thermodynamical characteristics of multi-planet extrasolar systems

Extrasolar system

Spectral class and type

kM

kR

μ r / i Teff , K Theoretical

dependence

Tcor , K

Number of planets

n

Average specific angular momentum

Regression dependence

Tcor , K

an vn , m 2 s

Ups And HD10180 Solar

F8V G1V G2V

1.27 1.06 1

1.63 1 1

1 1 1

6212 1.1679⋅106 4 5911 1.5899⋅106 9 5778 1.5 ⋅106 8

6.35⋅1015 3.723⋅1015 5.316⋅1015

1.1896⋅106 1.5261⋅106 1.5045⋅106

Kepler-20 G8 55 Cnc K0IV-V Gliese 876 M4V

0.912 0.905 0.334

0.944 0.943 0.36

1 1 1

5466 5196 3350

1.4492⋅106 5 1.4396⋅106 5 1.3916⋅106 4

1.415⋅1015 3.599⋅1015 9.92⋅1014

1.4606⋅106 1.5124⋅106 1.2512⋅106

Really, according to the Kepler’s 3-rd law: an3Ω n2 = γM ,

(90)

where an is a major semi-axis of planetary orbit, Ω n is a angular velocity of rotation of a planet on its orbit. Supposing that vn = an Ω n is the Kepler’s velocity of movement of a planet, let’s use the Kepler’s law in the form of Utting [60]: an ⋅ vn2 = γM . (91) In turn, the right part of eq. (91) can be expressed from the equation (46) of state of an ideal stellar substance taking into account a designation (45): 3 π kB i T 9 π kBi γM = ⋅ ⋅ ≈ ⋅ RT . (92) μr α mp mp μ r Comparing (91) with (92), we obtain:

T=

mp μ r α 3 π k Bi

⋅ a n v n2 ≈

mp μ r 9 π k Bi R

⋅ a n v n2 ,

(93)

whence

a nT =

mp μ r α

⋅ (a n v n ) 2 .

(94) 3 π k Bi Substituting (71) in the formula (94) we rewrite it as follows: mp μ r 4 A T α a n Teff = ⋅ (a n v n ) 2 . (95) 3 π k Bi Introducing the following notation: mp μ r 4 AT α . (96) K= 3 π k Bi we obtain a theoretical dependence confirming the mentioned regression equations (89а)-(89d) of Pintr– Peřinová–Lukš–Pathak (PPLP-equations) as a whole (97a) a nTeff = K⋅ (a n v n ) 2 , though comparison (89c), (89d) with (95) reveals an approximation lack in degree of an vn . Taking into account the estimation (75) let’s calculate the value of theoretical coefficient K for the Sun which is a representative of stars of the spectral class G: 1.67248 ⋅ 10 −27 ⋅ 1 ⋅ 3.852 ⋅ 10 −3 2.297 ⋅ 10 −19 K Sun = ≈ 0.4206 ⋅ 10 −16 (K⋅ s 2 m 3 ) . (97b) − 23 3 3.14159265 ⋅ 1.38049 ⋅ 10 ⋅ 1 The derived theoretical estimation (97b) well corresponds the heuristic PPLP - dependence (89a) for stars of the spectral class F and satisfactorily corresponds (89b) for stars of the spectral class G. On the other hand, taking into account (91) and (93) it is not difficult to see that for any star

mp μ r

T=

9 π k Bi R

⋅ γM ,

so that Teff =

mp μ r

9 π k Bi R

⋅ 4 AT ⋅ γM .

(98)

Using (91) let’s represent the heuristic PPLP-dependences (89а)-(89d) by analogy with (98) in the form: Teff F ≈

Teff G ≈ Teff K ≈ Teff M ≈

4 ⋅ 10 −17 (an vn ) 0.0037

⋅ γM ;

1 ⋅ 10 −16 ⋅ γM ; (a n v n ) 0.024 4 ⋅ 10 −14 ( an vn ) 0.193 5 ⋅ 10 −14 (an vn ) 0.186

4

4

AT G ≈ AT K ≈

AT M ≈

1 ⋅10−16 (an vn )

0.024

4 ⋅10 −14 ( an vn )

0.193

5 ⋅ 10 −14 ( a n vn )

0.186

⋅

(99b)

⋅ γM ;

(99c)

⋅ γM .

(99d)

Comparing (98) with (99a-d) shows that the following heuristic estimations are valid: 4 ⋅10 −17 9 π k Bi R 4 A ≈ ⋅ ; TF 0.0037 mp μr ( a n vn ) 4

(99а)

(100a)

⋅

9 π kBi R ; mp μr

(100b)

⋅

9 π k Bi R ; mp μr

(100c)

9 π k Bi R . m p μr

(100d)

Calculating the common constant in (100a)-(100d) separately: 9 π k B 9 ⋅ 3.14159265 ⋅ 1.38049 ⋅ 10 −23 = = 1.31670892 ⋅ 10 5 (J K⋅ kg ) , − 27 mp 1.67248 ⋅ 10 let’s rewrite the formulas (100a)-(100d) in the form: 5.266835678 ⋅ 10 −12 i R ⋅ ; μr (an vn ) 0.0037

(101a)

1.31670892 ⋅ 10−11 i R ⋅ ; μr (an vn ) 0.024

(101b)

5.266835678 ⋅10 −9 i R ⋅ ; μr (an vn ) 0.193

(101c)

6.5835446 ⋅ 10 −9 i R ⋅ . μr ( an vn ) 0.186

(101d)

4

AT F ≈

4

AT G ≈

4

AT K ≈

4

AT M ≈

Using (71) and (101a)-(101d) we can estimate the average temperature of the stellar corona for the spectral classes of stars F, G, K and M:

(a n v n ) 0.0037 μ ⋅ r ; −12 iR 5.266835678 ⋅ 10 0.024 μ (an vn ) = Teff G ⋅ ⋅ r ; −11

Tcor F = Teff F ⋅ Tcor G

1.31670892 ⋅10

Tcor K = Teff K ⋅

(an vn )0.193 5.266835678 ⋅10

Tcor M = Teff M ⋅

(an vn ) 0.186 6.5835446 ⋅ 10

(102a) (102b)

iR

−9

−9

μ ⋅ r; iR

(102c)

μ ⋅ r. iR

(102d)

Using heuristic dependence (99b) we can suppose that this relation is also valid for the Sun belonging to the spectral class G, i.e. 1 ⋅ 10 −16 (103) ⋅ γM Sun . Teff Sun = (a n v n ) 0.024 Dividing both parts of Eq. (99b) on the respective parts of (103) we obtain: Teff G M = = kM . (104) Teff Sun M Sun

According to Eq. (102b) we can estimate the average value of the temperature of the solar corona because the Sun belongs to the spectral class G: (a n v n ) 0.024 1 Tcor Sun = Teff Sun ⋅ ⋅ . (105) −11 1 ⋅ RSun 1.31670892 ⋅ 10 Analogously dividing Eq. (102b) on Eq. (105) we find: Tcor G Teff G μ r RSun Teff G μ r 1 = ⋅ ⋅ = ⋅ ⋅ . R Tcor Sun Teff Sun i Teff Sun i k R

(106)

Taking into account (104) we derive again from (106) the mentioned above formula (88b). Thus, the regression dependences for multiplanet extrasolar systems confirm the obtained result in Section 5 completely. Example 6.1. Let’s estimate temperature of stellar corona for the star υ Andromedae which is a representative of the spectral type F8. Taking into account that a n v n ≈ 6.35 ⋅ 1015 (m 2 / s), n = 4 [4] we obtain in accord with (102a): (6.35 ⋅1015 ) 0.0037 1 Tcor F = 6212 ⋅ ⋅ = 1.18959969 9 ⋅106 (K) . −12 5.266835678 ⋅10 1.1343605 ⋅10 − 9 Since the theoretical estimation of temperature of stellar corona (88a) for the star υ Andromedae is equal Tcor υAnd = 1.167995095 ⋅106 K (see Table 4) then relative error of discrepancy

[(

)

]

δ Tcor = Tcor υAnd − Tcor F / Tcor υAnd ⋅ 100% = 1.85% . Of course, the heuristic estimation Tcor F is more exact

under comparison with the preliminary estimation in Example ~ 3 2 6 Tcor υAnd ≈ 6.212 ⋅ 10 ⋅ 2.596 ⋅ 10 = 1.6126352 ⋅ 10 (K) (in this case δT cor = −38% ).

4.7

where

Example 6.2. Let’s calculate the estimation of temperature of stellar corona for the star Kepler-20 which is a representative of the spectral class G (type G8). According to the formula (102b), Example 4.1 and the average value for the Kepler-20 a n v n ≈ 1.4148 ⋅ 1015 (m 2 / s), n = 5 [4] we obtain: Tcor G = 5466 ⋅

(1.4148 ⋅1015 )0.024 1.31670892 ⋅10

−11

⋅

1 8

6.56552 ⋅10

= 1.460586904 ⋅106 (K) .

Taking into account that according to Table 4 the theoretic estimation of temperature of its stellar corona Tcor Kepler- 20 = 1.449 ⋅ 106 K we find that a relative error of discrepancy is

δ Tcor = [(Tcor Kepler - 20 − Tcor G )/ Tcor Kepler - 20 ]⋅ 100% = −0.8% ; for comparison, for a preliminary estimation in

~ Example 4.1 Tcor Kepler -20 ≈ 5466 ⋅ 2.596 ⋅ 10 2 = 1.41897 ⋅ 10 6 (K) the relative error is equal δT cor = 2.1%

respectively. Example 6.3. Let’s estimate temperature of stellar corona for the star 55Cnc belonging to the spectral class K (type KOIV-V). Taking into account that a n v n ≈ 3.5998 ⋅ 1015 (m 2 / s), n = 5 [4] as well as Example 4.5 we obtain in accord with (102c): (3.5998 ⋅ 1015 ) 0.193 1 Tcor K = 5196 ⋅ ⋅ = 1.512419515 ⋅ 10 6 (K). −9 5.266835678 ⋅ 10 6.558565 ⋅ 10 8 Taking into account Table 4 the theoretic estimation of temperature of its stellar corona Tcor 55Cnc = 1.439554612 ⋅ 106 K we can find a relative error δ Tcor = (Tcor 55Cnc − Tcor K )/ Tcor 55Cnc ⋅ 100% = −5.1% .

[

]

~

According to Example 4.5 a preliminary estimation is Tcor 55Cnc ≈ 5196 ⋅ 2.596 ⋅102 = 1.34899 ⋅106 (K), so that the respective error δT cor = 6.3% . Example 6.4. Let’s calculate the estimation of temperature of stellar corona for the star Gliese 876 belonging to the spectral class M (type M4V). According to the formula (102d), Example 4.8 and the average value for the Gliese 876 a n v n ≈ 9.92 ⋅ 1014 (m 2 / s), n = 4 [4] we obtain: Tcor M = 3350 ⋅

(9.92 ⋅1014 )0.186 6.5835446 ⋅10

−9

⋅

1 2.5038 ⋅108

= 1.251228307 ⋅106 (K) .

Since the theoretic estimation of temperature of stellar corona (88a) for the star Gliese 876 is equal (see Table 4) then relative error of discrepancy Tcor Gl 876 = 1.391666667 ⋅106 K

δ Tcor = [(Tcor Gl 876 − Tcor M )/ Tcor Gl 876 ]⋅ 100% = 10% . Nevertheless, the heuristic estimation Tcor M is more exact

under comparison with the preliminary estimation in ~ 2 5 Tcor Gl 876 ≈ 3350 ⋅ 2.596 ⋅ 10 = 8.6966 ⋅ 10 (K) (in this case δT cor = 37.5% ).

Example

4.8

where

Thus, the heuristic estimations (102a)-(102d) confirm enough satisfactory the derived theoretic formulas (88a,b) for estimation of temperature of stellar corona of multi-planet extrasolar systems (see Table 5 as well as the derivation (88b) on the basis of (103)-(106) ). 7. Conclusion

This paper derives the equation of state (48) of an ideal stellar substance based on conception of gravitating spheroidal body as far as cosmic interstellar space is not empty (in particular, the mean mass density of substance in the neighborhood of the Sun is 6 ⋅ 10−24 g сm3 whereas into interstellar space it is equal to 3 ⋅ 10−24 g сm3 ). Using this equation, the universal stellar law (USL) for the planetary systems (connecting temperature, size and mass of each of stars) has been obtained in Section 3. Analysis of USL (48) and its version (61) has shown that most part of them corresponds to category of ideal (or classical) stars (and, respectively, planetary systems) independent of spectral belonging to О, В, А, F, G, К, М classes (see Table 1, Table 3 and Fig. 1). The ordinary classical stars satisfy USL relatively their temperatures, sizes and masses and possess maximal mass densities in the stellar centers according to (4). Nevertheless, there exists a subclass of stars called group of red giants (for example, 18 Delphini, ξ Aquilae, HD 81688, 47 Ursae Majoris, Betelgeusе etc.), for which some characteristics, among them temperature of stellar corona (see Table 2), differ essentially from analogous characteristics of classical stars. In this connection due to instable process of stellar diameters changing it is assumed cavities forming inside of them (it is well-known that the diameter of Betelgeuse (α Orionis) has decreased systematically by 15% during 1993-2009 time period [61]).

This work also considers the solar corona in the connection with USL. Then it is accounting under calculation of the ratio (60) of temperature of the solar corona Tcor to effective temperature Teff of the Sun’ surface and modification of USL. To test justice of the modified USL (61) for different types of stars entirely, temperature of the stellar corona Tcor can be estimated approximately (see the formulas (88a), (88b) and Table 4). Using the modified USL (61) some predictions of star’s parameters can be made. In particular, for the star HD 181433 (see [43]) the modified USL (61) gives the following estimation of unknown radius: RHD181433 ≈ 6.102823 ⋅ 108 m as well as for τ Gem [43] the modified USL (61) permits to find unknown effective temperature Teff τ Gem ≈ 5582 K (moreover, using the formula (88a, b) it becomes possible to estimate the temperature of its stellar corona Tcor τ Gem ≈ 1.553 ⋅ 106 K ). Let’s note that a variant of the Hertzsprung–Russell’s dependence can be obtained from (61) directly. Really, if we use the Stefan-Boltzmann law and calculate the luminosity of a star in the form: 4 L = 4πR 2σTeff , (107) where R is the stellar radius, σ is the Stefan-Boltzmann constant, we can formulate the modified USL (61) through L : mpM Ξ / 2 αR ⋅ 4 = 4 . (108) πσ L Taking into account that according to (1) the square root of the parameter of gravitational compression is equal approximately α ≈ 1 / 3R we can get from (108) that ⎛ mp L ≈ 4πσ ⎜⎜ ⎝ 3Ξ

4 4 ⎞ M ⎟ ⋅ . ⎟ 2 ⎠ R

(109)

Remarking that ( α )3 = (2π )3 / 2 ρ0 / M according to (4), where ρ0 is a density in the center of a spheroidal body, we obtain from (109) the variant of Hertzsprung–Russell’s dependence: 4 8π 2 2 / 3 ⎛ mp ⎞ L≈ ⋅ σρ0 ⎜⎜ ⎟⎟ ⋅ M 10 / 3 . (110) 9 ⎝ Ξ ⎠ As follows from (110) the obtained dependence confirms the Hertzsprung–Russell’s law completely in the case of s = 3.3 , i.e. for stars of big masses ( 0.6 < lg M / M Sun < 1.7 ). As concerns the other types of stars (small and medium masses) it should be used initial USL (48) (not its approximate version in the form of modified USL (61)) together with the extended variants of the formulas (88a,b) for estimation of temperature of the stellar corona Tcor for not only the spectral class G but for F, К, М etc. Section 6 of this paper also shows that knowledge of some characteristics for multi-planet extrasolar systems permits us to refine own parameters of stars. In this connection, comparison with estimations of temperatures using of the regression dependences for multi-planet extrasolar systems testifies the obtained results entirely. Acknowledgments

I would like to thank anonymous reviewers for many stimulating recommendations and comments. The author would also like to express thankfulness to Dr. P. Pintr (Joint Laboratory of Optics, Palacky University, RCPTM, Olomouc, Czech Republic) for his fruitful comments and discussions concerning statistical theory of formation of gravitating cosmogonic bodies. I dedicate this paper to the memory of my dear mother Polina Adamovna Kulagina for her powerful vital support, attention and help. References

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Highlights:

We have developed the statistical theory of gravitating spheroidal bodies. We have introduced the universal stellar law (USL) for the planetary systems. We have estimated temperature of the stellar corona for different types of stars. We have derived the known Hertzsprung–Russell’s dependence based on USL. We have testified the obtained results for multi-planet extrasolar systems.